# Turbulent transport of radiation in the solar convective zone

I. Rogachevskii, N. Kleeorin
Department of Mechanical Engineering, Ben-Gurion University of Negev, POB 653,
8410530 Beer-Sheva, Israel
Nordita, KTH Royal Institute of Technology and Stockholm University, Hannes
Alfvens 12, 10691 Stockholm, Sweden

###### Abstract

A turbulent transport of radiation in the solar convective zone is
investigated. The mean-field equation for the irradiation intensity is
derived. It is shown that due to the turbulent effects, the effective
penetration length of radiation can be increased in several times in
comparison with the mean penetration length of radiation (defined as an
inverse mean absorption coefficient). Using the model of the solar convective
zone based on the mixing length theory, where the mean penetration length of
radiation is usually much smaller than the turbulent correlation length, it is
demonstrated that the ratio of the effective penetration length to the mean
penetration length of radiation increases in 2.5 times in the vicinity of the
solar surface. The main reason are the compressibility effects that become
important in the vicinity of the solar surface where temperature and density
fluctuations increase towards the solar surface, enhancing fluctuations of the
radiation absorption coefficient and increasing the effective penetration
length of radiation.

###### keywords:

Sun: interior — turbulence — radiative transfer

##  1 Introduction

Turbulent transport of temperature, particles and magnetic fields have been
studied analytically, in laboratory and field experiments and in numerical
simulations for more than a century (see, e.g., Monin & Yaglom, 1971, 1975;
McComb, 1990; Frisch, 1995; Lesieur, 2008; Davidson, 2013; Rogachevskii,
2021), however, some fundamental questions remain. This is particularly true
in applications to astrophysics, where the governing parameter values are too
extreme to be modelled either experimentally or numerically.

In astrophysical turbulent flows radiative transport can be affected by
turbulence. This effect is different for optically thick and optically thin
regimes of the radiative transport. For optically thick regime, the mean free
path of photons is much smaller in comparison with the typical scales of the
flow. In the opposite case, i.e., for optically thin regime, the mean free
path of photons is much larger than the fluid motion scales, so the photons
propagate over large distances before they are absorbed and re-emitted again
(see, e.g., Chandrasekhar, 1960; Mihalas & Mihalas, 1984; Apresyan & Kravtsov,
1996; Liou, 2002; Howell et al., 2010).

The relaxation time of small temperature perturbations by radiative diffusion
has been determined by Spiegel (1957), where a time-dependent equation for the
temperature field of a medium with deviations from radiative equilibrium has
been derived assuming that the medium is graylike and there are no internal
motions or compressional effects, and heat is exchanged only radiatively. It
has been shown that perturbations of small amplitude imposed on a homogeneous
medium decay exponentially in time, and the decay time depends on a
characteristic length of the perturbations (Spiegel, 1957).

Turbulent diffusion can be increased by a radiative diffusion, i.e., by a
photon diffusion. For instance, the decay rates of sinusoidal large-scale
temperature perturbations in the optically thick and thin regimes have been
determined by Brandenburg & Das (2021) using radiative hydrodynamic direct
numerical simulations of forced turbulence. It has been shown there that the
rate of decay increases with the wavenumber. However, this effect is much
weaker in comparison with the effect of the standard turbulent diffusion
(Brandenburg & Das, 2021).

In the present study, we investigate the turbulent transport of radiation in
the solar convective zone. We derive a mean-field equation for the irradiation
intensity, and show that the effective penetration length of radiation can be
increased by turbulence in several times in comparison with the mean
penetration length of radiation which is defined as an inverse mean absorption
coefficient. This effect has been tested using the model of the solar
convective zone (Spruit, 1974) based on the mixing length theory. According to
this model, the mean penetration length of radiation is much less than the
turbulent correlation length. We show that the ratio of the effective
penetration length to the mean penetration length of radiation increases in
2.5 times in the vicinity of the solar surface.

This paper is organized as follows. In Section 2, we discuss a general concept
of turbulent transport of radiation, and in Section 3 we derive the mean-field
radiation transport equation. In Section 4, we derive expression for the
effective penetration length of radiation in turbulent flows, which depends on
the ratio of fluctuations of the radiation absorption coefficient to the mean
penetration length of radiation. To calculate the effective penetration length
of radiation, in Section 4 we determine fluctuations of the radiation
absorption coefficient which are caused by fluctuations of fluid temperature
and density, and the temperature-density correlations. In Section 5, we apply
obtained results to the solar convective zone. Finally, in Section 6, we
discuss our results and draw conclusions. In Appendix A we derive expressions
for the level of temperature and density fluctuations as well as
temperature–density correlations which allow us to determine fluctuations of
the radiation absorption coefficient.

##  2 General concept of turbulent transport of radiation

In solar and stellar convective zones, the convective transport of energy is
more effective than the radiative transfer. The Schwarzschild criterion for
the onset of convection does not take into account an effect of turbulence on
radiative transfer. However, the absorption coefficient of radiation depends
on temperature and density, and there are strong fluctuations of the fluid
temperature and density in solar and stellar convective zones. These
fluctuations affect the absorption coefficient of radiation, and therefore
they affect turbulent transfer of radiation.

In solar and stellar convective zones, the characteristic times of turbulent
motions are much larger than the radiation time, and the integral turbulent
scales are much larger than the mean penetration length of radiation defined
as the inverse absorption coefficient of radiation. The latter implies that
turbulent eddies are optically thick, and inhomogeneities in the fluid
temperature and density can strongly affect the radiation transfer.

To describe the radiation transfer, we use the radiation transport equation.
This equation represents a steady state version of the equation for
electromagnetic energy transfer, since the time of photon propagation (the
radiation time) is very short. This equation is characterized by the
absorption coefficient of radiation and the black-body radiation intensity of
the gas. The radiative transport equation for the intensity
$I({\bm{r}},\hat{\bm{s}},\omega)$ reads (see, e.g., Chandrasekhar, 1960;
Mihalas & Mihalas, 1984; Apresyan & Kravtsov, 1996; Liou, 2002; Howell et al.,
2010):

$\displaystyle\left(\hat{\bm{s}}{\bm{\cdot}}{\bm{\nabla}}\right)I({\bm{r}},\hat{\bm{s}},\omega)=-\kappa({\bm{r}},\omega)\left[I-I_{b}(T,\omega)\right],$
(1)

where ${\bm{r}}$ is the position vector, $\hat{\bm{s}}={\bm{k}}/k$ is the unit
vector in the direction of radiation, ${\bm{k}}$ is the wave vector,
$\kappa({\bm{r}},\omega)=\rho\,\kappa_{\rm op}$ is the absorption coefficient
of gas, $\kappa_{\rm op}=\kappa_{0}\,\rho^{a}\,T^{b}$ is the opacity of the
gas, $T$ and $\rho$ are the gas temperature and density, $I_{b}(T,\omega)$ is
the black-body radiation intensity of the gas, and $\omega$ is the radiation
frequency. Here we take into account the radiation absorption in gases and
neglect the radiation scattering in gases. The function $I_{b}(T,\omega)$ in a
local thermodynamic equilibrium is given by

$\displaystyle
I_{b}(T,\omega)={\hbar\omega^{3}\over\pi^{2}c^{3}}\,\left[\exp\left({\hbar\omega\over
k_{\rm B}T}\right)-1\right]^{-1},$ (2)

where $\hbar$ is Planck’s constant, $c$ is the speed of light and $k_{\rm B}$
is the Boltzman constant. The integral $\int
I_{b}(T,\omega)\,d\omega\propto\sigma T^{4}$ yields the Stefan-Boltzmann law.

Our goal is to derive effective radiation transport equation with effective
transport coefficients: effective absorption coefficient of radiation and the
effective source of the radiation intensity. To take into account the
turbulence effects, we apply a mean-field approach and average the radiation
transport equation (1) over ensemble of fluctuations. In the framework of the
mean-field approach, all quantities are decomposed into the mean and
fluctuating parts: $I=\overline{I}+I^{\prime}$,
$I_{b}=\overline{I}_{b}+I^{\prime}_{b}$ and
$\kappa=\overline{\kappa}+\kappa^{\prime}$. We adopt the Reynolds averaging,
where $\overline{I}=\langle I\rangle$, $\overline{I}_{b}=\langle
I_{b}\rangle$, $\overline{\kappa}=\langle\kappa\rangle$ are the mean fields,
and $I^{\prime},I^{\prime}_{b},\kappa^{\prime}$ are the fluctuating fields
with zero mean, and the angular brackets denote ensemble averaging. To derive
the mean-field radiation transport equation, we adopt a method applied by
Kliorin et al. (1989); Liberman et al. (2017, 2018).

The obtained mean-field equation contains the correlation function for
fluctuations of the absorption coefficient of radiation $\kappa^{\prime}$ and
the radiation intensity $I^{\prime}$, i.e.,
$\langle\kappa^{\prime}\,I^{\prime}\rangle$. This correlation is due to
fluctuations of temperature and density. This equation also contains the
correlation function for fluctuations of the absorption coefficient of
radiation $\kappa^{\prime}$ and the black-body radiation intensity of the gas
$\langle\kappa^{\prime}\,I^{\prime}_{b}\rangle$ due to fluctuations of
temperature.

To determine the correlation functions,
$\langle\kappa^{\prime}\,I^{\prime}\rangle$ and
$\langle\kappa^{\prime}\,I^{\prime}_{b}\rangle$, we derive equation for
fluctuations of the radiation intensity $I^{\prime}$ by subtracting the
obtained mean-field equation from the radiation transport equation (1). Since
the equation for fluctuations of the radiation intensity $I^{\prime}$ is a
linear equation, we solve this equation exactly. However, the solution of this
equation is nonlinear in fluctuations of $\kappa^{\prime}$. This causes
appearance of the high-order moments in fluctuations of $\kappa^{\prime}$ in
the expression for the correlation function
$\langle\kappa^{\prime}\,I^{\prime}\rangle$. We assume that fluctuations of
$\kappa^{\prime}$ are essentially less than the mean absorption coefficient of
radiation. This allows us to obtain the closed results.

The main expected result of this study is that the derived mean-field equation
for the radiation transfer with the effective transport coefficients yields
the effective penetration length of radiation. When the effective penetration
length of radiation is larger than the mean penetration length of radiation,
the absorbtion coefficient decreases and an observer can see more deeper
layers inside the stars. The reasons for the increase of the effective
penetration length of radiation in turbulent flows are caused by the
correlation between fluctuations of the radiation absorption coefficient
$\kappa^{\prime}$ and fluctuations of the irradiation intensity $I^{\prime}$.
We show below that this correlation function should be negative, because an
increase of the absorption of radiation decreases the radiation intensity and
wise versa. We also demonstrate in this study that this effect is essential in
the vicinity of the solar surface.

##  3 Mean-field radiation transport equation

In this section we derive the mean-field radiation transport equation.
Ensemble averaging of equation (1) yields the equation for the mean radiation
intensity $\overline{I}$:

$\displaystyle\left(\hat{\bm{s}}{\bm{\cdot}}{\bm{\nabla}}\right)\overline{I}=-\overline{\kappa}\left(\overline{I}-\overline{I}_{b}\right)-\langle\kappa^{\prime}\,I^{\prime}\rangle+\langle\kappa^{\prime}\,I^{\prime}_{b}\rangle.$
(3)

This equation contains unknown correlation functions,
$\langle\kappa^{\prime}\,I^{\prime}\rangle$ and
$\langle\kappa^{\prime}\,I^{\prime}_{b}\rangle$. To determine these
correlation functions, we derive equation for fluctuations of the radiation
intensity $I^{\prime}$. To this end, we subtract the mean-field radiation
transport equation (3) from equation (1), so that the equation for
fluctuations of $I^{\prime}$ reads:

$\displaystyle\left(\hat{\bm{s}}{\bm{\cdot}}{\bm{\nabla}}+\overline{\kappa}+\kappa^{\prime}\right)I^{\prime}({\bm{r}},\hat{\bm{s}})=I_{\rm
source},$ (4)

where the source term $I_{\rm source}$ is given by

$\displaystyle I_{\rm source}$ $\displaystyle=$
$\displaystyle-\kappa^{\prime}\,\left(\overline{I}-\overline{I}_{b}\right)+\langle\kappa^{\prime}\,I^{\prime}\rangle+\left(\overline{\kappa}+\kappa^{\prime}\right)I^{\prime}_{b}-\langle\kappa^{\prime}\,I^{\prime}_{b}\rangle.$

The solution of equation (4) reads

$\displaystyle I^{\prime}({\bm{r}},\hat{\bm{s}})$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}\exp\left[-\left|\int_{s^{\prime}}^{s}\left[\overline{\kappa}+\kappa^{\prime}(s^{\prime\prime})\right]\,ds^{\prime\prime}\right|\right]$
(6) $\displaystyle\times I_{\rm source}(s^{\prime})\,ds^{\prime},$

where $s={\bm{r}}{\bm{\cdot}}\hat{\bm{s}}$. This solution is nonlinear in
fluctuations of $\kappa^{\prime}$. The latter causes appearance of the high-
order moments in fluctuations of $\kappa^{\prime}$ in the expression for the
correlation function $\langle\kappa^{\prime}\,I^{\prime}\rangle$. The high-
order moments are much less than the lower-order moments, because
$\kappa^{\prime}\ll\overline{\kappa}$. This allows us to expand the function,
$\exp\left[-\int_{s^{\prime}}^{s}\kappa^{\prime}(s^{\prime\prime})\,ds^{\prime\prime}\right]$,
in equation (6) in Taylor series:

$\displaystyle\exp\left[-\int_{s^{\prime}}^{s}\kappa^{\prime}(s^{\prime\prime})\,ds^{\prime\prime}\right]=1-\int_{s^{\prime}}^{s}\kappa^{\prime}(s^{\prime\prime})\,ds^{\prime\prime}+{\rm
O}\left(\kappa^{\prime 2}\right).$ (7)

Therefore, equation (6) can be rewritten as:

$\displaystyle I^{\prime}({\bm{r}},\hat{\bm{s}})$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}I_{\rm
source}(s^{\prime})\,\exp\left(-\overline{\kappa}|s-s^{\prime}|\right)$ (8)
$\displaystyle\times\left(1-\int_{s^{\prime}}^{s}\kappa^{\prime}(s^{\prime\prime})\,ds^{\prime\prime}\right)\,ds^{\prime}+{\rm
O}\left(\kappa^{\prime 2}\right).$

Multiplying equation (8) by $\kappa^{\prime}$ and averaging over the ensemble,
we obtain expression for the one-point correlation function
$\langle\kappa^{\prime}\,I^{\prime}\rangle$:

$\displaystyle\langle\kappa^{\prime}\,I^{\prime}\rangle\,\biggl{[}1+\int_{-\infty}^{\infty}\biggl{(}\int_{s^{\prime}}^{s}\langle\kappa^{\prime}(s)\kappa^{\prime}(s^{\prime\prime})\rangle\,ds^{\prime\prime}\biggr{)}$
$\displaystyle\quad\times\exp\biggl{(}-\overline{\kappa}|s-s^{\prime}|\biggr{)}\,\,ds^{\prime}\biggr{]}=-\biggl{[}\int_{-\infty}^{\infty}\langle\kappa^{\prime}(s)\kappa^{\prime}(s^{\prime})\rangle$
$\displaystyle\quad\times\exp\biggl{(}-\overline{\kappa}|s-s^{\prime}|\biggr{)}\,\,ds^{\prime}\biggr{]}\,\Big{(}\overline{I}-\overline{I}_{b}\Big{)},$
(9)

where we neglect the third-order and higher-order moments in fluctuations of
$\kappa^{\prime}$. Equation (9) can be rewritten as

$\displaystyle\langle\kappa^{\prime}\,I^{\prime}\rangle=-\overline{\kappa}\Big{(}\overline{I}-\overline{I}_{b}\Big{)}\,{2\overline{\kappa}J_{1}\over
1+2\overline{\kappa}J_{2}},$ (10)

where the integrals $J_{1}$ and $J_{2}$ in equation (10) are defined as:

$\displaystyle J_{1}=\int_{0}^{\infty}\Phi(Z)\exp(-\overline{\kappa}Z)\,dZ,$
(11) $\displaystyle
J_{2}=\overline{\kappa}\int_{0}^{\infty}\left(\int_{0}^{Z}\Phi(Z^{\prime})\,dZ^{\prime}\right)\,\exp(-\overline{\kappa}Z)\,dZ,$
(12)

the function $\Phi(Z)$ is defined as
$\Phi(Z)=\langle\kappa^{\prime}(s)\kappa^{\prime}(s^{\prime})\rangle$ and
$Z=|s-s^{\prime}|$.

Substituting equation (10) into the mean-field equation (3), we arrive at the
mean-field radiation transport equation as

$\displaystyle\left(\hat{\bm{s}}{\bm{\cdot}}{\bm{\nabla}}\right)\overline{I}=-\kappa_{\rm
eff}\,\left(\overline{I}-I_{b}^{\rm eff}\right),$ (13)

where the effective absorption coefficient $\kappa_{\rm eff}$ is given by

$\displaystyle\kappa_{\rm
eff}=\overline{\kappa}\,\left(1-{2\overline{\kappa}J_{1}\over
1+2\overline{\kappa}J_{2}}\right),$ (14)

and the effective radiation intensity is

$\displaystyle I_{b}^{\rm
eff}=\overline{I}_{b}+{\langle\kappa^{\prime}\,I^{\prime}_{b}\rangle\over\kappa_{\rm
eff}}.$ (15)

The function $\overline{I}_{b}$ is expanded in Taylor series as

$\displaystyle\overline{I}_{b}=\left[I_{b}+{\langle\theta^{2}\rangle\over
2}\,{\partial^{2}I_{b}\over\partial T^{2}}\right]_{T=\overline{T}},$ (16)

where the fluid temperature is decomposed into the mean $\overline{T}$ and
fluctuating $\theta$ parts: $T=\overline{T}+\theta$. Solution of the mean-
field radiation transport equation (13) for the mean irradiation intensity
$\overline{I}({\bm{r}},\hat{\bm{s}},\omega)$ is given by

$\displaystyle\overline{I}({\bm{r}},\hat{\bm{s}},\omega)=\int_{-\infty}^{\infty}I_{b}^{\rm
eff}({\bm{r}}^{\prime},\omega)\,\exp\left[-\left|\tau({\bm{r}},{\bm{r}}^{\prime},\hat{\bm{s}})\right|\right]\,\hat{\bm{s}}{\bm{\cdot}}\,d{\bm{r}}^{\prime},$
(17)

where
$\tau({\bm{r}},{\bm{r}}^{\prime},\hat{\bm{s}})=\int_{{\bm{r}}}^{{\bm{r}}^{\prime}}\kappa_{\rm
eff}({\bm{r}}^{\prime\prime})\,\hat{\bm{s}}{\bm{\cdot}}\,d{\bm{r}}^{\prime\prime}$
is the optical depth.

##  4 Effective penetration length of radiation and fluctuations of absorption
coefficient

In this section we determine the effective penetration length of radiation in
turbulent flows, defined as $L_{\rm eff}=\kappa_{\rm eff}^{-1}$. Since the
main contribution to the second moment
$\langle\kappa^{\prime}(s)\kappa^{\prime}(s^{\prime})\rangle$ of fluctuations
of the absorption coefficient is from the integral scale of turbulence
$\ell_{0}$, we assume that this correlation functions have the form:

$\displaystyle\langle\kappa^{\prime}(s)\kappa^{\prime}(s^{\prime})\rangle=\left\langle\kappa^{\prime\,2}\right\rangle\,\exp\left(-{|s-s^{\prime}|\over\ell_{0}}\right).$
(18)

Using equations (11)–(12) and (18), we calculate the integrals $J_{1}$ and
$J_{2}$ as

$\displaystyle
J_{1}=J_{2}={\left\langle\kappa^{\prime\,2}\right\rangle\over\overline{\kappa}^{3}}\,\left(1+{L_{r}\over\ell_{0}}\right)^{-1},$
(19)

where $L_{r}=\overline{\kappa}^{\,-1}$ characterises the mean penetration
length of radiation in turbulent flow. Therefore, equations (14) and (19)
allow us to determine the effective penetration length $L_{\rm
eff}=\kappa_{\rm eff}^{-1}$ of radiation in a turbulent flow as

$\displaystyle L_{\rm
eff}=L_{r}\,\left[1+{2\,\left\langle\kappa^{\prime\,2}\right\rangle\over\overline{\kappa}^{2}}\,\left(1+{L_{r}\over\ell_{0}}\right)^{-1}\right].$
(20)

We consider two limiting cases:

(i) $\ell_{0}\ll L_{r}$, the effective penetration length $L_{\rm eff}$ is

$\displaystyle L_{\rm
eff}=L_{r}\,\left[1+{2\,\left\langle\kappa^{\prime\,2}\right\rangle\over\overline{\kappa}^{2}}\,{\ell_{0}\over
L_{r}}\right],$ (21)

(ii) $\ell_{0}\gg L_{r}$, the effective penetration length $L_{\rm eff}$ is

$\displaystyle L_{\rm
eff}=L_{r}\,\left[1+{2\,\left\langle\kappa^{\prime\,2}\right\rangle\over\overline{\kappa}^{2}}\right].$
(22)

Equation (22) implies that for $\ell_{0}\gg L_{r}$, the effective penetration
length $L_{\rm eff}$ can increase in 3 times in comparison with the mean
penetration length $L_{r}$ of radiation due to the turbulence effects when
$\left\langle\kappa^{\prime\,2}\right\rangle\sim\overline{\kappa}^{2}$.

The mechanism of increase of the effective penetration length $L_{\rm eff}$ in
turbulent flows is related to the correlation between fluctuations of the
radiation absorption coefficient $\kappa^{\prime}$ and fluctuations of the
irradiation intensity $I^{\prime}$. The correlation
$\langle\kappa^{\prime}\,I^{\prime}\rangle$ is negative because an increase of
the absorption of radiation decreases the radiation intensity and wise versa.
Fluctuations of the radiation absorption coefficient are caused by
fluctuations of fluid temperature and density in turbulent flow.

Now we determine fluctuations of the radiation absorption coefficient. For
simplicity, we assume that the opacity of gas is $\kappa_{\rm
op}=\kappa_{0}\,\rho^{a}\,T^{b}$, so that the absorption coefficient of gas is
$\kappa=\rho\,\kappa_{\rm op}=\kappa_{0}\,\rho^{a+1}\,T^{b}$. According to the
Schwarzschild stability criterion, the case $a=1$ and $b=0$ corresponds to the
marginally stable regime, while the case $a=1$ and $b=1$ corresponds to the
unstable regime (Barekat & Brandenburg, 2014). The equation
$\kappa=\kappa_{0}\,\rho^{a+1}\,T^{b}$ allows us to determine the ratio of
fluctuations of the absorption coefficient $\kappa^{\prime}$ to the mean value
of $\overline{\kappa}$ as

$\displaystyle{\kappa^{\prime}\over\overline{\kappa}}=(a+1)\,{\rho^{\prime}\over\overline{\rho}}+b\,{\theta\over\overline{T}},$
(23)

where $\rho^{\prime}$ are density fluctuations and $\overline{\rho}$ is the
mean fluid density.

Using equation (23), we determine the correlation function
$\langle\kappa^{\prime}\,I^{\prime}_{b}\rangle$ as:

$\displaystyle\langle\kappa^{\prime}\,I^{\prime}_{b}\rangle=\overline{\kappa}\,\left[(a+1)\,{\langle\rho^{\prime}\theta\rangle\over\overline{\rho}}+b\,{\langle\theta^{2}\rangle\over\overline{T}}\right]\,\left({\partial
I_{b}\over\partial T}\right)_{T=\overline{T}},$ (24)

where we take into account that $I^{\prime}_{b}=\theta\,(\partial
I_{b}/\partial T)_{T=\overline{T}}$. To find the effective penetration length
of radiation, we determine the level of fluctuations
$\left\langle\kappa^{\prime\,2}\right\rangle$ of the absorption coefficient as

$\displaystyle{\left\langle\kappa^{\prime\,2}\right\rangle\over\overline{\kappa}^{2}}$
$\displaystyle=$
$\displaystyle(a+1)^{2}\,{\left\langle\rho^{\prime\,2}\right\rangle\over\overline{\rho}^{2}}+b^{2}\,{\left\langle\theta^{2}\right\rangle\over\overline{T}^{2}}+2b\,(a+1)\,{\left\langle\theta\,\rho^{\prime}\right\rangle\over\overline{\rho}\,\overline{T}}.$

The intensity of temperature fluctuations in a developed compressible
turbulence for large Péclet and Reynolds numbers is given by

$\displaystyle{\left\langle\theta^{2}\right\rangle\over\overline{T}^{2}}$
$\displaystyle=$ $\displaystyle 8\,f_{c}\,(\gamma-1)^{2}\left({\sigma_{c}\over
1+\sigma_{c}}\right)^{3}\,\biggl{[}1-{(\lambda\,\ell_{0})^{2}\over
9}\biggr{]}$ (26) $\displaystyle+{8\over
9}\,\ell_{0}^{2}\,\biggl{[}{{\bm{\nabla}}\overline{T}\over\overline{T}}+(\gamma-1)\,{\bm{\lambda}}\biggr{]}^{2},$

(see Appendix A), where $\gamma=c_{\rm p}/c_{\rm v}$ is the ratio of specific
heats, ${\bm{\lambda}}=-{\bm{\nabla}}\ln\overline{\rho}$ characterizes the
inhomogeneity of the mean fluid density, the parameter

$\displaystyle\sigma_{c}={\left\langle(\mbox{\boldmath$\nabla$}\cdot\,{\bm{u}})^{2}\right\rangle\over\left\langle(\mbox{\boldmath$\nabla$}\times{\bm{u}})^{2}\right\rangle}$
(27)

is the degree of compressibility of the turbulent velocity field ${\bm{u}}$
and $\ell_{0}$ is the integral scale of turbulence. The function
$f_{c}(q,q_{c},\sigma_{c})$ depends on the degree of compressibility and the
exponents of spectra $q$ and $q_{c}$ for the incompressible and compressible
parts of velocity fluctuations (see Appendix A):

$\displaystyle f_{c}$ $\displaystyle=$ $\displaystyle{q_{c}-1\over
3q_{c}-5}+{2(q_{c}-1)\over\sigma_{c}(q+2q_{c}-5)}+{q_{c}-1\over\sigma_{c}^{2}(2q+q_{c}-5)}.$

Equation (26) is different from that derived by Rogachevskii & Kleeorin
(2021). In this study, we take into account a strong density stratification.
The latter is important in view of applications to the solar convective zone,
where the fluid density in radial direction is changed by seven orders of
magnitude. We also neglect the gradient of the turbulent diffusion that is
changed very slowly in the solar convective zone. The first term in the right
hand side of equation (26) determines a compressibility contribution of
velocity fluctuations to temperature fluctuations, while the second term in
equation (26) is proportional to the squared gradient of the mean entropy.

The level of fluid density fluctuations in a developed compressible turbulence
for large Péclet and Reynolds numbers is given by

$\displaystyle{\left\langle\rho^{\prime\,2}\right\rangle\over\overline{\rho}^{2}}=8\,f_{c}\,\left({\sigma_{c}\over
1+\sigma_{c}}\right)^{3}\,\biggl{[}1-{(\lambda\,\ell_{0})^{2}\over
9}\biggr{]},$ (29)

and the cross correlations $\left\langle\theta\,\rho^{\prime}\right\rangle$ is

$\displaystyle{\left\langle\theta\,\rho^{\prime}\right\rangle\over\overline{\rho}\,\overline{T}}$
$\displaystyle=$ $\displaystyle 8\,f_{c}\,(\gamma-1)\,\left({\sigma_{c}\over
1+\sigma_{c}}\right)^{3}\,\biggl{[}1-{(\lambda\,\ell_{0})^{2}\over
9}\biggr{]}.$ (30)

Equations (26), (29) and (30) are valid for small $\sigma_{c}<1$. The latter
condition is typical for developed turbulence and turbulent convection.
Derivation of equations (26), (29) and (30) is given in Appendix A.

Therefore, the level of fluctuations
$\left\langle\kappa^{\prime\,2}\right\rangle$ of the absorption coefficient is

$\displaystyle{\left\langle\kappa^{\prime\,2}\right\rangle\over\overline{\kappa}^{2}}=8\,f_{c}\,\left({\sigma_{c}\over
1+\sigma_{c}}\right)^{3}\,\biggl{[}1-{(\lambda\,\ell_{0})^{2}\over
9}\biggr{]}\,\biggl{[}a+1$
$\displaystyle\quad+b\,(\gamma-1)\biggr{]}^{2}+{8\over
9}\,\ell_{0}^{2}\,b^{2}\,\biggl{[}{{\bm{\nabla}}\overline{T}\over\overline{T}}+(\gamma-1)\,{\bm{\lambda}}\biggr{]}^{2}.$
(31)

For $a=1$ and $b=1$, the level of fluctuations
$\left\langle\kappa^{\prime\,2}\right\rangle$ of the absorption coefficient is
given by

$\displaystyle{\left\langle\kappa^{\prime\,2}\right\rangle\over\overline{\kappa}^{2}}=8\,f_{c}\,\left({\sigma_{c}\over
1+\sigma_{c}}\right)^{3}\,\biggl{[}1-{(\lambda\,\ell_{0})^{2}\over
9}\biggr{]}\,(\gamma+1)^{2}$ $\displaystyle\quad+{8\over
9}\,\ell_{0}^{2}\,b^{2}\,\biggl{[}{{\bm{\nabla}}\overline{T}\over\overline{T}}+(\gamma-1)\,{\bm{\lambda}}\biggr{]}^{2}.$
(32)

Note that for nearly isentropic flows where
${\bm{\nabla}}\ln\overline{T}\approx(\gamma-1)\,{\bm{\nabla}}\ln\overline{\rho}$,
the last term in equations (31)–(32) is small. This term is proportional to
the gradient of the mean entropy ${\bm{\nabla}}\overline{S}=c_{\rm
v}[{\bm{\nabla}}\ln\overline{T}-(\gamma-1)\,{\bm{\nabla}}\ln\overline{\rho}]$.

##  5 Application to the solar convective zone

In this section, we apply the obtained results to the solar convective zone.
We use the model of the solar convective zone by Spruit (1974) based on the
mixing length theory. According to this model, the mean penetration length of
radiation is much less than the turbulent correlation length. Indeed, in Figs.
1–3 we show the radial profiles of the ratio of the mean penetration length of
radiation to the integral scale of turbulence $L_{r}(r)/\ell_{0}$, the ratio
$\ell_{\rm m}/H_{\rho}$ of the mixing length $\ell_{\rm m}$ to the density
stratification length $H_{\rho}=\lambda^{-1}$ and the Reynolds number
Re$(r)=u_{0}\,\ell_{0}/\nu$ for the solar convective zone based on the model
by Spruit (1974). The radius $r$ is measured in units of the solar radius
$R_{\odot}$. The mixing length $\ell_{\rm m}$ is identified with the size of
the solar granulations, while the ratio $\ell_{\rm m}/\ell_{0}=5-7$ is
justified by the results of analytical study (Elperin et al., 2002, 2006) and
laboratory experiments (Bukai et al., 2009), which show that the integral
scale $\ell_{0}$ of the turbulent convection is smaller in 5 - 7 times in
comparison with the size of the large-scale circulations. These turbulent
parameters increase towards the solar surface.

Figure 1:  The radial profile of the ratio $L_{r}/\ell_{0}$ for the solar
convective zone. Figure 2:  The profile of the ratio $\ell_{\rm m}/H_{\rho}$
of the mixing length $\ell_{\rm m}$ to the density stratification length
$H_{\rho}$ versus $r/R_{\odot}$ that is based on the model of the solar
convective zone. Figure 3:  The radial profile of the Reynolds number
Re$(r)=u_{0}\,\ell_{0}/\nu$ for the solar convective zone.

To determine the effective penetration length of radiation, we estimate the
degree of compressibility of the turbulent fluid flow for small Mach numbers
as (Rogachevskii & Kleeorin, 2021)

$\displaystyle\sigma_{c}\sim{\rm Ma}^{5}\,{\rm Re}^{1/4},$ (33)

where ${\rm Ma}=u_{0}/c_{\rm s}$ is the Mach number,
$u_{0}=\left\langle{\bm{u}}^{2}\right\rangle^{1/2}$ and $c_{\rm
s}=(\gamma\overline{P}/\overline{\rho})^{1/2}$ is the sound speed, ${\rm
Re}=u_{0}\,\ell_{0}/\nu$ is the Reynolds number and $\nu$ is the kinematic
viscosity. The estimate (33) is obtained assuming that the effect of
compressibility to the viscous heating $\overline{J}_{\nu}^{\,({\rm c})}$ is
of the order of the radiative wave energy density $E_{\rm w}$. In particular,
turbulence can generate acoustic waves, and the rate of the energy radiated by
the acoustic waves per unit mass for small Mach numbers is given by
(Lighthill, 1952, 1954; Proudman, 1952)

$\displaystyle E_{\rm
w}={\left\langle{\bm{u}}^{2}\right\rangle\over\tau_{0}}\,{\rm Ma}^{5},$ (34)

where $\tau_{0}=\ell_{0}/u_{0}$ is the turbulent correlation time. The
compressibility contribution $\overline{J}_{\nu}^{\,({\rm c})}$ to the rate of
the viscous heating is (Rogachevskii & Kleeorin, 2021)

$\displaystyle\overline{J}_{\nu}^{\,({\rm
c})}={\left\langle{\bm{u}}^{2}\right\rangle\over\tau_{0}}\,{\sigma_{c}\over
1+\sigma_{c}}\,{\rm Re}^{-1/4}.$ (35)

Equations (34)–(35) yield the estimate (33) for the degree of compressibility
$\sigma_{c}$ for small Mach numbers.

In Figs. 4–5 we show the radial profiles of the Mach number
Ma$(r)=u_{0}/c_{\rm s}$ and the degree of compressibility $\sigma_{c}$ of the
fluid velocity field for the solar convective zone based on the model by
Spruit (1974). The degree of compressibility $\sigma_{c}$ increases to the
surface because the decrease of the sound speed in the vicinity of the solar
surface. In Fig. 6 we plot the radial profile of the r.m.s. of temperature
fluctuations $\theta_{\rm rms}$ measured in the units of the mean temperature
$\overline{T}$ using the above parameters for the solar convective zone.
Temperature fluctuations increase towards to the solar surface due to the
compressibility effects.

Similar behaviour is observed for fluid density fluctuations
$\left\langle\rho^{\prime\,2}\right\rangle$ and the temperature-density
correlations $\left\langle\theta\,\rho^{\prime}\right\rangle$ [see equations
(29)–(30)]. In particular, these second moments enhance towards to the solar
surface, resulting in increase of fluctuations of the radiation absorption
coefficient and the effective penetration length of radiation. This is seen in
Fig. 7, where we show the radial profile of the ratio of the turbulence
induced effective penetration length of radiation to the mean radiation
penetration length $L_{\rm eff}/L_{r}$ for the solar convective zone based on
the model by Spruit (1974). The ratio $L_{\rm eff}/L_{r}$ increases in 2.5
times in the vicinity of the solar surface.

Figure 4:  The radial profile of the Mach number Ma$(r)=u_{0}/c_{\rm s}$ for
the solar convective zone. Figure 5:  The radial profile of the degree of
compressibility $\sigma_{c}$ for the solar convective zone. Figure 6:  The
radial profile of the r.m.s. of temperature fluctuations $\theta_{\rm rms}$
measured in the units of the mean temperature $\overline{T}$ for the solar
convective zone. Figure 7:  The radial profile of the ratio $L_{\rm
eff}/L_{r}$ for $a=b=1$ for the solar convective zone.

##  6 Conclusions

We study a turbulent transport of radiation in the solar convective zone. To
this end, we derive a mean-field equation for the irradiation intensity and
show that due to the turbulent effects the effective penetration length of
radiation is increased in several times in comparison with the mean
penetration length of radiation that is defined as an inverse mean absorption
coefficient. To demonstrate this effect, we adopt a model of the solar
convective zone based on the mixing length theory. The mean penetration length
of radiation in this model is much smaller than the turbulent integral scale.
We have shown that the ratio of the effective penetration length of radiation
to the mean penetration length of radiation is increased in 2.5 times in the
vicinity of the solar surface.

This effect can be explained by the compressibility effects that become
important in the vicinity of the solar surface, so that the level of
temperature and density fluctuations is increased towards the solar surface.
It causes an increase of fluctuations of the radiation absorption coefficient
and the effective penetration length of radiation. Since the effective
penetration length of radiation is changed only in the vicinity of the solar
surface (at the depth $\sim 2000$ km), the effect of turbulence on the
radiation transport is not strong for the solar type stars. However, this
effect can be essential for cold stars (like M3-M5 stars), for which Mach
number is larger than that for the sun.

Data availability

There are no new data associated with this article.

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##  Appendix A Temperature and density fluctuations

In this Appendix we derive expression for the level of temperature
fluctuations $\left\langle\theta^{2}\right\rangle$, density fluctuations
$\left\langle\rho^{\prime\,2}\right\rangle$ as well as temperature–density
correlations $\left\langle\theta\,\rho^{\prime}\right\rangle$ using the method
described by Rogachevskii et al. (2018); Rogachevskii & Kleeorin (2021);
Rogachevskii (2021). The temperature field $T(t,{\bm{r}})$ in a compressible
fluid velocity field ${\bf U}(t,{\bm{r}})$ is described by (Landau & Lifshits,
1987)

$\displaystyle{\partial T\over\partial t}+({\bf
U}\cdot{\bm{\nabla}})T+(\gamma-1)T({\bm{\nabla}}\cdot{\bm{U}})=D\Delta
T+J_{\nu},$ (36)

where $D$ is the molecular thermal conductivity, $\gamma=c_{\rm p}/c_{\rm v}$
is the ratio of specific heats and $J_{\nu}$ is the heating source due to a
viscous dissipation.

We study turbulent flows with large Reynolds $({\rm Re}=u_{0}\,\ell_{0}/\nu\gg
1)$ and Péclet $({\rm Pe}=u_{0}\,\ell_{0}/D\gg 1)$ numbers, where $u_{0}$ is
the characteristic turbulent velocity in the integral scale $\ell_{0}$ of
turbulence. Equations for the intensity of temperature fluctuations is derived
by means of the mean-field approach, where the temperature
$T=\overline{T}+\theta$, pressure $P=\overline{P}+p$, density
$\rho=\overline{\rho}+\rho^{\prime}$ and velocity
${\bm{U}}=\overline{\bm{U}}+{\bm{u}}$ are decomposed into mean and fluctuating
parts, with $\overline{T}=\langle T\rangle$ being the mean fluid temperature,
$\overline{P}=\langle P\rangle$ being the mean fluid pressure,
$\overline{\rho}=\langle\rho\rangle$ being the mean fluid density, and
$\overline{\bm{U}}=\langle{\bm{U}}\rangle$ being the mean fluid velocity. Here
$\theta$, $p$, $\rho^{\prime}$ and ${\bm{u}}$ are fluctuations of temperature,
pressure, density and velocity, respectively, and the angular brackets denote
an ensemble averaging. Application of the mean-field approach implies that
there is a separation of spatial ($\ell_{0}\ll H_{T}$) and temporal
($\tau_{0}\ll t_{T}$) scales, where $H_{T}$ and $t_{T}$ are the characteristic
spatial and temporal scales characterizing the variations of the mean
temperature field, and $\tau_{0}=\ell_{0}/u_{0}$.

Ensemble averaging of equation (36) yields the mean temperature field:

$\displaystyle{\partial\overline{T}\over\partial
t}+\mbox{\boldmath$\nabla$}{\bf\cdot}\left\langle\theta\,{\bm{u}}\right\rangle=-(\gamma-2)\,\left\langle\theta\,(\mbox{\boldmath$\nabla$}{\bf\cdot}{\bm{u}})\right\rangle+D\,\Delta\overline{T}+\overline{J}_{\nu},$
(37)

where $\left\langle\theta\,{\bm{u}}\right\rangle$ is the turbulent heat flux,
and $\overline{J}_{\nu}$ is the mean heating source caused by the viscous
dissipation of the turbulent kinetic energy. Here the case
$\overline{\bm{U}}=0$ is studied for simplicity. By means of equations (36)
and (37), we obtain equation for temperature fluctuations,
$\theta({\bm{x}},t)=T-\overline{T}$:

$\displaystyle{\partial\theta\over\partial t}+{\cal
Q}-D\Delta\theta=-({\bm{u}}{\bm{\cdot}}\mbox{\boldmath$\nabla$})\overline{T}-(\gamma-1)\,\overline{T}\,\mbox{\boldmath$\nabla$}{\bm{\cdot}}\,{\bm{u}},$
(38)

where ${\cal
Q}=\mbox{\boldmath$\nabla$}{\bm{\cdot}}\,(\theta{\bm{u}}-\langle{\bm{u}}\,\theta\rangle)+(\gamma-2)\,[\theta\,\mbox{\boldmath$\nabla$}{\bm{\cdot}}\,{\bm{u}}-\langle\theta\,\mbox{\boldmath$\nabla$}{\bm{\cdot}}\,{\bm{u}}\rangle]$
is the nonlinear term and temperature fluctuations are caused by the source
$-({\bm{u}}{\bm{\cdot}}\mbox{\boldmath$\nabla$})\overline{T}-(\gamma-1)\,\overline{T}\,\mbox{\boldmath$\nabla$}{\bm{\cdot}}\,{\bm{u}}$.
For simplicity we describe the effect of turbulence on the temperature field,
and neglect the feedback effect of the temperature on the turbulence.

We use two-point second-order correlation functions taking into account small-
scale properties of the turbulence, where the turbulent correlation time and
the turbulent kinetic energy spectrum are related via the Kolmogorov scalings
(Monin & Yaglom, 1971, 1975; McComb, 1990; Frisch, 1995). We adopt the multi-
scale approach (Roberts & Soward, 1975), and rewrite the two-point second-
order correlation functions as:

$\displaystyle\left\langle\theta({\bm{x}},t)\,\theta({\bm{y}},t)\right\rangle=\int\,d{\bm{k}}_{1}\,d{\bm{k}}_{2}\left\langle\theta({\bm{k}}_{1},t)\theta({\bm{k}}_{2},t)\right\rangle$
$\displaystyle\times\exp\big{[}i({\bm{k}}_{1}{\bm{\cdot}}{\bm{x}}+{\bm{k}}_{2}{\bm{\cdot}}{\bm{y}})\big{]}=\int\Theta^{(II)}({\bm{k}},{\bm{R}},t)\exp[i{\bm{k}}{\bm{\cdot}}{\bm{r}}]\,d{\bm{k}},$
(39)

where

$\displaystyle\Theta^{(II)}({\bm{k}},{\bm{R}},t)=\int\left\langle\theta({\bm{k}}_{1},t)\,\theta({\bm{k}}_{2},t)\right\rangle\,\exp[i{\bm{K}}{\bm{\cdot}}{\bm{R}}]\,d{\bm{K}},$
(40)

and we use large-scale variables: ${\bm{R}}=({\bm{x}}+{\bm{y}})/2$,
$\,{\bm{K}}={\bm{k}}_{1}+{\bm{k}}_{2}$, as well as small-scale variables:
${\bm{r}}={\bm{x}}-{\bm{y}}$, $\,{\bm{k}}=({\bm{k}}_{1}-{\bm{k}}_{2})/2$. Here
${\bm{k}}_{1}={\bm{k}}+{\bm{K}}/2$, and ${\bm{k}}_{2}=-{\bm{k}}+{\bm{K}}/2$.
Mean-fields depend on the large-scale variables, while fluctuations depend on
the small-scale variables.

The procedure of the derivations of the expressions for the intensity of
temperature fluctuations implies the following steps:

  * •

derivation of equations for the second-order moments in the Fourier space
using the multi-scale approach;

  * •

application of the spectral $\tau$ approach (see below) which relates the
deviations of the third-order moments from those of the background turbulence
with the corresponding deviations of the second-order moments;

  * •

solution of the obtained equations for the second-order moments in the Fourier
space;

  * •

inverse transformation to the physical space to derive expressions for the
intensity of temperature fluctuations.

By means of equation (38) for temperature fluctuations $\theta$ and the
Navier-Stokes equation for velocity fluctuations ${\bm{u}}$ rewritten in
Fourier space, we derive an equation for the second-order moment
$\left\langle\theta({\bm{k}}_{1})\,\theta({\bm{k}}_{2})\right\rangle$ as

$\displaystyle{\partial\over\partial
t}\left\langle\theta({\bm{k}}_{1})\,\theta({\bm{k}}_{2})\right\rangle=-\Big{[}\left\langle
u_{i}({\bm{k}}_{1})\,\theta({\bm{k}}_{2})\right\rangle$
$\displaystyle\quad+\left\langle\theta({\bm{k}}_{1})\,u_{i}({\bm{k}}_{2})\right\rangle\Big{]}\,\nabla_{i}\overline{T}-(\gamma-1)\,\Big{[}\left\langle({\rm
div}{\bm{u}})_{{\bm{k}}_{1}}\,\theta({\bm{k}}_{2})\right\rangle$
$\displaystyle\quad+\left\langle\theta({\bm{k}}_{1})\,({\rm
div}{\bm{u}})_{{\bm{k}}_{2}}\right\rangle\Big{]}\overline{T}+\hat{\cal
M}\Theta^{(III)},$ (41)

where $\hat{\cal M}\Theta^{(III)}$ are the third-order moment terms caused by
the nonlinear terms in the equation for temperature fluctuations. Temperature
and velocity fluctuations $\theta({\bm{k}}_{1,2},t)$ and
$u_{i}({\bm{k}}_{1,2},t)$ depend also on $t$, and the mean temperature
$\overline{T}(t,{\bm{R}})$ depend on $t$ and ${\bm{R}}$ as well. For brevity
of notations, we do not show these dependencies hereafter.

Equation (41) for the second-order moments includes the third-order moments
$\hat{\cal M}\Theta^{\rm(III)}$, and the closure problem arises, i.e., how to
express the third-order moments $\hat{\cal M}\Theta^{\rm(III)}$ through the
lower-order moments (Monin & Yaglom, 1971, 1975; McComb, 1990). We adopt the
spectral $\tau$ approach which postulates that the deviations of the third-
moment terms, $\hat{\cal M}\Theta^{\rm(III)}({\bm{k}})$, from those afforded
by the background turbulence, $\hat{\cal M}\Theta^{\rm(III,0)}({\bm{k}})$, can
be expressed through similar deviations of the second-order moments,
$\Theta^{\rm(II)}({\bm{k}})-\Theta^{\rm(II,0)}({\bm{k}})$ as (Orszag, 1970;
Pouquet et al., 1976; Kleeorin et al., 1990)

$\displaystyle\hat{\cal M}\Theta^{\rm(III)}({\bm{k}})-\hat{\cal
M}\Theta^{\rm(III,0)}({\bm{k}})=-{\Theta^{\rm(II)}({\bm{k}})-\Theta^{\rm(II,0)}({\bm{k}})\over\tau_{r}(k)},$
(42)

where $\tau_{r}(k)$ is the scale-dependent relaxation time which can be
identified with the correlation time $\tau(k)$ of the turbulent velocity field
for large Reynolds and Péclet numbers. Since the functions with superscript
$(0)$ describe the background turbulence with a zero turbulent heat flux,
equation (42) is reduced to $\hat{\cal
M}\Theta^{\rm(III)}=-\left\langle\theta({\bm{k}}_{1})\,\theta({\bm{k}}_{2})\right\rangle/\tau(k)$.
We apply the $\tau$ approximation only for the deviations from the background
turbulence, while the background turbulence is assumed to be known (see
below). Validation of the $\tau$ approximation for different problems has been
performed in various numerical simulations (Brandenburg et al., 2004;
Brandenburg & Subramanian, 2005a, b, c; Brandenburg et al., 2008; Rogachevskii
et al., 2011; Rädler et al., 2011; Haugen et al., 2012; Rogachevskii et al.,
2018; Elperin et al., 2017).

Since the characteristic times of variation of the second-order moment
$\Theta^{\rm(II)}$ are much larger than the correlation time $\tau(k)$ in all
turbulence scales, we use the steady-state solution of equation (41) as

$\displaystyle\left\langle\theta({\bm{k}}_{1})\,\theta({\bm{k}}_{2})\right\rangle=-\tau(k)\biggl{\\{}\Big{[}\left\langle
u_{i}({\bm{k}}_{1})\,\theta({\bm{k}}_{2})\right\rangle$
$\displaystyle\quad+\left\langle\theta({\bm{k}}_{1})\,u_{i}({\bm{k}}_{2})\right\rangle\Big{]}\nabla_{i}\overline{T}+(\gamma-1)\,\Big{[}\left\langle({\rm
div}{\bm{u}})_{{\bm{k}}_{1}}\,\theta({\bm{k}}_{2})\right\rangle$
$\displaystyle\quad+\left\langle\theta({\bm{k}}_{1})\,({\rm
div}{\bm{u}})_{{\bm{k}}_{2}}\right\rangle\Big{]}\overline{T}\biggr{\\}}.$ (43)

Similarly, we derive expression for the second moments entering in equation
(43), i.e., for $\left\langle
u_{i}({\bm{k}}_{1})\,\theta({\bm{k}}_{2})\right\rangle$ and
$\left\langle\theta({\bm{k}}_{1})\,u_{j}({\bm{k}}_{2})\right\rangle$ as

$\displaystyle\left\langle
u_{i}({\bm{k}}_{1})\,\theta({\bm{k}}_{2})\right\rangle=-\tau(k)\biggl{[}\left\langle
u_{i}({\bm{k}}_{1})\,u_{j}({\bm{k}}_{2})\right\rangle\,\nabla_{j}\overline{T}$
$\displaystyle\quad+(\gamma-1)\,\left\langle u_{i}({\bm{k}}_{1})\,({\rm
div}{\bm{u}})_{{\bm{k}}_{2}}\right\rangle\overline{T}\biggr{]},$ (44)
$\displaystyle\left\langle\theta({\bm{k}}_{1})\,u_{j}({\bm{k}}_{2})\right\rangle=-\tau(k)\biggl{[}\left\langle
u_{i}({\bm{k}}_{1})\,u_{j}({\bm{k}}_{2})\right\rangle\,\nabla_{i}\overline{T}$
$\displaystyle\quad+(\gamma-1)\,\left\langle({\rm
div}{\bm{u}})_{{\bm{k}}_{1}}\,u_{j}({\bm{k}}_{2})\right\rangle\overline{T}\biggr{]},$
(45)

and for the second moments $\left\langle({\rm
div}{\bm{u}})_{{\bm{k}}_{1}}\,\theta({\bm{k}}_{2})\right\rangle$ and
$\left\langle\theta({\bm{k}}_{1})\,({\rm
div}{\bm{u}})_{{\bm{k}}_{2}}\right\rangle$:

$\displaystyle\left\langle({\rm
div}{\bm{u}})_{{\bm{k}}_{1}}\,\theta({\bm{k}}_{2})\right\rangle=-\tau(k)\biggl{[}\left\langle({\rm
div}{\bm{u}})_{{\bm{k}}_{1}}\,u_{j}({\bm{k}}_{2})\right\rangle\,\nabla_{j}\overline{T}$
$\displaystyle\quad+(\gamma-1)\,\left\langle({\rm
div}{\bm{u}})_{{\bm{k}}_{1}}\,({\rm
div}{\bm{u}})_{{\bm{k}}_{2}}\right\rangle\overline{T}\biggr{]},$ (46)
$\displaystyle\left\langle\theta({\bm{k}}_{1})\,({\rm
div}{\bm{u}})_{{\bm{k}}_{2}}\right\rangle=-\tau(k)\biggl{[}\left\langle
u_{i}({\bm{k}}_{1})\,({\rm
div}{\bm{u}})_{{\bm{k}}_{2}}\right\rangle\,\nabla_{i}\overline{T}$
$\displaystyle\quad+(\gamma-1)\,\left\langle({\rm
div}{\bm{u}})_{{\bm{k}}_{1}}\,({\rm
div}{\bm{u}})_{{\bm{k}}_{2}}\right\rangle\overline{T}\biggr{]}.$ (47)

Substituting equations (44)–(47) into equation (43), we obtain

$\displaystyle\left\langle\theta({\bm{k}}_{1})\,\theta({\bm{k}}_{2})\right\rangle=2\tau^{2}(k)\biggl{\\{}\left\langle
u_{i}({\bm{k}}_{1})\,u_{j}({\bm{k}}_{2})\right\rangle\,(\nabla_{i}\overline{T})(\nabla_{j}\overline{T})$
$\displaystyle+(\gamma-1)\,\Big{[}\left\langle({\rm
div}{\bm{u}})_{{\bm{k}}_{1}}\,u_{j}({\bm{k}}_{2})\right\rangle+\left\langle
u_{j}({\bm{k}}_{1})\,({\rm div}{\bm{u}})_{{\bm{k}}_{2}}\right\rangle\Big{]}$
$\displaystyle\times\overline{T}\,\nabla_{j}\overline{T}+(\gamma-1)^{2}\,\left\langle({\rm
div}{\bm{u}})_{{\bm{k}}_{1}}\,({\rm
div}{\bm{u}})_{{\bm{k}}_{2}}\right\rangle\overline{T}^{2}\biggr{\\}}.$ (48)

Since all terms in equations (43) and (48) are proportional to either
$({\bm{\nabla}}\overline{T})^{2}$, or $({\bm{\nabla}}\overline{\rho})^{2}$, or
$({\bm{\nabla}}\overline{T})({\bm{\nabla}}\overline{\rho})$ (see below), and
we consider homogeneous density stratified turbulence, we do not need to
perform additional Taylor expansions over small parameters $\ell_{0}/H_{T}$
and $\ell_{0}/H_{\rho}$ in these terms (Rogachevskii & Kleeorin, 2021;
Rogachevskii, 2021), where $H_{T}$ and $H_{\rho}$ are the characteristic
scales of variations of the mean temperature and mean density, respectively.
In particular, hereafter we neglect small terms $\sim{\rm
O}[(\ell_{0}/H_{T})^{3},(\ell_{0}/H_{\rho})^{3}]$. This implies that we
replace ${\bm{k}}_{1}$ by ${\bm{k}}$ and ${\bm{k}}_{2}$ by $-{\bm{k}}$ in all
second moments in equation (48).

In equation (48) we take into account a one way coupling, i.e., we neglect the
feedback effect of the mean temperature gradients on the turbulent velocity
field. This implies that we replace the correlation function
$f_{ij}=\left\langle u_{i}({\bm{k}})\,u_{j}(-{\bm{k}})\right\rangle$ in
equation (48) by $f_{ij}^{(0)}=\left\langle
u_{i}({\bm{k}})\,u_{j}(-{\bm{k}})\right\rangle^{(0)}$ for the background
turbulence with a zero turbulent heat flux. Similarly we replace
$\left\langle({\rm div}{\bm{u}})_{{\bm{k}}}\,u_{j}(-{\bm{k}})\right\rangle$,
$\left\langle u_{j}({\bm{k}})\,({\rm div}{\bm{u}})_{-{\bm{k}}}\right\rangle$
and $\left\langle({\rm div}{\bm{u}})_{{\bm{k}}}\,({\rm
div}{\bm{u}})_{-{\bm{k}}}\right\rangle$ in equation (48) by the corresponding
correlation functions for the background turbulence with a zero turbulent heat
flux.

To find the intensity of temperature fluctuations
$\left\langle\theta^{2}\right\rangle$ for large Péclet numbers, we adopt a
model for the background turbulence, $f_{ij}^{(0)}({\bm{k}})=\left\langle
u_{i}({\bm{k}})\,u_{j}(-{\bm{k}})\right\rangle^{(0)}$, that is a statistically
stationary density-stratified compressible turbulence given by (Elperin et
al., 1995; Amir et al., 2017; Rogachevskii, 2021):

$\displaystyle f_{ij}^{(0)}({\bm{k}})={1\over
8\pi\,k^{2}\,(1+\sigma_{c})}\,\biggl{\\{}E(k)\,\biggl{[}(\delta_{ij}-k_{ij})\,\left(1-{\lambda^{2}\over
k^{2}}\right)$ $\displaystyle\quad+{\lambda^{2}\over
k^{2}}\,\big{(}\delta_{ij}-\lambda_{ij}\big{)}\biggr{]}+{{\rm i}\over
k^{2}}\,\left[E(k)+\sigma_{c}\,E_{c}(k)\right]\,\Big{(}k_{j}\lambda_{i}$
$\displaystyle\quad-
k_{i}\lambda_{j}\Big{)}+2\sigma_{c}\,E_{c}(k)\,k_{ij}\biggr{\\}}\left\langle{\bm{u}}^{2}\right\rangle,$
(49)

where $k_{ij}=k_{i}\,k_{j}/k^{2}$,
$\lambda_{ij}=\lambda_{i}\,\lambda_{j}/\lambda^{2}$, and
${\bm{\lambda}}=-{\bm{\nabla}}\ln\overline{\rho}$ characterizes the fluid
density stratification. This model is different from that derived by
Rogachevskii et al. (2018); Rogachevskii & Kleeorin (2021). In particular,
this model takes into account a strong density stratification. In addition,
the turbulent flux $\langle\rho^{\prime}{\bm{u}}\rangle$ is very small
$[\sim{\rm O}(\lambda\,\ell_{0})^{3}]$.

The background turbulence is of Kolmogorov type with a constant energy flux
over the spectrum, i.e., the turbulent kinetic energy spectrum for the
incompressible part of turbulence in the inertial range $k_{0}<k<k_{\nu}$ is
$E(k)=-d\tilde{\tau}(k)/dk$. Here $\tilde{\tau}(k)=(k/k_{0})^{1-q}$ with
$1<q<3$ is the exponent of the turbulent kinetic energy spectrum. Similarly,
the turbulent kinetic energy spectrum for the compressible part of turbulence
is $E_{c}(k)=-d\tilde{\tau}_{c}(k)/dk$, where
$\tilde{\tau}_{c}(k)=(k/k_{0})^{1-q_{c}}$ with $1<q_{c}<3$. For example, the
exponent of the incompressible part of the spectrum, $q=5/3$, corresponds to
the Kolmogorov spectrum, while the exponent of the compressible part of the
spectrum, $q_{c}=2$, describes the Burgers turbulence with shock waves. These
exponents of the spectra are observed in numerical simulations in compressible
turbulence (Kritsuk et al., 2007; Federrath, 2013). The correlation time for a
compressible turbulence in the Forier space is (Rogachevskii & Kleeorin, 2021)

$\displaystyle\tau(k)={2\tau_{0}\over
1+\sigma_{c}}\,\Big{[}\tilde{\tau}(k)+\sigma_{c}\,\tilde{\tau}_{c}(k)\Big{]}.$
(50)

To determine the level of temperature fluctuations
$\left\langle\theta^{2}\right\rangle=\int\tau^{2}(k)\left\langle\theta({\bm{k}})\,\theta(-{\bm{k}})\right\rangle\,d{\bm{k}}$
for large Péclet numbers, we use equations (48)–(50). To this end, we
calculate the following integrals:

$\displaystyle\int\tau^{2}(k)\left\langle
u_{i}({\bm{k}})\,u_{j}(-{\bm{k}})\right\rangle^{(0)}\,d{\bm{k}}={4\over
9}\,\ell_{0}^{2}\,\delta_{ij},$ (51)

$\displaystyle\int\tau^{2}(k)\left\langle u_{i}({\bm{k}})\,({\rm
div}{\bm{u}})_{-{\bm{k}}}\right\rangle^{(0)}\,d{\bm{k}}={4\over
9}\,\ell_{0}^{2}\,\lambda_{i},$ (52)

$\displaystyle\int\tau^{2}(k)\left\langle({\rm div}{\bm{u}})_{{\bm{k}}}\,({\rm
div}{\bm{u}})_{-{\bm{k}}}\right\rangle^{(0)}\,d{\bm{k}}={4\over
9}\,(\ell_{0}\,\lambda)^{2}$
$\displaystyle\quad+4f_{c}\,\left({\sigma_{c}\over
1+\sigma_{c}}\right)^{3}\,\left[1-{1\over 9}\,(\ell_{0}\,\lambda)^{2}\right].$
(53)

For the integration in ${\bm{k}}$ space in equations (51)–(53), we use the
following identities:

$\displaystyle\int_{k_{0}}^{k_{\nu}}\tau^{2}(k)\,\left[E(k)+\sigma_{c}\,E_{c}(k)\right]\,dk={4\over
3}\,\tau_{0}^{2}\,(1+\sigma_{c}),$ (54)

$\displaystyle\int_{k_{0}}^{k_{\nu}}\tau^{2}(k)\,k^{2}\,E_{c}(k)\,dk=4f_{c}\,\left({\tau_{0}\over\ell_{0}}\right)^{2}\,\left({\sigma_{c}\over
1+\sigma_{c}}\right)^{2}.$ (55)

Therefore, equations (48) and (51)–(53) yield the level of temperature
fluctuations $\left\langle\theta^{2}\right\rangle$ for large Péclet numbers
given by equation (26).

Derivation of equations (29) and (30) for
$\left\langle\rho^{\prime\,2}\right\rangle$ and
$\left\langle\theta\,\rho^{\prime}\right\rangle$ is performed in a similar
way. In particular, using the continuity equation for the fluid density
fluctuations $\rho^{\prime}$ written in the Fourier space, we obtain the
evolutionary equation for the second moment
$\left\langle\rho^{\prime}({\bm{k}}_{1})\,\rho^{\prime}({\bm{k}}_{2})\right\rangle$
as

$\displaystyle{\partial\over\partial
t}\left\langle\rho^{\prime}({\bm{k}}_{1})\,\rho^{\prime}({\bm{k}}_{2})\right\rangle=-\Big{[}\left\langle
u_{i}({\bm{k}}_{1})\,\rho^{\prime}({\bm{k}}_{2})\right\rangle$
$\displaystyle\quad+\left\langle\rho^{\prime}({\bm{k}}_{1})\,u_{i}({\bm{k}}_{2})\right\rangle\Big{]}\,\nabla_{i}\overline{\rho}-\Big{[}\left\langle({\rm
div}{\bm{u}})_{{\bm{k}}_{1}}\,\rho({\bm{k}}_{2})\right\rangle$
$\displaystyle\quad+\left\langle\rho({\bm{k}}_{1})\,({\rm
div}{\bm{u}})_{{\bm{k}}_{2}}\right\rangle\Big{]}\overline{\rho}+\hat{\cal
M}\rho^{(III)},$ (56)

where $\hat{\cal M}\rho^{(III)}$ are the third-order moment terms related to
nonlinear terms in the equation for density fluctuations.

Applying the spectral $\tau$ approach, we obtain expression for the second
moment
$\left\langle\rho^{\prime}({\bm{k}}_{1})\,\rho^{\prime}({\bm{k}}_{2})\right\rangle$
as

$\displaystyle\left\langle\rho^{\prime}({\bm{k}}_{1})\,\rho^{\prime}({\bm{k}}_{2})\right\rangle=-\tau(k)\biggl{\\{}\Big{[}\left\langle
u_{i}({\bm{k}}_{1})\,\rho^{\prime}({\bm{k}}_{2})\right\rangle$
$\displaystyle\quad+\left\langle\rho^{\prime}({\bm{k}}_{1})\,u_{i}({\bm{k}}_{2})\right\rangle\Big{]}\nabla_{i}\overline{\rho}+\Big{[}\left\langle({\rm
div}{\bm{u}})_{{\bm{k}}_{1}}\,\rho^{\prime}({\bm{k}}_{2})\right\rangle$
$\displaystyle\quad+\left\langle\rho^{\prime}({\bm{k}}_{1})\,({\rm
div}{\bm{u}})_{{\bm{k}}_{2}}\right\rangle\Big{]}\overline{\rho}\biggr{\\}}.$
(57)

Similarly, we derive expression for the second moments $\left\langle
u_{i}({\bm{k}}_{1})\,\rho^{\prime}({\bm{k}}_{2})\right\rangle$ and
$\left\langle\rho^{\prime}({\bm{k}}_{1})\,u_{j}({\bm{k}}_{2})\right\rangle$ as

$\displaystyle\left\langle
u_{i}({\bm{k}}_{1})\,\rho^{\prime}({\bm{k}}_{2})\right\rangle$
$\displaystyle=$ $\displaystyle-\tau(k)\biggl{[}\left\langle
u_{i}({\bm{k}}_{1})\,u_{j}({\bm{k}}_{2})\right\rangle\,\nabla_{j}\overline{\rho}$
(58) $\displaystyle\quad+\left\langle u_{i}({\bm{k}}_{1})\,({\rm
div}{\bm{u}})_{{\bm{k}}_{2}}\right\rangle\overline{\rho}\biggr{]},$
$\displaystyle\left\langle\rho^{\prime}({\bm{k}}_{1})\,u_{j}({\bm{k}}_{2})\right\rangle$
$\displaystyle=$ $\displaystyle-\tau(k)\biggl{[}\left\langle
u_{i}({\bm{k}}_{1})\,u_{j}({\bm{k}}_{2})\right\rangle\,\nabla_{i}\overline{\rho}$
(59) $\displaystyle+\left\langle({\rm
div}{\bm{u}})_{{\bm{k}}_{1}}\,u_{j}({\bm{k}}_{2})\right\rangle\overline{\rho}\biggr{]},$

and for $\left\langle({\rm
div}{\bm{u}})_{{\bm{k}}_{1}}\,\rho^{\prime}({\bm{k}}_{2})\right\rangle$ and
$\left\langle\rho^{\prime}({\bm{k}}_{1})\,({\rm
div}{\bm{u}})_{{\bm{k}}_{2}}\right\rangle$:

$\displaystyle\left\langle({\rm
div}{\bm{u}})_{{\bm{k}}_{1}}\,\rho^{\prime}({\bm{k}}_{2})\right\rangle$
$\displaystyle=$ $\displaystyle-\tau(k)\biggl{[}\left\langle({\rm
div}{\bm{u}})_{{\bm{k}}_{1}}\,u_{j}({\bm{k}}_{2})\right\rangle\,\nabla_{j}\overline{\rho}$
(60) $\displaystyle+\left\langle({\rm div}{\bm{u}})_{{\bm{k}}_{1}}\,({\rm
div}{\bm{u}})_{{\bm{k}}_{2}}\right\rangle\overline{\rho}\biggr{]},$
$\displaystyle\left\langle\rho^{\prime}({\bm{k}}_{1})\,({\rm
div}{\bm{u}})_{{\bm{k}}_{2}}\right\rangle$ $\displaystyle=$
$\displaystyle-\tau(k)\biggl{[}\left\langle u_{i}({\bm{k}}_{1})\,({\rm
div}{\bm{u}})_{{\bm{k}}_{2}}\right\rangle\,\nabla_{i}\overline{\rho}$ (61)
$\displaystyle+\left\langle({\rm div}{\bm{u}})_{{\bm{k}}_{1}}\,({\rm
div}{\bm{u}})_{{\bm{k}}_{2}}\right\rangle\overline{\rho}\biggr{]}.$

Substituting equations (58)–(61) into equation (57) we obtain

$\displaystyle\left\langle\rho^{\prime}({\bm{k}}_{1})\,\rho^{\prime}({\bm{k}}_{2})\right\rangle=2\tau^{2}(k)\biggl{\\{}\left\langle
u_{i}({\bm{k}}_{1})\,u_{j}({\bm{k}}_{2})\right\rangle\,(\nabla_{i}\overline{\rho})(\nabla_{j}\overline{\rho})$
$\displaystyle\quad+\Big{[}\left\langle({\rm
div}{\bm{u}})_{{\bm{k}}_{1}}\,u_{j}({\bm{k}}_{2})\right\rangle+\left\langle
u_{j}({\bm{k}}_{1})\,({\rm
div}{\bm{u}})_{{\bm{k}}_{2}}\right\rangle\Big{]}\overline{\rho}\,\nabla_{j}\overline{\rho}$
$\displaystyle\quad+\left\langle({\rm div}{\bm{u}})_{{\bm{k}}_{1}}\,({\rm
div}{\bm{u}})_{{\bm{k}}_{2}}\right\rangle\overline{\rho}^{2}\biggr{\\}}.$ (62)

We do not take into account small terms $\sim{\rm O}[(\ell_{0}/H_{\rho})^{3}]$
in equations (57) and (62). Since all terms in equations (57) and (62) are
proportional to $({\bm{\nabla}}\overline{\rho})^{2}$ and we consider
homogeneous density stratified turbulence, we do not need to perform
additional Taylor expansions over small parameter $\ell_{0}/H_{\rho}$ in these
terms. Thus, we replace ${\bm{k}}_{1}$ by ${\bm{k}}$ and ${\bm{k}}_{2}$ by
$-{\bm{k}}$ in all second moments in equation (62). Using equations (51)–(53)
and (62), we determine the level of density fluctuations
$\left\langle\rho^{\prime\,2}\right\rangle=\int\tau^{2}(k)\left\langle\rho^{\prime}({\bm{k}})\,\rho^{\prime}(-{\bm{k}})\right\rangle\,d{\bm{k}}$
for large Péclet numbers given by equation (29).

Now we derive the evolutionary equation for the second moment
$\left\langle\theta({\bm{k}}_{1})\,\rho^{\prime}({\bm{k}}_{2})\right\rangle$
using the continuity equation for the fluid density fluctuations and the
equation for the temperature fluctuations written in the Fourier space:

$\displaystyle{\partial\over\partial
t}\left\langle\theta({\bm{k}}_{1})\,\rho^{\prime}({\bm{k}}_{2})\right\rangle=-\left\langle
u_{i}({\bm{k}}_{1})\,\rho^{\prime}({\bm{k}}_{2})\right\rangle\,\nabla_{i}\overline{T}$
$\displaystyle\quad-\left\langle\theta({\bm{k}}_{1})\,u_{j}({\bm{k}}_{2})\right\rangle\nabla_{j}\overline{\rho}-(\gamma-1)\,\left\langle({\rm
div}{\bm{u}})_{{\bm{k}}_{1}}\,\rho^{\prime}({\bm{k}}_{2})\right\rangle\,\overline{T}$
$\displaystyle\quad-\left\langle\theta({\bm{k}}_{1})\,({\rm
div}{\bm{u}})_{{\bm{k}}_{2}}\right\rangle\overline{\rho}+\hat{\cal
M}\Theta_{\rho}^{(III)},$ (63)

where $\hat{\cal M}\Theta_{\rho}^{(III)}$ are the third-order moment terms
related to the nonlinear terms in the equations for temperature and density
fluctuations. Applying the spectral $\tau$ approach, we obtain expression for
the second moment
$\left\langle\theta({\bm{k}}_{1})\,\rho^{\prime}({\bm{k}}_{2})\right\rangle$
as

$\displaystyle\left\langle\theta({\bm{k}}_{1})\,\rho^{\prime}({\bm{k}}_{2})\right\rangle=-\tau(k)\biggl{[}\left\langle
u_{i}({\bm{k}}_{1})\,\rho^{\prime}({\bm{k}}_{2})\right\rangle\,\nabla_{i}\overline{T}$
$\displaystyle\quad+\left\langle\theta({\bm{k}}_{1})\,u_{j}({\bm{k}}_{2})\right\rangle\nabla_{j}\overline{\rho}+(\gamma-1)\,\left\langle({\rm
div}{\bm{u}})_{{\bm{k}}_{1}}\,\rho^{\prime}({\bm{k}}_{2})\right\rangle\,\overline{T}$
$\displaystyle\quad+\left\langle\theta({\bm{k}}_{1})\,({\rm
div}{\bm{u}})_{{\bm{k}}_{2}}\right\rangle\overline{\rho}\biggr{]}.$ (64)

Substituting equations (45), (47), (58), (60) into equation (64), we obtain

$\displaystyle\left\langle\theta({\bm{k}}_{1})\,\rho^{\prime}({\bm{k}}_{2})\right\rangle=2\tau^{2}(k)\biggl{\\{}\left\langle
u_{i}({\bm{k}}_{1})\,u_{j}({\bm{k}}_{2})\right\rangle\,(\nabla_{i}\overline{T})\,(\nabla_{j}\overline{\rho})$
$\displaystyle+(\gamma-1)\,\Big{[}\left\langle({\rm
div}{\bm{u}})_{{\bm{k}}_{1}}\,({\rm
div}{\bm{u}})_{{\bm{k}}_{2}}\right\rangle\,\overline{\rho}+\left\langle({\rm
div}{\bm{u}})_{{\bm{k}}_{1}}\,u_{j}({\bm{k}}_{2})\right\rangle$
$\displaystyle\times(\nabla_{j}\overline{\rho})\Big{]}\,\overline{T}+\left\langle
u_{i}({\bm{k}}_{1})\,({\rm
div}{\bm{u}})_{{\bm{k}}_{2}}\right\rangle\,\overline{\rho}\,(\nabla_{i}\overline{T})\biggr{\\}}.$
(65)

We neglect small terms $\sim{\rm
O}[(\ell_{0}/H_{T})^{3},(\ell_{0}/H_{\rho})^{3}]$ in equations (64) and (65).
Since all terms in equations (64) and (65) are proportional to either
$({\bm{\nabla}}\overline{\rho})^{2}$ or
$({\bm{\nabla}}\overline{T})({\bm{\nabla}}\overline{\rho})$, and we consider
homogeneous density stratified turbulence, we do not need to perform
additional Taylor expansions over small parameters $\ell_{0}/H_{T}$ and
$\ell_{0}/H_{\rho}$ in these terms. Therefore, we replace ${\bm{k}}_{1}$ by
${\bm{k}}$ and ${\bm{k}}_{2}$ by $-{\bm{k}}$ in all second moments in equation
(65). Using equations (51)–(53) and (65), we determine
$\left\langle\theta\,\rho^{\prime}\right\rangle=\int\tau^{2}(k)\left\langle\theta({\bm{k}})\,\rho^{\prime}(-{\bm{k}})\right\rangle\,d{\bm{k}}$
for large Péclet numbers given by equation (30).