The diffusion region in collisionless magnetic reconnectionMichael Hesse, Karl Schindler, Joachim Birn, and Masha Kuznetsova Citation: Physics of Plasmas (1994-present) 6, 1781 (1999); doi: 10.1063/1.873436 View online: http://dx.doi.org/10.1063/1.873436 View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/6/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Reversible collisionless magnetic reconnection Phys. Plasmas 20, 102116 (2013); 10.1063/1.4826201 The inner structure of collisionless magnetic reconnection: The electron-frame dissipation measure and Hallfields Phys. Plasmas 18, 122108 (2011); 10.1063/1.3662430 Effect of inflow density on ion diffusion region of magnetic reconnection: Particle-in-cell simulations Phys. Plasmas 18, 111204 (2011); 10.1063/1.3641964 Model of electron pressure anisotropy in the electron diffusion region of collisionless magnetic reconnection Phys. Plasmas 17, 122102 (2010); 10.1063/1.3521576 Structures of diffusion regions in collisionless magnetic reconnection Phys. Plasmas 17, 052103 (2010); 10.1063/1.3403345

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The diffusion region in collisionless magnetic reconnection *Michael Hesse†,a)

Electrodynamics Branch, NASA Goddard Space Flight Center, Greenbelt, Maryland 20771

Karl SchindlerTheoretische Physik IV, Ruhr-Universita¨t, D-44780 Bochum, Germany

Joachim BirnLos Alamos National Laboratory, Los Alamos, New Mexico 87545

Masha KuznetsovaRaytheon STX, NASA Goddard Space Flight Center, Greenbelt, Maryland 20771

~Received 16 November 1998; accepted 17 February 1999!

The structure of the dissipation region in collisionless magnetic reconnection is investigated bymeans of kinetic particle-in-cell simulations and analytical theory. Analyses of simulations ofreconnecting current sheets without guide magnetic field, which keep all parameters fixed with theexception of the electron mass, exhibit very similar large scale evolutions and time scales. Adetailed comparison of two runs with different electron masses reveals very similar large scaleparameters, such as ion flow velocities and magnetic field structures. The electron-scale phenomenain the reconnection region proper, however, appear to be quite different. The scale lengths of theseprocesses are best organized by the trapping length of bouncing electrons in a field reversal region.The dissipation is explained by the electric field generated by nongyrotropic electron pressure tensoreffects. In the reconnection region, the relevant electron pressure tensor components exhibitgradients which are independent of the electron mass. The similarities of the gradients as well as thebehavior of the electron flow velocity can be derived from the electron trapping scale and theelectron mass independence of the reconnection electric field. A further model which includes asignificant guide magnetic field exhibits almost identical behavior. The explanation of this result liesin a Hall-type electric field which locally eliminates the magnetizing effect on the electrons of theguide magnetic field. The resulting electron dynamics is nearly identical to the one found in themodel without guide magnetic field. This result strongly supports the hypothesis that the localphysics in the dissipation region adjusts itself to the demands of the large-scale evolution. A furtherverification of this notion is provided by Hall-magnetohydrodynamic simulations which employsimple resistive dissipation models in otherwise similar large-scale models. These results alsopertain to the inclusion of local reconnection physics in larger scale simulation models. ©1999American Institute of Physics.@S1070-664X~99!97405-0#

I. INTRODUCTION

Magnetic reconnection is widely recognized as a funda-mental transport mechanism in collisionless plasmas~e.g.,Schindler,1 Vasyliunas,2 Galeev3!. Magnetic reconnectionsupports large scale plasma transport by localized diffusiveeffects, rather than global diffusion which would otherwisebe required~Atkinson,4 Schindleret al.5!. Evidence of mag-netic reconnection is found in all relevant plasma environ-ments, ranging from particle acceleration in galaxies~e.g.,Kahn and Brett6! to laboratory plasmas~e.g., Baum andBratenahl,7 Stenzelet al.,8 Yamadaet al.9!. Intermediate sce-narios include processes in the solar corona, where reconnec-tion is recognized as the fundamental agent supporting theenergy conversion required to power fast dynamical pro-cesses associated with flares~Priest,10 Antiochoset al.11! and

coronal mass ejections~Goslinget al.12!. In the geospace en-vironment, magnetic reconnection at the dayside magneto-pause facilitates the input of mass, energy, and momentuminto the magnetosphere~Sonnerup et al.,13 Paschmannet al.14!, and magnetic reconnection processes on the night-side are fundamentally involved in storm and substorm dy-namics~e.g., Fairfield,15 Baker et al.,16 Birn et al.,17 Nagaiet al.18!.

Magnetic reconnection relies on the presence of a dissi-pation mechanism in a localized region of space, the so-called diffusion region. Here, dissipative processes generatean electric field, either in a region of very small magneticfield, or exhibiting a prominent magnetic field-aligned com-ponent. The presence of this electric field implies a violationof the frozen-flux condition, which allows the reconnectionprocess to proceed. This electric field impacts and needs tobe self-consistently supported by any particle species, in-cluding all kinds of ions and electrons. The dissipativemechanism responsible for the generation of these electricfields depends on the parameters of the plasma.2 In a colli-

*Paper D2I2.2 Bull. Am. Phys. Soc.43, 1699~1998!.†Invited speaker.a!Electronic mail: hesse@gsfc.nasa.gov

PHYSICS OF PLASMAS VOLUME 6, NUMBER 5 MAY 1999

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sional system, such as often found in laboratory plasmas andin the solar photosphere, binary Coulomb collisions providea viable dissipation mechanism~e.g., Krall andTrivelpiece19!. In a kinetic analysis, this process manifestsitself in the presence of a collision term in the kinetic equa-tion, which typically leads to a resistive contribution in themoment equation.19 Thus this basically Ohmic mechanismdissipates currents and thus generates resistive electric fields,which when included, easily support magnetic reconnectionas evidenced by the success of resistive magnetohydrody-namic ~MHD! models in plasma physics.

In collisionless plasmas, which we identify with plasmaswhere collisions are sufficiently infrequent to provide thenecessary dissipation on the time and spatial scales underconsideration, the nature of the dissipation underlying mag-netic reconnection is less obvious. While it is clear that mag-netic reconnection and the dissipative electric field requires aviolation ~for each speciess! of the idealness condition

E1vs3B50, ~1!

the lack of sufficiently frequent collisions mandates a furtheranalysis of relevant terms on the right-hand side of Eq.~1!.We denote by ‘‘dissipative’’ the effects on any term on theright-hand side of~1!. Therefore, our notion of dissipativedoes not necessarily coincide with the thermodynamic con-cept of dissipation. The most straightforward way to obtaininsight here is to calculate the full moment equation from thecollisionless Vlasov equation. Without approximation, oneobtains:

E1vs3B51

nsqs“–Ps1

ms

qsS ]vs

]t1vs–“vsD . ~2!

Here vs , ns , qs , ms , Ps denote velocity, number density,charge, mass, and pressure tensor of speciess, respectively.This simple expression shows that the nonideal mechanismhas to be based either on thermal, i.e., pressure based@thefirst term on the right-hand side~RHS! of ~2!#, or inertialeffects @the second term on the RHS of~2!#. Of these, thepressure-based dissipation might rely on magnetic field-aligned pressure gradients, or, in regions of sufficiently smallmagnetic field, on nongyrotropies of the distributionfunction.2,20,21We again emphasize that Eq.~2! has to hold,self-consistently, for all plasma species, including the elec-trons. Because of the vastly larger ion mass alone, supportingthis electric field is much easier for the ions than it is for thelight electrons. For the electrons, strong magnetic fieldsmight gyrotropize the distribution, and the small mass mightrequire very small spatial scales or very fast time scales forthe inertial terms to be important~Hesseet al.,22 Cai andLee,23 Hesse and Winske21!.

Previous analyses of time-dependent magnetic reconnec-tion ~Hewett et al.,24 Pritchett,25 Tanaka,26,27 Hesseet al.,22

Kuznetsovaet al.,28 Hesse and Winske,21 Shay et al.,29,30

Horiuchi and Sato,31,32 Cai and Lee23! have therefore begunto shed light on the electron behavior in different parameterregimes, primarily in the regions of low magnetic field. Hereit was found that, for current sheets of ion inertial length

thickness, deviations from gyrotropy in the electron distribu-tion function can give rise to reconnection electric fields vianongyrotropic electron pressures

Ey521

neeS ]Pxye

]x1

]Pyze

]z D . ~3!

This process can be understood as an inertial effect of ther-mal electrons which bounce in the field reversal region. Forcurrent sheets of reduced thicknesses, down to the collision-less skin depthc/ve , electron inertial effects become impor-tant and might generate very fast reconnection rates.

Clearly, a strong magnetic field in the reconnection re-gion tends to magnetize the electrons and thereby likely gen-erates gyrotropic distribution functions. As a result, thepressure-based dissipation mechanism ceases to operate un-less the electron velocities are large enough and the spatialscales of the reconnection process are small enough. This isso because the pressure nongyrotropic anisotropy scales like

sp5ve

LVe, ~4!

whereve is a typical electron flow velocity,L is a typicaldimension of the dissipation region, andVe is the local elec-tron cyclotron frequency.28 This implies that fast electronflows with small spatial scales can render pressure effectsrelevant if the electron cyclotron frequency is not too large.

If the parametersp is too small, the only remaining waysto maintain magnetic reconnection should be by electron in-ertial effects, or magnetic field-aligned electron pressure gra-dients. Of these, the latter cannot generate larger scale recon-nection effects and significant particle acceleration, since theintegrated reconnection electric field, and thus the particleenergy, is limited by the thermal energy of the electron popu-lation. Thus inertia is a more likely contributor, albeit onvery small spatial and temporal scales.

From a theoretical point of view, one might first wonderwhat effect different electron masses might have on the col-lisionless dissipation process in the reconnection region.Lighter electrons traverse the reconnection region faster andmight therefore be less prone to acceleration in the reconnec-tion electric fields. Therefore, dissipation based on bulk in-ertia might be more important for larger electron masses andrapid time dependence. Thus it seems likely that the recon-nection physics should vary if the electron mass is varied.Furthermore, different physics in the diffusion region mightlead to different dissipation, thereby influencing and poten-tially changing the larger scale behavior of the system underinvestigation. Thus a detailed study of the electron physics inthe dissipation region of the collisionless magnetic reconnec-tion region is required to develop an understanding of thedissipation process proper and its impact on the large scaleevolution. In the following, we will describe the design andresults of a study addressing these questions.

In order to isolate the dissipation region physics from theeffects of the ion-scale processes and from those of theboundary conditions, our modeled systems will be identicalin size and all physical parameters save those impacted bythe electron mass. Therefore, the only control parameter is

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the electron mass. Potential differences in the evolutions arethus attributable to different dissipation rates supported byelectrons of different mass, and to the coupling of thosevery-small-scale processes to the larger ion-scale processes.

We will also extend the investigation to a model whichincludes a magnetic guide field along the main current direc-tion. If this guide field is set up large enough, the electronsare magnetized and, by implication, the thermal inertia-baseddissipation process should be turned off. Since bulk inertiaand thermal inertia exhibit very different scale sizes and ef-fectiveness, an impact of the guide field on the large scaleevolution should be expected, with a potential slow-down ofthe evolution, even if the magnetic pressure associated withthe guide field is negligible.

In order to account for all potential kinetic effects, aswell as high-order moments such as heat flux, it becomesclear that particle-in-cell simulations are required to investi-gate and understand the dissipative underpinnings of the re-connection process. Furthermore, such simulations provide ameans to address self-consistently the acceleration and heat-ing of particles by the reconnection electric field.

In the present study we restrict the investigation to trans-lationally invariant models. While it appears evident thatthree-dimensional effects are likely important in real sys-tems, we here adopt the approach of trying to understand thesimpler systems first. We expect that knowledge of the re-connection physics in two-and-a-half-dimensional modelswill aid later analyses of more complicated models, where awealth of additional phenomena will likely complicate theanalysis. Accordingly, the present simulations are based ontwo-and-a-half-dimensional versions of our three-dimensional particle-in-cell, fully electromagnetic simulationcodes, which have been applied to a number of problems inthe magnetosphere of the Earth. These codes have recentlybeen developed further to include an implicit treatment of theelectromagnetic fields, which allows applications to consid-erably larger problems in a reasonable amount of simulationtime.

Finally, guided by the results of the kinetic investiga-tions, we will use a Hall-magnetohydrodynamics~MHD!model to study potential ways the local, electron-based,physics of the dissipation region can be incorporated intolarger scale models. This is particularly important for modelswhich aim to address the large scale dynamics quantitativelycorrectly. Because of the sheer size of systems like the mag-netosphere of the Earth these models are unable to coverboth large and small scales comprehensively. Therefore, aquantitative description of the dynamics of such very largesystems requires finding some adequate way to include ki-netic processes in these models without posing overly largedemands on computational effort. We will address this prob-lem by means of the last part of the investigations presentedhere.

In Sec. II, we will describe the numerical method em-ployed for the kinetic investigations, as well as the initialconditions to be employed for all simulations describedherein. In Sec. III we present the results of kinetic modelingwithout guide magnetic field. Particular emphasis here is onthe impact of different electron masses on the overall evolu-

tion, and on the structure of the dissipation region. The re-sults of a simulation with guide magnetic field and resultingchanges to the overall and local model structure will be dis-cussed in Sec. IV. An attempt to represent the kinetic resultsby means of transport models in Hall MHD is the subject ofSec. V. Finally, Sec. VI contains a summary and outlook.

II. NUMERICAL APPROACH AND INITIALCONDITIONS

For the purpose of the present investigation, we use atwo-and-a-half-dimensional version of our fully electromag-netic particle-in-cell code. The scheme is based on the Bun-eman layout of currents and fields on a rectangular grid~e.g.,Villasenor and Buneman33!. Particles are advanced by asecond-order leapfrog algorithm. Densities and fluxes are ac-cumulated on the grid, using a rectangular particle shapefunction. Charge conservation is guaranteed by the iterativeapplication of a Langdon–Marder-type~Langdon34! correc-tion to the electric field. The electromagnetic fields are inte-grated implicitly to avoid the Courant constraint on thepropagation of light waves. The light wave damping in theimplicit scheme also allows us to use simple reflectingboundary conditions for the electromagnetic fields at thezboundaries~periodicity is assumed inx!. Comparisons be-tween runs performed with the explicit and implicit algo-rithms showed excellent agreement.

Ions are assumed to be protons in the following investi-gations. Further, we normalize lengths to the ion inertiallengthsc/v i5c(e2n0 /e0mi)

21/2 using a current sheet den-sity n0 , and times are normalized to the inverse of the ioncyclotron frequencyV i5eB0 /mi in the asymptotic magneticfield B0 unless noted otherwise. The system dimensions andinitial conditions follow the ‘‘GEM Reconnection Chal-lenge’’ ~Drakeet al., in preparation!. In the present calcula-tions, the system dimensions areLz512.8c/v i and Lx

525.6c/v i with the number of cells depending on ion/electron mass ratio. A time step of an inverse electronplasma frequencyveDt51 is used.

The focus of the research presented here is on themechanisms of collisionless dissipation supporting the mag-netic reconnection process rather than on the question onhow magnetic reconnection starts. Thus we set up the simu-lations with anX-type neutral point of the poloidal magneticfield. Accordingly, the initial equilibrium configuration waschosen as a Harris-sheet equilibrium in thex–z plane, withthe initial current densityy-aligned

Bx5tan~z/l!, ~5!

with an additional perturbation of the form

Bxp5a0p

Lzcos~2px/Lx!sin~pz/Lz! ~6a!

and

Bzp522a0p

Lxsin~2px/Lx!cos~pz/Lz! ~6b!

and a ‘‘guide field’’ of

By05B0 . ~7!

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Here the perturbation amplitudea050.1 results in aBz am-plitude of about 2.5% of the asymptotic magnetic fieldstrength. The sheet half-thickness, in terms of ion scalelengths, is adopted to bel50.5 for all runs.

Four particle species, two of ions and electrons each,were integrated in each run. The first set of ion and electronspecies establishes the pressure and currents demanded by~5! and ~6!. The second set of species constitutes a constantdensity level backgroundnb50.2. Background temperaturesare identical to the temperatures of the current carrying spe-ciesTi1Te50.5.

The total particle number depends on the ion/electronmass ratio. Because of the falloff of the foreground densityaway from the current sheet, a large number of simulationparticles is needed to ensure that all cells are well populated.Thus, the particle number per cell is several hundred near thecenter of the current sheet.

Periodic boundary conditions were employed atx50andx525.8. At the top and bottom boundaries, particles arespecularly reflected. The electron–ion temperature ratio inall runs is chosen asTe /Ti50.2. The mass ratio, the essen-tial parameter in the present investigation, is varied frommi /me59 to mi /me5100. Using the asymptotic magneticfield to define the electron cyclotron frequency,Ve

5eB0 /me , and the plasma sheet densityn0 for the definitionof the electron plasma frequencyve5(e2n0 /e0me)

1/2, wesetve /Ve55.

All simulations were performed on a dedicated SUNULTRA 2300 workstation with a suitably large~1.3 Gbytes!amount of memory. The code was parallelized by frequentparticle sorting to effectively run on the two processors ofthis shared-memory architecture. Run times ranged betweenabout 1 h for the smallest, and about 10 days for the largestmass ratios.

III. ELECTRON MASS DEPENDENCE OF THERECONNECTION PROCESS FOR UNMAGNETIZEDELECTRONS

A. Overview

From earlier work~e.g., Hesse and Winske35! and thediscussion above it is evident that the electron dynamics inthe dissipation region facilitates the reconnection process.For the purpose of avoiding electron magnetization in thediffusion region, we set the guide field to zero, i.e.,By0

50. If we assume unmagnetized electrons, i.e., the absenceof a guide field~5!, previous investigations found that thedissipation is likely provided by electron pressure effects, ifthe sheet thickness is of the order of the ion inertiallength27,21and an ion/electron mass ratio of 25 was assumed.Since the pressure-based, or quasiviscous, dissipation de-pends on unmagnetized electrons, its effectiveness should beinfluenced by the choice of electron masses.

In order to test the dependence of the reconnection pro-cess on the assumed electron mass, we performed a set ofsimulations with varying electron masses, ranging frommi /me59 to mi /me5100. Table I shows the parameters foreach of these runs. In the following, we will discuss run 2,

i.e., the run with mass ratiomi /me525 in detail. The com-parison with the other runs will be performed by means oftypical parameters of the evolutions.

Figure 1 shows the magnetic field evolution for run 2with the current density color coded. Figure 1 shows thatmagnetic reconnection, initiated by the initial perturbation,causes large changes to the magnetic field and current den-sity distribution. The evolution continues beyondV i t520,but leads to a significantly reduced reconnection rate due tothe depletion of the magnetic flux in the source regions ad-jacent to the current sheet, or the buildup of pressure in theclosed field region enveloping theO-type magnetic neutralpoint. Figure 1 demonstrates two features: The current sheetthickness in the reconnection region is, at all times, some-what larger than the electron skin depth~which is equal to0.2c/V i for this mass ratio!, and the current density exhibitsa saddle point at the location of the reconnection region. Thelatter feature becomes most prominent at later times. Similarfeatures were also found in hybrid and particle simulations ofa similar system.

The diffusion region is the site of the processes whichlead to the disconnection and reconnection of magnetic flux.Here nonideal processes generate electric fields which devi-ate from ideal electric fields, which are equal to the ion orelectron convection electric fields~1!. It is apparent from thediscussion above that these processes have to rely on theinertia of individual electrons, which contributes to all of thefluid terms on the RHS of Eq.~2!. Heavier electrons shouldspend more time in the region of low magnetic field, leadingto more acceleration and thus stronger reconnection electricfields. Intuitively, one might expect that the electron massshould have a significant impact on the evolution of the sys-tem. Figure 2 proves that this expectation is essentially in-correct.

Figure 2 shows, for each of the runs described above, thetime evolution of the reconnected magnetic flux, defined by

F~ t !5EBz.0

dx Bz~z50!, ~8!

where the integral is taken between the majorX and Opoints, if there are more than one each. Each graph consistsof an initial slow growth, typically for the first seven to eightion cyclotron times, followed by a rapid time evolution forsome 15– 20V i

21. After that time, the evolution slows downconsiderably. This is due to a depletion of the magnetic fluxin the source regions adjacent to the current sheet whichreduces the energy available to power the reconnection pro-cess, and a buildup of plasma and magnetic pressure in themagnetic island, which reduces the ‘‘pulling’’ of plasmaaway from the reconnection region.

TABLE I. Parameters for all four runs.

Run number mi /me nx nz npforeground np background

1 9 240 120 7.23105 3.63105

2 25 400 200 2.03106 1.03106

3 64 640 320 5.123106 2.563106

4 100 800 400 7.03106 3.53106

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A comparison between the different graphs reveals a sur-prising result: With the exception of a small difference in theduration of the slow evolution, all graphs look essentiallyalike. This result suggests that the electron mass appears tobe of little importance to the large scale evolution, as hasbeen suggested on the basis on hybrid simulations with elec-tron inertia28,36 and particle simulations of a single ionspecies.30 In fact, a comparison of other parameters such asvelocities, and current densities shows striking similaritiesalso. We will return to this point below.

Figure 2 suggests that the electron physics in the dissi-pation region adjusts itself to accommodate the large scaleevolution. The ions exhibit a much larger mass. Conse-quently, the large scale evolution should be controlled by theinertia of the ions and might therefore occur on similar timescales independent of the local electron physics. In the fol-lowing, we will follow this line of thought further by inves-tigating the means by which the electron physics accommo-dates the large scale evolution. As a first step, we will studythe differences of the structure of the dissipation region intwo different runs.

FIG. 2. Time evolution of the magnetic flux normal to the current sheet forall runs without a guide magnetic field.

FIG. 3. Comparison between magnetic fields and current densities at thesame state of the evolution for mass ratiosmi /me59, andmi /me5100.

FIG. 1. Magnetic field evolution and current density~color coded! evolutionfor mass ratiomi /me525. Strong changes brought about by magnetic re-connection, initiated by the initial perturbation are shown.

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B. The diffusion region for two electron masses

In order to shed light on the behavior of the electrondissipation and its variation with electron mass we will in thefollowing compare the structure of the dissipation region fortwo different electron masses, i.e., for runs 1 and 4. In par-ticular, we will compare properties of the reconnection re-gion for times at which both simulations have led to similaramounts of newly reconnected magnetic fluxF. For this fluxlevel, we select a value ofF'1.8, corresponding to a time ofV i t518 for run 1, and ofV i t516 for run 4. Figure 3 showsthe magnetic fields and current densities for both runs at theselected times in a format similar to a single panel of Fig. 1.

Figure 3 demonstrates that run 1 exhibits the same sym-metry as run 2 with a reconnection site on the right, whereasthe reconnection site for run 4 appears to be shifted to theleft. We believe that this is due to the random noise in thecalculation which allows one of the two reconnection sitesevident in Fig. 1 to become dominant. Because of the peri-odicity of the model this difference is of no physical signifi-cance. Further, Fig. 3 shows that the overall structure of theevolution looks similar, despite the fact that run 4 for massratio 100 exhibits more detail than run 1. Both runs do showa thin current sheet extended in thex direction. However, inion scales, this current sheet is more extended in bothx andz directions for run 1.

In the following, we will compare the features of thedissipation regions for both simulations primarily throughthe investigation of cuts along thex axis. In order to facilitatethese comparisons, we will shift thex axis ~by 1.7c/v i! forrun 1 such that the reconnection sites are essentially at thesame position. Thus the plots in the following use the properx coordinate from run 4, and the shifted coordinate fromrun 1.

Figure 4 displays the larger scale features of both runs.The normal magnetic field componentBz , depicted in thetop panel, exhibits a very similar structure in both runs. Thesmall differences can be attributed to the slight mismatch inthe amount of reconnected flux, i.e., the state of the evolu-tion, at the times selected from both runs. The middle panelshows thex component of the ion flow velocity. On the left,low x side of the reconnection region the two graphs exhibitan almost exact overlap, whereas the peak ion velocity on theright-hand side is slightly larger for run 4. In both cases theion velocity is a significant fraction of the Alfve´n speed~nor-malized to one in this plot!. The electron density profile onthex axis is shown in the bottom panel. Here again a remark-able similarity exists between the two runs. In both cases, theevolution has carved out a density depletion region aroundthe reconnection site, with an extent along thex direction ofaboutDx'7c/v i . The residual density in the central regionis of the order of the initial background densitynb50.2. Wenote that an obvious reason why both electron densities ap-pear to be similar despite the mass difference is because ofthe electrostatic coupling between ions and electrons. Thefact that the ions behave similarly implies that the electrondensities show similar features as well. Thus we find thatFig. 4 supports the results of the analysis in that the large-

scale aspects of the two evolutions are very similar, if one ofthem is shifted in time by a small amount.

The electron scale features should be different, however.This expectation is verified by Fig. 5. The top panel showsthe x component of the electron flow velocity. The panelshows striking differences. In both cases the peak electronflow velocity exceeds the Alfve´n velocity. In the case of run1 the peak velocity is twice, and in the case of run 4 it is fourtimes, the Alfven velocity. Beyond this obvious velocity dif-ference, it is noteworthy that the gradient scale length is alsoquite different. In the high-electron mass case~run 1! thepeak-to-peak distance, i.e., twice the gradient scale length,amounts to aboutDx'5.5c/v i , whereas the much morecontracted shape for run 4 features a peak-to-peak distanceof only aboutDx'3.0c/v i . It is noteworthy that the elec-tron convection electric field, given by~1! with vs5ve , isagain very similar in the two cases. This is due to the factthat the large electron flow velocity formi /me5100 occursprimarily in a region of substantially reducedBz ~see Fig. 4!.

FIG. 4. Cuts along thex axis of the normal magnetic field~the top panel!, ofthe x component of the ion velocity~center panel!, and of the electrondensity ~the bottom panel!. Dotted lines are for mass ratiomi /me59 atV i t518, and solid lines formi /me5100 atV i t516. The former graphs areshifted inx to line up the reconnection regions for both models.

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The electron contribution to the total current density inthe y direction is shown in the center panel. Here again wefind that the overall structure appears to be quite similar,with a current sheet extending about ten ion inertial lengthsin the x direction and a peak structure at the location of thecurrent density peaks in Fig. 3. The central region of thiscurrent sheet, however, exhibits a much more localized peakin the reconnection region for run 4. Scale sizes of this peakare very similar to the ones for the electron velocity above.Accordingly, it again appears that scale sizes are about twiceas large for run 1 as they are for run 4, with higher ampli-tudes for the latter.

The bottom panel depictsPxye, one of the off-diagonalcomponents of the electron pressure tensor which contributeto the reconnection electric field. The arrow indicates theapproximate position of the reconnection region. At the lo-cation of the arrow, the pressure tensor components for both

runs exhibit a negative derivative, thereby generating a posi-tive contribution to the reconnection electric field@see Eq.~3!#. Matching the other panels, it is also evident from thebottom panel that the gradient scale length of the electronpressure tensor is similar to the other electron quantitiesshown in Fig. 5. The amplitudes of this tensor componentshow a drastic dependence on electron mass. The graph in-dicates that the lower mass ratio run produces anisotropieswhich are larger by a factor of about 3 when compared to thevalues formi /me5100. The contribution to the reconnectionelectric field by these tensor components is determined bytheir gradient, however. Here the plot suggests that the gra-dients are about equal in the vicinity of the reconnectionregion, indicating approximately equal contributions to thereconnection electric field. Using a density ofn50.2, thecontribution to the reconnection electric field by this pressuretensor component can be estimated asEr ,x'0.1, whichamounts to about half the slope of the graphs in Fig. 2. Theother contribution is provided by thez derivative of Pyze,which exhibits a behavior similar to that of thexy componentdiscussed here, albeit on smaller scales~see below!.

Thus we find that the structure of the dissipation regiondepends quite strongly on the electron mass. At this point thequestion arises as to which process controls the spatial scalesas a function of the electron mass. Noting that thex profile ofBz is virtually identical for both runs, it appears likely thatthe scale sizes should be determined by the trapping lengthof electrons in a field reversal. This length has been deter-mined in the past based on the analysis of electron orbits infield reversals~e.g., Biskamp and Schindler37!

lx5F 2meTe

e2S ]Bz

]x D 2G 1/4

. ~9!

Similar conclusion were recently drawn by Kuznetsovaet al.38 Inserting parameters from run 4 yieldslx'1.1,which is in acceptable agreement with the scale lengths ob-tained above. Thus the length scales of the electron dissipa-tion region are proportional to the fourth root of the electronmass. In our comparison, the ratio of these lengths shouldroughly be

lx~me51/9!

lx~me1/100!'1.83, ~10!

which is close to the value of about two found above.The results from this study can be used to estimate the

reconnection electric field, or, if the latter is assumed to befixed, provide an explanation of the difference in electronflow velocities for different electron masses. The contribu-tion to the reconnection electric field from gradients ofPxye

can be estimated as

Ey1521

nee

]Pxye

]x'

1

neelxPxye. ~11!

In the region where the electron cyclotron time is shorterthan the growth time of the system thexy component of theelectron pressure tensor can be approximately written as28

FIG. 5. Cuts along thex axis of thex component of the electron velocity~the top panel!, of the electron current density~the center panel!, and of theelectron pressure tensorPxye component~the bottom panel!. Dotted lines arefor mass ratiomi /me59 at V i t518, and solid lines formi /me5100 atV i t516. The former graphs are shifted inx to line up the reconnectionregions for both models.

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Pxye'pe

2Ve

]vxe

]x. ~12!

Here pe denotes the isotropic part of the electron pressuretensor, andVe the electron cyclotron frequency. Combining~10! and~11! and setting]Bz /]x5B8 andBz'B8lx leads to

Ey1'1

e

]vxe

]xAmeTe

2. ~13!

Since our simulations indicate that the reconnection electricfield appears to be independent of the electron mass, Eq.~13!indicates that the gradient of the electron flow velocity needsto scale like the inverse square root of the electron mass.This result is consistent with the scales shown in Fig. 5.

Similar results can be found for thez direction. Here wefind again that contributions to the reconnection electric fieldgenerated by the pressure tensor componentPyze are essen-tially independent of electron mass, and the length scale tobe given by~9! with z replacingx. Assuming approximatelyconstant density with a slow time evolution we can use anapproximate continuity equation

]vx

]x'2

]vz

]z

to relate the derivatives ofvx andvz . Thus the total recon-nection electric field is about twice the value of~13! andamounts to

Ey,rec'1

e

]vxe

]xA2meTe. ~14!

This is the same result found by Kuzne`tsovaet al.38 for theaverage electric field in the region of unmagnetized elec-trons. Our results indicate that this electric field is essentiallyconstant in the electron diffusion region.

The scales in thez direction in the vicinity of theX pointare shown explicitly in Fig. 6 for run 4 andV i t516. Herethe top panel illustrates the electron current density~the dot-ted line!, ion current density~the solid line!, and total currentdensity~the dashed line!. It is evident that the current densityin the reconnection region is provided almost exclusively bythe electrons. By examination of the full width at the half-maximum value we determine that the current scale size is ofthe order of a few electron inertial lengthsc/ve (c/ve

50.1c/v i in this simulation!.The center panel of Fig. 6 shows thez component of the

electron flow velocity. The panel demonstrates a rapid inflowat more than the Alfve´n velocity in the vicinity of the recon-nection site. A comparison with Fig. 5 also demonstrates thatthe continuity approximation used in the derivation of~14! isjustified. Last, the bottom panel of Fig. 6 depicts theyzcom-ponent of the electron pressure tensor. As required for a posi-tive contribution to the reconnection electric field, it exhibitsa negativez derivative atz50. This panel also confirms theexpectation that theyz electron pressure tensor componentcontributes equally to the reconnection electric field by itsderivative in thez direction.

The results of this section and Sec. III A confirm that theelectron physics does not appear to be a determining factor

in the large scale rate of magnetic reconnection, at least if themagnetic field in the reconnection region is small enough forthe electrons to be unmagnetized. Instead, the electron phys-ics in the diffusion region adjusts itself to the requirementsof the large scale evolution which is dominated by ion ef-fects. Here the evolution spatial scales are clearly dominatedby the electron nonadiabatic scalel. Thus the question re-mains what happens if that freedom is removed by electronmagnetization. While hybrid simulations with finite electronmass39,29 indicate that in this situation the electron mass maybe unimportant, a particle simulation which treats all specieskinetically is required to test this hypothesis. In the followingwe present the results of such a study.

IV. MAGNETIC RECONNECTION WITH A GUIDEFIELD

The next step in this study is to investigate the results ofa run with a magnetic guide field large enough to magnetize

FIG. 6. Cuts along thez direction at the location of the reconnection regionof the y component of the current densities~the top panel!, of thez compo-nent of the electron velocity~the center panel!, and of the electron pressuretensorPyze component~the bottom panel!. Dotted lines are for mass ratiomi /me59 at V i t518, and solid lines formi /me5100 at V i t516. Theformer graphs are shifted inx to line up the reconnection regions for bothmodels.

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the electrons. It is beneficial to use as small an electron massas possible for the guide field to magnetize electrons but tobe small enough not to contribute to the overall dynamics bymagnetic pressure effects. For guide fields large enough tosignificantly modify the magnetic pressure in the system,simulations not shown here indicate a strong reduction of thereconnection rate, commensurate with laboratoryexperiments.9 Instead, we here strive to leave the large-scalesystem virtually unaffected while attempting to significantlymodify the electron physics in the dissipation region. In or-der to accomplish this goal, we choose an electron massme50.01, and a guide field magnitude ofBy050.3, implyingan electron cyclotron frequency ofVe530 in units of the ioncyclotron frequency in the asymptotic magnetic field. Thusthe electron cyclotron period everywhere is much larger thanthe dynamical time scales of the runs discussed in Sec. III.All other parameters of the run discussed here are identical torun 4 above.

Figure 7 displays the growth of the magnetic flux normalto thex axis ~the dotted line!. Also shown are the time evo-lution of the normal flux taken from run 4 above~the solidline!, and of the inertia contribution to the normal flux~8! ofthe present run~dashed line!, defined by:

F inertia52me

eDvyeux , ~15!

where Dvyeux denotes the difference between the electronflow velocities at the dominantX point at timest50 and theevolution time under study. Figure 7 shows a surprising re-sult: The two evolution curves are virtually identical. Thisindicates that the guide magnetic field apparently does notchange the evolution appreciably.

Further, Fig. 7 proves that the dominant inertia contribu-tion, which is proportional to the time derivative of the elec-tron flow velocity,21 is much too small to contribute signifi-cantly to the total reconnection rate~note the difference inscales!. While this is, at first, quite unexpected, a secondlook at the setup of this system explains this behavior. Con-

sidering~15!, it is trivial to calculate the final electron veloc-ity for a given amount of reconnected magnetic flux, assum-ing all dissipation is provided by bulk electron acceleration.For a flux amount ofF inertia54 and our adopted electronmass, we findDvyeux'400 in units of the Alfve´n speed,which is completely unrealistic~the speed of light in theseunits isc550!.

How does the system accomplish the required reconnec-tion rate? A clue to the answer of this question is providedby Fig. 8, which compares the magnetic field and currentdensity structure in the present run and run 4 forV i t513.Although there appears to be a difference in the evolution inform of a small magnetic island, which is indicative of aninfluence of the guide field, the structure of the current sheetin the reconnection region is quite similar. In particular, thetwo current sheets in the reconnection regions are of similarthicknesses.

Figure 8 seems to indicate that the dissipation processunderlying the dynamics with a guide magnetic field is alsoprovided by pressure-based dissipation, although it is unclearhow this process operates in the presence of a guide mag-netic field. This expectation is confirmed by Fig. 9, whichshows in the forms ofx axis profiles thex component of theelectron flow velocity and thexy pressure tensor component~top panel!, the electron density and the value of the presentguide magnetic field~center panel!, and the values of thezcomponent of the total electric field and its convection com-ponent ~bottom panel! at V i t513 for the present run. Acomparison with Fig. 5 shows a striking similarity in elec-tron flow velocities, the values of the pressure tensor, and ofthe electron density in the vicinity of the reconnection regionat x'13. Thus we find that despite the large guide magnetic

FIG. 7. Time evolution of the normal magnetic flux for runs with~the dottedline!, and without~the solid line! guide magnetic field. Both runs adopted amass ratio ofmi /me5100. For comparison, the time integral of the totalinertia electric field at theX point for the run with guide field is shown also~the dashed line!. The difference in scales for the total normal flux and theinertia contribution proves that the inertia term is unimportant.

FIG. 8. Magnetic field andy current density for runs with and without guidefield at V i t513.

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field, magnetic reconnection still appears to be supported bynongyrotropic electron pressures.

The guide magnetic field, on the other hand, appears tobe slightly depressed compared to its original value, but stillof substantial value in the reconnection region, and enhancedoutside of it. A typical Larmor radius in this magnetic fieldcomponent is of the order ofL'0.3, i.e., much smaller thanthe x scale length of the dissipation region. This raises theimmediate question of the reason this dissipation process canwork effectively under the present conditions. The explana-tion lies in two facts: First, the nature of the electric field inthe neighborhood of the reconnection region is quite differ-ent from run 4. The bottom panel of Fig. 9 depicts a strong,localized enhancement of thez component of the electricfield, which well matches thez component of2ve3B closeto the reconnection site. This strong electric field modifiesparticle orbits such that the deflection around the magneticfield of an electron moving in thex direction is approxi-mately cancelled. This~Hall-type! electric field is generatedby charge separation effects between ions and electronswhich become deflected in opposite directions around theycomponent of the magnetic field. As a result, the electron

flow is essentially unchanged compared to the case withoutguide magnetic field. On the basis of the evolution equationsof the full electron pressure tensor22,28 it is possible to showthat the relevant tensor components evolve similarly to thecase without the guide field if the system exhibits only asmall deviation from symmetry about thex axis and the elec-tron flow field is similar. This is fulfilled to a very goodapproximation in the present result~errors less than some10%!. The slight tilt of the current sheet in the top panel ofFig. 8 leads to the emergence ofz derivatives in the pressuretensor evolution equations, which are just large enough toprovide the needed anisotropies. Accordingly, the small tilt~which becomes larger for larger guide fields! and the Hallelectric field play a crucial enabling role for the reconnectionprocess.

A second effect supporting the pressure-based dissipa-tion in this case is evident in Fig. 10. Figure 10 compares theelectron temperature evolution at the reconnection site forthe models with and without magnetic guide field. While theoverall evolution is quite similar, exhibiting elevated elec-tron temperatures during the fast evolution, the run withguide field generates hotter electrons. This difference of upto 50% is found primarily during the initial phase of the fastevolution, where the guide field based magnetization gener-ates some field-aligned electron acceleration in the reconnec-tion region~see Fig. 7!. The thermalization of these veloci-ties contributes to the overall temperature increase morestrongly than the acceleration of unmagnetized particlesbouncing in the reconnection region of run 4. The elevatedelectron temperature contributes to the formation of off-diagonal electron pressure components by elevating the iso-tropic pressure in~10!.

The above analysis demonstrates conclusively thatpressure-based dissipation can operate also in the presence ofa sizable magnetic guide field. While this might at first seemsurprising a closer look revealed that the system self-

FIG. 9. Cuts along thex axis of thex component of the electron velocity~the solid line!, and of Pxye ~the dotted line! ~top panel!, of the electrondensity ~the solid line! and guide magnetic field~the dotted line! ~centerpanel!, and of theEz electric field component~the solid line! and the con-vection electric field component2vexBy ~dotted line! ~bottom panel!. Allplots are for an initial guide field ofBy050.3 and mass ratiomi /me5100 atV i t516.

FIG. 10. Time evolution of the electron temperature in the reconnectionregion with ~the solid line! and without~the dashed line! guide magneticfield. The former exhibits higher temperatures during the initial phase of theevolution, where the thermalization of electron velocities gained bymagnetic-field aligned acceleration is likely to contribute to an overallhigher temperature.

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consistently forms the features needed to support this dissi-pation mode in the absence of sufficient bulk inertia baseddissipation. This observation raises the question whetherthere is a limit of guide magnetic fields above whichpressure-based dissipation ceases to be functional. This ques-tion is beyond the scope of the present investigation. Similarto the results of our above studies varying the electron masswe conclude, however, that the system apparently finds away to support the reconnection rate demanded by the largerscale evolution, i.e., in essence by the ions. An extrapolationof this conclusion would be to expect thatany suitable dis-sipation process would lead to similar reconnection ratesprovided the physics on intermediate and large scales is rep-resented correctly. In the extreme case, a Hall-MHD modelwith resistive dissipation should generate reconnection ratessimilar to what we find here for the fully kinetic models.This would also bear on means to represent collisionless dis-sipation in large scale models. We will study this suppositionin Sec. V.

V. HALL MHD SIMULATIONS

The equations of Hall MHD constitute a substantial sim-plification of the kinetic models used and analyzed in theprevious sections. Within Hall MHD, no kinetic effects areconsidered at all, excluding also the effects of anisotropicpressures for both ions and electrons. Pressure equations aresolved only for the ions, where a polytropic pressure lawwith resistive heating terms is commonly adopted. Dissipa-tion is based on ad-hoc resistivity models, although in somecases electron inertial effects are also included. Denoting byr the total mass density, byv the total flow velocity, byp thetotal pressure, byg55/3 the polytropic index, and byh theresistivity, the Hall-MHD equations used in our investigationassume the following dimensionless form:equation of continuity

]r

]t1“–~rv!50, ~16!

momentum equation

rS ]v

]t1v–“vD52“p1 j3B, ~17!

Faraday’s law

]B

]t52“3E, ~18!

Ohm’s law

E52v3B1h j11

rj3B1

me

r

] j

]t, ~19!

energy equation

]p

]t52v–“p2gp“–v1~g21!h j 2. ~20!

With the exception of the last two terms on the RHS ofOhm’s law~19! these equations are identical to the standardresistive MHD model. The addition of the Hall termj3B

changes the wave characteristics by adding Whistler modesto the spectrum of the equations. The last term on the RHS isa reduced form of the complete electron inertia term in~2!. Itis included here in order to stabilize the calculations by add-ing a cutoff to the Whistler modes at the electron cyclotronfrequency. Without the addition of this term or other meansto suppress Whistler fluctuations, performing Hall-MHDsimulations becomes computationally prohibitively expen-sive. For the purpose of the present investigations we choosean electron mass ofme51/25. The results presented in Sec.IV indicate that the dissipation provided by this term will beinsufficient to enable magnetic reconnection by itself. Wewill thus adopt different resistive or quasiresistive models inthe following.

Equations~16!–~20! are integrated by a standard leap-frog technique which includes a flux limiting routine and anelliptical solver for the electron inertial effects. The initialconditions for the ion fluid and the electromagnetic fields areidentical to the ones described for the kinetic models in Sec.II, with the sole exception being that the initial current den-sity is entirely provided by ion flow in they direction. A gridof 2003100 cells is used for the calculations.

While our above results suggest that the exact form ofthe dissipation might be unimportant, it is clear that a suffi-cient amount needs to be available to facilitate the evolution.In our case, this implies the need to select a sufficiently largeresistivity h such that at the neutral point a significant elec-tric field magnitudeE5h j can be reached by the systemwithout an excessively large current density. Therefore, it isclear that some sort of localization of the resistive dissipationis needed for the system not to become overly diffusive out-side the reconnection region proper. A localization of thedissipation is also suggested by the plots in Fig. 1, whichindicate a relatively localized nature of the dissipation pro-cess. Keeping this in mind we design our resistive model tobe current dependent. The analytical form is:

h5h0~112.5~ j 22!2! if j .2 ~21a!

or

h5h0 for j <2. ~21b!

The constanth050.001 corresponds to an overall Lundquistnumber ofS5103.

Figure 11 displays, in the same format as Fig. 1, the timeevolution of the magnetic field and current density in thisHall-MHD run. Flow vectors are shown also. Figure 11shows some striking similarities with Fig. 1. Differences arefound in the actual time when the evolution has reached acertain state~compare, e.g., the second panel of Fig. 1 withthe first panel of Fig. 11!, and in the persistence of the smallmagnetic island which formed in both simulations. Whereasthis island is completely dissipated atV i t520 in the kineticsimulation, it is still visible at a matching time ofV i t525 inthe Hall-MHD model. Following the evolution further showsa clear tendency of the Hall-MHD model to dissipate thisisland also, but numerical problems associated with the for-mation of low density regions prohibit this from being com-pleted during this run.

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The time evolution of the reconnected magnetic flux isshown in Fig. 12, together with the kinetic result forme

50.01. Also shown, for comparison, are the evolution graphsfor two other Hall-MHD runs; one where the only alloweddissipation is purely numerical, and another, where a con-stant resistivity ofh050.005 was adopted. Figure 12 dem-onstrates two features: First, both Hall-MHD runs with

current-dependent resistivity, and with purely numerical dis-sipation reproduce the growth behavior of the kinetic modelwell, albeit with a time delay of some seven ion cyclotrontimes. The constant resistivity model performs less well inthat the time delay appears to be even larger, and the growthrate of the reconnected flux is reduced by a factor of about 2.We attribute the latter to the larger global dissipation, whichtends to reduce gradients in the magnetic field and currentdensity, thereby reducing the accelerating force acting on theion fluid.

The delay in onset times of the fast evolution of theHall-MHD runs when compared to the kinetic model canlikely be attributed to the time each system requires to set upthe spatial scaled demanded by the assumed~or physical!dissipation mechanism. Here it appears that the fluid modelrequires a longer evolution, likely because a thinner currentsheet with higher current density needs to form.

Thus we find that it appears possible with relativelysimple resistive dissipation models to reproduce the largerscale fluid features of fully kinetic models. This includes, inparticular, the amount of magnetic flux that has undergonereconnection. Differences in the onset times of the fast evo-lution in the kinetic and the Hall-MHD model are likelyattributable to the time the system requires to set up therequired dissipation process. In the kinetic case, this entailsthe formation of the electron flow patterns which support thegeneration of pressure anisotropies via quasiviscous pro-cesses. In the resistive fluid model, the current sheet needs tothin far enough to generate the required dissipation. The dif-ference in time for this to occur is of the order of some 5–10ion cyclotron periods, which, for example, corresponds toabout the same amount of time in seconds in the magnetotailof the Earth.

VI. SUMMARY AND CONCLUSIONS

The studies presented in the present paper aimed at twoissues. The first was the question of the detailed structure ofthe dissipation region in collisionless magnetic reconnection.The second topic, which emerged as a consequence of theinvestigations of the former, dealt with possible representa-tions of collisionless dissipation in larger scale models,which lack the capability to properly represent the micro-physics of collisionless magnetic reconnection.

Investigations of the first topic built upon results of pre-vious works. These include modeling results to which indi-cate that the electron mass might not determine the recon-nection rate if the latter is provided by electron inertia.39,29Afactor of similar importance are recent results regarding thenature of the dissipation in systems with different currentsheet width.21 We used our fully self-consistent electromag-netic particle-in-cell algorithm21,40 in order to include impor-tant kinetic physics, which was not included in previous hy-brid models. This code has recently been modified to includean implicit solver for the electromagnetic fields, which al-lows us to perform much larger runs than otherwise possible.

The basic system under investigation consisted of aHarris-type equilibrium including a homogeneous plasmabackground with a magnetic field perturbation and system

FIG. 11. Magnetic field evolution and current density~color coded! evolu-tion for a Hall-MHD run with current dependent resistivity. Strong changesbrought about by magnetic reconnection, and similar features to the kineticresult depicted in Fig. 1 are shown.

FIG. 12. Time evolution of the magnetic flux normal to the current sheet forrun 4 ~the dotted line!, and Hall-MHD simulations with numerical dissipa-tion ~the dash-dotted line!, current dependent resistivity~the solid line!, andconstant resistivity~the dashed line!.

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dimensions which were independent of the electron massemployed in the simulations. This simple system representsreconnection in a plasma pressure dominated environment,such as found, e.g., in the magnetotail of the Earth~e.g., Ref.15!, or in laboratory experiments involving reconnection ofnear antiparallel magnetic fields.9

The same initial condition was also adopted for Hall-MHD simulations, which were performed for comparison.The time evolution of the reconnected magnetic flux provedto be similar for a set of simulations exhibiting differentelectron masses. In particular, similar evolutions of the re-connected flux imply similar reconnection electric fields.This result prompted us to investigate the question as to whyand how similar electric fields can be produced in modelswith different electron masses.

We thus investigated the spatial structure of the dissipa-tion region proper and its dependence on the electron mass.We found the existence of two different spatial scales ofrelevant processes in the models. The first larger scale char-acterizes processes which one would commonly attribute toone-fluid, MHD-like quantities. Examples of these quantitiesare the spatial variation of the magnetic fieldBz componentnormal to the current sheet, the ion flow velocity, and thespatial density variations. These quantities proved, at match-ing stages of the evolution, to be independent of electronmass to a very good approximation.

The second spatial scale is associated with electron iner-tial effects in the vicinity of theX-type neutral point. Spatialscales were smaller for smaller electron mass. Affectedquantities were the electron flow velocity, which exhibitedsteeper gradients and higher peak values for smaller electronmass. Furthermore, the largely electron-supported currentdensity appears to be more strongly localized for smallerelectron mass. The electron pressure tensor exhibits verysimilar gradients for both runs but substantially larger ampli-tudes for larger electron mass. The similarity of the gradi-ents, however, combined with the virtually identical electrondensities resulted in identical reconnection rates.

The difference in spatial scales can be explained by thelength scale of electrons confined in a field reversal region.An investigation of thez profiles of electron quantities in thevicinity of the X point revealed matching scaling laws, andan about equal contribution ofz derivatives ofPyze to thetotal reconnection electric field. The thickness of the electroncurrent layer in the reconnection region was found to be afew collisionless skin depths. Using the above estimate forthe length scales, we could derive a simple analytical expres-sion which relates the reconnection electric field to the elec-tron mass and temperature and the derivative of the electronflow velocity. This result has recently been derived by Kuz-netsovaet al.28 using averaging techniques. Assuming equalreconnection electric fields, this expression explains the dif-ferent electron flow velocity profiles for the two electronmasses considered in this study in bothx andz directions.

Thus we found that the small-scale features of the dissi-pation process self-adjust to provide the reconnection electricfield demanded by the large-scale evolution, presumablycontrolled by the attraction of parallel electric currents. Thisinsight led to the design of the next simulation to be studied.

In order to prevent the pressure-based quasiviscous dissipa-tion process from operating, we set up a simulation withsmall electron mass (me50.01) and a sizable guide magneticfield of By050.3. We considered the possibility that therapid electron cylotron motion about this magnetic fieldwould effectively destroy nongyrotropies in the electron dis-tribution function. Furthermore, we performed a simple esti-mate of the electron acceleration required to activate the onlyremaining, inertia-based, dissipation process. Based on thisestimate we demonstrated that highly unrealistic electron ac-celeration would be required for this run to exhibit featuressimilar to those of the runs without magnetized electrons.Thus it might have been expected that the lack of electrondissipation should quench the evolution.

An analysis of the actual evolution revealed, however,that the evolution proceeded virtually identically to the runwithout guide magnetic field, with the exception of smalldifferences in the magnetic field structure around theX pointat intermediate times. A detailed analysis of characteristicelectron quantities in the neighborhood of theX pointshowed profiles very similar to the ones found in the runwithout guide field but with the same electron mass. Thisalso included the electron flow velocity profile, and the elec-tron pressure tensor, which exhibited the same anisotropiesas in the unmagnetized case. The answer to this puzzle wasprovided by an inspection of the electric field. We found thatthe diverging motions of electron and ions away from thereconnection site generated by the guide field set up a Hall-type space-charge electric field, which allows the bulk ofions and electrons to be unaffected by the cyclotron motionabout the guide field. This electric field locally effectivelycancels the impact of the guide magnetic field on electron~and ion! flow velocities, thereby allowing electrons to set upnongyrotropic distribution functions by their interaction withthe normal magnetic field. The validity of this conclusioncan also be proven from an analysis of the electron pressuretensor evolution equation28,35 in the presence of symmetryabout thex axis. Therefore, the system finds a way to supportthe requirements imposed on the local electron physics bythe large scale, ion dominated, evolution. This study pro-vides further evidence that the local reconnection rate is ef-fectively determined by the large scale physics, which isdominated by ion dynamics.

In the last part of the present studies, we took this sup-position one step further. If it is indeed true that the specificnature of the local dissipation process is unimportant~as longas it is sufficiently large and satisfiesE–j.0! for the large-scale evolution, then a simple fluid model might be sufficientto adequately represent its large scale consequences. If thiswere found to be correct, it would be appreciably simpler toincorporate collisionless dissipation effects into large scalemodels. Following this reasoning further, we surmised that aHall-MHD based model might be entirely adequate for thispurpose if a suitable resistive dissipation model is employed.

In selecting simple resistive dissipation models for Hall-MHD simulations we found remarkable agreement betweenthe kinetic and fluid models. While the best agreement in thegrowth of the reconnected magnetic flux was obtained withsimple numerical dissipation, the normal magnetic field dis-

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tribution along thex axis of the kinetic models was less wellrepresented. Localized dissipation models, on the other hand,showed much better agreement between the magnetic fielddistributions but a somewhat slower evolution. Models withconstant resistivity appeared to be too diffusive and exhib-ited slower evolutions.

Neither of the adopted resistive models, however, led toa completely different, i.e., much faster, or much slower evo-lution. Thus our Hall-MHD simulation confirms the expec-tation that the nature of the local dissipation might not be ofmajor importance for the large scale evolution. This resultprovides hope for large scale numerical models, e.g., MHDmodels aimed at representing magnetospheric dynamics~e.g., Fedder and Lyon,41 Raederet al.42!.

The apparent independence of the system evolution onthe actual dissipation in the diffusion region~as long asE–j.0! might lead to the conclusion that electron physicaleffects are irrelevant. While the results of these studiesclearly support such a point of view, viewing this questionthrough fundamental physics glasses leads to a different an-swer. Indeed, the quest for basic physical understanding in-cludes the desire to know the small-scale facilitators of thelarge-scale evolution. In this regard, the electron physics and,in particular, dissipation based on the electron pressure ten-sor is fundamental to the reconnection process, at least in thepresent simplified geometry.

We also point out that the present study focused on thenonlinear stage of the magnetic reconnection process. Whilethe system evolution at these times appears to be indepen-dent of the dissipation process in the diffusion region, it isvery likely that the linear stage of the reconnection processwill depend, potentially strongly, on the nature of the elec-tron dissipation. This is true in particular in the situationwhere reconnection starts in a configuration with a finitemagnetic field normal to the initial current sheet. In that situ-ation the time when and potentially the location where thenormal magnetic field reduces to zero will likely depend onthe physical nature of the dissipation process.

Finally, we remark that our research still leaves open avery important feature of many plasma physical systems.Our simulations and investigations were based on two-and-a-half-dimensional considerations and therefore limited tomodes with wave vectors in the simulation plane. It is evi-dent that modes with wave vectors in the current direction,such as kink modes~e.g., Pritchettet al.,43 Daughton,44 La-penta and Brackbill45! Kelvin-Helmholtz-like modes~e.g.,Hesseet al., 1998!,40 or lower-hybrid drift modes~Huba andDrake46! are likely to influence the evolution of the recon-nection process. Interaction of these modes and of associatedturbulence with magnetic reconnection might influence theeffectiveness of magnetic reconnection on large scales, oreven support reconnection initiation in current sheets withfinite normal magnetic field components. Investigations ofthese effects are necessary to provide a final understandingof how collisionless dissipation operates in large scale sys-tems, and what it takes to comprehensively include dissipa-tion effects. These questions promise to be rewarding topicsof future research.

ACKNOWLEDGMENTS

This work was supported by the National Aeronauticsand Space Administration Sun-Earth-Connection Theory andSupporting Research and Technology Programs.

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