diamagnetic suppression of component magnetic reconnection...

Diamagnetic suppression of component

magnetic reconnection at the magnetopause

M. SwisdakInstitute for Research in Electronics and Applied Physics, University of Maryland, College Park, Maryland, USA

B. N. RogersDepartment of Physics, Dartmouth College, Hanover, New Hampshire, USA

J. F. Drake and M. A. ShayInstitute for Research in Electronics and Applied Physics, University of Maryland, College Park, Maryland, USA

Received 4 October 2002; revised 10 January 2003; accepted 13 March 2003; published 31 May 2003.

[1] We present particle-in-cell simulations of collisionless magnetic reconnection in asystem (like the magnetopause) with a large density asymmetry across the current layer.In the presence of an ambient component of the magnetic field perpendicular to thereconnection plane the gradient creates a diamagnetic drift that advects the X-line withthe electron diamagnetic velocity. When the relative drift between the ions andelectrons is of the order the Alfven speed the large scale outflows from the X-linenecessary for fast reconnection cannot develop and the reconnection is suppressed.We discuss how these effects vary with both the plasma b and the shear angle of thereconnecting field and discuss observational evidence for diamagnetic stabilization atthe magnetopause. INDEX TERMS: 7835 Space Plasma Physics: Magnetic reconnection; 2724

Magnetospheric Physics: Magnetopause, cusp, and boundary layers; 7843 Space Plasma Physics:

Numerical simulation studies; 2740 Magnetospheric Physics: Magnetospheric configuration and dynamics;

7827 Space Plasma Physics: Kinetic and MHD theory; KEYWORDS: magnetic reconnection, magnetopause,

diamagnetic drift, numerical simulation

Citation: Swisdak, M., B. N. Rogers, J. F. Drake, and M. A. Shay, Diamagnetic suppression of component magnetic reconnection at

the magnetopause, J. Geophys. Res., 108(A5), 1218, doi:10.1029/2002JA009726, 2003.

1. Introduction

[2] Magnetic reconnection, the breaking and reforming ofmagnetic field lines with a concomitant transfer of energyfrom the field to the surrounding plasma, is thought to drivesuch phenomena as solar flares, magnetospheric substorms,and tokamak sawtooth crashes. Despite the suspected link-age, the resistive magnetohydrodynamic (MHD) model ofreconnection due to Sweet [1958] and Parker [1957] (thefavored theoretical model for many years) predicts recon-nection rates in each of these systems that are typicallyorders of magnitude too low to be physically relevant. Thediscrepancy arises because the Sweet-Parker reconnectionrate varies with the Spitzer resistivity which in turn dependson particle interactions that are rare in these collisionlessplasmas.[3] Recent work suggests that terms important at small

length scales but usually ordered away in resistive MHD,notably the Hall term in Ohm’s law, lead to reconnectionrates that are consistent with the observations [Shay et al.,1999]. With the addition of this new physics the speed ofthe electron flows away from the reconnection site (the X-line) is no longer bounded by the ion Alfven speed, instead

scaling inversely with the width of the current layer formedby the reconnecting magnetic fields. As a result, the out-ward electron flux remains large even as the width, con-trolled by nonideal effects, becomes small. Downstreamfrom the reconnection site the super-Alfvenic electrons slowto rejoin the ions and expand outwards in a wide layerreminiscent of the Petschek [1964] model. This behaviorhas been confirmed by a variety of numerical simulations:two-fluid, hybrid (fluid electrons and particle ions), and fullparticle. In the GEM (Geospace Environment Modeling)reconnection challenge each of several independent codesmodeling an identical (two-dimensional) system foundfeatures similar to those described above [Birn et al.,2001 and references therein]. Laboratory experiments inthe relevant regimes are challenging, but recent results[Brown, 1999; Ji et al., 1999] support some aspects of thispicture.[4] Although the GEM challenge established a possible

mechanism for fast collisionless reconnection, it did so for arelatively simple geometry. Experimental evidence suggeststhat in nature reconnection is likely to be more complex. Forinstance, at the Earth’s dayside magnetopause the magneto-sphere, a region of low density but strong magnetic field,abuts the magnetosheath and its high density but weakermagnetic field. Still, early single-point spacecraft measure-ments of the magnetopause [Paschmann et al., 1979;

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 108, NO. A5, 1218, doi:10.1029/2002JA009726, 2003

Copyright 2003 by the American Geophysical Union.0148-0227/03/2002JA009726$09.00

SMP 23 - 1

Sonnerup et al., 1981] detected field signatures and particledistribution functions suggesting the occurrence of recon-nection. A more recent study using data from multiplespacecraft [Phan et al., 2000] found, at least in one case,the bidirectional jets that are a reconnection hallmark.Similar cross-field pressure gradients occur in laboratoryfusion experiments where their associated diamagneticdrifts are thought to control the onset of reconnection duringthe sawtooth crash [Levinton et al., 1994].[5] The first theoretical explorations of asymmetric

reconnection [Levy et al., 1964; Petschek and Thorne,1967] were based on incompressible MHD and predictan outflow region comprising a combination of slow MHDshocks and slow and intermediate MHD waves. Two-dimensional computer simulations using compressibleMHD [Hoshino and Nishida, 1983; Scholer, 1989] hybrid[Lin and Xie, 1997; Omidi et al., 1998; Krauss-Varban etal., 1999; Nakamura and Scholer, 2000] and particle[Okuda, 1993] codes have addressed these predictions.Common features include the preferential growth of themagnetic island (O-line) toward the magnetosheath and apronounced density drop on the magnetospheric side of theboundary layer. Also, consistent with the picture of Levy etal. [1964], the current layer abutting the magnetosheath isgenerally stronger than that bordering the magnetosphereand takes the form of a series of discontinuities. There isdisagreement, however, concerning the precise form ofthese discontinuities, possibly because the nonlocalityassociated with the motion of collisionless particles par-allel to the shock front makes the two-dimensional (2-D)simulations more complicated than the 1-D models.Simulations have also been performed with an initialout-of-plane component of the magnetic field (nonzeroBy in GSM coordinates) [Hoshino and Nishida, 1983;Krauss-Varban et al., 1999; Nakamura and Scholer,2000]. This guide field tends to slow, but not completelysuppress, reconnection and alter the structure of the shocktransitions.[6] In our investigations we begin with a simple magneto-

pause equilibrium model with an ambient pressure differ-ence and perform two-dimensional fully kinetic numericalsimulations. We reproduce the qualitative features seen inprevious work but also find that diamagnetic drifts producedby the pressure gradient advect the X-line. When thediamagnetic velocity, v* = �(c/qnB2)rp � B (c is thespeed of light; q, n, and p are the species charge, density,and pressure; B is the magnetic field), is comparable to theAlfven speed, reconnection is completely suppressed. Thesediamagnetic effects have not been previously reported in thecontext of magnetopause reconnection. They are orderedout of the MHD model [Hoshino and Nishida, 1983] andwere not seen in the hybrid simulations, perhaps because thespatially localized resistivity used to trigger fast reconnec-tion effectively prohibits X-line advection. In spite of this,diamagnetic effects can be significant at the magnetopause.Taking n � 10 cm�3, B � 10�4 G, kBT � 100 eV, andgradients of order an inverse ion inertial length, one findsv*/vA � 1. In the fusion context, previous analytic work hassuggested that diamagnetic drifts can stabilize the tearingmode and lower the rate of reconnection [Biskamp, 1981;Rogers and Zakharov, 1995]. However, since this workconsidered a reduced MHD model (equivalent to the limit of

a large guide field), its applicability to magnetosphericapplications has not been established.[7] Unlike the configuration in the magnetotail, the mag-

netic field on opposite sides of the magnetopause is usuallynot equal in magnitude and antiparallel. Reconnection insuch a system does not occur at a unique spatial location,even in a model with a 1-D equilibrium, since componentsof the magnetic field reverse direction at any location acrossthe current layer. Because of the possibility that reconnec-tion can occur at multiple spatial locations, it has beensuggested that the magnetopause magnetic field is stochas-tic [Galeev et al., 1986; Lee et al., 1993]. The exploration ofreconnection in a 2-D model must then be carried out withcare since stability at a particular surface does not necessa-rily imply stability at all surfaces. In exploring the impact ofdiamagnetic drifts on reconnection we therefore mustaddress both stability at a single plane and explore whichplane is expected to dominate the dynamics of a full 3-Dsystem. We suggest on the basis of analytic arguments andsimulations that the strongest reconnection occurs at thesurface where the reversed field components have equalmagnitudes.[8] In section 2 of the paper we present our computational

scheme and initial conditions. Section 3 is an overview ofreconnection at the magnetopause, both without and with adiamagnetic drift. Section 4 discusses how varying thestrength of the out-of-plane field changes the diamagneticeffects while section 5 addresses the question of determin-ing the dominant reconnection plane. Finally, in section 6we summarize our results and discuss their implications forunderstanding magnetic reconnection at the magnetopause.

2. Computational Methods

[9] Although diamagnetic drifts are present in two-fluidand hybrid models, a complete calculation in the collision-less limit requires a careful, and assumption-filled, treatmentof the full pressure tensor. To avoid this difficulty, wesacrifice the benefits of a fluid simulation for a morecomputationally demanding, but mathematically straightfor-ward, full particle description.

2.1. The Code

[10] The simulations are done with p3d, a massivelyparallel kinetic code that can evolve up to �109 particleson current Cray T3E and IBM SP systems [Zeiler et al.,2002]. The Lorentz equation of motion for each particle isevolved by a Boris algorithm (E accelerates for half atimestep, followed by a rotation of v by B, and then theother half-step acceleration by E). The electromagneticfields are advanced in time with an explicit trapezoidal-leapfrog method using second-order spatial derivatives.Poisson’s equation constrains the system; if r � E 6¼ 4pr,a multigrid algorithm recursively corrects the electric field.Although the code permits other choices, we work withfully periodic boundary conditions.[11] The code is written in normalized units: lengths to

the ion inertial length c/wpi = di, times to the inverse ioncyclotron frequency �ci

�1 , velocities to the Alfven speed vA,masses to the ion mass mi, and temperatures to mivA

2 . Unlessotherwise noted, the asymptotic reconnecting field isassumed to have a magnitude of 1 on both sides of the

SMP 23 - 2 SWISDAK ET AL.: DIAMAGNETIC SUPPRESSION OF RECONNECTION

magnetopause (the reason for this choice will be made clearin section 5) and the asymptotic density in the magneto-sheath is taken to be 1.[12] To conserve computational resources, yet assure a

sufficient separation of spatial and temporal scales, we takethe electron mass to be 0.005 = 1/200 and the speed of lightto be 20. Our simulations are performed in a box of 1024 �1024 gridpoints that is 25.6 on a side, corresponding to agrid scale of 0.025 and so �3 gridpoints per electron inertiallength. A typical timestep is 1.5 � 10�3.

2.2. Initial Conditions

[13] Reconnection occurs in the x-y plane in our coor-dinate system. For reference, this is equivalent to the z-xplane in GSM coordinates. The initial equilibrium is con-structed from two (modified) Harris current sheets centeredat Ly/4 and 3Ly/4, where Ly is the box size in the y direction;to allow periodicity the current is parallel to z in one sheetand antiparallel in the other. The current sheets produce amagnetic field Bx = tanh[(y � Ly/4)/w0] � tanh[(y � 3Ly/4)/w0] � 1, where w0 = 0.5. The density profile is similar, n =n0(tanh[(y � Ly/4)/w0] � tanh[(y � 3Ly/4)/w0]) + nmin,where unless otherwise specified, n0 = 0.45 and nmin =0.1. This profile gives an asymptotic density of n = 1 in themagnetosheath and 0.1 in the magnetosphere. The initial ionand electron temperatures are Ti = 2 and Te = 1, implyingthermal speeds of �1.4 and 14, respectively. The guide(out-of-plane) field Bz is assigned a specific asymptoticvalue Bz0 on the magnetosheath side and is calculatedelsewhere by assuming pressure balance. Note that at thereversal surface, where Bx = 0, pressure balance implies that@yn(Ti + Te) / Bz@yBz, and hence a density gradient must beaccompanied by a gradient in Bz. At t = 0 a 5% perturbationin the magnetic flux function places the X- and O-line ineach sheet.[14] Unlike a conventional Harris sheet, this initial state is

not a kinetic (Vlasov) equilibrium. Although it has beenshown [Quest and Coroniti, 1981] that the presence of aguide field reduces the growth rate of the tearing modeinstability in the linear regime, our initial perturbation islarge enough that the system begins to reconnect non-linearly before any other perturbations become important.We must also emphasize that our system characterizes themagnetopause only in a rough approximation. In particular,magnetospheric plasma is typically not isothermal butinstead has temperature gradients comparable in magnitude,but antiparallel, to those of the plasma density. If this effectis included the overall pressure gradient, as well as thediamagnetic effects we discuss here, will be smaller.

3. Simulation Overview

[15] In the simplest description of fast collisionless recon-nection three length scales are relevant: the system size andthe ion and electron inertial lengths. (Depending on thestrength of the initial out-of-plane field, the ion Larmorradius can also enter as a parameter [Kleva and Drake,1995].) The dynamic couplings between these scales, whichmust exist for fast reconnection to occur, are discussed indetail by Biskamp et al. [1997] and Shay et al. [1998]. Atlarge scales both species are frozen into the magnetic fieldand flow towards the X-line with a velocity vin that is some

fraction of the upstream Alfven speed. Roughly c/wpi awayfrom the current sheet the inertia term in the ion equation ofmotion becomes important and the ions, but not the elec-trons, decouple from the field. Deflected outwards, the ionsflow Alfvenically away from the X-line while the electronstravel further inward until they too decouple and traveloutward at velocities near the electron Alfven speed. Down-stream, the electrons slow to join the ionic outflow andreform an MHD fluid. In what follows we show thatdiamagnetic drifts can disrupt this structure by stronglyaltering the MHD outflows, thereby inhibiting the couplingof the ions to the X-line and halting the formation of a large-scale reconnection geometry.

3.1. Antiparallel Reconnection

[16] To document certain features and establish contactwith previous work, Figure 1 shows the X-line structurefrom a magnetopause simulation similar to those describedby Krauss-Varban et al. [1999] and Nakamura and Scholer[2000]. The upper section of each panel (y/di > 3.2)contains low-density plasma and a large magnetic field likethe magnetosphere, while the lower section has a highdensity plasma and a weak field like the magnetosheath.There are a few differences between the initial conditionsfor this simulation and the case with a guide field discussedin section 2.2 (and used in the rest of this paper). Thereversed field is given by a shifted hyperbolic tangent suchthat the magnitudes of the asymptotic fields are unequal,Bxsp = 1.5 and Bx

sh = 0.5, where magnetosphere andmagnetosheath quantities are denoted with the superscripts‘‘sp’’ and ‘‘sh,’’ respectively. The guide field Bz is zero.Pressure balance then fixes the density profile as a gradient

Figure 1. Reconnection in a magnetopause-like config-uration with no guide field. In each panel t = 13.5 �ci

�1 . (a)The out-of-plane current density Jz, (b) the in-plane fieldlines, and (c) the ion velocity vectors, averaged over a scaleof �di. Note the flows associated with the X-line. Thelongest vector corresponds to a velocity of �0.15 vA.

SWISDAK ET AL.: DIAMAGNETIC SUPPRESSION OF RECONNECTION SMP 23 - 3

from 1 to 2/3 across the current layer with a relativemaxima where Bx = 0.[17] The general features of Figure 1 are consistent with

earlier calculations. One consequence of the asymmetry infield strengths across the magnetopause is the evidenttendency for the magnetic island to grow preferentially inthe direction of the magnetosheath. Since equal amounts offlux must reconnect from either side in a given time, Bspvin

sp =Bshvin

sh, and so vinsh = (Bsp/Bsh)vin

sp vinsp . The larger inflow on

the magnetosheath side of the reversal surface is evident inthe ion velocity vectors shown in Figure 1c. With thestronger inflow, the (frozen-in) magnetosheath field is morestrongly advected toward the X-line than its magnetosphericcounterpart, leading to the wider opening angle (i.e., thebulge) on the magnetosheath side of the current layer. Inearlier MHD simulations it was shown that the asymmetry ofthe island did not appear when the amplitudes of the reversedfield on either side of the magnetopause are equal [Scholer,1989].[18] Because of the simulation’s relatively small size

clearly separated MHD and kinetic discontinuities do notexist, making comparisons with previous work difficult.Nevertheless, in cuts of the reconnection outflow (seeFigure 2), the self-generated, out-of-plane magnetic fieldis evident and the positive (negative) value of Bz on themagnetosheath (magnetosphere) side of the current layeris consistent with earlier predictions. The perturbation,however, is much stronger on the magnetosheath side;the antisymmetry of Bz seen in systems with a symmet-ric pressure does not extend to reconnection at themagnetopause.[19] The sharp density drop at the magnetosphere edge of

the current layer (y � 3.4 di) is a consequence of the

different inflow speeds on the two sides of the layer andthe plasma mixing across the island. Since the magneto-spheric field is much stronger, and therefore the inflowvelocity from this side of the current layer is weak, only asmall amount of magnetospheric plasma crosses the sepa-ratrices. The result is an outflow region almost completelycomprising magnetosheath plasma. This effect was evidentin earlier hybrid simulations [Krauss-Varban et al., 1999;Nakamura and Scholer, 2000] and has also been seen insatellite crossings of the magnetopause [Eastman et al.,1996].

3.2. Component Reconnection and DiamagneticPropagation

[20] In the presence of a magnetic field, any nonparallelpressure gradient produces a diamagnetic drift,

v*j

¼ �c

#

pj � B

qjnB2; ð1Þ

where pj = nTj is the thermal pressure and qj is the charge ofspecies j. Due to the charge dependence, ions and electronsdrift in opposite directions. To have diamagnetic flowsparallel to the current layer in our geometry the guide fieldBz and the gradient of the pressure must be nonzero whereBx = 0.[21] Even though diamagnetic drifts do not correspond to

actual particle motions they nevertheless advect the mag-netic field [Coppi, 1965; Scott and Hassam, 1987]. In twodimensions the magnetic field can be written as B = z � #y (x, y) + Bz(x, y)z where y is the magnetic flux function.Taking the cross product of Faraday’s law with z yields@try � crEz = 0, or Ez = @ty/c. Next, dotting the electronfluid equation with z gives

Ez ¼ � 1

cz � ve � Bð Þ � me

e

dvez

dt: ð2Þ

The last term represents the inertial effects of electrons andbreaks the frozen-in condition. Since it is usually small weignore it and substitute for B to get Ez = �ve �

#

y/c and aconvection equation for the flux [Coppi, 1965],

@tyþ ve � ry ¼ 0: ð3Þ

Hence the electron fluid velocity, which includes a diamag-netic component given by equation (1), advects magneticstructures.[22] Strictly speaking, equations (1) and (3) are fluid

results and need not describe the dynamics at the X-lineof our simulations where small-scale structures leave thefluid approximation on unsure ground. However, as can beseen in Figure 3, they are still good descriptions of thesystem. The out-of-plane current density essentially mapsthe magnetic field lines (see Figure 1b) so the island(centered at x/di � 5) grows robustly during the time shown.Simultaneously, the X-line propagates to the left, the direc-tion of the electron diamagnetic drift. The locations of theX-line in the three panels of Figure 3 are x/di = 19.2, 14.8and 10, respectively, corresponding to a drift speed of 0.61vA and consistent with the value calculated from the initialparameters of v*e = 0.56. As flux reconnects the magnetic

Figure 2. Cuts at x = 20 di of the outflow region of Figure1. Each plot has been averaged over 0.25 di in the horizontaldirection to reduce the noise. (a)–(c) The reconnectingfield, the out-of-plane field, and the density, respectively.


island widens, decreasing the pressure gradient across itscenter, slowing its drift, and allowing the X-line to overtakeit. This stabilizes the island’s growth because the plasmaoutflow from the X-line towards the near end of the islandmust slow due to the increase in magnetic pressure. Theactual collision of the X-line with the island, which occurslater in time, may be unphysical since in any real system theinitial separation of the X- and O-lines will be much greaterthan the 12.8 di of the present model. Thus the effects of thecollision will not be discussed further.[23] A surprise is that the diamagnetic propagation

evident in Figure 3 has not been seen in previous hybridsimulations of the magnetopause [Krauss-Varban et al.,1999; Nakamura and Scholer, 2000] even though thesemodels should include the relevant electron diamagneticdrifts. In some cases the simulation parameters may besuch as to minimize diamagnetic effects (e.g., a lowermagnetospheric temperature). Diamagnetic drifts may alsobe absent if the simulations were performed with aninitially uniform out-of-plane field. According to the forcebalance condition, the local pressure gradient (and hencethe drift) must then be zero at the reversal surface, althoughwhether it would remain zero as the system evolves isunclear. In section 5 we discuss further why a uniformguide field may not be representative of reconnection at themagnetopause. A final possibility occurs for simulationsusing a fixed, spatially localized resistivity to break thefrozen-in condition at the X-line. How such a resistivitymodel would affect an X-line that is trying to propagate isunclear, but it is nevertheless evident from our results that

such models may be inappropriate for simulating guidefield reconnection at the magnetopause when diamagneticdrifts are large.[24] Figure 4a shows the current density for the simula-

tion of Figure 3. Unlike the example shown in Figure 1, themagnetic island does not bulge into the magnetosphere sideof the current layer. This is because the reconnectingmagnetic field Bx is antisymmetric across the current layerand thus, as discussed earlier, both sides contribute equalamounts of magnetic flux to the reconnection. Clearlyevident in Figure 4a, as well as the last two panels ofFigure 3, is the left-right asymmetry of the opening angle ofthe magnetic field lines near the X-line. This is caused bythe interplay of the X-line’s diamagnetic drift, controlled byjust the electrons, and the reconnection outflow, which mustalso involve the ions.[25] Consider the relative motion of the X-line and the

ions. Downstream from the reconnection site the J � Bforce accelerates frozen-in ions up to the Alfven speed,±vAx. To this must be added the relative velocity betweenthe diamagnetic drift of the ions and the X-line, v* = jv*ij +jv*ej, with the result being a right-left asymmetry in theoutflow velocities, ±vAx + v*. Since, from continuity, theopening angle of the magnetic field downstream from theX-line is roughly vin/vout, and vin is roughly constant due tothe symmetric reconnecting field, we expect the rightwardopening angle to be much smaller than its counterpart,consistent with Figure 4b.[26] By a similar argument, normal ion outflow from an

X-line only develops when vAx > v*. For the simulationspresented in Figures 3 and 4 the electron and ion diamag-netic drifts at the reversal surface are �0.56 and 1.12,respectively, compared with vAx = 1.35 (based on a densityof 0.55, corresponding to a mixture of the magnetopauseand magnetosheath plasma). Thus for the parameters of this

Figure 3. The out-of-plane current density at three timesfor a magnetopause geometry similar to Figure 1 butincluding an out-of-plane guide field and therefore an in-plane diamagnetic drift. The initial conditions are thosedescribed in section 2.2 with an asymptotic guide field of3.0 in the magnetosheath and 3.8 in the magnetosphere. Thedensity drops from 1 to 0.1 across the current layer.

Figure 4. Structure of (a) the out-of-plane current densityJz and (b) ion flow vectors at t = 16.0 �ci

�1 for thesimulation shown in Figure 3. The large flows at the currentlayer are the ion diamagnetic drifts. The longest vectorcorresponds to a velocity of �1.2 vA.


simulation, the reconnection should be marginally unable toreverse the ion flow to the left of the X-line. The ion flowvectors are shown in Figure 4b at the same time as the currentdensity shown in Figure 4a. The strong ion flow on the rightside of the X-line results from the combination of the iondiamagnetic velocity with the direct acceleration by thereconnected magnetic field. The reconnection generatedforces almost reverse the diamagnetic flow just to the leftof the X-line.[27] We again emphasize that it is the relative drift of the

ions and electrons, i.e., v* = jv*ij + jv*ej, that must besmaller than the in-plane Alfven speed for normal recon-nection to proceed. Reconnection was suppressed in anearly indistinguishable fashion in a series of simulationswith the same total temperature Te + Ti (and thereforeidentical v*) but differing values of Te and Ti (Ti = 1/2,Te = 5/2 and Ti = 5/2, Te = 1/2). Since Ti Te at themagnetopause, stabilization can still occur even if the driftof the X-line, which is due only to v

*e, is negligible.

[28] The cuts in Figure 5 show that for this simulation thereconnection field Bx is symmetric across the layer whilethe pressure drop across the magnetopause is supported bythe jump in the guide field Bz. In contrast to the densityprofile in the case with no guide field, the density dropacross the layer is nearly centered on the magnetic island.Large drops occur on either edge of the current layer with aweak plateau in the middle of the island. The inflowvelocity into the island from either side of the current layeris now roughly equal because of the near antisymmetry ofBx. The plasma in the core of the island is therefore a nearlyequal mix of magnetosphere and magnetosheath plasma and

is not dominated by the magnetosheath, in contrast to thecase shown in section 3.1.

4. Suppression of Component Reconnection

[29] We have shown for a single case that consistent withequation (3), X-lines convect with the diamagnetic velocityof the electrons in the reversal region and suggested acriterion for the suppression of reconnection from diamag-netic drifts, v* > vAx. We now more fully explore thisstabilization criterion by varying the strength of the out-of-plane field to vary the ratio of v* to vAx. For ourmagnetopause model the diamagnetic velocity at the surfacewhere Bx = 0 can be written as

v*j

¼ cTj

qjLpjBz

; ð4Þ

where the pressure scale length is given by Lp�1 = j@ypj/p.

Varying the amplitude of the out-of-plane field Bz whilekeeping all the other parameter fixed alters v*j but keeps vAxconstant. As can be seen in Figure 6a, a large guide field

Figure 5. Cuts at x = 13.8 di of the outflow region ofFigure 4. Each plot has been averaged over 0.1 di in thehorizontal direction to reduce the noise. (a)–(c) Thereconnecting field, out-of-plane field, and density,respectively.

Figure 6. The top panel shows the position of the X-lineversus time for several runs differing only in the strength ofthe guide field Bz. The solid line is for the run of section 3.1and experiences no diamagnetic motion. The dotted,dashed, and dash-dotted lines correspond to Bz0 = 3, 1.5,and 0.5, respectively. The bottom panel is a comparison ofthe measured X-line velocities and the electron diamagneticvelocity at the reversal surface. The straight line has a slopeof one and is plotted to guide the eye. In both panels theX-point location corresponds to a saddle point in y.


implies a small v*. Despite the occasional glitches(discontinuities in the motion arise when a new X-linetriggered by random perturbations begins to dominate), thedrift velocity is remarkably constant.[30] The bottom panel of Figure 6 shows the initial

electron diamagnetic velocity at the reversal surface versusthe drift speed of the X-line measured in the simulation.Although the agreement for sub-Alfvenic velocities is quitegood, as v

*eapproaches vAx another effect becomes impor-

tant. Rather than move the ions at super-Alfvenic velocitiesthe system develops an electric field Ey transverse to thecurrent layer that, combined with the guide field Bz, gen-erates an E � B drift opposing the ion motion and adding tothe electron velocity. There is no change in the net currentbut the electrons, and hence the X-line, move faster than justthe diamagnetic velocity would suggest.[31] Since diamagnetic flows of sufficient strength disrupt

the large-scale flows characteristic of reconnection,increases in v

*should correlate with decreases in the

amount of reconnected flux. Figure 7 depicts the extent towhich a diamagnetic drift can alter the reconnection rate of asystem. The reconnected flux is plotted versus time for fourdifferent simulations: one with no diamagnetic drift, as inFigure 1, and three with varying asymptotic guide fields,Bz0 = 1.5, 1.0, and 0. For the latter cases the associated driftspeeds (diamagnetic plus E � B) at the reversal surface arev* = 0.8, 1.1 and 1.5, respectively. Reconnection is almostcompletely suppressed for the largest drifts. The plateau inreconnected flux seen in each of the bottom three curves isan artifact (discussed earlier) of our system’s periodicboundary conditions that allow the X-line to impinge onthe more slowly propagating magnetic islands.[32] For large guide fields the diamagnetic drift speed

approaches zero and the stabilization is minimized. Theconverse limit is more subtle since pressure balance imposesrestrictions on the system’s configuration. If the profiles ofthe density and the reconnecting field are specified thenthere is a minimal possible value of the guide field in thecurrent layer; Bz cannot be taken to be zero and the driftcannot grow without bound. Hence although (for example)

the GEM challenge simulations had zero guide field theydid not have a large diamagnetic drift. Rather, Bz ! 0 andLp ! 1, keeping v* = 0.[33] To confirm the hypothesis that the reconnection

process cannot reverse the ion diamagnetic drift and driveoutward flows from the X-line for large values of thediamagnetic velocity, we show Jz and the ion velocities froma simulation with an asymptotic out-of-plane field Bz = 0.5 inFigure 8. In this case the island of reconnected flux remainssmall even at late time because of the weak growth shown inFigure 7. The flows are actually smaller than the peak of theexpected diamagnetic velocity because of the developmentof the electric field Ey across the layer. A slight reduction ofthe ion flow speed to the left of the X-line indicates that thereconnected field lines exert a small leftward directed forceon the ions in this region, but the result is only a smallperturbation of the ion flow pattern. Thus the X-line isunable to couple effectively to the ions, strong reconnectiondoes not develop, and the island amplitude saturates.

5. Dependence on the Reconnection Plane

[34] Although two-dimensional simulations are computa-tionally cheaper than their three-dimensional counterparts,this simplicity comes with drawbacks. Perhaps the mostimportant when considering diamagnetic drifts is that theorientation of the X-line with respect to the magnetic fieldconfiguration is externally imposed, rather than being free todevelop. Different simulation planes have different recon-necting and out-of-plane fields and hence varying amountsof diamagnetic stabilization. In this section we explore thiseffect, albeit only by considering rotations around the y-axisto avoid introducing stabilizing By components into thesystem. We will assume that the favored plane (i.e., theone the system would choose if allowed to evolve in three

Figure 7. Reconnected flux versus time for four simula-tions. The reconnection rate is the slope of each curve. Thesolid line is a reference run with no diamagnetic drift. Thedotted, dashed, and dash-dotted lines are from simulationswith differing values of Bz0 (1.5, 1.0 and 0, respectively)and therefore v*.

Figure 8. (a) The out-of-plane current density Jz and (b)ion flow vectors at t = 13.0 �ci

�1 from a simulation with anasymptotic Bz = 0.5 and a strong diamagnetic drift. Thelarge flows to the right at the current layer are the result ofthe ion diamagnetic drift. The longest vector corresponds toa velocity of �0.95 vA.


dimensions) is the one where the reconnection rate ismaximal but note that the results are not a replacement formore ambitious 3-D simulations. We further assume that thediamagnetic effects are playing the dominant role in deter-mining the plane where reconnection is most robust. Thuswe are considering systems in which v* � vAx.[35] Three-dimensional particle simulations have already

addressed this issue for the basic case of a system with noguide field and no pressure drop across the current layer[Hesse et al., 2001; Pritchett, 2001]. Random perturbationswere used as seeds to avoid imposing a plane a priori.Although X-lines were free to develop with any orientation,they preferentially formed parallel to the current layer,consistent with our assumption of treating only rotationsabout y. Moreover they remained quasi-two-dimensionaleven at late times. However, the simplicity of B makes thiscase degenerate since the reversal surface (the place whereBx = 0) must be at the center of the current layer for anyrotation about y. This restriction does not exist in morecomplicated configurations.[36] The basic geometry of magnetic fields at the mag-

netopause is shown in Figure 9. In order to agree with oursimulation coordinates, the magnetospheric field Bsp istaken to be in the positive x direction while the magneto-sheath field Bsh is inclined at an angle q with respect to thenegative x axis. Thus the rotation angle across the magneto-pause is p–q. The y direction is perpendicular to the surfaceof the magnetopause. For a two-dimensional simulation inthe x-y plane the reconnecting components of the magneticfield will be parallel to x,

Bspx ¼ Bsp ð5Þ

Bshx ¼ �Bsh cos q; ð6Þ

and the guide field to z,

Bspz ¼ 0 ð7Þ

Bshz ¼ Bsh sin q: ð8Þ

The primed axes in Figure 9 are rotated clockwise by anangle a with respect to the unprimed system and representanother possible reconnection plane. In a simulation carriedout in the x0-y0 plane the reconnecting components of themagnetic field will be

Bspx0 ¼ Bsp cosa ð9Þ

Bshx0 ¼ �Bsh cos qþ að Þ; ð10Þ

while the guide field components are

Bspz0 ¼ �Bsp sina ð11Þ

Bshz0 ¼ Bsh sin qþ að Þ: ð12Þ

[37] As shown in section 4, stabilization via a diamagneticdrift is inversely proportional to the value of the guide fieldBz in the center of the current layer. Thus the expectation isthat the plane with the maximum value of the guide field atthe surface where reconnection takes place will dominate.Unfortunately, the asymptotic fields determine neither thelocation of the reversal surface nor the value of the guidefield there. Furthermore, the actual guide field at the reversalsurface may vary as reconnection proceeds. To simplifymatters, we adopt the crude hypothesis that the guide fieldat the reversal surface Bg is the mean of the asymptotic guidefields, which is [Bshsin(q + a) � Bspsina]/2 in the rotatedsystem. With this ansatz the optimal plane for reconnectionis that where Bg is maximized with respect to the angle a,from which we derive the condition

Bsp cosa ¼ Bsh cos qþ að Þ: ð13Þ

Interestingly, this corresponds to the plane where thereconnecting field components are equal in magnitude butoppositely directed (see equations (9) and (10)). Figure 10

Figure 9. Schematic of the asymptotic magnetic fields inthe magnetosphere, Bsp, and magnetosheath, Bsh. Thereconnecting and out-of-plane components of the magneticfield depend on the orientation of the reconnection plane:x-z versus x0-z0.

Figure 10. Reconnected flux versus time for threesimulations. The solid line is a simulation with anasymptotic guide field of 1.5 (the dotted line in Figure 7).The dashed and dotted lines are simulations in planesrotated ±15� about y.


shows the effect of rotating our simulation plane on thereconnection rate and suggests that when diamagnetic driftsare strong, reconnection in the antisymmetric plane isfavored.

6. Summary and Discussion

[38] Diamagnetic drifts can be an important part ofmagnetopause reconnection in certain parameter regimes.In our simulations the plasma b (approximately 1) and themagnetosheath-magnetosphere density constrast (a factor of10) are roughly consistent with observational constraints.However our initial conditions also neglect both temper-ature gradients and scale lengths larger than an ion inertiallength. For other magnetopause models that include thesefeatures the diamagnetic drifts and associated stabilizationmay be smaller.[39] When diamagnetic drifts are significant an X-line

will advect with the electron fluid velocity, causing theseparatrices to have an asymmetry in their opening angleswith the wider facing the direction of motion. For largeenough flows, v* = jv*ej + jv*ij � vA, reconnection can becompletely suppressed since the large-scale outflows fromthe X-line needed for fast reconnection cannot develop. Thereconnected field lines are not strong enough to reverse theion diamagnetic flows toward the X-line.[40] The diamagnetic effects presented here introduce a b

dependence that was absent in the symmetric systemsstudied earlier. The diamagnetic stabilization conditionv*> vA can be rewritten as a condition on b:

bx >Bz

Bx

2Lp

c=wpi

; ð14Þ

where Bx is the amplitude of the reconnecting field, bx =8pnT/Bx

2 , and Bz and Lp are evaluated at the current layer.For Bz � Bx and for pressure scale lengths of the order of theion inertial length c/wpi, the threshold for the suppression ofreconnection is b � 1.[41] A statistical study of accelerated flow events was

done by Scurry et al. [1994] to investigate the dependenceof dayside reconnection on magnetosheath b and the shearangle of the magnetic field across the magnetopause. Theyfound that for low values of b the accelerated flow eventsspread over a broad range of clock angles while at highvalues of b they were strongly correlated with large clockangles. The implication is that magnetic reconnection in thepresence of a significant guide field is suppressed at high b.Thus these observations are consistent with the criteriongiven by equation (14) for the diamagnetic suppression ofreconnection.[42] When Bz Bx, equation (14) indicates that it is

difficult to stabilize reconnection with diamagnetic effects,just as is demonstrated in section 4. We do not address whathappens when the guide field becomes very large, butRogers et al. [2001] examined the effect of b on reconnec-tion in a symmetric system (no pressure drop across thecurrent layer) using theoretical arguments as well as numer-ical simulations. They argued that fast reconnection asdescribed in section 3 can take place whenever dispersivewaves (e.g., whistlers or kinetic Alfven waves) exist in asystem at small spatial scales. Only under extreme con-

ditions (Bz � Bx

ffiffiffiffiffiffiffiffiffiffiffiffiffimi=me

p, bx = 8pnT/Bx

2 � 1) is reconnec-tion inhibited by a guide field Bz.[43] A final curious feature of equation (14) occurs in the

limit where Bz is small, as the expression would seem toimply that reconnection is inhibited even for low values ofbx. This is not the case because there is a linkage betweenthe pressure scale length Lp and the value of Bz at thereversal surface. Since the reconnection field Bx = 0 at thislocation, pressure balance requires that p/Lp = BzBz

0 withBz0 = j@yBzj. This condition implies that BzLp = p/Bz

0 and sothat if the right-hand side is to remain finite then Lp !1 asBz ! 0. In the absence of a guide field the pressure gradientmust be zero at the reversal surface. For nonconstant Bz theminimal initial value of the reversal guide field is deter-mined by the details of the initial conditions.[44] Unlike systems with simple reversed fields, recon-

nection at the magnetopause is not limited to a single plane,a fact that has led to the prediction that the magnetopausefield might become stochastic [Galeev et al., 1986; Lee etal., 1993]. Diamagnetic drifts offer a way to triggerreconnection at multiple surfaces since they can directlyaffect which planes dominate reconnection. A rough argu-ment suggests that the maximal guide field (and hence theminimal drift and minimal stabilization) occurs at thesurface where the components of the field that reconnectare equal and opposite. Hence we expect this to be thedominant location of ‘‘component reconnection’’ at themagnetopause.[45] The direction of the X-line advection can in principle

be predicted from the local pressure gradients and magneticfields. Under appropriate conditions this effect should bedetectable by spacecraft at the magnetopause, althoughdefinitive results are certainly more likely with multiplespacecraft missions like Cluster or the future Magneto-spheric Multiscale Mission.

[46] Acknowledgments. This work was supported in part by theNASA Sun Earth Connection Theory and Supporting Research andTechnology programs and by the NSF.[47] Shadia Rifai Habbal thanks Dietmar Krauss-Varban and another

referee for their assistance in evaluating this paper.

ReferencesBirn, J., et al., Geospace environmental modeling (GEM) magnetic recon-nection challenge, J. Geophys. Res., 106, 3715–3719, 2001.

Biskamp, D., Nonlinear theory of the m = 1 mode in hot tokamak plasmas,Phys. Rev. Lett., 46, 1522–1525, 1981.

Biskamp, D., E. Schwarz, and J. F. Drake, Two-fluid theory of collisionlessmagnetic reconnection, Phys. Plasmas, 4, 1002–1009, 1997.

Brown, M. R., Experimental studies of magnetic reconnection, Phys. Plas-mas, 6, 1717–1724, 1999.

Coppi, B., Current-driven instabilities in configurations with sheared mag-netic fields, Phys. Fluids, 8, 2273, 1965.

Eastman, T. E., S. A. Fuselier, and J. T. Gosling, Magnetopause crossingswithout a boundary layer, J. Geophys. Res., 101, 49–57, 1996.

Galeev, A. A., M. M. Kuznetsova, and L. M. Zeleny, Magnetopausestability threshold for patchy reconnection, Space Sci. Rev., 44, 1–41,1986.

Hoshino, M., and A. Nishida, Numerical simulation of the dayside recon-nection, J. Geophys. Res., 88, 6926–6936, 1983.

Ji, H., M. Yamada, S. Hsu, R. Kulsrud, T. Carter, and S. Zaharia, Magneticreconnection with Sweet-Parker characteristics in two-dimensional la-boratory plasmas, Phys. Plasmas, 6, 1743–1750, 1999.

Kleva, R. G., J. F. Drake, and F. L. Waelbroeck, Fast reconnection in hightemperature plasmas, Phys. Plasmas, 2, 23–34, 1995.

Krauss-Varban, D., H. Karimabadi, and N. Omidi, Two-dimensional struc-ture of the co-planar and non-coplanar magnetopause during reconnec-tion, Geophys. Res. Lett., 26, 1235–1238, 1999.


Lee, L. C., Z. W. Ma, Z. F. Fu, and A. Otto, Topology of magnetic fluxropes and formation of fossil flux transfer events and boundary layerplasmas, J. Geophys. Res., 98, 3943–3951, 1993.

Levinton, F. M., L. Zakharov, S. H. Batha, J. Manickam, and M. C. Zarn-storff, Stabilization and onset of sawteeth in TFTR, Phys. Rev. Lett., 72,2895–2898, 1994.

Levy, R. H., H. E. Petschek, and G. L. Siscoe, Aerodynamic aspects of themagnetospheric flow, AIAA J., 2, 2065–2076, 1964.

Lin, Y., and H. Xie, Formation of reconnection layer at the dayside mag-netopause, Geophys. Res. Lett., 24, 3145–3148, 1997.

Nakamura, M., and M. Scholer, Structure of the magnetopause reconnec-tion layer and of flux transfer events: Ion kinetic effects, J. Geophys. Res.,105, 23,179–23,191, 2000.

Okuda, H., Numerical simulation of the subsolar magnetopause currentlayer in the Sun-Earth meridian plane, J. Geophys. Res., 98, 3953–3962, 1993.

Omidi, N., H. Karimabadi, and D. Krauss-Varban, Hybrid simulation of thecurved dayside magnetopause during southward IMF, Geophys. Res.Lett., 25, 3273–3276, 1998.

Parker, E. N., Sweet’s mechanism for merging magnetic fields in conduct-ing fluids, J. Geophys. Res., 62, 509–520, 1957.

Paschmann, G., et al., Plasma acceleration at the Earth’s magnetopause:Evidence for reconnection, Nature, 282, 243–246, 1979.

Petschek, H. E., Magnetic field annihilation, in Proc. AAS-NASA Symp.Phys. Solar Flares, NASA-SP 50, 425–439, 1964.

Petschek, H. E., and R. M. Thorne, The existence of intermediate waves inneutral sheets, Ap. J., 147, 1157–1163, 1967.

Phan, T. D., et al., Extended magnetic reconnection at the Earth’s magne-topause from detection of bi-directional jets, Nature, 404, 848–850,2000.

Quest, K. B., and F. V. Coroniti, Linear theory of tearing in a high-b plasma,J. Geophys. Res., 86, 3299–3305, 1981.

Rogers, B., and L. Zakharov, Nonlinear w*stabilization of the m = 1 mode

in tokamaks, Phys. Plasmas, 2, 3420–3428, 1995.

Rogers, B. N., R. E. Denton, J. F. Drake, and M. A. Shay, The role ofdispersive waves in collisionless magnetic reconnection, Phys. Rev. Lett.,87, 195004, 2001.

Scholer, M., Asymmetric time-dependent and stationary magnetic recon-nection at the dayside magnetopause, J. Geophys. Res., 94, 15,099–15,111, 1989.

Scott, B. D., and A. B. Hassam, Analytical theory of nonlinear drift-tearingmode stability, Phys. Fluids, 30, 90–101, 1987.

Scurry, L., C. T. Russell, and J. T. Gosling, Geomagnetic activity and thebeta dependence of the dayside reconnection rate, J. Geophys. Res., 99,14,811–14,814, 1994.

Shay, M. A., J. F. Drake, R. E. Denton, and D. Biskamp, Structure of thedissipation region during collisionless magnetic reconnection, J. Geo-phys. Res., 103, 9165–9176, 1998.

Shay, M. A., J. F. Drake, B. N. Rogers, and R. E. Denton, The scaling ofcollisionless, magnetic reconnection for large systems, Geophys. Res.Lett., 26, 2163–2166, 1999.

Sonnerup, B. U. O., G. Paschmann, I. Papamastorakis, N. Sckopke,G. Haerendel, S. J. Bame, J. R. Asbridge, J. T. Gosling, and C. T. Russell,Evidence for magnetic field reconnection at the Earth’s magnetopause,J. Geophys. Res., 86, 10,049–10,067, 1981.

Sweet, P. A., Electromagnetic Phenomena in Cosmical Physics, p. 123,Cambridge Univ. Press, New York, 1958.

Zeiler, A., D. Biskamp, J. F. Drake, B. N. Rogers, M. A. Shay, andM. Scholer, Three-dimensional particle simulations of collisionless mag-netic reconnection, J. Geophys. Res., 107(A9), 1230, doi:10.1029/2001JA000287, 2002.

��J. F. Drake, M. A. Shay, and M. Swisdak, IREAP, University of

Maryland, College Park, MD 20742-3511, USA. ([email protected];[email protected]; [email protected])B. N. Rogers, Department of Physics, Dartmouth College, Hanover, NH

03755-3528, USA. ([email protected])


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