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Edited by: Vladimir I. Kolobov, CFD Research Corporation, United States

Reviewed by: Giovanni Lapenta, KU Leuven, Belgium; Maria Elena Innocenti, Jet Propulsion Laboratory, United States

This article was submitted to Plasma Physics, a section of the journal Frontiers in Physics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Magnetic reconnection is essentially a multi-scale phenomenon, driven by kinetic process in microscopic region and enabling explosive energy conversion from magnetic field energy to plasma kinetic energy in large area. It has been poorly understood how the kinetic process around the x-line connects to the magnetohydrodynamics (MHD) scale process in the reconnection downstream region. The present study has investigated the energy conversion process in the region far downstream of the x-line, by means of the particle-in-cell (PIC) simulation with the adaptive mesh refinement (AMR). The AMR-PIC model enables efficient kinetic simulation of multi-scale phenomena using dynamically adaptive meshes. It is found that the ion energy gain dominates in the reconnection region and is carried out mainly in the exhaust center rather than the exhaust boundaries. The simulation results suggest that the energy conversion process in collisionless magnetic reconnection is significantly different from that in the MHD reconnection model in which most energy conversion occurs at slow mode shocks formed at the exhaust boundaries.

Magnetic reconnection is a natural energy converter that allows explosive energy release of the magnetic field energy into plasma kinetic energy. The reconnection process is essentially multi-scale. The magnetic dissipation driving the reconnection process takes place in a localized region formed around the x-line, while it has a significant impact on large-scale dynamics of the planetary magnetosphere, leading to the global change of the field line configuration and the global plasma convection.

In collisionless reconnection, the dissipation (i.e., the effective resistivity) around the x-line is caused by the transport of the electron momentum in the diffusion region scaled by the electron kinetic scales [

So far, a number of particle-in-cell (PIC) simulations [

The question arising here is how the kinetic process around the x-line connects to the MHD-scale process far downstream of the x-line. Several simulation models have been recently developed to investigate the multi-scale dynamics of collisionless reconnection [

The previous PIC simulations with the adaptive mesh refinement (AMR) have suggested that the acceleration mechanisms of the ions and electrons in collisionless reconnection could be significantly different from those expected in the MHD reconnection model even far downstream of the x-line [

Because of the multi-scale nature of magnetic reconnection, numerical simulation is a strong and promising tool to lead to the comprehensive understanding of the underlying physics. It was the MHD simulation that pioneered simulation study of reconnection in a self-consistent manner [

The hybrid PIC simulation is also widely used in describing reconnection, where the particle ions are coupled with the mass-less fluid electrons. In the hybrid simulation, the ion kinetic physics is fully included as well as the Hall term in the generalized Ohm's law. However, the electron kinetic equations are not explicitly solved, so that the magnetic dissipation around the x-line is artificial and/or numerical. More serious problem in this model is that it can be contaminated by significant numerical noise [e.g.,

The full PIC simulation has been the most effective method to describe the evolution of collisionless reconnection [

Figure

Typical spatial and temporal scales in the Earth's magnetotail together with the ranges that each simulation model can cover. Here, λ_{De} is the electron Debye length, ρ_{s} = _{th, s}/ω_{cs} the gyro radius of species _{s} = _{ps} the inertia length of _{C} the collision mean free path with _{C} the collision time. The scales are estimated, based on _{i} = 1keV, _{e} = 100eV, ^{−3}, and

In order to bridge this gap, the present study has employed the adaptive mesh refinement (AMR) to the PIC code. The AMR technique subdivides computational cells locally in space and dynamically in time, achieving dynamically adaptive meshes to increase the dynamic range to be solved. For simulation of the current sheet evolution, the AMR-PIC code is designed to provide fine meshes only around the central current sheet where the plasma density is high, i.e., the electron scales are small, as represented by the electron inertia length λ_{e} = _{pe} (_{pe} are the speed of light and the electron plasma frequency, respectively). In most of the simulation domain outside the central current sheet, much coarser meshes are used, which enables much more efficient PIC simulation than the usual one, where all the simulation domain must be covered by the finest meshes. It is worth noticing that similar efforts have been made to bridge the gap between the electron and MHD scales, by developing a multi-level multi-domain method [

The AMR-PIC code uses the same equations as the usual explicit PIC code, where the Maxwell equations are solved on the staggering grids and the particle velocities and locations are advanced by the Buneman-Boris method. The main difference between the AMR-PIC and the usual PIC is in the data structure for the quantities defined on the grids. In the usual PIC code, the data is treated on an array with multiple dimensions consistent with spatial dimensions of the simulation domain. Each data location on the array corresponds to the spatial location in the simulation domain. Such the data structure has a great advantage in solving the finite differential equations, since the access to the data on the neighboring grids is very easy. In the AMR-PIC code, on the other hand, the data is aligned on a 1D array independent of the spatial location in the simulation domain. The 1D data structure enables flexible allocation of the data on the spatial grids, regardless of the grid size, so that it facilitates the implementation of the dynamically adaptive meshes. In order to solve the finite differential equations, each data residing closely in space is connected by a set of pointers that are used to identify the neighboring grids.

The fine cells are generated on the coarser cells. The Maxwell equations are solved on each cell level separately. In the region where different cell levels are overlapped, the numerical solutions from fine level cells are employed on coarse level cells. It is also necessary to define the boundary conditions for each cell level. They are provided due to an interpolation of the solutions from the coarser level cells, so that the magnetic fluxes through the cell surfaces are conserved. The particle velocities and locations are advanced using the field data on the finest level cells accessible at each particle's location. The AMR-PIC code performs particle splitting, when the particles move into finer cell region, and also particle coalescence, when they move out from the region. The particle splitting and coalescence are needed to control the number of particles per cell and to suppress the numerical noise raised due to low statistics of the particles.

Figure _{x}(_{0}tanh(_{b} and δ are the background density and the half width of the current sheet, respectively, and are set as _{b} = 0.044_{0} and δ = 0.5λ_{i} with λ_{i} = _{pi} the ion inertia length based on _{0}. The ion-to-electron mass ratio is _{i}/_{e} = 100. The other parameters are _{i}/_{e} = 3.0 and _{th, e}/_{L} ≥ 2.0λ_{De} or _{ey} ≥ 2.0_{A} at the center of each cell with the size Δ_{L}, where λ_{De} is the local electron Debye length and _{A} is the Alfvén velocity based on _{0} and _{0}. The criterion for the electron bulk velocity is helpful to pick up the locations where the electron-scale structures are likely formed due to the local super-Alfvénic acceleration. The number of the total refinement levels allowed in the simulation is fixed through the run to be four. The normalization parameters are _{i} for mass, _{i} for length, and _{A} for velocity. As seen in Figure

Results from 2D AMR-PIC simulation in the

The present study focuses on a large-scale evolution of collisionless magnetic reconnection. The simulation is performed in the 2D _{i} in this simulation. The boundary conditions employed are “open” both in the inflow (_{x} × _{z} = 655λ_{i} × 328λ_{i}, which is entirely covered by base-level cells (coarsest cells) with Δ_{LB} = 0.08λ_{i} and can be locally subdivided up to the dynamic range level with Δ_{LD} = 0.02λ_{i}. Thus, the highest resolution in space, evaluated by the effective number of the finest cells, is 32, 768 × 16, 384. The maximum number of particles used is ~10^{10} for each species, indicating that the simulation is carried out with only ~1TB memory. The analyses in this paper are performed for the base level cells.

The simulation is initiated with a small island to the out-of-plane component of the vector potential. This island provides a small perturbation to the magnetic field around the center of the simulation domain and facilitates the onset of magnetic reconnection. Once the reconnection process has been triggered, the rate of reconnection quickly reaches a quasi-steady value of _{R}~0.1, where _{R} is evaluated by the out-of-plane electric field at the x-line and is normalized to the instantaneous values in the inflow region. During the fast reconnection, the electron current layer formed around the x-line is elongated in the outflow direction and is subject to plasmoid formations. Repeating the electron layer elongation [_{i}.

In Figure _{ix}|≫|_{iz}|, is clearly separated from the inflow region characterized by the vertical flow with |_{ix}|≪|_{iz}|. Therefore, the isoline of |_{ix}|−|_{iz}| = 0, indicated by the thick black curves, can be a good indicator of the interface between the inflow and outflow regions. The local energy exchange between the electric and magnetic field and plasma is carried out through

where _{s} and

Results from 2D AMR-PIC simulation of magnetic reconnection at _{ci} = 140 in the _{i} = 485 and _{i} = 320. The locations of the 1D profiles in _{ix}|−|_{iz}| = 0. These locations are also indicated by dashed lines in

Figures _{i}, the energy conversion

It is interesting to note that the electrons lose energy in the most part of the exhaust except for the very localized region near the exhaust center (Figures _{e}, the electrons can be unmagnetized, so that they are effectively accelerated due to the reconnection electric field, leading to

In the region near the x-line, where the distance from the x-line is on the order of 10λ_{i}, it is found that the

The ion energy gain _{i} ≈ 306) for the same reason as in the exhaust. Instead of _{e} from the x-line), while most energy conversion occurs in the exhaust as modeled by Petschek [

It has been stated in earlier simulations with relatively smaller system [e.g., _{i}. Figure

Results from the same simulation run and at the same time as in Figure

The effect of plasmoid is also small in terms of

The present study has investigated multi-scale processes of collisionless magnetic reconnection by means of large-scale kinetic simulations. In particular, we have focused on the energy conversion process of the magnetic field energy into the plasma kinetic energy through reconnection. The goal of this study is to reveal how the kinetic process around the x-line connects to large-scale process far downstream of the x-line where the MHD approximation is believed to be valid. For this purpose, we have developed a multi-scale PIC simulation code with the AMR, which employs the dynamically adaptive meshes on the full particle simulation and enables efficient simulation of multi-scale kinetic processes. We have confirmed that the AMR-PIC code can dramatically save the computer resources for the simulation of the current sheet evolution, so that it works as a strong tool for simulating collisionless magnetic reconnection.

The large-scale AMR-PIC simulations have suggested that the energy conversion process in collisionless reconnection differs from that in the MHD reconnection model represented by the Petschek's model [

Schematic diagrams showing the ion acceleration process (red arrows) in

It is interesting to note that, near the x-line (on the order of 10λ_{i} from the x-line), the electrons are strongly accelerated at the exhaust boundaries, which is not dominant in the exhaust far downstream of the x-line (on the order of 100λ_{i} from the x-line). The electron acceleration is mainly due to an electrostatic potential hump (a double layer) that is locally formed in the separatrix regions. The further investigation will be needed to understand why such the double layers are formed only around the x-line and not in the far downstream region.

The simulation data are not available in public because of the large data size and can be obtained by contacting the author (

KF has developed the massively parallelized AMR-PIC code, carried out the large-scale simulations on a supercomputer system, analyzed the simulation data on a local workstation, and derived the scientific interpretation.

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The simulations were performed on the CX400 supercomputer system at Information Technology Center, Nagoya University, Japan.