Prolonged Kelvin–Helmholtz Waves at Dawn and Dusk Flank Magnetopause: Simultaneous Observations by MMS and THEMIS

The Kelvin–Helmholtz (K-H) waves predominantly excited at the Earth’s low-latitude magnetopause were suggested to be dawn–dusk asymmetric. We report a prolonged simultaneous observations of the K-H waves on the dawn and dusk magnetopause by Magnetospheric Multiscale (MMS) and THEMIS-A (THA) spacecraft, respectively. The quasi-periodic K-H waves on both flanks have unambiguous low-density and high-speed patterns. The wave periods vary gradually on both flanks, with similar average periods (303 ± 107 s for MMS and 266 ± 102 s for THA). The lag time between the variations of the wave periods is close to the wave propagation time from THA to MMS, which suggests that the K-H waves generate and propagate quasi-symmetrically on both flanks. Larger local magnetic shear angles are observed on the trailing edges by MMS than by THA, which is probably due to the strong magnetic field distortion during the tailward propagation. The increased magnetic shear may excite magnetic reconnection, thus contributing to the formation of the low-latitude boundary layer.


Introduction
The Kelvin-Helmholtz (K-H) instability could be excited at a plasma boundary where large velocity shear exists. The K-H waves have been widely studied in various plasma environments, including solar corona (Foullon et al. 2011), magnetosphere (e.g., Hasegawa et al. 2004), and potentially in heliospheric boundaries (Borovikov & Pogorelov 2014). The unstable condition of the K-H instability can be satisfied at the flank magnetopause where the fast-moving anti-sunward magnetosheath flow meets the quasi-stagnant magnetosphere plasma (Chandrasekahar 1961). The K-H instability is considered to be an important mechanism for the transport of solar wind particles and energy into the Earth's magnetosphere (Hasegawa et al. 2004(Hasegawa et al. , 2009Fairfield et al. 2007). Both numerical simulations (Nykyri & Otto 2001;Nakamura & Fujimoto 2005;Nakamura et al. 2013Nakamura et al. , 2017 and in situ observations (Hasegawa et al. 2009;Eriksson et al. 2016;Li et al. 2016) have shown that the magnetic reconnection induced by the K-H waves breaks the frozen-in condition of plasma during the large-scale evolution of the waves, allowing plasma transport from the solar wind into the magnetosphere.
The K-H unstable condition is more likely to be satisfied if plasma on the boundary has large velocity shear and magnetic fields are perpendicular to propagating direction (Chandrasekahar 1961). Previous works have shown that the K-H waves are more likely to be excited under northward interplanetary magnetic field (IMF; Miura 1995;Farrugia et al. 1998;) and fast solar wind speed (Otto & Fairfield 2000;Li et al. 2013;Kavosi & Raeder 2015), although a few K-H events during southward IMF have been reported (Hwang et al. 2011;Yan et al. 2014). The relationships between solar wind conditions and properties of the K-H waves have also been extensively studied (e.g., Farrugia et al. 1998Farrugia et al. , 2003Li et al. 2013;Lin et al. 2014). A comprehensive statistical study by Lin et al. (2014) shows that periods of the K-H waves decrease with solar wind speed, while they increase with the clock angle of IMF. During the convective propagation of the K-H waves from dayside to nightside magnetopause, the wavelengths and the phase speeds gradually increase (Li et al. 2012;Lin et al. 2014), and the waves themselves develop from a quasi-linear to nonlinear stage, generating rolled-up vortices around the leading edges of the K-H waves (Hasegawa et al. 2004Takagi et al. 2006) that distort the local magnetic fields and create conditions for magnetic reconnection (e.g., Nykyri & Otto 2001).
The transport of magnetosheath plasma contributes to the formation of the cold dense plasma sheet (CDPS) adjacent to the magnetopause boundary layer under prolonged northward IMF (e.g., Terasawa et al. 1997). The plasmas of magnetosheath origin inside the CDPS are denser and hotter by 30%-40% at the dawnside than the duskside (Wing et al. 2005). The dawn-dusk asymmetry of ion temperatures of magnetosheath plasma is inadequate for the asymmetric CDPS (Dimmock et al. 2015), which suggests that the asymmetry of the K-H waves may contribute to the asymmetric CDPS. The distributions of the K-H waves have been extensively studied. For example, Taylor et al. (2012) pointed out that about 62% of the K-H waves (21 out of 34) are found on the duskside, while Lin et al. (2014) found no clear asymmetry in a collection of 56 K-H waves. Moreover, a recent study by Henry et al. (2017) has shown that the K-H waves have preference on the dawnside during Parker-spiral IMF. The mechanism is suggested to be the asymmetric K-H growth rate at both flanks (Nykyri 2013).
The multispacecraft observations can help us understand the K-H waves simultaneously propagating at both flanks (Nishino et al. 2011;Ling et al. 2018). In this paper, we report a case study of the prolonged K-H waves observed simultaneously on the dawn and dusk magnetopause by the Magnetospheric Multiscale (MMS; Burch et al. 2015) and Time History of Events and Macroscale Interactions during Substorms (THE-MIS; Angelopoulos 2008) spacecraft. The symmetry and correlations of the K-H waves on both flanks are analyzed in detail.

Instruments
The in situ measurements of plasma and magnetic field from MMS and THEMIS spacecraft are used in this paper. For MMS data, we use the fast-mode (4.5 s) and burst-mode (0.15 s) ion data from the Fast Plasma Investigation (Pollock et al. 2016) and the survey-mode (16 Hz) magnetic field data from Fluxgate Magnetometer (FGM; Torbert et al. 2014). For THEMIS data, we use ion data from the Electrostatic Analyzer (McFadden et al. 2008) and magnetic field data from the FGM (Auster et al. 2008), both sampled at the spin resolution (3 s). The vectors are presented in the Geocentric Solar Magnetospheric (GSM) coordinate system. High-resolution (1 minute) OMNI data are used to get the solar wind and IMF parameters.
During the time interval of interest, MMS is located at the dawnside magnetopause. The average distance of four MMS spacecraft is about 50 km, which is much smaller than the typical wavelengths of the K-H waves (several R E ; see Lin et al. 2014). Thus, the MMS observations of this event are shown representatively by MMS1 data. Meanwhile, three of the five THEMIS spacecraft (A, D, and E) are located at the duskside magnetopause, in which THEMIS-A (THA) spacecraft has the longest simultaneous observation with MMS. The following sections provide detailed analysis of the K-H waves by MMS and THA.

K-H Waves Observed by MMS and THA
The K-H waves at the Earth's magnetopause are characterized by quasi-periodic fluctuations of the plasma and magnetic field parameters. We adopt the empirical criteria proposed by Hasegawa et al. (2006) to identify the K-H wave events from the in situ measurements: (1) The plasma and the magnetic field on the magnetosheath side of the magnetopause and/or that in the upstream solar wind are quasi-steady, and the orientation of the magnetic field is northward throughout the time interval of interest.
(2) The fluctuations of the plasma and magnetic field parameters during the magnetopause boundary crossings are quasi-periodic with periods between 1 and 5 minutes.
(3) The low-density and high-speed (LDHS) pattern  has sufficient data points with density less than half of that on the magnetosheath side and anti-sunward speed higher than that of the magnetosheath plasma.
Figures 1(a)-(d) show the solar wind condition during the interval of 04:00-08:00 UT on 2017 May 29. The solar wind and IMF conditions are generally quasi-steady during this time. The average solar wind speed is 346 km s −1 . The dynamic pressure is around 2.8 nPa with a few pulses. The IMF has a strong B y component of about 10 nT, while the B z component holds northward most of the time, with a varying amplitude from −1 to 6 nT. The IMF clock angle f=tan −1 B y /B z varies between 30°and 102°, with an average clock angle of 70°.
The trajectories of MMS and THA spacecraft are shown in Figures 1(e)-(f), with magnetopause position given by Shue et al. (1997). MMS crossed the dawnside flank magnetopause at [−15.0, −16.3, 3.5] R E , while THA crossed the duskside magnetopause at [3.9, 11.7, −3.7] R E . Plasma and magnetic field data from MMS and THA spacecraft are shown in Figures 1(g)-(r). The ion bulk velocity and magnetic field data are present in the local LMN coordinates, where N is the magnetopause normal direction determined by the minimum variance direction of the magnetic field, M is the cross product of N and the maximum variance direction of the ion velocity, and L completes the right-handed orthogonal coordinate system, which is approximately opposite to the magnetosheath flow direction. Quasi-periodic fluctuations in the ion moments and magnetic fields are observed by MMS and THA during the magnetopause boundary crossings from the hot and tenuous magnetosphere to the cold and dense magnetosheath. The typical periods are roughly estimated to be 3-5 minutes. Detailed analysis of wave periods will be presented in the following section. The time interval of quasi-periodic fluctuations by MMS is mainly between 04:30 UT and 07:00 UT, and between 05:00 UT and 07:30 UT for THA observations. Thus, the time intervals of the quasi-periodic fluctuations observed by MMS and THA overlap for about 2 hr. The main differences between the dawn and dusk magnetopause crossings are: (1) The L direction has a large Z component on the duskside, which means that the magnetosheath has a southward flow. This is consistent with the fact that the magnetosheath environments are asymmetric under B y -dominant IMF . (2) The V L component of the magnetosheath ion flow is larger at the dayside than the nightside. (3) The variations of the total magnetic field amplitude is stronger at the dawnside.
Figures 1(s) and (t) show the scatter plots of N i versus V L for the waves on both flanks. In each panel, the top left and the bottom right parts correspond to the magnetosphere and magnetosheath, respectively, while the data points between them correspond to the boundary layer. There are sufficient data points with density less than half of that on the magnetosheath side and speed higher than that of the magnetosheath plasma. These LDHS patterns are consistent with the patterns of the rolled-up vortices shown by Takagi et al. (2006) and Hasegawa et al. (2006), except that the V L components of the LDHS plasma does not significantly exceed the magnetosheath flow speed. This is probably because the K-H waves have not reached the highly nonlinear stage at the MMS and THA locations due to relatively slow solar wind speed during this event.
To sum up, the quasi-periodic fluctuations observed by MMS and THA satisfy the criteria of the K-H waves proposed by Hasegawa et al. (2006). The following sections will focus on detailed analysis of the boundary conditions and properties of the waves for both events.

Magnetosphere-Magnetosheath Boundary Conditions of the K-H Waves
To obtain the magnetosphere-magnetosheath boundary conditions for the K-H waves on both flanks, we adopt a transition parameter (Hapgood & Bryant 1992)    The boundary conditions are given in Table 1 using the ion and magnetic field data within the magnetosphere and magnetosheath domains. In addition, the angles between B msp , B msh , and velocity shear ΔV=V msh −V msp are given in Table 1. In general, the magnetic shear angles between the magnetosphere and the magnetosheath are 21°. 7 and 30°.1 for the K-H waves by MMS and THA, respectively.

Properties of K-H Waves
The shapes of the K-H waves on the magnetopause are mostly nonsinusoidal (e.g., Hasegawa et al. 2004;Hwang et al. 2011). The leading (anti-sunward) edges of the magnetopause boundary have steepened shapes, and are widely suggested to be rolled-up during the propagation of the K-H waves (e.g., Nykyri & Otto 2001;Nakamura & Fujimoto 2005). The trailing (sunward) edges are mostly sharp due to the centrifugal force and compression from the magnetosheath flow (Hasegawa et al. 2009). The trailing edges of the K-H waves by MMS and THA are highlighted by the black vertical lines in Figures 3(a)-(c) and Figures 3(e)-(g). These trailing edges are identified by sharp increases in N i and V L , as well as the increases of total (ion plus magnetic) pressure. Altogether 30±3 trailing edges are found within ∼2.5 hr in the K-H wave by MMS, and 24±2 trailing edges are found within ∼1.7 hr in the K-H wave by THA. The uncertainties of the numbers come from some ambiguous boundary crossings.
The periods of the K-H waves are estimated by the average periods of all the identified trailing edges, which are 303±107 s and 266±102 s for the K-H waves by MMS and THA, respectively. Thus, the K-H waves on both flanks have similar wave periods despite their different local times. Figures 3(d) and (h) show the wavelet power spectra of total pressure, which denote that the wave periods vary gradually with time. This can also be implied by the variation of the partial wave periods marked in Figures 3(c) and (g), which are estimated by the average periods of every five adjacent trailing edges. A cross-correlation analysis is performed on the partial wave periods of the trailing edges by MMS and THA, showing that the correlation coefficient reaches the maximum when the lag time between the THA and MMS data is ∼18 minutes. Considering the phase speeds of the K-H waves in the L direction (∼166 km s −1 for MMS and ∼171 km s −1 for THA; see Table 1) and the distance between the location of MMS and the mirror point of the location of THA along the magnetopause (∼22 R E ), it takes about 13-14 minutes for the K-H waves to propagate from THA to MMS. The consistency between the lag time and the propagation time suggests that the K-H waves propagate quasi-symmetrically on both flanks. A statistical study by Henry et al. (2017) shows a dawn-dusk asymmetry in the occurrence rates of the K-H instability under Parker-spiral IMF, which can be explained by a larger growth rate at the dawnside (Nykyri 2013). However, K-H waves can be excited simultaneously at both flanks with different growth rates, as is presented in Nykyri's paper and in this case. According to Lin et al. (2014), the wavelengths of the K-H waves are associated with IMF clock angles. It is suggested that the variations of the wavelengths might be modulated simultaneously at the symmetric locations on both flanks by the varying IMF, and roughly preserved during the propagation of the waves due to the dominated plasma dynamic pressure at the low-latitude boundary layer.
The nonlinear evolution of the K-H waves can compress the trailing edges down to ion inertial scale and distort the magnetic fields on both sides of the magnetopause to increase the local magnetic shear, which provides suitable conditions for magnetic reconnection. The recent MMS results have discovered the evidence of magnetic reconnection on the trailing edges of the K-H waves (e.g., Eriksson et al. 2016;Li et al. 2016). For the K-H waves in this study, the local magnetic shear angles of the trailing edges are analyzed and compared to the large-scale background magnetic shear on both sides.  Notes. a a b , á ñdenotes the angle between vectors a and b. b ΔV=V msh −V msp denotes the velocity shear between the magnetosheath and the magnetosphere.
The local magnetic shear angles of the trailing edges shown in Figures 4(a) and (b) are 53°. 7 and 25°.2, respectively.
Figures 4(c) and (d) show the local magnetic shear angles for the identified trailing edges. The average local magnetic shear angle of trailing edges observed by MMS is 37°.5, which is about 66% larger than the background magnetic shear (22°.5). Comparatively, the average local magnetic shear angle of trailing edges observed by THA is 25°. 9, which is close to the background magnetic shear (30°.1). The local shear angles of MMS trailing edges distribute in a wider range than those of THA, in which the largest local shear angle is 98°. The histograms of all the local magnetic shear angles are shown in Figures 4(e) and (f). The results show a trend of stronger magnetic field distortion on the trailing edges during the tailward propagation of the K-H waves. The large magnetic shears on the nightside trailing edges favor the magnetic reconnection, which may transport significant solar wind plasma into the magnetosphere. Some of the MMS trailing edges have fast ion flows, which could be the reconnection outflow (Li et al. 2016). Further studies will analyze the burstmode data to confirm this.

Summary
We report the first simultaneous observations of the Kelvin-Helmholtz waves by MMS and THA on the dawn and dusk flank magnetopause on 2017 May 29. The solar wind condition is quasi-steady. The solar wind speed is relatively low (∼346 km s −1 ), and the IMF has a northward B z and a dominant B y component. The dawnside K-H waves are observed by MMS at [−15.0, −16.3, 3.5] R E , while the duskside K-H waves are observed by THEMIS-A at [3.9, 11.7, −3.7] R E . The average periods are 303±107 s for the MMS K-H wave and 266±102 s for the THA K-H wave. The wave periods of the MMS and THA K-H waves vary gradually on both the dawn and dusk flanks. The variations of wave periods have good correlation when the lag time from THA to MMS meets the time for propagation of the K-H waves, which  suggests that the evolutions of the K-H waves and the variations of their wave periods are quasi-symmetric on both flanks. The large-scale magnetic shear angles between the magnetosphere and the magnetosheath are 22°.5 for the MMS K-H wave and 30°.1 for the THA K-H wave. On the trailing edges, MMS observed an average local magnetic shear 66% larger than the background magnetic shear, with a maximum local magnetic shear of 98°, while THA observed local magnetic shear angles similar to the background magnetic shear. The tailward propagation of the K-H waves distorts the magnetic field on the trailing edges. The significantly increased local magnetic shear favors the magnetic reconnection and plasma transport from the magnetosheath into the magnetosphere.