From 1,200 to 1300 UT on 20 November 2001 (Figures 1A–C) Cluster was located in the duskward magnetopause boundary layer. The four Cluster spacecraft (C1-4) were in a tetrahedral configuration and were separated by ∼1968 km on average (Figure 1D) with its barycenter at ∼[−5.3, 17.9, 3.2] Earth radii (R_E) in Geocentric Solar Ecliptic (GSE) coordinates. [GSE coordinates correspond to the boundary normal coordinates (LMN) obtained from Shue et al. (1997) model in this event.] The IMF was mostly northward during this period. Figure 1 shows (A) the magnetic field (B) averaged over the four spacecraft and (B, C) the electric current density (J) obtained using the curlometer technique (x, y, and z components in blue, green, and red in GSE) and decomposed into parallel (red) and perpendicular (blue) components with respect to B. Both B and J show quasi-periodic fluctuations with a period of ∼8–15 min that are most likely to be attributed to magnetopause KHWs.

To test if these fluctuations resulted from nonlinear KHWs, we expanded the C1 data from 1200 UT to 1215 UT in Figures 1E–J. On the top of Figure 1E, we denoted a more-magnetospheric region in a blue bar as characterized by a relatively larger B _z (Figure 1E), more flux of high-energy (≳1 keV) ions (Figure 1F showing the ion energy spectrogram), reduced anti-sunward flow velocity (V _x shown in blue; Figure 1G) and ion density (black in Figure 1I). The region of a smaller B _z accompanied by more flux of low-energy ions (<1 keV), increases in anti-sunward velocity and ion density represents a more-magnetosheath side (red bar). [Note that the energy spectrogram indicates that the boundary layer was rather in a mixed/turbulent state.] We marked the magnetosphere-to-magnetosheath transitions by ‘A’, ‘B’, ‘C’, and ‘D’ at the top of Figure 1E and vertical solid black lines. The magnetosheath-to-magnetosphere transitions are marked by ‘a’, ‘b’, ‘c’, and ‘d’ and vertical dashed black lines.

In a steady state of KHVs, the centrifugal force is balanced by the pressure force. Since the centrifugal force is radially outward in a rolled-up vortex, the pressure force should point inward to the vortex center. The high total pressure, then, builds up at the boundary from the more-magnetospheric side into the more-magnetosheath side crossing by Cluster (see Figure 19 of Hasegawa, 2012). And the total pressure is minimized close to the boundary from the more-magnetosheath side to the more-magnetospheric side crossing. Figure 1H shows this trend. The total pressure (black), i.e., the sum of plasma (red) and magnetic (blue) pressures, often peaks at boundaries toward the more-magnetosheath side (‘H’ letters in Figure 1H) and decreases at boundaries toward the more-magnetospheric side (‘L’).

Another characteristic of KHVs is the so-called density reversal (Hasegawa et al., 2004). For the rolled-up vortices, the density profile away from the nominal magnetopause (i.e., ∼along + y _GSE for duskward KHV events) shows a layer of a higher density (of magnetosheath origin) sandwiched between layers of a lower density (of magnetosphere origin). As a result, the spacecraft can detect a lower-density magnetosphere-origin layer located outward of a higher-density magnetosheath-origin layer. Figure 1I shows the ion density measured by C1 (black), C3 (green), and C4 (blue). Figure 1J presents these observations by color with C1, C3, and C4 data arranged in terms of their distance away from the magnetopause, i.e., along + y _GSE. Red arrows in Figure 1J indicate the times when the density observed by C1 (closest to the earth) is higher than that observed by C3 or C4 (further away from the earth).

Both features of total pressure (Figure 1H) and density reversal (Figure 1J) support the identification of KHVs. For the duskward KHV event such as Figure 1, we expect the development of the antiparallel current or, equivalently, upward FAC in the northern ionosphere. Figure 1C, indeed, shows that J is donimantly antiparallel throughout the event.

The geomagnetic field models (Tsyganenko, 1989; Tsyganenko, 1995) predict that the magnetic field lines encountered by Cluster during the Figure 1 event are mapped to the ionosphere at ∼73° LAT and ∼339° LON in geographic coordinates (GEO). FAST spacecraft fortuitously passed the northern ionosphere at/near the footprint of Cluster’s location around the time of the event. Figure 2 shows the energy spectrograms of down-going (with pitch angles of 0° ± 45°; A, C) and up-going (with pitch angles of 180° ± 45°; B, D) electrons (A, B) and ions (C, D). Precipitating fluxes are larger than up-going fluxes for both electrons and ions. The difference between the ion and electron flux gives rise to the current density along B (Figure 2E) that is mostly upward from the ground, i.e., antiparallel to B. The perturbed magnetic field (dB; Figure 2G) also gives rise to a negative J (Figure 2F) from Ampere’s law, demonstrating that FAST traversed the upward FAC region. J calculated from both particles and dB ranges from hundreds to ∼2000 nA/m².

During ∼1759–1809 UT on 1 October 2015, the MMS quartet with its barycenter at ∼[3.9, 9.2, −4.1] R_E in Geocentric Solar Magnetospheric (GSM) coordinates encountered magnetopause fluctuations shown in Figure 3. The IMF was mostly southward during the period. A more-magnetosheath region is then identified by a mostly negative B _z (Figure 3A), more flux of <2 keV-energy ions (Figure 3B), larger anti-sunward flow (Figure 3C), enhanced ion density (Figure 3D), and reduced ion temperature (Figure 3E). We denoted such repeated regions by red bars on the top of Figure 3A (although the region between ‘E’ and ‘e’ at the top of Figure 1 is a mixed region exhibiting a magnetospheric field and a magnetosheath plasma). Opposite trends represent a more-magnetospheric region as indicated by a blue bar.

We, again, marked magnetosphere-to-magnetosheath transitions by ‘A’, ‘B’, …, ‘F’ at the top of Figure 3A with vertical solid black lines and magnetosheath-to-magnetosphere transitions by ‘a’, ‘b’, …, ‘f’ with vertical dashed black lines. Figure 3F shows that the total pressure generally rises at/near magnetosphere-to-magnetosheath boundaries (‘H’ in Figure 3F) and lowers at/near magnetosheath-to-magnetosphere boundaries (‘L’). This supports that the observed fluctuations are attributed to KHVs.

The average spacecraft separation of ∼31 km during this event prevents us from testing the density reversal. Instead, we performed boundary normal analyses. As shown in Figure 3K (Hwang et al., 2011, 2020), boundaries of typical KHVs tilt from the initially-undisturbed magnetopause with its normal along n, showing a more-gentle waveform at the trailing edges (see black arrows at ‘A’, ‘B’, …, ‘F’ in Figure 3K) and a steeper waveform at the leading edges (black arrows at ‘a’, ‘b’, …. ‘f’). Also, since KHVs propagate tailward along the magnetopause (along − m or k-vector seen by a white arrow), normal propagation velocities (magenta arrows) are more aligned to the ± n direction with smaller speed at the trailing edges, and more perpendicular to n (or more parallel to k-vector) with larger speed at the leading edges. To test this, we determined the nominal boundary normal coordinates (LMN) derived from minimum variance analysis (MVA) (Sonnerup and Scheible, 1998; Siscoe and Suey 1972) for the magnetopause-crossing period from 1710 UT to 1730 UT prior to the occurrence of KHVs: l = [0.31, 0.34, 0.89], m = [0.59, −0.80, 0.10], and n = [0.75, 0.50, −0.44] in GSM. Table 1 lists the normal propagation velocities derived from a four-spacecraft timing analysis (Paschmann and Daly, 1998) and the MVA (using B)-derived boundary normals in LMN together with the medium-to-minimum eigenvalue ratio.

Figure 3J displays the mn-plane projections of boundary normals shown as black arrows and normal propagation velocities as magenta arrows. Both the normals and normal propagation vectors generally show a repetitive pattern between leading and trailing edges, consistent with Figure 3K. This confirms the identification of KHVs for the Figure 3 event.

Figure 3G shows J caculated from particle data (it is consistent with the curlometer-derived J) and Figure 3H shows parallel and perperdicular components of J. Both J _z and J _|| , although fluctuating around zero, are mostly negative with magenta shades in Figure 3H representing B _z < 0 periods. Therefore, J mainly pointed opposite to the geomagnetic B, consistent with the upward FAC in the northern ionosphere for the duskward KHV event.

Figure 3I shows the ion vorticity, Ω . As expected in Section 2 (Eq. 3), the z component of Ω . is relatively positive, indicating the counter-clockwise rotation of the duskward KHV. A quantitative test of Eq. 3 requires another spacecraft quartet simultaneously crossing the KHV flux tube above/below the near-equatorial plane where MMS traversed. Figures 3G–I, however, demonstrate the sign/sense of J and Ω . We note that the larger inertia on the faster magnetosheath side than the magnetospheric side generally leads to a larger |J| and smaller | Ω . |

The geomagnetic field model (Tsyganenko, 1989) predicts that the magnetic field lines encountered by MMS during the Figure 3 event are mapped to the ionosphere at ∼67° LAT and ∼302° LON. We use the gMAG data to derive the equivalent ionospheric current (EIC) and spherical elementary current (SEC). The result is shown in Figure 4, where the MMS footprint is denoted by a green dot. A counter-clockwise rotation of EIC around the green dot (upper panel) indicates upward FACs. SEC (lower), a proxy of the vertical current for an altitude of 100 km, shows the upward FAC of ∼18,400 A at the green dot. We note a bead-like structure in SEC indicative of upward FACs elongated in the east-west direction, possibly implying their generation via the duskward KHVs (Figure 3).

From ∼0250 UT to ∼0430 UT on 28 July 2006 Cluster located at ∼[−13, −13, −3.0] R_E in GSE observed KHV-induced magnetopause fluctuations (Figure 5) as reported by Hwang et al. (2011). [GSE coordinates that were close to GSM in this event correspond to the boundary normal coordinates (LMN) obtained from Shue et al. (1997) model.] The IMF was fluctuating with B _z ≤ 0. On the top of Figure 5A, blue (red) bars represent a more-magnetospheric (more-magnetosheath) region with a larger (smaller or negative) B _z (Figure 5A), more (less) flux of high-energy ions (Figure 5B), reduced (enhanced) anti-sunward flow (Figure 5C). We, again, denoted the magnetosphere-to-magnetosheath transitions by ‘A’, ‘B’, …, ‘I’ with vertical solid black lines and the magnetosheath-to-magnetosphere transitions by ‘a’, ‘b’, …, ‘j’ with vertical dashed black lines.

The total pressure (black in Figure 5D) is maximized at/near the boundaries toward the more-magnetosheath region (‘H’) and minimized at boundaries toward the more-magnetospheric region (‘L’). The four Cluster spacecraft in a tetrahedron were separated by > 1 R_E on average (Figure 5H), which enables us to test the density reversal. Figure 5E shows the ion density in color measured by C1/3/4 arranged in terms of their distance away from the magnetopause. Red arrows in Figure 5H mark the density-reversal times when the density observed by C4 or C3 (closer to the earth) is larger than that observed by C1 (further away from the earth). These observations confirm the dawnward-magnetopause KHVs for the Figure 5 event.

For the dawnward KHV event, we expect the development of the parallel current or, equivalently, downward FAC in the northern ionosphere (Section 2). J _z is, indeed, mostly positive (Figure 5F) and J _|| is mainly parallel (Figure 5G) although the anti-paralell component becomes significant during later (near-) magnetosheath-side crossings (‘G’-‘h’, ‘H’-‘i’, around ‘j’). The overall trend is consistent with the prediction.

The geomagnetic field models (Tsyganenko, 1989; Tsyganenko, 1995) predict that the footprint of the magnetic field lines encountered by Cluster at ∼0425 UT on 28 July 2006 falls at −69° LAT and ∼72° LON in GEO. FAST spacecraft fortuitously passed the conjugate southern ionosphere. Figure 6 shows the energy spectrograms of up-going (A, C) and down-going (B, D) electrons (A, B) and ions (C, D). During 0424:30-0426 UT precipitating fluxes of electrons are larger than up-going fluxes, and vice versa for ions. The difference between the ion and electron flux gives rise to the up-flowing FAC (Figure 6E) reaching −12,500 nA/m² (dB data is not available). The up-flowing FAC in the southern ionosphere corresponds to the down-flowing FAC in the northern ionosphere, as expected for the dawnward KHVs of Figure 5.

During ∼1833–2015 UT on 6 February 2016, MMS observed the dawnward magnetopause/low-latitude boundary layer to be fluctuating. We focus on 7-min (1947–1954 UT) data when MMS with its average spacecraft separation of ∼17 km was located in the boundary layer at ∼[3.6, −9.3, −5.4]R_E in GSM (Figure 7). The IMF was mostly duskward for the period. On the top of Figure 7A, blue (red) bars represent a more-magnetospheric (more-magnetosheath) region with more (less) flux of high-energy ions and electrons (Figures 7B,C), reduced (enhanced) anti-sunward flow and ion density (Figures 7D,E), and enhanced (reduced) ion temperature (Figure 7F). The magnetosphere-to-magnetosheath transitions are denoted by ‘A’, ‘B’, ‘C’, and ‘D’ with vertical solid black lines and the magnetosheath-to-magnetosphere transitions by ‘a’, ‘b’, ‘c’, and ‘d’ with vertical dashed black lines.

The total pressure (black in Figure 7G) generally shows the typical ‘H’/‘L’ trend at magnetosphere-to-magnetosheath/magnetosheath-to-magnetosphere boundaries. To test the unique signature of leading vs. trailing edges of KHVs, we determined the nominal boundary normal coordinates (LMN) using Shue et al. (1997) model: l = [0.27, −0.26, 0.93], m = [−0.70, −0.72, 0.00], and n = [0.66, −0.65, −0.38] in GSM. Table 2 lists the normal propagation velocities derived using the four-spacecraft timing analysis and the MVA-derived boundary normals in LMN together with the medium-to-minimum eigenvalue ratio.

Figure 7K shows the mn-plane projection of boundary normals (black arrows) and normal propagation velocities (magenta arrows). Both the normals and normal propagation vectors are more aligned to − —n with smaller normal-propagation speed at the trailing edges, and more aligned to k-vector (white arrow in Figure 7L) with larger speed at the leading edges. Agreement with Figure 7L confirms the identification of KHVs.

Figures 7H,I shows J caculated from the curlometer technique (decomposed into parallel and perperdicular components). Both J _z and J _|| are mostly positive, in particular, during ‘A’-‘a’ (red arrows between Figures 7H,I). J mainly points due the geomagnetic B, consistent with the downward FAC in the northern ionosphere for the dawnward KHV event.

The ion vorticity, Ω . (Figure 7J; note that there is no burst-mode ion data for this vent), although fluctuating around zero, shows— Ω . _z during ‘A’-‘a’ (red arrows between Figures 7I,J). This corresponds to the clockwise rotation of the dawnward KHV. Although a quantitative test of Eq. 3 is not available, Figures 7H–J indicates a linkage (red arrows) between the FAC and the vorticity.

Figure 8 shows the EIC (upper) and SEC (lower) at 1951 UT (corresponding to the Figure 7 event) using the data from gMAG. The footprint of the magnetic field lines encountered by MMS at 1804 UT, i.e., prior to the Figure 7 event is predicted to sit on the ionosphere at ∼70° LAT and ∼230° LON in GEO from the Tsyganenko (1989) model (mapping failed after 1804 UT on 6 February 2016). A green dot in Figure 8 denotes the MMS footprint at 1804 UT. A generally clockwise rotation of EIC around the green dot indicates downward FACs. SEC shows an azimuthally-extended band of downward FACs at/around the green dot. Considering ∼1.8 h interval between 1804 UT and 1951 UT, it is likely that the footprint of MMS at 1951 UT falls within the downward FAC band (a green arrow), where the magnitude of downward FACs ranges from— − 7800 A to − —1080 A. Again, a bead-like structure in SEC/FACs elongated in the east-west direction possibly implies the generation of the downward FACs via the dawnward KHVs (Figure 7).