MHD turbulence. Five important papers are due to Robinson, Rusbridge &. * Permanent address: College of'r"illiam and )Iary. Williamsburg, Virginia 23185.
J. Pla8maPhysics (1983),vol. 29, part 3, pp. 525-547 Printed in GreatBritain
525
Anisotropy in MHD turbulence due to a mean magnetic field By JOHN Westinghouse
Electric
Corporation,
WILLIAM NASAjGQddard
Space
(Received
Flight
of Maryland,
16 August
Oceanic
Division,
Annapolis,
H. MA TTHAE
AND DAVID University
V. SHEBALINt
Center,
Greenbelt,
Park,
21404
US* Maryland
MONTGOMER College
Maryland
Maryland
20771
Y* 20742
1982 and in revised form 20 December 1982)
The developmentof amsotropy in an imtially isotropic spectrum is studied numerically for two-dimensional magnetohydrodynamic turbulence. The amsotropydevelopsthrough the combinedeffectsof an externally imposedd.c. magneticfield and viscousand resistive dissipation at high wavenumbers.The effectis m~stpronouncedat high mechamcaland magneticReynoldsnumbers. The anisotropyis greater at the higher wavenumbers. 1. Introduction In the last several years, a systematic theory of magnetohydrodynamic (MHD) turbulence has developed, but it has dealt largely with the casein which no mean d.c. magnetic field is present. The presence of a mean d.c. magnetic field renders MHD turbulence inherently anisotropic, and one can no longer take advantage of the contraction of the statistical description that the isotropy provides. Here, we report :J. beginning on the problem of describing MHD turbulent phenomena in the presence of the anisotropy introduced by a mean d.c. magnetic field. \Ve proceed numerically, using a two-dimensional, spectralmethod, incompressible MHD code which has been modified to include a mean d.c. magnetic field. It would, of course, have been preferable to have used three-dimensional numerics. However, codes with adequate resolution to study three-dimensional MHD turbulence at high mechanical and magnetic Re}"nolds numbers are rare, and groups with adequate computer resources to study MHD turbulence with them systematically are rarer still. It has been possible to represent in two dimensions several of the features which we believe to be central to the threedimensional case. There are relatively few published results of laboratory measurements of MHD turbulence. Five important papers are due to Robinson, Rusbridge & * Permanent address: College of'r"illiam and )Iary. Williamsburg, Virginia 23185. t Present address: Kentron Technical Ccnter, Hampton, Virginia 23666.
526
J. V. Shebalin, JV. H. Matthaeus and D.1Jilontgomery
Saunders (1968), Rusbridg~ (1969), Robinson & Rusbridge (1971), Zweben, ~Ienyuk & Taylor (1979) and Zweben & Taylor (1981). The first three of these report measurements on the Culham-Harwell ZETA toroidal Z-pinch, and the last two, measurements on the UCLA Macrotor Tokamak. The most important two features that emerged from the magnetic fluctuation measurements on both machineswere: (i) the single-point frequency spectra were broad-band, extending from a few hundred kilohertz down to the lower limit of the frequency resolution, with a notable absenceof spikes or peaks; and (ii) the wavenumber spectra were peaked sharply in the direction perpendicular to the mean field, which manifested itself as a large ratio of parallel to perpendicular correlation lengths (~ 10). Both of these behaviours will be seen below to emerge from the evolution of initially isotropic spectra in two dimensions. The presence of a similar anisotropy is essential, in one form or another, to a theoretical derivation of the Strauss equations (Strauss 1976; Montgomery 1982). The Strauss equations are a reduced set of MHD equations, intermediate between two and three dimensions, which have found wide applicability to tokamak dynamics and may have considerably broader utility. It is unclear how to proceed in their derivatjon if the strong anisotropy is not present initially, and it would be reassuring to know that an initially isotropic spectrum would relax to an appropriately anisotropic one. The code used here has evolved from the spectral-method techniques given by Orszag (1971) and Patterson & Orszag (1971). Fyfe, Joyce & Montgomery (1977a, b) and Matthaeus & Montgomery (1980,1981) have used earlier versions of the present code to study forced dissipative MHD turbulence, selective decay processes,and the evolution of the sheet pinch, all in the presence of periodic boundary conditions. Orszag & Tang (1979) have used a similar code to study small-scale effects in two dimensions, and Pouquet (1978) has reported closure calculations for the same geometry. A general review of two-dimensional turbulence has been given by Kraichnan & Montgomery (1980). None of the above wor~ addressesthe case of a finite mean d.c. magnetic field. The outline of the present paper is as follows. In §2, the dynamical equations are described: they are only a slight modification of those used previously. The computational technique is briefly remarked upon in §3. Results from the computations are presented in §§4, 5 and 6. The most important effect consistently observed is the development of strong anisotropy in the spectra from isotropic initial conditions. The anisotropy develops towards the kinds of spectra which seem to prevail in the reported measurements in ZET.-\. and }Iacrotor. The development of the anisotropy depends upon the combined effect of the external d.c. magnetic field strength and the size of the Reynolds numbers in an unexpected, but ultimately simple, way. .-\.simple model of the effect is offered in § 7, where the results are briefly summarized and further directions for research are indicated.
Anisotropy in MHD turbulence
~
D
2"I
2. Dynamical equations The incompressible, dissipative i\lIHD equations in two dimensions are used. The magnetic field consists of a constant mean field part Bo = Boez,plus a timedependent zero-mean turbulent part B = (Bz,BlI' 0) = V x (eza).The magnetic vector potential is eza(x,y, t) so that the Coulomb gauge is employed. For all variables, alaz = O. The velocity field v = (vz,vll' 0) = V x (ez1jr)is expressed in terms of a stream function 1jr = 1jr(x,y, t) and has zero mean. In what has become a standard set of dimensionless variables; the vorticity (J)= (llez= v xv, so that (ll = -V21jr. Similarly, the electric current j = j~, with V2a = -j. Magnetic fields are measured in terms of the initial root mean square turbulent field strength B. Velocities are measured in units of the Alfven speed corresponding to B. The dimensionless viscosity and resistivity v and 1]are the reciprocals of mechanical and magnetic Reynolds numbers, respectively. The simplest form of the dynamical equations is, in the two-dimensional geometry, O(ll a-i+v.V(ll
.oj = B.VJ+VV2(ll+Bo8X'
oa
o1jr "{ji+v .Va = 1]V2a+Boa;;-.
(1) (2)
Without the Bo terms, equations (1) and (2) become those considered previously. In a recent derivation (Montgomery 1982), the Strauss (1976) equations were re-derived using a perturbation expansion of the full set of incompressible MHD equations in three dimensions, in powers of B I Bo. It was necessary to assumein the derivation that the time derivatives alot remained of 0(1), or that no zeroth-order population of Alfven waves was present. (The linearized solutions for the three-dimensional case,as for (1) and (2), is just a superposition of Alfven waves with angular frequencies (ll(k) = :t:k.Bo, where k is the wavenumber.) An objective of the present computation is to see how a spectrum evolves which does contain an initially isotropic spectrum of Alfven waves. It will be shown that the spatial dependenceof such a spectrum on the parallel spatial co-ordinate x becomes progressively relatively weaker with time.
3. Computational technique The essenceof the computational method is that all physical fields are expanded in truncated Fourier series.The Fourier coefficients are stepped forward in time and are sa..ed at predetermined time steps to provide a history of the dynamical e\'olution of the field variables. To be explicit, Fourier representations of I1Jand a (for example) are (see, for example, Fyfe et at. 197i a, b): l1J(x,t) = ~kl1J(k,t)exp(ik.x),
(3)
a(x,t) = ~ka(k,t)exp(ik.x),
(4)
where k = (kz, kv) and kz and kv are integers. Thus the dimension of the square box is chosen for convenience to be 21T.The Fourier coefficients retained lie in the range km'n = 1 ~ Ikl ~ kmax'where kmaxis essentially limited by available
528
J. V. Shebalin, W. H. J.lfatthaeu8and D. Montgomery
computer time. Limitations on the Reynolds numbers are provided by the requirements that v and 1/be large enough EOthat the Fourier coefficients for Ikl ~ kmaxare suppressed. The first test of the code was to run it with the dissipative terms removed. Conservation laws can then be tested and comparisons can be made between the computed behaviour of the Fourier coefficients and the predictions of the (unphysical) absolute equilibrium ensemble theory (see, for example, Kraichnan & Montgomery 1980). The second step was to compare dissipative results from two runs, one having twice the other's value of kmax'and thus having twice the spatial resolution. The third step was to run several caseswith varying values for v and 1/at a particular value of Eo, to determine the influence of the Reynolds numbers. Finally, the last step was to run a number of cases which differed only in the value of Eo, thereby investigating the effect that varying mean magnetic field strength has on the dynamics. Initial Fourier coefficients were chosen so that they would be non-zero only within a given annulus in k space. The lU(k, t) were initially non-zero only for k1 ~ k ~ k2, and the a(k, t) were non-zero only for k3 ~ k ~ k~. Within their respective annuli, the values of the lU(k,O) were chosen so that all Iv(k, 0)1 = IlU(k, O)I/kwereequal, and thea(k, 0) were chosensothatallIB(k, 0)\ = kla(k,O)1 were equal. The phases of the a(k,O) and lU(k, 0) were assigned randomly. The specification of the initial Fourier coefficients was completed by giving values for the magnetic energy EB = iLkIB(k, t)12and the kinetic energy Eu = tLklv(k, t)12att=0. The total energy is E = EB+E", the 'crosshelicity' is P = iLk v(k, t) .B*(k, e). The mean square vector potential is A = !Lkla(k, t)12.E, P, and A are significant quantities in the theory of two-dimensional MHD turbulence in the absence of a mean field: they are the only known non-dissipative invariants which remain invariant under truncation of the Fourier-expanded (v = 0 = 1/) version of (1) and (2) with Eo = o. If Bo .." 0, E and P still have this status, but A does not. We may define R = EB/Eu as the ratio of the energies. .;\'bout a dozen different sets of initial Fourier coefficients have been used for various runs. Many of the sets exhibited similar behaviour. The number of different sets of initial conditions presented in this paper has been kept to a minimum, but for the runs discussed,there are others unreported for which the behaviour was similar. Details of the runs explicitly discussedhere are collected in tables 1 and 2.
4. Non-dissipative tests To test the spectral code, several cases were run ,nth v = 17= 0, and the results compared with absolute equilibrium ensemble theory. Such results are, of course, unphysical, and should be regarded as preliminary to the dissipative results presented in §5. Equilibrium ensemble theory for the Bo = 0 case ,vas gi,.en by Fyfe & iYlontgomery (1976); the caseEo :j:: 0 is recovered by simply deleting the third 'rugged'
Time
Anisotropy in lJIIHD turbulence
Initial
average
529 % change (initial-fino.l)
E
1.000
Final .~1(Bo = 0) 0.9714
P
0.2354
0.2167
0-2227
7.9
A
0.03288
0.03288
0-06
1.0062
1.5
0-9782
E
1.0000
0-03286 _~2(Bo= 1) 1.0150
P
0.2354
0.2277
0.2313
A
0'03288
0.01497
0.01489
A1 A2
Initial 1.0000 1.0000
2.9
3.3
54,5
R = EB/E. Final 1.0058 0.9825
Time average 1.0608 1.0004
Theoretical 1.0484 1.0000
TABLE 1. Non-dissipative runs (v = 0 = 1/) for comparison with absolute equilibrium ensemble theory. Time step = (256)-1. Total no. of time steps = 12800. Averages taken over last 11520 time steps.
constant of the motion A. The prediction is a simple equipartition, (Iv(k)12) =
