Proc. Nati. Acad. Sci. USA Vol. 80, pp. 5798-5802, September 1983 Physics

Viscous, resistive magnetohydrodynamic stability computed by spectral techniques (transition to /"tearing modes"/dissipative magnetohydrodynamic ) 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. . 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 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 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 (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 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 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. Particularly extensive investigations were carried out for the where Tn(y) is the nth Chebyshev polynomial of the first kind, profile B`(y) for various values of a, S, M, and the "stretching and Bn, in, and bn are the respective expansion coefficients. parameter" y. Fig. 1 is a plot of BU(y) and its associated current The equations satisfied by the (unknown) expansion coeffi- profile j"'(y) =-DBU" as a function of y for y = 10. This case cients are obtained by substituting the N -> oo expansions of will be used to illustrate the results in Figs. 2-9. Eq. 5 into Eqs. 3 and 4, each of which produces a countably Fig. 2 shows typical eigenfunctions, stable and unstable, infinite number of equations in the expansion coefficients for computed from the B'I of Fig. 1. Fig. 2A applies to S = M = n = 0, 1, 2, ..., when the orthogonality and recursion relations 10, for the least-damped eigenvalue wi = -0.1695. For this (3) are used. We then set all coefficients beyond n = N to zero case, Wr = 0, a = 1.0, b = ibi is purely imaginary, and v = Vr and use the n = 0 to n = N - 4 equations from Eq. 3, the n is purely real. This last property always applies to the eigen- - 0 to N - 2 equations from Eq. 4, and the boundary con- function with the greatest wi. Fig. 2B shows bi, Vr for an ei- ditions Xn=O fn =0,(- 1 0° n=,_n Vn = 0, genfunction immediately above the instability threshold, with $n= 1 (- 1)'n2 n = 0, ;nT=o bn = 0, and Xn=o (- 1)" bn = 0. This method of truncation is called the "tau approximation" after 1.4 Lanczos (15); Gottlieb and Orszag (16) give a general discussion of the use of the tau method in the Chebyshev case. A -1 This spectral discretization process yields a generalized ei- Ilz0 genvalue problem that can be written as Ax = wBx, where the 5 vector x = (i0, t1, ... VN, bo, bi, ... bN), and A and B are non- symmetric (2N + 2) by (2N + 2) square matrices. lo As is customary for this type of stability problem, either global or local methods are used to determine the eigenvalues. The global method is based on the QR algorithm [Wilkinson (17), Gary and Helgason (18)] and produces a full spectrum of ei- genvalues. It is employed when no good guess for the least sta- ble (or most unstable) eigenvalue is available. The local method employs inverse Rayleigh iteration (17) and converges to the 1.0 eigenvalue (and its associated eigenfunction) closest to the ini- y tial guess for the eigenvalue. The global method may be used FIG. 1. Bo = Bg(y) and its associated current profilejo = -DB to identify the eigenvalue with the largest imaginary part, and for v = 10. Downloaded by guest on September 29, 2021 Oamooo Physics: Dahlburg et al. Proc. Natl. Acad. Sci. USA 80 (1983)

y 1.0 0.5 0 0.5 1.0 0.5 bi FIG. 3. Highly damped complex eigenfunctions forM = S = 20. 0 S = M = 20, oi = 0.06940, a = 1.o, w, = 0. Fig. 2C shows wbi,r Well above the threshold, with S = M = 1,000, a = 1.0, -0.5 Ur/ o = 0.19687, and (or = 0. Fig. 2D shows the case S = 10, M = 105, with a = 1.0, wi = 0.4397, and (Or = 0. Fig. 2E shows -1.0 the case S = 105, M = 10, wi = 0.002537, a = 1.0, (Or = 0 and illustrates the (perhaps unsurprising) result that viscosity is bet- 1.0 -0.5 0 0.5 1.0 ter at suppressing unstable growth than resistivity. Y The damped modes have, in general, both wi and 0r finite. }'cC Typical eigenfunctions for a highly damped mode for B" are 1.0 Hoshown in Fig. 3 ((Or = -0.40492, wi = -6.01303, S = M = /5 1/ X \20, a = 1.0). At the stability threshold ( = 0, the scaling v' vM'12, b' / I -bS-112 reduces Eqs. 3 and 4 to a pair of equations for v' and b' that depend upon S and M only in the combination (SM)1"2 -0.5 \ / _ (and, of course, upon a). The neutral stability curve in the SM plane, across which wi becomes positive for some a, is therefore -1.0 a hyperbola, approximately SM = 231.9. The computed loca- I tion of this hyperbola (for B" with y = 10) is shown in Fig. 4. -1.0 -0.5 0 0.5 1.0 The relatively low values of the critical Reynolds numbers (two Y orders of magnitude or more below the corresponding hydro- ''|D dynamic ones for shear flows) are perhaps the most significant 1.0 feature of this graph. It is also interesting that for low enough I \ \values of either , stability will always result 0.5 by/i / \\ for any fixed value of the other, but because SM increases with / \

-1.0 J 2 80 | \ Unstable ~60

-1.0 -0.5 0 0.5 1.0 Y ~~~Stabe~ FIG. 2. (A) Eigenfunctions b ibi,u=v, for the least-damped ei- 0 genvalues forM = 10, S = 10. (B) Eigenfunctions b = ibi, v = vrslightly 0 20 40 60 80 100 above the instability threshold: M = 20, S = 20. (C) Unstable eigen- SC functions b = ibi, v = urfor S = M = 1,000. (D) Unstable eigenfunctions b = ibi, v = Ur for S = 10, M = 105. (E) Unstable eigenfunctions b = ibi, FIG. 4. Locus of critical Reynolds-like numbers S = S,, M = MC in v = vforS = 105, M = 10. the MS plane, as determined by computation. Downloaded by guest on September 29, 2021 Physics: Dahlburg et A Proc. Natl. Acad. Sci. USA 80 (1983) 5801

7 S = 100 M= 1,000 0.4- S =10 6' 0.3 - 5_ 0.2 - M= 10 4 1 0.1 - 3 - E 3 3 / ~~~~M=1 _ -0.1 - 2 -0.2 - 1 -0.32 0-= ______-0.4 L 0 200 400 600 800 1,000 1 10 102 103 104 105 S M FIG. 5. The neutral stability curve w = 0 in the a, S plane forM = FIG. 7. Growth rate wi vs. M for fixed S = 10, 100, and 1,000, a = 1, 10, and 1,000 for B'I and y = 10. 1.0, y = 10 on Bo To make a comparison with traditional (1) hydrodynamic plots, netic field lines B'0) + B"1' for the BU equilibrium plus a 20% Fig. 5 exhibits a set of marginal stability curves: wt = 0 in the admixture of the eigenfunction shown in Fig. 2B. Fig. 8 Lower a, S plane for fixed M for BU and 'y = 10. The first unsta- shows the streamlines of the velocity field for the same eigen- ble a (which, for the reasons previously noted, must be the function. Fig. 9 shows the growth rate c,)i vs. a for several val- same for all such curves) is a = a, = 1.184 ± 0.005. We have ues of S and M. generated the same curves (not shown) in the a, M plane for The results presented have all been computational and are fixed S. not the result of an asymptotic "tearing layer" analysis. The Fig. 6A shows a growth rate (wi vs. S) curve for M = 10. Fig. marginal stability curves such as Fig. 5 or the stability hyper- 6B shows a logarithmic plot for large values of S, illustrating the bola of Fig. 4 could only have been obtained numerically. De- S-315 regime identified analytically by Furth et al. (8). Fig. 7 spite these results, we are well aware of other potentially im- shows the somewhat different behavior of wi vs. M for fixed S. Fig. 8 Upper shows a contour plot of two periods of the mag-

0.5 A 0.4 M 1oo X 0.3 X 0 ' 0.2 0.1 M1 0 -0.1 -0.2 - / _ -0.3 / -0.4 / l| 1 10 102 103 104 105 S 0.1 T-M1oo B i \ \ = ~~~~~~~~~1,000 0.05

S 0.01 0.005

0.001 1 102 103 104 105 S FIG. 8. (Upper) Contour plot of magnetic field lines for a typical un- FIG. 6. (A) Growth rate coi vs. S for a = 1.0 (onBU with y = 10), M stable eigenfunction near the threshold (S = M = 20). Field lines are = 10, 100, and 1,000. (B) Growth rate cHI vs. S for fixed M, a = 1.0, equilibrium plus 20% admixture of eigenfunction. (Lower) Velocity BU, with y = 10. streamlines corresponding to eigenfunction represented in Upper. Downloaded by guest on September 29, 2021 5802 Physics: Dahlburg et al. Proc. Natl. Acad. Sci. USA 80 (1983)

NAG-1-109 at the College of William and Mary. The work of M.Y.H. 0.6 - M= 1,000,S= 100 was supported at the Institute for Computer Applications in Science 0.4 and Engineering by National Aeronautics and Space Administration _ ~~~~~~~~~500 Contracts NAS1-16394 and NAS1-15810. 0.2 1. Lin, C. C. (1955) The Theory of Hydrodynamic Stability (Cam- M = S 1 M100,S=100= 0,- 00tbridgeR= S =Univ. Press, Cambridge, England), pp. 1-75. 0 ,02. Maslowe, S. A. (1981) in Hydrodynamic Instabilities and the Transition to Turbulence, eds. Swinney, H. L. & Gollub, J. P. -0.2 (Springer, Berlin), pp. 181-228. 3. Orszag, S. A. (1971) J. Mech. 50, 689-703. -0.4 4. Michael, D. H. (1953) Proc. Cambridge Philos. Soc. 49, 166-168. 5. Stuart, J. T. (1954) Proc. R. Soc. London Ser. A 221, 189-206. -0.6 6. Betchov, R. & Criminale, W. O., Jr. (1967) Stability of Parallel L_ (Academic, York), pp. 0 1 2 3 4 5 7. Montgomery, D. (1983) in Spectral Methods in Fluid Mechanics, a eds. Gottlieb, D., Hussaini, M. Y. & Voigt, R. (SIAM, Philadel-

in press. Ltewiparalelwaenumbera for M FIG. 9. Growth rate wi vs. s.parallel wavenumber a forBWB,, M = Sphia),S = 8. Furth, H. P., Killeen, J. & Rosenbluth, M. N. (1963) Phys. 100; M = S = 500; M = 1,000, S = 100; M = 100, S = 1,000. 6, 459-484. 9. Furth, H. P., Rutherford, P. H. & Selberg, H. (1973) Phys. Fluids portant physical eff ects that have been left out of the analysis, 16, 1054-1063. such as compressibility or the effects of spatially varying vis- 10. Wesson, J. (1966) Nucl. Fusion 6, 130-134. cosity and resistivityy tensors. Nevertheless, we believe there to 11. Coppi, B., Greene, J. M. & Johnson, J. L. (1966) Nucl. Fusion 6, ringingamoreclassialhydodynamc per-101-117. be some value in b]ringing a more classical hydrodynamic per- 12. Dibiase, J. A. & Killeen, J. (1977) J. Comp. Phys. 24, 158-185. spective to bear on this highly idealized problem. It is not im- 13. Braginskii, S. I. (1965) in Reviews of Plasma Physics, ed. Leon- possible that subcriitical instability thresholds from nonlinear tovich, M. A. (Consultant's Bureau, New York), Vol. 1, pp. 205- three-dimensional computations may be identified for this 309. problem as they have been for shear flows [see, e.g., Orszag 14. Reid, W H. (1965) in Basic Developments in , ed. and Kells (19) or Oreszag and Patera (20)]. Further computations Holt, M. (Academic, New York), pp. 249-307. . b rqie. t . possibility; 15. Lanczos, C. (1956) Applied Analysis (Prentice-Hall, New York). of a more elaborate kind will be required to test this possibility; 16. Gottlieb, D. & Orszag, S. A. (1977) Numerical Analysis of Spec- for further discussiotn of the hydrodynamic antecedents, see the tral Methods: Theory and Application (SIAM, Philadelphia). review by Herbert (21). 17. Wilkinson, J. H. (1965) The Algebraic Eigenvalue Problem (Ox- Univ. Press, London). .atthausad ahalaare gate-ford Conversations with Drs.Drs.W. H. H.Matthaeus and G. Vahala are grate- 18. Gary, J. & Helgason, R. (1970) J. Comp. Phys. 5, 169-187. fully acknowledged. T'he work of two of us (R.B. D. and D.M.) was sup- 19. Orszag, S. A. & Kells, L. C. (1980)J. Fluid Mech. 96, 159-205. ported in part by Nati()nal Aeronautics and Space Administration Grant 20. Orszag, S. A. & Patera, T. (1980) Phys. Rev. Lett. 45, 989-993. NSG-7416 and by U. S. Department of Energy Contract DE-AS05- 21. Herbert, T. (1983) Stability of Plane Poiseuille Flow-Theory and 76ET53045 at the Co]Ilege of William and Mary. The work of T.A.Z. Experiment (Virginia Polytechnic Inst. and State Univ., Blacks- was supported by Natiional Aeronautics and Space Administration Grant burg), Rep. VPI-E-81-35. Downloaded by guest on September 29, 2021