2.1 The diffusivity–current correlation

The diffusivity–current correlation $\boldsymbol{{\mathcal{J}}}$ from (2.4) is now analysed in detail. Fourier transformed fluctuations of the electric current are

(2.12)$$\begin{eqnarray}\text{curl}_{i}\,\hat{\boldsymbol{B}}=-\frac{\unicode[STIX]{x1D716}_{isp}k_{j}k_{s}}{-\text{i}\unicode[STIX]{x1D714}+\bar{\unicode[STIX]{x1D702}}k^{2}}(\hat{u} _{p}B_{j}+\hat{\unicode[STIX]{x1D702}}(B_{pj}-B_{jp})).\end{eqnarray}$$

Multiplication with the (negative) Fourier transform of the diffusivity fluctuation, $\hat{\unicode[STIX]{x1D702}}$, leads to

(2.13)$$\begin{eqnarray}\hat{{\mathcal{J}}}_{i}=\frac{\unicode[STIX]{x1D716}_{isp}k_{j}k_{s}}{-\text{i}\unicode[STIX]{x1D714}+\bar{\unicode[STIX]{x1D702}}k^{2}}(\hat{U} _{p}B_{j}+\hat{V}(B_{pj}-B_{jp})).\end{eqnarray}$$

Here, $\hat{V}$ is here the spectral function of the autocorrelation function $V=\langle \unicode[STIX]{x1D702}^{\prime }(\boldsymbol{x},t)\unicode[STIX]{x1D702}^{\prime }(\boldsymbol{x}+\unicode[STIX]{x1D743},t+\unicode[STIX]{x1D70F})\rangle$ of the diffusivity fluctuations. Equations (2.10) and (2.13) provide $\boldsymbol{{\mathcal{J}}}=-\unicode[STIX]{x1D6FE}\boldsymbol{g}\times \bar{\boldsymbol{B}}$ with

(2.14)$$\begin{eqnarray}\unicode[STIX]{x1D6FE}=\frac{1}{3}\iint \frac{\bar{\unicode[STIX]{x1D702}}k^{4}u_{1}}{\unicode[STIX]{x1D714}^{2}+\bar{\unicode[STIX]{x1D702}}^{2}k^{4}}\,\text{d}\boldsymbol{k}\,\text{d}\unicode[STIX]{x1D714},\end{eqnarray}$$

representing a turbulent transport of the magnetic background field (‘pumping’) anti-parallel to $\boldsymbol{g}$. For positive $u_{1}$ (i.e. for positive correlation of $\unicode[STIX]{x1D702}^{\prime }$ and $u_{z}^{\prime }$) the pumping goes downwards as $\boldsymbol{g}$ is the vertical unit vector. We note that formally the integral in (2.14) also exists in the high-conductivity limit $\bar{\unicode[STIX]{x1D702}}\rightarrow 0$ so that for small $\bar{\unicode[STIX]{x1D702}}$ it does not depend on the magnetic Reynolds number

(2.15)$$\begin{eqnarray}Rm=\frac{u_{\text{rms}}\ell }{\bar{\unicode[STIX]{x1D702}}},\end{eqnarray}$$

(with $\ell$ as the correlation length) for large $Rm$. In this limit $\unicode[STIX]{x1D6FE}$ is linear in the correlation function $u_{1}$. For small $Rm$ the integral in (2.14) linearly grows with $Rm$, which follows after application of the extremely steep correlation function $\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D714})$ as a proxy of the low-conductivity limit.

On the other hand, the term with $\hat{V}$ in (2.13) leads to

(2.16)$$\begin{eqnarray}\boldsymbol{{\mathcal{J}}}=\cdots +\frac{2}{3}\iint \frac{k^{2}\hat{V}}{-\text{i}\unicode[STIX]{x1D714}+\bar{\unicode[STIX]{x1D702}}k^{2}}\,\text{d}\boldsymbol{k}\,\text{d}\unicode[STIX]{x1D714}\,\text{curl}\,\bar{\boldsymbol{B}},\end{eqnarray}$$

which provides an extra contribution to the magnetic field dissipation. The question is whether this term reduces or enhances the eddy diffusivity $\unicode[STIX]{x1D702}_{\text{t}}$ which is due to the turbulence without $\unicode[STIX]{x1D702}$-fluctuations. For homogeneous turbulence one finds from (2.7)

(2.17)$$\begin{eqnarray}{\mathcal{E}}_{i}=-\unicode[STIX]{x1D716}_{ijp}\iint \left(B_{pn}\hat{Q}_{jn}+\frac{2\bar{\unicode[STIX]{x1D702}}k_{l}k_{m}}{-\text{i}\unicode[STIX]{x1D714}+\bar{\unicode[STIX]{x1D702}}k^{2}}B_{lm}\hat{Q}_{jp}\right)\frac{\text{d}\,\boldsymbol{k}\,\text{d}\unicode[STIX]{x1D714}}{-\text{i}\unicode[STIX]{x1D714}+\bar{\unicode[STIX]{x1D702}}k^{2}}.\end{eqnarray}$$

The spectral tensor $\hat{Q}_{ij}$ for isotropic turbulence is

(2.18)$$\begin{eqnarray}\hat{Q}_{ij}(\boldsymbol{k},\unicode[STIX]{x1D714})=\frac{E(k,\unicode[STIX]{x1D714})}{16\unicode[STIX]{x03C0}k^{2}}\left(\unicode[STIX]{x1D6FF}_{ij}-\frac{k_{i}k_{j}}{k^{2}}\right)-\text{i}\unicode[STIX]{x1D716}_{ijk}k_{k}H(k,\unicode[STIX]{x1D714}),\end{eqnarray}$$

where the positive–definite spectrum $E$ gives the energy

(2.19)$$\begin{eqnarray}u_{\text{rms}}^{2}=\int _{0}^{\infty }\int _{0}^{\infty }E(k,\unicode[STIX]{x1D714})\,\text{d}k\,\text{d}\unicode[STIX]{x1D714}\end{eqnarray}$$

and $H$ is the helical part of the turbulence field. From (2.17) follows $\boldsymbol{{\mathcal{E}}}=-\unicode[STIX]{x1D702}_{\text{t}}\,\text{curl}\,\bar{\boldsymbol{B}}$ with the positive eddy diffusivity

(2.20)$$\begin{eqnarray}\unicode[STIX]{x1D702}_{\text{t}}=\frac{1}{24\unicode[STIX]{x03C0}}\iint \frac{\bar{\unicode[STIX]{x1D702}}E}{\unicode[STIX]{x1D714}^{2}+\bar{\unicode[STIX]{x1D702}}^{2}k^{4}}\,\text{d}\boldsymbol{k}\,\text{d}\unicode[STIX]{x1D714}.\end{eqnarray}$$

For the sum of the turbulence-originated terms in (2.5) one obtains

(2.21)$$\begin{eqnarray}\displaystyle \boldsymbol{{\mathcal{E}}}+\boldsymbol{{\mathcal{J}}} & = & \displaystyle -\left(\frac{1}{24\unicode[STIX]{x03C0}}\iint \frac{\bar{\unicode[STIX]{x1D702}}E}{\unicode[STIX]{x1D714}^{2}+\bar{\unicode[STIX]{x1D702}}^{2}k^{4}}\,\text{d}\boldsymbol{k}\,\text{d}\unicode[STIX]{x1D714}-\frac{2}{3}\iint \frac{\bar{\unicode[STIX]{x1D702}}k^{4}\hat{V}}{\unicode[STIX]{x1D714}^{2}+\bar{\unicode[STIX]{x1D702}}^{2}k^{4}}\,\text{d}\boldsymbol{k}\,\text{d}\unicode[STIX]{x1D714}\right)\text{curl}\,\bar{\boldsymbol{B}}\nonumber\\ \displaystyle & & \displaystyle -\,\frac{1}{3}\iint \frac{\bar{\unicode[STIX]{x1D702}}k^{4}u_{1}}{\unicode[STIX]{x1D714}^{2}+\bar{\unicode[STIX]{x1D702}}^{2}k^{4}}\,\text{d}\boldsymbol{k}\,\text{d}\unicode[STIX]{x1D714}\quad \boldsymbol{g}\times \bar{\boldsymbol{B}},\end{eqnarray}$$

indicating the total turbulent diffusivity as reduced by the conductivity fluctuations. On the other hand, the pumping term in the second line of this equations only exists if these conductivity fluctuations are correlated with the flow component in a preferred direction within the fluid. All terms in (2.21) also exist in the high-conductivity limit, $\bar{\unicode[STIX]{x1D702}}\rightarrow 0$.

The modified eddy diffusivity is

(2.22)$$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x1D702}_{\text{t}}^{\text{eff}}}{\bar{\unicode[STIX]{x1D702}}}=\frac{1}{24\unicode[STIX]{x03C0}}\iint \frac{E}{\unicode[STIX]{x1D714}^{2}+\bar{\unicode[STIX]{x1D702}}^{2}k^{4}}\,\text{d}\boldsymbol{k}\,\text{d}\unicode[STIX]{x1D714}-\frac{2}{3}\iint \frac{k^{4}\hat{V}}{\unicode[STIX]{x1D714}^{2}+\bar{\unicode[STIX]{x1D702}}^{2}k^{4}}\,\text{d}\boldsymbol{k}\,\text{d}\unicode[STIX]{x1D714} & & \displaystyle\end{eqnarray}$$

(see Krause & Roberts Reference Krause and Roberts1973). For large $Rm$ both terms grow linearly with $Rm$ while for small $Rm$ both terms formally grow with $Rm^{2}$. If the second expression is considered a function of $\unicode[STIX]{x1D702}_{\text{rms}}/\bar{\unicode[STIX]{x1D702}}$ then it runs with $1/Rm$ for large $Rm$ and with $Rm^{0}$ for small $Rm$. As it should, the reduction of the eddy diffusivity by conductivity fluctuations disappears in the high-conductivity limit.

Discussing possible dynamo effects in hot Jupiter atmospheres, Rogers & McElwaine (Reference Rogers and McElwaine2017) considered variable molecular diffusivities which form patterns in the vertical direction and the horizontal plane. In the horizontal plane the quasi-two-dimensional velocity field existed without being correlated with the diffusivity. In consequence, the effective magnetic diffusivity is also reduced as in (2.22) but the pumping term (2.14) does not appear.

2.2 Direct numerical simulations

To test theoretical predictions, we run fully nonlinear numerical simulations with the Pencil CodeFootnote ¹. We solved the equations of compressible magnetohydrodynamics

(2.23)$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\boldsymbol{A}}{\unicode[STIX]{x2202}t}=\boldsymbol{u}\times \boldsymbol{B}-(\bar{\unicode[STIX]{x1D702}}+\unicode[STIX]{x1D702}^{\prime })\unicode[STIX]{x1D707}_{0}\,\boldsymbol{J}+\boldsymbol{{\mathcal{E}}}_{0}, & \displaystyle\end{eqnarray}$$

(2.24a,b)$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{D}\ln \unicode[STIX]{x1D70C}}{\text{D}t}=-\text{div}\,\boldsymbol{u},\quad \frac{\text{D}\boldsymbol{u}}{\text{D}t}=-c_{\text{s}}^{2}\ln \unicode[STIX]{x1D70C}+\boldsymbol{F}^{\text{visc}}+\boldsymbol{F}^{\text{force}}, & \displaystyle\end{eqnarray}$$

where $\boldsymbol{A}$ is the magnetic vector potential and $\boldsymbol{B}=\text{curl}\,\boldsymbol{A}$ is the magnetic field, $\boldsymbol{J}=\unicode[STIX]{x1D707}_{0}^{-1}\text{curl}\,\boldsymbol{B}$ is the current density, $\text{D}/\text{D}t=\unicode[STIX]{x2202}/\unicode[STIX]{x2202}t+\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}$ is the advective time derivative, $\unicode[STIX]{x1D70C}$ is the density and $c_{\text{s}}$ is the constant speed of sound. The last term of (2.23) describes an imposed EMF $\boldsymbol{{\mathcal{E}}}_{0}=\hat{{\mathcal{E}}}_{0}\sin (k_{1}x)\hat{\boldsymbol{e}}_{z}$, that is used to introduce a large-scale magnetic field $\bar{B}_{y}(x)$ to the system. Furthermore, the fluctuating component of the magnetic diffusivity is given by $\unicode[STIX]{x1D702}^{\prime }=c_{u}u_{z}$, where $c_{u}$ is used to control the strength of the correlation. We use $\unicode[STIX]{x1D702}_{\text{rms}}=c_{u}u_{z,\text{rms}}$ to quantify the amplitude of the fluctuating part of the diffusivity.

The viscous force is given by the standard expression

(2.25)$$\begin{eqnarray}\displaystyle \boldsymbol{F}^{\text{visc}}=\unicode[STIX]{x1D708}\left(\unicode[STIX]{x1D6FB}^{2}\boldsymbol{u}+{\textstyle \frac{1}{3}}\unicode[STIX]{x1D735}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}\right), & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D708}$ is the kinematic viscosity. The fluid is forced with an external body force $\boldsymbol{F}^{\text{force}}(\boldsymbol{x},t)=Re\{N\boldsymbol{f}_{\boldsymbol{k}(t)}\exp [\text{i}\boldsymbol{k}(t)\boldsymbol{\cdot }\boldsymbol{x}-\text{i}\unicode[STIX]{x1D719}(t)]\}$, where $\boldsymbol{x}$ is the position vector, $N=f_{0}c_{\text{s}}(kc_{\text{s}}/\unicode[STIX]{x1D6FF}t)^{1/2}$ is a normalization factor where $f_{0}$ is the non-dimensional amplitude, $k=|\boldsymbol{k}|$, $\unicode[STIX]{x1D6FF}t$ is the length of the time step and $-\unicode[STIX]{x03C0}<\unicode[STIX]{x1D719}(t)<\unicode[STIX]{x03C0}$ is a random delta-correlated phase. The vector $\boldsymbol{f}_{\boldsymbol{k}}$ describes non-helical transversal waves.

The simulation domain is a fully periodic cube with volume $(2\unicode[STIX]{x03C0})^{3}$. The units of length and time are $[x]=k_{1}^{-1},[t]=(c_{\text{s}}k_{1})^{-1}$ where $k_{1}$ is the wavenumber corresponding to the system size. The simulations are characterized by the magnetic Reynolds number (2.15) with $u_{\text{rms}}$ volume averaged and $\ell =(k_{\text{f}})^{-1}$. The flows under consideration are weakly compressible with Mach number $Ma=u_{\text{rms}}/c_{\text{s}}\approx 0.1$. All of the simulations use $k_{\text{f}}/k_{1}=30$ and a grid resolution of $288^{3}$.

We first run the simulations with each $Rm$ with $\unicode[STIX]{x1D702}^{\prime }=0$ sufficiently long that a stationary large-scale magnetic field $\bar{B}_{y}(x)$ due to the imposed $\boldsymbol{{\mathcal{E}}}_{0}$ is established. The amplitude of the resulting magnetic field is typically of the order of $10^{-3}$ of the equipartition strength such that its influence on the flow is negligible. Then we branch new simulations from snapshots of these runs with different levels of diffusivity fluctuations $\unicode[STIX]{x1D702}^{\prime }$ and switch off the imposed EMF, i.e. $\boldsymbol{{\mathcal{E}}}_{0}=0$. Without the EMF the large-scale magnetic field decays. Measuring the decay rate of the magnetic field, the effective turbulent diffusion can be computed. At least five decay experiments with each value of $Rm$ and $\unicode[STIX]{x1D702}^{\prime }$ were made and the averaged decay rate was used in the computation of $\unicode[STIX]{x1D702}_{\text{t}}^{\text{eff}}/\bar{\unicode[STIX]{x1D702}}$. The error bars in figure 1 indicate the standard deviation divided by the square root of the number of experiments.

Figure 1. The specific diffusivity $\unicode[STIX]{x1D702}_{\text{t}}^{\text{eff}}/\bar{\unicode[STIX]{x1D702}}$ (a) and the eddy diffusivity ratio $\unicode[STIX]{x1D702}_{\text{t}}^{\text{eff}}/\unicode[STIX]{x1D702}_{\text{t}}$ (b) as functions of the normalized diffusivity fluctuation $\unicode[STIX]{x1D702}_{\text{rms}}/\bar{\unicode[STIX]{x1D702}}$. In the high-conductivity limit ($Rm\gg 1$) the influence of the conductivity fluctuations disappears.

Going back to (2.22), we note that if the second expression is considered then its argument $(\unicode[STIX]{x1D702}_{\text{rms}}/\bar{\unicode[STIX]{x1D702}})^{2}$ must be multiplied with $1/Rm$ for large $Rm$ and with $Rm^{0}$ for small $Rm$. It is thus clear that the diffusivity reduction by conductivity fluctuations disappears in the high-conductivity limit, which is confirmed by the numerical results, see figure 1(a). Figure 1(b) shows the numerical results for the ratio $\unicode[STIX]{x1D702}_{\text{t}}^{\text{eff}}/\unicode[STIX]{x1D702}_{\text{t}}$ of the terms in (2.22) which, of course, is unity for vanishing $\unicode[STIX]{x1D702}_{\text{rms}}$. It is also unity for large $Rm$ as the $\unicode[STIX]{x1D702}$-fluctuation-induced second term in (2.22) vanishes with $1/Rm$. Its role, however, becomes more important for small $Rm$. In this case, the first term loses its dominance and the total diffusivity $\unicode[STIX]{x1D702}_{\text{t}}^{\text{eff}}$ is reduced. If the numbers of figure 1(b) are multiplied with $\unicode[STIX]{x1D702}_{\text{t}}/\bar{\unicode[STIX]{x1D702}}$ then panel (a) results where the total magnetic diffusivity normalized with the microscopic value $\bar{\unicode[STIX]{x1D702}}$ is given. The influence of the conductivity fluctuations vanishes for large $Rm$ while the fluctuations provide smaller effective diffusivities $\unicode[STIX]{x1D702}_{\text{t}}^{\text{eff}}$ so that the cycle frequencies of oscillating dynamo models are reduced (Roberts Reference Roberts1972), also characteristic growth and decay times become longer.

3.1 Quasilinear approximation

We start with (2.12) and include the influence of rotation by the transformation $\hat{u} _{p}=D_{pq}\hat{u} _{q}$ with the rotation operator

(3.1)$$\begin{eqnarray}D_{ij}=\unicode[STIX]{x1D6FF}_{ij}+\frac{(2\boldsymbol{k}\boldsymbol{\cdot }\unicode[STIX]{x1D734}/k)}{-\text{i}\unicode[STIX]{x1D714}+\unicode[STIX]{x1D708}k^{2}}\unicode[STIX]{x1D716}_{ijp}\frac{k_{p}}{k}\end{eqnarray}$$

in the linear approximation (Kitchatinov, Pipin & Rüdiger Reference Kitchatinov, Pipin and Rüdiger1994). The second term gives the influence of the basic rotation in the Fourier representation. As it should be, it is even in the wavenumber and odd in the angular velocity. The Levi-Civita tensor ensures that the term is invariant with respect to the transformation of the coordinate system. It follows that $\hat{{\mathcal{J}}}_{i}=\unicode[STIX]{x1D716}_{isp}k_{j}k_{s}D_{pq}\hat{U} _{q}B_{j}/(-\text{i}\unicode[STIX]{x1D714}+\bar{\unicode[STIX]{x1D702}}k^{2})+\cdots$, and finally

(3.2)$$\begin{eqnarray}\boldsymbol{{\mathcal{J}}}=-\unicode[STIX]{x1D6FE}\boldsymbol{g}\times \boldsymbol{B}+\unicode[STIX]{x1D6FC}(4(\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D734})\boldsymbol{g}-(\boldsymbol{g}\boldsymbol{\cdot }\boldsymbol{B})\unicode[STIX]{x1D734}-(\boldsymbol{g}\boldsymbol{\cdot }\unicode[STIX]{x1D734})\boldsymbol{B}),\end{eqnarray}$$

where $\unicode[STIX]{x1D6FE}$ is given by (2.14) and for the coefficient $\unicode[STIX]{x1D6FC}$ (related but not identical to the tensor $\unicode[STIX]{x1D6FC}$ in (2.8)) one finds

(3.3)$$\begin{eqnarray}\unicode[STIX]{x1D6FC}=\frac{2}{15}\iint \frac{(\unicode[STIX]{x1D708}\bar{\unicode[STIX]{x1D702}}k^{4}-\unicode[STIX]{x1D714}^{2})k^{2}u_{1}}{(\unicode[STIX]{x1D714}^{2}+\bar{\unicode[STIX]{x1D702}}^{2}k^{4})(\unicode[STIX]{x1D714}^{2}+\unicode[STIX]{x1D708}^{2}k^{4})}\,\text{d}\boldsymbol{k}\,\text{d}\unicode[STIX]{x1D714}.\end{eqnarray}$$

For $\unicode[STIX]{x1D708}=\bar{\unicode[STIX]{x1D702}}$ and for frequency spectra which monotonically decrease for increasing $\unicode[STIX]{x1D714}$ the frequency integral in (3.3) has the same sign as $u_{1}$ while it vanishes a for a spectrum (‘white noise’) which does not depend on the frequency $\unicode[STIX]{x1D714}$. Correlations of a white-noise spectrum possess zero correlation times so that, indeed, the rotational influence should vanish. The $\unicode[STIX]{x1D6FC}$ effect after (3.2) is highly anisotropic, its last term is the rotation-induced standard $\unicode[STIX]{x1D6FC}$ expression.

Both quantities $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FE}$ are linearly running with the ratio $\unicode[STIX]{x1D702}_{\text{rms}}/\bar{\unicode[STIX]{x1D702}}$. In the low-conductivity limit ($Rm<1$) they are

(3.4a,b)$$\begin{eqnarray}\frac{\unicode[STIX]{x1D6FE}}{u_{\text{rms}}}\simeq \frac{\unicode[STIX]{x1D702}_{\text{rms}}}{\bar{\unicode[STIX]{x1D702}}},\quad \frac{\unicode[STIX]{x1D6FE}}{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FA}}\simeq \frac{1}{Rm~(\unicode[STIX]{x1D70F}_{\text{corr}}\unicode[STIX]{x1D6FA})}\end{eqnarray}$$

while in the high-conductivity limit ($Rm>1$)

(3.5a,b)$$\begin{eqnarray}\frac{\unicode[STIX]{x1D6FE}}{u_{\text{rms}}}\simeq \frac{\unicode[STIX]{x1D702}_{\text{rms}}}{\bar{\unicode[STIX]{x1D702}}}\frac{1}{Rm},\quad \frac{\unicode[STIX]{x1D6FE}}{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FA}}\simeq \frac{1}{\unicode[STIX]{x1D70F}_{\text{corr}}\unicode[STIX]{x1D6FA}}.\end{eqnarray}$$

Both relations for the $\unicode[STIX]{x1D6FC}$ terms taken for all $Pm=\unicode[STIX]{x1D708}/\bar{\unicode[STIX]{x1D702}}\leqslant 1$. The $\unicode[STIX]{x1D6FC}$ effect always needs rotation; both of the given coefficients are small. The dimensionless ratio $\hat{\unicode[STIX]{x1D6FE}}=\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FA}$ of the pumping term $\unicode[STIX]{x1D6FE}$ and the $\unicode[STIX]{x1D6FC}$ effect indicate the ratio of off-diagonal and diagonal elements in the alpha tensor. For $\hat{\unicode[STIX]{x1D6FE}}>1$, dynamo operation can highly be disturbed. For a standard disk dynamo Rüdiger, Elstner & Schultz (Reference Rüdiger, Elstner and Schultz1993) demonstrated with numerical simulations that large values of $|\hat{\unicode[STIX]{x1D6FE}}|$ suppress the dynamo action. In spherical dynamo models the $\unicode[STIX]{x1D738}$ term plays the role of an upward buoyancy (Moss, Tuominen & Brandenburg Reference Moss, Tuominen and Brandenburg1990) or even a strong downward turbulent pumping (Brandenburg, Moss & Tuominen Reference Brandenburg, Moss, Tuominen and Harvey1992). In order to be relevant for dynamo excitation, the $\unicode[STIX]{x1D6FC}$ effect should numerically exceed the value $\unicode[STIX]{x1D6FE}$ of the pumping term. As the pumping effect exists even for $\unicode[STIX]{x1D6FA}=0$, the ratio $\hat{\unicode[STIX]{x1D6FE}}$ should decrease for faster rotation. With extensive numerical simulations Gressel et al. (Reference Gressel, Ziegler, Elstner and Rüdiger2008) derived values of order unity for interstellar turbulence driven by collective supernova explosions. For rotating magnetoconvection Ossendrijver, Stix & Brandenburg (Reference Ossendrijver, Stix and Brandenburg2001), Ossendrijver et al. (Reference Ossendrijver, Stix, Brandenburg and Rüdiger2002) also found $\hat{\unicode[STIX]{x1D6FE}}\simeq 1$, where both $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FE}$ reached approximately 10 % of the root-mean-square value of the convective velocity. In their simulations of turbulent magnetoconvection Käpylä, Korpi & Brandenburg (Reference Käpylä, Korpi and Brandenburg2009) also reached typical values of order unity for $\hat{\unicode[STIX]{x1D6FE}}$.

3.2 Turbulent transport of electric current

We shall demonstrate why the existence of a diamagnetic pumping and an $\unicode[STIX]{x1D6FC}$ effect for rotating but unstratified fluids with fluctuating diffusivity (in a fixed direction) is not too surprising. We start with the flow–current correlation $\langle \boldsymbol{u}^{\prime }\boldsymbol{\cdot }\text{curl}\,\boldsymbol{B}^{\prime }\rangle$ describing a turbulent transport of electric current fluctuations which after (2.12) for non-rotating turbulence certainly vanishes. This is not true for rotating turbulence as $\langle \boldsymbol{u}^{\prime }\boldsymbol{\cdot }\text{curl}\,\boldsymbol{B}^{\prime }\rangle \propto \bar{\boldsymbol{B}}\boldsymbol{\cdot }\unicode[STIX]{x1D734}$ is a possible construction for isotropic turbulence fields which only vanishes for $\bar{\boldsymbol{B}}\bot \unicode[STIX]{x1D734}$. Moreover, the tensor $\langle u_{i}^{\prime }\text{curl}_{j}\,\boldsymbol{B}^{\prime }\rangle$ for rotating isotropic turbulence may be written as

(3.6)$$\begin{eqnarray}\langle u_{i}^{\prime }\,\text{curl}_{j}\,\boldsymbol{B}^{\prime }\rangle =\unicode[STIX]{x1D705}_{1}\unicode[STIX]{x1D6FA}_{i}\bar{B}_{j}+\unicode[STIX]{x1D705}_{2}\unicode[STIX]{x1D6FA}_{j}\bar{B}_{i}+\unicode[STIX]{x1D705}_{3}(\unicode[STIX]{x1D734}\boldsymbol{\cdot }\bar{\boldsymbol{B}})\unicode[STIX]{x1D6FF}_{ij}.\end{eqnarray}$$

In opposition to the tensors forming the helicity, the current helicity and the cross-helicity, the tensor (3.6) is not a pseudo-tensor and there is no reason that the dimensionless coefficients $\unicode[STIX]{x1D705}_{i}$ identically vanish. The correlation $\langle u_{r}^{\prime }\,\text{curl}_{\unicode[STIX]{x1D719}}\,\boldsymbol{B}^{\prime }\rangle$ describes the up- or downward flow of azimuthal electric current fluctuations in a rotating magnetized turbulence. Imagine that $u_{r}^{\prime }$ is correlated (or anticorrelated) with fluctuations $\unicode[STIX]{x1D702}^{\prime }$ of the magnetic diffusivity, i.e. $\langle u_{r}^{\prime }\,\text{curl}_{\unicode[STIX]{x1D719}}\,\boldsymbol{B}^{\prime }\rangle \propto \langle \unicode[STIX]{x1D702}^{\prime }\,\text{curl}_{\unicode[STIX]{x1D719}}\,\boldsymbol{B}^{\prime }\rangle$ which is proportional to ${\mathcal{J}}_{\unicode[STIX]{x1D719}}$. If this quantity occurs for rotating turbulence under the influence of an azimuthal magnetic background $\bar{B}_{\unicode[STIX]{x1D719}}$ field then the existence of a new $\unicode[STIX]{x1D6FC}$ effect has been proven.

The calculation on the basis of (2.7) and (3.1) for rotating and magnetized but otherwise isotropic turbulence leads to the tensor expression

(3.7)$$\begin{eqnarray}\langle u_{i}^{\prime }\,\text{curl}_{j}\,\boldsymbol{B}^{\prime }\rangle =\unicode[STIX]{x1D705}(\unicode[STIX]{x1D6FA}_{i}\bar{B}_{j}+\unicode[STIX]{x1D6FA}_{j}\bar{B}_{i}-4(\unicode[STIX]{x1D734}\boldsymbol{\cdot }\bar{\boldsymbol{B}})\unicode[STIX]{x1D6FF}_{ij}),\end{eqnarray}$$

which is symmetric in its indices. One finds

(3.8)$$\begin{eqnarray}\unicode[STIX]{x1D705}=\frac{1}{30}\iint \frac{(\unicode[STIX]{x1D708}\bar{\unicode[STIX]{x1D702}}k^{4}-\unicode[STIX]{x1D714}^{2})k^{2}E}{(\unicode[STIX]{x1D714}^{2}+\bar{\unicode[STIX]{x1D702}}^{2}k^{4})(\unicode[STIX]{x1D714}^{2}+\unicode[STIX]{x1D708}^{2}k^{4})}\,\text{d}k\,\text{d}\unicode[STIX]{x1D714}.\end{eqnarray}$$

The dimensionless $\unicode[STIX]{x1D705}$ is almost identical to the integral (3.3); it is also positive for monotonically decreasing frequency spectra (at least for $\unicode[STIX]{x1D708}=\bar{\unicode[STIX]{x1D702}}$). For $\bar{\unicode[STIX]{x1D702}}\rightarrow 0$ one formally finds for the integrals $\unicode[STIX]{x1D705}\simeq St^{2}/15$ where the Strouhal number $St=u_{\text{rms}}\unicode[STIX]{x1D70F}_{\text{corr}}/\ell$, with $\ell$ being the correlation length. In the low-conductivity limit it runs with $Rm^{2}$.

On the other hand, without rotation the tensor (3.7) of the homogeneous turbulence can simply be written as $\langle u_{i}^{\prime }\,\text{curl}_{j}\,\boldsymbol{B}^{\prime }\rangle =\unicode[STIX]{x1D705}^{\prime }\unicode[STIX]{x1D716}_{jik}\bar{B}_{k}$. As it should, the tensor is invariant against the simultaneous transformation $i\rightarrow j$ and $\bar{B}_{k}\rightarrow -\bar{B}_{k}$. Then $\langle (\boldsymbol{g}\boldsymbol{\cdot }\boldsymbol{u}^{\prime })\text{curl}\,\boldsymbol{B}^{\prime }\rangle =\unicode[STIX]{x1D705}^{\prime }\boldsymbol{g}\times \bar{\boldsymbol{B}}$ for all directions $\boldsymbol{g}$, hence

(3.9)$$\begin{eqnarray}\langle u_{r}^{\prime }\text{curl}_{\unicode[STIX]{x1D703}}\,\boldsymbol{B}^{\prime }\rangle =-\unicode[STIX]{x1D705}^{\prime }\bar{B}_{\unicode[STIX]{x1D719}}\end{eqnarray}$$

for azimuthal background fields. After the heuristic replacement of $u_{r}^{\prime }$ by $\unicode[STIX]{x1D702}^{\prime }$, $\unicode[STIX]{x1D705}^{\prime }$ in (3.9) stands for the new pumping term discussed above. The coefficient

(3.10)$$\begin{eqnarray}\unicode[STIX]{x1D705}^{\prime }=\frac{1}{15}\iint \frac{\bar{\unicode[STIX]{x1D702}}k^{4}E}{\unicode[STIX]{x1D714}^{2}+\bar{\unicode[STIX]{x1D702}}^{2}k^{4}}\,\text{d}k\,\text{d}\unicode[STIX]{x1D714},\end{eqnarray}$$

which is of the dimension of the inverse of the correlation time, is positive–definite. In the formal limit $\bar{\unicode[STIX]{x1D702}}\rightarrow 0$ the integral yields $\unicode[STIX]{x1D705}^{\prime }\simeq (2/15)St^{2}/\unicode[STIX]{x1D70F}_{\text{corr}}$ whereas in the low-conductivity limit it runs with $Rm$. We shall further demonstrate by numerical simulations that the correlations (3.7) and (3.9) indeed exist and that the coefficients $\unicode[STIX]{x1D705}$ and $\unicode[STIX]{x1D705}^{\prime }$ are positive.

3.3 Rotating magnetoconvection

A nonlinear numerical simulation with an existing code demonstrates the existence of the scalar quantities $\unicode[STIX]{x1D705}^{\prime }$ and $\unicode[STIX]{x1D705}$ and, therefore, the existence of the pumping term (2.14) and the new $\unicode[STIX]{x1D6FC}$ effect. To this end the correlations $\langle u_{r}^{\prime }\text{curl}_{\unicode[STIX]{x1D703}}\,\boldsymbol{B}^{\prime }\rangle$ and $\langle u_{r}^{\prime }\text{curl}_{\unicode[STIX]{x1D719}}\,\boldsymbol{B}^{\prime }\rangle$ are calculated without and with rotation, yielding $\unicode[STIX]{x1D705}^{\prime }$ and $\unicode[STIX]{x1D705}$. As the latter correlation needs global rotation to exist, $\unicode[STIX]{x1D705}$ and, therefore, the $\unicode[STIX]{x1D6FC}$ effect, also need global rotation to exist.

A convectively unstable Cartesian box penetrated by an azimuthal magnetic field (fulfilling pseudo-vacuum boundary conditions at the top and bottom of the box) is considered with both density and temperature stratifications of (only) 10 %. A detailed description of the magnetoconvection code has been published earlier (Rüdiger & Küker Reference Rüdiger and Küker2016). The box is flat: two units in the vertical direction and four units in the two horizontal directions, there are $128\times 256\times 256$ grid points. In code units the molecular diffusivity is $\unicode[STIX]{x1D702}\simeq 6\times 10^{-3}$ and the resulting turbulence intensity $u_{\text{rms}}\simeq 0.7$. The convection cells are characterized by $\unicode[STIX]{x1D70F}_{\text{corr}}\simeq 0.6$, hence $Rm\lesssim 50$. The values are not varied for the various simulation runs. After the definitions the magnetic field $B_{\unicode[STIX]{x1D719}}=1$ would take 40 % of the equipartition value $B_{\text{eq}}=\sqrt{\unicode[STIX]{x1D707}_{0}\unicode[STIX]{x1D70C}}u_{\text{rms}}$.

Figure 2(a) gives the results of a numerical simulation for a non-rotating box penetrated by an azimuthal magnetic field. We find $\unicode[STIX]{x1D705}^{\prime }>0$ in accordance with the result (3.10) obtained within the quasi-linear approximation. If additionally $u_{r}^{\prime }$ and $\unicode[STIX]{x1D702}^{\prime }$ are (say) positively correlated then (3.10) provides positive values of $\unicode[STIX]{x1D6FE}$ in accordance to (2.14). Multiplication of the numerical result in figure 2(a) with the computed correlation time leads to $\unicode[STIX]{x1D70F}_{\text{corr}}\unicode[STIX]{x1D705}^{\prime }\simeq 0.5$, in good agreement with the analytical result (3.10).

Figure 2. Snapshots of the turbulence-induced coefficients $\unicode[STIX]{x1D705}^{\prime }$ after (3.9) (a) and the correlation (3.11) (b) from simulations of non-rotating convection with an azimuthal magnetic field. The convectively unstable region is located between the two vertical dashed lines, the red curves denote time averages and the yellow curves characterize the expectation value of the fluctuations. For non-rotating convection the correlation $\langle u_{r}^{\prime }\text{curl}_{\unicode[STIX]{x1D703}}\,\boldsymbol{B}^{\prime }\rangle$ exists but $\langle u_{r}^{\prime }\,\text{curl}_{\unicode[STIX]{x1D719}}\,\boldsymbol{B}^{\prime }\rangle$ vanishes. $B_{\unicode[STIX]{x1D719}}=1$, $\unicode[STIX]{x1D6FA}=0$, $Pm=0.1$.

From (3.7) it also follows that the tensor trace $\langle \boldsymbol{u}^{\prime }\boldsymbol{\cdot }\text{curl}\,\boldsymbol{B}^{\prime }\rangle =-10\unicode[STIX]{x1D705}(\unicode[STIX]{x1D734}\boldsymbol{\cdot }\bar{\boldsymbol{B}})$ has a sign opposite to that of the correlation $\langle (\boldsymbol{g}\boldsymbol{\cdot }\boldsymbol{u}^{\prime })\text{curl}\,\boldsymbol{B}^{\prime }\rangle =\unicode[STIX]{x1D705}(\boldsymbol{g}\boldsymbol{\cdot }\unicode[STIX]{x1D734})\bar{\boldsymbol{B}}$, which we now consider for the hemisphere where $\boldsymbol{g}\boldsymbol{\cdot }\unicode[STIX]{x1D734}>0$. One finds that fluctuations of electric currents in the direction of the large-scale magnetic background field are correlated with the velocity component $\boldsymbol{g}\boldsymbol{\cdot }\boldsymbol{u}^{\prime }$, provided $\boldsymbol{g}$ is not perpendicular to the rotation axis. Hence,

(3.11)$$\begin{eqnarray}\langle u_{r}^{\prime }\,\text{curl}_{\unicode[STIX]{x1D719}}\,\boldsymbol{B}^{\prime }\rangle =\unicode[STIX]{x1D705}\cos \unicode[STIX]{x1D703}\unicode[STIX]{x1D6FA}\bar{B}_{\unicode[STIX]{x1D719}},\end{eqnarray}$$

which means that in a rotating but otherwise isotropic turbulence with an azimuthal background field the radial flow fluctuations will always be correlated with azimuthal electric current fluctuations. The correlation (3.11) runs with $\cos \unicode[STIX]{x1D703}$, it is thus antisymmetric with respect to the equator and it vanishes there. An upflow motion provides a positive (negative) azimuthal electric current fluctuation while a downflow motion provides a negative (positive) azimuthal electric current fluctuation so that the products of $u_{r}^{\prime }$ and $\text{curl}_{\unicode[STIX]{x1D719}}\,\boldsymbol{B}^{\prime }$ have the same sign in both cases. Replace now $u_{r}^{\prime }$ by $\unicode[STIX]{x1D702}^{\prime }$ and the existence of correlations such as $\langle \unicode[STIX]{x1D702}^{\prime }\text{curl}_{\unicode[STIX]{x1D719}}\,\boldsymbol{B}^{\prime }\rangle$ becomes obvious in rotating isotropic turbulence fields magnetized with an azimuthal background field. Just this finding is formulated by (2.4) and (3.2). Hence, the dimensionless coefficient $\unicode[STIX]{x1D705}$ in (3.11) is a proxy of an $\unicode[STIX]{x1D6FC}$ effect which appears when $u_{r}^{\prime }$ and $\unicode[STIX]{x1D702}^{\prime }$ are correlated or anticorrelated.

Figure 3. The values of $\unicode[STIX]{x1D705}$ after (3.11) for rotating magnetoconvection with azimuthal magnetic field $B_{\unicode[STIX]{x1D719}}=\pm 1$ (a), $B_{\unicode[STIX]{x1D719}}=2$ (b) and $B_{\unicode[STIX]{x1D719}}=3$ (c). $\unicode[STIX]{x1D6FA}=3$, $Pm=0.1$, $\unicode[STIX]{x1D703}=45^{\circ }$.

We calculate $\unicode[STIX]{x1D705}$ for different magnetic background fields for a fixed rotation rate. Figure 2(b) confirms that the correlation (3.11) vanishes for $\unicode[STIX]{x1D6FA}=0$. The three examples given in figure 3 have been computed with the rotation rate $\unicode[STIX]{x1D6FA}=3$ which corresponds to a Coriolis number $2\unicode[STIX]{x1D70F}_{\text{corr}}\unicode[STIX]{x1D6FA}=3.6$. The preferred direction $\boldsymbol{g}$ has been fixed to $\unicode[STIX]{x1D703}=45^{\circ }$ corresponding to mid-latitudes in a spherical geometry. These are natural choices as equator and poles as the two extremes are excluded. At the equator we do not expect finite correlations to exist while the simulations often meet complications at the poles. As expected, the resulting $\unicode[STIX]{x1D705}$ is positive and does not depend on the sign of the magnetic field. It is approximately ${\lesssim}0.1$ for weak magnetic fields. Due to magnetic suppression an increase of the field by a factor of three reduces the $\unicode[STIX]{x1D705}$ by the same factor. An estimation of the analytical result (3.8) yields $\unicode[STIX]{x1D705}\simeq (1/15)St^{2}$ for $\unicode[STIX]{x1D708}=\bar{\unicode[STIX]{x1D702}}$. Hence, $\unicode[STIX]{x1D705}\lesssim 0.1$ for a Strouhal number of unity derived from the analytical expressions is confirmed by numerical calculations. For the effective pumping $\hat{\unicode[STIX]{x1D6FE}}=\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FA}$ the simulations provide the numerical value of $O(10)$ in (rough) accordance with the estimates (3.5).

For our argument only standard mean-field electrodynamics in turbulent media is needed. We note that the $\unicode[STIX]{x1D6FC}$ term in (3.2) is turbulence originated but it does not need a prescribed helicity in stratified turbulent media; the helicity parameter $H$ from (2.18) does not occur in the calculations. In order to ensure the $\unicode[STIX]{x1D6FC}$ tensor is a pseudo-tensor the new $\unicode[STIX]{x1D6FC}$ effect only exists in rotating media which, however, are no longer required to be stratified in density and/or turbulent intensity.

The dynamo number $C_{\unicode[STIX]{x1D6FC}}=|\unicode[STIX]{x1D6FC}|R/\unicode[STIX]{x1D702}_{\text{t}}$ for large $Rm$ is

(3.12)$$\begin{eqnarray}C_{\unicode[STIX]{x1D6FC}}\lesssim \frac{St}{Rm}\frac{\unicode[STIX]{x1D702}_{\text{rms}}}{\bar{\unicode[STIX]{x1D702}}}\frac{\unicode[STIX]{x1D6FA}R}{u_{\text{rms}}},\end{eqnarray}$$

with $R$ the characteristic size of the dynamo domain. That $C_{\unicode[STIX]{x1D6FC}}$ exceeds unity, which is necessary for dynamo excitation in $\unicode[STIX]{x1D6FC}^{2}$ models, cannot be excluded for sufficiently rapidly rotating large volumes. Applying the characteristic values of the geodynamo with $Rm\simeq 100$ and $u_{\text{rms}}\simeq 0.05~\text{cm}~\text{s}^{-1}$ would provide $\unicode[STIX]{x1D702}_{\text{rms}}/\bar{\unicode[STIX]{x1D702}}\gtrsim 10^{-4}$ as the excitation condition of an $\unicode[STIX]{x1D6FC}^{2}$-dynamo. We shall see below that in the outer core of the Earth such (large) values are not realistic. In the solar convection zone the equatorial velocity $\unicode[STIX]{x1D6FA}R$ slightly exceeds the maximal convection velocity but the very large $Rm$ will prevent sufficiently high values of $C_{\unicode[STIX]{x1D6FC}}$. The smallness of the presented $\unicode[STIX]{x1D6FC}$ effect does not prevent the operation of $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FA}$ dynamo models if sufficiently strong differential rotation exists. The standard solutions of these models, however, are oscillating with time scales of the order of the diffusion time.

Finally, it might be underlined that (3.11) describes a general turbulence-induced radial transport of azimuthal electric current fluctuations which vanishes for $\unicode[STIX]{x1D6FA}=0$. It exists for all rotating homogeneous turbulence fields without another preferred direction beyond the rotation axis and magnetic field direction.