The EDR inflow region of a reconnecting current sheet in the geomagnetic tail

Coupling of solar-wind energy into the Earth's magnetosphere occurs via magnetic reconnection, which creates open field lines that are swept into the geomagnetic tail. To complete the process, reconnection occurs again in the tail, thereby converting the open field lines to closed field lines, which convect toward the inner magnetosphere resulting in magnetospheric substorms and the aurora. Magnetic reconnection also plays a major energy-conversion role in solar physics, astrophysics, and laboratory plasma physics (Burch and Drake, 2009). The fundamental importance of magnetic reconnection for solar-wind magnetosphere interactions was the justification for the NASA Magnetospheric Multiscale (MMS) mission (Burch et al., 2016a). One of the objectives of MMS is to determine the normalized reconnection rate or the inflow speed normalized by the outflow speed of reconnected field lines. The reconnection process occurs within an electron diffusion region (EDR), which surrounds an X line and within which electrons become demagnetized. Based on the time scales of solar and magnetospheric disturbances, Parker (1973) suggested a universal reconnection rate of at least 0.1 V_A, where V_A is the Alfvén speed in the upstream region, or a normalized reconnection rate of 0.1. Computer simulations leading up to the MMS launch in 2015 indicated that a reconnection rate of ∼0.1 is a universal property of reconnection (Shay et al., 1999), so tests of this conclusion became an important goal of the mission. Cassak et al. (2017) provide a historical view of how the maximum reconnection rate of 0.1 was derived from numerous experimental and theoretical studies and why it persists under a variety of assumptions from Hall MHD to resistive MHD to collisionless plasmas. With MMS, reconnection rates above 0.1 (up to 0.2) have been derived in the magnetotail (Genestreti et al., 2018a) and in magnetosheath electron-only reconnection events (Burch et al., 2020), while rates from 0.05 to 0.14 have been observed for magnetopause reconnection by Burch et al. (2020). It is now clear that there is not a universal rate or maximum rate of 0.1 and that much work is left to be done to determine what determines the actual rate.

There are at least three different ways of measuring the reconnection rate (Burch and Lewis, 1999): (1) measure the inflow velocity at the edge of the reconnection diffusion region, (2) measure the reconnection electric field, and (3) measure the aspect ratio of the diffusion region. These measurements require an accurate determination of the boundary-normal coordinate system with L along the reconnecting magnetic field, N normal to L in the plane of reconnection, and M along the reconnection X line. In these coordinates, the inflow velocity is V_±N; the reconnection electric field is E_−M at the magnetopause and E_M in the magnetotail, corresponding to a right-handed system; and the aspect ratio of the EDR is dL/dN.

As shown by Burch et al. (2020) for asymmetric magnetopause reconnection, when the inflow velocities at the edge of the EDR are scaled to the electron Alfvén speed (V_Ae), they provide reconnection rates (Karimabadi et al., 2013; Klimas, 2015) because near the EDR, the magnetic field is advected by the electrons (Cassak et al., 2005; Tsiklauri, 2008). This method of determining the reconnection rate has not been reported for tail reconnection, possibly because of the very low inflow velocities observed in the published events. For the tail reconnection event reported in this study, higher inflow velocities occurred, and this method is shown to be useful.

Methods (2) and (3) have been employed for the 11 July 2017 tail reconnection event by Genestreti et al. (2018a), Nakamura et al. (2018; 2019), and Torbert et al. (2018) with normalized reconnection rates between 0.1 and 0.2. For the 6 July 2017 event reported here, we have used data taken at times during which the four MMS spacecrafts were closely spaced at an average separation of about 17 km (∼1.5 d_e) within a secondary EDR located within the exhaust region of a primary reconnection region. The EDR was identified by electron crescent distributions (Hesse et al., 2014; Burch et al., 2016b), out-of-plane current, and energy conversion seen simultaneously from all four spacecrafts. Because of the close spacing within the EDR, we were able to estimate fairly accurately the terms in the generalized Ohm's law with the result that the reconnection electric field was primarily formed by divergence of the electron pressure tensor. We used methods (1) and (2) to estimate the normalized reconnection rate. With method (1), the inflow velocity normalized to the electron Alfvén speed at the edge of the EDR gave reconnection rates between 0.11 and 0.14. With method (2), the reconnection electric field gave rates between 0.14 and 0.17 when normalized to the product of the ion Alfvén speed and reconnecting magnetic field in the tail lobe (as was done by Genestreti et al., 2018a and Nakamura et al., 2018).

On 6 July 2017, MMS encountered a reconnecting current sheet in the geomagnetic tail. Orbit plots and the spacecraft (s/c) tetrahedron configuration are shown in Figs. 1 and 2, respectively. As shown in Fig. 1, the MMS constellation, denoted by the cyan diamond, was northward of the neutral sheet at a geocentric distance of about 22 R_E. Although not shown, MMS was located in the pre-midnight region at about 2300 MLT (magnetic local time). Figure 2 shows the spatial configuration of the four s/c, which had an average separation of ∼17 km, one of the smallest used to date in the tail region. While this small separation perhaps prevented the s/c from sampling both sides of the X line, it did allow for simultaneous four-point measurements of inflow velocities and reconnection electric fields in the EDR.

A 12-s summary plot of MMS2 data on 6 July 2017 is shown in Fig. 3 where the EDR event of interest is bracketed by the two maroon-colored vertical lines. As shown in Figs. 3(a) and 3(b), the L component of B was positive throughout the interval, indicating that the spacecraft was north of the neutral sheet. Two near magnetic nulls occurred just before the EDR was encountered. Also notable is that the ion velocity was positive (earthward) throughout the interval, indicating that the EDR was part of a secondary reconnection site located in the earthward exhaust of a primary reconnection site tailward of the MMS constellation. The most prominent feature shown in Figs. 3(g) and 3(h) is the peak of electron velocity in the −M direction and the associated current in the M direction. This feature is the reconnection out-of-plane current, which, along with significant J_ME_M [Fig. 3(l)] and electron crescent distributions (shown later), strongly indicates the encounter with an EDR. Between the EDR and the vertical blue line labeled “lobe,” MMS2 passed through a turbulent region, which we identify as the plasma-sheet boundary layer (PSBL), until it emerged into the lobe. Although there is energy dissipation (J E) and strong wave activity in the PSBL, there are no indications of EDRs. The magnetic field strength and plasma density in the lobe are used in Sec. III B to determine the electric field in the large-scale inflow region of reconnection.

In order to identify the context of the observed reconnecting current sheet within the larger region of tail reconnection, we performed a simulation of the type described by Hesse et al. (2018) for the conditions encountered in our event with the results shown in Fig. 4. The 2.5-dimensional particle-in-cell (PIC) simulation starts from a Harris sheet with an additional density of 0.2 and a small, X-type perturbation. While periodic boundary conditions are used in the simulation, the resulting recirculating flows do not reach the region of interest during the simulation run. The top panel of Fig. 4 shows the out-of-plane (y component) current density with a primary X line apparent at about 50 d_i (di = c/ω_pi, the ion inertial length within the Harris sheet) and secondary X lines to the left and right of it. The secondary X line that we identify with our region of interest is the one on the Earthward (right-hand) side of the primary X line between 60 and 70 d_i. In the bottom panel, Fig. 4 is plotted B_z, v_ix, and v_ex along z = 0, and the most prominent feature being the large electron jet reversal at the primary X line. A smaller electron jet reversal shifted toward positive v_ex occurs at the secondary X line, noted by the horizontal blue bar. The ion velocity, v_ix, remains positive across the secondary X line, which is consistent with it being located in the exhaust of a primary X line. Reference to Figs. 3(f) and 3(g) shows that the same pattern of v_ix and v_ex, (blue traces) occurred at the EDR region denoted by the two maroon-colored vertical lines. A closer look at these ion and electron velocities is shown later.

In order to determine the configuration of magnetic field lines in the vicinity of the X line, we used the method of Torbert et al. (2020) and Denton et al. (2020). The method of Denton et al. (2020) was used to obtain the results shown in Figs. 5 and 6. Shown in Fig. 5 is a red square with a dimension of 50 km, which is ∼5 d_e or 3.3 r_g for a 500-eV electron, where d_e is the electron inertial length (c/ω_pe) and r_g is the electron gyro-radius. Figure 5 shows that the four MMS spacecraft were all located northward of the X line within about 4 d_e or 2.5 r_g of the X line.

Figure 6 shows the temporal development of the X line reconstruction at 0.03s intervals from 08:37:07.05 to 08:37:07.30 UT. The sequence of plots from 08:37:07.05 to 08:37:07.3 UT shows a movement of the X line toward the left, which is the tailward direction, of about 260 km/s as compared to the earthward ion flow of about 700 km/s shown in Fig. 3(f).

Evaluation of the generalized Ohm's law for dayside reconnection events sampled with MMS has been performed by Torbert et al. (2016) and Genestreti et al. (2018b). Since the average spacecraft separation of ∼17 km for this event was much smaller than the typical separation of ∼40 km in the magnetotail, it is expected that the sources of the reconnection electric field could be determined more accurately than for other tail events because all four spacecraft were simultaneously located within the EDR. For the analysis of the 6 November 2017 event, we followed the approach of Genestreti et al. (2018a) by computing the barycentric current with curlB from the four MMS spacecraft and determining the electric field resulting from the FPI measurements of the divergence of the electron pressure tensor (−∇·$P¯$_e/en) and the inertial electric field [−m_e∇·(v_ev_e)/en]. Figure 7 shows the total J·E′ in black, J·E′ from the inertial electric field at the barycenter of the four spacecraft (J·E′_Inertial) in blue and J·E′ from the divergence of the electron pressure tensor at the barycenter (J·E′_DivPe) in red. We note that the total J·E′ compares favorably with that shown for MMS2 in Fig. 3. The inertial term is very small, while the pressure divergence term is larger by almost a factor of ten. Between 7.18 s and 7.28 s, J·E′ is positive.

Shown in Fig. 8 is the measured electron velocities and electric fields for all four spacecraft as they passed through the EDR, which is bounded by maroon vertical lines. Also shown in Figs. 8(a)–8(c) is the ion velocity averaged over the four s/c (dashed magenta curves).

Within the EDR, the MMS3 M and N components of electric fields and electron velocities deviated significantly from those of the other three s/c, which agreed fairly closely with each other. Again, this difference is attributed to the fact that MMS3 was about twice as far north of the X line than the other three s/c. Figure 8(a) shows that both v_eL and v_iL remained positive before the EDR encounter, which is consistent with the flow from a primary X line located tailward of the MMS position. However, as shown in Fig. 8(a), v_eL dropped to near zero within the EDR as the flow turned toward the X line, with v_eN becoming negative in Fig. 8(c). Figure 8(b) shows the out-of-plane electron velocity, which reached negative values greater than 3000 km/s for MMS 1, 2, and 4 with somewhat lower values for MMS3, while v_iM maintained a negative value of only a few hundred km. Figure 8(c) shows v_eN, the inflow velocity, which was negative (southward) and peaked strongly in the latter part of the EDR (between 07.2 and 07.3 s). Again, the traces are very similar for MMS 1, 2, and 4 with a significant difference for MMS3. Figures 8(d)–8(f) shows the electric field measurements with small values along L, peak values along M in the same time 07.2–07.3 s time period noted for the inflow velocity, and strong negative values along N (the Hall electric field). Because of the peaks in v_eM and E_M, we identify the time period 07.2–07.3 s as the inner EDR and will use this time period to derive reconnection rates in sections that follow.

Figure 9 provides field and plasma data from all four MMS spacecraft during the EDR encounter. We note first, as in Fig. 8, the strong similarity of all parameters for MMS1, 2, and 4 with some notable differences for MMS3, because of its position farther from the X line. For the line plots in Figs. 9(a)–9(e), the vertical dashed lines indicate the EDR. Figure 9(a) shows the magnetic field components in the LMN coordinate system shown in the caption of Fig. 2. Figure 9(b) shows the electron bulk velocity with the L component decreasing to small values within the EDR as noted in Fig. 8(a), the M component being most prominent and accounting for the out-of-plane current, and the N component having small negative values as expected for the reconnection inflow velocity. Figure 9(c) shows small L components of E, somewhat larger positive values of E_M (the reconnection E field), and the largest component, E_N, as the Hall electric field. Figure 9(d) contains J_M·E_M′, showing that the energy conversion associated with the out-of-plane components of J and E′ is tightly constrained within the EDR for MMS1, 2, and 4 but with a profile for MMS3 that is broadened and shifted toward earlier times.

Figures 9(f)–9(h) show reduced electron velocity distribution functions (VDFs) for MMS 1, 2, and 4, summed over the axis orthogonal to each displayed pane and measured within the EDR every 30 ms after 08:37:07 UT as noted by the red numbers in each frame, which denote the start time of each VDF. The VDFs [panels (f)–(h)] are plotted in the v_⊥1–v_⊥2 plane and show crescent distributions along the v_⊥1 axis.

Figure 10 shows (a) the magnetic-field LMN components, (b) electron density, (c) electron inflow velocity (N direction), (d) outflow velocity from primary X line (L direction), and (e) out-of-plane electron velocity (M direction). All velocities are normalized to the electron Alfvén speed. As used by Burch et al. (2020) for a dayside magnetopause reconnection event, the normalized inflow velocity gives the normalized reconnection rate. For MMS 1, 2, and 4, the minimum value of v_eN/v_eA was between 0.15 and 0.2 and was located within the region of strong out-of-plane velocity, v_eM/v_eA, which is the best measure of closest approach to the X line.

Another measurement of the reconnection rate is the reconnection electric field, E_M, with the normalized reconnection rate being E_M/E_o, where E_o is the convection electric field of the inflow. One approach to obtaining E_o is determining the reconnecting magnetic field (B_L) and the ion Alfvén speed in the lobe region so that E_o = v_iALB_L, as was done by Nakamura et al. (2018) and Genestreti et al. (2018a) for another magnetotail reconnection event. Another approach is to evaluate E_o at the edge of the EDR with E_o = v_eALB_L, as was done by Burch et al. (2020) for a magnetopause reconnection event. We used both approaches after determining E_M in the EDR by eliminating contamination from E_N, as described by Genestreti et al. (2018a), and also from E_L.

Figure 11 shows magnetic and electric fields from MMS1 and MMS4 for the time period 08:37:06.8–07.4 UT on 6 July 2017. Figures 10(a)–10(d) shows the LMN components using the transformation in the caption of Fig. 2. As noted by Genestreti et al. (2018a), E_M is normally the smallest electric-field component, particularly when compared to E_N and so is subject to contamination from the other two components. While Genestreti et al. (2018a) tested numerous different coordinate transforms to find an optimum one with minimal contamination from E_N, we started with the transformation used up until now and used trial and error to minimize the contamination as was done by Burch et al. (2020) for a magnetopause reconnection event. The new transform, L^*M^*N^*, is shown in the caption of Fig. 11. As shown in Figs. 11(e) and 11(f), the E_M analysis was performed over the zoomed-in time interval 7.21–7.29 s where E_M remained positive. That the same time interval and the same L^*M^*N^* coordinate transform applied to both MMS1 and MMS4 are consistent with their near co-location in the L–N plane and separation of about 20 km along M (parallel to the X line).

Figure 12 shows scatterplots of E_M vs E_L and E_M vs E_N for MMS1 and MMS4 for the original LMN transform and the optimized L^*M^*N^* transform. Noted in Fig. 12 caption is the original average of E_M and its optimized value, which for MMS1 was 2.35 mV/m and for MMS4 was 1.94 mV/m. Referring to Fig. 3, where the lobe encounter of MMS2 is noted by the vertical blue line, we find that E_o = v_iAB_L= 14.1 mV/m. Using this value, the normalized reconnection rates for MMS1 and MMS4 were 0.17 and 0.14, respectively. The alternate approach, using E_o = v_eAB_L in the EDR inflow region, gives E_o= 38.2 mV/m and normalized reconnection rates for MMS1 and MMS4 of 0.06 and 0.05, respectively. While both approaches give results within a factor of two of 0.1, the significant differences are not unexpected because of different inflow regimes of the lobe and the EDR inflow region. The fact that using the inflow velocity yielded normalized reconnection rates intermediate between the rates determined by the two E_M methods is perhaps evidence that all three methods are valid considering the uncertainties in both the geometry of the X-line structure and the depth of penetration of MMS into the tail lobe.

A unique configuration of MMS spacecraft in the tail region on 6 July 2017 was investigated in this study. The event was unique because the average s/c separation was small (∼17 km), and the X line, which was located within the Earthward exhaust of a primary X line, was moving tailward at about 260 km/s.

The normalized reconnection rate was determined by two different methods, electron inflow velocity and reconnection electric field, with results in the range of theoretical prediction. First, by measuring the electron inflow velocity (v_eN) for the first time in a tail reconnection event, we found, after normalizing the inflow velocity to the product of the electron Alfvén speed and the reconnecting magnetic reconnection rates between 0.16 and 0.2 for the four spacecraft. Second, the reconnection electric field (E_M) was measured accurately by eliminating contamination from the L and N components with an optimized boundary normal coordinate (LMN) transform. When E_M was normalized to the lobe inflow velocity estimate (v_iAB_L), reconnection rates for MMS1 and MMS4, which were closest to the X line, were found to be 0.17 and 0.14, respectively, in good agreement with the values determined by the electron inflow velocity. When, alternatively, E_M was normalized to v_eAB_L in the inflow region, lower reconnection rates of 0.06 and 0.05 were found. Further refinement of these rates should be possible through analysis of other events.

This work was supported by NASA under Contract No. NNG04EB99C at SwRI (Southwest Research Institute). The OMNI data were obtained from the Goddard Space Flight Center/SPDF (Space Physics Data Facility) OMNIWeb interface at https://omniweb.gsfc.nasa.gov.

The authors have no conflicts to disclose.

The entire MMS dataset 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 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 (see MMS Science Data Center).