Thermal electron anisotropy driven by kinetic Alfv́en waves in the Earth’s magnetotail

Thermal and subthermal electron populations in the Earth’s magnetotail are usually characterized by pronounced field-aligned anisotropy that contributes to generation of strong electric currents within the magnetotail current sheet. Formation of this anisotropy requires electron field-aligned acceleration, and thus likely involves field-aligned electric fields. Such fields can be carried by various electromagnetic waves generated by fast plasma flows interacting with ambient magnetotail plasma. In this paper we consider one of the most intense observed wave emissions, kinetic Alfv́en waves, that often accompany fast 5 plasma flows in the magnetotail. Using two tail seasons (2017, 2018) of MMS observations we have collected statistics of 80 fast plasma flows (or bursty bulk flows) events with distinctive enhancement of intensity of broadband electromagnetic waves (kinetic Alfv́en waves). We show correlation the intensity of electric fields of kinetic Alfv́en waves and characteristics of electron anisotropy distributions: the parallel electron anisotropy increases with magnitude of the wave parallel electric field. Also the energy range of this electron anisotropic population is well within the expected acceleration range for assumed kinetic 10 Alfv́en waves. Our results indicate an important role of KAWs in production of thermal field-aligned electron population typically observed in the Earth’s magnetotail.

Although the typical KAW phase speed (about Alfven speed, that is v A ∼ 100 − 500 km/s in the plasma sheet) is insufficiently high for Landau resonance with plasma sheet electrons (m e v 2 A /2 < 1eV), the large effective wave potential ∼ E /k ∼ 1 keV (Damiano et al., 2015;Artemyev et al., 2015;Wang et al., 2019a) results in resonance widening (Palmadesso, 1972;Karimabadi et al., 1990). Electrons fast bouncing along magnetic field line can be trapped into such potential (Watt and Rankin, 2009) or 50 reflect from it (Kletzing, 1994), and both effects would result in electron heating along magnetic field lines. However, so far there were no statistical evidence of KAWs contribution to field-aligned electron anisotropy in the plasma sheet. In this study we use observations of Magnetospheric Multiscale (MMS) Mission (Burch et al., 2016) in the Earth's magnetotail to confirm the key KAWs' role in shaping plasma sheet electron anisotropy.
Following Chaston et al. (2012), we consider bursty bulk flows (Baumjohann et al., 1990;Angelopoulos et al., 1992) as a 55 main source of KAWs. For statistics of such flows collected in 2017-2018 MMS tail seasons, we identify KAWs from their dispersion properties (e.g. Chaston et al., 2014) and investigate correlations of KAWs characteristics and thermal electron anisotropy.

Spacecraft instruments and dataset
We focus on the KAW signatures associated with the fast plasma flows in the Earth's magnetotail at radial distances of ∼ 60 15 − 30R E . We pick up all intervals lasting more than 10 minutes and associated with increased plasma bulk velocities. Then we compare observed magnetic and electric field spectra with the theoretical predictions for KAW (see details below) and keep in dataset only intervals that are characterized by: We use burst-mode fields and plasma data measured on board MMS1 spacecraft (Burch et al., 2016): magnetic field data with 8ms time resolution from the fluxgate magnetometer (Russell et al., 2016); electric field data with 1ms time resolution from 70 Spin-plane Double Probe instrument/Axial Double Probe instrument(SDP/ADP) (Ergun et al., 2016;Lindqvist et al., 2016;Torbert et al., 2016); plasma moments, energy and pitch-angle distributions from the Fast Plasma Investigation (FPI) (Pollock et al., 2016) with 150ms and 30ms for ions and electrons, respectively. Hereinafter in our calculations we use the field-aligned coordinate system: the parallel direction ( ) is along background magnetic field vector (we use low-pass filtering with the cutoff frequency of 0.01 Hz), the first perpendicular component (⊥ 1 ) is along ion bulk velocity component perpendicular to 75 the background magnetic field, and the second perpendicular component (⊥ 2 ) completes the right-hand system. We subtract background magnetic field from the measured magnetic field vector to estimate > 0.01Hz magnetic field perturbations (see example on Fig. 2). The same procedure is applied to the electric field vector. We also construct continuous wavelet spectra (using Morlet wavelets) of the parallel and perpendicular components of magnetic and electric field perturbations.
In the rest reference frame KAWs have frequencies less than the ion cyclotron frequency ω i , whereas plasma flow speed 80 in our study usually exceeds 100 km/s (with mean value of ∼ 400 km/s). Thus, we expect that in the spacecraft frame all the spectrum properties will be dominated by the Doppler shift, i.e. ω sc = ω + k · v ω and the upper frequency limit of the interest will be well higher than ω i . So in our analysis we consider wave perturbations with frequencies up to the lower hybrid frequency (see details below). It has been shown (see Stasiewicz et al., 2000;Chaston et al., 2012) that for warm plasma in the low frequency limit (ω ω i ) KAWs show following polarization properties: and where v A is local Alfven speed, ρ i is ion gyroradius, ρ s is ion acoustic gyroradius, k ⊥ is the transverse component of the wave vector k, β = 4πn(T e + T i )/B 2 0 , B 0 is the magnitude of background magnetic field, B ⊥1 and B ⊥2 are two transverse wave 90 magnetic field components.
Note that we deal with quite inhomogeneous plasma and magnetic fields of fast plasma flows, but we use a local dispersion relation of KAWs (Chaston et al., 2012) and omit the effects of plasma gradients (Bashir et al., 2019) and nonlocal wave properties (Huang et al., 2018;Wang et al., 2019b). These effects can result in deviation of theoretical dispersion predictions from the observed wave characteristics. To estimate a possible role of plasma inhomogeneity we evaluate theoretical equations 95 both for local plasma parameters and for parameters at some distance from the equatorial plane. We use the vertical stress balance condition p = B 2 /8π +n(T i +T e ) = const and definition of the magnetotail lobe pressure p = B 2 lobe /8π to determine magnetic fields at the distance where plasma pressure is 30% of the equatorial value: 0.3p. We additionally assume n = const (based on statistics of current sheets in the middle tail, see Runov et al. (2006)) and T e /T i = const (based on comparison of T e (B x ) and T i (B x ) profiles in the middle tail, see Artemyev et al. (2011b, a)). Thus, 100 we provide theoretical estimates for B / B 2 ⊥1 + B 2 ⊥2 at some distance from the equatorial plane together with estimates for local plasma parameters (see yellow curve in the Fig. 5, 6). Estimates at some distance from equatorial plane are used only to show the range of theoretical expectations for wave dispersion across the plasma sheet.
We aim at statistically analyzing the relation of electron flux (thermal) anisotropy and KAW electric field intensity. Such analysis should reveal a possible KAW role in shaping of the anisotropic electron populations. For such statistical comparison, 105 we consider normalized electron energy spectra and electric field spectra. We use mean absolute value of convectional electric field |V i × B 0 | for electric field magnitude normalization, mean value of electron temperature T e and mean absolute value of KAW electrostatic potential |φ| for electron energy normalization (averaging is performed for the MMS1 probe within the time interval of enhanced ion bulk velocity; |V x | > 200 km/s, typically lasting for ∼ 5 minutes). To estimate φ we consider the parallel component of wave vector k : Thus, knowing the transverse component k ⊥ and wave frequency in the spacecraft frame ω, we estimate the parallel k from equation (Chaston et al., 2008): and α is pitch-angle. We use SPEDAS software (Angelopoulos et al., 2019) to calculate F ⊥ , F fluxes (note FPI pitch-angle resolution is 6°). The energy range of [0, ∼ 200]eV is contaminated by photoelectrons. To exclude this population from the consideration, we use electron data only for > 100eV energy range. Figure 4 shows parameters of magnetic and electric field perturbations in the field-aligned coordinate system after subtraction 130 of the low-pass filtered background field. Black vertical lines here bound time intervals which we use for spectral analysis.
Previous investigations of KAWs with Cluster (Escoubet et al., 2001) and THEMIS (Angelopoulos, 2008) measurements (Chaston et al., 2005(Chaston et al., , 2008(Chaston et al., , 2012(Chaston et al., , 2013 rely on theoretical relations that provide an estimate of the parallel electric field component from the measured perpendicular one. In our work we use direct measurements of the parallel electric component available on MMS Ergun et al., 2016).
There is a clear enhancement of the wave activity in the wide frequency range for the electric field components (see panels (d,e)) and mainly up to ion cyclotron frequency for the magnetic field components (see panels (f,g)). Time-averaged wave spectra are shown in Figs. 5, 6: up to the ∼ 0.1Hz magnetic and electric spectral densities (transverse components) have almost the same profiles (left panels of Figs. 5, 6), whereas E ⊥ /B ⊥ ∼ const ∼ v A , i.e. we deal with Alfvenic perturbations.
The ratio E ⊥ /B ⊥ grows with frequency, as it is expected for KAWs (Chaston et al., 2012). Huang et al. (2016)). Thus, we consider electric and magnetic wave characteristics only below the lower-hybrid frequency.

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At the same time, some compressional modes are presented in the ultra-low frequency range (B /B ⊥ > 1). Most likely KAWs are mixed with some ultra-low frequency compressional (diamagnetic perturbations) that are typical for high plasma pressure magnetotail (Volwerk et al., 2003;Runov et al., 2014). Observed frequencies of such perturbations are dominated by the Doppler shift and we cannot distinguish them from KAWs. However, contrast to KAWs, ultra-low frequency diamagnetic perturbations are MHD modes that are not expected to carry any significant field-aligned fields. Thus, we can assume that The third (highest) energy range is characterised by perpendicular anisotropy that is presumably associated with betatron electron acceleration at the dipolarization front (Fu et al., 2013;Birn et al., 2012Birn et al., , 2013. Note, that in the considered frequency range perpendicular electric field component usually have larger values than parallel one (both for our observations and for the theoretical predictions), so right panels on Figs. 7 and 8 cover a wider range of 180 amplitudes than left panels do. Moreover, events with larger parallel field are rarer, i.e in the rightmost bin on the left panels we have very poor statistics.
In this study we focus on the second (thermal and subthermal) electron population. Figures 7, 8 show that the field-aligned anisotropy of this population increases with the growth of the electric field amplitude. Note we use both parallel and transverse fields: whereas the best correlation of anisotropy is expected for field-aligned fields, accuracy of this field measurements in The electron acceleration by KAWs is controlled by wave field-aligned electric fields E , whereas typical accelerated energy is scaled with the wave scalar potential φ that can be defined as E = −∇ φ (Damiano et al., 2015;Artemyev et al., 2015).

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Note that effective potential describes a combined contributions of electrostatic and electromagnetic fields to E (Stasiewicz et al., 2000). To compare energies of field-aligned electron population with model predictions of electron acceleration by KAWs, we determine φ from E spectra and KAW dispersion relation (see Eq. (3)

Statistical analysis
In this section we analyze of flux anisotropy and KAWs' electric fields for the entire statistics of 81 events (see Fig. 1). Fig.   9 shows averaged wave spectra and ratios of wave spectra for the entire statistics. The ratio B /B ⊥ ∼ 1 for a wide frequency range, i.e. there is some contribution of compressional waves in the low frequency range (∼ 10 −2 − 10 −1 Hz where measured 200 compressional magnetic field fluctuations well exceed expectations from KAWs' dispersion relation, see Fig. 9 (c)). Such low frequency compressional fluctuations are quite typical for high-beta plasma sheet, especially during strong plasma flows (Bauer  , 1995;Zelenyi et al., 2015). Besides flapping waves (Fruit et al., 2002;Volwerk et al., 2004b;Runov et al., 2005), there is a wide range of magnetic holes, localized dips of magnetic field magnitude, contributing to compressional fluctuations on the low frequency range (Ergun et al., 2020). Due to Doppler shift of all these low frequency fluctuations, these are well mixed 205 with KAWs and we cannot distinguish KAWs contribution to B in the low frequency range. However, already at frequency > 0.1 Hz KAWs' contribution dominates and observed ratio B /B ⊥ follows theoretical expectations.
The effect of non-KAWs contribution to magnetic and electric field fluctuations at low frequencies is also seen in (Fig. 9(b)). Transverse electric field fluctuations are stronger than one would expect for Alfven waves, and these additional fluctuation intensity likely comes from convection of 210 compressional magnetic field structures, i.e. strong plasma flow converts almost stationary magnetic field fluctuations (low frequency flapping waves, magnetic holes) to electric field fluctuations (Volwerk et al., 2003;Zelenyi et al., 2015). For larger frequency (> 0.1 Hz) the ratio E ⊥ /B ⊥ follows well the prediction of KAWs dispersion, i.e. transverse electric field fluctuations grows with frequency (dominated by the Doppler shift) faster than magnetic field fluctuations (Chaston et al., 2005(Chaston et al., , 2012. Thus, Fig. 9 suggests that magnetic and electric fields (for frequency > 0.1 Hz) in selected intervals are dominated by KAWs. occupies E/T e < 1 energy range, that well coincides with the E/|φ| < 1 range expected from models of electron field-aligned acceleration by KAWs (Damiano et al., 2015;Damiano et al., 2016;Artemyev et al., 2015).

Discussion
We show that electron interaction with KAWs is the perspective candidate for generation of the field-aligned anisotropic electron population. This population is often observed in the Earth's magnetotail (Walsh et al., 2011;Artemyev et al., 2014), but its origins remain unestablished. Energies of field-aligned electrons can reach several keV Yushkov et al., 2017), whereas the ionosphere outflow is expected to be limited to sub-keV energies (Walsh et al., 2013). Thus, some acceler- 2020) but observations suggest that field-aligned anisotropic electrons are not necessary seen around spatially localized reconnection region (Artemyev et al., 2020). Fast plasma flows are known as primary regions of charged particle acceleration in the magnetotail (Birn et al., 2013(Birn et al., , 2014Ukhorskiy et al., 2017;Gabrielse et al., 2014Gabrielse et al., , 2016, but these flows are mainly associated 230 with compressional transverse electron heating (Khotyaintsev et al., 2011;Zhang et al., 2019). Our results suggest that KAWs associated with fast plasma flows (Chaston et al., 2012) can support the mechanism of the field-aligned electron acceleration.
Thus, fast flows are indeed the primary acceleration regions, but field-aligned electron acceleration is not directly resulted from electron interaction with dipolarization fronts (fast plasma flow fronts) and require KAWs' generation and further dissipation due to Landau resonance with electrons (Sharma Pyakurel et al., 2018;Gurram et al., 2020).

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KAWs as a natural mode of Alfven waves in hot ion plasma (Stasiewicz et al., 2000) are suggested to be generated by several different mechanisms, e.g., mode conversion (Hasegawa and Chen, 1975;Hasegawa, 1976;Lin et al., 2012;Johnson and Cheng, 1997), phase mixing (Guo et al., 2015) and magnetic reconnection (Chaston et al., 2005Liang et al., 2016;Wang et al., 2019b;Cheng et al., 2020). These waves are believed to be responsible for energy transfer from the magnetotail to the ionosphere (Lysak and Song, 2003;Keiling et al., 2003) and ionospheric electron outflow can be a response for such energy 240 deposition (Lysak, 1990;Keiling, 2009). Thus, KAW electric field correlation with the field-aligned electron population can be, at least partially, due to the outflow mechanism. However, such outflow is expected to be north-south asymmetric (due to asymmetry of ionosphere properties Haaland et al. (e.g., 2017);Laundal et al. (e.g., 2017)) and low-energy (< 100 eV). Indeed, MMS FPI measurements show parallel/anti-parallel asymmetry for low energy electrons for a wider energy range up to one keV. Presence of accelerated field-aligned anisotropic electrons and good correlation of their energy range with the upper limit 245 of KAWs acceleration, the scalar potential amplitude φ, suggests that electron acceleration by KAWs are not less important in production of the electron field-aligned anisotropy than more traditionally considered ionosphere outflow (Walsh et al., 2013).

Conclusions
In this study we have collected statistics of 81 fast plasma flow events accompanied by observations of KAWs. Case and statistical studies show a correlation between the KAWs' electric field magnitudes and thermal/subthermal electron (∼ 1 −