2.1 Governing equations

The magnetic field $\boldsymbol{B}(\boldsymbol{x},t)$ is governed by the induction equation:

(2.1) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}\boldsymbol{B}}{\unicode[STIX]{x2202}t}=\unicode[STIX]{x1D735}\times (\boldsymbol{U}\times \boldsymbol{B}-\unicode[STIX]{x1D702}\unicode[STIX]{x1D735}\times \boldsymbol{B}), & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D702}$ is the magnetic diffusivity due to the electrical conductivity of the fluid, and fluid velocity $\boldsymbol{U}(\boldsymbol{x},t)$ is determined by the Navier–Stokes equation,

(2.2) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70C}\left(\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}+\boldsymbol{U}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\right)\boldsymbol{U}-\unicode[STIX]{x1D735}\boldsymbol{\cdot }(2\unicode[STIX]{x1D708}\unicode[STIX]{x1D70C}\unicode[STIX]{x1D64E}^{(\text{U})})=\frac{1}{\unicode[STIX]{x1D707}_{0}}(\unicode[STIX]{x1D735}\times \boldsymbol{B})\times \boldsymbol{B}-\unicode[STIX]{x1D735}P, & & \displaystyle\end{eqnarray}$$

where $P(\boldsymbol{x},t)$ is the fluid pressure, $\unicode[STIX]{x1D61A}_{ij}^{(\text{U})}=1/2(U_{i,j}+U_{j,i})-1/3\unicode[STIX]{x1D6FF}_{ij}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{U}$ are the components of the traceless rate-of-strain tensor $\unicode[STIX]{x1D64E}^{(\text{U})}$ , commas denote partial differentiation and $\unicode[STIX]{x1D708}$ is the kinematic viscosity.

We apply a mean-field approach and use Reynolds averaging (Krause & Rädler Reference Krause and Rädler1980). In the framework of this approach, velocity and magnetic fields, fluid density and pressure are decomposed into mean (denoted by overbars) and fluctuating parts (lowercase symbols). Ensemble averaging of (2.1) yields an equation for the mean magnetic field $\overline{\boldsymbol{B}}(\boldsymbol{x},t)$ :

(2.3) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}\overline{\boldsymbol{B}}}{\unicode[STIX]{x2202}t}=\unicode[STIX]{x1D735}\times (\overline{\boldsymbol{U}}\times \overline{\boldsymbol{B}}+\overline{\boldsymbol{{\mathcal{E}}}}-\unicode[STIX]{x1D702}\unicode[STIX]{x1D735}\times \overline{\boldsymbol{B}}), & & \displaystyle\end{eqnarray}$$

where $\overline{\boldsymbol{U}}(\boldsymbol{x},t)$ is the mean velocity and $\overline{\boldsymbol{{\mathcal{E}}}}=\overline{\boldsymbol{u}\times \boldsymbol{b}}$ is the mean electromotive force. For simplicity, we consider in this study the case $\overline{\boldsymbol{U}}=\mathbf{0}$ . We determine $\overline{\boldsymbol{{\mathcal{E}}}}$ for compressible inhomogeneous and helical turbulence. The procedure of the derivation of the equation for the mean electromotive force is as follows. The momentum and induction equations for fluctuations are given by

(2.4) $$\begin{eqnarray}\displaystyle & \displaystyle \overline{\unicode[STIX]{x1D70C}}\,\frac{\unicode[STIX]{x2202}\boldsymbol{u}}{\unicode[STIX]{x2202}t}-\unicode[STIX]{x1D735}\boldsymbol{\cdot }(2\unicode[STIX]{x1D708}\overline{\unicode[STIX]{x1D70C}}\unicode[STIX]{x1D64E}^{\text{(u)}})=\frac{1}{\unicode[STIX]{x1D707}_{0}}[(\boldsymbol{b}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\overline{\boldsymbol{B}}+(\overline{\boldsymbol{B}}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{b}]-\unicode[STIX]{x1D735}p_{\text{tot}}+\boldsymbol{u}^{(\text{N})}, & \displaystyle\end{eqnarray}$$

(2.5) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\boldsymbol{b}}{\unicode[STIX]{x2202}t}-\unicode[STIX]{x1D702}\unicode[STIX]{x0394}\boldsymbol{b}=(\overline{\boldsymbol{B}}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{u}-(\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\overline{\boldsymbol{B}}-\overline{\boldsymbol{B}}(\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u})+\boldsymbol{b}^{(\text{N})}, & \displaystyle\end{eqnarray}$$

where $\boldsymbol{u}(\boldsymbol{x},t)$ and $\boldsymbol{b}(\boldsymbol{x},t)$ are the fluctuations of velocity and magnetic fields, $\boldsymbol{u}^{(\text{N})}$ and $\boldsymbol{b}^{(\text{N})}$ are terms that are nonlinear in the fluctuations, $p_{\text{tot}}=p+\unicode[STIX]{x1D707}_{0}^{-1}(\overline{\boldsymbol{B}}\boldsymbol{\cdot }\boldsymbol{b})$ are the fluctuations of the total pressure, $\unicode[STIX]{x1D64E}^{\text{(u)}}\equiv \unicode[STIX]{x1D61A}_{ij}^{\text{(u)}}=1/2(u_{i,j}+u_{j,i})-1/3\unicode[STIX]{x1D6FF}_{ij}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}$ , and $p(\boldsymbol{x},t)$ is the fluctuation of the fluid pressure. We rewrite (2.4) and (2.5) in $\boldsymbol{k}$ space. In the following, we derive equations for the mean electromotive force for small and large magnetic Reynolds numbers.

2.2 Small magnetic Reynolds numbers

We use a quasi-linear approach (also known in the literature as ‘first-order smoothing’ or ‘second-order correlation approximation’), which is valid for small hydrodynamic and magnetic Reynolds numbers, $Re=\ell _{0}\sqrt{\overline{\boldsymbol{u}^{2}}}/\unicode[STIX]{x1D708}\ll 1$ and $Rm=\ell _{0}\sqrt{\overline{\boldsymbol{u}^{2}}}/\unicode[STIX]{x1D702}\ll 1$ , respectively. Here $\sqrt{\overline{\boldsymbol{u}^{2}}}$ is the characteristic turbulent velocity in the integral turbulent scale $\ell _{0}$ . In the high conductivity limit (very small microscopic magnetic diffusivity  $\unicode[STIX]{x1D702}$ ), the quasi-linear approach is only valid for small Strouhal numbers, which is defined as the ratio of turbulent correlation time, $\unicode[STIX]{x1D70F}_{0}$ , to turnover time, $\ell _{0}/\sqrt{\overline{\boldsymbol{u}^{2}}}$ . This implies that the quasi-linear approach in this limit is only valid for very short turbulent correlation times. Often (see, e.g. Krause & Rädler Reference Krause and Rädler1980) the quasi-linear approach is applied only to the induction equation and the origin of the velocity field is not discussed. In this case the evolution of the magnetic field is considered for a prescribed velocity field, and formally the validity of the quasi-linear approach only requires small magnetic Reynolds numbers.

2.2.1 Multi-scale approach

To determine the mean electromotive force, which is a one-point correlation function, and to take into account small-scale properties of the turbulence, e.g. the turbulent spectrum, one must use two-point correlation functions. For fully developed turbulence, scalings for the turbulent correlation time and energy spectrum are related via the Kolmogorov scalings (McComb Reference McComb1990; Frisch Reference Frisch1995; Monin & Yaglom Reference Monin and Yaglom2013). This is the reason why for calculations of the mean electromotive force in fully developed turbulence, we use instantaneous two-point correlation functions. On the other hand, for a random flow with small $Re$ and $Rm$ , there are no universal scalings for the correlation time and energy spectrum. This is the reason why we have to use non-instantaneous two-point correlation functions in this case.

In the framework of the mean-field approach, we assume that there is a separation of spatial and temporal scales, i.e. $\ell _{0}\ll L_{B}$ and $\unicode[STIX]{x1D70F}_{0}\ll t_{B}$ , where $L_{B}$ and $t_{B}$ are the characteristic spatial and temporal scales characterizing the variations of the mean magnetic field. The mean fields depend on ‘slow’ variables, while fluctuations depend on ‘fast’ variables. Separation into slow and fast variables is widely used in theoretical physics, and all calculations are reduced to Taylor expansions of all functions using small parameters $\ell _{0}/L_{B}$ and $\unicode[STIX]{x1D70F}_{0}/t_{B}$ . The findings are further truncated to leading-order terms.

Separation to slow and fast variables is performed by means of a standard multi-scale approach (Roberts & Soward Reference Roberts and Soward1975). In the framework of this approach, the non-instantaneous two-point second-order correlation function of $\boldsymbol{b}$ and $\boldsymbol{u}$ is written as follows:

(2.6) $$\begin{eqnarray}\displaystyle \overline{b_{i}(\boldsymbol{x},t_{1})u_{j}(\boldsymbol{y},t_{2})} & = & \displaystyle \int \text{d}\unicode[STIX]{x1D714}_{1}\,\text{d}\unicode[STIX]{x1D714}_{2}\,\text{d}\boldsymbol{k}_{1}\,\text{d}\boldsymbol{k}_{2}\,\overline{b_{i}(\boldsymbol{k}_{1},\unicode[STIX]{x1D714}_{1})u_{j}(\boldsymbol{k}_{2},\unicode[STIX]{x1D714}_{2})}\exp [\text{i}(\boldsymbol{k}_{1}\boldsymbol{\cdot }\boldsymbol{x}+\boldsymbol{k}_{2}\boldsymbol{\cdot }\boldsymbol{y})\nonumber\\ \displaystyle & & \displaystyle +\,\text{i}(\unicode[STIX]{x1D714}_{1}t_{1}+\unicode[STIX]{x1D714}_{2}t_{2})]=\int g_{ij}(\boldsymbol{k},\unicode[STIX]{x1D714},t,\boldsymbol{R})\exp [\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{r}+\text{i}\unicode[STIX]{x1D714}\,\tilde{\unicode[STIX]{x1D70F}}]\,\text{d}\unicode[STIX]{x1D714}\,\text{d}\boldsymbol{k},\qquad\end{eqnarray}$$

where

(2.7) $$\begin{eqnarray}\displaystyle g_{ij}(\boldsymbol{k},\unicode[STIX]{x1D714},\boldsymbol{R},t) & = & \displaystyle \int \overline{b_{i}(\boldsymbol{k}_{1},\unicode[STIX]{x1D714}_{1})u_{j}(\boldsymbol{k}_{2},\unicode[STIX]{x1D714}_{2})}\exp [\text{i}\unicode[STIX]{x1D6FA}t+\text{i}\boldsymbol{K}\boldsymbol{\cdot }\boldsymbol{R}]\,\text{d}\unicode[STIX]{x1D6FA}\,\text{d}\boldsymbol{K}.\end{eqnarray}$$

Here we introduced large-scale variables: $\boldsymbol{R}=(\boldsymbol{x}+\boldsymbol{y})/2$ , $\boldsymbol{K}=\boldsymbol{k}_{1}+\boldsymbol{k}_{2}$ , $t=(t_{1}+t_{2})/2$ , $\unicode[STIX]{x1D6FA}=\unicode[STIX]{x1D714}_{1}+\unicode[STIX]{x1D714}_{2}$ , and small-scale variables: $\boldsymbol{r}=\boldsymbol{x}-\boldsymbol{y}$ , $\boldsymbol{k}=(\boldsymbol{k}_{1}-\boldsymbol{k}_{2})/2$ , $\tilde{\unicode[STIX]{x1D70F}}=t_{1}-t_{2}$ , $\unicode[STIX]{x1D714}=(\unicode[STIX]{x1D714}_{1}-\unicode[STIX]{x1D714}_{2})/2$ . This implies that $\unicode[STIX]{x1D714}_{1}=\unicode[STIX]{x1D714}+\unicode[STIX]{x1D6FA}/2$ , $\unicode[STIX]{x1D714}_{2}=-\unicode[STIX]{x1D714}+\unicode[STIX]{x1D6FA}/2$ , $\boldsymbol{k}_{1}=\boldsymbol{k}+\boldsymbol{K}/2$ and $\boldsymbol{k}_{2}=-\boldsymbol{k}+\boldsymbol{K}/2$ . Mean fields depend on the large-scale variables, while fluctuations depend on the small-scale variables. We have used here the Fourier transformation:

(2.8) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F7}(\boldsymbol{x},t)=\int \unicode[STIX]{x1D6F7}(\boldsymbol{k},\unicode[STIX]{x1D714})\exp [\text{i}(\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{x}+\unicode[STIX]{x1D714}t)]\,\text{d}\unicode[STIX]{x1D714}\,\text{d}\boldsymbol{k}. & & \displaystyle\end{eqnarray}$$

Similarly to (2.6)–(2.7), we obtain

(2.9) $$\begin{eqnarray}\displaystyle f_{ij}(\boldsymbol{k},\unicode[STIX]{x1D714},\boldsymbol{R},t) & = & \displaystyle \int \overline{u_{i}(\boldsymbol{k}_{1},\unicode[STIX]{x1D714}_{1})u_{j}(\boldsymbol{k}_{2},\unicode[STIX]{x1D714}_{2})}\exp [\text{i}\unicode[STIX]{x1D6FA}t+\text{i}\boldsymbol{K}\boldsymbol{\cdot }\boldsymbol{R}]\,\text{d}\unicode[STIX]{x1D6FA}\,\text{d}\boldsymbol{K}.\end{eqnarray}$$

After separation into slow and fast variables and calculating the function $g_{ij}(\boldsymbol{k},\unicode[STIX]{x1D714},\boldsymbol{R},t)$ , expression (2.6) allows us to determine the cross-helicity tensor in physical space and to calculate the limit of $\boldsymbol{r}\rightarrow \mathbf{0}$ and $\tilde{\unicode[STIX]{x1D70F}}\rightarrow 0$ , which yields

(2.10) $$\begin{eqnarray}\displaystyle \overline{b_{i}(\boldsymbol{x},t)u_{j}(\boldsymbol{x},t)}=\int g_{ij}(\boldsymbol{k},\unicode[STIX]{x1D714},\boldsymbol{R},t)\,\text{d}\unicode[STIX]{x1D714}\,\text{d}\boldsymbol{k}. & & \displaystyle\end{eqnarray}$$

For brevity of notations we omit below the large-scale variables $t$ and $\boldsymbol{R}$ in the correlation functions $f_{ij}(\boldsymbol{k},\unicode[STIX]{x1D714},\boldsymbol{R},t)$ and $g_{ij}(\boldsymbol{k},\unicode[STIX]{x1D714},\boldsymbol{R},t)$ .

2.2.2 Cross-helicity tensor

In the framework of the quasi-linear approach, we neglect nonlinear terms, but keep molecular dissipative terms in (2.5) for the magnetic fluctuations. This allows us to obtain the solution of (2.5) in the limit of small magnetic Reynolds number. Using this solution, we derive the equation for the correlation function $g_{ij}(\boldsymbol{k},\unicode[STIX]{x1D714})$ :

(2.11) $$\begin{eqnarray}\displaystyle g_{ij}(\boldsymbol{k},\unicode[STIX]{x1D714}) & = & \displaystyle G_{\unicode[STIX]{x1D702}}\left\{\overline{B}_{p}\left[\text{i}k_{p}f_{ij}+\frac{1}{2}\unicode[STIX]{x1D735}_{p}f_{ij}-\unicode[STIX]{x1D702}k^{2}G_{\unicode[STIX]{x1D702}}\,k_{sp}\unicode[STIX]{x1D735}_{s}f_{ij}\right]\right.\nonumber\\ \displaystyle & & \displaystyle -\,\overline{B}_{i}\left[\text{i}k_{p}f_{pj}+\frac{1}{2}\unicode[STIX]{x1D735}_{p}f_{pj}-\unicode[STIX]{x1D702}k^{2}G_{\unicode[STIX]{x1D702}}\,k_{sp}\unicode[STIX]{x1D735}_{s}f_{pj}\right]-(\unicode[STIX]{x1D735}_{s}\overline{B}_{p})\left[\frac{1}{2}k_{p}\frac{\unicode[STIX]{x2202}f_{ij}}{\unicode[STIX]{x2202}k_{s}}+\unicode[STIX]{x1D702}k^{2}G_{\unicode[STIX]{x1D702}}\,k_{sp}f_{ij}\right]\nonumber\\ \displaystyle & & \displaystyle +\left.\frac{1}{2}(\unicode[STIX]{x1D735}_{s}\overline{B}_{i})\left[-f_{sj}+k_{p}\frac{\unicode[STIX]{x2202}f_{pj}}{\unicode[STIX]{x2202}k_{s}}+2\unicode[STIX]{x1D702}k^{2}G_{\unicode[STIX]{x1D702}}\,k_{sp}f_{pj}\right]\right\},\end{eqnarray}$$

where $k_{ij}=k_{i}\,k_{j}/k^{2}$ and $G_{\unicode[STIX]{x1D702}}\equiv G_{\unicode[STIX]{x1D702}}(\boldsymbol{k},\unicode[STIX]{x1D714})=(\unicode[STIX]{x1D702}\boldsymbol{k}^{2}+\text{i}\unicode[STIX]{x1D714})^{-1}$ . The derivation of (2.11) is given in appendix A. Here, for brevity of notation, we omit the large-scale variables $\boldsymbol{R}$ and $t$ in the mean magnetic field $\overline{B}_{p}$ and correlation functions $g_{ij}$ and $f_{ij}$ . When $\overline{\boldsymbol{B}}^{2}/4\unicode[STIX]{x03C0}\ll \overline{\unicode[STIX]{x1D70C}}\overline{\boldsymbol{u}^{2}}$ , the correlation function $f_{ij}(\boldsymbol{k},\unicode[STIX]{x1D714})$ in (2.11) should be replaced by the correlation function $f_{ij}^{(0)}(\boldsymbol{k},\unicode[STIX]{x1D714})$ given below by (2.12) for the random flow with a zero mean magnetic field, called background flow.

2.2.3 A model for the background random flow for $Re\ll 1$

In the next step of the derivation, we need a model for the background random flow with a zero mean magnetic field. We consider a random, statistically stationary, density-stratified, inhomogeneous, compressible and helical background flow, which is determined by the following correlation function in Fourier space:

(2.12) $$\begin{eqnarray}\displaystyle f_{ij}^{(0)}(\boldsymbol{k},\unicode[STIX]{x1D714},\boldsymbol{R}) & = & \displaystyle \frac{\unicode[STIX]{x1D6F7}_{u}(\unicode[STIX]{x1D714})E(k)}{8\unicode[STIX]{x03C0}\,k^{2}(1+\unicode[STIX]{x1D70E}_{c})}\left\{\left[\unicode[STIX]{x1D6FF}_{ij}-k_{ij}+\frac{\text{i}}{k^{2}}(k_{j}\unicode[STIX]{x1D706}_{i}-k_{i}\unicode[STIX]{x1D706}_{j})+2\unicode[STIX]{x1D70E}_{c}\,k_{ij}\right.\right.\nonumber\\ \displaystyle & & \displaystyle +\left.\left.(1+2\unicode[STIX]{x1D70E}_{c})\frac{\text{i}}{2k^{2}}(k_{i}\unicode[STIX]{x1D735}_{j}-k_{j}\unicode[STIX]{x1D735}_{i})\right]\overline{\boldsymbol{u}^{2}}-\frac{\text{i}}{k^{2}}\,\unicode[STIX]{x1D700}_{ijp}\,k_{p}\unicode[STIX]{x1D712}\right\}.\end{eqnarray}$$

Here $\unicode[STIX]{x1D740}=-\unicode[STIX]{x1D735}\ln \overline{\unicode[STIX]{x1D70C}}$ characterizes the fluid density stratification, $\unicode[STIX]{x1D6FF}_{ij}$ is the Kronecker tensor, $\unicode[STIX]{x1D700}_{ijn}$ is the fully antisymmetric Levi-Civita tensor, $\unicode[STIX]{x1D712}=\overline{\boldsymbol{u}\boldsymbol{\cdot }(\unicode[STIX]{x1D735}\times \boldsymbol{u})}$ is the kinetic helicity and the parameter

(2.13) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70E}_{c}=\frac{\overline{(\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u})^{2}}}{\overline{(\unicode[STIX]{x1D735}\times \boldsymbol{u})^{2}}} & & \displaystyle\end{eqnarray}$$

is the degree of compressibility of the turbulent velocity field. In (2.12), we neglect higher-order effects ${\sim}O(\unicode[STIX]{x1D706}^{2}\,\overline{\boldsymbol{u}^{2}},\unicode[STIX]{x1D6FB}^{2}\,\overline{\boldsymbol{u}^{2}},\unicode[STIX]{x1D706}_{i}\unicode[STIX]{x1D735}_{i}\overline{\boldsymbol{u}^{2}})$ , i.e. we assume that $\ell _{0}\ll H$ and $\ell _{0}\ll L_{B}$ , where $\ell _{0}$ is the maximum scale of random motions, $H=|\unicode[STIX]{x1D740}|^{-1}$ is the mean density variation scale, which is assumed to be constant. This means that in (2.12) we only take into account leading effects, which are linear in stratification ( $\propto \ell _{0}/H$ ) and inhomogeneity of turbulence ( $\propto \ell _{0}/L_{B}$ ). Generally, stratification also contributes to div  $\boldsymbol{u}$ , and therefore it contributes to the parameter $\unicode[STIX]{x1D70E}_{c}$ . However, this contribution is small [ ${\sim}O(\unicode[STIX]{x1D706}^{2}\,\overline{\boldsymbol{u}^{2}})$ ], and neglected in (2.12). This implies that we separate effects of the arbitrary Mach number, characterized by the parameter $\unicode[STIX]{x1D70E}_{c}$ , and density stratification, described by $\unicode[STIX]{x1D740}$ . Note that the degree of compressibility, $\unicode[STIX]{x1D70E}_{c}$ , depends on the Mach number, but this dependence is not known for arbitrary Mach numbers and has to be determined in direct numerical simulations.

We also do not consider here effects related to the non-uniformity of kinetic helicity ( $\propto \unicode[STIX]{x1D735}\unicode[STIX]{x1D712}$ ) or the combined effect of stratification and kinetic helicity ( $\propto \unicode[STIX]{x1D740}\unicode[STIX]{x1D712}$ ), because their contributions to the mean electromotive force are much smaller in comparison with the standard contributions to the mean electromotive force caused by the turbulent motions (they are of the order of the terms we neglected in the present study). The non-uniformity of kinetic helicity (Yokoi & Yoshizawa Reference Yokoi and Yoshizawa1993; Yokoi & Brandenburg Reference Yokoi and Brandenburg2016; Kleeorin & Rogachevskii Reference Kleeorin and Rogachevskii2018) or the combined effect of stratification and kinetic helicity (Kleeorin & Rogachevskii Reference Kleeorin and Rogachevskii2018) contribute to the generation of large-scale vorticity or large-scale shear motions.

In (2.12), we assume that helical and non-helical parts of random flow have the same power-law spectrum $E(k)=k_{0}^{-1}(q-1)(k/k_{0})^{-q}$ for the wavenumber range $k_{0}<k<k_{\unicode[STIX]{x1D708}}$ , where $k_{\unicode[STIX]{x1D708}}=1/\ell _{\unicode[STIX]{x1D708}}$ is the wavenumber based on the viscous scale $\ell _{\unicode[STIX]{x1D708}}$ , and $k_{0}=1/\ell _{0}\ll k_{\unicode[STIX]{x1D708}}$ . We assume that there are no random motions for $k<k_{0}$ . This implies that $\ell _{0}$ is the maximum scale of random motions, and $E(k)=0$ for $k<k_{0}$ . For simplicity we also assume that compressible and incompressible parts of the random flow have the same spectra. We consider the frequency function $\unicode[STIX]{x1D6F7}_{u}(\unicode[STIX]{x1D714})$ in the form of a Lorentz profile: $\unicode[STIX]{x1D6F7}_{u}(\unicode[STIX]{x1D714})=[\unicode[STIX]{x03C0}\unicode[STIX]{x1D70F}_{0}(\unicode[STIX]{x1D714}^{2}+\unicode[STIX]{x1D70F}_{0}^{-2})]^{-1}$ , where $\unicode[STIX]{x1D70F}_{0}=\ell _{0}/\sqrt{\overline{\boldsymbol{u}^{2}}}$ is the correlation time and $\sqrt{\overline{\boldsymbol{u}^{2}}}$ is the characteristic turbulent velocity at scale $\ell _{0}$ . This model for the frequency function corresponds to the correlation function

(2.14) $$\begin{eqnarray}\displaystyle \overline{u_{i}(t)u_{j}(t+\unicode[STIX]{x1D70F})}\propto \exp (-\unicode[STIX]{x1D70F}/\unicode[STIX]{x1D70F}_{0}). & & \displaystyle\end{eqnarray}$$

We take into account that for small magnetic Reynolds numbers, $\unicode[STIX]{x1D70F}_{0}\gg (\unicode[STIX]{x1D702}k^{2})^{-1}$ for all turbulent scales.

In spite of the large-scale effects caused by stratification $\unicode[STIX]{x1D740}$ and inhomogeneous turbulence $\unicode[STIX]{x1D735}\overline{\boldsymbol{u}^{2}}$ , the model (2.12) describes a weakly anisotropic approximation. Indeed, the contributions of these effects to $f_{ij}^{(0)}$ are small because $\ell _{0}\ll H$ and $\ell _{0}\ll L_{B}$ . Since we consider only linear effects in $\unicode[STIX]{x1D740}$ and $\unicode[STIX]{x1D735}\overline{\boldsymbol{u}^{2}}$ , the second rank tensor $f_{ij}^{(0)}$ is constructed as a linear combination of symmetric tensors with respect to the indexes $i$ and $j$ , $\unicode[STIX]{x1D6FF}_{ij}$ , $k_{ij}$ , non-symmetric tensors: $k_{i}\unicode[STIX]{x1D706}_{j}$ , $k_{j}\unicode[STIX]{x1D706}_{i}$ , $k_{i}\unicode[STIX]{x1D735}_{j}\overline{\boldsymbol{u}^{2}}$ , $k_{j}\unicode[STIX]{x1D735}_{i}\overline{\boldsymbol{u}^{2}}$ and fully antisymmetric tensor $\unicode[STIX]{x1D700}_{ijp}\,k_{p}$ . To determine unknown coefficients multiplying by these tensors, we use the following conditions in the derivation of (2.12):

1. (i) $\int f_{ii}^{(0)}(\boldsymbol{k},\unicode[STIX]{x1D714},\boldsymbol{K})\exp (\text{i}\boldsymbol{K}\boldsymbol{\cdot }\boldsymbol{R})\,\text{d}\boldsymbol{k}\,\text{d}\unicode[STIX]{x1D714}\,\text{d}\boldsymbol{K}=\overline{\boldsymbol{u}^{2}}$ ;

2. (ii) $i\unicode[STIX]{x1D700}_{ipj}\int (-k_{p}+K_{p}/2)f_{ij}^{(0)}(\boldsymbol{k},\unicode[STIX]{x1D714},\boldsymbol{K})\exp (\text{i}\boldsymbol{K}\boldsymbol{\cdot }\boldsymbol{R})\,\text{d}\boldsymbol{k}\,\text{d}\unicode[STIX]{x1D714}\,\text{d}\boldsymbol{K}=\unicode[STIX]{x1D712}$ ;

3. (iii) $f_{ij}^{(0)}(\boldsymbol{k},\unicode[STIX]{x1D714},\boldsymbol{K})=f_{ji}^{\ast (0)}(\boldsymbol{k},\unicode[STIX]{x1D714},\boldsymbol{K})=f_{ji}^{(0)}(-\boldsymbol{k},\unicode[STIX]{x1D714},\boldsymbol{K})$ .

4. (iv) $\int (k_{i}+K_{i}/2)(k_{j}-K_{j}/2)f_{ij}^{(0)}(\boldsymbol{k},\unicode[STIX]{x1D714},\boldsymbol{K})\exp (\text{i}\boldsymbol{K}\boldsymbol{\cdot }\boldsymbol{R})\,\text{d}\boldsymbol{k}\,\text{d}\unicode[STIX]{x1D714}\,\text{d}\boldsymbol{K}=\overline{(\text{div}\,\boldsymbol{u})^{2}}$ ;

5. (v) for very low Mach number and when the parameter $\unicode[STIX]{x1D70E}_{c}$ is very small, the continuity equation can be written in the anelastic approximation, $\text{div}(\overline{\unicode[STIX]{x1D70C}}\,\boldsymbol{u})=0$ , which implies that $(\text{i}k_{i}+\text{i}K_{i}/2-\unicode[STIX]{x1D706}_{i})f_{ij}^{(0)}(\boldsymbol{k},\unicode[STIX]{x1D714},\boldsymbol{K})=0$ and $(-\text{i}k_{j}+\text{i}K_{j}/2-\unicode[STIX]{x1D706}_{j})f_{ij}^{(0)}(\boldsymbol{k},\unicode[STIX]{x1D714},\boldsymbol{K})=0$ .

Different contributions to (2.12) have been discussed by Batchelor (Reference Batchelor1953), Elperin, Kleeorin & Rogachevskii (Reference Elperin, Kleeorin and Rogachevskii1995), Rädler, Kleeorin & Rogachevskii (Reference Rädler, Kleeorin and Rogachevskii2003), Rogachevskii & Kleeorin (Reference Rogachevskii and Kleeorin2018) and Kleeorin & Rogachevskii (Reference Kleeorin and Rogachevskii2018). Note that for a purely potential flow, equation (2.12) for the background turbulence is used in the limit $\unicode[STIX]{x1D70E}_{c}\rightarrow \infty$ , which yields

(2.15) $$\begin{eqnarray}\displaystyle f_{ij}^{(0)}(\boldsymbol{k},\unicode[STIX]{x1D714},\boldsymbol{R})=\frac{\unicode[STIX]{x1D6F7}_{u}(\unicode[STIX]{x1D714})E(k)}{4\unicode[STIX]{x03C0}\,k^{2}}\left[k_{ij}+\frac{\text{i}}{2k^{2}}(k_{i}\unicode[STIX]{x1D735}_{j}-k_{j}\unicode[STIX]{x1D735}_{i})\right]\overline{\boldsymbol{u}^{2}}. & & \displaystyle\end{eqnarray}$$

2.2.4 Mean electromotive force for $Rm\ll 1$

The mean electromotive force is given by

(2.16) $$\begin{eqnarray}\displaystyle \overline{{\mathcal{E}}}_{i} & = & \displaystyle \unicode[STIX]{x1D700}_{inm}\int \overline{b_{m}(\boldsymbol{k},\unicode[STIX]{x1D714})u_{n}(-\boldsymbol{k},-\unicode[STIX]{x1D714})}\,\text{d}\boldsymbol{k}\,\text{d}\unicode[STIX]{x1D714}=\unicode[STIX]{x1D700}_{inm}\int g_{mn}(\boldsymbol{k},\unicode[STIX]{x1D714})\,\text{d}\boldsymbol{k}\,\text{d}\unicode[STIX]{x1D714}.\end{eqnarray}$$

Integrating in $\unicode[STIX]{x1D714}$ space and $\boldsymbol{k}$ space in (2.16) and neglecting higher-order effects ${\sim}O(\unicode[STIX]{x1D706}^{2}\,\overline{\boldsymbol{u}^{2}},\unicode[STIX]{x1D6FB}^{2}\,\overline{\boldsymbol{u}^{2}},\unicode[STIX]{x1D706}_{i}\unicode[STIX]{x1D735}_{i}\overline{\boldsymbol{u}^{2}})$ , we arrive at an equation for the mean electromotive force, $\overline{\boldsymbol{{\mathcal{E}}}}=\overline{\boldsymbol{u}\times \boldsymbol{b}}$ , for compressible, density-stratified, inhomogeneous and helical background turbulence:

(2.17) $$\begin{eqnarray}\displaystyle \overline{\boldsymbol{{\mathcal{E}}}}=\unicode[STIX]{x1D6FC}\overline{\boldsymbol{B}}+\unicode[STIX]{x1D738}\times \overline{\boldsymbol{B}}-\unicode[STIX]{x1D702}_{\text{t}}\unicode[STIX]{x1D735}\times \overline{\boldsymbol{B}}, & & \displaystyle\end{eqnarray}$$

where the $\unicode[STIX]{x1D6FC}$ effect, the turbulent magnetic diffusivity $\unicode[STIX]{x1D702}_{\text{t}}$ , and the pumping velocity $\unicode[STIX]{x1D738}$ for $Rm\ll 1$ are given by

(2.18) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FC} & = & \displaystyle -\frac{(q-1)}{3(q+1)}\,\frac{\unicode[STIX]{x1D70F}_{0}\,\overline{\boldsymbol{u}\boldsymbol{\cdot }(\unicode[STIX]{x1D735}\times \boldsymbol{u})}}{1+\unicode[STIX]{x1D70E}_{c}}\,Rm,\end{eqnarray}$$

(2.19) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D702}_{\text{t}} & = & \displaystyle \frac{(q-1)}{3(q+1)}\,\unicode[STIX]{x1D70F}_{0}\,\overline{\boldsymbol{u}^{2}}\left(1-\frac{2\unicode[STIX]{x1D70E}_{c}}{1+\unicode[STIX]{x1D70E}_{c}}\right)Rm\equiv \frac{(q-1)}{3(q+1)}\left(\frac{1-\unicode[STIX]{x1D70E}_{c}}{1+\unicode[STIX]{x1D70E}_{c}}\right)Rm^{2}\unicode[STIX]{x1D702},\end{eqnarray}$$

(2.20) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D738} & = & \displaystyle -\frac{1}{2}\,\unicode[STIX]{x1D735}\unicode[STIX]{x1D702}_{\text{t}},\end{eqnarray}$$

and $\unicode[STIX]{x1D70F}_{0}=\ell _{0}/\sqrt{\overline{\boldsymbol{u}^{2}}}$ . Let us analyse the obtained results. Since $\unicode[STIX]{x1D70F}_{0}\,Rm=\ell _{0}^{2}/\unicode[STIX]{x1D702}$ , the turbulent transport coefficients given by (2.18)–(2.20) are determined only by the resistive time scale, $\ell _{0}^{2}/\unicode[STIX]{x1D702}$ . The effective pumping velocity $\unicode[STIX]{x1D738}$ of the large-scale magnetic field is independent of $\unicode[STIX]{x1D740}$ (see detailed derivation of (2.20) and discussion in appendix B). We also have shown that there are no contributions of $\unicode[STIX]{x1D740}$ to the effective pumping velocity even for strong stratifications (see appendix B).

Equations (2.18)–(2.20) imply that for small magnetic Reynolds numbers, compressibility effects characterized by the parameter $\unicode[STIX]{x1D70E}_{c}$ decrease the $\unicode[STIX]{x1D6FC}$ effect, turbulent magnetic diffusion, and the effective pumping velocity. However, the total magnetic diffusivity, $\unicode[STIX]{x1D702}+\unicode[STIX]{x1D702}_{\text{t}}$ , cannot be negative, because for $Rm\ll 1$ the magnetic diffusivity, $\unicode[STIX]{x1D702}$ , is much larger than the turbulent value, i.e. $\unicode[STIX]{x1D702}\gg |\unicode[STIX]{x1D702}_{\text{t}}|$ . The decrease of turbulent magnetic diffusivity by compressible flows is consistent with the results obtained by Krause & Rädler (Reference Krause and Rädler1980) and Rädler et al. (Reference Rädler, Brandenburg, Del Sordo and Rheinhardt2011) for homogeneous random flow at small magnetic Reynolds numbers. Detailed comparison with these results is discussed in § 4.

As follows from (2.17)–(2.19), the resulting mean electromotive force $\overline{{\mathcal{E}}}_{i}$ is determined by the isotropic $\unicode[STIX]{x1D6FC}$ effect and turbulent diffusivity. The reason for that is as follows. (i) We consider the case of weak mean magnetic field, i.e. the energy of the mean magnetic field is much smaller than the turbulent kinetic energy. For large mean magnetic field, when the energy of the mean magnetic field is of the order of the turbulent kinetic energy, the $\unicode[STIX]{x1D6FC}$ tensor and turbulent diffusivity are anisotropic (Rogachevskii & Kleeorin Reference Rogachevskii and Kleeorin2000, Reference Rogachevskii and Kleeorin2001). (ii) We do not consider the effects of uniform rotation or large-scale shear. In the case of rotating turbulence (Rädler et al. Reference Rädler, Kleeorin and Rogachevskii2003; Kleeorin & Rogachevskii Reference Kleeorin and Rogachevskii2003) or turbulence with large-scale shear (Rogachevskii & Kleeorin Reference Rogachevskii and Kleeorin2003, Reference Rogachevskii and Kleeorin2004), the $\unicode[STIX]{x1D6FC}$ tensor and turbulent diffusivity are anisotropic. (iii) For anisotropic background turbulence, e.g. for turbulent convection (Kleeorin & Rogachevskii Reference Kleeorin and Rogachevskii2003), or for magnetically driven turbulence with relativistic particles (Rogachevskii et al. Reference Rogachevskii, Kleeorin, Brandenburg and Eichler2012, Reference Rogachevskii, Ruchayskiy, Boyarsky, Fröhlich, Kleeorin, Brandenburg and Schober2017), the $\unicode[STIX]{x1D6FC}$ tensor and turbulent diffusivity are also anisotropic. (iv) The model (2.12) of background random flow is weakly anisotropic, because $\ell _{0}\ll H$ and $\ell _{0}\ll L_{B}$ .

2.3 Large fluid and magnetic Reynolds numbers

In this section we consider the case of large fluid and magnetic Reynolds numbers, so that turbulence is fully developed, the Strouhal number is of the order of unity, and the turbulent correlation time is scale dependent, as in Kolmogorov type turbulence (see, e.g. McComb Reference McComb1990; Frisch Reference Frisch1995; Monin & Yaglom Reference Monin and Yaglom2013). In this case, we perform the Fourier transformation only in $\boldsymbol{k}$ space (but not in $\unicode[STIX]{x1D714}$ space), as is usually done in studies of turbulent transport in a fully developed Kolmogorov-type turbulence. We take into account the nonlinear terms in (2.4) and (2.5) for velocity and magnetic fluctuations and apply the $\unicode[STIX]{x1D70F}$ approach.

The $\unicode[STIX]{x1D70F}$ approach is a universal tool in turbulent transport for strongly nonlinear systems that allow us to obtain closed results and compare them with the results of laboratory experiments, observations and numerical simulations (see, e.g. Orszag Reference Orszag1970; Pouquet, Frisch & Léorat Reference Pouquet, Frisch and Léorat1976; Kleeorin, Rogachevskii & Ruzmaikin Reference Kleeorin, Rogachevskii and Ruzmaikin1990; Rogachevskii & Kleeorin Reference Rogachevskii and Kleeorin2004; Brandenburg & Subramanian Reference Brandenburg and Subramanian2005; Rogachevskii & Kleeorin Reference Rogachevskii and Kleeorin2007; Rogachevskii et al. Reference Rogachevskii, Kleeorin, Käpylä and Brandenburg2011). The $\unicode[STIX]{x1D70F}$ approximation reproduces many well-known phenomena found by other methods in turbulent transport of particles (see, e.g. Elperin et al. Reference Elperin, Kleeorin and Rogachevskii1996, Reference Elperin, Kleeorin and Rogachevskii1997; Pandya & Mashayek Reference Pandya and Mashayek2002; Blackman & Field Reference Blackman and Field2003; Reeks Reference Reeks2005; Sofiev et al. Reference Sofiev, Sofieva, Elperin, Kleeorin, Rogachevskii and Zilitinkevich2009) and magnetic fields, in turbulent convection (see, e.g. Elperin et al. Reference Elperin, Kleeorin, Rogachevskii and Zilitinkevich2002, Reference Elperin, Kleeorin, Rogachevskii and Zilitinkevich2006) and stably stratified turbulent flows (see, e.g. Elperin et al. Reference Elperin, Kleeorin, Rogachevskii and Zilitinkevich2002; Zilitinkevich et al. Reference Zilitinkevich, Elperin, Kleeorin, Lvov and Rogachevskii2009, Reference Zilitinkevich, Elperin, Kleeorin, Rogachevskii and Esau2013) for large fluid and magnetic Reynolds and Péclet numbers. This approach is different from the quasi-linear approach applied in the high-conductivity limit. The latter approach is only valid for small Strouhal numbers, i.e. for very short correlation time. Therefore, the final results found with this approach are different from those obtained using the $\unicode[STIX]{x1D70F}$ approach.

Using (2.5) for the magnetic fluctuations $\boldsymbol{b}$ and the Navier–Stokes equation (2.4) for the velocity fluctuations $\boldsymbol{u}$ in Fourier space, we derive an evolution equation for the cross-helicity tensor,

(2.21) $$\begin{eqnarray}\displaystyle g_{ij}(\boldsymbol{k},\boldsymbol{R},t)=\int \overline{b_{i}(\boldsymbol{k}+\boldsymbol{K}/2,t)u_{j}(-\boldsymbol{k}+\boldsymbol{K}/2,t)}\exp [\text{i}\boldsymbol{K}\boldsymbol{\cdot }\boldsymbol{R}]\,\text{d}\boldsymbol{K}, & & \displaystyle\end{eqnarray}$$

which depends on the velocity correlation function:

(2.22) $$\begin{eqnarray}\displaystyle f_{ij}(\boldsymbol{k},\boldsymbol{R},t)=\int \overline{u_{i}(\boldsymbol{k}+\boldsymbol{K}/2,t)u_{j}(-\boldsymbol{k}+\boldsymbol{K}/2,t)}\exp [\text{i}\boldsymbol{K}\boldsymbol{\cdot }\boldsymbol{R}]\,\text{d}\boldsymbol{K}. & & \displaystyle\end{eqnarray}$$

The evolution equation for the cross-helicity tensor reads:

(2.23) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}g_{ij}(\boldsymbol{k})}{\unicode[STIX]{x2202}t}=\text{i}(\boldsymbol{k}\boldsymbol{\cdot }\overline{\boldsymbol{B}})f_{ij}(\boldsymbol{k})-\text{i}k_{m}\overline{B}_{i}\,f_{mj}(\boldsymbol{k})+I_{ij}(\boldsymbol{k})+g_{ij}^{(\text{N})}(\boldsymbol{k}), & & \displaystyle\end{eqnarray}$$

where, for brevity of notation, we omit the large-scale variables $t$ and $\boldsymbol{R}$ in the functions $f_{ij}(\boldsymbol{k},\boldsymbol{R},t)$ , $g_{ij}(\boldsymbol{k},\boldsymbol{R},t)$ , $g_{ij}^{(\text{N})}(\boldsymbol{k},\boldsymbol{R},t)$ and $\overline{\boldsymbol{B}}(\boldsymbol{R},t)$ . Here the term $g_{ij}^{(\text{N})}$

(2.24) $$\begin{eqnarray}\displaystyle g_{ij}^{(\text{N})}(\boldsymbol{k}) & = & \displaystyle \int \left[\overline{b_{i}(\boldsymbol{k}_{1})\frac{\unicode[STIX]{x2202}u_{j}(\boldsymbol{k}_{2})}{\unicode[STIX]{x2202}t}}+\overline{b_{i}^{(\text{N})}(\boldsymbol{k}_{1})u_{j}(\boldsymbol{k}_{2})}\right]\exp [\text{i}\boldsymbol{K}\boldsymbol{\cdot }\boldsymbol{R}]\,\text{d}\boldsymbol{K}\end{eqnarray}$$

is determined by the third-order moments appearing due to the nonlinear terms. Note that since the Navier–Stokes equation is nonlinear, it appears only in the third-order moments $g_{ij}^{(\text{N})}$ . Here we took into account that the fluid pressure is nonlinear in the fluctuations. For large magnetic and fluid Reynolds numbers, the dissipative terms caused by the kinematic viscosity $\unicode[STIX]{x1D708}$ and magnetic diffusivity $\unicode[STIX]{x1D702}$ in (2.23) are negligibly small in comparison with the nonlinear terms. The term, $I_{ij}$ , which contains the large-scale spatial derivatives of both, the mean magnetic field and the turbulent intensity, is given by

(2.25) $$\begin{eqnarray}\displaystyle I_{ij}(\boldsymbol{k})=\frac{1}{2}\left[(\overline{\boldsymbol{B}}\boldsymbol{\cdot }\unicode[STIX]{x1D735})f_{ij}-k_{n}\left(\frac{\unicode[STIX]{x2202}f_{ij}}{\unicode[STIX]{x2202}k_{s}}\unicode[STIX]{x1D735}_{s}\overline{B}_{n}-\frac{\unicode[STIX]{x2202}f_{nj}}{\unicode[STIX]{x2202}k_{s}}\unicode[STIX]{x1D735}_{s}\overline{B}_{i}\right)-\unicode[STIX]{x1D735}_{n}(\overline{B}_{i}f_{nj})\right]. & & \displaystyle\end{eqnarray}$$

This term determines the turbulent magnetic diffusivity and effects of inhomogeneous turbulence. The derivation of (2.23) is given in appendix C. We consider the case of weak mean magnetic fields, i.e. the energy of the mean magnetic field is much less than the turbulent kinetic energy. This implies that in the present study we do not investigate quenching of the turbulent transport coefficients. Therefore, we do not need evolution equations for the second moments of velocity, $\langle u_{i}u_{j}\rangle$ , and magnetic field, $\langle b_{i}b_{j}\rangle$ (see, e.g. Rogachevskii & Kleeorin Reference Rogachevskii and Kleeorin2000, Reference Rogachevskii and Kleeorin2001, Reference Rogachevskii and Kleeorin2004). This implies that we consider linear effects in the mean magnetic field. The nonlinear mean-field theory for the mean electromotive force in a turbulent compressible fluid flow is the subject of a separate ongoing study.

2.3.1 The $\unicode[STIX]{x1D70F}$ approach

Equation (2.23) for the second moment includes first-order spatial differential operators $\hat{{\mathcal{M}}}$ applied to the third-order moments $F^{(\text{III})}$ . The problem arises in how to close (2.23), i.e. how to express the third-order term $\hat{{\mathcal{M}}}F^{(\text{III})}$ through the lower moments $F^{(\text{II})}$ (McComb Reference McComb1990; Monin & Yaglom Reference Monin and Yaglom2013). We use the spectral $\unicode[STIX]{x1D70F}$ approximation which postulates that the deviations of the third moment terms, $\hat{{\mathcal{M}}}F^{(\text{III})}(\boldsymbol{k})$ , from the contributions to these terms afforded by the background turbulence, $\hat{{\mathcal{M}}}F^{(\text{III},0)}(\boldsymbol{k})$ , can be expressed through similar deviations of the second moments, $F^{(\text{II})}(\boldsymbol{k})-F^{(\text{II},0)}(\boldsymbol{k})$ :

(2.26) $$\begin{eqnarray}\displaystyle \hat{{\mathcal{M}}}F^{(\text{III})}(\boldsymbol{k})-\hat{{\mathcal{M}}}F^{(\text{III},0)}(\boldsymbol{k})=-\frac{1}{\unicode[STIX]{x1D70F}_{r}(k)}[F^{(\text{II})}(\boldsymbol{k})-F^{(\text{II},0)}(\boldsymbol{k})], & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D70F}_{r}(k)$ is the scale-dependent relaxation time, which can be identified with the correlation time $\unicode[STIX]{x1D70F}(k)$ of the turbulent velocity field for large fluid and magnetic Reynolds numbers (Orszag Reference Orszag1970; Pouquet et al. Reference Pouquet, Frisch and Léorat1976; Kleeorin et al. Reference Kleeorin, Rogachevskii and Ruzmaikin1990; Rogachevskii & Kleeorin Reference Rogachevskii and Kleeorin2004). The functions with the superscript $(0)$ correspond to the background turbulence with zero mean magnetic field. We take into account that $g_{ij}^{(0)}=0$ , because when the mean magnetic field is zero, the electromotive force vanishes. We do not take into account magnetic fluctuations caused by a small-scale dynamo (i.e. a dynamo with zero mean magnetic field). Therefore, equation (2.26) for $g_{ij}(\boldsymbol{k})$ reduces to $\hat{{\mathcal{M}}}F_{i}^{(\text{III})}(\boldsymbol{k})=-F_{i}(\boldsymbol{k})/\unicode[STIX]{x1D70F}(k)$ . Validation of the $\unicode[STIX]{x1D70F}$ approximation for different situations has been performed in various numerical simulations and analytical studies (Brandenburg & Subramanian Reference Brandenburg and Subramanian2005; Rogachevskii & Kleeorin Reference Rogachevskii and Kleeorin2007; Rogachevskii et al. Reference Rogachevskii, Kleeorin, Käpylä and Brandenburg2011, Reference Rogachevskii, Kleeorin, Brandenburg and Eichler2012; Brandenburg et al. Reference Brandenburg, Gressel, Käpylä, Kleeorin, Mantere and Rogachevskii2012a ; Käpylä et al. Reference Käpylä, Brandenburg, Kleeorin, Mantere and Rogachevskii2012).

2.3.2 Model of background turbulence for $Re\gg 1$

In the next step of the derivation we need a model for the background turbulence. We use statistically stationary, density-stratified, inhomogeneous, compressible and helical background turbulence, which is determined by the following correlation function in $\boldsymbol{k}$ space:

(2.27) $$\begin{eqnarray}\displaystyle f_{ij}^{(0)}(\boldsymbol{k}) & = & \displaystyle \frac{E(k)}{8\unicode[STIX]{x03C0}\,k^{2}(1+\unicode[STIX]{x1D70E}_{c})}\left\{\left[\unicode[STIX]{x1D6FF}_{ij}-k_{ij}+\frac{\text{i}}{k^{2}}(k_{j}\unicode[STIX]{x1D706}_{i}-k_{i}\unicode[STIX]{x1D706}_{j})+2\unicode[STIX]{x1D70E}_{c}\,k_{ij}\right.\right.\nonumber\\ \displaystyle & & \displaystyle +\left.\left.(1+2\unicode[STIX]{x1D70E}_{c})\frac{\text{i}}{2k^{2}}(k_{i}\unicode[STIX]{x1D735}_{j}-k_{j}\unicode[STIX]{x1D735}_{i})\right]\overline{\boldsymbol{u}^{2}}-\frac{\text{i}}{k^{2}}\,\unicode[STIX]{x1D700}_{ijp}\,k_{p}\unicode[STIX]{x1D712}\right\}.\end{eqnarray}$$

Note that most statements and conditions used for the derivation of the tensorial structure of (2.12) are also valid for (2.27). We assume here that the background turbulence is of Kolmogorov type with constant fluxes of energy and kinetic helicity over the spectrum, i.e. the kinetic energy spectrum for the range of wavenumbers $k_{0}<k<k_{\unicode[STIX]{x1D708}}$ is $E(k)=-\text{d}\bar{\unicode[STIX]{x1D70F}}(k)/\text{d}k$ , the function $\bar{\unicode[STIX]{x1D70F}}(k)=(k/k_{0})^{1-q}$ with $1<q<3$ being the exponent of the kinetic energy spectrum ( $q=5/3$ for a Kolmogorov spectrum). The condition $q>1$ corresponds to finite kinetic energy for very large fluid Reynolds numbers, while $q<3$ corresponds to finite dissipation of the turbulent kinetic energy at the viscous scale (see, e.g. McComb Reference McComb1990; Frisch Reference Frisch1995; Monin & Yaglom Reference Monin and Yaglom2013). The turbulent correlation time in $\boldsymbol{k}$ space is $\unicode[STIX]{x1D70F}(k)=2\unicode[STIX]{x1D70F}_{0}\,\bar{\unicode[STIX]{x1D70F}}(k)$ , where $\unicode[STIX]{x1D70F}_{0}=\ell _{0}/\sqrt{\overline{\boldsymbol{u}^{2}}}$ is the turbulent correlation time in physical space, and $\sqrt{\overline{\boldsymbol{u}^{2}}}$ is the characteristic turbulent velocity at scale $\ell _{0}$ . Note that for fully developed Kolmogorov like turbulence, $\unicode[STIX]{x1D70E}_{c}<1$ (Chassaing et al. Reference Chassaing, Antonia, Anselmet, Joly and Sarkar2013).

2.3.3 Mean electromotive force for $Rm\gg 1$

We use the spectral $\unicode[STIX]{x1D70F}$ approach and assume that the characteristic time of variation of the mean magnetic field $\overline{\boldsymbol{B}}$ is substantially larger than the correlation time $\unicode[STIX]{x1D70F}(k)$ for all turbulence scales. This allows us to get a stationary solution of (2.23). We consider linear effects in the mean magnetic field, so that the function $f_{ij}(\boldsymbol{k})$ in (2.23) and (2.25) should be replaced by $f_{ij}^{(0)}(\boldsymbol{k})$ . The mean electromotive force is determined by $\overline{{\mathcal{E}}}_{i}=\unicode[STIX]{x1D700}_{inm}\int g_{mn}(\boldsymbol{k})\,\text{d}\boldsymbol{k}$ . We take into account that the terms with symmetric tensors with respect to the indexes $m$ and $n$ in $g_{mn}(\boldsymbol{k})$ do not contribute to the mean electromotive force. Therefore, the equation for the mean electromotive force is given by

(2.28) $$\begin{eqnarray}\displaystyle \overline{{\mathcal{E}}}_{m} & = & \displaystyle \unicode[STIX]{x1D700}_{mji}\int \unicode[STIX]{x1D70F}(k)\left\{\text{i}(\boldsymbol{k}\boldsymbol{\cdot }\overline{\boldsymbol{B}})f_{ij}^{(0)}-\overline{B}_{i}\left(\text{i}k_{n}\,f_{nj}^{(0)}+\frac{1}{2}\unicode[STIX]{x1D735}_{n}f_{nj}^{(0)}\right)\right.\nonumber\\ \displaystyle & & \displaystyle -\left.\left[f_{nj}^{(0)}+\frac{k}{2}\left(\frac{\text{d}\ln \unicode[STIX]{x1D70F}}{\text{d}k}\right)k_{np}\,f_{pj}^{(0)}\right]\unicode[STIX]{x1D735}_{n}\overline{B}_{i}\right\}\,\text{d}\boldsymbol{k}.\quad\end{eqnarray}$$

Using the model of background turbulence given by (2.27) and integrating in $\boldsymbol{k}$ -space (2.28), we arrive at equation for the mean electromotive force (2.17), where the $\unicode[STIX]{x1D6FC}$ effect, the turbulent magnetic diffusivity $\unicode[STIX]{x1D702}_{\text{t}}$ and the pumping velocity $\unicode[STIX]{x1D738}$ for large magnetic Reynolds numbers $(Rm\gg 1)$ are given by

(2.29) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FC} & = & \displaystyle -\frac{1}{3}\,\frac{\unicode[STIX]{x1D70F}_{0}\,\overline{\boldsymbol{u}\boldsymbol{\cdot }(\unicode[STIX]{x1D735}\times \boldsymbol{u})}}{1+\unicode[STIX]{x1D70E}_{c}},\end{eqnarray}$$

(2.30) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D702}_{\text{t}} & = & \displaystyle \frac{\unicode[STIX]{x1D70F}_{0}\,\overline{\boldsymbol{u}^{2}}}{3}\left[1-\frac{(q-1)\unicode[STIX]{x1D70E}_{c}}{2(1+\unicode[STIX]{x1D70E}_{c})}\right],\end{eqnarray}$$

(2.31) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D738} & = & \displaystyle -\frac{1}{6}\unicode[STIX]{x1D735}(\unicode[STIX]{x1D70F}_{0}\,\overline{\boldsymbol{u}^{2}}).\end{eqnarray}$$

Equations (2.29) and (2.30) imply that for large magnetic Reynolds numbers, compressibility decreases both the $\unicode[STIX]{x1D6FC}$ effect and turbulent magnetic diffusivity. Since $1<q<3$ , the turbulent magnetic diffusivity is always positive for $Rm\gg 1$ , even for very large compressibility, $\unicode[STIX]{x1D70E}_{c}\gg 1$ (e.g. for irrotational flows). Note that for an incompressible flow the turbulent magnetic diffusivity is independent of $q$ . On the other hand, compressibility of fluid flow does not affect the pumping velocity $\unicode[STIX]{x1D738}$ of the mean magnetic field for $Rm\gg 1$ , similarly to the case of $Rm\ll 1$ .

In the framework of mean-field dynamo theory, the threshold for the generation of a mean magnetic field is formulated in terms of the dynamo number. In the case of an $\unicode[STIX]{x1D6FC}^{2}$ dynamo, the dynamo number is given by $R_{\unicode[STIX]{x1D6FC}}=\unicode[STIX]{x1D6FC}L_{B}/\unicode[STIX]{x1D702}_{\text{t}}$ . Using (2.29) and (2.30), we find that, for a compressible flow, $R_{\unicode[STIX]{x1D6FC}}$ is given by

(2.32) $$\begin{eqnarray}\displaystyle R_{\unicode[STIX]{x1D6FC}}=R_{\unicode[STIX]{x1D6FC}}^{(\text{in})}\left(1+\frac{3-q}{2}\,\unicode[STIX]{x1D70E}_{c}\right)^{-1}, & & \displaystyle\end{eqnarray}$$

where $R_{\unicode[STIX]{x1D6FC}}^{(\text{in})}$ applies to the corresponding incompressible flow. This implies that compressibility decreases the dynamo number.

3.1 Turbulent transport coefficients for $Pe\ll 1$

We derive an equation for the turbulent flux of particles,

(3.4) $$\begin{eqnarray}\displaystyle \overline{n(\boldsymbol{x},t)u_{j}(\boldsymbol{x},t)}=\int F_{j}(\boldsymbol{k},\unicode[STIX]{x1D714},\boldsymbol{R},t)\,\text{d}\unicode[STIX]{x1D714}\,\text{d}\boldsymbol{k}, & & \displaystyle\end{eqnarray}$$

for small Péclet numbers using a quasi-linear approach, where

(3.5) $$\begin{eqnarray}\displaystyle F_{j}(\boldsymbol{k},\unicode[STIX]{x1D714},\boldsymbol{R},t)=\int \overline{n(\boldsymbol{k}_{1},\unicode[STIX]{x1D714}_{1})u_{j}(\boldsymbol{k}_{2},\unicode[STIX]{x1D714}_{2})}\exp [\text{i}\unicode[STIX]{x1D6FA}t+\text{i}\boldsymbol{K}\boldsymbol{\cdot }\boldsymbol{R}]\,\text{d}\unicode[STIX]{x1D6FA}\,\text{d}\boldsymbol{K}. & & \displaystyle\end{eqnarray}$$

For brevity of notations we omit below the large-scale variables $t$ and $\boldsymbol{R}$ in $F_{j}(\boldsymbol{k},\unicode[STIX]{x1D714},\boldsymbol{R},t)$ and $N(\boldsymbol{R},t)$ . We neglect the nonlinear term $\boldsymbol{{\mathcal{Q}}}$ , but keep the molecular diffusion term in (3.3). We rewrite (3.3) in Fourier space and find the solution of this equation (see (A 20) in appendix A). Here we apply the same approach that was used in § 2.2, i.e. we assume that there is a separation of spatial and time scales, $\ell _{0}\ll L_{N}$ and $\unicode[STIX]{x1D70F}_{0}\ll t_{N}$ , where $L_{N}$ and $t_{N}$ are the characteristic spatial and time scales of the mean particle number density variations. We perform calculations presented in appendix A, so that equation for the turbulent flux of particles is given by

(3.6) $$\begin{eqnarray}\displaystyle F_{j} & = & \displaystyle -\int G_{D}\left[N\left(\text{i}k_{i}f_{ij}+\frac{1}{2}\unicode[STIX]{x1D735}_{i}f_{ij}-Dk^{2}G_{D}k_{im}\unicode[STIX]{x1D735}_{m}f_{ij}\right)\right.\nonumber\\ \displaystyle & & \displaystyle +\left.\frac{1}{2}\left(f_{mj}-k_{i}\frac{\unicode[STIX]{x2202}f_{ij}}{\unicode[STIX]{x2202}k_{m}}-2Dk^{2}G_{D}\,k_{im}f_{ij}\right)\unicode[STIX]{x1D735}_{m}N\right]\,\text{d}\boldsymbol{k}\,\text{d}\unicode[STIX]{x1D714},\end{eqnarray}$$

where $G_{D}\equiv G_{D}(\boldsymbol{k},\unicode[STIX]{x1D714})=(D\boldsymbol{k}^{2}+\text{i}\unicode[STIX]{x1D714})^{-1}$ and $f_{ij}=f_{ij}(\boldsymbol{k},\unicode[STIX]{x1D714},\boldsymbol{R},t)$ . Since we consider one way coupling, the correlation function $f_{ij}$ in (3.6) should be replaced by $f_{ij}^{(0)}$ for the background random flow with zero turbulent flux of particles. Using the model of background random flow given by (2.12) with zero kinetic helicity, and integrating in $\unicode[STIX]{x1D714}$ and $\boldsymbol{k}$ -space, we arrive at an equation for the turbulent flux of particles:

(3.7) $$\begin{eqnarray}\displaystyle \boldsymbol{F}=N\,\boldsymbol{V}^{\text{eff}}-D_{\text{t}}\,\unicode[STIX]{x1D735}N, & & \displaystyle\end{eqnarray}$$

where for small Péclet numbers $(Pe\ll 1)$ , the turbulent diffusivity $D_{\text{t}}$ and the effective pumping velocity $\boldsymbol{V}^{\text{eff}}$ are given by

(3.8) $$\begin{eqnarray}\displaystyle D_{\text{t}} & = & \displaystyle \frac{(q-1)}{3(q+1)}\,\unicode[STIX]{x1D70F}_{0}\,\overline{\boldsymbol{u}^{2}}\left(1-\frac{2\unicode[STIX]{x1D70E}_{c}}{1+\unicode[STIX]{x1D70E}_{c}}\right)Pe\equiv \frac{(q-1)}{3(q+1)}\left(\frac{1-\unicode[STIX]{x1D70E}_{c}}{1+\unicode[STIX]{x1D70E}_{c}}\right)Pe^{2}\,D,\end{eqnarray}$$

(3.9) $$\begin{eqnarray}\displaystyle \boldsymbol{V}^{\text{eff}} & = & \displaystyle \frac{(q-1)}{3(q+1)}\,\unicode[STIX]{x1D70F}_{0}\,\overline{\boldsymbol{u}^{2}}\left[\frac{1}{(1+\unicode[STIX]{x1D70E}_{c})}\unicode[STIX]{x1D735}\ln \overline{\unicode[STIX]{x1D70C}}+\frac{3\unicode[STIX]{x1D70E}_{c}}{2(1+\unicode[STIX]{x1D70E}_{c})}\unicode[STIX]{x1D735}\ln \overline{\boldsymbol{u}^{2}}\right]Pe.\end{eqnarray}$$

Since $\unicode[STIX]{x1D70F}_{0}\,Pe=\ell _{0}^{2}/D$ , the turbulent transport coefficients given by (3.8) and (3.9) are determined only by the microphysical diffusion time scale, $\ell _{0}^{2}/D$ . Remarkably, equation (3.8) for the turbulent diffusivity of passive scalars $D_{\text{t}}$ coincides with (2.19) for the turbulent magnetic diffusivity $\unicode[STIX]{x1D702}_{\text{t}}$ after replacing $Pe$ by $Rm$ .

Equation (3.8) implies that for small Péclet numbers, compressibility effects decrease the turbulent diffusivity. However, the total (effective) diffusivity, $D+D_{\text{t}}$ , cannot be negative, because for $Pe\ll 1$ , the molecular diffusivity is much larger than the turbulent one, $D\gg |D_{\text{t}}|$ . This result is consistent with that of Rädler et al. (Reference Rädler, Brandenburg, Del Sordo and Rheinhardt2011), where it has been demonstrated that the total mean-field diffusivity for passive scalar transport in irrotational flows is smaller than the molecular diffusivity (see § 4 for a detailed comparison).

The physics related to different terms in (3.9) will be discussed in the next section. In the present study of turbulent transport of passive scalar and non-inertial particles, we consider non-helical turbulence, because there is no effect of the kinetic helicity on the particle flux – at least in the system investigated here (without rotation, the large-scale shear and neglecting effects ${\sim}O(\unicode[STIX]{x1D706}^{2}\,\overline{\boldsymbol{u}^{2}},\unicode[STIX]{x1D6FB}^{2}\,\overline{\boldsymbol{u}^{2}},\unicode[STIX]{x1D706}_{i}\unicode[STIX]{x1D735}_{j}\overline{\boldsymbol{u}^{2}})$ ).

By means of the equation of state for a perfect gas, $\unicode[STIX]{x1D735}\ln \overline{p}=\unicode[STIX]{x1D735}\ln \overline{\unicode[STIX]{x1D70C}}+\unicode[STIX]{x1D735}\ln \overline{T}$ , we rewrite (3.9) for the effective pumping velocity as:

(3.10) $$\begin{eqnarray}\displaystyle \boldsymbol{V}^{\text{eff}}=\frac{(q-1)}{3(q+1)}\,\unicode[STIX]{x1D70F}_{0}\,\overline{\boldsymbol{u}^{2}}\left[\frac{1}{(1+\unicode[STIX]{x1D70E}_{c})}(\unicode[STIX]{x1D735}\ln \overline{p}-\unicode[STIX]{x1D735}\ln \overline{T})+\frac{3\unicode[STIX]{x1D70E}_{c}}{2(1+\unicode[STIX]{x1D70E}_{c})}\unicode[STIX]{x1D735}\ln \overline{\boldsymbol{u}^{2}}\right]Pe, & & \displaystyle\end{eqnarray}$$

where $\overline{T}$ and $\overline{p}$ are the mean temperature and the mean pressure, respectively. Note that, since the density–temperature correlation $\overline{\unicode[STIX]{x1D70C}^{\prime }\unicode[STIX]{x1D717}}$ is much smaller than $\overline{\unicode[STIX]{x1D70C}}\,\overline{T}$ (Chassaing et al. Reference Chassaing, Antonia, Anselmet, Joly and Sarkar2013), the equation of state for perfect gas is also valid for the mean quantities, where $\unicode[STIX]{x1D70C}^{\prime }$ and $\unicode[STIX]{x1D717}$ are fluctuations of the fluid density and temperature.

3.2 Turbulent transport coefficients for $Pe\gg 1$

In this section we determine the turbulent flux of particles for large Péclet and Reynolds numbers. Similar to the study performed in § 2.3, we consider fully developed turbulence, where the Strouhal number is of the order of unity and the turbulent correlation time is scale dependent, so we apply the Fourier transformation only in $\boldsymbol{k}$ space. Using (3.3) for the fluctuations $n$ and the Navier–Stokes equation for the velocity $\boldsymbol{u}$ written in Fourier space, we derive an equation for the instantaneous two-point correlation function

(3.11) $$\begin{eqnarray}\displaystyle F_{j}(\boldsymbol{k}) & = & \displaystyle \int \overline{n(\boldsymbol{k}+\boldsymbol{K}/2)u_{j}(-\boldsymbol{k}+\boldsymbol{K}/2)}\exp [\text{i}\boldsymbol{K}\boldsymbol{\cdot }\boldsymbol{R}]\,\text{d}\boldsymbol{K}.\end{eqnarray}$$

For brevity of notation we omit the large-scale variables $t$ and $\boldsymbol{R}$ in the function $F_{j}(\boldsymbol{k},\boldsymbol{R},t)$ and the mean number density $N(\boldsymbol{R},t)$ . To derive an evolution equation for $F_{j}(\boldsymbol{k})$ , we perform the calculations presented in appendix C, which yields

(3.12) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}F_{j}}{\unicode[STIX]{x2202}t}=-N\left(\text{i}k_{i}f_{ij}+\frac{1}{2}\unicode[STIX]{x1D735}_{i}f_{ij}\right)-\frac{1}{2}\left(f_{ij}-k_{m}\,\frac{\unicode[STIX]{x2202}f_{mj}}{\unicode[STIX]{x2202}k_{i}}\right)\unicode[STIX]{x1D735}_{i}N+\hat{{\mathcal{M}}}F_{i}^{(\text{III})}(\boldsymbol{k}), & & \displaystyle\end{eqnarray}$$

where

(3.13) $$\begin{eqnarray}\displaystyle \hat{{\mathcal{M}}}F_{j}^{(\text{III})}(\boldsymbol{k}) & = & \displaystyle \int \left[\,\overline{n(\boldsymbol{k}_{1})\frac{\unicode[STIX]{x2202}u_{j}(\boldsymbol{k}_{2})}{\unicode[STIX]{x2202}t}}-\overline{{\mathcal{Q}}(\boldsymbol{k}_{1})u_{j}(\boldsymbol{k}_{2})}\right]\exp [\text{i}\boldsymbol{K}\boldsymbol{\cdot }\boldsymbol{R}]\,\text{d}\boldsymbol{K}\end{eqnarray}$$

are the third-order moment terms in $\boldsymbol{k}$ space appearing due to the nonlinear terms.

We use the spectral $\unicode[STIX]{x1D70F}$ approximation (2.26), where functions with superscript $(0)$ correspond to background turbulence with zero turbulent particle flux. Therefore, equation (2.26) reduces to $\hat{{\mathcal{M}}}F_{i}^{(\text{III})}(\boldsymbol{k})=-F_{i}(\boldsymbol{k})/\unicode[STIX]{x1D70F}(k)$ . We also assume that the characteristic time of variation of the second moment $F_{i}(\boldsymbol{k})$ is substantially larger than the correlation time $\unicode[STIX]{x1D70F}(k)$ on all turbulence scales. Therefore, the particle flux is given by

(3.14) $$\begin{eqnarray}F_{j}(\boldsymbol{k})=-\unicode[STIX]{x1D70F}(k)\left\{\left(\text{i}k_{i}f_{ij}^{(0)}+\frac{1}{2}\unicode[STIX]{x1D735}_{i}f_{ij}^{(0)}\right)N+\left[f_{mj}^{(0)}+\frac{k}{2}\left(\frac{\text{d}\ln \unicode[STIX]{x1D70F}}{\text{d}k}\right)k_{im}\,f_{ij}^{(0)}\right]\unicode[STIX]{x1D735}_{m}N\right\}.\end{eqnarray}$$

Using the model of background turbulence for $f_{ij}^{(0)}$ given by (2.27) with a zero kinetic helicity, and integrating (3.14) in $\boldsymbol{k}$ -space, we obtain the turbulent flux of particles (3.7), where the turbulent diffusivity $D_{\text{t}}$ and the effective pumping velocity $\boldsymbol{V}^{\text{eff}}$ of non-inertial particles for $Pe\gg 1$ are given by

(3.15) $$\begin{eqnarray}\displaystyle D_{\text{t}} & = & \displaystyle \frac{\unicode[STIX]{x1D70F}_{0}\,\overline{\boldsymbol{u}^{2}}}{3}\left[1-\frac{(q-1)\unicode[STIX]{x1D70E}_{c}}{2(1+\unicode[STIX]{x1D70E}_{c})}\right],\end{eqnarray}$$

(3.16) $$\begin{eqnarray}\displaystyle \boldsymbol{V}^{\text{eff}} & = & \displaystyle \frac{\unicode[STIX]{x1D70F}_{0}\,\overline{\boldsymbol{u}^{2}}}{3}\left[\frac{1}{(1+\unicode[STIX]{x1D70E}_{c})}\unicode[STIX]{x1D735}\ln \overline{\unicode[STIX]{x1D70C}}+\frac{\unicode[STIX]{x1D70E}_{c}}{2(1+\unicode[STIX]{x1D70E}_{c})}\unicode[STIX]{x1D735}\ln \overline{\boldsymbol{u}^{2}}\right].\end{eqnarray}$$

Equation (3.15) for the turbulent diffusivity of particles $D_{\text{t}}$ coincides with (2.30) for the turbulent magnetic diffusivity $\unicode[STIX]{x1D702}_{\text{t}}$ after replacing $Pe$ by $Rm$ . Equation (3.15) implies that for large Péclet numbers, compressibility decreases the turbulent diffusivity of particles. Since $1<q<3$ , the turbulent diffusivity is always positive when $Pe\gg 1$ , even for very strong compressibility, $\unicode[STIX]{x1D70E}_{c}\gg 1$ .

Figure 1. Dependence of $V^{\text{eff}}$ (dotted line) and $\unicode[STIX]{x1D6FE}$ (dashed line) (here in the $z$ direction) as a function of the inverse scale height $H$ , normalized by the wavenumber $k_{\text{f}}$ of the energy-carrying eddies. The turbulent pumping velocity $\unicode[STIX]{x1D6FE}$ caused by the non-uniformity of the turbulent r.m.s. velocity (solid line) in a non-stratified turbulence. Here the symbol $H$ is used either for the density scale height, $H=\unicode[STIX]{x1D706}^{-1}$ , or for the characteristic scale of the non-uniformity of the turbulent r.m.s. velocity, $H^{-1}=\unicode[STIX]{x1D735}u_{\text{rms}}/u_{\text{rms}}$ .

Equation (3.16) determines effective pumping velocity, $\boldsymbol{V}^{\text{eff}}$ , of passive scalars and non-inertial particles. The first term in (3.16), and likewise in (3.9) for $Pe\ll 1$ , describes pumping of passive scalars caused by density stratification. This effect for low Mach numbers has been studied theoretically using various analytical approaches (Elperin et al. Reference Elperin, Kleeorin and Rogachevskii1995, Reference Elperin, Kleeorin and Rogachevskii1996, Reference Elperin, Kleeorin and Rogachevskii1997, Reference Elperin, Kleeorin, Rogachevskii and Sokoloff2000, Reference Elperin, Kleeorin, Rogachevskii and Sokoloff2001; Pandya & Mashayek Reference Pandya and Mashayek2002; Reeks Reference Reeks2005; Amir et al. Reference Amir, Bar, Eidelman, Elperin, Kleeorin and Rogachevskii2017). It has also been detected in the direct numerical simulations of Brandenburg, Rädler & Kemel (Reference Brandenburg, Rädler and Kemel2012b ), where the density stratification was caused by gravity. Their result is reproduced in figure 1 and compared with the corresponding turbulent pumping velocities of mean magnetic field, which is close to zero. This demonstrates very clearly the different natures of pumping of particles and magnetic fields in one and the same simulation. Here, the relevant components of the effective pumping velocities of particles, $\boldsymbol{V}^{\text{eff}}$ , and magnetic field, $\unicode[STIX]{x1D738}$ , have been determined using the test-field method for axisymmetric turbulence described in Brandenburg et al. (Reference Brandenburg, Rädler and Kemel2012b ). The simulations have been carried out for approximately 200 turnover times, for $Pe=Re=u_{\text{rms}}/\unicode[STIX]{x1D708}k_{\text{f}}=22$ and $k_{\text{f}}/k_{1}=5$ , where $k_{1}$ is the wavenumber based on the size of the computational domain and $k_{\text{f}}$ is the forcing scale of turbulence. Error bars have been determined using any one third of the full time series of the instantaneous, but spatially averaged mean values of $\boldsymbol{V}^{\text{eff}}$ and $\unicode[STIX]{x1D738}$ . Within error bars, the result for $\unicode[STIX]{x1D738}$ is not quite compatible with zero, but this is entirely explicable as a consequence of a small upward increase in the root-mean-square (r.m.s.) velocity with height and thus a small positive gradient. Also, the deviation of the theoretical from the numerical results for larger $(k_{\text{f}}H)^{-1}$ is caused by the fact that the parameter $\ell _{0}/H$ is no longer very small, as was assumed in the theory. The effective pumping velocity of particles $\boldsymbol{V}^{\text{eff}}$ for arbitrary stratifications has been determined analytically by Amir et al. (Reference Amir, Bar, Eidelman, Elperin, Kleeorin and Rogachevskii2017). They have shown that there is a quenching of the pumping velocity caused by the strong stratification. This tendency is already seen in figure 1 (dotted line) for larger density stratifications.

Let us discuss the physics related to the first term in the right-hand side of (3.16) caused by the fluid density stratification, considering density-stratified homogeneous turbulence. Substituting (3.7) into (3.2), we obtain the equation for the mean number density, $N$ :

(3.17) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}N}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }[N\,\boldsymbol{V}^{\text{eff}}-(D+D_{\text{t}})\unicode[STIX]{x1D735}N]=0. & & \displaystyle\end{eqnarray}$$

The steady-state solution of (3.17) for the mean number density of non-inertial particles is given by $N/N_{0}=[\overline{\unicode[STIX]{x1D70C}}/\overline{\unicode[STIX]{x1D70C}}_{0}]^{D_{\text{t}}/(D+D_{\text{t}})}$ , where subscripts $0$ represent the values in the region with homogeneous fluid density. Here we used (3.8) and (3.9) or (3.15) and (3.16) for homogeneous turbulence. This solution implies that small particles are accumulated in the vicinity of the maximum of the fluid density, because the non-diffusive flux of particles, $N\,\boldsymbol{V}^{\text{eff}}$ , is directed toward the maximum of the fluid density.

The accumulation of non-inertial particles in the vicinity of the maximum of the mean fluid density can be explained as follows (Elperin et al. Reference Elperin, Kleeorin and Rogachevskii1995, Reference Elperin, Kleeorin and Rogachevskii1997; Haugen et al. Reference Haugen, Kleeorin, Rogachevskii and Brandenburg2012). Let us assume that the mean fluid density is inhomogeneous along the $x$ axis, and the mean density $\overline{\unicode[STIX]{x1D70C}}_{2}$ at point $2$ is larger than the mean density $\overline{\unicode[STIX]{x1D70C}}_{1}$ at point $1$ . Consider two small control volumes ‘ $\mathsf{a}$ ’ and ‘ $\mathsf{b}$ ’ located between these two points, and let the direction of the local turbulent velocity in volume ‘ $\mathsf{a}$ ’ at some instant be the same as the direction of the mean fluid density gradient $\unicode[STIX]{x1D735}\,\overline{\unicode[STIX]{x1D70C}}$ (e.g. along the $x$ axis toward point 2). Let the local turbulent velocity in volume ‘ $\mathsf{b}$ ’ at this instant be directed opposite to the mean fluid density gradient (i.e. toward point 1). In a fluid flow with an imposed mean fluid density gradient, one of the sources of particle number density fluctuations, $n\propto -\unicode[STIX]{x1D70F}_{0}\,N(\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u})$ , is caused by a non-zero $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}\approx -\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\ln \overline{\unicode[STIX]{x1D70C}}\not =0$ (see the first term on the right-hand side of (3.3)). Since fluctuations of the fluid velocity $u_{x}$ are positive in volume ‘ $\mathsf{a}$ ’ and negative in volume ‘ $\mathsf{b}$ ’, $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}<0$ in volume ‘ $\mathsf{a}$ ’ and $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}>0$ in volume ‘ $\mathsf{b}$ ’. Therefore, the fluctuations of the particle number density $n\propto -\unicode[STIX]{x1D70F}_{0}\,N(\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u})$ are positive in volume ‘ $\mathsf{a}$ ’ and negative in volume ‘ $\mathsf{b}$ ’. However, the flux of particles $n\,u_{x}$ is positive in volume ‘ $\mathsf{a}$ ’ (i.e. it is directed toward point 2), and it is also positive in volume ‘ $\mathsf{b}$ ’ (because both fluctuations of fluid velocity and number density of particles are negative in volume ‘ $\mathsf{b}$ ’). Therefore, the mean flux of particles $\overline{n\,\boldsymbol{u}}$ is directed, as is the mean fluid density gradient $\unicode[STIX]{x1D735}\,\overline{\unicode[STIX]{x1D70C}}$ , toward point 2. This forms large-scale heterogeneous structures of non-inertial particles in regions with a mean fluid density maximum.

Indirect manifestations of the pumping effect caused by the density stratification through an observed increase of small-scale clustering of particles can be identified in the direct numerical simulations of van Aartrijk & Clercx (Reference van Aartrijk and Clercx2008), who studied the evolution of inertial particles in stably stratified turbulence. This pumping effect increases the small-scale clustering of inertial particles by forming inhomogeneous distributions of the mean particle number density. Note also that tangling of the mean particle number density, $N$ , by compressible velocity fluctuations (described by the source term $-N\,\text{div}\,\boldsymbol{u}$ in (3.3)), can increase the level of particle number density fluctuations, $n$ , when the source term $-N\,\text{div}\,\boldsymbol{u}>0$ . In this case $\unicode[STIX]{x2202}n/\unicode[STIX]{x2202}t>0$ , see (3.3). This can amplify the particle clustering (Eidelman et al. Reference Eidelman, Elperin, Kleeorin, Melnik and Rogachevskii2010; Elperin et al. Reference Elperin, Kleeorin, Liberman and Rogachevskii2013).

Using the equation of state for a perfect gas, we rewrite (3.16) for the pumping effective velocity in the form

(3.18) $$\begin{eqnarray}\displaystyle \boldsymbol{V}^{\text{eff}} & = & \displaystyle \frac{\unicode[STIX]{x1D70F}_{0}\,\overline{\boldsymbol{u}^{2}}}{3}\left[\frac{1}{(1+\unicode[STIX]{x1D70E}_{c})}(\unicode[STIX]{x1D735}\ln \overline{p}-\unicode[STIX]{x1D735}\ln \overline{T})+\frac{\unicode[STIX]{x1D70E}_{c}}{2(1+\unicode[STIX]{x1D70E}_{c})}\unicode[STIX]{x1D735}\ln \overline{\boldsymbol{u}^{2}}\right].\end{eqnarray}$$

The first term on the right-hand side of (3.18) describes turbulent barodiffusion (Elperin et al. Reference Elperin, Kleeorin and Rogachevskii1997), while the second term characterizes the phenomenon of turbulent thermal diffusion (Elperin et al. Reference Elperin, Kleeorin and Rogachevskii1996, Reference Elperin, Kleeorin, Rogachevskii and Sokoloff2000; Amir et al. Reference Amir, Bar, Eidelman, Elperin, Kleeorin and Rogachevskii2017). The last term in (3.18), $\propto \unicode[STIX]{x1D70E}_{c}\unicode[STIX]{x1D735}\overline{\boldsymbol{u}^{2}}$ , describes the compressible turbophoresis for non-inertial particles.

The classical turbophoresis (Caporaloni et al. Reference Caporaloni, Tampieri, Trombetti and Vittori1975; Reeks Reference Reeks1983, Reference Reeks1992; Elperin, Kleeorin & Rogachevskii Reference Elperin, Kleeorin and Rogachevskii1998a ; Mitra, Haugen & Rogachevskii Reference Mitra, Haugen and Rogachevskii2018) exists only for inertial particles and causes an additional mean particle velocity that is proportional to $-\unicode[STIX]{x1D70F}_{s}\unicode[STIX]{x1D735}\overline{\boldsymbol{u}^{2}}$ , where $\unicode[STIX]{x1D70F}_{s}=m_{p}/6\unicode[STIX]{x03C0}\unicode[STIX]{x1D70C}\,\unicode[STIX]{x1D708}a_{p}$ is the particle Stokes time, i.e. the characteristic time of coupling between small spherical particles (of radius $a_{p}$ and mass $m_{p}$ ) and surrounding fluid. To illustrate the classical turbophoresis effect, let us consider small inertial particles in an isotropic incompressible turbulence and use equation of motion for a particle: $\text{d}\boldsymbol{v}^{(\text{p})}/\text{d}t=-(\boldsymbol{v}^{(\text{p})}-\boldsymbol{u})/\unicode[STIX]{x1D70F}_{s}$ , where $\boldsymbol{v}^{(\text{p})}$ is the particle velocity. This equation represents a balance of particle inertia with the fluid drag force produced by the motion of the particle relative to the surrounding fluid. Solution of this equation for small Stokes time is given by: $\boldsymbol{v}^{(\text{p})}=\boldsymbol{u}-\unicode[STIX]{x1D70F}_{s}[\unicode[STIX]{x2202}\boldsymbol{u}/\unicode[STIX]{x2202}t+(\boldsymbol{u}\cdot \unicode[STIX]{x1D735})\boldsymbol{u}]+O(\unicode[STIX]{x1D70F}_{s}^{2})$ (Maxey Reference Maxey1987). Averaging this equation over statistics of fluid velocity fluctuations, we determine the mean particle velocity: $\overline{v}_{j}^{(\text{p})}=-\unicode[STIX]{x1D70F}_{s}\unicode[STIX]{x1D735}_{j}\overline{u_{i}u_{j}}$ . We take into account that for isotropic turbulence $\overline{u_{i}u_{j}}=(1/3)\overline{\boldsymbol{u}^{2}}\unicode[STIX]{x1D6FF}_{ij}$ . Therefore, the mean particle velocity is $\overline{\boldsymbol{v}}^{(\text{p})}=-(\unicode[STIX]{x1D70F}_{s}/3)\unicode[STIX]{x1D735}\overline{\boldsymbol{u}^{2}}$ (for details, see Elperin et al. Reference Elperin, Kleeorin and Rogachevskii1998a ). This turbophoresis effect results in the formation of large-scale clusters of inertial particles at local minima of the mean-squared turbulent velocity. Note that the compressible turbophoresis for non-inertial particles originates from the turbulent particle flux $\overline{n\,\boldsymbol{u}}$ , i.e. it describes a collective statistical phenomenon, while classical turbophoresis is not related to the correlations between velocity and number density fluctuations and is obtained from the expression for the mean velocity of inertial particles.

In the case of rapidly rotating turbulence, another pumping effect of inertial particles arises which can accumulate particles (Hodgson & Brandenburg Reference Hodgson and Brandenburg1998; Elperin, Kleeorin & Rogachevskii Reference Elperin, Kleeorin and Rogachevskii1998b ). This effect can be related to the combined action of particle inertia and non-zero mean kinetic helicity in rotating turbulence.

4.1 Small magnetic Reynolds numbers

Substituting (4.3) into (2.16) and using (2.11), we find that the turbulent magnetic diffusivity for small magnetic Reynolds numbers reads:

(4.8) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D702}_{\text{t}}=\frac{1}{3\unicode[STIX]{x1D702}}(\overline{\unicode[STIX]{x1D74D}^{2}}-\overline{\unicode[STIX]{x1D719}^{2}}). & & \displaystyle\end{eqnarray}$$

This result agrees with that of Krause & Rädler (Reference Krause and Rädler1980). Substituting (4.3) into (3.6), we find that the turbulent diffusion coefficient for passive scalars at small Péclet numbers is given by:

(4.9) $$\begin{eqnarray}\displaystyle D_{\text{t}}=\frac{1}{3D}(\overline{\unicode[STIX]{x1D74D}^{2}}-\overline{\unicode[STIX]{x1D719}^{2}}). & & \displaystyle\end{eqnarray}$$

This result also agrees with that of Rädler et al. (Reference Rädler, Brandenburg, Del Sordo and Rheinhardt2011).

4.2 Large magnetic Reynolds numbers

In the case of large magnetic Reynolds numbers, substituting (4.3) into (2.28), we find for the turbulent magnetic diffusivity

(4.10) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D702}_{\text{t}}=\frac{\unicode[STIX]{x1D70F}_{0}\,\overline{\boldsymbol{u}^{2}}}{3}\left[1-\frac{(q-1)}{2}\left(\frac{\overline{\unicode[STIX]{x1D719}^{2}}}{\overline{\unicode[STIX]{x1D74D}^{2}}+\overline{\unicode[STIX]{x1D719}^{2}}}\right)\right]. & & \displaystyle\end{eqnarray}$$

This result is in good agreement with (2.30) for $\unicode[STIX]{x1D702}_{\text{t}}$ and with (3.15) for $D_{\text{t}}$ .