Forced magnetic field line reconnection in electron magnetohydrodynamicsK. Avinash, S. V. Bulanov, T. Esirkepov, P. Kaw, F. Pegoraro, P. V. Sasorov, and A. Sen Citation: Physics of Plasmas (1994-present) 5, 2849 (1998); doi: 10.1063/1.873005 View online: http://dx.doi.org/10.1063/1.873005 View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/5/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Magnetic reconnection in a comparison of topology and helicities in two and three dimensional resistivemagnetohydrodynamic simulations Phys. Plasmas 21, 032121 (2014); 10.1063/1.4869333 Resistive magnetohydrodynamic simulations of X-line retreat during magnetic reconnection Phys. Plasmas 17, 112310 (2010); 10.1063/1.3494570 Scaling of forced magnetic reconnection in the Hall-magnetohydrodynamical Taylor problem with arbitrary guidefield Phys. Plasmas 11, 3961 (2004); 10.1063/1.1768956 Ion acceleration, magnetic field line reconnection, and multiple current filament coalescence of a relativisticelectron beam in a plasma Phys. Plasmas 9, 2959 (2002); 10.1063/1.1484156 Force-free equilibria and reconnection of the magnetic field lines in collisionless plasma configurations Phys. Plasmas 8, 759 (2001); 10.1063/1.1344196

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Forced magnetic field line reconnection in electron magnetohydrodynamicsK. AvinashInstitute for Plasma Research, Gandhinagar, India

S. V. BulanovGeneral Physics Institute-RAS, Moscow, Russia

T. EsirkepovMoscow Institute for Physics and Technology, Russia

P. KawInstitute for Plasma Research, Gandhinagar, India

F. PegoraroUniversity of Pisa-INFM, Pisa, Italy

P. V. SasorovInstitute for Theoretical and Experimental Physics, Moscow, Russia

A. SenInstitute for Plasma Research, Gandhinagar, India

~Received 29 December 1997; accepted 6 May 1998!

The forced reconnection of magnetic field lines within the framework of electronmagnetohydrodynamics~EMHD! has been investigated. A broad class of solutions that describestationary reconnection have been found. The time evolution of the plasma and of the magnetic fieldwhen perturbations are imposed from the boundary of a high conductivity plasma slab are alsostudied. The initial magnetic field has a null surface. Following this discussion, the so-calledTaylor’s problem for EMHD in which the perturbations cause a change in the topology of themagnetic field has been solved. The plasma and the magnetic field are seen to evolve with the timescale of the linear tearing mode. Their time evolution is described by exponential dependences.Analytic and numerical simulation results of the nonlinear regime of forced magnetic reconnectionin EMHD are also presented. Finally, the above results are compared with a case where thereconnection is mediated by dissipative electron viscosity effects. ©1998 American Institute ofPhysics.@S1070-664X~98!01808-4#

I. INTRODUCTION

Reconnection of magnetic field lines, a fundamentalproblem of contemporary plasma physics,1 is known to ac-quire new features when considered in the framework ofelectron magnetohydrodynamics~EMHD!.2 The EMHD sys-tem corresponds to a plasma with highly magnetized elec-trons and static ions (vci!v!vce) and in which the elec-trons are treated as a magnetohydrodynamic~MHD! fluidand the ions serve as a fixed neutralizing background. Theseequations are appropriate for fast switching of plasma cur-rents in a plasma when disturbances at whistler frequenciesdominate the collective electron dynamics. Electron magne-tohydrodynamics has found various applications in the phys-ics of electric plasma discharges and electric switches~seereview Ref. 3!, in the explanation of ultrafast penetration ofthe magnetic field into collisionless plasmas, as well as inlaser plasmas, describing the evolution of the quasistaticmagnetic field.4 Since the EMHD system provides the sim-plest description of collisionless reconnection, it is of con-siderable interest,5–8 to reexamine in this new frameworkvarious aspects of magnetic field line reconnection that arewell known in the standard MHD model. In the present paperwe carry out such an investigation. In particular we study

stationary motions of the electron fluid in a magnetic fieldwith critical points and separatrices which correspond to sta-tionary magnetic reconnection in EMHD. We then investi-gate the evolution of perturbations imposed from the plasmaslab boundaries. We demonstrate that the electric currentpiles up at the neutral surface. We next solve the so-called‘‘Taylor problem’’ in the framework of the EMHD model. Inthis problem a simple slab equilibrium is perturbed from theboundary, leading to the formation of magnetic islands at thecenter of the slab. This problem was solved in the frameworkof the MHD description by Hahm and Kulsrud,9 who inves-tigated the sequential development of different regimes offorced reconnection and found that, in the final regime, theplasma evolves on the time scale of the linear tearingmode.10 We describe the plasma behavior under perturba-tions in the frequency range that corresponds to the helicon~whistler! modes. This frequency range is bounded on thelow-frequency side by the ion cyclotron frequencyvBi

5ZieB/mic and, on the high-frequency side, by the electroncyclotron frequencyvBe5eB/mec. Heremi andme are theion and electron masses andZi is the ion charge number. Inthis frequency range the plasma is described as a single elec-tron fluid ~EMHD3! where ions are assumed to be at rest, and

PHYSICS OF PLASMAS VOLUME 5, NUMBER 8 AUGUST 1998

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electrons behave in such a way as to keep the plasmaquasineutral under the conditionvBe!vpe , wherevpe is theLangmuir wave frequency. In the above treatment the non-ideal effects are provided by electron inertia which strictlyspeaking is a nondissipative term. The resulting equationsthus have a Hamiltonian structure which are shown to admitlinear solutions with rapid~exponential! rates of forced mag-netic field line reconnection and eventually a nonlinear satu-ration of the islands. In order to explore this forced recon-nection phenomena further we also introduce real dissipationvia electron viscosity. In the linear regime we again recoveran exponential rate of reconnection~in contrast to the alge-braic growth observed for the dissipative problem in theMHD description!. In the nonlinear regime the exponentialgrowths give way to algebraic growths. We validate a sub-stantial portion of our EMHD results~in the nondissipativecase! through extensive numerical EMHD simulations ofboth the forced and spontaneous magnetic reconnection phe-nomena in a slab geometry.

The paper is organized as follows. In Sec. II, we de-scribe the basic equations of the EMHD model and discussits salient properties vis-a-vis the usual MHD model. SectionIII is devoted to stationary solutions of the EMHD equationswhere we present some fairly general analytic results. In Sec.IV we take a detailed look at the natural EMHD disturbancesin a slab geometry and discuss their stability properties. Sec-tion V investigates the forced reconnection problem, the so-called Taylor problem, both for the nondissipative and dissi-pative~electron viscosity! cases. We present linear as well asnonlinear growth rate results in these regimes. The numericalsimulation results for the forced reconnection problem arepresented in Sec. VI. Our main results are summarized inSec. VII. In the Appendix we discuss some additional nu-merical results pertaining to the nonlinear development ofthe EMHD tearing mode in a current sheet.

II. BASIC EQUATIONS

An important similarity between ordinary magnetohy-drodynamics, MHD, and EMHD arises from the fact that inboth theories the magnetic flux is conserved when ‘‘non-ideal’’ effects, such as resistive dissipation and/or electroninertia, are neglected. In the case of MHD the magnetic fieldis frozen in the plasma flow, whereas in EMHD it is frozenin the flow of the electron component. In addition, the whis-tler mode in EMHD plays a role that corresponds to that ofthe Alfven mode in MHD. However, the basic spatial sym-metry of the EMHD equations is not equivalent to that of theMHD equations. The basic equation of electron magnetohy-drodynamics can be written in the form

] t~B2de2DB!5

c

4pe¹3F1

n~¹3B!3~B2de

2DB!G ,~1!

wherede5c/vpe is the inertial skin depth. Equation~1! isderived by taking the curl of the collisionless Ohm’s law,E1v3B/c5(m/ne2)dJ/dt and noting thatJ52env and¹3B5(4p/c)J ~because ion current and displacement cur-rents are negligible in the relevant range of frequencies!.

Equation~1! implies that the vector fieldB2de2DB is frozen

in the electron flow with electron velocityve52¹3B.Introducing the Lagrange coordinatesxi

0,t we can writethe formal solution of Eq.~1! in the well-known form

Bi~x,t !2DBi~x,t !5M i j „Bj~x0,0!2DBj~x0,0!…. ~2!

Here summation over repeated indices is assumed andM i j

5]xi /]xj0 is the matrix of the transformation from the

Lagrange to the Euler coordinates which are related by

xi5xi01j i~x0,t !, with v i5] tj i . ~3!

This is an implicit solution of Eq.~1!, since the matrix ofdeformationsMi j is a function of the plasma motion, i.e., itdepends on the magnetic field.

In the case of two-dimensional~2-D! configurations,where the magnetic field depends on two spatial coordinatesx andy and on timet, it is convenient to use the two scalarfunctionsc(x,y,t), which is thez component of the vectorpotential, andb(x,y,t), which is thez component of themagnetic field, instead ofB. Then we writeB5b'1bez

5]ycex2]xcey1bez . We normalize time by (vBe)21

5mec/eB0 and space coordinates in units of the electroninertial skin depthde5c/vpe . The ion density is assumed tobe uniform. The EMHD equations~1! then take the form

] tq2$b,q%50, ~4!

] tb2$b,b%52$c,q%. ~5!

Hereq5c2Dc, b5b2Db, and

$ f ,g%5]xf ]yg2]yf ]xg ~6!

are the Poisson brackets. A uniformz component,b0 , doesnot count in the above equations. We note that the typicalvalue ofc can be estimated to be equal toc'B0a, with athe size of the region measured in unitsde , and that thetypical value of the magnetic field at the boundary equalsB05b' . Thus in order to haveb of the order ofb' we musttakec@b for a@1.

The standard resistive MHD equations in two dimen-sions give instead

] tc1$f,c%5hDc, ~7!

] tDf1$f,Df%52$c,Dc%, ~8!

with h the electric resistivity, andf the plasma stream func-tion.

It is convenient to rewrite Eqs.~4! and ~5! in the form

] tc2$b,c%5] tDc2$b,Dc%, ~9!

] tb2$c,Dc%5] tDb2$b,Db%, ~10!

where the terms on the right-hand sides are due to electroninertia effects that allow for magnetic field line reconnection.

In the Lagrange coordinates,x0 ,y0 ,t, Eqs. ~4! and ~5!can be solved in the form

q~x,y,t !5q~x0,y0,0![q0~x0,y0!, ~11!

b~x,y,t !5b~x0,y0,0!2]q0

]x0

]z

]y0 1]q0

]y0

]z

]x0 . ~12!

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wherex, y, and z are functions of time due to Eq.~3! andtheir dependence on time must be obtained from the equa-tions

x5]b

]y, y52

]b

]x, z52Dc. ~13!

We see thatq(x,y,t) is conserved locally~Lagrangian in-variant!. It arises from the conservation of thez componentof the generalized momentum in a cold plasma which fol-lows from thez independence of the problem under consid-eration.

This implies that the topology of the linesq(x,y,t)5const cannot change. However, in general, the topology ofthe linesc(x,y,t)5const can change. This means that, de-spite the conservation of the topology ofq(x,y,t), the to-pology of the magnetic field lines can change and thus thatmagnetic reconnection can occur.

Collisionless reconnection due to the electron inertia wasfirst studied in a seminal paper by Coppi.11 A detailed inves-tigation of this type of reconnection in the case of the so-calledm51 instability~internal kink mode! was presented inRef. 12. The Hamiltonian nature of the EMHD equationswas pointed out in Ref. 3. It is natural to expect that theproperty of being Hamiltonian must change the scenario ofreconnection compared to that in standard resistive MHD.Collisionless reconnection in the MHD frequency range inthe two-dimension approximation, when electron inertia, ionmotion, and the finite ion-Larmor-radius effects are takeninto account, was investigated in Ref. 13 and shown to obeyHamiltonian equations. The role of electron inertia on themagnetic field generation and on the formation of thin cur-rent layers was discussed in Refs. 14 and 15. We note thatmagnetic reconnection in collisionless plasmas due to Lan-dau damping16 differs from that due to electron inertia be-cause Landau damping plays the role of dissipation, whereaselectron inertia effects leads to models that have a naturalHamilton structure. See also the discussion of this questionin Ref. 17. The main goal of the present paper is to clarifythe elementary mechanism of magnetic field line reconnec-tion in EMHD, since this system of equations provides uswith the simplest model which includes the property of beingHamiltonian. For the sake of comparison we also carry outcalculations for a case when the reconnection is mediated bydissipative electron viscosity effects.

In addition to the local conservation of~any arbitraryfunction of! q(x,y,t), the integral

C5ES~x,y,t !

b~x,y,t !G„q~x,y,t !…dx dy ~14!

is also conserved. Here the integration is performed over aLagrange surfaceS(x,y,t) enclosed by a closed contourq(x,y,t)5const andG is an arbitrary function. In order toprove this conservation, we note that the Poisson brackets$ f ,g% can be considered as the Jacobian]( f ,g)/](x,y).Thus integrating the second term in the right-hand side ofEq. ~12!

ES0~x0,y0!

$q0 ,z%dx0 dy0, ~15!

we obtain

2 Rlq0S ]z

]x0 dx01]z

]y0 dy0D52 Rlq0dz, ~16!

wherel is theq05const contour that encloses the surfaceS.Then, the above integral vanishes@the same proof holdswhen the integrand is multiplied byG(q0)#, and

C5ES~x,y,t !

b~x,y,t !G~q~x,y,t !!dx dy

5ES~x0,y0,!

b0G~q0!dx0 dy05const. ~17!

III. STATIONARY SOLUTIONS

For stationary solutions,]i50, we obtain from Eqs.~9!

$b,q%50, ~18!

and ~10!

$b,Db%5$c,Dc%, ~19!

respectively. The term$b,Db% arises from electron inertiaand is important for stationary configurations with spatialvariations on thede scale. For such configurations we obtain

b5b~q! ~20!

and

c1G~q!5F db

dq G2

Dq1F db

dqGF d2b

dq2G~¹q!2. ~21!

In the case of solutions that do not vary on thede scale, theright-hand side~RHS! of Eq. ~21! is negligible and we re-cover the standard relationshipDc5I (c). HereG and I arearbitrary functions.

However, solutions with] t50 are not appropriate to de-scribe generic EMHD equilibria because of the connectionbetween electric current and plasma motion. In fact, if elec-tron inertia effects are neglected, the above solutions onlyallow for field-aligned motions~and currents!. In order todescribe a richer class of equilibria that incorporate crossfield motions, we introduce an external irrotational anddivergence-free electric field and analyze configurations with

c52Et1c~x,y!, ~22!

i.e., q52Et1q(x,y), where the electric fieldE is directedalong thez axis and is supposed to be uniform. In this case,instead of Eq.~18!, we have

E1$b,q%50. ~23!

This stationary reconnection problem now becomessimilar to the Sweet–Parker reconnection problem in resis-tive magnetohydrodynamics. Electric field brings the elec-tron fluid across field lines into the null region,c2Dc5qremains invariant but thec lines reconnect. The flow is fi-

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nally directed outwards along the lines. In factq accumu-lates at theX point and separatrices and drives a logarithmicsingularity as shown later.

The solution for equilibria that do not vary on thede

scale are given byDc5I (c) while b is determined by theequation

¹b3¹c52Eez , ~24!

which gives a linear inhomogeneous relationship betweenbandE

¹b5Eez3¹c

u¹cu21a¹c. ~25!

Here a5a(x,y)5b8(c)1a8(x,y). The consistency rela-tionship

¹3¹b50 ~26!

leads to

¹a8~x,y!3¹c52E¹3ez3¹c

u¹cu2~27!

which can be solved by an iteration procedure based on theinequality

E¹3ez3¹c

u¹cu2 5E

b'

,1. ~28!

This procedure breaks near critical points where a logarith-mic singularity appears in the first term.

Special solutions that incorporate the effect of electroninertia can be obtained by an expansion in powers ofE andby assuming that, to lowest order inE, only q or only b donot vanish. In this approximation we can seek a solution ofthe form

q5q0~x,y!2Et1E2q2~x,y!1¯ , ~29!

b5Eb1~x,y!1E2b2~x,y!1¯ . ~30!

To first order inE, from Eqs.~29! and ~30! it follows that

Dc05I ~ c0!, ~31!

whereI (c) is an arbitrary function, andb1 satisfies the equa-tion

11$b1 ,q0%50. ~32!

For givenq05q0(x,y), the solution of Eq.~32! is

b1521

2 S E dx

]q0 /]yU

q05const

2E dy

]q0 /]xU

q05constD .

~33!

If we chooseI (c)5k221, with k5const, the solutions ofEqs.~31! and ~32! are of the form

q05x22k2y2

21k221, b15

1

2klnS x1ky

x2kyD , ~34!

which correspond to anX-point configuration and have beenobtained in Ref. 18 and discussed in Ref. 7. Choosing insteadI (c0)52a2c0 , we find

q05A cosh~qx!cos~ky!1B sinh~qx!cos~ky!, ~35!

where q5(k22a2)1/2. If we consider a solution odd inxwith A50, B51, we obtain

b1521

2 S E dx

k„sinh2~qx!2q02…

1/2

1E dy

q„cos2~qy!1q02…

1/2D52

1

2kqXFS arcsin

1

coshqx,wD1

1

wFS ky,

1

wD C.~36!

Herew5(11q02)1/25@11sinh2(qx)cos2(ky)#1/2, andF(w,k)

is the elliptic integral of the first kind.The generalization of this procedure to solutions that de-

scribe stationary magnetic reconnection in EMHD isstraightforward. If, for example, we chooseI (c0)5exp(c0), Eq. ~31! becomes Liouville’s equation whose so-lutions are well known. In particular, they describe a mag-netic configuration with a current sheet and with a sequenceof magnetic islands.

We see that on the separatrix of the magnetic field thevalue ofb has a logarithmic singularity; a similar behavior isexhibited by the shear Alfve´n waves investigated in Ref. 19.

Instead of Eqs.~30!, for E!1, we can also seek solu-tions of the form

q5E„q1~x,y!2t…1E2q2~x,y!1¯ , ~37!

b5b0~x,y!1Eb1~x,y!1¯ . ~38!

To the leading order inE, from Eqs.~37! and~38! we obtain

Db05W~ b0!. ~39!

which arises from the effect of electron inertia. HereW(b0)is an arbitrary function, whileq1 obeys the equation

11$b0 ,q1%50. ~40!

For givenb0(x,y), the solution of Eq.~40! is

q1521

2 S E dx

]b0/]yU

b0˜5const

2E dy

]b0/]xU

b0˜5const

D .

~41!

The functionq1 has a logarithmic singularity in the vicinityof the critical points and the separatrices on the planeb0(x,y)5const. On these planes the approximation given byrelationships~30! or ~37! and ~38! breaks down and oneneeds to take into account nonstationary and nonperturbativeeffects inE. An analogous problem arises in Eq.~33! at thecritical points ofq0 . An approach which allows us to de-scribe both nonstationary and nonperturbative effects is touse self-similar solutions for the EMHD equations~1!. Self-similar solutions with uniform deformation tensorMi j werefound in Ref. 2. These solutions describe both configurationsthat oscillate in time and configurations where the evolutionof the electric current and of the magnetic field near the

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critical points of the magnetic field leads to the formation ofquasi-one-dimensional singularities in a finite time.

IV. LINEAR SLAB DISTURBANCES

This section basically looks at the natural EMHD distur-bances in a slab with sheared field. We consider a slab ge-ometry configuration in the region2a,x,a with a neutralplane of the magnetic field atx50. The unperturbed mag-netic field is taken to be a linear function ofx, i.e., c0

52x2/2, andb050, which givesB05B0ey5xey . Assum-ing as boundary conditions atx56a that bothc1 and b1

vanish, we can demonstrate that this configuration is stableagainst perturbations which depend onx, y, andt.

Linearization of the EMHD equations forc5c01c1

andb5b01b1 , gives, forb050

] t~c12Dc1!2$b1 ,~c02Dc0!%50, ~42!

] t~b12Db1!5$c0 ,Dc1%. ~43!

For c052x2/2 and perturbations of the formc1

5c(x)exp(gt)cosky and b15b(x)exp(gt)sinky, Eqs. ~42!and ~43! can be written as

c92~11k2!c52kx

gb, ~44!

b92~11k2!b2S kx

g D 2

b5kx

gc. ~45!

Here and below a prime denotes differentiation with respectto x. Multiplying Eq. ~44! by c and ~45! by b and addingthem we obtain

c9c1b9b2~11k2!~c21b2!5S kx

g D 2

b2. ~46!

Then, integrating Eq.~46! over the region2a,x,a andtaking into account the boundary conditions, we find thatg2

is equal to

g252*2a

a ~kx!2b2dx

*2aa

„uc8u21ub8u21~11k2!~c21b2!…dx. ~47!

Both the numerator and the denominator are real and posi-tive. Thus g2 is negative, the configuration is stable, andreconnection of the magnetic field lines can only occur dueto a perturbation imposed from the boundary.

Similar results are well known in resistive MHD10 forconfigurations with constant electric current and in the sta-bility theory of shear flows in an Eulerian fluid20 when theshear is uniform.

A. Ideal solution

Neglecting the electron inertia effects in Eqs.~42! and~43!, we obtain

] tc2kxb50, ] tb2kxDc50 ~48!

from which it follows thatc obeys the equation

] ttc2k2x2~]xxc2k2c!50. ~49!

This is a 2-D wave equation with a propagation velocity thatdepends linearly on thex coordinate.

1. Long wavelength solution

Far enough from the neutral plane, where we can esti-mate]xx!k2, Eq. ~49! reduces to

] ttc1k4x2c50 ~50!

and its solution is

c~x,t !5A~x!sin k2xt1B~x!cosk2xt, ~51!

whereA(x) andB(x) are arbitrary functions ofx.The solution given by Eq.~51! exhibits phase mixing. If

we write c(x,t) as c(x,t)5exp„iu(x,t)…, with u(x,t)5kxt the eikonal, we obtain for the wave frequency and thewave number

v52] tu52k2x, q5]xu5k2t. ~52!

We see that in this approximation the wave has a continuousspectrum, and thatq;t, i.e., that small scale structures growlinearly with time.

2. Short wavelength solution

For ]xx@k2, Eq. ~49! takes the form

] ttc2k2x2]xxc50, ~53!

which can be rewritten as

] ttc2] l l c1] lc50, ~54!

wherel 5 ln x. Assuming a dependence of the solution on thecoordinatel and timet of the formc( l ,t)5A exp(nl2ivt),from Eq. ~54! we obtain the dispersion equation

v21k2~n22n!50, ~55!

with solution

n51

26F1

42S v

k D 2G1/2

. ~56!

The electric current density is

j 5c9[S v

k D 2

@A6x~221n6!#exp~2 ivt !. ~57!

If uvu,k/2, we see thatn2,1, and that both they and zcomponents of the perturbed magnetic field

by52]xc;x~211n2!, b5] tc

kx; i

v

kx~211n2!, ~58!

and the current density tend to infinity whenx→0. If uvu.k/2 we have

n651

26 i F S v

k D 2

21

4G1/2

. ~59!

For uvu@k/2, it follows from Eq.~59! that n6'1/26 iv/k,and c5A6x1/2 exp„2 iv(t6k21 ln x)…, j ;x23/2, b;x21/2. We see that the electric current density and the am-plitude of the magnetic field perturbation tend to infinity.The energy density of the perturbations also tends to infinitywhenx→0 as

E;ub'u21b2;x21. ~60!

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The relationships obtained above show the occurrence ofphase mixing with the effective wave number growing expo-nentially in time. Thus in ideal EMHD the natural distur-bances will simply phase mix away.

B. ‘‘Nonideal’’ solution

The limit u]xx2k2u@1 corresponds to the case when theslab widtha and/or the wavelength of perturbations are muchsmaller thande5c/vpe . In this case from Eqs.~42! and~43!we obtain

] tDc1kxb50, ] tDb1kxDc50, ~61!

i.e.,

] t j 5kxb, ] tDb5kx j, ~62!

and

] tt„~]xx2k2!b…2k2x2b50. ~63!

Here we look for disturbances which have a real periodicfrequencyv. We then get,

]xxb2Xk22S kx

v D 2Cb50. ~64!

The solution of this equation can be written in terms of theparabolic cylinder functionsDp(z):21

b~x!5C1D„ik2~v/2!2~1/2!…S k1/2x

v1/2 D1C2D

„ik2~v/2!2~1/2!…S 2k1/2x

v1/2 D . ~65!

Here

Dp~z!52p1~1/2!

p1/2 expS 2 ip

2p1

z2

4 D3E

2`

`

xpexp~22x212izx!dx, ~66!

with Re$p%.21. For low frequencies, we neglect the termk2v compared to 1/2. The solution of Eq.~64! is expressedin this limit in terms of Bessel functions:

b~x!5k2jS x

aD 1/2FA1

J1/4~kx2/2v!

vJ1/4~ka2/2v!

1A2

J21/4~kx2/2v!

vJ21/4~ka2/2v!G . ~67!

We see that these solutions correspond to localized perturba-tions that decay exponentially outside a layer the width ofwhich decreases asv decreases. We can estimate this widthasde'(v/k)1/2. That means that, for these localized pertur-bations, the gradient of the magnetic field, i.e., the electriccurrent density, increases whenv→0.

C. Nonstationary current sheet

In the expressions obtained above the effect of electroninertia is dominant and, in this sense, the solution given byexpression~65! describes ‘‘nonideal’’ modes which corre-

spond to the purely resistive localized modes that are foundin standard resistive MHD. In these expressions we havesupposed that the frequencyv is a given real number. Solu-tions that are nonperiodic in time are also of interest. In thelimit ]xx@k2, 1 from Eqs.~42! and ~43! we obtain

] txxc1kxb50, ~68!

] txxb1kx]xxc50. ~69!

Since in this approximation the electric current density isequal to j 52]xxc, we rewrite Eqs.~68! and ~69! as @seealso Eq.~62!#

] t j 5kxb, ] txxb5kx j. ~70!

We seek a solution of these equations in the form

j 5taJ~j!, b5tbB~j!, ~71!

wherej is the self-similar variable

j5kxt1/2. ~72!

The functionsJ(j) and B(j) obey the ordinary differentialequations

a J1j

2J85jB, ~73!

~b11!B91j

2B-5j J, ~74!

with a2b51/2. We look for solutions with a current den-sity even inx, and an odd magnetic fieldb, and expand bothJ(j) and B(j) in the vicinity of the x50 surface in theseries

J5c01c2j21¯ , ~75!

B5c1j1c3j31¯ . ~76!

Substituting these expressions into Eqs.~73! and ~74!, weobtain

a50, b521

2, c25c1[h, c35

c0

3[

J0

3, ¯ ,

~77!

i.e.,

J5J01hj21¯ , B5hj11

3J0j31¯ . ~78!

As a result we have

j ~x,t !'J01hk2x2t, b~x,t !'hkx1J0k3x3t. ~79!

We chooseJ0 positive andh negative. Then Eqs.~79! de-scribe a perturbation with an electric current and thez com-ponent of the magnetic fieldb localized in a region whosewidth decreases asx* ;(1/t)1/2 when t→`.

The z component of the total electric current, integratedover the current layer,'J0x* decreases as;(1/t)1/2, whilethe z component of the magnetic field decreases asb;t21/2. This means that they component of the total electriccurrent tends to zero ast21/2, while they component of theelectric current density is constant int.

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In the relationships obtained above, the valueJ0

[ j (0,t)'Dc(0)5c(0)2q(0) is constant in time. Thus tosatisfy the conditionq(0)50, we must have a finite nonva-nishing value ofc~0!. This can only occur if magnetic fieldline reconnection occurs.

In order to investigate the long time evolution of themagnetic configuration we shall use the approximationadopted in the theory of the tearing mode and used in Ref. 9to investigate the solution of the Taylor problem in the frameof the standard MHD description.

V. TAYLOR’S PROBLEM

A. Boundary conditions. Solution in the externalregion

Following the standard formulation of Taylor’s problem,we perturb the boundaries of the configuration as follows:

x56aS 12j~ t !

acoskyD , ~80!

wherej is the perturbation amplitude andk its wave number,and assume the ratioj/a to be small,j/a!1.

As in the usual approach adopted for tearing modes, wedivide the slab into two sub-regions: an external regionwhere the adiabatic approximation is valid, i.e., where wecan set] t50 in Eqs.~42! and ~43!, and the internal regionwhere we must take into account electron inertia. The mag-netic field perturbation in the external region is described by

x~c92k2c!50. ~81!

This equation is equivalent to an inhomogeneous equationwith a singular electric current on its right-hand side:

c92k2c5 j ~0!d~x!, ~82!

with d(x) the Dirac delta function, andj (0) the current am-plitude. The solution in the external region is the same as inthe Hahm and Kulsrud paper.9 There are two types of solu-tions with even symmetry. The first one hasc(0)50 andcan be written as

c~x!5ajsinh kuxusinh ka

. ~83!

This solution gives the surface electric current on the neutralplanex50 in Eq. ~82!, with

j ~0!52akj

sinh ka. ~84!

According to Eq.~44! the perturbations of thez compo-nent of the magnetic field is

b52ga sinh kuxukx sinh ka

. ~85!

In the vicinity of the neutral plane it behaves as

b'ga

sinh kasgn~x!. ~86!

This corresponds, for anygÞ0, to the presence of a singularcurrent sheet atx50 directed along thex direction.

The second type of solution is given by

c1~x!5ajcoshkx

coshka. ~87!

In this case there is no surface electric current and thex andy components of the magnetic field are continuous with

by~x!52akj sinh kx

coshka. ~88!

The widthw of the magnetic island that is formed undera perturbation of this type is given, forkw!1, by

w5S 2ja

coshkaD 1/2

. ~89!

The z component of the magnetic field diverges forx→0 as1/x for any gÞ0.

Combining expressions~83! and~87! we write the solu-tion in the external region as

c~x!5c1~0!S coshkx2sinh kuxutanhka D1aj

sinh kuxusinh ka

5c1~0!coshkx1j ~0!

2ksinh kuxu. ~90!

For c1(0)50 we recover the solution given by Eq.~83!,whereas forc1(0)5aj/coshka, i.e., for j (0)50, we obtainthe solution given by Eq.~88!.

B. Constant c approximation

The dependence ofc~0! on time in thec5const ap-proximation can be obtained by matching the solutions in theinternal and the external regions. Following Hahm andKulsrud9 @see Eqs.~34!, etc.#, we write the relationship be-tween c~0,g!, the amplitude of the perturbationj, and thefunctionD8 which gives the jump of the logarithmic deriva-tive of c through the internal region as

c~0,g!52kj

g~2k coshka1D8~g!sinh ka!, ~91!

whereg is the variable in the Laplace transform with respectto time

c~0,g!5E0

`

c~0,t !exp~2gt !dt. ~92!

The factorg in the denominator of Eq.~91! is due to the factthat j is taken to be zero fort,0. In Eq. ~91!, the functionD8(g) is a function ofg that must be found by matching thesolutions in the internal and external regions.

In order to obtain the long time behavior of the magneticconfiguration under the perturbation imposed from theboundary, we must invert the Laplace transform~92!. Thuswe must first determine the dependence ofD8 on g all overthe complexg plane. In the case of oscillatory EMHDmodes, Re$g%50, contrary to the case of standard resistiveMHD and to the case of EMHD instabilities growing expo-nentially in time, this analysis is complicated by the modebehavior in the short wavelength limit. To describe these

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modes properly we must take into account the effects thatcan regularize their short wavelength limit. Spitzer resistivitydoes not regularize this limit, while electron viscosity doeswith a mechanism similar to the regularization of the drifttearing mode due to ion viscosity.22

The addition of the electron viscosityn modifies Eqs.~4!and ~5! in the form

gq2$b,q%1nDDc50, ~93!

gb2$b,b%1$c,q%1nDDb50. ~94!

To solve these equations inside the internal region,uxu,de , we use the technique developed in Ref. 23 and used inRef. 2. Referring to thez-componentJ of the perturbed cur-rent density, we define

J~q!5deE2`

1`

JS x

deDexp~ iqx!dS x

deD , ~95!

whereJ(x) has been expressed in terms of the ‘‘stretched’’variablex/de over which the Fourier integral in the internalregion is performed. Recalling that inside the internal layerJ52c9, to leading order inde , we obtain from Eqs.~93!and ~94!,

d

dq S 1

q2~g1nq2!

dJ

dqD5~g1nq2!J. ~96!

This equation has to be solved subject to the boundary con-dition for q→0

J~q!5D8~g!2pg2q3

31¯ . ~97!

The asymptotic solutions of~96! for q→` are real exponen-tials. If we choose the solution that is exponentially decay-ing, we see that electron viscosity ensures the regularity ofthe perturbed current density independently of the phase ofg. In addition, for smalln, electron viscosity does not affectthe mode dispersion relation significantly. This is easilyseen, for example, in the case of perturbations withg nega-tive and almost real, in which case a turning point appearsdue to viscosity on the realq axis atqt

252g/n. To the leftof the turning point,q2,qt

2, we can express the solution ofEq. ~96! via modified Bessel functions as

J~q!5q3/2FAK3/4~2q2g!

K3/4~2qt2g!

1BI 3/4~2q2g!

I 3/4~2qt2g!G , ~98!

where the choice of the minus sign inside the arguments isconventional and allows us to find simple asymptotics wheng is negative. The coefficientsA andB in Eq. ~98! are of thesame order of magnitude and are determined by the condi-tion that on the right of the turning point the solution isexponentially decaying. From the properties of the modifiedBessel functions,24 we see that at smallq, the term propor-tional to B is exponentially small. As a result, close to thenegative realg axis we find

D8~g!5~2g!1/2

uku1/2

2pG~3/4!

G~1/4!. ~99!

For g close to the real positive axis, a similar proceduregives the standard result used in Ref. 2

D8~g!5~g!1/2

uku1/2

2pG~3/4!

G~1/4!. ~100!

Equations~99! and ~100! are limiting expressions, respec-tively, along the negative and the positive realg axis, of afunction regular over the complexg plane outside a smallviscosity region near the origin. This region gives a contri-bution to the inverse Laplace transform that leads to a non-exponential asymptotic long term behavior of the solutiondetermined by electron viscosity. In the following we restrictourselves to the intermediate asymptotic regime correspond-ing to collisionless magnetic reconnections due to electroninertia. In dimensional units Eq.~99! reads

D8~g!5S 2g

vBeD 1/2 a

~ ukude3!1/2

2pG~3/4!

G~1/4!, ~101!

wherevBe5eB0(a)/mc.Renormalizing variables, we can write the inverse

Laplace transform as

c~0,t !51

2p i RC

exp~pt!dp

p~11~2p!1/2!, ~102!

where the integration contourC has been chosen such thatthe form ofD8 given by Eq.~99! holds. We have neglectedthe contribution of the region near the origin. The integrandhas two poles atp50 andp521. Calculating the residueswe find

c~0,t !5„12exp~2t !…. ~103!

In dimensional units, we obtain

c~0,t !'B0j

coshkaX12expS 2

G~1/4!

2p tanhkaG~3/4!

3S ukude3vBet

a2 D D C. ~104!

C. Nonlinear regime of forced reconnection

We now consider the evolution in the case when theisland is driven to nonlinearly large amplitudes. In this case,the nonlinearJ3B terms dominate the inertia terms and de-termine the saturated amplitude of the island. The outer re-gion can still be treated in terms ofD8. Following the pre-scription of Rutherford,25 D8 can be determined from thenonlinear equations,

D8c524p

c E2`

` d2c

dx2 dx54p

c K coskyE J dxL~105!

i.e.,

D8]c

]t5

4p

c K coskyE2`

` ]J

]tdxL , ~106!

where^¯& indicates an average over they coordinate. Theabove relation gives~for the inertial case!,

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D8a5A

25/2

a2vp2

c2 W, ~107!

where

A5E21

`

dx^cosky/~x2cosky!1/2&2

^~x2cosky!21/2&'0.7,

and W5„ca2/(2c09)…1/2 is the dimensionless island width.

Substituting for (D8a) we get

A

25/2

a2W

de2 5S j

aW2 coshka21D 2ka

tanhka. ~108!

If we define (j/a coshka)1/25WS as the maximum islandwidth andWN5(25/2/A)(de

2/a2)(2ka/tanhka) we get

W

WN5S WS

2

W221D . ~109!

The solutions to this equation give the nonlinearly saturatedisland width. These are

W5 H ~WNWS2!1/3 when WS /WN@1

WS when WS /WN<1.~110!

D. Effect of electron viscosity

We have seen above that for the nondissipative Hamil-tonian problem, the reconnection ofc takes place exponen-tially rapidly and that the island acquires a nonlinearly satu-rated widthW. It is of interest to investigate the case whenfinite dissipation effects arising from viscosity are retained inthe electron equation of motion. We carry out such a calcu-lation below.

Restoring the units ofB0 , etc., Eq.~91! may be rewrittenas

c~0,g!52kjB0

g„2k coshka1D8~g!sinh ka…, ~111!

whereg is the variable in the Laplace transform with respectto time. From linear theory, we can easily obtain the depen-dence ofD8 on g as,

D8~g!5gtn

1/4tc3/4

a, ~112!

wheretc5vBe21 is the electron cyclotron time andtn5a4/n

is the viscous time. Therefore

c~0,t !51

2p i RC

c~0,g!exp~gt !dg

51

2p i

jB0

coshka RC

exp~pt!dp

p~11lp!, ~113!

where l5tanhka/(ka), p5gtn1/4tc

3/4, and t5t/(tn1/4tc

3/4),giving

c~0,t !5jB0

coshkaX12expS 2

t

l D C. ~114!

Note that asymptotically one still has an exponential decay~with a rate that is different from the previous inertial rate!leading to an eventual saturation amplitude of Hahm andKulsrud.

For the nonlinear problem we follow Ref. 26 to write

D85bE K cosky

~c2cosky!1/2L Jdc

522bE ^cosky~c2cosky!1/2&dJ

dcdc, ~115!

where

dJ

dc52a

^cosky~c2cosky!1/2&

^~c2cosky!1/2&, ~116!

a5c2c~0,t !

8pvpe2 nc09

]c~0,t !

]t, ~117!

and

b58pS 2

c09c~0,t ! D1/2

. ~118!

Substituting fordJ/dc we can write

D852baKV , ~119!

where

KV5E ^cosky~c2cosky!1/2&2

^~c2cosky!1/2&dc. ~120!

Substituting forb anda we get

c

c093/2

d

dtc1/25

D8

25/2

vpe2 n

c2KV. ~121!

Defining the dimensionless width of the islandW5(ca2/2c09)

1/2 and dimensionless timet5(tn/a4) we find

W2dW

dt5

D8a

2KV5

ka

2KV tanhka S j

W2a coshka21D ,

~122!

where we have eliminatedD8 in terms ofj from Eq. ~91!.This equation shows thatW increases initially ast1/5 whenthe first term on the right side dominates and eventually satu-rates at W;WS5„j/(a coshka)…1/2, the Hahm–Kulsrudmaximum island width. The general solution may be writtenas

1

2lnS 11W/WS

12W/WSD2

W

WSS W2

3WS2 11D 5

t

ts, ~123!

where

ts5~ka/2KV tanhka!S a coshka

j D 3/2

. ~124!

WhenW!WS , we get

W

WS'S 5t

tsD 1/5

. ~125!

and, whenW→WS , we get

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W'WSX122 expS 22t

tsD C. ~126!

The saturation timets is a function of the driving amplitudesj, as it is seen from Eq.~125! and tends to asj→0, clearlyshowing the nonlinear nature of this driven reconnection pro-cess.

VI. NUMERICAL SIMULATION OF FORCED EMHDRECONNECTION

The results of a numerical simulation of forced magneticfield reconnection of a high conductivity plasma slab withinthe EMHD description are shown in Fig. 1. The equilibriummagnetic field is linear in they coordinate and vanishes aty50, with c512y2, i.e.,c2Dc532y2, andb50, and isstable against tearing modes. This configuration is forced atuyu51 with an x-dependent perturbation with wavelengthequal to 4 and amplitudesdc and db that decay exponen-tially in time with time constantT50.25. The isolines ofb,b2Db, c, c2Dc are shown in Fig. 1 from top to bottom.The field lines of the magnetic field in the (x,y) plane, whichcorrespond to thec5const curves, reconnect while the La-grangian invariant is conserved and its field lines do notreconnect as shown by the bottom frame. The field lines ofbdevelop a convection cell structure which corresponds to ahyperbolic electron fluid motion in the (x,y) plane aroundthe origin wherec-field lines reconnect. Close to they50line the field lines ofb develop a small spatial scale whichcorrespond to the acceleration of the incompressible electronfluid leaving the reconnection layer. This small spatial scale

is not present in the distribution ofb2Db close to they50 line. The decrease with time of the scale length of theLagrangian invariantc2Dc around theX point of c isshown in Fig. 2. This decrease is due to the formation of thecurrent layer which effectively decouplesc2Dc from c.

In a Hamiltonian system, the memory of the initial per-turbation is preserved at all times. This is evident in theevolution ofc2Dc which shows that the effect of the per-turbation imposed from the boundary does not decay withtime and develops increasingly small spatial scales. On thecontrary, the effect of the perturbation on the ‘‘integrated’’quantityc tends to saturate with time.

VII. CONCLUSIONS

We have investigated forced reconnection of the mag-netic field lines in a plasma described within the frameworkof electron-magnetohydrodynamics where the Hall effect isdominant in Ohm’s law.

We have found a broad class of solutions that describestationary reconnection. We have studied the time evolutionof the plasma and of the magnetic field when perturbationsare imposed from the boundary of a high conductivityplasma slab. The initial magnetic field has a null surface. Theperturbations change the topology of the magnetic field dueto the effect of electron inertia.

We have solved the Taylor’s~Hahm and Kulsrud! prob-lem and have determined the time dependence of the growthof the magnetic islands near the null surface. When the per-turbations propagate toward the null surface, their wave vec-tor and their associated perturbed current density increase.On the null surface a singular electric current layer isformed. The current density in the layer remains constantwhile the layer thickness decreases with time in such a waythat the total electric current in the layer vanishes. This al-lows for the conservation of the Lagrange invariant, whilemagnetic field lines reconnect. Asymptotically, the plasmaand magnetic field evolve with the time scale of the lineartearing mode. The time evolution is described by an expo-nential dependence. The exponential dependence in the lin-ear regime persists even when we introduce true dissipationthrough electron viscosity effects. In the nonlinear regimethe exponential rates give way to less rapid algebraicgrowths.

FIG. 1. Forced EMHD reconnection: from top to bottom isolines ofb,b2Db, c,c2Dc in the nonlinear phase at timet56, dimensionless unitsare used.

FIG. 2. Time dependence of the width of the distribution of the Lagrangianinvariantc2Dc near thex point of c.

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To understand the above behavior it is worth discussingthe energetics of the reconnection process in the abovemodel. First of all it is important to note that the fast recon-nection in the Taylor–Hahm–Kulsrud~THK! model is dif-ferent from other standard models like the Sweet–Parker orPetschek models. In these latter models the boundary condi-tions are different; energy is brought in by flow coming infrom infinity and also after reconnection goes to infinityalong the current sheet. In the THK model on the other handreconnection is forced by deformation of boundaries andthere are no incoming or outgoing flows at infinity. The dif-ference between the EMHD and the resistive MHD versionsof the THK model arises because of the reversible nature ofelectron inertial effects. Thus in our model energy is fed intothe system through an elastic deformation of the plasmaboundary~see Sec. V C!. The source pushing the boundary isdoing work which is stored~used up! in producing the cur-rents which generate the magnetic island by accelerating theelectrons against inertia. The whole process is reversible andif we undo the deformation, the original energy can be re-covered. The saturated states, Eq.~110!, arise as a result ofanharmonic or nonlinear deformation. When viscosity is in-troduced~see Sec. V D! energy is irretrievably lost into vis-cous heating of the electrons. Thus if we removed the defor-mation in this case some of the energy would not berecovered. This case is closer to the resistive MHD versionof the THK model. The final steady state in both cases~non-viscous and viscous!, however, remains unchanged, it alwaysattains the Hahm and Kulsrud saturated amplitude. Viscosityonly affects the time scale of saturation.

To conclude, we have investigated some fundamentalaspects of the reconnection mechanism in the EMHD model.Since the EMHD system represents the simplest model of acollisionless system our results should be of value in under-standing reconnection phenomena in a host of physical situ-ations including the physics of electric plasma dischargesand electric switches, ultrafast penetration of the magneticfield into collisionless plasmas, and the evolution of quasi-static magnetic fields in laser plasmas.

APPENDIX A: NUMERICAL SIMULATION OF THENONLINEAR DEVELOPMENT OF THE EMHD TEARINGMODE IN A CURRENT SHEET

In Refs. 2, 5, and 6 it was shown that electron inertiamakes a reversed magnetic field configuration unstableagainst tearing modes when the~ideally conducting! bound-aries are located at a sufficiently large distance and the cur-rent density is localized around the region of field reversal.The development of these modes results in the reconnectionof the magnetic field lines. A slab configuration with a mag-netic field given by B05B0zez1B0x(y/L)ey, whereB0x(y/L) describes the magnetic field component generatedby the current sheet, is unstable with respect to perturbationsof the form f (y)exp(gt1ikx) with kL,1. In such a configu-ration (k–B0)50 at the surfacey50. In dimensional unitsthe growth rate of the tearing mode instability,g'(D8)2k/L, follows from Eq.~104! whereD8 is the jump inthe logarithmic derivative calculated in Ref. 10. Here the

effects of electron inertia play the formal role of Joule dissi-pation in the standard resistive MHD theory of the tearingmode instability. However, the nonlinear stage of the EMHDtearing mode differs drastically from that of the resistiveMHD tearing mode. This is mainly due to the restrictionsimposed by the requirement that the curl of the generalizedmomentum, i.e., the generalized vorticity, remains frozen inthe electron fluid.

In Fig. 3 the results of a numerical solution of the non-

FIG. 3. Nonlinear stage of the development of the EMHD tearing modeinstability in a current sheet withL51: ~a! c(x,y) at t50; ~b! c(x,y) att58; ~c! c2Dc at t50; ~d! c2Dc at t58. The wavelength of perturba-tion is 2p, dimensionless units are used

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linear evolution of the EMHD tearing mode in a two-dimensional geometry with magnetic fieldB(x,y,t)5(]c/]y)ex2(]c/]x)ey1bez are shown. The unperturbedconfiguration is chosen to be a current sheet, infinite in thexdirection, that separates two regions with opposite magneticfield polarities. The thickness of the current sheet isL51. InFigs. 3~a! and 3~c! the initial distribution of thez componentof the vector potentialc(x,y) and of the Lagrange invariantc2Dc are shown as functions of the coordinatesx andy. Att50 perturbations are imposed, with wavelength along thexaxis, such thatc(x,y,0) is symmetrical andb(x,y,0) anti-symmetrical with respect to the changex→2x. The distri-bution ofc(x,y) is shown in Fig. 3~b!, and that ofc2Dc inFig. 3~d!, at a time well within the strongly nonlinear stageof the development of the tearing mode. We recall that thelines c(x,y)5const correspond to the lines of the magneticfield in the (x,y) plane. The structure ofc(x,y) showsclearly that magnetic islands are formed inside the currentsheet. The magnetic field topology changes, as is seen fromFig. 3~b!. The resulting pattern of the electron motion istypical of the process of magnetic field line reconnection:electrons reach the vicinity of theX line in two quadrants andexit from the two other quadrants. The condition that thegeneralized vorticity is frozen in the motion of the electroncomponent leads to a distribution of the Lagrangian invariantc2Dc, and thus of the electric current density, with a non-uniformity scale much smaller than the scale of nonunifor-mity of the magnetic field. The latter is determined by elec-tron inertia and is approximately given by the value of thecollisionless skin depth. We see that magnetic field line re-connection appears despite the freezing of the generalizedvorticity in the electron flow. In the course of this process thethickness of the singular layer decreases with time.

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