url
stringlengths 15
1.13k
| text
stringlengths 100
1.04M
| metadata
stringlengths 1.06k
1.1k
|
|---|---|---|
https://www.physicsforums.com/threads/special-relativity-and-magnetism.875993/ | # Special relativity and magnetism
• I
• Start date
• #1
84
1
I've read in various places that magnetism can be explained in terms of the effects of special relativity. However, all of the explanations of this only mentioned the case of current flowing in a wire. Can special relativity explain the magnetism of free flowing electrons and other moving charged objects?
• #2
Orodruin
Staff Emeritus
Homework Helper
Gold Member
2021 Award
18,512
8,416
In order to answer your question, you must specify what you mean by "explained in terms of". The entire framework of Maxwell's equations was instrumental in the development of special relativity. Magnetic and electric fields mix under Lorentz transformations due to being different components of the same anti-symmetric rank two tensor.
Likes vanhees71
• #3
84
1
In order to answer your question, you must specify what you mean by "explained in terms of". The entire framework of Maxwell's equations was instrumental in the development of special relativity. Magnetic and electric fields mix under Lorentz transformations due to being different components of the same anti-symmetric rank two tensor.
Okay, "explained in terms of" is probably the wrong way to put it then, what I mean is, the special relativity picture of magnetism says that the magnetic field in one reference frame is actually an electric field in another reference frame (in the current in a wire situation). I'm asking, can the magnetic field of a moving electron or other moving charge in free space be explained as an electric field in another reference frame?
• #4
Nugatory
Mentor
13,847
7,240
Okay, "explained in terms of" is probably the wrong way to put it then, what I mean is, the special relativity picture of magnetism says that the magnetic field in one reference frame is actually an electric field in another reference frame (in the current in a wire situation). I'm asking, can the magnetic field of a moving electron or other moving charge in free space be explained as an electric field in another reference frame?
Yes. That's basically an English-language restatement of how the Faraday tensor Orodruin referred to transforms from one frame to another.
You're not likely to encounter this treatment at the undergraduate level though. There you'll find the special case of the current-carrying wire because that case is good enough to get the general idea across and can be taken on by students who have been through only an intro course on special relativity.
You will find much more discussion here if you look for threads referencing "Purcell", who will be the Edward Purcell who wrote an intro E&M textbook that has been widely used for decades and which uses the wire example.
Last edited:
• #5
Orodruin
Staff Emeritus
Homework Helper
Gold Member
2021 Award
18,512
8,416
Okay, "explained in terms of" is probably the wrong way to put it then, what I mean is, the special relativity picture of magnetism says that the magnetic field in one reference frame is actually an electric field in another reference frame (in the current in a wire situation). I'm asking, can the magnetic field of a moving electron or other moving charge in free space be explained as an electric field in another reference frame?
Yes and no. As I already said, the components of the electric and magnetic fields mix under Lorentz transformations. However, it is not certain that you can always find a frame where the electric field is zero, or where the magnetic field is zero.
You can easily find the electric and magnetic fields of a moving electron by performing a Lorentz transformation of the field configuration for the stationary charge (for which there is no magnetic field (if you disregard the magnetic moment of the electron)).
• #6
84
1
Yes and no. As I already said, the components of the electric and magnetic fields mix under Lorentz transformations. However, it is not certain that you can always find a frame where the electric field is zero, or where the magnetic field is zero.
You can easily find the electric and magnetic fields of a moving electron by performing a Lorentz transformation of the field configuration for the stationary charge (for which there is no magnetic field (if you disregard the magnetic moment of the electron)).
In the case of current in a wire there are positive charges in the wire that are needed for the effect to occur. Given that there are no positive charges near or around the electron as it moves through free space I can't see how its magnetic field can be attributed to an electric field in another reference frame. Is there any explanation that involves positive charges??
• #7
Orodruin
Staff Emeritus
Homework Helper
Gold Member
2021 Award
18,512
8,416
In the case of current in a wire there are positive charges in the wire that are needed for the effect to occur. Given that there are no positive charges near or around the electron as it moves through free space I can't see how its magnetic field can be attributed to an electric field in another reference frame. Is there any explanation that involves positive charges??
Any moving charge implies a current regardless of whether the overall charge is zero or not. This goes into Maxwell's equations and generally results in a magnetic field. I do not see why you think a positive charge is necessary. The positive charges are not needed.
• #8
84
1
Any moving charge implies a current regardless of whether the overall charge is zero or not. This goes into Maxwell's equations and generally results in a magnetic field. I do not see why you think a positive charge is necessary. The positive charges are not needed.
Consider moving a negative charge in the direction of current of a wire. This will cause the stationary positive charges in the wire to appear to move in the reference frame of the moving negative charge, this will cause length contraction on the positive charges and creates a net positive charge to appear on the wire in the reference frame of the negative charge and will cause the charge to attract to the wire. In this example the positive charges are required for any magnetism to occur
• #9
Orodruin
Staff Emeritus
Homework Helper
Gold Member
2021 Award
18,512
8,416
In this example the positive charges are required for any magnetism to occur
No they are not. The magnetic field would be there regardless of the positive charges. The positive charges are necessary only to make the electric field of the wire zero (in the wire frame).
• #10
1,559
132
Consider moving a negative charge in the direction of current of a wire. This will cause the stationary positive charges in the wire to appear to move in the reference frame of the moving negative charge, this will cause length contraction on the positive charges and creates a net positive charge to appear on the wire in the reference frame of the negative charge and will cause the charge to attract to the wire. In this example the positive charges are required for any magnetism to occur
Consider moving a negative charge in the direction of current of a wire. This will cause the stationary positive charges in the wire to appear to move in the reference frame of the moving negative charge, this will cause length contraction on the positive charges and creates a net positive charge to appear on the wire in the reference frame of the negative charge and will cause the charge to report that it feels a force F.
The motion of the charge has one more effect: We will transform the force reported by the charge, using a Lorentz transformation formula, which is: ## F'= \frac {F}{\gamma}## , when the force is perpendicular to the motion. The force in our frame is the force measured by the charge divided by gamma.
• #11
pervect
Staff Emeritus
10,147
1,298
I've read in various places that magnetism can be explained in terms of the effects of special relativity. However, all of the explanations of this only mentioned the case of current flowing in a wire. Can special relativity explain the magnetism of free flowing electrons and other moving charged objects?
Maxwell's equations, which include magnetism, are perfectly adequate to handle currents flowing in a wire, and also moving charges that are not in a wire. Maxwell's equations are also perfectly compatible with special relativity.
It's probably over-simple to say that magnetism "can be explained in terms of special relativity", though it is motiviationally helpful to realize that Maxwell's equations "with magnetism removed" would not compatible with special relativity. I say "motivationally helpful" because there is a certain lack of rigor in what it might mean to "remove magnetism from Maxwell's equations".
If one desires a purely classical (non-quantum) theory that will handle currents in wires, and also handle moving charges that are not in wires, in all generality, then theories one wants to sue are Maxwell's equations and relativistic mechanics.
Hopefully this answers your question? Or are you concerned with the issues that would arise if you tried to use Maxwell's equations with Newtonian mechanics (i.e. without special relativity)?
• #12
1,559
132
Consider moving a negative charge in the direction of current of a wire. This will cause the stationary positive charges in the wire to appear to move in the reference frame of the moving negative charge, this will cause length contraction on the positive charges and creates a net positive charge to appear on the wire in the reference frame of the negative charge and will cause the charge to report that it feels a force F.
The motion of the charge has one more effect: We will transform the force reported by the charge, using a Lorentz transformation formula, which is: ## F'= \frac {F}{\gamma}## , when the force is perpendicular to the motion. The force in our frame is the force measured by the charge divided by gamma.
Special relativity's distance transformation has an effect on forces.
Special relativity's force transformation has an effect on forces.
Magnetism includes all special relativity's effects on forces.
• Last Post
Replies
20
Views
2K
• Last Post
Replies
20
Views
3K
• Last Post
Replies
10
Views
4K
• Last Post
Replies
52
Views
5K
• Last Post
Replies
29
Views
7K
• Last Post
Replies
7
Views
1K
• Last Post
Replies
3
Views
1K
• Last Post
Replies
1
Views
1K
• Last Post
Replies
4
Views
2K
• Last Post
Replies
3
Views
1K | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8465994000434875, "perplexity": 284.4246894943929}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662534773.36/warc/CC-MAIN-20220521014358-20220521044358-00211.warc.gz"} |
http://math.stackexchange.com/questions/276161/how-to-solve-m-t-x-kx-0-sturm-liouville-equation-with-bessel-funct | How to solve $(m_{(t)} x')' + kx = 0$ Sturm Liouville equation with bessel functions
I have been working on this problem for a while now and think I need assistance. I am trying to solve with respect to $x_{(t)}$ over the interval $t = [0, \infty]$:
$$(m_{(t)} x')' + kx = 0$$ $$m_{(t)} = \frac{m_o}{1 + \alpha t}$$
Subject to he boundary conditions:
$$x(o) = x_o$$ $$\dot x(0) = 0$$
This is clearly of the Sturm - Liouville form which is unweighted as the weight function $P_{(t)} = 1$
$(r(x)y')' + \left( q(x) + \lambda P(x) \right)y = 0$
$k, \alpha, m_o$ are all constants, and I know that the solution Involves Bessel functions so I have been trying to get the Differential Equation into the form of a Bessel equation:
$$(x^ry')' + (ax^s + bx^{r-2})y = 0$$
Which has the solution if $(1 - r)^2 > = 4b$ and either $s >(r-2)$ or $b = 0$
$$y = x^\alpha \left[ C_1 J_\nu( \lambda x^ \gamma) + C_2 Y_\nu(\lambda x^ \gamma) \right]$$
where $\alpha = \frac{1 - r}{2}$ $\gamma = \frac{2 - r +s}{2}$ $\lambda = 2 \frac{\sqrt{a}}{2-r+s}$ $\nu = \frac{ \sqrt{(1 - r)^2 - 4b}}{2 - r +s}$
I am not sure if this is the right road to go down because I have tried just about everything I can think off to solve this thing. A poke in the right direction would not be soon forgotten. Thank you for your time.
-
Not sure how one gets Bessels from your equation, as it is a 1st order equation. Are you sure the 2nd term is $k x'$, or is it $k x$? – Ron Gordon Jan 12 '13 at 3:04
your right i made a mistake – Cactus BAMF Jan 12 '13 at 3:14
You can simplify things a bit by setting $u=1+\alpha t$ and rearranging things a bit to get
$$\left [ u^{-1} y' \right ]' + \frac{k}{m_0 \alpha^2} y = 0$$
where $y(u) = x \left ( \frac{u-1}{\alpha} \right ) = x(t)$. You can map this equation (which is really not much more than what you already have) onto the parameter values $r=-1$, $s=0$, $a =\frac{k}{m_0 \alpha^2}$, and $b=0$. You should be able to take it form here.
In passing, I note that the index $\nu = \frac{2}{3}$ with an argument of that Bessel being raised to the power $\frac{3}{2}$ is indicative of an Airy function, which you may wish to investigate a little.
-
Thank you very much, I had started to make the substitution you have there, but I did not see any use in it at the time. – Cactus BAMF Jan 12 '13 at 4:31 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9406734704971313, "perplexity": 141.72900566501147}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-06/segments/1422122328486.60/warc/CC-MAIN-20150124175848-00115-ip-10-180-212-252.ec2.internal.warc.gz"} |
http://www.gradesaver.com/textbooks/math/calculus/calculus-10th-edition/chapter-3-applications-of-differentiation-3-1-exercises-page-167/30 | ## Calculus 10th Edition
Over the specified interval, the function has an absolute maximum equal to $5$ and an absolute minimum equal to $0.$
Using the rule $\dfrac{d}{dx}|z(x)|=z'(x)\times\dfrac{|z(x)|}{z(x)}\to$ $g'(x)=\dfrac{|x+4|}{x+4}$ which is undefined for $x=-4.$ The interval's boundaries and $x=-4$ are possible candidates for absolute extrema. $g(-7)=3.$ $g(-4)=0.$ $g(1)=5.$ Over the specified interval, the function has an absolute maximum equal to $5$ and an absolute minimum equal to $0.$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8685317039489746, "perplexity": 174.97755518104742}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891813322.19/warc/CC-MAIN-20180221024420-20180221044420-00596.warc.gz"} |
https://math.stackexchange.com/questions/317479/a-closed-set-in-a-metric-space-is-g-delta | # A closed set in a metric space is $G_\delta$
How do I prove that a closed set $F$ in the metric $(X,d)$ is $G_\delta$.
Let $n\in \mathbb{N}$.
I consider $B_n={F}=\bigcup_ {x\in F} B(x,{1\over n})$, which is a collection of an open ball. Then I guess what I have to show next is the intersection of all these open balls is the closed set F.
If $x\in \bigcap_n B_n$, then for each $n$, by the definition of $B_n$, there is some $x_n\in F$ with $d(x_n,x)<1/n$. Then $(x_n)$ converges to $x$, so as $F$ is closed $x\in F$.
• so this shows that $\bigcap {B_n} \subset F$ right? Is $F\subset \bigcap {B_n}$ a fact since $F \subset B_n$ for each n? – Akaichan Mar 1 '13 at 3:54 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9982539415359497, "perplexity": 66.6810662698232}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575541288287.53/warc/CC-MAIN-20191214174719-20191214202719-00486.warc.gz"} |
http://physics.stackexchange.com/users/14474/formlesscloud | FormlessCloud
less info
reputation
5
bio website location Italy age member for 1 year, 11 months seen Oct 10 at 15:06 profile views 10
Hi, feel free to contact me if you like!
8 Questions
2 Einstein Equation at the Singularity. Understanding the Dr. Michio Kaku's explanation 2 Bra–ket notation & Schrödinger's cat intepretation 2 Parabolic motion and air drag 1 Calculate coefficient of friction from deceleration with and without conservation of energy 1 Which is the most accurate way to model the fundamental atmospheric parameters of the Earth?
158 Reputation
+5 Which is the most accurate way to model the fundamental atmospheric parameters of the Earth? +5 Calculate coefficient of friction from deceleration with and without conservation of energy +10 Einstein Equation at the Singularity. Understanding the Dr. Michio Kaku's explanation +5 Bra–ket notation & Schrödinger's cat intepretation
This user has not answered any questions
21 Tags
0 electromagnetism × 2 0 projectile 0 general-relativity × 2 0 newtonian-gravity 0 newtonian-mechanics × 2 0 curvature 0 drag 0 gravity 0 maxwell-equations 0 singularities
15 Accounts
TeX - LaTeX 2,842 rep 1046 Mathematics 301 rep 211 Mathematica 161 rep 16 Physics 158 rep 5 Biology 151 rep 5 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9493079781532288, "perplexity": 3040.0901092558493}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-42/segments/1413507445159.36/warc/CC-MAIN-20141017005725-00339-ip-10-16-133-185.ec2.internal.warc.gz"} |
https://www.aanda.org/articles/aa/full_html/2012/06/aa19123-12/aa19123-12.html | Subscriber Authentication Point
Free Access
Issue A&A Volume 542, June 2012 A75 7 Atomic, molecular, and nuclear data https://doi.org/10.1051/0004-6361/201219123 12 June 2012
## 1. Introduction
We studied the Stark broadening of the He I 492.2 nm line in a plasma with computer simulations. The results were compared with the available experimental data and with previous theories and approximate calculations. This line appears as a result of the transition between the 1s4d 1D and the 1s2p 1P° states and shows one allowed component and two forbidden components, the transitions between the 1s4p 1P° and the 1s2p 1P°, and between the 1s4f 1F° and the 1s2p 1P° states. Owing to the quasi degenerate set of upper states of the transition, the line shape strongly depends on the density of charged particles in a plasma, therefore the analysis of this dependence can be useful for plasma diagnostics.
Two characteristics of the line are especially relevant for obtaining the electron density, Ne, of the plasma: the distance from the 1s4f 1F°–1s2p 1P° forbidden component to the allowed one and the intensity ratio between these two line components.
The results of this work are shown mainly as tables of line profiles to obtain the plasma conditions through comparison with experimental data and approximate formula obtained by fitting the simulation results for plasma diagnostics. These are the final aims of this study.
The shape analysis of the 492.2 line was used to obtain information on He abundances (Leone & Lanzafame 1998; Castelli & Hubrig 2004; Lyubimkov et al. 2004; Catanzaro 2008), He stratification (Catanzaro 2008) or radial velocities (Jeffery et al. 2001). Owing to a lack of available information, some authors were in some cases only able to use line widths obtained from approximate models (Castelli & Hubrig 2004) instead of using the full profiles for comparison with the experimental data, though several authors already pointed out the better results obtained by comparing the full theoretical and experimental profiles instead of using only the line widths (Leckrone 1971). Other authors using full profiles for comparison with experimental data employed old calculations (Heasley & Wolff 1981; Jeffery et al. 2001) to include the influence of Stark broadening in the profiles. The discrepancies in those and other comparisons neccessitated more advances in line broadening calculations of the He I 492.2 nm line (Lyubimkov et al. 2004; Castelli & Hubrig 2004).
In this work we only considered the broadening and shifting of the spectral line caused by the Stark effect. Other effects such as Doppler broadening or van der Waals broadening, which are noticeable at very low electron densities, will not be taken into account. The electron density, the free electrons and ions temperatures and the ions mass will be used as the parameters that characterize the state of the plasma. Our treatment permits us to consider non-equilibrium conditions in which the plasma ions and electrons have different temperatures and the calculation technique includes the so-called “ion dynamics effects” in a natural way.
This work completes an earlier publication dedicated to the He I 447.1 nm line (Gigosos & González 2009) that has similar characteristics. In that work the special properties of these two spectral lines (492.2 and 447.1), which make them very suitable for plasma diagnostics, were already given, and will not be repeated here. Then, after a brief reminder of the calculation method, the dependences of several line parameters against electron temperature and density are shown and compared with previous experimental data.
Our calculations were obtained following the same technique as described in Gigosos & González (2009). Therefore we only recall some details here and refer the interested reader to that work.
Following Anderson (1949), the Stark broadening of the spectral lines emitted by a plasma is calculated considering the electric field generated by the charged particles, electrons and ions, in the plasma. Under the usual approximations (Gigosos et al. 2003; Gigosos & González 2009) the emitters in the plasma are considered as statistically independent particles that produce incoherent emissions, and the plasma is considered as a thermal bath whose evolution is not affected by the emission processes. Then, the line profile calculations are performed considering a typical emitter subjected to the electric fields of the charges in the plasma and averaging the results over the possible configurations of perturbers in the plasma and over the emitter states.
As in previous works, the emission profile is obtained as the Fourier transform of the autocorrelation function of the emitter dipole moment: where U(t) is the emitter time evolution operator, which is obtained in the calculation by solving the evolution equation (4)which includes the unperturbed emitter structure through the hamiltonian H0, and the plasma perturbation through the dipole interaction with the electric field, produced by the plasma charges on the emitter.
As discussed in Gigosos & González (2009), only a few emitter states need to be considered in Eq. (4) to obtain the line profiles with good accuracy. We took two groups of atomic states into account, an upper group with 16 states and a lower group with 3 states. The upper states include the 1s4d 1D states, source of the dipole allowed 492.19 nm transition, and the 1s4p 1P° and 1s4f 1F°, origin of the 491.08 and 492.06 nm dipole forbidden transitions together with the 1s4s 1S state. These last states are coupled to the 1s4d 1D through the plasma electric field, giving rise to the Stark broadening. The lower group consists of only the 1s2p 1P° states, which is destination of all the aforementioned transitions, as can be seen in Fig. 1. Because the distance in energies between the upper and lower group of states is considered to be very large compared with the typical energies of the plasma perturbations, the evolution of both sets of states are obtained independently. Table 1 shows the atomic data relevant to the upper group of states whose coupling through the plasma electric field gives rise to the line broadening and to the appearance of the forbidden components.
Fig. 1 Diagram of the levels involved in the transition studied in this work.
Table 1
Data of the helium I atomic structure used in this work.
As described in Gigosos & González (2009), our calculations were performed with computer simulations that reproduce the electric field on the emitter in a plasma. Inserting this electric field into Eqs. (1)–(4) permits one to obtain the broadened line profile corresponding to a single plasma configuration. The final profile is obtained after averaging over many (typically over 30 000 samples in this work) plasma configurations. Therefore, the final profile is calculated from the averaged autocorrelation {C(t)} obtained considering the autocorrelations corresponding to each field sequence.
Our computer simulation technique is a molecular dynamics technique with independent particles that uses the so-called μ-ion model (Seidel & Stamm 1982). More details have been given elsewhere (Gigosos et al. 2003; Gigosos & González 2009). This technique is valid only for lowly coupled plasmas. Accordingly, the cases of plasma electron density, Ne, and temperature, Te, considered in these calculations which correspond to high values of the coupling parameter must be considered as extrapolations of this method (see Dufour et al. 2005, for a discussion of this approximation) and serve only as an illustration of the results for different conditions.
## 3. Results
### 3.1. Shape of the spectral line
As mentioned before, to obtain different plasma conditions one usually compares the experimental measurements to calculated line profiles. Accodingly, we begin discussing our results by comparing full profiles obtained in the simulations with those from previous calculations as well as experimental results.
Figure 2 shows the comparison of two simulation calculations with a previous analytical result (Barnard et al. 1975). Though Barnard et al. (1975) used a calculation that included ion dynamical effects, for densities higher than 1022 m-3 the calculation considered a static theory. Figure 2 compares their profile at Ne = 1022 m-3 and Te = 10 000 K with two simulation results, one for perturbers with a high reduced mass (or low ionic temperature) whose behaviour would be close to static perturbers and another with H+ perturbers. Clearly, although the overall shape of the line is very similar for the three calculations, ion dynamics effects still affect the central part of the profile. While the theoretical static calculation and the high reduced mass simulation result agree, the profile obtained for full dynamic perturbers shows much less structure in the centre of the line.
Fig. 2 Comparison of a previous theoretical calcualtion (Barnard et al. 1975) with two simulation calculations. Ion dynamics effects are noticeable in the centre of the line.
### 3.2. Ion dynamics effects
Ion dynamics effects are also shown in Fig. 3. Here simulated profiles are compared with the experimental measurements in Richter & Piel (1985), performed for different plasma compositions with H+, He+ or Ar+ as perturbers to study ion dynamics effects in the line. In Richter & Piel (1985), plasma density was obtained with interferometry and for the profiles in the figure it was established to be Ne = 1021 m-3, Te through relative line intensities and from line to continuum ratio and gas temperature using Doppler broadening. The simulated profiles in Fig. 3 are those that agree best with the experimental ones. These comparisons show that computer simulations reproduce ion dynamic effects well. But it is also important to notice that the conditions of these calculations correspond to values of electron density and temperature similar to those obtained independently in the experiment.
Fig. 3 Comparison of experimental (Richter & Piel 1985) and simulated profiles to study the influence of ion dynamics effects on the line profile. The measured electronic density is 1021 m-3.
### 3.3. General comparison with experimental data
Another comparison with previous calculations (Barnard et al. 1975; Schöning 1994) and experiment (Richter & Piel 1985) is shown in Fig. 4. Here the simulated profile was obtained for the values of Ne and Te given in the experiment. Simulations and experiment again agree well.
Fig. 4 Comparison of different theoretical profiles (Barnard et al. 1975; Schöning 1994) with computer simulations and an experimental result from Richter & Piel (1985).
Figure 5 shows a last comparison between experimental profiles and simulations. The experimental data in this figure correspond to measurements at four different radial positions (Birkeland et al. 1971), so that a dependence of the line shape with the electron density can be observed in the figure. Radial distribution of the electron density in (Birkeland et al. 1971) was determined from the Stark broadening of three He I lines and from the integrated intensity of the He I 501.6 nm line. Close but not equal values were obtained through the different lines. The ranges of the experimentally determined electron densities are represented with vertical dashed lines in the square within Fig. 5 for the four radial positions corresponding to the profiles in the figure. The simulated profiles shown in the figure correspond to those that agree best with the experimental ones. Evidently, computer simulations satisfactorily reproduce the change in the shape of the line as the electron density increases. But is more interesting for diagnostics purposes that the electron densities corresponding to these calculated profiles are well within the ranges determined experimentally for each radial position. As can be seen in the inserted square, these values of electron density, which are represented with black full squares, fall near the centre of the experimentally determined range of Ne for most of the cases.
Fig. 5 Comparison between experimental and simulated full line profiles for different conditions of the plasma electron density corresponding to four radial positions in the experiment. The square within the figure shows with dashed lines the range of electron densities obtained from different methods in the experiment, while the squared point corresponds to the electron density of the calculated profile for each position.
### 3.4. FWHM and electronic impact broadening
Now we study the behaviour of different line parameters to analyse if they are suitable in plasma diagnostics. The parameters studied here are the full width at half maximum (FWHM) the ratio of the forbidden component to the allowed one, IF/IA, the ratio of the central valley intensity to the allowed peak intensity, ID/IA, and the distance between the allowed and forbidden peaks, s. These parameters are illustrated in Fig. 6 with a profile obtained in the simulation.
Fig. 6 Definition of the line parameters.
The dependence of the line FWHM versus the electron density, Ne, is shown in Fig. 7. The results obtained in two types of simulations are shown in this figure. The first are simulations considering a full plasma, ions and electrons, for a pure He plasma in thermodynamical equilibrium with Te = Tgas, which corresponds to a value of the reduced mass μ = 2. Evidently, the FWHM shows a small step upwards around Ne = 1022 m-3, which approximately corresponds to the density at which the intensity of the forbidden component reaches one half of the allowed one, so that the FWHM now includes the width of both components, while at lower densities only the allowed component width contributes to the FWHM. The second set of simulations are one-component plasma simulations, OCP, considering only electrons. These were performed to obtain the electronic impact broadening and to analyse the range of validity of the impact approximation.
Two different electron temperatures were considered in these OCP simulations to study the influence of Te in the impact behaviour. For the lower values of Ne considered in the simulations (which were performed down to electron densities of 1019 m-3, beyond the limits of the figure) the electronic width shows a linear dependence on Ne, which proves the validity of the impact approximation, but as the electron density increases, the electronic width behaviour starts to depart from that linear trend. Of course, the lower the temperature, the more noticeable this departure will be. Two dashed straight lines with unity slope corresponding to the electronic broadening at the lowest densities are represented to better illustrate the inaccuracy of impact approximation at high densities.
Table 2 shows a comparison of the simulation results for the OCP calculation with those obtained using standard theoretical models considering only the broadening caused by the electronic impact. The results for the FWHM are very close for the three different calculations while the shift shows higher discrepancies. Simulation results were obtained through a calculation for Ne = 1019m-3 and were extrapolated to the density of reference Ne = 1022m-3 (the extrapolation lines are represented with dashed lines in Fig. 7).
Fig. 7 Dependence of the FWHM on the electron density for a full plasma simulation considering ions and electrons in the case of pure He plasma and for an one-component plasma considering only electrons. Different temperatures were considered in both cases. Two dashed lines with the unity slope corresponding to the electronic broadening at low densities illustrate the behaviour predicted by the impact approximation. The difference between the pure electronic plasma results and the impact estimation illustrates the departure of the impact behaviour.
Table 2
FWHM and peak shift in nm for an only electron broadening calculation normalised at Ne = 1022 m-3.
It is very useful to have simple expressions obtained by fitting the relationship between the electronic density and the line width obtained in the simulations. At low densities (Ne < 1022m-3), when the measured width corresponds only to the allowed component (see Fig. 7), the simulation results fit quite well to a function of the type (5)where the full widths due to both ions and electrons are taken into account and all data corresponding to different temperatures for the case μ = 2 have been considered jointly. If one requires a more accurate fitting, the expression (6)can be used, with the same units as in (5), and for the same electron densities.
Table 3
Values of the fitting parameters a and b of the fit log 10(FWHM/(nm)) = alog 10(Ne/m-3) + b for values of electron density 1022 < Ne < 1024m-3.
When the electron density is higher than 1022 m-3 the allowed and forbidden components overlap (see Fig. 7) so that the parameter FWHM includes both components. For these cases, the influence of temperature is noticeable and it must be considered in the fittings. Table 3 gives the values of the fitting parameters to a function of the type (7)for the temperatures considered in this work.
Figure 8 shows a comparison between the calculated and experimental FWHM. Computer simulations correspond to three different temperatures, 5000, 20 000, and 40 000 K to check its influence on the FWHM and to cover the different conditions of the experiments. While the experiments shown in the figure generally agree with the simulations, the two older experiments increasingly disagree as the electron density decreases. Similar discrepancies are found for these two experiments in the comparisons with other parameters shown below.
Fig. 8 Comparison of the simulated and measured FWHM.
### 3.5. Ratio of allowed and forbidden component intensities
A line characteristic that can be used to obtain the plasma electron density is the ratio between the most intense forbidden component and the allowed one, IF/IA. While at low densities the forbidden component is very weak, at medium or high densities it can be easily recorded and the value of IF/IA obtained. Figure 9 shows a comparison of this parameter obtained in the simulations and in different experiments covering wide ranges of electron density and temperature. For the simulations three different cases are shown to check the influence of plasma composition or temperature in this parameter: for pure He plasma (μ = 2) at Te = Tgas = 5000 and 20 000 K and for a He-H+ plasma (μ = 0.5) with Te = Tgas = 5000 K. A good agreement is found along the wide range of electron densities shown in the figure in spite of the different conditions of all the experiments shown here. Clearly, the ratio IF/IA is quite insensitive to the electron temperature for the ranges shown in the figure, which supports its suitability for determining the electron density value in a plasma.
Fig. 9 Comparison experiment-simulation for the ratio of intensities between the forbidden and the allowed line components.
### 3.6. Intensity ratio of the central dip and the allowed component
Computer simulations also allow us to study plasmas far from equilibrium. In these simulations the situations where the gas temperature are lower than the electrons temperature in the plasma are represented with higher values of the reduced mass μ than the one that would correspond to the plasma (González & Gigosos 2009). The existence of this imbalance can affect the line shapes and, in turn, the line parameters used in plasma diagnostics. This is the case of the ratio, ID/IA, between the intensities of the central dip of the line and the allowed component. Imbalances in gas and electronic temperatures can notably affect the values of ID/IA, therefore its measurement is no reliable diagnostic for the plasma density. This is shown in Fig. 10 where we compare the experimental measurements (Nelson & Barnard 1971; Baravian et al. 1975; Diatta et al. 1976; Pérez et al. 1996) and the values of ID/IA obtained from the simulated profiles. Evidently, experiment and simulation agree well. However, the most interesting point in this figure is the strong dependence of this parameter on the electron temperature at the higher densities considered in the figure, when its measurement is easier, so that, as mentioned before, the measurement of this parameter is no accurate diagnostics method.
Fig. 10 Comparison between experiments and simulations of the ratio ID/IA for different temperatures and values of μ.
Fig. 11 Allowed-forbidden peaks distance and allowed peak shift versus the electron density of the plasma for a pure He plasma. Results for three temperatures are displayed in each case. These parameters values were measured on the simulated full profiles.
### 3.7. Distance between the allowed and the forbidden components
On the other hand, a parameter that can easily be employed in plasma diagnostics is the distance between the allowed and the forbidden components, s. In a previous work (Ivković et al. 2010) the dependence of this parameter on the electron density was used to obtain a simple expression that permits one to calculate Ne from an experimental measurement. An equivalent expression has been obtained here for the He I 492.2 nm line. Figure 11 shows the dependence obtained in the simulations for the distance between the allowed and the forbidden line components and for the shift of the allowed component on the electron density for different temperatures. As can be seen there, the value of s depends on the electron density and temperature, therefore the fitting must take both parameters into account. However the reduced mass μ has a minor influence, as also seen in Gigosos & González (2009) for the 447.1 nm line, which implies that the same fitting can be used for different plasma compositions and for conditions far from equilibrium when electron and gas temperature are different.
The good agreement between the value of parameter s obtained in the calculations and experimental measurements, shown in Fig. 12, reinforces the reliability of using this parameter as a mean to obtain the electron density in the plasma. As can be seen in that figure, simulation results for a pure He plasma in thermodynamical equilibrium and different electron temperatures have been compared to results from different experiments (Nelson & Barnard 1971; Baravian et al. 1975; Diatta et al. 1976; Adler & Piel 1991; Pérez et al. 1996) and a general good agreement is found. As happened before in other comparisons, the worst agreement corresponds to the comparison with the results of Nelson & Barnard (1971) and Baravian et al. (1975).
Fig. 12 Comparison of experimental and simulated values of the distance between the allowed and the forbidden peaks of the line. The simulated values were obtained measuring that parameter in the full profiles obtained in the simulations.
To obtain a practical expression for use in plasma diagnostics, we performed a fitting of the simulation results obeying a function of the type (8)or, which is the same, (9)The criterion followed to chose this type of fitting is the same as employed in Ivković et al. (2010), where a more complete description of the details is given. The procedure combines minimal squares and MinMax fittings1. The most convenient result is given by the expressions with Ne in m-3, T in K and s in nm.
### 3.8. Tables of line profiles
The results of this work are given as tables of full line profiles in electronic format. These tables (numbered as Tables 4 to 16) are available via anonymous ftp to cdsarc.u-srasbourg.fr. These tables form a set that covers a range of electron densities between 1020 and 1024 m-3, temperatures between 5000 and 40 000 K and values of the parameter μ, reduced mass of the emitter-perturber pair, between 0.8 (He – H+ plasma in thermodynamic equilibrium) and 10.0, which corresponds to a situation where the gas temperature is much lower than the electronic temperature.
1
In that publication there is a printing mistake: the captions of Tables 1 and 2 are interchanged.
## Acknowledgments
This work has been partially financed by Spanish Ministerio de Educación y Ciencia through grant ENE2007-63386/FTN, by the Spanish Ministerio de Ciencia e Innovación under grant ENE2010-19542 and by the Junta de Castilla y León through grant VA007A09.
## References
2. Anderson, P. W. 1949, Phys. Rev., 76, 647 [NASA ADS] [CrossRef] [Google Scholar]
3. Baravian, G., Bretagne, J., Godart, J., & Sultan, G. 1975, Z. Phys. B, 20, 255 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
4. Barnard, A. J., Cooper, J., & Smith, E. W. 1975, J. Quant. Spectrosc. Radiat. Transf., 15, 429 [Google Scholar]
5. Birkeland, J. W., Bacon, M. E., & Braun, W. G. 1971, Phys. Rev. A, 3, 354 [NASA ADS] [CrossRef] [Google Scholar]
6. Castelli, F., & Hubrig, S. 2004, A&A, 425, 263 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
8. Diatta, C. S., Czernichowski, A., & Chapelle, J. 1976, Physica, 84C, 425 [Google Scholar]
9. Dimitrijević, M. S., & Sahal-Bréchot, S. 1990, A&AS, 82, 519 [NASA ADS] [Google Scholar]
10. Dufour, E., Calisti, A., Talin, B., et al. 2005, Phys. Rev. E, 71, 066409 [NASA ADS] [CrossRef] [Google Scholar]
11. Gigosos, M. A., & González, M. Á. 2009, A&A, 503, 293 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
12. Gigosos, M. A., González, M. Á., & Cardeñoso, V. 2003, Spectrochim. Acta Part B, 58, 1489 [Google Scholar]
13. González, M. Á., & Gigosos, M. A. 2009, Plasma Sources Sci. Technol., 18, 034001 [NASA ADS] [CrossRef] [Google Scholar]
15. Heasley, J. N., & Wolff, S. C. 1981, ApJ, 245, 977 [NASA ADS] [CrossRef] [Google Scholar]
16. Ivković, M., González, M. Á., Jovićević, S., Gigosos, M. A., & Konjević, N. 2010, Spectrochim. Acta Part B, 65, 234 [Google Scholar]
17. Jeffery, C. S., Wolff, V. M., & Pollacco, D. L. 2001, A&A, 376, 497 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
18. Leckrone, D. S. 1971, A&A, 11, 387 [NASA ADS] [Google Scholar]
19. Leone, F., & Lanzafame, A. C. 1998, A&A, 330, 306 [NASA ADS] [Google Scholar]
20. Lyubimkov, L. S., Rostopchin, S. I., & Lambert, D. L. 2004, MNRAS, 351, 745 [Google Scholar]
21. Nelson, R. H., & Barnard, A. J. 1971, J. Quant. Spectrosc. Radiat. Transf., 11, 161 [Google Scholar]
22. Pérez, C., de la Rosa, I., Aparicio, J. A., Mar, S., & Gigosos, M. A. 1996, Jpn. J. Appl. Phys., 35, 4073 [NASA ADS] [CrossRef] [Google Scholar]
23. Richter, H., & Piel, A. 1985, J. Quant. Spectrosc. Radiat. Transf., 33, 615 [Google Scholar]
24. Schöning, T. 1994, J. Phys. B: At. Mol. Opt. Phys., 27, 4501 [NASA ADS] [CrossRef] [Google Scholar]
25. Seidel, J., & Stamm, R. 1982, J. Quant. Spectrosc. Radiat. Transf., 27, 499 [Google Scholar]
## All Tables
Table 1
Data of the helium I atomic structure used in this work.
Table 2
FWHM and peak shift in nm for an only electron broadening calculation normalised at Ne = 1022 m-3.
Table 3
Values of the fitting parameters a and b of the fit log 10(FWHM/(nm)) = alog 10(Ne/m-3) + b for values of electron density 1022 < Ne < 1024m-3.
## All Figures
Fig. 1 Diagram of the levels involved in the transition studied in this work. In the text
Fig. 2 Comparison of a previous theoretical calcualtion (Barnard et al. 1975) with two simulation calculations. Ion dynamics effects are noticeable in the centre of the line. In the text
Fig. 3 Comparison of experimental (Richter & Piel 1985) and simulated profiles to study the influence of ion dynamics effects on the line profile. The measured electronic density is 1021 m-3. In the text
Fig. 4 Comparison of different theoretical profiles (Barnard et al. 1975; Schöning 1994) with computer simulations and an experimental result from Richter & Piel (1985). In the text
Fig. 5 Comparison between experimental and simulated full line profiles for different conditions of the plasma electron density corresponding to four radial positions in the experiment. The square within the figure shows with dashed lines the range of electron densities obtained from different methods in the experiment, while the squared point corresponds to the electron density of the calculated profile for each position. In the text
Fig. 6 Definition of the line parameters. In the text
Fig. 7 Dependence of the FWHM on the electron density for a full plasma simulation considering ions and electrons in the case of pure He plasma and for an one-component plasma considering only electrons. Different temperatures were considered in both cases. Two dashed lines with the unity slope corresponding to the electronic broadening at low densities illustrate the behaviour predicted by the impact approximation. The difference between the pure electronic plasma results and the impact estimation illustrates the departure of the impact behaviour. In the text
Fig. 8 Comparison of the simulated and measured FWHM. In the text
Fig. 9 Comparison experiment-simulation for the ratio of intensities between the forbidden and the allowed line components. In the text
Fig. 10 Comparison between experiments and simulations of the ratio ID/IA for different temperatures and values of μ. In the text
Fig. 11 Allowed-forbidden peaks distance and allowed peak shift versus the electron density of the plasma for a pure He plasma. Results for three temperatures are displayed in each case. These parameters values were measured on the simulated full profiles. In the text
Fig. 12 Comparison of experimental and simulated values of the distance between the allowed and the forbidden peaks of the line. The simulated values were obtained measuring that parameter in the full profiles obtained in the simulations. In the text
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.83677738904953, "perplexity": 1237.1612823921075}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662562410.53/warc/CC-MAIN-20220524014636-20220524044636-00041.warc.gz"} |
http://blog.logicalphalluses.net/2017/02/25/math-crankery-with-john-gabriel-calculus-101-convergence-and-derivatives/ | (Previous post on John Gabriel: The Dunning-Krüger Effect in a Nutshell)
Few people can ever begin to match my intelligence and depth of insight. I am not arrogant or deluded.
John Gabriel
Yeah. That’s an actual quote. I’ve been made aware of Gabriel’s LinkedIn page, where he wrote hilarious posts about his new calculus and his axioms for arithmetic. And (as someone on Mathematical Mathematics Memes pointed out), it’s becoming increasingly plausible that this guy has some mild form of mental illness, or at least a personality disorder. I mean that without a hint of irony – the narcissism and ignorance of this guy even dwarf Donald Trump. Here are just some further choice quotes:
“After Euclid and before me, not a single mathematics academic, ever understood what is a number. That’s quite a big statement, but I have proved it.”
“Georg Cantor, whom I consider one of the greatest fools in mathematics and the reason so many have problems with math.”
“I loathe mainstream academia and it’s hard for me to restrain myself. My tolerance for stupid people has long ceased to exist.”
“I realised many years later, they rejected my discoveries for several reasons, but the one that stood out is the fact that they did not like me personally. Truth or proofs had little to do with the rejection. They decided to libel and defame me, rather than study my ingenious work which is worthy not of one Abel prize, but of ten Abel prizes.”
“One would think that given I am helping future generations of aspiring young mathematicians, they would be grateful and welcome this new knowledge I reveal. But no, my life has all but been destroyed by the efforts and attacks of the most vile scum in mainstream academia.”
“The NC is the first and only rigorous formulation of calculus in human history. That is an incredible accomplishment given that no one before me was able to do this – not even the so-called greats such as Archimedes, Newton or anyone else. It is no longer debatable, but proven fact.”
John Gabriel – Jesus, Aristotle, Newton and Einstein all rolled into one. Praise him.
Yeah. Verbatim, people, verbatim. And I don’t think he’s a troll either – he’s been doing this for years, if not decades, and he takes every piece of criticism as a personal attack. He really seems to think he is god’s gift to humanity. So, let’s continue to take him down a notch.
In the second video on John Gabriel’s YouTube channel, he starts ranting about how calculus (unlike his new calculus, which is perfect in every way!) is wrong, which means I might as well use this opportunity to explain why it’s not and in general how this stuff actually works. Unfortunately (or fortunately, depending on your aesthetics) that means getting into serious math territory – many things that Gabriel gets wrong have to do with the fundamental definitions of e.g. convergence, the real numbers etc. However, if we want to see how wrong Gabriel really is, we first need to make sure that we all agree what the “official” (i.e. “right”) definitions of all those concepts are, what motivates these definitions and what their implications are.
Disclaimer: I will assume that we all know and somewhat agree that rational numbers are, like, a thing – that is, numbers that can be expressed as fractions of integers $$\pm\frac ab$$. The set of all rational numbers is denoted as \)\mathbb Q\), the set of all natural numbers – i.e. the numbers $$1,2,3,\ldots$$ – as $$\mathbb N$$. I mention this, because I will have to talk about what “real numbers” really are in modern mathematics – something that Gabriel really doesn’t seem to grasp. Also: Usually I prefer to have $$0$$ to be a natural number, but I specifically exclude it from $$\mathbb N$$ here, just for convenience – it allows me to e.g. define a sequence $$(\frac1n)$$ without needing to worry about the case $$n=0$$.
Second disclaimer: I’m not a historian. I might get some, many or all of the historical details wrong. I’m writing this pretty much from the top of my head. The same holds for all definitions, proofs etc. With respect to the historical stuff, it doesn’t even matter – after all, almost everything to do with actual mathematics has changed since then, and what’s important is the motivation behind this stuff, not the precise historical development, which is why I can’t be bothered to fact check this in detail. With respect to the actual math: It’s waaay more fun to redevelop all the concepts from the top of my head, rather than looking everything up in textbooks. So don’t believe anything, check everything for yourself and see whether it works out. I’m still, like 90% sure that all my definitions are either standard or equivalent to standard definitions, so don’t reject everything I say out of hand either.
## The Origins of Calculus
Calculus was developed by Isaac Newton and Gottfried Leibniz. It’s not quite clear who invented it first; it’s not unlikely that they invented it independently of each other, inspired by similar problems. What we do know is that Leibniz published his calculus first and it’s his notations that we still use today. Newton (of course) claimed he invented it first and he used it to prove, that an inverse square law like the one in his theory of gravity would in fact imply elliptical planetary orbits. It’s an astonishing feat of intellect – this guy basically came up with a working, mathematical theory of gravity to explain planetary orbits, and invented completely new mathematics just to prove that it works.
Calculus is (quote Wikipedia) the mathematical study of continuous change. Its basic objects of interest are continuous functions on the real numbers (often described as “functions whose graph can be drawn in one stroke without lifting the pen”) and its most important notions (besides continuity) are derivatives and (basically the inverse to derivatives) integrals. Nowadays, we define the latter using limits of sequences, and those we define using $$\epsilon$$-$$\delta$$-criteria, which we have to thank Augustin-Louis Cauchy and Karl Weierstrass for.
However, in Newton’s and Leibniz’ times the “limit of a sequence” wasn’t yet a well-defined notion; instead, they used infinitesimal numbers in the development of their theories. So here’s approximately their thought process:
Assume we have some continuous function $$f$$. As an example, let’s say $$f(x)=\frac{1}{10}x^2$$. Its graph looks like this:
Question: What is the slope of that function at the point $$x=4$$? I mean, obviously the function is increasing to the right, but how fast is it increasing? Obviously it’s not increasing “at the same speed” everywhere – otherwise the graph would just be a straight line. So, how can we find out “how fast” the function is increasing at the specific point $$x=4$$ – and what does that even mean?
Well, let’s look at two points instead: e.g. $$x_1=4$$ and $$x_2=6$$. How fast does the function grow in the interval from $$x_1$$ to $$x_2$$? Now this we can answer: we know $$f(x_1)=\frac{1}{10}\cdot 4^2=\frac85$$ and $$f(x_2)=\frac{1}{10}\cdot 6^2=\frac{18}{5}$$. So the function has grown by $$f(x_2)-f(x_1)=\frac{10}{5}=2$$. That’s an absolute growth of 2 in the interval of length $$x_1-x_2=2$$.
Which means: on average the function grows by a factor of $$\frac{f(x_2)-f(x_1)}{x_2-x_1}=1$$ in that interval:
That’s how we measure speeds in practice: Note at which time $$x_1$$ e.g. a car passes a fixed point $$y_1$$, at which time $$x_2$$ it passes a second point $$y_2$$ and divide the distance $$y_2-y_1$$ by the time it took, i.e. $$x_2-x_1$$. This will give you the average speed in the time period from $$x_1$$ to $$x_2$$.
But of course, it doesn’t give you the exact slope at the singular point $$x_1=4$$. But it might give you an idea how to get there: If we decrease the distance between $$x_1$$ and $$x_2$$ (assuming the function doesn’t do weird stuff in between), we will be somewhat closer to the exact slope. For example, if we pick $$x_2=5$$, then $$f(x_2)=\frac{5}{2}$$ and thus the average growth is $$\frac{f(x_2)-f(x_1)}{x_2-x_1}=\frac{9}{10}$$.
And here’s Newton’s and Leibniz’ mental leap: If we decrease the distance between $$x_1$$ and $$x_2$$ to the point where it is infinitesimally small, then we will get the exact slope of $$f$$ at the point $$x_1$$ (or $$x_2$$ – the difference between the slopes will also be infinitesimally small)!
So, let’s assume we have some infinitesimally small $$\partial x$$ (whatever that means), then the derivative $$f'(x)$$ of $$f$$ (i.e. the slope of $$f$$ at the point $$x$$) is given by $$\displaystyle\frac{f(x+\partial x)-f(x)}{\partial x}$$.
For our function, that means:
$$\displaystyle f'(x)=\frac{f(x+\partial x)-f(x)}{\partial x}=\frac{\frac{1}{10}(x+\partial x)^2-\frac{1}{10}x^2}{\partial x}=\frac{1}{5}x + \frac 1{10}\partial x$$
…and (so the reasoning goes) since $$\partial x$$ is just an infinitesimally small number and hence ultimately negligible, we can ignore it and get $$f'(x)=\frac 15 x$$, and hence we finally get the exact value $$f'(4)=\frac 45$$.
Obviously, there are problems with that reasoning: What the hell are those “infinitesimal numbers” that are suddenly introduced, that I can can apparently add and multiply and divide by (I mean – I can’t divide by zero, but I can divide by something that’s “infinitely close” to zero?), but then in the end I just ignore them? What’s that all about? Is this supposed to make sense? And if $$\partial x$$ is “infinitesimally small”, shouldn’t that mean that $$\frac 1{\partial x}$$ would have to be infinitely large? Does that still make sense? What’s going on here? Aaaaaaaah!
Well… the thing is… it sort-of works. At least for relatively simple functions as the exemplary one I used it yields meaningful results, regardless of how weird the reasoning to justify the method is. But infinitesimals were never quite satisfactory, which is why Cauchy and Weierstrass tried to put the whole thing on a more solid basis.
Interestingly enough, this whole infinitesimal stuff was actually formally grounded in a rigorous way in the 20th century (and resurrected as “non-standard calculus”). But the way “standard” mathematicians interpret and think about calculus and real numbers in general is in terms of cauchy sequences, limits and $$\epsilon$$-$$\delta$$-criteria, so let’s explain the modern foundation for calculus now.
## Sequences, Limits and Differentiability
Definition: A sequence of rationals is simply a function $$a:\mathbb N\to\mathbb Q$$ – i.e. a function that maps each natural number $$n\in\mathbb N$$ to some rational number $$a(n)\in\mathbb Q$$.
Sequences are usually denoted as $$(a_n)_{n\in\mathbb N}$$ (or in short just $$(a_n)$$) and the individual elements as $$a_i$$ (instead of $$a(i)$$ – i.e. we just write the function argument as an index).
So, why are sequences interesting? Consider the following two examples:
1. $$(a_n) = (1,2,3,4\ldots)$$ (i.e. $$a(n):=n$$) and
2. $$(b_n)=(1,\frac12,\frac13,\frac14,\ldots)$$ (i.e. $$b(n):=\frac1n$$).
There’s something fundamentally different about the two: Obviously, if we increase $$n$$, the first sequence $$(a_n)$$ will strictly increase as well, while the second one $$(b_n)$$ strictly decreases. Okay, that’s not too interesting, but if we look closer, we notice that the first sequence is also unbounded: Pick an arbitrarily large number $$r$$ – at some point the first sequence will grow larger than $$r$$ (just pick any natural number $$k$$ larger than $$r$$, then $$a_k=k>r$$). For the second sequence however, we can give a lower bound; e.g. $$-1$$. Even though $$(b_n)$$ strictly decreases, it will never become smaller than $$-1$$.
But of course, we can give a “better” lower bound than $$-1$$ – namely $$0$$. This is also a lower bound, because all the elements of $$b_n$$ are strictly positive; hence no element will ever be $$\leq0$$. In fact, $$0$$ is the largest lower bound (or infimum) of the sequence, and the larger a natural number $$k$$ we choose, the closer the sequence element $$b_k$$ will be to $$0$$.
It’s consequently not completely absurd to suggest that the sequence $$(b_n)$$ approaches $$0$$ in such a way, that we may meaningfully say that $$0$$ is the limit of the sequence $$(b_n)$$. In contrast, $$(a_n)$$ does not seem to have such a limit – the sequence just gets larger and larger with no bound in sight (we could say that the limit of the sequence is “infinity”, but infinity is not a number per se, and infinities are – without a careful formal treatment! – problematic anyway). We say the sequence $$(b_n)$$ converges towards $$0$$, and the sequence $$(a_n)$$ diverges. Now let’s properly define those two terms:
Definition: Let $$(a_n)$$ be a sequence of rationals. Assume there is some rational number $$L$$ such that the following holds:
For any arbitrarily small rational number $$\epsilon>0$$ there is some index $$n_\epsilon\in\mathbb N$$ such that for any index $$k>n_\epsilon$$ the distance $$\mid a_k – L\mid$$ is smaller than $$\epsilon$$. In logical notation: $\forall\varepsilon>0\;\exists n_\varepsilon\in\mathbb N\;\forall k>n_\varepsilon\; \mid a_k – L\mid <\epsilon$
Then we say the sequence $$(a_n)$$ converges to $$L$$ and write $$\displaystyle \lim_{n\mapsto\infty}a_n=L$$ or $$\lim a_n=L$$.
If no such $$L$$ exists, we say the sequence diverges.
Okay, this looks a bit complicated, so let’s explain it in more detail: We say a sequence $$(a_n)$$ converges to some number $$L$$, if we can get “arbitrarily close” to $$L$$ by making the index $$n$$ of our sequence larger. This “arbitrarily close” we can express formally by thinking about it as a kind of game: You tell me how close to $$L$$ you want to be, by giving me an (arbitrarily small) distance $$\epsilon\in\mathbb Q$$. Then I’ll give you an index $$n_\epsilon$$ in return, such that all subsequent elements in the sequence are closer to $$L$$ than your chosen distance $$\epsilon$$ – i.e. for all subsequent indices $$k>n_\epsilon$$, we have $$\mid a_k-L\mid<\epsilon$$. If I can always give you such an initial index, no matter how small a distance $$\epsilon$$ you choose, then I can adequately say that the sequence converges towards $$L$$.
Alright? So far, so good. Now we can use limits of sequences to define the limit of a function $$f$$ at a point $$p_0$$. Why should we? Well, look at the function $$f(x)=\frac{x^2-1}{x-1}$$, for example. This function is not well-defined at $$x=1$$, because then the denominator would be $$0$$ – i.e. $$f(1)$$ “doesn’t exist”. But, you know, here’s what this function looks like:
In fact, the function is equal to the function $$x+1$$ everywhere except at $$x=1$$! Annoying, but if we build a sequence $$(a_n)$$ that converges to $$1$$ (for example the sequence $$(1+1,1+\frac{1}{2},1+\frac{1}{3},1+\frac{1}{4},\ldots$$), then we can define $$f(1)$$ as the limit of the sequence $$(f(a_n))$$ (the resulting, now everywhere-defined, function is called the continuous extension of $$f$$), which happens to work out nicely and give us $$f(1)=2$$. Problem solved!
…eeeexcept, of course, that this only makes sense if the sequence $$(f(a_n))$$ converges at all, and – more notably – that the limit does not depend on the specific sequence $$(a_n)$$. So instead of defining the limit of a function using sequences, we will use another $$\epsilon$$-$$\delta$$-criterion:
Definition: Let $$f:A\to\mathbb Q$$ be a function on rationals (i.e. $$A\subseteq\mathbb Q$$) and $$p_0\in\mathbb Q$$. If there is some number $$L$$ such that the following holds:
For every arbitrarily small rational number $$\epsilon>0$$, there exists some $$\delta_\epsilon>0$$ such that for every $$x\in A$$ with $$\mid x-p_0\mid<\delta_\epsilon$$ we have $$\mid f(x)-L\mid<\epsilon$$. In logical Notation: $\forall\epsilon>0\;\exists\delta_\epsilon>0\;\forall x\in A\; (\mid x-p_0\mid<\delta_\epsilon \;\Longrightarrow\; \mid f(x)-L\mid<\epsilon).$
Then we call $$L$$ the limit of $$f$$ at $$p_0$$ and write $$\displaystyle \lim_{x\mapsto p_0}f(x)=L$$.
The idea being a similar game as in the definition of convergence for sequences: You tell me any arbitrarily small distance $$\epsilon\in\mathbb Q$$ to the (supposed) limit $$L$$ you want to have, and in return I will give you a distance $$\delta_\epsilon$$, such that if any $$x\in A$$ is closer to $$p_0$$ than $$\delta_\epsilon$$, then $$f(x)$$ will be closer to $$L$$ than $$\epsilon$$. If I can always give you such a $$\delta$$, no matter which $$\epsilon$$ you pick, then I win and $$L$$ is indeed the limit of $$f$$ at $$p_0$$.
Alright, and now we can finally define derivatives using function limits – the idea being, that instead of picking an “infinitesimal number”, we take the function limit of the quotients:
Definition: Let $$f:A\to\mathbb Q$$ be a function on rationals and $$p_0\in\mathbb Q$$. If the limit $\lim_{x\mapsto p_0}\frac{f(x)-f(p_0)}{x-p_0}$
exists, we call $$f$$ differentiable at $$p_0$$. If $$f$$ is differentiable at every point in $$A$$ we call $$f$$ differentiable, and the function $f’ : A \to \mathbb Q\qquad f'(x):=\lim_{y\mapsto x}\frac{f(y)-f(x)}{y-x}=\lim_{h\mapsto 0}\frac{f(x+h)-f(x)}h$
the derivative of $$f$$.
You’ll note, that this is exactly what Newton and Leibniz did; except that we got rid of those pesky infinitesimals and only used notions, that are formally and rigorously defined – there’s no room for ambiguity anymore. Furthermore, all of this works perfectly and beautifully – for example, all of the following highly desirable properties (assuming all the occuring limits exist) hold and can be easily proven using the above definitions (left as an exercise):
1. The limit of a convergent sequence is unique,
2. $$\displaystyle \left(\lim a_n\right) + \left(\lim b_n\right) = \lim (a_n + b_n)$$
3. $$\displaystyle \left(\lim a_n\right) \cdot \left(\lim b_n\right) = \lim (a_n \cdot b_n)$$
4. $$\displaystyle \left( \lim_{x\mapsto x_0} f(x) \right) + \left( \lim_{x\mapsto x_0} g(x) \right) = \lim_{x\mapsto x_0} (f(x) + g(x))$$
5. $$\displaystyle \left( \lim_{x\mapsto x_0} f(x) \right) \cdot \left( \lim_{x\mapsto x_0} g(x) \right) = \lim_{x\mapsto x_0} (f(x) \cdot g(x))$$
6. $$\lim a_n = \lim b_n$$ if and only if $$\lim (a_n – b_n) = 0$$
7. $$\displaystyle \frac 1{\lim a_n} = \lim\frac{1}{a_n} \qquad \frac{1}{\lim_{x\mapsto x_0}f(x)}=\lim_{x\mapsto x_0}\frac 1{f(x)}$$
…and we didn’t even touch the real numbers yet!
(Next post on John Gabriel: Calculus 102 (Cauchy Sequences and the Real Numbers)) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9609425663948059, "perplexity": 287.7340575694548}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780056752.16/warc/CC-MAIN-20210919065755-20210919095755-00021.warc.gz"} |
http://tex.stackexchange.com/questions/77909/space-between-lines | # Space between lines
I try to typeset title pages and use LARGE and bold for this. The problem is that the lines seem to be too close to each other. I think it looks bad. Is this the way it is supposed to be? Is there a better way to do this?
\begin{document}
\begin{center}
{\LARGE\bf An \textit{a priori} Typology of Sentential Negation from an HPSG Perspective}\\[\baselineskip]
\end{center}
\end{document}
Edit: OK what I really wanted isa title page like the stuff below. I wanted to have space between the lines. This was the reason for baselineskip. And the whole reason for messing around with the font sizes are the descriptions by google scholar that tell pdf producers to use fonts in different sizes for title and author and rest.
\documentclass[11pt,a4paper,fleqn,draft]{article}
\begin{document}
\begin{center}
{\LARGE\bf An \textit{a priori} Typology of Sentential Negation from\\[1mm] an HPSG Perspective}\\[\baselineskip]
{\large Joshua Crowgey}\\[\baselineskip]
University of Washington\\[3\baselineskip]
Proceedings of the HPSG 2012 Conference\\[\baselineskip]
Department of Linguistics, Chungnam National University Daejeon, South Korea\\[\baselineskip]
Stefan M{\"u}ller (Editor)\\[\baselineskip]
2012\\[\baselineskip]
CSLI Publications\\[\baselineskip]
http://csli-publications.stanford.edu/
\end{center}
\newpage
\end{document}
-
related (but not quite duplicate) questions: Incorrect line spacing when using \Large in a center environment, Inconsistent line spacing. i've added the tag [line-spacing] for better retrieval. – barbara beeton Oct 16 '12 at 19:21
I've added an implementation of your updated post. The concepts still remain the same - using \par whenever you're in a group that has localized font formatting, and adding whitespace to simulate \par. Adding \bigskip or \medskip of \vspace*{3\bigskipamount} are just other ways of increasing the whitespace between document elements. – Werner Oct 16 '12 at 21:14
Insert \par inside the {\LARGE...} group and remove the line break at the end, since "There's no line here to end." Or, just drop the group formatting entirely, letting the font selection be scoped by the center environment:
\documentclass{article}
\begin{document}
\begin{center}
{\LARGE\bfseries An \textit{a priori} Typology of Sentential Negation from an HPSG Perspective}
\end{center}
\begin{center}
\LARGE\bfseries An \textit{a priori} Typology of Sentential Negation from an HPSG Perspective
\end{center}
\end{document}
The motivation here is that TeX only sets a paragraph once it's fully read it. So, without a blank line or an indication like \par the paragraph isn't set correctly with the change in \baselineskip.
Please note that the \it, \bf, etc. font macros are deprecated because they do not use the new font selection scheme introduced with LaTeX2e. Please use {\itshape ..}, {\bfseries ..} or \textit{..}, \textbf{..} instead. See Does it matter if I use \textit or \it, \bfseries or \bf, etc. and Will two-letter font style commands (\bf, \it, …) ever be resurrected in LaTeX? for more information.
With your updated post, here are some things to try, using the concepts described above:
\documentclass[11pt,a4paper,fleqn,draft]{article}
\usepackage{url}% http://ctan.org/pkg/url
\begin{document}
\begin{center}
{\LARGE\bf An \textit{a priori} Typology of Sentential Negation from\par an HPSG Perspective\par}
\bigskip
{\large Joshua Crowgey\par}
University of Washington
\vspace*{3\bigskipamount}
Proceedings of the HPSG 2012 Conference
\bigskip
Department of Linguistics, Chungnam National University Daejeon, South Korea
\medskip
Stefan M{\"u}ller (Editor)
\medskip
2012
\medskip
CSLI Publications
\medskip
\url{http://csli-publications.stanford.edu/}
\end{center}
\newpage
\end{document}
-
or just remove the {} group and let LARGE be scoped by the center environment which will insert the paragraph end automatically. – David Carlisle Oct 16 '12 at 20:18
@DavidCarlisle: Thanks. That's a tad cleaner. – Werner Oct 16 '12 at 20:25
1. Add \usepackage{setspace}.
2. To change line spacing for whole document use \setstretch{baselinestretch}. For some custom text use environment: \begin{spacing}{2.5} ... \end{spacing}.
That's, I think, what's you need.
-
It depends on the document class you are using but you can also use the \title{}. And see Does it matter if I use \textit or \it, \bfseries or \bf, etc
\documentclass{article}
\title{\textbf{\LARGE An \emph{a priori} Typology of Sentential Negation from an HPSG Perspective}}
\date{}
\begin{document}
\maketitle
\vspace{-3\baselineskip} % If you wish to remove the space for date and author
Start of the text part ....
\end{document}
- | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9350269436836243, "perplexity": 4945.736792158277}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-49/segments/1416400380368.73/warc/CC-MAIN-20141119123300-00215-ip-10-235-23-156.ec2.internal.warc.gz"} |
http://www.scholarpedia.org/article/Talk:Phase_model | # Talk:Phase model
## Round 2 of review
This is a great article. I only have a few small suggestions for now: - Add units for phase in the third panel of Fig. 1. It seems that phase is in terms of the period T of oscillation. Since there are three definitions of phase of oscillation mentioned in the main text, maybe the figure caption should state which one is used. - The phase of oscillation is actually the geometric phase. - Although largely arbitrary, the zero-phase point can be chosen to correspond to an event of interest.
Authors: Done. thank you.
I think this is a very nice article. It is concise, and offers the most important facts about phase models. A couple of suggestions: the user is pointed to Fig. 1 for the definition of the phase of oscillation, but the symbol for the phase does not appear in the figure, so this may be confusing.
Authors: The caption was changed to reflect that; thank you.
The models reviewed are very nice - however, I really missed pulse coupled oscillators, or perhaps something more like Haken's lighthouse model. As these are very useful, it may be good to mention them explicitly.
Possible extension: a phase reduction sometimes also makes sense in chaotic systems, and it may be worth mentioning that the assumption of periodicity may be relaxed. The same is true for stochastic system (Stratonovich has some nice chapters on this). However, if the goal is to be concise, these can probably be omitted.
Authors: We mention that the phase of oscillation could be defined for chaotic oscillators and make a link to Pikovsky et al. book.). Mentioning stochastic systems is way outside the scope of this article.
Carmen's contribution "They are stable if the slope of the graph is negative at the intersection." The previous statement is given without explanation. Please explain why a negative slope implies stability..
Authors: Carmen, the stability of $$x'=H(x)$$ follows from the condition $$H'<0$$. I am not sure what is confusing in this statement. I changed it a bit, hoping that it would be clear what slope of what graph I mean here.
H(chi) = -omega Eugene the above is only true for small delta phi.
Authors: The wording is changed.
When you measure the phase resetting it is in units of time, or normalized time. This is not equal to frequency or normalized frequency.
Authors: This is true. We removed the comment that the H function corresponds to phase resetting (inserted by a reviewer). The H function has the units of phase/time, i.e., the frequency.
Say you have a normalized intrinsic period of 1 and an intrinsic frequency of 1. Then say you have a phase reset (delta phi) of 0.1 so that your new period is 0.9. Then your change in frequency is not delta phi but delta w: (delta phi)/(1 - delta phi) = 0.1111 not 0.1 However, for weak coupling , delta phi and delta w are both small so it works, but why not write it in terms of delta phi or at least state that delta phi and delta w are the same for small delta phi/delta w
Authors: These points are valid for pulse-coupled oscillators (and for the Windree's method of finding Q), but, as you say, they are not important when the coupling strength is infinitezimal. The article Phase response curve is a better place to discuss these issues.
Reviewer A response: If you convolve the iPRC with the coupling waveform (synaptic current), what you get is the phase resetting curve obtained using the current waveform as the perturbation, provided that you are in the linear weak coupling regime. Netoff et al Beyond two-cell networks: experimental measurements of neuronal responses to multiple synaptic inputs, J Computational Neuroscience, clearly show this ( see the equation top left on page 290) and should optimally be cited. This is an important point that aids intuitive understanding and should not be omitted, in my opinion, and other than that the article looks great. It is also clear from this article that the appropriate units for the H function are phase, and based on argument 2 above, the units for the H function is not frequency but phase, however as you say in the article if w is small it doesn't matter. This is a required assumption for the analysis to work. Some confusion results because the units of the iPRC are different from those of the PRC itself. Preyer and Butera Physical Review letters 95, 138103 (2005) make the same point in their Equation 2 except that their iPRC is computed with respect to conductance. This is the only thing I feel very strongly about, that the relationship of the H function to the PRC should be made clear. Netoff et al did an excellent job on that point.
Author (EMI): I added a reference to Netoff's article. However, the equation on page 290 does not have normalization by time (the 1/T factor), so the units are phases, not phase/time=frequency. In any case, I added a paragraph (starting "Computational neuroscience...") to mention the meaning of H in computational neuroscience.
My major suggestion is that I think that it would be much clearer to use an illustrative example with Q(phi_i), g(phi_i,phi_j) and H(chi) all illustrated as in Bard and John's chapter in the Koch book on methods in computational neuroscience. The units need to be clear and labeled on every graph.
Authors: Different components of the vector Q(theta) have different units of measument. This could be seen from the normalization condition Q f = 1. Since each component of f has its own units/time, the vector Q has units (unit1/time, unit2/time, ... unitn/time). It probably be too confusing to plot all of them in the figures. The function H has units of frequency, so the caption to fig.3 is modified accordingly.
This is just a comment, but another thing I have never understood is why you allow g(phi_i,phi_j) to depend on both the forced and forcing oscillator. To be consistent with linear systems theory, it seems like it should not depend upon the system that produces the impulse response. Wouldn't it be better to look at an infinitessimal perturbation in conductance rather than current, because the conductance depends only on the presynaptic oscillator and not the postsynaptic oscillator? I realize that you then could not utilize the adjoint, but you could still use Winfree's criterion.
Authors: The linear coupling is a particular case of the general case, and the theory works for the general case. In the example that you sugges, i.e., perturbation to the conductance, the effect depends on the timing of the perturbation and on the voltage of the postsynaptic neuron, as the conductance is multipled by the (V-E) term, where E is the Nerst potential for the ionic current whose conductancei s perturbed. Thus, even this simple case requires both variables, pre- and post-synaptic. By the way, the adjoint approach works whether or not the coupling involves the post-synaptic (receiving) oscilaltor. The coupling function does not participate in the adjoint equation, but only in the intergral for H.
Reviewer A response: The effect of a perturbation in conductance does depend upon both the pre and postsynaptic voltage, but the effect of the postsynaptic voltage is built in when you use a perturbation in conductance, if you assume that since the perturbation is given at a fixed phase, the membrane potential at that phase for the postsynaptic oscillator will be the same as it was when the phase resetting curve was generated. This is the assumption in Preyer and Butera and in Maran and Canavier, J. computational Neuroscience 2008. Therefore you do not have to consider the voltage in the postsynaptic oscillator in the simplest case, and you do not need the adjoint because you use the iPRC to conductance rather than current.
Author (EMI): This is an important point. We do not consider the simplest case, but the most general case. However, even in the simplest case, you still need the adjoint, or at least its voltage component, because it is the iPRC in response to the infinitezimal perturbation of the voltage equation. The iPRC is the same whether you perturb conductance, current, or anything else in the voltage equation. The nature of perturbation will only affect what you are convolving with the iPRC. Let me illustrate this using the equation on page 290 of Netoff et al. It is
${\rm PRC}(t)=\int_0^T {\rm iPRC}(\tau)g_{\rm syn}(\tau-t)[V_m(\tau)-V_s]d\tau$
Using the notations of this article, $Q={\rm iPRC}$ and $g_{ij}=g_{\rm syn}(\tau-t)[V_m(\tau)-V_s]$ Apparently, the function $$g_{ij}$$ must depend on the pre- and post-synaptic (forcing and forced) oscillator. If you require that $$g_{ij}$$ be independent of the post-synaptic oscillator, then you will only consider additive influences (current perturbation). | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9062618017196655, "perplexity": 603.3252226164627}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267156192.24/warc/CC-MAIN-20180919102700-20180919122700-00192.warc.gz"} |
http://www.thespectrumofriemannium.com/tag/neutrino-mixing/ | ## LOG#136. Flavor ν mixtures.
Neutrino oscillations are one of the most surprising “sounds” in the whole Universe. Since neutrinos do oscillate/mix, they are massive. And due to mass, they can experiment “mixing” or “changes” of flavor (mass and flavor basis are different!). Even more, … Continue reading
## LOG#127. Basic Neutrinology(XII).
When neutrinos pass through matter or they propagate in a medium (not in the vacuum), a subtle and potentially important effect occurs. This is called the MSW effect (Mikheyev-Smirnov-Wolfenstein effect). It is pretty similar to a refraction of light in … Continue reading
## LOG#126. Basic Neutrinology(XI).
Why is the case of massive neutrinos so relevant in contemporary physics? The full answer to this question would be very long. In fact, I am making this long thread about neutrinology in order you understand it a little bit. … Continue reading
## LOG#120. Basic Neutrinology(V).
Supersymmetry (SUSY) is one of the most discussed ideas in theoretical physics. I am not discussed its details here (yet, in this blog). However, in this thread, some general features are worth to be told about it. SUSY model generally … Continue reading
## LOG#116. Basic Neutrinology(I).
This new post ignites a new thread. Subject: the Science of Neutrinos. Something I usually call Neutrinology. I am sure you will enjoy it, since I will keep it elementary (even if I discuss some more advanced topics at some … Continue reading | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8818796873092651, "perplexity": 2014.1867473502173}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891812405.3/warc/CC-MAIN-20180219052241-20180219072241-00619.warc.gz"} |
https://www.iaa.es/seminars/extreme-blazars | # Extreme Blazars
Blazars are jetted active Galactic Nuclei with the jet axis oriented close to the line of sight of the observer. Non-thermal emission processes in the jet cover the whole electromagnetic spectrum from radio wavelengths to TeV gamma rays, with a characteristic double-humped Spectral Energy Distribution (SED). Relativistic amplification effects on the observed fluxes make their jets ideal candidates for detection at any wavelength. A physically motivated “blazar sequence” has been proposed years ago, although is still heavily disputed as being due to observational selection biases. The sequence relates bolometric power to peak frequencies and luminosities of the two-humped Spectral Energy Distribution of blazars: the frequencies shift to higher energies ,and luminosities dim, along with decreasing power of the sources. At the low end of the luminosity range a subclass of blazars with extreme properties lies, so far poorly populated given that it eludes the best survey of the gamma-ray sky that is currently available, the one built over the data from the Fermi/LAT satellite. Nevertheless these extreme blazars are of the utmost scientific interest, standing at the crossroads of high energy astrophysics of the sources, astro-particle physics, cosmology and fundamental physics.
Fecha:
29/04/2021 - 12:30
Conferenciante:
Dr. Giacomo Bonnoli
Filiación: | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9482918977737427, "perplexity": 2990.6524434393446}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103947269.55/warc/CC-MAIN-20220701220150-20220702010150-00422.warc.gz"} |
https://physics.stackexchange.com/questions/498255/wave-equation-derivation/498259 | # Wave Equation derivation
I'm curious about part of the derivation of the wave equation as is done in all references that I've seen so far (I'm gonna reproduce only the part that's puzzling me).
We apply Newton's second law to the motion of a piece of vibrating string in the vertical direction. Let's call the vertical direction $$y$$, and the horizontal direction $$x$$. Then we limit ourselves to an infinitesimal part of the string, and we write down the mass of this piece as $$\rho \Delta x$$. The fact that we could replace the length of this portion of the string $$l$$ with $$\Delta x$$ is because we assumed small oscillations: $$l\approx \sqrt{(\Delta x)^2+(\Delta y)^2}\approx \Delta x$$, as the displacement of string from the equilibrium position $$\Delta y$$ was assumed to be very small.
Now the equation $$\partial^2y/\partial t^2=c (\partial^2y/\partial x^2)$$ is derived after some more calculations ($$c$$ is just some constant). This equation is derived based on the assumption that the oscillation is small ($$\Delta y\approx 0$$) but it's obviously satisfied for non small oscillations (e.g. sine and cosine type of waves) as well.
So how does the proof generalize to the case of non-small oscillations? In other words, why does a string oscillation which is not necessarily small, satisfies the wave equation as well?
• – jacob1729 Aug 22 '19 at 21:27
Here are derivations of the 1D wave equation that do not depend on the physics of a string (or even physics at all). So there are no approximations.
If the solutions of the wave equation are specified to be of the form f(x-ct), f(x+ct), then the wave equation can be derived from that alone:
https://physics.stackexchange.com/a/403761/45664
Or just by examining the geometry alone without using physics the wave equation can be derived:
https://physics.stackexchange.com/a/110842/45664
So then its a matter of seeing whether the physics of a particular situation fits the wave equation.
The wave equation for a string is indeed only true for small heights and is, as a result, only an approximation. There perhaps exists a more accurate model with a slightly altered wave equation for large heights but this is the simplest case to show how the wave equation can manifest itself in even everyday application.
There is also an important concept that must be understood here. The wave equation is a mathematical equation i.e. it represents the equation that equates the Laplacian of some quantity to its second time derivative:
$$$$\nabla^2 \Psi =\frac{1}{c^2}\frac{\partial^2 \Psi}{\partial t^2}$$$$
where $$c$$ is some characteristic speed. For your case, the 1D string, $$\Psi$$ represents string height, and the laplacian becomes $$\frac{\partial^2 \Psi}{\partial x^2}$$. Many other physical systems can be said to similarly follow this equation or variations of it. As a result, you don't derive it but show that a system follows it.
Students are usually introduced to the wave equation by analyzing a vibrating string, because this can be done using only Newtonian mechanics. The resulting equation applies only to small-amplitude vibrations of the string.
However, that equation, generalized to three dimensions, applies to electromagnetic waves of any amplitude in vacuum, at least in classical electromagnetism. That wave equation, derivable from Maxwell’s equations, is not an approximation. And, unless you are a musician, EM waves are more interesting and important than vibrating violin strings.
In fact, the propagation of all non-interacting elementary particles, not just photons, in vacuum is described by the wave equation, without it being a small-amplitude approximation.
The wave amplitude does have to be small for the simple wave equation to hold for a wave on a taut string. Sinusoidal waves also have to be of small amplitude (specifically, $$A\ll\lambda$$ in which $$A$$ is amplitude and $$\lambda$$ is wavelength).
When we derive the wave equation for a taut string we assume that the tension in the string is constant (as implied by your analysis leading to $$\ell=\Delta x$$). We also make various small angle approximations. These only hold for small amplitude waves.
why does a string oscillation which is not necessarily small, satisfies the wave equation as well?
The wave equation as written in the Question doesn’t work for all physical waves. Some waves have propagation speeds that vary with frequency or amplitude, for example.
Simplifications and approximations went into the model-creation that underlies that wave equation. When those are good, the equation models the physical situation well. But when not, not. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 18, "wp-katex-eq": 0, "align": 0, "equation": 1, "x-ck12": 0, "texerror": 0, "math_score": 0.976601779460907, "perplexity": 297.42086771435544}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655881984.34/warc/CC-MAIN-20200703091148-20200703121148-00336.warc.gz"} |
https://www.physicsforums.com/threads/momentum-and-center-of-mass.298648/ | # Homework Help: Momentum and Center of Mass
1. Mar 10, 2009
### TwinGemini14
1. Ball 1, ball 2, and ball 3 have masses 1 kg, 2 kg, and 3 kg respectively, and are initially arranged at x = 5 m, x = 10 m, and x = 15 m respectively (see picture). y = 5 m for all three balls. Now suppose that a net force of 1 N is applied to ball 1 in the +y direction and a net force of 2 N is applied to ball 2 in the -y direction. The net force on ball 3 is zero.
What is the magnitude of the acceleration of the center of mass of the three-ball system?
A) 0.00 m/s2
B) 0.17 m/s2
C) 0.33 m/s2
D) 1.00 m/s2
E) 1.33 m/s2
----
So I first drew out the picture of the masses. Since the net force on the system is 1N downward, I did...
1N / (m1+m2+m3) = 1/6 = 0.16666. So the answer is B?
============
A block of mass 50 kg is sliding along in the +x direction (to the right) with initial magnitude of momentum p0 = 2450 kg*m/s on a horizontal frictionless surface.
Then a constant force F = 14 N is applied as shown in the above drawing, where the angle of the force above horizontal is 20°.
2) One second after the force is applied, the momentum of the block in the ground frame will have:
A) decreased in magnitude
B) been conserved
C) increased in magnitude
----
The momentum has increased in magnitude since the object is accelerating due to a net force, correct? C?
----
3) After the force is applied, the total mechanical energy of the block will:
A) be less than it was before the force was applied
B) remain the same as it was before the force was applied
C) be greater than it was before the force was applied
----
C. Since the kinetic energy is gradually increasing due to a constant acceleration given by the applied force, the total mechanical energy is increasing.
----
I already submitted this to a previous forum but reworked the problems. Does anybody see anything wrong with these? Thanks in advance.
2. Mar 11, 2009
### Kurdt
Staff Emeritus
The first looks ok to me. The second two are harder to tell without the diagram but they seem reasonable. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9785338640213013, "perplexity": 488.88736982158053}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583510853.25/warc/CC-MAIN-20181016155643-20181016181143-00261.warc.gz"} |
https://en.wikibooks.org/wiki/Real_Analysis/Foreword | # Real Analysis/Foreword
## Foreword
### What is Analysis?
Prof. Elliot Lieb of Princeton University defines analysis as the "art of taking limits", and further adds that "estimates are the heart and soul of analysis". Mathematics is often roughly subdivided into analysis, algebra and topology, so the coverage of each of these fields is quite broad. This book is concerned in particular with analysis in the context of the real numbers — there are many other fields of analysis, such as complex analysis, functional analysis and harmonic analysis. It will first develop the basic concepts needed for the idea of functions, then move on to the more analysis-based topics.
### Analysis or Calculus?
Analysis is concerned with primarily the same topics as Calculus, such as limits, derivatives, and integrals, but in a mathematical way rather than in a simply practical way. Before you study analysis, you may want to study calculus; you will end up repeating much of the same material when you come back to analysis, but you will understand its practical significance. It may seem like a wasteful duplication of effort, but you will feel much more comfortable with many of the basic concepts of analysis.
On the other hand, when studying calculus you may be dismayed at the frequent statement of rules for performing various operations with little or no justification. The study of analysis puts all these on a formal basis and provides that justification.
### Preliminaries
In much of analysis, arguments must be constructed very carefully, and it must be possible to make statements very precisely. To this end, it is important to be familiar with the notation of mathematical logic, in particular the 'for all' ( ${\displaystyle \forall }$ ) and 'there exists' ( ${\displaystyle \exists }$ ) notations. This notation will be used in definitions and proofs throughout the book, and it is essential to understand what these symbols mean and how they relate to each other before attempting to understand the material presented.
The core of real analysis uses very little from other areas of mathematics. Some general fundamentals that will be useful include Set Theory, especially the sections on infinite sets and cardinality. In many places, but particularly for work on Sequences, an understanding of induction and recursion (on ${\displaystyle \mathbb {N} }$) is important; however, nothing will be required that goes beyond a standard introduction to proof technique. For some of the more advanced topics, some knowledge of Topology may be helpful. Familiarity with the concepts of Calculus will make this book easier to get through, but not necessary as all relevant concepts will be defined in the text when needed.
Note also that some minimal background in Linear Algebra is assumed in the chapter on Multivariable analysis, but this is optional as no other chapter is dependent on this one. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 3, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9032425880432129, "perplexity": 298.76609496195795}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128320865.14/warc/CC-MAIN-20170626203042-20170626223042-00643.warc.gz"} |
http://tex.stackexchange.com/questions/42347/two-equal-acronym-what-should-i-do?answertab=oldest | # Two equal acronym - What should I do?
I'm using the `acronym` package.
I have in two chapters the acronym MAC one time for "Media Access Control" and in the second chapter as "Message Authentication Code".
How can I use both MACs?
Is a other package better?
-
Maybe the answer to this question will answer yours too: tex.stackexchange.com/questions/42214/… – Philipp Jan 26 '12 at 12:10
You definitely should use different acronyms/abbreviations for the two. Otherwise, it wouldn't be possible to distinguish between the two.
You could typeset one in small caps (`\textsc{}`) and one in a monospaced typeface (`\texttt{}`) but it's not ideal.
Another solution is using `\textsc{mac}` for media access control and `\textsc{ma}u\textsc{c}` for message authentication code.
BTW, your problem isn't that uncommon. For example, consider the abbreviation `CSP`. I've found the following different usages:
• Communicating Sequential Process;
• Constraint Satisfaction Problem;
• Constant Sign Partition;
• 192 more abbreviations omitted
-
For small caps, use `\textsc{mac}`, not `\textsc{MAC}`. The latter looks just like `MAC`. – Michael Palmer Jan 26 '12 at 16:35
@MichaelPalmer Thanks. I've corrected it. – Marc van Dongen Jan 26 '12 at 19:43 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8655184507369995, "perplexity": 4610.751392252216}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-06/segments/1422122123549.67/warc/CC-MAIN-20150124175523-00032-ip-10-180-212-252.ec2.internal.warc.gz"} |
https://www.degruyter.com/view/j/ans.2016.16.issue-1/ans-2015-5015/ans-2015-5015.xml | Show Summary Details
More options …
IMPACT FACTOR 2018: 1.650
CiteScore 2018: 1.49
SCImago Journal Rank (SJR) 2018: 1.422
Source Normalized Impact per Paper (SNIP) 2018: 0.865
Mathematical Citation Quotient (MCQ) 2018: 1.19
Online
ISSN
2169-0375
See all formats and pricing
More options …
GO
Volume 16, Issue 1
# On the Blow-Up of Solutions to Liouville-Type Equations
Tonia Ricciardi
• Corresponding author
• Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli Federico II, Via Cintia, Monte S. Angelo, 80126 Napoli, Italy
• Email
• Other articles by this author:
/ Gabriella Zecca
• Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli Federico II, Via Cintia, Monte S. Angelo, 80126 Napoli, Italy
• Email
• Other articles by this author:
Published Online: 2015-12-08 | DOI: https://doi.org/10.1515/ans-2015-5015
## Abstract
We estimate some complex structures related to perturbed Liouville equations defined on a compact Riemannian 2-manifold. As a byproduct, we obtain a quick proof of the mass quantization and we locate the blow-up points.
MSC 2010: 35J20; 35J60
## 1 Introduction
In the article [6], Nagasaki and Suzuki considered the Liouville-type problem
$\left\{\begin{array}{cccc}\hfill -\mathrm{\Delta }u& =\rho f\left(u\right)\hfill & & \hfill \text{in}\mathrm{\Omega },\\ \hfill u& =0\hfill & & \hfill \text{on}\partial \mathrm{\Omega },\end{array}$(1.1)
where $\mathrm{\Omega }\subset {ℝ}^{2}$ is a smooth bounded domain, $\rho >0$, and $f:ℝ\to ℝ$ is a smooth function such that
$f\left(t\right)={e}^{t}+\phi \left(t\right)\mathit{ }\text{with}\phi \left(t\right)=o\left({e}^{t}\right)\text{as}t\to +\mathrm{\infty }.$(1.2)
Equations of the form (1.1) are of actual interest in several contexts, including turbulent Euler flows, chemotaxis, and the Nirenberg problem in geometry; see, e.g., [5] and the references therein. A recent example is given by the mean field equation
$\left\{\begin{array}{cccc}\hfill -\mathrm{\Delta }u& =\lambda {\int }_{\left[-1,1\right]}\frac{\alpha {e}^{\alpha u}\mathcal{𝒫}\left(d\alpha \right)}{{\iint }_{\left[-1,1\right]×\mathrm{\Omega }}{e}^{\alpha u}\mathcal{𝒫}\left(d\alpha \right)𝑑x}\hfill & & \hfill \text{in}\mathrm{\Omega },\\ \hfill u& =0\hfill & & \hfill \text{on}\partial \mathrm{\Omega },\end{array}$(1.3)
which was derived in [7] for turbulent flows with variable intensities, where $\mathcal{𝒫}\in \mathcal{ℳ}\left(\left[-1,1\right]\right)$ is a probability measure related to the vortex intensity distribution. In this case, setting
$f\left(t\right)={\int }_{\left[-1,1\right]}\alpha {e}^{\alpha t}\mathcal{𝒫}\left(d\alpha \right),\rho =\lambda {\left({\iint }_{\left[-1,1\right]×\mathrm{\Omega }}{e}^{\alpha u}\mathcal{𝒫}\left(d\alpha \right)𝑑x\right)}^{-1},$
it is readily seen that if $\mathcal{𝒫}\left(\left\{1\right\}\right)>0$, then along a blow-up sequence, (1.3) is of the form (1.1). See [10, 11, 13, 12] for more details, where the existence of solutions by variational arguments and blow-up analysis are also considered. Blow-up solution sequences for (1.3) have also been recently constructed in [9] following the approach introduced in [4].
In [6], Nagasaki and Suzuki derived a concentration-compactness principle for (1.1), mass quantization, and the location of blow-up points, under some additional technical assumptions for f. More precisely, they assumed that
$|\phi \left(t\right)-{\phi }^{\prime }\left(t\right)|\le \mathcal{𝒢}\left(t\right)\mathit{ }\text{for some}\mathcal{𝒢}\in {C}^{1}\left(ℝ,ℝ\right)\text{satisfying}\mathcal{𝒢}\left(t\right)+|{\mathcal{𝒢}}^{\prime }\left(t\right)|\le C{e}^{\gamma t}\text{with}\gamma <\frac{1}{4}$(1.4)
and
$f\left(t\right)\ge 0\mathit{ }\text{for all}t\ge 0.$(1.5)
By a complex analysis approach, they established the following result.
#### ([6])
Let f satisfy assumptions (1.2), (1.4), and (1.5). Let ${u}_{n}$ be a solution sequence to (1.1) with $\rho \mathrm{=}{\rho }_{n}\mathrm{\to }\mathrm{0}$. Suppose ${u}_{n}$ converges to some nontrivial function ${u}_{\mathrm{0}}$. Then,
${u}_{0}\left(x\right)=8\pi \sum _{j=1}^{m}{G}_{\mathrm{\Omega }}\left(x,{p}_{j}\right)$
for some ${p}_{\mathrm{1}}\mathrm{,}\mathrm{\dots }\mathrm{,}{p}_{m}\mathrm{\in }\mathrm{\Omega }$, $m\mathrm{\in }ℕ$, where ${G}_{\mathrm{\Omega }}$ denotes the Green’s function for the Dirichlet problem on Ω. Furthermore, at each blow-up point ${p}_{j}$, $j\mathrm{=}\mathrm{1}\mathrm{,}\mathrm{\dots }\mathrm{,}m$, there holds that
${\nabla \left[{G}_{\mathrm{\Omega }}\left(x,{p}_{j}\right)+\frac{1}{2\pi }\mathrm{log}|x-{p}_{j}|\right]|}_{x={p}_{j}}+\nabla \left[\sum _{i\ne j}{G}_{\mathrm{\Omega }}\left({p}_{j},{p}_{i}\right)\right]=0.$
The original estimates in [6] are involved and require the technical assumption $\gamma \in \left(0,\frac{1}{4}\right)$. It should be mentioned that this assumption was later weakened to the natural assumption $\gamma \in \left(0,1\right)$ in [14], by taking a different viewpoint on the line of [1].
Here, we are interested in revisiting the complex analysis framework introduced in [6]. In particular, we study the effect of the lower-order terms which naturally appear when the equation is considered on a compact Riemannian 2-manifold. We observe that, although the very elaborate key ${L}^{\mathrm{\infty }}$-estimate obtained in [6], namely, Proposition 1.2 below, may be extended in a straightforward manner to the case of manifolds (see Appendix A for the details), the lower-order terms are naturally estimated only in ${L}^{1}$. Therefore, we are led to consider an ${L}^{1}$-framework, which turns out to be significantly simpler and which holds under the weaker assumption $\gamma \in \left(0,\frac{1}{2}\right)$. As a byproduct, we obtain a quick proof of mass quantization and blow-up point location for the case $\gamma \in \left(0,\frac{1}{2}\right)$.
In order to state our results, for a function $u\in {C}^{2}\left(\mathrm{\Omega }\right)$, we define the quantity
$S\left(u\right)=\frac{{u}_{z}^{2}}{2}-{u}_{zz},$(1.6)
where
${\partial }_{z}=\frac{{\partial }_{x}-i{\partial }_{y}}{2},{\partial }_{\overline{z}}=\frac{{\partial }_{x}+i{\partial }_{y}}{2}.$
Then, if u is a solution to (1.1), we have
${\partial }_{\overline{z}}\left[S\left(u\right)\right]=-\frac{\rho }{4}{u}_{z}\left[f\left(u\right)-{f}^{\prime }\left(u\right)\right]=\frac{\rho }{4}{u}_{z}\left[\phi \left(u\right)-{\phi }^{\prime }\left(u\right)\right].$
In particular, in the Liouville case $f\left(u\right)={e}^{u}$, the function $S\left(u\right)$ is holomorphic. Therefore, the complex derivative ${\partial }_{\overline{z}}\left[S\left(u\right)\right]$ may be viewed as an estimate of the “distance” between the equation in (1.1) and the standard Liouville equation.
We recall that the main technical estimate in [6] is given by the following proposition.
#### ([6])
Let ${u}_{\rho }$ be a blow-up sequence for (1.1). Assume (1.2), (1.4), and (1.5). Then,
${\parallel {\partial }_{\overline{z}}S\left(u\right)\parallel }_{{L}^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)}=\frac{\rho }{4}{\parallel \nabla {u}_{\rho }\left({f}^{\prime }\left({u}_{\rho }\right)-f\left({u}_{\rho }\right)\right)\parallel }_{{L}^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)}\to 0.$
It is natural to expect that corresponding results should hold on a compact Riemannian 2-manifold $\left(M,g\right)$ without boundary. We show that, in fact, the ${L}^{\mathrm{\infty }}$-convergence as stated in Proposition 1.2 still holds true on M (see Proposition A.1 in Appendix A). However, a modified point of view is needed in order to suitably locally define a function S corresponding to (1.6), such that the lower-order terms may be controlled, as well as to prove its convergence to a holomorphic function in some suitable norm, so that the mass quantization and the location of the blow-up points may be derived. As we shall see, our point of view holds under the weaker assumption $\gamma \in \left(0,\frac{1}{2}\right)$ and is significantly simpler than the original ${L}^{\mathrm{\infty }}$-framework.
More precisely, on a compact Riemannian 2-manifold without boundary $\left(M,g\right)$, we consider the problem
$\left\{\begin{array}{cccc}\hfill -{\mathrm{\Delta }}_{g}u& =\rho f\left(u\right)-{c}_{\rho }\hfill & & \hfill \text{in}M,\\ \hfill {\int }_{M}u𝑑x& =0,\hfill & & \end{array}$(1.7)
where ${c}_{\rho }=\rho {|M|}^{-1}{\int }_{M}f\left(u\right)𝑑x\in ℝ$, dx denotes the volume element on M, and ${\mathrm{\Delta }}_{g}$ denotes the Laplace–Beltrami operator. We assume that $f\left(t\right)={e}^{t}+\phi \left(t\right)$ satisfies (1.2) and, moreover, that
$|\phi \left(t\right)-{\phi }^{\prime }\left(t\right)|\le \mathcal{𝒢}\left(t\right)\mathit{ }\text{for some}\mathcal{𝒢}\in {C}^{1}\left(ℝ,ℝ\right)\text{satisfying}\mathcal{𝒢}\left(t\right)+|{\mathcal{𝒢}}^{\prime }\left(t\right)|\le C{e}^{\gamma t}\text{with}\gamma <\frac{1}{2}$(1.8)
and
$f\left(t\right)\ge -C\mathit{ }\text{for all}t\ge 0.$(1.9)
In the spirit of [3], we assume that along a blow-up sequence we have
$\rho {\int }_{M}f\left(u\right)𝑑x\le C.$(1.10)
In particular, without loss of generality, we may assume that
${c}_{\rho }\to {c}_{0}\mathit{ }\text{as}\rho \to {0}^{+}.$(1.11)
We note that (1.9) implies that $u\ge -C$. We now define the modified quantity corresponding to $S\left(u\right)$. Let $\mathcal{𝒮}=\left\{{p}_{1},\mathrm{\dots },{p}_{m}\right\}$ denote the blow-up set. Let $p\in \mathcal{𝒮}$ and denote $X=\left({x}_{1},{x}_{2}\right)$. We consider a local isothermal chart $\left(\mathrm{\Psi },\mathcal{𝒰}\right)$ such that ${\mathcal{ℬ}}_{\epsilon }\left(p\right)\subset \mathcal{𝒰}$, $\mathrm{\Psi }\left(p\right)=0$, ${\mathcal{ℬ}}_{\epsilon }\left(p\right)\cap \mathcal{𝒮}=\mathrm{\varnothing }$, $g\left(X\right)={e}^{\xi \left(X\right)}\left(d{x}_{1}^{2}+d{x}_{2}^{2}\right)$, and $\xi \left(0\right)=0$. For the sake of simplicity, we identify here functions on M with their pullback functions to $B=B\left(0,r\right)=\mathrm{\Psi }\left({\mathcal{ℬ}}_{\epsilon }\left(p\right)\right)$. We denote by ${G}_{B}\left(X,Y\right)$ the Green’s function of ${\mathrm{\Delta }}_{X}={\partial }_{{x}_{1}}^{2}+{\partial }_{{x}_{2}}^{2}$ on B. We set
$K\left(X\right)=-{\int }_{B}{G}_{B}\left(X,Y\right){e}^{\xi \left(Y\right)}𝑑Y+{c}_{1}z$(1.12)
with ${c}_{1}\in ℂ$ defined by
${{\partial }_{z}\left[\xi \left(z,\overline{z}\right)+{c}_{0}K\left(z,\overline{z}\right)\right]|}_{z=0}=0,$(1.13)
where ${c}_{0}$ is defined in (1.11). Let u denote a solution sequence to (1.7). We define $w\left(z\right)=u-{c}_{\rho }K$, so that $-\mathrm{\Delta }w={e}^{\xi }\rho f\left(u\right)$ in B. Finally, consider $S\left(w\right)$, where S is defined in (1.6). Our main estimate is given in the following theorem.
Assume that $f\mathit{}\mathrm{\left(}t\mathrm{\right)}\mathrm{=}{e}^{t}\mathrm{+}\phi \mathit{}\mathrm{\left(}t\mathrm{\right)}$ satisfies (1.2), (1.8), and (1.9). Let ${u}_{\rho }$ be a blow-up solution sequence for (1.7). Then,
• (i)
for every $1\le s<{\left(\gamma +\frac{1}{2}\right)}^{-1}$,
$\rho {\parallel \nabla {u}_{\rho }\left({f}^{\prime }\left({u}_{\rho }\right)-f\left({u}_{\rho }\right)\right)\parallel }_{{L}^{s}\left(M\right)}\to 0\mathit{ }\mathit{\text{as}}\rho \to {0}^{+};$
• (ii)
for every blow-up point $p\in \mathcal{𝒮}$ , the function $S\left(w\right)\to {S}_{0}$ in ${L}^{1}\left(B\right)$ as $\rho \to {0}^{+}$ , where ${S}_{0}$ is holomorphic in B.
Consequently, we derive the following corollary.
Assume that $f\mathit{}\mathrm{\left(}t\mathrm{\right)}\mathrm{=}{e}^{t}\mathrm{+}\phi \mathit{}\mathrm{\left(}t\mathrm{\right)}$ satisfies (1.2), (1.8), and (1.9). Suppose ${u}_{n}$ converges to some nontrivial function ${u}_{\mathrm{0}}$. Then,
${u}_{0}\left(x\right)=8\pi \sum _{j=1}^{m}{G}_{M}\left(x,{p}_{j}\right).$(1.14)
Moreover, for all $p\mathrm{\in }\mathcal{𝒮}$, we have the relation
${\left[{\nabla }_{X}\left(\sum _{q\in \mathcal{𝒮}\setminus \left\{p\right\}}{G}_{M}\left({\mathrm{\Psi }}^{-1}\left(X\right),q\right)+{G}_{M}\left({\mathrm{\Psi }}^{-1}\left(X\right),p\right)+\frac{1}{2\pi }\mathrm{log}|X|+\frac{1}{8\pi }\xi \left(X\right)\right)\right]|}_{X=0}=0.$(1.15)
We provide the proofs of Theorem 1.3 and Corollary 1.4 in Section 2. For the sake of completeness and in order to readily allow a comparison with the ${L}^{\mathrm{\infty }}$-framework employed in [6], in Appendix A we extend Proposition 1.2 to the case of Riemannian 2-manifolds without boundary.
Throughout this note, we denote by $C>0$ a constant whose actual value may vary from line to line.
## 2 Proof of Theorem 1.3
We begin by establishing the following result.
Let u be a solution to (1.7). For every $r\mathrm{>}\mathrm{0}$, we have
$r{\int }_{M}{e}^{-ru}{|\nabla u|}^{2}𝑑x\le C,$(2.1)
where $C\mathrm{=}C\mathit{}\mathrm{\left(}r\mathrm{,}M\mathrm{,}\phi \mathrm{,}{c}_{\mathrm{0}}\mathrm{\right)}$.
#### Proof.
We multiply the equation $-{\mathrm{\Delta }}_{g}u=\rho f\left(u\right)-{c}_{\rho }$ by ${e}^{-ru}$. Integrating, we have
$r{\int }_{M}{e}^{-ru}{|\nabla u|}^{2}𝑑x={\int }_{M}{e}^{-ru}{\mathrm{\Delta }}_{g}u𝑑x$$=-\rho {\int }_{M}{e}^{-ru}f\left(u\right)𝑑x+{c}_{\rho }{\int }_{M}{e}^{-ru}𝑑x$$\le \rho {\int }_{M}{e}^{-ru}|\phi \left(u\right)|𝑑x+{c}_{\rho }{\int }_{M}{e}^{rC}𝑑x$$\le \rho {\int }_{M}{e}^{-ru}|\phi \left(u\right)|𝑑x+{c}_{\rho }{e}^{rC}|M|,$
since $u\ge -C$. Using the assumptions on φ, there exists ${t}_{0}>0$ such that $|g\left(u\right)|<{e}^{u}$ for $u>{t}_{0}$, so that
$r{\int }_{M}{e}^{-ru}{|\nabla u|}^{2}𝑑x\le C+\rho \left({\int }_{\left\{u>{t}_{0}\right\}}{e}^{\left(1-r\right)u}𝑑x+{\int }_{\left\{u\le {t}_{0}\right\}}{e}^{-ru}|\phi \left(u\right)|𝑑x\right)\le C+\rho \left({\int }_{M}{e}^{u}𝑑x+{\int }_{\left\{u\le {t}_{0}\right\}}{e}^{-ru}|\phi \left(u\right)|𝑑x\right),$
and the claim follows using again the fact that $u\ge -C$. ∎
The following proposition proves Theorem 1.3(i).
Let u be a solution to (1.7). Then, for every $\mathrm{1}\mathrm{\le }s\mathrm{<}{\mathrm{\left(}\gamma \mathrm{+}\frac{\mathrm{1}}{\mathrm{2}}\mathrm{\right)}}^{\mathrm{-}\mathrm{1}}$ and for every $\epsilon \mathrm{>}\mathrm{0}$, we have
${\parallel \nabla u\left({f}^{\prime }\left(u\right)-f\left(u\right)\right)\parallel }_{{L}^{s}\left(M\right)}\le C{\rho }^{-\gamma -\epsilon }$
for $\mathrm{0}\mathrm{<}\rho \mathrm{<}\mathrm{1}$.
#### Proof.
In view of (1.8), we have
$0\le |f\left(u\right)-{f}^{\prime }\left(u\right)|\le C{e}^{\gamma u}.$
Hence,
${\parallel \left(f\left(u\right)-{f}^{\prime }\left(u\right)\right)\nabla u\parallel }_{{L}^{s}}\le C{\parallel {e}^{\gamma u}\nabla u\parallel }_{{L}^{s}}.$(2.2)
Moreover, (1.10) implies that
${\int }_{M}{e}^{u}𝑑x\le c{\rho }^{-1}.$
Then, for every $1\le q<{\gamma }^{-1}$, using Hölder’s inequality we have
${\parallel {e}^{\gamma u}\parallel }_{{L}^{q}\left(M\right)}\le C{|M|}^{\frac{1}{q}-\gamma }{\rho }^{-\gamma }.$(2.3)
Let $0. By Lemma 2.1, for
$q=\frac{s+\frac{r}{\gamma }}{1-\frac{s}{2}}<\frac{1}{\gamma },$
using Hölder’s inequality again, we have
${\parallel {e}^{\gamma u}\nabla u\parallel }_{{L}^{s}\left(M\right)}^{s}={\int }_{M}{e}^{\left(s\gamma +r\right)u}\left({e}^{-ru}{|\nabla u|}^{s}\right)𝑑x\le {\left({\int }_{M}{e}^{\gamma uq}𝑑x\right)}^{1-\frac{s}{2}}{\left({\int }_{M}{e}^{-2ru}{|\nabla u|}^{2}𝑑x\right)}^{\frac{s}{2}}\le C{\parallel {e}^{\gamma u}\parallel }_{{L}^{q}\left(M\right)}^{s+\frac{r}{\gamma }}.$(2.4)
Then, by (2.3) and (2.4) we have
${\parallel {e}^{\gamma u}\nabla u\parallel }_{{L}^{s}\left(M\right)}\le C{\rho }^{-\gamma -\frac{r}{s}}.$(2.5)
Combining (2.2) and (2.5), the claim is proved. ∎
Let $p\in \mathcal{𝒮}$. We denote by $\left(\mathrm{\Psi },\mathcal{𝒰}\right)$ an isothermal chart satisfying
$\overline{\mathcal{𝒰}}\cap \mathcal{𝒮}=\left\{p\right\},\mathrm{\Psi }\left(\mathcal{𝒰}\right)=\mathcal{𝒪}\subset {ℝ}^{2},\mathrm{\Psi }\left(p\right)=0,g\left(X\right)={e}^{\xi \left(X\right)}\left(d{x}_{1}^{2}+d{x}_{2}^{2}\right),\xi \left(0\right)=0,$
where $X=\left({x}_{1},{x}_{2}\right)$ denotes a coordinate system on $\mathcal{𝒪}$. We consider $\epsilon >0$ sufficiently small so that $\mathcal{ℬ}\left(p,\epsilon \right)⋐\mathcal{𝒰}$ and let $B=B\left(0,r\right)=\mathrm{\Psi }\left(\mathcal{ℬ}\left(p,\epsilon \right)\right)$. The Laplace–Beltrami operator ${\mathrm{\Delta }}_{g}$ is then mapped to the operator ${e}^{-\xi \left(X\right)}{\mathrm{\Delta }}_{X}$ on $\mathcal{𝒪}$, where ${\mathrm{\Delta }}_{X}={\partial }_{{x}_{1}^{2}}^{2}+{\partial }_{{x}_{2}^{2}}^{2}$. By ${G}_{B}\left(X,Y\right)$ we denote the Green’s function of ${\mathrm{\Delta }}_{X}$ on B, namely,
$\left\{\begin{array}{cccc}\hfill -{\mathrm{\Delta }}_{X}{G}_{B}\left(X,Y\right)& ={\delta }_{Y}\hfill & & \hfill \text{in}B,\\ \hfill {G}_{B}\left(X,Y\right)& =0\hfill & & \hfill \text{on}\partial B.\end{array}$
We recall from (1.12) that
$K\left(X\right)=-{\int }_{B}{G}_{B}\left(X,Y\right){e}^{\xi \left(Y\right)}𝑑Y+{c}_{1}z$
with ${c}_{1}$ the constant defined by (1.13), namely,
${{\partial }_{z}\left[\xi \left(z,\overline{z}\right)+{c}_{0}K\left(z,\overline{z}\right)\right]|}_{z=0}=0,$
where ${c}_{0}={lim}_{\rho \to 0}{c}_{\rho }$. Then, $K\in {C}^{\mathrm{\infty }}\left(B\right)$ and
${\mathrm{\Delta }}_{X}K={e}^{\xi }\mathit{ }\text{in}\overline{B}.$
Let ${u}_{\rho }$ be a blow-up solution sequence for (1.7). As $\rho \to 0$, $u\to {u}_{0}$ in ${C}_{\mathrm{loc}}^{\mathrm{\infty }}\left(M\setminus \mathcal{𝒮}\right)$, $u-{u}_{0}\in {W}^{1,q}\left(M\right)$ for $1\le q<2$, and $f\left(u\right)\to f\left({u}_{0}\right)$ in ${C}_{\mathrm{loc}}^{\mathrm{\infty }}\left(M\setminus \mathcal{𝒮}\right)$, we have ${\mathrm{\Delta }}_{g}u\to {\mathrm{\Delta }}_{g}{u}_{0}$ in ${C}_{\mathrm{loc}}^{\mathrm{\infty }}\left(M\setminus \mathcal{𝒮}\right)$, so that
${\mathrm{\Delta }}_{g}{u}_{0}={c}_{0}\mathit{ }\text{in}M\setminus \mathcal{𝒮}.$
We consider the following functions defined in B:
$\stackrel{~}{u}=u\circ {\mathrm{\Psi }}^{-1},$${\stackrel{~}{u}}_{0}={u}_{0}\circ {\mathrm{\Psi }}^{-1},$$w\left(z\right)=\stackrel{~}{u}-{c}_{\rho }K,$${w}_{0}\left(z\right)={\stackrel{~}{u}}_{0}-{c}_{0}K,$$S\left(w\right)={w}_{zz}-\frac{1}{2}{w}_{z}^{2},$${S}_{0}={w}_{0zz}-\frac{1}{2}{w}_{0z}^{2}.$
The following proposition proves Theorem 1.3(ii).
The complex function ${S}_{\mathrm{0}}$ defined in (2.6) is holomorphic in B and $S\mathrm{\to }{S}_{\mathrm{0}}$ in ${L}^{\mathrm{1}}\mathit{}\mathrm{\left(}B\mathrm{\right)}\mathrm{.}$
#### Proof.
By (2.6) we have
$-{\mathrm{\Delta }}_{X}w=\rho f\left(\stackrel{~}{u}\right){e}^{\xi }\mathit{ }\text{and}\mathit{ }{w}_{z}={\stackrel{~}{u}}_{z}-{c}_{\rho }{K}_{z}.$
Then, using ${\mathrm{\Delta }}_{X}=4{\partial }_{z\overline{z}}$ we compute
${\partial }_{\overline{z}}\left[S\left(w\right)\right]=\frac{1}{4}\left({\partial }_{z}{\mathrm{\Delta }}_{X}w-{w}_{z}{\mathrm{\Delta }}_{X}w\right)$$=-\frac{\rho }{4}{e}^{\xi }\left(f\left(\stackrel{~}{u}\right){\xi }_{z}+{\stackrel{~}{u}}_{z}{f}^{\prime }\left(\stackrel{~}{u}\right)\right)+\frac{\rho }{4}{e}^{\xi }f\left(\stackrel{~}{u}\right)\left({\stackrel{~}{u}}_{z}-{c}_{\rho }{K}_{z}\right)$$=\frac{\rho }{4}{e}^{\xi }\left(f\left(\stackrel{~}{u}\right)-{f}^{\prime }\left(\stackrel{~}{u}\right)\right){\stackrel{~}{u}}_{z}-\frac{\rho }{4}{e}^{\xi }f\left(\stackrel{~}{u}\right)\left({\xi }_{z}+{c}_{0}{K}_{z}\right)+\left({c}_{0}-{c}_{\rho }\right)\frac{\rho }{4}{e}^{\xi }f\left(\stackrel{~}{u}\right){K}_{z}.$(2.7)
Using (2) we derive that
${\partial }_{\overline{z}}S\to 0\mathit{ }\text{in}{L}^{1}\left(B\right).$(2.8)
Indeed, this follows by Proposition 2.2, (1.13), and by the fact that $|\rho f\left(\stackrel{~}{u}\right)|\stackrel{*}{\to }a{\delta }_{0}\left(dx\right)$ for some $a>0$. On the other hand, by (2.6), since $u\to {u}_{0}$ in ${C}_{\mathrm{loc}}^{\mathrm{\infty }}\left(M\setminus \mathcal{𝒮}\right)$, we have
$w\to {w}_{0}\mathit{ }\text{in}{C}_{\mathrm{loc}}^{\mathrm{\infty }}\left(\overline{B}\setminus \left\{0\right\}\right)$
and then
$S\to {S}_{0}\mathit{ }\text{in}{C}_{\mathrm{loc}}^{\mathrm{\infty }}\left(\overline{B}\setminus \left\{0\right\}\right).$(2.9)
At this point, we set $\mathrm{\Xi }=\left({\xi }_{1},{\xi }_{2}\right)$ and $\zeta ={\xi }_{1}+i{\xi }_{2}$ and we observe that by the Cauchy integral formula we may write
$\left[S\left(w\right)\right]\left(\zeta \right)=\frac{1}{\pi }{\int }_{B}\frac{{\partial }_{\overline{z}}S\left(z\right)}{\zeta -z}𝑑X+\frac{i}{2\pi }{\int }_{+\partial B}\frac{\left[S\left(w\right)\right]\left(z\right)}{\zeta -z}𝑑z=g\left(\zeta \right)+h\left(\zeta \right).$(2.10)
We have
$h\left(\zeta \right)\to {h}_{0}\left(\zeta \right)=\frac{i}{2\pi }{\int }_{+\partial B}\frac{{S}_{0}\left(z\right)}{\zeta -z}𝑑z\mathit{ }\text{in}{C}_{\mathrm{loc}}^{0}\left(B\right)$(2.11)
and ${h}_{0}$ is holomorphic in B. On the other hand, we have
$g\to 0\mathit{ }\text{in}{L}^{1}\left(B\right).$(2.12)
To prove (2.12), it is sufficient to observe that for every $z\in B=B\left(0,r\right)$, we have $B\subset B\left(z,2r\right)$ and then
${\parallel g\parallel }_{{L}^{1}\left(B\right)}\le {\iint }_{B×B}|{\partial }_{\overline{z}}S\left(z\right)|\frac{1}{|\zeta -z|}𝑑X𝑑\mathrm{\Xi }\le {\int }_{B}|{\partial }_{\overline{z}}S\left(z\right)|\left({\int }_{B\left(z,2r\right)}\frac{1}{|\zeta -z|}𝑑\mathrm{\Xi }\right)𝑑X=4\pi r{\int }_{B}|{\partial }_{\overline{z}}S\left(z\right)|𝑑X,$
which tends to zero by (2.8). Combining (2.10), (2.11), and (2.12), we have
$S\to {h}_{0}\mathit{ }\text{in}{L}^{1}\left(B\right)\text{as}\rho \to 0,$
and hence, up to subsequences,
$S\to {h}_{0}\mathit{ }\text{a.e. in}B\text{as}\rho \to 0,$
so that by (2.9),
${S}_{0}\left(\zeta \right)={h}_{0}\left(\zeta \right)\mathit{ }\text{for all}\zeta \in B\setminus \left\{0\right\}.$
This completes our proof. ∎
Finally, we use the following result from [2].
#### ([2])
For $B\mathrm{=}B\mathit{}\mathrm{\left(}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{\right)}\mathrm{\subset }{ℝ}^{n}$, $n\mathrm{\ge }\mathrm{2}$, the conditions $v\mathrm{\in }{W}^{\mathrm{1}\mathrm{,}p}\mathit{}\mathrm{\left(}B\mathrm{\right)}$, $\mathrm{1}\mathrm{<}p\mathrm{<}\mathrm{\infty }$, and $\mathrm{\Delta }\mathit{}v\mathrm{=}\mathrm{0}$ in $B\mathrm{\setminus }\mathrm{\left\{}\mathrm{0}\mathrm{\right\}}$ imply that $H\mathrm{=}v\mathrm{-}\mathrm{\ell }\mathit{}E$ is harmonic in B, where $\mathrm{\ell }$ is some constant and
$E\left(x\right)=\left\{\begin{array}{cc}{|x|}^{2-n}\hfill & \mathit{\text{if}}n>2,\hfill \\ \mathrm{log}|x|\hfill & \mathit{\text{if}}n=2.\hfill \end{array}$
Now, we are ready to prove Corollary 1.4. By ${G}_{M}$ we denote the Green’s function on the manifold M, defined by
$\left\{\begin{array}{cc}\hfill -{\mathrm{\Delta }}_{g}{G}_{M}\left(x,y\right)& ={\delta }_{y}-\frac{1}{|M|}\hfill \\ \hfill {\int }_{M}{G}_{M}\left(x,y\right)𝑑x& =0.\hfill \end{array}$
#### Proof of Corollary 1.4.
Assume that $p\in \mathcal{𝒮}$. Let us start by observing that ${w}_{0}$ in (2.6) is harmonic in $B\setminus \left\{0\right\}$ by definition and that ${w}_{0}\in {W}^{1,q}\left(B\right)$ for all $1. Hence, also by using Proposition 2.4, we have
${w}_{0}\left(z\right)=\mathrm{\ell }\mathrm{log}\frac{1}{|z|}+H\left(z\right),$
where H is harmonic in B and $\mathrm{\ell }\ne 0$. Then, using the fact that
${\partial }_{z}\mathrm{log}|z|=\frac{1}{2}{\partial }_{z}\mathrm{log}\left(z\overline{z}\right)={\left(2z\right)}^{-1},$
we compute
${w}_{0z}=-\frac{\mathrm{\ell }}{2z}+{H}_{z},{w}_{0zz}=\frac{\mathrm{\ell }}{2{z}^{2}}+{H}_{zz}.$
Therefore,
${S}_{0}={w}_{0zz}-\frac{1}{2}{w}_{0z}^{2}=\frac{\mathrm{\ell }}{2{z}^{2}}+{H}_{zz}-\frac{1}{2}{\left(\frac{\mathrm{\ell }}{2z}-{H}_{z}\right)}^{2}=\frac{\mathrm{\ell }\left(4-\mathrm{\ell }\right)}{8{z}^{2}}+\frac{\mathrm{\ell }}{2z}{H}_{z}+{H}_{zz}-\frac{1}{2}{H}_{z}^{2}.$
By Proposition 2.3, we know that ${S}_{0}$ is holomorphic. Hence, we can conclude that $\mathrm{\ell }=4$ and ${H}_{z}\left(0\right)=0$. Since
$H={w}_{0}-4\mathrm{log}\frac{1}{|z|}$(2.13)
is harmonic in B, we have
${\mathrm{\Delta }}_{X}\left({\stackrel{~}{u}}_{0}-4\mathrm{log}\frac{1}{|z|}\right)={c}_{0}{e}^{\xi }\mathit{ }\text{in}B\left(0,r\right)$
and, therefore,
${\mathrm{\Delta }}_{g}\left({u}_{0}\left(x\right)-8\pi {G}_{M}\left(x,p\right)\right)={c}_{0}-\frac{8\pi }{|M|}+{h}_{p}\mathit{ }\text{in}\mathcal{ℬ}\left(p,\epsilon \right)$
for some harmonic function ${h}_{p}$. Arguing similarly for each $p\in \mathcal{𝒮}=\left\{{p}_{1},{p}_{2},\mathrm{\dots },{p}_{m}\right\}$, we conclude that
${\mathrm{\Delta }}_{g}\left({u}_{0}\left(x\right)-8\pi \sum _{j=1}^{m}{G}_{M}\left(x,{p}_{j}\right)\right)={c}_{0}-\frac{8\pi m}{|M|}\mathit{ }\text{in}M.$
In particular, we obtain
${u}_{0}\left(x\right)-8\pi \sum _{j=1}^{m}{G}_{M}\left(x,{p}_{j}\right)=\text{constant}\mathit{ }\text{in}M.$
Observing that ${\int }_{M}{u}_{0}=0$, this completes the proof of (1.14). To obtain (1.15) it is sufficient to observe that, in view of (2.13) and (1.13), we have
$0={\frac{1}{8\pi }{\partial }_{z}H\left(X\right)|}_{X=0}$$={{\partial }_{z}\left[\sum _{q\in \mathcal{𝒮}}{G}_{M}\left({\mathrm{\Psi }}^{-1}\left(X\right),q\right)+\frac{1}{2\pi }\mathrm{log}|X|\right]|}_{X=0}-{\left[\frac{m}{|M|}{\partial }_{z}K\left(X\right)\right]|}_{X=0}$$={{\partial }_{z}\left[\sum _{q\in \mathcal{𝒮}}{G}_{M}\left({\mathrm{\Psi }}^{-1}\left(X\right),q\right)+\frac{1}{2\pi }\mathrm{log}|X|-\frac{1}{8\pi }\xi \left(X\right)\right]|}_{X=0}.$
Now, Corollary 1.4 is completely established. ∎
## A The ${L}^{\mathrm{\infty }}$-estimate on M
In this appendix, for the sake of completeness and in order to outline the original arguments in [6], so that the simplification of our ${L}^{1}$-approach may be seen, we check that Proposition 1.2 may be actually extended to (1.7) on a compact Riemannian 2-manifold $\left(M,g\right)$ without boundary with minor modifications. We consider a solution sequence for (1.7). We assume that f satisfies (1.2), (1.4), and (1.5). Moreover, we assume (1.10), so that ${c}_{\rho }\to {c}_{0}$ as $\rho \to {0}^{+}$. We show the following proposition.
Let u be a solution to (1.7). Then,
$\rho {\parallel \nabla u\left({f}^{\prime }\left(u\right)-f\left(u\right)\right)\parallel }_{{L}^{\mathrm{\infty }}\left(M\right)}\to 0\mathit{ }\mathit{\text{as}}\rho \to 0.$
The proof relies on the following relation, due to Obata.
#### ([8])
Let $w\mathrm{=}w\mathit{}\mathrm{\left(}x\mathrm{\right)}\mathrm{>}\mathrm{0}$ be a solution to
$\mathrm{\Delta }w=\frac{{|\nabla w|}^{2}}{w}+F\left(w\right)\mathit{ }\mathit{\text{on}}M,$(A.1)
where F is a ${C}^{\mathrm{1}}$-function. Then, there holds the identity
$divV=J+\frac{1}{2}{|\nabla w|}^{2}{w}^{-2}\left(F\left(w\right)+w{F}^{\prime }\left(w\right)\right),$(A.2)
where, in local coordinates,
${V}_{j}={w}^{-1}\left\{\nabla \left(\frac{\partial w}{\partial {x}_{i}}\right)\cdot \nabla w-\frac{1}{2}\frac{\partial w}{\partial {x}_{i}}\mathrm{\Delta }w\right\},j=1,2,$
and
$J={w}^{-1}\left\{\sum _{i,j=1}^{2}{\left(\frac{{\partial }^{2}w}{\partial {x}_{i}\partial {x}_{j}}\right)}^{2}-\frac{1}{2}{\left(\mathrm{\Delta }w\right)}^{2}\right\}\ge 0.$
Let u be a solution to (1.7). Then, for every $r\mathrm{>}\mathrm{0}$, there holds
$\rho {\int }_{M}{e}^{-ru}{|\nabla u|}^{2}\left(2rf\left(u\right)-{f}^{\prime }\left(u\right)\right)\le 2r{c}_{\rho }{\int }_{M}{e}^{-ru}{|\nabla u|}^{2}.$(A.3)
#### Proof.
Let u be a solution to (1.7). Denoting $w={e}^{-ru}$, it is easy to see that Obata’s assumption (A.1) is satisfied by the function w with
$F\left(w\right)=r{e}^{-ru}\left(\rho f\left(u\right)-{c}_{\rho }\right).$
On the other hand, we have
$F\left(w\right)+w{F}^{\prime }\left(w\right)=\rho {e}^{-ru}\left(2rf\left(u\right)-{f}^{\prime }\left(u\right)\right)-2r{e}^{-ru}{c}_{\rho }.$
In view of Obata’s identity (A.2), we conclude that
${\int }_{M}\frac{{|\nabla w|}^{2}}{{w}^{2}}\left(F\left(w\right)+wF\left(w\right)\right)\le 2{\int }_{M}divV=0.$
In particular, since
$\frac{\nabla w}{w}=-r\nabla u,$
by the last inequality we obtain
${\int }_{M}{r}^{2}{|\nabla u|}^{2}\left(F\left(w\right)+w{F}^{\prime }\left(w\right)\right)={r}^{2}\rho {\int }_{M}{e}^{-ru}{|\nabla u|}^{2}\left(2rf\left(u\right)-{f}^{\prime }\left(u\right)\right)-2{r}^{3}{c}_{\rho }{\int }_{M}{e}^{-ru}{|\nabla u|}^{2}\le 0.\mathit{∎}$
We note that combining (A.3) and (2.1), for $\frac{1}{2}, we obtain
$\rho {\int }_{M}{e}^{-ru}{|\nabla u|}^{2}f\left(u\right)\le C\left(1+\rho {\int }_{M}{e}^{-\left(r-\gamma \right)u}{|\nabla u|}^{2}\right).$(A.4)
Since $\gamma <\frac{1}{4}$, combining (2.1) and (A.4) we obtain
$\rho {\int }_{M}{e}^{-ru}{|\nabla u|}^{2}f\left(u\right)𝑑x\le C$
and then, since $u\ge -C$, using (2.1) again we have
$\rho {\int }_{M}{e}^{-ru}{|\nabla u|}^{2}|f\left(u\right)|𝑑x\le C\mathit{ }\text{if}\frac{1}{2}(A.5)
For $r>0$, we define
${G}_{r}\left(t\right)={\int }_{0}^{t}{e}^{-\frac{r}{2}s}\sqrt{|f\left(s\right)|}𝑑s.$
Then, (A.5) may be written in the form
${\parallel \nabla {G}_{r}\left(u\right)\parallel }_{{L}^{2}\left(M\right)}\le \frac{C}{\sqrt{\rho }}.$(A.6)
There holds
${\parallel {G}_{r}\left(u\right)\parallel }_{{L}^{1}\left(M\right)}\le \frac{C}{\sqrt{\rho }}.$(A.7)
#### Proof.
The proof can be easily obtained as in Lemma 2.1. Let us observe that in our assumption, for every $\frac{1}{2}, we have
${\int }_{\left\{x\in M:u\left(x\right)\ge 0\right\}}{G}_{r}\left(u\right)𝑑x\le \frac{2}{r}{\int }_{\left\{u\ge 0\right\}}\sqrt{|f\left(u\right)|}𝑑x\le C{\left({\int }_{M}|f\left(u\right)|𝑑x\right)}^{\frac{1}{2}}\le \frac{C}{\sqrt{\rho }}.$(A.8)
On the other hand, since $-u\le C$, we have
${\int }_{\left\{x\in M:u\left(x\right)\le 0\right\}}|{G}_{r}\left(u\right)|𝑑x\le C{\int }_{\left\{u\le 0\right\}}𝑑x{\int }_{u}^{0}{e}^{\frac{Cr}{2}}\le C{e}^{\frac{Cr}{2}}|M|\le C.$(A.9)
Combining (A.8) and (A.9), we conclude the proof of (A.7). ∎
Reducing (A.6) to
${\parallel \nabla {G}_{r}\left(u\right)\parallel }_{{L}^{p}\left(M\right)}\le \frac{C}{\sqrt{\rho }}\mathit{ }\text{for}1
and using (A.7) and the Sobolev embedding, we obtain
${\parallel {G}_{r}\left(u\right)\parallel }_{{L}^{{p}^{*}}\left(M\right)}\le \frac{C}{\sqrt{\rho }},\frac{1}{{p}^{*}}=\frac{1}{p}-\frac{1}{2}.$
Moreover, we have
${|f\left(t\right)|}^{\frac{1}{2\sigma }}\le C\left(|{G}_{r}\left(t\right)|+1\right)$
for $\sigma =\frac{1}{1-r}$ ($>2$). We choose $\frac{1}{2} such that
$\left(\gamma +\frac{1}{2}\right)\sigma <\frac{3}{2}.$(A.10)
Arguing as in [6], for every $\epsilon >0$, we obtain
${\parallel f\left(u\right)\parallel }_{{L}^{p}\left(M\right)}\le C{\rho }^{-\sigma +\frac{\sigma -1}{p}-\epsilon },1(A.11)
and, for $q>2$,
${\parallel \nabla u\parallel }_{{L}^{q}\left(M\right)}\le C{\rho }^{\left(-\frac{1}{2}+\frac{1}{q}\right)\left(\sigma -1\right)-\epsilon }.$(A.12)
Now, we conclude the proof of Proposition A.1.
#### Proof of Proposition A.1.
There holds
${\parallel \left({f}^{\prime }\left(u\right)-f\left(u\right)\right)\nabla u\parallel }_{{L}^{\mathrm{\infty }}\left(M\right)}\le C{\parallel {e}^{\gamma u}\nabla u\parallel }_{{L}^{\mathrm{\infty }}\left(M\right)}=\frac{C}{\gamma }{\parallel \nabla {e}^{\gamma u}\parallel }_{{L}^{\mathrm{\infty }}\left(M\right)}.$(A.13)
Moreover, by (1.7) we have
$-{\mathrm{\Delta }}_{g}{e}^{\gamma u}=-{\gamma }^{2}{e}^{\gamma u}{|\nabla u|}^{2}+\rho \gamma {e}^{\gamma u}f\left(u\right)-{c}_{\rho }\gamma {e}^{\gamma u}\mathit{ }\text{in}M.$
Hence, for $p>2$, we have
${\parallel \nabla {e}^{\gamma u}\parallel }_{{L}^{\mathrm{\infty }}\left(M\right)}\le C\left({\parallel {\mathrm{\Delta }}_{g}{e}^{\gamma u}\parallel }_{{L}^{p}\left(M\right)}+{\parallel {e}^{\gamma u}\parallel }_{{L}^{1}\left(M\right)}\right)\le C\left({\parallel {e}^{\gamma u}{|\nabla u|}^{2}\parallel }_{{L}^{p}\left(M\right)}+\rho {\parallel {e}^{\gamma u}f\left(u\right)\parallel }_{{L}^{p}\left(M\right)}+{\parallel {c}_{\rho }{e}^{\gamma u}\parallel }_{{L}^{p}\left(M\right)}\right).$
Now, observing that ${e}^{u}\le C\left(f\left(u\right)+1\right)$, by (A.11) we obtain
$\rho {\parallel {e}^{\gamma u}f\left(u\right)\parallel }_{{L}^{p}\left(M\right)}\le C\rho {\parallel {e}^{\left(\gamma +1\right)u}\parallel }_{{L}^{p}\left(M\right)}=C\rho {\parallel {e}^{u}\parallel }_{{L}^{p\left(\gamma +1\right)}\left(M\right)}^{\gamma +1}\le C{\rho }^{\tau -\epsilon }$(A.14)
for every $\epsilon >0$ with
$\tau =1+\left(\gamma +1\right)\left(\frac{\sigma -1}{p\left(\gamma +1\right)}-\sigma \right)=1+\frac{\sigma -1}{p}-\sigma \left(\gamma +1\right).$(A.15)
Hence, as $p↓2$, we have
$\tau ↑1+\frac{1}{2}\left(\sigma -1\right)-\sigma \left(\gamma +1\right)>-1$(A.16)
by (A.10). On the other hand, by (2.3), for $1\le p<\frac{1}{\gamma }$, we have
${\parallel {c}_{\rho }{e}^{\gamma u}\parallel }_{{L}^{p}\left(M\right)}\le C{\rho }^{-\gamma }.$
Moreover, if $q>\frac{1}{2\gamma }$ $\left(>2\right)$, then
${\parallel {e}^{\gamma u}{|\nabla u|}^{2}\parallel }_{{L}^{p}\left(M\right)}\le {{\parallel {e}^{\gamma u}\parallel }_{{L}^{pq}\left(M\right)}\parallel \nabla u|\parallel }_{{L}^{2p{q}^{\prime }}\left(M\right)}^{2},$
where $q{q}^{\prime }=q+{q}^{\prime }$. By (A.12), for every $\epsilon >0$ and since $2p{q}^{\prime }>2$, we have
${\parallel \nabla u\parallel }_{{L}^{2p{q}^{\prime }}\left(M\right)}^{2}\le C{\rho }^{\left(-1+\frac{1}{p{q}^{\prime }}\right)\left(\sigma -1\right)-\epsilon }.$
Using again (A.11), for every $\epsilon >0$, we have
${\parallel {e}^{\gamma u}\parallel }_{{L}^{pq}\left(M\right)}\le C{\parallel {e}^{u}\parallel }_{{L}^{pq\gamma }\left(M\right)}^{\gamma }\le C{\rho }^{-\gamma \sigma +\frac{\sigma -1}{pq}-\epsilon }.$
Then, for every $\epsilon >0$, we have
${\parallel {e}^{\gamma u}{|\nabla u|}^{2}\parallel }_{{L}^{p}\left(M\right)}\le C{\rho }^{\tau -\epsilon }$(A.17)
with τ defined by (A.15). Combining (A.13)–(A.14) and (A.16)–(A.17), we complete the proof. ∎
## References
• [1]
Bethuel F., Brezis H. and Hélein F., Ginzburg–Landau Vortices, Progr. Nonlinear Differential Equations Appl. 13, Birkhäuser, Boston, 1994. Google Scholar
• [2]
Brézis H. and Lions P. L., A note on isolated singularities for linear elliptic equations, Mathematical Analysis and Applications, Part A, Adv. in Math. Suppl. Stud. 7A, Academic Press, New York (1981), 263–266. Google Scholar
• [3]
Brezis H. and Merle F., Uniform estimates and blow-up behavior for solutions of $-\mathrm{\Delta }u=V\left(x\right){e}^{u}$ in two dimensions, Comm. Partial Differential Equations 16 (1991), no. 8–9, 1223–1253. Google Scholar
• [4]
Esposito P., Grossi M. and Pistoia A., On the existence of blowing-up solutions for a mean field equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005), no. 2, 227–257. Google Scholar
• [5]
Lin C.-S., An expository survey on the recent development of mean field equations, Discrete Contin. Dyn. Syst. 19 (2007), no. 2, 387–410. Google Scholar
• [6]
Nagasaki K. and Suzuki T., Asymptotic analysis for two-dimensional elliptic eigenvalue problems with exponentially-dominated nonlinearities, Asymptot. Anal. 3 (1990), no. 2, 173–188. Google Scholar
• [7]
Neri C., Statistical mechanics of the N-point vortex system with random intensities on a bounded domain, Ann. Inst. H. Poincaré Anal. Non Linéaire 21 (2004), no. 3, 381–399. Google Scholar
• [8]
Obata M., The conjectures on conformal transformations of Riemannian manifolds, J. Differential Geom. 6 (1971), no. 2, 247–258. Google Scholar
• [9]
Pistoia A. and Ricciardi T., Concentrating solutions for a Liouville type equation with variable intensities in 2D-turbulence, preprint 2015, http://arxiv.org/abs/1505.05304. Google Scholar
• [10]
Ricciardi T. and Suzuki T., Duality and best constant for a Trudinger–Moser inequality involving probability measures, J. Eur. Math. Soc. (JEMS) 16 (2014), no. 7, 1327–1348. Google Scholar
• [11]
Ricciardi T. and Zecca G., Blow-up analysis for some mean field equations involving probability measures from statistical hydrodynamics, Differential Integral Equations 25 (2012), no. 3–4, 201–222. Google Scholar
• [12]
Ricciardi T. and Zecca G., Mean field equations with probability measure in 2D-turbulence, Ric. Mat. 63 (2014), no. 1, suppl., S255–S264. Google Scholar
• [13]
Ricciardi T. and Zecca G., Mass quantization and minimax solutions for Neri’s mean field equation in 2D-turbulence, J. Differential Equations 260 (2016), no. 1, 339–369. Google Scholar
• [14]
Ye D., Une remarque sur le comportement asymptotique des solutions de $-\mathrm{\Delta }u=\lambda f\left(u\right)$, C. R. Acad. Sci. Paris Sér I Math. 325 (1997), no. 12, 1279–1282. Google Scholar
Accepted: 2015-05-19
Published Online: 2015-12-08
Published in Print: 2016-02-01
The first author acknowledges the support of FP7-MC-2009-IRSES-247486 “MaNEqui”.
Citation Information: Advanced Nonlinear Studies, Volume 16, Issue 1, Pages 75–85, ISSN (Online) 2169-0375, ISSN (Print) 1536-1365,
Export Citation
## Citing Articles
[1]
F. De Marchis and T. Ricciardi
Nonlinear Analysis: Real World Applications, 2017, Volume 38, Page 222
[2]
Tonia Ricciardi and Ryo Takahashi
Calculus of Variations and Partial Differential Equations, 2016, Volume 55, Number 6 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 287, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9939634203910828, "perplexity": 1717.3626330146337}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986684854.67/warc/CC-MAIN-20191018204336-20191018231836-00363.warc.gz"} |
https://docs.numbas.org.uk/en/latest/authoring/example-gallery/standard-integral.html | # Apply a standard integral¶
Calculus students are typically given a table of ‘standard integrals’, which they can apply when they identify a suitable function in an integration.
This leaflet from mathcentre contains a typical table of standard integrals.
## Planning¶
### What does this question assess?¶
We’d like to establish whether the student can recognise a function with a standard integral and produce the appropriate integral.
This is a prerequisite skill for more complicated integration methods such as integration by parts or integration by substitution.
### What does the student have to do?¶
Some ideas:
• Show a table of integrals with some blanks that the student must fill in.
• Pick the integral of a given function from a list of expressions.
• Show a function, and ask the student to write its integral.
We’ll go with the last option, since it gives the student the most scope to come up with the answer on their own.
They’ll be shown the definition of a function $$f(x)$$. They must write the integral $$F(x) = \int f(x) \, \mathrm{d}x$$ with a constant of integration $$C$$.
### How might the student get the answer wrong?¶
They might:
• differentiate instead of integrating;
• forget the constant of integration;
• use a different letter for the constant of integration;
• apply the wrong integral from a table of standard integrals.
We could detect these mistakes and give appropriate feedback. Other errors such as typos will be handled by Numbas’ standard feedback routines.
### Sketch the structure of the question¶
The statement will define $$f(x)$$, which has been chosen from the list of standard integrals. We could give an expression involving several terms, such as $$e^{4x} + \sin(2x) + x^5$$, but if the student makes an error with any one of the terms it will be hard to award partial credit. A later question could assess this kind of expression once we’ve established that the student can apply the standard integrals individually.
There will be a single mathematical expression part, with a prompt to write the integral of $$f(x)$$.
We must decide whether to explicitly tell the student to use $$C$$ as the constant of integration. If we do, then the student is less likely to forget it - a common mistake when performing an indefinite integral. If we don’t, then we have to do some work when marking to establish which variable name they have used, and compare it with the expected answer. This could be assessed in a separate question, so for convenience we’ll tell the student to use $$C$$.
The correct answer is the integral corresponding to the chosen function, with a constant $$C$$ added on.
We will add alternative answers corresponding to the expressions we’d expect to see if the student differentiated instead of integrating, or forgot the constant of integration.
The advice section could show the table of standard integrals, or just mention that this function is in the table the student has already been given, and then identify the given function along with its integral.
## Implementation¶
Create a new question called Recognise and apply a standard integral, in Show all parts mode.
We will build the question up in stages, adding complexity as we go.
### The simplest version of this question¶
To begin with, let’s make a non-randomised version of the question. In the statement, write:
Let $f(x) = x^3$.
Then create a mathematical expression part, with the following prompt:
What is $\int f(x) \, \mathrm{d}x$?
Use $C$ for the constant of integration.
On the Marking settings tab, set the correct answer to x^4/4 + C.
Next, we’ll make alternative answers for each of the mistakes we expect to see. Click the Add an alternative answer button under this part on the right-hand side of the screen.
In Marking settings, under correct answer, write x^4/4. This alternative will be used if the student forgets to include the constant of integration but otherwise applies the correct standard integral. You might want to give some partial credit if the student makes this mistake - enter 0.5 in Marks to award half a mark.
In Feedback message, under Message if this alternative is used, write:
Did you forget to include a constant of integration?
Add another alternative, with correct answer 3x^2 for the case when the student differentiates instead of integrating. A student with a mechanical understanding of the rules might add on a constant of integration to this, so add a third alternative with correct answer 3x^2 + C.
It’s a good idea to name alternatives so you can see from the parts list what each one is for. Name the first alternative “Forgot constant of integration” and the other two “Differentiated - no constant” and “Differentiated - with constant”. These names aren’t shown to the student.
Finally, in the question’s Advice tab, write a short explanation of the answer:
$f(x) = x^3$.
From the table of standard integrals, the integral of $x^k$ with respect to $x$ when $k \neq -1$ is $\frac{1}{k+1} x^{k+1}$.
This is an indefinite integral, so we add an arbitrary constant of integration $C$.
Here, $k = 3$.
So
$\int f(x) \, \mathrm{d}x = \frac{1}{3+1} x^{3+1} + C = \frac{1}{4} x^4 + C$
Try this question out by clicking Test run.
This question works, but it only ever asks about one function, so isn’t a good assessment of the whole table of standard integrals. We could randomly choose from
### Introduce randomisation¶
An easy way to add randomisation into this question is to randomise the power of $$x$$. We’ll define a variable k to represent the exponent, so the student has to integrate $$x^k$$.
In the Variables tab, click Add a variable.
In the Name field, write k, and from the Data type drop-down, pick “Random number from a range”.
The range of values that $$k$$ can take needs to be carefully chosen:
• The case $$k = 0$$ corresponds to $$f(x) = x^0 = 1$$, which students typically memorise as .
• The case $$k = 1$$ corresponds to $$f(x) = x^1 = x$$, which doesn’t look quite the same as other cases.
• The case $$k = -1$$ has integral $$\ln(x)$$, which students typically memorise as a separate rule.
Fill out the Value field so it reads “A random number between 2 and 9 (inclusive) with step size 1”.
Now we need to use this variable throughout the question.
Change the statement to:
Let $f(x) = \simplify{ x^{k} }$.
Change the term:correct answer for the mathematical expression part to 1/{k+1} * x^{k+1} + C, and make corresponding changes to the alternative answers.
Finally, change the advice to:
$f(x) = \simplify{ x^{k} }$.
From the table of standard integrals, the integral of $x^k$ with respect to $x$ when $k \neq -1$ is $\frac{1}{k+1} x^{k+1}$.
Here, $k = \var{k}$.
This is an indefinite integral, so we add an arbitrary constant of integration $C$.
So
$\int f(x) \, \mathrm{d}x = \simplify[basic]{ 1/({k}+1) * x^({k} + 1)} + C = \simplify{ 1/{k+1} * x^{k+1} + C$
Try this question out by clicking Test run. Confirm that everything works as you’d expect for different values of $$k$$.
Note
We’ve had to substitute randomised values in LaTeX maths notation. It’s not as straightforward as you might expect; see the page on Substituting variables into displayed maths for more detail.
We have introduced some randomisation but we’re still only asking about one entry in the table of standard integrals.
### Choose the function to integrate from a list¶
Let’s make the question randomly choose one of $$e^{kx}$$, $$x^k$$, $$\cos(kx)$$ or $$sin(kx)$$ as the function to integrate.
There are two ways of doing this, each with their own benefits and drawbacks.
The first way is to work throughout the question with a long expression of the form $$c_1 e^{kx} + c_2x^k + c_3\cos(kx) + c_4\sin(kx)$$, and define the $$c_i$$ coefficients so that only one of them has the value 1, and the rest 0. The simplifier will remove the terms with zero coefficient, leaving just one term to integrate. It’s easy to set up the question variables for this but you have to write out the long expression in terms of $$c_i$$ throughout the question, making it hard to read as a question author. If we want to add more options for functions, the expression gets even longer.
The second way is to use JME expression variables to represent the function and its integral, and randomly pick one from a list. These are easier to use in question text and marking settings but need some more work at the variable generation stage.
We also need to think again about the definition of the variable $$k$$, because each of these functions behaves differently as $$k$$ changes. Fortunately, these functions all behave similarly when $$k \gt 1$$, as we’ve already chosen, so the definition of $$k$$ doesn’t need to change.
#### Method 1: Zero coefficients for unwanted terms¶
We want four coefficients, only one of which should have the value 1.
Define a variable named c with the following JME code:
shuffle([1,0,0,0])
This will produce a list with one one and three zeros, in random order.
Change the question statement to:
Let $f(x) = \simplify{ {c[0]} * e^({k}x) + {c[1]} * x^{k} + {c[2]} * cos({k}x) + {c[3]}*sin({k}x) }$.
Change the correct answer for the mathematical expression part to:
{c[0]} * (1/{k}) * e^({k}x) + {c[1]} * (1/{k+1}) * x^{k+1} + {c[2]} * (1/{k}) * sin({k}x) + {c[3]} * (-1/{k}) * cos({k}x)
Make corresponding changes to the alternative answers and the advice.
To add another option for the function, we would have to add another zero to the list used in c, and add another term to each occurrence of the long expression.
#### Method 2: Sub-expressions¶
We want to randomly pick a function from a list of options, and substitute in the coefficient k.
Each option has three parts: the function to present as $$f(x)$$, its integral, and its derivative. (The derivative will be used for the alternative answer which catches the case where the student differentiates instead of integrating)
Define a variable scenarios, with the following JME code definition:
[
[ "x^k", "1/(k+1)*x^(k+1)", "k^x^(k-1)" ],
[ "e^(k*x)", "1/k * e^(k*x)", "k*e^(k*x)" ],
[ "sin(k*x)", "-1/k * cos(k*x)", "k*cos(k*x)" ],
[ "cos(k*x)", "1/k * sin(k*x)", "-k*sin(k*x)" ]
]
This variable has four entries, each of which is a list containing three strings of JME code.
Next, a variable scenario will pick one of these at random:
random(scenarios)
Construct a sub-expression representing $$f(x)$$ by defining a variable named function as follows:
substitute(
["k": k],
expression(scenario[0])
)
This takes the first element in the chosen scenario, converts it to a expression value, then substitutes the value of k into it.
Check that the variable preview shows an expression such as e^(2x) for the value of function.
Add two more variables, integral and derivative, with definitions similar to that of function but using scenario[1] and scenario[2] respectively.
Change the question statement:
Let $f(x) = \var{function}$.
Set the correct answer for the mathematical expression part to:
{integral} + C
Make corresponding changes to the alternative answers.
For the advice, it would be helpful to give the student the generic form of their function, as it would appear in the table of standard integrals. For this, define a new variable generic_function:
expression(scenario[0])
and another variable generic_integral:
expression(scenario[1])
Finally, rewrite the advice:
$f(x) = \var{ {function} }$.
The integral of $\var{generic_function}$ with respect to $x$ is $\var{generic_integral}$.
This is an indefinite integral, so we add an arbitrary constant of integration $C$.
Here, $k = \var{k}$, so
$\int f(x) \, \mathrm{d}x = \var{ {integral} } + C$
## Evaluation¶
This question shows the student a randomly-chosen function to integrate, and gives appropriate feedback in response to some common mistakes.
This question uses:
• A mathematical expression part to mark an expression entered by the student.
• The \var and \simplify commands to substitute randomised values into LaTeX code <simpification-rules>.
• Alternative answers to recognise answers corresponding to common mistakes and give appropriate feedback.
• JME Sub-expressions to randomly choose from a list of available functions, and substitute in a randomised value.
For students who have trouble answering this question, you could add an information only part as a step containing either a link to the table of standard integrals, or the table itself.
A good question to follow this one might give the student an expression involving multiple terms to integrate, or use variables other than $$x$$ to check the depth of the student’s understanding of symbolic integration. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9057753682136536, "perplexity": 750.7825131353655}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141176256.21/warc/CC-MAIN-20201124111924-20201124141924-00249.warc.gz"} |
https://infoscience.epfl.ch/record/217747 | Infoscience
Journal article
Dynamic cantilever magnetometry of individual CoFeB nanotubes
We investigate single CoFeB nanotubes with hexagonal cross section using dynamic cantilever magnetometry (DCM). We develop both an analytical model based on the Stoner-Wohlfarth approximation and a broadly applicable numerical framework for analyzing DCM measurements of magnetic nanostructures. Magnetometry data taken at 4.2 K show the presence of a uniformly magnetized configuration at high external fields with mu M-0(s) = 1.3 +/- 0.1 T and nonuniform configurations at low fields. In this low-field regime, comparison between numerical simulations and DCM measurements supports the existence of flux-closure configurations. Crucially, evidence of such configurations is only apparent because of the sensitivity of DCM to single nanotubes, whereas conventional measurements of ensembles are often obscured by sample-to-sample inhomogeneities in size, shape, and orientation. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8754256367683411, "perplexity": 4870.869017054376}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-09/segments/1487501171281.53/warc/CC-MAIN-20170219104611-00431-ip-10-171-10-108.ec2.internal.warc.gz"} |
http://physics.stackexchange.com/questions/50106/pair-production-mathematically | # Pair production - mathematically?
Allover the web i am only seeing a statement similar to this:
Pair production is not possible in vaccum, 3rd particle is needed so that conservation of momentum holds.
Well noone out of many writers shows, how to prove this matematically. So this is what interests me here.
First i wanted to know if pair production really cannot happen in vacuum, so i drew a picture and used equations for conservation of energy and conservation of momentum to calculate energy of a photon $(h \nu)$ needed for pair production.
It turns out $h\nu$ is different if i calculate it out of conservation of energy or conservation of momentum. And even more! It can never be the same because equallity would mean parts $v_1 \cos \alpha$ and $v_2 \cos \beta$ should equall speed of light $c$. Well that cannot happen.
Below is my derivation.
CONSERVATION OF ENERGY: $$\scriptsize \begin{split} W_{1} &= W_{2}\\ W_f &= W_{e^-} + W_{e^+}\\ h\nu &= W_{ke^-} + W_{0e^-} + W_{ke^+} + W_{0e^+}\\ h\nu &=\left[m_ec^2 \gamma(v_1) - m_ec^2\right] + m_ec^2 + \left[m_ec^2 \gamma(v_2) - m_ec^2\right] + m_ec^2\\ h\nu &=m_ec^2 \gamma(v_1) + m_ec^2 \gamma(v_2)\\ h\nu &=m_ec^2 \left[\gamma(v_1) + \gamma(v_2) \right]\\ \end{split}$$
CONSERVATION OF MOMENTUM:
$y$ direction: $$\scriptsize \begin{split} p_{1} &= p_{2}\\ 0 &= p_{e^-} \sin \alpha - p_{e^+} \sin \beta \\ 0 &= m_e v_{1} \gamma(v_{1}) \sin \alpha - m_e v_{2} \gamma(v_{2}) \sin \beta\\ &\text{If \boxed{\alpha = \beta} \Longrightarrow \boxed{v_1 = v_2} and:} \end{split}$$
$$\begin{split} \scriptsize 0 = 0 \end{split}$$
$x$ direction:
$$\scriptsize \begin{split} p_{1} &= p_{2}\\ \frac{h}{\lambda} &= p_{e^-} \cos \alpha + p_{e^+} \cos \beta \\ \frac{h \nu}{c} &= m_e v_{1} \gamma(v_{1}) \cos \alpha + m_e v_{2} \gamma(v_{2}) \cos \beta\\ h \nu &= m_e c \Big[ \gamma(v_1) \underbrace{v_{1} \cos \alpha}_{\neq c} + \gamma(v_{2}) \underbrace{v_{2} \cos \beta}_{\neq c} \Big] \end{split}$$
Alltogether:
Because momentum in $y$ direction equals 0 (holds for some combinations of $\alpha, \beta, v_1, v_2$) whole momentum equals just the momentum in $x$ direction. So if i add them i get:
$$\scriptsize h \nu = m_e c \Big[ \gamma(v_1) \underbrace{v_{1} \cos \alpha}_{\neq c} + \gamma(v_{2}) \underbrace{v_{2} \cos \beta}_{\neq c} \Big]$$
From this i can conclude only that i cannot sucessfully apply conservation of energy and conservation of momentum at the same time and therefore pair production in vacuum cannot happen.
QUESTION1: Why do writers state that 3rd particle is needed so that conservation of momentum holds? What if conservation of momentum holds and conservation of energy doesn't? How can we say which one holds and which one doesnt?
QUESTION2: Do writters actually mean that if a 3rd particle is included we can achieve $h \nu$ to match in both cases?
QUESTION3: Can someone show me mathematically how this is done? I mean it should right?
-
– luksen Jan 13 '13 at 12:18
I had no idea \boxed and \underbrace work with mathjax. Cool. – Chris White Jan 13 '13 at 14:36
Question 3:
One time and two space dimensions for simplicity (t, x, y). Photon travelling in +x direction. Photon four momentum is $(\frac{E}{c}, p_x, 0)$. It's null so $$\frac{E^2}{c^2}-p_x^2=0$$ So $$p_x = \frac{E}{c}$$ $E=h\nu$, so photon four momentum is $(\frac{h\nu}{c}, \frac{h\nu}{c}, 0)$
To keep it simple, assume the electron/positron are emitted at equal angles to the x axis. One of them (the electron say) has four momentum $$(\frac{E_e}{c}, p_ecos\phi, p_esin\phi)$$ which, being timelike, satisfies $$\frac{E_e^2}{c^2}-p_e^2 = m_e^2c^2$$ Conservation of momentum in the x direction: $$\frac{h\nu}{c} = 2p_ecos\phi = 2\gamma_vm_evcos\phi \ \ (0)$$ where $v$ is the electron/positron velocity magnitude. So $$h\nu = 2\gamma_vm_ev \ c \ cos\phi \ \ (1)$$ So clearly $$h\nu < 2 m_e c^2$$ However, the photon must supply at least the electron and positron rest masses, so $$h\nu > 2m_e c^2 \ \ (2)$$ Therefore for (1) to be satisfied, the x momentum must have gone somewhere else. The nuclear recoil accounts for this.
-
This is how i want it! But i still have to ask you 2 questions. (1) At first part of explaination u only wanted to show, that photon momentum can be calculated as $p_f = \frac{E}{c}$? (2) Why do you state so many times that $\frac{E^2}{c^2}-p^2 = m^2c^2$ holds for time like vectors? Is this very important? Why? – 71GA Jan 13 '13 at 21:34
Yes, the fact the photon four momentum is null lets you deduce that its x momentum is E/c. The timelike relation for the electron/positron is what you need to get to (1), and then looking at the magnitudes of all the factors in (1) gets you to $h\nu < 2m_ec^2$ – twistor59 Jan 14 '13 at 7:33
When you say "x momentum must have gone somewhere else" you mean x momentum of a photon before collision? – 71GA Jan 14 '13 at 9:21
Yes, by inequality (2) we know the photon momentum $\frac{h\nu}{c}$ must be at least $2m_ec$, so if it's this big initially, then to get it down to the value implied by equation (0), it must have given some to some other body. – twistor59 Jan 15 '13 at 7:43 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9571669697761536, "perplexity": 396.2892933265745}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609539493.17/warc/CC-MAIN-20140416005219-00476-ip-10-147-4-33.ec2.internal.warc.gz"} |
https://www.intechopen.com/online-first/the-black-hole-binary-gravitons-and-related-problems | Open access peer-reviewed chapter - ONLINE FIRST
# The Black Hole Binary Gravitons and Related Problems
By Miroslav Pardy
Submitted: August 27th 2018Reviewed: November 22nd 2018Published: January 30th 2019
DOI: 10.5772/intechopen.82659
## Abstract
The energy spectrum of graviton emitted by the black hole binary is calculated in the first part of the chapter. Then, the total quantum loss of energy is calculated in the Schwinger theory of gravity. In the next part, we determine the electromagnetic shift of energy levels of H-atom electrons by calculating an electron coupling to the black hole thermal bath. The energy shift of electrons in H-atom is determined in the framework of nonrelativistic quantum mechanics. In the last section, we determine the velocity of sound in the black hole atmosphere, which is here considered as the black hole photon sea. Derivation is based on the thermodynamic theory of the black hole photon gas.
### Keywords
• graviton
• Schwinger source theory
• spectrum of H-atom
• Coulomb potential
• black hole spectrum
• energy shift
• sound
## 1. The graviton spectrum of the black hole binary
In 1916, Schwarzschild published the solution of the Einstein field equations [1] that were later understood to describe a black hole [2, 3], and in 1963, Kerr generalized the solution to rotating black holes [4]. The year 1970 was the starting point of the theoretical work leading to the understanding of black hole quasinormal modes [5, 6, 7], and in the 1990s, higher-order post-Newtonian calculations [8] were performed and later the extensive analytical studies of relativistic two-body dynamics were realized [9, 10]. These advances, together with numerical relativity breaks through in the past decade [11, 12, 13]. Numerous black hole candidates have now been identified through electromagnetic observations [14, 15, 16]. The black hole binary and their rotation and mergers are open problem of the astrophysics, and it is the integral part of the binary black hole physics.
The binary pulsar system PSR B1913+16 (also known as PSR J1915+1606) discovered by Hulse and Taylor [17] and subsequent observations of its energy loss by Taylor and Weisberg [18] demonstrated the existence of gravitational waves [19].
By the early 2000s, a set of initial detectors was completed, including TAMA 300 in Japan, GEO600 in Germany, the Laser Interferometer Gravitational-Wave Observatory (LIGO) in the United States, and Virgo in Italy. In 2015, Advanced LIGO became the first of a significantly more sensitive network of advanced detectors (a second-generation interferometric gravitational wave detector) to begin observations [20].
Taylor and Hulse, working at the Arecibo Radiotelescope, discovered the radio pulsar PSR B1913+16 in a binary, in 1974, and this is now considered as the best general relativistic laboratory [21].
Pulsar PSR B1913+16 is the massive body of the binary system where each of the rotating pairs is 1.4 times the mass of the Sun. These neutron stars rotate around each other in an orbit not much larger than the Sun’s diameter, with a period of 7.8 h. Every 59 ms, the pulsar emits a short signal that is so clear that the arrival time of a 5 min string of a set of such signals can be resolved within 15 μs.
A pulsar model based on strongly magnetized, rapidly spinning neutron stars was soon established as consistent with most of the known facts [22]; its electrodynamical properties were studied theoretically [23] and shown to be plausibly capable of generating broadband radio noise detectable over interstellar distances. The binary pulsar PSR B1913+16 is now recognized as the harbinger of a new class of unusually short-period pulsars, with numerous important applications.
Because the velocities and gravitational energies in a high-mass binary pulsar system can be significantly relativistic, strong-field and radiative effects come into play. The binary pulsar PSR B1913+16 provides significant tests of gravitation beyond the weak-field, slow-motion limit [24, 25].
We do not repeat here the derivation of the Einstein quadrupole formula in the Schwinger gravity theory [26]. We show that just in the framework of the Schwinger gravity theory, it is easy to determine the spectral formula for emitted gravitons and the quantum energy loss formula of the binary system. The energy loss formula is general, including black hole binary, and it involves arbitrarily strong gravity.
Since the measurement of the motion of the black hole binaries goes on, we hope that sooner or later the confirmation of our formula will be established.
### 1.1 The Schwinger approach for the problem
Source methods by Schwinger are adequate for the solution of the calculation of the spectral formula of gravitons and energy loss of binary. Source theory [27, 28] was initially constructed to describe the particle physics situations occurring in high-energy physics experiments. However, it was found that the original formulation simplifies the calculations in the electrodynamics and gravity, where the interactions are mediated by photon and graviton, respectively. The source theory of gravity forms the analogue of quantum electrodynamics because, while in QED the interaction is mediated by the photon, the gravitational interaction is mediated by the graviton [29]. The basic formula in the source theory is the vacuum-to-vacuum amplitude [30]:
0+0=eiWS,E1
where the minus and plus symbols refer to any time before and after the region of space–time with action of sources. The exponential form is postulated to express the physically independent experimental arrangements, with result that the associated probability amplitudes multiply and the corresponding Wexpressions add [27, 28].
In the flat space-time, the field of gravitons is described by the amplitude (1) with the action (c=1in the following text) [31]
WT=4πGdxdxTμνxD+xxTμνx12TxD+xxTx,E2
where the dimensionality of WThas the dimension of the Planck constant and Tμνis the momentum and energy tensor that, for a particle trajectory x=xt, is defined by the equation [32]
Tμνx=pμpνEδxxt,E3
where pμis the relativistic four-momentum of a particle with a rest mass mand
pμ=EpE4
pμpμ=m2,E5
and the relativistic energy is defined by the known relation
E=m1v2,E6
where v is the three-velocity of the moving particle.
Symbol Txin Eq. (2) is defined as T=gμνTμν, and D+xxis the graviton propagator whose explicit form will be determined later.
### 1.2 The power spectral formula in general
It may be easy to show that the probability of the persistence of vacuum is given by the following formula [27]:
0+02=exp2ImW=dexpdtdω1ωPωt,E7
where the so-called power spectral function Pωthas been introduced [27]. For the extraction of the spectral function from Im W, it is necessary to know the explicit form of the graviton propagator D+xx. This propagator involves the graviton property of spreading with velocity c. It means that its mathematical form is identical with the photon propagator form. With regard to Schwinger et al. [33], the x-representation of Dkin Eq. (2) is as follows:
D+xx=dk2π4eikxxDk,E8
where
Dk=1k2k02iϵ,E9
which gives
D+xx=i4π20sinωxx'xx'ett.E10
Now, using Eqs. (2), (7), and (10), we get the power spectral formula in the following form:
Pωt=4πGωdxdxdtsinωxxxx'cosωtt×TμνxtTμν(xt)12gμνTμν(xt)gαβTαβ(xt).E11
### 1.3 The power spectral formula for the binary system
In the case of the binary system with masses m1and m2, we suppose that they move in a uniform circular motion around their centre of gravity in the xyplane, with corresponding kinematical coordinates:
x1t=r1icosω0t+jsinω0tE12
x2t=r2icosω0t+π+jsinω0t+πE13
with
vit=dxi/dt,ω0=vi/ri,vi=vii=12.E14
For the tensor of energy and momentum of the binary, we have
Tμνx=p1μp1νE1δxx1t+p2μp2νE2δxx2t,E15
where we have omitted the tensor tμνG, which is associated with the massless, gravitational field distributed all over space and proportional to the gravitational constant G[32].
After the insertion of Eq. (15) into Eq. (11), we get [33]
Ptotalωt=P1ωt+P12ωt+P2ωt,E16
where (tt=τ)
P1ωt=r1πsin2ωr1sinω0τ/2sinω0τ/2cosωτ×E12ω02r12cosω0τ12m142E12,E17
P2ωt=r2πsin2ωr2sinω0τ/2sinω0τ/2cosωτ×E22ω02r22cosω0τ12m242E22,E18
P12ωt=4πsinωr12+r22+2r1r2cosω0τ1/2r12+r22+2r1r2cosω0τ1/2cosωτ×E1E2ω02r1r2cosω0τ+12m12m222E1E2.E19
### 1.4 The quantum energy loss of the binary
Using the following relations
ω0τ=φ+2πl,φππ,l=0,±1,±2,E20
l=l=cos2πlωω0=l=ω0δωω0l,E21
we get for Piωt, with ωbeing restricted to positive:
Piωt=l=1δωω0lPilωt.E22
Using the definition of the Bessel function J2lz
J2lz=12πππcoszsinφ2cos,E23
from which the derivatives and their integrals follow, we get for P1land P2lthe following formulae:
Pil=2ri(Ei2vi21mi42Ei202vildxJ2lx+ 4Ei2vi21vi2J2l2vil+4Ei2vi4J2l2vil),i=1,2.E24
Using r2=r1+ϵ, where ϵis supposed to be small in comparison with radii r1and r2, we obtain
r12+r22+2r1r2cosφ1/22acosφ2,E25
with
a=r11+ϵ2r1.E26
So, instead of Eq. (19), we get
P12ωt=2sin2ωacosω0τ/2cosω0τ/2]cosωτ×E1E2ω02r1r2cosω0τ+12m12m222E1E2.E27
Now, we can approach the evaluation of the energy loss formula for the binary from the power spectral of Eqs. (24) and (27). The energy loss is defined by the relation
dUdt=Pω=i,lδωω0lPil+P12ω=ddtU1+U2+U12.E28
From [34] we have Kapteyn’s formula:
l=1J2l2lvl2=v22.E29
After differentiating the last relation with respect to v, we have
l=1lJ2l2lv=0.E30
From [34] we learn other Kapteyn’s formulae:
l=12lJ2l2lv=v1v22,E31
and
l=1l02lvJ2lxdx=v331v23.E32
So, after the application of Eqs. (30), (31) and (32) to Eqs. (24) and (28), we get
dUidt=Gmi2vi3ω03ri1vi2313vi215.E33
Instead of using Kapteyn’s formulae for the interference term, we will perform a direct evaluation of the energy loss of the interference term by the ω-integration in (27) [35]. So, after some elementary modification in the ω-integral, we get
with
A=G,B=E1E2,C=v1v2,D=m12m222E1E2.E35
Using the definition of the δ-function and its derivative, we have, instead of Eq. (34), with v=aω0
dU12dt=Aω0πdxBCcosx+12Dcosx/2×δx2vcosx/2δx+2vcosx/2.E36
According to the Schwinger article [36], we express the delta function as follows:
δx±2vcosx/2=n=0±2vcosx/2nn!ddxnδx.E37
Then
δx±2vcosx/2=n=0±2vcosx/2nn!ddxn+1δx=E38
and it means that
δx+2vcosx/2δx2vcosx/2cosx/2=2n=12v2n1cosx/22n12n1!ddx2nδxE39
Now, we can write Eq. (36) in the following form after some elementary operations:
dU12dt=Aω0πdxBCcosx+12D×2n=12v2n1cosx/22n12n1!ddx2nδx,E40
where BCcosx+12Dcan be written as follows:
BCcosx+12D=4BC2(cos4x/2+4CB4BC2(cos2x/2+BC22CB+BD.E41
After the application of the per partes method, we get from Eq. (40) the following mathematical object:
dU12dt=2A4BC2ω0πdxδxn=1ddx2n2v2n1cosx/22n+22n1! 2A4CB4BC2ω0πdxδxn=1ddx2n2v2n1cosx/22n2n1! 2ABC22CB+BDω0πdxδxn=1ddx2n2v2n1cosx/22n12n1!.E42
We get after some elementary operations δfx=f0
J1=n=1ddx2n2v2n1cosx/22n+22n1!x=0=n=0fnv2n=Fv2,E43
J2=n=1ddx2n2v2n1cosx/22n2n1!x=0=n=0gnv2n=Gv2E44
and
J3=n=1ddx2n2v2n1cosx/22n12n1!x=0=n=0hnv2n=Hv2E45
where f,g,h,F,G,Hare functions which must be determined.
So we get instead of Eq. (41) the following final form:
dU12dt=2A4BC2ω0πGv22A4CB4BC2ω0πFv2 2A2CB+BC2+BDω0πHv2E46
Let us remark that we can use simple approximation in Eq. (41) as follows: cosx/22n+2cosx/22,cosx/22ncosx/22,cosx/22n1cosx/22. Then, after using the well-known formula
ddx2ncos2x/2=12cosx+πnE47
and
12cosx+πnx=0=121n.E48
So, instead of Eq. (46), we have
dU12dt=Aω0π2BC+BC2+BDn=12v2n11n2n1!.E49
## 2. Energy shift of H-atom electrons due to the black hole thermal bath
We here determine the electromagnetic shift of energy levels of H-atom electrons by calculating an electron coupling to the black hole thermal bath. The energy shift of electrons in H-atom is determined in the framework of nonrelativistic quantum mechanics.
The Gibbons-Hawking effect is the statement that a temperature can be associated to each solution of the Einstein field equations that contain a causal horizon.
Schwarzschild space-time involves an event horizon associated with temperature Tof a black hole of mass M. We consider here the influence of the heat bath of the Gibbons-Hawking photons on the energy shift of H-atom electrons.
The analogical problems are solved in the scientific respected journals. There is a general conviction of an analogy between the black hole and the hydrogen atom. Corda [37] used the model where Hawking radiation is a tunneling process. In his article the emission is expressed in terms of the black hole quantum levels. So, the Hawking radiation and black hole quasinormal modes by Corda [38] are analogical to hydrogen atom by Bohr.
In this model [39] the corresponding wave function is written in terms of a unitary evolution matrix. So, the final state is a pure quantum state with no information loss. Black hole is defined as the quantum systems, with discrete quantum spectra, with Hooft’s assumption that Schrödinger equations are universal for all universe dynamics.
Thermal photons by Gibbons and Hawking are blackbody photons, with the Planck photon distribution law [40, 41, 42], derived from the statistics of the oscillators inside of the blackbody. Later Einstein [43] derived the Planck formula from the Bohr model of atom where photons and electrons have the discrete energies related with the Bohr formula ω=EiEf, Ei,Efbeing the initial and final energies of electrons.
Now, we determine the modification of the Coulomb potential due to blackbody photons. At the start, the energy shift in the H-atom is the potential V0x, generated by nucleus of the H-atom. The potential at point V0x+δxis [44, 45]
V0x+δx=1+δx+12δx2+V0x.E50
The average of the last equation in space enables the elimination of the so-called the effective potential:
Vx=1+16δxT2Δ+V0x,E51
where δxT2is the average value of the square coordinate shift caused by the thermal photons. The potential shift follows from Eq. (51):
δVx=16δxT2ΔV0x.E52
The shift of the energy levels is given by the standard quantum formula [44]:
δEn=16δxT2ψnΔV0ψn.E53
In case of the Coulomb potential, which is the case of the H-atom, we have
V0=e24πx.E54
Then for the H-atom we can write
δEn=2π3δxT2e24πψn02,E55
where we used the following equation for the Coulomb potential
Δ1x=4πδx.E56
The motion of electron in the electric field is evidently described by elementary equation:
δx¨=emET,E57
which can be transformed by the Fourier transformation into the following equation
δx2=12e2m2ω4E2,E58
where the index ωconcerns the Fourier component of the above functions.
Using Bethe idea [46] of the influence of vacuum fluctuations on the energy shift of electron, the following elementary relations were applied by Welton [45], Akhiezer et al. [44] and Berestetzkii et al. [47]:
12Eω2=ω2,E59
and in case of the thermal bath of the blackbody, the last equation is of the following form [48]:
E2=ϱω=ω3π2c31eωkT1,E60
because the Planck law in (60) was written as
ϱω=Gω<Eω>=ω2π2c3ωeωkT1,E61
where the term
<Eω>=ωeωkT1E62
is the average energy of photons in the blackbody and
Gω=ω2π2c3E63
is the number of electromagnetic modes in the interval ω,ω+.
Then,
δx2=12e2m2ω4ω3π2c31eωkT1,E64
where δx2involves the number of frequencies in the interval ωω+.
So, after some integration, we get
δxT2=ω1ω212e2m2ω4ω3π2c3eωkT1=12e2m2π2c3Fω2ω1,E65
where Fωis the primitive function of the omega-integral with
1ω1eωkT1,E66
which is not elementary, and it is not in the tables of integrals.
Frequencies ω1and ω2can be determined from the field of thermal photons. It was performed for the Lamb shift [44, 47] caused by the interaction of the Coulombic atom with the field fluctuations. The Bethe-Welton method is valid here too and so we take Bethe-Welton frequencies. It means an electron does not respond to the fluctuating field if the frequency is much less than the atom binding energy given by the Rydberg constant [49] ERydberg=α2mc2/2. So, the lower frequency limit is
ω1=ERydberg/=α2mc22,E67
where α1/137is so-called the fine structure constant.
The second frequency follows from the cutoff, determined by the neglection of the relativistic effect in our theory. So, we write
ω2=mc2.E68
If we express the thermal function in the form of the geometric series
1eωkT1=q1+q2+q3+..;q=eωkT,E69
ω1ω2q1+q2+q3+..1ω=lnω+k=1ωkTkk!k+.;q=eωkTE70
and the first thermal contribution is
Thermalcontribution=lnω2ω1kTω2ω1,E71
then, with Eq. (55)
δEn2π3e2m2π2c3lnω2ω1kTω2ω1ψn02,E72
where according to Sokolov et al. [50]
ψn02=1πn2a02E73
with
a0=2me2.E74
Let us only remark that the numerical form of Eq. (72) has deep experimental astrophysical meaning.
Haroche [51] and his group performed experiments with the Rydberg atoms in a cavity. We used here Gibbons-Hawking black hole for the determination of the energy shift of H-atom electrons in the black hole gas.
## 3. Velocity of sound in the black hole photon gas
We have seen that the black hole can be modeled by the blackbody, and it means that there is the velocity of sound in the Gibbons-Hawking black hole thermal bath. So, let us derive the sound velocity from the thermodynamics of photon gas and energy mass relation.
In order to be pedagogically clear, we start with the derivation of the speed of sound in the real elastic rod.
Let Abe the cross-section of the element Adxof a rod on the axis x. Let φxtbe the deflection of Adxat point xat time t. The shift of the Adxat point x+dxis evidently
φ+φxdx.E75
Now, we suppose that the force tension Fxtacting on the Adxof the rod is given by Hooke’s law:
Fxt=EAφx,E76
where Eis Young’s modulus of elasticity. We easily derive that
Fx+dxFxEA2φx2dx.E77
The mass of Adxis ϱAdx, where ϱis the mass density of the rod and the dynamical equilibrium is expressed by Newton’s law of force:
or
φttv2φxx=0,E79
where
v=Eϱ1/2E80
is the velocity of sound in the rod.
The complete solution of Eq. (79) includes the initial and boundary conditions. We suppose that Eq. (80) is of the universal validity also for gas in the cylinder tube. If ΔL/Lis the relative prolongation of a rod, then an analogue for the tube of gas is ΔV/V, FΔp, where Vis the volume of a gas and pis gas pressure. Then, the modulus of elasticity as the analogue of Eq. (76) is
E=dpdVV.E81
The sound in ideal gas is the adiabatic thermodynamic process with no heat exchange. This is the model of the sound spreading in the gas of blackbody photons. Such process is described by the thermodynamic equation:
pVκ=const,E82
where κis the Poisson constant defined as κ=cp/cv, with cp,cvbeing the specific heat under constant pressure and under constant volume.
After differentiation of Eq. (82), we get the following equation:
dpVκ+κVκ1dV=0,E83
or
dpdV=κpV.E84
After inserting Eq. (84) into Eq. (81), we get from Eq. (80) the so-called Newton-Laplace formula:
v=κpϱ,E85
with ϱbeing the gas mass density.
The equilibrium radiation density has the Stefan-Boltzmann form:
u=aT4;a=7,5657.1016JK4m3.E86
Then, with regard to the thermodynamic definition of the specific heat,
cv=uTV=4aT3.E87
Similarly, with regard to the general thermodynamic theory,
cp=cv+uVT+pVTp=cv,E88
because VTT=0for photon gas, and in such a way, κ=1for photon gas. According to the theory of relativity, there is a relation for mass and energy, namely, m=E/c2. At the same time, the pressure and the internal energy of the blackbody gas are related as p=u/3. So, in our case
ϱ=u/c2=aT4c2;p=u3.E89
So, after the insertion of formulae in Eq. (88) into Eq. (85), the final formula for the sound velocity in photon blackbody sea is the following:
v=cκ3=c33,E90
which was derived by Partovi [52] using the QED theory of the photon gas. We correctly derived v/c<1.
So, we have performed the derivation of the velocity of sound in the relic photon sea. It is not excluded that the relic sound can be detected by the special microphones of Bell Laboratories. If we use van der Waals equation of state or the Kamerlingh Onnes virial equation, the obtained results will be modified with regard to the basic results.
Our derivation of the light velocity in the blackbody photon gas was based on the classical thermodynamic model with the adiabatic process (δQ=0), controlling the spreading of sound in the gas. Partovi [52] derived additional radiation corrections to the Planck distribution formula and the additional correction to the speed of sound in the relic photon sea. His formula is of the form
vsound=188π2α22025TTe4c3,E91
where αis the fine structure constant and Te=5.9G Kelvin. We see that our formula is the first approximation in the Partovi expression.
There is the Boltzmann statistical theory of transport of sound energy in a gas [53]. After the application of this theory to the photon gas or relic photon gas, we can obtain results involving the cross-section of the photon-photon interaction [47]:
σγγ=4,7α4cω2;ωmc2,E92
and
σγγ=97310125πα2re2ωmc26;ωmc2,E93
where re=e2/mc2=2,818×1013cm is the classical radius of electron and α=e2/cis the fine structure constant with numerical value 1/α=137,04.
## 4. Discussion and summary
We have derived the spectral density of gravitons and the total quantum loss of energy of the black hole binary. The energy loss is caused by the emission of gravitons during the motion of the two black hole binaries around each other under their gravitational interaction. The energy loss formulae of the production of gravitons are derived here by the Schwinger method. Because the general relativity and theory of gravity do not necessarily contain the last valid words to be written about the nature of gravity and it is not, of course, a quantum theory [21], they cannot give the answer on the production of gravitons and the quantum energy loss, respectively. So, this article is the original text that discusses the quantum energy loss caused by the production of gravitons by the black hole binary system. It is evident that the production of gravitons by the binary system forms a specific physical situation, where a general relativity can be seriously confronted with the source theory of gravity.
This article is an extended version of an older article by the present author [33], in which only the spectral formulae were derived. Here we have derived the quantum energy loss formulae, with no specific assumption concerning the strength of the gravitational field. We hope that future astrophysical observations will confirm the quantum version of the energy loss of the binary black hole.
In the next part of the chapter, the electromagnetic shift of energy levels of H-atom electrons was determined by calculating an electron coupling to the Gibbons-Hawking electromagnetic field thermal bath of the black hole. The energy shift of electrons in H-atom is determined in the framework of nonrelativistic quantum mechanics.
In the last section, we have determined the velocity of sound in the blackbody gas of photons inside of the black hole. Derivation was based on the thermodynamic theory of the photon gas and the Einstein relation between energy and mass. The spectral form for the n-dimensional blackbody was not here considered. The text is based mainly on the author articles published in the international journals of physics [33, 54, 55].
There is the fundamental problem concerning the maximal mass of the black hole. The theory of the space–time with maximal acceleration constant was derived by authors [56, 57]. In this theory the maximal acceleration constant is the analogue of the maximal velocity in special theory of relativity. Maximal acceleration determines the maximal black hole mass where the mass of the black hole is restricted by maximal acceleration of a body falling in the gravity field of the black hole.
Another question is what is the relation of our formulae to the results obtained by LIGO (Laser Interferometer Gravitational-Wave Observatory)? LIGO is the largest and most sensitive interferometer facility ever built. It has been periodically upgraded to increase its sensitivity. The most recent upgrade, Advanced LIGO (2015), detected for the first time the gravitational wave, with sensitivity far above the background noise. The event with number GW150914 was identified with the result of a merger of two black holes at a distance of about 400 Mpc from Earth [58]. Two additional significant detections, GW151226 and GW170104, were reported later. We can say that at this time it is not clear if the LIGO results involve information on the spectrum of gravitons calculated in this chapter.
chapter PDF
## More
© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
## How to cite and reference
### Cite this chapter Copy to clipboard
Miroslav Pardy (January 30th 2019). The Black Hole Binary Gravitons and Related Problems [Online First], IntechOpen, DOI: 10.5772/intechopen.82659. Available from:
### More statistics for editors and authors
We are IntechOpen, the world's leading publisher of Open Access books. Built by scientists, for scientists. Our readership spans scientists, professors, researchers, librarians, and students, as well as business professionals. We share our knowledge and peer-reveiwed research papers with libraries, scientific and engineering societies, and also work with corporate R&D departments and government entities.
View all Books | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9523955583572388, "perplexity": 797.3215873931865}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540503656.42/warc/CC-MAIN-20191207233943-20191208021943-00082.warc.gz"} |
https://www.physicsforums.com/threads/does-ap-count-as-college.89515/ | # Does AP count as College?
1. Sep 17, 2005
### TheRhinoMan
Did I do this problem right? I didn't use some of the info...
Oh well... if it doesn't, it's moveable. :)
I'm new on here, and I was wondering if someone could help me with my physics HW; And forgive me if I don't get it right away. :)
Here's the problem...
A 40kg. wagon is towed up a hill inclined at 18.5(degrees) with respect to the horizontal. The tow rope is parallel to the incline and has a tension of 140N in it. Assume that the wagon starts from rest at the botton of the hill and neglect friction. How fast is the wagon going after moving 80m. up the hill?
I tried doing this one, and well...
m=40kg.
delta(x)=80m.
Angle=18.5
Force(net)=140N
vf=?
Just stating the known, and then I...
v(fin)^2=v(ini)^2 + 2a(delta)x
vf^2 = 0 +160a
F(net)=ma
140=40a
a=3.5
in which case...
vf^2= 160(3.5)
vf^2= 560
vf = 23.66
So, that's what I did; I'm sure it's wrong, just because I didn't use the angle.
So... that's my question... what did I do wrong? Or did I do it right?
Thanks a bundle!
P.S. Heheh, I have a whole lot of physics "?"s that I don't understand...
Last edited by a moderator: Sep 17, 2005
2. Sep 18, 2005
### Fermat
The wagon is on an incline. So the accelerating force isn't 140N.
You have to take away the component of the wagon's weight acting down the slope.
3. Sep 18, 2005
### TheRhinoMan
OOOooohhhh....
I see it now, Thanks Fermat!
I called a friend last night, and he was able to help me with this problem.... but I didn't understand his reasoning. Now I do; thanks!.
So.... Basically, I had to make a right triangle, force of gravity being the hypotenuse (sp?), normal force being the second largest side, and the base (or smallest side) being, essentially, the force reacting against the wagon as it goes up.
The wagon needed to overcome that force and then some in order to accelerate up the hill....
Yay! I think I get it. :) I'll try another one similar to it, and if I still don't get it, I'll be back. :D
Thanks for the help, fermat!
TheRhinoMan
4. Sep 18, 2005
### Fermat
Your setup should look something like in the attachment.
The force acting against the wagon is its component down the slope, i.e. Mg.sin(18.5).
So, the accelerating force is,
F = T - Mg.sin(18.5)
File size:
18.3 KB
Views:
36 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8517163991928101, "perplexity": 1361.9202380233078}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-50/segments/1480698543434.57/warc/CC-MAIN-20161202170903-00145-ip-10-31-129-80.ec2.internal.warc.gz"} |
https://lavelle.chem.ucla.edu/forum/viewtopic.php?f=145&t=28973 | ## Order?
$aR \to bP, Rate = -\frac{1}{a} \frac{d[R]}{dt} = \frac{1}{b}\frac{d[P]}{dt}$
Jesse_torres2H
Posts: 41
Joined: Thu Jul 27, 2017 3:00 am
### Order?
How do you determine what order a reaction is?
Chem_Mod
Posts: 18715
Joined: Thu Aug 04, 2011 1:53 pm
Has upvoted: 631 times
### Re: Order?
You could use initial rates method where you measure different initial rates with different initial concentrations.
Refer to class notes to determine which plot gives you a straight line graphing concentration vs time.
Use experimental data to solve for the exponents of your reactants. If you have rate=k[A]x[B]y, you solve for x and y and add them together to find the order.
Come to step-up sessions, discussion sections, office hours for more help! Make sure to ask someone from class for lecture notes in case you missed them.
Mitch Mologne 1A
Posts: 74
Joined: Fri Sep 29, 2017 7:04 am
### Re: Order?
To elaborate, essentially if you take two experiments and only change one initial concentration of reactant, you can observe how the rate of the rxn is affected, thus determining the order in respect to the reactant you altered.
Lindsay Kester 2L
Posts: 29
Joined: Thu Jul 27, 2017 3:00 am
### Re: Order?
You could also look at the slow step of a reaction and figure out it's molecularity. If it's A->B, then first order, A+A->B, second order, and so on. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 1, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8928279876708984, "perplexity": 3568.705261428286}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107896778.71/warc/CC-MAIN-20201028044037-20201028074037-00498.warc.gz"} |
https://arxiv.org/abs/astro-ph/0211361 | astro-ph
(what is this?)
(what is this?)
# Title: Ozone Depletion from Nearby Supernovae
Abstract: Estimates made in the 1970's indicated that a supernova occurring within tens of parsecs of Earth could have significant effects on the ozone layer. Since that time, improved tools for detailed modeling of atmospheric chemistry have been developed to calculate ozone depletion, and advances have been made in theoretical modeling of supernovae and of the resultant gamma-ray spectra. In addition, one now has better knowledge of the occurrence rate of supernovae in the galaxy, and of the spatial distribution of progenitors to core-collapse supernovae. We report here the results of two-dimensional atmospheric model calculations that take as input the spectral energy distribution of a supernova, adopting various distances from Earth and various latitude impact angles. In separate simulations we calculate the ozone depletion due to both gamma-rays and cosmic rays. We find that for the combined ozone depletion roughly to double the biologically active'' UV flux received at the surface of the Earth, the supernova must occur at <8 pc. Based on the latest data, the time-averaged galactic rate of core-collapse supernovae occurring within 8 pc is ~1.5/Gyr. In comparing our calculated ozone depletions with those of previous studies, we find them to be significantly less severe than found by Ruderman (1974), and consistent with Whitten et al. (1976). In summary, given the amplitude of the effect, the rate of nearby supernovae, and the ~Gyr time scale for multicellular organisms on Earth, this particular pathway for mass extinctions may be less important than previously thought.
Comments: 24 pages, 4 Postscript figures, to appear in The Astrophysical Journal, 2003 March 10, vol. 585 Subjects: Astrophysics (astro-ph) Journal reference: Astrophys.J. 585 (2003) 1169-1176 DOI: 10.1086/346127 Cite as: arXiv:astro-ph/0211361 (or arXiv:astro-ph/0211361v1 for this version)
## Submission history
From: John K. Cannizzo [view email]
[v1] Fri, 15 Nov 2002 16:05:31 GMT (526kb) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8529759049415588, "perplexity": 2109.517081797576}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988719638.55/warc/CC-MAIN-20161020183839-00172-ip-10-171-6-4.ec2.internal.warc.gz"} |
http://mathhelpforum.com/trigonometry/13555-help-sine.html | # Math Help - help with sine
1. ## help with sine
i hav no idea howw to do any of this maths homework so i need a lot of help.. plz also add me on msn ( [email protected] 0 if you are further willing to help..
if 0<a<2pie then the equation sin(a)=2 has how many solutions?what are they? Justify your answer. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8822275996208191, "perplexity": 4713.729662135729}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-14/segments/1427131305143.93/warc/CC-MAIN-20150323172145-00097-ip-10-168-14-71.ec2.internal.warc.gz"} |
http://www.thespectrumofriemannium.com/2020/08/ | ## LOG#249. Basic twistor theory.
This my last normal post. Welcome to those who read me. TSOR is ending and from its ashes will arise another project. That is inevitable. I want to use new TeX packages, and that is not easy here, to … Continue reading
## LOG#248. Basic string theory.
3 posts to finish the TSOR adventure! This blog post will introduce you to basic string theory from my own biased viewpoint. My blog, my rules. I think you concede that! What is the Universe made of? This really ancient … Continue reading
## LOG#247. Seesawlogy.
Introduction One of the big issues of Standard Model (SM) is the origin of mass (OM). Usually, the electroweak sector implements mass in the gauge and matter sector through the well known Higgs mechanism. However, the Higgs mechanism is not … Continue reading | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8006642460823059, "perplexity": 3078.2597247270387}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662625600.87/warc/CC-MAIN-20220526193923-20220526223923-00512.warc.gz"} |
https://www.physicsforums.com/threads/need-help-figuring-this-out.220101/ | # Homework Help: Need help figuring this out
1. Mar 5, 2008
### heathphysics
hi there i am new to this , and might be posting at the wrong place, this is the first time i've posted.i really need somehelp with this guys.
does any1 know how to answer this
"Suggest a graphical method to determine the mathematical relationship between two quantities which are inversely proportional to each other.".
also what is the relationship between avs 1/Mtot .... acceleration vs. the reciprocal of mass total.
pls reply ASAP, labs due tomorrow ;(
2. Mar 5, 2008
### cemar.
flip along the y=x axis.
The relationship is force. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8393560647964478, "perplexity": 3271.845293379558}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676589537.21/warc/CC-MAIN-20180717012034-20180717032034-00517.warc.gz"} |
http://haskellformaths.blogspot.co.uk/2012/03/chas-ii-hopf-algebra-ssym-of.html | ## Sunday, 25 March 2012
### CHAs II: The Hopf Algebra SSym of Permutations
Last time we looked at YSym, the (dual of the) Loday-Ronco Hopf algebra of binary trees. This time I want to look at SSym, the Malvenuto-Reutenauer Hopf algebra of permutations. (In due course we'll see that YSym and SSym are quite closely related.)
The fundamental basis for SSym is (indexed by) the permutations of [1..n] for n <- [0..]. As usual, we work in the free vector space over this basis. Hence a typical element of SSym might be something like [1,3,2] + 2 [4,1,3,2]. Here's some code:
```newtype SSymF = SSymF [Int] deriving (Eq)
instance Ord SSymF where
compare (SSymF xs) (SSymF ys) = compare (length xs, xs) (length ys, ys)
instance Show SSymF where
show (SSymF xs) = "F " ++ show xs
ssymF :: [Int] -> Vect Q SSymF
ssymF xs | L.sort xs == [1..n] = return (SSymF xs)
| otherwise = error "Not a permutation of [1..n]"
where n = length xs
```
(The "F" in SSymF stands for fundamental basis: we may have cause to look at other bases in due course.)
Let's try it out:
```\$ cabal update
\$ ghci
> :m Math.Algebras.Structures Math.Combinatorics.CombinatorialHopfAlgebra
> ssymF [1,3,2] + 2 * ssymF [4,1,3,2]
F [1,3,2]+2F [4,1,3,2]
```
Ok, so how can we define an algebra structure on this basis? How can we multiply permutations? (We want to consider permutations as combinatorial rather than algebraic objects here. So no, the answer isn't the group algebra.) One possibility would be to use the following shifted concatenation operation:
```shiftedConcat (SSymF xs) (SSymF ys) = let k = length xs in SSymF (xs ++ map (+k) ys)
```
For example:
```> shiftedConcat (SSymF [1,2]) (SSymF [2,1,3])
F [1,2,4,3,5]
```
This has the required properties. It's associative:
```> quickCheck (\x y z -> shiftedConcat (shiftedConcat x y) z == shiftedConcat x (shiftedConcat y z))
+++ OK, passed 100 tests.
```
And it's pretty obvious that the empty permutation, SSymF [], is a left and right identity. Hence we could form a monoid algebra using this operation.
However, for the Hopf algebra we're going to look at, we will use a slightly more complicated multiplication. We will retain the idea of shifting the second permutation, so that the two lists are disjoint. However, instead of just concatenating them, we will form the sum of all possible "shuffles" of the two lists. Here's the shuffle code:
```shuffles (x:xs) (y:ys) = map (x:) (shuffles xs (y:ys)) ++ map (y:) (shuffles (x:xs) ys)
shuffles xs [] = [xs]
shuffles [] ys = [ys]
```
So shuffles takes two input "decks of cards", and it outputs all possible ways that they can be shuffled together, while preserving the order between cards from the same deck. For example:
```> shuffles [1,2] [3,4,5]
[[1,2,3,4,5],[1,3,2,4,5],[1,3,4,2,5],[1,3,4,5,2],[3,1,2,4,5],[3,1,4,2,5],[3,1,4,5,2],[3,4,1,2,5],[3,4,1,5,2],[3,4,5,1,2]]
```
Notice how in each of the output shuffles, we have 1 before 2, and 3 before 4 before 5. This enables us to define an algebra structure on permutations as follows:
```instance (Eq k, Num k) => Algebra k SSymF where
unit x = x *> return (SSymF [])
mult = linear mult'
where mult' (SSymF xs, SSymF ys) =
let k = length xs
in sumv [return (SSymF zs) | zs <- shuffles xs (map (+k) ys)]
```
For example:
```> ssymF [1,2] * ssymF [2,1,3]
F [1,2,4,3,5]+F [1,4,2,3,5]+F [1,4,3,2,5]+F [1,4,3,5,2]+F [4,1,2,3,5]+F [4,1,3,2,5]+F [4,1,3,5,2]+F [4,3,1,2,5]+F [4,3,1,5,2]+F [4,3,5,1,2]
```
It's clear that ssymF [] is indeed a left and right unit for this multiplication. It's also fairly clear that this multiplication is associative (because both shifting and shuffling are). Let's just check:
```> quickCheck (prop_Algebra :: (Q, Vect Q SSymF, Vect Q SSymF, Vect Q SSymF) -> Bool)
+++ OK, passed 100 tests.
```
(The test code isn't exposed in the package, so you'll have to dig around in the source if you want to try this. It takes a minute or two, because every time we multiply we end up with lots of terms.)
What about a coalgebra structure? As I mentioned last time, in a combinatorial Hopf algebra, the comultiplication is usually a sum of the different ways to take apart our combinatorial object into two parts. In this case, we take a permutation apart by "deconcatenating" it (considered as a list) into two pieces. This is like cutting a deck of cards:
```deconcatenations xs = zip (inits xs) (tails xs)
```
For example:
```> deconcatenations [2,3,4,1]
[([],[2,3,4,1]),([2],[3,4,1]),([2,3],[4,1]),([2,3,4],[1]),([2,3,4,1],[])]
```
However, most of those parts are no longer permutations of [1..n] (for any n), because they are missing some numbers. In order to get back to permutations, we need to "flatten" each part:
```flatten xs = let mapping = zip (L.sort xs) [1..]
in [y | x <- xs, let Just y = lookup x mapping]
```
For example:
```> flatten [3,4,1]
[2,3,1]
```
Putting the deconcatenation and flattening together we get the following coalgebra definition:
```instance (Eq k, Num k) => Coalgebra k SSymF where
counit = unwrap . linear counit' where counit' (SSymF xs) = if null xs then 1 else 0
comult = linear comult'
where comult' (SSymF xs) = sumv [return (SSymF (flatten us), SSymF (flatten vs))
| (us, vs) <- deconcatenations xs]
```
For example:
```> comult \$ ssymF [2,3,4,1]
(F [],F [2,3,4,1])+(F [1],F [2,3,1])+(F [1,2],F [2,1])+(F [1,2,3],F [1])+(F [2,3,4,1],F [])
```
(Recall that the result should be read as F[]⊗F[2,3,4,1] + F[1]⊗F[2,3,1] + ...)
It's fairly clear that this comultiplication is coassociative. The counit properties are equally straightforward. Hence:
```> quickCheck (prop_Coalgebra :: Vect Q SSymF -> Bool)
+++ OK, passed 100 tests.```
Is it a bialgebra? Do the algebra and coalgebra structures commute with one another? In previous posts I've been a bit hand-wavy about this, so let's take a short time out to look at what this actually means. For example, what does it mean for mult and comult to commute? Well, it means that the following diagram commutes:
But hold on a sec - what is the mult operation on (B⊗B)⊗(B⊗B) - is B⊗B even an algebra? And similarly, what is the comult operation on B⊗B - is it even a coalgebra?
Yes they are. Given any algebras A and B, we can define an algebra structure on A⊗B via (a1⊗b1) * (a2⊗b2) = (a1*a2)⊗(b1*b2). In code:
```instance (Eq k, Num k, Ord a, Ord b, Algebra k a, Algebra k b) => Algebra k (Tensor a b) where
unit x = x *> (unit 1 `te` unit 1)
mult = (mult `tf` mult) . fmap (\((a,b),(a',b')) -> ((a,a'),(b,b')) )
```
Similarly, given coalgebras A and B we can define a coalgebra structure on A⊗B as follows:
```instance (Eq k, Num k, Ord a, Ord b, Coalgebra k a, Coalgebra k b) => Coalgebra k (Tensor a b) where
counit = unwrap . linear counit'
where counit' (a,b) = (wrap . counit . return) a * (wrap . counit . return) b
comult = nf . fmap (\((a,a'),(b,b')) -> ((a,b),(a',b')) ) . (comult `tf` comult)
```
(Recall that those pesky wrap and unwrap calls are the isomorphisms k <-> Vect k (). What the counit definition really says is that counit (a⊗b) = counit a * counit b.)
Notice how in both the mult and comult definitions, we have to swap the middle two terms of the fourfold tensor product over, in order to have something of the right type.
So how does this work for SSym? Well, we start in the top left with SSym⊗SSym. You can think of that as two decks of cards.
If we go along the top and down the right, then we first shuffle the two decks together (in all possible ways), and then deconcatenate the results (in all possible ways).
If we go down the left and along the bottom, then we first deconcatenate each deck independently (in all possible ways), and then shuffle the first parts of both decks together and separately shuffle the second parts together (in all possible ways).
You can kind of see that this is going to lead to the same result. It's just saying that it doesn't matter whether you shuffle before you cut or cut before you shuffle. (There's also shifting and flattening going on, of course, but it's clear that doesn't affect the result.)
For a bialgebra, we require that each of unit and mult commutes with each of counit and comult - so four conditions in all. The other three are much easier to verify, so I'll leave them as an exercise.
Just to check:
```> quickCheck (prop_Bialgebra :: (Q, Vect Q SSymF, Vect Q SSymF) -> Bool)
+++ OK, passed 100 tests.
```
Okay so what about a Hopf algebra structure? Is there an antipode operation?
Well, recall that when we looked last time at YSym, we saw that it was possible to give a recursive definition of the antipode map. This isn't always possible. The reason it was possible for YSym was:
- We saw that the comultiplication of a tree t is a sum of terms u⊗v, where with the exception of the term 1⊗t, all the other terms have a smaller tree on the right hand side.
- We saw that the counit is 1 on the smallest tree, and 0 otherwise.
It turns out that these are instances of a more general concept, of a graded and connected coalgebra.
- A graded vector space means that there is a concept of the size or degree of the basis elements. For YSym, the degree was the number of nodes in the tree. A graded coalgebra means that the comultiplication respects the degree, in the sense that if comult t is a sum of terms u⊗v, then degree u + degree v = degree t.
- A connected coalgebra means that the counit is 1 on the degree zero piece, and 0 otherwise.
(There is a basis-independent way to explain this, for the purists.)
Now, SSym is also a graded connected coalgebra:
- the degree of SSymF xs is simply length xs. Comultiplication respects the degree.
- counit is 1 on the degree zero piece, 0 otherwise.
Hence we can again give a recursive definition for the antipode:
```instance (Eq k, Num k) => HopfAlgebra k SSymF where
antipode = linear antipode'
where antipode' (SSymF []) = return (SSymF [])
antipode' x@(SSymF xs) = (negatev . mult . (id `tf` antipode) . removeTerm (SSymF [],x) . comult . return) x
```
For example:
```> antipode \$ ssymF [1,2,3]
-F [3,2,1]
> antipode \$ ssymF [1,3,2]
-F [2,1,3]-F [2,3,1]+F [3,1,2]
> antipode \$ ssymF [2,1,3]
-F [1,3,2]+F [2,3,1]-F [3,1,2]
```
It is possible to give an explicit expression for the antipode. However, it's a little complicated, and I haven't got around to coding it yet.
Let's just check:
```> quickCheck (prop_HopfAlgebra :: Vect Q SSymF -> Bool)
+++ OK, passed 100 tests.
```
However, this isn't quite satisfactory. I'd like a little more insight into what the antipode actually does.
Recall that the definition requires that (mult . (id ⊗ antipode) . comult) == (unit . counit).
The left hand side is saying:
- first cut (deconcatenate) the deck of cards into two parts (in all possible ways)
- next apply the antipode to just one of the two parts
- finally shuffle the two parts together (in all possible ways)
The right hand side, remember, sends the empty permutation SSymF [] to itself, and every other permutation to zero.
So what this is saying is, you cut a (non-empty) deck of cards, wave a wand over one part, shuffle the two parts together again, and the cards all disappear!
Or is it? No, not quite. Remember that comult (resp. mult) is a sum of all possible deconcatenations (resp. shuffles). So what this is saying is that the antipode arranges things so that when you sum over all possible deconcatenations and shuffles, they cancel each other out. Cool!
(This sounds like it might have something to do with renormalization in physics, where we want to get a bunch of troublesome infinities to cancel each other out. There is apparently a connection between Hopf algebras and renormalization, but I don't know if this is it. Unfortunately my understanding of physics isn't up to figuring this all out.)
So we've now seen how to define Hopf algebra structures on two different sets of combinatorial objects: trees and permutations. We'll see that these two Hopf algebras are actually quite closely related, and that there is something like a family of combinatorial Hopf algebras with interesting connections to one another.
My main source for this article was Aguiar, Sottile, Structure of the Malvenuto-Reutenauer Hopf algebra of permutations. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8899446129798889, "perplexity": 697.5734114454718}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1394010450813/warc/CC-MAIN-20140305090730-00067-ip-10-183-142-35.ec2.internal.warc.gz"} |
https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Book%3A_Electromagnetics_I_(Ellingson)/03%3A_Transmission_Lines/3.20%3A_Power_Flow_on_Transmission_Lines | # 3.20: Power Flow on Transmission Lines
$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$
It is often important to know the power associated with a wave on a transmission line. The power of the waves incident upon, reflected by, and absorbed by a load are each of interest. In this section we shall work out expressions for these powers and consider some implications in terms of the voltage reflection coefficient ($$\Gamma$$) and standing wave ratio (SWR).
Let’s begin by considering a lossless transmission line that is oriented along the $$z$$ axis. The time-average power associated with a sinusoidal wave having potential $$v(z,t)$$ and current $$i(z,t)$$ is $P_{av}(z) \triangleq \frac{1}{T}\int_{t_0}^{t_0+T}{ v(z,t)~i(z,t)~dt } \nonumber$ where $$T\triangleq 2\pi/f$$ is one period of the wave and $$t_0$$ is the start time for the integration. Since the time-average power of a sinusoidal signal does not change with time, $$t_0$$ may be set equal to zero without loss of generality.
Let us now calculate the power of a wave incident from $$z<0$$ on a load impedance $$Z_L$$ at $$z=0$$. We may express the associated potential and current as follows: $v^+(z,t) = \left|V_0^+\right|\cos\left(\omega t -\beta z +\phi\right) \nonumber$ $i^+(z,t) = \frac{\left|V_0^+\right|}{Z_0}\cos\left(\omega t -\beta z +\phi\right) \nonumber$ And so the associated time-average power is $$$\begin{split} P_{av}^+(z) &= \frac{1}{T}\int_0^T{ v^+(z,t)~i^+(z,t)~dt } \\ &= \frac{\left|V_0^+\right|^2}{Z_0} \cdot \frac{1}{T}\int_0^T{ \cos^2\left(\omega t -\beta z +\phi\right) ~dt } \end{split}$$ \label{m0090_ePav1}$ Employing a well-known trigonometric identity: $\cos^2\theta = \frac{1}{2} + \frac{1}{2}\cos 2\theta \nonumber$ we may rewrite the integrand as follows $\cos^2\left(\omega t -\beta z +\phi\right) = \frac{1}{2} + \frac{1}{2}\cos\left(2\left[\omega t -\beta z +\phi\right]\right) \nonumber$ Then integrating over both sides of this quantity $\int_0^T{ \cos^2\left(\omega t -\beta z +\phi\right) ~dt } = \frac{T}{2} + 0 \nonumber$ The second term of the integral is zero because it is the integral of cosine over two complete periods. Subsequently, we see that the position dependence (here, the dependence on $$z$$) is eliminated. In other words, the power associated with the incident wave is the same for all $$z<0$$, as expected. Substituting into Equation \ref{m0090_ePav1} we obtain: $\boxed{ P_{av}^+ = \frac{\left|V_0^+\right|^2}{2Z_0} } \label{m0090_ePav+}$ This is the time-average power associated with the incident wave, measured at any point $$z<0$$ along the line.
Equation \ref{m0090_ePav+} gives the time-average power associated with a wave traveling in a single direction along a lossless transmission line.
Using precisely the same procedure, we find that the power associated with the reflected wave is $P_{av}^- = \frac{\left|\Gamma V_0^+\right|^2}{2Z_0} = \left|\Gamma\right|^2 \frac{\left|V_0^+\right|^2}{2Z_0} \nonumber$ or simply $\boxed{ P_{av}^- = \left|\Gamma\right|^2 P_{av}^+ } \label{m0090_ePav-}$
Equation \ref{m0090_ePav-} gives the time-average power associated with the wave reflected from an impedance mismatch.
Now, what is the power $$P_L$$ delivered to the load impedance $$Z_L$$? The simplest way to calculate this power is to use the principle of conservation of power. Applied to the present problem, this principle asserts that the power incident on the load must equal the power reflected plus the power absorbed; i.e., $P_{av}^+ = P_{av}^- + P_L \nonumber$ Applying the previous equations we obtain: $\boxed{ P_L = \left(1-\left|\Gamma\right|^2\right) P_{av}^+ } \label{m0090_ePL}$
Equations \ref{m0090_ePL} gives the time-average power transferred to a load impedance, and is equal to the difference between the powers of the incident and reflected waves.
##### Example $$\PageIndex{1}$$: How important is it to match $$50~\Omega$$ to $$75~\Omega$$?
Two impedances which commonly appear in radio engineering are $$50~\Omega$$ and $$75~\Omega$$. It is not uncommon to find that it is necessary to connect a transmission line having a $$50~\Omega$$ characteristic impedance to a device, circuit, or system having a $$75~\Omega$$ input impedance, or vice-versa. If no attempt is made to match these impedances, what fraction of the power will be delivered to the termination, and what fraction of power will be reflected? What is the SWR?
###### Solution
The voltage reflection coefficient going from $$50~\Omega$$ transmission line to a $$75~\Omega$$ load is $\Gamma = \frac{75-50}{75+50} = +0.2 \nonumber$ The fraction of power reflected is $$\left|\Gamma\right|^2 = 0.04$$, which is 4%. The fraction of power transmitted is $$1-\left|\Gamma\right|^2$$, which is 96%. Going from a $$50~\Omega$$ transmission line to a $$75~\Omega$$ termination changes only the sign of $$\Gamma$$, and therefore, the fractions of reflected and transmitted power remain 4% and 96%, respectively. In either case (from Section 3.14): $\mbox{SWR} = \frac{1+\left|\Gamma\right|}{1-\left|\Gamma\right|} = 1.5 \nonumber$ This is often acceptable, but may not be good enough in some particular applications. Suffice it to say that it is not necessarily required to use an impedance matching device to connect $$50~\Omega$$ to $$75~\Omega$$ devices.
This page titled 3.20: Power Flow on Transmission Lines is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Steven W. Ellingson (Virginia Tech Libraries' Open Education Initiative) . | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 1, "x-ck12": 0, "texerror": 0, "math_score": 0.9628364443778992, "perplexity": 1147.7666563898408}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710734.75/warc/CC-MAIN-20221130092453-20221130122453-00281.warc.gz"} |
http://cp3-origins.dk/a/17930 | Dark Matter (DM) and Modified Newtonian Dynamics (MOND) lead to different geometries for galactic rotation curves in $(g_{bar},g_{obs})$-space ($g2$-space). Here $g_{obs}$ is the total (observed) centripetal acceleration of matter in a rotationally supported galaxy and $g_{bar}$ is that from the visible (baryonic) matter distribution. DM and MOND predict closed curves, $\mathcal{C}_{DM/MOND}$, in $g2$-space where the area spanned by the DM curve is non-zero, $\mathcal{A}(\mathcal{C}_{DM})\neq 0$, whereas the area spanned by the MOND curve vanishes, $\mathcal{A}(\mathcal{C}_{MOND})=0$. Galaxies from the SPARC database\cite{SPARC} are investigated in order to determine if $\mathcal{A}(\mathcal{C}_{data})\neq 0$ in current data. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8879146575927734, "perplexity": 928.0868871495701}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948514238.17/warc/CC-MAIN-20171212002021-20171212022021-00755.warc.gz"} |
https://www.cheenta.com/tifr-2015-problem-1-solution/ | Select Page
TIFR 2015 Problem 1 Solution is a part of TIFR entrance preparation series. The Tata Institute of Fundamental Research is India’s premier institution for advanced research in Mathematics. The Institute runs a graduate programe leading to the award of Ph.D., Integrated M.Sc.-Ph.D. as well as M.Sc. degree in certain subjects.
The image is a front cover of a book named Introduction to Linear Algebra by Gilbert Strang. This book is very useful for the preparation of TIFR Entrance.
Also Visit: College Mathematics Program of Cheenta
## Problem:
Let (A) be an invertible (10 \times 10) matrix with real entries such that the sum of each row is 1. Then
A. The sum of the entries of each row of the inverse of A is 1
B. The sum of the entries of each column of the inverse of A is 1
C. The trace of the inverse of A is non-zero.
D. None of the above.
## Discussion:
The sum of each row of (A) is 1, means that the sum of the columns of A is the vector ((1,1,…,1)^T ) .
Note that i-th column of (A) is given by (Ae_i ). Therefore, (\sum_{i=1}^{10} Ae_i = (1,1,…,1)^T ).
Since left multiplication by (A) is a linear transformation, the left-hand side of the last expression can be written as (A(\sum_{i=1}^{10}e_i)).
Now, (\sum_{i=1}^{10}e_i = (1,1,…,1)^T ).
Hence we get (A(1,1…,1)^T = (1,1,…,1)^T ).
Another way of saying the last expression is that the vector ( (1,1…,1)^T ) is fixed by A.
Since A is invertible, applying (A^{-1}) on both sides of the last expression we get ((1,…,1)^T = A^{-1}(1,1,…,1)^T ).
By the linearity argument as above, this gives ( (1,1…,1)^T = \sum_{i=1}^{10} A^{-1} (e_i) ). And the i-th term in right-hand side expression is the i-th column of (A^{-1}). Therefore, sum of columns of (A^{-1}) is the vector ((1,1,…,1)^T ). This is same as saying that sum of entries of each row of (A^{-1}) is 1.
## Helpdesk
• What is this topic: Linear Algebra
• What are some of the associated concept: Linear Transformation,Invertible Matrix
• Book Suggestions: Introduction to Linear Algebra by Gilbert Strang | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9310937523841858, "perplexity": 843.9298233435776}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400192778.51/warc/CC-MAIN-20200919142021-20200919172021-00160.warc.gz"} |
http://stats.stackexchange.com/questions/46404/mle-and-ordinal-probit-regression | # MLE and ordinal probit regression
Suppose we have a data set $X$. This data set consists of ordinal data (4 levels). To get estimates of the threshold coefficients and probit slope ($\beta_1, \dots, \beta_3$ and $\beta_4$ respectively), do most computational packages use maximum likelihood estimation? That is, given the data, MLE chooses the parameters that maximizes the probability of observing the data?
-
Small note: If you have 4 ordinal levels, you should have only 3 thresholds. – gung Dec 21 '12 at 19:36
Just to be clear, the number of $\beta$s depends on the number of explanatory variables, not on the number of levels. – Dimitriy V. Masterov Dec 21 '12 at 21:13
As far as I know, MLE is the only way to estimate these. – Dimitriy V. Masterov Dec 22 '12 at 1:57 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.942878007888794, "perplexity": 1056.2057915426753}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042989891.18/warc/CC-MAIN-20150728002309-00231-ip-10-236-191-2.ec2.internal.warc.gz"} |
https://www.undocumented-features.com/contact/ | Let me know what questions and/or feedback you have for me. I’ll get back to you as quickly as I can.
Thanks! I look forward to hearing from you. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.897227942943573, "perplexity": 246.80077457498328}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655886865.30/warc/CC-MAIN-20200705023910-20200705053910-00587.warc.gz"} |
https://www.physicsforums.com/threads/effects-of-propylene-glycol-on-heat-transfer-efficiency.590580/ | # Effects of propylene glycol on heat transfer efficiency
1. Mar 26, 2012
### MrDieselT
I'm trying to calculate the heat loss of efficiency due to increasing the mixture of propylene glycol in a heat transfer fluid compared to water.. For this I'm saying that water has a heat transfer efficiency of 100% (relative only)
Im looking for multiple data points to create an equation not just random numbers.
2. Mar 29, 2012
3. Apr 2, 2012
### MrDieselT
Thanks! I tried googling but was not using the right words!
Similar Discussions: Effects of propylene glycol on heat transfer efficiency | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8543553352355957, "perplexity": 3182.4948503389573}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891813571.24/warc/CC-MAIN-20180221063956-20180221083956-00285.warc.gz"} |
https://www.physicsforums.com/threads/tension-and-friction.93328/ | Tension and Friction
1. Oct 10, 2005
Comtrend
I have a question where a horizontal rope pulls on a 10kg wood sled over a frictionless surface. On that sled is a wood box. It is asking what the largest tension force in the rope would be where the box doesn't slip off.
Air resistance/Drag is neglected. Wouldn't the box remain on the sled indefinetely as long as it didn't accelerate to fast? If not then I would use the coefficient of static friction right?
2. Oct 10, 2005
Andrew Mason
You would need to know the coefficient of static friction between the box and sled to find the maximum static friction force. That is the maximum force that the sled can apply to the box. If the tension exceeds that force, the box will slide off.
AM
3. Oct 10, 2005
clive
Andrew is right, you need the frictional force between the sled and the box.
for the box:
$$F_f=m_{box} a$$
and for the sled
$$F-F_f=m_{sled} a$$
As we can see, we have 3 unknows and only 2 eqs. So you must provide some information about the friction at the contact surface.
4. Oct 10, 2005
Comtrend
I got 14.72N with a coefficient of static friction of 0.3. So is the Ff equal to the force of Tension because the sled is on a frictionless surface?
So then Tension equals 14.72N?
5. Oct 10, 2005
Comtrend
For force of friction for the box I used $$F_f=mN$$
I don't know how to get mew so it is just an m.
6. Oct 10, 2005
clive
The tension you need is $$F$$ (in my eqs). $$F_f$$ (the frictional force) and $$F$$ (the tension in the rope) are different.
You have to find F from the eq. below:
$$\frac{F-F_f}{F_f}=\frac{m_{sled}}{m_{box}}$$
and you'll obtain the tension in the rope. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.868261456489563, "perplexity": 765.5633668268941}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583658681.7/warc/CC-MAIN-20190117020806-20190117042806-00099.warc.gz"} |
https://www.r-bloggers.com/2020/02/quantile-regression-home-made-part-2/ | Want to share your content on R-bloggers? click here if you have a blog, or here if you don't.
A few months ago, I posted a note with some home made codes for quantile regression… there was something odd on the output, but it was because there was a (small) mathematical problem in my equation. So since I should teach those tomorrow, let me fix them.
Median
Consider a sample $$\{y_1,\cdots,y_n\}$$. To compute the median, solve$$\min_\mu \left\lbrace\sum_{i=1}^n|y_i-\mu|\right\rbrace$$which can be solved using linear programming techniques. More precisely, this problem is equivalent to$$\min_{\mu,\mathbf{a},\mathbf{b}}\left\lbrace\sum_{i=1}^na_i+b_i\right\rbrace$$with $$a_i,b_i\geq 0$$ and $$y_i-\mu=a_i-b_i$$, $$\forall i=1,\cdots,n$$. Heuristically, the idea is to write $$y_i=\mu+\varepsilon_i$$, and then define $$a_i$$‘s and $$b_i$$‘s so that $$\varepsilon_i=a_i-b_i$$ and $$|\varepsilon_i|=a_i+b_i$$, i.e. $$a_i=(\varepsilon_i)_+=\max\lbrace0,\varepsilon_i\rbrace=|\varepsilon|\cdot\boldsymbol{1}_{\varepsilon_i>0}$$and$$b_i=(-\varepsilon_i)_+=\max\lbrace0,-\varepsilon_i\rbrace=|\varepsilon|\cdot\boldsymbol{1}_{\varepsilon_i<0}[/latex]denote respectively the positive and the negative parts. Unfortunately (that was the error in my previous post), the expression of linear programs is[latex display="true"]\min_{\mathbf{z}}\left\lbrace\boldsymbol{c}^\top\mathbf{z}\right\rbrace\text{ s.t. }\boldsymbol{A}\mathbf{z}=\boldsymbol{b},\mathbf{z}\geq\boldsymbol{0}$$In the equation above, with the $$a_i$$‘s and $$b_i$$‘s, we’re not far away. Except that we have $$\mu\in\mathbb{R}$$, while it should be positive. So similarly, set $$\mu=\mu^+-\mu^-$$ where $$\mu^+=(\mu)_+$$ and $$\mu^-=(-\mu)_+$$.
Thus, let$$\mathbf{z}=\big(\mu^+;\mu^-;\boldsymbol{a},\boldsymbol{b}\big)^\top\in\mathbb{R}_+^{2n+2}$$and then write the constraint as $$\boldsymbol{A}\mathbf{z}=\boldsymbol{b}$$ with $$\boldsymbol{b}=\boldsymbol{y}$$ and $$\boldsymbol{A}=\big[\boldsymbol{1}_n;-\boldsymbol{1}_n;\mathbb{I}_n;-\mathbb{I}_n\big]$$And for the objective function$$\boldsymbol{c}=\big(\boldsymbol{0},\boldsymbol{1}_n,-\boldsymbol{1}_n\big)^\top\in\mathbb{R}_+^{2n+2}$$
To illustrate, consider a sample from a lognormal distribution,
n = 101
set.seed(1)
y = rlnorm(n)
median(y)
[1] 1.077415
For the optimization problem, use the matrix form, with $$3n$$ constraints, and $$2n+1$$ parameters,
library(lpSolve)
X = rep(1,n)
A = cbind(X, -X, diag(n), -diag(n))
b = y
c = c(rep(0,2), rep(1,n),rep(1,n))
equal_type = rep("=", n)
r = lp("min", c,A,equal_type,b)
head(r$solution,1) [1] 1.077415 It looks like it’s working well… Quantile Of course, we can adapt our previous code for quantiles tau = .3 quantile(y,tau) 30% 0.6741586 The linear program is now$$\min_{q^+,q^-,\mathbf{a},\mathbf{b}}\left\lbrace\sum_{i=1}^n\tau a_i+(1-\tau)b_i\right\rbrace$$with $$a_i,b_i,q^+,q^-\geq 0$$ and $$y_i=q^+-q^-+a_i-b_i$$, $$\forall i=1,\cdots,n$$. The R code is now c = c(rep(0,2), tau*rep(1,n),(1-tau)*rep(1,n)) r = lp("min", c,A,equal_type,b) head(r$solution,1)
[1] 0.6741586
So far so good…
Quantile Regression
Consider the following dataset, with rents of flat, in a major German city, as function of the surface, the year of construction, etc.
base=read.table("http://freakonometrics.free.fr/rent98_00.txt",header=TRUE)
The linear program for the quantile regression is now$$\min_{\boldsymbol{\beta}^+,\boldsymbol{\beta}^-,\mathbf{a},\mathbf{b}}\left\lbrace\sum_{i=1}^n\tau a_i+(1-\tau)b_i\right\rbrace$$with $$a_i,b_i\geq 0$$ and $$y_i=\boldsymbol{x}^\top[\boldsymbol{\beta}^+-\boldsymbol{\beta}^-]+a_i-b_i$$$$\forall i=1,\cdots,n$$ and $$\beta_j^+,\beta_j^-\geq 0$$ $$\forall j=0,\cdots,k$$. So use here
require(lpSolve)
tau = .3
n=nrow(base)
X = cbind( 1, base$area) y = base$rent_euro
K = ncol(X)
N = nrow(X)
A = cbind(X,-X,diag(N),-diag(N))
c = c(rep(0,2*ncol(X)),tau*rep(1,N),(1-tau)*rep(1,N))
b = base$rent_euro const_type = rep("=",N) r = lp("min",c,A,const_type,b) beta = r$sol[1:K] - r$sol[(1:K+K)] beta [1] 148.946864 3.289674 Of course, we can use R function to fit that model library(quantreg) rq(rent_euro~area, tau=tau, data=base) Coefficients: (Intercept) area 148.946864 3.289674 Here again, it seems to work quite well. We can use a different probability level, of course, and get a plot plot(base$area,base$rent_euro,xlab=expression(paste("surface (",m^2,")")), ylab="rent (euros/month)",col=rgb(0,0,1,.4),cex=.5) sf=0:250 yr=r$solution[2*n+1]+r$solution[2*n+2]*sf lines(sf,yr,lwd=2,col="blue") tau = .9 r = lp("min",c,A,const_type,b) tail(r$solution,2)
[1] 121.815505 7.865536
yr=r$solution[2*n+1]+r$solution[2*n+2]*sf
lines(sf,yr,lwd=2,col="blue")
And we can adapt the later to multiple regressions, of course,
X = cbind(1,base$area,base$yearc)
K = ncol(X)
N = nrow(X)
A = cbind(X,-X,diag(N),-diag(N))
c = c(rep(0,2*ncol(X)),tau*rep(1,N),(1-tau)*rep(1,N))
b = base$rent_euro const_type = rep("=",N) r = lp("min",c,A,const_type,b) beta = r$sol[1:K] - r\$sol[(1:K+K)]
beta
[1] -5542.503252 3.978135 2.887234
to be compared with
library(quantreg)
rq(rent_euro~ area + yearc, tau=tau, data=base)
Coefficients:
(Intercept) area yearc
-5542.503252 3.978135 2.887234
Degrees of freedom: 4571 total; 4568 residual | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9047895073890686, "perplexity": 3387.546898833544}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320305277.88/warc/CC-MAIN-20220127163150-20220127193150-00700.warc.gz"} |
https://www.physicsforums.com/threads/e-h-nu-from-relativity.139227/ | # E = h*nu from Relativity
1. Oct 20, 2006
### lightarrow
In the book: "Introduction to Special Relativity" by Robert Resnick, Appendix C3: Gravitational red shift (I have the italian version), talking about photons, he says:
"...Actually, it's not necessary to use the Quantistic Theory. We can show from relativity itself that E is proportional to nu, since from relativistic transforms of energy and momentum, it comes that the energy of an electromagnetic pulse varies with the same factor of which varies its frequency when it's observed in a different frame of reference (see question 3)".
Is someone able to show me this? I am very interested in.
Last edited: Oct 20, 2006
2. Oct 20, 2006
### Meir Achuz
E and nu are each the zeroth component of 4-vectors.
But they are different, unrelated 4-vectors, unless QM is introduced.
E and nu transform in the same way, but there is no reason in SR to assume they are proportional. RR is just making an illogical extension.
3. Oct 22, 2006
### lightarrow
So, what do you think he wanted to mean, that could easily be solved from a first year undergraduate student?
4. Oct 22, 2006
### pmb_phy
I don't buy it. The constant of proportionality, h = Planck's constant (all of which means that E is proportional to nu) is a quantum concept.
Seems to me that he just plain screwed up.
Best wishes
Pete
5. Oct 23, 2006
### Meir Achuz
Could there be a slight mistranslation from English to Italian to English?
Hello...Ciao... goodbye.
6. Oct 23, 2006
### lightarrow
I don't know from the original english version to the italian version, but surely not in my (even if not perfect) translation from italian into english. I asked in the thread because I couldn't believe it too: it seemed a Nobel Prize discovery to me!
I wish I could read the original english version of that book.
Last edited: Oct 23, 2006
7. Oct 23, 2006
Staff Emeritus
The value of the constant of proportionality, and the fact that it comes in minimal chunks, belongs to quantum mechanics. It is not obvious to me that the proportionality itself needs to. And so, without the peculiarities of h, the discovery seems interesting but not earthshaking.
8. Oct 24, 2006
### lightarrow
Ok, but, being able to establish that proportionality, it would be possible to determine the proportionality constant in a different way than from quantum mechanics, isnt'it?
9. Oct 26, 2006
### Ich
In the original paper, you find:
So a light complex has energy E=hf, with h being a constant that is proportional to "the amount of light in the light complex".
But you need another hypothesis outside SR: that the "amount of light" comes in quanta - photons, with a corresponding constant value of h. Maybe it's not a coincidence that Einstein postulated this hypothesis in the same year.
10. Oct 26, 2006
### lightarrow
I'm sorry, what is meant here as "light complex"?
11. Nov 3, 2006
### Ich
A certain "amount of light". Could be a photon, a light pulse of finite duration, or a defined region of a planar wave.
12. Nov 3, 2006
### lightarrow
So, from the phrase: "It is remarkable that the energy and the frequency of a light complex vary with the state of motion of the observer in accordance with the same law" it seems that it's possible to compute the energy of "a light complex" in another ref. frame in a different way than from doppler effect. Is this possible?
13. Nov 3, 2006
### actionintegral
I am thinking about trying this sometime. I would calculate the energy of an harmonic oscillator in the rest frame, and then switching to another frame relativistically.
(But I am secretly hoping someone else does this first....)
14. Nov 3, 2006
### Meir Achuz
The relativistic Doppler effect (derived in many textbooks) just follows from the Lorentz transformation of the four vector (nu;k) of an EM wave.
15. Nov 3, 2006
### actionintegral
If I am looking at an harmonic oscillator at rest with me, it takes a time T to complete a period. A moving guy will see a longer period than I will. He will see a lower frequency than I will. So a moving harmonic oscillator has a lower energy than a rest one.
16. Nov 4, 2006
### lightarrow
It is exactly for this reason that I wonder how the energy of an EM wave in different ref. frames can be determined in a way different from this.
What I mean is that, from what others have wrote, including R. Resnick, it WOULD SEEM that:
1. We know how an EM wave's frequency trasforms from one ref. frame to another (relativistic doppler effect).
2. We know how an EM wave's energy trasforms from one ref. frame to another (How?!!!)
3. They transforms in the same way --> Energy is proportional to frequency.
I don't think this is possible at all, this is my concern.
Last edited: Nov 4, 2006
17. Nov 6, 2006
### Ich
Just read Einstein's paper. When you transform the electromagnetic field of a wave in some finite region, you find that the Amplitude increases with speed. The region itself becomes smaller (but contains the same number of wave fronts), and A²*Size (of the region) - the energy of the light complex - increases in the same way as the frequency does. Remarkable.
18. Nov 6, 2006
### lightarrow
That's very interesting, Ich. Do you know where can I find that paper ?
19. Nov 7, 2006
### Ich
http://www.fourmilab.ch/etexts/einstein/specrel/www/" [Broken]
§§ 7,8.
Last edited by a moderator: May 2, 2017
20. Nov 7, 2006
### lightarrow
Thank you Ich.
Similar Discussions: E = h*nu from Relativity | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8730385303497314, "perplexity": 1330.2574223648753}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463608058.13/warc/CC-MAIN-20170525083009-20170525103009-00473.warc.gz"} |
https://www.aanda.org/articles/aa/full_html/2009/32/aa12524-09/aa12524-09.html | Subscriber Authentication Point
Free Access
Issue A&A Volume 503, Number 2, August IV 2009 545 - 557 Stellar atmospheres https://doi.org/10.1051/0004-6361/200912524 02 July 2009
Signatures of intrinsic Li depletion and Li-Na anti-correlation in the metal-poor globular cluster NGC 6397,
K. Lind1 - F. Primas1 - C. Charbonnel2,3 - F. Grundahl4 - M. Asplund5
1 - European Southern Observatory (ESO), Karl-Schwarzschild-Strasse 2, 857 48 Garching bei München, Germany
2 - Geneva Observatory, 51 chemin des Mailettes, 1290 Sauverny, Switzerland
3 - Laboratoire d'Astrophysique de Toulouse-Tarbes, CNRS UMR 5572, Université de Toulouse, 14 Av. E.Belin, 31400 Toulouse, France
4 - Department of Physics & Astr onomy, Aarhus University, Ny Munkegade, 8000 Aarhus C, Denmark
5 - Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Strasse 1, 857 41 Garching bei München, Germany
Received 18 May 2009 / Accepted 10 June 2009
Abstract
Context. To alleviate the discrepancy between the prediction of the primordial lithium abundance in the universe and the abundances observed in Pop II dwarfs and subgiant stars, it has been suggested that the stars observable today have undergone photospheric depletion of lithium.
Aims. To identify the cause of this depletion, it is important to accurately establish the behaviour of lithium abundance with effective temperature and evolutionary phase. Stars in globular clusters are ideal objects for such an abundance analysis, because relative stellar parameters can be determined precisely.
Methods. We conducted a homogeneous analysis of a very large sample of stars in the metal-poor globular cluster NGC 6397, covering all evolutionary phases from below the main sequence turn-off to high up on the red giant branch. Non-LTE Li abundances or abundance upper limits were obtained for all stars, and for a sizeable subset of the targets sodium abundances were also obtained. The Na abundances were used to distinguish stars formed out of pristine material from stars formed out of material affected by pollution from a previous generation of more massive stars.
Results. The dwarf, turn-off, and early subgiant stars in our sample form a thin abundance plateau, disrupted in the middle of the subgiant branch by the Li dilution caused by the first dredge-up. A second steep abundance drop is seen at the luminosity of the red giant branch bump. The turn-off stars are more Li-poor, by up to 0.1 dex, than subgiants that have not yet undergone dredge-up. In addition, hotter dwarfs are slightly more Li-poor than cooler dwarfs, which may be a signature of the so-called Li dip in the cluster, commonly seen among Pop I stars. The feature is however weak. A considerably wide spread in Na abundance confirms that NGC 6397 has suffered from intracluster pollution in its infancy and a limited number of Na-enhanced and Li-deficient stars strongly contribute to forming a significant anti-correlation between the abundances of Na and Li. It is nevertheless seen that Li abundances are unaffected by relatively high degrees of pollution. Lithium abundance trends with effective temperature and stellar luminosity are compared to predictions from stellar structure models including atomic diffusion and ad-hoc turbulence below the convection zone. We confirm previous findings that some turbulence, with strict limits to its efficiency, is necessary for explaining the observations.
Key words: stars: abundances - stars: Population II - globular clusters: general - techniques: spectroscopic - methods: observational - diffusion
1 Introduction
The detection of Li in the atmospheres of old and metal-poor Population II stars has opened up an observational link to the primordial Universe. 7Li is indeed one of four isotopes that were synthesised by nuclear reactions shortly after the Big Bang. With the most recent determination of the baryon density from the 5-year release of WMAP data, (Dunkley et al. 2009), an initial abundance of or is obtained from standard Big Bang nucleosynthesis (BBNS) (Cyburt et al. 2008). This is significantly higher than what is found for the Spite plateau'' (Spite & Spite 1982) in the Galactic halo, a well-defined Li abundance plateau consisting of metal-poor halo dwarfs and early subgiant branch (SGB) stars.
The plateau abundance in the field has been determined in a number of recent studies to span the range -2.4 for stellar metallicities -(-1.0), with a possible tendency toward increasing Li abundance with increasing metallicity (e.g. Bonifacio et al. 2007a; Charbonnel & Primas 2005; Aoki et al. 2009; Meléndez & Ramírez 2004; Asplund et al. 2006; Hosford et al. 2009; Ryan et al. 2001). Differences between the various analyses may arise from the choice of effective temperature scale and corrections for non-LTE effects. It is nevertheless clear that a discrepancy by a factor of 2-4 with the cosmological prediction is unavoidable, suggesting that the stars have undergone the corresponding surface depletion of Li. An important fact is that in all the recent observational analysis (i.e., 21st century) of halo field stars, no evidence of significant dispersion has been found along the plateau, except for a few stars with strong Li abnormalities (e.g. Ryan et al. 1999; Asplund et al. 2006).
Soon after the discovery of the Spite plateau, Michaud et al. (1984) predicted Li depletion under the combined action of gravitational settling and weak turbulence in the radiative zones of Population II stars. Using sophisticated stellar models that treat atomic diffusion and radiative acceleration from first principles, Richard et al. (2005) illustrate how the comparably low Li value found in old, metal-poor stars could be naturally explained, assuming rather strict limits to the turbulent transport extent and efficiency. However, as of today the nature of the underlying physical mechanism responsible for turbulence has not been definitively identified, the main difficulty for the conjectured processes (mass loss, rotation-induced mixing, ...) being to account for the negligible Li dispersion along the plateau.
Charbonnel & Primas (2005) revisited the literature Li data for halo field stars with particular focus on the evolutionary status of the sample stars. This study discovered for the first time that the mean Li value appear to be higher for the turn-off (TO) and SGB stars than for the dwarfs. This result, together with the finding that all halo stars with Li abnormalities (i.e., strong deficiency or high content) lie on, or originate from, the hot side of the plateau, lead the authors to suggest that the most massive (i.e., post-MS) of the halo stars still observable today have had a different Li history than their less massive dwarf counterparts. Charbonnel & Primas suggested that such a behaviour may be the signature of a transport process of chemical elements and angular momentum whose efficiency changes on the blue edge of the plateau. This behaviour corresponds to that of the generation and filtering of internal gravity waves in both Population II and I stars (Talon & Charbonnel 2003; Charbonnel & Talon 2008). Talon & Charbonnel (2004) describe how internal gravity waves coupled with rotation-induced mixing are expected to lead to higher Li homogeneity among the plateau dwarf stars than among the more massive, slightly evolved stars, a scenario which may explain the observational findings by Charbonnel & Primas. Speaking in favour of this model is its ability to simultaneously explain the internal solar rotation profile and the time evolution of the Li abundance at the surface of solar- and F-type stars as seen in Galactic open clusters (Charbonnel & Talon 2005).
By turning to stars in metal-poor Galactic globular clusters, the intrinsic stellar processes involved can be further constrained. This approach presents some obvious advantages. In particular, the evolutionary status of the observed stars can be determined unambiguously and, presumably, all stars in a cluster were born with the same metallicity (although the surface metallicities observed today may vary between stars in different evolutionary phases, due to the effects of atomic diffusion, e.g. Lind et al. 2008; Korn et al. 2007).
NGC 6397 is one of the most well-studied metal-poor globular clusters and its Li content has been documented down to the magnitude of the cluster TO point in several studies (Castilho et al. 2000; Korn et al. 2006; Bonifacio et al. 2002; Thévenin et al. 2001; Pasquini & Molaro 1996; Korn et al. 2007), however, with rather poor number statistics. Bonifacio et al. found for the mean abundance of twelve TO stars, whereas Korn et al. found for five TO stars and for two SGB stars. By comparing this observed abundance difference of Li (as well as Fe, Ca, Mg, and Ti) to predictions from the stellar-structure models by Richard et al. (2005) and references therein, Korn et al. empirically constrain the efficiency of slow macroscopic motions counteracting atomic diffusion below the convective envelope, which should ultimately provide clues on the origin of turbulence. Note that some slow macroscopic process is also required to reproduce the observed morphologies of globular cluster colour-magnitude diagrams (VandenBerg et al. 2002).
Interpreting abundance trends in globular clusters must however be done with great caution. Indeed, it is well-known that globular-cluster stars present striking anomalies in their light element content that are not seen among their field counterparts (for reviews see e.g. Charbonnel 2005; Gratton et al. 2004). More precisely, C, N, O, Na, Mg, and Al abundances show large star-to-star variations within individual clusters. C and N, O and Na, and Mg and Al are respectively anti-correlated, the abundances of C, O, and Mg being depleted in some stars while those of N, Na, and Al are enhanced. Importantly, the abundance of Li was found anti-correlated with that of Na and correlated with that of O in turnoff stars in the more metal-rich globular clusters NGC 6752 and 47 Tuc (Pasquini et al. 2005; Bonifacio et al. 2007b). These abundance patterns are explained by contamination of the star-forming gas by hydrogen-processed material ejected by a first generation of short-lived massive globular-cluster stars. This increases the Na abundances while lowering the O and Li abundances with respect to the pristine cluster composition in a second generation long-lived low-mass stars. In the framework of the self-enrichment scenario the Li content of globular-cluster stars is actually an important tool to quantify the dilution factor between the ejecta of the massive stars responsible for pollution and the pristine intra-cluster matter (Prantzos & Charbonnel 2006; Decressin et al. 2007a,b). Gratton et al. (2001) and Carretta et al. (2005) uncovered rather large variations in both O and Na, anti-correlated with each other, in SGB and dwarf stars in NGC 6397 (see also Castilho et al. 2000; Norris & Da Costa 1995, for bright giants). This cluster has thus suffered from internal pollution in its infancy.
In the present study we analyse Li and Na abundances for a large sample of stars in NGC 6397, in an attempt to disentangle the primordial value of Li, with effects of atomic diffusion and intrinsic stellar depletion on the one hand, and early cluster pollution on the other hand. Section 2 describes the observations and data reduction, Sect. 3 describes the determination of stellar parameters and the abundance analysis. In Sect. 4 we present the Li and Na abundances found, and in Sect. 5 we discuss signatures of Li depletion in the cluster. Section 6 summarises our conclusions.
2 Observations
The observations include spectroscopic data, described in Sect. 2.1, and photometric data, described in Sect. 2.2. All targets, coordinates, and photometry are listed in Table 2. The locations of the targets in the colour-magnitude diagram (MV-(v-y)) of the cluster are shown in the right-hand panel of Fig. 5.
2.1 High and medium-high resolution spectroscopy
Table 1: FLAMES observations.
All our targets have been selected from the Strömgren uvby photometric survey carried out by Grundahl, matching the (b-y) and c1 ranges spanned by the field stars analysed by Charbonnel & Primas (2005). In total, 349 stars were observed across the colour-magnitude diagram of NGC 6397, from just below the cluster TO point ( ) to the end of the red giant branch (RGB) ( ). Each evolutionary phase is well sampled with around 180 stars at the TO, 80 on the SGB, and 90 stars on the RGB. Here we present data from two observing runs with FLAMES on the VLT-UT2 (Pasquini et al. 2002), one that collected the Li data for the whole sample in June 2006 and a second run that completed the data set with Na abundance indicators for a subset of the targets in Aug. 2008. The dates, set-ups, exposure times, and atmospheric conditions are summarised in Table 1.
Both sets of observations have used the GIRAFFE+UVES combined mode of FLAMES, with Medusa fibres allocated to the faintest targets and UVES fibres allocated to the brightest end of the RGB. During the first observing campaign (2006), we selected the high-resolution GIRAFFE set-ups H679.7 (also known as HR15) and H627.3 (HR13), which yield respectively R=19 000 and 22 300, and spectral ranges of 660.7-696.5 nm and 612.0-640.5 nm. Both settings were observed in combination with the UVES Red 580 setting (R=47 000, spectral range: 480-680 nm, with a small gap in the middle). The first GIRAFFE set-up covers the 7Li resonance line at 670.7 nm, whereas the second setting covers the Na I doublet at 615.4/616.0 nm,.
Unfortunately, the detection of the Na I doublet at 615.4/616.0 nm proved to be realistic only for the brightest targets of our sample, therefore preventing conclusions on the existence of a Li-Na anti-correlation. To overcome this problem, we proposed to observe a stronger Na I doublet (at 818.3 nm and 819.4 nm,) and were granted 3 h through the Director General Discretionary Time channel in Aug. 2008. Because of the limited time request we carefully selected a sub-set of our initial sample, mainly including the brightest TO and SGB stars (in total, 117 stars), with both high and low Li. For these observations, we used the high-resolution GIRAFFE set-up H805.3A (also known as HR19A, characterised by R=14 000 and spectral range 774.5-833.5 nm), in combination with the UVES Red 860 set-up (R=47 000, spectral range 760-1000 nm).
Standard data reduction of the 2006 GIRAFFE observations was performed with the Geneva Base Line Data Reduction Software (girBLDRS), version 1.13.1. To correct for the highly elevated dark current in the upper corner of the old CCD a carefully scaled and smoothed 2D dark-frame was subtracted from each science frame. The 2008 data were obtained after the GIRAFFE CCD upgrade that took place in Apr. /May 2008 and contain negligible dark current. These data were reduced with the ESO GIRAFFE pipeline, version 2. 5. 3. UVES data reductions were performed with the FLAMES UVES pipeline, version 2. 9. 7.
After initial reduction followed sky-subtraction and radial-velocity correction. Through radial-velocity measurements we identified six non-members. Furthermore, we disregarded one star that appeared to be too metal-rich to be a cluster member, three stars that clearly fall off the cluster sequence, and yet three more stars for which the observed spectra are of too poor quality. All stars that were rejected are listed with comments in Table 2. The remaining sample consists of 32 stars with spectra obtained with the UVES Red 580 set-up (we hereafter refer to this set as the UVES sample'') and 284 stars with spectra obtained with the GIRAFFE H679.7 and H627.3 set-ups. The UVES sample and the GIRAFFE sample have 11 stars in common. A sub-sample of eight stars were observed in the UVES Red 860 set-up and 117 with the GIRAFFE H805.3A set-up. The UVES sample and GIRAFFE sample have two stars in common.
The mean barycentric radial velocity obtained for all cluster members is , in good agreement with Lind et al. (2008) ( ) and Milone et al. (2006) ( ).
2.2 Strömgren photometry
The Strömgren uvby photometry employed for this investigation was obtained at the Danish 1.54 m telescope on La Silla, Chile, and is identical to the data set used by Korn et al. (2007) and Lind et al. (2008). We reiterate the main points here. All uvby observations were collected during a two week run in May 1997, with a large number of observations of standard stars observed on the photometric nights. The field covered is roughly 9 in diameter and slightly west of the cluster centre. For the photometric reductions we used the same programs and procedures as described in Grundahl et al. (1998, 1999, 2000, 2002a). The photometric zero points in vby have errors of approximately .
The observed (b-y), (v-y), and c1 colours were corrected for the reddening of NGC 6397, using the relations , , and (Crawford 1975). We adopted a value of 0.179 for EB-V, following Anthony-Twarog & Twarog (2000), and 12.57 for the cluster distance modulus. The observed colours of our targets show a spread around the cluster sequence, of the order of 2.2% for (v-y) and larger, 3.7%, for (b-y). To reduce the uncertainty in the photometric colours we constructed colour-magnitude fiducials for the cluster, as described in Korn et al. (2007) and Lind et al. (2008). The observed colour is interpolated at constant V-magnitude onto the fiducial sequence. The main advantage of using this method is that relative effective temperatures are more precisely determined, which in turn lowers the spread in chemical abundances.
Figure 1: The and relations considered in the analysis. The vertical lines mark the area that is linearly interpolated (see text). The O09 scale for pre-TO stars fall above the one for post-TO stars and vice versa for A9699. Open with DEXTER
Table 2: Photometry and effective temperatures. The full table can be retrieved from CDS/Vizier.
3 Analysis
For the abundance analysis we used a grid of 1D, LTE, plane-parallel and spherical, MARCS model atmospheres (Gustafsson et al. 2008). LTE abundances of Li and Na were derived from spectrum synthesis, using the Uppsala code BSYN. Non-LTE abundance corrections were thereafter applied. LTE abundances of Fe and Ca were derived from equivalent width measurements, using the corresponding Uppsala code EQWIDTH.
3.1 Effective temperatures and surface gravities
The cluster uvby-photometry was used to calculate effective temperatures for the whole sample. We implemented a number of different calibrations based on the colour indices (v-y) and (b-y) and c1. Table 2 lists effective temperatures calculated using the relations published in Alonso et al. (1996) and Alonso et al. (1999, hereafter these two papers are referred to as A9699), Ramírez & Meléndez (2005, hereafter RM05), and Önehag et al. (2009, hereafter O09). The first three are calibrated on the infra-red flux method (IRFM), whereas the last is based on theoretical colours from MARCS model atmospheres (Gustafsson et al. 2008). For all stars we assumed . The IRFM colour- relations constructed for main sequence (MS) stars were calibrated on stars with . We accordingly set a limit in absolute visual magnitude, MV>3.3, for which the MS calibrations are trusted. Analogously, the giant calibrations were calibrated on stars with and we trust them for MV<2.8. This leaves a gap on the SGB where no calibration is suitable. In this range we interpolated linearly between each dwarf- and giant-calibration pair (see vertical lines in Fig. 1). To each of the obtained effective-temperature scales we calculated surface gravities from the relation between surface gravity, effective temperature, stellar mass, and luminosity. The details of the calculations are given in Lind et al. (2008).
O09 have calculated synthetic (b-y)-colours for a grid of effective temperatures, surface gravities, and metallicities. We interpolated in the grid constructed for , which spans the parameter-space of the sample (see Fig. 4 in O09). The interpolation was done iteratively until the parameters found for each star were fully consistent, i.e. the pair found based on the observed colour are also constrained by relation to mass and luminosity.
For the 32 RGB stars observed with UVES, we could also derive H-based effective temperatures, using BSYN. Stark broadening is based on the tabulations by Stehlé & Hutcheon (1999) and self-broadening of hydrogen follow Barklem et al. (2000). The fitting method is automated and based on metal line-free regions extending up to nm from the line centre (see Lind et al. 2008), avoiding the line core. The wings of the H-line become very narrow high up on the RGB, but some sensitivity to effective temperature remains also for these stars. With a high S/N-ratio it is possible to constrain also for the coolest stars in our sample.
Figure 2 shows a comparison between all and relations. The O09 (b-y)-calibration matches the slope of the corresponding IRFM-based calibrations well, but is offset to higher effective temperatures by approximately 100 K. However, a recent updated calibration based on the IRFM finds effective temperatures that are higher than the existing IRFM calibrations, by similar order of magnitude, at these metallicities (Casagrande et al. 2009, in preparation). Interestingly, the O09 scale predicts slightly higher effective temperatures for dwarfs compared to SGB stars of the same colour, whereas the IRFM of A9699 has the opposite behaviour (through the sensitivity to the c1-index). The interpolation between dwarf and giant calibrations appears more suitable for the (b-y)-relations. The two (v-y) MS scales are too flat to match the slope of the corresponding giant relations well. For this reason we regard the (b-y)-relations as more trustworthy and place greater weight on those. The H-based values are generally cooler than all photometric calibrations, but the relative agreement is satisfactory.
The right panel of Fig. 2 shows all (b-y)-based effective-temperature scales with corresponding surface gravity values. Apparently, the two IRFM scales agree well for all but the hottest stars, where the A9699 scale predicts cooler effective temperatures. Again it is seen that the theoretical MARCS scale is offset to higher effective temperatures. The left panel of Fig. 2 shows the cluster fiducial for (b-y)0-MV.
Figure 2: Left: the (b-y)0-MV fiducial used to derive stellar parameters. Right: the three (b-y)-based stellar-parameter sets. Open with DEXTER
3.2 Metallicity
To constrain the cluster metallicity and microturbulence values, we calculated iron abundances from Fe I and Fe II lines for the UVES sample. The ionisation and excitation equilibrium of Fe also provide an additional check of the derived stellar parameters. We selected 48 unblended Fe I lines that have oscillator strengths determined by O'Brian et al. (1991) and 14 Fe II lines that have oscillator strengths determined by Biemont et al. (1991), Raassen & Uylings (1998), or Fuhr et al. (1988). Equivalent widths were measured by fitting Gaussian line profiles to the observed, fixing the FWHM of lines weaker than 5 pm to the average value for each individual star. In the GIRAFFE spectra only very few Fe I and Fe II lines are available and it is not possible to obtain reliable measurements for the fainter half of the sample. Table 3 lists all Fe I and Fe II lines with references to the adopted values, as well as the equivalent widths and abundances derived for one RGB star. The full table, including all UVES targets, can be retrieved from the CDS. The table lists also the average abundance derived from Fe I and Fe II lines, respectively, with the error bar representing the standard deviation of the mean abundance.
Microturbulence ( ) values were determined by requiring that Fe abundances show no trend with reduced equivalent width ( ) of Fe I lines. We empirically constructed a linear parameterisation of microturbulence with surface gravity, using the UVES targets only, and adopted it for the whole sample. This involved extrapolation to the SGB stars and dwarfs, since our observations do not cover enough strong lines that are sensitive to microturbulence in these stars. The impact on the inferred Li abundances is however negligible. The largest equivalent width measured for the Li line is 6.18 pm, for the SGB star #13160. Varying the microturbulence between and lowers the Li abundance inferred for this star by only 0.01 dex.
Table 3: Iron lines measured in the UVES spectrum of the RGB star #17691. The full table can be retrieved from CDS/Vizier.
The excitation equilibrium of Fe is commonly used to constrain effective temperatures. However, 3D modelling of stellar atmospheres show that especially low excitation lines may give strongly overestimated abundances in 1D models of metal-poor stars (Asplund et al. 1999; Collet et al. 2007). In addition, non-LTE effects are likely to have different impact on lines of different excitation potential (e.g. Asplund 2005). The extent to which this could bias the excitation equilibrium is presently not clear. The left-hand histogram in Fig. 3 shows the slope of Fe abundances with excitation potential of Fe I lines, derived for the A9699 and O09 scales. Apparently, both scales are too hot to satisfy the equilibrium. The necessary corrections are of order (-100)-(-200) K.
The right-hand histogram in Fig. 3 shows the Fe abundance derived for singly ionised Fe lines minus the Fe abundance derived from neutral Fe lines. The A9699 scale shows an offset from zero of approximately -0.14 dex, whereas the O09 scale has a smaller offset, -0.05 dex. The mean Fe abundance, as determined from Fe II lines, is , using both scales, with a dispersion of 0.04 dex. Adopting a solar abundance (Asplund et al. 2009, in prep. ), we obtain . This is in good agreement with most recent estimates made for RGB stars in this cluster; Korn et al. (2007) find for their RGB sample, Gratton et al. (2001) , and Castilho et al. (2000) . Lind et al. (2008) report a lower value, , for stars at the base of RGB. The last study implements a cooler effective temperature-scale and base the abundances mainly on Fe I lines, which explains the offset.
The investigation is not indicating that one of the stellar-parameter sets is clearly preferable to the other, but we used the O09 scale to derive abundances of all other elements, because this is the most homogeneous. The changes in Li abundances when implementing the IRFM stellar-parameter scales are discussed in Sect. 5.2.
Figure 3: Left: histogram of the slope of Fe abundance with excitation potential of neutral and singly ionised Fe lines. Only the UVES sample is shown. Solid lines represent the O09 effective temperature scale and dotted lines the A9699 scale. Right: histogram of Fe abundance derived from Fe II lines minus the Fe abundance derived from Fe I lines. Open with DEXTER
Figure 4: Example fits of the Li I 670.7 nm line and the Na I 819.4 nm line. In each panel is indicated the identification number of the star, stellar parameters, and LTE abundance inferred from the given line, according to the syntax , where x is the corresponding element and e(x) the error in abundance. Top: A Li and Na-normal TO star. Second from the top: A Li-poor and Na-rich TO star. Second from the bottom: GIRAFFE giant spectrum. Bottom: UVES spectrum of same giant. Open with DEXTER
3.3 Lithium
The 7Li resonance line at 670.7 nm has two fine-structure components, separated only by 0.015 nm. At the resolution of GIRAFFE (R=19 300) they are indistinguishable, but a small line-asymmetry can be seen in the higher-resolution UVES spectra (R=47 000). Accurate measurements of the line require spectrum synthesis (see example fits of the line in Fig. 4). We adopted and -0.309 for the two components respectively, according to Lindgård & Nielson (1977).
To derive Li LTE abundances for the stars we first normalised each spectrum by iterative fitting of a first order polynomial, taking into consideration a wavelength region extending nm from the line centre. Synthetic line profiles were convolved with a Gaussian profile to account for the instrumental broadening and broadening due to rotation and macroturbulence. The width of the convolving profile was fixed for each star by measurements of single lines of other species. The Li abundance was then varied to find the best fitting synthetic profile for each star through -minimisation. In total, Li abundances or upper abundance limits could be obtained for 305 stars. Following Norris et al. (2001) and Barklem et al. (2005), errors due to observational uncertainties were estimated with the expression: , where n denotes the number of pixels spanning the full-width at half maximum for the line, R the spectral resolution, and S/N the signal-to-noise ratio per pixel.
Non-LTE abundance corrections for Li were calculated according to Lind et al. (2009), a study that is based on calculations on the same MARCS model atmospheres as here (Gustafsson et al. 2008). In contrast to earlier non-LTE analyses (e.g. Carlsson et al. 1994; Takeda & Kawanomoto 2005) the study makes use of rigorous quantum mechanical calculations (as opposed to the debated but common classical recipes) of cross-sections for collisions with neutral hydrogen, which have influence over the statistical equilibrium of Li. Lind et al. (2009) show that including the charge transfer reactions between Li and hydrogen in the non-LTE calculations gives abundance corrections that are lower by almost -0.1 dex for dwarfs at , as compared to neglecting them. For our sample, the MS, TO, and early SGB stars all have non-LTE corrections that are very similar, around -0.06 dex. With decreasing effective temperature and surface gravity the corrections change sign and reach a maximum of +0.13 dex for the coolest RGB star.
3.4 Sodium
Sodium abundances or upper abundance limits were derived from the 2008 observations for a subset of 117 stars, mainly TO stars and early SGB stars. The abundances were based on the Na I 818.3 nm and 819.4 nm doublet (when the former was too weak to be detected, only the latter was used). This doublet is second in strength after the resonance doublet at 588.5 nm and 589.0 nm, which is strongly affected by interstellar extinction for the cluster stars and is therefore not useful as abundance indicator. As mentioned above, the 568.2/568.8 nm and 615.4/616.0 nm doublets are too weak to be suitable for abundance analysis in the TO region at this metallicity. Despite its unfortunate location in the middle of a strong telluric band, the 818.3/819.4 nm Na I doublet does not suffer strongly from atmospheric blends that would bias the abundance analysis.
The line-fitting method is the same as described for Li and oscillator strengths by Kurucz & Peytremann (1975) were adopted. We apply non-LTE abundance corrections to the 818.3/819.4 nm Na I lines according to Mashonkina et al. (2000). The size of the corrections at this metallicity ranges from -0.14 dex for the hottest, highest surface gravity stars, to approximately -0.31 dex for the coolest giants.
Table 4: Adopted stellar parameters, equivalent widths, and Li, Na, and Ca abundances. The full table can be retrieved from CDS/Vizier.
3.5 Calcium
For the complete sample of 305 stars, only the equivalent widths of Ca lines 612.2 nm and 616.2 nm could be reliably measured. We based the Ca abundance on these two lines for 305 stars, adopting oscillator strengths from Smith & O'Neill (1975).
4 Results
Figure 5: Left: the abscissa shows non-LTE Li abundances inferred for our sample (the archival data are not included) and the ordinate absolute visual magnitude ( MV=V-12.57). Arrows mark Li upper limits. Right: the spectroscopic targets marked with black filled circles in the colour-magnitude diagram of NGC 6397. The two horizontal dashed lines mark the locations of rapid decrease in Li abundance caused by stellar evolution (see text). Open with DEXTER
4.1 Li abundances
Table 4 lists the adopted stellar parameters, equivalent widths, and abundances. Figure 5 shows the Li abundances inferred side-by-side with a colour-magnitude diagram where the spectroscopic targets are marked. In agreement with the expectations (see Sect. 1), there is a rather well defined abundance plateau among the least evolved stars, followed by a drastic drop in Li abundance in the middle of the SGB ( ). This pattern is caused by the so-called first dredge-up, i.e., the dilution of the external convective stellar layers with deeper hydrogen-processed material. A second steep drop occurs at , which corresponds to the luminosity of the RGB bump. A similar decrease in Li abundance have been identified in stars located around the RGB bump in NGC 6752 (Grundahl et al. 2002a) as well as in stars in the halo field (Gratton et al. 2000). Charbonnel & Zahn (2007) describe how this second Li abundance drop can be explained by a mixing process called thermohaline convection, which becomes efficient when the hydrogen-burning shell crosses the chemical discontinuity left behind by the first dredge-up, and which rapidly transports surface Li down to internal hotter regions where this fragile element burns.
The average Li abundance on the plateau, i.e. only including stars that have not undergone dilution due to the first dredge-up (MV>3.3), is , where the error is the standard deviation of the mean. The 1 dispersion is 0.09 dex.
4.2 Lithium data from the ESO archive
To establish a view of the Li content in NGC 6397 that is as complete as possible, we have searched the ESO archive for observations of more MS, TO, and SGB stars. Especially, to put constraints on the size and significance of a small increase in located just before the onset of the first dredge-up (see Sect. 5.1 and Fig. 6) we would benefit from having more stars. In previous analyses, Bonifacio et al. (2002) analyse Li in twelve TO stars (of which seven stars were previously analysed by Thévenin et al. 2001; and three stars by Pasquini & Molaro 1996) and Korn et al. (2007) analyse five TO stars and two SGB stars. We did not incorporate those stars in our analysis since it would only slightly have increased the number statistics. However, we have retrieved and analysed a set of FLAMES archival data from 2007 (079.D-0399(A), Gonzalez-Hernandez). These include observations in the GIRAFFE high-resolution set-up H665.0 (HR15N, R=17 000), covering the LiI 670.7 nm line for a sample of 80 dwarfs ( ) and 88 SGB stars ( ) in NGC 6397. The reduction of the data was performed with the girBLDRS pipeline. In total, 38 MS stars and 55 SGB stars were identified in our uvby photometric catalogue. For the remaining stars, we add a zero-point correction of 0.09 mag to the visual magnitude scale listed in the archival data, estimated from the targets that were cross-matched between the catalogues. Applying the correction, the two magnitude scales match to typically within 0.02 mag for individual targets. After establishing the visual magnitudes, colours are assigned to each target by interpolating V onto the fiducial cluster sequences, as was done for the main sample (see Sect. 2.2).
The results of the analysis of the archival data are shown together with our own data set in Fig. 6. One more very Li-poor star, marked with an arrow, was identified in the archive sample, for which only an upper limit to Li could be derived. Especially, the region around MV=3.2-3.5 is now much better sampled, with the archival data confirming the presence of the small abundance increase just before the steep drop. In addition, the plateau is extended to fainter magnitudes, . Including this archive sample we have Li abundances for 454 stars in the cluster, of which 346 have MV>3.3. Overall the quality of the data sets are similar and the abundances agree very well between the samples. In total, 14 SGB stars overlap and thus have observations of the Li line in the H665.0 as well as in the H679.7 setting. The mean difference in Li abundance between these 14 stars is dex, i.e. there is no systematic bias between the data sets.
The average Li abundance on the plateau (i.e., excluding stars with MV<3.3), including both our data and the archival data, is . Unfortunately Na information is not available for the archive targets.
Figure 6: Lithium abundance against visual magnitude. The targets of our observations are plotted with filled circles and the archival data are plotted with open squares. Information about evolutionary status and effective temperature for various MV intervals is also given. Open with DEXTER
4.3 The Li-Na anti-correlation
As discussed in Sect. 1, globular-cluster samples may suffer from bias due to early pollution of the intra-cluster gas by a first generation of more massive, faster evolving stars. It is thus important to discriminate between first-generation stars that formed out of pure pristine material and second-generation stars that formed out of polluted material. The former should have been born with the cosmological Li abundance and a low Na-abundance, similar to that of field stars of comparable overall metallicity. The latter may have inherited a lower Li abundance at birth and will also exhibit other signatures of H-burning at high temperature, in particular enhanced Na abundances resulting from the NeNa chain in the polluter stars.
In Fig. 7 we report the Li and Na abundances for 100 stars in our sample with MV>3.3, i.e., those that have not yet undergone Li dilution via the first dredge-up. Note that it is the first time that these quantities are determined simultaneously for such a large sample of TO and SGB stars in a globular cluster. This diagram clearly emphasises the extent of the star-to-star Na abundance variations (by up to 1 dex) within NGC 6397. This is in agreement with the findings by Carretta et al. (2005), which were based on six dwarfs and nine SGB stars. From the appearance of Fig. 7, we deduce that the most Li-poor targets indeed show the most elevated Na levels, implying an anti-correlation between the abundances of Na and Li. The spectrum of the most Li-poor and second most Na-rich star in the sample, #6685, is shown in Fig. 4. The abundance pattern displayed by this star and by at least three more stars, agrees well with the globular cluster self-enrichment scenario and we conclude that they have likely been formed from polluted gas. Field stars with similar metallicity to NGC 6397 have typical Na abundances in the range -3.9 (Andrievsky et al. 2007). It is thus reasonable to regard stars with as belonging to a first generation of non-polluted objects.
To find the statistical significance of the indicated Li-Na anti-correlation we performed linear regression between and , taking into account measurement errors in both quantities as well as upper limits on Na (IDL-routine linmix_err, see Kelly 2007). The analysis gave a correlation coefficient of -0.6 between the Na and Li abundances, which for a sample of this size is highly significant. The probability of two uncorrelated variables producing such a correlation coefficient is less than 0.05%. However, as can be realised from Fig. 7, the significance of the anti-correlation is dependent on the most Na-enhanced stars. In fact, performing the same linear regression for stars with no significant anti-correlation was found. The Li abundances among the plateau stars are thus not much affected by high degrees of pollution.
In Fig. 7 it is also indicated with a tilted arrow approximately how the location of a star would be affected by an erroneous effective temperature. An error of K corresponds to and . Errors in effective temperature therefore weakly correlate the abundances with each other, rather than anti-correlate them. Hence, it is clear that the anti-correlation itself is not an artifact from uncertainties in the stellar-parameter determination. Possibly even a tighter anti-correlation exists for all targets, but is being distorted by the tendency of effective-temperature errors to align the abundances. As the Li abundance as well as the Na abundance are based on lines from neutral atoms, the effective temperature is the stellar parameter with by far the greatest influence over the results. The uncertainties stemming from surface gravity, microturbulence, and metallicity are all negligible in the context.
The extent of the Li and Na variations found in the present study is similar to that found in NGC 6752 (Pasquini et al. 2005) and 47 Tuc (Bonifacio et al. 2007b), and thus appears to be independent of the metallicity. However, the Li-Na anti-correlations in NGC 6752 and 47 Tuc are based respectively on nine and four stars only, limiting the comparison. A more detailed discussion of the implications for the self-enrichment of NGC 6397 will be performed in a separate paper where O, Na, Mg, and Al abundance determinations will also be presented for a subset of RGB stars in our sample. In Sect. 5 we use the information about Na to constrain intrinsic stellar Li depletion processes.
Figure 7: Non-LTE Li and Na abundances for the sub-sample of 100 dwarfs and SGB stars with observations in the H805.3A setting and MV>3.3. The arrows mark upper limits to the Na abundance. If the effective temperature of a target has been underestimated by 100 K, the corresponding point in this diagram should be shifted approximately in the direction of the tilted arrow and in its inverse direction for overestimated effective temperatures. Open with DEXTER
5 Discussion
5.1 Signatures of intrinsic lithium depletion
A proposed explanation for the difference between the Li abundance found in the metal-poor halo and the BBNS prediction is intrinsic Li depletion due to the combined effect of gravitational settling and weak turbulence, the nature of which is still a matter of debate. In the case of NGC 6397, the stars on the Spite plateau would have to have been depleted by typically dex, corresponding to almost a factor of three. To constrain the physics involved in producing such depletion, it is important to accurately establish trends of Li abundance with evolutionary phase and effective temperature. In the following, we carry out our analysis based on sub-samples of data defined according to Na content. As discussed in Sect. 4.3, it is reasonable to consider that the stars born out of pristine material are those with .
Figure 8: Group-averaged Li abundances against visual magnitude. The full line is drawn for the whole sample (including the archival data), i.e., for both first-generation Na-poor stars and second-generation Na-rich stars. Each bin contains 40 stars. The dashed and dotted lines correspond to different sub-sets of our own sample, selected according to their corresponding Na abundances. Each bin contains 13 stars. In the upper left corner mean error bars are shown for each line. Note that the range between -4.8 is not sampled by the data. Open with DEXTER
Figure 8 shows Li abundance vs visual magnitude on the plateau, with abundances binned in groups of targets for clarity. In the lower panel of each plot, the 1 dispersions in the corresponding bins are shown. The full lines are drawn for the whole sample including both our data and the archival data, i.e., it does not discriminate between first-generation Na-poor stars and second-generation Na-rich objects. The single plateau-star for which only an upper limit could be inferred is not included in the binning. The dashed lines correspond to first-generation stars with and dotted lines to second-generation stars with . Mean error bars for each line are also indicated in the figure. Figure 9 shows the same plot, but using effective temperature as the reference scale instead of visual magnitude. To separate pre- and post-TO stars with the same effective temperature, the dwarfs are represented by open square symbols and the SGB stars with filled circles. The approximate evolutionary status of the stars can be recovered from Fig. 6. Stars with are dwarf stars below the TO, having effective temperatures reaching from the maximum 6400 K to 6200 K at . Stars with MV<4 are slightly more massive objects that have passed the TO, i.e., SGB stars with -5900 K. At the Li dilution process through the first dredge-up sets in.
As can be seen in Figs. 8 and 9, trends of Li abundance with evolutionary phase and effective temperature are not significantly different between first- and second-generation objects. The mean value of appears generally to be slightly higher for the first-generation stars (dashed lines) than for the second generation (dotted lines), in agreement with what is expected in the globular cluster self-enrichment framework, but the trends are fully compatible within the error bars. We conclude that the mean Li abundance trends of the full sample are not significantly biased by intra-cluster pollution in NGC 6397. On examination of Fig. 8, we especially note the presence of an upturn in just before the steep abundance drop at , which appears rather robust against varying in the sample. In a very limited magnitude-range the mean Li abundance increases by around 0.1 dex up to . As mentioned previously, a difference of this size between TO and SGB stars in NGC 6397 was identified by Korn et al. (2007).
The behaviour of Li abundance with is illustrated in Fig. 9. There is an overall increase with decreasing effective temperature in the range 6400-6100 K, where a maximum is reached, followed by a decrease again until K, where the first dredge-up sets in. The initial increase appears to be present both for pre- and post-TO stars. Speculatively, the lower value found for hotter dwarfs compared to cooler dwarfs, could be interpreted as the counterpart of the very right-shoulder of the so-called Li-dip observed in Population I stars (see references in Talon & Charbonnel 1998,2003). This feature appears in the present data at exactly the same effective temperature as in open clusters: In the Hyades, the mean Li abundance decreases by approximately 0.10-0.15 dex in the effective temperature range between 6200 and 6400 K. As discussed by Talon & Charbonnel (2004,1998), a similarity between Pop I and Pop II dwarfs is theoretically expected, since the depth of the stellar convective envelope, and thus the nature and the efficiency of the Li depletion mechanism, are expected to depend only on the stellar effective temperature. If this feature is real (see Sect. 5.2), it seems that we may have discovered for the first time in Population II stars the very beginning (in terms of effective temperature) of the Li-dip, which may be more shallow at this low metallicity. Note that one should be cautious with interpretations of Li trends for dwarfs in our sample. As mentioned in Sect. 2.1, due to observational limitations the majority of our sample stars with Na determination are SGB stars, only a handful being relatively warm dwarfs. No Na abundance determination has thus been carried out for the coolest dwarfs below the TO. However, we have seen that the bias in Li abundance trends among SGB stars due to pollution is weak, which is likely true also for dwarfs.
Figure 9: Group-averaged Li abundances against effective temperature. The lines correspond to the same sub-samples as in Fig. 8. Filled circles represent post-TO stars and open squares pre-TO stars. In the upper left corner mean error bars are shown for each line. Open with DEXTER
In Sect. 4 we concluded that the most Li-deficient stars in our sample likely belong to a second generation of stars, having experienced intra-cluster pollution. Among the first-generation stars with low Na levels, there are no stars with Li abundances lower than . The lower panels of Figs. 8 and 9 show that the abundance dispersion of the full sample is always rather low, below 0.1 dex, except for the bin at and K, which contains the most Li-poor star with an abundance detection (see Fig. 6) and therefore has a higher dispersion. Focusing on the first generation of Na-poor stars only, the typical abundance scatter is 0.09 dex, not following any obvious trends with effective temperature or visual magnitude. Not shown in the figures is the typical measurement error in abundance, stemming from photon noise, which ranges from 0.08 dex for the hottest stars to 0.05 dex for the coolest plateau stars. To this measurement error should be added the propagated uncertainty in stellar parameters. Assuming that the typical star-to-star error in effective temperature, which is the most influential parameter, lies in the range 50-100 K, a corresponding additional spread of 0.04-0.07 dex in Li abundance is expected. Given these basic estimates, we conclude that the observations are compatible with zero scatter in Li abundance among the first-generation stars.
5.2 Effects of stellar parameters and non-LTE
We now briefly discuss the influence on the abundance trends presented in Sect. 5.1 from the choice of effective-temperature scale and from the non-LTE treatment. Basically, both effects have a systematic influence over the Li data. If we were to adopt the IRFM-based effective-temperature scale by RM05 (see Sect. 3.1) the Li abundances of post-TO stars would be systematically lowered by -0.06 dex, thus aggravating further the overall discrepancy to the primordial value. The A9699 scale would act in the same direction, but also have a minor differential impact on the Li abundances since this effective-temperature scale is steeper in the range from the TO to further up the SGB. The TO star abundances would therefore be lowered by a greater amount (-0.1 dex) than the cooler SGB star abundances, which in turn would enhance the abundance upturn seen in Fig. 8 and the corresponding maximum abundance seen in Fig. 9 at K. Nothing would thus qualitatively change in the discussion of abundance trends for post-TO stars. However, implementing either of the two IRFM-based scales would lower the abundances found for cool dwarfs more than hot dwarfs. The difference between these two groups discussed in Sect. 5.1, which we speculated could be a signature of the Li-dip, would hence be erased.
As mentioned in Sect. 3.3, the non-LTE abundance corrections are similar, approximately -0.06 dex, for all plateau stars. The LTE abundances therefore have the same relative behaviour but are offset to a slightly higher values.
5.3 Comparison to other studies
Comparing the found Li abundances to other determinations made for NGC 6397, we find good agreement with Korn et al. (2007), who place their five TO stars with at and two SGB stars with at . The similarity between the abundances for TO and SGB stars is partly due to the cancellation between the effects from the cooler effective-temperature scale of Korn et al. and the fact that they do not correct for non-LTE. For RGB stars the two effects do not cancel (non-LTE corrections are positive for RGB stars) and Korn et al. thus find abundances that are approximately 0.1 dex lower than ours. The Li abundances reported by Bonifacio et al. (2002) for TO stars lie on average 0.1 dex higher than ours. This difference can be fully traced to their effective-temperature scale, which is hotter by approximately 50 K than the one adopted in this study, and to their use of different non-LTE corrections (Carlsson et al. 1994). For these stars, the corrections by Carlsson et al. are smaller in absolute value than the ones found by Lind et al. (2009). The Li abundances and abundance upper limits reported by Castilho et al. (2000) for 16 RGB stars are in good agreement with ours, except for the Li detections made for giants cooler than 4900 K, which are significantly higher than what we find.
In the extensive literature study of field stars carried out by Charbonnel & Primas (2005), the authors concluded that the stars in their TO and SGB samples generally are more Li-rich than their dwarf sample. Especially, dwarfs that are cooler than 6000 K show a distinct decrease in Li abundance with decreasing effective temperature. As the dwarfs in our sample are all hotter than 6000 K we cannot verify this result. However, with hindsight one may trace a tendency of increasing Li abundance with decreasing effective temperature in the range -6400 K among the MS, TO, and post-TO stars observed by Charbonnel & Primas (see their online Tables 5-9), similar to the one we find.
Figure 10: Top: comparison between bin-averaged Li abundances (red filled circles connected with solid lines) and the predictions from the stellar-structure models of Richard et al. (2005). T5.80 represents the model with lowest efficiency of turbulent transport, T6.00 intermediate efficiency, and T6.09 highest efficiency. The reference scale is logarithmic luminosities in units of solar luminosities. Bottom: the same plot for Ca abundances. A colour version of this figure is available in the online edition of the journal. Open with DEXTER
Figure 11: Top: same as in in Fig. 10 but with effective temperature as the reference scale. Here, only post-TO stars are included. Bottom: the same plot for Ca abundances. A colour version of this figure is available in the online edition of the journal. Open with DEXTER
5.4 Comparison to diffusion-turbulence models
Figures 10 and 11 show comparisons of our obtained Li abundances with the predictions from stellar-structure models including atomic diffusion from first principles and an ad-hoc recipe for turbulent mixing (Richard et al. 2005, and references therein). For clarity we show the same group-averaged Li abundances as in Figs. 8 and 9, including the whole sample, irrespectively of Na abundance. Only post-TO stars appear in Fig. 11. For comparison, the bottom panels of both figures show corresponding trends for binned Ca abundances. Visual magnitudes are converted to stellar luminosities using the bolometric corrections of Alonso et al. (1999). The same three models, T5.80, T6.00, and T6.09, with different efficiencies of turbulent transport, as displayed in Korn et al. (2007) and Lind et al. (2008) are shown. To ensure that the model -plane is in agreement with our study regarding the location of the TO-point, we add +80 K to the effective-temperature scale of Richard et al. and +0.05 dex to the logarithmic luminosity-scale. The absolute abundance scale of the models is shifted to agree with the observations for stars evolved beyond the onset of the first dredge-up.
The efficiency of turbulent mixing determines how much Li that is transported between the external convection zone to the region in the stellar interior where Li is destroyed, which in turn has influences over the appearance of the Li abundance plateau. In the T6.09 model, with highest efficiency, the plateau is flat, whereas the surface abundances predicted by the T5.80 model show a clear dependence on stellar luminosity and effective temperature. As seen in Fig. 10, the Li abundances of NGC 6397 strictly limit the efficiency of turbulent mixing, although no model perfectly reproduces the observed abundances. In terms of the size of the Li variations on the plateau, the T6.00 model appears to be the best choice, supporting the findings by Korn et al. (2007) and Lind et al. (2008). The observed Ca abundance trend is in good agreement with both higher-efficiency models (T6.00 and T6.09), and especially the location of the upturn at is matched. However, Fig. 11 shows clearly that the behaviour of the post-TO Li abundance with effective temperature is not reproduced by any of the models. Especially the location of the maximum occurs at higher effective temperature than the models predict.
The initial Li abundance predicted by the models, accounting for the adjustment we make in the vertical direction, is , which is significantly closer, but still not in full agreement with the latest predictions from BBNS of (Cyburt et al. 2008).
6 Conclusions
We have presented a comprehensive study of the Li content in a large sample, 454 stars in total, of MS, TO, SGB, and RGB stars in the metal-poor globular cluster NGC 6397. The cluster dwarfs and early SGB stars form a thin Li abundance plateau on the same level as the Spite plateau for field stars, whereas more evolved stars have undergone drastic Li depletion due to physical processes connected to low-mass stellar evolution. Two remarkably well-defined locations in absolute visual magnitudes, and respectively, are identified as the locations of corresponding steep Li abundance drops. They correspond respectively to the occurrence of the first dredge-up on the SGB and to the onset of thermohaline mixing at the RGB bump (Charbonnel & Zahn 2007).
Using information of Li and Na abundance for a sub-sample of 100 dwarfs and early SGB, we have for the first time identified a significant anti-correlation between the two elements in this cluster. This is interpreted as the signature of intra-cluster pollution from a previous generation of more massive stars. The spread in Na and Li abundances of stars not having undergone Li depletion due to dredge-up is very large, about one order-of-magnitude. However, only a handful of stars show significantly depleted Li levels and the identified anti-correlation depends critically on these objects. For Na-enhancements up to 0.7 dex, no corresponding Li-deficiency can be detected. The average Li abundance is thus not much affected by internal pollution. These abundance patterns have consequences for for the self-enrichment scenario in NGC 6397, which will be discussed in a forth-coming paper.
By dividing our sub-sample with Na determinations into two groups, consisting of first and second generation of stars according to the degree of pollution, we illustrate how especially trends of Li abundance with effective temperature and evolutionary phase are in reasonable agreement between the two generations. The average Li abundance is typically for stars located below and above the turn-off, and show a slight upturn of dex for stars in the middle of the subgiant branch. This difference agrees with with previous findings for the cluster as well as for field stars. We find no support for a significant Li dispersion among the first generation of stars.
The identification of a minor Li deficiency of the hottest dwarfs in our sample compared to slightly cooler dwarfs leads us to suggest that the very right-hand wing of the Li-dip may be visible also among Pop II stars. However, the signature is erased if a different effective temperature scale is adopted.
The detailed picture we have formed of the Li abundance trends in this globular cluster can be used to constrain the physics involved in depleting Li from the photospheres of low-mass metal-poor stars, as well as the extent of the depletion. To illustrate this, we compare our observational findings to predictions from stellar-structure models including atomic diffusion, with additional turbulence below the convection zone. We find that some turbulence, in a very limited efficiency-range, is indeed required to explain observations. However, these models fail to reproduce the behaviour of Li abundance with effective temperature along the plateau, suggesting that a detailed understanding of the physics responsible for depletion is still lacking.
Acknowledgements
F. G. acknowledges support from the Danish AsteroSeismology Centre, the Carlsberg Foundation, and the Instrument Center for Danish Astronomy (IDA). C. C. and K. L. acknowledge financial support from the Swiss National Science Foundation (FNS) and the french Programme National de Physique Stellaire (PNPS) of CNRS/INSU. This research has made use of the VizieR catalogue access tool, CDS, Strasbourg, France.
Footnotes
... NGC 6397
Based on data collected at European Southern Observatory (ESO), Paranal, Chile, under program IDs 077.A-0018(A) and 281.D-5028(A), as well as data collected with the Danish 1.54 m at European Southern Observatory (ESO), La Silla.
...
Full Tables 2-4 are only available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/503/545
...
.
... temperature
At a given effective temperature, Pop II stars along the plateau have lower masses than Pop I stars on the cool side of the Li dip. Regarding the depth of the convective envelope however, their lower metallicity compensates for the mass effect. See Talon & Charbonnel (2004) for more details.
All Tables
Table 1: FLAMES observations.
Table 2: Photometry and effective temperatures. The full table can be retrieved from CDS/Vizier.
Table 3: Iron lines measured in the UVES spectrum of the RGB star #17691. The full table can be retrieved from CDS/Vizier.
Table 4: Adopted stellar parameters, equivalent widths, and Li, Na, and Ca abundances. The full table can be retrieved from CDS/Vizier.
All Figures
Figure 1: The and relations considered in the analysis. The vertical lines mark the area that is linearly interpolated (see text). The O09 scale for pre-TO stars fall above the one for post-TO stars and vice versa for A9699. Open with DEXTER In the text
Figure 2: Left: the (b-y)0-MV fiducial used to derive stellar parameters. Right: the three (b-y)-based stellar-parameter sets. Open with DEXTER In the text
Figure 3: Left: histogram of the slope of Fe abundance with excitation potential of neutral and singly ionised Fe lines. Only the UVES sample is shown. Solid lines represent the O09 effective temperature scale and dotted lines the A9699 scale. Right: histogram of Fe abundance derived from Fe II lines minus the Fe abundance derived from Fe I lines. Open with DEXTER In the text
Figure 4: Example fits of the Li I 670.7 nm line and the Na I 819.4 nm line. In each panel is indicated the identification number of the star, stellar parameters, and LTE abundance inferred from the given line, according to the syntax , where x is the corresponding element and e(x) the error in abundance. Top: A Li and Na-normal TO star. Second from the top: A Li-poor and Na-rich TO star. Second from the bottom: GIRAFFE giant spectrum. Bottom: UVES spectrum of same giant. Open with DEXTER In the text
Figure 5: Left: the abscissa shows non-LTE Li abundances inferred for our sample (the archival data are not included) and the ordinate absolute visual magnitude ( MV=V-12.57). Arrows mark Li upper limits. Right: the spectroscopic targets marked with black filled circles in the colour-magnitude diagram of NGC 6397. The two horizontal dashed lines mark the locations of rapid decrease in Li abundance caused by stellar evolution (see text). Open with DEXTER In the text
Figure 6: Lithium abundance against visual magnitude. The targets of our observations are plotted with filled circles and the archival data are plotted with open squares. Information about evolutionary status and effective temperature for various MV intervals is also given. Open with DEXTER In the text
Figure 7: Non-LTE Li and Na abundances for the sub-sample of 100 dwarfs and SGB stars with observations in the H805.3A setting and MV>3.3. The arrows mark upper limits to the Na abundance. If the effective temperature of a target has been underestimated by 100 K, the corresponding point in this diagram should be shifted approximately in the direction of the tilted arrow and in its inverse direction for overestimated effective temperatures. Open with DEXTER In the text
Figure 8: Group-averaged Li abundances against visual magnitude. The full line is drawn for the whole sample (including the archival data), i.e., for both first-generation Na-poor stars and second-generation Na-rich stars. Each bin contains 40 stars. The dashed and dotted lines correspond to different sub-sets of our own sample, selected according to their corresponding Na abundances. Each bin contains 13 stars. In the upper left corner mean error bars are shown for each line. Note that the range between -4.8 is not sampled by the data. Open with DEXTER In the text
Figure 9: Group-averaged Li abundances against effective temperature. The lines correspond to the same sub-samples as in Fig. 8. Filled circles represent post-TO stars and open squares pre-TO stars. In the upper left corner mean error bars are shown for each line. Open with DEXTER In the text
Figure 10: Top: comparison between bin-averaged Li abundances (red filled circles connected with solid lines) and the predictions from the stellar-structure models of Richard et al. (2005). T5.80 represents the model with lowest efficiency of turbulent transport, T6.00 intermediate efficiency, and T6.09 highest efficiency. The reference scale is logarithmic luminosities in units of solar luminosities. Bottom: the same plot for Ca abundances. A colour version of this figure is available in the online edition of the journal. Open with DEXTER In the text
Figure 11: Top: same as in in Fig. 10 but with effective temperature as the reference scale. Here, only post-TO stars are included. Bottom: the same plot for Ca abundances. A colour version of this figure is available in the online edition of the journal. Open with DEXTER In the text | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8869756460189819, "perplexity": 2521.4668815818764}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323588216.48/warc/CC-MAIN-20211027150823-20211027180823-00600.warc.gz"} |
http://physics.stackexchange.com/questions/54335/why-isnt-average-speed-equal-to-displacement-over-time/54340 | # Why isn't average speed equal to displacement over time?
I'm in an introductory Physics course and I need help!
During a one-hour trip, a small boat travels 80.0km north and then travels 60.0km east. What is the boat's average speed during the one-hour trip?
I found that the displacement is 100km. But I did not arrive at the right answer by dividing 100km/1hour. Why doesn't this give me the average speed?
-
Hi Brijette, and welcome to Physics Stack Exchange! I edited your question a bit to make it clear that you're asking about a concept, not looking for an answer (since the latter is not allowed here). – David Z Feb 18 '13 at 20:46
What if it had come back to the start point? What would be the average speed by your definition? – Martin Beckett Feb 18 '13 at 20:51
I know that the answer is 140 km/h. I just don't see how? – graybrij Feb 18 '13 at 21:06
I see what you are saying Martin...I can't use displacement in determining speed. But then what do I use? – graybrij Feb 18 '13 at 21:08
Damn, that's a very very fast boat! – Plouf Feb 19 '13 at 9:57
You use the total amount of movement over time. So here that is|:
80km plus 60km equals 140km
Which gives you the correct answer.
Displacement, using Pythagoras, would be 100km, but you travelled 140km in that hour! You didn't travel along that hypoteneuse, so it is irrelevant here.
-
+1 for answering the question in a way that's relatively easy to understand, considering that the OP is an introductory student. – Kevin Feb 19 '13 at 7:31
The formula you wanted to use gives you the magnitude of the average velocity, not the average speed.
To get the magnitude of the average velocity, you take the total displacement (which is a vector!), divide by the total time, and find the magnitude of that vector. What you get is:
$$\text{Magnitude of Average Velocity}= \biggl| \frac{\sum_i \vec{d}_i} {\Delta t_{\text{total}}} \bigg|=\bigg| \frac{\sum_i \vec{v}_i \Delta t_i} {\Delta t_{\text{total}}} \bigg|$$ Where the $\vec{v}_i$ are the different velocities, $\Delta t_i$ are the amounts of time spent at each velocity, $\vec{d}_i$ are the individual displacements ,and $\Delta t_{\text{total}}$ is the total amount of time.
To get the average speed, you take the magnitude of the individual displacement vectors, then sum and average them, giving this formula: $$\text{Average Speed=} \frac{\sum_i |\vec{d}_i|} {\Delta t_{\text{total}}} = \frac{\sum_i |\vec{v}_i| \Delta t_i} {\Delta t_{\text{total}}}$$
The difference is the order in which you take the absolute value and do the sum. This makes a big difference in some cases: the average velocity of helium atoms in a stationary balloon is $0$, but the average speed may be hundreds of $m/s$ (depending on the temperature).
-
So you are looking for the average speed of a board during a one hour period. The total distance traveled by the boat is 140 km (80 + 60) in the course of an hour, giving the boat an average speed of 140 km/hr.
The confusion I feel in these types of problems, is from the 1 hour. When you get the answer merely by summing it dosent impart an understanding. The same problem with 2 hours instead of 1 may be clearer. In this case you have to sum the distances and divide by the time (which was done in the last problem but because you divided by 1 it had no effect.) So the answer in this case would be (80km + 60km)/2hr = 70 km/hr
- | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8651772737503052, "perplexity": 323.4192388110454}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1408500826343.66/warc/CC-MAIN-20140820021346-00314-ip-10-180-136-8.ec2.internal.warc.gz"} |
https://socratic.org/questions/how-do-you-solve-the-system-of-equations-x-y-14-and-x-y-16 | Algebra
Topics
# How do you solve the system of equations x+y=14 and -x+y=16?
Apr 4, 2018
$\left(x , y\right) = \left(- 1 , 15\right)$
#### Explanation:
The end goal is to get an equation in terms of one variable, which can be solved.
We can solve using substitution. We'll need to pick one of our equations and solve for $x$ or $y .$ It does not matter which equation is used for this purpose, nor does it matter which variable is solved for.
This is a nice system, solving for either variable in either equation will be equally easy. Let's take $x + y = 14$:
$y = 14 - x$
Now, we can replace all $y$ in the other equation, $- x + y = 16$, with $14 - x :$
$- x + y = 16$
$- x + 14 - x = 16$
$- 2 x = 16 - 14$
$- 2 x = 2$
$x = - 1$
To solve for $y ,$ we can just back-substitute the above result into $y = 14 - x :$
$y = 14 - \left(- 1\right)$
$y = 15$
Thus,
$\left(x , y\right) = \left(- 1 , 15\right)$
##### Impact of this question
250 views around the world | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 18, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.937725305557251, "perplexity": 329.2840448908039}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570987835748.66/warc/CC-MAIN-20191023173708-20191023201208-00109.warc.gz"} |
http://mathhelpforum.com/advanced-algebra/280850-number-sylow-subgroups.html | ## Number of Sylow subgroups
Okay, I'm trying to list all the Sylow 2-subgroups of $\displaystyle D_{12}$. I have that n2 = 1 or n2 = 3 and I know that $\displaystyle | Syl_2 ( D_{12} ) | = 4$. It happens that there is only one subgroup of $\displaystyle D_{12}$ of order 4 so n2 = 1 is correct. However I'd like a more general argument to rule out n2 = 3 if possible.
If n2 = 3 then we must have 3 subgroups of order 4 that are all conjugate to each other and I'm trying to use this and the class equation to rule it out but I'm missing something simple. The conjugacy classes of $\displaystyle D_{12}$ are $\displaystyle \{ 1 \}, ~ \{ r^3 \}, \{ r, ~ r^5 \}, ~ \{ r^2 , ~ r^4 \}, ~ \{ s, ~sr^3 \}, ~ \{sr^2 , ~ sr^4 \}$. I'm thinking that this can't be if we have 3 conjugate order 4 subgroups... I think there is a factor of 1/3 somewhere here but I can't figure out how to finish the argument.
Any thoughts?
Thanks!
-Dan | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8285312652587891, "perplexity": 99.50568069542203}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676593223.90/warc/CC-MAIN-20180722120017-20180722140017-00609.warc.gz"} |
http://clay6.com/qa/11005/a-particle-moves-in-xy-plane-the-position-vector-of-particle-at-any-time-t- | # A particle moves in xy plane. The position vector of particle at any time t is $\overrightarrow r=\{(2t) i+(2t^2)j\}\; m$. The rate of change of $\theta$ at time $t= 2\;$ seconds is ( $\theta$ is the angle which its velocity vector makes with positive x-axis )
$(a)\;\frac{2}{17}rad/s\quad (b)\;\frac{1}{14}rad/s \quad (c)\;\frac{4}{7}rad/s \quad (d)\;\frac{6}{5}rad/s$
$x=2t$
=>$v_x=\large\frac{dx}{dt}$$=2 y=2t^2 =>v_y=\large\frac{dy}{dt}$$=4t$
$\tan \theta=\large\frac{v_y}{v_x}$
$\qquad=\large\frac{4t}{2}$$=2t Differentiating with respect to t (\sec^2 \theta) \large\frac{d\theta}{dt}$$=2$
$(1+\tan ^2 \theta) \large\frac{d\theta}{dt}$$=2 (1+4t^2) \large\frac{d\theta}{dt}$$=2$
$\large\frac{d\theta}{dt}=\large\frac{2}{1+4t^2}$
at $t=2$
$\large\frac{d\theta}{dt}=\frac{2}{1+4(2)^2}=\frac{2}{17}$$rad/s$
Hence a is the correct answer.
edited Jan 25, 2014 by meena.p | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8547975420951843, "perplexity": 956.3216732369976}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818693866.86/warc/CC-MAIN-20170925235144-20170926015144-00587.warc.gz"} |
http://physics.stackexchange.com/tags/geophysics/new | Tag Info
1
Strictly speaking, $g$ (even more strictly speaking, $g_0$ or $g_n$) is a constant. It is exactly 9.80665 m/s$^2$, by definition. There are many places in science and engineering where it is very useful to have an exact (albeit arbitrary) defined constant for gravitation on the surface of the Earth. That said, gravitation on the surface of the Earth does ...
4
To two significant figures, the acceleration due to gravity is $g=9.8\:\mathrm{m/s^2}$ everywhere on Earth (at sea level). That is to say, if you use e.g. a pendulum to measure $g$ to two sig figs, you will get this value no matter where you are. In a sense, this is the precision to which the Earth is well-approximated by a uniform sphere of matter. The ...
1
You ask a lot of questions, I'll try to answer just one. As the Earth's magnetic field protects us from ionizing radiation, theoretically, there can be negative correlation between the Earth's magnetic field and cancer incidence. EDIT: see, e.g., Health Physics; v. 34(3) p. 237-247; ISSN 0017-9078; 1978 ...
0
Yes, the strength of the Earth's gravity changes (slightly) from place to place. However, you can't just treat Earth's gravity as a varying number; it is a vector. It mostly points down towards the center of the Earth, but those same gravitational anomalies which cause the general strength of gravity to vary also cause it to deviate from what you'd generally ...
0
I think that the problem here is that $U = mgh$ is an approximation that we can make by assuming a constant value of $g$. Even taking it a bit further and using $$U = -G\frac{m_1m_2}{r}$$ will make some assumptions about the density and shape of earth if you consider one of the masses to be that of the whole earth. In reality, you'd need to sum the ...
0
Let's do some rough estimation. Considering: CuriousOne's very good and fitting comment with regards to geometrical estimation given in question and the last but not least the human physiology ($10^{-12} \ W$ can hear only a full healthy man, with perception much less than a whisper and only on a frequency range improbable due to the diffraction) we ...
Top 50 recent answers are included | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9243828058242798, "perplexity": 257.19549604859424}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398445679.7/warc/CC-MAIN-20151124205405-00003-ip-10-71-132-137.ec2.internal.warc.gz"} |
https://math.libretexts.org/TextMaps/Precalculus_Textmaps/Map%3A_Elementary_Trigonometry_(Corral)/3%3A_Identities | $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$
Thumbnail: If $$\theta$$ is in QIII, then the legs of the right triangle formed by the reference angle have lengths $$|x|$$ and $$|y|$$ (we use absolute values because $$x$$ and $$y$$ are negative in QIII). | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8723699450492859, "perplexity": 98.10247672934877}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218189242.54/warc/CC-MAIN-20170322212949-00085-ip-10-233-31-227.ec2.internal.warc.gz"} |
https://jp.mathworks.com/help/physmod/simscape/ref/pipema.html | # Pipe (MA)
Rigid conduit for moist air flow
• Library:
• Simscape / Foundation Library / Moist Air / Elements
## Description
The Pipe (MA) block models pipe flow dynamics in a moist air network due to viscous friction losses and convective heat transfer with the pipe wall. The pipe contains a constant volume of moist air. The pressure and temperature evolve based on the compressibility and thermal capacity of this moist air volume. Liquid water condenses out of the moist air volume when it reaches saturation. Choked flow occurs when the outlet reaches sonic condition.
Caution
Air flow through this block can choke. If a Mass Flow Rate Source (MA) block or a Controlled Mass Flow Rate Source (MA) block connected to the Pipe (MA) block specifies a greater mass flow rate than the possible choked mass flow rate, the simulation generates an error. For more information, see Choked Flow.
The block equations use these symbols. Subscripts `a`, `w`, and `g` indicate the properties of dry air, water vapor, and trace gas, respectively. Subscript `ws` indicates water vapor at saturation. Subscripts `A`, `B`, `H`, and `S` indicate the appropriate port. Subscript `I` indicates the properties of the internal moist air volume.
$\stackrel{˙}{m}$ Mass flow rate Φ Energy flow rate Q Heat flow rate p Pressure ρ Density R Specific gas constant V Volume of moist air inside the pipe cv Specific heat at constant volume cp Specific heat at constant pressure h Specific enthalpy u Specific internal energy x Mass fraction (xw is specific humidity, which is another term for water vapor mass fraction) y Mole fraction φ Relative humidity r Humidity ratio T Temperature t Time
### Mass and Energy Balance
The net flow rates into the moist air volume inside the pipe are
`$\begin{array}{l}{\stackrel{˙}{m}}_{net}={\stackrel{˙}{m}}_{A}+{\stackrel{˙}{m}}_{B}-{\stackrel{˙}{m}}_{condense}+{\stackrel{˙}{m}}_{wS}+{\stackrel{˙}{m}}_{gS}\\ {\Phi }_{net}={\Phi }_{A}+{\Phi }_{B}+{Q}_{H}-{\Phi }_{condense}+{\Phi }_{S}\\ {\stackrel{˙}{m}}_{w,net}={\stackrel{˙}{m}}_{wA}+{\stackrel{˙}{m}}_{wB}-{\stackrel{˙}{m}}_{condense}+{\stackrel{˙}{m}}_{wS}\\ {\stackrel{˙}{m}}_{g,net}={\stackrel{˙}{m}}_{gA}+{\stackrel{˙}{m}}_{gB}+{\stackrel{˙}{m}}_{gS}\end{array}$`
where:
• $\stackrel{˙}{m}$condense is the rate of condensation.
• Φcondense is the rate of energy loss from the condensed water.
• ΦS is the rate of energy added by the sources of moisture and trace gas. ${\stackrel{˙}{m}}_{wS}$ and ${\stackrel{˙}{m}}_{gS}$ are the mass flow rates of water and gas, respectively, through port S. The values of ${\stackrel{˙}{m}}_{wS}$, ${\stackrel{˙}{m}}_{gS}$, and ΦS are determined by the moisture and trace gas sources connected to port S of the pipe.
Water vapor mass conservation relates the water vapor mass flow rate to the dynamics of the moisture level in the internal moist air volume:
`$\frac{d{x}_{wI}}{dt}{\rho }_{I}V+{x}_{wI}{\stackrel{˙}{m}}_{net}={\stackrel{˙}{m}}_{w,net}$`
Similarly, trace gas mass conservation relates the trace gas mass flow rate to the dynamics of the trace gas level in the internal moist air volume:
`$\frac{d{x}_{gI}}{dt}{\rho }_{I}V+{x}_{gI}{\stackrel{˙}{m}}_{net}={\stackrel{˙}{m}}_{g,net}$`
Mixture mass conservation relates the mixture mass flow rate to the dynamics of the pressure, temperature, and mass fractions of the internal moist air volume:
`$\left(\frac{1}{{p}_{I}}\frac{d{p}_{I}}{dt}-\frac{1}{{T}_{I}}\frac{d{T}_{I}}{dt}\right){\rho }_{I}V+\frac{{R}_{a}-{R}_{w}}{{R}_{I}}\left({\stackrel{˙}{m}}_{w,net}-{x}_{w}{\stackrel{˙}{m}}_{net}\right)+\frac{{R}_{a}-{R}_{g}}{{R}_{I}}\left({\stackrel{˙}{m}}_{g,net}-{x}_{g}{\stackrel{˙}{m}}_{net}\right)={\stackrel{˙}{m}}_{net}$`
Finally, energy conservation relates the energy flow rate to the dynamics of the pressure, temperature, and mass fractions of the internal moist air volume:
`${\rho }_{I}{c}_{vI}V\frac{d{T}_{I}}{dt}+\left({u}_{wI}-{u}_{aI}\right)\left({\stackrel{˙}{m}}_{w,net}-{x}_{w}{\stackrel{˙}{m}}_{net}\right)+\left({u}_{gI}-{u}_{aI}\right)\left({\stackrel{˙}{m}}_{g,net}-{x}_{g}{\stackrel{˙}{m}}_{net}\right)+{u}_{I}{\stackrel{˙}{m}}_{net}={\Phi }_{net}$`
The equation of state relates the mixture density to the pressure and temperature:
`${p}_{I}={\rho }_{I}{R}_{I}{T}_{I}$`
The mixture specific gas constant is
`${R}_{I}={x}_{aI}{R}_{a}+{x}_{wI}{R}_{w}+{x}_{gI}{R}_{g}$`
### Momentum Balance
The momentum balance for each half of the pipe models the pressure drop due to momentum flux and viscous friction:
`$\begin{array}{l}{p}_{A}-{p}_{I}={\left(\frac{{\stackrel{˙}{m}}_{A}}{S}\right)}^{2}\cdot \left(\frac{{T}_{I}}{{p}_{I}}-\frac{{T}_{A}}{{p}_{A}}\right){R}_{I}+\Delta {p}_{AI}\\ {p}_{B}-{p}_{I}={\left(\frac{{\stackrel{˙}{m}}_{B}}{S}\right)}^{2}\cdot \left(\frac{{T}_{I}}{{p}_{I}}-\frac{{T}_{B}}{{p}_{B}}\right){R}_{I}+\Delta {p}_{BI}\end{array}$`
where:
• p is the pressure at port A, port B, or internal node I, as indicated by the subscript.
• ρ is the density at port A, port B, or internal node I, as indicated by the subscript.
• S is the cross-sectional area of the pipe.
• ΔpAI and ΔpBI are pressure losses due to viscous friction.
The pressure losses due to viscous friction, ΔpAI and ΔpBI, depend on the flow regime. The Reynolds numbers for each half of the pipe are defined as:
`$\begin{array}{l}{\mathrm{Re}}_{A}=\frac{|{\stackrel{˙}{m}}_{A}|\cdot {D}_{h}}{S\cdot {\mu }_{I}}\\ {\mathrm{Re}}_{B}=\frac{|{\stackrel{˙}{m}}_{B}|\cdot {D}_{h}}{S\cdot {\mu }_{I}}\end{array}$`
where:
• Dh is the hydraulic diameter of the pipe.
• μI is the dynamic viscosity at the internal node.
If the Reynolds number is less than the value of the Laminar flow upper Reynolds number limit parameter, then the flow is in the laminar flow regime. If the Reynolds number is greater than the value of the Turbulent flow lower Reynolds number limit parameter, then the flow is in the turbulent flow regime.
In the laminar flow regime, the pressure losses due to viscous friction are:
`$\begin{array}{l}\Delta {p}_{A{I}_{lam}}={f}_{shape}\frac{{\stackrel{˙}{m}}_{A}\cdot {\mu }_{I}}{2{\rho }_{I}\cdot {D}_{h}^{2}\cdot S}\cdot \frac{L+{L}_{eqv}}{2}\\ \Delta {p}_{B{I}_{lam}}={f}_{shape}\frac{{\stackrel{˙}{m}}_{B}\cdot {\mu }_{I}}{2{\rho }_{I}\cdot {D}_{h}^{2}\cdot S}\cdot \frac{L+{L}_{eqv}}{2}\end{array}$`
where:
• fshape is the value of the Shape factor for laminar flow viscous friction parameter.
• Leqv is the value of the Aggregate equivalent length of local resistances parameter.
In the turbulent flow regime, the pressure losses due to viscous friction are:
`$\begin{array}{l}\Delta {p}_{A{I}_{tur}}={f}_{Darc{y}_{A}}\frac{{\stackrel{˙}{m}}_{A}\cdot |{\stackrel{˙}{m}}_{A}|}{2{\rho }_{I}\cdot {D}_{h}\cdot {S}^{2}}\cdot \frac{L+{L}_{eqv}}{2}\\ \Delta {p}_{B{I}_{tur}}={f}_{Darc{y}_{B}}\frac{{\stackrel{˙}{m}}_{B}\cdot |{\stackrel{˙}{m}}_{B}|}{2{\rho }_{I}\cdot {D}_{h}\cdot {S}^{2}}\cdot \frac{L+{L}_{eqv}}{2}\end{array}$`
where fDarcy is the Darcy friction factor at port A or B, as indicated by the subscript.
The Darcy friction factors are computed from the Haaland correlation:
`$\begin{array}{l}{f}_{Darc{y}_{A}}={\left[-1.8\mathrm{log}\left(\frac{6.9}{{\mathrm{Re}}_{A}}+{\left(\frac{{\epsilon }_{rough}}{3.7{D}_{h}}\right)}^{1.11}\right)\right]}^{-2}\\ {f}_{Darc{y}_{B}}={\left[-1.8\mathrm{log}\left(\frac{6.9}{{\mathrm{Re}}_{B}}+{\left(\frac{{\epsilon }_{rough}}{3.7{D}_{h}}\right)}^{1.11}\right)\right]}^{-2}\end{array}$`
where εrough is the value of the Internal surface absolute roughness parameter.
When the Reynolds number is between the Laminar flow upper Reynolds number limit and the Turbulent flow lower Reynolds number limit parameter values, the flow is in transition between laminar flow and turbulent flow. The pressure losses due to viscous friction during the transition region follow a smooth connection between those in the laminar flow regime and those in the turbulent flow regime.
The heat exchanged with the pipe wall through port H is added to the energy of the moist air volume represented by the internal node via the energy conservation equation (see Mass and Energy Balance). Therefore, the momentum balances for each half of the pipe, between port A and the internal node and between port B and the internal node, are assumed to be adiabatic processes. The adiabatic relations are:
`$\begin{array}{l}{h}_{A}-{h}_{I}={\left(\frac{{R}_{I}{\stackrel{˙}{m}}_{A}}{S}\right)}^{2}\left[{\left(\frac{{T}_{I}}{{p}_{I}}\right)}^{2}-{\left(\frac{{T}_{A}}{{p}_{A}}\right)}^{2}\right]\\ {h}_{B}-{h}_{I}={\left(\frac{{R}_{I}{\stackrel{˙}{m}}_{B}}{S}\right)}^{2}\left[{\left(\frac{{T}_{I}}{{p}_{I}}\right)}^{2}-{\left(\frac{{T}_{B}}{{p}_{B}}\right)}^{2}\right]\end{array}$`
where h is the specific enthalpy at port A, port B, or internal node I, as indicated by the subscript.
### Convective Heat Transfer
The convective heat transfer equation between the pipe wall and the internal moist air volume is:
`${Q}_{H}={Q}_{conv}+\frac{{k}_{I}{S}_{surf}}{{D}_{h}}\left({T}_{H}-{T}_{I}\right)$`
Ssurf is the pipe surface area, Ssurf = 4SL/Dh. Assuming an exponential temperature distribution along the pipe, the convective heat transfer is
`${Q}_{conv}=|{\stackrel{˙}{m}}_{avg}|{c}_{{p}_{avg}}\left({T}_{H}-{T}_{in}\right)\left(1-\mathrm{exp}\left(-\frac{{h}_{coeff}{S}_{surf}}{|{\stackrel{˙}{m}}_{avg}|{c}_{{p}_{avg}}}\right)\right)$`
where:
• Tin is the inlet temperature depending on flow direction.
• ${\stackrel{˙}{m}}_{avg}=\left({\stackrel{˙}{m}}_{A}-{\stackrel{˙}{m}}_{B}\right)/2$ is the average mass flow rate from port A to port B.
• ${c}_{{p}_{avg}}$ is the specific heat evaluated at the average temperature.
The heat transfer coefficient, hcoeff, depends on the Nusselt number:
`${h}_{coeff}=Nu\frac{{k}_{avg}}{{D}_{h}}$`
where kavg is the thermal conductivity evaluated at the average temperature. The Nusselt number depends on the flow regime. The Nusselt number in the laminar flow regime is constant and equal to the value of the Nusselt number for laminar flow heat transfer parameter. The Nusselt number in the turbulent flow regime is computed from the Gnielinski correlation:
`$N{u}_{tur}=\frac{\frac{{f}_{Darcy}}{8}\left({\mathrm{Re}}_{avg}-1000\right){\mathrm{Pr}}_{avg}}{1+12.7\sqrt{\frac{{f}_{Darcy}}{8}}\left({\mathrm{Pr}}_{avg}^{2/3}-1\right)}$`
where Pravg is the Prandtl number evaluated at the average temperature. The average Reynolds number is
`${\mathrm{Re}}_{avg}=\frac{|{\stackrel{˙}{m}}_{avg}|{D}_{h}}{S{\mu }_{avg}}$`
where μavg is the dynamic viscosity evaluated at the average temperature. When the average Reynolds number is between the Laminar flow upper Reynolds number limit and the Turbulent flow lower Reynolds number limit parameter values, the Nusselt number follows a smooth transition between the laminar and turbulent Nusselt number values.
### Saturation and Condensation
When the moist air volume reaches saturation, condensation may occur. The specific humidity at saturation is
`${x}_{wsI}={\phi }_{ws}\frac{{R}_{I}}{{R}_{w}}\frac{{p}_{wsI}}{{p}_{I}}$`
where:
• φws is the relative humidity at saturation (typically 1).
• pwsI is the water vapor saturation pressure evaluated at TI.
The rate of condensation is
where τcondense is the value of the Condensation time constant parameter.
The condensed water is subtracted from the moist air volume, as shown in the conservation equations. The energy associated with the condensed water is
`${\Phi }_{condense}={\stackrel{˙}{m}}_{condense}\left({h}_{wI}-\Delta {h}_{vapI}\right)$`
where ΔhvapI is the specific enthalpy of vaporization evaluated at TI.
Other moisture and trace gas quantities are related to each other as follows:
`$\begin{array}{l}{\phi }_{wI}=\frac{{y}_{wI}{p}_{I}}{{p}_{wsI}}\\ {y}_{wI}=\frac{{x}_{wI}{R}_{w}}{{R}_{I}}\\ {r}_{wI}=\frac{{x}_{wI}}{1-{x}_{wI}}\\ {y}_{gI}=\frac{{x}_{gI}{R}_{g}}{{R}_{I}}\\ {x}_{aI}+{x}_{wI}+{x}_{gI}=1\end{array}$`
### Choked Flow
The unchoked pressure at port A or B is the value of the corresponding Across variable at that port:
`$\begin{array}{l}{p}_{{A}_{unchoked}}=\text{A}\text{.p}\\ {p}_{{B}_{unchoked}}=\text{B}\text{.p}\end{array}$`
However, the port pressure variables used in the momentum balance equations, pA and pB, do not necessarily coincide with the pressure across variables `A.p` and `B.p` because the pipe outlet may choke. Choked flow occurs when the downstream pressure is sufficiently low. At that point, the flow depends only on the conditions at the inlet. Therefore, when choked, the outlet pressure (pA or pB, whichever is the outlet) cannot decrease further even if the pressure downstream, represented by `A.p` or `B.p`, continues to decrease.
Choking can occur at the pipe outlet, but not at the pipe inlet. Therefore, if port A is the inlet, then pA = `A.p`. If port A is the outlet, then
Similarly, if port B is the inlet, then pB = `B.p`. If port B is the outlet, then
The choked pressures at ports A and B are derived from the momentum balance by assuming the outlet velocity is equal to the speed of sound:
`$\begin{array}{l}{p}_{{A}_{choked}}-{p}_{I}={p}_{{A}_{choked}}\left(\frac{{p}_{{A}_{choked}}{T}_{I}}{{p}_{I}{T}_{A}}-1\right)\frac{{c}_{{p}_{A}}}{{c}_{{v}_{I}}}+\Delta {p}_{AI}\\ {p}_{{B}_{choked}}-{p}_{I}={p}_{{B}_{choked}}\left(\frac{{p}_{{B}_{choked}}{T}_{I}}{{p}_{I}{T}_{B}}-1\right)\frac{{c}_{{p}_{B}}}{{c}_{{v}_{I}}}+\Delta {p}_{BI}\end{array}$`
### Variables
To set the priority and initial target values for the block variables prior to simulation, use the Variables tab in the block dialog box (or the Variables section in the block Property Inspector). For more information, see Set Priority and Initial Target for Block Variables and Initial Conditions for Blocks with Finite Moist Air Volume.
### Assumptions and Limitations
• The pipe wall is perfectly rigid.
• The flow is fully developed. Friction losses and heat transfer do not include entrance effects.
• The effect of gravity is negligible.
• Fluid inertia is negligible.
• This block does not model supersonic flow.
## Ports
### Output
expand all
Physical signal output port that measures the rate of condensation in the pipe.
Physical signal output port that outputs a vector signal. The vector contains the pressure (in Pa), temperature (in K), moisture level, and trace gas level measurements inside the component. Use the Measurement Selector (MA) block to unpack this vector signal.
### Conserving
expand all
Moist air conserving port associated with the inlet or outlet of the pipe. This block has no intrinsic directionality.
Moist air conserving port associated with the inlet or outlet of the pipe. This block has no intrinsic directionality.
Thermal conserving port associated with the temperature of the pipe wall. The block includes the convective heat transfer between the moist air mixture inside the pipe and the pipe wall.
Connect this port to port S of a block from the Moisture & Trace Gas Sources library to add or remove moisture and trace gas. For more information, see Using Moisture and Trace Gas Sources.
#### Dependencies
This port is visible only if you set the Moisture and trace gas source parameter to `Controlled`.
## Parameters
expand all
### Main
Length of the pipe along the direction of flow.
Internal area of the pipe normal to the direction of the flow.
Diameter of an equivalent cylindrical pipe with the same cross-sectional area.
### Friction and Heat Transfer
Combined length of all local resistances present in the pipe. Local resistances include bends, fittings, armatures, and pipe inlets and outlets. The effect of the local resistances is to increase the effective length of the pipe segment. This length is added to the geometrical pipe length only for friction calculations. The moist air volume depends only on the pipe geometrical length, defined by the Pipe length parameter.
Average depth of all surface defects on the internal surface of the pipe, which affects the pressure loss in the turbulent flow regime.
Reynolds number above which flow begins to transition from laminar to turbulent. This number equals the maximum Reynolds number corresponding to the fully developed laminar flow.
Reynolds number below which flow begins to transition from turbulent to laminar. This number equals to the minimum Reynolds number corresponding to the fully developed turbulent flow.
Dimensionless factor that encodes the effect of pipe cross-sectional geometry on the viscous friction losses in the laminar flow regime. Typical values are 64 for a circular cross section, 57 for a square cross section, 62 for a rectangular cross section with an aspect ratio of 2, and 96 for a thin annular cross section [1].
Ratio of convective to conductive heat transfer in the laminar flow regime. Its value depends on the pipe cross-sectional geometry and pipe wall thermal boundary conditions, such as constant temperature or constant heat flux. Typical value is 3.66, for a circular cross section with constant wall temperature [2].
### Moisture and Trace Gas
Relative humidity above which condensation occurs.
Characteristic time scale at which an oversaturated moist air volume returns to saturation by condensing out excess moisture.
This parameter controls visibility of port S and provides these options for modeling moisture and trace gas levels inside the component:
• `None` — No moisture or trace gas is injected into or extracted from the block. Port S is hidden. This is the default.
• `Constant` — Moisture and trace gas are injected into or extracted from the block at a constant rate. The same parameters as in the Moisture Source (MA) and Trace Gas Source (MA) blocks become available in the Moisture and Trace Gas section of the block interface. Port S is hidden.
• `Controlled` — Moisture and trace gas are injected into or extracted from the block at a time-varying rate. Port S is exposed. Connect the Controlled Moisture Source (MA) and Controlled Trace Gas Source (MA) blocks to this port.
Select whether the block adds or removes moisture as water vapor or liquid water:
• `Vapor` — The enthalpy of the added or removed moisture corresponds to the enthalpy of water vapor, which is greater than that of liquid water.
• `Liquid` — The enthalpy of the added or removed moisture corresponds to the enthalpy of liquid water, which is less than that of water vapor.
#### Dependencies
Enabled when the Moisture and trace gas source parameter is set to `Constant`.
Water vapor mass flow rate through the block. A positive value adds moisture to the connected moist air volume. A negative value extracts moisture from that volume.
#### Dependencies
Enabled when the Moisture and trace gas source parameter is set to `Constant`.
Select a specification method for the moisture temperature:
• `Atmospheric temperature` — Use the atmospheric temperature, specified by the Moist Air Properties (MA) block connected to the circuit.
• `Specified temperature` — Specify a value by using the Temperature of added moisture parameter.
#### Dependencies
Enabled when the Moisture and trace gas source parameter is set to `Constant`.
Enter the desired temperature of added moisture. This temperature remains constant during simulation. The block uses this value to evaluate the specific enthalpy of the added moisture only. The specific enthalpy of removed moisture is based on the temperature of the connected moist air volume.
#### Dependencies
Enabled when the Added moisture temperature specification parameter is set to `Specified temperature`.
Trace gas mass flow rate through the block. A positive value adds trace gas to the connected moist air volume. A negative value extracts trace gas from that volume.
#### Dependencies
Enabled when the Moisture and trace gas source parameter is set to `Constant`.
Select a specification method for the trace gas temperature:
• `Atmospheric temperature` — Use the atmospheric temperature, specified by the Moist Air Properties (MA) block connected to the circuit.
• `Specified temperature` — Specify a value by using the Temperature of added trace gas parameter.
#### Dependencies
Enabled when the Moisture and trace gas source parameter is set to `Constant`.
Enter the desired temperature of added trace gas. This temperature remains constant during simulation. The block uses this value to evaluate the specific enthalpy of the added trace gas only. The specific enthalpy of removed trace gas is based on the temperature of the connected moist air volume.
#### Dependencies
Enabled when the Added trace gas temperature specification parameter is set to `Specified temperature`.
## References
[1] White, F. M., Fluid Mechanics. 7th Ed, Section 6.8. McGraw-Hill, 2011.
[2] Cengel, Y. A., Heat and Mass Transfer – A Practical Approach. 3rd Ed, Section 8.5. McGraw-Hill, 2007. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 31, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.869038999080658, "perplexity": 1327.7114848627323}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964362219.5/warc/CC-MAIN-20211202114856-20211202144856-00590.warc.gz"} |
http://export.arxiv.org/abs/1603.00426 | math.QA
(what is this?)
# Title: Finite noncommutative geometries related to $F_p[x]$
Abstract: It is known that irreducible noncommutative differential structures over $\Bbb F_p[x]$ are classified by irreducible monics $m$. We show that the cohomology $H_{\rm dR}^0(\Bbb F_p[x]; m)=\Bbb F_p[g_d]$ if and only if ${\rm Tr}(m)\ne 0$, where $g_d=x^{p^d}-x$ and $d$ is the degree of $m$. This implies that there are ${p-1\over pd}\sum_{k|d, p\nmid k}\mu_M(k)p^{d\over k}$ such noncommutative differential structures ($\mu_M$ the M\"obius function). Motivated by killing this zero'th cohomology, we consider the directed system of finite-dimensional Hopf algebras $A_d=\Bbb F_p[x]/(g_d)$ as well as their inherited bicovariant differential calculi $\Omega(A_d;m)$. We show that $A_d=C_d\otimes_\chi A_1$ a cocycle extension where $C_d=A_d^\psi$ is the subalgebra of elements fixed under $\psi(x)=x+1$. We also have a Frobenius-fixed subalgebra $B_d$ of dimension $\frac{1}{d} \sum_{k | d} \phi(k) p^\frac{d}{k}$ ($\phi$ the Euler totient function), generalising Boolean algebras when $p=2$. As special cases, $A_1\cong \Bbb F_p(\Bbb Z/p\Bbb Z)$, the algebra of functions on the finite group $\Bbb Z/p\Bbb Z$, and we show dually that $\Bbb F_p\Bbb Z/p\Bbb Z\cong\Bbb F_p[L]/(L^p)$ for a `Lie algebra' generator $L$ with $e^L$ group-like, using a truncated exponential. By contrast, $A_2$ over $\Bbb F_2$ is a cocycle modification of $\Bbb F_2((\Bbb Z/2\Bbb Z)^2)$ and is a 1-dimensional extension of the Boolean algebra on 3 elements. In both cases we compute the Fourier theory, the invariant metrics and the Levi-Civita connections within bimodule noncommutative geometry.
Comments: 25 pages ams latex no figures Subjects: Quantum Algebra (math.QA); Algebraic Geometry (math.AG) Cite as: arXiv:1603.00426 [math.QA] (or arXiv:1603.00426v2 [math.QA] for this version)
## Submission history
From: Shahn Majid [view email]
[v1] Tue, 1 Mar 2016 19:51:15 GMT (23kb)
[v2] Fri, 1 Feb 2019 20:12:00 GMT (28kb)
Link back to: arXiv, form interface, contact. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8693309426307678, "perplexity": 916.670036222072}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107874026.22/warc/CC-MAIN-20201020162922-20201020192922-00509.warc.gz"} |
http://math.stackexchange.com/users/16192/gt6989b?tab=activity&sort=all&page=5 | gt6989b
Reputation
15,311
Top tag
Next privilege 20,000 Rep.
Access 'trusted user' tools
Apr 14 comment You can always delete a vertex from a tree $G$ such that the remaining connected components have size at most $|V(G)|/2$. one hunch is try to a proof by induction, possibly deleting a leaf, but not sure if it will help. Apr 14 answered Biased coin toss problem - Understanding a problem correctly Apr 14 answered Eliminating Sine Apr 14 comment What is the domain and range of the sum of two random variables? @Did perhaps i see the difference -- you want to define $S(\omega) = X_1(\omega) + X_2(\omega)$ where all variables map from $\Omega^n$, but each $X_i$ only uses the $i$th coordinate. Why would this make impossible every simple operation? We just defined addition, you can define integration and scalar multiple and everything else analogously -- why does this cause a problem? Please understand, I am not arguing, just asking for clarification for my own education. Thanks Apr 14 comment What is the domain and range of the sum of two random variables? @Did i honestly don't understand how you can do it any other way -- the construction in the accepted answer uses product spaces as well, which is what i did here, and so do Artem Mavrin's remarks, and your comment seems to indicate that you agree. I fail to see the difference between that construction and the one in my answer. Apr 13 comment How to evaluate the integral $\int_0^5 3x^2 dx$? @AkivaWeinberger i did fix it, but in this setting, everyone will end up with the same one so the was not exactly misplaced Apr 13 revised How to evaluate the integral $\int_0^5 3x^2 dx$? deleted 1 character in body Apr 13 answered How to evaluate the integral $\int_0^5 3x^2 dx$? Apr 13 revised How to evaluate the integral $\int_0^5 3x^2 dx$? added 6 characters in body Apr 13 comment solve $\sin 2x + \sin x = 0$ using addition formula @N.F.Taussig fixed Apr 13 revised solve $\sin 2x + \sin x = 0$ using addition formula added 1 character in body Apr 12 comment orthonormal columns and linear independent vectors Gram-Schmidt orthogonalization? Apr 12 revised Cubic Spline for a function edited tags Apr 12 answered solve $\sin 2x + \sin x = 0$ using addition formula Apr 12 revised evaluate the numbers from the coefficient edited tags Apr 12 revised Brownian hitting time of a closed set added 4 characters in body Apr 12 comment What is the domain and range of the sum of two random variables? @user1770201 no contradiction, his last paragraph states in 5 lines what i wrote in 1 Apr 12 comment Finding the bound of a linear functional defined on $C[-1,1]$ oknp :) you can click edit and see how i did it so yours will look pretty next time. \left and \right constructs are floating in size and quite useful for such cases Apr 12 revised Finding the bound of a linear functional defined on $C[-1,1]$ added 41 characters in body Apr 12 answered Linear span of subspaces | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9271377325057983, "perplexity": 535.7324294115892}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461860121618.46/warc/CC-MAIN-20160428161521-00013-ip-10-239-7-51.ec2.internal.warc.gz"} |
https://dsp.stackexchange.com/questions/38718/general-form-of-dilations-and-translations-of-wavelet-function/38720 | # General form of dilations and translations of wavelet function
In some papers, the dilations and translations of a wavelet function is written as follows:
$$\psi_{j,k}=\frac {1}{\sqrt {2^j}} \psi\left({2^{-j}t-k}\right),\quad\text{where j and k are integers}$$
However, in some literature, the above formula is written in a different way:
$$\psi_{j,k}=\ {\sqrt {2^j}} \psi\left({2^{j}x-k}\right), \quad\text{where j and k are integers}$$
Are these two formula equivalent?
Yes, they are equivalent, since $j\in\mathbb{Z}$. Note that your first definition becomes equal to the second when $j$ is substituded by $-j$. It is just a convention, if you consider a positive $j$ to be a up- or downscaling.
They are equivalent, as long as $$j_1$$ (first formula) and $$-j_2$$ (second formula) span the same integer set. They are, if $$j_1\in\mathbb{Z}$$ and $$j_2\in\mathbb{Z}$$, or if $$j_1\in\mathbb{N}^+$$ (positive integers) and $$j_2\in\mathbb{N}^-$$ (negative integers).
They are, under the change of variable: $$t\to 2^{2j}x$$. Let us remind that if $$\psi(u)$$ is of unit norm(in $$L_2$$ norm), $$\frac{1}{\sqrt{a}}\psi(u/a)$$ is of unit norm as well, and the translation $$u\to u-k$$ does not change it. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 11, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.976189136505127, "perplexity": 286.08066283101016}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540544696.93/warc/CC-MAIN-20191212153724-20191212181724-00261.warc.gz"} |
https://www.physicsforums.com/threads/density-and-viscosity.661614/ | # Density and viscosity
1. Dec 30, 2012
### scientifico
Hello, what forces causes density and viscosity in a fluid ?
I thought oil was more dense than water but it is only more viscous. I suppose a very viscous liquid has strong forces interacting between the molecules so they became closer and shouldn't density increase too ?
Thank you
2. Dec 30, 2012 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9707095623016357, "perplexity": 2725.894171974394}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257651780.99/warc/CC-MAIN-20180325025050-20180325045050-00469.warc.gz"} |
http://en.allexperts.com/q/Advanced-Math-1363/2013/1/quadratic-simultaneous-equations.htm | You are here:
Question
QUESTION: Here are the equations:
x^2+y=62
x+y^2=176
I have tried to substitute the value of y from one equation in another. This leads to a quartic equation. Eliminating co-efficients seems to be difficult.
ANSWER: Is this from a book or something? If you sketch the two parabolas carefully you will see they intersect in 4 places, so it is indeed a quartic equation, which doesn't in general have any nice solutions. Do you need exact answers? You can certainly estimate them however accurately you like.
---------- FOLLOW-UP ----------
QUESTION: I was curious to understand whether this equation can be solved algebraically
without too much complexity. How to obtain real values of roots?
There's a formula for solving cubic and quartic equations but it is very messy. Occasionally you get lucky and a quartic factors nicely, but that is not the usual situation.
For degrees higher than 5 there is no such formula, this is an important theorem from Galois theory.
The formula is horrible, you can see it here:
http://planetmath.org/QuarticFormula.html
Of course a good graphing calculator or computer can quickly estimate the roots, for example using Newton's method.
Questioner's Rating
Rating(1-10) Knowledgeability = 9 Clarity of Response = 10 Politeness = 10 Comment Thank You. your answer has been very helpful.
Volunteer | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9569792747497559, "perplexity": 558.3479259347611}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-09/segments/1487501171632.91/warc/CC-MAIN-20170219104611-00094-ip-10-171-10-108.ec2.internal.warc.gz"} |
http://hepnp.ihep.ac.cn/article/1995/4 | ## 1995 Vol. 19, No. 4
Display Method: |
1995, 19(4): 289-295.
Abstract:
Based on 7.8×106 J/ψ events collected by the BES at BEPC, through study of the hadronic decays J/ψ→φπ+π and J/ψ→ωπ+π, the branching ratios of the processes and the mass positions and the widths of f0 are obtained. The angular distributions of J/ψ→φf0,f0→π+π, are fitted and the helicity amplitude ratio of this process is first given.
1995, 19(4): 296-304.
Abstract:
The pseudorapidity distribution of charged particles produced in pp collisions at 400GeV/c was measured by using LEBC films offered by CERN NA27 collaboration. The scaled factorial cumulant moments have been caloulated. Comparing with the results obtained from Monte Carlo events with the same single particle spectrum and no correlations, it is shown that for broad mixed multiplicity distributions, a significant part of K2 is coming from the single particle fluctuation due to the fluctuating multiplicity. The factorial cumulant moments obtained by using the star correlation integrals reduce significantly the statistical errors of the higher order cumulant moments, but the essential conclusions remain unchanged.
1995, 19(4): 305-309.
Abstract:
The angular distributions of the fragments in the range of 4π solid angle for the 600 MeV 40Ar+197Au reaction have been measured using the beam-off γ-ray spectroscopy technique. The features of the fragment angular distributions and the reaction mechanisms of the production of fragments have been discussed.
1995, 19(4): 310-319.
Abstract:
The contributions of the double Pomeron exchange to the J/ψ production inhigh energy p+p(p) collisions are calculated using different models for parton distribution functions inside the Pomeron(P). For the Ingelman-Schlein model in which gluons dominate in partons, the cross sections in high energy increase smoothly with as In2S or InS. The total cross section σ(S) is about 102－103nb in the TeV energies. For the Donnachie-Landshoff model in which the P are considered as something like isoscalar photons with C=+1 the cross section behavior increasing with energy is a little bit complicated. In same energy range, the cross sections are only 1-3nb which are smaller than that of the former case by 2—3 order of magnitude. So, if we assume the parameters of the above models are reliable, then these J/ψ production processes should be a good place for testing these models.
1995, 19(4): 320-326.
Abstract:
Starting from the transformation property of the action integral of a system under the local and non-local transformation, we derive the generalized Noether's identities connecting with non-local transformation. The applications of the theory to the Yang-Mills field with high-order derivatives are presented. A new conservative PBRS charge is found which differs from BRS conservative charge. The other conservative charge connecting with non-local transformation is also obtained.
1995, 19(4): 327-331.
Abstract:
0-heavy meson bound states are discussed by using the Bethe-Salpeter (B-S)equation at the instantaneous approximation. In terms of the heavy flavor mass expansion method, the masses for mesons D0, D±, B0, B± are calculated at first order approximation. Comparisons of our results with experimental data and with some theoretical calculations from other method are given.
1995, 19(4): 332-339.
Abstract:
In this paper, gauge constrained conditions and quantization of SU(N) gauge theories are analysed by means of Dirac's formalism. In the framework of algebraic dynamics, gauge invariance, Gauss law and Ward identities are discussed. With use of the version of conservation law in correlation dynamics, the conserved Gauss law and Ward identities related to residual gauge invariance can be transformed into initial value problems.
1995, 19(4): 340-347.
Abstract:
Based on interacting gluon model, leading particle rapidity distributions for 100 GeV p+Ag collisions are calculated and compared with experimental data. The results are in a good agreement with data when using a more sophisticated collision geometry and including final fragmentation process.
1995, 19(4): 348-353.
Abstract:
Using the Glauber Multiple scattering theory, we analyse the experimental elastic differential cross section of antiproton-4He scattering at low-energy. It is found that the character of multiple collision is still dominant as the antiproton energy is as low as 19.6 MeV. The same conclusion is obtained by comparison with the optical model calculation.
1995, 19(4): 354-358.
Abstract:
Identical bands of odd-A superdeformed nuclei in the A～190 mass region have been investigated using triaxial-particle-rotor model within the framework of the BCS pairing correlation. Three pairs of identical bands (191Au, 191Hg(b1)), (191Hg(b2),193Hg(b2)), and (191Hg(b3), 193Hg(b3)) are analyzed. It is pointed out that the identity in the γ-transition energies in these bands may be the result of the occupation of some special orbitals.
1995, 19(4): 359-364.
Abstract:
The properties of the band in even nuclei of 166Hf94170Hf98 are investigated using cranking shell model with PNC method. The band crossing frequency, the interaction intensity between yrast and yrare bands, the aligned momentum, and the moment of inertia are calculated. The comparison between the calculation and experimental data shows a good agreement with each other if Lund systematic parameter is used for the Nilsson potential.
1995, 19(4): 365-373.
Abstract:
We generalize the relativistic Vlasov-Uchling-Uhlenbeck model to include the kaon mean-field potential and the final-state rescatterings of kaons with nucleons,and use it to simulate the subthrcsbold kaon production from the collisions of two Ca nuclci at an incident energy of 1GeV/nucleon. The effects of the vector meanfield potential and the kaon rescattering on the final-state properties of kaons are analyzed. From the calculated results it is found that the effect of repulsive vector potential due to nucleons on the final-state kinetic energy spectrum of kaon is similar to that for the kaon final-state rescattering and leads to an significantly enhanced yield of kaons with large momenta in the laboratory system. However, the effect of the kaon rescattering on its kinetic energy spectrum is reduced considerably by including the repulsive vector potential. This indicates that these final-state interactions of kaons with nucleons are not negligible in order to evaluate reasonably thekaon energy spectrum and angular distribution.
1995, 19(4): 374-376.
Abstract:
An energy dependent optical potential for 12C+12C system is established based on the analyses of the angular distributions of the elastic scattering in the energy range of 70-160MeV. This potential can well reproduce the excitation function for 12C+12C elastic scattering in the same energy range.
1995, 19(4): 377-384.
Abstract:
The neutron dose equivalent produced by skyshine was calculated using MORSE program. For these calculations, the neutron sourse was enclosed in a hollow concrete cylinder. There were two cases, one was the cylinder with concrete roof shielding, the other was without a roof. The sourse energies of 2.38 MeV, 14 MeV and giant resonance (G. R) neutron were calculated separately. We have compared the effects of different models, such as different sourse emission solid angle and different scattering media. The results can be applied for the estimation of radiological impact to invironment and the design of radiation protection on the particle accelerators and other nuclear facilities. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9415474534034729, "perplexity": 1912.6616488928642}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499949.24/warc/CC-MAIN-20230201180036-20230201210036-00324.warc.gz"} |
https://www.physicsforums.com/threads/density-of-states.117855/ | # Density of states
1. ### drcrabs
46
upon recent studies of the Density of states and Specific heat capacities, ive found the Einstien and Debye Models to be very helpful, Debye being the more accurate of the two models at low temperatures as it takes into account the low frequency modes.
However, the realistic density of states e.g for solid Argon, seems to not fit either the Debye or the Einstien model at moderate temperatures. The realistic density of states seems to increase faster than any other model as the temperature increases.
Why is this? Are there additional modes we are not taking into account?
Last edited: Apr 17, 2006
2. ### Gokul43201
11,044
Staff Emeritus
Do you have a reference for the density of modes in solid argon ? It's not clear (to me) what exactly you are comparing. Are you really looking at the temperature dependence or the frequency dependence ? And are you comparing each of the 3 modes (2 transverse and one longitudinal) separately ?
3. ### drcrabs
46
Actuallty what im looking at is the temperature dependacy of the specific heat capacity
4. ### ZapperZ
30,731
Staff Emeritus
I think there needs to be a clearer distinction made here between the "regular" density of states, which is really the electronic density of states, versus the phonon density of states. The latter is what you are asking for.
Zz. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9009921550750732, "perplexity": 1001.2076692891432}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398448227.60/warc/CC-MAIN-20151124205408-00097-ip-10-71-132-137.ec2.internal.warc.gz"} |
https://www.science.gov/topicpages/a/anisotropic+lattice+fields.html | #### Sample records for anisotropic lattice fields
1. Structural transitions of the vortex lattice in anisotropic superconductors and fingering instability of electron droplets in an inhomogeneous magnetic field
Klironomos, Alexios
I present a derivation of the nondispersive elastic moduli for the vortex lattice within the anisotropic Ginzburg-Landau model. I derive an extension of the virial theorem for superconductivity for anisotropic superconductors, with the anisotropy arising from s-d mixing or an anisotropic Fermi surface. The structural transition from rhombic to square vortex lattice is studied within this model along with the effects of thermal fluctuations on the structural transition. The reentrant transition from square to rhombic vortex lattice for high fields and the instability with respect to rigid rotations of the vortex lattice, predicted by calculations within the nonlocal London model, are also present in the anisotropic Ginzburg-Landau model. I also study the fingering of an electron droplet in a special Quantum Hall regime, where electrostatic forces are weak. Performing Monte Carlo simulations I study the growth and fingering of the electron droplet in an inhomogeneous magnetic field as the number of electrons is increased. I expand on recent theoretical results and find excellent agreement between my simulations and the theoretical predictions.
2. Discrete vortices on anisotropic lattices
Chen, Gui-Hua; Wang, Hong-Cheng; Chen, Zi-Fa
2015-08-01
We consider the effects of anisotropy on two types of localized states with topological charges equal to 1 in two-dimensional nonlinear lattices, using the discrete nonlinear Schrödinger equation as a paradigm model. We find that on-site-centered vortices with different propagation constants are not globally stable, and that upper and lower boundaries of the propagation constant exist. The region between these two boundaries is the domain outside of which the on-site-centered vortices are unstable. This region decreases in size as the anisotropy parameter is gradually increased. We also consider off-site-centered vortices on anisotropic lattices, which are unstable on this lattice type and either transform into stable quadrupoles or collapse. We find that the transformation of off-sitecentered vortices into quadrupoles, which occurs on anisotropic lattices, cannot occur on isotropic lattices. In the quadrupole case, a propagation-constant region also exists, outside of which the localized states cannot stably exist. The influence of anisotropy on this region is almost identical to its effects on the on-site-centered vortex case.
3. The Ising-Anisotropic Kondo Lattice in a transverse field: EDMFT study and implications for the global phase diagram
Nica, Emilian Marius; Ingersent, Kevin; Si, Qimiao
2015-03-01
Heavy-fermion materials exhibit a rich variety of phase transitions. Of particular interest are quantum phase transitions and the associated breakdown of the Fermi liquid picture. A theoretical example of this is the Kondo destruction effect in the context of local quantum criticality. To capture this effect and others, a zero-temperature global phase diagram for heavy-fermion materials has been proposed. It incorporates the competition between the Kondo effect (promoted by exchange coupling JK) and the variable quantum fluctuations of the local-moment magnetism (parameterized by G). We investigate this competition in the Ising-anisotropic Kondo lattice with a transverse magnetic field, where the field serves to tune G. We determine a zero-temperature phase diagram of this model within the extended dynamical mean-field theory (EDMFT), and discuss the implications of our results for the global phase diagram of heavy-fermion systems.
4. Thermal D mesons from anisotropic lattice QCD
Kelly, Aoife; Skullerud, Jon-Ivar
2017-03-01
We present results for correlators and spectral functions of open charm mesons using 2+1 flavours of clover fermions on anisotropic lattices. The D mesons are found to dissociate close to the deconfinement crossover temperature Tc. Our preliminary results suggest a shift in the thermal D meson mass below Tc. Mesons containing strange quarks exhibit smaller thermal modifications than those containing light quarks.
5. Staggered Fermion Thermodynamics using Anisotropic Lattices
Levkova, L.
2003-05-01
Numerical simulations of full QCD on anisotropic lattices provide a convenient way to study QCD thermodynamics with fixed physics scales and reduced lattice spacing errors. We report results from calculations with 2-flavors of dynamical fermions where all bare parameters and hence the physics scales are kept constant while the temperature is changed in small steps by varying only the number of the time slices. The results from a series of zero-temperature scale setting simulations are used to determine the Karsch coefficients and the equation of state at finite temperatures.
6. SU(3) lattice gauge autocorrelations with anisotropic action
Draper, Terrence; Nenkov, Constantine; Peardon, Mike
1997-02-01
We report results of autocorrelation measurements in pure SU(3) lattice gauge theory. The computations are performed on the CONVEX SPP1200 parallel platform within the CANOPY programming environment. The focus of our analysis is on typical autocorrelation times and optimization of the mixing ratio between overrelaxation and pseudo-heatbath sweeps for generating gauge field configurations. We study second order tadpole-improved approximation of the Wilson action in the gluon sector, which offers the advantage on smaller lattices (8 3 × 16 and 6 3 × 12 - 30). We also make use of anisotropic lattices, with temporal lattice spacing smaller than the spatial spacing, which prove useful for calculating noisy correlation functions with large spatial lattice discretization (of the order of 0.4 fm).
7. Anisotropic lattice distortions in biogenic aragonite
Pokroy, Boaz; Quintana, John P.; Caspi, El'ad N.; Berner, Alex; Zolotoyabko, Emil
2004-12-01
Composite biogenic materials produced by organisms have a complicated design on a nanometre scale. An outstanding example of organic-inorganic composites is provided by mollusc seashells, whose superior mechanical properties are due to their multi-level crystalline hierarchy and the presence of a small amount (0.1-5 wt%) of organic molecules. The presence of organic molecules, among other characteristics, can influence the coherence length for X-ray scattering in biogenic crystals. Here we show the results of synchrotron high-resolution X-ray powder diffraction measurements in biogenic and non-biogenic (geological) aragonite crystals. On applying the Rietveld refinement procedure to the high-resolution diffraction spectra, we were able to extract the aragonite lattice parameters with an accuracy of 10 p.p.m. As a result, we found anisotropic lattice distortions in biogenic aragonite relative to the geological sample, maximum distortion being 0.1% along the c axis of the orthorhombic unit cell. The organic molecules could be a source of these structural distortions in biogenic crystals. This finding may be important to the general understanding of the biomineralization process and the development of bio-inspired 'smart' materials.
8. Lattice-Boltzmann hydrodynamics of anisotropic active matter
de Graaf, Joost; Menke, Henri; Mathijssen, Arnold J. T. M.; Fabritius, Marc; Holm, Christian; Shendruk, Tyler N.
2016-04-01
A plethora of active matter models exist that describe the behavior of self-propelled particles (or swimmers), both with and without hydrodynamics. However, there are few studies that consider shape-anisotropic swimmers and include hydrodynamic interactions. Here, we introduce a simple method to simulate self-propelled colloids interacting hydrodynamically in a viscous medium using the lattice-Boltzmann technique. Our model is based on raspberry-type viscous coupling and a force/counter-force formalism, which ensures that the system is force free. We consider several anisotropic shapes and characterize their hydrodynamic multipolar flow field. We demonstrate that shape-anisotropy can lead to the presence of a strong quadrupole and octupole moments, in addition to the principle dipole moment. The ability to simulate and characterize these higher-order moments will prove crucial for understanding the behavior of model swimmers in confining geometries.
9. Discrete solitons and vortices on anisotropic lattices.
PubMed
Kevrekidis, P G; Frantzeskakis, D J; Carretero-González, R; Malomed, B A; Bishop, A R
2005-10-01
We consider the effects of anisotropy on solitons of various types in two-dimensional nonlinear lattices, using the discrete nonlinear Schrödinger equation as a paradigm model. For fundamental solitons, we develop a variational approximation that predicts that broad quasicontinuum solitons are unstable, while their strongly anisotropic counterparts are stable. By means of numerical methods, it is found that, in the general case, the fundamental solitons and simplest on-site-centered vortex solitons ("vortex crosses") feature enhanced or reduced stability areas, depending on the strength of the anisotropy. More surprising is the effect of anisotropy on the so-called "super-symmetric" intersite-centered vortices ("vortex squares"), with the topological charge equal to the square's size : we predict in an analytical form by means of the Lyapunov-Schmidt theory, and confirm by numerical results, that arbitrarily weak anisotropy results in dramatic changes in the stability and dynamics in comparison with the degenerate, in this case, isotropic, limit.
10. Quark–gluon plasma phenomenology from anisotropic lattice QCD
SciTech Connect
Skullerud, Jon-Ivar; Kelly, Aoife; Aarts, Gert; Allton, Chris; Amato, Alessandro; Evans, P. Wynne M.; Hands, Simon; Burnier, Yannis; Giudice, Pietro; Harris, Tim; Ryan, Sinéad M.; Kim, Seyong; Lombardo, Maria Paola; Oktay, Mehmet B.; Rothkopf, Alexander
2016-01-22
The FASTSUM collaboration has been carrying out simulations of N{sub f} = 2 + 1 QCD at nonzero temperature in the fixed-scale approach using anisotropic lattices. Here we present the status of these studies, including recent results for electrical conductivity and charge diffusion, and heavy quarkonium (charm and beauty) physics.
11. Chern-Simons theory of the anisotropic quantum Heisenberg antiferromagnet on a square lattice
SciTech Connect
Lopez, A. ); Rojo, A.G. Department of Physics, University of Michigan, Ann Arbor, Michigan 48109-1120 ); Fradkin, E. )
1994-06-01
We consider the anisotropic quantum Heisenberg antiferromagnetic (with anistropy [lambda]) on a square lattice using a Chern-Simons (or Wigner-Jordan) approach. We show that the average field approximation (AFA) yields a phase diagram with two phases: a Neel state for [lambda][gt][lambda][sub [ital c
12. Two-flavor QCD thermodynamics using anisotropic lattices
Levkova, Ludmila; Manke, Thomas; Mawhinney, Robert
2006-04-01
Numerical simulations of full QCD on anisotropic lattices provide a convenient way to study QCD thermodynamics with fixed physics scales and reduced lattice spacing errors. We report results from calculations with two flavors of dynamical staggered fermions, where all bare parameters and the renormalized anisotropy are kept constant and the temperature is changed in small steps by varying only the number of time slices. Including results from zero-temperature scale-setting simulations, which determine the Karsch coefficients, allows for the calculation of the equation of state at finite temperatures.
13. Dipolar matter-wave solitons in two-dimensional anisotropic discrete lattices
Chen, Huaiyu; Liu, Yan; Zhang, Qiang; Shi, Yuhan; Pang, Wei; Li, Yongyao
2016-05-01
We numerically demonstrate two-dimensional (2D) matter-wave solitons in the disk-shaped dipolar Bose-Einstein condensates (BECs) trapped in strongly anisotropic optical lattices (OLs) in a disk's plane. The considered OLs are square lattices which can be formed by interfering two pairs of plane waves with different intensities. The hopping rates of the condensates between two adjacent lattices in the orthogonal directions are different, which gives rise to a linearly anisotropic system. We find that when the polarized orientation of the dipoles is parallel to disk's plane with the same direction, the combined effects of the linearly anisotropy and the nonlocal nonlinear anisotropy strongly influence the formations, as well as the dynamics of the lattice solitons. Particularly, the isotropy-pattern solitons (IPSs) are found when these combined effects reach a balance. Motion, collision, and rotation of the IPSs are also studied in detail by means of systematic simulations. We further find that these IPSs can move freely in the 2D anisotropic discrete system, hence giving rise to an anisotropic effective mass. Four types of collisions between the IPSs are identified. By rotating an external magnetic field up to a critical angular velocity, the IPSs can still remain localized and play as a breather. Finally, the influences from the combined effects between the linear and the nonlocal nonlinear anisotropy with consideration of the contact and/or local nonlinearity are discussed too.
14. Measuring the aspect ratio renormalization of anisotropic-lattice gluons
SciTech Connect
Alford, M.; Drummond, I. T.; Horgan, R. R.; Shanahan, H.; Peardon, M.
2001-04-01
Using tadpole-improved actions we investigate the consistency between different methods of measuring the aspect ratio renormalization of anisotropic-lattice gluons for bare aspect ratios {chi}{sub 0}=4,6,10 and inverse lattice spacing in the range a{sub s}{sup -1}=660--840 MeV. The tadpole corrections to the action, which are established self-consistently, are defined for two cases, mean link tadpoles in the Landau gauge and gauge invariant mean plaquette tadpoles. Parameters in the latter case exhibited no dependence on the spatial lattice size L, while in the former, parameters showed only a weak dependence on L easily extrapolated to L={infinity}. The renormalized anisotropy {chi}{sub R} was measured using both the torelon dispersion relation and the sideways potential method. There is general agreement between these approaches, but there are discrepancies which are evidence for the presence of lattice artifact contributions. For the torelon these are estimated to be O({alpha}{sub S}a{sub s}{sup 2}/R{sup 2}), where R is the flux-tube radius. We also present some new data that suggest that rotational invariance is established more accurately for the mean-link action than the plaquette action.
15. Hall Effect in the Vortex Lattice of d-Wave Superconductors with Anisotropic Fermi Surfaces
Kohno, Wataru; Ueki, Hikaru; Kita, Takafumi
2017-02-01
On the basis of the augmented quasiclassical theory of superconductivity with the Lorentz force, we study the magnetic field dependence of the charge distribution due to the Lorentz force in a d-wave vortex lattice with anisotropic Fermi surfaces. Owing to the competition between the energy-gap and Fermi surface anisotropies, the charge profile in the vortex lattice changes dramatically with increasing magnetic field because of the overlaps of each nearest vortex-core charge. In addition, the accumulated charge in the core region may reverse its sign as a function of magnetic field. This strong field dependence of the vortex-core charge cannot be observed in the model with an isotropic Fermi surface.
16. Spin Relaxation in Kondo Lattice Systems with Anisotropic Kondo Interaction
Belov, S. I.; Kutuzov, A. S.
2016-12-01
We study the influence of the Kondo effect on the spin relaxation in systems with anisotropic Kondo interaction at temperatures both high and low as compared with the static magnetic field. In the absence of the Kondo effect, the electron spin resonance linewidth is not narrowed in the whole temperature range due to the high anisotropy of the Kondo interaction. The Kondo effect leads to the universal energy scale, which regulates the temperature and magnetic field dependence of different kinetic coefficients and results in a mutual cancelation of their singular parts in a collective spin mode.
17. An anisotropic preconditioning for the Wilson fermion matrix on the lattice
SciTech Connect
Balint Joo, Robert G. Edwards, Michael J. Peardon
2010-01-01
A preconditioning for the Wilson fermion matrix on the lattice is defined which is particularly suited to the case when the temporal lattice spacing is much smaller than the spatial one. Details on the implementation of the scheme are given. The method is tested in numerical studies of QCD on anisotropic lattices.
18. Lattice QCD in Background Fields
SciTech Connect
William Detmold, Brian Tiburzi, Andre Walker-Loud
2009-06-01
Electromagnetic properties of hadrons can be computed by lattice simulations of QCD in background fields. We demonstrate new techniques for the investigation of charged hadron properties in electric fields. Our current calculations employ large electric fields, motivating us to analyze chiral dynamics in strong QED backgrounds, and subsequently uncover surprising non-perturbative effects present at finite volume.
19. Topics in lattice QCD and effective field theory
Buchoff, Michael I.
Quantum Chromodynamics (QCD) is the fundamental theory that governs hadronic physics. However, due to its non-perturbative nature at low-energy/long distances, QCD calculations are difficult. The only method for performing these calculations is through lattice QCD. These computationally intensive calculations approximate continuum physics with a discretized lattice in order to extract hadronic phenomena from first principles. However, as in any approximation, there are multiple systematic errors between lattice QCD calculation and actual hardronic phenomena. Developing analytic formulae describing the systematic errors due to the discrete lattice spacings is the main focus of this work. To account for these systematic effects in terms of hadronic interactions, effective field theory proves to be useful. Effective field theory (EFT) provides a formalism for categorizing low-energy effects of a high-energy fundamental theory as long as there is a significant separation in scales. An example of this is in chiral perturbation theory (chiPT), where the low-energy effects of QCD are contained in a mesonic theory whose applicability is a result of a pion mass smaller than the chiral breaking scale. In a similar way, lattice chiPT accounts for the low-energy effects of lattice QCD, where a small lattice spacing acts the same way as the quark mass. In this work, the basics of this process are outlined, and multiple original calculations are presented: effective field theory for anisotropic lattices, I=2 pipi scattering for isotropic, anisotropic, and twisted mass lattices. Additionally, a combination of effective field theory and an isospin chemical potential on the lattice is proposed to extract several computationally difficult scattering parameters. Lastly, recently proposed local, chiral lattice actions are analyzed in the framework of effective field theory, which illuminates various challenges in simulating such actions.
20. Phase Field Modeling of Directional Fracture in Anisotropic Polycrystals
DTIC Science & Technology
2015-02-01
ARL-RP-0518 ● FEBRUARY 2015 US Army Research Laboratory Phase Field Modeling of Directional Fracture in Anisotropic Polycrystals...0518 ● FEBRUARY 2015 US Army Research Laboratory Phase Field Modeling of Directional Fracture in Anisotropic Polycrystals by JD Clayton...
1. Symmetry analysis for anisotropic field theories
SciTech Connect
Parra, Lorena; Vergara, J. David
2012-08-24
The purpose of this paper is to study with the help of Noether's theorem the symmetries of anisotropic actions for arbitrary fields which generally depend on higher order spatial derivatives, and to find the corresponding current densities and the Noether charges. We study in particular scale invariance and consider the cases of higher derivative extensions of the scalar field, electrodynamics and Chern-Simons theory.
2. Momentum transport in strongly coupled anisotropic plasmas in the presence of strong magnetic fields
Finazzo, Stefano Ivo; Critelli, Renato; Rougemont, Romulo; Noronha, Jorge
2016-09-01
We present a holographic perspective on momentum transport in strongly coupled, anisotropic non-Abelian plasmas in the presence of strong magnetic fields. We compute the anisotropic heavy quark drag forces and Langevin diffusion coefficients and also the anisotropic shear viscosities for two different holographic models, namely, a top-down deformation of strongly coupled N =4 super-Yang-Mills theory triggered by an external Abelian magnetic field, and a bottom-up Einstein-Maxwell-dilaton (EMD) model which is able to provide a quantitative description of lattice QCD thermodynamics with (2 +1 ) flavors at both zero and nonzero magnetic fields. We find that, in general, energy loss and momentum diffusion through strongly coupled anisotropic plasmas are enhanced by a magnetic field being larger in transverse directions than in the direction parallel to the magnetic field. Moreover, the anisotropic shear viscosity coefficient is smaller in the direction of the magnetic field than in the plane perpendicular to the field, which indicates that strongly coupled anisotropic plasmas become closer to the perfect fluid limit along the magnetic field. We also present, in the context of the EMD model, holographic predictions for the entropy density and the crossover critical temperature in a wider region of the (T , B ) phase diagram that has not yet been covered by lattice simulations. Our results for the transport coefficients in the phenomenologically realistic magnetic EMD model could be readily used as inputs in numerical codes for magnetohydrodynamics.
3. High Statistics Analysis using Anisotropic Clover Lattices: (IV) The Volume Dependence of the Light Hadron Masses
SciTech Connect
Beane, S R; Detmold, W; Lin, H W; Luu, T C; Orginos, K; Parreno, A; Savage, M J; Torok, A; Walker-Loud, A
2011-07-01
The volume dependence of the octet baryon masses and relations among them are explored with Lattice QCD. Calculations are performed with nf = 2 + 1 clover fermion discretization in four lattice volumes, with spatial extent L ? 2.0, 2.5, 3.0 and 4.0 fm, with an anisotropic lattice spacing of b_s ? 0.123 fm in the spatial direction, and b_t = b_s/3.5 in the time direction, and at a pion mass of m_\\pi ? 390 MeV. The typical precision of the ground-state baryon mass determination is lattice gauge-field configurations. Finally, the volume dependence of the pion and kaon masses are analyzed with two-flavor and three-flavor chiral perturbation theory.
4. High Statistics Analysis using Anisotropic Clover Lattices: (III) Baryon-Baryon Interactions
SciTech Connect
Silas Beane; Detmold, William; Lin, Huey-Wen; Luu, Thomas C.; Orginos, Kostas; Savage, Martin; Torok, Aaron M.; Walker-Loud, Andre
2010-03-01
Low-energy baryon-baryon interactions are calculated in a high-statistics lattice QCD study on a single ensemble of anisotropic clover gauge-field configurations at a pion mass of m_pi ~ 390 MeV, a spatial volume of L^3 ~ (2.5 fm)^3, and a spatial lattice spacing of b ~ 0.123 fm. Luscher’s method is used to extract nucleon-nucleon, hyperon-nucleon and hyperon-hyperon scattering phase shifts at one momentum from the one- and two-baryon ground-state energies in the lattice volume. The N-Sigma interactions are found to be highly spin-dependent, and the interaction in the ^3 S _1 channel is found to be strong. In contrast, the N-Lambda interactions are found to be spin-independent, within the uncertainties of the calculation, consistent with the absence of one-pion-exchange. The only channel for which a negative energy-shift is found is Lambda-Lambda, indicating that the Lambda-Lambda interaction is attractive, as anticipated from model-dependent discussions regarding the H-dibaryon. The NN scattering lengths are found to be small, clearly indicating the absence of any fine-tuning in the NN-sector at this pion mass. This is consistent with our previous Lattice QCD calculation of the NN interactions. The behavior of the signal-to-noise ratio in the baryon-baryon correlation functions, and in the ratio of correlation functions that yields the ground-state energy splitting
5. High statistics analysis using anisotropic clover lattices: (III) Baryon-baryon interactions
SciTech Connect
Beane, S; Detmold, W; Lin, H; Luu, T; Orginos, K; Savage, M; Torok, A; Walker-Loud, A
2010-01-19
Low-energy baryon-baryon interactions are calculated in a high-statistics lattice QCD study on a single ensemble of anisotropic clover gauge-field configurations at a pion mass of m{sub {pi}} {approx} 390 MeV, a spatial volume of L{sup 3} {approx} (2.5 fm){sup 3}, and a spatial lattice spacing of b {approx} 0.123 fm. Luescher's method is used to extract nucleon-nucleon, hyperon-nucleon and hyperon-hyperon scattering phase shifts at one momentum from the one- and two-baryon ground-state energies in the lattice volume. The isospin-3/2 N{Sigma} interactions are found to be highly spin-dependent, and the interaction in the {sup 3}S{sub 1} channel is found to be strong. In contrast, the N{Lambda} interactions are found to be spin-independent, within the uncertainties of the calculation, consistent with the absence of one-pion-exchange. The only channel for which a negative energy-shift is found is {Lambda}{Lambda}, indicating that the {Lambda}{Lambda} interaction is attractive, as anticipated from model-dependent discussions regarding the H-dibaryon. The NN scattering lengths are found to be small, clearly indicating the absence of any fine-tuning in the NN-sector at this pion mass. This is consistent with our previous Lattice QCD calculation of NN interactions. The behavior of the signal-to-noise ratio in the baryon-baryon correlation functions, and in the ratio of correlation functions that yields the ground-state energy splitting is explored. In particular, focus is placed on the window of time slices for which the signal-to-noise ratio does not degrade exponentially, as this provides the opportunity to extract quantitative information about multi-baryon systems.
6. Nonperturbative study of the action parameters for anisotropic-lattice quarks
SciTech Connect
Foley, Justin; Cais, Alan O; Peardon, Mike; Ryan, Sinead M.
2006-01-01
A quark action designed for highly anisotropic-lattice simulations is discussed. The mass-dependence of the parameters in the action is studied and the results are presented. Applications of this action in studies of heavy quark quantities are described and results are presented from simulations at an anisotropy of six, for a range of quark masses from strange to bottom.
7. Ground states of the Ising model on an anisotropic triangular lattice: stripes and zigzags.
PubMed
Dublenych, Yu I
2013-10-09
A complete solution of the ground-state problem for the Ising model on an anisotropic triangular lattice with the nearest-neighbor interactions in a magnetic field is presented. It is shown that this problem can be reduced to the ground-state problem for an infinite chain with the interactions up to the second neighbors. In addition to the known ground-state structures (which correspond to full-dimensional regions in the parameter space of the model), new structures are found (at the boundaries of these regions), in particular, zigzagging stripes similar to those observed experimentally in colloidal monolayers. Though the number of parameters is relatively large (four), all the ground-state structures of the model are constructed and analyzed and therefore the paper can be considered as an example of a complete solution of a ground-state problem for classical spin or lattice-gas models. The paper can also help to verify the correctness of some results obtained previously by other authors and concerning the ground states of the model under consideration.
8. Chern-Simons theory of the anisotropic quantum Heisenberg antiferromagnet on a square lattice
Lopez, Ana; Rojo, A. G.; Fradkin, Eduardo
1994-06-01
We consider the anisotropic quantum Heisenberg antiferromagnetic (with anistropy λ) on a square lattice using a Chern-Simons (or Wigner-Jordan) approach. We show that the average field approximation (AFA) yields a phase diagram with two phases: a Neèl state for λ>λc and a flux phase for λ<λc separated by a second-order transition at λc<1. We show that this phase diagram does not describe the XY regime of the antiferromagnet. Fluctuations around the AFA induce relevant operators which yield the correct phase diagram. We find an equivalence between the antiferromagnet and a relativistic field theory of two self-interacting Dirac fermions coupled to a Chern-Simons gauge field. The field theory has a phase diagram with the correct number of Goldstone modes in each regime and a phase transition at a critical coupling λ*>λc. We identify this transition with the isotropic Heisenberg point. It has a nonvanishing Neèl order parameter, which drops to zero discontinuously for λ<λ*.
9. Melting of the Abrikosov flux lattice in anisotropic superconductors
NASA Technical Reports Server (NTRS)
Beck, R. G.; Farrell, D. E.; Rice, J. P.; Ginsberg, D. M.; Kogan, V. G.
1992-01-01
It has been proposed that the Abrikosov flux lattice in high-Tc superconductors is melted over a significant fraction of the phase diagram. A thermodynamic argument is provided which establishes that the angular dependence of the melting temperature is controlled by the superconducting mass anisotropy. Using a low-frequency torsional-oscillator technique, this relationship has been tested in untwinned single-crystal YBa2Cu3O(7-delta). The results offer decisive support for the melting proposal.
10. Far field expansion for anisotropic wave equations
NASA Technical Reports Server (NTRS)
Hariharan, S. I.; Hagstrom, Thomas
1989-01-01
A necessary ingredient for the numerical simulation of many time dependent phenomena in acoustics and aerodynamics is the imposition of accurate radiation conditions at artificial boundaries. The asymptotic analysis of propagating waves provides a rational approach to the development of such conditions. A far field asymptotic expansion of solutions of anisotropic wave equations is derived. This generalizes the well known Friedlander expansion for the standard wave operator. The expansion is used to derive a hierarchy of radiation conditions of increasing accuracy. Two numerical experiments are given to illustrate the utility of this approach. The first application is the study of unsteady vortical disturbances impinging on a flat plate; the second is the simulation of inviscid flow past an impulsively started cylinder.
11. High Statistics Analysis using Anisotropic Clover Lattices: (I) Single Hadron Correlation Functions
SciTech Connect
Will Detmold,Konstantinos Orginos,Silas R. Beane,Will Detmold,William Detmold,Thomas C. Luu,Konstantinos Orginos,Assumpta Parreno,Martin J. Savage,Aaron Torok,Andre Walker-Loud
2009-06-01
We present the results of high-statistics calculations of correlation functions generated with single-baryon interpolating operators on an ensemble of dynamical anisotropic gauge-field configurations generated by the Hadron Spectrum Collaboration using a tadpole-improved clover fermion action and Symanzik-improved gauge action. A total of 292,500 sets of measurements are made using 1194 gauge configurations of size 20^3 x 128 with an anisotropy parameter \\xi= b_s/b_t = 3.5, a spatial lattice spacing of b_s=0.1227\\pm 0.0008 fm, and pion mass of m_\\pi ~ 390 MeV. Ground state baryon masses are extracted with fully quantified uncertainties that are at or below the ~0.2%-level in lattice units. The lowest-lying negative-parity states are also extracted albeit with a somewhat lower level of precision. In the case of the nucleon, this negative-parity state is above the N\\pi threshold and, therefore, the isospin-1/2 \\pi N s-wave scattering phase-shift can be extracted using Luescher's method. The disconnected contributions to this process are included indirectly in the gauge-field configurations and do not require additional calculations. The signal-to-noise ratio in the various correlation functions is explored and is found to degrade exponentially faster than naive expectations on many time-slices. This is due to backward propagating states arising from the anti-periodic boundary conditions imposed on the quark-propagators in the time-direction. We explore how best to distribute computational resources between configuration generation and propagator measurements in order to optimize the extraction of single baryon observables.
12. Discrete solitons and vortices in anisotropic hexagonal and honeycomb lattices
Hoq, Q. E.; Kevrekidis, P. G.; Bishop, A. R.
2016-02-01
In the present work, we consider the self-focusing discrete nonlinear Schrödinger equation on hexagonal and honeycomb lattice geometries. Our emphasis is on the study of the effects of anisotropy, motivated by the tunability afforded in recent optical and atomic physics experiments. We find that multi-soliton and discrete vortex states undergo destabilizing bifurcations as the relevant anisotropy control parameter is varied. We quantify these bifurcations by means of explicit analytical calculations of the solutions, as well as of their spectral linearization eigenvalues. Finally, we corroborate the relevant stability picture through direct numerical computations. In the latter, we observe the prototypical manifestation of these instabilities to be the spontaneous rearrangement of the solution, for larger values of the coupling, into localized waveforms typically centered over fewer sites than the original unstable structure. For weak coupling, the instability appears to result in a robust breathing of the relevant waveforms.
13. Discrete solitons and vortices in anisotropic hexagonal and honeycomb lattices
DOE PAGES
Hoq, Q. E.; Kevrekidis, P. G.; Bishop, A. R.
2016-01-14
We consider the self-focusing discrete nonlinear Schrödinger equation on hexagonal and honeycomb lattice geometries. Our emphasis is on the study of the effects of anisotropy, motivated by the tunability afforded in recent optical and atomic physics experiments. We find that multi-soliton and discrete vortex states undergo destabilizing bifurcations as the relevant anisotropy control parameter is varied. Furthermore, we quantify these bifurcations by means of explicit analytical calculations of the solutions, as well as of their spectral linearization eigenvalues. Finally, we corroborate the relevant stability picture through direct numerical computations. In the latter, we observe the prototypical manifestation of these instabilitiesmore » to be the spontaneous rearrangement of the solution, for larger values of the coupling, into localized waveforms typically centered over fewer sites than the original unstable structure. In weak coupling, the instability appears to result in a robust breathing of the relevant waveforms.« less
14. Discrete solitons and vortices in anisotropic hexagonal and honeycomb lattices
SciTech Connect
Hoq, Q. E.; Kevrekidis, P. G.; Bishop, A. R.
2016-01-14
We consider the self-focusing discrete nonlinear Schrödinger equation on hexagonal and honeycomb lattice geometries. Our emphasis is on the study of the effects of anisotropy, motivated by the tunability afforded in recent optical and atomic physics experiments. We find that multi-soliton and discrete vortex states undergo destabilizing bifurcations as the relevant anisotropy control parameter is varied. Furthermore, we quantify these bifurcations by means of explicit analytical calculations of the solutions, as well as of their spectral linearization eigenvalues. Finally, we corroborate the relevant stability picture through direct numerical computations. In the latter, we observe the prototypical manifestation of these instabilities to be the spontaneous rearrangement of the solution, for larger values of the coupling, into localized waveforms typically centered over fewer sites than the original unstable structure. In weak coupling, the instability appears to result in a robust breathing of the relevant waveforms.
15. Anisotropic lattice thermal conductivity in chiral tellurium from first principles
SciTech Connect
Peng, Hua; Kioussis, Nicholas; Stewart, Derek A.
2015-12-21
Using ab initio based calculations, we have calculated the intrinsic lattice thermal conductivity of chiral tellurium. We show that the interplay between the strong covalent intrachain and weak van der Waals interchain interactions gives rise to the phonon band gap between the lower and higher optical phonon branches. The underlying mechanism of the large anisotropy of the thermal conductivity is the anisotropy of the phonon group velocities and of the anharmonic interatomic force constants (IFCs), where large interchain anharmonic IFCs are associated with the lone electron pairs. We predict that tellurium has a large three-phonon scattering phase space that results in low thermal conductivity. The thermal conductivity anisotropy decreases under applied hydrostatic pressure.
16. Green function method study of the anisotropic ferromagnetic Heisenberg model on a square lattice
Hu, Ai-Yuan; Chen, Yuan
2008-06-01
We study the phase diagram of the anisotropic ferromagnetic Heisenberg model on a square lattice. We use the double-time Green’s function method within the Callen decoupling approximation. The dependence of the Curie temperature Tc on the spin S and on the anisotropy parameter Δ ( Δ=0 and 1 correspond to the isotropic Heisenberg and Ising model, respectively) is obtained explicitly. Our results are in agreement with results obtained from other theoretical approaches.
17. Formation of Bragg band gaps in anisotropic phononic crystals analyzed with the empty lattice model
SciTech Connect
Wang, Yan -Feng; Maznev, Alexei; Laude, Vincent
2016-05-11
Bragg band gaps of phononic crystals generally, but not always, open at Brillouin zone boundaries. The commonly accepted explanation stems from the empty lattice model: assuming a small material contrast between the constituents of the unit cell, avoided crossings in the phononic band structure appear at frequencies and wavenumbers corresponding to band intersections; for scalar waves the lowest intersections coincide with boundaries of the first Brillouin zone. However, if a phononic crystal contains elastically anisotropic materials, its overall symmetry is not dictated solely by the lattice symmetry. We construct an empty lattice model for phononic crystals made of isotropic and anisotropic materials, based on their slowness curves. We find that, in the anisotropic case, avoided crossings generally do not appear at the boundaries of traditionally defined Brillouin zones. Furthermore, the Bragg "planes" which give rise to phononic band gaps, are generally not flat planes but curved surfaces. Lastly, the same is found to be the case for avoided crossings between shear (transverse) and longitudinal bands in the isotropic case.
18. Formation of Bragg band gaps in anisotropic phononic crystals analyzed with the empty lattice model
DOE PAGES
Wang, Yan -Feng; Maznev, Alexei; Laude, Vincent
2016-05-11
Bragg band gaps of phononic crystals generally, but not always, open at Brillouin zone boundaries. The commonly accepted explanation stems from the empty lattice model: assuming a small material contrast between the constituents of the unit cell, avoided crossings in the phononic band structure appear at frequencies and wavenumbers corresponding to band intersections; for scalar waves the lowest intersections coincide with boundaries of the first Brillouin zone. However, if a phononic crystal contains elastically anisotropic materials, its overall symmetry is not dictated solely by the lattice symmetry. We construct an empty lattice model for phononic crystals made of isotropic andmore » anisotropic materials, based on their slowness curves. We find that, in the anisotropic case, avoided crossings generally do not appear at the boundaries of traditionally defined Brillouin zones. Furthermore, the Bragg "planes" which give rise to phononic band gaps, are generally not flat planes but curved surfaces. Lastly, the same is found to be the case for avoided crossings between shear (transverse) and longitudinal bands in the isotropic case.« less
19. Quantum Phase Transition and Local Entanglement in Extended Hubbard Model on Anisotropic Triangular Lattices
Gao, Ji-Ming; Tang, Rong-An; Zhang, Zheng-Mei; Xue, Ju-Kui
2016-11-01
Using a mean-field theory based upon Hartree—Fock approximation, we theoretically investigate the competition between the metallic conductivity, spin order and charge order phases in a two-dimensional half-filled extended Hubbard model on anisotropic triangular lattice. Bond order, double occupancy, spin and charge structure factor are calculated, and the phase diagram of the extended Hubbard model is presented. It is found that the interplay of strong interaction and geometric frustration leads to exotic phases, the charge fluctuation is enhanced and three kinds of charge orders appear with the introduction of the nearest-neighbor interaction. Moreover, for different frustrations, it is also found that the antiferromagnetic insulating phase and nonmagnetic insulating phase are rapidly suppressed, and eventually disappeared as the ratio between the nearest-neighbor interaction and on-site interaction increases. This indicates that spin order is also sensitive to the nearest-neighbor interaction. Finally, the single-site entanglement is calculated and it is found that a clear discontinuous of the single-site entanglement appears at the critical points of the phase transition. Supported by National Natural Science Foundation of China under Grant Nos.11274255, 11475027 and 11305132, Specialized Research Fund for the Doctoral Program of Higher Education of China under Grant No. 20136203110001, and Technology of Northwest Normal University, China under Grants No. NWNU-LKQN-11-26
20. Working Group Report: Lattice Field Theory
SciTech Connect
Blum, T.; et al.,
2013-10-22
This is the report of the Computing Frontier working group on Lattice Field Theory prepared for the proceedings of the 2013 Community Summer Study ("Snowmass"). We present the future computing needs and plans of the U.S. lattice gauge theory community and argue that continued support of the U.S. (and worldwide) lattice-QCD effort is essential to fully capitalize on the enormous investment in the high-energy physics experimental program. We first summarize the dramatic progress of numerical lattice-QCD simulations in the past decade, with some emphasis on calculations carried out under the auspices of the U.S. Lattice-QCD Collaboration, and describe a broad program of lattice-QCD calculations that will be relevant for future experiments at the intensity and energy frontiers. We then present details of the computational hardware and software resources needed to undertake these calculations.
1. High Statistics Analysis using Anisotropic Clover Lattices: (II) Three-Baryon Systems
SciTech Connect
Beane, S; Detmold, W; Luu, T; Orginos, K; Parreno, A; Savage, M; Torok, A; Walker-Loud, A
2009-05-05
We present the results of an exploratory Lattice QCD calculation of three-baryon systems through a high-statistics study of one ensemble of anisotropic clover gauge-field configurations with a pion mass of m{sub {pi}} {approx} 390 MeV. Because of the computational cost of the necessary contractions, we focus on correlation functions generated by interpolating-operators with the quantum numbers of the {Xi}{sup 0}{Xi}{sup 0}n system, one of the least demanding three baryon systems in terms of the number of contractions. We find that the ground state of this system has an energy of E{sub {Xi}{sup 0}{Xi}{sup 0}n} = 3877.9 {+-} 6.9 {+-} 9.2 {+-} 3.3 MeV corresponding to an energy-shift due to interactions of {delta}E{sub {Xi}{sup 0}{Xi}{sup 0}n} = E{sub {Xi}{sup 0}{Xi}{sup 0}n} - 2M{sub {Xi}{sup 0}} - M{sub n} = 4.6 {+-} 5.0 {+-} 7.9 {+-} 4.2 MeV. There are a significant number of time-slices in the three-baryon correlation function for which the signal-to-noise ratio is only slowly degrading with time. This is in contrast to the exponential degradation of the signal-to-noise ratio that is observed at larger times, and is due to the suppressed overlap of the source and sink interpolating-operators that are associated with the variance of the three-baryon correlation function onto the lightest eigenstates in the lattice volume (mesonic systems). As one of the motivations for this area of exploration is the calculation of the structure and reactions of light nuclei, we also present initial results for a system with the quantum numbers of the triton (pnn). This present work establishes a path to multi-baryon systems, and shows that Lattice QCD calculations of the properties and interactions of systems containing four and five baryons are now within sight.
2. High Statistics Analysis using Anisotropic Clover Lattices: (II) Three-Baryon Systems
SciTech Connect
Andre Walker-Loud, Will Detmold, William Detmold, Aaron Torok, Konstantinos Orginos, Silas Beane, Tom Luu, Martin Savage, Assumpta Parreno
2009-10-01
We present the results of an exploratory Lattice QCD calculation of three-baryon systems through a high-statistics study of one ensemble of anisotropic clover gauge-field configurations with a pion mass of m_\\pi ~ 390 MeV. Because of the computational cost of the necessary contractions, we focus on correlation functions generated by interpolating-operators with the quantum numbers of the $\\Xi^0\\Xi^0 n$ system, one of the least demanding three baryon systems in terms of the number of contractions. We find that the ground state of this system has an energy of E_{\\Xi^0\\Xi^0n}= 3877.9\\pm 6.9\\pm 9.2\\pm3.3 MeV corresponding to an energy-shift due to interactions of \\delta E_{\\Xi^0\\Xi^0n}=E_{\\Xi^0\\Xi^0n}-2M_{\\Xi^0} -M_n=4.6\\pm 5.0\\pm 7.9\\pm 4.2 MeV. There are a significant number of time-slices in the three-baryon correlation function for which the signal-to-noise ratio is only slowly degrading with time. This is in contrast to the exponential degradation of the signal-to-noise ratio that is observed at larger times, and is due to the suppressed overlap of the source and sink interpolating-operators that are associated with the variance of the three-baryon correlation function onto the lightest eigenstates in the lattice volume (mesonic systems). As one of the motivations for this area of exploration is the calculation of the structure and reactions of light nuclei, we also present initial results for a system with the quantum numbers of the triton (pnn). This present work establishes a path to multi-baryon systems, and shows that Lattice QCD calculations of the properties and interactions of systems containing four and five baryons are now within sight.
3. Lattice thermal expansion and anisotropic displacements in -sulfur from diffraction experiments and first-principles theory.
PubMed
George, Janine; Deringer, Volker L; Wang, Ai; Müller, Paul; Englert, Ulli; Dronskowski, Richard
2016-12-21
Thermal properties of solid-state materials are a fundamental topic of study with important practical implications. For example, anisotropic displacement parameters (ADPs) are routinely used in physics, chemistry, and crystallography to quantify the thermal motion of atoms in crystals. ADPs are commonly derived from diffraction experiments, but recent developments have also enabled their first-principles prediction using periodic density-functional theory (DFT). Here, we combine experiments and dispersion-corrected DFT to quantify lattice thermal expansion and ADPs in crystalline α-sulfur (S8), a prototypical elemental solid that is controlled by the interplay of covalent and van der Waals interactions. We begin by reporting on single-crystal and powder X-ray diffraction measurements that provide new and improved reference data from 10 K up to room temperature. We then use several popular dispersion-corrected DFT methods to predict vibrational and thermal properties of α-sulfur, including the anisotropic lattice thermal expansion. Hereafter, ADPs are derived in the commonly used harmonic approximation (in the computed zero-Kelvin structure) and also in the quasi-harmonic approximation (QHA) which takes the predicted lattice thermal expansion into account. At the PPBE+D3(BJ) level, the QHA leads to excellent agreement with experiments. Finally, more general implications of this study for theory and experiment are discussed.
4. Quantum Domain Walls Induce Incommensurate Supersolid Phase on the Anisotropic Triangular Lattice
Zhang, Xue-Feng; Hu, Shijie; Pelster, Axel; Eggert, Sebastian
2016-11-01
We investigate the extended hard-core Bose-Hubbard model on the triangular lattice as a function of spatial anisotropy with respect to both hopping and nearest-neighbor interaction strength. At half-filling the system can be tuned from decoupled one-dimensional chains to a two-dimensional solid phase with alternating density order by adjusting the anisotropic coupling. At intermediate anisotropy, however, frustration effects dominate and an incommensurate supersolid phase emerges, which is characterized by incommensurate density order as well as an anisotropic superfluid density. We demonstrate that this intermediate phase results from the proliferation of topological defects in the form of quantum bosonic domain walls. Accordingly, the structure factor has peaks at wave vectors, which are linearly related to the number of domain walls in a finite system in agreement with extensive quantum Monte Carlo simulations. We discuss possible connections with the supersolid behavior in the high-temperature superconducting striped phase.
5. A study of symmetry restoration at finite temperature in the O(4) model using anisotropic lattices
Gavai, R. V.; Heller, U. M.; Karsch, F.; Plache, B.; Neuhaus, T.
Results of investigations of the O(4) spin model at finite temperature using anisotropic lattices are presented. In both the large N approximation and the numerical simulations using the Wolff cluster algorithm we find that the ratio of the symmetry restoration temperature TSR to the Higgs mass mH is independent of the anisotropy. We obtain a lower bound of 0.59 ± 0.04 for the ratio, T SR/m H, at m H ⋍ 0.5 , which is lowered furhter by about 10% at m Ha ⋍ 1 .
6. Anisotropic intrinsic lattice thermal conductivity of borophane from first-principles calculations.
PubMed
Liu, Gang; Wang, Haifeng; Gao, Yan; Zhou, Jian; Wang, Hui
2017-01-25
Borophene (boron sheet) as a new type of two-dimensional (2D) material was grown successfully recently. Unfortunately, the structural stability of freestanding borophene is still an open issue. Theoretical research has found that full hydrogenation can remove such instability, and the product is called borophane. In this paper, using first-principles calculations we investigate the lattice dynamics and thermal transport properties of borophane. The intrinsic lattice thermal conductivity and the relaxation time of borophane are investigated by solving the phonon Boltzmann transport equation (BTE) based on first-principles calculations. We find that the intrinsic lattice thermal conductivity of borophane is anisotropic, as the higher value (along the zigzag direction) is about two times of the lower one (along the armchair direction). The contributions of phonon branches to the lattice thermal conductivities along different directions are evaluated. It is found that both the anisotropy of thermal conductivity and the different phonon branches which dominate the thermal transport along different directions are decided by the group velocity and the relaxation time of phonons with very low frequency. In addition, the size dependence of thermal conductivity is investigated using cumulative thermal conductivity. The underlying physical mechanisms of these unique properties are also discussed in this paper.
7. Noncommutative anisotropic oscillator in a homogeneous magnetic field
Nath, D.; Roy, P.
2017-02-01
We study anisotropic oscillator in the presence of a homogeneous magnetic field and other related systems in the noncommutative plane. Energy values as function of the noncommutative parameter θ and the magnetic field B have been obtained. Some features of the spectrum, for example, formation of energy bands etc. have been examined. The effect of anisotropy on the energy levels has also been discussed.
8. Cluster Mott insulators and two Curie-Weiss regimes on an anisotropic kagome lattice
Chen, Gang; Kee, Hae-Young; Kim, Yong Baek
2016-06-01
Motivated by recent experiments on the quantum-spin-liquid candidate material LiZn2Mo3O8 , we study a single-band extended Hubbard model on an anisotropic kagome lattice with the 1/6 electron filling. Due to the partial filling of the lattice, the intersite repulsive interaction is necessary to generate Mott insulators, where electrons are localized in clusters rather than at lattice sites. It is shown that these cluster Mott insulators are generally U(1) quantum spin liquids with spinon Fermi surfaces. The nature of charge excitations in cluster Mott insulators can be quite different from conventional Mott insulator and we show that there exists a cluster Mott insulator where charge fluctuations around the hexagonal cluster induce a plaquette charge order (PCO). The spinon excitation spectrum in this spin-liquid cluster Mott insulator is reconstructed due to the PCO so that only 1/3 of the total spinon excitations are magnetically active. Based on these results, we propose that the two Curie-Weiss regimes of the spin susceptibility in LiZn2Mo3O8 may be explained by finite-temperature properties of the cluster Mott insulator with the PCO as well as fractionalized spinon excitations. Existing and possible future experiments on LiZn2Mo3O8 , and other Mo-based cluster magnets are discussed in light of these theoretical predictions.
9. Integrated digital inverters based on two-dimensional anisotropic ReS₂ field-effect transistors
DOE PAGES
Liu, Erfu; Fu, Yajun; Wang, Yaojia; ...
2015-05-07
Semiconducting two-dimensional transition metal dichalcogenides are emerging as top candidates for post-silicon electronics. While most of them exhibit isotropic behaviour, lowering the lattice symmetry could induce anisotropic properties, which are both scientifically interesting and potentially useful. Here we present atomically thin rhenium disulfide (ReS₂) flakes with unique distorted 1T structure, which exhibit in-plane anisotropic properties. We fabricated monolayer and few-layer ReS₂ field-effect transistors, which exhibit competitive performance with large current on/off ratios (~10⁷) and low subthreshold swings (100 mV per decade). The observed anisotropic ratio along two principle axes reaches 3.1, which is the highest among all known two-dimensional semiconductingmore » materials. Furthermore, we successfully demonstrated an integrated digital inverter with good performance by utilizing two ReS₂ anisotropic field-effect transistors, suggesting the promising implementation of large-scale two-dimensional logic circuits. Our results underscore the unique properties of two-dimensional semiconducting materials with low crystal symmetry for future electronic applications.« less
10. Integrated digital inverters based on two-dimensional anisotropic ReS₂ field-effect transistors
SciTech Connect
Liu, Erfu; Fu, Yajun; Wang, Yaojia; Feng, Yanqing; Liu, Huimei; Wan, Xiangang; Zhou, Wei; Wang, Baigeng; Shao, Lubin; Ho, Ching -Hwa; Huang, Ying -Sheng; Cao, Zhengyi; Wang, Laiguo; Li, Aidong; Zeng, Junwen; Song, Fengqi; Wang, Xinran; Shi, Yi; Yuan, Hongtao; Hwang, Harold Y.; Cui, Yi; Miao, Feng; Xing, Dingyu
2015-05-07
Semiconducting two-dimensional transition metal dichalcogenides are emerging as top candidates for post-silicon electronics. While most of them exhibit isotropic behaviour, lowering the lattice symmetry could induce anisotropic properties, which are both scientifically interesting and potentially useful. Here we present atomically thin rhenium disulfide (ReS₂) flakes with unique distorted 1T structure, which exhibit in-plane anisotropic properties. We fabricated monolayer and few-layer ReS₂ field-effect transistors, which exhibit competitive performance with large current on/off ratios (~10⁷) and low subthreshold swings (100 mV per decade). The observed anisotropic ratio along two principle axes reaches 3.1, which is the highest among all known two-dimensional semiconducting materials. Furthermore, we successfully demonstrated an integrated digital inverter with good performance by utilizing two ReS₂ anisotropic field-effect transistors, suggesting the promising implementation of large-scale two-dimensional logic circuits. Our results underscore the unique properties of two-dimensional semiconducting materials with low crystal symmetry for future electronic applications.
11. Topological spin liquids in the ruby lattice with anisotropic Kitaev interactions
Jahromi, Saeed S.; Kargarian, Mehdi; Masoudi, S. Farhad; Langari, Abdollah
2016-09-01
The ruby lattice is a four-valent lattice interpolating between honeycomb and triangular lattices. In this work we investigate the topological spin-liquid phases of a spin Hamiltonian with Kitaev interactions on the ruby lattice using exact diagonalization and perturbative methods. The latter interactions combined with the structure of the lattice yield a model with Z2×Z2 gauge symmetry. We mapped out the phase diagram of the model and found gapped and gapless spin-liquid phases. While the low-energy sector of the gapped phase corresponds to the well-known topological color code model on a honeycomb lattice, the low-energy sector of the gapless phases is described by an effective spin model with three-body interactions on a triangular lattice. A gap is opened in the spectrum in small magnetic fields, where we showed that the ground state has a finite topological entanglement entropy. We argue that the gapped phases could be possibly described by exotic excitations, and their corresponding spectrum is richer than the Ising phase of the Kitaev model.
12. Anomalies, gauge field topology, and the lattice
SciTech Connect
Creutz, Michael
2011-04-15
Motivated by the connection between gauge field topology and the axial anomaly in fermion currents, I suggest that the fourth power of the naive Dirac operator can provide a natural method to define a local lattice measure of topological charge. For smooth gauge fields this reduces to the usual topological density. For typical gauge field configurations in a numerical simulation, however, quantum fluctuations dominate, and the sum of this density over the system does not generally give an integer winding. On cooling with respect to the Wilson gauge action, instanton like structures do emerge. As cooling proceeds, these objects tend shrink and finally 'fall through the lattice.' Modifying the action can block the shrinking at the expense of a loss of reflection positivity. The cooling procedure is highly sensitive to the details of the initial steps, suggesting that quantum fluctuations induce a small but fundamental ambiguity in the definition of topological susceptibility.
13. Simulation of flow of mixtures through anisotropic porous media using a lattice Boltzmann model.
PubMed
Mendoza, M; Wittel, F K; Herrmann, H J
2010-08-01
We propose a description for transient penetration simulations of miscible and immiscible fluid mixtures into anisotropic porous media, using the lattice Boltzmann (LB) method. Our model incorporates hydrodynamic flow, advection-diffusion, surface tension, and the possibility for global and local viscosity variations to consider various types of hardening fluids. The miscible mixture consists of two fluids, one governed by the hydrodynamic equations and one by advection-diffusion equations. We validate our model on standard problems like Poiseuille flow, the collision of a drop with an impermeable, solid interface and the deformation of the fluid due to surface tension forces. To demonstrate the applicability to complex geometries, we simulate the invasion process of mixtures into wood spruce samples.
14. A quantum fidelity study of the anisotropic next-nearest-neighbour triangular lattice Heisenberg model.
PubMed
Thesberg, Mischa; Sørensen, Erik S
2014-10-22
Ground- and excited-state quantum fidelities in combination with generalized quantum fidelity susceptibilites, obtained from exact diagonalizations, are used to explore the phase diagram of the anisotropic next-nearest-neighbour triangular Heisenberg model. Specifically, the J'-J2 plane of this model, which connects the J1-J2 chain and the anisotropic triangular lattice Heisenberg model, is explored using these quantities. Through the use of a quantum fidelity associated with the first excited-state, in addition to the conventional ground-state fidelity, the BKT-type transition and Majumdar-Ghosh point of the J1-J2 chain (J'=0) are found to extend into the J'-J2 plane and connect with points on the J2=0 axis thereby forming bounded regions in the phase diagram. These bounded regions are then explored through the generalized quantum fidelity susceptibilities χρ, χ₁₂₀°, χD and χCAF which are associated with the spin stiffness, 120° spiral order parameter, dimer order parameter and collinear antiferromagnetic order parameter respectively. These quantities are believed to be extremely sensitive to the underlying phase and are thus well suited for finite-size studies. Analysis of the fidelity susceptibilities suggests that the J', J2≪J phase of the anisotropic triangular model is either a collinear antiferromagnet or possibly a gapless disordered phase that is directly connected to the Luttinger phase of the J1-J2 chain. Furthermore, the outer region is dominated by incommensurate spiral physics as well as dimer order.
15. Anisotropic lattice thermal diffusivity in olivines and pyroxenes to high temperatures
Harrell, Michael Damian
The anisotropic lattice thermal diffusivity of three olivines (Fo 0, Fo78, and Fo91), one orthopyroxene (En 91), and one clinopyroxene (Di72He9Jd3Cr 3Ts12) have been measured via impulsive stimulated light scattering, permitting the calculation of their lattice thermal diffusivity tensors to high temperatures. For Fo0 olivine, measurements extend from room temperature to 600°C, for Fo78 to 900°C, and for Fo91 to 1000°C, all in steps of 100°C. The orthopyroxene also was taken in steps to 1000°C, while the clinopyroxene was measured at room temperature. A limited set of room-temperature measurements to 5 GPa on a fourth olivine (Fo89) is also included. Diffusivities have been combined with calculations of density and specific heat to determine the lattice thermal conductivity tensors. An earlier theory that explains the observed behavior in terms of a positive lower bound on the phonon mean free path is discussed, and the data are used to constrain a model of thermal conductivity at high temperature. The relative contributions of optic and acoustic modes are evaluated from analysis of published dispersion curves. Five conclusions are reached: First, the anisotropy of lattice thermal conductivity remains essentially unchanged over the observed range of temperatures, indicating that anisotropy remains significant under upper-mantle conditions, and, in regions displaying preferred alignment, may account for observed lateral variations in the geotherm. Second, thermal conductivity departs significantly from earlier predictions of its temperature dependence; this may be understood in terms of a phonon mean free path that cannot diminish below 1.75 times the mean interatomic spacing. Third, for olivine, the optic modes have group velocities that are approximately one-third those of the acoustic modes, and do not dominate lattice conduction despite their greater number. Fourth, impurity scattering is significant along the olivine Fe-Mg solid solution series, but is not
16. Anisotropic thermal conduction with magnetic fields in galaxy clusters
Arth, Alexander; Dolag, Klaus; Beck, Alexander; Petkova, Margarita; Lesch, Harald
2015-08-01
Magnetic fields play an important role for the propagation and diffusion of charged particles, which are responsible for thermal conduction. In this poster, we present an implementation of thermal conduction including the anisotropic effects of magnetic fields for smoothed particle hydrodynamics (SPH). The anisotropic thermal conduction is mainly proceeding parallel to magnetic fields and suppressed perpendicular to the fields. We derive the SPH formalism for the anisotropic heat transport and solve the corresponding equation with an implicit conjugate gradient scheme. We discuss several issues of unphysical heat transport in the cases of extreme ansiotropies or unmagnetized regions and present possible numerical workarounds. We implement our algorithm into the cosmological simulation code GADGET and study its behaviour in several test cases. In general, we reproduce the analytical solutions of our idealised test problems, and obtain good results in cosmological simulations of galaxy cluster formations. Within galaxy clusters, the anisotropic conduction produces a net heat transport similar to an isotropic Spitzer conduction model with low efficiency. In contrast to isotropic conduction our new formalism allows small-scale structure in the temperature distribution to remain stable, because of their decoupling caused by magnetic field lines. Compared to observations, strong isotropic conduction leads to an oversmoothed temperature distribution within clusters, while the results obtained with anisotropic thermal conduction reproduce the observed temperature fluctuations well. A proper treatment of heat transport is crucial especially in the outskirts of clusters and also in high density regions. It's connection to the local dynamical state of the cluster also might contribute to the observed bimodal distribution of cool core and non cool core clusters. Our new scheme significantly advances the modelling of thermal conduction in numerical simulations and overall gives
17. Pairing symmetries in a Hubbard model on an anisotropic triangular lattice
Watanabe, Tsutomu; Yokoyama, Hisatoshi; Tanaka, Yukio; Inoue, Jun-ichiro
2007-10-01
To consider the paring symmetry formed in organic compounds κ-(BEDT-TTF)2X, we study the half-filled-band Hubbard model on an anisotropic triangular lattice (t in two bond directions and t‧ in the other), using an optimization VMC method. As trial states, we adopt a coexisting state of an antiferromagnetic (AF) order and the dx2-y2 -wave RVB gap, in addition to the d + id- and d + d-wave gap states. In these states, we take account of the effect of band (or Fermi surface) renormalization. Magnetic Mott transitions occur, and a regime of robust superconductivity could not be found, in contrast with our previous study. In the insulating regime, the coexisting state in which an AF order prevails is always the lowest-energy state up to remarkably large t‧/t (≲1.3), whereas a dxy-wave RVB state becomes predominant when t‧/t exceeds this value. In the insulating regime, the effective Fermi surface, determined by the renormalized value t˜‧ / t , is markedly renormalized into different directions according to t‧/t; for t‧/t ≲ 1.3, it approaches that of the square lattice (t˜‧ / t = 0) , whereas for t‧/t ≳ 1.3, it becomes almost one-dimensional (t˜‧ / t≫ 1) .
18. Graphene, Lattice Field Theory and Symmetries
SciTech Connect
Drissi, L. B.; Bousmina, M.; Saidi, E. H.
2011-02-15
Borrowing ideas from tight binding model, we propose a board class of lattice field models that are classified by non simply laced Lie algebras. In the case of A{sub N-1{approx_equal}}su(N) series, we show that the couplings between the quantum states living at the first nearest neighbor sites of the lattice L{sub suN} are governed by the complex fundamental representations N-bar and N of su(N) and the second nearest neighbor interactions are described by its adjoint N-bar x N. The lattice models associated with the leading su(2), su(3), and su(4) cases are explicitly studied and their fermionic field realizations are given. It is also shown that the su(2) and su(3) models describe the electronic properties of the acetylene chain and the graphene, respectively. It is established as well that the energy dispersion of the first nearest neighbor couplings is completely determined by the A{sub N} roots {alpha} through the typical dependence N/2+{Sigma}{sub roots} cos(k.{alpha} with k the wave vector.Other features such as the SO(2N) extension and other applications are also discussed.
19. Synthetic magnetic field effects on neutral bosonic condensates in quasi-three-dimensional anisotropic layered structures
SciTech Connect
Zaleski, T. A.; Polak, T. P.
2011-02-15
We discuss a system of dilute Bose gas confined in a layered structure of stacked square lattices (slab geometry). A derived phase diagram reveals a nonmonotonic dependence of the ratio of tunneling to on-site repulsion on the artificial magnetic field applied to the system. The effect is reduced when more layers are added, which mimics a two- to quasi-three-dimensional geometry crossover. Furthermore, we establish a correspondence between anisotropic infinite (quasi-three-dimensional) and isotropic finite (slab geometry) systems that share exactly the same critical values, which can be an important clue for choosing experimental setups that are less demanding, but still leading to the identical results. Finally, we show that the properties of the ideal Bose gas in a three-dimensional optical lattice can be closely mimicked by finite (slab) systems when the number of two-dimensional layers is larger than 10 for isotropic interactions, or even less when the layers are weakly coupled.
20. Anisotropic magnetic particles in a magnetic field
PubMed Central
Martchenko, Ilya; Mihut, Adriana M.; Bialik, Erik; Hirt, Ann M.; Rufier, Chantal; Menzel, Andreas; Dietsch, Hervé; Linse, Per
2016-01-01
We characterize the structural properties of magnetic ellipsoidal hematite colloids with an aspect ratio ρ ≈ 2.3 using a combination of small-angle X-ray scattering and computer simulations. The evolution of the phase diagram with packing fraction φ and the strength of an applied magnetic field B is described, and the coupling between orientational order of magnetic ellipsoids and the bulk magnetic behavior of their suspension addressed. We establish quantitative structural criteria for the different phase and arrest transitions and map distinct isotropic, polarized non-nematic, and nematic phases over an extended range in the φ–B coordinates. We show that upon a rotational arrest of the ellipsoids around φ = 0.59, the bulk magnetic behavior of their suspension switches from superparamagnetic to ordered weakly ferromagnetic. If densely packed and arrested, these magnetic particles thus provide persisting remanent magnetization of the suspension. By exploring structural and magnetic properties together, we extend the often used colloid-atom analogy to the case of magnetic spins. PMID:27722439
1. Extracting Nucleon Magnetic Moments and Electric Polarizabilities from Lattice QCD in Background Electric Fields
SciTech Connect
William Detmold; Tiburzi, Brian C.; Walker-Loud, Andre
2010-03-01
Nucleon properties are investigated in background electric fields. As the magnetic moments of baryons affect their relativistic propagation in constant electric fields, electric polarizabilities cannot be determined without knowledge of magnetic moments. We devise combinations of baryon two-point functions in external electric fields to isolate both observables. Using an ensemble of anisotropic gauge configurations with dynamical clover fermions, we demonstrate how magnetic moments and electric polarizabilities can be determined from lattice QCD simulations in background electric fields. We obtain results for both the neutron and proton. Our study is currently limited to electrically neutral sea quarks.
2. Improved gauge actions on anisotropic lattices I. Study of fundamental parameters in the weak coupling limit
Sakai, S.; Saito, T.; Nakamura, A.
2000-09-01
On anisotropic lattices with the anisotropy ξ=a σ/a τ the following basic parameters are calculated by perturbative method: (1) the renormalization of the gauge coupling in spatial and temporal directions, g σ and g τ, (2) the Λ parameter, (3) the ratio of the renormalized and bare anisotropy η=ξ/ξ B and (4) the derivatives of the coupling constants with respect to ξ, ∂g σ-2/∂ξ and ∂g τ-2/∂ξ . We employ the improved gauge actions which consist of plaquette and six-link rectangular loops, c 0P(1×1) μν+c 1P(1×2) μν. This class of actions covers Symanzik, Iwasaki and DBW2 actions. The ratio η shows an impressive behavior as a function of c 1, i.e., η>1 for the standard Wilson and Symanzik actions, while η<1 for Iwasaki and DBW2 actions. This is confirmed non-perturbatively by numerical simulations in weak coupling regions. The derivatives ∂g -2τ/∂ξ and ∂g -2σ/∂ξ also change sign as -c 1 increases. For Iwasaki and DBW2 actions they become opposite sign to those for standard and Symanzik actions. However, their sum is independent of the type of actions due to Karsch's sum rule.
3. Magnetic-Field-Induced Insulator-Conductor Transition in SU(2) Quenched Lattice Gauge Theory
SciTech Connect
Buividovich, P.V.; Kharzeev, D.; Chernodub, M.N., Kalaydzhyan, T., Luschevskaya, E.V., and M.I. Polikarpov
2010-09-24
We study the correlator of two vector currents in quenched SU(2) lattice gauge theory with a chirally invariant lattice Dirac operator with a constant external magnetic field. It is found that in the confinement phase the correlator of the components of the current parallel to the magnetic field decays much slower than in the absence of a magnetic field, while for other components the correlation length slightly decreases. We apply the maximal entropy method to extract the corresponding spectral function. In the limit of zero frequency this spectral function yields the electric conductivity of quenched theory. We find that in the confinement phase the external magnetic field induces nonzero electric conductivity along the direction of the field, transforming the system from an insulator into an anisotropic conductor. In the deconfinement phase the conductivity does not exhibit any sizable dependence on the magnetic field.
4. Magnetic field-induced changes of lattice parameters and thermal expansion behavior of the CoMnSi compound
SciTech Connect
Kou, R. H.; Gao, J.; Wang, G.; Liu, Y. D.; Wang, Y. D.; Ren, Y.; Brown, D. E.
2016-02-01
The crystal structure of the CoMnSi compound during zero-field cooling and field cooling from room temperature down to 200 K was studied using the synchrotron radiation X-ray diffraction technique. The results show that the lattice parameters and thermal expansion behavior of the sample are changed by the applied magnetic fields. The lattice contracts along the a axis, but expands along the b and c axes. Due to enlarged and anisotropic changes under a magnetic field of 6 T, the lattice shows an invar-like behavior along all three axes. Critical interatomic distances and bond angles also show large changes under the influence of such a high magnetic field. These magnetic field-induced changes of the lattice are discussed with respect to their contributions to the large magnetocaloric effect of the CoMnSi compound.
5. Extracting nucleon magnetic moments and electric polarizabilities from lattice QCD in background electric fields
SciTech Connect
Detmold, W.; Tiburzi, B. C.; Walker-Loud, A.
2010-03-01
Nucleon properties are investigated in background electric fields. As the magnetic moments of baryons affect their relativistic propagation in constant electric fields, electric polarizabilities cannot be determined without knowledge of magnetic moments. This is analogous to the experimental situation, for which determination of polarizabilities from the Compton amplitude requires subtraction of Born terms. With the background field method, we devise combinations of nucleon correlation functions in constant electric fields that isolate magnetic moments and electric polarizabilities. Using an ensemble of anisotropic gauge configurations with dynamical clover fermions, we demonstrate how both observables can be determined from lattice QCD simulations in background electric fields. We obtain results for the neutron and proton, however, our study is currently limited to electrically neutral sea quarks. The value we extract for the nucleon isovector magnetic moment is comparable to those obtained from measuring lattice three-point functions at similar pion masses.
6. Formation of magnetically anisotropic composite films at low magnetic fields
Ghazi Zahedi, Maryam; Ennen, Inga; Marchi, Sophie; Barthel, Markus J.; Hütten, Andreas; Athanassiou, Athanassia; Fragouli, Despina
2017-04-01
We present a straightforward two-step technique for the fabrication of poly (methyl methacrylate) composites with embedded aligned magnetic chains. First, ferromagnetic microwires are realized in a poly (methyl methacrylate) solution by assembling iron nanoparticles in a methyl methacrylate solution under heat in an external magnetic field of 160 mT. The simultaneous thermal polymerization of the monomer throughout the wires is responsible for their permanent linkage and stability. Next, the polymer solution containing the randomly dispersed microwires is casted on a solid substrate in the presence of a low magnetic field (20–40 mT) which induces the final alignment of the microwires into long magnetic chains upon evaporation of the solvent. We prove that the presence of the nanoparticles assembled in the form of microwires is a key factor for the formation of the anisotropic films under low magnetic fields. In fact, such low fields are not capable of driving and assembling dispersed magnetic nanoparticles in the same type of polymer solutions. Hence, this innovative approach can be utilized for the synthesis of magnetically anisotropic nanocomposite films at low magnetic fields.
7. Clocking the anisotropic lattice dynamics of multi-walled carbon nanotubes by four-dimensional ultrafast transmission electron microscopy.
PubMed
Cao, Gaolong; Sun, Shuaishuai; Li, Zhongwen; Tian, Huanfang; Yang, Huaixin; Li, Jianqi
2015-02-12
Recent advances in the four-dimensional ultrafast transmission electron microscope (4D-UTEM) with combined spatial and temporal resolutions have made it possible to directly visualize structural dynamics of materials at the atomic level. Herein, we report on our development on a 4D-UTEM which can be operated properly on either the photo-emission or the thermionic mode. We demonstrate its ability to obtain sequences of snapshots with high spatial and temporal resolutions in the study of lattice dynamics of the multi-walled carbon nanotubes (MWCNTs). This investigation provides an atomic level description of remarkable anisotropic lattice dynamics at the picosecond timescales. Moreover, our UTEM measurements clearly reveal that distinguishable lattice relaxations appear in intra-tubular sheets on an ultrafast timescale of a few picoseconds and after then an evident lattice expansion along the radial direction. These anisotropic behaviors in the MWCNTs are considered arising from the variety of chemical bonding, i.e. the weak van der Waals bonding between the tubular planes and the strong covalent sp(2)-hybridized bonds in the tubular sheets.
8. Clocking the anisotropic lattice dynamics of multi-walled carbon nanotubes by four-dimensional ultrafast transmission electron microscopy
Cao, Gaolong; Sun, Shuaishuai; Li, Zhongwen; Tian, Huanfang; Yang, Huaixin; Li, Jianqi
2015-02-01
Recent advances in the four-dimensional ultrafast transmission electron microscope (4D-UTEM) with combined spatial and temporal resolutions have made it possible to directly visualize structural dynamics of materials at the atomic level. Herein, we report on our development on a 4D-UTEM which can be operated properly on either the photo-emission or the thermionic mode. We demonstrate its ability to obtain sequences of snapshots with high spatial and temporal resolutions in the study of lattice dynamics of the multi-walled carbon nanotubes (MWCNTs). This investigation provides an atomic level description of remarkable anisotropic lattice dynamics at the picosecond timescales. Moreover, our UTEM measurements clearly reveal that distinguishable lattice relaxations appear in intra-tubular sheets on an ultrafast timescale of a few picoseconds and after then an evident lattice expansion along the radial direction. These anisotropic behaviors in the MWCNTs are considered arising from the variety of chemical bonding, i.e. the weak van der Waals bonding between the tubular planes and the strong covalent sp2-hybridized bonds in the tubular sheets.
9. Bicritical universality of the anisotropic Heisenberg model in a crystal field.
PubMed
Freire, R T S; Plascak, J A
2015-03-01
The bicritical properties of the three-dimensional classical anisotropic Heisenberg model in a crystal field are investigated through extensive Monte Carlo simulations on a simple cubic lattice, using Metropolis and Wolff algorithms. Field-mixing and multidimensional histogram techniques were employed in order to compute the probability distribution function of the extensive conjugate variables of interest and, using finite-size scaling analysis, the first-order transition line of the model was precisely located. The fourth-order cumulant of the order parameter was then calculated along this line and the bicritical point located with good precision from the cumulant crossings. The bicritical properties of this point were further investigated through the measurement of the universal probability distribution function of the order parameter. The results lead us to conclude that the studied bicritical point belongs in fact to the three-dimensional Heisenberg universality class.
10. Paramagnetic relaxation in anisotropic materials in zero and weak constant fields
SciTech Connect
Fokina, N. P.; Khalvashi, E. Kh.; Khutsishvili, K. O.
2014-12-21
Paramagnetic relaxation in strongly anisotropic materials is analytically investigated in zero and weak constant magnetic fields. The objectives of the microscopic analytical investigation are (i) the weak-field electron paramagnetic resonance (EPR) linewidth and (ii) the electron spin relaxation rates given by a calorimetric Gorter type experiment in the zero constant field at the arbitrary low-frequency field directions, respectively, to the sample crystallographic axes. The EPR linewidth is calculated under the suggestion of its spin-phonon nature at the one-phonon mechanism of the spin-lattice relaxation in the case of the strong isotropic exchange interaction for the arbitrary direction Z of the constant magnetic field. The EPR linewidth is presented as the half sum of the zero-field relaxation rates, measured by the Gorter experiment with the low-frequency field oriented along the X, Y axes. With the help of the macroscopic consideration, it is shown that the zero-field relaxation rates describe the relaxation of the X and Y magnetization components in a zero or weak constant magnetic field. The relaxation rates of the magnetizations created along a,b,c crystallographic axes by a low-frequency field in a Gorter type experiment follow the obtained expressions in the particular cases and are in the experimentally confirmed relations with the EPR linewidth.
11. High field dielectric properties of anisotropic polymer-ceramic composites
SciTech Connect
Tomer, V.; Randall, C. A.
2008-10-01
Using dielectrophoretic assembly, we create anisotropic composites of BaTiO{sub 3} particles in a silicone elastomer thermoset polymer. We study a variety of electrical properties in these composites, i.e., permittivity, dielectric breakdown, and energy density as function of ceramic volume fraction and connectivity. The recoverable energy density of these electric-field-structured composites is found to be highly dependent on the anisotropy present in the system. Our results indicate that x-y-aligned composites exhibit higher breakdown strengths along with large recoverable energy densities when compared to 0-3 composites. This demonstrates that engineered anisotropy can be employed to control dielectric breakdown strengths and nonlinear conduction at high fields in heterogeneous systems. Consequently, manipulation of anisotropy in high-field dielectric properties can be exploited for the development of high energy density polymer-ceramic systems.
12. Anisotropic non-Gaussianity from a two-form field
Ohashi, Junko; Soda, Jiro; Tsujikawa, Shinji
2013-04-01
We study an inflationary scenario with a two-form field to which an inflaton couples nontrivially. First, we show that anisotropic inflation can be realized as an attractor solution and that the two-form hair remains during inflation. A statistical anisotropy can be developed because of a cumulative anisotropic interaction induced by the background two-form field. The power spectrum of curvature perturbations has a prolate-type anisotropy, in contrast to the vector models having an oblate-type anisotropy. We also evaluate the bispectrum and trispectrum of curvature perturbations by employing the in-in formalism based on the interacting Hamiltonians. We find that the nonlinear estimators fNL and τNL are correlated with the amplitude g* of the statistical anisotropy in the power spectrum. Unlike the vector models, both fNL and τNL vanish in the squeezed limit. However, the estimator fNL can reach the order of 10 in the equilateral and enfolded limits. These results are consistent with the latest bounds on fNL constrained by Planck.
13. Lattice thermal expansion and anisotropic displacements in 𝜶-sulfur from diffraction experiments and first-principles theory
George, Janine; Deringer, Volker L.; Wang, Ai; Müller, Paul; Englert, Ulli; Dronskowski, Richard
2016-12-01
Thermal properties of solid-state materials are a fundamental topic of study with important practical implications. For example, anisotropic displacement parameters (ADPs) are routinely used in physics, chemistry, and crystallography to quantify the thermal motion of atoms in crystals. ADPs are commonly derived from diffraction experiments, but recent developments have also enabled their first-principles prediction using periodic density-functional theory (DFT). Here, we combine experiments and dispersion-corrected DFT to quantify lattice thermal expansion and ADPs in crystalline α-sulfur (S8), a prototypical elemental solid that is controlled by the interplay of covalent and van der Waals interactions. We begin by reporting on single-crystal and powder X-ray diffraction measurements that provide new and improved reference data from 10 K up to room temperature. We then use several popular dispersion-corrected DFT methods to predict vibrational and thermal properties of α-sulfur, including the anisotropic lattice thermal expansion. Hereafter, ADPs are derived in the commonly used harmonic approximation (in the computed zero-Kelvin structure) and also in the quasi-harmonic approximation (QHA) which takes the predicted lattice thermal expansion into account. At the PPBE+D3(BJ) level, the QHA leads to excellent agreement with experiments. Finally, more general implications of this study for theory and experiment are discussed.
14. Subsurface Stress Fields in FCC Single Crystal Anisotropic Contacts
NASA Technical Reports Server (NTRS)
Arakere, Nagaraj K.; Knudsen, Erik; Swanson, Gregory R.; Duke, Gregory; Ham-Battista, Gilda
2004-01-01
15. Field dependent spin transport of anisotropic Heisenberg chain
Rezania, H.
2016-04-01
We have addressed the static spin conductivity and spin Drude weight of one-dimensional spin-1/2 anisotropic antiferromagnetic Heisenberg chain in the finite magnetic field. We have investigated the behavior of transport properties by means of excitation spectrum in terms of a hard core bosonic representation. The effect of in-plane anisotropy on the spin transport properties has also been studied via the bosonic model by Green's function approach. This anisotropy is considered for exchange constants that couple spin components perpendicular to magnetic field direction. We have found the temperature dependence of the spin conductivity and spin Drude weight in the gapped field induced spin-polarized phase for various magnetic field and anisotropy parameters. Furthermore we have studied the magnetic field dependence of static spin conductivity and Drude weight for various anisotropy parameters. Our results show the regular part of spin conductivity vanishes in isotropic case however Drude weight has a finite non-zero value and the system exhibits ballistic transport properties. We also find the peak in the static spin conductivity factor moves to higher temperature upon increasing the magnetic field at fixed anisotropy. The static spin conductivity is found to be monotonically decreasing with magnetic field due to increase of energy gap in the excitation spectrum. Furthermore we have studied the temperature dependence of spin Drude weight for different magnetic field and various anisotropy parameters.
16. Entanglement production due to quench dynamics of an anisotropic XY chain in a transverse field
Sengupta, K.; Sen, Diptiman
2009-09-01
We compute concurrence and negativity as measures of two-spin entanglement generated by a power-law quench (characterized by a rate τ-1 and an exponent α ) which takes an anisotropic XY chain in a transverse field through a quantum critical point (QCP). We show that only spins separated by an even number of lattice spacings get entangled in such a process. Moreover, there is a critical rate of quench, τc-1 , above which no two-spin entanglement is generated; the entire entanglement is multipartite. The ratio of the entanglements between consecutive even neighbors can be tuned by changing the quench rate. We also show that for large τ , the concurrence (negativity) scales as α/τ (α/τ) , and we relate this scaling behavior to defect production by the quench through a QCP.
17. Subsurface Stress Fields In Single Crystal (Anisotropic) Contacts
NASA Technical Reports Server (NTRS)
Arakere, Nagaraj K.; Knudsen, Erik C.; Duke, Greg; Battista, Gilda; Swanson, Greg
2004-01-01
18. Investigation of the Geometry-Dependent Anisotropic Material Behavior of Filigree Struts in ALM-Produced Lattice Structures
Reinhart, Gunther; Teufelhart, Stefan; Riss, Fabian
Additive Layer Manufacturing (ALM) shows a great potential for the production of lightweight designed components. The use of lattice structures is one of the most common approaches for lightweight design in ALM because they show a high stiffness and strength combined with a small mass. To reach an optimum design, these structures should not have a periodical build up, but have to be optimized concerning their course and the strut's diameters. For the load dependent adaption of the diameters, the material properties of such filigree structures have to be known well. This geometry-dependent, anisotropic material behavior is described in the following paper.
19. Minimally coupled scalar field cosmology in anisotropic cosmological model
Singh, C. P.; Srivastava, Milan
2017-02-01
We study a spatially homogeneous and anisotropic cosmological model in the Einstein gravitational theory with a minimally coupled scalar field. We consider a non-interacting combination of scalar field and perfect fluid as the source of matter components which are separately conserved. The dynamics of cosmic scalar fields with a zero rest mass and an exponential potential are studied, respectively. We find that both assumptions of potential along with the average scale factor as an exponential function of scalar field lead to the logarithmic form of scalar field in each case which further gives power-law form of the average scale factor. Using these forms of the average scale factor, exact solutions of the field equations are obtained to the metric functions which represent a power-law and a hybrid expansion, respectively. We find that the zero-rest-mass model expands with decelerated rate and behaves like a stiff matter. In the case of exponential potential function, the model decelerates, accelerates or shows the transition depending on the parameters. The isotropization is observed at late-time evolution of the Universe in the exponential potential model.
20. Breakdown of antiferromagnetism and the Coulomb phase for RVB states on anisotropic three-dimensional lattices
Beach, K. S. D.
2015-03-01
Nearest-neighbor (NN) resonating-valence-bond (RVB) wave functions often serve as prototype ground states for various frustrated models in two dimensions because of their lack of long-range magnetic correlations. In three dimensions, these states are generally not featureless, and their tendency is toward antiferromagnetic order. On the cubic and diamond lattices, for example, the NN RVB state exhibits both antiferromagnetism and power law dimer correlations characteristic of the Coulomb phase'' (in analogy with classical hardcore dimer models). The introduction of strong spatial anisotropy, however, leads to the destruction of these long-range and algebraic correlations, leaving behind an apparent short-range spin liquid state. We characterize the critical exponents at the phase boundaries for wave functions built from products of SU(2) singlets as well as their SU(N) generalizations and discuss attempts to construct a field theory that describes the transitions.
1. Spin-liquid and magnetic phases in the anisotropic triangular lattice: The case of κ-(ET)2X
Tocchio, Luca F.; Parola, Alberto; Gros, Claudius; Becca, Federico
2009-08-01
The two-dimensional Hubbard model on the anisotropic triangular lattice, with two different hopping amplitudes t and t' , is relevant to describe the low-energy physics of κ-(ET)2X , a family of organic salts. The ground-state properties of this model are studied by using Monte Carlo techniques, on the basis of a recent definition of backflow correlations for strongly correlated lattice systems. The results show that there is no magnetic order for reasonably large values of the electron-electron interaction U and frustrating ratio t'/t=0.85 , suitable to describe the nonmagnetic compound with X=Cu2(CN)3 . On the contrary, Néel order takes place for weaker frustrations, i.e., t'/ ttilde 0.4-0.6 , suitable for materials with X=Cu2(SCN)2 , Cu[N(CN)2]Cl , or Cu[N(CN)2]Br .
2. Spin-liquid and magnetic phases in the anisotropic triangular lattice: the case of κ-(ET)2X
Becca, Federico; Tocchio, Luca; Parola, Alberto; Gros, Claudius
2010-03-01
The two-dimensional Hubbard model on the anisotropic triangular lattice, with two different hopping amplitudes t and t^', is relevant to describe the low-energy physics of κ-(ET)2X, a family of organic salts. The ground-state properties of this model are studied by using Monte Carlo techniques, on the basis of a recent definition of backflow correlations for strongly-correlated lattice systems. The results show that there is no magnetic order for reasonably large values of the electron-electron interaction U and frustrating ratio t^'/t = 0.85, suitable to describe the non-magnetic compound with X=Cu2(CN)3. On the contrary, N'eel order takes place for weaker frustrations, i.e., t^'/t ˜0.4 0.6, suitable for materials with X=Cu2(SCN)2, Cu[N(CN)2]Cl, or Cu[N(CN)2]Br.
3. A lower bound on {T SR}/{m H} in the O(4) model on anisotropic lattices
Gavai, R. V.; Heller, U. M.; Karsch, F.; Neuhaus, T.; Plache, B.
1992-11-01
Results of an investigation of the O(4) spin model at finite temperature using anisotropic lattices are presented. In both the large N approximation and numerical simulations using the Wolff cluster algorithm we find that the ratio of the symmetry restoration temperature TSR to the Higgs mass mH is independent of the anisotropy ξ. From the numerical simulations we obtain a lower bound of {T SR}/{m H}⋍ 0.58 ± 0.02 at a value for the Higgs mass m Ha s ⋍ 0.5 , which is lowered further by about 10% at m Ha s ⋍ 1 . Requiring certain timelike correlation functions to coincide with their spacelike counterparts, quantum and scaling corrections to the anisotropy are determined and are found to be small i.e., the anisotropy is found to be close to the ratio of spacelike and timelike lattice spacings.
4. Fernique-type inequalities and moduli of continuity for anisotropic Gaussian random fields
PubMed Central
Meerschaert, Mark M.; Wang, Wensheng; Xiao, Yimin
2013-01-01
This paper is concerned with sample path properties of anisotropic Gaussian random fields. We establish Fernique-type inequalities and utilize them to study the global and local moduli of continuity for anisotropic Gaussian random fields. Applications to fractional Brownian sheets and to the solutions of stochastic partial differential equations are investigated. PMID:24825922
5. YANG-MILLS FIELDS AND THE LATTICE.
SciTech Connect
CREUTZ,M.
2004-05-18
The Yang-Mills theory lies at the heart of our understanding of elementary particle interactions. For the strong nuclear forces, we must understand this theory in the strong coupling regime. The primary technique for this is the lattice. While basically an ultraviolet regulator, the lattice avoids the use of a perturbative expansion. I discuss some of the historical circumstances that drove us to this approach, which has had immense success, convincingly demonstrating quark confinement and obtaining crucial properties of the strong interactions from first principles.
6. Elasto-viscoplastic phase field modelling of anisotropic cleavage fracture
Shanthraj, P.; Svendsen, B.; Sharma, L.; Roters, F.; Raabe, D.
2017-02-01
A finite-strain anisotropic phase field method is developed to model the localisation of damage on a defined family of crystallographic planes, characteristic of cleavage fracture in metals. The approach is based on the introduction of an undamaged configuration, and the inelastic deformation gradient mapping this configuration to a damaged configuration is microstructurally represented by the opening of a set of cleavage planes in the three fracture modes. Crack opening is modelled as a dissipative process, and its evolution is thermodynamically derived. To couple this approach with a physically-based phase field method for brittle fracture, a scalar measure of the overall local damage is introduced, whose evolution is determined by the crack opening rates, and weakly coupled with the non-local phase field energy representing the crack opening resistance in the classical sense of Griffith. A finite-element implementation of the proposed model is employed to simulate the crack propagation path in a laminate and a polycrystalline microstructure. As shown in this work, it is able to predict the localisation of damage on the set of pre-defined cleavage planes, as well as the kinking and branching of the crack resulting from the crystallographic misorientation across the laminate boundary and the grain boundaries respectively.
7. Tailoring complex optical fields via anisotropic microstructures (Presentation Recording)
Lu, Yan-Qing; Hu, Wei; Cui, Guo-Xin
2015-10-01
In recent years, complex optical fields with spatially inhomogeneous phases, polarizations and optical singularities have drawn many research interests. Many novel effects have been predicted and demonstrated for light beams with these unconventional states in both linear and nonlinear optics regimes. Although local optical phase could be controlled directly or through hologram structures in isotropic materials such as glasses, optical anisotropy is still required for manipulating polarization states and wavelengths. The anisotropy could be either intrinsic such as in crystals/liquid crystals (LCs) or the induced birefringence from dielectric or metallic structures. In this talk, we will briefly review some of our attempts in tailoring complex optical fields via anisotropic microstructures. We developed a micro-photo-patterning system that could generate complex micro-images then further guides the arbitrary local LC directors. Due to the electro-optically (EO) tunable anisotropy of LC, various reconfigurable complex optical fields such as optical vortices (OVs), multiplexed OVs, OV array, Airy beams and vector beams are obtained. Different LC modes such as homogeneous alignment nematic, hybrid alignment nematic and even blue phase LCs are adopted to optimize the static and dynamic beam characteristics depending on application circumstances. We are also trying to extend our approaches to new wavelength bands, such as mid-infrared and even THz ranges. Some preliminary results are obtained. In addition, based on our recently developed local poling techniques for ferroelectric crystals, we will also discuss and demonstrate the nonlinear complex optical field conversion in Lithium Niobate wafers with patterned ferroelectric domain structures.
8. Computation of far radiation field of an arbitrarily oriented dipole above layered anisotropic half space
Cao, Le; Wei, Bing; Ge, De-Biao
2013-11-01
Based on the reciprocity theorem, the far field formulation of an arbitrarily oriented electric dipole located at the interface of layered anisotropic half space is deduced. Then, considering the optical path difference of the direct wave and reflected wave, the formulation of the electric dipole located above the interface of layered anisotropic half space is discussed, and the transmission matrix method for computing the reflection coefficients of anisotropic layered half space is introduced in detail. Finally, numerical examples of the field produced by an electric dipole located above layered anisotropic half space are given. The numerical results show that this method can be used in the fast computation of far radiation field of an arbitrarily oriented dipole above layered anisotropic half space.
9. Deformation fields near a steady fatigue crack with anisotropic plasticity
SciTech Connect
Gao, Yanfei
2015-11-30
In this work, from finite element simulations based on an irreversible, hysteretic cohesive interface model, a steady fatigue crack can be realized if the crack extension exceeds about twice the plastic zone size, and both the crack increment per loading cycle and the crack bridging zone size are smaller than the plastic zone size. The corresponding deformation fields develop a plastic wake behind the crack tip and a compressive residual stress field ahead of the crack tip. In addition, the Hill’s plasticity model is used to study the role of plastic anisotropy on the retardation of fatigue crack growth and the elastic strain fields. It is found that for Mode-I cyclic loading, an enhanced yield stress in directions that are inclined from the crack plane will lead to slower crack growth rate, but this retardation is insignificant for typical degrees of plastic anisotropy. Furthermore, these results provide key inputs for future comparisons to neutron and synchrotron diffraction measurements that provide full-field lattice strain mapping near fracture and fatigue crack tips, especially in textured materials such as wrought or rolled Mg alloys.
10. Deformation fields near a steady fatigue crack with anisotropic plasticity
DOE PAGES
Gao, Yanfei
2015-11-30
In this work, from finite element simulations based on an irreversible, hysteretic cohesive interface model, a steady fatigue crack can be realized if the crack extension exceeds about twice the plastic zone size, and both the crack increment per loading cycle and the crack bridging zone size are smaller than the plastic zone size. The corresponding deformation fields develop a plastic wake behind the crack tip and a compressive residual stress field ahead of the crack tip. In addition, the Hill’s plasticity model is used to study the role of plastic anisotropy on the retardation of fatigue crack growth andmore » the elastic strain fields. It is found that for Mode-I cyclic loading, an enhanced yield stress in directions that are inclined from the crack plane will lead to slower crack growth rate, but this retardation is insignificant for typical degrees of plastic anisotropy. Furthermore, these results provide key inputs for future comparisons to neutron and synchrotron diffraction measurements that provide full-field lattice strain mapping near fracture and fatigue crack tips, especially in textured materials such as wrought or rolled Mg alloys.« less
11. Subsurface Stress Fields in Single Crystal (Anisotropic) Contacts
NASA Technical Reports Server (NTRS)
Arakere, Nagaraj K.
2003-01-01
12. Superfluidity of fermions with repulsive on-site interaction in an anisotropic optical lattice near a Feshbach resonance
Wang, Bin; Duan, Luming
2008-03-01
We present numerical analysis of ground state properties of the one-dimensional general Hubbard model (GHM) with particle assisted tunnelling rates and repulsive on-site interaction (positive-U), which describes fermionic atoms in an anisotropic optical lattice near a wide Feshbach resonance. Our calculation uses the time evolving block decimation algorithm, which is an extension of the density matrix renormalization group and provides a well controlled method for one-dimensional systems. We show that the positive-U GHM, when hole doped from half-filling, shows up a phase with coexistence of quasi-long-range superfluid and charge-density-wave orders. This feature is different from the property of the conventional Hubbard model with positive-U, indicting the particle assisted tunneling in the GHM could bring in qualitatively new physics.
13. Superfluidity of fermions with repulsive on-site interaction in an anisotropic optical lattice near a Feshbach resonance
Wang, B.; Duan, L.-M.
2008-07-01
We present a numerical study on ground state properties of a one-dimensional (1D) general Hubbard model (GHM) with particle-assisted tunnelling rates and repulsive on-site interaction (positive-U), which describes fermionic atoms in an anisotropic optical lattice near a wide Feshbach resonance. For our calculation, we utilize the time evolving block decimation (TEBD) algorithm, which is an extension of the density matrix renormalization group and provides a well-controlled method for 1D systems. We show that the positive-U GHM, when hole-doped from half-filling, exhibits a phase with coexistence of quasi-long-range superfluid and charge-density-wave orders. This feature is different from the property of the conventional Hubbard model with positive-U, indicating the particle-assisted tunnelling mechanism in GHM brings in qualitatively new physics.
14. Magnetism and thermodynamics of the anisotropic frustrated spin-1 Heisenberg antiferromagnet on a body-centered cubic lattice
Mi, Bin-Zhou
2017-02-01
The magnetic and thermodynamic properties of anisotropic frustrated spin-1 Heisenberg antiferromagnet on a body-centered cubic lattice for Néel phase (the region of weak frustration) are systematically investigated by use of the double-time Green's function method within the random phase approximation and the Anderson and Callen's decoupling. The zero-temperature sublattice magnetization and Néel temperature increase with spin anisotropy strength and single-ion anisotropy strength, and decrease with frustration strength. This indicates that quantum fluctuation is suppressed by spin anisotropy and single-ion anisotropy, by contrast, is strengthened by frustration. It is possible to tune the quantum fluctuations by the competition of anisotropy strength and frustration strength to change the ground state properties of magnetic materials. Although we find that both the spin anisotropy and the single-ion anisotropy suppress the quantum fluctuations, but their respective effects on the thermodynamic quantities, especially the internal energy and free energy, are different at zero temperature and finite temperature. Furthermore, when these two kinds of anisotropic coexist, the effect of the spin anisotropy on the sublattice magnetization and internal energy is larger than that of the single-ion anisotropy.
15. High-Field Fractional Quantum Hall Effect in Optical Lattices
SciTech Connect
Palmer, R.N.; Jaksch, D.
2006-05-12
We consider interacting bosonic atoms in an optical lattice subject to a large simulated magnetic field. We develop a model similar to a bilayer fractional quantum Hall system valid near simple rational numbers of magnetic flux quanta per lattice cell. Then we calculate its ground state, magnetic lengths, fractional fillings, and find unexpected sign changes in the Hall current. Finally we study methods for detecting these novel features via shot noise and Hall current measurements.
16. Spin-1/2 Heisenberg Antiferromagnet on the Spatially Anisotropic Kagome Lattice
Schnyder, Andreas; Starykh, Oleg; Balents, Leon
2008-03-01
We study the quasi-one-dimensional limit of the Spin-1/2 quantum antiferromagnet on the Kagome lattice, a model Hamiltonian that might be of relevance for the mineral volborthite [1,2]. The lattice is divided into antiferromagnetic spin-chains (exchange J) that are weakly coupled via intermediate dangling'' spins (exchange J'). Using bosonization, renormalization group methods, and current algebra techniques we determine the ground state as a function of J'/J. The case of a strictly one-dimensional Kagome strip is also discussed. [1] Z. Hiroi, M. Hanawa, N. Kobayashi, M. Nohara, Hidenori Takagi, Y. Kato, and M. Takigawa, J. Phys. Soc. Japan 70, 3377 (2001). [2] F. Bert, D. Bono, P. Mendels, F. Ladieu, F. Duc, J.-C. Trumbe, and P. Millet, Phys. Rev. Lett. 95, 087203 (2005).
17. Magnetic field induced lattice ground states from holography
Bu, Yan-Yan; Erdmenger, Johanna; Shock, Jonathan P.; Strydom, Migael
2013-03-01
We study the holographic field theory dual of a probe SU(2) Yang-Mills field in a background (4 + 1)-dimensional asymptotically Anti-de Sitter space. We find a new ground state when a magnetic component of the gauge field is larger than a critical value. The ground state forms a triangular Abrikosov lattice in the spatial directions perpendicular to the magnetic field. The lattice is composed of superconducting vortices induced by the condensation of a charged vector operator. We perform this calculation both at finite temperature and at zero temperature with a hard wall cutoff dual to a confining gauge theory. The study of this state may be of relevance to both holographic condensed matter models as well as to heavy ion physics. The results shown here provide support for the proposal that such a ground state may be found in the QCD vacuum when a large magnetic field is present.
18. Recent Progress in Nuclear Lattice Simulations with Effective Field Theory
Lee, D.
2007-10-01
This proceedings article summarizes recent work presented at Chiral Dynamics 2006 on nuclear lattice simulations with chiral effective field theory for light nuclei. This work has been done in collaboration with Bubar {gra} Borasoy , Evgeny Epelbaum, Hermann Krebs, and Ulf-G. Meißner.
19. Topology of four-dimensional lattice gauge fields
Panagiotakopoulos, C.
1985-08-01
An extremely careful implementation of Woit's definition of the topological charge for SU(2) lattice gauge fields reveals a scaling violation by the topological susceptibility in the region 2.1<=β<=2.3. The result leaves open the possibility that Woit's charge approaches Luscher's charge at weak enough coupling.
20. Nuclear structure and reactions using lattice effective field theory
Rupak, Gautam
2016-09-01
Effective field theory (EFT) formulated on a space-time lattice provides a model-independent framework for ab initio nuclear structure and reaction calculations. The EFT interactions are rooted in quantum chromodynamics through low energy symmetry constraints. In this talk I present several recent developments in lattice EFT, in particular I present the so called adiabatic projection method that enables elastic and in-elastic reaction calculations. Bound state properties of atomic nuclei such as carbon and oxygen will also be presented. Partial support from US National Science Foundation Grant PHY-1307453 is acknowledged.
1. Flow and dispersion in anisotropic porous media: A lattice-Boltzmann study
Maggiolo, D.; Picano, F.; Guarnieri, M.
2016-10-01
Given their capability of spreading active chemical species and collecting electricity, porous media made of carbon fibers are extensively used as diffusion layers in energy storage systems, such as redox flow batteries. In spite of this, the dispersion dynamics of species inside porous media is still not well understood and often lends itself to different interpretations. Actually, the microscopic design of efficient porous media, which can potentially and effectively improve the performances of flow batteries, is still an open challenge. The present study aims to investigate the effect of fibrous media micro-structure on dispersion, in particular the effect of fiber orientation on drag and dispersion dynamics. Several lattice-Boltzmann simulations of flows through differently oriented fibrous media coupled with Lagrangian simulations of particle tracers have been performed. Results show that orienting fibers preferentially along the streamwise direction minimizes the drag and maximizes the dispersion, which is the most desirable condition for diffusion layers in flow batteries' applications.
2. Anisotropic invasion and its consequences in two-strategy evolutionary games on a square lattice
Szabó, György; Varga, Levente; Szabó, Mátyás
2016-11-01
We have studied invasion processes in two-strategy evolutionary games on a square lattice for imitation rule when the players interact with their nearest neighbors. Monte Carlo simulations are performed for systems where the pair interactions are composed of a unit strength coordination game when varying the strengths of the self-dependent and cross-dependent components at a fixed noise level. The visualization of strategy distributions has clearly indicated that circular homogeneous domains evolve into squares with an orientation dependent on the composition. This phenomenon is related to the anisotropy of invasion velocities along the interfaces separating the two homogeneous regions. The quantified invasion velocities indicate the existence of a parameter region in which the invasions are opposite for the horizontal (or vertical) and the tilted interfaces. In this parameter region faceted islands of both strategies shrink and the system evolves from a random initial state into the homogeneous state that first percolated.
3. In-plane uniaxial magnetic anisotropy induced by anisotropic strain relaxation in high lattice-mismatched Dy/Sc superlattices
Benito, L.; Ballesteros, C.; Ward, R. C. C.
2014-04-01
We report on the magnetic and structural characterization of high lattice-mismatched [Dy2nm/SctSc] superlattices, with variable Sc thickness tSc= 2-6 nm. We find that the characteristic in-plane effective hexagonal magnetic anisotropy K66,ef reverses sign and undergoes a dramatic reduction, attaining values of ≈13-24 kJm-3, when compared to K66=-0.76 MJm-3 in bulk Dy. As a result, the basal plane magnetic anisotropy is dominated by a uniaxial magnetic anisotropy (UMA) unfound in bulk Dy, which amounts to ≈175-142 kJm-3. We attribute the large downsizing in K66,ef to the compression epitaxial strain, which generates a competing sixfold magnetoelastic (MEL) contribution to the magnetocrystalline (strain-free) magnetic anisotropy. Our study proves that the in-plane UMA is caused by the coupling between a giant symmetry-breaking MEL constant Mγ ,22≈1 GPa and a morphic orthorhombiclike strain ɛγ ,1≈10-4, whose origin resides on the arising of an in-plane anisotropic strain relaxation process of the pseudoepitaxial registry between the nonmagnetic bottom layers in the superstructure. This investigation shows a broader perspective on the crucial role played by epitaxial strains at engineering the magnetic anisotropy in multilayers.
4. Photoinduced Enhancement of Anisotropic Charge Correlations on Triangular Lattices with Trimers
Yonemitsu, Kenji
2017-02-01
To explore nontrivial photoinduced modulations of charge correlations, we theoretically study photoinduced dynamics in quarter-filled extended Hubbard models with competing intersite repulsive interactions on triangular lattices with trimers, where the end points are crystallographically equivalent. The exact diagonalization method is used and the time-dependent Schrödinger equation is numerically solved during and after photoexcitation. Time-averaged double occupancy and intersite density-density correlations can be interpreted as due to effective on-site and intersite repulsive interactions, respectively, relative to transfer energies. In the case where the intersite repulsive interactions compete with each other, the anisotropy of their effective interactions can be enhanced with the help of the trimers, irrespective of whether the trimers are linear or bent. In particular, in the case where the arrangement of the trimers is close to that in α-(bis[ethylenedithio]-tetrathiafulvalene)2I3 [α-(BEDT-TTF)2I3] in the metallic phase, the effective on-site repulsion is enhanced relative to the transfer energies. The relevance of this theoretical finding to the experimentally observed optical freezing of charge motion is discussed.
5. Anisotropic behaviour of transmission through thin superconducting NbN film in parallel magnetic field
Šindler, M.; Tesař, R.; Koláček, J.; Skrbek, L.
2017-02-01
Transmission of terahertz waves through a thin layer of the superconductor NbN deposited on an anisotropic R-cut sapphire substrate is studied as a function of temperature in a magnetic field oriented parallel with the sample. A significant difference is found between transmitted intensities of beams linearly polarised parallel with and perpendicular to the direction of applied magnetic field.
6. Nonlinear Interaction of a Shock Wave with an Anisotropic Entropy Perturbation Field
Gorodnichev, K. E.; Kuratov, S. E.; Gorodnichev, E. E.
2017-01-01
The problem of the interaction of a shock wave with an anisotropic entropy perturbation field has been solved including second-order corrections to hydrodynamic quantities. It has been shown that nonlinear interactions between acoustic waves result in the localization of acoustic perturbations behind the shock front. This effect is observed when sound attenuation is absent in the linear approximation. The problem of the propagation of the shock wave in an incident sample, where the spatially anisotropic density perturbation field initially exists, has been numerically solved in application to the collision of two plates. Numerical calculations confirm the results of the theoretical analysis.
7. Magnetic fields in QCD vacuum: A lattice view
Buividovich, P. V.
2016-08-01
We review the basic phenomena in QCD subject to strong magnetic fields which are accessible in experiment and can be also studied in lattice QCD simulations: enhanced fluctuations of electric current and electric dipole moment, the negative magnetoresistivity and the inverse magnetic catalysis. We comment on the possibility of experimental detection of negative magnetoresistivity by analysing the angular distributions of dilepton pairs in off-central heavy-ion collisions.
8. Influence of heterogeneous and anisotropic tissue conductivity on electric field distribution in deep brain stimulation.
PubMed
Aström, Mattias; Lemaire, Jean-Jacques; Wårdell, Karin
2012-01-01
The aim was to quantify the influence of heterogeneous isotropic and heterogeneous anisotropic tissue on the spatial distribution of the electric field during deep brain stimulation (DBS). Three finite element tissue models were created of one patient treated with DBS. Tissue conductivity was modelled as (I) homogeneous isotropic, (II) heterogeneous isotropic based on MRI, and (III) heterogeneous anisotropic based on diffusion tensor MRI. Modelled DBS electrodes were positioned in the subthalamic area, the pallidum, and the internal capsule in each tissue model. Electric fields generated during DBS were simulated for each model and target-combination and visualized with isolevels at 0.20 (inner), and 0.05 V mm(-1) (outer). Statistical and vector analysis was used for evaluation of the distribution of the electric field. Heterogeneous isotropic tissue altered the spatial distribution of the electric field by up to 4% at inner, and up to 10% at outer isolevel. Heterogeneous anisotropic tissue influenced the distribution of the electric field by up to 18 and 15% at each isolevel, respectively. The influence of heterogeneous and anisotropic tissue on the electric field may be clinically relevant in anatomic regions that are functionally subdivided and surrounded by multiple fibres of passage.
9. Lattice vacancies in silicon film exposed to external electric field
Mao, Yuliang; Caliste, Damien; Pochet, Pascal
2013-07-01
Density functional calculations based on wavelet basis set are performed to investigate the structure, internal electric-charge distribution, and formation energy of lattice vacancies in silicon film under electric fields. It was found that the formation energies of vacancies both in JT⊥ (Jahn-Teller distortion orthogonal to electric field) and JT‖ (Jahn-Teller distortion parallel to electric field) distortions are decreased with the increasing of field strength, due to the charge polarization in the whole space of silicon film. For the split vacancy, it can lower its energy by moving further away from the split space to form a tetragonal JT⊥ vacancy under electric field. Our results also demonstrate the importance of the potential fluctuations induced by the electric fields on the charge redistribution within the vacancy defects.
10. Long-range interactions in lattice field theory
SciTech Connect
Rabin, J.M.
1981-06-01
Lattice quantum field theories containing fermions can be formulated in a chirally invariant way provided long-range interactions are introduced. It is established that in weak-coupling perturbation theory such a lattice theory is renormalizable when the corresponding continuum theory is, and that the continuum theory is indeed recovered in the perturbative continuum limit. In the strong-coupling limit of these theories one is led to study an effective Hamiltonian describing a Heisenberg antiferromagnet with long-range interactions. Block-spin renormalization group methods are used to find a critical rate of falloff of the interactions, approximately as inverse distance squared, which separates a nearest-neighbor-antiferromagnetic phase from a phase displaying identifiable long-range effects. A duality-type symmetry is present in some block-spin calculations.
11. Vortex liquid crystals in anisotropic type II superconductors.
PubMed
Carlson, E W; Castro Neto, A H; Campbell, D K
2003-02-28
In an isotropic type II superconductor in a moderate magnetic field, the transition to the normal state occurs by vortex lattice melting. In certain anisotropic cases, the vortices acquire elongated cross sections and interactions. Systems of anisotropic, interacting constituents generally exhibit liquid crystalline phases. We examine the possibility of a two step melting in homogeneous type II superconductors with anisotropic superfluid stiffness from a vortex lattice into first a vortex smectic and then a vortex nematic at high temperature and magnetic field. We find that fluctuations of the ordered phase favor an instability to an intermediate smectic-A in the absence of intrinsic pinning.
12. Toward a realistic low-field SSC lattice
SciTech Connect
Heifets, S.
1985-10-01
Three six-fold lattices for 3 T superferric SSC have been generated at TAC. The program based on the first order canonical transformation was used to compare lattices. On this basis the realistic race-track lattices were generated.
13. The unique effect of in-plane anisotropic strain in the magnetization control by electric field
Zhao, Y. Y.; Wang, J.; Hu, F. X.; Liu, Y.; Kuang, H.; Wu, R. R.; Sun, J. R.; Shen, B. G.
2016-05-01
The electric field control of magnetization in both (100)- and (011)-Pr0.7Sr0.3MnO3/Pb(Mg1/3Nb2/3)0.7Ti0.3O3(PSMO/PMN-PT) heterostructures were investigated. It was found that the in-plane isotropic strain induced by electric field only slightly reduces the magnetization at low temperature in (100)-PSMO/PMN-PT film. On the other hand, for (011)-PSMO/PMN-PT film, the in-plane anisotropic strain results in in-plane anisotropic, nonvolatile change of magnetization at low-temperature. The magnetization, remanence and coercivity along in-plane [100] direction are suppressed by the electric field while the ones along [01-1] direction are enhanced, which is ascribed to the extra effective magnetic anisotropy induced by the electric field via anisotropic piezostrains. More interestingly, such anisotropic modulation behaviors are nonvolatile, demonstrating a memory effect.
14. Formation of temperature fields in doped anisotropic crystals under spatially inhomogeneous light beams passing through them
SciTech Connect
Zaitseva, E. V.; Markelov, A. S.; Trushin, V. N. Chuprunov, E. V.
2013-12-15
The features of formation of thermal fields in potassium dihydrophosphate crystal doped with potassium permanganate under a 532-nm laser beam passing through it have been investigated. Data on the influence of birefringence on the temperature distribution in an anisotropic crystal whose surface is illuminated by a spatially modulated light beam are presented.
15. Near-field techniques for probing collective modes of anisotropic superconducting thin films
Stinson, H. T.; Wu, J. S.; Jiang, B. Y.; Fei, Z.; Rodin, A. S.; Chapler, B.; McLeod, A. S.; Castro-Neto, A.; Lee, Y. S.; Fogler, M. M.; Basov, D. N.
2014-03-01
We propose the use of scattering-type scanning near-field optical microscopy (s-SNOM) to characterize the collective mode spectrum of anisotropic superconductors. To probe the dispersion of collective modes with large in-plane momenta, specifically surface plasmons and guided wave modes, we model the real-space interference patterns of modes launched by the sharp s-SNOM tip and their reflections off physical and electronic boundaries. In addition, we show that s-SNOM spectroscopy allows for a direct probe of the c-axis superfluid density in underdoped anisotropic superconductors with nanoscale spatial resolution.
16. Anisotropic Finite Element Modeling Based on a Harmonic Field for Patient-Specific Sclera
PubMed Central
Zheng, Wanqiu; Zou, Beiji
2017-01-01
Purpose. This study examined the influence of anisotropic material for human sclera. Method. First, the individual geometry of patient-specific sclera was reproduced from a laser scan. Then, high quality finite element modeling of individual sclera was performed using a convenient automatic hexahedral mesh generator based on harmonic field and integrated with anisotropic material assignment function. Finally, comparison experiments were designed to investigate the effects of anisotropy on finite element modeling of sclera biomechanics. Results. The experimental results show that the presented approach can generate high quality anisotropic hexahedral mesh for patient-specific sclera. Conclusion. The anisotropy shows significant differences for stresses and strain distribution and careful consideration should be given to its use in biomechanical FE studies. PMID:28271067
17. Proposal for generating synthetic magnetic fields in hexagonal optical lattices
Tian, Binbin; Endres, Manuel; Pekker, David
2015-05-01
We propose a new approach to generating synthetic magnetic fields in ultra cold atom systems that does not rely on either Raman transitions nor periodic drive. Instead, we consider a hexagonal optical lattice produced by the intersection of three laser beams at 120 degree angles, where the intensity of one or more of the beams is spatially non-uniform. The resulting optical lattice remains hexagonal, but has spatially varying hopping matrix elements. For atoms near the Dirac points, these spatial variations appear as a gauge field, similar to the fictitious gauge field that is induced for for electrons in strained graphene. We suggest that a robust way to generate a gauge field that corresponds to a uniform flux is to aligning three gaussian beams to intersect in an equilateral triangle. Using realistic experimental parameters, we show how the proposed setup can be used to observe cyclotron motion of an atom cloud - the conventional Hall effect and distinct Landau levels - the integer quantum Hall effect.
18. Anisotropic polymer composites synthesized by immobilizing cellulose nanocrystal suspensions specifically oriented under magnetic fields.
PubMed
Tatsumi, Mio; Kimura, Fumiko; Kimura, Tsunehisa; Teramoto, Yoshikuni; Nishio, Yoshiyuki
2014-12-08
Novel polymer composites reinforced with an oriented cellulose nanocrystal (CNC) assembly were prepared from suspensions of CNC in aqueous 2-hydroxyethyl methacrylate (HEMA) via magnetic field application to the suspensions followed by polymerization treatment. The starting suspensions used at ∼6 wt % CNC separated into an upper isotropic phase and a lower anisotropic (chiral nematic) one in the course of quiescent standing. A static or rotational magnetic field was applied to the isolated isotropic and anisotropic phases. UV-induced polymerization of HEMA perpetuated the respective states of magnetic orientation invested for the CNC dispersions to yield variously oriented CNC/poly(2-hydroxyethyl methacrylate) composites. The structural characterization was carried out by use of X-ray diffractometry and optical and scanning electron microscopy. The result indicated that CNCs were aligned in the composites distinctively according to the static or rotational magnetic application when the anisotropic phases were used, whereas such a specific CNC orientation was not appreciable when the isotropic phases were sampled. This marks out effectiveness of a coherent response of CNCs in the mesomorphic assembly. In dynamic mechanical experiments in tensile or compressive mode, we observed a clear mechanical anisotropy for the polymer composites synthesized from wholly anisotropic suspensions under static or rotational magnetization. The higher modulus (in compression) was detected for a composite reinforced by locking-in the uniaxial CNC alignment attainable through conversion of the initial chiral nematic phase into a nematic phase in the rotational magnetic field.
19. Multiple-q states and the Skyrmion lattice of the triangular-lattice Heisenberg antiferromagnet under magnetic fields.
PubMed
Okubo, Tsuyoshi; Chung, Sungki; Kawamura, Hikaru
2012-01-06
Ordering of the frustrated classical Heisenberg model on the triangular lattice with an incommensurate spiral structure is studied under magnetic fields by means of a mean-field analysis and a Monte Carlo simulation. Several types of multiple-q states including the Skyrmion-lattice state is observed in addition to the standard single-q state. In contrast to the Dzyaloshinskii-Moriya interaction driven system, the present model allows both Skyrmions and anti-Skyrmions, together with a new thermodynamic phase where Skyrmion and anti-Skyrmion lattices form a domain state.
20. Long-Range Order of the Three-Sublattice Structure in theS=1 Heisenberg Antiferromagnet on a Spatially Anisotropic Triangular Lattice
Nakano, Hiroki; Todo, Synge; Sakai, Tôru
2013-04-01
We study the S=1 Heisenberg antiferromagnet on a spatially anisotropic triangular lattice by the numerical diagonalization method. We examine the stability of the long-range order of a three-sublattice structure observed in the isotropic system between the isotropic case and the case of isolated one-dimensional chains. It is found that the long-range-ordered ground state with this structure exists in the range of 0.7 \\simle J_2/J_1 \\le 1, where J_1 is the interaction amplitude along the chains and J_2 is the amplitude of other interactions.
1. Low Energy Continuum and Lattice Effective Field Theories
Elhatisari, Serdar
calculations we use a new lattice algorithm called impurity lattice Monte Carlo. This algorithm can be viewed as a hybrid technique which incorporates elements of both worldline and auxiliary-field Monte Carlo simulations.
2. Lattice field theory studies of magnetic catalysis in graphene
Winterowd, Christopher R.
Consisting of a single two-dimensional layer of Carbon atoms arranged in a hexagonal lattice, graphene represents one of the most exciting recent developments in condensed matter physics. With novel electronic and mechancial properties, graphene not only has great potential with respect to technological applications, but also displays phenomena that typically appear in relativistic quantum field theory. The low-energy electronic excitations of graphene consist of two identical species of massless Dirac particles. Due to the small Fermi velocity, these particles are strongly coupled through the Coulomb interaction. Although various perturbative approaches have succeeded in elucidating many of the electronic properties of graphene, one would still like a nonperturbative study to address various questions. In particular, the spontaneous breaking of chiral symmetry in the presence of an external magnetic field, commonly known as magnetic catalysis, is one of these questions. Early studies of this phenomenon in model relativistic field theories have posited the mechanism to be universal. More recently, this mechanism of spontaneous symmetry breaking has been studied in low-dimensional condensed matter systems. Due to the strongly-coupled nature of the low-energy effective field theory of graphene, nonperturbative methods of lattice gauge theory can be used which are well suited to studying chiral symmetry breaking. Most notably used to study the theory of the strong interactions, quantum chromodynamics, these methods have proven successful in elucidating nonperturbative phenomena in cases where perturbative methods fail. In this thesis, using these methods, evidence in favor of magnetic catalysis in the graphene effective field theory will be presented.
3. Anisotropic magnetohydrodynamic turbulence in a strong external magnetic field
NASA Technical Reports Server (NTRS)
Montgomery, D.; Turner, L.
1981-01-01
A strong external dc magnetic field introduces a basic anisotropy in incompressible MHD turbulence. The modifications that this is likely to produce in the properties of the turbulence are investigated for high Reynolds numbers. It is found that the turbulent spectrum splits into two parts: (1) an essentially two-dimensional spectrum with both the velocity field and the magnetic fluctuations perpendicular to the dc magnetic field, and (2) a generally weaker and more nearly isotropic spectrum of Alfven waves. These results are discussed in relation to measurements from the Culham-Harwell Zeta pinch device and the UCLA Macrorotor tokamak, as well as in relation to measurements of MHD turbulence in the solar wind.
4. Anisotropic magnetohydrodynamic turbulence in a strong external magnetic field
NASA Technical Reports Server (NTRS)
Montgomery, D.; Turner, L.
1981-01-01
A strong external dc magnetic field introduces a basic anisotropy into incompressible magnetohydrodynamic turbulence. The modifications that this is likely to produce in the properties of the turbulence are explored for the high Reynolds number case. The conclusion is reached that the turbulent spectrum splits into two parts: an essentially two dimensional spectrum with both the velocity field and magnetic fluctuations perpendicular to the dc magnetic field, and a generally weaker and more nearly isotropic spectrum of Alfven waves. A minimal characterization of the spectral density tensors is given. Similarities to measurements from the Culham-Harwell Zeta pinch device and the UCLA Macrotor Tokamak are remarked upon, as are certain implications for the Belcher and Davis measurements of magnetohydrodynamic turbulence in the solar wind.
5. Linear and nonlinear optical properties of anisotropic quantum dots in a magnetic field
Xie, Wenfang
2013-05-01
We have investigated the linear and nonlinear optical properties of a two-dimensional anisotropic quantum dot in a magnetic field. Based on the computed energies and wave functions, the linear, third-order nonlinear and total optical absorption coefficients as well as the refractive index changes have been examined. The results are presented as a function of the incident photon energy for the different cases of anisotropy, dot size and external magnetic field. The results show that the linear and nonlinear optical properties of anisotropic quantum dots are strongly affected by the degree of anisotropy, the dot size, the external magnetic field and the polarized direction of the incident electromagnetic wave. The result also shows that the size effect of anisotropy quantum dots on the optical absorptions is different from that of isotropic quantum dots.
6. The fifteen theorem for universal Hermitian lattices over imaginary quadratic fields
Kim, Byeong Moon; Kim, Ji Young; Park, Poo-Sung
2010-04-01
We will introduce a method to get all universal Hermitian lattices over imaginary quadratic fields Q(√{-m}) for all m . For each imaginary quadratic field Q(√{-m}) , we obtain a criterion on universality of Hermitian lattices: if a Hermitian lattice L represents 1, 2, 3, 5, 6, 7, 10, 13, 14 and 15, then L is universal. We call this the fifteen theorem for universal Hermitian lattices. Note that the difference between Conway-Schneeberger's fifteen theorem and ours is the number 13. In addition, we determine the minimal rank of universal Hermitian lattices for all imaginary quadratic fields.
7. Assessing the Structure of Isotropic and Anisotropic Turbulent Magnetic Fields
Fatuzzo, Marco; Holden, Lisa; Grayson, Lindsay; Wallace, Kirk
2016-10-01
Turbulent magnetic fields permeate our universe, impacting a wide range of astronomical phenomena across all cosmic scales. A clear example is the magnetic field that threads the interstellar medium (ISM), which impacts the motion of cosmic rays through that medium. Understanding the structure of magnetic turbulence within the ISM and how it relates to the physical quantities that characterize it can thus inform our analysis of particle transport within these regions. Toward that end, we probe the structure of magentic turbulence through the use of Lyapunov exponents for a suite of isotropic and nonisotropic Alfvénic turbulence profiles. Our results provide a means of calculating a “turbulence lengthscale” that can then be connected to how cosmic rays propagate through magentically turbulent environments, and we perform such an analysis for molecular cloud environments.
8. Lattice simulations of real-time quantum fields
Berges, J.; Borsányi, Sz.; Sexty, D.; Stamatescu, I.-O.
2007-02-01
We investigate lattice simulations of scalar and non-Abelian gauge fields in Minkowski space-time. For SU(2) gauge-theory expectation values of link variables in 3+1 dimensions are constructed by a stochastic process in an additional (5th) “Langevin-time.” A sufficiently small Langevin step size and the use of a tilted real-time contour leads to converging results in general. All fixed point solutions are shown to fulfil the infinite hierarchy of Dyson-Schwinger identities, however, they are not unique without further constraints. For the non-Abelian gauge theory the thermal equilibrium fixed point is only approached at intermediate Langevin-times. It becomes more stable if the complex time path is deformed towards Euclidean space-time. We analyze this behavior further using the real-time evolution of a quantum anharmonic oscillator, which is alternatively solved by diagonalizing its Hamiltonian. Without further optimization stochastic quantization can give accurate descriptions if the real-time extent of the lattice is small on the scale of the inverse temperature.
9. Broken symmetry approach to density functional calculation of zero field splittings including anisotropic exchange interactions
SciTech Connect
Kessler, Eva M. V.; Schmitt, Sebastian; Wüllen, Christoph van
2013-11-14
The broken symmetry approach to the calculation of zero field splittings (or magnetic anisotropies) of multinuclear transition metal complexes is further developed. A procedure is suggested how to extract spin Hamiltonian parameters for anisotropic exchange from a set of broken symmetry density functional calculations. For isotropic exchange coupling constants J{sub ij}, the established procedure is retrieved, and anisotropic (or pseudodipolar) exchange coupling tensors D{sub ij} are obtained analogously. This procedure only yields the sum of the individual single-ion zero field splitting tensors D{sub i}. Therefore, a procedure based on localized orbitals has been developed to extract the individual single-ion contributions. With spin Hamiltonian parameters at hand, the zero field splittings of the individual spin multiplets are calculated by an exact diagonalization of the isotropic part, followed by a spin projection done numerically. The method is applied to the binuclear cation [LCr(OH){sub 3}CrL]{sup 3+} (L = 1,4,7-trimethyl-1,4,7-triazanonane) for which experimental zero field splittings for all low-energy spin states are known, and to the single-molecule magnet [Fe{sub 4}(CH{sub 3}C(CH{sub 2}O){sub 3}){sub 2}(dpm){sub 6}] (Hdpm = 2,2,6,6-tetramethylheptane-3,5-dione). In both these 3d compounds, the single-ion tensors mainly come from the spin-orbit interaction. Anisotropic exchange is dominated by the spin-dipolar interaction only for the chromium compound. Despite the rather small isotropic exchange couplings in the iron compound, spin-orbit and spin-dipolar contributions to anisotropic exchange are of similar size here.
10. Ultrasonic field modeling in anisotropic materials by distributed point source method.
PubMed
2017-03-16
DPSM (distributed point source method) is a modeling technique which is based on the concept of Green's function. First, a collection of source and target points are distributed over the solution domain based on the problem description and solution requirements. Then, the effects from all source points are superimposed at the location of every individual target point. Therefore, a successful implementation of DPSM entails an effective evaluation of Green's function between many pairs of source and target points. For homogeneous and isotropic media, the Green's function is available as a closed-form analytical expression. But for anisotropic solids, the evaluation of Green's function is more complicated and needs to be done numerically. Nevertheless, important applications such as defect detection in composite materials require anisotropic analysis. In this paper, the DPSM is used for ultrasonic field modeling in anisotropic materials. Considering the prohibitive computational cost of evaluating Green's function numerically for a large number of points, a technique called "windowing" is suggested which employs the repetitive pattern of points in DPSM in order to considerably reduce the number of evaluations of Green's function. In addition, different resolutions of numerical integration are used for computing Green's function corresponding to different distances in order to achieve a good balance between time and accuracy. The developed anisotropic DPSM model equipped with windowing technique and multi-resolution numerical integration is then applied to the problem of ultrasonic wave modeling in a plate immersed in a fluid. The transducers are placed in the fluid on both sides of the plate. First an isotropic plate is considered for the sake of verification and rough calibration of numerical integration. Then a composite plate is considered to demonstrate applicability and effectiveness of the developed model for simulating ultrasonic wave propagation in anisotropic
11. A partially mesh-free scheme for representing anisotropic spatial variations along field lines
McMillan, Ben F.
2017-03-01
A common numerical task is to represent functions which are highly spatially anisotropic, and to solve differential equations related to these functions. One way such anisotropy arises is that information transfer along one spatial direction is much faster than in others. In this situation, the derivative of the function is small in the local direction of a vector field B. In order to define a discrete representation, a set of surfaces Mi indexed by an integer i are chosen such that mapping along the field B induces a one-to-one relation between the points on surface Mi to those on Mi+1. For simple cases Mi may be surfaces of constant coordinate value. On each surface Mi, a function description is constructed using basis functions defined on a regular structured mesh. The definition of each basis function is extended from the surface M along the lines of the field B by multiplying it by a smooth compact support function whose argument increases with distance along B. Function values are evaluated by summing contributions associated with each surface Mi. This does not require any special connectivity of the meshes used in the neighbouring surfaces M, which substantially simplifies the meshing problem compared to attempting to find a space filling anisotropic mesh. We explore the numerical properties of the scheme, and show that it can be used to efficiently solve differential equations for certain anisotropic problems.
12. Superconformal field theories from M-theory crystal lattices
Lee, Sangmin
2007-05-01
We propose a brane configuration for the (2+1)d, N=2 superconformal theories (CFT3) arising from M2 branes probing toric Calabi-Yau 4-fold cones, using a T-duality transformation of M theory. We obtain intersections of M5-branes on a three-torus which form a 3d bipartite crystal lattice in a way similar to the 2d dimer models for CFT4. The fundamental fields of the CFT3 are M2-brane discs localized around the intersections, and the superpotential terms are identified with the atoms of the crystal. The model correctly reproduces the Bogomol’nyi-Prasad-Sommerfield (BPS) spectrum of mesons.
13. Non-abelian gauge fields and topological insulators in shaken optical lattices.
PubMed
Hauke, Philipp; Tieleman, Olivier; Celi, Alessio; Olschläger, Christoph; Simonet, Juliette; Struck, Julian; Weinberg, Malte; Windpassinger, Patrick; Sengstock, Klaus; Lewenstein, Maciej; Eckardt, André
2012-10-05
Time-periodic driving like lattice shaking offers a low-demanding method to generate artificial gauge fields in optical lattices. We identify the relevant symmetries that have to be broken by the driving function for that purpose and demonstrate the power of this method by making concrete proposals for its application to two-dimensional lattice systems: We show how to tune frustration and how to create and control band touching points like Dirac cones in the shaken kagome lattice. We propose the realization of a topological and a quantum spin Hall insulator in a shaken spin-dependent hexagonal lattice. We describe how strong artificial magnetic fields can be achieved for example in a square lattice by employing superlattice modulation. Finally, exemplified on a shaken spin-dependent square lattice, we develop a method to create strong non-abelian gauge fields.
14. Anisotropic polyurethane magnetorheological elastomer prepared through in situ polycondensation under a magnetic field
Wu, Jinkui; Gong, Xinglong; Fan, Yanceng; Xia, Hesheng
2010-10-01
Highly filled polytetramethylene ether glycol (PTMEG)-based polyurethane (PU) magnetorheological elastomers (MREs) with anisotropic structure and good mechanical properties were prepared. The difficulty in dispersion and orientation of iron particles in the PU elastomer was overcome by ball milling mixing and further in situ one-step polycondensation under a magnetic field. The microstructure and properties of the composite were characterized in detail. Scanning electron microscopy (SEM) showed that a chain-like structure of carbonyl iron was formed in the PU matrix after orientation under a magnetic field of 1.2 T. The aligned chain-like structure of carbonyl iron in PU greatly enhanced the thermal conductivity, the compression properties and the magnetorheological (MR) effect of anisotropic PU MREs compared to that of the isotropic one. When the test frequency is 1 Hz, the maximum absolute and relative MR effect of anisotropic PU MREs with 26 wt% hard segment and 70 wt% carbonyl iron were ~ 1.3 MPa and ~ 21%, respectively.
15. Evaluation of three-dimensional anisotropic head model for mapping realistic electromagnetic fields of brain tissues
Jeong, Woo Chul; Wi, Hun; Sajib, Saurav Z. K.; Oh, Tong In; Kim, Hyung Joong; Kwon, Oh In; Woo, Eung Je
2015-08-01
Electromagnetic fields provide fundamental data for the imaging of electrical tissue properties, such as conductivity and permittivity, in recent magnetic resonance (MR)-based tissue property mapping. The induced voltage, current density, and magnetic flux density caused by externally injected current are critical factors for determining the image quality of electrical tissue conductivity. As a useful tool to identify bio-electromagnetic phenomena, precise approaches are required to understand the exact responses inside the human body subject to an injected currents. In this study, we provide the numerical simulation results of electromagnetic field mapping of brain tissues using a MR-based conductivity imaging method. First, we implemented a realistic three-dimensional human anisotropic head model using high-resolution anatomical and diffusion tensor MR images. The voltage, current density, and magnetic flux density of brain tissues were imaged by injecting 1 mA of current through pairs of electrodes on the surface of our head model. The current density map of anisotropic brain tissues was calculated from the measured magnetic flux density based on the linear relationship between the water diffusion tensor and the electrical conductivity tensor. Comparing the current density to the previous isotropic model, the anisotropic model clearly showed the differences between the brain tissues. This originates from the enhanced signals by the inherent conductivity contrast as well as the actual tissue condition resulting from the injected currents.
16. Variational wave functions for the S =1/2 Heisenberg model on the anisotropic triangular lattice: Spin liquids and spiral orders
Ghorbani, Elaheh; Tocchio, Luca F.; Becca, Federico
2016-02-01
By using variational wave functions and quantum Monte Carlo techniques, we investigate the complete phase diagram of the Heisenberg model on the anisotropic triangular lattice, where two out of three bonds have superexchange couplings J and the third one has instead J'. This model interpolates between the square lattice and the isotropic triangular one, for J'/J ≤1 , and between the isotropic triangular lattice and a set of decoupled chains, for J /J'≤1 . We consider all the fully symmetric spin liquids that can be constructed with the fermionic projective-symmetry group classification (Zhou and Wen, arXiv:cond-mat/0210662) and we compare them with the spiral magnetic orders that can be accommodated on finite clusters. Our results show that, for J'/J ≤1 , the phase diagram is dominated by magnetic orderings, even though a spin-liquid state may be possible in a small parameter window, i.e., 0.7 ≲J'/J ≲0.8 . In contrast, for J /J'≤1 , a large spin-liquid region appears close to the limit of decoupled chains, i.e., for J /J'≲0.6 , while magnetically ordered phases with spiral order are stabilized close to the isotropic point.
17. Band structure engineering via piezoelectric fields in strained anisotropic CdSe/CdS nanocrystals
PubMed Central
Christodoulou, Sotirios; Rajadell, Fernando; Casu, Alberto; Vaccaro, Gianfranco; Grim, Joel Q.; Genovese, Alessandro; Manna, Liberato; Climente, Juan I.; Meinardi, Francesco; Rainò, Gabriele; Stöferle, Thilo; Mahrt, Rainer F.; Planelles, Josep; Brovelli, Sergio; Moreels, Iwan
2015-01-01
Strain in colloidal heteronanocrystals with non-centrosymmetric lattices presents a unique opportunity for controlling optoelectronic properties and adds a new degree of freedom to existing wavefunction engineering and doping paradigms. We synthesized wurtzite CdSe nanorods embedded in a thick CdS shell, hereby exploiting the large lattice mismatch between the two domains to generate a compressive strain of the CdSe core and a strong piezoelectric potential along its c-axis. Efficient charge separation results in an indirect ground-state transition with a lifetime of several microseconds, almost one order of magnitude longer than any other CdSe/CdS nanocrystal. Higher excited states recombine radiatively in the nanosecond time range, due to increasingly overlapping excited-state orbitals. k̇p calculations confirm the importance of the anisotropic shape and crystal structure in the buildup of the piezoelectric potential. Strain engineering thus presents an efficient approach to highly tunable single- and multiexciton interactions, driven by a dedicated core/shell nanocrystal design. PMID:26219691
18. Lattice field theory applications in high energy physics
Gottlieb, Steven
2016-10-01
Lattice gauge theory was formulated by Kenneth Wilson in 1974. In the ensuing decades, improvements in actions, algorithms, and computers have enabled tremendous progress in QCD, to the point where lattice calculations can yield sub-percent level precision for some quantities. Beyond QCD, lattice methods are being used to explore possible beyond the standard model (BSM) theories of dynamical symmetry breaking and supersymmetry. We survey progress in extracting information about the parameters of the standard model by confronting lattice calculations with experimental results and searching for evidence of BSM effects.
19. Direct observation of anisotropic magnetic field response of the spin helix in FeGe thin films
Kanazawa, N.; White, J. S.; Rønnow, H. M.; Dewhurst, C. D.; Fujishiro, Y.; Tsukazaki, A.; Kozuka, Y.; Kawasaki, M.; Ichikawa, M.; Kagawa, F.; Tokura, Y.
2016-11-01
We report the observation by small-angle neutron scattering (SANS) of a magnetic helical structure confined in a thin film of the chiral lattice magnet FeGe. Twofold magnetic Bragg spots appearing below the magnetic transition temperature indicate the formation of a spin helix with a single propagation vector q aligned perpendicular to the film plane. Due to magnetic anisotropy, the direction of q is unaffected by an external magnetic field H . Instead we observe anisotropic deformations of the spin helix with respect to the H direction. In the configuration with H ⊥q , the helical pitch exhibits hysteretic elongation with H , while the system tends to maintain an integer number of spiral turns within the film thickness by continuously pushing out one turn. For H ∥q , the helix is smoothly distorted to a conical structure with minimal change in the magnetic period. The direct measurement of q by SANS establishes a correspondence between helix deformation and macroscopic features observed in magnetization and magnetoresistivity.
20. Electric field distribution and energy absorption in anisotropic and dispersive red blood cells.
PubMed
Sebastián, J L; Muñoz, S; Sancho, M; Alvarez, G; Miranda, J M
2007-12-07
We have studied the influence of the anisotropic and dispersive nature of the red blood cell structure on the energy absorption and electric field distribution within the cell exposed to electromagnetic fields of frequencies in the range from 50 kHz to 10 GHz. For this purpose we have generated a realistic model of a multilayered erythrocyte cell from a set of parametric equations in terms of Jacobi elliptic functions. The effect of dipole relaxations and anisotropic conductivities is taken into account in the dispersion equations for the conductivity and permittivity of each layer (cytoplasmic and extra-cellular bound waters, membrane, cytoplasm and external medium). Using a finite element numerical technique, we have found that the electric field distribution and the energy absorbed in the membrane show well-defined maxima for both normal and parallel orientations of the external field with respect to the symmetry axis of the cell. The normal and tangential conductivities and permittivities of the membrane are shown to be responsible for the different peak amplitudes and frequency shifts of the maxima. A previously unnoticed effect is that the cell shape combined with the dispersion of the membrane permittivity and the influence of bound water layers leads to a very high amplification factor (greater than 300) of the electric field in the membrane at frequencies in the megahertz range.
1. Thermodynamically consistent phase field theory of phase transformations with anisotropic interface energies and stresses
Levitas, Valery I.; Warren, James A.
2015-10-01
The main focus of this paper is to introduce, in a thermodynamically consistent manner, an anisotropic interface energy into a phase field theory for phase transformations. Here we use a small strain formulation for simplicity, but we retain some geometric nonlinearities, which are necessary for introducing correct interface stresses. Previous theories have assumed the free energy density (i.e., gradient energy) is an anisotropic function of the gradient of the order parameters in the current (deformed) state, which yields a nonsymmetric Cauchy stress tensor. This violates two fundamental principles: the angular momentum equation and the principle of material objectivity. Here, it is justified that for a noncontradictory theory the gradient energy must be an isotropic function of the gradient of the order parameters in the current state, which also depends anisotropically on the direction of the gradient of the order parameters in the reference state. A complete system of thermodynamically consistent equations is presented. We find that the main contribution to the Ginzburg-Landau equation resulting from small strains arises from the anisotropy of the interface energy, which was neglected before. The explicit expression for the free energy is justified. An analytical solution for the nonequilibrium interface and critical nucleus has been found and a parametric study is performed for orientation dependence of the interface energy and width as well as the distribution of interface stresses.
2. Fields induced by three-dimensional dislocation loops in anisotropic magneto-electro-elastic bimaterials
Han, Xueli; Pan, Ernie; Sangghaleh, Ali
2013-08-01
The coupled elastic, electric and magnetic fields produced by an arbitrarily shaped three-dimensional dislocation loop in general anisotropic magneto-electro-elastic (MEE) bimaterials are derived. First, we develop line-integral expressions for the fields induced by a general dislocation loop. Then, we obtain analytical solutions for the fields, including the extended Peach-Koehler force, due to some useful dislocation segments such as straight line and elliptic arc. The present solutions contain the piezoelectric, piezomagnetic and purely elastic solutions as special cases. As numerical examples, the fields induced by a square and an elliptic dislocation loop in MEE bimaterials are studied. Our numerical results show the coupling effects among different fields, along with various interesting features associated with the dislocation and interface.
3. Anisotropic heat transport in integrable and chaotic 3-D magnetic fields
SciTech Connect
Del-Castillo-Negrete, Diego B; Blazevski, D.; Chacon, Luis
2012-01-01
A study of anisotropic heat transport in 3-D chaotic magnetic fields is presented. The approach is based on the recently proposed Lagrangian-Green s function (LG) method in Ref. [1] that allows an efficient and accurate integration of the parallel transport equation applicable to general magnetic fields with local or non-local parallel flux closures. We focus on reversed shear magnetic field configurations known to exhibit separatrix reconnection and shearless transport barriers. The role of reconnection and magnetic field line chaos on temperature transport is studied. Numerical results are presented on the anomalous relaxation of radial temperature gradients in the presence of shearless Cantori partial barri- ers. Also, numerical evidence of non-local effective radial temperature transport in chaotic fields is presented. Going beyond purely parallel transport, the LG method is generalized to include finite perpendicular diffusivity, and the problem of temperature flattening inside a magnetic island is studied.
4. Crystal structure and anisotropic magnetic properties of new ferromagnetic Kondo lattice compound Ce(Cu,Al,Si)2
Maurya, A.; Thamizhavel, A.; Dhar, S. K.; Provino, A.; Pani, M.; Costa, G. A.
2017-03-01
Single crystals of the new compound CeCu0.18Al0.24Si1.58 have been grown by high-temperature solution growth method using a eutectic Al-Si mixture as flux. This compound is derived from the binary CeSi2 (tetragonal α-ThSi2-type, Pearson symbol tI12, space group I41/amd) obtained by partial substitution of Si by Cu and Al atoms but showing full occupation of the Si crystal site (8e). While CeSi2 is a well-known valence-fluctuating paramagnetic compound, the CeCu0.18Al0.24Si1.58 phase orders ferromagnetically at TC=9.3 K. At low temperatures the easy-axis of magnetization is along the a-axis, which re-orients itself along the c-axis above 30 K. The presence of hysteresis in the magnetization curve, negative temperature coefficient of resistivity at high temperatures, reduced jump in the heat capacity and a relatively lower entropy released up to the ordering temperature, and enhanced Sommerfeld coefficient (≈100 mJ/mol K2) show that CeCu0.18Al0.24Si1.58 is a Kondo lattice ferromagnetic, moderate heavy fermion compound. Analysis of the high temperature heat capacity data in the paramagnetic region lets us infer that the crystal electric field split doublet levels are located at 178 and 357 K, respectively, and Kondo temperature (8.4 K) is of the order of TC in CeCu0.18Al0.24Si1.58.
5. 11B and 27Al NMR spin-lattice relaxation and Knight shift of Mg1-xAlxB2: Evidence for an anisotropic Fermi surface
Papavassiliou, G.; Pissas, M.; Karayanni, M.; Fardis, M.; Koutandos, S.; Prassides, K.
2002-10-01
We report a detailed study of the 11B and 27Al NMR spin-lattice relaxation rates (1/T1) and the 27Al Knight shift (K) in Mg1-xAlxB2, 0<=x<=1. The evolution of (1/T1T) and K with x is in excellent agreement with the prediction of ab initio calculations of a highly anisotropic Fermi surface, consisting mainly of hole-type two-dimensional (2D) cylindrical sheets from bonding 2px,y boron orbitals. The density of states at the Fermi level also decreases sharply on Al doping and the 2D sheets collapse at x~0.55, where the superconducting phase disappears.
6. Multiple dynamic transitions in an anisotropic Heisenberg ferromagnet driven by polarized magnetic field.
PubMed
Acharyya, Muktish
2004-02-01
A uniaxially (along the Z axis) anisotropic Heisenberg ferromagnet, in the presence of time-dependent (but uniform over space) magnetic field, is studied by Monte Carlo simulation. The time-dependent magnetic field was taken as elliptically polarized where the resultant field vector rotates in the X-Z plane. The system is cooled (in the presence of the elliptically polarized magnetic field) from high temperature. As the temperature decreases, it was found that in the low anisotropy limit the system undergoes three successive dynamical phase transitions. These three dynamic transitions were confirmed by studying the temperature variation of dynamic "specific heat." The temperature variation of dynamic specific heat shows three peaks indicating three dynamic transition points.
7. Anisotropic high-field terahertz response of free-standing carbon nanotubes
Lee, Byounghwak; Mousavian, Ali; Paul, Michael J.; Thompson, Zachary J.; Stickel, Andrew D.; McCuen, Dalton R.; Jang, Eui Yun; Kim, Yong Hyup; Kyoung, Jisoo; Kim, Dai-Sik; Lee, Yun-Shik
2016-06-01
We demonstrate that unidirectionally aligned, free-standing multi-walled carbon nanotubes (CNTs) exhibit highly anisotropic linear and nonlinear terahertz (THz) responses. For the polarization parallel to the CNT axis, strong THz pulses induce nonlinear absorption in the quasi-one-dimensional conducting media, while no nonlinear effect is observed in the perpendicular polarization configuration. Time-resolved measurements of transmitted THz pulses and a theoretical analysis of the data reveal that intense THz fields enhance permittivity in carbon nanotubes by generating charge carriers.
8. Energy levels of an anisotropic three-dimensional polaron in a magnetic field
Brancus, D. E.; Stan, G.
2001-06-01
In the context of the improved Wigner-Brillouin theory, the energy levels are found of a Fröhlich polaron in a uniaxial anisotropic polar semiconductor with complex structure, placed in a magnetic field directed either along the optical axis or orthogonal to it. All sources of anisotropy that are contained in the shape of constant-energy surfaces of the bare electron, the electron-optical-phonon interaction, and the frequency spectrum of the extraordinary phonon modes are considered. Analytical results for the electron-phonon interaction correction to the Landau levels below the optical-phonon continuum are given and, numerical results for the magnetic-field dependence of the cyclotron resonance frequency at low temperature are presented for the particular case of the layered semiconductors InSe and GaSe. Although the interaction between the bare electron and quasitransverse optical-phonon modes is weak, these modes play an important role in the pinning of Landau levels. The results given by Das Sarma for a two-dimensional isotropic magnetopolaron are generalized to the anisotropic uniaxial case by taking formally m∥-->∞ in the expression of the perturbed Landau levels found when the magnetic field is directed along the optical axis, m∥ being the component of the bare-electron effective-mass tensor along the optical axis.
9. Effects of nonlinear plasma wake field on the dust-lattice wave in complex plasmas
Lee, Myoung-Jae; Jung, Young-Dae
2017-02-01
The influence of a nonlinear ion wake field on the dust-lattice wave is investigated in complex dusty plasmas. The dispersion relation for the dust-lattice wave is derived from the equation of motion including the contribution due to the nearest-neighbour dust grain interaction. The results show that the nonlinear wake-field effect increases the wave frequency, especially at the maximum peak positions. It is found that the oscillatory behaviour of the dust-lattice wave enhances with an increase of the spacing of the dust grains. It is also found that the amplitude of the dust-lattice wave significantly decreases with an increase of the inter-dust grain distance. In addition, it is found that the amplitude of the dust-lattice wave increases with increasing Debye length. The variation of the dust-lattice wave due to the Mach number and plasma parameters is also discussed.
10. The magnetic field inside a layered anisotropic spherical conductor due to internal sources
Nieminen, Jaakko O.; Stenroos, Matti
2016-01-01
Recent advances in neuronal current imaging using magnetic resonance imaging and in invasive measurement of neuronal magnetic fields have given a need for methods to compute the magnetic field inside a volume conductor due to source currents that are within the conductor. In this work, we derive, verify, and demonstrate an analytical expression for the magnetic field inside an anisotropic multilayer spherically symmetric conductor due to an internal current dipole. We casted an existing solution for electric field to vector spherical harmonic (VSH) form. Next, we wrote an ansatz for the magnetic field using toroidal-poloidal decomposition that uses the same VSHs. Using properties of toroidal and poloidal components and VSHs and applying magnetic scalar potential, we then formulated a series expression for the magnetic field. The convergence of the solution was accelerated by formulating the solution using an addition-subtraction method. We verified the resulting formula against boundary-element method. The verification showed that the formulas and implementation are correct; 99th percentiles of amplitude and angle differences between the solutions were below 0.5% and 0.5°, respectively. As expected, the addition-subtraction model converged faster than the unaccelerated model; close to the source, 250 terms gave relative error below 1%, and the number of needed terms drops fast, as the distance to the source increases. Depending on model conductivities and source position, field patterns inside a layered sphere may differ considerably from those in a homogeneous sphere. In addition to being a practical modeling tool, the derived solution can be used to verify numerical methods, especially finite-element method, inside layered anisotropic conductors.
11. Giant field enhancement in anisotropic epsilon-near-zero films (Conference Presentation)
Kamandi, Mohammad; Guclu, Caner; Capolino, Filippo
2016-09-01
We investigated anisotropic epsilon-near-zero (AENZ) films under TM-polarized plane wave incidence and found they possess peculiar properties. In particular we studied uniaxially anisotropic films where either the permittivity along the surface normal or along the transverse plane tends to zero while the other one does not. Previously, numerous applications of isotropic epsilon-near-zero (ENZ) films including radiation pattern tailoring, enhanced harmonic generation, optical bistability and energy squeezing have been studied. A notable property of these materials is the capability of enhancing electric field. In this paper the capability of AENZ films in local electric field enhancement has been quantified and several AENZ conditions are reported with superior performance in comparison to (isotropic) ENZ films. Specifically, sensitivity to film thickness and losses, and the range of angles of incidence have been elaborated with the aim of achieving large electric field enhancement in the film. It has been proved that in comparison to the (isotropic) ENZ case the AENZ film's field enhancement is not only much larger but it also occurs for a wider range of angles of incidence. Furthermore the field enhancement in AENZ does not exhibit significant dependence on the film thickness unlike the isotropic case. The effect of loss on the value of the field enhancement is also investigated emphasizing the advantages of AENZ versus ENZ. Realization of AENZ materials can be done by a multilayered media made of a stack of conductive and insulator layers or by stacking semiconductor layers. This giant field enhancement is an important target in nonlinear optics for applications like second harmonic generation and other applications like light generation
12. Phase field approach with anisotropic interface energy and interface stresses: Large strain formulation
Levitas, Valery I.; Warren, James A.
2016-06-01
A thermodynamically consistent, large-strain, multi-phase field approach (with consequent interface stresses) is generalized for the case with anisotropic interface (gradient) energy (e.g. an energy density that depends both on the magnitude and direction of the gradients in the phase fields). Such a generalization, if done in the "usual" manner, yields a theory that can be shown to be manifestly unphysical. These theories consider the gradient energy as anisotropic in the deformed configuration, and, due to this supposition, several fundamental contradictions arise. First, the Cauchy stress tensor is non-symmetric and, consequently, violates the moment of momentum principle, in essence the Herring (thermodynamic) torque is imparting an unphysical angular momentum to the system. In addition, this non-symmetric stress implies a violation of the principle of material objectivity. These problems in the formulation can be resolved by insisting that the gradient energy is an isotropic function of the gradient of the order parameters in the deformed configuration, but depends on the direction of the gradient of the order parameters (is anisotropic) in the undeformed configuration. We find that for a propagating nonequilibrium interface, the structural part of the interfacial Cauchy stress is symmetric and reduces to a biaxial tension with the magnitude equal to the temperature- and orientation-dependent interface energy. Ginzburg-Landau equations for the evolution of the order parameters and temperature evolution equation, as well as the boundary conditions for the order parameters are derived. Small strain simplifications are presented. Remarkably, this anisotropy yields a first order correction in the Ginzburg-Landau equation for small strains, which has been neglected in prior works. The next strain-related term is third order. For concreteness, specific orientation dependencies of the gradient energy coefficients are examined, using published molecular dynamics
13. Locating earthquakes in west Texas oil fields using 3-D anisotropic velocity models
SciTech Connect
Hua, Fa; Doser, D.; Baker, M. . Dept. of Geological Sciences)
1993-02-01
Earthquakes within the War-Wink gas field, Ward County, Texas, that have been located with a 1-D velocity model occur near the edges and top of a naturally occurring overpressured zone. Because the War-Wink field is a structurally controlled anticline with significant velocity anisotropy associated with the overpressured zone and finely layered evaporites, the authors have attempted to re-locate earthquakes using a 3-D anisotropic velocity model. Preliminary results with this model give the unsatisfactory result that many earthquakes previously located at the top of the overpressured zone (3-3.5 km) moved into the evaporites (1-1.5 km) above the field. They believe that this result could be caused by: (1) aliasing the velocity model; or (2) problems in determining the correct location minima when several minima exist. They are currently attempting to determine which of these causes is more likely for the unsatisfactory result observed.
14. Langevin simulation of scalar fields: Additive and multiplicative noises and lattice renormalization
Cassol-Seewald, N. C.; Farias, R. L. S.; Fraga, E. S.; Krein, G.; Ramos, Rudnei O.
2012-08-01
We consider the Langevin lattice dynamics for a spontaneously broken λϕ4 scalar field theory where both additive and multiplicative noise terms are incorporated. The lattice renormalization for the corresponding stochastic Ginzburg-Landau-Langevin and the subtleties related to the multiplicative noise are investigated.
15. CHARGED-PARTICLE TRANSPORT IN MAGNETIC TURBULENCE. I. A GLOBALLY ANISOTROPIC FIELD
SciTech Connect
Sun, P.; Jokipii, J. R.
2015-12-10
Collisionless magnetohydrodynamic Turbulence is common in large scale astrophysical environments. The determination of the transport of charged particles both parallel and perpendicular in such a system is of considerable interest. Quasi-linear analysis or direct numerical simulation can be used to find the effects of the turbulent magnetic field on the transport of charged particles. A number of different magnetic turbulence models have been proposed in the last several decades. We present here the results of studying particle transport in synthesized, anisotropic turbulence and compare the results with those obtained using the standard isotropic turbulence model in a series of papers. In this paper we consider the magnetic field turbulence model with global anisotropy.
16. Heat source reconstruction from noisy temperature fields using a gradient anisotropic diffusion filter
Beitone, C.; Balandraud, X.; Delpueyo, D.; Grédiac, M.
2017-01-01
This paper presents a post-processing technique for noisy temperature maps based on a gradient anisotropic diffusion (GAD) filter in the context of heat source reconstruction. The aim is to reconstruct heat source maps from temperature maps measured using infrared (IR) thermography. Synthetic temperature fields corrupted by added noise are first considered. The GAD filter, which relies on a diffusion process, is optimized to retrieve as well as possible a heat source concentration in a two-dimensional plate. The influence of the dimensions and the intensity of the heat source concentration are discussed. The results obtained are also compared with two other types of filters: averaging filter and Gaussian derivative filter. The second part of this study presents an application for experimental temperature maps measured with an IR camera. The results demonstrate the relevancy of the GAD filter in extracting heat sources from noisy temperature fields.
17. Acoustic propagation through anisotropic internal wave fields: transmission loss, cross-range coherence, and horizontal refraction.
PubMed
Oba, Roger; Finette, Steven
2002-02-01
Results of a computer simulation study are presented for acoustic propagation in a shallow water, anisotropic ocean environment. The water column is characterized by random volume fluctuations in the sound speed field that are induced by internal gravity waves, and this variability is superimposed on a dominant summer thermocline. Both the internal wave field and resulting sound speed perturbations are represented in three-dimensional (3D) space and evolve in time. The isopycnal displacements consist of two components: a spatially diffuse, horizontally isotropic component and a spatially localized contribution from an undular bore (i.e., a solitary wave packet or solibore) that exhibits horizontal (azimuthal) anisotropy. An acoustic field is propagated through this waveguide using a 3D parabolic equation code based on differential operators representing wide-angle coverage in elevation and narrow-angle coverage in azimuth. Transmission loss is evaluated both for fixed time snapshots of the environment and as a function of time over an ordered set of snapshots which represent the time-evolving sound speed distribution. Horizontal acoustic coherence, also known as transverse or cross-range coherence, is estimated for horizontally separated points in the direction normal to the source-receiver orientation. Both transmission loss and spatial coherence are computed at acoustic frequencies 200 and 400 Hz for ranges extending to 10 km, a cross-range of 1 km, and a water depth of 68 m. Azimuthal filtering of the propagated field occurs for this environment, with the strongest variations appearing when propagation is parallel to the solitary wave depressions of the thermocline. A large anisotropic degradation in horizontal coherence occurs under the same conditions. Horizontal refraction of the acoustic wave front is responsible for the degradation, as demonstrated by an energy gradient analysis of in-plane and out-of-plane energy transfer. The solitary wave packet is
18. Variational methods in supersymmetric lattice field theory: The vacuum sector
SciTech Connect
Duncan, A.; Meyer-Ortmanns, H.; Roskies, R.
1987-12-15
The application of variational methods to the computation of the spectrum in supersymmetric lattice theories is considered, with special attention to O(N) supersymmetric sigma models. Substantial cancellations are found between bosonic and fermionic contributions even in approximate Ansauml: tze for the vacuum wave function. The nonlinear limit of the linear sigma model is studied in detail, and it is shown how to construct an appropriate non-Gaussian vacuum wave function for the nonlinear model. The vacuum energy is shown to be of order unity in lattice units in the latter case, after infinite cancellations. 19. Ultrasonic field profile evaluation in acoustically inhomogeneous anisotropic materials using 2D ray tracing model: Numerical and experimental comparison. PubMed Kolkoori, S R; Rahman, M-U; Chinta, P K; Ktreutzbruck, M; Rethmeier, M; Prager, J 2013-02-01 Ultrasound propagation in inhomogeneous anisotropic materials is difficult to examine because of the directional dependency of elastic properties. Simulation tools play an important role in developing advanced reliable ultrasonic non destructive testing techniques for the inspection of anisotropic materials particularly austenitic cladded materials, austenitic welds and dissimilar welds. In this contribution we present an adapted 2D ray tracing model for evaluating ultrasonic wave fields quantitatively in inhomogeneous anisotropic materials. Inhomogeneity in the anisotropic material is represented by discretizing into several homogeneous layers. According to ray tracing model, ultrasonic ray paths are traced during its energy propagation through various discretized layers of the material and at each interface the problem of reflection and transmission is solved. The presented algorithm evaluates the transducer excited ultrasonic fields accurately by taking into account the directivity of the transducer, divergence of the ray bundle, density of rays and phase relations as well as transmission coefficients. The ray tracing model is able to calculate the ultrasonic wave fields generated by a point source as well as a finite dimension transducer. The ray tracing model results are validated quantitatively with the results obtained from 2D Elastodynamic Finite Integration Technique (EFIT) on several configurations generally occurring in the ultrasonic non destructive testing of anisotropic materials. Finally, the quantitative comparison of ray tracing model results with experiments on 32mm thick austenitic weld material and 62mm thick austenitic cladded material is discussed. 20. Charmonium excited state spectrum in lattice QCD SciTech Connect Jozef Dudek; Robert Edwards; Nilmani Mathur; David Richards 2008-02-01 Working with a large basis of covariant derivative-based meson interpolating fields we demonstrate the feasibility of reliably extracting multiple excited states using a variational method. The study is performed on quenched anisotropic lattices with clover quarks at the charm mass. We demonstrate how a knowledge of the continuum limit of a lattice interpolating field can give additional spin-assignment information, even at a single lattice spacing, via the overlap factors of interpolating field and state. Excited state masses are systematically high with respect to quark potential model predictions and, where they exist, experimental states. We conclude that this is most likely a result of the quenched approximation. 1. Anisotropic heat diffusion on stochastic magnetic field in the Large Helical Device NASA Astrophysics Data System (ADS) Suzuki, Yasuhiro 2016-10-01 The magnetic topology is a key issue in fusion plasma researches. An example is the Resonant Magnetic Perturbation (RMP) to control the transport and MHD activities in tokamak and stellarator experiments. However, the physics how the RMP affects the transport and MHD is not clear. One reason is a role of the magnetic topology is unclear. That problem is connecting to the identification of the magnetic topology in the experiment. In the experiment, the finite temperature gradient is observed on the stochastic field where is stochastized by the theoretical prediction. In a classical theory, the electron temperature gradient should be zero on the stochastic magnetic field. We need to study the stochastic magnetic field can keep the finite temperature gradient or not. In this study, we study the anisotropic heat diffusion equation to simulate the heat transport on the stochastic magnetic field. Changing a ratio of κ∥ and κ⊥, the distribution of the temperature on the stochastic magnetic field is obtained. Hudson et al. pointed out the KAM surface is a barrier to keep the finite temperature. We simulate those results in realistic magnetic field of the Large Helical Device. 2. Magnetic-Field-Assisted Assembly of Anisotropic Superstructures by Iron Oxide Nanoparticles and Their Enhanced Magnetism. PubMed Jiang, Chengpeng; Leung, Chi Wah; Pong, Philip W T 2016-12-01 Magnetic nanoparticle superstructures with controlled magnetic alignment and desired structural anisotropy hold promise for applications in data storage and energy storage. Assembly of monodisperse magnetic nanoparticles under a magnetic field could lead to highly ordered superstructures, providing distinctive magnetic properties. In this work, a low-cost fabrication technique was demonstrated to assemble sub-20-nm iron oxide nanoparticles into crystalline superstructures under an in-plane magnetic field. The gradient of the applied magnetic field contributes to the anisotropic formation of micron-sized superstructures. The magnitude of the applied magnetic field promotes the alignment of magnetic moments of the nanoparticles. The strong dipole-dipole interactions between the neighboring nanoparticles lead to a close-packed pattern as an energetically favorable configuration. Rod-shaped and spindle-shaped superstructures with uniform size and controlled spacing were obtained using spherical and polyhedral nanoparticles, respectively. The arrangement and alignment of the superstructures can be tuned by changing the experimental conditions. The two types of superstructures both show enhancement of coercivity and saturation magnetization along the applied field direction, which is presumably associated with the magnetic anisotropy and magnetic dipole interactions of the constituent nanoparticles and the increased shape anisotropy of the superstructures. Our results show that the magnetic-field-assisted assembly technique could be used for fabricating nanomaterial-based structures with controlled geometric dimensions and enhanced magnetic properties for magnetic and energy storage applications. 3. Analysis of periodic anisotropic media by means of split-field FDTD method and GPU computing NASA Astrophysics Data System (ADS) Francés, J.; Bleda, S.; Álvarez López, M. L.; Martínez, F. J.; Márquez, A.; Neipp, C.; Beléndez, A. 2012-10-01 The implementation of the Split-Field Finite Difference Time-Domain (SP-FDTD) method in Graphics Pro- cessing Units is described in this work. This formalism is applied to light wave propagation through periodic media with arbitrary anisotropy. The anisotropic media is modeled by means of a permittivity tensor with non-diagonal elements and absorbing boundary conditions are also considered. The split-field technique and the periodic boundary condition allow to consider a single period of the structure reducing the simulation grid. Nevertheless, the analysis of anisotropic media implies considering all the electromagnetic field components and the use of complex notation. These aspects reduce the computational efficiency of the numerical method compared to the isotropic and non-periodic implementation. With the upcoming of the new generation of General-Purpose Computing on Graphics Units many scientific applications have been accelerated and others are being developed into this new parallel digital computing architecture. Specifically, the implementation of the SP-FDTD in the Fermi family of GPUs of NVIDIA is presented. An analysis of the performance of this implementation is done and several applications have been considered in order to estimate the possibilities provided by both the formalism and the implementation into GPU. The formalism has been used for analyzing different structures and phenomena: binary phase gratings and twisted-nematic liquid crystal cells. The numerical predictions obtained by means of the FDTD method here implemented are compared with theoretical curves achieving good results, thus validating the accuracy and the potential of the implementation. 4. A FFT-based formulation for efficient mechanical fields computation in isotropic and anisotropic periodic discrete dislocation dynamics NASA Astrophysics Data System (ADS) Bertin, N.; Upadhyay, M. V.; Pradalier, C.; Capolungo, L. 2015-09-01 In this paper, we propose a novel full-field approach based on the fast Fourier transform (FFT) technique to compute mechanical fields in periodic discrete dislocation dynamics (DDD) simulations for anisotropic materials: the DDD-FFT approach. By coupling the FFT-based approach to the discrete continuous model, the present approach benefits from the high computational efficiency of the FFT algorithm, while allowing for a discrete representation of dislocation lines. It is demonstrated that the computational time associated with the new DDD-FFT approach is significantly lower than that of current DDD approaches when large number of dislocation segments are involved for isotropic and anisotropic elasticity, respectively. Furthermore, for fine Fourier grids, the treatment of anisotropic elasticity comes at a similar computational cost to that of isotropic simulation. Thus, the proposed approach paves the way towards achieving scale transition from DDD to mesoscale plasticity, especially due to the method’s ability to incorporate inhomogeneous elasticity. 5. Cooling achieved by rotating an anisotropic superconductor in a constant magnetic field: A new perspective NASA Astrophysics Data System (ADS) Phan, Manh-Huong; Mandrus, David 2016-12-01 A new type of rotary coolers based on the temperature change (Δ Trot ) of an anisotropic superconductor when rotated in a constant magnetic field is proposed. We show that at low temperature the Sommerfeld coefficient γ (B ,Θ ) of a single crystalline superconductor, such as MgB2 and NbS2, sensitively depends on the applied magnetic field (B) and the orientation of the crystal axis (Θ ) , which is related to the electronic entropy (SE) and temperature (T) via the expression: SE=γ T . A simple rotation of the crystal from one axis to one another in a constant magnetic field results in a change in γ and hence SE: Δ SE =Δ γ T . A temperature change -Δ Trot ˜ 0.94 K from a bath temperature of 2.5 K is achieved by simply rotating the single crystal MgB2 by 90° with respect to the c-axis direction in a fixed field of 2 T. Δ Trot can be tuned by adjusting the strength of B within a wide magnetic field range. Our study paves the way for development of new materials and cryogenic refrigerators that are potentially more energy-efficient, simplified, and compact. 6. Stable anisotropic plasma confinement in magnetic configurations with convex-concave field lines NASA Astrophysics Data System (ADS) Tsventoukh, M. M. 2014-02-01 It is shown that a combination of the convex and the concave part of a field line provides a strong stabilizing action against convective (flute-interchange) plasma instability (Tsventoukh 2011 Nucl. Fusion 51 112002). This results in internal peaking of the stable plasma pressure profile that is calculated from the collisionless kinetic stability criterion for any magnetic confinement system with combination of mirrors and cusps. Connection of the convex and concave field line parts results in a reduction of the space charge that drives the unstable E × B motion, as there is an opposite direction of the particle drift in a non-uniform field at convex and concave field lines. The pressure peaking arises at the minimum of the second adiabatic invariant J that takes place at the ‘middle’ of a tandem mirror-cusp transverse cross-section. The position of the minimum in J varies with the particle pitch angle that results in a shift of the peaking position depending on plasma anisotropy. This allows one to improve a stable peaked pressure profile at a convex-concave field by changing the plasma anisotropy over the trap cross-section. Examples of such anisotropic distribution functions are found that give an additional substantial enhancement in the maximal central pressure. Furthermore, the shape of new calculated stable profiles has a wide central plasma layer instead of a narrow peak. 7. Anisotropic Turbulent Advection of a Passive Vector Field: Effects of the Finite Correlation Time NASA Astrophysics Data System (ADS) Antonov, N. V.; Gulitskiy, N. M. 2016-02-01 The turbulent passive advection under the environment (velocity) field with finite correlation time is studied. Inertial-range asymptotic behavior of a vector (e.g., magnetic) field, passively advected by a strongly anisotropic turbulent flow, is investigated by means of the field theoretic renormalization group and the operator product expansion. The advecting velocity field is Gaussian, with finite correlation time and prescribed pair correlation function. The inertial-range behavior of the model is described by two regimes (the limits of vanishing or infinite correlation time) that correspond to nontrivial fixed points of the RG equations and depend on the relation between the exponents in the energy energy spectrum ɛ ∝ k⊥1-ξ and the dispersion law ω ∝ k⊥2-η . The corresponding anomalous exponents are associated with the critical dimensions of tensor composite operators built solely of the passive vector field itself. In contrast to the well-known isotropic Kraichnan model, where various correlation functions exhibit anomalous scaling behavior with infinite sets of anomalous exponents, here the dependence on the integral turbulence scale L has a logarithmic behavior: instead of power-like corrections to ordinary scaling, determined by naive (canonical) dimensions, the anomalies manifest themselves as polynomials of logarithms of L. Due to the presence of the anisotropy in the model, all multiloop diagrams are equal to zero, thus this result is exact. 8. Singularity free charged anisotropic solutions of Einstein-Maxwell field equations in general relativity NASA Astrophysics Data System (ADS) Singh, K. N.; Pant, N. 2016-07-01 In this paper, we present generalization of anisotropic analogue of charged Heintzmann's solution of the general relativistic field equations in curvature coordinates. These exact solutions are stable and well behaved in all respect for a wide range of anisotropy parameter and charge parameter. We have found that these new solutions are suitable for the modeling of super dense stars like neutron stars and quark stars because they yield a wide range of masses and radii with simple mathematical expressions. By tuning different values of the few parameters, we can model various neutron stars and quark stars which are compatible with the experimentally observed values of masses and radii. Therefore, we have synchronized our solution with the observed values of some of the compact stars XTE J1739 - 217, EXO 0748 - 676, PSR J1614 - 2230, PSR J0348 + 0432 and PSR B0943 + 10. 9. Near-field radiative heat transfer between graphene and anisotropic magneto-dielectric hyperbolic metamaterials NASA Astrophysics Data System (ADS) Song, Jinlin; Cheng, Qiang 2016-09-01 We numerically investigate the near-field radiative heat transfer (NFRHT) between graphene and anisotropic magneto-dielectric hyperbolic metamaterials (AMDHMs) according to the fluctuational dissipation theorem. In this configuration, multiple modes, including the p - and s -polarized surface phonon polaritons (SPhPs) and hyperbolic modes supported by AMDHMs as well as the high-frequency antisymmetric modes supported by graphene for p polarization, can be observed. These extraordinary propagating modes enable the total NFRHT flux between graphene and AMDHMs to exceed that between graphene and SiC nanowires by several times. Numerical results suggest that the hyperbolic modes and SPhPs for both polarizations effectively impact the NFRHT flux via tuning the geometry of AMDHMs and the conductivity of graphene. This study paves the way toward studying the NFRHT involving graphene and metamaterials and facilitates in-depth study of the s -polarized NFRHT. 10. The diagonal two-point correlations of the Ising model on the anisotropic triangular lattice and Garnier systems NASA Astrophysics Data System (ADS) Witte, N. S. 2016-01-01 The diagonal spin-spin correlations < {σ0,0}{σN,N}> of the Ising model on a triangular lattice with general couplings in the three directions are evaluated in terms of a solution to a three-variable extension of the sixth Painlevé system, namely a Garnier system. This identification, which is accomplished using the theory of bi-orthogonal polynomials on the unit circle with regular semi-classical weights, has an additional consequence whereby the correlations are characterised by a simple system of coupled, nonlinear recurrence relations in the spin separation N\\in {{{Z}}≥slant 0} . The later recurrence relations are an example of discrete Garnier equations which, in turn, are extensions to a ‘discrete Painlevé V’ system. 11. Driven optical lattices as strong-field simulators SciTech Connect Arlinghaus, Stephan; Holthaus, Martin 2010-06-15 We argue that ultracold atoms in strongly shaken optical lattices can be subjected to conditions similar to those experienced by electrons in laser-irradiated crystalline solids, but without introducing secondary polarization effects. As a consequence, one can induce nonperturbative multiphoton-like resonances due to the mutual penetration of ac-Stark-shifted Bloch bands. These phenomena can be detected with a combination of currently available laboratory techniques. 12. Ab initio nuclear structure from lattice effective field theory SciTech Connect Lee, Dean 2014-11-11 This proceedings article reviews recent results by the Nuclear Lattice EFT Collaboration on an excited state of the {sup 12}C nucleus known as the Hoyle state. The Hoyle state plays a key role in the production of carbon via the triple-alpha reaction in red giant stars. We discuss the structure of low-lying states of {sup 12}C as well as the dependence of the triple-alpha reaction on the masses of the light quarks. 13. Magnetic-field-driven crack formation in an evaporated anisotropic colloidal assembly NASA Astrophysics Data System (ADS) Lama, Hisay; Dugyala, Venkateshwar Rao; Basavaraj, Madivala G.; Satapathy, Dillip K. 2016-07-01 We report the effect of applied magnetic field on the morphology of cracks formed after evaporation of a colloidal suspension consisting of shape-anisotropic ellipsoidal particles on a glass substrate. The evaporation experiments are performed in sessile drop configuration, which usually leads to accumulation of particles at the drop boundaries, commonly known as the "coffee-ring effect." The coffee-ring-like deposits that accompany cracks are formed in the presence as well as in the absence of magnetic field. However, the crack patterns formed in both cases are found to differ markedly. The direction of cracks in the presence of the magnetic field is found to be governed by the orientation of particles and not solely by the magnetic field direction. Our experimental results show that at the vicinity of cracks the particles are ordered and oriented with their long-axis parallel to crack direction. In addition, we observe that the crack spacing in general increases with the height of the particulate film. 14. Logarithmic violation of scaling in strongly anisotropic turbulent transfer of a passive vector field. PubMed Antonov, N V; Gulitskiy, N M 2015-01-01 Inertial-range asymptotic behavior of a vector (e.g., magnetic) field, passively advected by a strongly anisotropic turbulent flow, is studied by means of the field-theoretic renormalization group and the operator product expansion. The advecting velocity field is Gaussian, not correlated in time, with the pair correlation function of the form ∝δ(t-t')/k(⊥)(d-1+ξ), where k(⊥)=|k(⊥)| and k(⊥) is the component of the wave vector, perpendicular to the distinguished direction ("direction of the flow")--the d-dimensional generalization of the ensemble introduced by Avellaneda and Majda [Commun. Math. Phys. 131, 381 (1990)]. The stochastic advection-diffusion equation for the transverse (divergence-free) vector field includes, as special cases, the kinematic dynamo model for magnetohydrodynamic turbulence and the linearized Navier-Stokes equation. In contrast to the well-known isotropic Kraichnan's model, where various correlation functions exhibit anomalous scaling behavior with infinite sets of anomalous exponents, here the dependence on the integral turbulence scale L has a logarithmic behavior: Instead of powerlike corrections to ordinary scaling, determined by naive (canonical) dimensions, the anomalies manifest themselves as polynomials of logarithms of L. The key point is that the matrices of scaling dimensions of the relevant families of composite operators appear nilpotent and cannot be diagonalized. The detailed proof of this fact is given for the correlation functions of arbitrary order. 15. Logarithmic violation of scaling in strongly anisotropic turbulent transfer of a passive vector field NASA Astrophysics Data System (ADS) Antonov, N. V.; Gulitskiy, N. M. 2015-01-01 Inertial-range asymptotic behavior of a vector (e.g., magnetic) field, passively advected by a strongly anisotropic turbulent flow, is studied by means of the field-theoretic renormalization group and the operator product expansion. The advecting velocity field is Gaussian, not correlated in time, with the pair correlation function of the form ∝δ (t -t') /k⊥d -1 +ξ , where k⊥=|k⊥| and k⊥ is the component of the wave vector, perpendicular to the distinguished direction ("direction of the flow")—the d -dimensional generalization of the ensemble introduced by Avellaneda and Majda [Commun. Math. Phys. 131, 381 (1990), 10.1007/BF02161420]. The stochastic advection-diffusion equation for the transverse (divergence-free) vector field includes, as special cases, the kinematic dynamo model for magnetohydrodynamic turbulence and the linearized Navier-Stokes equation. In contrast to the well-known isotropic Kraichnan's model, where various correlation functions exhibit anomalous scaling behavior with infinite sets of anomalous exponents, here the dependence on the integral turbulence scale L has a logarithmic behavior: Instead of powerlike corrections to ordinary scaling, determined by naive (canonical) dimensions, the anomalies manifest themselves as polynomials of logarithms of L . The key point is that the matrices of scaling dimensions of the relevant families of composite operators appear nilpotent and cannot be diagonalized. The detailed proof of this fact is given for the correlation functions of arbitrary order. 16. Dipole-field sums and Lorentz factors for orthorhombic lattices, and implications for polarizable molecules NASA Technical Reports Server (NTRS) Purvis, C. K.; Taylor, P. L. 1982-01-01 A method for computing the Lorentz tensor components in single crystals via rapidly convergent sums of Bessels functions is developed using the relationship between dipole-field sums and the tensor components. The Lorentz factors for simple, body-centered, and base-centered orthorhombic lattices are computed using this method, and the derivative Lorentz factors for simple orthorhombic lattices are also determined. Both the Lorentz factors and their derivatives are shown to be very sensitive to a lattice structure. The equivalent of the Clausius-Mossotti relation for general orthorhombic lattices is derived using the Lorentz-factor formalism, and the permanent molecular dipole moment is related to crystal polarization for the case of a ferroelectric of polarizable point dipoles. It is concluded that the polarization enhancement due to self-polarization familiar from classical theory may actually be a reduction in consequences of negative Lorentz factors in one or two lattice directions for noncubic crystals. 17. Hamiltonian Effective Field Theory Study of the N^{*}(1535) Resonance in Lattice QCD. PubMed Liu, Zhan-Wei; Kamleh, Waseem; Leinweber, Derek B; Stokes, Finn M; Thomas, Anthony W; Wu, Jia-Jun 2016-02-26 Drawing on experimental data for baryon resonances, Hamiltonian effective field theory (HEFT) is used to predict the positions of the finite-volume energy levels to be observed in lattice QCD simulations of the lowest-lying J^{P}=1/2^{-} nucleon excitation. In the initial analysis, the phenomenological parameters of the Hamiltonian model are constrained by experiment and the finite-volume eigenstate energies are a prediction of the model. The agreement between HEFT predictions and lattice QCD results obtained on volumes with spatial lengths of 2 and 3 fm is excellent. These lattice results also admit a more conventional analysis where the low-energy coefficients are constrained by lattice QCD results, enabling a determination of resonance properties from lattice QCD itself. Finally, the role and importance of various components of the Hamiltonian model are examined. 18. Hamiltonian Effective Field Theory Study of the N*(1535 ) Resonance in Lattice QCD NASA Astrophysics Data System (ADS) Liu, Zhan-Wei; Kamleh, Waseem; Leinweber, Derek B.; Stokes, Finn M.; Thomas, Anthony W.; Wu, Jia-Jun 2016-02-01 Drawing on experimental data for baryon resonances, Hamiltonian effective field theory (HEFT) is used to predict the positions of the finite-volume energy levels to be observed in lattice QCD simulations of the lowest-lying JP=1 /2- nucleon excitation. In the initial analysis, the phenomenological parameters of the Hamiltonian model are constrained by experiment and the finite-volume eigenstate energies are a prediction of the model. The agreement between HEFT predictions and lattice QCD results obtained on volumes with spatial lengths of 2 and 3 fm is excellent. These lattice results also admit a more conventional analysis where the low-energy coefficients are constrained by lattice QCD results, enabling a determination of resonance properties from lattice QCD itself. Finally, the role and importance of various components of the Hamiltonian model are examined. 19. Automatic NMO Correction and Full Common Depth Point NMO Velocity Field Estimation in Anisotropic Media NASA Astrophysics Data System (ADS) Sedek, Mohamed; Gross, Lutz; Tyson, Stephen 2017-01-01 We present a new computational method of automatic normal moveout (NMO) correction that not only accurately flattens and corrects the far offset data, but simultaneously provides NMO velocity (v_nmo) for each individual seismic trace. The method is based on a predefined number of NMO velocity sweeps using linear vertical interpolation of different NMO velocities at each seismic trace. At each sweep, we measure the semblance between the zero offset trace (pilot trace) and the next seismic trace using a trace-by-trace rather than sample-by-sample based semblance measure; then after all the sweeps are done, the one with the maximum semblance value is chosen, which is assumed to be the most suitable NMO velocity trace that accurately flattens seismic reflection events. Other traces follow the same process, and a final velocity field is then extracted. Isotropic, anisotropic and lateral heterogenous synthetic geological models were built to test the method. A range of synthetic background noise, ranging from 10 to 30 %, was applied to the models. In addition, the method was tested on Hess's VTI (vertical transverse isotropy) model. Furthermore, we tested our method on a real pre-stack seismic CDP gathered from a gas field in Alaska. The results from the presented examples show an excellent NMO correction and extracted a reasonably accurate NMO velocity field. 20. An asymptotic preserving method for strongly anisotropic diffusion equations based on field line integration NASA Astrophysics Data System (ADS) Tang, Min; Wang, Yihong 2017-02-01 In magnetized plasma, the magnetic field confines the particles around the field lines. The anisotropy intensity in the viscosity and heat conduction may reach the order of 1012. When the boundary conditions are periodic or Neumann, the strong diffusion leads to an ill-posed limiting problem. To remove the ill-conditionedness in the highly anisotropic diffusion equations, we introduce a simple but very efficient asymptotic preserving reformulation in this paper. The key idea is that, instead of discretizing the Neumann boundary conditions locally, we replace one of the Neumann boundary condition by the integration of the original problem along the field line, the singular 1 / ɛ terms can be replaced by O (1) terms after the integration, which yields a well-posed problem. Small modifications to the original code are required and no change of coordinates nor mesh adaptation are needed. Uniform convergence with respect to the anisotropy strength 1 / ɛ can be observed numerically and the condition number does not scale with the anisotropy. 1. Simulating Dirac fermions with Abelian and non-Abelian gauge fields in optical lattices SciTech Connect Alba, E.; Fernandez-Gonzalvo, X.; Mur-Petit, J.; Garcia-Ripoll, J.J.; Pachos, J.K. 2013-01-15 In this work we present an optical lattice setup to realize a full Dirac Hamiltonian in 2+1 dimensions. We show how all possible external potentials coupled to the Dirac field can arise from perturbations of the existing couplings of the honeycomb lattice pattern. This greatly simplifies the proposed implementations, requiring only spatial modulations of the intensity of the laser beams to induce complex non-Abelian potentials. We finally suggest several experiments to observe the properties of the quantum field theory in the setup. - Highlights: Black-Right-Pointing-Pointer This work provides a very flexible setup for simulating Dirac fermions. Black-Right-Pointing-Pointer The manuscript contains a detailed study of optical lattice deformations. Black-Right-Pointing-Pointer The link between lattice deformations and effective gauge Hamiltonians is studied. 2. Quantum spin-1 anisotropic ferromagnetic Heisenberg model in a crystal field: a variational approach. PubMed Carvalho, D C; Plascak, J A; Castro, L M 2013-09-01 A variational approach based on Bogoliubov inequality for the free energy is employed in order to treat the quantum spin-1 anisotropic ferromagnetic Heisenberg model in the presence of a crystal field. Within the Bogoliubov scheme an improved pair approximation has been used. The temperature-dependent thermodynamic functions have been obtained and provide much better results than the previous simple mean-field scheme. In one dimension, which is still nonintegrable for quantum spin-1, we get the exact results in the classical limit, or near-exact results in the quantum case, for the free energy, magnetization, and quadrupole moment, as well for the transition temperature. In two and three dimensions the corresponding global phase diagrams have been obtained as a function of the parameters of the Hamiltonian. First-order transition lines, second-order transition lines, tricritical and tetracritical points, and critical endpoints have been located through the analysis of the minimum of the Helmholtz free energy and a Landau-like expansion in the approximated free energy. Only first-order quantum transitions have been found at zero temperature. Limiting cases, such as isotropic Heisenberg, Blume-Capel, and Ising models, have been analyzed and compared to previous results obtained from other analytical approaches as well as from Monte Carlo simulations. 3. Twofold and Fourfold Symmetric Anisotropic Magnetoresistance Effect in a Model with Crystal Field NASA Astrophysics Data System (ADS) Kokado, Satoshi; Tsunoda, Masakiyo 2015-09-01 We theoretically study the twofold and fourfold symmetric anisotropic magnetoresistance (AMR) effects of ferromagnets. We here use the two-current model for a system consisting of a conduction state and localized d states. The localized d states are obtained from a Hamiltonian with a spin-orbit interaction, an exchange field, and a crystal field. From the model, we first derive general expressions for the coefficient of the twofold symmetric term (C2) and that of the fourfold symmetric term (C4) in the AMR ratio. In the case of a strong ferromagnet, the dominant term in C2 is proportional to the difference in the partial densities of states (PDOSs) at the Fermi energy (EF) between the dɛ and dγ states, and that in C4 is proportional to the difference in the PDOSs at EF among the dɛ states. Using the dominant terms, we next analyze the experimental results for Fe4N, in which |C2| and |C4| increase with decreasing temperature. The experimental results can be reproduced by assuming that the tetragonal distortion increases with decreasing temperature. 4. Asymptotic-preserving Lagrangian approach for modeling anisotropic transport in magnetized plasmas for arbitrary magnetic fields NASA Astrophysics Data System (ADS) Chacon, Luis; Del-Castillo-Negrete, Diego; Hauck, Cory 2012-10-01 Modeling electron transport in magnetized plasmas is extremely challenging due to the extreme anisotropy between parallel (to the magnetic field) and perpendicular directions (χ/χ˜10^10 in fusion plasmas). Recently, a Lagrangian Green's function approach, developed for the purely parallel transport case,footnotetextD. del-Castillo-Negrete, L. Chac'on, PRL, 106, 195004 (2011)^,footnotetextD. del-Castillo-Negrete, L. Chac'on, Phys. Plasmas, 19, 056112 (2012) has been extended to the anisotropic transport case in the tokamak-ordering limit with constant density.footnotetextL. Chac'on, D. del-Castillo-Negrete, C. Hauck, JCP, submitted (2012) An operator-split algorithm is proposed that allows one to treat Eulerian and Lagrangian components separately. The approach is shown to feature bounded numerical errors for arbitrary χ/χ ratios, which renders it asymptotic-preserving. In this poster, we will present the generalization of the Lagrangian approach to arbitrary magnetic fields. We will demonstrate the potential of the approach with various challenging configurations, including the case of transport across a magnetic island in cylindrical geometry. 5. Internal Radiation Field in the Nonlinear Transfer Problem for a One-Dimensional Anisotropic Medium. II NASA Astrophysics Data System (ADS) Pikichyan, H. V. 2016-06-01 It is shown that for the nonlinear boundary value problem of determining the radiation field inside a one-dimensional anisotropic medium illuminated from outside at its boundaries on both sides, the formulas for adding layers in semilinear systems of differential equations for radiative transfer, invariant embedding, and total Ambartsumyan invariance can be used to reduce the equations for the problem to separable equations with initial conditions. The fields travelling to the left and right are thereby found independently of one another. In addition, when one of them has been determined, the other can be found directly using an explicit expression. A general equivalence property of operators with respect to a certain mathematical form, expression, or functional is formulated mathematically. New equations, referred to as kinetic equations of equivalency, are derived from the mutual equivalence of the differential operators of the Boltzmann kinetic equation (the equations of radiative transfer) and the functional equation of the Ambartsumian's complete invariance. Besides separability, these new equations also have the property of linearity. Formulas are also introduced for special problems of single sided illumination of a medium that in this case serve as supplementary information in the initial conditions for formulating Cauchy problems. 6. A novel quark-field creation operator construction for hadronic physics in lattice QCD SciTech Connect Michael Peardon, Jozef Dudek, Robert Edwards, Huey-Wen Lin, David Richards, John Bulava, Colin Morningstar, Keisuke Juge 2009-09-01 A new quark-field smearing algorithm is defined which enables efficient calculations of a broad range of hadron correlation functions. The technique applies a low-rank operator to define smooth fields, that are to be used in hadron creation operators. The resulting space of smooth fields is small enough that all elements of the reduced quark propagator can be computed exactly at reasonable computational cost. Correlations between arbitrary sources, including multi-hadron operators can be computed {\\em a posteriori} without requiring new lattice Dirac operator inversions. The method is tested on realistic lattice sizes with light dynamical quarks. 7. Anomalously large anisotropic magnetoresistance in a perovskite manganite. PubMed Li, Run-Wei; Wang, Huabing; Wang, Xuewen; Yu, X Z; Matsui, Y; Cheng, Zhao-Hua; Shen, Bao-Gen; Plummer, E Ward; Zhang, Jiandi 2009-08-25 The signature of correlated electron materials (CEMs) is the coupling between spin, charge, orbital and lattice resulting in exotic functionality. This complexity is directly responsible for their tunability. We demonstrate here that the broken symmetry, through cubic to orthorhombic distortion in the lattice structure in a prototype manganite single crystal, La(0.69)Ca(0.31)MnO(3), leads to an anisotropic magneto-elastic response to an external field, and consequently to remarkable magneto-transport behavior. An anomalous anisotropic magnetoresistance (AMR) effect occurs close to the metal-insulator transition (MIT) in the system, showing a direct correlation with the anisotropic field-tuned MIT in the system and can be understood by means of a simple phenomenological model. A small crystalline anisotropy stimulates a "colossal" AMR near the MIT phase boundary of the system, thus revealing the intimate interplay between magneto- and electronic-crystalline couplings. 8. A field proof-of-concept of tomographic slug tests in an anisotropic littoral aquifer NASA Astrophysics Data System (ADS) Paradis, Daniel; Gloaguen, Erwan; Lefebvre, René; Giroux, Bernard 2016-05-01 Hydraulic tomography is increasingly recognized as a characterization approach that can image pathways or barriers to flow as well as their connectivity. In this study, we assess the performance of a transient analysis of tomographic slug test head data in estimating heterogeneity in horizontal hydraulic conductivity (Kh), hydraulic conductivity anisotropy (the ratio between vertical and horizontal hydraulic conductivity - Kv/Kh) and specific storage (Ss) under actual field conditions. The tomographic experiment was carried out between two wells in a moderately heterogeneous and highly anisotropic silt and sand littoral aquifer. In this field proof-of-concept, the inversion of the two-dimensional (2D) head dataset was computed with a 2D radial flow algorithm that considers Kh, Kv/Kh, Ss and wellbore storage effects. This study demonstrated that a transient analysis of tomographic slug tests is able to capture the key features of the littoral environment of the test: the vertical profiles of Kh and Kv are indeed in agreement with those from other field and laboratory tests, and Ss values exhibit physically plausible profiles. Furthermore, the simulation of independent inter-well hydraulic tests (slug and pumping tests screened over the entire aquifer) using resolved Kh, Kv/Kh and Ss tomograms produce responses very close to field observations. This study demonstrates that the effects of fine scale heterogeneity that induces K-anisotropy at larger scales can be captured through a transient analysis of tomographic slug tests, which are very difficult to quantify otherwise with conventional hydraulic tests, thus allowing a better representation of properties controlling flow and transport in aquifer systems. 9. Three-dimensional magnetic trap lattice on an atom chip with an optically induced fictitious magnetic field SciTech Connect Yan Hui 2010-05-15 A robust type of three-dimensional magnetic trap lattice on an atom chip combining optically induced fictitious magnetic field with microcurrent-carrying wires is proposed. Compared to the regular optical lattice, the individual trap in this three-dimensional magnetic trap lattice can be easily addressed and manipulated. 10. Three-dimensional magnetic trap lattice on an atom chip with an optically induced fictitious magnetic field NASA Astrophysics Data System (ADS) Yan, Hui 2010-05-01 A robust type of three-dimensional magnetic trap lattice on an atom chip combining optically induced fictitious magnetic field with microcurrent-carrying wires is proposed. Compared to the regular optical lattice, the individual trap in this three-dimensional magnetic trap lattice can be easily addressed and manipulated. 11. Hamiltonian effective field theory study of the N*(1440 ) resonance in lattice QCD NASA Astrophysics Data System (ADS) Liu, Zhan-Wei; Kamleh, Waseem; Leinweber, Derek B.; Stokes, Finn M.; Thomas, Anthony W.; Wu, Jia-Jun 2017-02-01 We examine the phase shifts and inelasticities associated with the N*(1440 ) Roper resonance, and we connect these infinite-volume observables to the finite-volume spectrum of lattice QCD using Hamiltonian effective field theory. We explore three hypotheses for the structure of the Roper resonance. All three hypotheses are able to describe the scattering data well. In the third hypothesis the Roper resonance couples the low-lying bare basis-state component associated with the ground-state nucleon with the virtual meson-baryon contributions. Here the nontrivial superpositions of the meson-baryon scattering states are complemented by bare basis-state components, explaining their observation in contemporary lattice QCD calculations. The merit of this scenario lies in its ability to not only describe the observed nucleon energy levels in large-volume lattice QCD simulations but also explain why other low-lying states have been missed in today's lattice QCD results for the nucleon spectrum. 12. Lattice Study of Magnetic Catalysis in Graphene Effective Field Theory NASA Astrophysics Data System (ADS) Winterowd, Christopher; Detar, Carleton; Zafeiropoulos, Savvas 2016-03-01 The discovery of graphene ranks as one of the most important developments in condensed matter physics in recent years. As a strongly interacting system whose low-energy excitations are described by the Dirac equation, graphene has many similarities with other strongly interacting field theories, particularly quantum chromodynamics (QCD). Graphene, along with other relativistic field theories, have been predicted to exhibit spontaneous symmetry breaking (SSB) when an external magnetic field is present. Using nonperturbative methods developed to study QCD, we study the low-energy effective field theory (EFT) of graphene subject to an external magnetic field. We find strong evidence supporting the existence of SSB at zero-temperature and characterize the dependence of the chiral condensate on the external magnetic field. We also present results for the mass of the Nambu-Goldstone boson and the dynamically generated quasiparticle mass that result from the SSB. 13. Vortex Formation of Rotating Bose-Einstein Condensates in Synthetic Magnetic Field with Optical Lattice NASA Astrophysics Data System (ADS) Zhao, Qiang 2016-02-01 Motivated by recent experiments carried out by Spielman's group at NIST, we study the vortex formation in a rotating Bose-Einstein condensate in synthetic magnetic field confined in a harmonic potential combined with an optical lattice. We obtain numerical solutions of the two-dimensional Gross-Pitaevskii equation and compare the vortex formation by synthetic magnetic field method with those by rotating frame method. We conclude that a large angular momentum indeed can be created in the presence of the optical lattice. However, it is still more difficult to rotate the condensate by the synthetic magnetic field than by the rotating frame even if the optical lattice is added, and the chemical potential and energy remain almost unchanged by increasing rotational frequency. 14. A Numerical Treatment of Anisotropic Radiation Fields Coupled with Relativistic Resistive Magnetofluids NASA Astrophysics Data System (ADS) Takahashi, Hiroyuki R.; Ohsuga, Ken 2013-08-01 We develop a numerical scheme for solving fully special relativistic, resistive radiation magnetohydrodynamics. Our code guarantees conservation of total mass, momentum, and energy. The radiation energy density and the radiation flux are consistently updated using the M-1 closure method, which can resolve an anisotropic radiation field, in contrast to the Eddington approximation, as well as the flux-limited diffusion approximation. For the resistive part, we adopt a simple form of Ohm's law. The advection terms are explicitly solved with an approximate Riemann solver, mainly the Harten-Lax-van Leer scheme; the HLLC and HLLD schemes are also solved for some tests. The source terms, which describe the gas-radiation interaction and the magnetic energy dissipation, are implicitly integrated, relaxing the Courant-Friedrichs-Lewy condition even in an optically thick regime or a large magnetic Reynolds number regime. Although we need to invert 4 × 4 matrices (for the gas-radiation interaction) and 3 × 3 matrices (for the magnetic energy dissipation) at each grid point for implicit integration, they are obtained analytically without preventing massive parallel computing. We show that our code gives reasonable outcomes in numerical tests for ideal magnetohydrodynamics, propagating radiation, and radiation hydrodynamics. We also applied our resistive code to the relativistic Petschek-type magnetic reconnection, revealing the reduction of the reconnection rate via radiation drag. 15. Viscous flux motion in anisotropic type-II superconductors in low fields SciTech Connect Hao, Zhidong; Clem, J.R. Iowa State Univ. of Science and Technology, Ames, IA . Dept. of Physics) 1990-01-01 The Bardeen-Stephen model of viscous flux motion in isotropic Type-II superconductors is extended to the anisotropic case characterized by a phenomenological effective mass tensor m{sub ij}. When the magnetic field is low and the vortex lines are aligned along one of the three principal axes, simple expressions for the viscosity tensor {eta}{sub ij} of the viscous flux motion are obtained as functions of m{sub ij} and the normal state conductivity tensor {sigma}{sub ij} for temperature T close to the critical temperature {Tc}. For the high-temperature oxide superconductors the theory predicts that {eta}{sub b}{sup (a)}:{eta}{sub b}{sup (c)}:{eta}{sub c}{sup (a)} {approx} 1:4{gamma}:3{gamma}{sup 2}, where {eta}{sub i}{sup (j)} is the viscosity for the motion along the i-axis of a vortex parallel to the j-axis and {gamma} = {radical}m{sub c}/m{sub a} is the anisotropy parameter (m{sub i}, i = a,b,c, are the principal values of the mass tensor satisfying m{sub a} {approx} m{sub b} {much lt} m{sub c}). 9 refs., 1 fig. 16. A NUMERICAL TREATMENT OF ANISOTROPIC RADIATION FIELDS COUPLED WITH RELATIVISTIC RESISTIVE MAGNETOFLUIDS SciTech Connect Takahashi, Hiroyuki R.; Ohsuga, Ken 2013-08-01 We develop a numerical scheme for solving fully special relativistic, resistive radiation magnetohydrodynamics. Our code guarantees conservation of total mass, momentum, and energy. The radiation energy density and the radiation flux are consistently updated using the M-1 closure method, which can resolve an anisotropic radiation field, in contrast to the Eddington approximation, as well as the flux-limited diffusion approximation. For the resistive part, we adopt a simple form of Ohm's law. The advection terms are explicitly solved with an approximate Riemann solver, mainly the Harten-Lax-van Leer scheme; the HLLC and HLLD schemes are also solved for some tests. The source terms, which describe the gas-radiation interaction and the magnetic energy dissipation, are implicitly integrated, relaxing the Courant-Friedrichs-Lewy condition even in an optically thick regime or a large magnetic Reynolds number regime. Although we need to invert 4 Multiplication-Sign 4 matrices (for the gas-radiation interaction) and 3 Multiplication-Sign 3 matrices (for the magnetic energy dissipation) at each grid point for implicit integration, they are obtained analytically without preventing massive parallel computing. We show that our code gives reasonable outcomes in numerical tests for ideal magnetohydrodynamics, propagating radiation, and radiation hydrodynamics. We also applied our resistive code to the relativistic Petschek-type magnetic reconnection, revealing the reduction of the reconnection rate via radiation drag. 17. Vortex dynamics in anisotropic traps SciTech Connect McEndoo, S.; Busch, Th. 2010-07-15 We investigate the dynamics of linear vortex lattices in anisotropic traps in two dimensions and show that the interplay between the rotation and the anisotropy leads to a rich but highly regular dynamics. 18. Random-field Ising model on isometric lattices: Ground states and non-Porod scattering. PubMed Bupathy, Arunkumar; Banerjee, Varsha; Puri, Sanjay 2016-01-01 We use a computationally efficient graph cut method to obtain ground state morphologies of the random-field Ising model (RFIM) on (i) simple cubic (SC), (ii) body-centered cubic (BCC), and (iii) face-centered cubic (FCC) lattices. We determine the critical disorder strength Δ_{c} at zero temperature with high accuracy. For the SC lattice, our estimate (Δ_{c}=2.278±0.002) is consistent with earlier reports. For the BCC and FCC lattices, Δ_{c}=3.316±0.002 and 5.160±0.002, respectively, which are the most accurate estimates in the literature to date. The small-r behavior of the correlation function exhibits a cusp regime characterized by a cusp exponent α signifying fractal interfaces. In the paramagnetic phase, α=0.5±0.01 for all three lattices. In the ferromagnetic phase, the cusp exponent shows small variations due to the lattice structure. Consequently, the interfacial energy E_{i}(L) for an interface of size L is significantly different for the three lattices. This has important implications for nonequilibrium properties. 19. Random-field Ising model on isometric lattices: Ground states and non-Porod scattering NASA Astrophysics Data System (ADS) Bupathy, Arunkumar; Banerjee, Varsha; Puri, Sanjay 2016-01-01 We use a computationally efficient graph cut method to obtain ground state morphologies of the random-field Ising model (RFIM) on (i) simple cubic (SC), (ii) body-centered cubic (BCC), and (iii) face-centered cubic (FCC) lattices. We determine the critical disorder strength Δc at zero temperature with high accuracy. For the SC lattice, our estimate (Δc=2.278 ±0.002 ) is consistent with earlier reports. For the BCC and FCC lattices, Δc=3.316 ±0.002 and 5.160 ±0.002 , respectively, which are the most accurate estimates in the literature to date. The small-r behavior of the correlation function exhibits a cusp regime characterized by a cusp exponent α signifying fractal interfaces. In the paramagnetic phase, α =0.5 ±0.01 for all three lattices. In the ferromagnetic phase, the cusp exponent shows small variations due to the lattice structure. Consequently, the interfacial energy Ei(L ) for an interface of size L is significantly different for the three lattices. This has important implications for nonequilibrium properties. 20. Observation of exceptional points in reconfigurable non-Hermitian vector-field holographic lattices NASA Astrophysics Data System (ADS) Hahn, Choloong; Choi, Youngsun; Yoon, Jae Woong; Song, Seok Ho; Oh, Cha Hwan; Berini, Pierre 2016-07-01 Recently, synthetic optical materials represented via non-Hermitian Hamiltonians have attracted significant attention because of their nonorthogonal eigensystems, enabling unidirectionality, nonreciprocity and unconventional beam dynamics. Such systems demand carefully configured complex optical potentials to create skewed vector spaces with a desired metric distortion. In this paper, we report optically generated non-Hermitian photonic lattices with versatile control of real and imaginary sub-lattices. In the proposed method, such lattices are generated by vector-field holographic interference of two elliptically polarized pump beams on azobenzene-doped polymer thin films. We experimentally observe violation of Friedel's law of diffraction, indicating the onset of complex lattice formation. We further create an exact parity-time symmetric lattice to demonstrate totally asymmetric diffraction at the spontaneous symmetry-breaking threshold, referred to as an exceptional point. On this basis, we provide the experimental demonstration of reconfigurable non-Hermitian photonic lattices in the optical domain and observe the purest exceptional point ever reported to date. 1. Observation of exceptional points in reconfigurable non-Hermitian vector-field holographic lattices PubMed Central Hahn, Choloong; Choi, Youngsun; Yoon, Jae Woong; Song, Seok Ho; Oh, Cha Hwan; Berini, Pierre 2016-01-01 Recently, synthetic optical materials represented via non-Hermitian Hamiltonians have attracted significant attention because of their nonorthogonal eigensystems, enabling unidirectionality, nonreciprocity and unconventional beam dynamics. Such systems demand carefully configured complex optical potentials to create skewed vector spaces with a desired metric distortion. In this paper, we report optically generated non-Hermitian photonic lattices with versatile control of real and imaginary sub-lattices. In the proposed method, such lattices are generated by vector-field holographic interference of two elliptically polarized pump beams on azobenzene-doped polymer thin films. We experimentally observe violation of Friedel's law of diffraction, indicating the onset of complex lattice formation. We further create an exact parity-time symmetric lattice to demonstrate totally asymmetric diffraction at the spontaneous symmetry-breaking threshold, referred to as an exceptional point. On this basis, we provide the experimental demonstration of reconfigurable non-Hermitian photonic lattices in the optical domain and observe the purest exceptional point ever reported to date. PMID:27425577 2. Color fields of the static pentaquark system computed in SU(3) lattice QCD NASA Astrophysics Data System (ADS) Cardoso, Nuno; Bicudo, Pedro 2013-02-01 We compute the color fields of SU(3) lattice QCD created by static pentaquark systems, in a 243×48 lattice at β=6.2 corresponding to a lattice spacing a=0.07261(85)fm. We find that the pentaquark color fields are well described by a multi-Y-type shaped flux tube. The flux tube junction points are compatible with Fermat-Steiner points minimizing the total flux tube length. We also compare the pentaquark flux tube profile with the diquark-diantiquark central flux tube profile in the tetraquark and the quark-antiquark fundamental flux tube profile in the meson, and they match, thus showing that the pentaquark flux tubes are composed of fundamental flux tubes. 3. Lattice Models for Granular-Like Velocity Fields: Hydrodynamic Description NASA Astrophysics Data System (ADS) Manacorda, Alessandro; Plata, Carlos A.; Lasanta, Antonio; Puglisi, Andrea; Prados, Antonio 2016-08-01 A recently introduced model describing—on a 1d lattice—the velocity field of a granular fluid is discussed in detail. The dynamics of the velocity field occurs through next-neighbours inelastic collisions which conserve momentum but dissipate energy. The dynamics is described through the corresponding Master Equation for the time evolution of the probability distribution. In the continuum limit, equations for the average velocity and temperature fields with fluctuating currents are derived, which are analogous to hydrodynamic equations of granular fluids when restricted to the shear modes. Therefore, the homogeneous cooling state, with its linear instability, and other relevant regimes such as the uniform shear flow and the Couette flow states are described. The evolution in time and space of the single particle probability distribution, in all those regimes, is also discussed, showing that the local equilibrium is not valid in general. The noise for the momentum and energy currents, which are correlated, are white and Gaussian. The same is true for the noise of the energy sink, which is usually negligible. 4. Boundary-field-driven control of discontinuous phase transitions on hyperbolic lattices. PubMed Lee, Yoju; Verstraete, Frank; Gendiar, Andrej 2016-08-01 The multistate Potts models on two-dimensional hyperbolic lattices are studied with respect to various boundary effects. The free energy is numerically calculated using the corner transfer matrix renormalization group method. We analyze phase transitions of the Potts models in the thermodynamic limit with respect to contracted boundary layers. A false phase transition is present even if a couple of the boundary layers are contracted. Its significance weakens, as the number of the contracted boundary layers increases, until the correct phase transition (deep inside the bulk) prevails over the false one. For this purpose, we derive a thermodynamic quantity, the so-called bulk excess free energy, which depends on the contracted boundary layers and memorizes additional boundary effects. In particular, the magnetic field is imposed on the outermost boundary layer. While the boundary magnetic field does not affect the second-order phase transition in the bulk if suppressing all the boundary effects on the hyperbolic lattices, the first-order (discontinuous) phase transition is significantly sensitive to the boundary magnetic field. Contrary to the phase transition on the Euclidean lattices, the discontinuous phase transition on the hyperbolic lattices can be continuously controlled (within a certain temperature coexistence region) by varying the boundary magnetic field. 5. Boundary-field-driven control of discontinuous phase transitions on hyperbolic lattices NASA Astrophysics Data System (ADS) Lee, Yoju; Verstraete, Frank; Gendiar, Andrej 2016-08-01 The multistate Potts models on two-dimensional hyperbolic lattices are studied with respect to various boundary effects. The free energy is numerically calculated using the corner transfer matrix renormalization group method. We analyze phase transitions of the Potts models in the thermodynamic limit with respect to contracted boundary layers. A false phase transition is present even if a couple of the boundary layers are contracted. Its significance weakens, as the number of the contracted boundary layers increases, until the correct phase transition (deep inside the bulk) prevails over the false one. For this purpose, we derive a thermodynamic quantity, the so-called bulk excess free energy, which depends on the contracted boundary layers and memorizes additional boundary effects. In particular, the magnetic field is imposed on the outermost boundary layer. While the boundary magnetic field does not affect the second-order phase transition in the bulk if suppressing all the boundary effects on the hyperbolic lattices, the first-order (discontinuous) phase transition is significantly sensitive to the boundary magnetic field. Contrary to the phase transition on the Euclidean lattices, the discontinuous phase transition on the hyperbolic lattices can be continuously controlled (within a certain temperature coexistence region) by varying the boundary magnetic field. 6. Lattice Effective Field Theory Calculations for A=3, 4, 6, 12 Nuclei SciTech Connect Epelbaum, Evgeny; Krebs, Hermann; Lee, Dean; Meissner, Ulf-G. 2010-04-09 We present lattice results for the ground state energies of tritium, helium-3, helium-4, lithium-6, and carbon-12 nuclei. Our analysis includes isospin breaking, Coulomb effects, and interactions up to next-to-next-to-leading order in chiral effective field theory. 7. Realization of the Harper Hamiltonian with Artificial Gauge Fields in Optical Lattices NASA Astrophysics Data System (ADS) Miyake, Hirokazu; Siviloglou, Georgios; Kennedy, Colin; Burton, William Cody; Ketterle, Wolfgang 2014-03-01 Systems of charged particles in magnetic fields have led to many discoveries in science-such as the integer and fractional quantum Hall effects-and have become important paradigms of quantum many-body physics. We have proposed and implemented a scheme which realizes the Harper Hamiltonian, a lattice model for charged particles in magnetic fields, whose energy spectrum is the fractal Hofstadter butterfly. We experimentally realize this Hamiltonian for ultracold, charge neutral bosonic particles of 87Rb in a two-dimensional optical lattice by creating an artificial gauge field using laser-assisted tunneling and a potential energy gradient provided by gravity. Laser-assisted tunneling processes are characterized by studying the expansion of the atoms in the lattice. Furthermore, this scheme can be extended to realize spin-orbit coupling and the spin Hall effect for neutral atoms in optical lattices by modifying the motion of atoms in a spin-dependent way by laser recoil and Zeeman shifts created with a magnetic field gradient. Major advantages of our scheme are that it does not rely on near-resonant laser light to couple different spin states and should work even for fermionic particles. Our work is a step towards studying novel topological phenomena with ultracold atoms. Currently at the RAND Corporation. 8. On the distribution of scaling hydraulic parameters in a spatially anisotropic banana field NASA Astrophysics Data System (ADS) Regalado, Carlos M. 2005-06-01 density function for the scaling parameters, αi. Some indications for the origin of these disagreements, in terms of population size and test constraints, are pointed out. Visual inspection of normal probability plots can also lead to erroneous results. The scaling parameters αθ and αK show a sinusoidal spatial variation coincident with the underlying alignment of banana plants on the field. Such anisotropic distribution is explained in terms of porosity variations due to processes promoting soil degradation as surface desiccation and soil compaction, induced by tillage and localized irrigation of banana plants, and it is quantified by means of cross-correlograms. 9. Quantum Fisher information for periodic and quasiperiodic anisotropic XY chains in a transverse field NASA Astrophysics Data System (ADS) Liu, X. M.; Du, Z. Z.; Liu, J.-M. 2016-04-01 In this work, the concept of quantum Fisher information (QFI) is used to characterize the quantum transitions and factorization transitions in one-dimensional anisotropic XY models with periodic coupling interaction and quasiperiodic one. For the periodic-two model, it is found that the Ising transition and anisotropic transition can be distinctively illustrated by the evolution of QFI and its first-order derivatives, confirmed additionally by the scaling behavior. For the quasiperiodic Fibonacci chain, the number of quantum phase transitions increases from one to the lth Fibonacci number Fl when the anisotropic parameter γ approaches zero. The phase diagram for the approximant Fl=8 is derived as an example. In addition, the factorization transition in the two models can be marked by the correlation quantity defined from the QFI. The present work demonstrates the implication of the QFI as a general fingerprint to characterize the quantum transitions and factorization transitions. 10. Time evolution of linearized gauge field fluctuations on a real-time lattice NASA Astrophysics Data System (ADS) Kurkela, A.; Lappi, T.; Peuron, J. 2016-12-01 Classical real-time lattice simulations play an important role in understanding non-equilibrium phenomena in gauge theories and are used in particular to model the prethermal evolution of heavy-ion collisions. Due to instabilities, small quantum fluctuations on top of the classical background may significantly affect the dynamics of the system. In this paper we argue for the need for a numerical calculation of a system of classical gauge fields and small linearized fluctuations in a way that keeps the separation between the two manifest. We derive and test an explicit algorithm to solve these equations on the lattice, maintaining gauge invariance and Gauss' law. 11. Method of making V-MOS field effect transistors utilizing a two-step anisotropic etching and ion implantation NASA Technical Reports Server (NTRS) Jhabvala, M. D. (Inventor) 1981-01-01 A method of making V-MOS field effect transistors is disclosed wherein a masking layer is first formed over a surface of a crystalline substrate. An aperture is then formed in the masking layer to expose the surface of the substrate. An anisotropic etchant is applied to the exposed surface so that a groove having a decreasing width within increasing depth is formed. However, the etch is not allowed to go to completion with the result that a partially formed V-shaped groove is formed. Ions are accelerated through the aperture for implantation in the crystalline substrate in the lower portion of the partially formed V-shaped groove. Thereafter, an anisotropic etchant is reapplied to the partially formed V-shaped groove, and the etch is allowed to go to completion. 12. Experimental Realization of Strong Effective Magnetic Fields in an Optical Lattice SciTech Connect Aidelsburger, M.; Atala, M.; Trotzky, S.; Chen, Y.-A.; Bloch, I.; Nascimbene, S. 2011-12-16 We use Raman-assisted tunneling in an optical superlattice to generate large tunable effective magnetic fields for ultracold atoms. When hopping in the lattice, the accumulated phase shift by an atom is equivalent to the Aharonov-Bohm phase of a charged particle exposed to a staggered magnetic field of large magnitude, on the order of 1 flux quantum per plaquette. We study the ground state of this system and observe that the frustration induced by the magnetic field can lead to a degenerate ground state for noninteracting particles. We provide a measurement of the local phase acquired from Raman-induced tunneling, demonstrating time-reversal symmetry breaking of the underlying Hamiltonian. Furthermore, the quantum cyclotron orbit of single atoms in the lattice exposed to the magnetic field is directly revealed. 13. Fractal Nature of the Electronic Structure of a Penrose Tiling Lattice in a Magnetic Field NASA Astrophysics Data System (ADS) Hatakeyama, Tetsuo; Kamimura, Hiroshi 1989-01-01 The one-electron energy spectrum of a Penrose tiling lattice in a magnetic field is studied with a tight-binding Hamiltonian. We show the following remarkable results characteristic of a Penrose lattice. (1) The density of states in a magnetic field has a central peak with zero width at zero energy. It is shown that the zero-energy states correspond to the ring states in which wavefunction has nonvanishing amplitudes only at the sites circling the origin. (2) The magnetic field dependence of the energy spectrum shows a butterfly shape caused by Landau quantization. (3) The magnetic field dependence of the energy spectrum also shows a fractal nature. In particular it is characterized by two periods whose ratio is equal to the golden mean (1+\\sqrt{5})/2, and two periods comprising a Fibonacci sequence. We have clarified the origin of this fractal behavior of the energy spectrum analytically. 14. Anisotropic multi-resolution analysis in 2D, application to long-range correlations in cloud mm-radar fields SciTech Connect Davis, A.B.; Clothiaux, E. 1999-03-01 Because of Earths gravitational field, its atmosphere is strongly anisotropic with respect to the vertical; the effect of the Earths rotation on synoptic wind patterns also causes a more subtle form of anisotropy in the horizontal plane. The authors survey various approaches to statistically robust anisotropy from a wavelet perspective and present a new one adapted to strongly non-isotropic fields that are sampled on a rectangular grid with a large aspect ratio. This novel technique uses an anisotropic version of Multi-Resolution Analysis (MRA) in image analysis; the authors form a tensor product of the standard dyadic Haar basis, where the dividing ratio is {lambda}{sub z} = 2, and a nonstandard triadic counterpart, where the dividing ratio is {lambda}{sub x} = 3. The natural support of the field is therefore 2{sup n} pixels (vertically) by 3{sup n} pixels (horizontally) where n is the number of levels in the MRA. The natural triadic basis includes the French top-hat wavelet which resonates with bumps in the field whereas the Haar wavelet responds to ramps or steps. The complete 2D basis has one scaling function and five wavelets. The resulting anisotropic MRA is designed for application to the liquid water content (LWC) field in boundary-layer clouds, as the prevailing wind advects them by a vertically pointing mm-radar system. Spatial correlations are notoriously long-range in cloud structure and the authors use the wavelet coefficients from the new MRA to characterize these correlations in a multifractal analysis scheme. In the present study, the MRA is used (in synthesis mode) to generate fields that mimic cloud structure quite realistically although only a few parameters are used to control the randomness of the LWCs wavelet coefficients. 15. Nonequilibrium random-field Ising model on a diluted triangular lattice. PubMed Kurbah, Lobisor; Thongjaomayum, Diana; Shukla, Prabodh 2015-01-01 We study critical hysteresis in the random-field Ising model on a two-dimensional periodic lattice with a variable coordination number z(eff) in the range 3≤z(eff)≤6. We find that the model supports critical behavior in the range 4field. 16. Observation of a magnetic field dependence of the lattice thermal conductivity NASA Astrophysics Data System (ADS) Jin, Hyungyu; Restrepo, Oscar; Antolin, Nikolas; Windl, Wolfgang; Barnes, Stewart; Heremans, Joseph 2014-03-01 Can phonons respond to magnetic fields? From the simple point of view of the classical lattice vibrations, there is no clue that phonons possess any magnetic characteristics. Here, we report for the first time that the lattice thermal conductivity can show a response to an external magnetic field in a non-magnetic semiconductor crystal. We observe a magnetic field dependence of the lattice thermal conductivity in a high quality 2x1015 Te doped single crystal of InSb. The electronic contribution is over 106 times smaller than the lattice. The effect is observed in the temperature regime where the Umklapp processes start appearing, and still mainly involve phonons with long mean free paths. A special thermal design is employed to obtain a high accuracy heat flux measurement. Detailed experimental procedures and results are presented along with a brief discussion about possible origins of the effect. HJ and JPH are supported by AFOSR MURI Cryogenic Peltier Cooling'' Contract #FA9550-10-1-0533; ODR and WW are supported by the Center for Emergent Materials, an NSF MRSEC at The Ohio State University (Grant DMR-0820414). 17. Neutron electric dipole moment with external electric field method in lattice QCD SciTech Connect Shintani, E.; Kanaya, K.; Aoki, S.; Ishizuka, N.; Kuramashi, Y.; Ukawa, A.; Yoshie, T.; Kikukawa, Y.; Okawa, M. 2007-02-01 We discuss a possibility that the neutron electric dipole moment (NEDM) can be calculated in lattice QCD simulations in the presence of the CP-violating {theta} term. In this paper we measure the energy difference between spin-up and spin-down states of the neutron in the presence of a uniform and static external electric field. We first test this method in quenched QCD with the renormalization group improved gauge action on a 16{sup 3}x32 lattice at a{sup -1}{approx_equal}2 GeV, employing two different lattice fermion formulations, the domain-wall fermion and the clover fermion for quarks, at relatively heavy quark mass (m{sub PS}/m{sub V}{approx_equal}0.85). We obtain nonzero values of the NEDM from calculations with both fermion formulations. We next consider some systematic uncertainties of our method for the NEDM, using 24{sup 3}x32 lattice at the same lattice spacing only with the clover fermion. We finally investigate the quark mass dependence of the NEDM and observe a nonvanishing behavior of the NEDM toward the chiral limit. We interpret this behavior as a manifestation of the pathology in the quenched approximation. 18. π0 pole mass calculation in a strong magnetic field and lattice constraints NASA Astrophysics Data System (ADS) Avancini, Sidney S.; Farias, Ricardo L. S.; Benghi Pinto, Marcus; Tavares, William R.; Timóteo, Varese S. 2017-04-01 The π0 neutral meson pole mass is calculated in a strongly magnetized medium using the SU(2) Nambu-Jona-Lasinio model within the random phase approximation (RPA) at zero temperature and zero baryonic density. We employ a magnetic field dependent coupling, G (eB), fitted to reproduce lattice QCD results for the quark condensates. Divergent quantities are handled with a magnetic field independent regularization scheme in order to avoid unphysical oscillations. A comparison between the running and the fixed couplings reveals that the former produces results much closer to the predictions from recent lattice calculations. In particular, we find that the π0 meson mass systematically decreases when the magnetic field increases while the scalar mass remains almost constant. We also investigate how the magnetic background influences other mesonic properties such as fπ0 and gπ0qq. 19. Transition to and from the skyrmion lattice phase by electric fields in a magnetoelectric compound PubMed Central Okamura, Y.; Kagawa, F.; Seki, S.; Tokura, Y. 2016-01-01 Dissipation-less electric control of magnetic state variable is an important target of contemporary spintronics. The non-volatile control of magnetic skyrmions, nanometre-sized spin-swirling objects, with electric fields may exemplify this goal. The skyrmion-hosting magnetoelectric chiral magnet Cu2OSeO3 provides a unique platform for the implementation of such control; however, the hysteresis that accompanies the first-order transition associated with the skyrmion phase is negligibly narrow in practice. Here we demonstrate another method that functions irrespective of the transition boundary. Combination of magnetic-susceptibility measurements and microwave spectroscopy reveals that although the metastable skyrmion lattice is normally hidden behind a more thermodynamically stable conical phase, it emerges under electric fields and persists down to the lowest temperature. Once created, this metastable skyrmion lattice remains without electric fields, establishing a bistability distinct from the transition hysteresis. This bistability thus enables non-volatile electric-field control of the skyrmion lattice even in temperature/magnetic-field regions far from the transition boundary. PMID:27580648 20. Transition to and from the skyrmion lattice phase by electric fields in a magnetoelectric compound NASA Astrophysics Data System (ADS) Okamura, Y.; Kagawa, F.; Seki, S.; Tokura, Y. 2016-09-01 Dissipation-less electric control of magnetic state variable is an important target of contemporary spintronics. The non-volatile control of magnetic skyrmions, nanometre-sized spin-swirling objects, with electric fields may exemplify this goal. The skyrmion-hosting magnetoelectric chiral magnet Cu2OSeO3 provides a unique platform for the implementation of such control; however, the hysteresis that accompanies the first-order transition associated with the skyrmion phase is negligibly narrow in practice. Here we demonstrate another method that functions irrespective of the transition boundary. Combination of magnetic-susceptibility measurements and microwave spectroscopy reveals that although the metastable skyrmion lattice is normally hidden behind a more thermodynamically stable conical phase, it emerges under electric fields and persists down to the lowest temperature. Once created, this metastable skyrmion lattice remains without electric fields, establishing a bistability distinct from the transition hysteresis. This bistability thus enables non-volatile electric-field control of the skyrmion lattice even in temperature/magnetic-field regions far from the transition boundary. 1. Nonequilibrium phase transitions in lattice systems with random-field competing kinetics NASA Astrophysics Data System (ADS) López-Lacomba, A. I.; Marro, J. 1992-10-01 We study a class of lattice interacting-spin systems evolving stochastically under the simultaneous operation of several spin-flip mechanisms, each acting independently and responding to a different applied magnetic field. This induces an extra randomness which may occur in real systems, e.g., a magnetic system under the action of a field varying with a much shorter period than the mean time between successive transitions. Such a situation-in which one may say in some sense that frustration has a dynamical origin- may also be viewed as a nonequilibrium version of the random-field Ising model. By following a method of investigating stationary probability distributions in systems with competing kinetics [P. L. Garrido and J. Marro, Phys. Rev. Lett. 62, 1929 (1989)], we solve one-dimensional lattices supporting different field distributions and transition rates for the elementary kinetical processes, thus revealing a rich variety of phase transitions and critical phenomena. Some exact results for lattices of arbitrary dimension, and comparisons with the standard quenched and annealed random-field models, and with a nonequilibrium diluted antiferromagnetic system, are also reported. 2. Gaseous bubble oscillations in anisotropic non-Newtonian fluids under influence of high-frequency acoustic field NASA Astrophysics Data System (ADS) Golykh, R. N. 2016-06-01 Progress of technology and medicine dictates the ever-increasing requirements (heat resistance, corrosion resistance, strength properties, impregnating ability, etc.) for non-Newtonian fluids and materials produced on their basis (epoxy resin, coating materials, liquid crystals, etc.). Materials with improved properties obtaining is possible by modification of their physicochemical structure. One of the most promising approaches to the restructuring of non-Newtonian fluids is cavitation generated by high-frequency acoustic vibrations. The efficiency of cavitation in non-Newtonian fluid is determined by dynamics of gaseous bubble. Today, bubble dynamics in isotropic non-Newtonian fluids, in which cavitation bubble shape remains spherical, is most full investigated, because the problem reduces to ordinary differential equation for spherical bubble radius. However, gaseous bubble in anisotropic fluids which are most wide kind of non-Newtonian fluids (due to orientation of macromolecules) deviates from spherical shape due to viscosity dependence on shear rate direction. Therefore, the paper presents the mathematical model of gaseous bubble dynamics in anisotropic non-Newtonian fluids. The model is based on general equations for anisotropic non-Newtonian fluid flow. The equations are solved by asymptotic decomposition of fluid flow parameters. It allowed evaluating bubble size and shape evolution depending on rheological properties of liquid and acoustic field characteristics. 3. Near-field investigations of the anisotropic properties of supported lipid bilayers NASA Astrophysics Data System (ADS) Johnson, Merrell A. 2011-12-01 The details of Polarization Modulation Near-Field Scanning Optical Microscopy (PM-NSOM) are presented. How to properly calibrate and align the system is also introduced. A measurement of Muscovite crystal is used to display the capabilities of the setup. Measurements of supported gel state 1,2-dipalmitoyl-sn-glycero-3-phosphocholine (DPPC) lipid bilayers are presented, emphasizing how it was tooled in exploiting the anisotropic nature of the acyl chains. A discussion of how the effective retardance (DeltaS = 2 pi (ne-no) t /lambda) and the direction of the projection of the acyl chains (φ) are measured simultaneously is given, (where t is the thickness of the bilayer and lambda is the wavelength of light used). It is shown from DeltaS the birefringence (ne-n o) of the bilayer is determined, by assuming the acyl chain tilt with respect to the membrane's normal to be approximately φ ≈ 32 degrees. Time varying experiments show lateral diffusions of ˜ 2x10 -12 (cm2)/s. Temperature controlled PM-NSOM is shown to be a viable way to determine the main phase transition temperature (Tm) for going from the gel to liquid disorder state of supported DPPC bilayers. A change DeltaS ˜ (3.8+/-0.3 mrad) at the main phase transition temperature Tm (≈ 41°C) is observed. This agrees well with previous values of ( ne-no) and translates to an assumed φ ˜ 32 degrees, when T < Tm and 0 when T > Tm. Evidence of supper heating and supper cooling will be presented, along with a discussion of the fluctuations that occur around Tm. Finally it is shown how physical parameters such as the polarizability are extracted from the data. Values of the transverse (alpha t) and longitudinal (alphal) polarizabilites of the acyl chains are shown to be, alphat = 44.2A3 and alphal = 94.4 A3, which correspond well with the theoretical values of a single palmitic acid (C16) alpha t = 25.14 A3 and alpha l = 45.8 A3. 4. Generation of uniform synthetic magnetic fields by split driving of an optical lattice NASA Astrophysics Data System (ADS) Creffield, C. E.; Sols, F. 2014-08-01 We describe a method to generate a synthetic gauge potential for ultracold atoms held in an optical lattice. Our approach uses a time-periodic driving potential based on quickly alternating two Hamiltonians to engineer the appropriate Aharonov-Bohm phases, and permits the simulation of a uniform tunable magnetic field. We explicitly demonstrate that our split-driving scheme reproduces the behavior of a charged quantum particle in a magnetic field over the complete range of field strengths, and obtain the Hofstadter butterfly band structure for the Floquet quasienergies. 5. Noncircular skyrmion and its anisotropic response in thin films of chiral magnets under a tilted magnetic field SciTech Connect Lin, Shi-Zeng; Saxena, Avadh 2015-11-03 Here we study the equilibrium and dynamical properties of skyrmions in thin films of chiral magnets with oblique magnetic field. The shape of an individual skyrmion is non-circular and the skyrmion density decreases with the tilt angle from the normal of films. As a result, the interaction between two skyrmions depends on the relative angle between them in addition to their separation. The triangular lattice of skyrmions under a perpendicular magnetic field is distorted into a centered rectangular lattice for a tilted magnetic field. For a low skyrmion density, skyrmions form a chain like structure. Lastly, the dynamical response of the non-circular skyrmions depends on the direction of external currents. 6. Noncircular skyrmion and its anisotropic response in thin films of chiral magnets under a tilted magnetic field DOE PAGES Lin, Shi-Zeng; Saxena, Avadh 2015-11-03 Here we study the equilibrium and dynamical properties of skyrmions in thin films of chiral magnets with oblique magnetic field. The shape of an individual skyrmion is non-circular and the skyrmion density decreases with the tilt angle from the normal of films. As a result, the interaction between two skyrmions depends on the relative angle between them in addition to their separation. The triangular lattice of skyrmions under a perpendicular magnetic field is distorted into a centered rectangular lattice for a tilted magnetic field. For a low skyrmion density, skyrmions form a chain like structure. Lastly, the dynamical response ofmore » the non-circular skyrmions depends on the direction of external currents.« less 7. Nonequilibrium evolution of strong-field anisotropic ionized electrons towards a delayed plasma-state. PubMed Pasenow, B; Moloney, J V; Koch, S W; Chen, S H; Becker, A; Jaroń-Becker, A 2012-01-30 Rigorous quantum calculations of the femtosecond ionization of hydrogen atoms in air lead to highly anisotropic electron and ion angular (momentum) distributions. A quantum Monte-Carlo analysis of the subsequent many-body dynamics reveals two distinct relaxation steps, first to a nearly isotropic hot nonequilibrium and then to a quasi-equilibrium configuration. The collective isotropic plasma state is reached on a picosecond timescale well after the ultrashort ionizing pulse has passed. 8. Dust Lattice Waves in Two-Dimensional Hexagonal Dust Crystals with an External Magnetic Field SciTech Connect Farokhi, B.; Shahmansouri, M. 2008-09-07 The influence of a constant magnetic field on the propagation of dust-lattice (DL) modes in a two-dimensional hexagonal strongly coupled plasma crystal formed by paramagnetic particles is considered. The expression for the wave dispersion relation clearly shows that high-frequency and low-frequency branches exist as a result of the coupling of longitudinal and transverse modes due to the Lorentz force acting on the dust particles. 9. Aharonov-Bohm photonic cages in waveguide and coupled resonator lattices by synthetic magnetic fields. PubMed Longhi, Stefano 2014-10-15 We suggest a method for trapping photons in quasi-one-dimensional waveguide or coupled-resonator lattices, which is based on an optical analogue of the Aharonov-Bohm cages for charged particles. Light trapping results from a destructive interference of Aharonov-Bohm type induced by a synthetic magnetic field, which is realized by periodic modulation of the waveguide/resonator propagation constants/resonances. 10. Anisotropic viscosity and fabric evolution from laboratory experiments and field observations NASA Astrophysics Data System (ADS) Hansen, Lars; Warren, Jessica; Zimmerman, Mark; Kohlstedt, David; Skemer, Philip; Hirth, Greg 2013-04-01 Crystallographic alignment of grains during solid-state deformation imparts anisotropic material properties to the bulk rock, which results in significant macroscopic anisotropy in viscosity. The majority of previous laboratory studies on geological materials have performed experiments on relatively untextured samples, making it difficult to quantify the magnitude of anisotropy. Here we present results of laboratory deformation experiments that first produce strong crystallographic fabrics and then test the viscosity of these textured aggregates in multiple stress states. Our results are used in a model for shear zone evolution to reproduce field measurements of strain variation across a natural shear zone. Two sets of deformation experiments were performed in a gas-medium apparatus at 1473 K and 300 MPa confining pressure. In the first set of experiments (Hansen et al., Nature, 2012), large-strain torsion imparts a fabric in which the dominant [100] orientation is parallel to the shear direction and the dominant [010] orientation is normal to the shear plane, typical of a fabric due to shear on the (010)[100] slip system. Subsequent tension parallel to the initial torsion axis occurs with most grains having unfavorable orientations for slip on available slip systems. In the second set of experiments, samples were initially deformed in tension and subsequently deformed in torsion, with the torsion axis parallel to the initial tensional load. Tension imparts a fabric in which the dominant [100] orientation is parallel to the tension direction, with girdles of [010] and [001] axes. Subsequent torsion occurs with some grains having favorable orientations for (100)[001] slip and other grains having unfavorable orientations for slip on available slip systems. Electron-backscatter diffraction maps of axial sections of samples reveal that the crystallographic fabric reorients into a more favorable orientation at a shear strain of ~1.5. In both sets of experiments the 11. Evidence for lattice-polarization-enhanced field effects at the SrTiO3-based heterointerface PubMed Central Li, Y.; Zhang, H. R.; Lei, Y.; Chen, Y. Z.; Pryds, N.; Shen, Baogen; Sun, Jirong 2016-01-01 Electrostatic gating provides a powerful approach to tune the conductivity of the two-dimensional electron liquid between two insulating oxides. For the LaAlO3/SrTiO3 (LAO/STO) interface, such gating effect could be further enhanced by a strong lattice polarization of STO caused by simultaneous application of gate field and illumination light. Herein, by monitoring the discharging process upon removing the gate field, we give firm evidence for the occurrence of this lattice polarization at the amorphous-LaAlO3/SrTiO3 interface. Moreover, we find that the lattice polarization is accompanied with a large expansion of the out-of-plane lattice of STO. Photo excitation affects the polarization process by accelerating the field-induced lattice expansion. The present work demonstrates the great potential of combined stimuli in exploring emergent phenomenon at complex oxide interfaces. PMID:26926433 12. Displacement Fields and Self-Energies of Circular and Polygonal Dislocation Loops in Homogeneous and Layered Anisotropic Solids SciTech Connect Gao, Yanfei; Larson, Ben C. 2015-06-19 There are large classes of materials problems that involve the solutions of stress, displacement, and strain energy of dislocation loops in elastically anisotropic solids, including increasingly detailed investigations of the generation and evolution of irradiation induced defect clusters ranging in sizes from the micro- to meso-scopic length scales. Based on a two-dimensional Fourier transform and Stroh formalism that are ideal for homogeneous and layered anisotropic solids, we have developed robust and computationally efficient methods to calculate the displacement fields for circular and polygonal dislocation loops. Using the homogeneous nature of the Green tensor of order -1, we have shown that the displacement and stress fields of dislocation loops can be obtained by numerical quadrature of a line integral. In addition, it is shown that the sextuple integrals associated with the strain energy of loops can be represented by the product of a pre-factor containing elastic anisotropy effects and a universal term that is singular and equal to that for elastic isotropic case. Furthermore, we have found that the self-energy pre-factor of prismatic loops is identical to the effective modulus of normal contact, and the pre-factor of shear loops differs from the effective indentation modulus in shear by only a few percent. These results provide a convenient method for examining dislocation reaction energetic and efficient procedures for numerical computation of local displacements and stresses of dislocation loops, both of which play integral roles in quantitative defect analyses within combined experimental–theoretical investigations. 13. Hall viscosity and momentum transport in lattice and continuum models of the integer quantum Hall effect in strong magnetic fields NASA Astrophysics Data System (ADS) Tuegel, Thomas I.; Hughes, Taylor L. 2015-10-01 The Hall viscosity describes a nondissipative response to strain in systems with broken time-reversal symmetry. We develop a method for computing the Hall viscosity of lattice systems in strong magnetic fields based on momentum transport, which we compare to the method of momentum polarization used by Tu et al. [Phys. Rev. B 88, 195412 (2013), 10.1103/PhysRevB.88.195412] and Zaletel et al. [Phys. Rev. Lett. 110, 236801 (2013), 10.1103/PhysRevLett.110.236801] for noninteracting systems. We compare the Hall viscosity of square-lattice tight-binding models in magnetic field to the continuum integer quantum Hall effect (IQHE) showing agreement when the magnetic length is much larger than the lattice constant, but deviation as the magnetic field strength increases. We also relate the Hall viscosity of relativistic electrons in magnetic field (the Dirac IQHE) to the conventional IQHE. The Hall viscosity of the lattice Dirac model in magnetic field agrees with the continuum Dirac Hall viscosity when the magnetic length is much larger than the lattice constant. We also show that the Hall viscosity of the lattice model deviates further from the continuum model if the C4 symmetry of the square lattice is broken to C2, but the deviation is again minimized as the magnetic length increases. 14. Entropic contributions in Langevin equations for anisotropic driven systems NASA Astrophysics Data System (ADS) de los Santos, Francisco; Garrido, Pedro L.; Muñoz, Miguel A. 2001-07-01 We report on analytical results for a series of anisotropic driven systems in the context of a recently proposed Langevin equation approach. In a recent paper (P.L. Garrido et al., Phys. Rev. E 61 (2000) R4683) we have pointed out that entropic contributions, over-looked in previous works, are crucial in order to obtain suitable Langevin descriptions of driven lattice gases. Here, we present a more detailed derivation and justification of the entropic term for the standard driven lattice gas, and also we extend the improved approach to other anisotropic driven systems, namely: (i) the randomly driven lattice gas, (ii) the two-temperature model and, (iii) the bi-layer lattice gas. It is shown that the two-temperature model and the lattice gas driven either by a random field or by an uniform infinite one are members of the same universality class. When the drive is uniform and finite the ‘standard’ theory is recovered. A Langevin equation describing the phenomenology of the bi-layer lattice gas is also presented. 15. Plane wave propagation in a rotating anisotropic medium with voids under the action of a uniform magnetic field NASA Astrophysics Data System (ADS) Maity, Narottam; Barik, S. P.; Chaudhuri, P. K. 2016-09-01 In this paper, plane wave propagation in a rotating anisotropic material of general nature under the action of a magnetic field of constant magnitude has been investigated. The material is supposed to be porous in nature and contains voids. Following the concept of [Cowin S. C. and Nunziato, J. W. [1983] “Linear elastic materials with voids,” J. Elasticity 13, 125-147.] the governing equations of motion have been written in tensor notation taking account of rotation, magnetic field effect and presence of voids in the medium and the possibility of plane wave propagation has been examined. A number of particular cases have been derived from our general results to match with previously obtained results in this area. Effects of various parameters on the velocity of wave propagation have been presented graphically. 16. Vortex cores and vortex motion in superconductors with anisotropic Fermi surfaces NASA Astrophysics Data System (ADS) Galvis, J. A.; Herrera, E.; Guillamón, I.; Vieira, S.; Suderow, H. 2017-02-01 Explaning static and dynamic properties of the vortex lattice in anisotropic superconductors requires a careful characterization of vortex cores. The vortex core contains Andreev bound states whose spatial extension depends on the anisotropy of the electronic band-structure and superconducting gap. This might have an impact on the anisotropy of the superconducting properties and on vortex dynamics. Here we briefly summarize basic concepts to understand anisotropic vortex cores and review vortex core imaging experiments. We further discuss moving vortex lattices and the influence of vortex core shape in vortex motion. We find vortex motion in highly tilted magnetic fields. We associate vortex motion to the vortex entry barrier and the screening currents at the surface. We find preferential vortex motion along the main axis of the vortex lattice. After travelling integers of the intervortex distance, we find that vortices move more slowly due to the washboard potential of the vortex lattice. 17. Entanglement entropy for a Maxwell field: Numerical calculation on a two-dimensional lattice NASA Astrophysics Data System (ADS) Casini, Horacio; Huerta, Marina 2014-11-01 We study entanglement entropy (EE) for a Maxwell field in (2 +1 ) dimensions. We do numerical calculations in two-dimensional lattices. This gives a concrete example of the general results of our recent work [1] on entropy for lattice gauge fields using an algebraic approach. To evaluate the entropies we extend the standard calculation methods for the entropy of Gaussian states in canonical commutation algebras to the more general case of algebras with center and arbitrary numerical commutators. We find that while the entropy depends on the details of the algebra choice, mutual information has a well defined continuum limit as predicted in [1]. We study several universal terms for the entropy of the Maxwell field and compare with the case of a massless scalar field. We find some interesting new phenomena: an "evanescent" logarithmically divergent term in the entropy with topological coefficient which does not have any correspondence with ultraviolet entanglement in the universal quantities, and a nonstandard way in which strong subadditivity is realized. Based on the results of our calculations we propose a generalization of strong subadditivity for the entropy on some algebras that are not in tensor product. 18. Depolarized light scattering from prolate anisotropic particles: The influence of the particle shape on the field autocorrelation function. PubMed Passow, Christopher; ten Hagen, Borge; Löwen, Hartmut; Wagner, Joachim 2015-07-28 We provide a theoretical analysis for the intermediate scattering function typically measured in depolarized dynamic light scattering experiments. We calculate the field autocorrelation function g1(VH)(Q,t) in dependence on the wave vector Q and the time t explicitly in a vertical-horizontal scattering geometry for differently shaped solids of revolution. The shape of prolate cylinders, spherocylinders, spindles, and double cones with variable aspect ratio is expanded in rotational invariants flm(r). By Fourier transform of these expansion coefficients, a formal multipole expansion of the scattering function is obtained, which is used to calculate the weighting coefficients appearing in the depolarized scattering function. In addition to translational and rotational diffusion, especially the translational-rotational coupling of shape-anisotropic objects is considered. From the short-time behavior of the intermediate scattering function, the first cumulants Γ(Q) are calculated. In a depolarized scattering experiment, they deviate from the simple proportionality to Q(2). The coefficients flm(Q) strongly depend on the geometry and aspect ratio of the particles. The time dependence, in addition, is governed by the translational and rotational diffusion tensors, which are calculated by means of bead models for differently shaped particles in dependence on their aspect ratio. Therefore, our analysis shows how details of the particle shape--beyond their aspect ratio--can be determined by a precise scattering experiment. This is of high relevance in understanding smart materials which involve suspensions of anisotropic colloidal particles. 19. An anisotropic phase-field model for solid-state dewetting and its sharp-interface limit NASA Astrophysics Data System (ADS) Dziwnik, Marion; Münch, Andreas; Wagner, Barbara 2017-04-01 We propose a two-dimensional phase field model for solid state dewetting where the surface energy is weakly anisotropic. The evolution is described by the Cahn–Hilliard equation with a bi-quadratic degenerate mobility together with a bulk free energy based on a double-well potential and a free boundary condition at the film-substrate contact line. We derive the corresponding sharp interface limit via matched asymptotic analysis involving multiple inner layers. We show that in contrast to the frequently used quadratic degenerate mobility, the resulting sharp interface model for the bi-quatratic mobility is consistent with the pure surface diffusion model. In addition, we show that natural boundary conditions at the substrate obtained from the first variation of the total free energy including contributions at the substrate imply a contact angle condition in the sharp-interface limit which recovers the Young–Herring equation in the anisotropic and Young’s equation in the isotropic case, as well as a balance of fluxes at the contact line (or contact point). 20. Full-field bulge test for planar anisotropic tissues: part I--experimental methods applied to human skin tissue. PubMed Tonge, Theresa K; Atlan, Lorre S; Voo, Liming M; Nguyen, Thao D 2013-04-01 The nonlinear anisotropic properties of human skin tissue were investigated using bulge testing. Full-field displacement data were obtained during testing of human skin tissues procured from the lower back of post-mortem human subjects using 3-D digital image correlation. To measure anisotropy, the dominant fiber direction of the tissue was determined from the deformed geometry of the specimen. Local strains and stress resultants were calculated along both the dominant fiber direction and the perpendicular direction. Variation in anisotropy and stiffness was observed between specimens. The use of stress resultants rather than the membrane stress approximation accounted for bending effects, which are significant for a thick nonlinear tissue. Of the six specimens tested, it was observed that specimens from older donors exhibited a stiffer and more isotropic response than those from younger donors. It was seen that the mechanical response of the tissue was negligibly impacted by preconditioning or the ambient humidity. The methods presented in this work for skin tissue are sufficiently general to be applied to other planar tissues, such as pericardium, gastrointestinal tissue, and fetal membranes. The stress resultant-stretch relations will be used in a companion paper to obtain material parameters for a nonlinear anisotropic hyperelastic model. 1. Intentional anisotropic strain relaxation in (112{sup ¯}2) oriented Al{sub 1−x}In{sub x}N one-dimensionally lattice matched to GaN SciTech Connect Buß, E. R. Rossow, U.; Bremers, H.; Hangleiter, A.; Meisch, T.; Caliebe, M.; Scholz, F. 2014-09-22 We report on (112{sup ¯}2) oriented Al{sub 1−x}In{sub x}N grown by low pressure metal organic vapor phase epitaxy on (112{sup ¯}2) GaN templates on patterned r-plane sapphire. The indium incorporation efficiency as well as the growth rate of (112{sup ¯}2) oriented layers are similar to c-plane oriented Al{sub 1−x}In{sub x}N layers. Deposition of thick Al{sub 1−x}In{sub x}N layers does not lead to additional roughening like in case of c-plane oriented Al{sub 1−x}In{sub x}N. Independent of the thickness, the degree of relaxation of layers lattice matched in m-direction is in the range of 33%–45% in [112{sup ¯}3{sup ¯}]-direction. Associated with the relaxation in [112{sup ¯}3{sup ¯}]-direction, there is a tilt of the Al{sub 1−x}In{sub x}N layers around the [11{sup ¯}00] axis due to slip of threading dislocations on the basal (0001)-plane. Relaxation in m-direction is not observable for layers lattice matched in [112{sup ¯}3{sup ¯}] direction. The possibility to adjust the lattice parameter of AlInN in [112{sup ¯}3{sup ¯}] direction without changing the lattice parameter in m-direction by anisotropic strain relaxation opens up opportunities for subsequent growth of optically active structures. One possibility is to form relaxed buffer layers for GaInN quantum well structures. 2. Dynamics of an Electron in a Magnetic Field and in a Periodic Lattice NASA Astrophysics Data System (ADS) Adorjan, A. J.; Kaufman, M. 1996-11-01 We study the trajectory and the time dependence of the velocity of an electron moving in a 2d crystal in the presence of a magnetic field. This model is relevant to artificial 2d lattices(T.Geisel, J.Wagenhuber, P.Niebauer, G.Obermair, Phys.Rev.Lett.64,1581(1990)). The model is analyzed numerically by approximating the differential equations of motion with difference equations. To perform the calculations we use the mathematical package MathCad. We plan to use this study in undergraduate classes as an as an example of a research topic of current interest. 3. Mean-field phase diagram of disordered bosons in a lattice at nonzero temperature NASA Astrophysics Data System (ADS) Krutitsky, K. V.; Pelster, A.; Graham, R. 2006-09-01 Bosons in a periodic lattice with on-site disorder at low but nonzero temperatures are considered within a mean-field theory. The criteria used for the definition of the superfluid, Mott insulator and Bose glass are analysed. Since the compressibility never vanishes at nonzero temperatures, it cannot be used as a general criterion. We show that the phases are unambiguously distinguished by the superfluid density and the density of states of the low-energy excitations. The phase diagram of the system is calculated. It is shown that even a tiny temperature leads to a significant shift of the boundary between the Bose glass and superfluid. 4. Nonlocal field correlation functions on a lattice in the HP{sub 1{sigma}} model SciTech Connect Orlovsky, V. D. Shevchenko, V. I. 2010-11-15 Connected two-point field-strength correlation functions are measured on a lattice in the quaternionic projective {sigma} model within pure SU(2) theory. The correlation lengths extracted from exponential fits for these correlation functions, {lambda}{sub 1}{sup -1} = 1.40(3) GeV and {lambda}{sup -1} = 1.51(3) GeV, are found to be in good agreement with the results of other known calculations. The dependence of bilocal functions on the connector shape is also studied. 5. GPU phase-field lattice Boltzmann simulations of growth and motion of a binary alloy dendrite NASA Astrophysics Data System (ADS) Takaki, T.; Rojas, R.; Ohno, M.; Shimokawabe, T.; Aoki, T. 2015-06-01 A GPU code has been developed for a phase-field lattice Boltzmann (PFLB) method, which can simulate the dendritic growth with motion of solids in a dilute binary alloy melt. The GPU accelerated PFLB method has been implemented using CUDA C. The equiaxed dendritic growth in a shear flow and settling condition have been simulated by the developed GPU code. It has been confirmed that the PFLB simulations were efficiently accelerated by introducing the GPU computation. The characteristic dendrite morphologies which depend on the melt flow and the motion of the dendrite could also be confirmed by the simulations. 6. Mean field study of a propagation-turnover lattice model for the dynamics of histone marking NASA Astrophysics Data System (ADS) Yao, Fan; Li, FangTing; Li, TieJun 2017-02-01 We present a mean field study of a propagation-turnover lattice model, which was proposed by Hodges and Crabtree [Proc. Nat. Acad. Sci. 109, 13296 (2012)] for understanding how posttranslational histone marks modulate gene expression in mammalian cells. The kinetics of the lattice model consists of nucleation, propagation and turnover mechanisms, and exhibits second-order phase transition for the histone marking domain. We showed rigorously that the dynamics essentially depends on a non-dimensional parameter κ = k +/ k -, the ratio between the propagation and turnover rates, which has been observed in the simulations. We then studied the lowest order mean field approximation, and observed the phase transition with an analytically obtained critical parameter. The boundary layer analysis was utilized to investigate the structure of the decay profile of the mark density. We also studied the higher order mean field approximation to achieve sharper estimate of the critical transition parameter and more detailed features. The comparison between the simulation and theoretical results shows the validity of our theory. 7. Evidence for columnar order in the fully frustrated transverse field Ising model on the square lattice. PubMed Wenzel, Sandro; Coletta, Tommaso; Korshunov, Sergey E; Mila, Frédéric 2012-11-02 Using extensive classical and quantum Monte Carlo simulations, we investigate the ground-state phase diagram of the fully frustrated transverse field Ising model on the square lattice. We show that pure columnar order develops in the low-field phase above a surprisingly large length scale, below which an effective U(1) symmetry is present. The same conclusion applies to the quantum dimer model with purely kinetic energy, to which the model reduces in the zero-field limit, as well as to the stacked classical version of the model. By contrast, the 2D classical version of the model is shown to develop plaquette order. Semiclassical arguments show that the transition from plaquette to columnar order is a consequence of quantum fluctuations. 8. Extracting electric polarizabilities from lattice QCD SciTech Connect Detmold, W.; Tiburzi, B. C.; Walker-Loud, A. 2009-05-01 Charged and neutral, pion and kaon electric polarizabilities are extracted from lattice QCD using an ensemble of anisotropic gauge configurations with dynamical clover fermions. We utilize classical background fields to access the polarizabilities from two-point correlation functions. Uniform background fields are achieved by quantizing the electric field strength with the proper treatment of boundary flux. These external fields, however, are implemented only in the valence quark sector. A novel method to extract charge particle polarizabilities is successfully demonstrated for the first time. 9. Extracting Electric Polarizabilities from Lattice QCD SciTech Connect Will Detmold, William Detmold, Brian Tiburzi, Andre Walker-Loud 2009-05-01 Charged and neutral, pion and kaon electric polarizabilities are extracted from lattice QCD using an ensemble of anisotropic gauge configurations with dynamical clover fermions. We utilize classical background fields to access the polarizabilities from two-point correlation functions. Uniform background fields are achieved by quantizing the electric field strength with the proper treatment of boundary flux. These external fields, however, are implemented only in the valence quark sector. A novel method to extract charge particle polarizabilities is successfully demonstrated for the first time. 10. Anisotropic Field Dependence of the Superconducting Transition in the Magnetic Molecular Superconductor κ-(BETS)2FeBr4 NASA Astrophysics Data System (ADS) Fukuoka, Shuhei; Yamashita, Satoshi; Nakazawa, Yasuhiro; Yamamoto, Takashi; Fujiwara, Hideki 2017-01-01 Angle-resolved heat capacity measurements of a π-d interacting system of κ-(BETS)2FeBr4 [BETS = bis(ethylenedithio)tetraselenafulvalene] with in-plane magnetic fields are performed. We observed a thermal anomaly in association with the superconducting transition of the π electrons in the π-d compound for the first time. By pursuing a systematic change in the thermal anomaly, we found that the thermodynamic feature of the superconducting state shows large anisotropy against in-plane magnetic fields. When the field is applied parallel to the c-axis, the thermal anomaly remains up to 2.6 T with a distinct peak structure. On the other hand, it is suppressed in synchrony with the decrease of the antiferromagnetic transition temperature, when the field is applied parallel to the a-axis. Our thermodynamic results indicate that the effect of the π-d interaction appears even when the π electrons are itinerant and that the anisotropic field-direction dependence of the superconducting transition originates from the correlation between superconductivity and magnetism. 11. Three-dimensional stress-induced magnetic anisotropic constitutive model for ferromagnetic material in low intensity magnetic field NASA Astrophysics Data System (ADS) Sun, Le; Liu, Xin'en; Jia, Dong; Niu, Hongpan 2016-09-01 Metal magnetic memory (MMM) technique is a promising tool for inspecting early damage in ferromagnetic components due to its high sensitivity to stress in weak geomagnetic field. However, the quantitative analysis methods for the MMM haven't been sufficiently studied yet for absence of a reasonable constitutive model. A three-dimensional stress-induced magnetic anisotropic constitutive model is proposed in this paper to study magneto-mechanical coupling effect of the MMM. The model is developed in principal stress space and a linear relation between magnetization and magnetic field is employed for low intensity magnetic field. As a result, stress-induced magnetic anisotropy is represented by stress dependence of magnetic permeability in different directions, which is simple and convenient for applications in the MMM technique. Based on the model, the effect of stress on magnetic permeability and surface magnetic field is computed and compared with experimental data for a tensioned ferromagnetic specimen in low intensity magnetic field. The good consistency implies the validity of the proposed model. 12. Orbital magnetic field driven metal-insulator transition in spinless extended Falicov-Kimball model on a triangular lattice NASA Astrophysics Data System (ADS) Yadav, Umesh K. 2017-01-01 Ground state properties of spinless, extended Falicov-Kimball model (FKM) on a finite size triangular lattice with orbital magnetic field normal to the lattice are studied using numerical diagonalization and Monte-Carlo simulation methods. We show that the ground state configurations of localized electrons strongly depend on the magnetic field. Magnetic field induces a metal to insulator transition accompanied by segregated phase to an ordered regular phase except at density nf = 1 / 2 of localized electrons. It is proposed that magnetic field can be used as a new tool to produce segregated phase which was otherwise accessible only either with correlated hopping or with large on-site interactions. 13. Spin-lattice relaxation within a dimerized Ising chain in a magnetic field SciTech Connect Erdem, Rıza E-mail: [email protected]; Gülpınar, Gül; Yalçın, Orhan; Pawlak, Andrzej 2014-07-21 A qualitative study of the spin-lattice relaxation within a dimerized Ising chain in a magnetic field is presented. We have first determined the time dependence of the deviation of the lattice distortion parameter δΔ from the equilibrium state within framework of a technique combining the statistical equilibrium theory based on the transfer matrix method and the linear theory of irreversible thermodynamics. We have shown that the time dependence of the lattice distortion parameter is characterized by a single time constant (τ) which diverges around the critical point in both dimerized (Δ≠0) and uniform (Δ=0) phase regions. When the temperature and magnetic field are fixed to certain values, the time τ depends only on exchange coupling between the spins. It is a characteristic time associated with the long wavelength fluctuations of distortion. We have also taken into account the effects of spatial fluctuations on the relaxation time using the full Landau-Ginzburg free energy functional. We have found an explicit expression for the relaxation time as a function of temperature, coupling constant and wave vector (q) and shown that the critical mode corresponds to the case q=0. Finally, our results are found to be in good qualitative agreement with the results obtained in recent experimental study on synchrotron x-ray scattering and muon spin relaxation in diluted material Cu{sub 1−y}Mg{sub y}GeO{sub 3} where the composition y is very close to 0.0209. These results can be considered as natural extensions of some previous works on static aspects of the problem. 14. Heterogeneous anisotropic magnetic susceptibility of the myelin-water layers causes local magnetic field perturbations in axons. PubMed Puwal, Steffan; Roth, Bradley J; Basser, Peter J 2017-04-01 One goal of MRI is to determine the myelin water fraction in neural tissue. One approach is to measure the reduction in T2 * arising from microscopic perturbations in the magnetic field caused by heterogeneities in the magnetic susceptibility of myelin. In this paper, analytic expressions for the induced magnetic field distribution are derived within and around an axon, assuming that the myelin susceptibility is anisotropic. Previous models considered the susceptibility to be piecewise continuous, whereas this model considers a sinusoidally varying susceptibility. Many conclusions are common in both models. When the magnetic field is applied perpendicular to the axon, the magnetic field in the intraaxonal space is uniformly perturbed, the magnetic field in the myelin sheath oscillates between the lipid and water layers, and the magnetic field in the extracellular space just outside the myelin sheath is heterogeneous. These field heterogeneities cause the spins to dephase, shortening T2 *. When the magnetic field is applied along the axon, the field is homogeneous within water-filled regions, including between lipid layers. Therefore the spins do not dephase and the magnetic susceptibility has no effect on T2 *. Generally, the response of an axon is given as the superposition of these two contributions. The sinusoidal model uses a different set of approximations compared with the piecewise model, so their common predictions indicate that the models are not too sensitive to the details of the myelin-water distribution. Other predictions, such as the sensitivity to water diffusion between myelin and water layers, may highlight differences between the two approaches. Copyright © 2016 John Wiley & Sons, Ltd. 15. Displacement Fields and Self-Energies of Circular and Polygonal Dislocation Loops in Homogeneous and Layered Anisotropic Solids DOE PAGES Gao, Yanfei; Larson, Ben C. 2015-06-19 There are large classes of materials problems that involve the solutions of stress, displacement, and strain energy of dislocation loops in elastically anisotropic solids, including increasingly detailed investigations of the generation and evolution of irradiation induced defect clusters ranging in sizes from the micro- to meso-scopic length scales. Based on a two-dimensional Fourier transform and Stroh formalism that are ideal for homogeneous and layered anisotropic solids, we have developed robust and computationally efficient methods to calculate the displacement fields for circular and polygonal dislocation loops. Using the homogeneous nature of the Green tensor of order -1, we have shown thatmore » the displacement and stress fields of dislocation loops can be obtained by numerical quadrature of a line integral. In addition, it is shown that the sextuple integrals associated with the strain energy of loops can be represented by the product of a pre-factor containing elastic anisotropy effects and a universal term that is singular and equal to that for elastic isotropic case. Furthermore, we have found that the self-energy pre-factor of prismatic loops is identical to the effective modulus of normal contact, and the pre-factor of shear loops differs from the effective indentation modulus in shear by only a few percent. These results provide a convenient method for examining dislocation reaction energetic and efficient procedures for numerical computation of local displacements and stresses of dislocation loops, both of which play integral roles in quantitative defect analyses within combined experimental–theoretical investigations.« less 16. Temperature and field dependence of the flux-line-lattice symmetry in V{sub 3}Si SciTech Connect Yethiraj, M.; Christen, D.K.; Gapud, A.A.; Paul, D. McK.; Crowe, S.J.; Dewhurst, C.D.; Cubitt, R.; Porcar, L.; Gurevich, A. 2005-08-01 In V{sub 3}Si, a first-order structural phase transition from hexagonal to square flux-line lattice occurs at approximately 1 T with H parallel to the a axis. In this paper, we demonstrate the reentrant structural transition in the flux-line lattice, which reverts to hexagonal symmetry as the magnetic field approached H{sub c2}(T). This behavior is described very well by a nonlocal London theory with thermal fluctuations. The phase diagram of the flux lattice topology is mapped out for this geometry. 17. Bogoliubov theory of interacting bosons on a lattice in a synthetic magnetic field SciTech Connect Powell, Stephen; Barnett, Ryan; Sensarma, Rajdeep; Das Sarma, Sankar 2011-01-15 We consider theoretically the problem of an artificial gauge potential applied to a cold atomic system of interacting neutral bosons in a tight-binding optical lattice. Using the Bose-Hubbard model, we show that an effective magnetic field leads to superfluid phases with simultaneous spatial order, which we analyze using Bogliubov theory. This gives a consistent expansion in terms of quantum and thermal fluctuations, in which the lowest order gives a Gross-Pitaevskii equation determining the condensate configuration. We apply an analysis based on the magnetic symmetry group to show how the spatial structure of this configuration depends on commensuration between the magnetic field and the lattice. Higher orders describe the quasiparticle excitations, whose spectrum combines the intricacy of the Hofstadter butterfly with the characteristic features of the superfluid phase. We use the depletion of the condensate to determine the range of validity of our approximations and also to find an estimate for the onset of the Mott insulator phase. Our theory provides concrete experimental predictions for both time-of-flight imagery and Bragg spectroscopy. 18. Spectrum of a lattice exciton in a transverse magnetic field: Emergence of full translational symmetry NASA Astrophysics Data System (ADS) Berciu, Mona 2014-06-01 We show that even in the presence of a transverse magnetic field, the eigenstates of an exciton remain invariant to the full lattice translation group. This is expected if the exciton is viewed as a neutral quasiparticle, but not if one views it as a bound electron-hole pair. Single electron and hole wave functions are invariant only to the magnetic translation group, and their momenta are restricted to the magnetic Brillouin zone; the associated folding is the origin of their Hofstadter butterfly spectra. We find that such folding is not necessary for exciton eigenstates, which are characterized by momenta in the full Brillouin zone and thus have higher symmetry than the Hamiltonian. The magnetic field can have a significant effect on the shape of the exciton dispersion, however. While similar effects have been noted in continuous models, we find qualitatively different behavior for Frenkel excitons, whose origin we clarify. We also derive an analytical solution for the Hofstadter butterfly on a square lattice and analyze its dispersion in the full Brillouin zone. 19. Anisotropic x-ray scattering and orientation fields in cardiac tissue cells NASA Astrophysics Data System (ADS) Bernhardt, M.; Nicolas, J.-D.; Eckermann, M.; Eltzner, B.; Rehfeldt, F.; Salditt, T. 2017-01-01 X-ray diffraction from biomolecular assemblies is a powerful technique which can provide structural information about complex architectures such as the locomotor systems underlying muscle contraction. However, in its conventional form, macromolecular diffraction averages over large ensembles. Progress in x-ray optics has now enabled to probe structures on sub-cellular scales, with the beam confined to a distinct organelle. Here, we use scanning small angle x-ray scattering (scanning SAXS) to probe the diffraction from cytoskeleton networks in cardiac tissue cells. In particular, we focus on actin-myosin composites, which we identify as the dominating contribution to the anisotropic diffraction patterns, by correlation with optical fluorescence microscopy. To this end, we use a principal component analysis approach to quantify direction, degree of orientation, nematic order, and the second moment of the scattering distribution in each scan point. We compare the fiber orientation from micrographs of fluorescently labeled actin fibers to the structure orientation of the x-ray dataset and thus correlate signals of two different measurements: the native electron density distribution of the local probing area versus specifically labeled constituents of the sample. Further, we develop a robust and automated fitting approach based on a power law expansion, in order to describe the local structure factor in each scan point over a broad range of the momentum transfer {q}{{r}}. Finally, we demonstrate how the methodology shown for freeze dried cells in the first part of the paper can be translated to alive cell recordings. 20. Rupture of a ferrofluid droplet in external magnetic fields using a single-component lattice Boltzmann model for nonideal fluids. PubMed Falcucci, G; Chiatti, G; Succi, S; Mohamad, A A; Kuzmin, A 2009-05-01 A nonisotropic tensorial extension of the single-component Shan-Chen pseudopotential Lattice Boltzmann method for nonideal fluids is presented. Direct comparison with experimental data shows that this extension is able to capture relevant features of ferrofluid behavior, such as the deformation and subsequent rupture of a liquid droplet as a function of an externally applied magnetic field. The present model offers an economic lattice-kinetic pathway to the simulation of complex ferrofluid hydrodynamics. 1. Locality and efficient evaluation of lattice composite fields: Overlap-based gauge operators NASA Astrophysics Data System (ADS) Alexandru, Andrei; Horváth, Ivan 2017-01-01 We propose a novel general approach to locality of lattice composite fields, which in case of QCD involves locality in both quark and gauge degrees of freedom. The method is applied to gauge operators based on the overlap Dirac matrix elements, showing for the first time their local nature on realistic path-integral backgrounds. The framework entails a method for efficient evaluation of such nonultralocal operators, whose computational cost is volume independent at fixed accuracy, and only grows logarithmically as this accuracy approaches zero. This makes computation of useful operators, such as overlap-based topological density, practical. The key notion underlying these features is that of exponential insensitivity to distant fields, made rigorous by introducing the procedure of statistical regularization. The scales associated with insensitivity property are useful characteristics of nonlocal continuum operators. 2. Influence of lattice vibrations on the field driven electronic transport in chains with correlated disorder NASA Astrophysics Data System (ADS) da Silva, L. D.; Sales, M. O.; Ranciaro Neto, A.; Lyra, M. L.; de Moura, F. A. B. F. 2016-12-01 We investigate electronic transport in a one-dimensional model with four different types of atoms and long-ranged correlated disorder. The latter was attained by choosing an adequate distribution of on-site energies. The wave-packet dynamics is followed by taking into account effects due to a static electric field and electron-phonon coupling. In the absence of electron-phonon coupling, the competition between correlated disorder and the static electric field promotes the occurrence of wave-packet oscillations in the regime of strong correlations. When the electron-lattice coupling is switched on, phonon scattering degrades the Bloch oscillations. For weak electron-phonon couplings, a coherent oscillatory-like dynamics of the wave-packet centroid persists for short periods of time. For strong couplings the wave-packet acquires a diffusive-like displacement and spreading. A slower sub-diffusive spreading takes place in the regime of weak correlations. 3. Electrodeposition Modeling Using Coupled Phase-Field and Lattice Boltzmann Approach NASA Astrophysics Data System (ADS) Patil, D. V.; Premnath, K. N.; Desai, D.; Banerjee, Sanjoy 2014-01-01 In this paper, a coupled phase-field (PF) and lattice Boltzmann method (LBM) is presented to model the multiphysics phenomenon involving electro-chemical deposition. The deposition (or dissolution) of the electrode is represented using variations of an order-parameter. The time-evolution of an order-parameter is proportional to the variation of a Ginzburg-Landau free-energy functional. Further, the free-energy densities of the two phases are defined based on a dilute or an ideal solution approximation. An efficient LBM is used to obtain the converged electro-static potential field for each physical time-step of the evolution of the PF variable. The coupled approach demonstrates the applicability of the LBM in a multiphysics scenario. The numerical validation for the coupled approach is performed by the simulation of the electrodeposition process of Cu from CuSO4 solution. 4. Amplification of resonant field enhancement by plasmonic lattice coupling in metallic slit arrays NASA Astrophysics Data System (ADS) Klarskov, Pernille; Tarekegne, Abebe T.; Iwaszczuk, Krzysztof; Zhang, X.-C.; Jepsen, Peter Uhd 2016-11-01 Nonlinear spectroscopic investigation in the terahertz (THz) range requires significant field strength of the light fields. It is still a challenge to obtain the required field strengths in free space from table-top laser systems at sufficiently high repetition rates to enable quantitative nonlinear spectroscopy. It is well known that local enhancement of the THz field can be obtained for instance in narrow apertures in metallic films. Here we show by simulation, analytical modelling and experiment that the achievable field enhancement in a two-dimensional array of slits with micrometer dimensions in a metallic film can be increased by at least 60% compared to the enhancement in an isolated slit. The additional enhancement is obtained by optimized plasmonic coupling between the lattice modes and the resonance of the individual slits. Our results indicate a viable route to sensitive schemes for THz spectroscopy with slit arrays manufactured by standard UV photolithography, with local field strengths in the multi-ten-MV/cm range at kHz repetition rates, and tens of kV/cm at oscillator repetition rates. 5. Amplification of resonant field enhancement by plasmonic lattice coupling in metallic slit arrays PubMed Central Klarskov, Pernille; Tarekegne, Abebe T.; Iwaszczuk, Krzysztof; Zhang, X.-C.; Jepsen, Peter Uhd 2016-01-01 Nonlinear spectroscopic investigation in the terahertz (THz) range requires significant field strength of the light fields. It is still a challenge to obtain the required field strengths in free space from table-top laser systems at sufficiently high repetition rates to enable quantitative nonlinear spectroscopy. It is well known that local enhancement of the THz field can be obtained for instance in narrow apertures in metallic films. Here we show by simulation, analytical modelling and experiment that the achievable field enhancement in a two-dimensional array of slits with micrometer dimensions in a metallic film can be increased by at least 60% compared to the enhancement in an isolated slit. The additional enhancement is obtained by optimized plasmonic coupling between the lattice modes and the resonance of the individual slits. Our results indicate a viable route to sensitive schemes for THz spectroscopy with slit arrays manufactured by standard UV photolithography, with local field strengths in the multi-ten-MV/cm range at kHz repetition rates, and tens of kV/cm at oscillator repetition rates. PMID:27886232 6. Hydraulic-fracture growth in dipping anisotropic strata as viewed through the surface deformation field SciTech Connect Holzhausen, G.R.; Haase, C.S.; Stow, S.H.; Gazonas, G. 1985-01-01 In 1983 and 1984 Oak Rdige National Laboratory conducted a series of precision ground deformation measurements before, during, and after the generation of several large hydraulic fractures in a dipping member of the Cambrian Conasauga Shale. Each fracture was produced by the injection of approximately 500,000 L of slurry on a single day. Injection depth was 300 m. Leveling surveys were run several days before and several days after the injections. An array of eight high-precision borehole tiltmeters monitored ground deformations continuously for a period of several weeks. Analysis of the leveling and the tilt measurements revealed surface uplifts as great as 25 mm and tilts of tens of microradians during each injection. Furthermore, partial recovery (subsidence) of the ground took place during the days following an injection, accompanied by shifts in the position of maximum resultant uplift. Interpretation of the tilt measurements is consistent with stable widening and extension of hydraulic fractures with subhorizontal orientations. Comparison of the measured tilt patterns with fracture orientations established from logging of observation wells suggests that shearing parallel to the fracture planes accompanied fracture dilation. This interpretation is supported by measured tilts and ground uplifts that were as much as 100 percent greater than those expected from fracture dilation alone. Models of elastically anisotropic overburden rock do not explain the measured tilt patterns in the absence of shear stresses in the fracture planes. This work represents the first large-scale hydraulic-fracturing experiment in which the possible effects of material anisotropy and fracture-parallel shears have been measured and interpreted. 7. Parent Hamiltonians for lattice Halperin states from free-boson conformal field theories NASA Astrophysics Data System (ADS) Hackenbroich, Anna; Tu, Hong-Hao 2017-03-01 We introduce a family of many-body quantum states that describe interacting spin one-half hard-core particles with bosonic or fermionic statistics on arbitrary one- and two-dimensional lattices. The wave functions at lattice filling fraction ν = 2 / (2 m + 1) are derived from deformations of the Wess-Zumino-Witten model su (3)1 and are related to the (m + 1 , m + 1 , m) Halperin fractional quantum Hall states. We derive long-range SU(2) invariant parent Hamiltonians for these states which in two dimensions are chiral t-J-V models with additional three-body interaction terms. In one dimension we obtain a generalisation to open chains of a periodic inverse-square t-J-V model proposed in [25]. We observe that the gapless low-energy spectrum of this model and its open-boundary generalisation can be described by rapidity sets with the same generalised Pauli exclusion principle. A two-component compactified free boson conformal field theory is identified as the low-energy effective theory for the periodic inverse-square t-J-V model. 8. ERL with non-scaling fixed field alternating gradient lattice for eRHIC SciTech Connect Trbojevic, D.; Berg, J. S.; Brooks, S.; Hao, Y.; Litvinenko, V. N.; Liu, C.; Meot, F.; Minty, M.; Ptitsyn, V.; Roser, T.; Thieberger, P.; Tsoupas, N. 2015-05-03 The proposed eRHIC electron-hadron collider uses a "non-scaling FFAG" (NS-FFAG) lattice to recirculate 16 turns of different energy through just two beam lines located in the RHIC tunnel. This paper presents lattices for these two FFAGs that are optimized for low magnet field and to minimize total synchrotron radiation across the energy range. The higher number of recirculations in the FFAG allows a shorter linac (1.322GeV) to be used, drastically reducing cost, while still achieving a 21.2 GeV maximum energy to collide with one of the existing RHIC hadron rings at up to 250GeV. eRHIC uses many cost-saving measures in addition to the FFAG: the linac operates in energy recovery mode, so the beams also decelerate via the same FFAG loops and energy is recovered from the interacted beam. All magnets will be constructed from NdFeB permanent magnet material, meaning chillers and large magnet power supplies are not needed. This paper also describes a small prototype ERL-FFAG accelerator that will test all of these technologies in combination to reduce technical risk for eRHIC. 9. Hopping on the Bethe lattice: Exact results for densities of states and dynamical mean-field theory NASA Astrophysics Data System (ADS) Eckstein, Martin; Kollar, Marcus; Byczuk, Krzysztof; Vollhardt, Dieter 2005-06-01 We derive an operator identity which relates tight-binding Hamiltonians with arbitrary hopping on the Bethe lattice to the Hamiltonian with nearest-neighbor hopping. This provides an exact expression for the density of states (DOS) of a noninteracting quantum-mechanical particle for any hopping. We present analytic results for the DOS corresponding to hopping between nearest and next-nearest neighbors, and also for exponentially decreasing hopping amplitudes. Conversely it is possible to construct a hopping Hamiltonian on the Bethe lattice for any given DOS. These methods are based only on the so-called distance regularity of the infinite Bethe lattice, and not on the absence of loops. Results are also obtained for the triangular Husimi cactus, a recursive lattice with loops. Furthermore we derive the exact self-consistency equations arising in the context of dynamical mean-field theory, which serve as a starting point for studies of Hubbard-type models with frustration. 10. The quantum anomalous Hall effect on a star lattice with spin-orbit coupling and an exchange field. PubMed Chen, Mengsu; Wan, Shaolong 2012-08-15 We study a star lattice with Rashba spin-orbit coupling and an exchange field and find that there is a quantum anomalous Hall effect in this system, and that there are five energy gaps at Dirac points and quadratic band crossing points. We calculate the Berry curvature distribution and obtain the Hall conductivity (Chern number ν) quantized as integers, and find that ν =- 1,2,1,1,2 when the Fermi level lies in these five gaps. Our model can be viewed as a general quantum anomalous Hall system and, in limit cases, can give what the honeycomb lattice and kagome lattice give. We also find that there is a nearly flat band with ν = 1 which may provide an opportunity for realizing the fractional quantum anomalous Hall effect. Finally, the chiral edge states on a zigzag star lattice are given numerically, to confirm the topological property of this system. 11. ANALYTIC SOLUTIONS FOR CURRENT SHEET STRUCTURE DETERMINED BY SELF-CONSISTENT, ANISOTROPIC TRANSPORT PROCESSES IN A GRAVITATIONAL FIELD SciTech Connect Goodman, Michael L. 2011-04-10 A Harris sheet magnetic field with maximum magnitude B{sub 0} and length scale L is combined with the anisotropic electrical conductivity, viscosity, and thermoelectric tensors for an electron-proton plasma to define a magnetohydrodynamic model that determines the steady state of the plasma. The transport tensors are functions of temperature, density, and magnetic field strength, and are computed self-consistently as functions of position x normal to the current sheet. The flow velocity, magnetic field, and gravitational force lie along the z-axis. The plasma is supported against gravity by the viscous force. Analytic solutions are obtained for temperature, density, and velocity. They are valid over a broad range of temperature, density, and magnetic field strength, and so may be generally useful in astrophysical applications. Numerical examples of solutions in the parameter range of the solar atmosphere are presented. The objective is to compare Joule and viscous heating rates, determine the velocity shear that generates viscous forces that support the plasma and are self-consistent with a mean outward mass flux comparable to the solar wind mass flux, and compare the thermoelectric and conduction current contributions to the Joule heating rate. The ratio of the viscous to Joule heating rates per unit mass can exceed unity by orders of magnitude, and increases rapidly with L. The viscous heating rate can be concentrated outside the region where the current density is localized, corresponding to a resistively heated layer of plasma bounded by viscously heated plasma. The temperature gradient drives a thermoelectric current density that can have a magnitude greater than that of the electric-field-driven conduction current density, so thermoelectric effects are important in determining the Joule heating rate. 12. Observation of the vortex lattice melting by NMR spin-lattice relaxation in the mixed state SciTech Connect Bulaevskii, L.N.; Hammel, P.C.; Vinokur, V.M. 1994-01-01 For anisotropic layered superconductors the effect of moving vortices on the nuclear spin magnetization is calculated. Current is supposed to flow along layers, and applied magnetic field is tilted with respect to c-axis. In the solid phase the motion of the vortex lattice produces an alternating magnetic field perpendicular to the applied field which causes the decay of the spin-echo amplitude. This decay rate will display an array of peaks as a function of frequency. In the liquid phase this alternating field contribute to the longitudinal relaxation rate W{sub 1} which has a single peak. 13. Quadratic Zeeman effect and spin-lattice relaxation of Tm3 +:YAG at high magnetic fields NASA Astrophysics Data System (ADS) Veissier, Lucile; Thiel, Charles W.; Lutz, Thomas; Barclay, Paul E.; Tittel, Wolfgang; Cone, Rufus L. 2016-11-01 Anisotropy of the quadratic Zeeman effect for the H36→H34 transition at 793 nm wavelength in 3+169Tm-doped Y3Al5O12 is studied, revealing shifts ranging from near zero up to +4.69 GHz/T 2 for ions in magnetically inequivalent sites. This large range of shifts is used to spectrally resolve different subsets of ions and study nuclear spin relaxation as a function of temperature, magnetic field strength, and orientation in a site-selective manner. A rapid decrease in spin lifetime is found at large magnetic fields, revealing the weak contribution of direct phonon absorption and emission to the nuclear spin-lattice relaxation rate. We furthermore confirm theoretical predictions for the phonon coupling strength, finding much smaller values than those estimated in the limited number of past studies of thulium in similar crystals. Finally, we observe a significant—and unexpected—magnetic field dependence of the two-phonon Orbach spin relaxation process at higher field strengths, which we explain through changes in the electronic energy-level splitting arising from the quadratic Zeeman effect. 14. Accurate boundary treatments for lattice Boltzmann simulations of electric fields and electro-kinetic applications NASA Astrophysics Data System (ADS) Oulaid, Othmane; Chen, Qing; Zhang, Junfeng 2013-11-01 In this paper a novel boundary method is proposed for lattice Boltzmann simulations of electric potential fields with complex boundary shapes and conditions. A shifted boundary from the physical surface location is employed in simulations to achieve a better finite-difference approximation of the potential gradient at the physical surface. Simulations are presented to demonstrate the accuracy and capability of this method in dealing with complex surface situations. An example simulation of the electrical double layer and electro-osmotic flow around a three-dimensional spherical particle is also presented. These simulated results are compared with analytical predictions and are found to be in excellent agreement. This method could be useful for electro-kinetic and colloidal simulations with complex boundaries, and can also be readily extended to other phenomena and processes, such as heat transfer and convection-diffusion systems. 15. Current-induced spin polarization in anisotropic spin-orbit fields. PubMed Norman, B M; Trowbridge, C J; Awschalom, D D; Sih, V 2014-02-07 The magnitude and direction of current-induced spin polarization and spin-orbit splitting are measured in In0.04Ga0.96 As epilayers as a function of in-plane electric and magnetic fields. We show that, contrary to expectation, the magnitude of the current-induced spin polarization is smaller for crystal directions corresponding to larger spin-orbit fields. Furthermore, we find that the steady-state in-plane spin polarization does not align along the spin-orbit field, an effect due to anisotropy in the spin relaxation rate. 16. Highly anisotropic exchange interactions of jeff=12 iridium moments on the fcc lattice in La2BIrO6 (B=Mg,Zn) DOE PAGES Aczel, A. A.; Cook, A. M.; Williams, T. J.; ... 2016-06-20 Here we have performed inelastic neutron scattering (INS) experiments to investigate the magnetic excitations in the weakly distorted face-centered-cubic (fcc) iridate double perovskites Lamore » $$_2$$ZnIrO$$_6$$ and La$$_2$$MgIrO$$_6$$, which are characterized by A-type antiferromagnetic ground states. The powder inelastic neutron scattering data on these geometrically frustrated $$j_{\\rm eff}=1/2$$ Mott insulators provide clear evidence for gapped spin wave excitations with very weak dispersion. The INS results and thermodynamic data on these materials can be reproduced by conventional Heisenberg-Ising models with significant uniaxial Ising anisotropy and sizeable second-neighbor ferromagnetic interactions. Such a uniaxial Ising exchange interaction is symmetry-forbidden on the ideal fcc lattice, so that it can only arise from the weak crystal distortions away from the ideal fcc limit. This may suggest that even weak distortions in $$j_{\\rm eff}=1/2$$ Mott insulators might lead to strong exchange anisotropies. More tantalizingly, however, we find an alternative viable explanation of the INS results in terms of spin models with a dominant Kitaev interaction. In contrast to the uniaxial Ising exchange, the highly-directional Kitaev interaction is a type of exchange anisotropy which is symmetry-allowed even on the ideal fcc lattice. The Kitaev model has a magnon gap induced by quantum order-by-disorder, while weak anisotropies of the Kitaev couplings generated by the symmetry-lowering due to lattice distortions can pin the order and enhance the magnon gap. In conclusion, our findings highlight how even conventional magnetic orders in heavy transition metal oxides may be driven by highly-directional exchange interactions rooted in strong spin-orbit coupling.« less 17. Highly anisotropic exchange interactions of jeff=1/2 iridium moments on the fcc lattice in La2B IrO6 (B =Mg ,Zn ) NASA Astrophysics Data System (ADS) Aczel, A. A.; Cook, A. M.; Williams, T. J.; Calder, S.; Christianson, A. D.; Cao, G.-X.; Mandrus, D.; Kim, Yong-Baek; Paramekanti, A. 2016-06-01 We have performed inelastic neutron scattering (INS) experiments to investigate the magnetic excitations in the weakly distorted face-centered-cubic (fcc) iridate double perovskites La2ZnIrO6 and La2MgIrO6 , which are characterized by A-type antiferromagnetic ground states. The powder inelastic neutron scattering data on these geometrically frustrated jeff=1/2 Mott insulators provide clear evidence for gapped spin-wave excitations with very weak dispersion. The INS results and thermodynamic data on these materials can be reproduced by conventional Heisenberg-Ising models with significant uniaxial Ising anisotropy and sizeable second-neighbor ferromagnetic interactions. Such a uniaxial Ising exchange interaction is symmetry forbidden on the ideal fcc lattice, so that it can only arise from the weak crystal distortions away from the ideal fcc limit. This may suggest that even weak distortions in jeff=1/2 Mott insulators might lead to strong exchange anisotropies. More tantalizingly, however, we find an alternative viable explanation of the INS results in terms of spin models with a dominant Kitaev interaction. In contrast to the uniaxial Ising exchange, the highly directional Kitaev interaction is a type of exchange anisotropy which is symmetry allowed even on the ideal fcc lattice. The Kitaev model has a magnon gap induced by quantum order by disorder, while weak anisotropies of the Kitaev couplings generated by the symmetry lowering due to lattice distortions can pin the order and enhance the magnon gap. Our findings highlight how even conventional magnetic orders in heavy transition metal oxides may be driven by highly directional exchange interactions rooted in strong spin-orbit coupling. 18. X-ray diffuse scattering measurements of chemical short-range order and lattice strains in a highly magnetostrictive Fe0.813Ga0.187alloy in an applied magnetic field SciTech Connect Du, Y.; Huang, M.; Lograsso, T. A.; McQueeney, R. J. 2012-06-28 The rapid growth of the magnetostriction coefficient of ferromagnetic Fe1-xGax alloys that occurs at a composition range from 0anisotropic precipitates are proposed to rotate in an applied magnetic field, thereby resulting in a large magnetoelastic response. X-ray diffuse scattering measurements sensitive to the atomic short-range ordering and lattice strain were performed on a quenched 18.7 at. % Ga alloy and show no dependence on the application of a magnetic field. This negative result sets limits on the role of nanoscale precipitates in magnetostriction. 19. Anisotropic four-state clock model in the presence of random fields NASA Astrophysics Data System (ADS) Salmon, Octavio D. Rodriguez; Nobre, Fernando D. 2016-02-01 A four-state clock ferromagnetic model is studied in the presence of different configurations of anisotropies and random fields. The model is considered in the limit of infinite-range interactions, for which the mean-field approach becomes exact. Both representations of Cartesian spin components and two Ising variables are used, in terms of which the physical properties and phase diagrams are discussed. The random fields follow bimodal probability distributions and the richest criticality is found when the fields, applied in the two Ising systems, are not correlated. The phase diagrams present new interesting topologies, with a wide variety of critical points, which are expected to be useful in describing different complex phenomena. 20. The polarization electric field and its effects in an anisotropic rotating magnetospheric plasma NASA Technical Reports Server (NTRS) Huang, T. S.; Birmingham, T. J. 1992-01-01 Spatial variations of density and temperature along a magnetic field line are evaluated for a plasma undergoing adiabatic motion in a rotating magnetosphere. The effects of centrifugal and gravitational forces are accounted for, as is anisotropy in the pitch angle distribution functions of individual species. A polarization electric field is invoked to eliminate the net electric charge density resulting from the aforementioned mass dependent forces and different anisotropies. The position of maximum density in a two-component, electron-ion plasma is determined both in the absence and in the presence of the polarization effect and compared. A scale height, generalized to include anisotropies, is derived for the density fall-off. The polarization electric field is also included in the parallel guiding center equation; equilibrium points are determined and compared in both individual and average senses with the position of density maximum. Finally a transverse (to magnetic field lines) electric component is deduced as a consequence of dissimilar charge neutralization on adjacent field lines. The E x B velocity resultant from such a 'fringing' electric field is calculated and compared with the magnitude of other drifts. 1. LDN 1780: A Translucent High-Galactic Latitude Interstellar Cloud in a UV-rich, Anisotropic Radiation Field NASA Astrophysics Data System (ADS) DeRoo, Casey T.; Witt, A. N.; Barnes, F. S., III; Vijh, U. P.; Gordon, K. D. 2010-01-01 We combined high-resolution optical imaging observations in 12 intermediate-band (BATC) filter and deep mid- and far-IR Spitzer maps of LDN 1780 to characterize the external radiation field illuminating this high-latitude (l = 359 deg; b = 36.5 deg; distance 100 pc) translucent cloud and the infrared emission of dust within LDN 1780 in response to this external illumination. The overall energy density of the incident radiation field is approximately equal to that of the ISRF near the Sun, resulting in a large dust grain equilibrium temperature ranging from 14.5 K -16.8 K. However, the incident radiation field is highly anisotropic, with the southern portions of LDN 1780 being most strongly illuminated, especially at shorter wavelengths. This anisotropy is a result of the cloud's proximity to the Sco OB2 association (est. center: l = 322 deg; b = 10 deg). The southwestern portion and the optically-thin eastern tail of LDN 1780 exhibit strong intensity excesses at 24 micron (Spitzer MIPS) and at 8 micron (Spitzer IRAC Ch. #4) compared to dust in the diffuse ISM of the Milky Way Galaxy. We interpret these excesses as enhanced emission from stochastically-heated very small grains (VSG) and from PAH ions, respectively. These excesses, however, are not necessarily the result a greater relative abundance of these two small-particle components but rather reflect the increased frequency of photon-grain interactions (e.g. heating, excitation, or ionization) within a UV-rich radiation field. This research has been supported by grants from NASA and the NSF to the University of Toledo as well as by contributions from corporate sponsors AstroDon, RC Optical Systems, Santa Barbara Instrument Group, Software Bisque, and New Mexico Skies, for which we are grateful. We especially acknowledge the NSF-REU program at the University of Toledo, which supported the analysis of these data. 2. Anisotropic heavy quark potential in strongly-coupled N =4 SYM theory in a magnetic field NASA Astrophysics Data System (ADS) Rougemont, R.; Critelli, R.; Noronha, J. 2015-03-01 In this work we use the gauge/gravity duality to study the anisotropy in the heavy quark potential in strongly coupled N =4 super-Yang Mills (SYM) theory (both at zero and nonzero temperature) induced by a constant and uniform magnetic field B . At zero temperature, the inclusion of the magnetic field decreases the attractive force between heavy quarks with respect to its B =0 value and the force associated with the parallel potential is the least attractive force. We find that the same occurs at nonzero temperature and, thus, at least in the case of strongly coupled N =4 SYM, the presence of a magnetic field generally weakens the interaction between heavy quarks in the plasma. 3. Anisotropic properties of unconventional superconductors in a magnetic field: testing the nodal stucture NASA Astrophysics Data System (ADS) Vorontsov, Anton; Vekhter, Ilya 2006-03-01 We present a calculation of electronic specific heat and heat conductivity in a vortex state of quasi-two dimensional d-wave superconductors. We employ quasiclassical theory and use the Brand-Pesch-Tewordt approximation to model the superconducting state at moderate to high magnetic fields. Within this framework we investigate the dependence of heat capacity and heat conductivity on the angle of rotation of magnetic field with respect to the nodal directions. We find that the fourfold anisotropy due to nodal structure in both quantities changes sign in the temperature-field plane. This result helps resolve the apparent disagreement about the gap symmetry reached from the specific heat and the thermal conductivity measurements in CeCoIn5. We comment on the physics behind the difference between our results and those obtained in the Doppler shift approximation. 4. Characterization of a random anisotropic conductivity field with Karhunen-Loeve methods SciTech Connect Cherry, Matthew R.; Sabbagh, Harold S.; Pilchak, Adam L.; Knopp, Jeremy S. 2014-02-18 While parametric uncertainty quantification for NDE models has been addressed in recent years, the problem of stochastic field parameters such as spatially distributed electrical conductivity has only been investigated minimally in the last year. In that work, the authors treated the field as a one-dimensional random process and Karhunen-Loeve methods were used to discretize this process to make it amenable to UQ methods such as ANOVA expansions. In the present work, we will treat the field as a two dimensional random process, and the eigenvalues and eigenfunctions of the integral operator will be determined via Galerkin methods. The Karhunen-Loeve methods is extended to two dimensions and implemented to represent this process. Several different choices for basis functions will be discussed, as well as convergence criteria for each. The methods are applied to correlation functions collected over electron backscatter data from highly micro textured Ti-7Al. 5. Field Tolerances for the Triplet Quadrupoles of the LHC High Luminosity Lattice SciTech Connect Nosochkov, Yuri; Cai, Y.; Jiao, Y.; Wang, M-H.; Fartoukh, S.; Giovannozzi, M.; Maria, R.de; McIntosh, E. 2012-06-25 It has been proposed to implement the so-called Achromatic Telescopic Squeezing (ATS) scheme in the LHC high luminosity (HL) lattice to reduce beta functions at the Interaction Points (IP) up to a factor of 8. As a result, the nominal 4.5 km peak beta functions reached in the Inner Triplets (IT) at collision will be increased by the same factor. This, therefore, justifies the installation of new, larger aperture, superconducting IT quadrupoles. The higher beta functions will enhance the effects of the triplet quadrupole field errors leading to smaller beam dynamic aperture (DA). To maintain the acceptable DA, the effects of the triplet field errors must be re-evaluated, thus specifying new tolerances. Such a study has been performed for the so-called '4444' collision option of the HL-LHC layout version SLHCV3.01, where the IP beta functions are reduced by a factor of 4 in both planes with respect to a pre-squeezed value of 60 cm at two collision points. The dynamic aperture calculations were performed using SixTrack. The impact on the triplet field quality is presented. 6. Lattice models for granular-like velocity fields: finite-size effects NASA Astrophysics Data System (ADS) Plata, C. A.; Manacorda, A.; Lasanta, A.; Puglisi, A.; Prados, A. 2016-09-01 Long-range spatial correlations in the velocity and energy fields of a granular fluid are discussed in the framework of a 1d lattice model. The dynamics of the velocity field occurs through nearest-neighbour inelastic collisions that conserve momentum but dissipate energy. A set of equations for the fluctuating hydrodynamics of the velocity and energy mesoscopic fields give a first approximation for (i) the velocity structure factor and (ii) the finite-size correction to the Haff law, both in the homogeneous cooling regime. At a more refined level, we have derived the equations for the two-site velocity correlations and the total energy fluctuations. First, we seek a perturbative solution thereof, in powers of the inverse of system size. On the one hand, when scaled with the granular temperature, the velocity correlations tend to a stationary value in the long time limit. On the other hand, the scaled standard deviation of the total energy diverges, that is, the system shows multiscaling. Second, we find an exact solution for the velocity correlations in terms of the spectrum of eigenvalues of a certain matrix. The results of numerical simulations of the microscopic model confirm our theoretical results, including the above described multiscaling phenomenon. 7. A phase-field model coupled with lattice kinetics solver for modeling crystal growth in furnaces SciTech Connect Lin, Guang; Bao, Jie; Xu, Zhijie; Tartakovsky, Alexandre M.; Henager, Charles H. 2014-02-02 In this study, we present a new numerical model for crystal growth in a vertical solidification system. This model takes into account the buoyancy induced convective flow and its effect on the crystal growth process. The evolution of the crystal growth interface is simulated using the phase-field method. Two novel phase-field models are developed to model the crystal growth interface in vertical gradient furnaces with two temperature profile setups: 1) fixed wall temperature profile setup and 2) time-dependent temperature profile setup. A semi-implicit lattice kinetics solver based on the Boltzmann equation is employed to model the unsteady incompressible flow. This model is used to investigate the effect of furnace operational conditions on crystal growth interface profiles and growth velocities. For a simple case of macroscopic radial growth, the phase-field model is validated against an analytical solution. Crystal growth in vertical gradient furnaces with two temperature profile setups have been also investigated using the developed model. The numerical simulations reveal that for a certain set of temperature boundary conditions, the heat transport in the melt near the phase interface is diffusion dominant and advection is suppressed. 8. Regional electric field induced by electroconvulsive therapy in a realistic finite element head model: Influence of white matter anisotropic conductivity PubMed Central Lee, Won Hee; Deng, Zhi-De; Kim, Tae-Seong; Laine, Andrew F.; Lisanby, Sarah H.; Peterchev, Angel V. 2012-01-01 We present the first computational study investigating the electric field (E-field) strength generated by various electroconvulsive therapy (ECT) electrode configurations in specific brain regions of interest (ROIs) that have putative roles in the therapeutic action and/or adverse side effects of ECT. This study also characterizes the impact of the white matter (WM) conductivity anisotropy on the E-field distribution. A finite element head model incorporating tissue heterogeneity and WM anisotropic conductivity was constructed based on structural magnetic resonance imaging (MRI) and diffusion tensor MRI data. We computed the spatial E-field distributions generated by three standard ECT electrode placements including bilateral (BL), bifrontal (BF), and right unilateral (RUL) and an investigational electrode configuration for focal electrically administered seizure therapy (FEAST). The key results are that (1) the median E-field strength over the whole brain is 3.9, 1.5, 2.3, and 2.6 V/cm for the BL, BF, RUL, and FEAST electrode configurations, respectively, which coupled with the broad spread of the BL E-field suggests a biophysical basis for observations of superior efficacy of BL ECT compared to BF and RUL ECT; (2) in the hippocampi, BL ECT produces a median E-field of 4.8 V/cm that is 1.5–2.8 times stronger than that for the other electrode configurations, consistent with the more pronounced amnestic effects of BL ECT; and (3) neglecting the WM conductivity anisotropy results in E-field strength error up to 18% overall and up to 39% in specific ROIs, motivating the inclusion of the WM conductivity anisotropy in accurate head models. This computational study demonstrates how the realistic finite element head model incorporating tissue conductivity anisotropy provides quantitative insight into the biophysics of ECT, which may shed light on the differential clinical outcomes seen with various forms of ECT, and may guide the development of novel stimulation 9. Anisotropic Artificial Impedance Surfaces NASA Astrophysics Data System (ADS) Quarfoth, Ryan Gordon Anisotropic artificial impedance surfaces are a group of planar materials that can be modeled by the tensor impedance boundary condition. This boundary condition relates the electric and magnetic field components on a surface using a 2x2 tensor. The advantage of using the tensor impedance boundary condition, and by extension anisotropic artificial impedance surfaces, is that the method allows large and complex structures to be modeled quickly and accurately using a planar boundary condition. This thesis presents the theory of anisotropic impedance surfaces and multiple applications. Anisotropic impedance surfaces are a generalization of scalar impedance surfaces. Unlike the scalar version, anisotropic impedance surfaces have material properties that are dependent on the polarization and wave vector of electromagnetic radiation that interacts with the surface. This allows anisotropic impedance surfaces to be used for applications that scalar surfaces cannot achieve. Three of these applications are presented in this thesis. The first is an anisotropic surface wave waveguide which allows propagation in one direction, but passes radiation in the orthogonal direction without reflection. The second application is a surface wave beam shifter which splits a surface wave beam in two directions and reduces the scattering from an object placed on the surface. The third application is a patterned surface which can alter the scattered radiation pattern of a rectangular shape. For each application, anisotropic impedance surfaces are constructed using periodic unit cells. These unit cells are designed to give the desired surface impedance characteristics by modifying a patterned metallic patch on a grounded dielectric substrate. Multiple unit cell geometries are analyzed in order to find the setup with the best performance in terms of impedance characteristics and frequency bandwidth. 10. Comparison of | Q|=1 and | Q|=2 gauge-field configurations on the lattice four-torus NASA Astrophysics Data System (ADS) Bilson-Thompson, Sundance O.; Leinweber, Derek B.; Williams, Anthony G.; Dunne, Gerald V. 2004-06-01 It is known that exactly self-dual gauge-field configurations with topological charge | Q|=1 cannot exist on the untwisted continuum four-torus. We explore the manifestation of this remarkable fact on the lattice four-torus for SU(3) using advanced techniques for controlling lattice discretization errors, extending earlier work of De Forcrand et al. for SU(2). We identify three distinct signals for the instability of | Q|=1 configurations, and show that these signals manifest themselves early in the cooling process, long before the would-be instanton has shrunk to a size comparable to the lattice discretization threshold. These signals do not appear for the individual instantons which make up our | Q|=2 configurations. This indicates that these signals reflect the truly global nature of the instability, rather than the local discretization effects which cause the eventual disappearance of the would-be single instanton. Monte-Carlo generated SU(3) gauge-field configurations are cooled to the self-dual limit using an O(a 4) -improved gauge action chosen to have small but positive O(a 6) errors. This choice prevents lattice discretization errors from destroying instantons provided their size exceeds the dislocation threshold of the cooling algorithm. Lattice discretization errors are evaluated by comparing the O(a 4) -improved gauge-field action with an O(a 4) -improved action constructed from the square of an O(a 4) -improved lattice field-strength tensor, thus having different O(a 6) discretization errors. The number of action-density peaks, the instanton size, and the topological charge of configurations is monitored. We observe a fluctuation in the total topological charge of | Q|=1 configurations, and demonstrate that the onset of this unusual behavior corresponds with the disappearance of multiple-peaks in the action density. At the same time discretization errors are minimal. 11. Anisotropic magnetic field observed at 300 K in citrate-coated iron oxide nanoparticles: effect of counterions NASA Astrophysics Data System (ADS) Misra, Sushil K.; Li, Lin; Mukherjee, Sudip; Ghosh, Goutam 2015-12-01 Iron oxide nanoparticles (IONPs) have been synthesized by chemical co-precipitation method and coated with three citrates, namely, tri-lithium citrate (TLC), tri-sodium citrate (TSC), or tri-potassium citrate (TKC). In these core-shell' structures, the core' is a cluster of average 3 IONPs which is enveloped by a shell' of citrate molecules and counterions, and thus called core-shell' nano-clusters (CS-NCs), of average size 20 to 22 nm. The counterions in the three CS-NCs differ in ionic radii (r_{{ion}}), in the order of Li+ < Na+ < K+. Our aim was to investigate the effect of counterions on magnetic interactions between CS-NCs in different powder samples at 300 K, using vibrating sample magnetometer and electron magnetic resonance (EMR) techniques. The hysteresis loops showed negligible coercivity field ( H c) in all samples. The saturation magnetization ( M S) was the highest for TLC-coated CS-NCs. The blocking temperature ( T B), obtained from zero-field-cooled measurements, was >300 K for TLC-coated CS-NCs and <300 K for TSC- and TKC-coated CS-NCs. The EMR linewidth (∆ B PP), measured at 300 K, was also the broadest for TLC-coated CS-NCs. At low temperatures, Δ B PP was found to increase more significantly for TSC- and TKC-coated CS-NCs than for TLC-coated CS-NCs. These results indicate a significant anisotropic field effect; arising due to thermal motion of counterions at 300 K, on the magnetic interactions in TLC-coated CS-NCs. To our knowledge, this is the first report on the effect of counterions on magnetic interactions between CS-NCs. 12. Multimodal Brain-Tumor Segmentation Based on Dirichlet Process Mixture Model with Anisotropic Diffusion and Markov Random Field Prior PubMed Central Lu, Yisu; Jiang, Jun; Chen, Wufan 2014-01-01 Brain-tumor segmentation is an important clinical requirement for brain-tumor diagnosis and radiotherapy planning. It is well-known that the number of clusters is one of the most important parameters for automatic segmentation. However, it is difficult to define owing to the high diversity in appearance of tumor tissue among different patients and the ambiguous boundaries of lesions. In this study, a nonparametric mixture of Dirichlet process (MDP) model is applied to segment the tumor images, and the MDP segmentation can be performed without the initialization of the number of clusters. Because the classical MDP segmentation cannot be applied for real-time diagnosis, a new nonparametric segmentation algorithm combined with anisotropic diffusion and a Markov random field (MRF) smooth constraint is proposed in this study. Besides the segmentation of single modal brain-tumor images, we developed the algorithm to segment multimodal brain-tumor images by the magnetic resonance (MR) multimodal features and obtain the active tumor and edema in the same time. The proposed algorithm is evaluated using 32 multimodal MR glioma image sequences, and the segmentation results are compared with other approaches. The accuracy and computation time of our algorithm demonstrates very impressive performance and has a great potential for practical real-time clinical use. PMID:25254064 13. Vortex lattice phases in bosonic ladders in the presence of gauge field NASA Astrophysics Data System (ADS) Piraud, Marie; Greschner, Sebastian; Kolley, Fabian; McCulloch, Ian P.; Schollwoeck, Ulrich; Heidrich-Meisner, Fabian; Vekua, Temo 2016-05-01 We study vortex lattices in the interacting Bose-Hubbard model defined on two- and three-leg ladder geometries in the presence of a homogeneous flux. Our work is motivated by recent experiments using laser assisted-tunneling in optical lattices and lattices in synthetic dimensions, which studied the regime of weak interactions. We focus on the effects arising from stronger interactions, in both the real space optical lattice and the synthetic dimension schemes. Based on extensive density matrix renormalization group simulations and a bosonization analysis, we show that vortex lattices form at certain commensurate vortex densities. We identify the parameter space in which they emerge, and study their properties. Very interestingly, an enlarged unit cell forms in the vortex lattice phases, which can lead to the reversal of the current circulation-direction in both geometries. We demonstrate this effect in weak coupling and at sufficiently low temperature, and show that it is significant for intermediate interactions. 14. The thermal instability in a sheared magnetic field - Filament condensation with anisotropic heat conduction. [solar physics NASA Technical Reports Server (NTRS) Van Hoven, G.; Mok, Y. 1984-01-01 The condensation-mode growth rate of the thermal instability in an empirically motivated sheared field is shown to depend upon the existence of perpendicular thermal conduction. This typically very small effect (perpendicular conductivity/parallel conductivity less than about 10 to the -10th for the solar corona) increases the spatial-derivative order of the compressible temperature-perturbation equation, and thereby eliminates the singularities which appear when perpendicular conductivity = 0. The resulting growth rate is less than 1.5 times the controlling constant-density radiation rate, and has a clear maximum at a cross-field length of order 100 times and a width of about 0.1 the magnetic shear scale for solar conditions. The profiles of the observable temperature and density perturbations are independent of the thermal conductivity, and thus agree with those found previously. An analytic solution to the short-wavelength incompressible case is also given. 15. Tensor of the nonlinear polarizability of anisotropic medium and local'' field method NASA Astrophysics Data System (ADS) Lavric, V. V.; Ovander, L. N.; Shunyakov, V. T. 1983-08-01 The nonlinear polarizability tensor (NPT) for a molecular crystal of arbitrary symmetry has been obtained within the framework of polariton theory. Use of the Göppert-Mayer unitary transformation for the Hamiltonian of the crystal plus quantized electromagnetic field system made it possible to represent finally the result for the NPT in a compact form and to compare with results of semiphenomenological calculation of the NPT and to go out of the framework of the Gaitler-London approximation. 16. Statistical thermodynamics of lattice models in zeolites: Implications of local versus global mean field interactions NASA Astrophysics Data System (ADS) Ayappa, K. G. 1999-09-01 The statistical thermodynamics of adsorption in caged zeolites is developed by treating the zeolite as an ensemble of M identical cages or subsystems. Within each cage adsorption is assumed to occur onto a lattice of n identical sites. Expressions for the average occupancy per cage are obtained by minimizing the Helmholtz free energy in the canonical ensemble subject to the constraints of constant M and constant number of adsorbates N. Adsorbate-adsorbate interactions in the Brag-Williams or mean field approximation are treated in two ways. The local mean field approximation (LMFA) is based on the local cage occupancy and the global mean field approximation (GMFA) is based on the average coverage of the ensemble. The GMFA is shown to be equivalent in formulation to treating the zeolite as a collection of interacting single site subsystems. In contrast, the treatment in the LMFA retains the description of the zeolite as an ensemble of identical cages, whose thermodynamic properties are conveniently derived in the grand canonical ensemble. For a z coordinated lattice within the zeolite cage, with ɛaa as the adsorbate-adsorbate interaction parameter, the comparisons for different values of ɛaa*=ɛaaz/2kT, and number of sites per cage, n, illustrate that for -1<ɛaa*<0 and n⩾10, the adsorption isotherms and heats of adsorption predicted with the two approaches are similar. In general, the deviation between the LMFA and GMFA is greater for smaller n and less sensitive to n for ɛaa*>0. We compare the isotherms predicted with the LMFA with previous GMFA predictions [K. G. Ayappa, C. R. Kamala, and T. A. Abinandanan, J. Chem. Phys. 110, 8714 (1999)] (which incorporates both the site volume reduction and a coverage-dependent ɛaa) for xenon and methane in zeolite NaA. In all cases the predicted isotherms are very similar, with the exception of a small steplike feature present in the LMFA for xenon at higher coverages. 17. Quantum mean-field approximation for lattice quantum models: Truncating quantum correlations and retaining classical ones NASA Astrophysics Data System (ADS) Malpetti, Daniele; Roscilde, Tommaso 2017-02-01 The mean-field approximation is at the heart of our understanding of complex systems, despite its fundamental limitation of completely neglecting correlations between the elementary constituents. In a recent work [Phys. Rev. Lett. 117, 130401 (2016), 10.1103/PhysRevLett.117.130401], we have shown that in quantum many-body systems at finite temperature, two-point correlations can be formally separated into a thermal part and a quantum part and that quantum correlations are generically found to decay exponentially at finite temperature, with a characteristic, temperature-dependent quantum coherence length. The existence of these two different forms of correlation in quantum many-body systems suggests the possibility of formulating an approximation, which affects quantum correlations only, without preventing the correct description of classical fluctuations at all length scales. Focusing on lattice boson and quantum Ising models, we make use of the path-integral formulation of quantum statistical mechanics to introduce such an approximation, which we dub quantum mean-field (QMF) approach, and which can be readily generalized to a cluster form (cluster QMF or cQMF). The cQMF approximation reduces to cluster mean-field theory at T =0 , while at any finite temperature it produces a family of systematically improved, semi-classical approximations to the quantum statistical mechanics of the lattice theory at hand. Contrary to standard MF approximations, the correct nature of thermal critical phenomena is captured by any cluster size. In the two exemplary cases of the two-dimensional quantum Ising model and of two-dimensional quantum rotors, we study systematically the convergence of the cQMF approximation towards the exact result, and show that the convergence is typically linear or sublinear in the boundary-to-bulk ratio of the clusters as T →0 , while it becomes faster than linear as T grows. These results pave the way towards the development of semiclassical numerical 18. Tunable Lattice Coupling of Multipole Plasmon Modes and Near-Field Enhancement in Closely Spaced Gold Nanorod Arrays PubMed Central Huang, Yu; Zhang, Xian; Ringe, Emilie; Hou, Mengjing; Ma, Lingwei; Zhang, Zhengjun 2016-01-01 Considering the nanogap and lattice effects, there is an attractive structure in plasmonics: closely spaced metallic nanoarrays. In this work, we demonstrate experimentally and theoretically the lattice coupling of multipole plasmon modes for closely spaced gold nanorod arrays, offering a new insight into the higher order cavity modes coupled with each other in the lattice. The resonances can be greatly tuned by changes in inter-rod gaps and nanorod heights while the influence of the nanorod diameter is relatively insignificant. Experimentally, pronounced suppressions of the reflectance are observed. Meanwhile, the near-field enhancement can be further enhanced, as demonstrated through surface enhanced Raman scattering (SERS). We then confirm the correlation between the near-field and far-field plasmonic responses, which is significantly important for maximizing the near-field enhancement at a specific excitation wavelength. This lattice coupling of multipole plasmon modes is of broad interest not only for SERS but also for other plasmonic applications, such as subwavelength imaging or metamaterials. PMID:26983501 19. Mean field lattice model for adsorption isotherms in zeolite NaA NASA Astrophysics Data System (ADS) Ayappa, K. G.; Kamala, C. R.; Abinandanan, T. A. 1999-05-01 Using a lattice model for adsorption in microporous materials, pure component adsorption isotherms are obtained within a mean field approximation for methane at 300 K and xenon at 300 and 360 K in zeolite NaA. It is argued that the increased repulsive adsorbate-adsorbate interactions at high coverages must play an important role in determining the adsorption behavior. Therefore, this feature is incorporated through a "coverage-dependent interaction" model, which introduces a free, adjustable parameter. Another important feature, the site volume reduction, has been treated in two ways: a van der Waal model and a 1D hard-rod theory [van Tassel et al., AIChE J. 40, 925 (1994)]; we have also generalized the latter to include all possible adsorbate overlap scenarios. In particular, the 1D hard-rod model, with our coverage-dependent interaction model, is shown to be in best quantitative agreement with the previous grand canonical Monte Carlo isotherms. The expressions for the isosteric heats of adsorption indicate that attractive and repulsive adsorbate-adsorbate interactions increase and decrease the heats of adsorption, respectively. It is concluded that within the mean field approximation, our simple model for repulsive interactions and the 1D hard-rod model for site volume reduction are able to capture most of the important features of adsorption in confined regions. 20. Replica symmetry breaking transition of the weakly anisotropic Heisenberg spin glass in magnetic fields. PubMed Imagawa, Daisuke; Kawamura, Hikaru 2004-02-20 The spin and the chirality orderings of the three-dimensional Heisenberg spin glass with the weak random anisotropy are studied under applied magnetic fields by equilibrium Monte Carlo simulations. A replica symmetry breaking transition occurs in the chiral sector accompanied by the simultaneous spin-glass order. The ordering behavior differs significantly from that of the Ising spin glass, despite the similarity in the global symmetry. Our observation is consistent with the spin-chirality decoupling-recoupling scenario of a spin-glass transition. 1. Molecular determinants for drug-receptor interactions. 8. Anisotropic and internal motions in morphine, nalorphine, oxymorphone, naloxone and naltrexone in aqueous solution by carbon-13 NMR spin-lattice relaxation times NASA Astrophysics Data System (ADS) Grassi, Antonio; Perly, Bruno; Pappalardo, Giuseppe C. 1989-02-01 Carbon-13 NMR spin-lattice relaxation times ( T1) were measured for morphine, oxymorphone, nalorphine, naloxone and naltrexone as hydrochloride salts in 2H 2O solution. The data refer to the molecules in the N-equatorial configuration. The experimental T1 values were interpreted using a model of anisotropic reorientation of a rigid body with superimposed internal motions of the flexible N-methyl, N-methyl-allyl and N-methyl-cyclopropyl fragments. The calculated internal motional rates were found to markedly decrease on passing from agonists to mixed (nalorphine) and pure (naloxone, naltrexone) antagonists. For these latter the observed trend of the internal flexibility about NC and CC bonds of the N-substituents is discussed in terms of a correlation with their relative antagonistic potencies. In fact, such an evidence of decreasing internal conformational dynamics in the order nalorphine, naloxone, naltrexone, appeared interestingly in line with the "two-state" model of opiate receptor operation mode proposed by Snyder. 2. Kubo number and magnetic field line diffusion coefficient for anisotropic magnetic turbulence. PubMed Pommois, P; Veltri, P; Zimbardo, G 2001-06-01 The magnetic field line diffusion coefficients Dx and D(y) are obtained by numerical simulations in the case that all the magnetic turbulence correlation lengths l(x), l(y), and l(z) are different. We find that the variety of numerical results can be organized in terms of the Kubo number, the definition of which is extended from R=(deltaB/B(0))(l(parallel)/l(perpendicular)) to R=(deltaB/B(0))(l(z)/l(x)), for l(x) > or = l(y). Here, l(parallel) (l(perpendicular)) is the correlation length along (perpendicular to) the average field B(0)=B(0)ê(z). We have anomalous, non-Gaussian transport for R less, similar 0.1, in which case the mean square deviation scales nonlinearly with time. For R greater, similar 1 we have several Gaussian regimes: an almost quasilinear regime for 0.1 less, similar R less, similar 1, an intermediate, transition regime for 1 less, similar R less, similar 10, and a percolative regime for R greater, similar 10. An analytical form of the diffusion coefficient is proposed, D(i)=D(deltaBl(z)/B(0)l(x))(mu)(l(i)/l(x))(nu)l(2)(x)/l(z), which well describes the numerical simulation results in the quasilinear, intermediate, and percolative regimes. 3. Anisotropic mechanical behaviour of sedimentary basins inferred by advanced radar interferometry above gas storage fields NASA Astrophysics Data System (ADS) Teatini, P.; Gambolati, G.; Ferretti, A. 2010-12-01 Natural gas is commonly stored underground in depleted oil and gas fields to provide safe storage capacity and deliverability to market areas where production is limited, or to take advantage of seasonal price swings. In response to summer gas injection and winter gas withdrawal the reservoir expands and contracts with the overlying land that moves accordingly. Depending on the field burial depth, a few kilometres of the upper lithosphere are subject to local three-dimensional deformations with the related cyclic motion of the ground surface being both vertical and horizontal. Advanced Persistent Scatterer Interferometry (PSI) data, obtained by combining ascending and descending RADARSAT-1 images acquired from 2003 to 2008 above gas storage fields located in the sedimentary basin of the Po river plain, Italy, provide reliable measurement of these seasonal vertical ups and downs as well as horizontal displacements to and from the injection/withdrawal wells. Combination of the land surface movements together with an accurate reconstruction of the subsurface geology made available by three-dimensional seismic surveys and long-time records of fluid pore pressure within the 1000-1500 m deep reservoirs has allowed for the development of an accurate 3D poro-mechanical finite-element model of the gas injection/removal occurrence. Model calibration based on the observed cyclic motions, which are on the range of 10-15 mm and 5-10 mm in the vertical and horizontal west-east directions, respectively, helps characterize the nonlinear hysteretic geomechanical properties of the basin. First, using a basin-scale relationship between the oedometric rock compressibility cM in virgin loading conditions versus the effective intergranular stress derived from previous experimental studies, the modeling results show that the ratio s between loading and unloading-reloading cM is about 4, consistent with in-situ expansions measured by the radioactive marker technique in similar reservoirs 4. Magnetic-field-induced vortex-lattice transition in HgBa2CuO4 +δ NASA Astrophysics Data System (ADS) Lee, Jeongseop A.; Xin, Yizhou; Stolt, I.; Halperin, W. P.; Reyes, A. P.; Kuhns, P. L.; Chan, M. K. 2017-01-01 Measurements of the 17O nuclear magnetic resonance (NMR) quadrupolar spectrum of apical oxygen in HgBa2CuO4 +δ were performed over a range of magnetic fields from 6.4-30 T in the superconducting state. Oxygen-isotope-exchanged single crystals were investigated with doping corresponding to superconducting transition temperatures from 74 K underdoped, to 78 K overdoped. The apical oxygen site was chosen since its NMR spectrum has narrow quadrupolar satellites that are well separated from any other resonance. Nonvortex contributions to the spectra can be deconvolved in the time domain to determine the local magnetic field distribution from the vortices. Numerical analysis using Brandt's Ginzburg-Landau theory was used to find structural parameters of the vortex lattice, penetration depth, and coherence length as a function of magnetic field in the vortex solid phase. From this analysis we report a vortex structural transition near 15 T from an oblique lattice with an opening angle of 73∘ at low magnetic fields to a triangular lattice with 60∘ stabilized at high field. The temperature for onset of vortex dynamics has been identified from spin-spin relaxation. This is independent of the magnetic field at sufficiently high magnetic field similar to that reported for YBa2Cu3O7 and Bi2Sr2CaCu2O8 +δ and is correlated with mass anisotropy of the material. This behavior is accounted for theoretically only in the limit of very high anisotropy. 5. Zigzag antiferromagnetic ground state with anisotropic correlation lengths in the quasi-two-dimensional honeycomb lattice compound N a2C o2Te O6 NASA Astrophysics Data System (ADS) Bera, A. K.; Yusuf, S. M.; Kumar, Amit; Ritter, C. 2017-03-01 The crystal structure, magnetic ground state, and the temperature-dependent microscopic spin-spin correlations of the frustrated honeycomb lattice antiferromagnet N a2C o2Te O6 have been investigated by powder neutron diffraction. A long-range antiferromagnetic (AFM) ordering has been found below TN˜24.8 K . The magnetic ground state, determined to be zigzag antiferromagnetic and characterized by a propagation vector k =(1 /2 0 0 ) , occurs due to the competing exchange interactions up to third-nearest neighbors within the honeycomb lattice. The exceptional existence of a limited magnetic correlation length along the c axis (perpendicular to the honeycomb layers in the a b planes) has been found even at 1.8 K, well below the TN˜24.8 K . The observed limited correlation along the c axis is explained by the disorder distribution of the Na ions within the intermediate layers between honeycomb planes. The reduced ordered moments mCo (1 )=2.77 (3 ) μB/C o2 + and mCo (2 )=2.45 (2 ) μB/C o2 + at 1.8 K reflect the persistence of spin fluctuations in the ordered state. Above TN˜24.8 K , the presence of short-range magnetic correlations, manifested by broad diffuse magnetic peaks in the diffraction patterns, has been found. Reverse Monte Carlo analysis of the experimental diffuse magnetic scattering data reveals that the spin correlations are mainly confined within the two-dimensional honeycomb layers (a b plane) with a correlation length of ˜12 Å at 25 K. The nature of the spin arrangements is found to be similar in both the short-range and long-range ordered magnetic states. This implies that the short-range correlation grows with decreasing temperature and leads to the zigzag AFM ordering at T ≤TN . The present study provides a comprehensive picture of the magnetic correlations over the temperature range above and below the TN and their relation to the crystal structure. The role of intermediate soft Na layers on the magnetic coupling between honeycomb planes is 6. Anisotropic ray trace NASA Astrophysics Data System (ADS) Lam, Wai Sze Tiffany anisotropic ray tracing. x. Chapter 4 presents the data reduction of the P matrix of a crystal waveplate. The diattenuation is embedded in the singular values of P. The retardance is divided into two parts: (A) The physical retardance induced by OPLs and surface interactions, and (B) the geometrical transformation induced by geometry of a ray path, which is calculated by the geometrical transform Q matrix. The Q matrix of an anisotropic intercept is derived from the generalization of s- and p-bases at the anisotropic intercept; the p basis is not confined to the plane of incidence due to the anisotropic refraction or reflection. Chapter 5 shows how the multiple P matrices associated with the eigenmodes resulting from propagation through multiple anisotropic surfaces can be combined into one P matrix when the multiple modes interfere in their overlapping regions. The resultant P matrix contains diattenuation induced at each surface interaction as well as the retardance due to ray propagation and total internal reflections. The polarization aberrations of crystal waveplates and crystal polarizers are studied in Chapter 6 and Chapter 7. A wavefront simulated by a grid of rays is traced through the anisotropic system and the resultant grid of rays is analyzed. The analysis is complicated by the ray doubling effects and the partially overlapping eigen-wavefronts propagating in various directions. The wavefront and polarization aberrations of each eigenmode can be evaluated from the electric field distributions. The overall polarization at the plane of interest or the image quality at the image plane are affected by each of these eigen-wavefronts. Isotropic materials become anisotropic due to stress, strain, or applied electric or magnetic fields. In Chapter 8, the P matrix for anisotropic materials is extended to ray tracing in stress birefringent materials which are treated as spatially varying anisotropic materials. Such simulations can predict the spatial retardance variation 7. Anisotropic lattice thermal expansion of PbFeBO4: A study by X-ray and neutron diffraction, Raman spectroscopy and DFT calculations DOE PAGES Murshed, M. Mangir; Mendive, Cecilia B.; Curti, Mariano; ... 2014-11-01 We present the lattice thermal expansion of mullite-type PbFeBO4 in this study. The thermal expansion coefficients of the metric parameters were obtained from composite data collected from temperature-dependent neutron and X-ray powder diffraction between 10 K and 700 K. The volume thermal expansion was modeled using extended Grüneisen first-order approximation to the zero-pressure equation of state. The additive frame of the model includes harmonic, quasi-harmonic and intrinsic anharmonic potentials to describe the change of the internal energy as a function of temperature. Moreover, the unit-cell volume at zero-pressure and 0 K was optimized during the DFT simulations. Harmonic frequencies ofmore » the optical Raman modes at the Γ-point of the Brillouin zone at 0 K were also calculated by DFT, which help to assign and crosscheck the experimental frequencies. The low-temperature Raman spectra showed significant anomaly in the antiferromagnetic regions, leading to softening or hardening of some phonons. Selected modes were analyzed using a modified Klemens model. The shift of the frequencies and the broadening of the line-widths helped to understand the anharmonic vibrational behaviors of the PbO4, FeO6 and BO3 polyhedra as a function of temperature.« less 8. Anisotropic lattice thermal expansion of PbFeBO4: A study by X-ray and neutron diffraction, Raman spectroscopy and DFT calculations SciTech Connect Murshed, M. Mangir; Mendive, Cecilia B.; Curti, Mariano; Nénert, Gwilherm; Kalita, Patricia E.; Lipinska, Kris; Cornelius, Andrew L.; Huq, Ashfia; Gesing, Thorsten M. 2014-11-01 We present the lattice thermal expansion of mullite-type PbFeBO4 in this study. The thermal expansion coefficients of the metric parameters were obtained from composite data collected from temperature-dependent neutron and X-ray powder diffraction between 10 K and 700 K. The volume thermal expansion was modeled using extended Grüneisen first-order approximation to the zero-pressure equation of state. The additive frame of the model includes harmonic, quasi-harmonic and intrinsic anharmonic potentials to describe the change of the internal energy as a function of temperature. Moreover, the unit-cell volume at zero-pressure and 0 K was optimized during the DFT simulations. Harmonic frequencies of the optical Raman modes at the Γ-point of the Brillouin zone at 0 K were also calculated by DFT, which help to assign and crosscheck the experimental frequencies. The low-temperature Raman spectra showed significant anomaly in the antiferromagnetic regions, leading to softening or hardening of some phonons. Selected modes were analyzed using a modified Klemens model. The shift of the frequencies and the broadening of the line-widths helped to understand the anharmonic vibrational behaviors of the PbO4, FeO6 and BO3 polyhedra as a function of temperature. 9. Magnetic translation algebra with or without magnetic field in the continuum or on arbitrary Bravais lattices in any dimension NASA Astrophysics Data System (ADS) Chamon, Claudio; Mudry, Christopher 2012-11-01 The magnetic translation algebra plays an important role in the quantum Hall effect. Murthy and Shankar, arXiv:1207.2133, have shown how to realize this algebra using fermionic bilinears defined on a two-dimensional square lattice. We show that, in any dimension d, it is always possible to close the magnetic translation algebra using fermionic bilinears, whether in the continuum or on the lattice. We also show that these generators are complete in even, but not odd, dimensions, in the sense that any fermionic Hamiltonian in even dimensions that conserves particle number can be represented in terms of the generators of this algebra, whether or not time-reversal symmetry is broken. As an example, we reproduce the f-sum rule of interacting electrons at vanishing magnetic field using this representation. We also show that interactions can significantly change the bare bandwidth of lattice Hamiltonians when represented in terms of the generators of the magnetic translation algebra. 10. Deleterious localized stress fields: the effects of boundaries and stiffness tailoring in anisotropic laminated plates PubMed Central Weaver, P. M. 2016-01-01 The safe design of primary load-bearing structures requires accurate prediction of stresses, especially in the vicinity of geometric discontinuities where deleterious three-dimensional stress fields can be induced. Even for thin-walled structures significant through-thickness stresses arise at edges and boundaries, and this is especially precarious for laminates of advanced fibre-reinforced composites because through-thickness stresses are the predominant drivers in delamination failure. Here, we use a higher-order equivalent single-layer model derived from the Hellinger–Reissner mixed variational principle to examine boundary layer effects in laminated plates comprising constant-stiffness and variable-stiffness laminae and deforming statically in cylindrical bending. The results show that zigzag deformations, which arise due to layerwise differences in the transverse shear moduli, drive boundary layers towards clamped edges and are therefore critically important in quantifying localized stress gradients. The relative significance of the boundary layer scales with the degree of layerwise anisotropy and the thickness to characteristic length ratio. Finally, we demonstrate that the phenomenon of alternating positive and negative transverse shearing deformation through the thickness of composite laminates, previously only observed at clamped boundaries, can also occur at other locations as a result of smoothly varying the material properties over the in-plane dimensions of the laminate. PMID:27843401 11. Dependence of the magnetic properties on the alignment magnetic field for NdFeB bonded magnets made from anisotropic HDDR powders NASA Astrophysics Data System (ADS) Gao, R. W.; Zhang, J. C.; Zhang, D. H.; Dai, Y. Y.; Meng, X. H.; Wang, Z. M.; Zhang, Y. J.; Liu, H. Q. 1999-01-01 The dependence of the hard magnetic properties on the alignment magnetic field for Nd(Fe,Co)B bonded magnets made from anisotropic HDDR powders is studied. The experimental results demonstrate that addition of a little Ga can induce a strong magnetic anisotropy in the HDDR magnetic powders. The application of an alignment magnetic field while the powders are bonded can increase the remanence, the coercivity and the maximum energy product in different degrees and the hard magnetic properties of the magnet are obviously improved with increasing alignment field. 12. Lattice model for amyloid peptides: OPEP force field parametrization and applications to the nucleus size of Alzheimer's peptides. PubMed Tran, Thanh Thuy; Nguyen, Phuong H; Derreumaux, Philippe 2016-05-28 Coarse-grained protein lattice models approximate atomistic details and keep the essential interactions. They are, therefore, suitable for capturing generic features of protein folding and amyloid formation at low computational cost. As our aim is to study the critical nucleus sizes of two experimentally well-characterized peptide fragments Aβ16-22 and Aβ37-42 of the full length Aβ1-42 Alzheimer's peptide, it is important that simulations with the lattice model reproduce all-atom simulations. In this study, we present a comprehensive force field parameterization based on the OPEP (Optimized Potential for Efficient protein structure Prediction) force field for an on-lattice protein model, which incorporates explicitly the formation of hydrogen bonds and directions of side-chains. Our bottom-up approach starts with the determination of the best lattice force parameters for the Aβ16-22 dimer by fitting its equilibrium parallel and anti-parallel β-sheet populations to all-atom simulation results. Surprisingly, the calibrated force field is transferable to the trimer of Aβ16-22 and the dimer and trimer of Aβ37-42. Encouraged by this finding, we characterized the free energy landscapes of the two decamers. The dominant structure of the Aβ16-22 decamer matches the microcrystal structure. Pushing the simulations for aggregates between 4-mer and 12-mer suggests a nucleus size for fibril formation of 10 chains. In contrast, the Aβ37-42 decamer is largely disordered with mixed by parallel and antiparallel chains, suggesting that the nucleus size is >10 peptides. Our refined force field coupled to this on-lattice model should provide useful insights into the critical nucleation number associated with neurodegenerative diseases. 13. Lattice model for amyloid peptides: OPEP force field parametrization and applications to the nucleus size of Alzheimer's peptides NASA Astrophysics Data System (ADS) Tran, Thanh Thuy; Nguyen, Phuong H.; Derreumaux, Philippe 2016-05-01 Coarse-grained protein lattice models approximate atomistic details and keep the essential interactions. They are, therefore, suitable for capturing generic features of protein folding and amyloid formation at low computational cost. As our aim is to study the critical nucleus sizes of two experimentally well-characterized peptide fragments Aβ16-22 and Aβ37-42 of the full length Aβ1-42 Alzheimer's peptide, it is important that simulations with the lattice model reproduce all-atom simulations. In this study, we present a comprehensive force field parameterization based on the OPEP (Optimized Potential for Efficient protein structure Prediction) force field for an on-lattice protein model, which incorporates explicitly the formation of hydrogen bonds and directions of side-chains. Our bottom-up approach starts with the determination of the best lattice force parameters for the Aβ16-22 dimer by fitting its equilibrium parallel and anti-parallel β-sheet populations to all-atom simulation results. Surprisingly, the calibrated force field is transferable to the trimer of Aβ16-22 and the dimer and trimer of Aβ37-42. Encouraged by this finding, we characterized the free energy landscapes of the two decamers. The dominant structure of the Aβ16-22 decamer matches the microcrystal structure. Pushing the simulations for aggregates between 4-mer and 12-mer suggests a nucleus size for fibril formation of 10 chains. In contrast, the Aβ37-42 decamer is largely disordered with mixed by parallel and antiparallel chains, suggesting that the nucleus size is >10 peptides. Our refined force field coupled to this on-lattice model should provide useful insights into the critical nucleation number associated with neurodegenerative diseases. 14. CP(N - 1) quantum field theories with alkaline-earth atoms in optical lattices NASA Astrophysics Data System (ADS) Laflamme, C.; Evans, W.; Dalmonte, M.; Gerber, U.; Mejía-Díaz, H.; Bietenholz, W.; Wiese, U.-J.; Zoller, P. 2016-07-01 We propose a cold atom implementation to attain the continuum limit of (1 + 1) -d CP(N - 1) quantum field theories. These theories share important features with (3 + 1) -d QCD, such as asymptotic freedom and θ-vacua. Moreover, their continuum limit can be accessed via the mechanism of dimensional reduction. In our scheme, the CP(N - 1) degrees of freedom emerge at low energies from a ladder system of SU(N) quantum spins, where the N spin states are embodied by the nuclear Zeeman states of alkaline-earth atoms, trapped in an optical lattice. Based on Monte Carlo results, we establish that the continuum limit can be demonstrated by an atomic quantum simulation by employing the feature of asymptotic freedom. We discuss a protocol for the adiabatic preparation of the ground state of the system, the real-time evolution of a false θ-vacuum state after a quench, and we propose experiments to unravel the phase diagram at non-zero density. 15. Magnetic field-induced spectroscopy of forbidden optical transitions with application to lattice-based optical atomic clocks. PubMed Taichenachev, A V; Yudin, V I; Oates, C W; Hoyt, C W; Barber, Z W; Hollberg, L 2006-03-03 We develop a method of spectroscopy that uses a weak static magnetic field to enable direct optical excitation of forbidden electric-dipole transitions that are otherwise prohibitively weak. The power of this scheme is demonstrated using the important application of optical atomic clocks based on neutral atoms confined to an optical lattice. The simple experimental implementation of this method--a single clock laser combined with a dc magnetic field--relaxes stringent requirements in current lattice-based clocks (e.g., magnetic field shielding and light polarization), and could therefore expedite the realization of the extraordinary performance level predicted for these clocks. We estimate that a clock using alkaline-earth-like atoms such as Yb could achieve a fractional frequency uncertainty of well below 10(-17) for the metrologically preferred even isotopes. 16. Anisotropic Nanoparticles and Anisotropic Surface Chemistry. PubMed Burrows, Nathan D; Vartanian, Ariane M; Abadeer, Nardine S; Grzincic, Elissa M; Jacob, Lisa M; Lin, Wayne; Li, Ji; Dennison, Jordan M; Hinman, Joshua G; Murphy, Catherine J 2016-02-18 Anisotropic nanoparticles are powerful building blocks for materials engineering. Unusual properties emerge with added anisotropy-often to an extraordinary degree-enabling countless new applications. For bottom-up assembly, anisotropy is crucial for programmability; isotropic particles lack directional interactions and can self-assemble only by basic packing rules. Anisotropic particles have long fascinated scientists, and their properties and assembly behavior have been the subjects of many theoretical studies over the years. However, only recently has experiment caught up with theory. We have begun to witness tremendous diversity in the synthesis of nanoparticles with controlled anisotropy. In this Perspective, we highlight the synthetic achievements that have galvanized the field, presenting a comprehensive discussion of the mechanisms and products of both seed-mediated and alternative growth methods. We also address recent breakthroughs and challenges in regiospecific functionalization, which is the next frontier in exploiting nanoparticle anisotropy. 17. Coherence of interacting bosons in optical lattices in synthetic magnetic fields with a large number of subbands NASA Astrophysics Data System (ADS) Grygiel, B.; Patucha, K.; Zaleski, T. A. 2016-05-01 We study the behavior of interacting ultracold bosons in optical lattices in synthetic magnetic fields with wide range of in-cell fluxes α =p /q . The problem is similar to the one of an electron moving in a tight-binding scheme in the magnetic field and becomes difficult to tackle for a growing number of magnetic subbands, q . To overcome this, we focus on the interplay of the width, shape, and number of the subbands on the formation of the coherent state of cold bosons. Using the quantum rotor approach, which goes beyond the mean-field approximation, we are able to pinpoint the elements of the band structure, which are the most significant in a proper theoretical description of the synthetic magnetic field in a bosonic lattice system. As a result, we propose a method of reconstruction of the Hofstadter butterfly spectrum by replacing the magnetic subbands with renormalized bands of a square lattice. This allows us to effectively investigate the properties of the studied system for a wide range of magnetic fluxes and their impact on the Mott-insulator-superfluid transition. 18. Field dependence of the magnon dispersion in the Kondo lattice CeCu2 up to 12 T NASA Astrophysics Data System (ADS) Schedler, R.; Witte, U.; Rotter, M.; Loewenhaupt, M.; Schmidt, W. 2005-05-01 CeCu2 can be classified as a Kondo lattice which shows antiferromagnetic (AF) order below TN=3.5K [R. Trump et al., J. Appl. Phys. 69, 4699 (1991)]. The orthorhombic crystal and the simple AF magnetic structure with two magnetic moments in the primitive unit cell requires two magnon modes which are observed in zero and low magnetic fields and well described by spin wave theory. However, at higher fields, at and above 3T, an unexpected, additional magnetic excitation is observed. In contrast to the two low-energy magnon modes, it exhibits a steeper (factor 2) field dependence and a flat dispersion. Its origin is unclear. 19. Dynamical band flipping in fermionic lattice systems: an ac-field-driven change of the interaction from repulsive to attractive. PubMed Tsuji, Naoto; Oka, Takashi; Werner, Philipp; Aoki, Hideo 2011-06-10 We show theoretically that the sudden application of an appropriate ac field to correlated lattice fermions flips the band structure and effectively switches the interaction from repulsive to attractive. The nonadiabatically driven system is characterized by a negative temperature with a population inversion. We numerically demonstrate the converted interaction in an ac-driven Hubbard model with the nonequilibrium dynamical mean-field theory solved by the continuous-time quantum Monte Carlo method. Based on this, we propose an efficient ramp-up protocol for ac fields that can suppress heating, which leads to an effectively attractive Hubbard model with a temperature below the superconducting transition temperature of the equilibrium system. 20. Line of Dirac Nodes in Hyperhoneycomb Lattices. PubMed Mullen, Kieran; Uchoa, Bruno; Glatzhofer, Daniel T 2015-07-10 We propose a family of structures that have "Dirac loops," closed lines of Dirac nodes in momentum space, on which the density of states vanishes linearly with energy. Those lattices all possess the planar trigonal connectivity present in graphene, but are three dimensional. We show that their highly anisotropic and multiply connected Fermi surface leads to quantized Hall conductivities in three dimensions for magnetic fields with toroidal geometry. In the presence of spin-orbit coupling, we show that those structures have topological surface states. We discuss the feasibility of realizing the structures as new allotropes of carbon. 1. Line of Dirac Nodes in Hyperhoneycomb Lattices NASA Astrophysics Data System (ADS) Mullen, Kieran; Uchoa, Bruno; Glatzhofer, Daniel T. 2015-07-01 We propose a family of structures that have "Dirac loops," closed lines of Dirac nodes in momentum space, on which the density of states vanishes linearly with energy. Those lattices all possess the planar trigonal connectivity present in graphene, but are three dimensional. We show that their highly anisotropic and multiply connected Fermi surface leads to quantized Hall conductivities in three dimensions for magnetic fields with toroidal geometry. In the presence of spin-orbit coupling, we show that those structures have topological surface states. We discuss the feasibility of realizing the structures as new allotropes of carbon. 2. Artificial Gauge Field and Topological Phase in a Conventional Two-dimensional Electron Gas with Antidot Lattices PubMed Central Shi, Likun; Lou, Wenkai; Cheng, F.; Zou, Y. L.; Yang, Wen; Chang, Kai 2015-01-01 Based on the Born-Oppemheimer approximation, we divide the total electron Hamiltonian in a spin-orbit coupled system into the slow orbital motion and the fast interband transition processes. We find that the fast motion induces a gauge field on the slow orbital motion, perpendicular to the electron momentum, inducing a topological phase. From this general designing principle, we present a theory for generating artificial gauge field and topological phase in a conventional two-dimensional electron gas embedded in parabolically graded GaAs/InxGa1−xAs/GaAs quantum wells with antidot lattices. By tuning the etching depth and period of the antidot lattices, the band folding caused by the antidot potential leads to the formation of minibands and band inversions between neighboring subbands. The intersubband spin-orbit interaction opens considerably large nontrivial minigaps and leads to many pairs of helical edge states in these gaps. PMID:26471126 3. Cylindrical periodic surface lattice as a metadielectric: Concept of a surface-field Cherenkov source of coherent radiation SciTech Connect Konoplev, I. V.; MacLachlan, A. J.; Robertson, C. W.; Cross, A. W.; Phelps, A. D. R. 2011-07-15 A two-dimensional (2D), cylindrical, periodic surface lattice (PSL) forming a surface field cavity is considered. The lattice is created by introducing 2D periodic perturbations on the inner surface of a cylindrical waveguide. The PSL facilitates a resonant coupling of the surface and near cutoff volume fields, leading to the formation of a high-Q cavity eigenmode. The cavity eigenmode is described and investigated using a modal approach, considering the model of a cylindrical waveguide partially loaded with a metadielectric. By using a PSL-based cavity, the concept of a high-power, 0.2-THz Cherenkov source is developed. It is shown that if the PSL satisfies certain defined conditions, single-mode operation is observed. 4. Results from a strong-coupling expansion for a lattice Yukawa model with a real scalar field SciTech Connect Abada, A.; Shrock, R.E. ) 1991-01-15 Results are presented from a strong bare Yukawa coupling expansion for a lattice Yukawa theory with a real scalar field. It is found that the effective action involves competing interactions, consistent with the existence of a ferrimagnetic phase at intermediate Yukawa coupling {ital y}. We also give evidence that the (bosonic) continuum theory defined at the ferromagnetic-paramagnetic phase boundary at large {ital y} is free. 5. Spectral properties and phase diagram of correlated lattice bosons in an optical cavity within bosonic dynamical mean-field theory NASA Astrophysics Data System (ADS) Panas, Jaromir; Kauch, Anna; Byczuk, Krzysztof 2017-03-01 We use the Bose-Hubbard model with an effective infinite-range interaction to describe the correlated lattice bosons in an optical cavity. We study both static and spectral properties of such system within the bosonic dynamical mean-field theory, which is the state-of-the-art method for strongly correlated bosonic systems. Both similarities and differences are found and discussed between our results and those obtained within different theoretical methods and experiment. 6. Experimental exploration of novel semimetal state in strong anisotropic Pyrochlore iridate Nd2lr2O7 under high magnetic field NASA Astrophysics Data System (ADS) Tian, Z. M.; Kohama, Y.; Tomita, T.; Ishikawa, J.; Mairo, H.; Kindo, K.; Nakatsuji, S. 2016-02-01 We report the anisotropic magnetotransport of Nd2Ir2O7 single crystal under high magnetic field (B) up to 50 T with B along various directions. Only for B // [001] direction, a novel semimetal state is realized under high magnetic field evidenced by a field-induced insulating-semimetalic phase transition with critical field BMI∼12 T related to the destruction of all-in-all-out (AIAO) state. In contrast, abnormal magnetotransport hysteresis behavior is observed for B // [111] direction below the metal-insulator transition temperature (TMI), and magnetotransport reveals the Ir spin structure keep in the AIAO state under high magnetic field with temperature just below TMI, in prospect to realize Weyl semimetal state. 7. Simulating the hydraulic stimulation of multiple fractures in an anisotropic stress field applying the discrete element method NASA Astrophysics Data System (ADS) Zeeb, Conny; Frühwirt, Thomas; Konietzky, Heinz 2015-04-01 Key to a successful exploitation of deep geothermal reservoirs in a petrothermal environment is the hydraulic stimulation of the host rock to increase permeability. The presented research investigates the fracture propagation and interaction during hydraulic stimulation of multiple fractures in a highly anisotropic stress field. The presented work was conducted within the framework of the OPTIRISS project, which is a cooperation of industry partners and universities in Thuringia and Saxony (Federal States of Germany) and was funded by the European Fond for Regional Development. One objective was the design optimization of the subsurface geothermal heat exchanger (SGHE) by means of numerical simulations. The presented simulations were conducted applying 3DEC (Itasca™), a software tool based on the discrete element method. The simulation results indicate that the main direction of fracture propagation is towards lower stresses and thus towards the biosphere. Therefore, barriers might be necessary to limit fracture propagation to the designated geological formation. Moreover, the hydraulic stimulation significantly alters the stresses in the vicinity of newly created fractures. Especially the change of the minimum stress component affects the hydraulic stimulation of subsequent fractures, which are deflected away from the previously stimulated fractures. This fracture deflection can render it impossible to connect all fractures with a second borehole for the later production. The results of continuative simulations indicate that a fracture deflection cannot be avoided completely. Therefore, the stage alignment was modified to minimize fracture deflection by varying (1) the pauses between stages, (2) the spacing's between adjacent stages, and (3) the angle between stimulation borehole and minimum stress component. An optimum SGHE design, which implies that all stimulated fractures are connected to the production borehole, can be achieved by aligning the stimulation 8. Optical trapping of anisotropic nanocylinder NASA Astrophysics Data System (ADS) Bareil, Paul B.; Sheng, Yunlong 2013-09-01 The T-matrix method with the Vector Spherical Wave Function (VSWF) expansions represents some difficulties for computing optical scattering of anisotropic particles. As the divergence of the electric field is nonzero in the anisotropic medium and the VSWFs do not satisfy the anisotropic wave equations one questioned whether the VSWFs are still a suitable basis in the anisotropic medium. We made a systematic and careful review on the vector basis functions and the VSWFs. We found that a field vector in Euclidean space can be decomposed to triplet vectors {L, M, N}, which as non-coplanar. Especially, the vector L is designed to represent non-zero divergence component of the vector solution, so that the VSWF basis is sufficiently general to represent the solutions of the anisotropic wave equation. The mathematical proof can be that when the anisotropic wave equations is solved in the Fourier space, the solution is expanded in the basis of the plan waves with angular spectrum amplitude distributions. The plane waves constitute an orthogonal and complete set for the anisotropic solutions. Furthermore, the plane waves are expanded into the VSWF basis. These two-step expansions are equivalent to the one-step direct expansion of the anisotropic solution to the VSWF basis. We used direct VSWF expansion, along with the point-matching method in the T-matrix, and applied the boundary condition to the normal components displacement field in order to compute the stress and the related forces and torques and to show the mechanism of the optical trap of the anisotropic nano-cylinders. 9. Experimental observation of multiple-Q states for the magnetic skyrmion lattice and skyrmion excitations under a zero magnetic field NASA Astrophysics Data System (ADS) Nagao, Masahiro; So, Yeong-Gi; Yoshida, Hiroyuki; Yamaura, Kazunari; Nagai, Takuro; Hara, Toru; Yamazaki, Atsushi; Kimoto, Koji 2015-10-01 Model calculations indicate that the magnetic skyrmion lattice (SkL) is represented by a superposition of three spin helices at an angle of 120∘ to each other, the so-called triple-Q state. Using Lorentz transmission electron microscopy, we investigated the relationship between the SkL and the helix in FeGe thin films. After the magnetic field is removed, the ordered skyrmions are trapped inside helimagnetic domain walls (HDWs) where the different helical Q vectors are encountered. In situ observation revealed an unexpected topological excitation under such a zero-field state: skyrmions are spontaneously formed at HDWs. 10. Quantum Switching at a Mean-Field Instability of a Bose-Einstein Condensate in an Optical Lattice SciTech Connect Shchesnovich, V. S.; Konotop, V. V. 2009-02-06 It is shown that bifurcation of the mean-field dynamics of a Bose-Einstein condensate can be related to the quantum phase transition of the original many-body system. As an example we explore the intraband tunneling in the two-dimensional optical lattice. Such a system allows for easy control by the lattice depth as well as for macroscopic visualization of the phase transition. The system manifests switching between two self-trapping states or from a self-trapping state to a superposition of the macroscopically populated self-trapping states with a steplike variation of the control parameter about the bifurcation point. We have also observed the magnification of the microscopic difference between the even and odd number of atoms to a macroscopically distinguishable dynamics of the system. 11. Three-dimensional modeling of a thermal dendrite using the phase field method with automatic anisotropic and unstructured adaptive finite element meshing NASA Astrophysics Data System (ADS) Sarkis, C.; Silva, L.; Gandin, Ch-A.; Plapp, M. 2016-03-01 Dendritic growth is computed with automatic adaptation of an anisotropic and unstructured finite element mesh. The energy conservation equation is formulated for solid and liquid phases considering an interface balance that includes the Gibbs-Thomson effect. An equation for a diffuse interface is also developed by considering a phase field function with constant negative value in the liquid and constant positive value in the solid. Unknowns are the phase field function and a dimensionless temperature, as proposed by [1]. Linear finite element interpolation is used for both variables, and discretization stabilization techniques ensure convergence towards a correct non-oscillating solution. In order to perform quantitative computations of dendritic growth on a large domain, two additional numerical ingredients are necessary: automatic anisotropic unstructured adaptive meshing [2,[3] and parallel implementations [4], both made available with the numerical platform used (CimLib) based on C++ developments. Mesh adaptation is found to greatly reduce the number of degrees of freedom. Results of phase field simulations for dendritic solidification of a pure material in two and three dimensions are shown and compared with reference work [1]. Discussion on algorithm details and the CPU time will be outlined. 12. Anisotropic universe with anisotropic sources SciTech Connect Aluri, Pavan K.; Panda, Sukanta; Sharma, Manabendra; Thakur, Snigdha E-mail: [email protected] E-mail: [email protected] 2013-12-01 We analyze the state space of a Bianchi-I universe with anisotropic sources. Here we consider an extended state space which includes null geodesics in this background. The evolution equations for all the state observables are derived. Dynamical systems approach is used to study the evolution of these equations. The asymptotic stable fixed points for all the evolution equations are found. We also check our analytic results with numerical analysis of these dynamical equations. The evolution of the state observables are studied both in cosmic time and using a dimensionless time variable. Then we repeat the same analysis with a more realistic scenario, adding the isotropic (dust like dark) matter and a cosmological constant (dark energy) to our anisotropic sources, to study their co-evolution. The universe now approaches a de Sitter space asymptotically dominated by the cosmological constant. The cosmic microwave background anisotropy maps due to shear are also generated in this scenario, assuming that the universe contains anisotropic matter along with the usual (dark) matter and vacuum (dark) energy since decoupling. We find that they contribute dominantly to the CMB quadrupole. We also constrain the current level of anisotropy and also search for any cosmic preferred axis present in the data. We use the Union 2 Supernovae data to this extent. An anisotropy axis close to the mirror symmetry axis seen in the cosmic microwave background data from Planck probe is found. 13. The right circular polarized waves in the three-dimensional anisotropic dispersive photonic crystals consisting of the magnetized plasma and uniaxial material as the Faraday effects considered SciTech Connect Zhang, Hai-Feng E-mail: [email protected]; Liu, Shao-Bin E-mail: [email protected]; Tang, Yi-Jun; Zhen, Jian-Ping 2014-03-15 In this paper, the properties of the right circular polarized (RCP) waves in the three-dimensional (3D) dispersive photonic crystals (PCs) consisting of the magnetized plasma and uniaxial material with face-centered-cubic (fcc) lattices are theoretically investigated by the plane wave expansion method, which the homogeneous anisotropic dielectric spheres (the uniaxial material) immersed in the magnetized plasma background, as the Faraday effects of magnetized plasma are considered (the incidence electromagnetic wave vector is parallel to the external magnetic field at any time). The equations for calculating the anisotropic photonic band gaps (PBGs) for the RCP waves in the first irreducible Brillouin zone are theoretically deduced. The anisotropic PBGs and a flatbands region can be obtained. The effects of the ordinary-refractive index, extraordinary-refractive index, anisotropic dielectric filling factor, plasma frequency, and plasma cyclotron frequency (the external magnetic field) on the properties of first two anisotropic PBGs for the RCP waves are investigated in detail, respectively. The numerical results show that the anisotropy can open partial band gaps in fcc lattices at U and W points, and the complete PBGs for the RCP waves can be achieved compared to the conventional 3D dispersive PCs composed of the magnetized plasma and isotropic material. It is also shown that the first two anisotropic PBGs can be tuned by those parameters as mentioned above. Those PBGs can be enlarged by introducing the uniaxial material into such 3D PCs as the Faraday effects are considered. 14. Pulsed flying spot with the logarithmic parabolas method for the estimation of in-plane thermal diffusivity fields on heterogeneous and anisotropic materials NASA Astrophysics Data System (ADS) Gaverina, L.; Batsale, J. C.; Sommier, A.; Pradere, C. 2017-03-01 A novel thermal non-destructive technique based on a Pulsed Flying Spot is presented here by considering in-plane logarithmic processing of the relaxing temperature field around the heat source spot. Recent progress made in optical control, lasers, and infrared cameras permits the acquisition of 2D temperature fields and localized thermal excitation on a small area instead of the entire recorded image. This study focuses on a new method based on spatial logarithm analysis of a temperature field to analyse and measure different parameters, such as the in-plane thermal diffusivity and localization of the spot. In this paper, this method is presented and the first results of heterogeneous anisotropic materials are depicted. The in-plane thermal diffusivity is estimated with an error lower than 4%, and the initial location of the heating spot is determined. 15. Honeycomb-Lattice Heisenberg-Kitaev Model in a Magnetic Field: Spin Canting, Metamagnetism, and Vortex Crystals NASA Astrophysics Data System (ADS) Janssen, Lukas; Andrade, Eric C.; Vojta, Matthias 2016-12-01 The Heisenberg-Kitaev model is a paradigmatic model to describe the magnetism in honeycomb-lattice Mott insulators with strong spin-orbit coupling, such as A2IrO3 (A =Na , Li ) and α -RuCl3 . Here, we study in detail the physics of the Heisenberg-Kitaev model in an external magnetic field. Using a combination of Monte Carlo simulations and spin-wave theory, we map out the classical phase diagram for different directions of the magnetic field. Broken SU(2) spin symmetry renders the magnetization process rather complex, with sequences of phases and metamagnetic transitions. In particular, we find various large-unit-cell and multi-Q phases including a vortex-crystal phase for a field in the [111 ] direction. We also discuss quantum corrections in the high-field phase. 16. Honeycomb-Lattice Heisenberg-Kitaev Model in a Magnetic Field: Spin Canting, Metamagnetism, and Vortex Crystals. PubMed Janssen, Lukas; Andrade, Eric C; Vojta, Matthias 2016-12-30 The Heisenberg-Kitaev model is a paradigmatic model to describe the magnetism in honeycomb-lattice Mott insulators with strong spin-orbit coupling, such as A_{2}IrO_{3} (A=Na, Li) and α-RuCl_{3}. Here, we study in detail the physics of the Heisenberg-Kitaev model in an external magnetic field. Using a combination of Monte Carlo simulations and spin-wave theory, we map out the classical phase diagram for different directions of the magnetic field. Broken SU(2) spin symmetry renders the magnetization process rather complex, with sequences of phases and metamagnetic transitions. In particular, we find various large-unit-cell and multi-Q phases including a vortex-crystal phase for a field in the [111] direction. We also discuss quantum corrections in the high-field phase. 17. Modulation of periodic field on the atomic current in optical lattices with Landau-Zener tunneling considered NASA Astrophysics Data System (ADS) Yan, Jie-Yun; Wang, Lan-Yu 2016-09-01 We investigate the atomic current in optical lattices under the presence of both constant and periodic external field with Landau-Zener tunneling considered. By simplifying the system to a two-band model, the atomic current is obtained based on the Boltzmann equations. We focus on three situations to discuss the influence of the Landau-Zener tunneling and periodic field on the atomic current. Numerical calculations show the atomic transient current would finally become the stable oscillation, whose amplitude and average value can be further adjusted by the periodic external field. It is concluded that the periodic external field could provide an effective modulation on the atomic current even when the Landau-Zener tunneling probability has almostly become a constant. 18. Anisotropic structural and magnetic properties of the field-aligned superconducting system SmFeAsO1-xFx (x = 0, 0.1, 0.2, 0.25 and 0.3) NASA Astrophysics Data System (ADS) You, Y. B.; Hsiao, T. K.; Chang, B. C.; Tai, M. F.; Hsu, Y. Y.; Ku, H. C.; Wei, Z.; Ruan, K. Q.; Li, X. G. 2011-01-01 Anisotropic structural and magnetic properties of the field-aligned superconducting system SmFeAsO1-xFx (x = 0, 0.1, 0.2, 0.25 and 0.3) are reported. Due to the Fe spin-orbital related anisotropic exchange coupling, all the tetragonal microcrystalline powders in epoxy were aligned at room temperature using the field-rotation method where the tetragonal ab-plane is parallel to the magnetic alignment field Ba of 0.9 T and the c-axis parallels to the rotating axis. Anisotropic magnetic properties are studied through low temperature magnetic measurements along the c-axis and paralleled to the ab-plane of aligned samples in both zero-field-cooled (ZFC) and field-cooled (FC) modes. The under-doped compound (x = 0.1) is not superconducting with an antiferromagnetic Néel temperature TN ~ 40 K, while the two optimum-doped compounds (x = 0.2 and 0.25) show high superconducting transition temperatures Tc of 49K and 50K, respectively. The variation of anisotropic structural and magnetic properties for this system are discussed and compared with the previously reported 52 K anisotropic superconductor Sm0.95La0.05FeAsO0.85F0.15. 19. Lattice calculation of thermal properties of low-density neutron matter with pionless NN effective field theory SciTech Connect Abe, T.; Seki, R. 2009-05-15 Thermal properties of low-density neutron matter are investigated by determinantal quantum Monte Carlo lattice calculations on 3+1 dimensional cubic lattices. Nuclear effective field theory (EFT) is applied using the pionless single- and two-parameter neutron-neutron interactions, determined from the {sup 1}S{sub 0} scattering length and effective range. The determination of the interactions and the calculations of neutron matter are carried out consistently by applying EFT power counting rules. The thermodynamic limit is taken by the method of finite-size scaling, and the continuum limit is examined in the vanishing lattice filling limit. The {sup 1}S{sub 0} pairing gap at T{approx_equal}0 is computed directly from the off-diagonal long-range order of the spin pair-pair correlation function and is found to be approximately 30% smaller than BCS calculations with the conventional nucleon-nucleon potentials. The critical temperature T{sub c} of the normal-to-superfluid phase transition and the pairing temperature scale T* are determined, and the temperature-density phase diagram is constructed. The physics of low-density neutron matter is clearly identified as being a BCS-Bose-Einstein condensation crossover. 20. Commensurate states on incommensurate lattices. [for superconducting arrays in magnetic fields NASA Technical Reports Server (NTRS) Grest, Gary S.; Chaikin, Paul M.; Levine, Dov 1988-01-01 A simple one-dimensional model related to flux quantization on superconducting networks or charged particles on a substrate is proposed to investigate whether commensurate states can exist on incommensurate lattices. For both periodic and quasi-crystalline patterns, a set of low-energy states is found which is related to decimation symmetry and periodicity. It is suggested that the present quasi-periodic arrays which possess a decimation operation can be generalized to more-dimensional quasi-crystalline systems. 1. Anisotropic magneto-resistance of 2D holes in GaAs/Al_xGa_1-xAs heterostructure under in-plane magnetic field NASA Astrophysics Data System (ADS) Noh, H.; Tsui, D. C.; Shayegan, M.; Yoon, Jongsoo 2000-03-01 We report on measurements of anisotropic in-plane magneto-resistance of the 2D hole system (2DHS) in a GaAs/AlGaAs (311)A heterostructure, which exhibits both zero-field and in-plane field induced metal-insulator transitions. For high hole densities, when the direction of B field is changed relative to the current(I), which is always kept in the high mobility direction, the resistivity with B allel I is larger at low field, while the resistivity with B ⊥ I becomes larger at high field. This behavior is consistent with recent measurements(S. J. Papadakis et al.), cond-mat/9911239. on a quantum well system. That the resistivity at high field is larger for B ⊥ I than for B allel I is also consistent with a recent theoretical argument(S. Das Sarma and E. H. Hwang, cond-mat/9909452.), though the difference is smaller than that from the theory. As the density is lowered, the resistivity with B allel I gets larger at high field, and it eventually becomes greater than that with B ⊥ I at all field ranges measured. This change in anisotropy has not been seen in previous measurements. The critical field B_c, beyond which the metallic phase becomes insulating, is also different for two different directions of B, while the change in I-V characteristics across Bc remains the same. 2. Thermoelectric Figure of Merit in Anisotropic Systems NASA Astrophysics Data System (ADS) Bies, W.; Radtke, R. J.; Ehrenreich, H. 1998-03-01 General expressions for the electrical conductivity, thermopower, and electronic thermal conductivity are derived for anisotropic materials including their full tensorial character and properly treating the effects of the sample boundaries. The thermoelectric figure of merit ZT constructed from these quantities is proved to be maximal only when the electric field (in thermoelectric coolers) or thermal gradient (in power generators) is applied along the direction of highest conductivity. Fields applied along directions for which the conductivity tensor is non-diagonal induce transverse electric fields and thermal gradients which may be larger in magnitude than the applied fields. These fields reduce ZT below that expected from anisotropy alone. Numerical results are presented for bulk n-type Bi_2Te3 and quantum well and quantum wire geometries using semiclassical transport theory in the effective mass and relaxation time approximations. The effects of multi-valley conduction and confinement-induced splitting of the valley degeneracy are included. Surprisingly, this model predicts generally that the thermopower and hence ZT are independent of the direction of the applied fields in the limit of vanishing lattice thermal conductivity. 3. Frustrated honeycomb-lattice bilayer quantum antiferromagnet in a magnetic field: Unconventional phase transitions in a two-dimensional isotropic Heisenberg model NASA Astrophysics Data System (ADS) Krokhmalskii, Taras; Baliha, Vasyl; Derzhko, Oleg; Schulenburg, Jörg; Richter, Johannes 2017-03-01 We consider the spin-1/2 antiferromagnetic Heisenberg model on a bilayer honeycomb lattice including interlayer frustration in the presence of an external magnetic field. In the vicinity of the saturation field, we map the low-energy states of this quantum system onto the spatial configurations of hard hexagons on a honeycomb lattice. As a result, we can construct effective classical models (lattice-gas as well as Ising models) on the honeycomb lattice to calculate the properties of the frustrated quantum Heisenberg spin system in the low-temperature regime. We perform classical Monte Carlo simulations for a hard-hexagon model and adopt known results for an Ising model to discuss the finite-temperature order-disorder phase transition that is driven by a magnetic field at low temperatures. We also discuss an effective-model description around the ideal frustration case and find indications for a spin-flop-like transition in the considered isotropic spin model. 4. Two topics in nonperturbative lattice field theories: The U(1) quantum link model and perfect actions for scalar theories NASA Astrophysics Data System (ADS) Tsapalis, Antonios S. This thesis deals with two topics in lattice field theories. In the first part we discuss aspects of renormalization group flow and non-perturbative improvement of actions for scalar theories regularized on a lattice. We construct a perfect action, an action which is free of lattice artifacts, for a given theory. It is shown how a good approximation to the perfect action-referred to as classically perfect-can be constructed based on a well-defined blocking scheme for the O(3) non-linear σ-model. We study the O(N) non- linear σ-model in the large-N limit and derive analytically its perfect action. This action is applied to the O(3) model on a square lattice. The Wolff cluster algorithm is used to simulate numerically the system. We perform scaling tests and discuss the scaling properties of the large- N inspired perfect action as opposed to the standard and the classically perfect action. In the second part we present a new formulation for a quantum field theory with Abelian gauge symmetry. A Hamiltonian is constructed on a four-dimensional Euclidean space-time lattice which is invariant under local transformations. The model is formulated as a 5- dimensional path integral of discrete variables. We argue that dimensional reduction will allow us to study the behavior of the standard compact U(1) gauge theory in 4-d. Based on the idea of the loop- cluster algorithm for quantum spins, we present the construction of a flux-cluster algorithm for the U(1) quantum link model for the spin-1/2 quantization of the electric flux. It is shown how improved estimators for Wilson loop expectation values can be defined. This is important because the Wilson loops are traditionally used to identify confining and Coulomb phases in gauge theories. Our study indicates that the spin-1/2 U(1) quantum link model is strongly coupled for all bare coupling values we examined. (Copies available exclusively from MIT Libraries, Rm. 14-0551, Cambridge, MA 02139-4307. Ph. 617-253-5668; Fax 617-253-1690.) 5. OH MASER SOURCES IN W49N: PROBING MAGNETIC FIELD AND DIFFERENTIAL ANISOTROPIC SCATTERING WITH ZEEMAN PAIRS USING THE VERY LONG BASELINE ARRAY SciTech Connect Deshpande, Avinash A.; Goss, W. M.; Mendoza-Torres, J. E. E-mail: [email protected] 2013-09-20 Our analysis of a Very Long Baseline Array 12 hr synthesis observation of the OH masers in the well-known star-forming region W49N has yielded valuable data that enable us to probe distributions of magnetic fields in both the maser columns and the intervening interstellar medium (ISM). The data, consisting of detailed high angular resolution images (with beam width ∼20 mas) of several dozen OH maser sources, or spots, at 1612, 1665, and 1667 MHz, reveal anisotropic scatter broadening with typical sizes of a few tens of milliarcseconds and axial ratios between 1.5 and 3. Such anisotropies have been reported previously by Desai et al. and have been interpreted as being induced by the local magnetic field parallel to the Galactic plane. However, we find (1) apparent angular sizes of, on average, a factor of about 2.5 less than those reported by Desai et al., indicating significantly less scattering than inferred previously, and (2) a significant deviation in the average orientation of the scatter-broadened images (by ∼10°) from that implied by the magnetic field in the Galactic plane. More intriguingly, for a few Zeeman pairs in our set, significant differences (up to 6σ) are apparent in the scatter-broadened images for the two hands of circular polarization, even when the apparent velocity separation is less than 0.1 km s{sup –1}. This may possibly be the first example of a Faraday rotation contribution to the diffractive effects in the ISM. Using the Zeeman pairs, we also study the distribution of the magnetic field in the W49N complex, finding no significant trend in the spatial structure function. In this paper, we present the details of our observations and analysis leading to these findings, discuss implications of our results for the intervening anisotropic magneto-ionic medium, and suggest possible implications for the structure of magnetic fields within this star-forming region. 6. Crystalline electric field and lattice contributions to thermodynamic properties of PrGaO3: specific heat and thermal expansion NASA Astrophysics Data System (ADS) Senyshyn, A.; Schnelle, W.; Vasylechko, L.; Ehrenberg, H.; Berkowski, M. 2007-04-01 The low-temperature heat capacity of perovskite-type PrGaO3 has been measured in the temperature range from 2 to 320 K. Thermodynamic standard values at 298.15 K are reported. An initial Debye temperature θD(0) = (480 ± 10) K was determined by fitting the calculated lattice heat capacity. The entropy of the derived Debye temperature functions agrees well with values calculated from thermal displacement parameters and from atomistic simulations. The thermal expansion and the Grüneisen parameter, arising from a coupling of crystal field states of Pr3+ ion and phonon modes at low temperature, were analysed. 7. The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: Effect of smoothing of density field on reconstruction and anisotropic BAO analysis. NASA Astrophysics Data System (ADS) Vargas-Magaña, Mariana; Ho, Shirley; Fromenteau, Sebastien.; Cuesta, Antonio. J. 2017-01-01 The reconstruction algorithm introduced by Eisenstein et al. (2007), which is widely used in clustering analysis, is based on the inference of the first order Lagrangian displacement field from the Gaussian smoothed galaxy density field in redshift space. The 2smoothing scale applied to the density field affects the inferred displacement field that is used to move the galaxies, and partially 2erases the nonlinear evolution of the density field. In this article, we explore this crucial step 2in the reconstruction algorithm. We study the performance of the reconstruction technique using two metrics: first, we study the performance using the anisotropic clustering, extending previous studies focused on isotropic clustering; second, we study its effect on the displacement field. We find that smoothing has a strong effect in the quadrupole of the correlation function and affects the accuracy and precision 2with which we can measure DA(z) and H(z). We find that the optimal smoothing scale to use in the reconstruction algorithm applied to BOSS-CMASS is between 5-10 h-1Mpc. Varying from the "usual" 15h-1Mpc to 5h-1Mpc 2shows ˜ 0.3% variations in DA(z) and ˜ 0.4% H(z) and uncertainties are also reduced by 40% and 30% respectively. We also find that the accuracy of velocity field reconstruction depends strongly on the smoothing scale used for the density field. We measure the bias and uncertainties associated with different choices of smoothing length. 8. Anisotropic resistivity tomography NASA Astrophysics Data System (ADS) Herwanger, J. V.; Pain, C. C.; Binley, A.; de Oliveira, C. R. E.; Worthington, M. H. 2004-08-01 Geophysical tomographic techniques have the potential to remotely detect and characterize geological features, such as fractures and spatially varying lithologies, by their response to signals passed through these features. Anisotropic behaviour in many geological materials necessitates the generalization of tomographic methods to include anisotropic material properties in order to attain high-quality images of the subsurface. In this paper, we present a finite element (FE) based direct-current electrical inversion method to reconstruct the conductivity tensor at each node point of a FE mesh from electrical resistance measurements. The inverse problem is formulated as a functional optimization and the non-uniqueness of the electrical inverse problem is overcome by adding penalty terms for structure and anisotropy. We use a modified Levenberg-Marquardt method for the functional optimization and the resulting set of linear equation is solved using pre-conditioned conjugate gradients. The method is tested using both synthetic and field experiments in cross-well geometry. The acquisition geometry for both experiments uses a cross-well experiment at a hard-rock test site in Cornwall, southwest England. Two wells, spaced at 25.7 m, were equipped with electrodes at a 1 m spacing at depths from 21-108 m and data were gathered in pole-pole geometry. The test synthetic model consists of a strongly anisotropic and conductive body underlain by an isotropic resistive formation. Beneath the resistive formation, the model comprises a moderately anisotropic and moderately conductive half-space, intersected by an isotropic conductive layer. This model geometry was derived from the interpretation of a seismic tomogram and available geological logs and the conductivity values are based on observed conductivities. We use the test model to confirm the ability of the inversion scheme to recover the (known) true model. We find that all key features of the model are recovered. However 9. Spark-plasma-sintering magnetic field assisted compaction of Co{sub 80}Ni{sub 20} nanowires for anisotropic ferromagnetic bulk materials SciTech Connect Ouar, Nassima; Schoenstein, Frédéric; Mercone, Silvana; Farhat, Samir; Jouini, Noureddine; Villeroy, Benjamin; Leridon, Brigitte 2013-10-28 We developed a two-step process showing the way for sintering anisotropic nanostructured bulk ferromagnetic materials. A new reactor has been optimized allowing the synthesis of several grams per batch of nanopowders via a polyol soft chemistry route. The feasibility of the scale-up has been successfully demonstrated for Co{sub 80}Ni{sub 20} nanowires and a massic yield of ∼97% was obtained. The thus obtained nanowires show an average diameter of ∼6 nm and a length of ∼270 nm. A new bottom-up strategy allowed us to compact the powder into a bulk nanostructured system. We used a spark-plasma-sintering technique under uniaxial compression and low temperature assisted by a permanent magnetic field of 1 T. A macroscopic pellet of partially aligned nanowire arrays has been easily obtained. This showed optimized coercive properties along the direction of the magnetic field applied during compaction (i.e., the nanowires' direction) 10. Superalloy Lattice Block Structures NASA Technical Reports Server (NTRS) Whittenberger, J. D.; Nathal, M. V.; Hebsur, M. G.; Kraus, D. L. 2003-01-01 In their simplest form, lattice block panels are produced by direct casting and result in lightweight, fully triangulated truss-like configurations which provide strength and stiffness [2]. The earliest realizations of lattice block were made from A1 and steels, primarily under funding from the US Navy [3]. This work also showed that the mechanical efficiency (eg., specific stiffness) of lattice block structures approached that of honeycomb structures [2]. The lattice architectures are also less anisotropic, and the investment casting route should provide a large advantage in cost and temperature capability over honeycombs which are limited to alloys that can be processed into foils. Based on this early work, a program was initiated to determine the feasibility of extending the high temperature superalloy lattice block [3]. The objective of this effort was to provide an alternative to intermetallics and composites in achieving a lightweight high temperature structure without sacrificing the damage tolerance and moderate cost inherent in superalloys. To establish the feasibility of the superalloy lattice block concept, work was performed in conjunction with JAMCORP, Inc. Billerica, MA, to produce a number of lattice block panels from both IN71 8 and Mar-M247. 11. Spin-1/2 kagome XXZ model in a field: Competition between lattice nematic and solid orders NASA Astrophysics Data System (ADS) Kshetrimayum, Augustine; Picot, Thibaut; Orús, Román; Poilblanc, Didier 2016-12-01 We study numerically the spin-1/2 XXZ model in a field on an infinite kagome lattice. We use different algorithms based on infinite projected entangled pair states (iPEPSs) for this, namely, (i) an approach with simplex tensors and a 9-site unit cell, and (ii) an approach based on coarse-graining three spins in the kagome lattice and mapping it to a square-lattice model with local and nearest-neighbor interactions, with the usual PEPS tensors, 6- and 12-site unit cells. Similarly to our previous calculation at the SU(2)-symmetric point (Heisenberg Hamiltonian), for any anisotropy from the Ising limit to the XY limit, we also observe the emergence of magnetization plateaus as a function of the magnetic field, at mz=1/3 using 6-, 9-, and 12-site PEPS unit cells, and at mz=1/9 ,5/9 , and 7/9 using a 9-site PEPS unit cell, the latter setup being able to accommodate √{3 }×√{3 } solid order. We also find that, at mz=1/3 , (lattice) nematic and √{3 }×√{3 } VBC-order states are degenerate within the accuracy of the nine-site simplex method, for all anisotropy. The 6- and 12-site coarse-grained PEPS methods produce almost-degenerate nematic and 1 ×2 VBC-solid orders. We also find that, within our accuracy, the six-site coarse-grained PEPS method gives slightly lower energies, which can be explained by the larger amount of entanglement this approach can handle, even in cases where the PEPS unit cell is not commensurate with the expected ground-state unit cell. Furthermore, we do not observe chiral spin liquid behaviors at and close to the XY point, as has been recently proposed. Our results are the first tensor network investigations of the XXZ model in a field and reveal the subtle competition between nearby magnetic orders in numerical simulations of frustrated quantum antiferromagnets, as well as the delicate interplay between energy optimization and symmetry in tensor network numerical simulations. 12. Field-wide flow simulation in fractured porous media within lattice Boltzmann framework NASA Astrophysics Data System (ADS) Benamram, Z.; Tarakanov, A.; Nasrabadi, H.; Gildin, E. 2016-10-01 In this paper, a generalized lattice Boltzmann model for simulating fluid flow in porous media at the representative volume element scale is extended towards applications of hydraulically and naturally fractured reservoirs. The key element within the model is the development of boundary conditions for a vertical well and horizontal fracture with minimal node usage. In addition, the governing non-dimensional equations are derived and a new set of dimensionless numbers are presented for the simulation of a fractured reservoir system. Homogenous and heterogeneous vertical well and fracture systems are simulated and verified against commercial reservoir simulation suites. Results are in excellent agreement to analytical and finite difference solutions. 13. Lattice fermions NASA Technical Reports Server (NTRS) Wilczek, Frank 1987-01-01 A simple heuristic proof of the Nielsen-Ninomaya theorem is given. A method is proposed whereby the multiplication of fermion species on a lattice is reduced to the minimal doubling, in any dimension, with retention of appropriate chiral symmetries. Also, it is suggested that use of spatially thinned fermion fields is likely to be a useful and appropriate approximation in QCD - in any case, it is a self-checking one. 14. Anisotropic 2.5D Inversion of Towed Streamer EM Data from Three North Sea Fields Using Parallel Adaptive Finite Elements NASA Astrophysics Data System (ADS) Key, K.; Du, Z. 2014-12-01 We present anisotropic inversion results from towed streamer electromagnetic (EM) surveys of the Bressay, Bentley and Kraken (BBK) heavy oil fields in the North Sea. The BBK discoveries pose several challenges to conventional controlled-source EM surveying since the relatively shallow water dampens the anomaly magnitudes due to airwave coupling, and the reservoirs are in close proximity to other resistive features. The 160 m spacing of the 44 receiver bipoles on the towed streamer offers much higher data density than is typically achieved with conventional seafloor receiver surveys. We tested the resolving capabilities of the towed-streamer by inverting the survey data using a new code based on a 2.5D parallel goal-oriented adaptive finite element method and a modified implementation of the Occam inversion algorithm. The inversion successfully images the 1-2 km wide Bressay and ~5 km wide Bentley reservoirs, illustrating that the high data density of the towed streamer offers improved resolution over sparsely sampled nodal seafloor receiver data. The results also demonstrate the importance of allowing for anisotropy when inverting data from this region. Whereas anisotropic inversion clearly recovers the lateral edges of the known reservoirs, isotropic inversion results in inter-bedding of resistive and conductive layers that conceal the reservoirs. 15. Micropolar dissipative models for the analysis of 2D dispersive waves in periodic lattices NASA Astrophysics Data System (ADS) Reda, H.; Ganghoffer, J. F.; Lakiss, H. 2017-03-01 The computation of the dispersion relations for dissipative periodic lattices having the attributes of metamaterials is an actual research topic raising the interest of researchers in the field of acoustics and wave propagation phenomena. We analyze in this contribution the impact of wave damping on the dispersion features of periodic lattices, which are modeled as beam-lattices. The band diagram structure and damping ratio are computed for different repetitive lattices, based on the homogenized continuum response of the initially discrete lattice architecture, modeled as Kelvin-Voigt viscoelastic beams. Three of these lattices (reentrant hexagonal, chiral diamond, hexachiral lattice) are auxetic metamaterials, since they show negative Poisson's ratio. The effective viscoelastic anisotropic continuum behavior of the lattices is first computed in terms of the homogenized stiffness and viscosity matrices, based on the discrete homogenization technique. The dynamical equations of motion are obtained for an equivalent homogenized micropolar continuum evaluated based on the homogenized properties, and the dispersion relation and damping ratio are obtained by inserting an harmonic plane waves Ansatz into these equations. The comparison of the acoustic properties obtained in the low frequency range for the four considered lattices shows that auxetic lattices attenuate waves at lower frequencies compared to the classical hexagonal lattice. The diamond chiral lattice shows the best attenuation properties of harmonic waves over the entire Brillouin zone, and the hexachiral lattice presents better acoustic properties than the reentrant hexagonal lattice. The range of validity of the effective continuum obtained by the discrete homogenization has been assessed by comparing the frequency band structure of this continuum with that obtained by a Floquet-Bloch analysis. 16. Pair correlations and structure factor of the J1-J2 square lattice Ising model in an external field NASA Astrophysics Data System (ADS) Guerrero, Alejandra I.; Stariolo, Daniel A. 2017-01-01 We compute the structure factor of the J1-J2 Ising model in an external field on the square lattice within the Cluster Variation Method. We use a four point plaquette approximation, which is the minimal one able to capture phases with broken orientational order in real space, like the recently reported Ising-nematic phase in the model. The analysis of different local maxima in the structure factor allows us to track the different phases and phase transitions against temperature and external field. Although the nematic susceptibility is not directly related to the structure factor, we show that because of the close relationship between the nematic order parameter and the structure factor, the latter shows unambiguous signatures of the presence of a nematic phase, in agreement with results from direct minimization of a variational free energy. The disorder variety of the model is identified and the possibility that the CVM four point approximation be exact on the disorder variety is discussed. 17. Field induced suppression of the vortex lattice melting transition in twinned YBa 2Cu 3O 7-δ NASA Astrophysics Data System (ADS) Langan, R. M.; Gordeev, S. N.; Oussena, M.; Pinfold, S.; de Groot, P. A. J.; Jansen, L.; Gagnon, R.; Taillefer, L. 1997-08-01 We present magneto-resistance data for a high quality, twinned YBa2Cu3O7-δ crystal, taken with the current applied along the ab plane. The crystal was examined at a number of angles to an applied magnetic field, of up to 20T, in order to observe the influence of correlated and point-like disorder on the vortex dynamics. When the applied field was orientated at 15° to the crystalline c-axis (θ=15°), for fields below H*=12T, we observed the kink in ϱ(T) associated with vortex lattice melting. We found that when the field exceeded this value, there was a complete suppression of this kink and the ϱ(T) curves resembled those of crystals with extensive point disorder. This suppression in melting occurs abruptly between 11T and 12T. The melting transition is recovered when the angle between the c-axis and the field is increased. An analysis of features around 12T in ϱ(T) and ϱ(θ) has been performed for a number of fields and angles. 18. Phase-field-based lattice Boltzmann model for incompressible binary fluid systems with density and viscosity contrasts. PubMed Zu, Y Q; He, S 2013-04-01 A lattice Boltzmann model (LBM) is proposed based on the phase-field theory to simulate incompressible binary fluids with density and viscosity contrasts. Unlike many existing diffuse interface models which are limited to density matched binary fluids, the proposed model is capable of dealing with binary fluids with moderate density ratios. A new strategy for projecting the phase field to the viscosity field is proposed on the basis of the continuity of viscosity flux. The new LBM utilizes two lattice Boltzmann equations (LBEs): one for the interface tracking and the other for solving the hydrodynamic properties. The LBE for interface tracking can recover the Chan-Hilliard equation without any additional terms; while the LBE for hydrodynamic properties can recover the exact form of the divergence-free incompressible Navier-Stokes equations avoiding spurious interfacial forces. A series of 2D and 3D benchmark tests have been conducted for validation, which include a rigid-body rotation, stationary and moving droplets, a spinodal decomposition, a buoyancy-driven bubbly flow, a layered Poiseuille flow, and the Rayleigh-Taylor instability. It is shown that the proposed method can track the interface with high accuracy and stability and can significantly and systematically reduce the parasitic current across the interface. Comparisons with momentum-based models indicate that the newly proposed velocity-based model can better satisfy the incompressible condition in the flow fields, and eliminate or reduce the velocity fluctuations in the higher-pressure-gradient region and, therefore, achieve a better numerical stability. In addition, the test of a layered Poiseuille flow demonstrates that the proposed scheme for mixture viscosity performs significantly better than the traditional mixture viscosity methods. 19. Cluster variation studies of the anisotropic exchange interaction model NASA Astrophysics Data System (ADS) King, T. C.; Chen, H. H. The cluster variation method is applied to study critical properties of the Potts-like ferromagnetic anisotropic exchange interaction model. Phase transition temperatures, order parameter discontinuities and latent heats of the model on the triangular and the fcc lattices are determined by the triangle approximation; and those on the square and the sc lattices are determined by the square approximation. 20. Fluctuating pancake vortices revealed by dissipation of Josephson vortex lattice. SciTech Connect Koshelev, A. E.; Buzdin, A. I.; Kakeya, I.; Yamamoto, T.; Kadowaki, K. 2011-06-01 In strongly anisotropic layered superconductors in tilted magnetic fields, the Josephson vortex lattice coexists with the lattice of pancake vortices. Due to the interaction between them, the dissipation of the Josephson vortex lattice is very sensitive to the presence of the pancake vortices. If the c-axis magnetic field is smaller than the corresponding lower critical field, the pancake stacks are not formed but the individual pancakes may exist in the fluctuational regime either near the surface in large-size samples or in the central region for small-size mesas. We calculate the contribution of such fluctuating pancake vortices to the c-axis conductivity of the Josephson vortex lattice and compare the theoretical results with measurements on small mesas fabricated out of Bi{sub 2}Sr{sub 2}CaCu{sub 2}O{sub 8+{delta}} crystals. A fingerprint of fluctuating pancakes is a characteristic exponential dependence of the c-axis conductivity observed experimentally. Our results provide strong evidence of the existence of the fluctuating pancakes and their influence on the Josephson vortex lattice dissipation. 1. Diffusion-controlled anisotropic growth of stable and metastable crystal polymorphs in the phase-field crystal model. PubMed Tegze, G; Gránásy, L; Tóth, G I; Podmaniczky, F; Jaatinen, A; Ala-Nissila, T; Pusztai, T 2009-07-17 We use a simple density functional approach on a diffusional time scale, to address freezing to the body-centered cubic (bcc), hexagonal close-packed (hcp), and face-centered cubic (fcc) structures. We observe faceted equilibrium shapes and diffusion-controlled layerwise crystal growth consistent with two-dimensional nucleation. The predicted growth anisotropies are discussed in relation with results from experiment and atomistic simulations. We also demonstrate that varying the lattice constant of a simple cubic substrate, one can tune the epitaxially growing body-centered tetragonal structure between bcc and fcc, and observe a Mullins-Sekerka-Asaro-Tiller-Grinfeld-type instability. 2. Transferable Anisotropic United-Atom Force Field Based on the Mie Potential for Phase Equilibrium Calculations: n-Alkanes and n-Olefins. PubMed Hemmen, Andrea; Gross, Joachim 2015-09-03 A new transferable force field parametrization for n-alkanes and n-olefins is proposed in this work. A united-atom approach is taken, where hydrogen atoms are lumped with neighboring atoms to single interaction sites. A comprehensive study is conducted for alkanes, optimizing van der Waals force field parameters in 6 dimensions. A Mie n-6 potential is considered for the van der Waals interaction, where for n-alkanes we simultaneously optimize the energy parameters ϵCH3 and ϵCH2 as well as the size parameters σCH3 and σCH2 of the CH3(sp(3)) and CH2(sp(3)) groups. Further, the repulsive exponent n of the Mie n-6 potential is varied. Moreover, we investigate the bond length toward the terminal CH3 group as a degree of freedom. According to the AUA (anisotropic united-atom) force field, the bond length between the terminal CH3 group and the neighboring interaction site should be increased by Δl compared with the carbon-carbon distance in order to better account for the hydrogen atoms. The parameter Δl is considered as a degree of freedom. The intramolecular force field parametrization is taken from existing force fields. A single objective function for the optimization is defined as squared relative deviations in vapor pressure and in liquid density of propane, n-butane, n-hexane, and n-octane. A similar study is also done for olefins, where the objective function includes 1-butene, 1-hexene, 1-octene, cis-2-pentene, and trans-2-pentene. Molecular simulations are performed in the grand canonical ensemble with transition-matrix sampling where the phase equilibrium properties are obtained with the histogram reweighting technique. The 6-dimensional optimization of strongly correlated parameters is possible, because the analytic PC-SAFT equation of state is used to locally correlate simulation results. The procedure is iterative but leads to very efficient convergence. An implementation is proposed, where the converged result is not affected (disturbed) by the 3. Anisotropic local modification of crystal field levels in Pr-based pyrochlores: a muon-induced effect modeled using density functional theory. PubMed Foronda, F R; Lang, F; Möller, J S; Lancaster, T; Boothroyd, A T; Pratt, F L; Giblin, S R; Prabhakaran, D; Blundell, S J 2015-01-09 Although muon spin relaxation is commonly used to probe local magnetic order, spin freezing, and spin dynamics, we identify an experimental situation in which the measured response is dominated by an effect resulting from the muon-induced local distortion rather than the intrinsic behavior of the host compound. We demonstrate this effect in some quantum spin ice candidate materials Pr(2)B(2)O(7) (B=Sn, Zr, Hf), where we detect a static distribution of magnetic moments that appears to grow on cooling. Using density functional theory we show how this effect can be explained via a hyperfine enhancement arising from a splitting of the non-Kramers doublet ground states on Pr ions close to the muon, which itself causes a highly anisotropic distortion field. We provide a quantitative relationship between this effect and the measured temperature dependence of the muon relaxation and discuss the relevance of these observations to muon experiments in other magnetic materials. 4. Prestack inversion based on anisotropic Markov random field-maximum posterior probability inversion and its application to identify shale gas sweet spots NASA Astrophysics Data System (ADS) Wang, Kang-Ning; Sun, Zan-Dong; Dong, Ning 2015-12-01 Economic shale gas production requires hydraulic fracture stimulation to increase the formation permeability. Hydraulic fracturing strongly depends on geomechanical parameters such as Young's modulus and Poisson's ratio. Fracture-prone sweet spots can be predicted by prestack inversion, which is an ill-posed problem; thus, regularization is needed to obtain unique and stable solutions. To characterize gas-bearing shale sedimentary bodies, elastic parameter variations are regarded as an anisotropic Markov random field. Bayesian statistics are adopted for transforming prestack inversion to the maximum posterior probability. Two energy functions for the lateral and vertical directions are used to describe the distribution, and the expectation-maximization algorithm is used to estimate the hyperparameters of the prior probability of elastic parameters. Finally, the inversion yields clear geological boundaries, high vertical resolution, and reasonable lateral continuity using the conjugate gradient method to minimize the objective function. Antinoise and imaging ability of the method were tested using synthetic and real data. 5. Fractures in anisotropic media NASA Astrophysics Data System (ADS) Shao, Siyi theory and experimental results in this report demonstrate that the presence of fractures in anisotropic material can be unambiguously interpreted if experimental measurements are made as a function of stress, which eliminates many fracture-generated discrete modes (e.g., interface waves, and leaky guided-modes). Orthogonal fracture networks that are often encountered in field exploration bring in additional challenges for seismic/acoustic data interpretation. An innovative wavefront imaging system with a bi-axial load frame was designed and implemented on orthogonally-fractured samples to determine the effect of fracture networks on elastic wave propagation. The effects of central wave guiding and extra time delays along a fracture intersection were observed in experiments and was analyzed. Interpreting data from media with intersecting fracture sets must account for fracture intersections and the non-uniformity of fracture properties caused by local tectonic conditions or other physical process such as non-uniform fluid distributions within a network and/or chemical alterations. 6. Fine structure of the spectra of the Kondo lattice model: Two-site cellular dynamical mean-field theory study NASA Astrophysics Data System (ADS) Osolin, Žiga; Žitko, Rok 2017-01-01 We study the antiferromagnetic and paramagnetic Kondo insulator phases of the Kondo lattice model on the cubic lattice at half filling using the cellular dynamical mean-field theory (CDMFT) with the numerical renormalization group (NRG) as the impurity solver, focusing on the fine details of the spectral function and self-energy. We find that the nonlocal correlations increase the gap in both the antiferromagnetic and Kondo insulator phases and shrink the extent of the antiferromagnetic phase in the phase diagram but do not alter any properties qualitatively. The agreement between the numerical CDMFT results and those within a simple hybridization picture, which adequately describes the overall band structure of the system but neglects all effects on the inelastic-scattering processes, is similar to that of the single-site DMFT results; there are deviations that are responsible for the additional fine structure, in particular for the asymmetric spectral resonances or dips that become more pronounced in the strong-coupling regime close to the antiferromagnet-paramagnetic quantum phase transition. These features appear broader in the CDMFT mostly due to numerical artifacts linked to more aggressive state truncation required in the NRG. 7. Logic gates realized by nonvolatile GeTe/Sb2Te3 super lattice phase-change memory with a magnetic field input NASA Astrophysics Data System (ADS) Lu, Bin; Cheng, Xiaomin; Feng, Jinlong; Guan, Xiawei; Miao, Xiangshui 2016-07-01 Nonvolatile memory devices or circuits that can implement both storage and calculation are a crucial requirement for the efficiency improvement of modern computer. In this work, we realize logic functions by using [GeTe/Sb2Te3]n super lattice phase change memory (PCM) cell in which higher threshold voltage is needed for phase change with a magnetic field applied. First, the [GeTe/Sb2Te3]n super lattice cells were fabricated and the R-V curve was measured. Then we designed the logic circuits with the super lattice PCM cell verified by HSPICE simulation and experiments. Seven basic logic functions are first demonstrated in this letter; then several multi-input logic gates are presented. The proposed logic devices offer the advantages of simple structures and low power consumption, indicating that the super lattice PCM has the potential in the future nonvolatile central processing unit design, facilitating the development of massive parallel computing architecture. 8. Zeeman-field-induced nontrivial topological phases in a one-dimensional spin-orbit-coupled dimerized lattice NASA Astrophysics Data System (ADS) Bahari, Masoud; Hosseini, Mir Vahid 2016-09-01 We study theoretically the interplay effect of Zeeman field and modulated spin-orbit coupling on the topological properties of a one-dimensional dimerized lattice, known as Su-Schrieffer-Heeger model. We find that in the weak (strong) modulated spin-orbit coupling regime, trivial regions or nontrivial ones with two pairs of zero-energy states can be turned into nontrivial regions by applying a uniform (staggered) perpendicular Zeeman field through a topological phase transition. Furthermore, the resulting nontrivial phase hosting a pair of zero-energy boundary states can survive within a certain range of the perpendicular Zeeman field magnitude. Due to the effective time-reversal, particle-hole, chiral, and inversion symmetries, in the presence of either a uniform or a staggered perpendicular Zeeman field, the topological class of the system is BDI, which can be characterized by Z index. We also examine the robustness of the nontrivial phase by breaking the underlying symmetries, which results in that inversion symmetry plays an important role. 9. Electric Polarization Induced by Spin Ordering under Magnetic Fields in Distorted Triangular Lattice Antiferromagnet RbCoBr3 NASA Astrophysics Data System (ADS) Nishiwaki, Yoichi; Tokunaga, Masashi; Sakakura, Ryo; Takeyama, Shojiro; Kato, Tetsuya; Iio, Katsunori 2017-04-01 Magnetization and electric polarization are measured for RbCoBr3 in the presence of an applied high magnetic field. The saturation of magnetization is recognized in the magnetization curve. The g-value of pseudospin and the nearest-neighbor intrachain exchange interaction of RbCoBr3, which has the properties of a quasi-one-dimensional Ising antiferromagnet, are evaluated. The electric polarization parallel to the c-axis under a magnetic field alone and also under the simultaneous application of electric and magnetic fields along the c-axis is observed to increase around the magnetic phase transition point from the ferrimagnetic low-temperature phase to the partially disordered high-temperature phase. Experimental results indicate that the electric polarization is induced through the rearrangement of the spin structure accompanied by the magnetic phase transition under an applied magnetic field. A probable reason for the enhancement of electric polarization is given from the viewpoint of the interplay between the distortion of the triangular lattice and the interchain exchange interactions. 10. Excitation dynamics in a lattice Bose gas within the time-dependent Gutzwiller mean-field approach SciTech Connect Krutitsky, Konstantin V.; Navez, Patrick 2011-09-15 The dynamics of the collective excitations of a lattice Bose gas at zero temperature is systematically investigated using the time-dependent Gutzwiller mean-field approach. The excitation modes are determined within the framework of the linear-response theory as solutions of the generalized Bogoliubov-de Gennes equations valid in the superfluid and Mott-insulator phases at arbitrary values of parameters. The expression for the sound velocity derived in this approach coincides with the hydrodynamic relation. We calculate the transition amplitudes for the excitations in the Bragg scattering process and show that the higher excitation modes make significant contributions. We simulate the dynamics of the density perturbations and show that their propagation velocity in the limit of week perturbation is satisfactorily described by the predictions of the linear-response analysis. 11. Nuclear spin-lattice relaxation at field-induced level crossings in a Cr8F8 pivalate single crystal NASA Astrophysics Data System (ADS) Yamamoto, Shoji 2016-01-01 We construct a microscopic theory for the proton spin-lattice relaxation-rate 1 / T1 measurements around field-induced level crossings in a single crystal of the trivalent chromium ion wheel complex [Cr8F8(OOCtBu)16] at sufficiently low temperatures [E. Micotti et al., Phys. Rev. B 72 (2005) 020405(R)]. Exactly diagonalizing a well-equipped spin Hamiltonian for the individual clusters and giving further consideration to their possible interactions, we reveal the mechanism of 1 / T1 being single-peaked normally at the first level crossing but double-peaked intriguingly around the second level crossing. We wipe out the doubt about poor crystallization and find out a solution-intramolecular alternating Dzyaloshinsky-Moriya interaction combined with intermolecular coupling of antiferromagnetic character, each of which is so weak as several tens of mK in magnitude. 12. A three-dimensional phase field model coupled with lattice kinetics solver for modeling crystal growth in furnaces with accelerated crucible rotation and traveling magnetic field SciTech Connect Lin, Guang; Bao, Jie; Xu, Zhijie 2014-11-01 In this study, which builds on other related work, we present a new three-dimensional numerical model for crystal growth in a vertical solidification system. This model accounts for buoyancy, accelerated crucible rotation technique (ACRT), and traveling magnetic field (TMF) induced convective flow and their effect on crystal growth and the chemical component's transport process. The evolution of the crystal growth interface is simulated using the phase field method. A semi-implicit lattice kinetics solver based on the Boltzmann equation is employed to model the unsteady incompressible flow. A one-way coupled concentration transport model is used to simulate the component fraction variation in both the liquid and solid phases, which can be used to check the quality of the crystal growth. 13. Synthetic Dimensions with Magnetic Fields and Local Interactions in Photonic Lattices. PubMed Ozawa, Tomoki; Carusotto, Iacopo 2017-01-06 We discuss how one can realize a photonic device that combines synthetic dimensions and synthetic magnetic fields with spatially local interactions. Using an array of ring cavities, the angular coordinate around each cavity spans the synthetic dimension. The synthetic magnetic field arises as the intercavity photon hopping is associated with a change of angular momentum. Photon-photon interactions are local in the periodic angular coordinate around each cavity. Experimentally observable consequences of the synthetic magnetic field and of the local interactions are pointed out. 14. Phase Transition of Bosons Driven by a Staggered Gauge Field in AN Optical Lattice NASA Astrophysics Data System (ADS) Cha, Min-Chul 2013-06-01 We have studied the ground state properties of hard-core bosons in a two-leg optical ladder in the presence of uniform and staggered frustrations due to an artificial gauge field. By calculating the ground state via the Lanczos method, we find first-order phase transitions tuned by the staggered gauge field between the Meissner and the vortex states. The momentum distributions show that the Meissner state has edge and staggered currents, while the vortex states have vortex-solid or vortex-glass phases in the presence of a staggered field. 15. Effect of anisotropy in the S=1 underscreened Kondo lattice NASA Astrophysics Data System (ADS) Thomas, Christopher; da Rosa Simões, Acirete S.; Lacroix, Claudine; Iglesias, José Roberto; Coqblin, Bernard 2014-12-01 We study the effect of crystal field anisotropy in the underscreened S=1 Kondo lattice model. Starting from the two orbital Anderson lattice model and including a local anisotropy term, we show, through Schrieffer-Wolff transformation, that local anisotropy is equivalent to an anisotropic Kondo interaction (J∥≠J⊥). The competition and coexistence between ferromagnetism and Kondo effect in this effective model is studied within a generalized mean-field approximation. Several regimes are obtained, depending on the parameters, exhibiting or not coexistence of magnetic order and Kondo effect. Particularly, we show that a re-entrant Kondo phase at low temperature can be obtained. We are also able to describe phases where the Kondo temperature is smaller than the Curie temperature (TK 16. Spectral-domain-based scattering analysis of fields radiated by distributed sources in planar-stratified environments with arbitrarily anisotropic layers NASA Astrophysics Data System (ADS) Sainath, Kamalesh; Teixeira, Fernando L. 2014-12-01 We discuss the numerically stable, spectral-domain computation and extraction of the scattered electromagnetic field excited by distributed sources embedded in planar-layered environments, where each layer may exhibit arbitrary and independent electrical and magnetic anisotropic response and loss profiles. This stands in contrast to many standard spectral-domain algorithms that are restricted to computing the fields radiated by Hertzian dipole sources in planar-layered environments where the media possess azimuthal-symmetric material tensors (i.e., isotropic, and certain classes of uniaxial, media). Although computing the scattered field, particularly when due to distributed sources, appears (from the analytical perspective, at least) relatively straightforward, different procedures within the computation chain, if not treated carefully, are inherently susceptible to numerical instabilities and (or) accuracy limitations due to the potential manifestation of numerically overflown and (or) numerically unbalanced terms entering the chain. Therefore, primary emphasis herein is given to effecting these tasks in a numerically stable and robust manner for all ranges of physical parameters. After discussing the causes behind, and means to mitigate, these sources of numerical instability, we validate the algorithm's performance against closed-form solutions. Finally, we validate and illustrate the applicability of the proposed algorithm in case studies concerning active remote sensing of marine hydrocarbon reserves embedded deep within lossy, planar-layered media. 17. Anderson lattice with explicit Kondo coupling revisited: metamagnetism and the field-induced suppression of the heavy fermion state. PubMed Howczak, Olga; Spałek, Jozef 2012-05-23 We apply the extended (statistically consistent, SCA) Gutzwiller-type approach to the periodic Anderson model (PAM) in an applied magnetic field and in the strong-correlation limit. The finite-U corrections are included systematically by transforming the PAM into the form with the Kondo-type interaction and the residual hybridization, both appearing at the same time and on equal footing. This effective Hamiltonian represents the essence of our Anderson-Kondo lattice model. We show that in ferromagnetic phases the low-energy single-particle states are strongly affected by the presence of the applied magnetic field. We also find that for large values of hybridization strength the system enters the so-called locked heavy fermion state introduced earlier. In this state the chemical potential lies in the majority-spin hybridization gap and, as a consequence, the system evolution is insensitive to further increase of the applied field. However, for a sufficiently strong magnetic field, the system transforms from the locked state to the fully spin-polarized phase. This is accompanied by a metamagnetic transition, as well as by a drastic reduction of the effective mass of the quasiparticles. In particular, we observe no effective mass enhancement in the fully polarized state. The findings are in overall agreement with experimental results for the Ce compounds in high magnetic fields. The mass enhancement for the spin-minority electrons may also diminish with the increasing field, unlike for the quasiparticle states in a single narrow band in the same limit of strong correlations. 18. Tri-critical behavior of the Blume-Emery-Griffiths model on a Kagomé lattice: Effective-field theory and Rigorous bounds NASA Astrophysics Data System (ADS) Santos, Jander P.; Sá Barreto, F. C. 2016-01-01 Spin correlation identities for the Blume-Emery-Griffiths model on Kagomé lattice are derived and combined with rigorous correlation inequalities lead to upper bounds on the critical temperature. From the spin correlation identities the mean field approximation and the effective field approximation results for the magnetization, the critical frontiers and the tricritical points are obtained. The rigorous upper bounds on the critical temperature improve over those effective-field type theories results. 19. Synthetic gauge field and pseudospin-orbit interaction in a stacked two-dimensional ring-network lattice NASA Astrophysics Data System (ADS) Ochiai, Tetsuyuki 2017-02-01 We study the effects of a synthetic gauge field and pseudospin-orbit interaction in a stacked two-dimensional ring-network model. The model was introduced to simulate light propagation in the corresponding ring-resonator lattice, and is thus completely bosonic. Without these two items, the model exhibits Floquet-Weyl and Floquet-topological-insulator phases with topologically gapless and gapped band structures, respectively. The synthetic magnetic field implemented in the model results in a three-dimensional Hofstadter-butterfly-type spectrum in a photonic platform. The resulting gaps are characterized by the winding number of relevant S-matrices together with the Chern number of the bulk bands. The pseudospin-orbit interaction is defined as the mixing term between two pseudospin degrees of freedom in the rings, namely, the clockwise and counter-clockwise modes. It destroys the Floquet-topological-insulator phases, while the Floquet-Weyl phase with multiple Weyl points can be preserved by breaking the space-inversion symmetry. Implementing both the synthetic gauge field and pseudospin-orbit interaction requires a certain nonreciprocity. 20. Triviality problem and high-temperature expansions of higher susceptibilities for the Ising and scalar-field models in four-, five-, and six-dimensional lattices. PubMed Butera, P; Pernici, M 2012-02-01 High-temperature expansions are presently the only viable approach to the numerical calculation of the higher susceptibilities for the spin and the scalar-field models on high-dimensional lattices. The critical amplitudes of these quantities enter into a sequence of universal amplitude ratios that determine the critical equation of state. We have obtained a substantial extension, through order 24, of the high-temperature expansions of the free energy (in presence of a magnetic field) for the Ising models with spin s≥1/2 and for the lattice scalar-field theory with quartic self-interaction on the simple-cubic and the body-centered-cubic lattices in four, five, and six spatial dimensions. A numerical analysis of the higher susceptibilities obtained from these expansions yields results consistent with the widely accepted ideas, based on the renormalization group and the constructive approach to Euclidean quantum field theory, concerning the no-interaction ("triviality") property of the continuum (scaling) limit of spin-s Ising and lattice scalar-field models at and above the upper critical dimensionality. 1. Acceleration of split-field finite difference time-domain method for anisotropic media by means of graphics processing unit computing NASA Astrophysics Data System (ADS) Francés, Jorge; Bleda, Sergio; Álvarez, Mariela Lázara; Martínez, Francisco Javier; Márquez, Andres; Neipp, Cristian; Beléndez, Augusto 2014-01-01 The implementation of split-field finite difference time domain (SF-FDTD) applied to light-wave propagation through periodic media with arbitrary anisotropy method in graphics processing units (GPUs) is described. The SF-FDTD technique and the periodic boundary condition allow the consideration of a single period of the structure reducing the simulation grid. Nevertheless, the analysis of the anisotropic media implies considering all the electromagnetic field components and the use of complex notation. These aspects reduce the computational efficiency of the numerical method compared with the isotropic and nonperiodic implementation. Specifically, the implementation of the SF-FDTD in the Kepler family of GPUs of NVIDIA is presented. An analysis of the performance of this implementation is done, and several applications have been considered in order to estimate the possibilities provided by both the formalism and the implementation into GPU: binary phase gratings and twisted-nematic liquid crystal cells. Regarding the analysis of binary phase gratings, the validity of the scalar diffraction theory is evaluated by the comparison of the diffraction efficiencies predicted by SF-FDTD. The analysis for the second order of diffraction is extended, which is considered as a reference for the transmittance obtained by the SF-FDTD scheme for periodic media. 2. High-magnetic-field lattice length changes in URu2Si2. PubMed Correa, V F; Francoual, S; Jaime, M; Harrison, N; Murphy, T P; Palm, E C; Tozer, S W; Lacerda, A H; Sharma, P A; Mydosh, J A 2012-12-14 We report high-magnetic-field (up to 45 T) ĉ-axis thermal-expansion and magnetostriction experiments on URu(2)Si(2) single crystals. The sample length change ΔL(c)(T(HO))/L(c) associated with the transition to the "hidden order" phase becomes increasingly discontinuous as the magnetic field is raised above 25 T. The reentrant ordered phase III is clearly observed in both the thermal expansion ΔL(c)(T)/L(c) and magnetostriction ΔL(c)(B)/L(c) above 36 T, in good agreement with previous results. The sample length is also discontinuous at the boundaries of this phase, mainly at the upper boundary. A change in the sign of the coefficient of thermal expansion α(c)=1/L(c)(∂ΔL(c)/∂T) is observed at the metamagnetic transition (B(M) ~ 38 T), which is likely related to the existence of a quantum critical end point. 3. Chiral Spin Liquids in Triangular-Lattice SU (N ) Fermionic Mott Insulators with Artificial Gauge Fields NASA Astrophysics Data System (ADS) Nataf, Pierre; Lajkó, Miklós; Wietek, Alexander; Penc, Karlo; Mila, Frédéric; Läuchli, Andreas M. 2016-10-01 We show that, in the presence of a π /2 artificial gauge field per plaquette, Mott insulating phases of ultracold fermions with SU (N ) symmetry and one particle per site generically possess an extended chiral phase with intrinsic topological order characterized by an approximate ground space of N low-lying singlets for periodic boundary conditions, and by chiral edge states described by the SU(N ) 1 Wess-Zumino-Novikov-Witten conformal field theory for open boundary conditions. This has been achieved by extensive exact diagonalizations for N between 3 and 9, and by a parton construction based on a set of N Gutzwiller projected fermionic wave functions with flux π /N per triangular plaquette. Experimental implications are briefly discussed. 4. Complex q-ANALYSIS and Scalar Field Theory on a q-LATTICE NASA Astrophysics Data System (ADS) Ubriaco, Marcelo R. We develop the basic formalism of complex q-analysis to study the solutions of second order q-difference equations which reduce, in the q → 1 limit, to the ordinary Laplace equation in Euclidean and Minkowski space. After defining an inner product on the function space we construct and study the properties of the solutions, and then apply this formalism to the Schrödinger equation and two-dimensional scalar field theory. 5. Lattice Boltzmann method for binary fluids based on mass-conserving quasi-incompressible phase-field theory NASA Astrophysics Data System (ADS) Yang, Kang; Guo, Zhaoli 2016-04-01 In this paper, a lattice Boltzmann equation (LBE) model is proposed for binary fluids based on a quasi-incompressible phase-field model [J. Shen et al., Commun. Comput. Phys. 13, 1045 (2013), 10.4208/cicp.300711.160212a]. Compared with the other incompressible LBE models based on the incompressible phase-field theory, the quasi-incompressible model conserves mass locally. A series of numerical simulations are performed to validate the proposed model, and comparisons with an incompressible LBE model [H. Liang et al., Phys. Rev. E 89, 053320 (2014), 10.1103/PhysRevE.89.053320] are also carried out. It is shown that the proposed model can track the interface accurately. As the stationary droplet and rising bubble problems, the quasi-incompressible LBE gives nearly the same predictions as the incompressible model, but the compressible effect in the present model plays a significant role in the phase separation problem. Therefore, in general cases the present mass-conserving model should be adopted. 6. Nematic phase in the J(1)-J(2) square-lattice Ising model in an external field. PubMed Guerrero, Alejandra I; Stariolo, Daniel A; Almarza, Noé G 2015-05-01 The J(1)-J(2) Ising model in the square lattice in the presence of an external field is studied by two approaches: the cluster variation method (CVM) and Monte Carlo simulations. The use of the CVM in the square approximation leads to the presence of a new equilibrium phase, not previously reported for this model: an Ising-nematic phase, which shows orientational order but not positional order, between the known stripes and disordered phases. Suitable order parameters are defined, and the phase diagram of the model is obtained. Monte Carlo simulations are in qualitative agreement with the CVM results, giving support to the presence of the new Ising-nematic phase. Phase diagrams in the temperature-external field plane are obtained for selected values of the parameter κ=J(2)/|J(1)| which measures the relative strength of the competing interactions. From the CVM in the square approximation we obtain a line of second order transitions between the disordered and nematic phases, while the nematic-stripes phase transitions are found to be of first order. The Monte Carlo results suggest a line of second order nematic-disordered phase transitions in agreement with the CVM results. Regarding the stripes-nematic transitions, the present Monte Carlo results are not precise enough to reach definite conclusions about the nature of the transitions. 7. Discussion of parameters, lattices and beam stability for a 200-TeV low-field collider SciTech Connect Neuffer, D. 1996-03-01 Recently, it has been suggested that improved technology and reduced costs in remotely-drilled small-diameter tunnels, coupled with improvements in robotic technology, may make the original concept of the desertron more realistic and affordable. In this concept, a long, small-diameter tunnel is drilled (<{approximately}1m diameter sewer pipe) and filled with long, low-cost magnets, which are installed and serviced robotically. To obtain high-energy then requires low cost magnets, which are iron-dominated superferric magnets (B{approximately}2 T). A large circumference is then required ({approximately}1000 km for {approximately}100 TeV/beam). Table 1 shows parameters for a 200 TeV proton-proton collider, based on the premise of a large low-cost ring with super-ferric magnets. While outline designs for a low-cost {approximately}2T dipole have been initiated, an accelerator requires beam stability, which means quadrupole fields for focusing, as well as sextupoles for chromatic correction, and further design tolerances and correctors to obtain sufficiently linear fields. Previously we have developed initial lattices and dynamic motion discussions for the earlier 40 TeV incarnation of the superferric supercollider. In this note we apply those results to initiate discussions of the dynamic requirements of this 200 TeV collider. 8. Quantum field theory for the three-body constrained lattice Bose gas. II. Application to the many-body problem SciTech Connect Diehl, S.; Daley, A. J.; Zoller, P.; Baranov, M. 2010-08-01 We analyze the ground-state phase diagram of attractive lattice bosons, which are stabilized by a three-body onsite hardcore constraint. A salient feature of this model is an Ising-type transition from a conventional atomic superfluid to a dimer superfluid with vanishing atomic condensate. The study builds on an exact mapping of the constrained model to a theory of coupled bosons with polynomial interactions, proposed in a related paper [S. Diehl, M. Baranov, A. Daley, and P. Zoller, Phys. Rev. B 82, 064509 (2010).]. In this framework, we focus by analytical means on aspects of the phase diagram which are intimately connected to interactions, and are thus not accessible in a mean-field plus spin-wave approach. First, we determine shifts in the mean-field phase border, which are most pronounced in the low-density regime. Second, the investigation of the strong coupling limit reveals the existence of a 'continuous supersolid', which emerges as a consequence of enhanced symmetries in this regime. We discuss its experimental signatures. Third, we show that the Ising-type phase transition, driven first order via the competition of long-wavelength modes at generic fillings, terminates into a true Ising quantum critical point in the vicinity of half filling. 9. Finite-volume scheme for anisotropic diffusion SciTech Connect Es, Bram van; Koren, Barry; Blank, Hugo J. de 2016-02-01 In this paper, we apply a special finite-volume scheme, limited to smooth temperature distributions and Cartesian grids, to test the importance of connectivity of the finite volumes. The area of application is nuclear fusion plasma with field line aligned temperature gradients and extreme anisotropy. We apply the scheme to the anisotropic heat-conduction equation, and compare its results with those of existing finite-volume schemes for anisotropic diffusion. Also, we introduce a general model adaptation of the steady diffusion equation for extremely anisotropic diffusion problems with closed field lines. 10. Shaped beam scattering by an anisotropic particle NASA Astrophysics Data System (ADS) Chen, Zhenzhen; Zhang, Huayong; Huang, Zhixiang; Wu, Xianliang 2017-03-01 An exact semi-analytical solution to the electromagnetic scattering from an optically anisotropic particle illuminated by an arbitrarily shaped beam is proposed. The scattered fields and fields within the anisotropic particle are expanded in terms of spherical vector wave functions. The unknown expansion coefficients are determined by using the boundary conditions and the method of moments scheme. For incidence of a Gaussian beam, zero-order Bessel beam and Hertzian electric dipole radiation, numerical results of the normalized differential scattering cross section are given to a uniaxial, gyrotropic anisotropic spheroid and circular cylinder of finite length. The scattering properties are analyzed concisely. 11. Lattice dynamics and external magnetic-field effects in Ni-Fe-Ga alloys NASA Astrophysics Data System (ADS) Pérez-Landazábal, J. I.; Recarte, V.; Sánchez-Alarcos, V.; Rodríguez-Velamazán, J. A.; Jiménez-Ruiz, M.; Link, P.; Cesari, E.; Chumlyakov, Y. I. 2009-10-01 Precursor phenomena were investigated in a Ni-Fe-Ga alloy close to the stoichiometric Heusler composition Ni2FeGa . In particular, the phonon-dispersion curves, the diffuse scattering and the magnetic properties of a single crystalline Ni51.5Fe21.5Ga27 alloy were measured as a function of temperature. The TA2 branch along the [110] direction of the L21 phase shows a significant phonon softening around ξ=0.35 resulting in a marked dip which becomes more pronounced as the temperature decreases. Diffuse neutron-scattering measurements performed along [ξ¯ξ0] direction around Bragg reflections also reveal the presence of small satellite peaks at ξ=0.33 whose intensity increases on approaching the martensitic transformation temperature. Both elastic and inelastic-scattering anomalies confirm the occurrence of premartensitic phenomena in Ni-Fe-Ga alloys. The influence of an external magnetic field (6 T) on the anomalous phonon is shown to be negligible and just a small shift of the transformation temperature takes place because of the magnetic field. 12. Anisotropic Particles in Turbulence NASA Astrophysics Data System (ADS) Voth, Greg A.; Soldati, Alfredo 2017-01-01 Anisotropic particles are common in many industrial and natural turbulent flows. When these particles are small and neutrally buoyant, they follow Lagrangian trajectories while exhibiting rich orientational dynamics from the coupling of their rotation to the velocity gradients of the turbulence field. This system has proven to be a fascinating application of the fundamental properties of velocity gradients in turbulence. When particles are not neutrally buoyant, they experience preferential concentration and very different preferential alignment than neutrally buoyant tracer particles. A vast proportion of the parameter range of anisotropic particles in turbulence is still unexplored, with most existing research focusing on the simple foundational cases of axisymmetric ellipsoids at low concentrations in homogeneous isotropic turbulence and in turbulent channel flow. Numerical simulations and experiments have recently developed a fairly comprehensive picture of alignment and rotation in these cases, and they provide an essential foundation for addressing more complex problems of practical importance. Macroscopic effects of nonspherical particle dynamics include preferential concentration in coherent structures and drag reduction by fiber suspensions. We review the models used to describe nonspherical particle motion, along with numerical and experimental methods for measuring particle dynamics. 13. Quantum Lattice Representation of Dark Solitons DTIC Science & Technology 2004-01-01 Gross - Pitaevskii equation, which for a highly anisotropic (cigar-shaped) magnetic trap reduces to a one-dimensional (ID) cubic NLS, in an external...solitons Vector dark-bright solitons Nonlinear Schrodinger equation Gross - Pitaevskii equation Quantum lattice representation 16. SECURITY CLASSIFICATION...condensate (BEC) is described by the Gross - Pitaevskii equation, which for a highly anisotropic (cigar-shaped) magnetic trap reduces to a one 14. The effects of degeneracy of the carrier ensemble on the energy loss rate and the high field mobility characteristics under the conditions of low lattice temperatures NASA Astrophysics Data System (ADS) Basu, A.; Das, B.; Middya, T. R.; Bhattacharya, D. P. 2017-02-01 The rate of loss of energy of the non-equilibrium electrons to the acoustic mode lattice vibration in a degenerate semiconductor is obtained under the condition, when the lattice temperature is low enough, so that the traditional approximations like the elastic nature of the electron-phonon collisions and the truncation of the phonon distribution to the equipartition law are not valid any more. Using the results of the energy loss rate, the non-ohmic mobility is then calculated. Evaluating the loss rate and the non-ohmic mobility in degenerate samples of Si and Ge we find that significant changes in both the characteristics have been effected compared to that in the non-degenerate samples, in the regime of lower energy and for relatively lower fields. The effected changes are more significant the lower the lattice temperature is. 15. Lattice QCD determination of states with spin 5/2 or higher in the spectrum of nucleons SciTech Connect Stephen Wallace; S. Basak; R. Edwards; George Fleming; J. Juge; A. Lichtl; C. Morningstar; D. Richards; I. Sato 2006-09-28 Energies for excited isospin 1/2 states that include the nucleon are computed using quenched, anisotropic lattices. Baryon interpolating field operators that are used include nonlocal operators that provide G{sub 2} irreducible representations of the octahedral group. States with spin 5/2 or higher are identified as degenerate energies that occur in irreducible representations of the octahedral group corresponding to the subduction of the continuum spin. 16. Anisotropic decoherence in quantum wells with arbitrary magnetic fields: Interplay of the spin-orbit coupling terms NASA Astrophysics Data System (ADS) Prada, M.; Pfannkuche, D. 2017-01-01 We present a theoretical study of the anisotropy of the spin relaxation and decoherence in typical quantum wells with an arbitrary magnetic field. In such systems, the orientation of the magnetic field relative to the main crystallographic directions is crucial, owing to the lack of spin-rotation symmetry. For typical high mobility samples, relaxation anisotropies owing to the interplay of Rashba and Dresselhaus spin-orbit coupling are calculated. We also include the effect of the cubic-in-momentum terms. Although commonly ignored in literature, the latter were experimentally evidenced by the observation of strong anisotropy in spin decoherence measurements by different experimental groups and has long remained unexplained. This work suggests a method to determine the relative strength of spin-orbit coupling terms by angular resolution of decoherence in electron spin resonance experiments. 17. Methods for conditioning anisotropic, operator-scaling, fractal random fields, and the effect on solute transport simulations NASA Astrophysics Data System (ADS) Revielle, J.; Benson, D. A. 2008-12-01 The fractal scaling of aquifer materials have been observed in many data sets. Typically, the scaling coefficient is different in different directions. To date, only unconditional realizations with these properties can be generated. We present and analyze two methods of creating conditional operator-scaling fractal random fields (OSFRF) which have the ability to condition any number and geometry of measurements into each realization. One method is based on the theory of Orthographic Projection (Feller, 1971) and requires the continuous checking of a conditional probability function. The other method uses a best linear unbiased estimate (i.e., a kriged mean surface between known points) and an unconditional realization to create each conditional field. These two methods are analyzed for computational difficulty and their ability to recreate the desired fractal scaling along different (eigenvector) directions. Finally these methods are applied to a transport experiment through a slab of Massillon sandstone to show the advantage of using conditional OSFRF in solute transport modeling. 18. Dynamic mean field theory for lattice gas models of fluid mixtures confined in mesoporous materials. PubMed Edison, J R; Monson, P A 2013-11-12 We present the extension of dynamic mean field theory (DMFT) for fluids in porous materials (Monson, P. A. J. Chem. Phys. 2008, 128, 084701) to the case of mixtures. The theory can be used to describe the relaxation processes in the approach to equilibrium or metastable equilibrium states for fluids in pores after a change in the bulk pressure or composition. It is especially useful for studying systems where there are capillary condensation or evaporation transitions. Nucleation processes associated with these transitions are emergent features of the theory and can be visualized via the time dependence of the density distribution and composition distribution in the system. For mixtures an important component of the dynamics is relaxation of the composition distribution in the system, especially in the neighborhood of vapor-liquid interfaces. We consider two different types of mixtures, modeling hydrocarbon adsorption in carbon-like slit pores. We first present results on bulk phase equilibria of the mixtures and then the equilibrium (stable/metastable) behavior of these mixtures in a finite slit pore and an inkbottle pore. We then use DMFT to describe the evolution of the density and composition in the pore in the approach to equilibrium after changing the state of the bulk fluid via composition or pressure changes. 19. Bulk viscosity of anisotropically expanding hot QCD plasma SciTech Connect Chandra, Vinod 2011-11-01 The bulk viscosity, {zeta} and its ratio with the shear viscosity, {zeta}/{eta} have been studied in an anisotropically expanding pure glue plasma in the presence of turbulent color fields. It has been shown that the anisotropy in the momentum distribution function of gluons, which has been determined from a linearized transport equation eventually leads to the bulk viscosity. For the isotropic (equilibrium) state, a recently proposed quasiparticle model of pure SU(3) lattice QCD equation of state has been employed where the interactions are encoded in the effective fugacity. It has been argued that the interactions present in the equation of state, significantly contribute to the bulk viscosity. Its ratio with the shear viscosity is significant even at 1.5T{sub c}. Thus, one needs to take in account the effects of the bulk viscosity while studying the hydrodynamic expansion of quark-gluon plasma in the Relativistic Heavy Ion Collider and the Large Hadron Collider. 20. Direct observation of an anisotropic in-plane residual stress induced by B addition as an origin of high magnetic anisotropy field of Ru/FeCoB film SciTech Connect Hirata, Ken-ichiro; Gomi, Shunsuke; Mashiko, Yasuhiro; Nakagawa, Shigeki 2010-05-15 Although boron-free FeCo films prepared on a Ru underlayer exhibits isotropic in-plane magnetic property, boron added FeCoB films prepared on Ru underlayer revealed large in-plane magnetic anisotropy with a high anisotropy field of 500 Oe. The effect of boron addition on the in-plane anisotropic residual stress in FeCoB film was investigated using sin{sup 2} {psi} method of x-ray diffraction analysis. Large isotropic compressive stress was observed in Ru/FeCo film. In contrast, anisotropic in-plane residual stress was observed in Ru/FeCoB film. The compressive stress along the easy axis of Ru/FeCoB film is released more than that along the hard axis. Such anisotropic residual stress is regarded as an origin of the in-plane magnetic anisotropy through inverse magnetostriction effect. Owing to the configuration of the facing targets sputtering system, boron atoms are sputtered and deposited anisotropically, and so they penetrate FeCo crystals and release the compressive stress along the incidence direction. 1. Testing the Standard Model and Fundamental Symmetries in Nuclear Physics with Lattice QCD and Effective Field Theory SciTech Connect Walker-Loud, Andre 2016-10-14 The research supported by this grant is aimed at probing the limits of the Standard Model through precision low-energy nuclear physics. The work of the PI (AWL) and additional personnel is to provide theory input needed for a number of potentially high-impact experiments, notably, hadronic parity violation, Dark Matter direct detection and searches for permanent electric dipole moments (EDMs) in nucleons and nuclei. In all these examples, a quantitative understanding of low-energy nuclear physics from the fundamental theory of strong interactions, Quantum Chromo-Dynamics (QCD), is necessary to interpret the experimental results. The main theoretical tools used and developed in this work are the numerical solution to QCD known as lattice QCD (LQCD) and Effective Field Theory (EFT). This grant is supporting a new research program for the PI, and as such, needed to be developed from the ground up. Therefore, the first fiscal year of this grant, 08/01/2014-07/31/2015, has been spent predominantly establishing this new research effort. Very good progress has been made, although, at this time, there are not many publications to show for the effort. After one year, the PI accepted a job at Lawrence Berkeley National Laboratory, so this final report covers just a single year of five years of the grant. 2. Probing protein hydration and aging of food materials by the magnetic field dependence of proton spin-lattice relaxation times. PubMed Godefroy, Sophie; Korb, Jean-Pierre; Creamer, Lawrence K; Watkinson, Philip J; Callaghan, Paul T 2003-11-15 Most cheeses can be considered as solid emulsions of milk fat in a matrix of water and proteins. Regions of each of the phases can be liquid during processing and maturation. Identifying these regions and monitoring changes in them is important as a prelude to controlling the structure of the final cheese. We concentrate on the behavior of water in the vicinity of proteins as a function of cheese aging. Our method utilizes nuclear magnetic relaxation dispersion (NMRD) associated with the frequency dependence of water spin-lattice relaxation rates using the field cycling NMR technique. This method provides insight into the dynamical behavior of water molecules on a very large time scale. Moreover, we can distinguish between molecular motion in bulk and motion in the vicinity of a source of relaxation, such as proteins. A fit of our dispersion data using a theory developed by J.-P. Korb and R.G. Bryant (J. Chem. Phys. 115 (2001) 23) allowed us to determine the degree of hydration of proteins as a function of aging. In particular, we find that protein hydration increases with ripening. 3. Modeling a tracer test at the Grimsel Test Site (GTS) using a lattice Boltzmann method and transmissivity field NASA Astrophysics Data System (ADS) Kim, J. W.; Lanyon, G. W.; Baik, M. H.; Blechschmidt, I. 2015-12-01 A series of tracer tests have been conducted in the Migration (MI) Shear Zone at the Grimsel Test Site (GTS) for the Colloid Formation and Migration Project (CFM). As a part of the series, a dipole test (Tracer Test Run 13-05) using radionuclides, colloids and conservative tracers was performed to determine the breakthrough between CRR99.002-i2 and BOMI87.010-i2. To date, the breakthrough data of only the conservative dye tracer (Amino-G acid) are available. In the preceding project, the Colloid and Radionuclide Retardation Project (CRR), a transmissivity field for the MI shear zone was obtained by the geostatistical inverse modeling approach. In this study, the breakthrough of the tracer was computed by a gray lattice Boltzmann method (LBM). The transmissivity field with finite elements grid was transformed to the effective fracture aperture or flow porosity according to the cubic law, and the grid was uniformalized by the interpolation. The uniform mesh of the effective aperture was utilized as the model domain of the gray LBM. In the gray LBM, the heterogeneity of the aperture was dealt with a partial-bounceback scheme. The profiles of hydraulic heads monitored at the boreholes nearby were used as the reference values in the calculation of the pressure distribution in the model domain. The modeling results could reveal a dominant pathway of tracers in the dipole test. The developed model can be utilized in the calculation of the reactive transports of radionuclides and colloids by coupling with a geochemical model, such as Phreeqc, the Geochemist's Workbench, etc. 4. Quasiparticle anisotropic hydrodynamics for central collisions NASA Astrophysics Data System (ADS) Alqahtani, Mubarak; Nopoush, Mohammad; Strickland, Michael 2017-03-01 We use quasiparticle anisotropic hydrodynamics to study an azimuthally symmetric boost-invariant quark-gluon plasma including the effects of both shear and bulk viscosities. In quasiparticle anisotropic hydrodynamics, a single finite-temperature quasiparticle mass is introduced and fit to the lattice data in order to implement a realistic equation of state (EoS). We compare results obtained by using the quasiparticle method with the standard method of imposing the EoS in anisotropic hydrodynamics and viscous hydrodynamics. Using these three methods, we extract the primordial particle spectra, total number of charged particles, and average transverse momentum for various values of the shear viscosity to entropy density ratio η /s . We find that the three methods agree well for small shear viscosity to entropy density ratio η /s , but differ at large η /s , with the standard anisotropic EoS method showing suppressed production at low transverse-momentum compared with the other two methods considered. Finally, we demonstrate explicitly that, when using standard viscous hydrodynamics, the bulk-viscous correction can drive the primordial particle spectra negative at large pT. Such behavior is not seen in either anisotropic hydrodynamics approach, irrespective of the value of η /s . 5. Anisotropic inflation in Gauss-Bonnet gravity SciTech Connect Lahiri, Sayantani 2016-09-19 We study anisotropic inflation with Gauss-Bonnet correction in presence of a massless vector field. In this scenario, exact anisotropic power-law inflation is realized when the inflaton potential, gauge coupling function and the Gauss-Bonnet coupling are exponential functions. We show that anisotropy becomes proportional to two slow-roll parameters of the theory and hence gets enhanced in presence of quadratic curvature corrections. The stability analysis reveals that anisotropic power-law solutions remain stable over a substantially large parameter region. 6. Scanning phononic lattices with ultrasound SciTech Connect Vines, R.E.; Wolfe, J.P.; Every, A.V. 1999-11-01 A method for probing the elastic properties of newly developed periodic structures using acoustic waves is introduced. Highly anisotropic transmission of surface acoustic waves is observed by continuously scanning the wave vector angle. Preliminary models of wave propagation through multilayers and two-dimensional lattices explain some of the experimental features, while other features can be attributed to the resonant excitation of interface waves. {copyright} {ital 1999} {ital The American Physical Society} 7. Quantum lattice representation of dark solitons NASA Astrophysics Data System (ADS) Vahala, George; Vahala, Linda; Yepez, Jeffrey 2004-08-01 The nonlinear Schrodinger (NLS) equation in a self-defocusing Kerr medium supports dark solitons. Moreover the mean field description of a dilute Bose-Einstein condensate (BEC) is described by the Gross-Pitaevskii equation, which for a highly anisotropic (cigar-shaped) magnetic trap reduces to a one-dimensional (1D) cubic NLS in an external potential. A quantum lattice algorithm is developed for the dark solitons. Simulations are presented for both black (stationary) solitons as well as (moving) dark solitons. Collisions of dark solitons are compared with the exact analytic solutions and coupled dark-bright vector solitons are examined. The quantum algorithm requires 2 qubits per scalar field at each spatial node. The unitary collision operator quantum mechanically entangles the on-site qubits, and this transitory entanglement is spread throughout the lattice by the streaming operators. These algorithms are suitable for a Type-II quantum computers, with wave function collapse induced by quantum measurements required to determine the coupling potentials. 8. Stress heterogeneities in anisotropic materials - their effect on dislocation fields and post-deformational recrystallization: Insights from combined experiments and numerical simulations of polycrystalline ice NASA Astrophysics Data System (ADS) Piazolo, S.; Montagnat, M.; Borthwick, V.; Evans, L.; Griera, A.; Grennerat, F.; Moulinec, H.; Wheeler, J. 2014-12-01 We present a coupled experimental and modeling approach to better understand the role of stress field heterogeneities on deformation and post-deformational behavior in material with a high viscoplastic anisotropy e.g. polycrystalline ice. We investigate: (1) Effect of stress heterogeneities on deformation behavior and microstructural development and, (2) effect of such microstructures on post-deformational recrystallization. (1) Full-field elasto-viscoplastic modelling (CraFT) is used to predict the local stress and strain field during transient creep in a polycrystalline ice sample. Modeling input includes the experimental starting microstructure and a validated slip system dependent flow law. EBSD measurements on selected areas are used to estimate the local dislocation field utilizing the Weighted Burgers Vector (WBV) analysis. Areas of local stress concentration correlate with triple junctions and grain boundaries, originating from strain incompatibilities between differently oriented grains. In these areas, the WBV analysis shows a non-negligible c-axis component that must be related to resolved shear stress in a prismatic plane, coherent with the predicted elevated stress levels. The resultant defect structure is necessary for the formation of the observed kink bands which have a well-defined crystallographic character, lattice distortions and subgrain development. (2) The microstructures arising from (1) significantly affect post-deformational behavior. Combined post-deformational annealing experiments and numerical simulations using the microdynamic modeling platform ELLE, allow prediction of the local microstructural evolution taking recovery within grains, grain boundary migration and nucleation into account. Results from this study, can explain several of the observed features in natural ice, and help to refine large scale models. 9. Inflation in anisotropic scalar-tensor theories NASA Technical Reports Server (NTRS) Pimentel, Luis O.; Stein-Schabes, Jaime 1988-01-01 The existence of an inflationary phase in anisotropic Scalar-Tensor Theories is investigated by means of a conformal transformation that allows us to rewrite these theories as gravity minimally coupled to a scalar field with a nontrivial potential. The explicit form of the potential is then used and the No Hair Theorem concludes that there is an inflationary phase in all open or flat anisotropic spacetimes in these theories. Several examples are constructed where the effect becomes manifest. 10. A fast procedure of stress state evaluation in magnetically anisotropic steels with the help of a probe with adjustable magnetizing field direction NASA Astrophysics Data System (ADS) Chmielewski, Marek; Piotrowski, Leszek; Augustyniak, Bolesław 2017-04-01 The paper presents a novel approach to the stress state evaluation issue. It deals with a strongly (magnetically) anisotropic materials for which a direct interpretation of the Barkhausen effect (BE) intensity would lead to erroneous results. In such a case one has to take into account both the measured BE intensity and the orientation of the magnetisation direction relative to the magnetic easy axis. For the in plane stress distribution evaluation one has to perform at least three measurements in the non-collinear directions. The application of an apparatus with automatically changing magnetizing field direction allows to obtain the angular distribution of the BE intensity in about 30 s (with the angular step of 10°). Thanks to the dedicated post-processing software the procedure of the measurement data processing, resulting in the full information on the stress distribution (main stress components and their orientation in all the investigated points) is almost instantaneous. Apart from the measurement results the stress determination procedure requires two additional pieces of information. The first one is the calibration data obtained for at least two applied strain directions (along easy and hard magnetisation axes)— the data for the intermediate orientations are usually interpolated. The second one is the ‘reference level’ of the BE intensity angular distribution. In the case of welded plates it is obtained by averaging the results obtained at the analysed points before welding. The way of results presentation proposed in that paper is very illustrative and shows an interesting feature of the stress distribution in welded plates—namely the appearance of a ‘vortex’ structure of main stress. 11. Lennard-Jones and lattice models of driven fluids. PubMed Díez-Minguito, M; Garrido, P L; Marro, J 2005-08-01 We introduce a nonequilibrium off-lattice model for anisotropic phenomena in fluids. This is a Lennard-Jones generalization of the driven lattice-gas model in which the particles' spatial coordinates vary continuously. A comparison between the two models allows us to discuss some exceptional, hardly realistic features of the original discrete system--which has been considered a prototype for nonequilibrium anisotropic phase transitions. We thus help to clarify open issues, and discuss on the implications of our observations for future investigation of anisotropic phase transitions. 12. Anisotropic exchange-interaction model: From the Potts model to the exchange-interaction model NASA Astrophysics Data System (ADS) King, T. C.; Chen, H. H. 1995-04-01 A spin model called the anisotropic exchange-interaction model is proposed. The Potts model, the exchange-interaction model, and the spin-1/2 anisotropic Heisenberg model are special cases of the proposed model. Thermodynamic properties of the model on the bcc and the fcc lattices are determined by the constant-coupling approximation. 13. Spectroscopy of doubly charmed baryons from lattice QCD SciTech Connect Padmanath, M.; Edwards, Robert G.; Mathur, Nilmani; Peardon, Michael 2015-05-06 This study presents the ground and excited state spectra of doubly charmed baryons from lattice QCD with dynamical quark fields. Calculations are performed on anisotropic lattices of size 16³ × 128, with inverse spacing in temporal direction at⁻¹=5.67(4) GeV and with a pion mass of about 390 MeV. A large set of baryonic operators that respect the symmetries of the lattice yet which retain a memory of their continuum analogues are used. These operators transform as irreducible representations of SU(3)F symmetry for flavor, SU(4) symmetry for Dirac spins of quarks and O(3) for spatial symmetry. The distillation method is utilized to generate baryon correlation functions which are analyzed using the variational fitting method to extract excited states. The lattice spectra obtained have baryonic states with well-defined total spins up to 7/2 and the pattern of low-lying states does not support the diquark picture for doubly charmed baryons. On the contrary the calculated spectra are remarkably similar to the expectations from models with an SU(6)×O(3) symmetry. Various spin-dependent energy splittings between the extracted states are also evaluated. 14. Spectroscopy of doubly charmed baryons from lattice QCD NASA Astrophysics Data System (ADS) Padmanath, M.; Edwards, Robert G.; Mathur, Nilmani; Peardon, Michael; Hadron Spectrum Collaboration 2015-05-01 We present the ground and excited state spectra of doubly charmed baryons from lattice QCD with dynamical quark fields. Calculations are performed on anisotropic lattices of size 1 63×128 , with inverse spacing in temporal direction at-1=5.67 (4 ) GeV and with a pion mass of about 390 MeV. A large set of baryonic operators that respect the symmetries of the lattice yet which retain a memory of their continuum analogues are used. These operators transform as irreducible representations of SU(3 ) F symmetry for flavor, SU(4) symmetry for Dirac spins of quarks and O(3) for spatial symmetry. The distillation method is utilized to generate baryon correlation functions which are analyzed using the variational fitting method to extract excited states. The lattice spectra obtained have baryonic states with well-defined total spins up to 7 /2 and the pattern of low-lying states does not support the diquark picture for doubly charmed baryons. On the contrary the calculated spectra are remarkably similar to the expectations from models with an SU (6 )×O (3 ) symmetry. Various spin-dependent energy splittings between the extracted states are also evaluated. 15. Phyllotaxis of flux lattices in layered superconductors SciTech Connect Levitov, L.S. ) 1991-01-14 The geometry of a flux lattice pinned by superconducting layers is studied. Under variation of magnetic field the lattice undergoes an infinite sequence of continuous transitions corresponding to different ways of selection of shortest distances. All possible lattices form a hierarchical structure identified as the hierarchy of Farey numbers. It is shown that dynamically accessible lattices are characterized by pairs of consecutive Fibonacci numbers. 16. Anisotropic electrical and lattice transport properties of ordered quaternary phases Cr2TiAlC2 and Mo2TiAlC2: A first principles study NASA Astrophysics Data System (ADS) Li, Y. F.; Ding, Y. C.; Xiao, B.; Cheng, Y. H. 2016-11-01 Electrical conductivities of Cr2TiAlC2 and Mo2TiAlC2 in a and c directions are calculated from semi-classic Boltzmann transport theory. The values are found to be σa = 5.68 ×105 S /m (6.56 ×105 S /m) and σc = 2.15 ×105 S /m (2.69 ×105 S /m) for Cr2TiAlC2 (Mo2TiAlC2) at 300 K. Using the phonon-mode Debye temperature and Slack-model, the lattice thermal conductivities in the two directions are also evaluated, and the values are κa = 18.71 W /m K (16.11 W/m K) and κc = 0.48 W /m K (0.25 W /m K) for Cr2TiAlC2 (Mo2TiAlC2) at room temperature. The anisotropy in lattice thermal conductivity is found to be stronger than that of electrical conductivity. The predicted Seebeck coefficients and thermoelectric figure of merit (ZT) indicate that they are poor thermoelectric materials. Due to the relatively high conductivities, they might be used to fabricate high temperature conductive components in aerospace industry. In addition, our results in a direction have the direct implications for the relevant properties of MXenes (Cr2TiC2 and Mo2TiC2), produced from their bulk phases. 17. Zero-field studies of spin-lattice relaxation processes in non-Kramers doublet of LiF:Ni2+ NASA Astrophysics Data System (ADS) Azamat, D. V.; Badalyan, A. G.; Dejneka, A.; Jastrabik, L.; Lančok, J. 2016-12-01 We use the inversion recovery technique with electron-spin-echo detection in order to study the non-resonant cross-relaxation of Ni2+-VLi with a faster relaxers—the exchange-coupled clusters of Ni2+ ions. An analysis of the results revealed a very high relaxation rate in non-Kramers doublet of LiF:Ni2+. The effect of a magnetic field on the spin-lattice relaxation of Ni2+ has estimated by comparing the results obtained for non-Kramers doublet around zero-magnetic field and for resonance at 394 mT (X-band microwave frequency). 18. Branes and integrable lattice models NASA Astrophysics Data System (ADS) Yagi, Junya 2017-01-01 This is a brief review of my work on the correspondence between four-dimensional 𝒩 = 1 supersymmetric field theories realized by brane tilings and two-dimensional integrable lattice models. I explain how to construct integrable lattice models from extended operators in partially topological quantum field theories, and elucidate the correspondence as an application of this construction. 19. Static magnetic susceptibility, crystal field and exchange interactions in rare earth titanate pyrochlores. PubMed Malkin, B Z; Lummen, T T A; van Loosdrecht, P H M; Dhalenne, G; Zakirov, A R 2010-07-14 The experimental temperature dependence (T = 2-300 K) of single crystal bulk and site susceptibilities of rare earth titanate pyrochlores R(2)Ti(2)O(7) (R = Sm, Eu, Gd, Tb, Dy, Ho, Er, Yb) is analyzed in the framework of crystal field theory and a mean field approximation. Analytical expressions for the site and bulk susceptibilities of the pyrochlore lattice are derived taking into account long range dipole-dipole interactions and anisotropic exchange interactions between the nearest neighbor rare earth ions. The sets of crystal field parameters and anisotropic exchange coupling constants have been determined and their variations along the lanthanide series are discussed. 20. Spin-lattice coupling in uranium dioxide probed by magnetostriction measurements at high magnetic fields (P08358-E001-PF) SciTech Connect Gofryk, K.; Jaime, M. 2014-12-01 Our preliminary magnetostriction measurements have already shown a strong interplay of lattice dynamic and magnetism in both antiferromagnetic and paramagnetic states, and give unambiguous evidence of strong spin- phonon coupling in uranium dioxide. Further studies are planned to address the puzzling behavior of UO2 in magnetic and paramagnetic states and details of the spin-phonon coupling. 1. Optimal lattice-structured materials NASA Astrophysics Data System (ADS) Messner, Mark C. 2016-11-01 This work describes a method for optimizing the mesostructure of lattice-structured materials. These materials are periodic arrays of slender members resembling efficient, lightweight macroscale structures like bridges and frame buildings. Current additive manufacturing technologies can assemble lattice structures with length scales ranging from nanometers to millimeters. Previous work demonstrates that lattice materials have excellent stiffness- and strength-to-weight scaling, outperforming natural materials. However, there are currently no methods for producing optimal mesostructures that consider the full space of possible 3D lattice topologies. The inverse homogenization approach for optimizing the periodic structure of lattice materials requires a parameterized, homogenized material model describing the response of an arbitrary structure. This work develops such a model, starting with a method for describing the long-wavelength, macroscale deformation of an arbitrary lattice. The work combines the homogenized model with a parameterized description of the total design space to generate a parameterized model. Finally, the work describes an optimization method capable of producing optimal mesostructures. Several examples demonstrate the optimization method. One of these examples produces an elastically isotropic, maximally stiff structure, here called the isotruss, that arguably outperforms the anisotropic octet truss topology. 2. Optimal lattice-structured materials DOE PAGES Messner, Mark C. 2016-07-09 This paper describes a method for optimizing the mesostructure of lattice-structured materials. These materials are periodic arrays of slender members resembling efficient, lightweight macroscale structures like bridges and frame buildings. Current additive manufacturing technologies can assemble lattice structures with length scales ranging from nanometers to millimeters. Previous work demonstrates that lattice materials have excellent stiffness- and strength-to-weight scaling, outperforming natural materials. However, there are currently no methods for producing optimal mesostructures that consider the full space of possible 3D lattice topologies. The inverse homogenization approach for optimizing the periodic structure of lattice materials requires a parameterized, homogenized material model describingmore » the response of an arbitrary structure. This work develops such a model, starting with a method for describing the long-wavelength, macroscale deformation of an arbitrary lattice. The work combines the homogenized model with a parameterized description of the total design space to generate a parameterized model. Finally, the work describes an optimization method capable of producing optimal mesostructures. Several examples demonstrate the optimization method. One of these examples produces an elastically isotropic, maximally stiff structure, here called the isotruss, that arguably outperforms the anisotropic octet truss topology.« less 3. Optimal lattice-structured materials SciTech Connect Messner, Mark C. 2016-07-09 This paper describes a method for optimizing the mesostructure of lattice-structured materials. These materials are periodic arrays of slender members resembling efficient, lightweight macroscale structures like bridges and frame buildings. Current additive manufacturing technologies can assemble lattice structures with length scales ranging from nanometers to millimeters. Previous work demonstrates that lattice materials have excellent stiffness- and strength-to-weight scaling, outperforming natural materials. However, there are currently no methods for producing optimal mesostructures that consider the full space of possible 3D lattice topologies. The inverse homogenization approach for optimizing the periodic structure of lattice materials requires a parameterized, homogenized material model describing the response of an arbitrary structure. This work develops such a model, starting with a method for describing the long-wavelength, macroscale deformation of an arbitrary lattice. The work combines the homogenized model with a parameterized description of the total design space to generate a parameterized model. Finally, the work describes an optimization method capable of producing optimal mesostructures. Several examples demonstrate the optimization method. One of these examples produces an elastically isotropic, maximally stiff structure, here called the isotruss, that arguably outperforms the anisotropic octet truss topology. 4. Emergence of string valence-bond-solid state in the frustrated J1-J2 transverse field Ising model on the square lattice NASA Astrophysics Data System (ADS) Sadrzadeh, M.; Haghshenas, R.; Jahromi, S. S.; Langari, A. 2016-12-01 We investigate the ground-state nature of the transverse field Ising model on the J1-J2 square lattice at the highly frustrated point J2/J1=0.5 . At zero field, the model has an exponentially large degenerate classical ground state, which can be affected by quantum fluctuations for nonzero field toward a unique quantum ground state. We consider two types of quantum fluctuations, harmonic ones by using linear spin-wave theory (LSWT) with single-spin-flip excitations above a long-range magnetically ordered background and anharmonic fluctuations, by employing a cluster-operator approach (COA) with multispin cluster-type fluctuations above a nonmagnetic cluster-ordered background. Our findings reveal that the harmonic fluctuations of LSWT fail to lift the extensive degeneracy as well as signaling a violation of the Hellmann-Feynman theorem. However, the string-type anharmonic fluctuations of COA are able to lift the degeneracy toward a string valence-bond-solid (VBS) state, which is obtained from an effective theory consistent with the Hellmann-Feynman theorem as well. Our results are further confirmed by implementing numerical tree tensor network simulation. The emergent nonmagnetic string VBS phase is gapped and breaks lattice rotational symmetry with only twofold degeneracy, which bears a continuous quantum phase transition at Γ /J1≅0.50 to the quantum paramagnet phase of high fields. The critical behavior is characterized by ν ≅1.0 and γ ≅0.33 exponents. 5. Field induced domain switching as the origin of anomalous lattice strain along non-polar direction in rhombohedral BiScO{sub 3}-PbTiO{sub 3} close to the morphotropic phase boundary SciTech Connect Lalitha, K. V.; Ranjan, Rajeev; Fancher, Chris M.; Jones, Jacob L. 2015-08-03 The lattice strain and domain switching behavior of xBiScO{sub 3}–(1-x)PbTiO{sub 3} (x = 0.40) was investigated as a function of cyclic field and grain orientation by in situ X-ray diffraction during application of electric fields. The electric field induced 200 lattice strain was measured to be five times larger than the 111 lattice strain in pseudorhombohedral xBiScO{sub 3}–(1-x)PbTiO{sub 3} (x = 0.40). It is shown that the anomalous 200 lattice strain is not an intrinsic phenomenon, but arises primarily due to stress associated with the reorientation of the 111 domains in dense polycrystalline ceramic. 6. The bulk, surface and corner free energies of the square lattice Ising model NASA Astrophysics Data System (ADS) Baxter, R. J. 2017-01-01 We use Kaufman’s spinor method to calculate the bulk, surface and corner free energies {f}{{b}},{f}{{s}},{f}{{s}}\\prime ,{f}{{c}} of the anisotropic square lattice zero-field Ising model for the ordered ferromagnetic case. For {f}{{b}},{f}{{s}},{f}{{s}}\\prime our results of course agree with the early work of Onsager, McCoy and Wu. We also find agreement with the conjectures made by Vernier and Jacobsen (VJ) for the isotropic case. We note that the corner free energy f c depends only on the elliptic modulus k that enters the working, and not on the argument v, which means that VJ’s conjecture applies for the full anisotropic model. The only aspect of this paper that is new is the actual derivation of f c, but by reporting all four free energies together we can see interesting structures linking them. 7. Short-range +/-J interaction Ising spin glass in a transverse field on a Bethe lattice: a quantum-spherical approach NASA Astrophysics Data System (ADS) Kope, T. K.; Usadel, K. D. 2006-02-01 We consider the short-range interaction disordered quantum Ising model with symmetric binary +/-J bond distribution on the Bethe lattice (with coordination number z). The system exhibits quantum phase transition separating the spin glass and disordered phases where the quantum effect are regulated by a param- eter describing the transverse field. By introducing a mapping of the quantum Hamiltonian of the model onto a soft-spin action we consider it truncated version in a form of the solvable quantized spherical model. Quantum dynamics is examined via various correlation functions on the infinite tree which are evaluated in a closed form. 8. Anisotropic magnetic and superconducting properties of aligned weak-ferromagnetic superconductor RuSr2GdCu2O8 NASA Astrophysics Data System (ADS) Ku, H. C.; Chang, B. C.; Hsu, C. H.; Chen, Y. F.; Tai, M. F. 2009-03-01 The RuSr2GdCu2O8 Ru-1212 cuprate is a weak-ferromagnetic superconductor with a magnetic ordering of Ru moments at TN(Ru) = 131 K, a superconducting transition in the CuO2 layers at Tc = 56 K, and a low temperature Gd antiferromagnetic ordering at TN(Gd) = 2.5 K. The c-axis aligned powder can be achieved at room temperature using the field-rotation method where the tetragonal c-axis is perpendicular to the aligned magnetic field Ba and along the rotation axis. The anisotropic temperature dependence of magnetic susceptibility for the aligned powder down to 2 K indicates weak anisotropy with Xc > Xab at room temperature due to strong anisotropic Gd contribution and Xc < Xab below 185 K where strong Ru anisotropic short-range exchange interaction overtakes the Gd contribution. Anisotropic diamagnetic superconducting intragrain shielding signal of aligned microcrystalline powder-in-epoxy below vortex lattice melting temperature at 39 K in 1-G field is much weaker than the intergrain polycrystalline bulk sample signal due to the small grain size (d ~ 1-10 μm), long penetration depth (λab ~ 0.6 μm, λc ~ 2 μm) and the two-dimensional (2D) character of CuO2 layers. 9. 1H-19F spin-lattice relaxation spectroscopy: proton tunnelling in the hydrogen bond studied by field-cycling NMR. PubMed Noble, D L; Aibout, A; Horsewill, A J 2009-12-01 Proton tunnelling in the hydrogen bonds of two fluorine substituted benzoic acid dimers has been investigated using field-cycling NMR relaxometry. The close proximity of the (19)F nuclei to the hydrogen bond protons introduces heteronuclear (19)F-(1)H dipolar interactions into the spin-lattice relaxation processes. This renders the (1)H magnetisation-recovery biexponential and introduces multiple spectral density components into the relaxation matrix characterised by frequencies that are sums and differences of the (19)F and (1)H Larmor frequencies. Using field-cycling NMR pulse sequences that measure the spin-lattice relaxation and cross-relaxation rates we demonstrate how some of these multiple spectral density components can be separately resolved. This leads to an accurate determination of the correlation times that characterise the proton tunnelling motion. A broad spectrum of relaxation behaviour is illustrated and explored in the chosen samples and the investigation is used to explore the theory and practise of field-cycling NMR relaxometry in cases where heteronuclear interactions are significant. 10. An investigation of meson spectroscopy on isotropic clover lattices at the SU(3) flavor-symmetric point SciTech Connect Richards, David G.; Orginos, Konstantinos 2014-06-23 We present an investigation of the excited meson spectrum at the N_f= 3 point obtained on isotropic clover lattices with a plaquette Wilson gauge action, and a NP-improved clover fermion action, at a lattice spacing of a \\simeq 0.08 fm, and compare with corresponding calculations on an anisotropic lattice at fine temporal lattice spacing but a spatial lattice spacing of a_s \\simeq 0.125 fm. The methodology adopted follows that employed in the calculation of the spectrum on anisotropic lattices, and we test the efficacy of that approach for isotropic lattices. In particular, we explore the extent to which rotational symmetry for predominantly single-hadron states is realized. By comparison of the energy levels with that obtained using the anisotropic lattice, we obtain an indication of discretization uncertainties in the single-hadron spectrum. 11. Magnetization of anisotropic Type II superconductors SciTech Connect Mints, R.G. 1989-04-10 Peculiarities of magnetization of anisotropic type II superconductors are of considerable interest in view of the discovery of high-T/sub c/ superconductors characterized by strongly asymmetric layered structure. Specifics of the penetration of magnetic flux into an anisotropic type II superconductor were discussed in the literature. This analysis gave the distribution of induction in an isolated vortex, its energy, and critical magnetic field H/sub c1/. However, the magnetization curve of anisotropic superconductors was not considered. This paper deals with the magnetic moment of uniaxial London superconductor in the interval H/sub c1/ /le/ H/sub 0/ << H/sub c2/, where H/sub 0/ is the external magnetic field strength. 12. Gravitational stresses in anisotropic rock masses USGS Publications Warehouse Amadei, B.; Savage, W.Z.; Swolfs, H.S. 1987-01-01 This paper presents closed-form solutions for the stress field induced by gravity in anisotropic rock masses. These rocks are assumed to be laterally restrained and are modelled as a homogeneous, orthotropic or transversely isotropic, linearly elastic material. The analysis, constrained by the thermodynamic requirement that strain energy be positive definite, gives the following important result: inclusion of anisotropy broadens the range of permissible values of gravity-induced horizontal stresses. In fact, for some ranges of anisotropic rock properties, it is thermodynamically admissible for gravity-induced horizontal stresses to exceed the vertical stress component; this is not possible for the classical isotropic solution. Specific examples are presented to explore the nature of the gravity-induced stress field in anisotropic rocks and its dependence on the type, degree and orientation of anisotropy with respect to the horizontal ground surface. ?? 1987. 13. Lattice harmonics expansion revisited NASA Astrophysics Data System (ADS) Kontrym-Sznajd, G.; Holas, A. 2017-04-01 The main subject of the work is to provide the most effective way of determining the expansion of some quantities into orthogonal polynomials, when these quantities are known only along some limited number of sampling directions. By comparing the commonly used Houston method with the method based on the orthogonality relation, some relationships, which define the applicability and correctness of these methods, are demonstrated. They are verified for various sets of sampling directions applicable for expanding quantities having the full symmetry of the Brillouin zone of cubic and non-cubic lattices. All results clearly show that the Houston method is always better than the orthogonality-relation one. For the cubic symmetry we present a few sets of special directions (SDs) showing how their construction and, next, a proper application depend on the choice of various sets of lattice harmonics. SDs are important mainly for experimentalists who want to reconstruct anisotropic quantities from their measurements, performed at a limited number of sampling directions. 14. Finite-size effects at critical points with anisotropic correlations: Phenomenological scaling theory and Monte Carlo simulations NASA Astrophysics Data System (ADS) Binder, Kurt; Wang, Jian-Sheng 1989-04-01 Various thermal equilibrium and nonequilibrium phase transitions exist where the correlation lengths in different lattice directions diverge with different exponents v ‖, v ⊥: uniaxial Lifshitz points, the Kawasaki spin exchange model driven by an electric field, etc. An extension of finite-size scaling concepts to such anisotropic situations is proposed, including a discussion of (generalized) rectangular geometries, with linear dimension L ‖ in the special direction and linear dimensions L ⊥ in all other directions. The related shape effects for L ‖≠ L ⊥ but isotropic critical points are also discussed. Particular attention is paid to the case where the generalized hyperscaling relation v ‖+( d-1) v ⊥=γ+2 β does not hold. As a test of these ideas, a Monte Carlo simulation study for shape effects at isotropic critical point in the two-dimensional Ising model is presented, considering subsystems of a 1024x1024 square lattice at criticality. 15. Periodic ordering of clusters and stripes in a two-dimensional lattice model. I. Ground state, mean-field phase diagram and structure of the disordered phases NASA Astrophysics Data System (ADS) PÈ©kalski, J.; Ciach, A.; Almarza, N. G. 2014-03-01 The short-range attraction and long-range repulsion between nanoparticles or macromolecules can lead to spontaneous pattern formation on solid surfaces, fluid interfaces, or membranes. In order to study the self-assembly in such systems we consider a triangular lattice model with nearest-neighbor attraction and third-neighbor repulsion. At the ground state of the model (T = 0) the lattice is empty for small values of the chemical potential μ, and fully occupied for large μ. For intermediate values of μ periodically distributed clusters, bubbles, or stripes appear if the repulsion is sufficiently strong. At the phase coexistences between the vacuum and the ordered cluster phases and between the cluster and the lamellar (stripe) phases the entropy per site does not vanish. As a consequence of this ground state degeneracy, disordered fluid phases consisting of clusters or stripes are stable, and the surface tension vanishes. For T > 0 we construct the phase diagram in the mean-field approximation and calculate the correlation function in the self-consistent Brazovskii-type field theory. 16. Development of an immersed boundary-phase field-lattice Boltzmann method for Neumann boundary condition to study contact line dynamics NASA Astrophysics Data System (ADS) Shao, J. Y.; Shu, C.; Chew, Y. T. 2013-02-01 The implementation of Neumann boundary condition in the framework of immersed boundary method (IBM) is presented in this paper to simulate contact line dynamics using a phase field-lattice Boltzmann method. Immersed boundary method [10] is known as an efficient algorithm for modelling fluid-solid interaction. Abundance of prominent works have been devoted to refine IBM [1,11,12]. However, they are mainly restricted to problems with Dirichlet boundary condition. Research that implements the Neumann boundary condition in IBM is very limited to the best of our knowledge. This deficiency significantly limits the application of IBM in computational fluid dynamics (CFD) since physical phenomena associated with Neumann boundary conditions are extremely diverse. The difficulty is attributed to the fact that implementation of Neumann boundary condition is much more complex than that of Dirichlet boundary condition. In the present work, we initiate the first endeavour to implement Neumann boundary condition in IBM with assistance of its physical interpretation rather than simple mathematical manipulation. Concretely speaking, rooted from physical conservation law, the Neumann boundary condition is considered as contribution of flux from the boundary to its relevant physical parameter in a control volume. Moreover, the link between the flux and its corresponding flow field variable is directly manipulated through the immersed boundary concept. In this way, the Neumann boundary conditions can be implemented in IBM. The developed method is applied together with phase field-lattice Boltzmann method to study contact line dynamics. The phase field method [27,39], which becomes increasingly popular in multiphase flow simulation, can efficiently capture complex interface topology and naturally resolve the contact line singularity. Meanwhile, the lattice Boltzmann method is known as an alternative to model fluid dynamics and holds good prospect to simulate multiphase flows with 17. Anisotropic intermediate valence in Yb2M3Ga9 (M = Rh, Ir) SciTech Connect Christianson, A.D.; Lawrence, J.M.; Lobos, A.M.; Aligia, A.A.; Bauer, E.D.; Moreno, N.O.; Booth, C.H.; Goremychkin, E.A.; Sarrao, J.L.; Thompson, J.D.; Batista, C.D.; Trouw, F.R.; Hehlen, M.P. 2005-04-26 The intermediate valence compounds Yb{sub 2}M{sub 3}Ga{sub 9} (M = Rh, Ir) exhibit an anisotropic magnetic susceptibility. We report measurements of the temperature dependence of the 4f occupation number, n{sub f}(T), for Yb{sub 2}M{sub 3}Ga{sub 9} as well as the magnetic inelastic neutron scattering spectrum S{sub mag}({Delta}E) at 12 and 300 K for Yb{sub 2}Rh{sub 3}Ga{sub 9}. Both n{sub f}(T) and S{sub mag}({Delta}E) were calculated for the Anderson impurity model with crystal field terms within an approach based on the non-crossing approximation. These results corroborate the importance of crystal field effects in these materials; they also suggest that Anderson lattice effects are important to the physics of Yb{sub 2}M{sub 3}Ga{sub 9}. 18. Anisotropic Elliott-Yafet theory and application to KC8 potassium intercalated graphite NASA Astrophysics Data System (ADS) Márkus, Bence G.; Szolnoki, Lénárd; Iván, Dávid; Dóra, Balázs; Szirmai, Péter; Náfrádi, Bálint; Forró, László; Simon, Ferenc 2016-12-01 We report Electron Spin Resonance (ESR) measurements on stage-I potassium intercalated graphite (KC_8). Angular dependent measurements show that the spin-lattice relaxation time is longer when the magnetic field is perpendicular to the graphene layer as compared to when the magnetic field is in the plane. This anisotropy is analyzed in the framework of the Elliott-Yafet theory of spin-relaxation in metals. The analysis considers an anisotropic spin-orbit Hamiltonian and the first order perturbative treatment of Elliott is reproduced for this model Hamiltonian. The result provides an experimental input for the first-principles theories of spin-orbit interaction in layered carbon and thus to a better understanding of spin-relaxation phenomena in graphene and in other layered materials as well. 19. Charge-regulation phase transition on surface lattices of titratable sites adjacent to electrolyte solutions: An analog of the Ising antiferromagnet in a magnetic field. PubMed Shore, Joel D; Thurston, George M 2015-12-01 We report a charge-patterning phase transition on two-dimensional square lattices of titratable sites, here regarded as protonation sites, placed in a low-dielectric medium just below the planar interface between this medium and a salt solution. We calculate the work-of-charging matrix of the lattice with use of a linear Debye-Hückel model, as input to a grand-canonical partition function for the distribution of occupancy patterns. For a large range of parameter values, this model exhibits an approximate inverse cubic power-law decrease of the voltage produced by an individual charge, as a function of its in-lattice separation from neighboring titratable sites. Thus, the charge coupling voltage biases the local probabilities of proton binding as a function of the occupancy of sites for many neighbors beyond the nearest ones. We find that even in the presence of these longer-range interactions, the site couplings give rise to a phase transition in which the site occupancies exhibit an alternating, checkerboard pattern that is an analog of antiferromagnetic ordering. The overall strength W of this canonical charge coupling voltage, per unit charge, is a function of the Debye length, the charge depth, the Bjerrum length, and the dielectric coefficients of the medium and the solvent. The alternating occupancy transition occurs above a curve of thermodynamic critical points in the (pH-pK,W) plane, the curve representing a charge-regulation analog of variation of the Néel temperature of an Ising antiferromagnet as a function of an applied, uniform magnetic field. The analog of a uniform magnetic field in the antiferromagnet problem is a combination of pH-pK and W, and 1/W is the analog of the temperature in the antiferromagnet problem. We use Monte Carlo simulations to study the occupancy patterns of the titratable sites, including interactions out to the 37th nearest-neighbor category (a distance of √74 lattice constants), first validating simulations through 20. Charge-regulation phase transition on surface lattices of titratable sites adjacent to electrolyte solutions: An analog of the Ising antiferromagnet in a magnetic field NASA Astrophysics Data System (ADS) Shore, Joel D.; Thurston, George M. 2015-12-01 We report a charge-patterning phase transition on two-dimensional square lattices of titratable sites, here regarded as protonation sites, placed in a low-dielectric medium just below the planar interface between this medium and a salt solution. We calculate the work-of-charging matrix of the lattice with use of a linear Debye-Hückel model, as input to a grand-canonical partition function for the distribution of occupancy patterns. For a large range of parameter values, this model exhibits an approximate inverse cubic power-law decrease of the voltage produced by an individual charge, as a function of its in-lattice separation from neighboring titratable sites. Thus, the charge coupling voltage biases the local probabilities of proton binding as a function of the occupancy of sites for many neighbors beyond the nearest ones. We find that even in the presence of these longer-range interactions, the site couplings give rise to a phase transition in which the site occupancies exhibit an alternating, checkerboard pattern that is an analog of antiferromagnetic ordering. The overall strength W of this canonical charge coupling voltage, per unit charge, is a function of the Debye length, the charge depth, the Bjerrum length, and the dielectric coefficients of the medium and the solvent. The alternating occupancy transition occurs above a curve of thermodynamic critical points in the (p H-p K ,W ) plane, the curve representing a charge-regulation analog of variation of the Néel temperature of an Ising antiferromagnet as a function of an applied, uniform magnetic field. The analog of a uniform magnetic field in the antiferromagnet problem is a combination of p H-p K and W , and 1 /W is the analog of the temperature in the antiferromagnet problem. We use Monte Carlo simulations to study the occupancy patterns of the titratable sites, including interactions out to the 37th nearest-neighbor category (a distance of √{74 } lattice constants), first validating simulations 1. Self-consistent Bogoliubov-de Gennes theory of the vortex lattice state in a two-dimensional strongly type-II superconductor at high magnetic fields NASA Astrophysics Data System (ADS) Zhuravlev, Vladimir; Duan, Wenye; Maniv, Tsofar 2017-01-01 A self-consistent Bogoliubov-de Gennes theory of the vortex lattice state in a 2D strong type-II superconductor at high magnetic fields reveals a novel quantum mixed state around the semiclassical Hc 2, characterized by a well-defined Landau-Bloch band structure in the quasiparticle spectrum and suppressed order-parameter amplitude, which sharply crossover into the well-known semiclassical (Helfand-Werthamer) results upon decreasing magnetic field. Application to the 2D superconducting state observed recently on the surface of the topological insulator Sb2Te3 accounts well for the experimental data, revealing a strong type-II superconductor, with unusually low carrier density and very small cyclotron mass, which can be realized only in the strong coupling superconductor limit. 2. Magnetic field induced anisotropy of 139La spin-lattice relaxation rates in stripe ordered La1.875Ba0.125CuO4 DOE PAGES S. -H. Baek; Gu, G. D.; Utz, Y.; ... 2015-10-26 We report 139La nuclear magnetic resonance studies performed on a La1.875Ba0.125CuO4 single crystal. The data show that the structural phase transitions (high-temperature tetragonal → low-temperature orthorhombic → low-temperature tetragonal phase) are of the displacive type in this material. The 139La spin-lattice relaxation rate T–11 sharply upturns at the charge-ordering temperature TCO = 54 K, indicating that charge order triggers the slowing down of spin fluctuations. Detailed temperature and field dependencies of the T–11 below the spin-ordering temperature TSO=40 K reveal the development of enhanced spin fluctuations in the spin-ordered state for H ∥ [001], which are completely suppressed for largemore » fields along the CuO2 planes. Lastly, our results shed light on the unusual spin fluctuations in the charge and spin stripe ordered lanthanum cuprates.« less 3. A combined dislocation fan-finite element (DF-FE) method for stress field simulation of dislocations emerging at the free surfaces of 3D elastically anisotropic crystals NASA Astrophysics Data System (ADS) Balusu, K.; Huang, H. 2017-04-01 A combined dislocation fan-finite element (DF-FE) method is presented for efficient and accurate simulation of dislocation nodal forces in 3D elastically anisotropic crystals with dislocations intersecting the free surfaces. The finite domain problem is decomposed into half-spaces with singular traction stresses, an infinite domain, and a finite domain with non-singular traction stresses. As such, the singular and non-singular parts of the traction stresses are addressed separately; the dislocation fan (DF) method is introduced to balance the singular traction stresses in the half-spaces while the finite element method (FEM) is employed to enforce the non-singular boundary conditions. The accuracy and efficiency of the DF method is demonstrated using a simple isotropic test case, by comparing it with the analytical solution as well as the FEM solution. The DF-FE method is subsequently used for calculating the dislocation nodal forces in a finite elastically anisotropic crystal, which produces dislocation nodal forces that converge rapidly with increasing mesh resolutions. In comparison, the FEM solution fails to converge, especially for nodes closer to the surfaces. 4. An Anisotropic Multiphysics Model for Intervertebral Disk PubMed Central Gao, Xin; Zhu, Qiaoqiao; Gu, Weiyong 2016-01-01 Intervertebral disk (IVD) is the largest avascular structure in human body, consisting of three types of charged hydrated soft tissues. Its mechanical behavior is nonlinear and anisotropic, due mainly to nonlinear interactions among different constituents within tissues. In this study, a more realistic anisotropic multiphysics model was developed based on the continuum mixture theory and employed to characterize the couplings of multiple physical fields in the IVD. Numerical simulations demonstrate that this model is capable of systematically predicting the mechanical and electrochemical signals within the disk under various loading conditions, which is essential in understanding the mechanobiology of IVD. PMID:27099402 5. On cracking of charged anisotropic polytropes NASA Astrophysics Data System (ADS) Azam, M.; Mardan, S. A. 2017-01-01 Recently in [1], the role of electromagnetic field on the cracking of spherical polytropes has been investigated without perturbing charge parameter explicitly. In this study, we have examined the occurrence of cracking of anisotropic spherical polytropes through perturbing parameters like anisotropic pressure, energy density and charge. We consider two different types of polytropes in this study. We discuss the occurrence of cracking in two different ways (i) by perturbing polytropic constant, anisotropy and charge parameter (ii) by perturbing polytropic index, anisotropy and charge parameter for each case. We conclude that cracking appears for a wide range of parameters in both cases. Also, our results are reduced to [2] in the absence of charge. 6. Anisotropic contrast optical microscope. PubMed Peev, D; Hofmann, T; Kananizadeh, N; Beeram, S; Rodriguez, E; Wimer, S; Rodenhausen, K B; Herzinger, C M; Kasputis, T; Pfaunmiller, E; Nguyen, A; Korlacki, R; Pannier, A; Li, Y; Schubert, E; Hage, D; Schubert, M 2016-11-01 An optical microscope is described that reveals contrast in the Mueller matrix images of a thin, transparent, or semi-transparent specimen located within an anisotropic object plane (anisotropic filter). The specimen changes the anisotropy of the filter and thereby produces contrast within the Mueller matrix images. Here we use an anisotropic filter composed of a semi-transparent, nanostructured thin film with sub-wavelength thickness placed within the object plane. The sample is illuminated as in common optical microscopy but the light is modulated in its polarization using combinations of linear polarizers and phase plate (compensator) to control and analyze the state of polarization. Direct generalized ellipsometry data analysis approaches permit extraction of fundamental Mueller matrix object plane images dispensing with the need of Fourier expansion methods. Generalized ellipsometry model approaches are used for quantitative image analyses. These images are obtained from sets of multiple images obtained under various polarizer, analyzer, and compensator settings. Up to 16 independent Mueller matrix images can be obtained, while our current setup is limited to 11 images normalized by the unpolarized intensity. We demonstrate the anisotropic contrast optical microscope by measuring lithographically defined micro-patterned anisotropic filters, and we quantify the adsorption of an organic self-assembled monolayer film onto the anisotropic filter. Comparison with an isotropic glass slide demonstrates the image enhancement obtained by our method over microscopy without the use of an anisotropic filter. In our current instrument, we estimate the limit of detection for organic volumetric mass within the object plane of ≈49 fg within ≈7 × 7 μm(2) object surface area. Compared to a quartz crystal microbalance with dissipation instrumentation, where contemporary limits require a total load of ≈500 pg for detection, the instrumentation demonstrated here improves 7. Anisotropic contrast optical microscope NASA Astrophysics Data System (ADS) Peev, D.; Hofmann, T.; Kananizadeh, N.; Beeram, S.; Rodriguez, E.; Wimer, S.; Rodenhausen, K. B.; Herzinger, C. M.; Kasputis, T.; Pfaunmiller, E.; Nguyen, A.; Korlacki, R.; Pannier, A.; Li, Y.; Schubert, E.; Hage, D.; Schubert, M. 2016-11-01 An optical microscope is described that reveals contrast in the Mueller matrix images of a thin, transparent, or semi-transparent specimen located within an anisotropic object plane (anisotropic filter). The specimen changes the anisotropy of the filter and thereby produces contrast within the Mueller matrix images. Here we use an anisotropic filter composed of a semi-transparent, nanostructured thin film with sub-wavelength thickness placed within the object plane. The sample is illuminated as in common optical microscopy but the light is modulated in its polarization using combinations of linear polarizers and phase plate (compensator) to control and analyze the state of polarization. Direct generalized ellipsometry data analysis approaches permit extraction of fundamental Mueller matrix object plane images dispensing with the need of Fourier expansion methods. Generalized ellipsometry model approaches are used for quantitative image analyses. These images are obtained from sets of multiple images obtained under various polarizer, analyzer, and compensator settings. Up to 16 independent Mueller matrix images can be obtained, while our current setup is limited to 11 images normalized by the unpolarized intensity. We demonstrate the anisotropic contrast optical microscope by measuring lithographically defined micro-patterned anisotropic filters, and we quantify the adsorption of an organic self-assembled monolayer film onto the anisotropic filter. Comparison with an isotropic glass slide demonstrates the image enhancement obtained by our method over microscopy without the use of an anisotropic filter. In our current instrument, we estimate the limit of detection for organic volumetric mass within the object plane of ≈49 fg within ≈7 × 7 μm2 object surface area. Compared to a quartz crystal microbalance with dissipation instrumentation, where contemporary limits require a total load of ≈500 pg for detection, the instrumentation demonstrated here improves 8. Quasiparticle equation of state for anisotropic hydrodynamics NASA Astrophysics Data System (ADS) Alqahtani, Mubarak; Nopoush, Mohammad; Strickland, Michael 2015-11-01 We present a new method for imposing a realistic equation of state in anisotropic hydrodynamics. The method relies on the introduction of a single finite-temperature quasiparticle mass which is fit to lattice data. By taking moments of the Boltzmann equation, we obtain a set of coupled partial differential equations which can be used to describe the 3+1-dimensional (3+1d) spacetime evolution of an anisotropic relativistic system. We then specialize to the case of a 0+1d system undergoing boost-invariant Bjorken expansion and subject to the relaxation-time approximation collisional kernel. Using this setup, we compare results obtained using the new quasiparticle equation of state method with those obtained using the standard method for imposing the equation of state in anisotropic hydrodynamics. We demonstrate that the temperature evolution obtained using the two methods is nearly identical and that there are only small differences in the pressure anisotropy. However, we find that there are significant differences in the evolution of the bulk pressure correction. 9. Supersymmetry on the Lattice NASA Astrophysics Data System (ADS) Schaich, David 2016-03-01 Lattice field theory provides a non-perturbative regularization of strongly interacting systems, which has proven crucial to the study of quantum chromodynamics among many other theories. Supersymmetry plays prominent roles in the study of physics beyond the standard model, both as an ingredient in model building and as a tool to improve our understanding of quantum field theory. Attempts to apply lattice techniques to supersymmetric field theories have a long history, but until recently these efforts have generally encountered insurmountable difficulties related to the interplay of supersymmetry with the lattice discretization of spacetime. In recent years these difficulties have been overcome for a class of theories that includes the particularly interesting case of maximally supersymmetric Yang-Mills (N = 4 SYM) in four dimensions, which is a cornerstone of AdS/CFT duality. In combination with computational advances this progress enables practical numerical investigations of N = 4 SYM on the lattice, which can address questions that are difficult or impossible to handle through perturbation theory, AdS/CFT duality, or the conformal bootstrap program. I will briefly review some of the new ideas underlying this recent progress, and present some results from ongoing large-scale numerical calculations, including comparisons with analytic predictions. 10. Lattice gauge theories NASA Astrophysics Data System (ADS) Weisz, Peter; Majumdar, Pushan 2012-03-01 Lattice gauge theory is a formulation of quantum field theory with gauge symmetries on a space-time lattice. This formulation is particularly suitable for describing hadronic phenomena. In this article we review the present status of lattice QCD. We outline some of the computational methods, discuss some phenomenological applications and a variety of non-perturbative topics. The list of references is severely incomplete, the ones we have included are text books or reviews and a few subjectively selected papers. Kronfeld and Quigg (2010) supply a reasonably comprehensive set of QCD references. We apologize for the fact that have not covered many important topics such as QCD at finite density and heavy quark effective theory adequately, and mention some of them only in the last section "In Brief". These topics should be considered in further Scholarpedia articles. 11. Lattice overview SciTech Connect Creutz, M. 1984-01-01 After reviewing some recent developments in supercomputer access, the author discusses a few areas where perturbation theory and lattice gauge simulations make contact. The author concludes with a brief discussion of a deterministic dynamics for the Ising model. This may be useful for numerical studies of nonequilibrium phenomena. 13 references. 12. Realignment of the flux-line lattice by a change in the symmetry of superconductivity in UPt3 PubMed Huxley; Rodiere; Paul; van Dijk N; Cubitt; Flouquet 2000-07-13 In 1957, Abrikosov described how quanta of magnetic flux enter the interior of a bulk type II superconductor. It was subsequently predicted that, in an isotropic superconductor, the repulsive forces between the flux lines would cause them to order in two dimensions, forming a hexagonal lattice. Flux-line lattices with different geometry can also be found in conventional (type II) superconductors; however, the ideal hexagonal lattice structure should always occur when the magnetic field is applied along a hexagonal crystal direction. Here we report measurements of the orientation of the flux-line lattice in the heavy-fermion superconductor UPt3, for this special case. As the temperature is increased, the hexagonal lattice, which is initially aligned along the crystal symmetry directions, realigns itself with the anisotropic superconducting gap. The superconductivity in UPt3 is unusual (even compared to unconventional oxide superconductors) because the superconducting gap has a lower rotational symmetry than the crystal structure. This special feature enables our data to demonstrate clearly the link between the microscopic symmetry of the superconductivity and the mesoscopic physics of the flux-line lattice. Moreover, our observations provide a stringent test of the theoretical description of the unconventional superconductivity in UPt3. 13. Evidence for a Bound $H$ Dibaryon from Lattice QCD SciTech Connect Beane, S. R.; Chang, E.; Detmold, W.; Joo, B.; Lin, H. W.; Luu, T. C.; Orginos, K.; Parreño, A.; Savage, M. J.; Torok, A.; Walker-Loud, A. 2011-04-20 We present evidence for the existence of a bound H-dibaryon, an I = 0, J = 0, s = -2 state with valence quark structure uuddss, at a pion mass of m_{\\pi}$~389 MeV. Extrapolating the results of lattice QCD calculations performed on four ensembles of anisotropic clover gauge-field configurations, with spatial extents of L ~ 2.0, 2.5, 3.0 and 3.9 fm at a spatial lattice spacing of bs ~ 0.123 fm, we find an H-dibaryon bound by B$H\\atop{\\infty}$= 16.6 ± 2.1 ± 4.6 MeV at a pion mass of m$_{\\pi}~ 389 MeV. 14. Dynamic mean field theory for lattice gas models of fluids confined in porous materials: higher order theory based on the Bethe-Peierls and path probability method approximations. PubMed Edison, John R; Monson, Peter A 2014-07-14 Recently we have developed a dynamic mean field theory (DMFT) for lattice gas models of fluids in porous materials [P. A. Monson, J. Chem. Phys. 128(8), 084701 (2008)]. The theory can be used to describe the relaxation processes in the approach to equilibrium or metastable states for fluids in pores and is especially useful for studying system exhibiting adsorption/desorption hysteresis. In this paper we discuss the extension of the theory to higher order by means of the path probability method (PPM) of Kikuchi and co-workers. We show that this leads to a treatment of the dynamics that is consistent with thermodynamics coming from the Bethe-Peierls or Quasi-Chemical approximation for the equilibrium or metastable equilibrium states of the lattice model. We compare the results from the PPM with those from DMFT and from dynamic Monte Carlo simulations. We find that the predictions from PPM are qualitatively similar to those from DMFT but give somewhat improved quantitative accuracy, in part due to the superior treatment of the underlying thermodynamics. This comes at the cost of greater computational expense associated with the larger number of equations that must be solved. 15. Dynamic mean field theory for lattice gas models of fluids confined in porous materials: Higher order theory based on the Bethe-Peierls and path probability method approximations SciTech Connect Edison, John R.; Monson, Peter A. 2014-07-14 Recently we have developed a dynamic mean field theory (DMFT) for lattice gas models of fluids in porous materials [P. A. Monson, J. Chem. Phys. 128(8), 084701 (2008)]. The theory can be used to describe the relaxation processes in the approach to equilibrium or metastable states for fluids in pores and is especially useful for studying system exhibiting adsorption/desorption hysteresis. In this paper we discuss the extension of the theory to higher order by means of the path probability method (PPM) of Kikuchi and co-workers. We show that this leads to a treatment of the dynamics that is consistent with thermodynamics coming from the Bethe-Peierls or Quasi-Chemical approximation for the equilibrium or metastable equilibrium states of the lattice model. We compare the results from the PPM with those from DMFT and from dynamic Monte Carlo simulations. We find that the predictions from PPM are qualitatively similar to those from DMFT but give somewhat improved quantitative accuracy, in part due to the superior treatment of the underlying thermodynamics. This comes at the cost of greater computational expense associated with the larger number of equations that must be solved. 16. Anisotropic eddy viscosity models NASA Technical Reports Server (NTRS) Carati, D.; Cabot, W. 1996-01-01 A general discussion on the structure of the eddy viscosity tensor in anisotropic flows is presented. The systematic use of tensor symmetries and flow symmetries is shown to reduce drastically the number of independent parameters needed to describe the rank 4 eddy viscosity tensor. The possibility of using Onsager symmetries for simplifying further the eddy viscosity is discussed explicitly for the axisymmetric geometry. 17. Lattice QCD: A Brief Introduction NASA Astrophysics Data System (ADS) Meyer, H. B. A general introduction to lattice QCD is given. The reader is assumed to have some basic familiarity with the path integral representation of quantum field theory. Emphasis is placed on showing that the lattice regularization provides a robust conceptual and computational framework within quantum field theory. The goal is to provide a useful overview, with many references pointing to the following chapters and to freely available lecture series for more in-depth treatments of specifics topics. 18. Anisotropic microstructure near the sun NASA Astrophysics Data System (ADS) Coles, W. A.; Grall, R. R.; Spangler, S. R.; Sakurai, T.; Harmon, J. K. 1996-07-01 Radio scattering observations provide a means of measuring a two-dimensional projection of the three-dimensional spatial spectrum of electron density, i.e., in the plane perpendicular to the line of sight. Earlier observations have shown that the microstructure at scales of the order of 10 km becomes highly field-aligned inside of 10 Rsolar [Armstrong et al., 1990]. Earlier work has also shown that density fluctuations at scales larger than 1000 km have a Kolmogorov spectrum, whereas the smaller scale structure has a flatter spectrum and is considerably enhanced above the Kolmogorov background'' [Coles et al., 1991]. Here we present new observations made during 1990 and 1992. These confirm the earlier work, which was restricted to one source on a few days, but they suggest that the anisotropy changes abruptly near 6 Rsolar which was not clear in the earlier data. The axial ratio measurements are shown on Figure 1 below. The new observations were made with a more uniform sampling of the spatial plane. They show that contours of constant correlation are elliptical. This is apparently inconsistent with the spatial correlation of the ISEE-3 magnetic field which shows a Maltese Cross'' shape [Matthaeus et al., 1990]. However this inconsistency may be only apparent: the magnetic field and density correlations need not have the same shape; the scale of the magnetic field correlations is at least 4 orders of magnitude larger; they are much further from the sun; and they are point measurements whereas ours are path-integrated. We also made two simultaneous measurements, at 10 Rsolar, of the anisotropy on scales of 200 to 4000 km. Significant anisotropy was seen on the smaller scales, but the larger scale structure was essentially isotropic. This suggests that the process responsible for the anisotropic microstructure is independent of the larger scale isotropic turbulence. It is then tempting to speculate that the damping of this anisotropic process inside of 6 Rsolar 19. Hemispherical anisotropic patterns of the Earth’s inner core PubMed Central Mattesini, Maurizio 2010-01-01 It has been shown that the Earth’s inner core has an axisymmetric anisotropic structure with seismic waves traveling ∼3% faster along polar paths than along equatorial directions. Hemispherical anisotropic patterns of the solid Earth’s core are rather complex, and the commonly used hexagonal-close-packed iron phase might be insufficient to account for seismological observations. We show that the data we collected are in good agreement with the presence of two anisotropically specular east and west core hemispheres. The detected travel-time anomalies can only be disclosed by a lattice-preferred orientation of a body-centered-cubic iron aggregate, having a fraction of their [111] crystal axes parallel to the Earth’s rotation axis. This is compelling evidence for the presence of a body-centered-cubic Fe phase at the top of the Earth’s inner core. PMID:20457937 20. Hemispherical anisotropic patterns of the Earth's inner core. PubMed Mattesini, Maurizio; Belonoshko, Anatoly B; Buforn, Elisa; Ramírez, María; Simak, Sergei I; Udías, Agustín; Mao, Ho-Kwang; Ahuja, Rajeev 2010-05-25 It has been shown that the Earth's inner core has an axisymmetric anisotropic structure with seismic waves traveling approximately 3% faster along polar paths than along equatorial directions. Hemispherical anisotropic patterns of the solid Earth's core are rather complex, and the commonly used hexagonal-close-packed iron phase might be insufficient to account for seismological observations. We show that the data we collected are in good agreement with the presence of two anisotropically specular east and west core hemispheres. The detected travel-time anomalies can only be disclosed by a lattice-preferred orientation of a body-centered-cubic iron aggregate, having a fraction of their [111] crystal axes parallel to the Earth's rotation axis. This is compelling evidence for the presence of a body-centered-cubic Fe phase at the top of the Earth's inner core. 1. Self-force on dislocation segments in anisotropic crystals. PubMed Fitzgerald, S P; Aubry, S 2010-07-28 A dislocation segment in a crystal experiences a 'self-force', by virtue of the orientation dependence of its elastic energy. If the crystal is elastically isotropic, this force is manifested as a couple acting to rotate the segment toward the lower energy of the pure screw orientation (i.e. acting to align the dislocation line with its Burgers vector). If the crystal is anisotropic, there are additional contributions to the couple, arising from the more complex energy landscape of the lattice itself. These effects can strongly influence the dynamic evolution of dislocation networks, and via their governing role in dislocation multiplication phenomena, control plastic flow in metals. In this paper we develop a model for dislocation self-forces in a general anisotropic crystal, and briefly consider the technologically important example of α-iron, which becomes increasingly anisotropic as the temperature approaches that of the α-γ phase transition at 912 °C. 2. Anisotropy of Skyrmion Lattice in Mn0.9Fe0.1Si probed by magnetic field orientation dependence of the topological Hall effect and magnetoresistance NASA Astrophysics Data System (ADS) Siegfried, Peter; Treglia, Andrew; Bornstein, Alexander; Wolf, Thomas; Lee, Minhyea We report the magnetic field orientation dependence of the topological Hall effect (THE) and magnetoresistance (MR) of Mn0.9Fe0.1Si in the A-phase within the applied magnetic field (H) - temperature (T) phase diagram. In the A-phase a two dimensional Skyrmion lattice is formed in the plane perpendicular to the direction of H, which is responsible for the observed THE signal. At a given T within the A-phase, we investigated the angular dependence of THE and MR at a fixed H to probe the boundaries of the A-phase region. We find the MR signal exhibits a unique H-direction dependence at the entering and exiting of the A-phase, whereas, in the middle H range, i.e. in the core of A-phase, the angular dependence is consistent with what is expected from a perfect 2D Skyrmion lattice. However, THE signals show extreme sensitivity upon entering the A-phase and unexpected angular dependence, yet did not leave any trace through exiting. The discrepancy between the angular dependence of MR and THE signals at the A-phase boundaries indicates a crucial role of Fe impurities as pinning centers for the Skyrmions. We will discuss further our H-orientation dependence of the THE, compared to sweeping H at a fixed angle in Fe doped MnSi. Work at the University of Colorado was supported by the US DOE Basic Energy Sciences under Award No. DE-SC0006888. 3. Bimaterial lattices as thermal adapters and actuators NASA Astrophysics Data System (ADS) Toropova, Marina M.; Steeves, Craig A. 2016-11-01 The goal of this paper is to demonstrate how anisotropic biomaterial lattices can be used in thermal actuation. Compared to other lattices with tailored thermal expansion, the anisotropy of these bimaterial lattices makes them uniquely suitable for use as thermal actuators. Each individual cell, and hence lattices consisting of such cells, can be designed with widely different predetermined coefficients of thermal expansion (CTE) in different directions, enabling complex shape changes appropriate for actuation with either passive or active control. The lattices are composed of planar non-identical cells that each consist of a skewed hexagon surrounding an irregular triangle. The cells and all members of any cell are connected to each other by pins so that they have no rotational constraints and are able to expand or contract freely. In this case, the skew angles of the hexagon and the ratio of the CTEs of the two component materials determine the overall performance of the lattice. At its boundaries, the lattice is connected to substrates by pins and configured such that the CTE between two neighboring lattice vertices coincides with the CTE of the adjacent substrate. Provided the boundary behavior of the lattice is matched to the thermal properties of the substrates, temperature changes in the structure produce thermal strains without producing any corresponding stresses. Such lattices can be used in three different ways: as adaptive elements for stress-free connection of components with different CTEs; for fine tuning of structures; and as thermally driven actuators. In this paper, we demonstrate some concepts for lattice configurations that produce thermally-driven displacements that enable several actuators: a switch, a valve and tweezers. 4. Elimination of spurious lattice fermion solutions and noncompact lattice QCD SciTech Connect Lee, T.D. 1997-09-22 It is well known that the Dirac equation on a discrete hyper-cubic lattice in D dimension has 2{sup D} degenerate solutions. The usual method of removing these spurious solutions encounters difficulties with chiral symmetry when the lattice spacing l {ne} 0, as exemplified by the persistent problem of the pion mass. On the other hand, we recall that in any crystal in nature, all the electrons do move in a lattice and satisfy the Dirac equation; yet there is not a single physical result that has ever been entangled with a spurious fermion solution. Therefore it should not be difficult to eliminate these unphysical elements. On a discrete lattice, particle hop from point to point, whereas in a real crystal the lattice structure in embedded in a continuum and electrons move continuously from lattice cell to lattice cell. In a discrete system, the lattice functions are defined only on individual points (or links as in the case of gauge fields). However, in a crystal the electron state vector is represented by the Bloch wave functions which are continuous functions in {rvec {gamma}}, and herein lies one of the essential differences. 5. Quantum states of charge carriers and longitudinal conductivity in double periodic n-type semiconductor lattice structures in electric field SciTech Connect Perov, A. A. Penyagin, I. V. 2015-07-15 Quantum states of charge carriers in double periodic semiconductor superlattices of n-type quantum dots with Rashba spin–orbit coupling in an electron gas have been calculated in the one-electron approximation in the presence of mutually perpendicular electric and magnetic fields. For these structures in weak constant electric field, the solution to the quasi-classical kinetic Boltzmann equation shows that the states of carriers in magnetic Landau minibands with negative differential conductivity are possible. 6. Vacancies in a 3D-Kitaev model on hyper-honeycomb lattice NASA Astrophysics Data System (ADS) Sreejith, G. J.; Bhattacharjee, Subhro; Moessner, Roderich We study the properties of isolated single and pairs of vacancies in an exactly solvable Kitaev model on a three dimensional hyper-honeycomb lattice. We show that each vacancy in the lattice is associated with a low energy spin like degree of freedom, similar to the case of previously studied honeycomb model. We calculate the contribution from these vacancy spin-moments to the low field magnetization response to a z-directed field. Isolated vacancies in the gapped phase act as free spins. In the gapless phase, these spins interact with the surrounding spin-liquid suppressing the low-field magnetization to 1/√{ ln [ 1 /hz ] }. Pair of vacancies have a sublattice-dependent, anisotropic, spin-liquid mediated interaction with each other. In the gapless phase, interaction between vacancies in the same (opposite) sublattice enhances (suppresses) the low-field magnetization, indicating a ferromagnetic (anti-ferromagnetic) nature. We also show that, unlike vacancies in the honeycomb lattice, the vacancies here do not bind a flux at low-energies. 7. Magnetization Switching in Anisotropic Nanoscale Ferromagnets: Algorithms and Applications NASA Astrophysics Data System (ADS) Novotny, Mark 1997-08-01 Since magnetic recording is approaching the limit of one bit of information stored per nanoscale magnetic grain, the understanding of magnetization reversal in single domain highly anisotropic ferromagnets becomes more critical. In addition to novel methods for obtaining well-characterized single-domain nanocrystals, recent experimental techniques such as Magnetic Force Microscopy (MFM) and μ-bridge Josephson Junctions allow measurements on individual nanocrystals of P_not(t), the probability that the magnetization has not switched. In this talk two novel dynamic Monte Carlo methods will be discussed. The first uses the Monte Carlo with Absorbing Markov Chains (MCAMC) method(M.A. Novotny, Phys. Rev. Lett. 74) 1 (1995); erratum 75,1424 (1995)., which allows simulations to span the time range of physical interest (from inverse phonon frequencies to human lifetimes). The second method(M. Kolesik, M.A. Novotny, P.A. Rikvold, and D.M. Townsley in Computer Simulations in Condensed Matter Physics X), ed. D.P. Landau, K.K. Mon, and H.-B. Schüttler, Springer Verlag, in press. uses histograms and transition probabilities to obtain results of dynamic simulations on large lattices. The data from these methods will be analyzed using droplet theory. Even in the simplest model (square lattice Ising model with periodic boundary conditions) there are four relevant length scales, leading to a variety of different physical regimes(H.L. Richards et al.), J. Magn. Magn. Mater. 150, 37 (1995).. The form for quantities such as P_not(t) in these different regimes will be discussed. The addition of demagnetizing fields and different boundary conditions will also be discussed briefly(H.L. Richards et al.), Phys. Rev. B 54, 4113 (1996); Phys. Rev. B 55, 11521 (1997).. 8. Parallel Anisotropic Tetrahedral Adaptation NASA Technical Reports Server (NTRS) Park, Michael A.; Darmofal, David L. 2008-01-01 An adaptive method that robustly produces high aspect ratio tetrahedra to a general 3D metric specification without introducing hybrid semi-structured regions is presented. The elemental operators and higher-level logic is described with their respective domain-decomposed parallelizations. An anisotropic tetrahedral grid adaptation scheme is demonstrated for 1000-1 stretching for a simple cube geometry. This form of adaptation is applicable to more complex domain boundaries via a cut-cell approach as demonstrated by a parallel 3D supersonic simulation of a complex fighter aircraft. To avoid the assumptions and approximations required to form a metric to specify adaptation, an approach is introduced that directly evaluates interpolation error. The grid is adapted to reduce and equidistribute this interpolation error calculation without the use of an intervening anisotropic metric. Direct interpolation error adaptation is illustrated for 1D and 3D domains. 9. DNA-nanoparticle superlattices formed from anisotropic building blocks SciTech Connect Jones, Matthew R.; Macfarlane, Robert John; Lee, Byeongdu; Zhang, Jian; Young, Kaylie L.; Senesi, Andrew J.; Mirkin, Chad A. 2010-10-03 Directional bonding interactions in solid-state atomic lattices dictate the unique symmetries of atomic crystals, resulting in a diverse and complex assortment of three-dimensional structures that exhibit a wide variety of material properties. Methods to create analogous nanoparticle superlattices are beginning to be realized, but the concept of anisotropy is still largely underdeveloped in most particle assembly schemes. Some examples provide interesting methods to take advantage of anisotropic effects, but most are able to make only small clusters or lattices that are limited in crystallinity and especially in lattice parameter programmability. Anisotropic nanoparticles can be used to impart directional bonding interactions on the nanoscale, both through face-selective functionalization of the particle with recognition elements to introduce the concept of valency, and through anisotropic interactions resulting from particle shape. In this work, we examine the concept of inherent shape-directed crystallization in the context of DNA-mediated nanoparticle assembly. Importantly, we show how the anisotropy of these particles can be used to synthesize one-, two- and three-dimensional structures that cannot be made through the assembly of spherical particles. 10. A high-resolution field-emission-gun, scanning electron microscope investigation of anisotropic hydrogen decrepitation in Nd-Fe-B-based sintered magnets NASA Astrophysics Data System (ADS) Soderznik, Marko; McGuiness, Paul; Zuzek-Rozman, Kristina; Škulj, Irena; Yan, Gaolin; Kobe, Spomenka 2010-05-01 In this investigation commercial magnets based on (Nd,Dy)14(Fe,Co)79B7 were prepared by a conventional powder-metallurgy route with a degree of alignment equal to ˜90% and then exposed to hydrogen at a pressure of 1 bar. The magnets, in the form of cylinders, were observed to decrepitate exclusively from the ends. High-resolution electron microscopy was able to identify the presence of crack formation within the Nd2Fe14B grains, with the cracks running parallel to the c axis of these grains. Based on the concentration profile for hydrogen in a rare-earth transition-metal material, it is clear that the presence of hydrogen-induced cracks running perpendicular to the ends of the magnet provides for a much more rapidly progressing hydrogen front in this direction than from the sides of the magnet. This results in the magnet exhibiting a macroscopic tendency to decrepitate from the poles of the magnet toward the center. This combination of microstructural modification via particle alignment as part of the sintering process and direct observation via high-resolution electron microscopy has led to a satisfying explanation for the anisotropic hydrogen-decrepitation effect. 11. Optical Lattice Clocks NASA Astrophysics Data System (ADS) Oates, Chris 2012-06-01 Since they were first proposed in 2003 [1], optical lattice clocks have become one of the leading technologies for the next generation of atomic clocks, which will be used for advanced timing applications and in tests of fundamental physics [2]. These clocks are based on stabilized lasers whose frequency is ultimately referenced to an ultra-narrow neutral atom transition (natural linewidths << 1 Hz). To suppress the effects of atomic motion/recoil, the atoms in the sample (˜10^4 atoms) are confined tightly in the potential wells of an optical standing wave (lattice). The wavelength of the lattice light is tuned to its `magic'' value so as to yield a vanishing net AC Stark shift for the clock transition. As a result lattice clocks have demonstrated the capability of generating high stability clock signals with small absolute uncertainties (˜ 1 part in 10^16). In this presentation I will first give an overview of the field, which now includes three different atomic species. I will then use experiments with Yb performed in our laboratory to illustrate the key features of a lattice clock. Our research has included the development of state-of-the-art optical cavities enabling ultra-high-resolution optical spectroscopy (1 Hz linewidth). Together with the large atom number in the optical lattice, we are able to achieve very low clock instability (< 0.3 Hz in 1 s) [3]. Furthermore, I will show results from some of our recent investigations of key shifts for the Yb lattice clock, including high precision measurements of ultracold atom-atom interactions in the lattice and the dc Stark effect for the Yb clock transition (necessary for the evaluation of blackbody radiation shifts). [4pt] [1] H. Katori, M. Takamoto, V. G. Pal'chikov, and V. D. Ovsiannikov, Phys. Rev. Lett. 91, 173005 (2003). [0pt] [2] Andrei Derevianko and Hidetoshi Katori, Rev. Mod. Phys. 83, 331 (2011). [0pt] [3] Y. Y. Jiang, A. D. Ludlow, N. D. Lemke, R. W. Fox, J. A. Sherman, L.-S. Ma, and C. W. Oates 12. Fractional lattice charge transport PubMed Central Flach, Sergej; Khomeriki, Ramaz 2017-01-01 We consider the dynamics of noninteracting quantum particles on a square lattice in the presence of a magnetic flux α and a dc electric field E oriented along the lattice diagonal. In general, the adiabatic dynamics will be characterized by Bloch oscillations in the electrical field direction and dispersive ballistic transport in the perpendicular direction. For rational values of α and a corresponding discrete set of values of E(α) vanishing gaps in the spectrum induce a fractionalization of the charge in the perpendicular direction - while left movers are still performing dispersive ballistic transport, the complementary fraction of right movers is propagating in a dispersionless relativistic manner in the opposite direction. Generalizations and the possible probing of the effect with atomic Bose-Einstein condensates and photonic networks are discussed. Zak phase of respective band associated with gap closing regime has been computed and it is found converging to π/2 value. PMID:28102302 13. Introduction to lattice gauge theory NASA Astrophysics Data System (ADS) Gupta, R. The lattice formulation of Quantum Field Theory (QFT) can be exploited in many ways. We can derive the lattice Feynman rules and carry out weak coupling perturbation expansions. The lattice then serves as a manifestly gauge invariant regularization scheme, albeit one that is more complicated than standard continuum schemes. Strong coupling expansions: these give us useful qualitative information, but unfortunately no hard numbers. The lattice theory is amenable to numerical simulations by which one calculates the long distance properties of a strongly interacting theory from first principles. The observables are measured as a function of the bare coupling g and a gauge invariant cut-off approx. = 1/alpha, where alpha is the lattice spacing. The continuum (physical) behavior is recovered in the limit alpha yields 0, at which point the lattice artifacts go to zero. This is the more powerful use of lattice formulation, so in these lectures the author focuses on setting up the theory for the purpose of numerical simulations to get hard numbers. The numerical techniques used in Lattice Gauge Theories have their roots in statistical mechanics, so it is important to develop an intuition for the interconnection between quantum mechanics and statistical mechanics. 14. Effective-medium theory for anisotropic magnetic metamaterials NASA Astrophysics Data System (ADS) Jin, Junfeng; Liu, Shiyang; Lin, Zhifang; Chui, S. T. 2009-09-01 We have developed an effective-medium theory within the coherent-potential approximation, which is especially suitable to retrieve the effective constitutive parameters (permittivity and permeability) of the anisotropic magnetic metamaterials consisting of the ferrite rods. The anisotropy originates from the gyromagnetic property of the ferrite material whose permeability is a tensor with nonzero off-diagonal components. To confirm the validity of our method the photonic band structures of the two-dimensional periodic magnetic metamaterials are calculated, which are in agreement with the effective-medium theory in the long wavelength limit, in addition, even when a/λ0˜0.4 the effective-medium theory can still be applied, where a and λ0 are the lattice constant and the vacuum wavelength, respectively. The simulations on the electric field patterns for a plane wave illuminated on the magnetic metamaterials and the equal-size effective scattering objects are performed, the results corroborate the effectiveness of the effective-medium theory once again. We also perform the simulation for the metamaterial composed of disordered ferrite rods, which is still in agreement with the effective-medium theory, suggesting the powerfulness of the effective-medium theory. Moreover, our results suggest that the anisotropy must be considered exactly in order to retrieve the effective constitutive parameters accurately. 15. Anisotropic Total Variation Filtering SciTech Connect Grasmair, Markus; Lenzen, Frank 2010-12-15 Total variation regularization and anisotropic filtering have been established as standard methods for image denoising because of their ability to detect and keep prominent edges in the data. Both methods, however, introduce artifacts: In the case of anisotropic filtering, the preservation of edges comes at the cost of the creation of additional structures out of noise; total variation regularization, on the other hand, suffers from the stair-casing effect, which leads to gradual contrast changes in homogeneous objects, especially near curved edges and corners. In order to circumvent these drawbacks, we propose to combine the two regularization techniques. To that end we replace the isotropic TV semi-norm by an anisotropic term that mirrors the directional structure of either the noisy original data or the smoothed image. We provide a detailed existence theory for our regularization method by using the concept of relaxation. The numerical examples concluding the paper show that the proposed introduction of an anisotropy to TV regularization indeed leads to improved denoising: the stair-casing effect is reduced while at the same time the creation of artifacts is suppressed. 16. Design of a Kagome lattice from soft anisotropic particles. PubMed Fejer, Szilard N; Wales, David J 2015-09-07 We present a simple model of triblock Janus particles based on discoidal building blocks, which can form energetically stabilized Kagome structures. We find 'magic number' global minima in small clusters whenever particle numbers are compatible with a perfect Kagome structure, without constraining the accessible three-dimensional configuration space. The preference for planar structures with two bonds per patch among all other possible minima on the landscape is enhanced when sedimentation forces are included. For the building blocks in question, structures containing three bonds per patch become progressively higher in energy compared to Kagome structures as sedimentation forces increase. Rearrangements between competing structures, as well as ring formation mechanisms are characterised and found to be highly cooperative. 17. CLUSTEREASY: A program for lattice simulations of scalar fields in an expanding universe on parallel computing clusters NASA Astrophysics Data System (ADS) Felder, Gary 2008-10-01 We describe an MPI C++ program that we have written and made available for calculating the evolution of interacting scalar fields in an expanding universe on parallel clusters. The program is a parallel programming extension of the simulation program LATTICEEASY. The ability to run these simulations on parallel clusters, however, greatly extends the range of scales and times that can be simulated. The program is particularly useful for the study of reheating and thermalization after inflation. The program and its full documentation are available on the Web at http://www.science.smith.edu/departments/Physics/fstaff/gfelder/latticeeasy/. In this paper we provide a brief overview of what the program does and what it is useful for. Catalogue identifier: AEBJ_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEBJ_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 7469 No. of bytes in distributed program, including test data, etc.: 613 334 Distribution format: tar.gz Programming language: C++/MPI Computer: Cluster. Must have the library FFTW installed Operating system: Any RAM: Typically 4 MB to 1 GB per processor Classification: 1.9 External routines: A single-precision version of the FFTW library (http://www.fftw.org/) must be available on the target machine. Nature of problem: After inflation the universe consisted of interacting fields in a high energy, nonthermal state [1]. The evolution of these fields cannot be described with standard approximation techniques such as linearization, kinetic theory, or Hartree expansion, and must thus be simulated numerically. Fortunately, the fields rapidly acquire large occupation numbers over a range of frequencies, so their evolution can be accurately modeled with classical field theory [2]. The specific fields and 18. Monte Carlo simulation with aspect-ratio optimization: Anomalous anisotropic scaling in dimerized antiferromagnets NASA Astrophysics Data System (ADS) Yasuda, Shinya; Todo, Synge 2013-12-01 We present a method that optimizes the aspect ratio of a spatially anisotropic quantum lattice model during the quantum Monte Carlo simulation, and realizes the virtually isotropic lattice automatically. The anisotropy is removed by using the Robbins-Monro algorithm based on the correlation length in each direction. The method allows for comparing directly the value of the critical amplitude among different anisotropic models, and identifying the universality more precisely. We apply our method to the staggered dimer antiferromagnetic Heisenberg model and demonstrate that the apparent nonuniversal behavior is attributed mainly to the strong size correction of the effective aspect ratio due to the existence of the cubic interaction. 19. Monte Carlo simulation with aspect-ratio optimization: anomalous anisotropic scaling in dimerized antiferromagnets. PubMed Yasuda, Shinya; Todo, Synge 2013-12-01 We present a method that optimizes the aspect ratio of a spatially anisotropic quantum lattice model during the quantum Monte Carlo simulation, and realizes the virtually isotropic lattice automatically. The anisotropy is removed by using the Robbins-Monro algorithm based on the correlation length in each direction. The method allows for comparing directly the value of the critical amplitude among different anisotropic models, and identifying the universality more precisely. We apply our method to the staggered dimer antiferromagnetic Heisenberg model and demonstrate that the apparent nonuniversal behavior is attributed mainly to the strong size correction of the effective aspect ratio due to the existence of the cubic interaction. 20. What is the Brillouin zone of an anisotropic photonic crystal? NASA Astrophysics Data System (ADS) Sivarajah, P.; Maznev, A. A.; Ofori-Okai, B. K.; Nelson, K. A. 2016-02-01 The concept of the Brillouin zone (BZ) in relation to a photonic crystal fabricated in an optically anisotropic material is explored both experimentally and theoretically. In experiment we used femtosecond laser pulses to excite THz polaritons and image their propagation in lithium niobate and lithium tantalate photonic crystal (PhC) slabs. We directly measured the dispersion relation inside PhCs and observed that the lowest band gap expected to form at the BZ boundary forms inside the BZ in the anisotropic lithium niobate PhC. Our analysis shows that in an anisotropic material the BZ—defined as the Wigner-Seitz cell in the reciprocal lattice—is no longer bounded by Bragg planes and thus does not conform to the original definition of the BZ by Brillouin. We construct an alternative Brillouin zone defined by Bragg planes and show its utility in identifying features of the dispersion bands. We show that for an anisotropic two-dimensional PhC without dispersion, the Bragg plane BZ can be constructed by applying the Wigner-Seitz method to a stretched or compressed reciprocal lattice. We also show that in the presence of the dispersion in the underlying material or in a slab waveguide, the Bragg planes are generally represented by curved surfaces rather than planes. The concept of constructing a BZ with Bragg planes should prove useful in understanding the formation of dispersion bands in anisotropic PhCs and in selectively tailoring their optical properties. 1. Anisotropic transport in modulation doped quantum well structures NASA Technical Reports Server (NTRS) Radulescu, D. C.; Wicks, G. W.; Schaff, W. J.; Calawa, A. R.; Eastman, L. F. 1987-01-01 The degree of anisotropy in the anisotropic electron transport that has been observed in GaAs modulation-doped quantum wells grown by MBE on Al(0.3)Ga(0.7)As is related to the thickness and growth parameters of this substrate, which is grown just prior to the inverted interface. It is presently observed that the inverted interface has an anisotropic roughness which affects the 77 K low field electron transport parallel to the interface, and gives rise to anisotropic electron scattering in the GaAs modulation-doped quantum well. 2. Wigner crystallization of a two-dimensional electron gas in a magnetic field: Single electrons versus electron pairs at the lattice sites NASA Astrophysics Data System (ADS) Taut, M. 2001-10-01 The ground state energy and the lowest excitations of a two-dimensional Wigner crystal in a perpendicular magnetic field with one and two electrons per cell is investigated. In the case of two electrons per lattice site, the interaction of the electrons within each cell is taken into account exactly (including exchange and correlation effects), and the interaction between the cells is in second order (dipole) van der Waals approximation. No further approximations are made, in particular Landau level mixing and incomplete spin polarization are accounted for. Therefore, our calculation comprises a, roughly speaking, complementary description of the bubble phase (in the special case of one and two electrons per bubble), which was proposed by Koulakov, Fogler, and Shklovskii on the basis of a Hartree Fock calculation. The phase diagram shows that in GaAs the paired phase is energetically more favorable than the single electron phase for, roughly speaking, filling factor f larger than 0.3 and density parameter rs smaller than 19 effective Bohr radii (for a more precise statement see Figs. 3 and 4). If we start within the paired phase and increase magnetic field or decrease density, the pairs first undergo some singlet-triplet transitions before they break. 3. Spin precession in anisotropic media NASA Astrophysics Data System (ADS) Raes, B.; Cummings, A. W.; Bonell, F.; Costache, M. V.; Sierra, J. F.; Roche, S.; Valenzuela, S. O. 2017-02-01 We generalize the diffusive model for spin injection and detection in nonlocal spin structures to account for spin precession under an applied magnetic field in an anisotropic medium, for which the spin lifetime is not unique and depends on the spin orientation. We demonstrate that the spin precession (Hanle) line shape is strongly dependent on the degree of anisotropy and on the orientation of the magnetic field. In particular, we show that the anisotropy of the spin lifetime can be extracted from the measured spin signal, after dephasing in an oblique magnetic field, by using an analytical formula with a single fitting parameter. Alternatively, after identifying the fingerprints associated with the anisotropy, we propose a simple scaling of the Hanle line shapes at specific magnetic field orientations that results in a universal curve only in the isotropic case. The deviation from the universal curve can be used as a complementary means of quantifying the anisotropy by direct comparison with the solution of our generalized model. Finally, we applied our model to graphene devices and find that the spin relaxation for graphene on silicon oxide is isotropic within our experimental resolution. 4. Magnetospheric equilibrium with anisotropic pressure SciTech Connect Cheng, C.Z. 1991-07-01 Self-consistent magnetospheric equilibrium with anisotropic pressure is obtained by employing an iterative metric method for solving the inverse equilibrium equation in an optimal flux coordinate system. A method of determining plasma parallel and perpendicular pressures from either analytic particle distribution or particle distribution measured along the satellite's path is presented. The numerical results of axisymmetric magnetospheric equilibrium including the effects of finite beta, pressure anisotropy, and boundary conditions are presented for a bi-Maxwellian particle distribution. For the isotropic pressure cases, the finite beta effect produces an outward expansion of the constant magnetic flux surfaces in relation to the dipole field lines, and along the magnetic field the toroidal ring current is maximum at the magnetic equator. The effect of pressure anisotropy is found to further expand the flux surfaces outward. Along the magnetic field lines the westward ring current can be peak away from the equator due to an eastward current contribution resulting from pressure anisotropy. As pressure anisotropy increases, the peak westward current can become more singular. The outer boundary flux surface has significant effect on the magnetospheric equilibrium. For the outer flux boundary resembling dayside compressed flux surface due to solar wind pressure, the deformation of the magnetic field can be quite different from that for the outer flux boundary resembling the tail-like surface. 23 refs., 17 figs. 5. Spectral element method for band structures of three-dimensional anisotropic photonic crystals NASA Astrophysics Data System (ADS) Luo, Ma; Liu, Qing Huo 2009-11-01 A spectral element method (SEM) is introduced for accurate calculation of band structures of three-dimensional anisotropic photonic crystals. The method is based on the finite-element framework with curvilinear hexahedral elements. Gauss-Lobatto-Legendre polynomials are used to construct the basis functions. In order to suppress spurious modes, mixed-order vector basis functions are employed and the Bloch periodic boundary condition is imposed into the basis functions with tangential components at the boundary by multiplying a Bloch phase factor. The fields and coordinates in the curvilinear hexahedral elements are mapped to the reference domain by covariant mapping, which preserves the continuity of tangential components of the field. Numerical results show that the SEM has exponential convergence for both square-lattice and triangular-lattice photonic crystals. The sampling density as small as 3.4 points per wavelength can achieve accuracy as high as 99.9%. The band structures of several modified woodpile photonic crystals are calculated by using the SEM. 6. Spectral element method for band structures of three-dimensional anisotropic photonic crystals. PubMed Luo, Ma; Liu, Qing Huo 2009-11-01 A spectral element method (SEM) is introduced for accurate calculation of band structures of three-dimensional anisotropic photonic crystals. The method is based on the finite-element framework with curvilinear hexahedral elements. Gauss-Lobatto-Legendre polynomials are used to construct the basis functions. In order to suppress spurious modes, mixed-order vector basis functions are employed and the Bloch periodic boundary condition is imposed into the basis functions with tangential components at the boundary by multiplying a Bloch phase factor. The fields and coordinates in the curvilinear hexahedral elements are mapped to the reference domain by covariant mapping, which preserves the continuity of tangential components of the field. Numerical results show that the SEM has exponential convergence for both square-lattice and triangular-lattice photonic crystals. The sampling density as small as 3.4 points per wavelength can achieve accuracy as high as 99.9%. The band structures of several modified woodpile photonic crystals are calculated by using the SEM. 7. On the relativistic anisotropic configurations NASA Astrophysics Data System (ADS) Shojai, F.; Kohandel, M.; Stepanian, A. 2016-06-01 In this paper we study anisotropic spherical polytropes within the framework of general relativity. Using the anisotropic Tolman-Oppenheimer-Volkov equations, we explore the relativistic anisotropic Lane-Emden equations. We find how the anisotropic pressure affects the boundary conditions of these equations. Also we argue that the behavior of physical quantities near the center of star changes in the presence of anisotropy. For constant density, a class of exact solution is derived with the aid of a new ansatz and its physical properties are discussed. 8. Magnetic Field Induced Phase Transitions in Gd5(Si1.95Ge2.05)Single Crystal and the Anisotropic Magnetocaloric Effect SciTech Connect H. Tang; V.K. Pecharsky; A.O. Pecharsky; D.L. Schlagel; T.A. Lograsso; K.A. Gschneidner,jr. 2004-09-30 The magnetization measurements using a Gd{sub 5}(Si{sub 1.95}Ge{sub 2.05}) single crystal with the magnetic field applied along three crystallographic directions, [001], [010] and [100], were carried out as function of applied field (0-56 kOe) at various temperatures ({approx}5-320 K). The magnetic-field induced phase transformations at temperature above the zero-field critical temperature, i.e. the paramagnetic (PM) {leftrightarrow} ferromagnetic (FM) transitions with application or removal of magnetic field, are found to be temperature dependent and hysteretic. The corresponding critical fields increase with increasing temperature. The magnetic field (H)-temperature (T) phase diagrams have been constructed for the Gd{sub 5}(Si{sub 1.95}Ge{sub 2.05}) single crystal with field along the three directions. A small anisotropy has been observed. The magnetocaloric effect (MCE) has been calculated from the isothermal magnetization data, and the observed anisotropy correlates with H-T phase diagrams. The results are discussed in connection with the magnetic-field induced martensitic-like structural transition observed in the Gd{sub 5}(Si{sub 2}Ge{sub 2})-type compounds. 9. On the electrodynamics of Josephson effect in anisotropic superconductors SciTech Connect Mints, R.G. 1989-01-01 Specificities of Josephson effect electrodynamics in anisotropic superconductors are of considerable interest for the study of high temperature superconductors with strongly anisotropic layered structure. In this paper the authors give the calculation for the tunnel Josephson contact of an isolated vortex, the law of dispersion of its low-amplitude oscillations, the critical field H/sub cl/ for the penetration of magnetic flux, and the maximum current across a rectangular contact. 10. Anisotropic Bianchi types VIII and IX locally rotationally symmetric cosmologies SciTech Connect Assad, M.J.D.; Soares, I.D. 1983-10-15 We present a class of exact cosmological solutions of Einstein-Maxwell equations, which are anisotropic and spatially homogeneous of Bianchi types VIII and IX, and class IIIb in the Stewart-Ellis classification of locally rotationally symmetric models. If we take the electromagnetic field equal to zero, a class of Bianchi types VIII/IX spatially homogeneous anisotropic cosmological solutions with perfect fluid is obtained. 11. Effect of inflation on anisotropic cosmologies SciTech Connect Jensen, L.G.; Stein-Schabes, J.A. 1986-03-01 The effects of anisotropic cosmologies on inflation are studied. By properly formulating the field equations it is possible to show that any model that undergoes sufficient inflation will become isotropic on scales greater than the horizon today. Furthermore, we shall show that it takes a very long time for anisotropies to become visible in the observable part of the Universe. It is interesting to note that the time scale will be independent of the Bianchi Model and of the initial anisotropy. 6 refs. 12. Anisotropic conducting films for electromagnetic radiation applications DOEpatents Cavallo, Francesca; Lagally, Max G.; Rojas-Delgado, Richard 2015-06-16 Electronic devices for the generation of electromagnetic radiation are provided. Also provided are methods for using the devices to generate electromagnetic radiation. The radiation sources include an anisotropic electrically conducting thin film that is characterized by a periodically varying charge carrier mobility in the plane of the film. The periodic variation in carrier mobility gives rise to a spatially varying electric field, which produces electromagnetic radiation as charged particles pass through the film. 13. Topological lattice actions NASA Astrophysics Data System (ADS) Bietenholz, W.; Gerber, U.; Pepe, M.; Wiese, U.-J. 2010-12-01 We consider lattice field theories with topological actions, which are invariant against small deformations of the fields. Some of these actions have infinite barriers separating different topological sectors. Topological actions do not have the correct classical continuum limit and they cannot be treated using perturbation theory, but they still yield the correct quantum continuum limit. To show this, we present analytic studies of the 1-d O(2) and O(3) model, as well as Monte Carlo simulations of the 2-d O(3) model using topological lattice actions. Some topological actions obey and others violate a lattice Schwarz inequality between the action and the topological charge Q. Irrespective of this, in the 2-d O(3) model the topological susceptibility {χ_t} = {{{left< {{Q^2}} rightrangle }} left/ {V} right.} is logarithmically divergent in the continuum limit. Still, at non-zero distance the correlator of the topological charge density has a finite continuum limit which is consistent with analytic predictions. Our study shows explicitly that some classically important features of an action are irrelevant for reaching the correct quantum continuum limit. 14. Excited state baryon spectroscopy from lattice QCD DOE PAGES Robert G. Edwards; Dudek, Jozef J.; Richards, David G.; ... 2011-10-31 Here, we present a calculation of the Nucleon and Delta excited state spectrum on dynamical anisotropic clover lattices. A method for operator construction is introduced that allows for the reliable identification of the continuum spins of baryon states, overcoming the reduced symmetry of the cubic lattice. Using this method, we are able to determine a spectrum of single-particle states for spins up to and includingJ = 7/2$, of both parities, the first time this has been achieved in a lattice calculation. We find a spectrum of states identifiable as admixtures of$SU(6) Ⓧ O(3)$representations and a counting ofmore » levels that is consistent with the non-relativistic$qqq$constituent quark model. This dense spectrum is incompatible with quark-diquark model solutions to the "missing resonance problem" and shows no signs of parity doubling of states.« less 15. Excited state baryon spectroscopy from lattice QCD SciTech Connect Robert G. Edwards; Dudek, Jozef J.; Richards, David G.; Wallace, Stephen J. 2011-10-31 Here, we present a calculation of the Nucleon and Delta excited state spectrum on dynamical anisotropic clover lattices. A method for operator construction is introduced that allows for the reliable identification of the continuum spins of baryon states, overcoming the reduced symmetry of the cubic lattice. Using this method, we are able to determine a spectrum of single-particle states for spins up to and including$J = 7/2$, of both parities, the first time this has been achieved in a lattice calculation. We find a spectrum of states identifiable as admixtures of$SU(6) Ⓧ O(3)$representations and a counting of levels that is consistent with the non-relativistic$qqq\$ constituent quark model. This dense spectrum is incompatible with quark-diquark model solutions to the "missing resonance problem" and shows no signs of parity doubling of states.
16. Lattice Strain Defects in a Ceria Nanolayer
PubMed Central
2016-01-01
An ultrathin two-dimensional CeO2 (ceria) phase on a Cu(110) surface has been fabricated and fully characterized by high-resolution scanning tunneling microscopy, photoelectron spectroscopy, and density functional theory. The atomic lattice structure of the ceria/Cu(110) system is revealed as a hexagonal CeO2(111)-type monolayer separated from the Cu(110) surface by a partly disordered Cu–O intercalated buffer layer. The epitaxial coupling of the two-dimensional ceria overlayer to the Cu(110)-O surface leads to a nanoscopic stripe pattern, which creates defect regions of quasi-periodic lattice distortions. The symmetry and lattice mismatch at the interface is clarified to be responsible for the topographic stripe geometry and the related anisotropic strain defect regions at the ceria surface. This ceria monolayer is in a fully oxidized and thermodynamically stable state. PMID:26988695
17. Lattice Strain Defects in a Ceria Nanolayer.
PubMed
Ma, Liying; Doudin, Nassar; Surnev, Svetlozar; Barcaro, Giovanni; Sementa, Luca; Fortunelli, Alessandro; Netzer, Falko P
2016-04-07
An ultrathin two-dimensional CeO2 (ceria) phase on a Cu(110) surface has been fabricated and fully characterized by high-resolution scanning tunneling microscopy, photoelectron spectroscopy, and density functional theory. The atomic lattice structure of the ceria/Cu(110) system is revealed as a hexagonal CeO2(111)-type monolayer separated from the Cu(110) surface by a partly disordered Cu-O intercalated buffer layer. The epitaxial coupling of the two-dimensional ceria overlayer to the Cu(110)-O surface leads to a nanoscopic stripe pattern, which creates defect regions of quasi-periodic lattice distortions. The symmetry and lattice mismatch at the interface is clarified to be responsible for the topographic stripe geometry and the related anisotropic strain defect regions at the ceria surface. This ceria monolayer is in a fully oxidized and thermodynamically stable state.
18. High anisotropic pitch
SciTech Connect
Dickakian, G. B.
1985-11-05
An improved process for preparing an optically anisotropic pitch which comprises heating a pitch feed material at a temperature within the range of about 350/sup 0/ C. to 450/sup 0/ C. while passing an inert gas therethrough at a rate of at least 2.5 SCFH/lb of pitch feed material and agitating said pitch feed material at a stirrer rate of from about 500 to 600 rpm to obtain an essentially 100% mesophase pitch product suitable for carbon production.
19. Anisotropic spinfoam cosmology
Rennert, Julian; Sloan, David
2014-01-01
The dynamics of a homogeneous, anisotropic universe are investigated within the context of spinfoam cosmology. Transition amplitudes are calculated for a graph consisting of a single node and three links—the ‘Daisy graph’—probing the behaviour a classical Bianchi I spacetime. It is shown further how the use of such single node graphs gives rise to a simplification of states such that all orders in the spin expansion can be calculated, indicating that it is the vertex expansion that contains information about quantum dynamics.
20. Anisotropic swelling behavior of the cornea.
PubMed
Matsuura, Toyoaki; Ikeda, Hitoe; Idota, Naokazu; Motokawa, Ryuhei; Hara, Yoshiaki; Annaka, Masahiko
2009-12-24
The phase equilibrium property and structural and dynamical properties of pig cornea were studied by macroscopic observation of swelling behavior, dynamic light scattering (DLS), and small-angle X-ray scattering (SAXS) under various conditions. It was found that the corneal gel collapses into a compact state isotropically or anisotropically depending on the external conditions. The corneal gel collapses uniformly into a compact state at a temperature above 55 degrees C because of the denaturation of collagen, whereas it collapses along an axis parallel to the optic axis with increasing NaCl concentration. Anisotropic deswelling was also observed during desiccation. SAXS measurements revealed that the periodicity of the collagen fiber of the cornea does not change even at higher NaCl concentration, which indicates that hydration and dehydration resulting from changes in salt concentration simply cause swelling and deswelling of the glycosaminoglycan (GAG), which is located between the regular two-dimensional lattices of collagen fibers, which obliges the change in thickness. From observations of the dynamics of light scattered by the corneal gel, intensity autocorrelation functions that revealed two independent diffusion coefficients were obtained. Divergent behavior in the measured total scattered light intensities and diffusion coefficients with varying temperature was observed. That is, a slowing of the dynamic modes accompanied by increased "static" scattered intensities was observed. This is indicative of the occurrence of a phase transition as a function of temperature. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 1, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8225936889648438, "perplexity": 1948.2067472161036}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-30/segments/1500549429548.55/warc/CC-MAIN-20170727222533-20170728002533-00411.warc.gz"} |
https://proofwiki.org/wiki/Max_and_Min_Operations_are_Distributive_over_Each_Other | # Max and Min Operations are Distributive over Each Other
## Theorem
The Max and Min operations are distributive over each other:
$\map \max {x, \map \min {y, z} } = \map \min {\map \max {x, y}, \map \max {x, z} }$
$\map \max {\map \min {x, y}, z} = \map \min {\map \max {x, z}, \map \max {y, z} }$
$\map \min {x, \map \max {y, z} } = \map \max {\map \min {x, y}, \map \min {x, z} }$
$\map \min {\map \max {x, y}, z} = \map \max {\map \min {x, z}, \map \min {y, z} }$
## Proof
To simplify our notation, let $\map \max {x, y}$ be (temporarily) denoted $x \overline \wedge y$, and let $\map \min {x, y}$ be (temporarily) denoted $x \underline \vee y$.
Note that, once we have proved:
$x \overline \wedge \paren {y \underline \vee z} = \paren {x \overline \wedge y} \underline \vee \paren {x \overline \wedge z}$
$x \underline \vee \paren {y \overline \wedge z} = \paren {x \underline \vee y} \overline \wedge \paren {x \underline \vee z}$
then the other results follow immediately from Max Operation is Commutative and Min Operation is Commutative.
There are the following cases to consider:
$(1): \quad x \le y \le z$
$(2): \quad x \le z \le y$
$(3): \quad y \le x \le z$
$(4): \quad y \le z \le x$
$(5): \quad z \le x \le y$
$(6): \quad z \le y \le x$
$(1): \quad$ Let $x \le y \le z$.
Then:
$\ds x \overline \wedge \paren {y \underline \vee z}$ $=$ $\ds x \overline \wedge y = y$ $\ds \paren {x \overline \wedge y} \underline \vee \paren {x \overline \wedge z}$ $=$ $\ds y \underline \vee z = y$ $\ds$ $\ds$ $\ds x \underline \vee \paren {y \overline \wedge z}$ $=$ $\ds x \underline \vee z = x$ $\ds \paren {x \underline \vee y} \overline \wedge \paren {x \underline \vee z}$ $=$ $\ds x \overline \wedge x = x$
$(2): \quad$ Let $x \le z \le y$.
Then:
$\ds x \overline \wedge \paren {y \underline \vee z}$ $=$ $\ds x \overline \wedge z = z$ $\ds \paren {x \overline \wedge y} \underline \vee \paren {x \overline \wedge z}$ $=$ $\ds y \underline \vee z = z$ $\ds$ $\ds$ $\ds x \underline \vee \paren {y \overline \wedge z}$ $=$ $\ds x \underline \vee y = x$ $\ds \paren {x \underline \vee y} \overline \wedge \paren {x \underline \vee z}$ $=$ $\ds x \overline \wedge x = x$
$(3): \quad$ Let $y \le x \le z$.
Then:
$\ds x \overline \wedge \paren {y \underline \vee z}$ $=$ $\ds x \overline \wedge y = x$ $\ds \paren {x \overline \wedge y} \underline \vee \paren {x \overline \wedge z}$ $=$ $\ds x \underline \vee z = x$ $\ds$ $\ds$ $\ds x \underline \vee \paren {y \overline \wedge z}$ $=$ $\ds x \underline \vee z = x$ $\ds \paren {x \underline \vee y} \overline \wedge \paren {x \underline \vee z}$ $=$ $\ds y \overline \wedge x = x$
$(4): \quad$ Let $y \le z \le x$.
Then:
$\ds x \overline \wedge \paren {y \underline \vee z}$ $=$ $\ds x \overline \wedge y = x$ $\ds \paren {x \overline \wedge y} \underline \vee \paren {x \overline \wedge z}$ $=$ $\ds x \underline \vee x = x$ $\ds$ $\ds$ $\ds x \underline \vee \paren {y \overline \wedge z}$ $=$ $\ds x \underline \vee z = z$ $\ds \paren {x \underline \vee y} \overline \wedge \paren {x \underline \vee z}$ $=$ $\ds y \overline \wedge z = z$
$(5): \quad$ Let $z \le x \le y$.
Then:
$\ds x \overline \wedge \paren {y \underline \vee z}$ $=$ $\ds x \overline \wedge z = x$ $\ds \paren {x \overline \wedge y} \underline \vee \paren {x \overline \wedge z}$ $=$ $\ds y \underline \vee x = x$ $\ds$ $\ds$ $\ds x \underline \vee \paren {y \overline \wedge z}$ $=$ $\ds x \underline \vee y = x$ $\ds \paren {x \underline \vee y} \overline \wedge \paren {x \underline \vee z}$ $=$ $\ds x \overline \wedge z = x$
$(6): \quad$ Let $z \le y \le x$.
Then:
$\ds x \overline \wedge \paren {y \underline \vee z}$ $=$ $\ds x \overline \wedge z = x$ $\ds \paren {x \overline \wedge y} \underline \vee \paren {x \overline \wedge z}$ $=$ $\ds x \underline \vee x = x$ $\ds$ $\ds$ $\ds x \underline \vee \paren {y \overline \wedge z}$ $=$ $\ds x \underline \vee y = y$ $\ds \paren {x \underline \vee y} \overline \wedge \paren {x \underline \vee z}$ $=$ $\ds y \overline \wedge z = y$
Thus in all cases it can be seen that the result holds.
$\blacksquare$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9981261491775513, "perplexity": 59.968059210641414}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964358078.2/warc/CC-MAIN-20211127013935-20211127043935-00566.warc.gz"} |
https://yufree.cn/en/2013/12/07/mean-and-variance/ | # Mean, Variance and Something about Sample
OK，I must clear the concepts of those terms because this is the third time I review my notes.
## Description of Data
If one wants to descript a collection of data, the Mode, Median and Mean are the first three terms to be learned. They are used to show the whole data in one parameter. And yes, one value is enough and the most important property of a dataset is the center. So where do the center come from? Here, our formula could be written as a distance between a parameter and the every values in the dataset.
So the Mode could be written as
$\sum_{}^{}(M_{para} - x_{i})^0$
And the Median could be written as
$\sum_{}^{}(M_{para} - x_{i})^1$
Then the Mean could be written as
$\sum_{}^{}(M_{para} - x_{i})^2$
Eh, I think the solution of the formula will show the Mean which make the distance smallest and yes, just make a diff. So here the description of the data is finished or just begin. Change the uppercase as you like. The origin idea came from John Myles White.
## Mean
Here we get Mean to show the center of dataset. And mean of discrete random variable X could be wright in the following formula:
$E[X] = \sum_x xp(x)$
For a continuous random variable,
$E[X] = \int_{-\infty}^\infty t f(t)dt$
Mean tell us the center of the distribution because we just use the smallest distance. Now we want to known the spread of the data. So we use variance.
## Variance
Here, we need to talk about moment in another view. The mean or the expected values is the first raw moment and the second central moment is variance. The moment is another discription method of the distribution. Here we only use the formula of variance:
$Var(X) = E[(X - \mu)^2]$
And the variance could be written as:
$Var(X) = E[X^2] - E[X]^2$
So how to understand the variance in probability?
### Chebyshev’s inequality
$P(|X - \mu| \geq k\sigma) \leq \frac{1}{k^2}$
Here we found the variance actually show the probaliblity of a observation in a distribution. That is just a description of spread of the distribution.
Ok, we talk enough about the distribution itself. Next we will see the sample.
## Sample
The independent and identically distributed random variables are the default model for random samples. Under iid, we could use the probaliblity to give out the description of sample. So if the values of variables is uncorrelated, then the variance of the sum is the sum of the variances.
$Var\left(\sum_{i=1}^n X_i \right) = \sum_{i=1}^n Var(X_i)$
So we will get the variance of the mean of the sample:
\begin{align} Var(\bar X) & = Var \left( \frac{1}{n}\sum_{i=1}^n X_i \right)\\ & = \frac{1}{n^2} Var\left(\sum_{i=1}^n X_i \right)\\ & = \frac{1}{n^2} \sum_{i=1}^n Var(X_i) \\ & = \frac{1}{n^2} \times n\sigma^2 \\ & = \frac{\sigma^2}{n} \end{align}
We also get the standard error of the sample mean: $$\sigma/\sqrt{n}$$,the sample mean has to be less variable than a single observation, therefore its standard deviation is divided.
For the sample mean which is the unbiased estimator of the population, nothing could be discussed. But for the sample variance need more attentions:
$S^2 = \frac{\sum_{i=1}^n (X_i - \bar X)^2}{n-1}$
We proof it:
\begin{align} E\left[\sum_{i=1}^n (X_i - \bar X)^2\right] & = \sum_{i=1}^n E\left[X_i^2\right] - n E\left[\bar X^2\right] \\ \\ & = \sum_{i=1}^n \left\{Var(X_i) + \mu^2\right\} - n \left\{Var(\bar X) + \mu^2\right\} \\ \\ & = \sum_{i=1}^n \left\{\sigma^2 + \mu^2\right\} - n \left\{\sigma^2 / n + \mu^2\right\} \\ \\ & = n \sigma^2 + n \mu ^ 2 - \sigma^2 - n \mu^2 \\ \\ & = (n - 1) \sigma^2 \end{align}
For the estimator $$S^2$$ of $$\sigma^2$$ is unbiased, the calculation of sample variance $$S^2$$ involves dividing by $$n-1$$. And $$S / \sqrt{n}$$ is called the sample standard error of the mean.
## Mean Again
If I want to descript a collection of samples, the mean is esay to get. But if we want to know the population behind the samples, we will use S as a description of the spread. But if we only care the mean, the $$S / \sqrt{n}$$ will show the description of the error when we calculate the mean. When the n is large enough, we will get a precise mean with small error. But this error have no effect of S, which is the description of the spread behind the sample. If you want an interval estimation of the sample mean instead of a point estimation, you may need to consider that t-value multiply the standard error.
Ok, finally I sort out those terms. I cite the work done by Professor Brian Caffo, thank you very much. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 2, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.998664140701294, "perplexity": 454.79899944767783}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-09/segments/1550247484928.52/warc/CC-MAIN-20190218094308-20190218120308-00625.warc.gz"} |
https://www.physicsforums.com/threads/125-gev-higgs-and-vacuum-instability.560232/ | # 125 GeV Higgs and Vacuum Instability
1. Dec 14, 2011
### Coin
125 GeV Higgs and "Vacuum Instability"
So the Higgs has been http://press.web.cern.ch/press/pressreleases/Releases2011/PR25.11E.html [Broken] (maybe). Nothing "beyond the standard model" about that of course; the Higgs is standard model. Except-- in the leadup to the LHC announcement, I repeatedly saw claims that the exact candidate mass of the Higgs, 125 GeV, is a strong sign that something beyond the standard model is going on, because a Higgs at that mass possibly indicates "vacuum instability". Here's a typical example of the claim:
So, this is exciting. It seems to me most recent physics theories are solutions looking for a problem and now we have a very large problem to solve.
Here are some things I am wondering. Assuming we don't get lucky (and just happen to get the correct top mass and SM parameters to keep the 125-GeV-Higgs universe stable):
1. Where can I read a more precise explanation of this negative Higgs self-coupling -> unstable vacuum idea?
2. Supersymmetry is usually the first theory cited as benefactor if the SM Higgs is found unstable. What other theories can also fix the problem? Can Little Higgs/Composite Higgs/Technicolor do it? Do the "Asymptotic Safety" models which are cited in other threads in this forum currently as producing a ~125 GeV Higgs have a way of solving the vacuum stability issue? Are there any other candidates?
3. Is there any technical reason, out of the gate, to prefer any one of these vacuum-stabilizing candidates over the other? My understanding is SUSY would be strongly preferred in any case due to the many other benefits it brings (solves certain mathematical problems, makes string theory possible, is "beautiful") but are there any known technical advantages of SUSY, or any other candidate, for the specific purpose of solving this particular problem (stabilizing the vacuum with a light Higgs)?
4. Again assuming the problem doesn't go away on its own with more accurate measurements of Higgs, top, etc-- what will be the next steps for discerning which of the vacuum-stabilizing candidate theories is real? Have any of the candidates had their parameter space significantly excluded by the LHC work so far, or do any of the candidates have important parameter space the LHC might be able to detect in future?
My very dull impression from blog comment sections is that the next step is to pick a supersymmetry model with a 125 GeV Higgs and start looking for whatever it predicts to be the lightest supersymmetric partner (the word "gluino" keeps getting kicked around)?
Last edited by a moderator: May 5, 2017
2. Dec 14, 2011
### tom.stoer
Re: 125 GeV Higgs and "Vacuum Instability"
If I understand the FRG analysis correctly, a 125 GeV Higgs and therefore the whole SM is perfectly valid w/o any 'beyond-the-standard-model-physics'.
3. Dec 14, 2011
### Coin
Re: 125 GeV Higgs and "Vacuum Instability"
Wait-- I'm now double checking the things I claim above (that 125 GeV Higgs implies vacuum instability) and am having trouble finding claims that this is the case but that do not originate with Philip Gibbs or Vixra. Further checking Gibbs seems to think of himself as a bit of an "outsider" and does not appear to be currently affiliated with an academic institution. This of course does not mean he is not right, but above I take things said on his blog at absolute face value where perhaps I should have been treating them as one person's informed opinion...
On a cursory check for real references: This published paper from 1989 seems like it would contain relevant information, but is behind a paywall:
Electroweak Higgs potential and vacuum stability
This paper was submitted to the Arxiv yesterday (!), it appears to have a good conventional academic pedigree but is obviously unpublished.
Higgs mass implications on the stability of the electroweak vacuum
This paper I will need to read more closely, but it looks to be quite good and possibly the answer to my question (1) above (i.e. where to find a good technical overview of how the vacuum instability stuff works). It agrees with what Gibbs' claims from his blog but cites a much higher (orders of magnitude higher) number for at what energy the instability would come into play.
A question I maybe should have asked earlier: Is the vacuum stability problem with a 125 GeV Higgs indeed something scientific consensus would likely treat as "real", or as a problem that needs solving?
(EDIT: This post was written before seeing Tom's reply.)
4. Dec 14, 2011
### torquil
Re: 125 GeV Higgs and "Vacuum Instability"
Even with the "maybe", it is an overstatement to say that the "Higgs has been found (maybe)".
At the recent seminar, they explicitly stressed that it has not been found and that it has not been excluded. They have reduced the range of allowable energies.
5. Dec 14, 2011
### Haelfix
Re: 125 GeV Higgs and "Vacuum Instability"
The answer is unfortunately somewhat ambiguous.
A relevant paper is the following:
http://arxiv.org/abs/0906.0954v2
If we were to assume that what we are seeing is indeed the standard model Higgs at 125 +- 2 GeV -and that's far from given-, the stability of the vacuum is going to be a problem that a lot of people will look at very seriously in the next few years.
The problem is that determining the exact details is quite fuzzy in the 120-130 GeV range, and we will need to make more accurate measurements of the top Quark mass and several other Electroweak parameters before we can say with great confidence what the story is.
Anyway, almost all new physics (and even understanding old physics more accurately) will change the story significantly. For instance, whatever it is that explains dark matter will probably alter the conclusions.
6. Dec 15, 2011
### marcus
Re: 125 GeV Higgs and "Vacuum Instability"
Coin, Here is a part of what you quoted. I was pleased to see the reference to the Shaposhnikov and Wetterich paper!
==Gibbs excerpt==
As it turns out a Higgs mass of 125 GeV is quite a borderline case... if the mass of the Higgs turns out to be 120 GeV despite present rumours to the contrary then the stability problem would be a big deal.
... If on the other hand the Higgs mass was found at 130 GeV or more, then the stability problem would be no issue.
125 GeV leaves us in the uncertain region where more research and better measurements of the top mass will be required...
At 126 GeV the vacuum might remain stable up to Plank energies (see e.g. Shaposhnikov and Wetterich). If this is the case then there is nothing to worry ...
==endquote==
Shaposhnikov and Wetterich predicted higgs mass 126 GeV back in 2009, from a model where gravity is asym. safe and no new physics beyond SM is needed. They proposed the 126 figure as a signature. If that figure is confirmed next year, or anything close, I expect their idea ("read my lips, no new physics up to Planck scale") is one that is bound to be taken seriously.
This is in line with Tom's post referring to Functional Renormalization Group analysis.
The way Shaposhnikov and Wetterich express this is to describe SM + asym.safe gravity working as a fundamental (rather than merely effective) theory all the way to planck scale.
I remember both Hermann Nicolai and Steven Weinberg discussing this scenario in Summer of 2009.
Nicolai in the XXV Max Born symposium at Wroclaw and Weinberg in a talk he gave at CERN in early July. It seems to me very likely that these people and others like them will be further studying this possibility now that hints are seen of the 126 GeV signature discussed earlier.
Last edited: Dec 15, 2011
7. Dec 15, 2011
### Coin
Re: 125 GeV Higgs and "Vacuum Instability"
Thanks all for the explanations/responses.
8. Dec 15, 2011
### Haelfix
Re: 125 GeV Higgs and "Vacuum Instability"
So there have been several papers in the past few days on this subject. The newest was this one today:
http://arxiv.org/abs/1112.3112
Which argues for an instability at the 10^9-10^11 GeV scale, which is roughly consistent with what you can eyeball from the Ellis paper or from the results of the Cern paper in this thread (which is excellent).
So the standard model is probably broadly inconsistent if you run it up all the way to the Planck scale by the renormalization group, however the obvious way out of this is to argue that the vacuum is metastable (which is the scenario which the CERN paper favors) and doesn't decay for at least the lifetime of the universe.
Anyway, the punchline is that this analysis is woefully incomplete. We need better resolution on the electroweak parameters, and ideally we need lattice studies.
9. Feb 19, 2013
### abledoc
Re: 125 GeV Higgs and "Vacuum Instability"
Big bang event was a case of one pseudo-vacuum tunnelling into a lower energy (?pseudo)vacuum and we know that this sort of thing happened. May be we are at the lowest energy level. Thinking this way, this sort of Higgs mass was destined from an anthropic viewpoint
10. Apr 27, 2014
### kneemo
http://arxiv.org/abs/1404.4709
The Higgs vacuum is unstable
Archil Kobakhidze, Alexander Spencer-Smith
(Submitted on 18 Apr 2014 (v1), last revised 22 Apr 2014)
Abstract:
So far, the experiments at the Large Hadron Collider (LHC) have shown no sign of new physics beyond the Standard Model. Assuming the Standard Model is correct at presently available energies, we can accurately extrapolate the theory to higher energies in order to verify its validity. Here we report the results of new high precision calculations which show that absolute stability of the Higgs vacuum state is now excluded. Combining these new results with the recent observation of primordial gravitational waves by the BICEP Collaboration, we find that the Higgs vacuum state would have quickly decayed during cosmic inflation, leading to a catastrophic collapse of the universe into a black hole. Thus, we are driven to the conclusion that there must be some new physics beyond the Standard Model at energies below the instability scale ΛI∼109 GeV, which is responsible for the stabilisation of the Higgs vacuum.
11. Apr 28, 2014
### Blackforest
What is exactly meant with vacuum instability? Do we have to understand that pair of words as if it would be related to a kind of chemical potential? As it would be related to a kind of thermodynamic (metastable) state?
Thanks in advance for elaborating a little bit around these questions (level: amateur) - and if it is not the correct forum to ask such question, please feel free to move it somewhere else! | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8778773546218872, "perplexity": 965.0327310728829}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794866326.60/warc/CC-MAIN-20180524131721-20180524151721-00154.warc.gz"} |
http://mathhelpforum.com/statistics/8050-probability-question-tv-show.html | # Math Help - Probability question from a TV show
1. ## Probability question from a TV show
There are five boxes, one of which contains a prize. You choose one without opening it, and the probability for a win is 1/5. OK. Now you open one of the other boxes, and you find out that the prize is NOT in that box. Can you now say that the probability for you to win is 1/4?
The show is "Deal or no Deal" btw.
2. Originally Posted by changing_seasons
There are five boxes, one of which contains a prize. You choose one without opening it, and the probability for a win is 1/5. OK. Now you open one of the other boxes, and you find out that the prize is NOT in that box. Can you now say that the probability for you to win is 1/4?
The show is "Deal or no Deal" btw.
As long as that box was chosen at random, and not pre-selected to be one
without the prize. If the latter you have a variant of the Monte-Hall problem,
and the probability that the box you have contains the prize is still 1/5.
RonL
(A Bayesian analysis also gives the p=1/4 result confirming intuition,
but note 1 in 5 times the box opened will contain the prize) | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.890250027179718, "perplexity": 858.5716911807602}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-42/segments/1413507443869.1/warc/CC-MAIN-20141017005723-00165-ip-10-16-133-185.ec2.internal.warc.gz"} |
https://www.allaboutcircuits.com/technical-articles/analysis-of-four-dc-dc-converters-in-equilibrium/ | Technical Article
# Analysis of Four DC-DC Converters in Equilibrium
June 06, 2015 by Editorial Team
## This technical article deals with the analysis of the four fundamental DC-DC converters (or choppers) in equilibrium. The four basic DC-DC converters considered for analysis are the following: Buck Converter, Boost Converter, Buck-Boost Converter and Ćuk Converter.
This technical article deals with the analysis of the four fundamental DC-DC converters (or choppers) in equilibrium. The four basic DC-DC converters considered for analysis are: Buck Converter, Boost converter, Buck-Boost Converter and Ćuk Converter.
Beginner
#### Introduction
With the voltage and current relationship, influencing factors for the equilibrium of the converter such as the capacitor’s voltage ripple and inductor voltage-second balance condition have been derived.
Similar analysis can also be done for the other types of converters. Basic circuit diagram of all the fundamental converters are shown in Figure 1. They consist of the same basic elements. The building blocks of these converters are DC supply Vs, load, diode D, power electronics switch S, inductor L, and capacitor C.
##### Figure 1. Basic converters
It is worth noticing that any converters work in two distinct modes with respect to the inductor current: the continuous conduction mode (CCM) and discontinuous conduction mode (DCM). When the inductor current is always greater than zero, it is in CCM. When the average inductor current is too low due to the high-load resistance or low-switching frequency, then the converter is in DCM.
The CCM is preferable for high efficiency and efficient use of semiconductor switches and passive components. The DCM requires a special control since the dynamic order of the converter is reduced. Thus, it is required to find out the minimum value of the inductor to maintain the CCM.
Assume that the inductor and capacitor are pure (i.e. no resistive component). However, there is still what we call a small-ripple approximation. In an efficient converter, the output voltage ripple is small. It is assumed that the load is resistive and the DC component of the output voltage has no ripples, or simply the DC output has a fixed value as shown in Fig.2 for making the analysis easier.
So,
∆VO = 0 and VO = V.
#### Buck Converter
##### Figure 3. Buck converter circuit diagram
As shown in Fig. 3, the buck converter consists of a DC supply or a rectified AC output, two switches i.e. D (diode) and S (can be semi-controlled or fully-controlled power electronics switches), two-pole low-pass filter (L and C) and a load. Let the duty ratio of switch S be
$$D\,=\,\frac{T_{ON}}{T}$$
where
T=TON+TOFF.
Buck converter is mostly used for DC drives systems e.g. electric vehicles, electric traction and machine tools.
This circuit can be studied in two different modes. The first mode is when the switch S is on while the second mode is when the switch S is off. The circuit diagrams when the switch S is on and off are given in Fig. 4 and Fig. 5 respectively.
$$Voltage\,across\,the\,Inductor\,=\,V_{L} \,=\,L\frac{dI}{dt}$$ where I=IC+IO.
$$Load\,Current\,=\,I_{O}=\frac{V_{O}}{R}$$
##### Figure 4. Buck converter circuit when switch S is on (Mode-I)
When the switch S is on and applying the Kirchhoff‘s voltage law (KVL), we can get,
$$V_{S}=V_{L}+V_{O}$$
$$\Rightarrow V_{S}\,=\,L\frac{di}{dt} + V_{O}\;\;and\;\;V_{O}=V_{C}.$$
##### Figure 5. Buck converter circuit when switch S is off (Mode-II)
When the switch S is off, the KVL in Fig.5 gives,
$$V_{L} + V_{O} = 0$$
$$\Rightarrow V_{O}\,=\,-L\frac{di}{dt}.$$
As the output voltage is assumed constant by the small-ripple approximation,
$$\Rightarrow L\frac{di}{dt}\,=\,constant$$
$$\Rightarrow \frac{di}{dt}\,=\,constant$$
$$\Rightarrow slope \,of\,the\, inductor \,current\, is\, constant.$$
Waveforms of the voltage and current during the one-cycle period are shown in Fig. 6.
##### Figure 6. Supply current Is, diode current ID, inductor current I, and inductor voltage VL waveforms respectively (buck converter)
As
$$V_{L}=L\frac{di}{dt},$$
$$\Rightarrow (I_{max}-I_{min})_{SWITCH-ON}\,=\,\frac{V_{S}-V_{O}}{L}DT$$ [Equation 1]
and
$$(I_{min}-I_{max})_{SWITCH-OFF}\,=\,-\frac{V_{O}}{L}(1-DT)$$
$$\Rightarrow Average\, Inductor\, Current=\frac{I_{max}+I_{min}}{2}=I$$
From Fig. 6 and from the steady-state perspective, magnitude of the inductor current increment during switch on is equal to the inductor current decrement during switch off; i.e. $$\left | (I_{max}-I_{min})_{SWITCH-ON} \right |=\left| (I_{max}-I_{min})_{SWITCH-OFF}\right|.$$
$$\Rightarrow |\frac{V_{S}-V_{O}}{L}DT|\,=\,|-\frac{V_{O}}{L}(1-DT)|$$
$$\Rightarrow V_{O}\,=\,DV_{S}$$
In the case of the buck converter, output voltage is directly dependent on the duty cycle and the input voltage.
Alternatively, this can also be derived as follows:
As the net inductor current is nil i.e. $$I(T)-I(0)=0;$$
$$\Rightarrow \frac{1}{L}\int_{0}^{T}V_{L}dt=0;$$
$$\Rightarrow$$ Area under the inductor voltage and time is zero. This is called the inductor volt-sec balance for the one cycle of voltage with respect to time under steady-state condition.
$$\Rightarrow (V_{S}-V_{O}) DT\,-\, V_{O}(1-DT)=0$$
$$\Rightarrow V_{O}=DV_{s}$$ [Equation 2]
Now, the average capacitor current is zero for the complete cycle due to the capacitor charge balance.
We know that
$$I_{C}\,=\,C\frac{dv}{dt};$$
For a steady-state condition, net change in the capacitor voltage must be nil.
By integrating the above equation for a complete cycle, we can get,
$$V_{C}(T)\,-\,V_{C}(0)\,=\,\int_{0}^{T}\frac{I_{C}}{C}dt=0;$$
Hence, the total area under the capacitor current and time curve must be zero. This implies that for a complete cycle, average capacitor current IC=0;
$$\Rightarrow I_{O}\,=\,\frac{V_{O}}{R}\,=\,I+I_{C}=I{;}$$
$$\Rightarrow I\,=\,I_{O}\,=\,\frac{V_{O}}{R}\,=\,\frac{I_{max}+I_{min}}{2}$$ [Equation 3]
From Equations 1 and 3, we can get,
$$I_{max}\,=\,DV_{S}(\frac{1}{R}\,+\,\frac{1-D}{2L}T){;}$$ [Equation 4]
and
$$I_{min}\,=\,DV_{S}(\frac{1}{R}\,-\,\frac{1-D}{2L}T){;}$$ [Equation 5]
The capacitor current is alternating for the continuous inductor current as given in Fig. 7.
##### Figure 7. Capacitor current
The ripple voltage in the capacitor which depends on the area under the current-time curve is obtained. The charge accumulated during the half cycle is equal to the area under the curve for the capacitor current and time.
$$\Rightarrow ∆Q\,=\,\frac{1}{2}\frac{(I_{max}-I_{min})}{2}(\frac{T}{2})$$
$$\Rightarrow P_{k}\,-\,P_{k} output \,voltage \,ripple\,=\, ∆V_{C}\,=\,\frac{∆Q}{C}\,=\,\frac{D(1-D)V_{S}}{8CL}T^{2}$$ [Equation 6]
#### Boost Converter
The circuit diagram of the boost converter is shown in Fig. 8. One of the applications of the boost converter is for radar or ignition systems.
##### Figure 8. Boost converter circuit diagram
The equivalent circuit during switch on and off condition of the switch S is shown in Fig. 9 and Fig. 10 respectively.
##### Figure 10. Boost converter circuit when switch S is off (Mode-II)
When the switch S is on,
VS=VL
$$\Rightarrow L\frac{di}{dt}\,=\,V_{S}\,=\,Constant \,supply \,voltage$$
$$\frac{di}{dt}=constant$$
$$\Rightarrow Current \,increases\, with\, constant\, slope.$$
When the switch S is off,
$$\Rightarrow V_{S}\,=\,V_{L}\,+\,V_{C}$$
$$\Rightarrow L\frac{di}{dt}\,=\,V_{S}\,-\,V_{C};$$
$$\Rightarrow \frac{di}{dt}\,=\,\frac{(V_{S}-V_{C})}{L}.$$
Now, current decreases and must reach at a value equal to the value of the current at the initial stage when switch S is just switched on according to steady-state stability.
Current increment during switch on
$$=\,I_{max}\,-\,I_{min}\,=\,\frac{V_{S}}{L}DT$$ [Equation 7]
Current decrement during switch off
$$= \,I_{min}\,-\,I_{max}\,=\,\frac{V_{S}\,-\,V_{C}}{L}(1-D)T$$ [Equation 8]
##### Figure 11. Supply current, diode current, inductor current, and inductor voltage respectively (boost converter)
According to the inductor volt-sec balance,
$$V_{S}.DT=(V_{S}-V_{C}).(1-D)T$$
$$\Rightarrow V_{C}=\frac{V_{S}}{1-D}$$
$$\Rightarrow V_{O}=\frac{V_{S}}{1-D}$$ [Equation 9]
$$Average\,Inductor\,Current = \frac{I_{max}+I_{min}}{2}$$
$$Input\,Power,\,P_{IN}=\frac{I_{max}+I_{min}}{2}V_{S}$$
$$Output\,Power,\,P_{OUT}=\frac{V_{O}^{2}}{R}=\frac{V_{S}^{2}}{(1-D)^{2}R}$$ (from Equation 9) [Equation 10]
Assume there is no switching loss,
$$\Rightarrow P_{IN}=P_{OUT}$$
$$\Rightarrow I_{max}+I_{min}\,=\,2\frac{V_{S}}{R(1-D)^{2}}$$ [Equation 11]
From Equations 11 and 8, we can get,
$$I_{min}\,=\,\frac{V_{S}}{R{(1-D)}^{2}}-\frac{V_{S}}{2L}DT$$ [Equation 12]
$$I_{max}=\frac{V_{S}}{R(1-D)^{2}}+\frac{V_{S}}{2L}DT$$ [Equation 13]
For the continuous conduction mode (CCM), $$I_{min}=0;$$
$$\Rightarrow L_{min}=\frac{D(1-D)^{2}}{2}TR$$ [Equation 14]
Ripple voltage across the capacitor
$$= ∆V_{C}=\frac{∆Q}{C}$$
Where, ∆ Q is the charge accumulated during the switch-on condition.
$$\Rightarrow ∆V_{C}={DT}\frac{V_{O}}{R}.\frac{1}{C}$$
$$\Rightarrow ∆V_{C}=\frac{DTV_{S}}{(1-D)RC}$$ [Equation 15]
#### Buck-Boost Converter
This converter is an inverting DC-to-DC converter i.e. polarity of the output voltage is reversed compared to the input supply. Thus, it is a negative-output buck-boost converter.
##### Figure 12. Buck-boost converter circuit diagram
Let the capacitor be totally charged up before switching on the switch S. When the switch S is closed as shown in Fig. 13,
-VS + VL = 0
$$\Rightarrow V_{S}=V_{L}=L\frac{di}{dt}$$
##### Figure 13. Buck-boost converter circuit when switch S is on (Mode-I)
Also,
-VC + VO = 0
$$\Rightarrow V_{O}=V_{C}$$
##### Figure 14. Boost converter circuit when switch S is off (Mode-II)
Now, when the switch S is opened as given in Fig. 14,
+VL + VC = 0
$$L\frac{di}{dt}+V_{C}=0$$
$$\frac{di}{dt}=-\frac{V_{C}}{L}$$
Waveforms for the voltage and current for buck-boost converter are shown in Fig. 15.
##### Figure 15. Supply current, diode current, inductor current, and inductor voltage respectively (buck-boost converter)
Now, from the waveforms given in Fig.15,
When switch S is on,
$$Rise\,in\,the\,Inductor\,Current = I_{max}-I_{min}=\frac{V_{S}}{L}DT$$ [Equation 16]
And, when switch S is off,
$$Fall\,in\,the\,Inductor\,Current = I_{min}-I_{max}=-\frac{V_{C}}{L}(1-D)T$$ [Equation 17]
Equating the Equations 16 and 17, we can get,
$$\frac{V_{S}}{L}DT=\frac{V_{C}}{L}(1-D)T$$
$$\Rightarrow V_{O}=V_{C}=\frac{D}{1-D}V_{S}$$ [Equation 18]
$$Average\,Inductor\,Current = \frac{I_{min}+I_{max}}{2}$$
As there is no supply current during the switch-off condition,
$$Input\,Power = P_{IN}=\frac{I_{max}+I_{min}}{2}DV_{S}$$
And,
$$Output\,Power = P_{OUT}=\frac{{V_{O}}^{2}}{R}$$
If there is no switching loss,
PIN=POUT
$$\Rightarrow I_{max}+I_{min}=2D\frac{V_{S}}{R(1-D)^{2}}$$ [Equation 19]
From Equations 17 and 19, we can get,
$$I_{min}=D\frac{V_{S}}{R(1-D)^{2}}-\frac{V_{S}}{2L}D$$ [Equation 20]
and
$$I_{max}=D\frac{V_{S}}{R(1-D)^{2}}+\frac{V_{S}}{2L}DT$$ [Equation 21]
For CCM, Imin=0
$$\Rightarrow Value\,of\,the\,Minimum\,Inductance\,for\,CCM= L_{min}=\frac{{(1-D)}^{2}}{2}TR$$
$$Ripple\,Voltage\,Across\,the\,Capacitor = ∆V_{C}=\frac{∆Q}{C}=\frac{DTI_{O}}{C}=\frac{DTV_{O}}{RC}=\frac{{D}^{2}TV_{S}}{(1-D)RC}$$
as
$$V_{O}=\frac{D}{(1-D)}V_{S}$$
Note:
When D < 0.5, it acts as a step-down converter or a buck converter.
When D > 0.5, it acts as a step-up converter or a boost converter.
And when D = 0.5, input and output voltages are the same i.e.VO=VS.
That is why buck-boost converters are also called as DC transformers due to the same role in the case in AC.
The converter discussed above is a negative-output buck-boost converter. But in some applications, reversal of polarity is not allowed. In such cases, we require a positive-output converter whose configuration diagram is given below:
#### Ćuk Converter
##### Figure 17. Ćuk converter circuit diagram
A Ćuk converter can be obtained by cascading the boost converter followed by the buck converter. It also has a negative-output polarity as in the case of the simple buck-boost converter. But we have assumed here that the polarity of the output is positive.
##### Figure 18. Ćuk converter circuit when switch S is on (Mode-I)
When switch S is on, the circuit will be reduced as shown in Fig. 18.
From Fig.18,
VS=VL
$$\Rightarrow \frac{di}{dt}=\frac{V_{S}}{L}$$ (Current Increases)
$$\Rightarrow ∆I_{L}=V_{S}.\frac{DT}{L}$$ [Equation 22]
For inductor LO,
VC+VLo+VO=0
$$\Rightarrow \frac{dI_{Lo}}{dt}=-\frac{1}{L}(V_{C}+V_{O})$$ (Current Increases)
If the VC and VCO polarities are in the reverse direction of the loop considered and the current must increase, the current must actually increase in the opposite direction of the assumed direction.
##### Figure 19. Ćuk converter circuit when switch S is off (Mode-II)
When switch S is off, the circuit will reduced to the configuration shown in Fig.19.
From Fig. 19,
-VS+VL+VC=0
$$\Rightarrow \frac{di}{dt}=\frac{-(V_{C}-V_{S})}{L}$$ (Current decreases)
For inductor Lo,
VLo-VCo=0
$$\Rightarrow \frac{dI_{Lo}}{dt}=\frac{V_{Co}}{L}$$ (Current decreases)
Note that the current actually decreases in the opposite direction. So, $$\frac{di}{dt}$$ must be positive.
Also, from the Figure 19, it is clear that capacitor C is charging up for the time (1-D).T.
$$\Rightarrow ∆V_{C}=\frac{∆Q}{C}=\frac{1}{C}\int_{0}^{(1-D)T}I_{C}dt=I_{S}\frac{(1-D)T}{C}$$ [Equation 23]
(As IC=IS from Fig.19)
By volt-sec balance,
$$V_{S}.DT=(V_{C}-V_{S}).(1-D)T$$
$$\Rightarrow V_{C}=\frac{1}{(1-D)}V_{S}$$ [Equation 24]
And, $$(V_{C}+V_{Co})DT=-V_{Co}(1-D)T$$
$$\Rightarrow V_{C}D=-V_{Co}$$
From Equation 23,
$$\Rightarrow V_{Co}=-\frac{D}{1-D}V_{S}$$
Also, VO=-VCo
$$V_{O}=\frac{D}{(1-D)}V_{S}$$ [Equation 25]
From the capacitor charge balance condition and from the Fig. 19,
IL=IS and ILo=IO
Now, assume there is no switching loss within the converter,
PIN = VS IS
POUT = VO IO
$$\Rightarrow V_{S}I_{S}=\frac{D}{1-D}V_{S}I_{O}$$
$$\Rightarrow I_{S}=\frac{D}{(1-D)}I_{O}$$ [Equation 26]
Where $$I_{O} =\frac{ V_{O}}{ R}$$
According to the small-ripple approximation, ripples due to load itself are negligible. So,∆ICo=∆ILo.
##### Figure 20. Switch (S) voltage, capacitor (C) voltage, inductor (L) current, inductor (LO) current, capacitor (CO) voltage, capacitor (C) current, and load current respectively for the Ćuk converter
From Fig. 20, average charging current for CO which flows for $$\frac{T}{2}$$ duration $$=\frac{∆I_{Lo}}{4}.$$ It is also shown in Fig. 21. Thus, ripple voltage across the capacitor CO is given by
$$∆V_{Co}=\frac{1}{Co}\int_{0}^{\frac{T}{2}}I_{Co} dt=\frac{1}{Co}\int_{0}^{\frac{T}{2}}\frac{∆I_{Lo}}{4} dt =\frac{∆I_{Lo}}{8Co}T$$ [Equation 27]
##### Figure 21. Inductor (LO) current and capacitor (CO) voltage
As, $$∆I_{Lo}=\frac{V_{O}(1-D)T}{L_{Lo}}$$
$$∆V_{Co}=D\frac{V_{S}}{8Co L_{Lo}} T^{2}$$ [Equation 28]
##### Figure 22. Inductor( L) current
From Fig. 22, it is clear that for continuous inductor current i.e. for CCM, $$∆I_{L}=2I_{Lavg}= 2I_{S},$$ where ILavg is the average current through inductor L.
(Due to capacitor charge balance, IS=ILavg)
From Equations 26 and 22,
$$D\frac{V_{S}T}{L}=∆I_{L}=2I_{S}=2\frac{D}{1-D}I_{O}=2{(\frac{D}{1-D})}^{2}(\frac{V_{S}}{R})$$
$$\Rightarrow L_{min}=\frac{(1-D)^{2}R}{2D}T$$
Similarly, from Fig. 20, for CCM
$$∆I_{Lo}=2I_{Loavg}=2 I_{O}=2\frac{V_{O}}{R}$$
Where ILoavg is the average current through the inductor LO.
As,
$$∆I_{Lo}=DV_{S}\frac{T}{L_{Lo}}$$
$$\Rightarrow L_{Lo_{min}}=\frac{(1-D)RT}{2}$$
If VC is the capacitor average voltage across C, the CCM from Fig. 20,
Capacitor (C) Ripple Voltage = ∆V= 2 VO = 2 IO R
From Equation 23,
$$\Rightarrow \frac{I_{S}(1-D)T}{C}=2I_{O}R$$
$$C_{min}=\frac{DT}{2R}$$
In the CCM from Fig. 18, if VCo is the capacitor average voltage across CO,
Capacitor CO Ripple Voltage = ∆VCo = 2 VO = 2 IO R
From Equation 28,
$$\Rightarrow D\frac{V_{S}}{8C_{O}L_{O}}T^{2}=2V_{O}=2\frac{D}{1-D}V_{S}$$
$$\Rightarrow Co_{min}=\frac{1}{8R}T$$
The main benefit of the Ćuk converter is that you can control the continuous current at both in the input and output of the converter as it is based on the capacitor energy transfer. It has a low-switching loss making it more highly efficient. The downside of this converter is that it includes a high number of reactive components (L and C) and heavy current stresses on the components. And since the capacitor C provides a transfer of energy, ripples in the capacitor (C) current are high.
• Share
### You May Also Like
• T
trungkiendt9 October 01, 2016
Great article. Thank so lot.
Like.
• imre July 22, 2017
Thanks interesting article. I noticed a few typos in the equations:
After $$Equation\;1.$$ the SWITCH-OFF current delta should be: $$(I_{min}-I_{max})_{SWITCH-OFF}\,=\,-\frac{V_{O}}{L}(1-D)T$$
For the boost converter the voltage-sec balance before $Equation\;9.$ should be: $$V_{S}DT=-(V_{S}-V_{C})(1-D)T$$
Also I would like to better understand how we get the buck converter ripple voltage. In Figure 7. we have increasing capacitor current until $DT$ then decreasing current until $$T$$, but the following integration is done for the $$\frac{T}{2}$$ period. How do the two periods $$DT$$ and $$\frac{T}{2}$$ relate to each other? How does the ripple voltage-time curve look like, does it consist of a $$0..\frac{T}{2}$$ and a $$\frac{T}{2}..T$$ symmetric periods or a $$0..DT$$ and a $$DT..T$$ (asymmetric) period?
Thanks
Like. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8550867438316345, "perplexity": 2909.0162943432906}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496670743.44/warc/CC-MAIN-20191121074016-20191121102016-00264.warc.gz"} |
http://www.oalib.com/search?kw=Sharon%20M.%20Loverde&searchField=authors | Home OALib Journal OALib PrePrints Submit Ranking News My Lib FAQ About Us Follow Us+
Title Keywords Abstract Author All
Publish in OALib Journal
ISSN: 2333-9721
APC: Only $99 Submit 2020 ( 51 ) 2019 ( 657 ) 2018 ( 709 ) 2017 ( 703 ) Search Results: 1 - 10 of 402687 matches for " Sharon M. Loverde " All listed articles are free for downloading (OA Articles) Page 1 /402687 Display every page 5 10 20 Item Physics , 2006, DOI: 10.1103/PhysRevLett.98.237802 Abstract: We consider a mixture of one neutral and two oppositely charged types of molecules confined to a surface. Using analytical techniques and molecular dynamics simulations, we construct the phase diagram of the system and exhibit the coexistence between a patterned solid phase and a charge-dilute phase. The patterns in the solid phase arise from competition between short-range immiscibility and long-range electrostatic attractions between the charged species. The coexistence between phases leads to observations of stable patterned domains immersed in a neutral matrix background. Physics , 2006, DOI: 10.1063/1.2181573 Abstract: A binary mixture of oppositely charged components confined to a plane such as cationic and anionic lipid bilayers may exhibit local segregation. The relative strength of the net short range interactions, which favors macroscopic segregation, and the long range electrostatic interactions, which favors mixing, determines the length scale of the finite size or microphase segregation. The free energy of the system can be examined analytically in two separate regimes, when considering small density fluctuations at high temperatures, and when considering the periodic ordering of the system at low temperatures (F. J. Solis and M. Olvera de la Cruz, J. Chem. Phys. 122, 054905 (2000)). A simple Molecular Dynamics simulation of oppositely charged monomers, interacting with a short range Lennard Jones potential and confined to a two dimensional plane, is examined at different strengths of short and long range interactions. The system exhibits well-defined domains that can be characterized by their periodic length-scale as well as the orientational ordering of their interfaces. By adding salt, the ordering of the domains disappears and the mixture macroscopically phase segregates in agreement with analytical predictions. Physics , 2007, DOI: 10.1063/1.2888980 Abstract: We study ion condensation onto a patterned surface of alternating charges. The competition between self-energy and ion-surface interactions leads to the formation of ionic crystalline structures at low temperatures. We consider different arrangements of underlying ionic crystals, including single ion adsorption, as well as the formation of dipoles at the interface between charged domains. Molecular dynamic simulation illustrates existence of single and mixed phases. Our results contribute to understanding pattern recognition, and molecular separation and synthesis near patterned surfaces. Physics , 2011, DOI: 10.1088/1475-7516/2011/08/003 Abstract: Primordial non-Gaussianity has emerged as one of the most promising probes of the inflationary epoch. While the cosmic microwave background and large-scale halo bias currently provide the most stringent constraints on the non-Gaussian parameter fNL, the abundance of dark matter halos is a complementary probe which may allow tests of Gaussianity which are independent of the precise form of non-Gaussian initial conditions. We study the halo mass function in N-body simulations with a range of non-Gaussian initial conditions. In addition to the usual fNL model, we consider gNL Phi^3-type non-Gaussianity and models where the 4-point amplitude tauNL is an independent parameter. We introduce a new analytic form for the halo mass function in the presence of primordial non-Gaussianity, the "log-Edgeworth" mass function, and find good agreement with the N-body simulations. The log-Edgeworth mass function introduces no free parameters and can be constructed from first principles for any model of primordial non-Gaussianity. Physics , 2010, DOI: 10.1088/1475-7516/2011/11/009 Abstract: Large-scale clustering of highly biased tracers of large-scale structure has emerged as one of the best observational probes of primordial non-Gaussianity of the local type (i.e. f_{NL}^{local}). This type of non-Gaussianity can be generated in multifield models of inflation such as the curvaton model. Recently, Tseliakhovich, Hirata, and Slosar showed that the clustering statistics depend qualitatively on the ratio of inflaton to curvaton power \xi after reheating, a free parameter of the model. If \xi is significantly different from zero, so that the inflaton makes a non-negligible contribution to the primordial adiabatic curvature, then the peak-background split ansatz predicts that the halo bias will be stochastic on large scales. In this paper, we test this prediction in N-body simulations. We find that large-scale stochasticity is generated, in qualitative agreement with the prediction, but that the level of stochasticity is overpredicted by ~30%. Other predictions, such as \xi independence of the halo bias, are confirmed by the simulations. Surprisingly, even in the Gaussian case we do not find that halo model predictions for stochasticity agree consistently with simulations, suggesting that semi-analytic modeling of stochasticity is generally more difficult than modeling halo bias. Physics , 2011, DOI: 10.1088/1475-7516/2012/03/032 Abstract: A wide range of multifield inflationary models generate non-Gaussian initial conditions in which the initial adiabatic fluctuation is of the form (zeta_G + g_{NL} zeta_G^3). We study halo clustering in these models using two different analytic methods: the peak-background split framework, and brute force calculation in a barrier crossing model, obtaining agreement between the two. We find a simple, theoretically motivated expression for halo bias which agrees with N-body simulations and can be used to constrain g_{NL} from observations. We discuss practical caveats to constraining g_{NL} using only observable properties of a tracer population, and argue that constraints obtained from populations whose observed bias is <~ 2.5 are generally not robust to uncertainties in modeling the halo occupation distribution of the population. Physics , 2011, DOI: 10.1103/PhysRevLett.107.191301 Abstract: Models of inflation in which non-Gaussianity is generated outside the horizon, such as curvaton models, generate distinctive higher-order correlation functions in the CMB and other cosmological observables. Testing for violation of the Suyama-Yamaguchi inequality tauNL >= (6/5 fNL)^2, where fNL and tauNL denote the amplitude of the three-point and four-point functions in certain limits, has been proposed as a way to distinguish qualitative classes of models. This inequality has been proved for a wide range of models, but only weaker versions have been proved in general. In this paper, we give a proof that the Suyama-Yamaguchi inequality is always satisfied. We discuss scenarios in which the inequality may appear to be violated in an experiment such as Planck, and how this apparent violation should be interpreted. We analyze a specific example, the "ungaussiton" model, in which leading-order scaling relations suggest that the Suyama-Yamaguchi inequality is eventually violated, and show that the inequality always holds. Physics , 2011, Abstract: Cosmic inflation provides a mechanism for generating the early density perturbations that seeded the large-scale structures we see today. Primordial non-Gaussianity is among the most promising of few observational tests of physics at this epoch. At present non-Gaussianity is best constrained by the cosmic microwave background, but in the near term large-scale structure data may be competitive so long as the effects of primordial non-Gaussianity can be modeled through the non-linear process of structure formation. We discuss recent work modeling effects of a few types of primordial non-Gaussianity on the large-scale halo clustering and the halo mass function. More specifically, we compare analytic and N-body results for two variants of the curvaton model of inflation: (i) a "tauNL" scenario in which the curvaton and inflaton contribute equally to the primordial curvature perturbation and (ii) a "gNL" model where the usual quadratic fNL term in the potential cancels, but a large cubic term remains. Marilena LoVerde Physics , 2013, DOI: 10.1103/PhysRevD.89.023505 Abstract: If the entire post-inflationary patch is large compared to our Hubble volume even a small level of non-Gaussianity can cause statistics of the primordial curvature field in our Hubble volume to be biased by mode-coupling. We explicitly compute the variation of locally measured statistics of the primordial curvature$\zeta$from non-Gaussian mode coupling within a specific inflationary scenario: the curvaton model with a quadratic curvaton potential. This "super cosmic variance" is calculated in two ways: (i) as a super observer who has access to the curvature perturbation field across the entire post-inflationary patch and therefore sees local statistics as biased by mode coupling and (ii) as a local observer who sees the statistics of$\zeta$determined by the local values of quantities in their Hubble patch. The two calculations agree and show that in the quadratic curvaton model patch-to-patch differences in statistics of$\zeta\$ can be interpreted entirely as a shift in the value of the curvaton field at freeze out. Applying the same arguments to single-field slow-roll inflation gives a simple picture of how non-Gaussian mode-coupling between the curvature perturbations on very different physical scales must vanish in the attractor limit.
Marilena LoVerde Physics , 2014, DOI: 10.1103/PhysRevD.90.083530 Abstract: The large-scale distribution of cold dark matter halos is generally assumed to trace the large-scale distribution of matter. In a universe with multiple types of matter fluctuations, as is the case with massive neutrinos, the relation between the halo field and the matter fluctuations may be more complicated. We develop a method for calculating the bias factor relating fluctuations in the halo number density to fluctuations in the mass density in the presence of multiple fluctuating components of the energy density. In the presence of massive neutrinos we find a small but pronounced feature in the halo bias near the neutrino free-streaming scale. The neutrino feature is a small step with amplitude that increases with halo mass and neutrino mass density. The scale-dependent halo bias lessens the suppression of the small-scale halo power spectrum and should therefore weaken constraints on neutrino mass from the galaxy auto-power spectrum and correlation function. On the other hand, the feature in the bias is itself a novel signature of massive neutrinos that can be studied independently.
Page 1 /402687 Display every page 5 10 20 Item | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9464377164840698, "perplexity": 1330.5396780403928}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250592565.2/warc/CC-MAIN-20200118110141-20200118134141-00489.warc.gz"} |
https://www.mathworks.com/help/matlab/ref/quad2d.html | Numerically evaluate double integral — tiled method
## Syntax
``q = quad2d(fun,a,b,c,d)``
``q = quad2d(fun,a,b,c,d,Name,Value)``
``[q,E] = quad2d(___)``
## Description
example
````q = quad2d(fun,a,b,c,d)` approximates the integral of `fun(x,y)` over the planar region $a\le x\le b$ and $c\left(x\right)\le y\le d\left(x\right)$. The bounds `c` and `d` can each be scalars or function handles.```
example
````q = quad2d(fun,a,b,c,d,Name,Value)` specifies additional options with one or more `Name,Value` pair arguments. For example, you can specify `'AbsTol'` and `'RelTol'` to adjust the error thresholds that the algorithm must satisfy.```
````[q,E] = quad2d(___)` also returns an approximate upper bound on the absolute error, E = | q - I |, where I is the exact value of the integral.```
## Examples
collapse all
Integrate
`$y\mathrm{sin}\left(x\right)+x\mathrm{cos}\left(y\right)$`
over $-\pi \le x\le 2\pi$ and $0\le y\le \pi$.
```fun = @(x,y) y.*sin(x)+x.*cos(y); Q = quad2d(fun,pi,2*pi,0,pi)```
```Q = -9.8696 ```
Compare the result to the true value of the integral, $-{\pi }^{2}$.
`-pi^2`
```ans = -9.8696 ```
Integrate the function
`${\left[{\left(x+y\right)}^{1/2}{\left(1+x+y\right)}^{2}\right]}^{-1}$`
over the region $0\le x\le 1$ and $0\le y\le 1-x$. This integrand is infinite at the origin (0,0), which lies on the boundary of the integration region.
```fun = @(x,y) 1./(sqrt(x + y) .* (1 + x + y).^2 ); ymax = @(x) 1 - x; Q = quad2d(fun,0,1,0,ymax)```
```Q = 0.2854 ```
The true value of the integral is $\pi /4-1/2$.
`pi/4 - 0.5`
```ans = 0.2854 ```
`quad2d` begins by mapping the region of integration to a rectangle. Consequently, it may have trouble integrating over a region that does not have four sides or has a side that cannot be mapped smoothly to a straight line. If the integration is unsuccessful, some helpful tactics are leaving `Singular` set to its default value of `true`, changing between Cartesian and polar coordinates, or breaking the region of integration into pieces and adding the results of integration over the pieces.
For instance:
```fun = @(x,y)abs(x.^2 + y.^2 - 0.25); c = @(x)-sqrt(1 - x.^2); d = @(x)sqrt(1 - x.^2); quad2d(fun,-1,1,c,d,'AbsTol',1e-8,... 'FailurePlot',true,'Singular',false);```
```Warning: Reached the maximum number of function evaluations (2000). The result fails the global error test. ```
The failure plot shows two areas of difficulty, near the points `(-1,0)` and `(1,0)` and near the circle ${x}^{2}+{y}^{2}=0.25$.
Changing the value of `Singular` to `true` will cope with the geometric singularities at `(-1,0)` and `(1,0)`. The larger shaded areas may need refinement but are probably not areas of difficulty.
```Q = quad2d(fun,-1,1,c,d,'AbsTol',1e-8, ... 'FailurePlot',true,'Singular',true);```
```Warning: Reached the maximum number of function evaluations (2000). The result passes the global error test. ```
From here you can take advantage of symmetry:
```Q = 4*quad2d(fun,0,1,0,d,'Abstol',1e-8,... 'Singular',true,'FailurePlot',true)```
```Q = 0.9817 ```
However, the code is still working very hard near the singularity. It may not be able to provide higher accuracy:
```Q = 4*quad2d(fun,0,1,0,d,'Abstol',1e-10,... 'Singular',true,'FailurePlot',true);```
```Warning: Reached the maximum number of function evaluations (2000). The result passes the global error test. ```
At higher accuracy, a change in coordinates may work better.
```polarfun = @(theta,r) fun(r.*cos(theta),r.*sin(theta)).*r; Q = 4*quad2d(polarfun,0,pi/2,0,1,'AbsTol',1e-10);```
It is best to put the singularity on the boundary by splitting the region of integration into two parts:
```Q1 = 4*quad2d(polarfun,0,pi/2,0,0.5,'AbsTol',5e-11); Q2 = 4*quad2d(polarfun,0,pi/2,0.5,1,'AbsTol',5e-11); Q = Q1 + Q2;```
## Input Arguments
collapse all
Function to integrate, specified as a function handle. The function `Z = fun(X,Y)` must accept 2-D matrices `X` and `Y` of the same size and return a matrix `Z` of corresponding values. Therefore, the function must be vectorized (that is, you must use elementwise operators such as `.^` instead of matrix operators such as `^`). The inputs and outputs of the function must be either single or double precision.
Example: `@(x,y) x.^2 - y.^2`
Data Types: `function_handle`
x limits of integration, specified as scalars.
Data Types: `single` | `double`
Complex Number Support: Yes
y limits of integration, specified as scalars or function handles. Each limit can be specified as a scalar or a function handle. If the limits are specified as function handles, then they are functions of the x limit of integration ```ymin = @x c(x)``` and `ymax = @(x) d(x)`. The function handles `ymin` and `ymax` must accept matrices and return matrices of the same size with the corresponding values. The inputs and outputs of the functions must be either single or double precision.
Data Types: `single` | `double` | `function_handle`
Complex Number Support: Yes
### Name-Value Arguments
Specify optional comma-separated pairs of `Name,Value` arguments. `Name` is the argument name and `Value` is the corresponding value. `Name` must appear inside quotes. You can specify several name and value pair arguments in any order as `Name1,Value1,...,NameN,ValueN`.
Example: `quad2d(@(x,y) x.*y.^2, 0, 1, 0, 2, 'AbsTol',1e-3)` specifies the absolute tolerance for the integration as `1e-3`.
Absolute error tolerance, specified as the comma-separated pair consisting of `'AbsTol'` and a scalar.
`quad2d` attempts to satisfy ```ERRBND <= max(AbsTol,RelTol*|Q|)```. This is absolute error control when `|Q|` is sufficiently small and relative error control when `|Q|` is larger. A default tolerance value is used when a tolerance is not specified. The default value of `AbsTol` is 1e-5. The default value of `RelTol` is `100*eps(class(Q))`. This is also the minimum value of `RelTol`. Smaller `RelTol` values are automatically increased to the default value.
Relative error tolerance, specified as the comma-separated pair consisting of `'RelTol'` and a scalar.
`quad2d` attempts to satisfy ```ERRBND <= max(AbsTol,RelTol*|Q|)```. This is absolute error control when `|Q|` is sufficiently small and relative error control when `|Q|` is larger. A default tolerance value is used when a tolerance is not specified. The default value of `AbsTol` is 1e-5. The default value of `RelTol` is `100*eps(class(Q))`. This is also the minimum value of `RelTol`. Smaller `RelTol` values are automatically increased to the default value.
Maximum number of evaluations of `fun`, specified as the comma-separated pair consisting of `'MaxFunEvals'` and a scalar. Use this option to limit the number of times `quad2d` evaluates the function `fun`.
Toggle to generate failure plot, specified as the comma-separated pair consisting of `'FailurePlot'` and a numeric or logical `1` (`true`) or `0` (`false`). Set `FailurePlot` to `true` or `1` to generate a graphical representation of the regions needing further refinement when `MaxFunEvals` is reached. No plot is generated if the integration succeeds before reaching `MaxFunEvals`. The failure plot contains (generally) 4-sided regions that are mapped to rectangles internally. Clusters of small regions indicate the areas of difficulty in the integration.
Toggle to transform boundary singularities, specified as the comma-separated pair consisting of `'Singular'` and a numeric or logical `1` (`true`) or `0` (`false`). By default, `quad2d` employs transformations to weaken boundary singularities for better performance. Set `'Singular'` to `false` or `0` to turn these transformations off, which can provide a performance benefit on some smooth problems.
## Output Arguments
collapse all
Calculated integral, returned as a scalar.
Error bound, returned as a scalar. The error bound provides an upper bound on the error between the calculated integral q and the exact value of the integral I such that E = | q - I |.
## References
[1] L.F. Shampine, "MATLAB Program for Quadrature in 2D." Applied Mathematics and Computation. Vol. 202, Issue 1, 2008, pp. 266–274. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 11, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9398460388183594, "perplexity": 890.2829029228461}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585439.59/warc/CC-MAIN-20211021164535-20211021194535-00263.warc.gz"} |
http://www.dsplog.com/2008/03/18/symbol-error-rate-for-16psk/ | (4 votes, average: 4.50 out of 5)
# Symbol Error Rate for 16PSK
by on March 18, 2008
In this post, let us try to derive the symbol error rate for 16-PSK (16-Phase Shift Keying) modulation.
Consider a general M-PSK modulation, where the alphabets,
$alpha_{16PSK} =\sqrt{E_s}\left\{1,\ e^{\frac{j2\pi}{M}},\ e^{\frac{j4\pi}{M}},\ \ldots,\ e^{\frac{j2\pi(M-1)}{M}} \right}$ are used.
(Refer example 5-38 in [DIG-COMM-BARRY-LEE-MESSERSCHMITT])
Figure: 16-PSK constellation plot
## Deriving the symbol error rate
Let us the consider the symbol on the real axis, i.e
$s_0=\sqrt{E_s}$ .
The received symbol $y=\sqrt{E_s} + n$.
Where the additive noise $n$ follows the Gaussian probability distribution function,
$p(x) = \frac{1}{\sqrt{2\pi\sigma^2}}e^{\frac{-(x-\mu)^2}{2\sigma^2}$ with $\mu=0$ and $\sigma^2 = \frac{N_0}{2}$.
The conditional probability distribution function (PDF) of received symbol $y$ given $s_0$ was transmitted is:
$p(y|s_0)=\frac{1}{\sqrt{\pi N_0}}e^{-\frac{(x-\sqrt{E_s})^2}{N_0}}$.
As can be seen from the figure above, due to the addition of noise, the transmitted symbol gets spreaded. However, if the received symbol is present with in the boundary defined by the magenta lines, then the symbol will be demodulated correctly.
To derive the symbol error rate, the objective is to find the probability that the phase of the received symbol lies within this boundary defined by the magenta lines i.e. from $-\frac{\pi}{M}$ to $+\frac{\pi}{M}$.
For simplifying the derivation, let us make the following assumptions:
(a) The signal to noise ratio, $\frac{Es}{N_0}$ is reasonably high.
For a reasonably high value of $\frac{Es}{N_0}$, then the real part of the received symbol is not afected by noise i.e.,
$\Re{y}\approx\sqrt{E_s}$ and
the imaginary part of the received symbol is equal to noise, i.e.
$\Im{y}=n$.
(b) The value of M is reasonably high (typically M >4 suffice)
For a reasonably high value of M, the constellation points are closely spaced. Given so, the distance of the constellation point $s_0$ to the magenta line can be approximated as $\sqrt{E_s}sin(\frac{\pi}{M})$.
Figure: Distance between constellation points
Given the above two assumptions, it can be observed that the symbol $s_0$ will be decoded incorrectly, if the imaginary component of received symbol $y$ isgreater than $\sqrt{E_s}sin(\frac{\pi}{M})$. The probability of $y$ being greater than $\sqrt{E_s}sin(\frac{\pi}{M})$ is,
$p\left(\Im(y)>\sqrt{E_s}sin(\frac{\pi}{M})\right)=\frac{1}{\sqrt{\pi N_0}}\int_{\sqrt{E_s}sin(\frac{\pi}{M})}^{\infty}e^{-\frac{{\Im{y}}^2}{N_0}}dy$.
Changing the variable to $u=\frac{\Im{y}}{\sqrt{N_0}}$,
$p\left(\Im(y)>\sqrt{E_s}sin(\frac{\pi}{M})\right)=\frac{1}{\sqrt{\pi}}\int_{\sqrt{\frac{E_s}{N_0}}sin(\frac{\pi}{M})}^{\infty}e^{-u^2}du=\frac{1}{2}erfc\left[\sqrt{\frac{E_s}{N_0}}sin(\frac{\pi}{M})\right]$.
Note: The complementary error function, $erfc(x) = \frac{2}{\sqrt{\pi}}\int_x^\infty e^{-x^2}dx$.
Similarly, the symbol $s_0$ will be decoded incorrectly, if the imaginary component of received symbol $y$ is less than $-\sqrt{E_s}sin(\frac{\pi}{M})$. The probability of $y$ being less than $-\sqrt{E_s}sin(\frac{\pi}{M})$ is,
$p(\Im(y) <-\sqrt{E_s}sin(\frac{\pi}{M})=\frac{1}{\sqrt{\pi N_0}}\int_{-\infty}^{-\sqrt{E_s}sin(\frac{\pi}{M})}e^{\frac{{-\Im y}^2}{N_0}}dy=\frac{1}{2}erfc\left[\sqrt{\frac{E_s}{N_0}}sin(\frac{\pi}{M})\right]$.
The total probability of error given $s_0$ was transmittd is,
$p(e|s_0) =erfc\left[\sqrt{\frac{E_s}{N_0}}sin(\frac{\pi}{M})\right]$.
Total symbol error rate
The symbol will be in error, if atleast one of the symbol gets decoded incorrectly. Hence the total symbol error rate from M-PSK modulation is,
$P_{e,MPSK}=erfc\left[\sqrt{\frac{E_s}{N_0}}sin(\frac{\pi}{M})\right]$.
## Simulation model
Simple Matlab/Octave script for simulating transmission and recepetion of an M-PSK modulation is attached. It can be observed that the simulated symbol error rate compares well with the theoretical symbol error rate.
Figure: Symbol Error rate curve for 16PSK modulation
Hope this helps.
Krishna
References
D id you like this article? Make sure that you do not miss a new article by subscribing to RSS feed OR subscribing to e-mail newsletter. Note: Subscribing via e-mail entitles you to download the free e-Book on BER of BPSK/QPSK/16QAM/16PSK in AWGN.
abdelfatah elkhalifa abdelati April 26, 2012 at 12:25 pm
please help me in MATLAB simulink for rain attenuation and graphiical user interface . at least the block diagram . am MSC student
Krishna Sankar April 27, 2012 at 5:44 am
@abdelfatah : sorry, i do not have simulink and have not also not studied about rain attenuation
imran April 25, 2012 at 9:41 pm
please help me by giving me the Symbol Error Rate (SER)/ Bit error rate formula/Equation for 16-APSK modulation. because i need the theoretical equation for plotting and compare simulation result.
waiting someones feedback
Krishna Sankar April 26, 2012 at 5:53 am
@imran: sorry, i have not discussed 16APSK schemes in the posts
Ravi March 25, 2012 at 9:38 pm
Dear Mr. Krishna
I want to develop a MATLAB code for MIMO using MMSE equalizer for 4X4 antenna configuration and using BPSK, QPSK and 16-QAM modulation. Suggest me any MATLAB code for reference.
I hope you will reply me.
Krishna Sankar March 26, 2012 at 5:36 am
@Ravi: You can check out posts on MIMO at
http://www.dsplog.com/tag/mimo
Most of them discuss 2Tx-2Rx IID Rayleigh channel system with BPSK case
NH February 4, 2012 at 6:07 am
Hi,
Why did you use N=10^5 a big number?
Krishna Sankar February 4, 2012 at 11:15 am
@NH: To get a good average in symbol error rate….
abhijeet October 29, 2011 at 9:44 pm
sir,
i m doing b.tech in ece branch from bit mesra. i need a code for finding symbol error rate of bfsk signal in presence of additive gaussian noise.
Krishna Sankar October 30, 2011 at 7:23 pm
@abhijeet: Does this post on frequency shift keying with coherent demodulation help?
http://www.dsplog.com/2007/08/30/bit-error-rate-for-frequency-shift-keying-with-coherent-demodulation/
lyla February 22, 2010 at 9:38 am
Hello, I have a problem that I need to solve about a
modulation algorithm. I’m working on a assignment and need a code in Matlab to the
in-phase and quadrature components of a 16 QAM. Afterwards, I will run the
program in a Vector Signal Analyzer and I will have to see the I and Q waveforms
and the constellation.
I have been trying out by creating an array with 16 values (0 to 15) and
converting them to binary (0000 to 1111), afterwards, I have created the I and Q
signals, as a sine or cosine of the array values mentioned respectively using a
for loop, thus creating the different values for the phase, but I don’t know how
to express the amplitude levels (-3, -1, 1, 3).
I would appreciate if someone could lend me a hand in this problem.
Krishna Sankar March 31, 2010 at 5:32 am
@lyla: You can generate random 16QAM constellation points, pass them through a pulse shaping filter and then feed to the VSA
Moongeun August 19, 2009 at 1:12 pm
hi~~Krishna
i have a question~~~
in the matbla source code~
ipPhasehat=2*pi/M*round(opPhase/(2*pi/M))
i dont know why you use this code~~~
anyway thank you for your efforts~
Krishna Sankar August 22, 2009 at 4:56 am
@Moongeun: To round the received phase to the nearest constellation point.
Tukuben April 19, 2009 at 6:31 am
Hi Krishna,
im an MSc communication systems engineering student in the university of portsmouth, i would love to be a member of this forum and there is this problem i want to solve:
Analysis & Simulation of 16PSK Using
MATLAB.
OBJECTIVES:
· Demonstrate the ability to learn MATLAB for the purpose of this
coursework
· Write a technical report describing the principle, theory and applications
of 16PSK. Your system will be tested over the AWGN Channel.
· Implement a MATLAB simulation of 16PSK modulator and
demodulator system.
· Measurement of the symbol error rate and bit error rate as a function of
the ratios Eb / No and Es / No achieved by the system in the presence of
to the theoretical estimates that you should have derived from first
principles.
· Show that you are able to identify and use technical references.
But for now am learning MATLAB and the problem is giving me though time, i wish you can help me by putting me through with the rest apart from the MATLAB. Please sir i need your help.
Regards.
Tukuben
Krishna Sankar April 19, 2009 at 6:54 am
abdulmaleek January 29, 2009 at 3:40 pm
I am an MSc mobile, personal and satellite communication student in university of westminster, london. i like to be member of this forum
Krishna Sankar January 30, 2009 at 5:44 am
@abdulmaleek: There are two options:
1. You can subscribe to the email newsletter by providing you email address in the form on the top right of the page. Email subscription will entitle you to receive the free e-book on error rates in AWGN.
2. Further, you can join via Google FriendConnect option provided in the bottom of the page. Google FriendConnect enables you to share the contents of this blog to your friends/colleagues in an easier fasion.
Hope this helps.
gautam January 19, 2009 at 1:26 pm
sir
i am doing ME in digital communication. please refer me any project in MATLAB coding related to my subject.also suggest any reference material for MATLAB study
Krishna Sankar January 22, 2009 at 3:24 am
@gautam: Maybe you can try modeling the transmitter, receiver and channel per the 3GPP LTE specifications? It includes MIMO, advanced coding scheme, OFDMA etc. Quite a lot to learn.
Krishna Sankar April 10, 2008 at 4:33 am
@Rajesh:
1. If the device is non-linear, one would expect that the signal at the output will have more frequencies than the signal at the input (generation of harmonics dye to x^2, x^3 terms etc).
The out-of-band frequencies maybe filtered out, however nothing much can be done about the harmonics in the in-band. It may be that the measured power includes both out-band and in-band harmonics (along with the desired signal power), hence the C/N does not degrade. However, since the in-band harmonics are not deisrable for demodulation, there is degradation in the MER.
2. When you say ‘devices not matched’, I would think that you are referring to impedence mismatch between the devices (resulting in reflection…). May I suggest the generation of in-band and out-band frequencies due to impedence mismatch as a probable reason for MER degradation.
Does this perspective makes sense? Thanks for these nice questions…
Rajesh Kher April 8, 2008 at 4:58 pm
Dear Krishna
Thanks for the reply. While you are right, however we will have to add the noise level. Now the noise level at room temp will be about 3 dBuV while that for QPSK with MER of 15 dB at Signal strength of 86dBuV will be approx 71 dBuV. So that the additive White noise is negligible. We also find that C/N does not get deterioted as we pass thru varios devices.
However we have noticed that MER doe reduce by 0.7-1.0dB.
Question is
1. If thye device contains diodes then if at the point of operation diode is non linear it can add to the reduction of MER but not of C/N. How MER is related to non-linear CSO and CRB.
2. If the device is poorly matched then agian MER measurement will get affected. How does the mismatch between the device and the instruments affects the MER?
Reagards
Rajesh Kher
Krishna Sankar April 3, 2008 at 8:30 pm
@ Rajesh:
1. By definition modulation error ratio (and not rate) is,
MER, dB = 10*log10(average transmit symbol power/average error power) where
error is the distance between the transmitted and received constellation.
If we consider that the received symbol Y is
Y = S + N, where S is the transmitted symbol, N the noise.
then error = Y-S = N.
From the above equation, it is reasonably intuitive that modulation error ratio (MER) is comparable to signal to noise ratio (SNR) in noise only scenario.
In the presence of other impairments, MER will reflect the distortions in the constellation due to phase noise, IQ imbalance, frequency offset etc.
Slides 20-22 from http://chapters.scte.org/cascade/SCTE%20CNR%20vs%20SNR.pdf will be useful
2. Is your question the following: If we pass a received symbol with a noise of xdB
through a splitter having resistive loss of 1dB, how much will be the resultant noise
power. Did I interpret correctly?
I would say the resultant noise will be 10*log10(10^(-x/10) + 10^(-1/10))
Did the linear addition of two noise components, do you agree?
Regards,
Krishna
Rajesh Kher April 3, 2008 at 12:01 pm
Krishna
1. The symbol error rate should also be related to Modulation Error Rate (MER). Since MER is what most Meters measure a correlation will be most helpful.
2. Most of the signal are distributed over a network. What one also needs is the MER impairments due to network elements.
For example if the signal is split into two parts. That splitter will have an insertion loss which will have a spatial component. For example a 2 way lossless splitter will reduce the signal and noise by 3 dB and hence will not impair the signal. However there is an associated resistive loss aboput 1 dB. How much will the MER deteoriate?
Regards
Rajesh Kher
Krishna Sankar March 28, 2008 at 10:20 am
@mahesh:
Yeah… i agree. it takes a bit of time to get used to 20*log10() or 10*log10() …
My rule of thumb is as follows:
(a) for voltage signals use 20*log10()
(b) for power signals use 10*log10().
In the code, as you observed,
(a) for scaling noise, which is a voltage signal, I used the 20*log10().
(b) for finding the theoretical symbol error rate, the Es/No in dB (which is signal power by noise power), the conversion used is 10*log10().
Hope this helps.
Regards,
Krishna
mahesh March 28, 2008 at 8:25 am
In your simulation, you have added noise voltage/level i.e taken 20log10(ns) whereas for calculating theoretical BER , you have used noise power. (10log10(ns^2)). I always tend to get confused between these. Do you know any study material clarifying these and some examples?
thanks
mahesh
Krishna Sankar March 28, 2008 at 4:30 am
@mahesh: true. Infact in the transmit and receive chain, there are various sources which can introduce DC (LO leakage etc). Given so, it makes it difficult to recover information present on DC subcarrier.
mahesh March 27, 2008 at 9:15 am
@ Rajesh, How do you get sync information from DC?
Krishna Sankar March 21, 2008 at 11:51 am
@ rajesh:
well, typically dc subcarrier is not used (as far as I have seen). Can you please point to the section in the spec (and the spec version).
Thanks,
krishna
RAJESH.N March 20, 2008 at 11:06 am
hi i am doing M.E wireless tech. now i am doing my final sem project is implimentation of synchronization algorithem for wimax ieee 802.16e standard.
in wimax ieee802.16e standard
for synchronization purpose they are using preamble .in that preamble they using dc subcarrier .why they are using dc subcarrier… please if any one knows clarify my dought…its urgent
with regards
rajesh neelakandan | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 36, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8649005889892578, "perplexity": 1743.8890729476216}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454701165302.57/warc/CC-MAIN-20160205193925-00122-ip-10-236-182-209.ec2.internal.warc.gz"} |
https://www.degruyter.com/document/doi/10.1515/math-2016-0006/html | # Parabolic oblique derivative problem with discontinuous coefficients in generalized weighted Morrey spaces
Vagif S. Guliyev and Mehriban N. Omarova
From the journal Open Mathematics
# Abstract
We obtain the global weighted Morrey-type regularity of the solution of the regular oblique derivative problem for linear uniformly parabolic operators with VMO coefficients. We show that if the right-hand side of the parabolic equation belongs to certain generalized weighted Morrey space Mp,ϕ(Q, w), than the strong solution belongs to the generalized weighted Sobolev- Morrey space W˙2,1p,φ(Q,ω).
MSC 2010: 35K20; 35D35; 35B45; 35R05
## 1 Introduction
We consider the regular oblique derivative problem in generalized weighted Sobolev- Morrey space W˙2,1p,φ(Q,ω) for linear non-divergence form parabolic equations in a cylinder
utaij(x)Diju=f(x) a.e. in Q,u(x',0)=0, on Ω,u/l=li(x)Diu=0 on S.
The unique strong solvability of this problem was proved in [38]. In [39] Softova studied the regularity of the solution in the Morrey spaces Lp,ƛ with p ∈ (1, ∞), ƛ ∈ (0, n + 2) and also its Hölder regularity. In [41] Softova extended these studies on generalized Morrey spaces Lp,ϕ with a Morrey function ϕ satisfying the doubling and integral conditions introduced in [27, 31]. The approach associated to the names of Calderón and Zygmund and developed by Chiarenza, Frasca and Longo in [7, 8] consists of obtaining of explicit representation formula for the higher order derivatives of the solution by singular and nonsingular integrals. Further the regularity properties of the solution follows by the continuity properties of these integrals in the corresponding spaces. In [39] and [40] the regularity of the corresponding operators in the Morrey and generalized Morrey spaces is studied, while in [38] we can find the corresponding results obtained in Lp by [9] and [5]. In recent works there have been studied the regularity of the solutions of elliptic and parabolic problems with Dirichlet data on the boundary in generalized Morrey spaces Mp,ϕ with a weight ϕ satisfying (10) with w ≡ 1 (cf. [18, 19]). Precisely, a boundedness in Mp,ϕ was obtained for sub-linear operators generated by singular integrals as the Calderon-Zygmund. More results concerning sub-linear operators in generalized Morrey spaces can be found in [3, 12, 40] see also the references therein.
After studying generalized Morrey spaces in detail, researchers passed to weighted Morrey spaces and generalized weighted Morrey spaces. Recently, Komori and Shirai [23] defined the weighted Morrey spaces and studied the boundedness of some classical operators, such as the Hardy-Littlewood maximal operator or the Calderón-Zygmund operator on these spaces. Also, Guliyev in [13] first introduced the generalized weighted Morrey spaces Mwp,ϕ and studied the boundedness of the sublinear operators and their higher order commutators generated by Calderón-Zygmund operators and Riesz potentials in these spaces (see, also [15, 17]). Note that, Guliyev [13] gave the concept of generalized weighted Morrey space which could be viewed as an extension of both Mp,ϕ and Lp,κ(w).
We call weight a positive measurable function defined on ℝn ×. ℝ+. In [29] Muckenhoupt shows that the maximal inequality holds in weighted Lebesgue spaces Lwq if and only if the weight w satisfies the following integral condition called parabolic Muckenhoupt condition or parabolic Aq-condition. We say that the measurable, nonnegative function w : ℝn → ℝ+ satisfies the parabolic Aq-condition for q ∈ (1, ∞) if
(1)supI(1|I|Iw(x,t)dxdt)(1|I|Iw(x,t)1q1dxdt)q1A<
for all parabolic cylinders ℐ in ℝn+1. Then w(ℐ) means the weighted measure of ℐ, that is
w(I)=Iw(x,t)dxdt.
This measure satisfies strong and reverse doubling property. Precisely, for each ℐ and each measurable subset 𝒜 ⊂ ℐ, there exist positive constants c1 and τ1 ∈ (0, 1) such that
(2)1[w]q(|A||I|)qw(A)w(I)c1(|A||I|)τ1,
where c1 and τ1 depend on n and q but not on ℐ and 𝒜.
Throughout this paper the following notations are to be used: x=(x',t)=(x",xn,t)n+1,R+n+1=x'n,t>0 and D+n+1=x"Rn1,xn>0,t>0,Diu=u/xi,Diju=2u/xixj,Dtu=ut=u/t stand for the corresponding derivatives while Du = (D1u, … , Dnu) and D2u=Dijui,j=1n mean the spatial gradient and the Hessian matrix of u. For any measurable function f and A ⊂ ℝn+1 we write
fp,A=(A|f(y)|pdy)1/p,fA=1|A|Af(y)dy
where |A| is the Lebesgue measure of A. Through all the paper the standard summation convention on repeated upper and lower indexes is adopted. The letter C is used for various constants and may change from one occurrence to another.
## 2 Definitions and statement of the problem
Let Ω ⊂ ℝn, n ≥ 1 be a bounded C1,1-domain, Q = Ω × (0, T) be a cylinder in +n+1, and S = ∂Ω × (0, T) stands for the lateral boundary of Q. We consider the problem
(3)Bu:=utaij(x)Diju=f a.e. in Q,Fu:=u(x',0)=0, on Ω,Bu:=u/l=li(x)Diu=0 on S,
under the following conditions:
1. (i)
The operator 𝔅 is supposed to be uniformly parabolic, i.e. there exists a constant Λ > 0 such that for almost all xQ
(4)Λ1|ξ|2aij(x)ξiξjΛ|ξ|2,ξn,aij(x)=aji(x), i,j=1,,n.
The symmetry of the coefficient matrix a=aiji,j=1n implies essential boundedness of aij’s and we set a,Q=i,j=1naij,Q.
2. (ii)
The boundary operator 𝔅 is prescribed in terms of a directional derivative with respect to the unit vector field l(x) = (l1(x), … , ln(x)), xS. We suppose that 𝔅 is a regular oblique derivative operator, i.e., the field l is never tangential to S:
(5)l(x)n(x)=li(x)ni(x)>0 on S,liLip(S¯).
Here Lip(S) is the class of uniformly Lipschitz continuous functions on S and n(x) stands for the unit outward normal to Ω.
In the following, besides the parabolic metric Q(x) = max(|x′|, |t|1/2) and the defined by it parabolic cylinders
Ir(x)=yn+1:|x'y'|<r,|tτ|<r2, |Ir|=Crn+2.
we use the equivalent one ρ(x)=(|x'|2+|x'|4+4t22)12 (see [9]). The balls with respect to this metric are ellipsoids
εr(x)=yn+1:|x'y'|2r2+|tτ|2r4<1,|εr|=Crn+2.
Because of the equivalence of the metrics all estimates obtained over ellipsoids hold true also over parabolic cylinders and in the following we shall use this without explicit references.
Definition 2.1
([20, 37]). LetaLIoc1(n+1),denoted by
ηa(R)=supεr,rR1|εr|εr|f(y)fεr|dy, foreveryR>0
whererranges over all ellipsoids inn+1. The Banach space BMO (bounded mean oscillation) consists of functions for which the following norm is finitea*=supR>0ηa(R)<.
A function a belongs to VMO (vanishing mean oscillation) with VMO- modulus ηa(R) providedlimR0ηa(R)=0.
For any bounded cylinder Q we define BMO(Q) and VMO(Q) taking aL1(Q) and Qr = Q ⋂ ℐrinstead ofrin the definition above.
According to [1, 21] having a function aBMO/VMO(Q) it is possible to extend it in the whole ℝn+1 preserving its BMO-norm or VMO-modulus, respectively. In the following we use this property without explicit references.
For this goal we recall some well known properties of the BMO functions.
Lemma 2.2
(John-Nirenberg lemma, [20]). Let aBMO and p ∈ (1, ∞). Then for any ballthere holds
(1|B|B|a(y)aB|pdy)1pC(p)a*.
As an immediate consequence of Lemma 2.2 we get the following property.
Corollary 2.3.
Let aBMO then for all 0 < 2r < t holds
(6)|aBr+aBt+|Ca*lntr
where the constant is independent of a, x, t and r.
As mentioned before, we call weight a positive measurable function defined on ℝn × ℝ+. Given a weight w and a measurable set 𝕊 we denote by
w(S)=Sw(x)dx
the w-measure of 𝕊. A weight w belongs to the Muckenhoupt class Ap, 1 < p < ∞, if
(7)[w]Ap:=supB(1|B|Bw(x)dx)(1|B|Bw(x)p'/pdx)p/p'<,
where 1p+1p'=1. Note that, for any ball we have (see [11])
(8)[w]Ap(B)1/p=|B|1wL1(B)1/pw1/pLp'(B)1.
In case p = 1, we say that wA1 if
1|B|Bw(x)dxA essBinfw(x)
and [w]A1 is the smallest A for which the above inequality holds. It is an immediate consequence of (7) that whenever wAp than it satisfies the doubling property, precisely
(9)w(2Br)C(n,p)w(Br).
The following lemma collects some of the most important properties of the Muckenhoupt weights.
Lemma 2.4
([11]). We have the following:
1. (1)
If wApfor some 1 ≤ p < ∞, then for all ƛ > 1 we have
w(λB)λnp[w]Apw(B).
2. (2)
The following equality is valid: A = ⋃1≤p<∞Ap.
3. (3)
If wA, then for all ƛ > 1 we havew(λB)2λn[w]Aλnw(B).
4. (4)
If wApfor some 1 ≤ p ≤ ∞, then there exist C > 0 and δ > 0 such that for any balland a measurable set 𝕊 ⊂ ℬ,
1[w]Ap(|S||B|)w(S)w(B)C(|S||B|)δ.
Lemma 2.5
([30, Theorem 5]). Let wA. Then the norm of BMO(w) is equivalent to the norm of BMO(ℝn), where
BMO(w)={a:a*,w=supxn,r>01w(Br(x))Br(x)|a(y)aBr(x),w|w(y)dy<}
and
aBr(x),w=1w(Br(x))Br(x)a(y)w(y)dy.
Lemma 2.6
(The John-Nirenberg inequality). Let aBMO,
1. (1)
there exist constants C1, C2 > 0, such that for all β > 0
|xB:|a(x)aB|>β|C1|B|eC2β/a*, Bn;
2. (2)
for all p ∈ (1, ∞)
a*=CsupB(1|B|B|a(y)aB|pdy)1/p;
3. (3)
for all p ∈ [1, ∞) and wAa*=CsupB(1w(B)B|a(y)aB|pw(y)dy)1/p.
Definition 2.7.
Let ϕ(x, r) be weight in ϕ: ℝn × ℝ+ → ℝ+and ωAp, p ∈ [1, ∞). The generalized weighted Morrey space Mp,ϕ(ℝn, ω) or Mp,ϕ(ω) consists of all functionsfLp,ωloc(n)such thatfp,φ;ω=supxn,r>0φ(x,r)1(ω(εr(x))1εr(x)|f(y)|pω(y)dy)1/p<.
The space Mp,ϕ(Q, ω) consists ofLωp(Q)functions provided the following norm is finite
fp,φ,w;Q=supxn,r>0φ(x,r)1(ω(Qr(x))1Qr(x)|f(y)|pω(y)dy)1/p.
The generalized Sobolev-Morrey spaceconsist of all Sobolev functionsuW2,1p(Q,ω)with distributional derivativesDtlDxsuMp,φ(Q,ω),02l+|s|2,endowed by the norm
uW2,1p,φ(Q,ω)=utp,φ,ω;Q+|s|2Dsup,φ,ω;Q
and
W˙2,1p,φ(Q,ω)={uW2,1p,φ(Q,ω):u(x)=0,xQ},uW˙2,1p,φ(Q,ω)=uW2,1p,φ(Q,ω)
where ∂Q means the parabolic boundary Ω ∪ ﹛Ω × (0, T)﹜.
Theorem 2.8.
(Main result) Let (i) and (ii) hold,aVMO(Q, ω) and uW˙2,1p(Q,ω), ω), p(1, ∞), ωApbe a strong solution of (3). If fMp,ϕ (Q, ω) with ϕ(x, r) being measurable positive function satisfying
(10)r(1+lnsr)ess infφ(x,ς)ω(Qς(x))1ps<ς<ω(Qs(x))1pdssC
for each (x, r) ∈ Q × ℝ+, then uW˙2,1p,φ(Q,ω)and
(11)uW˙2,1p,φ(Q,ω)Cfp,φ,ω;Q
withC=C(n,p,[ω]Ap,Λ,Ω,T,a;Q,ηa)andηa=i,j=1nηaij.
If ϕ(x, r) = r(ƛ−n−2)/p, then Mp,ϕLp,ƛ and the condition (10) holds with a constant depending on n, p and ƛ. If ϕ(x, r) = ω(x, r)1/pr−(n+2)/p with ω : ℝn+1 × ℝ+ → ℝ+ satisfying the conditions
k1ω(x0,s)ω(x0,r)k2 x0n+1,rs2rrω(x0,s)sdsk3ω(x0,r)ki>i=1,2,3
than we obtain the spaces Lp,ω studied in [27, 31]. The following results are obtained in [19] and treat continuity in Mp,ϕ(ℝn+1, ω) of certain singular and nonsingular integrals.
Definition 2.9.
A measurable function 𝔎(x; ζ) : ℝn+1 × ℝn+1 \ ﹛0﹜ → ℝ is called variable parabolic Calderon-Zygmund kernel (PCZK) if:
1. i)
𝔎(x; ·) is a PCZK for a.a. xn+1:
1. a)
𝔎(x; ·)C(ℝn+1 \ ﹛0﹜),
2. b)
𝔎(x; μζ) = μ−(n+2) 𝔎(x; ζ) ∀μ > 0,
3. c)
SnK(x;ξ)dσξ=0, Sn|K(x;ξ)|dσξ<+.
2. ii)
DξβK;n+1×SnM(β)<for each multi-index β.
Consider the singular integrals
(12)Kf(x)=P.V.n+1K(x;xy)f(y)dy,[a,f](x)=P.V.n+1K(x;xy)[a(y)a(x)]f(y)dy.
Theorem 2.10.
For any fMp,ϕ(ℝn+1, ω) with (p, ϕ) as in Theorem 2.8 and aBMO there exist constants depending on n, p, ϕ, ω and the kernel such that
(13)Kfp,φ,ω;n+1C[ω]Ap1pfp,φ,ω;n+1[a,f]p,φ,ω;n+1C[ω]Ap1pa*fp,φ,ω;n+1.
Corollary 2.11.
Let Q be a cylinder in+n+1,fMp,φ(Q,ω),aBMO(Q)andK(x,ξ):Q×+n+1\0.Then the operators (12) are bounded in Mp,ϕ (Q, ω) with p, ϕ, and ω as in Theorem 2.10. Then
(14)Kfp,φ,ω;QC[ω]Ap1pfp,φ,ω;Q,[a,f]p,φ,ω;QC[ω]Ap1pa*fp,φ,ω;Q
withC=C(n,p,φ,[ω]Ap,|Ω|,K).
Corollary 2.12.
Let aVMO and (p, ϕ) be as in Theorem 2.8. Then for any ϵ > 0 there exists a positive number r0 = r0(ϵ, ηa) such that for anyr(x0) with a radius r(0, r0) and all fMp,ϕ(ℰr(x0), ω)
(15)[a,f]p,φ,ω;εr(x0)Cεfp,φ,ω;εr(x0)
where C is independent of ϵ, f, r and x0.
For any x'+n and any fixed t > 0 define the generalized reflection
(16)T(x)=(T'(x),t), T(x)=x'2xnan(x',t)ann(x',t)
where an(x) is the last row of the coefficients matrix a(x) of (3). The function 𝒯′(x) maps +n into n and the kernel 𝔎(x; 𝒯(x) − y) = 𝒦(x; 𝒯′(x) − y′, t − τ) is a nonsingular one for any x,yD+n+1. Taking x = (ʺ, −xn, t) there exist positive constants k1 and k2 such that
(17)k1ρ(x˜y)ρ(T(x)y)k2ρ(x˜y)
For anyfMp,φ(D+n+1,ω) with a norm
fp,φ,ω;D+n+1=supxD+n+1,r>0φ(x,r)1(ω(εr(x))1εr(x)|f(y)|pω(y)dy)1/p
and aBMO(D+n+1,ω) defines the nonsingular integral operators
(18)K˜f(x)=D+n+1K(x;T(x)y)f(y)dy˜[a,f](x)=D+n+1K(x;T(x)y)[a(x)a(y)f(y)dy.]
Theorem 2.13.
LetaBMO(D+n+1),ωApandfMp,φ(D+n+1,ω)with (p, ϕ) as in Theorem 2.8. Then the operatorsK˜fand˜[a,f]are continuous inMp,φ(D+n+1,ω)and
(19) K˜fp,φ,ω;D+n+1Cfp,φ,ω;D+n+1,˜[a,f]p,φ,ω;D+n+1C[ω]Ap1pa*fp,φ,ω;D+n+1.
with a constant independend of a and f.
Corollary 2.14.
Let a2 ϵ > 0 there exists a positive number r0 = r0(ϵ, ηa) such that for anyεr+(x0)=εr(x0)D+n+1with a radius r(0, r0) and center x0 = (ʺ, 0, 0) and for allfMp,φ(εr+(x0),ω)holds
(20)˜[a,f]p,φ,ω;εr+(x0)Cεfp,φ,ω;εr+(x0),
where C is independent of ϵ, f, r and x0.
## 3 Proof of the main result
As it follows by [39], the problem (3) is uniquely solvable in W˙2,1p(Q,ω) .
We are going to show that fMp,ϕ(Q, ω) implies uW˙2,1p,φ(Q,ω). For this goal we obtain an a priori estimate of u. Following the method used by Chiarenza, Frasca and Longo in [7] and [8], we prove the results considering two steps.
Interior estimate. For any x0+n+1 consider the parabolic semi-cylinders 𝒞r(x0) = ℬr(x′0) × (t0r2, t0). Let ν𝒞0(𝒞r) and suppose that ν(x, t) = 0 for t ≤ 0. According to [[5], Theorem 1.4] for any xsupp ν the following representation formula for the second derivatives of ν holds true
(21)Dijυ(x)=P.V.n+1Γij(x;xy)[ahk(y)ahk(x)]Dhkυ(y)dy+P.V.n+1Γij(x;xy)Bυ(y)dy+Bυ(x)SnΓj(x;y)vidσy,
where ν(ν1, … , νn+1) is the outward normal to 𝕊n. Here Γ(x; ζ) is the fundamental solution of the operator 𝔅 and Γ(x; ζ) = ∂2Γ(x; ζ)/∂ζiζj.
Because of density arguments the representation formula (21) still holds for any νW2, 1p(𝒞r(x0), ω). The properties of the fundamental solution (cf. [5, 25, 38]) imply Γij are Calderon-Zygmund kernels in the sense of Definition 2.9. We denote by 𝒦ij and Cij the singular integrals defined in (12) with kernels 𝔎(x; x − y) = Γij (x; x − y). Then we can write that
(22)Dijυ(x)=ij[ahk,Dhkυ](x)+Kij(Bυ)(x)+Bυ(x)SnΓj(x;y)vidσy.
Because of Corollaries 2.11 and 2.12 and the equivalence of the metrics we get
D2υp,φ,ω;Cr(x0)C(εD2υp,φ,ω;Cr(x0)+Bup,φ,ω;Cr(x0))
for some r small enough. Moving the norm of D2ν on the left-hand side we get
D2υp,φ,ω;Cr(x0)CBυp,φ,ω;Cr(x0)
with a constant depending on n, p, ηa(r), ║a║∞,Q and ║DΓ║∞,Q. Define a cut-off function ϕ(x) = ϕ1(x′)ϕ2(t), with ϕ1C0(ℬr(x′0)), ϕ2C0(ℝ) such that
ϕ1(x')=1x'Bθr(x0')0x'Bθ'r(x0'),ϕ2(t)=1 t(t0(θr)2,t0]0t<t0(θ'r)2
with θ(0, 1), θ′ = θ(3 − θ)/2 > θ and |Dsϕ| ≤ C ∈ [θ(1 − θ)r]−s, s = 0, 1, 2, |ϕt| ∼ |D2ϕ|. For any solution uW2,1p(Q, ω) of (3) define ν(x) = ϕ(x)u(x)W2,1p(𝒞r, ω). Then we get
D2up,φ,ω;Cθr(x0)D2υp,φ,ω;Cθ',r(x0)CBυp,φ,ω;Cθ'r(x0)C(fp,φ,ω;Cθ'r(x0)+Dup,φ,ω;Cθ'r(x0)θ(1θ)r+up,φ,ω;Cθ'r(x0)[θ(1θ)r]2).
By the choice of θ′ it holds θ(1 − θ) ≤ 2θ′(1 − θ′) which leads to
[θ(1θ)r]2D2up,φ,ω;Cθr(x0)C(r2fp,φ,ω;Q+θ(1θ)rDup,φ,ω;Cθr(x0)+up,φ,ω;Cθr(x0)).
Introducing the semi-norms
Θs=sup0<θ<1[θ(1θ)r]sD2up,φ,ω;Cθr(x0)s=0,1,2
and taking the supremo with respect to θ and θ′ we get
(23)Θ2C(r2fp,φ,ω;Q+Θ1+Θ0).
The interpolation inequality [26, Lemma 4.2] gives that there exists a positive constant C independent of r such that
Θ1εΘ2+CεΘ0 for anyε(0,2).
Thus (23) becomes
[θ(1θ)r]2D2up,φ,ω;Cθr(x0)Θ2C(r2fp,φ,ω;Q+Θ0)
for each θ(0, 1). Taking θ = 1/2 we get the Caccioppoli-type estimate
D2up,φ,ω;Cr/2(x0)C(fp,φ,ω;Q+1r2up,φ,ω;Cr(x0)).
To estimate ut we exploit the parabolic structure of the equation and the boundedness of the coefficients
utp,φ,ω;Cr/2(x0)a,QD2up,φ,ω;Cr/2(x0)+fp,φ,ω;Cr/2(x0)C(fp,φ,ω;Q+1r2up,φ,ω;Cr(x0)).
Consider cylinders Q′ = Ω′ × (0, T) and = Ωʺ × (0, T) with Ω′ ⋐ Ωʺ ⋐ Ω, by standard covering procedure and partition of the unity we get
(24)uW2,1p,φ(Q',ω)C(fp,φ,ω;Q+up,φ,ω;Q").
where C depends on n,p,[ω]Ap12Λ,TDΓ;Q,ηa(r),a,Q and dist(Ω′, ∂Ωʺ).
Boundary estimates. For any fixed R > 0 and x0 = (ʺ, 0, 0) define the semi-cylinders
CR+(x0)=CR(x0)D+n+1.
Without lost of generality we can take x0 = (0, 0, 0). Define ℬR+ = ﹛|x′| < R, xn > 0﹜, SR+ = ﹛|ʺ| < R, xn = 0, t(0, R2)﹜ and consider the problem
(25)Bu:=utaij(x)Diju=f(x) a.e. in CR+,u:=u(x',0)=0, on BR+,Bu:=li(x)Diu=0 on SR+.
Let uW2, 1p (CR+, ω) with u = 0 for t ≤ 0 and xn ≤ 0, then the following representation formula holds (see [26, 38])
Diju(x)=Iij(x)Jij(x)+Hij(x),
where
Iij(x)=P.V.CR+Γij(x;xy)F(x;y)dy+f(x)SnΓj(x;y)vidσy, i,j=1,,n;Jij(x)=CR+Γij(x;T(x)y)F(x;y)dy;Jin(x)=Jni(x)=CR+Γil(x;T(x)y)(T(x)xn)lF(x;y)dy,i,j=1,,n1Jnn(x)=CR+Γls(x;T(x)y)(T(x)xn)l(T(x)xn)sF(x;y)dy;F(x;y)dy=f(y)+[ahk(y)ahk(x)]Dhku(y),Hij(x)=(Gij*2g)(x)+g(x",t)SnGj(x;y",xn,τ)nidσ(y",τ), i,j=1,,n,T(x)xn=(2an1(x)ann(x),,2ann1(x)ann(x),1).
Here the kernel G = Γ 𝒬, is a byproduct of the fundamental solution and a bounded regular function 𝒬. Hence its derivatives Gij behave as Γij and the convolution that appears in Hij is defined as follows
(Gij*2g)(x)=P.V.SR+Gij(x;x"y",xn,tτ)g(y",0,τ)dy"dτ,g(x",0,t)=[(lk(0)lk(x",0,t))Dkulk(0)(Γk*F)]|xn=0(x",0,t),(Γk*F)(x)=CR+Γk(x;xy)F(x;y)dy.
Here Iij are a sum of singular integrals and bounded surface integrals hence the estimates obtained in Corollaries 2.11 and 2.12 hold true. On the nonsingular integrals Jij we apply the estimates obtained in Theorem 2.13 and Corollary 2.14 that give
(26)Iijp,φ,ω;CR++Jijp,φ,ω;CR+C(fp,φ,ω;CR++ηa(R)D2up,φ,ω;CR+)
for all i, j = 1, … , n. To estimate the norm of Hij we suppose that the vector field l is extended in 𝒞R+ preserving its Lipschitz regularity and the norm. This automatically leads to extension of the function g in 𝒞R+ that is
(27)g(x)=(lk(0)lk(x))Dku(x)lk(0)(Γk*F)(x).
Applying the estimates for the heat potentials [[25], Chapter 4] and the trace theorems in Lp [[2], Theorems 7.48, 7.53] (see also [[38], Theorem 1]) we get
CR+|(Gij*2g)(y)|pw(y)dyC(CR+|g(y)|pw(y)dy+CR+|Dg(y)|pw(y)dy).
Taking a parabolic cylinder ℐr(x) centered in some point x𝒞R+ we have
CR+Ir(x)|Gij*2g(y)|pw(y)dyCω(Ir(x))φ(x,r)p(φ(x,r)pω(Ir(x))CR+Ir(x)|g(y)|pw(y)dy+φ(x,r)pω(Ir(x))CR+Ir(x)|Dg(y)|pw(y)dy)Cω(Ir(x))φ(x,r)p(gp,φ,ω;CR+p+Dgp,φ,ω;CR+p).
Moving φ(x,r)pω(Ir(x)) on the left-hand side and taking the supremo with respect to (x, r)𝒞R+ × ℝ+ we get
Gij*2gp,φ,ω;CR+pC(gp,φ,ω;CR+p+Dgp,φ,ω;CR+p).
An immediate consequence of (27) is the estimate
gp,φ,ω;CR+[lk(0)lk()]Dkup,φ,ω;CR++lk(0)(Γk*F)p,φ,ω;CR+CRlLip(S¯)Dup,φ,ω;CR++Γk*fp,φ,ω;CR++Γk*[ahk()ahk(x)]Dhkup,φ,ω;CR+.
The convolution Γk * f is a Riesz potential. On the other hand
|(Γk*f)(x)|CCR+|f(y)|ρ(xy)n+1dyCRCR+|f(y)|ρ(xy)n+2dyCCR+|f(y)|ρ(xy)n+2dy
with a constant depending on T and diam Ω. Apply [[16], Theorem 4.8, [13], Theorem 3.1] that gives
Γk*fp,φ,ω,CR+Cfp,φ,ω,CR+.
Analogously
|Γk*[ahk()ahk(x)]Dhku()|CCR+|ahk(y)ahk(x)||Dhku(y)|ρ(xy)n+2dy
with a constant depending on diam Ω and T. The kernel ρ(x − y)−(n+2) is a nonnegative singular one and applying again the results for sub-linear integrals [[13], Theorem 3.7] we get
|Γk*[ahk()ahk(x)]Dhku()|ρ,ω,φ,CR+Ca*D2uρ,ω,φ,CR+.
Hence
(28)gρ,ω,φ,CR+C(RlLip(S¯)Dup,φ,ω;CR++fp,φ,ω;CR++Rηa(R)D2up,φ,ω;CR+).
Further, the Rademacher theorem asserts existence almost everywhere of the derivatives DhlkL, thus
Dhg(x)=Dhlk(x)Dku(x)+[lk(0)lk(x)]Dkhulk(0)(Γkh*F)(x).
The Mp,ϕ(ω) norm of the last term is estimated as above and
(29)Dgp,φ,ω;CR+C(Dl:SDup,φ,ω;CR++RlLip(S¯)D2up,φ,ω;CR++fp,φ,ω;CR++ηa(R)D2up,φ,ω;CR+).
Finally unifying (26), (28) and (29) we get
D2up,φ,ω;CR+C(fp,φ,ω;Q+(1+R)Dup,φ,ω;CR++(R+ηa(R)+Rηa(R))D2up,φ,ω;CR+)
with a constant depending on known quantities and ║lLip(S) and ║Dl∞;𝒮. Direct calculations lead to an interpolation inequality in Mp,ϕ(ω) analogous to [[25], Lemma 3.3] (cf. [41])
Dup,φ,ω;CR+δD2up,φ,ω;CR++Cδup,φ,ω;CR+,δ(0,R).
Taking 0<δ=RR+1<R we get
D2up,φ,ω;CR+C(fp,φ,ω;Q+RD2up,φ,ω;CR++CRup,φ,ω;CR++(R+ηa(R)+Rηa(R))D2up,φ,ω;CR+).
Choosing R small enough and moving the terms containing the norm of D2u on the left-hand side we get
D2up,φ,ω;CR+C(fp,φ,ω;CR++1Rup,φ,ω;CR+).
Because of the parabolic structure of the equation analogous estimate holds also for ut. Further the Jensen inequality applied to gives
up,φ,ω;CR+CR2utp,φ,ω;CR+C(R2fp,φ,ω;CR++Rup,φ,ω;CR+).
Choosing R smaller, if necessary, we get up,φ,ω;CR+Cfp,φ,ω;CR+ and therefore
(30)uW2,1p,φ(CR+,ω)Cfp,φ,ω;CR+Cfp,φ,ω;CR+.
Making a covering ﹛Cα+﹜, α𝒜 such that Q\QαACα+, considering a partition of unity subordinated to that covering and applying (30) for each 𝒞α+ we get
(31)uW2,1p,φ(Q\Q',ω)Cfp,φ,ω;Q
with a constant depending on n,p,[ω]Ap12,Λ,T,diamΩ, and ║Dl∞,𝒮. The estimate (11) follows from (24) and (31).
# Acknowledgement
The authors are thankful to the referee for very valuable comments. The research of V.S. Guliyev and M.N. Omarova is partially supported by the grant of Science Development Foundation under the President of the Republic of Azerbaijan, Grant EIF-2013-9(15)-46/10/1 and by the grant of Presidium Azerbaijan National Academy of Science 2015.
### References
[1] Acquistapace P., On BMO regularity for linear elliptic systems, Ann. Mat. Pura Appl., 1992, 161, 231-27010.1007/BF01759640Search in Google Scholar
[3] Akbulut A., Guliyev V.S., Mustafayev R., On the boundedness of the maximal operator and singular integral operators in generalized Morrey spaces, Math. Bohem., 2012, 137(1), 27-4310.21136/MB.2012.142786Search in Google Scholar
[4] Bramanti M., Commutators of integral operators with positive kernels, Le Matematiche, 1994, 49, 149-168Search in Google Scholar
[5] Bramanti M., Cerutti M.C., Wp1,2 solvability for the Cauchy-Dirichlet problem for parabolic equations with VMO coefficients, Comm. Partial Diff. Eq., 1993, 18, 1735-176310.1080/03605309308820991Search in Google Scholar
[6] Chiarenza F., Frasca M., Morrey spaces and Hardy-Littlewood maximal function, Rend. Mat., 1987, 7, 273-279Search in Google Scholar
[7] Chiarenza, F., Frasca, M., Longo, P., Interior W2,p-estimates for nondivergence ellipic equations with discontinuous coefficients, Ricerche Mat., 1991, 40, 149-168Search in Google Scholar
[8] Chiarenza F., Frasca M., Longo P., W2,p-solvability of Dirichlet problem for nondivergence ellipic equations with VMO coefficients, Trans. Am. Math. Soc., 1993, 336, 841-85310.1090/S0002-9947-1993-1088476-1Search in Google Scholar
[9] Fabes E.B., Rivière N., Singular integrals with mixed homogeneity, Studia Math., 1996, 27, 19-3810.4064/sm-27-1-19-38Search in Google Scholar
[10] Gilbarg D., Trudinger N.S., Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer-Verlag, Berlin, 1983Search in Google Scholar
[11] Grafakos L., Classical and Modern Fourier Analysis, Pearson Education, Inc. Upper Saddle River, New Jersey, 2004Search in Google Scholar
[12] Guliyev V.S., Aliyev S.S., Karaman T., Shukurov P., Boundedness of sublinear operators and commutators on generalized Morrey spaces, Int. Eq. Oper. Theory, 2011, 71(3), 327-35510.1007/s00020-011-1904-1Search in Google Scholar
[13] Guliyev V.S., Generalized weighted Morrey spaces and higher order commutators of sublinear operators, Eurasian Math. J., 2012, 3(3), 33-61Search in Google Scholar
[14] Guliyev V.S., Alizadeh F.Ch., Multilinear commutators of Calderon-Zygmund operator on generalized weighted Morrey spaces, Journal of Function Spaces, 2014, http://dx.doi.org/10.1155/2014/710542Search in Google Scholar
[15] Guliyev V.S., Karaman T., Mustafayev R.Ch., Serbetci A., Commutators of sublinear operators generated by Calderón-Zygmund operator on generalized weighted Morrey spaces, Czechoslovak Math. J., 2014, 64(139)(2), 365-38610.1007/s10587-014-0107-8Search in Google Scholar
[16] Guliyev V.S., Karaman T., Serbetci A., Boundedness of sublinear operators generated by Calderon-Zygmund operators on generalized weighted Morrey spaces, Scienti c Annals of “AI.I. Cuza” University of lasi, 2014, 60(1), 227-24410.2478/aicu-2013-0009Search in Google Scholar
[17] Guliyev V.S., Omarova M.N., Higher order commutators of vector-valued intrinsic square functions on vector-valued generalized weighted Morrey spaces, Azerb. J. Math., 2014, 4(2), 64-8510.1186/1029-242X-2014-258Search in Google Scholar
[18] Guliyev V.S., Softova L.G., Global regularity in generalized Morrey spaces of solutions to non-divergence elliptic equations with VMO coeficients, Potential Anal., 2013, 38 (3), 843-86210.1007/s11118-012-9299-4Search in Google Scholar
[19] Guliyev V.S., Softova L.G., Generalized Morrey regularity for parabolic equations with discontinuity data, Proc. Edinb. Math. Soc., 2015, (2) 58(1), 199-21810.1017/S0013091513000758Search in Google Scholar
[20] John F., Nirenberg L.,On functions of bounded mean oscillation, Commun. Pure Appl. Math., 1961, 14, 415-42610.1002/cpa.3160140317Search in Google Scholar
[21] Jones P.W., Extension theorems for BMO, Indiana Univ. Math. J., 1980, 2(9), 41-6610.1512/iumj.1980.29.29005Search in Google Scholar
[22] Kim, D., Parabolic equations with partially BMO coefficients and boundary value problems in Sobolev spaces with mixed norms, Potential Anal., 2010, 33 (1), 17-4610.1007/s11118-009-9158-0Search in Google Scholar
[23] Komori Y., Shirai S., Weighted Morrey spaces and a singular integral operator, Math. Nachr., 2009, 282(2), 219-23110.1002/mana.200610733Search in Google Scholar
[24] Krylov N.V., Parabolic equations with VMO coefficients in Sobolev spaces with mixed norms, J. Funct. Anal., 2007, 250 (2), 521-55810.1016/j.jfa.2007.04.003Search in Google Scholar
[25] Ladyzhenskaya O.A., Solonnikov V.A., Ural’tsev N.N., Linear and Quasilinear Equations of Parabolic Type. Transl. Math. Monographs 23, Amer. Math. Soc., Providence, R.I., 1968Search in Google Scholar
[26] Maugeri A., Palagachev D.K., Softova L.G., Elliptic and Parabolic Equations with Discontinuous Coeficients, Wiley-VCH, Berlin, 200010.1002/3527600868Search in Google Scholar
[27] Mizuhara T., Boundedness of some classical operators on generalized Morrey spaces. Harmonic Anal., Proc. Conf., Sendai/Jap., 1990, ICM-90 Satell. Conf. Proc., 1991, 183-18910.1007/978-4-431-68168-7_16Search in Google Scholar
[28] Morrey C.B., On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc., 1938, 43, 126-16610.1090/S0002-9947-1938-1501936-8Search in Google Scholar
[29] Muckenhoupt B., Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc., 1972, 165, 207-22610.1090/S0002-9947-1972-0293384-6Search in Google Scholar
[30] Muckenhoupt B., Wheeden R., Weighted bounded mean oscillation and the Hilbert transform, Studia Math., 1976, 54, 221-23710.4064/sm-54-3-221-237Search in Google Scholar
[31] Nakai E., Hardy-Littlewood maximal operator, singular integral operators and the Riesz potentials on generalized Morrey spaces, Math. Nachr., 1994, 166, 95-10310.1002/mana.19941660108Search in Google Scholar
[32] Palagachev D.K., Softova L.G., Singular integral operators, Morrey spaces and fine regularity of solutions to PDE’s, Potential Anal., 2004, 20, 237-26310.1023/B:POTA.0000010664.71807.f6Search in Google Scholar
[33] Palagachev D.K., Ragusa M.A., Softova L.G., Regular oblique derivative problem in Morrey spaces, Electron. J. Differential Equations, 2000, 39, http://ejde.math.txstate.edu/Volumes/2000/39/palagachev.pdfSearch in Google Scholar
[34] Palagachev D.K., Ragusa M.A., Softova L.G., Cauchy-Dirichlet problem in Morrey spaces for parabolic equations with discontinuous coefficients, Boll. Unione Mat. Ital., 2003, 6 B (3), 667-684Search in Google Scholar
[35] Ragusa M.A., Cauchy-Dirichlet problem associated to divergence form parabolic equations, Communications in Contemporary Mathematics, 2004, 6 (3), 377-39310.1142/S0219199704001392Search in Google Scholar
[36] Ragusa M.A., The Cauchy-Dirichlet problem for parabolic equations with VMO coefficients, Mathematical and Computer Modelling, 2005, 42 (11-12), 1245-125410.1016/j.mcm.2005.03.004Search in Google Scholar
[37] Sarason D., On functions of vanishes mean oscillation, Trans. Amer. Math. Soc., 1975, 207, 391-40510.1090/S0002-9947-1975-0377518-3Search in Google Scholar
[38] Softova L.G., Oblique derivative problem for parabolic operators with VMO coeficients, Manuscr. Math., 2002, 103 (2), 203-220Search in Google Scholar
[39] Softova L.G., Parabolic equations with VMO coefficients in Morrey spaces, Electron. J. Differ. Equ., 2001, 51, 1-25Search in Google Scholar
[40] Softova L.G., Singular integrals and commutators in generalized Morrey spaces, Acta Math. Sin., Engl. Ser., 2006, 22, 757-76610.1007/s10114-005-0628-zSearch in Google Scholar
[41] Softova L.G., Morrey-type regularity of solutions to parabolic problems with discontinuous data, Manuscr. Math., 2011, 136(3-4), 365-38210.1007/s00229-011-0447-8Search in Google Scholar | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9788480997085571, "perplexity": 3930.675741665136}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662593428.63/warc/CC-MAIN-20220525182604-20220525212604-00260.warc.gz"} |
https://kerodon.net/tag/01L5 | # Kerodon
$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
Proposition 4.6.5.9. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category. For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the pinch inclusion morphisms
$\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}(X,Y) \xrightarrow { \iota ^{\mathrm{L}}_{X,Y} } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \xleftarrow { \iota ^{\mathrm{R}}_{X,Y} } \operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}(X,Y)$
are homotopy equivalences of Kan complexes.
Proof. We will prove that the left-pinch inclusion morphism $\iota ^{\mathrm{L}}_{X,Y}$ is a homotopy equivalence; the proof for the right-pinch inclusion morphism $\iota ^{\mathrm{R}}_{X,Y}$ is similar. Note that we have a commutative diagram of $\infty$-categories
$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{X/} \ar [r] \ar [d] & \{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\ar [d] \\ \operatorname{\mathcal{C}}\ar [r]^-{\operatorname{id}} & \operatorname{\mathcal{C}}, }$
where the horizontal maps are equivalences of $\infty$-categories (Corollary 4.6.4.17) and the vertical maps are left fibrations (Propositions 4.3.6.1 and 4.6.4.10), hence isofibrations (Example 4.4.1.10). Applying Corollary 4.5.4.7, we deduce that the induced map of fibers
$\iota ^{\mathrm{L}}_{X,Y}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}(X,Y) = (\operatorname{\mathcal{C}}_{X/} ) \times _{\operatorname{\mathcal{C}}} \{ Y\} \rightarrow \{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ Y\} = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$
is an equivalence of $\infty$-categories, hence a homotopy equivalence of Kan complexes (Remark 4.5.1.4). $\square$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9964368939399719, "perplexity": 889.9218392587305}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057091.31/warc/CC-MAIN-20210920191528-20210920221528-00483.warc.gz"} |
http://physicshelpforum.com/kinematics-dynamics/11527-work-energy-theorem-problem.html | Physics Help Forum Work-energy theorem problem
Kinematics and Dynamics Kinematics and Dynamics Physics Help Forum
Jan 4th 2016, 03:43 AM #1 Member Join Date: Jan 2015 Posts: 96 Work-energy theorem problem Hi everyone, I have a problem that has me stumped and would appreciate some pointers as to where I am going wrong and maybe point me in the right direction for solving the problem. The problem is in essence to use the "Work-Energy Theorem" to find the co-efficient of kinetic friction in a pulley system. Problem - We have an 8.00 kg-block on flat horizontal tabletop attached via a rope and pulley to a hanging 6.00 kg-block. The rope and pulley have negligible mass and the pulley is friction-less. Initially the 6.00 kg-block is moving downward and the 8.00 kg-block is moving to the right, both with a speed of 0.900 m/s. The blocks come to rest after moving 2.00 meters. Use the Work-Energy Theorem to calculate the coefficient of Kinetic friction between the 8.00 kg-block and the tabletop. Basically I used the two main equations of the Work-Energy Theorem to try to solve this. I first calculated the Work used in moving each block using the difference in kinetic energy over the distance travelled that is ... W= K(2) - K(1) = 1/2 mv(2) sqrd - 1/2 mv(1) sqrd for each block, and since both blocks come to rest, each of the equations above reduce to just 1/2mv(1) sqrd for each block. I then took the difference in the values for the work each block expended to be the work expended by friction force. Now since (for a constant force) WORK also equals Force x distance, I equated the Work difference above to be equal to the work expended by the Kinetic Friction Force. And so Work difference = Work(Friction force)=Kinetic friction x distance moved. From my calculations I got W(8kg-block)=3.24 J, W(6kg-block)=2.43 J giving difference of 0.81 J as the Work of Friction force. Now since W=f x distance then f=w/distance =0.81/2.0 = 0.405 J I now have a value for Friction force (f) and I then used the relationship Friction = Coefficient of Friction x Normal force .. in this case 0.405=coefficient x mg (Normal force for 8kg-block = weight of block ie. 'mg') So we have coefficient = w/mg = 0.405/(8x9.8) = 0.405/78.8 = 0.02 . However the answer at the back of the book gives coefficient = 0.786. I have tried doing this in slightly different ways and the nearest I get to the correct answer is .. 0.75 (which if you notice is just the mass-ratio between the 2 blocks) ??? So where have I gone wrong ? can anyone help ?
Jan 6th 2016, 08:50 AM #2 Physics Team Join Date: Jun 2010 Location: Morristown, NJ USA Posts: 2,312 The work done by friction must equal the change in KE of the two blocks plus the change in PE of the 6 Kg block as it falls 2 meters under gravity: W_f = 1/2(m1+m2)v_i^2 + m_2 g h = (1/2) (6+8)Kg(0.9 m/s)^2 + (6 Kg)(9.8m/s^2)(2m) = 123.3 J That amount of work = force of friction times distance, so: F_f = 123.3J/2m = 61.6 N Can you take it from here?
Tags problem, theorem, workenergy
,
,
,
### the rope and the pulley have neglible mass and the pulley is frictionless
Click on a term to search for related topics.
Thread Tools Display Modes Linear Mode
Similar Physics Forum Discussions Thread Thread Starter Forum Replies Last Post jackthehat Kinematics and Dynamics 1 Mar 25th 2016 09:18 AM jackthehat Kinematics and Dynamics 6 Mar 6th 2016 12:40 PM chippysteve Energy and Work 3 Aug 18th 2013 01:57 PM chocolatelover Advanced Mechanics 1 Oct 14th 2008 10:59 AM chocolatelover Kinematics and Dynamics 1 Oct 14th 2008 10:56 AM | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9338793158531189, "perplexity": 1031.137341785531}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583517376.96/warc/CC-MAIN-20181023195531-20181023221031-00021.warc.gz"} |
http://www.ck12.org/algebra/Estimation-of-Square-Roots/flashcard/user:13InternE/Square-Roots/ | <meta http-equiv="refresh" content="1; url=/nojavascript/">
# Estimation of Square Roots
## Evaluate square roots that are not perfect by estimating a decimal equivalent.
0%
Progress
Practice Estimation of Square Roots
Progress
0%
Square Roots
This will help you study evaluation of square roots, evaluation of perfect square roots, simplification of radical expressions, equations with square roots, and estimation of square roots. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8405675888061523, "perplexity": 3299.3912168695424}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042987628.47/warc/CC-MAIN-20150728002307-00021-ip-10-236-191-2.ec2.internal.warc.gz"} |
https://www.mathbootcamps.com/supplementary-angles-and-examples/ | # Supplementary angles and examples
Supplementary angles are angles whose measures sum to 180°. In the lesson below, we will review this idea along with taking a look at some example problems.
In the image below, you see one of the common ways in which supplementary angles come up. The angles with measures $$a$$° and $$b$$° lie along a straight line. Since straight angles have measures of 180°, the angles are supplementary.
## Example problems with supplementary angles
Let’s look at a few examples of how you would work with the concept of supplementary angles.
### Example
In the figure, the angles lie along line $$m$$. What is the value of $$x$$?
### Solution
The two angles lie along a straight line, so they are supplementary. Therefore: $$x + 118 = 180$$. Solving this equation:
\begin{align}x + 118 &= 180\\ x &= \boxed{62} \end{align}
### Example
The angles $$A$$ and $$B$$ are supplementary. If $$m\angle A = (2x)^{\circ}$$ and $$m\angle B = (2x-2)^{\circ}$$, what is the value of $$x$$?
### Solution
Since the angles are supplementary, their measures add to 180°. In other words: $$2x + (2x – 2) = 180$$. Solving this equation gives the value of $$x$$.
\begin{align}2x + (2x – 2) &= 180\\ 4x – 2 &= 180\\ 4x &= 182\\ x &= \boxed{45.5} \end{align}
The previous example could have asked for some different information. Let’s look at a similar example that asks a slightly different question.
### Example
The angles $$A$$ and $$B$$ are supplementary. If $$m\angle A = (2x+5)^{\circ}$$ and $$m\angle B = (x-20)^{\circ}$$, what is $$m \angle A$$?
### Solution
This time you are being asked for the measure of the angle and not just $$x$$. But, the value of $$x$$ is needed to find the measure of the angle. So, first set up an equation and find $$x$$.
\begin{align}2x+5 + x – 20 &= 180\\ 3x-15 &= 180 \\ 3x &= 195\\ x&= 65\end{align}
The measure of angle $$A$$ is then:
$$m\angle A = (2x+5)^{\circ}$$ and $$x = 65$$
$$m\angle A = (2(65)+5)^{\circ} = \boxed{135^{\circ}}$$
## Summary
There isn’t much to working with supplementary angles. You just have to remember that their sum is 180° and that any set of angles lying along a straight line will also be supplementary. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 3, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9870471954345703, "perplexity": 514.5794757069577}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-09/segments/1550247522457.72/warc/CC-MAIN-20190222180107-20190222202107-00492.warc.gz"} |
https://brilliant.org/problems/back-to-basics-3/ | # Back to basics
Geometry Level 3
True or false:
If $$\vec{u}$$ and $$\vec{v}$$ are two vectors in $$\mathbb{R}^n$$, then the arrow drawn starting at the tip of $$\vec{u}$$ ending at the tip of $$\vec{v}$$ is a translated representation of the vector $$\vec{u}-\vec{v}$$.
× | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9498571157455444, "perplexity": 364.26671311836134}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218189583.91/warc/CC-MAIN-20170322212949-00014-ip-10-233-31-227.ec2.internal.warc.gz"} |
http://www.physicsforums.com/showpost.php?p=4210165&postcount=1 | View Single Post
P: 590 Hi all, I've been having a look at Thidé's treatment of magnetostatics in the free ebook www.plasma.uu.se/CED/Book/EMFT_Book.pdf A couple of questions, about embarrassingly simple matters that I've forgotten: 1) In equation (1.11), is it a mistake that the magnetic force between two current-carrying loops depends on on the position vector x? It looks to me as if it's supposed to be an integral over all the "two-body" forces between infinitesmal current elements in the two loops, but this surely shouldn't depend on the location of an arbitrary point on the first loop- or am I misunderstanding the meaning of this equation? 2) He defines the magnetic B-field as the current->0 limit of the magnetic force at a point, in equation (1.15). Is the right way to think about this firstly to think about the force that acts on a current element located at x, and then take the limit of that current going to zero? Thanks in advance. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8908981680870056, "perplexity": 327.08963749753156}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1398223206120.9/warc/CC-MAIN-20140423032006-00454-ip-10-147-4-33.ec2.internal.warc.gz"} |
https://asmedigitalcollection.asme.org/HT/proceedings-abstract/HT2007/42762/903/323890 | The mechanism of cellular damage associated with freezing of biological cells is discussed by summarizing the author’s recent studies that consists of four different types of experiments. The “solution effects” that designate the influence of elevated concentration of electrolytes during freezing is examined first by a nonfreezing experiment that exposes cells to hypertonic solutions using a perfusion microscope. The cell damage due to the solution effect is evaluated directly from a pseudo-freezing experiment, where cells were subjected to the milieu that simulated a freeze-thaw process in the absence of ice. Contribution of ice formed in the extracellular solution is then estimated from the difference in cell survival between the pseudo-freezing experiment and a corresponding freezing experiment. The cellular injury by the mechanical stress is also examined independently by a cell deformation experiment, which mimicked the situation that cells are compressed and deformed between ice crystals. This experiment was designed to examine a complex effect of mechanical stress from ice and elevated concentration of electrolytes. Based on all these experiments, the role of concentrated solutes and ice is revealed as a function of freezing conditions.
This content is only available via PDF. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8365489840507507, "perplexity": 1566.5457282892446}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986660231.30/warc/CC-MAIN-20191015182235-20191015205735-00397.warc.gz"} |
http://mathhelpforum.com/business-math/204382-economics-problem-help.html | 1. ## Economics Problem Help
See Attachment. I am the TA for a macroeconomics course. My professor provided me with the answer for this problem (3a.) but I am having a difficult time deriving it using the partial derivatives of the current period C(t) and future period C(t+1). The Euler equation format she has taught the students to use is u'(c1)= (1+r)B*u'(c2) where B is the discount rate and r is the interest rate. Please help!!
I have been finding the derivative of t=1 and t=2 and then trying to plug these into the Euler equation. For t=1 I get Ct^(-omega) and for t=2 I get B(Ct+1)^(-omega). Plugging these into the Euler equation gives me a B^2 term, which I know is not correct.
Attached Thumbnails
2. ## Re: Economics Problem Help
Hey nyson13.
Just to clarify, is c2 = c1 + 1? I'll assume this is the case right now (from the context of your post).
The basic Euler equation is u(x+a) = u(a) + u'(a)(x-a), but you actually have two derivative terms so I'll go off the basis that you are going to use two Euler expansions and then collect the terms. Let a be c1 and b be c2 where u(x+b) = u(b) + u'(b)(x-b). (These are of course approximations not real equalities). Take (2) - (1) and we get:
u(x+b) - u(x+a) = u(b) - u(a) + u'(b)(x-b) - u'(a)(x-a).
= u(b) - u(a) + x*(u'(b) - u'(a)) + u'(a)*a - u'(b)*b
if b = a + 1 then this means
= u(a+1) - u(a) + x*(u'(a+1) - u'(a)) + u'(a)*a - u'(a+1)*(a+1)
If x = 0 we get cancellations:
u(a+1) - u(a) = u(a+1) - u(a) + 0 + u'(a)*a - u'(a+1)*(a+1) which gives
u'(a)*a - u'(a+1)*(a+1) = 0 which gives
u'(a)*a = (1+a)u'(a+1) or
u'(a) = [1+a]/a * u'(a+1) or
u'(a) = [1/a + 1]*u'(a+1) with the Euler approximation.
Now do you have values for u'(a) and for u'(a+1) in terms of u's and a's (your notation is a little confusing) and does it match this? (Or have I screwed up somewhere along the lines?) | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8707745671272278, "perplexity": 2029.0072645512935}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886109803.8/warc/CC-MAIN-20170822011838-20170822031838-00148.warc.gz"} |
https://www.physicsforums.com/threads/about-an-example-related-to-angular-momentum-in-a-textbook.585712/ | # About an example related to angular momentum in a textbook
1. Mar 10, 2012
Hey I either have the worst physics textbook in the history of physic textbooks or i can't understand a topic.I'd appreciate if you help me decide which is it.
(Pic related is the image of the example)
First let's agree on the symbols lets call angular velocity: $\omega = \frac{rxv}{||r||^2}$ , angular momentum $L=rxmv$ and Lineal Momentum:$P=mv$. Being that defined, the book attemps to proof that $L$ and $\omega$ have generally different directions, here is the first problem i encountered, for what i know $m||r||^2\omega = L$ being $m||r||^2$ That means that Angular Velocity and Angular Momentum are parallel therefore have the same direction, the example proceed as following: 2 puntual mass are united by a rigid bar of despicable mass, therefore both have the same angular momentum. the momentum of body 1 is $L_1 = r_1xm_1v_1$ and it explicitly say that it's easy to see that it's contained in the plane given by $P_1 P_2 \omega$ (which i fail to see) being the direction perpendiculat to the line that unites $P_1$ with $P_2$ and that $L_2$ Has the same direction that $L_1$. Therefore the Angular Impulse make an angle of $\frac{\pi}{2}-\alpha$ with the axis of rotation($\omega$). Well that'd be an example i'd really appreciate if someone can clarify my doubts, i don't have a teacher to ask since i study by my own.
Here is a picture of the example:
Last edited: Mar 10, 2012
Can you offer guidance or do you also need help?
Draft saved Draft deleted
Similar Discussions: About an example related to angular momentum in a textbook | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.948769211769104, "perplexity": 410.13361697520895}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948595342.71/warc/CC-MAIN-20171217093816-20171217115816-00280.warc.gz"} |
https://math.stackexchange.com/questions/2370423/if-a-is-normal-and-aa-x-lambda-x-then-is-x-an-eigenvector-of-a | # If $A$ is normal and $AA^* x = \lambda x$, then is $x$ an eigenvector of $A$?
Let $H$ be a Hilbert space and let $A$ be a normal linear transform. Let $A^*$ be the adjoint of $A$. If $(\theta, x)$ is an eigenpair of $A$, then it is easy to prove that $(\lvert \theta \rvert^2, x)$ is an eigenpair of $AA^*$, because $$A x = \theta x \rightarrow A^*A x = \theta A^*x=\theta \bar{\theta} x = \lvert \theta \rvert^2 x$$. However, I am interested to know if the converse holds true, that is, if $AA^* x = \lambda x$ for some $x \in H$, then is $x$ an eigenvector of $A$?
• Already there's a simple counter-example in two dimensions: let $Ae_1=\mu\cdot e_1$ and $Ae_2=\overline{\mu}\cdot e_2$. Then $AA^*$ is the identity... – paul garrett Jul 24 '17 at 19:17
• Would it be true in infinite dimensions ($e.g. H = L^2(R)$, for example)? – user38397 Jul 24 '17 at 19:25
• Yes, take the shift on $l^2$. – Tsemo Aristide Jul 24 '17 at 19:27 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9869164228439331, "perplexity": 74.75247823518421}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141186761.30/warc/CC-MAIN-20201126055652-20201126085652-00240.warc.gz"} |
https://www.physicsforums.com/threads/euclids-formula.373883/ | # Homework Help: Euclid's formula
1. Jan 30, 2010
### imprank6
1. The problem statement, all variables and given/known data
Proof and show that in Euclid's formula for perfect numbers, n must be prime.
2. Relevant equations
(2^(n-1))((2^n)-1))
3. The attempt at a solution
I can show it by plugging in the number but I cannot prove it...any ideas?
2. Jan 30, 2010
### VeeEight
Try showing that if 2n-1 is prime, then n is prime. You can do this by assuming it is composite (n=ab), and finding a divisor of 2n-1 (namely 2a-1)
3. Jan 30, 2010
### mnc34829
Sounds like you are in History of Mathematics...I'm trying to figure this problem out (among many others)..to bad the hint in the back of the book says the same thing as what was posted previously..and that doesn't help
4. Jan 31, 2010
### sutupidmath
I am not sure whether the problem you have is to show that n does not divide 2^n-1 ( that is they are relatively prime) or to show that if 2^n-1 is prime than n is prime ( in which case i am not sure that the latter even holds). However, if it is the former here is another approach to this problem: Suppose the contrary: that is let $$2^n-1$$ be a prime number and n a number not relatively prime to it. Now, let $$p_1$$ be a prime divisor of n, and let k be the smallest positive integer for which $$p_1$$ divides $$2^k-1$$. Now from Fermat's little theorem we would get that $$p_1$$ is also a factor of $$2^{p_1-1}-1$$. Hence $$k\leq p_1-1<p_1.$$
Now your task is to prove that q divides n. Assume the contrary, by expressing n=hq+r.
I will stop here before i get in trouble from PF Moderators! | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9459515810012817, "perplexity": 313.3893552460295}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267158045.57/warc/CC-MAIN-20180922044853-20180922065253-00133.warc.gz"} |
https://www.easy-due.com/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%E4%BB%A3%E5%86%99-kernel-and-range-of-a-matrix/ | 1. 矩阵的核
” ” #
” ” \$
a1,1×1 + a1,2×2 +… + a1,nxn =0
a2,1×1 + a2,2×2 +… + a2,nxn =0
………………
am,1×1 + am,2×2 +… + am,nxn =0
!” 0 是 Rm 的零向量，A 是系统的系数矩阵。
！” x = ！” 0
（即 x1 = x2 = … = xn = 0）总是一个解。
%
21
11
&
A的核是系统的解集：
2×1 + x2 =0
x1 + x2 =0
x1 = x2 =0。
!” x 是零向量。所以这里，A 的核只包含
！”0}。
1
1. The kernel of a matrix
Definition 1.1. A system of linear equations is called homogeneous if the con-
stant term in each equation is equal to zero, that is, if the system can be written
as:
!
” ” #
” ” \$
a1,1×1 + a1,2×2 +… + a1,nxn =0
a2,1×1 + a2,2×2 +… + a2,nxn =0
… … … … …
am,1×1 + am,2×2 +… + am,nxn =0
A homogeneous system correponds to the matrix equation:
A!” x = !” 0
where
!” 0 is the zero vector of Rm and A is the coe!cient matrix of the system. The
last column of the augmented matrix B is zero. Such a system is always consistent,
because the zero vector, corresponding to
!” x = !” 0
(that is x1 = x2 = … = xn = 0) is always a solution.
Proposition 1.2. A homogeneous system of linear equations either has only one
solution, the zero vector of Rn, which is called the trivial solution, or it has
infinitely many solutions.
Definition 1.3. Let A be a m # n matrix. The kernel or the nullspace of the
matrix A is the space of solutions of the homogeneous system A!” x = 0. It is denoted
by Ker(A).
Example 1.4. Set
A =
%
21
11
&
The kernel of A is the set of solutions of the system:
2×1 + x2 =0
x1 + x2 =0
We solve this system and we easily obtain:
x1 = x2 =0.
In other words,
!” x is the zero vector. So here, the kernel of A contains only the
zero vector:
Ker(A)= {
!” 0 }.
1
EasyDue™ 支持PayPal, AliPay, WechatPay, Taobao等各种付款方式!
E-mail: [email protected] 微信:easydue
EasyDue™是一个服务全球中国留学生的专业代写公司 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9485427141189575, "perplexity": 4133.749737559393}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623488289268.76/warc/CC-MAIN-20210621181810-20210621211810-00524.warc.gz"} |
https://alphalerts.com/documentation/cryptocurrencies/price-change-percentage-24h | ## Documentation
### Price Change Percentage 24h
PRICE_CHANGE_PERCENTAGE_24H refers to the percentage change in price over the past 24 hours. Suppose, the price of a coin was $100 yesterday and it is$110 today. Since the percentage increase is 10%, PRICE_CHANGE_PERCENTAGE_24H is +10%.
This is a good indicator is how volatile a certain cryptocurrency is from day to day. Related terms include PRICE_CHANGE_PERCENTAGE. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8691133260726929, "perplexity": 2155.340012845968}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296950110.72/warc/CC-MAIN-20230401160259-20230401190259-00500.warc.gz"} |
http://math.stackexchange.com/questions/177332/can-the-range-of-this-operator-be-closed | # Can the range of this operator be closed?
Given an operator $T:H\rightarrow L$, where $H$ is a finite-dimensional Hilbert space and $L$ an infinitedimensional one, is the range of $T$, $T(H)$, a closed set in $L$ ? I know that the image doesn't have to be closed if $H$ were infinite, but I'm not sure in this case.
-
$T(H)$ is a finite dimensional subspace and any finite dimensional subspace is closed since it is isomorphic to $\mathbb R^n$ for some $n$ and hence complete.
- | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9688682556152344, "perplexity": 64.06566909199063}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-26/segments/1466783403823.74/warc/CC-MAIN-20160624155003-00073-ip-10-164-35-72.ec2.internal.warc.gz"} |
http://renormalization.com/17a2/ | ### Recent Papers
We study the perturbative unitarity of the Lee-Wick models, formulated as nonanalytically Wick rotated Euclidean theories. The complex energy plane is divided into disconnected regions and the values of a loop integral in the various regions are related to one another by a nonanalytic procedure. We show that the one-loop diagrams satisfy the expected, unitary cutting equations in each region: only the physical degrees of freedom propagate through the cuts. The goal can be achieved by working in suitable subsets of each region and proving that the cutting equations can be analytically continued as a whole. We make explicit calculations in the cases of the bubble and triangle diagrams and address the generality of our approach. We also show that the same higher-derivative models violate unitarity if they are formulated directly in Minkowski spacetime.
PDF
Phys. Rev. D 96 (2017) 045009 | DOI: 10.1103/PhysRevD.96.045009
arXiv: 1703.05563 [hep-th]
### Search this site
Support Renormalization
If you want to support Renormalization.com you can spread the word on social media or make a small donation
### Book
14B1 D. Anselmi
Renormalization
PDF
Last update: May 9th 2015, 230 pages
Contents: Preface | 1. Functional integral | 2. Renormalization | 3. Renormalization group | 4. Gauge symmetry | 5. Canonical formalism | 6. Quantum electrodynamics | 7. Non-Abelian gauge field theories | Notation and useful formulas | References
Course on renormalization, taught in Pisa in 2015. (More chapters will be added later.) | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8427508473396301, "perplexity": 1544.2839443347143}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267164469.99/warc/CC-MAIN-20180926081614-20180926102014-00117.warc.gz"} |
http://tex.stackexchange.com/questions/46700/r-sweave-hmisc-pretty-print-numbers | # R, Sweave, Hmisc: pretty-print numbers
I'm creating a LaTeX tables based on a matrix in R using the 'latex' function of the Hmisc package.
dat = matrix(c(1000, 100, 10000, 10000), 2)
latex(dat, file='')
This works (as expected) perfectly.
Additionally I would like to use
prettyNum(dat, '.')
to format the numbers in a more readable way. For example:
100 --> 100
1000 --> 1.000
Is there a simple way to combine these functions without breaking the (automatic!) table alignment ('r' for numbers instead of 'l' for character)?
-
knitr has more flexibility than Sweave. Have you looked to see if it supports something like this? – scottkosty Mar 4 '12 at 5:44
I stumbled over knitr some weeks ago, but build my current documents mainly with Sweave. Perhaps I'll try to switch in future projects. +1 for this tip :) – FloE Mar 4 '12 at 15:56
@scottkosty knitr looks very interesting, although I don't think it solves this problem directly; the problem here lies with Hmisc's (unavoidable) dependence on particular LaTeX packages. But I may be missing something. If you're familiar with knitr maybe you could work up a solution using it? – Alan Munn Mar 4 '12 at 17:37
@AlanMunn I'm just starting to dig below the surface of knitr so I won't be able to make an attempt. My guess is that you're right. – scottkosty Mar 5 '12 at 3:42
## 2 Answers
UPDATE
The Hmisc package has been updated, and now allows for arbitrary column specifications. Please see Boris' answer for a simple solution.
Original Answer
One of the problems with the way most R packages generate tables is that they are not easy to adapt to changing functionality within LaTeX. The standard for pretty printing numbers and tables within LaTeX is the siunitx package, which Hmisc doesn't support. There is no simple way around this if you are generating the tables using Sweave unless you post-edit your resultant .tex file.
However, if you are willing to do that, it's not that difficult to generate tables with Hmisc and then replace its r columns with the S column type defined by siunitx.
Here's a example:
.Rnw file
\documentclass{report}
\usepackage[utf8]{inputenc}
\usepackage{siunitx,booktabs}
\sisetup{group-separator={.},group-minimum-digits={3},output-decimal-marker={,}}
\usepackage[noae]{Sweave}
\begin{document}
<<>>=
library("Hmisc")
dat <- matrix(c(1000, 100, 10000, 10000,3.145,1700.42), 2)
@
<<echo=false,results=tex>>=
latex(dat,table=F,center='centering',file='',
booktabs=T,numeric.dollar=F,colheads=c("Col A","Col B","Col C"),colnamesTexCmd="bfseries")
@
\end{document}
Output .tex file
When you Sweave this file, you produce the .tex file which contains the following line:
\begin{tabular}{rrr}
If you manually change this to:
\begin{tabular}{SSS}
the final .tex file looks like this:
\documentclass{report}
\usepackage[utf8]{inputenc}
\usepackage{siunitx,booktabs}
\sisetup{group-separator={.},group-minimum-digits={3},output-decimal-marker={,}}
\usepackage[noae]{Sweave}
\begin{document}
\begin{Schunk}
\begin{Sinput}
> library("Hmisc")
> dat <- matrix(c(1000, 100, 10000, 10000,3.145,1700.42), 2)
\end{Sinput}
\end{Schunk}
% latex.default(dat, table = F, center = "centering", file = "", booktabs = T, numeric.dollar = F, colheads = c("Col A", "Col B", "Col C"), colnamesTexCmd = "bfseries")
%
\centering
\begin{tabular}{SSS}
\toprule
\multicolumn{1}{c}{\bfseries Col A}&\multicolumn{1}{c}{\bfseries Col B}&\multicolumn{1}{c}{\bfseries Col C}\tabularnewline
\midrule
1000&10000& 3.145\tabularnewline
100&10000&1700.420\tabularnewline
\bottomrule
\end{tabular}
\end{document}
Now siunitx can do its magic, and the output is the following:
-
It is much simpler with the new version of Hmisc - see my answer... – Boris Apr 16 '12 at 18:01
@Boris. That's great. It makes Hmisc even more useful. Thanks for posting your answer. – Alan Munn Apr 16 '12 at 18:25
Actually the new version of Hmisc has the parameter col.just, so you do not need to edit the tex file
\documentclass{report}
\usepackage[utf8]{inputenc}
\usepackage{siunitx,booktabs}
\sisetup{group-separator={.},group-minimum-digits={3},
output-decimal-marker={,}}
\usepackage[noae]{Sweave}
\begin{document}
<<>>=
library("Hmisc")
dat <- matrix(c(1000, 100, 10000, 10000,3.145,1700.42), 2)
@
<<echo=false,results=tex>>=
latex(dat,table=F,center='centering',file='',
booktabs=T,numeric.dollar=F,colheads=c("Col A","Col B","Col C"),
colnamesTexCmd="bfseries", col.just=c("S","S","S"))
@
\end{document}
- | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.807413637638092, "perplexity": 4948.2529479334235}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-27/segments/1435375098685.20/warc/CC-MAIN-20150627031818-00250-ip-10-179-60-89.ec2.internal.warc.gz"} |
https://www.authorea.com/users/4552/articles/4753-manifold-warping-manifold-alignment-over-time/_show_article | # Manifold Warping: Manifold Alignment Over Time
Abstract
Knowledge transfer is computationally challenging, due in part to the curse of dimensionality, compounded by source and target domains expressed using different features (e.g., documents written in different languages). Recent work on manifold learning has shown that data collected in real-world settings often have high-dimensional representations, but lie on low-dimensional manifolds. Furthermore, data sets collected from similar generating processes often present different high-dimensional views, even though their underlying manifolds are similar. The ability to align these data sets and extract this common structure is critical for many transfer learning tasks. In this paper, we present a novel framework for aligning two sequentially-ordered data sets, taking advantage of a shared low-dimensional manifold representation. Our approach combines traditional manifold alignment and dynamic time warping algorithms using alternating projections. We also show that the previously-proposed canonical time warping algorithm is a special case of our approach. We provide a theoretical formulation as well as experimental results on synthetic and real-world data, comparing manifold warping to other alignment methods.
# Introduction
The advent of large, often high-dimensional, digital data sets has made automated knowledge extraction a critical research focus in the field of machine learning. Often, we find real-world sequential data sets that encode the same information with disparate surface feature representations, such as sensor network data, activity and object recognition corpora, and audio/video streams. In these cases, an automated technique for discovering correlations between sets will allow easy transfer of knowledge from one domain to another, avoiding costly or infeasible re-learning. In this paper, we present a framework that combines manifold alignment \cite{ham2003MA,wang2009generalMAframework} and dynamic time warping (DTW) \cite{sakoe1978dtw} for aligning two such sequential data sets. We also show that the previously proposed method of canonical time warping (CTW) \cite{zhou2009ctw} is a special case of our approach.
Dynamic time warping has been used effectively for time-series alignment, but it requires an inter-set distance function, which usually implies that both input data sets must have the same dimensionality. DTW may also fail under arbitrary affine transformations of one or both inputs. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8296945095062256, "perplexity": 983.0491969551086}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218188924.7/warc/CC-MAIN-20170322212948-00383-ip-10-233-31-227.ec2.internal.warc.gz"} |
https://www.intechopen.com/books/chaos-theory/bifurcation-theory-of-dynamical-chaos | Open access peer-reviewed chapter
# Bifurcation Theory of Dynamical Chaos
By Nikolai A. Magnitskii
Submitted: June 23rd 2017Reviewed: September 13th 2017Published: March 28th 2018
DOI: 10.5772/intechopen.70987
## Abstract
The purpose of the present chapter is once again to show on concrete new examples that chaos in one-dimensional unimodal mappings, dynamical chaos in systems of ordinary differential equations, diffusion chaos in systems of the equations with partial derivatives and chaos in Hamiltonian and conservative systems are generated by cascades of bifurcations under universal bifurcation Feigenbaum-Sharkovsky-Magnitskii (FShM) scenario. And all irregular attractors of all such dissipative systems born during realization of such scenario are exclusively singular attractors that are the nonperiodic limited trajectories in finite dimensional or infinitely dimensional phase space any neighborhood of which contains the infinite number of unstable periodic trajectories.
### Keywords
• nonlinear systems
• dynamical chaos
• bifurcations
• singular attractors
• FShM theory
## 1. Introduction
Well-known, that chaotic dynamics is inherent practically in all nonlinear mappings and systems of differential equations having irregular attractors, distinct from stable fixed and singular points, limit cycles and tori. However, many years there was no clear understanding of that from itself represent irregular attractors and how they are formed. In this connection it was possible to find in the literature more than 20 various definitions of irregular attractors: stochastic, chaotic, strange, hyperbolic, quasiattractors, attractors of Lorenz, Ressler, Chua, Shilnikov, Chen, Sprott, Magnitskii and many others. It was considered that there are differences between attractors of autonomous and nonautonomous nonlinear systems, systems of ordinary differential equations and the equations with partial derivatives, and that the chaos in dissipative systems essentially differs from chaos in conservative and Hamiltonian systems. There was also an opinion which many outstanding scientists adhered, including Nobel prize winner I.R. Prigogine, that irregular attractors of complex nonlinear systems cannot be described by trajectory approach, that are systems of differential equations. And only in twenty-first century it has been proved and on numerous examples it was convincingly shown, that there is one universal bifurcation scenario of transition to chaos in nonlinear systems of mappings and differential equations: autonomous and nonautonomous, dissipative and conservative, ordinary, with private derivatives and with delay argument (see, for example, [1, 2, 3, 4, 5, 6, 7, 8, 9]). It is bifurcation Feigenbaum-Sharkovsky-Magnitskii (FShM) scenario, beginning with the Feigenbaum cascade of period-doubling bifurcations of stable cycles or tori and continuing from the Sharkovskii subharmonic cascade of bifurcations of stable cycles or tori of an arbitrary period up to the cycle or torus of the period three, and then proceeding to the Magnitskii homoclinic or heteroclinic cascade of bifurcations of stable cycles or tori. All irregular attractors born during realization of such scenario are exclusively singular attractors that are the nonperiodic limited trajectories in finite dimensional or infinitely dimensional phase space any neighborhood of which contains the infinite number of unstable periodic trajectories.
However, in the scientific literature many papers continue to appear in which authors, not understanding an essence of occurring processes, write about opened by them new attractors in nonlinear systems of differential equations. Such papers are, for example, papers [10, 11] which authors with surprise ascertain an existence of chaotic dynamics in nonlinear system of ordinary differential equations with one stable singular point and try to explain this phenomenon by presence in the system of Smale’s horseshoe. Numerous papers continue to be published also in which presence of chaotic attractor in the system of ordinary differential equations is connected with Lyapunov’s positive exponent found numerically, diffusion chaos in nonlinear system of equations with partial derivatives is explained by the Ruelle-Takens (RT) theory and is connected with birth of mythical strange attractor at destruction of three-dimensional torus, and presence of chaotic dynamics in Hamiltonian or conservative system is explained by the Kolmogorov-Arnold-Mozer (KAM) theory and is connected with consecutive destruction in the system of rational and mostly irrational tori of nonperturbed system.
The purpose of the present paper is once again to show on concrete new, not entered in [1, 2, 3, 4, 5, 6, 7, 8, 9], examples, that chaos in the system considered in Refs. [10, 11], and also chaos in one-dimensional unimodal mappings, dynamical chaos in systems of ordinary differential equations, diffusion chaos in systems of the equations with partial derivatives and chaos in Hamiltonian and conservative systems are generated by cascades of bifurcations under the FShM scenario. Thus, in any nonlinear system there can be an infinite number of various singular attractors, becoming complicated at change of bifurcation parameter in a direction of the cascade of bifurcations. Presence or absence in system of stable or unstable singular points, presence or absence of saddle-nodes or saddle-focuses, homoclinic or heteroclinic separatrix contours and Smale’s horseshoes and also positivity of the calculated senior Lyapunov’s exponent are not criteria of occurrence in system of chaotic dynamics. And the birth in the system of three-dimensional and even multi-dimensional stable torus leads not only to its destruction with birth of mythical strange attractor, but also to cascade of its period-doubling bifurcations along one of its frequencies or several frequencies simultaneously. Chaotic dynamics in Hamiltonian and conservative systems also is consequence of cascades of bifurcations of birth of new tori, instead of consequence of destruction of some already ostensibly existing mythical tori of nonperturbed system. Thus, for the analysis of chaotic dynamics of any nonlinear system, attempts of calculation of a positive Lyapunov’s exponent, application of КАМ and RT theories and the proof of existence of Smale’s horseshoes are absolutely senseless. Let us notice, that the results of Feigenbaum and Sharkovsky are received only for one-dimensional unimodal maps and then were transferred by Magnitskii at first on two-dimensional systems of differential equations with periodic coefficients, then on three-dimensional, multi-dimensional and infinitely dimensional dissipative and conservative autonomous systems of ordinary differential equations and then on systems of the equations with partial derivatives. Besides this, it is proved by Magnitskii, that the subharmonic cascade of Sharkovsky bifurcations can be continued by homoclinic or heteroclinic bifurcations cascade both in the differential equations, and in continuous one-dimensional unimodal mappings.
### 1.1. FShM-cascades of bifurcations of stable cycles and a birth of singular attractors in one-dimensional unimodal mappings
Let us give a summary of bifurcation FShM theory of chaos in one-dimensional continuous unimodal mappings. Detailed proof of statements of the present section can be found in Ref. [1].
#### 1.1.1. FShM-cascade of bifurcations in logistic mapping
Studying the properties of logistic mapping
fxμ=μx1x,x01,μ14E1
Feigenbaum proved that in this equation there is a cascade of period-doubling bifurcations of its cycles and found a sequence of values of the parameter μ at which these bifurcations occur. Further studies have shown that the complex chaotic dynamics of the logistic mapping is also characteristic of any continuous difference equation of а kind xn + 1 = f(xn, μ) in which one-dimensional mapping f : I → I is unimodal at corresponding choice of scale, that is, it has the only extremum on an interval I. Return mapping f−1 has in this case two branches on I.
Considering the map (1) on an interval x ∈ [0, 1], Feigenbaum has established, that there is the infinite sequence μn of parameter values μ converging with a speed of the geometrical progression with a denominator 1/δ ≈ 1/4.67 to value μ ≈ 3.57 in which period-doubling bifurcations of the cycles of logistic map occur. That is at all parameter values μn < μ < μn + 1 Eq. (1) has unique regular attractor—a stable cycle of the period 2n and a set of unstable cycles of all periods 2k, k = 0, …, n − 1. Thus, the first most simple and low-power singular attractor, born in unimodal one-dimensional continuous mapping at the end of the Feigenbaum period-doubling bifurcation cascade, is a nonperiodic trajectory consisting of points, any neighborhood of each contains points belonging to some unstable cycles of the periods 2n, n > 0. This attractor is called Feigenbaum attractor. It is, obviously, everywhere not dense set of points on an interval. In the case of logistic mapping (1), Feigenbaum attractor exists at the parameter value μ ≈ 3.57. However, logistic mapping is defined on the interval x ∈ [0, 1] at all parameter values μ ≤ 4. The answer to a question, that occurs with trajectories of logistic mapping and with any other unimodal continuous mapping at parameter values μ > μ, gives Sharkovsky theorem. It follows from this theorem that complication of structure of cycles of iterations of one-dimensional unimodal mappings, as a rule, does not come to the end with the cascade of Feigenbaum bifurcations and Feigenbaum attractor, and it is continued by more complex cascade of bifurcations according to the order established by Sharkovsky in his theorem.
Definition. Ordering in set of the natural numbers, looking like
12222n227225223272523753.E2
is called as Sharkovsky’s order. Theorem of Sharkovsky approves, that if continuous unimodal map f : I → I has a cycle of the period n then it has also all cycles of each period k, such that k ⊲ n in the sense of the order (2). Consequence of the theorem is the statement, that if map f has a cycle of the period 3, then it has cycles of all periods.
It also follows from the Sharkovsky theorem, that at change of values of bifurcation parameter, stable cycles in one-dimensional unimodal continuous mappings are obliged to be born according to the order (2). And their births occur in pairs together with unstable cycles as a result of saddle-node (tangent) bifurcations. Each stable cycle of Sharkovsky cascade, which has born thus, undergoes then the cascade of period-doubling bifurcations, generating its own window of periodicity (Figure 1). A limit of such cascade is more complex singular attractor—nonperiodic almost stable trajectory any neighborhood of which contains the infinite number of unstable periodic trajectories. Hence, the cascade of Feigenbaum bifurcations is an initial stage of the full subharmonic cascade of bifurcations, described by Sharkovsky order. In the case of logistic mapping (1) cycle of the period three is born at value μ ≈ 3.828 (Figure 1). Hence, the subharmonic cascade of Sharkovsky bifurcations does not cover all area of change of values of bifurcation parameter μ ≤ 4.
Behind subharmonic Sharkovsky cascade, homoclinic (heteroclinic) cascade of bifurcations lays, opened by Magnitskii at first in nonlinear systems of ordinary differential equations, and then found out in logistic and other unimodal continuous mappings. Homoclinic (heteroclinic) cascade of bifurcations consists of a consecutive birth of stable homoclinic (heteroclinic) cycles of the period n converging to a homoclinic (heteroclinic) contour. As a rule, it is a separatrix loop of a saddle-focus (heteroclinic separatrix contour) in nonlinear system of ordinary differential equations and a separatrix loop of a fixed point (heteroclinic separatrix contour) in one-dimensional unimodal mapping. Born before, unstable cycles and nonperiodic trajectories (singular attractors) remain in system, therefore dynamics of unimodal mapping in a neighborhood of a homoclinic (heteroclinic) contour is the most complex. The first cycles of homoclinic cascade are the most simple cycle of the period two of the Feigenbaum cascade and the most complex cycle of the period three of Sharkovsky cascade. In logistic mapping stable homoclinic cycle of the period four exists at μ = 3.9603, and the separatrix loop of the fixed point x = 0 exists at μ = 4, that completely covers all area of change of values of bifurcation parameter (Figure 1). So, in one-dimensional unimodal mappings at various parameter values stable periodic (regular) attractors and nonperiodic singular attractors can exist together with finite or infinite number of unstable periodic trajectories, and all such attractors are born as a result of cascades of soft bifurcations (saddle-node and period-doubling) in full accordance with the Feigenbaum-Sharkovsky-Magnitskii (FShM) theory.
## 2. Dynamical chaos in nonlinear dissipative systems of ordinary differential equations
Bases of the FShM theory with reference to nonlinear dissipative systems of ordinary differential equations are stated in Refs. [1, 2, 3, 7]. Thus in systems with strong dissipation it is realized both the full subharmonic cascade of Sharkovsky bifurcations, and full (or incomplete) homoclinic (or heteroclinic) cascade of Magnitskii bifurcations depending on, whether exists homoclinic (or heteroclinic) separatrix contour in the system. In systems with weak dissipation the FShM-order of bifurcations can be broken in its right part. Hence, attractors of such systems are regular attractors (stable singular points, stable cycles and stable tori of any dimension), or singular cyclic or toroidal attractors—limited nonperiodic almost stable trajectories or the toroidal manifolds, being limits of cascades of the period-doubling bifurcations of regular attractors (cycles, tori). In Refs. [1, 2, 3, 7] it is proved, that the FShM scenario of transition to chaos takes place in such classical two-dimensional dissipative systems with periodic coefficients, as systems of Duffing-Holmes, Mathieu, Croquette and Krasnoschekov; in three-dimensional autonomous dissipative systems, as systems of Lorenz, Ressler, Chua, Magnitskii, Vallis, Anishchenko-Astakhov, Rabinovich-Fabricant, Pikovskii-Rabinovich-Trakhtengertz, Sviregev, Volterra-Gause, Sprott, Chen, Rucklidge, Genezio-Tesi, Wiedlich-Trubetskov and many others; in multi-dimensional and infinitely dimensional autonomous dissipative systems, as systems of Rikitaki, Lorenz complex system, Mackey-Glass equation and many others. These systems describe processes and the phenomena in all areas of scientific researches. Lorenz system is a hydrodynamic system, Ressler system is a chemical system, Chua system describes the electro technical processes, Magnitskii system is a macroeconomic system, Widlich-Trubetskov system describes the social processes and phenomena, Mackey-Glass equation describes the processes of hematopoiesis.
### 2.1. Transition to chaos in the system with one stable singular point
In this chapter, let us consider the three-dimensional system of ordinary differential equations with one stable singular point which has been proposed in Ref. [10]
ẋ=yz+0.006,ẏ=x2y,ż=14xE3
This system has the only stable singular point (0.25, 0.0625, −0.096) of stable focus type as Jacobian matrix in the singular point has eigenvalues (−0.96069, −0.01966 ± 0.50975 i), where i2 = − 1. The system (3) has no neither saddle-focuses, nor a saddle-nodes and, hence, it has no homoclinic or heteroclinic contours, but it has strongly expressed chaotic dynamics (see in Ref. [10] and below in Figure 2). In Ref. [11] attempt is undertaken to explain chaos in system (3) by presence in it of Smale’s horseshoe. We shall show now, that transition to chaos in system (3) actually occurs in full accordance with universal bifurcation scenario of Feigenbaum-Sharkovsky-Magnitskii. For this purpose, it is necessary only to define correctly bifurcation parameter at which change the cascade of bifurcations under FShM scenario is realized in the system.
As bifurcation parameter we choose the parameter b and consider the system
ẋ=yz+0.006,ẏ=x2by,ż=14xE4
At b = 1, the system (4) obviously passes into system (3). We shall search stable cycles of the system (4) by numerical modeling of the system by the Runge-Kutta method of the fourth order. The system (4) remains dissipative at all parameter values b > 0. At values b < 0.39 there are no attractors in the system, except for a singular point of a stable focus type. At value b ≈ 0.39 there is a stable cycle in the system as a result a saddle-node bifurcation of births of stable and unstable cycles. This cycle exists up to the value b ≈ 0.8, when the stable cycle of the double period is born in the system . Further the cascade of Feigenbaum period-doubling bifurcations follows: the cycle of period 2 is observed up to value b ≈ 0.9, the cycle of the period 4—up to value b ≈ 0.926, generating a stable cycle of the period 8, etc. At the further increase in parameter values b, the next cycles have been found: of the period 7 at b ≈ 0.956, of the period 5 at b ≈ 0.965 and of the period 3 at b ≈ 0.982. This indicates the realization of full subharmonic cascade of Sharkovsky bifurcations in the system (4) (Figure 2). At b = 1 there exists a chaos in the system (4) and, hence, in the system (3), corresponding to an area of FShM scenario, which lies behind the Sharkovsky cascade. Homoclinic cascade in the system (4) is not found out, in view of absence in it of unstable singular points and homoclinic separatrix contours.
## 3. Dynamical chaos in Hamiltonian and conservative systems
Conservative system saves its volume at movement along the trajectories and, hence, cannot have attractors. Therefore studying of dynamical chaos in Hamiltonian and, especially, simply conservative systems is more a difficult task in comparison with the analysis of chaotic dynamics in dissipative systems which can be described by universal bifurcation FShM theory. The main problem solved by the modern classical theory of Hamiltonian systems (the Kolmogorov-Arnold-Moser theory) is the problem of integrability of such a system, that is, the problem of its reduction to the “action-angle” variables by constructing some canonical transformation. It is assumed that in such variables the motion in a Hamiltonian system is periodic or quasiperiodic and occurs on the surface of an n-dimensional torus. In this formulation, any non-integrable Hamiltonian system is considered as a perturbation of the integrable system, and the analysis of the dynamics of the perturbed system reduces to studying the problem of the destruction of the tori of an unperturbed system with increasing values of the perturbation parameter. But numerous examples of Hamiltonian and simply conservative systems, considered by the author in [4, 5, 6, 7], deny existence such classical KAM-scenario of transition to chaos.
One of the most effective approaches to the decision of a problem of the analysis of chaotic dynamics in conservative systems is offered by the author in Ref. [4] (see also [5, 6, 7]). The approach assumes consideration of conservative system in the form of limiting transition from corresponding extended dissipative system (in which the dissipative member is added) to initial conservative system. This approach can be evidently shown by means of construction of two-parametrical bifurcation diagram which corresponds to transition from dissipative state to conservative state. Attractors (stable cycles, tori and singular attractors) of extended dissipative system can be found numerically with use of results of universal bifurcation FShM theory. Further transition to chaos in conservative (Hamiltonian) system is carried out through cascades of bifurcations of attractors of extended dissipative system when dissipation parameter tends to zero. Areas of stability of stable cycles of the extended system at zero dissipation turn into tori of conservative (Hamiltonian) system around of its elliptic cycles into which stable cycles transform. Thus tori of conservative (Hamiltonian) system touch through hyperbolic cycles into which saddle cycles of extended dissipative system transform. In [4, 5, 6, 7] the considered above approach is described in detail with reference to Hamiltonian systems with one and a half, two, two and a half and three degrees of freedom, and also to simply conservative systems of differential equations, including the conservative Croquette equation, the equation of a pendulum with oscillating point of fixing, the conservative generalized Mathieu equation, well-known Hamiltonian system of Henon-Heiles equations. In Refs. [12, 13] the given approach has been applied and strictly proved by continuation along parameter of solutions from dissipative into conservative areas by means of the Magnitskii method of stabilization of unstable periodic orbits [1] at research bifurcations and chaos in the Duffing-Holmes equation
x¨+μẋδx+x3εcosωt=0,E5
and in the model of a space pendulum
x¨+μẋ+kx+εsin2πx=hcosωt.E6
Corresponding bifurcation diagrams in a plane (ε, μ) of existence of cycles of various periods down to a conservative case at μ = 0 are shown in [12, 13, 14]. Application of Magnitskii approach has revealed the essence of dynamical chaos in Hamiltonian and simply conservative systems. It became clear, that the chaos in such systems is not a result of destruction or non-destruction of some mythical tori of nonperturbed systems, as it follows from the KAM theory, but absolutely on the contrary, it is a consequence of limit transition of infinite number of cycles, tori and singular attractors, born according to the FShM theory as a result of cascades of bifurcations in extended dissipative system when dissipation parameter tends to zero.
### 3.1. Hamiltonian Yang-Mills-Higgs system with two degrees of freedom
In this chapter, let us illustrate Magnitskii approach by the example of Yang-Mills-Higgs system with two degrees of freedom and with Hamiltonian
H=ẋ2+ż2/2+x2z2/2+νx2+z2/2,E7
passing into classical system of the Yang-Mills equations at ν = 0. We shall consider four-dimensional phase space of the system with coordinates x,y=ẋ,z,r=ż:
ẋ=y,ẏ=xν+z2,ż=r,ṙ=zν+x2.E8
The system (8) has four sets of periodic solutions to which there correspond four basic cycles in phase space
Сx:z=r=0,y2+νx2=2;Cz:x=y=0,r2+νz2=2;C±:z=±x,y2+νx2+x4/2=1.E9
Assuming H = 1, we shall consider four-dimensional extended two-parametrical dissipative system of differential equations of a kind
ẋ=y,ẏ=xν+z2μy,ż=r+1Hxyzrz,ṙ=zν+x2,E10
where r=ż. Complication of solutions of Hamiltonian system (8) of the Yang-Mills-Higgs equations down to full chaotic dynamics occurs at ν → 0. In turn for each value ν > 0 the structure of solutions of Hamiltonian system (8) is completely determined by cascades of bifurcations of cycles of extended dissipative system (10) when dissipation parameter μ → 0. Stable cycles of dissipative system (10), born as a result of cascades of bifurcations, pass into elliptic cycles of Hamiltonian system (8), and their areas of stability—into tori around of these elliptic cycles. The contact of born tori of conservative system occurs on hyperbolic cycles in which corresponding unstable cycles of extended dissipative system transform. These unstable cycles are born in the dissipative system together with stable cycles as a result a saddle-node bifurcations, or at loss of stability of a cycle as a result of pitchfork bifurcation or period-doubling bifurcation. In neighborhoods of separatrix surfaces of hyperbolic cycles there is a formation of new more complex hyperbolic and elliptic cycles according to nonlocal effect of multiplication of cycles and tori in conservative systems (see [4, 5, 6, 7]). Last effect plays a key role in the system of Yang-Mills-Higgs equations at an initial stage of transition from regular dynamics to chaotic dynamics. At the same time, as numerical calculations show, the further complication of dynamics of solutions of system (10) at reduction of parameter value ν occurs not only by means of multiplication of elliptic and hyperbolic cycles and tori, but also by means of the cascade of period-doubling bifurcations of the basic cycles and by means of the subharmonic cascade of bifurcations. Initial cycles of the cascade of period-doubling bifurcations of the cycle C+ are presented in Figure 3. In Ref. [14] stabilization of unstable cycles of system (8) by modified Magnitskii method [1] is carried out.
Further, the process continues with the birth of infinitely folded heteroclinic separatrix manifold, stretched over separatrix Feigenbaum tree, both in extended dissipative system (10), and in Hamiltonian system (8) close to it. Accordions of corresponding heteroclinic separatrix zigzag fill the whole phase space of the system, however the limited accuracy of numerical methods does not allow to track this process up to the value ν = 0, corresponding to the initial system of the Yang-Mills equations.
## 4. Spatio-temporal chaos in nonlinear partial differential equations
Bases of FShM theory with reference to a wide class of nonlinear systems of partial differential equations are stated in Refs. [6, 7, 8, 9]. This class includes systems of the equations of reaction-diffusion type, describing various autowave oscillatory processes in chemical, biological, social and economic systems, including the well-known brusselator equations; the equations of FitzHugh-Nagumo type, describing processes of chemical and biological turbulence in excitable media; the equations of Kuramoto-Tsuzuki (or Time Dependent Ginzburg-Landau) type, describing complex autooscillating processes after loss of stability of a thermodynamic branch in reaction-diffusion systems; the systems of Navier-Stokes equations, describing laminar-turbulent transitions in hydrodynamical and gazodynamical mediums.
### 4.1. Diffusion chaos in reaction-diffusion systems
Wide class of physical, chemical, biological, ecological and economic processes and phenomena is described by reaction-diffusion systems of partial differential equations
ut=D1uxx+fuvμ,vt=D2vxx+guvμ,0xl,E11
depending on the scalar or vector parameter μ. The dynamics of the solutions of such a complex system of equations depends on the boundary conditions, the length of the spatial region, and the values of the diffusion coefficients. In many cases, there is a value of the system parameter μ0, such that for μ < μ0 the system (11) has a stable spatial homogeneous stationary solution (U, V), called the thermodynamic branch. In the case of loss of stability of the thermodynamic branch, when μ > μ0, solutions of the system (11) can be various homogeneous and inhomogeneous periodic solutions, spiral waves, running impulses, stationary dissipative structures, as well as nonstationary nonperiodic inhomogeneous solutions, called space-time or diffusion chaos.
The nonlinear processes occurring in so-called excitable media, are described by a special case of systems of the reaction-diffusion equations—FitzHugh-Nagumo type systems
ut=Duxx+fuvμ,vt=guvμ.E12
Solutions of the system (12) are: switching waves, traveling waves and running impulses, dissipative stationary spatially inhomogeneous structures, and diffusion chaos—nonstationary nonperiodic inhomogeneous structures, sometimes called biological or chemical turbulence. All such solutions can be analyze on a line by replacement ξ = x − c t and transition to three-dimensional system of ordinary differential equations
u̇=y,ẏ=cy+fuvμ/D,v̇=guvμ/c,E13
where the derivative is taken over the variable ξ. Therefore, the separatrix of the heteroclinic contour of system (13) describes the switching wave of the system (12), the limit cycle and the separatrix loop of the singular point of system (13) describe the traveling wave and the running impulse of system (12). And diffusion chaos in system (12) is described by singular attractors of the system of ordinary differential Eq. (13) in full accordance with the universal bifurcation Feigenbaum-Sharkovsky-Magnitskii theory. The greatest interest represents a case when c is a bifurcation parameter, describing a speed of wave distribution along an axis x, which is not included obviously into initial system (12). This case means, that system of a kind (12) with the fixed parameters can have infinitely number of various autowave solutions of any period running along a spatial axis with various speeds, and infinite number of modes of diffusion chaos. One of such system, describing chemical turbulence in autocatalytic chemical reactions, is studied in [6, 7, 15].
In this chapter, let us consider the system of a kind (12) describing distribution of nervous impulses in a cardiac muscle [16]:
ut=uxx+1εu1uu0.06+v0.75,vt=u3v.E14
where u is the activator, v—ingibitor, slowing down development of the activator, the parameter 1/ε defines time of restoration of the system after perturbation. Let us show, that transition to diffusion chaos in system (14), during complication of periodic fluctuations occurring in it, occurs according to universal bifurcation scenario of the FShM theory. We shall analyze solutions of system (14) by means of automodeling replacement of independent variables ξ = x − c t, having reduced the initial system of partial differential equations to three-dimensional system of ordinary differential equations
u̇=w,ẇ=cw+1εu1uu0.06+v0.75,v̇=vu3/c,E15
where derivative is taken with respect to the variable ξ. If (u(ξ), v(ξ), w(ξ)) is the solution of system of ODE (15) then (u(x − ct), v(x − ct), w(x − ct)) will be the solution of system in private derivatives (14). Thus running waves in system (14) are described by limit cycles of system (15), and running impulses—by separatrix loops of saddle-focuses. Let us carry out numerical research of system (15) in the field of where one of singular points is a saddle-focus. The greatest interest represents a case when c is the bifurcation parameter which describes a speed of waves distribution along an axis x and which is not included obviously into initial system (14). This case means, that the system (14) with the fixed parameters can have infinitely number of various autowave solutions of any period running along a spatial axis with various speeds, and infinite number of modes of diffusion chaos. Let us fix a parameter value ε : 1/ε = 17.4, and take the parameter c as bifurcation parameter. At c ∈ [1.6305, 1.6316] there is a stable cycle in the system (15). At c ≈ 1.6317 the cascade of Feigenbaum period-doubling bifurcations of the initial cycle begins, and at c ∈ [1.6317, 1.6331] the cycle of period 2 is observed, at c ∈ [1.6332, 1.6335]—the cycle of period 4, and at c ≈ 1.63375 the first singular attractor—Feigenbaum attractor is found out (Figure 4).
At the further reduction of values of parameter c, cycles of period 5 and period 3 are found out at c ≈ 1.6344 and at c ≈ 1.6347 (Figure 4). Thus, it is established, that in system (15) at change of parameter c, Feigenbaum cascade of period-doubling bifurcations of stable limit cycles and the full subharmonic Sharkovsky cascade of bifurcations of stable cycles according to the Sharkovsky order are realized. To the found cycles of system (15) there correspond running waves of system (14), some of which are represented in Figure 5.
### 4.2. Spatio-temporal chaos in autooscillating mediums
It is well-known that any solution of the reaction-diffusion system (11) in a neighborhood μ > μ0 of the thermodynamic branch can be approximated by some complex-valued solution W(r, τ) = u(r, τ) + iv(r, τ) of the Kuramoto-Tsuzuki (or Time Dependent Ginzburg-Landau) equation (see [1, 2, 6, 7]):
Wτ=W+1+ic1Wrr1+ic2W2W,E16
where r=εx,τ=ε2t,ε=μμ0,0rR,с1, c2—two real constants. Obviously, Eq. (16) has a spatial homogeneous solution W(τ) = exp(−i(c2τ + φ)) for an arbitrary phase φ. Consequently, each element of the medium (16) oscillates with a frequency c2. This solution is stable in a certain area of parameters c1 and c2. So, such media are called as autooscillating media. Research of solutions of the Kuramoto-Tsuzuki (Ginzburg-Landau) Eq. (16) directly in its phase space has shown, that actually in this equation there is subharmonic cascade of bifurcations of stable two-dimensional tori of any period according to the Sharkovsky order over each of frequencies and over two frequencies simultaneously. In [1, 2, 6, 7] solutions of the second boundary-value problem for the Eq. (16) on an interval are analyzed in detail. It has been constructed four-dimensional subspace (u(0), v(0), u(l/2), v(l/2)) of infinitely dimensional phase space of solutions of the problem, and its Poincare section by the plane u(l/2) = 0 for various values of bifurcation parameters c1 and c2 has been considered. Poincare’s method of the analysis of phase space of solutions of the Eq. (16) has allowed to find all cascades of bifurcations of two-dimensional tori in full accordance with the FShM theory.
In this chapter, we consider the problem of research of nonlinear effects in model of surface plasmon-polyariton. The passage of an electromagnetic wave through a configuration from three various environments dielectric-metal-dielectric can be described by following system of the equations in partial derivatives in the complex variables, turning out of Maxwell equations (see [17]):
iψpz+12β2ψpy2+ilΔβψp+κψa=0,iψaz+12β2ψay2+ilg+Δβψa+ψa2ψaκψp=0,E17
The system (17) represents two connected Ginzburg-Landau equations, ψp and ψa—complex-valued functions, y and z—independent variables. The role of time in Ginzburg-Landau equation in this case is played with spatial coordinate z. The equation for ψp corresponds to a wave on border of metal and passive dielectric, and for ψa—on border of metal and active nonlinear dielectric. Parameters l g, κ—accordingly dimensionless factors of losses, strengthening and connection between two borders. In Ref. [17] the following fixed values of parameters were considered: l = 0.0026, κ = 0.0028, Δβ = 0, β = 1.43, = f(Υ' + '') = 3.5 ⋅ 10−3(1 + 0.1i). We shall research dynamics of system (17) at various values of parameter g, and as boundary conditions on spatial variable y we shall consider periodic boundary conditions. In analysis of dynamics of the system (17) we use the real functions: u1, v1, u2, v2 instead of complex-valued functions ψp and ψa where ψp = u1 + iv1, ψa = u2 + iv2. And vector of independent variables is denoted x=u1v1u2v2T.
In the considered initial boundary-value problem it is possible to allocate a subclass of spatially homogeneous solutions, not dependent on a variable y. They can be found, solving the system of ordinary differential equations received from (17) by rejection of members, containing derivatives on y. The received system of ODE in coordinates xis
dxdz=lΔβ0κΔβlκ00κlgΔβκ0Δβgl×x+u22+v22f00Υ''u2+Υ'v2Υ''v2Υ'u2E18
Critical value of parameter is g = 0.0052. At smaller parameter values the zero solution is stable. At great values the solution loses stability, and the signs on the real parts are changed at once with four roots of the characteristic equation. Approximately at parameter value g = 0.00357 a pair of periodic solutions appears in system (18) as a result of the saddle-node bifurcation At parameter value g = 0.0052, when zero singular point loses its stability, the unstable periodic solution disappears as a result of subcritical Andronov-Hopf bifurcation. Thus, at g > 0.0052 there is a stable limit cycle in the system. Let us consider the scenario of complication of dynamics of solutions in system (17) at value L = 10, where L is the size of physical area on a variable y. Phase portraits of system we build in a point y = L/3: u1(L/3, z), v1(L/3, z), u2(L/3, z), v2(L/3, z) . In case of periodic boundary conditions, the first stages of complication go according to the Landau-Hopf scenario, that is, occurrence of periodic and quasiperiodic solutions of the increasing phase dimension have been found out. At parameter values g < 0.0095 the homogeneous cycle described above saves the stability, and at a parameter value g = 0.0096 he becomes non-homogeneous. The further complication of dynamics of system occurs at parameter value g ≈ 0.0105. At this value a quasiperiodic solution—torus of dimension two is born in the system as a result Andronov-Hopf bifurcation. A kind of this solution in phase space and its section by a plane u1 = 0 are represented in Figure 6. It is visible, that the section represents the closed curve.
Following bifurcation in the system (17) occurs in a range of parameter values g from 0.01385 till 0.01390. As a result of one more Andronov-Hopf bifurcation more complex quasiperiodic solution is formed in the system—it is torus of dimension three. A phase portrait of this torus at g = 0.0139 and its Poincare sections are represented in Figure 7. The first section u1 = 0 represents two-dimensional torus which in turn in section u2 = 0 gives two closed curves.
For the problem with Neumann’s homogeneous boundary conditions also it was possible to observe a non-homogeneous stable cycle at g = 0.0060. At g = 0.0095 stable two-dimensional torus is born from this cycle, and at g ≈ 0.013 stable three-dimensional torus is born from it as a result of the second Andronov-Hopf bifurcation. Thus, it is proved, that in complex nonlinear systems of partial differential equations stable three-dimensional tori can exist, that contradicts to the Ruelle-Takens theorem. The natural is not the decay of three-dimensional torus with forming uncertain mythical strange attractor, but further complication of dynamics of solutions as a result of following Andronov-Hopf bifurcation with forming four-dimensional torus, or as a result of period-doubling bifurcation of three-dimensional torus along one of its frequencies or along all frequencies simultaneously (that takes place in systems of Navier-Stokes equations).
### 4.3. Laminar-turbulent transition in Navier-Stokes equations
The problem of turbulence consists in explaining the nature of the disordered chaotic motion of a nonlinear continuous medium and in finding ways and methods of its adequate mathematical description. Originating more than a 100 years ago, the problem of turbulence is still one of the most complicated and most interesting problems in mathematical physics. It is in the list of seven mathematical millennium problems, named so by the Clay Institute of Mathematics [18]. In addition, the turbulence problem is formulated in the list of S.Smale's 18 most important mathematical problems of the twenty-first century [19]. The most important and interesting in the problem of turbulence is to elucidate the causes and mechanisms of chaos generation in a nonlinear continuous medium when passing from the laminar to the turbulent state. Currently, there are several mathematical models that claim to explain the mechanisms of generation of chaos and turbulence in nonlinear continuous media. The most famous among these models are: the Landau-Hopf model explaining turbulence by motion along an infinite-dimensional torus generated by an infinite cascade of Andronov-Hopf bifurcations; and the Ruelle-Takens model, which explains turbulence by moving along a strange attractor generated by the destruction of a three-dimensional torus. In recent years, the author and his pupils have proved (see [8, 9, 20, 21, 22]) that the universal bifurcation FShM mechanism for the transition to space-time chaos in nonlinear systems of partial differential equations through subharmonic cascades of bifurcations of stable cycles or two-dimensional and multi-dimensional tori also takes place in problems of laminar-turbulent transitions for Navier-Stokes equations
ut+uu=p+R1Δu+f,u=0,E19
where R is the bifurcation parameter (Reynolds number). The existence of stable two-dimensional tori of doubled period and stable three-dimensional tori and their further bifurcations is established for the problem of fluid flow from the ledge [20]. The existence of subharmonic cascades of bifurcations of stable cycles and two-dimensional tori is established for Rayleigh-Benard convection in Ref. [21]. A numerically complete subharmonic cascade of bifurcations of stable two-dimensional tori is found up to a torus of period three in the famous Kolmogorov problem in two-dimensional and three-dimensional spatial cases [22]. The features of compressible flow and instabilities triggered by Kelvin-Helmholtz (KH) and Rayleigh-Taylor (RT) mechanisms are considered in Ref [9]. The Kelvin-Helmholtz instability is the instability of the shear layer, which is a tangential discontinuity for the inviscid liquid and which arises when there is a velocity difference at the interface of two liquids or when there is a velocity shift in one of the liquids. Rayleigh-Taylor instability is the instability of the boundary between two liquids, where a lighter liquid supports a heavier fluid in a gravitational or external potential field, the gradient vector of which is directed from the heavier liquid to the lighter one. Light fluid can also push heavier one. Those two instabilities are often considered together. Indeed, RT instability causes movement of adjusted fluids in different directions with the appearance of the shear layer that is subject to KH instability.
In this chapter, we consider shortly the bifurcation scenario in coupled Kelvin-Helmholtz and Rayleigh-Taylor problem. This problem is solved in detail in Ref. [9]. We begin our consideration from the value of R = 1 for which the system has a stationary solution corresponding to a stable singular point in the phase space of solutions. Approximately for R = 10.5 the first bifurcation of the stationary solution occurs with the formation from the singular point of the stable limit cycle in the phase space of solutions. The next attractor that can be able to detect is the limited torus. Close resemblance to the cycle may indicate that this attractor was formed from the cycle as the result of Andronov-Hopf bifurcation. This indicates the existence of two irrational frequencies in the system. Further increase of the Reynolds number up to R = 516 resulted in the other Andronov-Hopf bifurcation with the formation of the three-dimensional invariant torus. This torus becomes singular (by period-doubling bifurcations along one of the frequencies). However this cascade of period-doubling bifurcations is reversed to the original 3D torus. The next bifurcation that could be traced at R = 520.5 is second Andronov-Hopf bifurcation leading to the formation of the four-dimensional invariant torus (Figure 8) . Further increase of the Reynolds number leads to the chaotic solution that corresponds to the dense field of points in phase subspace projections up to R = 2100. With the further increase of R, formation of inverse bifurcation cascades is observed. Thus, it seems reasonable, that there is no unified laminar-turbulent transition scenario in problems described by Navier-Stokes equations, it can be a cascade of stable limit cycles or cascade of stable two-dimensional or many dimensional tori, but all these scenarios lay in the frameworks of the FShM theory. However, the existence of computationally stable 4D invariant torus is a remarkable fact. It took 2.6 ⋅ 109 time samples to analyze and about 3.5 month to calculate this torus and its Poincare sections.
## 5. Conclusion
We make some general remarks on the chaotic dynamics of nonlinear systems of differential equations, since the very publication of papers [10, 11] and many similar papers, even in prestigious refereed journals, attests to a complete lack of understanding of the mechanism of transition to chaos in nonlinear systems of differential equations. In this chapter, on numerous examples, it is convincingly demonstrated that there exists one universal FShM bifurcation scenario of transition to chaos in all systems of nonlinear differential equations without exception: autonomous and nonautonomous, dissipative and conservative, ordinary, with partial derivatives and with delayed argument. All irregular attractors that arise during the implementation of this scenario are exclusively singular attractors. Each nonlinear system can have infinitely many different structurally unstable singular attractors for different values of the bifurcation parameter, which can enter implicitly into the equations of the system. Thus, neither the presence or absence of stable or unstable singular points in the system, nor the presence or absence of saddle-nodes or saddle-focuses, as well as homoclinic or heteroclinic separatrix contours, is not a criterion for the appearance of chaotic dynamics in the system. Also, neither the positivity of the senior Lyapunov exponent, nor the proof of existence of Smale’s horseshoe, nor the KAM (Kolmogorov-Arnold-Moser) theory, nor the theory of RT (Ruelle-Takens), are such criteria either. The positivity of the Lyapunov exponent is purely a consequence of computational errors, because due to the presence of an everywhere dense set of nonperiodic trajectories, numerical motion is possible only over the whole region occupied by the trajectory of the singular attractor, and not along its trajectory itself. In addition, the Lyapunov exponent will also be positive when moving along a stable periodic trajectory of a large period in the vicinity of some singular attractor. The presence of Smale’s horseshoe in the system testifies to the complex dynamics of the solutions, however, even in the neighborhood of the separatrix loop of saddle-focus, where by Shilnikov’s theorem there exists an infinite number of Smale’s horseshoes, the dynamics of solutions are determined not by horseshoes, but by a much more complex infinite set of unstable periodic solutions generated at all stages of all three cascades of bifurcations of the FShM scenario, whose homoclinic cascade of cycles ends in the limit precisely with the separatrix loop of saddle-focus. The only method that allows establishing reliably the presence of chaotic dynamics in the system is the numerical finding of stable cycles or tori of the FSM-cascades of bifurcations.
## Acknowledgments
Paper is supported by Russian Foundation for Basic Research (grants 14-07-00116-а).
## How to cite and reference
### Cite this chapter Copy to clipboard
Nikolai A. Magnitskii (March 28th 2018). Bifurcation Theory of Dynamical Chaos, Chaos Theory, Kais A. Mohamedamen Al Naimee, IntechOpen, DOI: 10.5772/intechopen.70987. Available from:
### chapter statistics
1Crossref citations
### Related Content
Next chapter
#### Structural Analysis of Deterministic Mass Fractals Using Small- Angle Scattering and Lacunarity Techniques
By Azat Mukhiddinuly Slyamov and Eugen Mircea Anitas
#### Numerical Simulations of Physical and Engineering Processes
Edited by Jan Awrejcewicz
First chapter
#### Numerical Solution of Many-Body Wave Scattering Problem for Small Particles and Creating Materials with Desired Refraction Coefficient
By M. I. Andriychuk and A. G. Ramm
We are IntechOpen, the world's leading publisher of Open Access books. Built by scientists, for scientists. Our readership spans scientists, professors, researchers, librarians, and students, as well as business professionals. We share our knowledge and peer-reveiwed research papers with libraries, scientific and engineering societies, and also work with corporate R&D departments and government entities.
View all Books | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8502091765403748, "perplexity": 863.973904215812}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875145742.20/warc/CC-MAIN-20200223001555-20200223031555-00520.warc.gz"} |
http://www.ic.sunysb.edu/Class/phy122ps/labs/dokuwiki/doku.php?id=phy124summer:lab_8&rev=1279052130&do=diff | # Differences
This shows you the differences between two versions of the page.
phy124summer:lab_8 [2010/07/13 16:15] (current)
Line 1: Line 1:
+<html><STYLE> #jsMath_Warning {display: none} </STYLE></html>
+====== PHY 124 Lab 8 - Measurement of e/m for the electron ======
+
+The purpose of this laboratory is the measurement of the charge (e) over mass (m) ratio e/m for the electron and to study qualitatively the motion of charged particles in a magnetic field.
+
+[[http://www.ic.sunysb.edu/Class/phy122ps/labs/dokuwiki/pdfs/124lab8worksheet.pdf|Important! You need to print out the 2 page worksheet you find by clicking on this link and bring it with you to your lab session.]]
+
+If you need the .pdf version of these instructions you can get them [[http://www.ic.sunysb.edu/Class/phy122ps/labs/dokuwiki/pdfs/phy124lab8.pdf|here]].
+
+===== Video =====
+<flashplayer width=640 height=480>file=http://www.ic.sunysb.edu/Class/phy122ps/labs/phy122vid/phy124lab8.flv</flashplayer>
+
+===== Equipment =====
+
+ * e/m aparatus
+ * meter stick
+ * black cloth
+
+There are kinds of 2 e/m apparatus. Apparatus A has built in power supplies for delivering the magnet current and the accelerating voltage. Apparatus B uses external power supplies for these tasks. The settings you use for the two apparatus will be slightly different. Please make sure you have identified correctly your apparatus and use the settings and instructions that apply to it
+
+{{124l8fig1.jpg?600}}
+===== Background to the experiment =====
+
+The principle of the e/m measurement is as follows (see the figure below): electrons are initially accelerated by an accelerating voltage V, and guided on a circular path by the magnetic field supplied by the two Helmholtz coils shown. The magnetic field B at the location of the spherical vacuum tube, in which the electrons move is approximately perpendicular to the plane of the coils (the paper plane). It causes a magnetic force (see Ch18 sheet 7) to act on the electrons and guides them on a circular path which is perpendicular to the magnetic field in the middle between the two magnet coils. You can visualize this by using the right-hand-rule (remember the electrons have negative charge!).
+
+{{124l8fig2.jpg}}
+
+\\ {{124l8sketch.jpg}}
+
+In the MapleTA prep assignment, you are going to derive the e/m relation you will use in your analysis for this lab. The kinetic energy KE and hence the velocity v of the electrons are determined by the accelerating voltage they have traversed (see Ch16 sheet 25). The radius r of the circular path can be related to the electron velocity and magnetic field (see Ch18 sheet 8,9). By combining these equations you will end up with a formula that relates e/m to the quantities you measure in this lab.
+
+Use the equations on sheet 25 Ch16 and the relation $KE=\frac{1}{2}mv^{2}$ , solving for the velocity to get a relation between the electron velocity v to the accelerating voltage V. Then use this relation and equation (18.2) for the radius r of the electron circular path (see Ch18 sheet 8,9). Substitute the equation for velocity into the equation for r and then solve for e/m. This will relate e/m to the accelerating voltage V and the magnetic field B. You should get the relation
+
+<WRAP column 35%>\\
+</WRAP>
+<WRAP column 45%>
+$\Large \frac{e}{m}=\frac{2V}{B^{2}r^{2}}$
+</WRAP>
+<WRAP column 10%>
+(8.1)
+</WRAP>
+\\
+
+In Chap. 18, sheet 24 an equation is given for the magnetic field in the center of the plane of a single loop. In the lab prep exercise you are asked to modify this equation for N loops. Make sure you refer to the coil radius with the letter a.
+
+To account for the 2 coils in our lab setup, we could approximately calculate the field by multiplying the field produced by 1 coil by 2 as you do in the lab prep exercise but this would not be quite right because the two coils are not stacked directly one on top of the other. A better equation for the magnetic field produced near the axis of a pair of Helmholtz coils at the location of the electron path is given by the equation (all units are SI)
+
+<WRAP column 35%>\\
+</WRAP>
+<WRAP column 45%>
+$\Large B=\frac{4\pi k'NI}{(5/4)^{3/2}a}$
+</WRAP>
+<WRAP column 10%>
+(8.2)
+</WRAP>
+\\
+
+
+\\ where
+\\
+N = the number of turns in each Helmholtz coil = 130\\
+a = the radius of the Helmholtz coils (see the figure above)\\
+k’ = 10<sup>-7</sup> in SI units, the universal magnetic constant (see Ch18 sheet 22)\\
+I = the current through the Helmholtz coils.\\
+
+Substituting equation (8.2) for the magnetic field, B, in equation (9.1) gives, for e/m
+
+<WRAP column 35%>\\
+</WRAP>
+<WRAP column 45%>
+$\Large \frac{e}{m}=\frac{2V(5/3)^{3}a^{2}}{(4\pi k'NIr)^2}$
+</WRAP>
+<WRAP column 10%>
+(8.3)
+</WRAP>
+\\
+
+\\
+You will use **this** expression (8.3) in the quantitative Part II.
+
+NOTE: In this lab you will be measuring two radii, the radius of the circular path of the electron and the radius of the Helmholtz coils. The radius of the electron path will be referred to by the symbol r and the radius of the coils will be referred to by the symbol a.
+
+===== Part I: Qualitative Exploration =====
+
+{{124l8fig3.jpg?600}}
+
+
+Before you use the equipment to determine e/m, you will first investigate the relation between the voltage and current controls of the apparatus and the radius of the path of the electron. In this part you explore how the accelerating voltage V and the magnet current I influence the radius r of the electron orbit. It is important to note that the voltage and current controls affect two different parts of the set up. The accelerating voltage, V, affects the electron beam and the magnet current, I, affects the magnetic field of the Helmholtz coils.
+==== Procedure ====
+
+^ Apparatus A ^ Apparatus B ^
+| Plug in the apparatus and turn the power on. Wait for 30 seconds during which the power supply runs a self-test. The power supply has dials (see the figure above) for the accelerating voltage V (on the left) and the magnet current I on the right. Set V to ~ 150 V and I to ~ 1.2 A.| The power supplies should already be connected before you begin. If not, ask your TA to do it for you. Plug in the power supplies and turn the power on for both units. Use the 500V adjust dial on the high voltage power supply to control the accelerating voltage V and the current adjust on the e/m apparatus to control the current magnet current I. Set V to ~ 250 V and I to ~ 1.5 A.|
+
+You should see the electron circular path as a “blue-green” glow. This glow comes from the ionization of a small amount of Helium left in the vacuum tube by the electron beam. Note that you will probably have to either cover the coil with a dark cloth or dim the room lights to see the glow.
+
+Keeping V constant at either 150 V or 250 V, depending on your setup, increase I in several steps and observe the radius r of the electron path. Explain your observation in terms of the coil current I , the electron velocity v and the relation (18.2) in Ch18 sheet 9.
+
+Keeping I constant at ~either 1.2 or 1.5 A, depending on your setup, increase V in several steps and observe r. Explain your observation in terms of the accelerating voltage V, the electron velocity v and the relation (18.2) in Ch18 sheet 9.
+===== Part II: Measurement of e/m =====
+
+In order to get an accurate coil radius, a, you will take several measurements and use their average for future calculations. Measure the vertical (from top to bottom) and the horizontal (from left to right) diameter (from the middle of the coil winding) for both coils and enter them on your worksheet . Notice that these values are referred to as 2a because they are diameters. Take the average of these 4 measurements to calculate the coil radius you will used in the analysis:
+
+$a=\frac{1}{2}(2a_{v1}+2a_{v2}+2a_{h1}+2a_{h2})/4$
+
+Estimate the error of one diameter and assume it is the same for each of the 4 measurements. Calculate the error of the average value of a using equation (E.5b) and (E.5a) in the error and uncertainty manual.
+==== Procedure ====
+
+^Apparatus A ^ Apparatus B ^
+|To measure the electron beam radius, r, you will use the marks etched on the glass scale in the vacuum tube. Be careful to notice that these marks are in cm and give the diameter of the beam path! You will take your data by keeping a fixed I and varying V. Keep I fixed at ~ 2 A and start with V ~ 150 V. Increase V in ~ 6 steps such that at each step the electron beam hits the cm marks etched on the glass scale in the center of the vacuum tube. It is more important that the beam hits these marks than to have equal V steps. For each step record V, I, r and your estimate of the error $\Delta$r in the table on your worksheet.\\ {{124l8fig4a.jpg}}| On apparatus B the globe can be rotated, altering the direction of the beam path relative to the magnetic field. Before you begin to make this measurement you should ensure it is oriented so that the beam forms a circle and not a spiral. To measure the electron beam radius, r, you will use the illuminated mirrored scale on the back of the apparatus and the ruler in front of the apparatus. The mirrored scale helps to reduce parallax error, you do this by moving your head until the edge of the reflection of the beam in the mirror aligns with the edge of the actual beam. (This means that you are looking straight on at beam.) Then position the metal triangle on the front ruler so it touch the sides of the beam path and measure the beam path radius from the scale on the mirrored ruler. Take an average of your measurement of the radius when you are looking from the left and from the right as the radius of the beam. You will take your data by keeping a fixed I and varying V. Keep I fixed at ~ 2 A and start with V ~ 200 V. Step V up in 6 steps of ~ 50V. For each step record V, I, r and your estimate of the error $\Delta$r . For each step record V, I, r and your estimate of the error $\Delta$r in the table on your worksheet.\\ {{124l8fig4b.jpg}}|
+
+When estimating $\Delta$r for either apparatus, consider things like the fuzziness or the brightness of the beam. Note that $\Delta$r does not have to be the same for each measurement.
+
+Assume that the errors of V and I are negligible. Solve equation (8.3) for V and write it such that it has the form $V=constant\times \frac{e}{m}\times (Ir)^2$ where “constant” stands for a factor containing only quantities that are constant during the experiment, including the magnet coil radius a. Calculate the value of “constant” and enter it on your worksheet. Notice it includes the radius a of the magnet coils, which has an uncertainty so you must propagate the error of a into the error of the constant using expressions (E.1) and (E.8) in Error and Uncertainty.
+
+Transfer your recorded data in to the form below.
+
+<html>
+<form method="post" action="http://mini.physics.sunysb.edu/~mdawber/plot/124lab8.php" target="_blank">
+
+V1  <input type="text" name="V1" size="10"/> V   I1  <input type="text" name="I1" size="10"/> A   r1  <input type="text" name="r1" size="10"/> m   Δr1  <input type="text" name="deltar1" size="10"/> m <br/>
+V2  <input type="text" name="V2" size="10"/> V   I2  <input type="text" name="I2" size="10"/> A   r2  <input type="text" name="r2" size="10"/> m   Δr2  <input type="text" name="deltar2" size="10"/> m<br/>
+V3  <input type="text" name="V3" size="10"/> V   I3  <input type="text" name="I3" size="10"/> A   r3  <input type="text" name="r3" size="10"/> m   Δr3  <input type="text" name="deltar3" size="10"/> m <br/>
+V4  <input type="text" name="V4" size="10"/> V   I4  <input type="text" name="I4" size="10"/> A   r4  <input type="text" name="r4" size="10"/> m   Δr4  <input type="text" name="deltar4" size="10"/> m <br/>
+V5  <input type="text" name="V5" size="10"/> V   I5  <input type="text" name="I5" size="10"/> A   r5  <input type="text" name="r5" size="10"/> m   Δr5  <input type="text" name="deltar5" size="10"/> m<br/>
+V6  <input type="text" name="V6" size="10"/> V   I6  <input type="text" name="I6" size="10"/> A   r6  <input type="text" name="r6" size="10"/> m   Δr6  <input type="text" name="deltar6" size="10"/> m<br/>
+
+
+<input type="submit" value="submit" name="submit" />
+</form>
+</html>
+
+
+When you click submit the computer calculates the quantity $(Ir)^2$ for each data point and then plots $V$ vs $(Ir)^2$. Why do we plot $V$ vs $(Ir)^2$ and not $V$ vs $(Ir)$? Since you are considering the errors of V and I to be negligible, all the error of your plot will come from the error in r, so you will only have horizontal error bars. The computer obtains the values for the error in $(Ir)^2$ by propagating the error of r into the error of the quantity $(Ir)^2$ using expression (E.1) and (E.8) in Error and Uncertainty. Write the slope of the graph and it's error on your worksheet.
+
+Next, set up an equation that relates the slope, s, to the “constant” and to e/m. Solve the equation for the ratio e/m. Using your errors for s and the “constant” calculate the error of e/m using expression (E.7) in Error and Uncertainty. Make sure you simplify units as much as possible. Compare your value to the established value which you can obtain by dividing the charge on the electron, $e=1.602\times10^{-19} C$ by it's mass $m_{e}=9.109\times10^{-31} kg$
+ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8947482705116272, "perplexity": 1492.8068497260067}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439738913.60/warc/CC-MAIN-20200812171125-20200812201125-00423.warc.gz"} |
http://mathhelpforum.com/pre-calculus/143324-polar-form-finding-all-solutions.html | # Math Help - Polar Form/Finding all solutions
1. ## Polar Form/Finding all solutions
How do I find all solutions of (2z + 6)^5= 1 + i
Do I have to find the polar form and determine what angle it has and then use
z=r(cosx + isinx)? Or is that not helpful in this situation?
Thanks.
2. Originally Posted by JollyJolly15
How do I find all solutions of (2z + 6)^5= 1 + i
Do I have to find the polar form and determine what angle it has and then use
z=r(cosx + isinx)? Or is that not helpful in this situation?
Thanks.
Oh, yes, yes: the polar form is very helpful in this case. So you have
$(2z+6)^5=1+i=\sqrt{2}\cdot e^{\displaystyle i\tfrac{\pi}{4}+i2n\pi}, n\in\mathbb{Z}$
(I am using $z=|z|\cdot e^{i\varphi}$ here as the "polar form" of z, instead of $z=|z|\cdot\big(\cos\varphi+i\sin\varphi\big)$, because it is easier to type.)
Next you can take the fifth root by dividing the exponents on both sides by 5, and you get
$2z+6=\sqrt[10]{2}\cdot e^{\displaystyle i\tfrac{(8n+1)\pi}{20}}, n\in \mathbb{Z}$
Finally subtract 6 and divide by 2: done. - Except that you still need to figure out how many different solutions you have now got... | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 4, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9595504999160767, "perplexity": 706.6268274185329}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-26/segments/1466783399106.96/warc/CC-MAIN-20160624154959-00147-ip-10-164-35-72.ec2.internal.warc.gz"} |
http://mathhelpforum.com/geometry/96162-need-help-finding-perimeter-square-rhombus.html | # Math Help - Need Help finding perimeter of square and rhombus
1. ## Need Help finding perimeter of square and rhombus
1. In square SQUA, the lengths of diagonal QA is 24 cm. In rhombus RHOM the length of diagonal MH is 24 cm and the measure of angle RHO = 60. Which quadrilateral has the greater perimeter? Show all the work to support your conclusion.
I'm taking an online geometry class right now and I having a really hard time understanding it. If someone could please explain to me how to solve this problem, that would be great!
Thanks
2. Originally Posted by sunshii17
1. In square SQUA, the lengths of diagonal QA is 24 cm. In rhombus RHOM the length of diagonal MH is 24 cm and the measure of angle RHO = 60. Which quadrilateral has the greater perimeter? Show all the work to support your conclusion.
If x is the side pf the square and d is the diagonal, using Pythagoras theorem, you can write 2*x^2 = d^2. D is given, Find x and perimeter which is 4x.
In the rhombus let o be the point of intersection of diagonals. In triangle RHO angle RHO = 30 degrees. If x is the side of the rhombus, OH = x*cos30. OH is given. Find x and perimeter which is again 4x.
3. ## Perimeter of square and rhombus
Hello sunshii17
Welcome to Math Help Forum!
Originally Posted by sunshii17
1. In square SQUA, the lengths of diagonal QA is 24 cm. In rhombus RHOM the length of diagonal MH is 24 cm and the measure of angle RHO = 60. Which quadrilateral has the greater perimeter? Show all the work to support your conclusion.
I'm taking an online geometry class right now and I having a really hard time understanding it. If someone could please explain to me how to solve this problem, that would be great!
Thanks
You don't need to do detailed calculations; all you need to know is a little about cosines.
Draw the two diagrams carefully, joining the diagonals in each case. Suppose the diagonals in each diagram meet at the point $X$.
Then (since the diagonals of a rhombus meet at right angles), all the angles in each diagram at $X$ are right angles, and so each figure - the square and the rhombus - consists of four identical right-angled triangles. In the case of the square, the other angles in one of the triangles (for instance, $\triangle SQX$) are each $45^o$. In the case of the rhombus the other angles in one of the triangles (for instance, $\triangle RHX$) are $\angle H=30^o$ and $\angle R = 60^o$.
So for the rhombus, we have:
$\cos 30^o = \frac{XH}{RH} \Rightarrow RH = \frac{12}{\cos 30^o}$
And for the square:
$\cos 45^o = \frac{XQ}{SQ}\Rightarrow SQ =\frac{12}{\cos 45^o}$
Now $\cos 30^o > \cos 45^o$ and so $RH < SQ$. This is because, when you calculate $RH$, you divide the $12$ by a larger number than when you calculate $SQ$.
So the perimeter of the rhombus ( $= 4\,RH$) is less than the perimeter of the square ( $= 4\,SQ$). | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 16, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9141122102737427, "perplexity": 363.5896657851708}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398449160.83/warc/CC-MAIN-20151124205409-00292-ip-10-71-132-137.ec2.internal.warc.gz"} |
https://epsproc.readthedocs.io/en/v1.2.6/methods/LF_AF_verification_tests_060720_tidy_100920.html | # ePSproc LF/AF function verification & tests¶
• 10/09/20 v5 AF code now working and verified.
• 27/08/20 v4 Revisiting again, updates to normalisation with degen factor. Now getting correct $$\beta$$ values, still some possible issues with total XS however.
• 06/07/20 v3 Updated plotting codes & added AF tests.
• 26/06/20 v2
• 19/06/20 v1
For LF and AF calculations, trying to get to the bottom of issues with magnitudes and/or phases and/or formalism differences with raw ePS matrix elements.
## Formalism¶
Test cases:
1. ePS matrix elements with formalism from [1], for LF cross-sections and $$\beta_{2}$$. Formalism with Clebsch-Gordan (CG) terms.
2. ePSproc AF calculations, for LF cross-sections and $$\beta_{2}$$ in isotropic case (or other terms in general cases). Usual 3j-symbol based formalism.
The AF calculations should reduce to the LF case for an isotropic ensemble, and both cases should match the “direct” ePS GetCro outputs (LF). Hopefully this should clear up any outstanding issues with normalisation, units, scale-factors, phase conventions etc. For details of the AF code, see the method dev notes.
(For MF verification, see the MFPADs and associated :math:beta_{LM} <https://epsproc.readthedocs.io/en/dev/demos/ePSproc_BLM_calc_demo_Sept2019.html#Benchmark-vs.->__ notebooks, where the numerics are verified for the NO2 test case, although the total cross-sections may still have issues (for more discussion, see the Matlab code release software paper). The geometric tensor version of the MF calculations is also verified against the same test case.)
[1] Cross section and asymmetry parameter calculation for sulfur 1s photoionization of SF6, A. P. P. Natalense and R. R. Lucchese, J. Chem. Phys. 111, 5344 (1999), http://dx.doi.org/10.1063/1.479794
[2] Reid, Katharine L., and Jonathan G. Underwood. “Extracting Molecular Axis Alignment from Photoelectron Angular Distributions.” The Journal of Chemical Physics 112, no. 8 (2000): 3643. https://doi.org/10.1063/1.480517.
[3] Underwood, Jonathan G., and Katharine L. Reid. “Time-Resolved Photoelectron Angular Distributions as a Probe of Intramolecular Dynamics: Connecting the Molecular Frame and the Laboratory Frame.” The Journal of Chemical Physics 113, no. 3 (2000): 1067. https://doi.org/10.1063/1.481918.
[4] Stolow, Albert, and Jonathan G. Underwood. “Time-Resolved Photoelectron Spectroscopy of Non-Adiabatic Dynamics in Polyatomic Molecules.” In Advances in Chemical Physics, edited by Stuart A. Rice, 139:497–584. Advances in Chemical Physics. Hoboken, NJ, USA: John Wiley & Sons, Inc., 2008. https://doi.org/10.1002/9780470259498.ch6.
### Formalism: LF case with CG terms¶
As given in ref. [1]. This is now implemented in implemented in ePSproc.lfblmGeom <https://epsproc.readthedocs.io/en/dev/modules/epsproc.geomFunc.lfblmGeom.html>__. NOTE - that the $$M$$ term here is an MF projection term, and should be summed over for the final LF result.
The matrix elements $$I_{\mathbf{k},\hat{n}}^{(L,V)}$$ of Eqs. (8) and (9) can be expanded in terms of the $$X_{lh}^{p\mu}$$ functions of Eq. (7) as$$^{14}$$
$$I_{\mathbf{k},\hat{n}}^{(L,V)}=\left[\frac{4\pi}{3}\right]^{1/2}\sum_{p\mu lhv}I_{lhv}^{p\mu(L,V)}X_{lh}^{p\mu}(\hat{k})X_{1v}^{p_{v}\mu_{v}}(\hat{n}).$$
{[}Note here the final term gives polarization (dipole) terms, with $$l=1$$, $$h=v$$, corresponding to a photon with one unit of angular momentum and projections $$v=-1,0,1$$, correlated with irreducible representations $$p_{v}\mu_{v}$$.{]}
The differential cross section is given by
$$\frac{d\sigma^{L,V}}{d\Omega_{\mathbf{k}}}=\frac{\sigma^{L,V}}{4\pi}[1+\beta_{\mathbf{k}}^{L,V}P_{2}(\cos\theta)],$$
where the asymmetry parameter can be written as$$^{14}$$
\begin{eqnarray} \beta_{\mathbf{k}}^{L,V} & = & \frac{3}{5}\frac{1}{\sum_{p\mu lhv}|I_{\mathbf{k},\hat{n}}^{(L,V)}|^{2}}\sum_{\stackrel{p\mu lhvmm_{v}}{p'\mu'l'h'v'm'm'_{v}}}(-1)^{m'-m_{v}}I_{\mathbf{k},\hat{n}}^{(L,V)}\nonumber \\ & \times & \left(I_{\mathbf{k},\hat{n}}^{(L,V)}\right)^{*}b_{lhm}^{p\mu}b_{l'h'm'}^{p'\mu'*}b_{1vm_{v}}^{p_{v}\mu_{v}}b_{1v'm'_{v}}^{p'_{v}\mu'_{v}*}\nonumber \\ & \times & [(2l+1)(2l'+1)]^{1/2}(1100|20)(l'l00|20)\nonumber \\ & \times & (11-m'_{v}m_{v}|2M')(l'l-m'm|2-M'), \end{eqnarray}
and the $$(l'lm'm|L'M')$$ are the usual Clebsch–Gordan coefficients. The total cross section is
$$\sigma^{L,V}=\frac{4\pi^{2}}{3c}E\sum_{p\mu lhv}|I_{\mathbf{k},\hat{n}}^{(L,V)}|^{2},$$
where c is the speed of light.
• In the current numerics, the XS is defined as $$\beta_0$$ calculated as per above, for $$L=0$$, or as $$1/3*\sum_{p\mu lhv}|I_{\mathbf{k},\hat{n}}^{(L,V)}|^{2}$$. These are identical.
• The normalisation factors of 1/3 and 1/5 appear to be $$1/(2L+1)$$ (or other degeneracy - possibly symmetry) terms - TBC.
• Computing $$\beta_0$$ with the full formalism gives the correct values here, while using the matrix elements directly requires the additional 1/3 normalisation term.
• For $$\beta_2$$, the 1/5 term is required in both cases.
### AF formalism¶
The original (full) form for the AF equations, as implemented in ePSproc.afblm <https://epsproc.readthedocs.io/en/dev/modules/epsproc.AFBLM.html>__ (NOTE - there are some corrections to be made here, which are not yet implemented in the base code, but are now in the geometric version):
\begin{eqnarray} \beta_{L,-M}^{\mu_{i},\mu_{f}} & = & \sum_{l,m,\mu}\sum_{l',m',\mu'}(-1)^{M}(-1)^{m}(-1)^{(\mu'-\mu_{0})}\left(\frac{(2l+1)(2l'+1)(2L+1)}{4\pi}\right)^{1/2}\left(\begin{array}{ccc} l & l' & L\\ 0 & 0 & 0 \end{array}\right)\left(\begin{array}{ccc} l & l' & L\\ -m & m' & -M \end{array}\right)\nonumber \\ & \times & I_{l,m,\mu}^{p_{i}\mu_{i},p_{f}\mu_{f}}(E)I_{l',m',\mu'}^{p_{i}\mu_{i},p_{f}\mu_{f}*}(E)\\ & \times & \sum_{P,R,R'}(2P+1)(-1)^{(R'-R)}\left(\begin{array}{ccc} 1 & 1 & P\\ \mu_{0} & -\mu_{0} & R \end{array}\right)\left(\begin{array}{ccc} 1 & 1 & P\\ \mu & -\mu' & R' \end{array}\right)\\ & \times & \sum_{K,Q,S}(2K+1)^{1/2}(-1)^{K+Q}\left(\begin{array}{ccc} P & K & L\\ R & -Q & -M \end{array}\right)\left(\begin{array}{ccc} P & K & L\\ R' & -S & S-R' \end{array}\right)A_{Q,S}^{K}(t) \end{eqnarray}
Where $$I_{l,m,\mu}^{p_{i}\mu_{i},p_{f}\mu_{f}}(E)$$ are the energy-dependent dipole matrix elements, and $$A_{Q,S}^{K}(t)$$ define the alignment parameters.
In terms of the geometric parameters, this can be rewritten as:
\begin{eqnarray} \beta_{L,-M}^{\mu_{i},\mu_{f}} & =(-1)^{M} & \sum_{P,R',R}{[P]^{\frac{1}{2}}}{E_{P-R}(\hat{e};\mu_{0})}\sum_{l,m,\mu}\sum_{l',m',\mu'}(-1)^{(\mu'-\mu_{0})}{\Lambda_{R'}(\mu,P,R')B_{L,-M}(l,l',m,m')}I_{l,m,\mu}^{p_{i}\mu_{i},p_{f}\mu_{f}}(E)I_{l',m',\mu'}^{p_{i}\mu_{i},p_{f}\mu_{f}*}(E)\sum_{K,Q,S}\Delta_{L,M}(K,Q,S)A_{Q,S}^{K}(t)\label{eq:BLM-tidy-prod-2} \end{eqnarray}
See the method dev notebook for more details. Both methods gave the same results for N2 test cases, so are at least consistent, but do not currently match ePS GetCro outputs for the LF case.
### Numerics¶
In both LF and AF cases, the numerics tested herein are based on the geometric tensor expansion code, which has been verified for the MF case as noted above (for PADs at a single energy).
A few additional notes on the implementations…
• The matrix elements used are taken from the DumpIdy output segments of the ePS output file, which provide “phase corrected and properly normalized dynamical coefs”.
• The matrix elements output by ePS are assumed to correspond to $$I_{lhv}^{p\mu(L,V)}$$ as defined above.
• The Scale Factor (SF) “to sqrt Mbarn” output with the matrix elements is assumed to correspond to the $$\frac{4\pi^{2}}{3c}E$$ term defined above, plus any other required numerical factors ($$4\pi$$ terms and similar).
• The SF is energy dependent, but not continuum (or partial wave) dependent.
• If correct, then using matrix elements * scale factor, should give correct results (as a function of $$E$$), while omitting the scale factor should still give correct PADs at any given $$E$$, but incorrect total cross-section and energy scaling.
• This may be incorrect, and some other assumptions are tested herein.
• EDIT: now verified as $$SF*\sqrt(4*pi)$$ to match LF/GetCro cross-section results.
• The AF and LF case should match for an isotropic distribution, defined as $$A^{0}_{0,0}=1$$. Additional normalisation required here…?
• A factor of $$\sqrt{(2K+1)}/8\pi^2$$ might be required for correct normalisation, although shouldn’t matter in this case. (See eqn. 47 in [4].)
• For the LF case, as defined above, conversion from Legendre-normalised $$\beta$$ to spherical harmonic normalised $$\beta$$ is required for comparison with the AF formalism, where $$\beta^{Sph}_{L,0} = \sqrt{(2L+1)/4\pi}\beta^{Lg}$$
### Status¶
09/09/20 - AF code now working correctly, and verified herein, ePSproc v1.2.5-dev.
## Set up¶
[1]:
# Imports
import numpy as np
import pandas as pd
import xarray as xr
# Special functions
# from scipy.special import sph_harm
import spherical_functions as sf
import quaternion
# Performance & benchmarking libraries
# from joblib import Memory
# import xyzpy as xyz
import numba as nb
# Timings with ttictoc or time
# https://github.com/hector-sab/ttictoc
# from ttictoc import TicToc
import time
# Package fns.
# For module testing, include path to module here
import sys
import os
if sys.platform == "win32":
modPath = r'D:\code\github\ePSproc' # Win test machine
else:
modPath = r'/home/femtolab/github/ePSproc/' # Linux test machine
sys.path.append(modPath)
import epsproc as ep
# TODO: tidy this up!
from epsproc.util import matEleSelector
from epsproc.geomFunc import geomCalc, geomUtils
from epsproc.geomFunc.lfblmGeom import lfblmXprod
# Plotters
from epsproc.plot import hvPlotters
* pyevtk not found, VTK export not available.
[2]:
hvPlotters.setPlotters()
# import bokeh
# import holoviews as hv
# hv.extension('bokeh') | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 3, "x-ck12": 0, "texerror": 0, "math_score": 0.950599193572998, "perplexity": 4620.444855085447}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038464065.57/warc/CC-MAIN-20210417222733-20210418012733-00341.warc.gz"} |
https://www.physicsforums.com/threads/restoring-force.837556/ | # Restoring Force
1. Oct 13, 2015
### Ibraheem
Hello,
In this simulation (link is below) the graph of the potential energy with respect to position U(x) when there is no friction on the ramp looks like the graph of potential energy of Hooke's Law spring U(x)=(1/2)KX^2). For the spring, the negative of the derivative is equal to the horizontal restoring force. So what restoring force is the negative derivative of potential energy graph for the skater in the simulation?I have included a picture of the simulation along with the potential energy U(x) graph.
Simulation:
#### Attached Files:
• ###### Untitled.jpg
File size:
31.6 KB
Views:
69
2. Oct 13, 2015
### LunaFly
Potential energy due to gravity is being plotted in this simulation. That being said, it is a tad bit complicated since the skater is on a ramp.
For example, an object in free-fall would have potential energy mgy. The force on the object would then be -d/dy (mgy) = -mg. Simple enough. Notice, however, that the only force acting on this object is that due to gravity.
For the ramp system, (assuming "frictionless" ramp used) the skater being acted upon by both the gravitational force and the normal force of the ramp at all times. This means that the net force on the skater is a sum of these two forces, and so the plotted potential energy function is related to both of these forces.
Similar Discussions: Restoring Force | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9700922966003418, "perplexity": 616.4494379347474}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187823350.23/warc/CC-MAIN-20171019160040-20171019180040-00812.warc.gz"} |
https://math.stackexchange.com/questions/435035/filling-a-conical-tank?noredirect=1 | # Filling a conical tank
I have been working on this problem for about 2 hours and I can't seem to get it, here is exactly what the question reads. "Water is poured into the top of a conical tank at the constant rate of 1 cubic inch per second and flows out of an opening at the3 bottom at a rate of .5 cubic inches per second. The tank has a height of 4 inches, and a radius of 2 inches at the top. How fast is the water level changing when the water is 2 inches high?"
• Welcome to Math.SE. Thank you for your question. We will better be able to help you if you share what you've got so far with your two hours of work. – vadim123 Jul 3 '13 at 3:36
• I don't really have much at all, because I am having most of my troubles just getting the problem started. – cschurman Jul 3 '13 at 3:39
Think about what is happening. The water (volume) is being poured in at a constant rate. This relates to how the water level ($h$) changes and how the width of the water in the tank at that level ($r$) changes. Further, $r$ and $h$ are related.
The volume of a cone is
$$V = \frac13 \pi \, r^2 \, h$$
How is $h$ related to $r$? You know that, at the top, the radius is $2$ and $h=4$. Because this is a cone, we can say that $r = h/2$ at all levels of the cone. Thus,
$$V(h) = \frac{1}{12} \pi \, h^3$$
We may then differentiate with respect to time; use the chain rule here
$$\frac{dV}{dt} = \frac{\pi}{4} h^2 \frac{dh}{dt}$$
You are given $dV/dt$ and the height $h$ at which to evaluate; solve for $dh/dt$.
• what would be dV/dt? is that the 1 cubic inches per second? – cschurman Jul 3 '13 at 3:51
• Well, it would be the net, because you have 0.5 flowing out from the bottom. – Ron Gordon Jul 3 '13 at 3:54
• well what i was asking, is to solve for dh/dt i would plug the 1 cubic inch per second in for dH/dt – cschurman Jul 3 '13 at 3:56
• You plug in the numbers for $dV/dt$ (+0.5 cubic inches/sec) and $h$ (2 inches) and solve for $dh/dt$. – Ron Gordon Jul 3 '13 at 3:57
• Okay, got it! Thanks you so much for your help! – cschurman Jul 3 '13 at 3:59 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8596756458282471, "perplexity": 255.2079445406574}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027331485.43/warc/CC-MAIN-20190826085356-20190826111356-00204.warc.gz"} |
https://www.physicsforums.com/threads/constant-of-proportionality.218265/ | # Homework Help: Constant of proportionality
1. Feb 26, 2008
### dranseth
1. The problem statement, all variables and given/known data
Can anyone translate what k would be? I have never had to deal with a constant of proportionality before, and am not quite sure how to deal with it..
The resulting aggregates therefore
each have a net charge Q that is directly proportional to their molecular weight M:
Q= K1 M
where K1 (and in subsequent steps of this derivation K ) is a constant of proportionality.
2. Feb 27, 2008
### blochwave
You've dealt with them, you just didn't know what they were called
When I say y is directly proportional to x, what I mean is that if I double x, I double y. If I halve x, I halve y. If I square root x, I square root y, etc.
so y=x? No, I said if I double x it doubles y, but I didn't say they're the SAME number! So if y and x are proportional you'd say y=k*x (or whatever letter) so all those things I said are true, if you double x, for that equation to still be equal you must double y
For a counterexample what if they AREN'T directly proportional? So say y=k*x^2, if I double x, for the equation to hold, I have to quadruple y!
You'll note that y=k*x results in the graph of a line with slope k, so you'll also hear that y and x are linearly related, which means the same thing as directly proportional
You'll also commonly hear "inversely proportional", which means y=k/x, so if I double x, I take half of y, if I triple x, a third of y
Counterexample again when it's NOT inversely proportional, y=k/x^2, if I double x, I take a fourth of y!
ADDENDUM: The constant of proportionality that you're likely most familiar with is probably pi!
What is pi exactly?(3! Wait no...) A bajillion years ago(1 jiggayear)some ancient Greeks or Carthaginians or whoever were looking at circles and thinking "hey the circumference and diameter are directly proportional!" Meaning if you took a circle with diameter of 1 m(well ancient greeks so like...1 hand?), it would have some circumference. If you took a circle with diameter 2(same unit), well the circumference is twice as much as before!
So C=k*d they figured. So what was k? They had to take an actual circle that they could measure the circumference and diameter of, and find the ratio(so if C=k*d, then k=C/d) and they found it was a little more than 3. About .14 more than 3. And they called it pi since they were Greeks and the latin letter k probably didn't exist then >_>
That's how Newton figured out his law of gravitation. He figured that the net force(which he had already figured out was directly proportional to acceleration, with that particular constant of proportionality being the mass)was directly proportional to the product of the two masses involved, and inversely proportional to the distance squared(so if you double either mass, you double the force, but if you double the distance between them you cut the force by a FOURTH)
And that's the meat of the law, the part that really matters. The constant of proportionality (G, so F=G*m1*m2/d^2) is just some fundamental constant which he didn't have the tools to measure at the time
So for the equation in your question, if you're not given K1, you're not really expected to care then. The POINT is that charge is directly proportional to M, so you understand the relationship between the two. As the molar mass of the aggregates increases, Q increases, and the nature of their relationship is linear
Last edited: Feb 27, 2008 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9079468250274658, "perplexity": 909.9457542175667}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794863923.6/warc/CC-MAIN-20180521023747-20180521043747-00277.warc.gz"} |
https://www.springerprofessional.de/en/non-ideal-compressible-fluid-dynamics-for-propulsion-and-power/18160208 | main-content
This book reports on advanced theories and methods aimed at characterizing the dynamics of non-ideal compressible fluids. A special emphasis is given to research fostering the use of non-ideal compressible fluids for propulsion and power engineering. Both numerical and experimental studies, as well as simulations, are described in the book, which is based on selected contributions and keynote lectures presented at the 2nd International Seminar on Non-Ideal Compressible-Fluid Dynamics for Propulsion & Power. Held on October 4-5 in Bochum, Germany, the seminar aimed at fostering collaborations between academics and professionals. The two perspectives have been gathered together in this book, which offers a timely guide to advanced fundamentals, innovative methods and current applications of non-ideal compressible fluids to developing turbomachines, and for propulsion and power generation.
### Experimental Investigations of Heat Transfer Processes in Cooling Channels for Cryogenic Hydrogen and Methane at Supercritical Pressure
Abstract
A rising demand for efficient and reusable rocket engines leads to the development of a new generation of methane fueled rocket engines. The most crucial part is the optimal design of the cooling system, with minimal hydrodynamic losses. Therefore a precise knowledge of the heat transfer processes in the combustion chamber and primarily in the cooling channels is necessary. Cooling channels with a high aspect ratio (height-to-width-ratio) in a wall material with high thermal conductivity are known to improve cooling efficiency with only moderate increase in hydrodynamic losses. In this paper tests will be presented, that were performed with a cylindrical combustion chamber. This chamber is divided into 4 sections around the circumference, each containing cooling channels with different aspect ratios (1.7, 3.5, 9.2 and 30). Cryogenic hydrogen and liquid methane at temperatures as low as 60 K for hydrogen and 130 K for methane respectively were used as cooling fluids. Results show a distinct thermal stratification for both coolants and a very high influence of changing fluid properties close to the critical point for methane.
Jan Haemisch, Dmitry Suslov, Michael Oschwald
### Efficient Handling of Cryogenic Equation of State for the Simulation of Rocket Combustion Chambers
Abstract
The simulation of cryogenic flows in rocket combustion chambers is challenging because we have to consider a reactive mixture over a wide temperature and density range. This necessitates the use of more advanced fluid models that impose additional computation overhead. In this study we compare two equation of state (EOS) mixture approximation approaches for cryogenic flows in rocket combustion chambers and present a computationally efficient implementation within a Reynolds Averaged Navier Stokes (RANS) context. The numerical study is validated with experimental results of a lab-scale rocket combustion chamber with optical access.
Stefan Fechter, Tim Horchler, Sebastian Karl, Klaus Hannemann, Dmitry Suslov, Justin Hardi, Michael Oschwald
### Non-equilibrium Model for Weakly Compressible Multi-component Flows: The Hyperbolic Operator
Abstract
We present a novel pressure-based method for weakly compressible multiphase flows, based on a non-equilibrium Baer and Nunziato-type model. Each component is described by its own thermodynamic model, thus the definition of a mixture speed of sound is not required. In this work, we describe the hyperbolic operator, without considering relaxation terms. The acoustic part of the governing equations is treated implicitly to avoid the severe restriction on the time step imposed by the CFL condition at low-Mach. Particular care is taken to discretize the non-conservative terms to avoid spurious oscillations across multi-material interfaces. The absence of oscillations and the agreement with analytical or published solutions is demonstrated in simplified test cases, which confirm the validity of the proposed approach as a building block on which developing more accurate and comprehensive methods.
Barbara Re, Rémi Abgrall
### Pressure-Based Solution Framework for Non-Ideal Flows at All Mach Numbers
Abstract
In this work, we present an all Mach number pressure-based solution framework suitable for the simulation of application-relevant high-pressure flow configurations. A cubic equation of state is applied for the accurate description of the thermodynamic state. Different formulations of the pressure equation for sub- and supersonic flows are discussed. The framework is employed to simulate one sub- and one supersonic test case, where experimental data are available. The comparison of the simulation results with experimental shadowgraphy and Schlieren images shows very good agreement.
Christoph Traxinger, Julian Zips, Matthias Banholzer, Michael Pfitzner
### Towards Direct Numerical Simulations of Shock-Turbulence Interaction in Real Gas Flows on GPUs: Initial Validation
Abstract
A better understanding of turbulent flows in presence of strong compressible and real gas effects is in high demand for future improvements in engineering applications, such as turbomachines for regenerative power production. Therefore, a solver for future investigation of shock-turbulence interaction in presence of strong real gas effects by means of direct numerical simulation with multi-parameter equations of state utilizing the computational power of GPUs is carefully designed and initially validated using the example of carbon dioxide. It relies on a hybrid energy-consistent WENO scheme based on a shock sensor. The implementation is validated with a set of test cases, comprising the Shu–Osher problem, the inviscid Taylor-Green vortex and the compressible decay of homogeneous isotropic turbulence with eddy shocklets. In absence of appropriate and well documented experimental results, the test cases are adapted to real gas in regions where the gas behavior can be deemed near to the ideal limit and results are compared with the validated ideal gas configurations.
Pascal Post, Francesca di Mare
### Direct Numerical Simulation of Turbulent Dense Gas Flows
Abstract
In order to assess the specific characteristics of turbulence in dense gas flows with respect to ideal gas flows, Direct Numerical Simulations are performed for both FC-70 described using Martin-Hou Equation of State (EoS) and a reference ideal gas, in the case of a forced Homogeneous Isotropic Turbulence (HIT) configuration and of a temporal compressible mixing layer configuration. The forced HIT shows that the statistically stationary turbulent kinetic energy (TKE) spectrum follows quite closely the one obtained in the incompressible case even when the turbulent Mach number is large. It is shown that the weakening of compressible dissipation of the TKE can be related to the decoupling of the density fluctuations from the velocity as well as to the strong weakening of compression shocklets in the dense gas case. The mixing layer case at a convective Mach number $$M_c=1.1$$ shows that the well-known compressibility-related reduction of the mixing layer momentum thickness growth rate is not significantly influenced by the EoS. Yet the unstable growth leading to the self-similarity phase is enhanced by the dense gas thermodynamics.
Alexis Giauque, Christophe Corre, Aurélien Vadrot
### Numerical Investigation of Supersonic Dense-Gas Boundary Layers
Abstract
A study of dense-gas effects on the laminar, transitional and turbulent characteristics of boundary layer flows is conducted. The laminar similarity solution shows that temperature variations are small due to the high specific heats of dense gases, leading to velocity profiles close to the incompressible ones. Nevertheless, the complex thermodynamics of the base flow has a major impact on unstable modes, which bear similarities with those obtained for a strongly cooled wall. Numerical simulations of spatially developing boundary layers yield turbulent statistics for the dense gas flow that remain closer to the incompressible regime than perfect gas ones despite the presence of strongly compressible structures.
Luca Sciacovelli, Donatella Passiatore, Xavier Gloerfelt, Paola Cinnella, Francesco Grasso
### Entropy Generation in Laminar Boundary Layers of Non-Ideal Fluid Flows
Abstract
This paper documents a numerical study on entropy generation in zero-pressure gradient, laminar boundary layers of adiabatic non-ideal compressible fluid flows. The entropy generation is expressed in terms of dissipation coefficient $$C_\mathrm {d}$$ and its dependency on free-stream Mach number, fluid molecular complexity, and flow non-ideality is investigated systematically by means of a boundary layer code extended to treat fluids modeled with arbitrary equations of state. The results of the study show that the trend of dissipation coefficient follows that of an incompressible flow for complex fluid molecules like siloxanes in all thermodynamic and flow conditions. For simpler fluids like CO$$_2$$ the trend becomes inversely proportional to the free-stream Mach number and the $$C_\mathrm {d}$$ value can significantly reduce in the non-ideal flow regime, where strong thermo-physical property gradients occur near the wall.
Matteo Pini, Carlo De Servi | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8197426795959473, "perplexity": 1379.3476061011622}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439737019.4/warc/CC-MAIN-20200806180859-20200806210859-00165.warc.gz"} |
http://physics.stackexchange.com/users/4160/jonathan?tab=activity | # Jonathan
less info
reputation
14
bio website math.utoronto.ca/jmfisher location Toronto age member for 2 years, 10 months seen Apr 14 at 13:11 profile views 36
I'm a PhD candidate interested in mathematics inspired by theoretical physics (QFT, strings, integrable systems, etc.).
# 22 Actions
Jan24 answered do quantum fields exist in superposition? Aug17 comment Free Particle Propagator Using Path Integrals Since the limits are $-\infty \to \infty$, each factor is an (analytically continued) Gaussian, which is explicitly computable with no special functions needed. This is exactly what happens in the piecewise-linear regularization, so it's a good sign that the right answer is sure to pop out (once the $N$-dependent normalization is determined). Aug17 comment What criteria distinguishes causality from retrocausality? But your comments are elaboration, and I agree with much of what you have said. I have more to say, which I might put into an answer. Since you have engaged in a healthy discussion and have defended your answer, I would be more than happy to remove my downvote. However, it seems that the SE software won't allow me to change my vote unless you edit your answer. Aug17 comment What criteria distinguishes causality from retrocausality? I don't mean to sound contrarian, and am absolutely sincere--if there is nothing to say, then why say it? Aug17 answered Free Particle Propagator Using Path Integrals Aug16 comment Why do some anomalies (only) lead to inconsistent quantum field theories I'm sure this has been said many times before, but "gauge symmetry" is absolutely a misnomer as what we really have is a gauge redundancy. Mathematically, we have a space of connections $\mathcal{A}$, and gauge theories are all about quantizing the Poisson space $T^\ast \mathcal{A} / \mathcal{G}$, rather than the symplectic space $T^\ast \mathcal{A}$ (which does have a gauge symmetry). From this point of view, of course an anomalous redundancy plays a different role, both conceptually and mathematically, than an anomalous symmetry. Aug16 awarded Critic Aug16 comment What criteria distinguishes causality from retrocausality? -1. "However, on the larger scale these many interactions form a system where time moves forward (i.e. positive) not backward (negative)." As far as I can tell, this is exactly what the questioner is asking and this response is a mindless regurgitation without any physical or mathematical content. If you could please elaborate and convince me otherwise I would be more than happy to remove my downvote (and possibly upvote instead!). Jul24 comment What does symplecticity imply? Even on a Poisson manifold, the dynamics given any particular initial condition are always constrained to a symplectic leaf, so you will have trouble finding examples that are "genuinely Poisson". You might find more interesting examples by looking at systems with constraints, but then even Poisson geometry is not entirely adequate for this--see, e.g., Dirac geometry. Jun23 awarded Yearling Apr25 comment Why/How is this Wick's theorem? I like this answer just the way it is, as it makes it very clear that Wick's theorem is not a theorem about QFT, but is rather a completely general theorem about algebras obeying a few simple axioms. The standard textbook treatment with fields and VEV's obfuscates what is really just basic algebra. Apr25 awarded Supporter Feb11 answered Are there any clear and expressive plainword sense of metric tensor components? Feb4 answered Can the Lorentz force expression be derived from Maxwell's equations? Sep29 comment Why is it important that Hamilton's equations have the four symplectic properties and what do they mean? The course would depend on the department and who's teaching it, but there are lots of good books. Arnold's Mathematical Methods of Classical Mechanics is a good companion to Goldstein (and is in some ways better), but places much more emphasis on the geometry. There is Foundations of Mechanics by Abraham and Marsden, and Symplectic Techniques in Physics by Guillemin and Sternberg. The are lecture notes by Cannas da Silva which are quite good and available online for free. But this is just the tip of the iceberg, as symplectic geometry is a huge field in modern mathematics. Sep29 answered Why is it important that Hamilton's equations have the four symplectic properties and what do they mean? Sep28 answered If the graph isomorphism problem can't be solved in polynomial time, do spin-networks in loop quantum gravity violate the Church-Turing thesis? Aug18 answered Four vectors from spinors Jul21 awarded Teacher Jul21 awarded Editor | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8528411388397217, "perplexity": 833.3106406975563}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609530895.48/warc/CC-MAIN-20140416005210-00325-ip-10-147-4-33.ec2.internal.warc.gz"} |
https://www.hepdata.net/search/?q=&phrases=Dijet+Production&sort_order=&page=1&sort_by=latest | Showing 25 of 208 results
#### Measurement of the cross section and longitudinal double-spin asymmetry for di-jet production in polarized $pp$ collisions at $\sqrt{s}$ = 200 GeV
The collaboration Adamczyk, L. ; Adkins, J.K. ; Agakishiev, G. ; et al.
No Journal Information, 2016.
Inspire Record 1493842
We report the first measurement of the longitudinal double-spin asymmetry $A_{LL}$ for mid-rapidity di-jet production in polarized $pp$ collisions at a center-of-mass energy of $\sqrt{s} = 200$ GeV. The di-jet cross section was measured and is shown to be consistent with next-to-leading order (NLO) perturbative QCD predictions. $A_{LL}$ results are presented for two distinct topologies, defined by the jet pseudorapidities, and are compared to predictions from several recent NLO global analyses. The measured asymmetries, the first such correlation measurements, support those analyses that find positive gluon polarization at the level of roughly 0.2 over the region of Bjorken-$x > 0.05$.
10 data tables
Data simulation comparison (with arbitrary normalization). Di-jet invariant mass.
Data simulation comparison (with arbitrary normalization). Difference between jet pseudorapidities.
Data simulation comparison (with arbitrary normalization). Difference between jet azimuthal angles.
More…
#### Measurement of electroweak production of a W boson and two forward jets in proton-proton collisions at $\sqrt{s}=8$ TeV
The collaboration Khachatryan, Vardan ; Sirunyan, Albert M ; Tumasyan, Armen ; et al.
JHEP 1611 (2016) 147, 2016.
Inspire Record 1477806
A measurement is presented of the cross section for the electroweak production of a W boson in association with two jets in proton-proton collisions at a center-of-mass energy of 8 TeV. The data set was collected with the CMS detector and corresponds to an integrated luminosity of 19.3 fb$^{−1}$. The measured fiducial cross section for W bosons decaying to electrons or muons and for p$_{T}^{j1}$ > 60 GeV, p$_{T}^{j2}$ > 50 GeV, |η$^{j}$| < 4.7, and m$_{jj}$ > 1000 GeV is 0.42 ± 0.04 (stat) ± 0.09 (syst) ± 0.01 (lumi) pb. This result is consistent with the standard model leading-order prediction of 0.50 ± 0.02 (scale) ± 0.02 (PDF) pb obtained with MadGraph5_amc@nlo 2.1 interfaced to pythia 6.4. This is the first cross section measurement for this process.
1 data table
The measured values for the EW W(-->env, munv)+2-jets fiducial cross section.
#### Distributions of Topological Observables in Inclusive Three- and Four-Jet Events in pp Collisions at sqrt(s) = 7 TeV
The collaboration Khachatryan, Vardan ; Sirunyan, Albert M ; Tumasyan, Armen ; et al.
Eur.Phys.J. C75 (2015) 302, 2015.
Inspire Record 1345159
This paper presents distributions of topological observables in inclusive three- and four-jet events produced in pp collisions at a centre-of-mass energy of 7 $\,\text {TeV}$ with a data sample collected by the CMS experiment corresponding to a luminosity of 5.1 $\,\text {fb}^{-1}$ . The distributions are corrected for detector effects, and compared with several event generators based on two- and multi-parton matrix elements at leading order. Among the considered calculations, MadGraph interfaced with pythia6 displays the overall best agreement with data.
7 data tables
CORRECTED NORMALIZED DISTRIBUTION OF THREE-JET MASS IN THE INCLUSIVE THREE-JET SAMPLE. THE PROVIDED UNCERTAINTY CORRESPONDS TO SYSTEMATIC UNCERTAINTY.
CORRECTED NORMALIZED DISTRIBUTION OF SCALED ENERGY OF THE LEADING-JET IN THE INCLUSIVE THREE-JET SAMPLE. THE PROVIDED UNCERTAINTY CORRESPONDS TO SYSTEMATIC UNCERTAINTY.
CORRECTED NORMALIZED DISTRIBUTION OF SCALED ENERGY OF THE SECOND-LEADING-JET IN THE INCLUSIVE THREE-JET SAMPLE. THE PROVIDED UNCERTAINTY CORRESPONDS TO SYSTEMATIC UNCERTAINTY.
More…
#### Search for heavy resonances decaying to two Higgs bosons in final states containing four b quarks
The collaboration Khachatryan, Vardan ; Sirunyan, Albert M ; Tumasyan, Armen ; et al.
Eur.Phys.J. C76 (2016) 371, 2016.
Inspire Record 1424833
A search is presented for narrow heavy resonances X decaying into pairs of Higgs bosons ( ${\mathrm{H}}$ ) in proton-proton collisions collected by the CMS experiment at the LHC at $\sqrt{s}=8\,\text {TeV}$ . The data correspond to an integrated luminosity of 19.7 $\,\text {fb}^{-1}$ . The search considers ${\mathrm{H}} {\mathrm{H}}$ resonances with masses between 1 and 3 $\,\text {TeV}$ , having final states of two b quark pairs. Each Higgs boson is produced with large momentum, and the hadronization products of the pair of b quarks can usually be reconstructed as single large jets. The background from multijet and ${\mathrm{t}}\overline{{\mathrm{t}}}$ events is significantly reduced by applying requirements related to the flavor of the jet, its mass, and its substructure. The signal would be identified as a peak on top of the dijet invariant mass spectrum of the remaining background events. No evidence is observed for such a signal. Upper limits obtained at 95 % confidence level for the product of the production cross section and branching fraction $\sigma ({{\mathrm{g}} {\mathrm{g}}} \rightarrow \mathrm {X})\, \mathcal {B}({\mathrm {X}} \rightarrow {\mathrm{H}} {\mathrm{H}} \rightarrow {\mathrm{b}} \overline{{\mathrm{b}}} {\mathrm{b}} \overline{{\mathrm{b}}} )$ range from 10 to 1.5 $\text {\,fb}$ for the mass of X from 1.15 to 2.0 $\,\text {TeV}$ , significantly extending previous searches. For a warped extra dimension theory with a mass scale $\Lambda _\mathrm {R} = 1$ $\,\text {TeV}$ , the data exclude radion scalar masses between 1.15 and 1.55 $\,\text {TeV}$ .
7 data tables
Observed $m_\mathrm{jj}$ spectrum (black points) compared with a background estimate (black line), obtained in background only hypothesis, for HPHP category. The simulated radion resonances of $m_\mathrm{X} = 1.5$ and 2 TeV are also shown.
Observed $m_\mathrm{jj}$ spectrum (black points) compared with a background estimate (black line), obtained in background only hypothesis, for HPLP category. The simulated radion resonances of $m_\mathrm{X} = 1.5$ and 2 TeV are also shown.
Observed $m_\mathrm{jj}$ spectrum (black points) compared with a background estimate (black line), obtained in background only hypothesis, for LPHP category. The simulated radion resonances of $m_\mathrm{X} = 1.5$ and 2 TeV are also shown.
More…
#### Production of exclusive dijets in diffractive deep inelastic scattering at HERA
The collaboration Abramowicz, H. ; Abt, I. ; Adamczyk, L. ; et al.
Eur.Phys.J. C76 (2016) 16, 2016.
Inspire Record 1372086
Production of exclusive dijets in diffractive deep inelastic $e^\pm p$ scattering has been measured with the ZEUS detector at HERA using an integrated luminosity of 372 pb$^{-1}$. The measurement was performed for $\gamma^*-p$ centre-of-mass energies in the range $90 < W < 250$ GeV and for photon virtualities $Q^2 > 25$ GeV$^2$. Energy and transverse-energy flows around the jet axis are presented. The cross section is presented as a function of $\beta$ and $\phi$, where $\beta=x/x_{\rm I\!P}$, $x$ is the Bjorken variable and $x_{\rm I\!P}$ is the proton fractional longitudinal momentum loss. The angle $\phi$ is defined by the $\gamma^*-$dijet plane and the $\gamma^*-e^\pm$ plane in the rest frame of the diffractive final state. The $\phi$ cross section is measured in bins of $\beta$. The results are compared to predictions from models based on different assumptions about the nature of the diffractive exchange.
3 data tables
Differential cross-section $d\sigma/d\beta$ in the kinematic range: $Q^2 > 25 GeV^2$, $90 < W < 250 GeV^2$, $x_{\rm I\!P} < 0.01$, $M_X > 5 GeV$ and $p_{T,jet} > 2 GeV$. The contribution from proton dissociation was subtracted. The uncertainty of the subtraction determines the uncertainty of the normalisation also given in the table.
Differential cross-section $d\sigma/d\phi$ in the kinematic range: $Q^2 > 25 GeV^2$, $90 < W < 250 GeV$, $x_{\rm I\!P} < 0.01$, $M_X > 5 GeV$ and $p_{T,jet} > 2 GeV$. The contribution from proton dissociation was subtracted. The uncertainty of the subtraction determines the uncertainty of the normalisation given in the table.
Results of the fit to the cross-section $d\sigma/d\phi$ in bins of $\beta$. The fitted function is proportional to $(1+A \rm{cos}2\phi)$. The uncertainty includes both statistical and systematical contributions (see text of paper).
#### Search for pair production of third-generation scalar leptoquarks and top squarks in proton–proton collisions at $\sqrt{s}$=8 TeV
The collaboration Khachatryan, Vardan ; Sirunyan, Albert M ; Tumasyan, Armen ; et al.
Phys.Lett. B739 (2014) 229-249, 2014.
Inspire Record 1309874
A search for pair production of third-generation scalar leptoquarks and supersymmetric top quark partners, top squarks, in final states involving tau leptons and bottom quarks is presented. The search uses events from a data sample of proton-proton collisions corresponding to an integrated luminosity of 19.7 inverse femtobarns, collected with the CMS detector at the LHC with sqrt(s) = 8 TeV. The number of observed events is found to be in agreement with the expected standard model background. Third-generation scalar leptoquarks with masses below 740 GeV are excluded at 95% confidence level, assuming a 100% branching fraction for the leptoquark decay to a tau lepton and a bottom quark. In addition, this mass limit applies directly to top squarks decaying via an R-parity violating coupling lambda'[333]. The search also considers a similar signature from top squarks undergoing a chargino-mediated decay involving the R-parity violating coupling lambda'[3jk]. Each top squark decays to a tau lepton, a bottom quark, and two light quarks. Top squarks in this model with masses below 580 GeV are excluded at 95% confidence level. The constraint on the leptoquark mass is the most stringent to date, and this is the first search for top squarks decaying via lambda'[3jk].
9 data tables
The estimated backgrounds, observed event yields, and expected number of signal events for the leptoquark search. For the simulation-based entries, the statistical and systematic uncertainties are shown separately, in that order.
The estimated backgrounds, observed event yields, and expected number of signal events for the top squark search. For the simulation-based entries, the statistical and systematic uncertainties are shown separately, in that order.
Selection efficiencies in % for the signal in the leptoquark search, estimated from the simulation.
More…
#### Dijet production in $\sqrt{s}=$ 7 TeV $pp$ collisions with large rapidity gaps at the ATLAS experiment
The collaboration Aad, Georges ; Abbott, Brad ; Abdallah, Jalal ; et al.
Phys.Lett. B754 (2016) 214-234, 2016.
Inspire Record 1402356
A $6.8 \ {\rm nb^{-1}}$ sample of $pp$ collision data collected under low-luminosity conditions at $\sqrt{s} = 7$ TeV by the ATLAS detector at the Large Hadron Collider is used to study diffractive dijet production. Events containing at least two jets with $p_\mathrm{T} > 20$ GeV are selected and analysed in terms of variables which discriminate between diffractive and non-diffractive processes. Cross sections are measured differentially in $\Delta\eta^F$, the size of the observable forward region of pseudorapidity which is devoid of hadronic activity, and in an estimator, $\tilde{\xi}$, of the fractional momentum loss of the proton assuming single diffractive dissociation ($pp \rightarrow pX$). Model comparisons indicate a dominant non-diffractive contribution up to moderately large $\Delta\eta^F$ and small $\tilde{\xi}$, with a diffractive contribution which is significant at the highest $\Delta\eta^F$ and the lowest $\tilde{\xi}$. The rapidity-gap survival probability is estimated from comparisons of the data in this latter region with predictions based on diffractive parton distribution functions.
6 data tables
The cross section differential in the forward rapidity gap size, DELTA(C=RAPGAP), for events with at least two jets of pt > 20 GeV found by the anti-kt jet algorithm with R=0.6.
The cross section differential in the forward rapidity gap size, DELTA(C=RAPGAP), for events with at least two jets of pt > 20 GeV found by the anti-kt jet algorithm with R=0.4.
The cross section differential in the fraction of the proton four-momentum carried by the Pomeron, LOG10(C=XI), for events with at least two jets of pt > 20 GeV found by the anti-kt jet algorithm with R=0.6.
More…
#### rivet Analysis Measurement of three-jet production cross-sections in $pp$ collisions at 7 TeV centre-of-mass energy using the ATLAS detector
The collaboration Aad, Georges ; Abbott, Brad ; Abdallah, Jalal ; et al.
Eur.Phys.J. C75 (2015) 228, 2015.
Inspire Record 1326641
Double-differential three-jet production cross-sections are measured in proton–proton collisions at a centre-of-mass energy of $\sqrt{s} = 7\mathrm \,TeV{}$ using the ATLAS detector at the large hadron collider. The measurements are presented as a function of the three-jet mass $(m_{jjj})$ , in bins of the sum of the absolute rapidity separations between the three leading jets $(\left| Y^{*}\right| )$ . Invariant masses extending up to 5 TeV are reached for $8< \left| Y^{*}\right| < 10$ . These measurements use a sample of data recorded using the ATLAS detector in 2011, which corresponds to an integrated luminosity of $4.51~\text{ fb }^{-1}$ . Jets are identified using the anti- $k_{t}$ algorithm with two different jet radius parameters, $R=0.4$ and $R=0.6$ . The dominant uncertainty in these measurements comes from the jet energy scale. Next-to-leading-order QCD calculations corrected to account for non-perturbative effects are compared to the measurements. Good agreement is found between the data and the theoretical predictions based on most of the available sets of parton distribution functions, over the full kinematic range, covering almost seven orders of magnitude in the measured cross-section values.
10 data tables
Measured three-jet double-differential cross sections as a function of M(3JET) in |Y*|<2 for anti-kt R=0.4 jets. The three columns correspond to nominal, stronger or weaker correlations between jet energy scale uncertainty components.
Measured three-jet double-differential cross sections as a function of M(3JET) in 2<=|Y*|<4 for anti-kt R=0.4 jets. The three columns correspond to nominal, stronger or weaker correlations between jet energy scale uncertainty components.
Measured three-jet double-differential cross sections as a function of M(3JET) in 4<=|Y*|<6 for anti-kt R=0.4 jets. The three columns correspond to nominal, stronger or weaker correlations between jet energy scale uncertainty components.
More…
#### Search for Dijet Resonances in 7 TeV pp Collisions at CMS
The collaboration Khachatryan, Vardan ; Sirunyan, Albert M. ; Tumasyan, Armen ; et al.
Phys.Rev.Lett. 105 (2010) 211801, 2010.
Inspire Record 871540
A search for narrow resonances in the dijet mass spectrum is performed using data corresponding to an integrated luminosity of 2.9 inverse pb collected by the CMS experiment at the LHC. Upper limits at the 95% confidence level (CL) are presented on the product of the resonance cross section, branching fraction into dijets, and acceptance, separately for decays into quark-quark, quark-gluon, or gluon-gluon pairs. The data exclude new particles predicted in the following models at the 95% CL: string resonances, with mass less than 2.50 TeV, excited quarks, with mass less than 1.58 TeV, and axigluons, colorons, and E_6 diquarks, in specific mass intervals. This extends previously published limits on these models.
2 data tables
Dijet mass spectrum. As this data is uncorrected for detector effects it is not tabulated here.
95% CL upper limits for dijet resonances for an acceptance value (A) of 0.6.
#### Jet Production in ep Collisions at Low Q**2 and Determination of alpha(s)
The collaboration Aaron, F.D. ; Aldaya Martin, M. ; Alexa, C. ; et al.
Eur.Phys.J. C67 (2010) 1-24, 2010.
Inspire Record 838435
The production of jets is studied in deep-inelastic e+p scattering at low negative four momentum transfer squared 5<Q^2<100 GeV^2 and at inelasticity 0.2<y<0.7 using data recorded by the H1 detector at HERA in the years 1999 and 2000, corresponding to an integrated luminosity of 43.5 pb^-1. Inclusive jet, 2-jet and 3-jet cross sections as well as the ratio of 3-jet to 2-jet cross sections are measured as a function of Q^2 and jet transverse momentum. The 2-jet cross section is also measured as a function of the proton momentum fraction xi. The measurements are well described by perturbative quantum chromodynamics at next-to-leading order corrected for hadronisation effects and are subsequently used to extract the strong coupling alpha_s.
13 data tables
Inclusive Jet Cross Section ${\rm\frac{d\sigma_{jet}}{dQ^2}}$.
2-Jet Cross Section ${\rm\frac{d\sigma_{2-jet}}{dQ^2}}$.
3-Jet Cross Section ${\rm\frac{d\sigma_{3-jet}}{dQ^2}}$.
More…
#### Jet Production in ep Collisions at High Q**2 and Determination of alpha(s)
The collaboration Aaron, F.D. ; Alexa, C. ; Alimujiang, K. ; et al.
Eur.Phys.J. C65 (2010) 363-383, 2010.
Inspire Record 818707
The production of jets is studied in deep-inelastic ep scattering at large negative four momentum transfer squared 150<Q^2<15000 GeV^2 using HERA data taken in 1999-2007, corresponding to an integrated luminosity of 395 pb^-1. Inclusive jet, 2-jet and 3-jet cross sections, normalised to the neutral current deep-inelastic scattering cross sections, are measured as functions of Q^2, jet transverse momentum and proton momentum fraction. The measurements are well described by perturbative QCD calculations at next-to-leading order corrected for hadronisation effects. The strong coupling as determined from these measurements is alpha_s(M_Z) = 0.1168 +/-0.0007 (exp.) +0.0046/-0.0030 (th.) +/-0.0016(pdf).
7 data tables
Normalised inclusive jet cross section in bins of $Q^{2}$.
Normalised 2-jet cross section in bins of $Q^{2}$.
Normalised 3-jet cross section in bins of $Q^{2}$.
More…
#### Search for massive resonances in dijet systems containing jets tagged as W or Z boson decays in pp collisions at $\sqrt{s}$ = 8 TeV
The collaboration Khachatryan, Vardan ; Sirunyan, Albert M ; Tumasyan, Armen ; et al.
JHEP 1408 (2014) 173, 2014.
Inspire Record 1294937
A search is reported for massive resonances decaying into a quark and a vector boson (W or Z), or two vector bosons (WW, WZ, or ZZ). The analysis is performed on an inclusive sample of multijet events corresponding to an integrated luminosity of 19.7 inverse femtobarns, collected in proton-proton collisions at a centre-of-mass energy of 8 TeV with the CMS detector at the LHC. The search uses novel jet-substructure identification techniques that provide sensitivity to the presence of highly boosted vector bosons decaying into a pair of quarks. Exclusion limits are set at a confidence level of 95% on the production of: (i) excited quark resonances q* decaying to qW and qZ for masses less than 3.2 TeV and 2.9 TeV, respectively, (ii) a Randall-Sundrum graviton G[RS] decaying into WW for masses below 1.2 TeV, and (iii) a heavy partner of the W boson W' decaying into WZ for masses less than 1.7 TeV. For the first time mass limits are set on W' to WZ and G[RS] to WW in the all-jets final state. The mass limits on q* to qW, q* to qZ, W' to WZ, G[RS] to WW are the most stringent to date. A model with a "bulk" graviton G[Bulk] that decays into WW or ZZ bosons is also studied.
9 data tables
DATA - Double W/Z tagged events in HIGH purity bin.
BACKGROUND - Double W/Z tagged background in HIGH purity bin estimated from a fit to data.
BACKGROUND PLUS - Double W/Z tagged background variation upward (1 sigma) in HIGH purity bin estimated from a fit to data.
More…
#### Measurement of dijet $k_T$ in p–Pb collisions at $\sqrt{s}_{NN}$=5.02 TeV
Phys.Lett. B746 (2015) 385-395, 2015.
Inspire Record 1351451
A measurement of dijet correlations in p–Pb collisions at sNN=5.02 TeV with the ALICE detector is presented. Jets are reconstructed from charged particles measured in the central tracking detectors and neutral energy deposited in the electromagnetic calorimeter. The transverse momentum of the full jet (clustered from charged and neutral constituents) and charged jet (clustered from charged particles only) is corrected event-by-event for the contribution of the underlying event, while corrections for underlying event fluctuations and finite detector resolution are applied on an inclusive basis. A projection of the dijet transverse momentum, kTy=pT,jetch+nesin(Δφdijet) with Δφdijet the azimuthal angle between a full and charged jet and pT,jetch+ne the transverse momentum of the full jet, is used to study nuclear matter effects in p–Pb collisions. This observable is sensitive to the acoplanarity of dijet production and its potential modification in p–Pb collisions with respect to pp collisions. Measurements of the dijet kTy as a function of the transverse momentum of the full and recoil charged jet, and the event multiplicity are presented. No significant modification of kTy due to nuclear matter effects in p–Pb collisions with respect to the event multiplicity or a PYTHIA8 reference is observed.
10 data tables
Dijet |k_{T,y}| distributions in p-Pb collisions in the 0-40% V0A multiplicity event class.
Dijet |k_{T,y}| distributions in p-Pb collisions in the 0-40% V0A multiplicity event class.
Dijet |k_{T,y}| distributions in p-Pb collisions in the 0-40% V0A multiplicity event class.
More…
#### Search for New Phenomena in Dijet Angular Distributions in Proton-Proton Collisions at $\sqrt{s} = 8$ TeV Measured with the ATLAS Detector
The collaboration Aad, Georges ; Abbott, Brad ; Abdallah, Jalal ; et al.
Phys.Rev.Lett. 114 (2015) 221802, 2015.
Inspire Record 1357594
A search for new phenomena in LHC proton-proton collisions at a center-of-mass energy of s=8 TeV was performed with the ATLAS detector using an integrated luminosity of 17.3 fb−1. The angular distributions are studied in events with at least two jets; the highest dijet mass observed is 5.5 TeV. All angular distributions are consistent with the predictions of the standard model. In a benchmark model of quark contact interactions, a compositeness scale below 8.1 TeV in a destructive interference scenario and 12.0 TeV in a constructive interference scenario is excluded at 95% C.L.; median expected limits are 8.9 TeV for the destructive interference scenario and 14.1 TeV for the constructive interference scenario.
7 data tables
mjj region 600 - 800 GeV. The observed systematic is the experimental uncertainty, while the SM prediction systematic is the theoretical uncertainty.
mjj region 800 - 1200 GeV. The observed systematic is the experimental uncertainty, while the SM prediction systematic is the theoretical uncertainty.
mjj region 1200 - 1600 GeV. The observed systematic is the experimental uncertainty, while the SM prediction systematic is the theoretical uncertainty.
More…
#### Measurement of the cross section ratio $\sigma_\mathrm{t \bar{t} b \bar{b}} / \sigma_\mathrm{t \bar{t} jj }$ in pp collisions at $\sqrt{s}$ = 8 TeV
The collaboration Khachatryan, Vardan ; Sirunyan, Albert M ; Tumasyan, Armen ; et al.
Phys.Lett. B746 (2015) 132-153, 2015.
Inspire Record 1328962
3 data tables
The number of events for each physics process and for each dilepton category after fitting to the data, their total, and the observed total number of events. The results are after the final event selection. The $Z/\gamma^* \to \ell\ell$ uncertainty is from data, while all other uncertainties include only the statistical uncertainties in the MC samples.
Summary of the systematic uncertainties from various sources contributing to $\sigma_{t\bar{t}b\bar{b}}$, $\sigma_{t\bar{t}jj}$, and the ratio $\sigma_{t\bar{t}b\bar{b}/t\bar{t}jj}$ for a jet pt threshold of $p_{\rm T}$ > 20 GeV in the visible phase space.
The measured cross sections $\sigma_{t\bar{t}b\bar{b}}$ and $\sigma_{t\bar{t}jj}$ and their ratio are given for the visible phase space (PS) defined as two leptons with $p_{\rm T}$ > 20 GeV and $|\eta|$ < 2.4 plus four jets, including two b jets with $p_{\rm T}$ > 20 GeV and $|\eta|$ < 2.5, and the full phase space, corrected for acceptance and branching fractions. The full phase-space results are given for jet thresholds of $p_{\rm T}$ > 20 and 40 GeV. The predictions of a NLO theoretical calculation for the full phase space and $p_{\rm T}$ > 40 GeV are also given.
#### Search for resonances and quantum black holes using dijet mass spectra in proton-proton collisions at $\sqrt{s} =$ 8 TeV
The collaboration Khachatryan, Vardan ; Sirunyan, Albert M ; Tumasyan, Armen ; et al.
Phys.Rev. D91 (2015) 052009, 2015.
Inspire Record 1340084
10 data tables
Inclusive dijet mass spectrum from wide jets (points) compared to a fit (solid curve) and to predictions including detector simulation of multijet events and signal resonances. The predicted multijet shape (QCD MC) has been scaled to the data (see text). The vertical error bars are statistical only and the horizontal error bars are the bin widths. For comparison,the signal distributions for a W resonance of mass 1900 GeV and an excited quark of mass 3.6 TeV are shown. The bin-by-bin fit residuals scaled to the statistical uncertainty of the data , (data - fit)/$\sigma_{data}$, are shown at the bottom and compared with the expected signal contributions.
Observed 95% CL upper limits on $\sigma B A$ for narrow qq, qg, and gg resonances, from the inclusive analysis for signal masses between 1.2 and 5.5 TeV.
Observed 95% CL upper limits on $\sigma B A$ for narrow gg/bb, qq/bb, and bg resonances from the b-enriched analysis, for signal masses between 1.2 and 4.0 TeV. The upper limits are given for different ratios $f_{bb}$ for gg/bb and qq/bb resonances, and for 100% branching fraction into bg.
More…
#### Measurement of the $WW+WZ$ cross section and limits on anomalous triple gauge couplings using final states with one lepton, missing transverse momentum, and two jets with the ATLAS detector at $\sqrt{\rm{s}} = 7$ TeV
The collaboration Aad, Georges ; Abbott, Brad ; Abdallah, Jalal ; et al.
JHEP 1501 (2015) 049, 2015.
Inspire Record 1324374
The production of a $W$ boson decaying to $e\nu$ or $\mu\nu$ in association with a $W$ or $Z$ boson decaying to two jets is studied using $4.6 \mathrm{fb}^{-1}$ of proton--proton collision data at $\sqrt{\rm{s}} = 7$ TeV recorded with the ATLAS detector at the LHC. The combined $WW+WZ$ cross section is measured with a significance of 3.4$\sigma$ and is found to be $68 \pm 7 \ \mathrm{(stat.)} \pm 19 \ \mathrm{(syst.)} \ pb$, in agreement with the Standard Model expectation of $61.1 \pm 2.2 \ \mathrm{pb}$. The distribution of the transverse momentum of the dijet system is used to set limits on anomalous contributions to the triple gauge coupling vertices and on parameters of an effective-field-theory model.
1 data table
The total and fiducial cross sections for the production of W(LEPTON NU) W(JET JET) or W(LEPTON NU) Z(JET JET). The cross sections are the sum of the WW and WZ processes.
#### Search for new phenomena in the dijet mass distribution using $p-p$ collision data at $\sqrt{s}=8$ TeV with the ATLAS detector
The collaboration Aad, Georges ; Abbott, Brad ; Abdallah, Jalal ; et al.
Phys.Rev. D91 (2015) 052007, 2015.
Inspire Record 1305096
14 data tables
The reconstructed dijet mass distribution (observed) fitted with a smooth functional form (expected) of 4-parameter or 5-parameter.
Dijet mass resolution obtained from fully simulated PYTHIA QCD Monte Carlo PYTHIA 8.175 [43], with the AU2 tune obtained from ATLAS data [45].
Total cross sections, branching ratios in qg only and acceptances for qg, qgamma, qW, qZ decays (A) for the q* model.
More…
#### Measurement of the combined rapidity and $p_T$ dependence of dijet azimuthal decorrelations in $p\bar{p}$ collisions at $\sqrt{s}=1.96\,$TeV
The collaboration Abazov, Victor Mukhamedovich ; Abbott, Braden Keim ; Acharya, Bannanje Sripath ; et al.
Phys.Lett. B721 (2013) 212-219, 2013.
Inspire Record 1206604
We present the first combined measurement of the rapidity and transverse momentum dependence of dijet azimuthal decorrelations, based on the recently proposed quantity $R_{\Delta \phi}$. The variable $R_{\Delta \phi}$ measures the fraction of the inclusive dijet events in which the azimuthal separation of the two jets with the highest transverse momenta is less than a specified value for the parameter $\Delta \phi_{\rm max}$. The quantity $R_{\Delta \phi}$ is measured in $p\bar{p}$ collisions at $\sqrt{s}=1.96\,$TeV, as a function of the dijet rapidity interval, the total scalar transverse momentum, and $\Delta \phi_{\rm max}$. The measurement uses an event sample corresponding to an integrated luminosity of $0.7\,$fb$^{-1}$ collected with the D0 detector at the Fermilab Tevatron Collider. The results are compared to predictions of a perturbative QCD calculation at next-to-leading order in the strong coupling with corrections for non-perturbative effects. The theory predictions describe the data, except in the kinematic region of large dijet rapidity intervals and large $\Delta \phi_{\rm max}$.
3 data tables
The results for $R_{\Delta\phi}$ with their relative uncertainties for $\Delta\phi_{\rm max}=7\pi/8$.
The results for $R_{\delta\phi}$ with their relative uncertainties for $\delta\phi_{\rm max}=5\pi/6$.
The results for $R_{\delta\phi}$ with their relative uncertainties for $\delta\phi_{\rm max}=3\pi/4$.
#### Double parton interactions in $\gamma+3$ jet and $\gamma+b/cjet+2$ jet events in $p \bar p$ collisions at $\sqrt s=1.96$ TeV
The collaboration Abazov, Victor Mukhamedovich ; Abbott, Braden Keim ; Acharya, Bannanje Sripath ; et al.
Phys.Rev. D89 (2014) 072006, 2014.
Inspire Record 1280531
We determine the fraction of events with double parton (DP) scattering in a single ppbar collision at sqrt{s}=1.96 TeV in samples of photon + 3 jet and photon + b/c jet + 2 jet events collected with the D0 detector and corresponding to an integrated luminosity of about 8.7 fb^{-1}. The DP fractions and effective cross sections (sigma_eff) are measured for both event samples using the same kinematic selections. The measured DP fractions range from 0.21 to 0.17, with effective cross sections in the photon + 3 jet and photon + b/c jet + 2 jet samples of sigma_eff^incl = 12.7 +- 0.2 (stat) +- 1.3 (syst) mb and sigma_eff^HF = 14.6 +- 0.6 (stat) +- 3.2 (syst) mb, respectively.
1 data table
The effective cross sections measured for the inclusive and heavy flavour event samples.
#### Search for $s$-Channel Single-Top-Quark Production in Events with Missing Energy Plus Jets in $p\bar p$ Collisions at $\sqrt s=$1.96 TeV
The collaboration Aaltonen, Timo Antero ; Amerio, Silvia ; Amidei, Dante E ; et al.
Phys.Rev.Lett. 112 (2014) 231805, 2014.
Inspire Record 1281537
The first search for single top quark production from the exchange of an $s$-channel virtual $W$ boson using events with an imbalance in the total transverse momentum, $b$-tagged jets, and no identified leptons is presented. The full data set collected by the Collider Detector at Fermilab, corresponding to an integrated luminosity of 9.45 fb$^{-1}$ from Fermilab Tevatron proton-antiproton collisions at a center of mass energy of 1.96 TeV, is used. Assuming the electroweak production of top quarks of mass 172.5 GeV/$c^2$ in the $s$-channel, a cross section of $1.12_{-0.57}^{+0.61}$ (stat+syst) pb, with a significance of 1.9 standard deviations, is measured. This measurement is combined with a previous result obtained from events with an imbalance in total transverse momentum, $b$-tagged jets, and exactly one identified lepton, yielding a cross section of $1.36_{-0.32}^{+0.37}$ (stat+syst) pb, with a significance of 4.2 standard deviations.
2 data tables
The s-channel single top quark cross section measured assuming top quarks of mass 172.5 GeV. The measurement uses a sample of events with large missing transverse energy, two or three jets of which one or more are b-tagged and no detected electron or muon candidates.
The combined s-channel single top quark cross section measurement assuming top quarks of mass 172.5 GeV. The measurement uses two samples of events. The first sample includes events with large missing transverse energy, two or three jets of which one or more are b-tagged and no detected electron or muon candidates. The second sample includes events with large missing transverse energy, one isolated muon or electron and two jets, at least one of which is b-tagged.
#### Invariant-mass distribution of jet pairs produced in association with a $W$ boson in $p \bar{p}$ collisions at $\sqrt{s}=1.96$ TeV using the full CDF Run II data set
The collaboration Aaltonen, T. ; Amerio, S. ; Amidei, D. ; et al.
Phys.Rev. D89 (2014) 092001, 2014.
Inspire Record 1282906
We report on a study of the dijet invariant-mass distribution in events with one identified lepton, a significant imbalance in the total event transverse momentum, and two jets. This distribution is sensitive to the possible production of a new particle in association with a $W$ boson, where the boson decays leptonically. We use the full data set of proton-antiproton collisions at 1.96 TeV center-of-mass energy collected by the Collider Detector at the Fermilab Tevatron and corresponding to an integrated luminosity of 8.9 fb$^{-1}$. The data are found to be consistent with standard-model expectations, and a 95$\%$ confidence level upper limit is set on the cross section for a $W$ boson produced in association with a new particle decaying into two jets.
2 data tables
The extracted cross section assuming that the new contribution (the excess over the expected background) has the same acceptance as that for a 140 GeV Higgs boson produced in association with a W boson.
The extracted cross section measured with a restriction on DELTAR(JET1 JET2) and assuming that the new contribution (the excess over the expected background) has the same acceptance as that for a 140 GeV Higgs boson produced in association with a W boson.
#### Measurement of the forward-backward asymmetry in the distribution of leptons in $t\bar{t}$ events in the lepton$+$jets channel
The collaboration Abazov, Victor Mukhamedovich ; Abbott, Braden Keim ; Acharya, Bannanje Sripath ; et al.
Phys.Rev. D90 (2014) 072001, 2014.
Inspire Record 1283842
14 data tables
Observed ASYMFB(LEPTON) as a function of PT(LEPTON) at reconstruction level.
Observed production-level ASYMFB(LEPTON) as a function of PT(LEPTON).
Observed production-level ASYMFB(LEPTON) as a function of ABS(YRAP(LEPTON)).
More…
#### Measurement of multijet production in $ep$ collisions at high $Q^2$ and determination of the strong coupling $\alpha _s$
The collaboration Andreev, V. ; Baghdasaryan, A. ; Begzsuren, K. ; et al.
Eur.Phys.J. C75 (2015) 65, 2015.
Inspire Record 1301218
20 data tables
Double-differential inclusive jet cross sections measured as a function of Q**2 and PT(JET) using the kT jet algorithm. The total systematic uncertainty sums all systematic uncertainties in quadrature, including the uncertainty due to the LAr noise of 0.5% and the total normalisation uncertainty of 2.9%. The correction factors on the theoretical cross sections C(HAD) and C(EW) are listed in the rightmost columns.
Double-differential dijet cross sections measured as a function of Q**2 and MEAN(PT(2JET)) using the kT jet algorithm. The total systematic uncertainty sums all systematic uncertainties in quadrature, including the uncertainty due to the LAr noise of 0.6% and the total normalisation uncertainty of 2.9%. The correction factors on the theoretical cross sections C(HAD) and C(EW) are listed in the rightmost columns.
Double-differential dijet cross sections measured as a function of Q**2 and XI(2) using the kT jet algorithm. The total systematic uncertainty sums all systematic uncertainties in quadrature, including the uncertainty due to the LAr noise of 0.6% and the total normalisation uncertainty of 2.9%. The correction factors on the theoretical cross sections C(HAD) and C(EW) are listed in the rightmost columns.
More…
#### Measurement of dijet photoproduction for events with a leading neutron at HERA
The collaboration Chekanov, S. ; Derrick, M. ; Magill, S. ; et al.
Nucl.Phys. B827 (2010) 1-33, 2010.
Inspire Record 831440
Differential cross sections for dijet photoproduction and this process in association with a leading neutron, e+ + p -> e+ + jet + jet + X (+ n), have been measured with the ZEUS detector at HERA using an integrated luminosity of 40 pb-1. The fraction of dijet events with a leading neutron was studied as a function of different jet and event variables. Single- and double-differential cross sections are presented as a function of the longitudinal fraction of the proton momentum carried by the leading neutron, xL, and of its transverse momentum squared, pT**2. The dijet data are compared to inclusive DIS and photoproduction results/ they are all consistent with a simple pion-exchange model. The neutron yield as a function of xL was found to depend only on the fraction of the proton beam energy going into the forward region, independent of the hard process. No firm conclusion can be drawn on the presence of rescattering effects.
13 data tables
The differential cross section as a function of jet transverse energy for dijet photon production both without and with a leading neutron, together with their ratio.
The differential cross section as a function of jet pseudorapidity for dijet photon production both without and with a leading neutron, together with their ratio.
The differential cross section as a function of x_photon, the fraction of the photon 4-momenta entering the hard scattering, for dijet photon production both without and with a leading neutron, together with their ratio.
More… | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9886902570724487, "perplexity": 3252.3126573190584}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-09/segments/1550247481612.36/warc/CC-MAIN-20190217031053-20190217053053-00542.warc.gz"} |
http://link.springer.com/article/10.1007%2FBF00127503 | , Volume 48, Issue 2, pp 81-87
# Investigation on early divergence between populations of Drosophila melanogaster kept at different temperatures
Rent the article at a discount
Rent now
* Final gross prices may vary according to local VAT.
An experiment was performed on Drosophila melanogaster populations kept at two temperatures (25°C and 28°C) with the aim of providing further evidence that:
1. 1
phenotypic differentiation between the two populations is already detectable in earlier generations of selection;
2. 2
the divergence is more related to a changed body shape than to body size;
3. 3
this divergence is correlated with fitness and, therefore, natural selection may operate on these differences.
The results obtained by univariate and multivariate statistical analysis imply that:
1. (i)
progress in the phenotypic divergence is observed from the first to the sixth generation as based on the discrimination by a linear function of nine metric traits of the wing, while, as expected, divergence is not detected on mean values and variance estimates;
2. (ii)
differential reproductive fitness is associated with these differences and not with single traits.
It is suggested that the kind of variation observed is the outcome of a rearrangement of the developmental pattern of the wing in the population kept at different temperatures and that the reproductive fitness values are more dependent on the developmental pattern than on the genetic basis of a given character.
The results are discussed in terms of population dynamics. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8850045204162598, "perplexity": 1427.2673168769154}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-27/segments/1435375091925.14/warc/CC-MAIN-20150627031811-00282-ip-10-179-60-89.ec2.internal.warc.gz"} |
https://www.jiskha.com/display.cgi?id=1364425796 | Geometry
posted by .
The value of y that minimizes the sum of the two distances from (3,5) to (1,y) and from (1,y) to (4,9) can be written as \frac{a}{b} where a and b are coprime positive integers. Find a + b.
Similar Questions
1. Math
Let S(n) denote the sum of digits of the integer n. Over all positive integers, the minimum and maximum values of S(n)/S(5n) are X and Y, respectively. The value of X+Y can be written as a/b , where a and b are coprime positive integers. …
2. Calulus
Given \displaystyle \int_0^{\frac{3\pi}{2}} x^2\cos x \, dx = a - \frac{b\pi^2}{c}, where a, b and c are positive integers and b and c are coprime, what is the value of a + b + c?
3. Geometry
Six standard six-sided die are rolled. Let p be the probability that the dice can be arranged in a row such that for 1\leq k \leq 6 the sum of the first k dice is not a multiple of 3. Then p can be expressed as \frac{a}{b} where a …
4. Algebra
Given the system of equations \begin{cases} x(x+y) &=& 9 \\ y(x+y) &=& 16 \end{cases} the value of xy can be written as \frac{a}{b} where a and b are positive coprime integers. Find a+b.
5. Algebra
Joe picks 2 distinct numbers from the set of the first 14 positive integers S = \{1,2,3,\ldots,14\}. The probability that the sum of the 2 numbers is divisible by 3 can be expressed as \frac{a}{b}, where a and b are coprime positive …
6. Calculus
Given f(x) = \frac{x^3-2x+5}{x+4} and f’(3) = \frac{a}{b}, where a and b are coprime positive integers, what is the value of a+b?
7. Algebra
The value of y that minimizes the sum of the two distances from (3,5) to (1,y) and from (1,y) to (4,9) can be written as \frac{a}{b} where a and b are coprime positive integers. Find a + b
8. Geometry
The value of y that minimizes the sum of the two distances from (3,5) to (1,y) and from (1,y) to (4,9) can be written as \frac{a}{b} where a and b are coprime positive integers. Find a + b
9. Maths
The value of y that minimizes the sum of the two distances from (3,5) to (1,y) and from (1,y) to (4,9) can be written as a/b where a and b are coprime positive integers. Find a + b.
10. Geometry
Let ƒÆ=sin −1 7 25 . Consider the sequence of values defined by a n =sin(nƒÆ) . They satisfy the recurrence relation a n+2 =k 1 a n+1 +k 0 a n ,n¸N for some (fixed) real numbers k 1 ,k 0 . The sum k 1 +k 0 can be written …
More Similar Questions | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9858285784721375, "perplexity": 359.2645456613116}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886106367.1/warc/CC-MAIN-20170820092918-20170820112918-00260.warc.gz"} |
http://mathhelpforum.com/pre-calculus/22869-help-please.html | # Math Help - Help please!
Consider the line that passes through the points (4,3) and -4,5
What is the slope of this line? also Write an equation in X and Y for this line. You do not have to simplify this equation or solve it.
last is to write an equation for the line that passes through the points 5,3 and 5,-2
2. Originally Posted by MathMack
Consider the line that passes through the points (4,3) and -4,5
What is the slope of this line? also Write an equation in X and Y for this line. You do not have to simplify this equation or solve it.
last is to write an equation for the line that passes through the points 5,3 and 5,-2
Given two points $(x_1, y_1) \text{ and } (x_2, y_2)$ the slope of the line that passes through them is
$m = \frac{y_2 - y_1}{x_2 - x_1}$
The slope-intercept form of a line is
$y = mx + b$
So let's take your first pair of points:
$m = \frac{5 - 3}{-4 - 4} = \frac{2}{-8} = -\frac{1}{4}$
So the form for the line is:
$y = -\frac{1}{4}x + b$
We need to find b. So plug either point into this formula. I'll choose the first one:
$3 = -\frac{1}{4} \cdot 4 + b$
$3 = -1 + b$
$b = 4$
$y = -\frac{1}{4}x + 4$
You can verify that the other point also satisfies this equation.
You do the other problem.
-Dan
3. ## noonin
change in y/change in x = m(slope)
m=Y2-y1/x2-x1
4. Hey guys thanks, i'm kinda lost on this part, "last is to write an equation for the line that passes through the points 5,3 and 5,-2"
5. Originally Posted by MathMack
Hey guys thanks, i'm kinda lost on this part, "last is to write an equation for the line that passes through the points 5,3 and 5,-2"
Hint: What is the slope of this line? Does it exist? If not, what does that mean about the line?
-Dan
6. For the 1st part I got y=1/4x+2
I did the y-y1=m(x-x1) which I figured and put in what I had, which is y-3=1/4(x-4) which then figures to y-3=1/4x-1 which figures to be y= -1/4x+2
I noticed you got y= -1/4x+4, where am I going wrong?
7. nevermind figured where I went wrong, forgot a minus sign in there! | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 9, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9063923358917236, "perplexity": 382.1840008920904}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1406510271654.40/warc/CC-MAIN-20140728011751-00156-ip-10-146-231-18.ec2.internal.warc.gz"} |
https://www.cabas.kyushu-u.ac.jp/indico/event/8/session/10/contribution/50 | 2019 Symposium on Nuclear Data 2019年度核データ研究会
28-30 November 2019
C-CUBE, Kyushu University Chikushi Campus 九州大学筑紫キャンパス総合研究棟 (C-Cube)
Asia/Tokyo timezone
Home > Timetable > Session details > Contribution details
Comparison between experimental and calculation neutron spectra of the 197Au ($\gamma$,n) reaction for 17 MeV polarized photon
Speakers
• Ms. Kim Tuyet TRAN
Abstract
Until now, neutron spectra were measured for 17MeV polarized photons on a thick Au target at angles from 30 to 150 degrees [1]. The result shows two components of neutrons, evaporation and direct. The angular distribution of the evaporation shows isotropic and that of the direct shows dependence on angle between photon polarization and neutron emission. These spectra are useful to evaluate models and parameters in theoretical calculation, however, comparison between of experiment and calculation has not been done. For this comparison, double differential cross section (DDX) is more preferable than thick target neutron spectra, because cumulative effect in the target can be eliminated. Therefore, we measured the DDX of the 197Au(g,n) for 17 MeV polarized photons on a thin target and compared the results with that of PHITS calculation. The experimental setup was identical except for target thickness, the reduction of which decreases the attenuation of photon and neutron. The polarization direction of photon was set to horizontal, that was parallel to the floor. The neutron detector, a NE213 organic liquid scintilla tor (12.7 cm x12.7 cm L), was placed at 90 degrees horizontally with respect to photon beam axis and 60 cm away from the target. PHITS code (version 3.12) has been used to calculate the double differential cross section. The result shows neutron spectrum only from evaporation model although two components were observed in experiment. The quantitative comparison and discussion will be presented at the symposium.
Keywords: differential double cross section photon neutron , 17MeV polarized photon, Au, PHITS calculation. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9624403119087219, "perplexity": 2697.494379937249}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882571597.73/warc/CC-MAIN-20220812075544-20220812105544-00029.warc.gz"} |
http://math.stackexchange.com/questions/157408/recurrence-relation-t-k1-2t-k-2/157409 | Recurrence relation $T_{k+1} = 2T_k + 2$
I have a series of number in binary system as following:
0, 10, 110, 1110, 11110, 111110, 1111110, 11111110, ...
I want to understand : Is there a general seri for my series?
I found this series has a formula as following:
(Number * 2) + 2
but i don't know this formula is correct or is there a general series (such as fibonacci) for my issue.
-
The series is...
$T_k = 2^k - 2$
-
thanks for your answer. Do have the seri any name? – MJM Jun 12 '12 at 15:17
no i figured it out because $2^k-1$ looks like all 1s, ie 111111 is a power of 2 less 1. So a number like 111110 is 1 less than this, so $2^k -2$ – Andrew Tomazos Jun 12 '12 at 15:18
$2^k-2$ is so short and succinct that the sequence does not need any shorter name. – Henning Makholm Jun 12 '12 at 15:22
$T_{k+1} = 2T_k + 2$. Adding $2$ to both sides, we get that $$\left(T_{k+1}+2 \right) = 2 T_k + 4 = 2 \left( T_k + 2\right)$$ Calling $T_k+2 = u_k$, we get that $u_{k+1} = 2u_k$. Hence, $u_{k+1} = 2^{k+1}u_0$. This gives us $$\left(T_{k}+2 \right) = 2^k \left( T_0 + 2\right) \implies T_k = 2^{k+1} - 2 +2^kT_0$$ Since, $T_0 = 0$, we get that $$T_k = 2^{k+1} - 2$$ where my index starts from $0$.
-
+1 for derivation of the closed form. – Shaktal Jun 12 '12 at 15:45 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9569724202156067, "perplexity": 410.3451739747173}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-26/segments/1466783400031.51/warc/CC-MAIN-20160624155000-00150-ip-10-164-35-72.ec2.internal.warc.gz"} |
https://www.whitman.edu/mathematics/calculus_online/section04.04.html | Now we can complete the calculation of the derivative of the sine: \eqalign{ {d\over dx}\sin x &= \lim_{\Delta x\to0} {\sin(x+\Delta x)-\sin x \over \Delta x}\cr &=\lim_{\Delta x\to0} \sin x{\cos \Delta x - 1\over \Delta x}+\cos x{\sin\Delta x\over \Delta x}\cr &=\sin x \cdot 0 + \cos x \cdot 1 = \cos x.\cr }
The derivative of a function measures the slope or steepness of the function; if we examine the graphs of the sine and cosine side by side, it should be that the latter appears to accurately describe the slope of the former, and indeed this is true:
Notice that where the cosine is zero the sine does appear to have a horizontal tangent line, and that the sine appears to be steepest where the cosine takes on its extreme values of 1 and $-1$.
Of course, now that we know the derivative of the sine, we can compute derivatives of more complicated functions involving the sine.
Example 4.4.1 Compute the derivative of $\ds \sin(x^2)$. $${d\over dx}\sin(x^2) = \cos(x^2)\cdot 2x = 2x\cos(x^2).$$ $\square$
Example 4.4.2 Compute the derivative of $\ds \sin^2(x^3-5x)$. \eqalign{ {d\over dx}\sin^2(x^3-5x)&={d\over dx}(\sin(x^3-5x))^2\cr &=2(\sin(x^3-5x))^1\cos(x^3-5x)(3x^2-5)\cr &=2(3x^2-5)\cos(x^3-5x)\sin(x^3-5x).\cr } $\square$
## Exercises 4.4
Find the derivatives of the following functions.
Ex 4.4.1 $\ds \sin^2(\sqrt{x})$ (answer)
Ex 4.4.2 $\ds \sqrt{x}\sin x$ (answer)
Ex 4.4.3 $\ds {1\over \sin x}$ (answer)
Ex 4.4.4 $\ds {x^2+x\over \sin x}$ (answer)
Ex 4.4.5 $\ds \sqrt{1-\sin^2x }$ (answer) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9994656443595886, "perplexity": 1298.8430504319792}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320306335.77/warc/CC-MAIN-20220128182552-20220128212552-00464.warc.gz"} |
http://happynear.wang/2016/03/28/Normalizing-All-Layers%EF%BC%9A-Back-Propagation/ | # 1.Introduction
In the last post, we discussed how to make all neurons of a neural network to have normal Gaussian distribution. However, as the Conclusion section claimed, we haven’t considered the back-propagation procedure. In fact, when we talk about the gradient vanishing or exploding problem, we usually refer to the gradients flow in the back-propagation procedure. Since this, the correct way seems to be normalizing the backward gradients of neurons, instead of the forward values.
In this post, we will discuss how to normalize all the gradients using a similar philosophy with the last post: for a given gradient $dy\sim N(0, I)$, normalizing the layer to make sure that $dx$ is expected to have zero mean and one standard deviation.
# 2. Parametric Layer
Consider the back-propagate fomulation of Convolution and InnerProdcut layer,
$dx = W dy,$
we will get a similar strategy of normalizing each row of $W$ to be on a $\ell 2$ unit ball. Please note that here we normalize through the fan-out dimension of $W$, not the fan-in dimension in the forward propagation.
# 3. Activation Layers
One problem that can’t be avoided when calculating the formulations of activations is that we should not only assume the distribution of the gradients, but also the forward input of the activation because the gradients of activations are usually dependent on the inputs. Here we assume that both the input $x$ and the gradient $dy$ follow the normal Gaussian distribution $N(0, I)$, and they are independent with each other.
## 1) ReLU
The forward formulation of ReLU is,
$y = max(0, x).$
Its backward gradients can be easily obtained:
$dx_i = dy_i * \left\{ \begin{array}{rcl} 1 & & {x_i > 0}\\ 0 & & {x_i \leq 0}. \end{array} \right.$
When $x\sim N(0, I)$, the gradient of the ReLU layer can be seen as a Bernoulli distribution with probability of 0.5, so the backward mean and standard deviation formulas are similar with those of Dropout layer,
$E[dx] = 0,$
$\sigma[dx]=\sqrt{\frac{1}{2}}.$
Here the question comes, now we have two different standard deviations, one for forward values and one for backward gradients, which one should be used to normalize the ReLU layer? My tendency is to use the $\sigma$ calculated by the backward gradients, because backward $\sigma$ is the real murderer of gradient vanishing. Moreover, since the bias term is not involved in the backward propagation, it is a good manner to substract the mean $\sqrt{\frac{1}{2\pi}}$ after ReLU activation to ensure zero mean.
## 2) Sigmoid
The backward gradient of Sigmoid activation is,
$dx = y \cdot (1-y).$
This time, I won’t attempt to calculate the close formulations of mean and std, it is really a tough work. I tend to directly use simulating to get the results.
We can get $E[dx] = 0$ and $\sigma[dx]=0.2123$. The same with ReLU, we should still minus the $E[y]=0.5$ after Sigmoid activation and use the $\sigma$ calculated by backward gradients, 0.2123.
# 4. Pooling Layer
The standard deviation of $3\times3$ average pooling can be simulated by,
It is $\frac{1}{9}$, and we can infer that the $\sigma$ for $2\times2$ average pooling is $\frac{1}{4}$.
For max pooling, we only pass the gradient to one of the neurons in the pooling window, so we have,
Running the script and we can get $\sigma$ for $3\times3$ is $\frac{1}{3}$ and $\sigma$ for $2\times2$ is $\frac{1}{2}$.
# 5. Dropout Layer
The backward formula for Dropout layer is almost the same with the forward one, we should still divide the preserved values by $\sqrt{q}$ to achieve 1 std for both forward and backward procedure.
# 6. Conclusion
In this post, we have discussed the normalization strategy that serves the gradient flow of the backward propagation. The mean and std values of forward and backward data flows are listed here:
Param Conv/IP ReLU Sigmoid $3\times3$ Max Pooling Ave Pooling Dropout
fp mean 0 $\sqrt{\frac{1}{2\pi}}$ $\frac{1}{2}$ 1.4850 0 0
fp std $\ell2$ fan-in $\sqrt{\frac{1}{2} - \frac{1}{2\pi}}$ 0.2083 0.5978 $\frac{1}{s}$ $\sqrt{\frac{1}{p}}$
bp std $\ell2$ fan-out $\sqrt{\frac{1}{2}}$ 0.2123 $\frac{1}{3}$ $\frac{1}{s^2}$ $\sqrt{\frac{1}{p}}$
However, here comes another problem that when we are using the std of backward gradients, the forward value scale would not be controlled well. Inhomogeneous(非齐次) activations, such as sigmoid and tanh, are not suitable for this method because their domain may not cover a sufficient non-linear part of the activation.
So maybe a good choice is to use a separate scaling method for forward and backward propagation? This idea conflicts with the back-propagation algorithm, so we should still carefully examine it through experiment. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9761162400245667, "perplexity": 680.1750236988853}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487613380.12/warc/CC-MAIN-20210614170602-20210614200602-00580.warc.gz"} |
https://www.physicsforums.com/threads/physics-help-on-rotational-inertia.147521/ | Homework Help: Physics help on rotational inertia
1. Dec 10, 2006
blackout85
First questionThree identical balls, with masses M, 2M, and 3M are fastened to a massless rod of length L as shown. The rotational inertia about the left end of the rod is: Thats the layout below. Would calculus be needed in this problem (intergration) because then Im in trouble. I know the rotation at the end of rod is I=ML^2/3. Could I use that formula.
3M-----L/2----2M----L/2-----M
work:
I came up with an answer of 3ML^2/2
does that look right. I added the two end mass and lengths using the equation I=mr^2--> simply plugging in the values and adding. Would that be correct.
second question:
If a wheel turns with a costant rotational speed then: each point on its rim moves with constant roational velocity, each point on its rim moves with constant translational acceleration, the wheel turns with constant translation acceleration, the wheel turns through equal angles in equal times, the angle through which the wheel turns in each second increases as time goes on, the angle through which the wheel turns in each second decreases as time goes on.
work
I thought if the wheel turns with a constant translational velocity along the rim because of the the equation v=wr. Am I right to think since along the rim will have the same radius as in a wheel.
I would appreciate any feedback on both questions. Thank you
Last edited by a moderator: May 9, 2015
2. Dec 10, 2006
Hootenanny
Staff Emeritus
The answer to your first question is correct. With regards to the second question, is that copied directly from your text or have you paraphrased it?
3. Dec 10, 2006
blackout85
physics
I rechecked it
If a wheel turns with a constant rotational speed:
each point on its rim moves with a constant translational velocity
each point on its rim moves with a constant translational acceleration
the wheel turns through equal angles in equal times
the angle through which the wheel turns in each second increases as times goes on
the angle through which the wheel turns in each second decreases as time goes on
work:
I thought if the wheel turns with a constant translational velocity along the rim because of the the equation v=wr. Am I right to think since along the rim will have the same radius as in a wheel. This equation connects rotational velocity to translational velocity. Since the radius is the same for the edge in a wheel I thought that the answer is constant translational velocity.
4. Dec 10, 2006
Hootenanny
Staff Emeritus
You are indeed correct. Although there is one further choice which is also correct... | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9558221697807312, "perplexity": 824.4361296106075}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583512395.23/warc/CC-MAIN-20181019103957-20181019125457-00415.warc.gz"} |
https://worldwidescience.org/topicpages/b/boundary+layer+separation.html | #### Sample records for boundary layer separation
1. Airfoil boundary layer separation and control at low Reynolds numbers
Yarusevych, S.; Sullivan, P.E. [University of Toronto, Department of Mechanical and Industrial Engineering, Toronto, ON (Canada); Kawall, J.G. [Ryerson University, Department of Mechanical and Industrial Engineering, Toronto, ON (Canada)
2005-04-01
The boundary layer separation on a NACA 0025 airfoil was studied experimentally via hot-wire anemometry and surface pressure measurements. The results provide added insight into periodic boundary layer control, suggesting that matching the excitation frequency with the most amplified disturbance in the separated shear layer is optimal for improving airfoil performance. (orig.)
2. Analysis and Modeling of Boundary Layer Separation Method (BLSM).
Pethő, Dóra; Horváth, Géza; Liszi, János; Tóth, Imre; Paor, Dávid
2010-09-01
Nowadays rules of environmental protection strictly regulate pollution material emission into environment. To keep the environmental protection laws recycling is one of the useful methods of waste material treatment. We have developed a new method for the treatment of industrial waste water and named it boundary layer separation method (BLSM). We apply the phenomena that ions can be enriched in the boundary layer of the electrically charged electrode surface compared to the bulk liquid phase. The main point of the method is that the boundary layer at correctly chosen movement velocity can be taken out of the waste water without being damaged, and the ion-enriched boundary layer can be recycled. Electrosorption is a surface phenomenon. It can be used with high efficiency in case of large electrochemically active surface of electrodes. During our research work two high surface area nickel electrodes have been prepared. The value of electrochemically active surface area of electrodes has been estimated. The existence of diffusion part of the double layer has been experimentally approved. The electrical double layer capacity has been determined. Ion transport by boundary layer separation has been introduced. Finally we have tried to estimate the relative significance of physical adsorption and electrosorption.
3. Vortex Formation During Unsteady Boundary-Layer Separation
Das, Debopam; Arakeri, Jaywant H.
1998-11-01
Unsteady laminar boundary-layer separation is invariably accompanied by the formation of vortices. The aim of the present work is to study the vortex formation mechanism(s). An adverse pressure gradient causing a separation can be decomposed into a spatial component ( spatial variation of the velocity external to the boundary layer ) and a temporal component ( temporal variation of the external velocity ). Experiments were conducted in a piston driven 2-D water channel, where the spatial component could be be contolled by geometry and the temporal component by the piston motion. We present results for three divergent channel geometries. The piston motion consists of three phases: constant acceleration from start, contant velocity, and constant deceleration to stop. Depending on the geometry and piston motion we observe different types of unsteady separation and vortex formation.
4. Streaming effect of wall oscillation to boundary layer separation
Wu, X. H.; Wu, J. Z.; Wu, J. M.
1991-01-01
This paper presents a preliminary theoretical result on the time averaged streaming effect of local forcing excitation to the boundary layer separation from smooth surface. The problem is formulated as a periodic disturbance to a basic steady breakaway separating flow, for which the data are taken from a numerical triple-deck solution. The ratio of Strouhal number St and Reynolds number Re plays an important role, both being assumed sufficiently high. The analytical and numerical results show that this streaming effect is quite strong at proper values of St/Re exp 1/4, which may delay or even suppress the separation.
5. Notes on the Prediction of Shock-induced Boundary-layer Separation
Lange, Roy H.
1953-01-01
The present status of available information relative to the prediction of shock-induced boundary-layer separation is discussed. Experimental results showing the effects of Reynolds number and Mach number on the separation of both laminar and turbulent boundary layer are given and compared with available methods for predicting separation. The flow phenomena associated with separation caused by forward-facing steps, wedges, and incident shock waves are discussed. Applications of the flat-plate data to problems of separation on spoilers, diffusers, and scoop inlets are indicated for turbulent boundary layers.
6. Decomposition Methods For a Piv Data Analysis with Application to a Boundary Layer Separation Dynamics
Václav URUBA
2010-12-01
Full Text Available Separation of the turbulent boundary layer (BL on a flat plate under adverse pressure gradient was studied experimentally using Time-Resolved PIV technique. The results of spatio-temporal analysis of flow-field in the separation zone are presented. For this purpose, the POD (Proper Orthogonal Decomposition and its extension BOD (Bi-Orthogonal Decomposition techniques are applied as well as dynamical approach based on POPs (Principal Oscillation Patterns method. The study contributes to understanding physical mechanisms of a boundary layer separation process. The acquired information could be used to improve strategies of a boundary layer separation control.
7. Separation prediction in two dimensional boundary layer flows using artificial neural networks
2003-01-01
In this article, the ability of artificial neural networks in prediction of separation in steady two dimensional boundary layer flows is studied. Data for network training is extracted from numerical solution of an ODE obtained from Von Karman integral equation with approximate one parameter Pohlhousen velocity profile. As an appropriate neural network, a two layer radial basis generalized regression artificial neural network is used. The results shows good agreements between the overall behavior of the flow fields predicted by the artificial neural network and the actual flow fields for some cases. The method easily can be extended to unsteady separation and turbulent as well as compressible boundary layer flows. (author)
8. On the Lagrangian description of unsteady boundary layer separation. Part 1: General theory
Vandommelen, Leon L.; Cowley, Stephen J.
1989-01-01
Although unsteady, high-Reynolds number, laminar boundary layers have conventionally been studied in terms of Eulerian coordinates, a Lagrangian approach may have significant analytical and computational advantages. In Lagrangian coordinates the classical boundary layer equations decouple into a momentum equation for the motion parallel to the boundary, and a hyperbolic continuity equation (essentially a conserved Jacobian) for the motion normal to the boundary. The momentum equations, plus the energy equation if the flow is compressible, can be solved independently of the continuity equation. Unsteady separation occurs when the continuity equation becomes singular as a result of touching characteristics, the condition for which can be expressed in terms of the solution of the momentum equations. The solutions to the momentum and energy equations remain regular. Asymptotic structures for a number of unsteady 3-D separating flows follow and depend on the symmetry properties of the flow. In the absence of any symmetry, the singularity structure just prior to separation is found to be quasi 2-D with a displacement thickness in the form of a crescent shaped ridge. Physically the singularities can be understood in terms of the behavior of a fluid element inside the boundary layer which contracts in a direction parallel to the boundary and expands normal to it, thus forcing the fluid above it to be ejected from the boundary layer.
9. On the Lagrangian description of unsteady boundary-layer separation. I - General theory
Van Dommelen, Leon L.; Cowley, Stephen J.
1990-01-01
Although unsteady, high-Reynolds number, laminar boundary layers have conventionally been studied in terms of Eulerian coordinates, a Lagrangian approach may have significant analytical and computational advantages. In Lagrangian coordinates the classical boundary layer equations decouple into a momentum equation for the motion parallel to the boundary, and a hyperbolic continuity equation (essentially a conserved Jacobian) for the motion normal to the boundary. The momentum equations, plus the energy equation if the flow is compressible, can be solved independently of the continuity equation. Unsteady separation occurs when the continuity equation becomes singular as a result of touching characteristics, the condition for which can be expressed in terms of the solution of the momentum equations. The solutions to the momentum and energy equations remain regular. Asymptotic structures for a number of unsteady 3-D separating flows follow and depend on the symmetry properties of the flow. In the absence of any symmetry, the singularity structure just prior to separation is found to be quasi 2-D with a displacement thickness in the form of a crescent shaped ridge. Physically the singularities can be understood in terms of the behavior of a fluid element inside the boundary layer which contracts in a direction parallel to the boundary and expands normal to it, thus forcing the fluid above it to be ejected from the boundary layer.
10. Experimental Investigation of Separated and Transitional Boundary Layers Under Low-Pressure Turbine Airfoil Conditions
Hultgren, Lennart S.; Volino, Ralph J.
2002-01-01
11. Boundary layer separation method for recycling of sodium ions from industrial wastewater.
Petho, Dóra; Horváth, Géza; Liszi, János; Tóth, Imre; Paor, Dávid
2010-12-01
The most effective technological solution for waste treatment is recycling. We have developed a new method for the treatment of industrial wastewaters and have called it the boundary layer separation method (BLSM). We have used the phenomenon that, on the surface of an electrically charged electrode, ions can be enriched in the boundary layer, as compared with the inside of the phase. The essence of the method is that, with an appropriately chosen velocity, the boundary layer can be removed from the wastewater, and the boundary layer, which is rich in ions, can be recycled. The BLSM can be executed as a cyclic procedure. The capacitance of the boundary layer was examined. The best mass transport can be achieved with the use of 1000 and 1200 mV polarization potentials in the examined system, with its value being 1200 mg/m2 per cycle. The necessary operation times were determined by the examination of the velocity of the electrochemical processes. When using 1000 mV polarization potential, the necessary adsorption time is at least 25 seconds, and the desorption time at least 300 seconds. The advantage of the procedure is that it does not use dangerous chemicals, only inert electrodes. The drawback is that it is not selective to ions, the achievable separation in one step is low, and the hydrogen that emerges during the electrolysis might be dangerous.
12. Effects of boundary-layer separation controllers on a desktop fume hood.
Huang, Rong Fung; Chen, Jia-Kun; Hsu, Ching Min; Hung, Shuo-Fu
2016-10-02
A desktop fume hood installed with an innovative design of flow boundary-layer separation controllers on the leading edges of the side plates, work surface, and corners was developed and characterized for its flow and containment leakage characteristics. The geometric features of the developed desktop fume hood included a rearward offset suction slot, two side plates, two side-plate boundary-layer separation controllers on the leading edges of the side plates, a slanted surface on the leading edge of the work surface, and two small triangular plates on the upper left and right corners of the hood face. The flow characteristics were examined using the laser-assisted smoke flow visualization technique. The containment leakages were measured by the tracer gas (sulphur hexafluoride) detection method on the hood face plane with a mannequin installed in front of the hood. The results of flow visualization showed that the smoke dispersions induced by the boundary-layer separations on the leading edges of the side plates and work surface, as well as the three-dimensional complex flows on the upper-left and -right corners of the hood face, were effectively alleviated by the boundary-layer separation controllers. The results of the tracer gas detection method with a mannequin standing in front of the hood showed that the leakage levels were negligibly small (≤0.003 ppm) at low face velocities (≥0.19 m/s).
13. Structural characteristics of the shock-induced boundary layer separation extended to the leading edge
Tao, Y.; Liu, W. D.; Fan, X. Q.; Zhao, Y. L.
2017-07-01
For a better understanding of the local unstart of supersonic/hypersonic inlet, a series of experiments has been conducted to investigate the shock-induced boundary layer separation extended to the leading edge. Using the nanoparticle-based planar laser scattering, we recorded the fine structures of these interactions under different conditions and paid more attention to their structural characteristics. According to their features, these interactions could be divided into four types. Specifically, Type A wave pattern is similar to the classic shock wave/turbulent boundary layer interaction, and Type B wave configuration consists of an overall Mach reflection above the large scale separation bubble. Due to the gradual decrease in the size of the separation bubble, the separation bubble was replaced by several vortices (Type C wave pattern). Besides, for Type D wave configuration which exists in the local unstart inlet, there appears to be some flow spillage around the leading edge.
14. Decomposition Methods For a Piv Data Analysis with Application to a Boundary Layer Separation Dynamics
Václav URUBA
2010-01-01
Separation of the turbulent boundary layer (BL) on a flat plate under adverse pressure gradient was studied experimentally using Time-Resolved PIV technique. The results of spatio-temporal analysis of flow-field in the separation zone are presented. For this purpose, the POD (Proper Orthogonal Decomposition) and its extension BOD (Bi-Orthogonal Decomposition) techniques are applied as well as dynamical approach based on POPs (Principal Oscillation Patterns) method. The study contributes...
15. Passive Flap Actuation by Reversing Flow in Laminar Boundary Layer Separation
Parsons, Chase; Lang, Amy; Santos, Leo; Bonacci, Andrew
2017-11-01
Reducing the flow separation is of great interest in the field of fluid mechanics in order to reduce drag and improve the overall efficiency of aircraft. This project seeks to investigate passive flow control using shark inspired microflaps in laminar boundary layer separation. This study aims to show that whether a flow is laminar or turbulent, laminar and 2D or turbulent and 3D, microflaps actuated by reversing flow is a robust means of controlling flow separation. In order to generate a controlled adverse pressure gradient, a rotating cylinder induces separation at a chosen location on a flat plate boundary layer with Re above 10000. Within this thick boundary layer, digital particle image velocimetry is used to map the flow. This research can be used in the future to better understand the nature of the bristling shark scales and its ability to passively control separation. Results show that microflaps successfully actuated due to backflow and that this altered the formation of flow separation. I would like to thank the NSF for REU Grant EEC 1659710 and the Army Research Office for funding this project.
16. A documentation of two- and three-dimensional shock-separated turbulent boundary layers
Brown, J. D.; Brown, J. L.; Kussoy, M. I.
1988-01-01
A shock-related separation of a turbulent boundary layer has been studied and documented. The flow was that of an axisymmetric turbulent boundary layer over a 5.02-cm-diam cylinder that was aligned with the wind tunnel axis. The boundary layer was compressed by a 30 deg half-angle conical flare, with the cone axis inclined at an angle alpha to the cylinder axis. Nominal test conditions were P sub tau equals 1.7 atm and M sub infinity equals 2.85. Measurements were confined to the upper-symmetry, phi equals 0 deg, plane. Data are presented for the cases of alpha equal to 0. 5. and 10 deg and include mean surface pressures, streamwise and normal mean velocities, kinematic turbulent stresses and kinetic energies, as well as reverse-flow intermittencies. All data are given in tabular form; pressures, streamwise velocities, turbulent shear stresses, and kinetic energies are also presented graphically.
17. Experimental Study of Unsteady Flow Separation in a Laminar Boundary Layer
Bonacci, Andrew; Lang, Amy; Wahidi, Redha; Santos, Leonardo
2017-11-01
Flow separation, caused by an adverse pressure gradient, is a major problem in many applications. Reversing flow near the wall is the first sign of incipient separation and can bristle shark scales which may be linked to a passive, flow actuated separation control mechanism. An investigation of how this backflow forms and how it interacts with shark skin is of interest due to the fact that this could be used as a bioinspired means of initiating flow control. A water tunnel experiment aims to study unsteady separation with a focus on the reversing flow development near the wall within a flat plate laminar boundary layer (Re on order of 105) as an increasing adverse pressure gradient is induced by a rotating cylinder. Unsteady reversing flow development is documented using DPIV. Funding was provided by the National Science Foundation under the Research Experience for Undergraduates (REU) program (EEC 1659710) and the Army Research Office.
18. Boundary Layer Separation and Reattachment Detection on Airfoils by Thermal Flow Sensors
Peter Busche
2012-10-01
Full Text Available A sensor concept for detection of boundary layer separation (flow separation, stall and reattachment on airfoils is introduced in this paper. Boundary layer separation and reattachment are phenomena of fluid mechanics showing characteristics of extinction and even inversion of the flow velocity on an overflowed surface. The flow sensor used in this work is able to measure the flow velocity in terms of direction and quantity at the sensor’s position and expected to determine those specific flow conditions. Therefore, an array of thermal flow sensors has been integrated (flush-mounted on an airfoil and placed in a wind tunnel for measurement. Sensor signals have been recorded at different wind speeds and angles of attack for different positions on the airfoil. The sensors used here are based on the change of temperature distribution on a membrane (calorimetric principle. Thermopiles are used as temperature sensors in this approach offering a baseline free sensor signal, which is favorable for measurements at zero flow. Measurement results show clear separation points (zero flow and even negative flow values (back flow for all sensor positions. In addition to standard silicon-based flow sensors, a polymer-based flexible approach has been tested showing similar results.
19. Thermophoretic motion behavior of submicron particles in boundary-layer-separation flow around a droplet.
Wang, Ao; Song, Qiang; Ji, Bingqiang; Yao, Qiang
2015-12-01
As a key mechanism of submicron particle capture in wet deposition and wet scrubbing processes, thermophoresis is influenced by the flow and temperature fields. Three-dimensional direct numerical simulations were conducted to quantify the characteristics of the flow and temperature fields around a droplet at three droplet Reynolds numbers (Re) that correspond to three typical boundary-layer-separation flows (steady axisymmetric, steady plane-symmetric, and unsteady plane-symmetric flows). The thermophoretic motion of submicron particles was simulated in these cases. Numerical results show that the motion of submicron particles around the droplet and the deposition distribution exhibit different characteristics under three typical flow forms. The motion patterns of particles are dependent on their initial positions in the upstream and flow forms. The patterns of particle motion and deposition are diversified as Re increases. The particle motion pattern, initial position of captured particles, and capture efficiency change periodically, especially during periodic vortex shedding. The key effects of flow forms on particle motion are the shape and stability of the wake behind the droplet. The drag force of fluid and the thermophoretic force in the wake contribute jointly to the deposition of submicron particles after the boundary-layer separation around a droplet.
20. Reversing flow causes passive shark scale actuation in a separating turbulent boundary layer
Lang, Amy; Gemmell, Bradford; Motta, Phil; Habegger, Laura; Du Clos, Kevin; Devey, Sean; Stanley, Caleb; Santos, Leo
2017-11-01
Control of flow separation by shortfin mako skin in experiments has been demonstrated, but the mechanism is still poorly understood yet must be to some extent Re independent. The hypothesized mechanisms inherent in the shark skin for controlling flow separation are: (1) the scales, which are capable of being bristled only by reversing flow, inhibit flow reversal events from further development into larger-scale separation and (2) the cavities formed when scales bristle induces mixing of high momentum flow towards the wall thus energizing the flow close to the surface. Two studies were carried out to measure passive scale actuation caused by reversing flow. A small flow channel induced an unsteady, wake flow over the scales prompting reversing flow events and scale actuation. To resolve the flow and scale movements simultaneously we used specialized optics at high magnification (1 mm field of view) at 50,000 fps. In another study, 3D printed models of shark scales, or microflaps (bristling capability up to 50 degrees), were set into a flat plate. Using a tripped, turbulent boundary layer grown over the long flat plate and a localized adverse pressure gradient, a separation bubble was generated within which the microflaps were placed. Passive flow actuation of both shark scales and microflaps by reversing flow was observed. Funding from Army Research Office and NSF REU site Grant.
1. The structure of a separating turbulent boundary layer. IV - Effects of periodic free-stream unsteadiness
Simpson, R. L.; Shivaprasad, B. G.; Chew, Y.-T.
1983-01-01
Measurements were obtained of the sinusoidal unsteadiness of the free stream velocity during the separation of the turbulent boundary layer. Data were gathered by single wire and cross-wire, anemometry upstream of flow detachment, by laser Doppler velocimetry to detect the movement of the flow in small increments, and by a laser anemometer in the detached zone to measure turbulence and velocities. The study was restricted to a sinusoidal instability frequency of 0.61 and a ratio of oscillation amplitude to mean velocity of 0.3. Large amplitude and phase variations were found after the detachment, with unsteady effects producing hysteresis in the relationships between flow parameters. The detached shear layer decreased in thickness with increasing free-stream velocity and increases in the Reynolds shear stress. Deceleration of the free stream velocity caused thickening in the shear layer and upstream movement of the flow reversal location. The results are useful for studies of compressor blade and helicopter rotors in transition.
2. Corner Separation Control by Boundary Layer Suction Applied to a Highly Loaded Axial Compressor Cascade
Yangwei Liu
2014-11-01
Full Text Available Control of corner separation has attracted much interest due to its improvement of performance and energy utilization in turbomachinery. Numerical studies have been performed under both design and off-design flow conditions to investigate the effects of boundary layer suction (BLS on corner separation in a highly loaded compressor cascade. Two new BLS slot configurations are proposed and a total of five suction slot configurations were studied and compared. Averaged static pressure rise, exit loss coefficient, passage blockage and flow turning angle have been given and compared systematically over a range of operation incidence angles. Distributions of significant loss removal, blade loading, exit deviation and total pressure loss at 3 degree and 7 degree incidence have also been studied. Under the same suction mass flows of 0.7% of the inlet mass flows, the pitchwise suction slot on the endwall shows a better optimal performance over the whole operation incidence among single suction slots. By using of the new proposed compound slot configuration with one spanwise slot on the blade suction side and one pitchwise slot on the endwall, the maximum reduction of total pressure loss at 7 degree incidence can be 39.4%.
3. Large-eddy simulation of separation and reattachment of a flat plate turbulent boundary layer
Cheng, W.
2015-11-11
© 2015 Cambridge University Press. We present large-eddy simulations (LES) of separation and reattachment of a flat-plate turbulent boundary-layer flow. Instead of resolving the near wall region, we develop a two-dimensional virtual wall model which can calculate the time- and space-dependent skin-friction vector field at the wall, at the resolved scale. By combining the virtual-wall model with the stretched-vortex subgrid-scale (SGS) model, we construct a self-consistent framework for the LES of separating and reattaching turbulent wall-bounded flows at large Reynolds numbers. The present LES methodology is applied to two different experimental flows designed to produce separation/reattachment of a flat-plate turbulent boundary layer at medium Reynolds number Reθ based on the momentum boundary-layer thickness θ. Comparison with data from the first case at demonstrates the present capability for accurate calculation of the variation, with the streamwise co-ordinate up to separation, of the skin friction coefficient, Reθ, the boundary-layer shape factor and a non-dimensional pressure-gradient parameter. Additionally the main large-scale features of the separation bubble, including the mean streamwise velocity profiles, show good agreement with experiment. At the larger Reθ = 11000 of the second case, the LES provides good postdiction of the measured skin-friction variation along the whole streamwise extent of the experiment, consisting of a very strong adverse pressure gradient leading to separation within the separation bubble itself, and in the recovering or reattachment region of strongly-favourable pressure gradient. Overall, the present two-dimensional wall model used in LES appears to be capable of capturing the quantitative features of a separation-reattachment turbulent boundary-layer flow at low to moderately large Reynolds numbers.
4. LARGE-EDDY SIMULATIONS OF A SEPARATION/REATTACHMENT BUBBLE IN A TURBULENT-BOUNDARY-LAYER SUBJECTED TO A PRESCRIBED UPPER-BOUNDARY, VERTICAL-VELOCITY PROFILE
Cheng, Wan
2015-06-30
We describe large-eddy simulations of turbulent boundary-layer flow over a flat plate at high Reynolds number in the presence of an unsteady, three-dimensional flow separation/reattachment bubble. The stretched-vortex subgrid-scale model is used in the main flow domain combined with a wall-model that is a two-dimensional extension of that developed by Chung & Pullin (2009). Flow separation and re-attachment of the incoming boundary layer is induced by prescribing wall-normal velocity distribution on the upper boundary of the flow domain that produces an adverse-favorable stream-wise pressure distribution at the wall. The LES predicts the distribution of mean shear stress along the wall including the interior of the separation bubble. Several properties of the separation/reattachment flow are discussed.
5. Large-eddy simulation of separation and reattachment of a flat plate turbulent boundary layer
Cheng, W.; Pullin, D. I.; Samtaney, Ravi
2015-01-01
© 2015 Cambridge University Press. We present large-eddy simulations (LES) of separation and reattachment of a flat-plate turbulent boundary-layer flow. Instead of resolving the near wall region, we develop a two-dimensional virtual wall model which
6. LARGE-EDDY SIMULATIONS OF A SEPARATION/REATTACHMENT BUBBLE IN A TURBULENT-BOUNDARY-LAYER SUBJECTED TO A PRESCRIBED UPPER-BOUNDARY, VERTICAL-VELOCITY PROFILE
Cheng, Wan; Pullin, D. I.; Samtaney, Ravi
2015-01-01
We describe large-eddy simulations of turbulent boundary-layer flow over a flat plate at high Reynolds number in the presence of an unsteady, three-dimensional flow separation/reattachment bubble. The stretched-vortex subgrid-scale model is used
7. Competitive separation of di- vs. mono-valent cations in electrodialysis: effects of the boundary layer properties.
Kim, Younggy; Walker, W Shane; Lawler, Desmond F
2012-05-01
In electrodialysis desalination, the boundary layer near ion-exchange membranes is the limiting region for the overall rate of ionic separation due to concentration polarization over tens of micrometers in that layer. Under high current conditions, this sharp concentration gradient, creating substantial ionic diffusion, can drive a preferential separation for certain ions depending on their concentration and diffusivity in the solution. Thus, this study tested a hypothesis that the boundary layer affects the competitive transport between di- and mono-valent cations, which is known to be governed primarily by the partitioning with cation-exchange membranes. A laboratory-scale electrodialyzer was operated at steady state with a mixture of 10mM KCl and 10mM CaCl(2) at various flow rates. Increased flows increased the relative calcium transport. A two-dimensional model was built with analytical solutions of the Nernst-Planck equation. In the model, the boundary layer thickness was considered as a random variable defined with three statistical parameters: mean, standard deviation, and correlation coefficient between the thicknesses of the two boundary layers facing across a spacer. Model simulations with the Monte Carlo method found that a greater calcium separation was achieved with a smaller mean, greater standard deviation, or more negative correlation coefficient. The model and experimental results were compared for the cationic transport number as well as the current and potential relationship. The mean boundary layer thickness was found to decrease from 40 to less than 10 μm as the superficial water velocity increased from 1.06 to 4.24 cm/s. The standard deviation was greater than the mean thickness at slower water velocities and smaller at faster water velocities. Copyright © 2012 Elsevier Ltd. All rights reserved.
8. Boundary-Layer Separation Control under Low-Pressure Turbine Airfoil Conditions using Glow-Discharge Plasma Actuators
Hultgren, Lennart S.; Ashpis, David E.
2003-01-01
Modem low-pressure turbines, in general, utilize highly loaded airfoils in an effort to improve efficiency and to lower the number of airfoils needed. Typically, the airfoil boundary layers are turbulent and fully attached at takeoff conditions, whereas a substantial fraction of the boundary layers on the airfoils may be transitional at cruise conditions due to the change of density with altitude. The strong adverse pressure gradients on the suction side of these airfoils can lead to boundary-layer separation at the latter low Reynolds number conditions. Large separation bubbles, particularly those which fail to reattach, cause a significant degradation of engine efficiency. A component efficiency drop of the order 2% may occur between takeoff and cruise conditions for large commercial transport engines and could be as large as 7% for smaller engines at higher altitude. An efficient means of of separation elimination/reduction is, therefore, crucial to improved turbine design. Because the large change in the Reynolds number from takeoff to cruise leads to a distinct change in the airfoil flow physics, a separation control strategy intended for cruise conditions will need to be carefully constructed so as to incur minimum impact/penalty at takeoff. A complicating factor, but also a potential advantage in the quest for an efficient strategy, is the intricate interplay between separation and transition for the situation at hand. Volino gives a comprehensive discussion of several recent studies on transition and separation under low-pressure-turbine conditions, among them one in the present facility. Transition may begin before or after separation, depending on the Reynolds number and other flow conditions. If the transition occurs early in the boundary layer then separation may be reduced or completely eliminated. Transition in the shear layer of a separation bubble can lead to rapid reattachment. This suggests using control mechanisms to trigger and enhance early
9. Reynolds stress structures in a self-similar adverse pressure gradient turbulent boundary layer at the verge of separation.
Atkinson, C.; Sekimoto, A.; Jiménez, J.; Soria, J.
2018-04-01
Mean Reynolds stress profiles and instantaneous Reynolds stress structures are investigated in a self-similar adverse pressure gradient turbulent boundary layer (APG-TBL) at the verge of separation using data from direct numerical simulations. The use of a self-similar APG-TBL provides a flow domain in which the flow gradually approaches a constant non-dimensional pressure gradient, resulting in a flow in which the relative contribution of each term in the governing equations is independent of streamwise position over a domain larger than two boundary layer thickness. This allows the flow structures to undergo a development that is less dependent on the upstream flow history when compared to more rapidly decelerated boundary layers. This APG-TBL maintains an almost constant shape factor of H = 2.3 to 2.35 over a momentum thickness based Reynolds number range of Re δ 2 = 8420 to 12400. In the APG-TBL the production of turbulent kinetic energy is still mostly due to the correlation of streamwise and wall-normal fluctuations, 〈uv〉, however the contribution form the other components of the Reynolds stress tensor are no longer negligible. Statistical properties associated with the scale and location of sweeps and ejections in this APG-TBL are compared with those of a zero pressure gradient turbulent boundary layer developing from the same inlet profile, resulting in momentum thickness based range of Re δ 2 = 3400 to 3770. In the APG-TBL the peak in both the mean Reynolds stress and the production of turbulent kinetic energy move from the near wall region out to a point consistent with the displacement thickness height. This is associated with a narrower distribution of the Reynolds stress and a 1.6 times higher relative number of wall-detached negative uv structures. These structures occupy 5 times less of the boundary layer volume and show a similar reduction in their streamwise extent with respect to the boundary layer thickness. A significantly lower percentage
10. Stable Boundary Layer Issues
Steeneveld, G.J.
2012-01-01
Understanding and prediction of the stable atmospheric boundary layer is a challenging task. Many physical processes are relevant in the stable boundary layer, i.e. turbulence, radiation, land surface coupling, orographic turbulent and gravity wave drag, and land surface heterogeneity. The development of robust stable boundary layer parameterizations for use in NWP and climate models is hampered by the multiplicity of processes and their unknown interactions. As a result, these models suffer ...
11. Removing Boundary Layer by Suction
Ackeret, J
1927-01-01
Through the utilization of the "Magnus effect" on the Flettner rotor ship, the attention of the public has been directed to the underlying physical principle. It has been found that the Prandtl boundary-layer theory furnishes a satisfactory explanation of the observed phenomena. The present article deals with the prevention of this separation or detachment of the flow by drawing the boundary layer into the inside of a body through a slot or slots in its surface.
12. Stable Boundary Layer Issues
Steeneveld, G.J.
2012-01-01
Understanding and prediction of the stable atmospheric boundary layer is a challenging task. Many physical processes are relevant in the stable boundary layer, i.e. turbulence, radiation, land surface coupling, orographic turbulent and gravity wave drag, and land surface heterogeneity. The
13. Development of boundary layers
Herbst, R.
1980-01-01
Boundary layers develop along the blade surfaces on both the pressure and the suction side in a non-stationary flow field. This is due to the fact that there is a strongly fluctuating flow on the downstream blade row, especially as a result of the wakes of the upstream blade row. The author investigates the formation of boundary layers under non-stationary flow conditions and tries to establish a model describing the non-stationary boundary layer. For this purpose, plate boundary layers are measured, at constant flow rates but different interferent frequency and variable pressure gradients. By introducing the sample technique, measurements of the non-stationary boundary layer become possible, and the flow rate fluctuation can be divided in its components, i.e. stochastic turbulence and periodical fluctuation. (GL) [de
14. Aerodynamics of wings at low Reynolds numbers: Boundary layer separation and reattachment
McArthur, John
Due to advances in electronics technology, it is now possible to build small scale flying and swimming vehicles. These vehicles will have size and velocity scales similar to small birds and fish, and their characteristic Reynolds number will be between 104 and 105. Currently, these flying and swimming vehicles do not perform well, and very little research has been done to characterize them, or to explain why they perform so poorly. This dissertation documents three basic investigations into the performance of small scale lifting surfaces, with Reynolds numbers near 104. Part I. Low Reynolds number aerodynamics. Three airfoil shapes were studied at Reynolds numbers of 1 and 2x104: a flat plate airfoil, a circular arc cambered airfoil, and the Eppler 387 airfoil. Lift and drag force measurements were made on both 2D and 3D conditions, with the 3D wings having an aspect ratio of 6, and the 2D condition being approximated by placing end plates at the wing tips. Comparisons to the limited number of previous measurements show adequate agreement. Previous studies have been inconclusive on whether lifting line theory can be applied to this range of Re, but this study shows that lifting line theory can be applied when there are no sudden changes in the slope of the force curves. This is highly dependent on the airfoil shape of the wing, and explains why previous studies have been inconclusive. Part II. The laminar separation bubble. The Eppler 387 airfoil was studied at two higher Reynolds numbers: 3 and 6x10 4. Previous studies at a Reynolds number of 6x104 had shown this airfoil experiences a drag increase at moderate lift, and a subsequent drag decrease at high lift. Previous studies suggested that the drag increase is caused by a laminar separation bubble, but the experiments used to show this were conducted at higher Reynolds numbers and extrapolated down. Force measurements were combined with flow field measurements at Reynolds numbers 3 and 6x104 to determine whether
15. The Bottom Boundary Layer.
Trowbridge, John H; Lentz, Steven J
2018-01-03
The oceanic bottom boundary layer extracts energy and momentum from the overlying flow, mediates the fate of near-bottom substances, and generates bedforms that retard the flow and affect benthic processes. The bottom boundary layer is forced by winds, waves, tides, and buoyancy and is influenced by surface waves, internal waves, and stratification by heat, salt, and suspended sediments. This review focuses on the coastal ocean. The main points are that (a) classical turbulence concepts and modern turbulence parameterizations provide accurate representations of the structure and turbulent fluxes under conditions in which the underlying assumptions hold, (b) modern sensors and analyses enable high-quality direct or near-direct measurements of the turbulent fluxes and dissipation rates, and (c) the remaining challenges include the interaction of waves and currents with the erodible seabed, the impact of layer-scale two- and three-dimensional instabilities, and the role of the bottom boundary layer in shelf-slope exchange.
16. The Bottom Boundary Layer
Trowbridge, John H.; Lentz, Steven J.
2018-01-01
The oceanic bottom boundary layer extracts energy and momentum from the overlying flow, mediates the fate of near-bottom substances, and generates bedforms that retard the flow and affect benthic processes. The bottom boundary layer is forced by winds, waves, tides, and buoyancy and is influenced by surface waves, internal waves, and stratification by heat, salt, and suspended sediments. This review focuses on the coastal ocean. The main points are that (a) classical turbulence concepts and modern turbulence parameterizations provide accurate representations of the structure and turbulent fluxes under conditions in which the underlying assumptions hold, (b) modern sensors and analyses enable high-quality direct or near-direct measurements of the turbulent fluxes and dissipation rates, and (c) the remaining challenges include the interaction of waves and currents with the erodible seabed, the impact of layer-scale two- and three-dimensional instabilities, and the role of the bottom boundary layer in shelf-slope exchange.
17. Boundary-layer theory
Schlichting (Deceased), Hermann
2017-01-01
This new edition of the near-legendary textbook by Schlichting and revised by Gersten presents a comprehensive overview of boundary-layer theory and its application to all areas of fluid mechanics, with particular emphasis on the flow past bodies (e.g. aircraft aerodynamics). The new edition features an updated reference list and over 100 additional changes throughout the book, reflecting the latest advances on the subject.
18. Superfluid Boundary Layer.
Stagg, G W; Parker, N G; Barenghi, C F
2017-03-31
We model the superfluid flow of liquid helium over the rough surface of a wire (used to experimentally generate turbulence) profiled by atomic force microscopy. Numerical simulations of the Gross-Pitaevskii equation reveal that the sharpest features in the surface induce vortex nucleation both intrinsically (due to the raised local fluid velocity) and extrinsically (providing pinning sites to vortex lines aligned with the flow). Vortex interactions and reconnections contribute to form a dense turbulent layer of vortices with a nonclassical average velocity profile which continually sheds small vortex rings into the bulk. We characterize this layer for various imposed flows. As boundary layers conventionally arise from viscous forces, this result opens up new insight into the nature of superflows.
19. The Atmospheric Boundary Layer
Garratt, J. R.
1994-05-01
A comprehensive and lucid account of the physics and dynamics of the lowest one to two kilometers of the Earth's atmosphere in direct contact with the Earth's surface, known as the atmospheric boundary layer (ABL). Dr. Garratt emphasizes the application of the ABL problems to numerical modeling of the climate, which makes this book unique among recent texts on the subject. He begins with a brief introduction to the ABL before leading to the development of mean and turbulence equations and the many scaling laws and theories that are the cornerstone of any serious ABL treatment. Modeling of the ABL is crucially dependent for its realism on the surface boundary conditions, so chapters four and five deal with aerodynamic and energy considerations, with attention given to both dry and wet land surfaces and the sea. The author next treats the structure of the clear-sky, thermally stratified ABL, including the convective and stable cases over homogeneous land, the marine ABL, and the internal boundary layer at the coastline. Chapter seven then extends this discussion to the cloudy ABL. This is particularly relevant to current research because the extensive stratocumulus regions over the subtropical oceans and stratus regions over the Arctic have been identified as key players in the climate system. In the final chapters, Dr. Garratt summarizes the book's material by discussing appropriate ABL and surface parameterization schemes in general circulation models of the atmosphere that are being used for climate stimulation.
20. Effect of Reynolds Number and Periodic Unsteady Wake Flow Condition on Boundary Layer Development, Separation, and Intermittency Behavior Along the Suction Surface of a Low Pressure Turbine Blade
Schobeiri, M. T.; Ozturk, B.; Ashpis, David E.
2007-01-01
The paper experimentally studies the effects of periodic unsteady wake flow and different Reynolds numbers on boundary layer development, separation and re-attachment along the suction surface of a low pressure turbine blade. The experimental investigations were performed on a large scale, subsonic unsteady turbine cascade research facility at Turbomachinery Performance and Flow Research Laboratory (TPFL) of Texas A&M University. The experiments were carried out at Reynolds numbers of 110,000 and 150,000 (based on suction surface length and exit velocity). One steady and two different unsteady inlet flow conditions with the corresponding passing frequencies, wake velocities, and turbulence intensities were investigated. The reduced frequencies chosen cover the operating range of LP turbines. In addition to the unsteady boundary layer measurements, surface pressure measurements were performed. The inception, onset, and the extent of the separation bubble information collected from the pressure measurements were compared with the hot wire measurements. The results presented in ensemble-averaged, and the contour plot forms help to understand the physics of the separation phenomenon under periodic unsteady wake flow and different Reynolds number. It was found that the suction surface displayed a strong separation bubble for these three different reduced frequencies. For each condition, the locations defining the separation bubble were determined carefully analyzing and examining the pressure and mean velocity profile data. The location of the boundary layer separation was dependent of the Reynolds number. It is observed that starting point of the separation bubble and the re-attachment point move further downstream by increasing Reynolds number from 110,000 to 150,000. Also, the size of the separation bubble is smaller when compared to that for Re=110,000.
1. Numerical and experimental modelling of transition in a separated boundary layer on the NACA63A421 Airfiol
Ďuriš, M.; Popelka, Lukáš; Příhoda, Jaromír; Šimurda, David
2010-01-01
Roč. 56, č. 3 (2010), s. 47-53 ISSN 1210-0471. [International Meeting of Departments of Fluid Mechanics and Thermomechanics held in Rožnov pod Radhoštěm /29./. Rožnov pod Radhoštěm, 23.06.2010-25.06.2010] R&D Projects: GA MŠk(CZ) 1M06031; GA ČR GA103/09/0977 Institutional research plan: CEZ:AV0Z20760514 Keywords : boundary layer transition * CFD * airfoil Subject RIV: BK - Fluid Dynamics
2. The Plasmasphere Boundary Layer
D. L. Carpenter
2004-12-01
Full Text Available As an inner magnetospheric phenomenon the plasmapause region is of interest for a number of reasons, one being the occurrence there of geophysically important interactions between the plasmas of the hot plasma sheet and of the cool plasmasphere. There is a need for a conceptual framework within which to examine and discuss these interactions and their consequences, and we therefore suggest that the plasmapause region be called the Plasmasphere Boundary Layer, or PBL. Such a term has been slow to emerge because of the complexity and variability of the plasma populations that can exist near the plasmapause and because of the variety of criteria used to identify the plasmapause in experimental data. Furthermore, and quite importantly in our view, a substantial obstacle to the consideration of the plasmapause region as a boundary layer has been the longstanding tendency of textbooks on space physics to limit introductory material on the plasmapause phenomenon to zeroth order descriptions in terms of ideal MHD theory, thus implying that the plasmasphere is relatively well understood. A textbook may introduce the concept of shielding of the inner magnetosphere from perturbing convection electric fields, but attention is not usually paid to the variety of physical processes reported to occur in the PBL, such as heating, instabilities, and fast longitudinal flows, processes which must play roles in plasmasphere dynamics in concert with the flow regimes associated with the major dynamo sources of electric fields. We believe that through the use of the PBL concept in future textbook discussions of the plasmasphere and in scientific communications, much progress can be made on longstanding questions about the physics involved in the formation of the plasmapause and in the cycles of erosion and recovery of the plasmasphere.
Key words. Magnetospheric physics (plasmasphere; plasma convection; MHD waves and instabilities
3. The Plasmasphere Boundary Layer
D. L. Carpenter
2004-12-01
Full Text Available As an inner magnetospheric phenomenon the plasmapause region is of interest for a number of reasons, one being the occurrence there of geophysically important interactions between the plasmas of the hot plasma sheet and of the cool plasmasphere. There is a need for a conceptual framework within which to examine and discuss these interactions and their consequences, and we therefore suggest that the plasmapause region be called the Plasmasphere Boundary Layer, or PBL. Such a term has been slow to emerge because of the complexity and variability of the plasma populations that can exist near the plasmapause and because of the variety of criteria used to identify the plasmapause in experimental data. Furthermore, and quite importantly in our view, a substantial obstacle to the consideration of the plasmapause region as a boundary layer has been the longstanding tendency of textbooks on space physics to limit introductory material on the plasmapause phenomenon to zeroth order descriptions in terms of ideal MHD theory, thus implying that the plasmasphere is relatively well understood. A textbook may introduce the concept of shielding of the inner magnetosphere from perturbing convection electric fields, but attention is not usually paid to the variety of physical processes reported to occur in the PBL, such as heating, instabilities, and fast longitudinal flows, processes which must play roles in plasmasphere dynamics in concert with the flow regimes associated with the major dynamo sources of electric fields. We believe that through the use of the PBL concept in future textbook discussions of the plasmasphere and in scientific communications, much progress can be made on longstanding questions about the physics involved in the formation of the plasmapause and in the cycles of erosion and recovery of the plasmasphere. Key words. Magnetospheric physics (plasmasphere; plasma convection; MHD waves and instabilities
4. Analysis of turbulent boundary layers
Cebeci, Tuncer
1974-01-01
Analysis of Turbulent Boundary Layers focuses on turbulent flows meeting the requirements for the boundary-layer or thin-shear-layer approximations. Its approach is devising relatively fundamental, and often subtle, empirical engineering correlations, which are then introduced into various forms of describing equations for final solution. After introducing the topic on turbulence, the book examines the conservation equations for compressible turbulent flows, boundary-layer equations, and general behavior of turbulent boundary layers. The latter chapters describe the CS method for calculati
5. Stochastic rocket dynamics under random nozzle side loads: Ornstein-Uhlenbeck boundary layer separation and its coarse grained connection to side loading and rocket response
Keanini, R.G.; Srivastava, N.; Tkacik, P.T. [Department of Mechanical Engineering, University of North Carolina at Charlotte, 9201 University City Blvd., Charlotte, NC 28078 (United States); Weggel, D.C. [Department of Civil and Environmental Engineering, University of North Carolina at Charlotte, 9201 University City Blvd., Charlotte, NC 28078 (United States); Knight, P.D. [Mitchell Aerospace and Engineering, Statesville, North Carolina 28677 (United States)
2011-06-15
A long-standing, though ill-understood problem in rocket dynamics, rocket response to random, altitude-dependent nozzle side-loads, is investigated. Side loads arise during low altitude flight due to random, asymmetric, shock-induced separation of in-nozzle boundary layers. In this paper, stochastic evolution of the in-nozzle boundary layer separation line, an essential feature underlying side load generation, is connected to random, altitude-dependent rotational and translational rocket response via a set of simple analytical models. Separation line motion, extant on a fast boundary layer time scale, is modeled as an Ornstein-Uhlenbeck process. Pitch and yaw responses, taking place on a long, rocket dynamics time scale, are shown to likewise evolve as OU processes. Stochastic, altitude-dependent rocket translational motion follows from linear, asymptotic versions of the full nonlinear equations of motion; the model is valid in the practical limit where random pitch, yaw, and roll rates all remain small. Computed altitude-dependent rotational and translational velocity and displacement statistics are compared against those obtained using recently reported high fidelity simulations [Srivastava, Tkacik, and Keanini, J. Appl. Phys. 108, 044911 (2010)]; in every case, reasonable agreement is observed. As an important prelude, evidence indicating the physical consistency of the model introduced in the above article is first presented: it is shown that the study's separation line model allows direct derivation of experimentally observed side load amplitude and direction densities. Finally, it is found that the analytical models proposed in this paper allow straightforward identification of practical approaches for: (i) reducing pitch/yaw response to side loads, and (ii) enhancing pitch/yaw damping once side loads cease. (Copyright copyright 2011 WILEY-VCH Verlag GmbH and Co. KGaA, Weinheim)
6. Boundary-Layer & health
Costigliola, V.
2010-09-01
It has long been known that specific atmospheric processes, such as weather and longer-term climatic fluctuations, affect human health. The biometeorological literature refers to this relationship as meteorotropism, defined as a change in an organism that is correlated with a change in atmospheric conditions. Plenty of (patho)physiological functions are affected by those conditions - like the respiratory diseases - and currently it is difficult to put any limits for pathologies developed in reply. Nowadays the importance of atmospheric boundary layer and health is increasingly recognised. A number of epidemiologic studies have reported associations between ambient concentrations of air pollution, specifically particulate pollution, and adverse health effects, even at the relatively low concentrations of pollution found. Since 1995 there have been over twenty-one studies from four continents that have explicitly examined the association between ambient air pollutant mixes and daily mortality. Statistically significant and positive associations have been reported in data from various locations around the world, all with varying air pollutant concentrations, weather conditions, population characteristics and public health policies. Particular role has been given to atmospheric boundary layer processes, the impact of which for specific patient-cohort is, however, not well understood till now. Assessing and monitoring air quality are thus fundamental to improve Europe's welfare. One of current projects run by the "European Medical Association" - PASODOBLE will develop and demonstrate user-driven downstream information services for the regional and local air quality sectors by combining space-based and in-situ data with models in 4 thematic service lines: - Health community support for hospitals, pharmacies, doctors and people at risk - Public information for regions, cities, tourist industry and sporting event organizers - Compliance monitoring support on particulate
7. Numerical and Experimental Modelling of Transition in a Separated Boundary Layer on the NACA63A421 Airfoil
Miroslav ĎURIŠ
2010-12-01
Full Text Available The article deals with design and numerical calculation of a variable test section for small supersonic wind tunnel. The supersonic wind tunnel is designed to be driven by a supersonic ejector. The test section, which is in focus, is considered to by placed on its suction inlet. Schlieren method will be used to investigate the flow within. The purpose of the test section is to demonstrate effects, which occur in supersonic flows, e.g. shock waves, interactions of shock waves with boundary layers etc. Proper demonstration of such phenomenon requires different conditions gained within test section. Internal parts of the device are designed to be interchangeable or variable to provide this capability. The work deals with investigation and design of construction of the variable test section. Consequently, shape of the supersonic inlet nozzles for chosen Mach numbers are carried out. Methods of characteristics and CFD are employed to manage this task. The construction of the test section and obtained numerical results are presented.
8. Sublayer of Prandtl Boundary Layers
Grenier, Emmanuel; Nguyen, Toan T.
2018-03-01
The aim of this paper is to investigate the stability of Prandtl boundary layers in the vanishing viscosity limit {ν \\to 0} . In Grenier (Commun Pure Appl Math 53(9):1067-1091, 2000), one of the authors proved that there exists no asymptotic expansion involving one of Prandtl's boundary layer, with thickness of order {√{ν}} , which describes the inviscid limit of Navier-Stokes equations. The instability gives rise to a viscous boundary sublayer whose thickness is of order {ν^{3/4}} . In this paper, we point out how the stability of the classical Prandtl's layer is linked to the stability of this sublayer. In particular, we prove that the two layers cannot both be nonlinearly stable in L^∞. That is, either the Prandtl's layer or the boundary sublayer is nonlinearly unstable in the sup norm.
9. Boundary Layer Control on Airfoils.
Gerhab, George; Eastlake, Charles
1991-01-01
A phenomena, boundary layer control (BLC), produced when visualizing the fluidlike flow of air is described. The use of BLC in modifying aerodynamic characteristics of airfoils, race cars, and boats is discussed. (KR)
10. The laminar boundary layer equations
Curle, N
2017-01-01
Thorough introduction to boundary layer problems offers an ordered, logical presentation accessible to undergraduates. The text's careful expositions of the limitations and accuracy of various methods will also benefit professionals. 1962 edition.
11. Tokamak plasma boundary layer model
Volkov, T.F.; Kirillov, V.D.
1983-01-01
A model has been developed for the limiter layer and for the boundary region of the plasma column in a tokamak to facilitate analytic calculations of the thickness of the limiter layers, the profiles and boundary values of the temperature and the density under various conditions, and the difference between the electron and ion temperatures. This model can also be used to analyze the recycling of neutrals, the energy and particle losses to the wall and the limiter, and other characteristics
12. Reducing the Drag and Damage of a High-Speed Train by Analyzing and Optimizing its Boundary Layer Separation and Roll-up into Wake Vortices
Jiang, Chung-Hsiang; Marcus, Philip
2012-11-01
We present numerical calculations of the boundary layers and shed wake vortices behind several aerodynamic bodies and generic models of high-speed trains. Our calculations illustrate new visual diagnostics that we developed that clearly show where the separation of a boundary layer occurs and where, how, and with what angles (with respect to the stream-wise direction) the wake vortices form. The calculations also illustrate novel 3D morphing and mesh pushing and pulling'' techniques that allow us to change the shapes of aerodynamic bodies and models in a controlled and automated manner without spurious features appearing. Using these tools we have examined the patterns of the shed vortices behind generic bodies and trains and correlated them with the changes in the drag as well as with the effects of the shed vortices on the environment. In particular, we have applied these techniques to the end car of a next-generation, high-speed train in order to minimize the drag and to minimize the adverse effects of the shed vortices on the track ballast.
13. Visualization of boundary layer separation and passive flow control on airfoils and bodies in wind-tunnel and in-flight experiments
Matejka Milan
2012-04-01
Full Text Available Infrared camera, Particle Image Velocimetry, smoke-wire, tuft filaments and oil-flow visualization techniques were used for wind-tunnel and in-flight investigation of boundary layer separation, both stall and separation bubbles, related to the low-Reynolds numbers transition mechanism. Airfoils of Wortmann FX66 series and FX66 series wing-fuselage interaction, as well as modern airfoils and their wing-fuselage geometry were subject to study. The presence of previously identified structures in the CFD modelling, such as horse-shoe vortices, was confirmed in the flow. Wind-tunnels and in-flight measurements on sailplanes were carried out and effect of passive flow control devices - vortex generators - was surveyed; namely counter-rotating vortex generators and Zig-zag type turbulators were applied. Separation suppression and consequent drag coefficient reduction of test aircrafts was reached. PIV investigation was further extended by Time-Resolved techniques. An important study on structure of the turbulent flow in the lower atmosphere, creating an environment of the soaring flight, was presented.
14. Visualization of boundary layer separation and passive flow control on airfoils and bodies in wind-tunnel and in-flight experiments
Popelka, Lukas; Kuklova, Jana; Simurda, David; Souckova, Natalie; Matejka, Milan; Uruba, Vaclav
2012-04-01
Infrared camera, Particle Image Velocimetry, smoke-wire, tuft filaments and oil-flow visualization techniques were used for wind-tunnel and in-flight investigation of boundary layer separation, both stall and separation bubbles, related to the low-Reynolds numbers transition mechanism. Airfoils of Wortmann FX66 series and FX66 series wing-fuselage interaction, as well as modern airfoils and their wing-fuselage geometry were subject to study. The presence of previously identified structures in the CFD modelling, such as horse-shoe vortices, was confirmed in the flow. Wind-tunnels and in-flight measurements on sailplanes were carried out and effect of passive flow control devices - vortex generators - was surveyed; namely counter-rotating vortex generators and Zig-zag type turbulators were applied. Separation suppression and consequent drag coefficient reduction of test aircrafts was reached. PIV investigation was further extended by Time-Resolved techniques. An important study on structure of the turbulent flow in the lower atmosphere, creating an environment of the soaring flight, was presented.
15. Asymptotic analysis and boundary layers
Cousteix, Jean
2007-01-01
This book presents a new method of asymptotic analysis of boundary-layer problems, the Successive Complementary Expansion Method (SCEM). The first part is devoted to a general comprehensive presentation of the tools of asymptotic analysis. It gives the keys to understand a boundary-layer problem and explains the methods to construct an approximation. The second part is devoted to SCEM and its applications in fluid mechanics, including external and internal flows. The advantages of SCEM are discussed in comparison with the standard Method of Matched Asymptotic Expansions. In particular, for the first time, the theory of Interactive Boundary Layer is fully justified. With its chapter summaries, detailed derivations of results, discussed examples and fully worked out problems and solutions, the book is self-contained. It is written on a mathematical level accessible to graduate and post-graduate students of engineering and physics with a good knowledge in fluid mechanics. Researchers and practitioners will estee...
16. Large field SPIV with separated sheets in a spanwise plane of a turbulent boundary layer with vortex generators
Foucaut, J.M.; Coudert, S.; Braud, C.
2012-01-01
The Stereoscopic PIV is nowadays a well established measurement technique for turbulent flows. However, the accuracy and the spatial resolution are still highly questionable in presence of complex flow with both strong gradients and out of plane motions. To give guidelines for both setup and meas......The Stereoscopic PIV is nowadays a well established measurement technique for turbulent flows. However, the accuracy and the spatial resolution are still highly questionable in presence of complex flow with both strong gradients and out of plane motions. To give guidelines for both setup...... of measurements by increasing the velocity dynamic range especially. It also presents the enhancement of accuracy due to the light sheets separation for characterizing streamwise vortices (i.e. perpendicular to the sheet). The present experiment was performed in the Laboratoire de Mécanique de Lille wind tunnel...
17. Magnetohydrodynamic boundary layer on a wedge
Rao, B.N.; Mittal, M.L.
1981-01-01
The effects of the Hall and ionslip currents on the gas-dynamic boundary layer are investigated in view of the increasing prospects for using the MHD principle in electric power generation. The currents are included in the analysis using the generalized Ohm's law (Sherman and Sutton, 1964), and the resulting two nonlinear coupled equations are solved using a modification in the method suggested by Nachtsheim and Swigert (1965), Dewey and Gross (1967), and Steinheuer (1968). Solutions are presented for the incompressible laminar boundary-layer equations in the absence and the presence of the load parameter, and for the pressure gradient parameter for flow separation
18. Modeling the summertime Arctic cloudy boundary layer
Curry, J.A.; Pinto, J.O. [Univ. of Colorado, Boulder, CO (United States); McInnes, K.L. [CSIRO Division of Atmospheric Research, Mordialloc (Australia)
1996-04-01
Global climate models have particular difficulty in simulating the low-level clouds during the Arctic summer. Model problems are exacerbated in the polar regions by the complicated vertical structure of the Arctic boundary layer. The presence of multiple cloud layers, a humidity inversion above cloud top, and vertical fluxes in the cloud that are decoupled from the surface fluxes, identified in Curry et al. (1988), suggest that models containing sophisticated physical parameterizations would be required to accurately model this region. Accurate modeling of the vertical structure of multiple cloud layers in climate models is important for determination of the surface radiative fluxes. This study focuses on the problem of modeling the layered structure of the Arctic summertime boundary-layer clouds and in particular, the representation of the more complex boundary layer type consisting of a stable foggy surface layer surmounted by a cloud-topped mixed layer. A hierarchical modeling/diagnosis approach is used. A case study from the summertime Arctic Stratus Experiment is examined. A high-resolution, one-dimensional model of turbulence and radiation is tested against the observations and is then used in sensitivity studies to infer the optimal conditions for maintaining two separate layers in the Arctic summertime boundary layer. A three-dimensional mesoscale atmospheric model is then used to simulate the interaction of this cloud deck with the large-scale atmospheric dynamics. An assessment of the improvements needed to the parameterizations of the boundary layer, cloud microphysics, and radiation in the 3-D model is made.
19. Benthic boundary layer modelling studies
Richards, K.J.
1984-01-01
A numerical model has been developed to study the factors which control the height of the benthic boundary layer in the deep ocean and the dispersion of a tracer within and directly above the layer. This report covers tracer clouds of horizontal scales of 10 to 100 km. The dispersion of a tracer has been studied in two ways. Firstly, a number of particles have been introduced into the flow. The trajectories of these particles provide information on dispersion rates. For flow conditions similar to those observed in the abyssal N.E. Atlantic the diffusivity of a tracer was found to be 5 x 10 6 cm 2 s -1 for a tracer within the boundary layer and 8 x 10 6 cm 2 s -1 for a tracer above the boundary layer. The results are in accord with estimates made from current meter measurements. The second method of studying dispersion was to calculate the evolution of individual tracer clouds. Clouds within and above the benthic boundary layer often show quite different behaviour from each other although the general structure of the clouds in the two regions were found to have no significant differences. (author)
20. Diamagnetic boundary layers: a kinetic theory
Lemaire, J.; Burlaga, L.F.
1976-01-01
A kinetic theory for boundary layers associated with MHD tangential 'discontinuities' in a collisionless magnetized plasma such as those observed in the solar wind is presented. The theory consists of finding self-consistent solutions of Vlasov's equation and Maxwell's equation for stationary, one-dimensional boundary layers separating two Maxwellian plasma states. Layers in which the current is carried by electrons are found to have a thickness of the order of a few electron gyroradii, but the drift speed of the current-carrying electrons is found to exceed the Alfven speed, and accordingly such layers are not stable. Several types of layers, in which the current is carried by protons are discussed; in particular, cases in which the magnetic field intensity and/or direction changed across the layer were considered. In every case, the thickness was of the order of a few proton gyroradii and the field changed smoothly , although the characteristics depended somewhat on the boundary conditions. The drift speed was always less than the Alfven speed, consistent with stability of such structures. The results are consistent with the observations of boundary layers in the solar wind near 1 AU. (Auth.)
1. Exploring the magnetospheric boundary layer
Hapgood, M.A.; Bryant, D.A.
1992-01-01
We show how, for most crossings of the boundary layer, one can construct a 'transition parameter', based on electron density and temperature, which orders independent plasma measurements into well-defined patterns which are consistent from case to case. We conclude that there is a gradual change in the balance of processes which determine the structure of the layer and suggest that there is no advantage in dividing the layer into different regions. We further conclude that the mixing processes in layer act in an organised way to give the consistent patterns revealed by the transition parameter. More active processes must sometimes take to give the extreme values (e.g. in velocity) which are seen in some crossings
2. RACORO Continental Boundary Layer Cloud Investigations: 3. Separation of Parameterization Biases in Single-Column Model CAM5 Simulations of Shallow Cumulus
Lin, Wuyin; Liu, Yangang; Vogelmann, Andrew M.; Fridlind, Ann; Endo, Satoshi; Song, Hua; Feng, Sha; Toto, Tami; Li, Zhijin; Zhang, Minghua
2015-01-01
Climatically important low-level clouds are commonly misrepresented in climate models. The FAst-physics System TEstbed and Research (FASTER) Project has constructed case studies from the Atmospheric Radiation Measurement Climate Research Facility's Southern Great Plain site during the RACORO aircraft campaign to facilitate research on model representation of boundary-layer clouds. This paper focuses on using the single-column Community Atmosphere Model version 5 (SCAM5) simulations of a multi-day continental shallow cumulus case to identify specific parameterization causes of low-cloud biases. Consistent model biases among the simulations driven by a set of alternative forcings suggest that uncertainty in the forcing plays only a relatively minor role. In-depth analysis reveals that the model's shallow cumulus convection scheme tends to significantly under-produce clouds during the times when shallow cumuli exist in the observations, while the deep convective and stratiform cloud schemes significantly over-produce low-level clouds throughout the day. The links between model biases and the underlying assumptions of the shallow cumulus scheme are further diagnosed with the aid of large-eddy simulations and aircraft measurements, and by suppressing the triggering of the deep convection scheme. It is found that the weak boundary layer turbulence simulated is directly responsible for the weak cumulus activity and the simulated boundary layer stratiform clouds. Increased vertical and temporal resolutions are shown to lead to stronger boundary layer turbulence and reduction of low-cloud biases.
3. Review: the atmospheric boundary layer
Garratt, J. R.
1994-10-01
An overview is given of the atmospheric boundary layer (ABL) over both continental and ocean surfaces, mainly from observational and modelling perspectives. Much is known about ABL structure over homogeneous land surfaces, but relatively little so far as the following are concerned, (i) the cloud-topped ABL (over the sea predominantly); (ii) the strongly nonhomogeneous and nonstationary ABL; (iii) the ABL over complex terrain. These three categories present exciting challenges so far as improved understanding of ABL behaviour and improved representation of the ABL in numerical models of the atmosphere are concerned.
4. A global boundary-layer height climatology
Dop, H. van; Krol, M.; Holtslag, B. [Inst. for Marine and Atmospheric Research Utrecht, IMAU, Utrecht (Netherlands)
1997-10-01
In principle the ABL (atmospheric boundary layer) height can be retrieved from atmospheric global circulation models since they contain algorithms which determine the intensity of the turbulence as a function of height. However, these data are not routinely available, or on a (vertical) resolution which is too crude in view of the application. This justifies the development of a separate algorithm in order to define the ABL. The algorithm should include the generation of turbulence by both shear and buoyancy and should be based on readily available atmospheric parameters. There is obviously a wide application for boundary heights in off-line global and regional chemistry and transport modelling. It is also a much used parameter in air pollution meteorology. In this article we shall present a theory which is based on current insights in ABL dynamics. The theory is applicable over land and sea surfaces in all seasons. The theory is (for various reasons) not valid in mountainous areas. In areas where boundary-layer clouds or deep cumulus convection are present the theory does not apply. However, the same global atmospheric circulation models contain parameterizations for shallow and deep convection from which separate estimates can be obtained for the extent of vertical mixing. (au)
5. Pitot-probe displacement in a supersonic turbulent boundary layer
Allen, J. M.
1972-01-01
Eight circular pitot probes ranging in size from 2 to 70 percent of the boundary-layer thickness were tested to provide experimental probe displacement results in a two-dimensional turbulent boundary layer at a nominal free-stream Mach number of 2 and unit Reynolds number of 8 million per meter. The displacement obtained in the study was larger than that reported by previous investigators in either an incompressible turbulent boundary layer or a supersonic laminar boundary layer. The large probes indicated distorted Mach number profiles, probably due to separation. When the probes were small enough to cause no appreciable distortion, the displacement was constant over most of the boundary layer. The displacement in the near-wall region decreased to negative displacement in some cases. This near-wall region was found to extend to about one probe diameter from the test surface.
6. Characteristics of the magnetospheric boundary layer and magnetopause layer as observed by Imp 6
Eastman, T.E.; Hones, E.W. Jr.
1979-01-01
Imp 6 observations of the low-latitude magnetospheric boundary layer indicate that the plasma within it is supplied primarily by direct entry of magnetosheath plasma across the magnetopause layer. We define the magnetopause layer as the current layer (separating the magnetosheath from the boundary layer) through which the magnetic field shifts in direction. High temporal resolution (3-s average) data reveal that in a majority of Imp 6 magnetopause crossing, no distinct changes in electron density or energry spectra are observed at the magne opause layer. In all Imp 6 crossings, some magnetosheathlike plasma is observed earthward of the magnetopause layer, implying the existence of a boundary layer. Boundary layer electron energy spectra are often virtually indistinguishable from the adjacent magnetosheath spectra. Low-latitude boundary layer bulk plasma flow as observed by Imp 6 almost always has an antisunward component and often has a significant cross-field component. The boundary layer thickness is highly variable and is generally much larger than the magnetopause layer thickness. Energetic electron pitch angle distributions indicate that the low-latitude boundary layers is normally on closed field lines. We conclude that diffusive as well as nondiffusive processes probably contribute to the entry of magnetosheath plasma into the boundary layer
7. Experimental investigation of wave boundary layer
Sumer, B. Mutlu
2003-01-01
A review is presented of experimental investigation of wave boundary layer. The review is organized in six main sections. The first section describes the wave boundary layer in a real-life environment and its simulation in the laboratory in an oscillating water tunnel and in a water tank...... with an oscillating seabed. A brief account is given of measured quantities, measurement techniques (LDA, PIV, flow visualization) and limitations/constraints in the experimental investigation of the wave boundary layer in the laboratory. The second section concentrates on uniform oscillating boundary layers...
8. Numerical Simulations of Hypersonic Boundary Layer Transition
Bartkowicz, Matthew David
Numerical schemes for supersonic flows tend to use large amounts of artificial viscosity for stability. This tends to damp out the small scale structures in the flow. Recently some low-dissipation methods have been proposed which selectively eliminate the artificial viscosity in regions which do not require it. This work builds upon the low-dissipation method of Subbareddy and Candler which uses the flux vector splitting method of Steger and Warming but identifies the dissipation portion to eliminate it. Computing accurate fluxes typically relies on large grid stencils or coupled linear systems that become computationally expensive to solve. Unstructured grids allow for CFD solutions to be obtained on complex geometries, unfortunately, it then becomes difficult to create a large stencil or the coupled linear system. Accurate solutions require grids that quickly become too large to be feasible. In this thesis a method is proposed to obtain more accurate solutions using relatively local data, making it suitable for unstructured grids composed of hexahedral elements. Fluxes are reconstructed using local gradients to extend the range of data used. The method is then validated on several test problems. Simulations of boundary layer transition are then performed. An elliptic cone at Mach 8 is simulated based on an experiment at the Princeton Gasdynamics Laboratory. A simulated acoustic noise boundary condition is imposed to model the noisy conditions of the wind tunnel and the transitioning boundary layer observed. A computation of an isolated roughness element is done based on an experiment in Purdue's Mach 6 quiet wind tunnel. The mechanism for transition is identified as an instability in the upstream separation region and a comparison is made to experimental data. In the CFD a fully turbulent boundary layer is observed downstream.
9. Boundary layers of the earth's outer magnetosphere
Eastman, T. E.; Frank, L. A.
1984-01-01
The magnetospheric boundary layer and the plasma-sheet boundary layer are the primary boundary layers of the earth's outer magnetosphere. Recent satellite observations indicate that they provide for more than 50 percent of the plasma and energy transport in the outer magnetosphere although they constitute less than 5 percent by volume. Relative to the energy density in the source regions, plasma in the magnetospheric boundary layer is predominantly deenergized whereas plasma in the plasma-sheet boundary layer has been accelerated. The reconnection hypothesis continues to provide a useful framework for comparing data sampled in the highly dynamic magnetospheric environment. Observations of 'flux transfer events' and other detailed features near the boundaries have been recently interpreted in terms of nonsteady-state reconnection. Alternative hypotheses are also being investigated. More work needs to be done, both in theory and observation, to determine whether reconnection actually occurs in the magnetosphere and, if so, whether it is important for overall magnetospheric dynamics.
10. Boundary layers of the earth's outer magnetosphere
Eastman, T.E.; Frank, L.A.
1984-01-01
The magnetospheric boundary layer and the plasma-sheet boundary layer are the primary boundary layers of the earth's outer magnetosphere. Recent satellite observations indicate that they provide for more than 50 percent of the plasma and energy transport in the outer magnetosphere although they constitute less than 5 percent by volume. Relative to the energy density in the source regions, plasma in the magnetospheric boundary layer is predominantly deenergized whereas plasma in the plasma-sheet boundary layer has been accelerated. The reconnection hypothesis continues to provide a useful framework for comparing data sampled in the highly dynamic magnetospheric environment. Observations of flux transfer events and other detailed features near the boundaries have been recently interpreted in terms of nonsteady-state reconnection. Alternative hypotheses are also being investigated. More work needs to be done, both in theory and observation, to determine whether reconnection actually occurs in the magnetosphere and, if so, whether it is important for overall magnetospheric dynamics. 30 references
11. Modelling stable atmospheric boundary layers over snow
Sterk, H.A.M.
2015-01-01
Thesis entitled:
Modelling Stable Atmospheric Boundary Layers over Snow
H.A.M. Sterk
Wageningen, 29th of April, 2015
Summary
The emphasis of this thesis is on the understanding and forecasting of the Stable Boundary Layer (SBL) over snow-covered surfaces. SBLs
12. Suction of MHD boundary layer flows
Rao, B.N.
1985-01-01
The boundary layer growth with tensor electrical conductivity and the transpiration number has been examined using local nonsimilarity solutions method. It is found that suction will cause the increase in wall shearing stress and decrease in thicknesses of the boundary layer. (Auth.)
13. Bristled shark skin: a microgeometry for boundary layer control?
Lang, A W; Hidalgo, P; Westcott, M; Motta, P
2008-01-01
There exists evidence that some fast-swimming shark species may have the ability to bristle their scales during fast swimming. Experimental work using a water tunnel facility has been performed to investigate the flow field over and within a bristled shark skin model submerged within a boundary layer to deduce the possible boundary layer control mechanisms being used by these fast-swimming sharks. Fluorescent dye flow visualization provides evidence of the formation of embedded cavity vortices within the scales. Digital particle image velocimetry (DPIV) data, used to evaluate the cavity vortex formation and boundary layer characteristics close to the surface, indicate increased momentum in the slip layer forming above the scales. This increase in flow velocity close to the shark's skin is indicative of boundary layer control mechanisms leading to separation control and possibly transition delay for the bristled shark skin microgeometry
14. Bristled shark skin: a microgeometry for boundary layer control?
Lang, A W; Hidalgo, P; Westcott, M [Aerospace Engineering and Mechanics Department, University of Alabama, Box 870280, Tuscaloosa, AL 35487 (United States); Motta, P [Biology Department, University of South Florida, 4202 East Fowler Avenue, Tampa, FL 33620 (United States)], E-mail: [email protected]
2008-12-01
There exists evidence that some fast-swimming shark species may have the ability to bristle their scales during fast swimming. Experimental work using a water tunnel facility has been performed to investigate the flow field over and within a bristled shark skin model submerged within a boundary layer to deduce the possible boundary layer control mechanisms being used by these fast-swimming sharks. Fluorescent dye flow visualization provides evidence of the formation of embedded cavity vortices within the scales. Digital particle image velocimetry (DPIV) data, used to evaluate the cavity vortex formation and boundary layer characteristics close to the surface, indicate increased momentum in the slip layer forming above the scales. This increase in flow velocity close to the shark's skin is indicative of boundary layer control mechanisms leading to separation control and possibly transition delay for the bristled shark skin microgeometry.
15. Structure of the low-latitude boundary layer
Sckopke, N.; Paschmann, G.; Haerendel, G.; Sonnerup, B.U.O.; Bame, S.J.; Forbes, T.G.; Hones, E.W. Jr.; Russell, C.T.
1981-01-01
Observations at high temporal resolution of the frontside magnetopause and plasma boundary layer, made with the Los Alamos Scientific Laboratory/Max-Planck-Institut, Institut fuer Extraterrestrische Physik, fast plasma analyzer on board the Isee 1 and 2 spacecraft, have revealed a complex quasi-periodic structure of some of the observed boundary layers: cool tailward streaming boundary layer plasma is seen intermittently, with intervening periods of hot tenuous plasma which has properties similar to the magnetospheric population. While individual encounters with the boundary layer plasma last only a few minutes, the total observation time may extend over 1 hour or more. One such crossing, at 0800 hours local time and 40 0 northern GSM latitude, is examined in detail, including a quantitative comparison of the boundary layer entry and exit times of the two spacecraft. The data are found to be compatible with a boundary layer that is always attached to the magnetopause but where the layer thickness has a large-scale spatial modulation pattern which travels tailward past the spacecraft. Included are periods when the thickness is essentially zero and others when it is of the order of 1 R/sub E/. The duration of these periods is highly variable but is typically in the range of 2--5 min, corresponding to a distance along the magnetopause of the order of 3--8 R/sub E/. The observed boundary layer features include a steep density gradient at the magnetopause, with an approximately constant boundary layer plasma density amounting to about 25% of the magnetosheath density, and a second abrupt density decrease at the inner edge of the layer. It also appears that the purely magnetospheric plasma is ocassionally separated from the boundary layer by a halo region in which the plasma density is somewhat higher, and the temperature somewhat lower, than in the magnetosphere. A tentative model is proposed
16. Contributions of the wall boundary layer to the formation of the counter-rotating vortex pair in transverse jets
SCHLEGEL, FABRICE; WEE, DAEHYUN; MARZOUK, YOUSSEF M.; GHONIEM, AHMED F.
2011-01-01
generation along the channel wall, captures unsteady interactions between the wall boundary layer and the jet - in particular, the separation of the wall boundary layer and its transport into the interior. For comparison, we also implement a reduced boundary
17. Measurements in a synthetic turbulent boundary layer
Arakeri, J. H.; Coles, D. E.
Some measurements in a synthetic turbulent boundary layer (SBL) are reported. The main diagnostic tool is an X-wire probe. The velocity of the large eddies is determined to be 0.842 times the freestream velocity. The mean properties of the SBL are reasonably close to those of a natural turbulent boundary layer. The large eddy in the SBL appears to be a pair of counterrotating eddies in the stream direction, inclined at a shallow angle and occupying much of the boundary-layer thickness.
18. Longitudinal vortices in a transitioning boundary layer
Anders, J.B.; Backwelder, R.F.
1980-01-01
Naturally occurring spanwise variations of the streamwise velocity component, characteristic of longitudinal vortices embedded in a transitioning boundary layer were explored using hot-wire anemometers. A vibrating ribbon introduced stable or unstable Tollmien-Schlichting waves into the laminar boundary layer. These damped or growing disturbances always developed a strong three-dimensional pattern even though no spanwise perturbations were artificially induced. Changing the radius of the leading edge and other modifications to the flat plate, wind tunnel and boundary layer did not alter the spanwise wavelength of the vortices. (orig.)
19. Separable boundary-value problems in physics
Willatzen, Morten
2011-01-01
Innovative developments in science and technology require a thorough knowledge of applied mathematics, particularly in the field of differential equations and special functions. These are relevant in modeling and computing applications of electromagnetic theory and quantum theory, e.g. in photonics and nanotechnology. The problem of solving partial differential equations remains an important topic that is taught at both the undergraduate and graduate level. Separable Boundary-Value Problems in Physics is an accessible and comprehensive treatment of partial differential equations i
20. A variable K - planetary boundary layer model
Misra, P.K.
1976-07-01
The steady-state, homogeneous and barotropic equations of motion within the planetary boundary layer are solved with the assumption that the coefficient of eddy viscosity varies as K(Z) = K 0 (1-Z/h)sup(p), where h is the height of the boundary layer and p a parameter which depends on the atmospheric stability. The solutions are compared with the observed velocity profiles based on the Wangara data. They compare favourably. (author)
1. Comments on Hypersonic Boundary-Layer Transition
1990-09-01
mechanism by which boundary-layer disturbance growth is generally initiated and establishes the initial distur- banca amplitude at the onset of disturbance...Patankar, S. V., and Spalding, P. B., Heat and Mass Transfer in Boundary Lavers, CRC Press , Cleveland, Ohio, 1968. 87. Neumann, R. D., and Patterson, .J. 1
2. Large eddy simulation of atmospheric boundary layer over wind farms using a prescribed boundary layer approach
Chivaee, Hamid Sarlak; Sørensen, Jens Nørkær; Mikkelsen, Robert Flemming
2012-01-01
Large eddy simulation (LES) of flow in a wind farm is studied in neutral as well as thermally stratified atmospheric boundary layer (ABL). An approach has been practiced to simulate the flow in a fully developed wind farm boundary layer. The approach is based on the Immersed Boundary Method (IBM......) and involves implementation of an arbitrary prescribed initial boundary layer (See [1]). A prescribed initial boundary layer profile is enforced through the computational domain using body forces to maintain a desired flow field. The body forces are then stored and applied on the domain through the simulation...... and the boundary layer shape will be modified due to the interaction of the turbine wakes and buoyancy contributions. The implemented method is capable of capturing the most important features of wakes of wind farms [1] while having the advantage of resolving the wall layer with a coarser grid than typically...
3. Numerical simulations of the stratified oceanic bottom boundary layer
Taylor, John R.
Numerical simulations are used to consider several problems relevant to the turbulent oceanic bottom boundary layer. In the first study, stratified open channel flow is considered with thermal boundary conditions chosen to approximate a shallow sea. Specifically, a constant heat flux is applied at the free surface and the lower wall is assumed to be adiabatic. When the surface heat flux is strong, turbulent upwellings of low speed fluid from near the lower wall are inhibited by the stable stratification. Subsequent studies consider a stratified bottom Ekman layer over a non-sloping lower wall. The influence of the free surface is removed by using an open boundary condition at the top of the computational domain. Particular attention is paid to the influence of the outer layer stratification on the boundary layer structure. When the density field is initialized with a linear profile, a turbulent mixed layer forms near the wall, which is separated from the outer layer by a strongly stable pycnocline. It is found that the bottom stress is not strongly affected by the outer layer stratification. However, stratification reduces turbulent transport to the outer layer and strongly limits the boundary layer height. The mean shear at the top of the boundary layer is enhanced when the outer layer is stratified, and this shear is strong enough to cause intermittent instabilities above the pycnocline. Turbulence-generated internal gravity waves are observed in the outer layer with a relatively narrow frequency range. An explanation for frequency content of these waves is proposed, starting with an observed broad-banded turbulent spectrum and invoking linear viscous decay to explain the preferential damping of low and high frequency waves. During the course of this work, an open-source computational fluid dynamics code has been developed with a number of advanced features including scalar advection, subgrid-scale models for large-eddy simulation, and distributed memory
4. Boundary-layer effects in droplet splashing
Riboux, Guillaume; Gordillo, Jose Manuel
2017-11-01
A drop falling onto a solid substrate will disintegrate into smaller parts when its impact velocity exceeds the so called critical velocity for splashing. Under these circumstances, the very thin liquid sheet ejected tangentially to the solid after the drop touches the substrate, lifts off as a consequence of the aerodynamic forces exerted on it and finally breaks into smaller droplets, violently ejected radially outwards, provoking the splash. Here, the tangential deceleration experienced by the fluid entering the thin liquid sheet is investigated making use of boundary layer theory. The velocity component tangent to the solid, computed using potential flow theory provides the far field boundary condition as well as the pressure gradient for the boundary layer equations. The structure of the flow permits to find a self similar solution of the boundary layer equations. This solution is then used to calculate the boundary layer thickness at the root of the lamella as well as the shear stress at the wall. The splash model presented in, which is slightly modified to account for the results obtained from the boundary layer analysis, provides a very good agreement between the measurements and the predicted values of the critical velocity for the splash.
5. Ground observations of magnetospheric boundary layer phenomena
McHenry, M.A.; Clauer, C.R.; Friis-Christensen, E.; Newell, P.T.; Kelly, J.D.
1990-01-01
Several classes of traveling vortices in the dayside ionospheric convection have been detected and tracked using the Greenland magnetometer chain (Friis-Christensen et al., 1988, McHenry et al., 1989). One class observed during quiet times consists of a continuous series of vortices moving generally anti-sunward for several hours at a time. The vortices strength is seen to be approximately steady and neighboring vortices rotate in opposite directions. Sondrestrom radar observations show that the vortices are located at the ionospheric convection reversal boundary. Low altitude DMSP observations indicate the vortices are on field lines which map to the inner edge of the low latitude boundary layer. Because the vortices are conjugate to the boundary layer, repeat in a regular fashion and travel antisunward, the authors argue that this class of vortices is caused by the Kelvin-Helmholtz instability of the inner edge of the magnetospheric boundary layer
6. Stability of spatially developing boundary layers
Govindarajan, Rama
1993-07-01
A new formulation of the stability of boundary-layer flows in pressure gradients is presented, taking into account the spatial development of the flow. The formulation assumes that disturbance wavelength and eigenfunction vary downstream no more rapidly than the boundary-layer thickness, and includes all terms of O(1) and O(R(exp -1)) in the boundary-layer Reynolds number R. Although containing the Orr-Sommerfeld operator, the present approach does not yield the Orr-Sommerfeld equation in any rational limit. In Blasius flow, the present stability equation is consistent with that of Bertolotti et al. (1992) to terms of O(R(exp -1)). For the Falkner-Skan similarity solutions neutral boundaries are computed without the necessity of having to march in space. Results show that the effects of spatial growth are striking in flows subjected to adverse pressure gradients.
7. Hundred years of the boundary layer – Some aspects
2005-08-02
Aug 2, 2005 ... from the wall and separation of the boundary layer, which in turn enables proper ... design, which performed better and consumed only one-third the power .... turbulent flow and also to free shear flows like wakes and jets.
8. On the modeling of electrical boundary layer (electrode layer) and ...
In the first part of the paper, equations and methodology are discussed and in the second, we discuss results. 2. Methodology. In the atmospheric electricity, the earth's surface is one electrode and electrode layer or electrical boundary layer is a region near the surface of the earth in which profiles of atmospheric electrical.
9. Unsteady turbulent boundary layers in swimming rainbow trout.
Yanase, Kazutaka; Saarenrinne, Pentti
2015-05-01
The boundary layers of rainbow trout, Oncorhynchus mykiss, swimming at 1.02±0.09 L s(-1) (mean±s.d., N=4), were measured by the particle image velocimetry (PIV) technique at a Reynolds number of 4×10(5). The boundary layer profile showed unsteadiness, oscillating above and beneath the classical logarithmic law of the wall with body motion. Across the entire surface regions that were measured, local Reynolds numbers based on momentum thickness, which is the distance that is perpendicular to the fish surface through which the boundary layer momentum flows at free-stream velocity, were greater than the critical value of 320 for the laminar-to-turbulent transition. The skin friction was dampened on the convex surface while the surface was moving towards a free-stream flow and increased on the concave surface while retreating. These observations contradict the result of a previous study using different species swimming by different methods. Boundary layer compression accompanied by an increase in local skin friction was not observed. Thus, the overall results may not support absolutely the Bone-Lighthill boundary layer thinning hypothesis that the undulatory motions of swimming fish cause a large increase in their friction drag because of the compression of the boundary layer. In some cases, marginal flow separation occurred on the convex surface in the relatively anterior surface region, but the separated flow reattached to the fish surface immediately downstream. Therefore, we believe that a severe impact due to induced drag components (i.e. pressure drag) on the swimming performance, an inevitable consequence of flow separation, was avoided. © 2015. Published by The Company of Biologists Ltd.
10. Problems of matter-antimatter boundary layers
Lehnert, B.
1975-01-01
This paper outlines the problems of the quasi-steady matter-antimatter boundary layers discussed in Klein-Alfven's cosmological theory, and a crude model of the corresponding ambiplasma balance is presented: (i) at interstellar particle densities, no well-defined boundary layer can exist in presence of neutral gas, nor can such a layer be sustained in an unmagnetized fully ionized ambiplasma. (ii) Within the limits of applicability of the present model, sharply defined boundary layers are under certain conditions found to exist in a magnetized ambiplasma. Thus, at beta values less than unity, a steep pressure drop of the low-energy components of matter and antimatter can be balanced by a magnetic field and the electric currents in the ambiplasma. (iii) The boundary layer thickness is of the order of 2x 0 approximately 10/BT 0 sup(1/4) meters, where B is the magnetic field strength in MKS units and T 0 the characteristic temperature of the low-energy components in the layer. (Auth.)
11. Boundary layer transition observations on a body of revolution with surface heating and cooling in water
Arakeri, V. H.
1980-04-01
Boundary layer flow visualization in water with surface heat transfer was carried out on a body of revolution which had the predicted possibility of laminar separation under isothermal conditions. Flow visualization was by in-line holographic technique. Boundary layer stabilization, including elimination of laminar separation, was observed to take place on surface heating. Conversely, boundary layer destabilization was observed on surface cooling. These findings are consistent with the theoretical predictions of Wazzan et al. (1970).
12. Beta limitation of matter-antimatter boundary layers
Lehnert, B.
1987-08-01
A model has earlier been proposed for a boundary layer which separates a cloud of matter from one of antimatter in a magnetized ambiplasma. In this model steady pressure equilibrium ceases to exist when a certain beta limit is exceeded. The latter is defined as the ratio between the ambiplasma and magnetic field pressures which balance each other in the boundary layer. Thus, at an increasing density, the high-energy particles created by annihilation within the layer are 'pumped up' to a pressure which cannot be balanced by a given magnetic field. The boundary layer then 'disrupts'. The critical beta limit thus obtained falls within the observed parameter ranges of galaxies and other large cosmical objects. Provided that the considered matter-antimatter balance holds true, this limit is thus expected to impose certain existence conditions on matter-antimatter boundary layers. Such a limitation may apply to certain cosmical objects and cosmological models. The maximum time scale for the corresponding disruption development has been estimated to be in the range from about 10 -4 to 10 2 seconds for boundary layers at ambiplasma particle densities in the range from 10 4 to 10 -2 m -3 , respectively. (author)
13. Boundary layer heights derived from velocity spectra
Hoejstrup, J.; Barthelmie, R.J. [Risoe National Lab., Roskilde (Denmark); Kaellstrand, B. [Univ. of Uppsala, Uppsala (Sweden)
1997-10-01
It is a well-known fact that the height of the mixed layer determines the size of the largest and most energetic eddies that can be observed in the unstable boundary layer, and consequently a peak can be observed in the power spectra of the along-wind velocity component at scales comparable to the mixed layer depth. We will now show how the mixed layer depth can be derived from the u-specta and the results will be compared with direct measurements using pibal and tethersonde measurements. (au)
14. Self-similar magnetohydrodynamic boundary layers
2010-10-15
The boundary layer created by parallel flow in a magnetized fluid of high conductivity is considered in this paper. Under appropriate boundary conditions, self-similar solutions analogous to the ones studied by Blasius for the hydrodynamic problem may be found. It is proved that for these to be stable, the size of the Alfven velocity at the outer flow must be smaller than the flow velocity, a fact that has a ready physical explanation. The process by which the transverse velocity and the thickness of the layer grow with the size of the Alfven velocity is detailed.
15. Self-similar magnetohydrodynamic boundary layers
Nunez, Manuel; Lastra, Alberto
2010-01-01
The boundary layer created by parallel flow in a magnetized fluid of high conductivity is considered in this paper. Under appropriate boundary conditions, self-similar solutions analogous to the ones studied by Blasius for the hydrodynamic problem may be found. It is proved that for these to be stable, the size of the Alfven velocity at the outer flow must be smaller than the flow velocity, a fact that has a ready physical explanation. The process by which the transverse velocity and the thickness of the layer grow with the size of the Alfven velocity is detailed.
16. Boundary layer for non-newtonian fluids on curved surfaces
Stenger, N.
1981-04-01
By using the basic equation of fluid motion (conservation of mass and momentum) the boundary layer parameters for a Non-Newtonian, incompressible and laminar fluid flow, has been evaluated. As a test, the flat plate boundary layer is first analized and afterwards, a case with pressure gradient, allowing separation, is studied. In the case of curved surfaces, the problem is first developed in general and afterwards particularized to a circular cylinder. Finally suction and slip in the flow interface are examined. The power law model is used to represent the stress strain relationship in Non-Newtonian flow. By varying the fluid exponent one can then, have an idea of how the Non-Newtonian behavior of the flow influences the parameters of the boundary layer. Two equations, in an appropriate coordinate system have been obtained after an order of magnitude analysis of the terms in the equations of motion is performed. (Author) [pt
17. Discussion of boundary-layer characteristics near the casing of an axial-flow compressor
Mager, Artur; Mahoney, John J; Budinger, Ray E
1951-01-01
Boundary-layer velocity profiles on the casing of an axial-flow compressor behind the guide vanes and rotor were measured and resolved into two components: along the streamline of the flow and perpendicular to it. Boundary-layer thickness and the deflection of the boundary layer at the wall were the generalizing parameters. By use of these results and the momentum-integral equations, the characteristics of boundary on the walls of axial-flow compressor are qualitatively discussed. Important parameters concerning secondary flow in the boundary layer appear to be turning of the flow and the product of boundary-layer thickness and streamline curvature outside the boundary layer. Two types of separation are shown to be possible in three dimensional boundary layer.
18. Diagnosis of boundary-layer circulations.
Beare, Robert J; Cullen, Michael J P
2013-05-28
Diagnoses of circulations in the vertical plane provide valuable insights into aspects of the dynamics of the climate system. Dynamical theories based on geostrophic balance have proved useful in deriving diagnostic equations for these circulations. For example, semi-geostrophic theory gives rise to the Sawyer-Eliassen equation (SEE) that predicts, among other things, circulations around mid-latitude fronts. A limitation of the SEE is the absence of a realistic boundary layer. However, the coupling provided by the boundary layer between the atmosphere and the surface is fundamental to the climate system. Here, we use a theory based on Ekman momentum balance to derive an SEE that includes a boundary layer (SEEBL). We consider a case study of a baroclinic low-level jet. The SEEBL solution shows significant benefits over Ekman pumping, including accommodating a boundary-layer depth that varies in space and structure, which accounts for buoyancy and momentum advection. The diagnosed low-level jet is stronger than that determined by Ekman balance. This is due to the inclusion of momentum advection. Momentum advection provides an additional mechanism for enhancement of the low-level jet that is distinct from inertial oscillations.
19. Diffusion processes in the magnetopause boundary layer
Tsurutani, B.T.; Thorne, R.M.
1982-01-01
Anomalous cross-field diffusion of magnetosheath ions and electrons is a direct consequence of cyclotron-resonant scattering by electrostatic and electromagnetic emissions which are continuously present within the magnetopause boundary layer. Expressions for the rate of cross-field diffusion involving either type of wave are developed and expressed in terms of the absolute upper limit referred to as Bohm diffusion. For the typical average intensity of waves observed in the boundary layer, resonant electron cross-field diffusion is always insignificant. However, magnetosheath ions, resonant with low frequency electrostatic waves, may be transported inward at a rate approaching one tenth the Bohm rate (D/sub perpendiculartsperpendicular/roughly-equal10 3 km 2 /s). While this is not the only mechanism capable of explaining the presence of the low latitude boundary layer it is adequate to account for the typical boundary layer thickness and it should occur at all local times and under all interplanetary conditions. It consequently provides a continuous mechanism for significant mass and momentum transfer across the magnetopause under conditions when field merging is inoperative
20. Nonlinear Transient Growth and Boundary Layer Transition
Paredes, Pedro; Choudhari, Meelan M.; Li, Fei
2016-01-01
Parabolized stability equations (PSE) are used in a variational approach to study the optimal, non-modal disturbance growth in a Mach 3 at plate boundary layer and a Mach 6 circular cone boundary layer. As noted in previous works, the optimal initial disturbances correspond to steady counter-rotating streamwise vortices, which subsequently lead to the formation of streamwise-elongated structures, i.e., streaks, via a lift-up effect. The nonlinear evolution of the linearly optimal stationary perturbations is computed using the nonlinear plane-marching PSE for stationary perturbations. A fully implicit marching technique is used to facilitate the computation of nonlinear streaks with large amplitudes. To assess the effect of the finite-amplitude streaks on transition, the linear form of plane- marching PSE is used to investigate the instability of the boundary layer flow modified by spanwise periodic streaks. The onset of bypass transition is estimated by using an N- factor criterion based on the amplification of the streak instabilities. Results show that, for both flow configurations of interest, streaks of sufficiently large amplitude can lead to significantly earlier onset of transition than that in an unperturbed boundary layer without any streaks.
1. Vortex sheet approximation of boundary layers
Chorin, A.J.
1978-01-01
a grid free method for approximating incomprssible boundary layers is introduced. The computational elements are segments of vortex sheets. The method is related to the earlier vortex method; simplicity is achieved at the cost of replacing the Navier-Stokes equations by the Prandtl boundary layer equations. A new method for generating vorticity at boundaries is also presented; it can be used with the earlier voartex method. The applications presented include (i) flat plate problems, and (ii) a flow problem in a model cylinder- piston assembly, where the new method is used near walls and an improved version of the random choice method is used in the interior. One of the attractive features of the new method is the ease with which it can be incorporated into hybrid algorithms
2. Characterization of the atmospheric boundary layer from radiosonde ...
In this paper, a comparison of two methods for the calculation of the height of atmospheric boundary layer (ABL) ... Boundary layer; GPS sonde; mixed layer height; turbulent flow depth. J. Earth Syst. ..... for her PhD research work. References.
3. Separation-induced boundary layer transition: Modeling with a non-linear eddy-viscosity model coupled with the laminar kinetic energy equation
Vlahostergios, Z.; Yakinthos, K.; Goulas, A.
2009-01-01
We present an effort to model the separation-induced transition on a flat plate with a semi-circular leading edge, using a cubic non-linear eddy-viscosity model combined with the laminar kinetic energy. A non-linear model, compared to a linear one, has the advantage to resolve the anisotropic behavior of the Reynolds-stresses in the near-wall region and it provides a more accurate expression for the generation of turbulence in the transport equation of the turbulence kinetic energy. Although in its original formulation the model is not able to accurately predict the separation-induced transition, the inclusion of the laminar kinetic energy increases its accuracy. The adoption of the laminar kinetic energy by the non-linear model is presented in detail, together with some additional modifications required for the adaption of the laminar kinetic energy into the basic concepts of the non-linear eddy-viscosity model. The computational results using the proposed combined model are shown together with the ones obtained using an isotropic linear eddy-viscosity model, which adopts also the laminar kinetic energy concept and in comparison with the existing experimental data.
4. Hairpin vortices in turbulent boundary layers
Eitel-Amor, G; Schlatter, P; Flores, O
2014-01-01
The present work addresses the question whether hairpin vortices are a dominant feature of near-wall turbulence and which role they play during transition. First, the parent-offspring mechanism is investigated in temporal simulations of a single hairpin vortex introduced in a mean shear flow corresponding to turbulent channels and boundary layers up to Re τ = 590. Using an eddy viscosity computed from resolved simulations, the effect of a turbulent background is also considered. Tracking the vortical structure downstream, it is found that secondary hairpins are created shortly after initialization. Thereafter, all rotational structures decay, whereas this effect is enforced in the presence of an eddy viscosity. In a second approach, a laminar boundary layer is tripped to transition by insertion of a regular pattern of hairpins by means of defined volumetric forces representing an ejection event. The idea is to create a synthetic turbulent boundary layer dominated by hairpin-like vortices. The flow for Re τ < 250 is analysed with respect to the lifetime of individual hairpin-like vortices. Both the temporal and spatial simulations demonstrate that the regeneration process is rather short-lived and may not sustain once a turbulent background has formed. From the transitional flow simulations, it is conjectured that the forest of hairpins reported in former DNS studies is an outer layer phenomenon not being connected to the onset of near-wall turbulence.
5. Boundary-layer height detection with a ceilometer at a coastal site in western Denmark
Hannesdóttir, Ásta; Hansen, Aksel Walle
in atmospheric transport- and dispersion models. A new method of filtering clouds from the ceilometer data is presented. This allows for the inclusion of more than half of the data in the subsequent analysis, as the presence of clouds would otherwise complicate the boundary-layer height estimations. The boundary....... The boundary-layer height estimates are then used to analyse the daily evolution of the boundary layer and to perform monthly and annual frequency distributions of the boundary-layer height. For westerly winds bi-modal distributions are often found, which may be separated by different criteria, while...
6. BOREAS AFM-6 Boundary Layer Height Data
Wilczak, James; Hall, Forrest G. (Editor); Newcomer, Jeffrey A. (Editor); Smith, David E. (Technical Monitor)
2000-01-01
The Boreal Ecosystem-Atmosphere Study (BOREAS) Airborne Fluxes and Meteorology (AFM)-6 team from National Oceanic and Atmospheric Adminsitration/Environment Technology Laboratory (NOAA/ETL) operated a 915-MHz wind/Radio Acoustic Sounding System (RASS) profiler system in the Southern Study Area (SSA) near the Old Jack Pine (OJP) site. This data set provides boundary layer height information over the site. The data were collected from 21 May 1994 to 20 Sep 1994 and are stored in tabular ASCII files. The boundary layer height data are available from the Earth Observing System Data and Information System (EOSDIS) Oak Ridge National Laboratory (ORNL) Distributed Active Archive Center (DAAC). The data files are available on a CD-ROM (see document number 20010000884).
7. Turbulent Helicity in the Atmospheric Boundary Layer
Chkhetiani, Otto G.; Kurgansky, Michael V.; Vazaeva, Natalia V.
2018-05-01
We consider the assumption postulated by Deusebio and Lindborg (J Fluid Mech 755:654-671, 2014) that the helicity injected into the Ekman boundary layer undergoes a cascade, with preservation of its sign (right- or alternatively left-handedness), which is a signature of the system rotation, from large to small scales, down to the Kolmogorov microscale of turbulence. At the same time, recent direct field measurements of turbulent helicity in the steppe region of southern Russia near Tsimlyansk Reservoir show the opposite sign of helicity from that expected. A possible explanation for this phenomenon may be the joint action of different scales of atmospheric flows within the boundary layer, including the sea-breeze circulation over the test site. In this regard, we consider a superposition of the classic Ekman spiral solution and Prandtl's jet-like slope-wind profile to describe the planetary boundary-layer wind structure. The latter solution mimics a hydrostatic shallow breeze circulation over a non-uniformly heated surface. A 180°-wide sector on the hodograph plane exists, within which the relative orientation of the Ekman and Prandtl velocity profiles favours the left rotation with height of the resulting wind velocity vector in the lowermost part of the boundary layer. This explains the negative (left-handed) helicity cascade toward small-scale turbulent motions, which agrees with the direct field measurements of turbulent helicity in Tsimlyansk. A simple turbulent relaxation model is proposed that explains the measured positive values of the relatively minor contribution to turbulent helicity from the vertical components of velocity and vorticity.
8. Boundary-layer theory. 9. ed.
Schlichting, Hermann [Technische Univ. Braunschweig (Germany). Inst. fuer Stroemungsmechanik; Gersten, Klaus [Bochum Univ. (Germany). Lehrstuhl fuer Thermodynamik und Stroemungsmechanik
2017-03-01
This new edition of the near-legendary textbook by Schlichting and revised by Gersten presents a comprehensive overview of boundary-layer theory and its application to all areas of fluid mechanics, with particular emphasis on the flow past bodies (e.g. aircraft aerodynamics). The new edition features an updated reference list and over 100 additional changes throughout the book, reflecting the latest advances on the subject.
9. Progress in modeling hypersonic turbulent boundary layers
Zeman, Otto
1993-01-01
A good knowledge of the turbulence structure, wall heat transfer, and friction in turbulent boundary layers (TBL) at high speeds is required for the design of hypersonic air breathing airplanes and reentry space vehicles. This work reports on recent progress in the modeling of high speed TBL flows. The specific research goal described here is the development of a second order closure model for zero pressure gradient TBL's for the range of Mach numbers up to hypersonic speeds with arbitrary wall cooling requirements.
10. Examination of uniform momentum zones in hypersonic turbulent boundary layers
Williams, Owen; Helm, Clara; Martin, Pino
2017-11-01
The presence of uniform momentum zones (UMZs) separated by regions of high shear is now well-established in incompressible flows, with the mean number of such zones increasing in a log-linear fashion with Reynolds number. While known to be present in supersonic and hypersonic boundary layers, the properties of these UMZs and the appropriate Reynolds number for comparison with incompressible results have not previously been investigated. A large, previously published DNS database of hypersonic boundary layers is used in this investigation, with Mach numbers up to 12 and wall temperatures from cold to adiabatic, resulting in a wide range of outer layer Reynolds numbers. UMZs are examined using a range of parameters in both conventional inner and semi-local scalings, and Reynolds number trends examined.
11. Coupled wake boundary layer model of windfarms
Stevens, Richard; Gayme, Dennice; Meneveau, Charles
2014-11-01
We present a coupled wake boundary layer (CWBL) model that describes the distribution of the power output in a windfarm. The model couples the traditional, industry-standard wake expansion/superposition approach with a top-down model for the overall windfarm boundary layer structure. Wake models capture the effect of turbine positioning, while the top-down approach represents the interaction between the windturbine wakes and the atmospheric boundary layer. Each portion of the CWBL model requires specification of a parameter that is unknown a-priori. The wake model requires the wake expansion rate, whereas the top-down model requires the effective spanwise turbine spacing within which the model's momentum balance is relevant. The wake expansion rate is obtained by matching the mean velocity at the turbine from both approaches, while the effective spanwise turbine spacing is determined from the wake model. Coupling of the constitutive components of the CWBL model is achieved by iterating these parameters until convergence is reached. We show that the CWBL model predictions compare more favorably with large eddy simulation results than those made with either the wake or top-down model in isolation and that the model can be applied successfully to the Horns Rev and Nysted windfarms. The Fellowships for Young Energy Scientists' (YES!) of the Foundation for Fundamental Research on Matter supported by NWO, and NSF Grant #1243482.
12. Turbulent fluxes in stably stratified boundary layers
L'vov, Victor S; Procaccia, Itamar; Rudenko, Oleksii
2008-01-01
We present here an extended version of an invited talk we gave at the international conference 'Turbulent Mixing and Beyond'. The dynamical and statistical description of stably stratified turbulent boundary layers with the important example of the stable atmospheric boundary layer in mind is addressed. Traditional approaches to this problem, based on the profiles of mean quantities, velocity second-order correlations and dimensional estimates of the turbulent thermal flux, run into a well-known difficulty, predicting the suppression of turbulence at a small critical value of the Richardson number, in contradiction to observations. Phenomenological attempts to overcome this problem suffer from various theoretical inconsistencies. Here, we present an approach taking into full account all the second-order statistics, which allows us to respect the conservation of total mechanical energy. The analysis culminates in an analytic solution of the profiles of all mean quantities and all second-order correlations, removing the unphysical predictions of previous theories. We propose that the approach taken here is sufficient to describe the lower parts of the atmospheric boundary layer, as long as the Richardson number does not exceed an order of unity. For much higher Richardson numbers, the physics may change qualitatively, requiring careful consideration of the potential Kelvin-Helmoholtz waves and their interaction with the vortical turbulence.
13. Towards Adaptive Grids for Atmospheric Boundary-Layer Simulations
van Hooft, J. Antoon; Popinet, Stéphane; van Heerwaarden, Chiel C.; van der Linden, Steven J. A.; de Roode, Stephan R.; van de Wiel, Bas J. H.
2018-02-01
We present a proof-of-concept for the adaptive mesh refinement method applied to atmospheric boundary-layer simulations. Such a method may form an attractive alternative to static grids for studies on atmospheric flows that have a high degree of scale separation in space and/or time. Examples include the diurnal cycle and a convective boundary layer capped by a strong inversion. For such cases, large-eddy simulations using regular grids often have to rely on a subgrid-scale closure for the most challenging regions in the spatial and/or temporal domain. Here we analyze a flow configuration that describes the growth and subsequent decay of a convective boundary layer using direct numerical simulation (DNS). We validate the obtained results and benchmark the performance of the adaptive solver against two runs using fixed regular grids. It appears that the adaptive-mesh algorithm is able to coarsen and refine the grid dynamically whilst maintaining an accurate solution. In particular, during the initial growth of the convective boundary layer a high resolution is required compared to the subsequent stage of decaying turbulence. More specifically, the number of grid cells varies by two orders of magnitude over the course of the simulation. For this specific DNS case, the adaptive solver was not yet more efficient than the more traditional solver that is dedicated to these types of flows. However, the overall analysis shows that the method has a clear potential for numerical investigations of the most challenging atmospheric cases.
14. Scaling the heterogeneously heated convective boundary layer
Van Heerwaarden, C.; Mellado, J.; De Lozar, A.
2013-12-01
We have studied the heterogeneously heated convective boundary layer (CBL) by means of large-eddy simulations (LES) and direct numerical simulations (DNS). What makes our study different from previous studies on this subject are our very long simulations in which the system travels through multiple states and that from there we have derived scaling laws. In our setup, a stratified atmosphere is heated from below by square patches with a high surface buoyancy flux, surrounded by regions with no or little flux. By letting a boundary layer grow in time we let the system evolve from the so-called meso-scale to the micro-scale regime. In the former the heterogeneity is large and strong circulations can develop, while in the latter the heterogeneity is small and does no longer influence the boundary layer structure. Within each simulation we can now observe the formation of a peak in kinetic energy, which represents the 'optimal' heterogeneity size in the meso-scale, and the subsequent decay of the peak and the development towards the transition to the micro-scale. We have created a non-dimensional parameter space that describes all properties of this system. By studying the previously described evolution for different combinations of parameters, we have derived three important conclusions. First, there exists a horizontal length scale of the heterogeneity (L) that is a function of the boundary layer height (h) and the Richardson (Ri) number of the inversion at the top of the boundary layer. This relationship has the form L = h Ri^(3/8). Second, this horizontal length scale L allows for expressing the time evolution, and thus the state of the system, as a ratio of this length scale and the distance between two patches Xp. This ratio thus describes to which extent the circulation fills up the space that exists between two patch centers. The timings of the transition from the meso- to the micro-scale collapse under this scaling for all simulations sharing the same flux
15. Convection Cells in the Atmospheric Boundary Layer
2017-04-01
In dry, shear-free convective boundary layers (CBLs), the turbulent flow of air is known to organise itself on large scales into coherent, cellular patterns, or superstructures, consisting of fast, narrow updraughts and slow, wide downdraughts which together form circulations. Superstructures act as transport mechanisms from the surface to the top of the boundary layer and vice-versa, as opposed to small-scale turbulence, which only modifies conditions locally. This suggests that a thorough investigation into superstructure properties may help us better understand transport across the atmospheric boundary layer as a whole. Whilst their existence has been noted, detailed studies into superstructures in the CBL have been scarce. By applying methods which are known to successfully isolate similar large-scale patterns in turbulent Rayleigh-Bénard convection, we can assess the efficacy of those detection techniques in the CBL. In addition, through non-dimensional analysis, we can systematically compare superstructures in various convective regimes. We use direct numerical simulation of four different cases for intercomparison: Rayleigh-Bénard convection (steady), Rayleigh-Bénard convection with an adiabatic top lid (quasi-steady), a stably-stratified CBL (quasi-steady) and a neutrally-stratified CBL (unsteady). The first two are non-penetrative and the latter two penetrative. We find that although superstructures clearly emerge from the time-mean flow in the non-penetrative cases, they become obscured by temporal averaging in the CBL. This is because a rigid lid acts to direct the flow into counter-rotating circulation cells whose axis of rotation remains stationary, whereas a boundary layer that grows in time and is able to entrain fluid from above causes the circulations to not only grow in vertical extent, but also to move horizontally and merge with neighbouring circulations. Spatial filtering is a useful comparative technique as it can be performed on boundary
16. Measurements of a Separating Turbulent Boundary Layer.
1980-04-01
2O 3.7402, 1!145 2 .,? , _, 3030 1.5^203 0 01 R;, L 3.2C742 I.?34P 2.t 172 1.:0000 .5nODD .,4 t 5 0 . tD ?;g -.1 421 4. .02 c; 3.50030 t .S...972 ., 5;o -. 193U ’i .4852 .1!! -. 093.. 2.Qg737 -,34513 3,S� ,53.1? .35033 . l53 !:.’,; 15 !’ .77& .7 ,C23 4 .420’. 5.67722 -.45!e7 5.7343e .40
17. The surface roughness and planetary boundary layer
Telford, James W.
1980-03-01
Applications of the entrainment process to layers at the boundary, which meet the self similarity requirements of the logarithmic profile, have been studied. By accepting that turbulence has dominating scales related in scale length to the height above the surface, a layer structure is postulated wherein exchange is rapid enough to keep the layers internally uniform. The diffusion rate is then controlled by entrainment between layers. It has been shown that theoretical relationships derived on the basis of using a single layer of this type give quantitatively correct factors relating the turbulence, wind and shear stress for very rough surface conditions. For less rough surfaces, the surface boundary layer can be divided into several layers interacting by entrainment across each interface. This analysis leads to the following quantitatively correct formula compared to published measurements. 1 24_2004_Article_BF00877766_TeX2GIFE1.gif {σ _w }/{u^* } = ( {2/{9Aa}} )^{{1/4}} ( {1 - 3^{{1/2}{ a/k{d_n }/z{σ _w }/{u^* }z/L} )^{{1/4}} = 1.28(1 - 0.945({{σ _w }/{u^* }}}) {{z/L}})^{{1/4 where u^* = ( {{tau/ρ}}^{{1/2}}, σ w is the standard deviation of the vertical velocity, z is the height and L is the Obukhov scale lenght. The constants a, A, k and d n are the entrainment constant, the turbulence decay constant, Von Karman's constant, and the layer depth derived from the theory. Of these, a and A, are universal constants and not empirically determined for the boundary layer. Thus the turbulence needed for the plume model of convection, which resides above these layers and reaches to the inversion, is determined by the shear stress and the heat flux in the surface layers. This model applies to convection in cool air over a warm sea. The whole field is now determined except for the temperature of the air relative to the water, and the wind, which need a further parameter describing sea surface roughness. As a first stop to describing a surface where roughness elements
18. Diffusive boundary layers over varying topography
Dell, R. W.; Pratt, L. J.
2015-01-01
Diffusive bottom boundary layers can produce upslope flows in a stratified fluid. Accumulating observations suggest that these boundary layers may drive upwelling and mixing in mid-ocean ridge flank canyons. However, most studies of diffusive bottom
19. The inland boundary layer at low latitudes
Garratt, J. R.
1985-08-01
Observations from the Koorin boundary-layer experiment in Australia (latitude 16 °S) were analysed in a study of the nocturnal jet development. For geostrophic winds in the range 10 20 m s-1, ageostrophic wind magnitudes of 5 10m s-1 were common above the surface layer near sunset, with cross-isobar flow angles of about 40 °. The jet that then developed by midnight was probably the result of these large ageostrophic winds, strong surface cooling and favourable baroclinity and sloping terrain. The analysis is supported by numerical model calculations with special emphasis on the role of long-wave radiative cooling on turbulent decay. Decay is rapid in the presence of radiation, although there is little influence on stress divergence levels. Evidence of sea-breeze influences on the jet evolution, and on features of deeply penetrating sea breezes in general, will be presented and discussed in part 2 of this study (submitted to Boundary-Layer Meteorol.).
20. A Coordinate Transformation for Unsteady Boundary Layer Equations
Paul G. A. CIZMAS
2011-12-01
Full Text Available This paper presents a new coordinate transformation for unsteady, incompressible boundary layer equations that applies to both laminar and turbulent flows. A generalization of this coordinate transformation is also proposed. The unsteady boundary layer equations are subsequently derived. In addition, the boundary layer equations are derived using a time linearization approach and assuming harmonically varying small disturbances.
1. Boundary layer flow past a circular cylinder in axial flow
Sawchuk, S.P.; Zamir, M.; Camiletti, S.E.
1985-01-01
This paper discusses a study of the laminar boundary layer on a semi-infinite circular cylinder in axial incompressible flow. Unlike previous studies, the present study investigates a full range of this boundary layer problem to determine skin friction, heat transfer and other integral properties of the boundary layer
2. Conference on Boundary and Interior Layers : Computational and Asymptotic Methods
Stynes, Martin; Zhang, Zhimin
2017-01-01
This volume collects papers associated with lectures that were presented at the BAIL 2016 conference, which was held from 14 to 19 August 2016 at Beijing Computational Science Research Center and Tsinghua University in Beijing, China. It showcases the variety and quality of current research into numerical and asymptotic methods for theoretical and practical problems whose solutions involve layer phenomena. The BAIL (Boundary And Interior Layers) conferences, held usually in even-numbered years, bring together mathematicians and engineers/physicists whose research involves layer phenomena, with the aim of promoting interaction between these often-separate disciplines. These layers appear as solutions of singularly perturbed differential equations of various types, and are common in physical problems, most notably in fluid dynamics. This book is of interest for current researchers from mathematics, engineering and physics whose work involves the accurate app roximation of solutions of singularly perturbed diffe...
3. The internal boundary layer — A review
Garratt, J. R.
1990-03-01
A review is given of relevant work on the internal boundary layer (IBL) associated with: (i) Small-scale flow in neutral conditions across an abrupt change in surface roughness, (ii) Small-scale flow in non-neutral conditions across an abrupt change in surface roughness, temperature or heat/moisture flux, (iii) Mesoscale flow, with emphasis on flow across the coastline for both convective and stably stratified conditions. The major theme in all cases is on the downstream, modified profile form (wind and temperature), and on the growth relations for IBL depth.
4. Boundary Layer Depth In Coastal Regions
Porson, A.; Schayes, G.
The results of earlier studies performed about sea breezes simulations have shown that this is a relevant feature of the Planetary Boundary Layer that still requires effort to be diagnosed properly by atmospheric models. Based on the observations made during the ESCOMPTE campaign, over the Mediterranean Sea, different CBL and SBL height estimation processes have been tested with a meso-scale model, TVM. The aim was to compare the critical points of the BL height determination computed using turbulent kinetic energy profile with some other standard evaluations. Moreover, these results have been analysed with different mixing length formulation. The sensitivity of formulation is also analysed with a simple coastal configuration.
5. Effect of boundary layer thickness on the flow characteristics around a rectangular prism
Ji, Ho Seong; Kim, Kyung Chun
2001-01-01
Effect of boundary layer thickness on the flow characteristics around a rectangular prism has been investigated by using a PIV(Particle Image Velocimetry) technique. Three different boundary layers (thick, medium and thin) were generated in the atmospheric boundary layer wind tunnel at Pusan National University. The thick boundary layer having 670mm thickness was generated by using spires and roughness elements. The medium thickness of boundary layer(δ=270mm) was the natural turbulent boundary layer at the test section with fully long developing length(18m). The thin boundary layer with 36.5mm thickness was generated by on a smooth panel elevated 70cm from the wind tunnel floor. The Reynolds number based on the free stream velocity and the height of the model was 7.9X10 3 . The mean velocity vector fields and turbulent kinetic energy distribution were measured and compared. The effect of boundary layer thickness is clearly observed not only in the length of separation bubble but also in the reattachment points. The thinner boundary layer thickness, the higher turbulent kinetic energy peak around the model roof. It is strongly recommended that the height ratio between model and approaching boundary layer thickness should be a major parameter
6. Boundary layer attenuation in turbulent sodium flows
Tenchine, D.
1994-01-01
Temperature fluctuations are produced in the sodium coolant of Liquid Metal Reactors when flows at different temperatures are mixing. That occurs in various areas of the reactor plant, in the primary and the secondary circuits. This paper deals with secondary circuit pipings, specifically the Superphenix steam generator outlet. The possibility of thermal striping in this area is studied because of the mixing of a main 'hot' flow surrounded by a smaller 'cold' flow in the vertical pipe located below the steam generator. This work was developed in the frame of a collaboration between CEA, EDF and FRAMATOME. The purpose of our study is to measure temperature fluctuations in the fluid and on the structures, on a sodium reduced scale model of the outlet region of the steam generator. We want to evidence the boundary layer attenuation by comparing wall and fluid measurements. From these experimental data, we shall propose a methodology to predict the boundary layer attenuation and the temperature fluctuations at the surface of the structure, for pipe flow configurations
7. Optimal Growth in Hypersonic Boundary Layers
Paredes, Pedro; Choudhari, Meelan M.; Li, Fei; Chang, Chau-Lyan
2016-01-01
The linear form of the parabolized linear stability equations is used in a variational approach to extend the previous body of results for the optimal, nonmodal disturbance growth in boundary-layer flows. This paper investigates the optimal growth characteristics in the hypersonic Mach number regime without any high-enthalpy effects. The influence of wall cooling is studied, with particular emphasis on the role of the initial disturbance location and the value of the spanwise wave number that leads to the maximum energy growth up to a specified location. Unlike previous predictions that used a basic state obtained from a self-similar solution to the boundary-layer equations, mean flow solutions based on the full Navier-Stokes equations are used in select cases to help account for the viscous- inviscid interaction near the leading edge of the plate and for the weak shock wave emanating from that region. Using the full Navier-Stokes mean flow is shown to result in further reduction with Mach number in the magnitude of optimal growth relative to the predictions based on the self-similar approximation to the base flow.
8. Active Boundary Layer Control on a Highly Loaded Turbine Exit Case Profile
Julia Kurz
2018-03-01
Full Text Available A highly loaded turbine exit guide vane with active boundary layer control was investigated experimentally in the High Speed Cascade Wind Tunnel at the University of the German Federal Armed Forces, Munich. The experiments include profile Mach number distributions, wake traverse measurements as well as boundary layer investigations with a flattened Pitot probe. Active boundary layer control by fluidic oscillators was applied to achieve improved performance in the low Reynolds number regime. Low solidity, which can be applied to reduce the number of blades, increases the risk of flow separation resulting in increased total pressure losses. Active boundary layer control is supposed to overcome these negative effects. The experiments show that active boundary layer control by fluidic oscillators is an appropriate way to suppress massive open separation bubbles in the low Reynolds number regime.
9. Flow Visualization in Supersonic Turbulent Boundary Layers.
Smith, Michael Wayne
This thesis is a collection of novel flow visualizations of two different flat-plate, zero pressure gradient, supersonic, turbulent boundary layers (M = 2.8, Re _theta ~ 82,000, and M = 2.5, Re_ theta ~ 25,000, respectively). The physics of supersonic shear flows has recently drawn increasing attention with the renewed interest in flight at super and hypersonic speeds. This work was driven by the belief that the study of organized, Reynolds -stress producing turbulence structures will lead to improved techniques for the modelling and control of high-speed boundary layers. Although flow-visualization is often thought of as a tool for providing qualitative information about complex flow fields, in this thesis an emphasis is placed on deriving quantitative results from image data whenever possible. Three visualization techniques were applied--'selective cut-off' schlieren, droplet seeding, and Rayleigh scattering. Two experiments employed 'selective cut-off' schlieren. In the first, high-speed movies (40,000 fps) were made of strong density gradient fronts leaning downstream at between 30^circ and 60^ circ and travelling at about 0.9U _infty. In the second experiment, the same fronts were detected with hot-wires and imaged in real time, thus allowing the examination of the density gradient fronts and their associated single-point mass -flux signals. Two experiments employed droplet seeding. In both experiments, the boundary layer was seeded by injecting a stream of acetone through a single point in the wall. The acetone is atomized by the high shear at the wall into a 'fog' of tiny (~3.5mu m) droplets. In the first droplet experiment, the fog was illuminated with copper-vapor laser sheets of various orientations. The copper vapor laser pulses 'froze' the fog motion, revealing a variety of organized turbulence structures, some with characteristic downstream inclinations, others with large-scale roll-up on the scale of delta. In the second droplet experiment, high
10. Bypass transition in compressible boundary layers
Vandervegt, J. J.
1992-01-01
Transition to turbulence in aerospace applications usually occurs in a strongly disturbed environment. For instance, the effects of free-stream turbulence, roughness and obstacles in the boundary layer strongly influence transition. Proper understanding of the mechanisms leading to transition is crucial in the design of aircraft wings and gas turbine blades, because lift, drag and heat transfer strongly depend on the state of the boundary layer, laminar or turbulent. Unfortunately, most of the transition research, both theoretical and experimental, has focused on natural transition. Many practical flows, however, defy any theoretical analysis and are extremely difficult to measure. Morkovin introduced in his review paper the concept of bypass transition as those forms of transition which bypass the known mechanisms of linear and non-linear transition theories and are currently not understood by experiments. In an effort to better understand the mechanisms leading to transition in a disturbed environment, experiments are conducted studying simpler cases, viz. the effects of free stream turbulence on transition on a flat plate. It turns out that these experiments are very difficult to conduct, because generation of free stream turbulence with sufficiently high fluctuation levels and reasonable homogeneity is non trivial. For a discussion see Morkovin. Serious problems also appear due to the fact that at high Reynolds numbers the boundary layers are very thin, especially in the nose region of the plate where the transition occurs, which makes the use of very small probes necessary. The effects of free-stream turbulence on transition are the subject of this research and are especially important in a gas turbine environment, where turbulence intensities are measured between 5 and 20 percent, Wang et al. Due to the fact that the Reynolds number for turbine blades is considerably lower than for aircraft wings, generally a larger portion of the blade will be in a laminar
11. The interaction of synthetic jets with turbulent boundary layers
Cui, Jing
In recent years, a promising approach to the control of wall bounded as well as free shear flows, using synthetic jet (oscillatory jet with zero-net-mass-flux) actuators, has received a great deal of attention. A variety of impressive flow control results have been achieved experimentally by many researchers including the vectoring of conventional propulsive jets, modification of aerodynamic characteristics of bluff bodies, control of lift and drag of airfoils, reduction of skin-friction of a flat plate boundary layer, enhanced mixing in circular jets, and control of external as well as internal flow separation and of cavity oscillations. More recently, attempts have been made to numerically simulate some of these flowfields. Numerically several of the above mentioned flow fields have been simulated primarily by employing the Unsteady Reynolds-Averaged Navier Stokes (URANS) equations with a turbulence model and a limited few by Direct Numerical Simulation (DNS). In simulations, both the simplified boundary conditions at the exit of the jet as well as the details of the cavity and lip have been included. In this dissertation, I describe the results of simulations for several two- and three-dimensional flowfields dealing with the interaction of a synthetic jet with a turbulent boundary layer and control of separation. These simulations have been performed using the URANS equations in conjunction with either one- or a two-equation turbulence model. 2D simulations correspond to the experiments performed by Honohan at Georgia Tech. and 3D simulations correspond to the CFD validation test cases proposed in the NASA Langley Research Center Workshop---"CFD Validation of Synthetic Jets and Turbulent Separation Control" held at Williamsburg VA in March 2004. The sources of uncertainty due to grid resolution, time step, boundary conditions, turbulence modeling etc. have been examined during the computations. Extensive comparisons for various flow variables are made with the
12. Role of residual layer and large-scale phenomena on the evolution of the boundary layer
Blay, E.; Pino, D.; Vilà-Guerau de Arellano, J.; Boer, van de A.; Coster, de O.; Faloona, I.; Garrouste, O.; Hartogensis, O.K.
2012-01-01
Mixed-layer theory and large-eddy simulations are used to analyze the dynamics of the boundary layer on two intensive operational periods during the Boundary Layer Late Afternoon and Sunset Turbulence (BLLAST) campaign: 1st and 2nd of July 2011, when convective boundary layers (CBLs) were observed.
13. Soot and radiation in combusting boundary layers
Beier, R.A.
1981-12-01
In most fires thermal radiation is the dominant mode of heat transfer. Carbon particles within the fire are responsible for most of this emitted radiation and hence warrant quantification. As a first step toward understanding thermal radiation in full scale fires, an experimental and theoretical study is presented for a laminar combusting boundary layer. Carbon particulate volume fraction profiles and approximate particle size distributions are experimentally determined in both free and forced flow for several hydrocarbon fuels and PMMA (polymethylmethacrylate). A multiwavelength laser transmission technique determines a most probable radius and a total particle concentration which are two unknown parameters in an assumed Gauss size distribution. A sooting region is observed on the fuel rich side of the main reaction zone. For free flow, all the flames are in air, but the free stream ambient oxygen mass fraction is a variable in forced flow. To study the effects of radiation heat transfer, a model is developed for a laminar combusting boundary layer over a pyrolyzing fuel surface. An optically thin approximation simplifies the calculation of the radiant energy flux at the fuel surface. For the free flames in air, the liquid fuel soot volume fractions, f/sub v/, range from f/sub v/ approx. 10/sup -7/ for n-heptane, a paraffin, to f/sub v/ approx. 10/sup -7/ for toluene, an aromatic. The PMMA soot volume fractions, f/sub v/ approx. 5 x 10/sup -7/, are approximately the same as the values previously reported for pool fires. Soot volume fraction increases monotonically with ambient oxygen mass fraction in the forced flow flames. For all fuels tested, a most probable radius between 20 nm and 80 nm is obtained which varies only slightly with oxygen mass fraction, streamwise position, or distance normal to the fuel surface. The theoretical analysis yields nine dimensionless parameters, which control the mass flux rate at the pyrolyzing fuel surface.
14. A preliminary investigation of boundary-layer transition along a flat plate with adverse pressure gradient
Von Doenhoff, Albert E
1938-01-01
Boundary-layer surveys were made throughout the transition region along a smooth flat plate placed in an airstream of practically zero turbulence and with an adverse pressure gradient. The boundary-layer Reynolds number at the laminar separation point was varied from 1,800 to 2,600. The test data, when considered in the light of certain theoretical deductions, indicated that transition probably began with separation of the laminar boundary layer. The extent of the transition region, defined as the distance from a calculated laminar separation point to the position of the first fully developed turbulent boundary-layer profile, could be expressed as a constant Reynolds number run of approximately 70,000. Some speculations are presented concerning the application of the foregoing concepts, after certain assumptions have been made, to the problem of the connection between transition on the upper surface of an airfoil at high angles of attack and the maximum lift.
15. Properties of the TEXTOR boundary layer
Bogen, P.; Hartwig, H.; Hintz, E.; Hoethker, K.; Lie, Y.T.; Pospieszczyk, A.; Samm, U.
1984-01-01
First measurements on the TEXTOR boundary layer are reported. The hydrogen recycling in front of the four limiter segments has been studied by means of a CCD-camera, which proved to be a good instrument to center the discharge for symmetric plasma-limiter contact. The composition of the neutral fluxes from the limiter have been measured: oxygen fluxes are about a factor of ten higher than the metal fluxes; within the error limits the composition does not change with varying limiter radius. Electron densities in the scrape-off layer away from the limiter have been determined by injecting an Li-atom beam from a thermal source and by observing its emission as a function of radius. Similar measurements have been made in front of the limiter with sputtered Cr and O atoms. Both methods gave for the magnetic surface of the limiter radius nsub(e) approx.= 1 x 10 12 /cm 23 . Infrared observations of a test limiter with a CCD-camera and a PbSe-detector have been performed to record the thermal loads. About 10% of the input power flows to the limiter. (orig.)
16. Transition to turbulence in the Hartmann boundary layer
Thess, A.; Krasnov, D.; Boeck, T.; Zienicke, E. [Dept. of Mechanical Engineering, Ilmenau Univ. of Tech. (Germany); Zikanov, O. [Dept. of Mechanical Engineering, Univ. of Michigan, Dearborn, MI (United States); Moresco, P. [School of Physics and Astronomy, The Univ. of Manchester (United Kingdom); Alboussiere, T. [Lab. de Geophysique Interne et Tectonophysique, Observatoire des Science de l' Univers de Grenoble, Univ. Joseph Fourier, Grenoble (France)
2007-07-01
The Hartmann boundary layer is a paradigm of magnetohydrodynamic (MHD) flows. Hartmann boundary layers develop when a liquid metal flows under the influence of a steady magnetic field. The present paper is an overview of recent successful attempts to understand the mechanisms by which the Hartmann layer undergoes a transition from laminar to turbulent flow. (orig.)
17. Analytical solution for the convectively-mixed atmospheric boundary layer
Ouwersloot, H.G.; Vilà-Guerau de Arellano, J.
2013-01-01
Based on the prognostic equations of mixed-layer theory assuming a zeroth order jump at the entrainment zone, analytical solutions for the boundary-layer height evolution are derived with different degrees of accuracy. First, an exact implicit expression for the boundary-layer height for a situation
18. Boundary layer control by a fish: Unsteady laminar boundary layers of rainbow trout swimming in turbulent flows.
Yanase, Kazutaka; Saarenrinne, Pentti
2016-12-15
The boundary layers of rainbow trout, Oncorhynchus mykiss [0.231±0.016 m total body length (L) (mean±s.d.); N=6], swimming at 1.6±0.09 L s -1 (N=6) in an experimental flow channel (Reynolds number, Re=4×10 5 ) with medium turbulence (5.6% intensity) were examined using the particle image velocimetry technique. The tangential flow velocity distributions in the pectoral and pelvic surface regions (arc length from the rostrum, l x =71±8 mm, N=3, and l x =110±13 mm, N=4, respectively) were approximated by a laminar boundary layer model, the Falkner-Skan equation. The flow regime over the pectoral and pelvic surfaces was regarded as a laminar flow, which could create less skin-friction drag than would be the case with turbulent flow. Flow separation was postponed until vortex shedding occurred over the posterior surface (l x =163±22 mm, N=3). The ratio of the body-wave velocity to the swimming speed was in the order of 1.2. This was consistent with the condition of the boundary layer laminarization that had been confirmed earlier using a mechanical model. These findings suggest an energy-efficient swimming strategy for rainbow trout in a turbulent environment. © 2016. Published by The Company of Biologists Ltd.
19. Stability of boundary layer flow based on energy gradient theory
Dou, Hua-Shu; Xu, Wenqian; Khoo, Boo Cheong
2018-05-01
The flow of the laminar boundary layer on a flat plate is studied with the simulation of Navier-Stokes equations. The mechanisms of flow instability at external edge of the boundary layer and near the wall are analyzed using the energy gradient theory. The simulation results show that there is an overshoot on the velocity profile at the external edge of the boundary layer. At this overshoot, the energy gradient function is very large which results in instability according to the energy gradient theory. It is found that the transverse gradient of the total mechanical energy is responsible for the instability at the external edge of the boundary layer, which induces the entrainment of external flow into the boundary layer. Within the boundary layer, there is a maximum of the energy gradient function near the wall, which leads to intensive flow instability near the wall and contributes to the generation of turbulence.
20. Slow Manifolds and Multiple Equilibria in Stratocumulus-Capped Boundary Layers
Junya Uchida
2010-12-01
Full Text Available In marine stratocumulus-capped boundary layers under strong inversions, the timescale for thermodynamic adjustment is roughly a day, much shorter than the multiday timescale for inversion height adjustment. Slow-manifold analysis is introduced to exploit this timescale separation when boundary layer air columns experience only slow changes in their boundary conditions. Its essence is that the thermodynamic structure of the boundary layer remains approximately slaved to its inversion height and the instantaneous boundary conditions; this slaved structure determines the entrainment rate and hence the slow evolution of the inversion height. Slow-manifold analysis is shown to apply to mixed-layer model and large-eddy simulations of an idealized nocturnal stratocumulus- capped boundary layer; simulations with different initial inversion heights collapse onto single relationships of cloud properties with inversion height. Depending on the initial inversion height, the simulations evolve toward a shallow thin-cloud boundary layer or a deep, well-mixed thick cloud boundary layer. In the large-eddy simulations, these evolutions occur on two separate slow manifolds (one of which becomes unstable if cloud droplet concentration is reduced. Applications to analysis of stratocumulus observations and to pockets of open cells and ship tracks are proposed.
1. Prediction of boundary-layer transition caused by crossflow disturbances
Nomura, Toshiyuki; 野村 聡幸
1999-01-01
A prediction system for boundary layer transition is developed which consists of the Navier-Stokes code computing a compressible boundary layer, the linear PSE (Parabolized Stability Equations) code computing the spatial growth of a disturbance, and the N-factor code integrating the growth rate. The system is applied to the case that the transition of the compressible boundary layer on a swept cylinder is caused by cross flow disturbances which have the same spanwise wavelength as observed in...
2. INCOMPRESSIBLE LAMINAR BOUNDARY LAYER CONTROL BY BLOWING AND SUCTION
AZZEDINE NAHOUI; LAKHDAR BAHI
2013-01-01
A two-dimensional incompressible laminar boundary layer and its control using blowing and suction over a flat plate and around the NACA 0012 and 661012 profiles, is studied numerically. The study is based on the Prandtl boundary layer model using the finite differences method and the Crank-Nicolson scheme. The velocity distribution, the boundary layer thickness and the friction coefficient, are determined and presented with and without control. The application of the control technique, has de...
3. Comparison of turbulence in a transitional boundary layer to turbulence in a developed boundary layer*
Park, G. I.; Wallace, J.; Wu, X.; Moin, P.
2010-11-01
Using a recent DNS of a flat-plate boundary layer, statistics of turbulence in transition at Reθ= 500 where spots merge (distributions of the mean velocity, rms velocity and vorticity fluctuations, Reynolds shear stress, kinetic energy production and dissipation rates and enstrophy) have been compared to these statistics for the developed boundary layer turbulence at Reθ= 1850. When the distributions in the transitional region, determined in narrow planes 0.03 Reθ wide, exclude regions and times when the flow is not turbulent, they closely resemble those in the developed turbulent state at the higher Reynolds number, especially in the buffer and sublayers. The skin friction coefficient, determined in this conditional manner in the transitional flow is, of course, much larger than that obtained by including both turbulent and non-turbulent information there, and is consistent with a value obtained by extrapolating from the developed turbulent region. We are attempting to perform this data analysis even further upstream in the transitioning flow at Reθ= 300 where the turbulent spots are individuated. These results add further evidence to support the view that the structure of a developed turbulent boundary layer is little different from its structure in its embryonic form in turbulent spots. *CTR 2010 Summer Program research.
4. Thin-layer approximation and algebraic model for separated turbulent flows
Baldwin, B.; Lomax, H.
1978-01-01
An algebraic turbulence model for two- and three-dimensional separated flows is specified that avoids the necessity for finding the edge of the boundary layer. Properties of the model are determined and comparisons made with experiment for an incident shock on a flat plate, separated flow over a compression corner, and transonic flow over an airfoil. Separation and reattachment points from numerical Navier-Stokes solutions agree with experiment within one boundary-layer thickness. Use of law-of-the-wall boundary conditions does not alter the predictions significantly. Applications of the model to other cases are contained in companion papers.
5. Pre-LBA Rondonia Boundary Layer Experiment (RBLE) Data
National Aeronautics and Space Administration — The atmospheric boundary layer (ABL) is the layer of air closest to the ground which is directly influenced on a daily basis by the heating and cooling of the...
6. Pre-LBA Rondonia Boundary Layer Experiment (RBLE) Data
National Aeronautics and Space Administration — ABSTRACT: The atmospheric boundary layer (ABL) is the layer of air closest to the ground which is directly influenced on a daily basis by the heating and cooling of...
7. CFD Analysis of a Centrifugal Fan for Performance Enhancement using Converging Boundary Layer Suction Slots
K. Vasudeva Karanth; N. Yagnesh Sharma
2009-01-01
Generally flow behavior in centrifugal fan is observed to be in a state of instability with flow separation zones on suction surface as well as near the front shroud. Overall performance of the diffusion process in a centrifugal fan could be enhanced by judiciously introducing the boundary layer suction slots. With easy accessibility of CFD as an analytical tool, an extensive numerical whole field analysis of the effect of boundary layer suction slots in discrete regions ...
8. Internal and external 2-d boundary layer flows
Crawford, M. E.; Kays, W. M.
1978-01-01
Computer program computes general two dimensional turbulent boundary-layer flow using finite-difference techniques. Structure allows for user modification to accommodate unique problems. Program should prove useful in many applications where accurate boundary-layer flow calculations are required.
9. High frequency ground temperature fluctuation in a Convective Boundary Layer
Garai, A.; Kleissl, J.; Lothon, M.; Lohou, F.; Pardyjak, E.; Saïd, F.; Cuxart, J.; Steeneveld, G.J.; Yaguë, C.; Derrien, S.; Alexander, D.; Villagrasa, D.M.
2012-01-01
To study influence of the turbulent structures in the convective boundary layer (CBL) on the ground temperature, during the Boundary Layer Late Afternoon and Sunset Turbulence (BLLAST) observational campaign, high frequency ground temperature was recorded through infra-red imagery from 13 June - 8
10. The turning of the wind in the atmospheric boundary layer
Pena Diaz, Alfredo; Gryning, Sven-Erik; Floors, Rogier Ralph
2014-01-01
Here we use accurate observations of the wind speed vector to analyze the behavior with height of the wind direction. The observations are a combination of tall meteorological mast and long-range wind lidar measurements covering the entire atmospheric boundary layer. The observations were performed...... winds underpredict the turning of the wind and the boundary-layer winds in general....
11. Transonic shock wave. Boundary layer interaction at a convex wall
Koren, B.; Bannink, W.J.
1984-01-01
A standard finite element procedure has been applied to the problem of transonic shock wave – boundary layer interaction at a convex wall. The method is based on the analytical Bohning-Zierep model, where the boundary layer is perturbed by a weak normal shock wave which shows a singular pressure
12. On hairpin vortices in a transitional boundary layer
Uruba Václav
2012-04-01
Full Text Available In the presented paper the results of experiments on transitional boundary layer are presented. The boundary layer was generated on smooth flat wall with zero pressure gradient forming one side of the channel of rectangular cross section. The hairpin vortices, packets of hairpin vortices, turbulent spots and calmed regions were experimentally investigated using time-resolved PIV technique.
13. Numerical simulation of tsunami-scale wave boundary layers
Williams, Isaac A.; Fuhrman, David R.
2016-01-01
This paper presents a numerical study of the boundary layer flow and properties induced by tsunami-scalewaves. For this purpose, an existing one-dimensional vertical (1DV) boundary layer model, based on the horizontal component of the incompressible Reynolds-averaged Navier–Stokes (RANS) equation...
14. Experimental study on effects of inlet boundary layer thickness and boundary layer fence in a turbine cascade
Jun, Y. M.; Chung, J. T.
2000-01-01
The working fluid from the combustor to the turbine stage of a gas turbine makes various boundary layer thickness. Since the inlet boundary layer thickness is one of the important factors that affect the turbine efficiency, It is necessary to investigate secondary flow and loss with various boundary layer thickness conditions. In the present study, the effect of various inlet boundary layer thickness on secondary flow and loss and the proper height of the boundary layer fences for various boundary layer thickness were investigated. Measurements of secondary flow velocity and total pressure loss within and downstream of the passage were taken under 5 boundary layer thickness conditions, 16, 36, 52, 69, 110mm. It was found that total pressure loss and secondary flow areas were increased with increase of thickness but they were maintained almost at the same position. At the following research about the boundary layer fences, 1/6, 1/3, 1/2 of each inlet boundary layer thickness and 12mm were used as the fence heights. As a result, it was observed that the proper height of the fences was generally constant since the passage vortex remained almost at the same position. Therefore once the geometry of a cascade is decided, the location of the passage vortex and the proper fence height are appeared to be determined at the same time. When the inlet boundary layer thickness is relatively small, the loss caused by the proper fence becomes bigger than end wall loss so that it dominates secondary loss. In these cases the proper fence height is decided not by the cascade geometry but by the inlet boundary layer thickness as previous investigations
15. Boundary Layer Control of Rotating Convection Systems
King, E. M.; Stellmach, S.; Noir, J.; Hansen, U.; Aurnou, J. M.
2008-12-01
Rotating convection is ubiquitous in the natural universe, and is likely responsible for planetary processes such magnetic field generation. Rapidly rotating convection is typically organized by the Coriolis force into tall, thin, coherent convection columns which are aligned with the axis of rotation. This organizational effect of rotation is thought to be responsible for the strength and structure of magnetic fields generated by convecting planetary interiors. As thermal forcing is increased, the relative influence of rotation weakens, and fully three-dimensional convection can exist. It has long been assumed that rotational effects will dominate convection dynamics when the ratio of buoyancy to the Coriolis force, the convective Rossby number, Roc, is less than unity. We investigate the influence of rotation on turbulent Rayleigh-Benard convection via a suite of coupled laboratory and numerical experiments over a broad parameter range: Rayleigh number, 10310; Ekman number, 10-6≤ E ≤ ∞; and Prandtl number, 1≤ Pr ≤ 100. In particular, we measure heat transfer (as characterized by the Nusselt number, Nu) as a function of the Rayleigh number for several different Ekman and Prandtl numbers. Two distinct heat transfer scaling regimes are identified: non-rotating style heat transfer, Nu ~ Ra2/7, and quasigeostrophic style heat transfer, Nu~ Ra6/5. The transition between the non-rotating regime and the rotationally dominant regime is described as a function of the Ekman number, E. We show that the regime transition depends not on the global force balance Roc, but on the relative thicknesses of the thermal and Ekman boundary layers. The transition scaling provides a predictive criterion for the applicability of convection models to natural systems such as Earth's core.
16. Destiny of earthward streaming plasma in the plasmasheet boundary layer
Green, J. L.; Horwitz, J. L.
1986-01-01
The dynamics of the earth's magnetotail have been investigated, and it has become clear that the plasmasheet boundary layer field lines map into the Region I Field-Aligned Currents (FAC) of the auroral zone. It is pointed out that the role of earthward streaming ions in the plasmasheet boundary layer may be of fundamental importance in the understanding of magnetotail dynamics, auroral zone physics, and especially for ionospheric-magnetospheric interactions. The present paper has the objective to evaluate propagation characteristics for the earthward streaming ions observed in the plasmasheet boundary layer. An investigation is conducted of the propagation characteristics of protons in the plasmasheet boundary layer using independent single particle dynamics, and conclusions are discussed. The density of earthward streaming ions found in the plasmasheet boundary layer should include the ring current as well as the auroral zone precipitaiton and inner plasmasheet regions of the magnetosphere.
17. Diffusive boundary layers over varying topography
Dell, R. W.
2015-03-25
Diffusive bottom boundary layers can produce upslope flows in a stratified fluid. Accumulating observations suggest that these boundary layers may drive upwelling and mixing in mid-ocean ridge flank canyons. However, most studies of diffusive bottom boundary layers to date have concentrated on constant bottom slopes. We present a study of how diffusive boundary layers interact with various idealized topography, such as changes in bottom slope, slopes with corrugations and isolated sills. We use linear theory and numerical simulations in the regional ocean modeling system (ROMS) model to show changes in bottom slope can cause convergences and divergences within the boundary layer, in turn causing fluid exchanges that reach far into the overlying fluid and alter stratification far from the bottom. We also identify several different regimes of boundary-layer behaviour for topography with oceanographically relevant size and shape, including reversing flows and overflows, and we develop a simple theory that predicts the regime boundaries, including what topographies will generate overflows. As observations also suggest there may be overflows in deep canyons where the flow passes over isolated bumps and sills, this parameter range may be particularly significant for understanding the role of boundary layers in the deep ocean.
18. Experimental research on crossing shock wave boundary layer interactions
Settles, G. S.; Garrison, T. J.
1994-10-01
An experimental research effort of the Penn State Gas Dynamics Laboratory on the subject of crossing shock wave boundary layer interactions is reported. This three year study was supported by AFOSR Grant 89-0315. A variety of experimental techniques were employed to study the above phenomena including planar laser scattering flowfield visualization, kerosene lampblack surface flow visualization, laser-interferometer skin friction surveys, wall static pressure measurements, and flowfield five-hole probe surveys. For a model configuration producing two intersecting shock waves, measurements were made for a range of oblique shock strengths at freestream Mach numbers of 3.0 and 3.85. Additionally, measurements were made at Mach 3.85 for a configuration producing three intersecting waves. The combined experimental dataset was used to formulate the first detailed flowfield models of the crossing-shock and triple-shock wave/boundary layer interactions. The structure of these interactions was found to be similar over a broad range of interaction strengths and is dominated by a large, separated, viscous flow region.
19. Rotor boundary layer development with inlet guide vane (IGV) wake impingement
Jia, Lichao; Zou, Tengda; Zhu, Yiding; Lee, Cunbiao
2018-04-01
This paper examines the transition process in a boundary layer on a rotor blade under the impingement of an inlet guide vane wake. The effects of wake strengths and the reduced frequency on the unsteady boundary layer development on a low-speed axial compressor were investigated using particle image velocimetry. The measurements were carried out at two reduced frequencies (fr = fIGVS0/U2i, fr = 1.35, and fr = 0.675) with the Reynolds number, based on the blade chord and the isentropic inlet velocity, being 97 500. At fr = 1.35, the flow separated at the trailing edge when the wake strength was weak. However, the separation was almost totally suppressed as the wake strength increased. For the stronger wake, both the wake's high turbulence and the negative jet behavior of the wake dominated the interaction between the unsteady wake and the separated boundary layer on the suction surface of the airfoil. The boundary layer displacement thickened first due to the negative jet effect. Then, as the disturbances developed underneath the wake, the boundary layer thickness reduced gradually. The high disturbance region convected downstream at a fraction of the free-stream velocity and spread in the streamwise direction. The separation on the suction surface was suppressed until the next wake's arrival. Because of the long recovery time at fr = 0.675, the boundary layer thickened gradually as the wake convected further downstream and finally separated due to the adverse pressure gradient. The different boundary layer states in turn affected the development of disturbances.
20. Investigation of 3D Shock-Boundary Layer Interaction: A Combined Approach using Experiments, Numerical Simulations and Stability Analysis
2015-12-02
layer , the non-reflecting boundary condition suggested by Poinsot and Lele is adopted.38 On the flat – plate surface, the no-penetration (v = 0) and the no...Introduction Shock-wave boundary layer interactions (SBLIs) occur in most supersonic flight applications and have been the subject of many studies...generator plate is emulated to create an oblique shock that impinges on the boundary layer causing separation. This is similar to the experimental
1. Three-dimensional turbulent boundary layers; Proceedings of the Symposium, Berlin, West Germany, March 29-April 1, 1982
Fernholz, H. H.; Krause, E.
Papers are presented on recent research concerning three-dimensional turbulent boundary layers. Topics examined include experimental techniques in three-dimensional turbulent boundary layers, turbulence measurements in ship-model flow, measurements of Reynolds-stress profiles in the stern region of a ship model, the effects of crossflow on the vortex-layer-type three-dimensional flow separation, and wind tunnel investigations of some three-dimensional separated turbulent boundary layers. Also examined are three-dimensional boundary layers in turbomachines, the boundary layers on bodies of revolution spinning in axial flows, the effect on a developed turbulent boundary layer of a sudden local wall motion, three-dimensional turbulent boundary layer along a concave wall, the numerical computation of three-dimensional boundary layers, a numerical study of corner flows, three-dimensional boundary calculations in design aerodynamics, and turbulent boundary-layer calculations in design aerodynamics. For individual items see A83-47012 to A83-47036
2. Boundary-Layer Characteristics Over a Coastal Megacity
Melecio-Vazquez, D.; Ramamurthy, P.; Arend, M.; Moshary, F.; Gonzalez, J.
2017-12-01
Boundary-layer characteristics over New York City are analyzed for various local and synoptic conditions over several seasons. An array of vertical profilers, including a Doppler LiDAR, a micro-pulse LiDAR and a microwave radiometer are used to observe the structure and evolution of the boundary-layer. Additionally, an urbanized Weather Research and Forecasting (uWRF) model coupled to a high resolution landcover/land-use database is used to study the spatial variability in boundary layer characteristics. The summer daytime averaged potential temperature profile from the microwave radiometer shows the presence of a thermal internal boundary layer wherein a superadiabatic layer lies underneath a stable layer instead of a mixed-layer. Both the winter daytime and nighttime seasonal averages show that the atmosphere remains unstable near the surface and does not reach stable conditions during the nighttime. The mixing ratio seasonal averages show peaks in humidity near 200-m and 1100-m, above instrument level, which could result from sea breeze and anthropogenic sources. Ceilometer measurements show a high degree of variability in boundary layer height depending on wind direction. Comparison with uWRF results show that the model tends to overestimate convective efficiency for selected summer and winter cases and therefore shows a much deeper thermal boundary layer than the observed profiles. The model estimates a less humid atmosphere than seen in observations.
3. Microbubble drag reduction in liquid turbulent boundary layers
Merkle, C.L.; Deutsch, S.
1992-01-01
The interactions between a dense cloud of small bubbles and a liquid turbulent boundary layer are reviewed on the basis of available experimental observations to understand and quantify their capability for reducing skin friction. Gas bubbles are generally introduced into the boundary layer by injection through a porous surface or by electrolysis. After injection, the bubbles stay near the wall in boundary-layer-like fashion giving rise to strong gradients in both velocity and gas concentration. In general, the magnitude of the skin friction reduction increases as the volume of bubbles in the boundary layer is increased until a maximum skin friction reduction of typically 80-90% of the undisturbed skin friction level is reached. The volumetric gas flow required for this maximum is nominally equal to the volume flow of the liquid in the boundary layer. Bubble size estimates indicate that in most microbubble experiments the bubbles have been intermediate in size between the inner and outer scales of the undisturbed boundary layer. Additional studies with other nondimensional bubble sizes would be useful. However, the bubble size is most likely controlled by the injection process, and considerably different conditions would be required to change this ratio appreciably. The trajectories of the bubble clouds are primarily determined by the random effects of turbulence and bubble-bubble interactions. The effects of buoyancy represent a weaker effect. The trajectories are unlike the deterministic trajectory of an individual bubble in a time-averaged boundary layer. Bubbles are most effective in high speed boundary layers and, for the bubble sizes tested to date, produce an effect that persists for some on hundred boundary layer thicknesses. Modeling suggests that microbubbles reduce skin friction by increasing the turbulence Reynolds number in the buffer layer in a manner similar to polymers
4. Inefficient Angular Momentum Transport in Accretion Disk Boundary Layers: Angular Momentum Belt in the Boundary Layer
Belyaev, Mikhail A.; Quataert, Eliot
2018-04-01
We present unstratified 3D MHD simulations of an accretion disk with a boundary layer (BL) that have a duration ˜1000 orbital periods at the inner radius of the accretion disk. We find the surprising result that angular momentum piles up in the boundary layer, which results in a rapidly rotating belt of accreted material at the surface of the star. The angular momentum stored in this belt increases monotonically in time, which implies that angular momentum transport mechanisms in the BL are inefficient and do not couple the accretion disk to the star. This is in spite of the fact that magnetic fields are advected into the BL from the disk and supersonic shear instabilities in the BL excite acoustic waves. In our simulations, these waves only carry a small fraction (˜10%) of the angular momentum required for steady state accretion. Using analytical theory and 2D viscous simulations in the R - ϕ plane, we derive an analytical criterion for belt formation to occur in the BL in terms of the ratio of the viscosity in the accretion disk to the viscosity in the BL. Our MHD simulations have a dimensionless viscosity (α) in the BL that is at least a factor of ˜100 smaller than that in the disk. We discuss the implications of these results for BL dynamics and emission.
5. INCOMPRESSIBLE LAMINAR BOUNDARY LAYER CONTROL BY BLOWING AND SUCTION
AZZEDINE NAHOUI
2013-12-01
Full Text Available A two-dimensional incompressible laminar boundary layer and its control using blowing and suction over a flat plate and around the NACA 0012 and 661012 profiles, is studied numerically. The study is based on the Prandtl boundary layer model using the finite differences method and the Crank-Nicolson scheme. The velocity distribution, the boundary layer thickness and the friction coefficient, are determined and presented with and without control. The application of the control technique, has demonstrated its positive effect on the transition point and the friction coefficient. Both control procedures are compared for different lengths, speeds and angles of blowing and suction.
6. The atmospheric boundary layer — advances in knowledge and application
Garratt, J. R.; Hess, G. D.; Physick, W. L.; Bougeault, P.
1996-02-01
We summarise major activities and advances in boundary-layer knowledge in the 25 years since 1970, with emphasis on the application of this knowledge to surface and boundary-layer parametrisation schemes in numerical models of the atmosphere. Progress in three areas is discussed: (i) the mesoscale modelling of selected phenomena; (ii) numerical weather prediction; and (iii) climate simulations. Future trends are identified, including the incorporation into models of advanced cloud schemes and interactive canopy schemes, and the nesting of high resolution boundary-layer schemes in global climate models.
7. Large Eddy Simulation of the ventilated wave boundary layer
Lohmann, Iris P.; Fredsøe, Jørgen; Sumer, B. Mutlu
2006-01-01
A Large Eddy Simulation (LES) of (1) a fully developed turbulent wave boundary layer and (2) case 1 subject to ventilation (i.e., suction and injection varying alternately in phase) has been performed, using the Smagorinsky subgrid-scale model to express the subgrid viscosity. The model was found...... slows down the flow in the full vertical extent of the boundary layer, destabilizes the flow and decreases the mean bed shear stress significantly; whereas suction generally speeds up the flow in the full vertical extent of the boundary layer, stabilizes the flow and increases the mean bed shear stress...
8. Change of Surface Roughness and Planetary Boundary Layer
Jensen, Niels Otto
1978-01-01
The ratio between upstream and far downstream surface friction velocities relative to a change in surface roughness is given on the basis of results from surface Rossby number similarity theory. By simple theories for the internal boundary layer, which are found to compare quite well with recent...... numerical results from higher-order closure models, it is found that, even at a downwind distance such that the internal boundary layer has grown to the full height of the planetary boundary layers, the surface stress still considerably exceeds the equilibrium value...
9. Effect of externally generated turbulence on wave boundary layer
Fredsøe, Jørgen; Sumer, B. Mutlu; Kozakiewicz, A.
2003-01-01
This experimental study deals with the effect of externally generated turbulence on the oscillatory boundary layer to simulate the turbulence in the wave boundary layer under broken waves in the swash zone. The subject has been investigated experimentally in a U-shaped, oscillating water tunnel...... results. The mean and turbulence quantities in the outer flow region are increased substantially with the introduction of the grids. It is shown that the externally generated turbulence is able to penetrate the bed boundary layer, resulting in an increase in the bed shear stress, and therefore...
10. Coherent structures in wave boundary layers. Part 2. Solitary motion
Sumer, B. Mutlu; Jensen, Palle Martin; Sørensen, Lone B.
2010-01-01
This study continues the investigation of wave boundary layers reported by Carstensen, Sumer & Fredsøe (J. Fluid Mech., 2010, part 1 of this paper). The present paper summarizes the results of an experimental investigation of turbulent solitary wave boundary layers, simulated by solitary motion...... the boundary-layer flow experiences a regular array of vortex tubes near the bed over a short period of time during the deceleration stage; and (iii) transitional regime characterized with turbulent spots, revealed by single/multiple, or, sometimes, quite dense spikes in the bed shear stress traces...
11. Size distributions of boundary-layer clouds
Stull, R.; Berg, L.; Modzelewski, H. [Univ. of Wisconsin, Madison, WI (United States)
1996-04-01
Scattered fair-weather clouds are triggered by thermals rising from the surface layer. Not all surface layer air is buoyant enough to rise. Also, each thermal has different humidities and temperatures, resulting in interthermal variability of their lifting condensation levels (LCL). For each air parcel in the surface layer, its virtual potential temperature and its LCL height can be computed.
12. Analysis of dimensionality effect on shock wave boundary layer interaction in laminar hypersonic flows
John, Bibin; Surendranath, Srikanth; Natarajan, Ganesh; Kulkarni, Vinayak
2016-01-01
13. Reactive boundary layers in metallic rolling contacts
Burbank, John
2016-05-01
more thorough investigation into the effects of residual austenite on the properties of this material. The high-performance alternative steels, 36NiCrMoV1-5-7 (hot working steel) and 45SiCrMo6 (spring steel), were heat treated as recommended by their respective manufacturers, and were not case-hardened. The selection of materials with and materials without case-hardening allows for an investigation into whether or not case-hardening is even necessary to deliver acceptable friction behaviour and wear performance. Elemental analyses were conducted by multiple methods to ensure accurate results. Residual austenite contents of the steels and the depth profiles of residual stresses were determined by X-Ray diffraction (XRD), for 20MnCr5 ranging from approximately 6 - 14 vol.%, and under 2 vol.% for the alternative alloys. Hardness profiles were taken from the testing surfaces into the material core. The carburization of 20MnCr5 led to higher hardness and the greater concentration of carbon in the carburization zone more representative of a hardened SAE E52100, or 100Cr6/102Cr6, than of a non-case-hardened 20MnCr5. Residual stresses from machining and case-hardening were measured directly at the sample surface. The high-performance steels fulfilled manufacturer expectations in terms of elemental content, with hardness values between 50 - 55 HRC and strongly martensitic microstructure character. With characterization of the chosen materials complete, the materials could then be subjected to pre-conditioning. The first pre-conditioning method involved targeted generation of cold work hardening as induced boundary layers to protect the contact zone against wear. Work hardening was identified both by variations in residual stress profiles, i.e. the introduction of beneficial compressive residual stresses, and hardness increases in the contact zone, providing enhanced wear resistance. Parameters for work hardening were further optimized to reduce damage to the surface substrates
14. Reactive boundary layers in metallic rolling contacts
Burbank, John
2016-01-01
thorough investigation into the effects of residual austenite on the properties of this material. The high-performance alternative steels, 36NiCrMoV1-5-7 (hot working steel) and 45SiCrMo6 (spring steel), were heat treated as recommended by their respective manufacturers, and were not case-hardened. The selection of materials with and materials without case-hardening allows for an investigation into whether or not case-hardening is even necessary to deliver acceptable friction behaviour and wear performance. Elemental analyses were conducted by multiple methods to ensure accurate results. Residual austenite contents of the steels and the depth profiles of residual stresses were determined by X-Ray diffraction (XRD), for 20MnCr5 ranging from approximately 6 - 14 vol.%, and under 2 vol.% for the alternative alloys. Hardness profiles were taken from the testing surfaces into the material core. The carburization of 20MnCr5 led to higher hardness and the greater concentration of carbon in the carburization zone more representative of a hardened SAE E52100, or 100Cr6/102Cr6, than of a non-case-hardened 20MnCr5. Residual stresses from machining and case-hardening were measured directly at the sample surface. The high-performance steels fulfilled manufacturer expectations in terms of elemental content, with hardness values between 50 - 55 HRC and strongly martensitic microstructure character. With characterization of the chosen materials complete, the materials could then be subjected to pre-conditioning. The first pre-conditioning method involved targeted generation of cold work hardening as induced boundary layers to protect the contact zone against wear. Work hardening was identified both by variations in residual stress profiles, i.e. the introduction of beneficial compressive residual stresses, and hardness increases in the contact zone, providing enhanced wear resistance. Parameters for work hardening were further optimized to reduce damage to the surface substrates of the
15. Rough-wall turbulent boundary layers with constant skin friction
Sridhar, A.; Pullin, D. I.; Cheng, W.
2017-01-01
A semi-empirical model is presented that describes the development of a fully developed turbulent boundary layer in the presence of surface roughness with length scale ks that varies with streamwise distance x . Interest is centred on flows
16. CFD simulation of the atmospheric boundary layer: wall function problems
Blocken, B.J.E.; Stathopoulos, T.; Carmeliet, J.
2007-01-01
Accurate Computational Fluid Dynamics (CFD) simulations of atmospheric boundary layer (ABL) flow are essential for a wide variety of atmospheric studies including pollutant dispersion and deposition. The accuracy of such simulations can be seriously compromised when wall-function roughness
17. Global instabilities and transient growth in Blasius boundary-layer ...
boundary-layer flow warrants attention. .... double prime indicates a dummy variable, while R and S respectively denote integration in the ..... (labelled) but it also features an unstable structural mode labelled S that ..... theory and experiment.
18. Accretion disc boundary layers - geometrically and optically thin case
Regev, Oded; Hougerat, A.A.
1988-01-01
The method of matched asymptotic expansions is applied to an optically and geometrically thin boundary layer between an accretion disc and the accreting star. Analytical solutions are presented for a particular viscosity prescription in the boundary layer. For a typical example we find that the disc closely resembles standard steady-disc theory. It is identical to it everywhere save a narrow boundary layer, where the temperature increases rapidly inward (by an order of magnitude), the angular velocity achieves maximum and decreases to its surface value and other variables also undergo rapid changes. This and previous work can now be used to calculate the emission from accretion discs including the boundary layers for a wide range of parameters. (author)
19. Boundary layer energies for nonconvex discrete systems
Scardia, L.; Schlömerkemper, A.; Zanini, C.
2011-01-01
In this work we consider a one-dimensional chain of atoms which interact through nearest and next-to-nearest neighbour interactions of Lennard-Jones type. We impose Dirichlet boundary conditions and in addition prescribe the deformation of the second and last but one atoms of the chain. This
20. Boundary-Layer Bypass Transition Over Large-Scale Bodies
2016-12-16
behaviour of the velocity and pressure changes with the curvature. This work aims to extend the results of the flat-plate boundary layer to a Rankine...example, consume an enormous amount of energy due to friction, many works have been directed to the suppression of transitional boundary layer disturbances...decrease of the enormous amount of energy consumed by airplanes during flight, moreover flight costs and aerodynamic noise could be reduced and number
1. On Hydromagnetic Stresses in Accretion Disk Boundary Layers
Pessah, Martin Elias; Chan, Chi-kwan
2012-01-01
Detailed calculations of the physical structure of accretion disk boundary layers, and thus their inferred observational properties, rely on the assumption that angular momentum transport is opposite to the radial angular frequency gradient of the disk. The standard model for turbulent shear...... of efficient angular momentum transport in the inner disk regions. This suggests that the detailed structure of turbulent MHD accretion disk boundary layers could differ appreciably from those derived within the standard framework of turbulent shear viscosity...
2. Control of Boundary Layers for Aero-optical Applications
2015-06-23
with some difficulty) from hot-wire velocity measurements, or computed directly from CFD results (e.g. Wang & Wang, 2012). Several different density...of experimental and computational research, especially applied to supersonic and hypersonic boundary layers; see Smits & Dussauge (1996), Spina et...Duan, L., Beekman, I. and Martin, M.P. (2010) Direct Numerical Simulation of Hypersonic Turbulent Boundary Layers. Part 2. Effect of Wall
3. MPLNET V3 Cloud and Planetary Boundary Layer Detection
Lewis, Jasper R.; Welton, Ellsworth J.; Campbell, James R.; Haftings, Phillip C.
2016-01-01
The NASA Micropulse Lidar Network Version 3 algorithms for planetary boundary layer and cloud detection are described and differences relative to the previous Version 2 algorithms are highlighted. A year of data from the Goddard Space Flight Center site in Greenbelt, MD consisting of diurnal and seasonal trends is used to demonstrate the results. Both the planetary boundary layer and cloud algorithms show significant improvement of the previous version.
4. Unequilibrium kinetic of collisionless boundary layers in binary plasmas
Kotelnikov, V.A.; Nikolaev, F.A.; Cherepanov, V.V.
1985-01-01
Relaxation processes of kinetic nonequilibrium collisionless boundary layers near spherical charged full absorbing surfaces in binary low-temperature plasmas are investigated. The effect of magnetic field on relaxation processes was neglected. The dynamics of components of the ionized gas was treated near the boundary layer. The potential distribution and the space dependence of concentration were calculated numerically. These results agree well with the experimental data. (D.Gy.)
5. Development of Robust Boundary Layer Controllers
Speyer, Jason
2002-01-01
.... The three-dimensional Navier-Stokes equations of channel flow, linearized about a Poisueille profile, and Oberbeck-Boussinesq equations of a layer of fluid, linearized about the no motion state...
6. Short climatology of the atmospheric boundary layer using acoustic methods
Schubert, J.F.
1975-06-01
A climatology of the boundary layer of the atmosphere at the Savannah River Laboratory is being compiled using acoustic methods. The atmospheric phenomenon as depicted on the facsimile recorder is classified and then placed into one of sixteen categories. After classification, the height of the boundary layer is measured. From this information, frequency tables of boundary layer height and category are created and then analyzed for the percentage of time that each category was detected by the acoustic sounder. The sounder also accurately depicts the diurnal cycle of the boundary layer and, depending on the sensitivity of the system, shows microstructure that is normally unavailable using other methods of profiling. The acoustic sounder provides a means for continuous, real time measurements of the time rate of change of the depth of the boundary layer. This continuous record of the boundary layer with its convective cells, gravity waves, inversions, and frontal system passages permits the synoptic and complex climatology of the local area to be compiled. (U.S.)
7. Turbulent boundary layer in high Rayleigh number convection in air.
du Puits, Ronald; Li, Ling; Resagk, Christian; Thess, André; Willert, Christian
2014-03-28
Flow visualizations and particle image velocimetry measurements in the boundary layer of a Rayleigh-Bénard experiment are presented for the Rayleigh number Ra=1.4×1010. Our visualizations indicate that the appearance of the flow structures is similar to ordinary (isothermal) turbulent boundary layers. Our particle image velocimetry measurements show that vorticity with both positive and negative sign is generated and that the smallest flow structures are 1 order of magnitude smaller than the boundary layer thickness. Additional local measurements using laser Doppler velocimetry yield turbulence intensities up to I=0.4 as in turbulent atmospheric boundary layers. From our observations, we conclude that the convective boundary layer becomes turbulent locally and temporarily although its Reynolds number Re≈200 is considerably smaller than the value 420 underlying existing phenomenological theories. We think that, in turbulent Rayleigh-Bénard convection, the transition of the boundary layer towards turbulence depends on subtle details of the flow field and is therefore not universal.
8. Turbulent Boundary Layers - Experiments, Theory and Modelling
1980-01-01
anemometry, London, Academic Press, 1976. 7. H.R.E. van Maanen, K. van der Molen and J. Blom, "Reduction of ambiguity noise in laser-Doppler...Raumfahrttechnik Hochschule der Bundeswehr München 8014 Neubiberg — Germany Professor Dr J.L. Van Ingen Dept. of Aerospace Engineering Delft...proximity to a solid boundary. J.Fluid Mech.12, 388 - 397, 1962. (21) Van Thin N., Messungen mit einem Hitzdraht in einer turbulenten Strömung in der
9. Plume meander and dispersion in a stable boundary layer
Hiscox, April L.; Miller, David R.; Nappo, Carmen J.
2010-11-01
Continuous lidar measurements of elevated plume dispersion and corresponding micrometeorology data are analyzed to establish the relationship between plume behavior and nocturnal boundary layer dynamics. Contrasting nights of data from the JORNADA field campaign in the New Mexico desert are analyzed. The aerosol lidar measurements were used to separate the plume diffusion (plume spread) from plume meander (displacement). Mutiresolution decomposition was used to separate the turbulence scale (90 s). Durations of turbulent kinetic energy stationarity and the wind steadiness were used to characterize the local scale and submesoscale turbulence. Plume meander, driven by submesoscale wind motions, was responsible for most of the total horizontal plume dispersion in weak and variable winds and strong stability. This proportion was reduced in high winds (i.e., >4 m s-1), weakly stable conditions but remained the dominant dispersion mechanism. The remainder of the plume dispersion in all cases was accounted for by internal spread of the plume, which is a small eddy diffusion process driven by turbulence. Turbulence stationarity and the wind steadiness are demonstrated to be closely related to plume diffusion and plume meander, respectively.
10. Shock-like structures in the tropical cyclone boundary layer
Williams, Gabriel J.; Taft, Richard K.; McNoldy, Brian D.; Schubert, Wayne H.
2013-06-01
This paper presents high horizontal resolution solutions of an axisymmetric, constant depth, slab boundary layer model designed to simulate the radial inflow and boundary layer pumping of a hurricane. Shock-like structures of increasing intensity appear for category 1-5 hurricanes. For example, in the category 3 case, the u>(∂u/∂r>) term in the radial equation of motion produces a shock-like structure in the radial wind, i.e., near the radius of maximum tangential wind the boundary layer radial inflow decreases from approximately 22 m s-1 to zero over a radial distance of a few kilometers. Associated with this large convergence is a spike in the radial distribution of boundary layer pumping, with updrafts larger than 22 m s-1 at a height of 1000 m. Based on these model results, it is argued that observed hurricane updrafts of this magnitude so close to the ocean surface are attributable to the dry dynamics of the frictional boundary layer rather than moist convective dynamics. The shock-like structure in the boundary layer radial wind also has important consequences for the evolution of the tangential wind and the vertical component of vorticity. On the inner side of the shock the tangential wind tendency is essentially zero, while on the outer side of the shock the tangential wind tendency is large due to the large radial inflow there. The result is the development of a U-shaped tangential wind profile and the development of a thin region of large vorticity. In many respects, the model solutions resemble the remarkable structures observed in the boundary layer of Hurricane Hugo (1989).
11. Coherent structures in wave boundary layers. Part 1. Oscillatory motion
Carstensen, Stefan; Sumer, B. Mutlu; Fredsøe, Jørgen
2010-01-01
This work concerns oscillatory boundary layers over smooth beds. It comprises combined visual and quantitative techniques including bed shear stress measurements. The experiments were carried out in an oscillating water tunnel. The experiments reveal two significant coherent flow structures: (i......) Vortex tubes, essentially two-dimensional vortices close to the bed extending across the width of the boundary-layer flow, caused by an inflectional-point shear layer instability. The imprint of these vortices in the bed shear stress is a series of small, insignificant kinks and dips. (ii) Turbulent...... spots, isolated arrowhead-shaped areas close to the bed in an otherwise laminar boundary layer where the flow ‘bursts’ with violent oscillations. The emergence of the turbulent spots marks the onset of turbulence. Turbulent spots cause single or multiple violent spikes in the bed shear stress signal...
12. Definition of Turbulent Boundary-Layer with Entropy Concept
Zhao Rui
2016-01-01
Full Text Available The relationship between the entropy increment and the viscosity dissipation in turbulent boundary-layer is systematically investigated. Through theoretical analysis and direct numerical simulation (DNS, an entropy function fs is proposed to distinguish the turbulent boundary-layer from the external flow. This approach is proved to be reliable after comparing its performance in the following complex flows, namely, low-speed airfoil flows with different wall temperature, supersonic cavity-ramp flow dominated by the combination of free-shear layer, larger recirculation and shocks, and the hypersonic flow past an aeroplane configuration. Moreover, fs is deduced from the point of energy, independent of any particular turbulent quantities. That is, this entropy concept could be utilized by other engineering applications related with turbulent boundary-layer, such as turbulence modelling transition prediction and engineering thermal protection.
13. Response of neutral boundary-layers to changes of roughness
Sempreviva, Anna Maria; Larsen, Søren Ejling; Mortensen, Niels Gylling
1990-01-01
boundary layer where again the drag laws can be used to estimate the surface wind. To study this problem, data have been sampled for two years from four 30-m meteorological masts placed from 0 to 30 km inland from the North Sea coast of Jutland in Denmark. The present analysis is limited to neutral......When air blows across a change in surface roughness, an internal boundary layer (IBL) develops within which the wind adapts to the new surface. This process is well described for short fetches, > 1 km. However, few data exist for large fetches on how the IBL grows to become a new equilibrium...... stratification, and the surface roughness is the main parameter. The analysis of wind data and two simple models, a surface layer and a planetary boundary layer (PBL) model, are described. Results from both models are discussed and compared with data analysis. Model parameters have been evaluated and the model...
14. Vortex Generators to Control Boundary Layer Interactions
Babinsky, Holger (Inventor); Loth, Eric (Inventor); Lee, Sang (Inventor)
2014-01-01
Devices for generating streamwise vorticity in a boundary includes various forms of vortex generators. One form of a split-ramp vortex generator includes a first ramp element and a second ramp element with front ends and back ends, ramp surfaces extending between the front ends and the back ends, and vertical surfaces extending between the front ends and the back ends adjacent the ramp surfaces. A flow channel is between the first ramp element and the second ramp element. The back ends of the ramp elements have a height greater than a height of the front ends, and the front ends of the ramp elements have a width greater than a width of the back ends.
15. Micro Ramps in Supersonic Turbulent Boundary Layers : An experimental and numerical study
Sun, Z.
2014-01-01
The micro vortex generator (MVG) is used extensively in low speed aerodynamic problems and is now extended into the supersonic flow regime to solve undesired flow features that are associated with shock wave boundary layer interactions (SWBLI) such as flow separation and associated unsteadiness of
16. Particle motion in atmospheric boundary layers of Mars and Earth
White, B. R.; Iversen, J. D.; Greeley, R.; Pollack, J. B.
1975-01-01
To study the eolian mechanics of saltating particles, both an experimental investigation of the flow field around a model crater in an atmospheric boundary layer wind tunnel and numerical solutions of the two- and three-dimensional equations of motion of a single particle under the influence of a turbulent boundary layer were conducted. Two-dimensional particle motion was calculated for flow near the surfaces of both Earth and Mars. For the case of Earth both a turbulent boundary layer with a viscous sublayer and one without were calculated. For the case of Mars it was only necessary to calculate turbulent boundary layer flow with a laminar sublayer because of the low values of friction Reynolds number; however, it was necessary to include the effects of slip flow on a particle caused by the rarefied Martian atmosphere. In the equations of motion the lift force functions were developed to act on a single particle only in the laminar sublayer or a corresponding small region of high shear near the surface for a fully turbulent boundary layer. The lift force functions were developed from the analytical work by Saffman concerning the lift force acting on a particle in simple shear flow.
17. Coupled vs. decoupled boundary layers in VOCALS-REx
C. R. Jones
2011-07-01
Full Text Available We analyze the extent of subtropical stratocumulus-capped boundary layer decoupling and its relation to other boundary-layer characteristics and forcings using aircraft observations from VOCALS-REx along a swath of the subtropical southeast Pacific Ocean running west 1600 km from the coast of Northern Chile. We develop two complementary and consistent measures of decoupling. The first is based on boundary layer moisture and temperature stratification in flight profiles from near the surface to above the capping inversion, and the second is based the difference between the lifted condensation level (LCL and a mean lidar-derived cloud base measured on flight legs at 150 m altitude. Most flights took place during early-mid morning, well before the peak in insolation-induced decoupling.
We find that the boundary layer is typically shallower, drier, and well mixed near the shore, and tends to deepen, decouple, and produce more drizzle further offshore to the west. Decoupling is strongly correlated to the "mixed layer cloud thickness", defined as the difference between the capping inversion height and the LCL; other factors such as wind speed, cloud droplet concentration, and inversion thermodynamic jumps have little additional explanatory power. The results are broadly consistent with the deepening-warming theory of decoupling.
In the deeper boundary layers observed well offshore, there was frequently nearly 100 % boundary-layer cloud cover despite pronounced decoupling. The cloud cover was more strongly correlated to a κ parameter related to the inversion jumps of humidity and temperature, though the exact functional relation is slightly different than found in prior large-eddy simulation studies.
18. Structure of reconnection boundary layers in incompressible MHD
Sonnerup, B.U.Oe.; Wang, D.J.
1987-01-01
The incompressible MHD equations with nonvanishing viscosity and resistivity are simplified by use of the boundary layer approximation to describe the flow and magnetic field in the exit flow regions of magnetic field reconnection configurations when the reconnection rate is small. The conditions are derived under which self-similar solutions exist of the resulting boundary layer equations. For the case of zero viscosity and resistivity, the equations describing such self-similar layers are then solved in terms of quadratures, and the resulting flow and field configurations are described. Symmetric solutions, relevant, for example, to reconnection in the geomagnetic tail, as well as asymmetric solutions, relevant to reconnection at the earth's magnetopause, are found to exist. The nature of the external solutions to which the boundary layer solutions should be matched is discussed briefly, but the actual matching, which is to occur at Alfven-wave characteristic curves in the boundary layer solutions, is not carried out. Finally, it is argued that the solutions obtained may also be used to describe the structure of the intense vortex layers observed to occur at magnetic separatrices in computer simulations and in certain analytical models of the reconnection process
Donald M. McEligot
2014-07-01
Full Text Available In an earlier paper in Entropy [1] we hypothesized that the entropy generation rate is the driving force for boundary layer transition from laminar to turbulent flow. Subsequently, with our colleagues we have examined the prediction of entropy generation during such transitions [2,3]. We found that reasonable predictions for engineering purposes could be obtained for flows with negligible streamwise pressure gradients by adapting the linear combination model of Emmons [4]. A question then arises—will the Emmons approach be useful for boundary layer transition with significant streamwise pressure gradients as by Nolan and Zaki [5]. In our implementation the intermittency is calculated by comparison to skin friction correlations for laminar and turbulent boundary layers and is then applied with comparable correlations for the energy dissipation coefficient (i.e., non-dimensional integral entropy generation rate. In the case of negligible pressure gradients the Blasius theory provides the necessary laminar correlations.
20. Rotor blade boundary layer measurement hardware feasibility demonstration
Clark, D. R.; Lawton, T. D.
1972-01-01
A traverse mechanism which allows the measurement of the three dimensional boundary layers on a helicopter rotor blade has been built and tested on a full scale rotor to full scale conditions producing centrifugal accelerations in excess of 400 g and Mach numbers of 0.6 and above. Boundary layer velocity profiles have been measured over a range of rotor speeds and blade collective pitch angles. A pressure scanning switch and transducer were also tested on the full scale rotor and found to be insensitive to centrifugal effects within the normal main rotor operating range. The demonstration of the capability to measure boundary layer behavior on helicopter rotor blades represents the first step toward obtaining, in the rotating system, data of a quality comparable to that already existing for flows in the fixed system.
1. Theoretical skin-friction law in a turbulent boundary layer
Cheskidov, A.
2005-01-01
We study transitional and turbulent boundary layers using a turbulent velocity profile equation recently derived from the Navier-Stokes-alpha and Leray-alpha models. From this equation we obtain a theoretical prediction of the skin-friction coefficient in a wide range of Reynolds numbers based on momentum thickness, and deduce the maximal value of c f max =0.0063 for turbulent velocity profiles. A two-parameter family of solutions to the equation matches experimental data in the transitional boundary layers with different free-stream turbulence intensity, while one-parameter family of solutions, obtained using our skin-friction coefficient law, matches experimental data in the turbulent boundary layer for moderately large Reynolds numbers
2. Vortex Generator Induced Flow in a High Re Boundary Layer
Velte, Clara Marika; Braud, C.; Coudert, S.
2014-01-01
Stereoscopic Particle Image Velocimetry measurements have been conducted in cross-planes behind three different geometries of Vortex Generators (VGs) in a high Reynolds number boundary layer. The VGs have been mounted in a cascade producing counter-rotating vortices and the downstream flow...... development was examined. Three VG geometries were investigated: rectangular, triangular and cambered. The various VG geometries tested are seen to produce different impacts on the boundary layer flow. Helical symmetry of the generated vortices is confirmed for all investigated VG geometries in this high...... Reynolds number boundary layer. From the parameters resulting from this analysis, it is observed at the most upstream measurement position that the rectangular and triangular VGs produce vortices of similar size, strength and velocity induction whilst the cambered VGs produce smaller and weaker vortices...
3. Vortex Generator Induced Flow in a High Re Boundary Layer
Velte, Clara Marika; Braud, C.; Coudert, S.
2012-01-01
Stereoscopic Particle Image Velocimetry measurements have been conducted in cross-planes behind three different geometries of Vortex Generators (VGs) in a high Reynolds number boundary layer. The VGs have been mounted in a cascade producing counter-rotating vortices and the downstream flow...... development was examined. Three VG geometries were investigated: rectangular, triangular and cambered. The various VG geometries tested are seen to produce different impacts on the boundary layer flow. Helical symmetry of the generated vortices is confirmed for all investigated VG geometries in this high...... Reynolds number boundary layer. From the parameters resulting from this analysis, it is observed at the most upstream measurement position that the rectangular and triangular VGs produce vortices of similar size, strength and velocity induction whilst the cambered VGs produce smaller and weaker vortices...
4. Defects and boundary layers in non-Euclidean plates
Gemmer, J A; Venkataramani, S C
2012-01-01
We investigate the behaviour of non-Euclidean plates with constant negative Gaussian curvature using the Föppl–von Kármán reduced theory of elasticity. Motivated by recent experimental results, we focus on annuli with a periodic profile. We prove rigorous upper and lower bounds for the elastic energy that scales like the thickness squared. In particular we show that are only two types of global minimizers—deformations that remain flat and saddle shaped deformations with isolated regions of stretching near the edge of the annulus. We also show that there exist local minimizers with a periodic profile that have additional boundary layers near their lines of inflection. These additional boundary layers are a new phenomenon in thin elastic sheets and are necessary to regularize jump discontinuities in the azimuthal curvature across lines of inflection. We rigorously derive scaling laws for the width of these boundary layers as a function of the thickness of the sheet. (paper)
5. Mixed convection boundary layer flow over a vertical surface embedded in a thermally stratified porous medium
Ishak, Anuar; Nazar, Roslinda; Pop, Ioan
2008-01-01
The mixed convection boundary layer flow through a stable stratified porous medium bounded by a vertical surface is investigated. The external velocity and the surface temperature are assumed to vary as x m , where x is measured from the leading edge of the vertical surface and m is a constant. Numerical solutions for the governing Darcy and energy equations are obtained. The results indicate that the thermal stratification significantly affects the surface shear stress as well as the surface heat transfer, besides delays the boundary layer separation
6. Boundary layer on a flat plate with suction
Favre, A.; Dumas, R.; Verollet, E.
1961-01-01
This research done in wind tunnel concerns the turbulent boundary layer of a porous flat plate with suction. The porous wall is 1 m long and begins 1 m downstream of the leading edge. The Reynolds number based on the boundary layer thickness is of the order of 16.300. The suction rate defined as the ratio of the velocity perpendicular to the wall to the external flow velocity ranges from 0 to 2 per cent. The pressure gradient can be controlled. The mean velocity profiles have been determined for various positions and suction rates by means of total pressure probes together with the intensities of the turbulent velocity fluctuations components, energy spectra and correlations by means of hot wire anemometers, spectral analyser and correlator. The stream lines, the values of the viscous and turbulent shear stresses, of the local wall friction, of the turbulent energy production term, with some information on the dissipation of the energy have been derived from these measurements. For these data the integral of equation of continuity in boundary layer have been drawn. The suction effects on the boundary layer are important. The suction thoroughly alters the mean velocity profiles by increasing the viscous shear stresses near the wall and decreasing them far from the wall, it diminishes the longitudinal and transversal turbulence intensities, the turbulent shear stresses, and the production of energy of turbulence. These effects are much stressed in the inner part of the boundary layer. On the other hand the energy spectra show that the turbulence scale is little modified, the boundary layer thickness being not much diminished by the suction. The suction effects can be appreciated by comparing twice the suction rate to the wall friction coefficient (assumed airtight), quite noticeable as soon as the rate is about unity, they become very important when it reaches ten. (author) [fr
7. Contributions of the wall boundary layer to the formation of the counter-rotating vortex pair in transverse jets
SCHLEGEL, FABRICE
2011-04-08
Using high-resolution 3-D vortex simulations, this study seeks a mechanistic understanding of vorticity dynamics in transverse jets at a finite Reynolds number. A full no-slip boundary condition, rigorously formulated in terms of vorticity generation along the channel wall, captures unsteady interactions between the wall boundary layer and the jet - in particular, the separation of the wall boundary layer and its transport into the interior. For comparison, we also implement a reduced boundary condition that suppresses the separation of the wall boundary layer away from the jet nozzle. By contrasting results obtained with these two boundary conditions, we characterize near-field vortical structures formed as the wall boundary layer separates on the backside of the jet. Using various Eulerian and Lagrangian diagnostics, it is demonstrated that several near-wall vortical structures are formed as the wall boundary layer separates. The counter-rotating vortex pair, manifested by the presence of vortices aligned with the jet trajectory, is initiated closer to the jet exit. Moreover tornado-like wall-normal vortices originate from the separation of spanwise vorticity in the wall boundary layer at the side of the jet and from the entrainment of streamwise wall vortices in the recirculation zone on the lee side. These tornado-like vortices are absent in the case where separation is suppressed. Tornado-like vortices merge with counter-rotating vorticity originating in the jet shear layer, significantly increasing wall-normal circulation and causing deeper jet penetration into the crossflow stream. © 2011 Cambridge University Press.
8. Receptivity of Hypersonic Boundary Layers over Straight and Flared Cones
Balakumar, Ponnampalam; Kegerise, Michael A.
2010-01-01
The effects of adverse pressure gradients on the receptivity and stability of hypersonic boundary layers were numerically investigated. Simulations were performed for boundary layer flows over a straight cone and two flared cones. The steady and the unsteady flow fields were obtained by solving the two-dimensional Navier-Stokes equations in axi-symmetric coordinates using the 5th order accurate weighted essentially non-oscillatory (WENO) scheme for space discretization and using third-order total-variation-diminishing (TVD) Runge-Kutta scheme for time integration. The mean boundary layer profiles were analyzed using local stability and non-local parabolized stability equations (PSE) methods. After the most amplified disturbances were identified, two-dimensional plane acoustic waves were introduced at the outer boundary of the computational domain and time accurate simulations were performed. The adverse pressure gradient was found to affect the boundary layer stability in two important ways. Firstly, the frequency of the most amplified second-mode disturbance was increased relative to the zero pressure gradient case. Secondly, the amplification of first- and second-mode disturbances was increased. Although an adverse pressure gradient enhances instability wave growth rates, small nose-tip bluntness was found to delay transition due to the low receptivity coefficient and the resulting weak initial amplitude of the instability waves. The computed and measured amplitude-frequency spectrums in all three cases agree very well in terms of frequency and the shape except for the amplitude.
9. An Innovative Flow-Measuring Device: Thermocouple Boundary Layer Rake
Hwang, Danny P.; Fralick, Gustave C.; Martin, Lisa C.; Wrbanek, John D.; Blaha, Charles A.
2001-01-01
An innovative flow-measuring device, a thermocouple boundary layer rake, was developed. The sensor detects the flow by using a thin-film thermocouple (TC) array to measure the temperature difference across a heater strip. The heater and TC arrays are microfabricated on a constant-thickness quartz strut with low heat conductivity. The device can measure the velocity profile well into the boundary layer, about 65 gm from the surface, which is almost four times closer to the surface than has been possible with the previously used total pressure tube.
10. Leading edge effect in laminar boundary layer excitation by sound
Leehey, P.; Shapiro, P.
1980-01-01
Essentially plane pure tone sound waves were directed downstream over a heavily damped smooth flat plate installed in a low turbulence (0.04%) subsonic wind tunnel. Laminar boundary layer disturbance growth rates were measured with and without sound excitation and compared with numerical results from spatial stability theory. The data indicate that the sound field and Tollmien-Schlichting (T-S) waves coexist with comparable amplitudes when the latter are damped; moreover, the response is linear. Higher early growth rates occur for excitation by sound than by stream turbulence. Theoretical considerations indicate that the boundary layer is receptive to sound excitation primarily at the test plate leading edge. (orig.)
11. Lower Atmospheric Boundary Layer Experiment (LABLE) Final Campaign Report
Klein, P [University of Oklahoma - School of Meteorology; Bonin, TA; Newman, JF [National Renewable Energy Laboratory; Turner, DD [National Oceanic and Atmospheric Administration; Chilson, P [University of Oklahoma; Blumberg, WG [University of Oklahoma; Mishra, S; Wainwright, CE; Carney, M [University of Oklahoma - School of Meteorology; Jacobsen, EP [University of Oklahoma; Wharton, S [Lawrence Livermore National Laboratory
2015-11-01
The Lower Atmospheric Boundary Layer Experiment (LABLE) included two measurement campaigns conducted at the Atmospheric Radiation Measurement (ARM) Southern Great Plains site in Oklahoma during 2012 and 2013. LABLE was designed as a multi-phase, low-cost collaboration among the University of Oklahoma, the National Severe Storms Laboratory, Lawrence Livermore National Laboratory, and the ARM program. A unique aspect was the role of graduate students in LABLE. They served as principal investigators and took the lead in designing and conducting experiments using different sampling strategies to best resolve boundary-layer phenomena.
12. Interactive boundary-layer calculations of a transonic wing flow
Kaups, Kalle; Cebeci, Tuncer; Mehta, Unmeel
1989-01-01
Results obtained from iterative solutions of inviscid and boundary-layer equations are presented and compared with experimental values. The calculated results were obtained with an Euler code and a transonic potential code in order to furnish solutions for the inviscid flow; they were interacted with solutions of two-dimensional boundary-layer equations having a strip-theory approximation. Euler code results are found to be in better agreement with the experimental data than with the full potential code, especially in the presence of shock waves, (with the sole exception of the near-tip region).
13. Analysis of diabatic flow modification in the internal boundary layer
Floors, Rogier; Gryning, Sven-Erik; Pena Diaz, Alfredo
2011-01-01
Measurements at two meteorological masts in Denmark, Horns Rev in the sea and Høvsøre near the coastline on land, are used to analyze the behaviour of the flow after a smooth-to-rough change in surface conditions. The study shows that the wind profile within the internal boundary layer is control......Measurements at two meteorological masts in Denmark, Horns Rev in the sea and Høvsøre near the coastline on land, are used to analyze the behaviour of the flow after a smooth-to-rough change in surface conditions. The study shows that the wind profile within the internal boundary layer...
14. Experimental demonstration of the Rayleigh acoustic viscous boundary layer theory.
Castrejón-Pita, J R; Castrejón-Pita, A A; Huelsz, G; Tovar, R
2006-03-01
Amplitude and phase velocity measurements on the laminar oscillatory viscous boundary layer produced by acoustic waves are presented. The measurements were carried out in acoustic standing waves in air with frequencies of 68.5 and 114.5 Hz using laser Doppler anemometry and particle image velocimetry. The results obtained by these two techniques are in good agreement with the predictions made by the Rayleigh viscous boundary layer theory and confirm the existence of a local maximum of the velocity amplitude and its expected location.
15. Effects of shock on hypersonic boundary layer stability
Pinna, F.; Rambaud, P.
2013-06-01
The design of hypersonic vehicles requires the estimate of the laminar to turbulent transition location for an accurate sizing of the thermal protection system. Linear stability theory is a fast scientific way to study the problem. Recent improvements in computational capabilities allow computing the flow around a full vehicle instead of using only simplified boundary layer equations. In this paper, the effect of the shock is studied on a mean flow provided by steady Computational Fluid Dynamics (CFD) computations and simplified boundary layer calculations.
16. Conference on Boundary and Interior Layers : Computational and Asymptotic Methods
2015-01-01
This volume offers contributions reflecting a selection of the lectures presented at the international conference BAIL 2014, which was held from 15th to 19th September 2014 at the Charles University in Prague, Czech Republic. These are devoted to the theoretical and/or numerical analysis of problems involving boundary and interior layers and methods for solving these problems numerically. The authors are both mathematicians (pure and applied) and engineers, and bring together a large number of interesting ideas. The wide variety of topics treated in the contributions provides an excellent overview of current research into the theory and numerical solution of problems involving boundary and interior layers. .
17. Rough-wall turbulent boundary layers with constant skin friction
Sridhar, A.
2017-03-28
A semi-empirical model is presented that describes the development of a fully developed turbulent boundary layer in the presence of surface roughness with length scale ks that varies with streamwise distance x . Interest is centred on flows for which all terms of the von Kármán integral relation, including the ratio of outer velocity to friction velocity U+∞≡U∞/uτ , are streamwise constant. For Rex assumed large, use is made of a simple log-wake model of the local turbulent mean-velocity profile that contains a standard mean-velocity correction for the asymptotic fully rough regime and with assumed constant parameter values. It is then shown that, for a general power-law external velocity variation U∞∼xm , all measures of the boundary-layer thickness must be proportional to x and that the surface sand-grain roughness scale variation must be the linear form ks(x)=αx , where x is the distance from the boundary layer of zero thickness and α is a dimensionless constant. This is shown to give a two-parameter (m,α) family of solutions, for which U+∞ (or equivalently Cf ) and boundary-layer thicknesses can be simply calculated. These correspond to perfectly self-similar boundary-layer growth in the streamwise direction with similarity variable z/(αx) , where z is the wall-normal coordinate. Results from this model over a range of α are discussed for several cases, including the zero-pressure-gradient ( m=0 ) and sink-flow ( m=−1 ) boundary layers. Trends observed in the model are supported by wall-modelled large-eddy simulation of the zero-pressure-gradient case for Rex in the range 108−1010 and for four values of α . Linear streamwise growth of the displacement, momentum and nominal boundary-layer thicknesses is confirmed, while, for each α , the mean-velocity profiles and streamwise turbulent variances are found to collapse reasonably well onto z/(αx) . For given α , calculations of U+∞ obtained from large-eddy simulations are streamwise
18. Experimental measurements and modelling of the WEGA boundary layer plasma
El Shaer, M.; Ichtchenko, G.
1983-02-01
The boundary layer of the WEGA Tokamak has been investigated by using specific diagnostics: movable 4 mm microwave interferometer, several types of movable and fixed probes, Katsumata probe, and multigrid electrostatic analyzer. During the RF heating at the lower hybrid frequency, some modifications in the parameters of the boundary layer are observed which are interpreted by the ponderomotive force effects. A comparison between the measured reflection coefficients of the grill waveguides and their predicted values by a coupling theory (not taking into account the real conditions facing the Grill) is presented. A diffusion model was also made to describe this particular region and to fit the experimental results
19. Atmospheric boundary layers in storms: advanced theory and modelling applications
S. S. Zilitinkevich
2005-01-01
Full Text Available Turbulent planetary boundary layers (PBLs control the exchange processes between the atmosphere and the ocean/land. The key problems of PBL physics are to determine the PBL height, the momentum, energy and matter fluxes at the surface and the mean wind and scalar profiles throughout the layer in a range of regimes from stable and neutral to convective. Until present, the PBLs typical of stormy weather were always considered as neutrally stratified. Recent works have disclosed that such PBLs are in fact very strongly affected by the static stability of the free atmosphere and must be treated as factually stable (we call this type of the PBL "conventionally neutral" in contract to the "truly neutral" PBLs developed against the neutrally stratified free flow. It is common knowledge that basic features of PBLs exhibit a noticeable dependence on the free-flow static stability and baroclinicity. However, the concern of the traditional theory of neural and stable PBLs was almost without exception the barotropic nocturnal PBL, which develops at mid latitudes during a few hours in the night, on the background of a neutral or slightly stable residual layer. The latter separates this type of the PBL from the free atmosphere. It is not surprising that the nature of turbulence in such regimes is basically local and does not depend on the properties of the free atmosphere. Alternatively, long-lived neutral (in fact only conditionally neutral or stable PBLs, which have much more time to grow up, are placed immediately below the stably stratified free flow. Under these conditions, the turbulent transports of momentum and scalars even in the surface layer - far away from the PBL outer boundary - depend on the free-flow Brunt-Väisälä frequency, N. Furthermore, integral measures of the long-lived PBLs (their depths and the resistance law functions depend on N and also on the baroclinic shear, S. In the traditional PBL models both non-local parameters N and S
20. Appraisal of boundary layer trips for landing gear testing
McCarthy, Philip; Feltham, Graham; Ekmekci, Alis
2013-11-01
Dynamic similarity during scaled model testing is difficult to maintain. Forced boundary layer transition via a surface protuberance is a common method used to address this issue, however few guidelines exist for the effective tripping of complex geometries, such as aircraft landing gears. To address this shortcoming, preliminary wind tunnel tests were performed at Re = 500,000. Surface transition visualisation and pressure measurements show that zigzag type trips of a given size and location are effective at promoting transition, thus preventing the formation of laminar separation bubbles and increasing the effective Reynolds number from the critical regime to the supercritical regime. Extension of these experiments to include three additional tripping methods (wires, roughness strips, CADCUT dots) in a range of sizes, at Reynolds number of 200,000 and below, have been performed in a recirculating water channel. Analysis of surface pressure measurements and time resolved PIV for each trip device, size and location has established a set of recommendations for successful use of tripping for future, low Reynolds number landing gear testing.
1. Surface influence upon vertical profiles in the nocturnal boundary layer
Garratt, J. R.
1983-05-01
Near-surface wind profiles in the nocturnal boundary layer, depth h, above relatively flat, tree-covered terrain are described in the context of the analysis of Garratt (1980) for the unstable atmospheric boundary layer. The observations at two sites imply a surface-based transition layer, of depth z *, within which the observed non-dimensional profiles Φ M 0 are a modified form of the inertial sub-layer relation Φ _M ( {{z L}} = ( {{{1 + 5_Z } L}} ) according to Φ _M^{{0}} ˜eq ( {{{1 + 5z} L}} )exp [ { - 0.7( {{{1 - z} z}_ * } )] , where z is height above the zero-plane displacement and L is the Monin-Obukhov length. At both sites the depth z * is significantly smaller than the appropriate neutral value ( z * N ) found from the previous analysis, as might be expected in the presence of a buoyant sink for turbulent kinetic energy.
2. Fast Fermi acceleration in the plasma sheet boundary layer
Wu, C.S.; Lui, A.T.Y.
1989-01-01
A longstanding question in the field of magnetospheric physics is the source of the energetic particles which are commonly observed along the plasma sheet boundary layer (PSBL). Several models have been suggested for the acceleration of these particles. We suggest a means by which the fast Fermi acceleration mechanism [Wu, 1984] can accelerate electrons at the plasma sheet and perhaps account for some of the observations. We propose the following: A localized hydromagnetic disturbance propagating through the tail lobe region impinges upon the PSBL deforming it and displacing it in towards the central plasma sheet. The boundary layer can then act like a moving magnetic mirror. If the disturbance is propagating nearly perpendicular to the layer then its velocity projected parallel to the layer (and the magnetic field) can be very large resulting in significant acceleration of reflected particles. copyright American Geophysical Union 1989
3. Interaction of Atmospheric Turbulence with Blade Boundary Layer Dynamics on a 5MW Wind Turbine using Blade-Boundary-Layer-Resolved CFD with hybrid URANS-LES.
Vijayakumar, Ganesh [National Renewable Energy Lab. (NREL), Golden, CO (United States); Pennsylvania State Univ., University Park, PA (United States); Brasseur, James [Pennsylvania State Univ., University Park, PA (United States); Univ. of Colorado, Boulder, CO (United States); Lavely, Adam; Jayaraman, Balaji; Craven, Brent
2016-01-04
We describe the response of the NREL 5 MW wind turbine blade boundary layer to the passage of atmospheric turbulence using blade-boundary-layer-resolved computational fluid dynamics with hybrid URANS-LES modeling.
4. Transitional boundary layer in low-Prandtl-number convection at high Rayleigh number
Schumacher, Joerg; Bandaru, Vinodh; Pandey, Ambrish; Scheel, Janet
2016-11-01
The boundary layer structure of the velocity and temperature fields in turbulent Rayleigh-Bénard flows in closed cylindrical cells of unit aspect ratio is revisited from a transitional and turbulent viscous boundary layer perspective. When the Rayleigh number is large enough the boundary layer dynamics at the bottom and top plates can be separated into an impact region of downwelling plumes, an ejection region of upwelling plumes and an interior region (away from side walls) that is dominated by a shear flow of varying orientation. This interior plate region is compared here to classical wall-bounded shear flows. The working fluid is liquid mercury or liquid gallium at a Prandtl number of Pr = 0 . 021 for a range of Rayleigh numbers of 3 ×105 Deutsche Forschungsgemeinschaft.
5. Study on Reflected Shock Wave/Boundary Layer Interaction in a Shock Tube
Kim, Dong Wook; Kim, Tae Ho; Kim, Heuy Dong [Andong Nat’l Univ., Andong (Korea, Republic of)
2017-07-15
The interaction between a shock wave and a boundary layer causes boundary layer separation, shock train, and in some cases, strong unsteadiness in the flow field. Such a situation is also observed in a shock tube, where the reflected shock wave interacts with the unsteady boundary layer. However, only a few studies have been conducted to investigate the shock train phenomenon in a shock tube. In the present study, numerical studies were conducted using the two-dimensional axisymmetric domain of a shock tube, and compressible Navier-Stokes equations were solved to clarify the flow characteristics of shock train phenomenon inside a shock tube. A detailed wave diagram was developed based on the present computational results, which were validated with existing experimental data.
6. Separate the inseparable one-layer mapping
Hu, Chia-Lun J.
2000-04-01
When the input-output mapping of a one-layered perceptron (OLP) does NOT meet the PLI condition which is the if-and- only-if, or 'IFF, condition that the mapping can be realized by a OLP, then no matter what learning rule we use, a OLP just cannot realize this mapping at all. However, because of the nature of the PLI, one can still construct a parallel- cascaded, two-layered perceptron system to realize this illegal' mapping. Theory and design example of this novel design will be reported in detail in this paper.
7. A numerical investigation of the impact of surface topology on laminar boundary layers
Beratlis, Nikolaos; Squires, Kyle; Balaras, Elias
2015-11-01
Surface topology, such as dimples or trip wires, has been utilized in the past for passive separation control over bluff bodies. The majority of the work, however, has focused on the indirect effects on the drag and lift forces, while the details of the impact on the boundary layer evolution are not well understood. Here we report a series of DNS of flow over a single row of spherical and hexagonal dimples, as well as, circular grooves. The Reynolds number and the thickness of the incoming laminar boundary layer is carefully controlled. In all cases transition to turbulence downstream of the elements comes with reorientation of the spanwise vorticity into hairpin like vortices. Although qualitatively the transition mechanism amongst different dimples and grooves is similar, important quantitative differences exist: two-dimensional geometries such as the groove, are more stable than three-dimensional geometries. In addition, it was found that the cavity geometry controls the initial thickness of the boundary layer and practically results in a shift of the virtual origin of the turbulent boundary layer. Important differences in the momentum transport downstream of the dimples exist, but in all cases the boundary layer evolves in a self-similar manner.
8. Temperature boundary layer profiles in turbulent Rayleigh-Benard convection
Ching, Emily S. C.; Emran, Mohammad S.; Horn, Susanne; Shishkina, Olga
2017-11-01
Classical boundary-layer theory for steady flows cannot adequately describe the boundary layer profiles in turbulent Rayleigh-Benard convection. We have developed a thermal boundary layer equation which takes into account fluctuations in terms of an eddy thermal diffusivity. Based on Prandtl's mixing length ideas, we relate the eddy thermal diffusivity to the stream function. With this proposed relation, we can solve the thermal boundary layer equation and obtain a closed-form expression for the dimensionless mean temperature profile in terms of two independent parameters: θ(ξ) =1/b∫0b ξ [ 1 +3a3/b3(η - arctan(η)) ] - c dη , where ξ is the similarity variable and the parameters a, b, and c are related by the condition θ(∞) = 1 . With a proper choice of the parameters, our predictions of the temperature profile are in excellent agreement with the results of our direct numerical simulations for a wide range of Prandtl numbers (Pr), from Pr=0.01 to Pr=2547.9. OS, ME and SH acknowledge the financial support by the Deutsche Forschungsgemeinschaft (DFG) under Grants Sh405/4-2 (Heisenberg fellowship), Sh405/3-2 and Ho 5890/1-1, respectively.
9. Conserved variable analysis of the marine boundary layer and air
The present study is based on the observed features of the MBL (Marine Boundary Layer) during the Bay of Bengal and Monsoon Experiment (BOBMEX) - Pilot phase. Conserved Variable Analysis (CVA) of the conserved variables such as potential temperature, virtual potential temperature, equivalent potential temperature ...
10. Axisymmetric free convection boundary-layer flow past slender bodies
Kuiken, H.K.
1968-01-01
Radial curvature effects on axisymmetric free convection boundary-layer flow are investigated for vertical cylinders and cones for some special non-uniform temperature differences between the surface and the ambient fluid. The solution is given as a power series expansion, the first term being equal
11. Exact solution of nonsteady thermal boundary layer equation
Dorfman, A.S.
1995-01-01
There are only a few exact solutions of the thermal boundary layer equation. Most of them are derived for a specific surface temperature distribution. The first exact solution of the steady-state boundary layer equation was given for a plate with constant surface temperature and free-stream velocity. The same problem for a plate with polynomial surface temperature distribution was solved by Chapmen and Rubesin. Levy gave the exact solution for the case of a power law distribution of both surface temperature and free-stream velocity. The exact solution of the steady-state boundary layer equation for an arbitrary surface temperature and a power law free-stream velocity distribution was given by the author in two forms: of series and of the integral with an influence function of unheated zone. A similar solution of the nonsteady thermal boundary layer equation for an arbitrary surface temperature and a power law free-stream velocity distribution is presented here. In this case, the coefficients of series depend on time, and in the limit t → ∞ they become the constant coefficients of a similar solution published before. This solution, unlike the one presented here, does not satisfy the initial conditions at t = 0, and, hence, can be used only in time after the beginning of the process. The solution in the form of a series becomes a closed-form exact solution for polynomial surface temperature and a power law free-stream velocity distribution. 7 refs., 2 figs
12. Exchange Processes in the Atmospheric Boundary Layer Over Mountainous Terrain
Stefano Serafin
2018-03-01
Full Text Available The exchange of heat, momentum, and mass in the atmosphere over mountainous terrain is controlled by synoptic-scale dynamics, thermally driven mesoscale circulations, and turbulence. This article reviews the key challenges relevant to the understanding of exchange processes in the mountain boundary layer and outlines possible research priorities for the future. The review describes the limitations of the experimental study of turbulent exchange over complex terrain, the impact of slope and valley breezes on the structure of the convective boundary layer, and the role of intermittent mixing and wave–turbulence interaction in the stable boundary layer. The interplay between exchange processes at different spatial scales is discussed in depth, emphasizing the role of elevated and ground-based stable layers in controlling multi-scale interactions in the atmosphere over and near mountains. Implications of the current understanding of exchange processes over mountains towards the improvement of numerical weather prediction and climate models are discussed, considering in particular the representation of surface boundary conditions, the parameterization of sub-grid-scale exchange, and the development of stochastic perturbation schemes.
13. A mathematical model of turbulence for turbulent boundary layers
Pereira Filho, H.D.V.
1977-01-01
Equations to the so called Reynolds stress-tensor (kinetic turbulent energy) and dissipation rate are developed and a turbulence flux approximation used. Our ideia here is to use those equations in order to develop an economical and fast numeircal procedure for computation of turbulent boundary layer. (author) [pt
14. Body surface adaptations to boundary-layer dynamics
Videler, J.J.
1995-01-01
Evolutionary processes have adapted nektonic animals to interact efficiently with the water that surrounds them. Not all these adaptations serve the same purpose. This paper concentrates on reduction of drag due to friction in the boundary layer close to the body surface. Mucus, compliant skins,
15. Wave boundary layer over a stone-covered bed
Dixen, Martin; Hatipoglu, Figen; Sumer, B. Mutlu
2008-01-01
This paper summarizes the results of an experimental investigation on wave boundary layers over a bed with large roughness, simulating stone/rock/armour block cover on the sea bottom. The roughness elements used in the experiments were stones the size of 1.4cm and 3.85cm in one group of experiments...
16. Stability of hypersonic boundary-layer flows with chemistry
Reed, Helen L.; Stuckert, Gregory K.; Haynes, Timothy S.
1993-01-01
The effects of nonequilibrium chemistry and three dimensionality on the stability characteristics of hypersonic flows are discussed. In two-dimensional (2-D) and axisymmetric flows, the inclusion of chemistry causes a shift of the second mode of Mack to lower frequencies. This is found to be due to the increase in size of the region of relative supersonic flow because of the lower speeds of sound in the relatively cooler boundary layers. Although this shift in frequency is present in both the equilibrium and nonequilibrium air results, the equilibrium approximation predicts modes which are not observed in the nonequilibrium calculations (for the flight conditions considered). These modes are superpositions of incoming and outgoing unstable disturbances which travel supersonically relative to the boundary-layer edge velocity. Such solutions are possible because of the finite shock stand-off distance. Their corresponding wall-normal profiles exhibit an oscillatory behavior in the inviscid region between the boundary-layer edge and the bow shock. For the examination of three-dimensional (3-D) effects, a rotating cone is used as a model of a swept wing. An increase of stagnation temperature is found to be only slightly stabilizing. The correlation of transition location (N = 9) with parameters describing the crossflow profile is discussed. Transition location does not correlate with the traditional crossflow Reynolds number. A new parameter that appears to correlate for boundary-layer flow was found. A verification with experiments on a yawed cone is provided.
Monty, J.P.; Harun, Z.; Marusic, I.
2011-01-01
There are many open questions regarding the behaviour of turbulent boundary layers subjected to pressure gradients and this is confounded by the large parameter space that may affect these flows. While there have been many valuable investigations conducted within this parameter space, there are still insufficient data to attempt to reduce this parameter space. Here, we consider a parametric study of adverse pressure gradient turbulent boundary layers where we restrict our attention to the pressure gradient parameter, β, the Reynolds number and the acceleration parameter, K. The statistics analyzed are limited to the streamwise fluctuating velocity. The data show that the mean velocity profile in strong pressure gradient boundary layers does not conform to the classical logarithmic law. Moreover, there appears to be no measurable logarithmic region in these cases. It is also found that the large-scale motions scaling with outer variables are energised by the pressure gradient. These increasingly strong large-scale motions are found to be the dominant contributor to the increase in turbulence intensity (scaled with friction velocity) with increasing pressure gradient across the boundary layer.
18. Chemical boundary layers in CVD II. Reversible reactions
Croon, de M.H.J.M.; Giling, L.J.
1990-01-01
In addition to irreversible reactions, which were treated in part I, reversible reactions in the gas phase have beenstudied using the concept of the chemical boundary layer. The analysis is given for the situations in which either the forwardor the back reaction is dominant. Two conceptual models
19. The use of a wave boundary layer model in SWAN
Du, Jianting; Bolaños, Rodolfo; Larsén, Xiaoli Guo
2017-01-01
A Wave Boundary Layer Model (WBLM) is implemented in the third-generation ocean wave model SWAN to improve the wind-input source function under idealized, fetch-limited condition. Accordingly, the white capping dissipation parameters are re-calibrated to fit the new wind-input source function...
20. Atmospheric boundary layer evening transitions over West Texas
A systemic analysis of the atmospheric boundary layer behavior during some evening transitions over West Texas was done using the data from an extensive array of instruments which included small and large aperture scintillometers, net radiometers, and meteorological stations. The analysis also comp...
1. Hair receptor sensitivity to changes in laminar boundary layer shape
Dickinson, B T
2010-01-01
Biologists have shown that bat wings contain distributed arrays of flow-sensitive hair receptors. The hair receptors are hypothesized to feedback information on airflows over the bat wing for enhanced stability or maneuverability during flight. Here, we study the geometric specialization of hair-like structures for the detection of changes in boundary layer velocity profiles (shapes). A quasi-steady model that relates the flow velocity profile incident on the longitudinal axis of a hair to the resultant moment and shear force at the hair base is developed. The hair length relative to the boundary layer momentum thickness that maximizes the resultant moment and shear-force sensitivity to changes in boundary layer shape is determined. The sensitivity of the resultant moment and shear force is shown to be highly dependent on hair length. Hairs that linearly taper to a point are shown to provide greater output sensitivity than hairs of uniform cross-section. On an order of magnitude basis, the computed optimal hair lengths are in agreement with the range of hair receptor lengths measured on individual bat species. These results support the hypothesis that bats use hair receptors for detecting changes in boundary layer shape and provide geometric guidelines for artificial hair sensor design and application.
2. Boundary Layer Flows in Porous Media with Lateral Mass Flux
Nemati, H; H, Bararnia; Noori, F
2015-01-01
Solutions for free convection boundary layers on a heated vertical plate with lateral mass flux embedded in a saturated porous medium are presented using the Homotopy Analysis Method and Shooting Numerical Method. Homotopy Analysis Method yields an analytic solution in the form of a rapidly...
3. Fifty Years of Boundary-Layer Theory and Experiment
Dryden, Hugh L.
1955-01-01
The year 1954 marked the 50th anniversary of the Prandtl boundary-layer theory from which we may date the beginning of man's understanding of the dynamics of real fluids. A backward look at this aspect of the history of the last 50 years may be instructive. This paper (1) attempts to compress the events of those 50 years into a few thousand words, to tell in this brief space the interesting story of the development of a new concept, its slow acceptance and growth, its spread from group to group within its country of origin, and its diffusion to other countries of the world. The original brief paper of Prandtl (2) was presented at the Third International Mathematical Congress at Heidelberg in 1904 and published in the following year. It was an attempt to explain the d'Alembert paradox, namely, that the neglect of the small friction of air in the theory resulted in the prediction of zero resistance to motion. Prandtl set himself the task of computing the motion of a fluid of small friction, so small that its effect could be neglected everywhere except where large velocity differences were present or a cumulative effect of friction occurred This led to the concept of boundary layer, or transition layer, near the wall of a body immersed in a fluid stream in which the velocity rises from zero to the free-stream value. It is interesting that Prandtl used the term Grenzsehicht (boundary layer) only once and the term Ubergangsschicht (transition layer) seven times in the brief article. Later writers also used Reibungsschicht (friction layer), but most writers today use Grenzschicht (boundary layer).
4. Pressure Fluctuations Induced by a Hypersonic Turbulent Boundary Layer
Duan, Lian; Choudhari, Meelan M.; Zhang, Chao
2016-01-01
Direct numerical simulations (DNS) are used to examine the pressure fluctuations generated by a spatially-developed Mach 5.86 turbulent boundary layer. The unsteady pressure field is analyzed at multiple wall-normal locations, including those at the wall, within the boundary layer (including inner layer, the log layer, and the outer layer), and in the free stream. The statistical and structural variations of pressure fluctuations as a function of wall-normal distance are highlighted. Computational predictions for mean velocity pro les and surface pressure spectrum are in good agreement with experimental measurements, providing a first ever comparison of this type at hypersonic Mach numbers. The simulation shows that the dominant frequency of boundary-layer-induced pressure fluctuations shifts to lower frequencies as the location of interest moves away from the wall. The pressure wave propagates with a speed nearly equal to the local mean velocity within the boundary layer (except in the immediate vicinity of the wall) while the propagation speed deviates from the Taylor's hypothesis in the free stream. Compared with the surface pressure fluctuations, which are primarily vortical, the acoustic pressure fluctuations in the free stream exhibit a significantly lower dominant frequency, a greater spatial extent, and a smaller bulk propagation speed. The freestream pressure structures are found to have similar Lagrangian time and spatial scales as the acoustic sources near the wall. As the Mach number increases, the freestream acoustic fluctuations exhibit increased radiation intensity, enhanced energy content at high frequencies, shallower orientation of wave fronts with respect to the flow direction, and larger propagation velocity.
5. Large Eddy Simulations of a Bottom Boundary Layer Under a Shallow Geostrophic Front
Bateman, S. P.; Simeonov, J.; Calantoni, J.
2017-12-01
The unstratified surf zone and the stratified shelf waters are often separated by dynamic fronts that can strongly impact the character of the Ekman bottom boundary layer. Here, we use large eddy simulations to study the turbulent bottom boundary layer associated with a geostrophic current on a stratified shelf of uniform depth. The simulations are initialized with a spatially uniform vertical shear that is in geostrophic balance with a pressure gradient due to a linear horizontal temperature variation. Superposed on the temperature front is a stable vertical temperature gradient. As turbulence develops near the bottom, the turbulence-induced mixing gradually erodes the initial uniform temperature stratification and a well-mixed layer grows in height until the turbulence becomes fully developed. The simulations provide the spatial distribution of the turbulent dissipation and the Reynolds stresses in the fully developed boundary layer. We vary the initial linear stratification and investigate its effect on the height of the bottom boundary layer and the turbulence statistics. The results are compared to previous models and simulations of stratified bottom Ekman layers.
6. Convective Cold Pool Structure and Boundary Layer Recovery in DYNAMO
Savarin, A.; Chen, S. S.; Kerns, B. W.; Lee, C.; Jorgensen, D. P.
2012-12-01
One of the key factors controlling convective cloud systems in the Madden-Julian Oscillation (MJO) over the tropical Indian Ocean is the property of the atmospheric boundary layer. Convective downdrafts and precipitation from the cloud systems produce cold pools in the boundary layer, which can inhibit subsequent development of convection. The recovery time is the time it takes for the boundary layer to return to pre convective conditions. It may affect the variability of the convection on various time scales during the initiation of MJO. This study examines the convective cold pool structure and boundary layer recovery using the NOAA WP-3D aircraft observations, include the flight-level, Doppler radar, and GPS dropsonde data, collected during the Dynamics of MJO (DYNAMO) field campaign from November-December 2011. The depth and strength of convective cold pools are defined by the negative buoyancy, which can be computed from the dropsonde data. Convective downdraft can be affected by environmental water vapor due to entrainment. Mid-level dry air observed during the convectively suppressed phase of MJO seems to enhance convective downdraft, making the cold pools stronger and deeper. Recovery of the cold pools in the boundary layer is determined by the strength and depth of the cold pools and also the air-sea heat and moisture fluxes. Given that the water vapor and surface winds are distinct for the convectively active and suppressed phases of MJO over the Indian Ocean, the aircraft data are stratified by the two different large-scale regimes of MJO. Preliminary results show that the strength and depth of the cold pools are inversely correlated with the surrounding mid-level moisture. During the convectively suppressed phase, the recovery time is ~5-20 hours in relative weak wind condition with small air-sea fluxes. The recovery time is generally less than 6 hours during the active phase of MJO with moist mid-levels and stronger surface wind and air-sea fluxes.
7. Mean flow structure of non-equilibrium boundary layers with adverse ...
According to them, an equilibrium boundary layer might exist if the pressure ... of adverse pressure gradient on the turbulent boundary layer at the flat plate for ..... of a constant-pressure turbulent layer to the sudden application of an sudden.
8. New Theories on Boundary Layer Transition and Turbulence Formation
Chaoqun Liu
2012-01-01
Full Text Available This paper is a short review of our recent DNS work on physics of late boundary layer transition and turbulence. Based on our DNS observation, we propose a new theory on boundary layer transition, which has five steps, that is, receptivity, linear instability, large vortex structure formation, small length scale generation, loss of symmetry and randomization to turbulence. For turbulence generation and sustenance, the classical theory, described with Richardson's energy cascade and Kolmogorov length scale, is not observed by our DNS. We proposed a new theory on turbulence generation that all small length scales are generated by “shear layer instability” through multiple level ejections and sweeps and consequent multiple level positive and negative spikes, but not by “vortex breakdown.” We believe “shear layer instability” is the “mother of turbulence.” The energy transferring from large vortices to small vortices is carried out by multiple level sweeps, but does not follow Kolmogorov's theory that large vortices pass energy to small ones through vortex stretch and breakdown. The loss of symmetry starts from the second level ring cycle in the middle of the flow field and spreads to the bottom of the boundary layer and then the whole flow field.
9. Linear segmentation algorithm for detecting layer boundary with lidar.
Mao, Feiyue; Gong, Wei; Logan, Timothy
2013-11-04
The automatic detection of aerosol- and cloud-layer boundary (base and top) is important in atmospheric lidar data processing, because the boundary information is not only useful for environment and climate studies, but can also be used as input for further data processing. Previous methods have demonstrated limitations in defining the base and top, window-size setting, and have neglected the in-layer attenuation. To overcome these limitations, we present a new layer detection scheme for up-looking lidars based on linear segmentation with a reasonable threshold setting, boundary selecting, and false positive removing strategies. Preliminary results from both real and simulated data show that this algorithm cannot only detect the layer-base as accurate as the simple multi-scale method, but can also detect the layer-top more accurately than that of the simple multi-scale method. Our algorithm can be directly applied to uncalibrated data without requiring any additional measurements or window size selections.
10. Provenance of the K/T boundary layers
Hildebrand, A.R.; Boynton, W.V.
1988-01-01
An array of chemical, physical and isotopic evidence indicates that an impact into oceanic crust terminated the Cretaceous Period. Approximately 1500 cu km of debris, dispersed by the impact fireball, fell out globally in marine and nonmarine environments producing a 2 to 4 mm thick layer (fireball layer). In North American locales, the fireball layer overlies a 15 to 25 mm thick layer of similar but distinct composition. This 15 to 25 mm layer (ejecta layer) may represent approximately 1000 cu km of lower energy ejecta from a nearby impact site. Isotopic and chemical evidence supports a mantle provenance for the bulk of the layers. The extraordinary REE pattern of the boundary clays was modelled as a mixture of oceanic crust, mantle, and approximately 10 percent continental material. The results are presented. If the siderophiles of the ejecta layer were derived solely from the mantle, a test may be available to see if the siderophile element anomaly of the fireball layer had an extraterrestrial origin. Radiogenic Os-187 is depleted in the mantle relative to an undifferentiated chondritic source. Os-187/Os-186 ratios of 1.049 and 1.108 were calculated for the ejecta and fireball layers, respectively
11. Active flow control insight gained from a modified integral boundary layer equation
Seifert, Avraham
2016-11-01
Active Flow Control (AFC) can alter the development of boundary layers with applications (e.g., reducing drag by separation delay or separating the boundary layers and enhancing vortex shedding to increase drag). Historically, significant effects of steady AFC methods were observed. Unsteady actuation is significantly more efficient than steady. Full-scale AFC tests were conducted with varying levels of success. While clearly relevant to industry, AFC implementation relies on expert knowledge with proven intuition and or costly and lengthy computational efforts. This situation hinders the use of AFC while simple, quick and reliable design method is absent. An updated form of the unsteady integral boundary layer (UIBL) equations, that include AFC terms (unsteady wall transpiration and body forces) can be used to assist in AFC analysis and design. With these equations and given a family of suitable velocity profiles, the momentum thickness can be calculated and matched with an outer, potential flow solution in 2D and 3D manner to create an AFC design tool, parallel to proven tools for airfoil design. Limiting cases of the UIBL equation can be used to analyze candidate AFC concepts in terms of their capability to modify the boundary layers development and system performance.
12. On the role of acoustic feedback in boundary-layer instability.
Wu, Xuesong
2014-07-28
In this paper, the classical triple-deck formalism is employed to investigate two instability problems in which an acoustic feedback loop plays an essential role. The first concerns a subsonic boundary layer over a flat plate on which two well-separated roughness elements are present. A spatially amplifying Tollmien-Schlichting (T-S) wave between the roughness elements is scattered by the downstream roughness to emit a sound wave that propagates upstream and impinges on the upstream roughness to regenerate the T-S wave, thereby forming a closed feedback loop in the streamwise direction. Numerical calculations suggest that, at high Reynolds numbers and for moderate roughness heights, the long-range acoustic coupling may lead to absolute instability, which is characterized by self-sustained oscillations at discrete frequencies. The dominant peak frequency may jump from one value to another as the Reynolds number, or the distance between the roughness elements, is varied gradually. The second problem concerns the supersonic 'twin boundary layers' that develop along two well-separated parallel flat plates. The two boundary layers are in mutual interaction through the impinging and reflected acoustic waves. It is found that the interaction leads to a new instability that is absent in the unconfined boundary layer. © 2014 The Author(s) Published by the Royal Society. All rights reserved.
13. The height of the atmospheric boundary layer during unstable conditions
Gryning, S.E.
2005-11-01
The height of the convective atmospheric boundary layer, also called the mixed-layer, is one of the fundamental parameters that characterise the structure of the atmosphere near the ground. It has many theoretical and practical applications such as the prediction of air pollution concentrations, surface temperature and the scaling of turbulence. However, as pointed out by Builtjes (2001) in a review paper on Major Twentieth Century Milestones in Air Pollution Modelling and Its Application, the weakest point in meteorology data is still the determination of the height of the mixed-layer, the so-called mixing height. A simple applied model for the height of the mixed-layer over homogeneous terrain is suggested in chapter 2. It is based on a parameterised budget for the turbulent kinetic energy. In the model basically three terms - the spin-up term and the production of mechanical and convective turbulent kinetic energy - control the growth of the mixed layer. The interplay between the three terms is related to the meteorological conditions and the height of the mixed layer. A stable layer, the so-called entrainment zone, which is confined between the mixed layer and the free air above, caps the mixed layer. A parameterisation of the depth of the entrainment zone is also suggested, and used to devise a combined model for the height of the mixed layer and the entrainment zone. Another important aspect of the mixed layer development exists in coastal areas where an internal boundary layer forms downwind from the coastline. A model for the growth of the internal boundary layer is developed in analogy with the model for mixed layer development over homogeneous terrain. The strength of this model is that it can operate on a very fine spatial resolution with minor computer resources. Chapter 3 deals with the validation of the models. It is based in parts on data from the literature, and on own measurements. For the validation of the formation of the internal boundary layer
14. Direct numerical simulation of thermally-stratified turbulent boundary layer subjected to adverse pressure gradient
Hattori, Hirofumi; Kono, Amane; Houra, Tomoya
2016-01-01
Highlights: • We study various thermally-stratified turbulent boundary layers having adverse pressure gradient (APG) by means of DNS. • The detailed turbulent statistics and structures in various thermally-stratified turbulent boundary layers having APG are discussed. • It is found that the friction coefficient and Stanton number decrease along the streamwise direction due to the effects of stable thermal stratification and APG, but those again increase due to the APG effect in the case of weak stable thermal stratification. • In the case of strong stable stratification with or without APG, the flow separation is observed in the downstream region. - Abstract: The objective of this study is to investigate and observe turbulent heat transfer structures and statistics in thermally-stratified turbulent boundary layers subjected to a non-equilibrium adverse pressure gradient (APG) by means of direct numerical simulation (DNS). DNSs are carried out under conditions of neutral, stable and unstable thermal stratifications with a non-equilibrium APG, in which DNS results reveal heat transfer characteristics of thermally-stratified non-equilibrium APG turbulent boundary layers. In cases of thermally-stratified turbulent boundary layers affected by APG, heat transfer performances increase in comparison with a turbulent boundary layer with neutral thermal stratification and zero pressure gradient (ZPG). Especially, it is found that the friction coefficient and Stanton number decrease along the streamwise direction due to the effects of stable thermal stratification and APG, but those again increase due to the APG effect in the case of weak stable thermal stratification (WSBL). Thus, the analysis for both the friction coefficient and Stanton number in the case of WSBL with/without APG is conducted using the FIK identity in order to investigate contributions from the transport equations, in which it is found that both Reynolds-shear-stress and the mean convection terms
15. Boundary Layer Effect on Behavior of Discrete Models.
Eliáš, Jan
2017-02-10
The paper studies systems of rigid bodies with randomly generated geometry interconnected by normal and tangential bonds. The stiffness of these bonds determines the macroscopic elastic modulus while the macroscopic Poisson's ratio of the system is determined solely by the normal/tangential stiffness ratio. Discrete models with no directional bias have the same probability of element orientation for any direction and therefore the same mechanical properties in a statistical sense at any point and direction. However, the layers of elements in the vicinity of the boundary exhibit biased orientation, preferring elements parallel with the boundary. As a consequence, when strain occurs in this direction, the boundary layer becomes stiffer than the interior for the normal/tangential stiffness ratio larger than one, and vice versa. Nonlinear constitutive laws are typically such that the straining of an element in shear results in higher strength and ductility than straining in tension. Since the boundary layer tends, due to the bias in the elemental orientation, to involve more tension than shear at the contacts, it also becomes weaker and less ductile. The paper documents these observations and compares them to the results of theoretical analysis.
16. Boundary layer studies related to fusion theory. Final report
1981-01-01
The described work studied the boundary between closed and open field lines in EBT geometry, with emphasis on the microstability properties. These properties were established primarily for drift waves in the lower hybrid range of frequencies. The transport due to these modes was evaluated by a self-consistent treatment, using quasilinear models in a plasma diffusion code. The model was benchmarked against the EDT experimental results from ORNL and the sensitivity to transport model established. Viscosity was estimated to be negligible compared with anomalous transport. Drift wave turbulence gave a boundary layer size much more consistent with experiment than either collisional transport or Bohm diffusion
17. Edge Plasma Boundary Layer Generated By Kink Modes in Tokamaks
Zakharov, L.E.
2010-01-01
This paper describes the structure of the electric current generated by external kink modes at the plasma edge using the ideally conducting plasma model. It is found that the edge current layer is created by both wall touching and free boundary kink modes. Near marginal stability, the total edge current has a universal expression as a result of partial compensation of the (delta)-functional surface current by the bulk current at the edge. The resolution of an apparent paradox with the pressure balance across the plasma boundary in the presence of the surface currents is provided.
18. Impurity production and transport in the boundary layer of tokamaks
McCracken, G.M.
1987-01-01
The processes by which impurities are produced and enter the discharge are discussed. Emphasis is placed on sputtering at the limiter and an analytical global model is described which incorporates the self-stabilizing effects whch control the edge temperature. Predictions of the scaling of edge temperature and of total radiated power are compared with experimental data from JET and other tokamaks operating with limiters. Under many conditions the scaling of the edge conditions and of the radiated power is accurately predicted. Impurity transport in the boundary and the question of how to control the boundary layer is then discussed. The example of the Impurity Control Limiter on DITE is described. (author)
19. Boundary layer turbulence in transitional and developed states
Park, George Ilhwan; Wallace, James M.; Wu, Xiaohua; Moin, Parviz
2012-03-01
Using the recent direct numerical simulations by Wu and Moin ["Transitional and turbulent boundary layer with heat transfer," Phys. Fluids 22, 85 (2010)] of a flat-plate boundary layer with a passively heated wall, statistical properties of the turbulence in transition at Reθ ≈ 300, from individual turbulent spots, and at Reθ ≈ 500, where the spots merge (distributions of the mean velocity, Reynolds stresses, kinetic energy production, and dissipation rates, enstrophy and its components) have been compared to these statistical properties for the developed boundary layer turbulence at Reθ = 1840. When the distributions in the transitional regions are conditionally averaged so as to exclude locations and times when the flow is not turbulent, they closely resemble the distributions in the developed turbulent state at the higher Reynolds number, especially in the buffer layer. Skin friction coefficients, determined in this conditional manner at the two Reynolds numbers in the transitional flow are, of course, much larger than when their values are obtained by including both turbulent and non-turbulent information there, and the conditional averaged values are consistent with the 1/7th power law approximation. An octant analysis based on the combinations of signs of the velocity and temperature fluctuations, u, v, and θ shows that the momentum and heat fluxes are predominantly of the mean gradient type in both the transitional and developed regions. The fluxes appear to be closely associated with vortices that transport momentum and heat toward and away from the wall in both regions of the flow. The results suggest that there may be little fundamental difference between the nonlinear processes involved in the formation of turbulent spots that appear in transition and those that sustain the turbulence when it is developed. They also support the view that the transport processes and the vortical structures that drive them in developed and transitional boundary
20. Acoustic Radiation From a Mach 14 Turbulent Boundary Layer
Zhang, Chao; Duan, Lian; Choudhari, Meelan M.
2016-01-01
Direct numerical simulations (DNS) are used to examine the turbulence statistics and the radiation field generated by a high-speed turbulent boundary layer with a nominal freestream Mach number of 14 and wall temperature of 0:18 times the recovery temperature. The flow conditions fall within the range of nozzle exit conditions of the Arnold Engineering Development Center (AEDC) Hypervelocity Tunnel No. 9 facility. The streamwise domain size is approximately 200 times the boundary-layer thickness at the inlet, with a useful range of Reynolds number corresponding to Re 450 ?? 650. Consistent with previous studies of turbulent boundary layer at high Mach numbers, the weak compressibility hypothesis for turbulent boundary layers remains applicable under this flow condition and the computational results confirm the validity of both the van Driest transformation and Morkovin's scaling. The Reynolds analogy is valid at the surface; the RMS of fluctuations in the surface pressure, wall shear stress, and heat flux is 24%, 53%, and 67% of the surface mean, respectively. The magnitude and dominant frequency of pressure fluctuations are found to vary dramatically within the inner layer (z/delta 0.< or approx. 0.08 or z+ < or approx. 50). The peak of the pre-multiplied frequency spectrum of the pressure fluctuation is f(delta)/U(sub infinity) approx. 2.1 at the surface and shifts to a lower frequency of f(delta)/U(sub infinity) approx. 0.7 in the free stream where the pressure signal is predominantly acoustic. The dominant frequency of the pressure spectrum shows a significant dependence on the freestream Mach number both at the wall and in the free stream.
1. Structure measurements in a synthetic turbulent boundary layer
Arakeri, Jaywant H.
1987-09-01
Extensive hot-wire measurements have been made to determine the structure of the large eddy in a synthejc turbulent boundary layer on a flat-plate model. The experiments were carried out in a wind tunnel at a nominal free-stream velocity of 12 m/s. The synthetic turbulent boundary layer had a hexagonal pattern of eddies and a ratio of streamwise scale to spanwise scale of 3.2:1. The measured celerity of the large eddy was 84.2 percent of the free-stream velocity. There was some loss of coherence, but very little distortion, as the eddies moved downstream. Several mean properties of the synthetic boundary layer were found to agree quite well with the mean properties of a natural turbulent boundary layer at the same Reynolds number. The large eddy is composed of a pair of primary counter-rotating vortices about five [...] long in the streamwise direction and about one [...] apart in the spanwise direction, where [...] is the mean boundary-layer thickness. The sense of the primary pair is such as to pump fluid away from the wall in the region between the vortices. A secondary pair of counter-rotating streamwise vortices, having a sense opposite to that of the primary pair, is observed outside of and slightly downstream from the primary vortices. Both pairs of vortices extend across the full thickness of the boundary layer and are inclined at a shallow angle to the surface of the flat plate. The data show that the mean vorticity vectors are not tangential to the large-eddy vortices. In fact, the streamwise and normal vorticity components that signal the presence of the eddy are of the same order of magnitude. Definite signatures are obtained in terms of the mean skin-friction coefficient and the mean wake parameter averaged at constant phase. Velocities induced by the vortices are partly responsible for entrainment of irrotational fluid, for transport of momentum, for generation of Reynolds stresses, and for maintenance of streamwise and normal vorticity in the outer
2. Simulation and optimal control of wind-farm boundary layers
Meyers, Johan; Goit, Jay
2014-05-01
In large wind farms, the effect of turbine wakes, and their interaction leads to a reduction in farm efficiency, with power generated by turbines in a farm being lower than that of a lone-standing turbine by up to 50%. In very large wind farms or deep arrays', this efficiency loss is related to interaction of the wind farms with the planetary boundary layer, leading to lower wind speeds at turbine level. Moreover, for these cases it has been demonstrated both in simulations and wind-tunnel experiments that the wind-farm energy extraction is dominated by the vertical turbulent transport of kinetic energy from higher regions in the boundary layer towards the turbine level. In the current study, we investigate the use of optimal control techniques combined with Large-Eddy Simulations (LES) of wind-farm boundary layer interaction for the increase of total energy extraction in very large infinite' wind farms. We consider the individual wind turbines as flow actuators, whose energy extraction can be dynamically regulated in time so as to optimally influence the turbulent flow field, maximizing the wind farm power. For the simulation of wind-farm boundary layers we use large-eddy simulations in combination with actuator-disk and actuator-line representations of wind turbines. Simulations are performed in our in-house pseudo-spectral code SP-Wind that combines Fourier-spectral discretization in horizontal directions with a fourth-order finite-volume approach in the vertical direction. For the optimal control study, we consider the dynamic control of turbine-thrust coefficients in an actuator-disk model. They represent the effect of turbine blades that can actively pitch in time, changing the lift- and drag coefficients of the turbine blades. Optimal model-predictive control (or optimal receding horizon control) is used, where the model simply consists of the full LES equations, and the time horizon is approximately 280 seconds. The optimization is performed using a
3. Thermocouple Rakes for Measuring Boundary Layer Flows Extremely Close to Surface
Hwang, Danny P.; Fralick, Gustave C.; Martin, Lisa C.; Blaha, Charles A.
2001-01-01
Of vital interest to aerodynamic researchers is precise knowledge of the flow velocity profile next to the surface. This information is needed for turbulence model development and the calculation of viscous shear force. Though many instruments can determine the flow velocity profile near the surface, none of them can make measurements closer than approximately 0.01 in. from the surface. The thermocouple boundary-layer rake can measure much closer to the surface than conventional instruments can, such as a total pressure boundary layer rake, hot wire, or hot film. By embedding the sensors (thermocouples) in the region where the velocity is equivalent to the velocity ahead of a constant thickness strut, the boundary-layer flow profile can be obtained. The present device fabricated at the NASA Glenn Research Center microsystem clean room has a heater made of platinum and thermocouples made of platinum and gold. Equal numbers of thermocouples are placed both upstream and downstream of the heater, so that the voltage generated by each pair at the same distance from the surface is indicative of the difference in temperature between the upstream and downstream thermocouple locations. This voltage differential is a function of the flow velocity, and like the conventional total pressure rake, it can provide the velocity profile. In order to measure flow extremely close to the surface, the strut is made of fused quartz with extremely low heat conductivity. A large size thermocouple boundary layer rake is shown in the following photo. The latest medium size sensors already provide smooth velocity profiles well into the boundary layer, as close as 0.0025 in. from the surface. This is about 4 times closer to the surface than the previously used total pressure rakes. This device also has the advantage of providing the flow profile of separated flow and also it is possible to measure simultaneous turbulence levels within the boundary layer.
4. Control of a shock wave-boundary layer interaction using localized arc filament plasma actuators
Webb, Nathan Joseph
Supersonic flight is currently possible, but expensive. Inexpensive supersonic travel will require increased efficiency of high-speed air entrainment, an integral part of air-breathing propulsion systems. Although mixed compression inlet geometry can significantly improve entrainment efficiency, numerous Shock Wave-Boundary Layer Interactions (SWBLIs) are generated in this configuration. The boundary layer must therefore develop through multiple regions of adverse pressure gradient, causing it to thicken, and, in severe cases, separate. The associated increase in unsteadiness can have adverse effects on downstream engine hardware. The most severe consequence of these interactions is the increased aerodynamic blockage generated by the thickened boundary layer. If the increase is sufficient, it can choke the flow, causing inlet unstart, and resulting in a loss of thrust and high transient forces on the engine, airframe, and aircraft occupants. The potentially severe consequences associated with SWBLIs require flow control to ensure proper operation. Traditionally, boundary layer bleed has been used to control the interaction. Although this method is effective, it has inherent efficiency penalties. Localized Arc Filament Plasma Actuators (LAFPAs) are designed to generate perturbations for flow control. Natural flow instabilities act to amplify certain perturbations, allowing the LAFPAs to control the flow with minimal power input. LAFPAs also have the flexibility to maintain control over a variety of operating conditions. This work seeks to examine the effectiveness of LAFPAs as a separation control method for an oblique, impinging SWBLI. The low frequency unsteadiness in the reflected shock was thought to be the natural manifestation of a Kelvin-Helmholtz instability in the shear layer above the separation region. The LAFPAs were therefore placed upstream of the interaction to allow their perturbations to convect to the receptivity region (near the shear layer origin/separation
5. Initializing a Mesoscale Boundary-Layer Model with Radiosonde Observations
Berri, Guillermo J.; Bertossa, Germán
2018-01-01
A mesoscale boundary-layer model is used to simulate low-level regional wind fields over the La Plata River of South America, a region characterized by a strong daily cycle of land-river surface-temperature contrast and low-level circulations of sea-land breeze type. The initial and boundary conditions are defined from a limited number of local observations and the upper boundary condition is taken from the only radiosonde observations available in the region. The study considers 14 different upper boundary conditions defined from the radiosonde data at standard levels, significant levels, level of the inversion base and interpolated levels at fixed heights, all of them within the first 1500 m. The period of analysis is 1994-2008 during which eight daily observations from 13 weather stations of the region are used to validate the 24-h surface-wind forecast. The model errors are defined as the root-mean-square of relative error in wind-direction frequency distribution and mean wind speed per wind sector. Wind-direction errors are greater than wind-speed errors and show significant dispersion among the different upper boundary conditions, not present in wind speed, revealing a sensitivity to the initialization method. The wind-direction errors show a well-defined daily cycle, not evident in wind speed, with the minimum at noon and the maximum at dusk, but no systematic deterioration with time. The errors grow with the height of the upper boundary condition level, in particular wind direction, and double the errors obtained when the upper boundary condition is defined from the lower levels. The conclusion is that defining the model upper boundary condition from radiosonde data closer to the ground minimizes the low-level wind-field errors throughout the region.
6. Bandgap tunability at single-layer molybdenum disulphide grain boundaries
Huang, Yu Li
2015-02-17
Two-dimensional transition metal dichalcogenides have emerged as a new class of semiconductor materials with novel electronic and optical properties of interest to future nanoelectronics technology. Single-layer molybdenum disulphide, which represents a prototype two-dimensional transition metal dichalcogenide, has an electronic bandgap that increases with decreasing layer thickness. Using high-resolution scanning tunnelling microscopy and spectroscopy, we measure the apparent quasiparticle energy gap to be 2.40±0.05 eV for single-layer, 2.10±0.05 eV for bilayer and 1.75±0.05 eV for trilayer molybdenum disulphide, which were directly grown on a graphite substrate by chemical vapour deposition method. More interestingly, we report an unexpected bandgap tunability (as large as 0.85±0.05 eV) with distance from the grain boundary in single-layer molybdenum disulphide, which also depends on the grain misorientation angle. This work opens up new possibilities for flexible electronic and optoelectronic devices with tunable bandgaps that utilize both the control of two-dimensional layer thickness and the grain boundary engineering.
7. Compressible stability of growing boundary layers using parabolized stability equations
Chang, Chau-Lyan; Malik, Mujeeb R.; Erlebacher, Gordon; Hussaini, M. Y.
1991-01-01
The parabolized stability equation (PSE) approach is employed to study linear and nonlinear compressible stability with an eye to providing a capability for boundary-layer transition prediction in both 'quiet' and 'disturbed' environments. The governing compressible stability equations are solved by a rational parabolizing approximation in the streamwise direction. Nonparallel flow effects are studied for both the first- and second-mode disturbances. For oblique waves of the first-mode type, the departure from the parallel results is more pronounced as compared to that for the two-dimensional waves. Results for the Mach 4.5 case show that flow nonparallelism has more influence on the first mode than on the second. The disturbance growth rate is shown to be a strong function of the wall-normal distance due to either flow nonparallelism or nonlinear interactions. The subharmonic and fundamental types of breakdown are found to be similar to the ones in incompressible boundary layers.
8. Grain boundary layer behavior in ZnO/Si heterostructure
Liu Bingce; Liu Cihui; Yi Bo
2010-01-01
The grain boundary layer behavior in ZnO/Si heterostucture is investigated. The current-voltage (I-V) curves, deep level transient spectra (DLTS) and capacitance-voltage (C-V) curves are measured. The transport currents of ZnO/Si heterojunction are dominated by grain boundary layer as high densities of interfacial states existed. The interesting phenomenon that the crossing of In I-V curves of ZnO/Si heterojunction at various measurement temperatures and the decrease of its effective barrier height with the decrement of temperature are in contradiction with the ideal heterojunction thermal emission model is observed. The details will be discussed in the following. (semiconductor physics)
9. Heat conduction boundary layers of condensed clumps in cooling flows
Boehringer, H.; Fabian, A.C.
1989-01-01
The structure of heat conduction boundary layers of gaseous condensations embedded in the hot intergalactic gas in clusters of galaxies is investigated by means of steady, one-dimensional, hydrodynamic models. It is assumed that heat conduction is effective only on scales much smaller than the total region of the cooling flow. Models are calculated for an arbitrary scaling factor, accounting for the reduction in heat conduction efficiency compared to the classical Spitzer case. The results imply a lower limit to the size spectrum of the condensations. The enhancement of cooling in the ambient medium due to heat conduction losses is calculated for a range of clump parameters. The luminosity of several observable emission lines, the extreme ultraviolet (EUV) and soft X-ray emission spectrum, and the column density of some important ions are determined for the model boundary layers and compared with observations. (author)
10. 3D LDV Measurements in Oscillatory Boundary Layers
Mier, J. M.; Garcia, M. H.
2012-12-01
The oscillatory boundary layer represents a particular case of unsteady wall-bounded flows in which fluid particles follow a periodic sinusoidal motion. Unlike steady boundary layer flows, the oscillatory flow regime and bed roughness character change in time along the period for every cycle, a characteristic that introduces a high degree of complexity in the analysis of these flows. Governing equations can be derived from the general Navier-Stokes equations for the motion of fluids, from which the exact solution for the laminar oscillatory boundary layer is obtained (also known as the 2nd Stokes problem). No exact solution exists for the turbulent case, thus, understanding of the main flow characteristics comes from experimental work. Several researchers have reported experimental work in oscillatory boundary layers since the 1960's; however, larger scale facilities and the development of newer measurement techniques with improved temporal and spatial resolution in recent years provides a unique opportunity to achieve a better understanding about this type of flows. Several experiments were performed in the Large Oscillatory Water and Sediment Tunnel (LOWST) facility at the Ven Te Chow Hydrosystems Laboratory, for a range of Reynolds wave numbers between 6x10^4 3D Laser Doppler Velocimetry (LDV) system was used to measure instantaneous flow velocities with a temporal resolution up to ~ 1,000 Hz. It was mounted on a 3-axis traverse with a spatial resolution of 0.01 mm in all three directions. The closest point to the bottom was measured at z = 0.2 mm (z+ ≈ 4), which allowed to capture boundary layer features with great detail. In order to achieve true 3D measurements, 2 probes were used on a perpendicular configuration, such that u and w components were measured from a probe on the side of the flume and v component was measured from a probe pointing down through and access window on top of the flume. The top probe was submerged in a water container, such that the
11. The Boundary Layer Flows of a Rivlin-Ericksen Fluid
Sadeghy, K.; Khabazi, N.; Taghavi, S. M.
The present work deals with the two-dimensional incompressible, laminar, steady-state boundary layer equations. First, we determine a family of velocity distributions outside the boundary layer such that these problems may have similarity solutions. We study the Falkner-Skan flow of a viscoelastic fluid governed by second order model, as the Reynolds number Re→ ∞. We obtain an ordinary forth order differential equation to obtain the stream function, velocity profile and the stress. The stream function is then governed by a generalized Falkner-Skan equation. In comparison with Newtonian Falkner-Skan equation that has two coefficients this new one has four coefficients that two of them represent elastic properties of the fluid. The effects of the elastic parameter on the velocity filed have been discussed. As it is shown in the figure there is a good agreement between numerical results and previous special cases confirm the validity of the presented algorithm.
12. Turbulent boundary layer under the control of different schemes.
Qiao, Z X; Zhou, Y; Wu, Z
2017-06-01
This work explores experimentally the control of a turbulent boundary layer over a flat plate based on wall perturbation generated by piezo-ceramic actuators. Different schemes are investigated, including the feed-forward, the feedback, and the combined feed-forward and feedback strategies, with a view to suppressing the near-wall high-speed events and hence reducing skin friction drag. While the strategies may achieve a local maximum drag reduction slightly less than their counterpart of the open-loop control, the corresponding duty cycles are substantially reduced when compared with that of the open-loop control. The results suggest a good potential to cut down the input energy under these control strategies. The fluctuating velocity, spectra, Taylor microscale and mean energy dissipation are measured across the boundary layer with and without control and, based on the measurements, the flow mechanism behind the control is proposed.
13. Flight Experiment Verification of Shuttle Boundary Layer Transition Prediction Tool
Berry, Scott A.; Berger, Karen T.; Horvath, Thomas J.; Wood, William A.
2016-01-01
Boundary layer transition at hypersonic conditions is critical to the design of future high-speed aircraft and spacecraft. Accurate methods to predict transition would directly impact the aerothermodynamic environments used to size a hypersonic vehicle's thermal protection system. A transition prediction tool, based on wind tunnel derived discrete roughness correlations, was developed and implemented for the Space Shuttle return-to-flight program. This tool was also used to design a boundary layer transition flight experiment in order to assess correlation uncertainties, particularly with regard to high Mach-number transition and tunnel-to-flight scaling. A review is provided of the results obtained from the flight experiment in order to evaluate the transition prediction tool implemented for the Shuttle program.
14. Diffusive boundary layers at the bottom of gaps and cracks
Etzold, Merlin A.; Landel, Julien R.; Dalziel, Stuart B.
2017-11-01
This work is motivated by the chemical decontamination of droplets of chemical warfare agents trapped in the gaps and cracks found in most man-made objects. We consider axial laminar flow within gaps with both straight and angled walls. We study the diffusive mass transfer from a source (e.g. a droplet surface) located at the bottom of the gap. This problem is similar to boundary layers and Graetz-type problems (heat transfer in pipe flow) with the added complication of a non-uniform lateral concentration profile due to the lateral variation of the velocity profile. We present 3D solutions for the diffusive boundary layer and demonstrate that a 2D mean-field model, for which we calculate series and similarity solutions, captures the essential physics. We demonstrate the immediate practical relevance of our findings by comparing decontamination of a droplet located in a gap and on an exposed surface.
15. Boundary layer development on turbine airfoil suction surfaces
Sharma, O. P.; Wells, R. A.; Schlinker, R. H.; Bailey, D. A.
1981-01-01
The results of a study supported by NASA under the Energy Efficient Engine Program, conducted to investigate the development of boundary layers under the influence of velocity distributions that simulate the suction sides of two state-of-the-art turbine airfoils, are presented. One velocity distribution represented a forward loaded airfoil ('squared-off' design), while the other represented an aft loaded airfoil ('aft loaded' design). These velocity distributions were simulated in a low-speed, high-aspect-ratio wind tunnel specifically designed for boundary layer investigations. It is intended that the detailed data presented in this paper be used to develop improved turbulence model suitable for application to turbine airfoil design.
16. Stereoscopic PIV measurement of boundary layer affected by DBD actuator
Procházka Pavel
2016-01-01
Full Text Available The effect of ionic wind generated by plasma actuator on developed boundary layer inside a narrow channel was investigated recently. Since the main investigated plane was parallel to the channel axis, the description of flow field was not evaluated credibly. This paper is dealing with cross-section planes downstream the actuator measured via 3D time-resolved PIV. The actuator position is in spanwise or in streamwise orientation so that ionic wind is blown in the same direction as the main flow or in opposite direction or perpendicularly. The interaction between boundary layer and ionic wind is evaluated for three different velocities of main flow and several parameters of plasma actuation (steady and unsteady regime, frequency etc.. Statistical properties of the flow are shown as well as dynamical behaviour of arising longitudinal vortices are discussed via phase-locked measurement and decomposition method.
17. Abyssal Upwelling and Downwelling and the role of boundary layers
McDougall, T. J.; Ferrari, R. M.
2016-02-01
The bottom-intensified mixing activity arising from the interaction of internal tides with bottom topography implies that the dianeutral advection in the ocean interior is downwards, rather than upwards as is required by continuity. The upwelling of Bottom Water through density surfaces in the deep ocean is however possible because of the sloping nature of the sea floor. A budget study of the abyss (deeper than 2000m) will be described that shows that while the upwelling of Bottom Water might be 25 Sv, this is achieved by very strong upwelling in the bottom turbulent boundary layer (of thickness 50m) of 100 Sv and strong downwelling in the ocean interior of 75 Sv. This downwelling occurs within 10 degrees of longitude of the continental boundaries. This near-boundary confined strong upwelling and downwelling clearly has implications for the Stommel-Arons circulation.
18. Simulation of hypersonic shock wave - laminar boundary layer interactions
2017-06-01
The capability of the Navier-Stokes equations with a perfect gas model for simulation of hypersonic shock wave - laminar boundary layer interactions is assessed. The configuration is a hollow cylinder flare. The experimental data were obtained by Calspan-University of Buffalo (CUBRC) for total enthalpies ranging from 5.07 to 21.85 MJ/kg. Comparison of the computed and experimental surface pressure and heat transfer is performed and the computed §ow¦eld structure is analyzed.
19. Numerical solution of the resistive magnetohydrodynamic boundary-layer equations
Glasser, A.H.; Jardin, S.C.; Tesauro, G.
1983-10-01
Three different techniques are presented for numerical solution of the equations governing the boundary layer of resistive magnetohydrodynamic tearing and interchange instabilities in toroidal geometry. Excellent agreement among these methods and with analytical results provides confidence in the correctness of the results. Solutions obtained in regimes where analytical medthods fail indicate a new scaling for the tearing mode as well as the existence of a new regime of stability
20. The curved kinetic boundary layer of active matter.
2018-01-03
A body submerged in active matter feels the swim pressure through a kinetic accumulation boundary layer on its surface. The boundary layer results from a balance between translational diffusion and advective swimming and occurs on the microscopic length scale . Here , D T is the Brownian translational diffusivity, τ R is the reorientation time and l = U 0 τ R is the swimmer's run length, with U 0 the swim speed [Yan and Brady, J. Fluid. Mech., 2015, 785, R1]. In this work we analyze the swim pressure on arbitrary shaped bodies by including the effect of local shape curvature in the kinetic boundary layer. When δ ≪ L and l ≪ L, where L is the body size, the leading order effects of curvature on the swim pressure are found analytically to scale as J S λδ 2 /L, where J S is twice the (non-dimensional) mean curvature. Particle-tracking simulations and direct solutions to the Smoluchowski equation governing the probability distribution of the active particles show that λδ 2 /L is a universal scaling parameter not limited to the regime δ, l ≪ L. The net force exerted on the body by the swimmers is found to scale as F net /(n ∞ k s T s L 2 ) = f(λδ 2 /L), where f(x) is a dimensionless function that is quadratic when x ≪ 1 and linear when x ∼ 1. Here, k s T s = ζU 0 2 τ R /6 defines the 'activity' of the swimmers, with ζ the drag coefficient, and n ∞ is the uniform number density of swimmers far from the body. We discuss the connection of this boundary layer to continuum mechanical descriptions of active matter and briefly present how to include hydrodynamics into this purely kinetic study.
1. Dynamical structure of the turbulent boundary layer on rough surface
Uruba, Václav; Jonáš, Pavel; Hladík, Ondřej
2011-01-01
Roč. 11, č. 1 (2011), s. 603-604 ISSN 1617-7061 R&D Projects: GA ČR GA101/08/1112; GA ČR GAP101/10/1230 Institutional research plan: CEZ:AV0Z20760514 Keywords : turbulent boundary layer * rough wall * hairpin vortex Subject RIV: BK - Fluid Dynamics http://onlinelibrary.wiley.com/doi/10.1002/pamm.201110291/abstract
2. The double layers in the plasma sheet boundary layer during magnetic reconnection
Guo, J.; Yu, B.
2014-11-01
We studied the evolutions of double layers which appear after the magnetic reconnection through two-dimensional electromagnetic particle-in-cell simulation. The simulation results show that the double layers are formed in the plasma sheet boundary layer after magnetic reconnection. At first, the double layers which have unipolar structures are formed. And then the double layers turn into bipolar structures, which will couple with another new weak bipolar structure. Thus a new double layer or tripolar structure comes into being. The double layers found in our work are about several ten Debye lengths, which accords with the observation results. It is suggested that the electron beam formed during the magnetic reconnection is responsible for the production of the double layers.
3. Experimental investigation of separated shear layer from a leading ...
Shear layer development over a thick flat plate with a semi-circular leading edge is investigated for a range of angles of attack under different pressure gradients for a Reynolds number of 2.44×105 (based on chord and free-stream velocity). The characteristics of the separated shear layer are very well documented through ...
4. Transitional and turbulent boundary layer with heat transfer
Wu, Xiaohua; Moin, Parviz
2010-08-01
We report on our direct numerical simulation of an incompressible, nominally zero-pressure-gradient flat-plate boundary layer from momentum thickness Reynolds number 80-1950. Heat transfer between the constant-temperature solid surface and the free-stream is also simulated with molecular Prandtl number Pr=1. Skin-friction coefficient and other boundary layer parameters follow the Blasius solutions prior to the onset of turbulent spots. Throughout the entire flat-plate, the ratio of Stanton number and skin-friction St/Cf deviates from the exact Reynolds analogy value of 0.5 by less than 1.5%. Mean velocity and Reynolds stresses agree with experimental data over an extended turbulent region downstream of transition. Normalized rms wall-pressure fluctuation increases gradually with the streamwise growth of the turbulent boundary layer. Wall shear stress fluctuation, τw,rms'+, on the other hand, remains constant at approximately 0.44 over the range, 800spots are tightly packed with numerous hairpin vortices. With the advection and merging of turbulent spots, these young isolated hairpin forests develop into the downstream turbulent region. Isosurfaces of temperature up to Reθ=1900 are found to display well-resolved signatures of hairpin vortices, which indicates the persistence of the hairpin forests.
5. Manipulation of Turbulent Boundary Layers Using Synthetic Jets
Berger, Zachary; Gomit, Guillaume; Lavoie, Philippe; Ganapathisubramani, Bharath
2015-11-01
This work focuses on the application of active flow control, in the form of synthetic jet actuators, of turbulent boundary layers. An array of 2 synthetic jets are oriented in the spanwise direction and located approximately 2.7 meters downstream from the leading edge of a flat plate. Actuation is applied perpendicular to the surface of the flat plate with varying blowing ratios and reduced frequencies (open-loop). Two-component large window particle image velocimetry (PIV) was performed at the University of Southampton, in the streamwise-wall-normal plane. Complementary stereo PIV measurements were performed at the University of Toronto Institute for Aerospace Studies (UTIAS), in the spanwise-wall-normal plane. The freestream Reynolds number is 3x104, based on the boundary layer thickness. The skin friction Reynolds number is 1,200 based on the skin friction velocity. The experiments at Southampton allow for the observation of the control effects as the flow propagates downstream. The experiments at UTIAS allow for the observation of the streamwise vorticity induced from the actuation. Overall the two experiments provide a 3D representation of the flow field with respect to actuation effects. The current work focuses on the comparison of the two experiments, as well as the effects of varying blowing ratios and reduced frequencies on the turbulent boundary layer. Funded Supported by Airbus.
6. Acoustic explorations of the upper ocean boundary layer
Vagle, Svein
2005-04-01
The upper ocean boundary layer is an important but difficult to probe part of the ocean. A better understanding of small scale processes at the air-sea interface, including the vertical transfer of gases, heat, mass and momentum, are crucial to improving our understanding of the coupling between atmosphere and ocean. Also, this part of the ocean contains a significant part of the total biomass at all trophic levels and is therefore of great interest to researchers in a range of different fields. Innovative measurement plays a critical role in developing our understanding of the processes involved in the boundary layer, and the availability of low-cost, compact, digital signal processors and sonar technology in self-contained and cabled configurations has led to a number of exciting developments. This talk summarizes some recent explorations of this dynamic boundary layer using both active and passive acoustics. The resonant behavior of upper ocean bubbles combined with single and multi-frequency broad band active and passive devices are now giving us invaluable information on air-sea gas transfer, estimation of biological production, marine mammal behavior, wind speed and precipitation, surface and internal waves, turbulence, and acoustic communication in the surf zone.
7. Thermocapillary Bubble Migration: Thermal Boundary Layers for Large Marangoni Numbers
Balasubramaniam, R.; Subramanian, R. S.
1996-01-01
The migration of an isolated gas bubble in an immiscible liquid possessing a temperature gradient is analyzed in the absence of gravity. The driving force for the bubble motion is the shear stress at the interface which is a consequence of the temperature dependence of the surface tension. The analysis is performed under conditions for which the Marangoni number is large, i.e. energy is transferred predominantly by convection. Velocity fields in the limit of both small and large Reynolds numbers are used. The thermal problem is treated by standard boundary layer theory. The outer temperature field is obtained in the vicinity of the bubble. A similarity solution is obtained for the inner temperature field. For both small and large Reynolds numbers, the asymptotic values of the scaled migration velocity of the bubble in the limit of large Marangoni numbers are calculated. The results show that the migration velocity has the same scaling for both low and large Reynolds numbers, but with a different coefficient. Higher order thermal boundary layers are analyzed for the large Reynolds number flow field and the higher order corrections to the migration velocity are obtained. Results are also presented for the momentum boundary layer and the thermal wake behind the bubble, for large Reynolds number conditions.
8. Coupling of magnetopause-boundary layer to the polar ionosphere
Wei, C.Q.; Lee, L.C.
1993-01-01
The authors develop a model which seeks to explain ultraviolet auroral images from the Viking satellite which show periodic bright regions which resemble open-quotes beadsclose quotes or open-quotes pearlsclose quotes aligned along the postnoon auroral oval. ULF geomagnetic pulsations observed in the cusp region are also addressed by this model. The model addresses plasma dynamics in the low-latitude boundary layer and interactions with the polar ionosphere by means of field-aligned current. The Kelvin-Helmholtz instability can develop in the presence of driven plasma flow, which can lead to the formation and growth of plasma vortices in the boundary layer. The finite conductivity of the earth ionosphere causes these vortices to decay. However regions of enhanced field-aligned power density in the postnoon auroral oval can be associated with field-aligned current filaments and boundary layer vortices. These structures may explain the observed bright spots. The authors also discuss the frequency spectrum and the polarization state of the pulsations
9. RANS Modeling of Benchmark Shockwave / Boundary Layer Interaction Experiments
Georgiadis, Nick; Vyas, Manan; Yoder, Dennis
2010-01-01
This presentation summarizes the computations of a set of shock wave / turbulent boundary layer interaction (SWTBLI) test cases using the Wind-US code, as part of the 2010 American Institute of Aeronautics and Astronautics (AIAA) shock / boundary layer interaction workshop. The experiments involve supersonic flows in wind tunnels with a shock generator that directs an oblique shock wave toward the boundary layer along one of the walls of the wind tunnel. The Wind-US calculations utilized structured grid computations performed in Reynolds-averaged Navier-Stokes mode. Three turbulence models were investigated: the Spalart-Allmaras one-equation model, the Menter Shear Stress Transport wavenumber-angular frequency two-equation model, and an explicit algebraic stress wavenumber-angular frequency formulation. Effects of grid resolution and upwinding scheme were also considered. The results from the CFD calculations are compared to particle image velocimetry (PIV) data from the experiments. As expected, turbulence model effects dominated the accuracy of the solutions with upwinding scheme selection indicating minimal effects.!
10. Cloud-Scale Numerical Modeling of the Arctic Boundary Layer
Krueger, Steven K.
1998-01-01
11. Large-eddy simulation of passive shock-wave/boundary-layer interaction control
Pasquariello, Vito; Grilli, Muzio; Hickel, Stefan; Adams, Nikolaus A.
2014-01-01
Highlights: • The present study investigates a passive flow-control technique for shock-wave/boundary-layer interaction. • The control configuration consists of local suction and injection through a pressure feedback duct. • Implicit LES have been conducted for three different suction locations. • Suction reduces the size of the separation zone. • Turbulence amplification and reflected shock dynamics can be significantly reduced. - Abstract: We investigate a passive flow-control technique for the interaction of an oblique shock generated by an 8.8° wedge with a turbulent boundary-layer at a free-stream Mach number of Ma ∞ =2.3 and a Reynolds number based on the incoming boundary-layer thickness of Re δ 0 =60.5×10 3 by means of large-eddy simulation (LES). The compressible Navier–Stokes equations in conservative form are solved using the adaptive local deconvolution method (ALDM) for physically consistent subgrid scale modeling. Emphasis is placed on the correct description of turbulent inflow boundary conditions, which do not artificially force low-frequency periodic motion of the reflected shock. The control configuration combines suction inside the separation zone and blowing upstream of the interaction region by a pressure feedback through a duct embedded in the wall. We vary the suction location within the recirculation zone while the injection position is kept constant. Suction reduces the size of the separation zone with strongest effect when applied in the rear part of the separation bubble. The analysis of wall-pressure spectra reveals that all control configurations shift the high-energy low-frequency range to higher frequencies, while the energy level is significantly reduced only if suction acts in the rear part of the separated zone. In that case also turbulence production within the interaction region is significantly reduced as a consequence of mitigated reflected shock dynamics and near-wall flow acceleration
12. Magnetohydrodynamic boundary layer flow past a porous substrate with Beavers-Joseph boundary condition
Jat, R.N.; Chaudhary, Santosh
2009-01-01
The flow of an electrically conducting fluid past a porous substrate attached to the flat plate with Beavers-Joseph boundary condition under the influence of a uniform transverse magnetic field has been studied. Taking suitable similar variables, the momentum equation is transformed to ordinary differential equation and solved by standard techniques. The energy equation is solved by considering two boundary layers, one in the porous substrate and the other above the porous substrate. The velocity and temperature distributions along with Nusselt number are discussed numerically and presented through graphs. (author)
13. Computational Study of Hypersonic Boundary Layer Stability on Cones
Gronvall, Joel Edwin
Due to the complex nature of boundary layer laminar-turbulent transition in hypersonic flows and the resultant effect on the design of re-entry vehicles, there remains considerable interest in developing a deeper understanding of the underlying physics. To that end, the use of experimental observations and computational analysis in a complementary manner will provide the greatest insights. It is the intent of this work to provide such an analysis for two ongoing experimental investigations. The first focuses on the hypersonic boundary layer transition experiments for a slender cone that are being conducted at JAXA's free-piston shock tunnel HIEST facility. Of particular interest are the measurements of disturbance frequencies associated with transition at high enthalpies. The computational analysis provided for these cases included two-dimensional CFD mean flow solutions for use in boundary layer stability analyses. The disturbances in the boundary layer were calculated using the linear parabolized stability equations. Estimates for transition locations, comparisons of measured disturbance frequencies and computed frequencies, and a determination of the type of disturbances present were made. It was found that for the cases where the disturbances were measured at locations where the flow was still laminar but nearly transitional, that the highly amplified disturbances showed reasonable agreement with the computations. Additionally, an investigation of the effects of finite-rate chemistry and vibrational excitation on flows over cones was conducted for a set of theoretical operational conditions at the HIEST facility. The second study focuses on transition in three-dimensional hypersonic boundary layers, and for this the cone at angle of attack experiments being conducted at the Boeing/AFOSR Mach-6 quiet tunnel at Purdue University were examined. Specifically, the effect of surface roughness on the development of the stationary crossflow instability are investigated
14. Separating grain boundary migration mechanisms in molecular dynamics simulations
Ulomek, Felix; Mohles, Volker
2016-01-01
In molecular dynamics (MD) simulations of grain boundary (GB) migration it is quite common to find a temperature dependence of GB mobility that deviates strongly from an Arrhenius-type dependence. This usually indicates that more than one mechanism is actually active. With the goal to separate different GB migration mechanisms we investigate a Σ7 <111> 38.2° GB by MD using an EAM potential for aluminium. To drive the GB with a well-known and adjustable force, the energy conserving orientational driving force (ECO DF) is used that had been introduced recently. The magnitude of the DF and the temperature are varied. This yielded a high and a low temperature range for the GB velocity, with a transition temperature that depends on the magnitude of the DF. A method is introduced which allows both a visual and a statistical characterization of GB motion on a per atom basis. These analyses reveal that two mechanisms are active in this GB, a shuffling mechanism and its initiation. These mechanisms operate in a sequential, coupled manner. Based on this, a simple model is introduced that describes all simulated GB velocities (and hence the mobility) very well, including the transition between the dominating mechanisms.
15. MHD Boundary Layer Flow of Dilatant Fluid in a Divergent Channel with Suction or Blowing
Bhattacharyya, Krishnendu; Layek, G. C.
2011-01-01
An analysis is carried out to study a steady magnetohydrodynamic (MHD) boundary layer flow of an electrically conducting incompressible power-law non-Newtonian fluid through a divergent channel. The channel walls are porous and subjected to either suction or blowing of equal magnitude of the same kind of fluid on both walls. The fluid is permeated by a magnetic field produced by electric current along the line of intersection of the channel walls. The governing partial differential equation is transformed into a self-similar nonlinear ordinary differential equation using similarity transformations. The possibility of boundary layer flow in a divergent channel is analyzed with the power-law fluid model. The analysis reveals that the boundary layer flow (without separation) is possible for the case of the dilatant fluid model subjected to suitable suction velocity applied through its porous walls, even in the absence of a magnetic field. Further, it is found that the boundary layer flow is possible even in the presence of blowing for a suitable value of the magnetic parameter. It is found that the velocity increases with increasing values of the power-law index for the case of dilatant fluid. The effects of suction/blowing and magnetic field on the velocity are shown graphically and discussed physically. (fundamental areas of phenomenology(including applications))
16. Turbulent boundary layer heat transfer experiments: Convex curvature effects including introduction and recovery
Simon, T. W.; Moffat, R. J.; Johnston, J. P.; Kays, W. M.
1982-01-01
Measurements were made of the heat transfer rate through turbulent and transitional boundary layers on an isothermal, convexly curved wall and downstream flat plate. The effect of convex curvature on the fully turbulent boundary layer was a reduction of the local Stanton numbers 20% to 50% below those predicted for a flat wall under the same circumstances. The recovery of the heat transfer rates on the downstream flat wall was extremely slow. After 60 cm of recovery length, the Stanton number was still typically 15% to 20% below the flat wall predicted value. Various effects important in the modeling of curved flows were studied separately. These are: the effect of initial boundary layer thickness, the effect of freestream velocity, the effect of freestream acceleration, the effect of unheated starting length, and the effect of the maturity of the boundary layer. An existing curvature prediction model was tested against this broad heat transfer data base to determine where it could appropriately be used for heat transfer predictions.
17. Experiments in a boundary layer subjected to free stream turbulence. Part 1: Boundary layer structure and receptivity
Westin, K.J.A.; Boiko, A.V.; Klingmann, B.G.B.; Kozlov, V.V.; Alfredsson, P.H.
1993-12-01
The modification of the mean and fluctuating characteristics of a flat plate boundary layer subjected to nearly isotropic free stream turbulence (FST) is studied experimentally using hot-wire anemometry. The study is focussed on the region upstream of the transition onset, where the fluctuations inside the boundary layer are dominated by elongated flow structures which grow downstream both in amplitude and length. Their downstream development and scaling is investigated, and the results are compared to those obtained by previous authors. This allows some conclusions about the parameters which are relevant for the modelling of the transition process. The mechanisms underlying the transition process and the relative importance of the Tollmien-Schlichting wave instability in this flow are treated in an accompanying paper. 25 refs
18. Interaction between plasma synthetic jet and subsonic turbulent boundary layer
Zong, Haohua; Kotsonis, Marios
2017-04-01
This paper experimentally investigates the interaction between a plasma synthetic jet (PSJ) and a subsonic turbulent boundary layer (TBL) using a hotwire anemometer and phase-locked particle imaging velocimetry. The PSJ is interacting with a fully developed turbulent boundary layer developing on the flat wall of a square wind tunnel section of 1.7 m length. The Reynolds number based on the freestream velocity (U∞ = 20 m/s) and the boundary layer thickness (δ99 = 34.5 mm) at the location of interaction is 44 400. A large-volume (1696 mm3) three-electrode plasma synthetic jet actuator (PSJA) with a round exit orifice (D = 2 mm) is adopted to produce high-speed (92 m/s) and short-duration (Tjet = 1 ms) pulsed jets. The exit velocity variation of the adopted PSJA in a crossflow is shown to remain almost identical to that in quiescent conditions. However, the flow structures emanating from the interaction between the PSJ and the TBL are significantly different from what were observed in quiescent conditions. In the midspan xy plane (z = 0 mm), the erupted jet body initially follows a wall-normal trajectory accompanied by the formation of a distinctive front vortex ring. After three convective time scales the jet bends to the crossflow, thus limiting the peak penetration depth to approximately 0.58δ99. Comparison of the normalized jet trajectories indicates that the penetration ability of the PSJ is less than steady jets with the same momentum flow velocity. Prior to the jet diminishing, a recirculation region is observed in the leeward side of the jet body, experiencing first an expansion and then a contraction in the area. In the cross-stream yz plane, the signature structure of jets in a crossflow, the counter-rotating vortex pair (CVP), transports high-momentum flow from the outer layer to the near-wall region, leading to a fuller velocity profile and a drop in the boundary layer shape factor (1.3 to 1.2). In contrast to steady jets, the CVP produced by the PSJ
19. High Reynolds number rough wall turbulent boundary layer experiments using Braille surfaces
Harris, Michael; Monty, Jason; Nova, Todd; Allen, James; Chong, Min
2007-11-01
This paper details smooth, transitional and fully rough turbulent boundary layer experiments in the New Mexico State high Reynolds number rough wall wind tunnel. The initial surface tested was generated with a Braille printer and consisted of an uniform array of Braille points. The average point height being 0.5mm, the spacing between the points in the span was 0.5mm and the surface consisted of span wise rows separated by 4mm. The wavelength to peak ratio was 8:1. The boundary layer thickness at the measurement location was 190mm giving a large separation of roughness height to layer thickness. The maximum friction velocity was uτ=1.5m/s at Rex=3.8 x10^7. Results for the skin friction co-efficient show that this surface follows a Nikuradse type inflectional curve and that Townsends outer layer similarity hypothesis is valid for rough wall flows with a large separation of scales. Mean flow and turbulence statistics will be presented.
20. Evaluation of the Atmospheric Boundary-Layer Electrical Variability
Anisimov, Sergey V.; Galichenko, Sergey V.; Aphinogenov, Konstantin V.; Prokhorchuk, Aleksandr A.
2017-12-01
Due to the chaotic motion of charged particles carried by turbulent eddies, electrical quantities in the atmospheric boundary layer (ABL) have short-term variability superimposed on long-term variability caused by sources from regional to global scales. In this study the influence of radon exhalation rate, aerosol distribution and turbulent transport efficiency on the variability of fair-weather atmospheric electricity is investigated via Lagrangian stochastic modelling. For the mid-latitude lower atmosphere undisturbed by precipitation, electrified clouds, or thunderstorms, the model is capable of reproducing the diurnal variation in atmospheric electrical parameters detected by ground-based measurements. Based on the analysis of field observations and numerical simulation it is found that the development of the convective boundary layer, accompanied by an increase in turbulent kinetic energy, forms the vertical distribution of radon and its decaying short-lived daughters to be approximately coincident with the barometric law for several eddy turnover times. In the daytime ABL the vertical distribution of atmospheric electrical conductivity tends to be uniform except within the surface layer, due to convective mixing of radon and its radioactive decay products. At the same time, a decrease in the conductivity near the ground is usually observed. This effect leads to an enhanced ground-level atmospheric electric field compared to that normally observed in the nocturnal stably-stratified boundary layer. The simulation showed that the variability of atmospheric electric field in the ABL associated with internal origins is significant in comparison to the variability related to changes in global parameters. It is suggested that vertical profiles of electrical quantities can serve as informative parameters on ABL turbulent dynamics and can even more broadly characterize the state of the environment.
1. Stability analysis of a boundary layer over a hump using parabolized stability equations
Gao, B; Park, D H; Park, S O, E-mail: [email protected] [Division of Aerospace Engineering, Korea Advanced Institute of Science and Technology, Gusong-dong, Yusong-gu, Daejeon 305-701 (Korea, Republic of)
2011-10-15
Parabolized stability equations (PSEs) were used to investigate the stability of boundary layer flows over a small hump. The applicability of PSEs to flows with a small separation bubble was examined by comparing the result with DNS data. It was found that PSEs can efficiently track the disturbance waves with an acceptable accuracy in spite of a small separation bubble. A typical evolution scenario of Tollmien-Schlichting (TS) wave is presented. The adverse pressure gradient and the flow separation due to the hump have a strong effect on the amplification of the disturbances. The effect of hump width and height is also examined. When the width of the hump is reduced, the amplification factor is increased. The height of the hump is found to obviously influence the stability only when it is greater than the critical layer thickness.
2. Stability analysis of a boundary layer over a hump using parabolized stability equations
Gao, B; Park, D H; Park, S O
2011-01-01
Parabolized stability equations (PSEs) were used to investigate the stability of boundary layer flows over a small hump. The applicability of PSEs to flows with a small separation bubble was examined by comparing the result with DNS data. It was found that PSEs can efficiently track the disturbance waves with an acceptable accuracy in spite of a small separation bubble. A typical evolution scenario of Tollmien-Schlichting (TS) wave is presented. The adverse pressure gradient and the flow separation due to the hump have a strong effect on the amplification of the disturbances. The effect of hump width and height is also examined. When the width of the hump is reduced, the amplification factor is increased. The height of the hump is found to obviously influence the stability only when it is greater than the critical layer thickness.
3. The effects of external conditions in turbulent boundary layers
Brzek, Brian G.
The effects of multiple external conditions on turbulent boundary layers were studied in detail. These external conditions include: surface roughness, upstream turbulence intensity, and pressure gradient. Furthermore, the combined effects of these conditions show the complicated nature of many realistic flow conditions. It was found that the effects of surface roughness are difficult to generalize, given the importance of so many parameters. These parameters include: roughness geometry, roughness regime, roughness height to boundary layer thickness, (k/delta), roughness parameter, ( k+), Reynolds number, and roughness function (Delta B+). A further complication, is the difficulty in computing the wall shear stress, tauw/rho. For the sand grain type roughness, the mean velocity and Reynolds stresses were studied in inner and outer variables, as well as, boundary layer parameters, anisotropy tensor, production term, and viscous stress and form drag contributions. To explore the effects of roughness and Reynolds number dependence in the boundary layer, a new experiment was carefully designed to properly capture the x-dependence of the single-point statistics. It was found that roughness destroys the viscous layer near the wall, thus, reducing the contribution of the viscous stress in the wall region. As a result, the contribution in the skin friction due to form drag increases, while the viscous stress decreases. This yields Reynolds number invariance in the skin friction, near-wall roughness parameters, and inner velocity profiles as k + increases into the fully rough regime. However, in the transitionally rough regime, (i.e., 5 component shows the largest influence of roughness, where the high peak near the wall was decreased and became nearly flat for the fully rough regime profiles. In addition, the Reynolds stresses in outer variables show self-similarity for fixed experimental conditions. However, as the roughness parameter, k +, increases, all Reynolds stress
4. Proceedings of the 17th and 18th NAL Workshops on Investigation and Control of Boundary-Layer Transition
National Aerospace Laboratory; 航空宇宙技術研究所
1996-01-01
The following topics were discussed: vortex shedding, laminar boundary layer measurement, vortex ring, turbulent flow measurement, high Reynolds number turbulence, pulsed flow, boundary layer instability, Ekman boundary layer, sound receptivity, Tollmien-Schlichting wave in supersonic boundary layer, flow field instability, turbulent flow pattern, vorticity distribution in shear flow, turbulence wedge, streamwise vortex mixing, thermal convection, oblique wave generation in boundary layer, in...
5. Four-parametric two-layer algebraic model of transition boundary layer at a planar plate
Labusov, A.N.; Lapin, Yu.V.
1996-01-01
Consideration is given to four-parametric two-layer algebraic model of transition boundary layer on a plane plate, based on generalization of one-parametric algebraic Prandtl-Loitsjansky-Klauzer-3 model. The algebraic model uses Prandtl formulas for mixing path with Loitsjansky damping multiplier in the internal region and the relation for turbulent viscosity, based on universal scales of external region and named the Klauzer-3 formula. 12 refs., 10 figs
6. Interaction between a normal shock wave and a turbulent boundary layer at high transonic speeds. II - Wall shear stress
Liou, M. S.; Adamson, T. C., Jr.
1980-01-01
Asymptotic methods are used to calculate the shear stress at the wall for the interaction between a normal shock wave and a turbulent boundary layer on a flat plate. A mixing length model is used for the eddy viscosity. The shock wave is taken to be strong enough that the sonic line is deep in the boundary layer and the upstream influence is thus very small. It is shown that unlike the result found for laminar flow an asymptotic criterion for separation is not found; however, conditions for incipient separation are computed numerically using the derived solution for the shear stress at the wall. Results are compared with available experimental measurements.
7. Radiative instabilities of atmospheric jets and boundary layers
Candelier, J.
2010-01-01
Complex flows occur in the atmosphere and they can be source of internal gravity waves. We focus here on the sources associated with radiative and shear (or Kelvin-Helmholtz) instabilities. Stability studies of shear layers in a stably stratified fluid concern mainly cases where shear and stratification are aligned along the same direction. In these cases, Miles (1961) and Howard (1961) found a necessary condition for stability based on the Richardson number: Ri ≥ 1/4. In this thesis, we show that this condition is not necessary when shear and stratification are not aligned: we demonstrate that a two-dimensional planar Bickley jet can be unstable for all Richardson numbers. Although the most unstable mode remains 2D, we show there exists an infinite family of 3D unstable modes exhibiting a radiative structure. A WKBJ theory is found to provide the main characteristics of these modes. We also study an inviscid and stratified boundary layer over an inclined wall with non-Boussinesq and compressible effects. We show that this flow is unstable as soon as the wall is not horizontal for all Froude numbers and that strongly stratified 3D perturbations behave exactly like compressible 2D perturbations. Applications of the results to the jet stream and the atmospheric boundary layer are proposed. (author) [fr
8. Dynamical Properties of Vortex Furrows in Transitioning Boundary Layers
Bernard, Peter
2011-11-01
A vortex filament simulation of the spatially growing transitional boundary layer reveals the presence of low speed streaks underlying furrow-like streamwise oriented folds in the surface vorticity layer (AIAA J. Vol. 48, 2010; Proc. ETC13, 2011). The putative hairpin vortices and packets widely observed in boundary layers are found to be an illusion created by assigning the status of structure to the visualized form of regions of rotational motion created by the vortex furrows. Thus, at best, hairpins roughly describe the shape taken by that part of the vorticity within the furrows that directly causes rotation while ignoring the invisible'' and considerable non-rotational part. The life history of the furrows is discussed here including a description of how they grow and the dynamics of the vorticity field within them. Long lived furrows represent factories'' within which initially spanwise vorticity progresses from arch to either one or two-lobed mushroom-like structures in a continuous stream. Furrows grow by this same process. At the heart of the furrow phenomenon is a self-reinforcing process by which streamwise vorticity begets more streamwise vorticity.
9. Vertical ozone characteristics in urban boundary layer in Beijing.
Ma, Zhiqiang; Xu, Honghui; Meng, Wei; Zhang, Xiaoling; Xu, Jing; Liu, Quan; Wang, Yuesi
2013-07-01
Vertical ozone and meteorological parameters were measured by tethered balloon in the boundary layer in the summer of 2009 in Beijing, China. A total of 77 tethersonde soundings were taken during the 27-day campaign. The surface ozone concentrations measured by ozonesondes and TEI 49C showed good agreement, albeit with temporal difference between the two instruments. Two case studies of nocturnal secondary ozone maxima are discussed in detail. The development of the low-level jet played a critical role leading to the observed ozone peak concentrations in nocturnal boundary layer (NBL). The maximum of surface ozone was 161.7 ppbv during the campaign, which could be attributed to abundant precursors storage near surface layer at nighttime. Vertical distribution of ozone was also measured utilizing conventional continuous analyzers on 325-m meteorological observation tower. The results showed the NBL height was between 47 and 280 m, which were consistent with the balloon data. Southerly air flow could bring ozone-rich air to Beijing, and the ozone concentrations exceeded the China's hourly ozone standard (approximately 100 ppb) above 600 m for more than 12 h.
10. On the nature of the plasma sheet boundary layer
Hones, E.W. Jr. (Mission Research Corp., Los Alamos, NM (USA) Los Alamos National Lab., NM (USA))
1990-01-01
The regions of the plasma sheet adjacent to the north and south lobes of the magnetotail have been described by many experimenters as locations of beams of energetic ions and fast-moving plasma directed primarily earthward and tailward along magnetic field lines. Measurements taken as satellites passed through one or the other of these boundary layers have frequently revealed near-earth mirroring of ions and a vertical segregation of velocities of both earthward-moving and mirroring ions with the fastest ions being found nearest the lobe-plasma sheet interface. These are features expected for particles from a distant tail source {bar E} {times} {bar B} drifting in a dawn-to-dusk electric field and are consistent with the source being a magnetic reconnection region. The plasma sheet boundary layers are thus understood as separatrix layers, bounded at their lobeward surfaces by the separatrices from the distant neutral line. This paper will review the observations that support this interpretation. 10 refs., 7 figs.
11. Estimates of the height of the boundary layer using SODAR and rawinsoundings in Amazonia
Fisch, G [Instituto de Aeronautica e Espaco (IAE/CTA), Sao Jose dos Campos, 12228-904 (Brazil); Santos, L A R dos [Instituto Nacional de Meteorologia (INMET), BrasIlia, 70680-900 (Brazil)], E-mail: [email protected], E-mail: [email protected]
2008-05-01
During the LBA campaign in Amazonia 2002, simultaneous measurements were made of the boundary layer using different instruments (rawinsoundings and SODAR). The profiles of potential temperature and humidity were used to estimates the height of the boundary layer using 3 different techniques. The SODAR's measurements did not capture the shallow morning boundary layer observed at the profiles.
12. Comments on deriving the equilibrium height of the stable boundary layer
Steeneveld, G.J.; Wiel, van de B.J.H.; Holtslag, A.A.M.
2007-01-01
Recently, the equilibrium height of the stable boundary layer received much attention in a series of papers by Zilitinkevich and co-workers. In these studies the stable boundary-layer height is derived in terms of inverse interpolation of different boundary-layer height scales, each representing a
13. Modelling the artic stable boundary layer and its coupling to the surface
Steeneveld, G.J.; Wiel, van de B.J.H.; Holtslag, A.A.M.
2006-01-01
The impact of coupling the atmosphere to the surface energy balance is examined for the stable boundary layer, as an extension of the first GABLS (GEWEX Atmospheric Boundary-Layer Study) one-dimensional model intercomparison. This coupling is of major importance for the stable boundary-layer
14. Numerical Investigation of a Heated, Sheared Planetary Boundary Layer
Liou, Yu-Chieng
1996-01-01
A planetary boundary layer (PBL) developed on 11 July, 1987 during the First International Satellites Land Surface Climatology Project (ISLSCP) Field Experiment (FIFE) is investigated numerically by a two dimensional and a three dimensional large eddy simulation (LES) model. Most of the simulated mean and statistical properties are utilized to compare or verify against the observational results extracted from single Doppler lidar scans conducted by Gal-Chen et al. (1992) on the same day. Through the methods of field measurements and numerical simulations, it is found that this PBL, in contrast to the well-known convective boundary layer (CBL), is driven by not only buoyancy but also wind shear. Large eddies produced by the surface heating, as well as internal gravity waves excited by the convection, are both present in the boundary layer. The most unique feature is that in the stable layer, the momentum flux ({overlinerm u^' w^'}), transported by the gravity waves, is counter-gradient. The occurrence of this phenomenon is interpreted by Gal-Chen et al. (1992) using the theory of critical layer singularity, and is confirmed by the numerical simulations in this study. Qualitative agreements are achieved between the model-generated and lidar-derived results. However, quantitative comparisons are less satisfactory. The most serious discrepancy is that in the stable layer the magnitudes of the observed momentum flux ({overlinerm u^ ' w^'}) and vertical velocity variance ({overlinerm w^'^2}) are much larger than their simulated counterparts. Nevertheless, through the technique of numerical simulation, evidence is collected to show inconsistencies among the observations. Thus, the lidar measurements of {overline rm u^' w^'} and {overlinerm w^ '^2} seem to be doubtful. A Four Dimensional Data Assimilation (FDDA) experiment is performed in order to connect the evolution of the model integration with the observations. The results indicate that the dynamical relaxation
15. Investigation of Materials for Boundary Layer Control in a Supersonic Wind Tunnel
Braafladt, Alexander; Lucero, John M.; Hirt, Stefanie M.
2013-01-01
During operation of the NASA Glenn Research Center 15- by 15-Centimeter Supersonic Wind Tunnel (SWT), a significant, undesirable corner flow separation is created by the three-dimensional interaction of the wall and floor boundary layers in the tunnel corners following an oblique-shock/ boundary-layer interaction. A method to minimize this effect was conceived by connecting the wall and floor boundary layers with a radius of curvature in the corners. The results and observations of a trade study to determine the effectiveness of candidate materials for creating the radius of curvature in the SWT are presented. The experiments in the study focus on the formation of corner fillets of four different radii of curvature, 6.35 mm (0.25 in.), 9.525 mm (0.375 in.), 12.7 mm (0.5 in.), and 15.875 mm (0.625 in.), based on the observed boundary layer thickness of 11.43 mm (0.45 in.). Tests were performed on ten candidate materials to determine shrinkage, surface roughness, cure time, ease of application and removal, adhesion, eccentricity, formability, and repeatability. Of the ten materials, the four materials which exhibited characteristics most promising for effective use were the heavy body and regular type dental impression materials, the basic sculpting epoxy, and the polyurethane sealant. Of these, the particular material which was most effective, the heavy body dental impression material, was tested in the SWT in Mach 2 flow, and was observed to satisfy all requirements for use in creating the corner fillets in the upcoming experiments on shock-wave/boundary-layer interaction.
16. Interaction of a Boundary Layer with a Turbulent Wake
Piomelli, Ugo
2004-01-01
The objective of this grant was to study the transition mechanisms on a flat-plate boundary layer interacting with the wake of a bluff body. This is a simplified configuration presented and designed to exemplify the phenomena that occur in multi-element airfoils, in which the wake of an upstream element impinges on a downstream one. Some experimental data is available for this configuration at various Reynolds numbers. The first task carried out was the implementation and validation of the immersed-boundary method. This was achieved by performing calculations of the flow over a cylinder at low and moderate Reynolds numbers. The low-Reynolds number results are discussed, which is enclosed as Appendix A. The high-Reynolds number results are presented in a paper in preparation for the Journal of Fluid Mechanics. We performed calculations of the wake-boundary-layer interaction at two Reynolds numbers, Re approximately equal to 385 and 1155. The first case is discussed and a comparison of the two calculations is reported. The simulations indicate that at the lower Reynolds number the boundary layer is buffeted by the unsteady Karman vortex street shed by the cylinder. This is shown: long streaky structures appear in the boundary layer in correspondence of the three-dimensionalities in the rollers. The fluctuations, however, cannot be self-sustained due to the low Reynolds-number, and the flow does not reach a turbulent state within the computational domain. In contrast, in the higher Reynolds-number case, boundary-layer fluctuations persist after the wake has decayed (due, in part, to the higher values of the local Reynolds number Re achieved in this case); some evidence could be observed that a self-sustaining turbulence generation cycle was beginning to be established. A third simulation was subsequently carried out at a higher Reynolds number, Re=3900. This calculation gave results similar to those of the Re=l155 case. Turbulence was established at fairly low
17. Numerical investigation of hypersonic flat-plate boundary layer transition mechanism induced by different roughness shapes
Zhou, Yunlong; Zhao, Yunfei; Xu, Dan; Chai, Zhenxia; Liu, Wei
2016-10-01
The roughness-induced laminar-turbulent boundary layer transition is significant for high-speed aerospace applications. The transition mechanism is closely related to the roughness shape. In this paper, high-order numerical method is used to investigate the effect of roughness shape on the flat-plate laminar-to-turbulent boundary layer transition. Computations are performed in both the supersonic and hypersonic regimes (free-stream Mach number from 3.37 up to 6.63) for the square, cylinder, diamond and hemisphere roughness elements. It is observed that the square and diamond roughness elements are more effective in inducing transition compared with the cylinder and hemisphere ones. The square roughness element has the longest separated region in which strong unsteadiness exists and the absolute instability is formed, thus resulting in the earliest transition. The diamond roughness element has a maximum width of the separated region leading to the widest turbulent wake region far downstream. Furthermore, transition location moves backward as the Mach number increases, which indicates that the compressibility significantly suppresses the roughness-induced boundary layer transition.
18. Hot-Film and Hot-Wire Anemometry for a Boundary Layer Active Flow Control Test
Lenahan, Keven C.; Schatzman, David M.; Wilson, Jacob Samuel
2013-01-01
Unsteady active flow control (AFC) has been used experimentally for many years to minimize bluff-body drag. This technology could significantly improve performance of rotorcraft by cleaning up flow separation. It is important, then, that new actuator technologies be studied for application to future vehicles. A boundary layer wind tunnel was constructed with a 1ft-x-3ft test section and unsteady measurement instrumentation to study how AFC manipulates the boundary layer to overcome adverse pressure gradients and flow separation. This unsteady flow control research requires unsteady measurement methods. In order to measure the boundary layer characteristics, both hot-wire and hot-film Constant Temperature Anemometry is used. A hot-wire probe is mounted in the flow to measure velocity while a hot-film array lays on the test surface to measure skin friction. Hot-film sensors are connected to an anemometer, a Wheatstone bridge circuit with an output that corresponds to the dynamic flow response. From this output, the time varying flow field, turbulence, and flow reversal can be characterized. Tuning the anemometers requires a fan test on the hot-film sensors to adjust each output. This is a delicate process as several variables drastically affect the data, including control resistance, signal input, trim, and gain settings.
19. Observations of the atmospheric boundary layer height over Abu Dhabi, United Arab Emirates: Investigating boundary layer climatology in arid regions
Marzooqi, Mohamed Al; Basha, Ghouse; Ouarda, Taha B. M. J.; Armstrong, Peter; Molini, Annalisa
2014-05-01
Strong sensible heat fluxes and deep turbulent mixing - together with marked dustiness and a low substrate water content - represent a characteristic signature in the boundary layer over hot deserts, resulting in "thicker" mixing layers and peculiar optical properties. Beside these main features however, desert ABLs present extremely complex local structures that have been scarcely addressed in the literature, and whose understanding is essential in modeling processes such as the transport of dust and pollutants, and turbulent fluxes of momentum, heat and water vapor in hyper-arid regions. In this study, we analyze a continuous record of observations of the atmospheric boundary layer (ABL) height from a single lens LiDAR ceilometer operated at Masdar Institute Field Station (24.4oN, 54.6o E, Abu Dhabi, United Arab Emirates), starting March 2013. We compare different methods for the estimation of the ABL height from Ceilometer data such as, classic variance-, gradient-, log gradient- and second derivation-methods as well as recently developed techniques such as the Bayesian Method and Wavelet covariance transform. Our goal is to select the most suited technique for describing the climatology of the ABL in desert environments. Comparison of our results with radiosonde observations collected at the nearby airport of Abu Dhabi indicate that the WCT and the Bayesian method are the most suitable tools to accurately identify the ABL height in all weather conditions. These two methods are used for the definition of diurnal and seasonal climatologies of the boundary layer conditional to different atmospheric stability classes.
20. Boundary layer polarization and voltage in the 14 MLT region
Lundin, R.; Yamauchi, M.; Woch, J.; Marklund, G.
1995-05-01
Viking midlatitude observations of ions and electrons in the postnoon auroral region show that field-aligned acceleration of electrons and ions with energies up to a few kiloelectron volts takes place. The characteristics of the upgoing ion beams and the local transverse electric field observed by Viking indicate that parallel ion acceleration is primarily due to a quasi-electrostatic field-aligned acceleration process below Viking altitudes, i.e., below 10,000-13,500 km. A good correlation is found between the maximum upgoing ion beam energy and the depth of the local potential well determined by the Viking electric field experiment within dayside 'ion inverted Vs.' The total transverse potential throughout the entire region near the ion inverted Vs. is generally much higher than the field-aligned potential and may reach well above 10 kV. However, the detailed mapping of the transverse potential out to the boundary layer, a fundamental issue which remains controversial, was not attempted here. An important finding in this study is the strong correlation between the maximum up going ion beam energy of dayside ion inverted Vs and the solar wind velocity. This suggests a direct coupling of the solar wind plasma dynamo/voltage generator to the region of field-aligned particle acceleration. The fact that the center of dayside ion inverted Vs coincide with convection reversals/flow stagnation and upward Birkeland currents on what appears to be closed field lines (Woch et al., 1993), suggests that field-aligned potential structures connect to the inner part of an MHD dyanmo in the low-latitude boundary layer. Thus the Viking observations substantiate the idea of a solar wind induced boundary layer polarization where negatively charged perturbations in the postnoon sector persistently develops along the magnetic field lines, establishing accelerating potential drops along the geomagnetic field lines in the 0.5-10 kV range.
1. Planetary Boundary Layer Dynamics over Reno, Nevada in Summer
Liming, A.; Sumlin, B.; Loria Salazar, S. M.; Holmes, H.; Arnott, W. P.
2014-12-01
Quantifying the height of the planetary boundary layer (PBL) is important to understand the transport behavior, mixing, and surface concentrations of air pollutants. In Reno, NV, located in complex, mountainous terrain with high desert climate, the daytime boundary layer can rise to an estimated 3km or more on a summer day due to surface heating and convection. The nocturnal boundary layer, conversely, tends to be much lower and highly stable due to radiative cooling from the surface at night and downslope flow of cool air from nearby mountains. With limited availability of radiosonde data, current estimates of the PBL height at any given time or location are potentially over or underestimated. To better quantify the height and characterize the PBL physics, we developed portable, lightweight sensors that measure CO2 concentrations, temperature, pressure, and humidity every 5 seconds. Four of these sensors are used on a tethered balloon system to monitor CO2 concentrations from the surface up to 300m. We will combine this data with Radio Acoustic Sounding System (RASS) data that measures vertical profiles of wind speed, temperature, and humidity from 40m to 400m. This experiment will characterize the diurnal evolution of CO2 concentrations at multiple heights in the PBL, provide insight into PBL physics during stability transition periods at sunrise and sunset, and estimate the nighttime PBL depth during August in Reno. Further, we expect to gain a better understanding of the impact of mixing volume changes (i.e., PBL height) on air quality and pollution concentrations in Reno. The custom portable sensor design will also be presented. It is expected that these instruments can be used for indoor or outdoor air quality studies, where lightness, small size, and battery operation can be of benefit.
2. Effect of nose bluntness on boundary layer stability and transition
Malik, M. R.; Spall, R. E.; Chang, C.-L.
1990-01-01
The effect of nose bluntness on boundary layer instability is studied theoretically for a Mach 8 flow past a 7 degree semivertex cone. The basic flow is computed by solving the parabolized Navier-Stokes equations. Linear stability analysis of the basic flow reveals that, with small amount of bluntness, the critical Reynolds number for the onset of instability increases by an order of magnitude compared to the sharp cone value. The computed second mode frequencies are also in reasonable agreement with the experimental results. The results are used to explain the effect of unit Reynolds number on transition present in the quiet aeroballistic range data.
3. Fluid Mechanics and Heat Transfer in Transitional Boundary Layers
Wang, Ting
2007-01-01
Experiments have been performed to investigate the effects of elevated free-stream turbulence and streamwise acceleration on flow and thermal structures in transitional boundary layers. The free-stream turbulence ranges from 0.5 to 6.4% and the streamwise acceleration ranges from K = 0 to 0.8 x 10(exp -6). The onset of transition, transition length and the turbulent spot formation rate are determined. The statistical results and conditionally sampled results of th streamwise and cross-stream velocity fluctuations, temperature fluctuations, Reynolds stress and Reynolds heat fluxes are presented.
4. Role of the vertical pressure gradient in wave boundary layers
Jensen, Karsten Lindegård; Sumer, B. Mutlu; Vittori, Giovanna
2014-01-01
By direct numerical simulation (DNS) of the flow in an oscillatory boundary layer, it is possible to obtain the pressure field. From the latter, the vertical pressure gradient is determined. Turbulent spots are detected by a criterion involving the vertical pressure gradient. The vertical pressure...... gradient is also treated as any other turbulence quantity like velocity fluctuations and statistical properties of the vertical pressure gradient are calculated from the DNS data. The presence of a vertical pressure gradient in the near bed region has significant implications for sediment transport....
5. Boundary-layer effects on cold fronts at a coastline
Garratt, J. R.
1986-07-01
The present note discusses one physical mechanism which may contribute to cold air channelling, manifest as a frontal bulge on a surface-analysis chart, in the coastal region of Victoria in southeast Australia. This involves the modification of boundary-layer air in both offshore (prefrontal) and onshore (postfrontal) flow, and the effect on cross-frontal thermal contrast. The problem is discussed in terms of a north-south-oriented cold front behaving as an atmospheric gravity current, propagating along an east-west-oriented coastline, in the presence of a prefrontal offshore stream.
6. Dynamics of turbulent spots in transitional boundary layer
Hladík, Ondřej; Jonáš, Pavel; Uruba, Václav
2011-01-01
Roč. 318, č. 032028 (2011), s. 1-5 E-ISSN 1742-6596. [European turbulence conference /13./. Warsaw, 12.09.2011-15.09.2011] R&D Projects: GA ČR GA101/08/1112; GA ČR GAP101/10/1230 Institutional research plan: CEZ:AV0Z20760514 Keywords : boundary layer transition * hairpin vortex * calmed region Subject RIV: BK - Fluid Dynamics http://iopscience.iop.org/1742-6596/318/3/032028?fromSearchPage=true
7. Numerical Simulation of Transition in Hypersonic Boundary Layers
2011-02-01
sile domes. AGARD Report CP 493. Advisory Group for Aerospace Research and Development. 273 Horvath, T. 2002 Boundary layer transition on slender...reference skin-friction coefficient cp , cv Specific heats at constant pressure and volume, respectively cph Phase speed in propagation direction e...y)) 73 and two-dimensional (W = 0): u = U (y) + u′ , (4.9a) v = v′ , (4.9b) w = w′ , (4.9c) p = 1 + p′ , (4.9d) T = T (y) + T ′ , (4.9e) ρ = 1 T (y
8. Asymptotically optimal unsaturated lattice cubature formulae with bounded boundary layer
Ramazanov, M D [Institute of Mathematics with Computing Centre, Ufa Science Centre, Russian Academy of Sciences, Ufa (Russian Federation)
2013-07-31
This paper describes a new algorithm for constructing lattice cubature formulae with bounded boundary layer. These formulae are unsaturated (in the sense of Babenko) both with respect to the order and in regard to the property of asymptotic optimality on W{sub 2}{sup m}-spaces, m element of (n/2,∞). Most of the results obtained apply also to W{sub 2}{sup μ}(R{sup n})-spaces with a hypoelliptic multiplier of smoothness μ. Bibliography: 6 titles.
9. Fluid-membrane tethers: minimal surfaces and elastic boundary layers.
Powers, Thomas R; Huber, Greg; Goldstein, Raymond E
2002-04-01
Thin cylindrical tethers are common lipid bilayer membrane structures, arising in situations ranging from micromanipulation experiments on artificial vesicles to the dynamic structure of the Golgi apparatus. We study the shape and formation of a tether in terms of the classical soap-film problem, which is applied to the case of a membrane disk under tension subject to a point force. A tether forms from the elastic boundary layer near the point of application of the force, for sufficiently large displacement. Analytic results for various aspects of the membrane shape are given.
10. Earth's magnetosphere formed by the low-latitude boundary layer
Heikkila, W J
2011-01-01
The author argues that, after five decades of debate about the interactive of solar wind with the magnetosphere, it is time to get back to basics. Starting with Newton's law, this book also examines Maxwell's equations and subsidiary equations such as continuity, constitutive relations and the Lorentz transformation; Helmholtz' theorem, and Poynting's theorem, among other methods for understanding this interaction. Includes chapters on prompt particle acceleration to high energies, plasma transfer event, and the low latitude boundary layer More than 200 figures illustrate the text Includes a color insert.
Inoue, M.; Pullin, D.I.; Harun, Z.; Marusic, I.
2013-01-01
12. Two-media boundary layer on a flat plate
Nikolay Ilyich Klyuev; Asgat Gatyatovich Gimadiev; Yuriy Alekseevich Kryukov
2014-01-01
The present paper provides a solution to the problem of a flow over a flat semi-infinite plate set at an angle to the horizon, and having a thin liquid film on its surface by external airflow. The film is formed by extrusion of liquid from the porous wall. The paper proposes a mathematical model of a two-media boundary layer flow. The main characteristics of the flow to a zero and a first approximation are determined. A drop of frictional stress is obtained.
13. Frequency effects of upstream wake and blade interaction on the unsteady boundary layer flow
Kang, Dong Jin; Bae, Sang Su
2002-01-01
Effects of the reduced frequency of upstream wake on downstream unsteady boundary layer flow were simulated by using a Navier-Stokes code. The Navier-Stokes code is based on an unstructured finite volume method and uses a low Reynolds number turbulence model to close the momentum equations. The geometry used in this paper is the MIT flapping foil experimental set-up and the reduced frequency of the upstream wake is varied in the range of 0.91 to 10.86 to study its effect on the unsteady boundary layer flow. Numerical solutions show that they can be divided into two categories. One is so called the low frequency solution, and behaves quite similar to a Stokes layer. Its characteristics is found to be quite similar to those due to either a temporal or spatial wave. The low frequency solutions are observed clearly when reduced frequency is smaller than 3.26. The other one is the high frequency solution. It is observed for the reduced frequency larger than 7.24. It shows a sudden shift of the phase angle of the unsteady velocity around the edge of the boundary layer. The shift of phase angle is about 180 degree, and leads to separation of the boundary layer flow from corresponding outer flow. The high frequency solution shows the characteristics of a temporal wave whose wave length is half of the upstream frequency. This characteristics of the high frequency solution is found to be caused by the strong interaction between unsteady vortices. This strong interaction also leads to destroy of the upstream wake stripe inside the viscous sublayer as well as the buffer layer
14. Interaction of a conical shock wave with a turbulent boundary layer
Teh, S. L.; Gai, S. L.
The paper reports an investigation on the interaction of an incident conical shock wave with a turbulent boundary layer. Although a conical shock theoretically creates a hyperbolic shock trace on the flat plate, the line joining all the experimental interaction origins takes a different form due to varying upstream influence. The existence of strong pressure gradients in the spanwise direction after the shock leads to the boundary-layer twist. A model based on the upstream influence of the shock when combined with McCabe's secondary-flow theory showed separation to occur at an external flow deflection of 11.8 deg. The oil flow measurements however show this to occur at 9.2 deg. This discrepancy is of the same order as that found by McCabe. Detailed data involving Schlieren and shadowgraph photography, surface-flow visualization, and surface-pressure measurements are presented.
15. Interactions between the thermal internal boundary layer and sea breezes
Steyn, D.G. [The Univ. of British Columbia, Dept. of Geography, Atmospheric Science Programme, Vancouver (Canada)
1997-10-01
In the absence of complex terrain, strongly curved coastline or strongly varying mean wind direction, the Thermal Internal Boundary Layer (TIBL) has well known square root behaviour with inland fetch. Existing slab modeling approaches to this phenomenon indicate no inland fetch limit at which this behaviour must cease. It is obvious however that the TIBL cannot continue to grow in depth with increasing fetch, since the typical continental Mixed Layer Depths (MLD) of 1500 to 2000 m must be reached between 100 and 200 km from the shoreline. The anticyclonic conditions with attendant strong convection and light winds which drive the TIBL, also drive daytime Sea Breeze Circulations (SBC) in the coastal zone. The onshore winds driving mesoscale advection of cool air are at the core of TIBL mechanisms, and are invariably part of a SBC. It is to be expected that TIBL and SBC be intimately linked through common mechanisms, as well as external conditions. (au)
16. Estimation of evaporation from equilibrium diurnal boundary layer humidity
Salvucci, G.; Rigden, A. J.; Li, D.; Gentine, P.
2017-12-01
Simplified conceptual models of the convective boundary layer as a well mixed profile of potential temperature (theta) and specific humidity (q) impinging on an initially stably stratified linear potential temperature profile have a long history in atmospheric sciences. These one dimensional representations of complex mixing are useful for gaining insights into land-atmosphere interactions and for prediction when state of the art LES approaches are infeasible. As previously shown (e.g. Betts), if one neglects the role of q in bouyancy, the framework yields a unique relation between mixed layer Theta, mixed layer height (h), and cumulative sensible heat flux (SH) throughout the day. Similarly assuming an initially q profile yields a simple relation between q, h, and cumulative latent heat flux (LH). The diurnal dynamics of theta and q are strongly dependent on SH and the initial lapse rates of theta (gamma_thet) and q (gamma q). In the estimation method proposed here, we further constrain these relations with two more assumptions: 1) The specific humidity is the same at the start of the period of boundary layer growth and at the collapse; and 2) Once the mixed layer reaches the LCL, further drying occurs proportionally to the deardorff convective velocity scale (omega) multiplied by q. Assumption (1) is based on the idea that below the cloud layer, there are no sinks of moisture within the mixed layer (neglecting lateral humidity divergence). Thus the net mixing of dry air aloft with evaporation from the surface must balance. Inclusion of the simple model of moisture loss above the LCL into the bulk-CBL model allows definition of an equilibrium humidity (q) condition at which the diurnal cycle of q repeats (i.e. additions of q from surface balance entrainment of dry air from above). Surprisingly, this framework allows estimation of LH from q, theta, and estimated net radiation by solving for the value of Evaporative Fraction (EF) for which the diurnal cycle of q
17. The time development of the plasma-glass boundary layer in a T-tube
Pavlov, M.; Djurovic, S.
1982-01-01
The refraction of a laser beam by a flat boundary layer between the plasma and the glass plate is analysed. A boundary layer with a constant gradient electron density is assumed. Results of the analysis for plasmas produced in a small T-tube show that the boundary layer thickness increases with time faster than linearly. This means that a relatively fast collapse due to cooling through the boundary layer happens at the second half of the reflected plasma life time, while the boundary layer is negligible thin during the first 2μs after the reflected shock front has passed the point of observation. (author)
18. Mixed convection boundary-layer flow from a horizontal circular cylinder with a constant surface heat flux
Nazar, R.; Amin, N. [Department of Mathematics, Universiti Teknologi Malaysia, 81310 Johor Bahru, Johor (Malaysia); Pop, I. [Faculty of Mathematics, University of Cluj, R-3400 Cluj, CP 253 (Romania)
2004-02-01
The laminar mixed convection boundary-layer flow of a viscous and incompressible fluid past a horizontal circular cylinder, which is maintained at a constant heat flux and is placed in a stream flowing vertically upward has been theoretically studied in this paper. The solutions for the flow and heat transfer characteristics are evaluated numerically for different values of the mixed convection parameter {lambda} with the Prandtl number Pr = 1 and 7, respectively. It is found, as for the case of a heated or cooled cylinder, considered by Merkin [5], that assisting flow delays separation of the boundary-layer and can, if the assisting flow is strong enough, suppress it completely. The opposing flow, on the other side, brings the separation point nearer to the lower stagnation point and for sufficiently strong opposing flows there will not be a boundary-layer on the cylinder. (orig.)
19. Boundary-Layer Control: In Memory of Bill Reynolds
Kim, John
2004-11-01
Professor Bill Reynolds (1933-2004) inspired many students and colleagues with his never-ending curiosity and thought-provoking ideas. Bill's relentless energy, together with his hallmark can-do character and do-it-yourself attitude, led to many seminal contributions to mechanical engineering in general, and fluid mechanics in particular. He has left a lasting impact on many of us, especially for those who had the privilege of working closely with him. Some of my current work on boundary-layer control, the use of neural networks in particular, were inspired by many discussions with Bill. He was among the first to see the potential of control-theoretic approaches for flow control, which has become the main thrust of my current research. Without his continued encouragement, I would not have been deeply involved in this line of research; and perhaps, we would not have seen the current flurry of research activities in applying modern control theories to flow control. In memory of Bill Reynolds, who himself has contributed much to flow control, an analysis of boundary-layer control from a linear system perspective will be presented.
20. On boundary layer modelling using the ASTEC code
Smith, B.L.
1991-07-01
The modelling of fluid boundary layers adjacent to non-slip, heated surface using the ASTEC code is described. The pricipal boundary layer characteristics are derived using simple dimensional arguments and these are developed into criteria for optimum placement of the computational mesh to achieve realistic simulation. In particular, the need for externally-imposed drag and heat transfer correlations as a function of the local mesh concentration is discussed in the context of both laminar and turbulent flow conditions. Special emphasis is placed in the latter case on the (k-ε) turbulence model, which is standard in the code. As far as possible, the analyses are pursued from first principles, so that no comprehensive knowledge of the history of the subject is required for the general ASTEC user to derive practical advice from the document. Some attention is paid to the use of heat transfer correlations for internal solid/fluid surfaces, whose treatment is not straightforward in ASTEC. It is shown that three formulations are possible to effect the heat transfer, called Explicit, Jacobian and Implicit. The particular advantages and disadvantages of each are discussed with regard to numerical stability and computational efficiency. (author) 18 figs., 1 tab., 39 refs
1. Aeromechanics Analysis of a Boundary Layer Ingesting Fan
Bakhle, Milind A.; Reddy, T. S. R.; Herrick, Gregory P.; Shabbir, Aamir; Florea, Razvan V.
2013-01-01
Boundary layer ingesting propulsion systems have the potential to significantly reduce fuel burn but these systems must overcome the challe nges related to aeromechanics-fan flutter stability and forced response dynamic stresses. High-fidelity computational analysis of the fan a eromechanics is integral to the ongoing effort to design a boundary layer ingesting inlet and fan for fabrication and wind-tunnel test. A t hree-dimensional, time-accurate, Reynolds-averaged Navier Stokes computational fluid dynamics code is used to study aerothermodynamic and a eromechanical behavior of the fan in response to both clean and distorted inflows. The computational aeromechanics analyses performed in th is study show an intermediate design iteration of the fan to be flutter-free at the design conditions analyzed with both clean and distorte d in-flows. Dynamic stresses from forced response have been calculated for the design rotational speed. Additional work is ongoing to expan d the analyses to off-design conditions, and for on-resonance conditions.
2. Wintertime Boundary Layer Structure in the Grand Canyon.
Whiteman, C. David; Zhong, Shiyuan; Bian, Xindi
1999-08-01
Wintertime temperature profiles in the Grand Canyon exhibit a neutral to isothermal stratification during both daytime and nighttime, with only rare instances of actual temperature inversions. The canyon warms during daytime and cools during nighttime more or less uniformly through the canyon's entire depth. This weak stability and temperature structure evolution differ from other Rocky Mountain valleys, which develop strong nocturnal inversions and exhibit convective and stable boundary layers that grow upward from the valley floor. Mechanisms that may be responsible for the different behavior of the Grand Canyon are discussed, including the possibility that the canyon atmosphere is frequently mixed to near-neutral stratification when cold air drains into the top of the canyon from the nearby snow-covered Kaibab Plateau. Another feature of canyon temperature profiles is the sharp inversions that often form near the canyon rims. These are generally produced when warm air is advected over the canyon in advance of passing synoptic-scale ridges.Wintertime winds in the main canyon are not classical diurnal along-valley wind systems. Rather, they are driven along the canyon axis by the horizontal synoptic-scale pressure gradient that is superimposed along the canyon's axis by passing synoptic-scale weather disturbances. They may thus bring winds into the canyon from either end at any time of day.The implications of the observed canyon boundary layer structure for air pollution dispersion are discussed.
3. Nonlinear evolution of Mack modes in a hypersonic boundary layer
Chokani, Ndaona
2005-01-01
In hypersonic boundary layer flows the nonlinear disturbance evolution occurs relatively slowly over a very long length scale and has a profound effect on boundary layer transition. In the case of low-level freestream disturbances and negligible surface roughness, the transition is due to the modal growth of exponentially growing Mack modes that are destabilized by wall cooling. Cross-bicoherence measurements, derived from hot-wire data acquired in a quiet hypersonic tunnel, are used to identify and quantify phase-locked, quadratic sum and difference interactions involving the Mack modes. In the early stages of the nonlinear disturbance evolution, cross-bicoherence measurements indicate that the energy exchange between the Mack mode and the mean flow first occurs to broaden the sidebands; this is immediately followed by a sum interaction of the Mack mode to generate the first harmonic. In the next stages of the nonlinear disturbance evolution, there is a difference interaction of the first harmonic, which is also thought to contribute to the mean flow distortion. This difference interaction, in the latter stages, is also accompanied by a difference interaction between Mack mode and first harmonic, and a sum interaction, which forces the second harmonic. Analysis using the digital complex demodulation technique, shows that the low-frequency, phase-locked interaction that is identified in the cross bicoherence when the Mack mode and first harmonic have large amplitudes, arises due to the amplitude modulation of Mack mode and first harmonic.
4. Transition Delay in Hypersonic Boundary Layers via Optimal Perturbations
Paredes, Pedro; Choudhari, Meelan M.; Li, Fei
2016-01-01
The effect of nonlinear optimal streaks on disturbance growth in a Mach 6 axisymmetric flow over a 7deg half-angle cone is investigated in an e ort to expand the range of available techniques for transition control. Plane-marching parabolized stability equations are used to characterize the boundary layer instability in the presence of azimuthally periodic streaks. The streaks are observed to stabilize nominally planar Mack mode instabilities, although oblique Mack mode disturbances are destabilized. Experimentally measured transition onset in the absence of any streaks correlates with an amplification factor of N = 6 for the planar Mack modes. For high enough streak amplitudes, the transition threshold of N = 6 is not reached by the Mack mode instabilities within the length of the cone, but subharmonic first mode instabilities, which are destabilized by the presence of the streaks, reach N = 6 near the end of the cone. These results suggest a passive flow control strategy of using micro vortex generators to induce streaks that would delay transition in hypersonic boundary layers.
5. Space Shuttle Boundary Layer Transition Flight Experiment Ground Testing Overview
Berger, Karen T.; Anderson, Brian P.; Campbell, Charles H.
2014-01-01
In support of the Boundary Layer Transition (BLT) Flight Experiment (FE) Project in which a manufactured protuberance tile was installed on the port wing of Space Shuttle Orbiter Discovery for STS-119, STS- 128, STS-131 and STS-133 as well as Space Shuttle Orbiter Endeavour for STS-134, a significant ground test campaign was completed. The primary goals of the test campaign were to provide ground test data to support the planning and safety certification efforts required to fly the flight experiment as well as validation for the collected flight data. These test included Arcjet testing of the tile protuberance, aerothermal testing to determine the boundary layer transition behavior and resultant surface heating and planar laser induced fluorescence (PLIF) testing in order to gain a better understanding of the flow field characteristics associated with the flight experiment. This paper provides an overview of the BLT FE Project ground testing. High-level overviews of the facilities, models, test techniques and data are presented, along with a summary of the insights gained from each test.
6. Boundary layer structure over areas of heterogeneous heat fluxes
Doran, J.C.; Barnes, F.J.; Coulter, R.L.; Crawford, T.L.
1993-01-01
In general circulation models (GCMs), some properties of a grid element are necessarily considered homogeneous. That is, for each grid volume there is associated a particular combination of boundary layer depth, vertical profiles of wind and temperature, surface fluxes of sensible and latent heat, etc. In reality, all of these quantities may exhibit significant spatial variations the grid area, and the larger the area the greater the likely variations. In balancing the benefits of higher resolution against increased computational time and expense, it is useful to consider what the consequences of such subgrid-scale variability may be. Moreover, in interpreting the results of a simulation, one must be able to define an appropriate average value over a grid. There are two aspects of this latter problem: (1) in observations, how does one take a set of discrete or volume-averaged measurements and relate these to properties of the entire domain, and (2) in computations, how can subgrid-scale features be accounted for in the model parameterizations? To address these and related issues, two field campaigns were carried out near Boardman, Oregon, in June 1991 and 1992. These campaigns were designed to measure the surface fluxes of latent and sensible heat over adjacent areas with strongly contrasting surface types and to measure the response of the boundary layer to those fluxes. This paper discusses some initial findings from those campaigns
7. Vertical transport of water in the Martian boundary layer
Zent, Aaron P.; Haberle, R. M.; Houben, Howard C.
1993-01-01
We are continuing our examination of the transport of H2O through the martian boundary layer, and we have written a one-dimensional numerical model of the exchange of H2O between the atmosphere and subsurface of Mars through the planetary boundary layer (PBL). Our goal is to explore the mechanisms of H2O exchange, and to elucidate the role played by the regolith in the local H2O budget. The atmospheric model includes effects of Coriolis, pressure gradient, and frictional forces for momentum, as well as radiation, sensible heat flux, and advection for heat. The model differs from Flasar and Goody by use of appropriate Viking-based physical constants and inclusion of the radiative effects of atmospheric dust. We specify the pressure gradient force or compute it from a simple slope model. The subsurface model accounts for conduction of heat and diffusion of H2O through a porous adsorbing medium in response to diurnal forcing. The model is initialized with depth-independent H2O concentrations (2 kg M(exp -3)) in the regolith, and a dry atmosphere. The model terminates when the atmospheric H2O column abundance stabilizes at 0.1 percent per sol.
8. Aerosol characteristics in the entrainment interface layer in relation to the marine boundary layer and free troposphere
2018-02-01
Full Text Available This study uses airborne data from two field campaigns off the California coast to characterize aerosol size distribution characteristics in the entrainment interface layer (EIL, a thin and turbulent layer above marine stratocumulus cloud tops, which separates the stratocumulus-topped boundary layer (STBL from the free troposphere (FT. The vertical bounds of the EIL are defined in this work based on considerations of buoyancy and turbulence using thermodynamic and dynamic data. Aerosol number concentrations are examined from three different probes with varying particle diameter (Dp ranges: > 3 nm, > 10 nm, and 0.11–3.4 µm. Relative to the EIL and FT layers, the sub-cloud (SUB layer exhibited lower aerosol number concentrations and higher surface area concentrations. High particle number concentrations between 3 and 10 nm in the EIL are indicative of enhanced nucleation, assisted by high actinic fluxes, cool and moist air, and much lower surface area concentrations than the STBL. Slopes of number concentration versus altitude in the EIL were correlated with the particle number concentration difference between the SUB and lower FT layers. The EIL aerosol size distribution was influenced by varying degrees from STBL aerosol versus subsiding FT aerosol depending on the case examined. These results emphasize the important role of the EIL in influencing nucleation and aerosol–cloud–climate interactions.
9. Aerosol characteristics in the entrainment interface layer in relation to the marine boundary layer and free troposphere
Dadashazar, Hossein; Braun, Rachel A.; Crosbie, Ewan; Chuang, Patrick Y.; Woods, Roy K.; Jonsson, Haflidi H.; Sorooshian, Armin
2018-02-01
This study uses airborne data from two field campaigns off the California coast to characterize aerosol size distribution characteristics in the entrainment interface layer (EIL), a thin and turbulent layer above marine stratocumulus cloud tops, which separates the stratocumulus-topped boundary layer (STBL) from the free troposphere (FT). The vertical bounds of the EIL are defined in this work based on considerations of buoyancy and turbulence using thermodynamic and dynamic data. Aerosol number concentrations are examined from three different probes with varying particle diameter (Dp) ranges: > 3 nm, > 10 nm, and 0.11-3.4 µm. Relative to the EIL and FT layers, the sub-cloud (SUB) layer exhibited lower aerosol number concentrations and higher surface area concentrations. High particle number concentrations between 3 and 10 nm in the EIL are indicative of enhanced nucleation, assisted by high actinic fluxes, cool and moist air, and much lower surface area concentrations than the STBL. Slopes of number concentration versus altitude in the EIL were correlated with the particle number concentration difference between the SUB and lower FT layers. The EIL aerosol size distribution was influenced by varying degrees from STBL aerosol versus subsiding FT aerosol depending on the case examined. These results emphasize the important role of the EIL in influencing nucleation and aerosol-cloud-climate interactions.
10. A stable boundary layer perspective on global temperature trends
McNider, R T; Christy, J R; Biazar, A
2010-01-01
One of the most significant signals in the thermometer-observed temperature record since 1900 is the decrease in the diurnal temperature range over land, largely due to warming of the minimum temperatures. While some data sets have indicated this asymmetrical warming has been reduced since 1979, regional analyses (e.g. East Africa) indicate that the nocturnal warming continues at a pace greater than daytime temperatures. The cause for this night time warming in the observed temperatures has been attributed to a variety of causes. Climate models have in general not replicated the change in diurnal temperature range well. Here we would like to try to distinguish between warming in the nocturnal boundary layer due to a redistribution of heat and warming due to the accumulation of heat. The temperature at night at shelter height is a result of competition between thermal stability and mechanical shear. If stability wins then turbulence is suppressed and the cooling surface becomes cut-off from the warmer air aloft, which leads to sharp decay in surface air temperature. If shear wins, then turbulence is maintained and warmer air from aloft is continually mixed to the surface, which leads to significantly lower cooling rates and warmer temperatures. This warming occurs due to a redistribution of heat. As will be shown by techniques of nonlinear analysis the winner of the stability and shear contest can be very sensitive to changes in greenhouse gas forcing, surface roughness, cloudiness, and surface heat capacity (including soil moisture). Further, the minimum temperatures measured in the nocturnal boundary layer represent only a very shallow layer of the atmosphere which is usually only a few hundred meters thick. It is likely that the observed warming in minimum temperature, whether caused by additional greenhouse forcing or land use changes or other land surface dynamics, is reflecting a redistribution of heat by turbulence-not an accumulation of heat. Because minimum
11. Predictions and Verification of an Isotope Marine Boundary Layer Model
Feng, X.; Posmentier, E. S.; Sonder, L. J.; Fan, N.
2017-12-01
A one-dimensional (1D), steady state isotope marine boundary layer (IMBL) model is constructed. The model includes meteorologically important features absent in Craig and Gordon type models, namely height-dependent diffusion/mixing and convergence of subsiding external air. Kinetic isotopic fractionation results from this height-dependent diffusion which starts as pure molecular diffusion at the air-water interface and increases linearly with height due to turbulent mixing. The convergence permits dry, isotopically depleted air subsiding adjacent to the model column to mix into ambient air. In δD-δ18O space, the model results fill a quadrilateral, of which three sides represent 1) vapor in equilibrium with various sea surface temperatures (SSTs) (high d18O boundary of quadrilateral); 2) mixture of vapor in equilibrium with seawater and vapor in the subsiding air (lower boundary depleted in both D and 18O); and 3) vapor that has experienced the maximum possible kinetic fractionation (high δD upper boundary). The results can be plotted in d-excess vs. δ18O space, indicating that these processes all cause variations in d-excess of MBL vapor. In particular, due to relatively high d-excess in the descending air, mixing of this air into the MBL causes an increase in d-excess, even without kinetic isotope fractionation. The model is tested by comparison with seven datasets of marine vapor isotopic ratios, with excellent correspondence; >95% of observational data fall within the quadrilateral area predicted by the model. The distribution of observations also highlights the significant influence of vapor from the nearby converging descending air on isotopic variations in the MBL. At least three factors may explain the affect the isotopic composition of precipitation. The model can be applied to modern as well as paleo- climate conditions.
12. FOREWORD: International Conference on Planetary Boundary Layer and Climate Change
Djolov, G.; Esau, I.
2010-05-01
One of the greatest achievements of climate science has been the establisment of the concept of climate change on a multitude of time scales. The Earth's complex climate system does not allow a straightforward interpretation of dependences between the external parameter perturbation, internal stochastic system dynamics and the long-term system response. The latter is usually referred to as climate change in a narrow sense (IPCC, 2007). The focused international conference "Planetary Boundary Layers and Climate Change" has addressed only time scales and dynamical aspects of climate change with possible links to the turbulent processes in the Planetary Boundary Layer (PBL). Although limited, the conference topic is by no means singular. One should clearly understand that the PBL is the layer where 99% of biosphere and human activity are concentrated. The PBL is the layer where the energy fluxes, which are followed by changes in cryosphere and other known feedbacks, are maximized. At the same time, the PBL processes are of a naturally small scale. What is the averaged long-term effect of the small-scale processes on the long-term climate dynamics? Can this effect be recognized in existing long-term paleo-climate data records? Can it be modeled? What is the current status of our theoretical understanding of this effect? What is the sensitivity of the climate model projections to the representation of small-scale processes? Are there significant indirect effects, e.g. through transport of chemical components, of the PBL processes on climate? These and other linked questions have been addressed during the conference. The Earth's climate has changed many times during the planet's history, with events ranging from ice ages to long periods of warmth. Historically, natural factors such as the amount of energy released from the Sun, volcanic eruptions and changes in the Earth's orbit have affected the Earth's climate. Beginning late in the 18th century, human activities
13. Non-linear processes in the Earth atmosphere boundary layer
Grunskaya, Lubov; Valery, Isakevich; Dmitry, Rubay
2013-04-01
The work is connected with studying electromagnetic fields in the resonator Earth-Ionosphere. There is studied the interconnection of tide processes of geophysical and astrophysical origin with the Earth electromagnetic fields. On account of non-linear property of the resonator Earth-Ionosphere the tides (moon and astrophysical tides) in the electromagnetic Earth fields are kinds of polyharmonic nature. It is impossible to detect such non-linear processes with the help of the classical spectral analysis. Therefore to extract tide processes in the electromagnetic fields, the method of covariance matrix eigen vectors is used. Experimental investigations of electromagnetic fields in the atmosphere boundary layer are done at the distance spaced stations, situated on Vladimir State University test ground, at Main Geophysical Observatory (St. Petersburg), on Kamchatka pen., on Lake Baikal. In 2012 there was continued to operate the multichannel synchronic monitoring system of electrical and geomagnetic fields at the spaced apart stations: VSU physical experimental proving ground; the station of the Institute of Solar and Terrestrial Physics of Russian Academy of Science (RAS) at Lake Baikal; the station of the Institute of volcanology and seismology of RAS in Paratunka; the station in Obninsk on the base of the scientific and production society "Typhoon". Such investigations turned out to be possible after developing the method of scanning experimental signal of electromagnetic field into non- correlated components. There was used a method of the analysis of the eigen vectors ofthe time series covariance matrix for exposing influence of the moon tides on Ez. The method allows to distribute an experimental signal into non-correlated periodicities. The present method is effective just in the situation when energetical deposit because of possible influence of moon tides upon the electromagnetic fields is little. There have been developed and realized in program components
14. Boundaries of the Realizability Region of Membrane Separation Processes
Tsirlin, A. M.; Akhrenemkov, A. A.
2018-01-01
The region of realizability of membrane separation systems having a constant total membrane area has been determined for a definite output of a final product at a definite composition of a mixture flow. The law of change in the pressure in the mixture, corresponding to the minimum energy required for its separation, was concretized for media close in properties to ideal gases and solutions.
15. Analysis of Ground Effects on Aerodynamic Characteristics of Aerofoils Using Boundary Layer Approximation
Takahashi, Yuji; Kikuchi, Masanori; Hirano, Kimitaka
A study of a new high-speed zero-emission transportation “Aerotrain” is being carried out in Tohoku University and the University of Miyazaki. Because the aerotrain utilizes the ground effect, research on the aerofoil section, which can harness the ground effect effectively, is important. The aerotrain moves along a U-shaped guideway, which has a ground and sidewalls, so it has many viscous interference elements. In an analysis of the ground effects on the aerodynamic characteristics of aerofoils, the boundary layers on the aerofoil surface must be considered. At first, velocity distributions on the surfaces of aerofoils in potential flows are computed using the vortex method, then the momentum integration equations of the boundary layer are solved with experimental formulas. This procedure has the following advantages: modifications of the aerofoil section are easy because it is not necessary to make complicated computational grids, boundary layer transition and separation can be predicted using empirical procedures. The aerodynamic characteristics of four types of aerofoil sections are investigated to clarify the relationship between aerofoil sections and ground effects. Computational results are compared with experimental results obtained using a towing wind tunnel to verify computational precisions. In addition, aerofoil characteristics at an actual cruise speed are analyzed.
16. Boundary Layer Flow Control by an Array of Ramp-Shaped Vortex Generators
Zaman, K. B. M. Q.; Hirt, S. M.; Bencic, T. J.
2012-01-01
Flow field survey results for the effect of ramp-shaped vortex generators (VG) on a turbulent boundary layer are presented. The experiments are carried out in a low-speed wind tunnel and the data are acquired primarily by hot-wire anemometry. Distributions of mean velocity and turbulent stresses as well as streamwise vorticity, on cross-sectional planes at various downstream locations, are obtained. These detailed flow field properties, including the boundary layer characteristics, are documented with the primary objective of aiding possible computational investigations. The results show that VG orientation with apex upstream, that produces a downwash directly behind it, yields a stronger pair of streamwise vortices. This is in contrast to the case with apex downstream that produces a pair of vortices of opposite sense. Thus, an array of VG s with the former orientation, usually considered for film-cooling application, may also be superior for mixing enhancement and boundary layer separation control. The data files can be found on a supplemental CD.
17. Factors influencing flow steadiness in laminar boundary layer shock interactions
Tumuklu, Ozgur; Levin, Deborah A.; Gimelshein, Sergey F.; Austin, Joanna M.
2016-11-01
The Direct Simulation Monte Carlo method has been used to model laminar shock wave boundary interactions of hypersonic flow over a 30/55-deg double-wedge and "tick-shaped" model configurations studied in the Hypervelocity Expansion Tube facility and T-ADFA free-piston shock tunnel, respectively. The impact of thermochemical effects on these interactions by changing the chemical composition from nitrogen to air as well as argon for a stagnation enthalpy of 8.0 MJ/kg flow are investigated using the 2-D wedge model. The simulations are found to reproduce many of the classic features related to Edney Type V strong shock interactions that include the attached, oblique shock formed over the first wedge, the detached bow shock from the second wedge, the separation zone, and the separation and reattachment shocks that cause complex features such as the triple point for both cases. However, results of a reacting air flow case indicate that the size of the separation length, and the movement of the triple point toward to the leading edge is much less than the nitrogen case.
18. The BLLAST field experiment: Boundary-Layer Late Afternoon and Sunset Turbulence
Lothon, M.; Lohou, F.; Pino, D.; Vilà-Guerau De Arellano, J.; Hartogensis, O.K.; Boer, van de A.; Coster, de O.; Moene, A.F.; Steeneveld, G.J.
2014-01-01
Due to the major role of the sun in heating the earth's surface, the atmospheric planetary boundary layer over land is inherently marked by a diurnal cycle. The afternoon transition, the period of the day that connects the daytime dry convective boundary layer to the night-time stable boundary
19. Investigation of transition scenarios in boundary-layer flows
Stolte, A.
1999-11-01
Laminar-turbulent transition mechanisms triggered by crossflow instability in three-dimensional, accelerated boundary-layer flows are investigated using numerical methods of stability analysis. The investigations are based on the DLR swept plate experiment, where stationary and traveling crossflow modes can be selectively introduced into the flow field. Nonlinear instability analyses employing the parabolized stability equations (PSE) show that unique saturation amplitudes do neither exist for stationary crossflow vortices nor for traveling crossflow waves. This phenomenon is explained by means of a spatial bifurcation model. Using Floquet theory, temporal secondary instability analyses are then performed for the mean flow distorted by primary disturbances. In these analyses, secondary high-frequency disturbances with high growth rates are found. The location of these disturbances correlates well with regions of high shear in the primarily distorted flow field, especially on the back of the primary crossflow vortices. (orig.)
20. On the secondary instability of three-dimensional boundary layers
Janke, E. [DLR Deutsches Zentrum fuer Luft- und Raumfahrt e.V., Goettingen (Germany). Inst. fuer Stroemungsmechanik; Balakumar, P. [Department of Aerospace Engineering, Old Dominion University, Norfolk, VA 23529 (United States)
2000-09-01
One of the possible transition scenarios in three-dimensional boundary layers, the saturation of stationary crossflow vortices and their secondary instability to high-frequency disturbances, is studied using the parabolized stability equations (PSE) and Floquet theory. Starting from nonlinear PSE solutions, we investigate the region where a purely stationary crossflow disturbance saturates for its secondary instability characteristics utilizing global and local eigenvalue solvers that are based on the implicitly restarted Arnoldi method and a Newton-Raphson technique, respectively. Results are presented for swept Hiemenz flow and the DLR swept flat plate experiment. The main focuses of this study are on the existence of multiple roots in the eigenvalue spectrum that could explain experimental observations of time-dependent occurrences of an explosive growth of traveling disturbances, on the origin of high-frequency disturbances, as well as on gaining more information about threshold amplitudes of primary disturbances necessary for the growth of secondary disturbances. (orig.)
1. Small particle transport across turbulent nonisothermal boundary layers
Rosner, D. E.; Fernandez De La Mora, J.
1982-01-01
The interaction between turbulent diffusion, Brownian diffusion, and particle thermophoresis in the limit of vanishing particle inertial effects is quantitatively modeled for applications in gas turbines. The model is initiated with consideration of the particle phase mass conservation equation for a two-dimensional boundary layer, including the thermophoretic flux term directed toward the cold wall. A formalism of a turbulent flow near a flat plate in a heat transfer problem is adopted, and variable property effects are neglected. Attention is given to the limit of very large Schmidt numbers and the particle concentration depletion outside of the Brownian sublayer. It is concluded that, in the parameter range of interest, thermophoresis augments the high Schmidt number mass-transfer coefficient by a factor equal to the product of the outer sink and the thermophoretic suction.
2. Numerical treatment of the unsteady hydromagnetic thermal boundary layer problem
Drymonitou, M.A.; Geroyannis, V.S.; Goudas, C.L.
1980-01-01
This paper presents a suitable numerical method for the treatment of the unsteady hydromagnetic thermal boundary layer problem for flows past an infinite porous flat plate, the motion of which is governed by a general time-dependent law, under the influence of a transverse externally set magnetic field. The normal velocity of suction/injection at the plate is also assumed to be time-dependent. The results obtained on the basis of numerical approximations seem to compare favourably with earlier results (Pande et al., 1976; Tokis, 1978). Analytical approximations are given for the cases of a plate (i) generally accelerated and (ii) harmonically oscillating. The direct numerical treatment is obviously advantageous since it allows handling of cases where the known methods for analytical approximations are not applicable. This problem is closely related to the motions and heat transfer occurring locally on the surfaces of stars. (orig.)
3. Microstructure of Turbulence in the Stably Stratified Boundary Layer
Sorbjan, Zbigniew; Balsley, Ben B.
2008-11-01
The microstructure of a stably stratified boundary layer, with a significant low-level nocturnal jet, is investigated based on observations from the CASES-99 campaign in Kansas, U.S.A. The reported, high-resolution vertical profiles of the temperature, wind speed, wind direction, pressure, and the turbulent dissipation rate, were collected under nocturnal conditions on October 14, 1999, using the CIRES Tethered Lifting System. Two methods for evaluating instantaneous (1-sec) background profiles are applied to the raw data. The background potential temperature is calculated using the “bubble sort” algorithm to produce a monotonically increasing potential temperature with increasing height. Other scalar quantities are smoothed using a running vertical average. The behaviour of background flow, buoyant overturns, turbulent fluctuations, and their respective histograms are presented. Ratios of the considered length scales and the Ozmidov scale are nearly constant with height, a fact that can be applied in practice for estimating instantaneous profiles of the dissipation rate.
4. Boundary layers and scaling relations in natural thermal convection
Shishkina, Olga; Lohse, Detlef; Grossmann, Siegfried
2017-11-01
We analyse the boundary layer (BL) equations in natural thermal convection, which includes vertical convection (VC), where the fluid is confined between two differently heated vertical walls, horizontal convection (HC), where the fluid is heated at one part of the bottom plate and cooled at some other part, and Rayleigh-Benard convection (RBC). For BL dominated regimes we derive the scaling relations of the Nusselt and Reynolds numbers (Nu, Re) with the Rayleigh and Prandtl numbers (Ra, Pr). For VC the scaling relations are obtained directly from the BL equations, while for HC they are derived by applying the Grossmann-Lohse theory to the case of VC. In particular, for RBC with large Pr we derive Nu Pr0Ra1/3 and Re Pr-1Ra2/3. The work is supported by the Deutsche Forschungsgemeinschaft (DFG) under the Grant Sh 405/4 - Heisenberg fellowship.
5. The large Reynolds number - Asymptotic theory of turbulent boundary layers.
Mellor, G. L.
1972-01-01
A self-consistent, asymptotic expansion of the one-point, mean turbulent equations of motion is obtained. Results such as the velocity defect law and the law of the wall evolve in a relatively rigorous manner, and a systematic ordering of the mean velocity boundary layer equations and their interaction with the main stream flow are obtained. The analysis is extended to the turbulent energy equation and to a treatment of the small scale equilibrium range of Kolmogoroff; in velocity correlation space the two-thirds power law is obtained. Thus, the two well-known 'laws' of turbulent flow are imbedded in an analysis which provides a great deal of other information.
6. Characterization of Alfvenic fluctuations in the magnetopause boundary layer
Rezeau, L.; Morane, A.; Perraut, S.; Roux, A.; Schmidt, R.
1989-01-01
The European Space Agency GEOS 2 spacecraft happened to cross the magnetopause several times, at various local times. Intense electric and magnetic fluctuations, in the ultralow-frequency (ULF) range (0-10 Hz) have been detected during each such crossing, with a peak at the magnetopause and still large amplitudes in the adjacent magnetosheath and magnetopause boundary layer. By applying spectral analysis and correlations to the electric and magnetic fluctuations, and a minimum variance analysis to the magnetic fluctuations, the authors investigate the nature of these fluctuations which appear as short-lasting bursts in the spacecraft frame. Having reviewed possible interpretations, they show that the observed electric and magnetic signatures are consistent with small-scale (L ∼ ion Larmor radius) Alfvenic field-aligned structures passing by the spacecraft at high speed. It is suggested that these structures correspond to nonlinear Alfvenic structures
7. Turbulent thermal boundary layer on a permeable flat plate
Vigdorovich, I. I.
2007-01-01
Scaling laws are established for the profiles of temperature, turbulent heat flux, rms temperature fluctuation, and wall heat transfer in the turbulent boundary layer on a flat plate with transpiration. In the case of blowing, the temperature distribution represented in scaling variables outside the viscous sublayer has a universal form known from experimental data for flows over impermeable flat plates. In the case of suction, the temperature distribution is described by a one-parameter family of curves. A universal law of heat transfer having the form of a generalized Reynolds analogy provides a basis for representation of the heat flux distributions corresponding to different Reynolds numbers and transpiration velocities in terms of a function of one variable. The results are obtained without invoking any special closure hypotheses
8. Boundary layer height estimation by sodar and sonic anemometer measurements
Contini, D; Cava, D; Martano, P; Donateo, A; Grasso, F M
2008-01-01
In this paper an analysis of different methods for the calculation of the boundary layer height (BLH) using sodar and ultrasonic anemometer measurements is presented. All the methods used are based on single point surface measurements. In particular the automatic spectral routine developed for Remtech sodar is compared with the results obtained with the parameterization of the vertical velocity variance, with the calculation of a prognostic model and with a parameterization based on horizontal velocity spectra. Results indicate that in unstable conditions the different methods provide similar pattern, with BLH relatively low, even if the parameterization of the vertical velocity variance is affected by a large scatter that limits its efficiency in evaluating the BLH. In stable nocturnal conditions the performances of the Remtech routine are lower with respect to the ones in unstable conditions. The spectral method, applied to sodar or sonic anemometer data, seems to be the most promising in order to develop an efficient routine for BLH determination
9. Buckling transition and boundary layer in non-Euclidean plates.
Efrati, Efi; Sharon, Eran; Kupferman, Raz
2009-07-01
Non-Euclidean plates are thin elastic bodies having no stress-free configuration, hence exhibiting residual stresses in the absence of external constraints. These bodies are endowed with a three-dimensional reference metric, which may not necessarily be immersible in physical space. Here, based on a recently developed theory for such bodies, we characterize the transition from flat to buckled equilibrium configurations at a critical value of the plate thickness. Depending on the reference metric, the buckling transition may be either continuous or discontinuous. In the infinitely thin plate limit, under the assumption that a limiting configuration exists, we show that the limit is a configuration that minimizes the bending content, among all configurations with zero stretching content (isometric immersions of the midsurface). For small but finite plate thickness, we show the formation of a boundary layer, whose size scales with the square root of the plate thickness and whose shape is determined by a balance between stretching and bending energies.
10. A preliminary assessment of the Titan planetary boundary layer
Allison, Michael
1992-01-01
Results of a preliminary assessment of the characteristic features of the Titan planetary boundary are addressed. These were derived from the combined application of a patched Ekman surface layer model and Rossby number similarity theory. Both these models together with Obukhov scaling, surface speed limits and saltation are discussed. A characteristic Akman depth of approximately 0.7 km is anticipated, with an eddy viscosity approximately equal to 1000 sq cm/s, an associated friction velocity approximately 0.01 m/s, and a surface wind typically smaller than 0.6 m/s. Actual values of these parameters probably vary by as much as a factor of two or three, in response to local temporal variations in surface roughness and stability. The saltation threshold for the windblown injection of approximately 50 micrometer particulates into the atmosphere is less than twice the nominal friction velocity, suggesting that dusty breezes might be an occassional feature of the Titan meteorology.
11. Combined core/boundary layer plasma transport simulations in tokamaks
Prinja, A.K.; Schafer, R.F. Jr.; Conn, R.W.; Howe, H.C.
1987-01-01
Significant new numerical results are presented from self-consistent core and boundary or scrape-off layer plasma simulations with 3-D neutral transport calculations. For a symmetric belt limiter it is shown that, for plasma conditions considered here, the pump limiter collection efficiency increases from 11% to 18% of the core efflux as a result of local reionization of blade deflected neutrals. This hitherto unobserved effect causes a significant amplification of upstream ion flux entering the pump limiter. Results from coupling of an earlier developed two-zone edge plasma model ODESSA to the PROCTR core plasma simulation code indicates that intense recycling divertor operation may not be possible because of stagnation of upstream flow velocity. This results in a self-consistent reduction of density gradient in an intermediate region between the central plasma and separatrix, and a concomitant reduction of core-efflux. There is also evidence of increased recycling at the first wall. (orig.)
12. Thorpe method applied to planetary boundary layer data
Gonzalez-Nieto, P.L.; Cano, J.L.; Tijera, M.; Cano, D.
2008-01-01
Turbulence affects the dynamics of atmospheric processes by enhancing the transport of mass, heat, humidity and pollutants. The global objective for our work is to analyze some direct turbulent descriptors which reflect the mixing processes in the atmospheric boundary layer (ABL). In this paper we present results related to the Thorpe displacements d Τ , the maximum Thorpe displacement (d Τ ) max and the Thorpe scale L Τ , the Ozmidov scale and their time evolution in the ABL during a day cycle. A tethered balloon was used to obtain vertical profiles of the atmospheric physical magnitudes up to 1000 m. We discuss the vertical and horizontal variability and how different descriptors are related to atmospheric mixing.
13. Rapid cycling of reactive nitrogen in the marine boundary layer.
Ye, Chunxiang; Zhou, Xianliang; Pu, Dennis; Stutz, Jochen; Festa, James; Spolaor, Max; Tsai, Catalina; Cantrell, Christopher; Mauldin, Roy L; Campos, Teresa; Weinheimer, Andrew; Hornbrook, Rebecca S; Apel, Eric C; Guenther, Alex; Kaser, Lisa; Yuan, Bin; Karl, Thomas; Haggerty, Julie; Hall, Samuel; Ullmann, Kirk; Smith, James N; Ortega, John; Knote, Christoph
2016-04-28
Nitrogen oxides are essential for the formation of secondary atmospheric aerosols and of atmospheric oxidants such as ozone and the hydroxyl radical, which controls the self-cleansing capacity of the atmosphere. Nitric acid, a major oxidation product of nitrogen oxides, has traditionally been considered to be a permanent sink of nitrogen oxides. However, model studies predict higher ratios of nitric acid to nitrogen oxides in the troposphere than are observed. A 'renoxification' process that recycles nitric acid into nitrogen oxides has been proposed to reconcile observations with model studies, but the mechanisms responsible for this process remain uncertain. Here we present data from an aircraft measurement campaign over the North Atlantic Ocean and find evidence for rapid recycling of nitric acid to nitrous acid and nitrogen oxides in the clean marine boundary layer via particulate nitrate photolysis. Laboratory experiments further demonstrate the photolysis of particulate nitrate collected on filters at a rate more than two orders of magnitude greater than that of gaseous nitric acid, with nitrous acid as the main product. Box model calculations based on the Master Chemical Mechanism suggest that particulate nitrate photolysis mainly sustains the observed levels of nitrous acid and nitrogen oxides at midday under typical marine boundary layer conditions. Given that oceans account for more than 70 per cent of Earth's surface, we propose that particulate nitrate photolysis could be a substantial tropospheric nitrogen oxide source. Recycling of nitrogen oxides in remote oceanic regions with minimal direct nitrogen oxide emissions could increase the formation of tropospheric oxidants and secondary atmospheric aerosols on a global scale.
14. The control effect in a detached laminar boundary layer of an array of normal synthetic jets
Valenzuela Calva, Fernando; Avila Rodriguez, Ruben
2016-11-01
In this work, 3D numerical simulations of an array of three normal circular synthetic jets embedded in an attached laminar boundary layer that separates under the influence of an inclined flap are performed for flow separation control. At the beginning of the present study, three cases are used to validate the numerical simulation with data obtained from experiments. The experimental data is chosen based on the cases which presented higher repeatability and reliability. Simulations showed reasonable agreement when compared with experiments. The simulations are undertaken at three synthetic jet operating conditions, i.e. Case A: L = 2, VR = 0.32; Case B: L = 4, VR = 0.64 and Case C: L = 6, VR = 0.96. The vortical structures produced for each synthetic jet operating condition are hairpin vortices for Case A and tilted vortices for Case B and C, respectively. By examining the spatial wall shear stress variations, the effect on the boundary layer prior to separation of the middle synthetic jet is evaluated. For effective flow control, produced at a relatively low the finding from this study suggests that hairpin vortical structures are more desirable structures. Universidad Nacional Autonoma de Mexico.
15. The Stokes boundary layer for a thixotropic or antithixotropic fluid
McArdle, Catriona R.
2012-10-01
We present a mathematical investigation of the oscillatory boundary layer in a semi-infinite fluid bounded by an oscillating wall (the so-called \\'Stokes problem\\'), when the fluid has a thixotropic or antithixotropic rheology. We obtain asymptotic solutions in the limit of small-amplitude oscillations, and we use numerical integration to validate the asymptotic solutions and to explore the behaviour of the system for larger-amplitude oscillations. The solutions that we obtain differ significantly from the classical solution for a Newtonian fluid. In particular, for antithixotropic fluids the velocity reaches zero at a finite distance from the wall, in contrast to the exponential decay for a thixotropic or a Newtonian fluid.For small amplitudes of oscillation, three regimes of behaviour are possible: the structure parameter may take values defined instantaneously by the shear rate, or by a long-term average; or it may behave hysteretically. The regime boundaries depend on the precise specification of structure build-up and breakdown rates in the rheological model, illustrating the subtleties of complex fluid models in non-rheometric settings. For larger amplitudes of oscillation the dominant behaviour is hysteretic. We discuss in particular the relationship between the shear stress and the shear rate at the oscillating wall. © 2012 Elsevier B.V.
16. Color surface-flow visualization of fin-generated shock wave boundary-layer interactions
Lu, F. K.; Settles, G. S.
1990-03-01
Kerosene-lampblack mixtures with addition of a ground colored chalk were used in an experiment on visualizing surface flows of swept shock boundary-layer interactions. The results show that contrasting colors intensify the visualization of different regions of the interaction surface, and help the eye in following the fine streaks to locate the upstream influence. The study confirms observations of the separation occurring at shock strength below accepted values. The superiority of the reported technique over the previous monochrome technique is demonstrated.
17. Transonic Shock-Wave/Boundary-Layer Interactions on an Oscillating Airfoil
Davis, Sanford S.; Malcolm, Gerald N.
1980-01-01
Unsteady aerodynamic loads were measured on an oscillating NACA 64A010 airfoil In the NASA Ames 11 by 11 ft Transonic Wind Tunnel. Data are presented to show the effect of the unsteady shock-wave/boundary-layer interaction on the fundamental frequency lift, moment, and pressure distributions. The data show that weak shock waves induce an unsteady pressure distribution that can be predicted quite well, while stronger shock waves cause complex frequency-dependent distributions due to flow separation. An experimental test of the principles of linearity and superposition showed that they hold for weak shock waves while flows with stronger shock waves cannot be superimposed.
18. Direct measurements of wall shear stress by buried wire gages in a shock-wave boundary-layer interaction region
Murthy, V. S.; Rose, W. C.
1977-01-01
Detailed measurements of wall shear stress (skin friction) were made with specially developed buried wire gages in the interaction regions of a Mach 2.9 turbulent boundary layer with externally generated shocks. Separation and reattachment points inferred by these measurements support the findings of earlier experiments which used a surface oil flow technique and pitot profile measurements. The measurements further indicate that the boundary layer tends to attain significantly higher skin-friction values downstream of the interaction region as compared to upstream. Comparisons between measured wall shear stress and published results of some theoretical calculation schemes show that the general, but not detailed, behavior is predicted well by such schemes.
19. Studies of planetary boundary layer by infrared thermal imagery
Albina, Bogdan; Dimitriu, Dan Gheorghe, E-mail: [email protected]; Gurlui, Silviu Octavian, E-mail: [email protected] [Alexandru Ioan Cuza University of Iasi, Faculty of Physics, Atmosphere Optics, Spectroscopy and Lasers Laboratory, 11 Carol I Blvd., 700506 Iasi (Romania); Cazacu, Marius Mihai [Alexandru Ioan Cuza University of Iasi, Faculty of Physics, Atmosphere Optics, Spectroscopy and Lasers Laboratory, 11 Carol I Blvd., 700506 Iasi, Romania and Department of Physics, Gheorghe Asachi Technical University of Iasi, 59A Mangeron Blvd., 700 (Romania); Timofte, Adrian [Alexandru Ioan Cuza University of Iasi, Faculty of Physics, Atmosphere Optics, Spectroscopy and Lasers Laboratory, 11 Carol I Blvd., 700506 Iasi, Romania and National Meteorological Administration, Regional Forecast Center Bacau, 1 Cuza Voda Str., 60 (Romania)
2014-11-24
The IR camera is a relatively novel device for remote sensing of atmospheric thermal processes from the Planetary Boundary Layer (PBL) based on measurements of the infrared radiation. Infrared radiation is energy radiated by the motion of atoms and molecules on the surface of aerosols, when their temperature is more than absolute zero. The IR camera measures directly the intensity of radiation emitted by aerosols which is converted by an imaging sensor into an electric signal, resulting a thermal image. Every image pixel that corresponds to a specific radiance is pre-processed to identify the brightness temperature. The thermal infrared imaging radiometer used in this study, NicAir, is a precision radiometer developed by Prata et al. The device was calibrated for the temperature range of 270–320 K and using a calibration table along with image processing software, important information about variations in temperature can be extracted from acquired IR images. The PBL is the lowest layer of the troposphere where the atmosphere interacts with the ground surfaces. The importance of PBL lies in the fact that it provides a finite but varying volume in which pollutants can disperse. The aim of this paper is to analyze the PBL altitude and thickness variations over Iasi region using the IR imaging camera as well as its behavior from day to night and thermal processes occurring in PBL.
20. Complementary aspects on matter-antimatter boundary layers
Lehnert, B.
1990-05-01
This paper gives some complementary aspects on the problems of the matter-antimatter metagalaxy model and its cellular structure, as being proposed by Klein and Alfven. A previously outlined one-dimensional model of a magnetized matter-antimatter boundary layer is updated and extended, by introducing amended nuclear annihilation data, and by making improved approximations of the layer structure and its dependence on relevant parameters. The critical beta value obtained from this model leads to critical plasma densities which are not high enough to become reconcilable with a cellular matter-antimatter structure within the volume of a galaxy. Additional investigations are required on the questions whether the obtained beta limit would still apply to cells of the size of a galaxy, and whether large modification of this limit could result from further refinement of the theory and from the transition to a three-dimensional model. Attention is called to the wide area of further research on ambiplasma physics, and on a three-dimensional cell structure with associated problems of equilibrium and stability. In particular, the high-energy ambiplasma component has to be further analysed in terms of kinetic theory, on account of the large Larmor radii of the corresponding electrons and positrons
1. Uncertainties in the CO2 buget associated to boundary layer dynamics and CO2-advection
Kaikkonen, J.P.; Pino, D.; Vilà-Guerau de Arellano, J.
2012-01-01
The relationship between boundary layer dynamics and carbon dioxide (CO2) budget in the convective boundary layer (CBL) is investigated by using mixed-layer theory. We derive a new set of analytical relations to quantify the uncertainties on the estimation of the bulk CO2 mixing ratio and the
2. Response of the Atmospheric Boundary Layer and Soil Layer to a High Altitude, Dense Aerosol Cover.
Garratt, J. R.; Pittock, A. B.; Walsh, K.
1990-01-01
The response of the atmospheric boundary layer to the appearance of a high-altitude smoke layer has been investigated in a mesoscale numerical model of the atmosphere. Emphasis is placed on the changes in mean boundary-layer structure and near-surface temperatures when smoke of absorption optical depth (AOD) in the, range 0 to 1 is introduced. Calculations have been made at 30°S, for different soil thermal properties and degrees of surface wetness, over a time period of several days during which major smoke-induced cooling occurs. The presence of smoke reduces the daytime mixed-layer depth and, for large enough values of AOD, results in a daytime surface inversion with large cooling confined to heights of less than a few hundred meters. Smoke-induced reductions in daytime soil and air temperatures of several degrees are typical, dependent critically upon soil wetness and smoke AOD. Locations near the coast experience reduced cooling whenever there is a significant onshore flow related to a sea breeze (this would also be the case with a large-scale onshore flow). The sea breeze itself disappears for large enough smoke AOD and, over sloping coastal terrain, a smoke-induced, offshore drainage flow may exist throughout the diurnal cycle.
3. Structure of a mushy layer at the inner core boundary
Deguen, R.; Huguet, L.; Bergman, M. I.; Labrosse, S.; Alboussiere, T.
2015-12-01
We present experimental results on the solidification of ammonium chloride from an aqueous solution, yielding a mushy zone, under hyper-gravity. A commercial centrifuge has been equipped with a slip-ring so that electric power, temperature and ultrasonic signals could be transmitted between the experimental setup and the laboratory. A Peltier element provides cooling at the bottom of the cell. Probes monitor the temperature along the height of the cell. Ultrasound measurements (2 to 6 MHz) is used to detect the position of the front of the mushy zone and to determine attenuation in the mush. A significant increase of solid fraction (or decrease of mushy layer thickness) and attenuation in the mush is observed as gravity is increased. Kinetic undercooling is significant in our experiments and has been included in a macroscopic mush model. The other ingredients of the model are conservation of energy and chemical species, along with heat/species transfer between the mush and the liquid phase: boundary-layer exchanges at the top of the mush and bulk convection within the mush (formation of chimneys). The outputs of the model compare well with our experiments. We have then run the model in a range of parameters suitable for the Earth's inner core, which has shown the role of bulk mush convection for the inner core and the reason why a solid fraction very close to unity should be expected. We have also run melting experiments: after crystallization of a mush, the liquid has been heated from above until the mush started to melt, while the bottom cold temperature was maintained. These melting experiments were motivated by the possible local melting at the inner core boundary that has been invoked to explain the formation of the anomalously slow F-layer at the bottom of the outer core or inner core hemispherical asymmetry. Oddly, the consequences of melting are an increase in solid fraction and a decrease in attenuation. It is hence possible that surface seismic velocity
4. Turbulent Boundary Layer Over Geophysical-like Topographies
Chamorro, L. P.; Hamed, A. M.; Castillo, L.
2016-12-01
An experimental investigation of the flow and the turbulence structure over 2D and 3D large-scale wavy walls was performed using high-resolution planar particle image velocimetry in a refractive-index-matching (RIM) channel. Extensive measurements were performed to characterize the developing and developed flows. The 2D wall is described by a sinusoidal wave in the streamwise direction with amplitude to wavelength ratio a/λx = 0.05, while the 3D wall has an additional wave superimposed in the spanwise direction with a/λy = 0.1. The flow over these walls was characterized at Reynolds numbers of 4000 and 40000, based on the bulk velocity and the channel half height. The walls have an amplitude to boundary layer thickness ratio a/δ99 ≈ 0.1 and resemble large-scale and geophysical-like roughnesses found in rivers beds and natural terrain. Instantaneous velocity fields and time-averaged turbulence quantities reveal strong coupling between large-scale topography and the turbulence dynamics near the wall. Turbulence statistics for both walls show the presence of a well-structured shear layer past the roughness crests. Analysis of the turbulent kinetic energy production rate suggests that the shear layer is responsible for the majority of turbulence production across both walls. However, the 3D wall exhibits preferential spanwise flows that are thought to result in the multiple distinctive flow features for the 3D wall including comparatively reduced spanwise vorticity and decreased turbulence levels. Further insight on the effect of roughness three-dimensionality and Reynolds number is drawn in both the developed and developing regions through proper orthogonal decomposition (POD) and quadrant analysis.
5. On the Unsteadiness of a Transitional Shock Wave-Boundary Layer Interaction Using Fast-Response Pressure-Sensitive Paint
Lash, E. Lara; Schmisseur, John
2017-11-01
Pressure-sensitive paint has been used to evaluate the unsteady dynamics of transitional and turbulent shock wave-boundary layer interactions generated by a vertical cylinder on a flat plate in a Mach 2 freestream. The resulting shock structure consists of an inviscid bow shock that bifurcates into a separation shock and trailing shock. The primary features of interest are the separation shock and an upstream influence shock that is intermittently present in transitional boundary layer interactions, but not observed in turbulent interactions. The power spectral densities, frequency peaks, and normalized wall pressures are analyzed as the incoming boundary layer state changes from transitional to fully turbulent, comparing both centerline and outboard regions of the interaction. The present study compares the scales and frequencies of the dynamics of the separation shock structure in different boundary layer regimes. Synchronized high-speed Schlieren imaging provides quantitative statistical analyses as well as qualitative comparisons to the fast-response pressure sensitive paint measurements. Materials based on research supported by the U.S. Office of Naval Research under Award Number N00014-15-1-2269.
6. Large artificially generated turbulent boundary layers for the study of atmospheric flows
Guimaraes, Joao Henrique D.; Santos Junior, Sergio J.F. dos; Freire, Atila P. Silva; Jian, Su
1999-01-01
The present work discusses in detail the experimental conditions for the establishment of thick artificially generated turbulent boundary layer which can be classified as having the near characteristics of an atmospheric boundary layer. The paper describes the experimental arrangement, including the features of the designed wind tunnel and of the instrumentation. the boundary layer is made to develop over a surface fitted with wedge generators which are used to yield a very thick boundary layer. The flow conditions were validated against the following features: growth, structure, equilibrium and turbulent transport momentum. Results are presented for the following main flow variables: mean velocity, local skin-friction coefficient, boundary layer momentum thickness and the Clauser factor. The velocity boundary layer characteristics were shown to be in good agreement with the expected trend in view of the classical expressions found in literature. (author)
7. High-energy X-ray production in a boundary layer of an accreting neutron star
Hanawa, Tomoyuki
1991-01-01
It is shown by Monte Carlo simulation that high-energy X-rays are produced through Compton scattering in a boundary layer of an accreting neutron star. The following is the mechanism for the high-energy X-ray production. An accreting neutron star has a boundary layer rotating rapidly on the surface. X-rays radiated from the star's surface are scattered in part in the boundary layer. Since the boundary layer rotates at a semirelativistic speed, the scattered X-ray energy is changed by the Compton effect. Some X-rays are scattered repeatedly between the neutron star and the boundary layer and become high-energy X-rays. This mechanism is a photon analog of the second-order Fermi acceleration of cosmic rays. When the boundary layer is semitransparent, high-energy X-rays are produced efficiently. 17 refs
8. Analysis of Windward Side Hypersonic Boundary Layer Transition on Blunted Cones at Angle of Attack
2017-01-09
correlated with PSE/LST N-Factors. 15. SUBJECT TERMS boundary layer transition, hypersonic, ground test 16. SECURITY CLASSIFICATION OF: 17. LIMITATION ...Maccoll) solution e condition at boundary layer edge w condition at wall, viscous ∞ condition in freestream Conventions LST Linear Stability Theory PSE...STATES AIR FORCE AFRL-RQ-WP-TP-2017-0169 ANALYSIS OF WINDWARD SIDE HYPERSONIC BOUNDARY LAYER TRANSITION ON BLUNTED CONES AT ANGLE OF ATTACK Roger
9. Boundary layer friction of solvate ionic liquids as a function of potential.
Li, Hua; Rutland, Mark W; Watanabe, Masayoshi; Atkin, Rob
2017-07-01
Atomic force microscopy (AFM) has been used to investigate the potential dependent boundary layer friction at solvate ionic liquid (SIL)-highly ordered pyrolytic graphite (HOPG) and SIL-Au(111) interfaces. Friction trace and retrace loops of lithium tetraglyme bis(trifluoromethylsulfonyl)amide (Li(G4) TFSI) at HOPG present clearer stick-slip events at negative potentials than at positive potentials, indicating that a Li + cation layer adsorbed to the HOPG lattice at negative potentials which enhances stick-slip events. The boundary layer friction data for Li(G4) TFSI shows that at HOPG, friction forces at all potentials are low. The TFSI - anion rich boundary layer at positive potentials is more lubricating than the Li + cation rich boundary layer at negative potentials. These results suggest that boundary layers at all potentials are smooth and energy is predominantly dissipated via stick-slip events. In contrast, friction at Au(111) for Li(G4) TFSI is significantly higher at positive potentials than at negative potentials, which is comparable to that at HOPG at the same potential. The similarity of boundary layer friction at negatively charged HOPG and Au(111) surfaces indicates that the boundary layer compositions are similar and rich in Li + cations for both surfaces at negative potentials. However, at Au(111), the TFSI - rich boundary layer is less lubricating than the Li + rich boundary layer, which implies that anion reorientations rather than stick-slip events are the predominant energy dissipation pathways. This is confirmed by the boundary friction of Li(G4) NO 3 at Au(111), which shows similar friction to Li(G4) TFSI at negative potentials due to the same cation rich boundary layer composition, but even higher friction at positive potentials, due to higher energy dissipation in the NO 3 - rich boundary layer.
10. Turbulent boundary layer noise : direct radiation at Mach number 0.5
Gloerfelt , Xavier; Berland , Julien
2013-01-01
International audience; Boundary layers constitute a fundamental source of aerodynamic noise. A turbulent boundary layer over a plane wall can provide an indirect contribution to the noise by exciting the structure, and a direct noise contribution. The latter part can play a significant role even if its intensity is very low, explaining why it is hardly measured unambiguously. In the present study, the aerodynamic noise generated by a spatially developing turbulent boundary layer is computed ...
11. MHD Boundary Layer Slip Flow and Heat Transfer over a Flat Plate
Bhattacharyya, Krishnendu; Mukhopadhyay, Swati; Layek, G. C.
2011-01-01
An analysis of magnetohydrodynamic (MHD) boundary layer flow and heat transfer over a flat plate with slip condition at the boundary is presented. A complete self-similar set of equations are obtained from the governing equations using similarity transformations and are solved by a shooting method. In the boundary slip condition no local similarity occurs. Velocity and temperature distributions within the boundary layer are presented. Our analysis reveals that the increase of magnetic and slip parameters reduce the boundary layer thickness and also enhance the heat transfer from the plate. (fundamental areas of phenomenology(including applications))
12. The vertical structure of the Saharan boundary layer: Observations and modelling
Garcia-Carreras, L.; Parker, D. J.; Marsham, J. H.; Rosenberg, P.; Marenco, F.; Mcquaid, J.
2012-04-01
The vertical structure of the Saharan atmospheric boundary layer (SABL) is investigated with the use of aircraft data from the Fennec observational campaign, and high-resolution large-eddy model (LEM) simulations. The SABL is one of the deepest on Earth, and crucial in controlling the vertical redistribution and long-range transport of dust in the Sahara. The SABL is typically made up of an actively growing convective region driven by high sensible heating at the surface, with a deep, near-neutrally stratified Saharan residual layer (SRL) above it, which is mostly well mixed in humidity and temperature and reaches a height of ~500hPa. These two layers are usually separated by a weak (≤1K) temperature inversion, making the vertical structure very sensitive to the surface fluxes. Large-eddy model (LEM) simulations initialized with radiosonde data from Bordj Bardji Mokhtar (BBM), southern Algeria, are used to improve our understanding of the turbulence structure of the stratification of the SABL, and any mixing or exchanges between the different layers. The model can reproduce the typical SABL structure from observations, and a tracer is used to illustrate the growth of the convective boundary layer into the residual layer above. The heat fluxes show a deep entrainment zone between the convective region and the SRL, potentially enhanced by the combination of a weak lid and a neutral layer above. The horizontal variability in the depth of the convective layer was also significant even with homogeneous surface fluxes. Aircraft observations from a number of flights are used to validate the model results, and to highlight the variability present in a more realistic setting, where conditions are rarely homogeneous in space. Stacked legs were performed to get an estimate of the mean flux profile of the boundary layer, as well as the variations in the vertical structure of the SABL with heterogeneous atmospheric and surface conditions. Regular radiosondes from BBM put
13. Optimally growing boundary layer disturbances in a convergent nozzle preceded by a circular pipe
Uzun, Ali; Davis, Timothy B.; Alvi, Farrukh S.; Hussaini, M. Yousuff
2017-06-01
We report the findings from a theoretical analysis of optimally growing disturbances in an initially turbulent boundary layer. The motivation behind this study originates from the desire to generate organized structures in an initially turbulent boundary layer via excitation by disturbances that are tailored to be preferentially amplified. Such optimally growing disturbances are of interest for implementation in an active flow control strategy that is investigated for effective jet noise control. Details of the optimal perturbation theory implemented in this study are discussed. The relevant stability equations are derived using both the standard decomposition and the triple decomposition. The chosen test case geometry contains a convergent nozzle, which generates a Mach 0.9 round jet, preceded by a circular pipe. Optimally growing disturbances are introduced at various stations within the circular pipe section to facilitate disturbance energy amplification upstream of the favorable pressure gradient zone within the convergent nozzle, which has a stabilizing effect on disturbance growth. Effects of temporal frequency, disturbance input and output plane locations as well as separation distance between output and input planes are investigated. The results indicate that optimally growing disturbances appear in the form of longitudinal counter-rotating vortex pairs, whose size can be on the order of several times the input plane mean boundary layer thickness. The azimuthal wavenumber, which represents the number of counter-rotating vortex pairs, is found to generally decrease with increasing separation distance. Compared to the standard decomposition, the triple decomposition analysis generally predicts relatively lower azimuthal wavenumbers and significantly reduced energy amplification ratios for the optimal disturbances.
14. Time-resolved stereo PIV measurements of shock-boundary layer interaction on a supercritical airfoil
Hartmann, Axel; Klaas, Michael; Schroeder, Wolfgang [RWTH Aachen University, Institute of Aerodynamics, Aachen (Germany)
2012-03-15
Time-resolved stereo particle-image velocimetry (TR-SPIV) and unsteady pressure measurements are used to analyze the unsteady flow over a supercritical DRA-2303 airfoil in transonic flow. The dynamic shock wave-boundary layer interaction is one of the most essential features of this unsteady flow causing a distinct oscillation of the flow field. Results from wind-tunnel experiments with a variation of the freestream Mach number at Reynolds numbers ranging from 2.55 to 2.79 x 10{sup 6} are analyzed regarding the origin and nature of the unsteady shock-boundary layer interaction. Therefore, the TR-SPIV results are analyzed for three buffet flows. One flow exhibits a sinusoidal streamwise oscillation of the shock wave only due to an acoustic feedback loop formed by the shock wave and the trailing-edge noise. The other two buffet flows have been intentionally influenced by an artificial acoustic source installed downstream of the test section to investigate the behavior of the interaction to upstream-propagating disturbances generated by a defined source of noise. The results show that such upstream-propagating disturbances could be identified to be responsible for the upstream displacement of the shock wave and that the feedback loop is formed by a pulsating separation of the boundary layer dependent on the shock position and the sound pressure level at the shock position. Thereby, the pulsation of the separation could be determined to be a reaction to the shock motion and not vice versa. (orig.)
15. Bubble and boundary layer behaviour in subcooled flow boiling
Maurus, Reinhold; Sattelmayer, Thomas [Lehrstuhl fuer Thermodynamik, Technische Universitaet Muenchen, 85747 Garching (Germany)
2006-03-15
Subcooled flow boiling is a commonly applied technique for achieving efficient heat transfer. In the study, an experimental investigation in the nucleate boiling regime was performed for water circulating in a closed loop at atmospheric pressure. The horizontal orientated test-section consists of a rectangular channel with a one side heated copper strip and good optical access. Various optical observation techniques were applied to study the bubble behaviour and the characteristics of the fluid phase. The bubble behaviour was recorded by the high-speed cinematography and by a digital high resolution camera. Automated image processing and analysis algorithms developed by the authors were applied for a wide range of mass flow rates and heat fluxes in order to extract characteristic length and time scales of the bubbly layer during the boiling process. Using this methodology, the bubbles were automatically analysed and the bubble size, bubble lifetime, waiting time between two cycles were evaluated. Due to the huge number of observed bubbles a statistical analysis was performed and distribution functions were derived. Using a two-dimensional cross-correlation algorithm, the averaged axial phase boundary velocity profile could be extracted. In addition, the fluid phase velocity profile was characterised by means of the particle image velocimetry (PIV) for the single phase flow as well as under subcooled flow boiling conditions. The results indicate that the bubbles increase the flow resistance. The impact on the flow exceeds by far the bubbly region and it depends on the magnitude of the boiling activity. Finally, the ratio of the averaged phase boundary velocity and of the averaged fluid velocity was evaluated for the bubbly region. (authors)
16. CFD simulation of neutral ABL flows; Atmospheric Boundary Layer
Xiaodong Zhang
2009-04-15
This work is to evaluate the CFD prediction of Atmospheric Boundary Layer flow field over different terrains employing Fluent 6.3 software. How accurate the simulation could achieve depend on following aspects: viscous model, wall functions, agreement of CFD model with inlet wind velocity profile and top boundary condition. Fluent employ wall function roughness modifications based on data from experiments with sand grain roughened pipes and channels, describe wall adjacent zone with Roughness Height (Ks) instead of Roughness Length (z{sub 0}). In a CFD simulation of ABL flow, the mean wind velocity profile is generally described with either a logarithmic equation by the presence of aerodynamic roughness length z{sub 0} or an exponential equation by the presence of exponent. As indicated by some former researchers, the disagreement between wall function model and ABL velocity profile description will result in some undesirable gradient along flow direction. There are some methods to improve the simulation model in literatures, some of them are discussed in this report, but none of those remedial methods are perfect to eliminate the streamwise gradients in mean wind speed and turbulence, as EllipSys3D could do. In this paper, a new near wall treatment function is designed, which, in some degree, can correct the horizontal gradients problem. Based on the corrected model constants and near wall treatment function, a simulation of Askervein Hill is carried out. The wind condition is neutrally stratified ABL and the measurements are best documented until now. Comparison with measured data shows that the CFD model can well predict the velocity field and relative turbulence kinetic energy field. Furthermore, a series of artificial complex terrains are designed, and some of the main simulation results are reported. (au)
17. Heat transfer through turbulent boundary layers - The effects of introduction of and recovery from convex curvature
Simon, T. W.; Moffat, R. J.
1979-01-01
Measurements have been made of the heat transfer through a turbulent boundary layer on a convexly curved isothermal wall and on a flat plate following the curved section. Data were taken for one free-stream velocity and two different ratios of boundary layer thickness to radius of curvature delta/R = 0.051 and delta/R = 0.077. Only small differences were observed in the distribution of heat transfer rates for the two boundary layer thicknesses tested, although differences were noted in the temperature distributions within the boundary layer
18. Mitigation of Adverse Effects Caused by Shock Wave Boundary Layer Interactions Through Optimal Wall Shaping
Liou, May-Fun; Lee, Byung Joon
2013-01-01
It is known that the adverse effects of shock wave boundary layer interactions in high speed inlets include reduced total pressure recovery and highly distorted flow at the aerodynamic interface plane (AIP). This paper presents a design method for flow control which creates perturbations in geometry. These perturbations are tailored to change the flow structures in order to minimize shock wave boundary layer interactions (SWBLI) inside supersonic inlets. Optimizing the shape of two dimensional micro-size bumps is shown to be a very effective flow control method for two-dimensional SWBLI. In investigating the three dimensional SWBLI, a square duct is employed as a baseline. To investigate the mechanism whereby the geometric elements of the baseline, i.e. the bottom wall, the sidewall and the corner, exert influence on the flow's aerodynamic characteristics, each element is studied and optimized separately. It is found that arrays of micro-size bumps on the bottom wall of the duct have little effect in improving total pressure recovery though they are useful in suppressing the incipient separation in three-dimensional problems. Shaping sidewall geometry is effective in re-distributing flow on the side wall and results in a less distorted flow at the exit. Subsequently, a near 50% reduction in distortion is achieved. A simple change in corner geometry resulted in a 2.4% improvement in total pressure recovery.
19. Plasmons in spatially separated double-layer graphene nanoribbons
Bagheri, Mehran; Bahrami, Mousa
2014-01-01
Motivated by innovative progresses in designing multi-layer graphene nanostructured materials in the laboratory, we theoretically investigate the Dirac plasmon modes of a spatially separated double-layer graphene nanoribbon system, made up of a vertically offset armchair and metallic graphene nanoribbon pair. We find striking features of the collective excitations in this novel Coulomb correlated system, where both nanoribbons are supposed to be either intrinsic (undoped/ungated) or extrinsic (doped/gated). In the former, it is shown the low-energy acoustical and the high-energy optical plasmon modes are tunable only by the inter-ribbon charge separation. In the later, the aforementioned plasmon branches are modified by the added doping factor. As a result, our model could be useful to examine the existence of a linear Landau-undamped low-energy acoustical plasmon mode tuned via the inter-ribbon charge separation as well as doping. This study might also be utilized for devising novel quantum optical waveguides based on the Coulomb coupled graphene nanoribbons
20. Numerical investigation on effects of induced jet on boundary layer and turbulent models around airfoils
Shojaeefard, M.H.; Pirnia, A.; Fallahian, M.A. [Iran University of Science and Technology, School of Mechanical Engineering, Tehran (Iran, Islamic Republic of); Tahani, M. [Iran University of Science and Technology, School of Mechanical Engineering, Tehran (Iran, Islamic Republic of); University of Tehran, Faculty of New Science and Technology, Tehran (Iran, Islamic Republic of)
2012-06-15
In this study the effects of induced jet at trailing edge of a two dimensional airfoil on its boundary layer shape, separation over surface and turbulent parameters behind trailing edge are numerically investigated and compared against a previous experimental data. After proving independency of results from mesh size and obtaining the required mesh size, different turbulent models are examined and RNG k-epsilon model is chosen because of good agreement with experimental data in velocity and turbulent intensity variations. A comparison between ordinary and jet induced cases, regarding numerical data, is made. The results showed that because of low number of measurement points in experimental study, turbulent intensity extremes are not captured. While in numerical study, these values and their positions are well calculated and exact variation of turbulent intensity is acquired. Also a study in effect of jet at high angles of attack is done and the results showed the ability of jet in controlling separation and reducing wake region. (orig.)
1. HEOS-2 observations of the boundary layer from the magnetopause to the ionosphere
Formisano, V.
1980-01-01
HEOS-2 low energy electron data (10 eV - 3.7 keV) from the LPS Frascati plasma experiment have been used to identify three different magnetospheric electron populations. Magnetosheath-like electron energy spectra (35-50 eV) are characteristic of the plasma mantle, entry layer and cusps from the magnetopause down to 2-3 Rsub(E). Plasma sheet electrons (energy > 1 keV) are found at all local times, with strong intensities in the early morning quadrant and weaker intensities in the afternoon quadrant. The plasma sheet shows a well defined inner edge at all local times and latitudes, the inner edge coinciding probably with the plasmapause. The plasma sheet does not reach the magnetopause, but it is separated from it by a boundary layer electron population that is very distinct from the other two electron populations, most electrons having energies 100-300 eV. These three electron populations are mapped from the magnetopause down to the high latitude near earth regions. The boundary layer extends along the magnetopause up to 5-7 Rsub(E) above the equator; at higher latitudes it follows the magnetic lines of force and it is found closer and closer to the earth. (author)
2. Three Dimensional Plenoptic PIV Measurements of a Turbulent Boundary Layer Overlying a Hemispherical Roughness Element
Johnson, Kyle; Thurow, Brian; Kim, Taehoon; Blois, Gianluca; Christensen, Kenneth
2016-11-01
Three-dimensional, three-component (3D-3C) measurements were made using a plenoptic camera on the flow around a roughness element immersed in a turbulent boundary layer. A refractive index matched approach allowed whole-field optical access from a single camera to a measurement volume that includes transparent solid geometries. In particular, this experiment measures the flow over a single hemispherical roughness element made of acrylic and immersed in a working fluid consisting of Sodium Iodide solution. Our results demonstrate that plenoptic particle image velocimetry (PIV) is a viable technique to obtaining statistically-significant volumetric velocity measurements even in a complex separated flow. The boundary layer to roughness height-ratio of the flow was 4.97 and the Reynolds number (based on roughness height) was 4.57×103. Our measurements reveal key flow features such as spiraling legs of the shear layer, a recirculation region, and shed arch vortices. Proper orthogonal decomposition (POD) analysis was applied to the instantaneous velocity and vorticity data to extract these features. Supported by the National Science Foundation Grant No. 1235726.
3. Is the boundary layer of an ionic liquid equally lubricating at higher temperature?
Hjalmarsson, Nicklas; Atkin, Rob; Rutland, Mark W
2016-04-07
Atomic force microscopy has been used to study the effect of temperature on normal forces and friction for the room temperature ionic liquid (IL) ethylammonium nitrate (EAN), confined between mica and a silica colloid probe at 25 °C, 50 °C, and 80 °C. Force curves revealed a strong fluid dynamic influence at room temperature, which was greatly reduced at elevated temperatures due to the reduced liquid viscosity. A fluid dynamic analysis reveals that bulk viscosity is manifested at large separation but that EAN displays a nonzero slip, indicating a region of different viscosity near the surface. At high temperatures, the reduction in fluid dynamic force reveals step-like force curves, similar to those found at room temperature using much lower scan rates. The ionic liquid boundary layer remains adsorbed to the solid surface even at high temperature, which provides a mechanism for lubrication when fluid dynamic lubrication is strongly reduced. The friction data reveals a decrease in absolute friction force with increasing temperature, which is associated with increased thermal motion and reduced viscosity of the near surface layers but, consistent with the normal force data, boundary layer lubrication was unaffected. The implications for ILs as lubricants are discussed in terms of the behaviour of this well characterised system.
4. Turbulent transport of large particles in the atmospheric boundary layer
Richter, D. H.; Chamecki, M.
2017-12-01
To describe the transport of heavy dust particles in the atmosphere, assumptions must typically be made in order to connect the micro-scale emission processes with the larger-scale atmospheric motions. In the context of numerical models, this can be thought of as the transport process which occurs between the domain bottom and the first vertical grid point. For example, in the limit of small particles (both low inertia and low settling velocity), theory built upon Monin-Obukhov similarity has proven effective in relating mean dust concentration profiles to surface emission fluxes. For increasing particle mass, however, it becomes more difficult to represent dust transport as a simple extension of the transport of a passive scalar due to issues such as the crossing trajectories effect. This study focuses specifically on the problem of large particle transport and dispersion in the turbulent boundary layer by utilizing direct numerical simulations with Lagrangian point-particle tracking to determine under what, if any, conditions the large dust particles (larger than 10 micron in diameter) can be accurately described in a simplified Eulerian framework. In particular, results will be presented detailing the independent contributions of both particle inertia and particle settling velocity relative to the strength of the surrounding turbulent flow, and consequences of overestimating surface fluxes via traditional parameterizations will be demonstrated.
5. Three-dimensional boundary layer stability and transition
Malik, M. R.; Li, F.
1992-01-01
Nonparallel and nonlinear stability of a three-dimensional boundary layer, subject to crossflow instability, is investigated using parabolized stability equations (PSEs). Both traveling and stationary disturbances are considered and nonparallel effect on crossflow instability is found to be destabilizing. Our linear PSE results for stationary disturbances agree well with the results from direct solution of Navier-Stokes equations obtained by Spalart (1989). Nonlinear calculations have been carried out for stationary vortices and the computed wall vorticity pattern results in streamwise streaks which resemble remarkably well with the surface oil-flow visualizations in swept-wing experiments. Other features of the stationary vortex development (half-mushroom structure, inflected velocity profiles, vortex doubling, etc.) are also captured in our nonlinear calculations. Nonlinear interaction of the stationary amplitude of the stationary vortex is large as compared to the traveling mode, and the stationary vortex dominates most of the downstream development. When the two modes have the same initial amplitude, the traveling mode dominates the downstream development owing to its higher growth rate, and there is a tendency for the stationary mode to be suppressed. The effect of nonlinear wave development on the skin-friction coefficient is also computed.
6. Non-parallel stability of compressible boundary layers
Chang, Chau-Lyan; Malik, Mujeeb R.
1993-01-01
Linear and nonlinear stability of compressible growing boundary layers is studied using parabolized stability equations (PSE). Linear PSE calculations are performed for Mach 1.6 and 4.5 plate-plate flow, and the results are compared with the predictions of the multiple-scales approach. In general, the nonparallel effect appears to be less significant for oblique waves near the lower neutral branch but it progressively becomes important at higher Reynolds numbers near the upper branch. In contrast, the nonparallel effect is more pronounced near the lower branch for two-dimensional first-mode waves. The PSE and multiple-scales results agree for the first mode waves, but in the first-second mode transition region, the latter approach tends to break down. Comparison with the first (oblique) and second mode growth rate data from Kendall's (1967) experiment shows good agreement; however, the peak second mode growth rate is over-predicted. Similar conclusions are drawn for the second mode experiment of Stetson et al. (1983) for Mach 8 flow past a sharp cone. We conjecture that the lower experimental growth rate is due to nonlinear saturation and provide supporting calculations.
7. Thermal catastrophe in the plasma sheet boundary layer
Smith, R.A.; Goertz, C.K.; Grossmann, W.
1986-01-01
This letter presents a first step towards a substorm model including particle heating and transport in the plasma sheet boundary layer (PSBL). The heating mechanism discussed is resonant absorption of Alfven waves. For some assumed MHD perturbation incident from the tail lobes onto the plasma sheet, the local heating rate in the PSBL has the form of a resonance function of the one-fluid plasma temperature. Balancing the local heating by convective transport of the heated plasma toward the central plasma sheet, and ''equation of state'' is found for the steady-state PSBL whose solution has the form of a mathematical catastrophe: at a critical value of a parameter containing the incident power flux, the local density, and the convection velocity, the equilibrium temperature jumps discontinuously. Associating this temperature increase with the abrupt onset of the substorm expansion phase, the catastrophe model indicates at least three ways in which the onset may be triggered. Several other consequences related to substorm dynamics are suggested by the simple catastrophe model
8. Subgrid-scale turbulence in shock-boundary layer flows
2015-04-01
Data generated by direct numerical simulation (DNS) for a Mach 2.75 zero-pressure gradient turbulent boundary layer interacting with shocks of different intensities are used for a priori analysis of subgrid-scale (SGS) turbulence and various terms in the compressible filtered Navier-Stokes equations. The numerical method used for DNS is based on a hybrid scheme that uses a non-dissipative central scheme in the shock-free turbulent regions and a robust monotonicity-preserving scheme in the shock regions. The behavior of SGS stresses and their components, namely Leonard, Cross and Reynolds components, is examined in various regions of the flow for different shock intensities and filter widths. The backscatter in various regions of the flow is found to be significant only instantaneously, while the ensemble-averaged statistics indicate no significant backscatter. The budgets for the SGS kinetic energy equation are examined for a better understanding of shock-tubulence interactions at the subgrid level and also with the aim of providing useful information for one-equation LES models. A term-by-term analysis of SGS terms in the filtered total energy equation indicate that while each term in this equation is significant by itself, the net contribution by all of them is relatively small. This observation is consistent with our a posteriori analysis.
9. Self-organization in cathode boundary layer discharges in xenon
Takano, Nobuhiko; Schoenbach, Karl H
2006-01-01
Self-organization of direct current xenon microdischarges in cathode boundary layer configuration has been studied for pressures in the range 30-140 Torr and for currents in the range 50 μA-1 mA. Side-on and end-on observations of the discharge have provided information on the structure and spatial arrangement of the plasma filaments. The regularly spaced filaments, which appear in the normal glow mode when the current is lowered, have a length which is determined by the cathode fall. It varies, dependent on pressure and current, between 50 and 70 μm. The minimum diameter is approximately 80 μm, as determined from the radiative emission in the visible. The filaments are sources of extensive excimer emission. Measurements of the cathode fall length have allowed us to determine the secondary emission coefficient for the discharge in the normal glow mode and to estimate the cathode fall voltage at the transition from normal glow mode to filamentary mode. It was found that the cathode fall voltage at this transition decreases, indicating the onset of additional electron gain processes at the cathode. The regular arrangement of the filaments, self-organization, is assumed to be due to Coulomb interactions between the positively charged cathode fall channels and positive space charges on the surface of the surrounding dielectric spacer. Calculations based on these assumptions showed good agreement with experimentally observed filament patterns
10. Benthic boundary layer. IOS observational and modelling programme
Saunders, P.M.; Richards, K.J.
1985-01-01
Near bottom currents, measured at three sites in the N.E. Atlantic, reveal the eddying characteristics of the flow. Eddies develop, migrate and decay in ways best revealed by numerical modelling simulations. Eddies control the thickness of the bottom mixed layer by accumulating and thickening or spreading and thinning the bottom waters. At the boundaries of eddies benthic fronts form providing a path for upward displacement of the bottom water. An experiment designed to estimate vertical diffusivity is performed. The flux of heat into the bottom of the Iberian basin through Discovery Gap is deduced from year long current measurements. The flux is supposed balanced by geothermal heating through the sea floor and diapycnal diffusion in the water. A diffusivity of 1.5 to 4 cm 2 s -1 is derived for the bottom few hundred meters of the deep ocean. Experiments to estimate horizontal diffusivity are described. If a tracer is discharged from the sea bed the volume of sea water in which it is found increases with time and after 20 years will fill an ocean basin of side 1000 km to a depth of only 1 to 2 km. (author)
11. Boundary layer parameterizations and long-range transport
Irwin, J.S.
1992-01-01
A joint work group between the American Meteorological Society (AMS) and the EPA is perusing the construction of an air quality model that incorporates boundary layer parameterizations of dispersion and transport. This model could replace the currently accepted model, the Industrial Source Complex (ISC) model. The ISC model is a Gaussian-plume multiple point-source model that provides for consideration of fugitive emissions, aerodynamic wake effects, gravitational settling and dry deposition. A work group of several Federal and State agencies is perusing the construction of an air quality modeling system for use in assessing and tracking visibility impairment resulting from long-range transport of pollutants. The modeling system is designed to use the hourly vertical profiles of wind, temperature and moisture resulting from a mesoscale meteorological processor that employs four dimensional data assimilation (FDDA). FDDA involves adding forcing functions to the governing model equations to gradually ''nudge'' the model state toward the observations (12-hourly upper air observations of wind, temperature and moisture, and 3-hourly surface observations of wind and moisture). In this way it is possible to generate data sets whose accuracy, in terms of transport, precipitation, and dynamic consistency is superior to both direct interpolation of synoptic-scale analyses of observations and purely predictive mode model result. (AB) ( 19 refs.)
12. Effect of free-stream turbulence on boundary layer transition.
Goldstein, M E
2014-07-28
This paper is concerned with the transition to turbulence in flat plate boundary layers due to moderately high levels of free-stream turbulence. The turbulence is assumed to be generated by an (idealized) grid and matched asymptotic expansions are used to analyse the resulting flow over a finite thickness flat plate located in the downstream region. The characteristic Reynolds number Rλ based on the mesh size λ and free-stream velocity is assumed to be large, and the turbulence intensity ε is assumed to be small. The asymptotic flow structure is discussed for the generic case where the turbulence Reynolds number εRλ and the plate thickness and are held fixed (at O(1) and O(λ), respectively) in the limit as [Formula: see text] and ε→0. But various limiting cases are considered in order to explain the relevant transition mechanisms. It is argued that there are two types of streak-like structures that can play a role in the transition process: (i) those that appear in the downstream region and are generated by streamwise vorticity in upstream flow and (ii) those that are concentrated near the leading edge and are generated by plate normal vorticity in upstream flow. The former are relatively unaffected by leading edge geometry and are usually referred to as Klebanoff modes while the latter are strongly affected by leading edge geometry and are more streamwise vortex-like in appearance. © 2014 The Author(s) Published by the Royal Society. All rights reserved.
13. Boundary Layer Instabilities Generated by Freestream Laser Perturbations
Chou, Amanda; Schneider, Steven P.
2015-01-01
A controlled, laser-generated, freestream perturbation was created in the freestream of the Boeing/AFOSR Mach-6 Quiet Tunnel (BAM6QT). The freestream perturbation convected downstream in the Mach-6 wind tunnel to interact with a flared cone model. The geometry of the flared cone is a body of revolution bounded by a circular arc with a 3-meter radius. Fourteen PCB 132A31 pressure transducers were used to measure a wave packet generated in the cone boundary layer by the freestream perturbation. This wave packet grew large and became nonlinear before experiencing natural transition in quiet flow. Breakdown of this wave packet occurred when the amplitude of the pressure fluctuations was approximately 10% of the surface pressure for a nominally sharp nosetip. The initial amplitude of the second mode instability on the blunt flared cone is estimated to be on the order of 10 -6 times the freestream static pressure. The freestream laser-generated perturbation was positioned upstream of the model in three different configurations: on the centerline, offset from the centerline by 1.5 mm, and offset from the centerline by 3.0 mm. When the perturbation was offset from the centerline of a blunt flared cone, a larger wave packet was generated on the side toward which the perturbation was offset. The offset perturbation did not show as much of an effect on the wave packet on a sharp flared cone as it did on a blunt flared cone.
14. The influence of viscosity stratification on boundary-layer turbulence
Lee, Jin; Jung, Seo Yoon; Sung, Hyung Jin; Zaki, Tamer A.
2012-11-01
Direct numerical simulations of turbulent flows over isothermally-heated walls were performed to investigate the influence of viscosity stratification on boundary-layer turbulence and drag. The adopted model for temperature-dependent viscosity was typical of water. The free-stream temperature was set to 30°C, and two wall temperatures, 70°C and 99°C, were simulated. In the heated flows, the mean shear-rate is enhanced near the wall and reduced in the buffer region, which induces a reduction in turbulence production. On the other hand, the turbulence dissipation is enhanced near the wall, despite the the reduction in fluid viscosity. The higher dissipation is attributed to a decrease in the smallest length scales and near-wall fine-scale motions. The combined effect of the reduced production and enhanced dissipation leads to lower Reynolds shear stresses and, as a result, reduction of the skin-friction coefficient. Supported by the Engineering and Physical Sciences Research Council (Grant EP/F034997/1) and partially supported by the Erasmus Mundus Build on Euro-Asian Mobility (EM-BEAM) programme.
15. Investigation of particle lift off in a turbulent boundary layer
Barros, Diogo; Tee, Yi Hui; Morse, Nicholas; Hiltbrand, Ben; Longmire, Ellen
2017-11-01
Entrainment and suspension of particles within turbulent flows occur widely in environmental and industrial processes. Three-dimensional particle tracking experiments are thus conducted in a water channel to understand the interaction of finite-size particles with a turbulent boundary layer. A neutrally buoyant sphere made of wax and iron oxide is first held in place on the bounding surface by a magnet before being released and tracked. The sphere is marked with dots to monitor rotation as well as translation. By setting up two pairs of cameras in a stereoscopic configuration, the trajectories of the sphere are reconstructed and tracked over a distance of 4 to 6 δ. Sphere diameters ranging from 40 to 130 wall units, initial particle Reynolds numbers of 600 to 2000 and friction Reynolds numbers of 500 to 1800 are considered. For this parameter set, the particle typically lifts off from the wall after release before falling back toward the wall. Aspects of both particle rotation and translation will be discussed. Supported by NSF (CBET-1510154).
16. Density effects on turbulent boundary layer structure: From the atmosphere to hypersonic flow
Williams, Owen J. H.
This dissertation examines the effects of density gradients on turbulent boundary layer statistics and structure using Particle Image Velocimetry (PIV). Two distinct cases were examined: the thermally stable atmospheric surface layer characteristic of nocturnal or polar conditions, and the hypersonic bounder layer characteristic of high speed aircraft and reentering spacecraft. Previous experimental studies examining the effects of stability on turbulent boundary layers identified two regimes, weak and strong stability, separated by a critical bulk stratification with a collapse of near-wall turbulence thought to be intrinsic to the strongly stable regime. To examine the characteristics of these two regimes, PIV measurements were obtained in conjunction with the mean temperature profile in a low Reynolds number facility over smooth and rough surfaces. The turbulent stresses were found to scale with the wall shear stress in the weakly stable regime prior relaminarization at a critical stratification. Changes in profile shape were shown to correlate with the local stratification profile, and as a result, the collapse of near-wall turbulence is not intrinsic to the strongly stable regime. The critical bulk stratification was found to be sensitive to surface roughness and potentially Reynolds number, and not constant as previously thought. Further investigations examined turbulent boundary layer structure and changes to the motions that contribute to turbulent production. To study the characteristics of a hypersonic turbulent boundary layer at Mach 8, significant improvements were required to the implementation and error characterization of PIV. Limited resolution or dynamic range effects were minimized and the effects of high shear on cross-correlation routines were examined. Significantly, an examination of particle dynamics, subject to fluid inertia, compressibility and non-continuum effects, revealed that particle frequency responses to turbulence can be up to an
17. Observations of the atmospheric electric field during two case studies of boundary layer processes
Piper, I M; Bennett, A J
2012-01-01
We present measurements of potential gradient (PG) with associated meteorological variables and cloud profiles for two examples of convective boundary layer processes. Aerosol acts as a tracer layer to show lofting of the convective boundary layer; the rising aerosol layer results in a decrease in PG. In foggy conditions, the PG is seen to increase during the fog and then reduce as the fog lifts, as expected. (letter)
18. Direct numerical simulation of a compressible boundary-layer flow past an isolated three-dimensional hump in a high-speed subsonic regime
De Grazia, D.; Moxey, D.; Sherwin, S. J.; Kravtsova, M. A.; Ruban, A. I.
2018-02-01
In this paper we study the boundary-layer separation produced in a high-speed subsonic boundary layer by a small wall roughness. Specifically, we present a direct numerical simulation (DNS) of a two-dimensional boundary-layer flow over a flat plate encountering a three-dimensional Gaussian-shaped hump. This work was motivated by the lack of DNS data of boundary-layer flows past roughness elements in a similar regime which is typical of civil aviation. The Mach and Reynolds numbers are chosen to be relevant for aeronautical applications when considering small imperfections at the leading edge of wings. We analyze different heights of the hump: The smaller heights result in a weakly nonlinear regime, while the larger result in a fully nonlinear regime with an increasing laminar separation bubble arising downstream of the roughness element and the formation of a pair of streamwise counterrotating vortices which appear to support themselves.
19. Polymer coatings as separator layers for microbial fuel cell cathodes
Watson, Valerie J.
2011-03-01
Membrane separators reduce oxygen flux from the cathode into the anolyte in microbial fuel cells (MFCs), but water accumulation and pH gradients between the separator and cathode reduces performance. Air cathodes were spray-coated (water-facing side) with anion exchange, cation exchange, and neutral polymer coatings of different thicknesses to incorporate the separator into the cathode. The anion exchange polymer coating resulted in greater power density (1167 ± 135 mW m-2) than a cation exchange coating (439 ± 2 mW m-2). This power output was similar to that produced by a Nafion-coated cathode (1114 ± 174 mW m-2), and slightly lower than the uncoated cathode (1384 ± 82 mW m-2). Thicker coatings reduced oxygen diffusion into the electrolyte and increased coulombic efficiency (CE = 56-64%) relative to an uncoated cathode (29 ± 8%), but decreased power production (255-574 mW m-2). Electrochemical characterization of the cathodes ex situ to the MFC showed that the cathodes with the lowest charge transfer resistance and the highest oxygen reduction activity produced the most power in MFC tests. The results on hydrophilic cathode separator layers revealed a trade off between power and CE. Cathodes coated with a thin coating of anion exchange polymer show promise for controlling oxygen transfer while minimally affecting power production. © 2010 Elsevier B.V. All rights reserved.
20. The boundary layer moist static energy budget: Convection picks up moisture and leaves footprints in the marine boundary layer
de Szoeke, S. P.
2017-12-01
Averaged over the tropical marine boundary layer (BL), 130 W m-2 turbulent surface moist static energy (MSE) flux, 120 W m-2 of which is evaporation, is balanced by upward MSE flux at the BL top due to 1) incorporation of cold air by downdrafts from deep convective clouds, and 2) turbulent entrainment of dry air into the BL. Cold saturated downdraft air, and warm clear air entrained into the BL have distinct thermodynamic properties. This work observationally quantifies their respective MSE fluxes in the central Indian Ocean in 2011, under different convective conditions of the intraseasonal (40-90 day) Madden Julian oscillation (MJO). Under convectively suppressed conditions, entrainment and downdraft fluxes export equal shares (60 W m-2) of MSE from the BL. Downdraft fluxes are more variable, increasing for stronger convection. In the convectively active phase of the MJO, downdrafts export 90 W m-2 from the BL, compared to 40 W m-2 by entrainment. These processes that control the internal, latent (condensation), and MSE of the tropical marine atmospheric BL determine the parcel buoyancy and strength of tropical deep convection.
1. Flat Plate Boundary Layer Stimulation Using Trip Wires and Hama Strips
Peguero, Charles; Henoch, Charles; Hrubes, James; Fredette, Albert; Roberts, Raymond; Huyer, Stephen
2017-11-01
Water tunnel experiments on a flat plate at zero angle of attack were performed to investigate the effect of single roughness elements, i.e., trip wires and Hama strips, on the transition to turbulence. Boundary layer trips are traditionally used in scale model testing to force a boundary layer to transition from laminar to turbulent flow at a single location to aid in scaling of flow characteristics. Several investigations of trip wire effects exist in the literature, but there is a dearth of information regarding the influence of Hama strips on the flat plate boundary layer. The intent of this investigation is to better understand the effects of boundary layer trips, particularly Hama strips, and to investigate the pressure-induced drag of both styles of boundary layer trips. Untripped and tripped boundary layers along a flat plate at a range of flow speeds were characterized with multiple diagnostic measurements in the NUWC/Newport 12-inch water tunnel. A wide range of Hama strip and wire trip thicknesses were used. Measurements included dye flow visualization, direct skin friction and parasitic drag force, boundary layer profiles using LDV, wall shear stress fluctuations using hot film anemometry, and streamwise pressure gradients. Test results will be compared to the CFD and boundary layer model results as well as the existing body of work. Conclusions, resulting in guidance for application of Hama strips in model scale experiments and non-dimensional predictions of pressure drag will be presented.
2. Shooting method for solution of boundary-layer flows with massive blowing
Liu, T.-M.; Nachtsheim, P. R.
1973-01-01
A modified, bidirectional shooting method is presented for solving boundary-layer equations under conditions of massive blowing. Unlike the conventional shooting method, which is unstable when the blowing rate increases, the proposed method avoids the unstable direction and is capable of solving complex boundary-layer problems involving mass and energy balance on the surface.
3. UAV-borne coherent doppler lidar for marine atmospheric boundary layer observations
Wu, Songhua; Wang, Qichao; Liu, Bingyi; Liu, Jintao; Zhang, Kailin; Song, Xiaoquan
2018-04-01
A compact UAV-borne Coherent Doppler Lidar (UCDL) has been developed at the Ocean University of China for the observation of wind profile and boundary layer structure in Marine Atmospheric Boundary Layer (MABL). The design, specifications and motion-correction methodology of the UCDL are presented. Preliminary results of the first flight campaign in Hailing Island in December 2016 is discussed.
4. Direct numerical simulation of stable and unstable turbulent thermal boundary layers
Hattori, Hirofumi; Houra, Tomoya; Nagano, Yasutaka
2007-01-01
This paper presents direct numerical simulations (DNS) of stable and unstable turbulent thermal boundary layers. Since a buoyancy-affected boundary layer is often encountered in an urban environmental space where stable and unstable stratifications exist, exploring a buoyancy-affected boundary layer is very important to know the transport phenomena of the flow in an urban space. Although actual observation may qualitatively provide the characteristics of these flows, the relevant quantitative turbulent quantities are very difficult to measure. Thus, in order to quantitatively investigate a buoyancy-affected boundary layer in detail, we have here carried out for the first time time- and space-developing DNS of slightly stable and unstable turbulent thermal boundary layers. The DNS results show the quantitative turbulent statistics and structures of stable and unstable thermal boundary layers, in which the characteristic transport phenomena of thermally stratified boundary layers are demonstrated by indicating the budgets of turbulent shear stress and turbulent heat flux. Even though the input of buoyant force is not large, the influence of buoyancy is clearly revealed in both stable and unstable turbulent boundary layers. In particular, it is found that both stable and unstable thermal stratifications caused by the weak buoyant force remarkably alter the structure of near-wall turbulence
5. On the Nature, Theory, and Modeling of Atmospheric Planetary Boundary Layers
Baklanov, Alexander A.; Grisogono, Branko; Bornstein, Robert
2011-01-01
The gap between our modern understanding of planetary boundary layer physics and its decades-old representations in current operational atmospheric models is widening, which has stimulated this review of the current state of the art and an analysis of the immediate needs in boundary layer theory......, measurements, and modeling....
6. The boundary layers as the primary transport regions of the earth's magnetotail
Eastman, T. E.; Frank, L. A.; Huang, C. Y.
1985-01-01
A comprehensive survey of ISEE and IMP LEPEDEA plasma measurements in the earth's magnetotail reveals that the magnetospheric boundary layer and the plasma sheet boundary layer are the primary transport regions there. These plasma measurements also reveal various components of the plasma sheet, including the central plasma sheet and plasma sheet boundary layer. A significant new result reported here is that of cold- and hot-plasma components that are spatially co-present within the central plasma sheet. Such plasma components cannot be explained merely by temporal variations in spectra involving the entire plasma sheet. Contributions to a low temperature component of the plasma sheet enter directly from the boundary layer located along the magnetotail flanks. Field-aligned flows predominate within the plasma sheet boundary layer which is almost always present and is located near the high- and low-latitude border of the plasma sheet. The plasma sheet boundary layer comprises highly anisotropic ion distributions, including counter-streaming ion beams, that evolve into the hot, isotropic component of the plasma sheet. Tailward acceleration regions generate these ion beams with plasma input from the magnetospheric boundary layer. Antisunward-flowing ion beams, at E/q less than 1 kV and of ionospheric composition, are frequently observed in the plasma sheet boundary layer and in tail lobes. These ion beams are likely accelerated at low altitude over the polar cap and especially along auroral field lines.
7. Stochastic Theory of Turbulence Mixing by Finite Eddies in the Turbulent Boundary Layer
Dekker, H.; Leeuw, G. de; Maassen van den Brink, A.
1995-01-01
Turbulence mixing is treated by means of a novel formulation of nonlocal K-theory, involving sample paths and a stochastic hypothesis. The theory simplifies for mixing by exchange (strong-eddies) and is then applied to the boundary layer (involving scaling). This maps boundary layer turbulence onto
8. A class of backward free-convective boundary-layer similarity solutions
Kuiken, H.K.
1983-01-01
This paper presents a class of backward free-convective boundary-layer similarity solutions. It is shown that these boundary layers can be produced along slender downward-projecting slabs of prescribed thickness variation, which are infinitely long. It is pointed out that these solutions can be used
9. Quantized vortex pair production in 4He films as a boundary-layer problem
McCauley, J.L. Jr.
1979-01-01
The author shows that the idea of a boundary layer for discrete vortices arises naturally from the equation of motion for the probability distribution of an interacting vortex pair. In contrast with classical hydrodynamics, this boundary layer is of statistical origin, and the method leads to a scaling law for the exact dissociation rate of a bound vortex pair. (Auth.)
10. Particle concentration and flux dynamics in the atmospheric boundary layer as the indicator of formation mechanism
Lauros, J.; Sogachev, Andrey; Smolander, S.
2011-01-01
the atmospheric boundary layer during nucleation event days shows a highly dynamical picture, where particle formation is coupled with chemistry and turbulent transport. We have demonstrated the suitability of our turbulent mixing scheme in reproducing the most important characteristics of particle dynamics...... within the boundary layer. Deposition and particle flux simulations show that deposition affects noticeably only the smallest particles...
11. Effects of micro-ramps on a shock wave/turbulent boundary layer interaction
Blinde, P.L.; Humble, R.A.; Van Oudheusden, B.W.; Scarano, F.
2009-01-01
Stereoscopic particle image velocimetry is used to investigate the effects of micro-ramp sub-boundary layer vortex generators, on an incident shock wave/boundary layer interaction at Mach 1.84. Single- and double-row arrangements of micro-ramps are considered. The micro-ramps have a height of 20% of
12. Representing the atmospheric boundary layer in climate models of intermediate compexity
Ronda, R.J.; Haarsma, R.J.; Holtslag, A.A.M.
2003-01-01
In this study the role of atmospheric boundary layer schemes in climate models is investigated. Including a boundary layer scheme in an Earth system model of intermediate complexity (EMIC) produces only minor differences in the estimated global distribution of sensible and latent heat fluxes over
13. Efficient modelling of aerodynamic flows in the boundary layer for high performance computing
Smith, L
2011-01-01
Full Text Available A unique technique to couple boundary-layer solutions with an inviscid solver is introduced. The boundary-layer solution is obtained using the two-integral method to solve displacement thickness with Newton’s method, at a fraction of the cost of a...
14. Collisional boundary layer analysis for neoclassical toroidal plasma viscosity in tokamaks
Shaing, K. C.; Cahyna, P.; Becoulet, M.; Park, J.-K.; Sabbagh, S. A.; Chu, M. S.
2008-01-01
It is demonstrated that the pitch angle integrals in the transport fluxes in the ν regime calculated in K. C. Shang [Phys. Plasmas 10, 1443 (2003)] are divergent as the trapped-circulating boundary is approached. Here, ν is the collision frequency. The origin of this divergence results from the logarithmic dependence in the bounce averaged radial drift velocity. A collisional boundary layer analysis is developed to remove the singularity. The resultant pitch angle integrals now include not only the original physics of the ν regime but also the boundary layer physics. The transport fluxes, caused by the particles inside the boundary layer, scale as √(ν)
15. Boundary-layer interactions in the plane-parallel incompressible flows
Nguyen, Toan T; Sueur, Franck
2012-01-01
We study the inviscid limit problem of incompressible flows in the presence of both impermeable regular boundaries and a hypersurface transversal to the boundary across which the inviscid flow has a discontinuity jump. In the former case, boundary layers have been introduced by Prandtl as correctors near the boundary between the inviscid and viscous flows. In the latter case, the viscosity smoothes out the discontinuity jump by creating a transition layer which has the same amplitude and thickness as the Prandtl layer. In the neighbourhood of the intersection of the impermeable boundary and of the hypersurface, interactions between the boundary and the transition layers must then be considered. In this paper, we initiate a mathematical study of this interaction and carry out a strong convergence in the inviscid limit for the case of the plane-parallel flows introduced by Di Perna and Majda (1987 Commun. Math. Phys. 108 667–89). (paper)
16. Planetary boundary layer and circulation dynamics at Gale Crater, Mars
Fonseca, Ricardo M.; Zorzano-Mier, María-Paz; Martín-Torres, Javier
2018-03-01
The Mars implementation of the Planet Weather Research and Forecasting (PlanetWRF) model, MarsWRF, is used here to simulate the atmospheric conditions at Gale Crater for different seasons during a period coincident with the Curiosity rover operations. The model is first evaluated with the existing single-point observations from the Rover Environmental Monitoring Station (REMS), and is then used to provide a larger scale interpretation of these unique measurements as well as to give complementary information where there are gaps in the measurements. The variability of the planetary boundary layer depth may be a driver of the changes in the local dust and trace gas content within the crater. Our results show that the average time when the PBL height is deeper than the crater rim increases and decreases with the same rate and pattern as Curiosity's observations of the line-of-sight of dust within the crater and that the season when maximal (minimal) mixing is produced is Ls 225°-315° (Ls 90°-110°). Thus the diurnal and seasonal variability of the PBL depth seems to be the driver of the changes in the local dust content within the crater. A comparison with the available methane measurements suggests that changes in the PBL depth may also be one of the factors that accounts for the observed variability, with the model results pointing towards a local source to the north of the MSL site. The interaction between regional and local flows at Gale Crater is also investigated assuming that the meridional wind, the dynamically important component of the horizontal wind at Gale, anomalies with respect to the daily mean can be approximated by a sinusoidal function as they typically oscillate between positive (south to north) and negative (north to south) values that correspond to upslope/downslope or downslope/upslope regimes along the crater rim and Mount Sharp slopes and the dichotomy boundary. The smallest magnitudes are found in the northern crater floor in a region that
17. Boundary layer models for calving marine outlet glaciers
C. Schoof
2017-10-01
Full Text Available We consider the flow of marine-terminating outlet glaciers that are laterally confined in a channel of prescribed width. In that case, the drag exerted by the channel side walls on a floating ice shelf can reduce extensional stress at the grounding line. If ice flux through the grounding line increases with both ice thickness and extensional stress, then a longer shelf can reduce ice flux by decreasing extensional stress. Consequently, calving has an effect on flux through the grounding line by regulating the length of the shelf. In the absence of a shelf, it plays a similar role by controlling the above-flotation height of the calving cliff. Using two calving laws, one due to Nick et al. (2010 based on a model for crevasse propagation due to hydrofracture and the other simply asserting that calving occurs where the glacier ice becomes afloat, we pose and analyse a flowline model for a marine-terminating glacier by two methods: direct numerical solution and matched asymptotic expansions. The latter leads to a boundary layer formulation that predicts flux through the grounding line as a function of depth to bedrock, channel width, basal drag coefficient, and a calving parameter. By contrast with unbuttressed marine ice sheets, we find that flux can decrease with increasing depth to bedrock at the grounding line, reversing the usual stability criterion for steady grounding line location. Stable steady states can then have grounding lines located on retrograde slopes. We show how this anomalous behaviour relates to the strength of lateral versus basal drag on the grounded portion of the glacier and to the specifics of the calving law used.
18. Global Distribution of Planetary Boundary Layer Height Derived from CALIPSO
Huang, J.
2015-12-01
The global distribution of planetary boundary layer (PBL) height, which was estimated from the attenuated back-scatter observations of Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observation (CALIPSO), is presented. In general, the PBL is capped by a temperature inversion that tends to trap moisture and aerosols. The gradient of back-scatter observed by lidar is almost always associated with this temperature inversion and the simultaneous decrease of moisture content. Thus, the PBL top is defined as the location of the maximum aerosol scattering gradient, which is analogous to the more conventional thermodynamic definition. The maximum standard deviation method, developed by Jordan et al. (2010), is modified and used to derive the global PBL heights. The derived PBL heights are not only consistent with the results of McGrath-Spangler and Denning (2012) but also agree well with the ground-based lidar measurements. It is found that the correlation between CALIPSO and the ground-based lidar was 0.73. The seasonal mean patterns from 4-year mid-day PBL heights over global are demonstrated. Also it is found that the largest PBL heights occur over the Tibetan Plateau and the coastal areas. The smallest PBL heights appear in the Tarim Basin and the northeast of China during the local winter. The comparison of PBL heights from CALIPSO and ECMWF under different land-cover conditions showed that, over ocean and forest surface, the PBL height estimated from the CALIPSO back-scatter climatology is larger than the ones estimated from ECMWF data. However, the PBL heights from ECMWF, over grass land and bare land surface in spring and summer are larger than the ones from CALIPSO.
19. Vortex dynamics of in-line twin synthetic jets in a laminar boundary layer
Wen, Xin; Tang, Hui; Duan, Fei
2015-08-01
An experimental investigation is conducted on the vortices induced by twin synthetic jets (SJs) in line with a laminar boundary layer flow over a flat plate. The twin SJs operating at four different phase differences, i.e., Δϕ = 0°, 90°, 180°, and 270°, are visualized using a stereoscopic color dye visualization system and measured using a two-dimensional particle image velocimetry (PIV) system. It is found that depending on the phase difference of twin SJs, three types of vortex structures are produced. At Δϕ = 90°, the two hairpin vortices interact in a very constructive way in terms of the vortex size, strength, and celerity, forming one combined vortex. At Δϕ = 270°, the two individual hairpin vortices do not have much interaction, forming two completely separated hairpin vortices that behave like doubling the frequency of the single SJ case. At Δϕ = 0° and 180°, the two hairpin vortices produced by the twin SJ actuators are close enough, with the head of one hairpin vortex coupled with the legs of the other, forming partially interacting vortex structures. Quantitative analysis of the twin SJs is conducted, including the time histories of vortex circulation in the mid-span plane as well as a selected spanwise-wall-normal plane, and the influence of the twin SJs on the boundary layer flow filed. In addition, dynamic mode decomposition analysis of the PIV data is conducted to extract representative coherent structures. Through this study, a better understanding in the vortex dynamics associated with the interaction of in-line twin SJs in laminar boundary layers is achieved, which provides useful information for future SJ-array applications.
20. Particle image velocimetry measurements of Mach 3 turbulent boundary layers at low Reynolds numbers
Brooks, J. M.; Gupta, A. K.; Smith, M. S.; Marineau, E. C.
2018-05-01
Particle image velocimetry (PIV) measurements of Mach 3 turbulent boundary layers (TBL) have been performed under low Reynolds number conditions, Re_τ =200{-}1000, typical of direct numerical simulations (DNS). Three reservoir pressures and three measurement locations create an overlap in parameter space at one research facility. This allows us to assess the effects of Reynolds number, particle response and boundary layer thickness separate from facility specific experimental apparatus or methods. The Morkovin-scaled streamwise fluctuating velocity profiles agree well with published experimental and numerical data and show a small standard deviation among the nine test conditions. The wall-normal fluctuating velocity profiles show larger variations which appears to be due to particle lag. Prior to the current study, no detailed experimental study characterizing the effect of Stokes number on attenuating wall-normal fluctuating velocities has been performed. A linear variation is found between the Stokes number ( St) and the relative error in wall-normal fluctuating velocity magnitude (compared to hot wire anemometry data from Klebanoff, Characteristics of Turbulence in a Boundary Layer with Zero Pressure Gradient. Tech. Rep. NACA-TR-1247, National Advisory Committee for Aeronautics, Springfield, Virginia, 1955). The relative error ranges from about 10% for St=0.26 to over 50% for St=1.06. Particle lag and spatial resolution are shown to act as low-pass filters on the fluctuating velocity power spectral densities which limit the measurable energy content. The wall-normal component appears more susceptible to these effects due to the flatter spectrum profile which indicates that there is additional energy at higher wave numbers not measured by PIV. The upstream inclination and spatial correlation extent of coherent turbulent structures agree well with published data including those using krypton tagging velocimetry (KTV) performed at the same facility.
1. White dwarf radii and boundary-layer constraints in three dwarf novae
Wood, J.H.
1990-01-01
The structure of the boundary layer between the accretion disc and white dwarf in three quiescent dwarf novae is explored with high signal-to-noise eclipse light curves obtained by phase folding 12-20 eclipses. Models of the eclipse shapes of various white dwarf/boundary layer configurations that might be at the centres of the accretion discs are calculated and compared with observations of the eclipses in Z Cha, OY Car and HT Cas. Possible models for the central objects are found to be a white dwarf with or without its lower hemisphere occulted by the disc, or a white dwarf with an optically thick boundary layer significantly extended in latitude up and down its sides. The most likely of these models for each system is an unocculted white dwarf with no boundary layer contributing significantly to the optical flux, or a white dwarf totally covered by an optically thick boundary layer. (author)
2. Current Challenges in Understanding and Forecasting Stable Boundary Layers over Land and Ice
Gert-Jan eSteeneveld
2014-10-01
Full Text Available Understanding and prediction of the stable atmospheric boundary layer is challenging. Many physical processes come into play in the stable boundary layer, i.e. turbulence, radiation, land surface coupling and heterogeneity, orographic turbulent and gravity wave drag. The development of robust stable boundary-layer parameterizations for weather and climate models is difficult because of the multiplicity of processes and their complex interactions. As a result, these models suffer from biases in key variables, such as the 2-m temperature, boundary-layer depth and wind speed. This short paper briefly summarizes the state-of-the-art of stable boundary layer research, and highlights physical processes that received only limited attention so far, in particular orographically-induced gravity wave drag, longwave radiation divergence, and the land-atmosphere coupling over a snow-covered surface. Finally, a conceptual framework with relevant processes and particularly their interactions is proposed.
3. Robust controller with adaptation within the boundary layer: application to nuclear underwater inspection robot
Park, Gee Yong; Yoon, Ji Sup; Hong, Dong Hee; Jeong, Jae Hoo
2002-01-01
In this paper, the robust control scheme with the improved control performance within the boundary layer is proposed. In the control scheme, the robust controller based on the traditional variable structure control method is modified to have the adaptation within the boundary layer. From this controller, the width of the boundary layer where the robust control input is smoothened out can be given by an appropriate value. But the improve control performance within the boundary layer can be achieved without the so-called control chattering because the role of adaptive control is to compensate for the uncovered portions of the robust control occurred from the continuous approximation within the boundary layer. Simulation tests for circular navigation of an underwater wall-ranging robot developed for inspection of wall surfaces in the research reactor, TRIGA MARK III, confirm the performance improvement
4. Prandtl boundary layer expansions of steady Navier-Stokes flows over a moving plate
Guo, Yan; Nguyen, Toan T.
2014-01-01
This paper concerns the validity of the Prandtl boundary layer theory in the inviscid limit for steady incompressible Navier-Stokes flows. The stationary flows, with small viscosity, are considered on $[0,L]\\times \\mathbb{R}_{+}$, assuming a no-slip boundary condition over a moving plate at $y=0$. We establish the validity of the Prandtl boundary layer expansion and its error estimates.
5. Electrons in the boundary layers near the dayside magnetopause
Hall, D.S.; Chaloner, C.P.; Bryant, D.A.; Lepine, D.R.; Tritakis, V.P.
1990-10-01
Entry of heated solar-wind plasma into the magnetosphere is examined using electron distributions. In particular, the angular structure of the electron distributions is studied within the transition region separating the magnetosheath from the inner magnetosphere. The measurements suggest that electrons in the outer part of the transition region originate in the magnetosheath, whilst the population closer to the Earth consists of electrons from the magnetosphere combined with an energized magnetosheath component. This energized component contains ''counterstreaming'' electrons, which are confined to directions closely parallel and anti-parallel to the magnetic field direction. The possibilities, that the energization of the counterstreaming electrons is cumulatively gained from either waves, electric fields perpendicular to the magnetic field, or quasi-Fermi acceleration, are discussed. It is not possible to identify the topology of the magnetic fields of the outer part of the region, but there is strong evidence that the inner part is on closed magnetic field lines, which map to the dayside auroral zone. The outer part of the transition region is a plasma depletion/magnetic-field compression layer. The structure of the transition region is similar to that surrounding flux transfer events, which leads to the deduction that the plasma and field signatures of flux transfer events may be the result of displacement of the transition region earthwards. Cases where the displacement is such that the field maximum of the depletion/compression region is encountered, may well explain ''crater'' flux transfer event signatures. (author)
6. Evolution of a Western Arctic Ice Ocean Boundary Layer and Mixed Layer Across a Developing Thermodynamically Forced Marginal Ice Zone
2016-09-01
heat and momentum transfer with the ice-ocean interface. These two observations demonstrate the intricate interplay between momentum, heat , and...summer evolution events: 1. Modulated shortwave radiative input to the ocean 2. Shoaled the ocean boundary layer increasing ocean heat storage 3... transfer in a stratified oceanic boundary layer. J. Geophys. Res., 92(C7), 6977–7986, doi:10.1029/JC092iC07p06977. McPhee, M. G., 1992: Turbulent heat
7. Extremely high wall-shear stress events in a turbulent boundary layer
Pan, Chong; Kwon, Yongseok
2018-04-01
The present work studies the fluctuating characteristics of the streamwise wall-shear stress in a DNS of a turbulent boundary layer at Re τ =1500 from a structural view. The two-dimensional field of the fluctuating friction velocity u‧ τ (x,z) is decomposed into the large- and small-scale components via a recently proposed scale separation algorithm, Quasi-bivariate Variational Mode Decomposition (QB-VMD). Both components are found to be dominated by streak-like structures, which can be regarded as the wall signature of the inner-layer streaks and the outer-layer LSMs, respectively. Extreme positive/negative wall-shear stress fluctuation events are detected in the large-scale component. The former’s occurrence frequency is nearly one order of magnitude higher than the latter; therefore, they contribute a significant portion of the long tail of the wall-shear stress distribution. Both two-point correlations and conditional averages show that these extreme positive wall-shear stress events are embedded in the large-scale positive u‧ τ streaks. They seem to be formed by near-wall ‘splatting’ process, which are related to strong finger-like sweeping (Q4) events originated from the outer-layer positive LSMs.
8. Coastal boundary layers in ocean modelling: an application to the Adriatic Sea
Malanotte Rizzoli, P.; Dell'Orto, F.
1981-01-01
Boundary layers play an important role in modelling geophysical fluid-dynamical flows, in as much as they constitute regions of ageostrophic dynamics in which the physical balances characterizing the main interior of the water mass break down. A short synopsis is given of important boundary layers in ocean circulation modelling with specific emphasis drawn upon side wall boundary layers, namely those adjacent to the coastlines of the considered basin. Application of boundary layer analysis is thereafter made for one specific phenomenological situation, namely the Northern Adriatic Sea and the problem posed by its wintertime seasonal circulation. The analysis furnishes a mathematical model fo the coastal strip adjacent to the Italian shoreline, treated as a boundary layer in the density field, starting from general model equations valid throughout the interior of the Northern Adriatic. The boundary layer model is consequently used to modify the side wall boundary condition for the interior density field. Related numerical experiments are shown and compared with previous standard experiments in which the boundary layer contribution to the density field has not been considered. (author)
9. Analytic Approximate Solutions to the Boundary Layer Flow Equation over a Stretching Wall with Partial Slip at the Boundary.
Ene, Remus-Daniel; Marinca, Vasile; Marinca, Bogdan
2016-01-01
Analytic approximate solutions using Optimal Homotopy Perturbation Method (OHPM) are given for steady boundary layer flow over a nonlinearly stretching wall in presence of partial slip at the boundary. The governing equations are reduced to nonlinear ordinary differential equation by means of similarity transformations. Some examples are considered and the effects of different parameters are shown. OHPM is a very efficient procedure, ensuring a very rapid convergence of the solutions after only two iterations.
10. Time-resolved PIV measurements of the atmospheric boundary layer over wind-driven surface waves
Markfort, Corey; Stegmeir, Matt
2017-11-01
Complex interactions at the air-water interface result in two-way coupling between wind-driven surface waves and the atmospheric boundary layer (ABL). Turbulence generated at the surface plays an important role in aquatic ecology and biogeochemistry, exchange of gases such as oxygen and carbon dioxide, and it is important for the transfer of energy and controlling evaporation. Energy transferred from the ABL promotes the generation and maintenance of waves. A fraction of the energy is transferred to the surface mixed layer through the generation of turbulence. Energy is also transferred back to the ABL by waves. There is a need to quantify the details of the coupled boundary layers of the air-water system to better understand how turbulence plays a role in the interactions. We employ time-resolved PIV to measure the detailed structure of the air and water boundary layers under varying wind and wave conditions in the newly developed IIHR Boundary-Layer Wind-Wave Tunnel. The facility combines a 30-m long recirculating water channel with an open-return boundary layer wind tunnel. A thick turbulent boundary layer is developed in the 1 m high air channel, over the water surface, allowing for the study of boundary layer turbulence interacting with a wind-driven wave field.
11. Magnetic Field Generation, Particle Energization and Radiation at Relativistic Shear Boundary Layers
Liang, Edison; Fu, Wen; Spisak, Jake; Boettcher, Markus
2015-11-01
Recent large scale Particle-in-Cell (PIC) simulations have demonstrated that in unmagnetized relativistic shear flows, strong transverse d.c. magnetic fields are generated and sustained by ion-dominated currents on the opposite sides of the shear interface. Instead of dissipating the shear flow free energy via turbulence formation and mixing as it is usually found in MHD simulations, the kinetic results show that the relativistic boundary layer stabilizes itself via the formation of a robust vacuum gap supported by a strong magnetic field, which effectively separates the opposing shear flows, as in a maglev train. Our new PIC simulations have extended the runs to many tens of light crossing times of the simulation box. Both the vacuum gap and supporting magnetic field remain intact. The electrons are energized to reach energy equipartition with the ions, with 10% of the total energy in electromagnetic fields. The dominant radiation mechanism is similar to that of a wiggler, due to oscillating electron orbits around the boundary layer.
12. Numerical study on effect of boundary layer trips on aerodynamic performance of E216 airfoil
B.K. Sreejith
2018-02-01
Full Text Available Simulation is carried out to find the performance of airfoil E216 using Transition γ-Reθ model at Reynolds number of 100,000. Flow behaviour and effect of angle of attack (AOA on laminar separation bubble (LSB formation are examined. The results are validated with wind tunnel experimental results. LSB formation is clearly spotted in the velocity vector plot and coefficient of pressure distribution over airfoil. LSB moved upstream towards the leading edge with increase in AOA. Effect of boundary layer trip on LSB formation over the airfoil and performance of airfoil are studied. Two different trip locations, 17% of chord and 10% of chord from leading edge, and different trip heights (0.3 mm, 0.5 mm, 0.7 mm, 1 mm are investigated in this study. Results showed that boundary layer trip could eliminate LSB partially or completely and improve aerodynamic performance of the airfoil. Maximum improvement in drag by 15.48% and lift to drag ratio by 21.62% are obtained at angle of attack of 60. In all the cases, improvement in performance is observed only up to trip height of 0.5 mm.
13. Instantaneous structure of a boundary layer subjected to free-stream turbulence
Hearst, R. Jason; de Silva, Charitha; Dogan, Eda; Ganapathisubramani, Bharathram
2017-11-01
A canonical turbulent boundary layer (TBL) has a distinct turbulent/non-turbulent interface (TNTI) separating the rotational wall-bounded fluid from the irrotational free-stream. If an intermittency profile is constructed separating the flow above and below the TNTI, this profile can be described by an error-function. Within the turbulent region, the flow is separated by interfaces that demarcate uniform momentum zones (UMZs). We observe that these characteristics of a TBL change if there is free-stream turbulence (FST). First, the entire flow is rotational, and thus a distinct TNTI does not exist. Nonetheless, it is possible to identify an interface that approximately separates the flow with mean zero vorticity from the distinctly wall-signed vorticity. This turbulent/turbulent interface is shown to be closer to the wall than the traditional TNTI, and the resulting intermittency profile is not an error-function. Also, UMZs appear to be masked by the free-stream perturbations. Despite these differences, a velocity field of a TBL with homogeneous, isotropic turbulence superimposed and weighted with the empirical intermittency profile, qualitatively reproduces the 1st and 2nd-order statistics. These findings suggest that a TBL subjected to FST may be described by a simple model. EPSRC, ERC, NSERC, Zonta International.
14. Fragmented coastal boundary layer induced by gap winds
Caldeira, Rui M. A.; Iglesias, Isabel; Sala, Iria; Vieira, Rui R.; Bastos, Luísa
2015-04-01
The oceanic impact of offshore-localized winds in the NW Iberian Peninsula was studied. Satellite and in situ observations showed the formation of plumes protruding offshore from the coast. To study the dynamics of such episodes tee Coupled-Ocean-Atmosphere-Wave- Sediment Transport Modeling System (COAWST) was used to reproduce the coastal conditions of the nortwestern Iberian Peninsula, allowing the concurrent representation of local winds, waves, currents, and rivers runoff. The use of coupled models is of outmost importance in order to accurately study the impact of the local winds on the coastal currents. The NW Iberian Peninsula has prominent capes, promontories and submarine canyons, which produce persistent hydrodynamic features. Thus far, the scientific literature shows that the western Iberian rivers produce a recurrent combined plume often denominated as the Western Iberian Buoyant Plume (WIBP) which increases the stratification of the water column and produces a vertical retention mechanism that keeps the biological material inshore. The WIBP extends northward along the coast (over the inner-shelf), and forms a front with the warmer and more saline surface (offshore) waters. However during episodes of strong offshore winds this coastal boundary layer is broken interrupting the WIBP. Coastal orography allows the formation of down-valley winds that produce coastal jets, promoting the offshore transport of pollutants, larvae and sediments. Acknowledgments: Acknowledgments: Numerical model solutions were calculated at CIIMARs HPC unit, acquired and maintained by FCT pluriannual funds (PesTC/Mar/LA0015/2013), and RAIA (0313-RAIA-1-E) and RAIA.co (0520-RAIACO-1-E) projects. The NICC (POCTI/CTA/49563/2002) project provided databases for this work. Rui Caldeira was supported by funds from the ECORISK project (NORTE-07-0124-FEDER-000054), co-financed by the North Portugal Regional Operational Programme (ON.2 - O Novo Norte), under the National Strategic Reference
15. Shipborne measurements of mercury in the marine boundary layer
Soerensen, A. L.; Skov, H.; Christensen, J.; Glasius, M.; Soerensen, B. T.; Steffen, A.; Jensen, B.; Christoffersen, C.; Madsen, H. W.; Johnson, M. S.
2008-12-01
Mercury accumulates in the human body, for example when consumed through fish and other aquatic animals. While it is poisonous to adults, only low doses are sufficient to cause severe effects in the development of foetuses where the source of mercury is through the mother's blood. From once being a problem restricted to certain populations, the negative effects of mercury consumption are becoming a global problem due to high anthropogenic emissions, long range transport in the atmosphere and bioaccumulation in the food chain after deposition. It is therefore important to understand the atmospheric photochemical pathways of mercury from source to sink. We have used a TEKRAN 2537A mercury vapor analyzer with a TEKRAN 1130 mercury speciation unit to measure gaseous elemental mercury (GEM) and reactive gaseous mercury (RGM) during an eight month circumnavigation of the Earth. This is the longest single track time series of mercury concentrations that we know of. This has offered the opportunity to give a global overview of the marine boundary layer (MBL) distribution of both GEM and RGM. Supplemented with earlier cruise measurements, we now have a broader knowledge of global GEM and RGM concentration in the MBL. The Galathea 3 cruise data offers new knowledge of the mechanisms causing the distribution patterns of GEM and RGM in the MBL. The first analysis of the Galathea 3 data indicates that measurements show a concentration difference between the northern and the southern hemispheres. In the northern hemisphere, the mean concentration in the free ocean is 2.06 ng/m3 and, including values down wind of Western Europe, an average value of 2.47 ng/m3 was found. Measurements in the southern hemisphere show a mean concentration of 1.24 ng/m3 and 1.57 ng/m3 where values close to the coast of West Africa are included. In contrast, the concentration levels of RGM are similar for the two hemispheres (northern hemisphere 3.40 pg/m3, southern hemisphere 3.95 pg/m3). Some
16. Inorganic bromine in the marine boundary layer: a critical review
R. Sander
2003-01-01
Full Text Available The cycling of inorganic bromine in the marine boundary layer (mbl has received increased attention in recent years. Bromide, a constituent of sea water, is injected into the atmosphere in association with sea-salt aerosol by breaking waves on the ocean surface. Measurements reveal that supermicrometer sea-salt aerosol is substantially depleted in bromine (often exceeding 50% relative to conservative tracers, whereas marine submicrometer aerosol is often enriched in bromine. Model calculations, laboratory studies, and field observations strongly suggest that the supermicrometer depletions reflect the chemical transformation of particulate bromide to reactive inorganic gases that influence the processing of ozone and other important constituents of marine air. Mechanisms for the submicrometer enrichments are not well understood. Currently available techniques cannot reliably quantify many Br containing compounds at ambient concentrations and, consequently, our understanding of inorganic Br cycling over the oceans and its global significance are uncertain. To provide a more coherent framework for future research, we have reviewed measurements in marine aerosol, the gas phase, and in rain. We also summarize sources and sinks, as well as model and laboratory studies of chemical transformations. The focus is on inorganic bromine over the open oceans outside the polar regions. The generation of sea-salt aerosol at the ocean surface is the major tropospheric source producing about 6.2 Tg/a of bromide. The transport of Br from continents (as mineral aerosol, and as products from biomass-burning and fossil-fuel combustion can be of local importance. Transport of degradation products of long-lived Br containing compounds from the stratosphere and other sources contribute lesser amounts. Available evidence suggests that, following aerosol acidification, sea-salt bromide reacts to form Br2 and BrCl that volatilize to the gas phase and photolyze in daylight
17. Inviscid/Boundary-Layer Aeroheating Approach for Integrated Vehicle Design
Lee, Esther; Wurster, Kathryn E.
2017-01-01
A typical entry vehicle design depends on the synthesis of many essential subsystems, including thermal protection system (TPS), structures, payload, avionics, and propulsion, among others. The ability to incorporate aerothermodynamic considerations and TPS design into the early design phase is crucial, as both are closely coupled to the vehicle's aerodynamics, shape and mass. In the preliminary design stage, reasonably accurate results with rapid turn-representative entry envelope was explored. Initial results suggest that for Mach numbers ranging from 9-20, a few inviscid solutions could reasonably sup- port surface heating predictions at Mach numbers variation of +/-2, altitudes variation of +/-10 to 20 kft, and angle-of-attack variation of +/- 5. Agreement with Navier-Stokes solutions was generally found to be within 10-15% for Mach number and altitude, and 20% for angle of attack. A smaller angle-of-attack increment than the 5 deg around times for parametric studies and quickly evolving configurations are necessary to steer design decisions. This investigation considers the use of an unstructured 3D inviscid code in conjunction with an integral boundary-layer method; the former providing the flow field solution and the latter the surface heating. Sensitivity studies for Mach number, angle of attack, and altitude, examine the feasibility of using this approach to populate a representative entry flight envelope based on a limited set of inviscid solutions. Each inviscid solution is used to generate surface heating over the nearby trajectory space. A subset of a considered in this study is recommended. Results of the angle-of-attack sensitivity studies show that smaller increments may be needed for better heating predictions. The approach is well suited for application to conceptual multidisciplinary design and analysis studies where transient aeroheating environments are critical for vehicle TPS and thermal design. Concurrent prediction of aeroheating
18. Implementation of a correction factor for the Pohlhausen laminar boundary layer applied on the CEVA curved wall jet model
Valeriu DRAGAN
2013-09-01
Full Text Available Curved wall jets have many technical applications, ranging from aeronautical circulation controlled wings to micro-fluidics and cryogenics. This paper addresses the issue of correctly estimating the boundary layer separation for laminar curved wall jets. For this, the Pohlhausen model was used in conjunction with the CEVA wall jet model with a semi-empirical modification which increases the accuracy for very thin jets. The method is therefore a mix of analytical equations with curve fitted experimental data in order to produce a simple yet effective way of estimating the boundary layer velocity profile along the curved wall. In order to cross-check the results, Newman’s empirical equation – which only provides a separation location but no information regarding the velocity profile - for boundary layer separation was used with good results. The hereby model could be used as a pre-design tool for rapid assessment of aeronautical high-lift applications such as Upper Surface Blown (USB or entrainment wings.
19. Algorithm based on regional separation for automatic grain boundary extraction using improved mean shift method
Zhenying, Xu; Jiandong, Zhu; Qi, Zhang; Yamba, Philip
2018-06-01
Metallographic microscopy shows that the vast majority of metal materials are composed of many small grains; the grain size of a metal is important for determining the tensile strength, toughness, plasticity, and other mechanical properties. In order to quantitatively evaluate grain size in metals, grain boundaries must be identified in metallographic images. Based on the phenomenon of grain boundary blurring or disconnection in metallographic images, this study develops an algorithm based on regional separation for automatically extracting grain boundaries by an improved mean shift method. Experimental observation shows that the grain boundaries obtained by the proposed algorithm are highly complete and accurate. This research has practical value because the proposed algorithm is suitable for grain boundary extraction from most metallographic images.
20. The Modelling of Particle Resuspension in a Turbulent Boundary Layer
Zhang, Fan
2011-01-01
lift and drag forces in turbulent boundary layers, the lift and drag we have con sidered and the impact of these data on predictions made by the non-Gaussian R'n'R model are compared with those based on O'Neill formula. The results indicate that, in terms of the long-term resuspension fraction, the difference is minor. It is concluded that as the particle size decreases the L and B method will lead to less-and-less long-term resuspension. Finally the ultimate model that has been developed in this work is a hybrid version of the R'n'R model adapted for application to multilayer deposits based on the Friess and Yadigaroglu multilayer approach. The deposit is modelled in several overlying layers where the coverage effect (masking) of the deposit layers has been studied; in the first instance a monodisperse deposit with a coverage ratio factor was modelled where this was subsequently replaced by the more general case of a polydisperse deposit with a particle size distribution. The results indicate that, in general, as the number of modelled layers increases the resuspension fraction of the whole deposit after a certain time decreases significantly. In other words, it takes a much longer time to re-suspend a thicker deposit. Taking account of the particle size distribution slightly increases the short-term resuspension. However, this change decreases the long-term resuspension significantly. The model results have been compared with data from the STORM SR11 test (ISP-40) and the BISE experiments. In general, both comparisons indicate that with smaller spread of the adhesive force distribution the new multilayer model agrees very well with the experimental data. It can be inferred that multilayer deposits lead to much narrower distributions of adhesive force
1. β-distribution for Reynolds stress and turbulent heat flux in relaxation turbulent boundary layer of compression ramp
Hu, YanChao; Bi, WeiTao; Li, ShiYao; She, ZhenSu
2017-12-01
A challenge in the study of turbulent boundary layers (TBLs) is to understand the non-equilibrium relaxation process after sep-aration and reattachment due to shock-wave/boundary-layer interaction. The classical boundary layer theory cannot deal with the strong adverse pressure gradient, and hence, the computational modeling of this process remains inaccurate. Here, we report the direct numerical simulation results of the relaxation TBL behind a compression ramp, which reveal the presence of intense large-scale eddies, with significantly enhanced Reynolds stress and turbulent heat flux. A crucial finding is that the wall-normal profiles of the excess Reynolds stress and turbulent heat flux obey a β-distribution, which is a product of two power laws with respect to the wall-normal distances from the wall and from the boundary layer edge. In addition, the streamwise decays of the excess Reynolds stress and turbulent heat flux also exhibit power laws with respect to the streamwise distance from the corner of the compression ramp. These results suggest that the relaxation TBL obeys the dilation symmetry, which is a specific form of self-organization in this complex non-equilibrium flow. The β-distribution yields important hints for the development of a turbulence model.
2. Simulation of Wind turbines in the atmospheric boundary layer
Chivaee, Hamid Sarlak; Sørensen, Jens Nørkær; Mikkelsen, Robert Flemming
as well as turbulent inflow condition. For generating turbulent inflow, a model is used in which a turbulent plane is introduced in the domain and convected in each time step, using Taylor's frozen hypothesis. The results of different simulations are analysed and compared in terms of mean values...... the computational costs scale rapidly with Reynolds number and domain size[1]. An approach to overcome these deficiencies is to use a wall modeling near the walls and then use a coarser grid at the first grid level above the ground. This could be performed by using simplified Navier-Stokes equations in the boundary...... condition is used in the bottom, a symmetry boundary on the top and periodic boundaries on the sides as well as inlet and outlet boundaries. For the temperature, a fixed value of 285 K is applied from the ground up to a height of 1 km and the temperature increases linearly with the rate of 3.5 degrees per...
3. Geologic Basin Boundaries (Basins_GHGRP) GIS Layer
U.S. Environmental Protection Agency — This is a coverage shapefile of geologic basin boundaries which are used by EPA's Greenhouse Gas Reporting Program. For onshore production, the "facility" includes...
4. Assessment of CFD Capability for Hypersonic Shock Wave Laminar Boundary Layer Interactions
Mehrnaz Rouhi Youssefi
2017-04-01
Full Text Available The goal of this study is to assess CFD capability for the prediction of shock wave laminar boundary layer interactions at hypersonic velocities. More specifically, the flow field over a double-cone configuration is simulated using both perfect gas and non-equilibrium Navier–Stokes models. Computations are compared with recent experimental data obtained from measurements conducted in the LENS XX (Large Energy National Shock Expansion Tunnel Version 2 at the Calspan University of Buffalo Research Center (CUBRC. Four separate cases of freestream conditions are simulated to examine the models for a range of stagnation enthalpies from 5.44 MJ/kg to 21.77 MJ/kg and Mach numbers from 10.9 to 12.82.
5. Unsteady Mixed Convection Boundary Layer from a Circular Cylinder in a Micropolar Fluid
Anati Ali
2010-01-01
Full Text Available Most industrial fluids such as polymers, liquid crystals, and colloids contain suspensions of rigid particles that undergo rotation. However, the classical Navier-Stokes theory normally associated with Newtonian fluids is inadequate to describe such fluids as it does not take into account the effects of these microstructures. In this paper, the unsteady mixed convection boundary layer flow of a micropolar fluid past an isothermal horizontal circular cylinder is numerically studied, where the unsteadiness is due to an impulsive motion of the free stream. Both the assisting (heated cylinder and opposing cases (cooled cylinder are considered. Thus, both small and large time solutions as well as the occurrence of flow separation, followed by the flow reversal are studied. The flow along the entire surface of a cylinder is solved numerically using the Keller-box scheme. The obtained results are compared with the ones from the open literature, and it is shown that the agreement is very good.
6. Mean motion and trajectories of heavy particles falling through a boundary layer
Stout, J.E.; Arya, S.P. [North Carolina State Univ., Raleigh, NC (United States)
1994-12-31
As particles fall through a turbulent boundary layer they experience a rather complex and unique time series of aerodynamic forces and, thus, each individual particle follows a rather complex and unique trajectory to the surface. For sufficiently large and heavy particles, the turbulence induced particle motion can be thought of as a small perturbation superimposed on the mean trajectory. By ignoring the turbulent contribution to particle motion it is possible to calculate the trajectory of a particle due to the mean flow alone. The mean trajectory provides an estimate of the ensemble-averaged path of a set of particles released from a given point in the atmosphere. The effect of turbulence on individual particle trajectories, the distribution of particle displacements from the mean trajectory, and their deposition patterns on the surface will be investigated in a separate study, using a random walk model.
7. The electron edge of the low latitude boundary layer during accelerated flow events
Gosling, J.T.; Thomsen, M.F.; Bame, S.J.; Onsager, T.G.; Russel, C.T.
1990-01-01
Magnetosheath plasma entering the Earth's magnetosphere to populate the low latitude boundary layer, LLBL, is often accelerated to speeds considerably greater than are observed in the adjacent magnetosheath. Measurements made during such accelerated flow events reveal separate electron and ion edges to the LLBL, with the electron edge being found earthward of the ion edge. Plasma electron velocity distributions observed at the earthward edge of the LLBL are often highly structured, exhibiting large asymmetries parallel and antiparallel, as well as perpendicular, to the local magnetic field. These features can consistently be interpreted as time-of-flight effects on recently reconnected field lines, and thus are strong evidence in support of the reconnection interpretation of accelerated plasma flow events
8. A numerical solution of a singular boundary value problem arising in boundary layer theory.
Hu, Jiancheng
2016-01-01
In this paper, a second-order nonlinear singular boundary value problem is presented, which is equivalent to the well-known Falkner-Skan equation. And the one-dimensional third-order boundary value problem on interval [Formula: see text] is equivalently transformed into a second-order boundary value problem on finite interval [Formula: see text]. The finite difference method is utilized to solve the singular boundary value problem, in which the amount of computational effort is significantly less than the other numerical methods. The numerical solutions obtained by the finite difference method are in agreement with those obtained by previous authors.
9. Boundary layer and fundamental problems of hydrodynamics (compatibility of a logarithmic velocity profile in a turbulent boundary layer with the experience values)
Zaryankin, A. E.
2017-11-01
The compatibility of the semiempirical turbulence theory of L. Prandtl with the actual flow pattern in a turbulent boundary layer is considered in this article, and the final calculation results of the boundary layer is analyzed based on the mentioned theory. It shows that accepted additional conditions and relationships, which integrate the differential equation of L. Prandtl, associating the turbulent stresses in the boundary layer with the transverse velocity gradient, are fulfilled only in the near-wall region where the mentioned equation loses meaning and are inconsistent with the physical meaning on the main part of integration. It is noted that an introduced concept about the presence of a laminar sublayer between the wall and the turbulent boundary layer is the way of making of a physical meaning to the logarithmic velocity profile, and can be defined as adjustment of the actual flow to the formula that is inconsistent with the actual boundary conditions. It shows that coincidence of the experimental data with the actual logarithmic profile is obtained as a result of the use of not particular physical value, as an argument, but function of this value.
10. Investigation of turbulent boundary layer over forward-facing step via direct numerical simulation
Hattori, Hirofumi; Nagano, Yasutaka
2010-01-01
This paper presents observations and investigations of the detailed turbulent structure of a boundary layer over a forward-facing step. The present DNSs are conducted under conditions with three Reynolds numbers based on step height, or three Reynolds numbers based on momentum thickness so as to investigate the effects of step height and inlet boundary layer thickness. DNS results show the quantitative turbulent statistics and structures of boundary layers over a forward-facing step, where pronounced counter-gradient diffusion phenomena (CDP) are especially observed on the step near the wall. Also, a quadrant analysis is conducted in which the results indicate in detail the turbulence motion around the step.
11. Stabilization of the hypersonic boundary layer by finite-amplitude streaks
Ren, Jie; Fu, Song; Hanifi, Ardeshir
2016-02-01
Stabilization of two-dimensional disturbances in hypersonic boundary layer flows by finite-amplitude streaks is investigated using nonlinear parabolized stability equations. The boundary-layer flows at Mach numbers 4.5 and 6.0 are studied in which both first and second modes are supported. The streaks considered here are driven either by the so-called optimal perturbations (Klebanoff-type) or the centrifugal instability (Görtler-type). When the streak amplitude is in an appropriate range, i.e., large enough to modulate the laminar boundary layer but low enough to not trigger secondary instability, both first and second modes can effectively be suppressed.
12. A Case Study of Offshore Advection of Boundary Layer Rolls over a Stably Stratified Sea Surface
Svensson, Nina; Sahlée, Erik; Bergström, Hans
2017-01-01
originate from boundary layer rolls generated over the convective air above Swedish mainland, also supported by visual satellite images showing the typical signature cloud streets. The simulations indicate that the rolls are advected and maintained at least 30–80 km off the coast, in agreement...... considerably for long times and over large areas in coastal regions. Although boundary layer rolls are a well-studied feature, no previous study has presented results concerning their persistence during situations with advection to a strongly stratified boundary layer. Such conditions are commonly encountered...
13. Interaction of discrete and continuous boundary layer modes to cause transition
Durbin, Paul A.; Zaki, Tamer A.; Liu Yang
2009-01-01
The interaction of discrete and continuous Orr-Sommerfeld modes in a boundary layer is studied by computer simulation. The discrete mode is an unstable Tollmien-Schlichting wave. The continuous modes generate jet-like disturbances inside the boundary layer. Either mode alone does not cause transition to turbulence; however, the interaction between them does. The continuous mode jets distort the discrete modes, producing Λ shaped vortices. Breakdown to turbulence is subsequent. The lateral spacing of the Λ's is sometimes the same as the wavelength of the continuous mode, sometimes it differs, depending on the ratio of wavelength to boundary layer thickness.
14. Receptivity of a high-speed boundary layer to temperature spottiness
Fedorov, A. V.; Ryzhov, A. A.; Soudakov, V. G.; Utyuzhnikov, S. V.
2013-01-01
Two-dimensional direct numerical simulation (DNS) of the receptivity of a flat-plate boundary layer to temperature spottiness in the Mach 6 free stream is carried out. The influence of spottiness parameters on the receptivity process is studied. It is shown that the temperature spots propagating near the upper boundary-layer edge generate mode F inside the boundary layer. Further downstream mode F is synchronized with unstable mode S (Mack second mode) and excites the latter via the inter-mod...
15. The effect of small streamwise velocity distortion on the boundary layer flow over a thin flat plate with application to boundary layer stability theory
Goldstein, M. E.; Leib, S. J.; Cowley, S. J.
1990-01-01
Researchers show how an initially linear spanwise disturbance in the free stream velocity field is amplified by leading edge bluntness effects and ultimately leads to a small amplitude but linear spanwise motion far downstream from the edge. This spanwise motion is imposed on the boundary layer flow and ultimately causes an order-one change in its profile shape. The modified profiles are highly unstable and can support Tollmein-Schlichting wave growth well upstream of the theoretical lower branch of the neutral stability curve for a Blasius boundary layer.
16. Layer-by-layer assembled biopolymer microcapsule with separate layer cavities generated by gas-liquid microfluidic approach.
Wang, Yifeng; Zhou, Jing; Guo, Xuecheng; Hu, Qian; Qin, Chaoran; Liu, Hui; Dong, Meng; Chen, Yanjun
2017-12-01
In this work, a layer-by-layer (LbL) assembled biopolymer microcapsule with separate layer cavities is generated by a novel and convenient gas-liquid microfluidic approach. This approach exhibits combined advantages of microfluidic approach and LbL assembly method, and it can straightforwardly build LbL-assembled capsules in mild aqueous environments at room temperature. In particular, using this approach we can build the polyelectrolyte multilayer capsule with favorable cavities in each layer, and without the need for organic solvent, emulsifying agent, or sacrificial template. Various components (e.g., drugs, proteins, fluorescent dyes, and nanoparticles) can be respectively encapsulated in the separate layer cavities of the LbL-assembled capsules. Moreover, the encapsulated capsules present the ability as colorimetric sensors, and they also exhibit the interesting release behavior. Therefore, the LbL-assembled biopolymer capsule is a promising candidate for biomedical applications in targeted delivery, controlled release, and bio-detection. Copyright © 2017 Elsevier B.V. All rights reserved.
17. The Modelling of Particle Resuspension in a Turbulent Boundary Layer
Zhang, Fan
2011-10-20
uncorrelated curve-fitted model. In view of recent numerical data for lift and drag forces in turbulent boundary layers, the lift and drag we have con sidered and the impact of these data on predictions made by the non-Gaussian R'n'R model are compared with those based on O'Neill formula. The results indicate that, in terms of the long-term resuspension fraction, the difference is minor. It is concluded that as the particle size decreases the L and B method will lead to less-and-less long-term resuspension. Finally the ultimate model that has been developed in this work is a hybrid version of the R'n'R model adapted for application to multilayer deposits based on the Friess and Yadigaroglu multilayer approach. The deposit is modelled in several overlying layers where the coverage effect (masking) of the deposit layers has been studied; in the first instance a monodisperse deposit with a coverage ratio factor was modelled where this was subsequently replaced by the more general case of a polydisperse deposit with a particle size distribution.
18. New formulations on the finite element method for boundary value problems with internal/external boundary layers
Pereira, Luis Carlos Martins
1998-06-01
New Petrov-Galerkin formulations on the finite element methods for convection-diffusion problems with boundary layers are presented. Such formulations are based on a consistent new theory on discontinuous finite element methods. Existence and uniqueness of solutions for these problems in the new finite element spaces are demonstrated. Some numerical experiments shows how the new formulation operate and also their efficacy. (author)
19. Assessment of boundary layer profiling formulas using tower, sodar and balloon data
Paine, R.J. [ENSR Consulting and Engineering, Inc., Acton, MA (United States); Kendall, S.B. [Phelps Dodge Corp., Phoenix, AZ (United States)
1994-12-31
The accuracy of an air quality dispersion model is largely dependent upon the availability of representative meteorological data for the simulation of plume rise, transport, and dispersion. In many cases where tall stacks and/or buoyant plumes are involved, the available meteorological measurements do not extend to plume height. Air quality models contend with these situations by either assuming no change of meteorological variables with elevation or by applying a profiling relationship based upon theoretical or empirical relationships. The latter treatment is employed in recently-developed models such as CTDMPLUS, and HPDM, and OML. In the well-mixed convective boundary layer, meteorological variables such as wind direction, wind speed, and turbulence do not vary substantially above the surface layer (about 0.1 z{sub i}, the mixed-layer height). Above the surface layer, behavior on an hourly average basis is fairly well parameterized by boundary-layer formulations. However, models are sensitive to the height of the convective boundary layer, z{sub i}, which affects the magnitude of the convective velocity scale, w., and is important for simulating plume trapping and plume penetration into the stable layer aloft. In the stable boundary layer, plumes are often released at heights above the stable boundary layer, the height of which is often hard to define. Models are sensitive to the manner in which wind direction, wind speed, temperature and turbulence are profiled with height in stable conditions.
20. Spectral Gap Energy Transfer in Atmospheric Boundary Layer
Bhushan, S.; Walters, K.; Barros, A. P.; Nogueira, M.
2012-12-01
Experimental measurements of atmospheric turbulence energy spectra show E(k) ~ k-3 slopes at synoptic scales (~ 600 km - 2000 km) and k-5/3 slopes at the mesoscales (theory, it is expected that a strong backward energy cascade would develop at the synoptic scale, and that circulation would grow infinitely. To limit this backward transfer, energy arrest at macroscales must be introduced. The most commonly used turbulence models developed to mimic the above energy transfer include the energy backscatter model for 2D turbulence in the horizontal plane via Large Eddy Simulation (LES) models, dissipative URANS models in the vertical plane, and Ekman friction for the energy arrest. One of the controversial issues surrounding the atmospheric turbulence spectra is the explanation of the generation of the 2D and 3D spectra and transition between them, for energy injection at the synoptic scales. Lilly (1989) proposed that the existence of 2D and 3D spectra can only be explained by the presence of an additional energy injection in the meso-scale region. A second issue is related to the observations of dual peak spectra with small variance in meso-scale, suggesting that the energy transfer occurs across a spectral gap (Van Der Hoven, 1957). Several studies have confirmed the spectral gap for the meso-scale circulations, and have suggested that they are enhanced by smaller scale vertical convection rather than by the synoptic scales. Further, the widely accepted energy arrest mechanism by boundary layer friction is closely related to the spectral gap transfer. This study proposes an energy transfer mechanism for atmospheric turbulence with synoptic scale injection, wherein the generation of 2D and 3D spectra is explained using spectral gap energy transfer. The existence of the spectral gap energy transfer is validated by performing LES for the interaction of large scale circulation with a wall, and studying the evolution of the energy spectra both near to and far from the wall
1. Computational methods for investigation of surface curvature effects on airfoil boundary layer behavior
Xiang Shen
2017-03-01
Full Text Available This article presents computational algorithms for the design, analysis, and optimization of airfoil aerodynamic performance. The prescribed surface curvature distribution blade design (CIRCLE method is applied to a symmetrical airfoil NACA0012 and a non-symmetrical airfoil E387 to remove their surface curvature and slope-of-curvature discontinuities. Computational fluid dynamics analysis is used to investigate the effects of curvature distribution on aerodynamic performance of the original and modified airfoils. An inviscid–viscid interaction scheme is introduced to predict the positions of laminar separation bubbles. The results are compared with experimental data obtained from tests on the original airfoil geometry. The computed aerodynamic advantages of the modified airfoils are analyzed in different operating conditions. The leading edge singularity of NACA0012 is removed and it is shown that the surface curvature discontinuity affects aerodynamic performance near the stalling angle of attack. The discontinuous slope-of-curvature distribution of E387 results in a larger laminar separation bubble at lower angles of attack and lower Reynolds numbers. It also affects the inherent performance of the airfoil at higher Reynolds numbers. It is shown that at relatively high angles of attack, a continuous slope-of-curvature distribution reduces the skin friction by suppressing both laminar and turbulent separation, and by delaying laminar-turbulent transition. It is concluded that the surface curvature distribution has significant effects on the boundary layer behavior and consequently an improved curvature distribution will lead to higher aerodynamic efficiency.
2. Velocity Spectra in the Unstable Planetary Boundary Layer
Højstrup, Jørgen
1982-01-01
Models for velocity spectra of all three components in the lower half of the unstable PBL are presented. The model spectra are written as a sum of two parts, nS(n) = A(fi, z/zi)w*2 + B(f, z/zi)u*02, a mixed layer part with a stability dependence, and a surface layer part without stability...
3. Experimental study of boundary-layer transition on an airfoil induced by periodically passing wake
Jeon, W.P. [Center for Turbulence and Flow Control Research Institute of Advanced Machinery and Design, Seoul National University (Korea); Park, T.C.; Kang, S.H. [School of Mechanical and Aerospace Engineering, Seoul National University (Korea)
2002-02-01
Hot-wire measurements are performed in boundary-layer flows developing on a NACA 0012 airfoil over which wakes pass periodically. The periodic wakes are generated by rotating circular cylinders clockwise or counterclockwise around the airfoil. The time- and phase-averaged mean streamwise velocities and turbulence fluctuations are measured to investigate the phenomena of wake-induced transition. Especially, the phase-averaged wall shear stresses are evaluated using a computational Preston tube method. The passing wakes significantly change the pressure distribution on the airfoil, which has influence on the transition process of the boundary layer. The orientation of the passing wake alters the pressure distribution in a different manner. Due to the passing wake, the turbulent patches are generated inside the laminar boundary layer on the airfoil, and the boundary layer becomes temporarily transitional. The patches propagate downstream at a speed smaller than the free-stream velocity and merge together further downstream. Relatively high values of phase-averaged turbulence fluctuations in the outer part of the boundary layer indicate the possibility that breakdown occurs in the outer layer away from the wall. It is confirmed that the phase-averaged mean velocity profile has two dips in the outer region of the transitional boundary layer for each passing cycle. (orig.)
4. New-particle formation events in a continental boundary layer: first results from the SATURN experiment
F. Stratmann
2003-01-01
Full Text Available During the SATURN experiment, which took place from 27 May to 14 June 2002, new particle formation in the continental boundary layer was investigated. Simultaneous ground-based and tethered-balloon-borne measurements were performed, including meteorological parameters, particle number concentrations and size distributions, gaseous precursor concentrations and SODAR and LIDAR observations. Newly formed particles were observed inside the residual layer, before the break-up process of the nocturnal inversion, and inside the mixing layer throughout the break-up of the nocturnal inversion and during the evolution of the planetary boundary layer.
5. Analysis of Boundary Layer Meteorological Data Collected at the White Sands Missile Range
O'Brien, Sean; Tofsted, David; Yarbrough, Jimmy; Elliott, D. S; Quintis, David
2007-01-01
... Sands Missile Range (WSMR). Our primary motivation for collecting these measurements is to refine the accuracy of outer and inner scale effects models for optical, thermal, and absolute humidity turbulence for the desert boundary layer...
6. Application of a Novel Laser-Doppler Velocimeter for Turbulence: Structural Measurements in Turbulent Boundary Layers
Lowe, Kevin T; Simpson, Roger L
2006-01-01
An advanced laser-Doppler velocimeter (LDV), deemed the 'comprehensive LDV', is designed to acquire fully-resolved turbulence structural measurements in high Reynolds number two- and three-dimensional turbulent boundary layers...
7. Direct Numerical Simulation and Experimental Validation of Hypersonic Boundary-Layer Receptivity and Instability
Zhong, Xiaolin
2007-01-01
.... During the three-year period, we have conducted extensive DNS studies on the receptivity of hypersonic boundary layer flows over a sharp wedge, a flat plate, a blunt cone, and the FRESH aeroshell...
8. Stability characteristics of compressible boundary layers over thermo-mechanically compliant walls
Dettenrieder, Fabian; Bodony, Daniel
2017-11-01
Transition prediction at hypersonic flight conditions continues to be a challenge and results in conservative safety factors that increase vehicle weight. The weight and thus cost reduction of the outer skin panels promises significant impact; however, fluid-structure interaction due to unsteady perturbations in the laminar boundary layer regime has not been systematically studied at conditions relevant for reusable, hypersonic flight. In this talk, we develop and apply convective and global stability analyses for compressible boundary layers over thermo-mechanically compliant panels. This compliance is shown to change the convective stability of the boundary layer modes, with both stabilization and destabilization observed. Finite panel lengths are shown to affect the global stability properties of the boundary layer.
9. Budget of Turbulent Kinetic Energy in a Shock Wave Boundary-Layer Interaction
Vyas, Manan A.; Waindim, Mbu; Gaitonde, Datta V.
2016-01-01
Implicit large-eddy simulation (ILES) of a shock wave/boundary-layer interaction (SBLI) was performed. Quantities present in the exact equation of the turbulent kinetic energy transport were accumulated and used to calculate terms like production, dissipation, molecular diffusion, and turbulent transport. The present results for a turbulent boundary layer were validated by comparison with direct numerical simulation data. It was found that a longer development domain was necessary for the boundary layer to reach an equilibrium state and a finer mesh resolution would improve the predictions. In spite of these findings, trends of the present budget match closely with that of the direct numerical simulation. Budgets for the SBLI region are presented at key axial stations. These budgets showed interesting dynamics as the incoming boundary layer transforms and the terms of the turbulent kinetic energy budget change behavior within the interaction region.
10. Profiles of Wind and Turbulence in the Coastal Atmospheric Boundary Layer of Lake Erie
Wang, H; Barthelmie, R J; Crippa, P; Doubrawa, P; Pryor, S C
2014-01-01
Prediction of wind resource in coastal zones is difficult due to the complexity of flow in the coastal atmospheric boundary layer (CABL). A three week campaign was conducted over Lake Erie in May 2013 to investigate wind characteristics and improve
11. RACORO Extended-Term Aircraft Observations of Boundary-Layer Clouds
Vogelmann, Andrew M.; McFarquhar, Greg M.; Ogren, John A.; Turner, David D.; Comstock, Jennifer M.; Feingold, Graham; Long, Charles N.; Jonsson, Haflidi H.; Bucholtz, Anthony; Collins, Don R.;
2012-01-01
Small boundary-layer clouds are ubiquitous over many parts of the globe and strongly influence the Earths radiative energy balance. However, our understanding of these clouds is insufficient to solve pressing scientific problems. For example, cloud feedback represents the largest uncertainty amongst all climate feedbacks in general circulation models (GCM). Several issues complicate understanding boundary-layer clouds and simulating them in GCMs. The high spatial variability of boundary-layer clouds poses an enormous computational challenge, since their horizontal dimensions and internal variability occur at spatial scales much finer than the computational grids used in GCMs. Aerosol-cloud interactions further complicate boundary-layer cloud measurement and simulation. Additionally, aerosols influence processes such as precipitation and cloud lifetime. An added complication is that at small scales (order meters to 10s of meters) distinguishing cloud from aerosol is increasingly difficult, due to the effects of aerosol humidification, cloud fragments and photon scattering between clouds.
12. Howard, R. M.; Miley, S. J.; Holmes, B. J.
1985-01-01
A research program is in progress to study the effects of the propeller slipstream on natural laminar flow. Flight and wind tunnel measurements of the wing boundary layer have been made using hot-film velocity sensor probes. The results show the boundary layer, at any given point, to alternate between laminar and turbulent states. This cyclic behavior is due to periodic external flow turbulence originating from the viscous wake of the propeller blades. Analytic studies show the cyclic laminar/turbulent boundary layer to result in a significantly lower wing section drag than a fully turbulent boundary layer. The application of natural laminar flow design philosophy yields drag reduction benefits in the slipstream affected regions of the airframe, as well as the unaffected regions.
13. Influence of relaxation processes on the structure of a thermal boundary layer in partially ionized argon
Dongen, M.E.H. van; Eck, R.B. van P. van; Hagebeuk, H.J.L.; Hirschberg, A.; Hutten-Mansfeld, A.C.B.; Jager, H.J.; Willems, J.F.H.
1981-01-01
A model for the unsteady thermal boundary-layer development at the end wall of a shock tube, in partially ionized atmospheric argon, is proposed. Consideration is given to ionization and thermal relaxation processes. In order to obtain some insight into the influence of the relaxation processes on the structure of the boundary layer, a study of the frozen and equilibrium limits has been carried out. The transition from a near-equilibrium situation in the outer part of the boundary layer towards a frozen situation near the wall is determined numerically. Experimental data on the electron and atom density profiles obtained from laser schlieren and absorption measurements are presented. A quantitative agreement between theory and experiment is found for a moderate degree of ionization (3%). At a higher degree of ionization the structure of the boundary layer is dominated by the influence of radiation cooling, which has been neglected in the model. (author)
14. Atmospheric Boundary Layer Modeling for Combined Meteorology and Air Quality Systems
Atmospheric Eulerian grid models for mesoscale and larger applications require sub-grid models for turbulent vertical exchange processes, particularly within the Planetary Boundary Layer (PSL). In combined meteorology and air quality modeling systems consistent PSL modeling of wi...
15. Marine boundary layer and turbulent fluxes over the Baltic Sea: Measurements and modelling
Gryning, Sven-Erik; Batchvarova, E.
2002-01-01
Two weeks of measurements of the boundary-layer height over a small island (Christianso) in the Baltic Sea are discussed. The meteorological conditions are characterised by positive heat flux over the sea. The boundary-layer height was simulated with two models, a simple applied high-resolution (2...... km x 2 km) model, and the operational numerical weather prediction model HIRLAM (grid resolution of 22.5 km x 22.5 km). For southwesterly winds it was found that a relatively large island (Bornholm) lying 20-km upwind of the measuring site influences the boundary-layer height. In this situation...... the high-resolution simple applied model reproduces the characteristics of the boundary-layer height over the measuring site. Richardson-number based methods using data from simulations with the HIRLAM model fail, most likely because the island and the water fetch to the measuring site are about the size...
16. Two-phase gas bubble-liquid boundary layer flow along vertical and inclined surfaces
Cheung, F.B.; Epstein, M.
1985-01-01
The behavior of a two-phase gas bubble-liquid boundary layer along vertical and inclined porous surfaces with uniform gas injection is investigated experimentally and analytically. Using argon gas and water as the working fluids, a photographical study of the two-phase boundary layer flow has been performed for various angles of inclination ranging from 45 0 to 135 0 and gas injection rates ranging from 0.01 to 0.1 m/s. An integral method has been employed to solve the system of equations governing the two-phase motion. The effects of the gas injection rate and the angle of inclination on the growth of the boundary layer have been determined. The predicted boundary layer thickness is found to be in good agreement with the experimental results. The calculated axial liquid velocity and the void fraction in the two-phase region are also presented along with the observed flow behavior
17. A fast Linear Complementarity Problem (LCP) solver for separating fluid-solid wall boundary Conditions
Andersen, Michael; Abel, Sarah Maria Niebe; Erleben, Kenny
2017-01-01
We address the task of computing solutions for a separating fluid-solid wall boundary condition model. We present an embarrassingly parallel, easy to implement, fluid LCP solver.We are able to use greater domain sizes than previous works have shown, due to our new solver. The solver exploits matr...
18. Modeling Turbulence Generation in the Atmospheric Surface and Boundary Layers
2015-10-01
hydrostatic equation: dP dz = −ρa g −→ ∫ ZI 0 ρa dz = − 1 g ∫ dP = + 1 g [P (0)− P (ZI)]. (6.14) The pressure at the surface is... surface pressure is estimated, we can compute a vertical pressure profile using the hydrostatic equation and a selected temperature profile based on dP... surface -layer atmosphere. By surface layer what is intended is a layer of foliage plus the surface itself. That is, a flat ground surface that
19. Uncertainty Assessments of 2D and Axisymmetric Hypersonic Shock Wave - Turbulent Boundary Layer Interaction Simulations at Compression Corners
Gnoffo, Peter A.; Berry, Scott A.; VanNorman, John W.
2011-01-01
This paper is one of a series of five papers in a special session organized by the NASA Fundamental Aeronautics Program that addresses uncertainty assessments for CFD simulations in hypersonic flow. Simulations of a shock emanating from a compression corner and interacting with a fully developed turbulent boundary layer are evaluated herein. Mission relevant conditions at Mach 7 and Mach 14 are defined for a pre-compression ramp of a scramjet powered vehicle. Three compression angles are defined, the smallest to avoid separation losses and the largest to force a separated flow engaging more complicated flow physics. The Baldwin-Lomax and the Cebeci-Smith algebraic models, the one-equation Spalart-Allmaras model with the Catrix-Aupoix compressibility modification and two-equation models including Menter SST, Wilcox k-omega 98, and Wilcox k-omega 06 turbulence models are evaluated. Each model is fully defined herein to preclude any ambiguity regarding model implementation. Comparisons are made to existing experimental data and Van Driest theory to provide preliminary assessment of model form uncertainty. A set of coarse grained uncertainty metrics are defined to capture essential differences among turbulence models. Except for the inability of algebraic models to converge for some separated flows there is no clearly superior model as judged by these metrics. A preliminary metric for the numerical component of uncertainty in shock-turbulent-boundary-layer interactions at compression corners sufficiently steep to cause separation is defined as 55%. This value is a median of differences with experimental data averaged for peak pressure and heating and for extent of separation captured in new, grid-converged solutions presented here. This value is consistent with existing results in a literature review of hypersonic shock-turbulent-boundary-layer interactions by Roy and Blottner and with more recent computations of MacLean.
20. Modeling mode interactions in boundary layer flows via the Parabolized Floquet Equations
Ran, Wei; Zare, Armin; Hack, M. J. Philipp; Jovanović, Mihailo R.
2017-01-01
In this paper, we develop a linear model to study interactions between different modes in slowly-growing boundary layer flows. Our method consists of two steps. First, we augment the Blasius boundary layer profile with a disturbance field resulting from the linear Parabolized Stability Equations (PSE) to obtain the modified base flow; and, second, we combine Floquet analysis with the linear PSE to capture the spatial evolution of flow fluctuations. This procedure yields the Parabolized Floque... | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8619582056999207, "perplexity": 1954.0472867746519}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439735881.90/warc/CC-MAIN-20200804161521-20200804191521-00141.warc.gz"} |
http://mathhelpforum.com/math-topics/123907-area.html | # Math Help - area
1. ## area
hi, my question is to find the area of this rectangle, im not sure how to work it out so could anyone help?
i think the answer would be 2?
2. Originally Posted by andyboy179
hi, my question is to find the area of this rectangle, im not sure how to work it out so could anyone help?
i think the answer would be 2?
$(a+b)(c-d) = ac - ad + bc - bd$
$(1+\sqrt{3})(2-\sqrt{3}) = 2 - \sqrt{3} + 2\sqrt{3} - 3 = \sqrt{3} - 1$
3. could you explain how you got that please?
4. Originally Posted by andyboy179
could you explain how you got that please?
I thought I did ...
5. well i don't understand how you got that, how did (1+root3)(-2root3) end up as 2- root3 + 2 root3-3
6. Originally Posted by andyboy179
well i don't understand how you got that, how did (1+root3)(-2root3) end up as 2- root3 + 2 root3-3
read a lesson and find out how ...
Adding (and Subtracting) Square Roots
7. don't worry i will try and find out at school | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 2, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9355171322822571, "perplexity": 649.9609909711457}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-14/segments/1427131298660.78/warc/CC-MAIN-20150323172138-00132-ip-10-168-14-71.ec2.internal.warc.gz"} |
http://mathhelpforum.com/calculus/139198-general-multiple-integration-question.html | # Math Help - General multiple integration question
1. ## General multiple integration question
Ok there's something simple I don't get here. When one double integrates a circle in polar coordinates or even cylindrical coordinates, sometimes one subs in x=rcostheta and y=rsintheta but sometimes x=3costheta and y=3sintheta even when we know both times that the radius of he circle is 3. Can anyone tell me how one knows when to plug in the radius and when just to leave it as r? I just got a question wrong because I put r instead of 3...
2. Originally Posted by Zirtco
Ok there's something simple I don't get here. When one double integrates a circle in polar coordinates or even cylindrical coordinates, sometimes one subs in x=rcostheta and y=rsintheta but sometimes x=3costheta and y=3sintheta even when we know both times that the radius of he circle is 3. Can anyone tell me how one knows when to plug in the radius and when just to leave it as r? I just got a question wrong because I put r instead of 3...
If you leave the radius as r then then this solves for the general case. You can then substitute 3 for r at the end and find out the result. One advantage of this is that if then you had to solve the same problem with radius 2 you could simply substitute 2 for r and the work would be less.
If that doesn't solve it then perhaps you should tell us what the question was that you got wrong?
3. ## Re:
No idea why the cross product was required. There's another way to do it isn't there? There's the solution to part a as well.
Attached Thumbnails | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8894146084785461, "perplexity": 535.7945094540365}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246659254.83/warc/CC-MAIN-20150417045739-00003-ip-10-235-10-82.ec2.internal.warc.gz"} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.