physical processes of driven magnetic reconnection in collisionless plasmas zero guide field case

Physical processes of driven magnetic reconnection in collisionless plasmas: Zero guide field case C Z Cheng, S Inoue, Y Ono, and R Horiuchi Citation: Phys Plasmas 22, 101205 (2015); doi: 10.1063/1.4932337 View online: http://dx.doi.org/10.1063/1.4932337 View Table of Contents: http://aip.scitation.org/toc/php/22/10 Published by the American Institute of Physics PHYSICS OF PLASMAS 22, 101205 (2015) Physical processes of driven magnetic reconnection in collisionless plasmas: Zero guide field case C Z Cheng,1,2 S Inoue,1 Y Ono,1 and R Horiuchi3 Graduate School of Frontier Sciences, University of Tokyo, Tokyo, Japan Institute of Space and Plasma Sciences, National Cheng Kung University, Tainan, Taiwan National Institute for Fusion Science, Toki, Japan (Received 29 January 2015; accepted 22 June 2015; published online October 2015; publisher error corrected October 2015) The key physical processes of the electron and ion dynamics, the structure of the electric and magnetic fields, and how particles gain energy in the driven magnetic reconnection in collisionless plasmas for the zero guide field case are presented The key kinetic physics is the decoupling of electron and ion dynamics around the magnetic reconnection region, where the magnetic field is reversed and the electron and ion orbits are meandering, and around the separatrix region, where electrons move mainly along the field line and ions move mainly across the field line The decoupling of the electron and ion dynamics causes charge separation to produce a pair of in-plane ~es ) pointing toward the neutral sheet in the magnetic bipolar converging electrostatic electric field (E ~ field reversal region and the monopolar E es around the separatrix region A pair of electron jets emanating from the reconnection current layer generate the quadrupole out-of-plane magnetic field, ~ind to accelerate particles along the magnetic ~jj ) from E which causes the parallel electric field (E field We explain the electron and ion dynamics and their velocity distributions and flow structures during the time-dependent driven reconnection as they move from the upstream to the downstream In particular, we address the following key physics issues: (1) the decoupling of electron and ion dynamics due to meandering orbits around the field reversal region and the generation of a pair of ~es ) around the reconnection region; (2) the slowconverging bipolar electrostatic electric field (E down of electron and ion inflow velocities due to acceleration/deceleration of electrons and ions by ~es as they move across the neutral sheet; (3) how the reconnection current layer is enhanced and E ~ind ; (4) why how the orbit meandering particles are accelerated inside the reconnection region by E the electron outflow velocity from the reconnection region reaches super-Alfvenic speed and the ion outflow velocity reaches Alfvenic speed; (5) how the quadrupole magnetic field is produced ~jj around the separatrix ~jj is produced; (6) how electrons and ions are accelerated by E and how E region; (7) why electrons have a flat-top parallel velocity distribution in the upstream just outside the reconnection region as observed in the magnetotail; (8) how electron and ion dynamics decouple and how the monopolar electrostatic electric field is produced around the separatrix region; (9) how ions gain energy as they move across the separatrix region into the downstream and how the ion velocity distribution is thermalized in the far downstream; and (10) how electrons move across the separatrix region and in the downstream and how the electron velocity distribution is thermalized in the far downstream Finally, the main energy source for driving magnetic reconnection and particle acceleration/heating is the inductive electric field, which accelerates both electrons and C 2015 Author(s) All article ions around the reconnection current layer and separatrix regions V content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License [http://dx.doi.org/10.1063/1.4932337] I INTRODUCTION Magnetic reconnection plays the crucial role of changing the magnetic field topology and converting the electric and magnetic (EM) field energy into the plasma energy Magnetic reconnection has been observed in laboratory experiments1–5 and space observations.6–14 The magnetic reconnection problem has been extensively studied in the last 40 years using the single-fluid magneto-hydrodynamic (MHD) model, the Hall-MHD model, the two-fluid model, and the full kinetic particle-in-cell (PIC) simulation model.15–28 Based on the single fluid MHD model in the two-dimensional merging magnetic field, there are two well1070-664X/2015/22(10)/101205/21 known steady-state models of magnetic reconnection: the Sweet-Parker model29,30 and the Petschek model.31 In the Sweet-Parker model, both electrons and ions flow together, and they flow from the upstream region through the magnetic reconnection diffusion layer into the downstream region so that the outflow velocity speeds up to the Alfven speed (defined with the upstream magnetic field and plasma density) by the reconnection electric field The reconnection rate, which is computed from the ratio of the inflow velocity to the Alfven speed, is proportional to the square root of the plasma resistivity and is too low to account for the observations if only classical resistivity is considered To overcome the slow reconnection rate problem, the Petschek model 22, 101205-1 C Author(s) 2015 V 101205-2 Cheng et al proposes that the plasma (both electrons and ions together) does not need to flow through the dissipative current sheet region, instead the plasma flows across the separatrix field line into the downstream and the plasma outflow velocity is ~ force (J~ is the plasma current denaccelerated by the J~  B ~ sity and B is the magnetic field) at the slow mode shock in the downstream region Then, the magnetic reconnection rate (or the plasma inflow velocity) can be enhanced to realistic value Later, using the Hall-MHD model, the electron flow decouples from the ion flow, and the electron flow velocity perpendicular to the magnetic field is governed by the ~ is the total electric field) The ~  B=B ~ drift velocity (E cE ~ and pressure graion flow velocity is controlled by the J~  B dient forces However, the charge quasi-neutrality is assumed, the plasma pressure is assumed to be isotropic and obey the adiabatic pressure law, and there is no separate information on the electron and ion parallel flow velocities along the magnetic field The main feature of the Hall-MHD model is the generation of the quadrupole out-of-plane magnetic field because the electron perpendicular outflow velocity is much larger than the ion perpendicular outflow velocity in the downstream region In this paper, we show that many crucial kinetic features of magnetic reconnection physics based on kinetic simulations employing the PIC are missing in the fluid models Moreover, the main physics results of magnetic reconnection based on the fluid models are considerably different from the results based on the kinetic model as described in this paper However, even within kinetic simulation studies, there are also major differences Most kinetic simulations were carried out by perturbing the initially imposed current sheet to examine the nonlinearly relaxed states, and the simulation system is treated to be periodic in the downstream direction However, magnetic reconnection in most laboratory experiments and space plasma phenomena occurs due to merging of differently directed magnetic field lines together with plasma inflow caused by the driving inductive electric field in the upstream region, and in the downstream region the plasma and electric and magnetic fields freely go out to interact with the external plasma and magnetic field The state of driven magnetic reconnection can be significantly different from the relaxed state of magnetic reconnection due to large perturbation to the initially imposed current sheet There are several major differences between the driven reconnection physics and the results of reconnection caused by initially imposed large perturbations, which will be presented in the future Up to now, only limited simulations of magnetic field merging and reconnection caused by the driving inductive electric field have been carried out because of the difficulties in imposing the frozen-in condition for both electrons and ions and supplying particles at the driving upstream boundary We emphasize that the results presented in this paper were obtained from kinetic simulations of collisionless driven magnetic reconnection The particle dynamics and the development of electric and magnetic fields during driven magnetic reconnection are highly nonlinear Because the driven reconnection physical processes are quite complicated, we first present an overview of the physical pictures of key driven reconnection processes Phys Plasmas 22, 101205 (2015) in the Introduction The detailed explanations of the key physical processes are then presented in Sections III–IX When oppositely directed magnetic fields are driven by the ~ind  B=B ~ drift veloc~ind ) with cE inductive electric field (E ity, both the electrons and ions are magnetized and they convect together with the magnetic field toward the neutral sheet Around the neutral sheet region, where oppositely directed magnetic field lines merge, the particle motion becomes meandering because the magnetic field is weakened and reversed across the neutral sheet.23,32 The scale width of the orbit meandering region is determined by the particle gyroradius when it equals to the magnetic field gradient scale length Thus, the width of the ion orbit meandering region is larger than the electron meandering width by ðTi mi =Te me Þ1=4 for thermal particles,24 where Ti, Te, mi, and me are the temperature and mass of ions and electrons, respectively As the magnetized electrons and ions drift toward the neutral sheet, their drift velocity increases because of the weakening magnetic field but then decreases as the particles enter the orbit meandering region The electron inflow velocity can reach the Alfven speed when they flow to the boundary of the electron orbit meandering region Inside the orbit meandering region, the particle density accumulates and is roughly uniform Thus, the electron and ion dynamics decouple inside the ion orbit meandering region, and the ion density is larger (smaller) than the electron density outside (inside) the electron orbit meandering region The charge separation inside the ion orbit meandering region produces a pair of converg~es on both sides of the ing bipolar electrostatic electric field E neutral sheet and they point toward the neutral sheet in the ~es is mainly perpendicular to the reconnection plane.6,33 E ambient magnetic field because the fast moving electrons can smear out the charge separation along the field line Note ~ind Outside the electron ~es is also perpendicular to E that E orbit meandering region, the electrons are still magnetized ~es produces strong and the in-plane bipolar converging E ~ ~ cE es  B=B drift velocity to enhance the electron current layer Inside the electron orbit meandering region, the electrons are unmagnetized and their inflow velocity is deceler~es , but the electrons are accelerated by the ated by E ~ind Inside the ion orbit meandering inductive electric field E region, the ion orbit size is comparable to or larger than the ~es spatial localization width and thus the ions are accelerE ~es alternatively on both sides of the ated and decelerated by E neutral sheet depending on their velocity direction with ~es direction In the meantime, ions are accelrespect to the E ~ind Therefore, the pair erated by the inductive electric field E ~ind control ~ of E es together with the inductive electric field E the particle dynamics and the physical processes of plasma flows and the acceleration and heating of particles around the reconnection region Electrons flow from the upstream region mainly through the magnetic field reconnection region into the downstream region Although some ions flow through the reconnection current layer into the downstream, most ions flow across the field line separatrix into the downstream The electron outflow velocity from the reconnection region dominates over the ion outflow velocity in the downstream exhaust region, and thus a pair of currents flow inward toward the 101205-3 Cheng et al reconnection region in the reconnection plane and generate ~z ) in the direction perpenthe quadrupole magnetic field (B dicular to the reconnection plane The quadrupole magnetic field concentrates mainly around the separatrix region and ~ind •B ~z =B2 ÞB), ~ ~jj ¼ ðE causes finite parallel electric field (E which can accelerate particles along the magnetic field line ~jj points mainly outward from the reconnection Because E region toward the downstream direction, around the separa~jj to flow along the trix region, electrons are accelerated by E field lines toward the reconnection region It is to be noted that our physical pictures of how the quadrupole Bz magnetic ~jj field is generated and how electrons are accelerated by E around the separatrix region are different from the one presented by Uzdensky and Kulsrud,34 which is based on changes in the flux tube volume to explain the electron parallel flows near the separatrix Our physical picture of how electrons are accelerated toward the reconnection region along the field line around the separatrix region is also different from the electron surfing mechanism by Hoshino,18 ~jj effect which did not consider the E Around the separatrix region, electrons flow very fast mainly along the field line and ions flow mainly across the field line The decoupling of the electron and ion dynamics around the separatrix region produces the electrostatic elec~es pointing toward the downstream midplane tric field E direction Because the ion gyroradii are comparable to or larger than the spatial localization width of the perpendicular ~es , ions can be accelerated/decelelectrostatic electric field E ~ erated by E es depending on the gyrating ion velocity direc~es Thus, how ions move in the tion with respect to E ~ localized E es around the separatrix region is critical to the physical processes of ion flow and acceleration and heating The large gyro-orbit ions are also accelerated by the parallel ~jj around the separatrix region electric field E Because the physical processes of the collisionless driven magnetic reconnection described above involve multiple-scale interactions between the particles and the electric and magnetic fields, the physical mechanisms of how the particle dynamics and the electric and magnetic fields evolve during magnetic reconnection are still not fully thought out even after more than several decades of research Most interpretations of the PIC simulation results were made by employing the two-fluid model By calculating the macroscopic quantities such as the particle flow velocities and pressure tensors from the particle velocity distributions obtained from the simulations, the momentum equation of each particle species is analyzed to provide interpretation of the particle flow structure However, it is quite difficult to understand the physical mechanisms of magnetic reconnection from such analysis because how the pressure tensor is related to the particle dynamics is hard to understand intuitively However, it is possible, albeit difficult, to understand the physical mechanism of magnetic reconnection processes directly from the particle dynamics and the temporal and spatial development of the particle velocity distribution functions obtained from the kinetic simulations In this paper, we provide interpretation of the key dynamical processes of driven magnetic reconnection in collisionless plasmas from the first principle of particle Phys Plasmas 22, 101205 (2015) dynamics We will present highlights and explanations of PIC simulation results of the temporal and spatial development of the electric and magnetic fields and the particle distribution functions when the reconnection rate is quasisteady In particular, we address the following key physics issues: (1) the decoupling of electron and ion dynamics due to meandering orbits around the field reversal region and the generation of a pair of converging bipolar electrostatic elec~es ðx; y; tÞ) around the reconnection region; (2) the tric field (E slowdown of electron and ion inflow velocities due to accel~es ðx; y; tÞ as eration/deceleration of electrons and ions by E they move across the neutral sheet and the features of electron and ion velocity distributions in the current layer; (3) how the reconnection current layer is enhanced and how the orbit meandering particles are accelerated inside the recon~ind ; (4) why the electron outflow velocity nection region by E from the reconnection region reaches super-Alfvenic speed and the ion outflow velocity reaches Alfvenic speed; (5) how the quadrupole magnetic field (Bz ðx; y; tÞ) is produced and ~jj ðx; y; tÞ) is produced; (6) how the parallel electric field (E ~jj around the sephow electrons and ions are accelerated by E aratrix region; (7) why electrons have a flat-top parallel velocity (vjj) distribution in the upstream just outside the reconnection region as observed in the magnetotail; (8) how electron and ion dynamics decouple around the separatrix region and how the monopolar electrostatic electric field is produced; (9) how ions gain energy as they move across the separatrix region into the downstream and how the ion velocity distribution is thermalized in the far downstream; and (10) how electrons move across the separatrix region and flow in the downstream and how the electron velocity distribution is thermalized in the far downstream We shall examine these questions based on the first principle of particle dynamics under the influence of the inductive electric field ~ind ðx; y; tÞ and the electrostatic electric field E ~es ðx; y; tÞ in E ~ ~P ðx; y; tÞ the 3D reconnection magnetic field (Bx; y; tị ẳ B ~z ðx; y; tÞ) We will show the particle velocity distribution þB functions in the key regions of the reconnection plane and explain the main features of the particle velocity distributions from the first principle of particle dynamics We emphasize that several key physical processes and their interpretations presented in this paper have not been presented before In the following, we describe the 2-1/2D PIC simulation model of driven magnetic reconnection and the simulation parameters in Section II In Section III, we discuss the particle motions and meandering orbits under the influence of the inductive and electrostatic electric fields in the 2-1/2D magnetic reconnection geometry In Section IV, we discuss the mechanisms of charge separation due to decoupling of electron and ion dynamics and the generation of electrostatic electric field and describe the electron and ion flow structures In Section V, we discuss the orbit dynamics of electrons and ions under the influence of the inductive and electrostatic electric fields in the inflow region around the reconnection current layer and provide interpretation of different electron and ion flow velocity structures based on the 101205-4 Cheng et al electron and ion velocity distributions In Section VI, we first describe the electron and ion outflow dynamics and structures from the reconnection current layer to the reconnection exhaust region and discuss the generation mechanism of quadrupole magnetic field and the field-aligned electric field Then, we explain the electron and ion outflow dynamics and structures based on the particle dynamics and velocity distributions In Section VII, we describe the electron dynamics around the separatrix region and discuss the generation mechanism of the electrostatic electric field around the separatrix region In Section VIII, we discuss the ion orbit dynamics across the separatrix region into downstream under the influence of the electrostatic electric field and explain the ion flow structure in the downstream In Section IX, we discuss how and where electrons and ions gain energy by electric field acceleration during magnetic reconnection The summary and conclusion are given in Section X II 2-1/2D PIC SIMULATION OF DRIVEN MAGNETIC RECONNECTION The PIC simulations were carried out in the (x,y,z) rectangular coordinate system under the condition of zero guide field by using the 2–1/2 dimensional relativistic, electromagnetic PArticle Simulation code for investigating driven Magnetic reconnection in an Open system (PASMO).27 In the 2-1/2D simulation model, particle orbits are followed in the 3-dimensional space, and all vector quantities point in three dimensions, but all physical quantities vary only in the poloidal (x, y) plane Initially, the plasma distribution is assumed to be a one-dimensional Harris current sheet equilibrium with the current density J~ ẳ cB0 =4pLị sech2 y=Lị^ e z , where B0 is the magnetic intensity at the upstream boundary and L is the current sheet half-width The particle velocity distributions are taken to be drift-Maxwellians The ~ ¼ e^x B0 tanhðy=LÞ, the electron magnetic field intensity is B density is ne ẳ neb ỵ ne0 neb Þ sech2 ðy=LÞ, where ne0 is the electron density at the initial current sheet center and neb is the background electron density, the plasma pressure is P ¼ ne ðTi ỵ Te ị ẳ Pb ỵ P0 sech2 y=Lị with uniform electron and ion temperatures Te and Ti , and P0 ẳ ne0 neb ị Ti ỵ Te ị ¼ B20 =8p At the upstream boundaries (y ¼ 6yb ) of the computational domain, the temporal and spatial varying electric field ~z x; y ẳ 6yb ; tị, which is defined in the paper of Ohtani and E Horiuchi,27 is imposed to evolve from Ez0 =B0 ¼ at t ¼ to a constant value after some simulation time together with Ex ¼ and @Ey =@y ¼ Particles with drift-Maxwellian velocity distributions are supplied at the upstream boundaries externally The frozen-in condition for both electrons and ions is imposed at the upstream boundaries with high accuracy The boundary electric field penetrates into the computational domain to produce inductive electric field, which is mainly in ~ind ¼ E ~z ðx; y; tÞ Then, the particles drift tothe z-direction E ward the neutral sheet together with the merging magnetic field An open boundary condition is imposed at the downstream boundaries x ẳ 6xb ị with @Ex =@x ẳ @By =@x ¼ @Bz =@x ¼ and @fe =@x ¼ @fi =@x ¼ so that the magnetic Phys Plasmas 22, 101205 (2015) flux moves freely and particles go freely in or out through the downstream boundaries The boundary conditions are imposed with very low unphysical noise The oppositely directed magnetic fields merge toward the neutral sheet ðy ¼ 0Þ by the self~z ðx; y; tÞ consistently generated electric field E The magnetic field reconnection takes place in the (x,y) plane The poloidal magnetic flux reconnection rate is deter~z at the mined by the inductive reconnection electric field E magnetic reconnection X-point In the quasi-steady state, the reconnection electric field is approximately the same as the driving electric field imposed at the upstream boundaries, which penetrates inductively to the whole simulation domain In the simulation, the electron dynamics decouples from the ion dynamics in certain regions, where the charge separation occurs to produce the electrostatic electric field ~es ðx; y; tÞ at each time step, we ~es ðx; y; tÞ To calculate E E solve the Poison equation from the computed electron and ion densities by taking the boundary condition that the electrostatic potential is zero on the boundaries of the (x,y) com~ind ðx; y; tÞ is putation domain The inductive electric field E ~ y; tÞ then obtained by subtracting the total electric field Eðx; ~es ðx; y; tÞ Because the from the electrostatic electric field E magnetic field changes with time, the parallel electric field is ~ Á B=B ~ ÞB ~ ~jj ðx; y; tÞ ¼ ðE calculated at each time step by E In the simulations, the electron and ion thermal speeds are defined as vTe ẳ Te =me ị1=2 and vTi ẳ Ti =mi Þ1=2 , where Te and me are the electron temperature and mass, Ti and mi are the ion temperature and mass, respectively The electron and ion plasma frequencies xpe ¼ ð4pne0 e2 =me Þ1=2 and xpi ¼ ð4pne0 e2 =mi ị1=2 , the electron Debye length kDe ẳ vTe = xpe , and the ion skin depth kdi ¼ c=xpi are defined with the electron density at the initial Harris current sheet location The electron and ion cyclotron frequencies xce ¼ eB0 =me c and xci ¼ eB0 =mi c are defined with the magnetic field value at the upstream boundary Then, the electron gyroradius is defined as qe ¼ vTe =xce , the ion gyroradius qi ¼ vTi =xci The Alfven velocity defined with the magnetic field and the plasma density at the upstream boundary is VA ¼ B0 = ð4pneb mi Þ1=2 , and the ion plasma beta at the upstream boundary is bi  4pneb Ti =B20 ẳ neb =ne0 ịqi =kdi ị2 The simulations were performed with the following input parameters: mi/me ¼ 100, Ti =Te ¼ 1, xpe =xce ¼ 4, neb =ne0 ¼ 0:22, the time step xce Dt ¼ 0:02, and the spatial grid size Dx ¼ Dy ¼ kDe Then, vTe =c % 0:1416, kDe % 0:0354 c=xce , qe % 0:1416 c=xce , qi % 1:416 c=xce , kdi % 2:46 c=xce , VA =c ¼ 0:0533, and bi % 0:32 The driving electric field at the upstream boundary is chosen to be uniform on the upstream boundaries with Ez0 =B0 ¼ À0:04 after xce t ! 335 Thus, the E  B drift velocity (Vd =VA ¼ cEz0 =VA B0 ¼ 0:75) is smaller than the electron thermal velocity, but is larger than the ion thermal velocity The simulation box size Lx  Ly in the (x,y) poloidal plane is varied from ð40 c=xce Þ Â ð9 c=xce Þ to ð68 c=xce Þ Â ð18 c=xce Þ to make sure that the boundaries not affect the simulation results The total number of finitesized particles is on the order of 109 In the figures of simulation results shown in the paper, the time is normalized by xceÀ1, the distance is normalized by c/xce, the velocity is normalized by the speed of light c, the electric field and 101205-5 Cheng et al magnetic field are normalized by B0, the particle density is normalized by ne0, the particle pressure tensor is by normalized by ne0 me c2, and the current density is normalized by B0xce/c In the paper, the simulation results are shown for the time of xcet ¼ 481.91 when the reconnection rate is quasi-steady III PARTICLE MOTION IN 2-1/2D MAGNETIC RECONNECTION GEOMETRY The particle dynamics in the 2–1/2-dimensional magnetic reconnection can be categorized into three classes in terms of the particle orbit topology The first class of particles is the magnetized particles with orbits gyrating around the magnetic field lines, and the particle gyroradii are much smaller than the electric field and magnetic field gradient scale lengths The second class of particles is the large gyroorbit particles with particle gyroradii comparable to or larger than the electric field and magnetic field gradient scale lengths The third class is the orbit meandering particles in a magnetic field reversal region with the magnetic field gradient scale length comparable to or smaller than the particle gyroradius Because of the reversal of the magnetic field line direction, the particle orbit gyrates around the magnetic field lines in one direction and then reverses the gyration direction in the field reversal region The dynamics of the magnetized particles can be studied by using the drift kinetic model The dynamics of the second and third classes of particles must be studied by employing the full equation of motion ~z ỵ E ~es ỵ ~ ~ md~ v =dtị ẳ qE v B=cị, where q is the particle charge and m is the particle mass In the 2–1/2-dimensional driven magnetic reconnection system, the inductive electric field is mainly in the z-direction and the electric field in the poloidal direction is mainly ~es caused by charge due to the electrostatic electric field E ~es is approximately perpendicular to separation Note that E the magnetic field because the charge separation along the field line is neutralized by fast moving electrons to maintain charge quasi-neutrality Thus, the electric field compo~jj ẳ Eã ~ B=B ~ nent parallel to the magnetic field line is E ~ which is mainly contributed by the inductive % ðEz Bz =B2 ÞB, ~z , and the electric field component perpendicuelectric field E ~? ẳ E ~es ỵ B2z =B2 Þ lar to the magnetic field is given by E ~p due to both E ~es and E ~z ~z À ðEz Bz =B2 ÞB E For magnetized particles, their flow velocity can be understood from the particle motion perpendicular and paral~jj conlel to the magnetic field line Particle acceleration by E trols the particle flow velocity parallel to the magnetic field ~ ~jj ¼ Vjj B=B The perpendicular flow velocity is mainly due V ~ ~e? % cðE ~es  B ~P ~ to the cE ?  B=B2 drift velocity so that V ~es  B ~z þ E ~z  B ~P Þ=B2 If the particle gyroradius is on ỵE the order of the electric and magnetic field scale lengths, the finite ion gyroradius effect must be considered The flow velocities in the z-direction and in the poloidal plane are ~z ¼ Vjj B ~es  B ~P =B2 and V ~p ¼ Vjj B ~z =B ỵ cE ~P =B given by V ~ ~ ~ ~ ỵcE es B z ỵ E z  B P Þ=B , respectively Phys Plasmas 22, 101205 (2015) Around the magnetic field-reversal region (reconnection ~ becomes current layer), the particle orbit perpendicular to B meandering because the magnetic field is weakened and is reversed across the neutral sheet The orbit meandering width scale is determined by m ẳ qy ẳ m ị, where y is the distance from the neutral sheet and the local particle gyroradius q equals to the local magnetic field gradient scale length q ẳ vyị=xc yị ẳ j@lnB=@yj1 , where xc ¼ eB=mc, and v is the particle velocity perpendicular to the magnetic field For example, for B ¼ B0y/L and q0 ¼ v/(eB0/mc), the particle pffiffiffiffiffiffiffiffi orbit meandering width is ‘m ¼ q0 L Because mi =me ) 1, the ion meandering width is larger than the electron meandering width and their ratio is given by ‘mi =‘me $ ðTi mi =Te me Þ1=4 for thermal particles.26 Inside the orbit meandering region, particles can be accelerated or decelerated by ~es depending on the ion velocity direction with respect to E ~z ~es but are always accelerated by E E Around the separatrix region, the ion gyroradii are either ~es localization width comparable to or larger than the E Then, the ion velocity can be accelerated or decelerated by ~? depending on the ion the perpendicular electric field E ~? direction gyrating velocity direction with respect to the E ~jj However, the ions are continuously accelerated by E IV CHARGE SEPARATION AND ELECTROSTATIC ELECTRIC FIELD When the oppositely directed poloidal magnetic fields ~P ðx; y; tÞ (pointing toward the positive x-direction in the B upper plane) are driven by the self-consistently generated in~z ðx; y; tÞ, which is mainly in the negaductive electric field E tive z-direction, as shown in Fig 1(a) at xcet ¼ 481.91, they ~z penemerge toward the neutral sheet (y ¼ 0) Note that E trates to the whole poloidal plane including the neutral sheet to cause magnetic reconnection Magnetized particles con~ ~z  B=B vect together with the magnetic field with the cE drift velocity toward the neutral sheet region, where the particle number accumulates FIG The color maps show the poloidal structure of (a) the inductive electric field Ez, (b) the charge separation (ni – ne)/ne, and (c) the main compo~P e^z =BP at xcet ẳ 481.91 The ~es ãB nent of the electrostatic electric field E ~P ) lines contour lines indicate the poloidal magnetic field (B 101205-6 Cheng et al Ions drift to the ion meandering region ðjyj ‘mi Þ where the ion density accumulates roughly uniformly However, electrons drift to the electron meandering region where the electron density ne is accumulated roughly uniformly inside the electron meandering region, although there is density reduction around the current sheet center probably due to fast electron outflow, which will be discussed later Thus, the electron density is higher (smaller) than the ion density inside (outside) the electron meandering region near the reconnection current sheet as shown in Fig 1(b) To view the detailed variations of the physical quantities, we show in Figs 2(a)–2(d) the variations of (a) the poloidal magnetic field Bx, (b) the inductive electric field Ez, (c) the electrostatic electric field Ey, and (d) the charge separation (ni À ne)/ ne0, respectively, along the y-axis From Fig 2(d), the charge separation is confined inside jyj ‘mi around the reconnection layer, and from Fig 2(c), a pair of strong in-plane bipo~es ¼ Ey e^y are lar converging electrostatic electric field E produced around the field reversal region by the charge separation, and they are peaked around y % 6‘me with a fullwidth of several electron meandering widths ($6 ‘me ) and point toward the neutral sheet From the comparison of Figs 2(b) and 2(c), the bipolar converging electrostatic electric ~es is much larger than the driving inductive electric field E Phys Plasmas 22, 101205 (2015) field Ez as observed in space plasma by Wygant et al.6 Note ~es is roughly perpendicular to B ~P because charge sepathat E ration along the field line can be quickly smeared out by fast ~P  e^zz =BP is shown ~es •B electron motion and its amplitude E ~es is also called the Hall electric in Fig 1(c) The bipolar E field.28 Note that from Fig 1(b) the charge separation occurs also around the field line separatrix regions because electrons flow mainly along the field line and ions flow mainly across the field line from Fig 3, which shows the distribution of (a) the electron flow velocity and (b) the ion flow velocity in the poloidal plane at xcet ¼ 481.91 The poloidal flow velocity component is shown in arrows, and the z-component flow velocity is shown in color The contour lines indicate the poloidal magnetic field lines Also, Fig 1(b) shows small scale charge separations associated with high frequency electrostatic instabilities driven by the field-aligned electron beam, which will be discussed in the future publication Around the ~es points toward the mid-plane and its separatrix region, E localization width is wider than that around the current layer ~es generation around the separatrix will The mechanism of E be discussed in more details in Section VII V DYNAMICS OF ELECTRON AND ION INFLOWS INTO RECONNECTION CURRENT LAYER ~es and the penetration of E ~z to the The generation of E neutral sheet strongly affect the particle motion inside the particle orbit meandering region.25,35 As shown in Fig 3, the electron flow velocity structure is significantly different from the ion flow velocity structure mainly because the electron dynamics decouples from the ion dynamics In the following, we will present physical understanding of the electron and ion flow structures and how electrons and ions are accelerated A Electron inflow and dynamics As electrons drift toward the neutral sheet, the electron ~ed ¼ cE ~z  B ~P =B2 increases to Alfven inflow drift velocity V FIG The variation of (a) Bx, (b) Ez, (c) Ey, (d) (ni-ne)/ne0, (e) the electron inflow velocity Vey (blue curve), the ion inflow velocity Viy (green curve), and the y-component of the E  B drift velocity (red curve), and (f) the electron flow velocity Vez (black curve) and the ion flow velocity Viz (green curve), and the z-component of the E  B drift velocity (red curve) along the y-axis at xcet ¼ 481.91 The electron and ion meandering widths (‘me and ‘mi ) are also indicated FIG The flow velocity distribution in the poloidal plane for (a) electrons and (b) ions at xcet ¼ 481.91 The poloidal flow velocity component is shown in arrow, and the z-component is shown in color The contour lines are poloidal magnetic field lines 101205-7 Cheng et al ~P as they reach speed due to smaller poloidal magnetic field B the boundary of the electron orbit meandering region ðjyj ‘me Þ To see the detailed variation of the electron and ion inflow velocities, we show in Fig 2(e) the variation of the electron inflow velocity Vey (blue curve), the ion inflow velocity Viy (green curve), and the y-component of the E  B drift velocity (red curve) along the y-axis Inside the electron orbit meandering region, the electron inflow ve~ey (blue curve) slows down to much less than the locity V ~P =B2 drift velocity (red curve) due to deceleration ~z  B cE by the bipolar electrostatic electric field, which is in the same direction as the electron inflow velocity It is noted that ~ey reaches Alfven speed and the electron inflow velocity V ~ then is decelerated by E es in the electron orbit meandering region before the electrons move across the neutral sheet Figure 2(f) shows the variation of the electron flow velocity Vez (black curve) and the ion flow velocity Viz (green curve) and the z-component of the E  B drift velocity (red curve) along the y-axis Outside the electron orbit meander~y is significant, the electrons are ~es ¼ E ing region where E magnetized and experience the out-of-plane drift velocity ~es  B ~P =B2 , and inside the electron orbit meander~ey ’ cE V ~z Both effects ing region the electrons are accelerated by E contribute to the enhancement of the out-of-plane electron ~z direction current J~ez in the E To understand the electron inflow dynamics, we examine the electron velocity distribution fe ðvx ; vy ; vz Þ along the y~ points in the x-direction axis where the magnetic field B Figure shows fe ðvx ; vy ; vz Þ in several boxed areas (around the reconnection current layer) In the top boxed area, fe ðvx ; vy ; vz Þ is a drift-Maxwellian velocity distribution with Phys Plasmas 22, 101205 (2015) ~ey ’ À0:032c^ the inflow velocity V e y but with a larger veloc~ of $0.25c Note that ity spread in vx (which is parallel to B) the Vey drift velocity is much smaller than the thermal velocity (0.14c) of the inflowing electrons In the boxed areas closer to the X-point (the second and third from the top), Vey decreases, but the velocity spread in vx increases to form a flat-top parallel velocity distribution The flat-top parallel velocity distribution has small net parallel flow velocity in the immediate upstream of the reconnection region as observed during magnetic reconnection in the magnetotail.11,12 The mechanism of the flat-top parallel velocity distribution in the immediate upstream of the reconnection region is due to the electron acceleration by the parallel electric field around the separatrix region and will be discussed in Section VII As the electrons enter the electron orbit meandering region, the electrons in the third boxed area from the top in ~z , ~y but accelerated by E Fig start to be decelerated by E ~ and both electric field components are perpendicular to B Thus, the Vey flow velocity decreases, but the Vez flow velocity increases Figure shows schematically the electron ~y meandering orbits in the (y,z) plane under the influence of E ~ and E z Electrons entering the electron orbit meandering ~y and then region from the upstream are first decelerated by E ~y after they move across the mid-plane are accelerated by E to the other side of the orbit meandering region because their ~y However, the vy velocity is in the opposite direction to E ~z to gain orbit meandering electrons are accelerated by E ~z large velocity in the direction opposite to E In the box around the X-point in Fig 4, the electron distribution shows two converging electron beams in vy, and it is due to the inflowing electrons from both above and below FIG The electron velocity distribution fe ðvx ; vy ; vz Þ sampled in the boxed areas along the y-axis at xcet ¼ 481.91 101205-8 Cheng et al Phys Plasmas 22, 101205 (2015) ~ % B^ FIG The electron orbits perpendicular to the magnetic field B e x in the magnetic field reversal region under the influence of the localized bipolar ~y and the relatively uniform inductive converging electrostatic electric field E ~ ~z , both of which are perpendicular to B reconnection electric field E ~ % B^ FIG The ion orbits perpendicular to the magnetic field B e x in the magnetic field reversal region under the influence of the localized bipolar ~es and the relatively uniform inducconverging electrostatic electric field E ~ ~z , both of which are perpendicular to B tive reconnection electric field E the mid-plane The vy spread is large in comparison with the ~y These beam velocity due to acceleration/deceleration by E electron beams preserve large velocity spread in the vx-direc~ These two converging electron vy-beams tion (parallel to B) ~z in the positive vz-direction are further accelerated by E The physical pictures based on the electron dynamics and the electron velocity distribution discussed above explain the variations of the electron Vey and Vez flow velocities along the y-axis as the electrons move toward the neutral sheet as show in Figs 2(e) and 2(f) ions have a drift-Maxwellian distribution with an inflow velocity of $0.02 c (from Fig 2(e)) and the thermal spread of $0.015 c outside the ion orbit meandering region As the drifting ions move closer to the current layer, the magnetic field intensity weakens and the gyro-radius becomes larger than the ion orbit meandering width (‘mi $ c/xce) so that the ion orbit becomes meandering in the region of jyj ‘mi The ~es localization width ion gyro-radius is also larger than the E ($1.5 c/xce) Then, the inward drifting ion velocity is acceler~y ¼ Ey e^y (jE ~es j=B0 $ 0.2) ~es ¼ E ated by both the localized E ~z (jE ~z j=B0 $ 0.04), both of which and the relatively uniform E ~ are perpendicular to B % B^ e x , and the ion velocity direction is ~ centrifugal force Because the controlled by the q~ v  B=c ~ $ from Fig 2(a)) near the magnetic field is weak (jBj=B ~ centrif~es location, the z-component of the q~ v  B=c peaked E ugal force is compensated by the comparable and opposite ~z force, and thus the ion velocity is accelerated mainly by qE ~y as they move toward the neutral sheet After the ion ~es ¼ E E ~es , crosses the neutral sheet, their velocity is decelerated by E ~z to gain vz Because the ion velocity but still accelerated by E ~ centrifugal force is also larger becomes larger, the q~ v  B=c Then, the ion orbit starts to gyrate and turns around by the ~ centrifugal force due to the reversed B ~ direction q~ v  B=c Then, the ion orbit crosses the neutral sheet back to the region ~es , but above the mid-plane, and the ions are decelerated by E ~z After that, the ion velocity goes still accelerated by E ~es alterthrough the deceleration and acceleration cycles by E natively and the ion orbit oscillates in both sides of the neutral ~z all the time, sheet In the meantime, ions are accelerated by E and the ion vz is in the negative e^z direction In the meantime, ~jj if it is ions are accelerated by the parallel electric field E finite From the schematic ion orbits shown in Fig 6, we can understand the ion velocity distribution fi ðvx ; vy ; vz Þ shown in Fig in several boxed areas along the y-axis, where the ~ points mainly in the x-direction At the ymagnetic field B location outside the orbit meandering region, ions have a drift-Maxwellian velocity distribution centered at the drift ~z  B ~x =B2 , which points toward the neu~dy ¼ cE velocity V tral sheet direction Near the edge of the ion orbit meandering region (the top boxed area shown in Fig 7), the ion velocity distribution consists of the inward drift-Maxwellian B Ion inflow and dynamics As magnetized ions move with the merging field lines toward the reconnection region, the ion orbits become meandering around the field-reversal region Then, the ion inflow ~iy (green curve in Fig 2(e)) slows down to signifivelocity V ~dy ¼ cE ~z  B ~x =B2 (red cantly below the drift velocity V curve) inside the ion meandering region ðjyj ‘mi Þ as shown in Fig 2(e) Based on the fluid description, the reduction of ~iy inside the ion orbit meandering the ion inflow velocity V region is due to the contribution of ion inertia, the flow velocity gradient, and the divergence of the ion pressure tensor effects.22,23,25 However, it is difficult to understand the physical picture of how the ion inflow velocity slows down in the ion orbit meandering region from the fluid picture Due to ~iy is much more reduced than the electhe large ion mass, V ~ey and remains sub-Alfvenic speed tron inflow velocity V when the ions enter the ion orbit meandering region as shown in Fig 2(e) ~iz is localized inside the ion meandering region Also, V ~ ~iz as shown in Fig 2(f) and V ez is more localized than V ~ ~ Because jV ez j ) jV iz j, the reconnection current density J~z is dominated by the electron current J~ez contribution and is localized in a half-width of several ‘me around the neutral sheet for the simulation parameters Physically, the reduction of the ion inflow velocity in the ion meandering region and the inflowing ion velocity distribution fi ðvx ; vy ; vz Þ can be understood from the ion dynamics Figure shows schematically the ion orbits ~ when the ions move to the magnetic fieldperpendicular to B reversal region The typical ion orbits of the downward drifting ions (upward drifting ions from the other side of field reversal region) in the field reversal region are shown in blue (red) color in Fig For this particular simulation case, the 101205-9 Cheng et al Phys Plasmas 22, 101205 (2015) FIG The ion velocity distribution fi ðvx ; vy ; vz Þ sampled in the boxed areas along the y-axis at xcet ¼ 481.91 ions and the orbit meandering ions (inside the red circle in Fig 7) that originate from the other side of the field reversal region (red orbits in Fig 6) Ions with larger inward drifting ~es region so that velocity can cross the localized bipolar E ~es , and they are accelerthere is no net acceleration from E ~z as shown in Fig Because of the orbit gyraated only by E tion due to the centrifugal force, the meandering ion orbits have gyrated around the magnetic field so that their vy veloc~z , and the total ity is smaller than vz due to acceleration by E perpendicular velocity becomes larger than the inward drift velocity as shown in the ion velocity distribution (inside the red circle for the top boxed area in Fig 7) Deeper into the ion orbit meandering region (the boxed areas closer to the neutral sheet shown in Fig 7), the ion velocity distribution consists of the accelerated inward moving ions and higher density orbit meandering ions The orbit meandering ions have more velocity spread due to the combi~es , E ~z , and the q~ ~ centrifugal forces; jvzj is nation of E v  B=c small for larger positive vy, jvzj is large for small vy near 0, and jvzj is even larger for vy < The velocity spread can be understood from the ion orbits shown in Fig Above the mid-plane in the field reversal region, the upward moving ions (positive vy) have vy > jvzj, and for the downward moving ions (with negative vy), jvzj is even larger with jvzj > jvyj Inside the current layer (the boxed area in the neutral sheet around the X-point shown in Fig 7), the ion velocity distribution function consists of ions coming from both sides of upstream, and the ~es and E ~z ions are further accelerated by both E From the ion velocity distribution function shown in Fig 7, we can understand why the mean ion inflow velocity decreases in the ion orbit meandering region as shown in Fig 2(e) Because the orbit meandering ion vy is either smaller than or in the opposite direction to the inward ion ~dy , the mean ion inflow velocity Viy is reduced drift velocity V from Vdy At position closer to the reconnection current sheet, the meandering ion number density increases, and thus the net ion inflow velocity Viy is reduced further Note that the vz velocity of the orbit meandering ions is also acceler~z (in the negative e^z -direction) as shown in ated further by E the upstream ion velocity distributions At the center of the reconnection region (around the X-point), the ion velocity distribution in the boxed area is approximately the sum of the ion velocity distributions in the boxed areas just above and below the center boxed area, but with further ion acceleration of both inward drifting ions and meandering ions by ~z However, the net ion flow velocity in the y-direc~es and E E tion is zero because the inward flow velocities from both sides of the reconnection current layer are in opposite direction VI DYNAMICS OF ELECTRON AND ION OUTFLOW FROM RECONNECTION CURRENT LAYER— GENERATION OF QUADRUPOLE MAGNETIC FIELD AND FIELD-ALIGNED ELECTRIC FIELD Because the reconnection current layer is mainly contributed by the electron current J~ez , the merging magnetic field lines are reconnected inside the electron current layer to form an X-line topology The current layer thickness in the y-direction is on the order of several electron orbit meandering widths ($4–6 ‘me in the simulation case presented in the 101205-10 Cheng et al paper), which is much smaller than its width in the x-direction, both the electron and ion outflow speeds must be much larger than their corresponding inflow speeds to avoid much charge accumulation inside the reconnection current sheet A Electron and ion outflows Figure shows (a) the magnetic field By, (b) the inductive electric field Ez, (c) the electron flow velocity Vez (blue curve), the ion flow velocity Viz (green curve), and the z~dz ¼ cE ~x  B ~y =B2y component of the E  B drift velocity V (red curve), and (d) the electron outflow velocity Vex (blue curve), the ion outflow velocity Viy (green curve), and the ~z  B ~y =B2 drift velocity (red color curve) along the ~dx ¼ cE V ~ex is less than the x-axis The electron outflow velocity V ~ ~ cE z  B y =B drift velocity near the X-point, but as electrons ~ex increases and reaches super-Alfvenic flow further out, V ~dx ¼ cE ~z  B ~y =B2 drift velocspeed and is larger than the V ~ex slows down and ity Further out in the downstream, V ~ ~ approximately equals to cE z  B y =B drift velocity (for x ¼ 7–12 c/xce at xcet ¼ 481), which equals approximately the Alfven speed The electron current sheet region with ~ez extends to the super-Alfvenic V ~ex jet location and strong V is often referred as the electron diffusion region Because the ion inflow velocity into the reconnection current sheet region is much smaller than the electron inflow velocity as shown in ~ix is also much smaller Fig 2(e), the ion outflow velocity V ~ than the electron outflow velocity V ex ~ix is smaller From Fig 8(d), the ion outflow velocity V ~z  B ~y =B drift velocity in the reconnection than the cE FIG The variation of (a) By, (b) Ez, (c) the electron flow velocity Vez (blue curve), the ion flow velocity Viz (green curve), and the z-component of the E  B drift velocity (red curve), and (d) the electron outflow velocity Vex (blue curve), the ion outflow velocity Viy (green curve), and the x-component of the E  B drift velocity (red curve) along the x-axis at xcet ¼ 481.91 Phys Plasmas 22, 101205 (2015) ~ix continues current layer But, as the ions flow out further, V ~ ~ to increase and eventually reach the cE z  B y =B2 drift velocity, which approximately equals to the Alfven velocity In the region where both the electron and ion outflow velocities are similar ($Alfven velocity), a fast shock-like structure (x ¼ 9–12 c/xce at xcet ¼ 481.91) is formed together with the pileup of By flux (shown in Fig 8(a)) and plasma density In the fast shock-like region, both electrons and ions are magnetized and their velocities are thermalized around their drift velocity as will be shown in the ion velocity distributions It is to be noted that if the inflows are continuously driven from the upstream boundary, the fast shock-like structure will move further outward, which is similar to the magnetic field dipolarization front observed in the magnetosphere.13 The fast shock-like structure will eventually move out of the downstream computational boundary When the dense plasma moves out together with the fast shock-like structure, additional particles and field energy will move into the computational domain from the outflow boundaries This phenomenon may not be physical and needs to be fixed if one wants to carry out the simulation to a much longer physical time B Quadrupole magnetic field and field-aligned electric field In the immediate downstream region (x < c/xce; before the shock-like region), as shown in Fig 8(d), the electron outflow velocity Vex is much larger than the ion outflow velocity Vix and thus the current is mainly determined by the super Alfvenic Vex jet Then, a pair of currents flow toward the reconnection region in both downstream regions, which by Ampere’s law produce the quadrupole out-of-plane magnetic field Bz(x,y,t) as shown in Fig 9(a) at xcet ¼ 481.91 ~z field points in the negative zNote that the quadrupole B direction in the upper right (x > 0, y > 0) and lower left (x < 0, y < 0) quadrants and points in the positive z-direction in the upper left (x < 0, y > 0) and lower right (x > 0, y < 0) ~z field is quadrants The orientation of the quadrupole B FIG The distribution of (a) the quadrupole Bz field, (b) the total magnetic field B, and (c) the parallel electric field Ejj in the poloidal plane at xcet ¼ 481.91 The contour lines are poloidal magnetic field lines 101205-11 Cheng et al independent of the direction of the merging magnetic fields ~z field is concentrated around the separatrix The quadrupole B region and is quite weak in the reconnection current sheet region and around the mid-plane (y ¼ 0) The electric field in the simulation system consists of the ~es , which is perpendicular to the electrostatic electric field E magnetic field and lies in the poloidal plane, and the inductive ~z , which is in the negative z-direction After electric field E ~z has compo~z is generated, the inductive E the quadrupole B nents both perpendicular and parallel to the magnetic field because of the quadrupole Bz magnetic field: the parallel com~jj ẳ Ez Bz =Bị^ e B and the perpendicular component ponent E ~ e z À ðBz =BÞ^ e BP Š Then, the total E ?1 ẳ Ez BP =BịẵBP =Bị^ ~?1 ỵ E ~?2 , ~? ¼ E perpendicular electric field is given by E ~es In the upper-right quadrant of the poloidal ~?2 % E where E ~jj points in the B ~ direction plane, Ez < and Bz < and thus E as shown in Fig 9(c) However, it is to be noted that at ~jj points in the negative B ~ direction in a narxcet ¼ 481.91 E row layer just off the mid-plane (0 < xxce/c < and < yxce/ ~jj points in the c < 1) Similarly, in the lower-right quadrant, E ~ negative B direction except in a small layer (0 < xxce/c < ~jj points in the B ~ direction and > yxce/c > À1), where E C Electron dynamics and velocity distribution in outflow To understand how electrons gain super-Alfvenic outflow velocity in the reconnection current layer, we need to study the electron velocity distribution From the upstream through the reconnection current layer into the downstream, the direction of the poloidal magnetic field direction changes from mainly in the x-direction in the upstream to mainly in the y-direction in the downstream Thus, it is easier to examine the electron velocity distribution in the rectangular velocity coordinate Figure 10 shows the electron velocity Phys Plasmas 22, 101205 (2015) distribution fe ðvx ; vy ; vz Þ in the rectangular velocity coordinate in three boxed areas (x ¼ 1.1–5.8 c/xce) along the midplane (y ¼ 0) in the current layer and downstream at xcet ¼ 481 In these boxed areas, Bz is either zero or very small, and By > for x > In the current layer (x ¼ 1.1–4.2 c/xce), the magnetic field has sharp cusp structure and points mainly in the positive x-direction above the mid-plane and in the negative x-direction below the mid-plane The electron velocity distribution consists of electrons that flow into the reconnection current layer from both sides of the current layer In the upstream of the current layer, these electrons have a flattop vjj-distribution centered roughly at vjj ¼ (positive vjj is mainly in the positive x-direction above the midplane) and smaller drifting velocity in the vy-direction When these upstream electrons flow into the current layer, the electrons that have positive-vx move toward the downstream direction, and most electrons with negative vx move toward the reconnection X-point region along the field line and then move to the positive vx-direction in the other side of the midplane Thus, the number of negative-vx electrons is much smaller than that of positive-vx electrons as shown in the electron velocity distribution in the (vx,vy) phase space in ~z Fig 10 In the meantime, the electrons are accelerated by E to gain large positive vz because of the electron orbit meandering effect Further out to the boxed areas of x ¼ 2.6–5.8 c/xce, By becomes larger and the electrons become magnetized and ~P =B2 % ÀjcEz BP =B2 j^ ~z  B e ?2 drift velocpick up larger cE ity, which has a component in the positive x-direction Then, the electrons gain higher vx-velocity and the number of negative-vx electrons is further reduced The electron mean outflow velocity Vex peaks in the boxed area of x ¼ 4.2–5.8 c/xce, where Vex reaches super Alfvenic speed as shown in Fig 8(d) This process of converting the parallel electron velocity in the inflow region of the reconnection current layer into superAlfvenic electron outflow velocity in the downstream has not been presented before FIG 10 The electron velocity distribution fe ðvx ; vy ; vz Þ sampled in boxed areas (1 < xxce/c < 6) along the x-axis in the downstream region at xcet ¼ 481.91 101205-12 Cheng et al Further downstream (x > c/xce) along the mid-plane, ~ % By e^y becomes much stronger, the electrons start to B gyrate round the magnetic field, and the electron velocities in all three directions are thermalized around the drift velocity with thermal spread significantly larger than that in the upstream as shown in Fig The reason for the larger electron thermal velocity spread is because the magnetic field topology changes from the upstream to the downstream When electrons enter the reconnection current layer in the upstream, their perpendicular velocity (vy) and the parallel velocity (vx) in the upstream become the parallel velocity and the perpendicular velocity, respectively, in the downstream mid-plane region Thus, after the electrons move from upstream through the reconnection current layer into the downstream, the parallel velocity spread in the electron flat-top vjj-distribution in the immediate upstream is converted to the perpendicular velocity with large spread around the mean electron perpendicular drift velocity This is a critical mechanism of how electrons are thermalized and heated in the perpendicular velocity direction in the downstream Moreover, because of the adiabatic invariant of the electron magnetic moment, the electron velocity distribution in all three velocity directions has large thermal velocity spread, which is larger than the electron mean flow velocity D Ion dynamics and velocity distribution in outflow To understand how ions flow out from the reconnection current layer, we examine the ion dynamics from the upstream through the reconnection current layer to the downstream Figure 11 shows the ion velocity distribution fi ðvx ; vy ; vz Þ along the x-axis in several boxed areas Phys Plasmas 22, 101205 (2015) (x ¼ 1.1–7.5) along the mid-plane in the current layer and downstream at xcet ¼ 481 Inside the reconnection current layer (x ¼ 1.1–4.2 c/xce), the magnetic field has sharp cusp structure and points mainly in the positive (negative) x-direction above (below) the mid-plane Thus, the ion perpendicular velocity distributions fi vy ; vz ị in the x ẳ 1.1–4.2 c/xce region are quite similar to the ion perpendicular velocity distribution fi ðvy ; vz Þ around the current sheet center (around the X-point) as shown in Fig These ion velocity distributions consist of the orbit meandering ions and the inward drifting ions that move into the current layer from both sides of the upstream However, both the inward drifting ions and the orbit meandering ions are accelerated to gain positive vx Note that vx increases with x and inside the boxed area of 2.6 < xxce/c < 4.2, the ion outflow velocity reaches Vix $ 0.02 c for the inward drifting ions and Vix $ 0.05 c for the orbit meandering ions In the ion outflow exhaust area (in the right two boxed areas with 4.2 < xxce/c < 7.5), there are new ions (inside the red circles in the velocity space shown in Fig 11) with smaller vx and vy These new ions come from the upstream and move across the separatrix Other than these new ions, fi ðvx ; vy ; vz Þ is quite similar to those inside the current layer but with further acceleration in vx Because the new ions have smaller vx, the average ion outflow velocity is reduced even though other ions are still accelerated to higher vx These new ions are ions that gyrate around the magnetic field when they move across the separatrix region, where the magnetic field is not reversed and their gyro-radii are comparable to or larger than the localization width of the electrostatic electric field Thus, we can consider the edge region of the FIG 11 The ion velocity distribution fi ðvx ; vy ; vz Þ sampled in boxed areas (1.1 < xxce/c < 7.5) along the x-axis in the downstream region at xcet ¼ 481.91 101205-13 Cheng et al current layer (the ion outflow exhaust area) as the transition region between two different types of ion orbits moving under two different types of magnetic field topology We delay the discussion of these new orbit gyrating ions in Section VIII To understand how ions gain increasing vx and outflow velocity Vix inside the reconnection current layer, we investigate the ion velocity distributions fi ðv1 ; v2 ; vjj ị at xcet ẳ 481.91 in the field-aligned velocity coordinate, where ~P  e^z =BP Þ Â ðB=BÞ, ~ ~P  e^z =BP , and vjj v •ðB v2 ¼ ~ v •B v1 ¼ ~ ~ ¼~ v •B=B, in the boxed areas just outside the reconnection ~P  e^z =BP lies current layer as shown in Fig 12 Note that B in the poloidal plane and is perpendicular to the poloidal ~P  e^z =BP points mainly toward the midmagnetic field B plane in these boxed areas, which are just outside the current layer and above the mid-plane as shown in Fig 12 We note that in the left two boxed areas (x ¼ 1–4.2 c/xce), the ion perpendicular velocity distributions are quite similar because By ( Bx The inward drifting ions (inside the yellow circle) are ~es and are hardly accelerated by the mainly accelerated by E ~jj because there is no much time for parallel electric field E them to be accelerated However, the orbit meandering ions (inside the red circle) are accelerated to gain positive vjj by ~ direction in most of the upper-right ~jj , which points in the B E quadrant of the poloidal plane as shown in Fig 9(c) The orbit meandering ion vjj increases with x and reaches vjj $ 0.045 c at x $ c/xce When these ions move into the ~jj is current layer, their vjj will not change much because E quite weak inside the current layer Because By, Bz (Bx, then vx $ vjj Also, because By increases linearly with x toward the downstream, the perpendicular velocities of both Phys Plasmas 22, 101205 (2015) the inward drifting ions and the orbit meandering ions contribute to vx Therefore, inside the current layer, the ion outflow velocity Vix ẳ (nmvjjBx ỵ ndVdBy)/B(nm ỵ nd) $ 0.03 c at x $ c/xce, where nm is the orbit meandering ion density and nd is the drifting ion density (can be estimated from fi ðvx ; vy ; vz Þ shown in Fig 11), and thus the ion outflow velocity inside the current layer is contributed mainly by the orbit meandering ion vjj and less by the drift velocity of the inward drifting ions It should be emphasized that the accel~jj around the eration of the orbit meandering ion vjj by E reconnection current layer and their contribution to the ion outflow velocity have not been discussed before Further downstream (in the right two boxed areas with x ¼ 4.2–7.5 c/xce), fi ðv1 ; v2 ; vjj Þ shows significant change from those in the left two boxed areas (x ¼ 1–4.2 c/xce) The right two boxed areas can be considered as the transition region where two different types of ion orbits under two different types of magnetic field topology co-exist: (1) the orbit meandering ions that meander in the field reversal geometry ~z and (2) ~es and uniform E under the influence of localized E the large gyro-orbit ions that gyrate around the magnetic field when they move across the separatrix region, where the magnetic field is not reversed and their gyro-radii are compa~es localization width We will discuss the large rable to the E gyro-orbit ion dynamics in Section VIII VII ELECTRON DYNAMICS AROUND SEPARATRIX REGION—GENERATION OF ELECTROSTATIC ELECTRIC FIELD AROUND SEPARATION REGION Outside the reconnection current layer, electrons are ~jj along the magnetic magnetized and can be accelerated by E FIG 12 The ion velocity distribution fi ðv1 ; v2 ; vjj Þ in the field-aligned velocity coordinate sampled in the boxed areas just outside the reconnection current ~P  e^z =BP Þ Â B=Bị, ~ ~P e^z =BP , and vjj ẳ ~ ~ v ãB v2 ẳ ~ v ãB v •B=B layer at xcet ¼ 481.91, where v1 ¼ ~ 101205-14 Cheng et al field lines to high velocity Thus, around the field line separa~jj is significant, electrons are accelerated to trix region, where E form a single-beam distribution in the parallel velocity Figure 13 shows the electron velocity distribution fe v1 ; v2 ; vjj ị at xcet ẳ 481.91 in the field-aligned velocity coordinate, where ~P  e^z =BP Þ Â ðB=BÞ, ~ ~P  e^z =BP , and v1 ẳ ~ v ãB v2 ẳ ~ v •B ~ vjj ¼ ~ v •B=B, in several boxed areas in the upstream region just above the field line separatrix As shown in Fig 13, in the ~jj boxed area furthest away from the reconnection region, E ~ direction and the inward drifting elecpoints in the positive B ~jj to form a single beam distribution trons are accelerated by E ~ejj Þ pointing in the negain vjj with the parallel flow velocity ðV 19 ~ direction Because V ~ejj is much larger than the perpentive B ~e? ’ cE ~  B=B ~ Þ, the inward dicular electron flow velocity ðV drifting electrons flow mainly along the field lines with high velocity toward the reconnection region Because the magnetic field intensity forms a magnetic well around the reconnection region as shown in Fig 9(b), the electron parallel velocity spreads out due to the magnetic moment conservation and magnetic trapping when electrons move from the separatrix region into the immediate upstream region just outside the reconnection current layer The parallel inflowing electrons from both sides of the upstream field line around the separatrix combine with the vertically inward drifting electrons to form a flat-top vjj-distribution shown in Fig (note that vx is approximately the same as vjj in Fig 4) The flat-top parallel velocity distribution has small parallel flow velocity in the immediate upstream of the reconnection region, as observed during magnetic reconnection in the magnetotail.11,12 Phys Plasmas 22, 101205 (2015) In the boxed areas closer to the reconnection region (shown in Fig 13), the inward drifting electron vjj is further ~jj ; however, there is a second electron popuaccelerated by E lation that has large vjj spread and smaller mean hvjji The second electron population consists of (a) the newly arrived inward drifting electrons that come to the separatrix region together with the reconnected field line and (b) the positive vjj electrons of the flat-top vjj distribution that move away from the magnetic well along the field line toward the downstream direction These positive vjj electrons of the flat-top vjj distribution have sufficient positive vjj to overcome the ~jj to reach these boxed areas They are deceleration force of E ~jj to very small vjj and then bounce back first decelerated by E ~jj again toward the magnetic well and be accelerated by E The density of this electron population increases at location closer to the reconnection region and dominates over the inward drifting electron density Thus, the parallel electron ~ejj , which points toward the reconnecmean flow velocity V ~ejj ¼ at tion region, decreases to very small value and V x ¼ As the electrons flow from the upstream into the reconnection current layer together with the reconnected field lines and then to the downstream exhaust region, the feature of the electron flat-top vjj distribution is preserved Inside the current layer, the electrons flow outward (positive vx for x > 0) as shown in Fig 10, which means that vjj is positive (negative) above (below) the mid-plane along the cusp field line Thus, along the field lines just below the field line separatrix in the downstream, the electrons consists of two population: (1) the outflowing electrons (shown in Fig 10) that come from the current layer and (2) the inward drifting electrons (shown in Fig 13) that move with the merged field across FIG 13 The electron velocity distribution fe ðv1 ; v2 ; vjj Þ sampled in the boxed areas in the upstream region just above the field line separatrix at xcet ¼ 481.91, ~P  e^z =BP Þ Â ðB=BÞ, ~ ~P  e^z =BP , and vjj ¼ ~ ~ v ãB v2 ẳ ~ v ãB v ãB=B where v1 ¼ ~ 101205-15 Cheng et al Phys Plasmas 22, 101205 (2015) FIG 14 The electron velocity distribution fe ðv1 ; v2 ; vjj Þ sampled in the boxed areas along the field line just below the separatrix in the downstream region at ~P  e^z =BP Þ Â ðB=BÞ, ~ ~P  e^z =BP , and vjj ¼ ~ ~ v ãB v2 ẳ ~ v ãB v ãB=B xcet ¼ 481.91, where v1 ¼ ~ the separatrix into the downstream Figure 14 shows the electron velocity distribution fe ðv1 ; v2 ; vjj Þ sampled in the boxed areas along the field lines just below the separatrix in the downstream region In the lower-left two boxed areas, the density of positive vjj electrons (outflowing from the current layer) and the inward drifting electrons is larger than the density of negative vjj beam electrons (inside the red circles ~jj , and thus the electron in Fig 14) that are accelerated by E ~ejj is in the B ~ direction However, in parallel flow velocity V the upper-right two boxed areas, the number of electrons that can overcome the magnetic trapping and the deceleration ~jj to reach these two boxed areas is much reduced, force of E and thus only very few positive vjj electrons can reach the ~ejj top boxed area Thus, the electron parallel flow velocity V is mainly contributed by the inward drifting electron beam ~jj acceleration and is in the opposite that is produced by E ~ in these two boxed areas This explains why direction to B ~ejj changes direction the electron parallel flow velocity V along the field lines in these boxed areas To examine how the electron parallel velocity distribution changes around the separatrix region, we compute the electron parallel flow velocity Vejj from the electron velocity distribution and the Vejj distribution in the poloidal plane is shown in Figure 15(a) It is clear that in the upstream just outside the ~ejj is much larger than the perfield line separatrix, because V ~  B=B ~ Þ, elec~e? ’ cE pendicular electron flow velocity ðV trons flow mainly along the field lines with high velocity into the immediate upstream region just outside the reconnection current layer However, in the downstream region (inside the ~ejj flows along the field line outward toward the separatrix), V downstream direction in most of the downstream region Only in the far away region near the separatrix, where the ~ejj is outflowing electrons cannot reach along the field line, V contributed mainly by the inward drifting electron beam ~jj as shown in Fig 14 accelerated by E It is important to point out that our physical picture of how electrons are accelerated to the region around the reconnection current layer along the upstream field lines near the separatrix is different from the electron surfing mechanism ~jj effect proposed by Hoshino,18 which did not consider the E It is also to be noted that our physical pictures of how the quadrupole magnetic field is generated and how electrons are accelerated by Ejj to flow mainly along the field line around the separatrix region (above and below the field line separatrix) are different from the one presented by Uzdensky and Kulsrud,34 which is based on changes in the flux tube volume to explain the electron parallel flows near the separatrix Now, we can understand how the electrostatic electric ~es is produced around the separatrix region, where field E FIG 15 The distribution of (a) the electron parallel flow velocity Vejj and (b) the ion parallel flow velocity Vijj in the poloidal plane at xcet ¼ 481.91 101205-16 Cheng et al ~ejj flow velocity along the field line as electrons have large V shown in Fig 15(a) It is clear that in the upstream just outside the separatrix, electrons flow mainly along the field lines to the magnetic well region just upstream of the reconnection current layer Thus, along these upstream field lines, the electron density increases toward the magnetic well region Because these electrons flow from both separatrix regions along the field lines, they form a flat-top vjj distribution in the magnetic well region just outside the current layer Then, these flat-top vjj distribution electrons have higher density and flow through the reconnection current layer into the downstream exhaust region However, because the ion flow velocity is much smaller than the electron parallel flow velocity around the separatrix region, most ions must flow across the separatrix field lines into the downstream so that large charge non-neutrality (negative charge buildup) can be avoided in the downstream region Thus, the electron density ne is smaller than the ion density ni in the region outside the field line separatrix as shown in Fig 1(b) However, the flattop vjj distribution electrons that have moved into the current layer eventually flow outward along the reconnected field lines to the region inside the field line separatrix, and thus ne is larger than ni in this region Then, this charge imbalance around the separatrix region causes an in-plane electrostatic ~P  e^z =BP ~es , which points mainly in the B electric field E direction (toward the mid-plane direction) as shown in Fig 1(c) However, further downstream around the region outside ~es of the magnetic field pileup (or fast shock-like) region, E ~P =BP direcchanges direction and points mainly in the e^z  B ~es is mainly perpendicular to the magnetic tion Note that E field line because the charge quasi-neutrality is maintained along the field line by the fast electron parallel motion ~es formation Again, we emphasize that the mechanism of E around the separatrix region has not been discussed before VIII ION DYNAMICS ACROSS SEPARATRIX REGION Although some ions flow from the upstream through the reconnection current layer into the downstream, most ions flow across the magnetic field line separatrix into the downstream region Outside the separatrix region, the ions have a ~z ~d ¼ cE drift-Maxwellian distribution with drift velocity V ~ ~P  e^z =BP and the thermal spread of ÂB=B $ (0.03 c)B $ 0.015 c With the drift velocity, the ion gyroradius is ~es ¼ Ees B ~P  e^z =BP $1.5 c/xce, which is comparable to the E localization width of $2 c/xce (estimated from Fig 1(c)) Thus, the ion dynamics must be studied by considering the ~ are rather uniform ~z and B full orbit motion Because E ~z perpenaround the separatrix region, the components of E ~P e^z =BP ị ~ are E ~?1 ẳ Ez BP =BÞðB dicular and parallel to B ~ Â^ e B and E jj ẳ Ez Bz =Bị^ e B , which are also quite uniform Thus, around the separatrix region, the ion dynamics is con~es and E ~?1 trolled by the perpendicular electric fields E ~es ãE ~?1 ẳ 0) and the parallel electric field E ~jj (E Figure 16 shows schematically the ion orbits due to the ~?1 when the ions move ~es and E perpendicular electric field E across the separatrix region It is to be noted that across the Phys Plasmas 22, 101205 (2015) ~ ¼ B^ FIG 16 The ion orbits perpendicular to the magnetic field B e B as they move across the separatrix region under the influence of the localized elec~?1 ¼ ðEz BP =BÞðB ~P ~P  e^z =BP and E ~es ¼ Ees B trostatic electric field E ~z perpendicular to B ~ Â^ e z =BP Þ Â e^B , which is the component of E ~es , E ~?1 and B ~ change separatrix region the directions of E slowly during the quasi-steady reconnection stage The ion velocity is accelerated by the strong localized electrostatic ~es (Ees/B0 $ 0.1 from Fig 1(c)) and the weaker inducfield E ~?1 (jE ~?1 j=B0 $ 0.03 estimated from Figs tive electric field E 1(a) and 9(a) and 9(b)), and the ion perpendicular velocity ~ centrifugal force direction is controlled by the q~ v  B=c Because the magnetic field is strong (B=B0 $ from Fig ~es region and the incoming ions 9(b)) near the localized E ~es , the ion velocity drift velocity is in the same direction as E ~es as they move toward the mid-plane is accelerated by E However, as the ions are accelerated, their velocity becomes ~ centrifugal force, larger and they experience larger q~ v  B=c which is in the opposite direction to and is larger than the ~?1 force, so that their orbit starts to gyrate counterqE ~ Note that ions with larger inward driftclockwise around B ing velocity have larger gyro orbits As the ions gyrate back ~es , ~es region, their velocity is decelerated by E to the strong E ~?1 As the ions further gyrate to but are still accelerated by E ~es again and in the ~ the E es direction, they are accelerated by E ~?1 It should be noted that meantime are accelerated by E ~ ions are also accelerated by E jj , which points in the positive ~ direction in the upper right quadrant of the poloidal plane B ~jj j=B0 $ 0.02–0.03 ~jj is quite significant with jE Because E (from Fig 9(c)), it is expected that ions can be accelerated to high positive vjj From the ion orbits sketched in Fig 16, we can now understand the ion velocity distribution fi v1 ; v2 ; vjj ị at xcet ẳ 481.91 in several boxed areas across the separatrix ~P  e^z =BP ị region shown in Fig 17, where v1 ẳ ~ v •ðB ~ ~ ~ v •B P  e^z =BP , and vjj ẳ ~ v ãB=B Outside the B=Bị, v2 ẳ ~ separatrix region (the top boxed area in Fig 17), the ion velocity distribution fi ðv1 ; v2 ; vjj Þ is an inward drifting ~d $ ð0:025cÞB ~P Maxwellian distribution with drift velocity V Â^ e z =BP Around the separatrix region (the second boxed area from the top), the ion velocity distribution consists of the inward drifting ions and a smaller population of ions ~ (inside the red circles in Fig 17) that have gyrated around B ~jj are strong, ~es and E once Inside the separatrix region, E ~es to and the inward drifting ions are accelerated mainly by E 101205-17 Cheng et al Phys Plasmas 22, 101205 (2015) ~P  e^z =BP Þ FIG 17 The ion velocity distribution fi ðv1 ; v2 ; vjj Þ sampled in the boxed areas across the separatrix region at xcet ¼ 481.91, where v1 ¼ ~ v •ðB ~P  e^z =BP , and vjj ẳ ~ ~ ~ v ãB v ãB=B B=Bị, v2 ¼ ~ ~?1 because the gain large v2 , but are not accelerated by E ~ ~?1 force q~ v  B=c centrifugal force is opposite to the qE ~?1 effect Also, and approximately cancels out the qE because the inward drifting ions move quickly across the ~jj separatrix region, they are only weakly accelerated by E because the time for acceleration is short and thus their vjj is quite small However, the gyrating ions not gain energy ~es in one orbit gyration, but after one orbit gyration, from E ~es to gain v2 while they move furthey are accelerated by E ther toward the mid-plane In the meantime, they are acceler~?1 is in ~?1 to gain negative v1 (jv1 j > Vd ) because E ated by E ~ ~ the opposite direction to ðB P  e^z =BP Þ Â ðB=BÞ, and are ~jj to vjj > jv1 j Further inward to the downaccelerated by E stream in the third boxed area from the top, the inward drifting ion density decreases, and the density of gyrating ions increases to become the dominant population The gyrating ~?1 to gain ~es and less by E ions are accelerated mainly by E ~?1 directions, so that the gyrating ~es and E velocity in the E ion perpendicular velocity distribution has large velocity spread in v2 and smaller velocity spread in v1 due to acceler~ ~?1 ỵ~ v  B=c) Note that the gyrating ions also ation by (E ~jj because have large positive vjj due to acceleration by E there is sufficient time to accelerate these ions Further inside the downstream region (in the bottom boxed area in Fig 17), the gyrating ions are the dominant population and they are ~es to gain larger v2 , but their accelerated further mainly by E ~ centrifuv  B=c v1 velocity is about the same because the q~ ~ gal force approximately cancels out the qE ?1 force Thus, the gyrating ions form a cigar-shape distribution in the (v1 , v2 ) space with velocity spread mainly in v2 From the above discussion, the physical picture of how the ion velocity distribution changes across the separatrix region (shown in Fig 17) can be qualitatively explained by the schematic ion orbits shown in Fig 16 However, to perform quantitative analysis, we should consider the detailed structure of the electric and magnetic fields From Figs 16 and 17, we can estimate how ions gain energy and how the ion flow velocity changes Before the ions enter the separatrix region from the upstream, the ion velocity distribution is a drift-Maxwellian distribution with ~es direction at the drift velocity of $0.02 c in the E xcet ¼ 481.91 After crossing the separatrix region, the ion ~es direction is $0.03 c, which is mean flow velocity in the E slightly larger than the incoming ion drift velocity in the top boxed area, but has large thermal spread of $ 0.04 c Because the thermal spread in the cigar-shape distribution is large, the ion thermal energy is larger than the flow kinetic ~?1 direction, the ion flow ~es direction In the E energy in the E velocity increases from to $0.015 c with the thermal spread of $0.02 c Again, we emphasize that the thermal ~ results from the spread in the direction perpendicular to B ~?1 , and q~ ~ forces ~es , qE v  B=c combined effect of the qE ~es are to Thus, the roles of the electrostatic electric field E thermalize the inward-drifting ion perpendicular velocity and accelerate the ions as they move across the separatrix region into downstream However, the ion flow kinetic energy is smaller than the thermal energy in this case ~jj Also, the orbit gyrating ion vjj is accelerated by E ~ (jE jj j=B0 $ 0.01–0.03) from $0 to $0.05 c with large ther~?1 are components ~jj and E mal spread of $0.03 c Because E ~ of E z , the ions gain energy mostly from the inductive electric ~z The poloidal distribution of the ion parallel flow vefield E locity Vijj at xcet ¼ 481.91 is shown in Fig 15(b), and we note that the ion parallel flow velocity is Vijj > in the upper 101205-18 Cheng et al ~jj accelright quadrant of the downstream region due to the E eration as explained here The ion flow velocities in the z-direction and in the poloidal magnetic field direction are related to the ion flow ~iz velocity components in the field-aligned coordinate by V ~iBp ẳ ẵVijj BP ỵ Vi1 Bz ị=B^ ẳ ẵVijj Bz Vi1 BP Þ=BŠ^ e z and V e BP , respectively Before the drifting ions enter the separatrix region, Vijj $ 0, Vi1 $0, BP/B $ 1, and Bz/B $ À0.1, so that the ion flow velocity components are Vi2 $0.02 c, ViBP $ 0, and Viz $ After the ions cross the separatrix region into the downstream, Vijj $ 0.05 c, Vi1 $ 0.01 c, BP/B $ 0.95, and Bz/B $ À0.3, so that the mean flow velocity components are Vi2 $ 0.03 c, ViBP $ 0.045 c, and Viz $À0.025 c Therefore, in the upper-right quadrant of the poloidal plane of the downstream region, the ion outflow velocity in the poloidal plane is mainly in the positive x-direction IX HOW ELECTRONS AND IONS GAIN ENERGY From the particle velocity distribution, we can calculate the average particle flow kinetic energy Wk and the average particle thermal energy, which is defined by Wth ẳ WtotWk, é x ;~ v ịd3 v is the average partiwhere Wtot ẳ 1=nị mv2 =2ịf ~ cle total energy and n is the particle density The average electron and ion energy distributions in the poloidal plane at xcet ¼ 481.91 are shown in Fig 18 for (a) the average electron flow kinetic energy Wke, (b) the average electron thermal energy Wthe, (c) the average ion flow kinetic energy Wki, and (d) the average ion thermal energy Wthi Even though the electron flow velocity is much larger than the ion flow velocity, the average ion flow kinetic energy is much larger than the average electron flow kinetic energy due to the large ionto-electron mass ratio mi/me ) FIG 18 The poloidal distribution of (a) the average electron kinetic flow energy Wke, (b) the average electron thermal energy Wthe, (c) the average ion kinetic flow energy Wki, and (d) the average ion thermal energy Wthi The contour lines are poloidal magnetic field lines Phys Plasmas 22, 101205 (2015) Figure 18(a) shows that the average electron flow kinetic energy Wke is large in the reconnection current layer and the separatrix region and the immediate downstream from the reconnection current layer In the separatrix region, electrons ~jj to gain large parallel flow velocity toare accelerated by E ward the reconnection current layer direction as shown in Fig 15(a) Inside the current layer, electrons also drift with ~x =B2 velocity in the positive z-direction to gain flow ~es  B cE ~z as discussed in kinetic energy due to acceleration by E Section V As discussed in Section VI, electrons also gain super-Alfvenic outflow velocity from the inflowing electrons, which has a flat-top vjj-distribution Electrons in the flat-top vjj-distribution that have outward moving parallel velocity enter the downstream of current layer and flow toward the downstream direction to give net outflow velocity In the immediate downstream outflow exhaust region from the cur~P =B2 velocity in ~es  B rent layer, the electrons drift with cE ~z , and drift with the positive z-direction to gain energy from E ~ ~ ~es ~ cE z  B P =B in the E es direction to lose energy by E Figure 18(b) shows that the average electron thermal energy Wthe is large in the downstream region This is mainly because when fast electron outflow velocity from the current layer and separatrix region reaches the downstream, the electrons are thermalized by the strong magnetic field by converting the flow kinetic energy into the thermal energy Figure 18(c) shows the distribution of the average ion flow kinetic energy Wki in the poloidal plane Wki is large in the reconnection current layer due to the acceleration of the ~z direction ~z to large vz in the E orbit meandering ions by E and in the outflow exhaust region due to acceleration of the ~jj as they move across the ~es and E orbit meandering ions by E separatrix region into the downstream as discussed in Sections VI D and VIII But Wki is largest in the fast shock~z via the shock surfing like region due to acceleration by E mechanism Figure 18(d) shows the distribution of the average ion thermal energy Wthi in the poloidal plane Wthi is large in the reconnection current layer and the downstream region In the current layer, the ion thermal spread is due to ~z in the negathe acceleration of orbit meandering ions by E tive z-direction and acceleration/deceleration of orbit mean~P  e^z =BP direction The velocity ~es in the B dering ion by E ~ spread caused by E es also reduces the ion inflow velocity in ~P  e^z =BP direction In the downstream region, Wthi is the B ~z and large due to acceleration of orbit meandering ions by E ~ E es , but also due to thermalization by the centrifugal force of the stronger magnetic field How particles gain or lose energy can also be understood from the energy exchange between the plasma and the ~ J~ dissipation, where J~ is the curelectric field through the E• rent density We calculate the energy dissipation due to the electron and ion current densities J~e and J~i separately We also consider the contributions due to the inductive electric ~es separately Figure 19 ~z and electrostatic field E field E ~es •J~e , (c) E ~z •J~i , ~ shows the distributions of (a) E z •J~e , (b) E ~ ~ and (d) E es •J i in the poloidal plane Figure 19(a) shows that ~z in the electrons gain both kinetic and thermal energy by E reconnection current layer and the separatrix region and the downstream fast shock-like region Figure 19(b) shows that electrons give energy to produce and maintain the 101205-19 Cheng et al ~ J~) distribution in the poloidal plane due FIG 19 The energy dissipation (E• ~es and ~z and the electrostatic electric field E to the inductive electric field E ~z •J~e , (b) the current density for electrons J~e and ions J~i separately: (a) E ~z •J~i , and (d) E ~es •J~i The contour lines are poloidal magnetic ~es •J~e , (c) E E field lines ~es in the reconnection current layer and electrostatic field E the separatrix region and the immediate downstream region ~es from the reconnection current layer, but gain energy by E in the fast shock-like region ~z in the Figure 19(c) shows that ions gain energy from E reconnection current sheet and the downstream and the fast shock-like region Figure 19(d) shows that ions gain energy ~es in the separatrix regions in the downstream, but from E ~es in the fast shock-like region It is to be lose energy to E emphasized that when ions flow from the upstream through the reconnection current layer or the separatrix region into the downstream (including the fast shock-like region), they ~z Only gain energy mostly from the inductive electric field E a smaller part of the ion energy is gained from the electro~es in the downstream region static field E The picture of energy exchange between the plasma par~ J~ ticles and the electric field through the consideration of E• is consistent with the poloidal distribution of the average electron and ion energy distributions shown in Fig 18 and the physical pictures obtained from the electron and ion velocity distributions shown in Sections V–VIII X SUMMARY AND CONCLUSION In this paper, we have made a significant advancement in the theoretical understanding of the physical processes of the electric and magnetic field evolution and the electron and ion dynamics in the collisionless driven magnetic reconnection in the absence of the external guide field In particular, a comprehensive theoretical analysis is provided to explain the 2–1/2 dimensional particle-in-cell simulation results of (1) the structure and change of the electric and magnetic fields, (2) the electron and ion orbits, velocity distributions, and Phys Plasmas 22, 101205 (2015) flow structures, and (3) the acceleration/heating of electrons and ions by the electric and magnetic fields In the 2–1/2 dimensional driven magnetic reconnection, the magnetic flux reconnection rate is related to the inductive electric field at the X-point, which is determined mainly by the driving electric field imposed at the upstream boundaries In general, we can consider the electron dynamics to be magnetized in the entire reconnection domain except in the small electron orbit meandering region inside the reconnection current layer The ion dynamics is unmagnetized (or demagnetized) around (1) the reconnection current layer where the ion orbits meander in the magnetic field reversal region and (2) the separatrix region and the downstream exhaust region from the reconnection current layer where the ion gyro-radii are comparable to or larger than the electrostatic electric field localization width The ions are magnetized in the far upstream and in the far downstream where the magnetic flux piles up to form a fast shock-like structure (or dipolarization front) Electrons flow from the upstream region mainly through the reconnection current layer into the downstream region While some ions flow through the reconnection region into the downstream, most ions flow across the separatrix into the downstream Thus, from the current layer into the downstream, the electron outflow velocity must be much faster than the ion outflow velocity, so that more electrons can flow to the downstream to avoid large charge non-neutrality in the downstream Then, from the reconnection current layer to the outflow exhaust region, the plasma current is determined mainly by the electron outflow velocity The large electron outflow jets from the reconnection current layer produce a pair of currents flowing into the reconnection current layer, which generate the out-of-plane quadrupole Bz magnetic field that is concentrated around the separatrix regions Because of the generation of the quadrupole Bz, the inductive ~jj ’ ~z has the parallel electric field E electric field E ~ ðEz Bz =B ÞB component in the separatrix and downstream region except in the mid-plane Then, around the separatrix ~jj region, the inward drifting electrons are accelerated by E along the field line to form a single beam vjj-distribution flowing toward the reconnection current layer direction The electron beams from both sides of the field line outside the separatrix arrive in the magnetic well region just outside of the reconnection current layer to form a flat-top vjjdistribution Around the reconnection current layer, the electrons flow to the electron orbit meandering region and accumulate the electron density inside the electron meandering region However, the ions flow to the ion orbit meandering region and accumulate the ion density inside the ion meandering region Thus, the electron density is larger (smaller) than the ion density inside (outside) the electron orbit meandering region The charge separation produces a pair of converging ~es , which are perpendicubipolar electrostatic electric field E lar to the merging magnetic field line and point toward the mid-plane As the flat-top vjj-distribution electrons flow toward the reconnection current layer, the electrons experience strong ~P =B2 drift velocity outside the electron orbit ~es  B cE 101205-20 Cheng et al meandering region Inside the electron orbit meandering region before the electrons move across the neutral sheet, ~es , but the electrons their inflow velocity is decelerated by E ~ are also accelerated by E z to gain large vz Both the strong ~P =B2 drift velocity and the direct electron vz accel~es  B cE ~z contribute to the enhancement of the out-oferation by E ~z direction inside the E ~es plane electron current J~ez in the E localization region After the electrons flow into the reconnection current layer, their flat-top vjj-distribution is preserved (where vjj is approximately parallel to vx) Then, inside the reconnection current layer, the electron parallel velocity is positive (negative) vx for x > (x < 0) and thus ~ex in the current determines the electron outflow velocity V layer and the downstream exhaust In the downstream, where the electric field and the magnetic field topology is substantially different from that in the upstream, the electrons expe~ B ~ drift and are accelerated by E ~jj Further rience E downstream, the magnetic field becomes stronger and the outflowing electrons are thermalized due to orbit gyration around the magnetic field Around the separatrix region, the vjj-beam electrons flow much faster along the field line toward the immediate upstream of the reconnection current layer and ions flow at slower velocity across the separatrix field lines into the downstream This causes the electron density to be smaller than the ion density outside the separatrix Inside the separatrix, the outflowing electrons have fast parallel velocity flowing along the field line toward the downstream direction Thus, inside the separatrix, electrons have higher density than the ions that move across the separatrix Then, the charge separation produces an in-plane electrostatic electric ~es , which points toward the mid-plane direction and is field E mainly perpendicular to the magnetic field line because the charge quasi-neutrality is maintained along the field lines by ~es around the sepathe fast electron parallel motion Thus, E ratrix region is mainly sustained by the fast electron parallel motion along the field lines around the separatrix region The ion dynamics through the current layer into the downstream is determined by the ion orbit meandering ~es , and the change of the ~z and E effect, the electric fields E magnetic field topology from the upstream to the downstream When ions move across the field lines into the orbit meandering region, the ions are accelerated (decelerated) by ~es before (after) they move across the neutral sheet, and E ~es alternatively then the ions are accelerated/decelerated by E as their orbits become meandering in the field reversal region The ion inflow velocity is reduced because of the acceleration/deceleration of the orbit meandering ion veloc~es , which causes large ion velocity spread Inside the ity by E ion meandering region, the ions are directly accelerated by ~z to gain ion flow velocity in the inductive electric field E directions both perpendicular and parallel to the magnetic ~ix around the reconnection field The ion outflow velocity V region is contributed mainly by the ion parallel flow velocity ~ijj V Around the separatrix region, the ion gyro-radii are ~es localization comparable or large comparing with the E width As ions move across the separatrix region, the ion orbit gyrates around the magnetic field The ions not gain Phys Plasmas 22, 101205 (2015) ~es in one orbit gyration, and the ion gyration energy from E ~es direction But, motion thermalizes the ion velocity in the E ~es to after one orbit gyration, the ions are accelerated by E gain energy In the meantime, the ions are accelerated con~jj , which is a compotinuously by the parallel electric field E ~ nent of E z , to gain large parallel velocity Ions gain both flow kinetic energy and thermal energy in the current layer and the downstream In the further downstream fast shock-like region where the magnetic flux piles ~z via up, ions gain much larger flow kinetic energy due to E the shock surfing mechanism It is to be emphasized that when ions flow from the upstream through the reconnection current layer or the separatrix region into the downstream (including the fast shock-like region), they gain energy ~z Only a smaller mostly from the inductive electric field E ~es in part of ion energy is gained from the electrostatic field E the downstream separatrix region Therefore, the main energy source for particle acceleration/heating in driven magnetic reconnection is the driving ~z , which penetrates to the entire inductive electric field E reconnection domain Around the reconnection current layer, ~z , the electrons gain energy mainly from acceleration by E and the ion meandering motion allows ions to be accelerated ~ ~z in directions both perpendicular and parallel to B by E Around the separatrix region, the electrons are magnetized ~jj , which is and are accelerated by the parallel electric field E ~ a component of E z , and the ion gyration motion allows ions ~jj as they move to have sufficient time to be accelerated by E across the separatrix region The role of the electrostatic ~z and B, ~ is ~es , which is perpendicular to both E electric field E to accelerate/decelerate the inward drifting ions in the reconnection current layer and the separatrix region Because of the ion meandering motion around the reconnection current layer, ions from both upstream sides are accelerated/deceler~es inside the reconnection current layer to cause ated by E ~es direction large velocity spread in the ion velocity in the E so that the ion inflow velocity is slowed down Around the separatrix region, the ion gyroradii are comparable to or ~es spatial location width across the separalarger than the E trix region, and thus ions are accelerated/decelerated during ~es ~es to cause large velocity spread in the E one gyration by E direction After one gyration motion, the ions are accelerated ~es again, and if they move into the downstream, they by E ~es causes the velocity thergain the inflow velocity Thus, E malization of the inward drifting ions, but also accelerates these ions if they leave the separatrix region and move into the downstream In summary, we have presented comprehensive physical mechanisms of key driven magnetic reconnection processes in collisionless plasmas Many interpretations of the key processes are new and have not been discussed before Moreover, it is to be noted that our physical interpretation of how the quadrupole magnetic field is generated and how the electron parallel velocity is accelerated by the parallel electric field around the separatrix region is different from the one presented by Uzdensky and Kulsrud,34 who made use of the argument of changes in the flux tube volume to explain the electron parallel flow near the separatrix Our physical picture of how electrons are accelerated by the parallel 101205-21 Cheng et al electric field along magnetic field lines on the upstream side of the separatrix region is also different from the electron surfing mechanism proposed by Hoshino,18 which did not consider the parallel electric field effect associated with the quadrupole Bz generation Our results of ion acceleration/ heating mainly by the inductive electric field are also different from the conclusion drawn in the previous papers4,5 that ~es across the field line the ions are mainly accelerated by E separatrix into the downstream The physical processes of driven magnetic reconnection presented in this paper are based on the simulation plasma parameters of mi/me ¼ 100, Ti =Te ¼ 1, xpe =xce ¼ with the background plasma density and the driving electric field Ez0 at the upstream boundaries chosen to be neb =ne0 ¼ 0:22 and Ez0 =B0 ¼ À0:04 Thus, at the upstream boundaries, the Alfven velocity is VA =c ¼ 0:0533, the ion plasma beta is bi % 0:32, and the inflow E  B drift velocity is Vd =VA ¼ cEz0 =VA B0 ¼ À0:75 In our opinion, the physical processes of driven magnetic reconnection in collisionless plasmas presented in this paper will not drastically change if different simulation parameters are used However, quantitative details of particle dynamics and acceleration/heating will depend on the plasma density, the plasma beta, and the driving electric field Ez0 imposed at the upstream boundary Furthermore, particle collisions should be considered in low temperature plasmas when the particle collision frequency is not much smaller than the ion cyclotron frequency These issues should be investigated in the future Finally, the physical picture of magnetic reconnection processes and particle dynamics presented in this paper cannot be achieved from the fluid models It is difficult to understand the physics from the fluid description mainly because the knowledge of electron and ion pressure tensors is not readily available and cannot be intuitively understood and must be determined from the higher order fluid equations.22,23,25 The effect of particle pressure tensor is very significant when the particle orbit is comparable or large with respect to the spatial scale lengths of the electric and magnetic fields Moreover, the effects of ion meandering orbits in the magnetic field reversal region and the ion orbit coupling with the electrostatic electric field that has spatial localization width smaller than or comparable to the ion gyro orbit in the separatrix region cannot be studied by using the fluid models ACKNOWLEDGMENTS This work was supported by University of Tokyo, a Grant-in-Aid for Scientific Research (A) No 22246119 and (B) No 19340170, JSPS Core-to-Core Program No 22001, NIFS Collaboration Research Programs (NIFS06KOAH02 and NIFS08KUTR023) in Japan, and by National Cheng Kung University and a research Grant No 100-2111M-006006 from the Ministry of Science and Technology in Taiwan A part of the paper was written when C Z Cheng was supported by National Fusion Research Institute, Korea, for a short term visit Phys Plasmas 22, 101205 (2015) M Yamada, R M Kulsrud, and H Ji, Rev Mod Phys 82, 603 (2010) Y Ono, H Tanabe, Y Hayashi, T Ii, Y Narushima, T Yamada, M Inomoto, and C Z Cheng, Phys Rev Lett 107, 185001 (2011) Y Ren, M Yamada, H Ji, S P Gerhardt, and R M Kulsrud, Phys Rev Lett 101, 085003 (2008) J Yoo, M Yamada, H Ji, and C E Myers, Phys Rev Lett 110, 215007 (2013) M Yamada, J Yoo, J Jara-Almonte, H Ji, R M Kulsrud, and C E Myers, Nat Commun 5, 4774 (2014) J R Wygant, C A Cattell, R Lysak, Y Song, J Dombeck, J McFadden, F S Mozer, C W Carlson, G Parks, E A Lucek et al., J Geophys Res 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Greco, J Geophys Res 115, A02209, doi:10.1029/2009JA014398 (2010) ...PHYSICS OF PLASMAS 22, 101205 (2015) Physical processes of driven magnetic reconnection in collisionless plasmas: Zero guide field case C Z Cheng,1,2 S Inoue,1 Y Ono,1 and R Horiuchi3... energy in the driven magnetic reconnection in collisionless plasmas for the zero guide field case are presented The key kinetic physics is the decoupling of electron and ion dynamics around the magnetic. .. kinetic simulations In this paper, we provide interpretation of the key dynamical processes of driven magnetic reconnection in collisionless plasmas from the first principle of particle Phys Plasmas