Observation of Nonuniform Energy Dissipation in the Electron Diffusion Region of Magnetopause Reconnection

We use Magnetospheric Multiscale (MMS) data to investigate the energy dissipation in a magnetopause reconnection electron diffusion region (EDR) event with moderate guide field. The four MMS spacecraft were separated by about 10 km so that comparative study among spacecraft within the EDR can be implemented. Similar magnetic field and electric current properties at each spacecraft indicate the formation of a quasi‐homogeneous magnetic and current structure in the diffusion region. However, we find that the energy dissipations detected by each spacecraft are still different due to the temporal or spatial effect of the out‐of‐plane reconnection electric field ( EM ) within the dissipation region. Our study suggests that the nonuniform or unsteady energy dissipation in the reconnection EDR may be a universal process.

reconnection current sheet are asymmetric, a series of EDR events have been reported based on MMS observations (e.g., Cao et al., 2017;Dong et al., 2020;Eriksson et al., 2016;Khotyaintsev et al., 2016;Lavraud et al., 2016;Norgren et al., 2018;R. Wang et al., 2017;Webster et al., 2018;Zhong et al., 2018;Zhou et al., 2017), where many characteristics associated with the EDR have been observed, for example, a strong electron current, electron agyrotropy, a "crescent-shaped" electron velocity distribution, and strong energy dissipation. Moreover, using the First-Order Taylor Expansion (FOTE) method (Fu et al., 2015(Fu et al., , 2016, which can accurately resolve the magnetic null position and reconstruct the magnetic topology around the spacecraft tetrahedron, direct evidence of the X-line in some of these events has also been further confirmed . In the EDR, energy dissipation is a significant issue, which needs to be investigated further . Energy dissipation in the EDR is usually estimated by using nonideal energy transfer   j E rate where     e E E v B is the electric field in the electron rest frame (Zenitani et al., 2011).    0 j E corresponds to the energy transfer is from the field to the particles. Due to the asymmetric boundary condition at the magnetopause, the flow stagnation point is separated from the X-point and is located a little on the magnetospheric side of the separatrix (Malakit et al., 2013;Wang, Fu, Olshevsky, et al., 2020). Recently, MMS results of magnetopause EDRs show that large dissipation could occur both around the X-point and/or in the flow stagnation region, and may be related to the prevailing guide field conditions (Genestreti et al., 2017). Energy dissipation can be localized and oscillatory due to large amplitude waves . Furthermore, under a large guide field, the strong parallel electric field can play an important role in energy dissipation  during reconnection. In a turbulent environment, numerous current filaments associated with O-lines can appear inside the diffusion region and can be responsible for energy dissipation . By comparing the observation results between each MMS spacecraft, Cozzani et al. (2019) reported an event with inhomogeneous current densities and energy conversion within an EDR, suggesting that an EDR can be rather structured. Although many EDR events have been reported, EDR events with a quasi-homogeneous magnetic field and current structure, however, are rarely reported, so its physical property has not been fully understood.
In this paper, we report an EDR event at the subsolar magnetopause with a moderate guide field resulting from a near-radial (normal direction of reconnection current sheet) crossing of spacecraft. The four MMS spacecraft were separated by about 10 km, which is comparable to a few electron inertial lengths (d e = 1.5 km). Magnetic field and electric current properties are similar within the EDR at each spacecraft, suggesting that the magnetic field and current structure in the EDR is quasi-homogeneous. We still find that the energy dissipations detected by different spacecraft are different, however, due to a nonuniform out-ofplane reconnection electric field M E .

Observations
In this study, we use burst mode data from the MMS mission, where the magnetic field is from the Fluxgate Magnetometer (Russell et al., 2016), plasma data is from the Fast Plasma Investigation  and electric field is from the Electric Field Double Probe (Ergun, Tucker, et al., 2016;Lindqvist et al., 2016). All the vectors are transformed into the local magnetic normal (LMN) coordinate system, as derived from the minimum variance analysis on the magnetic field taken from 09:43:14-09:43:21 UT, where N is the normal direction of the magnetopause current layer. Here the maximum, medium and minimum variance vectors in Geocentric Solar Ecliptic (  is the geometric average magnetic field considering an asymmetric boundary condition, when the magnitude of the inflowing magnetosheath and magnetospheric boundary layer magnetic fields are 13 and 52 nT, respectively (Cassak & Shay, 2007). As MMS enters the magnetosheath, the plasma densities increase from 0.5 to 13 cm −3 and low-energy plasma also appears in the energy spectrogram (Figures 1b,1h and 1i). In particular, we can see a narrow strong current (up to 1.8 μA/m 2 ) embedded in the wider current sheet a little before magnetic field reversal (  0 L B ) and magnetic intensity dropout ( Figure 1e). From this strong current region, the rate of increase of plasma density obviously slows down (Figure 2b). This is consistent with the signature of an inflow stagnation point. Here, the current density is calculated from the plasma moment as     e n e i e j v v . This current is mainly carried by a fast electron jet with a speed of up to 1,020 km/s in −L and +M direction (Figure 1d), while the ion speed remains slow and steady in this region ( Figure 1c). The lack of an ion jet indicates that the spacecraft were crossing a non-reconnecting magnetopause or crossing a reconnection EDR along a near-radial direction . Around electron jet region, the electric field is mainly in the N direction ( Figure 1f). This is consistent with the predicted, unipolar, normal Hall electric field along the magnetospheric side of the separatrix in order to stop the inflow  of magnetosheath ions during asymmetric reconnection (Pritchett, 2008). From the blue vertical dashed line, ions exhibit a dispersive distribution ( Figure 1h) with bulk speeds of up to 190 km/s, mainly in M direction ( Figure 1c) and an agyrotropic velocity distribution ( Figure 1k). These are typical characteristics of a finite ion Larmor gyroradius effect of overshot magnetosheath ions into the magnetosphere near the EDR (Shay et al., 2016). In this region, the electron parallel temperature ( || T ) starts to increase corresponding to counter-streaming distribution, low energy (10-800 eV) electron populations at pitch angles 0° and 180° (Figures 1g and 1j). This kind of anisotropy is also consistent with the features of the magnetospheric inflow region near the EDR (Egedal et al., 2011). Figure 2 shows the detailed structures of the magnetic field, current and electric field from four spacecraft. The four spacecraft timing method, measured on peak points of the current, gives a velocity of 43.5 km/s along (0.12, 0.09, −0.99) LMN relative to MMS. We see that the crossing velocity is nearly along the normal direction (N) of current sheet and its magnitude is consistent with the measured bulk plasma velocity N V in Figure 1c, indicating the convection velocity from the timing results is consistent with the normal bulk velocity. Based on this velocity, the estimated width of main current sheet (traversed from 09:43:18-09:43:20 UT) is about 87 km and the narrow, intense current structure is about 8 km width. For reference, the ion and electron inertial length of the inflowing magnetosheath region are 64 and 1.5 km; so that the width of the whole current sheet and the narrow strong current structure are of order the ion and electron inertial scales, respectively. Figure 2c shows that the boundary normal magnetic field N B is small, but positive (∼2 nT), which suggests that the spacecraft are located south of the X-line. Such a magnetic field corresponds to a dimensionless reconnection rate of ∼0.1 ( Phan et al., 2001), which is consistent with theoretical predictions .
Focusing on the strong current region, we find that the large current is mainly in the perpendicular direction, while the parallel component is negligible, especially for MMS 2 and 3 (Figures 2d and 2e). Figure 2f shows the dominant electric field N E (solid), together with the N component of Hall term  /ne j B (dotted) and the electron pressure gradient term   e P /ne (dashed) in the generalized Ohm's law. The Hall term is calculated at each spacecraft, while the electron pressure gradient term is obtained from the four spacecraft method (Sonnerup et al., 1998). We do not show the ion convention term    N i v B because it can be neglected within the dissipation region. We see that the Hall term is larger than the electric field N E and the deviation between them can partly be attributed to the negative electron pressure gradient term. We note that the magnitude of electron pressure gradient term estimated here should be smaller than actual value because the requirement that the spatial scale of this structure be larger than spacecraft separation is not satisfied (e.g., Dunlop et al., 2016). This non-negligible electron pressure gradient indicates that electrons are not magnetized and that the electron diamagnetic current associated with this electron pressure gradient is an important source of the large electron current . Figure 2g shows the scalar electron agyrotropy index Q of each spacecraft (Swisdak, 2016). We see a clear enhancement up to 0.05 in all the spacecraft signatures during the sampling of the strong current region. This strong agyrotropy is consistent with a crescent-shaped electron velocity distribution in the perpendicular plane (see Figure 1l), which is a typical characteristic of the electron inflow stagnation region .
In order to resolve the spacecraft position relative to the X-line, we perform the FOTE method (Fu et al. 2015(Fu et al. , 2016 to reconstruct magnetic field structure at 09:43:19.55 UT, where the small calculated error (see Figure S1) suggests reliable reconstruction results. Figures 2i and 2j show the magnetic field topology around X-line and the position of each spacecraft in both three-dimensional coordinates and a two-dimensional projection. In order to view the topology better, a new coordinate system (e 1 , e 2 , e 3 ), where e 1 = (0.97, 0, −0.24), e 2 = (0, 1, 0) and e 3 = e 1 × e 2 = (0.24, 0, 0.97) LMN , is adopted. We can see that e 1 , e 2 , and e 3 correspond roughly to L, M, N, respectively. The tetrahedron with the solid lines shows the MMS location at 09:43:19.55 UT, where MMS locates on the southward and slightly on the magnetosheath side of the X-line, which is consistent with above conjecture from the magnetic field and electron velocity. The nearest spacecraft, MMS3 is about 13 km from the X-point. The tetrahedron with dashed lines in Figure 2j shows that the predicted position at the time the strong current region is encountered by MMS1/MMS4, using an estimated time shift from the solid position, where the black arrow indicates the direction of the spacecraft motion. The order in which the four spacecraft cross the magnetospheric side of the separatrix in the reconstruction result is generally consistent with the observation result. In terms of the MMS array sequence, MMS2 passes through first, then MMS1 and MMS4 pass through simultaneously and MMS3 passes through last.
From Figures 2a-2g, we find that the magnetic field, current structure, normal electric field and electron agyrotropy at each spacecraft are almost the same except for the time lag (simply convecting), which indicates that this reconnection EDR is quasi-homogeneous and time stationary during the short period, at least around the local region MMS crossed. The energy dissipation   j E during the strong current region, however, shows different results between each spacecraft (Figure 2h), that is, MMS2 and MMS3 show similar positive values (3 and 3.8 nW/m 3 ), while MMS1 and MMS4 show small negative values. These pairs of similar results rule out a possible stochastic uncertainty caused by the electric field. In this event, the dissipation region seems to be only located around electron stagnation point and it is absent near the magnetic field reversal region.
In order to investigate the specific reason for different dissipations, we show the detailed current, electric field and energy dissipation information of each spacecraft in Figure 3. The currents between each spacecraft are very similar and are mainly in the L and M directions during the dissipation region, while the N component can be neglected (Figures 3b-3d). This indicates that different dissipation is caused by the electric field, especially the L and M components of electric field. is negative, which is consistent with the predicted direction of reconnection electric field, the total energy dissipation is positive, that is, that energy is transferred from magnetic field to particles.

Discussion and Summary
By combining the detailed relative position of each spacecraft (Figure 4), we can investigate the reason for the different energy dissipation rates and the role of the out-of-plane electric field E M (  M E ) in the dissipation region. The current sheet (blue shaded region) is located in the LM plane and the crossing velocity is nearly along the N direction. In the N direction, MMS1 and MMS4 have nearly the same positions along N, MMS2 and MMS3 are positioned 9 and −7 km relative to MMS1. This configuration is consistent with the crossing sequence and time interval of each spacecraft. In the plane of current sheet (LM plane), we see that the locations of MMS2 and MMS3 are very close together, while MMS1 is nearer MMS2/MMS3 in the L-direction and MMS4 is near MMS2/MMS3 in the M-direction. One explanation of the temporal effects is that the negative electric field M E (  M E ) is unsteady and disappears when MMS1/MMS4 cross the current sheet. Such unsteady magnetic reconnection was previously suggested in Cluster observations (Fu, Cao, et al., 2013; and recently confirmed in MMS observations (Wang, Fu, Olshevsky, et al., 2020;Wang, Fu, Vaivads, et al., 2020). Another possible explanation for the variations across the spatial array is that  As an important parameter, the reconnection rate, as determined by external magnetic field configuration or the physical process in internal diffusion region is a matter of concern (Liu et al., 2017). Due to the nonuniform reconnection electric field ( M E ), calculating the overall reconnection rate associated with reconnection electric field seems difficult. However, we can try to use the maximum value of electric field in the dissipation region to estimate the instantaneous reconnection rate. The dimensionless reconnection rate is equal to normalized reconnection electric field obtained from the inflow Alfvén speed and magnetic ). For a magnetopause with asymmetric boundary conditions, the scaled inflow Alfvén speed is     , where subscripts 1 and 2 represent magnetosphere and magnetosheath, respectively (Cassak & Shay, 2007). For the maximum peak reconnection electric field of  1.28 mV/m M E in MMS2, the calculated dimensionless reconnection rates are ∼0.29, which is slightly larger than the result calculated by external magnetic field rec / N B B above and theoretical predictions of Birn et al. (2001). If we consider that only the maximum value of the reconnection electric field is used in the calculation, this larger value compared to the overall reconnection rate is reasonable. Our results suggest that the reconnection electric field in the EDR is rather unsteady or nonuniform due to spatial or temporal effect, which makes it difficult to calculate the overall reconnection rate by in situ measurements, while the reconnection rate calculated by external magnetic field configuration is more reliable.
It is worth noting here that we find the accuracy of the measured electric field data is sufficient for our analysis. First, the plasma moment data is reliable, as can be confirmed by the similar results obtained for the current density from both four-spacecraft plasma moments and from the magnetic field, via the curlometer . Figure S2 shows that small deviations occur only for the central peak and close to 09:43:20 UT where the current sheet scales are smaller than the spacecraft separation. Second, cross-calibration between the electric field data and the plasma moment data can be implemented (as carried out by Torbert et al., 2016). Near the magnetosheath region, such as for times from 09:43:20.8 UT,  E should be almost zero, and this is true for our results (Figures 3g and 3h, where values remain <∼0.5 mV/m). We therefore conclude that this level of error may affect the specific value calculated above, such as energy dissipation and reconnection rate, in only a small way without affecting the conclusion of this paper.
In summary, we have investigated the localized energy dissipation   j E around the EDR by using four-spacecraft measurements at the magnetopause. The finite Larmor gyroradius effect of ions; a bi-streaming distribution of electrons in the magnetospheric inflow region; strong current on electron scales; a crescent-shaped electron velocity distribution, and no ion outflow jet, all indicate that MMS crossed the EDR along a near-radial direction. This is further confirmed by the reconstruction of the FOTE method. A similar magnetic field and electric current behavior, observed by all four spacecraft, indicates the formation of a quasi-homogeneous magnetic field structure in the diffusion region. However, the energy dissipations are DONG ET AL.   nonuniform due to either a temporal or spatial effect from the out-of-plane reconnection electric field M E . Thus, our study suggests that the nonuniform or unsteady energy dissipation in the reconnection dissipation region is a universal phenomenon, even under diffusion region with quasi-homogeneous magnetic field and current structure. This makes it difficult for estimating the overall energy dissipation around the EDR by in situ measurements.