Vlasov simulations of electron holes driven by particle distributions from PIC reconnection simulations with a guide field

GEOPHYSICAL RESEARCH LETTERS, VOL. 35, L22109, doi:10.1029/2008gl035608, 2008 Vlasov simulations of electron holes driven by particle distributions from PIC reconnection simulations with a guide field Martin V. Goldman, 1 David L. Newman, 1 and Philip Pritchett 2 Received 6 August 2008; revised 7 October 2008; accepted 21 October 2008; published 27 November 2008. [1] Parallel velocity particle distributions taken from 2-D PIC simulations of magnetic reconnection with a guide field, B g are used to initialize 1-D and 2-D electrostatic Vlasov simulations which include the direction parallel to the (local) magnetic field, B. Electron holes develop near the separatrix from an electron-ion (e.g., Buneman) instability. Restriction of the destabilizing current to a narrow sheet perpendicular to B reduces the Buneman growth rate but leads to more stable holes. Near the x-point, B is almost parallel to B g. Here, electron-electron kinetic two-stream instabilities lead to holes moving parallel to B, which can modify the electron-ion interaction. This results in a second generation of (slower) electron phase space perturbations which can reduce the current. Citation: Goldman, M. V., D. L. Newman, and P. Pritchett (2008), Vlasov simulations of electron holes driven by particle distributions from PIC reconnection simulations with a guide field, Geophys. Res. Lett., 35, L22109, doi:10.1029/2008gl035608. 1. Introduction [2] Bipolar electric fields interpreted as electron phase space holes have been observed in both the magnetopause [Cattell et al., 2002] and the magnetotail [Cattell et al., 2005] during magnetic reconnection. The holes may affect the reconnection process or act as a diagnostic for reconnection events. Drake et al. [2003] showed in 3-D PIC reconnection simulations with a strong guide field that electron holes can be driven parallel to B g near the x-line by electrostatic Buneman (e-i) instability. The holes were invoked as a source of the electron drag required to expedite fast reconnection by facilitating transfer of electron momentum to ions. The initial conditions in these simulations included a (non-diamagnetic) electron drift which was already above the threshold for a modified Buneman instability. Preliminary 2D and 3D PIC simulations by Che et al. [2007] confirm the role of Buneman instability and turbulence near the x-line in expediting magnetic reconnection for low initial electron temperatures. [3] Starting from a Harris equilibrium, Cattell et al. [2005] found evidence for in-plane electron holes near the separatrix in 2-D PIC reconnection simulations with a weak guide field and suggested that parallel particle distributions should be unstable to Buneman instability. Three-dimensional particle-in-cell (PIC) simulations of magnetic reconnection 1 Department of Physics, University of Colorado, Boulder, Colorado, USA. 2 Department of Physics and Astronomy, University of California, Los Angeles, California, USA. Copyright 2008 by the American Geophysical Union. 0094-8276/08/2008GL035608 can encounter difficulties in detecting electron phase space holes in the direction parallel to B. This is partly due to the high noise level in PIC simulations in 3D and partly to the large disparity between the physical time scales for electromagnetic and electrostatic processes which must be spanned in such simulations. Starting from a Harris equilibrium with a guide field, PIC reconnection simulations have suggested the possible presence of Buneman waves, but not holes. Pritchett and Coroniti [2004] (in 3-D), and, to a greater degree, Pritchett [2005] (in 2-D, at higher resolution) have found waves at frequencies on the order of the local ion plasma frequency, consistent with Buneman instability. Neither has claimed to identify the isolated bipolar field structure which accompanies electron phase space holes. Both have presented particle distributions in the parallel direction which are described as good candidates for electrostatic instabilities. [4] This Letter addresses these issues by using high resolution electrostatic Vlasov simulations to study the formation of holes driven by electrostatically unstable particle distributions found from 2D PIC guide-field reconnection studies [Goldman et al., 2007a, 2007b]. The initial conditions in the Vlasov simulations employ particle distributions which are functions of the velocity parallel to B, based on those measured in Pritchett s 2-D PIC reconnection simulations. [5] The Vlasov simulations confirm that on the island separatrix strong Buneman instability occurs parallel to B and leads to electron phase space holes. The width of the narrow current sheet near the island separatrix is shown to influence both the Buneman instability linear growth rate and the formation of holes. [6] The Vlasov simulations also show that electron holes form near the x-line (moving parallel to the x-line and B g ) as a result of a weak (kinetic) electron two-stream (e-e) instability. This process had been conjectured [Pritchett and Coroniti, 2004; Pritchett, 2005] but not demonstrated until this Letter. Later, a weak Buneman (e-i) instability also comes into play and reduces the current. [7] In both the PIC and Vlasov simulations the ion-toelectron mass ratio is 64. In the PIC simulations the guide field, B g, is equal to the asymptotic reversing field, B 0,the ratio of the speed of light to the Alfven speed is c/v A =51 and the ratio of initial electron plasma frequency to electron cyclotron frequency based on B 0 is w e0 /W e0 = 6.4. The density in the Harris equilibrium is n(z) = n 0 [0.1 + sech 2 (2z/d i )], which includes a background density, 0.1 n 0. [8] The results of the 2-D PIC reconnection simulation, at a time typical of the reconnection stage of the simulation are shown in Figure 1. Figure 1a displays in-plane magnetic field lines superimposed on the contours of electron density. Two white rectangles are chosen where the rectangle- L22109 1of5

Figure 1. PIC x z plane reconnection results. (a) Variation of electron density and magnetic field lines in x z plane; (b) 3D orientation of B near separatrix; (c) parallel electron drift (v A units) in x z plane; (d) electron (blue) and ion (red) parallel velocity distributions averaged over solid white rectangle in Figures 1a and 1c; (e) same for dashed rectangle. averaged parallel velocity distributions turn out to be unstable. The dashed-line rectangle is at the separatrix and the solid one is near the x-line, in the reconnection region. In the separatrix rectangle, B has components both in-plane and out-of-plane, as shown in Figure 1b, whereas in the rectangle around the x-line the dominant component of B is the guide field, B g. Figure 1c shows the variation with inplane location of the average parallel electron drift, hv k i, (in units of the initial Alfven speed). It is evident that the rectangles contain some of the highest hv k i values. 2. Vlasov Simulations of Hole Formation Near the Separatrix [9] First consider the Buneman unstable electron and ion parallel velocity distributions from the PIC simulation averaged over the dashed white box at the separatrix (Figure 1). These are distinctly separated peaks. We have introduced smoothed distributions modeled after the distributions on the separatrix as initial conditions in 2-D electrostatic Vlasov simulations. In the Vlasov simulations z is parallel to B. The y-direction is in the plane perpendicular to B, across the separatrix. The parallel drift velocity from the PIC simulation, hv k i, falls off steeply across the separatrix, as can be seen in Figure 1c. Hence, we have restricted the separatrix current sheet thickness to a finite channel in the y-direction, of half-width, dy. This current channel is taken to have a spatially-gaussian y-profile in the parallel (z) drift velocity, with spatially uniform density. [10] Vlasov simulations with particle orbits along B have been carried out for various thicknesses of the current sheet (a variety of values of dy). The results are seen in Figures 2a 2d, where the parallel electrostatic field is plotted versus z and y, at two times, for current sheet thicknesses, dy = 0.26 d i0 and dy = 0.17 d i0, where d i0 is the initial ion inertial length, c/w i0, based on the initial ion density in the ion plasma frequency, w i0. Note that there are two different scales for y. The scaling y/d i0 is displayed on the right ordinate of each frame, while a scaling, y/l e with the local Debye length, l e, based on the local density of 0.02 n 0 at the indicated times, is shown on the left ordinate of each frame. During the linear phase of the Buneman instability the unstable wave is an eigenfunction localized to around the current sheet thickness. In the nonlinear phase, the parallel electrostatic field becomes a set of localized bipolar fields, corresponding to electron phase space holes. It is evident from Figure 2d that the bipolar fields are more clearly formed and the holes more coherent for the narrower current sheet. Parallel electron phase space, (z, v z ) is shown, at y = 45 l e in Figures 2e and 2f. At the earlier (linear) time the dashed line at v 0.3 v A corresponds to the phase speed, v f, of a cold-electron linearly unstable Buneman wave. At the later, nonlinear time, phase space electron holes on the tail of the electron distribution have spread in Dv z until they trap the higher velocity electrons in the main-body distribution. The dashed line in Figure 2f shows that the mean hole velocity (v 3.0 v A ) is an order of magnitude larger than the phase velocity of the linear Buneman wave in Figure 2e. In general, both the linear wave speed and the hole speed will depend on the channel size, the degree of magnetization and the mass ratio. [11] In Figure 2g, growth rates and saturation levels are displayed for a sequence of values of dy, as well as for the uniform current sheet case (1-D run in z). The narrower current sheets yield lower linear Buneman growth rates. This is consistent with earlier studies showing that shear in the current lowers the Buneman growth rates. [Goldman et al., 2007a, 2007b; Serizawa and Dum, 1992] The peak bipolar field after saturation of the linear instability for the uniform current is comparable in magnitude to that for the current sheet of width dy = 0.26 d i0, even though the narrower current sheet yields a lower growth rate. [12] In terms of the ion plasma frequency, w i, based on the local density, 0.02 n 0, the real frequencies and growth 2of5

Figure 2. Linear and nonlinear time snapshots of E k (y, z), zkb: (a and b) for dy = 0.26 d i0 ; (c and d) for dy = 0.17 d i0. (e and f) Parallel (v z, z) phase space at y = 45 l e.. Dashed lines show phase velocity of linear Buneman-unstable wave and speed of nonlinear holes, respectively. (g) Time dependence of logje max j 2 and linear growth rates in 1-D (1st curve on left) and for current sheet half-width, dy = 0.26 d i0,0.17d i0,0.13d i0 (d i0 = initial ion inertial length). rates of the fastest growing Buneman mode are expressed as follows: For current sheets with dy = 0.17 d i0 and dy = 0.26 d i, respectively, the real frequencies and growth rates are [w r = 0.47 w i, g = 0.18 w i ] and [w r = 0.55 w i, g = 0.35 w i ]. The frequencies are somewhat lower than the frequencies of E k oscillations near the separatrix stated in Pritchett s 2D simulations. The growth rates should be compared with Buneman growth rates in the absence of shear (i.e., for a parallel drift uniform in y), which is g =1.1w i. [13] In the separatrix region, B, has components both in the x z plane and in the out-of plane, y, direction. Hence the parallel distributions have projections onto the x z plane which can drive Buneman instability and holes in that plane. Thus, holes driven by Buneman instability can occur even in 2-D simulations in the separatrix region of the reconnection plane [Cattell et al., 2005; Pritchett, 2005]. 3. Vlasov Simulations of Hole Formation Near the X-Line (Reconnection Region) [14] Next, we have studied the stability of the particle velocity distributions parallel to B averaged within the solid white rectangle in the diffusion region, near the x-line in Figure 1. Since the B-field in this rectangle is in the diffusion region, it is essentially parallel to B g. Onedimensional Vlasov simulations with initial distributions based on those PIC distributions have been carried out, as shown in Figure 3a. The distributions are normalized to the density in this region, which is taken as 0.1 n 0. [15] The double-humped green electron distribution in Figure 3a is a fit to the blue curve in Figure 1d. In the first case studied, the solid red ion distribution in Figure 3a is a fit to the red curve in Figure 1d (on the tail of the lower electron hump). The simulations reveal that this distribution is weakly (i.e., kinetically) e-e unstable, and leads to the holes shown in parallel electron phase space in Figure 3b. These holes move parallel to B g and the x-line at speed v hole 8.3 v A, roughly equal to that of the valley in the combined double-humped electron distribution in Figure 3a. This is on the order of the thermal speed of the combined electron distribution. The holes have a spatial half-width of two or three local Debye lengths. An interesting ripple can be seen at speed v 2.6v A in phase space in Figure 3b. The ripple is interpreted as a secondary kinetic e-i disturbance which is nonlinearly destabilized [Schamel and Luque, 2005] after the appearance of electron holes arising from the e-e instability. [16] In order to study this second-stage Buneman instability under more general conditions, we have carried out further simulations in which the ion velocity distribution is moved further away from the mean electron velocity. We model the ions by an initial ion distribution corresponding to the first dashed curve to the right of the solid ion curve in 3of5

Figure 3. (a) Parallel particle distributions from PIC simulation modeled by solid green line for electrons and solid red line for ions. (b) Holes created when these distributions are introduced initially in Vlasov simulation. (c and d) Holes created when solid red line ion distribution is replaced by dashed red distributions at higher speeds. (e) Average electron current versus time in rectangle near x-line; b 0, for PIC electron and ion distributions; c 0 and d 0, ions displaced to higher velocities, corresponding to Figures 3b, 3c, and 3d. Figure 3a. Once again the e-e instability leads to holes, but now the e-i interaction is close enough to the Buneman instability threshold to produce slow holes which interact with the faster e-e holes and can coalesce with them (Figure 3c). When the initial ion distribution is moved further to the right (second dashed curve to right of the first) the Buneman instability is slightly above the linear threshold, but nonlinearly enhanced. As a result, the two instabilities produce holes which quickly merge into the large holes seen in Figure 3d. Thus, there is competition between e-e and e-i instabilities in the production of holes which is sensitive to the details of the parallel electron and ion distributions. [17] Figure 3e shows that the e-i instabilities and holes they produce can reduce the average electron current, whereas the e-e instabilities cannot. The top curve represents the evolution in time of the average electron current for the actual electron and ion distribution. There is no change in the current as a result of the e-e instability and the holes it produces. It is not until the second-stage e-i instability cuts in that the current begins to fall gradually. The reduction in the electron current is most likely due to momentum transfer from electrons to ions, first cited by Drake et al. [2003] as a mechanism through which Buneman instability and holes can increase the reconnection rate. Although it is not within the scope of this letter to determine whether or not e-i electrostatic instabilities and holes affect the reconnection rate, we have shown here that they can reduce the electron current. The two lower curves, corresponding to the displaced initial ion distributions in Figure 3a confirm that earlier Buneman instability implies earlier electron current reduction and that the stronger the instability the more the reduction. In all three cases the current continues to decrease slowly until the end of the run. 4. Conclusions [18] A better theoretical understanding of the origins of electron phase space holes detected during magnetic reconnection in space and laboratory plasmas is important for a variety of reasons. Holes might play a role in the reconnection process itself and they also can be used as a remote proxy for detecting and analyzing reconnection events. Most explanations for holes invoke linear instabilities in which a growing electrostatic wave traps electrons. In the presence of a guide field, B g, it can be difficult using 3D PIC reconnection simulations alone to resolve linear electrostatic instabilities and the ensuing electron phase space holes in the direction parallel to B. In this letter, electron and ion distributions from 2-D PIC reconnection simulations with B g =B 0 have been introduced as initial conditions in 1D and 2D higher-resolution Vlasov simulations which include the B-field direction. Near the separatrix, electrostatic instabilities leading to phase space holes parallel to a B-field with components both in and out of the reconnection plane have been found due to an electron-ion (e-i) Buneman instability. This Letter demonstrates that holes can also be formed in the diffusion region by e-e instability (parallel to B g ). In this region a subsequent e-i instability is found, indicating a new mode of nonlinear electrostatic evolution not previously reported in the diffusion region. Among other new results, the finite width of the parallel electron current sheet near the separatrix can reduce the growth rate of e-i instability but appears to improve the coherence of the holes. Near the x-line the parallel electron current is not 4of5

reduced during e-e instability and hole formation. However, it is reduced due to momentum transfer from electrons to ions after subsequent nonlinear e-i perturbations appear at kinetic Buneman wave phase velocities. [19] Details of the relation between electron holes (i.e., speeds, widths, maximum field, etc) and linear electrostatic instabilities will depend on many parameters. Among them are the ion-to-electron mass ratio, the guide field strength, the degree of particle magnetization and the current sheet ( channel ) thickness. These will be addressed in future work. [20] Acknowledgments. We acknowledge important conversations with J. Drake, W. Wan and G. Lapenta. This work was supported by NSF, DOE and NASA. Numerical simulations were performed at the National Center for Atmospheric Research (NCAR). References Cattell, C., J. Crumley, J. Dombeck, J. Wygant, and F. S. Mozer (2002), Polar observations of solitary waves at the Earth s magnetopause, Geophys. Res. Lett., 29(5), 1065, doi:10.1029/2001gl014046. Cattell, C., et al. (2005), Cluster observations of electron holes in association with magnetotail reconnection and comparison to simulations, J. Geophys. Res., 110, A01211, doi:10.1029/2004ja010519. Che, H., J. Drake, and M. Swisdak (2007), Ohm s Law in 3D turbulent magnetic reconnection, paper presented at the 49th Annual Meeting of the Division of Plasma Physics, Am. Phys. Soc., Orlando, Fla. Drake, J. F., M. Swisdak, C. Cattell, M. A. Shay, B. N. Rogers, and A. Zeiler (2003), Formation of electron holes and particle energization during magnetic reconnection, Science, 299, 873 877. Goldman, M. V., D. L. Newman, and P. Pritchett (2007a), Electron distributions from 2D reconnection PIC fed into Vlasov simulations to study hole formation in out-of-plane (guide field) direction, paper presented at the 2007 Cambridge Workshop on Magnetic Reconnection, Isaac Newton Inst., St. Michaels, Md., 10 14 Sept. Goldman, M. V., D. L. Newman, and P. Pritchett (2007b), Are electron distributions associated with reconnection electrostatically unstable and do they lead to electron holes?, Eos Trans. AGU, 88(52), Fall Meet. Suppl., Abstract SM53B-1280. Pritchett, P. L. (2005), Onset and saturation of guide-field magnetic reconnection, Phys. Plasmas, 12, 062301, doi:10.1063/1.1914309. Pritchett, P. L., and F. V. Coroniti (2004), Three-dimensional collisionless magnetic reconnection in the presence of a guide field, J. Geophys. Res., 109, A01220, doi:10.1029/2003ja009999. Schamel, H., and A. Luque (2005), Non-linear growth of trapped particle modes in linearly stable, current-carrying plasmas A fundamental process in plasma turbulence and anomalous transport, Space Sci. Rev., 121, 313 331, doi:10.1007/s11214-006-5382-8. Serizawa, Y., and C. T. Dum (1992), Nonlocal analysis of finite-beamdriven instabilities, Phys. Fluids B, 4, 2389 2401. M. V. Goldman and D. L. Newman, Department of Physics, University of Colorado, UCB 390, Boulder, CO 80309, USA. (goldman@colorado.edu) P. Pritchett, Department of Physics and Astronomy, University of California, 405 Hilgard Avenue, Los Angeles, CA 90024, USA. 5of5