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PHYSICS OF PLASMAS 15, 042306 共2008兲 The Hall fields and fast magnetic reconnection J. F. Drake,1,a兲 M. A. Shay,2 and M. Swisdak3 1 University of California, Berkeley, California 94720, USA University of Delaware, Newark, Delaware 19716, USA 3 University of Maryland, College Park, Maryland 20742, USA 2 共Received 21 November 2007; accepted 3 March 2008; published online 15 April 2008兲 The results of large-scale, particle-in-cell simulations are presented on the role of Hall electric and magnetic fields on the structure of the electron dissipation region and outflow exhaust during the collisionless magnetic reconnection of antiparallel fields. The simulations reveal that the whistler wave plays the key role in driving the electrons away from the magnetic x-line. Further downstream the electron outflow exhaust consists of a narrow super-Alfvénic jet, which remains collimated far downstream of the x-line, flanked by a pedestal whose width increases monotonically with increasing distance downstream. The open outflow exhaust, which is required for fast reconnection in large systems, is driven by the Hall electric and magnetic fields. Finally, it is the whistler that ultimately facilitates fast reconnection by diverting the electrons flowing toward the current layer into the outflow direction and thereby limiting the length of this layer. The results are contrasted with reconnection in an electron-positron plasma where the Hall fields are absent. The consequence of the expanding outflow exhaust is that, consistent with recent observations, the extended super-Alfvénic electron outflow jet carries a smaller and smaller fraction of the outflowing electrons with increasing distance downstream of the x-line. The results suggest that the structure of the electron current layer and exhaust in simulations might be sensitive to boundary conditions unless the simulation boundary along the outflow direction is sufficiently far from the x-line. © 2008 American Institute of Physics. 关DOI: 10.1063/1.2901194兴 I. INTRODUCTION Magnetic reconnection drives the release of magnetic energy in explosive events such as disruptions in laboratory experiments, magnetic substorms in the Earth’s magnetosphere, and flares in the solar corona. The rate of reconnection is controlled by a narrow boundary layer, the “dissipation region,” where strong currents are driven and dissipative processes enable magnetic field lines to reconnect. In the Sweet–Parker model of reconnection the plasma resistivity provides the dissipation required to break magnetic field lines and the tension associated with newly reconnected magnetic field lines drives plasma away from the x-line at the Alfvén speed cA.1,2 Since the 1950s, however, it was recognized that the long and narrow dissipation region that develops in the Sweet–Parker model throttles reconnection and the resulting rate of reconnection is inconsistent with the observed fast energy release. Computer simulations of reconnection based on the resistive MHD equations with constant resistivity confirmed the macroscopic elongation of the current layer of the Sweet–Parker model and the resultant slow rate of reconnection.3 On the other hand, in the high temperature plasmas where magnetic reconnection is active, the width of the dissipation region in the Sweet–Parker model ␦SP becomes comparable to kinetic scales such as the ion inertial length c / pi, where the MHD description of the plasma dynamics breaks down. At these small spatial scales electron and ion a兲 Permanent address: University of Maryland, College Park, Maryland 20742. 1070-664X/2008/15共4兲/042306/10/$23.00 motion decouples and the Alfvén wave which drives the outflow from the x-line in the MHD model is replaced by the whistler wave 共or the kinetic Alfvén wave in the case of reconnection with a guide field兲. The dynamics and structure of the outflow jet during reconnection is a complex nonlinear problem. Nevertheless, Fig. 1 illustrates how the basics of the outflow can be understood from a simple wave picture. The newly reconnected magnetic field line that drives the outflow from the x-line can be treated as a half-wavelength of a periodic standing wave. In the MHD description this standing wave is a conventional Alfvén wave and from the linear theory of the Alfvén wave the velocity vx that results from the relaxation of the bent magnetic field is simply cAx = Bx / 冑4min, with Bx the “upstream” magnetic field. This velocity is the usual MHD condition on the outflow from the x-line during reconnection. In the Hall reconnection model with no guide field the Alfvén wave is replaced by the whistler wave in the region within an ion inertial scale of the x-line. In this case the magnetic field is frozen-in 共or at least partially frozen-in兲 to the electrons carrying the current supporting the magnetic field of the standing wave. Rather than simply expanding outward, the in-plane magnetic field rotates out of the plane producing a Bz whose magnitude is nearly equal to Bx. The electron current in the x direction supporting Bz carries the field line away from the x-line. When the width of the current channel is the electron skin depth de = c / pe, which is universally seen in collisionless reconnection simulations, the magnitude of the outflow velocity supported by the whistler is given by the electron Alfvén speed cAe = Bx / 冑4men. In a Hall MHD model it was shown that fast reconnection develops if and only if the 15, 042306-1 © 2008 American Institute of Physics Downloaded 18 Jun 2008 to 128.175.189.215. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp 042306-2 Phys. Plasmas 15, 042306 共2008兲 Drake, Shay, and Swisdak Vin Vout y x FIG. 1. Schematic of the magnetic field lines and associated plasma flows around a magnetic x-line undergoing reconnection. The newly reconnected field lines can be represented by a half period of a simple standing wave to understanding how these newly reconnected field lines relax their stress and accelerate the plasma away from the x-line. plasma dynamics near the x-line is controlled by dispersive waves, either the whistler or kinetic Alfvén wave.4 In the GEM Reconnection Challenge various simulation models produced essentially identical reconnection rates as long as the Hall term and the associated whistler dynamics were included in the model.5 Similarly, the rate of collisionless reconnection was found to be independent of the electron mass,6,7 which is a consequence of the electron flux away from the x-line ⌫ = cAede being independent of mass; for smaller me the width of the outflow channel narrows but the outflow velocity goes up to compensate. Finally, in the Hall reconnection model the dissipation region is dramatically shorter than in the MHD model and produces fast reconnection independent of the system size.8,9 There remains some disagreement about the critical role that Hall physics plays in facilitating fast reconnection. It has been suggested that “ion kinetics” alone can drive fast reconnection.10 The authors found that even after eliminating the Hall term in Ohm’s law, magnetic reconnection in a hybrid model remained fast. The authors therefore concluded that “ion kinetics alone can give rise to fast reconnection” and that the “quadratic property of whistlers is not critical to fast reconnection in the kinetic regime.” However, in the simulations that led to these conclusions the authors used a spatially localized resistivity. It is well known that even in the resistive MHD model a localized resistivity produces fast reconnection.11 In recent hybrid simulations without the Hall term and without a localized resistivity the rate of reconnection remained small.12 Recent kinetic simulations in open boundary configurations have also called into question the fundamental tenets of the Hall reconnection model and therefore the current explanation of fast reconnection seen in nature.13,14 In these simulations the electron out-of-plane current layer stretches along the outflow direction, leading to an electron flow bottleneck which causes the rate of reconnection to drop. The development of secondary islands eventually halts the lengthening of the current layer and the associated drop in the rate of reconnection. On the other hand, the authors see periods of steady reconnection with no stretching of the electron current layer and no secondary island formation.14 The authors conclude that the Hall electric and magnetic fields do not play a sig- nificant role in the dynamics of the electron current layer and do not facilitate fast reconnection in large systems. In contrast, kinetic simulations carried out in very large systems to test the assertion that the limited simulation domains in earlier models were responsible for fast reconnection, reproduced the earlier results that reconnection was fast and independent of the system size.15 In the present paper we explore more carefully the dynamics of the electron current layer and the role of the Hall fields in controlling the structure of this layer and the outflow exhaust during collisionless reconnection. The results from large-scale particle-in-cell 共PIC兲 simulations demonstrate that the whistler wave controls the outflow of electrons from the x-line and that the opening of the electron outflow jet from the x-line, necessary for fast reconnection in large systems, results from fast, nearly field-aligned propagation of the Hall electric and magnetic field downstream of the x-line. The critical role that the Hall fields play in opening the electron outflow exhaust and the rapid propagation of these fields downstream suggests that care must be taken in any finite simulation to ensure that the proximity of the boundary does not adversely affect the structure of the Hall fields. II. COMPUTATIONAL MODEL Our simulations are performed with the particle-in-cell code p3d.16 The electromagnetic fields are defined on grid points and advanced in time with an explicit trapezoidalleapfrog method using second-order spatial derivatives. The Lorentz equation of motion for each particle is evolved by a Boris algorithm where the velocity v is accelerated by E for half a time step, rotated by B, and accelerated by E for the final half time step. To ensure that · E = 4 a correction electric field is calculated by inverting Poisson’s equation with a multigrid algorithm. Although the code permits other choices, we work with fully periodic boundary conditions. The results are presented in normalized units: the magnetic field to the asymptotic value of the reversed field, the density to the value at the center of the current sheet minus the uniform background density, velocities to the Alfvén speed vA, lengths to the ion inertial length di, times to the inverse ion cyclotron frequency ⍀−1 ci , and temperatures to mivA2 . We consider a system periodic in the x − y plane where flows into and away from the x-line are parallel to ŷ and x̂, respectively. The reconnection electric field is parallel to ẑ. The initial equilibrium consists of two Harris current sheets superimposed on a ambient population of uniform density. The reconnection magnetic field is given by Bx = tanh关共y − Ly / 4兲 / w0兴 − tanh关共y − 3Ly / 4兲 / w0兴 − 1, where w0 and Ly are the half-width of the initial current sheets and the box size in the ŷ direction. The electron and ion temperatures are initially uniform. The initial density profile is the usual Harris form plus a uniform background of 0.2. The simulations presented here are two-dimensional, i.e., / z = 0. Reconnection is initiated with a small initial magnetic perturbation that produces a single magnetic island on each current layer. We present the results of simulations with Te = 1 / 12, Ti = 5 / 12, and four mass ratios, mi / me = 1, 25, 100, and 400. Data from all but the mass-ratio unity case were presented Downloaded 18 Jun 2008 to 128.175.189.215. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp Phys. Plasmas 15, 042306 共2008兲 The Hall fields and fast magnetic reconnection (a) 10 jez 8 6 -1 4 -2 30 40 45 50 40 45 50 (b) 10 8 6 4 30 35 x/di 10 Bz y/di 8 0.2 6 0.0 4 -0.2 30 35 40 45 50 x/di (d) 0.05 o 0.00 -0.05 Bz We examine first the dynamics of the electric and magnetic fields and the associated motion of the electrons just downstream of the x-line. Arguments about the role of the whistler wave and its role in reconnection are based on simple arguments based on one-dimensional waves such as shown in Fig. 1. The structure of the x-line that develops during reconnection is two-dimensional and possibly even three-dimensional. Further, momentum transport in collisionless plasmas produces substantial dissipation 共the offdiagonal pressure tensor contribution to Ohm’s law兲,7,15 which can also affect the dynamics of the region around the x-line. We now seek to establish more quantitatively the role played by the dynamics of the whistler in the region around the x-line based on direct analysis of simulation data. Shown in Fig. 2 are plots of 共a兲 the out-of-plane electron current jez, 共b兲 the contours of the flux function, and 共c兲 the out-of-plane magnetic field Bz in a blowup around one of the x-lines from the me = 1 / 400 simulation. The electron current around the x-line is flowing in the negative z direction so Bx is positive above the current layer and negative below. In the conventional whistler picture the electrons flowing in the positive z direction drag the newly reconnected field lines in the positive z direction out of the plane producing the quadrupole field as shown in 共c兲.17,18 We now put this picture to the test. In Fig. 3 we show the profile of Bx along a cut through the x-line at x = 37.4. The electron current layer causes the magnetic field to increase sharply to around 0.2, after which Bx increases much more gradually, eventually approaching unity. If the whistler picture is correct, this magnetic field just upstream of the current layer should rotate out of the plane until it is pointing in the z direction and the magnitude of this z directed field should take on a value that is close to the upstream Bx, in this case 0.2. For steady-state reconnection the rotation of the field occurs as the plasma convects downstream. Shown in Fig. 3 is a vertical cut of Bz at the end of the current layer. Bz peaks at around 0.2. To further check that the upstream magnetic field Bxup rotates into the z direction as in a simple whistler, we show in Fig. 2共d兲 a hodogram of Bx and Bz plotted along the top edge of the current layer 35 x/di (c) III. DYNAMICS OF THE INNER REGION 1 0 y/di previously, where it was shown that the rates of reconnection at late time were steady and close to 0.14 for all three mass ratios and independent of the size of the computational domain 共for domain sizes Lx ⫻ Ly greater than 51.2⫻ 25.6兲 and w0.15 All of the data shown is taken during times when the rate of reconnection is steady. For mi / me = 25, the simulation is on a computational domain of 204.8⫻ 102.4 with a grid scale ⌬ = 0.05, w0 = 1.5 and the speed of light c = 15. For mi / me = 100, the computational domain is 102.4⫻ 51.2 with a grid ⌬ = 0.025, w0 = 1.1 and c = 20. For mi / me = 400, the computational domain is 51.2⫻ 25.6 with ⌬ = 0.0125, w0 = 0.6 and c = 40. The mass unity run was on a domain of 200⫻ 100 with a grid of 0.195, had Te = Ti = 0.25, c = 5, w0 = 2.0 and a steady rate of reconnection of around 0.2. y/di 042306-3 -0.10 -0.15 -0.20 -0.05 0.00 y = 6.54 0.05 0.10 0.15 0.20 Bx FIG. 2. 共Color online兲 Blowup around the x-line of 共a兲 the electron outplane current Jez, 共b兲 the in-plane magnetic field lines 共Bx positive above the current layer and negative below兲, 共c兲 out-of-plane magnetic field Bz and 共d兲 a hodogram of Bx and Bz along the white line in 共c兲, which lies just above the electron current layer. The 0 in 共d兲 marks beginning of the hodogram 共close to the x-line兲. along the horizontal line in Fig. 2共c兲 共37.4⬍ x ⬍ 45.0, y = 6.54兲. The “0” marks the starting point just upstream of the x-line where Bz = 0 and Bx has its maximum value. The magnetic field rotates in the right-hand sense 共By is positive兲 as in 0.8 0.6 0.4 Bz , x=40.0 0.2 0.0 -0.2 -0.4 Bx , x=37.4 -0.6 -0.8 4 5 6 7 8 y/di FIG. 3. From the simulation shown in Fig. 2, Bx along a vertical cut through the x-line at x = 37.4 and Bz along a vertical cut at x = 40.0, the downstream edge of the out-of-plane current layer. Downloaded 18 Jun 2008 to 128.175.189.215. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp 042306-4 Phys. Plasmas 15, 042306 共2008兲 Drake, Shay, and Swisdak Ex = FIG. 4. 共Color兲 From the simulation shown in Fig. 2. In 共a兲 Ex close to the x-line. In 共b兲 cuts through the x-line in the outflow direction of the x components of Ohm’s law: Ex in black, vezBy / c in red, −共1 / ne兲 peyx / y in blue, 2 / x in green. In 共c兲 cuts of −共1 / ne兲 pexx / x in yellow, and −共1 / 2e兲 mevex the field line velocity, vBx = −Ez / By in black and vBz = Ex / By in red. a simple whistler. Further, as expected from the cuts shown in Fig. 3, the maximum amplitudes of Bx and Bz are nearly equal. In an ideal whistler the out-of-plane electron flow vez is produced by an electric field Ex = vezBy / c, which would be positive 共negative兲 to the right 共left兲 of the x-line in Figs. 1 and 2. In Fig. 4共a兲 we show a blowup of Ex around the x-line averaged over the time interval ⍀it = 41.0 to 41.2 from the simulation of Fig. 2. Although noisy, Ex has the expected symmetry. Ex is also shown in the black line in Fig. 4共b兲 along a cut through the x-line along the outflow direction. The ridges of large Ex above and below the symmetry line in Fig. 4共a兲 correspond to the x component of the traditional Hall electric field, which is perpendicular to the magnetic field and maps the magnetic separatrices during collisionless reconnection.19 To understand the mechanism for the development of Ex, it is helpful to look at the field line velocity vB in the vicinity of the x-line. This velocity is notoriously difficult to define, especially in a region where the parallel electric field E储 = b · E is nonzero. A line-preserving velocity cE ⫻ B / B2 can be defined only if20 b ⫻ ⫻ 共bE储兲 = 0. 共1兲 Because of the symmetry of antiparallel reconnection, this relation is satisfied along the outflow symmetry line and the components of the field-line velocity are given by 共−cEz / By , 0 , cEx / By兲. The relation vBz By c 共2兲 means that Ex is simply the motional electric field associated with the out-of-plane motion of By. The nonzero velocities vBx 共black兲 and vBz 共red兲 are shown along the outflow direction in Fig. 4共c兲. The x component of the velocity of the field line diverges at the x-line, which is required since the continuity of the flow of magnetic flux toward and away from the x-line must be preserved, which along the outflow direction requires that vBxBy remain finite. The out-of-plane motion of the field lines vBz is nearly constant in a region around the x-line 兵note that the apparent singularity close to the x-line is an artifact of the noise in Ex in the simulation 关black curve in Fig. 4共b兲兴其 since the symmetry of the various terms in Eq. 共3兲 for Ex below requires that Ex → 0 where By → 0. The key result of Fig. 4共c兲 is that vBz is much less than the local out-of-plane electron velocity vez ⬃ 8. The difference between these two velocities implies that the dynamics downstream of the x-line does not correspond to an ideal whistler, for which the particle and field-line velocity would be equal. The driving mechanism for Ex in the complex environment of the x-line can be understood by writing the x component of the electron equation of motion along the symmetry line of the outflow, which in steady state becomes 1 1 1 1 2 peyx − pexx − mevex . Ex = vezBy − c ne y ne x 2e x 共3兲 The source terms for Ex are plotted along with Ex in Fig. 4共b兲 with vezBy / c in red, −共1 / ne兲 peyx / y in blue, 2 / x in green. −共1 / ne兲 pexx / x in yellow, and −共1 / 2e兲 mevex The dominant source terms on the right-hand side of Eq. 共3兲 are the electron motional electric field vezBy / c and the offdiagonal pressure tensor contribution, which corresponds to a transport of x-directed momentum along y. The sum of all of the contributions on the right-hand side of Eq. 共2兲 approximately matches Ex. To understand the dynamics of the outflowing electrons, consider their motion in a frame moving with their velocity vez in the z direction. In this frame the electric field Ex⬘ seen by the electrons is Ex⬘ = Ex − vezBy / c, which points toward the x-line and accelerates the electrons outward from the x-line. In the absence of inertia or other retarding force in the x direction, Ex would have to increase until the field-line velocity in the z direction matched that of the electrons, forcing Ex⬘ → 0. However, the y-directed flux of the electron momentum density nmevex, which is given by peyx 关convert Eq. 共3兲 into a momentum transport equation by multiplying through by ne兴, effectively acts as a drag on vex. This drag reduces Ex so that the electrons can move faster than the field lines in the out-of-plane direction. There has been discussion in the recent literature on the possible role that Ex could play in trapping the electrons in the vicinity of the x-line since Ex points outwards from the x-line.14 However, it is clear that the effective electric field seen by the moving electrons Ex⬘ is large and points inwards. Thus, the bulk of the electrons will experience a repulsive rather than a trapping force. Downloaded 18 Jun 2008 to 128.175.189.215. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp Phys. Plasmas 15, 042306 共2008兲 The Hall fields and fast magnetic reconnection Bz,out ␦ y / de nout vex cAex ␦x vey veyc 25 0.35 0.36 3.3 0.14 4.3 4.5 7.2 0.36 0.4 100 400 0.23 0.20 0.21 0.18 2.8 3.0 0.11 0.11 6.5 10.0 7.1 11.4 3.5 2.0 0.55 0.88 0.52 0.75 The electron outflow velocity vex develops as the z-directed flow of the electrons is rotated by the reconnected magnetic field By into the x direction. Since the electron outflow velocity also produces the current Jex = −nevex and therefore the magnetic field Bz, the electron outflow velocity is constrained by Bz, i.e., the rotation is a collective wave phenomenon and not simply single particle motion. Since we have already shown from the simulation data that Bz and Bx are linked we establish the usual Alfvénic relation for the electron outflow velocity, vex = cAexde / ␦y, where cAex is the electron Alfvén speed based on Bx and ␦y is the half width of the electron current layer. Since the current layer width scales with de 共as long as the electron upstream  is less than unity兲, the electron outflow velocity scales with the electron Alfvén speed. This was demonstrated earlier in PIC and finite-electron-mass hybrid simulations.18,21 In the present simulation vex = 10.0 at x = 40.0, which with a density of 0.12 and ␦y = 6.54− 6.4= 0.14 from Ampère’s law yields a magnetic field of 0.17, consistent with the amplitude of Bz. In Table I we summarize the data from the proximity of the x-line from the mi / me = 400 simulation along with similar data from mi / me = 100 and 25. Shown are the values of the magnetic field just upstream 共a distance ␦y兲 of the electron current layer Bx,up, the out-of-plane magnetic field Bz,out evaluated just downstream of the electron out-of-plane current layer 共where By reaches a plateau15 and a distance ␦y above the symmetry line兲, the density in the outflow channel nout, the electron outflow velocity vex and the electron Alfvén speed cAex based on Bx,up. In all cases the magnetic fields Bx,up and Bz,out match, consistent with the rotation of the Bx,up field into the out-of-plane direction although the value of both of these fields decreases with decreasing electron mass. The electron outflow velocity very closely tracks the electron Alfvén speed. These results confirm the idea that the whistler drives the electron outflow from the x-line just as the Alfvén wave drives the Alfvénic outflow downstream of the ion dissipation region. IV. OPENING OF THE ELECTRON OUTFLOW EXHAUST Fast reconnection requires a mechanism for efficiently breaking the frozen-in condition and for opening the outflow exhaust downstream from the x-line so that the flow into and y/di vez 18 16 14 12 10 8 6 4 2 0 18 16 14 12 10 8 6 18 16 14 12 10 8 6 18 16 14 12 10 8 6 90 100 6 4 2 0 -2 -4 -6 70 80 90 100 x/di v(ExB)x 4 2 0 -2 -4 70 80 90 100 x/di 1.5 Ey 1.0 0.5 0.0 -0.5 -1.0 -1.5 60 (e) 20 80 vex 60 (d) 20 70 x/di 60 (c) 20 8 6 60 y/di Bx,up 18 16 14 12 10 8 6 (b) 20 y/di mi / me (a) 20 y/di TABLE I. Parameters near the x-line: the mass ratio mi / me, the magnetic field just upstream of the electron current layer Bx,up 共a distance ␦y兲, the magnetic field just downstream of the electron out-of-plane current layer Bz,out 共also evaluated a distance ␦y above the symmetry line兲, the half-width along y of the electron out-of-plane current layer ␦y 共normalized to the electron skin depth de to emphasize its well-known scaling with this parameter兲, the electron outflow velocity vex, the electron Alfvén speed based on Bx,up, the half-length along the outflow direction ␦x of the electron current layer, the electron inflow velocity vey, and the inflow velocity expected from flow continuity veyc. y/di 042306-5 70 80 90 100 x/di 0.4 Bz 0.2 0.0 -0.2 -0.4 60 70 80 90 100 x/di FIG. 5. 共Color online兲 Blowup around the x-line of 共a兲 the electron outplane velocity vez, 共b兲 the electron outflow velocity vex, 共c兲 the x component of the E ⫻ B velocity v共E⫻B兲x and the Hall fields Ey and Bz from a simulation with mi / me = 100. out of the dissipation region does not act as a bottleneck, constraining the rate of reconnection. The momentum transport associated with the off-diagonal pressure tensor easily balances the reconnection electric field in collisionless plasma and therefore does not appear to constrain the rate of reconnection.7 The opening of the outflow exhaust downstream of the x-line was central to Petschek’s original idea for fast reconnection in the MHD model. In the case of collisionless reconnection this requirement extends to both species and was the key ingredient in the argument that Hall reconnection remains fast even in very large systems.8,9 Whether the electron outflow exhaust opens up downstream is also central to the recent discussion of whether collisionless reconnection in large systems remains fast or is constrained as in the MHD model.13–15 Thus, the question is whether the electron outflow jet opens up downstream of the x-line and whether it is the Hall generated fields or another mechanism that opens the exhaust. In Fig. 5 we show data in the vicinity of the x-line from the simulation with mi / me = 100, including vez, vex, the x component of the E ⫻ B drift v共E⫻B兲x and the Hall fields Ey Downloaded 18 Jun 2008 to 128.175.189.215. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp 042306-6 Phys. Plasmas 15, 042306 共2008兲 Drake, Shay, and Swisdak y/di (a) 20 1.0 18 16 14 12 10 8 6 0.5 0.0 -0.5 -1.0 60 70 80 90 100 x/di (b) 1.0 jex 0.5 0.0 -0.5 8 10 12 14 16 y/di FIG. 6. 共Color online兲 From the same simulation as in Fig. 5 the electron outflow current jex in 共a兲. The vertical lines 共in the y direction兲 at intervals of 4di mark the location of cuts of jex that appear in 共b兲. and Bz. The out-of-plane electron velocity forms a localized current layer around the x-line. The electron velocity vex consists of a narrow super-Alfvénic jet flowing away from the x-line13–15 flanked by a slower-moving pedestal that becomes broader with increasing distance downstream. The structure of this outflow jet is shown in greater detail in Fig. 6共a兲, where vertical 共y direction兲 lines at intervals of 4di in a plot of jex mark the location of cuts of jex that appear in Fig. 6共b兲. The outflow exhaust consists of the central high-speed jet flanked by a widening pedestal. The reversal of jex at the flanks of the pedestal marks the magnetic separatrix. We now explore in detail what controls the structure of the outflow exhaust and in particular the broad pedestal. In Fig. 5 the spatial structure of the outflow exhaust in 共b兲 matches the x component of the E ⫻ B drift in 共c兲. In Fig. 7共a兲 we show the x components of the perpendicular electron velocity v⬜ex 共solid兲 and the E ⫻ B drift 共dashed兲 along vertical cuts through the outflow exhaust at x = 65.0 in Fig. 5. The two velocities are essentially identical except at the location of the jet in the center of the current sheet. The electrons making up the jet decouple from the local magnetic field and stream faster than the local field lines.15 Thus, within the jet the electron motion is not given by the local E ⫻ B drift. In Fig. 7共b兲 we show similar cuts of Ey and Bz. Except around the symmetry line at y = 12.8di the Hall fields Ey and Bz within the exhaust typically exceed their counterparts Ez ⬃ 0.14 and By ⬃ 0.07 by a large margin so the E ⫻ B drift along x is dominated by the Hall fields. Thus, the opening of the electron outflow exhaust downstream seen in Fig. 5 is produced by the Hall fields Ey and Bz. As discussed earlier, these fields are generated by the cross field motion of electrons around the x-line and rapidly propagate parallel to the magnetic field away from the x-line. As can be seen in Fig. 2, the peaks of both jez and Bz closely map the magnetic separatrix. The proximity of the upstream FIG. 7. Along a vertical cut at x = 65.0 from the simulation in Fig. 5, we show in 共a兲 the x components of the perpendicular electron velocity vēx 共solid兲 and the E ⫻ B drift 共dashed兲 and in 共b兲 the Hall fields Ey and Bz. The dotted lines mark the separatrix locations. edge of the Hall fields to the separatrix can be seen more clearly in cuts of Ey and Bz in Fig. 7, where the vertical dotted lines mark the location of the separatrices. We can obtain an estimate of the parallel propagation speed of the Hall fields by examining the separation between the separatrix and the Hall field structure downstream: this separation increases with distance downstream at a rate that is controlled by the propagation speed of the fields—higher propagation speeds yield smaller separations. In Fig. 8 we show a 2D plot of Ey around the x-line and separatrix from the mi / me = 400 run. This run produces a larger separation in velocities between dispersive waves and the Alfvén wave than the mass-ratio 100 and 25 runs. The separatrix, shown as a white line in this figure, lies slightly upstream from the boundary layer formed by Ey. The spatial separation ⌬y of the separatrix and the first peak in Ey along the left-hand boundary of Fig. 8 is 0.39, which since Bx ⯝ 0.75, corresponds to a magnetic flux separation of 0.29. The rate of reconnection at the time shown is steady and is close to FIG. 8. 共Color online兲 The structure of the Hall field Ey around the x-line and separatrix 共white line兲 from the simulation of Fig. 2. Downloaded 18 Jun 2008 to 128.175.189.215. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp Phys. Plasmas 15, 042306 共2008兲 The Hall fields and fast magnetic reconnection (a) 0.8 vez 90 0.6 y/di 80 0.4 70 0.2 60 0.0 50 0 20 40 60 x/di 80 100 20 40 60 x/di 80 100 (b) 90 80 y/di 0.14,15 so the field line corresponding to the peak of Ey reconnected at a time 2.1 earlier. The distance along the field line from the x-line to the left boundary is 28.3 so the mean parallel propagation speed of the Hall fields away from the x-line is around 13. The Hall fields propagate along the magnetic field either as a whistler or a kinetic Alfvén disturbance. The total pressure, including magnetic and plasma, is nearly constant across the boundary layer, which eliminates the whistler as the propagation mechanism, since it is on the fast-mode branch of the dispersion curve. The kinetic Alfvén wave has a peak parallel propagation speed given by the electron sound speed cse = 冑共Te + Ti兲 / me ⬃ 15, close to the observed propagation rate. Thus, it is the kinetic Alfvén wave, whose speed greatly exceeds the inflow velocities associated with reconnection, that is responsible for the near field aligned structure of Ey and Bz. Because these fields also drive the electron exhaust, the boundaries of the exhaust also closely map the separatrices. The broad electron outflow exhaust and associated Hall fields were seen in observations of a magnetopause reconnection event in Polar spacecraft data.22 This event was a crossing of the ion diffusion region so the ion motion within the crossing was distinctly sub-Alfvénic. The E ⫻ B motion, which was dominated by the Hall fields, produced the electron outflow, and the associated current supported the Hall out-of-plane magnetic field. No super-Alfvénic electron jet was seen in these data, perhaps because the jet does not extend the full length of the ion diffusion region or because the irregular motion of the magnetopause 共which is evident in the data兲 masked the signatures of the jet. In reconnection simulations density depletion layers are seen close to the separatrices during antiparallel reconnection.18 These dips are evident in the Polar data so we take these dips as a proxy for the location of the separatrices. The dips lie just upstream from the Hall fields and the associated electron exhaust. The electron outflow exhaust, including the localized superAlfvénic jet and broader pedestal, was recently seen in Cluster data from a magnetosheath reconnection event.23 In these data, which were estimated to be 63di downstream from the x-line, only 4% of the electron flux in the exhaust was carried by the super-Alfvénic jet. Thus data from the Polar and Cluster satellite supports the conclusion that the electron exhaust is driven by the Hall fields and extends nearly to the magnetic separatrices. We contrast the structure of the electron exhaust in the case of reconnection with mi me with that in an equalmass, electron-positron plasma. In the equal mass case the out-of-plane velocities of the electrons and positrons have equal magnitudes but opposite directions in the vicinity of the x-line and therefore pull the newly reconnected field lines in opposite directions out of the plane. As a result there are no traditional Hall electric and magnetic fields.24,25 Thus, we expect the structure of the electron out-of-plane current layer and the exhaust to be very different than the case with mi me. Shown in Fig. 9 are plots of vez, the magnetic flux and vex. The out-of-plane current layer is no longer localized in the outflow direction25 but scales with the system size as 70 60 50 0 (c) vex 90 1.0 0.5 80 y/di 042306-7 0.0 70 -0.5 60 -1.0 50 0 20 40 60 x/di 80 100 FIG. 9. 共Color online兲 Blowup around the x-line of 共a兲 the electron out-ofplane velocity vez, 共b兲 the in-plane magnetic field lines and 共c兲 the electron outflow velocity vex from a simulation with mi = me. The white contour in 共c兲 is the separatrix. in the Sweet–Parker model 共in very large systems the outflow exhaust becomes turbulent and the current layer again becomes localized along the outflow direction兲.26 Because of the length of the current layers in the equal mass case, secondary magnetic islands regularly form and complicate the interpretation. In Fig. 9 there is an island near the x-line and another that is being ejected near the right boundary. The electron exhausts display no evidence of the central jet and no evidence of the widening pedestal whose boundary maps the magnetic field lines. The exhaust on the left, which is less affected by the islands, is localized well within the magnetic separatrix 共white contour line兲. These data further confirms the key role played by the Hall fields in opening the electron outflow exhaust downstream of the x-line in the case of reconnection with mi me. V. THE TRANSITION REGION We have shown that whistler dynamics around the x-line causes the magnetic field just upstream of the electron current layer to bend into the out-of-plane direction producing the out-of-plane magnetic field Bz. This magnetic field is produced by the current associated with the electrons being ejected at high velocity from the x-line. An important question is what dynamics causes the electron outflow jet to transition from a collimated beam close to the x-line to the expanding outflow exhaust that develops further downstream. Downloaded 18 Jun 2008 to 128.175.189.215. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp 042306-8 Drake, Shay, and Swisdak Phys. Plasmas 15, 042306 共2008兲 At this location the Bx field has been dragged into the z direction and as a result 兩Bz 兩 兩Bx兩 in a substantial region upstream of the symmetry line. The reduction of Bx essentially shuts off the flow of electrons into the current layer since the convective outflow velocity along x, vex = c共EyBz − EzBy兲 / B2, now dominates. Thus, it is the location downstream of the x-line where the reduction of Bx by the whistler enables the electron outflow exhaust to begin expanding that marks the end of the inner electron diffusion region. VI. SUMMARY AND DISCUSSION FIG. 10. In 共a兲 a blowup around the x-line of the electron flow vectors from the mi / me = 400 simulation of Fig. 2. In 共b兲 vertical cuts of vey through the x-line at x = 37.4di 共solid兲 and x = 41.2di 共dotted兲 and in 共c兲 vertical cuts of −Bz 共solid兲 and Bx 共dotted兲 at x = 41.2di. We focus on the physics of this region by showing in Fig. 10共a兲 the electron flow pattern in the vicinity of the x-line from the simulation of Fig. 2. This data has been averaged over a time interval of 2⍀−1 i just prior to the data shown in Fig. 2. The x-line at this time is at x ⯝ 37.5di. In a region around the x-line spanning several di the electrons flow into the current layer without being significantly deflected into the outflow direction. This behavior can be seen more clearly in the cut of vey through the x-line along the inflow direction 共solid line兲 in Fig. 10共b兲. The flow is given by the y component of the E ⫻ B drift, cEzBx / B2 until the particles become demagnetized in the current layer. Further downstream, however, the electron flow toward the current layer is deflected into the downstream direction. The dotted line in Fig. 10共b兲 is a similar cut of vey in the downstream region at x = 41.2di. At this location downstream the electrons flowing toward the current layer are deflected well upstream of the current layer. It is evident from Fig. 10 that, consistent with the expanding exhaust shown in Fig. 5, the y location of the deflection of the electron motion expands upstream with increasing distance downstream of the x-line. The important question is how far downstream from the x-line do the electrons flowing toward the current layer begin to be deflected prior to reaching the unmagnetized region. This location marks the end of the collimated electron outflow channel, which, depending on its elongation, might constrain the rate of reconnection. To understand the physics behind the deflection of the electrons, we show in Fig. 10共c兲 vertical cuts of −Bz 共solid兲 and Bx at x = 41.2di, the location where the electrons show significant deflection in Fig. 10共b兲. We have explored the role of the Hall fields in the dynamics and structure of the electron outflow exhaust during reconnection in collisionless plasma. Just downstream of the x-line the newly reconnected magnetic fields expand outwards as a convecting whistler wave: the magnetic field Bx,in just upstream of the electron current layer rotates in a righthanded sense producing an out-of-plane magnetic field Bz,out with a magnitude nearly equal to that of Bx,in. The electron outflow velocity is equal to the electron Alfvén speed based on Bx,in. Downstream of the electron out-of-plane current layer the electron outflow exhaust consists of a narrow superAlfvénic jet, which remains collimated far downstream from the x-line,13–15 flanked by a pedestal whose width increases monotonically with increasing distance downstream. Thus, the super-Alvénic jet, while very interesting, does not appear to act to constrain the rate of reconnection. The broader outflow exhaust is driven by the Hall electric and magnetic fields at a velocity vex ⬃ cEyBz / B2, which is comparable to the local field line velocity cEz / By. The Hall fields are generated by the out-of-plane motion of the electrons near the x-line17 and propagate with high velocity nearly parallel to B downstream as a kinetic Alfvén disturbance. The rapid parallel propagation of the Hall fields in comparison to the reconnection inflows causes the Hall fields to closely map the magnetic field lines. In particular, the Hall fields are strongest just downstream of the magnetic separatrix. Since the electron outflow exhaust is driven by the Hall fields, the boundaries of the electron exhaust also closely map the magnetic separatrix. The structure of this exhaust differs greatly from that of the MHD model, where the outflow exhaust is driven by a slow shock that is well inside of the magnetic separatrix 共the angle that the slow shock makes with the symmetry axis is approximately half of that of the magnetic separatrix in the Petschek model兲. Thus, the opening of the electron exhaust, which prevents the electrons from throttling reconnection, is controlled by the Hall fields. The dominance of the Hall fields in driving the electron outflow as well as the conclusion that the boundaries of electron outflow exhaust closely map the magnetic separatrices are supported by Polar observations of antiparallel reconnection at the magnetopause.22 The two-scale structure of the electron exhaust, including the localized super-Alfvénic jet and broader Alfvénic pedestal and associated Hall fields, was seen in a recent magnetosheath reconnection event.27 In these observations only 4% of the outward flux of electrons was carried by the super-Alfvénic jet, the remaining 96% Downloaded 18 Jun 2008 to 128.175.189.215. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp 042306-9 The Hall fields and fast magnetic reconnection being carried by the broader outflow driven dominantly by the E ⫻ B motion in the Hall fields. Thus, while the superAlfvénic jet is certainly an interesting feature of reconnection, it appears to play only a minimal role in convecting electrons away from the x-line. The role of electrons in limiting the rate of reconnection is ultimately determined by the length of the intense electron out-of-plane current layer that develops in the vicinity of the x-line 关see Fig. 2共a兲兴. This region was denoted as the “inner electron diffusion region”15 to distinguish it from the superAlfvénic outflow jet that extends far downstream of the x-line. In the present simulations the end of this current layer and the simultaneous broadening of the electron outflow channel are controlled by the whistler. The whistler rotates the in-plane Bx field in the vicinity of the current layer into the z direction, creating the out-of-plane field Bz. The resulting reduction of Bx self-consistently reduces the electron outof-plane current. Of greater significance is the resulting diversion of the electron flow toward the current layer into the outflow direction 共Fig. 10兲 and thereby broadening the electron outflow channel. The rotation of the fields by the whistler forces Bz Bx in the current layer and just upstream. The electron inflow vey = cEzBx / B2 is therefore locally suppressed and the outflow velocity vex ⯝ cEyBz / B2 strongly enhanced. The resultant broadening of the electron outflow channel marks the end of the inner electron diffusion region. In light of these results on the structure of the electron current layer and outflow exhaust, a fundamental question is whether the electrons play a significant role in limiting the rate of reconnection. The insensitivity of the rate of reconnection to the electron mass in PIC simulations6,7,15 seems to suggest that the electrons simply adjust their dynamics to match the rate of reconnection determined by the ions. The earlier explanations for the insensitivity of the rate of reconnection to electron dissipation5,6 are borne out by the simulations reported here. In Table I the data from simulations with mi / me = 25, 100 and 400 indicate that the outflow velocity of the electrons from the x-line is given by the electron Alfvén speed based on the upstream magnetic field with the width ␦y of the outflow channel scaling with c / pe. The net flux from the x-line ⌫ = vex␦y is therefore independent of me. The length of the electron current layer ␦x along the outflow direction 共based on the half-width, half-maximum of vez兲 is also shown in Table I along with the measured electron inflow velocity just upstream of the x-line vey and the corresponding inflow velocity veyc based on the continuity of the flow. The measured and expected inflow velocities match. The continued increase of vey as me decreases seems to suggest that the current layer is not acting as a bottleneck for reconnection. Simulations of magnetic reconnection in electronpositron plasma, where, because of the equality of the masses of the two species, there are no Hall fields parallel to those discussed here, provide an interesting contrast24–26 to the results presented here. Reconnection remains fast even in the absence of the Hall fields. On this basis it was suggested that the Hall fields are not important in electron-ion reconnection.25 Since we have shown here that the Hall fields in our simulations are responsible for fast, electron-proton Phys. Plasmas 15, 042306 共2008兲 reconnection, what is producing fast reconnection in the electron-positron simulations? For systems with scale sizes Lx ⬍ 400di Swisdak et al. have shown that Sweet–Parker current layers form which lengthen with increasing system size, as expected in the absence of the Hall fields. In this regime the exhaust remains collimated well within the magnetic separatrix as in the MHD model. However, for systems with Lx ⬎ 400di the outflow exhaust develops a strong pressure anisotropy and the resulting Weibel instability produces a turbulent outflow jet that broadens with increasing distance downstream. As a result, the rate of reconnection of the electron-positron system remains fast even in very large simulation domains. In light of the critical role that the Hall fields play in controlling the structure of the electron exhaust during reconnection, it is important that simulation domains used for studying reconnection be chosen carefully to minimize the influence of boundary conditions on the development of the Hall fields. The challenge is that the Hall fields propagate along the magnetic field as kinetic Alfvén structures at velocities that peak at the electron sound speed cse, which for values of  exceeding me / mi, exceed the Alfvén speed. The Hall magnetic field Bz is produced as electrons near the x-line drag the in-plane field in the out-of-plane direction. It is critical that the magnetic field downstream is anchored by the inertia of the downstream plasma since it is the relative out-of-plane motion of the field line at the x-line versus downstream that generates Bz. In the case of periodic simulations the effective boundary condition on Bz is at the middle of the magnetic island where by symmetry Bz = 0. In periodic systems it is possible that the amplitude of Bz is higher than a real system where Bz asymptotes to zero further downstream. We have checked how the periodic boundaries affect the development of Bz by carrying out a set of simulations with mi / me = 25 in three different simulation domains: 204.8di ⫻ 102.4di, 102.4di ⫻ 51.2di, and 51.2di ⫻ 25.6di.15 The reconnection rates are indistinguishable in the three cases as are the lengths of the out-of-plane current layers. In Fig. 11共a兲 we show Bz in a blowup around the x-line from the run in the 102.4di ⫻ 51.2di domain. In Fig. 11共b兲 we show the profile of Bz 共dotted line兲 along the vertical lines in Fig. 11共a兲. Because of the symmetry across x = 0 the data along the two cuts were averaged by subtracting the values along the two cuts and dividing by two. The two cuts are just downstream of the out-of-plane electron current layer, 8.5di on either size of the x-line, which is where Bz begins to broaden the electron exhaust. Also in Fig. 11共b兲 are cuts from the two other simulation domains: the solid for the 51.2di ⫻ 25.6di domain and the dashed for the 204.8di ⫻ 102.4di domain. All of the cuts were taken 8.5di downstream of the x-line and when the rate of reconnection was steady. The cuts reveal that the location of the peaks of the Hall magnetic fields lie at the same distance from the symmetry line at y = 0 for all three simulations. This is especially evident in the two largest simulations. This indicates that the opening angle of the electron exhaust is the same in all of the simulations, which is consistent with the reconnection rate being the same. The magnitude of Bz decreases modestly with increas- Downloaded 18 Jun 2008 to 128.175.189.215. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp 042306-10 y/di (a) 6 0.4 Bz 4 2 0 -2 -4 -6 -0.2 -20 ACKNOWLEDGMENTS 0.2 0.0 -30 (b) Phys. Plasmas 15, 042306 共2008兲 Drake, Shay, and Swisdak -10 0 x/di 10 20 -0.4 0.4 0.2 J.F.D. acknowledges helpful discussions with Dr. J. Eastwood, Dr. T. D. Phan, and Dr. F. S. Mozer. This work has been supported by NSF Grant Nos. ATM0613782 and ATM0645271, NASA Grant Nos. NNG05GL74G and NNG05GM98G, and DOE Grant Nos. ER54197 and ER54784. Computations were carried out at the National Energy Research Scientific Computing Center. 1 0.0 -0.2 -0.4 -4 -2 0 y/di 2 4 FIG. 11. 共Color online兲 In 共a兲 the Hall magnetic field Bz from a simulation with mi / me = 25 on a 102.4di ⫻ 51.2di grid. The vertical lines a distance of 8.5di on either side of the x-line show the location of the cuts of Bz displayed in 共b兲. Cuts of Bz a distance of 8.5di on either size of the x-line from three mi / me = 25 simulations differing only in the size of their computational domain: 204.8di ⫻ 102.4di 共dashed兲; 102.4di ⫻ 51.2di 共dotted兲; and 51.2di ⫻ 25.6di 共solid兲. ing domain size by around 23% between the two smaller simulations and around 16% between the two larger simulations. Ideally, there should be no change but we emphasize that this is a very stringent test since the overall electron outflow exhausts in the larger domains are much longer than in the smaller domains yet the cuts are taken at a fixed distance downstream from the x-line. In open boundary simulations the condition on Bz on the outflow boundary is typically dBz / dx = 0.13,19,28,29 This allows Bz to take on any value at the boundary. However, if the electron out-of-plane velocity was nearly constant along the magnetic field 共i.e., nearly a function of , the in-plane magnetic flux兲 the field line would effectively be free floating rather than anchored and could move in and out of the reconnection plane without producing the Hall field Bz. Thus, Bz could be weaker than in an infinite system, thereby inhibiting the opening of the electron outflow exhaust and causing the electrons to throttle the rate of reconnection. Such behavior was seen in a recent open boundary models13 although whether the proximity of the x-line to the boundary produced this behavior is unclear. In any case the importance of the Hall fields to the structure of the electron exhaust and the rapid rate of propagation of the Hall fields downstream, suggests that care must be taken to ensure that the boundary conditions in open boundary models are not influencing the physics. A demonstration such as that in Fig. 11 that differing simulation domains do not influence the structure of the Hall fields is essential to be sure that physics results do not depend on boundary conditions. P. A. Sweet, in Electromagnetic Phenomena in Cosmical Physics, edited by B. Lehnert 共Cambridge University Press, New York, 1958兲, p. 123. 2 E. N. Parker, J. Geophys. Res. 62, 509, DOI: 10.1029/JZ062i004p00509 共1957兲. 3 D. Biskamp, Phys. Fluids 29, 1520 共1986兲. 4 B. N. Rogers, R. E. Denton, J. F. Drake, and M. A. Shay, Phys. Rev. Lett. 87, 195004 共2001兲. 5 J. Birn, J. F. Drake, M. A. Shay, B. N. Rogers, R. E. Denton, M. Hesse, M. Kuznetsova, Z. W. Ma, A. Bhattacharjee, A. Otto, and P. L. Pritchett, J. 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J. 共to be published兲. 27 T. D. Phan, G. Paschmann, C. Twitty, F. S. Mozer, J. T. Gosling, J. P. Eastwood, M. Øieroset, H. Reme, and E. A. Lucek, Geophys. Res. Lett. 34, L14104, DOI: 10.1029/2007GL030343 共2007兲. 28 J. D. Huba and L. I. Rudakov, Phys. Rev. Lett. 93, 175003 共2004兲. 29 A. V. Divin, M. I. Sitnov, M. Swisdak, and J. F. Drake, Geophys. Res. Lett. 34, L09109, DOI: 10.1029/2007GL029292 共2007兲. Downloaded 18 Jun 2008 to 128.175.189.215. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp