In-situ observations of the magnetothermodynamic evolution of electron-only reconnection

Abstract

Field-particle energy exchange is important to the magnetic reconnection process, but uncertainties regarding the time evolution of this exchange remain. We investigate the temporal dynamics of field-particle energy exchange during magnetic reconnection, using Magnetospheric Multiscale mission observations of an electron-only reconnection event in the magnetosheath. The electron energy is in local minimum at the x-line due to a density depletion, while the magnetic energy is in local maximum due to a guide field enhancement. The electromagnetic energy transport comes almost entirely from guide field contributions and is confined within the reconnection plane, while the most significant contribution to electron energy transport is independent of the drift velocity with additional out-of-plane signatures. Multi-spacecraft analysis suggests that the guide field energy is decreasing while the electron density is increasing, both evolving such that the system is moving toward a more uniform distribution of magnetic and thermal energy.

Introduction

Field-particle energy exchange processes in collisionless plasmas are often associated with the evolution of highly non-Maxwellian phase space structures, suggesting a thermodynamic process where the path that energy takes toward dissipation involves localized reductions in kinetic entropy^1,2. Despite the lack of an obvious dissipative mechanism, collisionless reconnection is irreversible³ and collisionless systems tend to exhibit “collisional-like” behavior⁴ in particle-in-cell (PIC) simulations. Recent theoretical insights regarding the pressure strain interaction term have also provided a more detailed picture of energy conversion processes in systems far from local thermodynamic equilibrium (LTE)^5,6.

Despite recent insights on the thermodynamic processes of collisionless reconnection, it is still unclear how the evolution of the local magnetic field influences local thermodynamics. One way to address this problem is by directly measuring the flow of energy and its relationship to the time evolution of the energy density structures in a reconnecting system. Energy continuity dictates a direct relationship between the local time evolution of energy density and the various energy fluxes that may be present.

$$\frac{\partial }{\partial t}\left({u}_{t}+{u}_{k}+{u}_{em}\right)=-\nabla \cdot \left(\overrightarrow{H}+\overrightarrow{K}+\overrightarrow{Q}+\overrightarrow{S}\right)$$

where u_t,u_k, and u_em are the thermal (pressure), kinetic (bulk motion), and electromagnetic energy densities, respectively. The energy fluxes $\overrightarrow{H}$,$\overrightarrow{K}$,$\overrightarrow{Q}$,and $\overrightarrow{K}$ are the enthalpy flux (thermal convection), kinetic energy flux (bulk flow), heat flux (asymmetry in the distribution function), and Poynting flux (electromagnetic energy flow), respectively. In the non-relativistic limit, the standard energy densities and energy fluxes are defined^7,8 by

$$\begin{array}{c}{u}_{k}=\frac{nm{v}^{2}}{2}\qquad {u}_{t}=\frac{Tr\left(\overleftrightarrow{P}\right)}{2}\qquad {u}_{em}=\frac{{B}^{2}}{2{\mu }_{0}}+\frac{{\epsilon }_{0}{E}^{2}}{2}\\ \overrightarrow{K}=\frac{nm{v}^{2}}{2}\overrightarrow{v}\qquad \overrightarrow{H}=\frac{\overrightarrow{v}\,Tr\left(\overleftrightarrow{P}\right)}{2}+\overrightarrow{v}\cdot \overleftrightarrow{P}\qquad \overrightarrow{S}=\frac{\overrightarrow{E}\times \overrightarrow{B}}{{\mu }_{0}}\end{array}$$

where n is the particle density, m is the particle mass, $\overrightarrow{v}$ is the velocity, $\overleftrightarrow{P}$ is the pressure tensor, $\overrightarrow{E}$ is the electric field, and $\overrightarrow{B}$ is the magnetic field. Previous studies^9,10 have presented in-situ observations of the various energy fluxes associated with reconnection and found that the ion enthalpy flux typically dominates in the outflow with smaller, more filamentary contributions from electron enthalpy flux. Measurements of heat flux are not presented in this article for reasons explained in the methods.

Modern reconnection models emphasize the importance of charge separation effects in the dynamics and rate of magnetic reconnection. Due to their different masses, and therefore different gyroradii, ions and electrons demagnetize at different length scales and create an electron diffusion region (EDR) embedded within a larger ion diffusion region (IDR). As a consequence, additional Hall electromagnetic fields can develop in the IDR due to current-carrying electrons following the field lines while ions are demagnetized. These Hall currents are an important kinetic mechanism underlying the source of the electric fields associated with fast reconnection in the IDR^{11,12,13,14,15}, and the Hall electromagnetic fields can influence the Poynting flux structure in the reconnection exhaust and allow more rapid energy transport and higher reconnection rates^16,17. Direct in-situ observations of the EDR at the magnetopause¹⁸ and in the magnetotail¹⁹ have been previously reported and have demonstrated the importance of electron-scale kinetic physics to the transfer of electromagnetic energy to particles. In-situ measurements of the terms in the generalized Ohm’s law found that the divergence of the electron pressure tensor is the most significant contribution to energy dissipation in the EDR, though there are also contributions from electron inertial effects²⁰.

New observations have complicated the Hall reconnection model and the concept of embedded EDR-IDR structures. Within the last few years a unique form of magnetic reconnection has been observed in which only the electrons take part in the reconnection process without the presence of a traditional IDR. Recent studies^21,22 have suggested that electron-only reconnection may be an early stage of traditional reconnection, after onset but before the ions begin to respond to the local Hall fields and gain energy. This is also supported by an earlier analysis²³ which found that ion participation can be prevented if the domain size is sufficiently small in PIC simulations of reconnection. An electron-only reconnection event was first observed²⁴ by the Magnetospheric Multiscale (MMS) mission in the turbulent magnetosheath, including an interval in which both inflows and outflows of the x-line were observed simultaneously by the four spacecraft. Due to the relative positions of the MMS spacecraft and the x-line, this magnetosheath event²⁴ provides a unique opportunity to study the energy transport processes of an electron-only x-line. This event is also well-suited to the analysis of non-equilibrium reconnection dynamics because it exists within a turbulent environment which may provide highly variable inflow conditions.

The following study is a detailed analysis of the electromagnetic and thermal evolution of an electron-only reconnection event²⁴. In the Methods section, the methods behind our analysis is presented in more detail. In the Results section, we present the observed energy density structure of the x-line and various other terms relevant to the electromagnetic and electron thermal energy densities (“Energy Density Structure” subsection), we describe how energy is being transported near the x-line by examining the local energy flux structures, their divergences, and the primary terms contributing to them (“Energy Flow Structure” subsection), and we use multi-spacecraft techniques to take local measurements of the time-evolution of various terms relevant to energy conversion and local thermodynamic evolution of the electrons (“Local Time Evolution” subsection). In the Discussion section, we discuss the results of the previous sections and what they imply about the relationship between collisionless reconnection and the “arrow of time”. The Conclusions section summarizes our findings.

Methods

MMS Instrumentation

This study uses high-time resolution burst mode data from the suite of particle and field instruments aboard MMS. Magnetic field data come from the fluxgate magnetometers (FGM), which measure magnetic field vectors at a cadence of 128 samples per second²⁵. Electric field data come from the electric field double probes (EDPs), which measure spin plane²⁶ and axial²⁷ components of the electric field at 8,192 samples per second. Particle data come from the electron and ion spectrometers part of the fast plasma investigation (FPI)²⁸.

The most significant sources of error that may affect the results in the following sections come from the FPI moments data. Within the time interval in this article the electron velocity ${\overrightarrow{v}}_{e}$, temperature $\overleftrightarrow{{T}_{e}}$, and density n_e have approximate relative errors of 0.1, 0.01, and 0.005, respectively. The errors associated with each parameter are taken directly from the FPI data products²⁸.

X-line Reference Frame

All vector quantities in this study are presented in a previously defined²⁴ current sheet normal LMN coordinate system, and are transformed into the x-line reference frame to isolate energy fluxes associated with the reconnection process from energy fluxes associated with the overall motion of the x-line relative to MMS. The electric field $\overrightarrow{E}$ and electron velocity ${\overrightarrow{v}}_{e}$ are transformed into their x-line frame equivalent with the x-line velocity ${\overrightarrow{v}}_{xl}$, which is taken to be the ion velocity measured at the center of the EDR crossing as is the case in other similar studies⁹.

$${\overrightarrow{E}}^{{\prime} }=\overrightarrow{E}+{\overrightarrow{v}}_{xl}\times \overrightarrow{B}\qquad {\overrightarrow{v}}_{e}^{{\prime} }={\overrightarrow{v}}_{e}-{\overrightarrow{v}}_{xl}$$

While it is possible that the ion reference frame and the x-line reference frame may not be explicitly identical, the ion reference frame is still an appropriate choice at the center of the x-line since it represents the local center-of-mass frame and it isolates the electron energy fluxes from the background (non-reconnecting) ion flow. The above relations can be used to transform any of the energy fluxes or energy densities mentioned in the introduction. In the non-relativistic regime, quantities such as $n,Tr(\overleftrightarrow{P})$, and $\overrightarrow{B}$ are frame-independent, and therefore do not need to be altered from the direct MMS data products. In the following sections, the heat flux $\overrightarrow{Q}$ will be neglected, due to the lack of significant signal that can be distinguished from the noise.

Calculating Time Derivatives

This work also requires evaluations of time derivatives to isolate the temporal evolution from the spatial evolution as MMS observes a passing structure. The standard method of evaluating the time derivative of an arbitrary scalar γ in a co-moving fluid frame is to use a convective derivative:

$$\frac{\partial \gamma }{\partial t}=\frac{d\gamma }{dt}-{\overrightarrow{v}}_{xl}\cdot \nabla \gamma$$

where $\frac{d\gamma }{dt}$ is the time derivative in the MMS frame, and ∇ γ is calculated using multi-spacecraft techniques²⁹. For variables with multiple contributions to the product, such as u_te ∝ n_eT_e, the temporal derivative is broken down into its separate contributions.

$$\frac{\partial {u}_{te}}{\partial t}=\frac{1}{2}\frac{\partial }{\partial t}\left({n}_{e}Tr(\overleftrightarrow{{T}_{e}})\right)=\frac{1}{2}\left(\langle Tr(\overleftrightarrow{{T}_{e}})\rangle \frac{\partial {n}_{e}}{\partial t}+\langle {n}_{e}\rangle \frac{\partial Tr(\overleftrightarrow{{T}_{e}})}{\partial t}\right)$$

The terms in angled brackets represent a simple four-spacecraft average, while the partial derivatives are determined via standard multi-spacecraft methods²⁹.

Validity of Linear Gradient Approximation

This work also utilizes multi-spacecraft techniques²⁹ to calculate energy flux divergences and the directional contributions to their sum. However, there are several limitations to these methods that should be mentioned. The first is that they assume that similar measurements vary linearly between any two spacecraft. A common test for the reliability of this assumption is to see how well the curlometer current density ${\overrightarrow{J}}_{curl}=\frac{\nabla \times \overrightarrow{B}}{{\mu }_{0}}$ matches the spacecraft-averaged moments current density ${\overrightarrow{J}}_{mom}$ calculated by FPI. In this case there is reasonable agreement between the current density via both methods (see Figure S1 in supplementary note 1), suggesting that the barycentric quantities in this study are reasonably well-resolved. The second limitation has to do with the determination of divergences, and how the net divergence of a given quantity may be affected due to the timing of encounters with the inflows or outflows between spacecraft, even if the separate component contributions to the sum are reliable. An illustration explaining this is also provided in Figure S2 in supplementary Note 1.

Results

Energy Density Structure

The left panel of Fig. 1 presents the electromagnetic energy density (u_em), electron thermal energy density (u_te) and their sum observed during the encounter by MMS2. u_em is larger than u_te, and it is in a local maximum compared to its surroundings, whereas u_te is in a local minimum. The local enhancement in u_em is larger than the u_te depletion and results in a local enhancement in the sum of the two terms.

Fig. 1: Magnetic and Thermal Structure.

(a) Electromagnetic and (b) electron thermal energy densities from MMS2, (c) Magnetic field components from MMS2, Diagonal components of the (d) electron pressure tensor, (e) temperature tensor, and (f) the electron density. Separate L, M, N contributions to the terms in panels c–e are colored blue, green, and red, respectively. The shaded intervals represent the full x-line crossing and the dashed lines represent the times of the first and last encounter of the x-line center (where B_L = 0) by MMS4 and MMS2, respectively.

Figure 1 further highlights the source of these energy densities. The most significant contribution to u_em and its local enhancement is the large guide field B_M associated with the event. Depletion in u_te comes primarily from a depletion in n_e, rather than T_e. The electron pressure P_e and its depletion are not fully isotropic, but instead are largely within the LN plane, with a negligible P_eMM contribution parallel to the guide field.

The large guide field B_M in this event causes the local field curvature radius, defined as ${R}_{c}=\frac{1}{| \hat{b}\cdot \nabla \hat{b}| }$ ($\hat{b}$ is the magnetic unit vector) to be relatively large. Figure 2 compares the local R_c with the thermal gyroradii of both the ion and electron populations (R_i and R_e, respectively). R_c is larger than R_i and much larger than R_e, suggesting that the electrons are well-magnetized even near the x-line.

Fig. 2: Magnetic curvature radius and thermal gyroradii.

Comparison of magnetic curvature radius (R_c) with the electron and ion gyroradii (R_e and R_i, respectively). The shaded interval represents the x-line crossing.

Energy Flow Structure

The energy flux densities observed by all the MMS spacecraft are shown in Fig. 3. The Poynting flux $\overrightarrow{S}$ is predominantly along the L and N axes, with simultaneous observations of both inflow S_N components by MMS 2 and 4 and of both outflow S_L components by MMS 1 and 3. Unlike $\overrightarrow{S}$, there is a large H_eM component observed by all spacecraft, which is approximately parallel to the local magnetic field due to the large B_M present over the interval. There are also smaller contributions by H_eL and H_eN. The kinetic energy flux structure has spikes in K_eL and K_eM, but they are an insignificant contribution compared to the overall energy transport in the system.

Fig. 3: Observed Energy Flux Structure.

Poynting flux (panels a1-a4), electron enthalpy flux (b1-b4) and electron kinetic energy flux (c1-c4) components observed by each spacecraft with a diagram depicting the approximate trajectory of MMS through the energy flow structures. The L, M, N components of all terms are colored blue, green, and red, respectively. The Poynting and electron enthalpy structures in the diagram are depicted in black and green, respectively, while the MMS spacecraft are depicted as red dots with red dashed arrows indicating their direction of motion relative to the x-line.

Figure 4 presents the breakdown of the individual components of $\overrightarrow{S}$ from MMS 2 into separate contributions from different field components, as well as a comparison of ${\overrightarrow{H}}_{e}$ without the ‘drift’ enthalpy flux H_eD, which is defined here as the electron enthalpy flux associated with the electron $\overrightarrow{E}\times \overrightarrow{B}$ drift velocity.

$${H}_{eD}=\frac{{\overrightarrow{v}}_{d}\,Tr\left(\overleftrightarrow{{P}_{e}}\right)}{2}+{\overrightarrow{v}}_{d}\cdot \overleftrightarrow{{P}_{e}}$$

where ${v}_{d}=\frac{\overrightarrow{E}\times \overrightarrow{B}}{{B}^{2}}$. The results indicate that the H_eM contribution is largely independent of the $\overrightarrow{E}\times \overrightarrow{B}$ drift, and that $\overrightarrow{S}$ is dominated by guide field contributions along with electric fields in the LN plane.

Fig. 4: Contributions to Thermal and Electromagnetic Transport.

a Total electron enthalpy flux from MMS2 and (b) the enthalpy flux excluding the drift contribution, where the L, M, N contributions in panels a, b are colored blue, green, and red, respectively. Contributions to the (c) L and (d) N components of Poynting flux, where the terms in green represent contributions by in-plane electric fields and the guide field, and the terms in red represent contributions from the reconnecting magnetic field components and the reconnection electric field.

Figure 5 shows the calculated divergences of the energy fluxes from Fig. 3 using standard multi-spacecraft techniques²⁹. $\nabla \cdot \overrightarrow{S}$ is the most significant term with somewhat comparable ∂_LS_L > 0 and negative ∂_NS_N < 0, though the ∂_NS_N contribution is generally larger and produces a net negative $\nabla \cdot \overrightarrow{S}$ between the first and last encounter of the x-line center. The net $\nabla \cdot \overrightarrow{S}$ consists of a local minimum magnitude of $| \nabla \cdot \overrightarrow{S}| \approx 10-30\,nW/{m}^{3}$ adjacent to two larger spikes of $| \nabla \cdot \overrightarrow{S}| \approx 70-100\,nW/{m}^{3}$. $\nabla \cdot {\overrightarrow{H}}_{e}$ consists of roughly balanced ∂_LH_eL > 0 and ∂_NH_eN < 0, with an additional ∂_MH_eM < 0 contribution resulting in a net negative $\nabla \cdot {\overrightarrow{H}}_{e}$, which peaks in magnitude at $| \nabla \cdot {\overrightarrow{H}}_{e}| \approx 50\,nW/{m}^{3}$ around the same time of the local minimum magnitude $\nabla \cdot \overrightarrow{S}$. At its peak, the magnitude of enthalpy flux divergence exceeds that of the Poynting flux divergence $| \nabla \cdot {\overrightarrow{H}}_{e}| > | \nabla \cdot \overrightarrow{S}|$ (interval highlighted in orange in Fig. 5). Though the magnitude of $\nabla \cdot {\overrightarrow{K}}_{e}$ is negligible, it still exhibits some interesting structure that distinguishes it from $\nabla \cdot {\overrightarrow{H}}_{e}$. $\nabla \cdot {\overrightarrow{K}}_{e}$ has net positive spikes, one before the first encounter of the x-line center and one near the last encounter, with a comparably small negative spike in between. $\nabla \cdot {\overrightarrow{K}}_{e}$ is comparably small in the center due to roughly balanced ∂_LK_eL > 0 and ∂_MK_eM < 0, whereas the positive spikes mostly come from an imbalance between ∂_MK_eM > 0 and ∂_LK_eL < 0.

Fig. 5: Energy Flux Divergence and Directional Contributions.

Includes divergence of (a) Poynting flux, (b) electron enthalpy flux, and (c) electron kinetic energy flux, where the totals are in black and the L, M, N contributions are in blue, green, and red, respectively. The grey shaded interval represents the full x-line crossing, the dashed lines represent the times of the first and last encounter of the x-line center, and the orange shaded interval represents the observation of the local $| \nabla \cdot {\overrightarrow{H}}_{e}|$ maximum and $| \nabla \cdot \overrightarrow{S}|$ minimum.

Local Time Evolution

Figure 6 presents the local time evolution of the electron thermal u_te and electromagnetic u_em energy densities, as well as the power density of energy transfer $\overrightarrow{J}\cdot \overrightarrow{E}$ broken down into its separate contributions. u_em is locally decreasing with time at a rate similar to the spacecraft averaged $\overrightarrow{J}\cdot \overrightarrow{E}$ at its peak. Further decomposition of u_em into separate magnetic field component contributions suggests that the magnetic energy loss at the center of the reversal is almost entirely due to a loss in guide field energy density.

Fig. 6: Power Density of Energy Density Time Evolution.

Power density of (a) field-to-particle energy transfer, b magnetic energy evolution, and c electron thermal energy evolution. The separate L, M, N contributions to the terms in panels a–b are colored blue, green, and red, respectively. The separate contributions by density evolution and temperature evolution in panel c are colored blue and red, respectively. The grey-shaded region represents the x-line crossing interval, the dotted lines represent the center of the B_L reversal observed by MMS 4 and 2, and the orange shaded interval represents the observation of the local $| \nabla \cdot {\overrightarrow{H}}_{e}|$ maximum and $| \nabla \cdot \overrightarrow{S}|$ minimum from Fig. 5.

Figure 6 also shows the temporal evolution of u_te and its separate contributions. The results suggest that the local u_te is increasing, but at a smaller magnitude than that of the guide field energy loss. The increasing u_te is largely due to the increase in n_e rather than T_e. The electrons are therefore being locally compressed with minimal temperature enhancement while the local guide field is decreasing. A comparison of the separate contributions to the total convective derivative can be found in supplementary Note 2, Figure S3.

Discussion

This particular event has both a strong background guide field B_M and a local B_M enhancement at its center, whereas the electron density n_e is a local minimum compared to its surroundings. This results in a u_te depletion within a u_em enhancement, though it should be emphasized that these two features do not appear to balance one another (Fig. 1) due to the u_em enhancement being larger than the u_te depletion. The result is an electron-scale feature that has a non-uniform distribution of energy density. The thermal energy density u_te and the scalar pressure P_e are directly proportional to one another where P_e = 2u_te/3. Therefore, P_e is smaller than u_te and the feature is also not in local pressure balance. The “Energy Flow Structure” and “Local Time Evolution” subsections in the Results provided a detailed look at the mechanisms that influence the system’s path toward a local equilibrium.

The strong B_M responsible for the locally enhanced energy density also heavily influences the local energy flow structures by causing $\overrightarrow{E}\times \overrightarrow{B}$ to largely be confined to the LN plane. This influences both the Poynting flux $\overrightarrow{S}$ and the ‘drift’ enthalpy flux ${\overrightarrow{H}}_{eD}$ (Figs. 3 and 4). The decomposition of S_L and S_N (Fig. 4) demonstrate that magnetic energy transport comes almost entirely from B_M contributions and in-plane electric fields, suggesting that the reconnection electric field E_M is not a significant mechanism of electromagnetic energy transport in this case (although this does not mean E_M is unimportant to the reconnection process). In contrast, H_eM is typically the largest component of ${\overrightarrow{H}}_{e}$ (Fig. 3). The H_eL and H_eN terms can be significant, but are largely due to the strong ${\overrightarrow{H}}_{eD}$ that comes from the $\overrightarrow{E}\times \overrightarrow{B}$ drift. The H_eM contribution is mostly independent of ${\overrightarrow{H}}_{eD}$ (Fig. 4), and distinct from $\overrightarrow{E}\times \overrightarrow{B}$ drift effects. The electric and magnetic field structures of this event have been previously examined in detail³⁰, but not the energy flow structures as we have examined here.

The energy flux divergences in Fig. 5 highlight the spatial variability of energy transport. Overall, the most significant term is $\nabla \cdot \overrightarrow{S} < 0$, in particular the strong ∂_NS_N < 0 and ∂_LS_L > 0 contributions associated with the electromagnetic energy inflows and outflows, respectively. The transport of electron thermal energy also has these inflow and outflow signatures. There are ∂_NH_eN < 0 and ∂_LH_eL > 0 contributions that approximately balance each other out, but unlike $\nabla \cdot \overrightarrow{S}$ there is also a strong ∂_MH_eM < 0 contribution that makes $\nabla \cdot {\overrightarrow{H}}_{e} < 0$. That is, the negative $\nabla \cdot \overrightarrow{S}$ is confined to the reconnection plane while the negative $\nabla \cdot {\overrightarrow{H}}_{e}$ is significantly influenced by its out-of-plane contribution. Since all four spacecraft measured H_eM in the same direction (Fig. 3), this signature of ∂_MH_eM < 0 is not indicative of converging thermal flows in the x-line frame, but it may be due to more thermal energy flowing along M into one side of the tetrahedron than out the other. Though it is insignificant in magnitude, $\nabla \cdot {\overrightarrow{K}}_{e}$ also has a considerable ∂_MK_eM < 0 term that almost balances its ∂_LK_eL > 0 contribution due to the outflows. The contributions to $\nabla \cdot {\overrightarrow{K}}_{e}$, though negligible, are confined to the plane of the current sheet. It is not entirely clear what role variability along M could play, but the existence of 3D structure in this particular event is consistent with an earlier study³⁰, and it is also consistent with another study⁹ of an EDR at the magnetopause that found significant out-of-plane electron energy flux structures suggesting the existence of mesoscale 3D effects. A recent statistical analysis³¹ of EDRs observed at the dayside magnetopause also found that significant positive and negative ∂_MH_eM contributions are common, suggesting that variable thermal energy transport along the M direction may be a ubiquitous and important aspect of reconnection generally. The 3D effects may also be important to the reconnection rate of electron-only reconnection, as it has been shown³² that 3D electron-only reconnection can allow for higher reconnection rates than in 2D due to the differential mass flux along the M direction allowing for faster inflows.

At its largest, $| \nabla \cdot {\overrightarrow{H}}_{e}| > | \nabla \cdot \overrightarrow{S}|$ where $| \nabla \cdot \overrightarrow{S}|$ is in a local minimum (orange shaded interval in Fig. 5). This brief interval occurs just after the first encounter of the x-line center by MMS 4 coinciding with the maximum $\overrightarrow{J}\cdot \overrightarrow{E} > 0$ and with the largest power density associated with $\frac{\partial {u}_{te}}{\partial t} > 0$, but not with the maximum $\frac{\partial {u}_{em}}{\partial t} < 0$ near the last encounter of the x-line center by MMS 2 (Fig. 6). Within this brief interval, $\frac{\partial {u}_{em}}{\partial t}\approx 0$ and Poynting’s theorem is nearest to closure with $-\nabla \cdot \overrightarrow{S}\approx 10-30\,nW/{m}^{3}$ and $\overrightarrow{J}\cdot \overrightarrow{E}\approx 5\,nW/{m}^{3}$. The averaged $\overrightarrow{J}\cdot \overrightarrow{E}$ shown in Fig. 6 is smaller than the individual measurements by the MMS spacecraft since they measure its peak at different times. If we were to consider the maximum magnitudes observed by individual spacecraft, which are closer to $\overrightarrow{J}\cdot \overrightarrow{E}\approx 10-20\,nW/{m}^{3}$^24,30, then Poynting’s theorem could be close to balanced $(\overrightarrow{J}\cdot \overrightarrow{E}\approx -\nabla \cdot \overrightarrow{S})$ in the short interval with the maximum$\frac{\partial {u}_{te}}{\partial t} > 0$ (Figs. 5 and 6). Although they do not occur simultaneously (or in the same exact location on the x-line), the maximum $\overrightarrow{J}\cdot \overrightarrow{E}$ early in the interval is comparable to the maximum $\frac{\partial {u}_{em}}{\partial t} < 0$ late in the interval, which is mainly due to the loss of guide field energy.

$${(\overrightarrow{J}\cdot \overrightarrow{E})}_{max}\approx -\frac{\partial }{\partial t}{\left(\frac{{B}_{M}^{2}}{2{\mu }_{0}}\right)}_{max}$$

The mechanism that relates magnetic topological changes to the observed electron compression (Fig. 6) at the x-line is worth further consideration. It has been shown that the electrons are well-magnetized (Fig. 2) due to the strong B_M that reduces magnetic curvature across the x-line. Local reduction in B_M is equivalent to local reduction in the magnetic curvature radius R_c and a contraction in the magnetic topology along M. One would expect the magnetized electrons to closely follow such topology changes, and compress to higher density as the magnetic field contracts. A simple illustration of this mechanism is presented in Fig. 7. While the magnitude of magnetic energy loss may not be fully accounted for by this mechanism, this serves as a potential explanation for why compressive work and guide field reduction are related processes in sheared magnetic fields.

Fig. 7: Compression via Curvature Enhancement.

Diagram depicting the isothermal compression of magnetized electrons due to the local reduction of the radius of curvature, where B_M is the guide field, R_c is the magnetic radius of curvature, and n_e is the electron density. The blue dots are representative of magnetized electrons along the field line, and the red arrows indicate how the structure evolves with time.

It is not clear what may fully account for the loss rate of guide field energy or the similar $\overrightarrow{J}\cdot \overrightarrow{E}$ contribution if not the electron compression. Additional contributions could come from ions if they undergo a similar compressive mechanism, which is possible since the local R_c is larger than R_i by roughly an order of magnitude at the x-line center (Fig. 2). There is insufficient FPI ion data along the interval to reliably determine their temporal evolution and spatial derivatives via techniques described in the methods. However, if it is assumed that the plasma is locally quasi-neutral during the compression, then $\frac{\partial {n}_{i}}{\partial t}\approx \frac{\partial {n}_{e}}{\partial t}$. $\frac{\partial {u}_{ti}}{\partial t}$ could therefore be estimated based on the temperature ratio for this event, which is approximately T_i/T_e ≈ 10. The power density associated with ion compression would therefore be roughly an order of magnitude larger than that of the electron density compression $\frac{\partial {u}_{ti}}{\partial t}\approx 10-15\,nW/{m}^{3}$, which is larger than the averaged $\overrightarrow{J}\cdot \overrightarrow{E}$ presented here, but is similar in magnitude to the maximum measured $\overrightarrow{J}\cdot \overrightarrow{E}$ in previous studies of this event^24,30. Since there is scale separation between the ions and electrons (R_i > R_e), their dynamics may differ. If local fluctuations in electron density are smaller than ion density fluctuations, there may be small-scale non-neutral regions that exhibit electric fields due to local charge separation. Analysis of multi-species compression and its relationship to energy dissipation may be a worthy topic for future studies.

The purpose of this study was to address three questions: 1) What is the distribution of energy for this event? 2) Where is the energy flowing? 3) How is the structure changing in time? All of the previous analysis is in service of a more fundamental question: Why is the system evolving this particular way? Macroscopic systems tend to have a preferred direction in phase space along which they evolve, despite the laws of physics being fully reversible at kinetic scales. This emergent “arrow of time” explains classic thermodynamic phenomena as a consequence of the mean behavior of many colliding particles. Despite the lack of the traditional collisional mechanism, evidence suggests that collisionless dissipation exhibits statistical properties similar to those in collisional systems⁴, and that collisionless reconnection is an entropy-producing¹ and irreversible³ process. It is therefore worth reflecting on the results of the preceding sections from this perspective.

The results of the preceding sections suggest that as a density depletion is being filled in, a local guide field enhancement is being reduced (Fig. 6), thereby pushing the ratio of electron thermal pressure to magnetic pressure (β_e = P_e/u_em) closer to its upstream value while the locally enhanced total pressure decreases. The observed trend toward a local equilibrium begs the question of how a local non-equilibrium state as observed in Fig. 1 developed in the first place. Since this is an in-situ measurement over a short time interval, the preceding dynamics that led to the observed state of the system in the “Energy Density Structure” subsection of the Results can only be left to speculation. Due to the turbulent environment in which this x-line exists, the upstream conditions may be highly variable. Such variability in the local conditions would mean that there is variability in what the local equilibrium state should be. For example, it may be the case that local pressure balance was disrupted due to a reduction in the upstream guide field strength and/or an increase in the upstream density, leaving behind a guide field enhancement within a density depletion. Previous studies have demonstrated considerable variability in the guide field associated with electron-only reconnection in the magnetosheath, and have suggested that electron-only reconnection may be more prevalent in turbulent environments^33,34. A more detailed analysis of non-equilibrium reconnection dynamics may be needed, and will likely require the aid of PIC simulations in future studies. Recent progress has been made with a study that suggests non-equilibrium current sheet dynamics can play an important role in reconnection onset³⁵.

The nearly isothermal electron compression at the x-line is a form of compressive work⁶ on the local electron population. Isothermal processes are associated with an exchange of thermodynamic heat Q with the surroundings in order to sustain a constant temperature. In this case the electron population is compressed, which reduces the local position-space kinetic entropy and must result in a loss of heat from the electrons in the compressed region to the surroundings ΔQ_e = − ΔQ_surr < 0. The most significant transport of thermal energy that is not associated with drift velocity is the H_eM component of electron enthalpy flux, which is parallel to the negligible component of the anisotropic pressure P_eMM. For reasons mentioned in the methods, the measurements of heat flux ${\overrightarrow{Q}}_{e}$ are not included. Previous studies have found heat flux signatures comparable in magnitude to the enthalpy flux^9,31, suggesting that heat flux can significantly influence the energy budget of reconnection. While the heat flux could play an important role in the transfer and transport of energy during reconnection, it is unclear whether it is significant for this event.

This system appears to undergo a multi-step process of field-to-thermal energy transfer where excess guide field energy loss drives local isothermal compression, which increases the local pressure, expels heat into its surroundings, drives thermal energy transport down the local pressure gradient and pushes the system toward a more balanced configuration. It implies a relationship between energy transfer and entropy production in a system that is collisionless. A simple diagram depicting this magneto-thermodynamic system is included in Fig. 8.

Fig. 8: Magneto-Thermodynamic Pathway.

Diagram depicting the magneto-thermodynamic process observed, where the boxes represent the interacting “systems”: the local electromagnetic fields, the local electrons, and the surroundings. Magnetic compressive mechanism (red arrows) does work to increase the local electron thermal energy density u_te, decreasing the magnetic energy density u_em and guide field B_M within the diffusion region (dotted circle) while heat, Q (green arrows) is expelled from the local compressed region, resulting in a local reduction in position-space kinetic entropy.

It is important to consider whether the observed time variation in this study is fundamentally related to the reconnection process or whether is merely a consequence of the turbulent environment. This is a difficult question to answer based on the results of this study. One should expect any current sheet, reconnecting or not, to evolve in such a way that the system tries to approach a more rather than less balanced state. However, reconnection may expedite this process by allowing for the rapid convection of excess guide field energy along L (Fig. 4 due to the emergence of the x-line geometry and the influence of Hall E_N structures. Distinguishing characteristics of reconnection from those of turbulence may also be a particularly difficult problem as both processes can influence each other either in the form of reconnection-driven turbulence or turbulence-driven reconnection, making it difficult to make any generally applicable statement about which is the “cause” and which is the “effect”^34,36,37,38.

Conclusion

We have analyzed the dynamics of energy transport and transfer for an electron-only reconnection event observed in the magnetosheath. First, we determined the energy density structure of the system, exhibited by a local depletion in electron thermal energy (via density) with an enhancement in the magnetic energy density (via the guide field) at the center of the x-line. Next, we performed an analysis of the various means by which magnetic and electron thermal energy flowed in the system, finding that the transport of electromagnetic energy came in the form of guide field energy transport via in-plane electric fields while the largest contribution to electron thermal energy transport was roughly parallel to the guide field and independent of the $\overrightarrow{E}\times \overrightarrow{B}$ drift motion. We also examined the various contributions to the divergence of the energy fluxes and found that a negative $\nabla \cdot \overrightarrow{S}$ was largest and confined to the LN plane while there was also a considerable negative $\nabla \cdot {\overrightarrow{H}}_{e}$ with contributions in all 3 dimensions. Lastly, we measured the temporal evolution of magnetic and electron thermal energy density in the ion reference frame, finding that the magnetic energy was decreasing via guide field reduction while the electron thermal energy was increasing via density enhancement.