Abstract
Energetic electron precipitation from Earth’s outer radiation belt heats the upper atmosphere and alters its chemical properties. The precipitating flux intensity, typically modelled using inputs from highaltitude, equatorial spacecraft, dictates the radiation belt’s energy contribution to the atmosphere and the strength of spaceatmosphere coupling. The classical quasilinear theory of electron precipitation through moderately fast diffusive interactions with plasma waves predicts that precipitating electron fluxes cannot exceed fluxes of electrons trapped in the radiation belt, setting an apparent upper limit for electron precipitation. Here we show from lowaltitude satellite observations, that ~100 keV electron precipitation rates often exceed this apparent upper limit. We demonstrate that such superfast precipitation is caused by nonlinear electron interactions with intense plasma waves, which have not been previously incorporated in radiation belt models. The high occurrence rate of superfast precipitation suggests that it is important for modelling both radiation belt fluxes and spaceatmosphere coupling.
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Introduction
Earth’s outer radiation belt, a torusshaped region close to the planet, is filled with energetic electrons^{1}. Because electron fluxes increase dramatically during geomagnetic storms, threatening satellites in that region^{2}, they have been studied theoretically and observationally^{3} throughout the entire history of space exploration. These fluxes are controlled by a delicate balance between multiple accelerations and loss processes in Earth’s magnetosphere^{4}. Whistler waves are particularly important for accelerating electrons to relativistic energies^{5,6,7,8} and for scattering them in pitch angle, α (the angle between electron speed and magnetic field direction), causing their precipitation into Earth’s atmosphere^{9,10,11}. Figure 1a depicts this process of electron scattering by waves and the resulting precipitation. Whistlerwave interaction with electrons has been traditionally modelled in the quasilinear regime^{12}, as random jumps in electron energy and pitch angle due to a superposition of lowamplitude, randomly phased wave packets. The randomphase assumption may not be valid for intense wave packets^{13}, and coherent nonlinear interactions, such as advection and trapping, can lead to a much faster electron transport in pitch angle and energy than in the quasilinear regime^{14,15,16}. However, such faster nonlinear processes can still transport electrons to small enough pitch angles to eventually reach the dense atmosphere and precipitate, i.e., lose their entire energy through collisions with neutrals. Electrons within the loss cone, the pitchangle range corresponding to such precipitation (see Fig. 1a), heat the upper atmosphere and alter its chemical properties^{17}.
There are two opposite limits in quasilinear diffusion theory^{12}, as outlined in Fig. 1b. The first corresponds to weak diffusion, in which electrons are slowly scattered into the loss cone on timescales much longer than the bounce period (the maximum time it can take a losscone electron to be lost in the atmosphere), evidenced by a nearly empty loss cone (see the solid grey curve in Fig. 1b showing electron phase space density inside the loss cone, shaded in yellow). It corresponds to weak losses. The second is strong (or fast) diffusion, in which electrons are scattered into the loss cone so quickly that their atmospheric loss rate simply matches the pitchangle diffusion rate near the loss cone. In that limit, the electron flux within the loss cone is replenished by diffusion quickly such that it matches the flux just outside the loss cone (see the solid black curve in Fig. 1b). Note that the latter population remains trapped by Earth’s magnetic field. It is evident that pitchangle diffusive processes, by definition, cannot result in losscone fluxes that exceed the strong diffusion limit. Thus, in quasilinear diffusion theory, electron fluxes within the loss cone (j_{prec}) will be always lower than, or at most equal to those just outside the loss cone (trapped fluxes, j_{trap}).
Here, we demonstrate that, contrary to expectation from classical quasilinear diffusion theory, precipitation of energetic electrons often occurs faster than prescribed by the strong diffusion limit, exhibiting losscone fluxes greater than trapped fluxes. Such a losscone overfilling is depicted in Fig. 1c. We show through numerical simulations that this superfast precipitation, observed by lowaltitude spacecraft (see Fig. 1a), is caused by electron nonlinear interactions with intense oblique whistler waves measured by conjugate highaltitude spacecraft.
Results
Observations
In Earth’s dipole field (B), electrons bounce along magnetic field lines while conserving their magnetic moment \(\mu ={{{{{{{\mathcal{E}}}}}}}}{\sin }^{2}\alpha /B\), where \({{{{{{{\mathcal{E}}}}}}}}\) is the electron energy. The losscone size (the pitchangle range of electrons to be lost) is defined as \(\sin {\alpha }_{{{{{{{{\rm{LC}}}}}}}}}=\sqrt{B/{B}_{00}}\) (B_{00} is the magnetic field at the altitude of electron losses). In the outer radiation belt, around the magnetic equator (minimum B), α_{LC} is only a few degrees, which makes it challenging for nearequatorial spacecraft to measure electron losses directly^{10}. Spacecraft at low altitudes (where B is large and α_{LC} reaches 60–70°), however, can measure electron fluxes within the loss cone. Conjugate spacecraft measurements of nearequatorial waves responsible for electron scattering into the loss cone and lowaltitude electron fluxes within it, are needed to quantify electron precipitation rates^{18}. We use such a combination of observations from the equatorial Time History of Events and Macroscale Interactions during Substorms (THEMIS) spacecraft (see ref. ^{19} and “Methods”, subsection “Modelling technique”) and the lowaltitude Electron Losses and Fields Investigation (ELFIN) spacecraft^{20}—see Fig. 1a. ELFIN consists of two spinning CubeSats (ELFINA/B) that provide highresolution pitchangle and energyresolved measurements of energetic electron fluxes at all latitudes at altitudes of about 400–450 km.
Figures 2 and 3 show an overview of an event with conjugate measurements from ELFIN and THEMIS (probe E). From approximately 03:01–03:04 UT on 23 November 2020, ELFIN crossed the outer radiation belt (indicated by a j_{trap} increase up to 1 MeV in Fig. 2a) in the southern hemisphere and observed very strong electron precipitation (j_{prec}~j_{trap} for <200 keV in Fig. 2b) at L ~ 7.5–10 (distance from Earth in Earth radii). At the same time, THEMIS was also at L ~ 7.5–10, in the same noon magnetic local time (MLT) sector as ELFIN, very close to the magnetic equator (where the magnetic field is dominated by GSM B_{z}, as in Fig. 3a). It is worth emphasizing that THEMIS continuously observed strong whistlermode waves with frequencies f ∈ [0.1, 0.3] of the electron gyrofrequency (f_{ce}, as in Fig. 3b, c) from 02:00 UT to 04:00 UT. Prior THEMIS statistics have shown that the typical correlation length of the source region of whistlermode waves is about 1.4 h in MLT and 1.5 Earth radii radially at L ~ 7.5–10 near 11–12 MLT^{21}. Accordingly, an approximate conjunction (within 1.4 MLT, 1.5 in L, and 35 min UT) between the observed ELFIN precipitation and THEMIS waves occurred between 02:26 UT and 03:16 UT in Fig. 3. Therefore, it is reasonable to assume that the precipitation detected by ELFIN at 03:02 UT is driven by the whistlermode waves measured by THEMIS near 03:00 UT in Fig. 3 (as well as near 02:30 UT in Fig. 4). According to ELFIN measurements, j_{prec} not only reached j_{trap}, but actually exceeded it for electron energies <200 keV, see Fig. 2d. Because such strong electron precipitation cannot be explained by the classical pitchangle diffusion theory, we seek an alternate mechanism that could be responsible for it.
Measurements from THEMIS show that the observed whistlermode waves carry strong fieldaligned electric fields and have an elliptical polarization (see Fig. 4). These are typical properties of waves propagating at very large angles relative to the background magnetic field^{22,23}. Such very oblique waves have been previously observed in the same (L, MLT) range by THEMIS^{24}, and at similarly low frequencies by Van Allen Probes^{25}, although they are more common at higher frequencies^{26}. Such oblique waves may interact with electrons through Landau resonance when the electron fieldaligned velocity is equal to the wave phase velocity (ratio of wave frequency to wavenumber), v_{∥} = 2πf/k_{∥}. Oblique whistlermode waves are not as common as the most intense fieldaligned whistlermode waves^{25,27}. Specific plasma conditions are required for these oblique waves to survive the strong damping caused by suprathermal electrons: the electron distribution function must have a reduced gradient within the v_{∥} range corresponding to equatorial Landau resonance (see ref. ^{23} and references therein). Such an electron distribution, commonly observed on the dayside, is also present in our event (see “Methods”, subsection “Statistics of precipitation with losscone overfilling”). In addition, the observed oblique whistlermode waves are sufficiently intense (with electric field amplitude reaching approximately 10 mV m^{−1}, see examples of wave packets in Fig. 4) to trap electrons in the wave potential via Landau resonance^{23,28}. Such intense waves can therefore interact with electrons nonlinearly, leading to a fast acceleration and pitchangle decrease of phasetrapped electrons^{15,22}. Landau trapping by oblique waves differs from the commonly investigated trapping by fieldaligned waves (those waves move accelerated electrons away from the loss cone^{29}, not into the loss cone as oblique ones do in our case). The combined electron acceleration (~10–30 keV electrons are accelerated to ~100 keV) and pitchangle decrease by Landau trapping can result in a large increase of the electron flux within the loss cone (Fig. 1d) due to the large phase space density of the 10–30 keV source electrons, thus explaining the losscone overfilling (j_{prec} > j_{trap}) observed by ELFIN in Fig. 2d.
Comparison with numerical simulations
Next, we performed numerical simulations to verify our hypothesis that the losscone overfilling is caused by nonlinear Landau resonance of ~10–100 keV electrons with oblique whistlermode waves. We combined the observed equatorial electron spectrum, wave intensities, and frequencies from THEMIS to evaluate the evolution of the distribution in phase space^{30} and derive the expected electron energy and pitchangle distribution at ELFIN (see “Methods”, subsection “Modelling technique”). Figure 5 shows a comparison of electron pitchangle distributions observed by ELFIN with those obtained from numerical simulations. ELFIN measurements show a clear increase in electron fluxes at about 100 keV from the trapped pitchangle range (90° < α < α_{LC} ≃ 115°) to the precipitating pitchangle range (α > α_{LC}). That this pattern of pitchangle distributions is observed during multiple spin periods of ELFIN implies that this observation is not due to time aliasing, and that losscone overfilling persists for long temporal (3–10 s) and spatial scales (20–100 km in the ionosphere).
Landau trapping can result in a rapid equatorial pitchangle reduction from α_{eq} ~ 4–10° to α_{eq} ~ 1. 5° < α_{eq,LC} during a single resonant interaction: waves trap electrons and quickly transport them directly into the loss cone rather than slowly diffusing them toward it during multiple scatterings with waves^{23}. This trapping is accompanied by an energy increase that can be modelled by magnetic moment conservation of electrons in Landau resonance, \({{{{{{{\mathcal{E}}}}}}}}{\sin }^{2}{\alpha }_{{{{{{{{\rm{eq}}}}}}}}}={{{{{{{\rm{const.}}}}}}}}\) and \({{\Delta }}{{{{{{{\mathcal{E}}}}}}}}/{{{{{{{\mathcal{E}}}}}}}} \sim {(\sin {\alpha }_{{{{{{{{\rm{eq}}}}}}}}}/\sin {\alpha }_{{{{{{{{\rm{eq,LC}}}}}}}}})}^{2}\gg 1\) for initial \(\sin {\alpha }_{{{{{{{{\rm{eq}}}}}}}}} \; > \; 2\sin {\alpha }_{{{{{{{{\rm{eq,LC}}}}}}}}}\). Trapped electrons can typically gain an energy of tens of keV before being released from the resonance into the loss cone (see “Methods”, subsection “Modelling technique”, and Supplementary Fig. 6). Therefore, the losscone overfilling with approximately 100 keV electrons observed by ELFIN likely results from a nonlinear acceleration of electrons with initial energies of about 10–35 keV around the equator. Because of the slightly higher trapping probability associated with the lower range of initial energies, more electrons are released from trapping at smaller equatorial pitch angles (deeper into the loss cone). This effect of higher fluxes at lower pitch angles is further enhanced by the presence of a flat initial (preaccelerated) pitchangle distribution \(\sim\!{\sin }^{1/6}({\alpha }_{{{{{{{{\rm{eq}}}}}}}}})\), in the nearequatorial THEMIS observations. The latitudinal confinement of oblique whistlermode waves^{22,23} limits the efficiency of Landau trapping acceleration: ~10 keV electrons trapped near the equator and released from trapping at a higher latitude can gain at most 100–200 keV. And, indeed, pitchangle distributions measured by ELFIN and those obtained from numerical simulations do not show any losscone overfilling (j_{prec} > j_{trap}) feature above 300 keV (see Supplementary Fig. 7).
A peculiarity of Landau trapping acceleration is that the final electron energy increases with latitude, in both theory and simulations^{15,23}. Near the equator, this overfilling effect is therefore expected to be present only at lower energies, below 30–40 keV. Equatorial observations of electron fluxes within the loss cone have been impossible in the past due to the small losscone size at the equator. Recently, however, the Exploration of energization and Radiation in Geospace (ERG/Arase^{31}) spacecraft has enabled such observations in the inner magnetosphere^{10}. During the event shown in Figs. 2 and 3, ERG, which was in the dayside outer radiation belt, detected electron fluxes (see instrument details in^{32}) encompassing the loss cone (see Fig. 5g). The strong precipitation at ERG with j_{prec} ≥ j_{trap} at 10–20 keV, an energy range at which electrons resonate with oblique whistlermode waves at the equator, further supports our hypothesis of losscone overfilling by nonlinear Landau resonant acceleration (note that to increase the low counting statistics, we averaged ERG measurements for about 20 min; zero count data are omitted from the average because they correspond to times when there are no whistlermode waves). These electrons were accelerated by trapping near the equator (ERG was below ~25° of magnetic latitude). The further acceleration up to 60–140 keV observed by ELFIN at high latitudes and reproduced in our simulations, however, requires electron trapping up to midlatitude (∈ [30, 40]°)^{15,23}. Therefore, even when lossconeresolving measurements are used aboard nearequatorial spacecraft, such as ERG, they can only observe <20 keV precipitating fluxes from this effect.
Discussion
Our analysis of strong precipitation measured by ELFIN demonstrates that these observations are associated with THEMIS equatorial measurements of whistlermode waves. Assuming that these waves are propagating very obliquely to the background magnetic field (THEMIS measurements in the same region 30 min before ELFIN’s orbit definitely show such very oblique waves), we show that such oblique whistlermode waves can significantly enhance electron losses and create strong fluxes of ~100 keV electrons precipitating into the atmosphere. Although these oblique waves are fairly common^{25,27} and their potential effect on precipitation has been discussed^{23,33}, they have been excluded from most radiation belt models due to their low average magnetic field intensity^{34}. So how often can these oblique waves produce losscone overfilling for ~100 keV electrons? To address this question, we examine five months of ELFIN observations in the dayside inner magnetosphere, where oblique whistlermode waves are most common^{27}. We find that the occurrence rate of losscone overfilling (with j_{prec}/j_{trap} > 1) is 10% of all precipitation events (those with j_{prec}/j_{trap} > 0.05) at L ∈ [10, 12]. We have also found that during the several fortuitous conjunctions with THEMIS, the losscone overfilling observations are associated with very oblique whistlermode waves on the equator, consistent with the case study reported herein. The occurrence rate remains as high as 5% at L ∈ [6, 9], and decreases further inward of that distance, down to nearzero at L < 4 (see “Methods”, subsection “Statistics of precipitation with losscone overfilling”). Because losscone overfilling corresponds to much stronger electron losses than at other times, an occurrence rate of even 5 to 10% makes it an important contributor to ~50–200 keV electron precipitation. Such strong losses can suppress the source electron fluxes in the ~10–30 keV range and prevent them from acting as a seed population for relativistic energies. Additionally, precipitating ~100 keV electrons can penetrate the atmosphere down to an altitude of about 75 km, where they can significantly alter atmospheric properties and even influence local winter climate^{17}. Therefore, the reported effect of losscone overfilling is likely important for both radiation belt dynamics and magnetosphereatmosphere coupling.
Methods
Statistics of precipitation with losscone overfilling
Figure 1 in the main text shows a typical losscone overfilling event observed by ELFIN, with j_{prec} > j_{trap}. From five months of ELFIN observations in the dayside magnetosphere (Lshell below 12, MLT=915), we identified 166 orbits with such events (among a total of 465 orbits) and a total number of 943 ELFIN spins with j_{prec} > j_{trap}. Supplementary Figure 1 shows that the occurrence rate of losscone overfilling can be significant over a wide Lshell range and for sufficiently intense trapped fluxes >10^{6} keV cm^{−2} s^{−1} sr^{−1} MeV^{−1}.
The observed statistics of losscone overfilling are most likely due to electron Landau resonance with very oblique whistlermode waves, because the more investigated cyclotron resonance cannot explain the formation of the j_{prec} > j_{trap} feature, despite the fact that lowfrequency waves present at high L = 8–10 in the dawnnoon sector where a plasma frequency to gyrofrequency ratio of f_{pe}/f_{ce} ~ 5 can allow cyclotron resonance with >40 keV electrons there^{35,36}. To show this, let us consider resonance curves in (energy, pitchangle) space: the waveparticle resonant interactions move electrons along such curves^{37,38}:
for Landau resonance (conservation of the magnetic moment) and
for cyclotron resonance, where ω is the wave frequency, Ω_{ce,eq} the equatorial gyrofrequency, and γ the Lorentz factor. The cyclotron phase trapping may change the electron energy significantly and create a local gradient in the (energy, pitchangle) space, but for the energy range of interest (<300 keV) the cyclotron phase trapping results in a pitchangle increase, whereas electron losses occur primarily due to phase bunching^{38}, which is associated with an energy decrease. Moreover, phase trapping is stronger at lower energies^{39}, which would produce stronger peaks of J_{prec}/J_{trap} at lower energies, contrary to the present ELFIN observations. Cyclotron phase bunching will move particles gradually toward the loss cone with an energy loss, but it cannot create new gradients in pitchangle space, because to reach a certain final (precipitating) energy, electrons at smaller pitch angles should start with a larger energy and would therefore result in a smaller flux. In contrast to cyclotron resonance, Landau resonance curves have a very strong gradient around the loss cone (\(\sim\!1/{\sin }^{2}{\alpha }_{{{{{{{{\rm{eq}}}}}}}}}\)), i.e., all electrons moving toward small pitch angles should gain energy. Thus, the Landau trapping responsible for pitchangle decrease can move lowenergy (associated with the large fluxes) electrons from moderate pitch angles into the loss cone, and this transport will be associated with an energy gain. The Landau resonance is effective only for quite oblique whistlermode waves^{23}, and to further support our conjecture that the losscone overfilling is due to Landau trapping, we check wave observations for events in Supplementary Fig. 1 with THEMIS conjunctions. Supplementary Figures 2–4 show three examples (in addition to the event shown in the main text) where ELFIN observations of losscone overfilling are associated with THEMIS conjugate observations of oblique (elliptically polarized and having large fieldaligned electric field) waves.
Note that the radial distribution of events in Supplementary Fig. 1 further supports the key role of the Landau resonance. As the Landau resonant energy quickly increases for lower background plasma density, (\({{{{{{{{\mathcal{E}}}}}}}}}_{{{{{{{{\rm{res}}}}}}}}} \sim {(\omega /{k}_{\parallel })}^{2} \sim {f}_{{{{{{{{\rm{pe}}}}}}}}}^{2}\)), losscone overfilling for higher energy (>100 keV) electrons is more often observed at higher L, where the plasma density is lower. Although here we restricted our dataset to the dayside, to exclude effects of isotropic electron precipitation from the plasma sheet^{40}, ELFIN captures losscone overfilling on the nightside as well. On the nightside, however, strongly nonlinear electrostatic solitary waves (socalled time domain structures consisting of electron holes, electronacoustic solitons, and double layers^{41,42}) may provide the same nonlinear Landau trapping for <300 keV electrons^{43} as very oblique whistlermode waves. Therefore, this phenomenon of superfast losses of ~100 keV electrons can affect a wide range of local times and can be responsible for rapid electron losses from the plasma sheet injection region associated with intense whistlermode waves and time domain structures^{44}.
ELFIN data and THEMIS data have been loaded and analyzed using plugins to the Space Physics Environment Data Analysis Software (SPEDAS) framework.
ERG measurements
We use ERG mediumenergy particle experimentselectron analyzer (MEPe) measurements of j_{prec}/j_{trap} in Fig. 5g. The precipitating flux j_{prec} is calculated based on the electron count during the time period when one of the MEPe scopes captures the loss cone (time step = 15.6 ms for an energy bin, see ref. ^{32}). To determine the trapped flux j_{trap}, we use the pitchangle range of [5, 15]°. This pitch angle range can be scanned multiple times by multiple telescopes during an ERG spin (8 s). The average of these multiple flux values is used to calculate j_{trap}. ERG data have been analyzed using plugins to the SPEDAS framework.
Modelling technique
ELFIN lowaltitude observations of losscone overfilling are obtained in conjunction with highaltitude equatorial observations of electrons and whistlermode waves at THEMISE. These waves, almost electrostatic, propagate in a very oblique whistler mode^{45} and may survive strong Landau damping by suprathermal electrons because of peculiarities in the electron distribution function^{46}. And, indeed, the THEMIS Electrostatic Analyzer^{47} clearly shows a plateau in both parallel (pitch angles < 30°) and perpendicular (pitch angles ∈ [75, 105]°) directions over the entire wave interval (see Supplementary Fig. 5a–c). The energies of this enhanced electron population (approximately 1–5 keV) are close to the Landau resonant energy at the equator for the observed wave frequencies and wavenormal angle. Such electron distribution functions with reduced velocity gradient in the vicinity of Landau resonant energies significantly reduce Landau damping and provide favourable conditions for the generation of veryoblique whistlermode waves^{23,48}.
To simulate waveparticle interactions, we used combined measurements from the Electrostatic Analyzer (<25 keV) and the Solid State Telescope (∈ [50, 500] keV) with an interpolation over the energy gap^{49}. We obtained the distribution of wave characteristics \({{{{{{{\mathcal{P}}}}}}}}({B}_{{{{{{{{\rm{w}}}}}}}}},\omega )\) from THEMIS fff wave spectra^{50}, as measured by electric field antennas^{51} and converted them to wave magnetic field using the cold plasma dispersion relation^{52} (as calibration of the searchcoil measurements for this recent date has not yet taken place). Supplementary Figure 5d shows the distribution of \({{{{{{{\mathcal{P}}}}}}}}({B}_{{{{{{{{\rm{w}}}}}}}}},f/{f}_{{{{{{{{\rm{ce}}}}}}}}})\) collected during the THEMISELFIN nearconjunction interval. Electric to magnetic field spectral power conversion and simulation of waveparticle interactions also require information about wave normal angles, θ. Using THEMIS electric field waveform data (see Fig. 4), we estimated θ to be large, with θ ∈ [θ_{r} − 10°, θ_{r} − 5°] (where θ_{r} is the resonance cone angle) for these very oblique whistlermode waves^{53}; these estimates are consistent with general whistlermode statistics showing the existence of two wellseparated whistlermode wave branches: quasiparallel whistlermode waves with θ below the Gendrin angle and very oblique whistlermode waves with θ near the resonance cone angle^{25}. In simulations, we therefore use a uniform θ distribution within the [θ_{r} − 10°, θ_{r} − 5°] range. Although Supplementary Fig. 5d contains slightly lower f/f_{ce} values than Fig. 4, note that the Landau resonant parallel electron energy \({{{{{{{{\mathcal{E}}}}}}}}}_{{{{{{{{\rm{res}}}}}}}}}\), most important in simulations, can be written as
showing that it is nearly independent of f/f_{ce} for 0.14 < f/f_{ce} < 0.25 in this θ range.
We use the mapping technique^{30} to test the hypothesis that nonlinear Landau resonance is responsible for the losscone overfilling observed by ELFIN. This technique is based on the theoretical model of nonlinear waveparticle interactions characterized by three main parameters: the probability of trapping into the resonance \({{\Pi }}({{{{{{{\mathcal{E}}}}}}}},{\alpha }_{{{{{{{{\rm{eq}}}}}}}}})\), the energy change due to trapping \({{\Delta }}{{{{{{{{\mathcal{E}}}}}}}}}_{{{{{{{{\rm{trap}}}}}}}}}({{{{{{{\mathcal{E}}}}}}}},{\alpha }_{{{{{{{{\rm{eq}}}}}}}}})\), and the energy change due to nonlinear scattering \({{\Delta }}{{{{{{{{\mathcal{E}}}}}}}}}_{{{{{{{{\rm{scat}}}}}}}}}({{{{{{{\mathcal{E}}}}}}}},{\alpha }_{{{{{{{{\rm{eq}}}}}}}}})\). The probability of trapping determines the ratio of trapped particles to the total number of particles passing through a single resonance (see analytical theory for Π in^{54,55,56}). The stochastization of the wave phase ξ between resonances^{57,58} makes individual nonlinear resonant interactions independent^{59}. Thus, each electron trajectory can be traced in time as: t → t + τ_{bounce}/2, \({{{{{{{\mathcal{E}}}}}}}}\to {{{{{{{\mathcal{E}}}}}}}}+{{\Delta }}{{{{{{{\mathcal{E}}}}}}}}\), where \({{\Delta }}{{{{{{{\mathcal{E}}}}}}}}={{\Delta }}{{{{{{{{\mathcal{E}}}}}}}}}_{{{{{{{{\rm{trap}}}}}}}}}\) for ξ ∈ 2π[0, Π) and \({{\Delta }}{{{{{{{\mathcal{E}}}}}}}}={{\Delta }}{{{{{{{{\mathcal{E}}}}}}}}}_{{{{{{{{\rm{scat}}}}}}}}}\) for ξ ∈ 2π[Π, 1], and ξ is given by a random uniform distribution ∈ [0, 1]. The corresponding pitchangle changes are calculated from the conservation of the magnetic moment \({{{{{{{\mathcal{E}}}}}}}}{\sin }^{2}{\alpha }_{{{{{{{{\rm{eq}}}}}}}}}/{B}_{{{{{{{{\rm{eq}}}}}}}}}\) in the Landau resonance. This approach has been developed and verified against test particle simulations in^{30,60}, whereas analytical equations for \({{\Pi }}({{{{{{{\mathcal{E}}}}}}}},{\alpha }_{{{{{{{{\rm{eq}}}}}}}}})\), \({{\Delta }}{{{{{{{{\mathcal{E}}}}}}}}}_{{{{{{{{\rm{trap}}}}}}}}}({{{{{{{\mathcal{E}}}}}}}},{\alpha }_{{{{{{{{\rm{eq}}}}}}}}})\), \({{\Delta }}{{{{{{{{\mathcal{E}}}}}}}}}_{{{{{{{{\rm{scat}}}}}}}}}({{{{{{{\mathcal{E}}}}}}}},{\alpha }_{{{{{{{{\rm{eq}}}}}}}}})\) have been derived and verified against test particle simulations in^{61,62}. Five main characteristics of the nonlinear waveparticle interaction (Π, \({{\Delta }}{{{{{{{{\mathcal{E}}}}}}}}}_{{{{{{{{\rm{trap,scat}}}}}}}}}\), Δα_{trap,scat}) depend on the preresonance energy and pitch angle \(({{{{{{{\mathcal{E}}}}}}}},{\alpha }_{{{{{{{{\rm{eq}}}}}}}}})\), the wave characteristics (frequency ω, wavenormal angle θ(s) and wave amplitude B_{w}(s) profiles along magnetic field lines), and background plasma/magnetic field characteristics ω_{pe}/Ω_{ce}, B(s). We use the distribution of observed wave equatorial characteristics \({{{{{{{\mathcal{P}}}}}}}}\left({B}_{{{{{{{{\rm{w}}}}}}}}},f/{f}_{{{{{{{{\rm{ce}}}}}}}}}\right)\) (see Supplementary Fig. 5d), dipole magnetic field model, and several equatorial values of f_{pe}/f_{ce} within the observed range (see Fig. 3a). The profile of f_{pe} along magnetic field lines is adopted from the^{63} model, and the profile of B_{w}(s) for oblique whistlermode waves is from the^{22} model. The wavenormal angle variation along magnetic field lines is set at a constant deviation from the resonance cone angle, \(\cos (\theta )=qf/{f}_{{{{{{{{\rm{ce}}}}}}}}}\) (according to statistical results from^{53}), with ten q such that θ is uniformly distributed over [θ_{r} − 10°, θ_{r} − 5°] at the equator. Wave electromagnetic field components are obtained from the cold plasma dispersion relation^{64} for given B_{w}(s), θ(s), f, and f_{pe}/f_{ce}. For each resonant interaction (half the bounce period), we use the observed \({{{{{{{\mathcal{P}}}}}}}}\left({B}_{{{{{{{{\rm{w}}}}}}}}},f/{f}_{{{{{{{{\rm{ce}}}}}}}}},\theta \right)\) distribution and the uniform q distribution to select wave characteristics, and then use Π, \({{\Delta }}{{{{{{{{\mathcal{E}}}}}}}}}_{{{{{{{{\rm{trap,scat}}}}}}}}}\), Δα_{trap,scat} precalculated for all wave characteristics. Such mapping repeats for 10^{9} orbits for t ∈ [0, 10] min^{30} of wave activity with several bursts of observed intense waves (the temporal profile of wave intensity is derived from THEMIS observations around the moment of conjunction with ELFIN). Then these orbits are used to transform the initial energy and pitchangle distribution (from THEMIS equatorial measurements) into the final distributions mapped at ELFIN’s altitude.
The characteristics of waveparticle nonlinear interactions (Π, \({{\Delta }}{{{{{{{{\mathcal{E}}}}}}}}}_{{{{{{{{\rm{trap,scat}}}}}}}}}\), Δα_{trap,scat}) cannot be averaged over wave characteristics (as is traditionally done for diffusion rates in quasilinear theory, see^{65}), because each nonlinear trapping can change the electron energy by a magnitude of about their initial energy (\({{\Delta }}{{{{{{{{\mathcal{E}}}}}}}}}_{{{{{{{{\rm{trap}}}}}}}}} \sim {{{{{{{\mathcal{E}}}}}}}}\)). Thus, the mapping technique is based on a probabilistic approach operating with an ensemble of Π, \({{\Delta }}{{{{{{{{\mathcal{E}}}}}}}}}_{{{{{{{{\rm{trap,scat}}}}}}}}}\), Δα_{trap,scat} distributions in \(({{{{{{{\mathcal{E}}}}}}}},{\alpha }_{{{{{{{{\rm{eq}}}}}}}}})\) space. Supplementary Fig. 6 illustrates these distributions for the observed \({{{{{{{\mathcal{P}}}}}}}}\left({B}_{{{{{{{{\rm{w}}}}}}}}},f/{f}_{{{{{{{{\rm{ce}}}}}}}}},\theta \right)\):
Lowenergy electrons have a higher probability of being trapped into Landau resonance and transported to the low pitchangle range (Δα_{trap} < 0), with significant acceleration (\({{\Delta }}{{{{{{{{\mathcal{E}}}}}}}}}_{{{{{{{{\rm{trap}}}}}}}}} \; > \; 0\)); see three example trajectories from the mapping technique in Supplementary Fig. 6f. These trajectories contribute to the losscone overfilling observed by ELFIN and reproduced in numerical simulations in Fig. 5 for electrons of 60–150 keV. Conversely, in Supplementary Fig. 7, losscone overfilling above 300 keV is absent. All plots with model data are obtained from theoretical equations provided in cited references.
Data availability
THEMIS and ELFIN data used in this study are available in public repository at http://themis.ssl.berkeley.edu and https://data.elfin.ucla.edu/ela. ERG (Arase) data are available from the ERG Science Center operated by ISAS/JAXA and ISEE/Nagoya University (https://ergsc.isee.nagoyau.ac.jp/index.shtml.en,^{66}). The present study analyzed MEPeL2 data v01.01^{67}, and MGFL2 data v04.04^{68}. The source data used to produce figures in this study are publicly accessible at https://doi.org/10.6084/m9.figshare.19200305.v1. The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.
Code availability
Data analysis was done using Space Physics Environment Data Analysis Software (SPEDAS) V4.1, available at https://spedas.org/. The computer code of numerical simulations in this study is available upon request to the corresponding author.
References
van Allen, J. A. & Frank, L. A. Radiation around the earth to a radial distance of 107,400 km. Nature 183, 430–434 (1959).
Horne, R. B. et al. Space weather impacts on satellites and forecasting the Earth’s electron radiation belts with SPACECAST. Space Weather 11, 169–186 (2013).
Li, W. & Hudson, M. K. Earth’s Van Allen radiation belts: from discovery to the Van Allen probes era. J. Geophys. Res. (Space Phys.) 124, 8319–8351 (2019).
Turner, D. L. et al. The effects of geomagnetic storms on electrons in Earth’s radiation belts. Geophys. Res. Lett. https://doi.org/10.1002/2015GL064747 (2015).
Horne, R. B. et al. Wave acceleration of electrons in the Van Allen radiation belts. Nature 437, 227–230 (2005).
Thorne, R. M. et al. Rapid local acceleration of relativistic radiationbelt electrons by magnetospheric chorus. Nature 504, 411–414 (2013).
Miyoshi, Y. et al. Highspeed solar wind with southward interplanetary magnetic field causes relativistic electron flux enhancement of the outer radiation belt via enhanced condition of whistler waves. Geophys. Res. Lett. 40, 4520–4525 (2013).
Li, W. et al. Radiation belt electron acceleration by chorus waves during the 17 March 2013 storm. J. Geophys. Res. 19, 4681–4693 (2014).
Nishimura, Y. et al. Identifying the driver of pulsating aurora. Science 330, 81–84 (2010).
Kasahara, S. et al. Pulsating aurora from electron scattering by chorus waves. Nature 554, 337–340 (2018).
Miyoshi, Y. et al. Relativistic electron microbursts as highenergy tail of pulsating aurora electrons. Geophys. Res. Lett. 47, e90360 (2020).
Kennel, C. F. & Petschek, H. E. Limit on stably trapped particle fluxes. J. Geophys. Res. 71, 1–28 (1966).
Tsurutani, B. et al. Lowerband “monochromatic” chorus riser subelement/wave packet observations. J. Geophys. Res. 125, e2020JA028090 (2020).
Lakhina, G. S., Tsurutani, B. T., Verkhoglyadova, O. P. & Pickett, J. S. Pitch angle transport of electrons due to cyclotron interactions with the coherent chorus subelements. J. Geophys. Res. 115, A00F15 (2010).
Hsieh, Y.K. & Omura, Y. Nonlinear dynamics of electrons interacting with oblique whistler mode chorus in the magnetosphere. J. Geophys. Res. 122, 675–694 (2017).
Allanson, O., Watt, C. E. J., Allison, H. J. & Ratcliffe, H. Electron diffusion and advection during nonlinear interactions with whistler mode waves. J. Geophys. Res. 126, e28793 (2021).
Matthes, K., Dudok de Wit, T. & Lilensten, J. Earth’s climate response to a changing Sun. in EDP Sciences. ISBN 9782759818495. https://doi.org/10.1051/9782759818495 (2021).
Li, W. et al. Constructing the global distribution of chorus wave intensity using measurements of electrons by the POES satellites and waves by the Van Allen Probes. Geophys. Res. Lett. 40, 4526–4532 (2013).
Angelopoulos, V. The THEMIS mission. Space Sci. Rev. 141, 5–34 (2008).
Angelopoulos, V. et al. The ELFIN mission. Space Sci. Rev. 216, 103 (2020).
Agapitov, O. V. et al. Spatial extent and temporal correlation of chorus and hiss: statistical results from multipoint THEMIS observations. J. Geophys. Res. 123, 8317–8330 (2018).
Agapitov, O. V., Artemyev, A. V., Mourenas, D., Mozer, F. S. & Krasnoselskikh, V. Nonlinear local parallel acceleration of electrons through Landau trapping by oblique whistler mode waves in the outer radiation belt. Geophys. Res. Lett. 42, 10 (2015).
Artemyev, A. V. et al. Oblique whistlermode waves in the Earth’s inner magnetosphere: energy distribution, origins, and role in radiation belt dynamics. Space Sci. Rev. 200, 261–355 (2016).
Gao, X. et al. Observational evidence of generation mechanisms for very oblique lower band chorus using THEMIS waveform data. J. Geophys. Res. 123, 6732–6748 (2016).
Li, W. et al. New chorus wave properties near the equator from Van Allen Probes wave observations. Geophys. Res. Lett. 43, 4725–4735 (2016).
Taubenschuss, U. et al. Wave normal angles of whistler mode chorus rising and falling tones. J. Geophys. Res. 123, 9567–9578 (2014).
Agapitov, O. V. et al. Statistics of whistler mode waves in the outer radiation belt: cluster STAFFSA measurements. J. Geophys. Res. 118, 3407–3420 (2013).
O’Neil, T. M., Winfrey, J. H. & Malmberg, J. H. Nonlinear interaction of a small cold beam and a plasma. Phys. Fluids 14, 1204–1212 (1971).
Omura, Y., Furuya, N. & Summers, D. Relativistic turning acceleration of resonant electrons by coherent whistler mode waves in a dipole magnetic field. J. Geophys. Res. 112, A06236 (2007).
Artemyev, A. V. et al. Longterm dynamics driven by resonant waveparticle interactions: from Hamiltonian resonance theory to phase space mapping. J. Plasma Phys. 87, 835870201 (2021).
Miyoshi, Y. et al. Geospace exploration project: Arase (ERG). J. Phys. Confer. Ser. 869, 012095. (2017).
Kasahara, S. et al. Mediumenergy particle experimentselectron analyzer (MEPe) for the exploration of energization and radiation in geospace (ERG) mission. Earth Planets Space 70, 69 (2018).
Li, W. et al. Evidence of stronger pitch angle scattering loss caused by oblique whistlermode waves as compared with quasiparallel waves. Geophys. Res. Lett. 41, 6063–6070 (2014).
Santolík, O., Macúšová, E., Kolmašová, I., CornilleauWehrlin, N. & Conchy, Y. Propagation of lowerband whistlermode waves in the outer Van Allen belt: systematic analysis of 11 years of multicomponent data from the Cluster spacecraft. Geophys. Res. Lett. 41, 2729–2737 (2014).
Li, W. et al. THEMIS analysis of observed equatorial electron distributions responsible for the chorus excitation. J. Geophys. Res. 115, A00F11 (2010).
Gao, Z. et al. Intense lowfrequency chorus waves observed by Van Allen probes: fine structures and potential effect on radiation belt electrons. Geophys. Res. Lett. 43, 967–977 (2016).
Summers, D., Thorne, R. M. & Xiao, F. Relativistic theory of waveparticle resonant diffusion with application to electron acceleration in the magnetosphere. J. Geophys. Res. 103, 20487–20500 (1998).
Shklyar, D. R. & Matsumoto, H. Oblique whistlermode waves in the inhomogeneous magnetospheric plasma: resonant interactions with energetic charged particles. Surv. Geophys. 30, 55–104 (2009).
Artemyev, A. V. et al. Theoretical model of the nonlinear resonant interaction of whistlermode waves and fieldaligned electrons. Phys Plasmas 28, 052902 (2021).
Yahnin, A. G., Sergeev, V. A., Gvozdevsky, B. B. & Vennerstrøm, S. Magnetospheric source region of discrete auroras inferred from their relationship with isotropy boundaries of energetic particles. Annal Geophys 15(August), 943–958 (1997).
Mozer, F. S. et al. Time domain structures: what and where they are, what they do, and how they are made. Geophys. Res. Lett. 42, 3627–3638 (2015).
Vasko, I. et al. Electronacoustic solitons and double layers in the inner magnetosphere. Geophys. Res. Lett. 44, 4575–4583 (2017).
Mozer, F. S., Artemyev, A., Agapitov, O., Mourenas, D. & Vasko, I. Nearrelativistic electron acceleration by Landau trapping in time domain structures. Geophys. Res. Lett. 43, 508–514 (2016).
Malaspina, D. M., Ukhorskiy, A., Chu, X. & Wygant, J. A census of plasma waves and structures associated with an injection front in the inner magnetosphere. J. Geophys. Res. 123, 2566–2587 (2018).
Sazhin, S. WhistlerMode Waves in a Hot Plasma (Cambridge University Press, 1993).
Ma, Q. et al. Very oblique whistler mode propagation in the radiation belts: effects of hot plasma and landau damping. Geophys. Res. Lett. 44, 12057–12066 (2017).
McFadden, J. P. et al. The THEMIS ESA plasma instrument and inflight calibration. Space Sci. Rev. 141, 277–302 (2008).
Li, W. et al. Unraveling the excitation mechanisms of highly oblique lower band chorus waves. Geophys. Res. Lett. 438867–438875. https://doi.org/10.1002/2016GL070386 (2016).
Turner, D. L. et al. Radial distributions of equatorial phase space density for outer radiation belt electrons. Geophys. Res. Lett. 39, L09101 (2012).
Cully, C. M., Ergun, R. E., Stevens, K., Nammari, A. & Westfall, J. The THEMIS digital fields board. Space Sci. Rev. 141, 343–355 (2008).
Bonnell, J. W. et al. The electric field instrument (EFI) for THEMIS. Space Sci. Rev. 141, 303–341 (2008).
Ni, B., Thorne, R. M., Meredith, N. P., Shprits, Y. Y. & Horne, R. B. Diffuse auroral scattering by whistler mode chorus waves: Dependence on wave normal angle distribution. J. Geophys. Res. 116, A10207 (2011).
Agapitov, O. V. et al. The quasielectrostatic mode of chorus waves and electron nonlinear acceleration. J. Geophys. Res. 119, 1606–1626 (2014).
Neishtadt, A. Passage through a separatrix in a resonance problem with a slowlyvarying parameter. J. Appl. Math. Mech. 39, 594–605 (1975).
Shklyar, D. R. Stochastic motion of relativistic particles in the field of a monochromatic wave. Sov. Phys. JETP 53, 1197–1192 (1981).
Cary, J. R., Escande, D. F. & Tennyson, J. L. Adiabaticinvariant change due to separatrix crossing. Phys. Rev. A 34, 4256–4275 (1986).
Karpman, V. I. Nonlinear effects in the ELF waves propagating along the magnetic field in the magnetosphere. Space Sci. Rev. 16, 361–388 (1974).
Albert, J. M. Diffusion by one wave and by many waves. J. Geophys. Res. 115, A00F05 (2010).
Artemyev, A. V., Neishtadt, A. I. & Vasiliev, A. A. A map for systems with resonant trappings and scatterings. Regul. Chaotic Dyn. 25, 2–10 (2020).
Artemyev, A. V., Neishtadt, A. I. & Vasiliev, A. A. Mapping for nonlinear electron interaction with whistlermode waves. Phys. Plasmas 27, 042902 (2020).
Artemyev, A. V., Vasiliev, A. A., Mourenas, D., Agapitov, O. & Krasnoselskikh, V. Nonlinear electron acceleration by oblique whistler waves: Landau resonance vs. cyclotron resonance. Phys. Plasmas 20, 122901 (2013).
Vainchtein, D. et al. Evolution of electron distribution driven by nonlinear resonances with intense fieldaligned chorus waves. J. Geophys. Res. (Space Phys.) 123, 8149–8169 (2018).
Denton, R. E. et al. Distribution of density along magnetospheric field lines. J. Geophys. Res. 111, A04213 (2006).
Tao, X. & Bortnik, J. Nonlinear interactions between relativistic radiation belt electrons and oblique whistler mode waves. Nonlinear Processes Geophys. 17, 599–604 (2010).
Glauert, S. A. & Horne, R. B. Calculation of pitch angle and energy diffusion coefficients with the PADIE code. J. Geophys. Res. 110, A04206 (2005).
Miyoshi, Y. et al. The ERG science center. Earth Planets Space 70, 96 (2018).
Kasahara, S. et al. The Mepe Instrument Level2 Omnidirectional Flux Data of Exploration of Energization and Radiation in Geospace (ERG) Arase Satellite (ERG Science Center, Institute for SpaceEarth Environmental Research, Nagoya University, 2018).
Matsuoka, A. et al. The MGF Instrument Level2 Highresolution Magnetic Field Data of Exploration of Energization and Radiation in Geospace (ERG) Arase Satellite (ERG Science Center, Institute for SpaceEarth Environmental Research, Nagoya, 2018).
Tsyganenko, N. A. & Sitnov, M. I. Modeling the dynamics of the inner magnetosphere during strong geomagnetic storms. J. Geophys. Res. 110, A03208 (2005).
Acknowledgements
X.J.Z., A.A., V.A., E.T., and C.W. acknowledge support by National Aeronautics and Space Administration (NASA) awards NNX14AN68G, 80NSSC20K1578, 80NSSC21K0729, NAS502099 and National Science Foundation (NSF) awards AGS1242918, AGS2019950. We are grateful to NASA’s CubeSat Launch Initiative programme for successfully launching the ELFIN satellites in the desired orbits under ELaNa XVIII. We thank the Air Force Office of Scientific Research (AFOSR) for early support of the ELFIN programme under its University Nanosatellite Program, UNP8 project, contract number FA945312D0285. We also thank the California Space Grant programme for student support during the project’s inception. We acknowledge the hardware contributions and technical assistance of Mr. David Hinkley and The Aerospace Corporation. We acknowledge contributions of volunteers in the ELFIN team to the routine operations of the ELFIN mission. We thank Ms. Judith Hohl for text editing.
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X.J.Z., A.A., V.A., D.M., S.K., and Y.M. discussed the concepts, X.J.Z. performed data analysis of ELFIN and THEMIS, A.A. performed simulations, S.K. performed data analysis of ERG, D.M. made theoretical estimates for data/model comparison, E.T. and C.W. performed ELFIN EPD calibration and data preparation, S.K., S.Y., K.K., T.H., Y.M., I.S., and A.M. performed ERG calibration and data preparation, X.J.Z., A.A., V.A., and D.M. wrote the manuscript.
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Zhang, XJ., Artemyev, A., Angelopoulos, V. et al. Superfast precipitation of energetic electrons in the radiation belts of the Earth. Nat Commun 13, 1611 (2022). https://doi.org/10.1038/s41467022292918
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DOI: https://doi.org/10.1038/s41467022292918
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