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THE ASTROPHYSICAL JOURNAL, 482:230244, 1997 June 101997. The American Astronomical Society. All rights reserved. Printed in U.S.A.(

THE MHD KELVIN-HELMHOLTZ INSTABILITY. II. THE ROLES OF WEAK ANDOBLIQUE FIELDS IN PLANAR FLOWS

T. W. JOSEPH B. DONGSU AND ADAMJONES,1,2,3 GAALAAS,1,2,4 RYU,5,6,7 FRANK1,8,9Received 1996 November 13; accepted 1997 January 8

ABSTRACTWe have carried out high-resolution MHD simulations of the nonlinear evolution of Kelvin-

Helmholtz unstable ows in dimensions. The modeled ows and elds were initially uniform except212for a thin shear layer with a hyperbolic tangent velocity prole and a small, normal mode perturbation.

These simulations extend work by Frank et al. and Malagoli, Bodo, & Rosner. They consider periodicsections of ows containing magnetic elds parallel to the shear layer, but projecting over a full range ofangles with respect to the ow vectors. They are intended as preparation for fully three-dimensional cal-culations and to address two specic questions raised in earlier work: (1) What role, if any, does theorientation of the eld play in nonlinear evolution of the MHD Kelvin-Helmholtz instability in 21

2dimensions? (2) Given that the eld is too weak to stabilize against a linear perturbation of the ow,how does the nonlinear evolution of the instability depend on strength of the eld? The magnetic eldcomponent in the third direction contributes only through minor pressure contributions, so the ows areessentially two-dimensional. In Frank et al. we found that elds too weak to stabilize a linear pertur-bation may still be able to alter fundamentally the ow so that it evolves from the classical Cats Eye vortex expected in gasdynamics into a marginally stable, broad laminar shear layer. In that process themagnetic eld plays the role of a catalyst, briey storing energy and then returning it to the plasmaduring reconnection events that lead to dynamical alignment between magnetic eld and ow vectors. Inour new work we identify another transformation in the ow evolution for elds below a criticalstrength. That we found to be D10% of the critical eld needed for linear stabilization in the cases westudied. In this very weak eld regime, the role of the magnetic eld is to enhance the rate of energydissipation within and around the Cats Eye vortex, not to disrupt it. The presence of even a very weakeld can add substantially to the rate at which ow kinetic energy is dissipated.

In all of the cases we studied magnetic eld amplication by stretching in the vortex is limited bytearing mode, fast reconnection events that isolate and then destroy magnetic ux islands within thevortex and relax the elds outside the vortex. If the magnetic tension developed prior to reconnection iscomparable to Reynolds stresses in the ow, that ow is reorganized during reconnection. Otherwise, theprimary inuence on the plasma is generation of entropy. The eective expulsion of ux from the vortex

is very similar to that shown by Weiss for passive elds in idealized vortices with large magneticReynolds numbers. We demonstrated that this expulsion cannot be interpreted as a direct consequenceof steady, resistive diusion, but must be seen as a consequence of unsteady fast reconnection.Subject headings: instabilities MHD plasmas turbulence

1. INTRODUCTIONWeak magnetic elds threading conducting uid media

can play vital dynamical roles, even when traditional cri-teria, such as relative magnetic and gas pressures, suggestthe elds are entirely negligible. Perhaps the best example ofthis is the destabilizing inuence of a vanishingly smallmagnetic eld crossing a Keplerian accretion disk (Balbus& Hawley where the mere presence of the eld seems1991),fundamentally to alter the local ow properties. Other

examples abound, however, that could be particularlyimportant in astrophysics. Among them, we would include

1 Department of Astronomy, University of Minnesota, Minneapolis,MN 55455.

2 Minnesota Supercomputer Institute, University of Minnesota,Minneapolis, MN 55455.

3 twj=astro.spa.umn.edu.4 gaalaas=msi.umn.edu.5 Department of Astronomy and Space Science, Chungnam National

University, Daejeon 305-764, Korea.6 Department of Astronomy, University of Washington, Box 351580,

Seattle, WA 98195-1580.7 ryu=sirius.chungnam.ac.kr.8 Department of Physics and Astronomy, University of Rochester,

Rochester, NY 14627-0171.9 afrank

=alethea.pas.rochester.edu.

weak elds penetrating turbulent or otherwise stronglyunstable ows, where such elds signicantly alter evolu-tion and transport properties (e.g., & WelterBiskamp 1989;

& Vainstein et al.Cattaneo 1991 ; Nordlund 1992 ; Jun,Norman, & Stone Sheared motion is a critical1995).common element in many of these ows, and the conse-quent stretching of a weak, but a large-scale eld can lead toa locally enhanced role for the eld. The ows will alsofrequently lead to current sheets and associated magneticeld topologies unstable to reconnection, and that is centralto the nonlinear evolution of the systems (e.g., Biskamp

Through these processes the elds can1993; Parker 1994).also have more global consequences. Study of the nonlinearevolution of the classical Kelvin-Helmholtz (KH) instabilitycould be particularly useful as a well-dened example ofstrongly sheared ows. Further, since KH unstable bound-ary layers are probably common, the behavior of the insta-bility is important for its own sake.

Although the KH instability is fairly well studied inordinary hydrodynamics (e.g., & ShermanCorcos 1984),comparable study has been much slower in magnetohydro-dynamics (MHD). That is because the magnetic eld sub-stantially complicates the physics itself and also because

230

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MHD KELVIN-HELMHOLTZ INSTABILITY. II. 231

computational methods and resources needed for suchstudies are only recently up to the task. The linear analysisof the MHD KH instability is relatively straightforwardand was long ago carried out for a number of simple owand eld congurations (e.g., Chandrasekhar 1961; Miura& Pritchett Generally, and especially if the velocity1982).change is not supersonic, the ordinary uid shear layer isunstable to perturbations with wavevectors in the plane of

the shear layer and with wavelengths greater than the thick-ness of the layer (e.g., When there is a eldMiura 1990).component projecting onto the ow eld, magnetic tensionprovides a stabilizing inuence. A simple vortex sheet isstabilized against linear perturbations wheneverthe magnetic eld strength is sufficient that c

A[ o (k U

0)/

where is the velocity dierence between the(2k B0) o, U

0two layers, is the Alfven speed, k is the perturbationcAwavevector, and is the direction of the magnetic eldB

0(Chandrasekhar 1961).et al. hereafter and Bodo,Frank (1996, Paper I) Malagoli,

& Rosner hereafter recently presented comple-(1996, MBR)mentary nonlinear analyses of the MHD KH instability inmildly compressible ows based on two-dimensional

numerical simulations carried out with new (and dierent)Riemann-solverbased MHD codes. While not the rstnumerical studies of the MHD KH instability, they rep-resented big improvements over previous calculations inboth numerical resolution and extent to which ow evolu-tion was followed toward asymptotic states (readers arereferred to for additional, earlier citations). Con-Paper Isidering perturbed two-dimensional ows that wereuniform except for a thin, smooth velocity transition layer,those two papers emphasized the qualitatively dierentbehaviors in the nonlinear evolution of unstable owsdepending on how close the eld strength is to its criticalstrength for stabilization. For elds only slightly below thecritical value, enhancements in the tension of the eld

through linear growth can stabilize the ow before itdevelops distinctly nonlinear characteristics. For weakerelds, however, the initial evolution of the instability is verysimilar to that for the ordinary KH instability. That resultsin the formation of eddies, and hence, to substantial stretch-ing of the magnetic eld lines as well as reconnection. Paper

emphasized the remarkable fact that in a case with a eldI2.5 times weaker than critical, reconnection can lead to self-organization in the ow and fairly rapid relaxation to aquasi-steady laminar and marginally stable ow. pre-MBRsented summaries of simulations extending to somewhatweaker elds showing evidence for similar behaviors.

Neither nor however, explored thePaper I MBR,problem in sufficient depth to establish the conditions

necessary for the previously mentioned self-organization. Inaddition, it is very important in this situation to understandhow the magnetic elds behave when they are very weak(a concept whose denition needs clarication, in fact). Aclosely related matter is what dierences, if any, existbetween the behavior of a truly weak eld and a strongereld whose projection onto the ow vectors is weak. Alter-natively stated, are there dierences between the nonlineartwo-dimensional MHD KH instability and the 21

2-

MHD KH instability? Answers to thosedimensional basic questions are the objective of this paper. We nd: (1)for the cases we have considered with an initially uniformeld that the magnetic eld transverse to the plane is unim-portant and (2) there is a transition from the role of the

magnetic eld as a catalyst to ow self-organization to arole as an added source of energy dissipation that shouldvanish directly as the initial magnetic eld strength project-ed onto the plane vanishes. Ultimately, we must understandthe full three-dimensional problem, in which the pertur-bation wavevector also lies outside the ow direction. Onthe other hand, it has been difficult to carry out three-dimensional MHD simulations with sufficient numerical

resolution to be condent of the results in complex owssuch as these. In addition, it will be useful to compare fullythree-dimensional behaviors with two-dimensional ows.We hope that the current work is a signicant, constructivestep toward a full understanding of this problem.

The paper plan is as follows. In we will summarize the2problem setup and relevant results from Paper I. Section 3contains a discussion of new results, while provides a 4brief summary and conclusion. We also include an

presenting an analytical model for diusive uxAppendixexpulsion from a steady vortex, in order to contrast thatphysics with what we observe in the eddies that form in oursimulations.

2.BACKGROUND

In order to focus on specic, important physical issues wehave chosen to explore an idealization of the MHD KHproblem, reserving for the future the more general problem.We present in this section only a bare outline of our methodand some key results from previous work. A full discussionof the computational setup along with several tests of suchissues as adequate numerical resolution and geometry of thecomputational box can be found in Paper I.

2.1. Problem DenitionThe geometry of the computations is shown in Figure 1.

The only dierence from is that in those earlierPaper Icomputations we assumed an aligned eld, h\ cos~1

whereas we now relax that constraint tooB0 U0 o\ 0,include magnetic elds oblique or orthogonal to the owplane. We assume the ow to be periodic in x and that the yboundaries are reecting (i.e., neither ow nor eld linescross the y boundaries). This was a conguration used ini-tially by and we followed it in toMiura (1984), Paper Ienable a direct comparison with his results. The inuencesof those boundary choices are discussed fully in InPaper I.brief, the periodic boundary limits coalescence of structuresto scales equal to the box dimension, L. That is more signi-cant than the existence of the reecting boundaries, whichseem to have only minor inuence on dynamics in the(narrow) central regions where ow organization and dissi-pation is largely determined.

The initial background ow has uniform density, o\ 1,gas pressure, p \ 0.6, and an adiabatic index, c\ 5/3, sothat the sound speed, The magneticc

s\ (cp/o)1@2\ 1.0.

eld, is also uniform. InB0

\ B0(x cos h] z sin h), Paper I

we considered cases with (and h\ 0), so thatB0

\ 0.4, 0.2since those were studied by WeM

A\ 2.5, 5, Miura (1984).

now add to these a number of new cases as outlined inTo facilitate comparisons we identify the simula-Table 1.

tions from as cases 1 and 2 in with the newPaper I Table 1,simulations following. The velocity in the background stateis antisymmetric about y \ L/2 according to the relation

u0

\ u0(y)x \ [

U0

2tanh

Ay[L/2a

Bx , (1)

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232 JONES ET AL. Vol. 482

FIG. 1.Cartoon illustrating the computational setup for these simula-tions. There is a central shear layer separating uniform ows. Except forthe initial perturbation the ow conditions are otherwise uniform. Themagnetic eld projects onto the computational plane at an angle, h.

with This describes a smoothly varying ow withinU0

\ 1.a shear layer of full width 2a. For all our simulations pre-sented here a \ L/25, chosen to make the interactions with

the reecting boundaries small. The square computationalbox has L\ 2.51. Flow is to the left in the top half-planeand to the right below that. To this state we add a pertur-bation, d(o, p, u, B), dened to be a normal mode foundfrom the linearized MHD equations appropriate for thechosen background, periodic in x and evanescent in y, with

period equal to the length of the computational box, L. Thiswas done exactly as in We note that whenPaper I. hD 0there are perturbations in all three vector components ofthe magnetic and velocity elds. Under the conditions weused the computational frame is comoving with the KHwaves. All ow and eld quantities are either symmetric orantisymmetric around two points, which happen with ourchoice of phases in d(o, p, u, B) to be at y \ L/2, x \

L/4, 3L/4. Because the velocity eld is antisymmetricaround these points they are the places where strong vor-tices tend to form.

The equations we solve numerically are those of idealcompressible MHD; namely,

Lo

Lt]$ (ou) \ 0 , (2)

Lu

Lt] u $u]

1o$p [

1o

($B)B\ 0 , (3)

Lp

Lt] u $p ] cp$ u\ 0 , (4)

LB

Lt[$ (uB) \ 0 , (5)

along with the constraint $ B\ 0 imposed to account forthe absence of magnetic monopoles (e.g., ThePriest 1984).isentropic gas equation of state is pP oc. Standard symbolsare used for common quantities. Here, we have chosenrationalized units for the magnetic eld so that the magneticpressure and the Alfven speed is simplyp

b\ B2/2 c

A\

B/o1@2.These equations were solved using a multidimensional

MHD code based on the explicit, nite dierence totalvariation diminishing or TVD scheme. The method isan MHD extension of the second-order nite-dierence,

upwinded, conservative gasdynamics scheme of Hartenas described by & Jones The multidi-(1983), Ryu (1995).

mensional version of the code, along with a description ofvarious one and two-dimensional ow tests is contained in

Jones, & Frank The code contains an t-basedRyu, (1995).routine that maintains the $ B\ 0 condition at each time

TABLE 1

SUMMARY OF MHD KH SIMULATIONS

h EndCasea B

0\ c

A/c

sb (deg) B

po\ B

0cos h M

Ab b

0b t

gc Timec N

xd

1e . . . . . . . 0.4 0 0.4 2.5 7.5 3.79 20q 512e2e . . . . . . . 0.2 0 0.2 5.0 30 1.86 20q 512e

3 . . . . . . . . 0.2 45 0.14 5.0 30 1.71 30q 5124 . . . . . . . . 0.14 0 0.14 7.07 60 1.71 30q 5125 . . . . . . . . 0.2 90 0.0 5.0 30 1.59 20q 2566h . . . . . . 0.2 85 0.02 5.0 30 1.59 50q 5126m . . . . . . 0.2 85 0.02 5.0 30 1.59 30q 2567 . . . . . . . . 0.02 0 0.02 50 3000 1.59 30q 2568 . . . . . . . . 0.04 0 0.04 25 750 1.67 30q 2569 . . . . . . . . 0.07 0 0.07 14.3 245 1.67 20q 256

a All models have used c\ 5/3, L\ 2.51, and a \ L/25.M \ U0/c

s\ 1, c

s\ 1,

b The Alfve n speed, Alfve n Mach number, and here refer to thecA

\ B/o1@2, b0

\pg/p

b\ (2/c)(M

A/M)2

total initial magnetic eld strength, not just that projected onto the plane of the ow.c The growth time is an approximation to the inverse linear growth rate ; namely t

g\ 1/! ; q\ t/t

g.

d Computations were carried out on a square grid of the size indicated, with PeriodicNy

\ Nx.

boundaries were assumed for x and reecting boundaries were assumed for y.e Cases 1 and 2 were presented and discussed in They are cited here for reference. Each wasPaper I.

computed with two or more numerical resolutions, with the largest listed here.

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No. 1, 1997 MHD KELVIN-HELMHOLTZ INSTABILITY. II. 233

step within machine accuracy. That step does not compro-mise the other conservation relations.

Numerical solution of equations on a discrete grid(2)(5)leads, through truncation errors, to diusion of energy, andmomentum, as well as to entropy generation. Of course,such eects are also present in nature and are important todening the character of the ows. The existence of eectivenumerical resistivity is necessary, for example, to allow

magnetic reconnection to occur in the calculations. Ourmethods exactly conserve total energy, as well as mass,momentum, and magnetic ux, so the exchange betweenkinetic, thermal and magnetic energies along with entropyproduction is internally consistent. There is fairly good evi-dence that conservative monotonic schemes, as this one is,do a good job of approximately representing physicalviscous and resistive dissipative processes that are expectedto take place on scales smaller than the grid (e.g., &PorterWoodward For the astrophysical environments1994).being simulated the expected dissipative scales are likelyvery much smaller than those that can be modeled directly.Recent numerical studies of reconnection suggest in theMHD limit with large kinetic and magnetic Reynolds

numbers that the local energy dissipation rate throughreconnection becomes independent of the value of the resis-tivity in complex ows (e.g., Biskamp However,1993, 1994).when we depend on numerical dissipation, we must be cau-tious about the possible role of uncaptured dynamicalstructures that could be expected on scales smaller than thegrid scale, or about magnetic eld structures on such scalesthat could enhance the number of reconnection sites. Infact, we shall see that for our KH instability-induced owsinvolving weak elds (where reconnection topologies areformed on many scales) total energy dissipation is slightly enhanced when the grid is ner. This is opposite to whatwe expect from the eects of reduced numerical diusionalone, but consistent with an increase in the number of

reconnection sites, or X points, allowed when smallerscale eld structures can be resolved cleanly. Further, wend that for a xed numerical resolution, addition of a veryweak magnetic eld substantially enhances the rate ofenergy dissipation, again as one expects in response toreconnection (e.g., Lesch, & Birk Thus, weZimmer, 1997).see strong evidence for unsteady, local reconnection as inhigh Reynolds number, MHD turbulence, but also that ournumerical solutions are not quite converged in terms oftotal dissipation.

Below we will express all of our results in time unitsdened by the growth time of the linear instability, t

g\

!~1, as estimated from graphs presented by & Prit-Miurachett That is, we express time as We nd(1982). q\ t/t

g.

that especially the initial saturation of the instability, butalso the relaxation processes are fairly uniformly expressedin these units. The time units, are listed for each case int

g,

along with the duration of the simulation in unitsTable 1,ofq. The ows examined in the present paper all have t

gB

1.61.7. For comparison the sound crossing time in the boxis (since and the Alfve n wave crossingt

s\ L\ 2.51 c

s\ 1)

time is (since The normalized turnovertA

\ MA

ts

U0

\ 1).time for a large eddy is roughly t

ED L/(U

0/2)D2t

s.

2.2. Magnetic Field EvolutionSince it enters prominently into our later discussions, it is

helpful here to remind readers in a simple way of what wecan expect for the local evolution of the magnetic eld in

these simulations. A full analysis of all the subtleties isbeyond our scope here, so readers are referred to detaileddiscussions such as those by Moatt (1978), Priest (1984),and follows elds when theBiskamp (1993). Equation (5)resistivity is exactly zero. As already mentioned, our nitedierence code will introduce eects that mimic a nite res-istivity, although it is not possible to dene an exact valuefor the resistivity, g. The eective resistivity will also depend

on grid resolution, decreasing roughly as N~2 withinsmooth ows. Despite these limitations we can make goodheuristic use of the resistive MHD extension of the induc-tion For our purposes it is interesting to castequation (5).that equation in the following form:

dln ( oB o/o)dt

\12

dlnpb

dt[

dlnodt

\B [(B $)u] [ gj2 ] g$ (jB)

B2, (6)

where d/dt is the Lagrangian time derivative, the rst termon the right represents eld amplication by stretching, and

the last two terms containing the resistivity account formagnetic annihilation and diusion. We have alsoused The current density, j\$]B.equation (2).The term gj2 is a dissipative term that balances the Jouleheating in the analogous energy equation for the gas (e.g.,

The last term is written as an expression involvingPaper I).the Lorentz force, jB, to show that it represents thetransport of momentum ux in response to resistivity; i.e.,the slippage of eld lines.

A frozen-in eld results when g\ 0. That leaves only thetime derivatives and one term on the right of Ifequation (6).there is ow compression or expansion only perpendicularto Bthen the right side of vanishes and leads toequation (6)oB o/o\ constant or Those are the most commonp

bPo2.

statements of eld compression. However, more generallyone needs to include the other ideal MHD term on the rightthat accounts for stretching. In fact, eld enhancementsdue to compression are much more limited than those dueto stretching, especially in mildly compressive ows, such asthose we are studying.

We also can see from that resistive inuencesequation (6)on magnetic energy are associated with both Joule heatingand momentum transport. In fact reconnection leads toboth irreversible heating of the local plasma and to its accel-eration. The physics of reconnection is complex and beyondthe scope of this paper. However, it may be helpful to esti-mate the dissipation rate using and the Sweet-equation (6)Parker description of reconnection (e.g., Biskamp (1994);

Lesch, & Birk Assuming anParker (1994) ; Zimmer, 1997).incompressible ow steadily carrying oppositely directedelds into a current sheet of thickness, d, givesequation (6)us a magnetic energy annihilation rate and associated dissi-pation rate per unit volume, where u isQ \ gj2D 1

2(u/d)B2,

the inow speed. In this picture plasma ows out from thereconnection region at the Alfven speed, so mass conserva-tion leads to the relation between the current sheet thick-ness, d, and its width, Then, So,l\ d(c

A/u). QD 1

2B2(c

A/l).

the integrated dissipation rate through reconnectiondepends on the eld energy advected into reconnection sitesand the summed volumes of all the reconnection sites. Thecurrent density within the current sheet can be estimated as

jBB/d, so that the aspect ratio of the reconnection region

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234 JONES ET AL. Vol. 482

is where is known as the Lund-d/lD 1/NL1@2, N

L\ (lc

A)/g

quist number of the plasma and is obviously related to themagnetic Reynolds number. Consequently, the dissipativevolume scales as In two dimensions, dissipativel2/N

L1@2.

reconnection regions form out of tearing mode instabilitieswithin a current sheet when the aspect ratio, d/l, is small

and hence, when the magnetic Reynolds(Biskamp 1994),number is large. Thus, reconnection is not really steady, and

the number of sites and their individual volumes willdepend on the magnetic Reynolds numbers in the ow.Those are, indeed, the behaviors we see in our simulations.

2.3. IssuesWe already alluded in the introduction to the basic char-

acter of nonlinear MHD KH instability properties foundfrom previous work. Our intent here is to explore more fullythe behaviors of weak magnetic elds in this situation. Inpreparation for that we note from past work several keyfeatures for two-dimensional symmetry:

1. When there is no magnetic eld or if the eld isorthogonal to the ow direction a shear layer will role up.

For the periodic ows considered here the result is a stable cats eye vortex whose length equals the imposed period-icity on the space and whose height the length. As longD1

3as the ow is subsonic or submagnetosonic no shocks areinvolved and the vortex decays only through viscous diu-sion. We will not deal with the supersonic or super-magnetosonic cases here (for some work on those see, e.g.,

& Woodward and referencesPedelty 1991 ; Miura 1990,therein).In many astrophysical applications the kinetic Rey-nolds number of the ow is very large, so we wish to con-sider similar cases, so that dissipative decay times are long.For our simulations, the empirical viscous decay time of theow is at least 4 orders of magnitude longer than the dura-tion of our computations (see case 5 energy evolution curves

in below). Thus, the cats eye represents a quasi-Fig. 3steady relaxed state, in which the shear layer has spreadvertically by horizontal localization of vorticity and becomestable. Of course, for nonperiodic systems there will be con-tinued spreading due to additional vortex mergers, while inthree dimensions the ows will be unstable to perturbationsdirected along the third direction. Those inuences arebeyond the scope of our present investigation, however.

2. In the other extreme, if the magnetic tension force pro-duced by a perturbation in the shear layer exceeds the lift force produced by the perturbation, the perturbed ow isstabilized. For wavevectors aligned with the ow that con-dition exists in a linear perturbation whenever the eld isstrong enough that ThenM

AA\ U

0/(B

0cos h)\M

Ac\ 2.

only viscous diusion contributes to spreading of the orig-inal shear layer. As before, we can neglect that inuenceover nite times and describe the shear layer as remainingin a quasi-steady relaxed state from the start. For thepresent computations the critical magnetic eld to stabilizea linear perturbation is elds slightly weakerB

c\ 0.5.For

than critical, a small, but nite amplitude perturbation maylead to the same stabilization condition, perhaps after asmall amount of quasi-linear growth to the instability. Suchwas the result obtained for the so-called strong-eld caseof where That calculation is listed asPaper I, M

A\ 2.5.

case 1 in here. Under these conditions the ow isTable 1,never far from laminar and there is a modest amount ofspreading in the shear layer before it also reaches a quasi-

steady relaxed state consisting of a broadened, laminarshear layer (a point made both in and by InPaper I MBR).either case 1 or case 2, the total magnetic energy changedvery little during the ow evolution. In we pointedPaper Iout that for the symmetry imposed the mean vector mag-netic eld is time invariant; i.e., SBT\ constant, so anyrelaxed state with a relatively uniform eld will automati-cally contain magnetic energy close to that of the initial

conditions. This condition just reects the conservation ofmagnetic ux on the grid. The relaxed magnetic energy isvery slightly enhanced, because the eld is not quiteuniform at the end. A small amount of kinetic energy dissi-pation takes place in case 1 before the ow becomes relaxed.That amount is mandated by total energy conservation, theapproximate magnetic energy conservation in this case, andthe fact that on a xed space the kinetic energy in a broad,symmetric shear layer is less than in a thin one. So, anyevolution leading to a broadened, laminar shear layerrequires an increased thermal energy determined by thenal width of the layer, independent of how it got there. Inour idealization the mass in the box is also exactly con-served, so the mean density is constant. Consequently, to

rst order there is no change in thermal energy by way ofreversible, adiabatic processes. Most of the increasedthermal energy must result from entropy-producing dissi-pation of some kind. Putting it simply, for these ows torelax entropy must be generated. These energy consider-ations apply to all of the calculations in our study if therelaxed state is a laminar ow. (It turns out that the kineticenergy of the cats eye vortex is less than that for the initialow, as well. So, its formation must also generate entropy.)

3. For initial elds too weak to prevent formation of thecats eye vortex through magnetic tension, the shear layerwill role up as for unmagnetized uid ow. In Paper I,however, we saw that when the initial eld is only a fewtimes weaker than the critical value for linear stabilization,

there follows a dramatic transformation in the ow as thecats eye develops. considered a ow withPaper I MA

\ 5.We list it in the current as case 2. As the vortex rolesTable 1up in such cases, magnetic eld lines are stretched around it,thus increasing magnetic energy at the expense of kineticenergy of the ow. The greatest magnetic pressures are pro-duced in thin ux tubes formed between the vortex and itstwins in periodic extensions of the space (see Figs. and4a 6below). Especially within the vortex and around its perim-eter, this evolution leads to magnetic reversals, beginningafter about one turnover time for the vortex. The reversedelds are unstable to tearing mode reconnection. So, oncethat happens the magnetic eld quickly reorganizes itself,releasing the magnetic stresses and stored magnetic energy.

For case 2 this leads, as well, to disruption of the cats eyeand eventually to an almost steady, laminar ow, after for-mation and disruption of several weaker vortices. That,nal, quasi-steady relaxed state, was similar to the initialconditions, except that the shear layer was broad enough tobe stable against perturbations on scales that t within theperiodic box. The relaxed shear layer had a linear velocityprole (see for similar points). There are additionalMBRinteresting characteristics of the relaxed state. Remaininguctuations in the magnetic and velocity elds were almostexactly correlated, so that they could be described as lin-early polarized Alfve n waves. The magnetic energy returnedto a level slightly above the initial conditions, with the nalexcess representing a pair of apparent magnetic ux

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No. 1, 1997 MHD KELVIN-HELMHOLTZ INSTABILITY. II. 235

tubes bounding a hot central core of the shear layer con-taining most of the entropy generated during the relaxationprocess. This nal condition was reached by about q\ 20.The relaxed shear layer was broader in this case than case 1(see also so that the kinetic energy was also smaller.MBR),Thus, even accounting for the slight increase in magneticenergy, we could correctly predict that case 2 with a weaker

eld was necessarily more dissipative than case 1. That extra

dissipation could come about only through the eects ofreconnection. Other aspects of this case will be visited pre-sently, since we will encounter them again in some newcases examined here.

Thus, from the calculations reported in and inPaper Iit is obvious that weak magnetic elds can play aMBR

major role in the evolution of MHD KH unstable ows. Itis also apparent that these roles involve the exchange ofenergy and momentum from the gas to the magnetic eldand then back to the gas through Maxwell stresses andreconnection. But it is not yet clear what are the crucialsteps in that exchange, nor how it depends on the initialstrength of the eld, so that the ordinary KH behaviorresults, if it does, in the limit that the magnetic eld becomesvanishingly small. In addition, although the linear MHDKH instability is not aected by the presence of a (possiblystrong) eld transverse to the ow (but still aligned to theplane of the shear), a eld oblique to the ow can carry circularly polarized Alfven waves. Since we found thatthe two-dimensional version of the problem generated lin-early polarized Alfven waves, it may be important to see ifthe nonlinear problem depends at all on the orientation ofthe eld, or just on the strength of the eld projected ontothe plane as in the linear problem.

3. RESULTS

To meet the objectives at the end of the previous section

we have carried out a set of seven new simulations. They areoutlined in as cases 39. (Once again, cases 1 and 2Table 1were discussed in The new simulations werePaper I.)designed to cover a wide range of strengths for the magneticeld in the computational plane. They include ows thathave exactly the same total eld strength as case 2, but inwhich the initial eld is oblique to the computational plane;i.e., (cases 3, 5, and 6), as well as ows in which thehD 0eld is entirely in the computational plane but is weakerthan that in case 2 (namely cases 4, 7, 8, and 9). The plane-projected eld strength for cases 6 and 7 is an order ofmagnitude weaker than for case 2. Note that cases 3 and 4are paired to have the same initial planar eld strengths, asare cases 6 and 7. This enables us to compare efficiently any

distinct roles of eld strength and orientation with respectto the plane. In case 5 the eld is orthogonal to the plane, sowe expect (and see) no important role for the magnetic eld.As others have noted before, compressible inuences arecontrolled in that case by magnetosonic waves rather thanpure sound waves, so there is a very slight modication inresponse to that in case[M

ms~1 \ (M~2 ] M

A~2)1@2\ 1.02

5]. We will not concern ourselves at all with ows in whichthe eld is strong, by which we mean situations where mag-netic tension precludes the nonlinear development of theMHD KH instability.

Figures and provide a broad overview of the evolu-2 3tion of the new models we computed. They illustrate thetime variation of energy components (thermal, kinetic, and

magnetic), as well as the pressure minimum ratio, bmin

\on the grid at each time. displays results(p/p

b)min

, Figure 2for the runs that were computed with whileN

x\ 512,

shows the cases computed with In addi-Figure 3 Nx

\ 256.tion, to provide a sense of the inuence of numericalresolution for very weak eld cases both case 6h (N

x\ 512)

and case 6m are shown together in(Nx

\ 256) Figure 2.Resolution issues were discussed in detail for cases 1 and 2

in where similar energy plots were also given. WePaper I,shall add a few additional relevant comments below.There is one distinctive detail about Figures and that2 3

is important to their interpretation. It was apparent to us bycomparing animations of the important dynamical quan-tities that cases 3 and 4 were virtually indistinguishablefrom one another, and likewise for cases 6m and 7. Thus, theimportant issues are somewhat easier to see if we eliminate

to rst order in Figures and We plot a reducedBz

2 3.energy for the cases with h 0. The magnetic and kineticDenergy plotted include only the planar components; i.e.,

where andEb@ \ / (1/2)B

p2 dxdy, B

p\ (B

x2 ] B

y2)1@2, E

k@ \

To compensate, the total energy is/ (1/2)o(ux2 ] u

y2)dxdy.

also reduced as where is theETot@ \ E

b@ ] E

t] E

k@ , E

t

thermal energy. Thus, we ignore the almost constant energycontributions from and Note, however, it is the totalBz

vz.

energy, not the reduced total energy, that is conserved. Inthe denition of total magnetic pressure from all threeb

min,

components was used, since it is the entire pressure thatexerts a force on the plasma.

We can see that the reduced energy evolution in cases 3and 4 are almost identical in and the same is trueFigure 2,of cases 6m and 7 in Thus, we observe inFigure 3. 21

2dimensions, at least, that the transverse magnetic eld com-ponent plays no signicant role in nonlinear evolution(B

z)

of the instability. The rationale for some role comes fromthe observation that nite and enable circularly pol-B

zvzarized Alfven waves in dimensions, while only linearly21

2polarized Alfven waves are allowed in two dimensions. Inother words, there are twice as many degrees of freedom in

dimensions for Alfven waves to help disperse pertur-212bations. However, there are two arguments to support our

observation of a minimal role. First, the group velocityvector for Alfven waves along which physical informationpropagates is i.e., aligned with the magnetic

g\B/o1@2 ;

eld (see, e.g., Lifshitz, & Pitaevskii WithLandau, 1984).invariance of quantities along the z-direction, only thegroup velocity components projected onto the x-y plane arerelevant, and they are independent of The second pointB

z.

is that even though and are nite in dimensions, theBz

vz

212above symmetry restricts their contributions to those of

magnetic pressure (see eqs. i.e., to the total pres-[2][5]);

sure gradient and the fast and slow wave speeds. In thepresent case with weak elds, even that has little impor-tance. For example, the evolution of is almost the sameb

minfor cases 3 and 4, where stretching of the eld in the plane isdominant in case 3. For cases 6m and 7 the values ofb

minare always large, because neither the planar nor the trans-verse elds are very strong. But at qD 5 when the stretchingof the planar eld is greatest, the values of are stillb

minsimilar. So, our results indicate that for weak elds the extradegrees of freedom allowed in dimensions have no sig-21

2nicant eect on the KH instability. These ows are essen-tially two-dimensional.

In one of the most striking ndings from these simula-tions, we nd two distinctive evolutionary behaviors for

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0 10 20 30 40 50

.86

.88

.9

.92

.94

.96

.98

3

6m

6h

4

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.01

.1

1

0 10 20 30 40 50

.0001

.001

.01

.1

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.1

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10

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236 JONES ET AL. Vol. 482

FIG. 2.Time evolution of the high-resolution simulations (cases 3, 4, 6h), plus the medium resolution simulation case 6m. Shown are the normalizedtotal thermal, kinetic, and magnetic energies, as well as the minimum value of the plasma b parameter at each time. To emphasize the minimal importance ofthe transverse magnetic eld component, reduced energies are shown that excludeB

z, B

z.

weak eld ows, depending on the eld strength. When theplanar magnetic elds are very weak, it turns out that the

evolution of the two-dimensional MHD KH instability isqualitatively similar to the gasdynamic version of the insta-bility. That is, the cats eye vortex continues to exist as longas we extend the simulation. The magnetic eld does,through unsteady reconnection, substantially enhanceenergy dissipation of the vortex over that from the gas-dynamical case, however. This can be seen by comparingcase 5 in with case 6m or case 7(B

p\ 0) Figure 3 (B

p\

0.02). We will characterize such ows as having very weakelds, and the role of the eld as dissipative. On theother hand, if the initial elds approach the critical eld forlinear stabilization within a factor of a few, the cats eye isdisrupted and the eld causes the ow to reorganize into alaminar form (as in case 2). We call this simply the weak-

eld regime, and the eld disruptive. In either situationthe magnetic energy peaks about the time of the initialreconnection instability. For the disruptive cases there is afairly prompt return of the magnetic energy near to itsinitial value (see Figs. and In eect, the magnetic eld2 3).plays the role of a catalyst, storing energy temporarily andthrough it modifying the plasma ow. That role is fairlydramatic, since reconnection leads to dynamical alignmentand self-organization in the ow. For dissipative cases, themagnetic energy declines much more slowly, and in thehigh-resolution case 6h, seems mostly to oscillate. Thatbehavior results from the fact that the cats eye vortex con-tinues to capture magnetic ux, amplifying it and thencedissipating it.

The existence of two qualitatively distinct evolutionarypatterns is also quite apparent in the histories of the thermal

and kinetic energies displayed in Figures and The2 3.kinetic energy decay in case 6 is almost exponential, with atime constant that can be roughly estimated as Byq

dD104.

contrast, cases 3 and 4 show before qD 10 a sharp drop inalong with an accompanying increase in followed byE

k@ E

t@,

a slow, possibly exponential evolution similar to case 6. Forall cases in the magnetic energy is always smallTable 1(although it can briey increase by factors between 6 and 20it never contributes more than a few percents to the totalenergy), so on the face of it the magnetic energy would notseem to be important. In fact, it can be crucial, as we havealready outlined and will discuss more fully, below. Figure

which shows a wider range of projected eld strengths,3,shows similar patterns with some variation in the abrupt-

ness of the early energy transition. We also conclude that asthe evolution of the two-dimensional MHD KHBp]0

instability will smoothly approach that for ordinary gas-dynamics.

3.1. Very Weak Fields: DissipativeLet us now describe in more detail the characteristics of

the two weak-eld regimes and also establish the physicalboundary between them. Case 6 and case 7 had the weakestnite planar eld, that we considered, and they demon-B

po,

strate very weak eld patterns. summarizes owFigure 4properties for case 6h at times q\ 5 and 30. Figure 4ashows the ow vorticity component out of the plane, u

z\

while shows the magnetic eld lines as($ u)z, Figure 4b

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0 5 10 15 20 25 30

.86

.88

.9

.92

.94

.96

.98

5

6m 7

98

0 5 10 15 20 25 30

.01

.1

1

0 5 10 15 20 25 30

.0001

.001

.01

.1

0 5 10 15 20 25 30

1

10

100

1000

10000

7

6m

8

9

5

No. 1, 1997 MHD KELVIN-HELMHOLTZ INSTABILITY. II. 237

FIG. 3.Same as except for medium resolution simulations (cases 5, 6m 7, 8, 9)Fig. 2,

they project onto the plane. Except for minor ne structurethat aligns with the planar eld, the vorticity at q\ 5 is thesame as for a ow with of case 5. It also is fairly closeB

p\ 0

to that for all our simulations, except case 1 at this earlytime. Primarily, this image illustrates the formation of thecats eye and how that concentrates vorticity. Note that the corners of adjacent cats eyes overlap, with a shear layerbetween them. It is in that shear layer where magnetic eldsare most strongly enhanced, as seen for this time in Figure

Extensions of that feature around the perimeter also4b.contain concentrations of magnetic ux. These regions rep-resent ows where gas is accelerated out of a stagnationpoint midway between vortex centers. Frozen-in elds arethus pulled out or stretched and amplied. At q\ 30, thecats eye vortex is still largely the same, except for somecomplex, low-level vorticity structures outside the mainvortex. Their origins are made clearer by examination of the

magnetic eld structure at this time. Those same regionsoutside the cats eye also contain isolated magnetic uxislands and eld reversal regions. Such magnetic featuresreveal an environment where magnetic reconnection isactive. The relationship is that reconnective processes notonly reorganize the magnetic eld topology and releasemagnetic energy, but they also accelerate the local plasma,and that contributes to the local vorticity. By contrast tocase 2 (or as we shall see cases 3 and 4, as well), however,magnetic stresses in cases 6 or 7 produce only small modi-cations to peripheral ows and are far too weak to disruptthe cats eye.

The vortex interior shows initial signs in ofFigure 4bmagnetic ux expulsion by q\ 5, a well-known phenome-

non (e.g., At this early stage, just as the vortexWeiss 1966).is fully formed, there is still some magnetic ux that threadsthrough the eye. That does not seem to be the case at the

later time. In a close examination of eld structures withinthe vortex we can nd no evidence after about q\ 10 thatany magnetic ux threads the vortex. Instead, the eldbreaks into ux islands within the vortex, and those areannihilated through mergers. This is, of course, just whathigh Reynolds number (both kinetic and magnetic), nonlin-ear resistive MHD ows are expected to do in response toreconnection (e.g., Biskamp The eld structure1993, 1994).in the bottom panel of is qualitatively very similarFigure 4bto that found by from a classic passive eldWeiss (1966)simulation in a steady vortex with magnetic Reynoldsnumber but very dierent from that for smallR

m\ 103,

magnetic Reynolds number Our own estimates(Rm

\ 20).of eective magnetic Reynolds numbers in the case 6h simu-

lation give numbers in excess of 103 et al.(Ryu 1995; Franket al. so the comparison is very reasonable.1996),We emphasize, however, that the eld behaviors seen here,

and presumably by Weiss, result from localized, inherentlytime-dependent reconnection, not simple ux diusion. Todemonstrate that we compute in the the equi-Appendixlibrium passive magnetic elds for a simple vortex in aresistive uid. displays two examples of magneticFigure 5eld structures predicted by this steady state resistive MHDtheory. The velocity structure for this model vortex issimilar to that observed in the simulated ows for case 6.The core, has a constant vorticity, A (seer\ r

0/2, Fig. 4a),

while there is an outer ange in which the velocity decreasesto zero at Solutions depend only on the eectiver \ r

0.

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FIG.

4a

FIG.

4b

FIG.

4.

(a)Gray-scaleimagesofthecase6hvorticitycomponentnormaltothecomputationalplaneatq

\

5(top)andq\

30(bottom).Tofacilitatevisualizationofstru

ctures,t

heperiodicspacehas

beenrepeatedonce.T

hevorticityiseverywherepositive.H

ighvaluestakehightones.(b)Case6hmagneticeldlinesprojectedontothecomputationalplaneatq

\

5(top)andq

\

30(bottom).

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MHD KELVIN-HELMHOLTZ INSTABILITY. II. 239

FIG. 5.Magnetic eld lines from an analytical model to study diu-sive ux expulsion from a steady eddy with q \ 10 (top) and q \ 1000(bottom). See the text for the denition ofq.

magnetic Reynolds number, within the vortex.q \ 2Ar02/g,

This parameter roughly measures the ratio of the timescalefor magnetic eld diusion to the rotation period of thevortex. When q is a few (q \ 10 in the top panel of thegure), ux diuses quickly enough to almost atten outeld lines to their external pattern except in the vortex core.At the center eld lines are actually concentrated by rota-

tion into a quasi-dipole pattern. That dipole results becausereconnection into a simpler topology is not permitted.The steady state pattern is very dierent when q is large,

since eld is almost frozen into the vortex. Above qD 102the eld forms a spiral pattern in the vortex core whosepitch depends on q. Shear is greatest in the ange region ofthis vortex and ux becomes concentrated increasingly intothe vortex perimeter. That particular behavior does resem-ble case 6 in some ways. There are multiple eld reversals inthe perimeter, with that number determined in the steadystate calculation by the balance between eld diusion andaddition of new turns to the eld through rotation. Thebottom panel in shows the steady state solution forFigure 5q \ 103. Despite some supercial similarities, there is an

obvious and essential dierence between the truly steadystate solution in and the quasi-steady eld in theFigure 5two time-dependent simulations (ours and Weisss). For thesteady solutions the eld line that was originally throughthe vortex center remains and is wrapped into a spiralpattern within the core giving an almost constant eldstrength there. In the time-dependent cases, reconnectionchanges the eld structure inside the vortex. The same state-

ment applies to Weisss simulation. If the eective magneticReynolds number within the ow is large enough for topol-ogies to develop that are tearing-mode unstable, eld withinthe vortex breaks o from the external eld and isdestroyed. Thus, we nd good agreement with Weiss in thenature of eddy ux expulsion and see that it cannot beviewed as a steady magnetic diusion.

The value of never drops to zero inside the vortex ofBpany of our very weak eld cases, although it is as much as 2

orders of magnitude smaller than along the vortex perim-eter. Action around the perimeter episodically injects newmagnetic ux into the vortex interior, so this process con-tinues as long as we have followed the ow. Examination ofthe magnetic energy, for case 6 in showsE

b@ , Figure 2

related, episodic peaks on rough intervals, *qD 2.53. Thatcorresponds to about half a turn over time for the vortex,and represents the interval on which elds along the vortexperimeter are stretched until they become folded, so as tomake them subject to tearing mode reconnection. So, what-ever ux is caught in this ow becomes amplied, increasingthe magnetic energy, before reconnection rearranges theeld lines. Those lines outside the vortex are relaxed towardthe initial eld conguration, often through multiple recon-nection events, while those inside are isolated into closedislands. Note that the total magnetic ux threading our boxis exactly conserved and exactly zero in these computations.Thus, any eld line entering on the left boundary must at alltimes extend continuously to the right boundary or exit

again on the left. Closed ux islands can exist, but nothingprevents them from being destroyed, since they contributeno net ux.

Reconnection, and most obviously ux-island destruc-tion, is irreversible, so that it must be accompanied byentropy generation. That outcome is very apparent inFigures and Recall that to rst-order total thermal2 3.energy changes in these, closed-system simulations reectnonadiabatic processes. Compare rst the evolution ofthermal energy in cases 6m, 7, and 5. We see that after someviscous dissipation necessary to form the cats eye, the case5 with has almost constant total thermal energy toB

p0\ 0

the end of the simulation. By contrast cases 6m and 7 showa steady rise in with small amplitude oscillations associ-E

t,

ated with major reconnection events as described above.Thus, the main dynamical impact of the eld is enhanceddissipation. The total magnetic energy remains small in allcases, but for the dissipative cases that reects a nearbalance between the rate at which kinetic energy is beingtransferred to magnetic energy and the rate at which mag-netic energy is being dissipated. For example, from initialeld energy evolution in we can crudely estimateFigure 3for case 6m that magnetic energy is generated at a rate

which is very close to the mean slope ofdEb@/dtD3] 10~4,

the thermal energy curve, Et.

Looking next at the case 6m,h plots in we seeEt

Figure 2,that energy dissipation is greater in the higher resolutioncase 6h. That results despite a smaller numerical resistivity

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240 JONES ET AL. Vol. 482

for the higher resolution simulation. This is because in case6h a larger amount of ux is caught in the vortex, so morereconnection sites develop. We estimate that D10% of thetotal magnetic ux is attached to the vortex at the end of thecalculation (q\ 50) in case 6h, while only D3% is attachedat the end of the calculation (q\ 30) in case 6m. As theeective magnetic Reynolds number increases there is a ten-dency for decreased resistivity to be countered by an

increase in the number of reconnection sites, as our earlierdiscussion of reconnection theory would suggest. If wecompare simple estimates for rates of magnetic energy gen-eration for cases 6m and 6h to the rates of thermal energyincrease, we see that they are consistent, as we found earlierfor case 6m, alone. The increased dissipation for the higherresolution simulation is relatively modest, however. Somestudies of resistive MHD turbulence suggest, in fact, that atvery large Reynolds numbers total reconnection rate in acomplex ow will be insensitive to the value of the resis-tivity (Biskamp If conrmed more generally,1993, 1994).that could provide a practical measure of convergence instudies of the present kind.

3.2. W eak Fields : DisruptiveLooking again at thermal energy evolution in Figures 2

and we can see a clear behavioral transformation in the3,sequence: case 5 ] 6(7)] 8 ] 9 ]3(4). This grouping isarranged in order of increased There is a sharp riseB

p0.

through the sequence in the amount of dissipation associ-ated with the initial formation of the cats eye vortex, as wellas a more modest increased slope to the subsequent long-term dissipation rate. The physical character of this trans-formation is apparent if one compares withFigure 4b

The latter shows for case 3 the magnetic pressureFigure 6.distribution at three times, q\ 5, 10, and 30. Behaviors forthis simulation are qualitatively similar to case 2, as dis-

cussed in detail in and outlined in Here wePaper I 2.3.note that, while the eld appears wrapped around thevortex at q\ 5, it has a laminar appearance at q\ 30. Thevelocity eld undergoes a similar transition; i.e., the vortexis completely disrupted. The magnetic eld at the interme-diate time, q\ 10, shows aspects of both the other times.Curiously, however, the ow in the dominant vortexpattern there has the opposite vorticity to the original ow.That feature is short-lived. We can understand this owtransition from rotational to laminar by examining Figures

and also relating to case 3. In we show the7, 8, 9, Figure 7log of with magnetic eld lines overlaid at q\ 5.b\p/p

bThe minimum b\ 0.55 in the strong ux tube connectingvortices, but the magnetic eld is dynamically signicant

most of the way around the perimeter of the vortex. Themaximum bD 106 in small regions where magnetic recon-nection has begun and the eld strength has decreased tovery small values.

To understand how the magnetic eld disrupts the vortexconsider the forces involved. The centripetal force associ-ated with motion around the cats eye is ou

2/RD 1

2oU

02D

since we observe that and RD 1 for the12

, uD1/21@2

vortex. Indeed, we also conrm that when the pres-Bp0

\ 0,sure gradient force within the vortex $pB0.5 and isdirected toward the vortex center, so that it supplies thenecessary force. On the other hand, in we displayFigure 8the magnetic eld lines along with vectors representing themagnetic tension force; i.e.,

FIG. 6.Gray-scale snapshots of the magnetic pressure distributionsfor case 3 at q\ 5, 10, and 30, showing the stages in disruption of the catseye vortex by the magnetic eld. High values take high tones.

T\ (B $)B , (7)

at q\ 7.5 for case 3. It is apparent that magnetic tensionforces are concentrated where the eld has been pulled intoloops by the ow and that they are directed toward thecenter of the vortex. The peak value of oTo\ 1.85. Atq\ 7.5 in case 3, however, the total pressure gradient is

small and actually of the wrong sign to eect signicantlythe motion of the plasma. The total force vector eld is verysimilar to the tension force eld shown in At theFigure 8.same time X-points in the eld topology show thatreconnection is underway that will isolate the associatedeld lines. Subsequently, ux islands are formed along the axis of the cats eye, and the magnetic tension pulls theplasma frozen into those loops toward the original center ofthe vortex, disrupting its rotation. Field line segmentsreconnecting outside the vortex core will tend to relaxtoward the original eld topology.

These observations allow us to estimate simply whatminimum initial eld, should lead to vortex disruption.B

p0Since the early evolution of a weak eld is self-similar, we

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No. 1, 1997 MHD KELVIN-HELMHOLTZ INSTABILITY. II. 241

FIG. 7.Inverted gray-scale image for case 3 at q\ 5. Projected magnetic eld lines are overlaid. The minimum bD 0.55 in the strong uxof log b\p/pbtube between vortices, while the maximumbD 106 near the center of the cats eye.

can use the behavior from case 3 to estimate oToBat the time of the rst major reconnection in1.85(B

p0/0.14)2

all weak-eld cases. Our earlier discussion requires oTofor disruption, leading to the constraint12

U02o\ 1

2Bp0

0.05, or alternately, where is the critical eldBp0 0.1B

c, B

cfor stabilization of original instability. Indeed, as Figure 3demonstrates, the transformation between dissipative anddisruptive evolution occurs for conditions between those ofcases 8 and 9; i.e., for initial eld values between 0.04 and0.07.

illustrates why the vortex disruption process isFigure 9also highly dissipative. The top panel shows at q\ 7.5 incase 3 the gas entropy distribution, while the lower paneldisplays the electric current density, ojo with the eld linesoverlaid. From this we can see that excess entropy is con-centrated into regions where reconnection is currently

active (highlighted by ojo ) or recently active. Ohmic heating(Pj2) is partly responsible for the irreversible energyexchange. The remainder should be viscous dissipation ofsmall-scale, disordered motions. As we discussed in Paper I,the nal laminar ow that results in this class of owincludes a central sheet of hot gas containing most of theexcess entropy produced through the self-organization ofthe ow.

4. SUMMARY AND CONCLUSION

We have carried out a series of high-resolution MHDsimulations of Kelvin-Helmholtz unstable ows in 212dimensions. All of these simulations involve magnetic elds

initially too weak to stabilize the ows in the linear regime;i.e., Thus, since simulations are performed on aB

p0\B

c.

periodic space, ows all begin formation of a single catseye vortex. If the eld lying in the computational plane isabsent or very weak the cats eye structure becomes apersistent, stable feature that represents a quasi-steady equilibrium. When there are very weak magnetic elds inthe plane they become wrapped into the vortex and ampli-ed by stretching. However, within a single turn of thevortex they are subject to tearing mode instabilities leadingto magnetic reconnection. That reconnection isolates some

magnetic ux within the vortex, which is eventually annihi-lated. This is the process through which ux is eectivelyexpelled from a vortex. As long as the vortex persists thisprocess will repeat. Since reconnection is irreversible, thisprocess is also dissipative and leads to an increase overviscous eects in conversion from kinetic to thermal energy.We nd in this regime that as the initial magnetic eldwithin the computational plane is increased the dissipation

FIG. 8.Magnetic eld structure for case 3 at q\ 7.5, just as substantial magnetic reconnection is underway. Projected eld lines are shown, along witharrows that represent magnetic tension forces. The maximum magnetic tension force is 1.74.

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242 JONES ET AL. Vol. 482

FIG. 9.Inverted gray-scale image of the gas entropy (top) and electrical current (bottom) distributions for case 3 at q\ 7.5. Magnetic eld lines are alsolaid on top of the current distribution to emphasize the relationships. High values take low tones.

rate increases in a similar manner. Likewise, as we use aner numerical grid, thus reducing the eective numericalresistivity and viscosity, the dissipation rate increases, reec-ting the increased ability of our code to capture small-scalereconnection events. This trend is backward from what onewould expect if simple magnetic diusion were primarilyresponsible for the reconnection. It suggests, perhaps that ifwe had been able to extend these calculations to even higherresolution the energy dissipation rate might have converged

to a value independent of the eective resistivity, just assome studies of resistive MHD turbulence nd. The recon-nection and expulsion of ux within vortices in our simula-tions are similar to those in a classic study by Weiss ofvortex ux expulsion in large-magnetic-Reynolds-numberows.

If the initial magnetic eld is strong enough that within asingle turn of the vortex it is amplied around the vortexperimeter to dynamical strength (B2DoU2), then thereconnection described in the previous paragraph releasesstresses that are capable of disrupting the vortex entirely.This can happen in a single event or, if the eld is onlymarginally strong enough through a suc-[B

p0D (1/10)B

c],

cession of dynamical realignment events. In either case the

net result is a laminar but marginally stable ow in whichthe original shear layer is greatly broadened. Thus, as wediscussed fully in such elds can have a remarkablePaper I,stabilizing inuence. This is despite the fact that their totalenergy content is a minor fraction of the total, so that theyare nominally too weak to be important, according to theusual criteria.

We considered cases in which the magnetic eld wasentirely within the ow plane and others in which the eld

was oblique to that plane, in order to examine the role innonlinear ows of the component out of the plane. For theows we have studied, only the eld com-21

2-dimensional

ponents in the ow plane have any dynamical signicance.In fully three-dimensional ows, however, we expect furtherevolution of the quasi-steady relaxed states of both veryweak eld (or dissipative) cases and weak eld (disruptive)cases. The cats eye vortex of very weak eld cases is subjectto a three-dimensional instability known as the ellipticalinstability unless the ow(Pierrehumbert 1986; Bayly 1986)lines around the vortex follow perfect circles. The planarshear ow of weak eld cases is stable against linear pertur-bations but unstable to three-dimensional nite-amplitudeperturbations Orszag, & Herbert Thus, it will(Bayly, 1988).

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No. 1, 1997 MHD KELVIN-HELMHOLTZ INSTABILITY. II. 243

be important to extend the present study to the fully three-dimensional regime, and we are preparing to do that.

This work by T. W. J., J. B. G., and A. F. was supported inpart by the NSF through grants AST 93-18959 and INT-

9511654, through NASA grants NAGW-2548 and NAG5-5055 and by the University of Minnesota SupercomputerInstitute. The work by D. R. was supported in part bySeoam Scholarship Foundation. We are grateful to B. I. Junfor stimulating and helpful discussions about these results.

APPENDIX A

DIFFUSIVE FLUX EXPULSION FROM A STEADY VORTEX

In order to show that the ux expulsion from vortex shown in requires localized, unsteady reconnection, not justFigure 4bdiusion, here we study the steady state passive eld solutions of resistive MHD. Under two- and symmetries,21

2-dimensional

the planar magnetic eld can be written as where t is the magnetic ux function. Then, the induction equation inB\ z$t,with constant resistivity becomes(5)

Lt

Lt] u $t\ g$2t . (A1)

We consider the evolution of an initially uniform magnetic eld, or within a vortex. We approximateB\ B0x t\ B

0rei,

that the vortex has azimuthal velocity

v

\

72Ar ,2A(r

0[ r) ,

0 ,

for 0\ r\ r0/2 ,

for r0/2\ r\ r

0,

for r[ r0

,(A2)

and zero radial velocity. A equals the (constant) vorticity within the vortex core. This is roughly the velocity eld of the vortexin Then, the evolution of the magnetic eld is described by with the boundary conditionsFigure 4. equation (A1) t(r

0) \

and at In a steady state, we can set t\ F(r)ein since the coefficients of the equation do notB0

r0

ei dt(r0)/dr \ B

0ei r \ r

0.

depend explicitly on /. Note that with the given initial magnetic eld and boundary conditions, only the solution with n \ 1 isallowed. Then, the equation for F is given as

dF

dr2]

1r

dF

dr[

1r2

F[iQ

r02

F \ 0 , (A3)

where

Q \

7q ,q(r

0/r [ 1) ,

0 ,

for 0\ r\ r0/2

for r0/2\ r\ r

0for r[ r

0

(A4)

with Here, q is the magnetic Reynolds number within a factor of 2.q \ 2Ar02/g.

We solve for F numerically. In we plot the resulting magnetic eld lines for the case with q \ 10equation (A3) Figure 5,(highly diusive case) and for the case with q \ 103 (quasi-adiabatic case). From dissipation tests we estimate the magneticReynolds number of the vortex in our time-dependent simulations to be larger than 103 (see Jones, & FrankRyu, 1995 ; Paper

Indeed, the elds in resemble those in a vortex simulation with magnetic Reynolds number 103 reported inI). Figure 4b WeissThe steady state solution we just described allows only for diusion of magnetic ux, since no change in the eld(1966).

topology is permitted. It allows us to see that such a steady state, diusive description does not account for ux expulsion

from the vortex. If that were the case the magnetic eld lines in should have a structure like those in the bottom ofFigure 4bActually, their topologies are fundamentally dierent in the sense that eld in threads completely throughFigure 5. Figure 5the vortex, while it does not in (or any similar gure showing additional eld lines for this simulation). The reason isFigure 4bthat the eld in the time-dependent simulation is subjected to reconnective instabilities that isolate and then destroy magneticux in the interior of the vortex.

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