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

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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Before 08:24:10 UT, MMS is located on the magnetosphere side, characterized by a steady northward magnetic field (B_L > 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⁻¹ 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 reconnection. 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_LH ≈ 20 Hz (Figure 1(g)) during the jet crossing. The properties of these fluctuations are analyzed next.

Figure 2 displays a detailed 1-minute view of the reconnection ion jets and associated LHWs. As illustrated in the schematic of panel (i), magnetosheath inflow ions are accelerated across the reconnection layer and form reconnection 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 current perpendicular to B and ∇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.

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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 β 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).

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 δ ϕ_E and δ ϕ_B are obtained as

$$\begin{eqnarray}&&\delta {\phi }_{E}=\int \delta {\boldsymbol{E}}\cdot {{\boldsymbol{V}}}_{\mathrm{ob}}{dt}\end{eqnarray} \tag{ 1 }$$

and

$$\begin{eqnarray}&&\delta {\phi }_{B}=\displaystyle \frac{| {{\boldsymbol{B}}}_{0}| \delta {B}_{\parallel }}{{{en}}_{e}{\mu }_{o}},\end{eqnarray} \tag{ 2 }$$

where B ₀ is the background magnetic field, δ B_∥ is the fluctuation of wave magnetic field aligned with B ₀, e is the electron charge, n_e is the electron density, and δ E are the fluctuations of the wave electric field. δ ϕ_E and δ ϕ_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 ϕ_B and ϕ_E from different wave angles. LHWs in cases 1 and 2 propagate mainly in the − M direction in the spacecraft frame. Using f = 20 Hz, the wavelengths λ = V_ob/f are 13.4 and 9.2 km. The wavelengths are larger than the thermal electron gyroradius ρ_e ∼ 430 m and smaller than the thermal ion gyroradius ρ_i ∼ 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 = ∇p × B /qnB² is estimated using the data from mms2 and mms3. Using n = 2.5 cm⁻³, B = 30 nT, V _di = − 250 km s⁻¹, and V _de =19 km s⁻¹ (case 1), the estimated current density j _d = ne( V _di − V _de) ∼ −9.2 × 10⁻⁸ A m⁻² is approximately consistent with the observed values j_M ∼ −2 × 10⁻⁸ A m⁻² (Figure 2(e)). Using a density n = 2.0 cm⁻³, B = 30 nT, diamagnetic drift V _di = 210 km s⁻¹, and V _de = −13 km s⁻¹ (case 2), the estimated j _d is ∼5.8 × 10⁻⁸ A m⁻², approximately consistent with the observed value (j_M ∼ 2 × 10⁻⁸ A m⁻²) in Figure 2(e).

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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 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)

$$\begin{eqnarray}\begin{array}{rcl}0 & = & 1-\displaystyle \frac{{\omega }_{\mathrm{pi}}^{2}}{{k}^{2}{v}_{i}^{2}}{Z}^{{\prime} }\left(\displaystyle \frac{\omega -{k}_{\perp }{V}_{{Di}}}{{{kv}}_{i}}\right)+\displaystyle \frac{2{\omega }_{\mathrm{pe}}^{2}}{{k}^{2}{v}_{e}^{2}}\\ & & \times \left(1+\displaystyle \frac{\omega -{k}_{\perp }{V}_{\mathrm{De}}}{({k}_{\parallel }\cos \alpha +{k}_{\perp }\sin \alpha ){v}_{e}}{J}_{0}(b){I}_{0}(\lambda ){e}^{-\lambda }Z(\xi )\right),\end{array}\end{eqnarray} \tag{ 3 }$$

where $b=(({k}_{\parallel }\sin \alpha -{k}_{\perp }\cos \alpha ){V}_{\mathrm{De}}\cos \alpha )/{{\rm{\Omega }}}_{\mathrm{ce}}$, $\lambda \,=({({k}_{\parallel }\sin \alpha -{k}_{\perp }\cos \alpha )}^{2}{v}_{e}^{2})/2{{\rm{\Omega }}}_{\mathrm{ce}}^{2}$, $\xi =\omega /({k}_{\parallel }\cos \alpha +{k}_{\perp }\sin \alpha ){v}_{e}$, Z is the plasma dispersion function, ω_pi and ω_pe are the ion and electron plasma frequencies, v_i and v_e are the thermal speeds of ion and electron, J₀ and I₀ are the Bessel function of the first kind and the modified Bessel function of the first kind in zero order, and α is the angle between the guide field B_G and B₀. For the case of no guide field B_G , sinα = B_G /B₀ = 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 and growth rate of the LHDI. The LHDI is unstable for the wave-normal angle θ ≳ 88° (cases 1 and 2). The mode propagating perpendicular to B ₀ reaches a peak growth rate at ω_rest ≈ 0.6ω_LH, similar to the values reported in the ion diffusion (Graham et al. 2017, 2019). The solutions of the dispersion relation that yield the ${\gamma }_{\max }$ (θ = 90°) are plotted in panels (c1) and (c2). The predicted ranges of ω_SC/ω_LH (0.5–4ω_LH) are in agreement with the observations in the spacecraft frame in Figure 2(g).

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Panels (d1), (d2), (e1), and (e2) display the parallel phase speeds ω/k_∥ of LHDI. The value of ω/k_∥ is useful in the context of Landau resonance. According to the resonance condition, electrons with a parallel resonance speed v_∥ = ω/k_∥ experience Landau resonance. The predicted energy range for electrons to participate in Landau resonance for this event is v_∥ ≳ 6000 km s⁻¹ (∼100 eV; panels (f1) and (f2)).

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 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).

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An important feature to notice is the large-scale electron temperature anisotropy (T_e⊥/T_e∥ < 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 display in situ evidence of wave−particle interactions (the Landau resonance) using high-resolution 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° ≲ θ_PA ≲ 40° or 140° ≲ θ_PA ≲ 180°. The dynamic spectrum in Figures 5(b)–(e) shows the intense fluctuations of electron flux around ∼20 Hz (∼1.0f_LH). The most intense fluctuations are in the field-aligned direction. At the same time, there is enhanced power in E_∥ 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 δ v and δ N (and thus electron flux δ 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 ω/k_∥ in the antiparallel direction in subinterval 4. In subintervals 1, 2, and 3, the flux oscillations due to wave−particle interaction are strong in the parallel diction. Accordingly, ω/k_∥ of the waves are expected in the parallel direction in subintervals 1, 2, and 3.

Figure 7 displays the phase relation between δF and δ E_∥ during the wave−particle interaction. As marked by the shaded region, δF and δ E_∥ 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):

$$\begin{eqnarray}&&\displaystyle \frac{\partial \delta f}{\partial t}+{\boldsymbol{v}}\cdot {\rm{\nabla }}\delta f-\displaystyle \frac{q}{m}\delta {E}_{z}\cdot \displaystyle \frac{\partial {f}_{o}}{\partial {\boldsymbol{v}}}=0.\end{eqnarray} \tag{ 4 }$$

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Fourier transforming the linearized equation, $\tfrac{\partial }{\partial t}=-i\omega$, $\tfrac{\partial }{\partial z}={{ik}}_{z}$, we obtain

$$\begin{eqnarray}&&\delta \tilde{f}=\displaystyle \frac{{iq}(\delta \tilde{{E}_{z}})}{{m}_{e}}\displaystyle \frac{\partial {f}_{o}/\partial {v}_{z}}{\omega -{k}_{z}{v}_{z}}.\end{eqnarray} \tag{ 5 }$$

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 ω − k_z v_z is either positive or negative, leading to a ±90° phase difference between the flux and electric field. Near the Landau resonance condition ω − k_z v_z = 0, the denominator approaches 0 unless we introduce an imaginary part of the frequency ω = ω_r + i γ. This is equivalent to saying that the waves can grow or decay. Near the resonance condition ω_r − k_z v_z = 0, Equation (5) becomes

$$\begin{eqnarray}&&\delta \tilde{f}\sim \displaystyle \frac{q(\delta \tilde{{E}_{z}})}{{m}_{e}}\displaystyle \frac{\partial {f}_{o}/\partial {v}_{z}}{\gamma }.\end{eqnarray} \tag{ 6 }$$

Hence, the electrons in Landau resonance are in phase (0°) or antiphase (180°) with the parallel wave electric field. This is distinctly different from the ±90° 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 ω 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 (γ → 0⁺). Considering a parallel-propagating wave (∂f_o /∂v_z < 0), the electron flux shall be 180⁰ 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.