The role of three-dimensional transport in driving enhanced electron acceleration during magnetic reconnection
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The role of three-dimensional transport in driving enhanced electron accelerationduring magnetic reconnection
J. T. Dahlin, a) J. F. Drake, b) and M. Swisdak Institute for Research in Electronics and Applied Physics,University of Maryland, College Park, Maryland 20742,USA
Magnetic reconnection is an important driver of energetic particles in many astro-physical phenomena. Using kinetic particle-in-cell (PIC) simulations, we explore theimpact of three-dimensional reconnection dynamics on the efficiency of particle accel-eration. In two-dimensional systems, Alfv´enic outflows expel energetic electrons intoflux ropes where they become trapped and disconnected from acceleration regions.However, in three-dimensional systems these flux ropes develop axial structure thatenables particles to leak out and return to acceleration regions. This requires a finiteguide field so that particles may move quickly along the flux rope axis. We showthat greatest energetic electron production occurs when the guide field is of the sameorder as the reconnecting component: large enough to facilitate strong transport, butnot so large as to throttle the dominant Fermi mechanism responsible for efficientelectron acceleration. This suggests a natural explanation for the envelope of electronacceleration during the impulsive phase of eruptive flares.PACS numbers: 52.35.Vd,94.30.cp,52.65.Rr,96.60.Iv a) University Corporation for Atmospheric Research, Boulder, CO, USA; NASA Goddard Space Flight Cen-ter, Greenbelt, Maryland 20771, USA; Jack Eddy Postdoctoral Fellow; [email protected] b) Department of Physics, University of Maryland, College Park, Maryland 20742, USA; Institute for PhysicalScience and Technology, University of Maryland, College Park, Maryland 20742, USA . INTRODUCTION Magnetic reconnection is thought to be an important driver of energetic particles inastrophysical plasmas, releasing stored magnetic energy via efficient acceleration of a non-thermal population. Reconnection-associated energetic particle production has been well-observed in solar flares and magnetospheric storms . Solar flare observations in particularindicate that reconnection-driven acceleration can be very efficient, driving a large non-thermal electron component with a total energy content comparable to that of the energy inthe initial magnetic field . This mechanism therefore provides a promising explanation for avariety of astrophysical phenomena characterized by energetic particle production, includingstellar flares, gamma-ray bursts , and gamma-ray flares in pulsar wind nebulae .Electron acceleration by magnetic reconnection has attracted significant interest, e.g. .Two specific processes have received the most attention. The first is acceleration by elec-tric fields parallel to the local magnetic field ( E k ) . However, the number of electronsthat can be accelerated through this mechanism can be limited because during magneticreconnection non-zero E k typically only occur near X-lines and separatrices. Additionally,acceleration by parallel electric fields has a weak energy-scaling ( ∼ ǫ / with ǫ the par-ticle energy) and characteristically drives bulk electron heating rather than a non-thermalcomponent.In the second process , charged particles gain energy as they reflect from the ends ofcontracting magnetic islands. (An analogous process occurs during the acceleration of cosmicrays by the first-order Fermi mechanism.) In contrast to the localization of E k , this canoccur wherever there are contracting field lines, including the merging of magnetic islandsand in the outflows of single X-line reconnection and in turbulent reconnectingsystems where magnetic field lines are stochastic and conventional islands do not exist .This mechanism is therefore volume-filling and can accelerate a large number of particles.This mechanism scales strongly with the particle energy ( ∼ ǫ ) and preferentially energizesnon-thermal particles .Several recent studies of two-dimensional reconnection found that the guide field(the magnetic component parallel to the reconnection axis) controls which mechanisms con-tribute to electron energy gain. In the antiparallel (small guide field) regime Fermi reflectiondominates , whereas in reconnection where the guide field is much larger than the recon-2ecting component parallel electric fields drive essentially all of the electron energy gain .In the latter (strong guide field) regime energetic electron production is weak, indicatingthat parallel electric fields are ineffective drivers of energetic electrons in reconnection .Studies of particle acceleration in reconnection have primarily been based on 2D simula-tions, in which accelerated particles are typically localized near the X-line, along magneticseparatrices and within magnetic islands . There are some observations with small ambi-ent guide fields that support such a picture. A notable exception are Wind observationsin which energetic electrons up to 300 keV are seen for more than an hour in an extendedregion around the reconnection region . These observations correspond to reconnection witha strong guide field.Two-dimensional simulations impose limitations on the magnetic topology as well as theavailable spectrum of instabilities. In the presence of an ambient guide field, 3D reconnectioncan become turbulent as a result of the generation of magnetic islands along separatricesand adjacent surfaces . While test particle trajectories in MHD fields have been used toexplore acceleration in such systems , the absence of feedback of energetic particles on thereconnection process in such models limits their applicability to physical systems. Recent 3Dstudies of kinetic reconnection examined particle acceleration in pair plasmas . However,these studies focused on relativistic regimes where the magnetic energy per particle exceedsthe rest mass energy and included no ambient guide field.In a recent kinetic study of nonrelativistic reconnection, we showed that energetic electronproduction was greatly enhanced in three-dimensional systems . This occurs because two-dimensional magnetic islands trap particles, limiting energy gain, whereas three-dimensionalreconnection generates a stochastic field that enables electrons to access volume-filling ac-celeration regions. The relative enhancement was found to increase with the size of thesimulation domain, suggesting that that the impact of three-dimensional dynamics is robustfor astrophysical characterized by spatial and temporal scales that are much larger thankinetic scales.In this article, we extend this study in several key ways. We begin by reviewing the theoryof particle acceleration in reconnection (section II) and describing our kinetic particle-in-cell(PIC) simulations (section III). We then explore the physics of two and three-dimensionalreconnection to highlight the remarkable similarity of many of the bulk properties (sec-tion IV). In section V, we review the physics of electron acceleration enhancement in 3D3econnection and demonstrate, by varying the spatial length in the third dimension, thattransport and enhanced acceleration are intrinsically linked. We show in section VI thata magnetic guide field plays an important role in facilitating three-dimensional transport,and in section VII introduce an ‘injection criterion’ that explains why energetic electronsare enhanced but protons are not. We discuss the astrophysical implications of these resultsin section VIII. II. PARTICLE ACCELERATION IN THE GUIDING-CENTER LIMIT
In order to examine electron acceleration we assume a guiding-center approximationrelevant for a strong guide field . In this limit, the evolution of the kinetic energy ǫ of asingle electron can be written as: dǫdt = qE k v k + µγ (cid:18) ∂B∂t + u E · ∇ B (cid:19) + γm e v k ( u E · κ ) (1)where E k = E · b is the parallel electric field, µ = m e γ v ⊥ / B is the magnetic moment, u E = c E × B /B , and κ = b · ∇ b is the magnetic curvature. The velocity components parallel andperpendicular to the magnetic field are v k and v ⊥ , respectively; γ is the relativistic Lorentzfactor, and b is the unit vector in the direction of the local magnetic field.The first term on the right-hand-side of the equation corresponds to acceleration byparallel electric fields, which are typically localized near the reconnection X-line and alongseparatrices. The second term corresponds to betatron acceleration associated with µ con-servation in a temporally and spatially varying magnetic field. Because reconnection releasesa system’s magnetic energy, this typically causes electron cooling . The last term corre-sponds to Fermi reflection of particles from contracting magnetic field lines . Both E k and Fermi reflection change the parallel energy of the particles, while betatron accel-eration changes the perpendicular energy. The term u E · κ corresponds to local field linecontraction: u E · κ = − ˙ ℓ/ℓ (where ℓ is the field line length) and is linked to the conservationof the parallel adiabatic invariant R v k dℓ . The guiding-center approximation given in Eq. (1)is accurate when electrons are well-magnetized. In the weak-guide field regime, other termssuch as the polarization drift may be significant (compare Li et al. ). However, the polar-ization drift gives the change in the electron bulk flow energy which is typically small for arealistic electron-to-ion mass ratio. 4 II. PARTICLE-IN-CELL SIMULATIONS
We explore particle acceleration in reconnection via simulations using the massively par-allel 3D particle-in-cell (PIC) code p3d . Particle trajectories are calculated using therelativistic Newton-Lorentz equation, and the electromagnetic fields are advanced usingMaxwell’s equations. The time and space coordinates are normalized, respectively, to theproton cyclotron time Ω − ci = m i c/eB x and inertial length d i = c/ω pi . The typical grid cellwidth ∆ = d e /
4, where d e = d i p m e /m i is the electron inertial length. The time step is dt = Ω − ce /
4, where Ω ce = ( m i /m e )Ω ci is the electron cyclotron frequency.All simulations are initialized with a force-free configuration and use periodic boundaryconditions. This is chosen as the most generic model for large-scale systems such as thesolar corona where the density jump between the current layer and upstream plasma is notexpected to be important. The magnetic field is given by B x = B x tanh( y/w ) and B z = q (1 + b g ) B x − B x , corresponding to an asymptotic guide field B z = b g B x . We include twocurrent sheets at y = L y / L y / w = 1 . d e .This initial configuration is not a kinetic equilibrium, which would require a temperatureanisotropy , but is in pressure balance. We use at least 50 particles per cell per species.The initial electron and proton temperatures are equal and isotropic with T = 0 . m i c A ,and the initial density n and pressure P are constant so that β x = 8 πP/B x = 0 .
5. Thespeed of light is c = 3 c A p m i /m e , where c A = B x / √ πm i n is the Alfv´en speed based onthe reconnecting component of the magnetic field.Table I lists the simulation configurations discussed in this paper. We focus on configura-tions SM (‘medium’) and SL (‘large’) with spatial dimensions L x × L y = 51 . d i × . d i and L x × L y = 102 . d i × . d i , respectively. The larger simulations (SL) have more magneticflux to reconnect, and can therefore run for a longer time and generate many more energeticelectrons. However, the SM configuration is ∼ b g = 1 and the three-dimensional simulations use L z /d i = 25 . ). The simulation with b g = 1 . /d e = Ω ce dt = 1 /
6. Theelectron-positron (pair plasma) configuration (S1) uses Ω ce dt = 1 /
20 and ∆ /d e = 1 / ame L x /d i L y /d i L z /d i m i /m e b g SM 51.2 25.6 2D, 1.6 - 25.6 25 0 - 1.5SL 102.4 51.2 2D, 1.6, 25.6 25 0, 0.5, 1.0S1 102.4 51.2 1.6, 51.2 1 1.0S100 51.2 25.6 0.8, 25.6 100 1.0TABLE I. Simulation Parameters. tions with L z ∼ . d i ≪ L x , L y so that the numerical heating is the same as in 3D, yet thereconnection physics remains essentially 2D (see Section V and Fig. 7). To reduce compu-tational expense, configurations SM and SL have an artificial proton-to-electron mass ratio m i /m e = 25. Simulations with m i /m e = 100 (S100) and m i /m e = 1 (S1) are also presentedto explore the impact of the separation between electron and proton scales. IV. OVERVIEW OF RECONNECTION IN 2D AND 3D DOMAINS
Reconnection develops from particle noise via the tearing instability, which generatesinteracting flux ropes that grow and merge until they reach the system size. These tearingmodes grow wherever k · B = 0 (Ref. 40). In a slab equilibrium with B = B x ( y ) x + B z ( y ) z ,such modes are characterized by a wavevector k = k x x + k z z and grow on flux surfacesdefined by B x /B z = − k z /k x . The pitch of the unstable mode is given by a characteristicangle θ = arctan( k z /k x ) with respect to the reconnecting field B x . In a two-dimensionalsystem where k z = d/dz = 0, tearing modes are constrained to grow at the center of thecurrent sheet where B x = 0. However, in three-dimensional systems the nonlinear interactionof modes with different pitches destroys flux surfaces and generates a stochastic, turbulentmagnetic field that facilitates enhanced particle transport .This stochastic magnetic structure is illustrated in Figs. 1a-b, which show isosurfaces ofone component of the electron current density J ez in configuration SM. At t = 12, severaltearing structures with k z = 0 are visible (compare Fig. 1 in Daughton et al. 2011 ). Thefilamentary current distribution at t = 50 showcases the late-time nonlinear development. Adifferent view of the filamentary structure, which emphasizes the stochastic structure of themagnetic field, can be observed in the Poincar´e surface-of-section shown in Figure 1c. Thereis a clear boundary between the stochastic reconnecting region (disordered punctures) and6he asymptotic, laminar field.The stochastic 3D dynamics do not substantially impact the magnetic energy release(see Fig. 2a), as has been noted by Daughton et al. Another diagnostic for the energyrelease is the field-line contraction u E · κ that drives Fermi acceleration according to Eq. (1).The spatial average h u E · κ i , ( calculated over the stochastic reconnection region describedbelow) is shown in Fig. 2b. Although the 2D simulation is relatively bursty, the overalltime-evolution is comparable to that in 3D. In both simulations, h u E · κ i decreases in timeas islands grow and the typical radius of curvature R c = | κ | − increases. Figure 2c shows theprobability distribution function (pdf) of u E · κ inside the stochastic reconnecting region atΩ ci t = 40. The pdf in the 3D system is symmetric for small values of u E · κ , and is consistentwith a double-exponential distribution ∝ exp[ −| u E · κ | τ A / .
39] where τ A = L x /c A is theAlfv´en crossing time. The symmetric component can only produce net acceleration througha second-order Fermi process. The positive mean value h u E · κ i τ A ≈ . u E · κ ) τ A >
1. Although the characteristic scales for first and second-order components are comparable, the first-order mechanism is far more efficient, and henceis the dominant driver of particle acceleration in this system. The symmetric (second-orderFermi) component is consistent with Alfv´enic fluctuations where the flow and curvature areout of phase corresponding to no net field-line contraction, i.e. h u E · κ i = 0.It has been shown previously that the development of pressure anisotropy with P k ≫ P ⊥ causes the cores of magnetic islands to approach firehose marginal stability, where the tensiondriving magnetic reconnection ceases, thereby throttling reconnection . Figure 3a,cshow that a significant anisotropy P k > P ⊥ develops, as is the typical case in 2D (notshown). This suggests that the plasma heating and energization occurs in similar waysin 2D and 3D, and that the turbulent magnetic field generated in 3D does not isotropizethe plasma. Phase space plots of temperature anisotropy T k /T ⊥ and β k = 8 πP k /B areshown in Fig. 4 for three values of the guide field b g in 2D and 3D, along with marginalstability boundaries for the firehose and mirror instabilities (bottom and top, respectively)in a similar format to that used previously in analyzing solar wind data . For the strongguide field cases (4e,f) there is insufficient free energy in the reconnecting field to drive thesystem either firehose or mirror unstable. However, for the simulations with b g = 0 . , . . The presenceof reconnection-accelerated electrons is therefore a useful proxy for the reconnection region.This is similar to the “electron mixing” described by Daughton et al. (Ref. 42). Indeed,the volume defined by P e k ,nt ≥ . n T e (where P e k ,nt is the parallel energy density ofelectrons exceeding ǫ = 0 . m e c ) corresponds well to the reconnecting region as indicatedby the electron pressure and current density (see Fig. 3). Using this marker for designatingthe reconnection domain allows us to estimate the reconnected volume V r and can be usedto determine a characteristic width in the 3D system: L r = V r /L x L z . The 2D analogue isthe area inside the separatrices of the primary X-line. A mean inflow velocity can then bedetermined from v in = ˙ L r /
2, yielding comparable v in /c A ≈ . , .
045 for the 3D and 2Dsimulations respectively ( ˙ L r is averaged from Ω ci t = 8 to Ω ci t = 125). A calculation of the2D reconnection rate, determined from the time rate of change of the flux function at theprimary X-line, yields a nearly identical inflow velocity v in /c A ≈ . R c ≈ h κ i − = B / | B · ∇ B | ≈ ( L r / B /B x ) ≈ L r . In summary, thetotal magnetic energy conversion, field line contraction (Fermi drive), and reconnection rateprovide strong evidence for the remarkable simlarity of bulk properties of the 2D and 3Dreconnection. V. ELECTRON ACCELERATION
In a previous study we found that electron acceleration was enhanced in a 3D recon-necting system. These results are summarized in Fig. 5. Although there is substantialacceleration in both systems, the fraction of electrons with energy exceeding 0 . m e c ismore than an order of magnitude larger than in the 2D simulation (Fig. 5a). However, asnoted in Section IV, the magnetic energy dissipation is comparable in 2D and 3D systems.This suggests that the increased energetic electron production in the 3D system is due toenhanced acceleration efficiency rather than an increase in the total energy imparted tothe plasma. According to equation (1) the acceleration mechanisms have different scalingswith the particle energy: the Fermi reflection term is second-order in the parallel velocity,whereas the parallel electric field term is only first-order. The instantaneous average acceler-8tion per particle for both E k and Fermi reflection in configuration SM, ( b g = 1) is shown inFig. 5b. The bulk thermal electrons (low energies) are primarily accelerated by E k , whereasFermi reflection is more important at high energies. The energetic electrons are primarilyaccelerated in the parallel direction so that the momentum distribution f ( p k ) exceeds f ( p ⊥ )(Fig. 5c), consistent with acceleration via Fermi reflection and E k . To summarize: electronacceleration, primarily driven by field-line contraction, is enhanced in 3D systems. How-ever, this is not due to greater energy release, so must instead be due to enhancement of theacceleration efficiency.As was discussed in Section IV, the stochastic structure of the magnetic field in 3Dsystems allows field-line-following particles to wander throughout the chaotic reconnectingregion . However, in 2D systems reconnected field lines form closed loops (islands) thattrap particles. The impact of topology on transport is reflected in the spatial distributionof the most energetic particles (shown in Fig. 6a,b). These particles occupy narrow bandswell inside the islands in the 2D simulation, but are distributed throughout the reconnectingregion in the 3D simulation. The most efficient electron acceleration regions are near theX-lines and at the ends of islands (Fig. 6c,d). In the 2D system, trapped energetic practiclesare unable to access these regions, and the overall acceleration efficiency is suppressed withrespect to the 3D system where the energetic particles wander the reconnecting region andundergo continuous acceleration. Figure 6e,f shows the resulting rate of Fermi energizationfor electrons with ǫ > . m e c , revealing an order-of-magnitude difference between the twosystems.In order to examine the transition from 2D to 3D reconnection, we performed a setof simulations with different values of L z , from a quasi-2D system with L z = 1 . d i to asimulation with L z = L y = 25 . d i . Figure 7a-c shows the spatial distribution of the energeticelectrons ( ǫ > . m e c ) for several of these simulations. Surprisingly, there is a sharptransition at L z = 6 . d i : below this threshold, energetic electrons are trapped inside islands,whereas above this threshold the energetic electrons are space-filling. Electron energy spectraexhibit the same transition (Fig. 7d). Simulations with L z < . d i do not show enhancementwith respect to the 2D result, whereas simulations L z ≥ . d i are consistent with the 3Dresult. This reinforces the correlation between enhanced transport and acceleration in 3Dsystems. 9 I. THE ROLE OF THE GUIDE FIELD
In a recent study of two-dimensional kinetic reconnection (Dahlin et al., 2016 ), we foundthat the magnetic guide field was a vital parameter that controls the efficiency of electronacceleration. In a system with a guide field much smaller than the reconnecting component,the dominant electron accelerator was a Fermi-type mechanism that preferentially energizesthe most energetic particles. In the strong guide field regime, however, the field-line con-traction that drives Fermi reflection was negligible. Instead, parallel electric fields ( E k ) wereprimarily responsible for driving electron heating. Electron acceleration was suppressed inthe systems with a strong guide field. We argued that this was due to the the weaker energyscaling of E k acceleration.To probe the role of the guide field in three-dimensional transport and particle accel-eration, we performed several three-dimensional simulations in the configuration SM with0 ≤ b g ≤ . L z = 1 . d i ).Selected electron energy spectra from these simulations are shown in Figure 8a-c. Figure8d shows the number of electrons exceeding 30 T , revealing that energetic electron produc-tion varies strongly with the guide field. The efficiency of the Fermi mechanism that drivesenergetic electron production weakens with increasing guide field . The quasi-2D spectra(Fig. 8c) are consistent with this result, explaining why the energetic electron productiondiminishes as b g >
1. The decreasing energetic electron production for b g ≪ f D /f D ) increases with b g until it saturates above b g = 1 (there is little difference between b g = 1 , . b g = 0, the field lines approximately close on themselves.This structure inhibits particle escape from islands, similar to the 2D structure shown inFig. 10a, where flux surfaces are closed and particles become topologically disconnected fromacceleration regions. In 3D, the guide field plays a role in breaking the 2D symmetry and10llowing particles to escape along the flux rope axis. This explains why the three-dimensionalenhancement increases with the guide field. The saturation above b g = 1 can be explainedby noting that magnetic structures are typically elongated along the guide field for b g > b g ≈ .
6. Resultsfrom a set of 3D simulations with L x × L y × L z = 102 . × . × . f D /f D increases as the spectra extend to higher energies, suggestingthat three-dimensional transport will be even more important in physical systems such asthe corona where the length scales L ≫ d i . The most efficient guide field, in these simula-tions b g ≈ .
6, will likely depend both on the system size and on other plasma parameterssuch as the plasma beta, which can impact the relative efficiency of Fermi and E k -drivenacceleration. VII. AN ‘INJECTION CRITERION’ FOR ENHANCED ACCELERATION
A limitation of the present simulations is the use of an artificial mass ratio, which re-duces the separation between proton and electron scales. To examine how the mass ratioimpacts particle acceleration, we performed simulations with m i /m e = 1 , ,
100 (configu-rations S1, SM, and S100) and b g = 1. Figure 11 shows the relative enhancement of theenergy spectra in the three-dimensional simulations ( f D /f D ). For the electron-positroncase ( m i /m e = 1), there is only a slight enhancement ( ∼
2) in the energetic tail for bothspecies. For the electron-proton cases ( m i /m e = 25 , v/c A ≫ m i /m e >
1, and both electrons and positrons for m i /m e = 1) are responsible for the bulkinertia and hence the reconnection outflow velocity c A . The characteristic velocity of theseparticles is therefore of the same order as the Alfv´en speed ( v ∼ c A ), so that the bulk parti-cles do not meet the injection criterion and do not experience enhanced 3D acceleration. Afew particles in the tail of the distribution satisfy the criterion in the electron-positron case,explaining the small ( ∼
2) enhancement at high energies. For the electron-proton simula-tions ( m i /m e = 25 , v th,e /c A ∼ p β x ( m i /m e ) ≫ β ≪
1, reconnection heats electrons to an appreciable fraction of the available magneticenergy density : ∆ T e ≈ . m e c Ae , corresponding to v th,e /c A ≈ p . m i /m e ≈
10 sothat essentially all reconnection-heated electrons will satisfy the criterion, independent ofthe initial temperature. In contrast, ions are typically sub-Alf´enic and would require an in-jection mechanism (e.g. ) to undergo continuous acceleration. However, the suppressionof ion acceleration in 3D systems is surprising. While the relative increase in energy goinginto energetic electrons may play some role, it is not clear that this should preferentiallyimpact the energetic ions. Further treatment is beyond the scope of this paper.The injection criterion may require modification in the large guide field limit ( B z /B ≫ z -direction, so the relevant velocityto compare to the outflow speed is is v x ≈ vB x /B ≈ v/b g so the injection criterion becomes v/c A ≫ b g . In strongly relativistic systems, all velocities approach c , so that the injectioncriterion cannot be met. This suggests that enhanced 3D acceleration should not occur forrelativistic reconnection in either pair or electron-proton plasmas. VIII. DISCUSSION
Electron acceleration in three-dimensional systems is a complex problem that intrinsicallydepends on the transport properties of reconnection-generated magnetic fields. The picturethat emerges from this set of simulations is that particle acceleration is efficient in a three-dimensional system when the energetic population can freely access acceleration sites and12hereby achieve continuous energy gain. This requires both topological access to energyrelease regions and a super-Alfv´enic particle velocity in order to explore the open topologyat a faster time scale than the system evolves (most easily understood as the ejection of fluxfrom the X-line). While electrons satisfy this condition rather easily, heavy species such asprotons would require an injection mechanism in order to be able to propagate upstreamagainst the reconnection outflow.Efficient transport requires a strong guide field. The field structure in antiparallel re-connection is quasi-laminar, so that energetic particles are still well-trapped in islands.Propagation upstream against the Alfv´en velocity is not possible in strongly relativistic re-connection, where all characteristic velocities approach the speed of light. This is consistentwith studies by Guo et al. , that exhibit no substantial difference in energetic particleproduction between two and three-dimensional simulations in the relativistic regime.The nonthermal electron spectra in both simulations do not assume a power law form asis frequently observed in nature. The maximum energy gain is limited due to the modest sizeof the simulations; previous 2D simulations have shown that the total energy gain is greaterin larger systems . An additional issue is that these simulations have periodic boundaryconditions that prevent particle loss from the system. Solar observations suggest that elec-trons are confined in regions of energy release in the corona ; possible mechanisms for thisconfinement include mirroring and double layers . This could suggest that particle loss isnot an important concern. On the other hand, it has been suggested that the developmentof a power law requires a loss mechanism in addition to an energy drive . However, severalrecent simulations suggest that power-law spectra may still develop in the absence of a lossmechanism . The set of conditions under which power-law spectra form in kineticreconnection simulations remains an open issue.The simulations have a number of numerical limitations. These include comparativelysmall spatial and temporal scales, and an artificial electron-to-proton mass ratio. However,as was discussed in Section VII, the greater characteristic velocity of a realistically ‘light’electron facilitates transport in the stochastic topology; the efficiency of Fermi accelerationdoes not directly dependent on the particle mass. The largest simulations show that three-dimensional dynamics are increasingly important at larger scales; in contrast the diminishingfrequency of island mergers leads to suppression of further acceleration in two-dimensionalsystems. 13he role of the guide field in magnetic reconnection has broad implications for reconnection-driven particle acceleration in astrophysical systems. Electron acceleration is most efficientin the regime where both (a) the Fermi mechanism operates and (b) strong three-dimensionaltransport exists. The former requires b g .
1, the latter b g >
0, suggesting that reconnectionwith a magnetic field of the same order as the reconnecting component ( b g ∼
1) will yieldthe most efficient energetic electron production. This result is especially relevant for solarflares, where the shear in the magnetic configuration typically diminishes during the impul-sive phase (see Fletcher et al. and references therein). This corresponds to a transitionfrom strong to weak guide field reconnection, and could explain why electron acceleration(as inferred from hard X-ray emissions) is typically confined to the impulsive phase. ACKNOWLEDGMENTS
This work has been supported by NSF Grants AGS1202330 and PHY1102479, and NASAgrants NNX11AQ93H, APL-975268, NNX08AV87G, NAS 5- 98033, and NNX08AO83G.J.T.D. acknowledges support from the NASA LWS Jack Eddy Fellowship administered bythe University Corporation for Atmospheric Research in Boulder, Colorado. Simulationswere carried out at the National Energy Research Scientific Computing Center.14
IG. 1. Results for configuration SM. (a,b) Isosurfaces of J ez at Ω ci t = 12 ,
50 respectively, illus-trating the nonlinear filamentary current structure. (c) Poincar´e surface-of-section at Ω ci t = 50.We trace a set of field lines beginning at x = 0, 0 < y < . z = 0 and plot where they puncturethe plane z = 0. The surface-of-section shows a clear boundary between the stochastic field linesinside the reconnecting region and the laminar unreconnected fields. IG. 2. Results for configuration SL: 3D (green) and 2D (red). (a) Magnetic energy release vs.time. (b) Spatially averaged ( u E · κ ) τ A over the reconnecting region, where τ A = L x /c A is theAlfv´en crossing time. (c) Probability distribution function of ( u E · κ ) τ A at Ω ci t = 40, with asuperimposed double-exponential fit (blue dotted line). (d) Reconnection region half-width L r andradius of curvature R c . IG. 3. Results for configuration SL. (a)(c) z = 0 slices at Ω ci t = 100 of parallel and perpendicularelectron pressure. (b) (d) z -directed electron current density in planes defined by z = 0 and x = 0,respectively. White and black outlines indicate the contour P e k ,nt = 0 . n T e , a marker for theregion of reconnected magnetic field. IG. 4. Results for configuration SM. Phase space distribution of temperature ratio T ⊥ /T k vs. β k .Black lines indicate marginal stability conditions for the ideal firehose (bottom) and mirror (top)instabilities. IG. 5. Results for configuration SL: 3D (green) and 2D (red). (a) Energetic spectra at Ω ci t = 50(dashed) and 125 (solid) (b) Average energy gain due to Fermi acceleration (red) and E k (blue) insimulation 4a at Ω ci t = 100. (c) Parallel and perpendicular momentum spectra at Ω ci t = 0 , IG. 6. Results for configuration SL: 3D (left) and 2D (right) at Ω ci t = 100 in the plane z = 0.(a-b) Parallel energy density for electrons with ǫ > . m e c . (c-d) Field-line contraction rate u E · κ .(e-f) Fermi acceleration for electrons with ǫ > . m e c . The similarity between (c) and (e) reflectsthe nearly uniform spatial distribution of the energetic electrons in the reconnecting region. Dashedboxes outline one X-line in each panel. IG. 7. Results for configuration SM, b g = 1. (a-c) Spatial distribution of electrons with ǫ > . m e c . (d) Electron energy spectra normalized to the 2D spectrum at Ω ci t = 50 . IG. 8. Results for configuration SM at Ω ci t = 50. Three-dimensional electron energy spectra f D normalized to the initial spectrum f (a) and quasi-2D spectra f D (b). (c) quasi-2D spectra f D normalized to the initial spectrum f . (d) Electrons exceeding 30 T e vs. guide field. The systemwith b g = 0 .
65 generates the greatest number of energetic electrons. IG. 9. Results for configuration SL. Three-dimensional electron energy spectra normalized tothe initial spectrum (a) and quasi-2D spectra (b) at Ω ci t = 125 (solid) and Ω ci t = 50 (dashed).(c) quasi-2D spectra normalized to the initial ( t = 0) spectrum. (d) Electrons exceeding 30 T e atΩ ci t = 50 vs. guide field. The system with b g = 0 . IG. 10. Results from configuration SM. Single field lines (blue) for simulations with differentguide fields. Two-dimensional ( z = 0) slices of the energetic electron density are shown at thebottom of each panel. IG. 11. Results from configurations SM, S1, and S100. Three-dimensional enhancement( f D /f D ) for simulations with different mass ratios m i /m e . For the pair plasma ( m i /m e = 1),both species exhibit a small enhancement ∼
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