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# Parallel Electron Heating through Landau Resonance with Lower Hybrid Waves at the Edge of Reconnection Ion Jets

Yong Ren, Lei Dai, Chi Wang, Benoit Lavraud

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Yong Ren, Lei Dai, Chi Wang, Benoit Lavraud. Parallel Electron Heating through Landau Resonance with Lower Hybrid Waves at the Edge of Reconnection Ion Jets. The Astrophysical Journal, 2022, 928 (1), pp.5. 10.3847/1538-4357/ac53fb. hal-04889372

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# Parallel Electron Heating through Landau Resonance with Lower Hybrid Waves at the Edge of Reconnection Ion Jets

Yong Ren<fnref n="1" />, Lei Dai<fnref n="1" /> Ⓒ, Chi Wang<fnref n="1" />, and Benoit Lavraud<fnref n="2" />,<fnref n="3" /> Ⓒ

Received 2021 November 18; revised 2022 January 26; accepted 2022 February 9; published 2022 March 22

## Abstract

We investigate lower hybrid waves in magnetic reconnection at the flank magnetopause using Magnetospheric Multiscale data. Intense emissions of lower hybrid waves are observed at the density boundary of the reconnection ion jet. Associated with the lower hybrid waves, electrons exhibit signatures of heating in the direction parallel to the magnetic field. Near the Landau resonance energy, the electron fluxes parallel to the magnetic field show oscillations at the same frequency as the lower hybrid waves. The electron flux oscillations are in phase or antiphase with the wave parallel electric field. These observations provide direct evidence for Landau resonance. Our analysis indicates that the density gradient at the edge of the ion jets provides free energy for the lower hybrid waves that further contribute to electron heating through Landau damping. These results shed light on the role of wave-particle interactions in the energy conversion chain of reconnection ion jets.

*Unified Astronomy Thesaurus concepts: Solar magnetic reconnection (1504); Planetary magnetospheres (997)*

## 1. Introduction

Magnetic reconnection transforms magnetic field energy into plasma flow and thermal energies. During this highly dynamic process, many types of plasma waves can develop and provide feedback on the reconnection through wave-particle interactions (Dai 2009; Fujimoto et al. 2011; Duan et al. 2016; Dai et al. 2017; Liang et al. 2017; Dai 2018; Khotyaintsev et al. 2019; Ren et al. 2019; Li et al. 2020; Ren et al. 2021). The investigation of plasma waves may provide perspectives on the chain of energy conversion in the reconnection process.

Lower hybrid waves (LHWs) are around the lower hybrid frequency ($f_{LH}$) and frequently observed in association with magnetic reconnection. In the presence of plasma $\beta \ll 1$, LHWs are usually observed to be quasi-electrostatic (Zhou et al. 2009; Norgren et al. 2012; Graham et al. 2017; Yoo et al. 2020). At the magnetopause, LHWs are detected frequently in ion diffusion regions and separatrices of reconnection (Bale et al. 2002; Pritchett et al. 2012; Graham et al. 2016; Khotyaintsev et al. 2016; Graham et al. 2017; Zhou et al. 2018; Tang et al. 2020). In the magnetotail, intense LHWs are observed at the front of magnetotail reconnection jets (Khotyaintsev et al. 2011; Divin et al. 2015; Le Contel et al. 2017; Greco et al. 2017).

The cross-field relative drift between electrons and ions is an important free energy source for LHWs. Such electron-ion relative drift is common in the presence of a density gradient. The lower hybrid drift instability (LHDI) is driven by the density gradient (Krall & Liewer 1971; Mcbride et al. 1972; Davidson et al. 1977; Yoon & Lui 2004, 2008; Stasiewicz & Eliasson 2020). LHDI is most effective in regions of large density gradients such as the separatrix regions, the ion diffusion regions, and jet fronts (Bale et al. 2002; Graham et al. 2016; Khotyaintsev et al. 2016; Graham et al. 2017;

Zhou et al. 2018). LHWs can also be driven through the modified two-stream instability (MTSI; Mcbride et al. 1972; Yoon & Lui 2004). MTSI is effective when unmagnetized ions have relative drift with electrons in the cross-field direction (Graham et al. 2017, 2019).

The outward ion jets represent a principle channel of energy conversion in magnetic reconnection (Dai et al. 2021). The reconnection jet is one of the main free energy sources that is suitable for wave generation (Khotyaintsev et al. 2011; Fu et al. 2014). Using high-resolution data of the Magnetospheric Multiscale (MMS) mission, here we provide a detailed analysis of LHW at the edge of the reconnection ion jet, with the focus on the role of LHW in the energy conversion chain.

## 2. Observations

Our study uses data from the MMS spacecraft (Burch et al. 2016). The three-dimensional electric field data are from the electric field double probes (EDPs; Ergun et al. 2016; Lindqvist et al. 2016). The magnetic field data are from the fluxgate magnetometer (FGM) at 128 samples $s^{-1}$ and the search-coil magnetometer (SCM) at 8192 samples $s^{-1}$ (Le Contel et al. 2016; Russell et al. 2016). We use the particle data from the fast plasma investigation (FPI; Pollock et al. 2016) and Electron Drift Instrument (EDI; Torbert et al. 2016). The time resolutions are 4.5 s for fast mode, 150 ms for burst mode of ions, and 30 ms for burst mode (FPI) of electrons. We use the high time resolution (1000 Hz) fluxes of 250 eV electrons from EDI.

### 2.1. Overview

Figure 1 displays an overview of MMS1 crossing of a reconnection ion jet at the flank magnetopause. The location of the MMS spacecraft is (7.2, 11.6, 1.9) Re (Earth radii) in Geocentric Solar Magnetospheric (GSM) coordinates. We transform the vector quantities into LMN coordinates using minimum variance analysis of the magnetic field **B** from 08:20 UT to 08:33 UT ($L = (-0.15, 0.27, 0.95)$, $M = (0.45, -0.84, 0.31)$, and $N = (0.88, 0.47, 0.00)$ in GSM).

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Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.

<footnote n="1">¹State Key Laboratory of Space Weather, National Space Science Center, Chinese Academy of Sciences, People's Republic of China; ldai@spaceweather.ac.cn</footnote>

<footnote n="2">²Laboratoire d'Astrophysique de Bordeaux, Université de Bordeaux, CNRS, Pessac, France</footnote>

<footnote n="3">³Institut de Recherche en Astrophysique et Planétologie, Université de Toulouse, CNRS, UPS, CNES, Toulouse, France</footnote>
<figure><img src="image_5.png" /><figcaption>**Figure 1.** Overview of a reconnection crossing observed on 2019 November 16 by MMS1. (a) Magnetic fields *B*. (b) The ion velocity *V*ᵢ, (c, d) Ion and electron omnidirectional differential energy flux, respectively. (e) Ion and electron densities *n*ᵢ and *n*ₑ, (f, g) The dynamic spectra of *B* and *E*, respectively. (h) Sketch of reconnection ion jets at the flank magnetopause. (i, j) 2D cuts of the ion distributions in the *V*ᵢ|*B* × *V*ᵢ, and *E* × *B*, where *E* is calculated from −*V*ᵢ × *B*. The white, magenta, and black lines in panels (f) and (g) are electron cyclotron frequency *f*ₑ, ion plasma frequency *f*ᵢ, and lower hybrid frequency *f*<sub>LH</sub>, respectively. The black dashed lines mark the region of reconnection ion jets.</figcaption></figure>

Before 08:24:10 UT, MMS is located on the magnetosphere side, characterized by a steady northward magnetic field (*B*<sub>L</sub> > 0) and hot plasma (Figures 1(a)–(e)). Some boundary layer (BL) plasmas are observed before 08:29:30 UT, when MMS starts crossing the main magnetopause current layer. Near the current layer, MMS observed an ion jet as large as 150 km s<sup>-1</sup> in the −L direction during 08:28:00–08:29:40 UT. After 08:32:00 UT, MMS exits in the magnetosheath, characterized by relatively stable flows and dense plasmas. The location and the trajectory of MMS are shown in the schematic in Figure 1(h).

The ion jets from 08:28:00 UT to 08:29:40 UT between
vertical dashed lines have a large -L component (-150 km
s⁻¹) that is opposite to the direction of the stable flow (+100
km s⁻¹) in the magnetosheath. The jet speed (150 km s⁻¹) is
comparable to the magnetosphere Alfvén speed (139 km s⁻¹),
suggesting that the southward jet is from magnetic reconnec-
tion. To confirm the observation of reconnection jets, we
further examine the ion distribution inside the jet. The ion
distributions in the jets display a classical D-shape (Figures 1(i)
and (j)), consistent with the expected velocity distribution
function in magnetopause reconnection jets (Cowley 1982;

Phan et al. 2016). Notice that the reconnection jet also has a
strong duskward (− *M*) component in this event (Figure 1(h)).
This is consistent with the tangential momentum balance, as the
magnetosheath ions in the flank are accelerated across the
reconnection layer. This feature is different from that in the
subsolar region.

Increased power density is observed in the dynamic spectra
of the electric field **E** around the lower hybrid frequency
*f*<sub>LH</sub> ≈ 20 Hz (Figure 1(g)) during the jet crossing. The proper-
ties of these fluctuations are analyzed next.

2.2. Lower Hybrid Waves at the Edge of Reconnection Ion Jets

Figure 2 displays a detailed 1-minute view of the reconnec-
tion ion jets and associated LHWs. As illustrated in the
schematic of panel (i), magnetosheath inflow ions are
accelerated across the reconnection layer and form reconnec-
tion jets that go into the BL. This is the typical scenario for
magnetopause reconnection (Sonnerup et al. 1981). Around
08:28:20 UT (marked by the first gray region), MMS crosses
the edge and associated density gradient of the reconnection
ion jet. The density gradient is expected to drive a diamagnetic
<figure><img src="image_6.png" /><figcaption>**Figure 2.** LHWs at the edge of reconnection ion jets. (a) Ion and electron omnidirectional differential energy flux. (b) Ion velocity $V_i$. (c) Ion number density $N_i$. (d) Plasma beta. (e) **M** component of the current computed from $\nabla \times \mathbf{B}/\mu_0$. (f, g) The dynamic spectra of **B** and electric field **E**, respectively. (h) The total electric field power in the frequency range between 10 and 80 Hz. (i) Sketch of MMS spacecraft trajectory (red line) during the crossing of reconnection ion jets. The yellow shaded region indicates the BL.</figcaption></figure>

current perpendicular to **B** and $\nabla n$. This is consistent with the **M** component of the electric current in panel (e). In the following minute, MMS moves back and forth, crossing several density gradients of the reconnection ion jets. The reconnection may strongly disturb the BL structure in this event (Nakamura 2021). Around the edge near 08:29:19 UT (marked by the second gray region), MMS observes another jet edge as indicated by the significant decrease of the flow speed and magnetosheath populations.

At the edges of the ion jets, MMS observes an enhancement of LHW power as seen in the electric field power density (panels (g) and (h)). The most intense activity of LHWs is observed near 08:28:30 UT and 08:29:20 UT. The enhanced power density of LHWs is mostly in the 10–30 Hz (approximately 0.5–1.5 $f_{LH}$) frequency band, and the total electric field power of waves can reach as large as 20 ($mV/m)^2$ (panels (g) and (h)). By contrast, little enhancement is observed in the dynamic spectra of **B** around the $f_{LH}$, which indicates that these LHWs are quasi-electrostatic (Graham et al. 2016; Zhou et al. 2018). Furthermore, the plasma $\beta$ is less than 1 at the edge of the ion jets (panel (d)), consistent with the condition for an electrostatic treatment of the LHDI (Krall & Liewer 1971; Davidson et al. 1977; Graham et al. 2019).

## 2.3. Properties and Instability Analysis of Lower Hybrid Waves

We analyze the properties of LHWs at the edge of the reconnection ion jets (gray shaded regions in Figure 2) using the single-spacecraft method as described in Norgren et al. (2012). The wave potentials $\delta\phi_E$ and $\delta\phi_B$ are obtained as

$$ \delta\phi_E = \int \delta E \cdot V_{\text{ob}} dt \quad (1) $$

and

$$ \delta\phi_B = \frac{|\mathbf{B}_0|\delta\mathbf{B}_\|}{en_e\mu_o}, \quad (2) $$

where $\mathbf{B}_0$ is the background magnetic field, $\delta\mathbf{B}_\|$ is the fluctuation of wave magnetic field aligned with $\mathbf{B}_0$, $e$ is the electron charge, $n_e$ is the electron density, and $\delta E$ are the fluctuations of the wave electric field. $\delta\phi_E$ and $\delta\phi_B$ of waves for this event are shown in Figure 3. The direction of the wave propagation and the phase speed are obtained by finding the maximum cross-correlation between $\phi_B$ and $\phi_E$ from different wave angles. LHWs in cases 1 and 2 propagate mainly in the $-\mathbf{M}$ direction in the spacecraft frame. Using $f=20$ Hz, the wavelengths $\lambda = V_{\text{ob}}/f$ are 13.4 and 9.2 km. The wavelengths
<figure><img src="image_7.png" /><figcaption>**Figure 3.** Electric potential $\delta\phi_E$ and $\delta\phi_B$ (>10 Hz) of LHW packets at the edge of reconnection ion jets. The phase velocities $V_{ob}$ in LMN coordinates are obtained by the best fit between $\delta\phi_E$ and $\delta\phi_B$.</figcaption></figure>

are larger than the thermal electron gyroradius $\rho_e \sim 430$ m and smaller than the thermal ion gyroradius $\rho_i \sim 68$ km. These properties are similar to those found in previous observations in the ion diffusion region (Graham et al. 2016). The two potentials in Figure 3 are not in good agreement at some times. This suggests that the phase velocity of the waves may change at some parts of the edge. The parameters of the waves and local plasmas for cases 1 and 2 in Figure 2 are summarized in Table 1. We assume that the density gradient is in the N direction and make an order-of-magnitude estimate of the density gradient. The diamagnetic drift $V_d = \nabla p \times B/qnB^2$ is estimated using the data from mms2 and mms3. Using $n = 2.5 \text{ cm}^{-3}$, $B = 30$ nT, $V_{di} = -250 \text{ km s}^{-1}$, and $V_{de} = 19 \text{ km s}^{-1}$ (case 1), the estimated current density $j_d = ne(V_{di} - V_{de}) \sim -9.2 \times 10^{-8} \text{ A m}^{-2}$ is approximately consistent with the observed values $j_M \sim -2 \times 10^{-8} \text{ A m}^{-2}$ (Figure 2(e)). Using a density $n = 2.0 \text{ cm}^{-3}$, $B = 30$ nT, diamagnetic drift $V_{di} = 210 \text{ km s}^{-1}$, and $V_{de} = -13 \text{ km s}^{-1}$ (case 2), the estimated $j_d$ is $\sim 5.8 \times 10^{-8} \text{ A m}^{-2}$, approximately consistent with the observed value ($j_M \sim 2 \times 10^{-8} \text{ A m}^{-2}$) in Figure 2(e).

We now analyze the instabilities at these two edges (the gray shaded region). As seen from Figure 2, the power spectral density of LHWs around $f_{LH}$ increases at the edges of ion jets, where the density gradients are large. This is consistent with the LHDI (Yoon & Lui 2008; Divin et al. 2015), in which the diamagnetic drift acts as a free energy source.

The electrostatic dispersion equation of LHDI is (Yoon & Lui 2008; Divin et al. 2015)

$$0 = 1 - \frac{\omega_{\text{pi}}^2}{k^2 v_i^2} Z' \left( \frac{\omega - k_{\perp} V_{\text{Di}}}{k v_i} \right) + \frac{2 \omega_{\text{pe}}^2}{k^2 v_e^2} \times \left( 1 + \frac{\omega - k_{\perp} V_{\text{De}}}{(k_{\parallel} \cos \alpha + k_{\perp} \sin \alpha) v_e} J_0(b) I_0(\lambda) e^{-\lambda Z(\xi)} \right), \quad (3)$$

where $b = ((k_{\parallel} \sin \alpha - k_{\perp} \cos \alpha) V_{\text{De}} \cos \alpha) / \Omega_{\text{ce}}$, $\lambda = ((k_{\parallel} \sin \alpha - k_{\perp} \cos \alpha)^2 v_e^2) / (2\Omega_{\text{ce}}^2)$, $\xi = \omega / (k_{\parallel} \cos \alpha + k_{\perp} \sin \alpha) v_e$, Z is the plasma dispersion function, $\omega_{\text{pi}}$ and $\omega_{\text{pe}}$ are the ion and electron plasma frequencies, $v_i$ and $v_e$ are the thermal speeds of ion and electron, $J_0$ and $I_0$ are the Bessel function of the first kind and the modified Bessel function of the first kind in zero order, and $\alpha$ is the angle between the guide field $B_G$ and $B_0$. For the case of no guide field $B_G$, $\sin \alpha = B_G/B_0 = 0$. The parameters used in Equation (3) are in Table 1. The $|V_{Di}|$ is much larger than $|V_{De}|$ in our cases, so the dispersion equation is approximately in the electron rest frame. In our event, we apply Equation (3) in the electron rest frame and transform the results to the spacecraft frame.

The results of the instability analysis are shown in Figure 4. Panels (a1), (a2), (b1), and (b2) show the normalized frequency
<table><caption>**Table 1**
Lower Hybrid Wave Properties in Subintervals</caption><thead><tr><th rowspan="2">Case</th><th rowspan="2">Time Range</th><th colspan="2">Observations</th><th rowspan="2">λ(km)</th><th colspan="7">Parameters for Equation (3) in the Frame at Rest to the Discontinuity</th></tr><tr><th>|V<sub>ob</sub>| (km s<sup>-1</sup>)</th><th>Direction (LMN)</th><th>C<sub>φ</sub></th><th>V<sub>de</sub> (km s<sup>-1</sup>)</th><th>V<sub>di</sub> (km s<sup>-1</sup>)</th><th>B (nT)</th><th>T<sub>i</sub> (eV)</th><th>T<sub>e</sub> (eV)</th><th>n<sub>i</sub> (cm<sup>-3</sup>)</th></tr></thead><tbody><tr><td>1</td><td>08:28:18<br/>to 08:28:31</td><td>269</td><td>[-0.40, -0.93, -0.05]</td><td>13.4</td><td>0.78</td><td>19</td><td>-250</td><td>30</td><td>400</td><td>30</td><td>2.5</td></tr><tr><td>2</td><td>08:29:19<br/>to 08:29:27</td><td>184</td><td>[-0.34, -0.93, -0.13]</td><td>9.2</td><td>0.83</td><td>-13</td><td>210</td><td>30</td><td>400</td><td>25</td><td>2.0</td></tr></tbody></table>

$n_e = n_i$.

Note. $C_φ$ is the cross-correlation of $\delta\phi_E$ and $\delta\phi_B$ in the wave propagation direction.
<figure><img src="image_8.png" /><figcaption>**Figure 4.** Dispersion of the LHWs based on mms1 data for the two cases at the boundary of reconnection ion jets. Panels (a)–(c) show the dispersion relation of LHDI in the electron rest frame ($\omega_{\text{rest}}$) and spacecraft frame ($\omega_{\text{SC}}$). The real parts ($\omega_{\text{rest}}$) in panel (c) are Doppler shifted into the spacecraft frame ($\omega_{\text{SC}}$; Song & Russell 1999) under the condition of a perpendicular electron flow velocity of –300 km s⁻¹ (case 1) and –500 km s⁻¹ (case 2). The blue shaded regions in panel (c) indicate the regime of observed LHWs. Panels (d)–(f) represent the parallel phase speed of the calculated LHW with positive growth rate $\gamma > 0$ and $\gamma > 0.5 \gamma_{\text{max}}$.</figcaption></figure>

and growth rate of the LHDI. The LHDI is unstable for the wave-normal angle $\theta \gtrsim 88^{\circ}$ (cases 1 and 2). The mode propagating perpendicular to $B_0$ reaches a peak growth rate at $\omega_{\text{rest}} \approx 0.6\omega_{\text{LH}}$, similar to the values reported in the ion diffusion (Graham et al. 2017, 2019). The solutions of the dispersion relation that yield the $\gamma_{\text{max}}$ ($\theta = 90^{\circ}$) are plotted in panels (c1) and (c2). The predicted ranges of $\omega_{\text{SC}}/\omega_{\text{LH}}$ (0.5–4$\omega_{\text{LH}}$) are in agreement with the observations in the spacecraft frame in Figure 2(g).

Panels (d1), (d2), (e1), and (e2) display the parallel phase speeds $\omega/k_\|$ of LHDI. The value of $\omega/k_\|$ is useful in the context of Landau resonance. According to the resonance condition, electrons with a parallel resonance speed $v_\| = \omega/k_\|$ experience Landau resonance. The predicted energy range for electrons to participate in Landau resonance for this event is $v_\| \gtrsim 6000 \text{ km s}^{-1}$ (~100 eV; panels (f1) and (f2)).

## 2.4. Parallel Electron Heating through Landau Damping

To investigate the electron heating due to wave–particle interactions, we investigate properties of electrons and LHWs associated with the density boundary of ion jets in four subintervals. In particular, subintervals 1 and 3 are during the crossing of the Earthward and sunward edge of the reconnection jet as marked in Figure 2.

Figure 5 shows distributions of electrons in the presence of strong LHWs at the density boundary. Strong wave powers of LHWs with amplitude as large as 10–20 mV m⁻¹ are observed, associated with the density gradient. In panel (d) of Figure 5, we compare the parallel velocity distribution of electrons during wave–particle interactions with those that are at the high-speed and high-density side of the ion jets and not in the presence of strong LHWs. The high-speed and high-density side of the jets represents a good reference of plasma
<figure><img src="image_9.png" /><figcaption>Sub-interval 2</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Sub-interval 3</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Sub-interval 4</figcaption></figure>

<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /></figure>

<figure><img src="image_15.png" /></figure>

<figure><img src="image_16.png" /></figure>

**Figure 5.** Electron temperature and distributions during the ion jets crossing in the four subintervals. (a1–a4) The parallel and perpendicular electron temperatures, $T_{e\parallel}$ and $T_{e\perp}$ (b1–b4) Electron number density $N_e$. (c1–c4) The total electric field power in the frequency range between 10 and 80 Hz. (d1–d4) The phase-space densities $f_e$ plotted as a function of energy (E) for pitch angles 0° or 180°. The red and gray lines are the averages of the regions marked by the red bar and gray shaded region, respectively.

population that is directly from reconnection and not yet affected by wave–particle interactions. The phase-space densities of electrons during the region of wave–particle interactions (Figure 6) display a relative enhancement at energies larger than ~60 eV (panel (d)), suggesting that electron distributions associated with intense LHWs are heated at this energy range. Notice that the energy range of the phase-space density enhancement is in good agreement with that of the parallel resonance energy of the LHWs as indicated in Figure 4(f).

An important feature to notice is the large-scale electron temperature anisotropy ($T_{e\perp}/T_{e\parallel} < 1$) that appears during most of the reconnection jet. This large-scale anisotropy and parallel electron heating are probably due to electron trapping and electric fields associated with reconnection itself (Egedal et al. 2008, 2011, 2013). In addition to this heating mechanism, the wave–particle interactions can further contribute to the electron heating at the density boundary of the ion jets.

Figure 6 displays in situ evidence of wave–particle interactions (the Landau resonance) using high-resolution
<figure><img src="image_17.png" /><figcaption>**Figure 6.** The power spectra of electron flux oscillation and parallel wave electric field in the four subintervals. (a1–a4) The fluxes of 250 eV electron for different pitch angles. (b1–b4, c1–c4, d1–d4, e1–e4, and f1–f4) The dynamic spectra of the 250 eV electron fluxes for different pitch angles. (g1–g4) The dynamic spectra of the $E_\parallel$.</figcaption></figure>

electron data from EDI. EDI provides high time resolution electron fluxes at energy 250 eV. Panels (a1)–(a4) display the fluxes for electrons at the energy ~250 eV, and the electron motion directions vary in the pitch-angle range 0° ~ $\theta_{PA}$ ~ 40° or 140° ~ $\theta_{PA}$ ~ 180°. The dynamic spectrum in Figures 5(b)–(e) shows the intense fluctuations of electron flux around ~20 Hz (~1.0fLH). The most intense fluctuations are in the field-aligned direction. At the same time, there is enhanced power in $E_\parallel$ around ~20 Hz (panels (g1)–(g4)). The observed electron flux oscillations are strong in the parallel direction and decay in the large pitch angle (30°–40°/140°–150°).

The oscillation in electron velocity distribution functions (VDFs) suggests the occurrence of Landau resonance interactions with LHWs at the same frequency band. During Landau

resonance interactions, the electrons in the parallel direction shall move back and forth under the influence of the oscillating parallel electric fields of waves. The linearized solutions of Landau damping predict intense oscillations in $\delta v$ and $\delta N$ (and thus electron flux $\delta F$) in the vicinity of the Landau resonance region (Dawson 1961; Stix 1992; Chen 2016). As shown in previous MMS observations, the most intense oscillations in the electron VDFs are expected to be near the resonance energy (Oka et al. 2017).

In subinterval 4, the electron flux oscillations are strong in the antiparallel direction instead of the parallel direction. This feature suggests that the dominant wave component that interacts with electrons has a $\omega/k_\parallel$ in the antiparallel direction in subinterval 4. In subintervals 1, 2, and 3, the flux oscillations
<figure><img src="image_18.png" /></figure>

<figure><img src="image_19.png" /></figure>

<figure><img src="image_20.png" /></figure>

<figure><img src="image_21.png" /><figcaption>**Figure 7.** The phase relation between electron flux oscillations and parallel wave electric fields in subintervals 1–4. (a1–a4) The flux oscillation of 250 eV electron, $\delta F_{250\ eV}$. (b1–b4) The wave parallel electric field $\delta E_{\parallel}$, $C_{\delta E_{\parallel}\delta F}$ is the linear Pearson correlation coefficient of $\delta E_{\parallel}$ and $\delta F_{250\ eV}$.</figcaption></figure>

due to wave–particle interaction are strong in the parallel direction. Accordingly, $\omega/k_{\parallel}$ of the waves are expected in the parallel direction in subintervals 1, 2, and 3.

Figure 7 displays the phase relation between $\delta F$ and $\delta E_{\parallel}$ during the wave–particle interaction. As marked by the shaded region, $\delta F$ and $\delta E_{\parallel}$ are in good correlation with each other. The electron flux oscillations are roughly in phase or antiphase with the wave parallel electric field. This is direct evidence of Landau resonance as shown in the following analysis. For simplicity, we consider the linearized Vlasov equation for the interaction of the electrons with wave parallel electric fields in the Z direction (Stix 1992; Chen 2016):

$$ \frac{\partial \delta f}{\partial t} + \mathbf{v} \cdot \nabla \delta f - \frac{q}{m} \delta E_z \cdot \frac{\partial f_o}{\partial \mathbf{v}} = 0. \quad (4) $$

Fourier transforming the linearized equation, $\frac{\partial}{\partial t} = -i\omega$, $\frac{\partial}{\partial z} = ik_z$, we obtain

$$ \delta \tilde{f} = \frac{i q (\delta \tilde{E}_z) \frac{\partial f_o}{\partial v_z}}{m_e \omega - k_z v_z}. \quad (5) $$

The phase relation between the flux oscillation (proportional to $\delta \tilde{f}$) and the wave electric field is specified by Equation (5). For nonresonance electrons interacting with the waves, the term $\omega - k_z v_z$ is either positive or negative, leading to a $\pm 90^{\circ}$ phase difference between the flux and electric field. Near the Landau resonance condition $\omega - k_z v_z = 0$, the denominator approaches 0 unless we introduce an imaginary part of the frequency $\omega = \omega_r + i\gamma$. This is equivalent to saying that the waves can grow or decay. Near the resonance condition $\omega_r - k_z v_z = 0$, Equation (5) becomes

$$ \delta \tilde{f} \sim \frac{q(\delta \tilde{E}_z) \frac{\partial f_o}{\partial v_z}}{m_e \gamma}. \quad (6) $$

Hence, the electrons in Landau resonance are in phase ($0^{\circ}$) or antiphase ($180^{\circ}$) with the parallel wave electric field. This is distinctly different from the $\pm 90^{\circ}$ phase difference for nonresonance electrons. The phase relation for resonance electrons indicates a net energy exchange between waves and particles, allowing waves to grow or damp. The above

treatment and phase relation for Landau resonance are similar to those in the analysis of drift resonance for radiation belt particles (Southwood & Kivelson 1981; Dai et al. 2013).

As in the treatment in Equations (8)–(35) of Stix (1992), Equations (4)–(5) in fact deal with the responses of electrons to the electric field in a manner like the test-particle calculation. Notice that the $\omega$ in Equations (4)–(5) is from the Fourier transform and describes the temporal behavior of the $\delta \tilde{f}$. Under the impact of the wave electric field, the amplitude of electron oscillations $\delta \tilde{f}$ near the resonance energy shall gradually grow ($\gamma \to 0^+$). Considering a parallel-propagating wave ($\partial f_o / \partial v_z < 0$), the electron flux shall be $180^{\circ}$ out of phase with the wave electric field, corresponding to an energy transfer to the electrons. This phase relation is verified with the observations in Figure 7.

The electron heating by LHWs is through a multiple-stage process. In the linear stage of the LHW instability, waves keep growing and the free energy source is from the diamagnetic drift of the plasma. Then, Landau damping is turned on, but not fast enough to damp all the waves from the LHW instability. The LHW instability and Landau damping process may operate together, but only one is dominant at one particular time and location. After the initial growth, the LHDI may saturate in realistic observations.

### 3. Conclusions

Using MMS spacecraft data, we examined the role of LHWs in the energy conversion chain at the edge of magnetic reconnection ion jets at the flank magnetopause.

Intense LHWs are observed at the edge of the reconnection ion jets. The density gradient at the boundary of the reconnection jets is associated with the diamagnetic drift in the cross-field direction, providing the free energy for the LHWs through the LHDI.

Electrons associated with the intense LHWs show signatures of heating at energies above ~100 eV in the direction parallel to the magnetic field. High time resolution data from MMS show oscillations in electron parallel fluxes at the same frequency (near the $f_{LH}$) as the parallel electric fields of the LHWs. The electron flux oscillations are in phase or antiphase
with the wave parallel electric field. These observations provide direct evidence that LHWs contribute to parallel electron heating through the Landau damping.

Our results suggest that LHWs can contribute to the energy conversion chain of reconnection through wave–particle interactions. In a multiple-stage process, the free energy in the diamagnetic drifts associated with the jets is transformed to wave energy and then to electron thermal energy. If we trace the energy transfer between different plasma populations, the energy goes from the low-energy ions/electrons (which contribute most to the diamagnetic drift) to the parallel random motion of higher-energy (~100 eV) electrons. This electron heating through wave–particle interactions can operate in addition to the parallel heating due to electron trapping associated with reconnection itself.

We appreciate the MMS team and the MMS Science Data Center (https://lasp.colorado.edu/mms/sdc/public/) for providing the MMS data for this study. Computer codes used for the calculations in this paper are freely available at http://spedas.org/blog/ and https://github.com/irfu/irfu-matlab. The work at NSSC was supported by NNSFC grants (41874175, 42174207, 41731070), the Specialized Research Fund for State Key Laboratories of China, the Strategic Pionner Program on Space Science II, Chinese Academy of Sciences, grants XDA15350201 and XDA15052500, and the Preliminary Study Program on Civil Aerospace Technology, No. D050103.

## ORCID iDs

Lei Dai https://orcid.org/0000-0002-5122-3066

Benoit Lavraud https://orcid.org/0000-0001-6807-8494

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