Electron and Proton Heating in Transrelativistic Guide Field Reconnection

In Figure 1 we show snapshots covering the time range $t/{t}_{{\rm{A}}}=0.1\mbox{--}3.1$ for a simulation with ${\beta }_{{\rm{i}}}=5\times {10}^{-4}$ and a moderate guide field, ${b}_{{\rm{g}}}=1$ (run b5e-4.bg1 in Table 1). In the first, second, and third columns, respectively, we show the number density n (in units of total upstream density, $2{n}_{0}$), the degree of charge non-neutrality (i.e., the ratio of charge density to particle number density, $({n}_{{\rm{i}}}-{n}_{{\rm{e}}})/({n}_{{\rm{i}}}+{n}_{{\rm{e}}})$), and the z-component of current density, normalized to the initial value in the current layer, ${j}_{z}/{j}_{z0};$ the gray contours show magnetic field lines.

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The time evolution is illustrated from the first to the fourth row (i.e., panels A–D). After reconnection is triggered at $x\sim 0$, an X-point forms. From the central X-point, two reconnection fronts, dragged by magnetic tension, recede from the center; for the simulation shown in Figure 1, the speed of recession is $0.31\,c\approx \sqrt{{\sigma }_{w}}\,c$, so the expected Alfvén limit is saturated. Because we use periodic boundary conditions, the receding fronts meet at $x=\pm 1080\,c/{\omega }_{\mathrm{pe}}$ after about one Alfvénic crossing time (second row), and merge into a volume of particles and magnetic flux that continues to grow as reconnection proceeds. As anticipated, we refer to this structure as the primary island; it is the main site where we extract our heating measurements because this is where particles ejected from the outflow region eventually end up. Up to the run times of our simulations, the primary island tends to maintain an oblong shape (elongated along x), a feature that is more prominent for stronger guide fields.

Secondary islands, as opposed to the primary island, form frequently at low ${\beta }_{{\rm{i}}}$ in the exhaust region (or equivalently, in the outflow region); the formation of secondary islands is suppressed at high ${\beta }_{{\rm{i}}}$ (Daughton & Karimabadi 2007; Uzdensky et al. 2010; RSN17). We find that simulations with high guide fields are characterized by a relative absence of secondary islands compared to simulations with the same ${\beta }_{{\rm{i}}}$ but weaker guide fields.

The current layer in guide field reconnection is characterized by left-right and top-bottom asymmetry, especially in the exhaust region, immediately downstream of the central X-point. Electrons and protons are ejected from the X-point toward different directions: for our magnetic geometry, electrons to the upper-left and lower-right quadrants, whereas protons are sent to the upper-right and lower-left ones (see panels E–H, which zoom into the central region of panels A–D; Zenitani & Hoshino 2008). The z-current (third column) is inhomogeneous in the immediate downstream (see panels I–L); there is some enhancement along the walls of the exhaust (at the interface with the upstream), in particular along the directions that electrons leave the X-point.

To assess the heating efficiency at late times, we focus on the change in particle internal energy, as particles travel from the inflow region (i.e., the upstream) to the far downstream, and eventually enter the primary island (these different regions are defined in more detail below). Internal energy and temperature in each cell of the simulation domain are calculated as in RSN17; here we briefly review the method, but we refer to RSN17 for more details.

The internal energy is computed by treating the plasma as a perfect, isotropic fluid,⁵ whose stress-energy tensor is

$$\begin{eqnarray}&&{T}_{s}^{\mu \nu }=\left({e}_{s}+{p}_{s}\right){U}_{s}^{\mu }{U}_{s}^{\nu }-{p}_{s}{g}^{\mu \nu },\end{eqnarray} \tag{ 9 }$$

where ${e}_{s}={\overline{n}}_{s}{m}_{s}{c}^{2}+{u}_{s},{p}_{s},{U}_{s}^{\mu }$, and ${g}^{\mu \nu }$ are the rest-frame energy density, pressure, dimensionless four-velocity, and flat-space Minkowski metric, and the subscript $s$ denotes the particle species; ${\overline{n}}_{s}$ is the rest-frame particle number density. From Equation (9), the dimensionless internal energy per particle in the fluid rest frame ${\upsilon }_{s}$ can be written as

$$\begin{eqnarray}&&{\upsilon }_{s}=\displaystyle \frac{({T}_{s}^{00}/{n}_{s}{m}_{s}{c}^{2}-{{\rm{\Gamma }}}_{s}){{\rm{\Gamma }}}_{s}}{1+{\hat{\gamma }}_{s}({\upsilon }_{s})({{\rm{\Gamma }}}_{s}^{2}-1)}.\end{eqnarray} \tag{ 10 }$$

Here,⁶ T⁰⁰_s is the laboratory-frame energy density, n_s is the laboratory-frame particle number density, ${{\rm{\Gamma }}}_{s}$ is the Lorentz factor computed from the local fluid velocity, ${\hat{\gamma }}_{s}$ is the adiabatic index, and the subscript $s\in \{{\rm{e}},{\rm{i}}\}$ indicates the type of particle (electron or proton). Note that the adiabatic index ${\hat{\gamma }}_{s}({\upsilon }_{s})$ is a function of the specific internal energy. Given a mapping between the specific internal energy and adiabatic index, Equation (10) can be solved iteratively for ${\upsilon }_{s}$. For the adiabatic index, we use a function of the form

$$\begin{eqnarray}&&{\hat{\gamma }}_{s}({\upsilon }_{s})=\displaystyle \frac{A\,+\,B{\upsilon }_{s}}{C\,+\,D{\upsilon }_{s}},\end{eqnarray} \tag{ 11 }$$

where $A\approx 1.176,B\approx 1.258,C\approx 0.706$, and $D\approx 0.942$. The numerical coefficients satisfy $A/C=5/3$ and $B/D=4/3$ in the nonrelativistic (${\upsilon }_{s}\to 0$) and ultra-relativistic (${\upsilon }_{s}\to \infty$) limits, respectively; see Equation (14) of RSN17 for additional details. The adiabatic index in Equation (11) is used to convert between specific dimensionless internal energy and dimensionless temperature: ${\theta }_{s}=[{\hat{\gamma }}_{s}({\upsilon }_{s})-1]{\upsilon }_{s}$.

In the following, we refer to "downstream" as the combination of the outflow region and the primary island. We select only part of the downstream to compute the late-time particle heating, in particular, part of the primary island, which is far from the central X-point. The region is selected based on three criteria: (i) the mixing between particles originating from the top ($y\gt 0$) and bottom ($y\lt 0$) of the domain must exceed a chosen threshold (and be lower than the complementary threshold): ${d}_{\mathrm{th}}\lt {n}_{\mathrm{top}}/{n}_{\mathrm{tot}}\lt 1-{d}_{\mathrm{th}}$ (RSN17), (ii) the z-component of the magnetic vector potential must exceed a value ${A}_{z}\gt {A}_{z,\mathrm{th}}$ that is related to the mixing threshold identified in (i) (Li et al. 2017; Ball et al. 2018), and (iii) cells containing particles that were part of the hot, overdense population initialized in the current sheet (see Section 2) are excluded because their properties depend on arbitrary choices at initialization. The use of the above criteria for selection of the "island" region is illustrated in Figure 2. Panel A shows the density ratio of particles originating from the top of the domain, ${n}_{\mathrm{top}}$, to the total density ${n}_{\mathrm{tot}}$. The part of the downstream that has mixed, according to (i) above, is shown in panel C by the combination of gray, yellow, and tan regions.

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To select the island area, which is a subset of the mixed region, we find cells at the boundary $x=\pm 1080\,c/{\omega }_{\mathrm{pe}}$ that satisfy ${d}_{\mathrm{th}}\lt {n}_{\mathrm{top}}/{n}_{\mathrm{tot}}\lt 1-{d}_{\mathrm{th}}$, and of these cells, we select those at the upper and lower edges of the island (along $\pm y$). In these cells, we compute the average value of the vector potential ${A}_{z,\mathrm{th}}$, to serve as a second threshold for selection of the island region (panel B). The island cells are then identified as those where there is sufficient mixing (criterion (i)), and where ${A}_{z}\gt {A}_{z,\mathrm{th}}$ (criterion (ii)). We also impose a strict spatial cutoff on the island region to ensure that it is distinct from the exhaust even at late times (see RSN17 for details). This criterion corresponds in panel C of Figure 2 to excluding the tan regions at $| x| \lt 430\,c/{\omega }_{\mathrm{pe}}$. Finally, from the island region, we exclude any cells where the density of the hot, overdense particles used for initialization (see Section 2) is greater than zero (criterion (iii)), so that the measured heating does not depend on particles whose properties are set by hand as initial conditions. These initial particles generally reside in the island center (see the gray core in panel C). The region that satisfies all our criteria (which we call "island region" for brevity) is shown in panel C of Figure 2 as the yellow area.

The method of island selection outlined here is a robust and consistent way of selecting cells that are far downstream of the central X-point for all guide field strengths we consider, and it is relatively insensitive to the choice of threshold value ${d}_{\mathrm{th}}$. For example, ${A}_{z,\mathrm{th}}$ differs by no more than 10% for ${d}_{\mathrm{th}}$ in the range 0.003–0.3; the overall measured values of particle heating in the island show a comparable sensitivity to the choice of ${d}_{\mathrm{th}}$ at a level of around 15% for ${d}_{\mathrm{th}}$ in the range 0.003–0.3. For the island selection, we find that ${d}_{\mathrm{th}}=0.3$ is suitable.

To assess particle heating, we measure the change in particle internal energies as they travel from the inflow to the island region (described above). The upstream region is defined such that ${n}_{\mathrm{top}}/{n}_{\mathrm{tot}}\lt {d}_{\mathrm{th},\mathrm{up}}$ or ${n}_{\mathrm{top}}/{n}_{\mathrm{tot}}\gt 1-{d}_{\mathrm{th},\mathrm{up}}$ (so, a complementary definition to the mixing criterion (i) above). We employ a threshold value ${d}_{\mathrm{th},\mathrm{up}}=3\times {10}^{-5};$ the fact that ${d}_{\mathrm{th},\mathrm{up}}\lt {d}_{\mathrm{th}}$ provides a thin (∼few $\,c/{\omega }_{\mathrm{pe}}$) buffer region between the downstream (tan and yellow areas combined in panel C of Figure 2) and the upstream (navy region in panel C of Figure 2). We further select only the upstream cells that lie within $\pm 100\,c/{\omega }_{\mathrm{pe}}$ of y = 0 (as delimited by the dashed white contours in panel C of Figure 2).

With the inflow and island regions suitably identified, overall heating fractions can be computed as the difference between the dimensionless internal energies in the island and inflow regions, normalized to the inflowing magnetic energy per particle (Shay et al. 2014; RSN17):

$$\begin{eqnarray}&&{M}_{u{\rm{e}},\mathrm{tot}}\equiv \displaystyle \frac{{\upsilon }_{{\rm{e}},\mathrm{isl}}-{\upsilon }_{{\rm{e}},\mathrm{up}}}{{\sigma }_{{\rm{i}}}{m}_{{\rm{i}}}/{m}_{{\rm{e}}}},\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&{M}_{u{\rm{i}},\mathrm{tot}}\equiv \displaystyle \frac{{\upsilon }_{{\rm{i}},\mathrm{isl}}-{\upsilon }_{{\rm{i}},\mathrm{up}}}{{\sigma }_{{\rm{i}}}}.\end{eqnarray} \tag{ 13 }$$

These dimensionless ratios indicate the fraction of magnetic energy per particle in the inflow that is converted to particle heating by the time the particle reaches the island, far downstream of the central X-point. As in RSN17, the heating fractions in Equations (12) and (13) can be decomposed into adiabatic-compressive and irreversible components,

$$\begin{eqnarray}&&{M}_{u{\rm{e}},\mathrm{tot}}={M}_{u{\rm{e}},\mathrm{ad}}+{M}_{u{\rm{e}},\mathrm{irr}},\end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}&&{M}_{u{\rm{i}},\mathrm{tot}}={M}_{u{\rm{i}},\mathrm{ad}}+{M}_{u{\rm{i}},\mathrm{irr}}.\end{eqnarray} \tag{ 15 }$$

The adiabatic heating fractions represent the heating that results solely from an increase in internal energy due to adiabatic compression of the plasma as it travels from the inflow to the island; for electrons, the adiabatic heating fraction is approximately⁷ (RSN17)

$$\begin{eqnarray}&&{M}_{u{\rm{e}},\mathrm{ad}}\approx \displaystyle \frac{1}{2}{\beta }_{{\rm{i}}}\displaystyle \frac{{T}_{{\rm{e}}0}}{{T}_{{\rm{i}}0}}\left[{\left(\displaystyle \frac{{n}_{\mathrm{isl}}}{{n}_{0}}\right)}^{{\hat{\gamma }}_{{\rm{e}}}-1}-1\right].\end{eqnarray} \tag{ 16 }$$

Here ${n}_{\mathrm{isl}}$ is the typical electron density in the island. The irreversible heating fractions are associated with a genuine increase in the entropy of the particles, and are of primary interest to us. The measured heating fractions we present in Section 4 are typically time-averaged over one Alfvénic crossing time ($\approx 7100\,{\omega }_{\mathrm{pe}}^{-1}$).

A representative temporal evolution of electron and proton irreversible heating fractions, ${M}_{u{\rm{e}},\mathrm{irr}}$ and ${M}_{u{\rm{i}},\mathrm{irr}}$, is shown in Figure 3. The time evolution of the heating fractions is shown from $t/{t}_{{\rm{A}}}=0$ to $t/{t}_{{\rm{A}}}\approx 3.5;$ at late times, the heating fractions achieve a steady state (i.e., both the electron and proton irreversible heating fractions are relatively flat after $t/{t}_{{\rm{A}}}\approx 2.5$). Time-averaged heating fractions are computed during this steady state; the points used for time-averaging are indicated by the shaded region in Figure 3.

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Electron and proton heating via reconnection shows substantial differences in the limits of a strong and weak guide field. Figure 4 shows 2D snapshots at $t/{t}_{{\rm{A}}}=2.7$ of electron (panels A–C) and proton (panels D–F) temperature⁸ and corresponding 1D profiles (panels G–I) for three simulations with a relatively low ${\beta }_{{\rm{i}}}=0.03$ and guide field strengths ${b}_{{\rm{g}}}=0.3,1,$ and 6, increasing from left to right. The simulations here correspond to runs b3e-2.bg3e-1, b3e-2.bg1, and b3e-2.bg6 in Table 1.

The first and second rows show the spatial dependence of electron (panels A–C) and proton (panels D–F) heating. At low ${b}_{{\rm{g}}}$ (run b3e-2.bg3e-1), the electron and proton temperatures are relatively uniform in the exhaust and island regions. For intermediate guide field strengths (run b3e-2.bg1), the electron and proton heating is less uniform in the island and shows a marked asymmetry in the exhaust region (see panels B and E, in between the cyan lines). For the strong guide field case (run b3e-2.bg6), electrons reach a maximum temperature of roughly ${k}_{{\rm{B}}}{T}_{{\rm{e}}}/{m}_{{\rm{i}}}{c}^{2}\approx 0.02$ along the upper-left and lower-right edges of the outflow; on the other hand, proton heating along the exhaust is essentially isolated to the upper-right and lower-left edges. Throughout the entire downstream (for run b3e-2.bg6), the proton temperature rarely exceeds ${k}_{{\rm{B}}}{T}_{{\rm{i}}}/{m}_{{\rm{i}}}{c}^{2}\approx 5\times {10}^{-3}$.

The 2D plots in panels A–F also illustrate that the primary island becomes more oblong with increasing guide field. For ${b}_{{\rm{g}}}=0.3$, the aspect ratio of the island (length along the layer to width orthogonal to it) is about 7:4, whereas at ${b}_{{\rm{g}}}=6$, it is twice as large, 7:2. In the cases with strong guide field, the primary islands do not circularize up to the run times of our simulations.

The bottom row of Figure 4 shows the 1D profiles of electron (blue) and proton (red) temperatures, both in units of ${m}_{{\rm{i}}}{c}^{2}$, averaged along y for cells within the downstream region (including the yellow and tan regions in panel C of Figure 2, and excluding the gray area in the island core that contains particles left over from initialization). The edges of the primary island are shown by vertical cyan lines. Horizontal black dashed lines indicate the initial temperature of particles in the far upstream. In the weak guide field case ${b}_{{\rm{g}}}=0.3$ (panel G), protons are heated substantially more than electrons, similar to the case of antiparallel reconnection (see Melzani et al. 2014; Werner et al. 2016, RSN17). As the strength of the guide field increases (panels H and I), proton heating in both the exhaust region and the primary island is strongly suppressed. The electron temperature, on the other hand, is largely unaffected; for ${b}_{{\rm{g}}}=0.3,1,$ and 6, the electron temperature in the island is always around ${k}_{{\rm{B}}}{T}_{{\rm{e}}}/{m}_{{\rm{i}}}{c}^{2}\approx 5\times {10}^{-3}$.

Figure 5 is similar to Figure 4, but corresponds to a set of simulations with ${\beta }_{{\rm{i}}}=2$ (runs b2.bg3e-1, b2.bg1, and b2.bg6 in Table 1). As we discuss below, this value of ${\beta }_{{\rm{i}}}=2$ is close to ${\beta }_{{\rm{i}},\max }=2.5$ (see Equation (18)), impliying that electrons and protons both start with relativistic temperatures. In stark contrast to the low ${\beta }_{{\rm{i}}}$ case, at ${\beta }_{{\rm{i}}}=2$ the electron and proton temperatures in the island region are roughly equal, regardless of the guide field strength (${b}_{{\rm{g}}}=0.3\mbox{--}6$). Still, the 2D temperature structure within the island differs between low and high guide field cases. At high ${\beta }_{{\rm{i}}}$ and low or intermediate guide field (runs b2.bg3e-1 and run b2.bg1), the electron and proton temperatures in the island are typically uniform (similar to the low ${\beta }_{{\rm{i}}}$, low ${b}_{{\rm{g}}}$ case in panels A and D of Figure 4). However, at high ${\beta }_{{\rm{i}}}$ and high guide field (run b2.bg6), the electron and proton temperatures are less uniform (relative to runs b2.bg3e-1 and b2.bg1; see panels C and F of Figure 4), with electron and proton temperatures greatest near the interfaces between the primary island and the outflows (i.e., $x=\pm 700\,c/{\omega }_{\mathrm{pe}}$).

Figure 6 shows the ${\beta }_{{\rm{i}}}$ and ${b}_{{\rm{g}}}$ dependence of the reconnection rate, $| {v}_{\mathrm{in}}| /{v}_{{\rm{A}}}$. The inflow speed $| {v}_{\mathrm{in}}|$ is computed as a spatial average over a specific region of the upstream,⁹ and temporal average from $t/{t}_{{\rm{A}}}\approx 0.7$ to 1 (when reconnection is roughly in steady state). Each point corresponds to the measurement from a different simulation, and those with the same guide field strength ${b}_{{\rm{g}}}$ are connected by a solid line. For these simulations, ${m}_{{\rm{i}}}/{m}_{{\rm{e}}}=1836$, ${T}_{{\rm{e}}0}{/T}_{{\rm{i}}0}=1$, and ${\sigma }_{w}=0.1$.

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In most cases, reconnection proceeds at or below the value often reported in the literature, i.e., $| {v}_{\mathrm{in}}| /{v}_{{\rm{A}}}\lesssim 0.1$ (Cassak et al. 2017); however, for low ${\beta }_{{\rm{i}}}$ and weak guide field (${b}_{{\rm{g}}}\lesssim 0.3$), the reconnection rate exceeds this fiducial value, with $| {v}_{\mathrm{in}}| /{v}_{{\rm{A}}}$ in the range 0.1–0.15. For ${b}_{{\rm{g}}}\lesssim 0.3$, the reconnection rate shows a relatively weak scaling with ${\beta }_{{\rm{i}}}$, decreasing from $| {v}_{\mathrm{in}}| /{v}_{{\rm{A}}}\,\approx 0.1\mbox{--}0.15$ to $| {v}_{\mathrm{in}}| /{v}_{{\rm{A}}}\approx 0.05$, only a factor of 2–3, as ${\beta }_{{\rm{i}}}$ increases from $5\times {10}^{-4}$ to 2 (Numata & Loureiro 2015, RSN17, Ball et al. 2018). For guide fields ${b}_{{\rm{g}}}\gtrsim 1$, the ${\beta }_{{\rm{i}}}$ dependence of the reconnection rate is even weaker, and $| {v}_{\mathrm{in}}| /{v}_{{\rm{A}}}$ typically varies from 0.01 to 0.07. We find that the presence of a guide field tends to suppress the reconnection rate, which is a dependence similar to that found by Melzani et al. (2014) for electron-ion relativistic reconnection, Ricci et al. (2003), Huba (2005), TenBarge et al. (2013) and Liu et al. (2014) for electron-ion nonrelativistic reconnection, and Hesse & Zenitani (2007) and Werner & Uzdensky (2017) for electron-positron plasma. The decrease in the reconnection rate with ${b}_{{\rm{g}}}$ is more pronounced at lower values of ${\beta }_{{\rm{i}}}$.

Figure 7 shows the ${b}_{{\rm{g}}}$ and ${\beta }_{{\rm{i}}}$ dependence of electron (panel A) and proton (panel B) dimensionless temperature. Each solid line shows the volume-averaged temperature in the island for a set of simulations with the same value of ${b}_{{\rm{g}}}$, and the black diamonds, connected with a dashed line, show the upstream temperature for each value of ${\beta }_{{\rm{i}}}$ (for simulations with fixed ${\beta }_{{\rm{i}}}$, the upstream temperature is the same, independent of ${b}_{{\rm{g}}}$). As discussed in Section 2, the numerical resolution is sufficient to keep numerical heating under control; temperatures measured in the inflow region are about the same as those at initialization throughout the duration of our simulations.¹⁰ The simulations presented here are the same as those in Section 4.2, so they employ ${m}_{{\rm{i}}}/{m}_{{\rm{e}}}=1836$, ${T}_{{\rm{e}}0}{/T}_{{\rm{i}}0}=1$, and ${\sigma }_{w}=0.1$.

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The upstream electron dimensionless temperatures range from nonrelativistic, ${\theta }_{{\rm{e}}}\approx 0.08$, up to ultra-relativistic, ${\theta }_{{\rm{e}}}\approx 700;$ the temperatures in the downstream region range from moderately to ultra-relativistic, ${\theta }_{{\rm{e}}}\approx 6\mbox{--}700$.¹¹ For all values of ${\beta }_{{\rm{i}}}$, the electron temperature in the island appears to be nearly independent of the guide field strength (Figure 7, panel A). The guide field simulations show the same scaling with ${\beta }_{{\rm{i}}}$ as the antiparallel case (blue circles), with the electron temperature increasing from about ${\theta }_{{\rm{e}}}\approx 6$ at ${\beta }_{{\rm{i}}}\sim 5\times {10}^{-4}$ up to ${\theta }_{{\rm{e}}}\approx 700$ at ${\beta }_{{\rm{i}}}\sim 2$. Additionally, the electron temperature shows only a relatively weak dependence on ${\beta }_{{\rm{i}}}$ for ${\beta }_{{\rm{i}}}\lesssim 0.5$. From ${\beta }_{{\rm{i}}}\sim 5\times {10}^{-4}$ up to 0.5, the electron temperature in the island changes by no more than a factor of 10 (the dependence is even weaker for ${\beta }_{{\rm{i}}}\lesssim 3\times {10}^{-2}$). At high ${\beta }_{{\rm{i}}}$, the electron temperature in the island appears to be nearly the same as that in the upstream. However, the increase in temperature from upstream to downstream corresponds to a substantial fraction (typically ∼30%) of the inflowing magnetic energy per electron (see Figure 8 below, panel A). These two statements are not in contradiction because for high ${\beta }_{{\rm{i}}}$ the available magnetic energy is only a small fraction of the initial thermal energy.

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In Figure 7, panel B, we show the proton dimensionless temperature in the island for the same simulations as shown in panel A. At low ${\beta }_{{\rm{i}}}$, protons show a clear decrease in island temperature with increasing guide field strength; for the antiparallel reconnection (${b}_{{\rm{g}}}=0$) and ${\beta }_{{\rm{i}}}\sim 5\times {10}^{-4}$, the proton dimensionless temperature in the island is ${\theta }_{{\rm{i}}}\approx 0.02,$ but decreases to ${\theta }_{{\rm{i}}}\approx 6\times {10}^{-4}$ for a strong guide field, ${b}_{{\rm{g}}}=6$. As ${\beta }_{{\rm{i}}}$ increases, the proton heating in the island shows a weaker dependence on the guide field strength. Similar to electron heating in the island at high ${\beta }_{{\rm{i}}}$, ${\theta }_{{\rm{i}}}$ is nearly independent of ${b}_{{\rm{g}}}$ at high ${\beta }_{{\rm{i}}}$. The proton dimensionless temperatures in both the upstream and the island region are generally nonrelativistic, ${\theta }_{{\rm{i}}}\lesssim 1$. In summary, when comparing panels A and B, a striking difference is that the electron dimensionless temperature in the island is independent of the guide field strength, whereas the proton dimensionless temperature appreciably decreases with increasing ${b}_{{\rm{g}}}$.

In Figure 8 we present the scaling of electron and proton heating with guide field strength ${b}_{{\rm{g}}}$ and proton-${\beta }_{{\rm{i}}}$. The first and second rows show the electron and proton heating fractions, respectively (see Equations (12)–(15)); the total heating (first column) is decomposed into adiabatic-compressive and irreversible components, shown in the second and third columns, respectively. In each panel, the corresponding heating fraction is plotted as a function of ${\beta }_{{\rm{i}}}$ for guide field strengths in the range 0–6.

The first row in Figure 8 shows the scaling of the electron total, adiabatic, and irreversible heating fractions (${M}_{u{\rm{e}},\mathrm{tot}},{M}_{u{\rm{e}},\mathrm{ad}}$, and ${M}_{u{\rm{e}},\mathrm{irr}}$) with respect to ${b}_{{\rm{g}}}$ and ${\beta }_{{\rm{i}}}$. At low ${\beta }_{{\rm{i}}}$, the electron total heating fraction within the island does not show a strong scaling with the strength of the guide field (consistent with Figure 7). For ${\beta }_{{\rm{i}}}\lesssim 0.03$, ${M}_{u{\rm{e}},\mathrm{tot}}\sim 0.1$. At high ${\beta }_{{\rm{i}}}$, the total heating is suppressed by strong guide fields, ${b}_{{\rm{g}}}\gtrsim 3$. Some insight into this trend is provided by decomposing the total heating fraction ${M}_{u{\rm{e}},\mathrm{tot}}$ into adiabatic and irreversible parts, ${M}_{u{\rm{e}},\mathrm{ad}}$ (panel B) and ${M}_{u{\rm{e}},\mathrm{irr}}$ (panel C). For low ${\beta }_{{\rm{i}}}$, compressive heating is negligible; however, at higher values of ${\beta }_{{\rm{i}}}$, compressive heating is more significant, but tends to decrease with stronger guide fields, which is in qualitative agreement with Li et al. (2018). This result is physically intuitive, as the plasma becomes less compressible when the magnetic pressure of the guide field is higher (and in fact, we note that the primary island is less dense for stronger guide fields).

To summarize, we find that the electron compressive heating fraction in panel (B) steadily increases with ${\beta }_{{\rm{i}}}$ and strongly decreases with ${b}_{{\rm{g}}}$. Both trends for ${M}_{u{\rm{e}},\mathrm{ad}}$ can be easily understood from Equation (16), given that stronger guide fields give weaker density compressions. In contrast, the electron irreversible heating fraction (panel C) is largely independent of both ${b}_{{\rm{g}}}$ and ${\beta }_{{\rm{i}}}$, and it is around ${M}_{u{\rm{e}},\mathrm{irr}}\sim 0.1$. The combination of irreversible and compressive heating explains why the total heating at low ${\beta }_{{\rm{i}}}$ is independent of both ${\beta }_{{\rm{i}}}$ and ${b}_{{\rm{g}}}$, whereas at high ${\beta }_{{\rm{i}}}$ it is lower for larger ${b}_{{\rm{g}}}$ (due to the corresponding trend in compressive heating).

The second row in Figure 8 shows the proton heating fractions ${M}_{u{\rm{i}},\mathrm{tot}},{M}_{u{\rm{i}},\mathrm{ad}}$, and ${M}_{u{\rm{i}},\mathrm{irr}}$ (panels D, E, and F). The proton total heating in the island differs sharply from the electron total heating (panel A). The proton total heating shows a strong dependence on the strength of the guide field; for antiparallel reconnection, ${M}_{u{\rm{i}},\mathrm{tot}}\approx 0.3$ regardless of ${\beta }_{{\rm{i}}}$, but the total heating is significantly suppressed as ${b}_{{\rm{g}}}$ increases. For ${b}_{{\rm{g}}}=6$ and ${\beta }_{{\rm{i}}}\lesssim 0.5$, ${M}_{u{\rm{i}},\mathrm{tot}}$ is negligible. The proton compressive heating (panel E) shows a trend similar to that of the electron compressive heating (panel B); for both electrons and protons, the compressive heating is controlled by the density in the upstream, the density in the island region, and the upstream temperature (here, we focus on the case ${T}_{{\rm{e}}0}{/T}_{{\rm{i}}0}=1$); because these quantities are similar for electrons and protons, the compressive heating for both species shows the same trend. The proton irreversible heating (panel F) is similar to the proton total heating (panel A) for ${\beta }_{{\rm{i}}}\lesssim 0.03$ because compressive heating is negligible in this regime. For ${\beta }_{{\rm{i}}}\gtrsim 0.03$, the proton irreversible heating is less sensitive to the guide field strength, and by ${\beta }_{{\rm{i}}}=2$, ${M}_{u{\rm{i}},\mathrm{irr}}\approx 0.08$ regardless of ${b}_{{\rm{g}}}$, similarly to the electron irreversible heating.

The electron and proton irreversible heating fractions ${M}_{u{\rm{e}},\mathrm{irr}}$ and ${M}_{u{\rm{i}},\mathrm{irr}}$ can be used to compute the ratio of electron irreversible heating to total irreversible particle heating liberated during reconnection (RSN17),

$$\begin{eqnarray}&&{q}_{u{\rm{e}},\mathrm{irr}}\equiv \displaystyle \frac{{M}_{u{\rm{e}},\mathrm{irr}}}{{M}_{u{\rm{e}},\mathrm{irr}}+{M}_{u{\rm{i}},\mathrm{irr}}}.\end{eqnarray} \tag{ 17 }$$

In Figure 9 we present the ${\beta }_{{\rm{i}}}$ and ${b}_{{\rm{g}}}$ dependence of ${q}_{u{\rm{e}},\mathrm{irr}}$, the electron irreversible heating efficiency. For all ${\beta }_{{\rm{i}}}\lesssim 2$, ${q}_{u{\rm{e}},\mathrm{irr}}$ increases with the guide field strength. For antiparallel reconnection, electrons ultimately receive ∼18% of the irreversible heat transferred to particles. As the guide field increases, so does the fraction of irreversible heating transferred to electrons; for ${b}_{{\rm{g}}}=1,{q}_{u{\rm{e}},\mathrm{irr}}\approx 45 \%$, and by ${b}_{{\rm{g}}}=6$, electrons receive the vast majority of magnetic energy that is converted into irreversible particle heating, with ${q}_{u{\rm{e}},\mathrm{irr}}\approx 93 \%$. At ${\beta }_{{\rm{i}}}=2\sim {\beta }_{{\rm{i}},\max }$, ${q}_{u{\rm{e}},\mathrm{irr}}\approx 50 \%$, independently of ${b}_{{\rm{g}}};$ ${\beta }_{{\rm{i}},\max }$ is the maximum possible value of ${\beta }_{{\rm{i}}}$, given ${\sigma }_{w}$ and ${T}_{{\rm{e}}0}{/T}_{{\rm{i}}0}$, and is defined as

$$\begin{eqnarray}&&{\beta }_{{\rm{i}},\max }=\displaystyle \frac{0.5}{{\sigma }_{w}+{\sigma }_{w}{T}_{{\rm{e}}0}{/T}_{{\rm{i}}0}}.\end{eqnarray} \tag{ 18 }$$

This equation is derived by expressing ${\beta }_{{\rm{i}}}$ as a function of ${T}_{{\rm{e}}0}{/T}_{{\rm{i}}0},{\sigma }_{w}$, and ${\theta }_{{\rm{i}}}$, then taking the limit ${\theta }_{{\rm{i}}}\to \infty$. For the simulations presented here, with ${m}_{{\rm{i}}}/{m}_{{\rm{e}}}=1836$, ${T}_{{\rm{e}}0}{/T}_{{\rm{i}}0}=1$, and ${\sigma }_{w}=0.1$, we find ${\beta }_{{\rm{i}},\max }=2.5$. Note that for ${\beta }_{{\rm{i}}}\sim {\beta }_{{\rm{i}},\max }$, electrons and protons start relativistically hot in the upstream, and the scale separation $(\,c/{\omega }_{\mathrm{pe}})/(\,c/{\omega }_{\mathrm{pi}})$ is of order unity (RSN17); in this case, electrons and protons behave nearly the same, which explains why for ${\beta }_{{\rm{i}}}\sim {\beta }_{{\rm{i}},\max }$ we obtain energy equipartition, i.e., we find that ${q}_{u{\rm{e}},\mathrm{irr}}\approx 50 \%$, independently of ${b}_{{\rm{g}}}$.

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For simplicity, we focused in Section 4.3 on electron heating for cases with representative magnetization ${\sigma }_{w}=0.1$ and temperature ratio ${T}_{{\rm{e}}0}{/T}_{{\rm{i}}0}=1$. A full exploration of the dependence of electron and proton heating on ${\beta }_{{\rm{i}}}$, ${b}_{{\rm{g}}}$, ${\sigma }_{w}$, and ${T}_{{\rm{e}}0}{/T}_{{\rm{i}}0}$ is beyond the scope of this work. Nevertheless, for a limited range of ${b}_{{\rm{g}}}$ and ${\beta }_{{\rm{i}}}$, we present in Figure 10 the electron irreversible heating efficiencies when we vary the electron-to-proton temperature ratio ${T}_{{\rm{e}}0}{/T}_{{\rm{i}}0}$ in the range 0.1–1 (panels A and B), as well as for several simulations with ${\sigma }_{w}=1$ (panel C). The physical parameters of these runs are given in Table 2.

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The effect of varying the initial electron-to-proton temperature ratio for antiparallel reconnection (${b}_{{\rm{g}}}=0$) is demonstrated in panel A of Figure 10. At low ${\beta }_{{\rm{i}}}$, the electron irreversible heating efficiency shows nearly no dependence on ${\beta }_{{\rm{i}}}$ or temperature ratio. At high ${\beta }_{{\rm{i}}}$, the dependence on temperature ratio can be understood via the dependence of ${\beta }_{{\rm{i}},\max }$ on ${T}_{{\rm{e}}0}/{T}_{{\rm{i}}0}$. According to Equation (18), decreasing the temperature ratio for fixed ${\sigma }_{w}$ leads to an increase in ${\beta }_{{\rm{i}},\max }$, and so (as discussed in Section 4.2) in the value of ${\beta }_{{\rm{i}}}\sim {\beta }_{{\rm{i}},\max }$ where equipartition between electrons and protons is realized.

The effect of varying the temperature ratio for ${b}_{{\rm{g}}}=0.3$ and ${b}_{{\rm{g}}}=6$ is shown in panel B. As for antiparallel reconnection, there is no significant dependence on ${T}_{{\rm{e}}0}/{T}_{{\rm{i}}0}$ at low ${\beta }_{{\rm{i}}}$ for each of the two ${b}_{{\rm{g}}}$ values. While ${\beta }_{{\rm{i}},\max }=2.5$ for ${T}_{{\rm{e}}0}{/T}_{{\rm{i}}0}=1$, for ${T}_{{\rm{e}}0}{/T}_{{\rm{i}}0}=0.3$ we expect ${\beta }_{{\rm{i}},\max }\approx 3.85$, so equipartition between electrons and protons, which should hold regardless of ${b}_{{\rm{g}}}$ at ${\beta }_{{\rm{i}}}\sim {\beta }_{{\rm{i}},\max }$, is expected at a higher ${\beta }_{{\rm{i}}}$ than is probed in panel (B).

The effect of varying the magnetization and guide field strength is shown in panel C of Figure 10. At low ${\beta }_{{\rm{i}}}$, the electron irreversible heating efficiency has a weaker dependence on the guide field for ${\sigma }_{w}=1$ than for ${\sigma }_{w}=0.1$. For ${\beta }_{{\rm{i}}}\sim {\beta }_{{\rm{i}},\max }\propto {\sigma }_{w}^{-1}$ (see Equation (18)), irreversible heating of electrons and protons is in equipartition, and this conclusion holds regardless of ${\sigma }_{w}$ or ${b}_{{\rm{g}}}$.

For use as a sub-grid model of electron heating in magnetohydrodynamic simulations (as in Ressler et al. 2017; Chael et al. 2018; Ryan et al. 2018), we provide the following fitting formula, motivated by the simulation results presented in Sections 4.3 and 4.4:

$$\begin{eqnarray}&&\begin{array}{l}{q}_{u{\rm{e}},\mathrm{irr},\mathrm{fit}}({\beta }_{{\rm{i}}},{b}_{{\rm{g}}},{T}_{{\rm{e}}0}{/T}_{{\rm{i}}0},{\sigma }_{w})=\displaystyle \frac{1}{2}\left(\tanh \left(0.33\,{b}_{{\rm{g}}}\right)-0.4\right)\\ \quad \times \,1.7\tanh \left(\displaystyle \frac{{\left(1-{\beta }_{{\rm{i}}}/{\beta }_{{\rm{i}},\max }\right)}^{1.5}}{(0.42+{T}_{{\rm{e}}0}{/T}_{{\rm{i}}0}){\sigma }_{w}^{0.3}}\right)+\displaystyle \frac{1}{2},\end{array}\end{eqnarray} \tag{ 19 }$$

where ${\beta }_{{\rm{i}},\max }$ is in Equation (18) in terms of ${\sigma }_{w}$ and ${T}_{{\rm{e}}0}/{T}_{{\rm{i}}0}$.

The fitting function in Equation (19) has the following limits: for low ${\beta }_{{\rm{i}}}$, ${q}_{u{\rm{e}},\mathrm{irr},\mathrm{fit}}$ asymptotes to a (${\sigma }_{w}$- and ${b}_{{\rm{g}}}$-dependent) value that does not depend on ${\beta }_{{\rm{i}}}$. The asymptotic low-${\beta }_{{\rm{i}}}$ limit tends to the equipartition value ${q}_{u{\rm{e}},\mathrm{irr},\mathrm{fit}}\sim 0.5$ for ${\sigma }_{w}\gg 1$ (i.e., in the limit of ultra-relativistic reconnection), regardless of ${b}_{{\rm{g}}}$. Still, at ${\beta }_{{\rm{i}}}\ll 1$, electrons receive most of the irreversible heat if ${b}_{{\rm{g}}}\gtrsim 1.3$. For ${b}_{{\rm{g}}}\gg 1$, ${\sigma }_{w}\ll 1$ and ${\beta }_{{\rm{i}}}\ll 1$, we get ${q}_{u{\rm{e}},\mathrm{irr},\mathrm{fit}}\approx 1.0$, i.e., all of the irreversible heat goes to electrons. At ${\beta }_{{\rm{i}}}\sim {\beta }_{{\rm{i}},\max }$, the fitting function returns ${q}_{u{\rm{e}},\mathrm{irr},\mathrm{fit}}\approx 0.5$, independent of ${b}_{{\rm{g}}}$, ${\sigma }_{w}$, and ${T}_{{\rm{e}}0}{/T}_{{\rm{i}}0}$. For ${b}_{{\rm{g}}}$ in the range 0–6, the fitting function in Equation (19) is plotted in Figures 9 and 10 as dotted lines, showing that it matches the trends obtained from the simulations well.

Predictions of the reconnection-mediated heating model presented here differ from those of heating via a Landau-damped turbulent cascade (Howes 2010; Zhdankin et al. 2018; Kawazura et al. 2019). In Figure 11 we show a comparison between reconnection-based heating (Equation (19)) for the antiparallel (${b}_{{\rm{g}}}=0$, panel A) and strong guide field (${b}_{{\rm{g}}}=6$, panel B) cases, and the turbulence-based heating prescription of Kawazura et al. (2019; panel C) over the range of plasma conditions we have investigated. First, one notes that turbulence-based heating is much more similar to heating via reconnection in the strong guide field limit than in the antiparallel case. In fact, for the latter (in contrast to the first two), protons are heated much more than electrons at low ${\beta }_{{\rm{i}}}$. However, some differences persist even between turbulent heating and heating via strong guide field reconnection. In fact, the turbulence-based heating model is nearly insensitive to the initial temperature ratio ${T}_{{\rm{e}}0}{/T}_{{\rm{i}}0}$, whereas for guide field reconnection, an increase in ${T}_{{\rm{e}}0}{/T}_{{\rm{i}}0}$ decreases ${\beta }_{{\rm{i}},\max }$ (see Equation (18)), which in turn decreases the value of ${\beta }_{{\rm{i}}}$ at which electrons and protons achieve equipartition, i.e., ${q}_{u{\rm{e}},\mathrm{irr}}\sim 0.5$. More generally, relativistic effects leave a unique fingerprint in our results at ${\beta }_{{\rm{i}}}\sim {\beta }_{{\rm{i}},\max }$,¹² where both species start as relativistically hot, and in the limit ${\sigma }_{w}\gg 1$. In either case, protons and electrons receive an equal amount of the dissipated energy, i.e., ${q}_{u{\rm{e}},\mathrm{irr}}\sim 0.5$, regardless of the guide field strength.

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Guide field reconnection can result in highly anisotropic electron distribution functions at late times (Dahlin et al. 2014; Numata & Loureiro 2015). To determine the dimensionless internal energy per particle in the fluid rest frame, however, we have assumed an isotropic stress-energy tensor at every location in the upstream and in the downstream. Equation (10) relies on this assumption. In addition, we have implicitly assumed isotropy in our prediction for the amount of adiabatic heating.

To assess whether isotropy is a reasonable assumption, we show in Figure 12 the electron temperature anisotropy ${T}_{{\rm{e}},\parallel }/{T}_{{\rm{e}},\perp }$ in the island (∥ and ⊥ refer to orientations relative to the local magnetic field); the simulations here are similar to the production runs listed in Table 1, but cover ${\beta }_{{\rm{i}}}$ more densely in the range $8\times {10}^{-3}$ up to 2.

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For weak guide fields (${b}_{{\rm{g}}}\lesssim 0.3$), the electron temperature is isotropic, ${T}_{{\rm{e}},\parallel }/{T}_{{\rm{e}},\perp }\approx 1$ (see RSN17). For ${b}_{{\rm{g}}}\gtrsim 0.6$ and ${\beta }_{{\rm{i}}}\lesssim 0.5$, we find substantial anisotropy, with temperature ratios in the range ${T}_{{\rm{e}},\parallel }/{T}_{{\rm{e}},\perp }\approx 2\mbox{--}27$. In these cases, isotropy is certainly not a valid assumption. As discussed above, this will affect our inferred internal energy (because in principle, Equation (10) cannot be employed) and the predicted degree of adiabatic heating. With regard to the internal energy, in a few cases we have calculated all the components of the stress-energy tensor in the simulation frame. By transforming into the comoving frame, we do not need to rely on any assumption of isotropy. In general, we have found that the inferred internal energies differ from Equation (10) only at the $\sim 10 \%$ level.

With regard to adiabatic heating, we have discussed in Section 4.3 that compressive heating is suppressed by strong guide fields, as well as at low values of ${\beta }_{{\rm{i}}}$. Therefore, in the majority of cases that show substantial temperature anisotropy, adiabatic heating constitutes a negligible fraction of the total heating, so the degree of anisotropy only has a negligible effect on the inferred irreversible heating. A notable exception here is the run with ${\beta }_{{\rm{i}}}\approx 0.1$ and ${b}_{{\rm{g}}}=1$, for which compressive heating accounts for about 33% of the total heating, and the measured anisotropy in the island is non-negligible, ${T}_{{\rm{e}},\parallel }/{T}_{{\rm{e}},\perp }\approx 2;$ of all our simulations, this one has the greatest systematic uncertainty on the compressive heating and consequently, on the inferred irreversible heating.

The orbit of a charged particle in electromagnetic fields may be approximated as the superposition of two motions: fast circular motion about a point, the guiding center, and a slow drift of the guiding center itself. This approximation is valid when the particle's gyroperiod is shorter than the variation timescale of the fields, and also when the particle's Larmor radius is smaller than the field gradient length scale. When valid, the guiding center approximation can provide valuable insight into the mechanisms responsible for particle energization (see, e.g., Dahlin et al. 2014; Sironi & Narayan 2015; Wang et al. 2016). In this section, we use the guiding center approximation to investigate the mechanisms of electron heating for ${\beta }_{{\rm{i}}}\sim 0.01$ as a function of the guide field strength. Details of the guiding center decomposition are discussed in Appendix B.

We track $\sim {10}^{4}$ electrons starting initially in the upstream region (see Figure 13, panel A) and compute the contributions ${\rm{\Delta }}{\varepsilon }_{{E}_{\parallel }}$ and ${\rm{\Delta }}{\varepsilon }_{\mathrm{curv}}$, which correspond to energy changes due to the parallel electric field and curvature drift, respectively. For clarity we focus in our discussion only on the E-parallel and curvature drift terms, which tend to dominate for the cases we investigate here (we have directly verified this, and it agrees with findings of Dahlin et al. (2014) for nonrelativistic reconnection). While the simulation time-step is ${\rm{\Delta }}t\,\approx 0.1\,{\omega }_{\mathrm{pe}}^{-1}$, the time interval we use here for outputs of the field and electron properties for the guiding center analysis is around ${\rm{\Delta }}{t}_{\mathrm{out}}\approx 3\,{\omega }_{\mathrm{pe}}^{-1}$. To ensure that this time resolution is sufficient for a guiding center reconstruction, we compare the actual evolution of the electron energy (computed on the fly by the simulation) to the value calculated from the downsampled field and particle information. For the time range over which we track particles ($\sim 3700\,{\omega }_{\mathrm{pe}}^{-1}\approx 1\,{t}_{{\rm{A}}}$ for these simulations), the energy gain computed from downsampled field and particle information shows excellent agreement with the actual value evolved at the time resolution of the simulation.

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To study electron heating via the guiding center theory, we use four simulations for which ${\beta }_{{\rm{i}}}=7.8\times {10}^{-3}$ and ${b}_{{\rm{g}}}\in \{0.1,0.3,0.6,1\}$. Here, we use a smaller box size, ${L}_{x}\approx 1080\,c/{\omega }_{\mathrm{pe}}$ (the domain size dependence of our results is discussed in Appendix A). Except for the domain size, the parameters are the same as in the main guide field simulations (i.e., ${m}_{{\rm{i}}}/{m}_{{\rm{e}}}=1836$, ${\sigma }_{w}=0.1$, ${T}_{{\rm{e}}0}{/T}_{{\rm{i}}0}=1$, and $\,c/{\omega }_{\mathrm{pe}}=4$ cells, ${N}_{\mathrm{ppc}}=16$). The heating fractions extracted from these simulations are roughly the same as in the production runs of Table 1.

Electrons are tracked from $t/\,{t}_{{\rm{A}}}\approx 0.9$ to 1.9 (equivalently, $t{\omega }_{\mathrm{pe}}\approx 3330$ to 7030). The tracked particles are selected at the initial time to lie in the upstream region, within roughly $\pm 50\,c/{\omega }_{\mathrm{pe}}$ of y = 0 (see Figure 13, panel A; gray contours show magnetic field lines). The selected electrons are tracked for $\sim 3700\,{\omega }_{\mathrm{pe}}^{-1}\approx 1\,{t}_{{\rm{A}}}$, at which point they typically reside in the island region (panel B).

Figure 14 shows the time evolution of electron energy gains for guide fields ${b}_{{\rm{g}}}=0.1,0.3,0.6$, and 1 in panels A, B, C, and D, respectively (the strength of the guide field increases from left to right). The energy gain is presented in dimensionless form with rest mass subtracted, i.e., $({\varepsilon }_{{\rm{e}}}-{m}_{{\rm{e}}}{c}^{2})/{m}_{{\rm{e}}}{c}^{2}\cong {\upsilon }_{{\rm{e}}}$.¹³ In each panel, the blue line corresponds to the electron energy gain measured directly in the simulations. The E-parallel and curvature terms are shown in green and red, respectively, and the yellow dashed curve is their sum. The good agreement between dashed yellow and blue lines is an indication that our output time resolution is adequate for the guiding center reconstruction. The black dashed line shows the specific internal energy in the far upstream (${\upsilon }_{{\rm{e}}0}\approx 1.6$), which matches the starting point of the curves well.

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For weak guide fields, ${b}_{{\rm{g}}}\lesssim 0.6$, the energy gains due to E-parallel and curvature terms are comparable, consistent with the findings of Dahlin et al. (2014). For strong guide fields, energization due to the parallel electric field dominates; in this case, the magnetic field in the current sheet is approximately straight (because it is dominated by the out-of-plane field), so heating due to the curvature term is negligible. Though the mechanisms responsible for energization of electrons differ for weak and strong guide fields, the overall energy gain is about the same in all cases, $({\varepsilon }_{{\rm{e}}}-{m}_{{\rm{e}}}{c}^{2})/{m}_{{\rm{e}}}{c}^{2}\cong {\upsilon }_{{\rm{e}}}\approx 14.5$ (see also Figure 7, panel A). The temporal evolution of electron heating (both the total heating and E-parallel and curvature contributions) saturates at late times, when most of the particles reside in the primary island.

Figure 15 shows the 2D spatial distribution of power associated with the E-parallel (panels A–D) and curvature (panels E–H) energization terms. For every tracked electron at each time, we deposit the corresponding E-parallel and curvature powers at the location where the particle instantaneously resides (power is deposited into spatial bins of length and width equal to $2\,c/{\omega }_{\mathrm{pe}};$ note that the color bar range in Figure 15 depends on this binning, so the units are arbitrary), and then we average over the number of tracked electrons. Gray lines show the magnetic field lines at $t/{t}_{{\rm{A}}}\approx 1.9$, for reference. For weak guide fields (${b}_{{\rm{g}}}\lesssim 0.3$), energization due to the parallel electric field is patchy (Dahlin et al. 2014), with heating spread over the exhaust region as well as the island. On average, there is a net energy gain, but parallel electric fields can also locally cool the electrons (blue patches in panel A). Heating due to the curvature drift is localized predominantly along the walls of the exhaust, in particular on the upper left and lower right (as in panel B, for ${b}_{{\rm{g}}}=0.3$), where outflowing electrons tend to become focused (see also Figure 1, panel F).

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As the strength of the guide field increases, the relative importance of the curvature drift energization decreases (panels G–H), and the E-parallel heating becomes dominant (panels C–D). A substantial amount of heating due to the parallel electric field is localized in the exhaust region, but energization continues into the island.

While the guiding center formalism makes no distinction between adiabatic and irreversible heating, we can infer based on our results for low ${\beta }_{{\rm{i}}}$ guide field reconnection (see Figure 8, first row) that the E-parallel and curvature drift terms in this case (having ${\beta }_{{\rm{i}}}\sim 0.01$) predominantly contribute to the irreversible heating of electrons. Since compressive heating is negligible at ${\beta }_{{\rm{i}}}\approx 8\times {10}^{-3}$ (see Figure 8, panel B; also, Equation (16)), irreversible heating in the low ${\beta }_{{\rm{i}}}$ regime represents the main contribution to total electron heating. It follows that in this low ${\beta }_{{\rm{i}}}$ regime, the guiding center decomposition assesses contributions to irreversible heating.

To clarify the spatial dependence of E-parallel and curvature drift heating, we show in the last row of Figure 15 (panels I–L) the 1D cumulative sum along $\pm x$ (as in Dahlin et al. 2014), starting from the vertical dashed line, of the E-parallel (solid green) and curvature (solid red) energization rates displayed in the first and second rows; their sum is shown by the dashed yellow line. This shows that heating continues throughout the exhaust region and at the interface between the outflow and the primary island. Little additional heating occurs inside the primary island.