Turbulent transport coefficients and residual energy in mean-field dynamo theory

PHYSICS OF PLASMAS 15, 022302 2008 Turbulent transport coefficients and residual energy in mean-field dynamo theory Fujihiro Hamba 1,a and Hisanori Sato 2 1 Institute of Industrial Science, University of Tokyo, omaba, Meguro-ku, Tokyo 153-8505, Japan 2 Japan Patent Office, asumigaseki, 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 a large-scale magnetic field in magnetohydrodynamic turbulence at high Reynolds number. Using a statistical theory for inhomogeneous turbulence, model expressions for transport coefficients appearing in the turbulent electromotive force are derived including the coefficient and the turbulent diffusivity. In particular, as one of the dynamo effects, the pumping effect is investigated and a model expression for the pumping term is obtained. It is shown that the pumping velocity is closely related to the gradient of the turbulent residual energy, or the difference between the turbulent kinetic and magnetic energies. The production terms in the transport equation for the turbulent electromotive force are also examined and the validity of the model expression is assessed by comparing with earlier results concerning the isotropic coefficient. The mean magnetic field in a rotating spherical shell is calculated using a turbulence model to demonstrate the pumping effect. 2008 American Institute of Physics. DOI: 10.1063/1.2839767 I. INTRODUCTION pic helical turbulence using a helical forcing to estimate the coefficients of the effect and of the turbulent diffusivity. Ossendrijver et al. 16 performed a DNS of magnetoconvection to examine the dependence of the coefficient on the rotation and the magnetic field strength. In addition to the effect, it is known that the pumping effect term can be derived from the tensorial form ij B j for the turbulent electromotive force B j is the mean magnetic field. 3 The mean magnetic field can be transported through the turbulence medium with the pumping velocity even though there is no mean motion. The effect and the pumping effect originate in the isotropic and antisymmetric components of ij, respectively. Rädler et al. 21 and Brandenburg and Subramanian 22 theoretically investigated turbulent transport coefficients including the pumping effect. The coefficients of the pumping effect were also evaluated using DNS. 17,20 In order to predict the mean magnetic field in realistic MHD turbulent flows, it is necessary to evaluate the turbulent transport coefficients such as the coefficient and the turbulent diffusivity. For a specific problem such as the solar magnetic field, some appropriate spatial distribution of the transport coefficients can be prescribed. However, in developing a general MHD turbulence model, model expressions for the transport coefficients are necessary to close the system of mean-field equations. For non-mhd turbulence, the Reynolds stress is often modeled using the eddy viscosity approximation to predict the mean velocity field. 23 To evaluate the turbulent viscosity, two-equation models have been widely used such as the - model that treats the transport equations for the turbulent kinetic energy and its dissipation rate. A statistical theory called the two-scale directinteraction approximation TSDIA was developed to theoretically derive and improve the eddy viscosity model for non-mhd turbulence. 24 The TSDIA was shown to successfully derive a nonlinear eddy-viscosity model. For MHD tura Electronic mail: hamba@iis.u-tokyo.ac.jp. The generation of a large-scale magnetic field by a turbulent flow of an electrically conducting fluid is an important problem in astrophysics and plasma physics. In general, turbulent velocity fluctuations enhance the effective magnetic diffusivity. Dynamo action is necessary to generate and sustain a large-scale magnetic field against the turbulent diffusivity. The effect in the mean-field dynamo theory is well known as one of the dynamo mechanisms and has been applied to solar and Earth s magnetic fields. 1 3 The dynamo was also invoked to explain the sustainment of the magnetic field in controlled fusion devices such as the reversed field pinch RFP. 4,5 From the physical point of view, it is very interesting and important to investigate the dynamo effect commonly seen in various magnetohydrodynamic MHD turbulent flows. Fundamental properties of MHD turbulence have been studied theoretically and numerically. Owing to the effect of Alfvén waves, the energy spectrum of isotropic homogenous MHD turbulence is expected to show k 3/2 wavenumber dependence in contrast to the k 5/3 spectrum of non-mhd turbulence. 6,7 The spectra of kinetic and magnetic energies have been investigated using turbulence theories such as the eddy-damped quasinormal Markovian approximation and the Lagrangian renormalized approximation. 8 10 Moreover, direct numerical simulation DNS of MHD turbulence is a powerful tool to examine the energy spectrum. 11 14 Haugen et al. 13 showed that the asymptotic spectrum is suggested to be k 5/3. Mason et al. 14 evaluated the velocity and magneticfield alignment to provide an explanation for the k 3/2 spectrum in the plane perpendicular to the guiding magnetic field. Dynamo action such as the effect was also studied using DNS. 15 20 Brandenburg 15 carried out a DNS of isotro- 1070-664X/2008/152/022302/12/$23.00 15, 022302-1 2008 American Institute of Physics

022302-2 F. Hamba and H. Sato Phys. Plasmas 15, 022302 2008 bulence, a few attempts including the TSDIA have been made to construct a model for transport equations. 25 29 The TSDIA is expected to be useful for deriving a universal MHD turbulence model. Using the TSDIA, Yoshizawa 26 examined the pumping effect and the turbulent diffusivity; he extended the - model to MHD turbulence is the sum of the turbulent kinetic and magnetic energies. In his work, the effect was not examined because the coefficient is a pseudoscalar and cannot be expressed in terms of scalars and only. Yoshizawa and Hamba 30 paid attention to the turbulent residual helicity as a pseudoscalar in modeling the coefficient; they proposed a three-equation model by adopting the turbulent residual helicity as a third model variable. The three-equation model was used to simulate the magnetic fields in the RFP Ref. 31 and in the Earth. 32 In addition to the effect parallel to the mean magnetic field, Yoshizawa 33 derived a new dynamo term called the cross-helicity dynamo parallel to the mean vorticity. The cross-helicity dynamo has been applied to several turbulent flows in astrophysical and engineering phenomena. 23,34 38 Recently, Yokoi 39 paid attention to the turbulent residual energy, the difference between the turbulent kinetic and magnetic energies, to investigate the solar-wind turbulence. A turbulence model for the turbulent residual energy equation was solved to evaluate the radial dependence of the turbulent quantities in solar wind. 40 However, the effect of the turbulent residual energy on the turbulent electromotive force has not been fully explored yet. It must be interesting to investigate the mean-field dynamo theory in more detail using the TSDIA and to examine the effect of the turbulent residual energy. In this work, using the TSDIA we theoretically calculate a model expression for the turbulent electromotive force. In the present analysis, the formulation is improved so that the frame-invariance of the model expression under rotating transformations can be satisfied. 41 The Green s functions that represent the response of the velocity and magnetic-field fluctuations to a disturbance at a previous time are treated in more accurate manner. In the perturbation expansion of the TSDIA, we calculate a few additional terms for the turbulent electromotive force compared to the previous work. 23 We derive the effect, the pumping effect, the turbulent diffusivity, and the cross-helicity dynamo effect, and we examine the transport coefficients appearing in these terms. This paper is organized as follows: In Sec. II, we explain the turbulent dynamo and diffusivity terms appearing in the turbulent electromotive force. In Sec. III, we apply the TSDIA to derive a model expression for the turbulent electromotive force. In Sec. IV, we compare the present result with the previous work. To assess the validity of the model expression, we examine the transport equation for the turbulent electromotive force. To demonstrate the pumping effect, we apply the model to the simulation of the magnetic field in a rotating spherical shell. Concluding remarks will be given in Sec. V. II. MEAN-FIELD EQUATIONS AND DYNAMO EFFECTS A. Mean-field equations In this paper we adopt Alfvén velocity units and replace the magnetic field b/ 0 b, the electric current density j/ /0 j, and the electric field e/ 0 e, is the fluid density and 0 is the magnetic permeability. The Navier-Stokes equation for a viscous, incompressible, electrically conducting fluid and the induction equation for the magnetic field in a rotating system are written as follows: u t = uu bb p M + 2 u 2 F u, u =0, 1 2 b t = e, e = u b + Mj, 3 b =0. Here, u is the velocity, p M =p+ F x 2 /2+b 2 /2 is the total pressure, F is the system rotation rate, is the kinematic viscosity, M is the magnetic diffusivity, and j=b. We use ensemble averaging to divide a quantity into the mean and fluctuating parts as f = F + f, F f, 5 f =u,b, p M,e,j, and =u is the vorticity. The evolution equations for the mean velocity U and the mean magnetic field B are written as U t = UU BB R P M + 2 U 2 F U, U =0, 4 6 7 B t = E, E = U B E M + M J, 8 B =0, R ij =u i u j b i b j is the Reynolds stress and E M =ub is the turbulent electromotive force. In order to calculate the time evolution of the mean velocity and the mean magnetic field, we need to evaluate R ij and E M ; some modeling is necessary for the two quantities. Many turbulence models for the Reynolds stress have been developed for non-mhd turbulence and some of them can be extended to MHD turbulence. On the other hand, the turbulent electromotive force is treated only in MHD turbulence. In this paper we focus on modeling the turbulent electromotive force to investigate the dynamo effect. B. Turbulent electromotive force In the mean-field dynamo theory, the turbulent electromotive force can be expressed in a tensorial form 9

022302-3 Turbulent transport coefficients Phys. Plasmas 15, 022302 2008 E Mi = ij B j + ijk B j x k, 10 2 A = C 1 C 2 2, = C 2 17 summation convention is used for repeated indices. If isotropic components of the coefficients ij = ij, ijk = ijk 11 ij is the ronecker delta symbol and ijk is the unit alternating tensor are considered, the turbulent electromotive force can be written as E M = B J. 12 The first term on the right-hand side of Eq. 12 represents the effect, as in the second term is the turbulent diffusivity. For homogeneous turbulence an expression for the coefficient was derived using the spectrum of the kinetic helicity u in the wavenumber space in the meanfield theory. 2,3 Pouquet et al. 8 showed that the coefficient can be expressed in terms of the spectrum of the turbulent residual helicity H= u +b j. The coefficient was also evaluated in the DNS of isotropic helical turbulence. 15 17,20 In general, the coefficient ij can be decomposed into the symmetric and antisymmetric components S ij = ij + ji /2, A ij = ij ji /2. 13 The antisymmetric component can be expressed in terms of a vector A defined as A i = ijk A jk /2. 14 If the antisymmetric component of ij is incorporated in addition to the isotropic components, Eq. 10 can be written as E M = B + A B J. 15 The term A B represents the pumping effect; that is, the mean magnetic field can be transported through the turbulence medium with the velocity A even though there is no mean motion. 3 The pumping velocity A is expected to be proportional to u 2 and the pumping effect can be considered as the turbulence-induced diamagnetism. 3 Using the tau approximation Rädler et al. 21 and Brandenburg and Subramanian 22 showed that the pumping velocity is proportional to the gradient of the difference between the kinetic and magnetic energies, u 2 b 2. Using the TSDIA, Yoshizawa 26 proposed a two-equation MHD turbulence model. This model is an extension of the non-mhd - model to MHD turbulence. As basic variables the turbulent MHD energy =u 2 +b 2 /2 and its dissipation rate =2ss ij + ij M j 2 were adopted s=u ij i /x j +u/x j i /2 is the strain-rate fluctuation. The turbulent electromotive force was modeled as E M = A B J, 16 and C 1, C 2, and C are nondimensional model constants. The pumping velocity A depends on the gradients of the turbulent MHD energy and its dissipation rate. The turbulent diffusivity is modeled in a similar form to the turbulent viscosity for non-mhd turbulence. Using the pumping term, Yoshizawa 26 discussed the mechanism of the sustainment of the mean magnetic field in the RFP. However, this model does not involve the well-known effect because a pseudoscalar cannot be expressed in terms of scalars and only. Adopting the turbulent residual helicity H as a third model variable, Yoshizawa and Hamba 30 proposed a threeequation model for the effect. The coefficient in Eq. 12 can be modeled in terms of H in a straightforward manner because H is also a pseudoscalar. Since then, the effect has been treated instead of the pumping effect in MHD studies using the TSDIA. In addition, Yoshizawa 33 pointed out the importance of the cross helicity W=u b and proposed a new dynamo term proportional to the mean vorticity.asa result, the turbulent electromotive force in a rotating frame can be modeled as 23 E M = B J + +2 F F, = C H, = C 2, = F = C W. 18 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 the one-point closure approximation, the two coefficients are modeled in the same form as shown in Eq. 19 and the cross-helicity dynamo term can be written as +2 F. Since the turbulent electromotive force is objective, or frame-invariant under rotating transformations, its model expression should also be objective. 41 If = F, the model expression given by Eq. 18 is objective because the mean absolute vorticity +2 F is objective. However, if F, then Eq. 18 cannot be objective. Therefore, at the beginning of the formulation the TSDIA must have some inaccurate procedure that violates the frame-invariance. In this work we improve the TSDIA formulation to derive model expressions satisfying the frame-invariance. On the other hand, in Eq. 19 the turbulence intensity is expressed in terms of, the sum of the turbulent kinetic and magnetic energies. Recently, Yokoi 39 pointed out the importance of the turbulent residual energy R =u 2 b 2 /2 in the solar-wind turbulence and investigate its evolution equation. A four-equation model treating,, W, and R was proposed and solved to evaluate turbulent statistics in the solar wind. 40 In the present work, we will examine the relation of the residual energy to the turbulent electromotive force.

022302-4 F. Hamba and H. Sato Phys. Plasmas 15, 022302 2008 III. THEORETICAL FORMULATION In this section we briefly explain the procedure of the TSDIA for deriving a model expression for the turbulent electromotive force. Compared with the previous TSDIA, we improve the formulation in the two respects: the treatment of the rotating frame and of the Green s functions. In the previous method, the fluctuating field is expanded in terms of the mean-field gradient such as U i /x j and of the system rotation rate F independently. This independent treatment leads to the difference between the forms of and F in the wavenumber space as will be shown in Sec. IV. In the present method, we expand the fluctuating field so that the mean field can always take an objective form such as the mean absolute vorticity. In addition, in the previous formulation, two Green s functions Ĝ and Ĝ are treated =u+b and =u b are Elsässer variables. The Green s function Ĝ Ĝ represents the response of to a disturbance in the equation. The Green s functions Ĝ and Ĝ are assumed to be zero although the cross response can exist. In the present formulation, we adopt primitive variables u and b and treat four Green s functions Ĝ uu, Ĝ bb, Ĝ ub, and Ĝ bu. Although the choice of the variables is arbitrary, the neglect of the Green s functions for the cross response may lead to a different result. We expect that the present method can give more accurate expressions. 0 = F /, D u i DT = u i T + U u i j + jik 0k u. X j j 24 We should note that the mean-field gradient such as U i /X j is of the order. To keep the terms involving the mean velocity gradient objective, we assume that the system rotation rate is also of the order and is set to F = 0. The parenthesis involving the mean velocity gradient in Eq. 22 can then be written as the sum of the mean strain-rate and absolute-vorticity tensors which are both objective, U i + jik 0k = 1 X j 2 U i + U j X j X i + 1 2 U i X j U j X i +2 jik 0k. 25 In addition, the material derivative term such as Du i /DT =u i /T+U j u i /X j violates the frame-invariance. To make the term objective, we adopt the co-rotational derivative D u i /DT defined as Eq. 24. The co-rotational derivative of a vector is objective and adequately describes an unsteady behavior of the fluctuating field. 41,42 As a result, each term on the right-hand side of Eq. 22 can be expressed in an objective form. The induction equation for b i can be derived in a similar manner. This formulation guarantees the frameinvariance of model expressions derived later. A. Two-scale variables and frame-invariance First, we introduce two space and time variables using a scale-expansion parameter as =x, X=x, =t, T=t. 20 Here, the fast variables and describe the rapid variation of the fluctuating field as the slow variables X and T describe the slow variation of the mean field. A quantity f can be written as f = FX;T + f,x;,t. 21 The equations for the velocity fluctuation u i can be written as u i + U u i j + uu j j i bb j i + p M 2 u i j i j j b i B j = b B i j u j X j j U i + jik X j 0k b i + B j X j D u i DT uu X j i bb j i uu j i bb j i p M j X i, u i + u i =0, i X i 22 23 B. Perturbation expansion and Green s function Using the Fourier transform with respect to we express a fluctuating field f as f,x;,t = dkfk,x;,texp ik U. 26 Hereafter, the dependence of fk,x;,t on X and T is not written explicitly. We expand the fluctuation fk; in powers of. We also solve the fluctuation iteratively with respect to the mean magnetic field B by assuming that B is small. 23 The former and latter expansions are denoted by indices n and m, respectively. For example, the velocity fluctuation can be written as u i k; = n=0 m=0 n u nmi k; n=0 m=0 n+1 i k i k 2 u nmj k;, X Ij 27 = exp ik U expik U. 28 X Ii X i The second term on the right-hand side of Eq. 27 is added so that the continuity equation given by Eq. 23 can be satisfied. In this work, we expand the fluctuating field up to the following order:

022302-5 Turbulent transport coefficients Phys. Plasmas 15, 022302 2008 u i k; = u 00i k; + u 01i k; + u 10i k; i k i k 2 u 00j k; i k i X Ij k 2 u 01j k;. 29 X Ij The first through third terms on the right-hand side were treated in the previous TSDIA, as the fourth and fifth terms are newly examined in the present work. Applying the Fourier transform to the evolution equations for u and b, expanding the fluctuations as Eq. 27, and equating quantities in each order of, we obtain the equations for u nmi k; and b nmi k;. For example, the equations for u 01i k; and b 01i k; are given by u 01ik; + k 2 u 01i k; 2iM ijk u 00j p;u 01k q; kpq C. Calculation of turbulent electromotive force The turbulent electromotive force can be written as E Mi ijk u j b k = ijk dku j k;b k k;/k + k. 37 From Eq. 29, the correlation in the integrand in Eq. 37 is expanded as u j b k = u 00j b 00k + u 01j b 00k + u 00j b 01k + u 10j b 00k + u 00j b 10k +. 38 The correlations of the basic fields u 00i and b 00i are assumed to be isotropic and are written as 00i k; 00j k;/k + k b 00j p;b 01k q; = ik j B j b 00i k;, 30 = D ij kq k;, + i 2 k 2 ijkh k;,, k k 39 b 01ik; + M k 2 b 01i k; in ijk u 00j p;b 01k q; kpq b 00j p;u 01k q; = ik j B j u 00i k;, respectively, 31 Q and H are basic correlations and and represent either u or b. The mean Green s function is also expressed as Ĝ ij k;, = ij G k;,. 40 For example, the spectra Q uu and H uu correspond to the turbulent kinetic energy and the turbulent kinetic helicity, respectively, as pq =dpdqk p q, 32 u 00 /2= 2 dkq uu k;,, 41 M ijk k = 1 2 k jd ik k + k k D ij k, N ijk k = k j ik k k ij. D ij k = ij k ik j k 2, 33 34 The right-hand sides of Eqs. 30 and 31 can be considered as external forces for u 01i k; and b 01i k;, respectively. By introducing four Green s functions for the velocity and the magnetic field, we can formally solve the equations for u 01i k; and b 01i k; as u 01i k; = ik j B j b 01i k; = ik j B j d 1 Ĝ uu ik k;, 1 b 00k k; 1 + Ĝ ub ik k;, 1 u 00k k; 1, d 1 Ĝ bu ik k;, 1 b 00k k; 1 35 u 00 00 = dkh uu k;,. 42 Substituting the formal solutions for the higher-order terms up to the order given by Eq. 29 into Eq. 38, wecan express the turbulent electromotive force in terms of the basic correlations, the mean Green s functions, and the mean field. The resulting expression is given by E M = B J + +2 F B, 43 = 1/3 IG bb,h uu + IG uu,h bb IG bu,h ub + IG ub,h bu, 44 = 1/3IG bb,q uu + IG uu,q bb IG bu,q ub IG ub,q bu, 45 + Ĝ bb ik k;, 1 u 00k k; 1. 36 The governing equations for the Green s functions are given by Eqs. A3 A6 in the Appendix. In a similar manner, formal solutions for u 10i k; and b 10i k; can also be obtained see the Appendix for detail. = 1/3IG bb,q ub + IG uu,q bu IG bu,q uu IG ub,q bb, = 1/3IG bb,q uu IG uu,q bb + IG bu,q ub IG ub,q bu, 46 47

022302-6 F. Hamba and H. Sato Phys. Plasmas 15, 022302 2008 = + = 2/3IG bb,q uu IG ub,q bu. 48 Here, we introduce the following abbreviation of wavenumber and time integration: IA,B = d 1 Ak;, 1 Bk;, 1. 49 dk The third term on the right-hand side of Eq. 43 represents the cross-helicity dynamo. In contrast to Eq. 18, it is proportional to the mean absolute vorticity; the model expression derived at this stage is exactly objective. In addition, the fourth term represents the pumping effect and the pumping velocity is given by A =. 50 As a result of the present stage of the TSDIA, the turbulent electromotive force is expressed in terms of wavenumber and time integrals of the basic correlations and the mean Green s functions. Such expressions are too complicated to use in practical simulations; some simplification is necessary. For non-mhd turbulence, specific forms of the basic correlations and the Green s functions are often assumed using the olmogorov spectrum in the wavenumber space and an exponential decay in time. For MHD turbulence, spectra of the turbulent MHD energy and the turbulent residual helicity have not been established yet although several theoretical and numerical studies have been done. The form of the Green s functions is also difficult to assume. In future work, some forms suggested by theoretical and numerical studies should be substituted into Eqs. 44 48 to make full use of the present result. Instead, in this work, we introduce the following simplification to obtain a one-point closure model. First, we assume that the Green s functions satisfy G uu k;, = G bb k;,, G ub k;, = G bu k;, =0. 51 Moreover, we replace the time integral with the Green s function by the turbulent time scale T as d 1 G uu k;, 1 fk;, 1 = T fk;,, T = C, 52 C is a nondimensional constant. This simplification was also made in the previous TSDIA. 23 The resulting forms can be easily integrated in wavenumber and expressed in terms of the turbulent statistics in the physical space. As a result, the transport coefficients appearing in Eq. 43 are expressed in terms of the turbulent statistics as follows: = C H, = C W, u =u 2 /2 and = C + R =2C u, = C R, 53 54 C = C = C = C =C /3. 55 Each transport coefficient in Eqs. 53 and 54 is expressed as the product of the turbulent time scale / and a characteristic statistical quantity such as H and u. The value of the model constants C, C, C, and C is expected to be approximately 0.1 and needs to be optimized using numerical simulations. 23 IV. DISCUSSION A. Comparison with previous TSDIA We compare the present result with the previous one in more detail. 23 The turbulent electromotive force derived in the previous TSDIA was given in Eq. 18. The transport coefficients appearing in Eq. 18 were expressed in terms of wavenumber and time integrals as follows: = 1/3IG S, H uu + H bb IG A,H ub H bu, = 1/3IG S,Q uu + Q bb IG A,Q ub + Q bu, = 1/3IG S,Q ub + Q bu IG A,Q uu + Q bb, F = 2/3IG S,Q bu IG A,Q uu. 56 57 58 59 Here the Green s functions for Elsässer variables are used; they are related to those used in the present analysis as follows: G S G + G /2=G uu + G bb /2, 60 G A G G /2=G ub + G bu /2. 61 As mentioned before, the expression for given by Eq. 58 is different from that for F given by Eq. 59. This defect is cured by the introduction of the frame-invariant form of the mean field in the present analysis. In addition to the difference between and F, the expression for the turbulent diffusivity is altered as shown in Eqs. 19 and 53. We can see that besides the turbulent time scale /, the turbulent diffusivity in Eq. 19 is proportional to the turbulent MHD energy = u + b, as in Eq. 53 it is proportional to the turbulent kinetic energy u. Therefore, if the turbulent magnetic energy b =b 2 /2 is much greater than u as is expected in the Earth s outer core, 43 the model expression 19 may overestimate the turbulent diffusivity. Rädler et al. 21 and Brandenburg and Subramanian 22 also showed that the turbulent diffusivity is proportional to u. This was a consequence of a cancellation arising from the pressure gradient term. This cancellation corresponds to the addition of the term in Eq. 48 in our analysis. The major difference between Eqs. 18 and 43 lies in the pumping effect B appearing in Eq. 43 only; this term was not treated in the recent TSDIA. 23 In the mean-field dynamo theory, 3 the pumping velocity A was originally expected to be proportional to the gradient of the turbulent kinetic energy, u.inthe- model of Yoshizawa, 26 A has a term proportional to the gradient of the turbulent MHD energy,. In the present work, it is shown that A is

022302-7 Turbulent transport coefficients Phys. Plasmas 15, 022302 2008 closely related to the gradient of the turbulent residual energy, R, because is proportional to R. The same result was also obtained theoretically by Rädler et al. 21 and Brandenburg and Subramanian. 22 This result is expected to be more appropriate in the following reason. Both the effect and the pumping effect originate in the tensorial form ij B j. The coefficient corresponding to the isotropic component of ij was shown to be expressed in terms of the turbulent residual helicity H, the difference between the kinetic and current helicities. 8 Therefore, it is natural that the pumping velocity A corresponding to the antisymmetric component of 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 also the modification of the turbulent diffusivity are directly related to the quantity given by Eq. 47. InEq.48 the coefficient is expressed as the sum of and ; the coefficient given by Eq. 45 corresponds to given by Eq. 57 in the previous result. If is neglected in the present result, the same expression as the previous result is recovered. In fact, the quantity stems from the fourth and fifth terms in Eq. 29 that are newly treated in the present TSDIA. B. Equation for turbulent electromotive force Although a new model expression for the turbulent electromotive force is derived using the TSDIA, it is not yet justified by numerical simulation or observation. In this subsection, to assess the validity of the model expression, we examine the transport equation for the turbulent electromotive force. For non-mhd turbulence it is known that the production terms in the transport equation for a correlation such as the Reynolds stress are closely related to the eddyviscosity-type model for the correlation. For example, the correlation can be modeled as the product of the production terms and the turbulent time scale. A similar method called the minimal tau approximation was used for MHD turbulence. 44,45 This type of approximation was also introduced in the TSDIA as the Markovian method. 23 Therefore, the investigation of the production terms gives a clue to modeling the turbulent electromotive force. The transport equation for E M in a rotating frame can be written as D E M Dt t + U + F E M = P E1 + P E2 + P E3 + R E, 62 three production terms are given by P E1i = B m ijk u k u j x m +b k b j x m, P E2i = ijk u j u m + b j b m B k x m, 63 64 P E3i = ijk bu j m + ub j m U k + mkn Fn, 65 x m and the detailed expression for the remaining part R E is omitted. Each term in Eq. 62 is expressed in an objective form. The three production terms P E1, P E2, and P E3 involve the mean velocity or the mean magnetic field; they represent the effect of the mean field on the turbulent electromotive force. To show the correspondence of the production terms to the terms in the model expression 43, we first assume an isotropic homogeneous turbulent field and replace the correlations appearing in Eqs. 63 65 as follows: i j = 1 3 ij, i k j = 1 x 6 ijk. Then, we can obtain the following expressions: 66 P E1 = 1 3 HB, P E2 = 2 3 J, P E3 = 2 3 W +2 F, 67 which correspond to the first, second, and third terms on the right-hand side of Eq. 43, respectively. Therefore, it is shown that the effect, the turbulent diffusivity, and the cross-helicity dynamo results from the corresponding production terms. The pumping effect term has not been obtained yet because the coefficient requires an inhomogeneous turbulent field. Next, instead of the isotropic homogeneous assumption 66, we assume a turbulent field that is locally isotropic and weakly inhomogeneous. Using the solenoidal condition for u i and b i the production term P E1 can be exactly rewritten as P E1i = ipq u k u k p +b x k b p q x kb kpq u k u p x i +b k b p x ib q. 68 Although the summation convention is applied to repeated indices including p, we consider here only the part in which p=k. Then, we can see that the second parenthesis in Eq. 68 does not contribute because of kpq =0 and the correlations in the first parenthesis can be approximated as u u p p p = 1 u 2 x 2 x p 1 u 2. 69 p 6 x p As a result, the production term P E1 can be expressed as P E1 = 1 3 R 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 the sense that they have corresponding production terms in the transport equation. In particular, the production term P E1 given by Eq. 63 involves the difference between the velocity correlation u k u j /x m and the magnetic-field correlation b k b j /x m ; this fact accounts for the dependence of the pumping effect on the turbulent residual helicity.

022302-8 F. Hamba and H. Sato Phys. Plasmas 15, 022302 2008 FIG. 1. Pumping effect in a spherical region of large residual energy. Solid lines represent the original uniform magnetic field and dashed lines stand for the magnetic field transported by the pumping effect. FIG. 2. Profiles of the mean magnetic field in a rotating spherical shell without pumping effect; a poloidal field B p and b toroidal field B. C. Pumping effect in rotating spherical shell The physical meaning of the pumping effect is illustrated in Fig. 1. Here we consider a uniform magnetic field B 0 plotted by solid lines and a spherical region of large residual energy given by = 0 exp r 2 /r 0 2, 71 is defined as Eq. 54. The pumping velocity is directed outward in the radial direction. The electromotive force B 0 and the resulting current J have a toroidal component. As a result, the magnetic field is transported in the outward direction as plotted by dashed lines. This is also called the turbulence-induced diamagnetism because the magnetic field in the central region is decreased. 3 To demonstrate the behavior of the pumping effect in a more realistic problem, we investigate the magnetic field in a rotating spherical shell similar to the Earth s outer core. Hamba 32 numerically simulated the axisymmetric mean magnetic field in a rotating spherical shell using a Reynoldsaveraged MHD turbulence model. In the model, the turbulent electromotive force is given by E M = B J, 72 = C f H, = C 2 f. 73 Here, model constants are set to C =C =0.09 and correction coefficients f 0 f 1 and f 0 f 1 are introduced to satisfy the realizability condition for the turbulent electromotive force, E M. To concentrate on the estimate of the pumping effect, the turbulent diffusivity is modeled in the same form as that of Hamba; 32 the modified expression 53 is not adopted here. In addition to the induction equation for the mean magnetic field, the transport equations for,, and H are solved to evaluate the transport coefficients and. In the simulation, physical quantities are nondimensionalized by the typical velocity U 0 =510 4 ms 1, the radius of the outer boundary r out =3.4810 6 m, and the fluid density =1.0910 4 kg m 3. The magnetic field and the time scale are then normalized by 0 U 0 =0.585 G and r out /U 0 = 221 year, respectively. The ratio of radii of the inner and outer boundaries is set to r in /r out =0.35. The outer regions at 0rr in and at rr out are assumed to be insulators. The angular velocity of the system rotation is set to F =510 5. The time evolution of the magnetic field is calculated until a steady state is reached see Hamba 32 for details. First, we briefly explain the result of the simulation of Hamba 32 in which the pumping effect is not treated. Figure 2 shows the profiles of the mean magnetic field. The poloidal field B p =B r,b plotted in Fig. 2a is approximately a dipole field as the toroidal field B plotted in Fig. 2b is negative positive in the northern southern hemisphere. It was shown in Hamba 32 that the dipole field is sustained owing to the 2 dynamo; that is, the toroidal and poloidal fields induce each other via the effect. Figure 3 shows the profiles of the turbulent MHD energy and the turbulent residual helicity H. The turbulent MHD energy is produced by the buoyancy effect. Since the value of has a maximum at rr out and at =/2, the gradient / is positive negative in the northern southern hemisphere. The transport equation for H involves the production term F whose

022302-9 Turbulent transport coefficients Phys. Plasmas 15, 022302 2008 FIG. 3. Profiles of turbulent statistics in a rotating spherical shell without pumping effect; a turbulent MHD energy and b turbulent residual helicity H. FIG. 4. Magnetic-field difference due to the pumping effect with R =0.1; a poloidal field B p and b toroidal field B. value is positive negative in the northern southern hemisphere. As a result, the turbulent residual helicity H of the same sign as F is produced as shown in Fig. 3b. The large value of H enhances the effect. Next, to examine the pumping effect we add the pumping term to Eq. 72 as follows: E M = B J B, = C f R. 74 75 Here we adopt C =0.09 and f = f. To accurately predict the profile of R, we need to solve its transport equation. However, its model equation including model constants has not been established yet. 40 In order to examine a qualitative effect of the pumping term, we assume here that R is proportional to as follows: R = R, 76 R is a small nondimensional constant; R =0.1 is adopted here. A minus sign is included in Eq. 76 because R is expected to be negative in the Earth s outer core. 43 We should note that Eq. 76 is just an assumption made for a qualitative estimate; the transport equation for R needs to be modeled in future work. To evaluate the effect of the pumping term B on the magnetic field, we calculate the difference between the original and newly obtained magnetic fields B = B 1 B 0, 77 B 0 is the original field shown in Fig. 2 and B 1 is the field obtained by solving the same model as that of Hamba 32 except for the pumping term with R =0.1. Figure 4 shows the difference B between the two magnetic fields. The profile of the poloidal field plotted in Fig. 4a is simple; it is approximately a dipole field and the direction is opposite to the original field. It is shown that the pumping effect reduces the original dipole field in this case. On the other hand, the toroidal field plotted in Fig. 4b is rather complicated. In the region near the equator, the difference B is positive negative in the northern southern hemisphere; these signs are opposite to the corresponding original field. In this region the toroidal field also decreases owing to the pumping effect. These profiles of B can be roughly explained by considering the directions of the pumping velocity and of the original magnetic field B 0 as illustrated in Fig. 5. In Fig. 5a we consider the toroidal component of the pumping term, B in the region near the equator. The pumping velocity points in the same direction as, as the poloidal field B p is directed opposite to the system rotation F. The pumping term B is then positive in this region. As a result, the pumping term induces a positive toroidal current J corresponding to the dipole field B p plotted in Fig. 4a. Therefore, the magnetic field B p is induced in the same manner as the example illustrated in Fig. 1. On the other hand, in Fig. 5b we consider the poloidal component of the pumping effect, B p. The pumping velocity is in the positive negative direction and the toroidal field B has a negative positive value in the north-

022302-10 F. Hamba and H. Sato Phys. Plasmas 15, 022302 2008 ern southern hemisphere. The pumping term B p then points in the negative r direction in both the northern and southern hemispheres and induces J p in the same direction. Such a current in the negative r direction near the equator can account for the negative gradient of the toroidal field, B /, in the region near the equator shown in Fig. 4b. Therefore, in this simulation of a rotating spherical shell, the pumping term has the effect of reducing the original magnetic field induced by the effect. In future work, we need to solve the transport equation for R to evaluate the pumping effect more accurately. V. CONCLUSIONS To predict the mean magnetic field in MHD turbulence at high Reynolds number, it is necessary to model the turbulent electromotive force. In this work, a new model expression for the turbulent electromotive force was derived using the TSDIA. We improved the formulation of the TSDIA so that obtained expressions can satisfy the frame-invariance under rotating transformations. We adopted four Green s functions to take into account the effect of the mean field on the fluctuations more accurately. The resulting expression for the turbulent electromotive force consists of terms representing the effect, the turbulent diffusivity, the cross-helicity dynamo, and the pumping effect. The transport coefficients in the turbulent electromotive force are expressed in terms of turbulent statistics such as the turbulent MHD energy and the turbulent residual helicity. It was shown that the turbulent diffusivity is proportional to the turbulent kinetic energy. Moreover, the pumping velocity was shown to be closely related to the gradient of the turbulent residual energy. These results confirm earlier findings of Rädler et al. 21 and Brandenburg and Subramanian 22 using the tau approximation. To assess the validity of the model expression, we examined the transport equation for the turbulent electromotive force. Using the isotropic homogeneous assumption, the three production terms were shown to correspond to the terms representing the effect, the turbulent diffusivity, and the cross-helicity dynamo. Using the weakly inhomogeneous assumption, the pumping effect term was also derived from one of the production terms. To demonstrate the pumping effect, we simulated the magnetic field in a rotating spherical shell. It was shown that in this case the pumping term has the effect of reducing the magnetic field induced by the dynamo. In future work, the transport equation for the turbulent residual energy should be solved to evaluate the pumping effect more accurately. It is also important to validate the pumping effect using data obtained from three-dimensional simulations of MHD turbulence. ACNOWLEDGMENTS F.H. would like to thank Dr. N. Yokoi for valuable discussion on the turbulent residual energy. This work was partially supported by the Grant-in-Aid for Scientific Research of Japan Society for the Promotion of Science 19560159. FIG. 5. Pumping effect in a rotating spherical shell; a poloidal component and b toroidal component. APPENDIX: GOVERNING EQUATIONS FOR GREEN S FUNCTIONS as The equations for the basic fields u 00i and b 00i are written u 00ik; + k 2 u 00i k; im ijk u 00j p;u 00k q; kpq b 00j p;b 00k q; =0, b 00ik; + M k 2 b 00i k; A1 in ijk kpq u 00j p;b 00k q; =0. A2 Since these equations are the same as those for isotropic turbulence, we assume that the basic fields u 00i and b 00i are isotropic and that the anisotropic and inhomogeneous effects can be incorporated through higher-order terms such as u 01i and u 10i. The equations for u 01i and b 01i are given by Eqs. 30 and 31, respectively. The left-hand sides of Eqs. 30 and 31 are linear with respect to u 01i and b 01i. The right-hand sides of Eqs. 30 and 31 can be formally considered as

022302-11 Turbulent transport coefficients Phys. Plasmas 15, 022302 2008 external forces for u 01i and b 01i. We then introduce the Green s functions satisfying the following system of equations: Ĝ uu ij k;, + k 2 Ĝ uu ij k;, 2iM ikm u 00k p;ĝ mj kpq uu q;, Ĝ ub ij k;, + k 2 Ĝ ub ij k;, 2iM ikm u 00k p;ĝ mj kpq ub q;, b 00k p;ĝ bb mj q;, =0, Ĝ bb ij k;, + M k 2 Ĝ bb ij k;, A5 b 00k p;ĝ bu mj q;, = ij, A3 in ikm u 00k p;ĝ mj kpq bb q;, and Ĝ bu ij k;, + M k 2 Ĝ bu ij k;, in ikm u 00k p;ĝ mj kpq bu q;, b 00k p;ĝ uu mj q;, =0, A4 b 00k p;ĝ ub mj q;, = ij. A6 The Green s functions Ĝ uu ij and Ĝ bu ij represent the response of the velocity and the magnetic field, respectively, to a disturbance at time in the velocity equation. On the other hand, Ĝ ub ij and Ĝ bb ij represent the response to a disturbance in the magnetic field equation. Using these Green s functions, we can obtain formal solutions for u 01i and b 01i given by Eqs. 35 and 36, respectively. In a similar manner, the solutions for u 10i and b 10i can be written as u 10i k; d 1 Ĝ ij = uu k;, 1 D jk k B k b 00m k; 1 D jk X m k U k + mkn 0nu 00m k; 1 + B k b 00j k; 1 X m X Ik D jk k D DT I u 00k k; 1 + d 1 Ĝ ub ij k;, 1 D jk k B k u 00m k; 1 + D jk X m k U k + mkn 0nb 00m k; 1 X m + B k X Ik u 00j k; 1 D jk k D DT I b 00k k; 1, A7 b 10i k; d 1 Ĝ ij = bu k;, 1 D jk k B k b 00m k; 1 D jk X m k U k + mkn 0nu 00m k; 1 + B k b 00j k; 1 X m X Ik D jk k D DT I u 00k k; 1 + d 1 Ĝ bb ij k;, 1 D jk k B k u 00m k; 1 + D jk X m k U k + mkn 0nb 00m k; 1 X m + B k X Ik u 00j k; 1 D jk k D DT I b 00k k; 1, A8 D D = exp ik U expik U. DT I DT A9 Using solutions 35, 36, A7, and A8 we can express the turbulent electromotive force in terms of the basic correlations, the mean Green s functions, and the mean field as shown in Eqs. 43 48. 1 E. N. Parker, Astrophys. J. 122, 293 1955. 2 H.. Moffatt, Magnetic Field Generations in Electrically Conducting Fluids Cambridge University Press, Cambridge, 1978. 3 F. rause and.-h. Rädler, Mean-Field Magnetohydrodynamics and Dynamo Theory Pergamon, Oxford, 1980. 4 H. A. B. Bodin and A. A. Newton, Nucl. Fusion 20, 12551980. 5 C. G. Gimblett and M. L. Watkins, in Proceedings of the Seventh European Conference on Controlled Fusion and Plasma Physics E. P. S., Petit-Lancy, 1975, Vol. 1, p. 103. 6 P. S. Iroshnikov, Sov. Appl. Mech. 7, 566 1964. 7 R. H. raichnan, Phys. Fluids 8, 13851965. 8 A. Pouquet, U. Frisch, and J. Léorat, J. Fluid Mech. 77, 3211976.

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