In the ANGIE3D code, the ions (protons) are treated as discrete, fully kinetic particles, and the electrons are treated as a massless fluid. Quasi charge neutrality is assumed. Detailed descriptions of the equations for ion particle motion, electric and magnetic fields and assumptions used in the ANGIE3D code are given in Lin et al. (2014). The code is valid for low-frequency physics with ω ∼Ω_i and kρ _i ∼1 (wavelength λ ∼6ρ_i), where ω is the wave frequency, k is the wave number, Ω_i is the ion gyrofrequency, and ρ_i is the ion Larmor radius.

The simulation domain is 25 ≥ X ≥ −60, 60 ≥ Y ≥ −35, 35 ≥ Z ≥ −45 R_E in the geocentric solar magnetospheric (GSM) coordinates. Inflow time-dependent boundary conditions for the solar wind are specified at the sunward boundary and open boundary conditions are used for the rest of the outer boundaries. An inner boundary is assumed at the geocentric distance of r ≈ 3 R_E. This inner boundary is composed of a zigzag grid line approximating the spherical surface as in global MHD simulations. For the region of the inner magnetosphere, a cold, incompressible ion fluid is assumed to be dominant in r <6 R_E, which coexists with particle ions, since this simulation focuses on the dynamics and ion kinetic physics in the outer magnetosphere. The inclusion of the cold ion fluid in the inner magnetosphere simplifies the conditions for the fluid-dominant low-altitude, inner boundary. A combination of spherical and Cartesian coordinates is used at the inner boundary. We let particles be reflected at exactly r = 3 R_E. This simple reflection of the ion parallel velocity means that loss cone effects are omitted. The E and B fields at the boundary reside on the Cartesian boundary approximating the spherical boundary, which are extrapolated to an extra grid point inside the r = 3 R_E surface. The B field is assumed to maintain the dipole field values at the inner boundary.

The ionospheric conditions (1,000 km altitude) are incorporated into the ANGIE3d code. The FACs, calculated within the inner boundary, are mapped along the geomagnetic field lines into the ionosphere as input to compute ionospheric potential. For this simulation, simplified ionospheric conductance with uniform Pederson conductance of 10 siemens and Hall conductance of 5 siemens is specified.

The TD is specified as a planar IMF discontinuity with a half-width of 0.12 R_E and the normal direction of (−0.5, 0.86, 0). The TD propagates with a velocity of (−400, 0, 33.7) km/s. At t = 0, the TD plane intersects the Y = 0 axis at X = 185 R_E. Unless otherwise noted, downstream (upstream) of the TD in this paper indicates the anti-sunward (sunward) side of the TD. The downstream IMF direction is (3, 1.7, 0) nT and upstream IMF is (0, 0, −3.4) nT. Constant solar wind density of 5 cm⁻³ and isotropic solar wind ion temperature of 10 eV are used. The solar wind velocities are (−370.7, 16.8, 33.7) km/s downstream and (−400, 0, 0) km/s upstream. The average solar wind Alfvén Mach number is M_A = 11.8. These solar wind values are within the typically observed ranges. To accomplish this large-scale simulation with the available computing resources and can still produce physical results, we choose the solar wind d _i to be 0.1 R_E (about 6 times larger than the realistic value) and the cell dimensions to be n_x × n_y × n_z = 502 × 507 × 400. Also, we use time-independent nonuniform cell sizes (ranging from ∼0.1 to 0.5 R_E) so that we can appropriately assign cell sizes comparable to the d _i values in different key regions from the solar wind to the outer magnetosphere. The bow shock and magnetopause form self-consistently by the interaction of the solar wind with the geomagnetic dipole. Before the arrival of the TD, the bow shock nose is at X ∼14 R_E and the magnetopause nose is at X ∼10 R_E, similar to the realistic locations.

Figures 1A–C show the 2D profiles of the magnetic field strength (|B|), ion density (N), and ion bulk flow speed (|V|), respectively, in the X-Y plane at Z = 0 at four different times from t = 53.4–75.3 min (see also Supplementary Movie S1 in Supplementary Material). The simulated magnetopause and bow shock are disturbed, so we also add in the t = 53.4 min plots two smooth model boundaries, the magnetopause locations predicted by Roelof and Sibeck (1993) and the bow shock locations predicted by Peredo et al. (1995), as visual references to help readers discern the magnetosheath perturbations. In this stimulation, before the arrival of the TD, the foreshock is mainly on the duskside extending from the dayside to the nightside. Note that there are weak perturbations in the foreshock and the magnetosheath due to the foreshock ULF waves. The TD first encounters the foreshock ions just outside the dayside bow shock at t ∼44 min and foreshock transient perturbations are formed (see Wang B. et al. (2021) for more details about the initiation of the foreshock transient). The foreshock transient perturbations consist of a core with lower density, higher temperature, lower magnetic field strength, and lower anti-sunward bulk flow speed than the values of the solar wind. An edge with relatively higher density and higher magnetic field strength is on the upstream side of the core. As the TD (indicated by the black or white dashed straight lines) propagates tailward, it continues to interact with the foreshock ions and generate perturbations around the TD (the low-density core is indicted by magenta arrows in Figure 1B). The perturbations newly generated just outside the bow shock subsequently enter the magnetosheath via their anti-sunward flows and continue to propagate anti-sunward. Note that these magnetosheath perturbations associated with the foreshock transients are the focus of this paper, not the pre-existing perturbations associated with the foreshock ULF waves.

Figure 1 shows the tailward propagation of the magnetosheath plasma and magnetic field perturbations resulting from foreshock transients. In the near-Earth region, as shown in the t = 53.4 and 59.7 min plots, the structures of magnetosheath perturbations are approximately aligned with the TD plane (the black or white dashed line). The perturbations seen closer to the magnetopause are associated with the foreshock transient perturbations that are generated and enter the magnetosheath earlier, while those seen closer to the bow shock are associated with the foreshock transient perturbations that are generated and enter the magnetosheath more recently. The newer perturbations coming into the magnetosheath interact nonlinearly with those further inside, leading to changes in the spatial structures of the perturbations across the magnetosheath. In this simulation, the foreshock region extends to the nightside. Thus, as the TD propagates from the near-Earth to the midtail, as shown in the t = 66 and 72.3 min plots, there are still new foreshock transient perturbations being continuously added into the flank magnetosheath. As a result, the magnetosheath perturbations are still strong in the midtail. Compared to the earlier magnetosheath perturbations in the near-Earth flank shown in the t = 59.7 min plots, which are more spatially confined around the TD plane and have well-defined structures, the spatial size of the mid-tail magnetosheath perturbations shown in the t = 72.3 min plots have become larger and their spatial structures become complex because of the nonlinear interaction described above.

Figure 2 compares the dayside magnetosheath and magnetopause before the arrival of the TD at t = 45.6 min with those associated with the magnetosheath perturbations at t = 52.8 min. As shown in Figures 2A–E for the X-Y distributions at Z = 0, at t = 45.6 min, there are small and localized perturbations in both the magnetosheath plasma and the magnetopause shape (black or white curves) associated with the foreshock ULF waves. The dayside magnetopause locations are determined by tracing magnetic field lines from Z = 0 and the field lines in the dayside magnetosphere are closed (both ends of the field lines are in the ionosphere). At t = 52.8 min, the low-density core and high-density edge can be seen in the new perturbations forming outside the bow shock as well as in the magnetosheath perturbations that have entered the magnetosheath earlier (Figure 2B). The magnetic field strength is lower inside the core and higher at the edge (Figure 2A). Figure 2C shows different flow speeds and directions for the core and edge, which would later cause the spatial extents of the core and edge regions to change as they propagate tailward. As a result of the lower density and flow speed within the core than at the edge, both the thermal pressure (P_th) and the dynamic pressure along the direction normal to the magnetopause (P_dyn,n) (the magnetopause normal direction in this paper is estimated using the model magnetopause of Roelof and Sibeck (1993)) are relatively lower within the core and higher at the edge. As shown in Figure 2D, the dayside magnetopause and magnetosphere intrude locally outward for ∼3 R_E into the magnetosheath in response to the lower P_n (P_n = P_th + P_dyn,n) of the core and are distorted locally inward for ∼1 R_E by the stronger P_n of the edge. The outward intruding magnetosphere is indicated by the plasma with relatively higher magnetic field strength (Figure 2A) and lower density (Figure 2B) than the surrounding magnetosheath plasma. Figure 2E shows the perpendicular current density. It shows that the magnetosheath perturbations at t = 52.8 min results in strong perpendicular currents along the distorted magnetopause. Figures 2F,G show the 2D X(Y)-Z profiles along the white dashed line indicated in Figure 2A (the TD plane at t = 52.8 min). The magnetopause outward distortion is seen mainly in the region of |Z| < ∼ 5 R_E with the maximum distortion near Z = 0. The 1D profiles at Z = 0 along the white dashed line indicated in Figure 2A are shown in Figures 2H–K. Comparing the 1D profiles between t = 45.6 and 52.8 min clearly show the changes in magnetic field components, flow velocity components, and pressure components outside the magnetopause (vertical magenta dashed lines) associated with the low-density core.

Figure 3 compares the X-Y distributions of the nightside magnetosheath and magnetosphere at Z = 0 at t = 45.6 with those at t = 60 min when the magnetosheath perturbations have propagated to the nightside around X = –10 R_E. The magnetosheath perturbations at t = 60 min are seen to be around the TD line (white dashed line). Similar to the dayside magnetopause distortion shown in Figure 2, the magnetopause (indicated by white dashed line) intrudes locally outward into the magnetosheath around X = –10 R_E in response to the low-density core of the magnetosheath perturbations while it is distorted inward around X = –7 R_E in response to the high-density edge. In determining the nightside magnetopause boundaries shown in Figure 3 and later in Figures 4, 5, we investigate the magnetosonic Mach number from the magnetosheath to the magnetosphere and use the location of a quick drop in the Mach number values to below a certain threshold as the approximate location for the magnetopause boundary. The outward intruding magnetosphere can be seen by the plasma with relatively higher magnetic field strength (Figure 3A), lower density (Figure 3B), and higher temperature (Figure 3C) than the surrounding magnetosheath plasma. Different from the slow-flowing plasma deep within the magnetosphere, the intruding magnetospheric plasma has a strong tailward flow speed (Figure 3D). Figure 3E shows the changes in the perpendicular current density within the magnetosphere associated with the distorted magnetopause. This results in FACs flowing into and out of the ionosphere, as described later in Impact on the Ionosphere. Figure 3F shows the 3D view of the number density distributions at t = 60 min from three different viewing angles together with the magnetic field lines. As indicated by the closed magnetic field lines (red), the plasma sheet is seen within the outward intruding magnetosphere. The field lines in the magnetosheath tailward of the intruding magnetosphere are open field lines (purple, with one end connecting to the Earth) due to open flank magnetopause resulting from the duskward IMF downstream of the discontinuity, while those earthward of the intruding magnetosphere are IMF field lines (light pink) corresponding to the southward IMF upstream of the discontinuity.

The 3D structure of the outward intruding magnetosphere at t = 60 min shown in Figure 3 can be better constructed with the 2D Y-Z and X-Z distributions cutting through the intrusion shown in Figures 4A,B,G,H, respectively (see also Supplementary Movie S2 in Supplementary Material). The magnetopause is distorted mainly in the region from Z ∼ −10 to 10 R_E with the maximum outward distortion at Z ∼0 (Figures 4A,B) so that the cross-section in the X direction is the widest near Z = 0 (Figures 4G,H). The Y-profiles of plasma and magnetic field along the cutting plane at Z = 0 are shown in Figures 4C–F. As indicated by the vertical magenta dashed line, the magnetopause boundary moves outward from Y ∼18 to 24 R_E during the distortion. Figures 4I–M show the X-profiles at Z = 0 along Y = 21 R_E. The X scale of the intruding magnetosphere is ∼6 R_E.

Figures 5A–C show the time sequence of the flank magnetopause (white solid curves) distortion in the X-Y, X-Z, and Y-Z planes, respectively. The white dotted curves in Figures 5A,C indicate the magnetopause at t = 45.6 min. Note that the magnetopause boundary shape can appear filamentary at some locations. This is associated with fine structures of the magnetosheath perturbations in the magnetic field strength and flow speed, which resulting in fine structures in the magnetosonic Mach number distributions used in determining the approximate magnetopause boundary. Figure 5 shows that as the magnetosheath perturbations move tailward from X ∼ −10 to X ∼ −40 R_E, they continue to distort the magnetopause. As described in Magnetosheath Perturbations and Tailward Propagation, the spatial structures of magnetosheath perturbations change substantially as they propagate tailward, thus the 3D structure of the outward intruding magnetosphere in the midtail (t = 70.4 min plot) is quite different from the earlier structure in the near-Earth tail (t = 60 min plot). The maximum outward intrusion remains around Z = 0 and it extends farther out in the Y direction with increasing downtail distances. The localized structure of the outward distortion shown in Figure 5 indicates that a satellite in the magnetosheath may observe the outward intruding magnetosphere with the probability strongly depending on the satellite locations.

Figure 6 shows the temporal profiles of magnetic field components, number density, ion temperature, and ion bulk flow velocities that would be observed by a virtual satellite in the magnetosheath at three downtail distances at Z ∼0. Because of the passing of the localized outward magnetopause distortion, the virtual satellite would observe transient appearance of the magnetosphere, as indicated by the magnetic field strength, density, and temperature changing from the magnetosheath values to the magnetospheric values and then return to the magnetosheath values. These temporal profiles are qualitatively similar to the perturbations observed in the midtail magnetosheath at X = −54 R_E reported by Wang et al. (2018). Another observation event in the flank magnetosheath closer to the Earth is shown in An Event for Flank Magnetopause Distortion.

The localized and transient magnetopause distortion affects the magnetic field within the magnetosphere. Figure 7 shows a time sequence of the Y-Z distributions at X = −10 R_E from the dusk flank to midnight for number density (Figure 7A), magnetic field strength (Figure 7B), amplitudes of the magnetic field perturbations in the parallel direction (Figure 7C), and perpendicular current strength (Figure 7D). The magnetic field perturbations shown in Figure 7C are obtained by subtracting the 10 min running averages. To better show the perturbations associated with waves propagating through a relatively uniform background, only the perturbations in the northern lobe where B_x >15 nT are plotted in Figure 7C. As shown in the t = 45.6 min plot for before the arrival of the magnetopause distortion, there are weak magnetic field perturbations within the magnetosphere. These are due to the small magnetopause disturbances associated with the foreshock ULF waves, like that seen on the dayside as shown in Figure 2A for t = 45.6 min. As the magnetopause distortion passes through X = –10 R_E, as shown in the t = 59.7 – t = 61.6 min plots in Figure 7, the magnetic field perturbations within the magnetosphere are enhanced. The enhancements are seen to extend from the dusk flank into the magnetosphere. Compared to the enhancements when the magnetopause is distorting outward around t ∼60 min, the perturbations generated by the inward magnetopause distortion around t = 61.3 min are stronger and deeper into the magnetosphere. This shows that the magnetopause distortion driven by foreshock transients can launch compressional waves within the magnetosphere, which qualitatively explains the observed enhancements in magnetospheric ULF waves associated with foreshock transients (e.g., Hartinger et al., 2013; Wang et al., 2017; Wang et al., 2018b; Wang et al., 2019; Wang B. et al., 2020).

As shown in Figure 7B, the inward and outward motion of the distorted magnetopause alters the magnetospheric magnetic field near the flank in Y >∼10 R_E. This causes transient changes in the perpendicular currents in the flank magnetosphere shown Figure 7D as well as FACs flowing into or out of the ionosphere in order to maintain current continuity, establishing impact on the ionosphere. The resulting FAC perturbations in the ionosphere are shown in Impact on the Ionosphere.

Figures 8A,B show the FACs and FAC perturbations at t = 60 min, respectively, in the Northern Hemisphere (N.H.) ionosphere (positive value indicates FACs flowing into the N.H. Ionosphere). The FAC perturbations are obtained by subtracting the 10 min averages of the FACs in the ionosphere. The FAC spatial distribution shown in Figure 8A has currents flowing into (out of) the ionosphere on the dawnside (duskside), which is the large-scale region-1 FACs connecting to the magnetosphere near the magnetopause. Figure 8B shows that the FAC perturbations are spatially localized. Figure 8C shows the time sequence of the ionospheric FAC perturbations in N.H. as a function of MLT and MLAT. Figure 8D shows the time series of N.H. FAC perturbations at different duskside MLT locations along MLAT = 73.5. Figures 8C,D show that the region of enhanced FAC perturbations moves anti-sunward from near noon toward later MLTs, which is consistent with the tailward propagation of the flank magnetopause distortion. At t = 60 min, FAC perturbations have moved to nightside at ∼18–20 MLT when the magnetopause distortion has propagated to nightside at X ∼ −10 R_E. The FAC perturbations would result in perturbations in the horizontal currents flowing in the ionosphere due to the current continuity, both would generate magnetic field perturbations on the ground.

Note that simplified and spatially uniform ionospheric conductance is used in this simulation and we do not further evaluate the simulated ionospheric horizontal currents. The spatial distributions of the simulated ionospheric potential pattern and FACs corresponding to this uniform conductance do not have day-night and dawn-dusk asymmetries as realistic as those corresponding to non-uniform conductance that accounts for EUV and aurora contribution (Ridley et al., 2004). We expect that using realistic EUV- and aurora-generated conductance would shift the MLT and MLAT locations as well as the amplitudes of the perturbations in FAC and horizontal currents, but it would not affect their physical connection with the flank magnetopause distortion presented above. The simulated FAC perturbations seen at a fixed ionospheric location shown in Figure 8D should still provide a qualitative explanation for the observed ground magnetic field perturbations associated with foreshock transients (e.g., Shen et al., 2018). An observation event for ground magnetic field perturbations propagating to the nightside is shown in An Event for the Ionospheric Disturbances.

We present in Figure 9 an observation event for transient flank magnetopause distortion driven by a foreshock transient on May 31, 2018. Figures 9A,B show that Geotail was in the solar wind, Cluster was in the dawnside magnetosheath at X ∼ 0 (data from Cluster C4 probe are used), and MMS was also in the dawnside magnetosheath further down the tail at X ∼ −18 R_E (data from MMS-3 probe are used). Both Cluster and MMS were near Z = 0. Figures 9C,D show that Geotail observed two IMF directional discontinuities (no change in the IMF strength) at ∼21:50 and 21:54 UT (indicated by the two vertical dashed lines), respectively. There were no changes in the solar wind density (Figure 9D), temperature (Figure 9E), and flow speed (Figure 9F) across the discontinuities. The IMF B_x was positive and IMF B_y was negative between the two discontinuities. The same discontinuities were also observed earlier at ∼21:05 UT by WIND at X ∼ 200 R_E (not shown) and the normal direction of the discontinuities estimated using the WIND-Geotail pair is (−0.85, 0.12, 0.5). This IMF condition would result in a foreshock cavity on the dawnside. The discontinuities later arrived at Cluster at ∼22:05 UT (Figure 9G). The ∼15 min delay from Geotail to Cluster is expected from the propagation of the discontinuities being slowed down after they entered the dayside magnetosheath (for example, see Figure 3A of Wang C. P. et al. (2020) for the propagation of an RD in the magnetosheath). Between the discontinuities, Cluster observed perturbations (yellow shaded region) with a core of low density (Figure 9H) and low magnetic field strength (Figure 9G), slight flow deflection (a slight decrease in |V_x| and increase in |V_y|) (Figure 9I), and some superthermal ions at ∼ 10 keV (Figure 9J). An edge of slightly higher magnetic field strength and density was seen next to the core (red shaded region at ∼22:08 UT in Figures 9G,H). These confirm the magnetosheath perturbations associated with the expected foreshock transient. Even though the type of the foreshock transient in this event is different from that of this simulation, the observed magnetosheath perturbations are qualitatively similar to the simulated perturbations shown in Figure 2 in the dayside magnetosheath. This is expected since, as described in Introduction, almost all types of foreshock transients exhibit the same characteristics in their density and magnetic field perturbations.

As the discontinuities and the magnetosheath perturbations observed at the Cluster location moved to the MMS location at ∼22:13 UT (Figure 9K), MMS observed transient appearance of the magnetosphere (yellow shaded region). The magnetosphere is indicated by that the values for the low density (Figure 9M) and high temperature (Figure 9N) within the yellow shaded region are typical for magnetospheric plasma. This change from the magnetosheath to the magnetosphere can also be seen in the sharp increases of ion fluxes at >10 keV and decreases at <2 keV shown in Figure 9P. This magnetospheric plasma seen intruding outward into the magnetosheath has substantial tailward flow speed, which is qualitatively consistent with the simulations shown in Figure 6B.

We present in Figure 10 an observation event for ground magnetic field perturbations associated with a foreshock transient on January 20, 2010. This event has been reported by Wang et al. (2018) and they have shown simultaneous satellite observations of the event in the solar wind, foreshock, and flank magnetosheath. For this event, the driver discontinuity was observed by WIND in the solar wind. Geotail was on the dayside in the foreshock (the location is indicated in Figure 10A) and observed a transient low-density core with higher temperature and deflected flows (see Figure 4 of Wang et al. (2018) for the WIND and Geotail observations). The density perturbations observed by Geotail are shown in Figure 10A with the time of the discontinuity observed at the Geotail location indicated by the vertical magenta line. As shown in Figures 10B,C for P2 and P1, respectively, artemis P1 and P2 were both in the dawnside magnetosheath (their locations are indicated in the plots) with P2 closer to the Earth at X ∼ −27 R_E and P1 further down the tail at X ∼ −50 R_E. Figures 10A–C show that the discontinuity and the associated low-density core observed at the Geotail location at 09:04 UT propagated to P2 at ∼09:34 UT then to P1 at ∼09:39 UT.

Figures 10D–F show the ground magnetic field perturbations (obtained by subtracting the 10 min running averages) in the north-south direction observed by three magnetometer stations. The three stations were on the dawnside at similar magnetic latitudes (∼73^°–76^°) but at different MLTs from the dayside to the nightside (their MLTs and MLATs at 09:10 UT are indicated in the plots). The aurora image in N.H. from DMSP F17 satellite around 09:11 UT (not shown) indicates that the three stations were within diffuse aurora so that they were mapped to the closed field-line region of the magnetosphere. The ground magnetic field perturbations were enhanced at the three stations within the interval when the foreshock transient perturbations propagated from Geotail on the dayside to P2 and P1 on the nightside. The enhanced perturbations were first observed at ∼11 MLT, then at 07 MLT, and then 04 MLT. These simultaneous observations of the tailward propagating magnetosheath perturbations and the anti-sunward propagating ground perturbations are qualitatively consistent with the simulated anti-sunward propagating FAC perturbations in the ionosphere shown in Figure 8 generated by the simulated tailward propagating magnetopause distortion shown in Figures 1–5.