Lower Hybrid Waves at the Magnetosheath Separatrix Region

Lower hybrid waves are investigated at the magnetosheath separatrix region in asymmetric guide field reconnection by using the Magnetospheric Multiscale (MMS) mission. Three of the four MMS spacecraft observe clear wave activities around the lower hybrid frequency across the magnetosheath separatrix, where a density gradient is present. The observed waves are consistent with generation by the lower hybrid drift instability. The characteristic properties of these waves include the following: (1) the waves propagate toward the x‐line in the spacecraft frame due to the large out‐of‐plane magnetic field, which is in the same direction of the diamagnetic drift of the x‐line; (2) the wave potential is about 20% of the electron temperature. These drift waves effectively produce cross‐field particle diffusion, enabling the transport of magnetosheath electrons into the exhaust region. At last, we suggest that the lower hybrid waves at the magnetosheath separatrix region represent some unique features of asymmetric guide field reconnection, which is different from that widely observed at the magnetospheric side of magnetopause reconnection.


Introduction
Magnetic reconnection is a fundamental process in plasma physics, which rapidly converts the magnetic field energy into plasma energy. At Earth's magnetopause, reconnection is generally asymmetric, where the magnetosheath plasma (with a weaker magnetic field and a larger plasma density) reconnects with the magnetospheric plasma (with a stronger magnetic field and a smaller plasma density), and thus, the reconnection differs significantly from symmetric reconnection (e.g., the magnetotail reconnection). The quadrupolar Hall magnetic field structure can become more bipolar, and the bipolar Hall electric field tends to become monopolar (Pritchett, 2008). The stagnation point is shifted to the low-density magnetospheric side of the x-line (Cassak & Shay, 2007). Electron trapping and associated parallel heating becomes asymmetric, primarily occurring on the lower density magnetospheric inflow region (Egedal et al., 2011;Graham et al., 2016). Plasma waves (including large-amplitude parallel electrostatic waves, whistler mode waves, and lower hybrid [LH] waves) are identified most typically on the magnetospheric side Wilder et al., 2019). In particular, the frequently observed LH waves (Bale et al., 2002;Graham et al., 2016Graham et al., , 2017Graham et al., , 2019Khotyaintsev et al., 2016) are taken as a basic feature of 3-D asymmetric reconnection (Le et al., 2017;Price et al., 2016;Roytershteyn et al., 2012). The frequency of LH waves is found near the LH frequency (f LH ≈ (f ce f ci ) 1/2 , where f ce and f ci are electron and ion cyclotron frequency). In this frequency range electrons remain approximately frozen in, while ions are almost unmagnetized. These waves are driven by lower hybrid drift instability (LHDI) at the steep density gradient or by the modified two-stream instability due to the entry of the finite gyroradius magnetosheath ions into the magnetosphere, and the energy source of instability is the cross-field current at the magnetopause . In reconnection, the role of LH waves remains an outstanding question (Bale et al., 2002;Mozer et al., 2011), but they are suggested to play an important role, which can contribute to anomalous resistivity and anomalous viscosity (Davidson & Gladd, 1975;Le et al., 2017;Price et al., 2016Price et al., , 2017Silin et al., 2005;Zhou et al., 2012), diffusive cross-field particle transport from the magnetosheath to the magnetospheric side (Graham et al., 2017;Treumann et al., 1991;Vaivads et al., 2004), and electron heating (Cairns & McMillan, 2005;Zhou et al., 2014).
When a finite guide field appears in asymmetric reconnection, the reconnection structure can be further modified. The reconnection electric field has a component parallel to the magnetic field in the vicinity of the x-line, which leads to strong electron beams. These beams are unstable for electron streaming instabilities, contributing to significant electron thermalization Khotyaintsev et al., 2020). The guide field can also cause the diamagnetic drift of the x-line  and affect the shape of electron crescent distributions (Bessho et al., 2019). LH waves at the low-density magnetospheric side are reported  as the cases revealed in other reconnection events (Le et al., 2018), and due to the presence of the guide field, the propagation of these waves can have a component along the outflow direction (Zhou et al., 2018). The imposing of the positive/negative bipolar Hall magnetic field to the guide field can enhance/reduce the out-of-plane magnetic field in the two different exhausts, leading to asymmetry of the fields and plasma in both reconnection exhausts (Mozer et al., 2008), and at the magnetosheath separatrix of the exhaust with enhanced out-of-plane magnetic fields, a density gradient is revealed due to the force balance ( Figure 5 in Mozer et al., 2008). In this study, we find clear evidence of LH waves at such a sharp density gradient across the magnetosheath separatrix in asymmetric guide field reconnection from Magnetospheric Multiscale (MMS) mission . The properties of these waves are further presented and compared with that at the magnetospheric side, indicating some unique features of asymmetric guide field reconnection.

Observations
We present an outbound magnetopause crossing of MMS near the subsolar point on 21 December 2017. The four MMS spacecraft are located at [10.39, -1.36, 1.57] Earth radii in geocentric solar ecliptic (GSE) coordinates, with an average separation of ∼30 km or 26.5 electron inertial lengths (d e ) considering a typical magnetosheath number density of ∼22 cm −3 . We use magnetic field data from the fluxgate magnetometer (Russell et al., 2016), electric field data from the electric field double probes Lindqvist et al., 2016), and particle data from the fast plasma investigation (Pollock et al., 2016). This event has been used to investigate the electron two-stream instability in the reconnection exhaust (Tang et al., 2020). During this outbound magnetopause crossing, we find that different MMS spacecraft observe significantly different plasma and magnetic field (Figures 1f-1i  A zoom-in of the MMS observations at the magnetosheath separatrix is presented in Figure 2, where the variation of the magnetic field and the plasma density is identified (Figures 2a1 and 2b1). An electric field normal to this boundary (E N⟂ ) is also revealed, with a magnitude of 5-10 mV m −1 (Figure 2d1), and similar electric field structures have also been reported in another reconnection separatrix (Yu et al., 2019;Zhou et al., 2019). In this event, this electric field is primarily balanced by the electron drift motion, suggesting that electrons are almost magnetized. The power spectral density of the electric field ( Figure 2f1) and magnetic field (Figure 2g1) shows enhanced perturbations near the LH frequency, which are electric perturbations perpendicular to the local magnetic field (Figure 2h1) and the parallel magnetic field perturbations (Figure 2i1). These observational features are consistent with LH waves. Similar density variations, E N⟂ structure and wave perturbations have been found at MMS 1 and MMS 3, but they are not obvious at MMS 2.
At LH time scales, electrons remain approximately frozen in, while ions are almost unmagnetized. If further assuming the current density perturbation J = −en e v e , the wave potential Φ B of the LH waves can be calculated from B || and the local plasma parameters (Norgren et al., 2012), using The wave potential peaks at ∼5-8 V as shown in Figures 2j1-2j3. The phase velocity v ph of LH waves is found by fitting Φ E = ∫ Edt ⋅ v ph to Φ B . The best fitted Φ E agrees well with Φ B , with a correlation coefficient (C Φ ) larger than 0.8 as listed in Table 1. The estimated phase speed is about 50-90 km s −1 in the spacecraft frame, Recently, a new single-spacecraft method has been developed to determine LH wave properties , which is written as where d e is the electron inertial length and W e ( ) and W B ( ) are electron kinetic energy and magnetic field energy computed in the frequency domain. This method requires the sample rate of electron moments to the LH frequency, and in this study, n e and v e, ⟂ are estimated from the spacecraft potential and the measured electric field ( v e,⟂ = E × B∕|B| 2 ). The top panels of Figure 3 show the dispersion relation estimated from Equation 2. The characteristic frequency, wave number, and wave phase speed (v ph ) of LH waves estimated by different MMS spacecraft are indicated by the maximum W E /W E, max , and the values can be found in Table 1. Overall, the computed wave properties are consistent the estimation from Norgren et al. (2012) except that the wave number is larger.   To investigate the instability of the observed waves, a local dispersion equation of LHDI which includes the finite plasma beta ( ) effect in the ion frame is considered (Davidson et al., 1977) where Z ′ is the derivative of the plasma dispersion function, pi, e are the ion and electron plasma frequencies, v i, e are the ion and electron thermal speeds, Ω ce is the electron cyclotron frequency, V E is the electron drift speed due to the electric field, and V de is the electron diamagnetic speed. The effect of the electron density gradient is included through V de (V de = −B × ∇ ⋅ P e ∕(B 2 n e e)). Figures 3d-3f show the predicted wave frequency, growth rate, and phase speed as a function of k ⟂ e . We use B = 22 nT, n e = 13 cm −3 , T e = 32 eV, T i = 500 eV, and i = 5.4, based on the observed plasma conditions. Due to the variation of the observed electron speeds at different spacecraft (Figures 2e1-2e3), two groups of V E and V de are considered, which are (1) V de = 20 km s −1 , while V E = 120 km s −1 (pink), 150 km s −1 (orange), and 200 km s −1 (magenta) and (2) V de = 50 km s −1 , while V E = 150 km s −1 (cyan), 200 km s −1 (purple), and 250 km s −1 (green). For comparison, we shift the waves into the ion rest frame as shown inside the parentheses in Table 1, and we find that the ion motion is relatively small, suggesting the ion E × B drift is approximately balanced by the ion diamagnetic drift . The LH wave properties estimated by different methods (black for Norgren et al., 2012, and blue for Graham et al., 2019) from different spacecraft (triangle, diamond, and square) are also presented (Figures 3d and 3f), and it is shown that the waves observed at the magnetosheath separatrix are in good agreement with theoretical LHDI predictions.

Discussion and Summary
In this study, we have presented new MMS observations of the LH waves at the magnetosheath separatrix in asymmetric guide field reconnection. These waves are found to spatially coincide with the density gradient and enhanced Hall electric field across the separatrix, which is responsible for the cross-field current, the free energy source of the LH drift instability. A schematic summary of the observed LH waves is presented in Figure 4. Different with the widely observed LH waves at the magnetospheric side, the waves at the magnetosheath separatrix can only develop in limited regions where there is the density gradient. As the density gradient becomes weaker at the further downstream region, it is more difficult to allow the waves to grow. In the observation, MMS 2 does not observe clear density gradient and the wave activities around the LH frequency are not obvious. Therefore, the LH waves reported in this study are less frequently to be observed than that at the magnetospheric side. Meanwhile, the density gradient revealed here is responsible to balance the enhanced out-of-plane B M in the exhaust, which could be significant when a guide field is present. So the resulting LH waves at the magnetosheath separatrix are potentially a characteristic feature for asymmetric guide field reconnection.
We have shown that the observed waves are consistent with predication of LHDI (Equation 3), which includes the finite plasma beta effect and assumes that electrons are relatively cold. The cold electron limit is valid when (Davidson et al., 1977), and we find that it is reasonably satisfied in this study. The finite plasma beta effect would reduce the maximum growth rate of electrostatic LH waves by a factor (1 + i /2) −1/2 , if T e ≪ T i and V E < v i (Davidson et al., 1977), but it does not stabilize the instability. Therefore, LH waves can survive at the magnetosheath separatrix with a finite plasma beta at ∼5.4, and LH waves under similar plasma beta conditions have been reported in the magnetosheath reconnection (Vörös et al., 2019) and in the magnetotail plasma sheet (Zhou et al., 2014).
The estimated wave potential of LH waves at the magnetosheath side is about 5-8 V, which is much smaller than the waves at the magnetospheric side (>100 V) . Considering the relatively lower electron temperature (∼32 eV), the corresponding eΦ/k B T e is ∼ 15-25%, suggesting that the electrons could be effectively scattered by the wave electric field. The cross-field diffusion coefficient (D ⟂ = n e v e,N ( n e ∕ N) −1 ) is shown in Figure 2k. Throughout the wave interval, D ⟂ is generally negative, corresponding to particle diffusion from the magnetosheath to the exhaust. The peak magnitude of D ⟂ reaches to ∼ − 3 × 10 8 m 2 s −1 , and the averaged value is −1 ∼ − 3 × 10 7 m 2 s −1 from MMS 1 and MMS 3. The estimated D ⟂ is consistent with theoretical predictions (∼4 × 10 7 m 2 s −1 ) (Treumann et al., 1991), which is at least 1 order of magnitude smaller than that at the magnetospheric side (Graham et al., 2017;Treumann et al., 1991;Vaivads et al., 2004). However, it implies a diffusion time of several seconds over a diffusion region with its width at one wave length (T ∼ 2 LH ∕D ⟂ , where LH is the the wave length of LH waves), which is sufficient for the broadening the density gradient across the separatrix. We note that D ⟂ estimated from MMS 4 is much smaller, but the reason is not clear. Whether it is caused by the uncertainty of v e estimation, which does not include the electron diamagnetic drift, or by other processes still needs further investigations.
The LH waves are shown to propagate in the -M direction and toward the x-line (+L), which is in the same direction of the E × B and electron diamagnetic drift direction. It is noted that the x-line is predicted to advect with the electron diamagnetic velocity , but its speed (V drift ∼ (p e, msh − p e, msp )/Ln e eB g ∼20 km s −1 , where the scale length L is approximately equal to d i ) in the spacecraft frame is smaller than the estimated LH wave phase speed. Then whether the LH waves can propagate into the x-line vicinity becomes an interesting issue. Although the LHDI has been suggested to be quenched near the x-line during antiparallel reconnection due to the large plasma beta in previous studies (Bale et al., 2002;Roytershteyn et al., 2012), the oscillation of magnetic nulls has been detected to be related to the perturbations of LH waves (Xiao et al., 2007), indicating the survival of LH waves in the x-line vicinity. There are several possible explanations for this discrepancy. First, the presence of a guide field would effectively reduce the plasma beta in the central diffusion region, so that the instability would not be stabilized (e.g., Zhou et al., 2018). Second, electromagnetic LH waves can develop in the center of a current sheet with a longer wavelength (Daughton, 2003;Zhou et al., 2009). Third, other effects such as the magnetic curvature drift could also destabilize the growth of LH waves (Shinohara et al., 1998;Ueno, 2001). Overall, if LH waves can propagate into the x-line region, more investigations focusing on the dynamics related to LH waves (Chen et al., 2020) should be performed in the future.

Data Availability Statement
The 2-D simulation for magnetic reconnection used LANL Institutional Computing resources and the data are available online (https://doi.org/10.5281/zenodo.3696305).