Particle acceleration in kinetic simulations of non-relativistic magnetic reconnection with different ion-electron mass ratio

TL;DR

This study uses kinetic simulations to examine how varying the ion-electron mass ratio affects magnetic reconnection and particle acceleration, revealing that the mass ratio significantly impacts electron acceleration but not ion energization.

Contribution

It demonstrates that the ion-electron mass ratio influences electron acceleration processes during magnetic reconnection, providing insights into scale separation effects in kinetic simulations.

Findings

Mass ratio does not significantly affect reconnection rate or ion energization.

Electron acceleration efficiency decreases with increasing mass ratio.

High-energy electron acceleration via Fermi mechanism is less effective at larger mass ratios.

Abstract

By means of fully kinetic particle-in-cell simulations, we study whether the proton-to-electron mass ratio $m_i/m_e$ influences the energy spectrum and underlying acceleration mechanism during magnetic reconnection. While kinetic simulations are essential for studying particle acceleration during magnetic reconnection, a reduced $m_i/m_e$ is often used to alleviate the demanding computing resources, which leads to artificial scale separation between electron and proton scales. Recent kinetic simulations with high-mass-ratio have suggested new regimes of reconnection, as electron pressure anisotropy develops in the exhaust region and supports extended current layers. In this work, we study whether different $m_i/m_e$ changes the particle acceleration processes by performing a series of simulations with different mass ratio ($m_i/m_e=25-400$) and guide-field strength in a low-$\beta$ plasma. We find that mass ratio does not strongly influence reconnection rate, magnetic energy conversion, ion internal energy gain, plasma energization processes, ion energy spectra, and the acceleration mechanisms for high-energy ions. Simulations with different mass ratios are different in electron acceleration processes, including electron internal energy gain, electron energy spectrum and the acceleration efficiencies for high-energy electrons. We find that high-energy electron acceleration becomes less efficient when the mass ratio gets larger because the \textit{Fermi}-like mechanism associated with particle curvature drift becomes less efficient. These results indicate that when particle curvature drift dominates high-energy particle acceleration, the further the particle kinetic scales are from the magnetic field curvature scales ($\sim d_i$), the weaker the acceleration will be.

Tables

Table 1. Table 1: Comparing particle description and fluid description of the energization processes

Particle description	Fluid description (Equ. A1)
inertial drift (Equ. B3)	curvature drift + part of flow inertial term
curvature drift (part of inertial drift Equ. B3)	curvature drift
gradient drift (Equ. B2)	gradient drift
parallel drift + betatron acceleration (Equ. B4 and B5)	magnetization
polarization drift (Equ: B6)	part of flow inertial term
parallel guiding-center velocity	parallel flow velocity
$𝑬\times 𝑩$ drift	$𝑬\times 𝑩$ drift

Equations

$${\mathbf{j}}_{s\perp }=p_{s\parallel }\frac{\mathbf{B}\times (\mathbf{B}\cdot \nabla )\mathbf{B}}{B^4}+p_{s\perp }\frac{\mathbf{B}\times \nabla B}{B^3}-{\left[\nabla \times \frac{p_{s\perp }\mathbf{B}}{B^2}\right]}_{\perp }+{\rho }_s\frac{\mathbf{E}\times \mathbf{B}}{B^2}-n_s{m}_s\frac{d{\mathbf{u}}_s}{dt}\times \frac{\mathbf{B}}{B^2},$$

$${\mathbf{j}}_{s\perp }\cdot {\mathbf{E}}_{\perp }$$

$${⟨{\mathbf{v}}_g⟩}_{\varphi }=v_{\parallel }\mathbf{b}+{\mathbf{v}}_E+\frac{M}{q_s}\frac{\mathbf{B}\times \nabla B}{B^2}+\frac{m_s{v}_{\parallel }}{q_{s}B}\mathbf{b}\times \frac{d\mathbf{b}}{dt}+\frac{M}{q_s}(\nabla \times \mathbf{b}{)}_{\parallel }+\frac{m_s}{q_{s}B}\mathbf{b}\times \frac{d{\mathbf{v}}_E}{dt}$$

$$\int q_s\frac{M}{q_s}\frac{\mathbf{B}\times \nabla B}{B^2}f{d}^{3}p=p_{s\perp }\frac{\mathbf{B}\times \nabla B}{B^3}.$$

$$\int q_s\frac{m_s{v}_{\parallel }}{q_{s}B}\mathbf{b}\times \frac{d\mathbf{b}}{dt}f{d}^{3}p$$

$$=$$

$$\int q_s\frac{M}{q_s}(\nabla \times \mathbf{b}{)}_{\parallel }f{d}^{3}p$$

$$=$$

$$\left(\nabla \times \frac{p_{s\perp }\mathbf{B}}{B^2}\right)\cdot \mathbf{E}$$

$$=$$

$$\int q\frac{m_s}{qB}\mathbf{b}\times \frac{d{\mathbf{v}}_E}{dt}f{d}^{3}p$$

$$\int q_s{⟨{\mathbf{v}}_g⟩}_{\varphi }\cdot \mathbf{E}f{d}^{3}p+\frac{p_{s\perp }}{B}\frac{\partial B}{\partial t}$$

Full text

acceleration of particles — magnetic reconnection — Sun: flares — Sun: corona

††journal: ApJ

1 Introduction

In many solar, space and astrophysical systems, magnetic reconnection is a major mechanism for energizing plasmas and accelerating nonthermal particles (Zweibel & Yamada, 2009). A remarkable example is solar flares, where reconnection is observed to trigger efficient magnetic energy release (Lin & Hudson, 1976), heats the coronal plasma (e.g. Caspi & Lin, 2010; Longcope et al., 2010), and accelerates both electrons and ions into nonthermal power-law energy distributions (Krucker et al., 2010; Krucker & Battaglia, 2014; Oka et al., 2013, 2015; Shih et al., 2009). However, how particles are accelerated over a large-scale reconnection region is still not well understood.

The dynamics of magnetic reconnection is believed to involve both the macroscopic scales ($>10^{6}$ m in solar flares) and the kinetic scales ($<10$ m in solar flares) (Daughton et al., 2009; Ji & Daughton, 2011), and thus a multi-scale approach is essential for understanding particle acceleration during reconnection. Starting from the kinetic scales, kinetic simulations (fully kinetic or hybrid) are often used to study how particles are accelerated and coupled with background fluids (e.g. Drake et al., 2006). Various models are then developed to capture these processes for studying the macroscopic particle acceleration (Drake et al., 2018; le Roux et al., 2015, 2016, 2018; Li et al., 2018b; Montag et al., 2017; Zank et al., 2014, 2015) and are applied in explaining local particle acceleration between interacting flux ropes in the solar wind (Zhao et al., 2018, 2019; Adhikari et al., 2019). Previous kinetic simulations have identified that the reconnection X-line region (Hoshino et al., 2001; Drake et al., 2005; Fu et al., 2006; Oka et al., 2010; Egedal et al., 2012, 2015; Wang et al., 2016) and contracting and merging magnetic islands (Drake et al., 2006; Oka et al., 2010; Liu et al., 2011; Drake et al., 2013; Nalewajko et al., 2015) are the major particle acceleration sites during reconnection. Under the guiding-center approximation, recent simulations have also identified particle curvature drift motion along the motional electric field as the major particle acceleration mechanism (Dahlin et al., 2014; Guo et al., 2014, 2015; Li et al., 2015, 2017). Li et al. (2018a) further showed that the flow compression and shear effects well capture the primary particle energization, as in the standard energetic particle transport theory (Parker, 1965; Zank, 2014; le Roux et al., 2015). Fluid compression and shear effects have also been used to quantify plasma energization during island coalescence problem (Du et al., 2018). The connection between particle acceleration associated with particle drift motion and that related to fluid motion is summarized in Appendix of current paper. Li et al. (2018a) also found that flow compression and shear are suppressed as the guide field increases. To alleviate the computational cost, these previous simulations were mostly carried out using a reduced proton-to-electron mass ratio $m_{i}/m_{e}=25$.

A higher mass ratio ($m_{i}/m_{e}\geqslant 400$), however, can potentially change the plasma energization and particle acceleration processes, because different magnetic field, currents, and pressure anisotropy structures emerge as $m_{i}/m_{e}$ becomes larger (e.g. Egedal et al., 2013, 2015; Le et al., 2013). By performing kinetic simulations of reconnection with different mass ratio, guide field, and plasma $\beta$, Le et al. (2013) demonstrated that the magnetic fields and currents fall into four regimes, and that the transition guide field between different regimes changes with the mass ratio and plasma $\beta$. They also identified a new regime with an extended current layer only when $m_{i}/m_{e}\geqslant 400$. Those works were mostly focused on the dynamics and structures in the reconnection layer. Therefore, it is worthwhile to understand how the mass ratio influences the plasma energization and particle acceleration processes.

In this paper, we focus on the consequences of having a disparity between the energy releasing scale (the radius of magnetic curvature $\sim$ the ion inertial length $d_{i}$) and the plasma kinetic scales (the electron gyroradius $\rho_{e}$ to $d_{i}$). The scale separation between electrons and protons becomes larger as the mass ratio approaches the realistic value. For example, $\rho_{e}/d_{i}=\sqrt{\beta_{i}}\sqrt{T_{e}/T_{i}}\sqrt{m_{e}/m_{i}}$ decreases with $m_{i}/m_{e}$, where the ion plasma $\beta_{i}$ and the temperature ratio $T_{e}/T_{i}$ are usually fixed.

Here we perform fully kinetic particle-in-cell simulations with $m_{i}/m_{e}=25$, 100, and 400 to study whether the mass ratio changes the plasma energization and particle acceleration processes during magnetic reconnection. For each mass ratio, we perform four runs with different guide field: 0, 0.2, 0.4, and 0.8 times of the reconnection magnetic field, so the series of simulations covers all the regimes studied by Le et al. (2013). In Section 2, we describe the simulation parameters. In Section 3, we present the results on how the energy conversion, reconnection rate, particle energy spectra, plasma energization processes, and particle acceleration rates change with the mass ratio and the guide field strength. In Section 4, we discuss the conclusions and the implications based on our simulation results.

2 Numerical Simulations

We carry out 2D kinetic simulations using the VPIC particle-in-cell code (Bowers et al., 2008), which solves Maxwell’s equations and the relativistic Vlasov equation. The simulations start from a force-free current sheet with $\bm{B}=B_{0}\tanh(z/\lambda)\hat{x}+B_{0}\sqrt{\operatorname{sech}^{2}(z/\lambda)+b_{g}^{2}}\hat{y}$, where $B_{0}$ is the strength of the reconnecting magnetic field, $b_{g}$ is the strength of the guide field $B_{g}$ normalized by $B_{0}$, and $\lambda$ is the half-thickness of the current sheet. Note that for this paper we will use $B_{g}$ and $b_{g}$ interchangeably when it does not cause confusion. We preform simulations with $B_{g}=0.0$, 0.2, 0.4, and 0.8 in three mass ratios: 25, 100, and 400. All simulations have the same Alfven speed $v_{\text{A}}$ ($=B_{0}/\sqrt{4\pi n_{0}m_{i}}$) and electron beta $\beta_{e}=8\pi nkT_{e}/B_{0}^{2}$ defined using reconnecting component of the magnetic field. We choose $\lambda=d_{i}$ for all simulations, where $d_{i}=c/\omega_{\text{pi}}=c/\sqrt{4\pi n_{i}e^{2}/m_{i}}$ is the ion inertial length. The initial particle distributions are Maxwellian with uniform density $n_{0}$ and temperature $T_{i}=T_{e}=T_{0}$. The temperature is taken to be $kT_{0}=6.25\times 10^{-4}m_{e}c^{2}$, $0.0025m_{e}c^{2}$, and $0.01m_{e}c^{2}$ for runs with $m_{i}/m_{e}=25$, 100, and 400, respectively, where $m_{e}c^{2}$ is fixed for runs with different mass ratio. Electrons are set to have a bulk velocity drift $U_{e}$ so the Ampere’s law is satisfied. The ratio of electron plasma frequency and electron gyrofrequency $\omega_{pe}/\Omega_{ce}=4$, 2, and 1 for runs with $m_{i}/m_{e}=25$, 100, and 400, respectively. The resulting Alfvén speed is $0.05c$ and the electron beta is 0.02 for all simulations. The domain sizes are $L_{x}\times L_{z}=100d_{i}\times 50d_{i}$, and the grid sizes are $8192\times 4096$ for all simulations. Figure 1 shows that the electron kinetic scales ($\rho_{e}$ and $d_{e}$) deviate more from the energy releasing scale ($\sim d_{i}$) as the mass ratio becomes larger. We use 400 particles per cell per species in all simulations. As the mass ratio increases, both the plasma skin depth and gyroradius are at scales shorter than one ion skin depth. For electric and magnetic fields, we employ periodic boundaries along the $x$-direction and perfectly conducting boundaries along the $z$-direction. For particles, we employ periodic boundaries along the $x$-direction and reflecting boundaries along the $z$-direction. Initially, a long wavelength perturbation with $B_{z}=0.02B_{0}$ is added to induce reconnection (Birn et al., 2001). The simulations are terminated around $t\Omega_{ci}=100$ (one Alfvén crossing time) to minimize the effect of the periodic boundary conditions along the $x$-direction.

3 Results

3.1 Current Layer Structures

As the simulations proceed, current layers are unstable to the tearing instability, leading to fractional sheets filled with magnetic islands. Figure 2 shows the out-of-plane current density $j_{y}$ for runs with three mass ratios $25-400$ with different guide fields from $B_{g}=0$ to $0.8$. The time steps shown are $t\Omega_{ci}=80$, 83, and 86 for $m_{i}/m_{e}=25$, 100, and 400, respectively. We choose slightly different time frames because reconnection onsets slightly faster in the runs with a lower mass ratio. Overall, the current layers vary in length and are oriented along different directions depending on the guide-field strength. In the low guide-field regime, an elongated current layer emerges because of an unmagnetized electron jet formed in the electron diffusion region (panels (b), (e), (f), and (i)). Since there is a finite $B_{y}$ field in the center of a force-free current sheet even when $B_{g}=0$, electrons could be magnetized in the low guide-field regime, and localized current layers are formed instead (panels (a) and (j)). A new regime, first studied by Le et al. (2013), emerges with extended current layers embedded in the reconnection exhaust when $m_{i}/m_{e}\geqslant 400$ (panel (k)). These current layers can extend over $20d_{i}$ and therefore might affect particle energization processes. In contrast, the current layers are shorter in runs with a lower mass ratio (panels (c) and (d)). As the guide field gets even stronger (panels (d), (h), and (l)), the electrons become well magnetized, and the current density tends to peak at one of the diagonal branches of the reconnection separatrix. Le et al. (2013) studied the scaling extensively and found that these structures are regulated by the electron pressure anisotropy and the properties of the electron orbits, which depend on the mass ratio and guide field. The scaling in our simulations does not exactly match with the diagram by Le et al. (2013) (Figure 3 in their paper). This is because the force-free current sheet (different from the Harris current sheet used in Le et al. (2013)) has a finite magnetic field along the guide-field direction in the center of the current sheet even when $B_{g}=0$, and also because these structures are dynamic and can be destroyed as the simulations evolve. In summary, different mass ratio results in different types of current layers, especially when $B_{g}\leqslant 0.4$. In the following sections we will study whether the mass-ratio dependence influences the mechanisms for plasma energization and particle acceleration processes.

3.2 Reconnection Rate

Before diving into the energization processes, we check whether the different mass ratio changes the reconnection rate. Following Daughton et al. (2009), we evaluate the normalized reconnection rate $E_{R}\equiv(\partial\psi/\partial t)/(B_{0}v_{\text{A}})$, where $\psi=\max(A_{y})-\min(A_{y})$ along $z=0$, $A_{y}$ is the $y$ component of the vector potential, and $v_{\text{A}}\equiv B_{0}/\sqrt{4\pi n_{0}m_{i}}$ is the Alfvén speed defined by $B_{0}$ and the initial particle number density $n_{0}$. Figure 3 shows that the reconnection rate for various cases. Since we do not average the rate over a long time interval (Daughton et al., 2009), the rate fluctuates rapidly. Figure 3 shows that the reconnection onset is faster in the runs with a lower mass ratio. In the following analysis, unless specified otherwise, we will shift the runs with $m_{i}/m_{e}=100$ by $-3\Omega_{ci}^{-1}$ and the runs with $m_{i}/m_{e}=400$ by $-6\Omega_{ci}^{-1}$ to match the reconnection onset. Figure 3 shows that the reconnection rate is roughly the same for runs with different mass ratio. The peak reconnection rate is about 0.1 for runs with $B_{g}\leq 0.4$, consistent with previous kinetic simulations (e.g. Birn et al., 2001). The peak rate does not sustain, because the periodic boundary conditions limit the simulation durations (Daughton et al., 2006), and because we use the upstream plasma parameters ($B_{0}$ and $n_{0}$) instead of that near the dominant reconnection $x$ point (Daughton & Karimabadi, 2007). At $t\Omega_{ci}=100$ (one Alfvén crossing time), $E_{R}$ decreases to about $0.06$. We will terminate our analysis at $t\Omega_{ci}=100$, when only a few large islands and smaller secondary islands are left in the simulations.

The evolution of reconnection rate shows that the runs are similar in the reconnected magnetic fluxes. Previous kinetic simulations have shown that the converted magnetic energy can be channelled into plasma kinetic energy preferentially by the parallel electric field $\bm{E}_{\parallel}$ near the reconnection $X$-line and by the Fermi-like mechanism associated with contracting and merging magnetic islands (e.g. Dahlin et al., 2014; Guo et al., 2014; Li et al., 2015, 2017). $\bm{E}_{\parallel}$ accelerates particles proportionally to their velocities; the Fermi-like mechanisms accelerate particles proportionally to their energies. The dominant mechanism could be different for particles with different energies and for electrons and ions. The mass ratio could change the relative importance of these mechanisms, leading to different particle energy distributions and energy partition between electrons and ions. The following analysis will show how the mass ratio changes the plasma energization and particle acceleration processes.

3.3 Energy Conversion

We start investigating the energization processes by examining the energy evolution in the simulations. Figure 4 shows the energy conversion in these simulations till $t\Omega_{ci}=100$. Panel (a) shows the time evolution of the change of the magnetic energy $\Delta\varepsilon_{B}$, the electron energy gain $\Delta K_{e}$, and the ion energy gain $\Delta K_{i}$ in the runs with $B_{g}=0.2$ . We normalize them by the initial energy of the $x$ component (reconnecting component) of the magnetic field $\varepsilon_{Bx0}=B_{x}^{2}/(8\pi)$. Similar fraction of magnetic energy (11% of $\varepsilon_{Bx0}$) is converted into plasma kinetic energy in all runs. Panel (b) shows that slightly more magnetic energy is converted in runs with $m_{i}/m_{e}=25$ and 400 when $B_{g}=0.0$ or 0.8, and similar fraction of magnetic energy is converted for the other cases. Panel (a) shows that as $m_{i}/m_{e}$ gets larger, electrons gain less energy, resulting in about 31%, 28%, and 21% of $\Delta\varepsilon_{B}$ going into electrons in the runs with $m_{i}/m_{e}=25$, 100, and 400, respectively. Panel (b) shows that the difference gets smaller as $B_{g}$ increases. When $B_{g}=0.8$, electrons gain a similar fraction of converted magnetic energy in runs with different mass ratio. Panel (a) also shows that ions gain less energy first and then more energy to the end of the simulation with $m_{i}/m_{e}=400$. Panel (b) shows that ions do gain more energy in runs with $m_{i}/m_{e}=400$ than the other runs, except when $B_{g}=0.8$, ions gain most energy in the run with $m_{i}/m_{e}=25$. The guide-field dependence of different energies shown in panel (b) is consistent for different mass ratio despite the differences in their actual values.

Since the reconnection outflow is about the Alfvén speed $0.05c$, the ion bulk energy is significant in our simulations. Panel (c) shows that, depending on the guide field, the ion bulk energy is comparable with or even larger than the ion internal energy. Panel (c) also shows that the ion bulk energy is larger in the runs with $m_{i}/m_{e}=400$ except when $B_{g}=0.8$, and that it does not change much with the guide field when $m_{i}/m_{e}=25$, while it generally gets weaker as $m_{i}/m_{e}$ becomes larger. In contrast, the ion internal energy always decreases as $B_{g}$ gets larger, and the difference between different mass ratio is subtle. As a result, $\Delta K_{i}/\Delta K_{e}$ does not show clear dependence on the guide field, while $\Delta U_{i}/\Delta U_{e}$ decreases as $B_{g}$ becomes larger (panel (d)). When $B_{g}=0.8$, $\Delta U_{i}/\Delta U_{e}$ approaches one for the cases with $m_{i}/m_{e}=100$ or 400 and becomes even smaller in the run with $m_{i}/m_{e}=25$. Panel (d) also shows that $\Delta U_{i}/\Delta U_{e}$ is much larger in runs with a higher mass ratio, especially in the low guide-field cases. We expect $\Delta U_{i}/\Delta U_{e}$ will be even larger in simulations with a real $m_{i}/m_{e}=1836$. In summary, a lower mass ratio helps reconnection to convert more magnetic energy into electron kinetic energy and a similar amount of magnetic energy into ion internal energy, which changes the energy partition between electrons and ions. Then, the next question is whether a different mass ratio results in different electron distributions but similar ion distributions, which we now discuss.

3.4 Particle Energy Distributions

Figure 5 shows the normalized electron energy spectra for all electrons at $t\Omega_{ci}=40$, 60, and 94. Electrons are accelerated to over 100 times of the initial thermal energy $\varepsilon_{\text{th}}$ in all runs. The accelerated electrons develop a significant high-energy tail ($>10\varepsilon_{\text{th}}$), which contains 0.7–4% of all electrons and 7–38% of the total electron kinetic energy to the end of the simulations ($t\Omega_{ci}=100$). Top panels show that electrons quickly reach $100\varepsilon_{\text{th}}$, and that the acceleration is faster in the runs with $m_{i}/m_{e}=100$ or 400. As studied by previous kinetic simulations, parallel electric field $\bm{E}_{\parallel}$ plays a key role in the acceleration, for that $E_{\parallel}$ not only accelerates most electrons near the reconnection X-line (Li et al., 2017; Lu et al., 2018) but also forms pseudo electric potential wells, which trap electrons so that they can be further accelerated by perpendicular electric field $\bm{E}_{\perp}$ (Egedal et al., 2015). As a result, most electrons near the X-line are accelerated to develop flat spectra that appear to be hard power-law distributions for $\varepsilon\in[20,50]\varepsilon_{\text{th}}$ (Li et al., 2015). But these spectra are usually transient, because they only contain less than 10% of the high-energy electrons ($>10\varepsilon_{\text{th}}$) at $t\Omega_{ci}=100$, and because these electrons are trapped near the center of the primary magnetic islands (Li et al., 2017). As the simulations evolve to $t\Omega_{ci}=60$ (panels (e)–(h)), the electron acceleration in the runs with $m_{i}/m_{e}=25$ catches up and becomes the strongest especially in runs with $B_{g}=0.4$ or 0.8 (panels (g) and (h)). The spectra appear to be power-law distributions with a power index -3.5 (dashed lines) for $\varepsilon\in[10,100]\varepsilon_{\text{th}}$, especially in the runs with $B_{g}=0$ and 0.2 (panels (e) and (f)). But these spectra are actually the superposition of a series of thermal-like distributions in different sectors of a 2D magnetic island (Li et al., 2017). To the end of the simulations (panels (i)–(l)), the separation between different mass ratio becomes even larger. The spectra in the runs with $m_{i}/m_{e}=25$ still appear to be power-laws with as power index -3.5, and the spectra are much steeper in the runs with higher mass ratios. These results indicate that a lower proton-to-electron mass ratio tends to overestimate the high-energy electron acceleration.

Figure 6 shows the normalized ion energy spectra for all ions at $t\Omega_{ci}=40$, 60, and 94. Ions are accelerated up to $500\varepsilon_{\text{th}}$, higher than electrons. The accelerated ions develop significant high-energy tails. At the beginning ($t\Omega_{ci}=40$), ions are quickly accelerated to the reconnection outflow speed $\approx v_{\text{A}}$. This process does not increase the ion internal energy much but rather accelerates all ions in the reconnection exhausts to a bulk kinetic energy of $0.5m_{i}v_{\text{A}}^{2}$. We find that the acceleration is associated with particle polarization drift instead of the parallel electric field as for electrons (more discussion in Figure 9). As the simulations evolve to $t\Omega_{ci}=60$ (panels (e)–(h)), the spectra in the runs with $m_{i}/m_{e}=25$ and 100 are close to each other, and the fluxes of high-energy ions in the runs with $m_{i}/m_{e}=400$ are still lower. The spectra appear to be power-laws $\propto\varepsilon^{-1}$ for $\varepsilon$ around $10\varepsilon_{\text{th}}$. The high-energy tail is likely a drift Maxwellian distribution with a drift energy $\approx 0.5m_{i}v_{\text{A}}^{2}$, because the break point of the spectra is about $0.5m_{i}v_{\text{A}}^{2}$ (vertical solid line in panel (f)). To the end of the simulations (panels (i)–(l)), the low-energy part is still a hard power low $\propto\varepsilon^{-1}$, and the high-energy tail becomes harder and resembles a power-law $\propto\varepsilon^{-6}$. The spectra in the runs with $m_{i}/m_{e}=400$ are still steeper when $B_{g}=0.0$ or 0.2, but the difference is obvious only at the highest energies ($\varepsilon>200\varepsilon_{\text{th}}$). The spectra in the runs with $B_{g}=0.4$ or 0.8 are close to each other. We find that high-energy particles ($>0.5m_{i}v_{\text{A}}^{2}$) are further accelerated by the Fermi-like mechanism associated with particle curvature drift (more discussion later). The maximum ion energy keeps increasing because of the Fermi-like mechanism but is limited by the simulation duration ($100\Omega_{ci}^{-1}\approx$ 16 ion gyro-period). We expect that ions can be accelerated to higher energies and develop an even harder high-energy tail in larger simulations. In summary, ions develop similar energy spectra for different mass ratio, and the spectra have a hard low-energy part and a steep high-energy part, separating by the reconnection bulk flow energy $0.5m_{i}v_{\text{A}}^{2}$.

3.5 Plasma Energization

Plasma energization analysis based on the guiding-center drift description has been routinely carried out in kinetic simulations for studying particle acceleration mechanisms (Dahlin et al., 2014; Li et al., 2015, 2017, 2018a; Wang et al., 2016). Figure 7 shows multiple plasma energization terms associated with the parallel or perpendicular electric field, flow compression or flow shear (see A2 for their definitions), curvature drift or gradient drift (see A1 for their definitions), flow inertia or magnetization (see A1 for their definitions), and gyrotropic or agyrotropic pressure tensors. For electrons, a mass ratio of 25 tends to overestimates the contributions by $\boldsymbol{E}_{\perp}$ (panel (a)), flow compression and shear (panel (b)), flow inertia (panel (d)), and gyrotropic pressure tensor (panel (e)), but the guide-field dependence is consistent across runs with different mass ratio. Among these terms, the inertia term is mostly overestimated in the runs with $m_{i}/m_{e}=25$, but it only contributes to the bulk energization. For ions, Figure 7 (g) shows that $m_{i}/m_{e}=25$ tends to overestimate the contribution by flow shear when $B_{g}\leq 0.2$, and that $m_{i}/m_{e}=25$ or 100 tends to overestimate the contribution by flow compression when $B_{g}\leq 0.2$; Figure 7 (j) shows that ions are more gyrotropic in the runs with $m_{i}/m_{e}=25$ than that in the runs with a higher mass ratio. This is because ions become less well-magnetized when its gyroradius $\rho_{i}/d_{e}=v_{\text{thi}}\omega_{pe}/(c\Omega_{ci})=\sqrt{m_{i}/m_{e}}(v_{\text{thi}}/v_{A})$ gets larger with the mass ratio, where $v_{\text{thi}}/v_{A}=\sqrt{\beta_{i}}$ is the same for all runs.

Since other energization terms were more or less studied before, we summarize the results shown in Figure 7 without going into details. For electrons, panel (a) shows that most energization is done by $\bm{E}_{\perp}$ when $B_{g}<0.5$, and that the energization by $\bm{E}_{\parallel}$ becomes comparable with that by $\bm{E}_{\perp}$ when $B_{g}=0.8$; panel (b) shows that flow compression energization dominates flow shear energization ($\propto$ pressure anisotropy), although the former keeps decreasing with the guide field, and the latter slightly increases until $B_{g}=0.4$ because of an increasing pressure anisotropy (Li et al., 2018a); panels (c) and (d) show that the energization associated with curvature drift dominates the other energization terms by $\bm{E}_{\perp}$, and that the energization associated with flow inertia contributes significantly only when $m_{i}/m_{e}=25$; panels (e) shows that the energization associated with the gyrotropic pressure tensor always dominates the energization associated with the agyrotropic pressure tensor, indicating that most electrons are well-magnetized in the simulations. For ions, panel (f) shows that most energization is done by $\bm{E}_{\perp}$, and that this does not change much with the guide field; panel (g) shows that compression energization always dominates shear energization, and that both terms gradually decrease with the guide field, which is different from that for electrons; panels (h)–(j) show that the energization associated with curvature drift and flow inertia are two most important terms for ions besides the energization associated with the agyrotropic pressure tensor, and that curvature drift dominates when $B_{g}<0.4$ and flow inertia dominates when $B_{g}\geqslant 0.4$. In summary, plasma energization is similar in runs with different mass ratio, so a lower mass ratio (e.g. 25) is still useful for studying particle acceleration mechanisms and their scaling with the guide field.

3.6 Particle Acceleration Rates

To further reveal the difference between runs with different mass ratio, we use all particles to evaluate the particle acceleration rates $\alpha(\varepsilon,t)\equiv\left<\dot{\varepsilon}(\varepsilon,t)/\varepsilon(t)\right>$ associated with $\bm{E}_{\parallel}$, $\bm{E}_{\perp}$, curvature drift, gradient drift, parallel drift, inertial drift, polarization drift, and betatron acceleration. Figure 8 shows the two largest terms for electrons: $\bm{E}_{\parallel}$ and curvature drift, for the runs with $B_{g}=0.2$. Since the simulation duration is $100\Omega_{ci}^{-1}$ for all runs, in order to compare among the runs with different mass ratio, we normalize $\alpha$ by $\Omega_{ci}$. We find that $\bm{E}_{\parallel}$ is efficient at accelerating electrons early in the simulation ($t\Omega_{ci}=40$), but it does not accelerate or even decelerates energetic electrons ($>10\varepsilon_{\text{th}}$) latter. The right panels of Figure 8 show particle curvature drift generally leads to acceleration. It gradually decreases as the simulation evolves and approaches zero for high-energy electrons ($>30\varepsilon_{\text{th}}$) in the runs with $m_{i}/m_{e}=100$ or 400 but stays finite in the run with $m_{i}/m_{e}=25$. Combining the negative acceleration rate due to $\bm{E}_{\parallel}$, we find that high-energy electrons are decelerated latter in the runs with $m_{i}/m_{e}=100$ or 400. In contrast, the high-energy electrons are continuously accelerated in the run with $m_{i}/m_{e}=25$, so the “power-law” can survive, as show in Figure 5. Note that these results still hold for runs with a different guide field that are not shown here. In summary, as the mass ratio gets larger, high-energy electron acceleration becomes less efficient, because the acceleration rate by $\bm{E}_{\parallel}$ becomes negative, and because the Fermi-like mechanism becomes less efficient.

Figure 9 shows the acceleration rates for ions. We find that the acceleration rates associated with particle inertial drift, polarization drift, and curvature drift are most important for ions. Since the inertial drift contains particle curvature drift, we subtract the curvature drift from the inertial drift and call the residue the inertial’ drift in the left panels. The acceleration rate associated with the inertial’ drift is negative for energetic ions with tens of $\varepsilon_{\text{th}}$, indicating that the acceleration process associated with the inertial’ drift decelerates ions. The middle panels of Figure 9 show that $\alpha$ associated with polarization drift is efficiently at accelerating ions at different energies early in the simulations but peaks around $5\varepsilon_{\text{th}}$ and approaches zero when $\varepsilon>20\varepsilon_{\text{th}}$ latter in the simulations. This indicates that particle polarization drift along $\bm{E}_{\perp}$ is efficient at driving the reconnection bulk flow. In contrast, the right panels of Figure 9 show that the Fermi-like mechanism associated with particle curvature drift preferentially accelerates ions at high energies ($>20\varepsilon_{\text{th}}$), and that it is still strong to the end of the simulations. We expect that ions can be accelerated to higher energies and develop an even harder high-energy spectra in larger simulations. The right panels show that the acceleration associated with curvature drift is slightly smaller in the run with $m_{i}/m_{e}=400$ than that in the runs with lower mass ratios. This explains why the high-energy ion fluxes are lower in the runs with $m_{i}/m_{e}=400$, as shown in Figure 6. These results on the ion acceleration rates are consistent among the runs with different mass ratio, suggesting that we could use a lower mass ratio (e.g. 25 or 100) to study ion acceleration in low-$\beta$ reconnection.

4 Conclusions and Discussions

In this work, we study whether and how the proton-to-electron mass ratio affects the particle acceleration processes in kinetic simulations of magnetic reconnection through performing simulations with different mass ratio and guide-field strength. The simulations show different current layer structures that depend on the mass ratio and guide-field strength, consistent with earlier studies (e.g. Le et al., 2013). We find that simulations with different mass ratios are similar in reconnection rate, magnetic energy conversion, ion internal energy gain, plasma energization processes, ion energy spectra, and the acceleration mechanisms for high-energy ions, but simulations show different electron internal energy gain, electron energy spectrum, and the acceleration efficiencies for high-energy electrons. We find that electrons gain more energy (internal or kinetic) in runs with a lower mass ratio. As a result, the ion-to-electron energy partition increases with the mass ratio, e.g. from 1.5 for $m_{i}/m_{e}=25$ to 2.25 for $m_{i}/m_{e}=400$ when $B_{g}=0$. We find that the electron spectrum gets steeper as the mass ratio gets larger. By calculating the particle acceleration rates due to different particle guiding-center drift motions, we find that as the mass ratio increases, high-energy electron acceleration becomes less efficient because parallel electric field tends to decelerate high-energy electrons, and because the Fermi-like mechanism associated with particle curvature drift becomes less efficient.

The simulations also show that the total plasma energization associated with the guiding-center drift motions and flow compression and shear is similar for the runs with different mass ratio. A lower mass ratio tends to overestimate some of the energization terms, but the guide-field dependence is consistent across runs with different mass ratio. By subtracting the gyrotropic pressure tensor from the whole pressure tensor, we find that most electrons are well magnetized even when $B_{g}=0$, and that the agyrotropic ion distributions contribute over 15% of the total ion energization when $m_{i}/m_{e}=400$ and $B_{g}\geqslant 0.2$. This indicates that ions are not well-magnetized when $m_{i}/m_{e}$ is large. These results suggest a lower mass ratio is still good for studying energy conversion mechanisms during magnetic reconnection.

The ion acceleration rates show that the acceleration terms associated with ion inertial drift, polarization drift, and curvature drift are most important for ions. Ion inertial drift (with curvature drift being subtracted) decelerates high-energy ions ($>20$ times of the initial thermal energy). Ion polarization drift tends to drive the reconnection bulk flow and is mostly efficient for low-energy ions (around 5 times of the initial thermal energy). We find that high-energy ions are accelerated by the Fermi-like mechanism associated with particle curvature drift along the motional electric field.

The ion energy distributions show that ions are accelerated to form Alfvénic reconnection outflow when they enter the reconnection layer. The thermalisation processes (e.g. compression and shear) result in a much hotter plasma than the inflow plasma. Similar processes could occur in solar flares, where $v_{\text{A}}\approx 10^{3}$ km/s and the ion thermal speed $v_{\text{thi}}\approx 10^{2}$ km/s in the lower solar corona. As indicated by observations (e.g. Liu et al., 2013), the coronal plasma can be heated from 1 million Kelvin (MK) to tens of MK in a flare region. The superposition of such multicomponent super-hot plasmas can even produce the observed coronal hard X-ray emission, as predicted in simulations by Cheung et al. (2018).

We have carried another set of simulations, in which we fix the electron thermal velocity and $\omega_{pe}/\Omega_{ce}$ (effectively varying the Alfvén speed for different mass ratio). This is typical when using a lower mass ratio to save the computationally costs. We find that the above conclusions still hold for this new set of simulations. The consistency between the two sets of simulations suggests once the scale separation between electrons and ions are fixed, the acceleration processes of a single species will be similar. Below is our explanation of similar ion acceleration and different electron acceleration in runs with different mass ratio. The particle acceleration rates (Figure 8 and 9) show particle curvature drift as the dominant high-energy acceleration mechanism, and the curvature drift acceleration is most efficient in the reconnection exhaust ($\sim d_{i}$). We can treat the $d_{i}$ as the energy-containing scale. The closer is particle gyromotion scale to $d_{i}$, the stronger high-energy acceleration do we expect. For ions, $\rho_{i}/d_{i}$ (0.1 in our simulations) is larger than $\rho_{e}/d_{i}$, so ions tend to be accelerated to higher energies than electrons; $\rho_{i}/d_{i}$ is constant for different mass ratio, so the ion spectra are similar for different mass ratio. For electrons, $\rho_{e}/d_{i}$ gets smaller as the mass ratio increases, so high-energy electron acceleration gets weaker when $m_{i}/m_{e}$ gets larger.

Although we present results only for runs with a guide field up to $0.8B_{0}$, we have performed simulations with a stronger guide field (1.6, 3.2, and $6.4B_{0}$), which are more relevant to particle acceleration due to quasi-2D turbulence or interacting small-scale flux ropes in the inner heliosphere (Smith et al., 2006; Zank, 2014; le Roux et al., 2015; Hu et al., 2018). We find that ion acceleration is still similar for runs with different mass ratio, and electron acceleration is still less efficient as the mass ratio gets larger. As the guide field becomes larger than $B_{0}$, the parallel electric field becomes the dominant energization mechanism for electrons, but it is inefficient at accelerating energetic electrons, resulting in a much lower high-energy electron fluxes; the acceleration associated polarization drift becomes the dominate energization mechanism for ions, but it drives ion bulk flow instead of accelerating high-energy ions. For both electrons and ions, the acceleration rate associated with particle curvature drift becomes lower as the guide field gets stronger, indicating that the acceleration time scale becomes longer. To fully evaluate the effect of curvature drift in the strong guide-field reconnection, we need much larger simulations that runs for a much longer time. We defer these studies to a future work.

Our simulations have a few limitations. First, we perform simulations only in low-$\beta$ plasmas with the same temperature for electrons and ions, which are suitable for studying particle acceleration at the reconnection site of a solar flare. Conclusions on the mass-ratio dependence might change at the reconnection sites in Earth’s magnetosphere or the accretion disk corona, where ions can be much hotter than electrons, and the plasma $\beta$ can be larger than 0.1. Second, the simulation duration is limited by the box sizes and the periodic boundary conditions. A larger simulation with more realistic open boundary conditions could change the relative importance of the acceleration near the reconnection X-line and the acceleration associated with magnetic islands. Third, high-energy particles are confined in the 2D magnetic islands and cannot be further accelerated. The self-generated turbulence in 3D reconnection could change the acceleration processes and their dependence on the mass ratio.

To conclude, we find that different mass ratios are similar in reconnection rate, magnetic energy conversion, ion internal energy gain, plasma energization processes, ion energy spectra, and the acceleration mechanisms for high-energy ions. We find that ion acceleration is similar for different mass ratio because the dominant acceleration mechanism for energetic ions is due to particle curvature drift, and it does not change much with the mass ratio. Runs with different mass ratios are different in electron internal energy gain, electron energy spectrum, and the acceleration efficiencies for high-energy electrons. We find that high-energy electron acceleration becomes less efficient when the mass ratio gets larger because parallel electric field tends to decelerate high-energy electrons, and because the Fermi-like mechanism associated with particle curvature drift becomes less efficient. These results indicate that when particle curvature drift dominates high-energy particle acceleration, the further the particle kinetic scale are from the magnetic field curvature scales ($\sim d_{i}$), the weaker the acceleration will be, at least in 2D.

Appendix A Fluid description of plasma energization

Li et al. (2018a) described the plasma energization processes in term of $\bm{j}_{s}\cdot\bm{E}$, where the perpendicular component of the current density $\bm{j}_{s}$ for any species is

[TABLE]

where $p_{s\parallel}=\left<(\bm{v}_{\parallel}-\bm{v}_{s\parallel})\cdot(\bm{p}_{\parallel}-\bm{p}_{s\parallel}/n_{s})\right>$ and $p_{s\perp}=0.5\left<(\bm{v}_{\perp}-\bm{v}_{s\perp})\cdot(\bm{p}_{\perp}-\bm{p}_{s\perp}/n_{s})\right>$ are parallel and perpendicular pressures w.r.t the local magnetic field, respectively, $\rho_{s}$ is the charge density, $n_{s}$ is particle number density, $m_{s}$ is particle mass, and $d/dt=\partial_{t}+\bm{v}_{s}\cdot\nabla$. In the language of particle drifts, the plasma energization is then associated with parallel electric field, curvature drift, gradient drift, magnetization, and flow inertia. Li et al. (2018a) reorganized $\bm{j}_{s\perp}\cdot\bm{E}_{\perp}$ as

[TABLE]

where $\bm{v}_{E}$ is the $\bm{E}\times\bm{B}$ drift velocity, $\sigma_{ij}=0.5(\partial_{i}v_{Ej}+\partial_{j}v_{Ei}-(2\nabla\cdot\bm{v}_{E}\delta_{ij})/3)$ is the shear tensor, and $p_{s}\equiv(p_{s\parallel}+2p_{s\perp})/3$ is the effective scalar pressure. Then, plasma energization is associated with parallel electric field, flow compression, flow shear, and flow inertia.

Appendix B Drift description of particle acceleration

Gyrophase-averaged particle guiding center velocity is (Northrop, 1963; Webb et al., 2009; le Roux & Webb, 2009; le Roux et al., 2015)

[TABLE]

where $d/dt=\partial/\partial t+(v_{\parallel}\bm{b}+\bm{v}_{E})\cdot\nabla$, $M=m_{s}(\bm{v}_{\perp}-\bm{v_{E}})^{2}/2B$ is particle magnetic moment in the plasma frame where $\bm{v}_{E}=0$. The terms on the right are the parallel guiding-center velocity, $\bm{E}\times\bm{B}$ drift, gradient drift, inertial drift (including curvature drift), parallel drift, and polarization drift. Assuming the perpendicular flow velocity $\bm{v}_{s\perp}\approx\bm{v}_{E}$ and particles are non-relativistic ($\bm{v}\approx\bm{p}/m_{s}$), the current density associated with particle gradient drift is

[TABLE]

The current density associated with particle inertial drift is

[TABLE]

where we get the current density associated with curvature drift and the flow inertial effect associated with the parallel component of the flow velocity. The current density associated with particle parallel drift is

[TABLE]

where the first term is the current density associated with magnetization, and the dot product of the second term with $\bm{E}$ gives

[TABLE]

where we used the Maxwell-Faraday equation, the first term cancels the first term on the right in Equation A2, and the second term cancels betatron acceleration. Finally, the energization associated with particle polarization drift is

[TABLE]

which contributes to the flow inertial term. Combining Equation B2 to B6, we can reproduce Equation A1. The total plasma energization is

[TABLE]

which is different from Equation A2 because of the terms in Equation B5. Table 1 compares the two descriptions.

This work was supported by NASA grant NNH16AC60I. HL and FG acknowledgess the support by DOE/OFES. We also acknowledge support by the DOE through the LDRD program at LANL. We gratefully acknowledge our discussions with Bill Daughton, Ari Le, Adam Stanier, and Patrick Kilian. Simulations were performed with LANL institutional computing.