Energy Dissipation in Electron-only Reconnection

Magnetic reconnection is a fundamental process in space and astrophysical plasmas that converts magnetic energy to particle energy. Recently, a novel kind of reconnection, called electron-only reconnection, has been observed in Earth's magnetosheath plasma. A defining characteristic of electron-only reconnection is that electron jets are observed but ion jets are absent. This is in contrast with traditional ion-coupled reconnection, where both ions and electrons exhibit outflowing velocity jets. Findings from the Magnetospheric Multiscale mission observations and particle-in-cell simulations show clear signatures of electron heating in electron-only reconnection events, while ions are not heated or cooled in these events. This result is unlike ion-coupled reconnection, where both ions and electrons are heated to varying degrees. The ratio of electron to ion dissipation increases with the local magnetic curvature, indicating that the partition of heat into ions and electrons is dependent on the current-sheet thickness.


Introduction
Magnetic reconnection is a prevalent phenomenon, observed in astrophysical plasmas that converts energy stored in the magnetic fields into kinetic energy of the constituent particles.In the standard model of reconnection, both ions and electrons participate in the reconnection process and are energized.However, recently some events were observed where the electrons participate in the reconnection but the ions do not (Phan et al. 2018).This new kind of reconnection was termed "electron-only reconnection" because of the absence of ion participation.In this study, we investigate the transition from the standard ion-coupled reconnection to electron-only reconnection from the energy dissipation perspective, measured by the pressure-strain interaction (PS; Yang et al. 2017).The paper is structured as follows.
Section 2 provides the relevant theoretical background.In Section 3, we present two case studies of PS analysis in magnetic reconnection events observed by the Magnetospheric Multiscale (MMS) spacecraft (Burch et al. 2016); one of these cases is an electron-only event and the other is an ion-coupled event.In the ion-coupled event, we find that there are signals in both the ion and electron PS interaction rates.However, in the electron-only event, we detect a signal only in the electron PS.This indicates that the PS rates may be used to distinguish whether a reconnection event is electron only or not.To explore the transition from ion-coupled to the electron-only regime, we perform a statistical study of the variation of the ratio of heating rates obtained using pressure strain versus the magnetic field curvature, which is related to the magnetic island size.For the statistical study, we use ∼250 reconnection events identified by Stawarz et al. (2022).We find that the electron-toion heating ratio increases with the magnetic field curvature, implying that the events with higher magnetic field curvature are more likely to be electron only.
In Section 4, we further analyze the PS rates for ions and electrons in several 2.5D particle-in-cell (PIC) simulations and explore the dependence of the electron-to-ion heating ratio as it varies with the simulation box length as well as the magnetic field curvature near the X-points.Conclusions and final discussion are given in Section 5.

Background
In studies of reconnection events, j • E is commonly used as the dissipation measure (Zenitani et al. 2011).However, another approach is available if the interest is in the production of internal energy, defined as the second central moment in velocity space of the velocity distribution function.This moment may unambiguously be equated with nT, where n is the density and T is (defined as) the direction-averaged temperature.This characterization holds in kinetic theory in general, whether or not the plasma is in equilibrium, whether it is collisional or not, and whether it is isotropic or anisotropic.Defined this way, internal energy is conventionally also known as thermal energy, with no implication of equilibrium conditions.
By examining the moments of the Vlasov-Maxwell equations (Marshall 1957;Cerri 2016;Yang et al. 2017), one finds that the PS interaction term, −(P • ∇) • u, is actually responsible for the interconversion of the bulk flow energy ( s f  ) into the internal (thermal) energy ( s th  ) of the particles, where s denotes the species.We also recall that pressure strain can be meaningfully decomposed into a pressure-dilatation term, pθ, and a gyroviscous term, often called Pi-D.Mathematically, this can be expressed as , and D ij s The second term, on the right-hand side in Equation (1) is commonly referred to as Pi-D, with the negative sign included.In addition, each of these terms can be computed separately for each species s.Other kinds of decomposition of the pressure strain have been proposed (Cassak & Barbhuiya 2022).However, here we will use the approach shown in Equation (1) to carry out several complementary observational and simulation studies of the electron-only reconnection phenomenon.

Magnetospheric Multiscale Observations
The MMS mission (Burch & Phan 2016) has observed several electron-only reconnection events in recent years (Phan et al. 2018;Huang et al. 2021;Stawarz et al. 2022).Previous works have analyzed the nature of ion and electron dissipation at ion-coupled reconnection events (Bandyopadhyay et al. 2021;Pezzi et al. 2021;Burch et al. 2023).Here, we calculate the PS rates to understand the key differences and similarities of dissipation mechanisms between electron-only and ioncoupled reconnection events to make comparison.Figure 1 shows an electron-only reconnection event, while an ioncoupled reconnection event is shown in Figure 2. We find that for the electron-only reconnection event the electron PS values are enhanced near the electron diffusion region (or EDR), but no such enhancement can be seen for the ions.However, in the ion-coupled reconnection event enhancements in the PS rates for both ions and electrons are observed.The errors in the PS rates are computed following Roberts et al. (2023).Whenever the PS values cross the uncertainty level, we interpret that as an accurate signal of enhancement.Therefore, we conclude that there is a nonzero contribution to the dissipation rate if the signal lies outside the shaded region, which indicates the associated error.
Sharma Pyakurel et al. (2019) noted that the magnetic bubble (or island) size is an important parameter controlling the extent of ion participation in reconnection.Smaller bubbles will have smaller average radius of curvature of the field lines.Therefore, a quantity closely related to bubble size is the curvature of the magnetic field lines, and this can be easily measured to a good approximation in MMS observations.The magnetic field curvature is defined by where b = B/|B| is the unit vector in the local magnetic field direction.Therefore, a well-motivated hypothesis is that a higher value of curvature (smaller radius of curvature) would impede the ions from participating in the reconnection process.
In Figure 3, we utilize MMS observations to plot the ratio of electron to ion PS rates as a function of the average magnetic field curvature near the reconnection sites.To compute the average curvature, we use a window of 2 s around the reconnecting current sheet.
We see that the ratio of electron to ion heating rates increases as a function of the curvature, in accordance with our expectations that as the curvature increases, the ions would be less likely to participate.It is observed that the reconnection transitions from ion-coupled to electron only when the local We also explored additional parameters related to spacecraft observations that might be associated with the transition from ion-coupled to electron-only dynamics.One of these relevant parameters is the current-sheet thickness as a measure of the size of the reconnection region.However, we did not find a strong correlation between the heating ratio and the currentsheet thickness.It is important to emphasize here that the observations provide only limited access to regional diagnostics, even with the four MMS spacecraft.Simulations afford an opportunity to further verify these relations, as we see in the next section.

Simulations
We employ 2.5D PIC simulations to study the variation of the energy dissipating on ions versus that on electrons, as computed through the PS interaction.For this purpose, we use six reconnection simulations, the details of which are given in the Appendix, Table 1.Magnetic fields are normalized to B 0 , densities to n 0 , length scales to d i (ion inertial length), and timescales to ci 1 w -(inverse ion cyclotron frequency).Velocities and temperatures are normalized to the Alfvén speed v A and m v , A i 2 respectively.These simulations were performed using the P3D code (Zeiler et al. 2002).
To reduce the PIC noise, we smooth the simulation quantities (currents, densities, and the pressure tensors for ions and electrons) four times recursively using Gaussian filtering and then compute the pressure strain rates, which we then smooth five times.We also perform time-averaging on the spatially smoothed pressure strain rates to further reduce noise.The spatial smoothing was done by averaging over the eight neighboring points for each cell.Then, we further apply a Gaussian filter of standard deviation equal to 3 before plotting 2D maps of the pressure-strain interaction rates.The compressive component of the PS, p s θ s , is still dominated by the noise.Therefore, we only analyze the incompressive part,

P
, from the simulations.As described in Sharma Pyakurel et al. (2019), the simulations are initialized with two current sheets, with the magnetic field along x given by B , where w 0 is the half-width of the initial current sheets and B up is the inflowing reconnecting magnetic field.n up is the density outside the current sheets.The density is varied to maintain total pressure balance.A local magnetic perturbation initiates reconnection.The initial currents are due solely to electron flows.Temperatures are initially uniform.There is uniform large guide field B z = 8B up , with B up = 1 in code units, for all the simulations, except the first one, for which B z = 1B up .
The strategy we employ is to change the spacing between magnetic bubbles by changing the simulation box size.This determines the length of the current sheet, which then, as we argue, controls the regime of reconnection.Parameters for the simulations, including domain size, are shown in Table 1.More details about the runs can be found in Sharma Pyakurel et al. (2019).
The main simulation results are shown in Figures 4, 5, 6, and 7. We show the values of from two simulations performed in domain sizes of 2.56 d i (in Figure 4) and 41.92 d i (in Figure 5).We see that for the first simulation, the ion Pi-D is mostly without any clear signal, but for electrons there is a strong enhancement near the X-point.To elucidate this point, we take a cut along x = 0.65 d i and plot . In the second simulation, with larger box size, we notice enhancements of Pi-D for both ions and electrons.However, the enhancement for ions is smaller than for electrons.
To quantify the variation of the relative heating as a function of the box size, we integrate the PS rates over the reconnection region, and then compute the ratio P P and examine it as a function of the box size for all simulations reported in Table 1.The reconnection region is defined by enhancements of Pi-D and j • E, which tend to be well correlated.This is shown in Figure 6.We observe that as the box size is increased, the ratio of electron to ion heating decreases, indicating that for smaller box sizes, where the type of reconnection is electron only, the PS rates for electrons is larger than that for ions.However, as the box size is increased, the ratio becomes smaller, meaning that the ion heating gradually starts to dominate and eventually becomes larger than the electron heating around 10-15 d i .
Finally, we examine the dependence of electron to proton heating rate with the magnetic field curvature near the X-line in our simulations.To compute this, we integrate the magnetic field curvature in the reconnection region, and then plot the ratio as a function of this integrated curvature.The result is shown in Figure 3.A transition from standard ioncoupled reconnection to electron-only reconnection is observed , as a function of the magnetic field curvature at the X-line, computed from MMS observations.The data points have been binned into six equally spaced logarithmic bins.The error bars are computed from the standard deviation within each bin.A transition from reconnection to electron-only reconnection is observed near κ = 7 × 10 −3 km −1 .when the local magnetic field curvature is roughly This corresponds to a radius of curvature, R = 1/κ = 100 d i .It is clear that relatively stronger electron heating occurs when the magnetic curvature at the reconnection region is higher.This is consistent with the results from MMS shown above.

Discussion
We have explored the dissipation occurring in ion-coupled and electron-only reconnection events, and how this dissipated energy is distributed between protons and electrons.We quantify the dissipation using the PS interaction.This is in contrast to looking for enhancement in temperature or the electromagnetic dissipation measure, j E • ¢, which are frequently used to describe dissipation in reconnection events (e.g., Shi et al. 2022).The advantage of using the PS rates is that we can directly observe the rate of change of internal energies of ions and electrons.This is a key element in the present task of diagnosing the electron-only reconnection regime.The PS interaction is also agnostic of the mechanisms actually causing the dissipation, and only gives us a quantitative measure of the change in internal energy.In the current study, we have used the full PS rates for the MMS observations, but only the incompressive part, Pi-D, for simulations.This is because the difference between these two quantities, the pressure-dilatation term, pθ, is usually found to be very small in simulations, and mostly dominated by noise.However, in the Earth's magnetosheath, it has been observed that the pθ term is usually much larger than the Pi-D term (Roy et al. 2022).It is understood that the magnetosheath experiences large-scale compressions due to streams penetrating the bow shock, consequently followed by expansion as the plasma convects away toward the flanks of the magnetosphere.The possible influence of this effect has been commented on in Wang et al. (2021).Due to the absence of such large-scale compressive effects in our simulations, it is reasonable to expect that the pθ term will be much smaller in comparison with the Pi-D term.By examining results from both MMS observations and from simulations, we are able to provide complementary pictures of the transition to electron-only reconnection.For example, simulations have the advantage of providing a quantifiable global context, a feature rarely available to observations.On the other hand, the range of possible scale sizes of the interacting magnetic structures is clearly less constrained in the case of observations.Moreover, typical simulations such as those we present have the inevitable disadvantage of requiring less than optimal (i.e., unrealistic) parameters such as proton-to-electron mass ratio and speed of light.Even with these differences, a consistent picture has emerged, notably from the similarity of the variations of electron-to-proton heating ratios with local magnetic curvature, a quantity that is accessible in both simulations and in MMS observations.The present results, based on a new approach employing separate evaluations of electron and proton pressure strain interactions, is quite consistent with earlier analyses (Phan et al. 2018;Sharma Pyakurel et al. 2019;Califano et al. 2020;Guan et al. 2023) based on appearance of jets and electromagnetic work.The results obtained in our present study indicate that the PS interaction as well as the local magnetic field curvature may be used to identify electron-only reconnection sites in MMS observations.In addition to the local curvature, it would also be interesting to explore the effect of other parameters, such as the local guide field and plasma beta in the dynamics and occurrence of these events.There have been some studies which explore the role these parameters play in electron-only reconnection events.The electron heating in these events is not found to have a clear dependence on the guide field (Shi et al. 2022), and they seem to occur at both low and high values of plasma beta (Vega et al. 2020).Recently, it has been suggested that the relevant parameter controlling the occurrence of electron-only reconnection is the ratio of the simulation box size to the ion gyroradius (Guan et al. 2023).Future multi-spacecraft missions capable of resolving lengths approaching electron scales in other systems such as the solar wind might help us understand the relevance of electron-only reconnection events in different plasma regimes, and the role they play in contributing to dissipation in the solar wind.

Figure 1 .
Figure 1.Electron-only reconnection event observed by Phan et al. (2018).Panel (a) shows the magnetic field components in GSE coordinates, along with the magnitude of the magnetic field.Panels (b) and (c) show the ion and electron velocities, respectively.Panel (d) shows j E• ¢.Panels (e) and (f) show the PS rates for ions and electrons, respectively.The shaded regions in panels (e) and (f) indicate the error associated with the measurement of the PS rates.

Figure 2 .
Figure 2. Ion-coupled reconnection in the magnetosheath observed by Wilder et al. (2018).Panel (a) shows the magnetic field components in GSE coordinates, along with the magnitude of the magnetic field.Panels (b) and (c) show the ion and electron velocities, respectively.Panel (d) shows j E • ¢.Panels (e) and (f) show the PS rates for ions and electrons, respectively.The shaded regions in panels (e) and (f) indicate the estimated uncertainties.

Figure 3 .
Figure3.Ratio of electron to ion PS rates, (P (e) • ∇) • u (e) /(P (i) • ∇) • u(i) , as a function of the magnetic field curvature at the X-line, computed from MMS observations.The data points have been binned into six equally spaced logarithmic bins.The error bars are computed from the standard deviation within each bin.A transition from reconnection to electron-only reconnection is observed near κ = 7 × 10 −3 km −1 .

Figure 4 .
Figure 4. Smaller simulation with box size 2.56 d i .This case, as expected, exhibits strong signatures of electron-only reconnection.Panel (a) shows a map of Pi-D for ions; panel (b) shows a map of Pi-D for electrons.Panel (c) shows a map of j • E. Panel (d) shows a line plot of electron Pi-D through an x-direction cross section of the reconnection layer.Panel (e) shows a similar cross-section plot of Pi-D for ions.Note the change in scale.Panel (f) shows a line plot of j • E through the same cross section of the reconnection layer.

Figure 6 .
Figure 6.Ratio of the incompressive electron to ion heating rates, D D ij ij ij ij e e i i | | | | ( ) ( ) (( ) ( ) P P as a function of the simulation box size L. The simulation parameters are given inTable 1.

Figure 7 .
Figure 7. Ratio of electron to proton Pi-D values vs. the magnetic field curvature in the reconnection region for the simulations in Table 1.Only the incompressive (Pi-D) heating rates are included in the ratio.A transition from ion-dominated reconnection to electron-dominated reconnection can be seen around d 10 2 i

Figure 5 .
Figure 5.A large simulation, box size 40.96d i .Here ion-coupled reconnection activity is evident, as both ions and electrons participate.Panel (a) shows a map of Pi-D for ions; panel (b) shows a map of Pi-D for electrons.Panel (c) shows a map of j • E. Panel (d) shows a line plot of electron Pi-D through an x-direction cross section of the reconnection layer.Panel (e) shows a similar cross section plot of Pi-D for ions.Note the change in scale relative to Figure 4. Reconnection activity is now found in both ions and electrons.Panel (f) shows a line plot of j • E through the same cross section of the reconnection layer.