Ion kinetic effects in magnetic reconnection:

Transcription

1 JOURNAL OF GEOPHYSCAL RESEARCH, VOL. 103, NO. A3, PAGES , MARCH 1, 1998 on kinetic effects in magnetic reconnection: Hybrid simulations R.-F. Loermoser and M. Scholer Max-Planck-nstitut fiir Extraterrestrische Physik, Garching, Germany A. P. Matthews Department of Physics, University of Natal, Durban, Republic of South Africa Abstract. Magnetic reconnection in a low beta plasma is studied by using a largescale two-dimensional hybrid simulation code that treats the ions fully kinetically and the electrons as a massless fluid. Reconnection is realized by localizing an anomalous resistivity in the the center of a Harris-type current sheet. The results are applicable to reconnection in the geomagnetic tail. Within a distance of about 200 ion inertial lengths from the neutraline the fast reconnection jet is characterized over most of the region by partial shell type ion distributions; i.e., the distribution in the central current sheet is not thermalized. nward drifting cold lobe ions perform in the center of the reconnection wedge due to the small magnetic field curvature Speiser-type orbits and are subsequently ejected again onto lobe field lines. This stadium is similar to the collisionless reconnection scenario described by Hill [1975]. The cross-tail current of the thin current sheet is' supported by the drifting ions. This current sheet warps and an instability develops, which leads further away from the neutral line to a filamentation of the cross tail current. These simulations suggested that after near-earth reconnection proceeds to lobe magnetic field lines a postplasmoid plasma sheet with a thin current sheet builds up, which is of a boundary layer type. The thin current sheet is instable. At about 200 ion inertial lengths away from the neutral line the instability has reached a nonlinear state 'and the cross-tail current becomes patchy; the incoming and ejected cold lobe ions are isotropized in the ensuing magnetic field and thus constitute a hot plasma sheet distribution. This can then supporthe occurrence of slow mode shocks at the boundary of the reconnection layer with backstreaming ions upstream of the shocks. 1. ntroduction 1996]. Such simulations have confirmed the Petschek [1964] model of reconnection with an inflow region, a Magnetic reconnection plays an essential role in de- wedge shaped outflow jet, and a small central diffutermining the magnetic field configuration in astrophys- sion region [e.g., Sato, 1979; Scholer, 1989]. n parical and space plasmas and is responsible for the fast ticular, slow mode shocks bounding the outflow region conversion of magnetic field energy into thermal and ki- have been found in the MHD simulations of reconnecnetic energy of the plasma. For example in the Earth's tion. As suggested by Petschek's [1964] model, the mamagnetosphere reconnection takes place at the magne- jority of the energy conversion and of the field reversal topause [e.g., Sonnerup e! al., 1995], where it leads to is achieved by these slow mode shocks. solar wind mass, momentum, and energy transfer, and n order to determine the actual structure and conis believed to be important during geomagnetic sub- figuration of steady state reconnection in a collisionless storms for the release of stored energy in the magne- plasma is necessary to take into account kinetic effects totail [e.g., Baker et al., 1996]. Dynamical aspects of of the ions. Kinetic studies in the past were mainly conreconnection concerning both the magnetopause and cerned with the question which dissipation mechanism the geomagnetic tail have been studied extensively in can support magnetic reconnection by providing electric the past by large-scale magnetohydrodynamic (MHD) fields in the diffusion region in the case of a collisionless simulations [e.g., Lee, 1995; Scholer, 1995; Birn et al., plasma. Some of these studies include both ion and electron dynamics [e.g., Swift, 1986; Hoshino, 1987], Copyright 1998 by the American Geophysical Union. kinetic ions and fluid electrons with a model for the full electron pressure tensor [Hesse and Winske, 1994], Paper number 97JA or one plasma species with the electrons simply as a /98/97 JA

2 4548 LOTTERMOSER ET AL.: HYBRD SMULATON OF RECONNECTON charge neutralizing fluid [e.g., Terasawa, 1981; Prichett et al., 1991; Cat ei al., 1994]. n case the main interest is not in the collisionless dissipation process, but in ferent from the one envisaged by Peischek [1964] and closely resembhng the collisionless merging configuration without slow mode shocks proposed by Hill [1975]. the structure of the reconnection layer in a collisionless n the past, ion distribution functions during reconnecplasma, it is sufficient to treat the ions fully kinetically tion have often been derived from test particle trajecand to assume that the electrons are a massless fluid tory calculations in the fields obtained from MHD simuwith isotropic pressure. n such hybrid simulations the electric field in the diffusion region has to be supplied explicitly by a resistive term in the electron momentum equation. lations. The results of this study show that in order to predict ion distribution functions during magnetic reconnection the self-consistent kinetic treatment of the ions is indispensable. Large-scale hybrid simulations relevant to magnetotail reconnection have first been performed by Krauss- Vatban and Omidi [1995]. These authors initiated re- 2. Simulation Model connection in a system with antiparallel magnetic fields in the two halves by a localized resistivity in the middle of the current sheet and by imposing an inflow from top and bottom perpendicular to the magnetic field. They found that after a transitional period, during which small-scale plasmoids emerged and were convected out of the system, a steady-state reconnection structure did build up with two pairs of thin transition layers attached to an X point, which divert and accelerate the flow. Upstream of this boundary layer fast ion beams were found, while in the center of the current sheet the plasma was heated and thermalized. These results are consistent with the features of fast Petschek-type reconnection. n the hybrid simulations of magnetotail The simulations are done with a two-dimensional hybrid code with macroparticle ions and an inertialess electron fluid. The code employs a current advance method and a cyclic leapfrog and is described by Matthews [1994]. The electron fluid is assumed to have a finite, but isotropic pressure, which follows an adiabatic law. All variables are functions of time t and two variables z and y. The time is expressed in units of the inverse of the ion gyrofrequency gi -- ebo/m, where e is the magnitude of the electron charge, m the ion mass, and Bo the constant magnetic field (in the z.direction) outside of the one-dimensional current sheet, i.e., in the lobe. Distances are expressed in units of the ion inertial length Ao -- c/o3pi, where c is the speed of reconnection by Lin and Swift [1996] reconnection was light and O3pi is the ion plasma frequency in the lobe. not driven but spontaneous by imposing a finite resistiv- The unit velocity is then the lobe Alfvdn velocity v t. ity in the center of the simulation domain. These simu- The initial profile of the z component of the magnetic lations were started with a Harris-type neutral sheet of field is given by finite thickness. Lin and Swift [1996] obtain two slow mode shocks bounding the outflow layer: they found Bx(y) - Bo tanh(y/5) that the ratios of normal magnetic fields and of normal flow velocities taken outside of the reconnection layer where Bois the lobe magnetic field and 5 is the halfwidth and in the center of the reconnection layer are consis- of the initial current sheet. n the present simulations tent with the Rankine-Hugoniot relations for a steady - 2.4Ao, i.e., we assume initially a thin plasma sheet. slow mode shock, although the increase of the tangen- The number density is assumed to be given by tial velocity across the discontinuity is considerably less than that predicted by the jump conditions. Recently, n(y) - no + n cosh-2(y/$) Nakamura ei al. [1997] have studied the ion dynamics in hybrid simulations of reconnection in a double where no is the lobe density and n is the sheet density. periodic system (two current sheets). They also start The temperature is determined from the total pressure with a Harris-sheet equilibrium and initiate reconnec- balance across the current sheet. The number density tion by imposing a localized resistivity at the center of is normalized in units of no, and nx/no = 2.5. The ion the simulation box. n contrast to the results of Kvauss- plasma beta (particle to magnetic field pressure) in the Vatban and Omidi 1995] and Lin and Swift [1996] they lobe is assumed to be/3i - 5. Since we allow for a found that the ions in the center of the current sheet finite electron temperature, we set/3e The spaare not fully thermalized, but rather perform modified tial profile of the resistivity imposed in the simulations Speiser orbits in the region close to the reconnection is given by line, and have partial shell distributions in the region further away from the reconnection line. No indication 7-- 7o exp[-(z 2 + y )/c ] for a slow mode shock has been seen in these simulations. with c = 2.2Ao. We have chosen r/o in such a way n the present paper we report on simulations similar that the Lundquist number with respecto the current to the ones performed by Lin and Swift [1996] and by sheet halfwidth Rm - ( /r/ - 8. The cell size in both Nakamuva et al. [1997]. We run a 2-D large-scale hy- directions is chosen to be Az- Ay- 0.8Ao. brid simulation to study the kinetic effects of the ions within the reconnection layer. t is shown that the ion dynamics can sustain a reconnection configuration dif- The following boundary conditions have been used. The normal derivative of the normal component of the electric field is kept zero, de, /dn- O. The tangential

3 LOTTERMOSER ET AL.: HYBRD SMULATON OF RECONNECTON 4549 component of the electric field is calculated in such a flpt= 47 way that the total electric field is divergence free. For the particles we used "replacing" boundary conditions. This allows the plasma to enter freely at all boundaries but does not allow the plasma to exit. A particle is replaced when it exits through one of the boundaries. The replacing particle is loaded at z - zt+az (z -- zt--az), y -- yt when the particle exits the left (right) hand boundary, where (zt, t) is the last position outside of the simulation domain. Similarly, the replacing particle is loaded at z - zt, - t- A (y -- + A ) when the particle exits through the top (bottom) boundary. oct= 9õ When a particle moves from the last cell further into the simulation system a new particle is placed at position z - zt-az, --, where (z, y ) is the position of the particle entering the simulation domain. Similarly, particles are injected at the other boundaries. The velocity of the replacing particles is taken to be equal to the velocity of the removed particles and the velocities of the injected particles is equal to the velocities of the inward moving particles. We have performed two runs differing in box size and oct= 142 number of particles per cell. The length of the simulation system for run is -L _ z _ L, -Ly _ _ Ly, with L - 240Ao and L - 60Ao. The ions are initialized as drifting Maxwellians corresponding to the two-dimensional kinetic equilibrium [e.g., Pri che e -60 al., 1991]. There are initially 40 particles in each lobe cell. For run 2 we considerably increased the system size (L - 480Ao, L - 60Ao) and compromised by decreasing the number of particle in the lobe cells to 30. We will first discuss run 1. As will be seen, the results of this run led us to perform the second run in the larger X/Xo system, so that we could follow the development of the reconnection layer to later times. Assuming a lobe den- Figure 1. Magnetic field (contour lines of the vector sity of 5 particles/cm 3 and a distant lobe magnetic potential a ) and stream lines in the z- plane at three field strength of 7 nt, there is about 6 o per R, and different times, 2gi - 47, f gi - 95, and f i i x - s. The whole size of the numerical box cor- responds to about 80R, x 20R, and 160R, x 20R,, respectively, in the geomagnetic tail. At the end of the runs we have 6.0 x 106 and 8.2 x 106 particles in the numerical box. 3. Simulation Results The smaller simulation (run 1) is run for a time of 166f i. Figure 1 shows magnetic field lines and bulk flow streamlines at three different times. The field lines are equivalent to contour lines of the z component of the vector potential. Note that the plasma is compressible so that a display of the bulk velocity by streamlines is only representative of the momentary velocity field. Also indicated by heavy lines in all panels are the separatrices. They are defined as the lines of constant vector potential of the magnetic field lines which cross the center z -- O, y -- O. As shown in the top two panels of Figure 1, reconnection sets in at the center of the current sheet at early times: magnetic field lines are convected into the central diffusion region, and reconnected field lines are convected out along the current sheet. No de- velopment of small-scale plasmoids is observed in our simulations. Such transient plasmoids have been seen in the hybrid simulations of Krauss-Verban and Ornidi [1995]. n a test run, we have decreased the number of particles in the lobes to 20 particles per cell. This results in considerably larger electric field fluctuations within the initial current sheet, so that reconnection sets in at several points, which leads to the development of transient plasmoids in the early phase of reconnection. As can be seen from the 3 panels exhibiting the streamlines a distinct feature of the simulations is a strongly converging flow toward the field line reversal region, indicating a fast mode expansion. The flow does not turn at the separatrix, i.e., there is no indication for a separatrix boundary layer. So far, the global structure seems not to be different from that obtained in MHD simulations of reconnection. First differences of the structure of reconnection become evident when analyzing the electrical current. Plate 1 shows from top to bottom magnetic field lines, the component of the current density perpendicular to

4 4550 LOTTERMOSER ET AL.' HYBRD SMULATON OF RECONNECTON 18 oct = o X/Xo Figure 2. From top to bottom: magnetic field, contour lines of the cross-tail current density, contour lines of the vorticity in part of the simulation system at 12git X = B, [ O; t = [1[ tl B 0.51 ' ' ' VL. ', 0' , øz'[8, ^', -0.5, l. 6,/ the simulation plane, jz, in the z-y plane, the streamlines of the bulk flow, and the z component of the vorticity f- 7 x v at the end of the run in part of the simulation system. The z component of the vorticity exhibits a quadrupol structure within the region bounded by the separatrices. n a region of about z - +40Ao from the X point the jz current coincides closely with the z component of the vorticity. Beyond a distance of about z - +60Ao from the X line, the maximum current flows in the center of the current sheet, and does not have its maximum at the boundary of the wedgeshaped high speed flow layer as would be expected for the occurrence of slow mode shocks. We will return later to the current pattern and analyze it in terms of the ion and electron contribution. The current sheet exhibits a global instability' Figure 2 shows from top to bottom magnetic field lines, contour lines of the cross tail current density, and contour lines of the z component of the vorticity in a selected part of the simulation system. t can be seen that the whole current sheet is warped with a wavelength of about 30Ao. Profiles of bulk parameters across the reconnection layer are shown in Figure 3. The left five columns show the magnetic field components, the magnetic field magnitude, and the jz component of the current at t flgi as a function of y at z - -85Ao. Dashed vertical lines indicate the positions of the separatrices. The current exhibits small maxima inside of the separatrices; Y/Xo Y/Xo Figure 3. Spatial profiles (left column) Bz, By, B, magnetic field strength B, current density component j, and (right column) velocity components V, Vy, V, density p, and temperature anisotropy.

5 LOTTERMOSER ET AL.- HYBRD SMULATON OF RECONNECTON 4551 however, the maximum current and the steepest gradi- collected in the small boxes shown in the figure Also ent in Bx occurs in the center of the current sheet. The most important difference to MHD simulations of reshown by lines from the origin are the projections of the magnetic field (heavy lines) and of the bulk flow velocconnection is the occurrence of a magnetic field compo- ity (thin lines) onto the respective planes. Within the nent in the invariant direction. This component is due separatrices, but outside of the high-speed flow layer, to the Hall effect, which results into a poloidal current the cold lobe distribution is observed to stream parallel density jp, i.e., a current in the ß - y plane. This Hall to the magnetic field (Figure 5a). When approaching magnetic field component has been seen before in par- the reconnection layer closer, in addition to the lowticle/hybrid simulations of reconnection [e.g., Heweft et speed lobe ions an additional population is seen which al., 1988; Mandt et al., 1995; Hesse and Winske, 1994; has a bulk flow speed away from the reconnection layer Nakamura et al., 1997], but the important point here is into the lobe which somewhat larger than the Alfv n that this component is seen throughout the reconnec- speed (Figure 5b). This backstreaming ion beam has tion layer and has a magnitude of-- 50% of the lobe field. Note the occurrence of smaller amplitude waves in B upstream of the separatrices. The right hand panels of Figure 3 show profiles of the components of the fist = 166 bulk flow velocity, of the density, and of the temper- 25 ature anisotropy Tii/Tñ, the ratio of the temperatures parallel and perpendicular to the magnetic field. The density profile also does not show any indication of slow mode shocks: the density increases almost continuously toward the center of the current sheet. n contrast, the high speed flow in the reconnection wedge has a steplike profile. The temperature anisotropy is largest near the edges of the high speed flow regime; there is no plateau-like structure within the reconnection wedge, which would indicate a uniform downstream state. Figure 4 shows in the top panel magnetic field lines and in the bottom panel contours of B in a part of the simulationsystem. Positive values of B are indicated Jz by solid contour lines, negative values by dashed contour lines. The vorticity is superimposed on the panel with the magnetic field lines by grey-shading. The magnetic field in the invariant direction exhibits the typical quadrupolar structure. As pointed out by Hesse and Winske [1994], because of jp - (1// o) ' B z x e z the contours of Bz are instantaneous flow lines of the poloidal current. The Hall current loop is similar to the v one shown by Nakamura et al. [1997]' the current flows along the current sheet toward the X point and returns in a thin layer at the edge of the outflow jet. The layer of outward flowing current coincides with the region of maximum vorticity. This structure is somewhat differ- -25 ent from the poloidal current system found by Winske and Hesse [1994]. These authors found in their simulations a current convergence onto the X point in the y direction and a divergence in. Presumably, they did not run the simulation long enough. The magnetic field perturbation produced by the poloidal current system propagates essentially with the intermediate speed along the current sheet and results at late times in a steady state Hall magnetic field along the whole reconnection layer. We will now discuss the ion distribution functions in :2,5,.,,.,,,,, ß. greater detail. n the upper panel of Figure 5 we depict the magnetic field lines in the left hand part of Plate 1. From top to bottom at git ' Magnetic the simulation system. The lower panels show reduced field B (contour lines of the vector potential a ), the distribution functions in the v -vx plane (left panels) z component of the electric current density j in color and in the vz -Vx plane (right panels) at three differ- code, stream lines of the bulk flow in the x - y plane, ent positions across the reconnection layer at and the z component of the vorticity (fl ) in color code O0 0 1 O0 X/Xo

6 4552 LOTTERMOSER ET AL.' HYBRD SMULATON OF RECONNECTON 60 Q t = 142 ß N -- N o.o v Q t = O N - N - 1o3-1, Plate 2. Positions of particles which are at figit at a certain position in the center of the reconnection layer at two earlier times, git and git Particles which have at git a velocity component vx < -0.3 are drawn in red, particles with vx > -0.6 at git are drawn in blue. Also shown are the distributions in v : - vz velocity space of the red and blue particles at the two times in the respective colors.

7 ., OTTERMOSER ET A,.: HYBRD SMU,ATON OF RECONNECTON i t-- 166,z --7!'"'... o... ß -:... :.:....tl.'.:.?.-.,..:. ;.:....,:::.:..:-. :- : ' :-..:' '"::.:.'t:'... _.._..:.....:..,. '..-,,..-. : '._--:..: Z. --_:. _..-.,, ,. BZ oo o 1 oo X/Xo Figure 4. (top) Magnetic field (contour lines of the vector potential a ). Superimposed is by greyshading the vorticity component fl, where shading runs from negative values (white) to positive values (black). (bottom) Contour plot of the magnetic field component B. Contours with negative values are dashed, contours with positive values are shown as solid lines. also been found in 2-D hybrid simulations of reconnec- be interpreted in terms of particles escaping from the tion by, e.g., Krauss-Varban and Omidi [1995] and œin heated population downstream of a slow mode shock, and Swift [1996]. t can clearly be seen that the lobe ion population has a negative average velocity v due to the Hall component of the magnetic field; the backstreaming ions and the incoming lobe ions are clearly field-aligned in the rest frame of the plasma. The projections of the distribution in the center of the current sheet are shown in Figure 5c. This distribution in the but are the particles which have performed Speiser-type orbits in the current sheet and are ejected after half a gyro-orbit determined by the normal magnetic field component with about zero velocity vz along the magnetic field. We have calculated the curvature parameter determined by the ratio of the minimmn radius R,,,i,, of curvature of the magnetic field in the current sheet to center of the current sheet at a rather large distance maxi nmn gyroradius p,,,,, t - (Rmi,,/p,,, ) /2. Up from the X point is clearly not a thermalized distribution. n the vu- vx plane the distribution seems to to Ao is within 0.1 and 0.2, i.e., the ion motion can indeed be specified by a multi-bounce motion in the consist of two different parts: a distribution with zero direction and a half-gyro orbit in the z-'z plane. velocity in the z direction and a second part streaming with vx --1.SvA away from the X point. However, as can be seen from the projection onto the v - v plane, Figure 6 shows reduced distribution functions at three different positions in the center of the reconnection layer. t can be seen that the partial ring-type distrithe distribution is actually a partial ring-type distribu- bution is characteristic in the central current sheet for tion. The magnitude of the velocity of the particles is, a large distance away kom the X point and dominates apart from the thermal spread, approximately constant the distribution in the reconnection layer. The distriin the plasma rest frame. This finding is different from bution function is different in the vicinity of the neutral the results of Krauss-Vatban and Omidi [1995] and œin line and in the magnetic field pile-up region at the rear and Swift [1996], who observed fairly isotropic distributions in the center, which they interpreted in terms of heating by the slow mode shocks. However, our results obtained during this run are consistent with the distri- end of the plasmoid near the left end of the simulation box. The distribution in the latter region (not shown) is similar to the counter-streaming ion distribution reported by Fujimoro et al. [1996]. As can be seen kom Figure 6 a partial shell distribution is observed near the X point. We interpret this type of distribution as being due to two effects: first, the radius of curvature of the magnetic field lines near the X point is large, bution obtained by Nakamura et al. [1997] in a region 40 o away from the neutral line. These authors have shown that the distribution results from Speiser-type orbits [Speiser, 1965] when the thickness of the current layer is larger than the ion Lamor radius. The orbits are and is equal or exceeds the gyroradius of the lobe ions. modified by the magnetic field component in the invari- Second, near the X line the 2-D effect of the reconnecant direction. The backstreaming ion beam observed tion layer becomes important in that there are no ions away from the center of the reconnection layer cannot which have been residing in the jet long enough to have,

8 4554 LOTTERMOSER ET AL.: HYBRD SMULATON OF RECONNECTON 6O o.o fi t X/Xo , nent vz ) -0.3 (painted blue). The latter are, as we will see, lobe ions newly entering the current layer. Plate 2 shows the positions of the "red" and "blue" particles at earlier times, i.e., at t = 142f i z and t - 156fi i z, respectively. Also below the pane[ with the magnetic field are velocity distribution functions in the vz- vz plane of the two different populations. t can be seen how the cloud of blue particles moves along the field lines toward the center of the current sheet and then arrive at t - 166f i all at the same position. The red particles have entered the current layer much earlier and at larger distances from the observation point. They have performed drift orbits, as can be seen from the distribution function in the plane of the current sheet. Particles entering as far as 70Xo from the observation point contribute to the distribution function at that point. This shows that in an extended region away from the neutral line 2-D effects will influence the ion distribution function. 6O fi9 t C" ' ' -C >' o.o."/q o.o _/. - - C ' X/'ko - t ,, - V,, ::>r > o.o - Figure 5. (top) Magnetic field at figit (bottom) ion velocity distributions v -v and vz-v at the three position indicated by the boxes in the top part. after cross-current sheet drift, v close to 0 and being ejected. n the v - v plane near v -0 the two coun- terstreaming lobe distributions can be seen, which just enter the volume determined by the small box. Particles which had reached the current sheet at earlier time and closer to the X point began to perform Speiser-type orbits without being ejected. The bounce motion in the y direction exceeds the extend of the sampling box, so that the region near the current sheet is void of particles with large negative vz and v close to zero (% - 0 at the turning point of the bounces). We return to the partial ring distributions which occupy most part of the central reconnection layer. n order to demonstrate the time history of ions observed at a certain position (z', y') (position c in Figure 5, top Vx Vx 0.6 panel) the particle trajectories are followed backward Figure 6. Same as Figure 5 with ion velocity distriin time. The distribution function at position butions at three positions obtained in the three boxes is divided in two parts: ions with a velocity component shown in the top part located in the center of the rev ( -0.6 (painted red) and ions with a velocity compo- connection layer. - -

9 LOTTERMOSER ET AL.: HYBRD SMULATON OF RECONNECTON Jez 0 Jiz Figure 7. (top)electron contribution j, (solid line) to the total cross tail current jz (dotted line) at y = 0 versus z. (bottom) on contribution jiz (solid line) to the total cross tail current j (dotted line) at y = 0 versus z. o is enhanced well before the drop in B, as would be expected for a boundary layer with backstreaming particles upstream of the slow mode shocks, but located within the separatrices. A further noticeable feature is the double wavetrain of the magnetic field component in the invariant direction, B:. Upstream of the separatrices there are small amplitude waves in B:. This indicates that the Hall current has developed into a illamentary structure within the current layer. The illamentation is due to the dispersive nature of the medium: the shear field in the current layer decays due to dispersion, which leads to a large wavelength wavetrain in the current layer and smaller wavelength upstream waves. Figure 9 displays in the top panel magnetic field lines in the z-y plane in the left hand part of the system and in the lower panels distribution functions f(v, v )and f(v:, v )in the three boxes indicated in the top panel. Note the vastly different scales used for z and y in the upper panel. nside the separatrices but upstream of the step like decrease of the magnetic field low-speed An important question is whether the electrons or the ions carry the cross-tail electrical current responsible for the sharp bend of the field lines in the center of the current sheet. n Plate 1 we have already seen that in the vicinity of the neutral line the current density has an X-shaped structure and has its maximu m value close to the region of maximum vorticity. This current system is responsible for the more gradual field line bend in the proximity of the X line. At distances larger Lhan z, +60, o the current density jz is concentrated in the center of the current sheet. Figure 7 shows the profile of the electrical current density along z between -96, o < z < 96, o at y - 0. The upper panel exhibits th? contribution je from the electrons to the total current density (solid line), and the lower panel exhibits the contribution ji from the ions to the total current density. The latter is shown in both panels by a dotted line. As can be seen, in an extended region of z, +40:ko the cross tail current is carried by the electrons. The same, although not shown here, is true for the X-shaped current along the vortex layer in that region. Beyond z +60, o the cross-taft current is mainly carried by the cross-taft drifting ions. Although the ion bulk velocity near the center of the current sheet is small, this is sufficien to produce in combination with the large density a high cross-tail ion current density. We will now report on the second run in an enlarged simulation system where we have reduced the number of particles to 30 per lobe cell. This run has been extended to t - 285f2 /. Figure 8 shows profiles various bulk parameters across the reconnection layer at the end of the run at z , o in the same format as Figure 3. From the B and the tolal magnetic field profiles it can be seen that a first step-like change occurs inside of the separatrices (dashed lines) which can be identified with slow mode shocks. The temperature anisotropy Tll/Tñ 0.6 Yl o X= t=285 'r/xo Go , -0., Pl,,,, i. i lr : i at,/t... ; o -Go o Go ¾/% Figure 8. Profiles of various parameters for run 2 in the same format as in Figure 3 at z and at f ait

10 4556 LOTTERMOSER ET AL.. HYBRD SMULATON OF RECONNECTON ;t = >- o.o - - X/Xo a i ' ' Vx Vx Figure 9. (top) Magnetic field at git (bottom) on velocity distributions vy-vx and vz-vx at the three position indicated by the small boxes in the top part. directly heated by the slow mode shocks. n the resulting magnetic field structure the freshly incoming low velocity lobe ions are accelerated and thefinalized. Determination of the curvature parameter in this region of space shows that c_ Thus, after break up of the current sheet the ion motion becomes stochastic [Biichner and Zelenyi, 1989]. n Figure 2 we have already seen that the current sheet in the center of the reconnection layer develops a global instability. n order to examine the spatial and temporal development of the instability we show in Figure 10 on the left-hand side isointensity contours of the cross tail current density jz and on the right hand side vector plots of the ion bulkflow (mean value subtracted) within a region of 24A0 x 81A0 in the y- x plane for different times as indicated on top of each panel. n the early state of the instability the selected region of these plots contains one wavelength (/k _ 50A0). The regions shown move with a speed of v 1.85Va in the +x direction (parallel to the current sheet). Since the wave is approximately standing in this frame, this speed corresponds to the phase velocity in the simulation frame. Thus, the wave propagates approximately with the lobe Alfv n speed along the current sheet and is, in addition convected with the outflowing plasma. As can be seen from Figure 10 the instability first broadens the current sheet and a mode of approximately half the wavelength of the initial one becomes subsequently dominant. The purely transverse distortion in the magnetic field components then enters a nonlinear state and leads to a filamentation of the current sheet. The ion bulkflow follows the same time development with a time delay of several gyroperiods. Therefore the extension of the central current sheet in the reconnection layer away from the X-point is deternfined by two competing processes' a reconnection layer with a Hill-type current sheet develops behind the ejected plasmoids and second the current layer warps and breaks up at larger distances due to the development of an instability. n our simulation the current sheet breaks up at xl > 200Ao and thus cannot be seen in a smaller system. lobe ions and a backstreaming ion beam is seen (Figure 9a). The beam has a larger perpendicular temperature than parallel temperature: the distribution is bean-like. Approaching the reconnection layer further (Figure 9b) the backstreaming beam distribution is seen to extend to lower energies. This change of the distribution when approaching the central current sheet is due to the well-known velocity filter effect: when particles are emitted from a point source in the reconnection layer higher-energy particles are found closer to the separatrices than lower-energy particles. The simple velocity filter effect is modified by the fact that the whole current layer emits the backstreaming particles. n the central current sheet (Figure 9c) the distribution is now isotropic, as would be expected downstream of a slow mode shock. However, in contrast to what has been found by Lin and Swift [1996] these ions are not 4. Discussion We have performed large-scale hybrid simulations of magnetic reconnection in a low beta plasma applicable to tail reconnection. As far as the technique is concerned we found it important that the number of particles per cell is sufficiently large and that the numerical resistivity is low enough. Less than about 30 particles per lobe cell led to spurious plasmoids in the current sheet in the initial state of reconnection. The results can be summarized as follows. 1. Up to about 200f i! the distribution for most part of the center of the reconnection layer is a partial ring. This distribution consists of incoming low velocity lobe ions, cross-tail drifting ions, and ions which are just about to be ejected along the reconnected field lines. The drifting ions carry the cross-tail electric current.

11 T,OTTERMOSER ET AT,.: HYBRD SMUT,ATON OF RECONNECTON 4557.o %t=.o 119 O t= Q = Q.t= O t= D = Dft= D =285 4 ' i - - X/o s7s Figure 10. (left) sointensity contours of the cross-tail current density jz and (right) vector plots of the ion bulk flow (mean value subtracted) in various regions of the simulation system at different times as indicated on top of each panel. The regions move with a constant velocity parallel to the axis, so that the wave in the box is at rest. X/o This current leads to the sharp bend in magnetic field in the center of the reconnection layer, which in turn self-consistently results in the Speiser type drift orbits. Particles ejected along the magnetic field lines constitute backstreaming ion beams. 2. Within about i40 o away from the neutral line the cross tail current has a X-shaped structure and is carried mainly by the electrons. Consequently, the bend in the field is less sharp. The distribution in this region is partial shell like and can be explained by a combination of the drift orbits in the field with a larger curvature and the close proximity of the X point, i.e., the limited extend of the current sheet in the ß direction. 3. The current in the simulation plane (Hall current) leads to a quadrupole like magnetic field component in the invariant direction. The poloidal Hall electrical current flow along the boundary of the high-speed flow layer away from the neutral line and returns within the reconnection layer proper. At larger distances, dispersion of the magnetic signal leads to wavetrains and consequently to a filamentation of the poloidal current. The dispersion leads to a small wavelength wavetrain upstream of the separatrices. 4. The central current sheet exhibits a global instability. This instability leads in its nonlinear stage at larger distances ( o) o ia men ion nd patchy pattern of the cross tail current density. 5. After t ~ 250f i 1 the incoming cold lobe ions are scattered in the fluctuating magnetic field in the distant reconnection layer. Wave-particle interaction of the freshly incoming lobe distribution and earlier drifting and picked up ions leads to a hot plasma sheet distribution. 6. At late times step-like decreases in [ B occur within the separatrices, which is the signature of slow mode shocks. The temperature anisotropy is drastically reduced in the region within these discontinuities. Up- stream of these discontinuities backstreaming ions are observed. The backstreaming ion beam has a larger temperature perpendicular to the magnetic field than parallel. According to these simulations the structure of the tail reconnection layer is for a large distance along the current sheet in the initial phase of reconnection determined by a thin cross tail current layer in which the ions perform Speiser-type orbits. The corresponding partial ting distributions are seen up to t ~ 200f -.1 which corresponds to about 5 min in teal time. First, we remark that this finding is rather different from results of MHD simulations of reconnection. t can of course not be obtained by simple MHD simulations since such nonisotropic distributions require the inclusion of a pressure tensor. Thus the method of reconstructing velocity distribution functions by following test particles in the magnetic and electric field obtained from MHD simulations of reconnection has to be considered with great caution. No slow mode shocks are found during this initial state. This is exactly the collisionless merging configuration envisaged by Hill [1975] wherein the flux of particles into the field reversal region and their sys-

12 4558 LOTTERMOSER ET AL.' HYBRD SMULATON OF RECONNECTON tematic displacement across the current sheet produces exactly the electric current that is required by the field geometry through Ampere's law. Hill has shown that this requirement is equivalent to the statement that the particles streaming from the lobe into the field reversal region must carry enough momentum to exactly balance the stress in the magnetic field in the steady state. This reconnection configuration has first been obtained in two-dimensional magnetoinductive particle simulations of reconnection by Lee and Ding [1987]. According to Hill [1975] is the outgoing plasma bounded by planes that make an angle with the field reversal given by tan = (1/2)tanx, where X is the angle of the field lines with the field reversal. At large distances kom the X point we obtain kom the simulations = 2.6ø; the angle between the separatfices and the field reversal region is X - 5.0ø. The situation is rather different in the vicinity of the X point. Within a distance given by one gyroradius in the normal field By lobe particles which have entered the current sheet have not yet performed thek full Speiser orbits and are not yet ejected along the magnetic field. These ions do not yet contribute to the cross tail electric current. We find that in this region the electron current determines the magnetic field configuration. As time progresses the thin current sheet further away from the X point becomes unstable: the current sheet warps and disrupts at nany places. Warping and instabilities of current sheets has been seen before in particle simulations of thin current sheets. Stoif-t [1992] has reported the warping of a current sheet when the system is driven by an external convection electric field, but offers no explanation. Furthermore, in his case the wavelength is of the same order of the numericurrence of slow mode shocks at the boundary of the reconnection layer. The post-plasmoid plasma sheet in the more distant tail consists then of a hot ion distribution in the central region and backstreaming beams cal box ( 6 ion inertial lengths) and is considerably upstream of slow lnode shocks. smaller than the wavelength observed in the present case. Pritchett and Coroniti [1992] attribute the instability in their simulation of a one-dimensional current The present silnulations are limited in several respects. First, ions observed at a certain position in the current sheet have been drifting across the sheet sheet to an anisotropic pressure distribution produced and come kom various cross-tail distances. This does by chaotic particle orbits in the field reversal region. The kinetic kink instability has been investigated by not matter in a two-dimensional simulation; however, in a 3-D situation certain drift orbits are not possi- means of full particle simulations by, e.g., Winske [1981] and by Zhu and Winglee [1996]. n this case the instability was driven by the diamagnetic current in a planar current sheet with wave vectors in the plane perpendicular to the magnetic field containing the current vector. n the reconnection layer with a thin central current sheet as found in our simulations there are basically two reservoirs of free energy. Lobe ions stream into the current sheet and ions ejected kom the current sheet constitute two penetrating cold beams with a velocity difference of about twice the lobe Alfv n velocity (equivalent to a plasma with a large parallel to perpendicular temperature anisotropy) and an electromagnetic nonresonant ion/ion beam instability (firehose-type instability) is possible. Another source of kee energy is the drift of the ions in the current sheet determined by the Hall magnetic field directed in the invariant direction. This drift is within the simulation plane and may lead to a kink-type instability. t is difficult to verify whether the instability seen by Swift [1992] is of the same type as the instability in the present simulation. The con- siderably smaller wavelength seen in the Swift [1992] simulation may reflect the small system size. n the silnulations of Swift [1992] the instability of the current sheet was ultilnately due to convection of cold particles koln the lobe toward a current sheet. Convection was imposed by an externally applied electric field. This situation is similar to the present reconnection si nula- tion: the self-consistently generated electric field in the reconnection simulation drives lobe ions into the current sheet. Based on these simulations we propose the following scenario. When near-earth reconnection has proceeded to lobe field lines a thin reconnection layer develops, which can extend several tens of Earth radii behind the departing plasmoid. The whole post-plasmoid plasma sheet has the character of a boundary layer with a thin central current sheet. At this stage no slow mode shocks are yet developed and the mode of reconnection closely resembles the Hill [1975] model for a low beta plasma. We note that in the near-earth tail, contrary to the distant tail, the search for slow mode shocks was unsuccessful [Cattell et al., 1995]. The thin current sheet is subject to a global instability. At about 200 ion inertial length kom the neutral line ( 30RE) the instability has reached a nonlinear state; the incoming and ejected cold lobe ions are isotropized and constitute a hot plasma sheet distribution. This supports the oc- ble. This will lead to significant dawn-dusk asymmetries of the initial reconnection layer. Second, we have not taken into account a realistic initial 2-D equilibrium of the plasma sheet. A 2-D equihbrium with dipole-like field lines at the earthward side will probably change the configuration of the reconnection layer earthward of the neutral line. Kuznetsova et al. [1996] have recently studied reconnection in a 2-D tail equilibrium by hybrid simulations. However, these authors were more concerned with the fate of the accelerated particles of the original plasma sheet. n any case, the present simulations have shown that it is important to perform large-scale kinetic simulations of reconnection, not only in order to infer the ion distribution functions, but also because the possibility of non. isotropic distributions (or off-diagonal terms of the pressure tensor) allows differ- ent modes of reconnection.

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