2.1. Governing equations

The budget equation for the instantaneous internal energy density $E=c_{v}T$ is (Landau & Lifshitz Reference Landau and Lifshitz1959)

(2.1) $$\begin{eqnarray}\displaystyle \frac{\partial ({\it\rho}E)}{\partial t}+\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }({\it\rho}\boldsymbol{U}W-\boldsymbol{U}P-K\boldsymbol{{\rm\nabla}}T)=Q, & & \displaystyle\end{eqnarray}$$

where $\boldsymbol{U}$ is the instantaneous velocity determined by the Navier–Stokes equation for fluid motion, ${\it\rho}$ , $T$ and $P$ are the instantaneous density, temperature and pressure, respectively, which satisfy the equation of state for a perfect gas, $K$ is the coefficient of molecular heat conductivity, $W=c_{p}T=c_{v}T+P/{\it\rho}=E+P/{\it\rho}$ is the instantaneous enthalpy, where $c_{v}$ and $c_{p}$ are the specific heats at constant volume and pressure, and $Q=-P\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{U}+\hat{{\it\sigma}}_{ij}(\boldsymbol{U}){\rm\nabla}_{j}U_{i}$ , where $\hat{{\it\sigma}}(\boldsymbol{U})=2{\it\nu}{\it\rho}\hat{{\mathcal{S}}}(\boldsymbol{U})$ , $\hat{{\mathcal{S}}}(\boldsymbol{U})={\mathcal{S}}_{ij}=(U_{i,j}+U_{j,i})/2-({\it\delta}_{ij}\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{U})/3$ , ${\it\nu}$ is the kinematic viscosity, and ${\it\delta}_{ij}$ is the Kronecker tensor. The instantaneous density ${\it\rho}$ is determined by the continuity equation:

(2.2) $$\begin{eqnarray}\displaystyle \frac{\partial {\it\rho}}{\partial t}+\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }({\it\rho}\,\boldsymbol{U})=0. & & \displaystyle\end{eqnarray}$$

The budget equation for the instantaneous kinetic energy density ${\it\rho}\boldsymbol{U}^{2}/2$ is (Landau & Lifshitz Reference Landau and Lifshitz1959)

(2.3) $$\begin{eqnarray}\displaystyle \frac{\partial }{\partial t}\left(\frac{1}{2}{\it\rho}\boldsymbol{U}^{2}\right)+\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\left[\boldsymbol{U}\left(\frac{1}{2}{\it\rho}\boldsymbol{U}^{2}+P\right)-\boldsymbol{U}\hat{{\it\sigma}}(\boldsymbol{U})\right]=-Q. & & \displaystyle\end{eqnarray}$$

The sum of (2.1) and (2.3) yields the conservation law for the instantaneous total (kinetic plus internal) energy densities $({\it\rho}\boldsymbol{U}^{2}/2+{\it\rho}\,E)$ (Landau & Lifshitz Reference Landau and Lifshitz1959):

(2.4) $$\begin{eqnarray}\displaystyle \frac{\partial }{\partial t}\left(\frac{1}{2}{\it\rho}\boldsymbol{U}^{2}+{\it\rho}\,E\right)+\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\left[\boldsymbol{U}\left(\frac{1}{2}{\it\rho}\boldsymbol{U}^{2}+{\it\rho}W\right)-\boldsymbol{U}\hat{{\it\sigma}}(\boldsymbol{U})-K\boldsymbol{{\rm\nabla}}T\right]=0. & & \displaystyle\end{eqnarray}$$

2.2. Turbulent convective flux and equation for mean internal energy

Averaging (2.1) over the ensemble we obtain the budget equation for the mean internal energy density $\overline{E}=c_{v}\overline{T}$ :

(2.5) $$\begin{eqnarray}\displaystyle \frac{\partial (\overline{{\it\rho}}\,\overline{E})}{\partial t}+\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }(\overline{{\it\rho}}\,\overline{\boldsymbol{U}}\,\overline{E}+\overline{{\it\rho}}\,\overline{\boldsymbol{u}w}-\overline{\boldsymbol{u}p}-K\boldsymbol{{\rm\nabla}}\overline{T})=\overline{Q}, & & \displaystyle\end{eqnarray}$$

where $\boldsymbol{u}$ is the velocity fluctuations, $w=\overline{T}\,s+p/\overline{{\it\rho}}$ are the enthalpy fluctuations, $s$ and $p$ are the entropy and pressure fluctuations, respectively, and $\overline{Q}=-\overline{P}\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\overline{\boldsymbol{U}}-\overline{p\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{u}}+\hat{{\it\sigma}}_{ij}(\overline{\boldsymbol{U}}){\rm\nabla}_{j}\overline{U}_{i}+\overline{\hat{{\it\sigma}}_{ij}(\boldsymbol{u}){\rm\nabla}_{j}u_{i}}$ . In the derivation of (2.5) we used the identity $\overline{W}=\overline{E}+\overline{P}/\overline{{\it\rho}}$ , and since we consider a low-Mach-number turbulent flow, we took into account that $|\overline{\boldsymbol{u}{\it\rho}}|/|\overline{{\it\rho}}|\ll |\overline{\boldsymbol{u}s}|/|\overline{S}|$ (Chassaing et al. Reference Chassaing, Antonia, Anselmet, Joly and Sarkar2002). The turbulent flux of enthalpy is

(2.6) $$\begin{eqnarray}\displaystyle \overline{\boldsymbol{u}w}=\overline{T}\,\overline{\boldsymbol{u}s}+\frac{\overline{\boldsymbol{u}p}}{\overline{{\it\rho}}}. & & \displaystyle\end{eqnarray}$$

Since $\overline{W}=\overline{E}+\overline{P}/\overline{{\it\rho}}$ we obtain that $\overline{{\it\rho}}\,\overline{\boldsymbol{U}}\,\overline{E}=\overline{\boldsymbol{U}}\,(\overline{{\it\rho}}\overline{W}-\overline{P})$ . Substituting the latter equation and (2.6) into (2.5), we obtain

(2.7) $$\begin{eqnarray}\displaystyle \frac{\partial \overline{{\it\rho}}\,\overline{E}}{\partial t}+\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }(\overline{{\it\rho}}\,\overline{\boldsymbol{U}}\,\overline{E}+\overline{T}\,\overline{{\it\rho}}\,\overline{\boldsymbol{u}s}-K\boldsymbol{{\rm\nabla}}\overline{T})=\overline{Q}. & & \displaystyle\end{eqnarray}$$

The mean fluid density $\overline{{\it\rho}}$ is determined by the continuity equation:

(2.8) $$\begin{eqnarray}\displaystyle \frac{\partial \overline{{\it\rho}}}{\partial t}+\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }(\overline{{\it\rho}}\,\overline{\boldsymbol{U}})=0. & & \displaystyle\end{eqnarray}$$

Equations (2.7) and (2.8) yield the following equation for the evolution of the mean internal energy:

(2.9) $$\begin{eqnarray}\displaystyle \overline{{\it\rho}}\left(\frac{\partial \overline{E}}{\partial t}+(\overline{\boldsymbol{U}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}})\overline{E}\right)+\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }(\overline{T}\,\overline{{\it\rho}}\,\overline{\boldsymbol{u}s}-K\boldsymbol{{\rm\nabla}}\overline{T})=\overline{Q}, & & \displaystyle\end{eqnarray}$$

where the turbulent convective flux of the mean internal energy is

(2.10) $$\begin{eqnarray}\displaystyle \boldsymbol{F}_{c}=\overline{T}\,\overline{{\it\rho}}\,\overline{\boldsymbol{u}s}. & & \displaystyle\end{eqnarray}$$

2.3. Equation for the sum of mean and turbulent kinetic energies and conservation law for total mean energy

Averaging (2.3) for the instantaneous kinetic energy density ${\it\rho}\boldsymbol{U}^{2}/2$ we obtain an equation for the sum of the mean and turbulent kinetic energies $\overline{{\it\rho}}\,\overline{\boldsymbol{U}^{2}}/2+\overline{{\it\rho}}\,\overline{\boldsymbol{u}^{2}}/2$ :

(2.11) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\partial }{\partial t}\left(\frac{1}{2}\overline{{\it\rho}}\,\overline{\boldsymbol{U}^{2}}+\frac{1}{2}\overline{{\it\rho}}\,\overline{\boldsymbol{u}^{2}}\right)+\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\left[\vphantom{\overline{\boldsymbol{u}\left(\frac{1}{2}{\it\rho}\boldsymbol{u}^{2}+p\right)}}\overline{\boldsymbol{U}}\left(\frac{1}{2}\overline{{\it\rho}}\overline{\boldsymbol{U}}^{2}+\overline{P}\right)-\overline{\boldsymbol{U}}\hat{{\it\sigma}}(\overline{\boldsymbol{U}})\right.\nonumber\\ \displaystyle & & \displaystyle \quad +\left.\,\overline{\boldsymbol{u}\left(\frac{1}{2}{\it\rho}\boldsymbol{u}^{2}+p\right)}-\overline{\boldsymbol{u}\hat{{\it\sigma}}(\boldsymbol{u})}\vphantom{\overline{\boldsymbol{u}\left(\frac{1}{2}{\it\rho}\boldsymbol{u}^{2}+p\right)}}\right]=-\overline{Q}.\end{eqnarray}$$

The sum of (2.7) and (2.11) yields the conservation law for the total mean energy $E_{\mathit{tot}}=\overline{{\it\rho}}\overline{\boldsymbol{U}}^{2}/2+\overline{{\it\rho}}\,\overline{\boldsymbol{u}^{2}}/2+\overline{{\it\rho}}\,\overline{E}$ :

(2.12) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\partial }{\partial t}\left(\frac{1}{2}\overline{{\it\rho}}\,\overline{\boldsymbol{U}}^{2}+\frac{1}{2}\overline{{\it\rho}}\,\overline{\boldsymbol{u}^{2}}+\overline{{\it\rho}}\,\overline{E}\right)+\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\left[\vphantom{\overline{\boldsymbol{u}\left(\frac{1}{2}{\it\rho}\boldsymbol{u}^{2}+p\right)}}\overline{\boldsymbol{U}}\left(\frac{1}{2}\overline{{\it\rho}}\overline{\boldsymbol{U}}^{2}+\overline{P}+\overline{{\it\rho}}\,\overline{E}\right)-\overline{\boldsymbol{U}}\hat{{\it\sigma}}(\overline{\boldsymbol{U}})\right.\nonumber\\ \displaystyle & & \displaystyle \quad -\left.\overline{\boldsymbol{u}\hat{{\it\sigma}}(\boldsymbol{u})}+\overline{\boldsymbol{u}\left(\frac{1}{2}{\it\rho}\boldsymbol{u}^{2}+p\right)}+\overline{T}\,\overline{{\it\rho}}\,\overline{\boldsymbol{u}s}-K\boldsymbol{{\rm\nabla}}\overline{T}\right]=0.\end{eqnarray}$$

This equation contains the turbulent convective flux $\boldsymbol{F}_{c}=\overline{T}\,\overline{{\it\rho}}\,\overline{\boldsymbol{u}s}$ . The conservation law (2.12) for the total mean energy $E_{\mathit{tot}}$ can be rewritten in terms of the mean, $\overline{\boldsymbol{U}}\,\overline{W}$ , and turbulent, $\overline{\boldsymbol{u}w}$ , fluxes of enthalpy (see (2.6)):

(2.13) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\partial }{\partial t}\left(\frac{1}{2}\overline{{\it\rho}}\overline{\boldsymbol{U}}^{2}+\frac{1}{2}\overline{{\it\rho}}\,\overline{\boldsymbol{u}^{2}}+\overline{{\it\rho}}\,\overline{E}\right)+\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\left[\vphantom{\overline{\boldsymbol{u}\left(\frac{1}{2}{\it\rho}\boldsymbol{u}^{2}+p\right)}}\overline{\boldsymbol{U}}\left(\frac{1}{2}\overline{{\it\rho}}\,\overline{\boldsymbol{U}}^{2}+\overline{{\it\rho}}\,\overline{W}\right)-\overline{\boldsymbol{U}}\hat{{\it\sigma}}(\overline{\boldsymbol{U}})\right.\nonumber\\ \displaystyle & & \displaystyle \quad -\,\left.\overline{\boldsymbol{u}\hat{{\it\sigma}}(\boldsymbol{u})}+\overline{\boldsymbol{u}\left(\frac{1}{2}{\it\rho}\boldsymbol{u}^{2}\right)}+\overline{{\it\rho}}\,\overline{\boldsymbol{u}w}-K\boldsymbol{{\rm\nabla}}\overline{T}\right]=0.\end{eqnarray}$$

2.4. Equation for mean entropy

The evolution equation for the instantaneous entropy $S=c_{v}\ln (P{\it\rho}^{-{\it\gamma}})$ is (Landau & Lifshitz Reference Landau and Lifshitz1959)

(2.14) $$\begin{eqnarray}\displaystyle \left(\frac{\partial }{\partial t}+\boldsymbol{U}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\right)S=\frac{1}{{\it\rho}T}[\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }(K\boldsymbol{{\rm\nabla}}T)+J], & & \displaystyle\end{eqnarray}$$

where ${\it\gamma}=c_{p}/c_{v}$ is the ratio of specific heats and $J$ is a source and/or sink of entropy (that also includes the viscous heating). Multiplying the equation for the entropy (2.14) by the fluid density  ${\it\rho}$ , and the continuity equation (2.2) by the fluid entropy  $S$ , and add them, we obtain the following equation:

(2.15) $$\begin{eqnarray}\displaystyle \frac{\partial ({\it\rho}\,S)}{\partial t}+\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }({\it\rho}\,\boldsymbol{U}S)=\frac{1}{T}[\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }(K\boldsymbol{{\rm\nabla}}T)+J]. & & \displaystyle\end{eqnarray}$$

The second term on the left-hand sides of (2.14) and (2.15), which contributes to the turbulent diffusion of the mean entropy, does not contain the temperature field. This is a reason why the turbulent flux of the mean entropy for stratified turbulence cannot contain the mean temperature.

Averaging (2.15) over the ensemble we obtain the equation for the mean entropy,  $\overline{S}$ :

(2.16) $$\begin{eqnarray}\displaystyle \frac{\partial (\overline{{\it\rho}}\,\overline{S})}{\partial t}+\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }(\overline{{\it\rho}}\,\overline{\boldsymbol{U}}\,\overline{S}+\overline{{\it\rho}}\,\overline{\boldsymbol{u}s})=\frac{1}{\overline{T}}[\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }(K\boldsymbol{{\rm\nabla}}\overline{T})+\overline{J}]. & & \displaystyle\end{eqnarray}$$

In the derivation of (2.16) we have taken into account that, for a low-Mach-number turbulent flow, $|\overline{{\it\rho}\,s}|\ll |\overline{{\it\rho}}|\,|\overline{S}|$ , $|\overline{{\it\rho}\,s}|/|\overline{{\it\rho}}|\ll |\overline{\boldsymbol{u}s}|/u_{\mathit{rms}}$ and $|\overline{\boldsymbol{u}{\it\rho}}|/|\overline{{\it\rho}}|\ll |\overline{\boldsymbol{u}s}|/|\overline{S}|$ (Chassaing et al. Reference Chassaing, Antonia, Anselmet, Joly and Sarkar2002). To get the simplest form of the molecular diffusion term and the source term on the right-hand side of (2.16) we assumed that: (a) the temperature fluctuations, ${\it\theta}$ , are much smaller than the mean fluid temperature, $\overline{T}$ , i.e.  $|{\it\theta}|\ll \overline{T}$ ; (b) in the framework of the mean-field theory there is a separation of scales, $\ell _{0}\ll L_{T}$ , where $L_{T}$ is the characteristic scale of the mean temperature variation and $\ell _{0}$ is the integral scale of turbulence (the random velocity field); (c) the coefficient of molecular heat conductivity $K$ is independent of the temperature fluctuations and (d) fluctuations of the source or sink of the entropy, $J^{\prime }$ , are independent of the temperature fluctuations. Equations (2.8) and (2.16) yield the following equation for the evolution of the mean entropy:

(2.17) $$\begin{eqnarray}\displaystyle \overline{{\it\rho}}\left(\frac{\partial \overline{S}}{\partial t}+(\overline{\boldsymbol{U}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}})\overline{S}\right)+\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }(\overline{{\it\rho}}\,\overline{\boldsymbol{u}s})=\frac{1}{\overline{T}}[\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }(K\boldsymbol{{\rm\nabla}}\overline{T})+\overline{J}], & & \displaystyle\end{eqnarray}$$

which coincides with (1.2), and the turbulent flux of the mean entropy for stratified turbulence with non-uniform profiles of the mean fluid temperature and density is

(2.18) $$\begin{eqnarray}\displaystyle \boldsymbol{F}_{s}=\overline{{\it\rho}}\,\overline{\boldsymbol{u}s}. & & \displaystyle\end{eqnarray}$$

Other forms of the turbulent flux of entropy (see (1.1)) used in the astrophysical and geophysical literature are incorrect. This is an exact statement for low-Mach-number temperature stratified turbulence and is independent of the model.

3.1. Small Péclet numbers

In order to study entropy fluctuations for small Péclet numbers we use a quasi-linear approach (Moffatt Reference Moffatt1978; Krause & Raedler Reference Krause and Raedler1980), that for a given velocity field is valid only for small Péclet numbers ( $\mathit{Pe}\ll 1$ ). In the framework of this approximation we neglect the nonlinear term and keep the molecular diffusion term in (3.1). We rewrite (3.1) in Fourier space and solve this equation. The solution is

(3.3) $$\begin{eqnarray}\displaystyle s({\it\omega},\boldsymbol{k})=G_{{\it\chi}}({\it\omega},\boldsymbol{k})I({\it\omega},\boldsymbol{k}), & & \displaystyle\end{eqnarray}$$

where $G_{{\it\chi}}({\it\omega},\boldsymbol{k})=({\it\chi}k^{2}+\text{i}{\it\omega})^{-1}$ , ${\it\omega}$ is the frequency and $\boldsymbol{k}$ is the wave vector. We apply a standard two-scale approach, whereby the non-instantaneous two-point second-order correlation function is written as follows:

(3.4) $$\begin{eqnarray}\displaystyle \overline{u_{i}(t_{1},\boldsymbol{x})\,s(t_{2},\boldsymbol{y})} & = & \displaystyle \int \overline{u_{i}({\it\omega}_{1},\boldsymbol{k}_{1})s({\it\omega}_{2},\boldsymbol{k}_{2})}\nonumber\\ \displaystyle & & \displaystyle \times \,\exp [\text{i}(\boldsymbol{k}_{1}\boldsymbol{\cdot }\boldsymbol{x}+\boldsymbol{k}_{2}\boldsymbol{\cdot }\boldsymbol{y})+\text{i}({\it\omega}_{1}t_{1}+{\it\omega}_{2}t_{2})]\,\text{d}{\it\omega}_{1}\,\text{d}{\it\omega}_{2}\,\text{d}\boldsymbol{k}_{1}\,\text{d}\boldsymbol{k}_{2}\nonumber\\ \displaystyle & = & \displaystyle \int F_{i}({\it\omega},\boldsymbol{k})\exp [\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{r}+\text{i}{\it\omega}\,\tilde{{\it\tau}}]\,\text{d}{\it\omega}\,\text{d}\boldsymbol{k},\end{eqnarray}$$

where we use large scale variables: $\boldsymbol{R}=(\boldsymbol{x}+\boldsymbol{y})/2$ , $\boldsymbol{K}=\boldsymbol{k}_{1}+\boldsymbol{k}_{2}$ , $t=(t_{1}+t_{2})/2$ , ${\it\Omega}={\it\omega}_{1}+{\it\omega}_{2}$ , and small scale variables: $\boldsymbol{r}=\boldsymbol{x}-\boldsymbol{y}$ , $\boldsymbol{k}=(\boldsymbol{k}_{1}-\boldsymbol{k}_{2})/2$ , $\tilde{{\it\tau}}=t_{1}-t_{2}$ , ${\it\omega}=({\it\omega}_{1}-{\it\omega}_{2})/2$ , and

(3.5) $$\begin{eqnarray}\displaystyle F_{i}({\it\omega},\boldsymbol{k})=\int \overline{u_{i}({\it\omega}_{1},\boldsymbol{k}_{1})\,s({\it\omega}_{2},\boldsymbol{k}_{2})}\exp [\text{i}{\it\Omega}t+\text{i}\boldsymbol{K}\boldsymbol{\cdot }\boldsymbol{R}]\,\text{d}{\it\Omega}\,\text{d}\boldsymbol{K}. & & \displaystyle\end{eqnarray}$$

Here ${\it\omega}_{1}={\it\omega}+{\it\Omega}/2$ , ${\it\omega}_{2}=-{\it\omega}+{\it\Omega}/2$ , $\boldsymbol{k}_{1}=\boldsymbol{k}+\boldsymbol{K}/2$ , and $\boldsymbol{k}_{2}=-\boldsymbol{k}+\boldsymbol{K}/2$ (see e.g. Roberts & Soward Reference Roberts and Soward1975). We assume here that there is a separation of scales, i.e. the maximum scale of random motions $\ell _{0}$ is much smaller than the characteristic scales of inhomogeneities of the mean entropy and fluid density. Equations (3.3)–(3.5) yield the velocity–entropy correlation $\overline{\boldsymbol{u}s}$ :

(3.6) $$\begin{eqnarray}\displaystyle \overline{u_{i}s}=\int \overline{u_{i}({\it\omega},\boldsymbol{k})I(-{\it\omega},-\boldsymbol{k})}G_{{\it\chi}}^{\ast }\,\text{d}{\it\omega}\,\text{d}\boldsymbol{k}=-({\rm\nabla}_{j}\overline{S})\int \overline{u_{i}({\it\omega},\boldsymbol{k})u_{j}(-{\it\omega},-\boldsymbol{k})}G_{{\it\chi}}^{\ast }\,\text{d}{\it\omega}\,\text{d}\boldsymbol{k}. & & \displaystyle\end{eqnarray}$$

We use the simple model for the second moments, $\overline{u_{i}({\it\omega},\boldsymbol{k})u_{j}(-{\it\omega},-\boldsymbol{k})}$ , of a random velocity field in Fourier space for inhomogeneous, isotropic and incompressible flow:

(3.7) $$\begin{eqnarray}\displaystyle \overline{u_{i}({\it\omega},\boldsymbol{k})u_{j}(-{\it\omega},-\boldsymbol{k})}=\frac{\tilde{E}(k){\it\Phi}({\it\omega})}{8{\rm\pi}k^{2}}\left[{\it\delta}_{ij}-\frac{k_{i}k_{j}}{k^{2}}+\frac{\text{i}}{2k^{2}}(k_{i}{\rm\nabla}_{j}-k_{j}{\rm\nabla}_{i})\right]\overline{\boldsymbol{u}^{2}}. & & \displaystyle\end{eqnarray}$$

This model is obtained using symmetry arguments (see e.g. Batchelor Reference Batchelor1971). Here ${\it\delta}_{ij}$ is the Kronecker tensor, the energy spectrum function is $\tilde{E}(k)=C_{E}k_{0}^{-1}\,(q-1)\,(k/k_{0})^{-q}$ for the range of wavenumbers $k_{0}<k<k_{d}$ , the wavenumber $k_{0}=1/\ell _{0}$ , the length $\ell _{0}$ is the maximum scale of random motions, the exponent $1<q<3$ , and the coefficient $C_{E}=[1-(k_{0}/k_{d})^{q-1}]^{-1}$ . We use the frequency function ${\it\Phi}({\it\omega})$ in the form of the Lorentz profile: ${\it\Phi}({\it\omega})=[{\rm\pi}{\it\tau}_{c}\,({\it\omega}^{2}+{\it\tau}_{c}^{-2})]^{-1}$ , where ${\it\tau}_{c}$ is the correlation time of a random velocity field. This model for the frequency function corresponds to the following non-instantaneous correlation function: $\overline{u_{i}(t)u_{j}(t+{\it\tau})}\propto \exp (-{\it\tau}/{\it\tau}_{c})$ .

We use (3.6) and (3.7), and after integration in ${\it\omega}$ -space and in $\boldsymbol{k}$ -space in (3.6) we obtain the formula for the velocity–entropy correlation $\overline{u_{i}\,s}$ :

(3.8) $$\begin{eqnarray}\displaystyle \overline{u_{i}\,s}=-{\it\chi}_{T}{\rm\nabla}_{i}\overline{S}, & & \displaystyle\end{eqnarray}$$

(3.9a,b ) $$\begin{eqnarray}\displaystyle {\it\chi}_{T}=C_{{\it\chi}}u_{0}\ell _{0}\,\mathit{Pe},\quad C_{{\it\chi}}=\frac{q-1}{3(q+1)}\left[\frac{1-(k_{0}/k_{d})^{q+1}}{1-(k_{0}/k_{d})^{q-1}}\right], & & \displaystyle\end{eqnarray}$$

where ${\it\chi}_{T}$ is the coefficient of turbulent diffusion of the mean entropy and $u_{0}=\sqrt{\overline{\boldsymbol{u}^{2}}}$ is the characteristic velocity in the maximum scale of random motions. Here we used that $I_{0}\equiv \int {\it\Phi}({\it\omega})G_{{\it\chi}}({\it\omega},\boldsymbol{k})\,\text{d}{\it\omega}={\it\tau}_{c}/(1+{\it\tau}_{c}{\it\eta}k^{2})$ , and for small Péclet numbers $I_{0}\approx ({\it\eta}k^{2})^{-1}$ . The coefficient $C_{{\it\chi}}=1/3$ for a narrow range of the random velocity field in the wavenumbers, $k_{d}-k_{0}\ll k_{d}$ , and $C_{{\it\chi}}=(q-1)/3(q+1)$ for a wide range in the wavenumbers, $k_{d}\gg k_{0}$ . Contributions (which are proportional to $\boldsymbol{{\rm\nabla}}$ in (3.7)) to the velocity–entropy correlation $\overline{u_{i}\,s}$ , after the integration over the angles in $\boldsymbol{k}$ -space, vanish. However, the coefficient of turbulent diffusion ${\it\chi}_{T}$ depends on the coordinates, due to the inhomogeneous turbulence. Equations (3.8) and (3.9a,b ) are in agreement with those obtained by means of dimensional arguments (Batchelor, Howells & Townsend Reference Batchelor, Howells and Townsend1959) and by the Lagrangian-history direct-interaction approximation (Kraichnan Reference Kraichnan1968).

3.2. Large Péclet numbers

In this subsection we derive a formula for the velocity–entropy correlation $\overline{\boldsymbol{u}s}$ using the spectral ${\it\tau}$ approach that is valid for large Péclet and Reynolds numbers ( $\mathit{Pe}\gg 1$ ). Using (3.1) written in Fourier space we derive an equation for the instantaneous two-point second-order correlation functions $F_{i}(t,\boldsymbol{k})=\overline{u_{i}(t,\boldsymbol{k})\,s(t,-\boldsymbol{k})}$ :

(3.10) $$\begin{eqnarray}\displaystyle \frac{\text{d}F_{i}}{\text{d}t}=\overline{u_{i}(t,\boldsymbol{k})\,I(t,-\boldsymbol{k})}+\hat{{\mathcal{M}}}F_{i}^{(III)}(\boldsymbol{k}), & & \displaystyle\end{eqnarray}$$

where $\hat{{\mathcal{M}}}F_{i}^{(III)}(\boldsymbol{k})=-[\overline{u_{i}\,{\mathcal{N}}}+\overline{(\partial u_{i}/\partial t)\,s}-{\it\chi}\overline{u_{i}\,{\rm\nabla}^{2}s}]_{\boldsymbol{k}}$ are the third-order moment terms appearing due to the nonlinear terms which also include the molecular diffusion term.

The equation for the second moment includes the first-order spatial differential operators applied to the third-order moments. A problem arises regarding how to close the system, i.e. how to express the third-order terms $\hat{{\mathcal{M}}}F^{(III)}$ through the lower moments $F^{(II)}$ (Orszag Reference Orszag1970; Monin & Yaglom Reference Monin and Yaglom1975; McComb Reference McComb1990). We use the spectral ${\it\tau}$ approximation which postulates that the deviations of the third-moment terms, $\hat{{\mathcal{M}}}F^{(III)}(\boldsymbol{k})$ , from the contributions to these terms by the background turbulence, $\hat{{\mathcal{M}}}F^{(III,0)}(\boldsymbol{k})$ , can be expressed through similar deviations of the second moments, $F^{(II)}(\boldsymbol{k})-F^{(II,0)}(\boldsymbol{k})$ (Orszag Reference Orszag1970; Pouquet, Frisch & Leorat Reference Pouquet, Frisch and Leorat1976):

(3.11) $$\begin{eqnarray}\displaystyle \hat{{\mathcal{M}}}F^{(III)}(\boldsymbol{k})-\hat{{\mathcal{M}}}F^{(III,0)}(\boldsymbol{k})=-\frac{1}{{\it\tau}_{r}(k)}[F^{(II)}(\boldsymbol{k})-F^{(II,0)}(\boldsymbol{k})], & & \displaystyle\end{eqnarray}$$

where ${\it\tau}_{r}(k)$ is the scale-dependent relaxation time, which can be identified with the correlation time ${\it\tau}(k)$ of the turbulent velocity field for large Reynolds and Péclet numbers. The functions with the superscript (0) correspond to the background turbulence with a zero gradient of the mean entropy. Validation of the ${\it\tau}$ approximation for different situations has been performed in numerous numerical simulations and analytical studies (see e.g. the review by Brandenburg & Subramanian Reference Brandenburg and Subramanian2005; and also discussions by Rogachevskii & Kleeorin Reference Rogachevskii and Kleeorin2007, Rogachevskii et al. Reference Rogachevskii, Kleeorin, Käpylä and Brandenburg2011).

Note that the contributions of the terms with the superscript (0) vanish because when the gradient of the mean entropy is zero, the turbulent heat flux and the entropy fluctuations vanish. Consequently, (3.11) for $\hat{{\mathcal{M}}}F_{i}^{(III)}(\boldsymbol{k})$ is reduced to $\hat{{\mathcal{M}}}F_{i}^{(III)}(\boldsymbol{k})=-F_{i}(\boldsymbol{k})/{\it\tau}(k)$ . We also assume that the characteristic time of variation of the second moment $F_{i}(\boldsymbol{k})$ is substantially larger than the correlation time ${\it\tau}(k)$ for all turbulence scales. Therefore, in a steady state (3.10) yields the following formula for the velocity–entropy correlation:

(3.12) $$\begin{eqnarray}\displaystyle F_{i}=\int {\it\tau}(k)\langle u_{i}(t,\boldsymbol{k})I(t,-\boldsymbol{k})\rangle \,\text{d}\boldsymbol{k}=-({\rm\nabla}_{j}\overline{S})\int {\it\tau}(k)\langle u_{i}(\boldsymbol{k})u_{j}(-\boldsymbol{k})\rangle \,\text{d}\boldsymbol{k}. & & \displaystyle\end{eqnarray}$$

We use the following simple model for the second moments, $\overline{u_{i}(\boldsymbol{k})u_{j}(-\boldsymbol{k})}$ , of a turbulent velocity field in Fourier space for inhomogeneous, isotropic and incompressible flow for large Reynolds numbers:

(3.13) $$\begin{eqnarray}\displaystyle \overline{u_{i}(\boldsymbol{k})u_{j}(-\boldsymbol{k})}=\frac{\tilde{E}(k)}{8{\rm\pi}k^{2}}\left[{\it\delta}_{ij}-\frac{k_{i}k_{j}}{k^{2}}+\frac{\text{i}}{2k^{2}}(k_{i}{\rm\nabla}_{j}-k_{j}{\rm\nabla}_{i})\right]\overline{\boldsymbol{u}^{2}}. & & \displaystyle\end{eqnarray}$$

This model is obtained using symmetry arguments (see e.g. Batchelor Reference Batchelor1971). After integration in $\boldsymbol{k}$ -space of (3.12) we arrive at an equation for the velocity–entropy correlation, $\langle u_{i}s\rangle$ :

(3.14) $$\begin{eqnarray}\displaystyle \langle u_{i}s\rangle =-{\it\chi}_{T}{\rm\nabla}_{i}\overline{S},\quad {\it\chi}_{T}=u_{0}\ell _{0}/3, & & \displaystyle\end{eqnarray}$$

where $u_{0}=\sqrt{\overline{\boldsymbol{u}^{2}}}$ is the characteristic turbulent velocity. In the derivation of (3.14) we used the following expression for the turbulent correlation time: ${\it\tau}(k)=2{\it\tau}_{0}(k/k_{0})^{1-q}$ , where ${\it\tau}_{0}=\ell _{0}/u_{0}$ is the characteristic turbulent time. Contributions (which are proportional to $\boldsymbol{{\rm\nabla}}$ in (3.13)) to the velocity–entropy correlation $\overline{u_{i}s}$ , after integration over the angles in $\boldsymbol{k}$ -space, vanish. However, the coefficient of turbulent diffusion ${\it\chi}_{T}$ depends on the coordinates, due to the inhomogeneous turbulence. Therefore, the formulae for the velocity–entropy correlation, $\langle u_{i}s\rangle$ , are similar for small and large Péclet numbers, while the coefficients of turbulent diffusion of the mean entropy are different in these two limiting cases. Equation (3.14) is in agreement with that derived by means of the path integral approach (Elperin, Kleeorin & Rogachevskii Reference Elperin, Kleeorin and Rogachevskii1995), by dimensional arguments and by the renormalization procedure used for large Péclet numbers (Elperin, Kleeorin & Rogachevskii Reference Elperin, Kleeorin and Rogachevskii1996).