2021

[1,2]Nobumitsu Yokoi

[1]Institute of Industrial Science, University of Tokyo, Komaba, Meguro,
Tokyo, 153-8505, Japan

2]Nordic Institute for Theoretical Physics, Stockholm University and KTH,
Hannes Alfvén väg, Stockholm, 106 91, Sweden

# Unappreciated cross-helicity effects in plasma physics: Anti-diffusion
effects in dynamo and momentum transport

<EMAIL_ADDRESS>*  [

###### Abstract

The cross helicity (velocity–magnetic-field correlation) effects in the
magnetic-field induction and momentum transport in the magnetohydrodynamic
(MHD) turbulence are investigated with the aid of the multiple-scale
renormalized perturbation expansion analysis, which is a theoretical framework
for the strongly non-linear and inhomogeneous turbulence. The outline of the
theory is presented with reference to the role of the cross-interaction
response functions between the velocity and magnetic field. In this
formulation, the expressions of the turbulent fluxes: the turbulent
electromotive force (EMF) in the mean induction equation and the Reynolds and
turbulent Maxwell stresses in the momentum equation are obtained. Related to
the expression of EMF, the physical origin of the cross-helicity effect in
dynamos, as well as other dynamo effects, is discussed. Properties of dynamo
and momentum transport are determined by the spatiotemporal distribution of
turbulence. In order to understand the actual role of the turbulent cross
helicity, its transport equations is considered. Several generation mechanisms
of cross helicity are discussed with illustrative examples. On the basis of
the cross-helicity production mechanisms, its effect in stellar dynamos is
discussed. The role of cross helicity in the momentum transport and global
flow generation is also argued. One of the situations where the cross-helicity
effect both in magnetic-field induction and global flow generation play an
important role is the turbulent magnetic reconnection. Characteristic features
of turbulence effects in fast reconnection are reviewed with special emphasis
on the role of cross helicity in localizing the effective resistivity.
Finally, a remark is addressed on an approach that elucidates the structure
generation and sustainment in extremely strong turbulence. An appropriate
formulation for the anti-diffusion effect, which acts against the usual
diffusion effect, is needed. Turbulence modeling approach based on such an
analytical formulation is also argued in comparison with the conventional
heuristic modeling. The importance of the self-consistent framework treating
the non-linear interaction between the mean field and turbulence is stressed
as well.

###### keywords:

Turbulence closure, Cross helicity, Dynamo, Flow generation

##  1 Introduction

Astrophysical and geophysical and plasma physics phenomena, such as the
formation of jets on giant planets, the eleven-year solar activity cycle, the
generation of Earth’s magnetism, the large-scale vortical and magnetic-field
structures in interstellar medium, and jets from massive black holes, are
characterized by the interaction processes of vast range of spatial and
temporal scales. As such these amazing phenomena are almost invariably
extremely turbulent. Magnetohydrodynamics (MHD) provides a good framework for
understanding such turbulence with the mutual interaction between the fluid
flows and magnetic fields. However, because of the vast range of scales it
contains from the largest energy-containing scales to the smallest dissipation
scales both for viscosity and resistivity, direct computations of the
turbulent astrophysical and geophysical flows are simply impossible, even with
using most sophisticated algorithms optimized for massively parallel
computers. Also, because of the strong non-linear coupling among the modes or
scales, we cannot expect any obvious scale separation in strong turbulence.
Then, for understanding the nature of turbulent MHD flows, developing a
sophisticated statistical analytical theory and modeling turbulence on the
basis of the theoretical results are of central importance. Here in this
article, an approach for developing such a theory will be introduced. Such a
theory will have a considerable impact on constructing a turbulence model that
predicts the properties of amazing phenomena we encounter in astrophysics,
geophysics, and plasma physics, beyond the conventional heuristic turbulence
modeling.

In the theoretical formulation of large-scale structure formation, breakage of
mirror-symmetry plays an essential role. Such breakages are represented by
pseudo-scalars such as the kinetic, magnetic (current), cross, and other
generalized helicities. The statistical properties of helical turbulence and
their consequences were reviewed in another review paper (Pouquet & Yokoi,
2022), and will not be treated in this paper. Also the basic characteristics
of the cross helicity including the conservation, topological interpretation,
pseudo-scalar properties, boundedness, relation to Alfvén waves, etc. were
treated in other article (Yokoi, 2013), so the descriptions related to such
characteristics will not be repeated in this article, either. Here, we focus
our attention on two aspects of the turbulent MHD transport with primary
importance: the magnetic-field induction (dynamos) and the linear and angular
momentum transport (global flow generation).

One of the main subjects in the dynamo studies is to explore the effects of
turbulence in the mean induction equation. This type of studies is often
called the mean-field dynamo theory, which has historically gained several
connotations. They include (i) kinematic approach; (ii) Ansatz for the
turbulent electromotive (EMF) expression; (iii) dynamo coefficients as
constant or prescribed parameters; (iv) exclusive $\alpha$–$\Omega$ dynamo
scenario for dynamo process; (v) azimuthal averaging as a substitute for the
ensemble averaging; and (vi) incompressible treatment of turbulent motions.
Some of these treatments have played a role to consider dynamos in a
simplified manner. However, none of them is a basic ingredient of the mean-
field dynamo theory. In the following, we briefly refer to these treatments
(assumptions and approximations) one by one.

(i) Kinematic approach: In the kinematic approach, the velocity field in the
induction equation is treated as a prescribed one. The evolution of the
magnetic field is subject to the induction equation with a fixed velocity. The
feedback of the magnetic field to the velocity through the Lorentz force
${\bf{j}\times{\bf{b}}}$ in the momentum equation is neglected [${\bf{b}}$:
magnetic field, ${\bf{j}}(=\nabla\times{\bf{b}})$: electric-current density].
This kinematic approach of the dynamo has a long history and has provided a
good tool for understanding the basic properties of dynamo actions, especially
in some simple configuration (Tobias, 2021). However, the dynamo process in
the real world is usually non-linear, and the back-reactions to the velocity
field by the generated magnetic fields play a crucial role in the turbulence
generation and consequently to the dynamo process. For instance, this
importance of the feed-back has been stressed in the context of the
magnetorotational instability (MRI). Even a very week magnetic field, coupled
with a specific configuration of rotation, can contribute to the turbulence
generation (Balbus & Hawley, 1998). Then, resultant dynamo properties are
determined by the turbulent motions at least at an developing stage of this
MRI process. In this sense, the kinematic approach has only limited relevance
to the dynamo process.

(ii) Ansatz for the turbulent electromotive force (EMF): Another assumption
which has been very often adopted in the mean-field dynamo studies is the
Ansatz that the turbulent EMF
$\langle{{\bf{u}}^{\prime}\times{\bf{b}}^{\prime}}\rangle$ is linearly related
to the mean magnetic field ${\bf{B}}$ and its spatial derivatives
(${\bf{u}}^{\prime}$: velocity fluctuation, ${\bf{b}}^{\prime}$: magnetic-
field fluctuation). This ansatz, based on the assumption of the linear
relationship between the fluctuating magnetic field ${\bf{b}}^{\prime}$ and
the mean magnetic field ${\bf{B}}$, greatly reduces the complexity of the mean
magnetic-field evolution, with the kinematic treatment. However, in order to
obtain an expression of the turbulent EMF
$\langle{{\bf{u}}^{\prime}\times{\bf{b}}^{\prime}}\rangle$ in fully non-linear
regime of turbulence, we have to directly treat the equations of the velocity
and magnetic-field fluctuations, ${\bf{u}}^{\prime}$ and ${\bf{b}}^{\prime}$.
These fluctuation fields depend not only on the mean magnetic field ${\bf{B}}$
and its spatial derivatives, but also on the mean velocity ${\bf{U}}$ and its
spatial derivatives. In addition, the system of equations of
${\bf{u}}^{\prime}$ and ${\bf{b}}^{\prime}$ is fully non-linear in the
presence of the fluctuating Lorentz force. As these considerations indicate,
the dynamo arguments based on the Ansatz are too simple, and their
applicability is limited to very specific simple situations such as the
kinematic cases at very low magnetic Reynolds number ($Rm=UL/\eta$, $U$:
characteristic velocity, $L$: characteristic length scale, $\eta$: magnetic
diffusivity or resistivity). This point will be argued in detail in Section
3.1.

(iii) Dynamo coefficients as constant or prescribed parameters: Related to the
Ansatz mentioned above, the proportional coefficients for the mean magnetic
field ${\bf{B}}$ and its derivatives, $\alpha$, $\eta_{\rm{T}}$, etc. are
often treated as adjustable parameters. However, this is nothing but the
crudest approximation. The transport coefficients for the turbulent fluxes
must reflect the statistical properties of turbulence. As this consequence,
these transport coefficients are not constants but vary in space and time
reflecting the spatiotemporal distributions of the turbulent statistical
quantities such as the turbulent energy, turbulent helicity, turbulent cross
helicity, the energy dissipation rate, etc.

(iv) $\alpha$–$\Omega$ framework: The usual mean-field dynamo model is
constructed in the framework of the $\alpha$–$\Omega$ dynamo. In this
framework, the turbulent EMF is assumed to be constituted by the $\alpha$
effect proportional to the mean magnetic field ${\bf{B}}$ and by the turbulent
magnetic diffusivity $\eta_{\rm{T}}$ effect proportional to the curl of
${\bf{B}}$. Another crucial ingredient is the inhomogeneous mean velocity that
induces the mean magnetic-field component vertical to the original field
($\Omega$ effect). As will be discussed later in Section 3.1, the
inhomogeneity of the mean velocity alters not only the mean magnetic-field
component but also the properties of fluctuation fields. The equations of the
velocity fluctuation ${\bf{u}}^{\prime}$ and the magnetic-field fluctuation
${\bf{b}}^{\prime}$ depend on the mean velocity shear. If we retain these mean
velocity shears in the ${\bf{u}}^{\prime}$ and ${\bf{b}}^{\prime}$ equations,
the cross-correlation between ${\bf{u}}^{\prime}$ and ${\bf{b}}^{\prime}$, or
the turbulent cross helicity
$\langle{{\bf{u}}^{\prime}\cdot{\bf{b}}^{\prime}}\rangle$ enters the turbulent
EMF expression as the coefficient of the mean velocity shear. We see a marked
difference between this cross-helicity effect and the usual $\alpha$ effect,
which is expressed by the correlation between the velocity and vorticity
fluctuations,
$\langle{{\bf{u}}^{\prime}\cdot\mbox{\boldmath$\omega$}^{\prime}}\rangle$
[$\mbox{\boldmath$\omega$}^{\prime}(=\nabla\times{\bf{u}}^{\prime})$:
vorticity fluctuation] and the correlation between the magnetic-field
fluctuation and the electric-current
$\langle{{\bf{b}}^{\prime}\cdot{\bf{j}}}\rangle$
[${\bf{j}}^{\prime}(=\nabla\times{\bf{b}}^{\prime})$: electric-current density
fluctuation] coupled with the mean magnetic-field $\bf{B}$. The cross-helicity
dynamo effect, which is based on the dependence of the velocity and magnetic-
field fluctuations on the mean velocity shear, may pave the way for expanding
the applicability of the mean-field dynamo theories to the inhomogeneous
turbulence.

(v) Azimuthal averaging: In the mean-field dynamo studies, the azimuthal
averaging is often adopted as a substitute for the ensemble averaging. This
sometimes severely limits the applicability of the mean-field theories to
situations where non-axisymmetric global structures play an essential role in
the dynamo process (Schüssler & Ferriz-Mas, 2003). One way to adopt an
averaging procedure other than the azimuthal averaging is to adopt a subgrid-
scale (SGS) filtering which allows us to retain large-scale non-axisymmetric
structures.

(vi) Incompressible treatment of turbulence: Another ubiquitous assumption in
the mean-field dynamo studies is assuming the incompressible turbulence. One
of the justifications for the treatment of incompressible turbulence lies in
the point that the magnetic induction equation does not explicitly depend on
the density. However, statistical properties and dynamic behavior of
turbulence is essentially different between the incompressible and
compressible cases. There is no reason to discard compressibility in dynamo on
phenomena with a high density variance $\langle{\rho^{\prime}{}^{2}}\rangle$
($\rho^{\prime}$: density fluctuation) (Yokoi, 2018a, b). Such a high density
variance corresponds to a high Mach number case. We ubiquitously encounter
astrophysical phenomena with a high Mach number, including the star-formation
region in the interstellar medium and the interior of massive stars.

None of the above assumptions/approximations are intrinsic condition of the
mean-field dynamo theory. Hereafter, the term “mean-field dynamo” theory
contains some of the above connotations, and leads to misunderstanding on the
scope of the theory. Therefore, we prefer using the term “turbulent dynamo” to
“mean-field dynamo” in the following.

Another important aspect of the turbulent MHD transport in the astrophysical
and geophysical plasma context is the turbulent momentum fluxes. The linear
and angular momentum transport due to turbulence is represented by the
Reynolds stress
$\langle{{\bf{u}}^{\prime}{\bf{u}}^{\prime}}\rangle=\\{\langle{u^{\prime}{}^{i}u^{\prime}{}^{j}}\rangle\\}$
and the turbulent Maxwell stress
$\langle{{\bf{b}}^{\prime}{\bf{b}}^{\prime}}\rangle=\\{{\langle{b^{\prime}{}^{i}b^{\prime}{}^{j}}\rangle}\\}$.
They are sole quantities directly express the linear and angular momentum
transports in the mean momentum equation. In astrophysical and geophysical as
well as fusion plasma flow phenomena, turbulence is considered to play an
important role in the angular momentum transport. An example is an accretion
disk surrounding a central massive astrophysical body such as black hole. In
order for the rotating plasma gas to accrete to the central object, the
angular momentum should be transported by turbulence (Shakura & Sunyaev,
1973).

The angular momentum transport in the stellar convection is another important
topic. It has been recently recognized that the current numerical simulations
do not capture some basic characteristics of the solar convection. The
convective velocity amplitude at large horizontal scales observed by
helioseismic investigations is much smaller than the one predicted by global
convection simulations (Hanasoge, Duvall & Sreenivasan, 2012; Hanasoge, Gizon
& Sreenivasan, 2016; Proxauf, 2021). The differential rotation profile
obtained by the global numerical simulations is not realistic in the sense
that the azimuthal rotational velocity in the equator region does not show the
solar-like prograde profile as compared with the one in the higher latitude
region, but does show the retrograde profile, if the solar values of
luminosity (energy transfer rate) and rotation rate are adopted in the
simulation. Also, if the large-scale convection motions are actually small,
how such weak flows can transfer the solar luminosity and mean differential
rotation rate observed in the helioseismology? These discrepancies between the
numerical simulations and observations are called the convection conundrum
(Schumacher & Sreenivasan, 2020).

In the stellar convection problems, the conventional expressions for the
turbulent fluxes (turbulent momentum, heat, and mass fluxes) on the basis of
the gradient-diffusion approximation are known to be inappropriate since they
give too destructive or dissipative effects on the fluxes. From the viewpoint
of theoretical investigation of the turbulent fluxes, there may be several
possibilities to resolve this convection conundrum. One possible way is to
introduce the effect of coherent fluctuating motions such as plumes, thermals,
and jets in the convective turbulence. Implementation of the coherent-
structure effect into the modeling of convective turbulence has been
considered important to capture the transport properties of turbulence (Rast,
1998; Brandenburg, 2016; Green, Vlaykov, Mellado, et al., 2020). Recently, the
relevance of the non-equilibrium effect of turbulence along the plume motions
has been pointed out in the context of the turbulence modeling of stellar
convection (Yokoi, Masada & Takiwaki, 2022). Since the non-equilibrium effect
along the coherent fluctuating motions alters the length scale and timescale
of turbulence, it directly affects the transport properties of turbulence.

The second possible way is to implement the rotation or global vorticity
effect. Rotation is considered to suppress the convective motion and enhance
the thermal transport at relatively smaller scales (Vasil, Julien &
Featherstone, 2021). One of the studies to incorporate the rotation effect
into the linear and angular momentum transport is through the kinetic-helicity
inhomogeneity. It was found that, in the Reynolds stress expression, the
absolute vorticity
$\mbox{\boldmath$\Omega$}_{\ast}(\equiv\mbox{\boldmath$\Omega$}+2\mbox{\boldmath$\omega$}_{\rm{F}})$
[$\mbox{\boldmath$\Omega$}(=\nabla\times{\bf{U}})$: the mean vorticity,
$\mbox{\boldmath$\omega$}_{\rm{F}}$: rotation] is coupled with the
inhomogeneous turbulent helicity represented by the gradient of turbulent
helicity,
$\nabla\langle{{\bf{u}}^{\prime}\cdot\mbox{\boldmath$\omega$}^{\prime}}\rangle$
(Yokoi & Yoshizawa, 1993). Unlike the eddy-viscosity effect, which arises from
the turbulent energy coupled with the mean velocity strain rate (the symmetric
part of the mean velocity shear) and destroys the large-scale inhomogeneous
flow structure by strong effective viscosity, this inhomogeneous turbulent
helicity effect arises from the presence of non-uniform kinetic helicity
coupled with the mean absolute vorticity (the anti-symmetric part of the mean
velocity shear). The latter effect is expected to contribute to generating and
sustaining non-trivial large-scale flow structures such as prograde
differential rotation in the solar convection zone (Yokoi, 2023). Since one of
the generation mechanisms of turbulent helicity is coupling of a rotation and
inhomogeneities of energy, density, etc. along the rotation axis,
inhomogeneous turbulent helicity (spatial distribution and segregation) is
ubiquitously present in a rotating spherical configuration (Duarte, Wicht,
Browning, et al., 2016; Ranjan & Davidson, 2023). In this sense, the
inhomogeneous helicity effect is expected to play an important role in the
angular momentum transport in the stellar convection zone.

The third possible way is to incorporate the magnetic shear effect into the
Reynolds and turbulent Maxwell stresses. In this effect, the transport
coefficient is expressed by the turbulent cross helicity
$\langle{{\bf{u}}^{\prime}\cdot{\bf{b}}^{\prime}}\rangle$. The presence of the
magnetic fluctuation ${\bf{b}}^{\prime}$ in statistical correlation with the
velocity fluctuation ${\bf{u}}^{\prime}$ is a crucial ingredient of this
cross-helicity effect in the linear and angular momentum transport.

These theoretical consideration suggests that, in addition to the usual eddy-
viscosity effect coupled with the mean velocity strain, the non-equilibrium
effect coupled with the coherent component of fluctuations (plumes, thermals,
and jets), the inhomogeneous helicity effect coupled with the mean absolute
vorticity (rotation and large-scale vortical motion), and the cross-helicity
effect coupled with the mean magnetic-field strain may play some role in the
linear and angular momentum transport. These effects have not been well
explored in the previous studies of the turbulent fluxes in the stellar
convection. The first two effects have been argued in the other papers: the
non-equilibrium effect along the plume motions in the stellar convection in
Yokoi, Masada & Takiwaki (2022) and the inhomogeneous turbulent helicity
effect in Yokoi & Yoshizawa (1993); Yokoi & Brandenburg (2016); Yokoi (2023).
In the present article, we will discuss the cross-helicity effect coupled with
the global magnetic-field shear in the momentum transport, later in Section 6.

The organization of this article is as follows. Following Introduction, the
theoretical formulation for the inhomogeneous MHD turbulence is presented in
Section 2. Our theoretical framework: the two-scale direct-interaction
approximation, a multiple-scale renormalized perturbation expansion theory is
briefly outlined. Special reference is put to the distinction of the simple
self-response Green’s functions and the cross-interaction Green’s functions.
As examples of the application of the theory, the analytical expressions for
the turbulent electromotive force in the mean magnetic induction equation and
the Reynolds and turbulent Maxwell stresses in the mean momentum equation are
presented. In Section 3, the cross-helicity effect in the dynamo is discussed.
First, the physical origin of the cross-helicity effect, as well as the
counterparts of the kinetic and electric-current helicity effects and the
turbulent magnetic diffusivity effect, will be argued. Some numerical
validations of the cross-helicity effect and related dynamo are presented. The
properties of cross-helicity dynamo depend on the spatiotemporal evolution of
the turbulent cross helicity. In order to better understand the cross helicity
evolution, the transport equation of the turbulent cross helicity is discussed
in Section 4. The representative production mechanisms of the turbulent cross
helicity are illustrated with special reference to the field configurations
leading to the cross-helicity generation. In addition, the evaluation of the
cross-helicity dissipation rate is presented through the derivation of the
equation of the cross-helicity dissipation rate. In Section 5, the cross
helicity effect is applied to the oscillatory stellar dynamo. Unlike the
kinetic and current helicities, the cross helicity is expected to change its
sign across the reversal of magnetic field. This property of cross helicity is
fully investigated in the framework of the mean-field dynamo equations.  In
Section 6, another important aspect of the cross helicity effects,
contribution in the momentum equation, is discussed. In the presence of non-
trivial mean magnetic-field configuration, the turbulent cross helicity plays
an important role in the linear and angular momenta transport. The physical
origin of the flow generation is discussed with reference to the role of the
fluctuating Lorentz force in the momentum equation. Both of these effects of
turbulent cross-helicity: the magnetic induction and flow generation are
expected to arise in the magnetic reconnection phenomena. In Section 7, the
turbulent effects in magnetic reconnection are argued. The cross helicity,
which counterbalances the turbulent magnetic diffusivity effect, contributes
to the localization of the effective diffusivity, leading to the fast
reconnection. The concluding remarks are given in Section 8, where a special
emphasis is given in the counter-diffusion effect in the turbulent EMF and the
Reynolds and turbulent Maxwell stresses.

##  2 Theoretical formulation

In this section, we present the basic procedure of a multiple-scale
renormalized perturbation expansion theory, the two-scale direct-interaction
approximation (TSDIA) (Yoshizawa, 1984; Yokoi, 2020). To be specific, we
discuss the incompressible magnetohydrodynamic (MHD) turbulence (Yoshizawa,
1990; Hamba & Sato, 2008; Yokoi, 2013), which is favored for presenting the
basic properties of MHD turbulent transport It should be understood, however,
the following formulation can be readily adapted to the strongly compressible
MHD turbulence as well. The calculations are cumbersome, but are
straightforward (Yokoi, 2018a, b).

###  2.1 Fundamental equations

In order to show the basic theoretical formulation for a fluid plasma
turbulence, we present here the fundamental equations of MHD for the
incompressible or non-variable density fluid.

In the non-variable density case, a plasma fluid obeys the equations of
incompressible magnetohydrodynamics, which are constituted by the momentum
equation:

$\frac{\partial{\bf{u}}}{\partial
t}+({\bf{u}}\cdot\nabla){\bf{u}}=({\bf{b}}\cdot\nabla){\bf{b}}-\nabla
p_{\rm{M}}-2\mbox{\boldmath$\omega$}_{\rm{F}}\times{\bf{u}}+\nu\nabla^{2}{\bf{u}},$
(1)

magnetic induction equation:

$\frac{\partial{\bf{b}}}{\partial
t}+({\bf{u}}\cdot\nabla){\bf{b}}=({\bf{b}}\cdot\nabla){\bf{u}}+\eta\nabla^{2}{\bf{b}},$
(2)

and the solenoidal conditions of the velocity and magnetic field:

$\nabla\cdot{\bf{u}}=\nabla\cdot{\bf{b}}=0,$ (3)

where ${\bf{u}}$ is the velocity, ${\bf{b}}$ the magnetic field,
$p_{\rm{M}}(=p+{\bf{b}}^{2}/2)$ the MHD pressure ($p$: gas pressure),
$\mbox{\boldmath$\omega$}_{\rm{F}}$ the angular velocity of a rotation, $\nu$
the kinematic viscosity, and $\eta$ the magnetic diffusivity. Here, the
magnetic field is measured in the Alfvén speed unit defined by
${\bf{b}}={\bf{b}}_{\ast}/(\mu_{0}\rho)^{1/2}$ (${\bf{b}}_{\ast}$: magnetic
field measured in the natural unit, $\mu$: magnetic permeability, $\rho$:
density of fluid).

###  2.2 Mean and fluctuation in multiple-scale analysis

####  2.2.1 Mean and fluctuation

In order to see the turbulent effect on the mean fields, we divide a field
quantity $f$ into the mean $F$ and the fluctuation around it, $f^{\prime}$, as

$f=F+f^{\prime},\;\;F=\langle{f}\rangle$ (4)

with

$f=({\bf{u}},{\bf{b}},p,\mbox{\boldmath$\omega$},{\bf{j}}),$ (5a)
$F=({\bf{U}},{\bf{B}},P,\mbox{\boldmath$\Omega$},{\bf{J}}),$ (5b)
$f^{\prime}=({\bf{u}}^{\prime},{\bf{b}}^{\prime},p^{\prime},\mbox{\boldmath$\omega$}^{\prime},{\bf{j}}^{\prime}).$
(5c)

Here, $\mbox{\boldmath$\omega$}(=\nabla\times{\bf{u}})$ is the vorticity,
${\bf{j}}(=\nabla\times{\bf{b}})$ is the electric-current density, and
$\langle{\cdots}\rangle$ denotes the ensemble average.

Under this decomposition, the equations for the mean velocity ${\bf{U}}$ and
the mean magnetic field ${\bf{B}}$ are written as

$\frac{\partial{\bf{U}}}{\partial
t}+({\bf{U}}\cdot\nabla){\bf{U}}=({\bf{B}}\cdot\nabla){\bf{B}}-\nabla\cdot\mbox{\boldmath${\cal{R}}$}-\nabla
P_{\rm{M}}-2\mbox{\boldmath$\omega$}_{\rm{F}}\times{\bf{U}}+\nu\nabla^{2}{\bf{U}},$
(6) $\frac{\partial{\bf{B}}}{\partial
t}+({\bf{U}}\cdot\nabla){\bf{B}}=({\bf{B}}\cdot\nabla){\bf{U}}-\langle{({\bf{u}}^{\prime}\cdot\nabla){\bf{b}}^{\prime}}\rangle+\langle{({\bf{b}}^{\prime}\cdot\nabla){\bf{u}}^{\prime}}\rangle+\eta\nabla^{2}{\bf{B}},$
(7)

with the solenoidal conditions

$\nabla\cdot{\bf{U}}=\nabla\cdot{\bf{B}}=0,$ (8)

Alternatively, the mean magnetic field equation (7) is rewritten in the
rotational form as

$\frac{\partial{\bf{B}}}{\partial
t}=\nabla\times({\bf{U}}\times{\bf{B}})+\nabla\times{\bf{E}}_{\rm{M}}+\eta\nabla^{2}{\bf{B}}.$
(9)

In (6), $\mbox{\boldmath${\cal{R}}$}=\\{{{\cal{R}}^{ij}}\\}$ is the Reynolds
and turbulent Maxwell stresses defined by

${\cal{R}}^{ij}\equiv\langle{u^{\prime}{}^{i}u^{\prime}{}^{j}-b^{\prime}{}^{i}b^{\prime}{}^{j}}\rangle$
(10)

and $P_{\rm{M}}$ is the mean part of the total MHD pressure $p_{\rm{M}}\equiv
p+{\bf{b}}^{2}/2$, defined by

$P_{\rm{M}}=P+{\bf{B}}^{2}/2+\langle{{\bf{b}}^{\prime}{}^{2}}\rangle/2.$ (11)

In (9), ${\bf{E}}_{\rm{M}}$ is the turbulent electromotive force (EMF) defined
by

${\bf{E}}_{\rm{M}}\equiv\langle{{\bf{u}}^{\prime}\times{\bf{b}}^{\prime}}\rangle.$
(12)

On the other hand, the equations of the velocity fluctuation
${\bf{b}}^{\prime}$ and the magnetic-field fluctuation ${\bf{b}}^{\prime}$ are
written as

$\displaystyle\frac{\partial{\bf{u}}^{\prime}}{\partial t}$ $\displaystyle+$
$\displaystyle({\bf{U}}\cdot\nabla){\bf{u}}^{\prime}=-({\bf{u}}^{\prime}\cdot\nabla){\bf{U}}+({\bf{B}}\cdot\nabla){\bf{b}}^{\prime}+({\bf{b}}^{\prime}\cdot\nabla){\bf{B}}$
(13) $\displaystyle-$
$\displaystyle({\bf{u}}^{\prime}\cdot\nabla){\bf{u}}^{\prime}+({\bf{b}}^{\prime}\cdot\nabla){\bf{b}}^{\prime}+\nabla\cdot\mbox{\boldmath${\cal{R}}$}-\nabla
p^{\prime}_{\rm{M}}-2\mbox{\boldmath$\omega$}_{\rm{F}}\times{\bf{u}}^{\prime}+\nu\nabla^{2}{\bf{u}}^{\prime},$
$\displaystyle\frac{\partial{\bf{b}}^{\prime}}{\partial t}$ $\displaystyle+$
$\displaystyle({\bf{U}}\cdot\nabla){\bf{b}}^{\prime}=-({\bf{u}}^{\prime}\cdot\nabla){\bf{B}}+({\bf{B}}\cdot\nabla){\bf{u}}^{\prime}+({\bf{b}}^{\prime}\cdot\nabla){\bf{U}}$
(14) $\displaystyle-$
$\displaystyle({\bf{u}}^{\prime}\cdot\nabla){\bf{b}}^{\prime}+({\bf{b}}^{\prime}\cdot\nabla){\bf{u}}^{\prime}+\langle{({\bf{u}}^{\prime}\cdot\nabla){\bf{b}}^{\prime}}\rangle-\langle{({\bf{b}}^{\prime}\cdot\nabla){\bf{u}}^{\prime}}\rangle+\eta\nabla^{2}{\bf{b}}^{\prime}$

with the solenoidal conditions

$\nabla\cdot{\bf{u}}^{\prime}=\nabla\cdot{\bf{b}}^{\prime}=0.$ (15)

In (13), $p_{\rm{M}}^{\prime}$ is the fluctuating MHD pressure defined by
$p_{\rm{M}}^{\prime}=p_{\rm{M}}-P_{\rm{M}}$.

####  2.2.2 Multiple-scale analysis

Considering that mean fields vary slowly at large scales while fluctuations do
fast at small scales, we introduce two scales: slow and fast variables as

$\mbox{\boldmath$\xi$}(={\bf{x}}),\;{\bf{X}}(=\delta{\bf{x}}),\;\tau(=t),\;T(=\delta
t),$ (16)

where $\delta$ is the scale parameter. If $\delta$ is small, ${\bf{X}}$ and
$T$ change substantially only when ${\bf{x}}$ and $t$ vary considerably. In
this sense, ${\bf{X}}$ and $T$, which are suitable for describing the slow and
large variations, are called slow variables. On the other hand, $\xi$ and
$\tau$ are called the fast variables. The scale parameter $\delta$ is not
necessarily small, but if $\delta$ is small ($\delta\ll 1$), there is a large
scale separation between the slow and fast variables. Because of the
introduction of two scales defined by (16), the spatial and temporal
derivatives are expressed as

$\nabla_{\bf{x}}=\nabla_{\mbox{\boldmath$\xi$}}+\delta\nabla_{\bf{X}}\mathchar
24635\relax\;\;\;\frac{\partial}{\partial
t}=\frac{\partial}{\partial\tau}+\delta\frac{\partial}{\partial T}.$ (17)

This means that the derivatives with respect to the slow variables ${\bf{X}}$
and $T$ show up with a scale parameter $\delta$. Expansions with respect to
$\delta$ are derivative expansions. With these slow and fast variables, a
field quantity $f$ is expressed as

$f({\bf{x}}\mathchar 24635\relax\;t)=F({\bf{X}}\mathchar
24635\relax\;T)+f^{\prime}(\mbox{\boldmath$\xi$},{\bf{X}}\mathchar
24635\relax\;\tau,T).$ (18)

Note that the fluctuating field $f^{\prime}$ depends on the slow variables
${\bf{X}}$ and $T$ as well as on the fast variables $\xi$ and $\tau$. Such a
dependence of fluctuating fields on the slow variables is of essential
importance for describing inhomogeneous turbulence.

####  2.2.3 Mean- and fluctuation-field equations

In this two-scale formulation, the equations of the fluctuating velocity
${\bf{u}}^{\prime}$ is written as

$\displaystyle\frac{\partial u^{\prime}{}^{i}}{\partial\tau}$ $\displaystyle+$
$\displaystyle U^{j}\frac{\partial
u^{\prime}{}^{i}}{\partial\xi^{j}}+\frac{\partial}{\partial\xi^{j}}(u^{\prime}{}^{j}u^{\prime}{}^{i}-b^{\prime}{}^{j}b^{\prime}{}^{i})+\frac{\partial
p^{\prime}_{\rm{M}}}{\partial\xi^{i}}-\nu\frac{\partial^{2}u^{\prime}{}^{i}}{\partial\xi^{j}\partial\xi^{j}}-B^{j}\frac{\partial
b^{\prime}{}^{i}}{\partial\xi^{j}}$ (19) $\displaystyle=$
$\displaystyle\delta\left[{b^{\prime}{}^{j}\frac{\partial B^{i}}{\partial
X^{j}}-u^{\prime}{}^{j}\left({\frac{\partial U^{i}}{\partial
X^{j}}+\epsilon^{jik}\Omega_{0}^{k}}\right)+B^{j}\frac{\partial
b^{\prime}{}^{i}}{\partial
X^{j}}-\frac{\widetilde{D}u^{\prime}{}^{i}}{DT}}\right.$
$\displaystyle\left.{\hskip 25.0pt-\frac{\partial}{\partial
X^{j}}\left({u^{\prime}{}^{j}u^{\prime}{}^{i}-b^{\prime}{}^{j}b^{\prime}{}^{i}-\langle{u^{\prime}{}^{j}u^{\prime}{}^{i}-b^{\prime}{}^{j}b^{\prime}{}^{i}}\rangle}\right)-\frac{\partial
p^{\prime}_{\rm{M}}}{\partial X^{i}}}\right],$

and the solenoidal condition:

$\frac{\partial u^{\prime}{}^{j}}{\partial\xi^{j}}+\delta\frac{\partial
u^{\prime}{}^{j}}{\partial X^{j}}=0.$ (20)

The counterparts of the fluctuating magnetic field ${\bf{b}}^{\prime}$ is
written as

$\displaystyle\frac{\partial b^{\prime}{}^{i}}{\partial\tau}$ $\displaystyle+$
$\displaystyle U^{j}\frac{\partial
b^{\prime}{}^{i}}{\partial\xi^{j}}+\frac{\partial}{\partial\xi^{j}}(u^{\prime}{}^{j}b^{\prime}{}^{i}-b^{\prime}{}^{j}u^{\prime}{}^{i})-\eta\frac{\partial^{2}b^{\prime}{}^{i}}{\partial\xi^{j}\partial\xi^{j}}-B^{j}\frac{\partial
u^{\prime}{}^{i}}{\partial\xi^{j}}$ (21) $\displaystyle=$
$\displaystyle\delta\left[{u^{\prime}{}^{j}\frac{\partial B^{i}}{\partial
X^{j}}+b^{\prime}{}^{j}\left({\frac{\partial U^{i}}{\partial
X^{j}}+\epsilon^{jik}\Omega_{0}^{k}}\right)+B^{j}\frac{\partial
u^{\prime}{}^{i}}{\partial
X^{j}}-\frac{\widetilde{D}b^{\prime}{}^{i}}{DT}}\right.$
$\displaystyle\left.{\hskip 25.0pt-\frac{\partial}{\partial
X^{j}}\left({u^{\prime}{}^{j}b^{\prime}{}^{i}-b^{\prime}{}^{j}u^{\prime}{}^{i}-\langle{u^{\prime}{}^{j}b^{\prime}{}^{i}-b^{\prime}{}^{j}u^{\prime}{}^{i}}\rangle}\right)}\right],$

and the solenoidal condition:

$\frac{\partial b^{\prime}{}^{j}}{\partial\xi^{j}}+\delta\frac{\partial
b^{\prime}{}^{j}}{\partial X^{j}}=0.$ (22)

Here, in order to keep the material derivatives to be objective, we adopt a
co-rotational derivative

$\frac{\widetilde{D}u^{\prime}{}^{i}}{DT}=\frac{\partial
u^{\prime}{}^{i}}{\partial t}+U^{j}\frac{\partial u^{\prime}{}^{i}}{\partial
x^{j}}+\epsilon^{jik}\Omega_{0}^{k}u^{\prime}{}^{i}$ (23)

with

$\mbox{\boldmath$\Omega$}_{0}=\mbox{\boldmath$\omega$}_{\rm{F}}/\delta,$ (24)

in place of the Lagrange or advective derivative

$\frac{Du^{\prime}{}^{i}}{DT}=\frac{\partial u^{\prime}{}^{i}}{\partial
t}+U^{j}\frac{\partial u^{\prime}{}^{i}}{\partial x^{j}},$ (25)

which is not objective with respect to a rotation. Note that the mean velocity
gradient in the fluctuation equations:

$\frac{\partial U^{i}}{\partial
X^{j}}+\epsilon^{jik}\Omega_{0}^{k}=\frac{1}{2}\left({\frac{\partial
U^{i}}{\partial X^{j}}+\frac{\partial U^{j}}{\partial
X^{i}}}\right)+\frac{1}{2}\left({\frac{\partial U^{i}}{\partial
X^{j}}-\frac{\partial U^{j}}{\partial
X^{i}}+2\epsilon^{jik}\Omega_{0}^{k}}\right)$ (26)

is objective since the both the strain-rate tensor and the absolute-vorticity
tensor are objective (Thiffeault, 2001; Hamba, 2006).

A field quantity $f^{\prime}(\mbox{\boldmath$\xi$},{\bf{X}}\mathchar
24635\relax\;\tau,T)$ is Fourier transformed with respect to the fast spatial
variable $\xi$ as

$f^{\prime}(\mbox{\boldmath$\xi$},{\bf{X}}\mathchar 24635\relax\;\tau,T)=\int
d{\bf{k}}f({\bf{k}},{\bf{X}}\mathchar
24635\relax\;\tau,T)\exp[-i{\bf{k}}\cdot(\mbox{\boldmath$\xi$}-{\bf{U}}\tau)],$
(27)

where the Fourier transform of the fast variable is taken in the frame co-
moving with the local mean velocity ${\bf{U}}$. Hereafter, for the sake of
simplicity of notation, the arguments of the slow variable for the fluctuation
field $f(\mbox{\boldmath$\xi$},{\bf{X}}\mathchar 24635\relax\;\tau,T)$ is
suppressed and just denoted as $f(\mbox{\boldmath$\xi$}\mathchar
24635\relax\;\tau)$.

We apply the Fourier transformation (27) to (19) and (21) and to the
solenoidal conditions (20) and (22). Then we obtain the system of two-scale
differential equations as

$\displaystyle\frac{\partial u^{i}({\bf{k}}\mathchar
24635\relax\;\tau)}{\partial\tau}$ $\displaystyle+$ $\displaystyle\nu
k^{2}u^{i}({\bf{k}}\mathchar 24635\relax\;\tau)$ (28) $\displaystyle-$
$\displaystyle iM^{ij\ell}({\bf{k}})\iint d{\bf{p}}d{\bf{q}}\
\delta({\bf{k}}-{\bf{p}}-{\bf{q}})\times$ $\displaystyle\hskip
60.0pt\left[{u^{j}({\bf{p}}\mathchar
24635\relax\;\tau)u^{\ell}({\bf{q}}\mathchar
24635\relax\;\tau)-b^{j}({\bf{p}}\mathchar
24635\relax\;\tau)b^{\ell}({\bf{q}}\mathchar 24635\relax\;\tau)}\right]$
$\displaystyle+$ $\displaystyle ik^{j}B^{j}b^{i}({\bf{k}}\mathchar
24635\relax\;\tau)$ $\displaystyle=$
$\displaystyle\delta\left[{-D^{ij}({\bf{k}})\frac{\widetilde{D}u^{j}({\bf{k}}\mathchar
24635\relax\;\tau)}{DT_{\rm{I}}}-D^{ij}({\bf{k}})u^{m}({\bf{k}}\mathchar
24635\relax\;\tau)\left({\frac{\partial U^{j}}{\partial
x^{m}}+\epsilon^{mj\ell}\Omega_{0}^{\ell}}\right)}\right.$
$\displaystyle\left.{\hskip 22.0pt+B^{j}\frac{\partial b^{i}({\bf{k}}\mathchar
24635\relax\;\tau)}{\partial X_{\rm{I}}^{j}}+b^{j}({\bf{k}}\mathchar
24635\relax\;\tau)\frac{\partial B^{i}}{\partial x^{j}}}\right],$
$-ik^{j}u^{j}({\bf{k}}\mathchar 24635\relax\;\tau)+\delta\frac{\partial
u^{j}({\bf{k}}\mathchar 24635\relax\;\tau)}{\partial X^{j}}=0,$ (29)
$\displaystyle\frac{\partial b^{i}({\bf{k}}\mathchar
24635\relax\;\tau)}{\partial\tau}$ $\displaystyle+$ $\displaystyle\eta
k^{2}b^{i}({\bf{k}}\mathchar 24635\relax\;\tau)$ (30) $\displaystyle+$
$\displaystyle iN^{ij\ell}({\bf{k}})\iint d{\bf{p}}d{\bf{q}}\
\delta({\bf{k}}-{\bf{p}}-{\bf{q}})\times$ $\displaystyle\hskip
60.0pt\left[{b^{j}({\bf{p}}\mathchar
24635\relax\;\tau)u^{\ell}({\bf{q}}\mathchar
24635\relax\;\tau)-u^{j}({\bf{p}}\mathchar
24635\relax\;\tau)b^{\ell}({\bf{q}}\mathchar 24635\relax\;\tau)}\right]$
$\displaystyle+$ $\displaystyle ik^{j}B^{j}u^{i}({\bf{k}}\mathchar
24635\relax\;\tau)$ $\displaystyle=$
$\displaystyle\delta\left[{-D^{ij}({\bf{k}})\frac{\widetilde{D}b^{j}({\bf{k}}\mathchar
24635\relax\;\tau)}{DT_{\rm{I}}}-D^{ij}({\bf{k}}\mathchar
24635\relax\;\tau)b^{m}({\bf{k}}\mathchar
24635\relax\;\tau)\left({\frac{\partial U^{j}}{\partial
x^{m}}+\epsilon^{mj\ell}\Omega_{0}^{\ell}}\right)}\right.$
$\displaystyle\left.{\hskip 22.0pt+B^{j}\frac{\partial u^{i}({\bf{k}}\mathchar
24635\relax\;\tau)}{\partial X_{\rm{I}}^{j}}+u^{j}({\bf{k}}\mathchar
24635\relax\;\tau)\frac{\partial B^{i}}{\partial X^{j}}}\right],$
$-ik^{j}b^{j}({\bf{k}}\mathchar 24635\relax\;\tau)+\delta\frac{\partial
b^{j}({\bf{k}}\mathchar 24635\relax\;\tau)}{\partial X^{j}}=0,$ (31)

where

$\left({\nabla_{{\bf{X}}{\rm{I}}},\frac{D}{DT_{\rm{I}}}}\right)=\exp\left({-i{\bf{k}}\cdot{\bf{U}}\tau}\right)\left({\nabla_{\bf{X}},\frac{D}{DT}}\right)\exp\left({i{\bf{k}}\cdot{\bf{U}}\tau}\right)$
(32)

is the differential operators in the interaction representation. Here in (28)
and (30),

$M^{ijk}({\bf{k}})=k^{j}D^{ik}({\bf{k}})+k^{k}D^{ij}({\bf{k}}),$ (33)

with the solenoidal projection operator

$D^{ij}({\bf{k}})=\delta^{ij}-\frac{k^{i}k^{j}}{k^{2}},$ (34)

and

$N^{ijk}({\bf{k}})=k^{j}\delta^{ik}-k^{k}\delta^{ij}.$ (35)

They represent the nonlinear interaction among the different modes.

###  2.3 Field equations

####  2.3.1 Scale-parameter expansion

We expand a field $f(\mbox{\boldmath$\xi$}\mathchar 24635\relax\;\tau)$ with
respect to the scale parameter $\delta$, and further expand each field by the
external field (the mean magnetic field in this particular case).

$\displaystyle f^{i}({\bf{k}}\mathchar 24635\relax\;\tau)$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\delta^{n}f_{n}^{i}({\bf{k}}\mathchar
24635\relax\;\tau)-\sum_{n=0}^{\infty}\delta^{n+1}i\frac{k^{i}}{k^{2}}\frac{\partial}{\partial
X_{\rm{I}}^{j}}f_{n}^{j}({\bf{k}}\mathchar 24635\relax\;\tau)$ (36)
$\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\delta^{n}f_{nm}^{i}({\bf{k}}\mathchar
24635\relax\;\tau)-\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\delta^{n+1}i\frac{k^{i}}{k^{2}}\frac{\partial}{\partial
X_{\rm{I}}^{j}}f_{nm}^{j}({\bf{k}}\mathchar 24635\relax\;\tau).$

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

Using the expansion (36), we write the equations of each order in matrix form.
With the abbreviated form of the spectral integral

$\int_{\Delta}=\iint d{\bf{p}}d{\bf{q}}\ \delta({\bf{k}}-{\bf{p}}-{\bf{q}}),$
(37)

the $f_{00}({\bf{k}}\mathchar 24635\relax\;\tau)$ equations are given as

$\displaystyle{\left({\begin{array}[]{cc}\frac{\partial}{\partial\tau}+\nu
k^{2}&0\\\ 0&\frac{\partial}{\partial\tau}+\eta
k^{2}\end{array}}\right)}\left({\begin{array}[]{cc}u_{00}^{i}({\bf{k}}\mathchar
24635\relax\;\tau)\\\ b_{00}^{i}({\bf{k}}\mathchar
24635\relax\;\tau)\rule{0.0pt}{21.52771pt}\end{array}}\right)$ (42)
$\displaystyle{+i\left({\begin{array}[]{cc}-M^{ij\ell}({\bf{k}})\int_{\Delta}u_{00}^{j}({\bf{p}}\mathchar
24635\relax\;\tau)&M^{ij\ell}({\bf{k}})\int_{\Delta}b_{00}^{j}({\bf{p}}\mathchar
24635\relax\;\tau)\\\
N^{ij\ell}({\bf{k}})\int_{\Delta}b_{00}^{j}({\bf{p}}\mathchar
24635\relax\;\tau)&-N^{ij\ell}({\bf{k}})\int_{\Delta}u_{00}^{j}({\bf{p}}\mathchar
24635\relax\;\tau)\end{array}}\right)\left({\begin{array}[]{c}u_{00}^{\ell}({\bf{q}}\mathchar
24635\relax\;\tau)\\\ b_{00}^{\ell}({\bf{q}}\mathchar
24635\relax\;\tau)\rule{0.0pt}{21.52771pt}\end{array}}\right)}$ (47)
$\displaystyle=\left({\begin{array}[]{c}0\\\
0\rule{0.0pt}{21.52771pt}\end{array}}\right),$ (50)

the $f_{01}({\bf{k}}\mathchar 24635\relax\;\tau)$ equations are given as

$\displaystyle{\left({\begin{array}[]{cc}\frac{\partial}{\partial\tau}+\nu
k^{2}&0\\\ 0&\frac{\partial}{\partial\tau}+\eta
k^{2}\end{array}}\right)}\left({\begin{array}[]{cc}u_{01}^{i}({\bf{k}}\mathchar
24635\relax\;\tau)\\\ b_{01}^{i}({\bf{k}}\mathchar
24635\relax\;\tau)\rule{0.0pt}{21.52771pt}\end{array}}\right)$ (55)
$\displaystyle{+i\left({\begin{array}[]{cc}-2M^{ij\ell}({\bf{k}})\int_{\Delta}u_{00}^{j}({\bf{p}}\mathchar
24635\relax\;\tau)&2M^{ij\ell}({\bf{k}})\int_{\Delta}b_{00}^{j}({\bf{p}}\mathchar
24635\relax\;\tau)\\\
N^{ij\ell}({\bf{k}})\int_{\Delta}b_{00}^{j}({\bf{p}}\mathchar
24635\relax\;\tau)&-N^{ij\ell}({\bf{k}})\int_{\Delta}u_{00}^{j}({\bf{p}}\mathchar
24635\relax\;\tau)\end{array}}\right)\left({\begin{array}[]{c}u_{01}^{\ell}({\bf{q}}\mathchar
24635\relax\;\tau)\\\ b_{01}^{\ell}({\bf{q}}\mathchar
24635\relax\;\tau)\rule{0.0pt}{21.52771pt}\end{array}}\right)}$ (60)
$\displaystyle=-ik^{j}B^{j}\left({\begin{array}[]{cc}0&1\\\
1&0\rule{0.0pt}{21.52771pt}\end{array}}\right)\left({\begin{array}[]{c}u_{00}^{i}({\bf{k}}\mathchar
24635\relax\;\tau)\\\ b_{00}^{i}({\bf{k}}\mathchar
24635\relax\;\tau)\rule{0.0pt}{21.52771pt}\end{array}}\right)\equiv\left({\begin{array}[]{c}F_{01u}^{i}\\\
F_{01b}^{i}\rule{0.0pt}{21.52771pt}\end{array}}\right),$ (67)

and the $f_{10}({\bf{k}}\mathchar 24635\relax\;\tau)$ equations are

$\displaystyle{\left({\begin{array}[]{cc}\frac{\partial}{\partial\tau}+\nu
k^{2}&0\\\ 0&\frac{\partial}{\partial\tau}+\eta
k^{2}\end{array}}\right)}\left({\begin{array}[]{cc}u_{10}^{i}({\bf{k}}\mathchar
24635\relax\;\tau)\\\ b_{10}^{i}({\bf{k}}\mathchar
24635\relax\;\tau)\rule{0.0pt}{21.52771pt}\end{array}}\right)$ (72)
$\displaystyle{+i\left({\begin{array}[]{cc}-2M^{ij\ell}({\bf{k}})\int_{\Delta}u_{00}^{j}({\bf{p}}\mathchar
24635\relax\;\tau)&2M^{ij\ell}({\bf{k}})\int_{\Delta}b_{00}^{j}({\bf{p}}\mathchar
24635\relax\;\tau)\\\
N^{ij\ell}({\bf{k}})\int_{\Delta}b_{00}^{j}({\bf{p}}\mathchar
24635\relax\;\tau)&-N^{ij\ell}({\bf{k}})\int_{\Delta}u_{00}^{j}({\bf{p}}\mathchar
24635\relax\;\tau)\end{array}}\right)\left({\begin{array}[]{c}u_{10}^{\ell}({\bf{q}}\mathchar
24635\relax\;\tau)\\\ b_{10}^{\ell}({\bf{q}}\mathchar
24635\relax\;\tau)\rule{0.0pt}{21.52771pt}\end{array}}\right)}$ (77)
$\displaystyle={B^{j}\frac{\partial}{\partial
X_{\rm{I}}^{j}}\left({\begin{array}[]{cc}0&1\\\
1&0\rule{0.0pt}{21.52771pt}\end{array}}\right)\left({\begin{array}[]{c}u_{00}^{i}({\bf{k}})\\\
b_{00}^{i}({\bf{k}})\rule{0.0pt}{21.52771pt}\end{array}}\right)}-D^{ij}({\bf{k}})\frac{\widetilde{D}}{DT_{\rm{I}}}{\left({\begin{array}[]{cc}1&0\\\
0&1\rule{0.0pt}{21.52771pt}\end{array}}\right)\left({\begin{array}[]{c}u_{00}^{j}({\bf{k}})\\\
b_{00}^{j}({\bf{k}})\rule{0.0pt}{21.52771pt}\end{array}}\right)}$ (86)
$\displaystyle+{\left({\begin{array}[]{cc}-D^{ij}({\bf{k}})\left({\frac{\partial
U^{j}}{\partial X^{\ell}}+\epsilon^{\ell
jn}\Omega_{0}^{n}}\right)&D^{ij}({\bf{k}})\frac{\partial B^{j}}{\partial
X^{\ell}}\\\ -D^{ij}({\bf{k}})\frac{\partial B^{j}}{\partial
X^{\ell}}&D^{ij}({\bf{k}})\left({\frac{\partial U^{j}}{\partial
X^{\ell}}+\epsilon^{\ell
jn}\Omega_{0}^{n}}\right)\end{array}}\right)\left({\begin{array}[]{c}u_{00}^{\ell}({\bf{k}}\mathchar
24635\relax\;\tau)\\\ b_{00}^{\ell}({\bf{k}}\mathchar
24635\relax\;\tau)\rule{0.0pt}{21.52771pt}\end{array}}\right)}$ (91)
$\displaystyle\equiv\left({\begin{array}[]{c}F_{10u}^{i}\\\
F_{10b}^{i}\rule{0.0pt}{21.52771pt}\end{array}}\right),$ (94)

where, $F_{01u}$, $F_{01b}$, $F_{10u}$, and $F_{10b}$ denote each component of
the second right-hand sides (r.h.s.) of (67) and (94). They can be regarded as
the forcing for the evolution equations of $f_{01}({\bf{k}}\mathchar
24635\relax\;\tau)$ and $f_{10}({\bf{k}}\mathchar 24635\relax\;\tau)$,
respectively.

####  2.3.2 Introduction of Green’s functions

For the purpose of solving these differential equations, we introduce the
Green’s functions. We consider the responses of the velocity and magnetic
field to infinitesimal perturbations of the velocity and magnetic field
$\delta{\bf{u}}$ and $\delta{\bf{b}}$. Such infinitesimal perturbations arise
from the external stirring force $\delta{\bf{F}}$, for instance for the
velocity perturbation $\delta{\bf{u}}$ as

$\delta u^{i}=\int
d{\bf{k}}^{\prime}\int_{-\infty}^{t}\\!\\!\\!\\!dt^{\prime}\
{\cal{G}}^{ij}({\bf{k}},{\bf{k}}^{\prime}\mathchar 24635\relax\;t,t^{\prime})\
\delta F^{j}({\bf{k}}^{\prime}\mathchar 24635\relax\;t^{\prime})$ (95)

with a Green’s function in the wave number space. As this form shows, the
Green’s function itself is a random variable, which changes from one
realization to realization of ${\bf{u}}$.

It follows from (50) that the equations of the infinitesimal perturbations can
be written as

$\displaystyle{\left({\begin{array}[]{cc}\frac{\partial}{\partial\tau}+\nu
k^{2}&0\\\ 0&\frac{\partial}{\partial\tau}+\eta
k^{2}\end{array}}\right)}\left({\begin{array}[]{cc}\delta
u_{00}^{i}({\bf{k}}\mathchar 24635\relax\;\tau)\\\ \delta
b_{00}^{i}({\bf{k}}\mathchar
24635\relax\;\tau)\rule{0.0pt}{21.52771pt}\end{array}}\right)$ (100)
$\displaystyle{+i\left({\begin{array}[]{cc}-2M^{ij\ell}({\bf{k}})\int_{\Delta}u_{00}^{j}({\bf{p}}\mathchar
24635\relax\;\tau)&2M^{ij\ell}({\bf{k}})\int_{\Delta}b_{00}^{j}({\bf{p}}\mathchar
24635\relax\;\tau)\\\
N^{ij\ell}({\bf{k}})\int_{\Delta}b_{00}^{j}({\bf{p}}\mathchar
24635\relax\;\tau)&-N^{ij\ell}({\bf{k}})\int_{\Delta}u_{00}^{j}({\bf{p}}\mathchar
24635\relax\;\tau)\end{array}}\right)\left({\begin{array}[]{c}\delta
u_{00}^{\ell}({\bf{q}}\mathchar 24635\relax\;\tau)\\\ \delta
b_{00}^{\ell}({\bf{q}}\mathchar
24635\relax\;\tau)\rule{0.0pt}{21.52771pt}\end{array}}\right)}$ (105)
$\displaystyle=\left({\begin{array}[]{c}0\\\
0\rule{0.0pt}{21.52771pt}\end{array}}\right).$ (108)

Here, we should note that the nonlinear convolution terms in (50) have led to
the linear contribution in the form

$\widehat{{\bf{u}}{\bf{u}}}\;\;\rightarrow\;\;2\widehat{{\bf{u}}\delta{\bf{u}}},\;\;\;\;\widehat{{\bf{b}}{\bf{b}}}\;\;\rightarrow\;\;2\widehat{{\bf{b}}\delta{\bf{b}}}.$
(109)

In order to treat mutual interaction among 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 left-hand side (l.h.s.) of (108) 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

$\displaystyle{\left({\begin{array}[]{cc}\frac{\partial}{\partial\tau}+\nu
k^{2}&0\\\ 0&\frac{\partial}{\partial\tau}+\eta
k^{2}\end{array}}\right)}\left({\begin{array}[]{cc}G_{uu}^{ij}&G_{ub}^{ij}\\\
G_{bu}^{ij}&G_{bb}^{ij}\rule{0.0pt}{21.52771pt}\end{array}}\right)$ (114)
$\displaystyle{+i\left({\begin{array}[]{cc}-2M^{ikm}\int_{\Delta}u_{00}^{k}&2M^{ikm}\int_{\Delta}b_{00}^{k}\\\
N^{ikm}\int_{\Delta}b_{00}^{k}&-N^{ikm}\int_{\Delta}u_{00}^{k}\end{array}}\right)\left({\begin{array}[]{cc}G_{uu}^{mj}&G_{ub}^{mj}\\\
G_{bu}^{mj}&G_{bb}^{mj}\rule{0.0pt}{21.52771pt}\end{array}}\right)}$ (119)
$\displaystyle=\delta^{ij}\delta(\tau-\tau^{\prime})\left({\begin{array}[]{cc}1&0\\\
0&1\rule{0.0pt}{21.52771pt}\end{array}}\right).$ (122)

Reflecting the structure of the MHD equations and the field expansion (36),
the left-hand sides (l.h.s.) of (67) and (94) or the differential operators to
the $f_{01}({\bf{k}}\mathchar 24635\relax\;\tau)$ and
$f_{10}({\bf{k}}\mathchar 24635\relax\;\tau)$ fields are in the same form as
the equations of fluctuations to the infinitesimal perturbations (108).
Considering that the r.h.s. of (67) and (94) 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

$\left({\begin{array}[]{c}u_{01}^{i}\\\
b_{01}^{i}\rule{0.0pt}{21.52771pt}\end{array}}\right)=\int_{-\infty}^{\tau}\\!\\!\\!d\tau_{1}\left({\begin{array}[]{cc}G_{uu}^{ij}&G_{ub}^{ij}\\\
G_{bu}^{ij}&G_{bb}^{ij}\rule{0.0pt}{21.52771pt}\end{array}}\right)\left({\begin{array}[]{c}F_{01u}^{j}\\\
F_{01b}^{j}\rule{0.0pt}{21.52771pt}\end{array}}\right),$ (123)

or explicitly written as

$\displaystyle u_{01}^{i}({\bf{k}}\mathchar 24635\relax\;\tau)$
$\displaystyle=$ $\displaystyle\int d\tau_{1}G_{uu}^{ij}F_{01u}^{j}+\int
d\tau_{1}G_{ub}^{ij}F_{01b}^{j}$ (124) $\displaystyle=$
$\displaystyle\int_{-\infty}^{\tau}\\!\\!\\!d\tau_{1}G_{ub}^{ij}({\bf{k}}\mathchar
24635\relax\;\tau,\tau_{1})\left[{-ik^{m}B^{m}b_{00}^{j}({\bf{k}}\mathchar
24635\relax\;\tau)}\right],$ $\displaystyle b_{01}^{i}({\bf{k}}\mathchar
24635\relax\;\tau)$ $\displaystyle=$ $\displaystyle\int d\tau_{1}\
G_{bu}^{ij}F_{01u}^{j}+\int d\tau_{1}\ G_{bb}^{ij}F_{01b}^{j}$ (125)
$\displaystyle=$
$\displaystyle\int_{-\infty}^{\tau}\\!\\!\\!d\tau_{1}G_{bu}^{ij}({\bf{k}}\mathchar
24635\relax\;\tau,\tau_{1})\left[{-ik^{m}B^{m}u_{00}^{j}({\bf{k}}\mathchar
24635\relax\;\tau)}\right].$

Note that ${\bf{u}}_{01}$ and ${\bf{b}}_{01}$ are expressed in terms of
${\bf{b}}_{00}$ and ${\bf{u}}_{00}$ coupled with the mean magnetic field
${\bf{B}}$, respectively. Consequently, ${\bf{u}}_{01}$ 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

$\left({\begin{array}[]{c}u_{10}^{i}\\\
b_{10}^{i}\rule{0.0pt}{21.52771pt}\end{array}}\right)=\int_{-\infty}^{\tau}\\!\\!\\!d\tau_{1}\left({\begin{array}[]{cc}G_{uu}^{ij}&G_{ub}^{ij}\\\
G_{bu}^{ij}&G_{bb}^{ij}\rule{0.0pt}{21.52771pt}\end{array}}\right)\left({\begin{array}[]{c}F_{10u}^{j}\\\
F_{10b}^{j}\rule{0.0pt}{21.52771pt}\end{array}}\right),$ (126)

or explicitly written as

$\displaystyle u_{10}^{i}({\bf{k}}\mathchar 24635\relax\;\tau)=\int d\tau_{1}\
G_{uu}^{ij}F_{10u}^{j}+\int d\tau_{1}\ G_{ub}^{ij}F_{10b}^{j}$
$\displaystyle=\int_{-\infty}^{\tau}\\!\\!\\!d\tau_{1}\
G_{uu}^{ij}({\bf{k}}\mathchar
24635\relax\;\tau,\tau_{1})\left[{-D^{jk}({\bf{k}})\frac{\widetilde{D}u_{00}^{k}({\bf{k}}\mathchar
24635\relax\;\tau_{1})}{DT_{\rm{I}}}}\right.$ $\displaystyle\hskip
90.0pt-D^{jk}({\bf{k}})\left({\frac{\partial U^{k}}{\partial
X^{m}}+\epsilon^{mkn}\Omega_{0}^{n}}\right)u_{00}^{m}({\bf{k}}\mathchar
24635\relax\;\tau_{1})$ $\displaystyle\hskip 90.0pt+\left.{B^{k}\frac{\partial
b_{00}^{i}({\bf{k}}\mathchar 24635\relax\;\tau_{1})}{\partial
X_{\rm{I}}^{k}}+D^{jk}({\bf{k}})\frac{\partial B^{k}}{\partial
X^{m}}b_{00}^{m}({\bf{k}}\mathchar 24635\relax\;\tau_{1})}\right]$
$\displaystyle+\int_{-\infty}^{\tau}\\!\\!\\!d\tau_{1}\
G_{ub}^{ij}({\bf{k}}\mathchar
24635\relax\;\tau,\tau_{1})\left[{B^{k}\frac{\partial
u_{00}^{i}({\bf{k}}\mathchar 24635\relax\;\tau_{1})}{\partial
X_{\rm{I}}^{k}}-D^{jk}({\bf{k}})\frac{\partial B^{k}}{\partial
X^{m}}u_{00}^{m}({\bf{k}}\mathchar 24635\relax\;\tau_{1})}\right.$
$\displaystyle\hskip
10.0pt\left.{-D^{jk}({\bf{k}})\frac{\widetilde{D}b_{00}^{k}({\bf{k}}\mathchar
24635\relax\;\tau_{1})}{DT_{\rm{I}}}+D^{jk}({\bf{k}})\left({\frac{\partial
U^{k}}{\partial
X^{m}}+\epsilon^{mkn}\Omega_{0}}\right)b_{00}^{m}({\bf{k}}\mathchar
24635\relax\;\tau_{1})}\right],$ (127) $\displaystyle
b_{10}^{i}({\bf{k}}\mathchar 24635\relax\;\tau)=\int
d\tau_{1}G_{bu}^{ij}F_{10u}^{j}+\int d\tau_{1}G_{bb}^{ij}F_{10b}^{j}$
$\displaystyle=\int_{-\infty}^{\tau}\\!\\!\\!d\tau_{1}\
G_{bu}^{ij}({\bf{k}}\mathchar
24635\relax\;\tau,\tau_{1})\left[{-D^{jk}({\bf{k}})\frac{\widetilde{D}u_{00}^{k}({\bf{k}}\mathchar
24635\relax\;\tau_{1})}{DT_{\rm{I}}}}\right.$ $\displaystyle\hskip
90.0pt-D^{jk}({\bf{k}})\left({\frac{\partial U^{k}}{\partial
X^{m}}+\epsilon^{mkn}\Omega_{0}^{n}}\right)u_{00}^{m}({\bf{k}}\mathchar
24635\relax\;\tau_{1})$ $\displaystyle\hskip 90.0pt+\left.{B^{k}\frac{\partial
b_{00}^{i}({\bf{k}}\mathchar 24635\relax\;\tau_{1})}{\partial
X_{\rm{I}}^{k}}+D^{jk}({\bf{k}})\frac{\partial B^{k}}{\partial
X^{m}}b_{00}^{m}({\bf{k}}\mathchar 24635\relax\;\tau_{1})}\right]$
$\displaystyle+\int_{-\infty}^{\tau}\\!\\!\\!d\tau_{1}\
G_{bb}^{ij}({\bf{k}}\mathchar
24635\relax\;\tau,\tau_{1})\left[{B^{k}\frac{\partial
u_{00}^{i}({\bf{k}}\mathchar 24635\relax\;\tau_{1})}{\partial
X_{\rm{I}}^{k}}-D^{jk}({\bf{k}})\frac{\partial B^{k}}{\partial
X^{m}}u_{00}^{m}({\bf{k}}\mathchar 24635\relax\;\tau_{1})}\right.$
$\displaystyle\hskip
10.0pt\left.{-D^{jk}({\bf{k}})\frac{\widetilde{D}b_{00}^{k}({\bf{k}}\mathchar
24635\relax\;\tau_{1})}{DT_{\rm{I}}}+D^{jk}({\bf{k}})\left({\frac{\partial
U^{k}}{\partial
X^{m}}+\epsilon^{mkn}\Omega_{0}}\right)b_{00}^{m}({\bf{k}}\mathchar
24635\relax\;\tau_{1})}\right].$ (128)

####  2.3.3 Statistical assumption on the basic fields

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

$\frac{\left\langle{\vartheta_{00}^{i}({\bf{k}}\mathchar
24635\relax\;\tau)\chi_{00}^{j}({\bf{k}}^{\prime}\mathchar
24635\relax\;\tau^{\prime})}\right\rangle}{\delta({\bf{k}}+{\bf{k}}^{\prime})}=D^{ij}({\bf{k}})Q_{\vartheta\chi}({\bf{k}}\mathchar
24635\relax\;\tau,\tau^{\prime})+\frac{i}{2}\frac{k^{\ell}}{k^{2}}\epsilon^{ij\ell}H_{\vartheta\chi}({\bf{k}}\mathchar
24635\relax\;\tau,\tau^{\prime}),$ (129)

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

$\langle{G_{\vartheta\chi}^{ij}({\bf{k}}\mathchar
24635\relax\;\tau,\tau^{\prime})}\rangle=D^{ij}({\bf{k}})G_{\vartheta\chi}({\bf{k}}\mathchar
24635\relax\;\tau,\tau^{\prime}).$ (130)

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

$\int d{\bf{k}}\ Q_{uu}(k\mathchar
24635\relax\;\tau,\tau)=\langle{{\bf{u}}^{\prime}_{00}{}^{2}}\rangle/2,$ (131)
$\int d{\bf{k}}\ Q_{bb}(k\mathchar
24635\relax\;\tau,\tau)=\langle{{\bf{b}}^{\prime}_{00}{}^{2}}\rangle/2,$ (132)
$\int d{\bf{k}}\ Q_{ub}(k\mathchar
24635\relax\;\tau,\tau)=\langle{{\bf{u}}^{\prime}_{00}\cdot{\bf{b}}^{\prime}_{00}}\rangle,$
(133) $\int d{\bf{k}}\ H_{uu}(k\mathchar
24635\relax\;\tau,\tau)=\langle{{\bf{u}}^{\prime}_{00}\cdot\mbox{\boldmath$\omega$}^{\prime}_{00}}\rangle,$
(134) $\int d{\bf{k}}\ H_{bb}(k\mathchar
24635\relax\;\tau,\tau)=\langle{{\bf{b}}^{\prime}_{00}\cdot{\bf{j}}^{\prime}_{00}}\rangle,$
(135) $\int d{\bf{k}}\ H_{ub}(k\mathchar
24635\relax\;\tau,\tau)=\langle{{\bf{u}}^{\prime}_{00}\cdot{\bf{j}}^{\prime}_{00}}\rangle,$
(136) $\int d{\bf{k}}\ H_{bu}(k\mathchar
24635\relax\;\tau,\tau)=\langle{{\bf{b}}^{\prime}_{00}\cdot\mbox{\boldmath$\omega$}^{\prime}_{00}}\rangle.$
(137)

###  2.4 Calculation of the electromotive force

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

$E_{\rm{M}}^{i}\equiv\epsilon^{ijk}\langle{u^{\prime}{}^{j}b^{\prime}{}^{k}}\rangle=\epsilon^{ijk}\int
d{\bf{k}}\ \langle{u^{j}({\bf{k}}\mathchar
24635\relax\;\tau)b^{k}({\bf{k}}^{\prime}\mathchar
24635\relax\;\tau)}\rangle/\delta({\bf{k}}+{\bf{k}}^{\prime}).$ (138)

Using the results of (123)-(128), we calculate the velocity–magnetic-field
correlation up to the $f_{01}g_{00}$ and $f_{10}g_{00}$ orders as

$\langle{u^{j}b^{k}}\rangle=\langle{u_{00}^{j}b_{00}^{k}}\rangle+\langle{u_{01}^{j}b_{00}^{k}}\rangle+\langle{u_{00}^{j}b_{01}^{k}}\rangle+\delta\langle{u_{10}^{j}b_{00}^{k}}\rangle+\delta\langle{u_{00}^{j}b_{10}^{k}}\rangle+\cdots.$
(139)

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 as

$\displaystyle
Q_{uu}\to\widetilde{Q}_{uu},\;\;Q_{bb}\to\widetilde{Q}_{bb},\;\;Q_{ub}\to\widetilde{Q}_{ub},\;\;$
$\displaystyle
H_{uu}\to\widetilde{H}_{uu},\;\;H_{bb}\to\widetilde{H}_{ub},\;\;H_{ub}\to\widetilde{H}_{ub},\;\;H_{bu}\to\widetilde{Q}_{bu}$
$\displaystyle
G_{uu}\to\widetilde{G}_{uu},\;\;G_{bb}\to\widetilde{G}_{bb},\;\;G_{ub}\to\widetilde{G}_{ub},\;\;G_{bu}\to\widetilde{G}_{bu}.$
(140)

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 the exact propagator will be
suppressed. Namely, the exact propagators are denoted without tilde.

Here we present the final results of the turbulent EMF as

$\langle{{\bf{u}}^{\prime}\times{\bf{b}}^{\prime}}\rangle=\alpha{\bf{B}}-(\beta+\zeta)\nabla\times{\bf{B}}-(\nabla\zeta)\times{\bf{B}}+\gamma\nabla\times{\bf{U}},$
(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 the spectral and time integral

$I\\{{A,B}\\}=\int
d{\bf{k}}\int_{-\infty}^{\tau}\\!\\!\\!d\tau_{1}A(k\mathchar
24635\relax\;\tau,\tau_{1})B(k\mathchar 24635\relax\;\tau,\tau_{1}).$ (146)

###  2.5 Reynolds and turbulent Maxwell stresses

The Reynolds and turbulent Maxwell stresses in the mean momentum equation is
defined by

${\cal{R}}^{ij}=\langle{u^{\prime}{}^{i}u^{\prime}{}^{j}}\rangle-\langle{b^{\prime}{}^{i}b^{\prime}{}^{j}}\rangle.$
(147)

In a similar manner as for the turbulent electromotive force
${\bf{E}}_{\rm{M}}$, ${\cal{R}}$ can be calculated from ${\bf{u}}^{\prime}$
and ${\bf{b}}^{\prime}$ with

$\langle{u^{j}u^{k}}\rangle=\langle{u_{00}^{j}u_{00}^{k}}\rangle+\langle{u_{01}^{j}u_{00}^{k}}\rangle+\langle{u_{00}^{j}u_{01}^{k}}\rangle+\delta\langle{u_{10}^{j}u_{00}^{k}}\rangle+\delta\langle{u_{00}^{j}u_{10}^{k}}\rangle+\cdots,$
(148)
$\langle{b^{j}b^{k}}\rangle=\langle{b_{00}^{j}b_{00}^{k}}\rangle+\langle{b_{01}^{j}b_{00}^{k}}\rangle+\langle{b_{00}^{j}b_{01}^{k}}\rangle+\delta\langle{b_{10}^{j}b_{00}^{k}}\rangle+\delta\langle{b_{00}^{j}b_{10}^{k}}\rangle+\cdots.$
(149)

The expression of ${\cal{R}}$ is given as

$\langle{u^{\prime}{}^{i}u^{\prime}{}^{j}-b^{\prime}{}^{i}b^{\prime}{}^{j}}\rangle_{\rm{D}}=-\nu_{\rm{K}}{\cal{S}}^{ij}+\nu_{\rm{M}}{\cal{M}}^{ij}+[\Gamma^{i}\Omega_{\ast}^{j}+\Gamma^{j}\Omega_{\ast}^{i}]_{\rm{D}},$
(150)

where
$\mbox{\boldmath$\Omega$}_{\ast}(=\nabla\times{\bf{U}}+2\mbox{\boldmath$\omega$}_{\rm{F}})$
is the absolute vorticity (the relative vorticity and the angular velocity of
rotation), the strain rate of the mean velocity
$\mbox{\boldmath${\cal{S}}$}=\\{{{\cal{S}}^{ij}}\\}$ and that of the mean
magnetic field $\mbox{\boldmath${\cal{M}}$}=\\{{{\cal{M}}^{ij}}\\}$ are
defined by

${\cal{S}}^{ij}=\frac{\partial U^{j}}{\partial x^{i}}+\frac{\partial
U^{i}}{\partial x^{j}}-\frac{2}{3}\delta^{ij}\nabla\cdot{\bf{U}},$ (151)
${\cal{M}}^{ij}=\frac{\partial B^{j}}{\partial x^{i}}+\frac{\partial
B^{i}}{\partial
x^{j}}-\frac{2}{3}\delta^{ij}\nabla\cdot{\bf{B}}=\frac{\partial
B^{j}}{\partial x^{i}}+\frac{\partial B^{i}}{\partial x^{j}},$ (152)

respectively, and the suffix ${\rm{D}}$ denotes the deviatoric part of a
tensor as

${\cal{A}}^{ij}_{\rm{D}}={\cal{A}}^{ij}-\frac{1}{3}\delta^{ij}{\cal{A}}^{\ell\ell}.$
(153)

The transport coefficients $\nu_{\rm{K}}$, $\nu_{\rm{M}}$, and $\Gamma$ in
(150) are expressed as

$\nu_{\rm{K}}=\frac{7}{5}\beta=\frac{7}{15}\left[{I\\{{G_{bb},Q_{uu}}\\}+I\\{{G_{uu},Q_{bb}}\\}-I\\{{G_{bu},Q_{ub}}\\}-I\\{{G_{ub},Q_{bu}}\\}}\right],$
(154)
$\nu_{\rm{M}}=\frac{7}{5}\gamma=\frac{7}{15}\left[{I\\{{G_{bb},Q_{ub}}\\}+I\\{{G_{uu},Q_{bu}}\\}-I\\{{G_{bu},Q_{uu}}\\}-I\\{{G_{ub},Q_{bb}}\\}}\right],$
(155)
$\mbox{\boldmath$\Gamma$}=\frac{1}{15}\left({I_{-1}\\{{G_{uu}+G_{bb},\nabla
H_{uu}}\\}-I_{-1}\\{{G_{ub},\nabla H_{bu}}\\}}\right).$ (156)

Here, $\nu_{\rm{K}}$ is the eddy viscosity coupled with the mean velocity
strain $\mbox{\boldmath${\cal{S}}$}=\\{{{\cal{S}}^{ij}}\\}$, and is determined
mainly by the turbulent MHD energy as the first two terms of the right-most
equation (154) show. On the other hand, $\nu_{\rm{M}}$ couples with the mean
magnetic-field strain $\mbox{\boldmath${\cal{M}}$}=\\{{{\cal{M}}^{ij}}\\}$,
and is mainly determined by the turbulent cross helicity
$\langle{{\bf{u}}^{\prime}\cdot{\bf{b}}^{\prime}}\rangle$ as the first two
terms of the right-most equation (155) show. The transport coefficient
$\Gamma$ couples with the absolute vorticity
$\mbox{\boldmath$\Omega$}_{\ast}$, and is determined by the gradient of the
turbulent helicity
$\langle{{\bf{u}}^{\prime}\cdot\mbox{\boldmath$\omega$}}\rangle$.

The expression of ${\cal{R}}$ (150) indicates that, in addition to the eddy or
turbulent viscosity $\nu_{\rm{K}}$ coupled with the mean velocity strain
(symmetric part of the mean velocity shear), we also have other effects in the
Reynolds and turbulent Maxwell stresses. One is the cross-helicity-related
effect $\nu_{\rm{M}}$ coupled with the mean magnetic-field strain, and the
other is the inhomogeneous kinetic helicity effect $\Gamma$ coupled with the
mean absolute vorticity (anti-symmetric part of the mean velocity shear).

The role of the Reynolds and turbulent Maxwell stresses and the consequence of
the expression (150) shall be further discussed later in Sections 6 and 7 with
special reference to the large-scale flow generation and the linear and
angular momentum transport.

###  2.6 Symmetric and anti-symmetric response function effects

####  2.6.1 Standard self-interaction response function effects

In our formulations, we have four Green’s functions, $G_{uu}$, $G_{bb}$,
$G_{ub}$, and $G_{bu}$, whose definitions are given in the evolution equations
(122). If we assume that the cross-interaction Green’s functions $G_{ub}$ and
$G_{bu}$ vanish as

$G_{ub}=G_{bu}=0,$ (157)

the usual EMF for the incompressible turbulence is recovered. As the simplest
possible model for the EMF, apart from the model constants, we have

$\alpha=-\tau_{b}\langle{{\bf{u}}^{\prime}\cdot\mbox{\boldmath$\omega$}}\rangle+\tau_{u}\langle{{\bf{b}}^{\prime}\cdot{\bf{j}}^{\prime}}\rangle\equiv\alpha_{\rm{S}},$
(158)
$\beta=\tau_{b}\langle{{\bf{u}}^{\prime}{}^{2}}\rangle/2+\tau_{u}\langle{{\bf{b}}^{\prime}{}^{2}}\rangle/2\equiv\beta_{\rm{S}},$
(159)
$\zeta=\tau_{b}\langle{{\bf{u}}^{\prime}{}^{2}}\rangle/2-\tau_{u}\langle{{\bf{b}}^{\prime}{}^{2}}\rangle/2\equiv\zeta_{\rm{S}},$
(160)
$\gamma=(\tau_{b}+\tau_{u})\langle{{\bf{u}}^{\prime}\cdot{\bf{b}}^{\prime}}\rangle\equiv\gamma_{\rm{S}},$
(161)

where $\tau_{u}$ and $\tau_{b}$ are timescales associated with the Green’s
function $G_{uu}$ and $G_{bb}$, and they are evaluated as

$\tau_{u}f({\bf{k}},\tau)\sim\int_{-\infty}^{\tau}\\!\\!\\!d\tau^{\prime}\
G_{uu}({\bf{k}}\mathchar 24635\relax\;\tau,\tau^{\prime})f({\bf{k}}\mathchar
24635\relax\;\tau^{\prime}),$ (162)
$\tau_{b}f({\bf{k}},\tau)\sim\int_{-\infty}^{\tau}\\!\\!\\!d\tau^{\prime}\
G_{bb}({\bf{k}}\mathchar 24635\relax\;\tau,\tau^{\prime})f({\bf{k}}\mathchar
24635\relax\;\tau^{\prime}),$ (163)

respectively. With special emphasis on the simplest dynamo model, we denote
these transport coefficients with suffix ${\rm{S}}$ as $\alpha_{\rm{S}}$,
$\beta_{\rm{S}}$, $\zeta_{\rm{S}}$, and $\gamma_{\rm{S}}$, as the right-most
sides of (158)-(161) show.

The alpha effect, $\alpha{\bf{B}}$, the first term in (141), depends on the
kinetic helicity density
$\langle{{\bf{u}}^{\prime}\cdot\mbox{\boldmath$\omega$}^{\prime}}\rangle$ and
the electric-current helicity density
$\langle{{\bf{b}}^{\prime}\cdot{\bf{j}}^{\prime}}\rangle$. The physical origin
of the kinetic helicity effect and the current helicity effect will be
discussed in the following section (Section 3). Equation (158) shows that, if
the kinetic helicity
$\langle{{\bf{u}}^{\prime}\cdot\mbox{\boldmath$\omega$}^{\prime}}\rangle$ and
current helicity $\langle{{\bf{b}}^{\prime}\cdot{\bf{j}}^{\prime}}\rangle$
have the same sign, their effects are suppressed with each other. This is the
reason why the current-helicity effect is often argued as a correction to the
alpha effect, leading to the suppression or saturation of the alpha effect.
However, this is no the case if the kinetic and current helicities are
generated by each production mechanism due to large-scale inhomogeneities.
This point will be discussed in Section 3. We should note that the timescale
associated with the kinetic helicity contribution is the magnetic one
$\tau_{b}$, while the counterpart associated with the electric-current
helicity is the kinetic one $\tau_{u}$.

The turbulent magnetic diffusivity effect coupled with the mean electric-
current density ${\bf{J}}(=\nabla\times{\bf{B}})$, the second term in (141),
arises from the turbulent kinetic and magnetic energies. Since the magnetic
energy $\langle{{\bf{b}}^{\prime}{}^{2}}\rangle$ contributions in $\beta$ and
$\zeta$ completely cancel with each other, the magnetic diffusivity
$\beta+\zeta$ depends solely on the turbulent kinetic energy
$\langle{{\bf{u}}^{\prime}{}^{2}}\rangle$. It has been theoretically pointed
out that the turbulent magnetic energy
$\langle{{\bf{b}}^{\prime}{}^{2}}\rangle$ contributes to the turbulent
magnetic diffusivity in anisotropic turbulence (Rogachevskii & Kleeorin, 2001)
and in the presence of compressibility (Yokoi, 2018a).

The magnetic pumping effect $-(\nabla\zeta)\times{\bf{B}}$, the third term in
(141), depends on the gradient of the MHD residual energy
$\langle{{\bf{u}}^{\prime}{}^{2}-{\bf{b}}^{\prime}{}^{2}}\rangle/2$. For the
same mean magnetic field ${\bf{B}}$, the direction of the pumping effect
alters depending on the direction of $\nabla\zeta$.

The cross helicity effect $\gamma\mbox{\boldmath$\Omega$}_{\ast}$, the fourth
term in (141), arises from the turbulent cross helicity
$\langle{{\bf{u}}^{\prime}\cdot{\bf{b}}^{\prime}}\rangle$. This effect is
expected to play an important role in dynamo in the presence of the global
vortical motion or rotation represented by the mean absolute vorticity
$\mbox{\boldmath$\Omega$}_{\ast}(=\nabla\times{\bf{U}}+2\mbox{\boldmath$\omega$}_{\rm{F}})$
($\mbox{\boldmath$\omega$}_{\rm{F}}$: angular velocity of rotation).

####  2.6.2 Effects of cross-interaction response functions

If we retain the contributions from the cross-interaction Green’s functions
$G_{ub}$ and $G_{bu}$, the third and fourth terms in each of (142)-(142), we
have additional contributions to the dynamo transport coefficients $\alpha$,
$\beta$, $\zeta$, and $\gamma$. In order to get a clear picture on the cross-
interaction Green’s functions, we first consider the counterpart of the cross-
interaction in the Elsässer-variable formulation (Yoshizawa, 1990; Yokoi,
2013).

In the Elsässer-variable formulation with
$\mbox{\boldmath$\phi$}={\bf{u}}+{\bf{b}}$ and
$\mbox{\boldmath$\psi$}={\bf{u}}-{\bf{b}}$, we introduce four Green’s
functions, $G_{\phi\phi}$, $G_{\psi\psi}$, $G_{\phi\phi}$, $G_{\phi\psi}$, and
$G_{\psi\phi}$. In this formulation, the transport coefficients of the
turbulent EMF are expressed as

$\alpha=-I\\{{G_{\rm{S}},H_{uu}}\\}+I\\{{G_{\rm{S}},H_{bb}}\\}-I\\{{G_{\rm{A}},H_{ub}}\\}+I\\{{G_{\rm{A}},H_{bu}}\\},$
(164)
$\beta=I\\{{G_{\rm{S}},Q_{uu}}\\}+I\\{{G_{\rm{S}},Q_{bb}}\\}-I\\{{G_{\rm{A}},Q_{ub}}\\}-I\\{{G_{\rm{A}},Q_{bu}}\\},$
(165)
$\zeta=I\\{{G_{\rm{S}},Q_{uu}}\\}-I\\{{G_{\rm{S}},Q_{bb}}\\}+I\\{{G_{\rm{A}},Q_{ub}}\\}-I\\{{G_{\rm{A}},Q_{bu}}\\},$
(166)
$\gamma=I\\{{G_{\rm{S}},Q_{ub}}\\}+I\\{{G_{\rm{S}},Q_{bu}}\\}-I\\{{G_{\rm{A}},Q_{uu}}\\}-I\\{{G_{\rm{A}},Q_{bb}}\\},$
(167)

where $G_{\rm{S}}$ and $G_{\rm{A}}$ denote the mirror-symmetric and anti-
mirror-symmetric parts of $G_{\phi\phi}$ and $G_{\psi\psi}$, defined by

$G_{\rm{S}}=\frac{G_{\phi\phi}+G_{\psi\psi}}{2},$ (168)
$G_{\rm{A}}=\frac{G_{\phi\phi}-G_{\psi\psi}}{2},$ (169)

respectively.

Comparing (164)-(167) with (142)-(145), we see that the self-interaction part
$G_{uu}$ and $G_{bb}$ corresponds to the symmetric part, $G_{\rm{S}}$, while
the cross-interaction part, $G_{ub}$ and $G_{bu}$, does the anti-symmetric
part, $G_{\rm{A}}$. We see from (169) that the anti-symmetric part
$G_{\rm{A}}$ is connected to the difference between the timescales associated
with the Alfvén waves propagating in the counter-parallel and pro-parallel
directions along the magnetic field. The imbalance between the counter- and
pro-propagating Alfvén waves results in a non-vanishing turbulent cross
helicity. These points suggest that non-vanishing $G_{ub}$ and $G_{bu}$ are
linked to the turbulent cross helicity.

Following the above consideration, the dynamo coefficients
$d=(\alpha,\beta,\zeta,\gamma)$ are constituted of $d_{\rm{S}}$ and
$d_{\rm{X}}$ as

$d=d_{\rm{S}}+d_{\rm{X}},$ (170)

where $d_{\rm{S}}$ is the standard $d$ coefficients defined by (158)-(161).
The cross-interaction parts of the dynamo coefficients, $\alpha_{\rm{X}}$,
$\beta_{\rm{X}}$, $\zeta_{\rm{X}}$, and $\gamma_{\rm{X}}$, are expressed from
the third and fourth terms of (142)-(145) as

$\alpha_{\rm{X}}=\frac{1}{3}\left[{-I\\{{G_{bu},H_{ub}}\\}+I\\{{G_{ub},H_{bu}}\\}}\right],$
(171)
$\beta_{\rm{X}}=\frac{1}{3}\left[{-I\\{{G_{bu},Q_{ub}}\\}-I\\{{G_{ub},Q_{bu}}\\}}\right],$
(172)
$\zeta_{\rm{X}}=\frac{1}{3}\left[{+I\\{{G_{bu},Q_{ub}}\\}-I\\{{G_{ub},Q_{bu}}\\}}\right],$
(173)
$\gamma_{\rm{X}}=\frac{1}{3}\left[{-I\\{{G_{bu},Q_{uu}}\\}+I\\{{G_{ub},Q_{bb}}\\}}\right].$
(174)

These cross-interaction dynamo coefficients may be modeled as

$\alpha_{\rm{X}}=-\tau_{{\rm{X}}bu}\Upsilon\langle{{\bf{u}}^{\prime}\cdot{\bf{j}}^{\prime}}\rangle+\tau_{{\rm{X}}ub}\Upsilon\langle{\mbox{\boldmath$\omega$}^{\prime}\cdot{\bf{b}}^{\prime}}\rangle,$
(175)
$\beta_{\rm{X}}=-\tau_{{\rm{X}}bu}\Upsilon\langle{{\bf{u}}^{\prime}\cdot{\bf{b}}^{\prime}}\rangle-\tau_{{\rm{X}}ub}\Upsilon\langle{{\bf{u}}^{\prime}\cdot{\bf{b}}^{\prime}}\rangle,$
(176)
$\zeta_{\rm{X}}=+\tau_{{\rm{X}}bu}\Upsilon\langle{{\bf{u}}^{\prime}\cdot{\bf{b}}^{\prime}}\rangle-\tau_{{\rm{X}}ub}\Upsilon\langle{{\bf{u}}^{\prime}\cdot{\bf{b}}^{\prime}}\rangle,$
(177)
$\gamma_{\rm{X}}=-\tau_{{\rm{X}}bu}\Upsilon\langle{{\bf{u}}^{\prime}{}^{2}}\rangle+\tau_{{\rm{X}}ub}\Upsilon\langle{{\bf{b}}^{\prime}{}^{2}}\rangle,$
(178)

where $\tau_{{\rm{X}}bu}$ and $\tau_{{\rm{X}}ub}$ are timescales associated
with the Green’s functions $G_{bu}$ and $G_{ub}$, respectively.

For instance, the transport coefficient $\alpha_{\rm{X}}$ as well as
$\alpha_{\rm{S}}$ is a pseudo-scalar. Since both
$\langle{{\bf{u}}^{\prime}\cdot{\bf{j}}^{\prime}}\rangle$ and
$\langle{\mbox{\boldmath$\omega$}^{\prime}\cdot{\bf{b}}^{\prime}}\rangle$ are
pure-scalars, we need a pseudo-scalar factor $\Upsilon$ in (175). This is a
direct consequence of the fact that the Green’s functions $G_{bu}$ and
$G_{ub}$ are pseudo-scalar functions as their definitions (122) show. As the
simplest possible candidate, we adopt a non-dimensional pseudoscalar quantity
defined by

$\Upsilon=\frac{\langle{{\bf{u}}^{\prime}\cdot{\bf{b}}^{\prime}}\rangle}{\langle{{\bf{u}}^{\prime}{}^{2}+{\bf{b}}^{\prime}{}^{2}}\rangle/2},$
(179)

or alternatively we may adopt

$\Upsilon=\frac{\langle{{\bf{u}}^{\prime}\cdot{\bf{b}}^{\prime}}\rangle}{\sqrt{\langle{{\bf{u}}^{\prime}{}^{2}}\rangle\langle{{\bf{b}}^{\prime}{}^{2}}\rangle}}.$
(180)

The $\alpha_{\rm{X}}$ expression (171) and its model (175) suggest that there
are some conditions for this cross-interaction response effect to work. First
we have to remark that the torsional cross correlations
$\langle{{\bf{u}}^{\prime}\cdot{\bf{j}}^{\prime}}\rangle$ and
$\langle{\mbox{\boldmath$\omega$}^{\prime}\cdot{\bf{b}}^{\prime}}\rangle$ are
related to each other as

$-\langle{{\bf{u}}^{\prime}\cdot{\bf{j}}^{\prime}}\rangle+\langle{\mbox{\boldmath$\omega$}^{\prime}\cdot{\bf{b}}^{\prime}}\rangle=\nabla\cdot\langle{{\bf{u}}^{\prime}\times{\bf{b}}^{\prime}}\rangle.$
(181)

In homogeneous turbulence, the r.h.s. of (181) vanishes, and we have

$\langle{{\bf{u}}^{\prime}\cdot{\bf{j}}^{\prime}}\rangle=\langle{\mbox{\boldmath$\omega$}^{\prime}\cdot{\bf{b}}^{\prime}}\rangle.$
(182)

In this case, we have no contribution to $\alpha_{\rm{X}}$ for
$\tau_{bu}=\tau_{ub}$. This suggests the conditions for the effect of the
cross-interaction response functions to work:

(i) Even if we have no timescale difference ($\tau_{bu}=\tau_{ub}$), non-zero
flux of the turbulent EMF across the boundary
$\nabla\cdot\langle{{\bf{u}}^{\prime}\times{\bf{b}}^{\prime}}\rangle$ may lead
to a finite $\alpha_{\rm{X}}$;

(ii) Even in homogeneous turbulence, where
$\langle{{\bf{u}}^{\prime}\cdot{\bf{j}}^{\prime}}\rangle=\langle{\mbox{\boldmath$\omega$}^{\prime}\cdot{\bf{b}}^{\prime}}\rangle$,
if there is a timescale difference between $\tau_{bu}$ and $\tau_{ub}$, we
have a non-zero $\alpha_{\rm{X}}$;

(iii) The pseudo-scalar factor $\Upsilon$ coupled with the timescale
$\tau_{bu}$ and $\tau_{ub}$, should represent $G_{bu}$ and $G_{ub}$. The
torsional cross correlations may be regarded as the combination of the cross
helicity and the helicities as

$\langle{{\bf{u}}^{\prime}\cdot{\bf{j}}^{\prime}}\rangle\propto\langle{{\bf{u}}^{\prime}\cdot{\bf{b}}^{\prime}}\rangle\langle{{\bf{b}}^{\prime}\cdot{\bf{j}}^{\prime}}\rangle,$
(183)
$\langle{\mbox{\boldmath$\omega$}^{\prime}\cdot{\bf{b}}^{\prime}}\rangle\propto\langle{\mbox{\boldmath$\omega$}^{\prime}\cdot{\bf{u}}^{\prime}}\rangle\langle{{\bf{u}}^{\prime}\cdot{\bf{b}}^{\prime}}\rangle,$
(184)

This suggests that the coexistence of the cross helicity and current helicity
and the coexistence of the cross helicity and kinetic helicity may be
favorable conditions for the cross-interaction response effect
$\alpha_{\rm{X}}$. In this sense, the turbulent cross helicity may play a key
role in this cross-interaction effect.  As an important possibility, the
cross-interaction response effect in the $\alpha$ effect, $\alpha_{\rm{X}}$,
is investigated in the non-equilibrium or non-stationary turbulence in
Mizerski, Yokoi & Brandenburg (2023).

We should note that this cross-interaction effect arises from the formulation
with the response functions. In addition to the velocity and magnetic-field
fluctuations and their correlation functions, the equations of the responses
are also treated. As a result of this formulation, the response of the
velocity fluctuation to the magnetic disturbance and the response of the
magnetic-field fluctuation to the velocity disturbance enter the expressions
of the turbulent fluxes. The structure of this formulation should be further
explored.

These cross-interaction effects may show a strong relevance especially under
some conditions, such as the turbulence with non-equilibrium, non-stationary,
and breakage of some symmetry. However, such conditions are rather specific,
we only treat the standard or self-interaction response-function effects in
the following sections. This does not deny the potential importance of the
cross-interaction effects.

##  3 Cross-helicity effect in dynamos

In the previous section, with the aid of analytical formulation for the
inhomogeneous turbulence: the multiple-scale renormalized perturbation
expansion theory, the expressions of the turbulent electromotive force (EMF)
$\langle{{\bf{u}}^{\prime}\times{\bf{b}}^{\prime}}\rangle$ in the mean
induction equation and the Reynolds and turbulent Maxwell stresses
$\langle{{\bf{u}}^{\prime}{\bf{u}}^{\prime}-{\bf{b}}^{\prime}{\bf{b}}^{\prime}}\rangle$
in the mean momentum equation, are derived. The turbulent cross-helicity
effect gets into the
$\langle{{\bf{u}}^{\prime}\times{\bf{b}}^{\prime}}\rangle$ expression coupled
with the inhomogeneous mean velocity, and into the
$\langle{{\bf{u}}^{\prime}{\bf{u}}^{\prime}-{\bf{b}}^{\prime}{\bf{b}}^{\prime}}\rangle$
expression coupled with the inhomogeneous mean magnetic field. In this
section, the cross-helicity effect in the mean magnetic-field induction is
argued. This argument is followed by the argument of the cross-helicity
evolution in Section 4, and the cross-helicity effect in stellar dynamos in
Section 5. The cross-helicity effect in the momentum equation will be argued
in Section 6.

###  3.1 Cross helicity and global flow effect in dynamos

The mean magnetic induction equation is written as

$\frac{\partial{\bf{B}}}{\partial
t}=\nabla\times({\bf{U}}\times{\bf{B}})+\nabla\times\langle{{\bf{u}}^{\prime}\times{\bf{b}}^{\prime}}\rangle+\eta\nabla^{2}{\bf{B}},$
(185)

or equivalently

$\frac{\partial{\bf{B}}}{\partial
t}+({\bf{U}}\cdot\nabla){\bf{B}}=({\bf{B}}\cdot\nabla){\bf{U}}-(\nabla\cdot{\bf{U}}){\bf{B}}+\nabla\times\langle{{\bf{u}}^{\prime}\times{\bf{b}}^{\prime}}\rangle+\eta\nabla^{2}{\bf{B}}.$
(186)

The first term on the right-hand side (r.h.s.) of (186) represents the
differential-rotation effect. If the mean velocity ${\bf{U}}$ is inhomogeneous
along the mean magnetic field ${\bf{B}}$, $({\bf{B}}\cdot\nabla){\bf{U}}$, it
contributes to the generation of the mean magnetic-field component in the
direction of the mean velocity (Fig. 1). This differential-rotation effect
plays an important role in dynamo process, and is called the $\Omega$ (Omega)
effect. It is considered to produce the azimuthal or toroidal component of the
mean magnetic field from the latitudinal or poloidal one (and vice versa)
through the non-uniform mean velocity effect.

Figure 1: Differential rotation effect ($\Omega$ effect).

The second term of the r.h.s. of (185) and the third term on the r.h.s. of
(186) contain the contribution of the turbulent electromotive force (EMF)
defined by

${\bf{E}}_{\rm{M}}\equiv\langle{{\bf{u}}^{\prime}\times{\bf{b}}^{\prime}}\rangle.$
(187)

This ${\bf{E}}_{\rm{M}}$ is the sole (direct) turbulence effect in the mean
magnetic induction equation [(185) and (186)], and is the quantity of central
importance in the turbulent dynamo study.

Unlike the treatment in the $\Omega$ effect, the non-uniform or inhomogeneous
mean velocity effect has been neglected in considering turbulence. As we saw
in (13) and (14), the equations of the fluctuating velocity and magnetic field
are given as

$\frac{\partial{\bf{u}}^{\prime}}{\partial
t}+({\bf{U}}\cdot\nabla){\bf{u}}^{\prime}=({\bf{B}}\cdot\nabla){\bf{b}}^{\prime}+({\bf{b}}^{\prime}\cdot\nabla){\bf{B}}-({\bf{u}}^{\prime}\cdot\nabla){\bf{U}}+\cdots,$
(188) $\frac{\partial{\bf{b}}^{\prime}}{\partial
t}+({\bf{U}}\cdot\nabla){\bf{b}}^{\prime}=({\bf{B}}\cdot\nabla){\bf{u}}^{\prime}-({\bf{u}}^{\prime}\cdot\nabla){\bf{B}}+({\bf{b}}^{\prime}\cdot\nabla){\bf{U}}+\cdots,$
(189)

The evaluation of EMF is obtained from the equations of the fluctuating
velocity and magnetic field [(188) and (189)]. We multiply (188) and (189) by
${\bf{b}}^{\prime}$ and ${\bf{u}}^{\prime}$ in the vector product manner,
respectively, and add them. After taking the ensemble average, we obtain the
evolution equation of the ELM as

$\displaystyle\frac{DE_{\rm{M}}^{i}}{Dt}$ $\displaystyle=$
$\displaystyle\left\langle{\frac{1}{\overline{\rho}}b^{\prime}{}^{k}\epsilon^{ijk}\frac{\partial
b^{\prime}{}^{j}}{\partial
x^{\ell}}-u^{\prime}{}^{k}\epsilon^{ijk}\frac{\partial
u^{\prime}{}^{j}}{\partial x^{\ell}}}\right\rangle
B^{\ell}-\left\langle{u^{\prime}{}^{\ell}u^{\prime}{}^{k}+\frac{b^{\prime}{}^{\ell}b^{\prime}{}^{k}}{\mu_{0}\overline{\rho}}}\right\rangle\mu_{0}\epsilon^{ijk}\frac{\partial
B^{j}}{\partial x^{\ell}}$ (190)
$\displaystyle+\left\langle{u^{\prime}{}^{\ell}b^{\prime}{}^{k}+b^{\prime}{}^{\ell}u^{\prime}{}^{k}}\right\rangle\epsilon^{ijk}\frac{\partial
U^{j}}{\partial x^{\ell}}+{\rm{H.T.}},$

Equation (190) with the inhomogeneous mean velocity or the
$\nabla{\bf{U}}$-term dropped corresponds to the Ansatz that the EMF is
expressed in terms of the mean magnetic field and its derivatives such as

$\langle{{\bf{u}}^{\prime}\times{\bf{b}}^{\prime}}\rangle^{i}=\alpha^{i\ell}B^{\ell}+\beta^{ij\ell}\frac{\partial
B^{j}}{\partial x^{\ell}}+\cdots,$ (191)

where $\alpha^{i\ell}$ and $\beta^{ij\ell}$ are transport coefficients. In
other words, the adoption of the usual Ansatz corresponds to the assumption of
the no mean velocity inhomogeneity effect in the EMF. As (188) and (189) show,
the velocity and magnetic-field fluctuations depend on the large-scale
inhomogeneity of the velocity. If we retain the mean velocity inhomogeneity
effects, the third or $\nabla{\bf{U}}$ terms should show up in the ELM
expression.

In Section 2, we obtained the expression of the turbulent EMF (141) in an
elaborated formulation. However, in order to get an intuitive view of the
physical origins of the dynamo effects, here we use much more simplified
arguments with assuming the simplest statistics on turbulence. If we assume
that turbulent field is homogeneous and isotropic, the two-point two-time
turbulent velocity and magnetic-field correlations are expressed in the
generic form as

$\displaystyle\langle{\phi^{\prime}{}^{i}({\bf{x}}\mathchar
24635\relax\;t)\psi^{\prime}{}^{j}({\bf{x}}^{\prime}\mathchar
24635\relax\;t^{\prime})}\rangle=\langle{\phi^{\prime}{}^{i}({\bf{0}}\mathchar
24635\relax\;t)\psi^{\prime}{}^{j}({\bf{r}}\mathchar
24635\relax\;t^{\prime})}\rangle$ $\displaystyle=g(r\mathchar
24635\relax\;t,t^{\prime})\delta^{ij}+\frac{f(r\mathchar
24635\relax\;t,t^{\prime})-g(r\mathchar
24635\relax\;t,t^{\prime})}{r^{2}}r^{i}r^{j}+h(r\mathchar
24635\relax\;t,t^{\prime})\epsilon^{ij\ell}\frac{r^{\ell}}{r},$ (192)

where $r=\|{\bf{r}}\|$ with ${\bf{r}}={\bf{x}}^{\prime}-{\bf{x}}$ is the
distance between two points, and $\mbox{\boldmath$\phi$}^{\prime}$ and
$\mbox{\boldmath$\psi$}^{\prime}$ denote either one of the velocity and
magnetic field, ${\bf{u}}^{\prime}$ and ${\bf{b}}^{\prime}$, respectively.
Here, $g$, $f$, and $h$ are the longitudinal, transverse, and cross
correlation functions, respectively.

If we substitute (192) with $\mbox{\boldmath$\phi$}^{\prime}$ and
$\mbox{\boldmath$\psi$}^{\prime}$ being ${\bf{u}}^{\prime}$ and
${\bf{b}}^{\prime}$ into (190), we have

$\displaystyle\frac{DE_{\rm{M}}^{i}}{Dt}=\frac{1}{3}\left\langle{\frac{1}{\overline{\rho}}b^{\prime}{}^{k}\epsilon^{k\ell
j}\frac{\partial b^{\prime}{}^{j}}{\partial
x^{\ell}}-u^{\prime}{}^{k}\epsilon^{k\ell j}\frac{\partial
u^{\prime}{}^{j}}{\partial x^{\ell}}}\right\rangle B^{i}$
$\displaystyle-\frac{1}{3}\left\langle{u^{\prime}{}^{k}u^{\prime}{}^{k}+\frac{b^{\prime}{}^{k}b^{\prime}{}^{k}}{\mu_{0}\overline{\rho}}}\right\rangle\mu_{0}\epsilon^{i\ell
j}\frac{\partial B^{j}}{\partial
x^{\ell}}+\frac{2}{3}\left\langle{u^{\prime}{}^{k}b^{\prime}{}^{k}}\right\rangle\epsilon^{i\ell
j}\frac{\partial U^{j}}{\partial x^{\ell}}+{\rm{H.T.}},$ (193)

This suggests that the EMF is expressed as

${\bf{E}}_{\rm{M}}=\alpha{\bf{B}}-\beta\mu_{0}{\bf{J}}+\gamma\mbox{\boldmath$\Omega$}.$
(194)

The transport coefficients $\alpha$, $\beta$, and $\gamma$ are expressed as

$\alpha=\tau_{\alpha}H,$ (195) $\beta=\tau_{\beta}K,$ (196)
$\gamma=\tau_{\gamma}W.$ (197)

Here, $H$, $K$, and $W$ are the turbulent residual helicity, the turbulent MHD
energy, and the turbulent cross helicity defined by

$H=\left\langle{-{\bf{u}}^{\prime}\cdot\mbox{\boldmath$\omega$}^{\prime}+{\bf{b}}^{\prime}\cdot{\bf{j}}^{\prime}/\overline{\rho}}\right\rangle,$
(198)
$K=\left\langle{{\bf{u}}^{\prime}{}^{2}+{\bf{b}}^{\prime}{}^{2}/(\mu_{0}\overline{\rho})}\right\rangle/2,$
(199) $W=\left\langle{{\bf{u}}^{\prime}\cdot{\bf{b}}^{\prime}}\right\rangle,$
(200)

and $\tau_{\alpha}$, $\tau_{\beta}$, and $\tau_{\gamma}$ are the time scales
associated with the turbulent residual helicity, turbulent MHD energy, and the
turbulent cross helicity, respectively. Equation (194) shows that in the
presence of the non-uniform mean velocity, the cross-helicity or $W$-related
term, in addition to the usual helicity or $H$-related term and the energy or
$K$-related term, should be included in the expression of the EMF.

Note that more thorough and detailed expressions of the EMF can be obtained by
more elaborated dynamo theories than the above simple argument with resorting
to (190) with the homogeneous and isotropic assumption (192). However, the
point here is that retaining the non-uniform mean velocity effects in the
fluctuating velocity and magnetic field, the turbulent cross-helicity coupled
with the mean vortical motion emerges in the EMF expression.

###  3.2 Physical origins of the turbulent effects in dynamos

If turbulence possesses some statistical properties, the effective fluxes
associated with the turbulence contribute to the dynamo-related transport
through the turbulent EMF in coupling with a mean-field configuration. In the
following, we scrutinize what physical processes are the essence of these
turbulence effects by examining each term of the evolution equation of the
velocity and magnetic-field fluctuations. Of course, we should be cautious in
the use of such an argument, since it relies on the consideration of one
particular term. It may occur that other terms completely or substantially
cancels the effect. However, it is true such an argument is useful to grasp a
feel of the physical origin of the effect.

####  3.2.1 $\alpha$ effect: Kinetic and current helicity effect

Let us consider a fluid element fluctuating in the mean magnetic field
${\bf{B}}$ (Fig. 2). Here, we assume a positive kinetic helicity in
turbulence,
$\langle{{\bf{u}}^{\prime}\cdot\mbox{\boldmath$\omega$}^{\prime}}\rangle>0$.
Namely, the fluctuating velocity and magnetic field are statistically aligned
with each other. In the presence of the mean magnetic field ${\bf{B}}$, the
velocity fluctuation associated with the fluctuating vorticity varies along
${\bf{B}}$. Due to this ${\bf{u}}^{\prime}$ variation along the mean magnetic
field ${\bf{B}}$, from the first term on the r.h.s. of (189), the magnetic-
field fluctuation $\delta{\bf{b}}^{\prime}$ is induced as

$\delta{\bf{b}}^{\prime}=\tau_{b}({\bf{B}}\cdot\nabla){\bf{u}}^{\prime},$
(201)

where $\tau_{b}$ is the time scale of ${\bf{b}}^{\prime}$ evolution. The
electromotive force (EMF) due to this effect is expressed as

$\langle{{\bf{u}}^{\prime}\times\delta{\bf{b}}^{\prime}}\rangle=\tau_{b}\langle{{\bf{u}}^{\prime}\times({\bf{B}}\cdot\nabla){\bf{u}}^{\prime}}\rangle,$
(202)

whose direction is antiparallel to the mean magnetic field ${\bf{B}}$ for the
positive turbulent kinetic helicity
$\langle{{\bf{u}}^{\prime}\cdot\mbox{\boldmath$\omega$}^{\prime}}\rangle>0$.
The direction of the EMF is parallel to ${\bf{B}}$ for the negative turbulent
kinetic helicity
$\langle{{\bf{u}}^{\prime}\cdot\mbox{\boldmath$\omega$}^{\prime}}\rangle<0$.

Figure 2: $\alpha$ effect due to the turbulent kinetic helicity.

On the other hand, if we assume a positive current helicity in turbulence,
$\langle{{\bf{b}}^{\prime}\cdot{\bf{j}}^{\prime}}\rangle>0$, the fluctuating
magnetic field ${\bf{b}}^{\prime}$ is statistically aligned with the electric-
current density ${\bf{j}}^{\prime}$ (Fig. 3). In an entirely similar manner as
in the turbulent kinetic helicity effect mentioned above, in association with
the fluctuating electric-current density ${\bf{j}}^{\prime}$, the fluctuating
magnetic field varies along the mean magnetic field ${\bf{B}}$. From the first
term or $({\bf{B}}\cdot\nabla){\bf{b}}^{\prime}$, the fluctuating velocity
$\delta{\bf{u}}^{\prime}$ is induced as

$\delta{\bf{u}}^{\prime}=\tau_{u}({\bf{B}}\cdot\nabla){\bf{b}}^{\prime},$
(203)

where $\tau_{u}$ is the time scale of ${\bf{u}}^{\prime}$ evolution due to
this mean magnetic field effect. In this case, we can alternatively consider
the effect of the fluctuating Lorentz force ${\bf{j}}^{\prime}\times{\bf{B}}$
associated with the mean magnetic field ${\bf{B}}$. The fluctuating velocity
$\delta{\bf{u}}^{\prime}$ is induced as

$\delta{\bf{u}}^{\prime}=\tau_{u}{\bf{j}}^{\prime}\times{\bf{B}}.$ (204)

This is equivalent to (203). This induced velocity fluctuation
$\delta{\bf{b}}^{\prime}$ combined with the fluctuating magnetic field
${\bf{b}}^{\prime}$ constitutes the electromotive force as

$\langle{\delta{\bf{u}}^{\prime}\times{\bf{b}}^{\prime}}\rangle^{i}=\tau_{u}\langle{({\bf{j}}^{\prime}\times{\bf{B}})\times{\bf{b}}^{\prime}}\rangle^{i}=\tau_{u}\langle{{\bf{b}}^{\prime}\cdot{\bf{j}}^{\prime}}\rangle
B^{i}.$ (205)

Here, we dropped the contribution from the fluctuations along the mean
magnetic field ${\bf{B}}$ as
$-\langle{({\bf{b}}^{\prime}\cdot{\bf{B}}){\bf{j}}^{\prime}}\rangle=-\langle{b^{\prime}_{\parallel
B}{\bf{j}}^{\prime}}\rangle\|{\bf{B}}\|$, where $b^{\prime}_{\parallel
B}={\bf{b}}^{\prime}\cdot{\bf{B}}/\|{\bf{B}}\|$ is the ${\bf{b}}^{\prime}$
component parallel to ${\bf{B}}$. The direction of the EMF is parallel to the
mean magnetic field ${\bf{B}}$ for positive turbulent current helicity
$\langle{{\bf{b}}^{\prime}\cdot{\bf{j}}^{\prime}}\rangle>0$, and is
antiparallel to ${\bf{B}}$ for negative turbulent current helicity
$\langle{{\bf{b}}^{\prime}\cdot{\bf{j}}^{\prime}}\rangle<0$. This current
helicity effect on the $\alpha$ dynamo was first pointed out by Pouquet,
Frisch & Léorat (1976) on the basis of a closure calculation of MHD turbulence
with the aid of the eddy-damped quasi-normal Markovianized (EDQNM)
approximation. In the derivation, the timescale for the kinetic-helicity
effect, $\tau_{b}$ (202), and the timescale for the current-helicity effect,
$\tau_{u}$ (205), were not distinguished.

Figure 3: $\alpha$ effect due to the turbulent electric-current helicity.

These physical pictures show that if the turbulent kinetic and current
helicities have a sign same with each other, these two helicity effects
counterbalance each other. In this sense, the current-helicity in the $\alpha$
dynamo is often argued that this magnetic correction to the $\alpha$ dynamo
represents the saturation of the $\alpha$ dynamo due to the magnetic-field
effect. However, the sign of the current helicity can be opposite to that of
the kinetic helicity. In such a case, the current-helicity effect may enhance
the magnetic field generated by the kinetic-helicity effect. In the case of
inhomogeneous turbulence, which is ubiquitous in the real world, the
spatiotemporal evolution of the current helicity
$\langle{{\bf{b}}^{\prime}\cdot{\bf{j}}^{\prime}}\rangle$ depends on the
production rates of the current helicity directly related to the mean-field
inhomogeneities. Actually, it is reported that, in the numerical simulation of
the solar convective zone, at some deep region, the magnitude of the turbulent
current helicity is dominantly larger than the counterpart of the turbulent
kinetic helicity. This suggests that mean magnetic field can be generated by
the current-helicity effect at the depth even in the absence of the turbulent
kinetic helicity there. Such arguments should be done on the basis of the
evolution equation of the kinetic and current helicities.

####  3.2.2 Turbulent magnetic diffusivity: Turbulent energy effect

Let us consider a fluid element located in a mean electric-current density
${\bf{J}}$ (Figure 3.4). In this case, the fluid element fluctuates in a non-
uniform mean magnetic field associated with the mean electric-current density
${\bf{J}}$. Because of the second term or
$-({\bf{u}}^{\prime}\cdot\nabla){\bf{B}}$ on the r.h.s. of (189), the
magnetic-field fluctuation $\delta{\bf{b}}^{\prime}$ is induced in the
direction of the non-uniform ${\bf{B}}$ but in the sense that it
counterbalances the increase of ${\bf{B}}$ as

$\delta{\bf{b}}^{\prime}=-\tau_{b}({\bf{u}}^{\prime}\cdot\nabla){\bf{B}}.$
(206)

The EMF arising from the non-uniform mean magnetic field is given by

$\langle{{\bf{u}}^{\prime}\times\delta{\bf{b}}^{\prime}}\rangle^{i}=-\tau_{b}\langle{{\bf{u}}^{\prime}\times[({\bf{u}}^{\prime}\cdot\nabla){\bf{B}}]}\rangle^{i}=-\tau_{b}\langle{u^{\prime}{}^{j}u^{\prime}{}^{\ell}}\rangle\epsilon^{ijk}\frac{\partial
B^{k}}{\partial x^{\ell}}.$ (207)

If we assume that the velocity fluctuation is isotropic as
$\langle{u^{\prime}{}^{j}u^{\prime}{}^{\ell}}\rangle=\langle{{\bf{u}}^{\prime}{}^{2}}\rangle\delta^{j\ell}/3$,
(207) is reduced to

$\langle{{\bf{u}}^{\prime}\times\delta{\bf{b}}^{\prime}}\rangle^{i}=-\frac{1}{3}\tau_{b}\langle{{\bf{u}}^{\prime}{}^{2}}\rangle\epsilon^{i\ell
k}\frac{\partial B^{k}}{\partial
x^{\ell}}=-\frac{1}{3}\tau_{b}\langle{{\bf{u}}^{\prime}{}^{2}}\rangle(\nabla\times{\bf{B}})^{i}.$
(208)

In the presence of velocity fluctuation, the turbulent electromotive force
(EMF) due to the non-uniform mean magnetic field $\nabla{\bf{B}}$ is in the
direction antiparallel to the mean electric-current density
${\bf{J}}(=\nabla\times{\bf{B}})$. The transport coefficient is proportional
to the intensity of fluctuation or the turbulent kinetic energy,
$\langle{{\bf{u}}^{\prime}{}^{2}}\rangle$.

Figure 4: Turbulent magnetic diffusivity due to the velocity fluctuation.

In a similar manner, we argue the effect of the magnetic fluctuation on the
turbulent magnetic diffusivity. First we consider turbulence with magnetic-
field fluctuation ${\bf{b}}^{\prime}$ in a non-uniform mean magnetic field
${\bf{B}}$ (Fig. 5). In the presence of the mean electric-current density
${\bf{J}}(=\nabla\times{\bf{B}})$, the fluid element is subject to the
fluctuating Lorentz force associated with the mean electric-current density,
${\bf{J}}\times{\bf{b}}^{\prime}$. Then the fluctuating velocity
$\delta{\bf{u}}^{\prime}$ due to ${\bf{J}}$ is induced as

$\delta{\bf{u}}^{\prime}=\tau_{u}{\bf{J}}\times{\bf{b}}^{\prime}.$ (209)

The EMF due to the magnetic-field fluctuation in the presence of ${\bf{J}}$ is
expressed as

$\langle{\delta{\bf{u}}^{\prime}\times{\bf{b}}^{\prime}}\rangle=\langle{\tau_{u}({\bf{J}}\times{\bf{b}}^{\prime})\times{\bf{b}}^{\prime}}\rangle=-\tau_{u}\langle{{\bf{b}}^{\prime}{}^{2}}\rangle{\bf{J}}.$
(210)

Here, we dropped the contribution from the fluctuating magnetic field
component parallel to the mean electric-current density,
$\tau_{u}({\bf{b}}^{\prime}\cdot{\bf{J}}){\bf{b}}^{\prime}=\tau_{u}\langle{b^{\prime}_{\parallel
J}{\bf{b}}^{\prime}}\rangle\|{\bf{J}}\|$, where $b^{\prime}_{\parallel
J}={\bf{b}}^{\prime}\cdot{\bf{J}}/\|{\bf{J}}\|$ is the fluctuating magnetic-
field component along the mean electric-current density ${\bf{J}}$. Equation
(210) implies that the magnetic fluctuation ${\bf{b}}^{\prime}$ as well as the
velocity fluctuation ${\bf{u}}^{\prime}$ contributes to the turbulent magnetic
diffusivity coupled with the mean electric-current density ${\bf{J}}$.

Figure 5: Turbulent magnetic diffusivity due to the magnetic fluctuation.

As for this magnetic fluctuation effect, however, we should note the following
point. As we see above, in the presence of the mean electric-current density
${\bf{J}}$, the part of fluctuating Lorentz force
${\bf{J}}\times{\bf{b}}^{\prime}$ induces the fluctuating velocity
$\delta{\bf{u}}^{\prime}$, which plays a key role for the turbulent magnetic
diffusivity due to the fluctuating magnetic field. However, this effect can be
at least partly canceled by the other part of the fluctuating Lorentz force
${\bf{j}}^{\prime}\times{\bf{B}}$. If the magnetic fluctuation has a spatial
distribution like Fig. 6, the fluctuating electric-current density associated
with ${\bf{b}}^{\prime}$, $\Delta{\bf{j}}$, also shows a spatial distribution.
The associated electric-current density $\Delta{\bf{j}}^{\prime}$ is in the
direction enhancing the original mean electric-current density ${\bf{J}}$ on
the side parallel to ${\bf{J}}\times{\bf{b}}^{\prime}$ with respect to
${\bf{b}}^{\prime}$ (parallel side), and $\Delta{\bf{j}}^{\prime}$ in the
direction reducing ${\bf{J}}$ on the side antiparallel to
${\bf{J}}\times{\bf{b}}^{\prime}$ (antiparallel side). Due to this
$\Delta{\bf{j}}^{\prime}$ effect, the magnetic pressure becomes higher in the
parallel side than the antiparallel side. In other words, the velocity
fluctuation $\delta{\bf{u}}$ induced by the fluctuating Lorentz force
$\Delta{\bf{j}}^{\prime}\times{\bf{B}}$ works for increasing the plasma
density on the parallel side, while the flow due to the fluctuating Lorentz
force,
$\delta{\bf{u}}^{\prime}=\tau_{u}\Delta{\bf{j}}^{\prime}\times{\bf{B}}$, works
for decreasing the plasma density on the antiparallel side. This magnetic
pressure effect in the combination of the magnetic fluctuation
${\bf{b}}^{\prime}$ and the non-uniform mean magnetic field ${\bf{B}}$,
counterbalances the effect of the Lorentz force
${\bf{J}}\times{\bf{b}}^{\prime}$. In this sense, the magnetic-field
fluctuations ${\bf{b}}^{\prime}$ effect on the turbulent magnetic diffusivity
is not obvious as the counter part of the velocity fluctuations
${\bf{u}}^{\prime}$. Actually, in the detailed analytical calculation shows
that in the solenoidal turbulence, the turbulent magnetic diffusivity does not
depend on the magnetic fluctuation energy
$\langle{{\bf{b}}^{\prime}{}^{2}}\rangle$. The magnetic-field fluctuation
energy becomes relevant to the turbulent magnetic diffusivity in the case of
compressible turbulence (Yokoi, 2018a) and anisotropic situations
(Rogachevskii & Kleeorin, 2001).

Figure 6: Cancellation of the turbulent magnetic energy effect due to the
magnetic pressure effect.

####  3.2.3 Turbulent cross-helicity effect

We consider a fluid element that fluctuates in a mean vorticity field
$\mbox{\boldmath$\Omega$}(=\nabla\times{\bf{U}})$ (Fig. 7). We assume the
turbulence has a positive cross helicity
$\langle{{\bf{u}}^{\prime}\cdot{\bf{b}}^{\prime}}\rangle>0$, namely, the
fluctuating velocity and magnetic field are statistically aligned with each
other. Because of the local angular momentum conservation, the fluid is
subject to the Coriolis-like force. Then, the velocity fluctuation is induced
by the mean vortical motion $\Omega$ as

$\delta{\bf{u}}^{\prime}=\tau_{u}{\bf{u}}^{\prime}\times\mbox{\boldmath$\Omega$}.$
(211)

The electromotive force constituted by this induced $\delta{\bf{u}}^{\prime}$
and the fluctuating magnetic-field component parallel to the magnetic field
${\bf{b}}^{\prime}$ is expressed by

$\langle{\delta{\bf{u}}^{\prime}\times{\bf{b}}^{\prime}}\rangle=\langle{\tau_{u}({\bf{u}}^{\prime}\times\mbox{\boldmath$\Omega$})\times{\bf{b}}^{\prime}}\rangle=\tau_{u}\langle{{\bf{u}}^{\prime}\cdot{\bf{b}}^{\prime}}\rangle\mbox{\boldmath$\Omega$}.$
(212)

Here, we dropped the contribution from the magnetic fluctuations along the
mean vorticity
$-\tau_{u}\langle{{\bf{u}}^{\prime}({\bf{b}}^{\prime}\cdot\mbox{\boldmath$\Omega$})}\rangle=-\tau_{u}\langle{b_{\parallel\Omega}}\rangle\|\mbox{\boldmath$\Omega$}\|$,
where $b^{\prime}_{\parallel\Omega}$ is the ${\bf{b}}^{\prime}$ component
parallel to $\Omega$ defined by
${\bf{b}}^{\prime}\cdot\mbox{\boldmath$\Omega$}/\|\mbox{\boldmath$\Omega$}\|$
since the statistical average of
$b^{\prime}_{\parallel\Omega}{\bf{u}}^{\prime}$ is expected to vanish.
Equation (212) suggests that EMF due to the cross helicity is aligned with the
mean vorticity $\Omega$. The direction of EMF is determined by the sign of the
turbulent cross helicity. The EMF is parallel to the mean vorticity $\Omega$
for positive turbulent cross helicity
$\langle{{\bf{u}}^{\prime}\cdot{\bf{b}}^{\prime}}\rangle>0$, and antiparallel
to $\Omega$ for negative turbulent cross helicity
$\langle{{\bf{u}}^{\prime}\cdot{\bf{b}}^{\prime}}\rangle<0$.

Figure 7: Turbulent cross-helicity effect due to the velocity fluctuation.

Next, we consider motion of a fluid element with magnetic fluctuations
${\bf{b}}^{\prime}$ in a non-uniform mean velocity $\nabla{\bf{U}}$ (Fig. 8).
Also in this case, we assume the cross helicity in turbulence is positive
($\langle{{\bf{u}}^{\prime}\cdot{\bf{b}}^{\prime}}\rangle$); the turbulent
velocity ${\bf{u}}^{\prime}$ and turbulent magnetic field ${\bf{b}}^{\prime}$
are statistically aligned and parallel with each other. We see from the third
term or $\tau_{b}({\bf{b}}^{\prime}\cdot\nabla){\bf{U}}$ on the r.h.s. of Eq.
(188) that the magnetic fluctuation $\delta{\bf{b}}^{\prime}$ is induced as

$\delta{\bf{b}}^{\prime}=\tau_{b}({\bf{b}}^{\prime}\cdot\nabla){\bf{U}}.$
(213)

This induced magnetic fluctuation $\delta{\bf{b}}^{\prime}$ is in the
direction parallel to ${\bf{U}}$ if the non-uniform ${\bf{U}}$ increases as
the fluctuating magnetic field ${\bf{b}}^{\prime}$ moves. Multiplying the
fluctuating velocity ${\bf{u}}^{\prime}$ in the vector product manner by Eq.
(213) and taking the ensemble averaging, we obtain

$\langle{{\bf{u}}^{\prime}\times\delta{\bf{b}}^{\prime}}\rangle^{i}=\langle{{\bf{u}}^{\prime}\times\tau_{b}({\bf{b}}^{\prime}\cdot\nabla){\bf{U}}}\rangle^{i}=\left\langle{\epsilon^{ijk}u^{\prime}{}^{j}\tau_{b}b^{\prime\ell}\frac{\partial
U^{k}}{\partial
x^{\ell}}}\right\rangle=\tau_{b}\langle{u^{\prime}{}^{j}b^{\prime}{}^{\ell}}\rangle\epsilon^{ijk}\frac{\partial
U^{k}}{\partial x^{\ell}}.$ (214)

If we assume the velocity and magnetic-field correlation is isotropic as
$\langle{u^{\prime}{}^{j}b^{\prime}{}^{\ell}}\rangle=\langle{{\bf{u}}^{\prime}\cdot{\bf{b}}^{\prime}}\rangle\delta^{j\ell}/3$,
(214) is reduced to

$\langle{{\bf{u}}^{\prime}\times\delta{\bf{b}}^{\prime}}\rangle^{i}=\tau_{b}\langle{{\bf{u}}^{\prime}\cdot{\bf{b}}^{\prime}}\rangle\epsilon^{ijk}\frac{\partial
U^{k}}{\partial
x^{j}}=\tau_{b}\langle{{\bf{u}}^{\prime}\cdot{\bf{b}}^{\prime}}\rangle(\nabla\times{\bf{U}})^{i}=\tau_{b}\langle{{\bf{u}}^{\prime}\cdot{\bf{b}}^{\prime}}\rangle\Omega^{i}.$
(215)

Figure 8: Turbulent cross-helicity effect due to the magnetic fluctuation.

Since the mean vorticity $\mbox{\boldmath$\Omega$}=\nabla\times{\bf{U}}$ is
locally equivalent to the angular velocity of a rotation
$\mbox{\boldmath$\omega$}_{\rm{F}}=\mbox{\boldmath$\Omega$}/2$, exactly the
same argument can be applied to the fluid element in a rotating system. Hence,
we can replace the mean vorticity $\Omega$ by the mean absolute vorticity
$\mbox{\boldmath$\Omega$}_{\ast}\equiv\mbox{\boldmath$\Omega$}+2\mbox{\boldmath$\omega$}_{\rm{F}}$.

Equations (210) and (215) show that in the presence of the mean absolute
vorticity $\mbox{\boldmath$\Omega$}_{\ast}$, the turbulent electromotive force
(EMF) is expressed as

$\langle{{\bf{u}}^{\prime}\times{\bf{b}}^{\prime}}\rangle_{\gamma}=\gamma\mbox{\boldmath$\Omega$}_{\ast},$
(216)

where $\gamma$ is the transport coefficient determined by the turbulent cross
helicity and turbulence timescale related to the cross helicity,
$\tau_{\gamma}$, expressed by

$\gamma=\tau_{\gamma}\langle{{\bf{u}}^{\prime}\cdot{\bf{b}}^{\prime}}\rangle,$
(217)

and $\mbox{\boldmath$\Omega$}_{\ast}$ is the mean absolute vorticity defined
by

$\mbox{\boldmath$\Omega$}_{\ast}\equiv\mbox{\boldmath$\Omega$}+2\mbox{\boldmath$\omega$}_{\rm{F}}$
(218)

with the mean relative vorticity
$\mbox{\boldmath$\Omega$}=\nabla\times{\bf{U}}$ and the angular velocity of
the system rotation $\mbox{\boldmath$\omega$}_{\rm{F}}$.

###  3.3 Numerical validation of the cross-helicity effect

####  3.3.1 Kolmogorov flow with imposed uniform magnetic field

In order to validate the cross-helicity effect in dynamo problems, we
performed a direct numerical simulation (DNS) of forced magnetohydrodynamic
(MHD) turbulence in a three-dimensional periodic box with a uniform magnetic
field imposed (Fig. 9) (Yokoi & Balarac, 2011). The size of the box is
$(L_{x},L_{y},L_{z})$. The imposed forcing is in the $x$ direction:

${\bf{f}}_{\rm{ext}}=(f^{x},f^{y},f^{z})=(f^{x},0,0)$ (219)

and is inhomogeneous in the $y$ direction:

$f^{x}(y)=f_{0}\sin\left({\frac{2\pi y}{L_{y}}}\right).$ (220)

In this setup, turbulence is generated and sustained by the velocity shear due
to the sinusoidal forcing (219) with (220). Because of the forcing, the
statistics of the turbulence is inhomogeneous in the $y$ direction, but is
homogeneous in the $x$ and $z$ directions. The statistics of quantities are
represented by averaging in the homogeneous ($x$-$z$) directions. This
configuration is known as the Kolmogorov turbulent flow. In order to see the
turbulence properties related to dynamo and its transport, we further impose a
uniform mean magnetic field in the inhomogeneous direction, namely in the $y$
direction as

${\bf{B}}_{0}=(B_{0}^{x},B_{0}^{y},B_{0}^{z})=(0,B_{0},0).$ (221)

The contour of the streamwise velocity is shown in Figure 9. Reflecting the
sinusoidal form of the forcing (219), the value of the streamwise velocity
component is positive and negative in the upper half domain ($y>0$) and lower
half domain ($y<0$), respectively.

Figure 9: Kolmogorov flow with imposed uniform magnetic field. The triple
periodic configuration with the imposed uniform magnetic field in the
direction of inhomogeneity ($y$ direction) (left). Sinusoidal forcing for
inhomogeneous velocity shear (middle). The contour of the streamwise velocity
(right).

With this setup, we calculate the turbulent correlations and the mean-field
quantities that are relevant to the mean-field dynamo. The expression of the
turbulent electromotive force (EMF) is theoretically obtained in Section 2,
and modeled as

$\langle{{\bf{u}}^{\prime}\times{\bf{b}}^{\prime}}\rangle=\alpha{\bf{B}}-\beta{\bf{J}}+\gamma\mbox{\boldmath$\Omega$},$
(222)

where the transport coefficients $\alpha$, $\beta$, and $\gamma$ are expressed
in terms of the turbulent residual helicity, the turbulent energy, and the
turbulent cross helicity.

As (222) shows, the relevant mean fields are the mean magnetic field
${\bf{B}}$, the mean electric-current density
${\bf{J}}(=\nabla\times{\bf{B}})$, and the mean relative vorticity
$\mbox{\boldmath$\Omega$}(=\nabla\times{\bf{U}})$. On the other hand, the
relevant turbulent correlations are the turbulent MHD energy $K$, its
dissipation rate $\varepsilon_{K}(\equiv\varepsilon)$, the turbulent residual
helicity $H$, and the turbulent cross helicity $W$. They are defined by

$K=\langle{{\bf{u}}^{\prime}{}^{2}+{\bf{b}}^{\prime}{}^{2}}\rangle/2,$ (223)
$\varepsilon_{K}=\nu\left\langle{\left({\frac{\partial
u^{\prime}{}^{j}}{\partial
x^{i}}}\right)^{2}}\right\rangle+\eta\left\langle{\left(\frac{\partial
b^{\prime}{}^{j}}{\partial x^{i}}\right)^{2}}\right\rangle\equiv\varepsilon,$
(224)
$H=\left\langle{-{\bf{u}}^{\prime}\cdot\mbox{\boldmath$\omega$}^{\prime}+{\bf{b}}^{\prime}\cdot{\bf{j}}^{\prime}}\right\rangle,$
(225) $W=\left\langle{{\bf{u}}^{\prime}\cdot{\bf{b}}^{\prime}}\right\rangle.$
(226)

In terms of these turbulent statistical quantities, the main dynamo transport
coefficients are expressed as

$\alpha=\tau H,$ (227) $\beta=\tau K,$ (228) $\gamma=\tau W,$ (229)

where $\tau$ is the timescale of turbulence evaluated by

$\tau=K/\varepsilon.$ (230)

Using the DNS results, we plot the spatial ($y$) distribution of the $x$
component of the EMF
$\langle{{\bf{u}}^{\prime}\times{\bf{b}}^{\prime}}\rangle$ as well as each
term of the r.h.s. of Eq. (222), $\alpha{\bf{B}}$, $\beta{\bf{J}}$, and
$\gamma\mbox{\boldmath$\Omega$}$ with (227)-(230) in Fig. 10. The DNS of the
EMF $\langle{{\bf{u}}^{\prime}\times{\bf{b}}^{\prime}}\rangle^{x}$ shows a
sinusoidal distribution in the $y$ direction; negative values for $y<0$ and
positive values for $y>0$. Among the model terms, the contribution of the
$(\alpha{\bf{B}})^{x}$ is almost negligible in the whole domain as compared
with the other terms, $-\beta{\bf{J}}$ and $\gamma\mbox{\boldmath$\Omega$}$.
On the other hand, the turbulent diffusivity $\beta{\bf{J}}$ term plays a
dominant role in constituting of the EMF. The spatial ($y$) distribution of
$-\beta{\bf{J}}$ is coarsely sinusoidal; negative for $y<0$ and positive for
$y>0$. The main balancer for the turbulent magnetic diffusivity
$-\beta{\bf{J}}$ is the cross-helicity effect
$\gamma\mbox{\boldmath$\Omega$}$. Its spatial distribution is basically
positive for $y<0$ and negative for $y>0$. It is very notable that the spatial
distribution of the sum of the turbulent magnetic diffusivity and the cross-
helicity effect, $-\beta{\bf{J}}+\gamma\mbox{\boldmath$\Omega$}$, roughly
agrees with the counterpart of the EMF
$\langle{{\bf{u}}^{\prime}\times{\bf{b}}^{\prime}}\rangle$. This DNS result
shows that, in this configuration, the turbulent EMF is constituted by the
turbulent magnetic diffusivity $\beta{\bf{J}}$ and the cross-helicity effect
$\gamma\mbox{\boldmath$\Omega$}$, and that the turbulent helicity or $\alpha$
effect does not play any substantial role in the turbulent transport. The
profiles of the EMF, whose sign is opposite to that of
$\gamma\mbox{\boldmath$\Omega$}$, is similar to the counterpart of the
$\beta{\bf{J}}$. This indicates that, in this simulation, the turbulent EMF as
a whole works for enhancing the effective diffusivity, not for dynamo.
However, it is worth noting that the increased transport or the field
destruction by the turbulent magnetic diffusivity $\beta{\bf{J}}$ is certainly
suppressed by the dynamo or field-generation/sustainment effect by the cross
helicity.

Figure 10: Comparison of the turbulent electromotive force (EMF) with the
model terms, $\alpha{\bf{B}}$, $\beta{\bf{J}}$, and
$\gamma\mbox{\boldmath$\Omega$}$ in Kolmogorov flow with imposed uniform
magnetic field. The spatial distribution with respect to the inhomogeneous
direction ($y$ direction) of
$\langle{{\bf{u}}^{\prime}\times{\bf{b}}^{\prime}}\rangle^{x}$ (black) is
compared with the $\alpha B^{x}$ (green), $\beta J^{x}$ (blue), and
$\gamma\Omega^{x}$ (red).

This numerical validation is suggestive for the interpretation of the result
of a dynamo experiment. In the liquid sodium dynamo experiment, the turbulent
EMF $\langle{{\bf{u}}^{\prime}\times{\bf{b}}^{\prime}}\rangle$ was for the
first time directly measured by simultaneously measuring three component of
the velocity and magnetic field (Rahbarnia, Brown, Clark, et al., 2012). By
comparing with the mean electric-current density term, it was shown that the
EMF tends to oppose the local mean electric current, and that the mean
magnetic field ${\bf{B}}$ tends to perpendicular to the direction of the EMF.
This experimental result also suggests that the helicity or $\alpha$ effect,
expressed by $\alpha{\bf{B}}$, does not contribute to the turbulent EMF. This
experimental result is consistent with the numerical result of the turbulent
transport (Fig. 11) showing the dominance of the turbulent magnetic
diffusivity and the irrelevance of the helicity or $\alpha$ effect in dynamo
action. We should note that in this liquid sodium experiment a large-scale
rotational or poloidal motion is also observed. Since there exists a mean
vortical motion and the three-dimensional data of velocity and magnetic field
are measured, it is interesting to re-examine the effect of the turbulent
cross helicity $\langle{{\bf{u}}^{\prime}\cdot{\bf{b}}^{\prime}}\rangle$ which
couples with the mean vortical motion
$\mbox{\boldmath$\Omega$}(=\nabla\times{\bf{U}})$ as
$\gamma\mbox{\boldmath$\Omega$}$.

Figure 11: Comparison of the measured turbulent electromotive force (EMF) with
the dynamo terms, $\alpha{\bf{B}}$, $\beta{\bf{J}}$, and
$\gamma\mbox{\boldmath$\Omega$}$ in Kolmogorov flow with imposed uniform
magnetic field. The spatial distribution with respect to the inhomogeneous
direction ($y$ direction) of
$\langle{{\bf{u}}^{\prime}\times{\bf{b}}^{\prime}}\rangle^{x}$ (black) is
compared with the $\alpha B^{x}$ (green) and $\beta J^{x}$ (blue). Redrawn
from Rahbarnia, Brown, Clark, et al. (2012).

####  3.3.2 Archontis flow

Another numerical validation was performed in the Archontis flow
configuration, which is a generalization of the Arnold–Beltrami–Childress flow
but with the cosine terms omitted (Sur & Brandenburg, 2009). As the result of
this setup, the Archontis flow configuration is a non-helical one, and
suitable to explore the dynamo due to the cross-helicity effect. A net cross
helicity with either sign is generated by an instability, and the sign of the
cross helicity depends on the initial condition. The direct numerical
simulations of this flow show that the cross helicity contributes to inducing
a large-scale magnetic field with exponential growth. It turns out that in
order to evaluate the cross-helicity effect in dynamo, the mean-field effect
should be also considered. This naturally leads us to explore the problem how
and how much turbulent cross helicity is generated by the effects of the mean
fields. This is the subject of Section 4.

####  3.3.3 Cross-helicity and differential rotation effect in spherical shell

In Section 3.1, we argued the cross helicity and global flow effect in
dynamos. We saw the treatment of global-flow inhomogeneity is fairly different
between the $\Omega$ (differential rotation) effect and turbulence. If we
consider the velocity shear effect also in turbulence, the cross-helicity
effect inevitably shows up. Since both the $\Omega$ effect and the cross-
helicity effect depend on the inhomogeneous large-scale flow, the relative
importance of of these effects in the magnetic field generation process is an
interesting subject of the cross-helicity effect in dynamos.

The spatial distributions of the total, mean, and fluctuating cross helicities
in the direct numerical simulation of a spherical shell mimicking the Sun are
plotted in contour in Fig. 12. Here, the total cross helicity is calculated
using the instantaneous velocity and magnetic fields, the mean cross helicity
is calculated under the azimuthal averaging, and the fluctuating cross
helicity is calculated by subtraction of the mean cross helicity from the
total one. The spatial distributions of cross helicity is basically
antisymmetric with respect to the midplane. This reflects pseudo-scalar
property of cross helicity. We also see the signs of the mean and fluctuating
cross helicity in each hemisphere are opposite each other. This is not
unreasonable since the cross helicity is not a positive-definite quantity and
its sign can be altered in scale.

Figure 12: Spatial distribution of the cross helicity in the meridional
surface. The total cross helicity (left), the mean cross helicity (middle),
and the fluctuating cross helicity (right). Provided by Mark Miesch.

As will be referred to in Section 4, we have the transport equations of the
turbulent and mean-field cross helicities. Comparison of the numerical data of
the cross helicity with the production, dissipation, and transport rates of
the turbulent and mean-field cross helicities in the global simulation of the
spherical shell geometry would give a way to understand the physical
mechanisms that determine the spatiotemporal distribution of the turbulent
cross helicity.

Next we evaluate the relative importance of cross-helicity effect to the
differential rotation effect in the mean induction equation by calculating

$\frac{(\mbox{cross-helicity effect})}{(\mbox{differential-rotation
effect})}=\frac{\|{\nabla\times(\gamma\nabla\times{\bf{U}})}\|}{\|{\nabla\times({\bf{U}}\times{\bf{B}})}\|}.$
(231)

This ratio is plotted against the radius $r$ in Fig. 13.

Figure 13: Relative magnitude of the cross-helicity effect
$\|{\nabla\times(\gamma\nabla\times{\bf{U}})}\|$ to the differential rotation
effect $\|{\nabla\times({\bf{U}}\times{\bf{B}})}\|$. Here, $R$ is the radius
of the Sun, and $r$ is the radial coordinate. Provided by Mark Miesch.

This plot implies that the relative importance of the cross-helicity effect to
the differential-rotation effect is negligibly small in the deeper region
($0.7<r/R\lesssim 0.85$), but it becomes $\sim 0.5$ in the upper middle layer
($0.85\lesssim r/R\lesssim 0.96$). This ratio raises to unity and much more
higher in the near surface layer ($0.96\lesssim r/R$). This increase is
because the large-scale vorticity becomes much stronger in the near surface
layer. This result shows that the cross-helicity effect is comparable to the
differential rotation effect, and is not negligible in the upper middle layer
of the Sun. This result further suggests that, in the near surface layer, the
cross-helicity effect plays a dominant role in the mean magnetic-field
induction.

The ratio (231) may be estimated as

$\displaystyle\frac{\|{\nabla\times(\gamma\nabla\times{\bf{U}})}\|}{\|{\nabla\times({\bf{U}}\times{\bf{B}})}\|}$
$\displaystyle\sim\frac{\langle{{\bf{u}}^{\prime}\cdot{\bf{b}}^{\prime}}\rangle}{D(\partial
U/\partial
r)B^{r}}\frac{\tau_{\rm{turb}}}{\tau_{\rm{mean}}}\sim\frac{\langle{{\bf{u}}^{\prime}\cdot{\bf{b}}^{\prime}}\rangle}{\delta
UB^{r}}Ro^{-1}=\frac{\langle{{\bf{u}}^{\prime}\cdot{\bf{b}}^{\prime}}\rangle}{\delta
UB^{r}}\frac{K/\varepsilon}{D/\delta U},$ (232)

where $D$ is the depth of the convection zone, $B^{r}$ the poloidal magnetic
field, $\delta U$ the magnitude of velocity differential rotation, $Ro$ the
Rossby number. This suggests that the relative importance of the cross-
helicity to the differential rotation can be evaluated by the value of the
turbulent cross helicity
$\langle{{\bf{u}}^{\prime}\cdot{\bf{b}}^{\prime}}\rangle$ and the eddy turn-
over time $\tau_{\rm{turb}}(=K/\varepsilon)$, as well as the observable
quantities such as the differential rotation velocity $\delta U$, the poloidal
magnetic field $B^{r}$, and the depth of the convection zone $D$. The
evaluation of this ratio in several simulation conditions may be interesting
subject to explore.

##  4 Evolution of the turbulent cross helicity

In the previous section, we saw that in the presence of cross helicity in
turbulence, a mean absolute vorticity (rotation and mean relative vorticity)
associated with the non-uniform mean velocity can produce the electromotive
force. However, how much cross helicity is present in turbulence is another
problem. In this section, we will examine how the turbulent cross helicity is
generated by considering the evolution equation of it.

###  4.1 Evolution equation of cross helicity

As is well known, the total amount of the cross helicity
$\int_{V}{\bf{u}}\cdot{\bf{b}}dV$, as well as the total amount of the MHD
energy $\int_{V}({\bf{u}}^{\prime}{}^{2}+{\bf{b}}^{\prime}{}^{2})/2dV$ and the
counterpart of the magnetic helicity $\int_{V}{\bf{a}}\cdot{\bf{b}}dV$, is an
inviscid invariant of the incompressible MHD equation, where magnetic field is
measured with the Alfvén speed unit as
${\bf{b}}={\bf{b}}_{\ast}/(\mu_{0}\rho)^{1/2}$ ($\mu_{0}$: magnetic
permeability, $\rho$: mass density), ${\bf{a}}$ is the magnetic potential, and
$\int_{V}$ denotes the integral throughout the volume considered, $V$.

We define the local density of the turbulent MHD energy and cross helicity as

$K\equiv\langle{{\bf{u}}^{\prime}{}^{2}+{\bf{b}}^{\prime}{}^{2}}\rangle/2,$
(233) $W\equiv\langle{{\bf{u}}^{\prime}\cdot{\bf{b}}^{\prime}}\rangle,$ (234)

respectively. Due to the conservation property of the total amount of the MHD
energy and cross helicity, the evolution equations of the turbulent MHD energy
density $K$ and the turbulent cross helicity density $W$ are written in a very
simple form as

$\frac{DF}{Dt}=\left({{\bf{U}}\cdot\nabla}\right)F=P_{F}-\varepsilon_{F}+T_{F},$
(235)

where $F=(K\mbox{ or }W)$. In (235), $P_{F}$, $\varepsilon_{F}$, $T_{F}$ are
the production, dissipation, and transport rates of $F$, respectively. The
production rate $P_{F}$ arises from the coupling of the turbulent correlations
and the mean-field inhomogeneities. The dissipation rate of $F$,
$\varepsilon_{F}$, comes from the molecular viscosity and magnetic diffusivity
or some alternatives such as wave interaction. The transport rate $T_{F}$
represents fluxes through the boundary. In the case of conservation-related
quantities, such fluxes are written in a divergence form.

The production, dissipation, and transport rates of the turbulent MHD energy
$K$ and turbulent cross helicity $W$ are expressed as

$P_{K}=-{\cal{R}}^{ij}\frac{\partial U^{j}}{\partial
x^{i}}-{\bf{E}}_{\rm{M}}\cdot{\bf{J}},$ (236)
$\varepsilon_{K}=\nu\left\langle{\left({\frac{\partial
u^{\prime}{}^{j}}{\partial
x^{i}}}\right)^{2}}\right\rangle+\eta\left\langle{\left({\frac{\partial
b^{\prime}{}^{j}}{\partial x^{i}}}\right)^{2}}\right\rangle\equiv\varepsilon,$
(237) $T_{K}={\bf{B}}\cdot\nabla W+\nabla\cdot{\bf{T}}^{\prime}_{K}\equiv
T_{K}^{(B)}+\nabla\cdot{\bf{T}}^{\prime}_{K},$ (238)
$P_{W}=-{\cal{R}}^{ij}\frac{\partial B^{j}}{\partial
x^{i}}-{\bf{E}}_{\rm{M}}\cdot\mbox{\boldmath$\Omega$},$ (239)
$\varepsilon_{W}=(\nu+\eta)\left\langle{\frac{\partial
u^{\prime}{}^{j}}{\partial x^{i}}\frac{\partial b^{\prime}{}^{j}}{\partial
x^{i}}}\right\rangle,$ (240) $T_{W}={\bf{B}}\cdot\nabla
K+\nabla\cdot{\bf{T}}^{\prime}_{W}\equiv
T_{W}^{(B)}+\nabla\cdot{\bf{T}}^{\prime}_{W},$ (241)

where $\mbox{\boldmath${\cal{R}}$}=\\{{{\cal{R}}^{ij}}\\}$ is the Reynolds and
turbulent Maxwell stress defined by

${\cal{R}}^{ij}=\langle{u^{\prime}{}^{i}u^{\prime}{}^{j}-b^{\prime}{}^{i}b^{\prime}{}^{j}}\rangle,$
(242)

and ${\bf{E}}_{\rm{M}}$ is the turbulent electromotive force (EMF) defined by
(187).

The production rates $P_{K}$ and $P_{W}$ are respectively related to the
transfer of the turbulent MHD energy and cross helicity between the large- and
small-scales. This point is clearly seen if we write the evolution equations
of the mean-field MHD energy ${\cal{K}}$ and the mean-field cross helicity
${\cal{W}}$ defined by

${\cal{K}}=({\bf{U}}^{2}+{\bf{B}}^{2})/2,$ (243)
${\cal{W}}={\bf{U}}\cdot{\bf{B}},$ (244)

respectively. The equations of ${\cal{K}}$ and ${\cal{W}}$ are written as

$\frac{D{\cal{F}}}{Dt}\equiv\left({\frac{\partial}{\partial
t}+{\bf{U}}\cdot\nabla}\right){\cal{F}}=P_{\cal{F}}-\varepsilon_{\cal{F}}+T_{\cal{F}},$
(245) $P_{\cal{K}}=+{\cal{R}}^{ij}\frac{\partial U^{j}}{\partial
x^{i}}+{\bf{E}}_{\rm{M}}\cdot{\bf{J}}=-P_{K},$ (246)
$\varepsilon_{\cal{K}}=\nu\left\langle{\left({\frac{\partial U^{j}}{\partial
x^{i}}}\right)^{2}}\right\rangle+\eta\left\langle{\left({\frac{\partial
B^{j}}{\partial x^{i}}}\right)^{2}}\right\rangle,$ (247)
$T_{\cal{K}}={\bf{B}}\cdot\nabla{\cal{W}}+\nabla\cdot{\bf{T}}^{\prime}_{\cal{K}}\equiv
T_{\cal{W}}^{(B)}+\nabla\cdot{\bf{T}}^{\prime}_{\cal{K}},$ (248)
$P_{\cal{W}}=+{\cal{R}}^{ij}\frac{\partial B^{j}}{\partial
x^{i}}+{\bf{E}}_{\rm{M}}\cdot\mbox{\boldmath$\Omega$}=-P_{W},$ (249)
$\varepsilon_{\cal{W}}=(\nu+\eta)\left\langle{\frac{\partial U^{j}}{\partial
x^{i}}\frac{\partial B^{j}}{\partial x^{i}}}\right\rangle,$ (250)
$T_{\cal{W}}={\bf{B}}\cdot\nabla{\cal{K}}+\nabla\cdot{\bf{T}}^{\prime}_{\cal{W}}\equiv
T_{\cal{W}}^{(B)}+\nabla\cdot{\bf{T}}^{\prime}_{\cal{W}}.$ (251)

Equations (246) and (249) show that the production rates of the mean-field MHD
energy ${\cal{K}}$ and the mean-field cross helicity ${\cal{W}}$ are exactly
the same as the counterpart of the turbulent MHD energy $K$ (236) and the
turbulent cross helicity $W$ (239), but with the opposite signs. As this
consequence, these production terms do not contribute to the total amount of
that quantity; the sum of the turbulent and mean-field quantities,
${\cal{K}}+K$ and ${\cal{W}}+W$. They contribute just to the transfer of the
quantity between the mean and turbulent components. If the sign of the
production of a turbulent quantity is positive (or negative), the turbulent
quantity increases (or decreases). At the same time, in this case, the sign of
the mean-field counterpart is negative (or positive) and the mean-field
quantity decreases (or increases). This means that the generation of the
turbulent quantity arising from the positive production rate is supplied by
the drain or sink of the mean-field counterpart. In this sense, production
rates of $K$ and $W$, $P_{K}$ and $P_{W}$, represent the cascade properties of
the turbulent MHD energy and cross helicity.

The dissipation rates of the mean-field MHD energy and cross helicity,
$\varepsilon_{\cal{K}}$ (247) and $\varepsilon_{\cal{W}}$ (250), are the
counterparts of $\varepsilon_{K}$ (237) and $\varepsilon_{W}$ (240). The
molecular viscosity $\nu$ and the magnetic diffusivity $\eta$ coupled with the
mean-field inhomogeneities $\nabla{\bf{U}}$ and $\nabla{\bf{B}}$ lead to the
dissipation. As compared with the inhomogeneities in small scales,
$\nabla{\bf{u}}^{\prime}$ and $\nabla{\bf{b}}^{\prime}$, these large-scale
inhomogeneities are considered to be small. However, in the region where the
large-scale field inhomogeneities are fairly large, such as near-wall region
and shock vicinity, these dissipation rates associated with the mean-field
inhomogeneities can be considerably large and not negligible.

Like the turbulent counterparts $T_{K}$ and $T_{W}$, the transport rates of
the mean-field quantities, $T_{\cal{K}}$ and $T_{\cal{W}}$ represent the flux
through the boundary. The first terms in $T_{K}$ (238) and $T_{W}$ (241) are
originally written in a divergence form as

$T_{K}^{({\rm{B}})}=\nabla\cdot(W{\bf{B}})=({\bf{B}}\cdot\nabla)W,$ (252)
$T_{W}^{({\rm{B}})}=\nabla\cdot(K{\bf{B}})=({\bf{B}}\cdot\nabla)K.$ (253)

However, in order to explicitly show that they are related to the turbulence
inhomogeneities along the mean magnetic field, they are written as the first
terms of (238) and (241).

The first term of (238) or (252), $({\bf{B}}\cdot\nabla)W$, is linked to the
Alfvén wave propagating along a mean magnetic field ${\bf{B}}$. In terms of
the Alfvén wave, a positive cross helicity $W>0$ represents a dominance of the
counter propagating Alfvén waves (Alfvén wave propagating in the direction
antiparallel to the mean magnetic field). Let us consider a domain bounded by
two surfaces (Fig. 14): one is the surface of the upstream direction with
respect to the mean magnetic field (up-flux boundary) and the other is the
surface of the downstream direction (down-flux boundary). If the magnitude of