2.1 Magnetohydrodynamic 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 the kinetic and magnetic energies to heat through 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 interest lies 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 Means and fluctuations

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

2.3 Mean equations

The mean-density equation is

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

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

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

Hereafter we drop the fluctuation part of the viscosity, thermal conductivity, magnetic diffusivity ( $\unicode[STIX]{x1D707}^{\prime }=\unicode[STIX]{x1D705}^{\prime }=\unicode[STIX]{x1D702}^{\prime }=0$ ). Here $R_{U}$ is

(2.17) $$\begin{eqnarray}R_{U}^{\unicode[STIX]{x1D6FC}}=-\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\langle \unicode[STIX]{x1D70C}^{\prime }{u^{\prime }}^{\unicode[STIX]{x1D6FC}}\rangle -\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x^{a}}\langle \unicode[STIX]{x1D70C}^{\prime }{u^{\prime }}^{a}{u^{\prime }}^{\unicode[STIX]{x1D6FC}}\rangle -(\unicode[STIX]{x1D6FE}_{\text{s}}-1)\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x^{\unicode[STIX]{x1D6FC}}}\langle \unicode[STIX]{x1D70C}^{\prime }q^{\prime }\rangle -\frac{1}{2\unicode[STIX]{x1D707}_{0}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x^{\unicode[STIX]{x1D6FC}}}\langle \boldsymbol{b}^{\prime 2}\rangle .\end{eqnarray}$$

The mean internal energy equation:

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

where $\overline{\unicode[STIX]{x1D705}}$ is the mean part of the thermal conductivity, and $R_{Q}$ is

(2.19) $$\begin{eqnarray}R_{Q}=-\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\langle \unicode[STIX]{x1D70C}^{\prime }q^{\prime }\rangle -\unicode[STIX]{x1D735}\boldsymbol{\cdot }\langle \unicode[STIX]{x1D70C}^{\prime }q^{\prime }\boldsymbol{u}^{\prime }\rangle -(\unicode[STIX]{x1D6FE}_{\text{s}}-1)\langle \unicode[STIX]{x1D70C}^{\prime }q^{\prime }\rangle \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{U}-(\unicode[STIX]{x1D6FE}_{\text{s}}-1)\langle \unicode[STIX]{x1D70C}^{\prime }q^{\prime }\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}^{\prime }\rangle .\end{eqnarray}$$

The mean magnetic induction equation is

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

( $\overline{\unicode[STIX]{x1D702}}$ : mean part of the magnetic diffusivity). Hereafter $R_{U}$ and $R_{Q}$ will be neglected since their contributions are considered to be small.

2.4 Fluctuation equations

Subtracting the mean-field equations from (2.1)–(2.3) and (2.11), we obtain the equations of the fluctuation fields as follows:

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

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

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

(2.24) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}\boldsymbol{b}^{\prime }}{\unicode[STIX]{x2202}t}+(\boldsymbol{U}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{b}^{\prime }+(\boldsymbol{u}^{\prime }\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{b}^{\prime }-(\boldsymbol{b}^{\prime }\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{u}^{\prime }-\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FB}^{2}\boldsymbol{b}^{\prime }-(\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{u}^{\prime }+\boldsymbol{B}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}^{\prime }\nonumber\\ \displaystyle & & \displaystyle \quad =-(\boldsymbol{u}^{\prime }\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{B}+(\boldsymbol{b}^{\prime }\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{U}-\boldsymbol{b}^{\prime }\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{U}+R_{b}^{\unicode[STIX]{x1D6FC}},\end{eqnarray}$$

where $\text{D}/\text{D}t=\unicode[STIX]{x2202}/\unicode[STIX]{x2202}t+\boldsymbol{U}\boldsymbol{\cdot }\unicode[STIX]{x1D735}$ . Hereafter, the terms expected to be small, $\boldsymbol{R}_{u}$ and $\boldsymbol{R}_{b}$ , will be neglected.

3.1 Introduction of two scales

We introduce two scales by the fast and slow variables as

(3.1a-d ) $$\begin{eqnarray}\unicode[STIX]{x1D743}=\boldsymbol{x},\quad \boldsymbol{X}=\unicode[STIX]{x1D6FF}\boldsymbol{x};\quad \unicode[STIX]{x1D70F}=t,\quad T=\unicode[STIX]{x1D6FF}t,\end{eqnarray}$$

where $\unicode[STIX]{x1D6FF}$ is the scale parameter. If $\unicode[STIX]{x1D6FF}$ is small, the variables $\boldsymbol{X}$ and $T$ represent long spatial and time scales since they are not negligible only when the original variables $\boldsymbol{x}$ and $t$ change considerably. In this sense, $\boldsymbol{X}$ and $T$ are called the slow variables. The field quantities are expressed as

(3.2) $$\begin{eqnarray}f=F(\boldsymbol{X};T)+f^{\prime }(\unicode[STIX]{x1D743},\boldsymbol{X};\unicode[STIX]{x1D70F},T).\end{eqnarray}$$

Here we assume the mean field depends on the slow variables whereas the fluctuation depends on both the fast and slow variables. Because of the chain rule for partial derivatives, the partial derivatives are written as

(3.3a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D735}_{\boldsymbol{x}}=\unicode[STIX]{x1D735}_{\unicode[STIX]{x1D743}}+\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D735}_{\boldsymbol{X}},\quad \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}=\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}+\unicode[STIX]{x1D6FF}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}T}\end{eqnarray}$$

( $\unicode[STIX]{x1D735}_{\boldsymbol{x}}^{\unicode[STIX]{x1D6FC}}=\unicode[STIX]{x2202}/\unicode[STIX]{x2202}x^{\unicode[STIX]{x1D6FC}}$ , $\unicode[STIX]{x1D735}_{\unicode[STIX]{x1D743}}^{\unicode[STIX]{x1D6FC}}=\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\unicode[STIX]{x1D6FC}}$ , and $\unicode[STIX]{x1D735}_{\boldsymbol{X}}^{\unicode[STIX]{x1D6FC}}=\unicode[STIX]{x2202}/\unicode[STIX]{x2202}X^{\unicode[STIX]{x1D6FC}}$ ). This is a derivative expansion: a derivative with respect to a slow variable appears as a higher-order term in $\unicode[STIX]{x1D6FF}$ . We seek solutions which are functions of both fast and slow variables, $(\boldsymbol{X};T)$ and $(\unicode[STIX]{x1D743};\unicode[STIX]{x1D70F})$ artificially treated as independent variables.

Under the two-scale descriptions, the equations of the fluctuation fields are written as

(3.4) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}^{\prime }}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}+U^{a}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}^{\prime }}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{a}}+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{a}}(\unicode[STIX]{x1D70C}^{\prime }{u^{\prime }}^{a})+\overline{\unicode[STIX]{x1D70C}}\frac{\unicode[STIX]{x2202}{u^{\prime }}^{a}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{a}}=\unicode[STIX]{x1D6FF}\left[-{u^{\prime }}^{a}\frac{\unicode[STIX]{x2202}\overline{\unicode[STIX]{x1D70C}}}{\unicode[STIX]{x2202}X^{a}}-\unicode[STIX]{x1D70C}^{\prime }\frac{\unicode[STIX]{x2202}U^{a}}{\unicode[STIX]{x2202}X^{a}}\right]+R_{\unicode[STIX]{x1D70C}}^{\prime },\end{eqnarray}$$

(3.5) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}{u^{\prime }}^{\unicode[STIX]{x1D6FC}}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}+U^{a}\frac{\unicode[STIX]{x2202}{u^{\prime }}^{\unicode[STIX]{x1D6FC}}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{a}}+{u^{\prime }}^{a}\frac{\unicode[STIX]{x2202}{u^{\prime }}^{\unicode[STIX]{x1D6FC}}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{a}}-\frac{1}{\overline{\unicode[STIX]{x1D70C}}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{a}}\overline{\unicode[STIX]{x1D707}}{s^{\prime }}^{a\unicode[STIX]{x1D6FC}}\nonumber\\ \displaystyle & & \displaystyle \qquad +\,(\unicode[STIX]{x1D6FE}_{\text{s}}-1)\left(\frac{\unicode[STIX]{x2202}q^{\prime }}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\unicode[STIX]{x1D6FC}}}+\frac{Q}{\overline{\unicode[STIX]{x1D70C}}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}^{\prime }}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\unicode[STIX]{x1D6FC}}}\right)-\frac{1}{\unicode[STIX]{x1D707}_{0}\overline{\unicode[STIX]{x1D70C}}}\left[{b^{\prime }}^{a}\frac{\unicode[STIX]{x2202}{b^{\prime }}^{\unicode[STIX]{x1D6FC}}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{a}}+B^{a}\frac{\unicode[STIX]{x2202}{b^{\prime }}^{\unicode[STIX]{x1D6FC}}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{a}}\right]\nonumber\\ \displaystyle & & \displaystyle \quad =\unicode[STIX]{x1D6FF}\left[-{u^{\prime }}^{a}\frac{\unicode[STIX]{x2202}U^{\unicode[STIX]{x1D6FC}}}{\unicode[STIX]{x2202}X^{a}}-(\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)+\frac{1}{\unicode[STIX]{x1D707}_{0}\overline{\unicode[STIX]{x1D70C}}}{b^{\prime }}^{a}\frac{\unicode[STIX]{x2202}B^{\unicode[STIX]{x1D6FC}}}{\unicode[STIX]{x2202}X^{a}}-\frac{\unicode[STIX]{x1D70C}^{\prime }}{\overline{\unicode[STIX]{x1D70C}}}\frac{\text{D}U^{\unicode[STIX]{x1D6FC}}}{\text{D}T}\right]+R_{u}^{\prime \unicode[STIX]{x1D6FC}},\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

(3.6) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}q^{\prime }}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}+U^{a}\frac{\unicode[STIX]{x2202}q^{\prime }}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{a}}+{u^{\prime }}^{a}\frac{\unicode[STIX]{x2202}q^{\prime }}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{a}}-\frac{1}{\overline{\unicode[STIX]{x1D70C}}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{a}}\frac{\overline{\unicode[STIX]{x1D705}}}{C_{V}}\frac{\unicode[STIX]{x2202}q^{\prime }}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{a}}+(\unicode[STIX]{x1D6FE}_{\text{s}}-1)Q\frac{\unicode[STIX]{x2202}{u^{\prime }}^{a}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{a}}\nonumber\\ \displaystyle & & \displaystyle \quad =\unicode[STIX]{x1D6FF}\left[-{u^{\prime }}^{a}\frac{\unicode[STIX]{x2202}Q}{\unicode[STIX]{x2202}X^{a}}-(\unicode[STIX]{x1D6FE}_{\text{s}}-1)\left(q^{\prime }+\frac{\unicode[STIX]{x1D70C}^{\prime }}{\overline{\unicode[STIX]{x1D70C}}}Q\right)\frac{\unicode[STIX]{x2202}U^{a}}{\unicode[STIX]{x2202}X^{a}}\right]+R_{q}^{\prime },\end{eqnarray}$$

(3.7) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}{b^{\prime }}^{\unicode[STIX]{x1D6FC}}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}+U^{a}\frac{\unicode[STIX]{x2202}{b^{\prime }}^{\unicode[STIX]{x1D6FC}}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{a}}+{u^{\prime }}^{a}\frac{\unicode[STIX]{x2202}{b^{\prime }}^{\unicode[STIX]{x1D6FC}}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{a}}-{b^{\prime }}^{a}\frac{\unicode[STIX]{x2202}{u^{\prime }}^{\unicode[STIX]{x1D6FC}}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{a}}-\unicode[STIX]{x1D702}\frac{\unicode[STIX]{x2202}^{2}{b^{\prime }}^{\unicode[STIX]{x1D6FC}}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{a}\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{a}}-B^{a}\frac{\unicode[STIX]{x2202}{u^{\prime }}^{\unicode[STIX]{x1D6FC}}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{a}}+B^{\unicode[STIX]{x1D6FC}}\frac{\unicode[STIX]{x2202}{u^{\prime }}^{a}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{a}}\nonumber\\ \displaystyle & & \displaystyle \quad =\unicode[STIX]{x1D6FF}\left[-{u^{\prime }}^{a}\frac{\unicode[STIX]{x2202}B^{\unicode[STIX]{x1D6FC}}}{\unicode[STIX]{x2202}X^{a}}+{b^{\prime }}^{a}\frac{\unicode[STIX]{x2202}U^{\unicode[STIX]{x1D6FC}}}{\unicode[STIX]{x2202}X^{a}}-{b^{\prime }}^{\unicode[STIX]{x1D6FC}}\frac{\unicode[STIX]{x2202}U^{a}}{\unicode[STIX]{x2202}X^{a}}\right]+R_{b}^{\prime },\end{eqnarray}$$

where $\text{D}/\text{D}T=\unicode[STIX]{x2202}/\unicode[STIX]{x2202}T+\boldsymbol{U}\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{\boldsymbol{X}}$ .

3.2 Fourier transform with respect to the fast variables

We use the Fourier representation with respect to the fast spatial variable $\unicode[STIX]{x1D743}$ as

(3.8) $$\begin{eqnarray}f^{\prime }(\unicode[STIX]{x1D743},\boldsymbol{X};\unicode[STIX]{x1D70F},T)=\frac{1}{(2\unicode[STIX]{x03C0})^{3}}\int \,\text{d}\boldsymbol{k}\,\hat{f}^{\prime }(\boldsymbol{k},\boldsymbol{X};\unicode[STIX]{x1D70F},T)\exp [-\text{i}\boldsymbol{k}\boldsymbol{\cdot }(\unicode[STIX]{x1D743}-\boldsymbol{U}\unicode[STIX]{x1D70F})].\end{eqnarray}$$

Here the factor $\text{i}\boldsymbol{k}\boldsymbol{\cdot }(\unicode[STIX]{x1D743}-\boldsymbol{U}\unicode[STIX]{x1D70F})$ means that the fast-varying motions of turbulence are treated in the frame moving with the velocity $\boldsymbol{U}$ .

The system of equations of MHD turbulence consists of

(3.9) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F})}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}-\text{i}k^{a}\iint \unicode[STIX]{x1D6FF}(\boldsymbol{k}-\boldsymbol{p}-\boldsymbol{q})\,\text{d}\boldsymbol{p}\,\text{d}\boldsymbol{q}\,{u^{\prime }}^{a}(\boldsymbol{p};\unicode[STIX]{x1D70F})\unicode[STIX]{x1D70C}^{\prime }(\boldsymbol{q};\unicode[STIX]{x1D70F})-\text{i}k^{a}\overline{\unicode[STIX]{x1D70C}}{u^{\prime }}^{a}(\boldsymbol{k};\unicode[STIX]{x1D70F})\nonumber\\ \displaystyle & & \displaystyle \quad =\unicode[STIX]{x1D6FF}\left[-{u^{\prime }}^{a}(\boldsymbol{k};\unicode[STIX]{x1D70F})\frac{\unicode[STIX]{x2202}\overline{\unicode[STIX]{x1D70C}}}{\unicode[STIX]{x2202}X^{a}}-\unicode[STIX]{x1D70C}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F})\unicode[STIX]{x1D735}_{\boldsymbol{X}}\boldsymbol{\cdot }\boldsymbol{U}-\frac{D\unicode[STIX]{x1D70C}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F})}{\text{D}T}-\overline{\unicode[STIX]{x1D70C}}\frac{\unicode[STIX]{x2202}{u^{\prime }}^{a}(\boldsymbol{k};\unicode[STIX]{x1D70F})}{\unicode[STIX]{x2202}X^{a}}\right],\end{eqnarray}$$

(3.10) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}{u^{\prime }}^{\unicode[STIX]{x1D6FC}}(\boldsymbol{k};\unicode[STIX]{x1D70F})}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}+\overline{\unicode[STIX]{x1D708}}k^{2}{u^{\prime }}^{a}(\boldsymbol{k};\unicode[STIX]{x1D70F})+\frac{1}{3}\overline{\unicode[STIX]{x1D708}}k^{\unicode[STIX]{x1D6FC}}k^{a}{u^{\prime }}^{a}(\boldsymbol{k};\unicode[STIX]{x1D70F})\nonumber\\ \displaystyle & & \displaystyle \qquad -\,\text{i}\iint \unicode[STIX]{x1D6FF}(\boldsymbol{k}-\boldsymbol{p}-\boldsymbol{q})\,\text{d}\boldsymbol{p}\,\text{d}\boldsymbol{q}M_{\text{c}}^{\unicode[STIX]{x1D6FC}ab}{u^{\prime }}^{a}(\boldsymbol{p};\unicode[STIX]{x1D70F}){u^{\prime }}^{b}(\boldsymbol{q};\unicode[STIX]{x1D70F})\nonumber\\ \displaystyle & & \displaystyle \qquad -\,\text{i}(\unicode[STIX]{x1D6FE}_{\text{s}}-1)k^{\unicode[STIX]{x1D6FC}}q^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F})-\text{i}(\unicode[STIX]{x1D6FE}_{\text{s}}-1)\frac{Q}{\overline{\unicode[STIX]{x1D70C}}}k^{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D70C}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F})\nonumber\\ \displaystyle & & \displaystyle \qquad -\,\frac{\text{i}}{\unicode[STIX]{x1D707}_{0}\overline{\unicode[STIX]{x1D70C}}}\iint \unicode[STIX]{x1D6FF}(\boldsymbol{k}-\boldsymbol{p}-\boldsymbol{q})\,\text{d}\boldsymbol{p}\,\text{d}\boldsymbol{q}\,M_{\text{c}}^{\unicode[STIX]{x1D6FC}ab}{b^{\prime }}^{a}(\boldsymbol{p};\unicode[STIX]{x1D70F}){b^{\prime }}^{b}(\boldsymbol{q};\unicode[STIX]{x1D70F})+\frac{\text{i}}{\unicode[STIX]{x1D707}_{0}\overline{\unicode[STIX]{x1D70C}}}k^{a}B^{a}{b^{\prime }}^{\unicode[STIX]{x1D6FC}}(\boldsymbol{k};\unicode[STIX]{x1D70F})\nonumber\\ \displaystyle & & \displaystyle \quad =\unicode[STIX]{x1D6FF}\left[-{u^{\prime }}^{a}(\boldsymbol{k};\unicode[STIX]{x1D70F})\frac{\unicode[STIX]{x2202}U^{\unicode[STIX]{x1D6FC}}}{\unicode[STIX]{x2202}X^{a}}-(\unicode[STIX]{x1D6FE}_{\text{s}}-1)\left(\frac{\unicode[STIX]{x1D70C}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F})}{\overline{\unicode[STIX]{x1D70C}}}\frac{\unicode[STIX]{x2202}Q}{\unicode[STIX]{x2202}X^{\unicode[STIX]{x1D6FC}}}+\frac{q^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F})}{\overline{\unicode[STIX]{x1D70C}}}\frac{\unicode[STIX]{x2202}\overline{\unicode[STIX]{x1D70C}}}{\unicode[STIX]{x2202}X^{\unicode[STIX]{x1D6FC}}}\right)\right.\nonumber\\ \displaystyle & & \displaystyle \qquad \left.-\,\frac{1}{\unicode[STIX]{x1D707}_{0}\overline{\unicode[STIX]{x1D70C}}}{b^{\prime }}^{a}(\boldsymbol{k};\unicode[STIX]{x1D70F})\frac{\unicode[STIX]{x2202}B^{\unicode[STIX]{x1D6FC}}}{\unicode[STIX]{x2202}X^{a}}-\frac{\unicode[STIX]{x1D70C}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F})}{\overline{\unicode[STIX]{x1D70C}}}\left(\frac{\unicode[STIX]{x2202}U^{\unicode[STIX]{x1D6FC}}}{\unicode[STIX]{x2202}T}+U^{a}\frac{\unicode[STIX]{x2202}U^{\unicode[STIX]{x1D6FC}}}{\unicode[STIX]{x2202}X^{a}}\right)\right],\end{eqnarray}$$

(3.11) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}q^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F})}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}+\overline{\unicode[STIX]{x1D706}}k^{2}q^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F})-\text{i}(\unicode[STIX]{x1D6FE}_{\text{s}}-1)Qk^{a}{u^{\prime }}^{a}(\boldsymbol{k};\unicode[STIX]{x1D70F})\nonumber\\ \displaystyle & & \displaystyle \qquad -\,\text{i}\iint \unicode[STIX]{x1D6FF}(\boldsymbol{k}-\boldsymbol{p}-\boldsymbol{q})\,\text{d}\boldsymbol{p}\,\text{d}\boldsymbol{q}q^{a}{u^{\prime }}^{a}(\boldsymbol{p};\unicode[STIX]{x1D70F})q^{\prime }(\boldsymbol{q};\unicode[STIX]{x1D70F})\nonumber\\ \displaystyle & & \displaystyle \quad =\unicode[STIX]{x1D6FF}\left[-{u^{\prime }}^{a}(\boldsymbol{k};\unicode[STIX]{x1D70F})\frac{\unicode[STIX]{x2202}Q}{\unicode[STIX]{x2202}X^{a}}-(\unicode[STIX]{x1D6FE}_{\text{s}}-1)\left(q^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F})+\frac{Q}{\overline{\unicode[STIX]{x1D70C}}}\unicode[STIX]{x1D70C}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F})\right)\frac{\unicode[STIX]{x2202}U^{a}}{\unicode[STIX]{x2202}X^{a}}\right.\nonumber\\ \displaystyle & & \displaystyle \qquad \left.-\,\frac{Dq^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F})}{\text{D}T}-\text{i}(\unicode[STIX]{x1D6FE}_{\text{s}}-1)Qk^{a}{u^{\prime }}^{a}(\boldsymbol{k};\unicode[STIX]{x1D70F})\right],\end{eqnarray}$$

(3.12) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}{b^{\prime }}^{\unicode[STIX]{x1D6FC}}(\boldsymbol{k};\unicode[STIX]{x1D70F})}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}+\overline{\unicode[STIX]{x1D702}}k^{2}{b^{\prime }}^{\unicode[STIX]{x1D6FC}}(\boldsymbol{k};\unicode[STIX]{x1D70F})\nonumber\\ \displaystyle & & \displaystyle \qquad -\,\text{i}\iint \unicode[STIX]{x1D6FF}(\boldsymbol{k}-\boldsymbol{p}-\boldsymbol{q})\,\text{d}\boldsymbol{p}\,\text{d}\boldsymbol{q}N_{\text{c}}^{\unicode[STIX]{x1D6FC}ab}{b^{\prime }}^{a}(\boldsymbol{p};\unicode[STIX]{x1D70F}){u^{\prime }}^{b}(\boldsymbol{q};\unicode[STIX]{x1D70F})+\text{i}B^{a}k^{a}{u^{\prime }}^{\unicode[STIX]{x1D6FC}}(\boldsymbol{k};\unicode[STIX]{x1D70F})+\text{i}B^{\unicode[STIX]{x1D6FC}}k^{a}{u^{\prime }}^{a}(\boldsymbol{k};\unicode[STIX]{x1D70F})\nonumber\\ \displaystyle & & \displaystyle \quad =\unicode[STIX]{x1D6FF}\left[-{u^{\prime }}^{a}(\boldsymbol{k};\unicode[STIX]{x1D70F})\frac{\unicode[STIX]{x2202}B^{\unicode[STIX]{x1D6FC}}}{\unicode[STIX]{x2202}X^{a}}+{b^{\prime }}^{a}(\boldsymbol{k};\unicode[STIX]{x1D70F})\frac{\unicode[STIX]{x2202}U^{\unicode[STIX]{x1D6FC}}}{\unicode[STIX]{x2202}X^{a}}+{b^{\prime }}^{\unicode[STIX]{x1D6FC}}(\boldsymbol{k};\unicode[STIX]{x1D70F})\frac{\unicode[STIX]{x2202}U^{a}}{\unicode[STIX]{x2202}X^{a}}\right.\nonumber\\ \displaystyle & & \displaystyle \qquad \left.-\,\frac{D{b^{\prime }}^{\unicode[STIX]{x1D6FC}}(\boldsymbol{k};\unicode[STIX]{x1D70F})}{\text{D}T}+B^{a}\frac{\unicode[STIX]{x2202}{u^{\prime }}^{\unicode[STIX]{x1D6FC}}(\boldsymbol{k};\unicode[STIX]{x1D70F})}{\unicode[STIX]{x2202}X^{a}}-B^{\unicode[STIX]{x1D6FC}}\frac{\unicode[STIX]{x2202}{u^{\prime }}^{a}(\boldsymbol{k};\unicode[STIX]{x1D70F})}{\unicode[STIX]{x2202}X^{a}}\right],\end{eqnarray}$$

where

(3.13) $$\begin{eqnarray}\displaystyle & \overline{\unicode[STIX]{x1D706}}=\displaystyle \frac{\overline{\unicode[STIX]{x1D705}}}{C_{\text{V}}\overline{\unicode[STIX]{x1D70C}}}, & \displaystyle\end{eqnarray}$$

(3.14) $$\begin{eqnarray}\displaystyle & M_{\text{c}}^{\unicode[STIX]{x1D6FC}ab}(\boldsymbol{p},\boldsymbol{q})={\textstyle \frac{1}{2}}(q^{a}\unicode[STIX]{x1D6FF}^{\unicode[STIX]{x1D6FC}b}+p^{b}\unicode[STIX]{x1D6FF}^{\unicode[STIX]{x1D6FC}a}), & \displaystyle\end{eqnarray}$$

(3.15) $$\begin{eqnarray}\displaystyle & N_{\text{c}}^{\unicode[STIX]{x1D6FC}ab}(\boldsymbol{p},\boldsymbol{q})={\textstyle \frac{1}{2}}(q^{a}\unicode[STIX]{x1D6FF}^{\unicode[STIX]{x1D6FC}b}-p^{b}\unicode[STIX]{x1D6FF}^{\unicode[STIX]{x1D6FC}a}). & \displaystyle\end{eqnarray}$$

3.3 Scale-parameter expansion

We apply the scale-parameter expansion

(3.16) $$\begin{eqnarray}f^{\prime }=\mathop{\sum }_{n=0}^{\infty }\unicode[STIX]{x1D6FF}^{n}f_{n}^{\prime }\end{eqnarray}$$

to (3.9)–(3.12). The zeroth-order $O(\unicode[STIX]{x1D6FF}^{0})$ equations are written as

(3.17) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}_{0}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F})}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}-\text{i}k^{a}\iint \unicode[STIX]{x1D6FF}(\boldsymbol{k}-\boldsymbol{p}-\boldsymbol{q})\,\text{d}\boldsymbol{p}\,\text{d}\boldsymbol{q}u_{0}^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{p};\unicode[STIX]{x1D70F})\unicode[STIX]{x1D70C}_{0}^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{q};\unicode[STIX]{x1D70F})-\text{i}k^{a}\overline{\unicode[STIX]{x1D70C}}u_{0}^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{k};\unicode[STIX]{x1D70F})=0,\end{eqnarray}$$

(3.18) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}u_{0}^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FC}}(\boldsymbol{k};\unicode[STIX]{x1D70F})}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}+\overline{\unicode[STIX]{x1D708}}k^{2}u_{0}^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{k};\unicode[STIX]{x1D70F})+\frac{1}{3}\overline{\unicode[STIX]{x1D708}}k^{\unicode[STIX]{x1D6FC}}k^{a}u_{0}^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{k};\unicode[STIX]{x1D70F})\nonumber\\ \displaystyle & & \displaystyle \quad -\,\text{i}\iint \unicode[STIX]{x1D6FF}(\boldsymbol{k}-\boldsymbol{p}-\boldsymbol{q})\,\text{d}\boldsymbol{p}\,\text{d}\boldsymbol{q}M_{\text{c}}^{\unicode[STIX]{x1D6FC}ab}u_{0}^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{p};\unicode[STIX]{x1D70F})u_{0}^{\prime }\hspace{0.0pt}^{b}(\boldsymbol{q};\unicode[STIX]{x1D70F})\nonumber\\ \displaystyle & & \displaystyle \quad -\,\text{i}(\unicode[STIX]{x1D6FE}_{\text{s}}-1)k^{\unicode[STIX]{x1D6FC}}q_{0}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F})-\text{i}(\unicode[STIX]{x1D6FE}_{\text{s}}-1)\frac{Q}{\overline{\unicode[STIX]{x1D70C}}}k^{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D70C}_{0}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F})\nonumber\\ \displaystyle & & \displaystyle \quad -\,\frac{\text{i}}{\unicode[STIX]{x1D707}_{0}\overline{\unicode[STIX]{x1D70C}}}\iint \unicode[STIX]{x1D6FF}(\boldsymbol{k}-\boldsymbol{p}-\boldsymbol{q})\,\text{d}\boldsymbol{p}\,\text{d}\boldsymbol{q}M_{\text{c}}^{\unicode[STIX]{x1D6FC}ab}b^{\prime }\hspace{0.0pt}_{0}^{a}(\boldsymbol{p};\unicode[STIX]{x1D70F})b_{0}^{\prime }\hspace{0.0pt}^{b}(\boldsymbol{q};\unicode[STIX]{x1D70F})\nonumber\\ \displaystyle & & \displaystyle \quad +\,\frac{\text{i}}{\unicode[STIX]{x1D707}_{0}\overline{\unicode[STIX]{x1D70C}}}k^{a}B^{a}b_{0}^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FC}}(\boldsymbol{k};\unicode[STIX]{x1D70F})=0,\end{eqnarray}$$

(3.19) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}q_{0}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F})}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}+\overline{\unicode[STIX]{x1D706}}k^{2}q_{0}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F})-\text{i}\iint \unicode[STIX]{x1D6FF}(\boldsymbol{k}-\boldsymbol{p}-\boldsymbol{q})\,\text{d}\boldsymbol{p}\,\text{d}\boldsymbol{q}q^{a}u_{0}^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{p};\unicode[STIX]{x1D70F})q_{0}^{\prime }(\boldsymbol{q};\unicode[STIX]{x1D70F})\nonumber\\ \displaystyle & & \displaystyle \quad -\,\text{i}(\unicode[STIX]{x1D6FE}_{\text{s}}-1)Qk^{a}u_{0}^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{k};\unicode[STIX]{x1D70F})=0,\end{eqnarray}$$

(3.20) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}b_{0}^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FC}}(\boldsymbol{k};\unicode[STIX]{x1D70F})}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}+\overline{\unicode[STIX]{x1D702}}k^{2}b_{0}^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FC}}(\boldsymbol{k};\unicode[STIX]{x1D70F})-\text{i}\iint \unicode[STIX]{x1D6FF}(\boldsymbol{k}-\boldsymbol{p}-\boldsymbol{q})\,\text{d}\boldsymbol{p}\,\text{d}\boldsymbol{q}~N_{\text{c}}^{\unicode[STIX]{x1D6FC}ab}u_{0}^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{p};\unicode[STIX]{x1D70F})b_{0}^{\prime }\hspace{0.0pt}^{b}(\boldsymbol{q};\unicode[STIX]{x1D70F})\nonumber\\ \displaystyle & & \displaystyle \quad +\,\text{i}B^{a}k^{a}u_{0}^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FC}}(\boldsymbol{k};\unicode[STIX]{x1D70F})+\text{i}B^{\unicode[STIX]{x1D6FC}}k^{a}u_{0}^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{k};\unicode[STIX]{x1D70F})=0.\end{eqnarray}$$

On the other hand, the first-order $O(\unicode[STIX]{x1D6FF}^{1})$ equations are

(3.21) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}_{1}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F})}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}-\text{i}k^{a}\iint \unicode[STIX]{x1D6FF}(\boldsymbol{k}-\boldsymbol{p}-\boldsymbol{q})\,\text{d}\boldsymbol{p}\,\text{d}\boldsymbol{q}\,u_{0}^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{p};\unicode[STIX]{x1D70F})\unicode[STIX]{x1D70C}_{1}^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{q};\unicode[STIX]{x1D70F})-\text{i}k^{a}\overline{\unicode[STIX]{x1D70C}}u_{1}^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{k};\unicode[STIX]{x1D70F})\nonumber\\ \displaystyle & & \displaystyle \quad =-u_{0}^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{k};\unicode[STIX]{x1D70F})\frac{\unicode[STIX]{x2202}\overline{\unicode[STIX]{x1D70C}}}{\unicode[STIX]{x2202}X^{a}}-\unicode[STIX]{x1D70C}_{0}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F})\unicode[STIX]{x1D735}_{\boldsymbol{X}}\boldsymbol{\cdot }\boldsymbol{U}-\frac{D\unicode[STIX]{x1D70C}_{0}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F})}{\text{D}T}-\overline{\unicode[STIX]{x1D70C}}\frac{\unicode[STIX]{x2202}u_{0}^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{k};\unicode[STIX]{x1D70F})}{\unicode[STIX]{x2202}X^{a}},\end{eqnarray}$$

(3.22) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}u_{1}^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FC}}(\boldsymbol{k};\unicode[STIX]{x1D70F})}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}+\overline{\unicode[STIX]{x1D708}}k^{2}u_{1}^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{k};\unicode[STIX]{x1D70F})+\frac{1}{3}\overline{\unicode[STIX]{x1D708}}k^{\unicode[STIX]{x1D6FC}}k^{a}u_{1}^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{k};\unicode[STIX]{x1D70F})\nonumber\\ \displaystyle & & \displaystyle \qquad -\,2\text{i}\iint \unicode[STIX]{x1D6FF}(\boldsymbol{k}-\boldsymbol{p}-\boldsymbol{q})\,\text{d}\boldsymbol{p}\,\text{d}\boldsymbol{q}\,M_{\text{c}}^{\unicode[STIX]{x1D6FC}ab}u_{0}^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{p};\unicode[STIX]{x1D70F})u_{1}^{\prime }\hspace{0.0pt}^{b}(\boldsymbol{q};\unicode[STIX]{x1D70F})\nonumber\\ \displaystyle & & \displaystyle \qquad -\,\text{i}(\unicode[STIX]{x1D6FE}_{\text{s}}-1)k^{\unicode[STIX]{x1D6FC}}q_{1}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F})-\text{i}(\unicode[STIX]{x1D6FE}_{\text{s}}-1)\frac{Q}{\overline{\unicode[STIX]{x1D70C}}}k^{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D70C}_{1}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F})\nonumber\\ \displaystyle & & \displaystyle \qquad -\frac{2\text{i}}{\unicode[STIX]{x1D707}_{0}\overline{\unicode[STIX]{x1D70C}}}\iint \unicode[STIX]{x1D6FF}(\boldsymbol{k}-\boldsymbol{p}-\boldsymbol{q})\,\text{d}\boldsymbol{p}\,\text{d}\boldsymbol{q}\,M_{\text{c}}^{\unicode[STIX]{x1D6FC}ab}b_{0}^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{p};\unicode[STIX]{x1D70F})b_{1}^{\prime }\hspace{0.0pt}^{b}(\boldsymbol{q};\unicode[STIX]{x1D70F})+\frac{\text{i}}{\unicode[STIX]{x1D707}_{0}\overline{\unicode[STIX]{x1D70C}}}k^{a}B^{a}b_{1}^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FC}}(\boldsymbol{k};\unicode[STIX]{x1D70F})\nonumber\\ \displaystyle & & \displaystyle \quad =-u_{0}^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{k};\unicode[STIX]{x1D70F})\frac{\unicode[STIX]{x2202}U^{\unicode[STIX]{x1D6FC}}}{\unicode[STIX]{x2202}X^{a}}-(\unicode[STIX]{x1D6FE}_{\text{s}}-1)\left(\frac{\unicode[STIX]{x1D70C}_{0}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F})}{\overline{\unicode[STIX]{x1D70C}}}\frac{\unicode[STIX]{x2202}Q}{\unicode[STIX]{x2202}X^{\unicode[STIX]{x1D6FC}}}+\frac{q_{0}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F})}{\overline{\unicode[STIX]{x1D70C}}}\frac{\unicode[STIX]{x2202}\overline{\unicode[STIX]{x1D70C}}}{\unicode[STIX]{x2202}X^{\unicode[STIX]{x1D6FC}}}\right)\nonumber\\ \displaystyle & & \displaystyle \qquad -\,\frac{1}{\unicode[STIX]{x1D707}_{0}\overline{\unicode[STIX]{x1D70C}}}b_{0}^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{k};\unicode[STIX]{x1D70F})\frac{\unicode[STIX]{x2202}B^{\unicode[STIX]{x1D6FC}}}{\unicode[STIX]{x2202}X^{a}}-\frac{\unicode[STIX]{x1D70C}_{0}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F})}{\overline{\unicode[STIX]{x1D70C}}}\left(\frac{\unicode[STIX]{x2202}U^{\unicode[STIX]{x1D6FC}}}{\unicode[STIX]{x2202}T}+U^{a}\frac{\unicode[STIX]{x2202}U^{\unicode[STIX]{x1D6FC}}}{\unicode[STIX]{x2202}X^{a}}\right),\end{eqnarray}$$

(3.23) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}q_{1}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F})}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}+\overline{\unicode[STIX]{x1D706}}k^{2}q_{1}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F})-\text{i}(\unicode[STIX]{x1D6FE}_{\text{s}}-1)Qk^{a}u_{1}^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{k};\unicode[STIX]{x1D70F})\nonumber\\ \displaystyle & & \displaystyle \qquad -\,\text{i}\iint \unicode[STIX]{x1D6FF}(\boldsymbol{k}-\boldsymbol{p}-\boldsymbol{q})\,\text{d}\boldsymbol{p}\,\text{d}\boldsymbol{q}q^{a}u_{0}^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{p};\unicode[STIX]{x1D70F})q_{1}^{\prime }(\boldsymbol{q};\unicode[STIX]{x1D70F})\nonumber\\ \displaystyle & & \displaystyle \quad =-u_{0}^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{k};\unicode[STIX]{x1D70F})\frac{\unicode[STIX]{x2202}Q}{\unicode[STIX]{x2202}X^{a}}-(\unicode[STIX]{x1D6FE}_{\text{s}}-1)\left(q_{0}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F})+\frac{Q}{\overline{\unicode[STIX]{x1D70C}}}\unicode[STIX]{x1D70C}_{0}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F})\right)\frac{\unicode[STIX]{x2202}U^{a}}{\unicode[STIX]{x2202}X^{a}}\nonumber\\ \displaystyle & & \displaystyle \qquad -\,\frac{Dq_{0}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F})}{\text{D}T}-\text{i}(\unicode[STIX]{x1D6FE}_{\text{s}}-1)Qk^{a}u_{0}^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{k};\unicode[STIX]{x1D70F}),\end{eqnarray}$$

(3.24) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}b_{1}^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FC}}(\boldsymbol{k};\unicode[STIX]{x1D70F})}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}+\overline{\unicode[STIX]{x1D702}}k^{2}b_{1}^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FC}}(\boldsymbol{k};\unicode[STIX]{x1D70F})-\text{i}\iint \unicode[STIX]{x1D6FF}(\boldsymbol{k}-\boldsymbol{p}-\boldsymbol{q})\,\text{d}\boldsymbol{p}\,\text{d}\boldsymbol{q}~N_{\text{c}}^{\unicode[STIX]{x1D6FC}ab}b_{1}^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{p};\unicode[STIX]{x1D70F})u_{0}^{\prime }\hspace{0.0pt}^{b}(\boldsymbol{q};\unicode[STIX]{x1D70F})\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\text{i}B^{a}k^{a}u_{1}^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FC}}(\boldsymbol{k};\unicode[STIX]{x1D70F})+\text{i}B^{\unicode[STIX]{x1D6FC}}k^{a}u_{1}^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{k};\unicode[STIX]{x1D70F})\nonumber\\ \displaystyle & & \displaystyle \quad =-\,u_{0}^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{k};\unicode[STIX]{x1D70F})\frac{\unicode[STIX]{x2202}B^{\unicode[STIX]{x1D6FC}}}{\unicode[STIX]{x2202}X^{a}}+b_{0}^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{k};\unicode[STIX]{x1D70F})\frac{\unicode[STIX]{x2202}U^{\unicode[STIX]{x1D6FC}}}{\unicode[STIX]{x2202}X^{a}}+b_{0}^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FC}}(\boldsymbol{k};\unicode[STIX]{x1D70F})\frac{\unicode[STIX]{x2202}U^{a}}{\unicode[STIX]{x2202}X^{a}}\nonumber\\ \displaystyle & & \displaystyle \qquad -\,\frac{\text{D}b_{0}^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FC}}(\boldsymbol{k};\unicode[STIX]{x1D70F})}{\text{D}T}+B^{a}\frac{\unicode[STIX]{x2202}u_{0}^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FC}}(\boldsymbol{k};\unicode[STIX]{x1D70F})}{\unicode[STIX]{x2202}X^{a}}-B^{\unicode[STIX]{x1D6FC}}\frac{\unicode[STIX]{x2202}u_{0}^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{k};\unicode[STIX]{x1D70F})}{\unicode[STIX]{x2202}X^{a}}.\end{eqnarray}$$

3.4 Basic-field expansion and introduction of the Green’s functions

In the TSDIA formalism, the systems of (3.17)–(3.20) and (3.21)–(3.24) are formally solved with the aid of the Green’s functions associated with (3.17)–(3.20). These Green’s functions are nonlinearly coupled to each other. In order to reduce this complexity, we introduce the basic fields. The zeroth-order fields $f_{0}=(\unicode[STIX]{x1D70C}_{0}^{\prime },\boldsymbol{u}_{0}^{\prime },q_{0}^{\prime },\boldsymbol{b}_{0}^{\prime })$ are expanded as

(3.25) $$\begin{eqnarray}f_{0}^{\prime }=f_{\text{B}}^{\prime }+\mathop{\sum }_{m=1}^{\infty }f_{0m}^{\prime },\end{eqnarray}$$

where $m$ is the number of iterations and the basic fields $f_{\text{B}}^{\prime }(\equiv f_{00}^{\prime })=(\unicode[STIX]{x1D70C}_{\text{B}}^{\prime },\boldsymbol{u}_{\text{B}}^{\prime },q_{\text{B}}^{\prime },\boldsymbol{b}_{\text{B}}^{\prime })$ are defined by the following equations:

(3.26) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}_{\text{B}}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F})}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}-\text{i}k^{a}\iint \unicode[STIX]{x1D6FF}(\boldsymbol{k}-\boldsymbol{p}-\boldsymbol{q})\,\text{d}\boldsymbol{p}\,\text{d}\boldsymbol{q}u_{\text{B}}^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{p};\unicode[STIX]{x1D70F})\unicode[STIX]{x1D70C}_{\text{B}}^{\prime }(\boldsymbol{q};\unicode[STIX]{x1D70F})=0,\end{eqnarray}$$

(3.27) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}u_{\text{B}}^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FC}}(\boldsymbol{k};\unicode[STIX]{x1D70F})}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}+\overline{\unicode[STIX]{x1D708}}k^{2}u_{\text{B}}^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{k};\unicode[STIX]{x1D70F})+\frac{1}{3}\overline{\unicode[STIX]{x1D708}}k^{\unicode[STIX]{x1D6FC}}k^{a}u_{\text{B}}^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{k};\unicode[STIX]{x1D70F})\nonumber\\ \displaystyle & & \displaystyle \quad -\,\text{i}\iint \unicode[STIX]{x1D6FF}(\boldsymbol{k}-\boldsymbol{p}-\boldsymbol{q})\,\text{d}\boldsymbol{p}\,\text{d}\boldsymbol{q}\,M_{\text{c}}^{\unicode[STIX]{x1D6FC}ab}u_{\text{B}}^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{p};\unicode[STIX]{x1D70F})u_{\text{B}}^{\prime }\hspace{0.0pt}^{b}(\boldsymbol{q};\unicode[STIX]{x1D70F})=0,\end{eqnarray}$$

(3.28) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}q_{\text{B}}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F})}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}+\overline{\unicode[STIX]{x1D706}}k^{2}q_{\text{B}}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F})-\text{i}\iint \unicode[STIX]{x1D6FF}(\boldsymbol{k}-\boldsymbol{p}-\boldsymbol{q})\,\text{d}\boldsymbol{p}\,\text{d}\boldsymbol{q}\,q^{a}u_{\text{B}}^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{p};\unicode[STIX]{x1D70F})q_{\text{B}}^{\prime }(\boldsymbol{q};\unicode[STIX]{x1D70F})=0,\end{eqnarray}$$

(3.29) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}b_{\text{B}}^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FC}}(\boldsymbol{k};\unicode[STIX]{x1D70F})}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}+\overline{\unicode[STIX]{x1D702}}k^{2}b_{\text{B}}^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FC}}(\boldsymbol{k};\unicode[STIX]{x1D70F})-\text{i}\iint \unicode[STIX]{x1D6FF}(\boldsymbol{k}-\boldsymbol{p}-\boldsymbol{q})\,\text{d}\boldsymbol{p}\,\text{d}\boldsymbol{q}\,N_{\text{c}}^{\unicode[STIX]{x1D6FC}ab}u_{\text{B}}^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{p};\unicode[STIX]{x1D70F})b_{\text{B}}^{\prime }\hspace{0.0pt}^{b}(\boldsymbol{q};\unicode[STIX]{x1D70F})=0.\end{eqnarray}$$

The basic fields correspond to the homogeneous and isotropic turbulent fields. For these basic fields, we introduce four Green’s functions, $G_{\unicode[STIX]{x1D70C}}^{\prime }$ , $G_{u}^{\prime }$ , $G_{q}^{\prime }$ and $G_{b}^{\prime }$ , defined by the equations

(3.30) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}G_{\unicode[STIX]{x1D70C}}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}^{\prime })}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}-\text{i}k^{a}\iint \unicode[STIX]{x1D6FF}(\boldsymbol{k}-\boldsymbol{p}-\boldsymbol{q})\,\text{d}\boldsymbol{p}\,\text{d}\boldsymbol{q}\,u_{\text{B}}^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{p};\unicode[STIX]{x1D70F})G_{\unicode[STIX]{x1D70C}}^{\prime }(\boldsymbol{q};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}^{\prime })=\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D70F}-\unicode[STIX]{x1D70F}^{\prime }), & & \displaystyle\end{eqnarray}$$

(3.31) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}G_{u}^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FC}a}(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}^{\prime })}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}+\overline{\unicode[STIX]{x1D708}}k^{2}G_{u}^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FC}a}(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}^{\prime })+\frac{1}{3}\overline{\unicode[STIX]{x1D708}}k^{\unicode[STIX]{x1D6FC}}k^{b}G_{u}^{\prime }\hspace{0.0pt}^{ba}(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}^{\prime })\nonumber\\ \displaystyle & & \displaystyle \quad -\,2\text{i}\iint \unicode[STIX]{x1D6FF}(\boldsymbol{k}-\boldsymbol{p}-\boldsymbol{q})\,\text{d}\boldsymbol{p}\,\text{d}\boldsymbol{q}\,M_{\text{c}}^{\unicode[STIX]{x1D6FC}bc}u_{\text{B}}^{\prime }\hspace{0.0pt}^{b}(\boldsymbol{p};\unicode[STIX]{x1D70F})G_{u}^{\prime }\hspace{0.0pt}^{ca}(\boldsymbol{q};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}^{\prime })=\unicode[STIX]{x1D6FF}^{\unicode[STIX]{x1D6FC}a}\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D70F}-\unicode[STIX]{x1D70F}^{\prime }),\end{eqnarray}$$

(3.32) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}G_{q}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}^{\prime })}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}+\overline{\unicode[STIX]{x1D706}}k^{2}G_{q}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}^{\prime })\nonumber\\ \displaystyle & & \displaystyle \quad -\text{i}\iint \unicode[STIX]{x1D6FF}(\boldsymbol{k}-\boldsymbol{p}-\boldsymbol{q})\,\text{d}\boldsymbol{p}\,\text{d}\boldsymbol{q}\,N_{\text{c}}^{\unicode[STIX]{x1D6FC}bc}q^{a}u_{\text{B}}^{\prime }\hspace{0.0pt}^{b}(\boldsymbol{p};\unicode[STIX]{x1D70F})G_{q}^{\prime }(\boldsymbol{q};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}^{\prime })=\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D70F}-\unicode[STIX]{x1D70F}^{\prime }),\end{eqnarray}$$

(3.33) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}G_{b}^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FC}a}(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}^{\prime })}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}+\overline{\unicode[STIX]{x1D702}}k^{2}G_{b}^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FC}a}(\boldsymbol{k};\unicode[STIX]{x1D70F})\nonumber\\ \displaystyle & & \displaystyle \quad -\,\text{i}\iint \unicode[STIX]{x1D6FF}(\boldsymbol{k}-\boldsymbol{p}-\boldsymbol{q})\,\text{d}\boldsymbol{p}\,\text{d}\boldsymbol{q}\,N_{\text{c}}^{\unicode[STIX]{x1D6FC}bc}u_{\text{B}}^{\prime }\hspace{0.0pt}^{b}(\boldsymbol{p};\unicode[STIX]{x1D70F})G_{b}^{\prime }\hspace{0.0pt}^{ca}(\boldsymbol{q};\unicode[STIX]{x1D70F})=\unicode[STIX]{x1D6FF}^{\unicode[STIX]{x1D6FC}a}\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D70F}-\unicode[STIX]{x1D70F}^{\prime }).\end{eqnarray}$$

With the Green’s functions, the zeroth-order field (3.17)–(3.20) can be solved in an iterative manner as

(3.34) $$\begin{eqnarray}\unicode[STIX]{x1D70C}_{0}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F})=\unicode[STIX]{x1D70C}_{\text{B}}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F})+\text{i}k^{a}\overline{\unicode[STIX]{x1D70C}}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}G_{\unicode[STIX]{x1D70C}}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})u_{\text{B}}^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{k};\unicode[STIX]{x1D70F}_{1}),\end{eqnarray}$$

(3.35) $$\begin{eqnarray}\displaystyle & & \displaystyle u_{0}^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FC}}(\boldsymbol{k};\unicode[STIX]{x1D70F})=u_{\text{B}}^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FC}}(\boldsymbol{k};\unicode[STIX]{x1D70F})+\text{i}(\unicode[STIX]{x1D6FE}_{\text{s}}-1)k^{a}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}\,G_{u}^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FC}a}(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})q_{\text{B}}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F}_{1})\nonumber\\ \displaystyle & & \displaystyle \quad +\,\text{i}(\unicode[STIX]{x1D6FE}_{\text{s}}-1)\frac{Q}{\overline{\unicode[STIX]{x1D70C}}}k^{a}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}G_{u}^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FC}a}(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})\unicode[STIX]{x1D70C}_{\text{B}}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F}_{1})\nonumber\\ \displaystyle & & \displaystyle \quad +\,\frac{\text{i}}{\unicode[STIX]{x1D707}_{0}\overline{\unicode[STIX]{x1D70C}}}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}G_{u}^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FC}a}(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})\iint \unicode[STIX]{x1D6FF}(\boldsymbol{k}-\boldsymbol{p}-\boldsymbol{q})\,\text{d}\boldsymbol{p}\,\text{d}\boldsymbol{q}\,M_{\text{c}}^{abc}b_{\text{B}}^{\prime }\hspace{0.0pt}^{b}(\boldsymbol{p};\unicode[STIX]{x1D70F}_{1})b_{\text{B}}^{\prime }\hspace{0.0pt}^{c}(\boldsymbol{q};\unicode[STIX]{x1D70F}_{1})\nonumber\\ \displaystyle & & \displaystyle \quad -\,\frac{\text{i}}{\unicode[STIX]{x1D707}_{0}\overline{\unicode[STIX]{x1D70C}}}B^{b}k^{b}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}G_{u}^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FC}a}(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})b_{\text{B}}^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{k};\unicode[STIX]{x1D70F}_{1}),\end{eqnarray}$$

(3.36) $$\begin{eqnarray}\displaystyle q_{0}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F})=q_{\text{B}}(\boldsymbol{k};\unicode[STIX]{x1D70F})+\text{i}(\unicode[STIX]{x1D6FE}_{\text{s}}-1)Qk^{a}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}\,G_{q}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})u_{\text{B}}^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{k};\unicode[STIX]{x1D70F}_{1}), & & \displaystyle\end{eqnarray}$$

(3.37) $$\begin{eqnarray}\displaystyle b_{0}^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FC}}(\boldsymbol{k};\unicode[STIX]{x1D70F}) & = & \displaystyle b_{\text{B}}^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FC}}(\boldsymbol{k};\unicode[STIX]{x1D70F})-\text{i}B^{b}k^{b}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}G_{b}^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FC}a}(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})u_{\text{B}}^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{k};\unicode[STIX]{x1D70F}_{1})\nonumber\\ \displaystyle & & \displaystyle +\,\text{i}B^{a}k^{b}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}G_{b}^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FC}a}(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})u_{\text{B}}^{\prime }\hspace{0.0pt}^{b}(\boldsymbol{k};\unicode[STIX]{x1D70F}_{1}).\end{eqnarray}$$

With the aid of Green’s functions, we can formally solve (3.21)–(3.24) to obtain

(3.38) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70C}_{1}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F}) & = & \displaystyle \text{i}k^{a}\overline{\unicode[STIX]{x1D70C}}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}G_{\unicode[STIX]{x1D70C}}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})u_{\text{B}}^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{k};\unicode[STIX]{x1D70F}_{1})-\frac{\unicode[STIX]{x2202}\overline{\unicode[STIX]{x1D70C}}}{\unicode[STIX]{x2202}X^{a}}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}G_{\unicode[STIX]{x1D70C}}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})u_{\text{B}}^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{k};\unicode[STIX]{x1D70F}_{1})\nonumber\\ \displaystyle & & \displaystyle -\,\frac{\unicode[STIX]{x2202}U^{b}}{\unicode[STIX]{x2202}X^{b}}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}G_{\unicode[STIX]{x1D70C}}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})\unicode[STIX]{x1D70C}_{\text{B}}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F}_{1})-\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}G_{\unicode[STIX]{x1D70C}}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})\frac{D\unicode[STIX]{x1D70C}_{\text{B}}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F}_{1})}{\text{D}T}\nonumber\\ \displaystyle & & \displaystyle -\,\overline{\unicode[STIX]{x1D70C}}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}\,G_{\unicode[STIX]{x1D70C}}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})\frac{\unicode[STIX]{x2202}u_{\text{B}}^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{k};\unicode[STIX]{x1D70F}_{1})}{\unicode[STIX]{x2202}X^{a}},\end{eqnarray}$$

(3.39) $$\begin{eqnarray}\displaystyle & & \displaystyle u_{1}^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FC}}(\boldsymbol{k};\unicode[STIX]{x1D70F})=\text{i}(\unicode[STIX]{x1D6FE}_{\text{s}}-1)k^{a}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}\,G_{u}^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FC}a}(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})q_{1}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F}_{1})\nonumber\\ \displaystyle & & \displaystyle \quad +\,\text{i}(\unicode[STIX]{x1D6FE}_{\text{s}}-1)\frac{Q}{\overline{\unicode[STIX]{x1D70C}}}k^{a}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}G_{u}^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FC}a}(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})\unicode[STIX]{x1D70C}_{1}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F}_{1})\nonumber\\ \displaystyle & & \displaystyle \quad +\,\frac{2\text{i}}{\unicode[STIX]{x1D707}_{0}\overline{\unicode[STIX]{x1D70C}}}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}G_{u}^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FC}a}(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})\iint \unicode[STIX]{x1D6FF}(\boldsymbol{k}-\boldsymbol{p}-\boldsymbol{q})\,\text{d}\boldsymbol{p}\,\text{d}\boldsymbol{q}\,M_{\text{c}}^{abc}b_{0}^{\prime }\hspace{0.0pt}^{b}(\boldsymbol{p};\unicode[STIX]{x1D70F}_{1})b_{1}^{\prime }\hspace{0.0pt}^{c}(\boldsymbol{q};\unicode[STIX]{x1D70F}_{1})\nonumber\\ \displaystyle & & \displaystyle \quad -\,\frac{\text{i}}{\unicode[STIX]{x1D707}_{0}\overline{\unicode[STIX]{x1D70C}}}B^{b}k^{b}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}G_{u}^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FC}a}(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})b_{1}^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{k};\unicode[STIX]{x1D70F}_{1})-\frac{\unicode[STIX]{x2202}U^{a}}{\unicode[STIX]{x2202}X^{b}}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}\,G_{u}^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FC}a}(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})u_{0}^{\prime }\hspace{0.0pt}^{b}(\boldsymbol{k};\unicode[STIX]{x1D70F}_{1})\nonumber\\ \displaystyle & & \displaystyle \quad -\,(\unicode[STIX]{x1D6FE}_{\text{s}}-1)\frac{1}{\overline{\unicode[STIX]{x1D70C}}}\frac{\unicode[STIX]{x2202}Q}{\unicode[STIX]{x2202}X^{a}}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}G_{u}^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FC}a}(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})\unicode[STIX]{x1D70C}_{0}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F}_{1})\nonumber\\ \displaystyle & & \displaystyle \quad -\,(\unicode[STIX]{x1D6FE}_{\text{s}}-1)\frac{1}{\overline{\unicode[STIX]{x1D70C}}}\frac{\unicode[STIX]{x2202}\overline{\unicode[STIX]{x1D70C}}}{\unicode[STIX]{x2202}X^{a}}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}G_{u}^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FC}a}(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})q_{0}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F}_{1})\nonumber\\ \displaystyle & & \displaystyle \quad -\,\frac{1}{\unicode[STIX]{x1D707}_{0}\overline{\unicode[STIX]{x1D70C}}}\frac{\unicode[STIX]{x2202}B^{a}}{\unicode[STIX]{x2202}X^{b}}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}G_{u}^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FC}a}(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})b_{0}^{\prime }\hspace{0.0pt}^{b}(\boldsymbol{k};\unicode[STIX]{x1D70F}_{1})\nonumber\\ \displaystyle & & \displaystyle \quad -\,\frac{1}{\overline{\unicode[STIX]{x1D70C}}}\left(\frac{\unicode[STIX]{x2202}U^{a}}{\unicode[STIX]{x2202}T}+U^{b}\frac{\unicode[STIX]{x2202}U^{a}}{\unicode[STIX]{x2202}X^{b}}\right)\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}G_{u}^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FC}a}(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})\unicode[STIX]{x1D70C}_{0}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F}_{1}),\end{eqnarray}$$

(3.40) $$\begin{eqnarray}\displaystyle & & \displaystyle q_{1}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F})=\text{i}(\unicode[STIX]{x1D6FE}_{\text{s}}-1)Qk^{a}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}G_{q}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})u_{1}^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{k};\unicode[STIX]{x1D70F}_{1})\nonumber\\ \displaystyle & & \displaystyle \quad -\,\frac{\unicode[STIX]{x2202}Q}{\unicode[STIX]{x2202}X^{a}}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}G_{q}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})u_{0}^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{k};\unicode[STIX]{x1D70F}_{1})\nonumber\\ \displaystyle & & \displaystyle \quad -\,(\unicode[STIX]{x1D6FE}_{\text{s}}-1)\frac{\unicode[STIX]{x2202}U^{a}}{\unicode[STIX]{x2202}X^{a}}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}G_{q}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})q_{0}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F}_{1})\nonumber\\ \displaystyle & & \displaystyle \quad -\,(\unicode[STIX]{x1D6FE}_{\text{s}}-1)\frac{Q}{\overline{\unicode[STIX]{x1D70C}}}\frac{\unicode[STIX]{x2202}U^{a}}{\unicode[STIX]{x2202}X^{a}}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}G_{q}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})\unicode[STIX]{x1D70C}_{0}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F}_{1})\nonumber\\ \displaystyle & & \displaystyle \quad -\,\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}G_{q}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})\frac{\text{D}q_{0}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F})}{\text{D}T}-(\unicode[STIX]{x1D6FE}_{\text{s}}-1)Q\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}G_{q}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})\frac{u_{0}^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{k};\unicode[STIX]{x1D70F}_{1})}{\unicode[STIX]{x2202}X^{a}},\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

(3.41) $$\begin{eqnarray}\displaystyle & & \displaystyle b_{1}^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FC}}(\boldsymbol{k};\unicode[STIX]{x1D70F})=-\text{i}B^{b}k^{b}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}G_{b}^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FC}a}(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})u_{1}^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{k};\unicode[STIX]{x1D70F}_{1})\nonumber\\ \displaystyle & & \displaystyle \quad +\,\text{i}B^{a}k^{b}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}G_{b}^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FC}a}(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})u_{1}^{\prime }\hspace{0.0pt}^{b}(\boldsymbol{k};\unicode[STIX]{x1D70F}_{1})-\frac{\unicode[STIX]{x2202}B^{a}}{\unicode[STIX]{x2202}X^{b}}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}G_{b}^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FC}a}(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})u_{0}^{\prime }\hspace{0.0pt}^{b}(\boldsymbol{k};\unicode[STIX]{x1D70F}_{1})\nonumber\\ \displaystyle & & \displaystyle \quad +\,\frac{\unicode[STIX]{x2202}U^{a}}{\unicode[STIX]{x2202}X^{b}}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}G_{b}^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FC}a}(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})b_{0}^{\prime }\hspace{0.0pt}^{b}(\boldsymbol{k};\unicode[STIX]{x1D70F}_{1})-\frac{\unicode[STIX]{x2202}U^{b}}{\unicode[STIX]{x2202}X^{b}}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}G_{b}^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FC}a}(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})b_{0}^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{k};\unicode[STIX]{x1D70F}_{1})\nonumber\\ \displaystyle & & \displaystyle \quad -\,\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}G_{b}^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FC}a}(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})\frac{\text{D}b_{0}^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{k};\unicode[STIX]{x1D70F}_{1})}{\text{D}T}\nonumber\\ \displaystyle & & \displaystyle \quad +\,B^{b}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}G_{b}^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FC}a}(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})\frac{\unicode[STIX]{x2202}u_{0}^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{k};\unicode[STIX]{x1D70F}_{1})}{\unicode[STIX]{x2202}X^{b}}-B^{a}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}G_{b}^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FC}a}(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})\frac{\unicode[STIX]{x2202}u_{0}^{\prime }\hspace{0.0pt}^{b}(\boldsymbol{k};\unicode[STIX]{x1D70F}_{1})}{\unicode[STIX]{x2202}X^{b}}.\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

3.5 Statistical assumptions on the basic fields

We assume the statistical property of the basic fields, $\unicode[STIX]{x1D70C}_{\text{B}}^{\prime }$ , $\boldsymbol{u}_{\text{B}}^{\prime }$ , $\boldsymbol{b}_{\text{B}}^{\prime }$ , and $q_{\text{B}}^{\prime }$ , as

(3.42) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\langle \unicode[STIX]{x1D70C}_{\text{B}}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F})\unicode[STIX]{x1D70C}_{\text{B}}^{\prime }(\boldsymbol{k}^{\prime };\unicode[STIX]{x1D70F}^{\prime })\rangle }{\unicode[STIX]{x1D6FF}(\boldsymbol{k}+\boldsymbol{k}^{\prime })}=\langle Q_{\unicode[STIX]{x1D70C}}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}^{\prime })\rangle =Q_{\unicode[STIX]{x1D70C}}(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}^{\prime }), & \displaystyle\end{eqnarray}$$

(3.43) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\langle q_{\text{B}}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F})q_{\text{B}}^{\prime }(\boldsymbol{k}^{\prime };\unicode[STIX]{x1D70F}^{\prime })\rangle }{\unicode[STIX]{x1D6FF}(\boldsymbol{k}+\boldsymbol{k}^{\prime })}=\langle Q_{q}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}^{\prime })\rangle =Q_{q}(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}^{\prime }), & \displaystyle\end{eqnarray}$$

(3.44) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\langle \unicode[STIX]{x1D717}_{\text{B}}^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FC}}(\boldsymbol{k};\unicode[STIX]{x1D70F})\unicode[STIX]{x1D712}_{\text{B}}^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FD}}(\boldsymbol{k}^{\prime };\unicode[STIX]{x1D70F}^{\prime })\rangle }{\unicode[STIX]{x1D6FF}(\boldsymbol{k}+\boldsymbol{k}^{\prime })}\nonumber\\ \displaystyle & & \displaystyle \quad =D^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}(\boldsymbol{k})Q_{\unicode[STIX]{x1D717}\unicode[STIX]{x1D712}\text{S}}(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}^{\prime })+\unicode[STIX]{x1D6F1}^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}(\boldsymbol{k})Q_{\unicode[STIX]{x1D717}\unicode[STIX]{x1D712}\text{C}}(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}^{\prime })+\frac{\text{i}}{2}\frac{k^{c}}{k^{2}}\unicode[STIX]{x1D716}^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}c}H_{\unicode[STIX]{x1D717}\unicode[STIX]{x1D712}}(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}^{\prime }),\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D751}^{\prime }$ and $\unicode[STIX]{x1D74C}^{\prime }$ represent either one of the velocity and magnetic field fluctuations, $\boldsymbol{u}^{\prime }$ and $\boldsymbol{b}^{\prime }$ , $D^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}$ and $\unicode[STIX]{x1D6F1}^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}$ are the solenoidal projection operator and its compressible counterpart defined by

(3.45a,b ) $$\begin{eqnarray}D^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}(\boldsymbol{k})=\unicode[STIX]{x1D6FF}^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}-\frac{k^{\unicode[STIX]{x1D6FC}}k^{\unicode[STIX]{x1D6FD}}}{k^{2}},\quad \unicode[STIX]{x1D6F1}^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}(\boldsymbol{k})=\frac{k^{\unicode[STIX]{x1D6FC}}k^{\unicode[STIX]{x1D6FD}}}{k^{2}}.\end{eqnarray}$$

Here, $Q_{\unicode[STIX]{x1D717}\unicode[STIX]{x1D712}\text{S}}$ and $Q_{\unicode[STIX]{x1D717}\unicode[STIX]{x1D712}\text{C}}$ are the spectral functions of the solenoidal and compressible parts.

On the other hand, for Green’s functions of the basic fields, the simplest possible forms are assumed as

(3.46) $$\begin{eqnarray}\displaystyle & \displaystyle \langle G_{\unicode[STIX]{x1D70C}}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}^{\prime })\rangle =G_{\unicode[STIX]{x1D70C}}(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}^{\prime }), & \displaystyle\end{eqnarray}$$

(3.47) $$\begin{eqnarray}\displaystyle & \displaystyle \langle G_{q}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}^{\prime })\rangle =G_{q}(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}^{\prime }). & \displaystyle\end{eqnarray}$$

(3.48) $$\begin{eqnarray}\displaystyle & \displaystyle \langle G_{u}^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}^{\prime })\rangle =D^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}(\boldsymbol{k})G_{u\text{S}}(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}^{\prime })+\unicode[STIX]{x1D6F1}^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}(\boldsymbol{k})G_{u\text{C}}(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}^{\prime }), & \displaystyle\end{eqnarray}$$

(3.49) $$\begin{eqnarray}\displaystyle & \displaystyle \langle G_{b}^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}^{\prime })\rangle =D^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}(\boldsymbol{k})G_{b}(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}^{\prime }). & \displaystyle\end{eqnarray}$$

In general, we should consider more complicated forms arising from the complicated nonlinear interactions between the field quantities. However, for the sake of simplicity, we just retain the simplest possible Green’s functions in (3.46)–(3.49).

Under the assumptions on the basic fields (3.42)–(3.44), we have

(3.50) $$\begin{eqnarray}\displaystyle & \displaystyle \langle \unicode[STIX]{x1D70C}_{\text{B}}^{\prime }\hspace{0.0pt}^{2}\rangle =\int Q_{\unicode[STIX]{x1D70C}}(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F})\,\text{d}\boldsymbol{k}, & \displaystyle\end{eqnarray}$$

(3.51) $$\begin{eqnarray}\displaystyle & \displaystyle \langle q_{\text{B}}^{\prime }\hspace{0.0pt}^{2}\rangle =\int Q_{q}(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F})\,\text{d}\boldsymbol{k}, & \displaystyle\end{eqnarray}$$

(3.52) $$\begin{eqnarray}\displaystyle & \displaystyle \langle \boldsymbol{u}_{\text{B}}^{\prime }\hspace{0.0pt}^{2}\rangle /2=\int Q_{u}(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F})\,\text{d}\boldsymbol{k}, & \displaystyle\end{eqnarray}$$

(3.53) $$\begin{eqnarray}\displaystyle & \displaystyle \langle \boldsymbol{b}_{\text{B}}^{\prime }\hspace{0.0pt}^{2}\rangle /2=\int Q_{b}(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F})\,\text{d}\boldsymbol{k}, & \displaystyle\end{eqnarray}$$

(3.54) $$\begin{eqnarray}\displaystyle & \displaystyle \langle \boldsymbol{u}_{\text{B}}^{\prime }\boldsymbol{\cdot }\boldsymbol{b}_{\text{B}}^{\prime }\rangle =\int Q_{w}(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F})\,\text{d}\boldsymbol{k}, & \displaystyle\end{eqnarray}$$

(3.55) $$\begin{eqnarray}\displaystyle & \displaystyle \langle \boldsymbol{u}_{\text{B}}^{\prime }\boldsymbol{\cdot }\unicode[STIX]{x1D74E}_{\text{B}}^{\prime }\rangle =\int H_{uu}(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F})\,\text{d}\boldsymbol{k}, & \displaystyle\end{eqnarray}$$

(3.56) $$\begin{eqnarray}\displaystyle & \displaystyle \langle \boldsymbol{b}_{\text{B}}^{\prime }\boldsymbol{\cdot }\boldsymbol{j}_{\text{B}}^{\prime }\rangle =\int H_{bb}(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F})\,\text{d}\boldsymbol{k}. & \displaystyle\end{eqnarray}$$

In the following, we also use the spectral functions associated with the solenoidal and compressional components of the basic field: $\boldsymbol{u}_{\text{B}}^{\prime }=\boldsymbol{u}_{\text{BS}}^{\prime }+\boldsymbol{u}_{\text{BC}}^{\prime }$ , such as

(3.57) $$\begin{eqnarray}\displaystyle & \displaystyle \langle \boldsymbol{u}_{\text{BS}}^{\prime }\hspace{0.0pt}^{2}\rangle /2=\int Q_{u\text{S}}(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F})\,\text{d}\boldsymbol{k}, & \displaystyle\end{eqnarray}$$

(3.58) $$\begin{eqnarray}\displaystyle & \displaystyle \langle \boldsymbol{u}_{\text{BC}}^{\prime }\hspace{0.0pt}^{2}\rangle /2=\int Q_{u\text{C}}(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F})\,\text{d}\boldsymbol{k}, & \displaystyle\end{eqnarray}$$

etc.

3.6 Calculation of correlations with renormalisation

Each order of the fields is expressed in terms of the correlation and response functions of the basic fields. We calculate the turbulent correlations with the zeroth- and first-order fields with the statistical properties of the basic fields (3.42)–(3.44) and (3.46)–(3.49). The correlation between $f^{\prime }$ and $g^{\prime }$ in the configuration space is calculated from their counterparts in the wavenumber space as

(3.59) $$\begin{eqnarray}\displaystyle \langle \,f^{\prime }(\boldsymbol{x};t)g^{\prime }(\boldsymbol{x};t)\rangle & = & \displaystyle \int \,\text{d}\boldsymbol{k}\langle \,f^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F})g^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F})\rangle /\unicode[STIX]{x1D6FF}(\mathbf{0})\nonumber\\ \displaystyle & = & \displaystyle \int \,\text{d}\boldsymbol{k}(\langle \,f_{0}^{\prime }g_{0}^{\prime }\rangle +\langle \,f_{0}^{\prime }g_{1}^{\prime }\rangle +\langle \,f_{1}^{\prime }g_{0}^{\prime }\rangle +\cdots )/\unicode[STIX]{x1D6FF}(\mathbf{0}).\end{eqnarray}$$

Following the direct-interaction approximation (DIA) formalism, the basic-field propagators are replaced with their exact counterparts (renormalisation). Note that the scale-expansion parameter $\unicode[STIX]{x1D6FF}$ automatically disappears through the replacement of $\boldsymbol{X}\rightarrow \unicode[STIX]{x1D6FF}\boldsymbol{x}$ and $T\rightarrow \unicode[STIX]{x1D6FF}t$ .

4.1 Theoretical expression for turbulent electromotive force

In the present work, we concentrate on the turbulent electromotive force

(4.1) $$\begin{eqnarray}\displaystyle \langle \boldsymbol{u}^{\prime }\times \boldsymbol{b}^{\prime }\rangle ^{\unicode[STIX]{x1D6FC}} & = & \displaystyle \unicode[STIX]{x1D716}^{\unicode[STIX]{x1D6FC}ab}\langle u^{\prime }\hspace{0.0pt}^{a}b^{\prime }\hspace{0.0pt}^{b}\rangle \nonumber\\ \displaystyle & = & \displaystyle \unicode[STIX]{x1D716}^{\unicode[STIX]{x1D6FC}ab}(\langle u_{0}^{\prime }\hspace{0.0pt}^{a}b_{0}^{\prime }\hspace{0.0pt}^{b}\rangle +\langle u_{0}^{\prime }\hspace{0.0pt}^{a}b_{1}^{\prime }\hspace{0.0pt}^{b}\rangle +\langle u_{1}^{\prime }\hspace{0.0pt}^{a}b_{0}^{\prime }\hspace{0.0pt}^{b}\rangle +\cdots )\nonumber\\ \displaystyle & = & \displaystyle \unicode[STIX]{x1D716}^{\unicode[STIX]{x1D6FC}ab} (\langle u_{\text{B}}^{\prime }\hspace{0.0pt}^{a}b_{\text{B}}^{\prime }\hspace{0.0pt}^{b}\rangle +\langle u_{\text{B}}^{\prime }\hspace{0.0pt}^{a}b_{01}^{\prime }\hspace{0.0pt}^{b}\rangle +\langle u_{\text{B}}^{\prime }\hspace{0.0pt}^{a}b_{10}^{\prime }\hspace{0.0pt}^{b}\rangle +\cdots \nonumber\\ \displaystyle & & \displaystyle +\,\langle u_{01}^{\prime }\hspace{0.0pt}^{a}b_{\text{B}}^{\prime }\hspace{0.0pt}^{b}\rangle +\cdots +\langle u_{10}^{\prime }\hspace{0.0pt}^{a}b_{\text{B}}^{\prime }\hspace{0.0pt}^{b}\rangle +\langle u_{10}^{\prime }\hspace{0.0pt}^{a}b_{01}^{\prime }\hspace{0.0pt}^{b}\rangle +\cdots ).\end{eqnarray}$$

From the formal solutions of the fluctuation velocity $\boldsymbol{u}_{0}^{\prime }$ (3.35) and $\boldsymbol{u}_{1}^{\prime }$ (3.39) and the fluctuation magnetic field $\boldsymbol{b}_{0}^{\prime }$ (3.37) and $\boldsymbol{b}_{1}^{\prime }$ (3.41), the expression of the turbulent electromotive force is calculated.

In this multiple-scale analysis with (3.3), the divergence of a fluctuating vector field $\boldsymbol{f}^{\prime }(\boldsymbol{x};t)$ is written as

(4.2) $$\begin{eqnarray}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{f}^{\prime }(\boldsymbol{x};t)=\frac{\unicode[STIX]{x2202}f^{\prime }\hspace{0.0pt}^{a}(\unicode[STIX]{x1D743},\boldsymbol{X};\unicode[STIX]{x1D70F},T)}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{a}}+\unicode[STIX]{x1D6FF}\frac{\unicode[STIX]{x2202}f^{\prime }\hspace{0.0pt}^{a}(\unicode[STIX]{x1D743},\boldsymbol{X};\unicode[STIX]{x1D70F},T)}{\unicode[STIX]{x2202}X^{a}},\end{eqnarray}$$

or equivalently in the wavenumber space

(4.3) $$\begin{eqnarray}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{f}^{\prime }(\boldsymbol{k},\boldsymbol{X};\unicode[STIX]{x1D70F},T)=-\text{i}k^{a}f^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{k},\boldsymbol{X};\unicode[STIX]{x1D70F},T)+\unicode[STIX]{x1D6FF}\frac{\unicode[STIX]{x2202}f^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{k},\boldsymbol{X};\unicode[STIX]{x1D70F},T)}{\unicode[STIX]{x2202}X_{\text{I}}^{a}},\end{eqnarray}$$

where

(4.4) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}X_{\text{I}}}=\exp (-\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{U}\unicode[STIX]{x1D70F})\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}X^{a}}\exp (\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{U}\unicode[STIX]{x1D70F})\end{eqnarray}$$

is the spatial derivative in the so-called interaction representation associated with (3.8). In order to secure the divergence-free condition for the solenoidal or incompressible field quantities, we introduce the solenoidal field $\boldsymbol{f}_{\text{S}}^{\prime }(\boldsymbol{k},\boldsymbol{X};\unicode[STIX]{x1D70F},T)$ as

(4.5) $$\begin{eqnarray}f_{\text{S}}^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FC}}(\boldsymbol{k},\boldsymbol{X};\unicode[STIX]{x1D70F},T)=f^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FC}}(\boldsymbol{k},\boldsymbol{X};\unicode[STIX]{x1D70F},T)+\text{i}\unicode[STIX]{x1D6FF}\frac{k^{\unicode[STIX]{x1D6FC}}}{k^{2}}\frac{\unicode[STIX]{x2202}f^{a}(\boldsymbol{k},\boldsymbol{X};\unicode[STIX]{x1D70F},T)}{\unicode[STIX]{x2202}X_{\text{I}}^{a}},\end{eqnarray}$$

which satisfies the solenoidal condition for each order of the solenoidal fields as

(4.6) $$\begin{eqnarray}k^{a}f_{\text{S}}^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{k},\boldsymbol{X};\unicode[STIX]{x1D70F},T)=k^{a}f^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{k},\boldsymbol{X};\unicode[STIX]{x1D70F},T)+\text{i}\unicode[STIX]{x1D6FF}\frac{k^{a}k^{a}}{k^{2}}\frac{\unicode[STIX]{x2202}f^{\prime }\hspace{0.0pt}^{b}(\boldsymbol{k},\boldsymbol{X};\unicode[STIX]{x1D70F},T)}{\unicode[STIX]{x2202}X_{\text{I}}^{b}}=0\end{eqnarray}$$

in the wavenumber space (Hamba Reference Hamba1987). In the theoretical analysis, each order of the solenoidal field is calculated by the combination of

(4.7) $$\begin{eqnarray}f^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FC}}(\boldsymbol{k},\boldsymbol{X};\unicode[STIX]{x1D70F},T)=\mathop{\sum }_{n=0}^{\infty }\mathop{\sum }_{m=0}^{\infty }f_{nm}^{\prime }\hspace{0.0pt}^{\unicode[STIX]{x1D6FC}}(\boldsymbol{k},\boldsymbol{X};\unicode[STIX]{x1D70F},T)-\mathop{\sum }_{n=0}^{\infty }\mathop{\sum }_{m=0}^{\infty }\text{i}\unicode[STIX]{x1D6FF}\frac{k^{\unicode[STIX]{x1D6FC}}}{k^{2}}\frac{\unicode[STIX]{x2202}f_{nm}^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{k},\boldsymbol{X};\unicode[STIX]{x1D70F},T)}{\unicode[STIX]{x2202}X_{\text{I}}^{a}}.\end{eqnarray}$$

The second terms may be called the solenoidal corrections to each order of field. These terms give rise to the contributions to the turbulent magnetic diffusivity and the magnetic pumping effect through the turbulent MHD residual energy (the difference between the turbulent kinetic and magnetic energies). See (4.13) and (4.15b ) shown later.

In addition to the usual solenoidal terms arising from the incompressible MHD turbulence, we have some terms originating from compressibility. Among them, the ones related to the fluctuation dilatation are important in strongly compressible turbulence. Those terms are written as

(4.8) $$\begin{eqnarray}\langle \boldsymbol{u}^{\prime }\times \boldsymbol{b}^{\prime }\rangle _{\text{dil}}=\boldsymbol{E}_{\text{M}\unicode[STIX]{x1D735}Q}+\boldsymbol{E}_{\text{M}\unicode[STIX]{x1D735}\overline{\unicode[STIX]{x1D70C}}}+\boldsymbol{E}_{\text{M}DU},\end{eqnarray}$$

where

(4.9a ) $$\begin{eqnarray}\displaystyle & & \displaystyle \boldsymbol{E}_{\text{M}\unicode[STIX]{x1D735}\overline{\unicode[STIX]{x1D70C}}}^{\unicode[STIX]{x1D6FC}}\equiv \int \,\text{d}\boldsymbol{k}\unicode[STIX]{x1D716}^{\unicode[STIX]{x1D6FC}ab}\langle u_{1}^{\prime }\hspace{0.0pt}^{a}b_{0}^{\prime }\hspace{0.0pt}^{b}\rangle _{\unicode[STIX]{x1D735}\overline{\unicode[STIX]{x1D70C}}}/\unicode[STIX]{x1D6FF}(\mathbf{0})\nonumber\\ \displaystyle & & \displaystyle \quad =-\unicode[STIX]{x1D716}^{\unicode[STIX]{x1D6FC}ab}(\unicode[STIX]{x1D6FE}_{\text{s}}-1)\frac{1}{\overline{\unicode[STIX]{x1D70C}}}\frac{\unicode[STIX]{x2202}\overline{\unicode[STIX]{x1D70C}}}{\unicode[STIX]{x2202}X^{c}}\int \,\text{d}\boldsymbol{k}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}\langle G_{u}^{\prime }\hspace{0.0pt}^{ac}(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})q_{0}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F}_{1})b_{0}^{\prime }\hspace{0.0pt}^{b}(-\boldsymbol{k};\unicode[STIX]{x1D70F}^{\prime })\rangle /\unicode[STIX]{x1D6FF}(\mathbf{0}),\quad \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

(4.9b ) $$\begin{eqnarray}\displaystyle & & \displaystyle \boldsymbol{E}_{\text{M}\unicode[STIX]{x1D735}Q}^{\unicode[STIX]{x1D6FC}}\equiv \int \,\text{d}\boldsymbol{k}\unicode[STIX]{x1D716}^{\unicode[STIX]{x1D6FC}ab}\langle u_{1}^{\prime }\hspace{0.0pt}^{a}b_{0}^{\prime }\hspace{0.0pt}^{b}\rangle _{\unicode[STIX]{x1D735}Q}/\unicode[STIX]{x1D6FF}(\mathbf{0})\nonumber\\ \displaystyle & & \displaystyle \quad =-\unicode[STIX]{x1D716}^{\unicode[STIX]{x1D6FC}ab}(\unicode[STIX]{x1D6FE}_{\text{s}}-1)\frac{1}{\overline{\unicode[STIX]{x1D70C}}}\frac{\unicode[STIX]{x2202}Q}{\unicode[STIX]{x2202}X^{c}}\int \,\text{d}\boldsymbol{k}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}\langle G_{u}^{\prime }\hspace{0.0pt}^{ac}(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})\unicode[STIX]{x1D70C}_{0}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F}_{1})b_{0}^{\prime }\hspace{0.0pt}^{b}(-\boldsymbol{k};\unicode[STIX]{x1D70F}^{\prime })\rangle /\unicode[STIX]{x1D6FF}(\mathbf{0}),\quad \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

(4.9c ) $$\begin{eqnarray}\displaystyle & & \displaystyle \boldsymbol{E}_{\text{M}DU}^{\unicode[STIX]{x1D6FC}}\equiv \int \,\text{d}\boldsymbol{k}\,\unicode[STIX]{x1D716}^{\unicode[STIX]{x1D6FC}ab}\langle u_{1}^{\prime }\hspace{0.0pt}^{a}b_{0}^{\prime }\hspace{0.0pt}^{b}\rangle _{DU}/\unicode[STIX]{x1D6FF}(\mathbf{0})\nonumber\\ \displaystyle & & \displaystyle \quad =-\unicode[STIX]{x1D716}^{\unicode[STIX]{x1D6FC}ab}\frac{\text{D}U^{c}}{\text{D}T}\int \,\text{d}\boldsymbol{k}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}\langle G_{u}^{\prime }\hspace{0.0pt}^{ac}(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})\unicode[STIX]{x1D70C}_{0}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F}_{1})b_{0}^{\prime }\hspace{0.0pt}^{b}(-\boldsymbol{k};\unicode[STIX]{x1D70F}^{\prime })\rangle /\unicode[STIX]{x1D6FF}(\mathbf{0}).\quad\end{eqnarray}$$

From the expressions of the zeroth-order density $\unicode[STIX]{x1D70C}_{0}^{\prime }$ (3.34), internal energy $q_{0}^{\prime }$ (3.36) and magnetic field $\boldsymbol{b}_{0}^{\prime }$ (3.37) with the statistical properties of the basic fields (3.46)–(3.49), these terms are expressed as

(4.10a ) $$\begin{eqnarray}\displaystyle \boldsymbol{E}_{\text{M}\unicode[STIX]{x1D735}\overline{\unicode[STIX]{x1D70C}}}^{\unicode[STIX]{x1D6FC}} & = & \displaystyle -\unicode[STIX]{x1D716}^{\unicode[STIX]{x1D6FC}ab}(\unicode[STIX]{x1D6FE}_{s}-1)\frac{1}{\overline{\unicode[STIX]{x1D70C}}}\frac{\unicode[STIX]{x2202}\overline{\unicode[STIX]{x1D70C}}}{\unicode[STIX]{x2202}X^{c}}\int \,\text{d}\boldsymbol{k}\left\langle \int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}~G_{u}^{\prime }\hspace{0.0pt}^{ac}(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})\right.\nonumber\\ \displaystyle & & \displaystyle \times \,\text{i}(\unicode[STIX]{x1D6FE}_{\text{s}}-1)Qk^{d}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{2}\,G_{q}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{2})u_{\text{B}}^{\prime }\hspace{0.0pt}^{d}(\boldsymbol{k};\unicode[STIX]{x1D70F}_{2})\nonumber\\ \displaystyle & & \displaystyle \left.\times \,B^{e}(-\text{i}k^{f})\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{3}\,G_{b}^{\prime }\hspace{0.0pt}^{be}(-\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{3})u_{\text{B}}^{\prime }\hspace{0.0pt}^{f}(-\boldsymbol{k};\unicode[STIX]{x1D70F}_{3})\right\rangle \bigg/\unicode[STIX]{x1D6FF}(\mathbf{0}),\end{eqnarray}$$

(4.10b ) $$\begin{eqnarray}\displaystyle \boldsymbol{E}_{\text{M}\unicode[STIX]{x1D735}Q}^{\unicode[STIX]{x1D6FC}} & = & \displaystyle -\unicode[STIX]{x1D716}^{\unicode[STIX]{x1D6FC}ab}(\unicode[STIX]{x1D6FE}_{s}-1)\frac{1}{\overline{\unicode[STIX]{x1D70C}}}\frac{\unicode[STIX]{x2202}Q}{\unicode[STIX]{x2202}X^{c}}\int \,\text{d}\boldsymbol{k}\left\langle \int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}~G_{u}^{\prime }\hspace{0.0pt}^{ac}(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})\right.\nonumber\\ \displaystyle & & \displaystyle \times \,\text{i}k^{d}\overline{\unicode[STIX]{x1D70C}}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{2}~G_{\unicode[STIX]{x1D70C}}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{2})u_{\text{B}}^{\prime }\hspace{0.0pt}^{d}(\boldsymbol{k};\unicode[STIX]{x1D70F}_{2})\nonumber\\ \displaystyle & & \displaystyle \left.\times \,(-\text{i}k^{f})B^{e}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{3}\,G_{b}^{\prime }\hspace{0.0pt}^{be}(-\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{3})u_{\text{B}}^{\prime }\hspace{0.0pt}^{f}(-\boldsymbol{k};\unicode[STIX]{x1D70F}_{3})\right\rangle \bigg/\unicode[STIX]{x1D6FF}(\mathbf{0}),\end{eqnarray}$$

(4.10c ) $$\begin{eqnarray}\displaystyle \boldsymbol{E}_{\text{M}DU}^{\unicode[STIX]{x1D6FC}} & = & \displaystyle -\unicode[STIX]{x1D716}^{\unicode[STIX]{x1D6FC}ab}\frac{DU^{c}}{\text{D}T}\int \,\text{d}\boldsymbol{k}\left\langle \int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}~G_{u}^{\prime }\hspace{0.0pt}^{ac}(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})\right.\nonumber\\ \displaystyle & & \displaystyle \times \,\text{i}k^{d}\overline{\unicode[STIX]{x1D70C}}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{2}~G_{\unicode[STIX]{x1D70C}}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{2})u_{\text{B}}^{\prime }\hspace{0.0pt}^{d}(\boldsymbol{k};\unicode[STIX]{x1D70F}_{2})\nonumber\\ \displaystyle & & \displaystyle \left.\times (-\text{i}k^{f})B^{e}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{3}\,G_{b}^{\prime }\hspace{0.0pt}^{be}(-\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{3})u_{\text{B}}^{\prime }\hspace{0.0pt}^{f}(-\boldsymbol{k};\unicode[STIX]{x1D70F}_{3})\right\rangle \bigg/\unicode[STIX]{x1D6FF}(\mathbf{0}).\end{eqnarray}$$

These contributions to the turbulent electromotive force can be expressed as

(4.11a ) $$\begin{eqnarray}\displaystyle \boldsymbol{E}_{\text{M}\unicode[STIX]{x1D735}\overline{\unicode[STIX]{x1D70C}}}^{\unicode[STIX]{x1D6FC}} & = & \displaystyle -\frac{1}{3}(\unicode[STIX]{x1D6FE}_{s}-1)^{2}\frac{Q}{\overline{\unicode[STIX]{x1D70C}}}(\unicode[STIX]{x1D735}\overline{\unicode[STIX]{x1D70C}}\times \boldsymbol{B})^{\unicode[STIX]{x1D6FC}}\int \,\text{d}\boldsymbol{k}\,k^{2}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}\int _{-\infty }^{\unicode[STIX]{x1D70F}_{1}}\,\text{d}\unicode[STIX]{x1D70F}_{2}\int _{-\infty }^{\unicode[STIX]{x1D70F}_{2}}\,\text{d}\unicode[STIX]{x1D70F}_{3}\nonumber\\ \displaystyle & & \displaystyle \times \left[G_{u\text{S}}(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})+G_{u\text{C}}(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})\right]G_{q}(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{2})G_{b}(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{3})Q_{u\text{C}}(k;\unicode[STIX]{x1D70F}_{2},\unicode[STIX]{x1D70F}_{3}),\qquad \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

(4.11b ) $$\begin{eqnarray}\displaystyle \boldsymbol{E}_{\text{M}\unicode[STIX]{x1D735}Q}^{\unicode[STIX]{x1D6FC}} & = & \displaystyle -\frac{1}{3}(\unicode[STIX]{x1D6FE}_{s}-1)(\unicode[STIX]{x1D735}Q\times \boldsymbol{B})^{\unicode[STIX]{x1D6FC}}\int \,\text{d}\boldsymbol{k}k^{2}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}\int _{-\infty }^{\unicode[STIX]{x1D70F}_{1}}\,\text{d}\unicode[STIX]{x1D70F}_{2}\int _{-\infty }^{\unicode[STIX]{x1D70F}_{2}}\,\text{d}\unicode[STIX]{x1D70F}_{3}\nonumber\\ \displaystyle & & \displaystyle \times \,G_{u\text{C}}(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})G_{\unicode[STIX]{x1D70C}}(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{2})G_{b}(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{3})Q_{u\text{C}}(k;\unicode[STIX]{x1D70F}_{2},\unicode[STIX]{x1D70F}_{3}),\end{eqnarray}$$

(4.11c ) $$\begin{eqnarray}\displaystyle \boldsymbol{E}_{\text{M}DU}^{\unicode[STIX]{x1D6FC}} & = & \displaystyle -\frac{1}{3}\left(\frac{D\boldsymbol{U}}{\text{D}T}\times \boldsymbol{B}\right)^{\unicode[STIX]{x1D6FC}}\int \,\text{d}\boldsymbol{k}k^{2}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}\int _{-\infty }^{\unicode[STIX]{x1D70F}_{1}}\,\text{d}\unicode[STIX]{x1D70F}_{2}\int _{-\infty }^{\unicode[STIX]{x1D70F}_{2}}\,\text{\{}\nonumber\\ \displaystyle & & \displaystyle \times \,G_{u\text{C}}(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})G_{\unicode[STIX]{x1D70C}}(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{2})G_{b}(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{3})Q_{u\text{C}}(k;\unicode[STIX]{x1D70F}_{2},\unicode[STIX]{x1D70F}_{3}).\end{eqnarray}$$

Here, use has been made of the identity relations such as

(4.12a,b ) $$\begin{eqnarray}\int \,\text{d}\boldsymbol{k}\frac{k^{a}k^{b}}{k^{2}}=\int \,\text{d}\boldsymbol{k}\frac{1}{3}\unicode[STIX]{x1D6FF}^{ab},\quad \int \,\text{d}\boldsymbol{k}\frac{k^{a}k^{b}k^{c}k^{d}}{k^{4}}=\int \,\text{d}\boldsymbol{k}\frac{1}{15}(\unicode[STIX]{x1D6FF}^{ab}\unicode[STIX]{x1D6FF}^{cd}+\unicode[STIX]{x1D6FF}^{ac}\unicode[STIX]{x1D6FF}^{bd}+\unicode[STIX]{x1D6FF}^{ad}\unicode[STIX]{x1D6FF}^{bc}).\end{eqnarray}$$

It should be noted that all the terms (4.11) are related to the magnitude of turbulent dilatation, represented by $(\unicode[STIX]{x1D735}_{\unicode[STIX]{x1D743}}\boldsymbol{\cdot }\boldsymbol{u}_{\text{B}}^{\prime })^{2}$ . We see from the second term of (3.34) that such a dilatation is directly related to the density variance. This point will be argued later in § 5.2.

With the results of the present analysis of inhomogeneous MHD turbulence, the expression for the turbulent electromotive force (EMF) is written as

(4.13) $$\begin{eqnarray}\displaystyle \langle \boldsymbol{u}^{\prime }\times \boldsymbol{b}^{\prime }\rangle & = & \displaystyle -(\unicode[STIX]{x1D6FD}+\unicode[STIX]{x1D701})\unicode[STIX]{x1D735}\times \boldsymbol{B}+\unicode[STIX]{x1D6FC}\boldsymbol{B}-(\unicode[STIX]{x1D735}\unicode[STIX]{x1D701})\times \boldsymbol{B}+\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D735}\times \boldsymbol{U}\nonumber\\ \displaystyle & & \displaystyle -\,\unicode[STIX]{x1D712}_{\overline{\unicode[STIX]{x1D70C}}}\unicode[STIX]{x1D735}\overline{\unicode[STIX]{x1D70C}}\times \boldsymbol{B}-\unicode[STIX]{x1D712}_{Q}\unicode[STIX]{x1D735}Q\times \boldsymbol{B}-\unicode[STIX]{x1D712}_{D}\frac{D\boldsymbol{U}}{\text{D}t}\times \boldsymbol{B}.\end{eqnarray}$$

Here, $\unicode[STIX]{x1D6FD}$ , $\unicode[STIX]{x1D701}$ , $\unicode[STIX]{x1D6FC}$ , $\unicode[STIX]{x1D6FE}$ , $\unicode[STIX]{x1D712}_{\overline{\unicode[STIX]{x1D70C}}}$ , $\unicode[STIX]{x1D712}_{Q}$ and $\unicode[STIX]{x1D712}_{D}$ are the transport coefficients. As shown below, the latter three terms related to the $\unicode[STIX]{x1D712}$ values have a genuine compressibility origin. For the sake of simplicity of description, we introduce the abbreviated integral forms of the time and wavenumber integrals as

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

(4.14b ) $$\begin{eqnarray}\displaystyle & & \displaystyle I_{n}\{A^{(1)},B^{(2)},C^{(3)};D^{(3)}\}\nonumber\\ \displaystyle & & \displaystyle \quad =\int \,\text{d}\boldsymbol{k}\,k^{2n}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{2}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{3}\,A(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})B(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{2})C(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{3})D(k;\unicode[STIX]{x1D70F}_{2},\unicode[STIX]{x1D70F}_{3}).\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

With these abbreviated forms, the transport coefficients in (4.13) are written as

(4.15a ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FD}={\textstyle \frac{1}{3}}I_{0}\{G_{b},Q_{u\text{S}}+Q_{u\text{C}}\}+{\textstyle \frac{1}{3}}I_{0}\{G_{u\text{S}}+G_{u\text{C}},Q_{b}\}, & \displaystyle\end{eqnarray}$$

(4.15b ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D701}={\textstyle \frac{1}{3}}I_{0}\{G_{b},Q_{u\text{S}}\}-{\textstyle \frac{1}{3}}I_{0}\{G_{u\text{S}},Q_{b}\}, & \displaystyle\end{eqnarray}$$

(4.15c ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FC}=-{\textstyle \frac{1}{3}}I_{0}\{G_{b},H_{uu}\}+{\textstyle \frac{1}{3}}I_{0}\{G_{u\text{S}},H_{bb}\}, & \displaystyle\end{eqnarray}$$

(4.15d ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FE}={\textstyle \frac{1}{3}}I_{0}\{G_{u\text{S}}+G_{u\text{C}}+G_{b},Q_{w\text{S}}\}+{\textstyle \frac{1}{3}}I_{0}\{G_{u\text{S}}+G_{b},Q_{w\text{C}}\}, & \displaystyle\end{eqnarray}$$

(4.15e ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D712}_{\overline{\unicode[STIX]{x1D70C}}}=\frac{1}{3}(\unicode[STIX]{x1D6FE}_{\text{s}}-1)^{2}\frac{Q}{\overline{\unicode[STIX]{x1D70C}}}I_{1}\{G_{u\text{C}}^{(1)},G_{q}^{(2)},G_{b}^{(3)};Q_{u\text{C}}^{(3)}\}, & \displaystyle\end{eqnarray}$$

(4.15f ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D712}_{Q}={\textstyle \frac{1}{3}}(\unicode[STIX]{x1D6FE}_{\text{s}}-1)I_{1}\{G_{u\text{C}}^{(1)},G_{\unicode[STIX]{x1D70C}}^{(2)},G_{b}^{(3)};Q_{u\text{C}}^{(3)}\}, & \displaystyle\end{eqnarray}$$

(4.15g ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D712}_{D}={\textstyle \frac{1}{3}}I_{1}\{G_{u\text{S}}^{(1)}+G_{u\text{C}}^{(1)},G_{\unicode[STIX]{x1D70C}}^{(2)},G_{b}^{(3)};Q_{u\text{C}}^{(3)}\}. & \displaystyle\end{eqnarray}$$

4.2 A few notes on turbulent transport coefficients

The coefficient $\unicode[STIX]{x1D701}$ (4.15b ) corresponds to the difference between the kinetic and magnetic energy or the turbulent MHD residual energy. This directly arises from the solenoidal corrections to the velocity and magnetic field fluctuations. The third term on the right-hand side of (4.13) represents the so-called magnetic pumping effect. The coefficient of the magnetic pumping is determined by the gradient of the residual energy, $\unicode[STIX]{x1D735}\unicode[STIX]{x1D701}$ .

With (4.15a ) and (4.15b ), the turbulent magnetic diffusivity or the coefficient for the mean electric current $\unicode[STIX]{x1D735}\times \boldsymbol{B}$ is expressed as

(4.16) $$\begin{eqnarray}\unicode[STIX]{x1D6FD}+\unicode[STIX]{x1D701}={\textstyle \frac{1}{3}}I_{0}\{G_{b},2Q_{u\text{S}}+Q_{u\text{C}}\}+{\textstyle \frac{1}{3}}I_{0}\{G_{u\text{C}},Q_{b}\}.\end{eqnarray}$$

This shows that, in the compressible case, the turbulent magnetic energy $\langle \boldsymbol{b}^{\prime 2}\rangle /2$ represented by $Q_{b}$ may contribute to the turbulent magnetic diffusivity in coupling with the compressible response $G_{u\text{C}}$ . How much the turbulent magnetic diffusivity depends on the magnetic fluctuation is to be determined by the magnitude of the magnetic energy spectrum $Q_{b}$ and the compressible response time of the velocity field, which should be reflected in the turbulent Mach number $M_{\text{t}}$ . Its dependence on $M_{\text{t}}$ is an interesting subject for future study.

In the solenoidal limit, with vanishing $Q_{u\text{C}}$ and $G_{u\text{C}}$ , the turbulent magnetic diffusivity depends only on the turbulent kinetic energy as

(4.17) $$\begin{eqnarray}\unicode[STIX]{x1D6FD}+\unicode[STIX]{x1D701}={\textstyle \frac{2}{3}}I_{0}\{G_{b},Q_{u\text{S}}\}\end{eqnarray}$$

in the calculation up to $O(\unicode[STIX]{x1D6FF}^{1})$ (4.1). As for the dependence of the turbulent magnetic diffusivity on the magnetic fluctuation $\langle \boldsymbol{b}^{\prime 2}\rangle$ , the reader is referred to Rogachevskii & Kleeorin (Reference Rogachevskii and Kleeorin2001, Reference Rogachevskii and Kleeorin2004). Another possibly interesting point is the sign of $\unicode[STIX]{x1D6FD}$ and $\unicode[STIX]{x1D701}$ . As can be seen in (4.15b ), the coefficient $\unicode[STIX]{x1D701}$ is related to the difference between the kinetic and magnetic energies (MHD residual energy), and can be negative. However, the total coefficient of $\unicode[STIX]{x1D735}\times \boldsymbol{B}$ , $\unicode[STIX]{x1D6FD}+\unicode[STIX]{x1D701}$ (4.16), is still positive in this formulation. As for the possibility of a negative turbulent magnetic diffusivity for intermediate magnetic Reynolds numbers, see Rädler et al. (Reference Rädler, Brandenburg, Del Sordo and Rheinhardt2011).

Notice that the final three terms, the terms related to $\unicode[STIX]{x1D712}_{\overline{\unicode[STIX]{x1D70C}}}$ , $\unicode[STIX]{x1D712}_{Q}$ and $\unicode[STIX]{x1D712}_{D}$ , in (4.13) represent genuine compressible effects. The transport coefficients $\unicode[STIX]{x1D712}_{\overline{\unicode[STIX]{x1D70C}}}$ (4.15e ), $\unicode[STIX]{x1D712}_{Q}$ (4.15f ), and $\unicode[STIX]{x1D712}_{D}$ (4.15g ) do not depend on the solenoidal part of the kinetic energy,  $Q_{u\text{S}}$ , but only on the compressible part of the energy spectrum, $Q_{u\text{C}}$ . In the following section, we will see that this compressible turbulent energy is linked to the dilatation and contraction of the fluctuating motion and directly connected to the density variance $\langle \unicode[STIX]{x1D70C}^{\prime 2}\rangle$ .

Equations (4.15e ) and (4.15f ) show that these effects do not show up in the isothermal process, where $\unicode[STIX]{x1D6FE}_{\text{s}}=1$ . As can be seen in § 5.4 for the physical pictures of these effects, the coupling of the internal energy fluctuation $q^{\prime }$ and density fluctuation $\unicode[STIX]{x1D70C}^{\prime }$ with the fluctuation motion $\boldsymbol{u}^{\prime }$ is essential for these effects. In the isothermal cycle, where the thermal energy cannot be derived as the kinetic energy in a cycle, such links between the heat and motion are missing (see (5.11) and (5.13)). Related to this point, see recent work by Rogachevskii, Kleeorin & Brandenburg (Reference Rogachevskii, Kleeorin and Brandenburg2018).

5.1 Choice of turbulent statistical quantities

In order to construct a system of turbulence model equations, we have to choose a set of turbulent statistical quantities that properly represent the statistical properties of turbulence. For instance, if we have decided to adopt an eddy-viscosity type model for the Reynolds stress as $\langle {u^{\prime }}^{\unicode[STIX]{x1D6FC}}{u^{\prime }}^{\unicode[STIX]{x1D6FD}}\rangle _{\text{D}}=-\unicode[STIX]{x1D708}_{t}{\mathcal{S}}^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}$ ( ${\mathcal{S}}^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}$ is the rate of strain tensor for the mean velocity, ${\mathcal{A}}_{\text{D}}$ denotes the deviatoric or traceless part of a tensor $\unicode[STIX]{x1D63C}$ : ${\mathcal{A}}_{\text{D}}^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}={\mathcal{A}}^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}-{\mathcal{A}}^{aa}\unicode[STIX]{x1D6FF}^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}/3$ ), we still need an expression for the eddy viscosity $\unicode[STIX]{x1D708}_{t}$ . Depending on the level of closure theory, there are several possibilities for the expression of $\unicode[STIX]{x1D708}_{t}$ . The most naïve one is to adopt a parameter value for $\unicode[STIX]{x1D708}_{t}$ . Another simple possibility is to adopt the mixing length $\ell$ and a characteristic turbulence velocity $v$ to express $\unicode[STIX]{x1D708}_{t}=v\ell$ . If we choose the turbulent energy $K$ instead of $v$ , the eddy viscosity in the mixing-length model can be expressed as $\unicode[STIX]{x1D708}_{t}=\ell K^{1/2}$ (Kolmogorov–Prandtl expression). If we choose the turbulent energy $K$ and its dissipation rate $\unicode[STIX]{x1D716}$ as the turbulent statistical quantities, the eddy viscosity can be expressed as $\unicode[STIX]{x1D708}_{t}=K^{2}/\unicode[STIX]{x1D716}$ (apart from the model constant) without resorting to using a mixing length. Instead of using the mixing length, a somewhat arbitrarily determined quantity, we solve the transport equations of $K$ and $\unicode[STIX]{x1D716}$ . As compared with the mixing-length approach, the transport equation approach makes it possible to construct a turbulence model which is based on the nonlinear dynamics of turbulence evolution. However, in any case, it is crucial to choose an appropriate set of turbulent statistical quantities in order to construct a closed system of model equations.

5.2 Density-variance effect

In a strong compressibility case, the dilatation and contraction of the fluctuating motion represented by the divergence of the fluctuation velocity $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}^{\prime }$ are among the most important field quantities. This point is clearly seen in the fourth term of the left-hand side of the original equation for the density fluctuation (2.21). In homogeneous compressible turbulence without large-scale inhomogeneity, this turbulent dilatation is the main source of the density fluctuation (see (3.17)), so the turbulence dilatation and contraction represented by $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}^{\prime }$ are considered to be equivalent to the density fluctuation as

(5.1) $$\begin{eqnarray}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}^{\prime }(\boldsymbol{x};t)=-\frac{1}{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D70C}}}\frac{\unicode[STIX]{x1D70C}^{\prime }(\boldsymbol{x};t)}{\overline{\unicode[STIX]{x1D70C}}},\end{eqnarray}$$

where $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D70C}}$ is the time scale of the density evolution.

In the present formulation, the lowest-order density fluctuation is given as (3.34), whose second term represents the density fluctuation directly related to the dilatation $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}^{\prime }$ as

(5.2) $$\begin{eqnarray}\unicode[STIX]{x1D70C}_{01}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F})=\text{i}k^{a}\overline{\unicode[STIX]{x1D70C}}\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}G_{\unicode[STIX]{x1D70C}}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})u_{\text{B}}^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{k};\unicode[STIX]{x1D70F}_{1}).\end{eqnarray}$$

Due to the strong compressibility, we assume that this part of density fluctuation is much larger than its counterpart due to the solenoidal turbulent eddies, $\unicode[STIX]{x1D70C}_{\text{B}}^{\prime }$ . It follows from (5.2) that the density variance $\langle \unicode[STIX]{x1D70C}^{\prime 2}\rangle$ is approximated as

(5.3) $$\begin{eqnarray}\langle \unicode[STIX]{x1D70C}^{\prime 2}\rangle =\overline{\unicode[STIX]{x1D70C}}^{2}I_{1}\{G_{\unicode[STIX]{x1D70C}}^{(1)},G_{\unicode[STIX]{x1D70C}}^{(2)};Q_{u\text{C}}^{(2)}\},\end{eqnarray}$$

with the abbreviated form of integral

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

In the case of strong compressibility, the correlation time of the density variation can be comparable to the eddy turnover time. The contribution of $\text{i}k^{a}u_{\text{B}}^{\prime }~^{a}(\boldsymbol{k};\unicode[STIX]{x1D70F}_{1})$ (the Fourier transform of $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}_{\text{B}}^{\prime }$ ) has fallen to zero from the past time that is older than $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D70C}}>t_{1}$ , where $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D70C}}$ is the characteristic time defined by

(5.5) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D70C}}=\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}\langle G_{\unicode[STIX]{x1D70C}}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})\rangle =\int _{-\infty }^{\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}_{1}G_{\unicode[STIX]{x1D70C}}(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1}).\end{eqnarray}$$

For example, this is equivalent to having

(5.6) $$\begin{eqnarray}G_{\unicode[STIX]{x1D70C}}(k;\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}_{1})=\unicode[STIX]{x1D6EF}(t-t_{1})\exp \left[-\frac{\unicode[STIX]{x1D70F}-\unicode[STIX]{x1D70F}_{1}}{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D70C}}(k)}\right],\end{eqnarray}$$

where $\unicode[STIX]{x1D6EF}$ is the Heaviside step function: $\unicode[STIX]{x1D6EF}(t)=1$ for $t>0$ and $\unicode[STIX]{x1D6EF}(t)=0$ for $t<0$ . With this approximation, equation (5.2) can be rewritten as

(5.7) $$\begin{eqnarray}\text{i}k^{a}u_{\text{B}}^{\prime }\hspace{0.0pt}^{a}(\boldsymbol{k};\unicode[STIX]{x1D70F})\simeq \frac{1}{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D70C}}\overline{\unicode[STIX]{x1D70C}}}\unicode[STIX]{x1D70C}_{0}^{\prime }(\boldsymbol{k};\unicode[STIX]{x1D70F}).\end{eqnarray}$$

This corresponds to the Markovianised approximation of the dilatation effect.

From (5.3), the compressible energy spectrum is related to the density variance as

(5.8a ) $$\begin{eqnarray}\langle \unicode[STIX]{x1D70C}^{\prime 2}\rangle =\overline{\unicode[STIX]{x1D70C}}^{2}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D70C}}^{2}\int \,\text{d}\boldsymbol{k}\,k^{2}Q_{u\text{C}}(k;\unicode[STIX]{x1D70F}_{1},\unicode[STIX]{x1D70F}_{2})\end{eqnarray}$$

or equivalently,

(5.8b ) $$\begin{eqnarray}\int \,\text{d}\boldsymbol{k}\,k^{2}Q_{u\text{C}}(k;\unicode[STIX]{x1D70F}_{1},\unicode[STIX]{x1D70F}_{2})=\frac{1}{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D70C}}^{2}}\frac{\langle \unicode[STIX]{x1D70C}^{\prime 2}\rangle }{\overline{\unicode[STIX]{x1D70C}}^{2}}.\end{eqnarray}$$

In other words, the compressible component of the turbulent motion appearing in (4.15e )–(4.15g ), which is directly related to the divergence of the fluctuating velocity, is represented by the density variance.

We should note that, from the viewpoint of turbulence modelling, it is very important to choose the density variance $\langle \unicode[STIX]{x1D70C}^{\prime 2}\rangle$ as a basic statistical quantity as well as the turbulent kinetic energy, magnetic energy and several helicities. As has been explained in § 5.1, in the turbulence modelling approach, a system of equations should be constructed in a closed form and a self-consistent manner. The mean-field equations with transport coefficients ( $\unicode[STIX]{x1D6FD}$ , $\unicode[STIX]{x1D701}$ , $\unicode[STIX]{x1D6FC}$ , $\unicode[STIX]{x1D6FE}$ , etc.) should be solved simultaneously with the equations of the turbulent statistical quantities, which represent the statistical properties of the turbulence and express the transport coefficients. In order to construct a closed system of equations in strongly compressible turbulence, we solve the transport equation of the density variance:

(5.9) $$\begin{eqnarray}\frac{\text{D}}{\text{D}t}\langle \unicode[STIX]{x1D70C}^{\prime 2}\rangle =-2\langle \unicode[STIX]{x1D70C}^{\prime }\boldsymbol{u}^{\prime }\rangle \boldsymbol{\cdot }\unicode[STIX]{x1D735}\overline{\unicode[STIX]{x1D70C}}-2\langle \unicode[STIX]{x1D70C}^{\prime 2}\rangle \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{U}-2\overline{\unicode[STIX]{x1D70C}}\langle \unicode[STIX]{x1D70C}^{\prime }\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}^{\prime }\rangle -\langle \unicode[STIX]{x1D70C}^{\prime 2}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}^{\prime }\rangle -\unicode[STIX]{x1D735}\boldsymbol{\cdot }\langle \unicode[STIX]{x1D70C}^{\prime 2}\boldsymbol{u}^{\prime }\rangle ,\end{eqnarray}$$

with the model expressions for the turbulent correlations such as the turbulent mass flux $\langle \unicode[STIX]{x1D70C}^{\prime }\boldsymbol{u}^{\prime }\rangle$ , the density–dilatation correlation $\langle \unicode[STIX]{x1D70C}^{\prime }\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}^{\prime }\rangle$ , etc. The details of the modelling of those correlations are suppressed here.

It should be also noted that the density variance normalised by the square of the mean density, $\langle \unicode[STIX]{x1D70C}^{\prime 2}\rangle /\overline{\unicode[STIX]{x1D70C}}^{2}$ , is strongly correlated to the turbulent Mach number $M_{\text{t}}$ , and considered to be one of the most important quantities representing the strong compressibility effect (Chassaing et al. Reference Chassaing, Antonia, Anselmet, Joly and Sarker2002).

5.3 Model of the turbulent electromotive force

With the aid of the analytical expressions of the turbulent transport coefficients (4.15), we can construct model expressions for the transport coefficients for the turbulent electromotive force (4.13) as

(5.10a ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FD}=\unicode[STIX]{x1D70F}_{b}\langle \boldsymbol{u}^{\prime 2}\rangle /2+\unicode[STIX]{x1D70F}_{u}\langle \boldsymbol{b}^{\prime 2}\rangle /(2\unicode[STIX]{x1D707}_{0}\overline{\unicode[STIX]{x1D70C}}), & \displaystyle\end{eqnarray}$$

(5.10b ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D701}=\unicode[STIX]{x1D70F}_{b}\langle \boldsymbol{u}^{\prime 2}\rangle /2-\unicode[STIX]{x1D70F}_{u}\langle \boldsymbol{b}^{\prime 2}\rangle /(2\unicode[STIX]{x1D707}_{0}\overline{\unicode[STIX]{x1D70C}}), & \displaystyle\end{eqnarray}$$

(5.10c ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FC}=-\unicode[STIX]{x1D70F}_{b}\langle \boldsymbol{u}^{\prime }\boldsymbol{\cdot }\unicode[STIX]{x1D74E}^{\prime }\rangle +\unicode[STIX]{x1D70F}_{u}\langle \boldsymbol{b}^{\prime }\boldsymbol{\cdot }\boldsymbol{j}^{\prime }\rangle /\overline{\unicode[STIX]{x1D70C}}, & \displaystyle\end{eqnarray}$$

(5.10d ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FE}=(\unicode[STIX]{x1D70F}_{u}+\unicode[STIX]{x1D70F}_{b})\langle \boldsymbol{u}^{\prime }\boldsymbol{\cdot }\boldsymbol{b}^{\prime }\rangle , & \displaystyle\end{eqnarray}$$

(5.10e ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D712}_{\overline{\unicode[STIX]{x1D70C}}}=(\unicode[STIX]{x1D6FE}_{\text{s}}-1)^{2}\frac{\unicode[STIX]{x1D70F}_{u}\unicode[STIX]{x1D70F}_{q}\unicode[STIX]{x1D70F}_{b}}{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D70C}}^{2}}\frac{Q}{\overline{\unicode[STIX]{x1D70C}}}\frac{\langle \unicode[STIX]{x1D70C}^{\prime 2}\rangle }{\overline{\unicode[STIX]{x1D70C}}^{2}}, & \displaystyle\end{eqnarray}$$

(5.10f ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D712}_{Q}=(\unicode[STIX]{x1D6FE}_{\text{s}}-1)\frac{\unicode[STIX]{x1D70F}_{u}\unicode[STIX]{x1D70F}_{b}}{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D70C}}}\frac{\langle \unicode[STIX]{x1D70C}^{\prime 2}\rangle }{\overline{\unicode[STIX]{x1D70C}}^{2}}, & \displaystyle\end{eqnarray}$$

(5.10g ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D712}_{D}=\frac{\unicode[STIX]{x1D70F}_{u}\unicode[STIX]{x1D70F}_{b}}{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D70C}}}\frac{\langle \unicode[STIX]{x1D70C}^{\prime 2}\rangle }{\overline{\unicode[STIX]{x1D70C}}^{2}}, & \displaystyle\end{eqnarray}$$

$\unicode[STIX]{x1D70F}_{u}$

$\unicode[STIX]{x1D70F}_{q}$

$\unicode[STIX]{x1D70F}_{b}$

$\unicode[STIX]{x1D6FD}$

$\unicode[STIX]{x1D6FC}$

$\unicode[STIX]{x1D6FE}$

$\unicode[STIX]{x1D712}_{\overline{\unicode[STIX]{x1D70C}}}$

$\unicode[STIX]{x1D712}_{Q}$

$\unicode[STIX]{x1D712}_{D}$

5.4 Physical origin of density-variance effect

The fifth term of (4.13) suggests that the obliqueness between the mean-density gradient $\unicode[STIX]{x1D735}\overline{\unicode[STIX]{x1D70C}}$ and magnetic field $\boldsymbol{B}$ contributes to the turbulent electromotive force in the presence of the density variance $\langle \unicode[STIX]{x1D70C}^{\prime 2}\rangle$ . Let us consider the physical origin of this effect and see a possible situation in which this effect plays an essential role.

The equations for the fluctuation velocity and magnetic field are written as (2.22) and (2.24), respectively. The relevant contributions may be written as

(5.11) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}\boldsymbol{u}^{\prime }}{\unicode[STIX]{x2202}t} & = & \displaystyle -(\unicode[STIX]{x1D6FE}_{\text{s}}-1)\frac{q^{\prime }}{\overline{\unicode[STIX]{x1D70C}}}\unicode[STIX]{x1D735}\overline{\unicode[STIX]{x1D70C}}+\cdots \nonumber\\ \displaystyle & = & \displaystyle +(\unicode[STIX]{x1D6FE}_{\text{s}}-1)^{2}\unicode[STIX]{x1D70F}_{q}\frac{Q}{\overline{\unicode[STIX]{x1D70C}}}(\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}^{\prime })\unicode[STIX]{x1D735}\overline{\unicode[STIX]{x1D70C}}+\cdots \,,\end{eqnarray}$$

(5.12) $$\begin{eqnarray}\unicode[STIX]{x2202}\boldsymbol{b}^{\prime }/\unicode[STIX]{x2202}t=-(\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}^{\prime })\boldsymbol{B}+\cdots \,.\end{eqnarray}$$

In (5.11), use has been made of the simplest expression for the internal energy fluctuation:

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

(see (3.36)). Equation (5.11) indicates that the direction of the velocity variation is parallel (anti-parallel) to the direction of the mean-density gradient $\unicode[STIX]{x1D735}\overline{\unicode[STIX]{x1D70C}}$ if the turbulent dilatation $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}^{\prime }$ is positive (negative) (see figure 1). On the other hand, we see from (5.12) that the direction of the magnetic field variation is antiparallel (parallel) to the mean magnetic field $\boldsymbol{B}$ if the dilatation $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}^{\prime }$ is positive (negative) (see figure 2).

Figure 1.

Turbulent velocity variation $\unicode[STIX]{x1D6FF}\boldsymbol{u}^{\prime }$ tends to be parallel (anti-parallel) to the mean-density gradient $\unicode[STIX]{x1D735}\overline{\unicode[STIX]{x1D70C}}$ , if the local turbulent dilatation $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}^{\prime }$ is positive (negative).

Figure 2.

Magnetoacoustic waves. The turbulent magnetic field variation $\unicode[STIX]{x1D6FF}\boldsymbol{b}^{\prime }$ is parallel to the mean magnetic field $\boldsymbol{B}$ in the case of local turbulent contraction $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\unicode[STIX]{x1D743}_{\bot }<0$ . Here $\unicode[STIX]{x1D743}_{\bot }$ is the displacement of plasma perpendicular to the mean magnetic field.

It follows from (5.11) and (5.12) that the turbulent electromotive force is expressed as

(5.14) $$\begin{eqnarray}\displaystyle \langle \boldsymbol{u}^{\prime }\times \boldsymbol{b}^{\prime }\rangle /\unicode[STIX]{x1D70F} & \simeq & \displaystyle (\unicode[STIX]{x1D6FE}_{\text{s}}-1)\frac{1}{\overline{\unicode[STIX]{x1D70C}}}\langle q^{\prime }\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}^{\prime }\rangle \unicode[STIX]{x1D735}\overline{\unicode[STIX]{x1D70C}}\times \boldsymbol{B}\nonumber\\ \displaystyle & = & \displaystyle -(\unicode[STIX]{x1D6FE}_{\text{s}}-1)^{2}\frac{\unicode[STIX]{x1D70F}_{q}}{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D70C}}}\frac{\langle \unicode[STIX]{x1D70C}^{\prime 2}\rangle }{\overline{\unicode[STIX]{x1D70C}}^{2}}\frac{Q}{\overline{\unicode[STIX]{x1D70C}}}\unicode[STIX]{x1D735}\overline{\unicode[STIX]{x1D70C}}\times \boldsymbol{B},\end{eqnarray}$$

where use has been made of (5.13) and the simplest expression for the density fluctuation, equation (5.1).

Equation (5.14) is equivalent to the fifth term in (4.13) with (5.10e ). This expression shows that, irrespective of the sign of $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}^{\prime }$ (dilatation or contraction), the density variance will contribute to the turbulent EMF in the direction of $\boldsymbol{B}\times \unicode[STIX]{x1D735}\overline{\unicode[STIX]{x1D70C}}$ (figure 3).

Similar arguments can be made for the other terms related to $\unicode[STIX]{x1D712}_{Q}$ and $\unicode[STIX]{x1D712}_{D}$ .

Figure 3.

Turbulent electromotive force due to the misalignment of the mean magnetic field from the gradient of mean density. Cases for dilatation (a) and contraction (b).

7.1 Summary

A system of equations for fully compressible MHD turbulence was investigated with the aid of the two-scale direct-interaction approximation (TSDIA) formulation: a combination of a renormalised perturbation closure scheme for very high Reynolds number turbulence by Kriachnan with a multiple-scale analysis, developed to analyse inhomogeneous turbulence. The main results in this work are the analytical expression for the turbulent EMF (4.13). In addition to the usual mean-field terms in the turbulent electromotive force in the incompressible case, some terms directly connected to the full compressibility have been derived from the fundamental equations. They are $\unicode[STIX]{x1D712}$ -related terms (4.11a )–(4.11c ), with the analytical expressions for the $\unicode[STIX]{x1D712}$ -related transport coefficients $\unicode[STIX]{x1D712}_{\overline{\unicode[STIX]{x1D70C}}}$ (4.15e ), $\unicode[STIX]{x1D712}_{Q}$ (4.15f ) and $\unicode[STIX]{x1D712}_{D}$ (4.15g ).

On the basis of these theoretical expressions, we proposed a simplified system of model expressions for the strong compressibility effects in terms of the density variance $\langle \unicode[STIX]{x1D70C}^{\prime 2}\rangle$ . The time scales in (5.10), $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D70C}}$ , $\unicode[STIX]{x1D70F}_{u}$ , $\unicode[STIX]{x1D70F}_{q}$ and $\unicode[STIX]{x1D70F}_{b}$ are determined by Green’s functions, which obey (3.30)–(3.33), respectively. If the time scales $\unicode[STIX]{x1D70F}_{q}$ , $\unicode[STIX]{x1D70F}_{u}$ , etc. can be treated as the same, the model expression becomes much simpler.

In the presence of a strong compressibility represented by the density variance $\langle \unicode[STIX]{x1D70C}^{\prime 2}\rangle$ , the mean-density inhomogeneity $\unicode[STIX]{x1D735}\overline{\unicode[STIX]{x1D70C}}$ , as well as the mean internal energy inhomogeneity $\unicode[STIX]{x1D735}Q$ and the non-equilibrium mean flow $D\boldsymbol{U}/DT$ , coupled with the oblique mean magnetic field $\boldsymbol{B}$ can contribute to the production of the turbulent electromotive force. Since they are mediated by the density variance, these effects are important in the region with a very strong turbulent dilatation and contraction. The vicinity of a shock front, where a strong density variation exists, is one of the places these effects are relevant. This mechanism is expected to work for fast magnetic reconnection by locally enhancing the intensity of turbulence across the shock front, as compared with the incompressible turbulence case.

7.2 Approaches for validation

For the validation or test of these strong compressibility effects, we can perform numerical simulations of inhomogeneous turbulence. The most straightforward approach is validations through direct numerical simulations (DNSs). The simulation set-up utilised in Yokoi & Balarac (Reference Yokoi and Balarac2011) for a Kolmogorov flow in MHD turbulence and the one in Yokoi & Brandenburg (Reference Yokoi and Brandenburg2016) based on Yokoi & Yoshizawa (Reference Yokoi and Yoshizawa1993) for rotating helical hydrodynamic (HD) turbulence may provide a basis of the numerical set-up. Since the normalised density variance $\langle \unicode[STIX]{x1D70C}^{\prime 2}\rangle /\overline{\unicode[STIX]{x1D70C}}^{2}$ is expected to be large only in the region where the mean-density stratification is very large, vicinity of a shock front is an appropriate location to check these effects. DNSs of a boxed turbulence with a strongly localised density stratification whose inhomogeneous direction is perpendicular to a uniform magnetic field may be a possible set-up. Then we can compare the DNS data of turbulent EMF, $\langle \boldsymbol{u}^{\prime }\times \boldsymbol{b}^{\prime }\rangle$ (the left-hand side of (4.13)) and the model expression (the right-hand side of (4.13)) with the transport coefficients evaluated by (5.10).

Another, less direct but more practical, approach is to perform turbulence model simulations in a simple configuration of the magnetic reconnection. Yokoi & Hoshino (Reference Yokoi and Hoshino2011) proposed a turbulence model for fast magnetic reconnection. There, in addition to the evolution of the mean-field quantities, the evolution of turbulence is solved through the transport equations of the turbulent statistical quantities such as the MHD energy, its dissipation rate, helicity, cross-helicity, etc. A series of applications of the turbulence model to simple configurations of the magnetic reconnection (Yokoi & Hoshino Reference Yokoi and Hoshino2011; Higashimori et al. Reference Higashimori, Yokoi and Hoshino2013; Yokoi et al. Reference Yokoi, Higashimori and Hoshino2013; Widmer, Büchner & Yokoi Reference Widmer, Büchner and Yokoi2016a ,Reference Widmer, Büchner and Yokoi b ) showed that the Petchek-type fast reconnection can be obtained and sustained for a long time as a stationary state in the framework of MHD if fully developed nonlinear turbulence is self-generated and sustained by the large-scale inhomogeneities of the mean fields. This approach is somewhat different from the previous studies of fast magnetic reconnection, such as fast magnetic reconnection in fully developed homogeneous turbulence that is externally injected by forcing, and the fast magnetic reconnection obtained by some highly inhomogeneous instability but not at the fully developed nonlinear stage. The turbulence model approach to fast magnetic reconnection has already given novel results (Higashimori et al. Reference Higashimori, Yokoi and Hoshino2013; Yokoi et al. Reference Yokoi, Higashimori and Hoshino2013; Widmer et al. Reference Widmer, Büchner and Yokoi2016a ,Reference Widmer, Büchner and Yokoi b ). But the turbulence solved there so far has been incompressible ( $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}^{\prime }=0$ ) while the mean fields are solved with a variable mean density ( $\unicode[STIX]{x1D735}\overline{\unicode[STIX]{x1D70C}}\neq 0$ ). In order to implement the present density-variance effects in this turbulence model for fast magnetic reconnection, we have to add at least the equation of the density variance $\langle \unicode[STIX]{x1D70C}^{\prime 2}\rangle$ with a model of the turbulent mass flux $\langle \unicode[STIX]{x1D70C}^{\prime }\boldsymbol{u}^{\prime }\rangle$ etc. (see (6.1)). From the arguments extended in § 6, the density-variance effects are expected to locally enhance the level of turbulence. These compressibility effects should be preferable for the realisation of more fast reconnection often observed in the solar flare reconnections. The validation of the turbulence model can be done in a similar manner as in Widmer et al. (Reference Widmer, Büchner and Yokoi2016b ). With the present analytical results shown in § 3, such an extension is very straightforward, and will be reported in a forthcoming paper.