Relating the Phases of Magnetic Reconnection Growth to Energy Transport Mechanisms in the Exhaust

The efficiency of energy conversion during magnetic reconnection is related to the reconnection rate. While the stable reconnection rate has been studied extensively, its growth between the time of reconnection onset and its peak has not been thoroughly discussed. We use a 2D particle‐in‐cell simulation to examine how the reconnection rate evolves during the growth process and how it relates to changes near the x‐line. We identify three phases of growth: (a) slow quasi‐linear growth, (b) rapid exponential growth, and (c) tapered growth followed by negative growth after the reconnection rate peaks. We associate phase 1 with the breaking of x‐line uniformity by a localized density depletion that changes the in‐plane electric field structure near the neutral line, followed by the expansion of the inflow region and the enhancement of inflow Poynting flux Sz associated with the out‐of‐plane electric field Ey in phase 2. We show how the Hall fields facilitate rapid growth in phase 2 by opening up the exhaust and relieving the electron‐scale bottleneck to allow rapid energy transport across the separatrices. We find that in phase 3, the inflow of electromagnetic energy accumulates until the downstream electromagnetic energy density saturates toward the initial upstream asymptotic value. Finally, we examine how the electron outflow and the downstream ion populations interact in phase 3 and how each species exchanges energy with the local field structures in the exhaust.


Introduction
The process of magnetic reconnection consists of a topological re-configuration of magnetic fields and a transfer of electromagnetic energy to plasma energy in the form of particle acceleration and heating (Sonnerup, 1979;Vasyliunas, 1975;Yamada, 2007).The initiation of this process is accompanied by a localized violation of the frozen-in flux condition and a de-coupling of electron motion from the convection of the local magnetic fields.
The non-ideal electric field that violates the frozen-in flux condition is given by E , and the energy conversion rate associated with this non-ideal electric field is given by J → ⋅ E → ′ (Zenitani et al., 2011).Signatures of large positive J → ⋅ E → ′ are often used to identify the electron diffusion region (EDR) in spacecraft observations of magnetic reconnection (Burch & Phan, 2016;Torbert et al., 2018).While the electron-frame electric field is necessary to initiate reconnection, it does not account for all of the work done to the plasma in the reconnection process, especially outside the central EDR (Yamada et al., 2014(Yamada et al., , 2015)).
The efficiency with which magnetic flux is reconnected and plasma is energized is described by the reconnection rate.Spacecraft observations and simulations show that the reconnection rate consistently stabilizes to a normalized value of approximately 0.1 v a B 0 /c, where v a is the upstream Alfvén velocity, B 0 is the upstream magnetic field, and c is the speed of light.A normalized reconnection rate of 0.1 is much faster than allowed by the Sweet-Parker reconnection model (Cassak et al., 2017;Parker, 1973).Recent progress has been made on the theoretical basis for this result based on pressure considerations at the reconnection site and the influence of Hall fields on the stabilized geometry of the exhaust (Liu et al., 2022).Measurements of the terms in Poynting's theorem in particle-in-cell (PIC) simulations and Magnetospheric Multiscale (MMS) data show that the EDR is typically in a steady-state energy balance, where the rate of electromagnetic flux convergence is balanced by the work rate such that the time evolution of the electromagnetic energy density is negligible (Genestreti et al., 2018;Payne et al., 2020a).This steady-state balance in Poynting's theorem does not necessarily correspond with the stabilization of the reconnection rate, as it has been shown in PIC simulations that the steady-state balance occurs well before the maximum reconnection rate is reached, soon after the development of the localized out-of-plane E′ y (Payne et al., 2020a).
In the present study, we do not focus on the steady-state reconnection rate, but instead on the growth of the reconnection rate during the early stages of reconnection to better understand the physical processes that accompany the transition from a quiet current sheet to stabilized fast reconnection.Some progress has been made relating the structural evolution of the x-line and of phase space to the temporal evolution of the reconnection rate, including the result that highly structured velocity distribution functions (VDFs) and temperature anisotropies develop in reconnection exhausts close to the occurrence of the maximum reconnection rate (Shuster et al., 2014(Shuster et al., , 2015)).Recent studies using MMS data (Hubbert et al., 2021(Hubbert et al., , 2022) ) and simulations (Lu et al., 2022) have argued that electron-only reconnection may be understood as a transitional phase between quiet current sheets and traditional electron-ion coupled reconnection in the magnetotail.While these studies did not focus specifically on the growth of the reconnection rate, they identified multiple parameters that appeared to characterize the transition of a time-evolving electron-only current sheet including an increase in perpendicular ion temperature, parallel electron temperature, Hall field strength, and the ratio of ion to electron temperature, which grows late in the reconnection process due to the delayed heating of ions compared to electrons.
The motivation behind the following content is to better understand how the reconnection rate evolves after onset and the physical processes that control its growth.Section 2 discusses the parameters of the simulation used in this study and the units associated with the variables used in the following sections.In Section 3, we discuss the growth of the reconnection electric field and characterize three separate phases of the growth interval.Section 4 examines how the x-line structure changes during the first two phases of reconnection growth.In Section 5, we investigate the role of Hall fields in the rapid transport of magnetic energy and in the eventual stabilization of the reconnection rate in the last phase of reconnection growth.Section 7 examines the coupling between electrons and ions in the reconnection exhaust during the last phase of growth.In Sections 8 and 9, we discuss our results in the context of recent literature and summarize our primary conclusions.

Simulation Parameters
This study includes a 2-D PIC simulation with an initial Harris current sheet configuration using the plasma simulation code (Germaschewski et al., 2016).The domain size in ion inertial lengths is L x × L z = 80d i × 20d i , with 64 grid points per d i and 300 particles per cell (ppc) at the center of the current sheet.The boundary conditions are periodic at the x boundaries while the z boundaries are conducting walls for the fields and reflect particles.The initial structure of the anti-parallel magnetic fields within the domain is given by B x = B 0 tanh(z), where B 0 = 0.5 is the initial asymptotic magnetic field.The initial plasma density is given by n = n 0 sech 2 (z) + n b , where n 0 is the density at the center of the current sheet, n b is the uniform background density, and the ratio of these densities is n b n 0 = 0.05.The initial thickness of the current sheet is L = 0.5d i .There is an ion to electron mass ratio of m i m e = 100 and an ion to electron temperature ratio T i T e = 5.The simulation starts with a small perturbation δB z at the center of the domain, which helps initiate and localize the x-line.The structure of the perturbation is similar to that of the Geospace environmental modeling Reconnection Challenge (Birn et al., 2001), except for the magnitude which is δB z B 0 = 0.03 in this study.The speed of light is c = 1, and the Alfvén velocity v a is defined based on the background density n b and is given by v a c = 0.223607.It should be noted that in most real physical systems of interest, the Alfvén velocity is not as large as it is in this study, meaning that the timescale of Magnetohydrodynamic (MHD) wave phenomena τ A is typically much larger in physical systems.
Here we note the units corresponding to quantities used in the following sections.All times presented in this paper are in units of the ion gyro-period Ω 1 ci , and every reference to a "time step" in the following sections refers to an interval of 1 Ω 1 ci .Lengths within the domain are expressed in d e .Electric field contributions are in units of v a B 0 c and magnetic fields are expressed in units of B 0 .Poynting flux is expressed as a product of E and B, without an additional factor.Ion temperatures are expressed in units of T 0 , where T 0 is the initial ion temperature in the current sheet.Finally, J

Growth Phases of the Reconnection Rate
is triggered by a perturbation in B z .This small B z reversal helps to localize the onset region and prevent the formation of many x-lines along the neutral line.
In Figure 1 we present various quantities measured at the center of the B z reversal as the system evolves from the beginning of the simulation to a few time steps beyond the occurrence of the maximum reconnection rate.These include the reconnection electric field, which we define here as the out-of-plane contribution to the non-ideal electric field E′ y , and its time derivative.We also include quantities relevant to energy transport: the work rate .All quantities are measured at three coordinates along the neutral line spaced 1d e apart, with the central coordinate chosen as the center of the B z reversal, and then averaged to obtain the values shown in the time series plots in Figure 1.By taking an average of the three measurements across the B z reversal, we limit the influence of small spatial fluctuations and better isolate the temporal evolution of the measured quantities in the EDR.Since ∇ ⋅ S → requires some volume in which it can be calculated, each individual measurement of the energy transport terms are determined for a small n x × n z = 2 × 2 grid cell box (or L x × L z = 0.31 × 0.31d e ) centered at the coordinate the measurement was taken.In the EDR, ∇ ⋅ S → is negative due to the converging magnetic energy flux of the inflow regions, but we present its magnitude here in order to more easily compare to in this region can therefore be thought of as a measure of the rate at which Poynting flux converges toward the x-point.
The onset of reconnection and the initial growth of E y ′ begins around t = 10-11 and reaches its maximum value near t = 18 (Figure 1).The evolution of dE y ′ dt from onset suggests that the initial growth up to t ≈ 13 is distinct from the growth during t ≈ 13-16, which is distinct from the few time steps just before and after the maximum reconnection rate, t ≈ 16-20.Going forward, we will refer to these intervals as phases 1, 2, and 3, respectively.It should be noted that these labels are meant to be a useful way to think about different stages of the reconnection growth process, and do not apply strictly within only certain intervals (for example, t = 13 does not necessarily belong to only phase 1 or only phase 2, but t = 12-14 can be considered a transition between phases 1 and 2).
For much of phase 1, dE y ′ dt is not quite constant, but is slowly decreasing with time, indicating that the reconnection rate is growing at a slightly sublinear rate.The growing E y and the imposed perturbation in B z initiate the outflow pattern as contributions from E y B z on either side of the B z reversal produce diverging S x components along the neutral line.Likewise, the E y B x contributions from the reconnecting fields produce the converging S z components toward the neutral line.The emergence of a consistent E y at the center of the current sheet also produces the initial enhancement of J To understand the physics that govern reconnection growth, it is important to relate the structural evolution of the x-line to the phases observed in the growth of the reconnection rate.In the following sections, we examine the structural evolution of parameters near the EDR during the phases of reconnection growth.

Electron Density Depletion in Phase 1
Figure 2 shows how the electron density evolves with the reconnection electric field E′ y and the outflow component of the electric field E x during phase 1 of reconnection growth.In phase 1, E′ y > 0 begins to emerge and a localized depletion in the electron density begins to develop as electrons start to pileup on either side of the depleted region.As the electron distribution along the current sheet becomes non-uniform, E x electric fields develop pointing away from the depleted region toward the electron pileup region due to the small charge imbalance that arises from the local electron depletion.The slower growth rate of phase 1 suggests that this process is not an efficient means of energy conversion and transport, but it is an important first step that allows the subsequent phases to follow.The structural changes to the electric fields will play an important role in the evolution of energy transport in phase 2.

Hall-Facilitated Energy Transport in Phase 2
In Figure 3, we show the separate contributions to S x and S z at t = 16 along a horizontal cut at z = 4 d e .The cut extends across the inflow region and terminates just beyond the separatrices to capture both outflows.For both S x and S z , the E y contribution plays a significant role, with E y B z contributing to S x away from the EDR and E y B x contributing to S z toward the EDR as discussed previously.These terms do not include Hall B y , but the Hall magnetic field contributions and in-plane electric fields play a non-trivial role in the production of both S x and S z .
In the case of S x , the contribution by E z B y is significant across the separatrices and has the same sign as the E y B z component, thus enhancing the magnitude of S x .In the case of S z , the contributions by E x B y are also significant across the separatrices, but have the opposite sign to the larger E y B x contribution.The E x structure in the exhaust therefore aids the redirection of electromagnetic energy transport by suppressing the normal component while the

Magnetic Saturation in Phase 3
As the Hall field structures develop, they facilitate more electromagnetic energy transport by relieving the bottleneck imposed by the small scale of the EDR. Figure 4 shows how this process relates to the overall transport of magnetic energy during the transition between phases 2 and 3 up to t = 18, when the reconnection rate peaks.
The color in Figure 4 represents the magnetic energy density B 2 8π and the arrows represent the direction and magnitude of the in-plane Poynting flux.The arrows in all three frames are scaled to the same value, so their lengths are directly analogous to their magnitudes as they change between frames.The Poynting flux across the separatrices continues to grow in magnitude, rapidly transporting upstream magnetic energy density into the exhaust, where it accumulates downstream of the EDR.The upstream magnetic energy depletes and the downstream magnetic energy accumulates as the magnetic energy density in the exhaust approaches the far upstream value.This transition to the exhaust becoming a local maximum of magnetic energy density appears to correspond with the transition to phase 3 and the time of the maximum reconnection rate.

Electron-Ion Coupling in the Exhaust
In addition to the downstream accumulation of magnetic energy, we are also interested in how the downstream ion heating evolves along with the exhaust electron velocity.The panels in the top row of Figure 5 show another sequence of frames from phase 3, this time showing the ion temperature (defined as one third of the trace of the ion temperature tensor) overlaid with arrows indicating the in-plane electron velocity.Just as in Figure 4, the arrows in Figure 5 are all scaled to the same value so their lengths are directly analogous to the magnitude of the in-plane electron velocity.Both the downstream ion temperature and the electron velocity in the outflow grow in time.On the approach to the time of maximum reconnection rate (t = 18), the region of strong ion heating starts to separate from the immediate edge of the electron jet with a region of relatively weak electron velocity in between.
To better understand energy transport and exchange in the Hall field region, we also show different contributions to the total J → ⋅ E → in the exhaust near the phase 2-3 transition at t = 16 and at t = 18, when the reconnection rate peaks (cutouts in Figure 5).The contributions are broken down by individual species and by components parallel or perpendicular to the local magnetic field.The electrons within the exhaust tend to lose energy parallel to the local magnetic field J e‖ E ‖ < 0 and gain energy perpendicular to the local magnetic field J e ⊥ E ⊥ > 0 with the exception of the narrow "outer" EDR region, where the breaking of super-Alfvénic electron jets can lead to a localized loss of electron energy near the neutral line (J e ⊥ E ⊥ < 0) .In contrast, the ions mostly gain energy in the exhaust, both parallel and perpendicular to the local fields.By t = 18, the parallel energization J ‖ E ‖ of both species reduces overall near the center of the exhaust, and there is also a co-located narrow localized reversal in the sign of J e‖ E ‖ and J i‖ E ‖ bordering the separatrices above and below the neutral line, where J e‖ E ‖ > 0 and J i‖ E ‖ < 0. Figure 6 shows the evolution of electron and ion temperature anisotropies and velocities during phase 3. The quantities are presented such that negative values indicate a perpendicular-dominant anisotropy (T ‖ < T ⊥ ) and positive values indicate a parallel-dominant anisotropy (T ‖ > T ⊥ ).During phase 3, the electron temperature anisotropy becomes more perpendicular-dominant in the exhaust, in contrast to the inflow regions where T e‖ ≫ T e⊥ .The background ion population is strongly perpendicular-dominant from the start, but that anisotropy reduces in the exhaust during phase 3, especially near the separatrices where T i‖ /T i⊥ approaches unity.

Discussion
Between the onset of reconnection at the electron scale and the development of mature Petschek-type x-line structures at the ion scale, there exists a sequence of distinct physical processes that influence the growth of the reconnection rate, and thus the efficiency of magnetic reconnection.While many of the x-line structures shown here have been observed in earlier studies, we argue that it is important to understand how and why the evolution of those structures is related to the reconnection rate and thus, the efficiency of the reconnection process.
In phase 1, the reconnection electric field in the central EDR begins to grow at a slightly sublinear rate and the plasma starts to gain energy from the fields (Figure 1).The energy going into the plasma particles in the central EDR is evenly distributed between the electrons and ions (J ) up until the beginning of phase 2. In phase 2, the reconnection rate grows exponentially and plasma energization is dominated by the electron contribution (J ) .This electron dominance only emerges after the initial growth phase, suggesting that both particle species play an important role in the initial annihilation of field energy that starts the reconnection process.In phase 3, the growth of the reconnection rate tapers and reaches its peak by t = 18.
The structural changes near the neutral line also shed light on the processes that influence the evolution of the reconnection rate in each phase.The imposed B z reversal provides the structure necessary to initiate reconnection onset.It helps localize the EDR and provides the diverging components of the Lorentz force (v y B z terms) that create the localized depletion of electron density (Figure 2), and provides the diverging components of Poynting flux via the E y B z terms.As phase 1 continues, the localized depletion of electron density at the neutral line corresponds to local changes in the surrounding electric field structure, leading to the buildup of a diverging E x pattern in the developing outflow regions (Figure 2).These results are consistent with earlier studies.For example, it has been shown that in a 2.5D Hall MHD simulation (with parameters based on observations by the WIND spacecraft (Øieroset et al., 2001)), a density depletion region spatially coincides with the reversal of both the ion outflow jets and a reversal in the B y component associated with Hall fields (Yang et al., 2006).It has recently been argued that pressure depletion at the current sheet (equivalent to density depletion under constant temperature) is responsible for the initial localization of the x-line as magnetic tension makes up for the force balance, providing the necessary ±B z components and the outflow geometry (Liu et al., 2020(Liu et al., , 2022)).In our case, the ±B z components preceded the density depletion only because they were imposed initially in order to trigger the reconnection process.
The Hall magnetic and electric fields that develop in the reconnection exhaust play an important role in the structural evolution of the x-line and the rapid increase of the reconnection rate.In Figure 3 we showed how multiple processes influence the transport of electromagnetic energy in the x-z plane.In the absence of any Hall fields, the energy transport through the region is governed by E y , B x , and B z .In the case of this simulation, there is a small B z component initially, so when E y begins to grow near the neutral line the pattern of converging and diverging Poynting flux emerges from the E y B x and E y B z terms, respectively.As the electron outflow and Hall currents begin to develop, the Hall fields associated with those currents change the local Poynting flux structure via the E x B y term suppressing the inflow component and the E z B y term enhancing the outflow component across the separatrices.As the Hall field structures continue to develop they effectively drag the Poynting flux vectors away from the central EDR and toward the outflow over time, allowing most of the Poynting flux to avoid the EDR all together.This facilitates more energy transport from the inflow toward the outflow because the bottleneck imposed by the small-scale EDR is relieved.A similar "bottleneck" analogy was previously used to describe a kinetic simulation of undriven reconnection at late time, when the EDR grew long enough that the reconnection rate was limited by the electron physics in the EDR (Daughton et al., 2006).We do not address the later stages of the reconnection process, but we still consider the EDR (before the Hall fields open the exhaust) as a "bottleneck" that limits how much energy can flow from the upstream fields to the downstream fields due to the limited scale size that the EDR imposes.As the scale size of the exhaust changes, the reconnection rate evolves toward Petschek type fast rate because the plasma pressure in the EDR no longer throttles the inflow velocity.This effect of the Hall fields on the direction of Poynting flux has previously been invoked as a means of localizing the diffusion region around an "energy void" at the x-line and leading to fast reconnection (Liu et al., 2022).Their study argued that within the ion diffusion region (IDR), ∇ ⋅ S → ≈ J → ⋅ E → Hall = 0 and ∂u ∂t ≈ 0 in the steady state, where u = B 2 8π is the magnetic energy density (Liu et al., 2022).This steady-state condition is determined for well-developed fast magnetic reconnection, when the reconnection rate has stabilized and the IDR and EDR have the same aspect ratios (Liu et al., 2022).In contrast, we are interested in the dynamics of the system before it reaches a steady state, while the exhaust structure is still changing.Until the reconnection rate reaches its peak, it may be the case that ∇ ⋅ S → < 0 and that ∂u ∂t > 0 in the IDR, at least until the Poynting flux can be fully diverted toward the outflow.This energy balance of the "Hall region" is presented in the diagram shown in Figure 7, which depicts the structural evolution of the electromagnetic fields and Poynting flux during each phase of reconnection growth.To illustrate the role of Hall fields in Figure 7, we consider the Hall region as having only Hall electric fields, even though reconnection exhausts can also have small contributions from other terms in generalized Ohm's law (Xiong et al., 2022).It is well known that the Hall term is critical to the modeling of fast reconnection (Birn et al., 2001), but here we have explored the co-evolution of the Hall field structures and the reconnection rate in time.
We have seen from Figure 4 that in the transition to phase 3, the growing Hall fields cause the magnetic energy density downstream to rapidly accumulate, eventually to energy densities larger than any present upstream as the reconnection rate reaches its peak at t = 18.The similar timing of the field energy pileup and the peak reconnection rate may suggest that they are related, and that the pileup of field energy acts as a mechanism to stabalize the reconnection rate.Details of the pileup region have been discussed in the context of dipolarization fronts (Sitnov et al., 2009) demonstrating that the downstream B z does eventually stabilize even without the presence of a neighboring plasmoid.In future studies, it may be useful to examine time-evolving magnetic energy transport and accumulation with quantities such as magnetic flux transport, which has been used to effectively identify xlines in simulations (Li et al., 2021) and spacecraft data (Qi et al., 2022).The energy exchange between particle species in the late stages of reconnection growth has also been explored here.In the top panels of Figure 5 we examined how the downstream ion heating and the electron flow in the exhaust co-evolve in phase 3. A previous study showed that regions of significant ion energization in reconnection exhausts appear downstream of the edge of the electron current sheet (Sitnov et al., 2009).Consistent with those results, we see that between the electron jet and the region of ion heating, there is a region with relatively small electron velocity and large Poynting flux due to the Hall fields as shown in Figure 4.At the phase 2-3 transition, electrons lose energy to E ‖ but gain energy from E ⊥ in the exhaust.This is not the case in the "outer" EDR, where B → ≈ B z , the local E ‖ is small, and the electron jet initially loses energy to the local E ⊥ .More detailed explanations of the electron dynamics in the outer EDR (Payne et al., 2021;Xiong et al., 2022) and the broader exhaust region (Wang et al., 2016) have been proposed, but will not be elaborated here.In contrast to the electrons, the ions at t = 16 in Figure 5 are energized by both E ‖ and E ⊥ within most of the exhaust region, though the region of the largest J i ⊥ E ⊥ seems to occur just beyond the outer EDR, concentrated near the same region where J e ⊥ E ⊥ > 0. These structures are likely a consequence of the different effects that Hall fields and parallel electric fields have on fast electrons leaving the EDR versus cold ions from the initial current sheet that have not yet been energized by the reconnection process.The Hall effect does not contribute to E ‖ , which is instead supported by anisotropic electron distributions and can have significant influence on electron dynamics in regions larger than a d e (Egedal et al., 2012).
The region between the EDR and IDR boundaries has a complex electric field structure associated with the electron density depletion, including E x components aligned with the electron outflows (visible by t = 13 in Figure 2).This field structure decelerates the fast electrons and suppresses the outflow motion, consistent with the result that J e‖ E ‖ < 0 in the exhaust, as fast electrons leaving the EDR lose energy to the parallel component of these electric fields.The relatively cold ion population from the initial current sheet responds to these fields and gains energy J i‖ E ‖ > 0) .By t = 18 the rate of parallel ion energization and parallel electron energy loss in the exhaust are both reduced and a region of J ‖ E ‖ ≈ 0 emerges for both species, suggesting a relationship between the minimization of parallel energy transfer in the exhaust and the stabilization of the reconnection rate.Overall, both electrons and ions gain energy in the exhaust due to the dominance of J ⊥ E ⊥ , but E ‖ helps mediate energy exchange between the electron jet and the downstream ion population through parallel electron energy loss J e‖ E e‖ < 0 and ion energy gain J i‖ E i‖ > 0 until both approach zero as the reconnection rate stabilizes.The consequence of these interactions is seen in the evolution of the temperature anisotropy of both species (Figure 6).As electrons lose energy and ions gain energy via parallel electric fields the electron anisotropy becomes more perpendicular dominant while the ion anisotropy becomes less perpendicular dominant and approaches unity.
It may be useful in the future to consider how the phases of reconnection growth and the energy transport mechanisms discussed here could relate to electron-only versus traditional reconnection.The sequence of events discussed in recent studies (Hubbert et al., 2021(Hubbert et al., , 2022;;Lu et al., 2022) are similar to the phases of reconnection growth discussed here, especially the increasing strength of the Hall fields, which they identify as a signature of transition from electron-only reconnecting current sheets to traditional reconnection.The increase in the Hall fields during the rapid growth of the reconnection rate in phase 2 is what allows electromagnetic energy to rapidly flow into the exhaust and overcome the bottleneck imposed by the small-scale EDR.As we also discussed here, the Hall field and E ‖ structures in the exhaust act as a means of energy exchange between energized electrons and cold ions further downstream.This is also consistent with results (Hubbert et al., 2021(Hubbert et al., , 2022) ) showing that ion heating in traditional reconnection can be preceded by electron heating in electron-only reconnection.
Another aspect worth exploring in the future is how temporal evolution of the x-line and the direction of energy flow relate to entropy, disorder, and the arrow of time.Studies using PIC simulations (Liang et al., 2020;Xuan et al., 2021) and MMS data (Argall et al., 2022) suggest that even in collisionless reconnection, the velocity-space kinetic entropy does increase and the reconnection process is irreversible.Within the central EDR, electromagnetic fields do work to lower the local plasma entropy, making the local electron VDFs more ordered and less Maxwellian before those electrons become thermalized in the exhaust and increase in entropy again (Argall et al., 2022).The VDF structures associated with the remagnetization of the electron jet in the outer EDR (Payne et al., 2021) also appear to form near the time of the maximum reconnection rate (Shuster et al., 2014(Shuster et al., , 2015)).Future research may benefit from exploring how the energy-entropy exchange across the EDR relates to the flow of energy and the general trend of the reconnection process in time.

Conclusions
We have examined the growth of the reconnection rate divided into three distinct phases characterized by slow quasi-linear growth, followed by rapid exponential growth, followed by a tapering and eventual reduction of the reconnection rate following its peak.Based on the structural changes early after onset, we found that phase 1 is associated with the breaking of uniform density along the neutral line, and the reconfiguration of the local in-plane electric fields near the onset region.Following phase 1, the inflow region grew in both the spatial extent and magnitude of inflow Poynting flux S z and E y , accompanied by an increase in the inflow velocity.By comparing different field contributions to the S x and S z components across the inflow region, we showed how the in-plane electric fields and Hall magnetic fields play a role in the rapid transport of magnetic energy downstream and thus the rapid growth of the reconnection rate.While the emergence of the Hall fields initiated phase 2, the accumulation and saturation of the energy they transported downstream of the EDR marked the transition to phase 3, limiting further growth of the reconnection rate.Finally, we found that in the exhaust, both particle species gain energy via E ⊥ overall, but E ‖ acts as a mediating influence between species until the parallel energization of both species approaches zero in phase 3 and the ion temperature anisotropy approaches unity.By examining the correlation of these various processes in time, we may gain a better understanding of the mechanisms that control the temporal evolution of the reconnection rate.Similar analysis may prove to be useful in the future for understanding the dynamics of x-lines after the stabilization of the reconnection rate or when the upstream conditions vary in time.
units of en 0 B 0 v 2 a .

Figure 1 .
Figure 1.Time evolution of parameters measured in the central electron diffusion region.Left: Evolution of B z between reconnection onset and the time of the maximum reconnection rate, including three overlapping black dots indicating where quantities are measured and then averaged.Top right: The reconnection electric field and its time derivative.Bottom right: Energy transport terms including the total work rate J → ⋅ E → alongside the separate electron and ion contributions and the magnitude of the approximately balanced by an equivalent enhancement in the magnitude of ∇ ⋅ S → .The relative agreement between the magnitudes of ∇ ⋅ S plasma is dissipating electromagnetic energy at the same rate that the converging fields can supply it, as seen in an earlier study(Payne et al., 2020a) using the same simulation.The separate contributions from electrons (J during phase 1, suggesting that the electrons and ions in the central EDR are energized at the same rate just after reconnection onset.In phase 2, dE y ′ dt grows quickly.The magnitudes of E′ y and dE y ′ dt are in close agreement from t = 14-16, suggesting that much of phase 2 is characterized by an exponential increase in the reconnection rate.During this phase, J in lockstep, but the energy dissipation becomes dominated by the electrons as J → 2, electrons start to dissipate electromagnetic energy more efficiently than ions in the central EDR.In the transition to phase 3, dE y ′ dt decouples from E′ y as the growth in the reconnection rate begins to slow down.Within 1Ω 1 ci of t = 18, E′ y , J maximum magnitudes.After t = 18, all three quantities start to decrease and dE y ′ dt < 0.

Figure 2 .
Figure 2. Evolution of the electron number density n e , the reconnection electric field E y ′ and the electric field along the outflow E x during phase 1 of reconnection growth.

Figure 3 .
Figure 3. Field contributions to the in-plane components of Poynting flux at t = 16.Right: Color plots of both S x and S z .Left: Plots of the in-plane Poynting flux components (black) and their separate contributions by Hall (red) and non-Hall (green) fields measured along the cuts shown in the color plots.

Figure 4 .
Figure 4. Evolution of Poynting flux and magnetic energy density during phase 3 of reconnection growth.The color represents the magnetic energy density and the arrows represent the in-plane Poynting flux.The expanding exhaust and the inflow regions are designated the dashed blue circle and the green arrows, respectively.

Figure 5 .
Figure 5. Top Row: Evolution of the electron velocity (arrows) and the ion temperature (color scale) during phase 3 of reconnection growth.Bottom Cutouts: Contributions to J → ⋅ E → in the exhaust, broken down by species and by parallel and perpendicular components for t = 16 and t = 18.

Figure 6 .
Figure 6.Top Row: Evolution of the electron (top) and ion (bottom) temperature anisotropies in phase 3 of reconnection growth.Regions in blue are dominated by perpendicular temperature, while regions in red are parallel-dominant.The arrows in the top and bottom plots represent the electron and ion velocity, respectively.

Figure 7 .
Figure 7. Diagram depicting the evolution of fields and energy transport in a reconnection outflow during the phases of reconnection growth.Since J → ⋅ E → Hall = 0 by definition, the balance of Poynting's theorem in the Hall region only comes from the ∇ ⋅ S → and ∂u ∂t terms.