Renormalization Group Analysis of Magnetohydrodynamic Turbulence with the Alfv?n Effect

全文

(1)Journal of the Physical Society of Japan Vol. 71, No. 6, June, 2002, pp. 1450–1462 #2002 The Physical Society of Japan. Renormalization Group Analysis of Magnetohydrodynamic Turbulence with the Alfveń Effect Chien C. C HANG and Bin-Shei L IN Institute of Applied Mechanics, College of Engineering, National Taiwan University Taipei 10764, Taiwan, Republic of China (Received December 21, 2001). In this study, we continue with a recursive renormalization group (RG) analysis of incompressible turbulence, aiming at investigating various turbulent properties of three-dimensional magnetohydrodynamics (MHD). In particular, we are able to locate the fixed point (i.e. the invariant effective eddy viscosity) of the RG transformation under the following conditions. (i) The mean magnetic induction is relatively weak compared to the mean flow velocity. (ii) The Alfveń effect holds, that is, the fluctuating velocity and magnetic induction are nearly parallel and approximately equal in magnitude. It is found under these conditions that re-normalization does not incur an increment of the magnetic resistivity, while the coupling effect tends to reduce the invariant effective eddy viscosity. Both the velocity and magnetic energy spectra are shown to follow the Kolmogorov k5=3 in the inertial subrange; this is consistent with some laboratory measurements and observations in astronomical physics. By assuming further that the velocity and magnetic induction share the same specified form of energy spectrum, we are able to determine the dependence of the (magnetic) Kolmogorov constant CK (CM ) and the model constant CS of the Smagorinsky model for large-eddy simulation on some characteristic wavenumbers. KEYWORDS: renormalization group analysis, magnetohydrodynamic turbulence, Alfveń effect, effective eddy viscosity, magnetic resistivity DOI: 10.1143/JPSJ.71.1450. 1.. Introduction. Recently, the authors1,2) carried out a recursive renormalization group (RG) analysis of incompressible turbulence for flow turbulence and thermal turbulent transport. In this study, we continue with this previous RG analysis for magneto-hydrodynamic (MHD) turbulence, aiming at investigating various transport properties, in particular, the coupling effects between the flow and magnetic induction fields on the kinetic energy spectrum and the effective eddy viscosity. The plasma science is widely applied to many areas from laser skill, thin film produce, nuclear rocket, even to astronomical physics (for example, solar wind, solar flares and coronal structures). Like in ordinary Newtonian fluids, MHD turbulence is expected to arise in plasma or magnetized fluids as the Reynolds number is increased beyond some critical value. In spite of the already scarce literature, the interest of MHD turbulence may further be divided into two-dimensional and three-dimensional turbulence. Kim and Yang3) studied the scaling behavior of the randomly stirred MHD plasma in two dimensions and were able to show existence of the scaling solution at the fixed point of the RG transformation and derive the dependence of the power exponent of the energy spectrum on the driving Gaussian noise. Liang and Diamond4) also presented their study for two-dimensional MHD turbulence by introducing the velocity stream function and the magnetic flux function in MHD equations. However, the latter authors showed no existence of a fixed point of the RG transformation and especially suggested that the applicability of RG method to turbulent system is intrinsically limited, especially in the case of systems with dual-direction energy transfer. In contrast to flow in two dimensions, the effect of dual-. direction energy transfer becomes weak in three dimensions (cf. McComb5)). It would therefore be legitimate to employ the RG analysis for MHD turbulence in three dimensions. In the literature, there are some measured evidences about the validity of the Kolmogorov spectrum for the three-dimensional MHD turbulence. Alemany et al.6) designed an equipment in the laboratory which produced turbulence by passing magnetized fluid to a mesh under an additional magnetic induction. In the area of astronomical physics, Matthaeus et al.7) measured the magnetic energy spectrum of the solar wind, while Leamon et al.8) measured the MHD turbulence within the coronal mass ejection. Both of their results suggested the Kolmogorov power law for the energy spectrum. Besides, Biskamp9) mentioned that the Kolmogorov constant depends on the precise definition of the average magnetic induction, and hence on the geometry of the large scale eddies. On the theoretical side, Hatori10) obtained the Kolmogorov spectrum for the three-dimensional MHD turbulence, but suggested that the Kolmogorov constant is universal. Verma11) constructed a self-consistent renormalization group procedure for MHD turbulence and also found that the energy spectrum for the velocity obeys the Kolmogorov spectrum. It is the purpose of the present study to provide a recursive renormalization group analysis for MHD turbulence in three dimensions with the specific points of interest as follows. We will obtain the energy spectra for both of the velocity and magnetic induction fields, look for the invariant effective eddy viscosity and determine the dependence of the (magnetic) Kolmogorov constant CK (CM ) and the model constant CS for the Smagorinsky model for large-eddy simulation (LES). Let us give a brief description of the present work. MHD is governed by a coupling set of equations, meanwhile, the MHD turbulence considered is further assumed to be isotropic, homogeneous and stationary. It is found con-. . E-mail: changcc@gauss.iam.ntu.edu.tw 1450.

(2) J. Phys. Soc. Jpn., Vol. 71, No. 6, June, 2002. venient to introduce the Elsa¨sser variables to write the equations for the velocity and magnetic induction fields in a symmetric form. In §3, a recursive RG analysis is carried out for the MHD equations in the wavenumber domain and a recursive relationship for the effective eddy viscosity n ðkÞ between two successive steps is established. The resulting expression is complicated enough and is apparently not amenable to further RG analysis. Instead, we restrict ourselves to the case when the following conditions hold. (i) The mean magnetic induction is relatively weak compared to the mean flow velocity. (ii) The Alfveń effect holds, that is, the fluctuating velocity and magnetic induction are nearly parallel and approximately equal in magnitude. As a matter of fact, the two conditions imply a negligible effect of the subgrid cross helicity between the velocity and magnetic fields. In spite of these restrictions, the present RG analysis still warrants a sufficient interest as we investigate several observations in the area of astronomical physics. In §4, the energy spectra of the velocity and the magnetic fields are determined through use of the RG transformation, i.e. the recursive relation. Both spectra are found to follow the Kolmogorov k5=3 law in the inertial subrange. The results are consistent with the experimental results of Alemany et al.,6) and the observational results of Matthaeus et al.7) From a different approach, Chen and Montgomery12) obtained the same power law in the inertial subrange by using some multiple-scale self-consistent calculations of turbulent MHD transport coefficients. In §5, the fixed point of the RG equation is located to give the invariant effective eddy viscosity ðkÞ and magnetic resistivity ðkÞ. By assuming further a combination form of the energy spectra proposed respectively by Pao13) and Quarini and Leslie,14) the invariant effective eddy viscosity is then employed in §6 to determine the dependence of the (magnetic) Kolmogorov constant CK (CM ) and the Smagorinsky constant CS on the cutoff wavenumber kc , the wavenumber ks of the largest eddies and the wavenumber kp that peaks in the energy spectrum. Finally, concluding remarks are drawn in §7. 2.. Magnetohydrodynamic Equations. In considering MHD turbulence, we shall take the SI units. The magnetized fluid is assumed to be incompressible (constant density ) and have a constant permeability 0 . It is convenient to simply set ¼ 1 and 0 ¼ 1. In doing do, the magnetic induction B and the velocity v have the same dimension because of the dimensional relationship ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½B ¼ ½0 ½. Let us start with the treatment of Alfveń16) and Cowling,17) and write down the following magnetohydrodynamic equations. (i) Continuity equation:. C. C. CHANG and B.-S. LIN. r

(3) Eþ. 1451. @B ¼ 0; @t. and r

(4) B¼J where E is the electric field, and we have neglected the displacement current. The Ohm’s law takes the form J ¼ 0 ðE þ v

(5) BÞ;. ð1Þ. where 0 is the electric conductivity. Let us take r

(6) on the both hand sides of eq. (1) and reorganize the electromagnetic equations to obtain @B ¼ r

(7) ðv

(8) BÞ þ 0 r2 B; @t where 0 ¼ 1=0 is the magnetic resistivity. In summary, we have the following MHD equations for use, ( @v=@t þ ðv rÞv ¼ rp þ ðr

(9) BÞ

(10) B þ 0 r2 v; ð2Þ @B=@t ¼ r

(11) ðv

(12) BÞ þ 0 r2 B; with the solenoidal equations r v ¼ 0; r B ¼ 0; where the gravitation is incorporated into p. It is convenient to introduce the Elsa¨sser variables (cf. ref. 18) for eq. (2), defined by ¼ v þ B; ¼ v B: Equation (2) can then be transformed to ( @ =@t þ ð rÞ ¼ rp þ 0 r2 þ 0 r2 ; @=@t þ ð rÞ ¼ rp þ 0 r2 þ 0 r2 ;. ð3Þ. where p ¼ p þ ðB BÞ=2, and we have set . 0 ¼ ð0 þ 0 Þ=2; 0 ¼ ð0 0 Þ=2: It is obvious from the definitions of , that they are also solenoidal, i.e. r ¼ r ¼ 0: Since RG analysis will be performed in the wavenumber domain, the next goal is to Fourier transform eq. (3) into the wavenumber domain. First of all, take r on the both sides of eq. (3) to obtain r 2 p ¼ . r v ¼ 0;. @2 : @x @x. ð4Þ. where v is the velocity. (ii) Momentum equation: dv ¼ rp þ J

(13) B þ 0 r2 v þ g; dt. Let us now introduce the following operators in the wavenumber space k k D ðkÞ ¼

(14) 2 ; k. where J is the electric current density, B the magnetic induction, 0 the molecular viscosity and g the gravitation. (iii) Electromagnetic equations:. and M ðkÞ ¼ ½k D ðkÞ þ k D ðkÞ=2i: By using the Elsa¨sser variables with the help of eq. (4), we.

(15) 1452. J. Phys. Soc. Jpn., Vol. 71, No. 6, June, 2002. C. C. C HANG and B.-S. LIN. can transform eq. (3) into the wavenumber domain as follows, Z ð j; tÞ ðk j; tÞ. ðk; tÞ 3 L< ðk; tÞ ; ¼ M ðkÞ d j ð j; tÞ ðk j; tÞ ðk; tÞ ð5Þ where the matrix L< ðk; tÞ is defined by L< ðk; tÞ ¼. @=@t þ 0 k2 0 k 2. ! 0 k 2 : @=@t þ 0 k2. For later use, we shall need the following statistical correlations: 8 0 0 > < hu ðk; tÞu ðk ; tÞi ¼ D ðkÞ

(16) ðk þ k ÞQðkÞ; hu ðk; tÞB ðk0 ; tÞi ¼ D ðkÞ

(17) ðk þ k0 ÞRðkÞ; ð6Þ > : 0 0 hB ðk; tÞB ðk ; tÞi ¼ D ðkÞ

(18) ðk þ k ÞSðkÞ; where QðkÞ is the kinetic energy spectrum, SðkÞ the magnetic energy spectrum and RðkÞ the cross energy spectrum. It is noted that these relationships are valid for isotropic, homogeneous and stationary turbulence; see, for example, McComb5) for the details. 3.. Renormalization Group Analysis for MHD Turbulence. The basic idea of recursive RG analysis is to divide the wavenumber space ð0; k0 Þ, where k0 is Kolmogorov’s scale, to a supergrid region ð0; kc Þ and a subgrid region ðkc ; k0 Þ. The subgrid modes are then removed shell by shell by taking the. L< ðk; tÞ. <. ðk; tÞ <. ðk; tÞ. !. Z ¼ M ðkÞ. 0. kc. kn. k2 k1. Fig. 1. The termini ki for recursive renormalization with a fixed cutoff ratio ¼ knþ1 =kn . Recursive renormalization analysis starts at the Kolmogorov’s scale k0 , and ends at the cutoff wavenumber kc .. subgrid average over a spherical shell ðknþ1 ; kn Þ, as shown in Fig. 1. At the present stage, the cutoff ratio, defined by ¼ knþ1 =kn is maintained a constant, and will be later set to tend to 1 as the differential version of the RG analysis leading to the invariant effective eddy viscosity is sought. In this section, we will follow the renormalization group analysis that we have developed in refs. 1 and 2. First of all, in order to distinguish the supergrid and subgrid modes, we introduce the following notations: (i) for field, <. ðk; tÞ for jkj < k1 ;. ðk; tÞ ¼ > jkj > k1 ;. ðk; tÞ for (ii) for field, ðk; tÞ ¼. . <. ðk; tÞ >. ðk; tÞ. for for. jkj < k1 ; jkj > k1 :. The momentum equations for the supergrid modes can be written. d3 j. < < > > > < ð j; tÞ ðk j; tÞ þ 2 ð j; tÞ ðk j; tÞ þ ð j; tÞ ðk j; tÞ < < > > > < ð j; tÞ ðk j; tÞ þ 2 ð j; tÞ ðk j; tÞ þ ð j; tÞ ðk j; tÞ. ! ;. and the momentum equations for the subgrid modes can be written ! Z > ð j; tÞ 0 0 L> ð jÞ ¼ M ð jÞ d3 j 0 ð j; tÞ >. ! < 0 0 < 0 > 0 > 0 > 0 0 ð j ; tÞ < 0 ð j j ; tÞ þ 2 0 ð j ; tÞ 0 ð j j ; tÞ þ 0 ð j ; tÞ 0 ð j j ; tÞ < 0 ; 0 < 0 > 0 > 0 > 0 0 ð j ; tÞ< 0 ð j j ; tÞ þ 2 0 ð j ; tÞ 0 ð j j ; tÞ þ 0 ð j ; tÞ 0 ð j j ; tÞ and we have assumed that Markovian approximation holds for the subgrid modes (Rose19) and McComb20)). Its physical ground is that the subgrid modes are considered to evolve much faster than the supergrid modes, which implies that the subgrid modes relax to the steady state while the supergrid modes are still evolving. The matrix L> ðjÞ is defined by . 0 0 2 L> ð jÞ ¼ j: 0 0 Before substantial progress can be made with the RG analysis, we shall make the following statistical hypotheses. (i) The MHD fields have ensemble-mean-zero fluctuation, > h >. ðk; tÞi ¼ h ðk; tÞi ¼ 0:. k0. ð7Þ. ð8Þ. (ii) Supergrid components are considered to be statistically independent of subgrid averaging (cf. McComb5) and Zhou15)), < h <. ðk; tÞi ¼ ðk; tÞ; < h<. ðk; tÞi ¼ ðk; tÞ:. This assumption is simple (but not void) though its validity may be restrictive. But RG theory based on this assumption has not been explored to its full strength. Indeed, as shown in,1,2) the RG results based on this assumption were found to be in remarkably close agreement with computed/measured data..

(19) J. Phys. Soc. Jpn., Vol. 71, No. 6, June, 2002. C. C. CHANG and B.-S. LIN. 1453. Performing subgrid averaging of eq. (7) with use of (i) and (ii), we may obtain the averaged equation for the supergrid modes, ! < Z < > > <. ðk; tÞ ð j; tÞ ðk j; tÞ þ h ð j; tÞ ðk j; tÞi 3 ; ð9Þ L< ðk; tÞ ¼ M ðkÞ d j < > > < < ð j; tÞ ðk j; tÞ þ h ð j; tÞ ðk j; tÞi. ðk; tÞ This subgrid averaged equation for the supergrid modes will be contrasted to eq. (5). The comparison between eqs. (5) and (9) suggests that the ensemble averaging terms on the right hand side of eq. (9) contribute respectively to the effective eddy viscosity and effective magnetic resistivity. Now we return to the original variables u and B by rotating the matrices L< and L> 45 degrees counterclockwise. If we choose a set of new base vectors which are the eigenvectors of matrices L< and L> , eq. (9) takes the following expression ! Z u<. ðk; tÞ L0 ðk; tÞ < ¼ M ðkÞ d3 j B ðk; tÞ ! < < < > > > > u ð j; tÞu< ðk j; tÞ B ð j; tÞB ðk j; tÞ þ hu ð j; tÞu ðk j; tÞ B ð j; tÞB ðk j; tÞi ð10Þ < < < > > > > B< ð j; tÞu ðk j; tÞ u ð j; tÞB ðk j; tÞ þ hB ð j; tÞu ðk j; tÞ u ð j; tÞB ðk j; tÞi where L0 ðk; tÞ ¼. @=@t þ 0 k2. 0. 0. @=@t þ 0 k2. ! :. Similarly, based on the new basis, eq. (8) for the subgrid modes can be transformed into the form, ! Z u> ð j; tÞ ¼ M 0 0 ð jÞ d3 j0 Gð jÞ > B ð j; tÞ " < 0 ! 0 < 0 < 0 u 0 ð j ; tÞu< 0 ð j j ; tÞ B 0 ð j ; tÞB 0 ð j j ; tÞ

(20) 0 < 0 < 0 < 0 B< 0 ð j ; tÞu 0 ð j j ; tÞ u 0 ð j ; tÞB 0 ð j j ; tÞ ! 0 > 0 < 0 > 0 u< 0 ð j ; tÞu 0 ð j j ; tÞ B 0 ð j ; tÞB 0 ð j j ; tÞ þ2 < 0 0 < 0 > 0 B 0 ð j ; tÞu> 0 ð j j ; tÞ u 0 ð j ; tÞB 0 ð j j ; tÞ !# 0 > 0 > 0 > 0 u> 0 ð j ; tÞu 0 ð j j ; tÞ B 0 ð j ; tÞB 0 ð j j ; tÞ þ ; 0 > 0 > 0 > 0 B> 0 ð j ; tÞu 0 ð j j ; tÞ u 0 ð j ; tÞB 0 ð j j ; tÞ where Gð jÞ ¼. 0 j2. 0. 0. 0 j2. ð11Þ. ! :. The next step is to obtain the four correlations in (10) by making use of eq. (11). First of all, multiplying the factor u> ðk j; tÞ on both hand sides of eq. (11), and then taking subgrid averaging yields u> ð j; tÞu> ðk j; tÞ Z. 0 0 Gð jÞ > ¼ 2M ð jÞ d3 j 0 B ð j; tÞu> ðk j; tÞ. ! > 0 < 0 > 0 > < 0 hu 0 ð j j ; tÞu> ðk j; tÞiu 0 ð j ; tÞ hB 0 ð j j ; tÞu ðk j; tÞiB 0 ð j ; tÞ : ð12Þ 0 > < 0 > 0 > < 0 hu> 0 ð j j ; tÞu ðk j; tÞiB 0 ð j ; tÞ hB 0 ð j j ; tÞu ðk j; tÞiu 0 ð j ; tÞ On the other hand, multiplying the factor B> ðk j; tÞ on both hand sides of eq. (11), and then taking subgrid averaging yields u> ð j; tÞB> ðk j; tÞ Z. Gð jÞ > ¼ 2M 0 0 ð jÞ d3 j0 B ð j; tÞB> ðk j; tÞ ! 0 > < 0 > 0 > < 0 hu> 0 ð j j ; tÞB ðk j; tÞiu 0 ð j ; tÞ hB 0 ð j j ; tÞB ðk j; tÞiB 0 ð j ; tÞ : ð13Þ 0 > < 0 > 0 > < 0 hu> 0 ð j j ; tÞB ðk j; tÞiB 0 ð j ; tÞ hB 0 ð j j ; tÞB ðk j; tÞiu 0 ð j ; tÞ Next, we rewrite eq. (11) by changing the index to , and changing the wavenumber j to k j, to obtain.

(21) 1454. J. Phys. Soc. Jpn., Vol. 71, No. 6, June, 2002. Gðjk jjÞ ". u> ðk j; tÞ. !. B> ðk j; tÞ. Z ¼M. 0 0 ðkjÞ. d3 j0. 0 < 0 < 0 < 0 u< 0 ð j ; tÞu 0 ðk j j ; tÞ B 0 ð j ; tÞB 0 ðk j j ; tÞ.

(22). 0 < 0 < 0 < 0 B< 0 ð j ; tÞu 0 ðk j j ; tÞ u 0 ð j ; tÞB 0 ðk j j ; tÞ. þ2. þ. C. C. C HANG and B.-S. LIN. 0 > 0 < 0 > 0 u< 0 ð j ; tÞu 0 ðk j j ; tÞ B 0 ð j ; tÞB 0 ðk j j ; tÞ. ! !. 0 > 0 < 0 > 0 B< 0 ð j ; tÞu 0 ðk j j ; tÞ u 0 ð j ; tÞB 0 ðk j j ; tÞ !# 0 > 0 > 0 > 0 u> 0 ð j ; tÞu 0 ðk j j ; tÞ B 0 ð j ; tÞB 0 ðk j j ; tÞ 0 > 0 > 0 > 0 B> 0 ð j ; tÞu 0 ðk j j ; tÞ u 0 ð j ; tÞB 0 ðk j j ; tÞ. :. Multiplying the factor u> ðj; tÞ on both sides of eq. (14), and then taking subgrid averaging yields u> ð j; tÞu> ðk j; tÞ Z. 0 0 ðk jÞ Gðjk jjÞ > d3 j 0 ¼ 2M ðk j; tÞ u ð j; tÞB>. ! > 0 < 0 > 0 > < 0 hu 0 ðk j j ; tÞu> ð j; tÞiu 0 ð j ; tÞ hB 0 ðk j j ; tÞu ð j; tÞiB 0 ð j ; tÞ 0 > < 0 > 0 > < 0 hu> 0 ðk j j ; tÞu ð j; tÞiB 0 ð j ; tÞ hB 0 ðk j j ; tÞu ð j; tÞiu 0 ð j ; tÞ. ð14Þ. :. Multiplying the factor B> ðj; tÞ on both sides of eq. (14), and then taking subgrid averaging yields B> ð j; tÞu> ðk j; tÞ Z. 0 0 Gðjk jjÞ > ¼ 2M ðk jÞ d3 j 0 B ð j; tÞB> ðk j; tÞ. ! > 0 < 0 > 0 > < 0 hu 0 ðk j j ; tÞB> ð j; tÞiu 0 ð j ; tÞ hB 0 ðk j j ; tÞB ð j; tÞiB 0 ð j ; tÞ 0 > < 0 > 0 > < 0 hu> 0 ðk j j ; tÞB ð j; tÞiB 0 ð j ; tÞ hB 0 ðk j j ; tÞB ð j; tÞiu 0 ð j ; tÞ. :. ð15Þ. ð16Þ. Let us focus on the first equations of (12) and (15). Recall the correlations defined in (6). If we take the proper rearrangement of the indices and make change of variables, it is quite straightforward to prove that the right hand sides of first equations of R (12) and (15) are identical under the operation of M d3 j. Applying the said operation and adding these two together yields Z Z Z M 0 0 ðk jÞ 3 > > 3 d3 j 0 M ðkÞ d jhu ð j; tÞu ðk j; tÞi ¼ 4M ðkÞ d j 2 0 j þ 0 jk jj2 > < 0 > 0 > < 0 ð17Þ hu 0 ðk j j0 ; tÞu> ð j; tÞiu 0 ð j ; tÞ hB 0 ðk j j ; tÞu ð j; tÞiB 0 ð j ; tÞ : > (i) For hu> ðj; tÞu ðk j; tÞi, it follows from eq. (17), Z Z > M ðkÞ d3 jhu> ð j; tÞu ðk j; tÞi ¼ 4M ðkÞ d3 j. . M 0 0 ðk jÞD 0 ðjÞD 0 ðkÞ < Qð jÞu<. ðk; tÞ Rð jÞB ðk; tÞ 2 2 0 j þ 0 jk jj Z . Lðk; k jÞ < ¼ 2 d3 j 2 Qð jÞu<. ðk; tÞ Rð jÞB ðk; tÞ ; 2 0 j þ 0 jk jj.

(23). ð18Þ. where we have used the relationships: M ðkÞD 0 ðkÞ ¼ M 0 ðkÞ; and Lðk; k jÞ ¼ 2M 0 ðkÞM 0 0 ðk jÞD 0 ð jÞ ¼. ðk4 2k3 j þ kj3 Þð1 2 Þ : jk jj2. ð19Þ. A similar procedure can be applied to obtain other correlations. > (ii) For hB> ðj; tÞB ðk j; tÞi, we have from the second equations of (13) and (16) Z > M ðkÞ d3 jhB> ð j; tÞB ðk j; tÞi Z ¼ 2. d3 j. . Lðk; k jÞ < Rð jÞB<. ðk; tÞ Sð jÞV ðk; tÞ ; 2 0 þ 0 jk jj j2. ð20Þ.

(24) J. Phys. Soc. Jpn., Vol. 71, No. 6, June, 2002. C. C. CHANG and B.-S. LIN. 1455. > (iii) For hB> ðj; tÞu ðk j; tÞi, we obtain from the second equation of (12) and the first equation of (16) Z > M ðkÞ d3 jhB> ð j; tÞu ðk j; tÞi. Z ¼ 2. d3 j. . Lðk; k jÞ < Qð jÞB<. ðk; tÞ Sð jÞB ðk; tÞ ; 2 0 þ 0 jk jj j2. ð21Þ. > (iv) For hu> ðj; tÞB ðk j; tÞi, we obtain from the first equation of (13) and the second equation of (15) Z > M ðkÞ d3 jhu> ð j; tÞB ðk j; tÞi. Z ¼ 2. d3 j. . Lðk; k jÞ < Qð jÞB<. ðk; tÞ Sð jÞB ðk; tÞ : 2 0 þ 0 jk jj j2. Collecting the above results (i)–(iv) by substituting eqs. (18), (20), (21) and (22) in eq. (10) yields ! u<. ðk; tÞ L0 ðk; tÞ < B ðk; tÞ ! Z ð j; tÞu< ðk j; tÞ B< ð j; tÞB< ðk j; tÞ u<. ¼ M ðkÞ d3 j < < < B ð j; tÞu< ðk j; tÞ u ð j; tÞB ðk j; tÞ 0 1 < < QðjÞu< RðjÞB<. ðk; tÞ RðjÞB ðk; tÞ. ðk; tÞ SðjÞu ðk; tÞ Z B C 0 j2 þ 0 jk jj2 0 j2 þ 0 jk jj2 B C 2 d3 jLðk; k jÞB C: < < < @ QðjÞB< QðjÞB ðk; tÞ SðjÞB ðk; tÞ A. ðk; tÞ SðjÞB ðk; tÞ 0 j2 þ 0 jk jj2 0 j2 þ 0 jk jj2. ð22Þ. ð23Þ. Apparently, eq. (23) is not amenable to renormalization because of the difficulty in singling out increments of the effective eddy viscosity and of the magnetic resistivity. To alleviate this problem, we consider the two conditions: (i) the mean magnetic induction is relatively weak compared to the mean flow velocity, and (ii) the Alfveń effect holds, that is the fluctuating velocity and magnetic induction are nearly parallel and approximately equal in magnitude. Those two conditions < < < directly imply that QðjÞu<. ðk; tÞ RðjÞB ðk; tÞ and SðjÞu ðk; tÞ RðjÞB ðk; tÞ, and then eq. (23) can be simplified as follows: ! u<. ðk; tÞ L1 ðk; tÞ < B ðk; tÞ ! < < < Z u< ð j; tÞu ðk j; tÞ B ð j; tÞB ðk j; tÞ 3 ; ð24Þ d j < ¼ M ðkÞ < < B ð j; tÞu< 1 ðk j; tÞ u ð j; tÞB ðk j; tÞ where L1 ðk; tÞ ¼. !. @=@t þ 1 ðkÞk2. 0. 0. @=@t þ 1 ðkÞk2. :. The effective eddy viscosity and magnetic resistivity after the first-step renormalization are given by 1 ðkÞ and 1 ðkÞ as follows 1 ðkÞ ¼ 0 þ

(25) 0 ðkÞ; and 1 ðkÞ ¼ 0 þ

(26) 0 ðkÞ; where. Z

(27) 0 ðkÞ ¼ 2. d3 j. Lðk; k jÞ QðjÞ SðjÞ þ ; k2 0 j2 þ 0 jk jj2 0 j2 þ 0 jk jj2. ð25Þ. d3 j. Lðk; k jÞ QðjÞ SðjÞ QðjÞ SðjÞ : k2 0 j2 þ 0 jk jj2 0 j2 þ 0 jk jj2. ð26Þ. 0. and. Z

(28) 0 ðkÞ ¼ 2 0. The integrals in (25) or (26) are performed over the intersection of two spherical shells:. 0 ðkÞ ¼ f jjk1 < j jj; jk jj < k0 g: Let us now discuss the validity of the conditions (i) and.

(29) 1456. J. Phys. Soc. Jpn., Vol. 71, No. 6, June, 2002. C. C. C HANG and B.-S. LIN. (ii). In light of their effects, these conditions amount to neglecting the effects of the subgrid cross helicity between the velocity and magnetic fields. Nevertheless, there are important cases of application of the present formulation. In astronomical physics, the typical velocity is often larger than the magnetic induction, for example, in solar wind and in solar flares. For comparison in correct dimension, we have pffiffiffiffiffiffiffiffi to recover the velocity v to v 0 . Inside the solar interior, the typical temperature at the core is about 1:5

(30) 107 K, and the density about 1:5

(31) 105 kg/m3 . In the outer edge of the core, about 1:75

(32) 105 km from the center, the density drops to 2

(33) 104 kg/m3 . In the Radiative Zone, the density drops from 2

(34) 104 kg/m3 to 2

(35) 102 kg/m3 . At solar surface the temperature has dropped to 5:7

(36) 103 K and the density is only 2

(37) 104 kg/m3 , and the magnetic induction is about 7:75

(38) 104 tesla. Near the solar surface, the solar wind bulk speed is typically from 2

(39) 105 m/s to 2

(40) 106 m/s. The hydrogen and helium are diamagnetic materials and their magnetic susceptibilities are all quite small (about 109 ), thus 0 ’ 0 (vacuum) for hydrogen and helium. From pffiffiffiffiffiffiffiffi these data, the order of v 0 is therefore in the range of 1– pffiffiffiffiffiffiffiffi 10. At solar surface, the velocity scaled as v 0 is larger than the magnetic induction B in magnitude by 4 to 5 orders. Z

(41) 0 ðkÞ ¼ Z

(42) 0 ðkÞ ¼ 0. 4.. Determination of the Energy Spectrum. For MHD turbulence, we shall consider two kinds of energy contribution with wavenumber vectors lying within the spherical shell between k and k þ dk: ( Ev ðkÞdk ¼ 4k2 QðkÞdk : ð27Þ EM ðkÞdk ¼ 4k2 SðkÞdk Substituting (27) into eqs. (25) and (26) respectively yields. d3 j. Lðk; k jÞ E0v ð jÞ E0M ð jÞ þ ; 2j2 k2 0 j2 þ 0 jk jj2 0 j2 þ 0 jk jj2. ð28Þ. d3 j. Lðk; k jÞ E0v ð jÞ E0M ð jÞ E0v ð jÞ E0M ð jÞ : 2j2 k2 0 j2 þ 0 jk jj2 0 j2 þ 0 jk jj2. ð29Þ. 0. and. As an another example, the highest recorded speed of solar flares is 1,500 km/s, but 100–300 km/s is more typical, with pffiffiffiffiffiffiffiffi sizable variation. The velocity scaled as v 0 is still several orders larger than B in magnitude. In deriving eq. (24), we consider also the Alfveń effect16) in the order analysis of the above discussion that assumes that the mean of magnetic induction B is not small, and the fluid takes large Reynolds number, then the Lorentz force will become important and make effect on the small scale velocity flucation v> , such that the small scale motions transform into Alfveń waves and this results in that v> ðx; tÞ

(43) B> ðx; tÞ ’ 0, and thus v> ðx; tÞ ’ B> ðx; tÞ.. Equations (28) and (29) are respectively the increments of the effective eddy viscosity and magnetic resistivity after the first step of renormalization. Repeating the RG procedure for n þ 1 times, we obtain the subgrid-averaged equations for the supergrid modes, ! u<. ðk; tÞ Lnþ1 ðk; tÞ < B ðk; tÞ ! < < < Z u< ð j; tÞu ðk j; tÞ B ð j; tÞB ðk j; tÞ 3 ; dj < ¼ M ðkÞ < < B ð j; tÞu< nþ1 ðk j; tÞ u ð j; tÞB ðk j; tÞ and the recursive relationship is established as follows: nþ1 ðkÞ ¼ n ðkÞ þ

(44) n ðkÞ; and nþ1 ðkÞ ¼ n ðkÞ þ

(45) n ðkÞ; where. Lðk; k jÞ Env ð jÞ EnM ð jÞ d j þ

(46) n ðkÞ ¼ ; 2j2 k2 n ð jÞj2 þ n ðk jÞjk jj2 n ð jÞj2 þ n ðk jÞjk jj2 n. ð30Þ. Lðk; k jÞ Env ð jÞ EnM ð jÞ Env ð jÞ EnM ð jÞ

(47) n ðkÞ ¼ d j : 2j2 k2 n ð jÞj2 þ n ðk jÞjk jj2 n ð jÞj2 þ n ðk jÞjk jj2 n. ð31Þ. Z. and. Z. 3. 3. with n ðkÞ ¼ f jjknþ1 < j jj; jk jj < kn g. The typical behavior of the increment of the effective eddy viscosity

(48) n ðkÞ is shown in Fig. 2, while the increment of the magnetic resistivity

(49) n ðkÞ is small due to large cancellation of the two integrands. There are two physical quantities in eqs. (30) and (31); they are the wavenumber k, the kinetic energy. dissipation rate "v and the magnetic energy dissipation rate "M . It is therefore natural to propose the following scaling laws respectively for Env and EnM :.

(50) J. Phys. Soc. Jpn., Vol. 71, No. 6, June, 2002. C. C. CHANG and B.-S. LIN. (. 0.35. 0.3. ð32Þ. where kp denotes the wavenumber that peaks in the energy spectrum, CK is the Kolmogorov constant and CM may be termed the magnetic Kolmogorov constant. Both scaling functions of n ð j=kp Þ, ’n ð j=kp Þ will be specified more precisely later. In analogy to (32) we may assume that there are dimensionless effective eddy viscosity ^n ð j=kp Þ and effective eddy resistivity ^n ð j=kp Þ, such that ( n ð jÞ ¼ CKe "vf jg ^n ð j=kp Þ; ð33Þ h i l n ð jÞ ¼ CM "M j ^n ð j=kp Þ:. 0.25. 0.2 viscous increment. Env ð jÞ ¼ CK "av jb n ð j=kp Þ; c d EnM ð jÞ ¼ CM "M j ’n ð j=kp Þ;. 1457. 0.15. 0.1. Next, set ¼ j=k and substitute the first equation of (33) into eq. (30); this yields. 0.05. 0 -6 10. -5. 10. -4. -3. 10. 10 k. -2. -1. 10. 0. 10. 10. Fig. 2. The typical behavior of the increment of the effective eddy viscosity

(51) versus the normalized wavenumber k (0 k 1), as the cutoff ratio ¼ kn =knþ1 is close to 1.. Z

(52) n ðkÞ ¼ . d3 . n . ð1 2 þ 3 Þð1 2 Þ 2ð1 þ 2 2Þ. 1h ci dl1 d2 CK1e "vaf kbg1 b2 ð j=kp Þ CM "M k ’ð j=kp Þ þ ; ^n ðÞ2 þ ^n ðÞð1 þ 2 2Þ ^n ðÞ2 þ ^n ðÞð1 þ 2 2Þ. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where we have used the shorthand ¼ 1 þ 2 2, and denotes the direction cosine between k and j. In order that both sides of eq. (34) are consistent in dimension, we must have the following relationships: 8 b g ¼ d l; > > > < e ¼ h ¼ 1=2; ð35Þ > f ¼ a=2; > > : g ¼ ðb 1Þ=2:. . þ. CKe vf kgþ1 g ^n ðÞð1. þ. 2. From (32) and (33), we have 8 Z > ð3bþ5Þ=2 3=2 3a=2 > > " ¼ 2k C " < v K v n > > 3=2 3c=2 > : "M ¼ 2knð3dþ5Þ=2 CM "M. 2Þ. c d d CK av kb b ð j=kp Þ CM "M k ’ð j=kp Þ f gþ1 g e h i 2 lþ1 l CK "v k ^n ðÞ þ CM M k ^n ðÞð1 þ 2 . So far, eqs. (35) and (37) contain eight independent relations, and we still need two more constraints to determine the overall ten exponents. For this, the eddy dissipation equations for both of the velocity and magnetic induction will be employed; they are 8 Z knþ1 > > > "v ¼ 2k2 Ev ðkÞdk; < 0 ð38Þ Z knþ1 > > 2 M > 2k E ðkÞdk: : "M ¼ 0. The same argument is applied to eq. (31), that is, Z ð1 2 þ 3 Þð1 2 Þ

(53) n ðkÞ ¼ d3 2ð1 þ 2 2Þ n c d d CK av kb b ð j=kp Þ CM "M k ’ð j=kp Þ h "i klþ1 l ^ ðÞ2 CM n M. ð34Þ. 2Þ. :. ð36Þ The consistency in dimension on both sides of eq. (36) gives another three independent relationships: 8 > < a ¼ c; b ¼ d; ð37Þ > : i ¼ c=2:. . ^n ðk~Þk~ð3bþ3Þ=2 dk~; 0. Z. ð39Þ. . ^n ðk~Þk~ð3dþ3Þ=2 dk~; 0. where we have set k~ ¼ k=knþ1 and ¼ knþ1 =kn . Dimensional consistency on both sides of eq. (39) gives a ¼ 2=3;. b ¼ 5=3:. Finally, we obtain by substituting these values in eq. (37), then in (35).

(54) 1458. J. Phys. Soc. Jpn., Vol. 71, No. 6, June, 2002. C. C. C HANG and B.-S. LIN. 8 c ¼ 2=3; > > > < d ¼ 5=3; > g ¼ l ¼ 4=3; > > : f ¼ i ¼ 1=3:. 0. 10. 10-1. With all the exponents determined, the kinetic energy spectrum and the magnetic energy spectrum take respectively the following expressions: ( 5=3 Env ðkÞ ¼ CK "2=3 n ðk=kp Þ; v k ð40Þ 2=3 5=3 M ’n ðk=kp Þ; En ðkÞ ¼ CM "M k. -2. E( k ). 10. Except the time dependence, our RG result is in good agreement with their experimental results. There are some. -4. 10. 3. 5=3 "2=3 v k : ½1 þ NðtÞ2=3. / =-5. Ev ðk; tÞ . pe. Equation (40) shows that the energy spectrum have the dependence of the power law of k5=3 which is exactly the Kolmogorov energy spectrum. Compared with laboratory experiments, the result is consistent with Alemany’s6) measurement in passing a magnetized fluid to a grid mesh. Part of their experimental results are shown in Fig. 3, for which, they provided a energy spectrum of the type:. S lo. and the effective eddy viscosity and the magnetic resistivity take respectively the expressions: ( 4=3 ^n ðk=kp Þ; n ðkÞ ¼ CK1=2 "1=3 v k ð41Þ 1=2 1=3 4=3 n ðkÞ ¼ CM "M k ^n ðk=kp Þ;. 10-3. -5. 10. -8. 10. -7. 10. -6. 10. -5. 10. -4. 10. wavenumber (1/km) Fig. 4. The observational results of magnetic energy spectrum of the solar wind at 2.8 AU. from Matthaeus et al.7) The solar wind velocity is 442 km/s, and the total fluctuation energy is 4:8

(55) 1012 erg/cm3 . EðkÞ has a power-law slope of k1:70:1 . It is also noted that Biskamp9) (p. 203) mentioned that the numerical simulations showed also that EM is close to k5=3 .. other evidences from observations in astronomical physics that also support this Kolmogorov spectrum law. Matthaeus et al.7) discovered that the magnetic energy spectrum measured in the solar wind is often found to be close to k5=3 , as shown in Fig. 4. Velli et al.21) investigated a new phenomenology which involves the solar wind fluctuations near the sun and leads to a kinetic power spectrum scaling as k where ’ 1 for the largest scales, and ’ 1:5{1:7 for the small scales. Moreover, the recent observations by Leamon et al.8) (the January 1997 event which involves the solar coronal mass ejections), also showed a power law, scaled as k1:67 . 5.. Fig. 3. The experimental results of energy spectrum of velocity from Alemany et al.,6) for various values of the magnetic induction (B ¼ 0 T; B ¼ 0:25 T) and velocity (V ¼ 10 cm/s; V ¼ 20 cm/s).. Equation of the Invariant Effective Eddy Viscosity. The purpose of this section is to look for the invariant effective eddy viscosity by pursuing a differential version of the recursive relationship. Recall that the basic idea underlining the recursive RG analysis is to divide the wavenumber space ð0; k0 Þ to a supergrid region ð0; kc Þ and a subgrid region ðkc ; k0 Þ; the subgrid modes are then removed piece by piece by taking subgrid averaging over a spherical shell ðknþ1 ; kn Þ. The result will certainly depend on the cutoff ratio ¼ knþ1 =kn ; and thus the invariant (limiting) effective eddy viscosity should be sought by taking the limiting operation ! 1. First of all, we rescale the wavenumber by setting k~ ¼ k=knþ1 and rewrite eqs. (30) and (31) by expressing the results of (40) and (41) in the form:.

(56) J. Phys. Soc. Jpn., Vol. 71, No. 6, June, 2002.

(57) n ðkÞ ¼ . 8=3 knþ1. C. C. CHANG and B.-S. LIN. Z ~n . d3 j~. 1459. ðk~4 2k~3 þ k~j~3 Þð1 2 Þ 2j~2 k~2 ðk~2 þ j~2 2k~j~Þ. ~5=3 ðj~=k~p Þ ~5=3 ’ðj~=k~p Þ CK "2=3 CM "2=3 v j M j þ ; ^n ð jÞj~2 þ ^n ðjk jjÞðk~2 þ j~2 2k~j~Þ ^n ð jÞj~2 þ ^n ðjk jjÞðk~2 þ j~2 2k~j~Þ. ð42Þ. and 8=3

(58) n ðkÞ ¼ knþ1. . Z. d3 j~ ~n . ðk~4 2k~3 þ k~j~3 Þð1 2 Þ 2j~2 k~2 ðk~2 þ j~2 2k~j~Þ. ~5=3 ðj~=k~p Þ CM "2=3 ~5=3 ’ðj~=k~p Þ CK "2=3 M j v j ^n ð jÞj~2 þ ^n ðjk jjÞðk~2 þ j~2 2k~j~Þ. ~5=3 ðj~=k~p Þ CM "2=3 ~5=3 ’ðj~=k~p Þ CK "2=3 M j v j ^n ð jÞj~2 þ ^n ðjk jjÞðk~2 þ j~2 2k~j~Þ. ð43Þ. According to eqs. (42) and (43), we may assume that n ðkÞ ¼ knt ~n ðk~Þ, and n ðkÞ ¼ knt ~n ðk~Þ where t is an undetermined parameter. With this scaling law, combining the recursive relationship of viscosity and eq. (42) gives ( t knþ1 ~nþ1 k~ ¼ knt ~n k~ þ kn8=3t

(59) ~n k~ ; ð44Þ kt ~nþ1 k~ ¼ kt ~n k~ þ k8=3t

(60) ~n k~ : nþ1. n. n. For consistency of the dimension on both sides of eq. (44), we must have t ¼ 8=3 t, and thus t ¼ 4=3. It follows by 4=3 on both sides of eq. (44), dividing by knþ1 8 4 < ~ k~ 4 3 ~n k~ ¼ 3

(61) ~n k~ ; nþ1 ð45Þ 4 : ~ k~ 4 3 ~ k~ ¼ 3

(62) ~ k~ : nþ1. n. n. Now we write ¼ 1 , and let n ! 1, equivalently, we have ! 0, ~n ! ~ and ~n ! ~. Then for n 1, eq. (45) becomes 4 4 Z 2 ~ ~3 ~3 ~ ~ d~ k~ 4 ~ ð1 Þ 3 3 ~ k 2k j þ kj ð1 Þ ~ k dj þ ~ k ¼ 3 2 ~n dk~ j~2 k~2 k~2 þ j~2 2k~j~ ~5=3 ðj~=k~p Þ ~5=3 ’ð j~=k~p Þ CK "2=3 CM "2=3 M j v j þ

(63) 2 2 ~ j~ j ~jj2 þ ~ jk~ ~jj ~k ~j ~ j~ j ~jj2 þ ~ jk~ ~jj ~k ~j ð46Þ þ O 2 ; and . 4 Z 4 2 3 ~ ~3 ~3 ~ ~ ~ k~ d 1 4 3 ~ k 2k j þ k j 1 ~ ~ ~ dj þ k ¼ k 3 2 ~n j~2 k~2 ðk~2 þ j~2 2k~j~Þ dk~ ( ~5=3 ð j~=k~p Þ CM "2=3 ~5=3 ’ð j~=k~p Þ CK "2=3 v j M j

(64) 2 2 ~ j~ j ~jj þ ~ jk~ ~jj ~k ~j ) ~5=3 ð j~=k~p Þ CM "2=3 ~5=3 ’ð j~=k~p Þ CK "2=3 M j v j þ O 2 ; 2 2 ~ j~ j ~jj þ ~ jk~ ~jj ~k ~j. ð47Þ. ~ n Þ denotes the measure of the set ~ n , under the limit of n ! 1, the measure of ~ n had evaluated in1) to be where ð ~ n ¼ 2k~ þ Oð2 Þ: Therefore in the limit of ! 0, eqs. (46) and (47) simply become 8 ~ 2 ~ ~ ~ > d~ðk~Þ 4 ~ CK "2=3 CM "2=3 k > M ’ð1=kp Þ k v ð1=kp Þ ~ > þ ~ k ¼ þ 1 ; <k 3 ~ð1Þ 4 2 ~ð1Þ dk~ > > d~ðk~Þ 4 ~ > : k~ þ ~ k ¼ 0: 3 dk~. ð48Þ. Notice that the right hand side of the second equation of (48) vanishes, since the two integrands in eq. (43) will cancel out each other exactly in the limit of n ! 1. The two equations in (48) can be readily solved to yield.

(65) 1460. J. Phys. Soc. Jpn., Vol. 71, No. 6, June, 2002. C. C. C HANG and B.-S. LIN. 4 4 135 CK "2=3 CM "2=3 M ’ðkc =kp Þ v ðkc =kp Þ ðkÞ ¼ ðkc Þkc3 þ k3 364 4ðkc Þkc4=3 4ðkc Þkc4=3 4 3 CK "2=3 CM "2=3 3 k 3 k v ðkc =kp Þ M ’ðkc =kp Þ þ kc3 ; 52 kc 7 kc 4ðkc Þkc4=3 4ðkc Þkc4=3. ð49Þ. and 1=2 1=3 4=3 4=3 "M kc k ðkÞ ¼ CM ^n ðkc =kp Þ;. where kc denotes the cutoff wavenumber. It is appropriate to term ðkÞ and ðkÞ the invariant effective eddy viscosity and the invariant effective magnetic resistivity, respectively. It is notable that the RG procedure does not incur an increment of the magnetic resistivity ðkÞ, which obeys the second equation of (48) and must scale as in eq. (50) being 4=3 proportional to "1=3 . On the other hand, because of the M k minus sign in front of the terms containing CM (or M ) in the expression (49), the effect of the magnetic effect on the (. ð50Þ. effective eddy viscosity is to reduce the latter in magnitude, but not to change its basic behavior. 6.. Evaluation of the Kolmogorov Constant and Smagorinsky Model. The results of §5 will be applied here to evaluate the Kolmogorov and Smagorinsky constants. First of all, we set 1. 1. ðkÞ ¼ CK2 "v3 FðkÞ, then (49) can be written. " #) 4 135 ðkc =kp Þ ’ðkc =kp Þ þ FðkÞ ¼ Fðkc Þkc k3 4=3 4=3 364 4Fðkc Þkc 4^ðkc =kp Þkc " #" # 4 ðkc =kp Þ ’ðkc =kp Þ 3 k 3 3 k þ kc3 ; 7 kc 4Fðkc Þkc4=3 4^ðkc =kp Þkc4=3 52 kc 4 3. where used the result of (50), and set ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiwe p pffiffiffiffiffiffiffiffiffiffiffiffi CM =CK 3 "M ="v . Let us now consider a cutoff kc for the first expression of (38) and then substitute the first expression of (40) and (51) in it; this yields Z kc 2k2 ðkÞEðkÞdk ks. Z. 3 2. ¼ 2CK "v. kc. 1 3. FðkÞk ðk=kp Þdk ks. ¼ "v ; where ks denotes the wavenumber of the largest eddy existing in the flow. Canceling out on both sides, we obtain the Kolmogorov constant CK in terms of the three characteristic wavenumbers kc , kp and ks : Z kc 2 3 1 ð52Þ 3 CK ¼ 2 FðkÞk ðk=kp Þdk : ks. ð51Þ. 1 4 k 3 1 3 2 C "v ðkÞk 3 ; exp ðk=kp Þ ¼ Ap kp 2 K and. ’ðk=kp Þ ¼ Ap. 1 4 k 3 1 3 2 3 C " ðkÞk : exp kp 2 M M. It follows immediately from (40) that k 3 1 v 2=3 5=3 4=3 2 1=3 exp CK "v ðkÞk C K "v k ; E ðkÞ ¼ Ap kp 2 ð54Þ and. k 3 1 2=3 5=3 4=3 2 1=3 E ðkÞ ¼ Ap exp CM "M ðkÞk CM "M k : kp 2 M. ð55Þ In these formulas, we have the factor. Similarly, substituting the second expression of (40) and (50) in the second expression of (38) gives 4 2 Z kc 3 1 3 ð53Þ CM ¼ 2kc ^ðkc =kp Þ k ’ðk=kp Þdk : ks. So far, we have not given a precise form for and ’. Let us assume further that both the velocity and magnetic induction fields share the same form of the energy spectrum which is a combination form of the scaling laws proposed respectively by Pao,13) and Leslie and Quarini,14) that is,. Ap ðxÞ ¼. xsþ5=3 ; 1 þ xsþ5=3. to take care of energy-containing eddies, where s is a flow parameter. If we consider the leading term of (51) and apply (54) to (52), we may rewrite (52) in a more precise form as follows: 2 2M 1:5M ðkc =kp Þsþ5=3 þ 1 3 CK ¼ e log ; ð56Þ s þ 5=3 ðks =kp Þsþ5=3 þ 1 where we denote.

(66) J. Phys. Soc. Jpn., Vol. 71, No. 6, June, 2002. C. C. CHANG and B.-S. LIN. 1461. 135 Ap ðk=kp Þ expð1:5FðkÞk4=3 Þ ’ðkc =kp Þ M Fðkc Þkc þ : 364 4Fðkc Þkc4=3 4^ðkc =kp Þkc4=3 . 4 3. Following the same calculations as in the above, we may also obtain 9 2 8 " # 3 3 3 k k > > c c > > 4 4 > > 2k exp 1:5k c ^ c ^ > > < kp kp ðkc =kp Þsþ5=3 þ 1 = : log CM ¼ > s þ 5=3 ðks =kp Þsþ5=3 þ 1 > > > > > > > ; :. ð57Þ. As a matter of fact, Biskamp9) indicated that the Kolmogorov constant depends on the precise definition of the averge magnetic induction, and hence on the geometry of the large scale eddies. Here, we have provided two relationships which show how the large-scale eddies can influence not only the Kolmogorov constant CK but also the magnetic Kolmogorov constant CM , as shown in (56) and (57). The large-scale eddies with the wavenumbers kp and ks indeed play an important role in deciding both of CK and CM . Here we recall that, kp denotes the 1/(geometric size) of the energy-containing eddies and ks denotes the 1/(geometric size) of the largest eddies in fluid. In order to carry out large eddy simulation (LES) for MHD turbulence, we need to evaluate the Smagorinsky constant for MHD turbulence by using (49). First of all, we suppose no matter whenever we perform the RG analysis, the cutoff kc is always very close to the Kolmogorov scale k0 , that is, we may replace kc by k0 in (49). In doing so, we evaluate the effective eddy viscosity at k ¼ kc which is far from k0 . Then (49) becomes 4 4 135 CK "2=3 CM "2=3 v ðk0 =kp Þ M ’ðk0 =kp Þ 3 ðkc Þ ¼ 0 k0 þ kc3 364 40 k04=3 40 k04=3 4 3 CK "2=3 CM "2=3 3 kc 3 kc v ðk0 =kp Þ M ’ðk0 =kp Þ þ ð58Þ k03 : 52 k0 7 k0 40 k04=3 40 k04=3 Since kc =k0 1, we can make the following approximation 4 4 135 CK "2=3 CM "2=3 M ’ðk0 =kp Þ v ðk0 =kp Þ ðkc Þ ’ 0 k03 þ kc3 364 40 k04=3 40 k04=3 4 135 ðk0 =kp Þ ’ðk0 =kp Þ ^ ¼ CK1=2 "1=3 ðk =k Þ þ kc3 ; 0 0 p v 364 4H 1 4H 2. ð59Þ. where 0 k04=3 H 1 ¼ CK1=2 "1=3 v Next, we express "v in the resolvable velocity,. and. 1=2 1=3 H 2 ¼ CM "M 0 k04=3 :. 2 @u< ðkc Þ @u< j i "v ¼ þ : 2 @xj @xi. Substituting it in eq. (59) yields #1 < 1" < 2 3 4 @u 135 ðk0 =kp Þ ’ðk0 =kp Þ ðk Þ @u c j i þ þ kc3 : ðkc Þ ¼ ^0 ðk0 =kp Þ CK2 364 4H 1 4H 2 2 @xj @xi . Solving the above algebraic equation for ðkc Þ and replacing kc by 2=$ where $ denotes the cutoff size, we obtain 1 3 < 2 @u< 1 135 ðk0 =kp Þ ’ðk0 =kp Þ j 2 @ui 2 ðkc Þ ¼ pffiffiffi þ þ ^0 ðk0 =kp Þ CK $ 364 4H 1 4H 2 @xj @xi 4 22 < < @u @uj ; CS $2 i þ @xj @xi where 1 135 ðk0 =kp Þ ’ðk0 =kp Þ CS ¼ pffiffiffi þ ^0 ðk0 =kp Þ 364 4H 1 4H 2 4 22 This is the Smagorinsky constant, where we have left two undetermined parameters H 1 and H 2 , which require two. 3 2. 3. CK4 :. ð60Þ. additional conditions to be fully determined. In summary, the closed-form solutions (49) and (50) for.

(67) 1462. J. Phys. Soc. Jpn., Vol. 71, No. 6, June, 2002. C. C. C HANG and B.-S. LIN. ðkÞ and ðkÞ have enabled derivation of the functional dependence of CK , CM and CS in (56), (57) and (60), respectively. In other words, these numbers CK , CM and CS are not genuine constants but dependent upon the characteristic wavenumbers kp and ks of the energy-containing eddies. Namely, the theory requires an input of the large-eddy wavenumbers kp and ks from observations and/or experiments. The value of ks is approximately that of kp . This was done in our early study for incompressible flow turbulence as well as in thermal-fluid turbulence; the range of variation of the relevant Kolmogorov’s and Batchelor’s constants were found in close agreement with experiments (cf. Chang et al.1) and Lin et al.2)).. the inertial subrange when the magnetic energy in the subinertial wavenumbers exceeds the total energy in the inertial subrange. Pouquet et al.23) had an intensive study on strong MHD helical turbulence and the nonlinear dynamo effect. Recently, Nakayama,24,25) obtained also the k3=2 energy spectrum in the inertial subrange by constructing a spectral theory of strong shear Alfveń turbulence anisotropized by the presence of a uniform mean magnetic field. Of particular interest, we refer to Yoshizawa et al.26) for reviewing the importance of the cross-helicity effect, and more generally for an extensive review of turbulence theories and modeling of fluids and plasmas.. 7.. The authors are grateful to an anonymous referee for his/ her helpful comments and pointing out some recent and important references. The work is supported in part by the National Science Council of the Republic of China under Contracts No. NSC89-2212-E002-067 and No. NSC902212-E-002-238.. Concluding Remarks. In this study, we have extended our previous RG analysis of incompressible flow turbulence to incompressible MHD turbulence. The Elsa¨sser variables are introduced to write the MHD equations for the velocity and magnetic induction fields in a symmetric form. RG analysis is then performed in the wavenumber domain. Taking subgrid averaging of the equation governing the supergrid modes yields a renormalizable form of the MHD equations. To proceed further with the RG transformation, we have to impose the following two assumptions. (i) The mean magnetic induction is relatively weak compared to the mean flow velocity. (ii) The Alfveń effect holds, that is, the fluctuating velocity and magnetic induction are nearly parallel and approximately equal in magnitude. That these conditions still warrant sufficient interest are illustrated by some available data from observations in astronomical physics. Under these conditions, renormalization does not incur an increment of the magnetic resistivity , while the coupling effect tends to reduce the invariant effective eddy viscosity ðkÞ. Both the velocity and magnetic energy spectra are shown to follow the Kolmogorov k5=3 in the inertial subrange; this is consistent with some available laboratory measurements and observations in astronomical physics. Furthermore, by assuming that the velocity and magnetic induction fields share the same combined form of the energy spectra proposed respectively by Pao, and Leslie and Quarini, we are able to determine the dependence of the Kolmogorov constant CK and the magnetic Kolmogorov constant CM on the characteristic wavenumbers kc , kp and ks . The results are applied to obtain the dependence of the Smagorinsky constant CS for large-eddy simulation, which however contains two undetermined constants to be resolved. In spite of the present success, it must be stressed upon that the imposed conditions (i) and (ii) imply a negligible effect of the subgrid cross helicity between the velocity and magnetic fields. There are cases where the effect is important and which may lead to quite different energy spectrum. In an early study, Kraichnan22) derived a k3=2 energy spectrum of. Acknowledgement. 1) C. C. Chang, B. S. Lin and C. T. Wang: submitted for publication. 2) B. S. Lin, C. C. Chang and C. T. Wang: Phys. Rev. E 63 (2000) 016304. 3) C. B. Kim and T. J. Yang: Phys. Plasmas 6 (1999) 2714. 4) W. Z. Liang and P. H. Diamond: Phys. Fluids B 5 (1993) 63. 5) W. D. McComb: The physics of fluid turbulence (Oxford University Press, New York, 1990). 6) A. Alemany, R. Moreau, P. L. Sulem and U. Frisch: J. de Meća. Theor. Appl. 18 (1979) 277. 7) W. H. Matthaeus, M. L. Goldstein and C. Smith: Phys. Rev. Lett. 48 (1982) 1256. 8) R. J. Leamon, C. W. Smith and N. F. Ness: Geophys. Res. Lett. 25 (1998) 2505. 9) D. Biskamp: Nonlinear Magnetohydrodynamics (Cambridge University Press, Cambridge, 1993). 10) T. Hatori: J. Phys. Soc. Jpn. 53 (1984) 2539. 11) M. K. Verma: Phys. Plasmas 6 (1999) 1455. 12) H. Chen and D. Montgomery: Plasma Phys. Control. Fusion 29 (1987) 205. 13) Y. H. Pao: Phys. Fluids 8 (1965) 1063. 14) D. C. Leslie and G. L. Quarini: J. Fluid Mech. 91 (1979) 65. 15) Y. Zhou, G. Vahala and Hossain: Phys. Rev. A 37 (1988) 2590. 16) H. Alfveń: Cosmical Electrodynamics (Oxford University Press, London, New York, 1950). 17) T. G. Cowling: Magnetohydrodynamics (John Wiley & Sons, New York, London, 1963) 3rd ed. 1963. 18) W. M. Elsa¨sser: Phys. Rev. 69 (1946) 106; ibid. 70 (1946) 202; ibid. 72 (1947) 821; ibid. 79 (1950) 183. 19) H. A. Rose: J. Fluid Mech. 81 (1977) 719. 20) W. D. McComb: Phys. Rev. A 26 (1982) 1078. 21) M. Velli, R. Grappin and A. Mangeney: Comput. Phys. Commun. 59 (1990) 153. 22) R. H. Kraichnan: Phys. Fluids 8 (1965) 1385. 23) A. Pouquet, U. Frisch and J. Leorat: J. Fluid Mech. 77 (1976) 321. 24) K. Nakayama: Astrophys. J. 523 (1999) 315. 25) K. Nakayama: Astrophys. J. 556 (2001) 1027. 26) A. Yoshizawa, S. Itoh, K. Itoh and N. Yokoi: Plasma Phys. Control. Fusion 43 (2001) R1..

(68)