#  Cross-helicity effect on $\alpha$-type dynamo
in non-equilibrium turbulence

Krzysztof A. Mizerski  Department of Magnetism, Institute of Geophysics,
Polish Academy of Sciences, Ksiecia Janusza 64, 01-452 Warsaw, Poland
<EMAIL_ADDRESS>Nobumitsu Yokoi  Institute of Industrial Science, University
of Tokyo, Komaba, Meguro, Tokyo 153-8505, Japan Axel Brandenburg  Nordita, KTH
Royal Institute of Technology and Stockholm University, Hannes Alfvéns väg 12,
10691 Stockholm, Sweden The Oskar Klein Centre, Department of Astronomy,
Stockholm University, AlbaNova, 10691 Stockholm, Sweden School of Natural
Sciences and Medicine, Ilia State University, 0194 Tbilisi, Georgia McWilliams
Center for Cosmology and Department of Physics, Carnegie Mellon University,
Pittsburgh, Pennsylvania 15213, USA

###### Abstract

Turbulence is typically not in equilibrium, i.e. mean quantities such as the
mean energy and helicity are typically time-dependent. The effect of non-
stationarity on the turbulent hydromagnetic dynamo process is studied here
with the use of the two-scale direct-interaction approximation (TSDIA), which
allows to explicitly relate the mean turbulent Reynolds and Maxwell stresses
and the mean electromotive force (EMF) to the spectral characteristics of
turbulence, such as e.g. the mean energy, as well as kinetic and cross-
helicity. It is demonstrated, that the non-equilibrium effects can enhance the
dynamo process when the magnetohydrodynamic (MHD) turbulence is both helical
and cross-helical. This effect is based on the turbulent infinitesimal-impulse
cross-response functions, which do not affect turbulent flows in equilibrium.
The evolution and sources of the cross-helicity in MHD turbulence is also
discussed.

##  1 Introduction

The effect of hydromagnetic dynamo action is ubiquitous in astrophysical
plasmas e.g. in stellar and planetary interiors, accretion discs or the
interstellar medium (cf. Roberts and Soward 1972, Brandenburg and Subramanian
2005, Dormy and Soward 2007, Roberts and King 2013, Balbus and Hawley
1991a,b). This is particularly important in view of the recent advancement of
tokamak devices, reaching very high plasma temperatures, thus giving hope for
the production of thermonuclear fusion power (cf. Li _et al._ 2019, Gibney
2022). The investigations of the large-scale dynamo mechanisms in
magnetohydrodynamic (MHD) turbulence, that is those that lead to generation of
large-scale magnetic fields, is mainly limited to equilibrium, i.e.
statistically stationary turbulence.

One of the widely known and often invoked mechanisms is the so-called
$\alpha$-effect, which requires chirality (lack of reflexional symmetry) in
the turbulent flow, and this requires some mechanism that breaks the ‘up-down’
symmetry of the system, cf. Krause and Rädler (1980), Dormy and Soward (2007),
Moffatt and Dormy (2019). A large-scale electromotive force (EMF) is then
generated and this leads to the amplification of magnetic energy. The lack of
reflectional symmetry is typically introduced by stratification and background
rotation and a useful measure of the flow chirality is the kinetic helicity,
$\langle\mathbf{U}\cdot\nabla\times\mathbf{U}\rangle$, where
$\langle\cdot\rangle$ denotes the ensemble mean. Another pseudoscalar quantity
of importance in dynamo theory is the cross-helicity
$\langle\mathbf{U}\cdot\mathbf{B}\rangle$, cf. e.g. Hamba and Tsuchiya (2010),
Yokoi (2013); see Yokoi (2023) for a review.

The aim of this paper can be shortly stated as a demonstration of the
influence of non-equilibrium effects in MHD turbulence on the $\alpha$-effect
and thereby on large-scale dynamos. This issue has already been investigated
in a series of papers by Mizerski (2018a,b, 2020, 2021, 2022), which however,
assumed that the turbulence was stirred by a Gaussian and helical forcing; the
physical properties of the forcing were then present in the expressions for
the $\alpha$ coefficient. On the contrary, here we apply the Two-Scale Direct-
Interaction Approximation (TSDIA), which allows to remove the stirring force,
but instead we need to assume some statistical properties of the background
turbulence. Nevertheless, this approach allows to explicitly relate the mean
electromotive force to kinetic and cross-helicities, through consideration of
the Green response functions, which describe the responses of the turbulent
flow and magnetic field to infinitesimal perturbations, cf. e.g. Yoshizawa
(1985, 1990, 1998), Yokoi (2013, 2018). We show that the infinitesimal-impulse
cross-responses affect the mean EMF through non-equilibrium effects in MHD
turbulence, and the $\alpha$-effect is potentially enhanced, provided that the
kinetic and cross-helicities are both non-zero. We also discuss the evolution
equation of the cross-helicity, its sources and sinks in MHD turbulence, hence
the possibility of a coexistence of the kinetic and cross-helicities; this
issue is also investigated numerically.

##  2 Mathematical formulation

To study the magnetohydrodynamic turbulence in an incompressible conducting
fluid we consider the following dynamical equations describing the evolution
of the velocity field of the fluid flow $\mathbf{U}(\mathbf{x},t)$ and the
magnetic field $\mathbf{B}(\mathbf{x}.t)$

$\frac{\partial\mathbf{U}}{\partial
t}+\left(\mathbf{U}\cdot\nabla\right)\mathbf{U}=-\nabla\Pi-2\boldsymbol{\Omega}\times\mathbf{U}+\left(\mathbf{B}\cdot\nabla\right)\mathbf{B}+\nu\nabla^{2}\mathbf{U},$
(1a) $\frac{\partial\mathbf{B}}{\partial
t}+\left(\mathbf{U}\cdot\nabla\right)\mathbf{B}=\left(\mathbf{B}\cdot\nabla\right)\mathbf{U}+\eta\nabla^{2}\mathbf{B},$
(1b) $\nabla\cdot\mathbf{U}=0\qquad\nabla\cdot\mathbf{B}=0,$ (1c)

where

$\Pi=\frac{p}{\rho}+\frac{B^{2}}{2}-\frac{1}{2}(\boldsymbol{\Omega}\times\mathbf{x})^{2},$
(2)

is the total pressure, $\rho$ is the density, $\boldsymbol{\Omega}$ is the
angular velocity, $\nu$ is the viscosity, $\eta$ is the magnetic diffusivity.
For the purpose of simplicity we rescaled the magnetic field in the following
way $\mathbf{B}/\sqrt{\mu_{0}\rho}\rightarrow\mathbf{B}$, where $\mu_{0}$ is
the vacuum permeability (so that the prefactor $1/\mu_{0}\rho$ in the Lorentz-
force term in the Navier-Stokes equation is lost); in the following we also
rescale the currents, $\sqrt{\mu_{0}/\rho}\mathbf{J}\rightarrow\mathbf{J}$, so
that $\mathbf{J}=\nabla\times\mathbf{B}$. Next, denoting by angular brackets
the ensemble mean,

$\left\langle\cdot\right\rangle-\textrm{ensemble mean}$

we put forward the standard decomposition

$\mathbf{U}=\left\langle\mathbf{U}\right\rangle+\mathbf{u}^{\prime},\quad\mathbf{B}=\left\langle\mathbf{B}\right\rangle+\mathbf{b}^{\prime},\quad
p=\left\langle p\right\rangle+p^{\prime},$ (3)

and write down separately the equations for the mean fields
$\left\langle\mathbf{U}\right\rangle$ and
$\left\langle\mathbf{B}\right\rangle$ and the turbulent fluctuations
$\mathbf{u}^{\prime}$ and $\mathbf{b^{\prime}}$; this yields

$\displaystyle\frac{\partial\left\langle\mathbf{U}\right\rangle}{\partial
t}+\left(\left\langle\mathbf{U}\right\rangle\cdot\nabla\right)\left\langle\mathbf{U}\right\rangle=$
$\displaystyle-\nabla\left\langle\Pi\right\rangle-2\boldsymbol{\Omega}\times\langle\mathbf{U}\rangle+\left(\left\langle\mathbf{B}\right\rangle\cdot\nabla\right)\left\langle\mathbf{B}\right\rangle+\nu\nabla^{2}\left\langle\mathbf{U}\right\rangle$
$\displaystyle-\nabla\cdot\left(\left\langle\mathbf{u}^{\prime}\mathbf{u}^{\prime}\right\rangle-\left\langle\mathbf{b}^{\prime}\mathbf{b}^{\prime}\right\rangle\right),$
(4a) $\frac{\partial\left\langle\mathbf{B}\right\rangle}{\partial
t}=\nabla\times\left(\left\langle\mathbf{U}\right\rangle\times\left\langle\mathbf{B}\right\rangle\right)+\nabla\times\left\langle\mathbf{u}^{\prime}\times\mathbf{b}^{\prime}\right\rangle+\eta\nabla^{2}\left\langle\mathbf{B}\right\rangle,$
(4b)
$\nabla\cdot\left\langle\mathbf{B}\right\rangle=0,\quad\nabla\cdot\left\langle\mathbf{U}\right\rangle=0,$
(4c)

where

$\boldsymbol{\mathcal{E}}=\left\langle\mathbf{u}^{\prime}\times\mathbf{b}^{\prime}\right\rangle,$
(5)

is the large-scale electromotive force (EMF) and

$\displaystyle\frac{\partial\mathbf{u}^{\prime}}{\partial
t}-\nu\nabla^{2}\mathbf{u}^{\prime}+2\boldsymbol{\Omega}\times\mathbf{u}^{\prime}+\left(\left\langle\mathbf{U}\right\rangle\cdot\nabla\right)\mathbf{u}^{\prime}+\left(\mathbf{u}^{\prime}\cdot\nabla\right)\left\langle\mathbf{U}\right\rangle-\left(\left\langle\mathbf{B}\right\rangle\cdot\nabla\right)\mathbf{b}^{\prime}-\left(\mathbf{b}^{\prime}\cdot\nabla\right)\left\langle\mathbf{B}\right\rangle$
$\displaystyle+\nabla\Pi^{\prime}=-\nabla\cdot\left(\mathbf{u}^{\prime}\mathbf{u}^{\prime}-\mathbf{b}^{\prime}\mathbf{b}^{\prime}\right)+\nabla\cdot\left(\left\langle\mathbf{u}^{\prime}\mathbf{u}^{\prime}\right\rangle-\left\langle\mathbf{b}^{\prime}\mathbf{b}^{\prime}\right\rangle\right),$
(6a) $\displaystyle\frac{\partial\mathbf{b}^{\prime}}{\partial
t}-\eta\nabla^{2}\mathbf{b}^{\prime}+\left(\left\langle\mathbf{U}\right\rangle\cdot\nabla\right)\mathbf{b}^{\prime}-\left(\left\langle\mathbf{B}\right\rangle\cdot\nabla\right)\mathbf{u}^{\prime}+\left(\mathbf{u}^{\prime}\cdot\nabla\right)\left\langle\mathbf{B}\right\rangle-\left(\mathbf{b}^{\prime}\cdot\nabla\right)\left\langle\mathbf{U}\right\rangle$
$\displaystyle=\nabla\times\left(\mathbf{u}^{\prime}\times\mathbf{b}^{\prime}-\left\langle\mathbf{u}^{\prime}\times\mathbf{b}^{\prime}\right\rangle\right),$
(6b) $\nabla\cdot\mathbf{b}^{\prime}=0,\quad\nabla\cdot\mathbf{u}^{\prime}=0.$
(6c)

##  3 Non-equilibrium effects in dynamo theory

Previous results of Mizerski (2018a,b, 2020, 2021, 2022), obtained in the
absence of the Coriolis force but with chiral stochastic forcing, in the
context of the geodynamo and galactic dynamos suggest that the non-stationary
$\alpha$-effect is proportional to the energy production rate resulting from
the presence of the forcing (e.g. stochastic buoyancy) and is oscillatory on
time scales induced by the forcing, which could be long (cf. also Mizerski _et
al_. 2012 for non-stationary dynamo in the context of the elliptical
instability). Here we utilize the Two-Scale Direct Interaction Approximation,
in order to extract the effect of non-stirred, non-equilibrium turbulence on
the large-scale hydromagnetic dynamo. In other words, the new approach allows
to study non-stationary MHD turbulence and the turbulent dynamo effect in the
absence of external stochastic forcing although with assumed statistical
properties of the background turbulence. We demonstrate, that in non-
equilibrium turbulence the quantity
$\left\langle\mathbf{u}^{\prime}\cdot\mathbf{j}^{\prime}\right\rangle$ plays a
significant role in generation of the large-scale EMF through the
$\alpha$-effect and the effect of
$\left\langle\mathbf{u}^{\prime}\cdot\mathbf{j}^{\prime}\right\rangle$
vanishes in stationary turbulence.

###  3.1 Application of the TSDIA method

Let us introduce a small parameter $\delta$ and define slow and fast spatial
and temporal variables

$\boldsymbol{\xi}=\mathbf{x},\quad\mathbf{X}=\delta\mathbf{x},\quad\tau=t,\quad
T=\delta t.$ (7)

The large-scale fields depend only on the slow variables,
$\left\langle\mathbf{U}\right\rangle(\mathbf{X},T)$ and the fluctuations
depend on both, $\mathbf{u}^{\prime}(\boldsymbol{\xi},\mathbf{X};\tau,T)$. We
also define the Fourier transform, involving Galilean transformation to the
frame moving with the velocity $\left\langle\mathbf{U}\right\rangle$

$u_{i}^{\prime}(\boldsymbol{\xi},\mathbf{X};\tau,T)=\int\mathrm{d}^{3}k\hat{u}_{i}^{\prime}(\mathbf{k},\mathbf{X;\tau},T)\mathrm{e}^{-\mathrm{i}\mathbf{k}\cdot(\boldsymbol{\xi}-\left\langle\mathbf{U}\right\rangle\tau)},$
(8)

but the explicit dependence on the slow variables $\mathbf{X}$ and $T$ will be
typically suppressed in notation for clarity. The details of the TSDIA
approach are provided in Appendix A (see also $\mathsection$ 9.6 of Yoshizawa
1998, Yoshizawa 1985, 1990 and Yokoi 2023) and here we present the major
results. The method involves introduction of the concept of background
turbulence with given statistical properties, uninfluenced by the large-scale
field and rotation, hence isotropic; this background turbulence is defined by
the following correlation functions

$\left\langle\hat{f}_{i}(\mathbf{k};\tau)\hat{g}_{j}(\mathbf{k}_{1};\tau_{1})\right\rangle=\left[P_{ij}(\mathbf{k})Q_{fg}\left(k;\tau,\tau_{1}\right)+\frac{1}{2}\mathrm{i}\epsilon_{ijk}\frac{k_{k}}{k^{2}}H_{fg}\left(k;\tau,\tau_{1}\right)\right]\delta(\mathbf{k}+\mathbf{k}_{1}),$
(9) $\left\langle
G_{fgij}^{\prime}(\mathbf{k};\tau,\tau_{1})\right\rangle=\delta_{ij}G_{fg}\left(k;\tau,\tau_{1}\right),$
(10)

where $f$ and $g$ represent one of the variables $\mathbf{u}_{00}^{\prime}$
and $\mathbf{b}_{00}^{\prime}$ and
$G_{fij}^{\prime}(\mathbf{k};\tau,\tau_{1})$ denote the Green’s functions
describing the system’s response to infinitesimal disturbances. It is useful
at this stage to write down explicitly the following quantity

$\displaystyle\left\langle\mathbf{u}_{00}^{\prime}(\mathbf{x},\tau)\cdot\mathbf{j}_{00}^{\prime}(\mathbf{x},\tau_{1})\right\rangle$
$\displaystyle\,\,=-\mathrm{i}\epsilon_{ijk}\int\mathrm{d}k\int\mathrm{d}k^{\prime}k^{\prime}_{j}\left\langle\hat{u}_{00i}^{\prime}(\mathbf{k};\tau)\hat{b}_{00k}^{\prime}(\mathbf{k}^{\prime};\tau_{1})\right\rangle\mathrm{e}^{-\mathrm{i}(\mathbf{k}+\mathbf{k}^{\prime})\cdot\mathbf{x}}$
$\displaystyle\,\,=\int\mathrm{d}kH_{ub}\left(k;\tau,\tau_{1}\right)=\int\mathrm{d}kH_{bu}\left(k;\tau_{1},\tau\right),$
(11)

since this quantity will play an important role in the theory of non-
equilibrium $\alpha$-effect, developed below.

The derivation of the formula for the EMF presented in Appendix A leads to

$\boldsymbol{\mathcal{E}}=\alpha\left\langle\mathbf{B}\right\rangle-\left(\beta+\zeta\right)\left\langle\mathbf{J}\right\rangle-\nabla\zeta\times\left\langle\mathbf{B}\right\rangle+\gamma\left(\left\langle\mathbf{W}\right\rangle+2\boldsymbol{\Omega}\right),$
(12)

where
$\mathbf{J}=\nabla\times\mathbf{B}=\left\langle\mathbf{J}\right\rangle+\mathbf{j}$’
and
$\mathbf{W}=\nabla\times\mathbf{U}=\left\langle\mathbf{W}\right\rangle+\mathbf{w}^{\prime}$
denote electric currents and the vorticity respectively. The statistically
stationary case has been studied in detail in Yoshizawa (1998) and Yokoi
(2013, 2018).

We now concentrate on the $\alpha$-effect, which can be decomposed into two
contributions,

$\alpha=\alpha_{S}+\alpha_{\rm X},$ (13)

the standard one, related to the so-called residual helicity

$\displaystyle\alpha_{S}=$
$\displaystyle\frac{1}{3}\int\mathrm{d}^{3}k\int_{-\infty}^{\tau}\mathrm{d}\tau_{1}\left[G_{uu}\left(k,\mathbf{X};\tau,\tau_{1},T\right)H_{bb}\left(k,\mathbf{X};\tau,\tau_{1},T\right)\right.$
$\displaystyle\left.\qquad\qquad\qquad\qquad-
G_{bb}\left(k,\mathbf{X};\tau,\tau_{1},T\right)H_{uu}\left(k,\mathbf{X};\tau_{1},\tau,T\right)\right],$
(14)

and a less obvious one, related to the cross helicity and the quantity
$\langle\mathbf{u}^{\prime}\cdot\mathbf{j}^{\prime}\rangle$ which takes the
form

$\displaystyle\alpha_{\rm X}=$
$\displaystyle-\frac{1}{3}\int\mathrm{d}^{3}k\int_{-\infty}^{\tau}\mathrm{d}\tau_{1}G_{bu}\left(k,\mathbf{X};\tau,\tau_{1},T\right)H_{ub}\left(k,\mathbf{X};\tau,\tau_{1},T\right)$
$\displaystyle+\frac{1}{3}\int\mathrm{d}^{3}k\int_{-\infty}^{\tau}\mathrm{d}\tau_{1}G_{ub}\left(k,\mathbf{X};\tau,\tau_{1},T\right)H_{bu}\left(k,\mathbf{X};\tau,\tau_{1},T\right).$
(15)

Since the helical functions of the background turbulence satisfy

$H_{bu}\left(\tau,\tau_{1}\right)=H_{ub}\left(\tau_{1},\tau\right),$ (16)

we obtain

$\displaystyle\alpha_{\rm X}=$
$\displaystyle-\frac{1}{3}\int\mathrm{d}^{3}k\int_{-\infty}^{\tau}\mathrm{d}\tau_{1}G_{bu}\left(k,\mathbf{X};\tau,\tau_{1},T\right)H_{ub}\left(k,\mathbf{X};\tau,\tau_{1},T\right)$
$\displaystyle+\frac{1}{3}\int\mathrm{d}^{3}k\int_{-\infty}^{\tau}\mathrm{d}\tau_{1}G_{ub}\left(k,\mathbf{X};\tau,\tau_{1},T\right)H_{ub}\left(k,\mathbf{X};\tau_{1},\tau,T\right).$
(17)

We now introduce the following symmetric and antisymmetric parts of $H_{ub}$
with respect to exchange of time variables

$H_{ub}^{(s)}\left(\tau,\tau_{1}\right)=\frac{1}{2}\left(H_{ub}\left(\tau,\tau_{1}\right)+H_{ub}\left(\tau_{1},\tau\right)\right),$
(18a)
$H_{ub}^{(a)}\left(\tau,\tau_{1}\right)=\frac{1}{2}\left(H_{ub}\left(\tau,\tau_{1}\right)-H_{ub}\left(\tau_{1},\tau\right)\right),$
(18b)

which allows to further separate the $\alpha_{\rm X}$ term into two
contributions

$\displaystyle\alpha_{\rm X}=$
$\displaystyle-\frac{1}{3}\int\mathrm{d}^{3}k\int_{-\infty}^{\tau}\mathrm{d}\tau_{1}\left[G_{ub}\left(k,\mathbf{X};\tau,\tau_{1},T\right)+G_{bu}\left(k,\mathbf{X};\tau,\tau_{1},T\right)\right]H_{ub}^{(a)}\left(k,\mathbf{X};\tau,\tau_{1},T\right)$
$\displaystyle+\frac{1}{3}\int\mathrm{d}^{3}k\int_{-\infty}^{\tau}\mathrm{d}\tau_{1}\left[G_{ub}\left(k,\mathbf{X};\tau,\tau_{1},T\right)-G_{bu}\left(k,\mathbf{X};\tau,\tau_{1},T\right)\right]H_{ub}^{(s)}\left(k,\mathbf{X};\tau_{1},\tau,T\right).$
(19)

The first term in equation (19), i.e.

$\alpha_{\rm
neq}=-\frac{1}{3}\int\mathrm{d}^{3}k\int_{-\infty}^{\tau}\mathrm{d}\tau_{1}\left[G_{ub}\left(k,\mathbf{X};\tau,\tau_{1},T\right)+G_{bu}\left(k,\mathbf{X};\tau,\tau_{1},T\right)\right]H_{ub}^{(a)}\left(k,\mathbf{X};\tau,\tau_{1},T\right),$
(20)

clearly constitutes a contribution from non-stationarity of the turbulence, as
the antisymmetric part $H_{ub}^{(a)}$ is clearly a non-equilibrium effect.

###  3.2 Physics of the non-equilibrium $\alpha_{\rm neq}$-effect

If we further assume that the function

$\mathcal{G}\left(\tau,\tau_{1}\right)=G_{ub}\left(\tau,\tau_{1}\right)+G_{bu}\left(\tau,\tau_{1}\right)$
(21)

is independent of $k$, the non-equilibrium $\alpha$-effect can be expressed as
follows:

$\alpha_{\rm
neq}=-\frac{1}{3}\int_{-\infty}^{\tau}\mathrm{d}\tau_{1}\mathcal{G}\left(\tau,\tau_{1}\right)\langle\mathbf{u}_{00}^{\prime}\cdot\mathbf{j}_{00}^{\prime}\rangle^{(a)}\left(\mathbf{x},\tau,\tau_{1}\right),$
(22)

where

$\langle\mathbf{u}_{00}^{\prime}\cdot\mathbf{j}_{00}^{\prime}\rangle^{(a)}\left(\mathbf{x},\tau,\tau_{1}\right)=\frac{1}{2}\left[\langle\mathbf{u}_{00}^{\prime}\left(\mathbf{x},\tau\right)\cdot\mathbf{j}_{00}^{\prime}\left(\mathbf{x},\tau_{1}\right)\rangle-\langle\mathbf{u}_{00}^{\prime}\left(\mathbf{x},\tau_{1}\right)\cdot\mathbf{j}_{00}^{\prime}\left(\mathbf{x},\tau\right)\rangle\right].$
(23)

The memory effect, expressed by the time integral in (23) is clearly crucial,
as
$\langle\mathbf{u}_{00}^{\prime}\cdot\mathbf{j}_{00}^{\prime}\rangle^{(a)}\left(\mathbf{x},\tau,\tau\right)=0$.
Next, inspection of the evolution equations for the Green’s functions leads to
the conclusion that $G_{ub}$ must be an odd function of
$\mathbf{b}_{00}^{\prime}$. This is expected, since the $\alpha_{\rm X}$
contribution to the $\alpha$-effect results from the action of the Lorentz
force, and since $H_{ub}$ is associated with the quantity
$\left\langle\mathbf{u}_{00}^{\prime}\cdot\mathbf{j}_{00}^{\prime}\right\rangle$,
i.e. $H_{ub}$ is linear in $\mathbf{b}_{00}^{\prime}$, it follows that
$G_{ub}$ must be an odd function of the latter. Moreover, since
$\left\langle\mathbf{u}_{00}^{\prime}\cdot\mathbf{j}_{00}^{\prime}\right\rangle$
is a scalar quantity (does not change sign under reflections), $G_{ub}$ must
be skew. The only dynamical quantity that is skew and odd in
$\mathbf{b}_{00}^{\prime}$ is the cross helicity,
$\langle\mathbf{u}_{00}^{\prime}\cdot\mathbf{b}_{00}^{\prime}\rangle$, hence
we expect that $G_{ub}\sim Q_{ub}$. Having in mind that the response function
$\mathcal{G}(\tau,\tau_{1})$ is non-dimensional we can now provide the
following rough estimate of the non-equilibrium $\alpha_{\rm neq}$-effect

$\alpha_{\rm
neq}\sim-\frac{2}{3}\int_{-\infty}^{\tau}\mathrm{d}\tau_{1}\Upsilon^{(s)}\left(\mathbf{x},\tau,\tau_{1}\right)\langle\mathbf{u}_{00}^{\prime}\cdot\mathbf{j}_{00}^{\prime}\rangle^{(a)}\left(\mathbf{x},\tau,\tau_{1}\right),$
(24)

where

$\Upsilon\left(\mathbf{x},\tau,\tau_{1}\right)=\frac{\langle\mathbf{u}_{00}^{\prime}\left(\mathbf{x},\tau\right)\cdot\mathbf{b}_{00}^{\prime}\left(\mathbf{x},\tau_{1}\right)\rangle}{\sqrt{\langle
u_{00}^{\prime 2}\rangle\left(\mathbf{x},\tau\right)\langle b_{00}^{\prime
2}\rangle\left(\mathbf{x},\tau_{1}\right)}},$ (25)
$\Upsilon^{(s)}\left(\mathbf{x},\tau,\tau_{1}\right)=\frac{1}{2}\left[\Upsilon\left(\mathbf{x},\tau,\tau_{1}\right)+\Upsilon\left(\mathbf{x},\tau_{1},\tau\right)\right],$
(26)

and the cross helicity has been normalized by the geometric mean of the
kinetic and magnetic fluctuational energies (see Yokoi 2011 for a discussion
of different cross-helicity normalizations). The latter equation expresses an
effect which results from the lack of equilibrium in the turbulent state.

The second term in (19) is likely to be small because of the factor
$G_{ub}(\tau,\tau_{1})-G_{bu}(\tau,\tau_{1})$. For example in the case when
$\nu=\eta$ the two response functions $G_{ub}$ and $G_{bu}$ are equal and
$\alpha_{\rm X}=\alpha_{\rm neq}$. This still holds approximately true, when
the diffusivities are unequal but weak,

$G_{ub}\approx G_{bu},\qquad\textrm{and}\qquad\alpha_{\rm X}\approx\alpha_{\rm
neq}.$ (27)

The same symmetry arguments as in the case of $\alpha_{\rm neq}$ can be also
applied to the second term in (19) which is therefore proportional to the non-
dimensional cross helicity $\Upsilon=\Upsilon(\mathbf{x},\tau,\tau)$ and the
quantity
$\langle\mathbf{u}_{00}^{\prime}\cdot\mathbf{j}_{00}^{\prime}\rangle=\langle\mathbf{u}_{00}^{\prime}\left(\mathbf{x},\tau\right)\cdot\mathbf{j}_{00}^{\prime}\left(\mathbf{x},\tau\right)\rangle$,
i.e. $\alpha_{\rm X}-\alpha_{\rm
neq}\sim\tau_{t}\Upsilon\left\langle\mathbf{u}_{00}^{\prime}\cdot\mathbf{j}_{00}^{\prime}\right\rangle$,
where $\tau_{t}$ is the turn over time of the most energetic turbulent eddies.
However, as remarked above this effect should be weak, when the diffusion is
weak or the magnetic Prandtl number $\mbox{Pr}_{M}=\nu/\eta\approx 1$.

Finally, we also expect the
$\langle\mathbf{u}^{\prime}_{00}\cdot\mathbf{j}^{\prime}_{00}\rangle$
correlations in fully turbulent flows to be proportional to the kinetic
helicity
$\langle\mathbf{u}^{\prime}_{00}\cdot\mathbf{w}^{\prime}_{00}\rangle$, since
typically the velocities and magnetic fields tend to align in such flows.
Again, the prefactor must be skew and odd in $\mathbf{b}^{\prime}_{00}$,
therefore we propose

$\langle\mathbf{u}^{\prime}_{00}\cdot\mathbf{j}^{\prime}_{00}\rangle\approx\Upsilon\langle\mathbf{u}^{\prime}_{00}\cdot\mathbf{w}^{\prime}_{00}\rangle=\frac{\langle\mathbf{u}^{\prime}_{00}\cdot\mathbf{b}^{\prime}_{00}\rangle\langle\mathbf{u}^{\prime}_{00}\cdot\mathbf{w}^{\prime}_{00}\rangle}{\sqrt{\left\langle
u^{\prime 2}_{00}\right\rangle\left\langle b^{\prime 2}_{00}\right\rangle}}.$
(28)

Introducing the latter relation into (24) leads to

$\alpha_{\rm
neq}\sim-\frac{2}{3}\int_{-\infty}^{\tau}\mathrm{d}\tau_{1}\left(\Upsilon^{(s)}\left(\mathbf{x},\tau,\tau_{1}\right)\right)^{2}\langle\mathbf{u}_{00}^{\prime}\cdot\mathbf{w}_{00}^{\prime}\rangle^{(a)}\left(\mathbf{x},\tau,\tau_{1}\right),$
(29)

which shows, that the non-equilibrium $\alpha_{\rm neq}$-effect relies on
coexistence of the kinetic and cross helicities and their history in MHD
turbulence (more precisely, in the case of kinetic helicity only the
antisymmetric part of the time correlations
$\langle\mathbf{u}_{00}^{\prime}\cdot\mathbf{w}_{00}^{\prime}\rangle^{(a)}\left(\mathbf{x},\tau,\tau_{1}\right)$
contributes to the new effect).

###  3.3 Calculation of the $\alpha_{\rm neq}$-effect

We will now investigate this dynamo mechanism in some more detail. In order to
calculate the effect of non-equilibrium turbulence we adopt a similar approach
to that in $\mathsection$ 7 of Yoshizawa (1998). In stationary turbulence the
functions $H_{fg}\left(k,\mathbf{X};\tau,\tau_{1},T\right)$ and
$G_{f}\left(k,\mathbf{X};\tau,\tau_{1},T\right)$ depend only on
$\left|\tau-\tau_{1}\right|$, hence to study the non-equilibrium effects we
postulate a similar formulae for these functions as those of Yoshizawa (1998)
(cf. formulae 6.53-6.54 of this book), but modified in order to introduce
simple explicit and distinct dependencies on $\tau$ and $\tau_{1}$

$H_{fg}\left(k,\mathbf{k}\cdot\left\langle\mathbf{B}\right\rangle,\mathbf{X};\tau,\tau_{1},T\right)=\sigma\left(k,\mathbf{X},T\right)\mathrm{e}^{-\varpi\left(k,\mathbf{X},T\right)\left|\tau-\tau_{1}\right|}\mathcal{H}\left(\tau\right)\mathcal{H}_{1}\left(\tau_{1}\right),$
(30)
$G_{fg}\left(k,\mathbf{X};\tau,\tau_{1},T\right)=\theta\left(\tau-\tau_{1}\right)\varsigma\left(k,\mathbf{X},T\right)\mathrm{e}^{-\varpi\left(k,\mathbf{X},T\right)\left|\tau-\tau_{1}\right|}\mathcal{G}\left(\tau\right)\mathcal{G}_{1}\left(\tau_{1}\right),$
(31)

for some functions $\mathcal{H}\left(\tau\right)$,
$\mathcal{H}_{1}(\tau_{1})$, $\mathcal{G}(\tau)$ and
$\mathcal{G}_{1}(\tau_{1})$. We can decompose these functions into Fourier
modes, which allows to adopt the following, simpler, generic model

$H_{fg}\left(\tau,\tau_{1}\right)=\sigma\mathrm{e}^{-\varpi\left|\tau-\tau_{1}\right|}\sin\left(\varpi_{h0}\tau\right)\sin\left(\varpi_{h1}\tau_{1}\right),$
(32a)
$G_{fg}\left(\tau,\tau_{1}\right)=\theta\left(\tau-\tau_{1}\right)\varsigma\mathrm{e}^{-\varpi\left|\tau-\tau_{1}\right|}\sin\left(\varpi_{g0}\tau\right)\sin\left(\varpi_{g1}\tau_{1}\right),$
(32b)

where the dependence on the slow variables and the wavenumber $k$ was
suppressed in notation for clarity; moreover $\varpi>0$ and to fix ideas we
also assume $\varpi_{h0}>0$, $\varpi_{g0}>0$, $\varpi_{h1}>0$ and
$\varpi_{g1}>0$. For the sake of simplicity we also assume

$G_{ub}\approx G_{bu}.$ (33)

The following calculation

$\displaystyle\int_{-\infty}^{\tau}\mathrm{d}\tau_{1}G_{fg}\left(\tau,\tau_{1}\right)H_{fg}\left(\tau,\tau_{1}\right)$
$\displaystyle=\frac{\sigma\varsigma}{4}\left(\cos\Delta_{0}\tau-\cos\Sigma_{0}\tau\right)\left[\frac{1}{4\varpi^{2}+\Delta_{1}^{2}}\left(2\varpi\cos\Delta_{1}\tau+\Delta_{1}\sin\Delta_{1}\tau\right)\right.\qquad\qquad\qquad\qquad\qquad\qquad\qquad$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\left.-\frac{1}{4\varpi^{2}+\Sigma_{1}^{2}}\left(2\varpi\cos\Sigma_{1}\tau+\Sigma_{1}\sin\Sigma_{1}\tau\right)\right],$
(34)

where

$\Delta_{i}=\varpi_{hi}-\varpi_{gi},\qquad\Sigma_{i}=\varpi_{hi}+\varpi_{gi},$
(35)

shows, that in non-equilibrium turbulence both contributions to the
$\alpha$-effect, the ‘standard’ $\alpha_{S}$ and the one associated with cross
helicity $\alpha_{X}$, are enhanced by non-stationarity. Since the frequencies
correspond to the fast oscillations of turbulent fluctuations in most of the
cases the cosines and sines do not contribute to large time scales (their time
average vanishes). Under the time average over long time scales $\delta^{-1}t$
the non-zero contribution comes from the cases $\varpi_{hi}=\varpi_{gi}$ (or
$\varpi_{hi}\approx\varpi_{gi}$). Therefore we pick
$(\varpi,\,\varpi_{h},\,\varpi_{g})$-modes such that the following relations
are satisfied

$\Delta_{i}\ll\varpi\ll\varpi_{hi},\,\varpi_{gi},\quad\textrm{for}\quad
i=0,1,$ (36)

in which case

$\int_{-\infty}^{\tau}\mathrm{d}\tau_{1}G_{fg}\left(\tau,\tau_{1}\right)H_{fg}\left(\tau,\tau_{1}\right)\approx\frac{\sigma\varsigma}{8\varpi};$
(37)

for comparison in the stationary case one obtains
$\sigma_{s}\varsigma_{s}/2\varpi_{s}$ with
$H_{fg}=\sigma_{s}\exp(-\varpi_{s}\left|\tau-\tau_{1}\right|)$,
$G_{fg}=\varsigma_{s}\exp(-\varpi_{s}\left|\tau-\tau_{1}\right|)$. However,
the influence of non-stationarity on the ‘standard’ $\alpha_{S}$ contribution
has been studied using different methods in Mizerski (2018a,b, 2020, 2021,
2022). Here we concentrate on the cross-helicity contribution $\alpha_{\rm
X}\approx\alpha_{\rm neq}$, which is apparent within the TSDIA approach.
Introduction of the formulae (32a,b) into (20) yields

$\alpha_{\rm neq}\approx-\frac{\pi}{6}\int\mathrm{d}k\frac{\sigma\varsigma
k^{2}}{\varpi}.$ (38)

According to our previous observations in the above we have
$\varsigma\sim\Upsilon$. We note that a very similar result is obtained if one
assumes a simpler non-stationary form of the $H_{ub}$ and $G_{ub}$ functions

$H_{ub}\left(\tau,\tau_{1}\right)=\sigma\mathrm{e}^{-\varpi\left|\tau-\tau_{1}\right|}\sin\left[\varpi_{h}\left(\tau-\tau_{1}\right)\right],$
(39a)
$G_{ub}\left(\tau,\tau_{1}\right)=\theta\left(\tau-\tau_{1}\right)\varsigma\mathrm{e}^{-\varpi\left|\tau-\tau_{1}\right|}\sin\left[\varpi_{g}\left(\tau-\tau_{1}\right)\right],$
(39b)

which satisfies $H_{ub}(\tau,\tau_{1})=-H_{ub}(\tau_{1},\tau)$, and considers
the limit (36).

In the above calculation we have used some standard models of the statistical
properties of turbulence in order to emphasize the importance of the history
of evolution of the helicities in the turbulent dynamo process. The
$\alpha_{\rm neq}$-effect, induced by the simultaneous presence of cross and
kinetic helicities, can be strong and depends on their magnitude.

##  4 Coexistence of the kinetic and cross helicities in turbulence

We now consider the question of the likelihood of coexistence of the cross and
kinetic helicities in developed turbulence. Although it is not possible to
draw definite conclusions in this matter, it is still instructive to study the
sources and sinks of the cross helicity in turbulent flows in order to develop
some intuition about its generation.

In the Appendix B we consider a stirred turbulence (with homogeneous,
isotropic, stationary and helical Gaussian forcing) and show that under the
first-order smoothing approximation the kinetic helicity is proportional to
the helicity of the forcing, whereas the cross-helicity is defined by the
product
$\langle\mathbf{f}\cdot\nabla\times\mathbf{f}\rangle(\langle\mathbf{B}\rangle\cdot\boldsymbol{\Omega})$.
In other words, within the FOSA approach the existence of the cross-helicity
is dependent on the existence of the mean field component parallel to the
background rotation vector.

A more general calculation is presented in the Appendix C, where we have
derived the general evolution equation for the cross-helicity (cf. also Yokoi
and Hamba 2007, Yokoi 2011, Yokoi and Balarac 2011, Yokoi and Hoshino 2011,
Yokoi 2013). This equation involves mean quantities such as the mean EMF
$\boldsymbol{\mathcal{E}}$ and the mean Reynolds and Maxwell stresses $\langle
u_{i}^{\prime}u_{j}^{\prime}-b_{i}^{\prime}b_{j}^{\prime}\rangle$. For the
former we utilize the result (12) and for the latter we take the expression
obtained also via the TSDIA approach in Yokoi and Hoshino (2011), i.e.

$-\langle
u_{i}^{\prime}u_{j}^{\prime}-b_{i}^{\prime}b_{j}^{\prime}\rangle\frac{\partial\langle
B\rangle_{i}}{\partial
x_{j}}=\frac{7}{10}\beta\mathcal{S}_{ij}\mathcal{M}_{ij}-\frac{7}{10}\gamma\mathrm{Tr}\left(\boldsymbol{\mathcal{M}}^{2}\right),$
(40)

where

$\mathcal{S}_{ij}=\frac{\partial\langle U\rangle_{i}}{\partial
x_{j}}+\frac{\partial\langle U\rangle_{j}}{\partial
x_{i}},\qquad\mathcal{M}_{ij}=\frac{\partial\langle B\rangle_{i}}{\partial
x_{j}}+\frac{\partial\langle B\rangle_{j}}{\partial x_{i}}.$ (41)

This leads to

$\displaystyle\frac{D}{Dt}\left\langle\mathbf{u}^{\prime}\cdot\mathbf{b}^{\prime}\right\rangle=$
$\displaystyle-\alpha\left(\left\langle\mathbf{B}\right\rangle\cdot\left\langle\mathbf{W}\right\rangle+2\left\langle\mathbf{B}\right\rangle\cdot\boldsymbol{\Omega}\right)+\left(\beta+\zeta\right)\left(\left\langle\mathbf{J}\right\rangle\cdot\left\langle\mathbf{W}\right\rangle+2\left\langle\mathbf{J}\right\rangle\cdot\boldsymbol{\Omega}\right)$
$\displaystyle-\gamma\left(\left\langle\mathbf{W}\right\rangle+2\boldsymbol{\Omega}\right)^{2}-\frac{7}{10}\gamma\mathrm{Tr}\left(\boldsymbol{\mathcal{M}}^{2}\right)$
$\displaystyle+\frac{7}{10}\beta\mathrm{Tr}\left(\boldsymbol{\mathcal{S}}\cdot\boldsymbol{\mathcal{M}}\right)+\left(\nabla\zeta\times\left\langle\mathbf{B}\right\rangle\right)\cdot\left(\left\langle\mathbf{W}\right\rangle+2\boldsymbol{\Omega}\right)$
$\displaystyle+\nabla\cdot\left[\left\langle\left(-\Pi^{\prime}+\frac{\mathbf{u}^{\prime
2}+\mathbf{b}^{\prime
2}}{2}\right)\mathbf{b}^{\prime}\right\rangle+\left\langle\frac{\mathbf{u}^{\prime
2}+\mathbf{b}^{\prime
2}}{2}\right\rangle\left\langle\mathbf{B}\right\rangle-\nu\left\langle\mathbf{w}^{\prime}\times\mathbf{b}^{\prime}\right\rangle+\eta\left\langle\mathbf{u}^{\prime}\times\mathbf{j}^{\prime}\right\rangle\right]$
$\displaystyle-\left(\nu+\eta\right)\left\langle\mathbf{w}^{\prime}\cdot\mathbf{j}^{\prime}\right\rangle.$
(42)

Of course if the turbulence is stirred with some forcing $\mathbf{f}$ there is
also another production term
$\left\langle\mathbf{f}\cdot\mathbf{b}^{\prime}\right\rangle$.

According to (29) the magnitude of the non-equilibrium $\alpha$-effect depends
on both, the kinetic and cross helicities and their history. The total
$\alpha$-effect consists of the two contributions $\alpha=\alpha_{\rm
S}+\alpha_{\rm X}$, where the standard one can be assumed proportional to the
kinetic helicity, $\alpha_{\rm
S}\approx-\tau_{t}\langle\mathbf{u}^{\prime}\cdot\mathbf{w}^{\prime}\rangle/3$.
The final balance between the two contributions $\alpha_{\rm S}$ and
$\alpha_{\rm X}$ determines whether the $\alpha$ coefficient has the same or
the opposite sign to the kinetic helicity. The effect of different terms in
the equation (42) has been studied in the aforementioned works of Yokoi and
Hamba (2007), Yokoi (2011), Yokoi and Balarac (2011), Yokoi and Hoshino (2011)
and Yokoi (2013) under some simplifying assumptions, in particular under the
neglect of the effects from the $G_{ub}$ and $G_{bu}$ response functions,
responsible for the non-equilibrium effects studied here. Assuming that
$\alpha=-\tau_{t}\langle\mathbf{u}^{\prime}\cdot\mathbf{w}^{\prime}\rangle/3$
they showed, that the first term
$-\alpha\langle\mathbf{B}\rangle\cdot\langle\mathbf{W}\rangle$ always leads to
destruction of the cross helicity. This is no longer true, when $\Upsilon\neq
0$ in non-equilibrium turbulence, since depending on the balance between the
$\alpha_{\rm S}$ and $\alpha_{\rm X}$ terms the term
$-\alpha\langle\mathbf{B}\rangle\cdot\langle\mathbf{W}\rangle$ in (42) may
either amplify or destroy the cross helicity. Furthermore, Yokoi and Hoshino
(2011) take $\beta+\zeta\sim\langle\mathbf{u}^{\prime 2}\rangle$ and
$\gamma\sim\langle\mathbf{u}^{\prime}\cdot\mathbf{b}^{\prime}\rangle$ which
allows them to identify another two terms that always lead to destruction of
the cross-helicity, namely

$-\gamma\left(\left\langle\mathbf{W}\right\rangle+2\boldsymbol{\Omega}\right)^{2}-\frac{7}{10}\gamma\mathrm{Tr}\left(\boldsymbol{\mathcal{M}}^{2}\right).$
(43)

In addition Yokoi and Hoshino (2011) have described various situations when
the terms
$\left(\beta+\zeta\right)\left\langle\mathbf{J}\right\rangle\cdot\left\langle\mathbf{W}\right\rangle$,
$\beta\mathrm{Tr}\left(\boldsymbol{\mathcal{S}}\cdot\boldsymbol{\mathcal{M}}\right)$
and $\nabla\cdot\left[\left\langle\mathbf{u}^{\prime 2}+\mathbf{b}^{\prime
2}\right\rangle\left\langle\mathbf{B}\right\rangle\right]$ may lead to
production of the cross-helicity in the geometry of the tokamak devices.
Finally, in the term
$-2\alpha\left\langle\mathbf{B}\right\rangle\cdot\boldsymbol{\Omega}$ we
recover the action of the mean field component parallel to the rotation
vector, as in the FOSA approach.

The action of all the other terms in (42) is difficult to predict and, in
general, they can either amplify or destroy the cross-helicity in developed
turbulence. The final balance on the right hand side of (42) depends on many
dynamical features of turbulence and is expected to be time dependent.
Therefore in order to demonstrate the possibility of coexistence of the cross-
and kinetic helicities in magnetized turbulence we have performed numerical
simulations of the compressible version of equations (1a–c) in the presence of
gravity, density stratification and an imposed magnetic field
$\mathbf{g}\parallel\nabla\rho\parallel\mathbf{B}_{0}\parallel\boldsymbol{\Omega}$
in a periodic box with the use of the Pencil Code (Pencil Code Collaboration)
with $256^{3}$ mesh points; stress-free and perfectly conducting boundary
conditions were imposed at the top and bottom boundaries; see Appendix D. The
action of rotation along the direction of stratification leads to kinetic
helicity (see figure 5 of Jabbari et al. 2014 for simulation results) and the
action of a magnetic field along the direction of stratification leads to
cross helicity (Rüdiger et al. 2011).

The values of the physical parameters are as follows: working again with the
unscaled magnetic field $B=0.01\,c_{\rm s}\sqrt{\mu_{0}\bar{\rho}}$ and
gravity $g=1\,c_{\rm s}^{2}k_{1}$ (these are varied in other runs), where
$c_{\rm s}$ is the speed of sound, $\Omega=0.5\,c_{\rm s}k_{1}$ is kept fixed
in all runs, $\bar{\rho}$ is the mean density and $k_{1}$ is the box
wavenumber; the remaining parameters, which are constant for all runs are
listed in table 1, where we used the Alfvén speed
$v_{A}=B/\sqrt{\mu_{0}\overline{\rho}}$ to quantify the strength of the
imposed and rms magnetic fields through $v_{A0}$ and $v_{A}^{\rm rms}$,
respectively. The results obtained for two values of the imposed magnetic
field which differ by an order of magnitude at variable gravity strength are
depicted in figure 1 and tables 1 and 2; see also Appendix E for additional
figures. The normalized helicities,
$\langle\mathbf{u}^{\prime}\cdot\mathbf{b}^{\prime}\rangle/\sqrt{\langle
u^{\prime 2}\rangle\langle b^{\prime 2}\rangle}$ and
$\langle\mathbf{u}^{\prime}\cdot\mathbf{w}^{\prime}\rangle/\sqrt{\langle
u^{\prime 2}\rangle\langle w^{\prime 2}\rangle}$ are plotted against time and
they are both clearly non-zero in all the considered cases; the cross-helicity
is plotted in red and the blue lines correspond to the kinetic helicity
whereas their time averages are marked with white lines. In addition, only for
the sake of reference, the figures also show the estimates of the non-
equilibrium effect in the form

$\alpha_{\rm
neq}\approx-\frac{1}{3}\frac{\langle\mathbf{u}^{\prime}\cdot\mathbf{b}^{\prime}\rangle}{\sqrt{\langle
u^{\prime 2}\rangle\langle b^{\prime
2}\rangle}}\int_{-\infty}^{\tau}\mathrm{d}\tau_{1}\left[\langle\mathbf{u}^{\prime}\left(\mathbf{x},\tau\right)\cdot\mathbf{j}^{\prime}\left(\mathbf{x},\tau_{1}\right)\rangle-\langle\mathbf{u}^{\prime}\left(\mathbf{x},\tau_{1}\right)\cdot\mathbf{j}^{\prime}\left(\mathbf{x},\tau\right)\rangle\right],$
(44)

which can be compared with the following standard estimate of the
$\alpha$-effect, associated with the presence of the kinetic and current
helicities

$\alpha_{S}\approx-\frac{1}{3}\tau_{t}\left(\langle\mathbf{u}^{\prime}\cdot\mathbf{w}^{\prime}\rangle-\langle\mathbf{b}^{\prime}\cdot\mathbf{j}^{\prime}\rangle\right),$
(45)

where $\tau_{t}=1/u_{\rm rms}k_{\rm f}$ is the turnover time of most energetic
turbulent eddies, with $u_{\rm rms}=\sqrt{\langle u^{\prime 2}\rangle}$ and
$k_{\rm f}=30\,k_{1}$ denoting the forcing the wavenumber ($k_{1}=2\pi/L$ is
the wavenumber of the box of length $L$).

Figure 1:  Results for Run A of numerical simulations of MHD turbulence in a
periodic box with the use of the Pencil Code. The upper panel shows the time
evolution of the normalized cross-helicity,
$\langle\mathbf{u}^{\prime}\cdot\mathbf{b}^{\prime}\rangle/\sqrt{\langle
u^{\prime 2}\rangle\langle b^{\prime 2}\rangle}$ (red) and the kinetic
helicity
$\langle\mathbf{u}^{\prime}\cdot\mathbf{w}^{\prime}\rangle/\sqrt{\langle
u^{\prime 2}\rangle\langle w^{\prime 2}\rangle}$ (blue); the time averages are
marked with the white continuous lines and the green line depicts the current
helicity
$\langle\mathbf{b}^{\prime}\cdot\mathbf{j}^{\prime}\rangle/\sqrt{\langle
u^{\prime 2}\rangle\langle w^{\prime 2}\rangle}$. The estimates of the
coefficients $\alpha_{\rm neq}$ (44) and $\alpha_{\rm S}$ (45) as functions of
time (normalized with $\alpha_{0}=u_{\rm rms}/3$) are provided in the bottom
panel in red and orange respectively; the continuous white line marks the time
averaged value of $\alpha_{\rm S}/\alpha_{0}$ and the dashed white line the
time average of $\alpha_{\rm neq}/\alpha_{0}$.

Although in the numerically studied cases the statistical non-stationarity of
turbulence is rather weak and the estimate of the $\alpha_{\rm neq}$
coefficient is always at least an order of magnitude weaker than $\alpha_{\rm
S}$, the former is clearly different from zero and its relative importance
seems to correlate with the magnitude of the cross-helicity. The relative
enhancement of the $\alpha_{\rm neq}$-effect visible for a stronger magnetic
field (Run B) and weaker gravity (Run E) corresponds to the enhancement of the
cross-helicity with respect to the kinetic one. Of course in the latter case
(see figure 2), although the $\alpha_{\rm neq}$ coefficient has the largest
relative magnitude it also has a different sign than $\alpha_{\rm S}$, hence
in this case the non-equilibrium effects tend to suppress the standard dynamo
effect. In figure 3 we see, that weak magnetic field and strong gravity have
suppressed the non-equilibrium effect to a very small relative magnitude.

| $\frac{g}{c_{\rm s}^{2}k_{1}}$ | $\frac{v_{A0}}{c_{\rm s}}$ | $\frac{\langle\mathbf{u}^{\prime}\cdot\mathbf{b}^{\prime}\rangle}{\sqrt{\langle u^{\prime 2}\rangle\langle b^{\prime 2}\rangle}}$ | $\frac{\langle\mathbf{u}^{\prime}\cdot\mathbf{w}^{\prime}\rangle}{\sqrt{\langle u^{\prime 2}\rangle\langle w^{\prime 2}\rangle}}$ | $\frac{\langle\mathbf{b}^{\prime}\cdot\mathbf{j}^{\prime}\rangle}{\sqrt{\langle u^{\prime 2}\rangle\langle w^{\prime 2}\rangle}}$ | $\frac{\alpha_{\rm neq}}{\alpha_{0}}$ | $\frac{\alpha_{\rm S}}{\alpha_{0}}$ | $\frac{u_{\rm rms}}{c_{\rm s}}$ | $\frac{v_{A}^{\rm rms}}{c_{\rm s}}$ |
---|---|---|---|---|---|---|---|---|---|---
C | 0.5 | 0.01 | $-9.8\times 10^{-3}$ | $-1.6\times 10^{-2}$ | $-2.0\times 10^{-4}$ | $7.8\times 10^{-4}$ | $1.8\times 10^{-2}$ | 0.10 | 0.03 |
A | 1.0 | 0.01 | $-1.7\times 10^{-2}$ | $-3.0\times 10^{-2}$ | $-3.3\times 10^{-4}$ | $1.1\times 10^{-3}$ | $3.5\times 10^{-2}$ | 0.11 | 0.04 |
D | 2.0 | 0.01 | $-2.0\times 10^{-2}$ | $-3.6\times 10^{-2}$ | $-2.8\times 10^{-4}$ | $6.1\times 10^{-4}$ | $4.1\times 10^{-2}$ | 0.16 | 0.04 |
E | 0.5 | 0.10 | $-5.5\times 10^{-2}$ | $-1.9\times 10^{-2}$ | $-6.2\times 10^{-4}$ | $-5.6\times 10^{-3}$ | $1.5\times 10^{-2}$ | 0.08 | 0.07 |
B | 1.0 | 0.10 | $-5.3\times 10^{-2}$ | $-3.2\times 10^{-2}$ | $-1.2\times 10^{-2}$ | $2.3\times 10^{-3}$ | $1.8\times 10^{-2}$ | 0.09 | 0.12 |
Table 1: Summary of the simulation results for Runs0A–E. | $k_{\rm f}$ | $\frac{\langle\mathbf{u}^{\prime}\cdot\mathbf{b}^{\prime}\rangle}{\sqrt{\langle u^{\prime 2}\rangle\langle b^{\prime 2}\rangle}}$ | $\frac{\langle\mathbf{u}^{\prime}\cdot\mathbf{w}^{\prime}\rangle}{\sqrt{\langle u^{\prime 2}\rangle\langle w^{\prime 2}\rangle}}$ | $\frac{\langle\mathbf{b}^{\prime}\cdot\mathbf{j}^{\prime}\rangle}{\sqrt{\langle u^{\prime 2}\rangle\langle w^{\prime 2}\rangle}}$ | $\frac{\alpha_{\rm neq}}{\alpha_{0}}$ | $\frac{\alpha_{\rm S}}{\alpha_{0}}$ | $\frac{u_{\rm rms}}{c_{\rm s}}$ | $\frac{v_{A}^{\rm rms}}{c_{\rm s}}$ |
---|---|---|---|---|---|---|---|---|---
A | 30 | $-1.7\times 10^{-2}$ | $-3.0\times 10^{-2}$ | $-3.3\times 10^{-4}$ | $1.1\times 10^{-3}$ | $3.5\times 10^{-2}$ | 0.11 | 0.04 |
A2 | 10 | $-1.3\times 10^{-1}$ | $-1.2\times 10^{-1}$ | $1.3\times 10^{-3}$ | $-1.7\times 10^{-2}$ | $6.9\times 10^{-2}$ | 0.12 | 0.12 |
A3 | 3 | $-6.4\times 10^{-2}$ | $-2.1\times 10^{-1}$ | $-3.0\times 10^{-2}$ | $-6.0\times 10^{-3}$ | $5.5\times 10^{-2}$ | 0.19 | 0.09 |
Table 2: Summary of the simulation results for Runs0A, A2, and A3.

At smaller scale separation, i.e., for smaller values of $k_{\rm f}$, we
expect the turbulence to be more intermittent and degree of non-stationarity
to be enhanced. To address this possibility, we have performed additional
simulations for smaller values of $k_{\rm f}$ with the other parameters being
the same as for Run A. The results shown in table 2 do show that $\alpha_{\rm
neq}$ is twice as large when $k_{\rm f}$ is reduced from 30 to 10, but an
additional decrease of $k_{\rm f}$ from 10 to 3 does not lead to an additional
increase of $\alpha_{\rm neq}$. To some extent, however, this is caused by the
normalization by $\alpha_{0}$, which has increased by about 60%.

We conclude, that in fully developed helical turbulence, that is in turbulence
with strong kinetic helicity, the cross-helicity is rather likely to be
produced as well and at least for some periods of time the two helicities can
coexist.

##  5 Conclusions

We have analysed the hydromagnetic dynamo process in non-equilibrium
turbulence. It was shown that in non-equilibrium MHD turbulence the effect of
the infinitesimal-impulse cross responses
$\mathbf{u}^{\prime}\leftrightarrow\mathbf{b}^{\prime}$ is pronounced, which
vanishes in stationary state. This creates additional terms in the expression
for the large-scale electromotive force.

The main conclusion is that the non-equilibrium effects in MHD turbulence
modify the $\alpha$-effect by introducing a correction dependent on the square
of the non-dimensional cross-helicity
$\Upsilon=\langle\mathbf{u}^{\prime}\cdot\mathbf{b}^{\prime}\rangle/\sqrt{\langle
u^{\prime 2}\rangle\langle b^{\prime 2}\rangle}$, the kinetic helicity and
their history in the MHD turbulence, which takes the form provided in (29).
This requires coexistence of both, the kinetic and cross-helicities in the
turbulent flow. The discussion of the production mechanisms of the cross-
helicity, provided in section 4 and the results of numerical simulations, lead
to a conclusion that such coexistence is possible and perhaps even ubiquitous
in many natural systems. Simple strong production mechanisms have been
identified already and thoroughly discussed in earlier works, e.g. Yokoi and
Hoshino (2011).

The non-equilibrium effects in turbulence affect also other components of the
mean EMF (12), that is the turbulent diffusivity $\beta$ and the coefficients
$\zeta$ and $\gamma$ in a non-trivial way, through the effect of the Green’s
cross-response functions $G_{ub}$ and $G_{bu}$. This interesting topic should
be investigated in more detail in future studies.

## Acknowledgements

We would like to thank the Isaac Newton Institute for Mathematical Sciences,
Cambridge, for support and hospitality during the programme “Frontiers in
dynamo theory: from the Earth to the stars” (DYT2) where much of the work on
this paper was undertaken.

## Funding

KAM was supported by a subsidy from the Polish Ministry of Education and
Science for the Institute of Geophysics, Polish Academy of Sciences. NY was
supported by the Japan Society of the Promotion of Science (JSPS) Grants-in-
Aid for Scientific Research JP18H01212. We also acknowledge the support of the
EPSRC grant no EP/R014604/1 and the Swedish Research Council (Vetenskapsrådet,
2019-04234). Nordita is sponsored by Nordforsk. We acknowledge the allocation
of computing resources provided by the Swedish National Allocations Committee
at the Center for Parallel Computers at the Royal Institute of Technology in
Stockholm and Linköping.

## Declaration of Interests

The authors report no conflict of interest.

## Data availability statement

The data that support the findings of this study are openly available on
Zenodo at doi:10.5281/zenodo.7683615 (v2023.02.28). All calculations have been
performed with the Pencil Code; DOI:10.5281/zenodo.3961647.

## Author ORCID

K. A. Mizerski, https://orcid.org/0000-0003-0106-675X
N. Yokoi, https://orcid.org/0000-0002-5242-7634
A. Brandenburg, https://orcid.org/0000-0002-7304-021X

##  Appendix A Outline of the two-scale direct-interaction approximation
(TSDIA) with self- and cross-interaction response functions for the velocity
and magnetic fields.

The two-scale direct-interaction approximation (TSDIA) is a combination of the
direct-interaction approximation (DIA) for strongly nonlinear homogeneous
isotropic turbulence and the multiple-scale analysis with the derivative
expansion with respect to the large-scale inhomogeneity. The TSDIA provides a
powerful tool for investigating strongly-nonlinear turbulence with large-scale
inhomogeneities. In applying the TSDIA scheme to the magnetohydrodynamic
turbulence, the Elsässer variable formulation has been often adopted. In this
formulation, symmetries of the velocity and magnetic-field equations are fully
utilized, which reduces the complexities in treating the original MHD
equations. The correspondence between the Elsässer variable formulation and
the usual velocity–magnetic-field formulation in the TSDIA has been discussed
in some literature (Yoshizawa 1998, Hamba & Sato 2008, Yokoi 2013). Here, we
present the outline of the TSDIA formulation under the velocity and magnetic-
field variables with special references to the self- and cross-interaction
response functions in the MHD turbulence. For the outline of the DIA in the
context of the TSDIA, the reader is referred to textbooks such as Yoshizawa
(1998) and Yokoi (2020).

#### Wave-number space equations

We introduce the Fourier representation concerning the fast space variable
$\xi$ as

$f^{\prime}(\mbox{\boldmath$\xi$},{\bf{X}};\tau,T)=\int
d{\bf{k}}f({\bf{k}},{\bf{X}};\tau,T)\exp[-i{\bf{k}}\cdot(\mbox{\boldmath$\xi$}-\langle{\bf{U}}\rangle\tau)],$
(46)

where the Fourier transform of the fast variable is taken in the frame co-
moving with the local mean velocity $\langle{\bf{U}}\rangle$. Hereafter, for
the sake of simplicity of notation, the arguments of the slow variable for the
fluctuation field $f(\mbox{\boldmath$\xi$},{\bf{X}};\tau,T)$ is suppressed and
just denoted as $f(\mbox{\boldmath$\xi$};\tau)$.

The system of two-scale differential equations under the velocity and
magnetic-field variables in the wavenumber space is written as

$\displaystyle\frac{\partial u^{i}({\bf{k}};\tau)}{\partial\tau}+\nu
k^{2}u^{i}({\bf{k}};\tau)+ik^{j}\langle B\rangle^{j}b^{i}({\bf{k}};\tau)$
$\displaystyle-iM^{ij\ell}({\bf{k}})\iint d{\bf{p}}d{\bf{q}}\
\delta({\bf{k}}-{\bf{p}}-{\bf{q}})\times\left[{u^{j}({\bf{p}};\tau)u^{\ell}({\bf{q}};\tau)-b^{j}({\bf{p}};\tau)b^{\ell}({\bf{q}};\tau)}\right]$
$\displaystyle\hskip
80.0pt=\delta\left[{-D^{ij}({\bf{k}})\frac{\widehat{D}u^{j}({\bf{k}};\tau)}{DT_{\rm{I}}}-D^{ij}({\bf{k}})u^{m}({\bf{k}};\tau)\left({\frac{\partial\langle
U\rangle^{j}}{\partial
X^{m}}+\epsilon^{mj\ell}\Omega_{0}^{\ell}}\right)}\right.$
$\displaystyle\hskip 80.0pt\left.{\hskip 22.0pt+\langle
B\rangle^{j}\frac{\partial b^{i}({\bf{k}};\tau)}{\partial
X_{\rm{I}}^{j}}+D^{ij}({\bf{k}})b^{m}({\bf{k}};\tau)\frac{\partial\langle
B\rangle^{j}}{\partial X^{m}}}\right],$ (47)
$-ik^{j}u^{j}({\bf{k}};\tau)+\delta\frac{\partial
u^{j}({\bf{k}};\tau)}{\partial X^{j}}=0,$ (48) $\displaystyle\frac{\partial
b^{i}({\bf{k}};\tau)}{\partial\tau}+\eta
k^{2}b^{i}({\bf{k}};\tau)+ik^{j}\langle B\rangle^{j}u^{i}({\bf{k}};\tau)$
$\displaystyle+iN^{ij\ell}({\bf{k}})\iint d{\bf{p}}d{\bf{q}}\
\delta({\bf{k}}-{\bf{p}}-{\bf{q}})\times\left[{b^{j}({\bf{p}};\tau)u^{\ell}({\bf{q}};\tau)-u^{j}({\bf{p}};\tau)b^{\ell}({\bf{q}};\tau)}\right]$
$\displaystyle\hskip
80.0pt=\delta\left[{-D^{ij}({\bf{k}})\frac{\widehat{D}b^{j}({\bf{k}};\tau)}{DT_{\rm{I}}}+D^{ij}({\bf{k}})b^{m}({\bf{k}};\tau)\left({\frac{\partial\langle
U\rangle^{j}}{\partial
X^{m}}+\epsilon^{mj\ell}\Omega_{0}^{\ell}}\right)}\right.$
$\displaystyle\hskip 80.0pt\left.{\hskip 22.0pt+\langle
B\rangle^{j}\frac{\partial u^{i}({\bf{k}};\tau)}{\partial
X_{\rm{I}}^{j}}-D^{ij}({\bf{k}})u^{m}({\bf{k}};\tau)\frac{\partial\langle
B\rangle^{j}}{\partial X^{m}}}\right],$ (49)
$-ik^{j}b^{j}({\bf{k}};\tau)+\delta\frac{\partial
b^{j}({\bf{k}};\tau)}{\partial X^{j}}=0,$ (50)

where

$\left({\nabla_{{\bf{X}}{\rm{I}}},\frac{D}{DT_{\rm{I}}}}\right)=\exp\left({-i{\bf{k}}\cdot{\bf{U}}\tau}\right)\left({\nabla_{\bf{X}},\frac{D}{DT}}\right)\exp\left({i{\bf{k}}\cdot{\bf{U}}\tau}\right)$
(51)

is the differential operators in the interaction representation. Here in (47)
and (49),

$M^{ijk}({\bf{k}})=\frac{1}{2}\left(k^{j}D^{ik}({\bf{k}})+k^{k}D^{ij}({\bf{k}})\right),$
(52)

with the solenoidal projection operator

$D^{ij}({\bf{k}})=\delta^{ij}-\frac{k^{i}k^{j}}{k^{2}},$ (53)

and

$N^{ijk}({\bf{k}})=k^{j}\delta^{ik}-k^{k}\delta^{ij}.$ (54)

The operators $M$ and $N$ are point vertices showing the wave-number
conservation among the nonlinear mode coupling with
$\delta({\bf{k}}-{\bf{p}}-{\bf{q}})$.

In (47) and (49), to keep the material derivatives objective (invariant with
respect to rotations), we adopt a co-rotational derivative

$\frac{\widehat{D}u^{\prime}{}^{i}}{DT}=\frac{\partial
u^{\prime}{}^{i}}{\partial T}+\langle U\rangle^{j}\frac{\partial
u^{\prime}{}^{i}}{\partial
X^{j}}+\epsilon^{jik}\Omega_{0}^{k}u^{\prime}{}^{j}$ (55)

with

$\mbox{\boldmath$\Omega$}_{0}=\mbox{\boldmath$\Omega$}/\delta$ (56)

in place of the Lagrange or advective derivative

$\frac{Du^{\prime}{}^{i}}{DT}=\frac{\partial u^{\prime}{}^{i}}{\partial
t}+\langle U\rangle^{j}\frac{\partial u^{\prime}{}^{i}}{\partial x^{j}},$ (57)

which is not objective with respect to a rotation.

#### Scale-parameter expansion

We expand a field $f({\bf{k}};\tau)$ with respect to the scale parameter
$\delta$, and further expand each field by the external field (the mean
magnetic field in the present case) as

$\displaystyle f^{i}({\bf{k}};\tau)$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\delta^{n}f_{n}^{i}({\bf{k}};\tau)-\sum_{n=0}^{\infty}\delta^{n+1}i\frac{k^{i}}{k^{2}}\frac{\partial}{\partial
X_{\rm{I}}^{j}}f_{n}^{j}({\bf{k}};\tau)$ (58) $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\delta^{n}f_{nm}^{i}({\bf{k}};\tau)-\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\delta^{n+1}i\frac{k^{i}}{k^{2}}\frac{\partial}{\partial
X_{\rm{I}}^{j}}f_{nm}^{j}({\bf{k}};\tau).$

In this two-scale formulation, inhomogeneities and anisotropies enter with the
scale parameter $\delta$ and the external parameters $\langle{\bf{B}}\rangle$
in higher-order fields. The lowest-order fields $f_{00}$ fields correspond to
the homogeneous and isotropic turbulence.

Using the expansion (58), we write the equations of each order in matrix form.
With the abbreviated form of the spectral integral

$\int_{\Delta}=\iint d{\bf{p}}d{\bf{q}}\ \delta({\bf{k}}-{\bf{p}}-{\bf{q}}),$
(59)

the $f_{00}({\bf{k}};\tau)$ equations are given as

$\displaystyle\left({\begin{array}[]{c}0\\\
0\rule{0.0pt}{21.52771pt}\end{array}}\right)={\left({\begin{array}[]{cc}\frac{\partial}{\partial\tau}+\nu
k^{2}&0\\\ 0&\frac{\partial}{\partial\tau}+\eta
k^{2}\end{array}}\right)}\left({\begin{array}[]{cc}u_{00}^{i}({\bf{k}};\tau)\\\
b_{00}^{i}({\bf{k}};\tau)\rule{0.0pt}{21.52771pt}\end{array}}\right)\hskip
142.26378pt$ (66)
$\displaystyle{+i\left({\begin{array}[]{cc}-M^{ij\ell}({\bf{k}})\int_{\Delta}u_{00}^{j}({\bf{p}};\tau)&M^{ij\ell}({\bf{k}})\int_{\Delta}b_{00}^{j}({\bf{p}};\tau)\\\
N^{ij\ell}({\bf{k}})\int_{\Delta}b_{00}^{j}({\bf{p}};\tau)&-N^{ij\ell}({\bf{k}})\int_{\Delta}u_{00}^{j}({\bf{p}};\tau)\end{array}}\right)\left({\begin{array}[]{c}u_{00}^{\ell}({\bf{q}};\tau)\\\
b_{00}^{\ell}({\bf{q}};\tau)\rule{0.0pt}{21.52771pt}\end{array}}\right)},$
(71)

the $f_{01}({\bf{k}};\tau)$ equations are given as

$\displaystyle{\left({\begin{array}[]{cc}\frac{\partial}{\partial\tau}+\nu
k^{2}&0\\\ 0&\frac{\partial}{\partial\tau}+\eta
k^{2}\end{array}}\right)}\left({\begin{array}[]{cc}u_{01}^{i}({\bf{k}};\tau)\\\
b_{01}^{i}({\bf{k}};\tau)\rule{0.0pt}{21.52771pt}\end{array}}\right)$ (76)
$\displaystyle{+i\left({\begin{array}[]{cc}-2M^{ij\ell}({\bf{k}})\int_{\Delta}u_{00}^{j}({\bf{p}};\tau)&2M^{ij\ell}({\bf{k}})\int_{\Delta}b_{00}^{j}({\bf{p}};\tau)\\\
N^{ij\ell}({\bf{k}})\int_{\Delta}b_{00}^{j}({\bf{p}};\tau)&-N^{ij\ell}({\bf{k}})\int_{\Delta}u_{00}^{j}({\bf{p}};\tau)\end{array}}\right)\left({\begin{array}[]{c}u_{01}^{\ell}({\bf{q}};\tau)\\\
b_{01}^{\ell}({\bf{q}};\tau)\rule{0.0pt}{21.52771pt}\end{array}}\right)}$ (81)
$\displaystyle\hskip 85.35826pt=-ik^{j}\langle
B\rangle^{j}\left({\begin{array}[]{cc}0&1\\\
1&0\rule{0.0pt}{21.52771pt}\end{array}}\right)\left({\begin{array}[]{c}u_{00}^{i}({\bf{k}};\tau)\\\
b_{00}^{i}({\bf{k}};\tau)\rule{0.0pt}{21.52771pt}\end{array}}\right)\equiv\left({\begin{array}[]{c}F_{01u}^{i}\\\
F_{01b}^{i}\rule{0.0pt}{21.52771pt}\end{array}}\right),\qquad\qquad\qquad$
(88)

and the $f_{10}({\bf{k}};\tau)$ equations are

$\displaystyle{\left({\begin{array}[]{cc}\frac{\partial}{\partial\tau}+\nu
k^{2}&0\\\ 0&\frac{\partial}{\partial\tau}+\eta
k^{2}\end{array}}\right)}\left({\begin{array}[]{cc}u_{10}^{i}({\bf{k}};\tau)\\\
b_{10}^{i}({\bf{k}};\tau)\rule{0.0pt}{21.52771pt}\end{array}}\right)$ (93)
$\displaystyle{+i\left({\begin{array}[]{cc}-2M^{ij\ell}({\bf{k}})\int_{\Delta}u_{00}^{j}({\bf{p}};\tau)&2M^{ij\ell}({\bf{k}})\int_{\Delta}b_{00}^{j}({\bf{p}};\tau)\\\
N^{ij\ell}({\bf{k}})\int_{\Delta}b_{00}^{j}({\bf{p}};\tau)&-N^{ij\ell}({\bf{k}})\int_{\Delta}u_{00}^{j}({\bf{p}};\tau)\end{array}}\right)\left({\begin{array}[]{c}u_{10}^{\ell}({\bf{q}};\tau)\\\
b_{10}^{\ell}({\bf{q}};\tau)\rule{0.0pt}{21.52771pt}\end{array}}\right)}$ (98)
$\displaystyle\hskip 14.22636pt={\langle B\rangle^{j}\frac{\partial}{\partial
X_{\rm{I}}^{j}}\left({\begin{array}[]{cc}0&1\\\
1&0\rule{0.0pt}{21.52771pt}\end{array}}\right)\left({\begin{array}[]{c}u_{00}^{i}({\bf{k}})\\\
b_{00}^{i}({\bf{k}})\rule{0.0pt}{21.52771pt}\end{array}}\right)}-D^{ij}({\bf{k}})\frac{\widehat{D}}{DT_{\rm{I}}}{\left({\begin{array}[]{cc}1&0\\\
0&1\rule{0.0pt}{21.52771pt}\end{array}}\right)\left({\begin{array}[]{c}u_{00}^{j}({\bf{k}})\\\
b_{00}^{j}({\bf{k}})\rule{0.0pt}{21.52771pt}\end{array}}\right)}$ (107)
$\displaystyle\hskip
28.45274pt+{\left({\begin{array}[]{cc}-D^{ij}({\bf{k}})\left({\frac{\partial\langle
U\rangle^{j}}{\partial X^{\ell}}+\epsilon^{\ell
jn}\Omega_{0}^{n}}\right)&D^{ij}({\bf{k}})\frac{\partial\langle
B\rangle^{j}}{\partial X^{\ell}}\\\ -D^{ij}({\bf{k}})\frac{\partial\langle
B\rangle^{j}}{\partial X^{\ell}}&D^{ij}({\bf{k}})\left({\frac{\partial\langle
U\rangle^{j}}{\partial X^{\ell}}+\epsilon^{\ell
jn}\Omega_{0}^{n}}\right)\end{array}}\right)\left({\begin{array}[]{c}u_{00}^{\ell}({\bf{k}};\tau)\\\
b_{00}^{\ell}({\bf{k}};\tau)\rule{0.0pt}{21.52771pt}\end{array}}\right)}$
(112) $\displaystyle\hskip
14.22636pt\equiv\left({\begin{array}[]{c}F_{10u}^{i}\\\
F_{10b}^{i}\rule{0.0pt}{21.52771pt}\end{array}}\right),$ (115)

where, $F_{01u}$, $F_{01b}$, $F_{10u}$, and $F_{10b}$ denote each component of
the second right-hand sides (r.h.s.) of (88) and (115). They can be regarded
as the forcing for the evolution equations of $f_{01}({\bf{k}};\tau)$ and
$f_{10}({\bf{k}};\tau)$, respectively.

#### Introduction of Green’s functions

For the purpose of solving these differential equations, we introduce the
Green’s functions associated with (71). We consider the response of the
turbulence to an infinitesimal disturbance. Reflecting the structure of the
MHD equations and the field expansion (58), the left-hand side of the
linearized differential equations for the Green’s function is in the same form
as the l.h.s. of (88) and (115) or the differential operators to the
$f_{01}({\bf{k}};\tau)$ and $f_{10}({\bf{k}};\tau)$ fields. In order to treat
mutual interaction between the velocity and magnetic field, we consider four
Green’s functions; the Green function $G_{uu}$ representing the response of
the velocity field ${\bf{u}}$ to the velocity perturbation ${\bf{u}}$,
$G_{ub}$ the response of ${\bf{u}}$ to the magnetic perturbation ${\bf{b}}$,
$G_{bu}$ the response of ${\bf{b}}$ to the velocity perturbation ${\bf{u}}$,
and $G_{bb}$ the response of magnetic field ${\bf{b}}$ to the magnetic
perturbation ${\bf{b}}$. From the l.h.s. of (88) and (115) we construct the
system of equations representing the responses to the infinitesimal forcing.
It follows that these four Green’s functions should be defined by their
evolution equations as

$\displaystyle{\left({\begin{array}[]{cc}\frac{\partial}{\partial\tau}+\nu
k^{2}&0\\\ 0&\frac{\partial}{\partial\tau}+\eta
k^{2}\end{array}}\right)}\left({\begin{array}[]{cc}G_{uu}^{ij}&G_{bu}^{ij}\\\
G_{ub}^{ij}&G_{bb}^{ij}\rule{0.0pt}{21.52771pt}\end{array}}\right)$ (120)
$\displaystyle{+i\left({\begin{array}[]{cc}-2M^{ikm}\int_{\Delta}u_{00}^{k}&2M^{ikm}\int_{\Delta}b_{00}^{k}\\\
N^{ikm}\int_{\Delta}b_{00}^{k}&-N^{ikm}\int_{\Delta}u_{00}^{k}\end{array}}\right)\left({\begin{array}[]{cc}G_{uu}^{mj}&G_{bu}^{mj}\\\
G_{ub}^{mj}&G_{bb}^{mj}\rule{0.0pt}{21.52771pt}\end{array}}\right)}=\delta^{ij}\delta(\tau-\tau^{\prime})\left({\begin{array}[]{cc}1&0\\\
0&1\rule{0.0pt}{21.52771pt}\end{array}}\right).$ (127)

Considering that the r.h.s. of (88) and (115) are the force terms, we formally
solve $f_{01}$ and $f_{10}$ fields with the aid of the Green’s functions. The
$f_{01}$ fields are expressed as

$\left({\begin{array}[]{c}u_{01}^{i}\\\
b_{01}^{i}\rule{0.0pt}{21.52771pt}\end{array}}\right)=\int_{-\infty}^{\tau}\\!\\!\\!d\tau_{1}\left({\begin{array}[]{cc}G_{uu}^{ij}&G_{ub}^{ij}\\\
G_{bu}^{ij}&G_{bb}^{ij}\rule{0.0pt}{21.52771pt}\end{array}}\right)\left({\begin{array}[]{c}F_{01u}^{j}\\\
F_{01b}^{j}\rule{0.0pt}{21.52771pt}\end{array}}\right).$ (128)

Note that ${\bf{u}}_{01}$ and ${\bf{b}}_{01}$ are expressed the
${\bf{b}}_{00}$ and ${\bf{u}}_{00}$ coupled with the mean magnetic field
$\langle{\bf{B}}\rangle$, respectively. As this result, ${\bf{u}}_{01}$ and
${\bf{b}}_{01}$ multiplied by ${\bf{b}}_{00}$ and ${\bf{u}}_{00}$ in an
external product manner will not contribute to the EMF.

On the other hand, the $f_{10}$ fields are expressed as

$\left({\begin{array}[]{c}u_{10}^{i}\\\
b_{10}^{i}\rule{0.0pt}{21.52771pt}\end{array}}\right)=\int_{-\infty}^{\tau}\\!\\!\\!d\tau_{1}\left({\begin{array}[]{cc}G_{uu}^{ij}&G_{ub}^{ij}\\\
G_{bu}^{ij}&G_{bb}^{ij}\rule{0.0pt}{21.52771pt}\end{array}}\right)\left({\begin{array}[]{c}F_{10u}^{j}\\\
F_{10b}^{j}\rule{0.0pt}{21.52771pt}\end{array}}\right).$ (129)

#### Statistical assumption on the basic fields

We assume that the basic or lowest-order fields are homogeneous and isotropic.

$\frac{\left\langle{\vartheta_{00}^{i}({\bf{k}};\tau)\chi_{00}^{j}({\bf{k}}^{\prime};\tau^{\prime})}\right\rangle}{\delta({\bf{k}}+{\bf{k}}^{\prime})}=D^{ij}({\bf{k}})Q_{\vartheta\chi}({\bf{k}};\tau,\tau^{\prime})+\frac{i}{2}\frac{k^{\ell}}{k^{2}}\epsilon^{ij\ell}H_{\vartheta\chi}({\bf{k}};\tau,\tau^{\prime}),$
(130)

where $\mbox{\boldmath$\vartheta$}_{00}$ and $\mbox{\boldmath$\chi$}_{00}$
represent one of ${\bf{u}}_{00}$ and ${\bf{b}}_{00}$, and the indices
$\vartheta$ and $\chi$ do one of $u$ and $b$. The Green’s functions are
written as

$\langle{G_{\vartheta\chi}^{ij}({\bf{k}};\tau,\tau^{\prime})}\rangle=D^{ij}({\bf{k}})G_{\vartheta\chi}({\bf{k}};\tau,\tau^{\prime}).$
(131)

The spectral functions, $Q_{uu}$, $Q_{bb}$, $Q_{ub}$, $H_{uu}$, $H_{bb}$,
$H_{ub}$, and $H_{bu}$, are related to the turbulent statistical quantities
(the turbulent kinetic energy, magnetic energy, cross helicity, kinetic
helicity, electric-current helicity, torsional correlations between velocity
and magnetic field) of the basic or lowest-order fields as

$\int d{\bf{k}}\
Q_{uu}(k;\tau,\tau)=\langle{{\bf{u}}^{\prime}_{00}{}^{2}}\rangle/2,$ (132)
$\int d{\bf{k}}\
Q_{bb}(k;\tau,\tau)=\langle{{\bf{b}}^{\prime}_{00}{}^{2}}\rangle/2,$ (133)
$\int d{\bf{k}}\
Q_{ub}(k;\tau,\tau)=\langle{{\bf{u}}^{\prime}_{00}\cdot{\bf{b}}^{\prime}_{00}}\rangle,$
(134) $\int d{\bf{k}}\
H_{uu}(k;\tau,\tau)=\langle{{\bf{u}}^{\prime}_{00}\cdot\mbox{\boldmath$\omega$}^{\prime}_{00}}\rangle,$
(135) $\int d{\bf{k}}\
H_{bb}(k;\tau,\tau)=\langle{{\bf{b}}^{\prime}_{00}\cdot{\bf{j}}^{\prime}_{00}}\rangle,$
(136) $\int d{\bf{k}}\
H_{ub}(k;\tau,\tau)=\langle{{\bf{u}}^{\prime}_{00}\cdot{\bf{j}}^{\prime}_{00}}\rangle,$
(137) $\int d{\bf{k}}\
H_{bu}(k;\tau,\tau)=\langle{{\bf{b}}^{\prime}_{00}\cdot\mbox{\boldmath$\omega$}^{\prime}_{00}}\rangle.$
(138)

#### Calculation of the electromotive force (EMF)

The turbulent electromotive force (EMF) is expressed in terms of the wave-
number representation of the velocity and magnetic-field as

$E_{\rm{M}}^{i}\equiv\epsilon^{ijk}\langle{u^{\prime}{}^{j}b^{\prime}{}^{k}}\rangle=\epsilon^{ijk}\int
d{\bf{k}}\
\langle{u^{j}({\bf{k}};\tau)b^{k}({\bf{k}}^{\prime};\tau)}\rangle/\delta({\bf{k}}+{\bf{k}}^{\prime}).$
(139)

Using the results of (128) and (129), we calculate the velocity–magnetic-field
correlation up to the $f_{01}g_{00}$ and $f_{10}g_{00}$ orders as

$\langle{u^{j}b^{k}}\rangle=\langle{u_{00}^{j}b_{00}^{k}}\rangle+\langle{u_{01}^{j}b_{00}^{k}}\rangle+\langle{u_{00}^{j}b_{01}^{k}}\rangle+\delta\langle{u_{10}^{j}b_{00}^{k}}\rangle+\delta\langle{u_{00}^{j}b_{10}^{k}}\rangle+\cdots.$
(140)

In the direct-interaction approximation (DIA) formalism, the lowest-order
spectral functions $Q_{uu}$, $Q_{bb}$, $Q_{ub}$, $H_{uu}$, $H_{bb}$, $H_{ub}$,
and $H_{bu}$, and the lowest-order Green’s functions $G_{uu}$, $G_{bb}$,
$G_{ub}$, and $G_{bu}$ are replaced with their exact counterparts,
$\tilde{Q}_{uu}$, $\tilde{Q}_{bb}$, $\cdots$, and $\tilde{G}_{uu}$,
$\tilde{G}_{bb}$, $\cdots$, respectively. Under this renormalization procedure
on the propagators (spectral and response functions), important turbulent
correlation functions are calculated. For the sake of simplicity, hereafter,
the tilde denoting an exact propagator will be omitted as $\tilde{Q}_{uu}\to
Q_{uu}$, $\tilde{G}_{uu}\to G_{uu}$, etc.

Here we present the final results of the turbulent EMF as

$\langle{\bf{u}}^{\prime}\times{\bf{b}}^{\prime}\rangle=\alpha\langle{\bf{B}}\rangle-(\beta+\zeta)\nabla\times\langle{\bf{B}}\rangle-(\nabla\zeta)\times\langle{\bf{B}}\rangle+\gamma\left(\langle{\bf{W}}\rangle+2\boldsymbol{\Omega}\right),$
(141)

where transport coefficients $\alpha$, $\beta$, $\zeta$, and $\gamma$ are
given as

$\alpha=\frac{1}{3}\left[{-I\\{{G_{bb},H_{uu}}\\}+I\\{{G_{uu},H_{bb}}\\}-I\\{{G_{bu},H_{ub}}\\}+I\\{{G_{ub},H_{bu}}\\}}\right],$
(142)
$\beta=\frac{1}{3}\left[{I\\{{G_{bb},Q_{uu}}\\}+I\\{{G_{uu},Q_{bb}}\\}-I\\{{G_{bu},Q_{ub}}\\}-I\\{{G_{ub},Q_{bu}}\\}}\right],$
(143)
$\zeta=\frac{1}{3}\left[{I\\{{G_{bb},Q_{uu}}\\}-I\\{{G_{uu},Q_{bb}}\\}+I\\{{G_{bu},Q_{ub}}\\}-I\\{{G_{ub},Q_{bu}}\\}}\right],$
(144)
$\gamma=\frac{1}{3}\left[{I\\{{G_{bb},Q_{ub}}\\}+I\\{{G_{uu},Q_{bu}}\\}-I\\{{G_{bu},Q_{uu}}\\}-I\\{{G_{ub},Q_{bb}}\\}}\right]$
(145)

with the abbreviate form of integral

$I\\{{A,B}\\}=\int
d{\bf{k}}\int_{-\infty}^{\tau}\\!\\!\\!d\tau_{1}A(k;\tau,\tau_{1})B(k;\tau,\tau_{1}).$
(146)

##  Appendix B Cross helicity and
$\left\langle\mathbf{u}^{\prime}\cdot\mathbf{j}^{\prime}\right\rangle$ under
FOSA

In the presence of the Coriolis force and under the ’first-order smoothing
approximation’ (FOSA), in the Fourier space the linearised equations take the
form

$\displaystyle\left(-\mathrm{i}\omega+\nu
k^{2}\right)\hat{u}_{i}(\mathbf{q})+2\Omega\epsilon_{i3j}\hat{u}_{j}(\mathbf{q})=$
$\displaystyle\hat{f}_{i}(\mathbf{q})+\mathrm{i}\mathbf{k}\cdot\left\langle\mathbf{B}\right\rangle\hat{b}_{i}(\mathbf{q}),$
(147) $\displaystyle\left(-\mathrm{i}\omega+\eta
k^{2}\right)\hat{b}_{i}(\mathbf{q})=$
$\displaystyle\mathrm{i}\mathbf{k}\cdot\left\langle\mathbf{B}\right\rangle\hat{u}_{i}(\mathbf{q}),$
(148)

where the the forcing is assumed Gaussian with zero mean, homogeneous,
stationary, and isotropic

$\left\langle\hat{f}_{i}(\mathbf{k},\omega)\hat{f}_{j}(\mathbf{k}^{\prime},\omega^{\prime})\right\rangle=\left[\frac{D_{0}}{k^{3}}P_{ij}(\mathbf{k})+\mathrm{i}\frac{D_{1}}{k^{5}}\epsilon_{ijk}k_{k}\right]\delta(\mathbf{k}+\mathbf{k}^{\prime})\delta(\omega+\omega^{\prime}),$
(149)

and $P_{ij}(\mathbf{k})=\delta_{ij}-k_{i}k_{j}/k^{2}$ is the projection
operator on the plane perpendicular to the wave vector $\mathbf{k}$.
Introducing

$\gamma_{\nu}=-\mathrm{i}\omega+\nu
k^{2},\qquad\gamma_{\eta}=-\mathrm{i}\omega+\eta k^{2},$ (150)

and considering the weak seed field limit defined by

$\left\langle\mathbf{B}\right\rangle^{2}\ll\left\langle\mathbf{U}\right\rangle^{2},\quad\textrm{hence
also}\quad\left\langle\mathbf{b}^{\prime
2}\right\rangle\ll\left\langle\mathbf{u}^{\prime 2}\right\rangle,$ (151)

the equations reduce to

$\hat{u}_{i}(\mathbf{q})\approx\mathfrak{G}_{ij}\hat{f}_{j}^{>}(\mathbf{q}),$
(152)
$\hat{b}_{i}(\mathbf{q})\approx\mathrm{i}\frac{\mathbf{k}\cdot\left\langle\mathbf{B}\right\rangle}{\gamma_{\eta}}\mathfrak{G}_{ij}\hat{f}_{j}(\mathbf{q}),$
(153)

where

$\mathfrak{G}_{ij}=\frac{1}{\gamma_{\nu}^{2}+4\Omega^{2}\frac{k_{z}^{2}}{k^{2}}}\left[\gamma_{\nu}\delta_{ij}-2\Omega\epsilon_{i3j}+2\Omega\frac{k_{i}k_{m}}{k^{2}}\epsilon_{jm3}\right].$
(154)

The cross-helicity takes the form

$\displaystyle\left\langle h_{ub}\right\rangle=$ $\displaystyle\left\langle
u_{i}^{\prime}(\mathbf{x},t)b_{i}^{\prime}(\mathbf{x},t)\right\rangle$
$\displaystyle=$
$\displaystyle\,\,\mathrm{i}\int\mathrm{d}^{4}q\int\mathrm{d}^{4}q^{\prime}\mathrm{e}^{\mathrm{i}\left[\left(\mathbf{k}+\mathbf{k}^{\prime}\right)\cdot\mathbf{x}-\left(\omega+\omega^{\prime}\right)t\right]}\frac{\mathbf{k}^{\prime}\cdot\left\langle\mathbf{B}\right\rangle}{\gamma_{\eta}\left(\mathbf{q}^{\prime}\right)}\mathfrak{G}_{ij}\left(\mathbf{q}\right)\mathfrak{G}_{ik}\left(\mathbf{q}^{\prime}\right)\left\langle\hat{f}_{j}(\mathbf{q})\hat{f}_{k}(\mathbf{q}^{\prime})\right\rangle$
$\displaystyle=$ $\displaystyle-\mathrm{i}\left\langle
B\right\rangle_{m}\int\mathrm{d}^{4}q\frac{k_{m}}{\gamma_{\eta}\left(-\mathbf{q}\right)}\mathfrak{G}_{ij}\left(\mathbf{q}\right)\mathfrak{G}_{ik}\left(-\mathbf{q}\right)\left[\frac{D_{0}}{k^{3}}P_{jk}(\mathbf{k})+\mathrm{i}\frac{D_{1}}{k^{5}}\epsilon_{jks}k_{s}\right]$
$\displaystyle=$ $\displaystyle-8\Omega\left\langle
B\right\rangle_{m}\int\frac{\mathrm{d}k}{k}\int_{-\infty}^{\infty}\mathrm{d}\omega\int_{0}^{2\pi}\mathrm{d}\varphi\int_{-1}^{1}\mathrm{d}X\frac{\omega^{2}D_{1}}{\left(\omega^{2}+\eta^{2}k^{4}\right)\mathcal{F}(\omega,X)}\frac{k_{z}k_{m}}{k^{2}}$
$\displaystyle=$
$\displaystyle-16\pi\left(\left\langle\mathbf{B}\right\rangle\cdot\boldsymbol{\Omega}\right)\int\frac{\mathrm{d}k}{k}\int_{-\infty}^{\infty}\mathrm{d}\omega\int_{-1}^{1}\mathrm{d}X\frac{\omega^{2}X^{2}D_{1}}{\left(\omega^{2}+\eta^{2}k^{4}\right)\mathcal{F}(\omega,X)}$
$\displaystyle=$ $\displaystyle-16\pi
D_{1}\mathcal{I}\left(\nu,\eta,\Omega,k_{\ell}\right)\left(\left\langle\mathbf{B}\right\rangle\cdot\boldsymbol{\Omega}\right),$
(155)

where

$\displaystyle\mathcal{F}(\omega,X)=$
$\displaystyle\left(\omega^{2}+\nu^{2}k^{4}\right)^{2}-8\Omega^{2}X^{2}\left(\omega^{2}-\nu^{2}k^{4}\right)+16\Omega^{4}X^{4}$
$\displaystyle=$
$\displaystyle\left(\omega^{2}-4\Omega^{2}X^{2}\right)^{2}+2\omega^{2}\nu^{2}k^{4}+8\nu^{2}k^{4}\Omega^{2}X^{2}+\nu^{4}k^{8}>0,$
(156)

and

$D_{1}\sim\left\langle\mathbf{f}\cdot\nabla\times\mathbf{f}\right\rangle,\qquad\mathcal{I}\left(\nu,\eta,\Omega,k_{\ell}\right)>0.$
(157)

On the other hand for the scalar quantity $\left\langle
s_{uj}\right\rangle=\left\langle\mathbf{u}^{\prime}\cdot\mathbf{j}^{\prime}\right\rangle$
this approach yields

$\displaystyle\left\langle s_{uj}\right\rangle=$ $\displaystyle\left\langle
u_{i}^{\prime}(\mathbf{x},t)j_{i}^{\prime}(\mathbf{x},t)\right\rangle$
$\displaystyle=$ $\displaystyle-\epsilon_{irt}\left\langle
B\right\rangle_{m}\int\mathrm{d}^{4}q\int\mathrm{d}^{4}q^{\prime}\mathrm{e}^{\mathrm{i}\left[\left(\mathbf{k}+\mathbf{k}^{\prime}\right)\cdot\mathbf{x}-\left(\omega+\omega^{\prime}\right)t\right]}\frac{k_{r}^{\prime}k_{m}^{\prime}}{\gamma_{\eta}\left(\mathbf{q}^{\prime}\right)}\mathfrak{G}_{ij}\left(\mathbf{q}\right)\mathfrak{G}_{tk}\left(\mathbf{q}^{\prime}\right)\left\langle\hat{f}_{j}(\mathbf{q})\hat{f}_{k}(\mathbf{q}^{\prime})\right\rangle$
$\displaystyle=$ $\displaystyle-\epsilon_{irt}\left\langle
B\right\rangle_{m}\int\mathrm{d}^{4}q\frac{k_{r}k_{m}}{\gamma_{\eta}\left(-\mathbf{q}\right)}\mathfrak{G}_{ij}\left(\mathbf{q}\right)\mathfrak{G}_{ik}\left(-\mathbf{q}\right)\left[\frac{D_{0}}{k^{3}}P_{jk}(\mathbf{k})+\mathrm{i}\frac{D_{1}}{k^{5}}\epsilon_{jks}k_{s}\right]$
$\displaystyle=$ $\displaystyle\,\,8\Omega\left\langle B\right\rangle_{m}\int
k\mathrm{d}k\int_{-\infty}^{\infty}\mathrm{d}\omega\int_{0}^{2\pi}\mathrm{d}\varphi\int_{-1}^{1}\mathrm{d}X\frac{\omega^{2}D_{0}}{\left(\omega^{2}+\eta^{2}k^{4}\right)\mathcal{F}(\omega,X)}\frac{k_{z}k_{m}}{k^{2}}$
$\displaystyle=$
$\displaystyle\,\,16\pi\left(\left\langle\mathbf{B}\right\rangle\cdot\boldsymbol{\Omega}\right)\int
k\mathrm{d}k\int_{-\infty}^{\infty}\mathrm{d}\omega\int_{-1}^{1}\mathrm{d}X\frac{\omega^{2}X^{2}D_{0}}{\left(\omega^{2}+\eta^{2}k^{4}\right)\mathcal{F}(\omega,X)}$
$\displaystyle=$ $\displaystyle\,\,16\pi
D_{0}\widetilde{\mathcal{I}}\left(\nu,\eta,\Omega,k_{\ell}\right)\left(\left\langle\mathbf{B}\right\rangle\cdot\boldsymbol{\Omega}\right),$
(158)

where

$D_{0}\sim\left\langle\mathbf{f}^{2}\right\rangle,\qquad\widetilde{\mathcal{I}}\left(\nu,\eta,\Omega,k_{\ell}\right)>0.$
(159)

In the above we have used

$\int\frac{k_{j}}{k}f\left(\cos^{2}\theta\right)\mathrm{d}\mathring{\varOmega}=0,\qquad\int\frac{k_{i}k_{j}k_{k}}{k^{3}}f\left(\cos^{2}\theta\right)\mathrm{d}\mathring{\varOmega}=0,$
(160)
$\int\frac{k_{j}k_{n}}{k^{2}}f\left(\cos^{2}\theta\right)\mathrm{d}\mathring{\varOmega}=\pi\int_{-1}^{1}f(X^{2})\left\\{\delta_{jn}\left(1-X^{2}\right)+\delta_{j3}\delta_{n3}\left(3X^{2}-1\right)\right\\}\mathrm{d}X.$
(161)

where $\mathring{\varOmega}$ denotes the solid angle and the spherical
coordinates $(k,\theta,\varphi)$ have been used (with a substitution
$X=\cos\theta$). Furthermore, in a similar way we can calculate the kinetic
helicity and turbulent energy

$\displaystyle\left\langle h_{kin}\right\rangle=$ $\displaystyle\left\langle
u_{i}(\mathbf{x},t)w_{i}(\mathbf{x},t)\right\rangle$ $\displaystyle=$
$\displaystyle\,\,\mathrm{i}\epsilon_{ijk}\int\mathrm{d}^{4}q\int\mathrm{d}^{4}q^{\prime}k_{j}^{\prime}\mathfrak{G}_{in}\left(\mathbf{q}\right)\mathfrak{G}_{km}\left(\mathbf{q}^{\prime}\right)\left\langle\hat{f}_{n}(\mathbf{q})\hat{f}_{m}(\mathbf{q}^{\prime})\right\rangle\mathrm{e}^{\mathrm{i}\left[\left(\mathbf{k}+\mathbf{k}^{\prime}\right)\cdot\mathbf{x}-\left(\omega+\omega^{\prime}\right)t\right]}$
$\displaystyle=$
$\displaystyle-\mathrm{i}\epsilon_{ijk}\int\mathrm{d}^{4}qk_{j}\mathfrak{G}_{in}\left(\mathbf{q}\right)\mathfrak{G}_{km}\left(-\mathbf{q}\right)\left[\frac{D_{0}}{k^{3}}P_{nm}(\mathbf{k})+\mathrm{i}\frac{D_{1}}{k^{5}}\epsilon_{nmp}k_{p}\right]$
$\displaystyle=$
$\displaystyle-\int\frac{\mathrm{d}k}{k}\int_{-\infty}^{\infty}\mathrm{d}\omega\int_{0}^{2\pi}\mathrm{d}\varphi\int_{-1}^{1}\mathrm{d}X\frac{2D_{1}\left(\omega^{2}+\nu^{2}k^{4}+4\Omega^{2}X^{2}\right)}{\left(\omega^{2}+\nu^{2}k^{4}\right)^{2}-8\Omega^{2}X^{2}\left(\omega^{2}-\nu^{2}k^{4}\right)+16\Omega^{4}X^{4}}$
$\displaystyle=$ $\displaystyle-4\pi
D_{1}\mathcal{I}_{u^{2}}\left(\nu,\Omega,k_{\ell}\right)$ (162)
$\displaystyle\left\langle\mathbf{u}^{\prime 2}\right\rangle=$
$\displaystyle\left\langle
u_{i}(\mathbf{x},t)u_{i}(\mathbf{x},t)\right\rangle$ $\displaystyle=$
$\displaystyle\int\mathrm{d}^{4}q\int\mathrm{d}^{4}q^{\prime}\mathfrak{G}_{in}\left(\mathbf{q}\right)\mathfrak{G}_{im}\left(\mathbf{q}^{\prime}\right)\left\langle\hat{f}_{n}(\mathbf{q})\hat{f}_{m}(\mathbf{q}^{\prime})\right\rangle\mathrm{e}^{\mathrm{i}\left[\left(\mathbf{k}+\mathbf{k}^{\prime}\right)\cdot\mathbf{x}-\left(\omega+\omega^{\prime}\right)t\right]}$
$\displaystyle=$
$\displaystyle\int\mathrm{d}^{4}q\mathfrak{G}_{in}\left(\mathbf{q}\right)\mathfrak{G}_{im}\left(-\mathbf{q}\right)\left[\frac{D_{0}}{k^{3}}P_{nm}(\mathbf{k})+\mathrm{i}\frac{D_{1}}{k^{5}}\epsilon_{nmp}k_{p}\right]$
$\displaystyle=$
$\displaystyle\,\,2\int\frac{\mathrm{d}k}{k}\int_{-\infty}^{\infty}\mathrm{d}\omega\int_{0}^{2\pi}\mathrm{d}\varphi\int_{-1}^{1}\mathrm{d}X\frac{D_{0}\left(\omega^{2}+\nu^{2}k^{4}+4\Omega^{2}X^{2}\right)}{\left(\omega^{2}+\nu^{2}k^{4}\right)^{2}-8\Omega^{2}X^{2}\left(\omega^{2}-\nu^{2}k^{4}\right)+16\Omega^{4}X^{4}}$
$\displaystyle=$ $\displaystyle\,\,4\pi
D_{0}\mathcal{I}_{u^{2}}\left(\nu,\Omega,k_{\ell}\right).$ (163)

Note, that in the weak seed field limit (151) the turbulent energy reduces to
$\left\langle\mathbf{u}^{\prime 2}\right\rangle+\left\langle\mathbf{b}^{\prime
2}\right\rangle\approx\left\langle\mathbf{u}^{\prime 2}\right\rangle$. We can
now utilize the above results to show

$\displaystyle\left\langle\mathbf{u}^{\prime}\cdot\mathbf{j}^{\prime}\right\rangle=$
$\displaystyle\left(\frac{\mathcal{I}_{uj}\left(\nu,\eta,\Omega,k_{\ell}\right)}{\mathcal{I}_{ub}\left(\nu,\eta,\Omega,k_{\ell}\right)}\frac{D_{0}^{2}}{D_{1}^{2}}\right)\frac{\left\langle\mathbf{u}^{\prime}\cdot\mathbf{b}^{\prime}\right\rangle}{\left\langle\mathbf{u}^{\prime
2}\right\rangle}\left\langle\mathbf{u}^{\prime}\cdot\mathbf{w}^{\prime}\right\rangle$
$\displaystyle\sim$
$\displaystyle\,\,\frac{\left\langle\mathbf{u}^{\prime}\cdot\mathbf{b}^{\prime}\right\rangle}{\left\langle\mathbf{u}^{\prime
2}\right\rangle}\left\langle\mathbf{u}^{\prime}\cdot\mathbf{w}^{\prime}\right\rangle$
(164)

##  Appendix C Evolution equations for
$\langle\mathbf{u}^{\prime}\cdot\mathbf{b}^{\prime}\rangle$ and
$\langle\mathbf{u}^{\prime}\cdot\mathbf{j}^{\prime}\rangle$

Utilizing the evolution equations

$\displaystyle\frac{D\mathbf{u}^{\prime}}{Dt}=-\nabla\Pi^{\prime}-2\boldsymbol{\Omega}\times\mathbf{u}^{\prime}-\left(\mathbf{u}^{\prime}\cdot\nabla\right)\left\langle\mathbf{U}\right\rangle+\left(\left\langle\mathbf{B}\right\rangle\cdot\nabla\right)\mathbf{b}^{\prime}+\left(\mathbf{b}^{\prime}\cdot\nabla\right)\left\langle\mathbf{B}\right\rangle+\nu\nabla^{2}\mathbf{u}^{\prime}$
$\displaystyle+\nabla\cdot\left(\mathbf{b}^{\prime}\mathbf{b}^{\prime}\right)+\nabla\cdot\left(\left\langle\mathbf{u}^{\prime}\mathbf{u}^{\prime}\right\rangle-\left\langle\mathbf{b}^{\prime}\mathbf{b}^{\prime}\right\rangle\right),$
(165)
$\frac{D\mathbf{b}^{\prime}}{Dt}=\left(\left\langle\mathbf{B}\right\rangle\cdot\nabla\right)\mathbf{u}^{\prime}-\left(\mathbf{u}^{\prime}\cdot\nabla\right)\left\langle\mathbf{B}\right\rangle+\left(\mathbf{b}^{\prime}\cdot\nabla\right)\left\langle\mathbf{U}\right\rangle+\eta\nabla^{2}\mathbf{b}^{\prime}+\left(\mathbf{b}^{\prime}\cdot\nabla\right)\mathbf{u}^{\prime}-\nabla\times\boldsymbol{\mathcal{E}},$
(166)

where

$\frac{D}{Dt}=\frac{\partial}{\partial
t}+\left(\langle\mathbf{U}\rangle+\mathbf{u}^{\prime}\right)\cdot\nabla,$
(167)

we arrive at

$\displaystyle\frac{D}{Dt}\left\langle\mathbf{u}^{\prime}\cdot\mathbf{b}^{\prime}\right\rangle=$
$\displaystyle-\boldsymbol{\mathcal{E}}\cdot\left(\left\langle\mathbf{W}\right\rangle+2\boldsymbol{\Omega}\right)-\left\langle
u_{i}^{\prime}u_{j}^{\prime}-b_{i}^{\prime}b_{j}^{\prime}\right\rangle\partial_{j}\left\langle
B\right\rangle_{i}$
$\displaystyle+\nabla\cdot\left[\left\langle\left(-\Pi^{\prime}+\frac{\mathbf{u}^{\prime
2}+\mathbf{b}^{\prime
2}}{2}\right)\mathbf{b}^{\prime}\right\rangle+\left\langle\frac{\mathbf{u}^{\prime
2}+\mathbf{b}^{\prime
2}}{2}\right\rangle\left\langle\mathbf{B}\right\rangle\right]$
$\displaystyle-\mu_{0}\left(\nu+\eta\right)\left\langle\mathbf{w}^{\prime}\cdot\mathbf{j}^{\prime}\right\rangle,$
(168)

and

$\displaystyle\frac{D}{Dt}\left\langle\mathbf{u}^{\prime}\cdot\mathbf{j}^{\prime}\right\rangle=$
$\displaystyle-\left\langle\mathbf{u}^{\prime}\times\mathbf{j}^{\prime}\right\rangle\cdot\left(\left\langle\mathbf{W}\right\rangle+2\boldsymbol{\Omega}\right)-\left\langle
w_{i}^{\prime}u_{j}^{\prime}-j_{i}^{\prime}b_{j}^{\prime}\right\rangle\partial_{j}\left\langle
B\right\rangle_{i}-\partial_{j}\left\langle
U\right\rangle_{m}\left\langle\epsilon_{ijk}u_{i}^{\prime}\partial_{m}b_{k}^{\prime}\right\rangle$
$\displaystyle+\left\langle\left[\left(\left\langle\mathbf{B}\right\rangle+\mathbf{b}^{\prime}\right)\cdot\nabla\right]\mathbf{b}^{\prime}\cdot\mathbf{j}^{\prime}+\left[\left(\left\langle\mathbf{B}\right\rangle+\mathbf{b}^{\prime}\right)\cdot\nabla\right]\mathbf{u}^{\prime}\cdot\mathbf{w}^{\prime}-u_{i}^{\prime}\epsilon_{ijk}\partial_{j}u_{m}^{\prime}\partial_{m}b_{k}^{\prime}\right\rangle$
$\displaystyle-\nabla\cdot\left\langle\Pi^{\prime}\mathbf{j}^{\prime}\right\rangle-\left(\nu-\eta\right)\left\langle\mathbf{w}^{\prime}\cdot\nabla^{2}\mathbf{b}^{\prime}\right\rangle,$
(169)

where in the last equation, apart from no-slip boundary conditions, we have
also assumed vanishing of the helical quantity
$\left\langle\mathbf{w}^{\prime}\cdot\mathbf{j}^{\prime}\right\rangle$ at the
boundaries.

##  Appendix D Basic equations used in the compressible case

In the numerical simulations, instead of equations (1a–c), we solve the
following set of equations for a compressible isothermal gas with constant
sound speed $c_{\rm s}$ for $\mathbf{U}$, $\rho$, and the magnetic vector
potential $\mathbf{A}$:

$\frac{\partial\mathbf{U}}{\partial
t}+\left(\mathbf{U}\cdot\nabla\right)\mathbf{U}=-c_{\rm
s}^{2}\nabla\ln\rho-2\boldsymbol{\Omega}\times\mathbf{U}+\frac{1}{\rho}\mathbf{J}\times\mathbf{B}-\nu\mathbf{Q}+\mathbf{g}+\mathbf{f},$
(170a) $\frac{\partial\rho}{\partial t}=-\nabla\cdot(\rho\mathbf{U}),$ (170b)
$\frac{\partial\mathbf{A}}{\partial
t}=\mathbf{U}\times\mathbf{B}-\eta\mu_{0}\mathbf{J},$ (171)

where

$\mathbf{Q}=-\nabla^{2}\mathbf{U}-\frac{1}{3}\nabla\nabla\cdot\mathbf{U}-\mbox{\boldmath${\sf
S}$}\nabla\ln\rho,$ (172)
$\mu_{0}\mathbf{J}=-\nabla^{2}\mathbf{A}+\nabla\nabla\cdot\mathbf{A},$ (173)
$\mathbf{B}=\mathbf{B}_{0}+\nabla\times\mathbf{A},$ (174)

and

$\mathsf{S}_{ij}=\frac{1}{2}(\partial_{i}U_{j}+\partial_{j}U_{i})-\frac{1}{3}\delta_{ij}\nabla\cdot\mathbf{U}$
(175)

are the components of the traceless rate-of-strain tensor and $\mathbf{f}$ is
a random forcing function consisting of plane unpolarized waves with typical
wavenumber $k_{\rm f}$ and an amplitude such that $u_{\rm rms}/c_{\rm
s}\approx 0.1$; see table 1. Here, $\boldsymbol{\Omega}=(0,0,\Omega)$ is the
angular velocity, $\mathbf{g}=(0,0,-g)$ is gravity,
$\mathbf{B}_{0}=(0,0,B_{0})$ is the imposed magnetic field, $\eta$ is the
magnetic diffusivity, and $\nu$ is the kinematic viscosity, whose value is
such that $u_{\rm rms}/\nu k_{1}\approx 1000$. A resolution of $N^{3}=256^{3}$
mesh points is then sufficient. Since we chose $k_{\rm f}/k_{1}=30$, we have
for the Reynolds number $\mbox{Re}\equiv u_{\rm rms}/\nu k_{\rm f}\approx 30$.
For the magnetic Prandtl number we chose, as in Jabbari et al. (2014) the
value $\mbox{Pr}_{M}\equiv\nu/\eta=0.5$, so the magnetic Reynolds number is
$\mbox{Re}_{M}\equiv u_{\rm rms}/\eta k_{\rm f}\approx 15$. The equilibrium
stratification is given by $\ln(\rho/\rho_{0})=-z/H_{\rho}$, where
$H_{\rho}=c_{\rm s}^{2}/g$ is the density scale height.

##  Appendix E Results for Runs B–E

In figure 2, we present the results for Runs B and E with a stronger magnetic
field: $B=0.1\,c_{\rm s}\sqrt{\mu_{0}\bar{\rho}}$, and two values of gravity
$g=1\,c_{\rm s}^{2}k_{1}$ and $g=0.5\,c_{\rm s}^{2}k_{1}$. Finally, in figure
3, we present the results for Run C and D with weaker magnetic field
$B=0.01\,c_{\rm s}\sqrt{\mu_{0}\bar{\rho}}$, and two values of gravity:
$g=0.5\,c_{\rm s}^{2}k_{1}$ and $g=2\,c_{\rm s}^{2}k_{1}$.

Figure 2: Same as figure 1, but for Run B and E with a stronger magnetic
field: $B=0.1\,c_{\rm s}\sqrt{\mu_{0}\bar{\rho}}$, and two values of gravity
$g=1\,c_{\rm s}^{2}k_{1}$ and $g=0.5\,c_{\rm s}^{2}k_{1}$.

Figure 3: Same as figure 1, but for Run C and D with weaker magnetic field
$B=0.01\,c_{\rm s}\sqrt{\mu_{0}\bar{\rho}}$, and two values of gravity:
$g=0.5\,c_{\rm s}^{2}k_{1}$ and $g=2\,c_{\rm s}^{2}k_{1}$.

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