# Physics of Magnetopause Reconnection: A Study of the Combined Effects of Density Asymmetry, Velocity Shear, and Guide Field

Authors: Kentaro G. Tanaka; Masaki Fujimoto; Iku Shinohara
Journal: International Journal of Geophysics
 (2010)

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## Abstract

Magnetopause reconnection would be characterized by the density jump across the current sheet, the flow shear across the boundary, and nonzero guide field. While effects of each of these elements have been studied, the effects arising from the combination of these are still unexplored. Two-dimensional full-particle simulations show that the combination of shear flow and/or guide field with density asymmetry induces the sliding motion of theX-line along the magnetopause. The direction of the X-line motion is controlled either by the ion flow at the X-line when the shear flow effects dominate or by the electron flow at the X-line when the guide field effects dominate. The shear flow effects and the guide field effects may counteract each other in determining the direction of the X-line motion and, in the close proximity of the subsolar region where the flow is slow, the X-line motion can be opposite to the flow direction.


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## 1. Introduction

Magnetic reconnection drives explosive phenomena in laboratory and astrophysical plasmas. It is one of the most important fundamental processes in the plasma universe (e.g., [1–3]). The magnetic reconnection is associated with a geometrical change in magnetic field structure, namely, the “X-line” formation.Numerous studies of magnetic reconnection have been carried out assuming symmetric current sheets, which nicely model some of the situations seen in the nature, such as those in the Earth’s magnetotail. Sharma et al. [4] recently reviewed the magnetotail reconnection topic. In general, however, the electromagnetic and thermodynamic states across a current sheet are not always symmetric. A good example is seen at the Earth’s dayside magnetopause, where there is a significant jump in the plasma properties (e.g., density and temperature) across the current sheet. In-situ observations show that the magnetopause reconnection indeed occurs under an asymmetric condition (e.g., [5–8]). Asymmetric reconnection is also observed in the solar wind (e.g., [9, 10]) and is expected in astrophysical situations [11, 12].Recently, Borovsky and Hesse [13] used resistive magnetohydrodynamics (MHD) simulations to study reconnection rates as a function of the density asymmetry. Cassak and Shay [14] performed a Sweet-Parker type analysis of the structure of the dissipation region to predict that the stagnation point is shifted towards the magnetospheric side of the X-line. This means that the plasma inflow along the current sheet normal is allowed to cross the X-line from the magnetosheath side to the magnetospheric side. In other words, the X-line site is filled by denser plasma of the magnetosheathorigin flowing towards the lessdense magnetospheric side. They have extended their theory to the Hall-MHD regime [15] to find good agreements between the theory and two-fluid simulation results.The first step in particle simulations aiming at the magnetopause reconnection is to superimpose a uniform guide field on the symmetric Harris [16] current sheet (e.g., [17–23]). Some simulations [24–26] studied the effects of diamagnetic drifts on reconnection by taking into account the asymmetries of both density and guide field intensity. The density asymmetry was also taken into account in the two-dimensional (2D) full-particle simulations utilized to assist data analyses of the magnetopause reconnection events [27, 28]. Tanaka et al. [27] not only reproduced the features identified by insitu observations but also succeeded in elucidating the electron dynamics associated with magnetopause reconnection, which is not fully resolved by the currently available dataset.In addition to density asymmetry and nonzero guide field, velocity shear across the magnetopause current sheet is another major item that characterizes the magnetopause reconnection. La Belle-Hammer et al.’s [29] investigated magnetic reconnection in the presence of sheared flow and density asymmetry by 2D resistive MHD simulations. They showed the X-line to move in the same direction as the magnetosheath flow. The solar wind reconnection under an asymmetric situation with a shear flow studied by Eriksson et al. [9] showed good agreements with the results given by La Belle-Hammer et al. simulation [29].Density asymmetry, nonzero guide field, and shear flow are the three major items that characterize the magnetopause reconnection situation. While some studies have been performed to see the effects of these elements, a systematic study using full-particle simulations is not available yet. As will be shown, anX-line in motion is one of the key characteristics of the magnetopause reconnection, where the density asymmetry is combined with more than one of the other effects. Then, it is best to retain full description of the X-line physics in the study, namely, to use full-particle simulations.Regarding the mechanism of theX-line motion along the current sheet, we propose the slanted (from the current sheet normal direction) ion inflow towards the X-line to be one of the mechanisms. Let us outline a simple argument that, in the presence of density asymmetry and shear flow or nonzero guide field, the ion flow at the X-line from the magnetosheath side should be slanted. Here, it should be noted that we are dealing with the ion flow and that only the sign of the guide field component matters in the argument. Unlike in the argument for the electron flow, the nonuniformity of the guide field intensity across the magnetopause (to be more precise, the local current direction right at the X-line) is not considered to be the essential factor in this line of thinking.The argument using Figure1 and the simulation runs shown here are performed in the frame where the flow is symmetric across the magnetopause (see (b)). This is the natural choice because the symmetry is recovered as soon as the density is made symmetric. While the magnetosphere rest frame will be used when the discussion is observer oriented, this frame will be used for discussion of the basic physics.A schematic diagram showing why anX-line should move in the magnetopause reconnection situation. (a) Shift of stagnation point towards the magnetospheric side in the presence of density asymmetry. Here, neither a velocity shear nor a guide field is imposed. (b) The effect of a velocity shear in a density symmetric case. The inflow towards the X-line is slanted from the current sheet normal direction. (c) The ion flow pattern at the X-line when density asymmetry and velocity shear are combined. The X-line site is occupied by magnetosheath ions having nonzero component along the current sheet. (d) The effect of nonzero guide field in a density symmetric case. Due to the Lorentz force near the diffusion region, the inflow towards the X-line becomes slanted from the current sheet normal direction.
(a)
density asymmetry,VMSH=0, BMo=0(b)
density symmetry,VMSH≠0, BMo=0(c)
density asymmetry,VMSH≠0, BMo=0(d)
density symmetry,VMSH=0, BMo≠0Panel (a) depicts the effect studied by Cassak and Shay [14]. The density asymmetry alone makes the ion stagnation point to be shifted toward the magnetospheric side. In the absence of any other effects, the ion flow from the magnetosheath is along the current sheet normal and does not induce an X-line motion. Panel (b) shows the effects of a shear flow. The ion inflow towards the X-line is now slanted. In the absence of density asymmetry, however, the two inflows meet at the X-line, and the ion flow component along the current sheet at the X-line is cancelled to become zero. Panel (c) shows the situation when both density asymmetry and shear flow are present. Now with the shift of the X-line toward the magnetospheric side, the ions filling the X-line site are those from the magnetosheath inflowing in a slanted manner. The ion flow component along the current sheet is nonzero at the X-line. There is a possibility for an X-line motion to be induced in the same direction as the magnetosheath ion flow. Panel (d) shows that a slanted inflow can be created by the effect of nonzero guide field. While the inflow is essentially normal to the current sheet outside of the current sheet, the flow becomes slanted near the center of it due to the Lorentz force (The same flow pattern is also found in [18, 25]. When density asymmetry is also present, the combined effects would produce a situation very similar to that depicted in Panel (c), and thus a sliding motion of the X-line.Regarding the situation where density asymmetry and nonzero guide field are combined, there may well be another explanation. Swisdak et al. [12, 24] proposed that it is the electron flow at the X-line that regulates the X-line motion. The cases to be studied here, unlike the ones studied by Swisdak et al. [24], have uniform guide field, and there is no current density arising from inhomogeneity of guide field (i.e., ∂BM/∂N=0. See the coordinates of Figure 1) along the magnetopause current sheet in the initial condition. As the magnetopause reconnection develops under the density asymmetry, however, there generated nonzero gradient of the guide field along the current sheet normal at the X-line, indicating that nonzero current density along the current sheet emerges at the X-line. The current density is sustained by the electron flow at the X-line along the current sheet. That is, when density asymmetry and nonzero guide field are combined, the electron flow at the X-line along the current sheet, which is another candidate to drive the X-line motion, is generated.The structure of this paper is such that the above two mechanisms for theX-line motion will be tested in a step-by-step manner using full-particle simulations. As will be shown, the systematic survey of the results from the cases that include all the elements characterizing magnetopause reconnection (density asymmetry, shear flow, and nonzero guide field) is useful in elucidating the issue.


## 2. Simulation Setup

We use 2 1/2-dimensional (two spatial dimensions (N,L) and three velocity components (vL,vM,vN)) electromagnetic full-particle simulation [30]. The LMN coordinates set the basic framework for magnetopause studies, with the L-axis pointing to the direction of the magnetospheric field and the N-axis pointing outward along the magnetopause normal. The Harris-type current sheet is used to ensure that the initial condition is not far from a Vlasov equilibrium. Full details are given below.Two plasma populations are set in the initial condition. One population constitutes the Harris current sheet [16]. The Harris current sheet is expressed as nCS=n0/cosh2(N/D) for the plasma density and as BL=-Botanh(L/D) (see Figure 2(f)) for the magnetic field, where D is the half-thickness of the current sheet. The plasma inside the current sheet has the ion-to-electron temperature ratio of Ti,cs=8Te,cs. The other plasma population is the background plasma having the number density of nBG1=n1{1-1/cosh2(N/D)} at xN>0 (the magnetosheath side) and nBG2=n2{1-1/cosh2(N/D)} at xN<0 (the magnetospheric side) (see Figure 2(e)). It is distributed outside the current sheet. In the present paper, the background plasma density is set as n1=n0 and n2=0.2n0, respectively. That is, there is a factor of five-density difference across the magnetopause current sheet. The background temperatures are set to be Ti,BG1=Te,BG1=Te,BG2=Te,CS. Then, Ti,BG2 (the ion temperature on the magnetospheric side) is determined to satisfy the pressure balance. After some easy calculation, the ion temperature is found to be Ti,BG2=9Te,CS. The initial plasma beta in the magnetosheath is 0.22. We note that this low beta magnetosheath is one of the limitations of our study.The initial configuration used in this study. From (a) to (d), the one-dimensional cuts of theL component magnetic field BL, the electron number density ne, the ion and electron pressure tensor components Pi,NN (solid) and Pe,NN (dashed), and the ion and electron temperature tensor components Ti,NN (solid) and Te,NN (dashed), respectively. (e) and (f) show the two-dimensional contours of the electron density and the BL  component. The coordinate system is depicted in (e). White arrows in (f) depict how the shear flow is given in this study. Black arrows in (f) show the directions of the magnetic field.
(a)(b)(c)(d)(e)(f)Hereinafter, magnetic field and density will be normalized byBo and n0, respectively. The ion inertial length λi=c/ωpi based on n0 and the inverse ion gyrofrequency Ωi-1 based on Bo are the units for spatial and time scales, respectively, and velocities will be measured in the magnetosheath ion Alfvén speed unit VAi=Bo/(4πmin0)1/2 (mi: the ion mass). The electric field is normalized by Bo*(VAi/c), where c is the light speed. In this normalization, the magnetosheath electron inertial length is λe=M-1/2 (M: the ion-to-electron mass ratio mi/me). We set the electron plasma frequency to electron gyrofrequency ratio to be τ=ωpe=/Ωe=1 to reduce the computational cost, whereas the realistic value is ~40. In the present study, we do not expect τ to play a major role in the two-dimensional cases, as was recently suggested by Haijima et al. [31]. M=100 and thus c/VAi=10 in our study.Figure2 shows the initial configuration of this study. Panels (a)~(d) show the one-dimensional cuts along the N-axis of various parameters. From top to bottom, the magnetic fields BL, the electron plasma density ne, the ion and electron NN-pressure tensor elements Pi,NN (solid) and Pe,NN (dashed), and the ion and electron NN-temperature tensor elements Ti,NN (solid) and Te,NN (dashed) are presented. Two-dimensional structures of electron number density ne and the reconnecting magnetic field BL are shown in Panels (e) and (f). The LMN-coordinates are shown in Panel (e).It is noted that the pressure balance initialization is different from that of Swisdak et al. [24]. While the initial pressure balance of Swisdak et al. [24] was under the varying magnetic field strength across the current layer and thus there was nonzero pressure gradient at the center of the current sheet, there is no pressure gradient at the center in our case.In some of the cases studied in this paper, we will have a sheared flow and/or nonzero out-of-plane magnetic field. Sheared flow is set up by giving oppositely directed background bulk flow±0.5VMSH in the simulation frame. A schematic of a sheared flow is given in Figure 2(f) by the white arrows. Here, VMSH denotes the magnetosheath flow velocity measured in the magnetosphere-rest frame. With this choice of the simulation frame of reference, the system is symmetric in the absence of the density asymmetry. The guide field BMo is uniformly added in whole simulation domain (as mentioned already, this is different from the cases studied in Swisdak et al. [24]).In addition to the Harris magnetic field, a Gaussian-type perturbation vector potential is introduced to set anX-line at the center of the box. The amplitude of the initial perturbation in the component normal to the current sheet is δBN=0.05.Periodic boundary conditions are imposed in theL direction while conducting walls are set at the N boundaries. The simulation box is [-LL/2:LL/2]×[-LN/2:LN/2] in the normalized unit, with LL=24 and LN=16. The boundary conditions do not have any effects on the results shown in this paper. The half-thickness of the current layer D is set to be 0.5. The spatial grid of Δx=0.9λDe (λDe: Debye length) and the time step of Δt=0.15625Ωe-1 (Ωe: the electron gyrofrequency) are adopted, respectively. 320 particles per grid represent the unit density.


## 3. Key Quantities to Be Inspected in This Study

The location of theX-line is denoted by (LXL, NXL). In the same way, the velocities of the X-line, ion velocity at the X-line, and the electron velocity at the X-line are denoted by (VL,XL, VN,XL), (ViL,XL, ViN,XL), and (VeL,XL, VeN,XL), respectively. Note that these velocities are measured in the simulation reference frame, and translation to the magnetosphere-rest frame requires correction by 0.5VMSH in the L-component. The reconnected flux will be often used to present the reconnection activity and is defined by ΨRX=0.5∫|BN(L,0)|dL.


## 4. Simulation Results

In the presence of density asymmetry, it is now known that the ion stagnation point is shifted to the magnetospheric (low density) side of the X-line [14, 15]. For a clarification purpose, this simplest case with neither shear flow nor guide field is introduced briefly at the beginning. Then more effects will be added in the subsections that follow.


### 4.1. TheVMSH=BMo=0 Case

Figure3 shows the results of the VMSH=BMo=0 case. Panel (a) shows the time history of reconnected flux Ψ. Panel (b) shows the time history of NXL. The X-line position starts to shift toward the magnetospheric side (N<0) at t~16. Then until t~30, the X-line continues to move at a roughly constant speed of VN,XL~0.023. This result obtained by our particle code is consistent with the previous studies using fluid models [14, 15].The result ofVMSH=BMo=0 case with density asymmetry. Temporal developments of (a) the reconnected flux and (b) N-coordinate of the X-line are shown. The X-line position shifts to the magnetospheric (negative N) side between t=15 and 30.
(a)(b)Figure4 shows the two-dimensional contour of the (a) ion density and (b) out-of-plane magnetic field BM at t=25. At this time, magnetic reconnection is still going on (Figure 3(a)). Only a small fraction around the X-line [L,N]=[-3.5:2.5,-2:2] is shown. The black and purple lines represent the field lines and the ion number density flux streamlines niVi, respectively. The streamlines clearly show the asymmetric inflow toward the X-line from the magnetosheath side. BM is positive at L>0 and vice versa. This dipolar feature of the Hall field is the characteristics of a density asymmetry case (e.g., [25, 27]) and has been seen in observations [32]. The red horizontal line is the path along which the N directional profiles of various parameters are sampled (see Figure 5).Zooming into theX-line region for the VMSH=BMo=0 case. (a) The density and (b) out-of-plane magnetic field BM patterns in the area of [LXL-3,LXL+3]×[NXL-2,NXL+2] are shown. The time is t=25. The black and purple lines show the field lines and the ion streamlines, respectively.
(a)(b)Figure 5
One-dimensional cuts of various parameters along theN-axis at the X-line position (along the red arrow shown in Figure 4). From top to bottom, (a) ne, (b) BL (red) and BN (blue, multiplied by 10), (c) niViL (red) and niViN (blue), and (d) EN, respectively. The dashed line at XN=-0.25 indicates the X-line position (where BL=0 in Panel (b)).The asymmetric inflow is shown in a more explicit manner. Shown in Figure5 are the N-directional profiles of various parameters at L=LXL. From top to bottom, (a) ne, (b) BL (red) and BN (blue, multiplied by 10), (c) niViL (red) and niViN (blue), and (d) EN are shown. The black vertical line depicts the location of the X-line NXL (determined in Panel b). In Panel (c), while the ion stagnation point is located between -1<N<-0.5, it is clear that the ion stagnation point is more to the magnetospheric side than the X-line.


### 4.2. Combination of Density Asymmetry and Sheared Flow

Section4.1 showed that the magnetosheath ions flow across the X-line into the magnetospheric side. On the other hand, we are aware that there is a flow component along the magnetopause (the L direction) in the magnetosheath side away from the equator. Then, it is natural to expect that the L component of the magnetosheath ion flow at the X-line site is not zero. In this section, we study how such a slanted ion inflow affects the properties of the X-line. Here, guide field is set to be zero. How the shear flow is set (i.e., the frame of reference) is described in Figure 2(f).Figure6 compares the results from various VMSH (the velocity jump across the magnetopause) cases. Panel (a) shows the time histories of ΨRX. The red, blue, and black lines represent the VMSH=0, 1, and 2 cases, respectively. One can see the reconnection activity reduces with increasing VMSH, but the effects are not very dramatic up to VMSH=2. Panel (b) shows the position of the X-line (LXL) versus time. The dotted lines depict the slopes and thus the speeds of the X-line motion VL,XL during the reconnection stage. VL,XL are 0.18 and 0.32 for the VMSH=1 and =2  cases, respectively. Comparison between Panels (a) and (b) suggests that the X-line sliding is an ongoing process of magnetic reconnection. Seen in the magnetospheric-rest frame, the speed of the moving X-lines is ~0.7 times of the magnetosheath flow speed in both cases (VL,XL=0.68 in the VMSH=1 case and =1.32 in the VMSH=2 case, resp.). Panel (c) shows the L component of the ion flow at the X-line ViL,XL, which is nonzero. The dotted black and blue horizontal lines depict VL,XL as references. One can see that there is nice agreement between ViL,XL and VL,XL. The slanted inflow of the magnetosheath ions makes nonzero ViL,XL at the X-line and induces the X-line to move in the same direction at the same speed, VL,XL~ViL,XL.The results of nonzeroVMSH cases without guide field. (a) Growth curves of reconnected flux. (b) The L-coordinates of the X-lines versus time. (c) The X-line speeds compared with the ion and electron flow at the X-line.
(a)(b)(c)Figure7 is the same format as Figure 4 except for the VMSH=1 case at t=20.25. The area covered by each panel is [-0.3:5.7]×[-2:2]. Comparison between Figures 4(b) and 7(b) presents that BM structure has less significant difference in the parameter range considered here. The red vertical arrow is the path along which the one-dimensional cuts at L=LXL shown in Figure 8 are made.Figure 7
Zooming into theX-line region for the VMSH=1 and BMo=0 case. (a) Density and (b) BM. The black and purple curves are the field lines and the ion streamlines, respectively. The spatial dimension is [LXL-3,LXL+3]×[NXL-2,NXL+2]. The time is t=20.25.Figure 8
Various parameters along the red arrow shown in Figure7. In the same format as Figure 5, Panel (c) clearly indicates that the ion flow component at the X-line ViL,XL is nonzero and positive.Figure8 shows the N-directional cuts at the X-line position L=LXL. The format is the same as Figure 5. In Panel (b), the X-line has shifted into the magnetospheric side at NXL=-0.1 (vertical black dashed line) while in Panel (c), the ion stagnation point is around N=-0.8.niViN shown in Panel (c) indicates that the magnetosheath ions flow across the X-line toward the magnetospheric side and niViL at the X-line has a nonzero positive value.


### 4.3. Combination of Density Asymmetry and out-of-Plane Magnetic Field

Section4.2 has shown that a slanted ion inflow to the X-line forces it to slide along the L-direction. The slanted inflow was realized by the combination of density asymmetry and shear flow. One may also expect that the combination of density asymmetry and nonzero guide field BM is to produce a slanted inflow of ions (See Figure 1). On the other hand, when nonzero guide field is present, the inplane current pattern is formed in the course of reconnection and there generated nonzero electron flow at the X-line, which may also drive the X-line to move. The case discussed in this subsection has the uniform guide field of BMo=-0.5 and is without a shear flow VMSH=0.Figure9 shows the results. Panels (a) and (b) show the time histories of ΨRX and LXL, respectively. In Panel (b), time-space evolution of the BN component along the reversing-BL curve (i.e., BL=0) is shown. For the convenience, the BL=0 curve is projected onto the L-axis. The black stripe elongated to the upright direction shows the time evolution of the region where BN=0 and thus shows the X-line motion. The X-line starts to slide at t~10 with the constant velocity of VL,XL~0.15 until t~30. Panel (c) compares this VL,XL (horizontal dashed line) with the ion and electron bulk flow velocities at the X-line, ViL,XL (dotted) and VeL,XL (dotted). All the three components are in the same direction (positive-L). The large difference in the velocity component between the ions and the electrons indicates that there is significant current density along the L-axis at the X-line, which is not present in the initial condition. The X-line speed VL,XL=0.15 is larger than the rather small ion velocity of ViL,XL=0.05. On the other hand, the electrons have a large flow of VeL,XL=1. While a definitive conclusion cannot be drawn only from these results, one would tend to think that the small ion flow is not sufficient to drive the faster X-line motion and that the role of the electrons cannot be neglected.Results of the nonzero guide fieldBMo=-0.5 case without velocity shear (VMSH=0). (a) Growth of reconnection flux, (b) gray-scale contour of BN along the BL=0 curve in the L-time diagram depicting the X-line motion, and (c) the X-line motion speed compared with ion and electron flow speed at the X-line.
(a)(b)(c)Figure10 shows the two-dimensional contours of (a) density and (b) BM at t=16. The format is the same as Figure 7. At the X-line, one can see that there is ∂BM/∂N>0. The cut along the N-axis at the X-line shown in Figure 11 tells this more explicitly. Panels (a)~(d) show the magnetic field, the ion flow, the electron flow, and ∂BM/∂N, respectively. The X-line, which, unlike the zero-guide field cases shown previously, somehow has shifted to the magnetosheath side, is filled with the magnetosheath ions having the small positive L-component velocity. At the same time, the negative L-component of the current density implied by ∂BM/∂N>0 at the X-line is sustained by the fast electron flow in the positive-L direction.Figure 10
Results from the (VMSH,BMo) = (0,-0.5) case shown in the same format as Figure 7. The spatial dimension is [LXL-3,LXL+3]×[NXL-2,NXL+2]. The time is t=16. The part of the upward jet having density higher than the magnetosheath value is accompanied by the band of BM~0 on the magnetosheath side of it.Figure 11
The cuts along theN-axis at the X-line of (a) B, (b) Vi, (d) Ve, and (d) ∂BM/∂N. The X-line is filled with the magnetosheath ions having the small positive L-component velocity. At the same time, the negative L-component of the current density implied by ∂BM/∂N>0 at the X-line is sustained by the fast electron flow in the positive-L direction.Coming back to Figure10, in the quadrant above the X-line and on the magnetosheath side, both the density and the BM component show peculiar features that appear through the combination of the density asymmetry and the nonzero guide field. Here, the density in the upward directed jet is larger than the magnetosheath density (=1). This signature is consistent with an asymmetric reconnection with a guide field work by Pritchett [25].  Both the density and BM perturbation patterns located near the separatrices are similar to what is understood in guide field magnetic reconnection with a symmetric density situation although these patterns are skewed somewhat due to the inflow asymmetry. Since the density on the magnetospheric side is 0.2, a natural expectation is that the jet density would be between 0.2 and 1. In addition, there is a band of weak BM located at the outer part of the jet (the reddish part with BM~-0.2). This is also counter to a natural expectation that the guide field component is compressed (<-0.5) in the jet. Figure 12 shows the one-dimensional cut along the upper red arrow depicted in Figure 10. The format is the same as Figure 5, except that Panels (c) and (d) are now for the ion and electron flows, respectively. Panel (a) shows that the density at N=0.5 increases up to ne=1.5. In this location, the ion jet ViL is nonnegligibly large. The high-density part of the jet is situated between the magnetosheath and the jet showing EN>0 (pointing outward to the magnetosheath side, Panel (e)). The ions coming into the jet from the magnetosheath side, when unmagnetized, would sense this EN>0 as the repelling force to push the ions back toward the magnetosheath side. Then the ions are confined in the magnetosheath side of the jet. This would explain the origin of the high-density part. In the second and fourth (first and third) quadrants of Figures 10(a) and 10(b), the density decreases (increases) and BM is compressed (rarefied). Previous guide field reconnection works with symmetric density also discussed these structures and suggested that this structure arises due to the physics of the kinetic Alfvén wave in association with intense electric fields [18, 33–35].Figure 12
The cuts along theN-axis of the high-density structure at L=4. In the same format as Figure 5 except that ion and electron flows are now shown in Panels (c) and (d), respectively. The high-density peak (at N=0.5) is between the two density dips (at N=-0.85 and 1.27) and is on the magnetosheath side of the region showing EN>0.TheX-line motion induced by the combination of density asymmetry and the uniform guide field is not necessary trivial, and we have double-checked the results by performing a run with a smaller M=25 but with a larger simulation box LL=192 (with D=1). Figure 13 shows the X-line motion in the same format as Figure 9(b). From the slope, one obtains VL,XL~0.15, which is in excellent agreement with the M=100 case. This run also enables us to confirm the constancy of the X-line speed for an extended time period in a larger box. Figure 14 shows two-dimensional contours of (a) number density and (b) out-of-plane magnetic field. One can clearly see that the peculiar features in the upward jet, namely, the higher density in the magnetosheath side half and the weak BM attached at the edge on the magnetosheath side, elongate away from the X-line implying that they are not the features local to the X-line region.Figure 13
The same as Figure9(b) but for the smaller mass ratio/larger simulation box case of (VMSH,BMo)=(0,-0.5). It shows that VL,XL is constant for an extended time interval.Figure 14
Two-dimensional contours of (a) density and (b) out-of-plane magnetic field for the low mass ratio/larger simulation box case. The peculiar features in density and inBM seen in the upward-directed jet are not local to the X-line but are extended away from it.


### 4.4. Combination of Density Asymmetry, Shear Flow, and out-of-Plane Magnetic Field

We have learned that combination of density asymmetry and shear flow (in Section4.2), or of density asymmetry and nonzero guide field (in Section 4.3), makes the X-line to slide along the L direction. While there may well be various other ways to create an X-line motion, the two items combined with the density asymmetry considered here are the ones well expected at the magnetopause. In other words, the X-lines driving the magnetopause reconnection are most likely to be in motion. The direction of the X-line motion is determined by (1) the flow direction in the magnetosheath for the shear flow only case and (2) the sign of the out-of-plane magnetic field for the nonzero guide field only case. Then, an interesting question is how an X-line behaves when both the shear flow and guide-field are simultaneously present, and especially when they counteract to each other in determining the direction of the X-line motion.Table1 is the summary of the results described so far that have relevance to the run studied in the next subsection. VXL,sim is the X-line speed in the simulation frame. BMo=±0.5 alone or VMSH=1 alone induces almost the same X-line sliding speed. The direction, however, can be opposite according to the sign of BMo (as mentioned, we think that the frame in which these velocities are measured is suited for this kind of discussion). The counter-acting case would be the most interesting one to study and the results are shown below.Table 1
Summary of the results from the simpler cases.BMo: guide field, VMSH: velocity jump across the magnetopause, VXL,sim: the X-line speed measured in the simulation frame.
BMoVMSHVXL,sim-0.50+0.15+0.50-0.1501+0.18


#### 4.4.1. The BMo = 0.5 and VMSH = 1 Case

Figure15 shows the results of the (VMSH,BMo) = (1, 0.5) case. Previous results show that the effects of VMSH=1 would push the X-line in the positive L-direction while those of BMo=0.5 in the negative L-direction. One can see in Panel (a) that the X-line starts to slide from t=10 in the positive L-direction with VL,XL=0.14. This X-line speed is almost the same as the VMSH=1 alone case (Table 1). Panel (b) shows ViL,XL=0.35 and VeL,XL=-0.5. The X-line motion is in the same direction as the ion flow and is opposite to the electron flow.The results from the(VMSH,BMo)=(1,0.5) case, where the guide field and the velocity shear are considered to counteract in determining the direction of the X-line motion (See Table 1). The X-line motion shown in Panel (a) indicates that the shear flow effects dominate. Discussion on the X-line properties shown in Panel (b) given in the text, however, suggests that there are some roles played by the electrons as well.
(a)(b)What is interesting here is the fact thatViL,XL is faster than the VMSH=1 alone case (0.18). The argument in Figure 1 implied that the guide field BMo>0 would make the ion flow to be slower, but this turns out not to be the case. What is more interesting, despite this accelerated ion flow, the X-line speed is slightly slower. This indicates that there should be something else other than the ion flow to determine the speed of the X-line motion. Indeed, the gradient of the guide field component at the X-line drives the fast electron flow in the opposite direction (-0.5) and this may be the slowing down agent. That is, both ion and electron flow at the X-line seem to play roles in determining the direction and speed of the X-line motion.


#### 4.4.2. VL,XL from Various (VMSH, BMo) Cases

When the sign ofBMo is flipped ((VMSH,BMo)=(1,-0.5) case), the combined effects are found to speed up the X-line motion to VL,XL=0.37. The flow velocities at the X-line are ViL,XL=0.42 and VeL,XL=1.3, respectively. The flipping of the BM component leads to a substantial change in VL,XL (from 0.14 to 0.37) with only slight change in ViL,XL (from 0.35 to 0.42). In contrast, there is a substantial change in VeL,XL (from -0.5 to 1.3). These again suggest possible roles played by the electrons in regulating the X-line motion. Here, we try to learn more by studying the results from various sets of (VMSH,BMo) in this subsection. Table 2 shows all the cases studied.Table 2
Summary of all the cases studied in this paper.BMo: guide field, VMSH: velocity jump across the magnetopause, VXL,sim: the X-line speed measured in the simulation frame, ViL,XL: the L-component of the ion flow at the X-line, VeL,XL: the L-component of the electron flow at the X-line, and  VXL,MSP: the X-line speed in the magnetosphere-rest frame.
BMoVMSHVXL,simViL,XLVeL,XLVXL,MSPVXL,MSP/VMSH-0.500.150.05110.15N/A-0.50.50.280.261.210.531.06-0.510.370.421.30.870.87010.180.180.40.680.68020.320.310.521.320.660.50-0.15-0.051-1-0.15N/A0.50.2-0.0880.047-0.970.0120.060.50.4-0.0270.1-0.840.170.430.50.50.00470.14-0.780.250.510.50.70.0720.22-0.690.420.60.510.140.35-0.50.640.64Figure16 shows the time history of reconnected flux for the four cases (VMSH,BMo)=(0,-0.5) (black), (1, 0) (blue), (1, 0.5) (greed), and (1, −0.5) (red), respectively. This figure is to confirm that there is no substantial difference in terms of reconnection history due to the shear flow, nonzero guide field, and their combinations within the range considered here.Figure 16
Growth curves of reconnection flux from various sets of (VMSH, BMo). There is little difference within the (VMSH, BMo) range surveyed.Figure17(a) plots VL,XL versus VMSH, with different BMo cases shown by different colors (blue, black, and red for BMo=-0.5, 0, and 0.5, resp.). The general trends are (1) VL,XL is larger for larger VMSH and (2) BMo=-0.5 (blue) cases give the fastest and BMo=0.5 (red) cases give the lowest VL,XL when VMSH is the same. The most interesting feature is that the X-line motion can be opposite to the magnetosheath flow when VMSH is small and BMo=0.5 (red). These clearly indicate that it is not only the ion flow that determines the direction of the X-line motion.The summary plot for the discussion on theX-line motion. Different color indicates different BMo cases (blue, black, and red for BMo=-0.5, 0, and 0.5). (a) VL,XL versus VMSH. While VL,XL is larger for larger VMSH in general, the X-line motion can be opposite to the magnetosheath flow when VMSH is small and BMo=+0.5. (b) Comparison of VL,XL with ViL,XL (solid circles) and with VeL,XL (open squares). The interpretation is that the electron effects set in with the nonzero guide field and can dominate to determine the direction of the X-line motion when the shear flow effects are small.
(a)(b)Figure17(b) compares VL,XL with ViL,XL (solid circles) and with VeL,XL (open squares). The most eye-catching feature is the behavior of the BMo=0.5 (red) cases. These have smaller VL,XL but it is not necessarily because these cases have smaller ViL,XL compared to the others. Furthermore VL,XL becomes negative as VMSH is made smaller, but this happens even when ViL,XL stays positive. Meanwhile VeL,XL is negative as long as BMo=0.5 (red open squares). These suggest that the role played by the electron flow at the X-line, which is enhanced in the presence of nonzero guide field, is not limited to controlling the X-line speed but can be as substantial as to determine its direction when the shear flow effects are weak.Comparing the three cases having mostly the sameX-line speed of VL,XL~0.15 would be illuminating in this regard. The three cases are (VMSH,BMo)=(0,-0.5), (1, 0), and (1, 0.5) (See Table 1). The X-line of the first case (blue symbols) has this speed even though the ion flow at the X-line is pretty small. Instead, the electron flow is fast in the same direction (positive-L). The third case (red symbols) has almost the same X-line speed despite the considerably larger ion flow. It is noted that the electrons at the X-line are flowing fast in the opposite direction (negative-L). In the second case (black symbols) without guide field, there is only small difference between the ion and the electron flow at the X-line. The X-line speed coincides with the ion speed that is made nonzero by the slanted in-flow from the magnetosheath side.


## 4.1. TheVMSH=BMo=0 Case

Figure3 shows the results of the VMSH=BMo=0 case. Panel (a) shows the time history of reconnected flux Ψ. Panel (b) shows the time history of NXL. The X-line position starts to shift toward the magnetospheric side (N<0) at t~16. Then until t~30, the X-line continues to move at a roughly constant speed of VN,XL~0.023. This result obtained by our particle code is consistent with the previous studies using fluid models [14, 15].The result ofVMSH=BMo=0 case with density asymmetry. Temporal developments of (a) the reconnected flux and (b) N-coordinate of the X-line are shown. The X-line position shifts to the magnetospheric (negative N) side between t=15 and 30.
(a)(b)Figure4 shows the two-dimensional contour of the (a) ion density and (b) out-of-plane magnetic field BM at t=25. At this time, magnetic reconnection is still going on (Figure 3(a)). Only a small fraction around the X-line [L,N]=[-3.5:2.5,-2:2] is shown. The black and purple lines represent the field lines and the ion number density flux streamlines niVi, respectively. The streamlines clearly show the asymmetric inflow toward the X-line from the magnetosheath side. BM is positive at L>0 and vice versa. This dipolar feature of the Hall field is the characteristics of a density asymmetry case (e.g., [25, 27]) and has been seen in observations [32]. The red horizontal line is the path along which the N directional profiles of various parameters are sampled (see Figure 5).Zooming into theX-line region for the VMSH=BMo=0 case. (a) The density and (b) out-of-plane magnetic field BM patterns in the area of [LXL-3,LXL+3]×[NXL-2,NXL+2] are shown. The time is t=25. The black and purple lines show the field lines and the ion streamlines, respectively.
(a)(b)Figure 5
One-dimensional cuts of various parameters along theN-axis at the X-line position (along the red arrow shown in Figure 4). From top to bottom, (a) ne, (b) BL (red) and BN (blue, multiplied by 10), (c) niViL (red) and niViN (blue), and (d) EN, respectively. The dashed line at XN=-0.25 indicates the X-line position (where BL=0 in Panel (b)).The asymmetric inflow is shown in a more explicit manner. Shown in Figure5 are the N-directional profiles of various parameters at L=LXL. From top to bottom, (a) ne, (b) BL (red) and BN (blue, multiplied by 10), (c) niViL (red) and niViN (blue), and (d) EN are shown. The black vertical line depicts the location of the X-line NXL (determined in Panel b). In Panel (c), while the ion stagnation point is located between -1<N<-0.5, it is clear that the ion stagnation point is more to the magnetospheric side than the X-line.


## 4.2. Combination of Density Asymmetry and Sheared Flow

Section4.1 showed that the magnetosheath ions flow across the X-line into the magnetospheric side. On the other hand, we are aware that there is a flow component along the magnetopause (the L direction) in the magnetosheath side away from the equator. Then, it is natural to expect that the L component of the magnetosheath ion flow at the X-line site is not zero. In this section, we study how such a slanted ion inflow affects the properties of the X-line. Here, guide field is set to be zero. How the shear flow is set (i.e., the frame of reference) is described in Figure 2(f).Figure6 compares the results from various VMSH (the velocity jump across the magnetopause) cases. Panel (a) shows the time histories of ΨRX. The red, blue, and black lines represent the VMSH=0, 1, and 2 cases, respectively. One can see the reconnection activity reduces with increasing VMSH, but the effects are not very dramatic up to VMSH=2. Panel (b) shows the position of the X-line (LXL) versus time. The dotted lines depict the slopes and thus the speeds of the X-line motion VL,XL during the reconnection stage. VL,XL are 0.18 and 0.32 for the VMSH=1 and =2  cases, respectively. Comparison between Panels (a) and (b) suggests that the X-line sliding is an ongoing process of magnetic reconnection. Seen in the magnetospheric-rest frame, the speed of the moving X-lines is ~0.7 times of the magnetosheath flow speed in both cases (VL,XL=0.68 in the VMSH=1 case and =1.32 in the VMSH=2 case, resp.). Panel (c) shows the L component of the ion flow at the X-line ViL,XL, which is nonzero. The dotted black and blue horizontal lines depict VL,XL as references. One can see that there is nice agreement between ViL,XL and VL,XL. The slanted inflow of the magnetosheath ions makes nonzero ViL,XL at the X-line and induces the X-line to move in the same direction at the same speed, VL,XL~ViL,XL.The results of nonzeroVMSH cases without guide field. (a) Growth curves of reconnected flux. (b) The L-coordinates of the X-lines versus time. (c) The X-line speeds compared with the ion and electron flow at the X-line.
(a)(b)(c)Figure7 is the same format as Figure 4 except for the VMSH=1 case at t=20.25. The area covered by each panel is [-0.3:5.7]×[-2:2]. Comparison between Figures 4(b) and 7(b) presents that BM structure has less significant difference in the parameter range considered here. The red vertical arrow is the path along which the one-dimensional cuts at L=LXL shown in Figure 8 are made.Figure 7
Zooming into theX-line region for the VMSH=1 and BMo=0 case. (a) Density and (b) BM. The black and purple curves are the field lines and the ion streamlines, respectively. The spatial dimension is [LXL-3,LXL+3]×[NXL-2,NXL+2]. The time is t=20.25.Figure 8
Various parameters along the red arrow shown in Figure7. In the same format as Figure 5, Panel (c) clearly indicates that the ion flow component at the X-line ViL,XL is nonzero and positive.Figure8 shows the N-directional cuts at the X-line position L=LXL. The format is the same as Figure 5. In Panel (b), the X-line has shifted into the magnetospheric side at NXL=-0.1 (vertical black dashed line) while in Panel (c), the ion stagnation point is around N=-0.8.niViN shown in Panel (c) indicates that the magnetosheath ions flow across the X-line toward the magnetospheric side and niViL at the X-line has a nonzero positive value.


## 4.3. Combination of Density Asymmetry and out-of-Plane Magnetic Field

Section4.2 has shown that a slanted ion inflow to the X-line forces it to slide along the L-direction. The slanted inflow was realized by the combination of density asymmetry and shear flow. One may also expect that the combination of density asymmetry and nonzero guide field BM is to produce a slanted inflow of ions (See Figure 1). On the other hand, when nonzero guide field is present, the inplane current pattern is formed in the course of reconnection and there generated nonzero electron flow at the X-line, which may also drive the X-line to move. The case discussed in this subsection has the uniform guide field of BMo=-0.5 and is without a shear flow VMSH=0.Figure9 shows the results. Panels (a) and (b) show the time histories of ΨRX and LXL, respectively. In Panel (b), time-space evolution of the BN component along the reversing-BL curve (i.e., BL=0) is shown. For the convenience, the BL=0 curve is projected onto the L-axis. The black stripe elongated to the upright direction shows the time evolution of the region where BN=0 and thus shows the X-line motion. The X-line starts to slide at t~10 with the constant velocity of VL,XL~0.15 until t~30. Panel (c) compares this VL,XL (horizontal dashed line) with the ion and electron bulk flow velocities at the X-line, ViL,XL (dotted) and VeL,XL (dotted). All the three components are in the same direction (positive-L). The large difference in the velocity component between the ions and the electrons indicates that there is significant current density along the L-axis at the X-line, which is not present in the initial condition. The X-line speed VL,XL=0.15 is larger than the rather small ion velocity of ViL,XL=0.05. On the other hand, the electrons have a large flow of VeL,XL=1. While a definitive conclusion cannot be drawn only from these results, one would tend to think that the small ion flow is not sufficient to drive the faster X-line motion and that the role of the electrons cannot be neglected.Results of the nonzero guide fieldBMo=-0.5 case without velocity shear (VMSH=0). (a) Growth of reconnection flux, (b) gray-scale contour of BN along the BL=0 curve in the L-time diagram depicting the X-line motion, and (c) the X-line motion speed compared with ion and electron flow speed at the X-line.
(a)(b)(c)Figure10 shows the two-dimensional contours of (a) density and (b) BM at t=16. The format is the same as Figure 7. At the X-line, one can see that there is ∂BM/∂N>0. The cut along the N-axis at the X-line shown in Figure 11 tells this more explicitly. Panels (a)~(d) show the magnetic field, the ion flow, the electron flow, and ∂BM/∂N, respectively. The X-line, which, unlike the zero-guide field cases shown previously, somehow has shifted to the magnetosheath side, is filled with the magnetosheath ions having the small positive L-component velocity. At the same time, the negative L-component of the current density implied by ∂BM/∂N>0 at the X-line is sustained by the fast electron flow in the positive-L direction.Figure 10
Results from the (VMSH,BMo) = (0,-0.5) case shown in the same format as Figure 7. The spatial dimension is [LXL-3,LXL+3]×[NXL-2,NXL+2]. The time is t=16. The part of the upward jet having density higher than the magnetosheath value is accompanied by the band of BM~0 on the magnetosheath side of it.Figure 11
The cuts along theN-axis at the X-line of (a) B, (b) Vi, (d) Ve, and (d) ∂BM/∂N. The X-line is filled with the magnetosheath ions having the small positive L-component velocity. At the same time, the negative L-component of the current density implied by ∂BM/∂N>0 at the X-line is sustained by the fast electron flow in the positive-L direction.Coming back to Figure10, in the quadrant above the X-line and on the magnetosheath side, both the density and the BM component show peculiar features that appear through the combination of the density asymmetry and the nonzero guide field. Here, the density in the upward directed jet is larger than the magnetosheath density (=1). This signature is consistent with an asymmetric reconnection with a guide field work by Pritchett [25].  Both the density and BM perturbation patterns located near the separatrices are similar to what is understood in guide field magnetic reconnection with a symmetric density situation although these patterns are skewed somewhat due to the inflow asymmetry. Since the density on the magnetospheric side is 0.2, a natural expectation is that the jet density would be between 0.2 and 1. In addition, there is a band of weak BM located at the outer part of the jet (the reddish part with BM~-0.2). This is also counter to a natural expectation that the guide field component is compressed (<-0.5) in the jet. Figure 12 shows the one-dimensional cut along the upper red arrow depicted in Figure 10. The format is the same as Figure 5, except that Panels (c) and (d) are now for the ion and electron flows, respectively. Panel (a) shows that the density at N=0.5 increases up to ne=1.5. In this location, the ion jet ViL is nonnegligibly large. The high-density part of the jet is situated between the magnetosheath and the jet showing EN>0 (pointing outward to the magnetosheath side, Panel (e)). The ions coming into the jet from the magnetosheath side, when unmagnetized, would sense this EN>0 as the repelling force to push the ions back toward the magnetosheath side. Then the ions are confined in the magnetosheath side of the jet. This would explain the origin of the high-density part. In the second and fourth (first and third) quadrants of Figures 10(a) and 10(b), the density decreases (increases) and BM is compressed (rarefied). Previous guide field reconnection works with symmetric density also discussed these structures and suggested that this structure arises due to the physics of the kinetic Alfvén wave in association with intense electric fields [18, 33–35].Figure 12
The cuts along theN-axis of the high-density structure at L=4. In the same format as Figure 5 except that ion and electron flows are now shown in Panels (c) and (d), respectively. The high-density peak (at N=0.5) is between the two density dips (at N=-0.85 and 1.27) and is on the magnetosheath side of the region showing EN>0.TheX-line motion induced by the combination of density asymmetry and the uniform guide field is not necessary trivial, and we have double-checked the results by performing a run with a smaller M=25 but with a larger simulation box LL=192 (with D=1). Figure 13 shows the X-line motion in the same format as Figure 9(b). From the slope, one obtains VL,XL~0.15, which is in excellent agreement with the M=100 case. This run also enables us to confirm the constancy of the X-line speed for an extended time period in a larger box. Figure 14 shows two-dimensional contours of (a) number density and (b) out-of-plane magnetic field. One can clearly see that the peculiar features in the upward jet, namely, the higher density in the magnetosheath side half and the weak BM attached at the edge on the magnetosheath side, elongate away from the X-line implying that they are not the features local to the X-line region.Figure 13
The same as Figure9(b) but for the smaller mass ratio/larger simulation box case of (VMSH,BMo)=(0,-0.5). It shows that VL,XL is constant for an extended time interval.Figure 14
Two-dimensional contours of (a) density and (b) out-of-plane magnetic field for the low mass ratio/larger simulation box case. The peculiar features in density and inBM seen in the upward-directed jet are not local to the X-line but are extended away from it.


## 4.4. Combination of Density Asymmetry, Shear Flow, and out-of-Plane Magnetic Field

We have learned that combination of density asymmetry and shear flow (in Section4.2), or of density asymmetry and nonzero guide field (in Section 4.3), makes the X-line to slide along the L direction. While there may well be various other ways to create an X-line motion, the two items combined with the density asymmetry considered here are the ones well expected at the magnetopause. In other words, the X-lines driving the magnetopause reconnection are most likely to be in motion. The direction of the X-line motion is determined by (1) the flow direction in the magnetosheath for the shear flow only case and (2) the sign of the out-of-plane magnetic field for the nonzero guide field only case. Then, an interesting question is how an X-line behaves when both the shear flow and guide-field are simultaneously present, and especially when they counteract to each other in determining the direction of the X-line motion.Table1 is the summary of the results described so far that have relevance to the run studied in the next subsection. VXL,sim is the X-line speed in the simulation frame. BMo=±0.5 alone or VMSH=1 alone induces almost the same X-line sliding speed. The direction, however, can be opposite according to the sign of BMo (as mentioned, we think that the frame in which these velocities are measured is suited for this kind of discussion). The counter-acting case would be the most interesting one to study and the results are shown below.Table 1
Summary of the results from the simpler cases.BMo: guide field, VMSH: velocity jump across the magnetopause, VXL,sim: the X-line speed measured in the simulation frame.
BMoVMSHVXL,sim-0.50+0.15+0.50-0.1501+0.18


### 4.4.1. The BMo = 0.5 and VMSH = 1 Case

Figure15 shows the results of the (VMSH,BMo) = (1, 0.5) case. Previous results show that the effects of VMSH=1 would push the X-line in the positive L-direction while those of BMo=0.5 in the negative L-direction. One can see in Panel (a) that the X-line starts to slide from t=10 in the positive L-direction with VL,XL=0.14. This X-line speed is almost the same as the VMSH=1 alone case (Table 1). Panel (b) shows ViL,XL=0.35 and VeL,XL=-0.5. The X-line motion is in the same direction as the ion flow and is opposite to the electron flow.The results from the(VMSH,BMo)=(1,0.5) case, where the guide field and the velocity shear are considered to counteract in determining the direction of the X-line motion (See Table 1). The X-line motion shown in Panel (a) indicates that the shear flow effects dominate. Discussion on the X-line properties shown in Panel (b) given in the text, however, suggests that there are some roles played by the electrons as well.
(a)(b)What is interesting here is the fact thatViL,XL is faster than the VMSH=1 alone case (0.18). The argument in Figure 1 implied that the guide field BMo>0 would make the ion flow to be slower, but this turns out not to be the case. What is more interesting, despite this accelerated ion flow, the X-line speed is slightly slower. This indicates that there should be something else other than the ion flow to determine the speed of the X-line motion. Indeed, the gradient of the guide field component at the X-line drives the fast electron flow in the opposite direction (-0.5) and this may be the slowing down agent. That is, both ion and electron flow at the X-line seem to play roles in determining the direction and speed of the X-line motion.


### 4.4.2. VL,XL from Various (VMSH, BMo) Cases

When the sign ofBMo is flipped ((VMSH,BMo)=(1,-0.5) case), the combined effects are found to speed up the X-line motion to VL,XL=0.37. The flow velocities at the X-line are ViL,XL=0.42 and VeL,XL=1.3, respectively. The flipping of the BM component leads to a substantial change in VL,XL (from 0.14 to 0.37) with only slight change in ViL,XL (from 0.35 to 0.42). In contrast, there is a substantial change in VeL,XL (from -0.5 to 1.3). These again suggest possible roles played by the electrons in regulating the X-line motion. Here, we try to learn more by studying the results from various sets of (VMSH,BMo) in this subsection. Table 2 shows all the cases studied.Table 2
Summary of all the cases studied in this paper.BMo: guide field, VMSH: velocity jump across the magnetopause, VXL,sim: the X-line speed measured in the simulation frame, ViL,XL: the L-component of the ion flow at the X-line, VeL,XL: the L-component of the electron flow at the X-line, and  VXL,MSP: the X-line speed in the magnetosphere-rest frame.
BMoVMSHVXL,simViL,XLVeL,XLVXL,MSPVXL,MSP/VMSH-0.500.150.05110.15N/A-0.50.50.280.261.210.531.06-0.510.370.421.30.870.87010.180.180.40.680.68020.320.310.521.320.660.50-0.15-0.051-1-0.15N/A0.50.2-0.0880.047-0.970.0120.060.50.4-0.0270.1-0.840.170.430.50.50.00470.14-0.780.250.510.50.70.0720.22-0.690.420.60.510.140.35-0.50.640.64Figure16 shows the time history of reconnected flux for the four cases (VMSH,BMo)=(0,-0.5) (black), (1, 0) (blue), (1, 0.5) (greed), and (1, −0.5) (red), respectively. This figure is to confirm that there is no substantial difference in terms of reconnection history due to the shear flow, nonzero guide field, and their combinations within the range considered here.Figure 16
Growth curves of reconnection flux from various sets of (VMSH, BMo). There is little difference within the (VMSH, BMo) range surveyed.Figure17(a) plots VL,XL versus VMSH, with different BMo cases shown by different colors (blue, black, and red for BMo=-0.5, 0, and 0.5, resp.). The general trends are (1) VL,XL is larger for larger VMSH and (2) BMo=-0.5 (blue) cases give the fastest and BMo=0.5 (red) cases give the lowest VL,XL when VMSH is the same. The most interesting feature is that the X-line motion can be opposite to the magnetosheath flow when VMSH is small and BMo=0.5 (red). These clearly indicate that it is not only the ion flow that determines the direction of the X-line motion.The summary plot for the discussion on theX-line motion. Different color indicates different BMo cases (blue, black, and red for BMo=-0.5, 0, and 0.5). (a) VL,XL versus VMSH. While VL,XL is larger for larger VMSH in general, the X-line motion can be opposite to the magnetosheath flow when VMSH is small and BMo=+0.5. (b) Comparison of VL,XL with ViL,XL (solid circles) and with VeL,XL (open squares). The interpretation is that the electron effects set in with the nonzero guide field and can dominate to determine the direction of the X-line motion when the shear flow effects are small.
(a)(b)Figure17(b) compares VL,XL with ViL,XL (solid circles) and with VeL,XL (open squares). The most eye-catching feature is the behavior of the BMo=0.5 (red) cases. These have smaller VL,XL but it is not necessarily because these cases have smaller ViL,XL compared to the others. Furthermore VL,XL becomes negative as VMSH is made smaller, but this happens even when ViL,XL stays positive. Meanwhile VeL,XL is negative as long as BMo=0.5 (red open squares). These suggest that the role played by the electron flow at the X-line, which is enhanced in the presence of nonzero guide field, is not limited to controlling the X-line speed but can be as substantial as to determine its direction when the shear flow effects are weak.Comparing the three cases having mostly the sameX-line speed of VL,XL~0.15 would be illuminating in this regard. The three cases are (VMSH,BMo)=(0,-0.5), (1, 0), and (1, 0.5) (See Table 1). The X-line of the first case (blue symbols) has this speed even though the ion flow at the X-line is pretty small. Instead, the electron flow is fast in the same direction (positive-L). The third case (red symbols) has almost the same X-line speed despite the considerably larger ion flow. It is noted that the electrons at the X-line are flowing fast in the opposite direction (negative-L). In the second case (black symbols) without guide field, there is only small difference between the ion and the electron flow at the X-line. The X-line speed coincides with the ion speed that is made nonzero by the slanted in-flow from the magnetosheath side.


## 4.4.1. The BMo = 0.5 and VMSH = 1 Case

Figure15 shows the results of the (VMSH,BMo) = (1, 0.5) case. Previous results show that the effects of VMSH=1 would push the X-line in the positive L-direction while those of BMo=0.5 in the negative L-direction. One can see in Panel (a) that the X-line starts to slide from t=10 in the positive L-direction with VL,XL=0.14. This X-line speed is almost the same as the VMSH=1 alone case (Table 1). Panel (b) shows ViL,XL=0.35 and VeL,XL=-0.5. The X-line motion is in the same direction as the ion flow and is opposite to the electron flow.The results from the(VMSH,BMo)=(1,0.5) case, where the guide field and the velocity shear are considered to counteract in determining the direction of the X-line motion (See Table 1). The X-line motion shown in Panel (a) indicates that the shear flow effects dominate. Discussion on the X-line properties shown in Panel (b) given in the text, however, suggests that there are some roles played by the electrons as well.
(a)(b)What is interesting here is the fact thatViL,XL is faster than the VMSH=1 alone case (0.18). The argument in Figure 1 implied that the guide field BMo>0 would make the ion flow to be slower, but this turns out not to be the case. What is more interesting, despite this accelerated ion flow, the X-line speed is slightly slower. This indicates that there should be something else other than the ion flow to determine the speed of the X-line motion. Indeed, the gradient of the guide field component at the X-line drives the fast electron flow in the opposite direction (-0.5) and this may be the slowing down agent. That is, both ion and electron flow at the X-line seem to play roles in determining the direction and speed of the X-line motion.


## 4.4.2. VL,XL from Various (VMSH, BMo) Cases

When the sign ofBMo is flipped ((VMSH,BMo)=(1,-0.5) case), the combined effects are found to speed up the X-line motion to VL,XL=0.37. The flow velocities at the X-line are ViL,XL=0.42 and VeL,XL=1.3, respectively. The flipping of the BM component leads to a substantial change in VL,XL (from 0.14 to 0.37) with only slight change in ViL,XL (from 0.35 to 0.42). In contrast, there is a substantial change in VeL,XL (from -0.5 to 1.3). These again suggest possible roles played by the electrons in regulating the X-line motion. Here, we try to learn more by studying the results from various sets of (VMSH,BMo) in this subsection. Table 2 shows all the cases studied.Table 2
Summary of all the cases studied in this paper.BMo: guide field, VMSH: velocity jump across the magnetopause, VXL,sim: the X-line speed measured in the simulation frame, ViL,XL: the L-component of the ion flow at the X-line, VeL,XL: the L-component of the electron flow at the X-line, and  VXL,MSP: the X-line speed in the magnetosphere-rest frame.
BMoVMSHVXL,simViL,XLVeL,XLVXL,MSPVXL,MSP/VMSH-0.500.150.05110.15N/A-0.50.50.280.261.210.531.06-0.510.370.421.30.870.87010.180.180.40.680.68020.320.310.521.320.660.50-0.15-0.051-1-0.15N/A0.50.2-0.0880.047-0.970.0120.060.50.4-0.0270.1-0.840.170.430.50.50.00470.14-0.780.250.510.50.70.0720.22-0.690.420.60.510.140.35-0.50.640.64Figure16 shows the time history of reconnected flux for the four cases (VMSH,BMo)=(0,-0.5) (black), (1, 0) (blue), (1, 0.5) (greed), and (1, −0.5) (red), respectively. This figure is to confirm that there is no substantial difference in terms of reconnection history due to the shear flow, nonzero guide field, and their combinations within the range considered here.Figure 16
Growth curves of reconnection flux from various sets of (VMSH, BMo). There is little difference within the (VMSH, BMo) range surveyed.Figure17(a) plots VL,XL versus VMSH, with different BMo cases shown by different colors (blue, black, and red for BMo=-0.5, 0, and 0.5, resp.). The general trends are (1) VL,XL is larger for larger VMSH and (2) BMo=-0.5 (blue) cases give the fastest and BMo=0.5 (red) cases give the lowest VL,XL when VMSH is the same. The most interesting feature is that the X-line motion can be opposite to the magnetosheath flow when VMSH is small and BMo=0.5 (red). These clearly indicate that it is not only the ion flow that determines the direction of the X-line motion.The summary plot for the discussion on theX-line motion. Different color indicates different BMo cases (blue, black, and red for BMo=-0.5, 0, and 0.5). (a) VL,XL versus VMSH. While VL,XL is larger for larger VMSH in general, the X-line motion can be opposite to the magnetosheath flow when VMSH is small and BMo=+0.5. (b) Comparison of VL,XL with ViL,XL (solid circles) and with VeL,XL (open squares). The interpretation is that the electron effects set in with the nonzero guide field and can dominate to determine the direction of the X-line motion when the shear flow effects are small.
(a)(b)Figure17(b) compares VL,XL with ViL,XL (solid circles) and with VeL,XL (open squares). The most eye-catching feature is the behavior of the BMo=0.5 (red) cases. These have smaller VL,XL but it is not necessarily because these cases have smaller ViL,XL compared to the others. Furthermore VL,XL becomes negative as VMSH is made smaller, but this happens even when ViL,XL stays positive. Meanwhile VeL,XL is negative as long as BMo=0.5 (red open squares). These suggest that the role played by the electron flow at the X-line, which is enhanced in the presence of nonzero guide field, is not limited to controlling the X-line speed but can be as substantial as to determine its direction when the shear flow effects are weak.Comparing the three cases having mostly the sameX-line speed of VL,XL~0.15 would be illuminating in this regard. The three cases are (VMSH,BMo)=(0,-0.5), (1, 0), and (1, 0.5) (See Table 1). The X-line of the first case (blue symbols) has this speed even though the ion flow at the X-line is pretty small. Instead, the electron flow is fast in the same direction (positive-L). The third case (red symbols) has almost the same X-line speed despite the considerably larger ion flow. It is noted that the electrons at the X-line are flowing fast in the opposite direction (negative-L). In the second case (black symbols) without guide field, there is only small difference between the ion and the electron flow at the X-line. The X-line speed coincides with the ion speed that is made nonzero by the slanted in-flow from the magnetosheath side.


## 5. Discussion

### 5.1. Summary

Table2 summarizes all the results shown in this paper. The key issues would be as follows. (1) Magnetopause reconnection situation, namely, density asymmetry combined with shear flow and/or nonzero guide field, makes the X-line to slide along the current sheet. (2) In the presence of a shear flow, the ion flow at the X-line has nonzero L-component because the ion flow stagnation point is shifted towards the magnetospheric side by the density asymmetry and because the ion inflow from the magnetosheath side is slanted. When the shear is substantial, this drives the X-line to move in the same direction as the ion flow. (4) In the presence of nonzero guide field, even if the guide field is uniform initially, nonzero L-component of the current density is generated at the X-line as reconnection develops. The current density is sustained by a fast electron flow at the X-line and, when the shear is weak, this can induce the X-line to move in the same direction as the electron flow. (5) When the shear flow effects dominate, the X-line motion can be opposite to the electron flow at the X-line. (6) When the shear flow effects are weakened, the guide field effects can dominate to move the X-line in the direction opposite to the magnetosheath flow. That is, in the close proximity of the subsolar region, an X-line may move opposite to the magnetosheath flow.


### 5.2. On theX-Line Motion

The results by Oka et al. [36] show the fact that the X-line motion can be in the opposite direction to that of the electron flow. It is not only the local physics at the X-line that determines the X-line motion, but also the mesoscale ion flow pattern surrounding the X-line can be one of the determining factors. On the other hand, there can be cases where the local X-line physics can dominate. Indeed, the case studied by Swisdak et al. [24] is now understood by the case where the electron flow effects can easily dominate. The case studied has nonzero current density along the L-direction already in the initial condition and does not have the shear flow effects.It would be observer friendly if we summarize our results in the magnetosphere-rest frame (the natural coordinates for observers). This is done in Figure18. Dealing the effects of guide field in terms of “error bars”, one can say that the X-line speed in the magnetosphere-rest frame is 0.73VMSH±0.15, where VMSH is the magnetosheath flow speed.Figure 18
VMSH versus VL,XL,MSP (VL,XL in the magnetosphere-rest frame). The X-line speed in the magnetosphere-rest frame is 0.73VMSH±0.15 if the guide field effects are taken as “error bars.”At this stage, it would be interesting to ask us how much interest there is in precisely determining theX-line speed in the context of magnetopause reconnection. The magnetosheath flow seems to have the largest influence as soon as we leave the close proximity of the subsolar region. There VXL,MSP/VMSH~0.7 is what we obtain from our results.


### 5.3. Anomalous High-Density in the Jet

When the density asymmetry and the nonzero guide field are combined, we have found that one of the jets ejected out of theX-line has density higher than the magnetosheath value. The jet showing this anomalous feature is the one flowing in the same direction as the X-line motion. We have also studied the effects of the shear flow on this anomalous density feature, and the results are as follows. (1) It is enhanced when the guide field counteracts against the flow in determining the X-line motion and is seen in the jet flowing in the same direction as the one the guide field would have driven the X-line to move. That is, it is seen in the jet directed opposite the actual X-line motion if the flow effect dominates. (2) It is almost invisible when the guide field joins the flow in accelerating the X-line motion.In an observer friendly manner, the anomalously high density would be easily detected (1) in the southward/stagnant jet in the northern hemisphere under dusk-to-dawn guide field or (2) in the northward/stagnant jet in the southern hemisphere under dawn-to-dusk guide field. While these imply a large chance for the anomalous jet to be detected, there is, to the best of the authors’ knowledge, neither report in the literature, nor clear event in our own survey of the THEMIS data. We think this discrepancy has to do with the low magnetosheath plasma beta we assume in this study. It is 0.22 here but it is >1 in the real situation. Ions of a higher temperature will suffer less from the repelling electric field upon crossing the current sheet, which is the mechanism we propose to explain the anomalous density. While the plasma beta on the magnetosheath side here is rather low (0.22), it is possible that more appropriate observational data may be obtained in the solar wind, where the averaged plasma beta is ~0.25 (e.g., [37]).


### 5.4. Limitations of the Present Study or Future Works

While we think there have been enough to learn from the present results, it is important to point out the limitations of the present study. The initial conditions used are limited such that they are close to the known Vlasov equilibrium. The drawbacks are as follows. (1) The plasma beta in the magnetosheath is rather low. It seems to have led to the prediction of anomalous density in the jets that would not apply to the real magnetopause situation. (2) The guide field (theM-component) is assumed to be uniform initially. To have the guide field to vary across the magnetopause (i.e., ∂BM/∂N is nonzero) is to set current density along the L-axis in the initial condition. This will enhance the electron effects in determining the direction of the X-line motion [24], and a larger area around the subsolar region may show X-lines moving opposite to the magnetosheath flow. (3) The reconnecting magnetic field (the L-component) is symmetric across the current sheet. If the magnetic field intensity of this component is unbalanced, it will lead to differential convection speed along the N-axis toward the X-line. This will lead to a different X-line structure and there may well be a class of new effects arising. Aside from the Vlasov equilibrium issue, (4) the velocity shear is only in the L-component while in reality it is in the M-component as well. The effects of VM shear are left open. (5) 3D effects are out of the scope. As seen in numerous observations, the density gradient at the magnetopause excites Lower-Hybrid Drift Instability (LHDI) (e.g., [38, 39]) and the coupling of this mode with reconnection has become a topic of intensive research [17, 19, 40, 41]. It has already been pointed out that the density gradient even steepens at the X-line [27], which makes us expect that there is a direct coupling between the LHDI mode and the physics of the X-line. (6) Limited parameter range. It would have been more informative to the observers if we could show a cutoff in the maximum flow shear allowed. While Swisdak et al. [24] showed a clear prediction about a maximum allowed flow shear, our simulations did not find clear cutoff of reconnection activity. It might be because parameter range we surveyed was still limited. Extended parametric survey will be studied in a future work.


## 5.1. Summary

Table2 summarizes all the results shown in this paper. The key issues would be as follows. (1) Magnetopause reconnection situation, namely, density asymmetry combined with shear flow and/or nonzero guide field, makes the X-line to slide along the current sheet. (2) In the presence of a shear flow, the ion flow at the X-line has nonzero L-component because the ion flow stagnation point is shifted towards the magnetospheric side by the density asymmetry and because the ion inflow from the magnetosheath side is slanted. When the shear is substantial, this drives the X-line to move in the same direction as the ion flow. (4) In the presence of nonzero guide field, even if the guide field is uniform initially, nonzero L-component of the current density is generated at the X-line as reconnection develops. The current density is sustained by a fast electron flow at the X-line and, when the shear is weak, this can induce the X-line to move in the same direction as the electron flow. (5) When the shear flow effects dominate, the X-line motion can be opposite to the electron flow at the X-line. (6) When the shear flow effects are weakened, the guide field effects can dominate to move the X-line in the direction opposite to the magnetosheath flow. That is, in the close proximity of the subsolar region, an X-line may move opposite to the magnetosheath flow.


## 5.2. On theX-Line Motion

The results by Oka et al. [36] show the fact that the X-line motion can be in the opposite direction to that of the electron flow. It is not only the local physics at the X-line that determines the X-line motion, but also the mesoscale ion flow pattern surrounding the X-line can be one of the determining factors. On the other hand, there can be cases where the local X-line physics can dominate. Indeed, the case studied by Swisdak et al. [24] is now understood by the case where the electron flow effects can easily dominate. The case studied has nonzero current density along the L-direction already in the initial condition and does not have the shear flow effects.It would be observer friendly if we summarize our results in the magnetosphere-rest frame (the natural coordinates for observers). This is done in Figure18. Dealing the effects of guide field in terms of “error bars”, one can say that the X-line speed in the magnetosphere-rest frame is 0.73VMSH±0.15, where VMSH is the magnetosheath flow speed.Figure 18
VMSH versus VL,XL,MSP (VL,XL in the magnetosphere-rest frame). The X-line speed in the magnetosphere-rest frame is 0.73VMSH±0.15 if the guide field effects are taken as “error bars.”At this stage, it would be interesting to ask us how much interest there is in precisely determining theX-line speed in the context of magnetopause reconnection. The magnetosheath flow seems to have the largest influence as soon as we leave the close proximity of the subsolar region. There VXL,MSP/VMSH~0.7 is what we obtain from our results.


## 5.3. Anomalous High-Density in the Jet

When the density asymmetry and the nonzero guide field are combined, we have found that one of the jets ejected out of theX-line has density higher than the magnetosheath value. The jet showing this anomalous feature is the one flowing in the same direction as the X-line motion. We have also studied the effects of the shear flow on this anomalous density feature, and the results are as follows. (1) It is enhanced when the guide field counteracts against the flow in determining the X-line motion and is seen in the jet flowing in the same direction as the one the guide field would have driven the X-line to move. That is, it is seen in the jet directed opposite the actual X-line motion if the flow effect dominates. (2) It is almost invisible when the guide field joins the flow in accelerating the X-line motion.In an observer friendly manner, the anomalously high density would be easily detected (1) in the southward/stagnant jet in the northern hemisphere under dusk-to-dawn guide field or (2) in the northward/stagnant jet in the southern hemisphere under dawn-to-dusk guide field. While these imply a large chance for the anomalous jet to be detected, there is, to the best of the authors’ knowledge, neither report in the literature, nor clear event in our own survey of the THEMIS data. We think this discrepancy has to do with the low magnetosheath plasma beta we assume in this study. It is 0.22 here but it is >1 in the real situation. Ions of a higher temperature will suffer less from the repelling electric field upon crossing the current sheet, which is the mechanism we propose to explain the anomalous density. While the plasma beta on the magnetosheath side here is rather low (0.22), it is possible that more appropriate observational data may be obtained in the solar wind, where the averaged plasma beta is ~0.25 (e.g., [37]).


## 5.4. Limitations of the Present Study or Future Works

While we think there have been enough to learn from the present results, it is important to point out the limitations of the present study. The initial conditions used are limited such that they are close to the known Vlasov equilibrium. The drawbacks are as follows. (1) The plasma beta in the magnetosheath is rather low. It seems to have led to the prediction of anomalous density in the jets that would not apply to the real magnetopause situation. (2) The guide field (theM-component) is assumed to be uniform initially. To have the guide field to vary across the magnetopause (i.e., ∂BM/∂N is nonzero) is to set current density along the L-axis in the initial condition. This will enhance the electron effects in determining the direction of the X-line motion [24], and a larger area around the subsolar region may show X-lines moving opposite to the magnetosheath flow. (3) The reconnecting magnetic field (the L-component) is symmetric across the current sheet. If the magnetic field intensity of this component is unbalanced, it will lead to differential convection speed along the N-axis toward the X-line. This will lead to a different X-line structure and there may well be a class of new effects arising. Aside from the Vlasov equilibrium issue, (4) the velocity shear is only in the L-component while in reality it is in the M-component as well. The effects of VM shear are left open. (5) 3D effects are out of the scope. As seen in numerous observations, the density gradient at the magnetopause excites Lower-Hybrid Drift Instability (LHDI) (e.g., [38, 39]) and the coupling of this mode with reconnection has become a topic of intensive research [17, 19, 40, 41]. It has already been pointed out that the density gradient even steepens at the X-line [27], which makes us expect that there is a direct coupling between the LHDI mode and the physics of the X-line. (6) Limited parameter range. It would have been more informative to the observers if we could show a cutoff in the maximum flow shear allowed. While Swisdak et al. [24] showed a clear prediction about a maximum allowed flow shear, our simulations did not find clear cutoff of reconnection activity. It might be because parameter range we surveyed was still limited. Extended parametric survey will be studied in a future work.



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*Source: 202583-2010-12-09.xml*