Turbulent transport coefficients and residual …hamba/pdf/hamba2008.pdfTurbulent transport coefficients and residual energy in mean-field dynamo theory Fujihiro Hamba1,a and Hisanori

Turbulent transport coefficients and residual energy in mean-fielddynamo theory

Fujihiro Hamba1,a� and Hisanori Sato2

1Institute of Industrial Science, University of Tokyo, Komaba, Meguro-ku, Tokyo 153-8505, Japan2Japan Patent Office, Kasumigaseki, Chiyoda-ku, Tokyo 100-8915, Japan

�Received 3 December 2007; accepted 14 January 2008; published online 25 February 2008�

The turbulent electromotive force in the mean-field equation needs to be modeled to predict alarge-scale magnetic field in magnetohydrodynamic turbulence at high Reynolds number. Using astatistical theory for inhomogeneous turbulence, model expressions for transport coefficientsappearing in the turbulent electromotive force are derived including the � coefficient and theturbulent diffusivity. In particular, as one of the dynamo effects, the pumping effect is investigatedand a model expression for the pumping term is obtained. It is shown that the pumping velocity isclosely related to the gradient of the turbulent residual energy, or the difference between theturbulent kinetic and magnetic energies. The production terms in the transport equation for theturbulent electromotive force are also examined and the validity of the model expression is assessedby comparing with earlier results concerning the isotropic � coefficient. The mean magnetic field ina rotating spherical shell is calculated using a turbulence model to demonstrate the pumpingeffect. © 2008 American Institute of Physics. �DOI: 10.1063/1.2839767�

I. INTRODUCTION

The generation of a large-scale magnetic field by a tur-bulent flow of an electrically conducting fluid is an importantproblem in astrophysics and plasma physics. In general, tur-bulent velocity fluctuations enhance the effective magneticdiffusivity. Dynamo action is necessary to generate and sus-tain a large-scale magnetic field against the turbulent diffu-sivity. The � effect in the mean-field dynamo theory is wellknown as one of the dynamo mechanisms and has been ap-plied to solar and Earth’s magnetic fields.1–3 The � dynamowas also invoked to explain the sustainment of the magneticfield in controlled fusion devices such as the reversed fieldpinch �RFP�.4,5 From the physical point of view, it is veryinteresting and important to investigate the dynamo effectcommonly seen in various magnetohydrodynamic �MHD�turbulent flows.

Fundamental properties of MHD turbulence have beenstudied theoretically and numerically. Owing to the effect ofAlfvén waves, the energy spectrum of isotropic homogenousMHD turbulence is expected to show k−3/2 wavenumber de-pendence in contrast to the k−5/3 spectrum of non-MHDturbulence.6,7 The spectra of kinetic and magnetic energieshave been investigated using turbulence theories such as theeddy-damped quasinormal Markovian approximation and theLagrangian renormalized approximation.8–10 Moreover, di-rect numerical simulation �DNS� of MHD turbulence is apowerful tool to examine the energy spectrum.11–14 Haugenet al.13 showed that the asymptotic spectrum is suggested tobe k−5/3. Mason et al.14 evaluated the velocity and magnetic-field alignment to provide an explanation for the k−3/2 spec-trum in the plane perpendicular to the guiding magnetic field.

Dynamo action such as the � effect was also studiedusing DNS.15–20 Brandenburg15 carried out a DNS of isotro-

pic helical turbulence using a helical forcing to estimate thecoefficients of the � effect and of the turbulent diffusivity.Ossendrijver et al.16 performed a DNS of magnetoconvectionto examine the dependence of the � coefficient on the rota-tion and the magnetic field strength. In addition to the �effect, it is known that the pumping effect term can be de-rived from the tensorial form �ijBj for the turbulent electro-motive force where Bj is the mean magnetic field.3 The meanmagnetic field can be transported through the turbulence me-dium with the pumping velocity even though there is nomean motion. The � effect and the pumping effect originatein the isotropic and antisymmetric components of �ij, respec-tively. Rädler et al.21 and Brandenburg and Subramanian22

theoretically investigated turbulent transport coefficients in-cluding the pumping effect. The coefficients of the pumpingeffect were also evaluated using DNS.17,20

In order to predict the mean magnetic field in realisticMHD turbulent flows, it is necessary to evaluate the turbu-lent transport coefficients such as the � coefficient and theturbulent diffusivity. For a specific problem such as the solarmagnetic field, some appropriate spatial distribution of thetransport coefficients can be prescribed. However, in devel-oping a general MHD turbulence model, model expressionsfor the transport coefficients are necessary to close the sys-tem of mean-field equations. For non-MHD turbulence, theReynolds stress is often modeled using the eddy viscosityapproximation to predict the mean velocity field.23 To evalu-ate the turbulent viscosity, two-equation models have beenwidely used such as the K-� model that treats the transportequations for the turbulent kinetic energy K and its dissipa-tion rate �. A statistical theory called the two-scale direct-interaction approximation �TSDIA� was developed to theo-retically derive and improve the eddy viscosity model fornon-MHD turbulence.24 The TSDIA was shown to success-fully derive a nonlinear eddy-viscosity model. For MHD tur-a�Electronic mail: [email protected].

PHYSICS OF PLASMAS 15, 022302 �2008�

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bulence, a few attempts including the TSDIA have beenmade to construct a model for transport equations.25–29 TheTSDIA is expected to be useful for deriving a universalMHD turbulence model.

Using the TSDIA, Yoshizawa26 examined the pumpingeffect and the turbulent diffusivity; he extended the K-�model to MHD turbulence where K is the sum of the turbu-lent kinetic and magnetic energies. In his work, the � effectwas not examined because the � coefficient is a pseudoscalarand cannot be expressed in terms of scalars K and � only.Yoshizawa and Hamba30 paid attention to the turbulent re-sidual helicity as a pseudoscalar in modeling the � coeffi-cient; they proposed a three-equation model by adopting theturbulent residual helicity as a third model variable. Thethree-equation model was used to simulate the magneticfields in the RFP �Ref. 31� and in the Earth.32 In addition tothe � effect parallel to the mean magnetic field, Yoshizawa33

derived a new dynamo term called the cross-helicity dynamoparallel to the mean vorticity. The cross-helicity dynamo hasbeen applied to several turbulent flows in astrophysical andengineering phenomena.23,34–38

Recently, Yokoi39 paid attention to the turbulent residualenergy, the difference between the turbulent kinetic and mag-netic energies, to investigate the solar-wind turbulence. Aturbulence model for the turbulent residual energy equationwas solved to evaluate the radial dependence of the turbulentquantities in solar wind.40 However, the effect of the turbu-lent residual energy on the turbulent electromotive force hasnot been fully explored yet. It must be interesting to investi-gate the mean-field dynamo theory in more detail using theTSDIA and to examine the effect of the turbulent residualenergy.

In this work, using the TSDIA we theoretically calculatea model expression for the turbulent electromotive force. Inthe present analysis, the formulation is improved so that theframe-invariance of the model expression under rotatingtransformations can be satisfied.41 The Green’s functions thatrepresent the response of the velocity and magnetic-fieldfluctuations to a disturbance at a previous time are treated inmore accurate manner. In the perturbation expansion of theTSDIA, we calculate a few additional terms for the turbulentelectromotive force compared to the previous work.23 Wederive the � effect, the pumping effect, the turbulent diffu-sivity, and the cross-helicity dynamo effect, and we examinethe transport coefficients appearing in these terms.

This paper is organized as follows: In Sec. II, we explainthe turbulent dynamo and diffusivity terms appearing inthe turbulent electromotive force. In Sec. III, we apply theTSDIA to derive a model expression for the turbulent elec-tromotive force. In Sec. IV, we compare the present resultwith the previous work. To assess the validity of the modelexpression, we examine the transport equation for the turbu-lent electromotive force. To demonstrate the pumping effect,we apply the model to the simulation of the magnetic field ina rotating spherical shell. Concluding remarks will be givenin Sec. V.

II. MEAN-FIELD EQUATIONS AND DYNAMO EFFECTS

A. Mean-field equations

In this paper we adopt Alfvén velocity units and replacethe magnetic field b /��0→b, the electric current densityj /�� /�0→ j, and the electric field e /��0→e, where � isthe fluid density and �0 is the magnetic permeability. TheNavier-Stokes equation for a viscous, incompressible, elec-trically conducting fluid and the induction equation for themagnetic field in a rotating system are written as follows:

�u

�t= − � · �uu − bb� − �pM + ��2u − 2�F � u , �1�

� · u = 0, �2�

�b

�t= − � � e, e = − u � b + �Mj , �3�

� · b = 0. �4�

Here, u is the velocity, pM�=p+ ��F�x�2 /2+b2 /2� is thetotal pressure, �F is the system rotation rate, � is the kine-matic viscosity, �M is the magnetic diffusivity, and j=��b.We use ensemble averaging �·� to divide a quantity into themean and fluctuating parts as

f = F + f�, F � �f� , �5�

where f = �u ,b , pM ,e , j ,�� and ��=��u� is the vorticity.The evolution equations for the mean velocity U and themean magnetic field B are written as

�U

�t= − � · �UU − BB� − � · R − �PM + ��2U

− 2�F � U , �6�

� · U = 0, �7�

�B

�t= − � � E, E = − U � B − EM + �MJ , �8�

� · B = 0, �9�

where Rij�=�ui�uj�−bi�bj�� is the Reynolds stress andEM�=�u��b�� is the turbulent electromotive force. In orderto calculate the time evolution of the mean velocity and themean magnetic field, we need to evaluate Rij and EM; somemodeling is necessary for the two quantities. Many turbu-lence models for the Reynolds stress have been developedfor non-MHD turbulence and some of them can be extendedto MHD turbulence. On the other hand, the turbulent elec-tromotive force is treated only in MHD turbulence. In thispaper we focus on modeling the turbulent electromotiveforce to investigate the dynamo effect.

B. Turbulent electromotive force

In the mean-field dynamo theory, the turbulent electro-motive force can be expressed in a tensorial form

022302-2 F. Hamba and H. Sato Phys. Plasmas 15, 022302 �2008�


EMi = �ijBj + �ijk�Bj

�xk, �10�

where summation convention is used for repeated indices. Ifisotropic components of the coefficients

�ij = �ij, �ijk = ��ijk �11�

�where ij is the Kronecker delta symbol and �ijk is the unitalternating tensor� are considered, the turbulent electromo-tive force can be written as

EM = �B − �J . �12�

The first term on the right-hand side of Eq. �12� representsthe � effect, whereas � in the second term is the turbulentdiffusivity. For homogeneous turbulence an expression forthe � coefficient was derived using the spectrum of the ki-netic helicity �u� ·�� in the wavenumber space in the mean-field theory.2,3 Pouquet et al.8 showed that the � coefficientcan be expressed in terms of the spectrum of the turbulentresidual helicity H�=�−u� ·��+b� · j��. The � coefficientwas also evaluated in the DNS of isotropic helicalturbulence.15–17,20

In general, the coefficient �ij can be decomposed intothe symmetric and antisymmetric components

�ijS = ��ij + � ji�/2, �ij

A = ��ij − � ji�/2. �13�

The antisymmetric component can be expressed in terms of avector �A defined as

�iA = − �ijk� jk

A /2. �14�

If the antisymmetric component of �ij is incorporated in ad-dition to the isotropic components, Eq. �10� can be written as

EM = �B + �A � B − �J . �15�

The term �A�B represents the pumping effect; that is, themean magnetic field can be transported through the turbu-lence medium with the velocity �A even though there is nomean motion.3 The pumping velocity �A is expected to beproportional to −��u�2� and the pumping effect can be con-sidered as the turbulence-induced diamagnetism.3 Using thetau approximation Rädler et al.21 and Brandenburg andSubramanian22 showed that the pumping velocity is propor-tional to the gradient of the difference between the kineticand magnetic energies, −��u�2−b�2�.

Using the TSDIA, Yoshizawa26 proposed a two-equationMHD turbulence model. This model is an extension of thenon-MHD K-� model to MHD turbulence. As basic variablesthe turbulent MHD energy K�=�u�2+b�2� /2� and its dissipa-tion rate ��=2��sij�sij� �+�M�j�2�� were adopted wheresij� �=��ui� /�xj +�uj� /�xi� /2� is the strain-rate fluctuation. Theturbulent electromotive force was modeled as

EM = �A � B − �J , �16�

where

�A = C�1K

�� K − C�2

K2

�2 � �, � = C�

K2

��17�

and C�1, C�2, and C� are nondimensional model constants.The pumping velocity �A depends on the gradients of theturbulent MHD energy and its dissipation rate. The turbulentdiffusivity � is modeled in a similar form to the turbulentviscosity for non-MHD turbulence. Using the pumping term,Yoshizawa26 discussed the mechanism of the sustainment ofthe mean magnetic field in the RFP. However, this modeldoes not involve the well-known � effect because a pseudo-scalar � cannot be expressed in terms of scalars K and �only.

Adopting the turbulent residual helicity H as a thirdmodel variable, Yoshizawa and Hamba30 proposed a three-equation model for the � effect. The � coefficient in Eq. �12�can be modeled in terms of H in a straightforward mannerbecause H is also a pseudoscalar. Since then, the � effect hasbeen treated instead of the pumping effect in MHD studiesusing the TSDIA. In addition, Yoshizawa33 pointed out theimportance of the cross helicity W�=�u� ·b�� and proposed anew dynamo term proportional to the mean vorticity �. As aresult, the turbulent electromotive force in a rotating framecan be modeled as23

EM = �B − �J + � + 2F�F, �18�

where

� = C�

K

�H, � = C�

K2

�, = F = C

K

�W . �19�

The third and fourth terms on the right-hand side of Eq. �18�represent the cross-helicity dynamo. In the TSDIA analysis,the expressions for the transport coefficients appearing in Eq.�18� are first derived in the wavenumber space; the forms of and F are slightly different from each other. After theone-point closure approximation, the two coefficients aremodeled in the same form as shown in Eq. �19� and thecross-helicity dynamo term can be written as ��+2�F�.

Since the turbulent electromotive force is objective, orframe-invariant under rotating transformations, its model ex-pression should also be objective.41 If =F, the model ex-pression given by Eq. �18� is objective because the meanabsolute vorticity �+2�F is objective. However, if �F,then Eq. �18� cannot be objective. Therefore, at the begin-ning of the formulation the TSDIA must have some inaccu-rate procedure that violates the frame-invariance. In thiswork we improve the TSDIA formulation to derive modelexpressions satisfying the frame-invariance.

On the other hand, in Eq. �19� the turbulence intensity isexpressed in terms of K, the sum of the turbulent kinetic andmagnetic energies. Recently, Yokoi39 pointed out the impor-tance of the turbulent residual energy KR�=�u�2−b�2� /2� inthe solar-wind turbulence and investigate its evolution equa-tion. A four-equation model treating K, �, W, and KR wasproposed and solved to evaluate turbulent statistics in thesolar wind.40 In the present work, we will examine the rela-tion of the residual energy to the turbulent electromotiveforce.

022302-3 Turbulent transport coefficients… Phys. Plasmas 15, 022302 �2008�


III. THEORETICAL FORMULATION

In this section we briefly explain the procedure of theTSDIA for deriving a model expression for the turbulentelectromotive force. Compared with the previous TSDIA, weimprove the formulation in the two respects: the treatment ofthe rotating frame and of the Green’s functions. In the pre-vious method, the fluctuating field is expanded in terms ofthe mean-field gradient such as �Ui /�xj and of the systemrotation rate �F independently. This independent treatmentleads to the difference between the forms of and F in thewavenumber space as will be shown in Sec. IV. In thepresent method, we expand the fluctuating field so that themean field can always take an objective form such as themean absolute vorticity. In addition, in the previous formu-

lation, two Green’s functions G�� and G�� are treated where��=u+b� and ��=u−b� are Elsässer variables. The Green’s

function G�� G�� represents the response of � �� to adisturbance in the � �� equation. The Green’s functions

G�� and G�� are assumed to be zero although the crossresponse can exist. In the present formulation, we adoptprimitive variables u and b and treat four Green’s functions

Guu, Gbb, Gub, and Gbu. Although the choice of the variablesis arbitrary, the neglect of the Green’s functions for the crossresponse may lead to a different result. We expect that thepresent method can give more accurate expressions.

A. Two-scale variables and frame-invariance

First, we introduce two space and time variables using ascale-expansion parameter as

��=x�, X�=x�, �=t�, T�=t� . �20�

Here, the fast variables � and describe the rapid variationof the fluctuating field whereas the slow variables X and Tdescribe the slow variation of the mean field. A quantity fcan be written as

f = F�X;T� + f��,X; ,T� . �21�

The equations for the velocity fluctuation ui� can be writtenas

�ui�

� + Uj

�ui�

�� j+

�

�� j�uj�ui� − bj�bi�� +

�pM�

��i− �

�2ui�

�� j�� j

− Bj

�bi�

�� j= bj�

�Bi

�Xj− uj� �Ui

�Xj+ � jik�0k� + Bj

�bi�

�Xj

−Dui�

DT−

�

�Xj�uj�ui� − bj�bi� − �uj�ui� − bj�bi�� −

�pM�

�Xi� ,

�22�

�ui�

��i+

�ui�

�Xi= 0, �23�

where

�0 = �F/,Dui�

DT=

�ui�

�T+ Uj

�ui�

�Xj+ � jik�0kuj�. �24�

We should note that the mean-field gradient such as �Ui /�Xj

is of the order . To keep the terms involving the meanvelocity gradient objective, we assume that the system rota-tion rate is also of the order and is set to �F=�0. Theparenthesis involving the mean velocity gradient in Eq. �22�can then be written as the sum of the mean strain-rate andabsolute-vorticity tensors which are both objective,

�Ui

�Xj+ � jik�0k =

1

2 �Ui

�Xj+

�Uj

�Xi�

+1

2 �Ui

�Xj−

�Uj

�Xi+ 2� jik�0k� . �25�

In addition, the material derivative term such as Dui� /DT=�ui� /�T+Uj�ui� /�Xj violates the frame-invariance. To makethe term objective, we adopt the co-rotational derivative

Dui� /DT defined as Eq. �24�. The co-rotational derivative ofa vector is objective and adequately describes an unsteadybehavior of the fluctuating field.41,42 As a result, each term onthe right-hand side of Eq. �22� can be expressed in an objec-tive form. The induction equation for bi� can be derived in asimilar manner. This formulation guarantees the frame-invariance of model expressions derived later.

B. Perturbation expansion and Green’s function

Using the Fourier transform with respect to � we expressa fluctuating field f� as

f��,X; ,T� = dkf�k,X; ,T�exp�− ik · �� − U �� .

�26�

Hereafter, the dependence of f�k ,X ; ,T� on X and T is notwritten explicitly. We expand the fluctuation f�k ; � in pow-ers of . We also solve the fluctuation iteratively with respectto the mean magnetic field B by assuming that B is small.23

The former and latter expansions are denoted by indices nand m, respectively. For example, the velocity fluctuation canbe written as

ui�k; � = �n=0

�

�m=0

�

nunmi�k; �

− �n=0

�

�m=0

�

n+1iki

k2

�

�XIjunmj�k; � , �27�

where

�

�XIi= exp�− ik · U �

�

�Xiexp�ik · U � . �28�

The second term on the right-hand side of Eq. �27� is addedso that the continuity equation given by Eq. �23� can besatisfied. In this work, we expand the fluctuating field up tothe following order:



ui�k; � = u00i�k; � + u01i�k; � + u10i�k; �

− iki

k2

�

�XIju00j�k; � − i

ki

k2

�

�XIju01j�k; � . �29�

The first through third terms on the right-hand side weretreated in the previous TSDIA, whereas the fourth and fifthterms are newly examined in the present work.

Applying the Fourier transform to the evolution equa-tions for u� and b�, expanding the fluctuations as Eq. �27�,and equating quantities in each order of , we obtain theequations for unmi�k ; � and bnmi�k ; �. For example, theequations for u01i�k ; � and b01i�k ; � are given by

�

� u01i�k; � + �k2u01i�k; �

− 2iMijk�k� pq

�u00j�p; �u01k�q; �

− b00j�p; �b01k�q; �� = − ikjBjb00i�k; � , �30�

�

� b01i�k; � + �Mk2b01i�k; �

− iNijk�k� pq

�u00j�p; �b01k�q; �

− b00j�p; �u01k�q; �� = − ikjBju00i�k; � , �31�

respectively, where

pq

= dpdq�k − p − q� , �32�

Mijk�k� =1

2�kjDik�k� + kkDij�k��, Dij�k� = ij −

kikj

k2 ,

�33�

Nijk�k� = kjik − kkij . �34�

The right-hand sides of Eqs. �30� and �31� can be consideredas external forces for u01i�k ; � and b01i�k ; �, respectively.By introducing four Green’s functions for the velocity andthe magnetic field, we can formally solve the equations foru01i�k ; � and b01i�k ; � as

u01i�k; � = − ikjBj −�

d 1�Gikuu�k; , 1�b00k�k; 1�

+ Gikub�k; , 1�u00k�k; 1�� , �35�

b01i�k; � = − ikjBj −�

d 1�Gikbu�k; , 1�b00k�k; 1�

+ Gikbb�k; , 1�u00k�k; 1�� . �36�

The governing equations for the Green’s functions are givenby Eqs. �A3�–�A6� in the Appendix. In a similar manner,formal solutions for u10i�k ; � and b10i�k ; � can also be ob-tained �see the Appendix for detail�.

C. Calculation of turbulent electromotive force

The turbulent electromotive force can be written as

EMi � �ijk�uj�bk�� = �ijk dk�uj�k; �bk�k�; ��/�k + k�� .

�37�

From Eq. �29�, the correlation in the integrand in Eq. �37� isexpanded as

�ujbk� = �u00jb00k� + �u01jb00k� + �u00jb01k� + �u10jb00k�

+ �u00jb10k� + ¯ . �38�

The correlations of the basic fields u00i and b00i are assumedto be isotropic and are written as

��00i�k; ��00j�k�; ��/�k + k��

= Dij�k�Q��k; , �� +i

2

kk

k2�ijkH��k; , �� , �39�

where Q�� and H�� are basic correlations and � and � rep-resent either u or b. The mean Green’s function is also ex-pressed as

�Gij��k; , �� = ijG

��k; , �� . �40�

For example, the spectra Quu and Huu correspond to the tur-bulent kinetic energy and the turbulent kinetic helicity, re-spectively, as

�u00�2�/2 = dkQuu�k; , � , �41�

�u00� · �00� � = dkHuu�k; , � . �42�

Substituting the formal solutions for the higher-orderterms up to the order given by Eq. �29� into Eq. �38�, we canexpress the turbulent electromotive force in terms of the ba-sic correlations, the mean Green’s functions, and the meanfield. The resulting expression is given by

EM = �B − �J + �� + 2�F� − �� B , �43�

where

� = �1/3��− I�Gbb,Huu� + I�Guu,Hbb� − I�Gbu,Hub�

+ I�Gub,Hbu�� , �44�

�� = �1/3��I�Gbb,Quu� + I�Guu,Qbb� − I�Gbu,Qub�

− I�Gub,Qbu�� , �45�

= �1/3��I�Gbb,Qub� + I�Guu,Qbu� − I�Gbu,Quu�

− I�Gub,Qbb�� , �46�

� = �1/3��I�Gbb,Quu� − I�Guu,Qbb� + I�Gbu,Qub�

− I�Gub,Qbu�� , �47�



� = �� + � = �2/3��I�Gbb,Quu� − I�Gub,Qbu�� . �48�

Here, we introduce the following abbreviation of wavenum-ber and time integration:

I�A,B� = dk −�

d 1A�k; , 1�B�k; , 1� . �49�

The third term on the right-hand side of Eq. �43� representsthe cross-helicity dynamo. In contrast to Eq. �18�, it is pro-portional to the mean absolute vorticity; the model expres-sion derived at this stage is exactly objective. In addition, thefourth term represents the pumping effect and the pumpingvelocity is given by

�A = − �� . �50�

As a result of the present stage of the TSDIA, the turbu-lent electromotive force is expressed in terms of wavenum-ber and time integrals of the basic correlations and the meanGreen’s functions. Such expressions are too complicated touse in practical simulations; some simplification is necessary.For non-MHD turbulence, specific forms of the basic corre-lations and the Green’s functions are often assumed using theKolmogorov spectrum in the wavenumber space and an ex-ponential decay in time. For MHD turbulence, spectra of theturbulent MHD energy and the turbulent residual helicityhave not been established yet although several theoreticaland numerical studies have been done. The form of theGreen’s functions is also difficult to assume. In future work,some forms suggested by theoretical and numerical studiesshould be substituted into Eqs. �44�–�48� to make full use ofthe present result. Instead, in this work, we introduce thefollowing simplification to obtain a one-point closure model.First, we assume that the Green’s functions satisfy

Guu�k; , �� = Gbb�k; , ��,�51�

Gub�k; , �� = Gbu�k; , �� = 0.

Moreover, we replace the time integral with the Green’sfunction by the turbulent time scale T as

−�

d 1Guu�k; , 1�f�k; , 1� = Tf�k; , �, T = C

K

�,

�52�

where C is a nondimensional constant. This simplificationwas also made in the previous TSDIA.23 The resulting formscan be easily integrated in wavenumber and expressed interms of the turbulent statistics in the physical space. As aresult, the transport coefficients appearing in Eq. �43� areexpressed in terms of the turbulent statistics as follows:

� = C�

K

�H, � = C�

K

��K + KR� = 2C�

K

�Ku, �53�

= C

K

�W, � = C�

K

�KR, �54�

where Ku= �u�2� /2 and

C� = C� = C = C��=C /3� . �55�

Each transport coefficient in Eqs. �53� and �54� is expressedas the product of the turbulent time scale K /� and a charac-teristic statistical quantity such as H and Ku. The value of themodel constants C�, C�, C, and C� is expected to be ap-proximately 0.1 and needs to be optimized using numericalsimulations.23

IV. DISCUSSION

A. Comparison with previous TSDIA

We compare the present result with the previous one inmore detail.23 The turbulent electromotive force derived inthe previous TSDIA was given in Eq. �18�. The transportcoefficients appearing in Eq. �18� were expressed in terms ofwavenumber and time integrals as follows:

� = �1/3��I�GS,− Huu + Hbb� − I�GA,Hub − Hbu�� , �56�

� = �1/3��I�GS,Quu + Qbb� − I�GA,Qub + Qbu�� , �57�

= �1/3��I�GS,Qub + Qbu� − I�GA,Quu + Qbb�� , �58�

F = �2/3��I�GS,Qbu� − I�GA,Quu�� . �59�

Here the Green’s functions for Elsässer variables are used;they are related to those used in the present analysis as fol-lows:

GS � �G�� + G��/2 = �Guu + Gbb�/2, �60�

GA � �G�� − G��/2 = �Gub + Gbu�/2. �61�

As mentioned before, the expression for given by Eq. �58�is different from that for F given by Eq. �59�. This defect iscured by the introduction of the frame-invariant form of themean field in the present analysis.

In addition to the difference between and F, the ex-pression for the turbulent diffusivity is altered as shown inEqs. �19� and �53�. We can see that besides the turbulent timescale K /�, the turbulent diffusivity � in Eq. �19� is propor-tional to the turbulent MHD energy K�=Ku+Kb�, whereas inEq. �53� it is proportional to the turbulent kinetic energy Ku.Therefore, if the turbulent magnetic energy Kb�=�b�2� /2� ismuch greater than Ku as is expected in the Earth’s outercore,43 the model expression �19� may overestimate the tur-bulent diffusivity. Rädler et al.21 and Brandenburg andSubramanian22 also showed that the turbulent diffusivity isproportional to Ku. This was a consequence of a cancellationarising from the pressure gradient term. This cancellationcorresponds to the addition of the � term in Eq. �48� in ouranalysis.

The major difference between Eqs. �18� and �43� lies inthe pumping effect −��B appearing in Eq. �43� only; thisterm was not treated in the recent TSDIA.23 In the mean-fielddynamo theory,3 the pumping velocity �A was originally ex-pected to be proportional to the gradient of the turbulentkinetic energy, �Ku. In the K-� model of Yoshizawa,26 �A

has a term proportional to the gradient of the turbulent MHDenergy, �K. In the present work, it is shown that �A is



closely related to the gradient of the turbulent residual en-ergy, �KR, because � is proportional to KR. The same resultwas also obtained theoretically by Rädler et al.21 and Bran-denburg and Subramanian.22 This result is expected to bemore appropriate in the following reason. Both the � effectand the pumping effect originate in the tensorial form �ijBj.The � coefficient corresponding to the isotropic componentof �ij was shown to be expressed in terms of the turbulentresidual helicity H, the difference between the kinetic andcurrent helicities.8 Therefore, it is natural that the pumpingvelocity �A corresponding to the antisymmetric componentof �ij is expressed in terms of the turbulent residual energy,the difference between kinetic and magnetic energies.

We should note that not only the pumping effect but alsothe modification of the turbulent diffusivity are directly re-lated to the quantity � given by Eq. �47�. In Eq. �48� thecoefficient � is expressed as the sum of �� and �; the coef-ficient �� given by Eq. �45� corresponds to � given byEq. �57� in the previous result. If � is neglected in the presentresult, the same expression as the previous result is recov-ered. In fact, the quantity � stems from the fourth and fifthterms in Eq. �29� that are newly treated in the presentTSDIA.

B. Equation for turbulent electromotive force

Although a new model expression for the turbulent elec-tromotive force is derived using the TSDIA, it is not yetjustified by numerical simulation or observation. In this sub-section, to assess the validity of the model expression, weexamine the transport equation for the turbulent electromo-tive force. For non-MHD turbulence it is known that theproduction terms in the transport equation for a correlationsuch as the Reynolds stress are closely related to the eddy-viscosity-type model for the correlation. For example, thecorrelation can be modeled as the product of the productionterms and the turbulent time scale. A similar method calledthe minimal tau approximation was used for MHDturbulence.44,45 This type of approximation was also intro-duced in the TSDIA as the Markovian method.23 Therefore,the investigation of the production terms gives a clue tomodeling the turbulent electromotive force.

The transport equation for EM in a rotating frame can bewritten as

DEM

Dt� �

�t+ U · � + �F � �EM

= PE1 + PE2 + PE3 + RE, �62�

where three production terms are given by

PE1i = Bm�ijk− �uk��uj�

�xm� + �bk�

�bj�

�xm�� , �63�

PE2i = − �ijk��uj�um� � + �bj�bm� ��Bk

�xm, �64�

PE3i = �ijk��bj�um� � + �uj�bm� �� Uk

�xm+ �mkn�Fn� , �65�

and the detailed expression for the remaining part RE isomitted. Each term in Eq. �62� is expressed in an objectiveform. The three production terms PE1, PE2, and PE3 involvethe mean velocity or the mean magnetic field; they representthe effect of the mean field on the turbulent electromotiveforce.

To show the correspondence of the production terms tothe terms in the model expression �43�, we first assume anisotropic homogeneous turbulent field and replace the corre-lations appearing in Eqs. �63�–�65� as follows:

��i�� j�� =1

3ij�� · ��,��i�

��k�

�xj� =

1

6�ijk�� · � � �� .

�66�

Then, we can obtain the following expressions:

PE1 = 13HB, PE2 = − 2

3KJ, PE3 = 23W�� + 2�F� , �67�

which correspond to the first, second, and third terms on theright-hand side of Eq. �43�, respectively. Therefore, it isshown that the � effect, the turbulent diffusivity, and thecross-helicity dynamo results from the corresponding pro-duction terms.

The pumping effect term has not been obtained yet be-cause the coefficient �� requires an inhomogeneous turbu-lent field. Next, instead of the isotropic homogeneous as-sumption �66�, we assume a turbulent field that is locallyisotropic and weakly inhomogeneous. Using the solenoidalcondition for ui� and bi� the production term PE1 can be ex-actly rewritten as

PE1i = �ipq− �uk��up�

�xk� + �bk�

�bp�

�xk��Bq

− �kpq− �uk��up�

�xi� + �bk�

�bp�

�xi��Bq. �68�

Although the summation convention is applied to repeatedindices including p, we consider here only the part in whichp=k. Then, we can see that the second parenthesis in Eq.�68� does not contribute because of �kpq=0 and the correla-tions in the first parenthesis can be approximated as

�up��up�

�xp� =

1

2

�

�xp�up�

2� �1

6

�

�xp�u�2� . �69�

As a result, the production term PE1 can be expressed as

PE1 = − 13 � KR � B , �70�

which corresponds to the pumping effect term. Therefore,not only the three terms already derived in the previous TS-DIA but also the pumping effect term can be justified in thesense that they have corresponding production terms in thetransport equation. In particular, the production term PE1

given by Eq. �63� involves the difference between the veloc-ity correlation �uk��uj� /�xm� and the magnetic-field correlation�bk��bj� /�xm�; this fact accounts for the dependence of thepumping effect on the turbulent residual helicity.



C. Pumping effect in rotating spherical shell

The physical meaning of the pumping effect is illustratedin Fig. 1. Here we consider a uniform magnetic field B0

plotted by solid lines and a spherical region of large residualenergy given by

� = �0 exp�− r2/r02� , �71�

where � is defined as Eq. �54�. The pumping velocity −�� isdirected outward in the radial direction. The electromotiveforce −��B0 and the resulting current �J have a toroidalcomponent. As a result, the magnetic field is transported inthe outward direction as plotted by dashed lines. This is alsocalled the turbulence-induced diamagnetism because themagnetic field in the central region is decreased.3

To demonstrate the behavior of the pumping effect in amore realistic problem, we investigate the magnetic field in arotating spherical shell similar to the Earth’s outer core.Hamba32 numerically simulated the axisymmetric meanmagnetic field in a rotating spherical shell using a Reynolds-averaged MHD turbulence model. In the model, the turbulentelectromotive force is given by

EM = �B − �J , �72�

where

� = C�f�

K

�H, � = C�f�

K2

�. �73�

Here, model constants are set to C�=C�=0.09 and correctioncoefficients f� �0� f��1� and f� �0� f��1� are introducedto satisfy the realizability condition for the turbulent electro-motive force, �EM��K. To concentrate on the estimate of the

pumping effect, the turbulent diffusivity � is modeled in thesame form as that of Hamba;32 the modified expression �53�is not adopted here. In addition to the induction equation forthe mean magnetic field, the transport equations for K, �, andH are solved to evaluate the transport coefficients � and �.In the simulation, physical quantities are nondimensionalizedby the typical velocity U0�=5�10−4 m s−1�, the radius of theouter boundary rout�=3.48�106 m�, and the fluid density��=1.09�104 kg m−3�. The magnetic field and the timescale are then normalized by ��0U0=0.585 G and rout /U0

=221 year, respectively. The ratio of radii of the innerand outer boundaries is set to rin /rout=0.35. The outer re-gions at 0�r�rin and at r�rout are assumed to be insula-tors. The angular velocity of the system rotation is set to�F=5�105. The time evolution of the magnetic field is cal-culated until a steady state is reached �see Hamba32 for de-tails�.

First, we briefly explain the result of the simulation ofHamba32 in which the pumping effect is not treated. Figure 2shows the profiles of the mean magnetic field. The poloidalfield Bp= �Br ,B�� plotted in Fig. 2�a� is approximately a di-pole field whereas the toroidal field B� plotted in Fig. 2�b� isnegative �positive� in the northern �southern� hemisphere. Itwas shown in Hamba32 that the dipole field is sustained ow-ing to the �2 dynamo; that is, the toroidal and poloidal fieldsinduce each other via the � effect. Figure 3 shows the pro-files of the turbulent MHD energy K and the turbulent re-sidual helicity H. The turbulent MHD energy is produced bythe buoyancy effect. Since the value of K has a maximum atr�rout and at �=� /2, the gradient �K /�� is positive �nega-tive� in the northern �southern� hemisphere. The transportequation for H involves the production term −�F ·�K whose

FIG. 1. Pumping effect in a spherical region of large residual energy. Solidlines represent the original uniform magnetic field and dashed lines stand forthe magnetic field transported by the pumping effect.

FIG. 2. Profiles of the mean magnetic field in a rotating spherical shellwithout pumping effect; �a� poloidal field Bp and �b� toroidal field B�.



value is positive �negative� in the northern �southern� hemi-sphere. As a result, the turbulent residual helicity H of thesame sign as −�F ·�K is produced as shown in Fig. 3�b�.The large value of H enhances the � effect.

Next, to examine the pumping effect we add the pump-ing term to Eq. �72� as follows:

EM = �B − �J − �� B , �74�

where

� = C�f�

K

�KR. �75�

Here we adopt C�=0.09 and f�= f�. To accurately predict theprofile of KR, we need to solve its transport equation. How-ever, its model equation including model constants has notbeen established yet.40 In order to examine a qualitative ef-fect of the pumping term, we assume here that KR is propor-tional to K as follows:

KR = − RK , �76�

where R is a small nondimensional constant; R=0.1 isadopted here. A minus sign is included in Eq. �76� becauseKR is expected to be negative in the Earth’s outer core.43 Weshould note that Eq. �76� is just an assumption made for aqualitative estimate; the transport equation for KR needs to bemodeled in future work.

To evaluate the effect of the pumping term −��B onthe magnetic field, we calculate the difference between theoriginal and newly obtained magnetic fields

�B = B1 − B0, �77�

where B0 is the original field shown in Fig. 2 and B1 is thefield obtained by solving the same model as that of Hamba32

except for the pumping term with R=0.1.Figure 4 shows the difference �B between the two mag-

netic fields. The profile of the poloidal field plotted in Fig.4�a� is simple; it is approximately a dipole field and thedirection is opposite to the original field. It is shown that thepumping effect reduces the original dipole field in this case.On the other hand, the toroidal field plotted in Fig. 4�b� israther complicated. In the region near the equator, the differ-ence �B is positive �negative� in the northern �southern�hemisphere; these signs are opposite to the correspondingoriginal field. In this region the toroidal field also decreasesowing to the pumping effect.

These profiles of �B can be roughly explained by con-sidering the directions of the pumping velocity −�� and ofthe original magnetic field B0 as illustrated in Fig. 5. In Fig.5�a� we consider the toroidal component of the pumpingterm, �−��B�� in the region near the equator. The pump-ing velocity −�� points in the same direction as �K, whereasthe poloidal field Bp is directed opposite to the system rota-tion �F. The pumping term �−��B�� is then positive inthis region. As a result, the pumping term induces a positivetoroidal current �J� corresponding to the dipole field �Bp

plotted in Fig. 4�a�. Therefore, the magnetic field �Bp isinduced in the same manner as the example illustrated in Fig.1. On the other hand, in Fig. 5�b� we consider the poloidalcomponent of the pumping effect, �−��B�p. The pumpingvelocity −�� is in the positive �negative� � direction and thetoroidal field B� has a negative �positive� value in the north-

FIG. 3. Profiles of turbulent statistics in a rotating spherical shell withoutpumping effect; �a� turbulent MHD energy K and �b� turbulent residualhelicity H.

FIG. 4. Magnetic-field difference due to the pumping effect with R=0.1;�a� poloidal field �Bp and �b� toroidal field �B�.



ern �southern� hemisphere. The pumping term �−��B�p

then points in the negative r direction in both the northernand southern hemispheres and induces �Jp in the same di-rection. Such a current in the negative r direction near theequator can account for the negative gradient of the toroidalfield, ��B� /��, in the region near the equator shown in Fig.4�b�. Therefore, in this simulation of a rotating sphericalshell, the pumping term has the effect of reducing the origi-nal magnetic field induced by the � effect. In future work,we need to solve the transport equation for KR to evaluate thepumping effect more accurately.

V. CONCLUSIONS

To predict the mean magnetic field in MHD turbulenceat high Reynolds number, it is necessary to model the turbu-lent electromotive force. In this work, a new model expres-sion for the turbulent electromotive force was derived usingthe TSDIA. We improved the formulation of the TSDIA sothat obtained expressions can satisfy the frame-invarianceunder rotating transformations. We adopted four Green’sfunctions to take into account the effect of the mean field onthe fluctuations more accurately. The resulting expression forthe turbulent electromotive force consists of terms represent-ing the � effect, the turbulent diffusivity, the cross-helicitydynamo, and the pumping effect. The transport coefficientsin the turbulent electromotive force are expressed in terms ofturbulent statistics such as the turbulent MHD energy and theturbulent residual helicity. It was shown that the turbulentdiffusivity is proportional to the turbulent kinetic energy.Moreover, the pumping velocity was shown to be closelyrelated to the gradient of the turbulent residual energy. Theseresults confirm earlier findings of Rädler et al.21 and Bran-denburg and Subramanian22 using the tau approximation.

To assess the validity of the model expression, we exam-ined the transport equation for the turbulent electromotiveforce. Using the isotropic homogeneous assumption, thethree production terms were shown to correspond to theterms representing the � effect, the turbulent diffusivity, andthe cross-helicity dynamo. Using the weakly inhomogeneousassumption, the pumping effect term was also derived fromone of the production terms. To demonstrate the pumpingeffect, we simulated the magnetic field in a rotating sphericalshell. It was shown that in this case the pumping term has theeffect of reducing the magnetic field induced by the � dy-namo. In future work, the transport equation for the turbulentresidual energy should be solved to evaluate the pumpingeffect more accurately. It is also important to validate thepumping effect using data obtained from three-dimensionalsimulations of MHD turbulence.

ACKNOWLEDGMENTS

F.H. would like to thank Dr. N. Yokoi for valuable dis-cussion on the turbulent residual energy.

This work was partially supported by the Grant-in-Aidfor Scientific Research of Japan Society for the Promotion ofScience �19560159�.

APPENDIX: GOVERNING EQUATIONS FOR GREEN’SFUNCTIONS

The equations for the basic fields u00i and b00i are writtenas

�

� u00i�k; � + �k2u00i�k; �

− iMijk�k� pq

�u00j�p; �u00k�q; �

− b00j�p; �b00k�q; �� = 0, �A1�

�

� b00i�k; � + �Mk2b00i�k; �

− iNijk�k� pq

u00j�p; �b00k�q; � = 0. �A2�

Since these equations are the same as those for isotropicturbulence, we assume that the basic fields u00i and b00i areisotropic and that the anisotropic and inhomogeneous effectscan be incorporated through higher-order terms such as u01i

and u10i.The equations for u01i and b01i are given by Eqs. �30�

and �31�, respectively. The left-hand sides of Eqs. �30� and�31� are linear with respect to u01i and b01i. The right-handsides of Eqs. �30� and �31� can be formally considered as

FIG. 5. Pumping effect in a rotating spherical shell; �a� poloidal componentand �b� toroidal component.



external forces for u01i and b01i. We then introduce theGreen’s functions satisfying the following system of equa-tions:

�

� Gij

uu�k; , �� + �k2Gijuu�k; , ��

− 2iMikm�k� pq

�u00k�p; �Gmjuu�q; , ��

− b00k�p; �Gmjbu�q; , �� = ij� − �� , �A3�

�

� Gij

bu�k; , �� + �Mk2Gijbu�k; , ��

− iNikm�k� pq

�u00k�p; �Gmjbu�q; , ��

− b00k�p; �Gmjuu�q; , �� = 0, �A4�

and

�

� Gij

ub�k; , �� + �k2Gijub�k; , ��

− 2iMikm�k� pq

�u00k�p; �Gmjub�q; , ��

− b00k�p; �Gmjbb�q; , �� = 0, �A5�

�

� Gij

bb�k; , �� + �Mk2Gijbb�k; , ��

− iNikm�k� pq

�u00k�p; �Gmjbb�q; , ��

− b00k�p; �Gmjub�q; , �� = ij� − �� . �A6�

The Green’s functions Gijuu and Gij

bu represent the response ofthe velocity and the magnetic field, respectively, to a distur-bance at time � in the velocity equation. On the other hand,

Gijub and Gij

bb represent the response to a disturbance in themagnetic field equation. Using these Green’s functions, wecan obtain formal solutions for u01i and b01i given by Eqs.�35� and �36�, respectively. In a similar manner, the solutionsfor u10i and b10i can be written as

u10i�k; � = −�

d 1Gijuu�k; , 1�Djk�k�

�Bk

�Xmb00m�k; 1� − Djk�k� �Uk

�Xm+ �mkn�0n�u00m�k; 1� + Bk

�

�XIkb00j�k; 1�

− Djk�k�D

DTIu00k�k; 1�� +

−�

d 1Gijub�k; , 1�− Djk�k�

�Bk

�Xmu00m�k; 1� + Djk�k� �Uk

�Xm+ �mkn�0n�b00m�k; 1�

+ Bk�

�XIku00j�k; 1� − Djk�k�

D

DTIb00k�k; 1�� , �A7�

b10i�k; � = −�

d 1Gijbu�k; , 1�Djk�k�

�Bk

�Xmb00m�k; 1� − Djk�k� �Uk

�Xm+ �mkn�0n�u00m�k; 1� + Bk

�

�XIkb00j�k; 1�

− Djk�k�D

DTIu00k�k; 1�� +

−�

d 1Gijbb�k; , 1�− Djk�k�

�Bk

�Xmu00m�k; 1� + Djk�k� �Uk

�Xm+ �mkn�0n�b00m�k; 1�

+ Bk�

�XIku00j�k; 1� − Djk�k�

D

DTIb00k�k; 1�� , �A8�

where

D

DTI= exp�− ik · U �

D

DTexp�ik · U � . �A9�

Using solutions �35�, �36�, �A7�, and �A8� we can express theturbulent electromotive force in terms of the basic correla-tions, the mean Green’s functions, and the mean field asshown in Eqs. �43�–�48�.

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