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H0 may. be demonstrated. that he result is another normal distribution ING.νήσο|στ) essentially by virtue of the lav of error combination. | It may be demonstrated, that the result is another normal distribution $N(\Delta,\sqrt{\sigma_0^2+\sigma_1^2})$, essentially by virtue of the law of error combination. |
By substituting A=3.628)2.7590.869 rad. we find that he observed velocity. O21 of V440. Per deviates from the "uncdamental mode sequence by 7.250. | By substituting $\Delta=3.628-2.759=0.869$ rad, we find that the observed velocity $\phi_{21}$ of V440 Per deviates from the fundamental mode sequence by $\sigma$. |
Thus. from the purely observational evidence we conclude that V440 Per does not oulsate in the fundamental mode. | Thus, from the purely observational evidence we conclude that V440 Per does not pulsate in the fundamental mode. |
Consequently. it must be an overtone Cepheic. | Consequently, it must be an overtone Cepheid. |
The argument presented: above critically depends. on he assumption that velocity σι (at à given. period) does not depend on the pulsation amplitude. which for V440 Per is lower than for the fundamental. mode Cepheids used as comparison. | The argument presented above critically depends on the assumption that velocity $\phi_{21}$ (at a given period) does not depend on the pulsation amplitude, which for V440 Per is lower than for the fundamental mode Cepheids used as comparison. |
Such an assumption is well justified by numerical computations Buehler et al. | Such an assumption is well justified by numerical computations Buchler et al. |
1990: Smolec Aloskalik 2008). nevertheless it should. be verified with available data. | 1990; Smolec Moskalik 2008), nevertheless it should be verified with available data. |
In repASC3a. we plot residuals from the mean fundamental mode P?οι relation amplitude stp. | In \\ref{ASC3a} we plot residuals from the mean fundamental mode $P-\phi_{21}$ relation amplitude $A_1$. |
In this case the mean relation is defined as the parabola fitted to all fundamental mode Cepheids clisplaved in reLASC3.. | In this case the mean relation is defined as the parabola fitted to all fundamental mode Cepheids displayed in \\ref{ASC3}. |
For 2, in the range of 10lv km/s there is no correlation of G2, residuals with pulsation amplitude. | For $A_1$ in the range of $10-17$ km/s there is no correlation of $\phi_{21}$ residuals with pulsation amplitude. |
For Aj«10 km/s there is à weak evidence of ó», increase. | For $A_1<10$ km/s there is a weak evidence of $\phi_{21}$ increase. |
As the data are very scarce. we are not convinced that this increase is significant. | As the data are very scarce, we are not convinced that this increase is significant. |
Hit were real. however. it would only strengthen our conclusion that V440 Per stronely deviates from the funcamental mode sequence. | If it were real, however, it would only strengthen our conclusion that V440 Per strongly deviates from the fundamental mode sequence. |
Classification of V4t0 Pep as à Dist overtone pulsator. discussed in the previous Section. is based solely on the morphological properties of Cepheicl velocity. curves. | Classification of V440 Per as a first overtone pulsator, discussed in the previous Section, is based solely on the morphological properties of Cepheid velocity curves. |
This mode identification can be strengthened. by comparing the velocity curve of V440. Per with those of hyedrocyvnamical Cepheicl models. | This mode identification can be strengthened by comparing the velocity curve of V440 Per with those of hydrodynamical Cepheid models. |
Such a comparison was first. performed bv Wienzleetal.(1999)... who used the unpublished radiative models of Schaller Buchler (1994). | Such a comparison was first performed by \citet{kie99}, who used the unpublished radiative models of Schaller Buchler (1994). |
They showed that the theoretical. progression of velocity. G2, supported the overtone classification of W440 Per. | They showed that the theoretical progression of velocity $\phi_{21}$ supported the overtone classification of V440 Per. |
However. their conclusion was somewhat weakened by a laree error of then available Oo, of this star. | However, their conclusion was somewhat weakened by a large error of then available $\phi_{21}$ of this star. |
In the present section. we confirm and extend the results of Ixienzleetal.(1999). | In the present section, we confirm and extend the results of \citet{kie99}. |
. H is easy to show. that the velocity curve of V4AL0 Per is incompatible with the fundamental mode Cepheid mocels. | It is easy to show, that the velocity curve of V440 Per is incompatible with the fundamental mode Cepheid models. |
Indeed. all publishecl models display velocity. G2, higher that rad.periods anc higher than 8.5rad at the period of V440 Per Buehler οἱ al. | Indeed, all published models display velocity $\phi_{21}$ higher that rad and higher than rad at the period of V440 Per Buchler et al. |
1990: Moskalik et al. | 1990; Moskalik et al. |
1992: Smolec AMoskalik 2008). | 1992; Smolec Moskalik 2008). |
This holds true both for the convective models and for the older racliative models. | This holds true both for the convective models and for the older radiative models. |
The computed velocity G2, is very robust ancl shows no sensitivity to the treatment of convection. choice of opacities or to details of the numerical code. | The computed velocity $\phi_{21}$ is very robust and shows no sensitivity to the treatment of convection, choice of opacities or to details of the numerical code. |
Most importantly. it is insensitive to the pulsation amplitude Figs.s. 11 of Smolee Aloskalik 2008). | Most importantly, it is insensitive to the pulsation amplitude 8, 11 of Smolec Moskalik 2008). |
Clearly. the only chance to match the observed. velocity curve of V440 Per is to search for an appropriate overtone moclel. | Clearly, the only chance to match the observed velocity curve of V440 Per is to search for an appropriate overtone model. |
With this goal in mind. we computed several sequences of the convective overtone Cepheicl models. | With this goal in mind, we computed several sequences of the convective overtone Cepheid models. |
We show that V44O Per fits theoretical o», progression of the first overtone Cepheids and that its velocity. Fourier. parameters can be accurately reproduced with hydrodynamical computations. | We show that V440 Per fits theoretical $\phi_{21}$ progression of the first overtone Cepheids and that its velocity Fourier parameters can be accurately reproduced with hydrodynamical computations. |
Modeling of such a long period overtone pulsator is not an easy task. | Modeling of such a long period overtone pulsator is not an easy task. |
Vhe satisfactory models have to reproduce Fouricr parameters ancl the long period. of this variable. | The satisfactory models have to reproduce Fourier parameters and the long period of this variable. |
The current. hyelrococles used. for modeling of the racial pulsationsadopt a time-dependent. convection models (e.g. Stellingwert. 1982. 1986). | The current hydrocodes used for modeling of the radial pulsationsadopt a time-dependent convection models (e.g. Stellingwerf 1982, 1986). |
These mocels introduce several dimensionless parameters. that should be adjusted | These models introduce several dimensionless parameters, that should be adjusted |
is louud at the location of the outer boundary. (defined in the linear coulterpart) in Figure 9bb for he vertical strucure. Le.. the are structure liaviug :c-intercep at cLr, (see also Fig. | is found at the location of the outer boundary (defined in the linear counterpart) in Figure \ref{fig:non}b b for the vertical structure, i.e., the arc structure having $x$ -intercept at $\sim4r_p$ (see also Fig. |
Tbb). | \ref{fig:nlv}b b). |
This oesence of the expected outer couuterpart suggests that the «xiegin of ttie. cliscoutinuity along the solid line Is not tie outer boundary but it is likely interprete as the mocified inner boundary. of he linear wake with the above explanation. | This presence of the expected outer counterpart suggests that the origin of the discontinuity along the solid line is not the outer boundary but it is likely interpreted as the modified inner boundary of the linear wake with the above explanation. |
As a result. the interior of the nonlinear wake is filed ip witliot ta celiuite inuer boundary. except [or a moderate iilerfereuce near the black solid lines in Figure 9bb (eq. [10]]). | As a result, the interior of the nonlinear wake is filled up without a definite inner boundary, except for a moderate interference near the black solid lines in Figure \ref{fig:non}b b (eq. \ref{equ:asy}] ]). |
The vertical structure of the nonlinear wake is more ellidtically shaped han circilarly shaped. | The vertical structure of the nonlinear wake is more elliptically shaped than circularly shaped. |
As tle extension of the solid curve in Figure Jaa. the naterial imimeciately beiind the object also [lows over the curved Mach cone shape. advauciug 6°..7 in angle alor& the orbit. | As the extension of the solid curve in Figure \ref{fig:non}a a, the material immediately behind the object also flows over the curved Mach cone shape, advancing .7 in angle along the orbit. |
This augle of he isotleal wake (+= 1) excellently matches the fit in Wiu(2010) for tje shock stancloll distauce in adiabatic gas (5= 5/3). | This angle of the isothermal wake $\gamma=1$ ) excellently matches the fit in \citet{kwt10} for the shock standoff distance in adiabatic gas $\gamma=5/3$ ). |
It implies that the shock ceaclhinent is a colΠο nature of nonlinear lows irespective of adiabatic index + as long as the sonic W.yeecl is definec as ο5sp/p)*7. | It implies that the shock detachment is a common nature of nonlinear flows irrespective of adiabatic index $\gamma$ as long as the sonic speed is defined as $\cs=(\gamma p/\rho)^{1/2}$. |
In he oocess of the overllow creating a new shock [οί in acValce. the [low is hiely uustable between he ok auc uew shock fronts. generating low deusity eccles in extremely 1onlinear cases. | In the process of the overflow creating a new shock front in advance, the flow is highly unstable between the old and new shock fronts, generating low density eddies in extremely nonlinear cases. |
According o Wit1&Wim(2009) for linear trajectory cases. the low density. eddies si‘vive ouly when the detaclec shock distauce from the object in tle lateral clirecion 15 larger tlali "M. | According to \citet{kim09} for linear trajectory cases, the low density eddies survive only when the detached shock distance from the object in the lateral direction is larger than $r_B\mach^{-2}$. |
> Lt is" checked hat tje lateral distance of the stock in Figure Jaa is also arger than tli5 Vaιο, corresponding to Q.lry. | It is checked that the lateral distance of the shock in Figure \ref{fig:non}a a is also larger than this value, corresponding to $0.1r_p$. |
Lastly. Figre Occ shows {1e density euliauceimuen [9] the wake in this extremely noulinear eeime. | Lastly, Figure \ref{fig:non}c c shows the density enhancement of the wake in this extremely nonlinear regime. |
The peaς values do not lave a systematic chauge TOL1 the linear predicti1 (solid curve). while the minima do not mate1i auy of the dashed curves iuiike in the lilear regime. | The peak values do not have a systematic change from the linear prediction (solid curve), while the minima do not match any of the dashed curves unlike in the linear regime. |
The minima π παν o the dotted clurve. a—rgircry) Lovedτη& the arm-interarm density contrast in the entire wake. | The minima instead increase to the dotted curve, $\alpha=r_B\,(r+r_p)^{-1}$, reducing the arm-interarm density contrast in the entire wake. |
The wake cdie to circular motion attains a hyclrodyuaic equilibrium state about he system center. | The wake due to circular motion attains a hydrodynamic equilibrium state about the system center. |
In contrast. tle wake due to rectilinear motiou has the equilibrium ceuter about he object itself. so that the aiipliude of gravitatioual influence is scale by rp/r, relevant to the object size re. | In contrast, the wake due to rectilinear motion has the equilibrium center about the object itself, so that the amplitude of gravitational influence is scaled by $r_B/r_s$ relevant to the object size $r_s$. |
The wake of a ciretlarly orbiting object actually tends to achieve quasi-equilibrium alxtut both the object aud its orbital center. | The wake of a circularly orbiting object actually tends to achieve quasi-equilibrium about both the object and its orbital center. |
However. the former iuflueices the localized area at disance from the object of at mos rp(Me—1) (i£ this distance is greate "than re. alic My>1: ]|xiu&Wim2009: 2010)). Wwule the latter affects the wake properties globally. | However, the former influences the localized area at distance from the object of at most $r_B(\mach^2-1)^{-1}$ (if this distance is greater than $r_s$ , and $\mach>1$; \citealp{kim09,kwt10}) ), while the latter affects the wake properties globally. |
Tje central valie of deusity enliaucement. taiηαμ» the value of riry independent o ‘thenonlinearity. thus readily explaining Ixims (2010) ]aim that the background ceusity aloug t1e orbit isproportional to hus value as numerically found α0.αυ... in equation (10) of Ixim (2010).. | The central value of density enhancement maintains the value of $r_B/r_p$ independent of thenonlinearity, thus readily explaining \citeauthor{kwt10}' 's \citeyearpar{kwt10} claim that the background density along the orbit isproportional to this value as numerically found $\alpha\approx0.46(r_B/r_p)^{1.1}$ in equation (10) of \citet{kwt10}. . |
one truncates the Edgeworth expansion after a finite number of terms. | one truncates the Edgeworth expansion after a finite number of terms. |
As we will see shortly, it now becomes possible to describe an entirely non-negative pdf with only fwi, and ονι,, circumventing the result of the previous section. | As we will see shortly, it now becomes possible to describe an entirely non-negative pdf with only $\fnl$ and $\gnl$, circumventing the result of the previous section. |
The disadvantage of the truncation is, of course, that the cumulants of the reconstructed pdf may not correspond exactly to those of the actual pdf, and thus the statistical significance of fwr, and gwr, is somewhat weakened. | The disadvantage of the truncation is, of course, that the cumulants of the reconstructed pdf may not correspond exactly to those of the actual pdf, and thus the statistical significance of $\fnl$ and $\gnl$ is somewhat weakened. |
For very short series of just a few terms, the interpretations of fwr, as skewness and gwr, as excess kurtosis are especially dubious. | For very short series of just a few terms, the interpretations of $\fnl$ as skewness and $\gnl$ as excess kurtosis are especially dubious. |
Given information on a finite number of cumulants, we shall investigate the sensitivity of the resulting pdf to the number of terms in the Edgeworth expansion. | Given information on a finite number of cumulants, we shall investigate the sensitivity of the resulting pdf to the number of terms in the Edgeworth expansion. |
This sensitivity has been alluded to by several works in the literature (???),, though we believe that our analysis goes beyond those works. | This sensitivity has been alluded to by several works in the literature \citep{juszkiewicz,loverde,desjacques}, though we believe that our analysis goes beyond those works. |
In particular, we shall argue that the truncated series cannot be used to deduce results for negative gwr, unless some higher-order non-Gaussianities are known. | In particular, we shall argue that the truncated series cannot be used to deduce results for negative $\gnl$, unless some higher-order non-Gaussianities are known. |
In this paper, we shall be using the form of the Edgeworth series given by ?,, who gave a method of calculating the Edgeworth series to arbitrarily high order. | In this paper, we shall be using the form of the Edgeworth series given by \cite{petrov}, who gave a method of calculating the Edgeworth series to arbitrarily high order. |
Given a non-Gaussian pdf with zero mean and variance 0%, we can express its deviation from Gaussianity as a product of the normal distribution and a Taylor series in og: D(9,)0, where N(dp) is the normal distribution and the coefficients E, in the Taylor series are given by We now explain the various components of the coefficient − | Given a non-Gaussian pdf with zero mean and variance $\sigma_R^2$, we can express its deviation from Gaussianity as a product of the normal distribution and a Taylor series in $\sigma_R$ : _R) = where $N(\delta_R)$ is the normal distribution and the coefficients $E_s$ in the Taylor series are given by We now explain the various components of the coefficient . |
− Firstly, the sum is taken over all distinct sets of non-negative integers {k,,}*,_, satisfying the Diophantine equation +sks = We also define Next, the function H,,(v) is the Hermite polynomial of degree n. | Firstly, the sum is taken over all distinct sets of non-negative integers $\{k_m\}_{m=1}^{s}$ satisfying the Diophantine equation +sk_s = We also define Next, the function $H_n(\nu)$ is the Hermite polynomial of degree $n$. |
They can be obtained by the Rodrigues’ formula H,(v))= ου For example, Ho(v)=1 and Hi(v)=v. | They can be obtained by the Rodrigues' formula )= (-1)^n For example, $H_0(\nu)=1$ and $H_1(\nu)=\nu$. |
Higher order polynomials can be easily obtained via the recurrence relation vH)-nHs Note that if p(óg) is Gaussian, the cumulants of order >3 vanish identically, and so do the expansion coefficients(59)),, as one might expect. | Higher order polynomials can be easily obtained via the recurrence relation )-n Note that if $p(\delta_R)$ is Gaussian, the cumulants of order $\geq3$ vanish identically, and so do the expansion coefficients, as one might expect. |
The Edgeworth expansion takes, as input, a sequence of cumulants {S,} which are combined with polynomials of various degrees up to order N. | The Edgeworth expansion takes, as input, a sequence of cumulants $\{S_n\}$ which are combined with polynomials of various degrees up to order $N$. |
Therefore, when using the Edgeworth expansion, there are two factors which will determine its accuracy, namely the number of available cumulants, and the order N. | Therefore, when using the Edgeworth expansion, there are two factors which will determine its accuracy, namely 1) the number of available cumulants, and 2) the order $N$. |
These two issues are separate in the sense that it is possible 1)to expand the Edgeworth series to arbitrarily2) high order given a limited number of cumulants. | These two issues are separate in the sense that it is possible to expand the Edgeworth series to arbitrarily high order given a limited number of cumulants. |
Boththeseissues must be analysed to properly monitor the sources of error. | Boththeseissues must be analysed to properly monitor the sources of error. |
vertical—required to model them is currently prohibitive. | vertical—required to model them is currently prohibitive. |
The waves important to the large-scale extrasolar planet atmospheric circulation have horizontal length scales ranging from approximately ~10° m to ~10? m and vertical wavelengths as small as ~10! m. Waves with periods of few hours can carry significant momentum and energy flux vertically. but the sources of these waves include processes that are not included or resolvable in current circulation models. | The waves important to the large-scale extrasolar planet atmospheric circulation have horizontal length scales ranging from approximately $\sim\! 10^5$ m to $\sim\! 10^7$ m and vertical wavelengths as small as $\sim\! 10^4$ m. Waves with periods of few hours can carry significant momentum and energy flux vertically, but the sources of these waves include processes that are not included or resolvable in current circulation models. |
The difficulty with representing gravity waves in GCMs exists even for the GCMs of the Earth. | The difficulty with representing gravity waves in GCMs exists even for the GCMs of the Earth. |
For example. the parameterization for convection is not aimed at producing realistic. gravity waves (Collinsetal.2004). | For example, the parameterization for convection is not aimed at producing realistic gravity waves \citep{Collins2004}. |
.. However. not representing gravity waves can affect the accuracy of the GCMs. | However, not representing gravity waves can affect the accuracy of the GCMs. |
The lack of gravity wave drag can lead to the overestimation of wind speeds. resulting in faster and narrower jets than observed (McLandress1998). | The lack of gravity wave drag can lead to the overestimation of wind speeds, resulting in faster and narrower jets than observed \citep{McLandress1998}. |
. Further. gravity waves introduce turbulence with subsequent mixing c£ond thermal transport (Fritts&Alexander2003). | Further, gravity waves introduce turbulence with subsequent mixing and thermal transport \citep{Fritts2003}. |
. This leads to greater homogenization of the atmosphere with a reduction in. for example. temperature gradients. | This leads to greater homogenization of the atmosphere with a reduction in, for example, temperature gradients. |
Gravity waves also interact with planetary waves. playing a role in important transient phenomena (such as sudden stratospheric warming). | Gravity waves also interact with planetary waves, playing a role in important transient phenomena (such as sudden stratospheric warming). |
In the absence of gravity waves. these phenomena are not accurately modeled (Richteretal.20100. | In the absence of gravity waves, these phenomena are not accurately modeled \citep{Richter2010}. |
There are many parametrization schemes currently incorporated or proposed for general circulation modeling AcLandress1998). | There are many parametrization schemes currently incorporated or proposed for general circulation modeling \citep{McLandress1998}. |
. In all of the schemes. the basic components are 1) specification of the characteristic of the waves at the source level. 2) wave propagation and evolution as a function of altitude. and 3) effects on and by the atmosphere. | In all of the schemes, the basic components are 1) specification of the characteristic of the waves at the source level, 2) wave propagation and evolution as a function of altitude, and 3) effects on and by the atmosphere. |
All of them are essentially linear and one-dimensional. in. that waves only propagate vertically and that only vertical variation in the background influence the propagation. | All of them are essentially linear and one-dimensional, in that waves only propagate vertically and that only vertical variation in the background influence the propagation. |
As seen in this work. linear theory. still requires information such as the wave's phase speed c and wavenumber K. for example. | As seen in this work, linear theory still requires information such as the wave's phase speed $c$ and wavenumber $\mathbi{k}$, for example. |
A more complete theory would need spatial and temporal spectral information. | A more complete theory would need spatial and temporal spectral information. |
Intermittency is another crucial feature that would need to be taken into account. | Intermittency is another crucial feature that would need to be taken into account. |
The primary differences in various schemes pertain to the treatment of nonlinearity and specificity of wave dissipation mechanisms. | The primary differences in various schemes pertain to the treatment of nonlinearity and specificity of wave dissipation mechanisms. |
Currently. all global circulation models of hot Jupiters suggest the presence of a low number of zonal jets. | Currently, all global circulation models of hot Jupiters suggest the presence of a low number of zonal jets. |
However. all the models do not have the resolution required to adequately resolve gravity waves and are subject to all of the limitations described above. | However, all the models do not have the resolution required to adequately resolve gravity waves and are subject to all of the limitations described above. |
This issue has been previously raised by Choetal.(2008).. in which they advocate caution against quantitative interpretation. of current model results. | This issue has been previously raised by \citet{Cho2008a}, in which they advocate caution against quantitative interpretation of current model results. |
For example. without the inclusion of the wave effects discussed in this work. high jet speeds and precise eastward shift of putative “hot spots” can be questioned (e.g..Cooper2007) Gravity wave propagation and momentum and energy deposition are complicated by the environment in which the wave propagates. | For example, without the inclusion of the wave effects discussed in this work, high jet speeds and precise eastward shift of putative “hot spots” can be questioned \citep[e.g.,][]{Cooper2005,Knutson2007,Langton2007} Gravity wave propagation and momentum and energy deposition are complicated by the environment in which the wave propagates. |
For example. spatial variability of the background winds causes the wave to be refracted. reflected. focused. and ducted. | For example, spatial variability of the background winds causes the wave to be refracted, reflected, focused, and ducted. |
Additionally. temporal variability of the background winds cause the wave to alter its phase speed. | Additionally, temporal variability of the background winds cause the wave to alter its phase speed. |
Still further complications arise due to the wave's ability to generate turbulence. which can modify the source or serve as a secondary source. and the wave's interaction with the vortical (rotational) mode of the atmosphere. | Still further complications arise due to the wave's ability to generate turbulence, which can modify the source or serve as a secondary source, and the wave's interaction with the vortical (rotational) mode of the atmosphere. |
Many of these issues are as yet not well-understood and are currently areas of active research. | Many of these issues are as yet not well-understood and are currently areas of active research. |
In this work. we have emphasized only some of these issues. | In this work, we have emphasized only some of these issues. |
We have shown that gravity waves propagate and transport momentum and heat in the atmospheres of hot extrasolar planets and that the waves play an important role in the atmosphere. | We have shown that gravity waves propagate and transport momentum and heat in the atmospheres of hot extrasolar planets and that the waves play an important role in the atmosphere. |
They modify the circulation through exertirisi accelerations on the mean flow whenever the wave encounters a critical level or saturates. | They modify the circulation through exerting accelerations on the mean flow whenever the wave encounters a critical level or saturates. |
They also transport heat to the upper stratosphere and thermosphere. causing significant heating in these regions. | They also transport heat to the upper stratosphere and thermosphere, causing significant heating in these regions. |
Moreover. through ducting. they also provide a mechanism for transporting heat from the dayside of tidally synchronized planets. | Moreover, through ducting, they also provide a mechanism for transporting heat from the dayside of tidally synchronized planets. |
Before relying on GCMs for quantitative descriptions of hot extrasolar planet atmospheric circulations. further work needs to be performed to ensure that the effects of important sub-scale phenomena. such as the gravity waves discussed here. are accurately parameterized and included in the GCMs. | Before relying on GCMs for quantitative descriptions of hot extrasolar planet atmospheric circulations, further work needs to be performed to ensure that the effects of important sub-scale phenomena, such as the gravity waves discussed here, are accurately parameterized and included in the GCMs. |
The authors thank Heidar Thrastarson for generously sharing information from his simulation work on extrasolar planet atmospheric circulation, | The authors thank Heidar Thrastarson for generously sharing information from his simulation work on extrasolar planet atmospheric circulation. |
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