MMS Observations of Oscillating Energy Conversion and Electron Vorticity in an Electron‐Scale Layer Within a Southward Magnetopause Reconnection Exhaust

The MMS satellites traversed a ∼6 di‐wide and ∼500 km/s southward reconnection exhaust at the dayside magnetopause on 6 December 2015 and ∼29 di from the associated X‐line region. A narrow ∼0.26–0.34 di layer of enhanced ±3.5 nW/m3 oscillating energy conversion perpendicular to the magnetic field resides in this exhaust. It contained two regions of diverging in‐plane electric fields in general agreement with two clockwise electron flow vortices and a proposed increase of the electron vorticity ∇ × Ve. The layer developed sunward of a unipolar Hall magnetic field for a duskward BM/BL ∼ 0.9 guide field. Each electron flow vortex supported a local ∆BM ∼ 10 nT strengthening of this Hall field. The presence of this electron‐scale layer in a southward exhaust for a duskward guide field is consistent with a two‐dimensional simulation of a similar structure that evolved from an X‐line into a northward exhaust for a similar strength dawnward guide field.


Introduction
Magnetic reconnection is a universal plasma energy conversion process that releases stored magnetic field energy at thin current sheets into plasma heating and particle acceleration.Most of this energy conversion typically occurs in a narrow electron-scale diffusion region (EDR).Many EDRs and their associated energy dissipation have been recorded at high time resolution by the Magnetospheric Multiscale (MMS) satellites (Burch et al., 2016) for different external plasma and field conditions across current sheets at the Earth's dayside magnetopause (Webster et al., 2018).
The plasmas on the two sides of the dayside magnetopause current sheet are characterized by an asymmetry of the magnetic field strength, plasma number density and temperature.The gradients of magnetic field and density result in an Earthward displacement of the magnetosheath flow stagnation point relative to the magnetic reconnection X-line (e.g., Cassak & Shay, 2007;Cassak et al., 2017).Dayside reconnection exhausts consequently shift toward the Earthward side of the magnetopause current sheet.The magnetic fields on the two sides of the magnetopause are typically associated with an arbitrary rotation angle.This angle represents an out-of-plane component of the magnetic field, which is also referred to as a guide magnetic field.The guide field and asymmetries across the dayside magnetopause can generate various instabilities across the separatrix and exhaust regions that are still not completely understood, including an electron Kelvin-Helmholtz velocity shear instability (KHI) and its associated electron vorticity (e.g., Fermo et al., 2012;Pritchett & Mozer, 2009).Pritchett and Mozer (2009) (hereafter PM9) modeled the reconnection dynamics in a two-dimensional particle-incell (PIC) numerical code for a dayside asymmetric magnetopause in the presence of a uniform and substantial dawnward (B M > 0) guide field with magnitude equal to the magnetosheath B L reversing field (B M /B L = 1).The corresponding magnetic field rotation across the simulated magnetopause is 117°.A significant dawnward guide field changes the north-south symmetry of the reconnection Hall magnetic field relative to the X-line.The out-of-plane Hall magnetic field inside the northward exhaust region increases substantially, while the Hall magnetic field of the southward exhaust is much reduced.The simulation produced an electron velocity shear flow layer at the edge of a northward electron jet emanating from the EDR.The shear flow layer was deemed to be unstable to the electron KHI and resulted in a train of northward propagating electron flow vortices that extended at least ∼10 d i northward from the X-line, where 1 d i = c/ω pi is the ion inertial length.The electron vortices, with an estimated ∼0.3 d i normal width, are predicted to modulate the amplitude of the out-of-plane B M magnetic field by as much as 30%.Moreover, the electron-scale flow vortices are predicted to be associated with a violation of E + V e × B = 0 and an alternating J • (E + V e × B) energy conversion dominated by the components of current density J and non-ideal electric field E + V e × B perpendicular to a local B.An opposite duskward (B M < 0) direction of the guide field is expected to result in a similar electron-scale structure in a southward exhaust region near a dominant X-line.
We present MMS observations across a dayside [8.33, 3.96, 0.80] GSM R E magnetopause boundary on 6 December 2015 as they traversed a southward reconnection exhaust for a duskward (B M < 0) guide magnetic field from 2329:02 UT to 2329:14 UT.This event presents a particularly compelling case study when comparing measured exhaust structures with those predicted to exist at electron-scales by PM9.In Section 2, we present an overview of this exhaust as observed by MMS-1 in GSM coordinates to the south of a dayside magnetopause Xline.This is followed by a set of MMS-4 observations in an appropriate LMN coordinate system that focuses on a localized region of enhanced energy conversion sunward of the B L = 0 center of the magnetopause current sheet.The LMN system is defined with N being the direction normal to the current sheet, L being the direction of the reversing component of the magnetic field, and N × L = M along the guide field direction.Finally, we compare observations at all four MMS satellites for a short 1.6 s interval surrounding an electron-scale layer of alternating energy conversion within the exhaust.Section 3 provides a discussion on the similarities and differences between these electron-scale layer observations by MMS and those first predicted by PM9 to develop in a northward exhaust for a similar guide field magnitude in a dawnward direction.

Observations
In this event study of exhaust structure and energy conversion, we analyze MMS observations of the magnetic field (B) from the FGM instrument (Russell et al., 2016) at ∼8 ms resolution, the electric field (E) from the EDP instrument (Ergun et al., 2016;Lindqvist et al., 2016) at 31.3 ms fast mode cadence, ion moments at 150 ms and electron moments at 30 ms from the FPI instrument (Pollock et al., 2016).
Figure 1 shows MMS-1 observations of a southward (Viz < 0) plasma exhaust across the magnetopause with a duskward flow deflection (Figure 1g) that lasted Δt exh ∼ 4.7-s from 2329:05.4 UT to 2329:10.1 UT, as indicated between a pair of black, vertical dotted lines.The ions and electrons were co-moving with Vz ∼ 400 km/s and Vy ∼ 300 km/s (Figures 1g and 1h) in the central 2329:08-2329:09 UT exhaust region.The MMS tetrahedron moved from a high-density, low-temperature magnetosheath (Figures 1a-1d) into a low-density, hightemperature magnetosphere as the magnetopause expanded sunward.B (Figures 1e and 1f) rotated from a southward (Bz < 0) and duskward (By > 0) direction in the magnetosheath to a northward and dawnward field inside the magnetosphere.A red vertical dotted line at 2329:06.5 UT marks a section of the exhaust near this Bz rotation, where E and current density J = eN(V i V e ) changed significantly (see Figures 1i and 1j).As demonstrated below, this region supports an electron-scale exhaust structure with substantial energy conversion and enhanced electron vorticity.
The magnetopause expansion velocity can be estimated as V HT = [220.9, 194.1, 177.5]    of the J M < 0 magnetopause current density (Figure 2h) around the B L = 0 crossing.Second, the B N normal component is weakly positive during most of the southward exhaust as expected.The exception is a region of weak B N < 0 in a sunward section of the Hall B M region, which is centered about ∼2329:06.5UT as marked by a red vertical dotted line.Third, the southward V eL electron flows (Figure 2e) are comparable in magnitude to V iL (Figure 2d) in the exhaust region, while displaying highly variable and fast V eL ∼ ±800 km/s flows at ∼2329:09.7 UT in the vicinity of the earthward exhaust boundary.Fourth, there is an earthward normal electric field E N < 0 (Figure 2f) due to a mostly frozen-in electron motion V e × B in this V eL < 0 exhaust region associated with a B G < 0 guide field up until ∼2329:09.5UT, where E N then turns positive in agreement with the presence of a polarization E N > 0 Hall electric field at the earthward-side reconnection separatrix layer (Swisdak et al., 2018).
There are a few regions across the exhaust where MMS-4 recorded a violation of the frozen-in condition E + V e × B = 0 for the electrons (Figure 2g).One is associated with a peak of the ion exhaust at 2329:09 UT and another coincides with the earthward separatrix at ∼2329:10 UT.However, the relatively weak current density J = eN(V i V e ) in both regions (Figure 2h) does not result in any considerable energy conversion (Figure 2i).In contrast, the local enhancements of both J and E + V e × B centered about 2329:06.5 UT support significant positive and negative fluctuations of J • (E + V e × B) that reach about ±3.5 nW/m 3 in the electron frame of reference at MMS-4.This exhaust region of interest, immediately sunward of a B L = 0 reversal and the maximum deflection of the Hall magnetic field, is characterized by a southward current deflection (J L < 0) due to electron flows directed toward the X-line (V eL > 0) despite an otherwise southward exhaust flow regime.This electron shear flow region inside the exhaust is also associated with a considerable deflection of the in-plane components of the electric field, E L > 0 and E N > 0, as compared with the immediate background values.Localized regions of oscillatory J • (E + V e × B) energy conversion of about ±2 nW/m 3 have also been measured by MMS at the EDR and the adjacent flow stagnation region (Burch et al., 2018) for a similar B G /B L ∼ 1 guide field in basic agreement with PIC simulations (Swisdak et al., 2018).

Geophysical Research Letters
10.1029/2024GL109878 (Figure 3a).First, the strongest fluctuations in J • (E + V e × B) are measured at MMS-4 and MMS-2 (Figure 3r) and they are dominated by the perpendicular (Figures 3g and 3p) J • (E + V e × B) component with weaker and mostly negative contributions from the parallel (Figure 3q) component of J • (E + V e × B).Second, the in-plane components of J and E + V e × B (Figures 3k-3n) dominate this fluctuation of energy conversion between the electrons and fields as compared with a lower amplitude of the out-of-plane components (Figures 3j and 3o) when the largest values of a perpendicular J • (E + V e × B) are observed.Third, the electric fields measured in the satellite frames of reference at MMS-4 and MMS-2 (Figures 3d-3f) demonstrate a particularly clear correlation (whether in-phase or out-of-phase) as compared with the other two satellites during a Δt ∼ 300 ms duration time interval at 2329:06.340-2329:06.640UT.This period of enhanced J • (E + V e × B) and variable E is also associated with enhanced electron vorticity, Ω e = ∇ × V e (Figure 3h), as obtained from the measurements of electron velocity and known positions of the four MMS satellites (Ahmadi et al., 2022).The assumption of the curlometer technique (Dunlop et al., 2002), which is used to obtain Ω e , could be associated with variability at scales smaller than the dimension of the MMS tetrahedron at this time, since ∇ • B = 0 (Figure 3i) is not perfectly satisfied at ∼2329:06.5 UT.However, a comparison of J from the FPI instruments and J obtained from a curlometer technique (not shown) suggests that small-scale variations are potentially of lesser importance, and that the region of strong J • (E + V e × B) fluctuations is likely also associated with enhanced Ω e .
Figure 4 (top left) displays the along-track measurements of B as observed by MMS-2 and MMS-4 in the NL-plane at 30-ms electron cadence.Time is transformed into spatial coordinates (x N ,y L ) = (V HTN ∆t,V HTL ∆t) in terms of the ion inertial length (d i ∼ 84 km) for the 2329:06.0-2329:07.0UT interval, where V HTN = 106.1 km/s and V HTL = 294.2km/s.The start coordinates of the two MMS satellites are displayed using their respective separations from MMS-1 at (x N ,y L ) = (0,0).The color-scale of the in-plane B (top left) reflects the measured range of the perpendicular contribution to J • (E + V e × B) at MMS-4 for this 1-s period (see Figure 3g).The Δt∼300-ms region of interest, where enhanced energy conversion and significant rotations of E were recorded at either MMS-2 or MMS-4, are shown as blue-colored dots for MMS-4 at 2329:06.450-2329:06.660UT with a total duration of Δt 4 = 210-ms.Similarly, the region is shown as red-colored dots for MMS-2 at 2329:06.360-2329:06.630UT with a Δt 2 = 270-ms duration.Figure 4 (bottom left) shows the same in-plane B, but colored-coded using the measured range of B M (see Figure 3b).The corresponding in-plane vectors of the interpolated E at MMS-2 and MMS-4 are shown in Figure 4 (top right) at 30-ms cadence.Taken together, these two satellite measurements display two inplane regions of diverging E. The center of the first region is located at about (x N ,y L ) = ( 0.47, 1.46) d i and the second is located at about (x N ,y L ) = ( 0.63, 1.90) d i , shown as two black circles.
The two proposed locations of diverging electric fields, which are separated by ∼0.47 d i in this plane, strongly indicate that MMS recorded a southward motion of two regions of positive charge density.An in-plane clockwise flow vortex of (mostly) frozen-in electrons in a magnetic field dominated by its negative out-of-plane B M < 0 component adjacent to a maximum Hall field deflection would explain a positive charge density through the formation of a diverging electric field from E = V e × B (Stawarz et al., 2018).This is directly supported by the in-plane components of the electron velocity (V e ) at MMS-2 and MMS-4 as shown in Figure 4 (bottom right).MMS-4 recorded the sunward section of the first vortex with a V e rotation from sunward, northward, and finally earthward.MMS-2 sampled the earthward section of this first electron vortex in five 30-ms measurements with a V e rotation from a sunward and northward V e to a mostly southward V e , and finally a southward and earthward V e .The center of the second electron flow vortex crossed near MMS-4 with three V e measurements first pointing sunward and northward, followed by one fast southward and earthward V e measurement.The fastest sunward and southward V e measurement by MMS-2 likely caught the southern edge of this second electron vortex.Each clockwise electron flow vortex is expected to generate a negative dB M /dt near its center under the assumption that it supports a mostly in-plane current density in the opposite direction.A strengthening of the background (negative) B M is expected during the passage of each structure.The observed J L ∼ 0.80 μA/m 2 and the estimated 10 km ∼ 5 d e normal widths of each individual electron flow vortex are in general agreement with Ampère's law and each of the two distinct ∆B M ∼ 10 nT decreases of the measured B M magnetic field (see Figures 3b and 4, bottom left), which are separated by a temporary B M ∼ 40 nT plateau.

Discussion and Conclusions
PM9 included a possible electron flow vortex observation by a THEMIS satellite.However, this relied on indirect E measurements, typically affected by significant spin axis uncertainties (Bonnell et al., 2008), to obtain an E × B/B 2 drift in the absence of a direct V e .This THEMIS measurement cannot unambiguously verify an electron vortex presence predicted to also coincide with a non-ideal E. In contrast, MMS records E with adequate double probe baselines (Ergun et al., 2016;Lindqvist et al., 2016) and 30-ms V e observations (Pollock et al., 2016) that allow a first direct detection of electron flow vortices and their mostly diverging E from multiple satellites at electron-scale separations.
In assuming that the observed structure of perpendicular and oscillatory J • (E + V e × B) is moving at the magnetopause V HTN ∼ 106 km/s normal speed, we can estimate a normal width of this active J • (E + V e × B) layer to range from ∼22 km at MMS-4 for a Δt 4 = 210-ms duration to ∼29 km at MMS-2 for a Δt 2 = 270-ms duration.This corresponds to ∼0.26-0.34d i in a normal direction or ∼11-15 d e for a background number density N ∼ 7.3 cm 3 of the exhaust plasma at this location.
The estimated normal width of the electron-scale layer is notably larger than the ∼8 d e separation of the MMS satellite formation (see Figure 2).The elevated Ω e obtained from the curlometer method (see Figure 3h) thus indicates a presence of an actual Ω e , since the scale size of the spacecraft tetrahedron is smaller than the structure itself.However, it is unlikely that Ω e thus obtained over the ∼8 d e tetrahedron can resolve the presence of the two ∼5 d e wide electron flow vortex structures that MMS encountered within this ∼0.3 d i wide energy conversion layer as shown in Figure 4 (bottom right).
PM9 proposed that a narrow ∼0.3 d i wide layer of electron flow vortices should form due to sheared electron flows and an associated electron KHI layer earthward of B L = 0 within a northward reconnection exhaust for a dawnward guide-magnetic field (B M > 0) in the range 0.33 < B G /B L < 1.The width of the observed J • (E + V e × B) activity channel on 6 December 2015 compares favorably with the predicted width for a very similar guide magnetic field 0.43 < B G /B L < 0.87 as observed by MMS across the magnetopause.The expectation from the simulation is that this electron vorticity channel should only form on the side of the X-line (north or south) where the Hall magnetic field enhances the background guide field.The fact that MMS observed an electron-scale structure within a southward ion-scale exhaust is consistent with a duskward guide field.Moreover, the presence of northward V eL > 0 electron flows that support an in-plane J L < 0 Hall current sunward of the V eL <0 exhaust flow from the X-line (see Figures 2e and 2h) in a dominant B M < 0 field could potentially support an onset of the electron KHI (Fermo et al., 2012;Pritchett & Mozer, 2009).
The electron vorticity layer is predicted to be associated with a violation of the frozen-in condition for electrons with a finite electric field E + V e × B in the electron frame of reference.However, whereas the two-dimensional PIC simulation emphasized an out-of-plane (E M ) component of this non-ideal electric field, MMS rather observed a dominant violation of the in-plane electric fields (E N and E L ) with a lower magnitude of the out-of-plane E M (see Figures 3m-3o) when MMS recorded the largest amplitude of a perpendicular J • (E + V e × B).
The narrow layer is predicted to be elongated by at least ∼10 d i from the X-line into the reconnection exhaust region.MMS can confirm that it is at least ∼8 d e long based on the strong and simultaneous J • (E + V e × B) activity observations at MMS-2 and MMS-1.However, under the assumption of a fast ∼0.1 reconnection rate, it is likely that MMS was ∼30 d i to the south of the corresponding X-line and we may expect a similar spatial extent of this electron energy conversion layer.
The predicted electron-scale layer is associated with alternating J • E energy conversion (positive and negative) between the electrons and the fields, and predominantly associated with the perpendicular components of J and the non-ideal E + V e × B. MMS can corroborate an oscillating and predominantly perpendicular J • (E + V e × B) energy conversion within this thin layer, while a smaller amplitude of the parallel J • (E + V e × B) conversion is consistently negative with energy being extracted from the particles to the fields.The perpendicular J • (E + V e × B) conversion is likely due to the presence of the dominant out-of-plane component of a Hall magnetic field.
Finally, PM9 proposed a substantial 30% change of the out-of-plane magnetic field magnitude relative to B L0 = (B L1 + B L2 /2) because of an electron flow vortex, where B L0 is defined as the average of the asymptotic values of the reversing magnetic field magnitudes on the magnetosheath (B L1 ) and magnetosphere (B L2 ) sides of the magnetopause current layer.From Figure 2, we note B L1 ∼ 30 nT and B L2 ∼ 60 nT at the edges of the exhaust region.The ∆B M ∼ 10 nT change across each electron flow vortex would correspond to ∼20% of this B L0 = 45 nT.
In summary, we conclude that MMS encountered a ∼0.3 d i wide electron-scale layer of alternating energy conversion inside a southward exhaust as predicted from a two-dimensional PIC simulation to exist in a southward exhaust for a duskward guide magnetic field.This is the first time that all the kinetic signatures of this electron-scale energy conversion layer have been measured directly at adequate time resolution from within a dayside reconnection exhaust region with each electron flow vortex contributing a substantial fraction toward a large-scale out-of-plane Hall magnetic field.
km/s (GSM) at 2329:05-2329:09 UT (see Figure 1, right) from a high-quality deHoffmann-Teller (HT) boundary moving frame of reference.A local LMN coordinate system is obtained for this magnetopause interval, where N GSM = [0.6742,0.6065, 0.4215] from MMS-4 measurements using the Burch et al. (2020) optimization criterion and M GSM = [ 0.7211, 0.4170, 0.5533] is obtained as the cross-product of N GSM and a maximum magnetic field variance L MVA = [ 0.1665, 0.6709, 0.7226] for MMS-4 observations at 2329:04.670-2329:11.190UT.A final L GSM = [ 0.1598, 0.6770, 0.7185] completes the LMN system from L GSM = M GSM × N GSM .The corresponding V HT = [ 294.2, 142.0, 106.1] km/s in this LMN system indicates a fast, outward V HTN = 106 km/s average magnetopause motion along N GSM .The associated width of the reconnection exhaust can be estimated from d cs = V HTN Δt exh or d cs ∼ 495 km.In using a plasma density N ∼ 7.3 cm 3 within the exhaust at 2329:06.6 UT to represent the spatial scale of all structures that MMS recorded across this layer, we have d i ∼ 84 km and d e ∼ 2 km, with 1 d e = c/ω pe being the electron inertial length.That is, the exhaust width is d cs ∼ 6 d i for this density.Under the assumption of a fast 0.1 reconnection rate, we may also estimate that MMS traversed this southward exhaust at a distance ∆L = 5d cs or ∆L ∼ 2,476 km from the X-line.This translates to a mere ∆L ∼ 29 d i distance.

Figure 2
Figure2displays MMS-4 observations in the LMN system across the southward V iL ∼ 500 km/s exhaust and the locations of the MMS satellites relative to MMS-1 at 2329:06.5 UT in terms of d e .We note that MMS-4 measured an average V iN ∼ 100 km/s after 2329:06 UT (Figure2d) in agreement with V HTN .The magnetopause is characterized by a field rotation of ∼110°from [B L , B M , B N ] ∼[ 30, 28, 15]  nT in the magnetosheath at ∼2329:04-2329:05 UT to [B L , B M , B N ] ∼ [60, 26, 5] nT in the magnetosphere at ∼2329:10 UT (Figures2b and 2c).The B M component demonstrates the presence of a large-scale and dusk-oriented guide magnetic field, B G ∼ 26 nT, across the magnetopause.The corresponding ratio of this B G with the reversing magnetic field is B G /B L ∼ 0.87 sunward of the magnetopause and B G /B L ∼ 0.43 on the earthward side of the boundary.There are several reconnection-associated signatures to note from Figure2.First, the observed B M confirms the presence of a strong and unipolar B M ∼ 50 nT Hall magnetic field enhancement at ∼2329:06-2329:08 UT directed toward dusk as expected for a duskward B G .This Hall B M region coincides with the most intense section

Figure 1 .
Figure 1.MMS-1 observations are shown in a GSM coordinate system (left) on 6 December 2015, at 2329:02-2329:14 UT as the magnetopause expanded sunward across MMS: (a) Ion energy-time spectrogram; (b) electron energy-time spectrogram; (c) plasma number density (N i : ion in black, N e : electron in red); (d) plasma temperatures perpendicular to B (T i : ion in black, T e : electron in blue) and parallel to B (T i : ion in green, T e : electron in red); (e) magnetic field strength B; (f) magnetic field B;(g) ion velocity V i ; (h) electron velocity V e ; (i) electric field E; (j) current density J = eN(V i V e ).A deHoffmann-Teller (HT) moving frame of reference is obtained (right) with V HT= [220.9, 194.1, 177.5] km/s (GSM) with x, y, z GSM components shown as black, green and red colored dots.

Figure 2 .
Figure 2. observations are shown in LMN components on 6 December 2015, at 2329:02-2329:10.5 UT (left): (a) Plasma density; (b) B L (black) and magnetic field amplitude (green); (c) B M (red) and B N (black); (d) ion velocity; (e) Lcomponent of electron velocity; (f) electric field; (g) electric field in the electron frame of reference; (h) current density J = eN(V i V e ) from the FPI instruments at 30-ms cadence; (i) energy conversion J • (E + V e × B).MMS satellite separations shown relative to MMS-1 in d e -scale (right).

Figure 3
Figure 3 compares MMS-4 observations of enhanced J • (E + V e × B) fluctuations and in-plane E rotations with those recorded by the other satellites during a 1.6-s interval centered at the region of interest for 20 < B L < 30 nT

Figure 3 .
Figure 3. MMS observations from 2329:05.8 UT to 2329:07.4UT on 6 December 2015 in LMN components with MMS-1 in black, MMS-2 in red, MMS-3 in green, and MMS-4 in blue: (a) B L ; (b) B M ; (c) V eL ; (d) E N ; (e) E L ; (f) E M ; (g) perpendicular J • (E + V e × B) energy conversion at 30-ms DES cadence; (h) amplitude of the electron vorticityΩ e = ∇ × V e ; (i) ∇ • B; (j) J M ; (k) J N ; (l) J L ; (m) [E + V e × B] N ; (n) [E + V e × B] L ; (o) [E + V e × B] M ; (p) perpendicular J • (E + V e × B); (q) parallel J • (E + V e × B); (r) total energy conversion J • (E + V e × B).The electric field (d)-(f) is shown at 31.3 ms fast cadence.The current density (j-l), electron-frame electric field [E + V e × B] (m-o) and J • (E + V e × B) (p-r) are shown at 30-ms electron cadence.Two red vertical dotted lines mark the region of interest between 2329:06.310UT and 2329:06.670UT when MMS observed perpendicular J • (E + V e × B) fluctuations and significant rotations of the in-plane electric field.
Figure 3. MMS observations from 2329:05.8 UT to 2329:07.4UT on 6 December 2015 in LMN components with MMS-1 in black, MMS-2 in red, MMS-3 in green, and MMS-4 in blue: (a) B L ; (b) B M ; (c) V eL ; (d) E N ; (e) E L ; (f) E M ; (g) perpendicular J • (E + V e × B) energy conversion at 30-ms DES cadence; (h) amplitude of the electron vorticityΩ e = ∇ × V e ; (i) ∇ • B; (j) J M ; (k) J N ; (l) J L ; (m) [E + V e × B] N ; (n) [E + V e × B] L ; (o) [E + V e × B] M ; (p) perpendicular J • (E + V e × B); (q) parallel J • (E + V e × B); (r) total energy conversion J • (E + V e × B).The electric field (d)-(f) is shown at 31.3 ms fast cadence.The current density (j-l), electron-frame electric field [E + V e × B] (m-o) and J • (E + V e × B) (p-r) are shown at 30-ms electron cadence.Two red vertical dotted lines mark the region of interest between 2329:06.310UT and 2329:06.670UT when MMS observed perpendicular J • (E + V e × B) fluctuations and significant rotations of the in-plane electric field.

Figure 4 .
Figure 4. In-plane measurements of magnetic field (left), electric field (top right) and electron velocity (bottom right) are displayed along the MMS-2 and MMS-4 trajectories during 1.0 s from 2329:06.0 UT on 6 December 2015.The color scales of the magnetic field vectors represent the dynamic ranges of the perpendicular component of J • (E + V e × B) at MMS-4 (top left) and B M (bottom left).The approximate centers of two clockwise electron vortices are displayed as black circles.