Statistics on the Magnetosheath Properties Related to Magnetopause Magnetic Reconnection

In this study, the THEMIS and Cluster satellite observations of the plasma and the magnetic field are analyzed. The THEMIS mission includes five low-latitude satellites, and the Cluster mission consists of four satellites of elliptical polar orbits. Their orbits swept the dayside magnetosphere and the solar wind during the summer and winter half-year, respectively. Data from the fluxgate magnetometers (FGMs; Balogh et al. 2001; Auster et al. 2008) and ion detectors (ESA, McFadden et al. 2008; HIA, Reme et al. 2001) on board Cluster-3 (C3) and Cluster-4 (C4) from 2001 to 2008 and THEMIS-P2 from 2007 to 2010 are applied in this study. The solar wind and IMF conditions just outside the bow shock are determined by the OMNI data with a time resolution of 1 minute obtained from the website of NASA's Space Physics Data Facility: cdaweb.gsfc.nasa.gov. In order to synchronize the data of Cluster, THEMIS, and OMNI, the 4 s Cluster data and the 3 s THEMIS data are averaged within a time period of 1 minute centered at the recording time of OMNI in our statistics. In other words, the time resolution of our data set is 1 minute.

To eliminate the effects of the variations of the incident solar wind direction, a new coordinate system, the solar wind magnetospheric (SWM) coordinate system, is introduced in this study. The vector data for the magnetosphere studies, including the magnetic fields, plasma velocities, and satellite positions, are usually provided in the geocentric solar ecliptic (GSE) coordinate system. The GSM is also often adopted to exclude the effect of the rotation of the dipole axis. The X-axes of both the GSE and GSM point to the Sun. The extending direction of the magnetosphere (tail), however, is related to the solar wind direction, which is not necessarily along the X direction of the GSE and GSM. The SWM is a dynamic coordinate system with the X-axis always pointing against the instantaneous solar wind direction (${\hat{X}}_{\mathrm{SWM}}=-{{\boldsymbol{V}}}_{\mathrm{SW}}/| {{\boldsymbol{V}}}_{\mathrm{SW}}|$). The Y-axis is determined by ${\hat{Y}}_{\mathrm{SWM}}=\hat{D}\times {\hat{X}}_{\mathrm{SWM}}/| \hat{D}\times {\hat{X}}_{\mathrm{SWM}}|$, where $\hat{D}$ is a unit vector along the Earth's dipole axis. Then, Z completes the orthogonal coordinate system by ${\hat{Z}}_{\mathrm{SWM}}={\hat{X}}_{\mathrm{SWM}}\times {\hat{Y}}_{\mathrm{SWM}}$. This coordinate system is basically very close to the GSM coordinate system, and the dipole axis is always located within the Y = 0 plane, but the solar wind is always along the −X direction in the SWM coordinate system. In this way, the biases or asymmetries produced by the variations of the solar wind's incident direction are alleviated. The inclination of the dipole axis in the dawn–dusk direction is always zero, and the possible asymmetries of sheath parameters related to the inclination of the dipole axis in the ±X direction (θ_d) can thus be examined in SWM.

Another coordinate system, named the solar wind interplanetary (SWI) magnetic field coordinate system, is adopted to eliminate the possible asymmetries produced by the dipole tilt angle θ_d, and the asymmetries related to the orientation of IMF are able to be examined. The definition of the SWI coordinate system is the same as that adopted by Dimmock & Nykyri (2013). The X-axis of this system always points against the instantaneous incident solar wind (${\hat{X}}_{\mathrm{SWI}}=-{{\boldsymbol{V}}}_{\mathrm{SW}}/| {{\boldsymbol{V}}}_{\mathrm{SW}}|$); the Z-axis is determined by ${\hat{Z}}_{\mathrm{SWI}}\,={\hat{X}}_{\mathrm{SWI}}\times {{\boldsymbol{B}}}_{\mathrm{SW}}/| {\hat{X}}_{\mathrm{SWI}}\times {{\boldsymbol{B}}}_{\mathrm{SW}}|$ if the IMF cone angle θ_CO > 0 and ${\hat{Z}}_{\mathrm{SWI}}={\hat{X}}_{\mathrm{SWI}}\times (-{{\boldsymbol{B}}}_{\mathrm{SW}})/| {\hat{X}}_{\mathrm{SWI}}\times {{\boldsymbol{B}}}_{\mathrm{SW}}|$ if θ_CO < 0; and the Y-axis completes the right-hand coordinate system through ${\hat{Y}}_{\mathrm{SWI}}\,={\hat{Z}}_{\mathrm{SWI}}\times {\hat{X}}_{\mathrm{SWI}}$. In addition, if θ_CO < 0, the signs of the observed magnetic field are reversed in this study, i.e., ${\boldsymbol{B}}$ to $-{\boldsymbol{B}}$ and ${{\boldsymbol{B}}}_{\mathrm{SW}}$ to $-{{\boldsymbol{B}}}_{\mathrm{SW}}$. In this way, the IMF ${{\boldsymbol{B}}}_{\mathrm{SW}}$ always points in the +X and +Y direction, and the parallel bow shock is thus always located on the +Y section and the perpendicular shock on the −Y section. The geometries of the incident solar wind (yellow arrows) and the embedded IMF flux tube (green rectangle) in the SWI coordinate system are schematically shown in Figure 1. The moving flux tube touches the bow shock first at the −Y flank; this portion of the flux tube is called the leading end, and the other end, at the +Y flank, is called the trailing end in this paper. Under these geometries, the possible sheath asymmetries produced by the orientation of the IMF (θ_CO and θ_CL) can be conveniently investigated. In this study, the magnetosheath property difference between the ±Y flanks in SWI is referred to as the ±Y asymmetry, and the property difference between the regions at the ±Y flanks and the regions at the ±Z flanks is referred to as the Y–Z asymmetry.

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To determine the X-axis of the SWM and SWI, the solar wind velocities obtained from the OMNI data set are not directly used. Solar wind velocities are originally measured in the rest frame of the Sun–Earth system; the data in the OMNI data set, however, are given in the rest frame of the Sun–star inertial system, and the aberration velocity induced by the Earth's revolution around the Sun has been removed (∼29.8 km s⁻¹ in the −Y direction in GSE). The solar wind velocities, which the magnetosphere feels, however, should be in the rest frame of the Sun–Earth system and include both the OMNI solar wind velocities and the revolution velocity. In this study, a 29.8 km s⁻¹ velocity has been added back to the +Y component of the solar wind velocity in the GSE coordinate system.

The distribution of a parameter in the magnetosphere–bow shock space is hard to conduct, since the locations of the bow shock and magnetopause corresponding to each data point in the data set are different. To avoid this, each data point in this study has been relocated into a space with an averaged magnetopause and bow shock. In this normalized space, the radial direction of each data point (the latitude ϕ and longitude φ) and the ratio of the radial distances of each data point to the averaged magnetopause ${D}_{\mathrm{MP}}^{{\prime} }$ and bow shock ${D}_{\mathrm{BS}}^{{\prime} }$ (${R}_{d}={D}_{\mathrm{MP}}^{{\prime} }/{D}_{\mathrm{BS}}^{{\prime} }$) remain the same as those in the original space.

The shape and location of the magnetopause and bow shock are described by the empirical models in this study. There are various magnetopause models, among which the axially symmetric model proposed by Shue et al. (1997) and the asymmetric model established by Lin et al. (2010) are the two popular ones. The latter can describe the asymmetric magnetopause with or without the polar cusp indentations, and the asymmetry is mainly induced by some intrinsic axial asymmetries, the polar indentations, and the dipole tilt. In our data set, 1573 magnetopause crossing events are identified by using the sharp changes in the plasma density, temperature, and velocity and the magnetic field. The distances to the X-axis, R_YZ, of all of the events are plotted as a function of the X-coordinates in Figure 2(a), and the green, red, and blue curves are the cuts at the Y = 0 and Z = 0 planes in SWM of the averaged magnetopause modeled by Shue et al. (1997) and Lin et al. (2010), respectively. The distances of these events to the Earth are also plotted in Figures 2(b)–(d) as a function of the distances modeled by Shue et al. (1997; Figure 2(b)) and Lin et al. (2010) without (Figure 2(c)) and with (Figure 2(d)) the polar cusp indentations. The correlation analyses of these observed and modeled distances reveal that Lin et al.'s model with the polar cusp indentations can provide the best estimation of the magnetopause (Figure 2(d)). Lin et al.'s model with the polar indentations is thus adopted to calculate the radial distance of each data point to the magnetopause, D_MP = r_data−r_MP, where r_data is the distance of a data point to the Earth and r_MP is the distance of the modeled magnetopause to the Earth along the radial direction of the data point.

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The simple hyperbolic model proposed by Fairfield (1971) is used (see Equations (8) and (9)) to calculate the radial distance of a given data point to the bow shock, D_BS = r_data−r_BS, where r_data is the distance of a data point to the Earth and r_BS is the distance of the modeled bow shock to the Earth along the radial direction of the data point. The standoff distance of the bow shock nose d_n determines the size of the bow shock through L = (d_n + x₀)(1 + ε), and the focus location of the hyperbolic surface (x₀, 0, 0) and the eccentricity ε determine the shape of the bow shock. The nose standoff distance d_n is found to be a function of the upstream solar wind quantities, such as the magnetosonic Mach number, β_SW, and the IMF orientation (Farris & Russell 1994), and the modeled nose standoff distances have been provided by the OMNI data set at a time resolution of 1 minute. The eccentricity ε and the focus location of the hyperbolic bow shock surface are generally considered to be constant and determined by seeking the best fit of the observed bow shock locations to the modeled locations. In our data set, we identified 1103 dayside bow shock crossing events by using the sharp changes in plasma density, temperature, and velocity and the magnetic field. The distances to the X-axis, R_YZ, of all of the events are plotted as a function of the X-coordinates in Figure 2(e), and the blue curve is the cut at the Y = 0 or Z = 0 plane in SWM of the averaged bow shock modeled by Fairfield (1971). These bow shock crossing events are limited on the dayside, and the best fit of this data set to the hyperbolic model shown in Figure 2(f) gives ε = 0 and x₀ = −12.4R_E. The distance of a data point to the bow shock D_BS is then calculated by adopting ε = 0 and x₀ = −12.4R_E, and the nose standoff distance d_n is obtained directly from the OMNI data set.

After obtaining the location of a data point in the original space, including the latitude ϕ, longitude φ, and radial distances to the bow shock D_BS and magnetopause D_MP, the data point is then relocated into the normalized space described by an averaged bow shock and magnetopause. The input parameters to determine the averaged models are the averaged ones for all of the data in our database. When conducting statistics in the SWI coordinate system, for example, in Section 3.2, the averaged magnetopause is calculated through Lin et al.'s model without the polar cusp indentations. This is just for displaying the statistical data conveniently, since in SWI, the polar cusp indentations of the magnetopause are randomly rotated around the X-axis. When the statistics in the SWM coordinate system are performed, i.e., in Section 3.3, the averaged magnetopause is calculated through Lin et al.'s model with the polar cusp indentations. In the normalized space, the latitude ϕ, longitude φ, and distance ratio R_d of a data point remain the same as those in the original space, through which the normalized distances of the data point to the averaged bow shock and magnetosphere, ${D}_{\mathrm{BS}}^{{\prime} }$ and ${D}_{\mathrm{MP}}^{{\prime} }$, can be obtained, and the data point is then relocated into the normalized space. Distributing parameters in this space are thus feasible, since all of the data points now have the identical averaged magnetopause and bow shock.

The solar wind parameters, including the IMF orientation (θ_CL and θ_CO), plasma β_SW, sound Mach number M_S, and dipole tilt angle of the Earth θ_d, are determinative of the magnetosheath properties. In the data set of this study, the total number of data points is 2,318,978. The distributions of all solar wind and geomagnetic parameters corresponding to each data point are shown in Figure 3. The vector parameters in this figure are in the SWM or GSM coordinate systems.

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The IMF cone angle θ_CO controls the bow shock type on the magnetopause and thus is a key parameter determinative of the magnetosheath properties. Figure 3(a) shows the probability density of the IMF cone angle θ_CO calculated in SWM. According to its working definition in this study, θ_CO = ±90° indicates that the IMF is antiparallel and parallel to the direction of the incident solar wind, respectively, and θ_CO = 0° means the IMF is perpendicular to the solar wind velocity. It is seen that the IMF in the data set is mainly perpendicular or oblique to the Sun–Earth line ($| {\theta }_{\mathrm{CO}}|$ < 60°) and that the radial IMF is relatively rare ($| {\theta }_{\mathrm{CO}}|$ > 60°). The mean of the absolute clock angle $\langle | {\theta }_{\mathrm{CO}}| \rangle$ is 36°.

The IMF clock angle θ_CL controls the magnetic shear θ_X on the magnetopause and thus is critical to the occurrence of MR. The distribution of θ_CL is shown in Figure 3(b). It is not surprising to see that the probability density peaks roughly at ±90°, and this is the main feature of the Parker spiral IMF. It is worth noticing that, according to this distribution, the Y-axis of the instantaneous SWI coordinate system has a high probability of pointing in the dawn–dusk direction, and thus the Z-axis is nearly in the south–north direction in most cases.

The solar wind plasma β_SW is another important parameter of the initiation and proceeding of MR on the magnetopause, and this parameter describes the relative contributions of the plasma and the magnetic field to plasma behaviors. The probability density of β_SW is shown in Figure 3(c), and it is seen that β_SW ranges mainly from 0 to 10 and peaks roughly at 1. This result suggests that in most cases, the plasma and the magnetic field in the solar wind are of equal significance.

The solar wind sound Mach number M_S determines the bow shock strength and thus is critical to the magnetosheath properties. Figure 3(d) shows the probability density of M_S in the data set. The M_S is mainly greater than 5, and the distribution peaks at ∼12. In this large-M_S situation, it is expected that the plasma compression and deceleration are almost the same for the different types of bow shocks as seen in Equations (6) and (7).

The dipole tilt angle θ_d, which sets the inner boundary conditions for the magnetosheath, is a key parameter to describe the symmetry of the magnetosphere. The black dots in Figure 3(e) show the occurrence probability of θ_d calculated in the GSM coordinate system (the angle between the Z-axis of GSM and the dipole axis), and it is seen that θ_d ranges between ±34° and peaks at roughly ±13°. When calculated in the SWM coordinate system, however, the range of θ_d expands from ±34° to more than ±50° (red dots in Figure 3(e)), although the occurrence probability of θ_d is very low at large angles. This is because the X-axis of the SWM is not fixed to the direction of the Sun but points against the instantaneous solar wind. That is to say, when the solar wind direction varies, the effective dipole tilt angle can be much larger than that expected in GSM, and the solar wind may hit the magnetosphere at a much higher magnetic latitude.

The SWI coordinate system is related to the instantaneous solar wind velocity and the IMF but not the dipole axis of the Earth. The polar cusp indentations of the magnetopause are not located near the XZ plane but randomly distributed around the X-axis. For this reason, Lin et al.'s (2010) magnetopause model without the polar indentations is applied to calculate the averaged magnetopause and display the statistical data. The input parameters for the averaged model are the median quantities of the parameters in the data set, including the solar wind dynamic pressure P_dyn = 1.59 nPa, IMF B_Z = 0.03 nT, IMF magnitude B_T = 4.79 nT, and dipole tilt angle θ_d = 1°. For the averaged bow shock, the median value of the bow shock standoff distance d_n = 13.6R_E is input. The cuts of the modeled bow shock and magnetosphere at the planes with Y = 0 and Z = 0 are plotted by the red and white curves in Figure 4, respectively.

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Figure 4 shows the cuts of the distributions of the plasma density, velocity, and field magnitude at Y = 0 (the XZ plane; Figures 4(a), (c), and (e)) and Z = 0 (the XY plane; Figures 4(b), (d), and (f)). All of the data are normalized by the corresponding solar wind quantities (${{N}_{i}/N}_{\mathrm{SW}}$, $| {\boldsymbol{V}}| /| {{\boldsymbol{V}}}_{\mathrm{SW}}|$, and $| {\boldsymbol{B}}| /| {{\boldsymbol{B}}}_{\mathrm{SW}}|$, where N_i, ${\boldsymbol{V}}$, and ${\boldsymbol{B}}$ are the observed ion density, velocity, and magnetic field and N_SW, ${{\boldsymbol{V}}}_{\mathrm{SW}}$, and ${{\boldsymbol{B}}}_{\mathrm{SW}}$ are the corresponding solar wind density, velocity, and magnetic field). According to the definition of the SWI coordinate system, the IMF is always parallel to the Z = 0 plane and points to the directions of +X and +Y. The average of the absolute cone angle $\langle | {\theta }_{\mathrm{CO}}| \rangle$ is 36°, which is denoted by the arrow in the XY planes in Figures 4(b), (d), and (f). It is seen that the magnetosheath, identified by the compression of both the plasma (Figures 4(a) and (b)) and the magnetic field (Figures 4(e) and (f)) and the plasma deceleration (Figures 4(c) and (d)), is embraced well by the averaged magnetopause (white curves) and bow shock (red curves). There are two main features in these distributions. First, both the deceleration and the compression are weaker at flanks or high latitudes than in the subsolar region, as seen in all of the distributions in Figure 4. This feature is produced by the parabolic shape of the bow shock, and the solar wind velocity component in the direction normal to the local magnetopause is smaller at flanks or high latitudes than in the subsolar region; i.e., the effective M_S is smaller there, and thus the bow shock is weaker. The other feature is that there is a clear ±Y asymmetry present in the Z = 0 distribution of the field magnitude (Figure 4(f)), and the magnetic field behind the bow shock is compressed more significantly at the Y < 0 flank (by a factor of >3.5) than at the Y > 0 flank (by a factor of 1–2). When approaching the magnetopause at both the ±Y flanks, the field magnitude continues to increase. Notice that the IMF geometry related to the local bow shock at the ±Y flanks is different. Here ϕ_BS, which describes the IMF orientation relative to the normal of the local bow shock, is closer to 0° or 180° at the +Y flank and 90° at the −Y flank. The different types of bow shock, the quasi-parallel and quasi-perpendicular bow shocks, make this ±Y asymmetry. As is already known, the SWI Y-axes for most of the data points in the data set are near the dawn–dusk direction; thus, the ±Y asymmetries are in fact the dawn–dusk asymmetries produced by the Parker spiral IMF, as observed by Walsh et al. (2012) and Dimmock & Nykyri (2013). It should be noted that there is no clear ±Y asymmetry seen in the distributions of either plasma compression (Figures 4(a) and (b)) or plasma deceleration (Figures 4(c) and (d)), although the strong plasma compression region extends a little bit farther in the +Y direction than in the −Y direction, as seen in the dark red region in Figure 4(b).

In order to further explore the asymmetries of the magnetosheath plasma and magnetic field related to the IMF orientations, the distributions of sheath parameters in the YZ planes are particularly examined in Figures 5–12. Since the IMF geometry is the key factor to determine the bow shock types and makes the ±Y asymmetries, two subsets of the different IMF cone angles in the data set, i.e., the subsets with $| {\theta }_{\mathrm{CO}}|$ < 30° and  >60°, are particularly compared. The distributions of the plasma density are shown in Figure 5. Figures 5(a) and (b) show the distributions of the normalized ion density (${{N}_{i}/N}_{\mathrm{SW}}$) in the YZ plane for the subset of $| {\theta }_{\mathrm{CO}}|$ < 30°. Figure 5(a) is the YZ projection of the distribution of the data located within a layer in the magnetosheath with a distance to the magnetopause greater than 1R_E but less than 2R_E. The blue and red curves are the intersections of the averaged magnetopause and bow shock with the X = 0 plane, denoting the terminator magnetopause and bow shock, respectively. The ±Y asymmetry can be seen in Figure 5(a), though it is very weak, and the plasma compression just in front of the terminator (the dayside magnetopause) is stronger at the +Y flank (marked in red) than that at the −Y flank (marked in yellow). This result is consistent with the data shown in Figure 4(b), where the strongest plasma compression region biases toward the +Y direction. Figure 5(c) is the same as Figure 5(a) but for the subset with $| {\theta }_{\mathrm{CO}}|$ > 60°. The ±Y asymmetry also appears in Figure 5(c), but this asymmetry is much weaker than the $| {\theta }_{\mathrm{CO}}|$ < 30° case, and it is able to be seen only behind the terminator (the nightside magnetopause): the green and yellow colors at the +Y flank denote a little stronger plasma compression than that at the −Y flank, marked in cyan and blue. The cuts of the distributions of ${{N}_{i}/N}_{\mathrm{SW}}$ at X = 0 are pretty uniform, as seen in Figures 5(b) and (d), indicating that this ±Y asymmetry is limited mainly in the region just outside of the magnetopause but not distant to it. The weak ±Y asymmetries in the plasma density distributions in Figures 5(a) and (c) cannot be produced by the different shock types, otherwise the plasma at the −Y flank could have been compressed more significantly than that at the +Y flank, according to Equations (6) and (7).

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The asymmetries of the ion velocity are investigated in Figures 6 and 7 for the two subsets with $| {\theta }_{\mathrm{CO}}|$ < 30° and$| {\theta }_{\mathrm{CO}}|$ >60°, respectively. The formats of these figures are the same as in Figure 5. The left column shows the YZ projections of the distributions of the magnitude and the components of the normalized ion velocity (${\boldsymbol{V}}/| {{\boldsymbol{V}}}_{\mathrm{SW}}|$) within the layer in the magnetosheath with a distance to the magnetopause greater than 1R_E but less than 2R_E, and the right column shows the cuts of the velocity distributions at X = 0. From top to bottom in Figures 6 and 7, the parameters are $| {V}_{X}| /| {V}_{\mathrm{SW}}|$, $| {\boldsymbol{V}}| /| {V}_{\mathrm{SW}}|$, ${V}_{{YZ}}=\sqrt{{V}_{Y}^{2}+{V}_{Z}^{2}}/| {V}_{\mathrm{SW}}|$, and ${\varphi }_{\mathrm{div}}\,={\tan }^{-1}\sqrt{{V}_{Y}^{2}+{V}_{Z}^{2}}/| {V}_{X}|$. The first two parameters describe the plasma deceleration, and the last two parameters describe the plasma deflection. Here φ_div = 0° indicates that the sheath plasma is not diverted and still flows along the incident solar wind direction; φ_div = 90° means the sheath flow is diverted to the direction perpendicular to the incident solar wind. There are clear Y–Z asymmetries seen in the distributions of both the plasma deceleration and deflection, and the deceleration is stronger and the deflection weaker at the ±Y flanks than those at the ±Z flanks. These Y–Z asymmetries cannot be produced by the different types of shocks, since ϕ_BS at the ±Z flanks must be between the quantities of ϕ_BS at the +Y flank (ϕ_BS ∼ 0°) and −Y flank (ϕ_BS ∼ 90°), and the magnitudes of the deceleration and deflection at the ±Z flanks should also be between those at the ±Y flanks, which is clearly not what is seen in Figure 6. An additional mechanism must be present within the magnetosheath to reaccelerate the plasma at the ±Z flanks (Figures 6(b), (b'), (c), and (c')). The plasma deceleration does not show any clear ±Y asymmetry, but a weak ±Y asymmetry presents in the deflection; V_YZ is smaller (Figures 6(c) and (c')), and, correspondingly, the deflection angle φ_div is smaller (Figures 6(d) and (d')) at the +Y flank than that at the −Y flank. The black arrows in Figures 6(c) and (c') show the direction of ${{\boldsymbol{V}}}_{{YZ}}$; they basically point radially outward, and no clear asymmetry is presented.

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For the large cone angle case in Figure 7, i.e., $| {\theta }_{\mathrm{CO}}|$ > 60°, the overall characteristic is that the deceleration is weaker than the small cone angle case if Figures 7(a), (a'), (b), and (b') are compared with Figures 6(a), (a'), (b), and (b'), respectively, while the flow deflection is stronger if Figures 7(c), (c'), (d), and (d') are compared with Figures 6(c), (c'), (d), and (d'), respectively. Both the Y–Z asymmetries and the ±Y asymmetries are not clear anymore in Figure 7, and only a weak ±Y asymmetry presents in Figure 7(c). The magnitude of the diverted flow at the +Y flank is weaker.

The comparison between Figures 6(b) and 7(b) indicates that the sheath flow is decelerated more significantly for the small IMF cone angle condition. As suggested by Cowley & Owen (1989), Cooling et al. (2001), and Swisdak et al. (2003), whether the sheath flow adjacent to the magnetopause is decelerated to be sub-Alfvénic is critical to the development of MR on the magnetopause. The "Alfvénicity" of the sheath flow within the layer with a distance to the magnetopause greater than 1R_E but less than 2R_E is examined in Figure 8 for the small IMF cone angle subset $| {\theta }_{\mathrm{CO}}|$ < 30° and the large cone angle subset $| {\theta }_{\mathrm{CO}}|$ > 60°. Figure 8(a) shows the YZ projection of the distribution of the occurrence probability of the super-Alfvénic flow for the low IMF cone angle condition, and the blue in the subsolar region indicates that the flow in this region has a very low probability (<20%) of being super-Aflvénic. In this condition, MR may be easily initiated or proceed in a steady state. For the large cone angle condition (Figure 8(b)), the low probability region (blue) shrinks, and most of the observed flows on the magnetopause are super-Aflvénic.

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The compression and bending of the magnetic field in the magnetosheath exhibit both the Y–Z and ±Y asymmetries for both of the two cone angle subsets. Figures 9 and 10 show the distributions of the magnetic field properties. The formats are also the same as in Figure 5. From top to bottom, the parameters are the normalized field magnitude $| {\boldsymbol{B}}| /| {{\boldsymbol{B}}}_{\mathrm{SW}}|$, the normalized YZ component of the magnetic field $| {{\boldsymbol{B}}}_{{YZ}}| /| {{\boldsymbol{B}}}_{\mathrm{SW}}|$, the normalized X component of the magnetic field $| {{\boldsymbol{B}}}_{X}| /| {{\boldsymbol{B}}}_{\mathrm{SW}}|$, and the field line bending angle in the XY plane ${\varphi }_{m,X}={\tan }^{-1}{B}_{X}/{B}_{Y}$. The arrows in the second row show the direction of the bent field in the YZ plane (φ_m,YZ). For the low cone angle case $| {\theta }_{\mathrm{CO}}|$ < 30°, the field compression exhibits clear Y–Z and ±Y asymmetries in Figures 9(a), (a'), (b), and (b'): the strongest field appears outside the dayside magnetopause in the Y < 0 region where, under the IMF geometry in SWI, the downstream of the quasi-perpendicular bow shock is, and the weaker field is found to be located in the Y > 0 region, downstream of the quasi-parallel bow shock. As seen in Figures 9(b) and (b'), in the region near the Y = 0 plane, either at the ±Z flanks or at the subsolar region, the arrows are basically along the +Y direction, which is the direction of the projection of the IMF on the YZ plane; the field lines are, however, divergent at the −Y flank and convergent at the +Y flank. These results show that when encountering the magnetosphere, an IMF flux tube in the solar wind (see the green region in Figures 1(a) and (b)) is locally split and bent by the magnetosphere (see the purple region in Figure 1(b)). The X component of the field $| {{\boldsymbol{B}}}_{X}| /| {{\boldsymbol{B}}}_{\mathrm{SW}}|$ and the angle φ_m,X in Figures 9(c), (c'), (d), and (d') describe the field line bending in the X direction. The magnitude of the X component is stronger at the ±Y flanks and almost zero near the Y = 0 plane. The distributions of the bending angles φ_m,X in Figures 9(d) and (d') exhibit a systematic pattern related to the IMF: at the −Y (+Y) flank the field line bends against (toward) the incident solar wind; in the region near Y = 0, the field lines are basically parallel to the YZ plane. Figure 9 is for the small IMF cone angle case, i.e., $| {\theta }_{\mathrm{CO}}|$ < 30°, and even this small cone angle makes a difference in the bending angles between the ±Y flanks. The magnitude of B_X is stronger, the area of the strong B_X is wider at the −Y flank than at the +Y flank (Figures 9(c) and (c')), and the zero-bent field region (the green region in Figures 9(d) and (d')) is also biased from the Y = 0 plane toward the +Y flank.

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When the IMF cone angle is large (Figure 10), i.e., $| {\theta }_{\mathrm{CO}}|$ > 60°, the compression of the magnetic field is weaker, but the asymmetries in the field compression and field line bending are much more significant than those for the small cone angle case. The maximum $| {\boldsymbol{B}}| /| {{\boldsymbol{B}}}_{\mathrm{SW}}|$ is about 3.5 in Figures 10(a) and (a'), which is not so large as that for the $| {\theta }_{\mathrm{CO}}|$ < 30° case (over 4 in Figures 9(a) and (a')). The strong compression occurs at the −Y flank. The field line bending in the YZ plane is significantly asymmetric, and at the −Y flank, the projections of the fields are basically along the +Y direction, while at the +Y flank, they converge significantly toward the Z = 0 plane (Figures 10(b) and (b')). The field line bending in the XY plane also exhibits a significant ±Y asymmetry, as shown in Figures 10(d) and (d'), and the remarkable characteristics of the distribution of φ_m,X are that the zero-bent field line region (green region) is biased significantly to the +Y flank and that at the Y = 0 plane, the bending angles are +20° to +30°.

The plasma and magnetic pressures, P_i and P_m, are critical to the physics on the magnetopause, and they are investigated in Figures 11 and 12 for the small and large cone angle subsets, respectively. The same formats are used in these two figures as in Figure 5. All pressures are normalized by the corresponding solar wind dynamic pressure, P_dyn. From top to bottom in Figures 11 and 12 are the ion thermal pressure P_i/P_dyn, the magnetic pressure P_m/P_dyn, the proxy for the total pressure P_t = (2P_i + P_m)/P_dyn (the electron pressure is not good enough to conduct the statistics, and 2P_i is used to represent the total plasma pressure, which is the sum of the ion and electron pressures), and the ion β = P_i/P_m. Figure 11(a) shows that the ion pressure P_i is strong in the subsolar region. A weak asymmetry can be seen between the ±Y flanks, and the strong P_i region (green region) shifts toward the +Y flank a little bit. The strong magnetic pressure P_m is mainly located in front of the terminator outside the dayside magnetopause and a significant asymmetry presents: in the subsolar region and at the +Y flank, the magnetic pressure is weaker than that elsewhere on the dayside magnetopause (Figure 11(b)). The total pressure is strong on the dayside magnetopause and almost centrally symmetric (Figures 11(c) and (c')). The ion β distribution is asymmetric and larger in the subsolar region and at the +Y flank (Figures 11(d) and (d')), and this ±Y asymmetry is induced by the asymmetry of the magnetic pressure P_m (Figures 11(b) and (b')).

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The same ±Y asymmetries appear for the large cone angle case in the distributions of P_m and β in Figure 12. In the subsolar region and at the +Y flank, P_m is weaker and β is stronger than those elsewhere on the dayside magnetopause. Compared with the small cone angle case in Figure 11, the differences are remarkable: the area of the strong ion pressure P_i region in Figure 12(a) (green) expands in the subsolar region if compared with that of the $| {\theta }_{\mathrm{CO}}|$ < 30° subset in Figure 11(a); the magnetic pressure is much weaker in Figure 12(b) (cyan and blue) than that in Figure 11(b) (green); the total pressure P_t is a little bit weaker in Figure 12(c), and the strong P_t region shrinks in Figure 12(c) compared with that in Figure 11(c), which is mainly due to the much weaker P_m in the region just before the terminator; the ion β is much larger over all of the magnetopause in Figures 12(d) and (d') (dark red) than that in Figures 11(d) and (d'); and the ±Y asymmetry is still present in Figures 12(d) and (d').

The variations of the plasma density, magnetic field magnitude, and plasma velocity along the X-axis in the subsolar magnetosheath (the distance to the X-axis is less than 6R_E) are investigated in Figures 13, 14 and 15, respectively. The vertical black lines in these figures denote the positions of the averaged bow shock and magnetopause at X = 13.6R_E and 11.1R_E, respectively. In order to explore the dependences of the variations on the outside solar wind and IMF conditions, the data in these analyses are divided into different subsets by the IMF cone angles as shown in panel (a) (subsets with $| {\theta }_{\mathrm{CO}}| \lt 30^\circ$, $30^\circ \lt | {\theta }_{\mathrm{CO}}| \lt 60^\circ$, and $| {\theta }_{\mathrm{CO}}| \gt 60^\circ$), the IMF clock angles as shown in panel (b) (subsets with $| {\theta }_{\mathrm{CL}}| \lt 45^\circ$, $45^\circ \lt | {\theta }_{\mathrm{CL}}| \lt 135^\circ$, and $| {\theta }_{\mathrm{CL}}| \gt 135^\circ$), the solar wind plasma β_SW as shown in panel (c) (subsets with β_SW < 1, 1 < β_SW <2, and β_SW > 2), and the solar wind convection electric field ${{\boldsymbol{E}}}_{\mathrm{SW}}$ shown in panel (d) (subsets with $\left|{{\boldsymbol{E}}}_{\mathrm{SW}}\right|\lt 1$, $1\lt \left|{{\boldsymbol{E}}}_{\mathrm{SW}}\right|\,\lt 2$, and $\left|{{\boldsymbol{E}}}_{\mathrm{SW}}\right|\gt 2$ mV m⁻¹). The solar wind sound Mach number M_S is not used to classify the data set because M_S values in the data set are basically greater than 5, as shown in Figure 3(d), and in this situation, the deceleration and compression of plasma through the bow shock in the subsolar region are almost independent of M_S according to Equations (6) and (7). This paper tries to touch on the questions of how fast and where magnetic flux is reconnected on the magnetopause. The solar wind convection electric field ${{\boldsymbol{E}}}_{\mathrm{SW}}$ is involved, since it describes how fast the IMF flux is transported toward the bow shock and the magnetospheric system. When the IMF cone angle θ_CO is large, that is, ${{\boldsymbol{B}}}_{\mathrm{SW}}$ is almost along or against the direction of ${{\boldsymbol{V}}}_{\mathrm{SW}}$, the uncertainty of the calculation of the solar wind convection electric field ${{\boldsymbol{E}}}_{\mathrm{SW}}=-{{\boldsymbol{V}}}_{\mathrm{SW}}\times {{\boldsymbol{B}}}_{\mathrm{SW}}$ increases. For this reason, only the data with $\left|{\theta }_{\mathrm{CO}}\right|\lt 60^\circ$ are involved in Figures 13(d), 14(d), and 15(d). The colored curves in these figures show the spline regression of the studied parameters on the X-coordinate for all subsets, and these regressions are performed by using the public R statistical software (R Core Team 2019).

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The overall profiles of the normalized ion density ${{N}_{i}/N}_{\mathrm{SW}}$ along the X-axis are almost identical for all but the subsets with different ${{\boldsymbol{E}}}_{\mathrm{SW}}$ in Figure 13. The ion density increases at the bow shock at X = 13.6R_E, which is due to the strong compression of the bow shock. Here ${{N}_{i}/N}_{\mathrm{SW}}$ continues to increase behind the bow shock and peaks roughly at X = 12R_E, and this increase results from the pileup of the shocked plasma against the magnetopause. Behind X = 12R_E but before the averaged magnetopause, the ion density drops suddenly, and this sudden decrease possibly stands for the depletion layer just ahead of the magnetopause, or it may just derive from the error of determining the location of the bow shock, since the error of the modeled bow shock location is about 1R_E (Figure 2(f)), that is to say, some of the data behind X = 12R_E and in front of the average magnetopause are actually located in the magnetosphere in the original space. These two possibilities cannot be distinguished in the present analysis. None of the outside conditions but the solar wind convection electric field ${{\boldsymbol{E}}}_{\mathrm{SW}}$ make a clear difference among their corresponding subsets. It is clear in Figure 13(d) that the plasma pileup effect is less significant for the large ${{\boldsymbol{E}}}_{\mathrm{SW}}$ subset, and the profile of ${{N}_{i}/N}_{\mathrm{SW}}$ for the large ${{\boldsymbol{E}}}_{\mathrm{SW}}$ becomes flat at the location very close to the bow shock (at X = 13R_E), as shown by the red curve in Figure 13(d). The pileup effect is most significant for the smallest ${{\boldsymbol{E}}}_{\mathrm{SW}}$ subset, and the density profile keeps increasing until deep into the magnetosheath and peaks roughly behind X = 12R_E. The profile with the median quantity of ${{\boldsymbol{E}}}_{\mathrm{SW}}$ (black curve) is located between the profiles with the large and small ${{\boldsymbol{E}}}_{\mathrm{SW}}$ in Figure 13(d).

The field magnitude jumps at the bow shock at X = 13.6R_E, continues to increase gradually when heading toward the magnetopause, and jumps abruptly again at roughly X = 12R_E for all subsets shown in Figure 14. As mentioned previously, the second abrupt increase in the field magnitude ∼1R_E ahead of the magnetopause may be attributed to the depletion layer or the error of estimating the location of the magnetopause. The IMF cone angle θ_CO makes the remarkable difference among the profiles of the field magnitude of the subsets shown in Figure 14(a): for the small cone angle case of $\left|{\theta }_{\mathrm{CO}}\right|\lt 30^\circ$, the quasi-perpendicular bow shock in the subsolar region compresses the magnetic field most efficiently by a factor of ∼4, and the whole profile of the field magnitude (black curve) is above those of all other subsets; for the large cone angle case of $\left|{\theta }_{\mathrm{CO}}\right|\gt 60^\circ$, the quasi-parallel shock in the subsolar region almost does not compress the magnetic field, and the whole profile is lowest (blue curve); for the middle cone angle case of $30^\circ \lt \left|{\theta }_{\mathrm{CO}}\right|\lt 60^\circ$, the field magnitude profile is in the middle (red curve). The solar wind plasma β_SW (Figure 14(c)) and convection electric field ${{\boldsymbol{E}}}_{\mathrm{SW}}$ (Figure 14(d)) also make differences among their subsets. When heading toward the magnetopause, the magnetic field pileup effect is least significant for the smallest β_SW or strongest ${{\boldsymbol{E}}}_{\mathrm{SW}}$: the field magnitude increases slower than the other subsets, as shown by the blue curve in Figure 14(c) and the red curve in Figure 14(d). These differences among the subsets do not begin right at the bow shock but appear in the magnetosheath on the way to the magnetopause, suggesting that the magnetosheath internal process or the process on the inner boundary, i.e., the magnetopause, may produce these differences.

The difference among the subsets made by the solar wind β_SW and the convection electric field ${{\boldsymbol{E}}}_{\mathrm{SW}}$ are also seen in the velocity profiles $| {V}_{X}| /| {V}_{\mathrm{SW}}|$ in Figure 15. Although the velocity profiles for all subsets in Figure 15 decrease all the way from the bow shock to the magnetopause, the velocity is faster at any location in the magnetosheath for the smaller β_SW or stronger ${{\boldsymbol{E}}}_{\mathrm{SW}}$ subsets: the velocity profiles for the smaller β_SW (β_SW < 1, as shown by the blue curve in Figure 15(c)) and the stronger ${{\boldsymbol{E}}}_{\mathrm{SW}}$ ($| {{\boldsymbol{E}}}_{\mathrm{SW}}| \gt 2$ mV m⁻¹, as shown by the red curve in Figure 15(d)) are above those of the other subsets.

It is traditionally thought that MR tends to occur or proceed faster when the IMF clock angle θ_CO is large, and it is thus expected to see that the pileup effect may also be less significant under a larger IMF clock angle condition (Phan et al. 1994). The data shown in Figures 13(b), 14(b), and 15(b) demonstrate, however, that there is no systematic difference among the profiles of the subsets of different IMF clock angles.

By taking advantage of the SWI coordinate system, a method is proposed to roughly estimate how much IMF magnetic flux goes through the magnetosphere via MR on the magnetopause. When a flux tube embedded in the solar wind encounters the magnetopause, the IMF field lines within the flux tube have two ways to return back to the solar wind. One way is that the IMF field lines reconnect with the magnetospheric field lines through MR on either the low- or high-latitude magnetopause, and this way can be termed "going through the magnetosphere"; the other way is that the IMF field lines are bent in front of the magnetopause and diverted into the flank magnetosheath, and in this way, the IMF field lines directly come back to the downstream solar wind. By calculating the total magnetic flux in the solar wind rushing toward the bow shock per unit time (F_SW) and the magnetic flux diverted into the flank magnetosheath per unit time (F_SH), the magnetic flux going through the magnetosphere via MR on the magnetopause can be estimated by F_R = F_SW − F_SH. The SWI coordinate system is convenient for performing this estimation. In the SWI coordinate system, the flux tube in the solar wind is aligned in the X and Y directions. Let us assume that the width of the flux tube in the solar wind is equal to the dimension of the terminator bow shock in the Z direction, L_SW (as shown in Figure 1(b)). This width can be estimated by the bow shock model for each data point in our data set. Assuming that the solar wind velocity and the IMF are uniform within the scale of L_SW, the total flux in the solar wind F_SW can be estimated directly via F_SW = V_SW,XB_SW,YL_SW. The width of the magnetosheath at the ±Z flanks in the terminator plane can be estimated via L_SH = L_SW − L_MS, where L_MS is the width of the terminator magnetopause in the ±Z directions. If the convection electric field in the sheath E_X = V_XB_Y is uniformly distributed along the ±Z directions at the terminator, the total flux flowing from the flank magnetosheath into the downstream solar wind per unit time can be estimated via F_SH = E_XL_SH. After that, the total magnetic flux going through the magnetosphere per unit time via MR can be calculated by F_R = F_SW − F_SH. It should be noted, however, that the dimension parameters involved in this calculation, i.e., the widths of the magnetopause and the bow shock along the Z-axis at the terminator plane, are modeled, and thus the errors may enter into the calculation.

To perform the aforementioned estimation of F_R, the key constraint is that the convection electric field in the sheath, E_X = V_XB_Y, should be constant along the ±Z directions, and only in this condition can the total flux be estimated simply via F_SH = E_XL_SH, instead of integrating E_X along the Z direction in the sheath. Figure 16 shows the cuts of the distributions of the electric field at the terminator plane X = 0. The data set is divided by the IMF cone angle θ_CO into two subsets with $| {\theta }_{\mathrm{CO}}| \lt 60^\circ$ and $\left|{\theta }_{\mathrm{CO}}\right|\gt 60^\circ$, and the two subsets are shown in the left and right columns, respectively. From top to bottom of each column are the electric field magnitude normalized by the solar wind convection electric field, $| {\boldsymbol{E}}| /| {{\boldsymbol{E}}}_{\mathrm{SW}}|$; the electric field component describing the flux transport in the X direction, ${E}_{X}/| {{\boldsymbol{E}}}_{\mathrm{SW}}| ={V}_{X}{B}_{Y}/| {{\boldsymbol{E}}}_{\mathrm{SW}}|$; and the electric field component describing the flux transport outward in the YZ plane, ${E}_{{YZ}}/| {{\boldsymbol{E}}}_{\mathrm{SW}}|$. Data in Figure 16 indicate that the transport of the magnetic flux is mainly in the X direction for both the large and the small cone angle subsets. Here $| {\boldsymbol{E}}| /| {{\boldsymbol{E}}}_{\mathrm{SW}}|$ and ${E}_{X}/| {{\boldsymbol{E}}}_{\mathrm{SW}}|$ exhibit clear Y–Z asymmetries, and they are stronger at the ±Z flanks than those at the ±Y flanks. Also presenting weak ±Y asymmetries are ${E}_{X}/| {{\boldsymbol{E}}}_{\mathrm{SW}}|$ and ${E}_{{YZ}}/| {{\boldsymbol{E}}}_{\mathrm{SW}}|$ , and they are a little bit stronger at the −Y flank. These asymmetries mainly derive from the asymmetries of the distributions of the magnetic field as shown in Figures 9(b') and 10(b'). The one important characteristic is that ${E}_{X}/| {{\boldsymbol{E}}}_{\mathrm{SW}}|$ is pretty constant along the ±Z directions at the ±Z flanks near Y = 0, at least for the low cone angle case as shown in Figure 16(b). For the large cone angle case shown in Figure 16(b'), however, the small angle between the incident solar wind and the IMF introduces an uncertainty in the calculation of ${E}_{X}/| {{\boldsymbol{E}}}_{\mathrm{SW}}|$, and ${E}_{X}/| {{\boldsymbol{E}}}_{\mathrm{SW}}|$ is not that constant along the ±Z directions at the ±Z flanks near Y = 0. The uniform ${E}_{X}/| {{\boldsymbol{E}}}_{\mathrm{SW}}|$ along the Z-axis allows us to calculate F_SH, and thus F_R.

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Figure 17 shows the distributions of the calculated F_R, the magnetic flux going through the magnetosphere per unit time, as functions of the outside solar wind and IMF conditions. The data used to calculate F_R are observed in the magnetosheath at the ±Z flanks with $| X|$ < 3R_E and $| Y|$ < 5R_E, and the distances to the magnetopause and bow shock are greater than 0.5R_E. In each plot of Figure 17, the red curves are the splines regression of F_R on the IMF cone angle θ_CO (Figure 17(a)), the absolute IMF clock angle $| {\theta }_{\mathrm{CL}}|$ (Figure 17(b)), the IMF strength $| {{\boldsymbol{B}}}_{\mathrm{SW}}|$ (Figure 17(c)), the solar wind plasma β_SW (Figure 17(d)), the solar wind electric field ${{\boldsymbol{E}}}_{\mathrm{SW}}$ (Figure 17(e)), and the solar wind sound Mach number M_S (Figure 17(f)). The regressions here are also performed by using the public R software (R Core Team 2019). Despite the significant scattering of the data, it is clearly seen that more magnetic flux goes through the magnetosphere when $| {\theta }_{\mathrm{CO}}|$ is smaller, $| {{\boldsymbol{B}}}_{\mathrm{SW}}|$ is stronger, β_SW is smaller, ${{\boldsymbol{E}}}_{\mathrm{SW}}$ is stronger, or M_S is smaller, as shown in Figures 17(a), (c), (d), (e), and (f). The control of the IMF clock angle, θ_CL, to F_R is seen in Figure 17(b): when $| {\theta }_{\mathrm{CL}}|$ > 90°, the larger the $| {\theta }_{\mathrm{CL}}|$, the more magnetic flux F_R is reconnected (at the top of Figure 17(b)). When $| {\theta }_{\mathrm{CL}}|$ < 90°, however, F_R tends to decease when $| {\theta }_{\mathrm{CL}}|$ increases toward 90° (at the bottom of Figure 17(b)). In fact, the high-latitude lobe reconnection may occur at high latitudes behind the polar cusps when θ_CL is small and the IMF flux may go through the magnetosphere via the dual high-latitude lobe MR (Song & Russell 1992).

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There are less data points distributed near θ_CL = 90° in Figure 17(b). The observing satellites, either Cluster or THEMIS, are typically located in the dawn or dusk flanks in the GSM (or SWM) coordinate system when they fly through the terminator. In this condition, when $| {\theta }_{\mathrm{CL}}|$ ∼ 90°, these satellites have little chance to be located at the ±Z flanks in the SWI coordinate system, since this region is actually the high-latitude terminator region in GSM (or SWM) where the observing satellite cannot enter.

The SWM coordinate system, similar to the GSM coordinate system, is related to the magnetic dipole axis of the Earth. The dipole axis always lies in the Y = 0 plane, it may be parallel or oblique to the Z-axis of SWM, and the angle between them is the dipole tilt angle θ_d. The magnetosheath asymmetries present in SWI should be smoothed out in the SWM coordinate system, since the IMF orientations are now randomly distributed. The geometry of the magnetic dipole thus can be a source to induce the magnetosheath asymmetries in SWM.

Figure 18 shows the YZ projections of the distributions of the pressures within a layer outside the magnetopause with distances to the magnetopause greater than 1R_E and less than 2R_E (left column) and another layer inside the magnetopause with distances to the magnetopause less than 1R_E (right column). All pressure quantities in Figure 18 are normalized by the corresponding solar wind dynamic pressures, P_dyn. The ion pressure outside the magnetopause, P_i/P_dyn, is shown in Figure 18(a). A strong P_i/P_dyn region is located at low latitudes (cyan and green), and this region extends wider along the dawn–dusk direction than the south–north direction. The maximum P_i/P_dyn occurs in the subsolar region, and its quantity reaches 0.45. Theoretically, the plasma thermal pressure P_th, i.e., the sum of the ion and electron pressures (P_i + P_e), should be ∼89% of the incident solar wind pressure (P_th/P_dym = ∼0.89; Margaret G. Kivelson 1995). The maximum quantity of 0.45 is about half the theoretical value, which is reasonable since only the ion pressure is involved. The magnetic pressure P_m/P_dyn just outside the magnetopause is below 0.2 (Figure 18(b)), and it is much weaker than the ion pressure. The strongest magnetic pressure is not present in the subsolar region but at the higher latitudes. This result is consistent with what is seen in SWI in Figures 11(b) and 12(b), and there the statistics show that the strongest magnetic pressure occurs at the −Y flank and the bias may be attributed to the nonzero IMF cone angle. The total pressure is hard to obtain in this study, since the electron pressure is not good enough, and P_t = (2P_i + P_m)/P_dyn is plotted in Figure 18(c) as a proxy for the total pressure. It is seen that the distribution pattern of P_t is similar to that of P_i in Figure 18(a). The ion β outside the magnetopause is plotted in Figure 18(d), and it shows that the maximum β is as large as 10 and the high β is limited within a belt at the low latitudes.

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Figures 18(a') and (b') show the magnetic and ion pressures inside the magnetopause, respectively. The strong magnetic pressure region is located at low latitudes and extends more in the dawn–dusk direction than in the south–north direction (Figure 18(a')), the pattern of which is similar to those of the ion and the total pressures outside the magnetopause. The maximum magnetic pressure occurs in the subsolar region, and the quantity reaches ∼0.9. The ion pressure inside the magnetopause is mostly below 0.15, as shown in Figure 18(b'), leading to the very low ion β just inside the magnetopause, as shown in Figure 18(d'). The total pressure inside the magnetopause is dominated by the magnetic pressure, and its magnitude is comparable with that of the total pressure outside the magnetopause, as shown in Figures 18(c) and (c').

The flow pattern in the magnetosheath is investigated in Figure 19. All of the data involved are in the layer outside the magnetopause. Figure 19(a) displays the distribution of the occurrence probability of the positive V_Z (hereafter referred to as the "V_Z polarity"). A V_Z polarity equal to 1 means that the flows at this location are all northward, and a zero V_Z polarity means that the flows there are all southward. Data in Figure 19(a) indicate that the V_Z polarity is almost 1 in the northern hemisphere (red) and zero in the southern hemisphere (blue). The transition region marked by cyan and green (the V_Z polarity is ∼0.5) between the red and blue regions is located at the equator and very narrow. This narrow transition region suggests that the location of the line separating the northward and southward plasma flows is fixed near the equator and is not controlled by the dipole tilt angles, otherwise the randomly distributed dipole tilt angle θ_d may make a wider transition region between the red and blue regions.

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The "Alfvénicity" of the sheath flow is examined in the SWM coordinate system, and the probability of the super-Alfvénic flow outside the magnetopause is shown in Figure 19(b). The data are the same as those in Figure 8. The lowest occurrence probability of the super-Alfvénic flow (<20%) appears around the subsolar region; at high latitudes near the noon meridian plane, the occurrence probability increases to above 50%; and at the dawn and dusk flanks ($| Y|$> ±7R_E), the sheath is almost always super-Alfvénic, and the occurrence probability is almost 100%.

The dependences on the dipole tilt angle of the patterns of the ion and magnetic pressure distributions are examined in Figure 20. The data involved in this figure are located within the two layers just outside or inside the magnetopause, and they are the same as in Figure 18 but are confined around the noon meridian with $| Y|$ < 10R_E. Here P_i/P_dyn, β, and the V_Z polarity outside the magnetopause and P_m/P_dyn both inside and outside the magnetopause are plotted as functions of the latitudes in SWM and the dipole tilt angle θ_d in Figure 20. A data gap is present in the southern hemisphere (negative latitudes) for all plasma data outside the magnetopause, and this is due to lack of plasma data on Cluster-4. A ridge clearly appears in Figure 20(a) for P_i/P_dyn outside the magnetopause (dark red), and when the dipole axis rotates toward the Sun (θ_d increases from −34° to 34°), the maximum pressure region on the magnetopause shifts southward from the northern hemisphere to the southern hemisphere (the latitudes of the maximum pressure region decrease). In the magnetic pressure plot in Figure 20(b), a valley is present near the magnetic equator (blue), and when the dipole axis rotates toward the Sun, the magnetic pressure valley outside the magnetopause also shifts southward along with the ion pressure ridge. Both the ion β outside the magnetopause and the magnetic pressure inside the magnetopause exhibit a ridge near the magnetic equator, and the same controls of the dipole tilt angle θ_d to these two quantities are seen clearly in Figures 20(c) and (d). The V_Z polarity outside the magnetopause shown in Figure 20(e) confirms the observations in Figure 19(a), and the line separating the northward and southward flows is always located at the equator plane of SWM, no matter what the dipole tilt angle is.

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The magnetosheath properties of both the plasma and the magnetic field are investigated in the SWI coordinate system. These parameters include ${\boldsymbol{B}}$, ρ, _Pi, β, V_A, and ${\boldsymbol{V}}$. There are two kinds of asymmetries existing in the distributions of these studied parameters, the ±Y asymmetries and the Y–Z asymmetries, and they are produced by different mechanisms.

The ±Y asymmetries are mainly present in the magnetic field quantities. The field magnitude $| {\boldsymbol{B}}|$ exhibits a clear ±Y asymmetry in Figure 4(f). This asymmetry is produced by the different types of bow shock at the opposite flanks. Due to the parabolic shape of the bow shock and the geometry of the IMF in the SWI coordinate system ($\langle | {\theta }_{\mathrm{CO}}| \rangle$ = 36°), ϕ_BS, which describes the IMF orientation relative to the normal direction of the local bow shock, is different at the ±Y flanks. At the +Y flank in SWI, ϕ_BS is closer to 0° or 180°; thus, the bow shock there is the quasi-parallel shock, which cannot compress the magnetic field significantly (see Equation (6)). At the −Y flank, ϕ_BS is closed to 90°, and the quasi-perpendicular bow can compress the magnetic field by a factor up to 4 (see Equation (7)). This asymmetry in the magnetic field compression creates the asymmetries in all of the magnetic field–related parameters, such as the field components $| {{\boldsymbol{B}}}_{X}|$ and $| {{\boldsymbol{B}}}_{{YZ}}|$, field bending angle φ_m,X, magnetic pressure P_m, and plasma β. Clearly, these asymmetries are all controlled by the IMF cone angle θ_CO if comparing Figures 9 and 11 with Figures 10 and 12, respectively.

The ±Y asymmetries for the plasma parameters, however, are very weak, and this is understandable because the solar wind Mach number M_S is large. Figure 3(d) shows that most of the M_S values in the current data set are greater than 5, and the occurrence probability density peaks at M_S ∼ 12. According to Equations (6) and (7), the compression and deceleration of plasma do not depend on either the type of the bow shock (ϕ_BS) or the Mach number (M_S) strongly if M_S is big enough. That is why the different types of bow shock have not decelerated and compressed the plasma differently at the ±Y flanks. In fact, according to the shock theory, the quasi-parallel bow shock at the +Y flank should make less decelerated (faster) and more tenuous plasma. This faster plasma has been seen at the +Y flank just inside the terminator shock (the red region inside the terminator shock at the +Y flank in Figures 7(a') and (b')). In this terminator region, the component of the incident solar wind velocity in the direction normal to the local magnetopause is much smaller than that in the subsolar region; thus, the effective Mach number (M_S) is small, and in this condition, the quantity of M_S acts in Equation (6). Except for this terminator region, the slightly denser and slower plasma is seen in the magnetosheath at the +Y flank (Figures 4(b), 5(a) and (c), 6(c) and (c'), 7(c)). The magnitude of the flow diversion is also less significant at the +Y flank than at the −Y flank (Figures 6(d) and (d')). These weak ±Y asymmetries can be attributed to the continuous deceleration, compression, and diversion processes within the magnetosheath when the shocked solar wind and IMF continue to head toward the magnetopause. The more bent magnetic field lines within the +Y flank sheath may play a role in further blocking the plasma there (Figures 9(d) and 10(d)). The strongly compressed magnetic field within the −Y flank sheath may squeeze the plasma out and reduce the plasma density there. In this case, i.e., the IMF cone angle θ_CO is not zero, the most significant depletion effect (Phan et al. 1994) may not be located in the subsolar region but biases toward the −Y flank in the SWI coordinate system (Figure 5(a)).

The magnetosheath internal processes can also create the ±Y asymmetries of the magnetic properties. Here φ_m,YZ, which describes the bending of the magnetosheath field line in the YZ plane and is determinative of the magnetic shear θ_X on the magnetopause, exhibits a clear ±Y asymmetry that is observed to be controlled by the IMF θ_CO but not related to the different types of bow shocks. For the zero $| {\theta }_{\mathrm{CO}}|$ case, it is expected to see that the field line bending in the Y–Z plane is symmetric at the ±Y flanks of the magnetopause. When an IMF flux tube with $\left|{\theta }_{\mathrm{CO}}\right|=0^\circ$ moves toward the magnetopause, it is initially split at the subsolar region, and the field lines diverge at the −Y flank and converge at the +Y flank. For the small $| {\theta }_{\mathrm{CO}}|$ case, the initial split point shifts a little bit toward the −Y flank, and it can be seen that at the +Y flank, the convergence of the sheath magnetic field lines is more significant than the divergence at the −Y flank (arrows in Figures 9(b) and (b')). For the large $| {\theta }_{\mathrm{CO}}|$ case in Figures 10(b) and (b'), this difference in the field line bending between the ±Y flanks is much more remarkable. When an IMF flux tube inclining toward the Sun (nonzero cone angle ${\theta }_{\mathrm{CO}}\ne 0$) encounters the bow shock (green rectangles in Figure 1), the leading end of the flux tube touches the magnetopause first at the −Y flank, and this portion of the flux tube is split into two flux tubes (purple region in Figure 1(b)). In this region, the field lines basically lie on the magnetopause and are bent a little in the Y–Z plane. That is why the field line bending in the YZ plane is always slight at the −Y flank regardless of whether the cone angle θ_CO is large or small. As the flux tube continues to move, more and more portions of the flux tube touch the magnetopause. The newly split point of the flux tube moves further toward the +Y flank and the trailing end of the flux tube, where the magnetic field lines of the flux tube are close to the normal direction of the local magnetopause, and thus are bent significantly by the magnetopause. That is why the field line convergence at the +Y flank is more remarkable than the divergence at the −Y flank, particularly when the IMF cone angle $| {\theta }_{\mathrm{CO}}|$ is large.

The magnetosheath internal processes also make the Y–Z asymmetries of the magnetosheath. All components of the plasma velocity, i.e., $| {V}_{X}|$, $| {V}_{{YZ}}|$, and $| {\boldsymbol{V}}|$, exhibit clear Y–Z asymmetries (e.g., Figure 6), and the plasma is faster at the ±Z flanks than at the ±Y flanks, indicating an additional accelerating mechanism present in the magnetosheath at the ±Z flanks, and the tension force exerted by the bent sheath magnetic field may have acted to reaccelerate the plasma there (Figures 9 and 10). The shocked plasma is continuously diverted in the magnetosheath, and it convects the magnetic flux toward the ±Z flanks; that is why the field magnitude is much stronger at the ±Z flanks than at the ±Y flanks (Figure 9). The faster plasma and stronger magnetic fields leads to the stronger electric field within the magnetosheath at the ±Z flanks (Figure 16).

It should be mentioned that the asymmetries presented here are discussed in the SWI coordinate system and related to the IMF flux tube directions. In this coordinate system, the level of the asymmetries is controlled only by the IMF cone angle θ_CO. If discussed in the SWM coordinate system, these asymmetries should still be related to the direction of the IMF flux tube, but they are determined by both the IMF clock angle θ_CL and the cone angle θ_CO.

Thus, one must be very cautious to use the single solar wind and IMF conditions as a substitute for the conditions in the magnetosheath adjacent to the magnetopause, particularly when trying to pursue the answers to questions like where MR occurs on the magnetopause or how fast MR goes. For example, the magnetic field bending can be very different over the magnetopause, and thus the magnetic shear θ_X varies significantly, particularly for an IMF with a large $| {\theta }_{\mathrm{CO}}| ;$ a strong asymmetry is also present in the distribution of the plasma β. Notice that both θ_X and β are critical to the initiation or proceeding of MR on the magnetopause (Sonnerup 1974; Swisdak et al. 2003; Trattner et al. 2007; Phan et al. 2013). The strong plasma β outside the subsolar magnetopause may suggest that it is not likely to contain fast MR.

A by-product that is the variation of the magnetopause flaring angle under the different IMF cone angle θ_CO is obtained by comparing the distributions for the small and large $| {\theta }_{\mathrm{CO}}|$. When the IMF cone angle increases from $\left|{\theta }_{\mathrm{CO}}\right|$ < 30° to >60°, it is seen that the total pressure P_t decreases and the strong P_t region shrinks in the subsolar region (Figures 11(c) and 12(c)). These variations of P_t are mainly because of the decrease in P_m (Figures 11(b) and 12(b)), and, in fact, the ion thermal pressure _Pi remains unchanged, and the strong plasma compression region (N_i and P_i) even expands (compare Figures 5(a) and 11(a) with Figures 5(c) and 12(a), respectively). The P_m decreases because the quasi-parallel bow shock dominates over a wider region around the subsolar region when the IMF changes to be radial ($| {\theta }_{\mathrm{CO}}|$ > 60°) and, as is well known, the quasi-parallel bow shock does not compress the magnetic field significantly. The decrease in P_m thus results in a larger β (Figures 11(d) and 12(d)). The plasma is observed to be decelerated less significantly ($| V|$ and $| {V}_{x}|$; Figures 6(a) and (b), 7(a) and (b)) but diverted more significantly, at least at the magnetopause behind the terminator ($| {V}_{{YZ}}|$ and φ_div; Figures 6(c) and (d), 7(c) and (d)). The stronger diversion of the sheath plasma adjacent to the magnetopause, i.e., φ_div, becomes larger on the magnetopause behind the terminator (Figures 6(d) and (d'), 7(d) and (d')), indicating that the magnetopause may become more flaring under a more radial IMF condition, e.g., $| {\theta }_{\mathrm{CO}}| \gt 60^\circ$. The decrease in the total pressure P_t at the subsolar region, the shrink of the strong P_t region, and the more flaring magnetopause behind the terminator all indicate that the whole magnetopause may expand outward when the IMF changes to be radial. This conclusion is consistent with the case study presented by Suvorova et al. (2010). In addition, under a more radial IMF condition, the larger β may reduce the MR rate or even cease MR on the magnetopause (Sonnerup 1974; Swisdak et al. 2003; Phan et al. 2013), which can also contribute to the expansion of the magnetopause (Aubry et al. 1970; Phan et al. 1996). At high latitudes or the flanks, the component of the solar wind velocity in the direction normal to the local magnetopause becomes larger because of a more flaring magnetopause under a radial IMF condition, and the higher effective Mach number produces a strong bow shock there. That is why the high thermal pressure region expands as seen in Figures 11(a) and 12(a).

In this section, the variations of parameters along the X-axis are first discussed. The subsolar region along the X-axis is basically near the plasma stream line connecting to the flow stagnation point on the magnetopause (Figures 19(a) and 20(e)), and the variations along the X-axis are determined by both the outer and inner boundary conditions of the magnetosheath at the subsolar bow shock and magnetopause, respectively.

The variations of three parameters, the plasma density N_i, the field magnitude $| {\boldsymbol{B}}|$, and the plasma velocity V_X, along the X-axis or plasma stream line connecting to the flow stagnation point on the magnetopause are analyzed in Figures 13–15, and the continuous increases in N_i and $| {\boldsymbol{B}}|$ and the decrease in V_X represent the further compression (pileup) and deceleration within the magnetosheath. In order to examine what outside conditions may control this pileup and deceleration, all data of an individual parameter are divided into different subsets by the outside conditions: the IMF cone angle $| {\theta }_{\mathrm{CO}}|$, the IMF clock angle $| {\theta }_{\mathrm{CL}}|$, the solar wind β_SW, and the solar wind convection electric field $\left|{{\boldsymbol{E}}}_{\mathrm{SW}}\right|$. The different IMF cone angles $| {\theta }_{\mathrm{CO}}|$ make the different profiles of $| {\boldsymbol{B}}|$ along the X-axis, and the differences among these subsets start right at the bow shock (Figure 14(a)), which indicates that the process on the bow shock produces these differences. The IMF cone angle acts through making different types of bow shocks at the outer boundary of the subsolar magnetosheath, and the quasi-perpendicular (quasi-parallel) bow shock compresses the magnetic field most (less) significantly under a smaller (larger) $| {\theta }_{\mathrm{CO}}|$ condition (Figure 14(a)). The bow shock types are determinative of the magnetic field–related parameters; the plasma parameters, however, are less dependent on the shock types (Figures 13(a) and 15(a)), since the effective Mach number is large.

Here β_SW and ${{\boldsymbol{E}}}_{\mathrm{SW}}$ also control the parameter profiles along the X-axis. When β_SW is smaller and/or ${{\boldsymbol{E}}}_{\mathrm{SW}}$ is stronger, the plasma in the sheath runs faster toward the magnetopause (Figures 15(c) and (d)), and both the magnetic field and the plasma are less compressed (Figures 13(c) and (d), 14(c) and (d)). The differences among the profiles distinguished by β_SW or ${{\boldsymbol{E}}}_{\mathrm{SW}}$ do not start at the bow shock but appear gradually on the way to the magnetopause, suggesting that the process on the inner boundary, i.e., on the magnetopause, may produce these differences. Magnetopause MR may be responsible for these observations: a faster MR on the magnetopause may create fewer pileup effects in the magnetosheath and make the magnetosphere less of an impediment to the incident solar wind.

How much IMF magnetic flux is reconnected on the magnetopause is estimated through a method established in this study by taking advantage of the magnetic field geometry in the SWI coordinate system. The dependence of the estimated reconnection flux on outside conditions is directly examined in Figure 17. The IMF cone angle $\left|{\theta }_{\mathrm{CO}}\right|$, IMF clock angle $\left|{\theta }_{\mathrm{CL}}\right|$, solar wind β_SW, convection electric field ${{\boldsymbol{E}}}_{\mathrm{SW}}$, IMF strength $\left|{B}_{\mathrm{SW}}\right|$, and solar wind Mach number M_S are all involved in Figure 17. It is seen that under the smaller $\left|{\theta }_{\mathrm{CO}}\right|$, lower β_SW, or stronger ${{\boldsymbol{E}}}_{\mathrm{SW}}$ conditions, the magnetopause MR tends to be faster if the MR rate is quantified by the total reconnected magnetic flux per unit time F_R. Both $\left|{B}_{\mathrm{SW}}\right|$ and M_S also control the efficiency of the magnetopause MR, and MR tends to be faster when $\left|{B}_{\mathrm{SW}}\right|$ is larger or M_S is smaller. In addition, the dependence of F_R on both the solar wind density N_SW and the solar wind velocity $| {{\boldsymbol{V}}}_{\mathrm{SW}}|$ has been examined in this study (not shown), and there is no clear dependence found.

In fact, the sheath plasma β right ahead of the magnetopause is a key parameter to control the proceeding of MR on the magnetopause (Sonnerup 1974; Swisdak et al. 2003; Phan et al.2013). Figures 11 and 12 already indicate that the ion β outside the magnetopause can be very different when the IMF cone angle θ_CO varies. The dependence of β on the outside solar wind and IMF conditions is further examined in Figure 21. The data involved in this figure are located in the subsolar magnetosheath, with the distance to the magnetopause more than 1R_E and less than 2R_E and the distance to the X-axis less than 6R_E. The logarithm magnetosheath ion β is plotted as a function of the IMF cone angles θ_CO (Figure 21(a)), IMF clock angles θ_CL (Figure 21(b)), IMF strength $| {B}_{\mathrm{SW}}|$ (Figure 21(c)), solar wind plasma β_SW (Figure 21(d)), solar wind convection electric field $| {{\boldsymbol{E}}}_{\mathrm{SW}}|$ (Figure 21(e)), and solar wind sound Mach number M_S (Figure 21(f)). The red curves are the spline regressions of β on the corresponding conditions.

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It is clear in Figure 21(a) that the IMF cone angle $\left|{\theta }_{\mathrm{CO}}\right|$ controls the plasma β outside the subsolar magnetopause: the larger the $\left|{\theta }_{\mathrm{CO}}\right|$, the larger the β, the same relation as already shown in Figures 11(d) and 12(d). Here $\left|{\theta }_{\mathrm{CO}}\right|$ acts mainly through the different types of bow shocks, and a stronger (weaker) compression of the magnetic field by the quasi-perpendicular (quasi-parallel) bow shock produces a smaller (larger) β under a smaller (larger) cone angle condition.

It is not surprising to see that the subsolar β is positively related to the plasma β_SW (Figure 21(d)) but negatively related to the convection electric field ${{\boldsymbol{E}}}_{\mathrm{SW}}$ (Figure 21(e)) and IMF strength $| {B}_{\mathrm{SW}}|$ (Figure 21(c)) in the incident solar wind, since β_SW and ${{\boldsymbol{E}}}_{\mathrm{SW}}$ are not independent of each other, and both of them are partially determined by the strength of the IMF $\left|{{\boldsymbol{B}}}_{\mathrm{SW}}\right|$: the larger $\left|{{\boldsymbol{B}}}_{\mathrm{SW}}\right|$ corresponds to the smaller β_SW and larger ${{\boldsymbol{E}}}_{\mathrm{SW}}$, and the larger $\left|{{\boldsymbol{B}}}_{\mathrm{SW}}\right|$ also corresponds to the larger $\left|{\boldsymbol{B}}\right|$ and thus the smaller β in the magnetosheath if the other IMF and solar wind conditions are the same. A stronger $\left|{{\boldsymbol{B}}}_{\mathrm{SW}}\right|$, smaller β_SW, or stronger ${{\boldsymbol{E}}}_{\mathrm{SW}}$ can produce a smaller β plasma in the magnetosheath, which is the favorite condition for MR to occur or proceed faster on the magnetopause (Sonnerup 1974; Swisdak et al. 2003; Phan et al. 2013). That interprets why the pileup of both the plasma and the magnetic field is less significant under the lower β_SW or stronger ${{\boldsymbol{E}}}_{\mathrm{SW}}$ conditions, as shown in Figures 13 and 14. Once more, according to this logic, it can be inferred that the subsolar magnetopause may not be a likely location for faster MR, since the plasma β is largest in this area no matter what the IMF cone angle is, as shown in Figures 11(d) and 12(d).

The solar wind sound Mach number is observed to not significantly control the subsolar plasma β in Figure 21(f), but it does control the MR rate in Figure 17(f). According to Equations (6) and (7), the compression and deceleration of plasma do not strongly depend on the Mach number (M_S) if M_S is big enough. In Figure 21(f), the involved solar wind sound Mach numbers are basically greater than 8, which leads to the almost constant compression efficiencies of the magnetic field and plasma in the subsolar magnetosheath. That is why the plasma β in the subsolar magnetosheath remains constant no matter how big the solar wind M_S is (Figure 21(f)). At high latitudes or in the flank magnetosheath, however, the component of the solar wind velocity in the normal direction to the local bow shock is much less than that in the subsolar region; thus, the effective M_S in this area is small. Under this situation, the effective M_S acts in Equation (6), and a smaller M_S in the solar wind may produce a less compressed and smaller β plasma in the high-latitude or flank magnetosheath, where MR may go faster than a larger M_S case. Here F_R describes the global efficiency of MR. Although the rate of the subsolar magnetopause MR does not vary much as M_S decreases, the faster MR at high latitudes or on the flank magnetopause may lead to the enhancement in F_R when the solar wind M_S decreases, as seen in Figure 17(f).

Figure 21(b) shows that the subsolar magnetosheath β does not vary with the IMF clock angle θ_CL, suggesting that the IMF clock angle θ_CL is independent of the magnetosheath β. Figure 17(b) shows that F_R increases when $| {\theta }_{\mathrm{CL}}|$ increases from 90° to 180°, and this result is consistent with all previous works (Sonnerup 1974; Wygant et al. 1983; Swisdak et al. 2003; Milan et al. 2012; Phan et al. 2013; Liu et al. 2018). When the IMF $| {\theta }_{\mathrm{CL}}|$ is less than 90°, however, F_R has almost the same level as that of $| {\theta }_{\mathrm{CL}}|$ > 90°, and F_R tends to increase when $| {\theta }_{\mathrm{CL}}|$ decreases from 90° to 0°. Although this result is totally different from what was expected in previous works (Sonnerup 1974; Wygant et al. 1983; Milan et al. 2012), it does not mean that our results are wrong. The difference mainly derives from the method used in this paper to calculate F_R.

Milan et al. (2012) used the variation of the area of the polar cap to estimate how much magnetic flux of the closed geomagnetic field is newly opened by the magnetopause MR and how much open flux is newly added into the polar cap. Their results are then parameterized by the solar wind properties in the form of $\bigtriangleup {U}_{R}={E}_{\mathrm{SW}}^{{\prime} }{L}_{\mathrm{eff}}$, where ${E}_{\mathrm{SW}}^{{\prime} }$ and L_eff are in the forms shown in Equations (4) and (5), respectively. In this study, the method used to calculate F_R takes advantage of the magnetic geometry in the SWI coordinate system and is particularly useful when estimating how much of the IMF magnetic flux passes through the magnetosphere via magnetopause MR. The magnetic fluxes F_R and $\bigtriangleup {U}_{R}$ for each data point in the data set are calculated by using the method proposed in this study and Equations (4) and (5), respectively. Figure 22 shows the distributions of the ratio of $\bigtriangleup {U}_{R}/{F}_{R}$ as a function of the IMF clock angle $| {\theta }_{\mathrm{CL}}|$.

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It is clear in Figure 22 that, no matter what $| {\theta }_{\mathrm{CL}}|$ is, F_R is always larger than $\bigtriangleup {U}_{R}$ because the ratio $\bigtriangleup {U}_{R}/{F}_{R}$ is always less than 1. Particularly when $| {\theta }_{\mathrm{CL}}|$ is greater than 90°, the ratio $\bigtriangleup {U}_{R}/{F}_{R}$ varies from 0.1 to 0.3. Here F_R may be overestimated because of the underestimation of the magnetic flux flowing through the flank magnetosheath F_SH, since F_R = F_SW − F_SH. The F_SH is calculated through F_SH = V_XB_YL_SH, where V_X and B_Y are the X component of the plasma velocity and the Y component of the magnetic field in the flank magnetosheath in the SWI coordinate system and L_SH is the dimension of the flank magnetosheath along the Z direction. The errors from the dimension L_SH are randomly distributed around the real quantities, as shown in Figures 2(d) and (f); thus, the systematic underestimation of F_SH might not derive from L_SH but from E_X = V_XB_Y. In our calculation, E_X is assumed to be constant along the Z direction in the flank magnetosheath. This assumption may be too strong, and as in the region close to the magnetopause, E_X may be enhanced, that might lead to the underestimation of F_SH and overestimation of F_R. In this sense, the quantities of F_R are only able to be used for a statistical study, and in this paper, their responses to the variations of the outside solar wind conditions are examined, such as in Figure 17.

When $| {\theta }_{\mathrm{CL}}|$ is less than 90°, the ratio $\bigtriangleup {U}_{R}/{F}_{R}$ is very small, and it decreases rapidly as $| {\theta }_{\mathrm{CL}}|$ decreases (Figure 22). In Figure 17(b), F_R is seen to not vary significantly with the variation of $| {\theta }_{\mathrm{CL}}|$, and thus the very small $\bigtriangleup {U}_{R}/{F}_{R}$ is due to the small quantity of $\bigtriangleup {U}_{R}$. According to the method to determine $\bigtriangleup {U}_{R}$, it can be understood that $\bigtriangleup {U}_{R}$ describes the newly opened magnetic flux added into the polar cap (Milan et al. 2012). However, F_R describes a different quantity related to the rate of the magnetopause MR. According to its definition, F_R estimates how much of the IMF fluxes go through the magnetosphere per unit time. These fluxes can go through the magnetosphere via the low-latitude MR on the magnetopause when $| {\theta }_{\mathrm{CL}}|$ is large, and they finally return to the downstream solar wind through MR in the distant tail. These fluxes can also go through the magnetosphere via the lobe MR at high latitudes when $| {\theta }_{\mathrm{CL}}|$ is small (Song & Russell 1992). In this situation, an IMF field line may reconnect twice at the high latitudes of the opposite pole of the magnetosphere and return back to the solar wind. These high-latitude MRs, typically referred to as "dual MRs," may not add any open flux into the polar cap but produce a newly closed field line wrapping the magnetopause forming the low-latitude boundary layer (Song & Russell 1992). These magnetic fluxes going through the magnetosphere, however, cannot have been involved in $\bigtriangleup {U}_{R}$ (Milan et al. 2012), and that is why F_R is much larger than $\bigtriangleup {U}_{R}$, particularly when $| {\theta }_{\mathrm{CL}}|$ is small. In Figure 17(b), it is seen that the rough estimation of F_R for the small $| {\theta }_{\mathrm{CL}}|$ case is equivalent to that for the larger $| {\theta }_{\mathrm{CL}}|$ case. When $| {\theta }_{\mathrm{CL}}|$ decreases from 90°, the magnetic shear at high latitudes may increase, leading to the increase in F_R as seen in Figure 17(b).

Efforts have always been exerted toward looking for the locations where MR is initiated on the magnetopause. It is reasonable to believe that these locations may have some special properties, since MR occurs there but not elsewhere. In the literature, many models have been established to predict the locations of MR or X lines. Usually, MR is treated as the development of a current-driven instability, i.e., the tearing mode instability, the growth rate of which is linked to the current properties across the current sheet (Pudovkin & Semenov 1985; Alexeev et al. 1998). It thus may be reasonable to relate the initiation locations of MR to the electric current ${{\boldsymbol{j}}}_{\mathrm{MP}}$ on the magnetopause.

In the present study, a maximum magnetosheath thermal pressure region is found to be located within a belt lying in the subsolar area (Figure 18(a)), and the strongest ion thermal pressure indicates the strongest magnetopause current density ${{\boldsymbol{j}}}_{\mathrm{MP}}$ there. Assuming a quasi-steady magnetopause, a balance is achieved through the equal total pressures P_t beside the magnetopause (as seen in Figures 18(c) and (c')), and the flows near the magnetopause are basically tangential to the boundary and thus do not contribute to this balance. The plasma thermal pressure dominates outside the magnetopause, while the magnetic pressure dominates inside the magnetopause (Figures 18(a), (a'), (b), and (b')). The current layer in the subsolar region should be thinnest over the whole magnetopause, and there is no reason to believe that the current layer here could be even thicker, since the total pressure is strongest and the magnetopause is compressed most significantly (Figures 18(c) and (c')). In this circumstance, the strongest thermal pressure gradient across the magnetopause is present in the subsolar region, and it produces the strongest diamagnetic current density. If the local current density ${{\boldsymbol{j}}}_{\mathrm{MP}}$ is great enough to excite the tearing mode instabilities (Pudovkin & Semenov 1985; Alexeev et al. 1998), MR may be initiated. In addition, the sheath flows present in the subsolar sheath region are sub-Alfvénic (as seen in Figures 7 and 19(b)), which is also a favorite condition for the steady MR on the magnetopause (Cowley & Owen 1989; Cooling et al. 2001).

There is another clue for the conjecture that MR is initiated in the strongest magnetopause current region, i.e., in the strongest thermal pressure region. By using the energy dispersion of particles ejected from MR, the location of an X line can be retrieved (Trattner et al. 2007; Zhu et al. 2015). Both the case and statistic studies have revealed that the initial X lines cross the subsolar region, and that their locations shift southward or northward when the dipole tilt axis inclines toward or away from the Sun. The normalized offsets of the retrieved MR sites away from the predicted X lines, which are anchored at the subsolar point, are plotted as a function of the Earth's dipole tilt angle in Figure 6 of Zhu et al. (2015). In this paper, the normalized offsets are converted to be the latitudinal shift angles in the GSM and SWM coordinate systems, and their dependences on the dipole tilt angle are plotted in Figures 23(a) and (b), respectively. It is seen that the latitudinal shift angles of the X lines are comparable to the corresponding dipole tile angles in GSM or SWM (the slopes of the regressing lines in Figure 23 are 1.3 in the GSM coordinate system and 1.2 in the SWM coordinate system). The thermal pressure ridge also shifts southward or northward when the dipole axis rotates toward or away from the Sun, and the slope of the thermal plasma ridge in Figure 20(a) is also about 1, which means that the shift latitudes of the ridge are also equal to the dipole tilt angles. Although the results in this study cannot assert where MR initiates on the magnetopause or distinguish which MR location model is best, the colocalization of the MR initiation sites and the thermal pressure ridge and their dependence on the dipole tilt angle θ_d suggest that MR tends to occur in association with the thermal pressure ridge, i.e., the strong magnetopause current.

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The background sheath flow can modify the X-line location, and it can carry an X line away from the subsolar region after the formation of the X line (Raeder 2006). Both Figures 19(a) and 20(e) show that the sheath flows are diverted on the magnetopause, and that the flow stagnation point is persistent at the subsolar point and does not shift away from the subsolar region, no matter what the dipole tilt angle θ_d is. This phenomenon may derive from the way of the magnetopause reshaping when θ_d varies. The magnetopause reshapes due to the variation of θ_d, which can be examined in the MHD simulations done by Liu et al. (2012). Their results show that the magnetopause nose stays persistently around the subsolar point, not depending on θ_d. The sheath flows are diverted at the nose when they encounter the magnetopause, and that is why the stagnation point remains unchanged as the dipole axis varies (Figure 20(e)). In this scenario, if θ_d is nonzero, MR will be initiated in the strongest current region away from the subsolar region where the background sheath flow is always present. The background sheath flow may carry the X line further away from the subsolar region, which interpret why the latitudinal shifting angles in Figure 23 are always larger than the corresponding dipole tile angles (the slopes of the regressing lines are greater than 1). One fact should be noted: the probability of θ_d = 0 is actually very low, and thus they are the most likely locations to investigate the MR triggering processes or initiation conditions.

The background sheath flows present at the MR sites help to produce FTEs. On the basis of MHD simulations, Raeder (2006) proposed that, if θ_d is nonzero, the background magnetosheath flow carries an X line away from its initiation location, and an FTE forms between the leaving X line and its initiation location when a second X line forms at this location. This scenario is supported by observations provided by Zhang et al. (2012), Zhong et al. (2013), and Pu et al. (2013). The size or duration of an FTE is found to be proportional to the size of the planet's magnetosphere; e.g., the size of the Earth's magnetopause is ∼20 times that of Mercury's magnetopause, and the size/duration of an FTE on the Earth's magnetopause is also 20 times that of an FTE on Mercury (Russell 1995). These observations can be understood in the scenario proposed by Raeder (2006), and the duration of an FTE is actually proportional to the time the first X line takes to move before the second X line forms or the length of the path along which the first X line has moved. Let us assume that the distributions of all kinds of parameters on the Earth's and Mercury's magnetopauses are similar but of different spatial scales. For example, the sheath flow and magnetopause current on the Earth's magnetopause have the same patterns as on Mercury's magnetopause, and the length or time for which the first X line has moved before the second X line forms are then only proportional to the magnetopause's dimensions: the bigger the magnetopause, the longer time/larger distance the first X line may move and the larger the FTE.