Modeling the turbulent cross-helicity evolution: Production, dissipation, and transport rates

# Modeling the turbulent cross-helicity evolution: Production, dissipation, and transport rates

N. Yokoi
Institute of Industrial Science, University of Tokyo
4-6-1, Komaba, Meguro-ku, Tokyo 153-8505, Japan
Nordic Institute for Theoretical Physics (NORDITA)
Roslagstullsbacken 23, 106 91 Stockholm, Sweden
Corresponding author. Email: nobyokoi@iis.u-tokyo.ac.jpGuest researcher at the National Astronomical Observatory of Japan (NAOJ)
23rd February, 2010, accepted 14th May, 2011
###### Abstract

It has been recognized that the turbulent cross helicity (correlation between the velocity and magnetic-field fluctuations) can play an important role in several magnetohydrodynamic (MHD) plasma phenomena such as the global magnetic-field generation, turbulence suppression, etc. Despite its relevance to the cross-helicity evolution, little attention has been paid to the dissipation rate of the turbulent cross helicity, . In this paper, we consider the model expression for the dissipation rate of the turbulent cross helicity. In addition to the algebraic model, an evolution equation of is proposed on the basis of the statistical analytical theory of inhomogeneous turbulence. A turbulence model with the modeling of is applied to the solar-wind turbulence. Numerical results on the large-scale evolution of the cross helicity is compared with the satellite observations. It is shown that, as far as the solar-wind application is concerned, the simplest possible algebraic model for is sufficient for elucidating the large-scale spatial evolution of the solar-wind turbulence. Dependence of the cross-helicity evolution on the large-scale velocity structures such as velocity shear and flow expansion is also discussed.

Keywords: Magnetohydrodynamic turbulence; turbulence model; cross helicity; dissipation rate; solar wind;

\doi

10.1080/14685240YYxxxxxxx \issn1468-5248 \jvol12 \jnum00 \jyear2011

\articletype

RESEARCH ARTICLE

] ] ]

## 1 Introduction

In the magnetohydrodynamic (MHD) turbulent flow at high magnetic Reynolds number (), magnetic fields are considered to be frozen in plasmas, and move with the flow.[1] In such a flow, the induced magnetic field is often much larger than the originally imposed field. Besides, MHD waves such as the Alfvén wave are considered to exist ubiquitously. The cross helicity, defined by the correlation between the velocity and magnetic field , is a possible describer of such MHD turbulence properties. Actually, the magnetic-field generation due to the turbulent cross helicity has been investigated.[2, 3, 4, 5, 6, 7, 8, 9]

As is well known, the total amount of cross helicity , as well as that of the MHD energy , is an inviscid invariant of the MHD equations. Because of this conservative property, the turbulent densities of the MHD energy and cross-helicity, and , may serve themselves as a good measure for characterizing the statistical properties of MHD turbulence (: velocity fluctuation, : magnetic-field fluctuation, : ensemble average).

The evolution equations of and are similar in form, and their mathematical structures are quite simple. The evolution of and are determined by three constitutes: the production, dissipation, and transport rates. Firstly, the production rate is expressed by the correlation of turbulence fields coupled with the mean-field inhomogeneity. We note that the production rate of turbulence quantities such as can be expressed exactly in the same as the counterpart of mean-field quantities such as , but with the opposite sign (: the mean velocity, : the mean magnetic field). This makes our interpretation possible that the drain of mean-field quantity gives rise to the generation of turbulence counterpart. So, the production rate represents how a quantity is supplied to turbulence by way of its cascading process (See Appendix A). Secondly, the dissipation rates of the turbulent MHD energy and cross helicity, and , whose definitions will be given shortly in Section 2, represent the effects of molecular viscosity and magnetic diffusivity coupled with the small-scale fluctuations. However, we stress the following point. The dissipation rates of the turbulent MHD energy and cross helicity, and , can be considered from another aspect. In the intermediate range of turbulence, called the inertial range, the energy and cross helicity supplied from the energy-containing range compensate the energy and cross-helicity lost in the dissipation range. For this cascade picture of turbulence, the energy and cross-helicity transfer from lower to higher wavenumber ranges are most important quantities. In an equilibrium turbulence, and represent these transfer rates of and , respectively. This makes the construction of the and equation possible as we show later. Finally, the transport rates express the flux of a quantity that enters the fluid volume through the boundary. The expression for the transport rates suggests in what situation the quantities considered can be supplied to turbulence.

Thanks to these clear-cut pictures associated with the evolution equation, the cross helicity (density) , as well as the turbulent MHD energy (density) , may play an important role in the turbulence modeling of MHD fluids. However, as compared with and other pseudoscalar turbulence quantities such as the turbulence kinetic and magnetic helicities, only a limited attention so far has been paid to the cross helicity.

In the context of homogeneous isotropic MHD turbulence, some important investigations have been made on the decaying rate of the cross helicity or . It was shown that if there is a prevailed sign of the cross helicity in the initial state, the system goes towards a dynamically aligned state. The cross helicity scaled by the MHD energy grows towards +1 or -1 depending on the initially prevailed sign of the cross helicity.[10, 11, 12, 13]

In the context of inhomogeneous MHD turbulence, the cross helicity has been investigated mostly in the solar-wind research. By using spacecraft observations, detailed spectra of cross helicity have been examined.[14, 15, 16] In order to explain the large-scale behavior of the solar-wind turbulence, several models have been proposed.[17, 16] However, investigations related to the cross helicity are mostly concentrated on arguments of its production rate, and effects of large-scale inhomogeneities such as the mean velocity shear have been discussed. Matthaeus and coworkers have employed a kind of algebraic model for the cross-helicity dissipation rate.[17, 18, 19, 20] Adopting this algebraic model of , Usmanov et al. have recently performed a series of elaborated numerical simulations on the large-scale evolution of solar-wind turbulence.[21] However, generally speaking, arguments concerning the dissipation rate of are still far from sufficient.

Also in the context of the turbulence dynamo, the transport equation for the turbulent cross helicity has been considered, where an algebraic model for the cross-helicity dissipation rate has been proposed.[2, 3] Sur and Brandenburg wrote down an evolution equation for the cross-helicity effect, and argued the cross-helicity destruction in the context of quenching mechanism.[22]

In order to examine the evolution of the turbulent cross helicity , it is indispensable to properly estimate the dissipation rate of , , as well as the cross-helicity production rate . We address this problem; modeling the cross-helicity dissipation rate on the basis of a statistical analytical theory.

Spacecraft observations of solar-wind turbulence have revealed detailed information on the large-scale behavior of turbulent statistical quantities, which includes the radial evolution of the cross helicity both in the low- and high-speed wind regions. Comparison of the satellite observations with the numerical simulation with the aid of a turbulence model provides a good test for the cross-helicity dissipation models. In this work, we will delve into the problem of cross-helicity dissipation modeling by using such comparisons.

The organization of this paper is as follows. After briefly showing the evolution equation of the cross helicity in Section 2, we present the exact equation of the cross-helicity dissipation rate in Section 3. Then, we consider two candidates for the model of in Section 4; one is the algebraic model and another is a transport-equation model. These expressions are systematically derived with the aid of a statistical analytical theory of inhomogeneous turbulence. Some features of the adopted turbulence model, which is an expansion of the hydrodynamic -type one-point turbulence model in the engineering field, are noted in Section 5. An application of the model to the solar wind is presented in Section 6. A brief summary is given in Section 7.

## 2 Equation for the cross helicity

The fluctuation velocity and magnetic field, and , in incompressible magnetohydrodynamic (MHD) flow are governed by

 (∂∂t+U⋅∇)u′ = −(u′⋅∇)U+(B⋅∇)b′+(b′⋅∇)B (1) −(u′⋅∇)u′−(b′⋅∇)b′+∇⋅% \boldmathR−∇p′M+ν∇2u′+f′,
 (∂∂t+U⋅∇)b′ = −(u′⋅∇)B+(B⋅∇)u′+(b′⋅∇)U (2) −(u′⋅∇)b′−(b′⋅∇)u′−∇×EM+λ∇2b′

with the solenoidal conditions

 ∇⋅u′=∇⋅b′=0 (3)

(: kinematic viscosity, : magnetic diffusivity). The primed quantities denote the deviations from the ensemble average as

 φ=¯¯¯¯φ+φ′,¯¯¯¯φ=⟨φ⟩ (4)

with

 φ=(ρ,u,\boldmathω,b,j,p,pM,f), (5a)
 ¯¯¯¯φ=(¯¯¯ρ,U,\boldmathΩ,B,J,P,PM,F), (5b)
 φ′=(ρ′,u′,\boldmathω′,b′,j′,p′,p′M,f′). (5c)

Here, is the density, the vorticity, the electric-current density, the gas pressure, the MHD pressure, and the external force. Note that the magnetic field etc. are measured in the Alfvén-speed unit. They are related to the ones measured in the original unit (asterisked) as

 b=b∗(μ0ρ)1/2,j=j∗(ρ/μ0)1/2,e=e∗(μ0ρ)1/2,p=p∗ρ (6)

(: magnetic permeability).

The Reynolds stress and the turbulent electromotive force represent the turbulence effects on the mean field. They are defined by

 Rαβ≡⟨u′αu′β−b′αb′β⟩, (7)
 EM≡⟨u′×b′⟩, (8)

respectively.

In order to describe the properties of turbulence, we consider several turbulent quadratic statistical quantities. Among them, the turbulent MHD energy and the turbulent cross helicity are defined by

 K≡12⟨u′2+b′2⟩, (9)
 W≡⟨u′⋅b′⟩. (10)

From Equations (1) and (2), the evolution equations of and are rightly obtained as

 DGDt≡(∂∂t+U⋅∇)G=PG−εG+TG (11)

with . Here, , , and are the production, dissipation, and transport rates of the turbulent statistical quantity . They are defined by

 PK=−Rab∂Ua∂xb−EM⋅J, (12a)
 εK=ν⟨(∂u′a∂xb)2⟩+λ⟨(∂b′a∂xb)2⟩≡ε, (12b)
 TK=B⋅∇W−∇⋅⟨(u′2+b′22+p′M)u′+(u′⋅b′)b′⟩+⟨f′⋅u′⟩, (12c)
 PW=−Rab∂Ba∂xb−EM⋅\boldmathΩ, (13a)
 εW=(ν+λ)⟨∂u′a∂xb∂b′a∂xb⟩, (13b)
 TW=B⋅∇K−∇⋅⟨(u′⋅b′)u′−(u′2+b′22−p′M)b′⟩+⟨f′⋅b′⟩. (13c)

As was already mentioned in Section 1 and Appendix A, the production rates and represent the supply of and , respectively, due to turbulence cascade. We see from the first term of Equation (12a) that the turbulent energy is sustained by the mean velocity shear . The second or -related term of Equation (12a) is related to the Joule dissipation of MHD turbulence. Equation (13a) shows that inhomogeneities of the mean fields coupled with the fluctuation correlations are essential for . One is the shear of the large-scale magnetic field, , coupled with the Reynolds stress :

 PWR=−Rab∂Ba∂xb. (14)

The other is the large-scale vortical motion coupled with the turbulent electromotive force :

 PWE=−EM⋅\boldmathΩ. (15)

Finite positive (or negative) values of and infer the generation of positive (or negative) turbulence cross helicity by way of cross-helicity cascade. The role of the cross-helicity production have been argued mainly in the context of turbulent dynamo and turbulent transport suppression.

Other important mechanisms that possibly supply the cross helicity to turbulence come from the first and last terms of Equation (13c):

 TWK=B⋅∇K, (16)
 TWF=⟨f′⋅b′⟩. (17)

Equation (16) shows that the inhomogeneity of along the large-scale magnetic field may contribute to the supply of the cross helicity. We often meet such situations in astrophysical phenomena where inhomogeneous turbulent plasma is threaded through by the ambient magnetic fields.

On the other hand, Equation (17) represents the turbulent cross-helicity generation due to the coupling of the external forcing and magnetic fluctuation. As we see in the evolution equation of the mean cross helicity [Equation (A9) in Appendix A], the coupling of the (mean) external forcing with the large-scale magnetic field provides the mean cross helicity. Actually, such cross-helicity generation due to an external forcing plays a crucial role in the dynamo action or magnetic-field generation mechanism in a generalized Arnold–Beltrami–Childress flow called the Archonitis flow.[22] In this sense, analysis of the forcing effect on the evolution of the cross helicity is very important. It may affect the expression of the cross-helicity flux, just as the body-force effects such as the buoyancy, frame rotation, etc. change the scalar transport. However, this topic is beyond the scope of this paper.

Unlike the production rates [Equation (12a)] and [Equation (13a)] and the first or -related terms of the transport rates [Equations (12c) and (13c)], the definitions of the dissipation rates, [Equation (12b)] and [Equation (13b)], do not directly contain any mean fields, but are expressed by the combination of the molecular viscosity and magnetic diffusivity with the inhomogeneity of the fluctuation fields. So, we need some specific treatments on them. We consider the cross-helicity dissipation rate in the following sections.

## 3 Equation for the cross-helicity dissipation rate

From the equations of the fluctuation velocity and magnetic field, we construct the exact equation for the dissipation rate of the cross helicity, [Equation (13b)], as

 DεWDt≡(∂∂t+U⋅∇)εW (18) =(ν+λ)⟨∂u′a∂xc∂b′b∂xc−∂b′a∂xc∂u′b∂xc⟩∂Ua∂xb +(ν+λ)⟨∂u′a∂xcb′b−u′b∂b′a∂xc⟩∂2Ua∂xb∂xc +(ν+λ)⟨∂u′b∂xa∂2u′b∂xa∂xc+∂b′b∂xa∂2b′b∂xa∂xc⟩Bc +(ν+λ)⟨∂u′c∂xa∂u′c∂xb−∂b′c∂xa∂b′c∂xb⟩∂Ba∂xb −(ν+λ)⟨∂u′a∂xc∂u′b∂xc−∂b′a∂xc∂b′b∂xc⟩∂Ba∂xb −(ν+λ)⟨∂u′a∂xcu′b−∂b′a∂xcb′b⟩∂2Ba∂xb∂xc −(ν+λ)⟨∂u′b∂xa∂u′c∂xa∂b′b∂xc⟩−(ν+λ)⟨∂b′b∂xa∂u′c∂xa∂u′b∂xc⟩ +(ν+λ)⟨∂u′b∂xa∂b′c∂xa∂b′b∂xc⟩+(ν+λ)⟨∂b′b∂xa∂b′c∂xa∂b′b∂xc⟩ −(ν+λ)⟨(u′c±b′c)∂∂xc(∂u′b∂xa∂b′b∂xa)⟩ +(ν+λ)∂∂xc⟨12b′c[∂∂xa(u′b±b′b)]2⟩ −(ν+λ)⟨∂b′b∂xa∂2p′M∂xa∂xb⟩+(ν+λ)∂2∂xc∂xcεW −(ν+λ)2⟨∂2u′b∂xa∂xc∂2b′b∂xa∂xc⟩

(both upper or both lower signs should be chosen in the double signs). This equation, lacking the connection with a conservative law, has a considerably complicated structure. This is in sharp contrast to the and equations [Equation (11)].

In the hydrodynamic case with an electrically non-conducting fluid, the energy dissipation rate

 ϵ≡ν⟨∂u′a∂xb∂u′a∂xb⟩, (19)

as well as the turbulent energy , plays a central role in turbulence modeling.[23] From the equation of the velocity fluctuation, we write the equation exactly as

 DϵDt≡(∂∂t+U⋅∇)ϵ (20) =−2ν⟨∂u′b∂xa∂u′c∂xa∂u′b∂xc⟩−2⟨(ν∂2u′b∂xa∂xc)2⟩ −2ν⟨u′b∂u′a∂xc⟩∂2Ua∂xb∂xc +∂∂xc[−ν⟨(∂u′b∂xa)2⟩−2ν⟨∂p′∂xb∂u′a∂xb⟩] +ν∂2ϵ∂xc∂xc.

The mathematical structure of this equation is also complicated because of the lack of the connection with a conservative law.

The energy dissipation itself is dominant at small scales. Using this, the length and velocity scales are estimated as

 |x|∼ν3/4ϵ−1/4,|u′|∼ν1/4ϵ1/4, (21)

respectively. Using Equation (21), we can estimate each term in Equation (20). In the flow at high Reynolds number (), two terms behaving as are dominant, and these two terms should balance each other[24, 5]

 −2ν⟨∂u′a∂xb∂u′a∂xc∂u′b∂xc⟩∼2⟨(ν∂2u′a∂xb∂xc)2⟩. (22)

In other word, in the modeling of the equation, it is of crucial importance to properly estimate the two terms in Equation (22).

In the hydrodynamic turbulence modeling, an empirical model equation for :

 DϵDt≡(∂∂t+U⋅∇)ϵ=Cϵ1ϵkPk−Cϵ2ϵkϵ+∇(νTσϵ∇ϵ) (23)

was proposed and has been widely accepted as useful.

Here, , , and are model constants. The values of these constants have been optimized through various applications of the model. Usually, the values of

 Cϵ1=1.4,Cϵ2=1.9,σϵ=1.0 (24)

are adopted.[23]

Equation (23) is based on an empirical inference that the dissipation should be larger where the turbulence is larger, and is derived by dimensional analysis. However, it is also pointed out that the system of constants given by Equation (24) is not a unique combination for the model constants because equation can take an infinite number of self-similar states.[25]

A similar argument using Equation (21) can be applied to the exact equation for the dissipation rate of the turbulent cross helicity [Equation (18)]. The difference between the molecular viscosity and the magnetic diffusivity is expressed by the magnetic Prandtl number:

 Pm≡ν/λ. (25)

If we estimate each term in Equation (18) under the simplest possible condition of , we have a dominant balance between the terms expressed by

 (ν+λ)⟨∂u′b∂xa∂u′c∂xa∂b′b∂xc⟩∼ν−1/2ε3/2, (26a)
 (ν+λ)2⟨∂2u′b∂xa∂xc∂2b′b∂xa∂xc⟩∼ν−1/2ε3/2 (26b)

in the equation at the high Reynolds number flow.

## 4 Models for the dissipation of the turbulent cross helicity

### 4.1 Algebraic model

As was mentioned in the previous section, the equation governing the dissipation rate of , , is very complicated. In such a situation, the simplest possible model for is the algebraic approximation as follows.

Using the turbulent MHD energy and its dissipation rate , we construct a characteristic time scale of turbulence as

 τ=K/ε. (27)

Other choices of time scale are possible. A large-scale magnetic field may alter the characteristics of turbulence. In the case of MHD turbulence, the Alfvén time associated with the magnetic field may modulate time scale of turbulence. This point will be referred to later at the end of Section 5.

With the aid of the time scale of Equation (27), the dissipation rate of can be modeled as

 εW=CWWτ=CWεKW, (28)

where is the model constant. Namely, we consider the dissipation of the turbulent cross helicity is proportional to the turbulent cross helicity divided by the time scale.

It is worth noting a mathematical constraint on the cross helicity. Namely, the magnitude of the turbulent cross helicity is bounded by the magnitude of the turbulent MHD energy as

 |W|K=|⟨u′⋅b′⟩|⟨u′2+b′2⟩/2≤1. (29)

This relation constrains the value of . Actually, from Equation (11), the turbulent cross helicity scaled by the turbulent MHD energy, , is subject to

 DDtWK=WK(1WDWDt−1KDKDt) (30) = WK(1WPW−1KPK)−WK(1WεW−1Kε) + WK(1WTW−1KTK).

Equation (30) is a very general expression for the evolution equation of the scaled cross helicity, , which should be satisfied with in any situations of turbulence flow. If we consider homogeneous turbulence, where the spatial variations of mean quantities vanish, we have neither production nor transport rate (). In such a case, Equation (30) is reduced to

 ∂∂tWK=−(1WεW−1Kε)WK. (31)

This may give a constraint on the values of the dissipation rates of the turbulent MHD energy and cross helicity, and . At least, Equation (31) provides us with a constraint for the turbulence modeling of and as we see in the following.

If the coefficient of or parenthesized quantity in Equation (31) does not depend on , which occurs naturally when we adopt simple algebraic models for and as below, in order for the magnitude of the scaled cross helicity to be bounded in time evolution, the parenthesized quantity should be positive. Then we have inequality

 |εW|ε>|W|K. (32)

Provided that the difference between the time scales of and is independent of , this is a necessary condition for in relation to arising from the upper-boundedness of [Equation (29)]. Actually, if we adopt simple algebraic models for and such as

 ε=Kτ,εW=CWWτ, (33)

the parenthesized quantity in Equation (31) does not depend on . Then we have inequality (32), which gives a constraint on the model constant as will be seen later in Equation (88) in Section 6.2.

In the context of homogeneous MHD turbulence, since the pioneering work by Dobrowolny, Mageney & Veltri[10, 11], it had been considered that the initially prevailing cross helicity grows with time due to the nonlinear interaction towards the (or ) value of the scaled cross helicity (Also see [12, 13]). However, further investigation by Ting et al.[26] and Stribling & Matthaeus[27] showed that MHD turbulence has diverse possibilities for long time evolution, including either growth or reduction of . In contrast to such homogeneous MHD turbulence, in inhomogeneous MHD turbulence with non-vanishing mean velocity shear, we have production, dissipation, and transport mechanisms of the turbulent cross helicity arising from or related to the inhomogeneity of mean fields. In the present work, we explore these mechanisms intrinsic to the inhomogeneities of MHD turbulence. There have been several works where the effects of mean velocity shear are incorporated into the evolution equations of turbulent quantities including the turbulent cross helicity. In this sense, this work is in the same line with [18, 19, 20], and also [28, 29, 30]. However, in the present work, special emphasis is placed on the theoretical derivation of the turbulent cross-helicity dissipation-rate equation as will be shown in the following sections.

Here, we had better remark on the notation of the scaled cross helicities. The turbulent cross helicity scaled by the turbulent MHD energy, [Equation (29)], is “dynamically” important in the context of turbulent dynamo etc. There is another scaled cross helicity

 Γ=⟨u′⋅b′⟩√⟨u′2⟩⟨b′2⟩ (34)

that is “kinematically” or geometrically important, since it represents the degree of alignment of the velocity and magnetic-field vectors. In the solar-wind turbulence community, the former, , is often referred to as “”. However, in the “general” turbulence community, this is not the case; Some use “” for this quantity, others do “”, and so on. Hence, in this paper, we confine ourselves to just mentioning these notation conventions.

Interestingly, the directional alignment expressed by has shown to be robust rather than the behaviour of .[31] This tendency is related to the fact that, unlike , may dynamically change its value depending on how much differently the turbulent MHD energy and cross helicity, and , are influenced by the large-scale shears. This point will be referred to later in Sec. 6.4.

### 4.2 Model equation for the cross-helicity dissipation rate (εW equation)

The equation of the dissipation rate, equation, can be written as

 ∂ε∂t+(U⋅∇)ε=Cε1εKPK−Cε2εKε+∇⋅(νKσε∇ε). (35)

This is a direct expansion of the hydrodynamic (HD) or non-MHD turbulence energy dissipation equation (23). The model constants in Equation (35) are same as the ones appearing in Equation (23). This is a consequence of the requirement that the equation for MHD turbulence should be reduced to the equation for the HD in the limit of the vanishing magnetic field ().

We see from Equation (11) that the equations of and are written in a similar form. So, it is natural to consider that the equation of the dissipation rate of , , can be expressed in a form similar to the equation of the dissipation rate . Then we may consider the equation for as

 DεWDt=CW1εKPW−CW2εKεW+∇⋅(νKσεW∇εW), (36)

where , , and are model constants.

In contrast to the MHD energy, the cross helicity, defined by the correlation between the velocity and magnetic field, vanishes in the limit of the vanishing magnetic field. As this result, as far as the equation is concerned, we can not make use of the knowledge accumulated in the history of HD turbulence modeling. To say nothing of the model constants , , and appearing in Equation (36), we have to consider the structure of equation itself from more fundamental basis. For this purpose, in the following we will construct the equation for the cross-helicity dissipation rate on the analytical theoretical basis.

From the theoretical analysis of inhomogeneous MHD turbulence, the cross-helicity density defined by is expressed as

 W=2I0{Qub}−I0{GS,DDt(Qub+Qbu)}, (37)

where we have used abbreviated forms of spectral and time integrals:

 In{A}=∫A(k,x;τ,τ,t)k2ndk, (38a)
 In{A,B}=∫dk k2n∫τ−∞dτ1A(k,x;τ,τ1,t)B(k,x;τ,τ1,t). (38b)

In Equation (37), is the Green’s function, and and are the correlation functions related to the basic or lowest-order fields and :

 (39)

For the derivation of Equation (37) and higher-order expression for , see Appendix B. Suggestions from the higher-order expressions for the cross-helicity dissipation model are also presented in Appendix C.

We assume that the correlation function and the Green’s function in the inertial range are expressed as

 Qub(k,x;τ,τ′,t)=σW(k,x;t)exp[−ωW(k,x;t)|τ−τ′|], (40)
 GS(k,x;τ,τ′,t)=θ(τ−τ′)exp[−ωS(k,x;t)(τ−τ′)], (41)

where is the Heaviside’s step function that is 1 and 0 for and , respectively. Here, is the power spectra of the turbulent cross helicity, and and represent the frequencies or time scales of fluctuations. As for the spectra in the inertial range, we assume

 σW(k,x;t)=σW0ε−1/3εW(x;t)k−11/3, (42)

and for the time scales,

 ωS(k,x;t)=ωS0ε1/3k2/3=τ−1S, (43)
 ωW(k,x;t)=ωW0ε1/3Wk2/3=τ−1W (44)

(, , and are numerical factors). Equation (42) arises from the assumption that the spectrum of the cross helicity is determined by the scale, energy and cross-helicity transfer rates; , and .

Using Equations (40) and (41), is calculated as

 W=2∫dk ε−1/3(x;t)εW(x;t)k−11/3 (45) −2∫dk[1ωS+ωWDσWDt−σW(ωS+ωW)2DωWDt].

From the inertial-range forms (42)-(44), this can be rewritten as

 W=4⋅2πε−1/3εW∫|k|≥kCdk k−5/3 (46) −4⋅2πσW0ωswε1/3sw∫|k|≥kCdk k4/3DDt[ε−1/3(x;t)εW(x;t)k−11/3] +4⋅2πσW0ωW0(ωswε1/3sw)2ε−1/3εW∫|k|≥kCdk k−3DDt[ε1/3W(x;t)k2/3]

(: cut-off wave number). Here, we have introduced a synthesized time scale defined by

 1τSW=1τS+1τW=(ωS0ε1/3+ωW0ε1/3W)k2/3≡ωswε1/3swk2/3. (47)

We should note that and will appear only in the combination of ; each of and has no definite meaning. For simplicity of notation, hereafter we denotes

 AW(ωS0,ωW0)≡ωW0ε1/3Wωswε1/3sw=ωW0ε1/3WωS0ε1/3+ωW0ε1/3W=τSWτW. (48)

In Equation (46), the lower bound of the spectral integral region, , is directly connected to the largest eddy size of turbulent motions as

 ℓC=2π/kC, (49)

which dominantly contributes to and determines the turbulent MHD energy and cross helicity. Strictly speaking, the correlation length of the cross helicity, , can be different from that of the MHD energy, . However, if we consider the fact that the characteristic lengths of turbulence represent the scales of the largest turbulent motion corresponding to the scales with largest magnitudes of the turbulent MHD-energy and cross-helicity spectral densities, we see that in order for the length scales of the MHD energy and cross helicity to be considerably different from each other, the spectral distributions of the MHD energy and cross helicity should be entirely different from each other. This is not the case, for instance, in the case of the solar-wind turbulence. Actually, in most cases of interests, these two length scale are similar to each other. In this sense, we can regard the characteristic lengths of the energy and cross helicity are approximately the same, and we denote hereafter. In this respect, extensive discussions of correlation length scales as well as time scales found in Matthaeus et al.[32] and Hossain et al.[33] are very important.

With this point in mind, we perform the Lagrange derivatives and calculate the spectral integrals in Equation (46). Denoting the scaled wave number

 s=k/kC, (50)

we have

 W=4⋅(2π)1/3σW0ε−1/3εWk2/3C∫s≥1ds s−5/3 (51) −4⋅2πσW0ωswε1/3swk7/3C∫s≥1ds s−7/3DDt[ε−1/3(x;t)εW(x;t)k−11/3C] +4⋅2πσW0ωW0(ωswε1/3sw)2ε−1/3εWk−2C∫s≥1ds s−7/3DDt[ε1/3W(x;t)k2/3C].

Using Equation (49), we calculate Equation (51) to obtain

 W=6⋅(2π)1/3σW0ε−1/3εWℓ2/3C (52) +1(2π)1/3σW0ωswε1/3swε−1/3εWℓ4/3C{1εDεDt−[3−AW(ωS0,ωW0)]1εWDεWDt +[11−2AW(ωS0,ωW0)]1ℓCDℓCDt}.

In hydrodynamic turbulence modeling, the turbulent energy , its dissipation rate , and the correlation or integral length are equivalently important. For the purpose of closing the system of model equations, we can select any combination of , , and . This is known as the transferability of the model with respect to , , and . In order to satisfy this transferability requirement, these quantities should be connected with each other in an algebraic relation:

 Ku=CKϵ2/3ℓ2/3C. (53)

This transferability requirement gives a theoretical foundation of the energy-dissipation-rate equation (23).[34]

Relation (53) itself has been argued for a long time. As for the theoretical and experimental basis for this relation, the reader is referred to classical papers [35, 36] and also to Batchelor’s book [37] and works cited therein. As for the simple physical arguments for this relation, see [38, 39, 40]. Here, we just point out that this relation is easily obtained if we assume the local equilibrium of turbulence; with the notion of the mixing length. Then we can estimate the dissipation rate as

 ϵ≃PKu=−⟨u′au′b⟩∂Ub∂xa∼uuuℓ (54)

(: characteristic intensity of turbulence, : mixing length). This is equivalent to Equation (53).

In the context of theoretical derivation of equation, it is the algebraic property of Equation (53) that is much more important. If the relation is not algebraic, we have no transferability among any combination of , , and at all.

We expand the transferability requirement to the model equation related to the cross helicity. For this purpose, we solve Equation (52) concerning in a perturbational manner. At the lowest-order, we have

 W=6(2π)1/3σW0ε−1/3εWℓ2/3C (55)

or

 ℓC=6−3/2(2π)−1/2σ−3/2W0ε1/2ε−3/2WW3/2. (56)

The transferability of model requires an algebraic relation among , , , and . Equations (55) and (56) correspond to such a relation.

Using Equation (56), we change expression (52) based on , , and into the one based on , , and . We require Equation (55) or (56). As a result, we have

 DεWDt=C1(ωS0,ωW0)εWεDεDt+C2(ωS0,ωW0)εWWDWDt (57)

with

 C1(ωS0,ωW0)=13−AW(ωS0,ωW0)39−8AW(ωS0,ωW0),C2(ωS0,ωW0)=33−6AW(ωS0,ωW