Proc. Nati. Acad. Sci. USA Vol. 80, pp. 5798-5802, September 1983 Physics Viscous, resistive magnetohydrodynamic stability computed by spectral techniques (transition to turbulence/"tearing modes"/dissipative magnetohydrodynamic instabilities) R. B. DAHLBURG*, T. A. ZANG*, D. MONTGOMERY*, AND M. Y. HUSSAINIt *College of William and Mary, Williamsburg, Virginia 23185; and Institute for Computer Applications in Science and Engineering, National Aeronautics and Space Administration Langley Research Center, Hampton, Virginia 23665 Communicated by Peter D. Lax, June 2, 1983 ABSTRACT Expansions in Chebyshev polynomials are used Both 7q and v are assumed to be scalars. The regime of most to study the linear stability of one-dimensional magnetohydro- interest is that in which S and M are both substantially greater dynamic quasiequilibria, in the presence of finite resistivity and than unity. viscosity. The method is modeled on the one used by Orszag in The boundary conditions are taken to be those appropriate accurate computation of solutions of the Orr-Sommerfeld equa- to a perfectly conducting, mechanically impenetrable wall tion. Two Reynolds-like numbers involving Alfven speeds, length bounding a viscous, resistive magnetofluid: v = 0, n' - B = 0, scales, kinematic viscosity, and magnetic diffusivity govern the and ni X (V X B) = 0, where n is the unit normal at the wall. stability boundaries, which are determined by the geometric mean We study the linear stability of the quasiequilibrium B(°) = of the two Reynolds-like numbers. Marginal stability curves, growth = between plane infinite rates versus Reynolds-like numbers, and growth rates versus par- (Bo(y),0,0) and v(°) (0,0,0) parallel, allel wave numbers are exhibited. A numerical result that appears boundaries at y = 1 and y =-1. The current density is in the general is that instability has been found to be associated with in- z direction only: jo = -DBO, where D d/dy. The configu- flection points in the current profile, though no general analytical ration described is not a true equilibrium, and the magnetic proof has emerged. It is possible that nonlinear subcritical three- field will resistively decay according to Bo(y,t) = exp(S-'tV2)' dimensional instabilities may exist, similar to those in Poiseuille Bo(y,0). The temporal variation will be assumed to be slow enough and Couette flow. to be negligible: Bo(y,t) Bo(y,0) = Bo(y). This implies that our stability boundaries will not be accurate in regions of small S; The linear stability of plane shear flows has been one of the there is in this feature a conceptual difference from the already most intensively studied hydrodynamic problems from the time much-studied (4-6) problem of a mean flow parallel to a uni- of Rayleigh, since it was thought to hold clues to the nature of form magnetic field with no current, which is a true equilib- turbulence [see, e.g., Lin (1) or Maslowe (2)]. Although the lin- rium, and from Hartmann flow (6). ear theory alone appears to be inadequate to predict the onset A linear expansion B = B(°) + B(l), v = v(l), is assumed, with of shear flow instabilities, it remains an important first step in products of v(1) and B(1) being discarded everywhere in the any discussion of the problem. We report here on an analogous equations of motion. Manipulating the components of the re- problem in incompressible magnetohydrodynamics (MHD). We sulting linear equations, we may prove a Squire's theorem (1), report numerical solutions of the quiescent-MHD analogue of which implies that for the location of the most unstable modes the Orr-Sommerfeld equation, using spectral methods devel- it suffices to consider the two-dimensional case: a/az may be oped by Orszag (3). set equal to zero throughout. All variations with the parallel We begin with the incompressible MHD equations in a fa- coordinate x and the time t-are assumed to be contained in a miliar dimensionless form: factor exp(iax - iwt), with a an arbitrary, real, parallel wave number and (0 = Cr + iwi a complex eigenvalue. Dahlburg and aB 1 Montgomery (7) have given the eigenvalue equations in the form -= Vx(v x B) +- V2B, [1] at S used here: av 1 (D2 - a2)2v = -iwjM(D2 - a2)v - = -v.Vv + B-VB - Vp + - V~, [2] at M - iaMBo(D2-a_2)b + iaM(D2Bo)b [3] supplemented by the conditions that V v = 0 and V-B = 0. B and is the magnetic field measured in units of a mean magnetic field magnitude ,. say. The velocity field is measured in units of the (D2 _ a2 + iwS)b = -iaSBov. [4] mean Alfven speed CA -(4rp)-1/2, where p is the mass den- sity, assumed uniform. The dimensionless pressure is p, and it Here b and v are the y components of B(') and v(l) and depend is determined by solving the Poisson equation that results from only upon y. The boundary conditions become v = 0, Dv = 0, taking the divergence of Eq. 2 and using V av/at = 0. The di- and b = 0 at y = 1 and y = -1. mensionless numbers S and M have the structure of Reynolds Eqs. 3 and 4 are the magnetostatic analogue of the Orr-Som- - numbers. S CAL/q is the Lundquist number, where q is the merfeld equation, which in the same notation (1-3) is (D2 magnetic diffusivity and L is a macroscopic length scale; M- a =)'v- iaR[(Uo - w/a)(D2 - a2)v - (D2Uo)V], where UO(y) CAL/v is a viscous analogue, where v is a kinematic viscosity. is a shear flow velocity profile in the x direction, R is the Rey- nolds number, and the boundary conditions are that v = 0, Dv The publication costs of this article were defrayed in part by page charge = 0 at y = + 1. payment. This article must therefore be hereby marked "advertise- ment" in accordance with 18 U.S.C. §1734 solely to indicate this fact. Abbreviation: MHD, magnetohydrodynamics. 5798 Downloaded by guest on September 29, 2021 Physics: Dahlburg et al. Proc. Natl. Acad. Sci. USA 80 (1983) 5799 Eqs. 3 and 4 are quite similar to eigenvalue problems arising the local method is useful when making a series of computa- in connection with confinement of thermonuclear plasmas. The tions in which either the wavenumber or the Reynolds numbers literature on "tearing modes" is extensive, and we may cite the are slowly varied. central papers of Furth et al. (8, 9), of Wesson (10), of Coppi For functions Bo(y) that are antisymmetric about y = 0, it is et al. (11), and of Dibiase and Killeen (12). Concern has fre- readily inferred from Eqs. 3 and 4 that v and b are of opposite quently been with the nonviscous (M = oo) case, which lowers parity when reflected about y = 0. We have confined attention the order of the differential equations. Viscous results from a to the case Bo(y) = -Bo(-y) with an associated current distri- linear initial-value computation have been reported by Dibiase butionjo = -DBo, which has even parity about y = 0: the clas- and Killeen (12) for the compressible case, and, to the extent sic "sheet pinch" configuration. This configuration [indeed, any that the results can be compared, ours do not appear to disagree Bo(y) profile] can be rigorously proved to be stable in the ideal with theirs. Because for plasmas of interest to date, the cal- limit (M = 00, S = 00); any instabilities must result from finite culated viscosity coefficients give estimates of v at as great as values of S or M or (in our case) both. those for 'i [see, e.g., Braginskii (13)], it seems desirable to re- We have solved for the several eigenfunctions corresponding tain viscous effects even at the price of raising the order of the to the largest values of wi for four different antisymmetric pro- eigenvalue problem to the point where results can be extracted files Bo(y): only numerically. Analytical information is difficult to extract even for the sim- B'(y) y - y3/3 pler Orr-Sommerfeld equation [e.g., Maslowe (2) or Reid (14)] BU1(y) tan-lyy _ yy(y2 + 1)-' and numerical solution is indicated here. We use a spectral ['I] technique closely patterned on the method used by Orszag (3) Bo"(y) y - /21 to calculate critical Reynolds numbers for Poiseuille flow to six- figure accuracy. It is to be expected that spectral methods will BIV(y) a sinh-lyy - ^yy(y2 + 1)-1/2. find further applications in plasma physics beyond the imme- Two numerical results have characterized all runs per- diate ones. formed, and, though we have been unable to prove either one The mean magnetic field Bo(y) and the perturbation quan- analytically, we suspect they are generally true: (i) as S or M is tities v and b are expanded in truncated Chebyshev series raised a first unstable eigenvalue always appears (wi > 0) at fi- N nite values of S and M with wr = 0; and (ii) a necessary con- Bo(y) = I B,, T,,(y) dition that instabilities appear is that the current profileJo shall n =O have an inflection point between y = -1 and y = +1. Steep current gradients alone seem insufficient to produce instability. N For example, Bt"(y), which has a large maximum current gra- v(y) = E ,n Tn(Y) [5] dient of 20 near the walls, was found to be stable up to M - n=O 104, S = 104 (for a = 1), whereas profiles such as BW, which N did contain inflection points, would characteristically be un- b(y) = E bn Tn(Y), stable for S and M no greater than a few tens, with considerably n=O smaller current gradients involved.