Electron Inflow Velocities and Reconnection Rates at Earth's Magnetopause and Magnetosheath

Electron inflow and outflow velocities during magnetic reconnection at and near the dayside magnetopause are measured using satellites from NASA's Magnetospheric Multiscale (MMS) mission. A case study is examined in detail, and three other events with similar behavior are shown, with one of them being a recently published electron‐only reconnection event in the magnetosheath. The measured inflow speeds of 200–400 km/s imply dimensionless reconnection rates of 0.05–0.25 when normalized to the relevant electron Alfvén speed, which are within the range of expectations. The outflow speeds are about 1.5–3 times the inflow speeds, which is consistent with theoretical predictions of the aspect ratio of the inner electron diffusion region. A reconnection rate of 0.04 ± 25% was obtained for the case study event using the reconnection electric field as compared to the 0.12 ± 20% rate determined from the inflow velocity.


Introduction
During asymmetric reconnection, as occurs at the dayside magnetopause (MP), the flow stagnation point does not coincide with the X line, as it does with symmetric reconnection (Cassak & Shay, 2007;Priest et al., 2000). Since there is no transport of magnetic flux across the X line, and no net transport of mass flux across the stagnation point, the two points coincide only for equal mass densities, inflow velocities, and magnetic field strengths within each inflow region. The higher densities on the magnetosheath side and the ©2020. The Authors. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. higher magnetic field strength on the magnetosphere side displace the stagnation point toward the Earth from the X line.
At kinetic scales there are separate electron and ion stagnation points resulting from the larger gyroradii of ions (Cassak & Shay, 2009). Hesse et al. (2014) used particle-in-cell simulation and theory to examine energy conversion and electron distribution functions (DFs) within the asymmetric reconnection electron diffusion region (EDR). Their prediction of crescent-shaped electron distributions was confirmed with data from the NASA Magnetospheric Multiscale (MMS) mission by . Pritchard et al. (2019) noted the occurrence of converging electron flows in a region of strong out-of-plane current and interpreted them as electron inflow velocities to the EDR. Inflow velocities are important because when scaled to the electron Alfvén speed (v Ae ), they provide the reconnection rate (R) (Karimabadi et al., 2013;Klimas, 2015). The use of v Ae instead of v Ai in this context is warranted by the fact that near the EDR the magnetic field is advected by the electrons (Cassak et al., 2005;Tsiklauri, 2008).
We report on a study of electron inflows associated with three MP reconnection events and one magnetosheath electron-only event. The inflow velocities (v eN ) range from 0.05 to 0.25v AeL (with v AeL = the asymmetric electron Alfvén speed based on the reconnecting magnetic field in the inflow regions). The outflow speeds (v eL ) have magnitudes 1-3 times v eN , and the out-of-plane velocities (v eM ) are about 3-4 times v eN . While the strong v eM is well known as the main current contribution to J·E (ohmic dissipation), v eN has not been analyzed before. We derive a reconnection rate of 0.04 ± 25% for one event using the reconnection electric field (E M ) as R = E M /(v AeL B L ), with B L the average L component of the magnetic field in the EDR, as compared to a value of 0.12 ± 20% for R = v eN /v AeL .

Data and Methods
MMS makes electron-scale measurements at four locations within or surrounding magnetic reconnection sites in the boundary regions of the magnetosphere. The measurements are summarized by , with details on plasma and electric and magnetic field data provided by Pollock et al. (2016) and Torbert et al. (2016). A major enabling factor for electron-scale reconnection studies is the unprecedented 30-ms time resolution of 3-D electron distributions by the Fast Plasma Investigation (FPI). For special studies, even faster measurements at 7.5 ms are possible because of the interleaved nature of the azimuthal sampling of FPI (Rager et al., 2018). These faster measurements are derived during ground processing and are used in parts of the current study.

15 April 2018 Event
MMS encountered the MP near 10 magnetic local time (MLT) at low negative geocentric solar magnetospheric (GSM) latitudes. Figure 1a shows the maximum shear model (Trattner et al., 2012) applied to this event with the MMS position noted by the square symbol. Figure 1b shows the S/C positions in boundary-normal coordinates.
Figures 1c-1h plot magnetic field, electron velocities, and average ion velocities for MMS1-4. There was a significant guide field (B M ) of about 0.4 times the magnetosphere reconnecting field. The method of Schwartz (1998) was used to estimate a reconnection structure velocity along N at −23 km/s. Following Denton et al. (2016), the structure velocity along L is estimated at −50 km/s based on <v iL > as plotted in Figure 1f. The most prominent features in the electron velocity plots (Figures 1f-1h) are the peaks in v eM in (g), which carry the out-of-plane current associated with the reconnection EDR. Using the peaks in v eM as the most prominent markers, we note in Figure 1h that three of them are spanned by bipolar signatures in the inflow velocities v eN . Only in MMS4 is the inflow velocity unipolar, which may indicate that it encountered just one side of the EDR. These bipolar signatures are most easily seen for MMS2 (red plots). Figure 1f shows that also overlapping the v eM peaks is a single or binary v eL jet. Such v eL peaks can also be seen in Figure 3 of Pritchard et al. (2019). Figure 2 shows results of a reconstruction of the reconnection magnetic field using the polynomial method of Denton et al. (2020). A very similar result was obtained using the method of Torbert et al. (2020).  Figures 3 and 4 show data for MMS2 and MMS3, respectively. MMS was inbound at the morning-side (~10 MLT) MP, but since the MP was moving inward at a higher velocity, the MP crossing was outbound as seen by the magnetic field plots in Figures 1-4. In these plots, the FPI electron velocities in panel (c) and pitch angle distributions (PADs) in panel (i) are derived from the 7.5-ms data. The E-field data in panel (d) are averaged to 7.5-ms resolution so that the J·E values in panels (g) and (h) are also at 7.5-ms resolution. As shown in Figures 3b, 3f, 4b, and 4f, the wave activity was predominantly electrostatic with frequency peaks between the electron cyclotron and plasma frequencies. Figures 3c and 4c both show a strong peak of electron velocity in the M direction; the width of these peaks at half maximum is~1.5 d e , where d e is the electron inertial length of~1.5 km.

Comparison of MMS2 and MMS3
We first focus on the primary energy conversion site, which for both S/C was within the electron stagnation region, although as seen in Figure 2b, MMS3 was near the magnetosphere separatrix. We define this location by the time of the J·E peaks (43.75 s for MMS2 and 43.61 s for MMS3).
Next, we note in Figures 3c and 4c that the bipolar patterns of v eN and v eL overlap the broad peak of v eM ; but there is a difference in that for MMS2 these patterns are shifted only slightly to a later time, while this shift is  (Figure 4d), on the other hand, detected a bipolar signature of higher values of E M , which was accompanied by a slightly shifted bipolar signature of E L . The maximum negative value of E M during this bipolar trace coincided with the peak of J·E. This type of pattern was reported before by Burch et al. (2018) at the boundary between open and closed field lines in an EDR, which is consistent with the MMS3 location shown in Figure 2b.
We now focus on the J·E plots in panels (g) and (h) of Figures 3 and 4. J·E is a scalar quantity, but it is instructive to plot separately the contributions from the L, M, and N components of J and E in panel (g). The plots in (g) and (h) are made in the rest frame of the reconnection structure so that E str = E + v str × B. In Figure 3g it is notable that the green trace (J M E M ) has a significant positive value through the broad v eM peak shown in panel (c). There is a smaller, mostly positive blue trace (J L E L ) but a strong negative red trace (J N E N ). This negative J N E N is caused by the positive E N and the positive v eN shown in panel (c), which is the inflow velocity. This alignment of E N and v eN agrees with the conclusion of Swisdak et al. (2018) that the oscillations in J·E arise from changes in the sign of v eN .
In the total J·E plot in Figure 3h, the negative J N E N is seen to cause net negative values. The J·E plots for MMS3 in Figures 4g and 4h show a much stronger and narrower J M E M peak and only small negative values in the total J·E plot just before the peak. In this case the negative total J·E was caused by the positive values of E M shown in panel (d) within the positive v eM peak shown in panel (c).  , and (j-n) electron distribution function (DF) top: in plane perpendicular to B with v ⊥1 along E × B and v ⊥2 along E; middle: in plane containing B and v ⊥1 ; bottom: v L versus v N . The yellow, red, black, and blue traces in the wave spectrograms plot F pe (electron plasma), F pi (ion plasma), F ce (electron cyclotron), and F lh (lower hybrid) frequencies.

Geophysical Research Letters
Looking now at the region between the J·E peak and the B L reversal, Figure 3g for MMS2 shows a peak in J L E L associated with the outflow v eL in panel (c) and a peak in J N E N associated with a localized increase in the flow along −N noted above. The MMS3 data in Figure 4 are different in that the only secondary J·E peak occurred near the B L reversal where there is a peak in v eL within a broader region of v eL > 0 and E L < 0. The peak in v eL > 0 just before the B L reversal, which is not seen by MMS2, is interpreted as electron flow toward the X line in the magnetosheath boundary layer as shown by the simulations of Swisdak et al. (2018) just to the −L side of the X line. Moving to the left in Figure 4c, we note an enhancement in −v eN (earthward flow) and a decrease in v eL , suggesting a clockwise turning of the flow from the X line to the Earth, which is complete between the second and third vertical dotted lines where v eL drops to zero.  Figures 5a-5e show the event on 15 April 2018, which was presented in detail in section 3.1. The inflow speeds approached 0.14v AeL , and the peak outflow speed was only slightly larger than the inflow speeds, but with the out-of-plane speed reaching~0.7v AeL , following the same pattern as the other events. Also shown for this event are the 8,175/s electric field components in the X-line frame and correlations between 10-point average E M and E N (panel g) and E L (panel h), which are discussed in section 3.3. The second event, on 14 December 2015 (panels l-p), has been the subject of previous publications by Chen et al. (2017) and Graham et al. (2017) among others. The inflow speed reached~0.12v AeL , implying a reconnection rate near 0.12 while the outflow speeds (panel e) were about twice the inflow speeds. Similar patterns are seen for the event on 9 December 2016 (panels q-u), which is the magnetosheath electron-only reconnection event reported by Phan et al. (2018). This event is included because, although not discussed by Phan et al., electron inflows in the form of bipolar v eN were present and provide another valuable comparison. In this case the inflow speeds reached~0.25v AeL , implying a reconnection rate of~0.25. MMS2 measured a unipolar outflow speed (panel u) of up to 0.4v AeL , while the highest speeds were observed in the out-of-plane flow (panel t), which reached nearly 0.8v AeL .

Electron Flow
Comparable patterns are seen for the event on 29 December 2016 (Figures 5v-5z), which occurred in the electron stagnation region. For this event, which was published by Pritchard et al. (2019), the inflow speeds reached~0.05v AeL , indicating a reconnection rate of~0.05.

Electric Field Data and Error Analysis
The tilt of the X-line structure in the L-N plane in the Figure 2 reconstruction could be due to a combination of a sub-optimal LMN transform and inaccuracies in the reconstruction code. As shown in Figures 3e and 4e, there is very good equivalence between E M and (−v e × B) M except, as expected, when MHD is violated in dissipation regions as identified by J·E str . Thus, we conclude as have others (e.g., Torbert et al., 2017) that the measurements of v e are very accurate and that errors in the LMN transformation represent the primary measurable source of error in the reconnection rate. Following Genestreti et al. (2018), we can assess the accuracy of the LMN transform by comparing the values of E M , which we assume are uniform within the EDR, with the larger values of E L and E N . Plots of these comparisons are shown in Figures 5g and 5h. The slopes of the fit lines in these plots give the tilt angle of the two axes in radians since for small angles cos(tilt)~tilt. From Figure 5g the tilt angle in the N,M plane between the ideal and actual M axes is 10.7°, while Figure 5h shows the tilt in the L,M plane to be~10.1°. Since our LMN transform requires orthogonality, we conclude that a similar error occurs in the L,N plane.
We have applied a rotation to the electric field data using the matrix shown in the Figure 5 caption. This rotation minimizes the contamination of E M values by E N and E L . The mean value of E M is −1.164 mV/m with a standard deviation of 0.376. The normalized reconnection rate derived from E M is <E M >/ (v AeL <B L >)~0.04 ± 25%. In order to estimate the error in the reconnection rate derived from v eN /v AeL , we applied the same correction of the LMN transform to the electron velocity data and found a reconnection rate of 0.12, which is~16% lower than shown at the vertical dashed lines in Figure 5c for the original LMN transform. Thus, we conclude that the relative error in the reconnection rate derived from the electron inflow velocity is about ±20%, which is similar to the error in E M .

Conclusions
We have presented electron velocities within an MP reconnection site on 15 April 2018. The observations were made at all four spacecraft with detailed data shown for MMS2, which was well within the electron stagnation region, and MMS3, which was near the magnetosphere separatrix. In both cases electron outflow jets (along L and/or −L) and inflow velocities (along ±N) were observed to span the region of highest out-ofplane velocity and J·E. These observations are compared to previously published events on 14 December 2015, 9 December 2016, and 29 December 2016 and found to have similar values in that the inflow velocities were in the range from 0.05 to 0.25v eA , which provide normalized reconnection rates. We used electric field data to estimate the inaccuracy of the LMN transform for the 15 April 2018 event to be~10.5°. We corrected the transform errors by applying an additional matrix rotation and obtained a normalized reconnection rate of 0.04 ± 25%. Applying the same LMN correction to the electron velocity data yielded a reconnection rate about 16% below that shown in Figure 5c or~0.12 with an estimated error of ±20%. The difference between the reconnection rates derived from E M and v eN is not understood and will be investigated for more events. One possible explanation is the existence of v eN components parallel to B, which would not contribute to the advection of the magnetic field and so would reduce the reconnection rate derived from the inflow velocity. The mean angle between v eN and B in Figure 5c is 87.6°± 19%, which places this potential error within the 20% range derived from the LMN transform uncertainty.
The outflow velocities along ±L ranged up to about three times the inflow velocities. This ratio is similar to the theoretical prediction of the aspect ratio of the inner EDR based on electron trapping length in a field reversal (Hesse et al., 1999). A similar result (aspect ratio of inner EDR~4) was obtained with MMS for a tail reconnection event by Nakamura et al. (2019) who found v eL~0 .1v Ae in the outflow region. It is predicted that such sub-Alfvénic outflow in the inner EDR will accelerate to v AeL as the electrons move toward the exhaust region as shown by the simulations of Shay et al. (2007) and Drake et al. (2008).
Similar investigations are conducted in the laboratory with MRX: Yamada et al. (2018) observed the high out-of-plane velocity (v eM ) at the stagnation point of asymmetric reconnection; Ren et al. (2008) observed outflow velocities at 0.11v eA , which are consistent with our measurements.

Data Availability Statement
The 15 April 2018 reconnection event was discovered in a database created for the International Space Science Institute Team 442, "Study of the physical processes in magnetopause and magnetosheath current sheets using a large MMS database." The entire MMS data set is available online (at https://lasp.colorado. edu/mms/sdc/public/links/). Fully calibrated data are placed online at this site within 30 days of their transmission to the MMS Science Operations Center. The data are archived in the NASA Common Data Format (CDF) and so can be plotted using a number of different data display software packages that can use CDF files. A very comprehensive system called the Space Physics Environment Data Analysis System (SPEDAS) is available by downloading (http://themis.ssl.berkeley.edu/socware/bleeding_edge/) and selecting (spdsw_latest.zip). Training sessions on the use of SPEDAS are held on a regular basis at space physics-related scientific meetings. All of the data plots in this paper were generated with SPEDAS software applied to the publicly available MMS database, so they can readily be duplicated.