Characterizing velocity–space signatures of electron energization in large-guide-field collisionless magnetic reconnection

I. INTRODUCTION

Vast amounts of energy can be stored in the magnetic field of space and astrophysical plasmas. Upon reconfiguration, this embedded field may undergo reconnection that releases substantial energy, energizing particles, sometimes explosively. Magnetic reconnection occurs in a host of plasma regimes from fusion device disruptions to the birth of the solar wind and solar flares. Additionally, magnetic reconnection occurs often in the dynamic solar wind, especially at interfaces with planetary magnetic fields. Identification and quantification of particle energization may help describe the physics needed to answer such questions as the coronal heating problem. 1 In diffuse plasmas that are nearly collisionless, as those composing the solar wind and present throughout most of the heliosphere, kinetic descriptions of energy transfer are necessary to understand reconnection at particle kinetic length scales. 2,3 Significant work through theoretical, numerical, and observational studies have developed a more complete picture of the various mechanisms potentially responsible for particle energization during magnetic reconnection in collisionless plasmas. [4][5][6][7][8][9][10][11][12] Fermi acceleration and direct E k particle acceleration have been identified to account for electron energization in collisionless reconnection. 6,13,14 Dahlin, Drake, and Swisdak 15 demonstrated that there is a clear transition between Fermi acceleration and direct electric field acceleration at a guide field of unity, B g =B R ¼ 1, where B g is the out-of-plane guide magnetic field and B R is the in-plane reconnecting magnetic field magnitude. This first order Fermi acceleration is nearly completely suppressed at values B g =B R > 1. Pucci et al. 10 also found electron energization switches from perpendicular (j ? E ? ) to parallel (j k E k ) energization at B g =B R ¼ 0:6 for driven reconnection. Guo et al. 16 identified evidence of perpendicular electric field energization of electrons in strong-guide-field steady reconnection, B g =B R ¼ 3. Using 2D particle-in-cell (PIC) simulations, they found perpendicular energization is due to polarization drifts, sustained by charge-separation generated electrostatic fields near the x-point and along the separatrices, and global curvature drifts. Additionally, they proposed chargeseparation is sustained long enough to break the frozen-in condition in the presence of a large guide field, allowing perpendicular energization mechanisms due to additional sustained non-ideal effects, which are not present in anti-parallel reconnection.

Numerous studies on anti-parallel reconnection with no guidefield have been carried out. However, large-guide-field investigations have received less attention in the literature. This neglect is likely due to historically few observations of very large guide fields in the heliosphere, (i.e., B g =B R > 5). Larger guide fields (B g =B R > 1) are expected to be found within the corona, which may be confirmed observationally as Parker Solar Probe completes its mission. In the last decade, larger guide fields have been found to exist during reconnection at the magnetosheath and magnetosphere, [17][18][19][20][21] and magnetotail. 22,23 Additionally, many previous studies have focused on categorizing and identifying reconnection events through their fields and local macroscopic plasma parameters in both numerical and spacecraft investigations. 21,[24][25][26] Some recent kinetic investigations have been enabled by advances in particle detection on recent spacecraft missions, which have identified evidence of crescent distributions that are likely common in asymmetric reconnection as found by the magnetospheric multiscale (MMS) mission. [27][28][29] Until recently, few investigations have focused directly on kinetic particle energization in the non-relativistic large-guide-field limit using a formal kinetic analysis framework.

Most in situ measurements in collisionless plasmas are performed by single spacecraft. Even in cases with multiple spacecraft, such as in the MMS mission, the few points of spatial information are insufficient to describe the larger scale spatial distribution of particle energization. Thus, analysis techniques must be structured to use localized (usually single-point) measurements to make observations and determinations of plasma behavior. Understanding the kinetic behavior of large-guide-field reconnection with single-point measurement techniques may help as a diagnostic tool for spacecraft identification of such events. We present an analysis of electron energization through the characterization of kinetic velocity-space signatures at specific spatial locations in a 2D gyrokinetic reconnection simulation. This work is built from a geometry proposed by Porcelli et al. 30 implemented in the astrophysical gyrokinetics Code AstroGK, as previously used by Numata and Loureiro. 31 Using the field-particle correlation (FPC) framework developed by Klein and Howes, 32 we identify and analyze the velocity-space signatures of particle energization that arise from strong-guide-field collisionless reconnection.

II. KINETIC MECHANISMS OF ELECTRON ENERGIZATION

In analyzing energy transfer in a weakly collisional plasma, it is important to point out that the energy transfer is inherently a two-step process. 33 First, collisionless interactions between the electromagnetic fields and the plasma particles serve to transfer energy between the fields and the particles. Energy transferred from the fields to the particles will generate fluctuations in the particle velocity distribution function (VDF). The collisionless energy transfer in this first step is inherently reversible, so the energy associated with those fluctuations is non-thermal. Subsequently, these fluctuations in the VDF can undergo linear 8,34 or nonlinear phase mixing 35 to sufficiently small scales in velocity space that arbitrarily weak collisions can serve to smooth out those fluctuations. This second collisional step is irreversible, effectively thermalizing the energy that was transferred to the particles, heating that plasma species and increasing the entropy.

In this investigation, we will focus on the first step in this process, whereby the electromagnetic fields do reversible work on the plasma particles. In magnetic reconnection, this process effectively releases magnetic energy and converts it into other forms (bulk plasma flows or non-thermal energization). Note that the collisionless energization can be facilitated through a resonant process, as in the case of Landau damping of kinetic Alfv en waves, or through non-resonant processes, e.g., direct particle acceleration by electric fields. The second collisional step of the particle energization in collisionless magnetic reconnection was the focus of the analysis by Numata and Loureiro. 31 In the analysis of the particle energization in weakly collisional plasma turbulence, the collisionless dynamics leads to the continual transfer of energy back and forth between the electromagnetic field fluctuations and the particles. If the turbulent fluctuation is undamped, this energy transfer is oscillatory and reversible, contributing no net particle energization. For example, an Alfv en wave in the MHD limit kq i ( 1 is undamped and involves an oscillatory transfer of energy between magnetic field energy and plasma bulk flow kinetic energy. If the energy transfer is not purely oscillatory, e.g., in the case of a resonant damping process, a portion of the energy transfer may contribute to a net particle energization, which we will define as the secular, or net, energy transfer. This secular energy transfer is manifested through the net increase in microscopic kinetic energy of plasma particles leading to perturbations of the VDF. In collisionless magnetic reconnection, as is studied in this paper, the dynamics are not typically oscillatory as the magnetic field is reconfigured, but reversible kinetic energy transfer is still possible. Therefore, it is important to investigate the net energization of the particles during the evolution of a plasma undergoing magnetic reconnection.

In order to analyze the collisionless energy transfer in the first step of particle energization, we begin with the generalized Boltzmann equation,

This equation represents the evolution of the 3D-3V velocity distribution function f s ðr; v; tÞ for a plasma species s. The species charge and mass are q s and m s , respectively, v is the velocity, E and B are the electric and magnetic fields, and c is the speed of light; and the right-hand side represents the collision operator. Combining the Boltzmann equation for each plasma species together with Maxwell's equations forms the closed set of Maxwell-Boltzmann equations that govern the nonlinear evolution of turbulent fluctuations in a magnetized kinetic plasma.

On the timescale of the energy transfer occurring in magnetic reconnection, the collisional term is negligible under typical conditions in space and astrophysical plasmas, so we may neglect the collision operator, recovering the Vlasov equation. The Vlasov equation describes the full phase-space dynamics of a collisionless magnetized plasma. The ballistic or advection term, second on the left-hand side of Eq. ( 1), represents the advection of particles. The third term is the classical Lorentz force term, which governs the self-consistent waveparticle interactions in the kinetic plasma. Therefore, we focus on the Lorentz term to characterize the energization of particles in collisionless magnetic reconnection.

III. METHODS A. Field-particle correlation technique

Developed by Klein and Howes, 32 the field-particle correlation (FPC) technique produces a velocity-space representation of the phase-space energy density transfer in a kinetic plasma applicable to a Vlasov-Maxwell description for a collisionless plasma. The FPC technique was initially developed to separate oscillatory energy transfer during a physical process from any secular energy transfer, by taking an average over a sufficiently long correlation intervals that the oscillatory transfer largely cancels out. This method has been used to mostly identify resonant processes leading to net positive particle energization occurring in wave damping in heliospheric plasmas, 32,34,36 broadband kinetic turbulence, [37][38][39][40][41][42] strong Alfv en wave collisions, 33 collisionless shocks, 43 and laboratory evidence of the electron energization responsible for the aurora. 44 Using the Vlasov equation for the evolution of the particle distribution function, we can define a new formulation that describes the phase-space energy density evolution for a given species s. Multiplying the Vlasov equation by kinetic energy m s v 2 =2, we cast into a form

which describes the rate of change of the phase-space energy density w e;s ¼ m s v 2 f s =2. Using this description for the rate of change of phase-space energy density, we can then define a field-particle correlation that produces the velocity-space signature of energization at a single spatial location.

For the application of this technique to data from our collisionless magnetic reconnection simulations using the astrophysical gyrokinetics code AstroGK, 45 we note that the gyrokinetic distribution function h s ðx; y; z; v ? ; v k Þ 46 is related to the total distribution function f s via f s ðr; v; tÞ ¼ F 0s ðvÞ 1 À q s /ðr; tÞ T 0s þ h s ðr; v k ; v ? ; tÞ:

Here, F 0s is the equilibrium Maxwellian distribution function, / is the scalar potential, and T 0s is the reference species temperature. The parallel and perpendicular directions are with respect to the local equilibrium magnetic field B 0 up to Oð 2 Þ in the gyrokinetic ordering. 46 As a technical step, we transform from the gyrokinetic distribution function h s to the complementary perturbed distribution function,

where hÁ Á Ái denotes the ring average taken at fixed guiding center R s . 46 The complementary distribution function g s describes perturbations to the background distribution in the frame of reference moving with the transverse oscillations of an Alfv en wave. Field-particle correlations calculated using h s or f s yield qualitatively and quantitatively similar results to those computed with g s . 37 Below, we present the correlations between the complementary perturbed distribution function and the parallel electric field E k at a single-point r 0 ,

The unnormalized, centered correlation C(A, B) is essentially a sliding time average, and is defined at time t i by

for quantities A and B, which together as a product represent a rate of change of energy density, which are measured at discrete times t j ¼ jDt, with their product averaged over the correlation interval of s NDt. 37 By averaging over a finite correlation interval s, oscillatory energy transfer between the electromagnetic fields and the plasma particles is averaged out, leaving only the net rate of energy transfer between the fields and the particles. In the 2D simulations of magnetic reconnection presented here, the flow remains generally laminar during the main phase of reconnection, so it is not necessary to time-average over a finite correlation interval to cancel out a large oscillatory component. Therefore, we chose to simply evaluate the field-particle correlation instantaneously, taking the correlation interval s ¼ 0. To analyze simulations of collisionless magnetic reconnection in 3D, which are often found to become turbulent, it may be necessary to employ a finite correlation interval s > 0.

The parallel electric field correlation defined in Eq. ( 5) describes the phase-space energy transfer rate to species s by E k at a single point in space r 0 and is a three dimensional function in gyrotropic phase space and time, ðv k ; v ? ; tÞ. We present the correlation in several standard ways to aid visualization of the particle energization in velocityspace and time. A gyrotropic plot of the correlation C E k ;s ðv k ; v ? Þ at a specific time t 0 shows how the rate of change of phase-space energy density varies in gyrotropic velocity space ðv k ; v ? Þ, as in Fig. 2(b). We generally refer to the pattern of energization seen in the gyrotropic plot as the velocity-space signature of the particle energization mechanism.

Alternatively, we can integrate the correlation over the perpendicular velocity,

to obtain the reduced parallel correlation, C E k ;s ðv k ; tÞ. A timestack plot presents this reduced parallel correlation as a function of v k and time, which is particularly useful to explore the rate energization of particles over the course of the main phase of magnetic reconnection in our simulations, as in the main panel of Fig. 2(c).

Integrating the reduced parallel correlation over full simulation duration T yields

a simple one-dimensional representation of net energization of particles as a function of v k over the course of the simulation, as in the lower panel of Fig. 2(c). This visualization facilitates the identification of the bipolar signatures that are indicative of collisionless resonant energization mechanisms, such as Landau damping. 38,41 Alternatively, one can integrate the reduced parallel correlation over v k to obtain the rate of particle energization at the single spatial point r 0 as a function of time, given by

as in the left-hand panel of Fig. 2(c). Note that this form shows that, when integrated over all velocity space, the parallel electric field correlation simply yields the rate of work done by the parallel electric field on the particle species s at position r 0 vs time.

B. Simulation

In this paper, we analyze 2D magnetic reconnection simulations (d=dz ¼ 0) with a strong out-of-plane guide field. The domain consists of a doubly-periodic slab geometry with an in-plane reconnection field. To solve the fully electromagnetic gyrokinetic equations for the electrons and ions, AstroGK employs a pseudo-spectral algorithm for the spatial coordinates (x, y), and Gaussian quadrature for velocity space integrals. The velocity grid is discretized into energy

, where B z0 is the constant, background (guide) magnetic field. Derivatives of velocity space in the collision operator are estimated using a first-order finite difference scheme on an unequally spaced grid according to the quadrature rules in Barnes et al. 47 We perform the same simulations as in Numata and Loureiro 31 with a fixed collisionality ei ¼ ee ¼ ii ¼ 1:0 Â 10 À4 for all simulations. The simulation is initialized with an unstable tearing mode for the in-plane magnetic field configuration as in Numata et al. 45,48 The equilibrium total magnetic field is given by

where B eq y is the in-plane, reconnecting component, with a maximum value B max y ¼ 1, determined from the parallel vector potential by B eq y ðxÞ ¼ À@A eq k =@x, is the gyrokinetic epsilon-a small expansion parameter defining scale separation in gyrokinetics (see Howes et al. 46 ) The background Maxwellian electron distribution is perturbed with a perturbation of the from df e / V k f e0 . To support the modified distribution function, the vector potential is defined as follows using a shape function S h ðxÞ to enforce periodicity (see Numata et al. 45 ) such that

A eq k arises from the parallel electron current that must satisfy Ampe `re's law. The dimensions of the simulation determine the scale lengths, with equilibrium current width a and L x the scale of the box in the x-direction, where L x =a ¼ 3:2p. For the y-direction, the width of the box is L y =a ¼ 2:5p. The tearing mode is imposed by a small sinusoidal perturbation to the equilibrium magnetic field, so that Ãk / cos ðk y yÞ with wave number k y a ¼ 2pa=L y ¼ 0:8, which yields D 0 a % 23:2 for the tearing instability parameter. The plasma considered is quasineutral, so that n 0i ¼ n 0e ¼ n 0 , with singly charged ions q i ¼ Àq e ¼ e.

The scale of the system is determined by the equilibrium magnetic field. Thus, we normalize time by the Alfv en time s A a=V A , where V A B max y = ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 4pn 0 m i p is the in-plane Alfv en velocity corresponding to the maximum initial B eq y . Additional fundamental parameters define the physical scales within the simulation: The mass ratio, l ¼ m e =m i , the equilibrium plasma temperature ratio, T 0i =T 0e 1, the ion plasma beta, b i ¼ b e , and the ratio of ion sound Larmor radius to the equilibrium scale length a (i.e., width of the current sheet), q se =a c se =ðX ci aÞ. The ion sound speed for cold ions is c se ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi T 0e =m i p , and the ion cyclotron frequency is X ci ¼ eB z0 =ðm i cÞ. The following parameters are fixed for all simulations throughout this paper,

These scale parameters require q i =a ¼ 0:25; q e =a ¼ 0:025, and b i ¼ b e . As explained by TenBarge et al., 49 the gyrokinetic expansion parameter is neither a fixed nor a user chosen parameter: under the gyrokinetic ordering 46 and using the normalization employed in AstroGK, 45 all quantities are scaled by to make the calculations of asymptotically small values using numerical computations of order Oð1Þ. To compare results to a particular system, it is necessary to specify a value of -e.g., choosing a specific ratio of the in-plane to the guide magnetic field. The ion plasma beta b i is defined using the outof-plane guide magnetic field, b i n 0i T 0i =ðB 2 z0 =8pÞ. To perform simulations with different values of b i , one can take the ratio of the guide magnetic field B z0 to the in-plane magnetic field B eq y to be constant, where B eq y =B z0 ¼ ( 1, and vary the ion temperature T 0i relative to these fixed quantities to change the plasma b i . Five simulations are performed using varying values of b i ¼ ð0:01; 0:03; 0:1; 0:3; 1:0Þ.

In these simulations, the electron current layer width d CS;e decreases with increasing b e , such that the electron Larmor radius q e ¼ l 1=2 q se ffiffi ffi 2 p approaches the electron skin depth d e ¼ b À1=2 e l 1=2 q se ffiffi ffi 2 p . Numata and Loureiro 31 demonstrate with linear simulations in the collisionless regime, the frozen-flux condition is broken by electron inertia for small b e . When b e is greater than unity, q e becomes larger than d e , such that electron finite-Larmor radius (FLR) effects, rather than electron inertia, lead to field line breaking. By varying the collisionality, they also find in the linear regime small ( e s A Շ 1 Â 10 À3 ), but finite collisionality results in asymptotic growth rates and current sheet width.

B. Locations of electron energization

The electron energization as a function of the position in the (x, y) plane is given by the work done on the electrons by the electric field, j e Á E. Our gyrokinetic simulations of collisionless magnetic reconnection are valid in the limits of strong guide field B z0 =B eq y $ À1 ) 1 and of the gyrokinetic approximation 46,49 with k ? ) k k , where the parallel direction is along the guide field in the out-of-plane, z direction. In all of our simulations, the summed contributions to the electron energization by the in-plane (perpendicular) components of the electric field are much smaller than that by the out-of-plane component, j x;e E x þ j y;e E y ( j k;e E k , as expected in the gyrokinetic limit. In the gyrokinetic approximation, the quasi-neutrality condition 46 Physics of Plasmas ARTICLE scitation.org/journal/php dictates that r Á j ¼ 0, so the limit k ? ) k k implies that k ? Á j ? ¼ 0 to lowest order in the gyrokinetic expansion parameter . Since the perpendicular electric field in the same k ? ) k k limit scales as k ? /, where / is the electrostatic potential, then the work done by the perpendicular electric field j ?;e Á E ? scales as j ?;e Á k ? / ' 0 to lowest order in . Therefore, in our analysis, here we focus strictly on the parallel contribution to the rate of electron energization, j k;e E k . In Fig. 1, for the b i ¼ 0:01 simulation, we plot the spatial distribution over the (x, y) plane of (a) the parallel electron current j k;e , (b) the parallel electric field E k , and (c) the work done on the electrons due to the reconnection electric field j k;e E k =Q 0 , where we normalize by the characteristic heating rate per unit volume,

We plot these quantities at the time t=s A ¼ 22:5 of the maximum spatially-integrated electron energization rate Ð dxdyj k;e E k =Q 0 , plotted in Fig. 1(d). The reconnection rate, estimated using the magnitude of the reconnecting (parallel) electric field E k at the x-point in the simulation cE k =ðv A;y B max y Þ, is plotted in Fig. 1(e), and its peak roughly coincides in time with the spatiallyintegrated electron energization rate during the main phase of reconnection, roughly spanning 10 Շ t=s A Շ 30. Note that, for the initial Porcelli equilibrium 30 employed in these simulations, there is a limited amount of upstream magnetic flux, so the main phase of reconnection eventually ceases once a majority of the initial flux has reconnected.

In Fig. 1(a), the parallel electron current in the þz direction peaks at the x-point and along the separatrices. The parallel electric field that drives the reconnection flow is fairly uniform throughout the region spanning 17 < x=q i < 23 and 7 < y=q i < 25, as shown in Fig. 1(b). The rate of change of electron energy density is given by the product of these two quantities, and we find that the positive electron energization occurs dominantly over a relatively small region at the x-point and along the separatrices within the ion diffusion region, as indicated by the bright red regions in Fig. 1(c). Since the field-particle

C. Energization at the x-point

To investigate the energization of the electrons at the x-point of the reconnection geometry, located at ðx=q i ; y=q i Þ ¼ ð20:1; 15:7Þ, we focus initially on the perturbations to the electron velocity distribution function and the parallel electric field at the time t=s A ¼ 22:5 when the electron energization rate j k;e E k peaks at the x-point. In Fig. 2(a), we plot the complementary perturbed electron distribution function g e ðv k ; v ? Þ over gyrotropic velocity space, along with the reduced parallel perturbed distribution g e ðv k Þ, obtained by integrating over v ? , in the lower panel. This perturbed velocity distribution leads to the parallel electron current j k;e needed to sustain the change in the B y component of the magnetic field across the midplane, as seen in Fig. 1.

To explore the electron energization at the x-point, we use g e ðv k ; v ? Þ and the parallel electric field E k to compute the fieldparticle correlation in gyrotropic velocity space C E k ;e ðv k ; v ? Þ, given by Eq. ( 5) and plotted in Fig. 2(b). In this figure, we see a symmetric (about v k Þ increase in the phase-space energy density w e of the electrons, dominantly occurring over the parallel velocity range 1 Շ jv k j=v te Շ 2, as made clear by the lower panel where the reduced parallel correlation C E k ðv k Þ, computed by integrating over v ? , is plotted. The vertical dashed black lines indicate the Alfv en velocity v A;z ¼ B z0 = ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 4pn 0 m i p in the parallel direction, 6v A;z =v te , for b i ¼ 0:01 and mass ratio m i =m e ¼ 100, where velocities are normalized by the electron thermal velocity

, with temperature given in units of energy. The gyrotropic correlation C E k ;e ðv k ; v ? Þ in panel (b) is the velocity-space signature of the electron energization at the x-point in this simulation of collisionless magnetic reconnection in the strongguide-field limit.

To probe how this electron energization as a function of v k evolves over time at the x-point, we plot in Fig. 2(c we see that both the peak and time-integrated energization rates have a nearly identical dependence on v k .

A simple model can be constructed that explains the qualitative and quantitative features of the reduced velocity-space signature observed in the lower panel of Fig. 2(b). The complementary perturbed distribution function g e ðv k Þ plotted in Fig. 2(a) is well approximated by an analytic form,

where the integration over this perturbation leads to zero density perturbation and a parallel flow given by the parameter U k;e . Neglecting the Boltzmann contribution Àq e /F 0;e =T e (which is proportional to F 0;e , and therefore yields zero net energization), the total electron velocity distribution is given by

, where F 0;e ðv k Þ is the equilibrium Maxwellian parallel velocity distribution and g e ðv k Þ is an approximate analytical form for the self-consistently evolved perturbed electron distribution in the simulation. In Fig. 3(a), we plot the equilibrium electron parallel velocity distribution F 0;e ðv k Þ (black) and the total parallel electron velocity distribution f e ðv k Þ (red).

Panel (b) shows the perturbed electron velocity distribution function g e ðv k Þ (blue) with parameter U k;e =v te ¼ À0:25, guided by the first velocity moment of the distribution in the out-of-plane direction. The factor of the reduced parallel correlation C E k ;e ðv k Þ that depends on the distribution function, þev 2 k =2ð@g e =@v k Þ, where we have substituted the electron charge q e ¼ Àe, is plotted in Fig. 3(c). This form of the reduced parallel correlation shows excellent agreement with the analysis of the simulations in the lower panels of Figs. 2(b) and 2(c), where there is a small loss of phase-space energy density at jv k j=v te 1= ffiffi ffi 2 p , and the bulk of the increase in the phase-space energy density occurs in the range 1 Շ jv k j=v te Շ 2. This peak location is due to the mathematical form of the correlation, which is weighted by v 2 k and the derivative of the perturbed complementary distribution function. Thus, the velocity-space signature of the electron energization at the x-point is simply due to the bulk acceleration of the electrons in the-z direction by the parallel electric field E k -where the parallel electron flow U k;e supports the parallel current arising due to the odd (in v k ) perturbation of g e ðv k Þ, with a form well modeled by Eq. ( 13). Thus, this velocity-space signature represents the bulk acceleration of the electrons in the out-of-plane direction by E k , with a net electron energization rate at the x-point simply given by j k;e E k ¼ Ð dv k C E k ;e ðv k Þ. In summary, for collisionless magnetic reconnection in the strong-guide-field limit, the electron energization at the x-point is dominated by bulk acceleration of the electrons by E k . The particular form of the distribution function arising in the simulation yields a positive rate of electron energization with a signature, that is, symmetric about v k ¼ 0 and that peaks over the velocity range 1 Շ jv k j=v te Շ 2. Note that this is not a resonant acceleration of the electrons, as would be expected for collisionless damping via the Landau resonance, but instead it is a bulk acceleration of the electrons, in agreement with the gyrokinetic ordering and previous analysis of electron energization by E k in the strong-guide-field limit of collisionless magnetic reconnection. 6,[13][14][15]

D. Energization in exhaust

In the same manner as for the x-point, we now select three points located at r=q i ¼ ðx=q i ; y=q i Þ, where r A =q i ¼ ð18:9; 11:8Þ; r B =q i ¼ ð20:1; 11:8Þ; r C =q i ¼ ð21:4; 11:8Þ to investigate the particle energization within the exhaust. As before, we begin with analyzing the perturbations to the electron velocity distribution function and electric field at the time t=s A ¼ 22:5. In Fig. 4(a), we show the total spatial electron energization rate j k;e E k again for reference. The points r A and r C , on either side of the midplane, are located just inside of the separatrix boundary at time t=s A ¼ 22:5. Each column in the lower two rows of Fig. 4 corresponds to labels A, B, and C in the exhaust of panel (a) from left to right, respectively.

At point r B , directly downstream from the x-point, along the midplane, we see a qualitatively similar perturbed complementary electron velocity distribution function to that at the x-point, shown in Fig. 4(c), with the reduced parallel perturbed distribution shown in the lower panel. The odd perturbation in v k is more confined in parallel velocity than at the x-point, where the bulk of the perturbed electron VDF is contained at jv k =v te j < 1. In contrast to this point B on the midplane shown in Fig. 4(c), we observe that, at the points A and C near the separatrix on either side of the midplane, there is an The model complementary perturbed electron velocity distribution function g e as in Eq. ( 13). (c) The corresponding reduced correlation representing the phase-space density signature of the perturbed electron velocity distribution function. To analyze the particle energization within the exhaust, we plot the reduced field-particle correlation C E k ðv k ; tÞ timestack shown in the bottom row of Fig. 4. In the left vertical panel for each point in the exhaust, we see the net energization rate @W e ðtÞ=@t peaks at a similar time of t=s A ¼ 22:5 for either side of the midplane (e) and (g). At the midplane in the exhaust (f), the net energization rate peaks slightly earlier at t=s A % 20:0. In the lower horizontal panel for each point, we again plot the time-integrated reduced parallel energization rate over the full interval. At the midplane in the exhaust, we again find a symmetric (about v k ) increase in phase-space energy density w e of the electrons. The energization extends over a range, that is, slightly narrower than at the x-point, 0:5 Շ jv k j=v te Շ 2:0, and is centered close to jv k j=v te ¼ 1. On either side of the midplane, there is an asymmetric increase in phase-space energy density over the velocity range 0:7 Շ jv k j=v te Շ 3. At point r A , the electrons traveling in the parallel direction v k > 0 are preferentially accelerated, gaining phase-space energy density. At point r C , on the other hand, the electrons traveling in the anti-parallel direction are preferentially gaining phase-space energy density. On either side of the midplane, there is a clear cutoff in the dominant energization signature at jv k j=v te ¼ 1. Panels (e)-(g) in Fig. 4 show the characteristic velocity-space signatures of electron energization in the exhaust region of strong-guide-field magnetic reconnection, a key result of this investigation. The parallel velocity ranges for the electron energization are apparent in the lower panels of (e), (f) and (g), showing the time-integrated reduced correlation Ð dtC E k ðv k ; tÞ.

Physics of Plasmas

The asymmetry in the perturbed distribution function and reduced parallel energization rate on either side of the midplane within the exhaust is at first surprising, given the symmetric signature of j k;e E k both in magnitude and sign. We hypothesize here that the asymmetric velocity-space signatures in the exhaust region, shown in Figs.

4(e) and 4(g), can be explained by the combination of a parallel electron flow with an electron density perturbation. If we look at the density perturbation in the (x, y) plane, shown in Fig. 12(a) of Appendix B, we see in the lower half of the simulation plane there is a decrease in electron density to the left of the midplane and an increase in density to the right of the midplane along each lower separatrix arm. A similar quadrupolar density pattern is well-known from previous hall-MHD and two-fluid simulations. [52][53][54] The perturbed electron velocity distributions shown in Figs. 4(b) and 4(d) are consistent with the density perturbations shown in Fig. 12(a). Since the velocity-space signature of energization depends on the details of the electron velocity distribution, it is expected that this density perturbation will influence the form of the observed velocity-space signature.

To demonstrate that a parallel electron flow combined with a density perturbation can indeed generate the asymmetric velocity-space signatures of electron energization seen in Figs. 4(e) and 4(g), we present a simple model with either (i) a shifted Maxwellian distribution for a bulk parallel electron flow U k;e , given by

or (ii) an electron density perturbation dn e

or (iii) a linear combination of the deviations from a Maxwellian distribution for both a shifted Maxwellian with flow U k;e and an electron density perturbation dn e . Here, our approximation of the complementary perturbed distribution can be computed by

We emphasize that the precise quantitative form in velocity space of the net parallel flow and density perturbation is not critical: 55 what is important is the general concept that the sum of a parallel flow with a density perturbation qualitatively leads to asymmetric signatures as seen in Fig. 4(g). For each simple model, we plot a column in Fig. 5 with the equilibrium parallel electron velocity distribution (black) and total parallel electron velocity distribution (red) in the top row, the perturbed velocity distribution in the middle row, and the form of the velocity dependence þev 2 k =2ð@g e =@v k Þ of the field-particle correlation C E k ðv k Þ in the bottom row.

Guided by the appropriate moments of the electron velocity distribution, in the left column (a), we plot the case for a shifted Maxwellian with bulk parallel electron flow U k;e =v te ¼ À0:25, showing that it results in an electron energization signature, that is, asymmetric in v k , with a larger signature of energization in the same direction as the bulk parallel flow. In the middle column (b), we plot the case for a density perturbation with dn e =n 0e ¼ þ0:35, showing

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scitation.org/journal/php this density perturbation alone leads to an energization signature, that is, odd in v k , meaning there is zero net energization of the electrons by the parallel electric field due to a density perturbation when integrated over velocity. In the right column (c), we plot the case with a superposition of the shifted Maxwellian with U k;e =v te ¼ À0:25 and the density perturbation with dn e =n 0e ¼ þ0:35. The combination of the flow and density perturbations leads to an energization signature as a function of v k that is, qualitatively similar to that seen at point r C in Fig. 4(g).

Although the detailed perturbed electron velocity distributions arising through the evolution of the simulation show modest quantitative differences from the forms used in this simple model, this example demonstrates that the combination of a parallel flow and a density perturbation can indeed lead to the asymmetric signatures of electron energization in the exhaust region of collisionless magnetic reconnection in the strong-guide-field limit seen in Fig. 4.

In summary, the electron energization in the exhaust region of collisionless magnetic reconnection in the strong-guide-field limit is caused by a bulk acceleration of the electrons by the parallel component of the electric field E k . Although, along the separatrices away from the midplane in the exhaust, we find an asymmetric signature of electron energization, since a density perturbation leads to zero net electron energization when integrated over v k (as seen in the lower middle panel of Fig. 5), the net electron energization is simply due to this bulk acceleration of the out-of-plane electron flow by the reconnection electric field. This asymmetric signature about v k ¼ 0 is indicative of the spatial location in the exhaust where the diagnostic is sampling velocity space distributions. On each side of the midplane, we see equal magnitudes of electron energization (increase in phase-space energy density). The asymmetry in the signature of electron energization motivates the possibility to identify observationally, the physics of electron energization by collisionless magnetic reconnection, using only single-point measurements of the electromagnetic fields and electron velocity distributions. This simulation demonstrates a characteristic velocity-space signature for electron acceleration through bulk parallel acceleration, by the reconnection electric field in the exhaust of collisionless magnetic reconnection in the strong-guidefield limit. Further evidence that this energization is not a resonant acceleration of electrons, is provided by investigating the variation of the electron energization signatures with the plasma beta parameter.

E. Variation of energization with plasma b i

The general qualitative picture of the energization rate for the remaining simulations is consistent as b i increases from 0:03 b i 1:0. The energization of electrons for all five simulations is largely dominated in the electron diffusion region (EDR) around the x-point and into the exhaust along the separatrices as in Fig. 1(c). The overall reconnection geometry persists as b i varies from 0.01 to 1, with similar energization signatures. However, there are some differences in the dynamic evolution of the reconnecting field. In the b i ¼ 1 case, there is a clear development of a secondary island at the original xpoint. This secondary island is formed as a consequence of the plasmoid instability. 56 As noted by Numata and Loureiro, 31 this secondary island eventually moves in the Ày direction due to numerical noise, and secondary reconnection commences, which allows renewed particle energization and plasma heating late in the simulation at a lower magnitude.

As b i increases, we see a thinning of the current sheet supporting the reconnection process, as shown in Fig. 6(b) for b i ¼ 1. In addition to these qualitative changes to the reconnection geometry and associated current sheets, the magnitude of both the current sheet and selfconsistent E k decrease in magnitude with increased b i .

In Fig. 7, we show the normalized reconnection rate cE k ðr X Þ=V A B max y for all five b i cases. The time evolution of the out-ofplane electric field at the x-point ðx=q i ; y=q i Þ ¼ ðL x =2; L y =2Þ is used as the measure of the reconnection rate. The peak reconnection rate decreases in magnitude as b i increase and occurs later in time as the tearing instability develops more slowly at higher b i. 31,57 Physically, the plasma thermal pressure resists the onset of the reconnection flow driven by the tearing instability, so as plasma beta increases, the

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scitation.org/journal/php growth rate decreases. In the higher b i runs, there is a steep drop in the electric field just after the reconnection rate peaks, and the field eventually reverses sign at the x-point. This is a consequence of the formation of the plasmoid instability, 56,58 leading to the conversion from an x-point to an o-point at the center of the reconnecting geometry.

At the x-point, the energization signatures stay qualitatively consistent in shape as b i increases, with the energization abruptly ceasing when the plasmoid instability causes the x-point to convert to an opoint in the b i ! 0:1 simulations. The magnitude of energization for electrons decreases with increasing b i , consistent with the lower magnitudes of j k;e and E k with increasing b i , and also consistent with the decreasing conversion of energy as shown in Fig. 11.

In the exhaust, there is markedly more variation in the electron energization signatures as b i increases. We show the reduced correlation timestack plots in the exhaust for point r C in Fig. 8 We suggest that this development of a loss of phase space energy density in the exhaust at r C for v k > 0, with increasing b i , is due to an incomplete cancelation of the contributions to the rate of electron energization from the parallel electron flow and the electron density perturbation. For the b i ¼ 0:01 simulation, we show in Fig. 5 then the sum of these two contributions will not cancel out, but rather will lead to a net loss of phase-space energy density at v k > 0. This idea an incomplete cancelation of the perturbations due to the parallel flow and density perturbation at v k > 0 leads to an increasing loss of electron energy appears to be consistent with the results shown for point r C in Fig. 8, where the rate of energization becomes increasingly negative at v k > 0 as b i increases.

It is important to emphasize that all of the asymmetric velocity-space energization signatures at point r C shown in Fig. 8 are due to the bulk acceleration of the electrons in the out-ofplane direction by the reconnection electric field E k . Although the lower jv k j boundary of the positive electron energization appears to decrease along with v A =v te (vertical dashed lines) as b i increases, this does not necessarily indicate a resonant process. The shift in the positive electron energization to lower jv k j with increasing b i is governed by a narrowing of the complementary perturbed distribution function g e ðv k Þ to smaller values of jv k j with increasing b i . In Fig. 9, we plot the reduced parallel complementary perturbed distribution function g e ðv k Þ at the peak of the electron energization at point r C for each b i simulation. This plot shows clearly that the perturbations are increasingly confined to a more narrow region around v k ¼ 0 as b i increases.

A resonant energization process, such as electron Landau damping, typically generates a velocity-space signature of energization, that is, more localized in v k around the resonant parallel phase velocity v A =v te , as seen in previous studies. 32,36,37 The velocity-space signatures shown in Fig. 8 are significantly more broad in v k than expected for a resonant energization mechanism and appear to indicate a bulk acceleration of the electrons, as our modeling demonstrates in Figs. 3 and5.

V. DISCUSSION

To understand the kinetic physics governing the energization of electrons in collisionless magnetic reconnection in the strong-guidefield limit, it is critical to recognize that the conversion of the initial magnetic energy into electron heat occurs through a two-step process: 33,36 (i) first, collisionless interactions transfer energy from electromagnetic fluctuations to microscopic kinetic energy of the electrons, a reversible process; and (ii) subsequently, the energy transferred to the electrons, which exists as free energy in the non-thermal fluctuations of the electron velocity distribution function (VDF), undergoes a linear 8,34 or nonlinear 35 phase mixing process to sufficiently small scales in velocity space that arbitrary weak collisions can thermalize those fluctuations, irreversibly converting the energy to electron heat. Using nonlinear gyrokinetic simulations of collisionless magnetic reconnection, we focus in this investigation on the first step of this process, and we show that work done on the electrons by the (out-of-plane) reconnection electric field dominates the electron energization through

The electron current j k;e , necessary to support the change in the in-plane magnetic field across the midplane of the simulation, peaks through the x-point and along the separatrices in the reconnection magnetic field geometry. The (out-of-plane) reconnection electric field, the E k component in the strong-guide-field limit, is fairly uniform throughout the region approximately spanning the range 17 < x=q i < 23 and 7 < y=q i < 25. When these fields are combined to determine the work done by E k on the electrons through j k;e E k , we find that the electron energization during the main phase of magnetic reconnection (from 10 Շ t=s A Շ 30 in the b i ¼ 0:01 simulation) occurs dominantly at the x-point and along the separatrices within the exhaust, as shown clearly in Fig. 1. Thus, we focus specifically on exploring the energization of the electrons by E k at the x-point and in the exhaust.

We use the field-particle correlation technique to determine the characteristic velocity-space signature C E k;e ðv ? ; v k Þ of the electron energization at the x-point and at three positions across the midplane in the exhaust. At the x-point, the velocity-space signature is well modeled by energization of the bulk out-of-plane electron flow U k;e (which provides the current required by Maxwell's equations to support the change in the in-plane magnetic field B y across the mid-plane) by the parallel electric field E k which drives the reconnection flow in the (x, y) plane. In the exhaust, the symmetric (about the midplane) spatial pattern of positive electron energization j k;e E k > 0 arises from a more complicated kinetic picture of the energization. The combination of the bulk out-of-plane electron flow U k;e with the well-known quadrupolar electron density variation dn e in guide-field magnetic reconnection 8,20,52,54,59,60 leads to a velocity-space signature, that is unexpectedly asymmetric across the midplane: in regions of a negative density perturbation (point A in Fig. 4), electrons with v k > 0 experience a net gain in energy; in regions of positive density perturbation (point C in Fig. 4), electrons with v k < 0 experience a net gain in energy. Note that, since a density perturbation leads to zero net energization when integrated over v k [see the lower panel of Fig. 5(b)], the net electron energization at all positions through the exhaust is simply due to the bulk acceleration of the out-of-plane electron flow by E k through the j k;e E k > 0 work.

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The velocity-space signatures of electron energization at the x-point, shown in Fig. 2(b), and within the exhaust, shown in Figs.

4(e)-4(g), are key results of this study. In particular, the asymmetric electron velocity-space signature within the exhaust region is potentially a valuable new way of identifying that one is probing along a trajectory through the exhaust of collisionless magnetic reconnection using only single-point measurements. This technique can be applied to either spacecraft observations from missions such as the magnetospheric multiscale (MMS) mission 17 or laboratory measurements, 61 and the possibility to probe the physics of particle energization in magnetic reconnection using single-point measurements is a key implication of this work.

By using the complementary perturbed distribution function (4) in AstroGK to compute the velocity-space signatures presented here, we are implicitly assuming a Maxwellian equilibrium distribution function. In a realistic space plasma, the equilibrium velocity distributions are not necessarily Maxwellian, but may take on more complex forms. In this case, the implementation of the field-particle correlation technique requires determining an "equilibrium distribution" through a temporal and/or spatial average of measurements, similar to what has been implemented in the exploration of observed space plasma turbulence using the field-particle correlation technique. 38,42 Alternatively, one can take the numerical results here for the perturbed velocity distributions and add the equilibrium distribution (which requires specifying a particular value of the gyrokinetic expansion parameter $ B eq y =B z0 ( 1) to predict the total velocity distribution. For example, in Fig. 10, we plot the predicted total parallel electron velocity distribution f e ðv k Þ in the exhaust region at (a) point A, (b) point B, and (c) point C using a gyrokinetic expansion parameter ¼ 0:15. This shows that the simulations predict a measurable decrease below the equilibrium in f e ðv k Þ at v k =v te ' þ1 at point A on the low density arm, and a measurable increase above the equilibrium in f e ðv k Þ at v k =v te ' À1 at point C on the high density arm. Evidence for such a peak (on the high density arm) in the predicted total electron velocity distribution has been recently measured in laboratory experiments of strong-guide-field magnetic reconnection in the phase space mapping (PHASMA) device at West Virginia University. 62 Future comparisons between PHASMA experiments and gyrokinetic simulations of collisionless magnetic reconnection in the strong-guide-field limit are a promising direction for improving our understanding of the resulting electron energization.

A. Resonant vs non-resonant energization

A major conclusion of our modeling of the velocity-space signatures is that the electron energization is due to bulk acceleration of the electron flow by the parallel electric field, rather than some resonant acceleration mechanism, in agreement with previous investigations of electron energization by E k in the strong-guide-field limit of collisionless magnetic reconnection. 6,[13][14][15] This finding differs from the interpretation of the electron energization in strong-guide-field magnetic reconnection by Numata and Loureiro 31 (hereafter NL15), where it was suggested that the location in v k of the fluctuations in the electron velocity distribution function implied a Landau resonant mechanism of energization. Below, we discuss these contrasting interpretations in more detail.

First, it is crucial to emphasize that while our study directly analyzes the work done on the electrons by the electric field-the first step in the two-step process of particle energization in weakly collisional plasmas 33,36,63 -the NL15 analysis focuses on the second step of the process, the collisional thermalization of energy in the electron velocity distribution. Note that the energy of the electrons changes in the first step when the electric field does reversible work on the electrons collisionlessly, whereas the second step is the irreversible conversion (through collisions) of the energy gained in the first step, from non-thermal free energy in the electron velocity distribution to thermal energy of the electrons. These two processes occur at different times and different spatial locations during the process of magnetic reconnection. Phase mixing is the bridge between these two steps, taking the energy transferred to the electrons in the first step, which is represented by fluctuations in the electron velocity distribution, and transporting these fluctuations to sufficiently small scales in velocity-space that arbitrary weak collisions can smooth out those fluctuations, irreversibly converting the electron energy into heat of the plasma species. To be specific, below we use the term "energization" to refer to the collisionless work done on electrons that changes their energy, and "heating" to refer to the collisional thermalization of that energy.

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scitation.org/journal/php Nonetheless, they argue that the observed range of parallel velocities NL15 report that little electron heating occurs during the main phase of reconnection at the x-point and in the reconnection exhaust. This is consistent with the weakly collisional conditions of the plasma, whereby Ohmic heating, via resistivity acting on the out-of-plane current, is small compared to the subsequent collisional thermalization of phase-mixed fluctuations in the velocity distribution that contain the energy previously transferred to the electrons by the parallel electric field. Although the NL15 analysis directly examines the electron velocity distributions at the later times and downstream positions where the collisional thermalization peaks, they use these observations to deduce an earlier stage of Landau resonant energization. NL15 suggest that a resonant transfer of energy to the electrons occurs due to the projection of the electron motion (along the total magnetic field, which is dominantly out-of-plane) in the (x, y) plane of the simulation, with a resonant condition on the parallel motion given by v k =v te $ ðv A =v te ÞðB z0 =B ? Þ $ 1 for b i ¼ 0:01 and mass ratio m i =m e ¼ 100. The electron heating is found to peak in the island after the reconnection phase has ended, and they suggest that the localization in v k of the linearly phase-mixed fluctuations at v k =v te $ 1 supports their interpretation of a resonant electron energization. For b i ¼ 1, the phase-mixed fluctuations are confined to within v k =v te < 1, qualitatively consistent with the resonant parallel phase velocity decreasing relative to v te as b i increases, which they argue is further evidence of a Landau resonant interaction with the electrons.

Several lines of argument support our interpretation that the electron energization instead is non-resonant in nature, and is simply a bulk acceleration of the electrons by E k .

First, Landau resonant energization implies that particles within a particular range of parallel velocities stay in phase with changes in the parallel component of the electric field. By staying in phase with the accelerating electric field, those resonant particles can experience a large gain in energy. Non-resonant particles, with velocities outside of that particular range, quickly fall out of phase with the accelerating electric field, and so those particles experience little net energy gain. Thus, a resonant energization mechanism leads to a significant particle energy gain, that is, localized to a limited region of velocity space. This process generally implies an accelerating electric field, that is, propagating with a phase velocity in the parallel direction, so that it can remain in phase with particles moving at nearly the same parallel velocity. A common example is the collisionless damping of plasma waves, such as kinetic Alfv en waves 36,37 or Langmuir waves. 32,34 In these reconnection simulations, on the other hand, the parallel electric field remains relatively constant in time during the main phase of reconnection, and is relatively uniform in space over the region spanning 17 < x=q i < 23 and 7 < y=q i < 25, as shown in Fig. 1(b). Thus, it is not clear that an interpretation of the particle energization as resonant applies in this case.

A second argument is an alternative explanation for the parallel velocity range of the phase-mixed fluctuations in the electron velocity distributions that are presented in NL15. The energization of electrons in the exhaust peaks on the magnetic field lines just inside the separatrix, which are swept downstream and ultimately constitute the closed field lines of the magnetic islands where NL15 find that the electron heating peaks. Since the energization is spatially non-uniform along these closed field lines, occurring primarily in the near exhaust region spanning 17 < x=q i < 23 and 7 < y=q i < 25, the fluctuations in the electron velocity distribution will subsequently phase mix linearly due to the advective term in the Vlasov equation. The resulting phasemixed fluctuations will have the largest amplitudes in the range of parallel velocities where the perturbed electron distribution g e ðv k Þ is the largest. In Fig. 9, we plot g e ðv k Þ at the peak of the electron energization at point r C for each b i simulation, showing that the perturbed distribution is more narrowly confined to an increasingly small range of jv k j about v k ¼ 0 as b i increases. This smaller range in v k is consistent with the lower normalized value of U k;e =v te needed to generate the current required by Maxwell's equations to support the in-plane magnetic field change across the mid-plane as b i is increased. For example, the parallel current required by the initial Porcelli equilibrium scales as

where a ¼ 4q i is the initial current sheet width. Thus, the more narrow localization of the phase-mixed fluctuations in v k with increased b i -cited by NL15 as evidence for a resonant energization mechanism-may simply be a consequence of the variation with b i of the perturbed velocity distributions that feed the linear phase mixing process.

A final argument is the fact that the velocity-space signatures of electron energization produced by applying the field-particle correlation technique, presented in Figs. 2 and4, are well modeled by a simple non-resonant bulk acceleration of the electrons by the reconnection electric field E k , as shown in Figs. 3 and5. Even for a bulk acceleration of all electrons, it is the specific mathematical form of the electron energization by E k in ( 5) that leads to a localization of the particle energization

The electron energization as a function of v k is proportional to the derivative @g e =@v k and is weighted by v 2 k , and these factors lead to the natural confinement of the energization over the observed range in v k .

A very simple way to explain this localization in velocity space is to consider the work done by E k on a charged particle with charge q s . The rate of work done on a single particle is q s E k v k , and this must be multiplied by the distribution of particles

te Þ lead to a localization of energization similar to what we observe in our FPC analysis. 64 A future extension of this work is to explore the dynamics of the phase-mixing process that transports the fluctuations in the electron velocity distribution to small velocity scales in the specific context of the reconnection exhaust and downstream island regions. Such a study would connect our analysis of the collisionless energization of electrons in magnetic reconnection to the collisional dissipation leading to electron heating studied by NL15, and should definitively answer the question of whether the electron energization is resonant or nonresonant through both stages of particle energization.

E nt s

where du s is the bulk flow velocity (first moment) of species s and collisionally thermalized particle energy,

where h s is the non-Boltzmann portion of the perturbed electron distribution function and F 0e is the equilibrium electron distribution function. 33,45,46 In the gyrokinetic limit, the electric field energy is negligible compared to the magnetic energy. 46 The following primary partition of energies are magnetic perpendicular energy E B? (green), parallel electron kinetic energy E u k ;e (cyan), perpendicular ion kinetic energy E u?;i (maroon), non-thermal electron energy E ðntÞ e

(blue), non-thermal ion energy E ðntÞ i (red), collisional ion energy E coll;i (light red), and collisional electron energy E coll;e (light blue). Other components are parallel magnetic field energy E B k (dark green), perpendicular electron kinetic energy E u?;e (light purple), parallel ion kinetic energy, and E u k ;i (medium purple). We show the fraction of the energy content at the end of each simulation (values տ 0:01) for both species and the magnetic field for each simulation in Table I. In Table I, RE i is the sum of the non-thermal and collisional ion energies, and RE e is the sum of the non-thermal and collisional electron energies.

The initial configuration energy consists of perpendicular magnetic energy and parallel electron flow due to the initial conditions of the Porcelli equilibrium. As demonstrated by (16), the parallel electron flow providing the current required by Maxwell's equations to support the initial magnetic configuration decreases with increasing b i , so the share of the initial energy in E u k ;e decreases as b i increases. The reconnection dynamics then releases some fraction of this initial magnetic energy, leading rapidly to non-thermal energization of the electrons and ions, and some perpendicular bulk acceleration of the ions. Once reconnection begins, the magnetic field energy is quickly transferred to the particles (almost exponential growth of energization), consistent with the fast ramp-up and decline of phase-space energy density rate shown by the field-particle correlation analysis, e.g., the left panel of Fig. 2(c) and left panels of Figs. 4(d), 4(f), and 4(g). The parallel bulk kinetic energy of the electrons stays fairly constant in time, even after the primary reconnection phase in each of the simulations.

It is not until well after the primary reconnection phase commences that thermalization processes begin and the collisionally thermalized energy of the particles (light blue and light red, above the solid black line) begins to increase. At this time, the reconnection has essentially ceased, except for b e ! 0:1, where the formation of a plasmoid at the x-point allows for secondary reconnection. However, the only partition affected during the secondary reconnection is the perpendicular ion bulk energy, which decreases as the perpendicular magnetic energy increases. Once all reconnection has ceased, there is little energization due to the fields. At late times, thermalization is ongoing, as evidenced by the black line, which indicates the total amount of energy in the simulations that has not been collisionally thermalized, retaining a non-zero slope in each plot of Fig. 11. The thermalization of ion energy is significantly slower than for electrons due to two factors. First, the linear phase mixing that drives non-thermal energy in the particle velocity distributions functions is proportional to the species thermal velocity, and is, therefore, a factor of ðm e =m i Þ 1=2 slower for the ions than for the electrons. Second, like-species collisions that dominate the thermalization of each species scale as ii = ee / ðm e =m i Þ 1=2 . Thus, at the end of each simulation, the ions have collisionally thermalized a (red), E coll;i (light red), and E coll;e (light blue). The other components are E B k (dark green), E u?;e (light purple), and E u k ;i (medium purple). The total perturbed energy in the plasma that has not been collisionally thermalized is dW (thick solid black line).