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aand the z=0 Cosmo25 halos does not arise froma different choice of cosmology/simulation parameters. | and the z=0 Cosmo25 halos does not arise from a different choice of cosmology/simulation parameters. |
For practical purposes, we compute a redshift-dependent fit to the specific angular momentum gas to dark matter ratio for halos with My;X10?Mo We emphasize that all the extra specific angular momentum brought in by the gas will not be devoted to spinning up the central galactic disc, given that its typical misalignment with the central gaseous disc is measured to be =40? for intermediate mass halos (Myir~10!! Mo) in the ssimulation. | For practical purposes, we compute a redshift-dependent fit to the specific angular momentum gas to dark matter ratio for halos with $\mvir \leq 10^{13}\msun$ We emphasize that all the extra specific angular momentum brought in by the gas will not be devoted to spinning up the central galactic disc, given that its typical misalignment with the central gaseous disc is measured to be $\approx 40^{\circ}$ for intermediate mass halos $\mvir\sim10^{11}\msun$ ) in the simulation. |
Moreover, in the vicinity of the central disc, hydrodynamical interactions with the circum-galactic gas will become important and may substantially redistribute angular momentum. | Moreover, in the vicinity of the central disc, hydrodynamical interactions with the circum-galactic gas will become important and may substantially redistribute angular momentum. |
A careful numerical investigation of angular momentum advection in the central region is therefore needed to determine whether or not a simple semi-analytic approach is capable of correctly describing the evolution of angular momentum of disc galaxies using the new initial conditions we provide with this fit. | A careful numerical investigation of angular momentum advection in the central region is therefore needed to determine whether or not a simple semi-analytic approach is capable of correctly describing the evolution of angular momentum of disc galaxies using the new initial conditions we provide with this fit. |
We also note that limited spatial resolution of our large volume cosmological simulations will lead to an artificial increase of the gas angular momentum, especially in low mass halos. | We also note that limited spatial resolution of our large volume cosmological simulations will lead to an artificial increase of the gas angular momentum, especially in low mass halos. |
Indeed, the softening of the gravitational force produces very extended discs at the centre of these halos, which would probably be contained within 0.1 aat higher resolution. | Indeed, the softening of the gravitational force produces very extended discs at the centre of these halos, which would probably be contained within 0.1 at higher resolution. |
However, comparing the spin of the gas in the NutCO halo (large empty stars in Fig. 8)) | However, comparing the spin of the gas in the NutCO halo (large empty stars in Fig. \ref{fig:lamratio}) ) |
at various redshifts with that measured for a sample of halos of comparable mass available in these cosmological simulations (solid circles and asterisks in Fig. 8)), | at various redshifts with that measured for a sample of halos of comparable mass available in these cosmological simulations (solid circles and asterisks in Fig. \ref{fig:lamratio}) ), |
we conclude that resolution effects most likely account for a minor fraction of the gas spin (~ 20%). | we conclude that resolution effects most likely account for a minor fraction of the gas spin $\sim20\%$ ). |
Ultimately, to decide whether or not our simulations yield realistic galaxies, we have to compare their properties with observations. | Ultimately, to decide whether or not our simulations yield realistic galaxies, we have to compare their properties with observations. |
Unfortunately, with the high resolution we need to properly address the issues related to angular momentum, cooling and feedback, we can only perform such a comparison at z=0 with one of our runs (NutCO) at the time being. | Unfortunately, with the high resolution we need to properly address the issues related to angular momentum, cooling and feedback, we can only perform such a comparison at $z=0$ with one of our runs (NutCO) at the time being. |
As this simulation does not model any feedback mechanism, by z=0 the simulated galaxy has formed too many stars (Mstar=6.8x107? Mo). | As this simulation does not model any feedback mechanism, by z=0 the simulated galaxy has formed too many stars $M_{\rm star}=6.8\times10^{10}\msun$ ). |
At z=0. the J-band bulge-to-disc ratio (B/D) of the central galaxy in the NutCO run is ~0.78 (see Fig. 9)). | At $z=0$, the $I$ -band bulge-to-disc ratio $B/D$ ) of the central galaxy in the NutCO run is $\simeq 0.78$ (see Fig. \ref{fig:galimage}) ). |
Such value is close to the typical B/D of Sa/Sb type spirals, but the disca scale length (2.0 kpc) turns out to be smaller than those observed (?).. | Such a value is close to the typical $B/D$ of Sa/Sb type spirals, but the disc scale length (2.0 kpc) turns out to be smaller than those observed \citep{graham08}. . |
Nevertheless, bearing this caveat in mind, we present in Fig. | Nevertheless, bearing this caveat in mind, we present in Fig. |
10 measurements of the j versus V relation for the NutCO galaxy at various redshifts, compared to the observational bright disc galaxy sample gathered by ? (at z~ 0). | \ref{fig:comp_obs} measurements of the $j$ versus $V$ relation for the NutCO galaxy at various redshifts, compared to the observational bright disc galaxy sample gathered by \citet{kassin06} (at $z \simeq 0$ ). |
Kassin et al. | Kassin et al. |
derived the specific angular momenta for these galaxies from Halpha + HI rotation curves and radial stellar mass distributions given in (?).. | derived the specific angular momenta for these galaxies from Halpha + HI rotation curves and radial stellar mass distributions given in \citep{kassin06}. |
The average errors in 7 measured from the data are £z60kpckms"!. | The average errors in $j$ measured from the data are $\approx 60 \,{\rm kpc\, km s^{-1}}$. |
As is done in the observations, both the amount of specific angular momentum and the velocity of the simulated galaxy (Vmax, maximum of the rotation curve) are estimated using the stellar component, except for the filled circle symbols where the velocity is measured as the circular velocity at the virial radius of the dark matter halo. | As is done in the observations, both the amount of specific angular momentum and the velocity of the simulated galaxy $V_{\rm
max}$, maximum of the rotation curve) are estimated using the stellar component, except for the filled circle symbols where the velocity is measured as the circular velocity at the virial radius of the dark matter halo. |
For a fair comparison with observations, we also include velocities measured at 2.2 times the disc scale length in the J band (V2.2) at z«1. | For a fair comparison with observations, we also include velocities measured at 2.2 times the disc scale length in the $I$ band $V_{\rm 2.2}$ ) at $z\leq1$. |
Looking at Fig. 10,, | Looking at Fig. \ref{fig:comp_obs}, |
one clearly sees that the velocity estimate plays a crucial role in our ability to assess whether simulated galaxy stars have the correct amount of angular momentum. | one clearly sees that the velocity estimate plays a crucial role in our ability to assess whether simulated galaxy stars have the correct amount of angular momentum. |
This is because the observed slope of the relation (« 2) is quite large so any error on the velocity will translate into a much larger error on the specific angular momentum. | This is because the observed slope of the relation $\approx$ 2) is quite large so any error on the velocity will translate into a much larger error on the specific angular momentum. |
More specifically, our simulation suffers from forming too many stars and as a result any velocity estimated in the central region of the halo is bound to be too large. | More specifically, our simulation suffers from forming too many stars and as a result any velocity estimated in the central region of the halo is bound to be too large. |
If, on the other hand, we use the circular velocity at the virial radius of the halo to bypass the problem (note that this is current practice, see e.g. ?)), we find that the level of angular momentum of the simulated stars is in fair agreement with the observations (Fig. 10)). | If, on the other hand, we use the circular velocity at the virial radius of the halo to bypass the problem (note that this is current practice, see e.g. \citealt{dutton09}) ), we find that the level of angular momentum of the simulated stars is in fair agreement with the observations (Fig. \ref{fig:comp_obs}) ). |
We point out that this is a consequence of the lossless transport of a large amount of specific angular momentum by gas from super halo scales right into the inner regions, followed by its redistribution in the vicinity of the disc, as discussed in the previous section. | We point out that this is a consequence of the lossless transport of a large amount of specific angular momentum by gas from super halo scales right into the inner regions, followed by its redistribution in the vicinity of the disc, as discussed in the previous section. |
As a result of this effect and the low efficiency of the star formation process on disc scales, stars end up with factor 2 to 3 less angular momentum than their dark matter halo acounterpart which is very close to the discrepancy between dark matter only simulations and Observations, as pointed out by ? and Kassin et al. ( | As a result of this effect and the low efficiency of the star formation process on disc scales, stars end up with a factor 2 to 3 less angular momentum than their dark matter halo counterpart which is very close to the discrepancy between dark matter only simulations and observations, as pointed out by \citet{navarro00} and Kassin et al. ( |
2011, prep.). | 2011, ). |
Finally, it is interesting to note that when we use the Vinax estimates for the velocity of the galaxy, although we do not match the zero point of the relation, simulated data points at different redshifts move along the observed sequence, suggesting that we would predict very little time evolution of the sequence. | Finally, it is interesting to note that when we use the $V_{\rm max}$ estimates for the velocity of the galaxy, although we do not match the zero point of the relation, simulated data points at different redshifts move along the observed sequence, suggesting that we would predict very little time evolution of the sequence. |
On the contrary, V. estimates suggest there exists a strong evolution ofthe relation with redshift as the flat velocity profile of the dark matter halo is already in place very early on. | On the contrary, $V_{\rm c}$ estimates suggest there exists a strong evolution ofthe relation with redshift as the flat velocity profile of the dark matter halo is already in place very early on. |
feature with Fe XXD and the EW range of 0 — 80 eV as motivated by the time-sliced data. | feature with Fe ) and the EW range of 0 – 80 eV as motivated by the time-sliced data. |
? detected a variable absorption feature at 7.7 keV in the spectrum of MCG-5-23-16. | \citet{braito07} detected a variable absorption feature at 7.7 keV in the spectrum of MCG-5-23-16. |
This feature is also indicated in Fig. 10.. | This feature is also indicated in Fig. \ref{fig_ewvsshift}, |
adopting the upper bound of on its EW range based on their analysis of the one of tive 20 ks time intervals in which the line was most clearly detected (see their table 4). | adopting the upper bound of on its EW range based on their analysis of the one of five 20 ks time intervals in which the line was most clearly detected (see their table 4). |
The velocity shift was computed assuming that Fe is dominant. | The velocity shift was computed assuming that Fe is dominant. |
In terms of shift and range of EW. the absorption features in PGI2114143. Mrk 766 and MCOG-5-23-16 lie close to the space explored by our grid of models (Fig. 103). | In terms of shift and range of EW, the absorption features in PG1211+143, Mrk 766 and MCG-5-23-16 lie close to the space explored by our grid of models (Fig. \ref{fig_ewvsshift}) ). |
Thus wind models such as those we have investigated are promising starting points for explaining the Ka absorption in these observations. | Thus wind models such as those we have investigated are promising starting points for explaining the $\alpha$ absorption in these observations. |
Blueshifted Kea absorption features have also been reported in NGC 1365 ¢?).. | Blueshifted $\alpha$ absorption features have also been reported in NGC 1365 \citep{risaliti05}. |
In this case. theshifts are considerably smaller. ~S5000 km 1 at most. | In this case, theshifts are considerably smaller, $\sim$ 5000 km $^{-1}$ at most. |
Although our grid of models does include some spectra with features at these low velocities (see Fig. 10). | Although our grid of models does include some spectra with features at these low velocities (see Fig. \ref{fig_ewvsshift}) ), |
the EWs in the computed spectra are below those observed by a significant factor (the combined Fe Ka EWs reported by ? are in the range 200 300 eV. lying off the scale of Fig. 109). | the EWs in the computed spectra are below those observed by a significant factor (the combined Fe $\alpha$ EWs reported by \citealt{risaliti05} are in the range 200 – 300 eV, lying off the scale of Fig. \ref{fig_ewvsshift}) ). |
Even highly-ionized— absorption has been reported in (222) and NGC 3516 ο. | Even lower-blueshift highly-ionized absorption has been reported in MCG-6-30-15 \citep{young05,miniutti07,miller08} and NGC 3516 \citep{turner08}. |
The simplest way in which such strong. relatively low-velocity features might be produced with our model would be by considering lower terminal flow velocities but it may also be possible to obtain deeper absorption by considering flows which are significantly clumped. | The simplest way in which such strong, relatively low-velocity features might be produced with our model would be by considering lower terminal flow velocities but it may also be possible to obtain deeper absorption by considering flows which are significantly clumped. |
Detailed study of these yossibilities Hes beyond the scope of this paper. but we do note hat even the NGC 1365 features are within a factor of ~5 in EW of those obtained with our grid of models. | Detailed study of these possibilities lies beyond the scope of this paper, but we do note that even the NGC 1365 features are within a factor of $\sim 5$ in EW of those obtained with our grid of models. |
In principle. the model spectra indicate that the absorption ine shapes are complex and therefore provide information on he outflow properties. | In principle, the model spectra indicate that the absorption line shapes are complex and therefore provide information on the outflow properties. |
However. given the limited sensitivity and spectral resolution of current X-ray telescopes. determining line shapes is virtually impossible — at best. only a rough limit on he characteristic linewidth is available. | However, given the limited sensitivity and spectral resolution of current X-ray telescopes, determining line shapes is virtually impossible – at best, only a rough limit on the characteristic linewidth is available. |
For comparison with such imits. Fig. | For comparison with such limits, Fig. |
||]. shows the FWHM determined from our spectra (by Gaussian fitting) versus the Fe blueshift for the same models as plotted in Fig. 10.. | \ref{fig_widvsshift} shows the FWHM determined from our spectra (by Gaussian fitting) versus the Fe blueshift for the same models as plotted in Fig. \ref{fig_ewvsshift}. |
The FWHM varies between about 3.510* and km |. with values around 5 — 7 10 km 7! being most common. | The FWHM varies between about $3.5 \times 10^{3}$ and $1.8 \times
10^4$ km $^{-1}$, with values around 5 – 7 $\times 10^3$ km $^{-1}$ being most common. |
For all our model spectra in which the blueshift is greater than 107 km |. the FWHM is smaller than the blueshift meaning that the absorption does not extend down to the rest energy of the line. | For all our model spectra in which the blueshift is greater than $^4$ km $^{-1}$, the FWHM is smaller than the blueshift meaning that the absorption does not extend down to the rest energy of the line. |
As one would expect. the largest linewidth CoL82110! Km ID occurs in model where ¢ has the smallest value considered (f=0.33) and for a viewing direction which has a large pathlength through the flow. | As one would expect, the largest linewidth $\sim 1.8 \times 10^4$ km $^{-1}$ ) occurs in model where $d$ has the smallest value considered $d = 0.33$ ) and for a viewing direction which has a large pathlength through the flow. |
To our knowledge. there are currently no firm measurements of linewidths for narrow blueshifted Fe Ka absorption features in AGN spectra. | To our knowledge, there are currently no firm measurements of linewidths for narrow blueshifted Fe $\alpha$ absorption features in AGN spectra. |
For PGI2114143. ?. placed a limit of kms 1 on the Fe Ka linewidth. which is consistent with the majority of the measurements from our spectra shown in Fig. IL.. | For PG1211+143, \cite{pounds03} placed a limit of $< 1.2 \times 10^4$ km $^{-1}$ on the Fe $\alpha$ linewidth, which is consistent with the majority of the measurements from our spectra shown in Fig. \ref{fig_widvsshift}. |
? found a best-fit line width of @=200+100 eV (FWHM =L8.10! kms +) for the absorption feature in MCG-5-23-16. | \cite{braito07} found a best-fit line width of $\sigma = 200 \pm
100$ eV (FWHM $= 1.8 \times 10^{4}$ km $^{-1}$ ) for the absorption feature in MCG-5-23-16. |
However. the fit in which they allowed the linewidth to vary Was not a significant improvement over that in which they pinned the width ato 100 eV. Thus. their line width constraints are also consistent with the majority of the spectra described by Fig. |L.. | However, the fit in which they allowed the linewidth to vary was not a significant improvement over that in which they pinned the width at $\sigma = 100$ eV. Thus, their line width constraints are also consistent with the majority of the spectra described by Fig. \ref{fig_widvsshift}. |
As discussed in Section |.. blueshifted Fe Ka absorption is the clearest signature of highly ionized outflowing gas and has therefore been the central topic of this paper | As discussed in Section \ref{sect_intro}, blueshifted Fe $\alpha$ absorption is the clearest signature of highly ionized outflowing gas and has therefore been the central topic of this paper. |
However. as mentioned in Section 4.. narrow blueshifted Fe Ka absorption is neither a necessary consequence nor the only signature of a highly ionized flow. | However, as mentioned in Section \ref{sect_example}, narrow blueshifted Fe $\alpha$ absorption is neither a necessary consequence nor the only signature of a highly ionized flow. |
Thus if such outflows are common in the AGN population it is important to consider other effects they may have on X-ray spectra. | Thus if such outflows are common in the AGN population it is important to consider other effects they may have on X-ray spectra. |
In this section. we briefly describe some of these effects and their possible implications with reference to our model spectra. | In this section, we briefly describe some of these effects and their possible implications with reference to our model spectra. |
Although the Fe « feature is typically the most prominent absorption line. Ka absorption by lighter elements is present in many of the spectra computed from our grid of models. particularly those with high Ai-values. | Although the Fe $\alpha$ feature is typically the most prominent absorption line, $\alpha$ absorption by lighter elements is present in many of the spectra computed from our grid of models, particularly those with high $\dot{M}$ -values. |
The strongest of these Ka features are those of S. οἱ and O -- the absorption EWS for these lines are up to 10. 10 and 3 eV respectively. | The strongest of these $\alpha$ features are those of S, Si and O – the absorption EWs for these lines are up to 10, 10 and 3 eV respectively. |
Their strengths correlate closely with each other and more loosely with the Fe Ka feature. | Their strengths correlate closely with each other and more loosely with the Fe $\alpha$ feature. |
At energies harder than Fe Ka. absorption by H-/He-like Ni Ke and by Fe K.i/K- occur. | At energies harder than Fe $\alpha$, absorption by H-/He-like Ni $\alpha$ and by Fe $\beta$ $\gamma$ occur. |
Individually. these features are all weaker than Fe Ka but they are often blended together producing fairly significant absorption which can extend up to around 10 keV. These additional absorption lines can. provide both confirmation of outflows detected via Fe Ka and supplementary constraints on the flow properties. | Individually, these features are all weaker than Fe $\alpha$ but they are often blended together producing fairly significant absorption which can extend up to around 10 keV. These additional absorption lines can provide both confirmation of outflows detected via Fe $\alpha$ and supplementary constraints on the flow properties. |
However. their reliability is limited since (i) in our models. they are always weaker and thus harder to detect than Fe Ko and (ii) with the exception of the hard energy Ni or Fe K.? lines. their identifications are less clear since there are more potential atomic transitions at soft energy. | However, their reliability is limited since (i) in our models, they are always weaker and thus harder to detect than Fe $\alpha$ and (ii) with the exception of the hard energy Ni or Fe $\beta$ lines, their identifications are less clear since there are more potential atomic transitions at soft energy. |
Thus. as diagnostics for highly ionized flows. these lines are always of subordinate value to Fe Ka. | Thus, as diagnostics for highly ionized flows, these lines are always of subordinate value to Fe $\alpha$. |
As pointed out in the reference spectra shown in Section 4.. the blueshifted absorption lines are often accompanied by broad emission which is centred close to the line rest energy. | As pointed out in the reference spectra shown in Section \ref{sect_example}, , the blueshifted absorption lines are often accompanied by broad emission which is centred close to the line rest energy. |
Since these emission features are formed as a result of line scattering and recombination emission in the outflow. their strength strongly depends on the density of the wind: thus although they are fairly weak in the spectra from the example model. they become more | Since these emission features are formed as a result of line scattering and recombination emission in the outflow, their strength strongly depends on the density of the wind; thus although they are fairly weak in the spectra from the example model, they become more |
to order « in the expansion of the ton equation of motion. the compression term V.V can be eliminated from (11)) and (8)) using the fact that p~ 6-. | to order $\alpha$ in the expansion of the ion equation of motion, the compression term $\nabla.\mathbf{V}$ can be eliminated from \ref{eq20}) ) and \ref{eq17}) ) using the fact that $p\simeq b_{z}$ . |
Therefore. defining Z=b-/cy. cy=ND/Y+6 and dy=cyd;. the following model is obtained Z | v:(UY+Z. More details concerning the derivation of this reduced two-fluid MHD model can be found in (?).. | Therefore, defining $Z=b_{z}/c_{\beta}$, $c_{\beta}=\sqrt{\beta/1+\beta}$ and $d_{\beta}=c_{\beta}d_{i}$, the following model is obtained: The system conserves the total energy $E=\int d^{3}r[(\nabla\phi)^{2}+v^{2}_{z}+(\nabla \psi)^{2}+Z^{2}].$ More details concerning the derivation of this reduced two-fluid MHD model can be found in \citep{BianTsiklauri2009}. |
For B.«1. the parallel flow dynamics decouples in the above reduced two-fluid MHD model. hence Z=ριω-. and therefore it simplifies to: with p,=C,/w,; being the ion sound gyroradius. | For $\beta\ll 1$, the parallel flow dynamics decouples in the above reduced two-fluid MHD model, hence $Z=-\rho_{s}\omega_{z}$, and therefore it simplifies to: with $\rho_{s}=C_{s}/\omega_{ci}$ being the ion sound gyroradius. |
The total energy takes the form: Notice that this model is very similar to the EMHD system when k_p,>>|. with the compressibility effect retained. while it reduces to the standard RMHD deseription of shear Alfvénn waves perturbations for οι«I. | The total energy takes the form: Notice that this model is very similar to the EMHD system when $k_{\perp}\rho_{s}\gg 1$, with the compressibility effect retained, while it reduces to the standard RMHD description of shear Alfvénn waves perturbations for $k_{\perp}\rho_{s}\ll 1$. |
Linearizing this two-field model yields the frequency of the kinetic Alfven wave: This shows that the low-frequency Alfven wave. with a frequency much smaller that the ion cyclotron frequency w< ωμ. becomes dispersive when the wavelength perpendicular to the background magnetic field is comparable or smaller than the ion sound gyroradius p,. Le. c)»=hyok_. this dispersion being similar to the one of the low-frequency whistler wave. | Linearizing this two-field model yields the frequency of the kinetic Alfven wave: This shows that the low-frequency Alfven wave, with a frequency much smaller that the ion cyclotron frequency $\omega<\omega_{ci}$ , becomes dispersive when the wavelength perpendicular to the background magnetic field is comparable or smaller than the ion sound gyroradius $\rho_{s}$, i.e. $\omega_{\pm}=\pm k_{\parallel}\rho_{s}k_{\perp}$, this dispersion being similar to the one of the low-frequency whistler wave. |
We further notice that the model given by (16))-(17)) ts the simplest subset of the so-called electromagnetic gvrofluid models (see e.g. ? and references therein) which are obtained às moments of the drift-kinetie equations. | We further notice that the model given by \ref{eq32}) \ref{eq33}) ) is the simplest subset of the so-called electromagnetic gyrofluid models (see e.g. \citet{Waelbroeck_etal2009} and references therein) which are obtained as moments of the drift-kinetic equations. |
From Equations (16))-(17)). a theory for KAW turbulence is now constructed along the same lines as the Goldreich-Sidrar theory (?) for Alfven wave turbulence (?).. | From Equations \ref{eq32}) \ref{eq33}) ), a theory for KAW turbulence is now constructed along the same lines as the Goldreich-Sidrar theory \citep{GoldreichSridhar1995}
for Alfven wave turbulence \citep{Kraichnan1965}. |
Some form of dissipation at small scales. balancing the energy input at large scales. is necessary for a steady cascade of energy to take place. | Some form of dissipation at small scales, balancing the energy input at large scales, is necessary for a steady cascade of energy to take place. |
It is assumed that the turbulent fluctuations are composed of KAWSs. hence. Focusing first on the perpendicular cascade. we can express the energy per wave number Ες from (18)) and use (20)) to obtain that Moreover. we adopt the standard assumption that the flux of turbulent energy at a given scale is determined by the turbulence at that scale and is a constant equal to the energy injection rate ε. | It is assumed that the turbulent fluctuations are composed of KAWs, hence, Focusing first on the perpendicular cascade, we can express the energy per wave number $E_{k_{\perp}}$ from \ref{en}) ) and use \ref{k}) ) to obtain that Moreover, we adopt the standard assumption that the flux of turbulent energy at a given scale is determined by the turbulence at that scale and is a constant equal to the energy injection rate $\epsilon$. |
Therefore. the expression for the energy cascade rate Is with the non-linear time scale being given by Ty,~V/IdyUtpzR7). which. using (20)). is equivalently expressed als Combining relations (21))-(22))-(23)) yields the scaling law for the energy spectrum: where C is a constant of the order of unity (2)... | Therefore, the expression for the energy cascade rate is with the non-linear time scale being given by $\tau_{NL} \sim 1/k^{2}_{\perp}\phi_{k_{\perp}}(1+\rho_{s}^{2}k_{\perp}^{2})$ , which, using \ref{k}) ), is equivalently expressed as Combining relations \ref{en2}) \ref{rat2}) \ref{nl2}) ) yields the scaling law for the energy spectrum: where $C$ is a constant of the order of unity \citep{Kraichnan1965}. |
This expression recovers the spectrum of Alfvenic turbulence. in the limitk p,<I. ie. Ej~CeX7*, while in the dispersive range. for κp,1. then. £j.~CeVDWt» | This expression recovers the spectrum of Alfvenic turbulence, in the limit $k_{\perp}\rho_{s}\ll1$, i.e., $E_{k_{\perp}}\sim C\epsilon^{2/3}k_{\perp}^{-5/3}$, while in the dispersive range, for $k_{\perp}\rho_{s}\gg1$ , then, $E_{k_{\perp}}\sim C\epsilon^{2/3}\rho_{s}^{-2/3}k_{\perp}^{-7/3}$. |
Implicit in the derivation of (24)) is the assumption that the fraction of the energy flux of Alfvenic turbulence which ts transferred from the MHD scales onto the dispersive scales 1s of the order unity. | Implicit in the derivation of \ref{spec}) ) is the assumption that the fraction of the energy flux of Alfvenic turbulence which is transferred from the MHD scales onto the dispersive scales is of the order unity. |
Notice that since from (21)). the magnetic energy spectrum is Ej.=ΚΕ. then (24)) is also equivalent to the scaling relation Now. we recall a fundamental ordering used in the derivation of the two-fluid reduced MHD system (16))-(17)) This ordering is not restrictive in the sense that we are interested in the inertial range and not in the outer scale of the Alfvenic turbulence. where 6B_ can be of order unity. | Notice that since from \ref{en2}) ), the magnetic energy spectrum is $E_{k_{\perp}}=k^{-1}_{\perp}\delta B^{2}_{\perp}$, then \ref{spec}) ) is also equivalent to the scaling relation Now, we recall a fundamental ordering used in the derivation of the two-fluid reduced MHD system \ref{eq32}) \ref{eq33}) ): This ordering is not restrictive in the sense that we are interested in the inertial range and not in the outer scale of the Alfvenic turbulence, where $\delta B_{\perp}$ can be of order unity. |
Using (25)). (26)) provides the scale dependent anisotropy of the turbulence: which recovers the original Goldreich-Sridhar critical balance relation Kjκὃν for Alfven wave turbulence when Kkpom1. while in the dispersive range. Ajekl* | Using \ref{db}) ), \ref{ord}) ) provides the scale dependent anisotropy of the turbulence: which recovers the original Goldreich-Sridhar critical balance relation $k_{\parallel}\propto k_{\perp}^{2/3}$, for Alfven wave turbulence when $k_{\perp}\rho_{s}\ll1$, while in the dispersive range, $k_{\parallel}\propto k_{\perp}^{1/3}$. |
In fact. it would have been equivalent to argue. following ?.. that the anisotropy of the turbulence ts fixed by the so-called critical balance condition. ie. to assume that the characteristic non-linear decorrelation time is of the order of the inverse KAW frequency. Le. cay~Tv. with w given by Eq.(19)). | In fact, it would have been equivalent to argue, following \citet{GoldreichSridhar1995}, that the anisotropy of the turbulence is fixed by the so-called critical balance condition, i.e. to assume that the characteristic non-linear decorrelation time is of the order of the inverse KAW frequency, i.e. $\omega ^{-1}_{KAW}\sim \tau _{NL}$, with $\omega$ given by \ref{dr}) ). |
The scaling relations obtained for the energy spectrum and anisotropy in the dispersive scales of kinetic Alfven wave turbulence (???) are similar to the ones of EMHD turbulence (?22??).. | The scaling relations obtained for the energy spectrum and anisotropy in the dispersive scales of kinetic Alfven wave turbulence \citep{CranmerVanBallegooijen2003,Howes_etal2008,Schekochihin_etal2009}
are similar to the ones of EMHD turbulence \citep{Biskamp_etal1999,Ng_etal2003,ChoLazarian2004,Cho2009}. |
Notice however that for an EMHD Ohm's law given by E=djxB. whistlers do not have a parallel electric field. | Notice however that for an EMHD Ohm's law given by $\mathbf{E}=d_{i}\mathbf{j}\times \mathbf{B}$, whistlers do not have a parallel electric field. |
Before concluding this section. few comments are due. | Before concluding this section, few comments are due. |
In deriving the energy spectrum for kinetic. Alfven wave turbulence we are relying on the existing theory developed by(?) for strong anisotropic and balanced Alfven turbulence. | In deriving the energy spectrum for kinetic Alfven wave turbulence we are relying on the existing theory developed by\citep{GoldreichSridhar1995} for strong anisotropic and balanced Alfven turbulence. |
The same approach was followed by (?) based on a compressible EMHD model to describe the dispersive range of Alfven turbulence. see also (2).. | The same approach was followed by \citep{Schekochihin_etal2009} based on a compressible EMHD model to describe the dispersive range of Alfven turbulence, see also \citep{CranmerVanBallegooijen2003}. |
It is ourframework to investigate the spectral structure of the turbulent parallel electric. field. | It is ourframework to investigate the spectral structure of the turbulent parallel electric field. |
This should however not suggest that there is one universal cascade | This should however not suggest that there is one universal cascade |
The Riemann-Christollel tensor 72,4, lor the background metric has the following independent components. | The Riemann-Christoffel tensor $R_{\alpha \beta \mu \nu}$ for the background metric has the following independent components. |
All the rest are either related to these by the algebraic properties of F2,4, OF are zero. — (((sie) E2 | LLLp p— | All the rest are either related to these by the algebraic properties of $R_{\alpha \beta \mu \nu}$ or are zero. |
uso= (( 2) | = —= — Royeym = = "(et ))2 | = ( )^2 ] = = = ( )^2 ] = = = = = = ( )^2 |
uso= (( 2) | = —= — Royeym = = "(et ))2; | = ( )^2 ] = = = ( )^2 ] = = = = = = ( )^2 |
One of the strongest. pieces of evidence for an evolving universe has long been the observed evolution in comoving space density of the quasar population (Schmidt 1968). | One of the strongest pieces of evidence for an evolving universe has long been the observed evolution in comoving space density of the quasar population (Schmidt 1968). |
Until recently. it had. been thought that the shape of the quasar optical luminosity function. and. its evolution over the redshift range 0.«z2 was well understood. and attempts to explain the physical causes of quasar evolution have relied on attempting to predict the observed evolution of the luminosity function (e.g. Llachnelt Rees 1993). | Until recently it had been thought that the shape of the quasar optical luminosity function and its evolution over the redshift range $0 < z < 2$ was well understood, and attempts to explain the physical causes of quasar evolution have relied on attempting to predict the observed evolution of the luminosity function (e.g. Haehnelt Rees 1993). |
llowever recent studies (Goldschmidt et al.. | However recent studies (Goldschmidt et al., |
1992. Llewett et al.. | 1992, Hewett et al., |
1993. Llawkins Vérron 1993 1995) have cast doubt upon the completeness of the surveys. used. to define the uninosity function. | 1993, Hawkins Vérron 1993 1995) have cast doubt upon the completeness of the surveys used to define the luminosity function. |
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