Plain Language Summary Determining radiation belt electron loss rates and mechanisms is a key aspect of modeling and predicting the highly dynamic near-Earth radiation environment; however, it is typically not well studied due to a dearth of measurements from a low-altitude, high-latitude vantage point. One such loss mechanism is the resonant interaction of electrons with whistler-mode waves, a type of electromagnetic wave. This interaction can knock electrons into Earth's atmosphere, thus causing them to be lost (i.e., "precipitated"). New measurements from NASA's Electron Losses and Fields INvestgation (ELFIN) mission (a pair of CubeSats built and operated by UCLA) can measure these precipitating electrons for the first time. When ELFIN and another equatorial satellite are on/near the same magnetic field line (a "conjunction"), we can measure two points in time during the time evolution of an electron distribution that eventually precipitates. Two such conjunctions are analyzed in this study with NASA's Time History of Events and Macroscale Interactions during Substorms and Magnetospheric MultiScale missions. High-performance particle simulations are employed based on equatorial measurements; modeled precipitating fluxes are found to agree well with ELFIN measurements when accounting for how far whistler waves propagate away from the equator. This suggests the potential use of just ELFIN's precipitation measurements to estimate the characteristics of the whistler-mode waves that scattered them.

O 'Brien, 2016;Breneman et al., 2017;O'Brien et al., 2004;Shumko et al., 2018;Thorne et al., 2005), pulsating auroras (tens of keV; see, e.g., Kasahara, Miyoshi, et al., 2018;Nishimura et al., 2010), and diffuse auroras (keV to tens of keV; see, e.g., Ni et al., 2016;Nishimura et al., 2020;Thorne et al., 2010). Since such resonant scattering is also associated with electron acceleration (e.g., Allison & Shprits, 2020;Li et al., 2014;Thorne et al., 2013), whistler-mode waves act as an efficient energy exchange mechanism between different electron populations on microphysical timescales (e.g., Shklyar, 2011;Shklyar & Matsumoto, 2009). These resonant interactions are key to the dynamics of Earth's radiation belts, provide an important space weather proxy (Horne et al., 2013), and are critical to incorporate in radiation belt flux models. In addition, recent studies show that relativistic precipitation can deposit energy deep into the D-region (Xu et al., 2020) and therefore affect atmospheric chemistry by way of ozone destruction (e.g., Lam et al., 2010;Thorne, 1980;Turunen et al., 2016).

Although electron resonant interactions with whistler-mode waves are believed to be a major driver of electron precipitation, it remains difficult to directly measure and associate precipitation with resonant wave-particle interactions. This is partly because precipitation occurs over a wide spatial and temporal range: while it is oftentimes slow and quasi-steady (e.g., Ni et al., 2016, and references therein), it can sometimes take the form of precipitation bands (which last from a few seconds to few minutes) or even erupt in the form of violent microbursts (which last a few hundred milliseconds; Blum et al., 2015;Mozer et al., 2018;O'Brien et al., 2004;Shumko, Blum, & Crew, 2021). These effects are difficult to resolve at the equatorial plane due to lack of observations with sufficiently high time and energy resolution required to follow the dynamics of electron fluxes. Much of the ambiguity stems from the inability to directly observe the spatiotemporal evolution of a single whistler-mode wave-packet propagating along a geomagnetic field line and its interaction with electrons. Yet, such evolution along field lines is key to understanding the elementary physics of resonant interactions (see discussion in Hiraga and Omura (2020) and Zhang, Agapitov, et al. (2020)). Observing the evolution of the waves away from their source location at the equator is difficult because a single whistler-mode wave-packet lasts for only a few hundred milliseconds (e.g., Zhang et al., 2018), yet, such propagation is essential to the evolution of the resonant energy along the raypath and the precipitation spectra resulting from the wave-electron interaction.

Statistical (Li et al., 2013;Ni et al., 2014) and case-study (e.g., Mozer et al., 2018;Nishimura et al., 2010) results have shown a good correlation between equatorial whistler-mode wave intensity and subrelativistic (∼10 keV) electron precipitation. In fact, this correlation has recently been confirmed by Kasahara, Miyoshi, et al. (2018) with the first direct equatorial measurements of electron precipitation driven by resonances with whistler-mode waves on the Exploration of energization and Radiation in Geospace (ERG/Arase, Miyoshi et al., 2017) mission. The loss cone resolving instrument on-board ERG (Kasahara, Yokota, et al., 2018) shows direct correlation of the loss cone filling by ∼20 keV electron fluxes with bursts of whistler-mode intensity. Recent studies also suggest that precipitation of ∼10-30 keV electrons associated with whistler-mode wave driven pulsating aurora is also well correlated with precipitation of energetic electrons (100-500 keV) and even relativistic (∼1 MeV) electron microbursts (Miyoshi et al., 2020(Miyoshi et al., , 2021;;Shumko, Gallardo-Lacourt, et al., 2021). Thus, energy ranges and precipitating electron spectra are likely controlled by whistler-mode wave-packet characteristics and ambient plasma conditions affecting the resonant energy of the interaction. In order to reveal which characteristics are truly responsible, it is necessary to embark on a more systematic investigation of the correlations between precipitating electron spectra and waves.

In this study, we examine a new data set of electron precipitation from the ELFIN CubeSats (Angelopoulos et al., 2020). The primary advancements ELFIN delivers are the high pitch angle and energy resolution measurements of electron fluxes at around 400-450 km altitude. Spinning at ∼21 RPM, ELFIN can resolve trapped (outside of the loss cone) and precipitating (within the loss cone) fluxes of 50-5,000 keV electrons. ELFIN conjunctions with equatorial missions, like Time History of Events and Macroscale Interactions during Substorms (THEMIS; Angelopoulos, 2008) and Magnetospheric Multiscale (MMS; Burch et al., 2016), provide a unique opportunity for analyzing whistler-driven electron precipitation.

We first describe the data sets and spacecraft instruments in Section 2.1 before presenting the two selected conjunction events in Sections 2.2 and 2.3. In the first conjunction, THEMIS-A and THEMIS-E showed bursts of intense whistler-mode waves propagating in the lower band chorus wave frequency range (0.1f ce -0.5f ce ), while ELFIN-B observed strong electron precipitation from tens of keV to beyond 300 keV. In the second event, MMS measured three spatially localized sources of intense whistler-mode waves distributed between the plasma sheet inner edge and the plasmasphere. Around the same time, ELFIN-A rapidly traversed three bursts of strong electron precipitation (tens to ∼300 keV) distributed across the outer radiation belt. To compare these data sets with the model of wave-particle resonant interactions, we use a test particle simulation introduced in Section 3.1, the implementation details of which are discussed in Section 3.2. Detailed results are shown in Section 4 and discussed in Section 5, where we show that the latitudinal extension of whistler-mode waves is a key parameter controlling the observed scattering of energetic electrons.

In this study, energetic electron fluxes, magnetic field, and/or wave data are used from three missions to analyze two events that reveal effects of electron scattering due to nonlinear resonance with whistler-mode waves. We use measurements from the ELFIN mission (Angelopoulos et al., 2020): two identical CubeSats, ELA and ELB, each equipped with an Energetic Particle Detector for Electrons (EPDE) instrument measuring pitch angle resolved fluxes from 50 keV to 5 MeV over 16 energy bins while spinning at roughly 21 RPM. The energy resolution (ΔE/E < 50%) and angular resolution (16 sectors per spin, or ∼22.5° spin phase resolution), allows ELFIN to adequately resolve the bounce loss cone in their low-altitude polar orbit and therefore distinguish between precipitating and trapped electron populations. These precipitating electron fluxes are then used for verification of theoretical models predicting equatorial electron scattering by whistler-mode waves.

In the first conjunction event, we utilize two spacecraft (TH-A and TH-E) from the THEMIS mission (Angelopoulos, 2008) for equatorial measurements. The background magnetic field is provided by the fluxgate magnetometer (Auster et al., 2008) and the electron differential fluxes and pitch angle distributions are provided by a pair of instruments: the electrostatic analyzer (McFadden et al., 2008) for <25 keV electrons and the solid state telescope (Angelopoulos et al., 2008) for 30-700 keV electrons. We use magnetic field and electric field measurements of waves between 10 Hz and 4 kHz obtained by the search-coil magnetometer (SCM; Le Contel et al., 2008) and the electric fields instrument (EFI; Bonnell et al., 2008), respectively. During this event, Fast Survey spectra (FFF data product) was available on THEMIS-A, whereas wave-burst waveforms (SCW and EFW data products,8,192 samples per second) from SCM and EFI were available from THEMIS-E.

For the second conjunction event, MMS (Burch et al., 2016) provides the equatorial measurements. The MMS mission consists of four spacecraft that are held in tight formation, so data from any one spacecraft provides near-identical results for our purposes; thus, we use measurements from MMS-1 here. The Fly's Eye Energetic Particle Spectrometer (FEEPS; Blake et al., 2016) provides differential flux measurements of electrons in the range of 25-600 keV. The analog and digital fluxgate magnetometers (Russell et al., 2016) provide background magnetic field information, and a search-coil magnetometer (Le Contel et al., 2016) provides magnetic field wave measurements in the range between 2 Hz and 6 kHz. Data were only available in survey mode at this time, so with only 32 samples per second on the SCM, we must infer some equatorial wave properties.

Figure 1 shows the orbital configuration of the conjunction between the THEMIS-A, THEMIS-E, and ELFIN-B that occurred on 29 April 2021. THEMIS-A and THEMIS-E begin at the plasmasphere around 03:00 UT in an outbound track through the outer radiation belt and into the plasma sheet. The energy spectra (Figure 2a) shows a gradual decrease of relativistic electron fluxes from L = 5.2 to L = 9 (L in IGRF field is used throughout our analysis). For the entirety of this 3 hour period, the FFF data set from THEMIS-A (Figure 2b) shows a clear presence of whistler-mode waves in the lower band chorus wave frequency range (i.e., 0.1-0.5 of the local electron gyrofrequency, f ce ). Wave bursts are observed for the entire L-shell range of the outer radiation belt; in particular, two intense wave bursts are seen at L ∈ [5.2, 5.5] and L ∈ [5.8, 6.0], close to the observed plasmapause at L ∼ 5.1 (not shown).

During this conjunction, a 6-min long equator-ward ELFIN-B crossing of the plasma sheet and outer radiation belts occurred around 03:15 UT. Figure 3a shows the distribution of trapped electron fluxes as a function of L-shell. The plasma sheet (prior to 03:13 UT) is characterized by <300 keV highly isotropic electrons (average fluxes of precipitating and trapped are about the same; see Figure 3c). The energy of trapped electrons increases as ELFIN moves from the plasma sheet to the outer radiation belt until it crosses the plasmapause at around L ∼ 4 (shortly after 03:16 UT), where we see a characteristic rapid decrease of energetic electron fluxes. Within the outer radiation belt (from 03:14 to 03:16 UT), ELFIN observed several bursts of electron precipitation (see electron fluxes in Figure 3b), with the most intense burst at L ∈ [5.1, 5.3] (between 03:14:45 and 03:15:00 UT, as marked by the red rectangle). In this burst, ELFIN measures electrons precipitating with energies up to 800 keV (see Figure 3b), with the most intense precipitating fluxes (where the precipitating-to-trapped flux ratios are near one) at 300 keV and below (see Figure 3c). We will use THEMIS and ELFIN measurements at this particular precipitation burst to compare with results from our theoretical model of wave-particle resonant interactions.

The second event occurred on 22 September 2020 and was a conjunction between MMS and ELFIN-A. MMS was on an inbound trajectory (Figure 4), leaving the plasma sheet at around 08:00 UT and entering the plasmasphere around 09:20 UT. During this crossing of the outer radiation belt (between 08:00 and 09:20 UT, as evidenced by the electron spectra measured by FEEPS in Figure 5a), MMS detected three bursts of whistler-mode waves at 08:28 UT, 08:44 UT, and 09:15 UT (Figure 5b). At around 09:16 UT, ELFIN-A observed three bursts of precipitation up to 300 keV (Figures 6b and 6c) while traversing the outer radiation belt in the southern hemisphere poleward. 6.1 3.5 -65.3 4.3 3.9 -60.0 L-igrf MLT-igrf MLAT-igrf 10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 1 10 2 10 3 10 4 10 5 10 6 10 7 0.01 0.10 1.00 Plasmasphere Rad. Belt 100 1000 100 1000 100 1000 100 1000 100 1000 100 1000 Trapped Energy [keV] Precipitating Energy [keV] Prec/Trap Energy [keV] 1/cm^2-sec-sr-keV Jprec/Jtrap ELFIN-B EPD-E South Ascending 0314 0316 2021 Apr 29 (a) (b) (c) Plasma Sheet 7.6 3.3 -68.0 0313 5.1 3.6 -63.9 0315 3.7 4.0 -57.4

The energetic electron spectrum seen by MMS during this 2-hr period (Figure 5a) shows a clear signature of a plasma sheet injection starting around 08:20 UT, evidenced by a rapid increase in electron fluxes up to 500 keV (compare this with the typical injection signatures described in, e.g., Gabrielse et al., 2014;Turner et al., 2016).

The first burst of intense whistler-mode waves was observed behind the injection, around 08:30 UT (Figure 5b, likely driven by thermal anisotropy of injected electrons, e.g., Le Contel et al., 2009;Zhang et al., 2018). At lower L-shells, MMS observed two other whistler-mode bursts (numbered (2) and (3) atop Figure 5b), with the last one occurring right before MMS crossed the plasmapause, around 09:20 UT. The plasmapause is seen as a sharp boundary separating discrete whistler-mode emissions in the low band chorus frequency range and broadband whistler-mode emissions in the hiss frequency range, f < 0.1f ce (see, e.g., Malaspina et al., 2017).

ELFIN-A crossed a wide L-shell range over a 5-min interval centered around 09:16 UT, seen in Figure 6. The precipitating-to-trapped electron flux ratios in panel (c), examined together with the trapped fluxes in panel (a), reveal three main regions: plasma sheet, corresponding to highly isotropized electrons below 300 keV; outer (blue) traced to the equator using the T96 model (Tsyganenko, 1995). The plus signs demarcate 1-min intervals for ELFIN-A and 1-hr intervals for MMS.

radiation belt, corresponding to energetic (50 keV to 1 MeV) electrons which do not exhibit continuous, but only occasional, isotropization; and the plasmasphere, corresponding to mostly relativistic (>500 keV) electrons. The plasmapause is likely crossed around 09:15:50 UT at L ∼ 5.6. In the plasmasphere, whistler-mode hiss waves scatter energetic <500 keV electrons and form the clear low energy boundary of electron fluxes with the observed inverse energy/L-shell dispersion (see model/data comparison in, e.g., Ma et al., 2016;Mourenas et al., 2017). Within the outer radiation belt (between 09:15:50 and 09:17:00 UT), ELFIN-A observes three bursts of electron precipitation (numbered 1-3 above Figure 6a). The precipitating fluxes during those bursts are limited to <300 keV (Figure 6b), but the precipitation is quite intense, with precipitating-to-trapped ratios of around 1 1/(cm -sec-sr-keV) 10 -8 10 -6 10 -4 nT /Hz 10 3 10 5 10 10 1 2 2 1000 SCM Freqeuncy [Hz] 100 FEEPS Energy [keV] 100 7.4 2.1 -21.3 0750 L-igrf MLT-igrf MLAT-igrf 2020 Sep 22 MMS-1 FEEPS and SCM Observations 400 0.1 fce 0.5 fce fce 0.35 fce 3 1 2 (a) (b) 7.1 2.3 -21.3 0800 6.7 2.4 -21.2 0810 6.0 2.9 -20.8 0830 6.4 2.6 -21.1 0820 5.6 3.1 -20.3 0840 5.2 3.4 -19.7 0850 4.8 3.7 -18.8 0900 4.0 4.4 -15.9 0920 4.4 4.0 -17.6 0910 3.6 4.9 -13.7 0930 Figure 6. Trapped (a) and precipitating (b) electron fluxes are shown for Electron Losses and Fields INvestgation's (ELFIN-A) 5 min crossing, with panel (c) showing the precipitating-to-trapped ratios. The three bursts of electron precipitation up to 300 keV are highlighted in red and numbered, likely corresponding to the three bursts of whistler activity seen by MMS in Figure 5. This study will focus on conjunction data from (1) at L = 7.

3 1 2 3.7 1.4 5.8 1.5 -64.6 10.3 1.8 -71.2 -57.4 L-igrf MLT-igrf MLAT-igrf ELFIN-A EPD-E South Descending Plasmasphere Rad. Belt 0916 0918 0914 2020 Sept 22 10 1 10 2 10 3 10 4 10 5 10 6 10 1 10 2 10 3 10 4 10 5 10 6 10 7 0.01 0.10 1.00 100 1000 100 1000 100 1000 100 1000 100 1000 100 1000 Trapped Energy [keV] Precipitating Energy [keV] Prec/Trap Energy [keV] 1/cm^2-sec-sr-keV Jprec/Jtrap (a) (b) (c) 10 7 Plasma Sheet 4.6 1.4 -61.0 0915 7.6 1.7 -68.0

(Figure 6c). We use the third burst of precipitation (numbered (1) in Figure 6, at L = 7 between 09:16:38 and 09:16:45 UT) to compare with our model.

To examine whether the observed precipitation patterns are consistent with our interpretation based on wave-particle resonant interaction theory, we employ a test particle simulation with a sufficiently large number of particles to numerically generate statistically significant precipitation spectra. We use the Hamiltonian formulation for wave-particle resonant interaction model (e.g., Albert et al., 2013;Vainchtein et al., 2018), which includes nonlinear effects such as phase bunching and phase trapping (see detailed description of these effects in, e.g., Bortnik et al., 2008;Demekhov et al., 2006;Katoh et al., 2008;Omura et al., 2007). We begin with a curvature-free dipole magnetic field (Bell, 1984) defined by

where z is the field-aligned coordinate (z = 0 is the equator) and x is the cross-field coordinate. The Hamiltonian of a relativistic electron (with rest mass m and charge -e) in such a vector potential with a wave vector potential of 𝐴𝐴 ⃗ 𝐴𝐴𝑤𝑤 can be described with just two pairs of conjugate variables. One pair is the field-aligned coordinate and momentum (z, p z ); since the z variables change much slower than gyrorotation, this allows us to introduce the other pair: gyrophase θ and magnetic moment 𝐴𝐴 𝐴𝐴𝑥𝑥 = (2π) -1 ∮p x d 𝐴𝐴 𝐴𝐴 . For a field-aligned whistler wave with phase ϕ, the Hamiltonian can be written as a sum of the unperturbed Hamiltonian 𝐴𝐴 0 and a wave perturbation component 𝐴𝐴 1 (e.g., Albert, 1993;Artemyev, Vasiliev, et al., 2015;Vainchtein et al., 2018):

where 𝐴𝐴 𝐴𝐴𝑥𝑥 = ℎsin 𝛼𝛼 2 𝑒𝑒𝑒𝑒 ∕2Ω-( 0) is defined based on the equatorial pitch angle α eq and particle energy h = mc 2 (γ -1), and Ω -= eB 0 (z)/mc is the electron gyrofrequency. Note the use of Cartesian coordinates, which implies that field curvature does not affect wave-particle interaction. However, B 0 (z), which can generally be defined arbitrarily, does describe the distribution of a dipole field along a field line; that is, we set B 0 = B eq b(λ) with

, where λ is the magnetic latitude and R = LR E .

The wave phase ϕ is defined by the wave number k(z) = ∂ϕ/∂z and wave frequency ω = -∂ϕ/∂t, whereas k is given by the cold plasma dispersion relation for an R-mode whistler wave at a constant frequency (Stix, 1962):

where ω p is the plasma frequency. We introduce a dimensionless wave number K:

where ω m = ω/Ω -(0) is the (whistler) wave frequency (normalized to the equatorial gyrofrequency); Ω p = ω p (0)/ Ω -(0) is the plasma frequency (also normalized to equatorial gyrofrequency); and an empirical model for the plasma density (Denton et al., 2006) has been used to account for the variation along the field line as: ω p / ω p (0) = cos -5/2 λ. We also convert the variables in Equation 2 to dimensionless ones with the following substitutions/normalizations: z → z/R, p z → p z /mc, t → tc/R, 𝐴𝐴  → ∕𝑚𝑚𝑚𝑚 2 , and 𝐴𝐴 𝐴𝐴𝑥𝑥 → 𝐴𝐴 𝐴𝐴𝑥𝑥 /mcR. For such normalized variables, there are two main system parameters: η = RΩ -/c, the normalized spatial scale of the background magnetic field inhomogeneity (so kR = ηK) and ϵ = 𝐴𝐴 𝐴𝐴𝑤𝑤 (0)/B 0 (0), the wave amplitude normal-ized to the equatorial background magnetic field. The wavefield obeys 𝐴𝐴 𝐴𝐴𝑤𝑤 (λ) = k(λ) 𝐴𝐴 𝐴𝐴𝑤𝑤 (λ). To parameterize the variation of the wavefield with latitude and incorporate wave modulation, we express the vector potential as 𝐴𝐴 𝐴𝐴𝑤𝑤 / mc 2 = ϵf(λ)g(ϕ)/K(λ), where f(λ) determines wave amplitude distribution along magnetic field lines and g(ϕ) describes the shape of wave-packets. Both functions f and g are sufficiently smooth, df/dz ≪ k and dg/dϕ ≪ 1, and are discussed further in Section 3.2. Finally, we introduce the combined phase of the electron gyrophase and whistler wave phase ζ = ϕ + θ, along with normalized magnetic moment μ = 𝐴𝐴 𝐴𝐴𝑥𝑥η. All combined, this yields the following equations of motion:

where ' = ∂/∂z and terms ∼ df/dz, dg/dϕ are neglected.

In Earth's radiation belt magnetic field, η ∼ 10 4 ≫ 1, whereas ϵ ∼ 10 -3 ≪ 1. Thus, ζ (in Equation 5) is the fastest changing variable since 𝐴𝐴 ζ𝜁 ∼ 𝜂𝜂 . For an accurate numerical integration of the equations of motion, the integration time step Δt should be at least one order of magnitude less than 1/η. To optimize the simulation procedure the integration code is written in Julia (Bezanson et al., 2017), a relatively new scientific computing language that is as performant as C and Fortran for most differential equation solvers (Rackauckas & Nie, 2017). Each test trajectory is integrated up to the loss cone crossing (or up to the end of the simulation time, whichever comes first). To account for particles within the loss cone that can be scattered to higher pitch angles due to anomalous trapping (Albert et al., 2021;Artemyev, Neishtadt, et al., 2021;Kitahara & Katoh, 2019), each trajectory which enters the loss cone is further integrated until p z = 0 (true mirror point) before being considered "lost".

Equation 5 show that for nonlinear wave-particle interaction the wave force ∼ϵη cos ζ should be comparable to (or even larger than) the mirror force ∼ μ 𝐴𝐴 𝐴𝐴 ′ /γ; that is, the wave amplitude should be large enough such that ϵη > 1 (see discussion of ϵη values for observed waves in Zhang et al. (2019)). For example, at L = 5, typical whistler-mode waves with B w ≥ 100 pT (Agapitov et al., 2013;Li, Bortnik, et al., 2011;Li, Thorne, et al., 2011) have ϵη > 1, and can thus interact with electrons nonlinearly.

The majority of whistler-mode chorus waves propagate in the form of short packets with 10-30 wave periods per packet (Zhang et al., 2018). Strong modulation of wave-packets can destroy long-term resonances (Allanson et al., 2020(Allanson et al., , 2021;;Tao et al., 2013) and lead to less efficient nonlinear wave resonance with electrons (e.g., An et al., 2022;Gan et al., 2020;Hiraga & Omura, 2020;Mourenas et al., 2018;Zhang, Mourenas, et al., 2020). To account for this effect, we use the function g(ϕ) which specifies the wave-packet modulation in accordance with available observations:

where a controls the depth of the modulation (wavefield drops to nearly zero between wave-packets if a ≫ 1) and δϕ controls the number of wavelengths within each wave-packet.

Another key parameter to explore in this simulation is the latitudinal distribution of the whistler wave intensity. Whistler-mode chorus waves are typically generated around the equator, then propagate toward higher latitudes (Helliwell, 1967). Such propagation is associated with an increase of wave obliquity (e.g., Breuillard et al., 2012;Watt et al., 2013) and resultant damping due to Landau resonance with suprathermal electrons (Bortnik et al., 2007;Chen et al., 2013). The exception to this wave evolution is the situation when, due to ducting, whistler-mode waves propagate to high latitudes while remaining quasi-parallel, thus avoiding damping (e.g., Karpman & Kaufman, 1982;Pasmanik & Trakhtengerts, 2005;Streltsov & Bengtson, 2020). Statistical data sets collected by off-equatorial spacecraft provide relevant latitudinal wave intensity models (Agapitov et al., 2018;Haque et al., 2010;Wang et al., 2019) which we can incorporate using the function f(λ):

where δλ 1 describes the spatial scale of the wave source region (δλ 1 ∼ few degrees; see estimates in Agapitov et al. (2018)) and δλ 2 describes the spatial scale of wave intensity decay due to damping/wavefield divergence (δλ 2 ∼ tens of degrees).

In the first event, we use B eq = 188 nT based on THEMIS FGM measurements (as also evident from f ce line in Figure 2b). During the pertinent portion of THEMIS-A observations (03:00-03:30 UT), the normalized wave frequency remained fairly constant around ω m = ω/Ω -≈ 0.35f ce , as did the normalized plasma frequency Ω p = ω p / Ω -≈ 8, obtained using density measurements inferred from the spacecraft potential measurements of the EFI instrument. THEMIS' electron phase space density (PSD) as a function of pitch angle and energy and its fits are shown in Figure 7, at around the time of the closest conjunction between THEMIS-A and ELFIN-B (L = 5.2). These fits determine the initial equatorial PSD for numerical simulations.

THEMIS-E SCM measurements for two out of the three magnetic field components were not fully calibrated at the time of this writing, so we used electric field measurements (and compared them with the one calibrated SCM Z component spectra) to estimate the wave-packet shape and magnetic field amplitude. Figure 8 shows examples of wave-packets captured by THEMIS-E SCM and EFI in Wave-Burst (WB) mode of operation. We find an average of δϕ ≈ 30 waves/packet, in line with previous statistical models and wave generation simulations (see Zhang et al., 2021, and references therein). We used 3D EFI electric field measurements to determine the magnetic field from the cold plasma dispersion relation (see details in, e.g., Agapitov et al., 2014;Ni et al., 2011;Tao & Bortnik, 2010). Using Minimum Variance Analysis (MVA; Sonnerup & Cahill, 1968), we can find E N along the minimum variation of the electric field in the LMN principle axis coordinate system. The other two components E L and E M have similar amplitudes, implying that these waves are field-aligned and circularly polarized. We then estimate the wave magnetic field amplitude

. Averaging over all wave-burst measurements gives 〈B w 〉 ≈ 162 pT with peak values for individual wave-packets commonly reaching 0.9 nT. Thus, we use wave-packets with packet-averaged wave amplitude of B w /2 ≈ 450 pT (〈g〉 ϕ ≈ 1/2, since we do not have waves roughly half the time) and normalize the wave occurrence rate to attain the average wave amplitude: 162/450 ≈ 36%.

To obtain the input wavefield distribution along magnetic field lines in our simulation, we use the empirical model of Agapitov et al. (2013) showing that in the range of 4 < L < 5.5 and for Kp < 3, the ratio of whistler wave occurrence rates between λ < 20° and λ ≈ 30°-40° is ∼5. Thus, we use three simulation runs with different δλ 2 and whistler-mode wave occurrence rates: δλ 2 = 20° and occurrence rate of 30%, δλ 2 = 30° and occurrence rate of 4%, δλ 2 = 40° and occurrence rate of 2% (so the total wave occurrence rate is 36%, while the net occurrence rate at <20° (30%) is 5 times that between 30° and 40° (6%)).

Figure 9 shows simulated trajectories of electrons in pitch angle and energy, as a function of time, for six example electrons. The initial pitch angles are fixed at 30°, but initial energies and δλ 2 were varied (as indicated in the plots) in order to demonstrate the variation of resonant energy with higher latitude propagation. The ensemble simulation was run for just over ∼100 bounce periods (∼11s with a typical bounce period of ∼0.1s), which is sufficiently long enough to densely sample the particle energy spectrum given the two resonances per bounce period (wave-particle interactions occur on both sides of the equator); however, the individual trajectories in Figure 9 were run for triple the length for illustration purposes (∼33 s). In Figures 9a and 9b, chorus waves are limited to λ < 20°, and are thus not expected to resonate with high energy electrons. While plenty of phase bunching and trapping are evident for the two example electrons at energies <100 keV, the higher-energy electron energies and pitch angle remain fairly constant, implying no abrupt changes from their interactions with the input fixed-frequency waves. The two resonant electron trajectories (<100 keV) exhibit evidence of the less common phase trapping (large energy and pitch angle increases) mixed with regular phase bunching (small energy and pitch angle decreases). The combination of multiple phase trapping and phase bunching generally tends to uniformly distribute electrons in energy/pitch angle space (not shown, but see also Hsieh & Omura, 2017;Vainchtein et al., 2018). As δλ 2 is increased to 30° (Figures 9c and 9d), even higher energies (up to 500 keV) efficiently resonate with waves; after consistent bunching and some phase trapping, they appear within the model loss cone, that is, wave-particle interaction results in electron precipitation. For δλ 2 = 40°, the same phase trappings and bunchings are seen but now also for relativistic electrons (Figures 9e and 9f). Figure 9 thus confirms the importance of parameter δλ 2 and strongly suggests the nonlinear nature of wave-particle resonances for this event.

To model the observed precipitation spectrum, we use an ensemble of electrons with initial energies ∈ [50, 1,000] keV and pitch angles ∈ [3°, 90°]. Each electron represents a fraction of its initial phase space density -E). Because two of the axes of the SCM were not fully calibrated, we use electric fields instrument (EFI) burst data to obtain the amplitude of the wave. We translate to LMN coordinates (using Minimum Variance Analysis, see Sonnerup & Cahill, 1968) and use the main electric field components to find a proxy magnitude of the wave magnetic field amplitude (see text for details).

E E w 1000 10 -9 10 -8 10 -7 10 -6 P Tot .300 .350 .400 0 30 E Apr 29 05:03:02 L M THEMIS-E SCM/EFI Wave Detail SCM-Z wavelet [Hz] EFI Ewave [mV/m] 2021 0.5 fce

which, by combining all test electrons, contributes to (and matches) the distribution function derived by fitting the THEMIS flux measurements seen in Figure 10 (top-most line). The simulation period is then divided into 10 segments; in each of these intervals, we evaluate the electron phase space density within the loss cone. Such phase space density is recalculated to electron precipitating fluxes, and we show both time-averaged precipitating flux and its standard deviation in Figure 10 from the three sets of simulations plus their sum (black line). The three runs, corresponding to different δλ 2 , show the importance of wave propagation to high latitudes for scattering of relativistic electrons: only the two runs with δλ 2 ≥ 30° provide electron precipitation with energies ≥600 keV.

It is evident from this figure that, albeit small, some fraction of whistler-mode waves must propagate to λ ∼ 40° (orange line) in order to produce the high energy precipitating tail seen in ELFIN's observations (see dark blue curve in Figure 10). However, this wave population alone does not occur frequently enough to account for ELFIN's observations of <300 keV precipitation, and higher occurrence rates of waves which are more confined around the plane are required. We accomplish this by incorporating wave populations at 30° and 20° as well. Note that Figure 10 also clearly shows that waves with δλ 2 = 30° cannot scatter energies >600 keV, and waves with δλ 2 = 20° cannot scatter energies >200 keV electrons.

Precipitating fluxes at ELFIN-B, shown in Figure 10 (blue line) were averaged over five spins from 03:14:45 to 03:15:00 UT (red lines in Figure 3), and error bars denote statistical variance (which are small due to fairly uniform fluxes over the short duration of precipitation). The precipitating fluxes obtained from the test particle simulation driven by the equatorial waves and electrons' measurements, agree well with ELFIN-B observations. The deviation in the lower energy range is most likely due to electron resonant interactions with lower-amplitude waves, which were not included since only monochromatic waves were utilized in the simulation. The simulation/ data comparison in Figure 10 and electron trajectory examples in Figure 9 emphasize the key role of nonlinear electron interactions with waves that can reach high latitudes. These are likely ducted waves (see discussion in Artemyev, Demekhov, et al. (2021), and references therein), since without ducting, whistler waves already become oblique at λ ∼ 20-30° (e.g., Breuillard et al., 2012;Katoh, 2014) and become quickly damped by Landau resonance with suprathermal electrons (Bortnik et al., 2007;Chen et al., 2013).

In the second event, we do not have waveform data because the MMS data set was only available in slow survey mode. Thus, we use δϕ = 30 (same as the first event and fairly typical for intense whistler-mode waves based statistics, see Zhang et al. (2018)) and vary latitudinal extension δλ 2 and modulation depth a to match ELFIN observations. We use B eq = 65 nT, Ω p = ω p /Ω -= 6, ω m = ω/Ω -= 0.3, and L = 6 based on MMS observations and a plasma density model (Sheeley et al., 2001). From MMS FEEPS measurements, we obtain the PSD fits shown in Figure 11 in a similar format to that of Figure 7.

Precipitating flux measurements from ELFIN show electron fluxes only up to 300 keV within the loss cone (see dark blue curve in Figure 12). When δλ 2 = 30°, there is a high energy tail beyond 300 keV of precipitating electron fluxes, which does not match ELFIN observations, and when waves are too confined around the equator, with δλ 2 = 15°, resonant wave scattering is incapable of providing the observed 300 keV electron fluxes. Thus, we can expect that equatorially confined whistler-mode waves with δλ 2 ∼ 20° (typical for night-side injections, see Agapitov et al., 2018;Meredith et al., 2012) are likely responsible for the observed precipitation.

Next, we investigate parameter a in Equation 6, which controls the depth of wave-packet modulation (e.g., a = 0 is an infinite sin wave). Deeper modulation (large a) can change phase trapping efficiency (An et al., 2022;Tao et al., 2012, 2013) by (a) increasing the probability of phase trapping due to sharper wave-packet edges, resulting in electron transport away from the loss cone (Hiraga & Omura, 2020), and (b) decreasing electron diffusive scattering to the loss cone due to the reduced intensity of low-amplitude waves away from wave-packet peaks. Indeed, for fixed δϕ, Figure 13 matches our expectations of reduced scattering into the loss cone as a increases: for a = 3 (max/min wavefield is 1/0.05), modeled fluxes match ELFIN observations well, whereas for a = 7 (max/ min wavefield is 1/10 -3 ), a combination of strong trapping and insufficient scattering results in the inability to precipitate >150 keV electrons. Figure 13 demonstrates that for the second event, a = 3 provides a reasonable explanation for ELFIN-measured precipitation (which agrees well with statistical wave observations, see Zhang et al., 2018). Figures 12 and 13 thus demonstrate that ELFIN observations of rapidly precipitating fluxes can be used to provide a good estimate of both the latitudinal distribution and wave modulation characteristics of electron scattering whistler-mode waves. 0 Figure 13. Similar to Figure 12, except this plot compares the depth of wave modulation by varying a. Packet lengths are kept to a constant δϕ = 30 wavelengths per packet.

In this study, we investigated energetic electron scattering by large-amplitude whistler-mode waves and the associated electron precipitation. Test particle simulations showed that nonlinear wave-particle interaction with whistler-mode waves can explain energetic electron precipitation seen by the low-altitude ELFIN CubeSats using conjugate observations of strong whistler waves near the equator. While a relatively simplified model for whistler-mode waves was used in this study, we still included several important wave characteristics which play a potentially significant role in wave-particle interaction. Specifically, in our model:

1. We incorporated the finite size of wave-packets, which mimics wave-packet modulation (Zhang et al., 2021) and restricts the efficiency of phase trapping (Hiraga & Omura, 2020;Kubota & Omura, 2018;Tao et al., 2013;Zhang, Agapitov, et al., 2020). Based on near-simultaneous conjugate THEMIS observations (consistent with previous statistical studies, Zhang et al., 2018), our simulated wave-packets contained 30 wavelengths. Such wave-packet modulation also reduces the efficiency of anomalous trapping (see Appendix in Mourenas et al. (2021)), which may block precipitation of <100 keV electrons when waves are especially intense (Albert et al., 2021;Artemyev, Neishtadt, et al., 2021;Gan et al., 2020;Kitahara & Katoh, 2019) 2. We did not include wave frequency variation within wave-packets (i.e., an approximation of ω = const), which may reduce efficiency of electron acceleration by trapping (see discussion of the importance of ∂ω/∂t ≠ 0 for trapping acceleration in, e.g., Demekhov et al., 2006;Katoh & Omura, 2007). This simplification should not affect the results of short-term electron precipitation modeling; however, the effects of ∂ω/∂t ≠ 0 should be considered for long-term electron acceleration due to nonlinear resonances (e.g., Hsieh & Omura, 2017;Omura et al., 2015) 3. We incorporated a realistic wave intensity decay as a function of distance from the equatorial plane 𝐴𝐴 𝐴𝐴𝑤𝑤 ∼ exp ( -(𝜆𝜆∕𝛿𝛿𝜆𝜆2) 2 ) , which mimics the effects of wave propagation in an inhomogeneous magnetic field (Breuillard et al., 2012;Katoh, 2014;Watt et al., 2013) and Landau damping at midlatitudes (Bortnik et al., 2007;Chen et al., 2013), both of which are often seen in whistler-mode wave empirical models (e.g., Agapitov et al., 2018;Wang et al., 2019). In order to reproduce ELFIN observations of relativistic electron precipitation, we showed that a small fraction of waves should be assumed to propagate without strong damping or divergence from the field lines up to λ ∼ 40°. This likely corresponds to ducted wave propagation (Chen et al., 2021;Demekhov et al., 2017;Martinez-Calderon et al., 2020) that can be supported by small plasma density variations (Hanzelka & Santolík, 2019;Hosseini et al., 2021;Ke et al., 2021;Shen et al., 2021) and is important for precipitation of relativistic electrons (Artemyev, Demekhov, et al., 2021;Chen et al., 2020Chen et al., , 2021) ) 4. We did not incorporate realistic phase decoherence for wave-packets: that is, there are no perturbations of the wave phase ϕ within each packet. While observed whistler-mode waves are typically highly coherent (Tsurutani et al., 2011(Tsurutani et al., , 2020)), frequency/phase jumps are often detected around packet edges (Santolík et al., 2014;Zhang, Mourenas, et al., 2020). Such wave decoherence affects the efficiency of trapping acceleration (Artemyev, Mourenas, et al., 2015;Brinca, 1978;Tao et al., 2013) and probably requires additional investigation in long-term electron flux dynamics modeling. The effect of wave decoherence on electron scattering due to the phase bunching around the loss cone, however, is expected to minimal.

To summarize, this study shows that ELFIN CubeSat measurements, when combined with equatorial missions like THEMIS and MMS, enable direct quantification of the contributions of nonlinear wave-particle interaction to energetic electron scattering and losses. By using a test particle approach, we have demonstrated the importance of whistler wave characteristics such as the latitudinal extent of wavefield. In a conjunction with THEMIS, we showed that the precipitation seen by ELFIN can only be explained if we take into account a small population of waves (present only 2% of the time) that propagate up to 40° in latitude. Such waves can scatter up to nearly 1 MeV electrons, as ELFIN shows. Comparing two events (with and without relativistic electron precipitation) shows that ELFIN observations are indeed quite capable of characterizing wave populations responsible for electron scattering. We find that even small populations of ducted waves (propagating to midlatitudes in the fieldaligned regime) may play a crucial role in shaping the spectra of precipitating electrons. Therefore, occurrence rate and characteristics of such wave populations should be first thoroughly described, then carefully prescribed into future global radiation belt models.