2.1 Fundamental equations

The equations of the density $\unicode[STIX]{x1D70C}$ , the velocity $\boldsymbol{u}$ , the internal energy $q$ and the magnetic field $\boldsymbol{b}$ are given by

(2.1) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D70C}\boldsymbol{u})=0, & \displaystyle\end{eqnarray}$$

(2.2) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\unicode[STIX]{x1D70C}u^{\unicode[STIX]{x1D6FC}}+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x^{a}}\unicode[STIX]{x1D70C}u^{a}u^{\unicode[STIX]{x1D6FC}}=-\frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}x^{\unicode[STIX]{x1D6FC}}}+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x^{a}}\unicode[STIX]{x1D707}s^{a\unicode[STIX]{x1D6FC}}+\frac{1}{\unicode[STIX]{x1D707}_{0}}(\boldsymbol{j}\times \boldsymbol{b})^{\unicode[STIX]{x1D6FC}}+f_{\text{ex}}^{\unicode[STIX]{x1D6FC}}, & \displaystyle\end{eqnarray}$$

(2.3) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\unicode[STIX]{x1D70C}q+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D70C}\boldsymbol{u}q)=\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D705}\unicode[STIX]{x1D735}\unicode[STIX]{x1D703})-p\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}+\unicode[STIX]{x1D719}, & \displaystyle\end{eqnarray}$$

(2.4) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\boldsymbol{b}}{\unicode[STIX]{x2202}t}=-\unicode[STIX]{x1D735}\times \boldsymbol{e}, & \displaystyle\end{eqnarray}$$

with Ohm’s law for moving media:

(2.5) $$\begin{eqnarray}\boldsymbol{j}=\frac{1}{\unicode[STIX]{x1D707}_{0}}\unicode[STIX]{x1D735}\times \boldsymbol{b}=\unicode[STIX]{x1D70E}(\boldsymbol{e}+\boldsymbol{u}\times \boldsymbol{b}),\end{eqnarray}$$

where $p$ is the plasma gas pressure, $\unicode[STIX]{x1D707}$ the viscosity, $\boldsymbol{j}$ the electric-current density, $\boldsymbol{e}$ the electric field, $\unicode[STIX]{x1D705}$ the thermal conductivity, $\unicode[STIX]{x1D707}_{0}$ the magnetic permeability, $\unicode[STIX]{x1D70E}$ the electric conductivity and $s^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}$ the deviatoric or traceless part of the velocity strain defined by

(2.6) $$\begin{eqnarray}s^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}=\frac{\unicode[STIX]{x2202}u^{\unicode[STIX]{x1D6FD}}}{\unicode[STIX]{x2202}x^{\unicode[STIX]{x1D6FC}}}+\frac{\unicode[STIX]{x2202}u^{\unicode[STIX]{x1D6FC}}}{\unicode[STIX]{x2202}x^{\unicode[STIX]{x1D6FD}}}-\frac{2}{3}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}\unicode[STIX]{x1D6FF}^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}.\end{eqnarray}$$

In (2.2), $\boldsymbol{f}_{\text{ex}}$ represents the external forces including the gravity force and external forcing. In what follows, we pay more attention to the theoretical formulation of the nonlinear turbulence dynamics that does not directly depend on specific forcing effects, so we neglect $\boldsymbol{f}_{\text{ex}}$ hereafter. In (2.3) $\unicode[STIX]{x1D719}$ is the dissipation function that represents the conversion of kinetic and magnetic energies to heat through the molecular effects:

(2.7) $$\begin{eqnarray}\unicode[STIX]{x1D719}=\unicode[STIX]{x1D707}s^{ab}\frac{\unicode[STIX]{x2202}u^{a}}{\unicode[STIX]{x2202}x^{b}}+\frac{1}{\unicode[STIX]{x1D70E}}\boldsymbol{j}^{2}.\end{eqnarray}$$

In the following theoretical calculations, $\unicode[STIX]{x1D719}$ is also neglected since our main interests lie in the $p$ -related effects in the turbulent transport. Note that these treatments do not deny the importance of the external forces and the dissipation function. Actually, in order to sustain turbulence and its evolution properties, including energies and helicities, those effects play important roles.

The pressure $p$ is related to the temperature $\unicode[STIX]{x1D703}$ and the internal energy $q$ as

(2.8) $$\begin{eqnarray}p=R\unicode[STIX]{x1D70C}\unicode[STIX]{x1D703}=(\unicode[STIX]{x1D6FE}_{\text{s}}-1)\unicode[STIX]{x1D70C}q,\end{eqnarray}$$

where

(2.9) $$\begin{eqnarray}q=C_{V}(\unicode[STIX]{x1D703})\unicode[STIX]{x1D703}.\end{eqnarray}$$

Here, $C_{V}$ is the specific heat at constant volume, $R$ is the gas constant and $\unicode[STIX]{x1D6FE}_{\text{s}}$ is the ratio of $C_{P}$ (the specific heat at constant pressure) to $C_{V}$ .

From (2.4) and (2.5), the induction equation of the magnetic field is written as

(2.10) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\boldsymbol{b}}{\unicode[STIX]{x2202}t}=\unicode[STIX]{x1D735}\times (\boldsymbol{u}\times \boldsymbol{b})+\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FB}^{2}\boldsymbol{b}\end{eqnarray}$$

or

(2.11) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\boldsymbol{b}}{\unicode[STIX]{x2202}t}+(\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{b}=(\boldsymbol{b}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{u}-\boldsymbol{b}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}+\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FB}^{2}\boldsymbol{b},\end{eqnarray}$$

where $\unicode[STIX]{x1D702}$ is the magnetic diffusivity defined as $\unicode[STIX]{x1D702}=1/(\unicode[STIX]{x1D70E}\unicode[STIX]{x1D707}_{0})$ .

2.2 Mean-field equations

In contrast to the constant density cases, the physical interpretation of the mean-field transport expressions is not unique when the density varies. Due to the additional correlations originating from the variable density, the system of the mean-field transport equations is much more complicated. The structure and the meaning of the transport expression; which turbulence correlations appear and what is their physical interpretation, depend on the averaging formulation. If we adopt mass-weighted averaging, the density-fluctuation correlations can be formally removed from the mean transport expressions. This allows the variable density case to be treated in a strong analogy with the constant density case. Because of this property, mass-weighted averaging is often adopted in compressible turbulence studies. As an example of the recent works with mass-weighted decomposition, Aluie (Reference Aluie2011) applied this formulation to the energy transfer function of compressible turbulence, and argued that the cascade of the mean kinetic energy occurs in a conservative and scale-local manner due to the statistical decoupling between the mean kinetic and internal-energy budgets.

Although all the different averaging formulations are algebraically equivalent to each other, their physical interpretations of the turbulence correlations can be entirely different. As for detailed descriptions on the statistical averaging in variable density fluid turbulent motion, the reader is referred to chapter 5 of Chassaing et al. (Reference Chassaing, Antonia, Anselmet, Joly and Sarkar2002), which includes the formal mathematical relationship and physical comparison between the mass-weighted and Reynolds averaging.

In the mass-weighted averaging formulation, all the density-fluctuation correlations are implicitly embedded in the mean values. This makes it very difficult to identify all the density-fluctuation correlation effects. In order to see the explicit effects of the density fluctuation in the mass and internal-energy equations, we adopt ensemble or Reynolds averaging in this work.

With the Reynolds averaging denoted by $\langle \cdots \!\rangle$ , a field quantity $f$ is divided into the mean $F$ and the fluctuation around it, $f^{\prime }$ , as

(2.12) $$\begin{eqnarray}f=F+f^{\prime },\quad F=\langle f\rangle ,\end{eqnarray}$$

with

(2.13a ) $$\begin{eqnarray}\displaystyle & f=(\unicode[STIX]{x1D70C},\boldsymbol{u},q,\unicode[STIX]{x1D703},\boldsymbol{b},\boldsymbol{j},\boldsymbol{e}), & \displaystyle\end{eqnarray}$$

(2.13b ) $$\begin{eqnarray}\displaystyle & F=(\overline{\unicode[STIX]{x1D70C}},\boldsymbol{U},Q,\unicode[STIX]{x1D6E9},\boldsymbol{B},\boldsymbol{J},\boldsymbol{E}), & \displaystyle\end{eqnarray}$$

(2.13c ) $$\begin{eqnarray}\displaystyle & f^{\prime }=(\unicode[STIX]{x1D70C}^{\prime },\boldsymbol{u}^{\prime },q^{\prime },\unicode[STIX]{x1D703}^{\prime },\boldsymbol{b}^{\prime },\boldsymbol{j}^{\prime },\boldsymbol{e}^{\prime }). & \displaystyle\end{eqnarray}$$

With the Reynolds decomposition (2.12), the mean density equation is given by

(2.14) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\overline{\unicode[STIX]{x1D70C}}}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\overline{\unicode[STIX]{x1D70C}}\boldsymbol{U})=-\unicode[STIX]{x1D735}\boldsymbol{\cdot }\langle \unicode[STIX]{x1D70C}^{\prime }\boldsymbol{u}^{\prime }\rangle .\end{eqnarray}$$

The mean velocity equation is written as

(2.15) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\overline{\unicode[STIX]{x1D70C}}U^{\unicode[STIX]{x1D6FC}}+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x^{a}}\overline{\unicode[STIX]{x1D70C}}U^{a}U^{\unicode[STIX]{x1D6FC}}=-(\unicode[STIX]{x1D6FE}_{s}-1)\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x^{\unicode[STIX]{x1D6FC}}}\overline{\unicode[STIX]{x1D70C}}Q+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x^{\unicode[STIX]{x1D6FC}}}\overline{\unicode[STIX]{x1D707}}{\mathcal{S}}^{a\unicode[STIX]{x1D6FC}}+(\boldsymbol{J}\times \boldsymbol{B})^{\unicode[STIX]{x1D6FC}}\nonumber\\ \displaystyle & & \displaystyle \quad -\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x^{\unicode[STIX]{x1D6FC}}}\left(\overline{\unicode[STIX]{x1D70C}}\langle u^{\prime a}u^{\prime \unicode[STIX]{x1D6FC}}\rangle -\frac{1}{\unicode[STIX]{x1D707}_{0}}\langle b^{\prime a}b^{\prime \unicode[STIX]{x1D6FC}}\rangle +U^{a}\langle \unicode[STIX]{x1D70C}^{\prime }u^{\prime \unicode[STIX]{x1D6FC}}\rangle +U^{\unicode[STIX]{x1D6FC}}\langle \unicode[STIX]{x1D70C}^{\prime }u^{\prime a}\rangle \right)+R_{U}^{\unicode[STIX]{x1D6FC}},\end{eqnarray}$$

where $\overline{\unicode[STIX]{x1D707}}$ is the mean part of the viscosity and $\unicode[STIX]{x1D64E}=\{{\mathcal{S}}^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}\}$ is the deviatoric or traceless part of the mean velocity strain tensor defined by

(2.16) $$\begin{eqnarray}{\mathcal{S}}^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}=\frac{\unicode[STIX]{x2202}U^{\unicode[STIX]{x1D6FD}}}{\unicode[STIX]{x2202}x^{\unicode[STIX]{x1D6FC}}}+\frac{\unicode[STIX]{x2202}U^{\unicode[STIX]{x1D6FC}}}{\unicode[STIX]{x2202}x^{\unicode[STIX]{x1D6FD}}}-\frac{2}{3}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{U}\unicode[STIX]{x1D6FF}^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}.\end{eqnarray}$$

The mean internal-energy equation is written as

(2.17) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\overline{\unicode[STIX]{x1D70C}}Q+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\overline{\unicode[STIX]{x1D70C}}\boldsymbol{U}Q)=\unicode[STIX]{x1D735}\boldsymbol{\cdot }\left(\frac{\overline{\unicode[STIX]{x1D705}}}{C_{V}}\unicode[STIX]{x1D735}Q\right)-\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\overline{\unicode[STIX]{x1D70C}}\langle q^{\prime }\boldsymbol{u}^{\prime }\rangle +Q\langle \unicode[STIX]{x1D70C}^{\prime }\boldsymbol{u}^{\prime }\rangle +\boldsymbol{U}\langle \unicode[STIX]{x1D70C}^{\prime }q^{\prime }\rangle )\nonumber\\ \displaystyle & & \displaystyle \quad -\,(\unicode[STIX]{x1D6FE}_{s}-1)(\overline{\unicode[STIX]{x1D70C}}Q\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{U}+\overline{\unicode[STIX]{x1D70C}}\langle {q^{\prime }\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}}^{\prime }\rangle +Q\langle \unicode[STIX]{x1D70C}^{\prime }\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}^{\prime }\rangle )+R_{Q},\end{eqnarray}$$

where $\overline{\unicode[STIX]{x1D705}}$ is the mean part of the diffusivity. And the mean magnetic induction equation is written as

(2.18) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}\boldsymbol{B}}{\unicode[STIX]{x2202}t}=\unicode[STIX]{x1D735}\times (\boldsymbol{U}\times \boldsymbol{B}+\langle \boldsymbol{u}^{\prime }\times \boldsymbol{b}^{\prime }\rangle )+\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FB}^{2}\boldsymbol{B}. & & \displaystyle\end{eqnarray}$$

In (2.15) and (2.17), $\boldsymbol{R}_{U}$ and $R_{Q}$ are the terms expected to be small, whose detailed expressions are suppressed here.

In the mean-field equations (2.14), (2.15), (2.17) and (2.18), we have the turbulent mass flux $\langle \unicode[STIX]{x1D70C}^{\prime }\boldsymbol{u}^{\prime }\rangle$ , the Reynolds stress $\langle {\boldsymbol{u}^{\prime }\boldsymbol{u}}^{\prime }\rangle$ , the turbulent Maxwell stress $\langle {\boldsymbol{b}^{\prime }\boldsymbol{b}}^{\prime }\rangle$ , the turbulent heat flux $\langle q^{\prime }\boldsymbol{u}^{\prime }\rangle$ , the turbulent electromotive force $\langle \boldsymbol{u}^{\prime }\times \boldsymbol{b}^{\prime }\rangle$ , etc. These turbulence correlations represent the turbulence effects and play key roles in determining the effective transports due to turbulence. The evaluation of these turbulent correlations is the central objective of this work.

2.3 Fluctuation equations

Subtracting the mean-field equations (2.14), (2.15), (2.17) and (2.18) from the fundamental equations (2.1)–(2.3) and (2.11), we obtain the equations of the fluctuations as

(2.19) $$\begin{eqnarray}\frac{\text{D}\unicode[STIX]{x1D70C}^{\prime }}{\text{D}t}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D70C}^{\prime }\boldsymbol{u}^{\prime })+\overline{\unicode[STIX]{x1D70C}}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}^{\prime }=-(\boldsymbol{u}^{\prime }\boldsymbol{\cdot }\unicode[STIX]{x1D735})\overline{\unicode[STIX]{x1D70C}}-\unicode[STIX]{x1D70C}^{\prime }\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{U}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }\langle \unicode[STIX]{x1D70C}^{\prime }\boldsymbol{u}^{\prime }\rangle ,\end{eqnarray}$$

(2.20) $$\begin{eqnarray}\displaystyle \frac{\text{D}u^{\prime \unicode[STIX]{x1D6FC}}}{\text{D}t} & = & \displaystyle -(\boldsymbol{u}^{\prime }\boldsymbol{\cdot }\unicode[STIX]{x1D735})u^{\prime \unicode[STIX]{x1D6FC}}+\frac{1}{\overline{\unicode[STIX]{x1D70C}}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x^{a}}\overline{\unicode[STIX]{x1D707}}{s^{\prime }}^{a\unicode[STIX]{x1D6FC}}-(\unicode[STIX]{x1D6FE}_{\text{s}}-1)\left(\frac{\unicode[STIX]{x2202}q^{\prime }}{\unicode[STIX]{x2202}x^{\unicode[STIX]{x1D6FC}}}+\frac{Q}{\overline{\unicode[STIX]{x1D70C}}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}^{\prime }}{\unicode[STIX]{x2202}x^{\unicode[STIX]{x1D6FC}}}\right)\nonumber\\ \displaystyle & & \displaystyle +\,\frac{1}{\unicode[STIX]{x1D707}_{0}\overline{\unicode[STIX]{x1D70C}}}[(\boldsymbol{b}^{\prime }\boldsymbol{\cdot }\unicode[STIX]{x1D735})b^{\prime \unicode[STIX]{x1D6FC}}+(\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735})b^{\prime \unicode[STIX]{x1D6FC}}]-(\boldsymbol{u}^{\prime }\boldsymbol{\cdot }\unicode[STIX]{x1D735})U^{\unicode[STIX]{x1D6FC}}-(\unicode[STIX]{x1D6FE}_{\text{s}}-1)\left(\frac{\unicode[STIX]{x1D70C}^{\prime }}{\overline{\unicode[STIX]{x1D70C}}}\frac{\unicode[STIX]{x2202}Q}{\unicode[STIX]{x2202}x^{\unicode[STIX]{x1D6FC}}}+\frac{q^{\prime }}{\overline{\unicode[STIX]{x1D70C}}}\frac{\unicode[STIX]{x2202}\overline{\unicode[STIX]{x1D70C}}}{\unicode[STIX]{x2202}x^{\unicode[STIX]{x1D6FC}}}\right)\nonumber\\ \displaystyle & & \displaystyle +\,\frac{1}{\unicode[STIX]{x1D707}_{0}\overline{\unicode[STIX]{x1D70C}}}(\boldsymbol{b}^{\prime }\boldsymbol{\cdot }\unicode[STIX]{x1D735})B^{\unicode[STIX]{x1D6FC}}-\frac{\unicode[STIX]{x1D70C}^{\prime }}{\overline{\unicode[STIX]{x1D70C}}}\frac{\text{D}U^{\unicode[STIX]{x1D6FC}}}{\text{D}t}+R_{u}^{\unicode[STIX]{x1D6FC}},\end{eqnarray}$$

(2.21) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\text{D}q^{\prime }}{\text{D}t}+(\boldsymbol{u}^{\prime }\boldsymbol{\cdot }\unicode[STIX]{x1D735})q^{\prime }-\frac{1}{\overline{\unicode[STIX]{x1D70C}}}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\left(\frac{\overline{\unicode[STIX]{x1D705}}}{C_{V}}\unicode[STIX]{x1D735}q^{\prime }\right)+(\unicode[STIX]{x1D6FE}_{\text{s}}-1)Q\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}^{\prime }\nonumber\\ \displaystyle & & \displaystyle \quad =-(\boldsymbol{u}^{\prime }\boldsymbol{\cdot }\unicode[STIX]{x1D735})Q-(\unicode[STIX]{x1D6FE}_{\text{s}}-1)\left(q^{\prime }+\frac{\unicode[STIX]{x1D70C}^{\prime }}{\overline{\unicode[STIX]{x1D70C}}}Q\right)\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{U},\end{eqnarray}$$

(2.22) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\text{D}\boldsymbol{b}^{\prime }}{\text{D}t}+(\boldsymbol{u}^{\prime }\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{b}^{\prime }-(\boldsymbol{b}^{\prime }\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{u}^{\prime }-\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FB}^{2}\boldsymbol{b}^{\prime }-(\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{u}^{\prime }+\boldsymbol{B}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}^{\prime }\nonumber\\ \displaystyle & & \displaystyle \quad =-(\boldsymbol{u}^{\prime }\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{B}+(\boldsymbol{b}^{\prime }\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{U}-\boldsymbol{b}^{\prime }\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{U}+R_{b}^{\unicode[STIX]{x1D6FC}},\end{eqnarray}$$

where $\text{D}/\text{D}t(=\unicode[STIX]{x2202}/\unicode[STIX]{x2202}t+\boldsymbol{U}\boldsymbol{\cdot }\unicode[STIX]{x1D735})$ is the mean-flow convective derivative, and $\boldsymbol{R}_{u}$ and $\boldsymbol{R}_{b}$ are the higher-order terms whose detailed expressions are suppressed here.

If the fluctuations or perturbations are small, the nonlinear products of them can be neglected in the evolution of the fluctuation fields. This quasi-linearity is often the starting point of the linear wave and instability studies. However, this is not the case for fully developed turbulent flows with astrophysical and geophysical interests. In realistic turbulent flows with large kinetic and magnetic Reynolds numbers, the nonlinear coupling terms, such as $\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D70C}^{\prime }\boldsymbol{u}^{\prime })$ in (2.19), $(\boldsymbol{u}^{\prime }\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{u}^{\prime }$ and $(\boldsymbol{b}^{\prime }\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{b}^{\prime }$ in (2.20), $(\boldsymbol{u}^{\prime }\boldsymbol{\cdot }\unicode[STIX]{x1D735})q^{\prime }$ in (2.21), $(\boldsymbol{u}^{\prime }\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{b}^{\prime }$ and $(\boldsymbol{b}^{\prime }\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{u}^{\prime }$ in (2.22), etc., are not small and cannot be neglected at all. Here, we have to simultaneously treat the effects of strong nonlinearity and inhomogeneity. In the following section, we present a framework of how to address such nonlinear and inhomogeneous turbulent flows.

4.1 Turbulent mass flux

The turbulent mass flux is calculated as

(4.1) $$\begin{eqnarray}\displaystyle \langle \unicode[STIX]{x1D70C}^{\prime }\boldsymbol{u}^{\prime }\rangle & = & \displaystyle \langle \unicode[STIX]{x1D70C}_{0}^{\prime }\boldsymbol{u}_{0}^{\prime }\rangle +\langle \unicode[STIX]{x1D70C}_{0}^{\prime }\boldsymbol{u}_{1}^{\prime }\rangle +\langle \unicode[STIX]{x1D70C}_{1}^{\prime }\boldsymbol{u}_{0}^{\prime }\rangle +\cdots \nonumber\\ \displaystyle & = & \displaystyle \langle \unicode[STIX]{x1D70C}_{\text{B}}^{\prime }\boldsymbol{u}_{\text{B}}^{\prime }\rangle +\langle \unicode[STIX]{x1D70C}_{\text{B}}^{\prime }\boldsymbol{u}_{01}^{\prime }\rangle +\langle \unicode[STIX]{x1D70C}_{\text{B}}^{\prime }\boldsymbol{u}_{10}^{\prime }\rangle +\cdots \nonumber\\ \displaystyle & & \displaystyle +\,\langle \unicode[STIX]{x1D70C}_{01}^{\prime }\boldsymbol{u}_{\text{B}}^{\prime }\rangle +\langle \unicode[STIX]{x1D70C}_{01}^{\prime }\boldsymbol{u}_{01}^{\prime }\rangle +\cdots +\langle \unicode[STIX]{x1D70C}_{10}^{\prime }\boldsymbol{u}_{\text{B}}^{\prime }\rangle +\cdots \,.\end{eqnarray}$$

From (3.16)–(3.23), each term in (4.1) is expressed as

(4.2) $$\begin{eqnarray}\displaystyle & \displaystyle \int \,\text{d}\boldsymbol{k}\langle \unicode[STIX]{x1D70C}_{\text{B}}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F})u_{\text{B}}^{\prime }\,^{\unicode[STIX]{x1D6FC}}(-\boldsymbol{k};\unicode[STIX]{x1D70F})\rangle /\unicode[STIX]{x1D6FF}(\mathbf{0})=0, & \displaystyle\end{eqnarray}$$

(4.3) $$\begin{eqnarray}\displaystyle & \displaystyle \int \,\text{d}\boldsymbol{k}\langle \unicode[STIX]{x1D70C}_{\text{B}}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F})u_{01}^{\prime }\,^{\unicode[STIX]{x1D6FC}}(-\boldsymbol{k};\unicode[STIX]{x1D70F})\rangle /\unicode[STIX]{x1D6FF}(\mathbf{0})=0, & \displaystyle\end{eqnarray}$$

(4.4) $$\begin{eqnarray}\displaystyle & & \displaystyle \int \,\text{d}\boldsymbol{k}\langle \unicode[STIX]{x1D70C}_{\text{B}}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F})u_{10}^{\prime }\,^{\unicode[STIX]{x1D6FC}}(-\boldsymbol{k};\unicode[STIX]{x1D70F})\rangle /\unicode[STIX]{x1D6FF}(\mathbf{0})\nonumber\\ \displaystyle & & \displaystyle \quad =-\frac{1}{3}(\unicode[STIX]{x1D6FE}_{\text{s}}-1)\frac{1}{\overline{\unicode[STIX]{x1D70C}}}\frac{\unicode[STIX]{x2202}Q}{\unicode[STIX]{x2202}X^{\unicode[STIX]{x1D6FC}}}\int \,\text{d}\boldsymbol{k}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}[2G_{u\text{S}}(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})+G_{u\text{C}}(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})]Q_{\unicode[STIX]{x1D70C}}(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})\nonumber\\ \displaystyle & & \displaystyle \qquad -\,\frac{1}{3}\frac{1}{\overline{\unicode[STIX]{x1D70C}}}\frac{\text{D}U^{\unicode[STIX]{x1D6FC}}}{\text{D}T}\int \,\text{d}\boldsymbol{k}~\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}[2G_{u\text{S}}(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})+G_{u\text{C}}(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})]Q_{\unicode[STIX]{x1D70C}}(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1}),\end{eqnarray}$$

(4.5) $$\begin{eqnarray}\int \,\text{d}\boldsymbol{k}\langle \unicode[STIX]{x1D70C}_{01}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F})u_{\text{B}}^{\prime }\,^{\unicode[STIX]{x1D6FC}}(-\boldsymbol{k};\unicode[STIX]{x1D70F})\rangle /\unicode[STIX]{x1D6FF}(\mathbf{0})=0,\end{eqnarray}$$

(4.6) $$\begin{eqnarray}\displaystyle & & \displaystyle \int \,\text{d}\boldsymbol{k}\langle \unicode[STIX]{x1D70C}_{1}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F})u_{\text{B}}^{\prime }\,^{\unicode[STIX]{x1D6FC}}(-\boldsymbol{k};\unicode[STIX]{x1D70F})\rangle /\unicode[STIX]{x1D6FF}(\mathbf{0})\nonumber\\ \displaystyle & & \displaystyle \quad =-\frac{1}{3}\frac{\unicode[STIX]{x2202}\overline{\unicode[STIX]{x1D70C}}}{\unicode[STIX]{x2202}X^{\unicode[STIX]{x1D6FC}}}\int \,\text{d}\boldsymbol{k}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}G_{\unicode[STIX]{x1D70C}}(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})[2Q_{u\text{S}}(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})+Q_{u\text{C}}(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})],\end{eqnarray}$$

(4.7) $$\begin{eqnarray}\displaystyle & & \displaystyle \int \,\text{d}\boldsymbol{k}\langle \unicode[STIX]{x1D70C}_{01}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F})u_{01}^{\prime }\,^{\unicode[STIX]{x1D6FC}}(-\boldsymbol{k};\unicode[STIX]{x1D70F})\rangle /\unicode[STIX]{x1D6FF}(\mathbf{0})\nonumber\\ \displaystyle & & \displaystyle \quad =-\frac{1}{3\unicode[STIX]{x1D707}_{0}}B^{\unicode[STIX]{x1D6FC}}\int \,\text{d}\boldsymbol{k}k^{2}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{2}G_{\unicode[STIX]{x1D70C}}(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})G_{u\text{C}}(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{2})Q_{w\text{C}}(k,\unicode[STIX]{x1D70F}_{1},\unicode[STIX]{x1D70F}_{2}).\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

Here, we have used identity relations based on the properties of the solenoidal and compressional projection operators, $D^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}$ and $\unicode[STIX]{x1D6F1}^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}$ (3.28), such as

(4.8) $$\begin{eqnarray}\displaystyle & \displaystyle k^{a}D^{\unicode[STIX]{x1D6FC}a}=0, & \displaystyle\end{eqnarray}$$

(4.9) $$\begin{eqnarray}\displaystyle & \displaystyle \int \,\text{d}\boldsymbol{k}k^{a}\unicode[STIX]{x1D6F1}^{\unicode[STIX]{x1D6FC}a}=\int \,\text{d}\boldsymbol{k}k^{a}\frac{k^{\unicode[STIX]{x1D6FC}}k^{a}}{k^{2}}=\int \,\text{d}\boldsymbol{k}k^{\unicode[STIX]{x1D6FC}}=0, & \displaystyle\end{eqnarray}$$

(4.10) $$\begin{eqnarray}\displaystyle & \displaystyle \int \,\text{d}\boldsymbol{k}D^{\unicode[STIX]{x1D6FC}a}=\frac{2}{3}\unicode[STIX]{x1D6FF}^{\unicode[STIX]{x1D6FC}a}\int \,\text{d}\boldsymbol{k}, & \displaystyle\end{eqnarray}$$

(4.11) $$\begin{eqnarray}\displaystyle & \displaystyle \int \,\text{d}\boldsymbol{k}\unicode[STIX]{x1D6F1}^{\unicode[STIX]{x1D6FC}a}=\frac{1}{3}\unicode[STIX]{x1D6FF}^{\unicode[STIX]{x1D6FC}a}\int \,\text{d}\boldsymbol{k}, & \displaystyle\end{eqnarray}$$

(4.12) $$\begin{eqnarray}\displaystyle & \displaystyle D^{\unicode[STIX]{x1D6FC}b}\unicode[STIX]{x1D6F1}^{ab}=0, & \displaystyle\end{eqnarray}$$

(4.13) $$\begin{eqnarray}\displaystyle & \displaystyle \int \,\text{d}\boldsymbol{k}k^{a}k^{e}\unicode[STIX]{x1D6F1}^{\unicode[STIX]{x1D6FC}b}\unicode[STIX]{x1D6F1}^{ab}=\frac{1}{3}\unicode[STIX]{x1D6FF}^{e\unicode[STIX]{x1D6FC}}\int \,\text{d}\boldsymbol{k}. & \displaystyle\end{eqnarray}$$

These relations are mathematically rigorous and make the theoretical calculations accurate and simple by eliminating several coupling terms that give identically null contributions.

For the sake of brevity of description, we adopt the abbreviated forms of spectral and time integrals as

(4.14) $$\begin{eqnarray}\displaystyle & \displaystyle I_{0}\{A,B\}=\int \,\text{d}\boldsymbol{k}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}A(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})B(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1}), & \displaystyle\end{eqnarray}$$

(4.15) $$\begin{eqnarray}\displaystyle & \displaystyle I_{2n}\{A^{(1)},B^{(2)},C^{(2)}\}=\int \,\text{d}\boldsymbol{k}k^{2n}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{2}A(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})B(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{2})C(k;\unicode[STIX]{x1D70F}_{1},\unicode[STIX]{x1D70F}_{2}). & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

It follows from (4.2)–(4.7) that the turbulent mass flux $\langle \unicode[STIX]{x1D70C}^{\prime }\boldsymbol{u}^{\prime }\rangle$ is expressed as

(4.16) $$\begin{eqnarray}\langle \unicode[STIX]{x1D70C}^{\prime }\boldsymbol{u}^{\prime }\rangle =-\unicode[STIX]{x1D705}_{\overline{\unicode[STIX]{x1D70C}}}\unicode[STIX]{x1D735}\overline{\unicode[STIX]{x1D70C}}-\unicode[STIX]{x1D705}_{Q}\unicode[STIX]{x1D735}Q-\unicode[STIX]{x1D705}_{D}\frac{\text{D}\boldsymbol{U}}{\text{D}T}-\unicode[STIX]{x1D705}_{B}\boldsymbol{B},\end{eqnarray}$$

with the transport coefficients:

(4.17) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D705}_{\overline{\unicode[STIX]{x1D70C}}}={\textstyle \frac{1}{3}}I_{0}\{G_{\unicode[STIX]{x1D70C}},2Q_{u\text{S}}+Q_{u\text{C}}\}, & \displaystyle\end{eqnarray}$$

(4.18) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D705}_{Q}=\frac{1}{3}(\unicode[STIX]{x1D6FE}_{\text{s}}-1)\frac{1}{\overline{\unicode[STIX]{x1D70C}}}I_{0}\{2G_{u\text{S}}+G_{u\text{C}},Q_{\unicode[STIX]{x1D70C}}\}, & \displaystyle\end{eqnarray}$$

(4.19) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D705}_{D}=\frac{1}{3}\frac{1}{\overline{\unicode[STIX]{x1D70C}}}I_{0}\{2G_{u\text{S}}+G_{u\text{C}},Q_{\unicode[STIX]{x1D70C}}\}, & \displaystyle\end{eqnarray}$$

(4.20) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D705}_{B}=\frac{1}{3\unicode[STIX]{x1D707}_{0}}I_{1}\{G_{\unicode[STIX]{x1D70C}}^{(1)},G_{u\text{C}}^{(2)},Q_{w\text{C}}^{(2)}\}. & \displaystyle\end{eqnarray}$$

The first or $\unicode[STIX]{x1D705}_{\overline{\unicode[STIX]{x1D70C}}}$ -related term in (4.16) corresponds to the so-called gradient-diffusion approximation of the turbulent mass flux. In the presence of the mean-density variation $\unicode[STIX]{x1D735}\overline{\unicode[STIX]{x1D70C}}$ , the mass flux due to turbulence occurs in the opposite direction to the mean-density gradient, and works so that it reduces the mean-density variation. The transport coefficient $\unicode[STIX]{x1D705}_{\overline{\unicode[STIX]{x1D70C}}}$ (4.17) is determined by the intensity of the turbulent motion represented by the spectra $Q_{u\text{S}}$ and $Q_{u\text{C}}$ . The turbulent motions, both the solenoidal and compressional ones, contribute to this gradient diffusion of the mass.

The second or $\unicode[STIX]{x1D705}_{Q}$ -related term corresponds to the effective mass flux due to the mean internal energy, or equivalently the mean temperature variation, $\unicode[STIX]{x1D735}Q$ , and may be called the cross-diffusion due to the internal-energy inhomogeneity. The transport coefficient $\unicode[STIX]{x1D705}_{Q}$ (4.18) is determined by the density-variance spectrum $Q_{\unicode[STIX]{x1D70C}}$ , and $\unicode[STIX]{x1D705}_{Q}$ as well as $\unicode[STIX]{x1D705}_{\overline{\unicode[STIX]{x1D70C}}}$ is always positive. This effect arises from strong compressibility. Depending on the relative configurations of the mean-density and internal-energy gradients, the cross-diffusion gives a deviation of the turbulent mass flux from the gradient diffusion due to the mean-density variation $\unicode[STIX]{x1D735}\overline{\unicode[STIX]{x1D70C}}$ . If the mean internal-energy or temperature gradient is in the opposite direction to the mean-density gradient, the mass flux might be effectively suppressed.

The third or $\unicode[STIX]{x1D705}_{D}$ -related term is the non-equilibrium mean velocity effect on the turbulent mass flux. A finite material derivative of the mean velocity $\text{D}\boldsymbol{U}/\text{D}t$ represents a non-equilibrium variation of the mean velocity. For instance, in the case of an impinging flow, $\text{D}\boldsymbol{U}/\text{D}t$ is negative in the direction of the flow impinging on the wall. Since the transport coefficient $\unicode[STIX]{x1D705}_{D}$ (4.19), which is determined by the density variance spectrum $Q_{\unicode[STIX]{x1D70C}}$ , is always positive, a negative $\text{D}\boldsymbol{U}/\text{D}t$ in impinging flow suggests an enhancement of the effective mass transport due to the density fluctuation.

The fourth or $\unicode[STIX]{x1D705}_{B}$ -related term represents the effective mass transport in the direction of the mean magnetic field $\boldsymbol{B}$ . The transport coefficient $\unicode[STIX]{x1D705}_{B}$ (4.20) is determined by the compressional cross-helicity (i.e. the cross-helicity projected by the compressive projection $\unicode[STIX]{x1D6F1}$ ) spectrum. This compressional cross-helicity corresponds to the part of the cross-helicity which is coupled with the dilatation/contraction fluctuation motions. Unlike $\unicode[STIX]{x1D705}_{\overline{\unicode[STIX]{x1D70C}}}$ , $\unicode[STIX]{x1D705}_{Q}$ and $\unicode[STIX]{x1D705}_{D}$ , $\unicode[STIX]{x1D705}_{B}$ can be negative depending on the sign of the compressional cross-helicity. This term suggests that the mass is effectively transported by turbulence along the mean magnetic field $\boldsymbol{B}$ . This effect also gives a possibility of the turbulent mass flux independent of the gradient of the mean density $\unicode[STIX]{x1D735}\overline{\unicode[STIX]{x1D70C}}$ . This effect will be further discussed in § 5.

4.2 Turbulent internal-energy flux

The turbulent heat flux is calculated as

(4.21) $$\begin{eqnarray}\displaystyle \langle q^{\prime }\boldsymbol{u}^{\prime }\rangle & = & \displaystyle \langle q_{0}^{\prime }\boldsymbol{u}_{0}^{\prime }\rangle +\langle q_{0}^{\prime }\boldsymbol{u}_{1}^{\prime }\rangle +\langle q_{1}^{\prime }\boldsymbol{u}_{0}^{\prime }\rangle +\cdots \nonumber\\ \displaystyle & = & \displaystyle \langle q_{\text{B}}^{\prime }\boldsymbol{u}_{\text{B}}^{\prime }\rangle +\langle q_{\text{B}}^{\prime }\boldsymbol{u}_{01}^{\prime }\rangle +\langle q_{\text{B}}^{\prime }\boldsymbol{u}_{10}^{\prime }\rangle +\cdots \nonumber\\ \displaystyle & & \displaystyle +\,\langle q_{01}^{\prime }\boldsymbol{u}_{\text{B}}^{\prime }\rangle +\langle q_{01}^{\prime }\boldsymbol{u}_{01}^{\prime }\rangle +\cdots +\langle q_{10}^{\prime }\boldsymbol{u}_{\text{B}}^{\prime }\rangle +\cdots \,.\end{eqnarray}$$

From (3.16)–(3.23), each term in (4.21) is expressed as

(4.22) $$\begin{eqnarray}\displaystyle & \displaystyle \int \,\text{d}\boldsymbol{k}\langle q_{\text{B}}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F})u_{\text{B}}^{\prime }\,^{\unicode[STIX]{x1D6FC}}(-\boldsymbol{k};\unicode[STIX]{x1D70F})\rangle /\unicode[STIX]{x1D6FF}(\mathbf{0})=0, & \displaystyle\end{eqnarray}$$

(4.23) $$\begin{eqnarray}\displaystyle & \displaystyle \int \,\text{d}\boldsymbol{k}\langle q_{\text{B}}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F})u_{01}^{\prime }\,^{\unicode[STIX]{x1D6FC}}(-\boldsymbol{k};\unicode[STIX]{x1D70F})\rangle /\unicode[STIX]{x1D6FF}(\mathbf{0})=0, & \displaystyle\end{eqnarray}$$

(4.24) $$\begin{eqnarray}\displaystyle & & \displaystyle \int \,\text{d}\boldsymbol{k}\langle \unicode[STIX]{x1D70C}_{\text{B}}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F})u_{10}^{\prime }\,^{\unicode[STIX]{x1D6FC}}(-\boldsymbol{k};\unicode[STIX]{x1D70F})\rangle /\unicode[STIX]{x1D6FF}(\mathbf{0})\nonumber\\ \displaystyle & & \displaystyle \quad =-\frac{1}{3}(\unicode[STIX]{x1D6FE}_{\text{s}}-1)\frac{1}{\overline{\unicode[STIX]{x1D70C}}}\frac{\unicode[STIX]{x2202}\overline{\unicode[STIX]{x1D70C}}}{\unicode[STIX]{x2202}X^{\unicode[STIX]{x1D6FC}}}\int \,\text{d}\boldsymbol{k}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}[2G_{u\text{S}}(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})+G_{u\text{C}}(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})]Q_{q}(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1}),\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

(4.25) $$\begin{eqnarray}\int \,\text{d}\boldsymbol{k}\langle q_{01}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F})u_{\text{B}}^{\prime }\,^{\unicode[STIX]{x1D6FC}}(-\boldsymbol{k};\unicode[STIX]{x1D70F})\rangle /\unicode[STIX]{x1D6FF}(\mathbf{0})=0,\end{eqnarray}$$

(4.26) $$\begin{eqnarray}\displaystyle & & \displaystyle \int \,\text{d}\boldsymbol{k}\langle q_{1}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F})u_{\text{B}}^{\prime }\,^{\unicode[STIX]{x1D6FC}}(-\boldsymbol{k};\unicode[STIX]{x1D70F})\rangle /\unicode[STIX]{x1D6FF}(\mathbf{0})\nonumber\\ \displaystyle & & \displaystyle \quad =-\frac{\unicode[STIX]{x2202}Q}{\unicode[STIX]{x2202}X^{\unicode[STIX]{x1D6FC}}}\frac{1}{3}\int \,\text{d}\boldsymbol{k}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}G_{q}(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})[2Q_{u\text{S}}(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})+Q_{u\text{C}}(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})],\qquad\end{eqnarray}$$

(4.27) $$\begin{eqnarray}\displaystyle & & \displaystyle \int \,\text{d}\boldsymbol{k}\langle q_{01}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F})u_{01}^{\prime }\,^{\unicode[STIX]{x1D6FC}}(-\boldsymbol{k};\unicode[STIX]{x1D70F})\rangle /\unicode[STIX]{x1D6FF}(\mathbf{0})=-\frac{1}{3}\frac{1}{\unicode[STIX]{x1D707}_{0}\overline{\unicode[STIX]{x1D70C}}}(\unicode[STIX]{x1D6FE}_{s}-1)Q\int \,\text{d}\boldsymbol{k}k^{2}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{2}\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,G_{q}(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})G_{u\text{C}}(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{2})Q_{w\text{C}}(k;\unicode[STIX]{x1D70F}_{1},\unicode[STIX]{x1D70F}_{2})B^{\unicode[STIX]{x1D6FC}}.\end{eqnarray}$$

It follows from (4.22)–(4.27) that the turbulent internal-energy flux $\langle q^{\prime }\boldsymbol{u}^{\prime }\rangle$ is expressed as

(4.28) $$\begin{eqnarray}\langle q^{\prime }\boldsymbol{u}^{\prime }\rangle =-\unicode[STIX]{x1D702}_{Q}\unicode[STIX]{x1D735}Q-\unicode[STIX]{x1D702}_{\overline{\unicode[STIX]{x1D70C}}}\unicode[STIX]{x1D735}\overline{\unicode[STIX]{x1D70C}}-\unicode[STIX]{x1D702}_{B}\boldsymbol{B},\end{eqnarray}$$

with the transport coefficients:

(4.29) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D702}_{Q}={\textstyle \frac{1}{3}}I_{0}\{G_{q},2Q_{u\text{S}}+Q_{u\text{C}}\}, & \displaystyle\end{eqnarray}$$

(4.30) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D702}_{\overline{\unicode[STIX]{x1D70C}}}=\frac{1}{3}(\unicode[STIX]{x1D6FE}_{\text{s}}-1)\frac{1}{\overline{\unicode[STIX]{x1D70C}}}I_{0}\{2G_{u\text{S}}+G_{u\text{C}},Q_{q}\}, & \displaystyle\end{eqnarray}$$

(4.31) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D702}_{B}=\frac{1}{3\unicode[STIX]{x1D707}_{0}\overline{\unicode[STIX]{x1D70C}}}(\unicode[STIX]{x1D6FE}_{\text{s}}-1)QI_{1}\{G_{q}^{(1)},G_{u\text{C}}^{(2)},Q_{w\text{C}}^{(2)}\}. & \displaystyle\end{eqnarray}$$

The first or $\unicode[STIX]{x1D702}_{Q}$ -related term in (4.28) represents the gradient diffusion of the internal energy. In the presence of the mean internal-energy gradient $\unicode[STIX]{x1D735}Q$ , the effective energy transport, mediated by the turbulent motions, is in the direction reducing the inhomogeneity of the internal energy. The transport coefficient $\unicode[STIX]{x1D702}_{Q}$ (4.29) is determined by the solenoidal and compressional components of the spectral functions of the turbulent motions.

The second or $\unicode[STIX]{x1D702}_{\overline{\unicode[STIX]{x1D70C}}}$ -related term represents the turbulent internal-energy flux due to the mean-density gradient $\unicode[STIX]{x1D735}\overline{\unicode[STIX]{x1D70C}}$ . In addition to the gradient diffusion due to $\unicode[STIX]{x1D735}Q$ , a gradient of the mean density, $\unicode[STIX]{x1D735}\overline{\unicode[STIX]{x1D70C}}$ , may contribute to the internal-energy transport through the fluctuation of the internal energy. This is a cross-diffusion term; depending on the relative configurations of $\unicode[STIX]{x1D735}\overline{\unicode[STIX]{x1D70C}}$ and $\unicode[STIX]{x1D735}Q$ , the turbulent internal-energy flux can deviate from the simple gradient diffusion proportional to $\unicode[STIX]{x1D735}Q$ . The transport coefficient $\unicode[STIX]{x1D702}_{\overline{\unicode[STIX]{x1D70C}}}$ (4.30) is determined by the spectral function of the internal-energy variance $Q_{q}$ , which is directly related to the density-variance spectrum $Q_{\unicode[STIX]{x1D70C}}$ or the density variance itself $\langle \unicode[STIX]{x1D70C}^{\prime 2}\rangle$ as we see for the modelling of this effect in § 4.3.

The third or $\unicode[STIX]{x1D702}_{B}$ -related term gives a possibility of enhancing the turbulent internal-energy flux along the mean magnetic field $\boldsymbol{B}$ . The transport coefficient $\unicode[STIX]{x1D702}_{B}$ (4.31) is determined by the spectral function of the compressional part of the turbulent cross-helicity. Unlike the spectral functions of the turbulent energy $Q_{u}$ and the turbulent internal-energy $Q_{q}$ , $Q_{w\text{C}}$ is not positive definite. Hence, the transport coefficient $\unicode[STIX]{x1D702}_{B}$ can be positive or negative depending on the sign of the turbulent compressional cross-helicity. This term makes it possible for the internal energy to transport independently of the gradient of the mean internal energy $\unicode[STIX]{x1D735}Q$ or the gradient of the mean density $\unicode[STIX]{x1D735}\overline{\unicode[STIX]{x1D70C}}$ . The physical origin of this effect is related to the magnetohydrodynamic (MHD) wave effect. We discuss this point later in § 5.

4.3 Modelling turbulent fluxes

In order to properly evaluate the mean-field quantities, $\overline{\unicode[STIX]{x1D70C}}$ , $\boldsymbol{U}$ , $Q$ and $\boldsymbol{B}$ , from their transport equations (2.14)–(2.15) and (2.17)–(2.18), we need the expressions for the turbulent correlations. For example, the turbulent mass flux $\langle \unicode[STIX]{x1D70C}^{\prime }\boldsymbol{u}^{\prime }\rangle$ and the turbulent internal-energy flux $\langle q^{\prime }\boldsymbol{u}^{\prime }\rangle$ provide essential contributions to the evolution of the mean internal energy (2.17). The theoretical results for $\langle \unicode[STIX]{x1D70C}^{\prime }\boldsymbol{u}^{\prime }\rangle$ (4.16) and $\langle q^{\prime }\boldsymbol{u}^{\prime }\rangle$ (4.28) with the analytical expressions of the transport coefficients (4.17)–(4.20) and (4.29)–(4.31) are written in terms of the spectral and time integrals of the Green’s functions and spectral functions of MHD turbulence. For practical applications these analytical expressions are too heavy, so we construct a turbulence model with one-point quantities on the basis of the analytical results.

The Green’s functions represent how much the past states affect the present states. The characteristic times of turbulence may be defined with the aid of the Green’s functions as

(4.32) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{s}=\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}\langle G_{s}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})\rangle =\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}G_{s}(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1}),\end{eqnarray}$$

with $s=(\unicode[STIX]{x1D70C},u,q,b)$ . Utilising these time scales, the transport coefficients $\unicode[STIX]{x1D705}_{\overline{\unicode[STIX]{x1D70C}}}$ (4.17), $\unicode[STIX]{x1D705}_{Q}$ (4.18), $\unicode[STIX]{x1D705}_{D}$ (4.19), $\unicode[STIX]{x1D705}_{B}$ (4.20), $\unicode[STIX]{x1D702}_{Q}$ (4.29), $\unicode[STIX]{x1D702}_{\overline{\unicode[STIX]{x1D70C}}}$ (4.30) and $\unicode[STIX]{x1D702}_{B}$ (4.31) may be modelled as

(4.33) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D705}_{\overline{\unicode[STIX]{x1D70C}}}=C_{\unicode[STIX]{x1D705}\unicode[STIX]{x1D70C}}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D70C}}\langle \boldsymbol{u}^{\prime 2}\rangle /2, & \displaystyle\end{eqnarray}$$

(4.34) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D705}_{Q}=C_{\unicode[STIX]{x1D705}Q}(\unicode[STIX]{x1D6FE}_{\text{s}}-1)\unicode[STIX]{x1D70F}_{u}\overline{\unicode[STIX]{x1D70C}}\frac{\langle \unicode[STIX]{x1D70C}^{\prime 2}\rangle }{\overline{\unicode[STIX]{x1D70C}}^{2}}, & \displaystyle\end{eqnarray}$$

(4.35) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D705}_{D}=C_{\unicode[STIX]{x1D705}D}\unicode[STIX]{x1D70F}_{u}\overline{\unicode[STIX]{x1D70C}}\frac{\langle \unicode[STIX]{x1D70C}^{\prime 2}\rangle }{\overline{\unicode[STIX]{x1D70C}}^{2}}, & \displaystyle\end{eqnarray}$$

(4.36) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D705}_{B}=C_{\unicode[STIX]{x1D705}B}\frac{1}{\unicode[STIX]{x1D707}_{0}\overline{\unicode[STIX]{x1D70C}}}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D70C}}\unicode[STIX]{x1D70F}_{uc}\overline{\unicode[STIX]{x1D70C}}\langle \boldsymbol{u}^{\prime }\boldsymbol{\cdot }\boldsymbol{b}^{\prime }\rangle _{\text{C}}, & \displaystyle\end{eqnarray}$$

(4.37) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D702}_{Q}=C_{\unicode[STIX]{x1D702}Q}\unicode[STIX]{x1D70F}_{q}\langle \boldsymbol{u}^{\prime 2}\rangle /2, & \displaystyle\end{eqnarray}$$

(4.38) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D702}_{\overline{\unicode[STIX]{x1D70C}}}=C_{\unicode[STIX]{x1D702}\unicode[STIX]{x1D70C}}(\unicode[STIX]{x1D6FE}_{\text{s}}-1)\unicode[STIX]{x1D70F}_{u}\frac{\langle q^{\prime 2}\rangle }{\overline{\unicode[STIX]{x1D70C}}}=C_{\unicode[STIX]{x1D702}\unicode[STIX]{x1D70C}}(\unicode[STIX]{x1D6FE}_{\text{s}}-1)^{3}\frac{\unicode[STIX]{x1D70F}_{u}\unicode[STIX]{x1D70F}_{q}^{2}}{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D70C}}^{2}}\frac{Q^{2}}{\overline{\unicode[STIX]{x1D70C}}}\frac{\langle \unicode[STIX]{x1D70C}^{\prime 2}\rangle }{\overline{\unicode[STIX]{x1D70C}}^{2}}, & \displaystyle\end{eqnarray}$$

(4.39) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D702}_{B}=C_{\unicode[STIX]{x1D702}B}\frac{1}{\unicode[STIX]{x1D707}_{0}\overline{\unicode[STIX]{x1D70C}}}(\unicode[STIX]{x1D6FE}_{\text{s}}-1)\unicode[STIX]{x1D70F}_{q}\unicode[STIX]{x1D70F}_{uc}Q\langle \boldsymbol{u}^{\prime }\boldsymbol{\cdot }\boldsymbol{b}^{\prime }\rangle _{\text{C}}. & \displaystyle\end{eqnarray}$$

In the second equality of (4.38), use has been made of the approximate relations:

(4.40) $$\begin{eqnarray}q^{\prime }\simeq -(\unicode[STIX]{x1D6FE}_{\text{s}}-1)\unicode[STIX]{x1D70F}_{q}Q\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}^{\prime }\end{eqnarray}$$

and

(4.41) $$\begin{eqnarray}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}^{\prime }\simeq -\frac{1}{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D70C}}}\frac{\unicode[STIX]{x1D70C}^{\prime }}{\overline{\unicode[STIX]{x1D70C}}}\end{eqnarray}$$

(see (2.19) and (2.21) with (3.16) and (3.18)).

The gradient-diffusion-related coefficients $\unicode[STIX]{x1D705}_{\overline{\unicode[STIX]{x1D70C}}}$ (4.33) and $\unicode[STIX]{x1D702}_{Q}$ (4.37) are determined by the turbulent energy $\langle \boldsymbol{u}^{\prime 2}\rangle /2$ , which includes both solenoidal and compressive fluctuation motions. These effects are present even in the solenoidal case. The cross-diffusion effects $\unicode[STIX]{x1D705}_{Q}$ (4.34) and $\unicode[STIX]{x1D702}_{\overline{\unicode[STIX]{x1D70C}}}$ (4.38) and the non-equilibrium effect $\unicode[STIX]{x1D705}_{D}$ (4.35) ddepend on the density variance $\langle \unicode[STIX]{x1D70C}^{\prime 2}\rangle$ , which is expected to be relevant in strongly compressible turbulence. The turbulent mass and internal-energy fluxes along the mean magnetic field $\boldsymbol{B}$ , $\unicode[STIX]{x1D705}_{B}$ (4.36) and $\unicode[STIX]{x1D702}_{B}$ (4.39), are determined by the compressional part of the turbulent cross-helicity, $\langle \boldsymbol{u}^{\prime }\boldsymbol{\cdot }\boldsymbol{b}^{\prime }\rangle _{\text{C}}$ . These compressional turbulent cross-helicity effects (i.e. cross-helicity effects coupled with the compressible motions) are further discussed in the following section (§ 5).

4.4 Correspondence to turbulent entropy flux

In the astrophysical and geophysical contexts, the entropy $s$ , in place of the internal energy $q$ , is often adopted as one of the key statistical quantities. From the first law of thermodynamics:

(4.42) $$\begin{eqnarray}\text{d}q=-pd(1/\unicode[STIX]{x1D70C})+\unicode[STIX]{x1D703}\,\text{d}s,\end{eqnarray}$$

the evolution equation of $s$ is written as

(4.43) $$\begin{eqnarray}\unicode[STIX]{x1D703}\frac{\text{D}s}{\text{D}t}=\frac{\text{D}q}{\text{D}t}+p\frac{\text{D}}{\text{D}t}\frac{1}{\unicode[STIX]{x1D70C}}=\frac{\text{D}q}{\text{D}t}-\frac{p}{\unicode[STIX]{x1D70C}^{2}}\frac{\text{D}\unicode[STIX]{x1D70C}}{\text{D}t}\end{eqnarray}$$

in relation with the equations of the internal energy and density, $q$ and $\unicode[STIX]{x1D70C}$ (Mihalas & Weibel-Mihalas Reference Mihalas and Weibel-Mihalas1984).

Applying ensemble or Reynolds decomposition of the variables $\unicode[STIX]{x1D703}$ , $s$ , $q$ , $p$ and $\unicode[STIX]{x1D70C}$ into (4.43) and taking an average, we obtain the evolution equation of the mean entropy, $S$ , which contains the convective flux $\langle s^{\prime }\boldsymbol{u}^{\prime }\rangle$ contribution. An elaborated analysis of the expression of the turbulent entropy flux $\langle s^{\prime }\boldsymbol{u}^{\prime }\rangle$ requires another paper, but from (4.43), the turbulent entropy flux may be approximated as

(4.44) $$\begin{eqnarray}\langle s^{\prime }\boldsymbol{u}^{\prime }\rangle =\frac{1}{\unicode[STIX]{x1D6E9}}\langle q^{\prime }\boldsymbol{u}^{\prime }\rangle -\frac{1}{\unicode[STIX]{x1D6E9}}\frac{P}{\overline{\unicode[STIX]{x1D70C}}^{2}}\langle \unicode[STIX]{x1D70C}^{\prime }\boldsymbol{u}^{\prime }\rangle .\end{eqnarray}$$

The expressions of the turbulent fluxes of the internal energy and mass, $\langle q^{\prime }\boldsymbol{u}^{\prime }\rangle$ (4.28) and $\langle \unicode[STIX]{x1D70C}^{\prime }\boldsymbol{u}^{\prime }\rangle$ (4.16), suggest that the primary part of the expression of $\langle s^{\prime }\boldsymbol{u}^{\prime }\rangle$ may be written as

(4.45) $$\begin{eqnarray}\langle s^{\prime }\boldsymbol{u}^{\prime }\rangle =-\unicode[STIX]{x1D702}_{Q}\frac{1}{\unicode[STIX]{x1D6E9}}\unicode[STIX]{x1D735}Q+\unicode[STIX]{x1D705}_{\overline{\unicode[STIX]{x1D70C}}}\frac{1}{\unicode[STIX]{x1D6E9}}\frac{P}{\overline{\unicode[STIX]{x1D70C}}^{2}}\unicode[STIX]{x1D735}\overline{\unicode[STIX]{x1D70C}}.\end{eqnarray}$$

With the expressions for the transport coefficients $\unicode[STIX]{x1D702}_{Q}$ (4.29) and $\unicode[STIX]{x1D705}_{\overline{\unicode[STIX]{x1D70C}}}$ (4.17), and their models (4.37) and (4.33), the turbulent entropy flux may be modelled as

(4.46) $$\begin{eqnarray}\langle s^{\prime }\boldsymbol{u}^{\prime }\rangle =-\unicode[STIX]{x1D712}_{S}\unicode[STIX]{x1D735}S,\end{eqnarray}$$

where $\unicode[STIX]{x1D712}_{S}$ is the transport coefficient proportional to the turbulent kinetic energy $\langle \boldsymbol{u}^{\prime 2}\rangle /2$ . This expression is similar to the one obtained by Rogachevskii & Kleeorin (Reference Rogachevskii and Kleeorin2015) for low Mach number flows. Note that, among several transport coefficients, the gradient-diffusion coefficients, $\unicode[STIX]{x1D702}_{Q}$ and $\unicode[STIX]{x1D705}_{\overline{\unicode[STIX]{x1D70C}}}$ , arise even in the case of no or weak compressibility. Of course, the full expressions for $\langle \unicode[STIX]{x1D70C}^{\prime }\boldsymbol{u}^{\prime }\rangle$ (4.28) and $\langle q^{\prime }\boldsymbol{u}^{\prime }\rangle$ (4.16) infer some additional terms for $\langle s^{\prime }\boldsymbol{u}^{\prime }\rangle$ . A more elaborated analysis in the context of astrophysical and geophysical applications will be reported in the future.

With (4.43), we construct the mean entropy $\overline{\unicode[STIX]{x1D70C}}S$ equation ( $S=\langle s\rangle$ ) (Braginsky & Roberts Reference Braginsky and Roberts1995; Rogachevskii & Kleeorin Reference Rogachevskii and Kleeorin2015). In the mean entropy equation, we have the turbulent flux of entropy $\overline{\unicode[STIX]{x1D70C}}\langle s^{\prime }\boldsymbol{u}^{\prime }\rangle$ . In some of the literature, the effect of the entropy flux is expressed by $(1/\unicode[STIX]{x1D6E9})\unicode[STIX]{x1D735}\boldsymbol{\cdot }\unicode[STIX]{x1D6E9}\overline{\unicode[STIX]{x1D70C}}\langle s^{\prime }\boldsymbol{u}^{\prime }\rangle$ , which coincides with the formulation with the turbulent flux of internal energy. From the definition of the entropy (4.42) and the equation of state (2.8) with (2.9), the internal-energy fluctuation is written as

(4.47) $$\begin{eqnarray}q^{\prime }=\left[s^{\prime }+C_{V}(\unicode[STIX]{x1D6FE}_{\text{s}}-1)\frac{\unicode[STIX]{x1D70C}^{\prime }}{\overline{\unicode[STIX]{x1D70C}}}\right]\unicode[STIX]{x1D6E9}.\end{eqnarray}$$

The convective flux $\unicode[STIX]{x1D6E9}\overline{\unicode[STIX]{x1D70C}}\langle s^{\prime }\boldsymbol{u}^{\prime }\rangle$ , which is often used in the current studies, corresponds to the second term on the right-hand side in (2.17) with the low Mach number approximation of (4.47) ( $q^{\prime }=\unicode[STIX]{x1D6E9}s^{\prime }$ ).

5.1 Dilatational cross-helicity effects

The fourth term in (4.16) and the third term in (4.28) imply possibilities of the turbulent mass and internal-energy fluxes along the mean magnetic field, respectively. They deserve further consideration, since they may lead to deviations from the gradient-diffusion approximation for the turbulent fluxes, expressed by $\unicode[STIX]{x1D735}\overline{\unicode[STIX]{x1D70C}}$ in $\langle \unicode[STIX]{x1D70C}^{\prime }\boldsymbol{u}^{\prime }\rangle$ and $\unicode[STIX]{x1D735}Q$ in $\langle q^{\prime }\boldsymbol{u}^{\prime }\rangle$ , in the compressible MHD turbulence.

As we saw in (4.20) and (4.31) and their models (4.36) and (4.39), the transport coefficients $\unicode[STIX]{x1D705}_{B}$ and $\unicode[STIX]{x1D702}_{B}$ are expressed and modelled as

(5.1) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D705}_{B} & = & \displaystyle \frac{1}{3\unicode[STIX]{x1D707}_{0}}\int \,\text{d}\boldsymbol{k}k^{2}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{2}G_{\unicode[STIX]{x1D70C}}(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})G_{uc}(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{2})Q_{w\text{C}}(k;\unicode[STIX]{x1D70F}_{1},\unicode[STIX]{x1D70F}_{2})\nonumber\\ \displaystyle & \equiv & \displaystyle \frac{1}{3\unicode[STIX]{x1D707}_{0}}I_{1}\{G_{\unicode[STIX]{x1D70C}}^{(1)},G_{u\text{C}}^{(2)},Q_{w\text{C}}^{(2)}\}\nonumber\\ \displaystyle & = & \displaystyle C_{\unicode[STIX]{x1D705}B}\frac{1}{\unicode[STIX]{x1D707}_{0}\overline{\unicode[STIX]{x1D70C}}}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D70C}}\unicode[STIX]{x1D70F}_{uc}\overline{\unicode[STIX]{x1D70C}}\langle \boldsymbol{u}^{\prime }\boldsymbol{\cdot }\boldsymbol{b}^{\prime }\rangle _{\text{C}}\end{eqnarray}$$

and

(5.2) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D702}_{B} & = & \displaystyle \frac{1}{3\unicode[STIX]{x1D707}_{0}}\int \,\text{d}\boldsymbol{k}k^{2}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{2}G_{q}(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})G_{uc}(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{2})Q_{w\text{C}}(k;\unicode[STIX]{x1D70F}_{1},\unicode[STIX]{x1D70F}_{2})\nonumber\\ \displaystyle & \equiv & \displaystyle \frac{1}{3\unicode[STIX]{x1D707}_{0}}I_{1}\{G_{q}^{(1)},G_{u\text{C}}^{(2)},Q_{w\text{C}}^{(2)}\}\nonumber\\ \displaystyle & = & \displaystyle C_{\unicode[STIX]{x1D702}B}\frac{1}{\unicode[STIX]{x1D707}_{0}\overline{\unicode[STIX]{x1D70C}}}(\unicode[STIX]{x1D6FE}_{\text{s}}-1)\unicode[STIX]{x1D70F}_{q}\unicode[STIX]{x1D70F}_{uc}Q\langle \boldsymbol{u}^{\prime }\boldsymbol{\cdot }\boldsymbol{b}^{\prime }\rangle _{\text{C}},\end{eqnarray}$$

where $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D70C}}$ , $\unicode[STIX]{x1D70F}_{q}$ and $\unicode[STIX]{x1D70F}_{uc}$ are the characteristic time scales of the density, internal energy and compressible velocity evolutions. The transport coefficients $\unicode[STIX]{x1D705}_{B}$ and $\unicode[STIX]{x1D702}_{B}$ coupled with the mean magnetic field $\boldsymbol{B}$ are constituted of the compressional or dilatational part of the turbulent cross-helicity, $\langle \boldsymbol{u}^{\prime }\boldsymbol{\cdot }\boldsymbol{b}^{\prime }\rangle _{\text{C}}$ . In order for these effects to work, both the compressibility and the cross-correlation between the velocity and magnetic-field fluctuations should present simultaneously.

For the compressional cross-helicity effects, fluid dynamics related to the dilatational motions $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}_{\text{C}}^{\prime }$ should play an essential role. To get a deeper understanding of these turbulent transports along the mean magnetic field, we first examine the turbulent mass flux $\langle \unicode[STIX]{x1D70C}^{\prime }\boldsymbol{u}^{\prime }\rangle$ by considering a simple situation of the density and velocity fluctuations. From the fluctuation equations of the density $\unicode[STIX]{x1D70C}^{\prime }$ (2.19) and the velocity $\boldsymbol{u}^{\prime }$ (2.20), the time evolution of $\unicode[STIX]{x1D70C}^{\prime }$ and $\boldsymbol{u}^{\prime }$ are respectively approximated as

(5.3) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{D}\unicode[STIX]{x1D70C}^{\prime }}{\text{D}t}\simeq -\overline{\unicode[STIX]{x1D70C}}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}^{\prime }, & \displaystyle\end{eqnarray}$$

(5.4) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{D}\boldsymbol{u}^{\prime }}{\text{D}t}\simeq \frac{1}{\unicode[STIX]{x1D707}_{0}\overline{\unicode[STIX]{x1D70C}}}(\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{b}^{\prime }. & \displaystyle\end{eqnarray}$$

These can be integrated with respect to time, and the variations of the density fluctuation and velocity fluctuation, $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D70C}^{\prime }$ and $\unicode[STIX]{x1D6FF}\boldsymbol{u}^{\prime }$ , are primarily expressed as

(5.5) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FF}\unicode[STIX]{x1D70C}^{\prime }\simeq -\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D70C}}\overline{\unicode[STIX]{x1D70C}}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}^{\prime }, & \displaystyle\end{eqnarray}$$

(5.6) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FF}\boldsymbol{u}^{\prime }\simeq \unicode[STIX]{x1D70F}_{u}\frac{1}{\unicode[STIX]{x1D707}_{0}\overline{\unicode[STIX]{x1D70C}}}(\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{b}^{\prime }, & \displaystyle\end{eqnarray}$$

with some other constraints such as the solenoidal condition of the magnetic-field fluctuation $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{b}^{\prime }=0$ . As we see from (2.19) and its expanded expression in the wavenumber space (3.16), the density fluctuation $\unicode[STIX]{x1D70C}^{\prime }$ is directly related to the turbulent dilatation/contraction $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}^{\prime }$ . Also the magnetic fluctuation being inhomogeneous along the mean magnetic field, $(\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{b}^{\prime }$ , may lead to the variation of the velocity fluctuation $\unicode[STIX]{x1D6FF}\boldsymbol{u}^{\prime }$ along the direction of the magnetic fluctuation variation. For evaluating the turbulent mass flux $\langle \unicode[STIX]{x1D70C}^{\prime }\boldsymbol{u}^{\prime }\rangle$ , we utilise these expressions.

We consider a compressible fluid plasma element located in a uniform mean magnetic field $\boldsymbol{B}$ (figure 1). Because of the compressibility, we have a finite turbulent dilatation/contraction ( $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}^{\prime }\neq 0$ ). Note that the magnitude of the density fluctuation depends on the strength of the compressibility. By considering the equation of the density variance $\langle \unicode[STIX]{x1D70C}^{\prime 2}\rangle$ , we expect a large density fluctuation at a location with a large mean-density variation (see arguments in § 6 of Paper 1). In the expansion or positive turbulent dilatation case ( $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}^{\prime }>0$ ), the variation of the density fluctuation $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D70C}$ is negative as

(5.7) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D70C}^{\prime }=-\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D70C}}\overline{\unicode[STIX]{x1D70C}}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}^{\prime }<0\end{eqnarray}$$

(figure 1 a,c).

Figure 1. Compressional cross-helicity effect. (a) Dilatation and positive compressional cross-helicity (CCH) ( $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}^{\prime }>0$ and $\langle \boldsymbol{u}^{\prime }\boldsymbol{\cdot }\boldsymbol{b}^{\prime }\rangle _{\text{C}}>0$ ); (b) contraction and positive CCH ( $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}^{\prime }<0$ and $\langle \boldsymbol{u}^{\prime }\boldsymbol{\cdot }\boldsymbol{b}^{\prime }\rangle _{\text{C}}>0$ ); (c) dilatation and negative CCH ( $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}^{\prime }>0$ and $\langle \boldsymbol{u}^{\prime }\boldsymbol{\cdot }\boldsymbol{b}^{\prime }\rangle _{\text{C}}<0$ ); (d) contraction and negative CCH ( $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}^{\prime }<0$ and $\langle \boldsymbol{u}^{\prime }\boldsymbol{\cdot }\boldsymbol{b}^{\prime }\rangle _{\text{C}}<0$ ).

Note that, in general, both $\boldsymbol{u}_{\text{C}}^{\prime }$ (compressible component of $\boldsymbol{u}^{\prime }$ ) and the solenoidal magnetic-field fluctuation $\boldsymbol{b}^{\prime }$ have components in the direction of the mean magnetic field $\boldsymbol{B}$ . First we assume that the turbulent cross-helicity associated with the turbulent dilatation is positive ( $\langle \boldsymbol{u}^{\prime }\boldsymbol{\cdot }\boldsymbol{b}^{\prime }\rangle _{\text{C}}>0$ ). This corresponds to a situation where the magnetic fluctuation $\boldsymbol{b}^{\prime }$ longitudinal to the mean magnetic field $\boldsymbol{B}$ is statistically aligned with the counterpart of the velocity fluctuation $\boldsymbol{u}^{\prime }$ (figure 1 a). Because of this assumption of positive compressional cross-helicity, the direction of the longitudinal magnetic fluctuation $\boldsymbol{b}_{\Vert }^{\prime }$ is statistically aligned with the longitudinal velocity fluctuation $\boldsymbol{u}_{\Vert }^{\prime }$ (the suffix $\Vert$ denotes the component along the mean magnetic field $\boldsymbol{B}$ ). For a positive compressional cross-helicity ( $\langle \boldsymbol{u}^{\prime }\boldsymbol{\cdot }\boldsymbol{b}^{\prime }\rangle _{\text{C}}>0$ ), an expansion for the plasma along $\boldsymbol{B}$ ( $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}_{\Vert }^{\prime }>0$ ) is statistically equivalent to $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{b}_{\Vert }^{\prime }>0$ . In this case, as shown by the thin dashed line in figure 1(a), the variation of the velocity fluctuation (5.6) is positive in the direction of $\boldsymbol{B}$ as

(5.8) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}u_{\Vert }^{\prime }=\unicode[STIX]{x1D70F}_{u}\frac{1}{\unicode[STIX]{x1D707}_{0}\overline{\unicode[STIX]{x1D70C}}}(\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735})b_{\Vert }^{\prime }>0.\end{eqnarray}$$

Note that because of the solenoidal property of the magnetic field, $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{b}^{\prime }=0$ should be always satisfied. It follows from (5.7) and (5.8) that the turbulent mass flux estimated by $\langle \unicode[STIX]{x1D6FF}\unicode[STIX]{x1D70C}^{\prime }\unicode[STIX]{x1D6FF}\boldsymbol{u}^{\prime }\rangle$ due to the compressional turbulent cross-helicity is along the mean magnetic field but antiparallel to $\boldsymbol{B}$ (thick dashed line in figure 1 a). In the contraction or negative dilatation case ( $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}^{\prime }<0$ ), $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D70C}^{\prime }$ in (5.7) turns out to be positive while $\unicode[STIX]{x1D6FF}u_{\Vert }^{\prime }$ in (5.8) becomes negative. As a result, $\langle \unicode[STIX]{x1D6FF}\unicode[STIX]{x1D70C}^{\prime }\unicode[STIX]{x1D6FF}\boldsymbol{u}^{\prime }\rangle$ is exactly the same as in the expansion or positive dilatation case (figure 1 b). Irrespective of the sign of the turbulent dilatation, the turbulent mass flux $\langle \unicode[STIX]{x1D70C}^{\prime }\boldsymbol{u}^{\prime }\rangle$ due to the coupling of the mean magnetic field $\boldsymbol{B}$ and compressional cross-helicity $\langle \boldsymbol{u}^{\prime }\boldsymbol{\cdot }\boldsymbol{b}^{\prime }\rangle _{\text{C}}$ is along but antiparallel to $\boldsymbol{B}$ as long as the $\langle \boldsymbol{u}^{\prime }\boldsymbol{\cdot }\boldsymbol{b}^{\prime }\rangle _{\text{C}}>0$ is positive (figure 1 a,b).

On the other hand, for the negative compressional cross-helicity ( $\langle \boldsymbol{u}^{\prime }\boldsymbol{\cdot }\boldsymbol{b}^{\prime }\rangle _{\text{C}}<0$ ), the turbulent mass flux due to the compressional cross-helicity effect is along and parallel to the mean magnetic field $\boldsymbol{B}$ (figure 1 c,d).

This is a simplified explanation for the fourth or $\unicode[STIX]{x1D705}_{B}$ -related term in (4.16) with the transport coefficient (4.20) or the model expression (4.36):

(5.9) $$\begin{eqnarray}\langle \unicode[STIX]{x1D70C}^{\prime }\boldsymbol{u}^{\prime }\rangle _{B}=-\unicode[STIX]{x1D705}_{B}\boldsymbol{B},\end{eqnarray}$$

with $\unicode[STIX]{x1D705}_{B}$ being proportional to the compressional cross-helicity (5.1).

Entirely similar arguments can be made for the turbulent internal-energy flux $\langle q^{\prime }\boldsymbol{u}^{\prime }\rangle$ . In this case, the equation of the turbulent internal energy (2.21) should be used in place of the equation of the turbulent density (2.19). Equation (5.3) is replaced by the time evolution of $q^{\prime }$ approximated by

(5.10) $$\begin{eqnarray}\frac{\text{D}q^{\prime }}{\text{D}t}\simeq -(\unicode[STIX]{x1D6FE}_{\text{s}}-1)Q\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}^{\prime }\end{eqnarray}$$

from (2.21). The variation of the turbulent internal-energy, $\unicode[STIX]{x1D6FF}q^{\prime }$ , can be estimated as

(5.11) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}q^{\prime }\simeq -\unicode[STIX]{x1D70F}_{q}(\unicode[STIX]{x1D6FE}_{\text{s}}-1)Q\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}^{\prime }\end{eqnarray}$$

from (3.18). Then, the turbulent internal-energy flux is given as

(5.12) $$\begin{eqnarray}\langle q^{\prime }\boldsymbol{u}^{\prime }\rangle _{B}=-\unicode[STIX]{x1D702}_{B}\boldsymbol{B},\end{eqnarray}$$

with $\unicode[STIX]{x1D702}_{B}$ being (5.2). As with the turbulent mass flux $\langle \unicode[STIX]{x1D70C}^{\prime }\boldsymbol{u}^{\prime }\rangle _{B}$ , the turbulent internal-energy flux $\langle q^{\prime }\boldsymbol{u}^{\prime }\rangle _{B}$ is along the mean magnetic field $\boldsymbol{B}$ in the opposite (or same) direction to $\boldsymbol{B}$ for the positive (or negative) dilatational cross-helicity. Irrespective of the sign of the turbulent dilatation, the direction of the internal-energy flux is determined by the sign of the dilatational turbulent cross-helicity. One interesting point is that in the isothermal process ( $\unicode[STIX]{x1D6FE}_{\text{s}}=1$ ) this internal-energy transport due to the dilatational cross-helicity does not show up, since in this case the internal-energy variation due to the turbulent dilatation disappears as (5.11).

Here we should note that, in the above physical pictures, the sign of the dilatational turbulent cross-helicity is just assumed or prescribed. In reality, turbulent fields including the velocity and magnetic-field fluctuations evolve subjected to the nonlinear dynamics of the MHD turbulence with inhomogeneous mean fields. A self-consistent treatment of the spatio-temporal evolution of the turbulent cross-helicity can be done only by solving the transport equation of the turbulent cross-helicity under the mutual influence of the mean and turbulence fields (e.g. see Yokoi & Hoshino Reference Yokoi and Hoshino2011; Yokoi Reference Yokoi2013). This point will be argued later in § 6. However, in the linearised MHD equations, the perturbations including the velocity and magnetic-field ones obey definite and specific modes following the dispersion relations. Since those linear wave arguments provide us with some understanding on the dilatational turbulent cross-helicity effect, we present such arguments in the following subsection, § 5.2.

Figure 2. Magnetohydrodynamic wave configuration.

5.2 Magnetohydrodynamic waves

In the presence of the mean magnetic field $\boldsymbol{B}$ , the time-dependent disturbances of the velocity and magnetic fields are propagated as magnetohydrodynamic (MHD) waves. If we restrict our attention to small-amplitude disturbances, we can consider small-scale perturbations as wave-like solutions of the linearised MHD equations.

As is shown in appendix A, the dispersion relation (A 12) has three roots. One is the transverse or shear Alfvén wave mode given by (A 16), and the other two are the fast and slow magnetoacoustic wave modes given by (A 17). In the following, these wave modes are argued in relation to the turbulent mass and internal-energy fluxes due to the dilatational cross-helicity effect.

In order to describe the eigenvector solutions of the velocity, magnetic-field and electric-field perturbations, $\unicode[STIX]{x1D6FF}\boldsymbol{u}$ , $\unicode[STIX]{x1D6FF}\boldsymbol{b}$ and $\unicode[STIX]{x1D6FF}\boldsymbol{e}$ , it is convenient to define a mutually orthogonal system of unit vectors. The unit vector in the direction of the mean magnetic field $\boldsymbol{B}$ is $\hat{\boldsymbol{b}}(=\boldsymbol{B}/|\boldsymbol{B}|)$ . The unit vector along the wave vector is $\hat{\unicode[STIX]{x1D73F}}(=\boldsymbol{k}/|\boldsymbol{k}|)$ . In addition, $\hat{\boldsymbol{a}}$ and $\hat{\unicode[STIX]{x1D749}}$ are the mutually orthogonal unit vectors to $\hat{\unicode[STIX]{x1D73F}}$ , which are defined by $\hat{\boldsymbol{a}}=-\boldsymbol{k}\times \boldsymbol{B}/|\boldsymbol{k}\times \boldsymbol{B}|$ and $\hat{\unicode[STIX]{x1D749}}=\hat{\boldsymbol{a}}\times \hat{\unicode[STIX]{x1D73F}}$ . Without losing generality, we have $\boldsymbol{B}$ lie in the $z$ direction and $\boldsymbol{k}$ vector in the $z$ – $x$ plane (figure 2). In this case, we have the unit vectors as

(5.13a ) $$\begin{eqnarray}\displaystyle & \hat{\boldsymbol{b}}=(0,0,1), & \displaystyle\end{eqnarray}$$

(5.13b ) $$\begin{eqnarray}\displaystyle & \hat{\unicode[STIX]{x1D73F}}=(\sin \unicode[STIX]{x1D703},0,\cos \unicode[STIX]{x1D703}), & \displaystyle\end{eqnarray}$$

(5.13c ) $$\begin{eqnarray}\displaystyle & \hat{\boldsymbol{a}}=(0,1,0), & \displaystyle\end{eqnarray}$$

(5.13d ) $$\begin{eqnarray}\displaystyle & \hat{\unicode[STIX]{x1D749}}=(\cos \unicode[STIX]{x1D703},0,-\sin \unicode[STIX]{x1D703}). & \displaystyle\end{eqnarray}$$

Transverse Alfvén mode

The transverse Alfvén mode given by (A 16) has eigenvectors

(5.14a ) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D6FF}\boldsymbol{u}\propto \unicode[STIX]{x1D6FF}\boldsymbol{b}\propto \hat{\boldsymbol{a}}=(0,1,0), & \displaystyle\end{eqnarray}$$

(5.14b ) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D6FF}\boldsymbol{e}\propto \cos \unicode[STIX]{x1D703}\hat{\unicode[STIX]{x1D749}}+\sin \unicode[STIX]{x1D703}\hat{\unicode[STIX]{x1D73F}}=(1,0,0) & \displaystyle\end{eqnarray}$$

$\boldsymbol{B}$

$\boldsymbol{k}$

$\unicode[STIX]{x1D6FF}\boldsymbol{u}$

$\unicode[STIX]{x1D6FF}\boldsymbol{b}$

$\hat{\unicode[STIX]{x1D73F}}\boldsymbol{\cdot }\unicode[STIX]{x1D6FF}\boldsymbol{u}=\hat{\unicode[STIX]{x1D73F}}\boldsymbol{\cdot }\unicode[STIX]{x1D6FF}\boldsymbol{b}=0$

$\hat{\unicode[STIX]{x1D73F}}\boldsymbol{\cdot }\unicode[STIX]{x1D6FF}\boldsymbol{u}=0$

$\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D70C}=0$

The group velocity of the transverse Alfvén mode is written as

(5.15) $$\begin{eqnarray}\boldsymbol{v}_{\text{g}}=\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D714}}{\unicode[STIX]{x2202}\boldsymbol{k}}=v_{\text{A}}\cos \unicode[STIX]{x1D703}\hat{\boldsymbol{b}},\end{eqnarray}$$

where $v_{\text{A}}$ is the Alfvén speed defined by (A 14). This implies that the material or mass is carried with this group velocity along the mean magnetic field $\boldsymbol{B}$ in the case of the transverse Alfvén mode.

The electric-field perturbation $\unicode[STIX]{x1D6FF}\boldsymbol{e}$ is not perpendicular to the wave vector ( $\hat{\unicode[STIX]{x1D73F}}\boldsymbol{\cdot }\unicode[STIX]{x1D6FF}\boldsymbol{e}\neq 0$ ), but is perpendicular to the mean magnetic field ( $\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D6FF}\boldsymbol{e}=0$ ). The Poynting flux due to the electric- and magnetic-field perturbations in the transverse Alfvén mode, $\boldsymbol{S}_{\text{A}}$ , is given by

(5.16) $$\begin{eqnarray}\boldsymbol{S}_{\text{A}}=\frac{\unicode[STIX]{x1D6FF}\boldsymbol{e}\times \unicode[STIX]{x1D6FF}\boldsymbol{b}}{\unicode[STIX]{x1D707}_{0}}\propto (\cos \unicode[STIX]{x1D703}\hat{\unicode[STIX]{x1D749}}+\sin \unicode[STIX]{x1D703}\hat{\unicode[STIX]{x1D73F}})\times \hat{\boldsymbol{a}}=\cos \unicode[STIX]{x1D703}\hat{\unicode[STIX]{x1D73F}}-\sin \unicode[STIX]{x1D703}\hat{\unicode[STIX]{x1D749}}=(0,0,1)=\hat{\boldsymbol{b}}.\end{eqnarray}$$

This shows that the wave energy is transferred along the mean magnetic field $\boldsymbol{B}$ . This effect is a purely incompressible effect, so not directly related to the compressional cross-helicity effects $-\unicode[STIX]{x1D705}_{B}\boldsymbol{B}$ or $-\unicode[STIX]{x1D702}_{B}\boldsymbol{B}$ . However, as will be referred to in § 6, the solenoidal and dilatational parts of the cross-helicity are coupled with each other. If the transverse Alfvén waves provide solenoidal but highly aligned velocity and magnetic-field perturbations, the dilatational cross-helicity may be also generated at the nonlinear stage of compressible MHD flow, resulting in the compressional or dilatational cross-helicity effects in the turbulent mass and internal-energy transports.

Magnetoacoustic modes

The fast and slow magnetoacoustic modes given by (A 17) have eigenvectors

(5.17a ) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D6FF}\boldsymbol{u}\propto \cos \unicode[STIX]{x1D703}(\unicode[STIX]{x1D714}^{2}-c_{\text{S}}^{2}k^{2})\hat{\unicode[STIX]{x1D749}}+\sin \unicode[STIX]{x1D703}\unicode[STIX]{x1D714}^{2}\hat{\unicode[STIX]{x1D73F}}=(\unicode[STIX]{x1D714}^{2}-c_{\text{S}}^{2}k^{2}\cos ^{2}\unicode[STIX]{x1D703},0,c_{\text{S}}^{2}k^{2}\cos \unicode[STIX]{x1D703}\sin \unicode[STIX]{x1D703}),\qquad & \displaystyle\end{eqnarray}$$

(5.17b ) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D6FF}\boldsymbol{b}\propto \hat{\unicode[STIX]{x1D749}}=(\cos \unicode[STIX]{x1D703},0,\,-\sin \unicode[STIX]{x1D703}), & \displaystyle\end{eqnarray}$$

(5.17c ) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D6FF}\boldsymbol{e}\propto \hat{\boldsymbol{a}}=(0,1,0) & \displaystyle\end{eqnarray}$$

$c_{\text{S}}$

$\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D70C}\neq 0$

$\unicode[STIX]{x1D735}\boldsymbol{\cdot }\unicode[STIX]{x1D6FF}\boldsymbol{u}\neq 0$

We see from (5.17a ) and (5.17b ) that the velocity perturbation $\unicode[STIX]{x1D6FF}\boldsymbol{u}$ has both perpendicular and parallel components to the wave vector $\unicode[STIX]{x1D73F}$ while the magnetic perturbation is only in the direction perpendicular to $\hat{\unicode[STIX]{x1D73F}}$ because of the solenoidal nature of $\unicode[STIX]{x1D6FF}\boldsymbol{b}$ ( $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\unicode[STIX]{x1D6FF}\boldsymbol{b}=0$ ). At the same time, there is a magnetic-field perturbation $\unicode[STIX]{x1D6FF}\boldsymbol{b}$ , as well as a velocity perturbation $\unicode[STIX]{x1D6FF}\boldsymbol{u}$ , that is parallel to the unperturbed magnetic field $\boldsymbol{B}$ .

If we take the inner product of $\unicode[STIX]{x1D6FF}\boldsymbol{u}$ (5.17a ) and $\unicode[STIX]{x1D6FF}\boldsymbol{b}$ (5.17b ) in the magnetoacoustic modes, we get

(5.18) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D6FF}\boldsymbol{b}\propto [\cos \unicode[STIX]{x1D703}(\unicode[STIX]{x1D714}^{2}-c_{\text{S}}^{2}k^{2})\hat{\unicode[STIX]{x1D749}}+\sin \unicode[STIX]{x1D703}\unicode[STIX]{x1D714}^{2}\hat{\unicode[STIX]{x1D73F}}]\boldsymbol{\cdot }\hat{\unicode[STIX]{x1D749}}=\cos \unicode[STIX]{x1D703}(\unicode[STIX]{x1D714}^{2}-c_{\text{S}}^{2}k^{2}).\end{eqnarray}$$

Since the magnetic perturbation $\unicode[STIX]{x1D6FF}\boldsymbol{b}$ has no components in the $\unicode[STIX]{x1D73F}$ direction (5.17b ), the cross-helicity due to the velocity and magnetic-field perturbations (turbulent cross-helicity) is generated only by the combination of the velocity and magnetic-field perturbations transverse to the wave vector $\boldsymbol{k}$ . The sign of the turbulent cross-helicity depends on the angle $\unicode[STIX]{x1D703}$ between the wave propagation $\boldsymbol{k}$ and the mean magnetic field $\boldsymbol{B}$ . The magnitude of the turbulent cross-helicity is expected to be maximum when the wave propagation is parallel or antiparallel to $\boldsymbol{B}$ ( $\unicode[STIX]{x1D703}=0$ or $\unicode[STIX]{x03C0}$ ). Also even in the case of $\unicode[STIX]{x1D703}=0$ or $\unicode[STIX]{x03C0}$ , if the phase velocity is represented by the sound speed: $u_{\unicode[STIX]{x1D719}}=c_{\text{S}}$ or equivalently $\unicode[STIX]{x1D714}^{2}=c_{\text{S}}^{2}k^{2}$ , the velocity perturbation along $\unicode[STIX]{x1D749}$ vanishes, then the velocity perturbation $\unicode[STIX]{x1D6FF}\boldsymbol{u}$ and the magnetic-field perturbation $\unicode[STIX]{x1D6FF}\boldsymbol{b}$ are orthogonal to each other. In this case, we have no turbulent cross-helicity. This is the case of the slow mode: the electromagnetic component completely disappears at $\unicode[STIX]{x1D703}=0,\unicode[STIX]{x03C0}$ and the slow mode wave is reduced to a sound wave. On the other hand, if the wave propagation direction is perpendicular to the unperturbed magnetic field ( $\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/2$ ), the velocity and magnetic perturbation are orthogonal to each other, then we have no turbulent cross-helicity. This is the case for the magnetoacoustic wave propagating in the perpendicular direction to $\boldsymbol{B}$ .

The Poynting flux due to the electric- and magnetic-field perturbations in the magnetoacoustic modes, $\boldsymbol{S}_{\text{M}}$ , is given by

(5.19) $$\begin{eqnarray}\boldsymbol{S}_{\text{M}}=\frac{\unicode[STIX]{x1D6FF}\boldsymbol{e}\times \unicode[STIX]{x1D6FF}\boldsymbol{b}}{\unicode[STIX]{x1D707}_{0}}\propto \hat{\boldsymbol{a}}\times \hat{\unicode[STIX]{x1D749}}=\hat{\boldsymbol{a}}\times (\hat{\boldsymbol{a}}\times \hat{\unicode[STIX]{x1D73F}})=-\hat{\boldsymbol{a}}^{2}\hat{\unicode[STIX]{x1D73F}}=-\hat{\unicode[STIX]{x1D73F}}.\end{eqnarray}$$

This shows that the wave energy is transferred along the wave vector in the direction opposite to $\boldsymbol{k}$ or $\unicode[STIX]{x1D73F}(\equiv \boldsymbol{k}/|\boldsymbol{k}|)$ .

These arguments suggest that the magnetoacoustic modes are, to some extent, the linearised counterpart of the compressive cross-helicity effect we obtained in §§ 4 and 5.1, since these modes have both compressibility and cross-helicity simultaneously. At the same time, the transverse Alfvén mode is very important for providing the turbulent cross-helicity. We should note that the cross-helicity is constituted only of the solenoidal motion and magnetic field both in the transverse Alfvén and magnetoacoustic modes as (5.14a ), (5.17a ), (5.17b ) and (5.18) show.

In the linearised MHD wave argument, mass and energy are considered to be carried in the direction of the mean magnetic field (transverse Alfvén mode) and in the direction to the wave vector (magnetoacoustic modes). Deviation from the aligned transport, i.e. a transfer in the direction oblique to $\boldsymbol{B}$ or $\boldsymbol{k}$ , may require some nonlinear, resistive or kinetic effects. On the other hand, in the arguments with strongly turbulent or nonlinear MHD processes, the gradient diffusion should be the effect of primary importance, and deviations from the gradient diffusion in the turbulent mass and energy transports show up with the strong density-variance effects in cross-diffusion and the compressional cross-helicity effects in transport along the mean magnetic field.

Both these effects need compressibility and the aligned components of the velocity and magnetic-field perturbations. However, there are several differences between these two effects. For example, as with the physical interpretation given in § 5.1, the compressive cross-helicity effect requires the inhomogeneity of the magnetic fluctuations along the mean magnetic field. In the magnetoacoustic modes, such magnetic fluctuation is not allowed to exist if the wave propagation is along the unperturbed magnetic field ( $\unicode[STIX]{x1D703}=0,\unicode[STIX]{x03C0}$ ) (see (5.17b )).