Statistical Characteristics of Electron Vortexes in the Terrestrial Magnetosheath

Utilizing the unprecedented high-resolution Magnetospheric Multiscale mission data from 2015 September to 2017 December, we perform a statistical study of electron vortexes in the turbulent terrestrial magnetosheath. On the whole, 506 electron vortex events are successfully selected. Electron vortexes can occur at four known types of magnetic structures, including 78, 42, 26, and 39 electron vortexes observed during the crossings of the current sheets, magnetic holes, magnetic peaks, and flux ropes, respectively. Except for the four types of structures, the rest of the electron vortexes are in the “Others” category, defined as unknown structures. The electron vortexes mainly occur in the subsolar region, with only a few in the flank region. The total occurrence rate of all electron vortexes is 4.86 hr–1, with, on average, 3.65 events hr−1 in the X-Y plane and 3.26 events hr−1 in the X-Z plane. The durations of most of the electron vortexes concentrate within 0.5–1.5 s and are 1.09 s on average. The electron vortexes are ion-scale structures owing to the average scale of 2.05 ion gyroradius. In addition, the means, medians, and maxima of the energy dissipation J · E′ in the electron vortexes are almost positive, implying that the electron vortex may be a potential coherent structure or channel for turbulent energy dissipation. All these results reveal the statistical characteristics of electron vortexes in the magnetosheath and improve our understanding of energy dissipation in astrophysical and space plasmas.

Plasma turbulence plays a critical role in energy transferring from large to small scales and energy conversion between the fields and the particles (Schekochihin et al. 2009;Zimbardo et al. 2010;Franci et al. 2017).Various small-scale coherent structures can be formed in a self-consistent manner in turbulence with energy transfer and conversion, and it is considered that these structures are directly related to the cascade and dissipation mechanism of turbulent energy (Naulin & Spatschek 1997;Grulke et al. 2001;Wu et al. 2013).Furthermore, a series of coherent structures at small scales have been detected in the space plasma environment, such as electron-scale magnetic holes (MHs), thin current sheets (CSs), and other discontinuities (Huang et al. 2017a(Huang et al. , 2017b(Huang et al. , 2019a(Huang et al. , 2021;;Yao et al. 2017;Liu et al. 2019).
The electron vortex is a fluid field structure where the electron motion is in the form of a vortex.The in situ observations of the electron vortex usually appear as the bipolar variations of electron velocity and the large electron vorticity (Huang et al. 2017b;Jiang et al. 2020).Electron vorticity is defined as the curl of the electron velocity vector, the magnitude of which indicates the strength of local rotation or parallel shear of the fluid motion.Previous studies have manifested the electron vortex in the MH through particle-in-cell (PIC) simulations (Haynes et al. 2015;Roytershteyn et al. 2015) and considered such holes as a new type of coherent structure formed in turbulent magnetized plasmas.Huang et al. (2017b) have reported on the direct in situ observations of an electron vortex MH (EVMH) in the turbulent magnetosheath plasma for the first time.Huang et al. (2017a) have investigated the features of kinetic-size MHs (KSMHs) on statistical overview and revealed that the detected KSMHs might be best explained as EVMHs.
In addition, except for the abovementioned electron vortexes in the MHs, previous studies have reported the electron vortexes in other structures.For example, in the tailward outflow of a magnetic reconnection (MR), an ion-scale electron vortex in the plane perpendicular to the axis is observed inside an ion-scale flux rope (FR) with a heavily tilted axis (Jiang et al. 2023).Yao et al. (2018) found one electron vortex in the magnetic peak (MP).Ergun et al. (2019) reported an electron vortex that increases the magnetic field relevant to the CS.Jiang et al. (2020) have identified an electron vortex at the dipolarization front in the magnetotail.
However, where electron vortexes can be observed and what the characteristics of electron vortexes are in the magnetosheath are still unclear, especially from a statistical view.In this study, the unprecedented high-resolution Magnetospheric Multiscale (MMS) mission data from 2015 September to 2017 December are utilized for performing a statistical investigation of the electron vortexes in the terrestrial magnetosheath.The statistical characteristics of electron vortexes are presented in Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
detail, including the spatial distributions, occurrence rate, durations, scales, and energy dissipation.We also classify the electron vortexes, as they are near several types of magnetic structures.This paper is organized as follows: The selection criteria for electron vortexes are described in Section 2. In Sections 3 and 4, the case studies and statistical results of electron vortexes are presented, respectively.Finally, we summarize and discuss our results in Section 5.

Selection Criteria of Electron Vortex
All data from the MMS mission utilized in this study are in burst mode.Specifically, the magnetic field from the Fluxgate Magnetometer (FGM) and the electric field from the Electric Double Probe (EDP) are the sampling of 128 Hz (Russell et al. 2016) and 8192 Hz (Ergun et al. 2016;Lindqvist et al. 2016), respectively, and the plasma measurements from the Fast Plasma Investigation (FPI) are the time resolution of 30 ms for electrons and 150 ms for ions (Pollock et al. 2016), respectively.Unless stated otherwise, the geomagnetic solar ecliptic (GSE) coordinates are applied to all parameters in this study.
For the present study, the selection criteria for electron vortexes are set as follows: 1. Therefore, the 506 electron vortex events were eventually selected in the magnetosheath from 2015 September to 2017 December.

Case Studies
Magnetic structures are abundant in the magnetosheath, which is related to the dynamic processes in the magnetosheath (Huang 2022).Here we find that the selected electron vortexes occur in or near different magnetic structures.Four known types of structures in which the electron vortexes are detected are shown in Figures 1-4, following CS, FR, MH, and MP, respectively.
Figure 1 shows two electron vortexes observed during a CS crossing on 2017 January 11.The CS is a two-dimensional structure in which the magnetic field direction alters obviously from one side to the other side; thus, there are bipolar variations of magnetic field components during the CS crossing.We perform minimum variance analysis (Sonnerup & Scheible 1998) approaching 90 s −1 ; Figure 1(g)), which implies that the electron vortex is in the center of the CS and this vortex lasts for 0.79 s.Given the averaged background velocity, V imean ∼170 km s −1 , the spatial scale of the electron vortex is ∼170 km s −1 × 0.79 s ≈ 134 km (0.98ρ i , ρ i ∼ 137 km is the ion gyroradius based on B t ∼ 23 nT and T i ∼479 eV) based on the assumption of moving together between electron vortex and plasma flow.Moreover, the same characteristics also occur in the interval from 05:57:36.10 to 05:57:36.59UT in the right border region.V em and V el change from negative to positive, and there is an obvious peak for |∇ × V e | (more than 100 s −1 ; Figure 1(g)), which implies that the electron vortex is in the border region of the CS and it lasts for 0.49 s.Then, the spatial scale of the electron vortex is estimated as ∼83 km ≈ 0.61ρ i .In particular, J • E′, calculated by using J = e(n e V e − n i V i ) and E′ = E +V e × B, can represent the energy conversion between the fields and the plasmas (Zenitani et al. 2011;Huang et al. 2022).Both electron vortexes have positive J • E′ during the durations (Figure 1(f)), indicating the energy from the fields to the plasmas.
Figure 2 shows the electron vortex event detected in the FR on 2017 January 28.The FR, characterized by the bipolar change of the magnetic field normal component and enhancement of the core field, widely exists in various magnetized plasma environments, such as the solar wind, interplanetary space, the magnetosphere of the planet, and laboratory plasmas (Tripathi & Gekelman 2010;Wang et al. 2010;Huang et al. 2016;Vinogradov et al. 2016;Sindhuja & Gopalswamy 2020;Jiang et al. 2021;Wang et al. 2023).Similarly, we perform a minimum variance analysis using the magnetic field during 06:57:51.74-06:57:51.Figure 3 displays the observations of the electron vortex in the MH on 2015 December 28.The MH, featured by the depression of the magnetic strength, has been widely reported in the solar wind, magnetosheath, and planetary magnetosphere (Tsurutani et al. 2005;Huang et al. 2017aHuang et al. , 2017bHuang et al. , 2021;;Yao et al. 2017;Liu et al. 2020;Yu et al. 2021Yu et al. , 2022)).From 02:44:11.26 to 02:44:14.80UT, B t decreases obviously from more than 15 nT to approximately 5 nT (Figure 3(a)), and the angle between the magnetic field vectors at the edges of the structure is about 4.49°, implying the existence of an MH.The electron density increases simultaneously during the decrease of B t (Figure 3(c)), and the parallel temperature increases and the perpendicular temperature decreases at the border of the electron vortex (Figure 3(d)).Here the interior of the MH is defined as the region where the B t is much weaker than the background magnetic field, and the definition of the exterior of the MH is the region where the B t almost recovers to the background magnetic field.One can see that the vorticity The MP is another kind of magnetic structure.Contrary to the MH, there is a sudden enhancement in the magnitude of the magnetic field for the MP.They are also widely found in the planetary magnetosheath, heliosheath, and terrestrial foreshock region (Horbury et al. 2004;Burlaga et al. 2006;Balikhin et al. 2010;Yao et al. 2018;Wang et al. 2021a).Figure 4 presents an electron vortex inside the MP.In this event, B t increases obviously from less than 20 nT to more than 40 nT (Figure 4(a)).The angle of the magnetic field between the two sides is about 8.73°, along with the decrease of electron density (Figure 4(c)), which is characteristic of the MP.The interior and the exterior of the MP are defined as the region where the B t is much stronger than the background magnetic field and where the B t almost decreases to the background magnetic field, respectively.In the interior of the MP, the electron vortex is identified by the vorticity peak (Figure 4(g)) and bipolar signal  of electron velocity V ez (Figure 4(e)) from 11:09:24.41 to 11:09:25.20UT (lasting for 0.79 s), during which the electron temperature decreases slightly (Figure 4(d)).Then, the spatial scale of the electron vortex is estimated as ∼265 km s −1 (averaged ambient ion velocity) × 0.79 s ≈209 km ≈1.69ρ i (ρ i ∼ 124 km, calculated from B t ∼ 24 nT and T i ∼ 409 eV).This electron vortex has positive energy conversion with J • E′ > 0 (up to 12 nW m −3 in Figure 4(f)).
Overall, the above electron vortexes are all at ion scales.A total of 185 electron vortexes are detected around four types of magnetic structures, including 78, 42, 26, and 39 electron vortex events observed during the crossings of the CS, MH, MP, and FR, respectively.
In addition, the remaining 321 electron vortexes are detected around some unknown structures that cannot be clearly defined and are different from the above four types of magnetic structures, such as the boundary layer (BL).These unknown structures are classified into the "Others" category in this study.

Spatial Distribution and Occurrence Rate of Electron Vortex
In the X-Y and X-Z planes, the 2D spatial distribution and occurrence rate of all 506 electron vortexes in the magnetosheath are displayed in Figure 6.It is shown that  there are more electron vortexes in the subsolar region, which is 1R E < X < 13R E , −11R E < Y < 13R E , and −2R E < Z < 2R E (R E is Earthʼs radius), and only a few electron vortexes are detected in the flank region (Figures 6(a) and (d)).
The coverage of MMS spacecraft during the terrestrial magnetosheath season should be considered for determining this preference.Figures 6(b) and (e) give the total dwell time of MMS in burst mode, and it has been shown that the MMS spacecraft mainly covers the subsolar and northern regions.Moreover, we can obtain the occurrence rate of electron vortexes (shown in Figures 6(c) and (f)) by dividing the number of electron vortex events in each bin by the dwell time of MMS for the corresponding bin.In the X-Y plane, the occurrence rate (about eight events per hour) mostly in the subsolar region is the highest, and some in the region The other peaks of occurrence rate are mostly in the dusk-side region In the north (Z = 5R E -6R E ) of the X-Z plane, there is a peak group of the occurrence rate, while the other peak groups more continuously concentrate in the regions with X = 1R E to 13R E and Z = −2R E to 3R E .The total occurrence rate of all electron vortex events is 4.86 hr −1 .On average, 3.65 and 3.26 electron vortex events for all bins in the X-Y and X-Z planes can occur per hour, respectively, suggesting that almost similar average occurrence rates are in the two planes and an electron vortex can be observed every 0.27-0.30hr in bins of the two planes.
Moreover, the spatial distribution of the 185 electron vortexes (excluding the "Others" category from total electron vortexes) is similar to that of 506 electron vortexes.The electron vortexes also concentrate in the subsolar region, and the peak groups of occurrence rate also continuously concentrate in the regions Only 25 electron vortexes were observed when MMS crossed the bow shock before and after the detection of these electron vortexes.By calculating the angles between the interplanetary magnetic field and the normal direction of local bow shock, it is found that 80% and 20% of electron vortexes are behind quasi-parallel and quasiperpendicular shocks, respectively.For the remaining 481 electron vortexes, we cannot obtain their locations because MMS did not cross the bow shock near the detection of the electron vortexes.Due to the limited number of electron vortexes that can be determined behind a quasi-parallel or quasi-perpendicular shock, one cannot investigate whether the spatial distribution and occurrence rate of electron vortexes depend on the geometry of the bow shock.

Durations and Spatial Scales of Electron Vortexes
The histograms in Figure 7 display detailed information regarding the durations (Figure 7(a)) and scales (Figure 7(b)) of electron vortexes.One can well perceive that most of the electron vortexes last for less than 1.5 s (81.82%) and primarily within 0.5-1.5 s, especially for the "CS" (85.90%) and "MH" (78.57%) categories.Nevertheless, the "FR" category mainly concentrates on 0.5-1 s and 1.5-2 s, which is different from other categories.Moreover, there are no electron vortexes with durations of more than 2 s except for the "Others" category.In total, the electron vortexes last for 1.09 s on average.Moreover, we used the L = (V imean × dt)/ρ i to estimate the spatial scales of electron vortexes, where V imean is the averaged background ion velocity, dt is the duration of the electron vortex (the interval of bipolar signature of electron velocity V e ), and ρ i is the ion gyroradius (computed from the averaged ambient parameters: total magnetic field B t , ion temperature T i ).The spatial scales of electron vortexes mainly concentrate on less than 4ρ i (90.32%).The average spatial scales of electron vortexes of "CS," "MH," "FR," "MP," and "Others" categories are 1.52ρ i , 2.40ρ i , 2.36ρ i , 1.89ρ i , and 2.11ρ i , respectively.In total, the electron vortexes are 2.05ρ i on average, implying that electron vortexes belong to ion-scale structures.

Energy Dissipation in Electron Vortexes
We also investigate energy dissipation in electron vortexes.Figure 8 displays the statistical results for the means (panel (a)), medians (panel (b)), and maxima (panel (c)) of J • E′ in the electron vortex events associated with several types of structures.Figure 8(a) shows that the events of positive J • E′ (65.22%) are more than those of negative J • E′, implying that the overall trend is the energy conversion from the fields to the plasmas for most electron vortexes.This trend always exists for the electron vortexes observed in all types of magnetic structures owing to more proportion of positive J • E′.According to statistical results for the means (Figure 8    detected on crossing the CS and FR show more significant energy dissipation.In addition, the proportion of positive J • E′ of electron vortexes detected during the crossings of CS is the largest for the "Others" category, suggesting that the CS is more closely related to turbulent energy dissipation.For the maxima of J • E′ (Figure 8(c)), they are all more than zero, and the proportions of the maxima of more than 1 of other magnetic structures are more except for the "Others" category.Overall, these results indicate that the electron vortex may be one type of potential channel for energy dissipation in the turbulent magnetosheath.

Conclusions and Discussions
In the present study, the electron vortex events are investigated statistically in the terrestrial magnetosheath by utilizing the high-resolution MMS data from 2015 September to 2017 December, and 506 electron vortex events are successfully selected.The following is summarized as the main conclusion for the statistical results.
1.Because of the coverage of MMS spacecraft, electron vortexes mainly occur in the subsolar region, and the groups of occurrence rate more continuously concentrate in the regions The total occurrence rate of all electron vortexes is 4.86 hr −1 .On average, 3.65 and 3.26 electron vortex events for all bins in the X-Y and X-Z planes can occur per hour, respectively.
2. Electron vortexes are observed in relation to four types of magnetic structures.It is found that there are 78, 39, 42, and 26 electron vortexes detected during the CS, FR, MH, and MP crossings, respectively.3. The durations of more electron vortexes are less than 1.5 s (with an average of 1.09 s), most of which are within 0.5-1.5 s.The average spatial scales of electron vortexes for all categories are within 1ρ i -3ρ i an average of 2.05ρ i ), suggesting ion-scale electron vortexes.4. The means, medians, and maxima of the J • E′ for the electron vortexes with several types of structures are mainly positive (i.e., the energy from the fields to the plasmas).
Coherent structures can be formed self-consistently in plasma turbulence.In the solar wind and magnetosheath, intermittent coherent structures are generally identified with the increase of plasma temperature, implying that the dissipation of coherent structures results in plasma heating (Osman et al. 2012a(Osman et al. , 2012b;;Chasapis et al. 2017Chasapis et al. , 2018;;Qudsi et al. 2020;Huang 2022).Previous results suggested that coherent structures at small scales are directly related to the cascade and dissipation mechanism of turbulence energy (Naulin & Spatschek 1997;Grulke et al. 2001;Wu et al. 2013).Moreover, Huang et al. (2017aHuang et al. ( , 2017b) ) reported that in the magnetosheath, electron temperature significantly increases inside kinetic-scale MHs, suggesting that these holes have an essential effect on the heating and acceleration of the electrons.Hou et al. (2021) have investigated that electron-scale MR events contribute to energy dissipation in turbulent plasmas.However, they found that MR might not play the most crucial role in the dissipation mechanisms of magnetosheath turbulence.Wang et al. (2021b) indicated that the MPs at ion scales are the coherent structures related to energy dissipation and electron heating in the magnetosheath.Our work shows that the ionscale electron vortexes are widely distributed in the turbulent plasma environment of terrestrial magnetosheath.Furthermore, it is found that most of the observed electron vortexes have positive energy dissipation, indicating that the electron vortexes make a certain impact on the energy dissipation in turbulent plasmas.It is well known that electron vortexes are ring-shaped structures formed by electron motion, thus resulting in the existence of strong currents.Then, the strong currents usually can lead to strong energy dissipation (Fermo et al. 2012;Huang et al. 2017b).Specifically, Fermo et al. (2012) demonstrated that in their simulations MR with a guide field produces elongated electron current layers that typically start out as electron vortexes within the CS, along with current growth.Zhong et al. (2019) reported a kinetic-scale electron vortex embedded within the MH and observed intense current and nonideal electric field, which led to strong energy dissipation. et al. (2019b) observed that strong energy dissipation occurs in an ion-scale FR embedded in an electron vortex.Stawarz et al. (2018) found an electron vortex associated with an MP, along with strong currents that lead to strong energy dissipation.Moreover, using the general-relativistic resistive magnetohydrodynamics simulations, Ripperda et al. (2020) studied MR and plasmoid formation in CSs embedded with a small-scale vortex in the black hole accretion disk and found the release of energy in the small-scale vortex.Moll et al. (2011) also reported small-scale vortexes with the occurrence of energy transfer in the convective solar surface.Thus, this suggests that the electron vortexes may be a potential coherent structure for turbulence energy dissipation.All these results contribute to our comprehension of turbulent energy dissipation in space and astrophysical plasmas.

From 05 :
57:33.80 to 05:57:35.60UT, B t decreases sharply (Figure 1(a)), and B l decreases from approximately 20 to −21 nT (Figure 1(b)) along with the increase of electron density (Figure 1(c)) and electron temperature (Figure 1(d)), indicating that MMS crossed one CS.Here the center of the CS is defined as the region where the B l component shows a bipolar variation and B t decreases, and the border of the CS is considered as the region where B t almost recovers to the background magnetic field.V em has a bipolar variation from 05:57:34.41 to 05:57:35.20UT (Figure 1(e)) along with the peak of the vorticity |∇ × V e | (almost 97 UT: L = [−0.3247,−0.6727, 0.6649], M = [−0.8556,−0.0907, −0.5097], and N = [0.4031,−0.7344, −0.5461] in GSE.The ratio of the maximum eigenvalue and the median eigenvalue is 20.B t has a peak (Figure 2(a)) along with the decrease of electron density (Figure 2(c)), while B n changes from negative to positive with the characteristics of bipolar variation, and B m also reaches the peak simultaneously (Figure 2(b)), implying that the MMS detected one FR.The region where the B t and B m simultaneously peak is defined as the interior of the FR, and the region where the B t gradually decreases to the background magnetic field is defined as the exterior of the FR.Here MMS detected an electron vortex in the FR: the components of electron velocity are from positive to negative from 06:57:51.78 to 06:57:52.58UT (Figure 2(e)), along with B t (more than 40 nT; Figure 2(a)), B m (almost approaching 40 nT; Figure 2(b)) and the vorticity |∇ × V e | (almost approaching 100 s −1 ; Figure 2(g)) all reaching the peaks.The electron vortex with a duration of 0.80 s is located in the interior of the FR, where the electron temperature increases slightly (Figure 2(d)).The spatial scale of the electron vortex is estimated as ∼ 221 km s −1 (averaged ambient ion velocity) × 0.80 s ≈ 177 km ≈ 1.88ρ i (ρ i ∼ 94 km, calculated from B t ∼28 nT and T i ∼324 eV).The positive J • E′ is also during the crossing of the electron vortex (Figure 2(f)), which indicates the energy from the fields to the plasmas.

Figure 1 .
Figure 1.Electron vortexes detected by MMS during the crossing of the CS.(a) The magnitude of the magnetic field; (b) three components of the magnetic field in LMN coordinates; (c) electron density; (d) parallel temperature (black) and perpendicular temperature (blue) of electrons; (e) three components of the electron velocity in LMN coordinates; (f) energy dissipation J • E¢; (g) magnitude of vorticity |∇ × V e |.Four vertical gray dashed lines mark two electron vortexes, and the black horizontal dashed line in panel (g) shows the threshold (〈x〉 + 3δ) of |∇ × V e | for selecting the electron vortexes.The color bars at the top of the figure mark the border region (blue) and center region (red) of the CS crossing, respectively.

Figure 2 .
Figure 2. Electron vortex detected by MMS during the crossing of the FR.The same format is shown in Figure 1.Two vertical gray dashed lines mark the electron vortex.The color bars at the top of the figure mark the exterior region (blue) and interior region (red) of the FR crossing, respectively.
Figure 5 manifests an electron vortex event observed during the crossing of one BL on 2016 October 24.From 09:59:31.32 to 09:54:35.60UT, significant variations can be detected in the magnitude (Figure 5(a)) and the direction (Figure 5(b)) of the magnetic field and the plasma parameters (such as electron density in Figure 5(c) and parallel temperature and perpendicular temperature in Figure 5(d)), implying the observation of one BL.Specifically, the center of the BL is defined as the region where the magnitude of the magnetic field and the plasma parameters have the strongest change.The other regions belong to the border of the BL.From 09:59:33.30to 09:59:33.84UT (along with the peak of the vorticity |∇ × V e | in Figure 5(g)), the V ez component has a noticeable bipolar variation (Figure 5(e)), suggesting the existence of an electron vortex with a duration of 0.54 s in the center of the BL.Meanwhile, energy conversion from the fields to the plasmas occurs during the electron vortex (Figure 5(f)).The spatial scale of the electron vortex is estimated as ∼190.7 km s −1 (averaged ambient ion velocity) × 0.54 s ≈ 103 km ≈ 1.02ρ i (ρ i ∼ 101 km, calculated from B t ∼ 57 nT and T i ∼ 1483 eV), indicating that this electron vortex belongs to an ion-scale structure.

Figure 3 .
Figure 3. Electron vortex detected by MMS during the crossing of the MH.(a) The magnitude of the magnetic field; (b) three components of the magnetic field in GSE coordinates; (c) electron density; (d) parallel temperature (black) and perpendicular temperature (blue) of electrons; (e) three components of the electron velocity in GSE coordinates; (f) energy dissipation J • E¢; (g) magnitude of vorticity ∇ × V e .Two vertical gray dashed lines mark the electron vortex, and the horizontal black dashed line in panel (g) shows the threshold (〈x〉 +3 δ) of |∇ × V e | for selecting the electron vortex.The color bars at the top of the figure mark the exterior region (blue) and interior region (red) of the MH crossing, respectively.

Figure 4 .
Figure 4. vortexes detected by MMS during the crossing of the MP.The format is the same as shown in Figure 3.The color bars at the top of the figure mark the exterior region (blue) and interior region (red) of the MP crossing, respectively.
(a)) and the medians (Figure 8(b)) of the J • E′, the electron vortexes

Figure 5 .
Figure 5. Electron vortexes detected by MMS during the crossing of the BL.The format is the same as shown in Figure 3.The color bars at the top of the figure mark the border region (blue) and center region (red) of the BL crossing, respectively.

Figure 6 .
Figure 6.Spatial distribution and occurrence rate of electron vortex in the magnetosheath.Statistical spatial distributions of all 506 electron vortex events are shown in panels (a) and (d), dwell time of orbit coverage of MMS from 2015 September to 2017 December is shown in panels (b) and (e), and occurrence rates of electron vortex events are shown in panels (c) and (f).The gray dashed lines and solid lines in panels (a)-(c) represent the bow shock and magnetopause positions computed using the paraboloidal bow shock model of Filbert & Kellogg (1979) and the magnetopause model of Sibeck et al. (1991), respectively.

Figure 7 .
Figure 7. Histograms of the durations (a) and the spatial scales (b) of electron vortexes when MMS crossed the CS, MH, FR, MP, and Others.

Figure 8 .
Figure 8. Statistical results of J • E′ inside electron vortexes associated with different structures: (a) the means of J • E′, (b) the medians of J • E′, and (c) the maxima of J • E′.The exploded pies in panel (a) are all J • E′ < 0, and the rest are J • E′ > 0.
The electron vorticity magnitude |∇ × V e | should be larger than the sum of the average value 〈x〉 of electron vorticity |∇ × V e | and three times the standard deviation 3δ of electron vorticity |∇ × V e | during the entire burst mode time interval, i.e., |∇ × V e | > 〈x〉 + 3δ. 2. The corresponding time of |∇ × V e | > 〈x〉 + 3δ needs to