Hall Fields and Current Systems of Magnetic Reconnection under Asymmetric Conditions

The Hall effect is a key component of collisionless reconnection. A previous study showed that a quadrupolar Hall pattern under symmetric conditions degenerates into a bipolar pattern under highly asymmetric conditions. We study the properties of Hall magnetic fields and current systems during three reconnection events observed by the Magnetospheric Multiscale mission at the dayside magnetopause. Although the asymptotic density ratio between the magnetosheath and magnetospheric plasmas is very high for all three events, clear quadrupolar Hall field patterns are observed in all three events. The quadrupolar Hall magnetic fields in the three events display their respective properties on the intensity asymmetry and the distributing location. Among these events, a quadrupolar Hall field pattern is observed for the first time in in situ observations: the magnetosheath Hall pattern occupies the whole midplane region, while the magnetosphere Hall pattern still exists under highly asymmetric conditions. Observations show that the plasma mixture modulates the density asymmetry in the Hall region, which can be very different from the asymptotic density asymmetry in the magnetosheath and magnetospheric inflow plasma. The analyses indicate that the different density asymmetry inside the Hall region, but not the asymptotic density asymmetry, is an exact indicator that explains the different observed Hall patterns. Based on the observed facts, we suggest that the three reconnection events studied here are in different phases of their development after they are triggered under highly asymmetric conditions. Our results provide new insights into how the Hall effect works with the evolution of asymmetry during reconnection.


Introduction
Magnetic reconnection is a key physical process that explosively converts magnetic energy into particle kinetic energy and thermal energy for a wide range of plasma conditions, such as space plasmas, astrophysical plasmas, and laboratory plasmas (Yamada et al. 2010).The space plasma in the Earthʼs magnetosphere is so tenuous that it is considered to be collisionless.One striking feature of collisionless reconnection is the Hall effect attributable to the decoupling of ions and electrons in the ion diffusion region (Sonnerup 1979;Deng & Matsumoto 2001).The particle, magnetic, and electric field signatures related to the Hall effect are often used to identify the ion diffusion region in space observations (Nagai et al. 2001;Borg et al. 2005;Wygant et al. 2005).In addition, simulation works have indicated that the inclusion of the Hall effect provides a fast reconnection rate (Birn et al. 2001).Understanding the Hall effect has significant implications for reconnection (Birn et al. 2001;Shay et al. 2001;Wang et al. 2001;Xiao et al. 2007;Dai 2009;Lu et al. 2013;Zhang 2016;Zhang et al. 2022).Dai (2009) and Dai et al. (2017) proposed a kinetic Alfvén wave (KAW) explanation of the Hall effect in magnetic reconnection.The ratio of the Hall electric field to the Hall magnetic field, on the order of the Alfvén speed, is a unique feature predicted by KAW physics.In situ observations provide evidence of the kinetic Alfvén eigenmode in the near-Earth magnetotail during the substorm expansion phase (Duan et al. 2016).Recent Magnetospheric Multiscale(MMS) mission observations recently found that the global pattern of the Hall system in the reconnection region is consistent with the KAW theory (Zhang et al. 2017).
The asymptotic asymmetry parameter (Phan & Paschmann 1996) is found to play an important role in determining the magnetic field and electric field pattern in the Hall effect (Oieroset et al. 2004;Cassak & Shay 2007;Mozer et al. 2008).In previous studies, the asymmetry is typically estimated using asymptotic (also called background) plasma parameters in the inflow regions of reconnection.Under symmetric conditions, the Hall effect operates separately on both sides of the reconnecting current sheet to produce symmetric Hall field patterns (Mozer et al. 2002;Dai 2009).Hall fields display asymmetric features as the asymptotic condition becomes asymmetric (Mozer et al. 2008;Shay et al. 2016).For example, at the dayside magnetopause with high plasma asymmetry, an overwhelming bipolar Hall magnetic field with a magnetosheath pattern under symmetric conditions is observed, while the magnetospheric counterpart is essentially invisible (Mozer et al. 2008;Pritchett 2008).With the presence of a moderate guide field, previous simulation results showed that Hall fields displayed a quadrupolar pattern with strong asymmetric intensities under asymptotic asymmetric conditions (Pritchett & Mozer 2009).Recent MMS observations showed that quadrupolar Hall fields can be present under asymptotic asymmetric conditions at the magnetopause and can display nearly symmetric intensities with or without a guide field (Peng et al. 2017;Wang et al. 2017;Zhang et al. 2017).The past results and recent results indicate that the asymptotic condition is not precise enough on its own to evaluate the features of the 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.
Hall effect.In this paper, we try to investigate Hall fields and current systems of magnetic reconnection under asymmetric conditions and find an indicator to mark the variable Hall effect associated with the asymmetry.
The MMS mission provides an opportunity to study the Hall effect with multipoint high-resolution measurements at the ion inertial scale and less (Burch et al. 2016).In this paper, we investigate the Hall field pattern and the fine structure of the related Hall current system under asymmetric conditions using the Fluxgate Magnetometer (FGM; Russell et al. 2016) and Fast Plasma Instrument (FPI) measurements (Pollock et al. 2016) from the MMS mission.We find that the plasma density in the Hall region is very different from that in the inflow region.When the density asymmetry inside the Hall region, instead of that in the inflow region, is chosen to estimate the features of the Hall effect, the asymmetry in the Hall region and the pattern of the quadrupolar Hall pattern display a better correlation.The study also explains how the evolution of plasma density asymmetry impacts the dynamic evolution of Hall patterns in magnetic reconnection.This paper is organized as follows: MMS observations of three reconnection events are presented in Section 2. We then present the analyses and discussion in Section 3. The conclusions are given in Section 4.

Reconnection Event A on 2015 December 6
Figure 1 displays the MMS 1 observations of the reconnection event at the dayside magnetopause at 23:29 UT on 2015 December 6, in local LMN coordinates derived using minimum variance analysis (Sonnerup & Scheible 1998).In the GSE coordinates, L points northward, M points downward, and N points sunward and perpendicular to the magnetopause in this work.Only the data from MMS 1 are used in this work unless otherwise stated, such as for the analyses based on multipoint observations.An inbound magnetopause crossing occurs at 23:29:06.8UT, as marked by the B L component becoming positive (vertical dashed line in Figure 1(a)) at the midplane.The most striking feature in the observations is that the ion flows in the -L direction during the magnetopause crossing (Figure 1(d)).The peaks in V L are −519 km s −1 on the magnetosphere side, which is 2 times the Alfvén speed (∼251 km s −1 ) in the magnetosheath (Figure 1(e)).Therefore, we infer that this flow is a southward ion jet accelerated by magnetic reconnection.The southward ion jet indicates that MMS 1 traverses the reconnection event in the southern outflow region, as shown in Figure 2(a).
The average of the magnetic field and plasma density data in the intervals labeled with two black bars at the top of Figure 1 are taken as representative of the magnetic field and plasma density in the magnetosphere and in the magnetosheath.The calculated asymptotic condition for this event can be summarized as follows: N SH /N SP = 17.4;B SP /B SH = 1.51 and B G = −11.8nT, where N SH and B SH are the plasma density and field strength in the magnetosheath, N SP and B SP are those in the magnetosphere and B G is the guide field.The ratio of field strength and plasma density indicates that this reconnection event occurs under highly asymmetric conditions.The asymptotic conditions for this event and the two later events are summarized in Table 1.
In Figure 1(a), B M displays a negative-positive variation in the outflow region after a guide field of −11.8 nT is subtracted.
B M is negative in the outflow associated with B L turning (shaded with dark gray in Figure 1(a)) and becomes positive in the outflow at the magnetosphere side (shaded with light gray in Figure 1(a)).The negative B M peak (−32 nT) collocated with the B L turning point at 23:29:06.8UT, which means that the negative B M completely occupies the midplane region at the magnetopause.This B M distribution is consistent with the unipolar Hall fields expected at the south side of the X line in asymmetric reconnection (Mozer et al. 2008;Pritchett 2008;Shay et al. 2016).However, as MMS 1 traveled further toward the magnetosphere away from the midplane, it continued to observe a weak Hall field (+B M shaded with light gray in Figure 1(a)) with a peak of +8.6 nT.The fact that MMS 1 in this light gray shaded region was still in the ion diffusion region is shown by the presence of electrons at magnetosheath energies (Figure 1(c)) and the decoupling of ion flow and electron flow (Figure 1(h)).Therefore, the bipolar B M at one side of the X line indicates that the Hall fields for this event display a quadrupolar pattern, although the intensity of the Hall field is highly asymmetric.Such a Hall pattern was first observed in Pritchett & Mozerʼs (2009) particle-in-cell simulations.In their simulation, the guide field is 30% of the asymptotic field and the plasma density asymmetry is 10, which are similar to those in this event.In this paper, we call the gray region with Hall field perturbation the Hall region.
Before going into the analysis of the currents associated with the Hall fields, we wish to state that the term "Hall current system" is used in this paper instead of "Hall currents" (Nagai et al. 2003;Alexeev et al. 2005;Zhang et al. 2017;Tang et al. 2021).Hall currents contributing to the Hall effect by the J × B term come from the decoupling of ions and electrons perpendicular to the magnetic field lines.To maintain the closure of these Hall currents, the field-aligned currents together with Hall currents form a Hall current system (Sonnerup 1979), which contributes to inducing Hall fields.As shown later, the current discussed in this paper is the fieldaligned part of the Hall current system.Figure 1(f) shows the currents calculated from the FPI moment measurements (Lavraud et al. 2016).In the midplane region (the dark gray region in Figure 2(f)), strong J L turns from negative to positive at the peak of −B M and the B L zero crossing point.This indicates that the Hall currents around the Hall fields flow from the magnetosphere (along the blue current line in Figure 2(a)) into the magnetosheath (along the green current line in Figure 2(a)).Figure 1(g) shows the components of the currents parallel and perpendicular to the local magnetic field in the reconnection plane.The strong Hall currents in the midplane region run mainly in the field-aligned direction, with minor perpendicular components.These Hall currents are carried by the electrons flowing from the magnetosheath into the magnetosphere along but antiparallel to the magnetic field (Figure 1(h)).On the magnetospheric side of the southern outer flow, the weak positive-negative Hall current cell around the Hall magnetic field is observable, as shown by J L in Figure 1

Reconnection Event B on 2015 October 30
Figure 3 displays MMS 1 observations for a reconnection event on 2015 October 30, at the dayside magnetopause in local LMN coordinates.MMS 1 travels across the magnetopause outbound at 05:15:48.6UT.The reconnection jet is along the −L direction and concentrated on the magnetospheric side of the magnetopause (Figure 3(d)).Relative to the magnetosheath flow, with typical V L = −120 km s −1 , the jet has a peak value of −62 km s −1 , which is 77.5% of the Alfvén speed of ∼80 km s −1 in the magnetosheath (Figure 3(e)).The agreement between the jet velocity and the Alfvén speed indicates that reconnection occurs (Paschmann et al. 1979).MMS 1 sampled the southern reconnection outflow, as shown in Figure 2(b).The asymptotic condition for this event in Table 1 indicates that this event is also highly asymmetric.After subtracting a −3 nT guide field, the B M component shows positive-negative variations from the magnetosphere to the magnetosheath (Figure 3(a)), in agreement with bipolar polar Hall magnetic fields.The peaks of the Hall B M field in the magnetosphere and in the magnetosheath are +9.1 and −14.4 nT, respectively.The separation point between the magnetosphere and magnetosheath Hall fields is at 05:15:47.5 UT and does not coincide with the B L zero crossing point at 05:15:48.6UT (as marked by the vertical dashed line).Thus, the magnetosheath Hall magnetic field is shifted toward the magnetosphere and traverses the magnetopause, yet its negative peak is still located on the magnetosheath side.
In Figure 3(f), on the magnetospheric side of the Hall region, J L turns from negative to positive at the +B M peak.J L maintains a positive direction across the magnetopause and becomes negative at the −B M peak on the magnetosheath side

Reconnection Event C on 2016 December 26
Figure 4 displays MMS 1 observations for a dayside magnetopause reconnection event on 2016 December 26, in local LMN coordinates.The reconnection ion jet along the +L direction has a peak velocity of +260 km s −1 relative to the magnetosheath flow (Figure 4(d)), which is 1.2 times the Alfvén speed of ∼220 km s −1 in the magnetosheath.The asymptotic condition for this event in Table 1 indicates that this event is highly asymmetric.Given the multiple magnetopause crossings observed after this event, the actual magnetosheath plasma conditions are deemed rather unstable until approximately 10:49:10 UT.Therefore, the magnetosheath parameters used here and in Table 1 later

Analyses and Discussion
As shown in the observational section, in all three highly asymmetric reconnection events, the Hall magnetic fields at one side of the X line display a bipolar variation, which indicates that a quadrupolar pattern exists in all events.However, the Hall currents and Hall magnetic fields of these three events display different features.First, the intensities of the magnetosheath and magnetospheric Hall magnetic fields are comparable in Events B and C, while the Hall magnetic fields are highly asymmetric in Event A. The ratio of the magnetosheath Hall magnetic fields to the magnetospheric Hall magnetic field for the three events is listed in the second column of Table 1.Second, the locations of the Hall magnetic fields are rather symmetrical relative to the midplane in Event C, while the magnetosheath Hall pattern is shifted into the magnetosphere in Events A and B (Figure 2).
Previous studies found that the asymptotic asymmetric conditions and the guide field are the two main factors determining Hall field patterns (Oieroset et al. 2004;Cassak & Shay 2007;Cassak & Shay 2008;Mozer et al. 2008;Pritchett 2008;Eastwood et al. 2010).The asymptotic asymmetry and the guide field for the three events are listed in Table 1.The ratio of the guide field to the reconnecting field (third column) is less than or equal to 20% for all three events.Simulations and laboratory experiments show that a guide field less than 20% can change the features of Hall magnetic fields compared to runs without guide fields (Karimabadi et al. 2005;Swisdak et al. 2005;Tharp et al. 2012).A guide field of 20% can produce a 1.5 times asymmetry in the intensities of Hall fields (Swisdak et al. 2005).For the higher asymmetric Hall field, it is found that the asymptotic asymmetric conditions contribute more (Pritchett & Mozer 2009).The three events chosen here are thus well suited for studying the impact of asymptotic asymmetry on the Hall effect.The values of the asymptotic asymmetries in the magnetic field and plasma density (fourth and fifth columns in Table 1) indicate that all three reconnection events occurred under highly asymmetric conditions.However, the very different features of the Hall effect in these events indicate that the asymptotic asymmetric conditions cannot explain the discrepancy between these features.In this study, we found that the plasma density asymmetry in the Hall region is a useful parameter to discuss the features of Hall fields.This parameter is defined as the ratio of the averaged plasma density in the dark gray shaded magnetosheath Hall region to that in the light gray shaded magnetosphere Hall region.In the sixth column, we give the plasma density asymmetry in the Hall region compared to the asymptotic values given in the fifth column.We can see that in all three events, the plasma density asymmetry in the Hall region decreased compared to the asymptotic values.However, the density asymmetry in the Hall region still retains a high value of 4.5 in Event A, while it has much lower values of 1.8 and 1.1 for Events B and C, respectively.
In Event A, the asymptotic density asymmetry and magnetic field asymmetry are high (Table 1).As observed by MMS 1, the Hall system currents at the midplane region flow along the open magnetic field from the magnetosphere into the magnetosheath, resulting in the peak of the Hall fields at the B L turning point.Therefore, in this case, the Hall fields at the midplane region display a negative unipolar, which agrees with previous observations and simulations under highly asymmetric conditions (Mozer et al. 2008;Pritchett 2008;Shay et al. 2016).However, in addition to such negative unipolar Hall fields at the midplane, other weaker positive unipolar Hall fields and the associated Hall current system on the magnetosphere side are present.Two unipolar regions at the midplane region and magnetosphere region compose a bipolar Hall pattern, which has been first reported in simulations (Pritchett & Mozer 2009), but has never been found in in situ observations before.The strongly asymmetric Hall fields in Event A observationally evidence a quadrupolar pattern on both sides of the X line despite its high asymptotic asymmetry and its highly distorted location compared to the quadrupolar Hall field under symmetric conditions.
The Hall effect originates from the decoupling of ions and electrons (Sonnerup 1979).Hall currents are determined by J = q(N i V i -N e V e ).In the frame of quasi-neutrality, J = q(N i V i -N e V e ) reduces to J = qN (V i -V e ), in which V i -V e represents the contribution of ion-electron decoupling and N represents the density of the particle carriers performing decoupling.In the following discussion of the field-aligned part of the Hall current system, ion-electron velocity separation is used to express V i -V e instead of the term "ion-electron decoupling" because ion-electron decoupling only occurs in the direction perpendicular to the magnetic field.In Table 1, the asymptotic density asymmetry is 17.4 in Event A, with a decreased density asymmetry of ∼4.5 in the Hall region.The high-density asymmetry in the Hall region can be seen in the electron spectrogram in Figure 1(c).Magnetosheath electrons with high fluxes in the middle energy bands (approximately 10 eV) are present in the midplane Hall region, and there is less electron mixture from the magnetosphere.In the magnetospheric Hall region, magnetospheric electrons with low fluxes in the high energy bands are dominant.The inflow-then-outflow electrons in that region (Figure 1(h)) are consistent with the Hall electrons in the same region.This indicates that the Hall effect operated on the magnetospheric side of the reconnection region in this event, although the plasma on the magnetospheric side was very tenuous.This is understandable.The ion-electron decoupling process depends on the particles' individual kinetic behavior on the ion scale but not on the plasma density (Sonnerup 1979), so the highdensity asymmetry cannot destroy the Hall process on the side with low plasma density.In the magnetospheric Hall region, the maximum ion-electron velocity separation reaches over +1500 km s −1 in the +L direction and −790 km s −1 in the -L direction (Figure 1(h)).In the midplane Hall region, the maximum ion-electron velocity separation is over +100 km s −1 in the +L direction and −510 km s −1 in the −L direction.It is thus obvious that despite the high-density asymmetry, the ion-electron velocity separation on both sides is strong and very weakly asymmetric.Based on the equation J = qN (V i -V e ), the high plasma density asymmetry and the weakly asymmetric ionelectron velocity separation together produce the highly asymmetric Hall current system for Event A (J L in Figure 1(f)).Using the Biot-Savart law, Zhang et al. (2017) presented quantitative measures of the Hall magnetic fields induced by this Hall current system.
The shifting of the magnetosheath Hall magnetic fields into the magnetosphere in Event A can be explained using Cassak-Shay theory (Cassak & Shay 2007, 2008).In this theory, the stagnation point between the magnetosheath and magnetospheric inflows is located on the magnetospheric side of the X line.The magnetosheath inflow plasma traverses the magnetopause and penetrates the magnetosphere to maintain momentum balance under such asymmetric conditions.In Event A, although the magnetic field asymmetry is low (∼1.51), the density asymmetry is very high.This high-density asymmetry of ∼17.5 is strong enough to significantly shift the stagnation point toward the magnetosphere (Cassak & Shay 2007, 2008).Thus, the magnetosheath inflowing electrons enter the magnetosphere across the magnetopause.In Event B, Hall currents and magnetic fields are observed on the magnetospheric side of the magnetopause.The asymptotic density asymmetry is 18.1 and similar to that in Event A, and the magnetic asymmetry is 2.36 stronger than that in Event A. Past simulation results suggest that a unipolar magnetosheath Hall field pattern should be present with its peak at the midplane under such asymmetric conditions (Pritchett 2008;Shay et al. 2016), such as that in Event A. In Event B, the magnetosheath Hall currents and magnetic fields are truly shifted into the magnetosphere, but the peak in Hall magnetic fields and the turning point of the Hall currents remain on the magnetosheath side.The observed bipolar Hall magnetic field has a weak asymmetry of 1.6.Therefore, the observed bipolar Hall magnetic field pattern in Event B is different from that in Event A. Considering the similar asymptotic asymmetry in Events A and B, this observational fact implies that the asymptotic asymmetry is not proper for interpreting the discrepancy in the Hall pattern.If one focuses on the Hall region for this event (gray-shaded region), however, the plasma density asymmetry is found to be much reduced, down to 1.8, in stark contrast with 4.5 in Event A. The spectrogram in Figure 3(c) shows that electrons throughout the whole Hall region have a similar distribution.In the magnetosphere Hall region, the magnetosheath electrons dominate.This is caused by the mixture of the magnetosphere and magnetosheath electrons.The sharply reduced plasma asymmetry in the Hall region indicates that the plasma mixture due to reconnection in this event is approaching a homogeneous state in the whole Hall region.In the magnetospheric Hall region, the maximum ion-electron velocity separation at times is greater than +480 km s −1 in the +L direction and -550 km s −1 in the −L direction.In the magnetosheath Hall region, the maximum ion-electron velocity separation is at times greater than +300 km s −1 in the +L direction and −150 km s −1 in the −L direction.Combined with the weak electron density asymmetry in the Hall region, the strong ion-electron velocity separation on the magnetospheric side results in Hall currents (Hall magnetic fields) in the magnetosphere that are weaker than but comparable to those in the magnetosheath.The magnetosheath Hall currents (Hall magnetic fields) do not shift into the magnetosphere as deeply as in Event A. The magnetic field asymmetry of ∼2.36 in this event is the strongest of all three events, but the density asymmetry in the Hall region is weak (1.8).Past simulation results imply that a density asymmetry of at least 3 is necessary to totally shift the magnetosheath Hall fields under such magnetic field asymmetry conditions (Pritchett 2008;Huang et al. 2014).The weak plasma density asymmetry in the Hall region of this event can only make the outflowing magnetosheath Hall electrons partially traverse the magnetopause (the dark gray region before the vertical dashed line Figure 3(h)), referring to the Cassak-Shay theory of momentum balance (Cassak & Shay 2007, 2008).Eventually, the magnetosheath Hall currents (and resulting Hall fields) are shifted toward the magnetosphere (Figures 3(g  explained by the density asymmetry in the Hall region than by the asymptotic density asymmetry. In Event C, the Hall magnetic fields and currents are symmetrically located on both sides of the magnetopause, consistent with the expected symmetrical bipolar Hall pattern of symmetric reconnection (Mozer et al. 2002;Borg et al. 2005).The asymptotic density asymmetry is 5 for this event, and the magnetic asymmetry is 1.55.However, the observed Hall magnetic field in the magnetosphere (−18 nT) is stronger than that in the magnetosheath (+11 nT).We also note that the density asymmetry is reduced to 1.1 in the Hall region for this event.This means that the plasma from the magnetosphere and the magnetosheath have totally mixed in the Hall region and reached a homogeneous state without a discrepancy in density.This total mixture can be seen in the nearly symmetric electron energy-flux spectrogram in the magnetosphere Hall region and the magnetosheath Hall region of Figure 4(c).As shown in Figure 4(h), in the magnetospheric Hall region, the maximum ion-electron velocity separation exceeds +650 km s −1 in the +L direction and −850 km s −1 in the −L direction.In the magnetosheath Hall region, the maximum ion-electron velocity separation exceeds +480 km s −1 in the +L directed electrons and −580 km s −1 for the −L direction.The ion-electron velocity separation is stronger on the magnetospheric side than on the magnetosheath side.Combined with the more symmetric electron density in the Hall region, these velocity separations result in stronger Hall magnetic fields on the magnetospheric side than those on the magnetosheath side.Additionally, due to the nearly symmetric density in the Hall region, the plasma momentum is balanced, and the magnetosheath Hall magnetic fields and currents do not show any shift toward the magnetospheric side.In this event, despite the highly asymptotic asymmetric conditions, the Hall magnetic fields are symmetric and thus contrary to expectations under such asymptotic conditions (Mozer et al. 2002;Pritchett 2008).These facts again imply that the density asymmetry in the Hall region, rather than the asymptotic density farther from each side, is an accurate indicator to explain the observed Hall field patterns.Compared to Event A, both events have highly asymptotic plasma asymmetry and the same asymptotic magnetic asymmetry but very different Hall field asymmetries and spatial patterns.The density symmetry in the Hall region in Event C enables this event to produce a symmetric Hall field.This further implies that the density asymmetries in the Hall region play an important role in causing the different observed Hall patterns.
In summary, the plasma density asymmetry in the Hall region of Events A-C increases gradually, while the asymmetry of the Hall effect intensity increases gradually, and the quadrupolar Hall pattern gradually shifts toward the magnetosphere.Electrons are the main carriers of Hall currents resulting in the Hall magnetic field, and their density will determine the intensity of Hall currents.The mixture of magnetospheric and magnetosheath electrons results from the opened magnetic field line in reconnection.Under asymmetric conditions, with the temporal development of reconnection, the plasma mixing gradually moves the dense magnetosheath electrons into the magnetosphere and reduces the density asymmetry in the Hall region accordingly.In the above analyses and discussions, when we choose the density ratio in the Hall region as an indicator for reconnection development, we suggest that the three events are in their respective phases of reconnection development after they are triggered under highly asymmetric conditions.Event A is in its early phase of the development of asymmetric reconnection.In this phase, once reconnection is triggered under highly asymmetric conditions, there is less plasma mixture, and the plasma density asymmetry in the Hall region is as high as the background density asymmetry.Thus, the resulting Hall current and Hall field pattern display a high asymmetry and significantly shift toward the magnetosphere.The dominant polar pattern shown in Figure 2(a) agrees with the Hall pattern from previous simulations and observations under a highly asymmetric background (Mozer et al. 2002;Pritchett 2008).Event B is in the middle phase of its development.In this phase, as the magnetosheath electrons and the magnetosphere electrons increasingly mix due to the reconnected fields with the development of reconnection, the plasma asymmetry across the midplane is reduced.Accordingly, the Hall current asymmetry and Hall field asymmetry are reduced accordingly, and the Hall field distribution gradually shifts toward the magnetosheath side (Zhang et al. 2017).Figure 2(b) corresponds to the Hall pattern in this phase.Event C is in its late phase when the mixture of the electrons on both sides of the magnetopause approaches a balance, the density asymmetry disappears in the Hall region, and the Hall field pattern eventually displays nearly symmetric features (Peng et al. 2017), as shown in Figure 2(c).

Conclusions
We analyze the Hall fields and Hall current system in three reconnection events observed by MMS 1 at the dayside magnetopause.All three events are observed with high asymptotic asymmetry between the magnetosphere plasma and the magnetosheath plasma, but have very different density asymmetry in the Hall region.The Hall patterns in the three events display distinct features in the asymmetry of the Hall field intensity and the Hall spatial distribution.The discrepancy of these features in the three events cannot be explained by asymptotic asymmetry, while plasma asymmetry in the Hall region is proposed to be a proper explanation.
The strong ion-electron velocity separation consistent with the Hall current system exists on the magnetospherical side of the magnetopause for all three events.Even under the high-density asymmetry in the Hall region of Event A, this behavior does not disappear, which may imply that the ion-electron decoupling on the magnetospheric side is a common feature of the Hall effect despite the asymmetric condition.The magnetospherical Hall fields in Event A are not prominent due to the tenuous electrons in the magnetospherical Hall region.The Hall pattern under highly asymmetric conditions in this event has not been reported because high-resolution particle measurements were lacking before the MMS era.In Events B and C, the mixture of the magnetosheath electrons with the magnetosphere electrons provides abundant carriers for the magnetosphere Hall currents, so the intensity of the magnetosphere Hall fields is comparable to the intensity of the magnetosheath Hall fields.For the observed Hall asymmetry in these two events, the asymptotic asymmetric condition is not a good indicator to explain their properties.The density asymmetry in the Hall region is more useful to explain the observed Hall intensity asymmetry.
The Hall region in Event A inherits the high asymptotic asymmetry between the magnetospheric plasma and the magnetosheath plasma.The inflow magnetosheath electrons traverse the magnetopause into the magnetosphere due to this high asymmetry and are accelerated to form the outflowing Hall electrons on the magnetospheric side; thus, the magnetosheath Hall currents change direction at the B L turning point, inducing the magnetosheath Hall fields to completely occupy the midplane.The medium density asymmetry in the Hall region of Event B does not allow the magnetosheath Hall currents and Hall fields to shift toward the magnetosphere as deeply as those in Event A. In Event C, the magnetosheath Hall currents and Hall fields are located on both sides of the magnetopause due to the near-density symmetry of the Hall region.The density asymmetry in the Hall region is also accurate in explaining the shift of Hall fields, but the asymptotic density asymmetry out of the Hall region cannot.
Based on the results in this paper, we attribute the different Hall patterns (Hall asymmetry and Hall shift) in the three reconnection events to the respective temporal phases of reconnection after they are triggered under the asymmetric condition.The Hall effect is a temporal process such that the observed Hall pattern is dominated not by the stable asymptotic asymmetry but by the local and dynamical plasma asymmetry in the Hall region due to the plasma mixture.The works in this paper provide a hint to interpret the different observational properties of the Hall pattern and to understand how the Hall effect evolves with the variations in asymmetry.The conclusion of this study is based on the analysis of three independent events.The statistical analysis of more events will help to establish a more solid conclusion.The systematic study of the dynamic evolution of the Hall effect in asymmetric reconnections depends on simulation works that consider the processes of plasma mixing.
(f) in the light gray region.It is interesting to note that at around 23:29:10 UT there is electron velocity shear flow embedded in the southern outer flow of the magnetospheric side (Figure 1(f)).The time span of such electron velocity shear flow is about 0.4 s.These signals are produced by small-scale electron vortices that propagated away from the X line (Pritchett & Mozer 2009).The electron vortices can generate perturbation in the Hall field.As shown by the corresponding variation of B M in Figure 1(a), B M has a decrease of 2nT compared to the background Hall field.

Figure 1 .
Figure 1.MMS 1 plasma and field observations for the reconnection event on 2015 December 6 in LMN coordinates.In GSE coordinates, L = [−0.153,−0.547, 0.823], M = [−0.250,−0.785, −0.568, and N = [0.956,−0.293, −0.016].(a) The three components of the magnetic field, (b) the ion and electron densities, (c) the electron energy-flux spectrogram vs. time, (d) the three components of the ion velocity, (e) the ion Alfvén speed, (f) the three components of the current density from the particle measurements, (g) the parallel and perpendicular components of the current density in the reconnection plane, and (h) the ion and electron velocity components in the L direction.The central current sheet and magnetospheric Hall regions are shaded with light gray and dark gray, respectively.
Figure4displays MMS 1 observations for a dayside magnetopause reconnection event on 2016 December 26, in local LMN coordinates.The reconnection ion jet along the +L direction has a peak velocity of +260 km s −1 relative to the magnetosheath flow (Figure4(d)), which is 1.2 times the Alfvén speed of ∼220 km s −1 in the magnetosheath.The asymptotic condition for this event in Table1indicates that this event is highly asymmetric.Given the multiple magnetopause crossings observed after this event, the actual magnetosheath plasma conditions are deemed rather unstable until approximately 10:49:10 UT.Therefore, the magnetosheath parameters used here and in Table 1 later are chosen during the time interval from 10:49:10-10:49:20 UT for this event, which are not shown in Figure4.Associated with the positive-negative turning of B L shown in Figure4(a) and the positive ion jet in the L direction shown in Figure4(d), B M shown in Figure4(a) displays a bipolar variation from −18 to +11 nT after a guide field of −3 nT is subtracted.These observations are consistent

Figure 2 .
Figure 2. Sketch depicting the Hall current system and Hall magnetic fields for the asymmetric reconnection Events A (a), B (b), and C (c), with the local LMN coordinates shown on the right.The vertical dashed red line indicates the central current sheet at the magnetopause, which separates the magnetosheath and the magnetosphere.The magnetosheath is to the left, and the magnetosphere is to the right.The solid black curves indicate magnetic field lines, and the solid green curves indicate the Hall current system.The blue lines indicate the magnetospheric part of the inflowing magnetosheath Hall currents.The red lines with an arrow indicate the MMS 1 trajectory across the reconnection region.
These magnetosheath inflowing electrons are accelerated by reconnection and form the magnetosheath outflowing Hall electrons on the magnetospheric side (the dark gray region after the vertical dashed line in Figure 1(c) and Figure 1(h)), thereby carrying the inflowing Hall current indicated by the blue line in Figure 2(a).Thus, the magnetosheath Hall currents turn the direction at the B L turning point, and the magnetosheath Hall fields are induced across the midplane region.
) and 3(b)), leaving the current turning point (the peak of Hall fields) in the magnetosheath (Figure 2(b)), but do not have the same pattern as in Event A. Comparing Events A and B, it is clear that the observed Hall pattern is better

Table 1
The Hall Symmetry and Asymptotic Conditions for Events A-C a Events |B H−SH /B H−SP | B G /B L−SP B SP /B SH N SH /N SP (N SH /N SP ) Hall Column |B H−SH /B H−SP | is the absolute value of the ratio of the Hall magnetic field in the magnetosheath to that in the magnetosphere.Column B G /B L−SP is the ratio of the guide field B G to the reconnecting field, in which B L−SP is B L on the magnetospheric side (taken as the reconnecting field).Column B SP /B SH is the asymptotic magnetic field asymmetry with B SP and B SH the means of the magnetospheric and magnetosheath fields in the reconnection in low regions.Column N SH /N SP is the asymptotic density asymmetry in the reconnection in low regions.Column (N SH /N SP ) Hall is the density asymmetry in the Hall region itself (see the text for details). a