Statistical properties of MHD turbulence and turbulent dynamoShigeo Kida, Shinichiro Yanase, and Jiro Mizushima Citation: Phys. Fluids A 3, 457 (1991); doi: 10.1063/1.858102 View online: http://dx.doi.org/10.1063/1.858102 View Table of Contents: http://pof.aip.org/resource/1/PFADEB/v3/i3 Published by the American Institute of Physics. Additional information on Phys. Fluids AJournal Homepage: http://pof.aip.org/ Journal Information: http://pof.aip.org/about/about_the_journal Top downloads: http://pof.aip.org/features/most_downloaded Information for Authors: http://pof.aip.org/authors

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Statistical properties of MHD turbulence and turbulent dynamo Shigeo Kida Research Institute for Mathematical Sciences, Kyoto Universityx Kyoto 606, Japan

Shinichiro Yanase Engineering Mathematics, Faculty of Engineering, Okayama University, Okayama 700, Japan

Jiro Mizushimaa) Department of Information Science, Sagami Institute of Technology, Fujisawa 251, Japan

(Received 29 September 1989; accepted 6 November 1990)

Statistical properties of MHD turbulence and the mechanism of turbulent dynamo are investigated by direct numerical simulations of three-dimensional MHD equations. It is assumed that the turbulent field has a high symmetry and that the fluid has hyperviscosity and hypermagnetic diffusivity. An external force is exerted on the fluid as kinetic energy and helicity sources. The main concern of the present study is whether magnetic fields of scales comparable to the dominant fluid motions can be generated or not. It is shown that the turbulent dynamo is effective if hypermagnetic diffusivity is smaller than a critical value. The total energy spectrum is close to the k - 5’3 power law in the inertial range. The energy transfer between kinetic and magnetic fields is discussed.

I. INTRODUCTION

A subject that has attracted a great amount of interest in physics, geophysics, and astrophysics is how large-scale magnetic fields were generated and are sustained in the G&Xy”* or in planets.3 Such a mechanism is called the “dynamo effects,” which has been studied extensively by many researchers (see Moffatt4). The dynamo theories are classified into two groups, one is the laminar dynamo and the other is the turbulent dynamo. The laminar dynamo, for which a lot of models have been proposed, has a com- mon feature among almost all models. It requires the lack of reflectional symmetry of the fields which is expressed mathematically as to have a nonzero helicity.

Motions of magnetohydrodynamic (MHD) fluids with gigantic scales in nature are usually turbulent and the tur- bulent dynamo is expected to play an important role. A kinematic theory of turbulent dynamo was proposed by Steenbeck et al.’ and developed further by Moffatt6 and others to explain the generation of a large-scale magnetic field. The basic method they employed is a two-scale anal- ysis, where a turbulent field is assumed to consist of a magnetic field of large scales and of velocity and magnetic fields of much smaller scales. A turbulent magnetic field of small scales is excited by a turbulent velocity field as a form of AlfvCn waves in the presence of a weak initial magnetic field of large scales. Then, turbulent velocity and magnetic fields of small scales strengthen and maintain the large- scale magnetic field. This process is called the “CX effect.” In order that this effect may work, it is crucial that the helicity of a turbulent velocity field has a nonzero value, i.e., the velocity field has no reflectional symmetry.

The kinematic theories revealed several important as- pects of turbulent dynamo. Montgomery and Hatori’ and Montgomery and Chen8 evaluated the turbulent transfer

rates from small to large-scale excitations using two-scale analysis. They showed that even in situations in which the alpha coefficients vanish, the beta coefficients can contrib- ute to amplification of the mean magnetic field. In other words, even if the total helicity of turbulence is zero, the temporal or spatial fluctuation of helicity can generate a large-scale magnetic field. The dynamo effect is classified into two groups, the fast dynamo and the slow dynamo. The growth rate of the large-scale magnetic field by slow (or fast) dynamo tends to zero (or remain finite) as the magnetic Reynolds number increases. Numerical results by Galloway and Frisch’ suggest that the ABC flow, which has a spatial symmetry, can act as a fast dynamo.

The kinematic theory with two-scale analysis, how- ever, has two major defects though it produced the several interesting results mentioned above. One is the scale sepa- ration of turbulence into two different scales. Since it is an essential character of turbulence that the flow field is com- posed of modes of continuous scales, such a scale- separation analysis gives a rather easy look at the highly nonlocal contributions of the small-scale field to the large scale field.’ The other is the assumption that the turbulent velocity field is given a priori and no reaction from a de- veloped magnetic field is taken into consideration.

Kraichnan” proposed a phenomenological theory by extending the Kolmogorov theory for the incompressible Navier-Stokes turbulence to derive EK( k) =EIM( k) a k- 3’2 in the nonhelical case, where EK( k) and E,(k) are the kinetic and the magnetic energy spectra, respec- tively. By employing an eddy-damped quasinormal Mar- kovian (EDQNM) approximation, Pouquet et al.” con- firmed Kraichnan’s result for the nonhelical case and argued the mechanism of the turbulent dynamo in a helical MHD turbulence. Saito et aZ.‘* derived another conclusion, that E,(k)=E,(k) cck- 5’3 for a nonhelical MHD turbu-

*)Present address: Faculty of Education, Wakayama University, Wakayama 640, Japan.

457 Phys. Fluids A 3 (3), March 1991 0899-8213/91/030457-09$02.00 0 1991 American Institute of Physics 457

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lence, by using the modified zero fourth-order cumulant approximation. The measurement of the magnetic field in the solar wind by Matthaeus et aLI shows that EM(k) = k- ‘.‘*‘.’ though it is not clear whether the tur- bulence is isotropic or not. A critical condition for the turbulent dynamo was investigated using EDQNM theory by Leorat et a1.I4 They obtained a critical magnetic Rey- nolds number above which the magnetic energy is gener- ated and showed that the generation of magnetic energy is independent of the kinetic Reynolds number.

Meneguzzi et aLI made a direct numerical simulation of three-dimensional MHD turbulence. The kinetic energy and helicity were injected into the turbulence. It was shown that the magnetic energy is generated near the in- jection wave number at first, and then the peak of the magnetic energy spectrum moves down to the minimum wave number. It then keeps growing linearly without sat- uration. The linear growth was shown to be consistent with a two-scale model by Kraichnan.i6 They also found that even if the total helicity is zero, the turbulent dynamo is still effective, which is consistent with results of Montgom- ery and Hatori

A three-dimensional self-organization process in an MHD turbulence is investigated by means of a direct nu- merical simulation by Horiuti and Sate.” They showed that the magnetic structure is transformed into a simpler one through magnetic reconnection. The peak of magnetic helicity spectrum drifts toward low wave numbers while the magnetic energy spectrum drifts toward high wave numbers. Effects of a dc magnetic field on MHD turbu- lence and a mean electric current were investigated by Dahlburg et aZ.18 They made a numerical simulation of MHD turbulence with a constant, uniform dc magnetic field and a mean electric current along the magnetic field to show that the average toroidal magnetic field at the bound- ary reverses the sign spontaneously. Other works on MHD turbulence, especially on two-dimensional turbulence, are reviewed by Kraichnan and Montgomery.”

In the present paper, we study dynamo effects numer- ically for the generation of magnetic field of scales cumpa- rable with that of the energy-carrying eddies of turbulence. Because the typical length and time scales of the kinetic and magnetic f’ields are comparable, the coupling of these two fields is much stronger and more complicated than that for the case of different scales. The two-scale analysis cannot be applied any more, nor can the concept of the a effect be introduced. To the authors’ knowledge, there seems no dynamo theory applied to this situation. Three- dimensional MHD equations are solved numerically in a cubic domain by using the spectral method, A high sym- metry (Kida2’) is imposed on the velocity and magnetic fields to save the necessary memory and computation time. High-symmetric nonmagnetic incompressible Navier- Stokes flows were investigated earlier by Kida and Mu- rakami.21122 They confirmed that the velocity field is statis- tically isotropic in the small scales and showed that statistical properties of the small-scale turbulence coincide with laboratory experiments. We expect that the symmetry may not break down the dynamo action if the dynamo

effect is originated from the interaction between the kinetic and magnetic components in the small scales (inertial range). The large-scale structures of the magnetic and ve- locity fields, however, may be affected by the assumption of high symmetry. For example, the total helicity is zero in a high-symmetric flow. The nonzero total helicity is very important for the inverse cascade (a effect) of the mag- netic energy. As will be shown later, the local helicity [or fluctuation helicity ( Kraichnan2j)] may also contribute to the inverse cascade. The critical condition of the turbulent dynamo is investigated and the statistical properties of MHD turbulence are studied in detail in terms of several integral quantities~ and energy spectra. We also examine how the large-scale magnetic field grows up, which has been explained by closure theories as the inverse cascade of magnetic energy or helicity.

II. DYNAMICAL EQUATIONS

Suppose that an MHD fluid is filled in a periodic cube with a side length of 2n and is driven by an external force. We assume that both the velocity and magnetic fields have high symmetry. Then, information for a single component of the velocity in the fundamental box (04x, y, z < 5~) is sufficient to describe the whole velocity field. The number of degrees of freedom of a high-symmetric flow is reduced by a factor of & compared with that of a general periodic flow. Furthermore, in order to increase Reynolds numbers to be simulated with a given resolution, we employ hyper- viscosity and hypermagnetic diffusivity instead of the usual viscosity and magnetic diffusivity. The inertial range dy- namics (and therefore the mechanisms of fast dynamo) may remain essentially unaffected by the choice of this modified viscosity and diffusivity (Meneguzzi et aZ.;15 Pas- sot and Pouquet;24 Farge25).

If the magnetic Geld is resealed to have the dimension of velocity, the dynamical equations are written as

al z i- (u*V)u= (b+V)b - Vp - YA*U + f, (1)

z + (u*V)b= (b*V)u - ;1A2b,

v-u=o, (3)

V.b=O, (4) where u is the velocity, b is the magnetic field, p is the total pressure, Y is the hyperviscosity, /z is the hypermagnetic diffusivity, and f is a driving force.

The driving force is applied in such a way that several Fourier components of velocity [see (6) below] always have prefixed amplitudes. For the x component of velocity we fix the following Fourier components:

ZZJl,*t,Al)= - Z,( - 1,&3,&l) = - B,(l,*l,*t)

= Zx( - 1,d=l,13) 1 . = --ml,

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Y c

DK’k’=Z k-l/2<lk’l<k+l/2 k4/ ii (k’) 12, (14)

H.(4,*2,0)= - H,( -4,*2,0)

= Z,(4,0,*2)

=- i2.J -4,0,+2)

=- ( qlmi, (5d

fJ2,*4,0)= - Zx( - 2,=‘=4,0)

= i1,.2,0,*4) = - il,( - 2,0,&4) =f i,

which are represented in the physical space as

u,=fsinx(cos3ycosz-cosycos3z) + (JCiC2)

X [sin 4x(cos 2y + cos 22) - 2 sin 2x(cos 4y

+ cos 42) 1. (5b)

The other two components, u,, and u, are given by replac- ing x, y, and z cyclically in (5). The time argument is omitted for brevity. This force excites Fourier modes of wave numbers of m z 3.3 and JzTj~4.5 so that the forc- ing wave number kf is roughly 4. Velocity field (5)) which satisfies the high-symmetry condition, was determined so as to maximize the helicity in the fundamental box [local he/icily, see ( 18) below].

The periodic velocity and magnetic fields are expanded in Fourier series as

U(X) = 1 ii (k)exp(lk*x), k

(6)

b(x) = c G (k)exp(&*x). k

(7)

The band-averaged spectra of kinetic energy E,(k) and magnetic energy E,(k) are, respectively, defined by

1 EK(k)=i k-1/2<Ik’l<k+1/2

c I ii (k’) 12, (8)

1

E.u(k)=Z k- 1/2<(k’l<k+ l/2 c I i; (k’) I29 (9)

where k=O,l,... . The sum of these two spectra gives the total energy spectrum

ET(k)=&(k) +-h(k). (10) The kinetic, magnetic, and total energies per unit mass are then calculated by summing up the respective energy spec- tra over all the wave numbers as

a,=; (1111~) = 1 E,(k), L k

DrM’f ( lb12> = 1 E&k), k

&=; (lu12> + ;(]b12) = 1 E,(k), k

(11)

(12)

(13)

where ( ) represents the spatial average over the periodic box.

The spectra of the kinetic and magnetic energy dissi- pations are defined in the same way as energies:

a D”‘k’=Z k- 1/2<lk’l<k+ l/2

c k41 i; (k’) 12. (15)

These spectra sum up to give the negatives of kinetic and magnetic energy dissipations, respectively, i.e.,

DK=~(lAu12) = c D,(k), k

(16)

DM=A( lAb12) = c D,(k). k

(17)

The kinetic and magnetic helicities are zero due to the high-symmetry condition if the average is taken over the whole domain. If the average is limited in the fundamental box, however, both helicities-can have nonzero values. We shall call them the local kinetic helicity

HK=f( (u-m)), (18)

and the local magnetic helicity

H,=i((a*b)). (19)

Here w = V x u is the vorticity and a is the vector potential of magnetic field (b = V X a), and ( ( ) ) denotes the spa- tial average over the fundamental box. We expect these local helicities to play an important role as the usual he- licities.

Ill. DEVELOPED TURBULENCE AND INITIAL CONDITION

The numerical integration of ( 1 )-( 4) is made by using the spectral method with 121S3 collocation points. The ali- asing interactions in the calculation of the nonlinear terms are eliminated by the $ rule so that the maximum wave- number k,, is 42 and the total number of interacting modes is 853 (Orszag26). The time marching is made by the Runge-Kutta-Gill scheme.

As the initial velocity field we take a fully developed turbulence, which is created by solving ( 1) and (3) with- out magnetic field for a sufficiently long period in which a statistically equilibrium state is attained. In Fig. 1, we plot the kinetic energy spectrum for Y = 2~ lo- 6, the velocity field of which is taken as the initial condition for the sub- sequent simulations of MHD turbulence for cases III and IV. We can see that the spectrum is well developed to the highest wave number and that there exists a wave-number region where EK( k) a k - 5’3.

The initial magnetic field is given as a small random seed having an energy spectrum of

EM(k) =Ak” exp [ - Bk’], (20)

where A=9X 1O-39 and B=5.50~ 10e6. The level of the initial magnetic spectrum (20) is very low compared with the kinetic energy spectrum over the whole wave-number range. Actually, E,(k) increases with k up to the maxi- mum wave number k,,,, but it is three orders of magni-

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2 I , t,r,l 3 4 5 6 78910

k

FIG. 1. Kinetic energy spectrum of nonmagnetic turbulence in a statis- tically equilibrium state for $6 = 2X 10e6. This velocity field is used for the initial condition of the subsequent simulations of MHD turbulence for cases III and IV.

tude less than EK( k) even at k= k,,,. The ratio of 8)M to $iy at t=o is 1.5x 10w4.

We made many runs with different combinations of hyperviscosity and hypermagnetic diffusiiity. As will be shown later, the magnetic field is generated or destroyed depending on the values of hyperviscosity and hypermag- netic diffusivity. In Table I, we show five cases in which the magnetic energy is generated, We also list the time incre- ment At used for the time advancement, the final time t,,, of the numerical simulation, and the time- $ when the MHD turbulence reaches a statistically equrhbrium state. By comparing tes with the large-scale eddy turnover time T, [see (24) and Table III], we can see that it takes a long time for the MHD turbulence to attain its stationary state: fes/TL = 62, 33, 17, 130, and 17 for cases I-V, respectively. Note that the saturation time is largest for case IV. The saturation time is longer for larger values of ;C/Y.

An important parameter that characterizes the veloc- ity field is the Reynolds number. In the present hypervis- cosity turbulence, we can define a microscale Reynolds number (analogous to the Taylor microscale Reynolds number for an ordinary viscosity) as follows. Take the square root of the turbulent kinetic energy and the micro-

TABLE I. Parameters used in the simulation. Here, Y is the hyperviscas- ity, h is the hypermagnetic diffusivity, r,,,,, is the final time of the simu- lation, fW is the time at which MHD turbulence enters a statistically equilibnum state, At is the time increment, Re the microscale Reynolds number, and Rm the magnetic Reynolds number.

Cases Y il fnm f At Re Rm

I 10-S 10-s 170 100 0.01 71 71 II 5x10-6 5x 10-6 100 so 0.02 80 80 III 2x 10-6 2x10-6 200 30 0.02 143 143 IV 2x 10-6 to-5 310 210 0,Ol 102 20 V 10-5 2x10-6 70 30 0.01 95 475

460 Phys. Fluids A, Vol. 3, No. 3, March 1991

TABLE II. Characteristic time scales. Here, ?“E is the eddy turnover time of energy-carrying motion, rS is the dissipation time, and ,5 is the expo- nential growth rate of magnetic field.

Cases

I 1.6 0.21 7.0 0.142 II 1*6 0.18 3.9 0.256 III 1.7 0.18 2.4 0.381 IV 1.6 0.15 6.3 0.158 V 1.7 0.25 2.5 0.397

scale (analogous to the Taylor microscale) for the charac- teristic Velocity and length, reSpe&ely, as u= mK and L = ( Y%‘~/D#) 1’4. Then the microscale Reynolds number is defined-by

Re= UL’/v= (2 $&vD~) “4, (21

Similarly, the magnetic Reynolds number is defined by 1

(22 1

In Table I, we list the values of Re and Rm for five cases. Two characteristic time scales of a turbulent field are

introduced here. One is the dissipation time

Ts= ( v/@~) "5, (23)

which represents a time scale of the smallest motion (cor- responding to the Kolmogorov time for the ordinary vis- cosity). Another characteristic time is related to the large- scale motion of turbulence. The fluid motion is driven by an external force of a wave number kfz4 [see (5)]. The characteristic length and velocity of the energy-carrying motion are estimated, respectively, by L=2n-/kf and U = a, By tak’ mg the ratio of L and U, we obtain the time scale of the dominant motion of turbulence as

T,= (2n/kJ/ a. As will be shown in the next section, even after the mag- netic field is generated, the form of the kinetic energy spec- trum does not change very much, nor the characteristic

FIG. 2. Time development of kinetic energy (solid line) and magnetic energy (dashed line) for case III.

Kida, Yanase, and Mizushima 460

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TABLE III. The mean values of energies and energy dissipation rates. Here, g, is the kinetic energy, gsu is the magnetic energy, 5, is the kinetic energy dissipation rate, and DY is the magnetic energy dissipation rate. The averages are taken over equilibrium periods f,<t<t,,,,.

cases 8, ZfiM EK B, ZM/ZjK BM/DK

I 0.451 0.0181 0.152 0.0401 0.0396 0.264 II 0.498 0.0166 0.169 0.0476 0.0333 0.282 III 0.414 0.0380 0.100 0.0822 0.0918 0.826 IV 0.484 0.0070 0.165 0.0238 0.0145 0.144 V 0.415 0.0373 0.100 0.083 1 0.0899 0.831

times. The values of the above two characteristic times are shown in Table II, where the time-averaged values are used for g4, and DK (see Table III).

IV. STATISTICAL PROPERTY OF MHD TURBULENCE

In this section, we investigate the statistical property of imum wavenumber is one, whereas the forcing wave num- the MHD turbulence mainly for case III, where Y and ,J ber kf is about 4. The closeness of these two wave numbers take the smallest values in the five cases listed in Table I. may inhibit the inverse cascade of magnetic energy.

A. Energies and energy dissipations

The time developments of kinetic energy gK and mag- netic energy gM for case III are plotted in a linear- logarithmic scale in Fig. 2. The kinetic energy does not change significantly throughout the whole time but only fluctuates around the mean value. The magnetic energy, on the other hand, increases exponentially in the initial period O(t 6 30, and then it fluctuates around its own mean value.

The time averages of the kinetic and magnetic energies, gK and gLU, after they reach a statistically equilibrium state, are listed in Table III for the five cases. The averages are taken over tqeq<t<t,,,. We also show the ratio of the magnetic to kinetic energies in the same table. It is found that the magnitude of @ ‘)K hardly depends on Y nor on 2, whereas the magnitude of g)M strongly depends on both Y and A. The value of magnetic energy for case IV is a few times lower than for the other cases. It seems that the magnetic field in case IV is not in a fully developed equi- librium state even at the final time of the simulation ( tmax/ TL = 192). We do not know why such a peculiar behavior of the magnetic energy is seen only for case IV. On the’ other hand, the value of magnetic energy for case I is larger than that for case II. It is likely that the magnitude of magnetic energy depends very sensitively on the individual character of each realization of turbulence slightly above the critical condition at which the dynamo effect is neutral (see Sec. IV B). The magnitude of magnetic energy and the ratio of the magnetic to kinetic energies are very similar in cases III and V and are greater than in the other cases. The ratio of the magnetic to kinetic energies, which is about l/10, is comparable with that for a nonhelical case by Meneguzzi et al. I5 They showed that the ratio goes up beyond 2 for their helical forcing case. The suppression of the increase of the magnetic ,energy in our case may be due to the following two reasons: the lack of net helicity, and that there is no room for magnetic energy to cascade back- ward. Because of the periodic boundary condition the m in-

10-7~“““““‘111’I”t~ 0 50 100

t 150 200

FIG. 3. Time development of kinetic energy dissipation (solid line) and magnetic energy dissipation (dashed line) for case III.

The time development of the kinetic and magnetic en- ergy dissipations, DK and DM, is plotted in. Fig. 3. It is interesting to note that the magnitudes of DK and DM are nearly equal after tz30 (an equipartition of the energy- dissipation rates). If we denote the time averages (over tq< t<t,,,) of DK and DM by Bk and EM, respectively, then D,/B K--,O.8. The values of 5,/D, for the other four cases are shown in Table III. The ratio of the two dissipation rates varies with parameters.

Only the kinetic energy is supplied to the MHD tur- bulence. The present results show that in case III the 55% [ = Bx/( zK + BM) x lOO%] of the kinetic energy sup- plied by an external force is transferred to higher wave- number components of kinetic energy and dissipated by viscosity, while the 45% [ = oM/(zK + BM) X lOO%] is transformed into the magnetic component by the nonlinear interaction between velocity and magnetic fields and dissi- pated by the magnetic diffusivity at higher wave numbers. We shall investigate the interaction between the velocity and magnetic fields in Sec. IV E.

6. Critical condition of turbulent dynamo

We observe in Fig. 2 (and in other cases also) that magnetic energy 8:M decreases abruptly at first due to the strong dissipation at high wave numbers. Then, it increases almost exponentially as

~.eew[Ptl (25)

for cases I-III and V. For case IV, 8:M increases and de- creases repeatedly before t=: 120 and then begins to grow exponentially. The growth rate j? is listed in Table II, which is evaluated from the data in the period of O(t<t, for cases I-III and V, and from those in the period of 120<t&t, for case IV. As seen in Table II, the growth rate j!l hardly depends on Y but it depends strongly on A. In Fig. 4, we plot the values of /I for cases I-III (in which Y = A). The least-squares fitting gives an approximately linear re- lation between p and log 1:

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2.0

P HM(t)

0

FIG. 4. A logarithmic-linear plot of the exponential growth rate p of magnetic energy versus hypermagnetic diffusivity 1. The straight line represents 0 = - 0.342 log,$. - 1.56.

FIG, 6. Time evolution of local magnetic helicity & for case III.

/3= - 0.342 loglO/z - 1.56. (26) By putting B = 0 in (26), we obtain the critical value

of the hypermagnetic diffusivity as /2, = 2.7 X 10 - 5. Using the values of Rm listed in Table I, we can estimate the corresponding critical magnetic Reynolds number as Rm,=;43.

We also made numerical simulations for ~=A.)2.7xlO-~. The results for Y = il = 10 - 4 and 5 x 10 - 5 show that the magnetic energy decreases to zero. For Y = il = 2 X 10 - 5, the magnetic energy grows and de- cays repeatedly for a long time without any clear exponen- tial growth.

In Table III, we give the characteristic time l/j3 of the exponential growth of magnetic energy. We can see that l/D is larger than the characteristic time TL of the domi- nant motion of turbulence. This implies that the fluid mo- tion varies appreciably during the period of magnetic field growth contrary to the assumption in the kinematic dy- namo theory in which feedback effects of the magnetic field to the fluid motion are not taken into account.

C. Kinetic and magnetic helicities

Helicities are regarded as one of the most important quantities in the dynamo theory. In Fig. 5, we plot the time development of local kinetic helicity HK [see ( 18) J for case III. The time-averaged value of HK is about 2.3. It is al- ways positive though its magnitude fluctuates very much. The positiveness of HK comes from some Fourier compo-

HK(t)

FIG. 5. Time evolution of local kinetic helicity HK for case III.

nents of the velocity field [see (5)] that are held constant in amplitudes which give a positive helicity in 04x, y, z<r/2.

In Fig. 6, we show the time development of local mag- netic helicity HM [see (19)] for case III, which gives us the most important information. It is seen that HM-O for ts22, but it begins to oscillate at f~ 30 and then takes negative values (opposite sign to HK) most of the time. The averaged value is - 4.30X 10a4. Unlike HK, the value of HM sometimes changes the sign. It was observed earlier by Pouquet et al.” and Meneguzzi et al-l5 that the magnetic helicity takes an opposite sign to the kinetic he- licity, An interpretation of the occurrence of the opposite sign is given by Pouquet et al.”

The local cross-helicity HK~ is defined by

&w=t ((wb)). (27) In Fig. 7, we plot the time variation of lyKM for case III. It oscillates around the time-averaged value - 1.8~ 10 - 4 which is very small. The local vorticity-magnetic field cor- relation C is defined by

C=; ((wb)), (28) The time variation of C for case III is plotted in Fig. 8. The time average of C is - 2.8X IO- 2. Since both the local cross-helicity and the local vorticity-magnetic field corre- lation oscillate around zero, there is no definite correlation between the velocity or vorticity field with the magnetic field. This implies that the magnetic field is not strongly coupled with the velocity field in the averaged sense.

FIG, 7, Time evolution of local cross-helicity HKM for case III.

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C(t) 0

-1.0

' 0 100 t 200

FIG. 8. Time evolution of local vorticity-magnetic field correlation C for case III.

Therefore, we can say that it is the magnetic helicity that reflects the characteristic properties of the MHD turbu- lence best.

From the data obtained for energies and helicities, we can discriminate two characteristic periods of evolution. Magnetic energy ZS’M increases exponentially during 0 < c 5 30. After a transitional period, 30 <, t 5 50, the MHD turbulence arrives at an equilibrium state and the magnetic energy oscillates around the mean value.

D. Energy spectra

Delicate properties of MHD turbulence are expressed better by the energy spectra of velocity and magnetic fields than the integrated quantities discussed in Sets. IV A and IV c.

In the initial period of evolution the magnetic energy is generated rapidly by the transfer of energy from the veloc- ity field. In Fig. 9, we plot the kinetic energy spectrum E,(k) and the magnetic energy spectrum EM(k) at f= 18 for case III, when the magnetic energy increases exponen- tially in time.

The kinetic energy spectrum EK(k), the magnetic en- ergy spectrum E,(k), and the total energy spectrum

10-l

10-2

IO“

I o-'

I o-"

I o-6

10-7

10-e

!o-p

-r - \ I \ /\

c- \ W-\,\,

‘\ EK( k) -\

‘1 r

B -s

EM(k) s- \/.\,,, /?//------.+-...

- ,f \.\,,,’ r ./ ?

I 4 3 4 5 678910 20 30 40

k

FIG. 9. The kinetic EK and magnetic EM energy spectra at t= 18 for case III.

1 o-1 5

2

1 o-2 5

2

10-3 5

2

I o-4 5

c3 4 5 6 7 8 910 20 30 40 k

FIG. 10. The total E,(k), kinetic E,(k), and magnetic EM(k) energy spectra averaged over 50<2<70 for case III.

E,(k) averaged over 50~~70 for case III are plotted in Fig. 10. Three straight lines represent the k- 5’3, k - ‘, and k” power spectra, respectively. It is seen that there exists a wave-number region where E,(k) a k - 5’3. By use of the least-squares method, we calculate the Kolmogorov con- stant A of the total energy spectrum averaged over 50<t<70 as

E,(k)=A(DK+ DM)2’3k-55/3, with A=2.1. (29)

Here the total energy dissipation rate, DK + D,, is also evaluated by averaging over 50<t<70. It is very interesting to note that the magnitude of Kolmogorov constant A for MHD turbulence is close to that obtained experimentally and numerically in fluid turbulence (Monin and Yaglom;” Kida and Murakami’l). It is reasonable to expect that the total energy spectrum of an MHD turbulence obeys the Kolmogorov k - 5’3 power law as the kinetic energy of a nonmagnetic fluid turbulence because the total energy is conserved by the nonlinear terms in the former as the ki- netic energy in the latter. It is also found that EK( k) a k - ’ and EM(k) a /co in the middle wave-number region. There- fore, the magnetic energy spectrum E,(k) has a form sim- ilar to the enstrophy spectrum k2EK(k). These results agree with Batchelor’s conjecture that the vorticity field behaves similar to the magnetic field if the magnetic energy is small enough.28

In the high wave-number region, however, Batchelor’s prediction does not hold because a vital energy transfer from velocity field to magnetic field occurs and the velocity field should be strongly affected by the presence of mag- netic field. It should be noted that E,(k) > EK( k) at the highest wave-number region, which agrees with the results obtained by Meneguzzi et al.” Thus there is no equiparti- tion between the kinetic and magnetic components. The energy spectra averaged over lOO<t<120 are similar to those in Fig. 10.

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0 0 100 t 200

FIG. 11. Time variation of energy transfer from velocity to magnetic components for case III.

E. Energy transfer between kinetic and magnetic fields

We now consider how the kinetic energy, which is injected to the velocity field at low wave numbers, is trans- ferred to the magnetic field.

The equations of the kinetic and magnetic energies are obtained by taking a spatial average of ( 1) multiplied by u and (2) multiplied by II, respectively:

f (i 1~12)= - T-~(lAul’) + (uifi>,

f ($ /b/l) =T - n(lAb12),

T= i, (~,b,~) j (32)

represents the exchange of energy between the kinetic and magnetic components. If T > 0, the kinetic energy is trans- formed into magnetic energy and vice versa. In Fig, 11, we plot the time variation of T for case III. After a transient period O~t6 30, T is always positive. Thus the energy is continuously transferred from velocity to magnetic fields.

A more detailed discussion of the energy exchange be- tween velocity and magnetic fields can be made by inves- tigating the energy transfer spectra Tb,(k) and Tub(k). Here, TbU( k) is defined as the increasing rate of the kinetic energy at wave number k through the interaction between velocity and magnetic fields, which is evaluated by sum- m ing up the Fourier coefficients of u.(b*V) b within a band of a shell containing k= Ik}. Likewise, Tub(k) is defined as the increasing rate of magnetic energy, which is evaluated from the Fourier coefficients of b*(b*V)u in the same man- ner. Note that

T= - c T,,(k)= c Tub(k), k k In the initial period of evolution, the magnetic energy

where the sum is taken over all the wave numbers, increases exponentially, Then, the M I-ID turbulence enters In Fig. 12, we plot Tb,(k) and T,,(k) for case III into a statistically equilibrium state where the magnetic

averaged over 50gtG70. We can see that Tbu( k) is nega- energy oscillates around the mean value. The energy is tive at low and m iddle wave numbers and is positive with continuously transferred from the velocity to magnetic smalf magnitude at high wave numbers. On the other components. The analysis of the transfer spectra shows

FIG. 12. The energy-transfer spectra averaged over 50(ts70 for case III.

hand, T,,(k) is positive almost everywhere except at sev- eral wave numbers. The magnitude of T&,(k) becomes larger at higher wave numbers. Therefore, it seems that energy flows mainly out of the low and m iddle wave- number regions of the velocity field to the high wave- number region of the magnetic field. It is also seen that a small amount of energy is transferred to the high wave- number region of the velocity field from the magnetic field. The energy transfer spectra Tbu( k) and Tub(k) averaged over lOO~t<120 are similar to those in Fig. 12.

The results of the present calculation supports the mechanism which has been supposed for a long time that a turbulent magnetic field of small scales is excited by a tur- bulent velocity field as a form of AlfvCn waves. It has been conjectured that energy is transferred from the velocity to magnetic fields at high wave numbers. The present results seem to be a little unexpected since the energy comes from the low and m iddle wave-number regions of the velocity field and since a small amount of energy is transferred to the high wave-number region of the velocity field. The ex- act mechanism of the energy transfer between the velocity and magnetic fields, however, cannot be fully understood only by the information of T,,(k) and T&(k). A more detailed investigation of the energy transfer mechanism is left for future work.

V. CONCLUDING REMARKS

The condition of the occurrence of turbulent dynamo and the characteristic property of MHD turbulence are studied by a direct numerical simuIation of MHD equa- tions. We found that it does not depend on kinematic vis- cosity but on magnetic diffusivity whether the magnetic fieId is excited or not.

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that the transfer causes a decrease of the kinetic energy spectrum at low and middle wave numbers and an increase of the magnetic energy spectrum at high wave numbers.

The present study includes two assumptions, one is high symmetry and the other is hyperviscosity and hyper- magnetic diffusivity. There is a fear that the results ob- tained here may be affected by these assumptions. How- ever, considering the substantial amount of experience in successfully applying these assumptions to the Navier- Stokes turbulence (see, for example, Kida and Mu- rakami;2’*22 Passot and Pouquet24), we expect that the es- sential feature of a dynamo mechanism may not be destroyed by them. Further investigation of MHD turbu- lence without these assumptions is necessary to clarify it.

Although we did not discuss it in this paper, we have observed the formation and destruction of coherent struc- tures of vorticity and magnetic fields at earlier times of evolution [e.g., for t5 120 for case III until the local mag- netic helicity takes large negative values (see Fig. 6)], and that these coherent structures die away at later times. The statistical properties of MHD turbulence are drastically affected by such a coherent structure. This subject will be reported in a separate paper soon.

ACKNOWLEDGMENTS

The authors express their cordial thanks to the Japan Institute of Plasma Physics for the use of VP2OOE.

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