Scaling of Magnetic Reconnection Electron Bulk Heating in the High-Alfvén-speed and Low-β Regime of Earth's Magnetotail

Figures 1(a)–(h) show MMS3 observations from 2017 July 17, 14:38–14:49 UT, when the spacecraft was located in the plasma sheet at XYZ_GSM = (−21.9, 6.8, 2.2)R_E and encountered a high-speed tailward-to-earthward flow reversal. This event was previously reported by Oka et al. (2022). An interval of tailward-directed (negative V_X ) high-speed flows was observed from 14:38:30 to 14:40:30 UT, followed by an interval of earthward (positive V_x ) high-speed flow at 14:45:30–14:49 UT (Figure 1(d)). Intervals in the tailward and earthward high-speed flow where the outflow velocity was relatively constant are marked with pairs of vertical red lines and discussed in more detail below. These observations of a tailward-to-earthward jet reversal with an associated predominantly negative to predominantly positive B_Z reversal (the red curve in Figure 1(a)) are consistent with a reconnection X-line moving tailward past the spacecraft, resulting in an approximate spacecraft trajectory as indicated in Figure 1(i).

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Between the tailward and earthward jets, the spacecraft encountered a region of large ∣B_X ∣ (the blue curve in Figure 1(a)), lower density (Figure 1(f)), and no perpendicular V_X flow (Figure 1(e)), suggesting that the spacecraft encountered the reconnection inflow region during the passage of the X-line (Figure 1(i)).

To estimate the electron heating in the two (tailward and earthward) reconnection exhausts, we first selected appropriate outflow intervals. We are interested in direct heating in the fully developed exhaust, which is where the outflow jet has reached a quasi-steady level. Selecting an interval of quasi-steady outflow helps minimize the effects of changing inflow conditions. Thus, for each exhaust, we chose an interval of relatively stable outflow speed. Furthermore, to ensure a fair comparison between exhaust intervals from different events, we restricted the outflow observations to near the neutral sheet, by requiring the outflow velocity to have a significant component perpendicular to the magnetic field and a small ∣B_X ∣ (<10 nT). The two selected intervals in the tailward and earthward exhausts are marked with pairs of red vertical lines. The average and standard deviation of the tailward and earthward exhaust electron temperatures were 928 ± 13 eV and 769 ± 15 eV, respectively.

Next, we identify an asymptotic (i.e., unmodified by reconnection) inflow interval that may correspond to the selected outflows. As mentioned above, the spacecraft likely encountered the inflow region between the tailward and earthward jets. The inflow interval should have large and stable ∣B_X ∣, density lower than in the outflows, and no field-aligned electron temperature anisotropy. The latter condition is to avoid the inflow regions that have been modified by reconnection, such as field-aligned electron heating (e.g., Chen et al. 2008; Egedal et al. 2008; Shay et al. 2016). The interval between the two vertical blue lines in Figures 1(a)–(h) satisfies these criteria. The relatively high inflow density of 0.086 cm⁻³ (which was still lower than the outflow density of ∼0.2 cm⁻³, thus satisfying the conditions stated above) and electron temperature (∼415 eV) in this event indicate that the inflow was plasma sheet plasma, not lobe. Furthermore, the inflow interval had T_e∣∣ ∼ T_e⊥. The inflow Alfvén speed was 1488 km s⁻¹, based on density ∼ 0.086 cm⁻³ and ∣B_X ∣ ∼ 20 nT, and the inflow β_e was 0.036.

The average electron temperature in the selected inflow interval was 415 eV (Figure 1(g)). Thus, the electron temperature increase, ΔT_e = T_e,outflow – T_e,inflow, was 510 ± 13 eV for the earthward exhaust and 350 ± 15 eV for the tailward exhaust.

A second example of a high-speed flow reversal event is shown in Figures 2(a)–(h). On 2019 August 8, at 13:54 UT, MMS was located in the magnetotail at XYZ_GSM = (−27.6, 0.5, 5.1)R_E when it observed a high-speed V_X flow reversal; a tailward (negative V_X ) jet (13:54–14:00 UT) followed by an earthward (positive V_X ) jet (14:15–14:27 UT; Figure 2(d)), with a correlated predominantly negative to predominantly positive B_Z reversal (Figure 2(a), red curve), consistent with a reconnection X-line moving tailward past the spacecraft. Based on B_X and V_X time profiles, the effective spacecraft trajectory is shown in Figure 2(k) (blue dashed curve). Both the earthward and tailward flows contained intervals where V_X had a component perpendicular to the magnetic field (Figure 2(e)) and ∣B_X ∣ < 10 nT (Figure 2(a)), implying that the spacecraft observed the jets in the vicinity of the neutral sheet, away from the plasma sheet boundary layer (PSBL) during those times. Electron temperatures of 500–600 eV were observed in the tailward jet, while the electron temperature in the earthward jet was higher, at 1–3 keV (Figure 2(g)).

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As with the previous example (Section 3.1), to evaluate the electron heating we selected exhaust intervals where the ion outflow velocity had reached a stable level, had a significant V_X component perpendicular to the magnetic field, and small ∣B_X ∣. The selected outflow intervals are marked with red vertical lines in Figures 2(a)–(h) in both the tailward and earthward exhaust.

The average electron temperature in the selected tailward exhaust interval was 521 ± 13 eV. To estimate the heating in this exhaust, we compare with the closest inflow interval. Between the tailward and earthward jet (∼14:00–14:15 UT), the spacecraft entered a region of large B_X values (mostly >10 nT), consistent with the spacecraft being in the vicinity of the inflow region (Figure 2(k)). The density and temperature in this interval varied, and earthward-directed (positive V_x ) field-aligned flows were observed intermittently, indicating that the spacecraft intermittently encountered the outer edge of the exhaust in the PSBL. However, there were two brief intervals (between pairs of vertical blue lines) within this large-B_X region where the peak electron energy flux was below ∼200 eV and density <0.05 cm⁻³, and with no earthward flows, indicating encounters with the cold magnetotail lobe.

One of the lobe intervals was observed adjacent to the tailward exhaust, and we use this interval to estimate the upstream conditions for the tailward exhaust interval. Level 2 electron moments are not available, due to large photoelectron effects in this cold and tenuous lobe. We therefore obtained the inflow electron temperature by fitting the average omnidirectional electron distribution measurements with a Maxwellian distribution, as shown in Figure 2(i). When fitting the distributions, we excluded data points contaminated by photoelectrons at low energies and data points at high energies where the error bars extend below the detection limit (the one-count level) of the instrument (Oka et al. 2022). The best-fit Maxwellian distribution corresponds to an electron temperature of 89 eV, which we used as an estimate for the inflow temperature, in lieu of electron moments. From this, we found the electron temperature increase in the tailward exhaust interval to be 432 ± 13 eV. The inflow Alfvén speed in this lobe interval was 2285 km s⁻¹, with 0.033 cm⁻³ inflow density and B_X = 18.9 nT. Inflow β_e was 0.0033.

The average electron temperature for the earthward exhaust interval was 1390 ± 272 eV. We avoid the PSBL interval, as discussed above, and use the second lobe encounter at ∼14:06:30–14:07:00 UT (marked by the vertical blue lines in Figures 2(a)–(h)) to estimate the upstream conditions for the earthward exhaust. Again, we estimate the electron temperature in this lobe interval by Maxwellian fitting (Figure 2(j)) and obtain a lobe inflow temperature of 83 eV. Thus, the estimated electron temperature increase was 1307 eV ± 272 for the earthward flow interval. For this lobe interval, the inflow Alfvén speed was 2128 km s⁻¹, the inflow density was 0.031 cm⁻³, B_X = 17.3 nT, and β_e was 0.0035.

The goal of the present study is to investigate direct heating by reconnection in the fully developed exhausts, not by secondary heating processes farther downstream of the X-line, e.g., in the flow-breaking region (e.g., Runov et al. 2011). To ensure that we are observing the exhaust away from the flow-breaking region, we restricted the data set to plasma jet reversals, like those presented above in Section 3. Such events are generally associated with the passage of a reconnection X-line, so that the observations are away from the flow-breaking region, which is located far downstream.

The events were identified based on the presence of tailward-to-earthward fast plasma sheet ion flow reversals with correlated negative to positive B_Z reversals. These signatures indicate the tailward retreat of a reconnecting X-line, which is a typical behavior of magnetotail X-lines (Hones 1979). The jet reversal events were identified using 4 yr of MMS3 magnetotail observations from 2017 to 2020. We found 74 events that satisfied these initial criteria. We then imposed several additional selection criteria to ensure that the selected outflow and inflow intervals were appropriate for the heating study.

For the outflow intervals, we required the following:

• 1.  
  The interval should be relatively close to the flow reversal to avoid effects from additional heating processes farther downstream.
• 2.  
  The interval should have a relatively stable outflow speed, to be in the fully developed exhaust and to avoid sampling the diffusion region.
• 3.  
  The interval should have a significant V_X component perpendicular to the magnetic field and ∣B_X ∣ < 10 nT. This ensures that the observations are obtained relatively close to the current sheet midplane for fair comparisons between exhaust intervals from different events.

For the inflow intervals, we required the following:

• 1.  
  The interval should be observed as close to the selected exhaust interval as possible, but should not contain any parts of the exhaust itself.
• 2.  
  The interval should have a large and stable ∣B_X ∣ > 10 nT.
• 3.  
  The density should be less than the exhaust density.
• 4.  
  The interval should have an isotropic electron temperature (T_e∣∣ ∼ T_e⊥) to ensure that one is sampling the asymptotic inflow, rather than the modified inflow, where preheating of electrons takes place (Chen et al. 2008; Egedal et al. 2008; Shay et al. 2016).

In some cases, the inflow was not observed, it was observed too far from the outflow, or two or more candidate inflow intervals with different plasma properties were observed near the outflow and we could not determine which one corresponded to the selected outflow interval. Such events were not included.

After applying these criteria, we were left with 14 high-speed flow reversals where at least one, and often both, of the bidirectional jets and their associated inflows satisfied the selection criteria. This resulted in a total of 21 exhaust events, where 10 of the exhausts were directed tailward and 11 earthward. Most (18 of 21) inflow observations, including the two examples shown above (Section 3), were obtained when the spacecraft entered the inflow region between the tailward and the earthward jets. The 21 events are listed in Table 1.

In the magnetotail, spacecraft typically pass through a reconnection event in an along-the-current sheet trajectory (as opposed to across the current sheet at the magnetopause). Combined with the dynamic nature of tail reconnection, this can make it challenging to deduce which inflow and outflow intervals are associated with each other. Even with the strict selection criteria, which left us with only 21 events, we cannot be 100% certain that every selected outflow is associated with the selected inflow or that the inflow did not change during the time between the two observations. These uncertainties are discussed in more detail below, in Section 4.4. The selected events (Table 1) thus constitute our best attempt to pair up inflow and outflow intervals.

Some of the events have been reported previously. For example, the exhausts in events 1 and 2 (Table 1) are from the flow reversal associated with the electron diffusion region encounter reported by Torbert et al. (2018). As mentioned above, we are interested in the fully developed reconnection jet downstream of the X-line in this study, and we do not include observations inside the diffusion region, which was the focus of the Torbert et al. (2018) study. The well-studied tail reconnection event of 2017 July 26 (Ergun et al. 2018, 2020a, 2020b) could not be included in our study because we were unable to identify an unperturbed inflow interval.

In near-Earth magnetotail reconnection, the guide field is typically small (<0.2), and the inflow is generally symmetric on the two sides of the magnetotail current sheet. Thus, the reconnection geometry in the events presented here is different from the magnetopause reconnection exhausts studied by Phan et al. (2013), where the inflow was asymmetric and the guide field varied between zero and one.

Figure 3(a) shows the electron temperature change from inflow to outflow, ΔT_e, versus the inflow Alfvén speed based on the X_GSM (roughly the reconnecting) component of the magnetic field (V_Ax,in) for the 21 magnetotail reconnection exhausts in the data set described above (Section 4.1; see also Table 1). Events with lobe inflow are marked in blue and are generally characterized by low inflow density (Figure 3(c)) and low inflow electron temperature (Figure 3(e)). Events with plasma sheet inflow are in black and are characterized by higher inflow electron temperatures. ΔT_e is well correlated with V_Ax,in, but the dependence is not linear. To deduce the power of the V_Ax,in dependence, we fitted the data to the function ΔT_e = constant ∗ ${{V}_{\mathrm{Ax},\mathrm{in}}}^{\mathrm{POWER}}$. The best fit (the black dashed curve in Figure 3(a)) gives the relation ΔT_e = 1.77 × 10⁻⁴ ${{V}_{\mathrm{Ax},\mathrm{in}}}^{2.02}$, indicating that ΔT_e scales with ${{V}_{\mathrm{Ax},\mathrm{in}}}^{2}$. Refitting the data to ${{V}_{\mathrm{Ax},\mathrm{in}}}^{2}$ gives ΔT_e = 2.06 × 10⁻⁴ ${{V}_{\mathrm{Ax},\mathrm{in}}}^{2}$ (the red dashed curve in Figure 3(a)), where ΔT_e and V_Ax,in are in units of eV and km s⁻¹, respectively.

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Since ${{m}_{{\rm{i}}}{V}_{\mathrm{Ax},\mathrm{in}}}^{2}$ represents the available magnetic energy per particle from both sides of the current sheet, the dependence on ${{V}_{\mathrm{Ax},\mathrm{in}}}^{2}$ means that there is a linear dependence between the available magnetic energy per particle and the temperature increase (Figure 3(b)). Linear fitting of the data gives the empirical relation:

$$\begin{eqnarray}&&{\rm{\Delta }}{T}_{{\rm{e}}}=0.020{m}_{{\rm{i}}}{{V}_{\mathrm{Ax},\mathrm{in}}}^{2}.\end{eqnarray} \tag{ 2 }$$

This relation is remarkably similar to the relation obtained using magnetopause observations, ΔT_e = 0.017 m_i ${{V}_{\mathrm{AL},\mathrm{asym}}}^{2}$ (Phan et al. 2013), which was for a much lower magnetic energy per particle regime and for highly asymmetric reconnection. The empirical relation in (2) implies that the electron temperature increase is ∼2% of the available magnetic energy per particle. In terms of energy partition, for the simplified case of isotropic plasmas, the percentage of the inflowing Poynting flux per particle density flux converted into electron enthalpy flux is given by [γ/(γ − 1)]ΔT_e/m_i ${{V}_{\mathrm{Ax},\mathrm{in}}}^{2}$ = 0.020 γ/(γ − 1) = 5%, where γ = 5/3 is the ratio of the specific heats.

We also performed linear least-squares fitting for the 12 data points with plasma sheet inflow and the nine events with lobe inflow separately. The linear fit for the plasma sheet events (black dots) gives the relation ΔT_e = 0.018 m_i ${{V}_{\mathrm{Ax},\mathrm{in}}}^{2}$, while for the lobe events (blue dots), one obtains ΔT_e = 0.020 ${m}_{{\rm{i}}}{{V}_{\mathrm{Ax},\mathrm{in}}}^{2}$. Thus, the scalings of the two separate groups of events are essentially the same as the combined result.

We have also examined how ΔT_e might depend on the individual inflow parameters B_x,in, N_in, and T_e,in. No discernible relationship is seen between ΔT_e and B_x,in (approximately the reconnecting field) in this data set (Figure 3(d)). However, the range of ∣B_x,in∣ values in this data set (∼15 nT to ∼30 nT) could be too small to reveal any dependency. The inflow density, on the other hand, covers more than one order of magnitude (Figure 3(c)) and shows an inverse relationship with the temperature increase: events with large electron temperature increases (say >1 keV) have inflow density <0.06 cm⁻³. This correlation is understandable, since a lower inflow density means more magnetic energy per particle is available to heat the particles.

Figure 3(e) shows the electron temperature change versus inflow electron temperature. The highest heating events are seen at low inflow electron temperatures. However, this apparent relationship should be viewed with care, since the high heating events correspond to the inflows being in the low-density lobe (marked in blue), where the inflow electron temperature tends to be low.

In terms of the dependence on inflow β_e, Figure 3(f) shows that there is an inverse relation between inflow β_e and ΔT_e.

We have also investigated the parallel and perpendicular electron temperature changes separately. Figure 3(g) shows that the parallel heating was slightly more than the perpendicular heating, with ΔT_e∣∣,out = 1.07 ΔT_e⊥,out (the dashed line in Figure 3(g)). The near-isotropic electron heating for this low-guide-field magnetotail data set is in agreement with previous studies showing low heating anisotropy for low-guide-field cases (Chen et al. 2008; Phan et al. 2013; Li et al. 2018).

When selecting inflow intervals, one criterion was isotropic inflow electron temperature, to avoid sampling any part of the inflow that has been modified (preheated) by reconnection (Chen et al. 2008; Egedal et al. 2008). Figure 3(h) confirms that T_e∣∣ ∼ T_e⊥ for all inflow intervals.

We examined (in Figure 4(a)) whether the percentage of heating relative to the available magnetic energy per particle is dependent on the inflow electron temperature and found no clear dependence. The data in Figure 4(a) also provide an alternative way to infer the coefficient in relation (2). The average and standard deviation of ΔT_e/m_i ${{V}_{\mathrm{Ax},\mathrm{in}}}^{2}$ are 0.019 ± 0.005. The standard deviation of 0.005 suggests that relation (2) is statistically the same as the magnetopause expression in relation (1).

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Figure 4(b) shows electron heating versus available magnetic energy per particle with both parameters normalized to the inflow temperature T_e,in. The linear fit (dashed curve) to the normalized data is excellent, with ΔT_e/T_e,in = 0.021 ${{m}_{{\rm{i}}}{V}_{\mathrm{Ax},\mathrm{in}}}^{2}$/T_e,in, i.e., nearly identical to the relation for the unnormalized values (Figure 3(b)). While the similarity between the normalized and unnormalized empirical relations is somewhat trivial, the positions of the data points are reordered in Figure 4(a) due to the normalization to the individual inflow T_e for each event.

There is also a linear dependence between ΔT_e/T_e,in and the inflow β_e, with ΔT_e/T_e,in inversely proportional to β_e (Figure 4(c)). This relation is equivalent to the relation in Figure 4(a) (with a factor of a half difference) since $1/{\beta }_{{\rm{e}},\mathrm{in}}={{m}_{{\rm{i}}}{V}_{\mathrm{Ax},\mathrm{in}}}^{2}$/2T_e,in.

Figure 4(d) shows electron heating versus the ratio of exhaust (outflow) density to inflow density. Interestingly, six of the lobe inflow events have a density ratio larger than 4. A density increase from inflow to exhaust by more than a factor of 4 is not expected of density compression across traditional switchoff slow shocks at the exhaust boundaries (Lin & Lee 1993). While we cannot entirely exclude the possibility that the large density ratios could be a side effect of mismatched inflow and outflow intervals (see Sections 4.1 and 4.4), another possible explanation is that the large density ratios are a consequence of additional density compression occurring in the low-β magnetotail exhaust. For example, it has been well established that when magnetic islands develop during reconnection, significant compression of the density takes place, leading to higher outflow densities (Drake et al. 2006; Li et al. 2018).

The error bars in Figure 3 represent the standard deviations for the data in the observed intervals. These error bars should be interpreted with caution, since there are additional uncertainties that cannot be quantified.

One important source of uncertainty arises from the fact that the outflow and the inflow are not observed simultaneously, but with a delay of 1–9 minutes (Table 1). In our analysis, we have assumed that the inflow conditions remained stable for this length of time; however, this stability cannot be checked. To minimize this uncertainty, we selected only intervals where the outflow speed is stable for several minutes.

Another source of uncertainty is the choice of inflow interval. In some events, two or more candidate inflow intervals were observed near the outflow. For consistency, we chose the inflow interval observed closest to the selected outflow interval.