Nonthermally Dominated Electron Acceleration during Magnetic Reconnection in a Low-beta Plasma
AAccepted for Publication in
The Astrophysical Journal Letters
Nonthermally Dominated Electron Acceleration during MagneticReconnection in a Low- β Plasma
Xiaocan Li , , , Fan Guo , Hui Li , Gang Li , ABSTRACT
By means of fully kinetic simulations, we investigate electron accelerationduring magnetic reconnection in a nonrelativistic proton–electron plasma withconditions similar to solar corona and flares. We demonstrate that reconnectionleads to a nonthermally dominated electron acceleration with a power-law energydistribution in the nonrelativistic low- β regime but not in the high- β regime,where β is the ratio of the plasma thermal pressure and the magnetic pressure.The accelerated electrons contain most of the dissipated magnetic energy in thelow- β regime. A guiding-center current description is used to reveal the role ofelectron drift motions during the bulk nonthermal energization. We find that themain acceleration mechanism is a Fermi -type acceleration accomplished by theparticle curvature drift motion along the electric field induced by the reconnectionoutflows. Although the acceleration mechanism is similar for different plasma β ,low- β reconnection drives fast acceleration on Alfv´enic timescales and developspower laws out of thermal distribution. The nonthermally dominated acceler-ation resulting from magnetic reconnection in low- β plasma may have strongimplications for the highly efficient electron acceleration in solar flares and otherastrophysical systems. Subject headings: acceleration of particles — magnetic reconnection — Sun:flares — Sun: corona Department of Space Science, University of Alabama in Huntsville, Huntsville, AL 35899, USA Center for Space Plasma and Aeronomic Research, University of Alabama in Huntsville, Huntsville, AL35899, USA Los Alamos National Laboratory, Los Alamos, NM 87545, USA a r X i v : . [ a s t r o - ph . S R ] S e p
1. Introduction
Magnetic reconnection is a fundamental plasma process during which the magneticfield restructures itself and converts its energy into plasma kinetic energies (e.g. Priest &Forbes (2000)). It occurs ubiquitously in laboratory, space, and astrophysical magnetizedplasmas. An important unsolved problem is the acceleration of nonthermal particles in thereconnection region. Magnetic reconnection has been suggested as a primary mechanismfor accelerating nonthermal particles in solar flares (Masuda et al. 1994; Krucker et al.2010; Lin 2011), Earth’s magnetosphere (Øieroset et al. 2002; Fu et al. 2011; Huang et al.2012), the sawtooth crash of tokamaks (Savrukhin 2001), and high-energy astrophysicalsystems (Colgate et al. 2001; Zhang & Yan 2011). In particular, observations of solar flareshave revealed an efficient particle energization with 10% −
50% of magnetic energy convertedinto energetic electrons and ions (Lin & Hudson 1976). The energetic particles usuallydevelop a power-law energy distribution that contains energy on the same order of thedissipated magnetic energy (Krucker et al. 2010; Oka et al. 2015). Some observations findthat the emission has no distinguishable thermal component, indicating that most of theelectrons are accelerated to nonthermal energies (Krucker et al. 2010; Krucker & Battaglia2014). This efficient production of energetic particles poses a challenge to current theoriesof particle acceleration.Particle acceleration associated with reconnection has been studied in reconnection-driven turbulence (Miller et al. 1996), at shocks in the outflow region (Tsuneta & Naito1998; Guo & Giacalone 2012), and in the reconnection layer (Drake et al. 2006; Fu et al.2006; Oka et al. 2010; Kowal et al. 2012; Guo et al. 2014; Zank et al. 2014). Previouskinetic simulations have examined various acceleration mechanisms during reconnection,including the
Fermi -type mechanism in magnetic islands (flux ropes in three-dimensionalsimulations; Drake et al. (2006); Guo et al. (2014)) and direct acceleration in the diffusionregion (Pritchett 2006; Huang et al. 2010). Most simulations focus on regimes with plasma β ≥ . σ = B πn e m e c (cid:29)
1) pair plasma found global power-law distributions withoutthe particle loss, although the loss mechanism may be important in determining the spectralindex (Guo et al. 2014, 2015). It is unknown whether or not this is valid for reconnection ina nonrelativistic proton–electron plasma, since its property is different from the relativisticreconnection (Liu et al. 2015).Motivated by the results of relativistic reconnection, here, we report fully kinetic simu- 3 –lations of magnetic reconnection in a nonrelativistic proton–electron plasma with a range ofelectron and ion β e = β i = 0 . − .
2. The low- β regime was previously relatively unexploreddue to various numerical challenges. We find that reconnection in the low- β regime drives ef-ficient energy conversion and accelerates electrons into a power-law distribution f ( E ) ∼ E − .At the end of the low- β cases, more than half of the electrons in number and 90% in en-ergy are in the nonthermal electron population. This strong energy conversion and particleacceleration led to a post-reconnection region with the kinetic energy of energetic particlescomparable to magnetic energy. Since most electrons are magnetized in the low- β plasma,we use a guiding-center drift description to demonstrate that the main acceleration processis a Fermi -type mechanism through the particle curvature drift motion along the electricfield induced by fast plasma flows. The development of power-law distributions is consistentwith the analytical model (Guo et al. 2014). The nonthermally dominated energization mayhelp explain the efficient electron acceleration in the low- β plasma environments, such assolar flares and other astrophysical reconnection sites.In Section 2, we describe the numerical simulations. In Section 3, we present simulationresults and discuss the conditions for the development of power-law distributions. We discussand conclude the results in Section 4.
2. Numerical simulations
The kinetic simulations are carried out using the VPIC code (Bowers et al. 2008),which solves Maxwell’s equations and follows particles in a fully relativistic manner. Theinitial condition is a force-free current sheet with a magnetic field B = B tanh( z/λ )ˆ x + B sech( z/λ )ˆ y , where λ = d i is the half thickness of the layer. Here, d i is the ion inertiallength. The plasma consists of protons and electrons with a mass ratio m i /m e = 25. Theinitial distributions for both electrons and protons are Maxwellian with uniform density n and temperature kT i = kT e = 0 . m e c . A drift velocity for electrons U e is added torepresent the current density that satisfies the Ampere’s law. The initial electron and ion β e = β i = 8 πn kT e /B are varied by changing ω pe / Ω ce , where ω pe = (cid:112) πn e /m e is theelectron plasma frequency and Ω ce = eB / ( m e c ) is the electron gyrofrequency. Quantities β e = 0 . ω pe / Ω ce = 0 .
6, 1, √ √
10, respectively.The domain sizes are L x × L z = 200 d i × d i . We use N x × N z = 4096 × x -direction,perfectly conducting boundaries for fields and reflecting boundaries for particles along the z -direction. A long wavelength perturbation is added to induce reconnection (Birn et al.2001). 4 –
3. Simulation results
Under the influence of the initial perturbation, the current sheet quickly thins down toa thickness of ∼ d e (electron inertial length c/ω pe ) that is unstable to the secondary tearinginstability (Daughton et al. 2009; Liu et al. 2013b). Fig. 1(a) and (b) show the evolution of theout-of-plane current density. The reconnection layer breaks and generates a chain of magneticislands that interact and coalesce with each other. The largest island eventually growscomparable to the system size and the reconnection saturates at t Ω ci ∼ x -direction (the reconnecting component) ε bx and the kinetic energy of electrons K e and ions K i for the case with β e = 0 .
02, respectively.Throughout the simulation, 40% of the initial ε bx is converted into plasma kinetic energy. Ofthe converted energy, 38% goes into electrons and 62% goes into ions. We have carried outsimulations with larger domains (not shown) to confirm that the energy conversion is stillefficient and weakly depends on system size. Since the free magnetic energy overwhelms theinitial kinetic energy, particles in the reconnection region are strongly energized. Eventually, K e and K i are 5.8 and 9.4 times their initial values, respectively. Fig. 1(d) shows the ratio ofthe electron energy gain ∆ K e to the initial electron energy K e (0) for different cases. Whilethe β e = 0 . β e give stronger energizationas the free energy increases.The energy conversion drives strong nonthermal electron acceleration. Fig. 2(a) showsthe final electron energy spectra over the whole simulation domain for the four cases.More electrons are accelerated to high energies for lower- β cases, similar to earlier simu-lations (Bessho & Bhattacharjee 2010). More interestingly, in the cases with β e = 0 .
02 and0 . f ( E ) ∼ E − p with the spectral index p ∼
1. This is similar to results from relativistic reconnection (Guo et al. 2014, 2015). Wehave carried out one simulation with m i /m e = 100 and β e = 0 .
02 and find a similar electronspectrum. In contrast, the case with β e = 0 . β cases. For example, when we subtract the thermal populationby fitting the low-energy distribution as Maxwellian, the nonthermal tail in the β e = 0 . E b ∼ E th for β e = 0 .
02, and extends to a higher energy for β e = 0 . β e = 0 . β e = 0 .
2. Fig. 2(c) and (d)show n acc /n e at t Ω ci = 125 and 400 for the case with β e = 0 .
02, where n acc is the numberdensity of accelerated electrons with energies larger than three times their initial thermalenergy, and n e is the total electron number density. The fraction of energetic electrons isover 40% and up to 80% inside the magnetic islands and reconnection exhausts, indicating a 5 – (a)(b) (c) (d) Fig. 1.— Out-of-plane current density for the case with β e = 0 .
02 at (a) t Ω ci = 62 . t Ω ci = 400. (c) The energy evolution for the β e = 0 .
02 case. ε bx ( t ) is the magnetic energy ofthe reconnecting component. ε e is the electric energy. K i and K e are ion and electron kineticenergies normalized by ε bx (0), respectively. (d) The ratio of electron energy gain ∆ K e to theinitial K e for different β e . 6 –bulk energization for most of electrons in the reconnection layer. The energetic electrons willeventually be trapped inside the largest magnetic island. The nonthermally dominated dis-tribution contains most of the converted magnetic energy, indicating that energy conversionand particle acceleration are intimately related.To study the energy conversion, Fig. 3(a) shows the energy conversion rate dε c /dt fromthe magnetic field to electrons through directions parallel and perpendicular to the localmagnetic field. We define dε c /dt = (cid:82) D j (cid:48) · E dV , where D indicates the simulation domainand j (cid:48) is j (cid:107) or j ⊥ . We find that energy conversion from the perpendicular directions gives ∼
90% of the electron energy gain. By tracking the trajectories (not shown) of a largenumber of accelerated electrons, we find various acceleration processes in the diffusion region,magnetic pile-up region, contracting islands, and island coalescence regions (Hoshino et al.2001; Hoshino 2005; Drake et al. 2006; Fu et al. 2006; Huang et al. 2010; Oka et al. 2010;Dahlin et al. 2014; Guo et al. 2014). The dominant acceleration is by particles bouncingback and forth through a
Fermi -like process accomplished by particle drift motions withinmagnetic structures (Li et al. 2015, in preparation). To reveal the role of particle driftmotions, we use a guiding-center drift description to study the electron energization for the β e = 0 .
02 case. The initial low β guarantees that this is a good approximation since thetypical electron gyroradius ρ e is smaller than the spatial scale of the field variation ( ∼ d i ).By ensemble averaging the particle gyromotion and drift motions, the perpendicularcurrent density for a single species can be expressed as (Parker 1957; Blandford et al. 2014) j ⊥ = P (cid:107) B × ( B · ∇ ) B B + P ⊥ (cid:18) B B (cid:19) × ∇ B − (cid:20) ∇ × P ⊥ B B (cid:21) ⊥ + ρ E × B B + ρ m B B × d u E dt using a gyrotropic pressure tensor P = P ⊥ I + ( P (cid:107) − P ⊥ ) bb , where P (cid:107) ≡ m e (cid:82) f v (cid:107) d v and P ⊥ ≡ . m e (cid:82) f v ⊥ d v , ρ is the particle charge density, and ρ m is the particle mass density.The terms on the right are due to curvature drift, ∇ B drift, magnetization, E × B drift,and polarization drift, respectively. The expression is simplified as j ⊥ = j c + j g + j m + j E × B + j p , in which j E × B has no direct contribution to the energy conversion. This givesan accurate description for j ⊥ if the pressure tensor is gyrotropic. To confirm this, wecalculate the electron pressure agyrotropy A Ø e ≡ | P ⊥ e − P ⊥ e | P ⊥ e + P ⊥ e , where P ⊥ e and P ⊥ e are thetwo pressure eigenvalues associated with eigenvectors perpendicular to the mean magneticfield direction (Scudder & Daughton 2008). A Ø e measures the departure of the pressuretensor from cylindrical symmetry about the local magnetic field. It is zero when the localparticle distribution is gyrotropic. Fig. 3(b) shows that the regions with nonzero A Ø e arelocalized to X-points. The small A Ø e indicates that the electron distributions are nearlygyrotropic in most regions. Therefore, the drift description is a good approximation forelectrons in our simulations even without an external guide field, which is required for this 7 – (a)(b)(c)(d) Fig. 2.— (a) Electron energy spectra f ( E ) at t Ω ci = 800 for different β e . The electronenergy E is normalized to the initial thermal energy E th . The black dashed line is theinitial thermal distribution. (b) Time evolution of the fraction of nonthermal electrons fordifferent initial β e . n nth is the number of nonthermal electrons obtained by subtracting thefitted thermal population from the whole particle distribution. The fraction of electronswith energies larger than three times the initial thermal energy at (c) t Ω ci = 125 and (d) t Ω ci = 400. 8 – (a)(b) Fig. 3.— (a) Energy conversion rate dε c /dt for electrons through the parallel and perpen-dicular directions with respect to the local magnetic field, compared with the energy changerate of electrons dK e /dt for the case with β e = 0 .
02. The shown values are integrals over thewhole simulation domain. (b) Electron pressure agyrotropy A Ø e at t Ω ci = 400 in the samecase. See the text for details.description in a high- β plasma (Dahlin et al. 2014).Fig. 4(a) and (b) show time-dependent dε c /dt and ε c from different current terms, where ε c = (cid:82) t ( dε c /dt ) dt . The contribution from polarization current and parallel current are smalland not shown. The curvature drift term is a globally dominant term of j ⊥ · E , the ∇ B termgives a net cooling, and the magnetization term is small compared to these two. Fig. 4(c)shows the spatial distribution of j c · E . When the flow velocity u is along the magneticfield curvature κ due to tension force, j c · E ≈ ( P (cid:107) B × κ /B ) · ( − u × B ) >
0. Theseregions are a few d i along the z -direction, but over 50 d i along the x -direction. The overalleffect of j c · E is a strong electron energization. Fig. 4(d) shows that j g · E is negative inmost regions because the strong ∇ B is along the direction out of the reconnection exhausts.Then, j g · E ∼ ( B × ∇ B ) · ( − u × B ) <
0. Note at some regions, j g · E can give strongacceleration. Fig. 4(e) shows the cumulation of the j c · E and j g · E along the x -direction.In the reconnection exhaust region( x = 60 − d i ), j c · E is stronger than j g · E , sothe electrons can be efficiently accelerated when going through these regions. In the pile-up region( x = 120 − d i ), κ , ∇ B and u are along the same direction, so both termsgive electron energization. In the island coalescence region( x ∼ d i ), j c · E gives electronheating, while j g · E gives strong electron cooling. Although the net effect is electron cooling,island coalescence can be efficient in accelerating electrons to the highest energies (Oka et al.2010).It has been shown that the curvature drift acceleration in the reconnection region cor-responds to a Fermi -type mechanism (Dahlin et al. 2014; Guo et al. 2014, 2015). To develop 9 – ... ... (cid:20) . u . B . ∇ B . ∇ B .(a) .(b) .(c) .(d) .(e) Fig. 4.— Analysis using a drift description for the case with β e = 0 .
02. (a) The energyconversion rate due to different types of current terms, compared with the electron energychange rate dK e /dt . j c · E , j g · E , and j m · E represent energy conversion due to curvaturedrift, ∇ B drift, and magnetization, respectively. (b) The converted magnetic energy due tovarious terms in (a), normalized to the initial magnetic energy of the reconnecting component ε bx (0). (c) Color-coded contours of energy conversion rate due to curvature drift at t =400Ω − ci . κ and u indicate the directions of the magnetic field curvature and the bulk flowvelocity. (d) Color-coded contours of energy conversion rate due to ∇ B drift at t = 400Ω − ci . B and ∇ B indicate the directions of the magnetic field and the gradient of | B | . Both j c · E and j g · E are normalized to the 0 . n m e c ω pe . (e) The cumulation of j c · E (blue) and j g · E (green) along the x -direction. The black line is the sum of these two. 10 –a power-law energy distribution for the Fermi acceleration mechanism, the characteristicacceleration time τ acc = 1 /α needs to be smaller than the particle injection time τ inj (Guoet al. 2014, 2015), where α = (1 /ε )( ∂ε/∂t ), and ∂ε/∂t is the energy change rate of particles.To estimate the ordering of acceleration rate from the single-particle drift motion, considerthe curvature drift velocity v c = v (cid:107) B × κ / (Ω ce B ) in a curved field where R c = | κ | − ,so the time for a particle to cross this region is ∼ R c /v (cid:107) and the electric field is mostlyinduced by the Alfv´enic plasma flow E ∼ − v A × B /c . The energy gain in one cycle is δε ∼ mv A v (cid:107) . The time for a particle to cross the island is L island /v (cid:107) . Then, the accelerationrate ∂ε/∂t ∼ εv A /L island for a nearly isotropic distribution. The characteristic accelera-tion time τ acc ∼ L island /v A . Taking L island ∼ d i and v A ∼ . c , the acceleration time τ acc ∼ − ci . The actual acceleration time may be longer because the outflow speed willdecrease from v A away from the X-points, and the ∇ B term gives a non-negligible cool-ing effect. Our analysis has also found that pre-acceleration and trapping effects at theX-line region can lead to more efficient electron acceleration by the Fermi mechanism andare worthwhile to investigate further (Hoshino 2005; Egedal et al. 2015; Huang et al. 2015).Taking the main energy release phase as the injection time τ inj ∼ − ci , the estimatedvalue of τ inj /τ acc ∼ .
2, well above the threshold. For the case with β e = 0 .
2, the ratio τ inj /τ acc ∼ . <
1, so there is no power-law energy distribution.
4. Discussion and conclusion
Nonthermal power-law distributions have rarely been found in previous kinetic simula-tions of nonrelativistic magnetic reconnection (Drake et al. 2010). We find that two essentialconditions are required for producing power-law electron distribution. The first is that thedomain should be large enough to sustain reconnection for a sufficient duration. A power-law tail develops as the acceleration accumulates long enough ( τ inj /τ acc > β must be low to form a nonthermally dominated power-law dis-tribution by providing enough free energy ( ∝ /β ) for nonthermal electrons. Assuming 10%of magnetic energy is converted into nonthermal electrons with spectral index p = 1, onecan estimate that β e is about 0.02 for half of the electrons to be accelerated into a powerlaw that extends to 10 E th . This agrees well with our simulation. We point out that a lossmechanism or radiation cooling can affect the final power-law index (Fermi 1949; Guo et al.2014) of nonthermal electrons. Consequently, including loss mechanisms in a large three-dimensional open system is important, for example, to explain the observed power-law indexin solar flares and other astrophysical processes. Another factor that may influence our re-sults is the presence of an external guide field B g . Our preliminary analysis has shown thatthe Fermi acceleration dominates when B g (cid:46) B . The full discussion for the cases includ- 11 –ing the guide field will be reported in another publication (Li et al. 2015, in preparation).A potentially important issue is the three-dimensional instability, such as kink instabilitythat may strongly influence the results. Unfortunately, the corresponding three-dimensionalsimulation is beyond the available computing resources. We note that results from three-dimensional simulations with pair plasmas have shown development of strong kink instabilitybut appear to have no strong influence on particle acceleration (Guo et al. 2014; Sironi &Spitkovsky 2014). The growth rate of the kink instability can be much less than the tearinginstability for a high mass ratio (Daughton 1999), and therefore the kink instability may beeven less important for electron acceleration in a proton–electron plasma.In our simulations, the low- β condition is achieved by increasing magnetic field strength(or equivalently decreasing density). We have carried out low- β simulations with the samemagnetic field but lower temperature and found a similar power-law distribution (Li et al.2015, in preparation).The energy partition between electrons and protons shows that more magnetic energyis converted into protons. For simulations with a higher mass ratio m i /m e = 100, theenergetic electrons still develop a power-law distribution, and the fraction of electron energyto the total plasma energy is about 33%, indicating that the energy conversion and electronacceleration are still efficient for higher mass ratios. Our results show that ions also develop apower-law energy spectrum for low- β cases and the curvature drift acceleration is the leadingmechanism. However, the ion acceleration has a strong dependence on the mass ratio m i /m e for our relatively small simulation domain ( ∼ d i ). We therefore defer the study of ionacceleration to a future work (Li et al., 2015, in preparation).The energetic electrons can generate observable X-ray emissions. As nonthermal elec-trons are mostly concentrated inside the magnetic islands, the generated hard X-ray flux canbe strong enough to be observed during solar flares in the above-the-loop-top region (Ma-suda et al. 1994; Krucker et al. 2010) and the reconnection outflow region (Liu et al.2013a). The nonthermal electrons may also account for the X-ray flares in the accretiondisk corona (Galeev et al. 1979; Haardt et al. 1994; Li & Miller 1997).In summary, we find that in a nonrelativistic low- β proton–electron plasma, magneticreconnection is highly efficient at converting the free energy stored in a magnetic shear intoplasma kinetic energy and accelerating electrons into nonthermal energies. The nonthermalelectrons contain more than half of the total electrons, and their distribution resembles power-law energy spectra with spectral index p ∼ β case, where no obvious power-law spectrum is observed. It is important toemphasize that the particle acceleration discussed here is distinct from the acceleration byshocks, where the nonthermal population contains only about 1% of particles (Neergaard 12 –Parker & Zank 2012).We gratefully thank William Daughton for providing access to the VPIC code and foruseful discussions. We also acknowledge the valuable discussions with Andrey Beresnyakand Yi-Hsin Liu. This work was supported by NASA Headquarters under the NASA Earthand Space Science Fellowship Program-Grant NNX13AM30H and by the DOE through theLDRD program at LANL and DOE/OFES support to LANL in collaboration with CMSO.Simulations were performed with LANL institutional computing. REFERENCES
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