Observations of Kelvin‐Helmholtz Waves in the Earth's Magnetotail Near the Lunar Orbit

Kelvin‐Helmholtz waves (KHWs), which have been widely observed at the magnetopause in the region near the Earth, play an essential role in the transport of solar wind plasma and energy into the magnetosphere under dominantly northward interplanetary magnetic field (IMF) conditions. In this study, we present simultaneous observations of KHWs under the northward IMF observed by both the Acceleration, Reconnection, Turbulence, and Electrodynamics of Moon's Interaction with the Sun (ARTEMIS) spacecraft in the Earth's magnetotail around the lunar orbit (at X ~ −50RE, Y ~ 30RE, dusk side) and the Geotail in the near‐Earth space (at X ~ −5RE, Y ~ −10RE, dawn side). The KHWs are quantitatively characterized by their dominant period, phase velocity, and wavelength, utilizing wavelet analysis and an approximation of their center‐of‐mass velocity. Our results suggest that the phase velocity and spatial scale of KHWs may increase as they propagate along the boundary layer toward the tail. Alternatively, the differences between the ARTEMIS and Geotail observations may indicate the possibility of dawn‐dusk asymmetry in the excited KHWs in this study. Our results strongly evidence the existence of the development of KHWs in terms of their wave frequency and scale size in the magnetotail and provide insight to the time evolution of KHWs along the magnetopause.


Introduction
The magnetopause, as the outer boundary of the terrestrial magnetosphere, is a critical region for entry of energy and particles from the solar wind into the magnetosphere. Usually, low-latitude magnetic reconnection along the dayside magnetopause facilitates the transportation of energy and mass during southward interplanetary magnetic field (IMF) periods (Dungey, 1962). However, the situation is less clear for periods of northward IMF, as dayside magnetic reconnection is less efficient under these conditions. Several possible mechanisms for the transport of solar wind plasma, momentum, and energy across the magnetopause under northward IMF have been proposed (e.g., Gou et al., 2016;Shi et al., 2013;Song & Russell, 1992;Terasawa et al., 1997). The Kelvin-Helmholtz instability (KHI), excited by velocity shear near the magnetopause, is one of the possible mechanisms of plasma entry (e.g., Chandrasekhar, 1961;Fairfield et al., 2000;Hasegawa et al., 2004;Liu et al., 2018). Besides the energy transport into the magnetosphere, KHI along the magnetopause also excites ultra low frequency (ULF) waves in the magnetosphere (e.g., Chen & Hasegawa, 1974;Rae et al., 2005;Wang, Thorne, et al., 2017), which play an important role in many dynamic processes, such as auroral activity and transport of radiation belt electrons. Recently, Kavosi and Raeder (2015) found that the occurrence probability of Kelvin-Helmholtz waves (KHWs) is~19% regardless of the solar wind conditions and also an important KH activity for southward IMF, when low-latitude magnetic reconnection dominates and prevents KHWs growth; hence, KHWs may be more important for the plasma entry than previously thought. Moreover, previous studies have indicated that the KHI contributes to the formation of the low-latitude boundary layer during periods of northward IMF (e.g., Mitchell et al., 1987;Taylor et al., 2008). Song and Russell (1992) predicted theoretically that high-latitude lobe reconnection would form the low-latitude boundary layer. However, Taylor et al. (2008) showed simultaneous observational evidence of high-latitude reconnection by the IMAGE spacecraft and ground-based instruments, as well as KHWs by the TC-1 spacecraft at the magnetopause, supporting the notion that not only high-latitude reconnection but also KHI makes nonnegligible contribution to the formation of the low-latitude boundary layer.
The KHWs have been studied extensively using both in situ observations and simulations. With observations by the four Cluster spacecraft, Hasegawa et al. (2004) found that KHWs can grow nonlinearly, and the resultant rolled-up vortices can mix the solar wind and magnetospheric plasmas. Takagi et al. (2006) showed that the detection of low-density and faster-than-sheath magnetospheric plasma can be taken as a marker of rolled-up vortices. This typical feature, together with quasi-periodic fluctuations (periods of 1-5 min) in the plasma parameters and magnetic fields under northward IMF, can be used as criteria to identify rolled-up vortices in single-spacecraft observations Taylor et al., 2012). The wavelength, phase velocity, and period were used to quantitatively characterize the KHWs by Foullon et al. (2008). Additionally, a statistical survey has been carried out to investigate the influence of solar wind conditions on the development of KHWs (Lin et al., 2014). The KHW period is found to be shorter for higher solar wind velocity and smaller IMF clock angle. Recently, the high-resolution plasma data obtained by the Magnetospheric Multiscale mission provides the first opportunity to directly investigate the evidence of reconnection due to KHWs, which leads to direct plasma transfer across the shear layer inside KH vortices (e.g., Eriksson, Lavraud, et al., 2016;Li et al., 2016). The time evolution of KHWs has long been discussed. Some computational studies have shown that in the nonlinear stage of the KHW evolution, small-scale vortices will be born within the larger scale parent KH vortices; thus, KH vortices are found to decay and break into turbulence (Matsumoto & Hoshino, 2004;Nakamura et al., 2004). However, other numerical simulations suggest that small vortices can merge along the magnetopause and evolve into large-scale vortices in the magnetotail (Miura, 1999). This so-called "inverse cascade" is a candidate mechanism to explain how wavelengths longer than predicted by linear theory can be observed (Belmont & Chanteur, 1989). Since the previous observations are mainly reported in the near-Earth region, the issue of how KHWs develop further downtail remains a subject of debate. Lately, Wang, Merkin, et al. (2017) reported events with transient perturbations in plasma and magnetic field parameters (B x ) by Acceleration, Reconnection, Turbulence and Electrodynamics of Moon's Interaction with the Sun (ARTEMIS) observations under steady northward IMF conditions and attributed them to KH vortices in the midtail. It is necessary to cover both terrestrial space and the magnetotail with more observations in order to provide more detailed benchmarks for theoretical modeling of nonlinear KH dynamics and plasma transport processes along the magnetopause.
In this article, we study an event observed by the ARTEMIS spacecraft in the magnetotail on 13 and 14 March 2014, and we also present simultaneous observations by the Geotail spacecraft in the region near the Earth. The paper is organized as follows. In section 2, we introduce the data used. Section 3 presents the observations and analysis results. Section 4 gives a discussion and summary.

Data
The Time History of Events and Macroscale Interactions during Substorms (THEMIS) spacecraft was launched on 17 February 2007 (Angelopoulos, 2008). The mission contains five probes (THA, THB, THC, THD, and THE). Since 2010, THB and THC were retasked to orbit the moon and have become the ARTEMIS mission, while the three remaining probes continue to orbit Earth. In this study, we use 3-s resolution data from the ARTEMIS mission. The fluxgate magnetometer (FGM) (Auster et al., 2008) onboard each probe provides magnetic field data, and thermal (5 eV-25 keV) ion data are taken from the electrostatic analyzer (ESA) (McFadden et al., 2008), while ion velocity data are taken from onboard plasma moments (MOM). The low-energy particle experiment (LEP) onboard Geotail (Mukai et al., 1994) offers plasma moment data, and the magnetic field experiment (MGF) (Kokubun et al., 1994) provides magnetic field data. In particular, we used Geotail data measured by the LEP-EA instrument in RAM-A mode only, which is more reliable in the boundary region . Time resolutions are approximately 12 and 3 s for the LEP and MGF data, respectively. The WIND spacecraft, which was 238R E away from the Earth (upstream), monitored IMF and solar wind conditions.

Observations
We have surveyed the ARTEMIS data from January 2010 to December 2014 to obtain a database of about 100 magnetopause crossings close to the lunar orbit and identified three candidate KHW events according to the criteria used by Hasegawa et al. (2006) and Taylor et al. (2012). In this study, we focus on an event on 13 and 14 March 2014, as there are simultaneous observations of three satellites (ARTEMIS P1, P2, and Geotail) in this event, which can help us understand the evolution of KHW along the magnetopause.  Figure 2. From 19:00 UT to 21:10 UT on 13 March, the ARTEMIS spacecraft was located in the magnetosheath, where the density was larger than 1 cm À3 (Figure 2d) and the ion energy ranges from 100 eV to 1 keV (Figures 2j and 2k). From 21:10 UT on 13 March to about 00:48 UT on 14 March, P1 and P2 entered a magnetospheric boundary layer region, which is characterized by lower density (Figure 2d), higher temperature (Figure 2e), and obvious quasi-periodic fluctuations of plasma parameters and magnetic fields (Figures 2a-2c). Meanwhile, the ion energy flux spectrogram shown in Figures 2j and 2k indicates a mixture of cold-dense magnetosheath-like plasma and hot-tenuous magnetosphere-like plasma. Around 00:48 UT on 14 March, ARTEMIS-P1 moved into the magnetosphere with a rapid decrease in the number density and velocity, and the probe remained in the magnetosphere for the remainder of the interval. ARTEMIS-P2 was 2-3R E earthward from P1 (see Figure 1) and entered the magnetosphere~15 min earlier than P1. The total pressure, as shown in Figure 2i, was not rigidly constant around the magnetopause. In the initial stage of the KHI, there would be no significant total pressure fluctuations around the magnetopause. As the KHI grows to a nonlinear phase and generates rolled-up vortices, the centrifugal forces push out plasmas from the center of the vortices, and thus, there is a minimum in the total pressure at the central part of the vortices and a maximum at the edge (e.g., Hasegawa, 2012). The total pressure variations mean that some of the vortices encountered were rolled up, and its observed level is consistent with observations of the KHWs (e.g., Hasegawa, 2012;Hasegawa et al., 2009).

Data Analysis
Here we make a close inspection of the interval of interest when the ARTEMIS and Geotail spacecraft detected clear KHW signals. As mentioned previously, the low-density and high-velocity feature seen in the data suggests the existence of rolled-up vortices . Following the method of Hasegawa et al.  Figures 4a and 4b are for ARTEMIS-P1 and Geotail, respectively. The magenta region is quantitatively defined using the method proposed by Taylor et al. (2012). The value N_MSH is defined as 80% of the maximum density during this period, and the upper threshold for density is set to be 70% of N_MSH. The threshold for ion velocity is obtained from the averaged velocity for all points with density higher than N_MSH. It should be noted that the plasma from the magnetospheric side of the velocity shear layer where the KH vortex is formed should have the faster-than-sheath signature. In order to provide a more stringent estimate of the magnetospheric plasma, a stricter threshold for density of 50% of N_MSH is also used, and the corresponding points are marked with red. Because the Geotail data have lower time resolution (12 s) and measured by the LEP-EA instrument in RAM-A mode only, the number of data points satisfying the criteria is only a few in Figure 4b but is acceptable. The detection of plasma with low density and high speed suggests that a vortex or "rolled up" structure is generated near the boundary.
Minimum variance analysis (MVA) on mass flux (MVA_NV) is used to further confirm the possibility of a vortex structure (Sonnerup & Scheible, 1998). The KHW properties are characterized by their frequency and phase velocity. The wavelength can be estimated by multiplying the phase velocity and wave period. Figure 6 shows the ARTEMIS-P1 data and wavelet spectrum analysis of plasma parameters and the magnetic field for the interval 23:35 UT, 13 March, to 00:13 UT, 14 March. The fluctuations have noticeable power at a frequency of~2 mHz (500 s) during the whole wave interval, and another peak in wave power is observed at a frequency of~4 mHz (250 s) from 23:50 UT on 13 March to 00:05 UT on 14 March. Figure 7 is in the same format as Figure 6, showing the magnetic field measured by Geotail and their wavelet spectra for the interval 23:35 UT, 13 March, to 00:05 UT, 14 March. The LEP-EA instrument on Geotail changed mode (RAM-A or RAM-B) multiple times during the interval of interest, so the wavelet analysis was performed only for the magnetic field data. Geotail observed waves with peaks in spectral power at two frequencies of 3 (333 s) and~6 mHz (167 s), as shown in Figure 7. The KHWs with two frequency ranges have been reported in previous observations (Hasegawa et al., 2009). For Geotail, the 6-mHz frequency is more common and thus could be the fundamental mode , and the 3-mHz mode may be a signature of inverse cascade. For ARTEMIS, the 2-mHz mode may be a signature of inverse-cascade or vortex pairing. This phenomenon is further discussed in section 4.
According to the linear MHD theory for KHWs on a zero-thickness surface, the phase velocity is equal to the center-of-mass velocity ("1" and "2" refer to the magnetosphere and magnetosheath, respectively; Chandrasekhar, 1961). However, in a layer with finite thickness, the phase velocity is in between the average velocity and center-of-mass velocity (Hasegawa et al., 2009). Considering the situation where the velocity of the background magnetosphere and magnetosheath are basically unavailable, we chose the center-of-mass velocity to estimate the phase velocity in this event. This approximation is not accurate when spacecraft trajectories deviate from the center of the vortex. The period, phase velocity, and spatial scale calculated from ARTEMIS and Geotail data are listed in Table 1. Comparing the results gained via data from the two satellites at different locations, both the phase velocity and the spatial scale of the KHWs tend to increase with distance from the subsolar point along the magnetopause. Figure 8 shows the magnetic field and plasma parameters observed by ARTEMIS-P1 during the interval of interest, with points that correspond to the magenta and red region in Figure 4a. Many of the red points are in higher-temperature regions, which suggests that they are from the magnetospheric side, suggesting the presence of rolled-up vortices. These points are mostly concentrated from 23:50 UT, 13 March, to 00:05 UT, 14 March, suggesting that the low-density and high-velocity flows correspond to waves with higher frequency, as shown in Figure 6. Therefore, 4 mHz is chosen as the dominant mode frequency of the KHWs recorded by ARTEMIS-P1, and the associated dominant wavelength of the KHWs is about 11.1R E in the tail.

Discussion and Summary
A global MHD simulation (simulation method, see Li et al., 2013) was performed (see Animation S1 in the supporting information) to simulate the magnetospheric response under the average solar wind conditions (V sw = 400 km/s, IMF B z = 5nT, N = 5 cm À3 ), which were observed by the WIND spacecraft as shown in Figure 3. Figure 9 shows the simulated X component of the velocity (V x ) in the equatorial plane. The KHWs are formed from the dayside magnetopause to magnetotail, and the continuous development of waves illustrates that the wavelength grows with increasing distance along the magnetopause from the subsolar point. The simulation results support our points obtained from two-point observations, and the increasing KH wavelength has been presented in previous simulations (e.g., Li et al., 2012). If Geotail and ARTEMIS saw the same KHWs, the frequency should be the same or be different by a factor of 2, 4, 8, etc. This is because even if the wavelength or propagation speed are different at the two locations, the oscillations should occur at the same frequency. However, comparison between Figures 6 and 7 shows that this is not the case. It should be noted that in this study, the ARTEMIS and Geotail spacecraft are on different sides of the magnetotail, as shown in Figure 1. It is widely accepted that KHWs can be driven at both flanks (Henry et al., 2017), but a dawn-dusk asymmetry in KHW may occur if ion effects are taken into account. The simulations of Huba (1996), for example, suggested a dawn-dusk asymmetry in the evolution of KHWs in the boundary layer. Hasegawa et al. (2006) reported a nearly symmetric distribution of rolled up KH vortices at both flanks. Taylor et al. (2012) suggested that a dawn-dusk asymmetry in the rolled up KH vortices exists on the dayside but disappears antisunward, Figure 7. Magnetic data recorded by Geotail and their wavelet spectra for interval 23:35 UT, 13 March, to 00:05 UT, 14 March; the format is the same as in Figure 5. The black dashed line marks the frequency of 3 and 6 mHz, respectively. Journal of Geophysical Research: Space Physics comparing TC1 events with Geotail events from Hasegawa et al. (2006). Observations from Nishino et al. (2011) showed that KHI vortices grow nearly simultaneously and symmetrically on both flanks, when Geotail and Cluster were located on the opposite sides of the near-Earth magnetotail at almost the same latitude. A dawn-dusk asymmetry was also observed by MESSENGER (e.g., Gershman et al., 2015;Sundberg et al., 2012) in Mercury's magnetosphere and was attributed to spatial scales comparable with an ion gyroradius. Finite Larmor radius effects should affect the instability and lead to asymmetries in the growth rates on the dawn and the dusk flanks (Sundberg et al., 2012). However, the effects of finite Larmor radius are less observable at the Earth since the thickness of the magnetopause current layer is many times the magnetosheath ion Larmor radius (Boardsen et al., 2010). Although the KH wave periods on the Earth's magnetosphere (hundreds of seconds) are many times the periods in MESSENGER KH observations (tens of seconds), Li et al. (2012) proposed that the KH wave periods in Earth's and Mercury's magnetopauses are in a similar band if the different physical dimensions (size, plasma, and magnetic field) of the planetary magnetospheres are taken into account. The simulation results in Figure 9 show rough symmetry of KHWs on the dawn and dusk sides. However, the event shown by Nishino et al. (2011) is for somewhat different solar wind conditions, and the simulation does not include many effects (kinetic, magnetosphere-ionosphere coupling, etc.), whose neglect may not be justifiable for all conditions. We thus cannot completely exclude the presence of dawn-dusk asymmetry in KH vortices in this study. There are two possibilities that may explain the difference in the two-point observations. One is the development of the KHWs as they propagate toward the tail, in consideration of tailward distance between ARTEMIS-P1 and Geotail. The other is the effect of dawn-dusk asymmetry in the KHWs, as the two spacecrafts are on the different sides of the magnetotail.
In the last section, we show that KHWs with two frequency ranges are observed by both ARTEMIS-P1 and Geotail. This suggests that there may be two spatial scales associated with the KH vortices. Figure 10 shows P1 observations in detail, and at the bottom is shown a  schematic of rolled-up KH vortices along the magnetopause in the tail. During this interval, P1 encountered the magnetosphere and magnetosheath repeatedly, and the 13 boundaries are characterized by positive (negative) B x and low (high) density, as marked with vertical black lines and red numbers in Figure 10a. Figure 10i shows a schematic of the KH vortices observed by P1; the upper and lower sides refer to the magnetosphere and magnetosheath, respectively, and a darker color indicates that the density is higher. A minimum variance analysis of the magnetic field (Boardsen et al., 2010) is applied to each boundary crossing to obtain the boundary normal direction, which is then projected onto the GSM x-y plane. These  Figure 10i by short gray lines. Figure 10h compares the unfiltered density observed by P1 (black curve) with two band-pass filtered density time series (red and blue curves), each with pass-bands of 1 mHz, centered at 2 and 4 mHz, respectively (corresponding to the two frequency ranges obtained from wavelet analysis in Figure 6).
From 23:35 UT to 23:48 UT on 13 March, the unfiltered density variation is consistent with the 2-mHz filtered data, and a 4-mHz component appears to become more significant for times after 23:50 UT on 13 March. This suggests that P1 may have passed a larger spatial structure at first, and then the satellite moved closer to the center of the vortex and encountered substructures. We provide two possible explanations for the formation mechanism of substructures within the larger vortex. First, the feature could be associated with the time evolution of the KHWs, as they develop along the magnetopause and evolve into the nonlinear stage where rolled-up vortices appear. At this stage of the KHW evolution process, the wavelength would increase; however, the waveform becomes multivalued. In other words, the smaller scale is observed because of strongly rolled-up vortices due to the nonlinear evolution of KHWs. The second possibility is that smaller structures associated with higher-frequency vortices become coupled and then form a larger one as the KHWs propagate tailward, supporting the so-called "inverse cascade" mechanism (Belmont & Chanteur, 1989).
In this paper, we present simultaneous observations of KHWs by the ARTEMIS spacecraft (P1 and P2) and Geotail at different locations when the IMF is dominantly northward and provide observational evidence for solar wind-magnetosphere interactions in the tail region near the lunar orbit. The KHWs are studied by using quantitative methods to calculate the dominant period, phase velocity, and wavelength. By a comparison of the results between these two satellites, we can infer that either (i) the phase velocity and the spatial scale of the KHWs tend to increase along the boundary layer when the waves propagate tailward or (ii) the difference between dawnside and duskside observations is due to the influence of dawn-dusk asymmetry. The KHWs with two frequency ranges are observed by these satellites, indicating multiscale development of KHWs in the nonlinear stage. The time evolution of KHWs and processes within rolled-up vortices that facilitate plasma transport and mixing still remain unclear. Therefore, more observational studies to clarify these unsolved questions will be required.