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In the Earth's radiation belts, an upper limit on the electron flux is expected to be imposed by the Kennel‐Petschek mechanism, through the generation of exponentially more intense whistler‐mode waves as the trapped flux increases above this upper limit, leading to fast electron pitch‐angle diffusion and precipitation into the atmosphere. Here, we examine a different upper limit, corresponding to a dynamical equilibrium in the presence of energetic electron injections and both pitch‐angle and energy diffusion by whistler‐mode chorus waves. We first show that during sustained injections, the electron flux energy spectrum tends toward a steady‐state attractor resulting from combined chorus wave‐driven energy and pitch‐angle diffusion. We derive simple analytical expressions for this steady‐state energy spectrum in a wide parameter range, in agreement with simulations. Approximate analytical expressions for the corresponding equilibrium upper limit on the electron flux are provided as a function of the strength of energetic electron injections from the plasma sheet. The analytical steady‐state energy spectrum is also compared with maximum electron fluxes measured in the outer radiation belt during several geomagnetic storms with strong injections, showing a good agreement at 100–600 keV.

Wave‐particle resonant interaction is a key process controlling energetic electron flux dynamics in the Earth's radiation belts. All existing radiation belt codes are Fokker‐Planck models relying on the quasi‐linear diffusion theory to describe the impact of wave‐particle interactions. However, in the outer radiation belt, spacecraft often detect waves sufficiently intense to interact resonantly with electrons in the nonlinear regime. In this study, we propose an approach for estimating and including the contribution of such nonlinear resonant interactions into diffusion‐based radiation belt models. We consider electron resonances with whistler‐mode wave‐packets responsible for injected plasma sheet (∼100 keV) electron acceleration to relativistic energies and/or for their precipitation into the atmosphere. Using statistics of chorus wave‐packet amplitudes and sizes (number of wave periods within one packet), we provide a rescaling factor for quasi‐linear diffusion rates, that accounts for the contribution of nonlinear interactions in long‐term electron flux dynamics. Such nonlinear effects may speed up 0.1–1 MeV electron diffusive acceleration by a factor of ×1.5–2 during disturbed periods. We discuss further applications of the proposed approach and the importance of nonlinear resonant interactions for long‐term radiation belt dynamics.

Electron diffusion by whistler‐mode chorus waves is one of the key processes controlling the dynamics of relativistic electron fluxes in the Earth's radiation belts. It is responsible for the acceleration of sub‐relativistic electrons injected from the plasma sheet to relativistic energies as well as for their precipitation and loss into the atmosphere. Based on analytical estimates of chorus wave‐driven quasi‐linear electron energy and pitch‐angle diffusion rates, we provide analytical steady‐state solutions to the corresponding Fokker‐Planck equation for the relativistic electron distribution and flux. The impact on these steady‐state solutions of additional electromagnetic ion cyclotron waves, and of ultralow frequency waves are examined. Such steady‐state solutions correspond to hard energy spectra at 1–4 MeV, dangerous for satellite electronics, and represent attractors for the system dynamics in the presence of sufficiently strong driving by continuous injections of 10–300 keV electrons. Therefore, these analytical steady‐state solutions provide a simple means for estimating the most extreme electron energy spectra potentially encountered in the outer radiation belt, despite the great variability of injections and plasma conditions. These analytical steady‐state solutions are compared with numerical simulations based on the full Fokker‐Planck equation and with relativistic electron flux spectra measured by satellites during one extreme event and three strong events of high time‐integrated geomagnetic activity, demonstrating a good agreement.

Since the advent of the Space Age, the importance of understanding and forecasting relativistic electron fluxes in the Earth’s radiation belts has been steadily growing due to the threat that such particles pose to satellite electronics. Here, we provide a model of long‐duration periods of high time‐integrated 2‐MeV electron flux deep inside the outer radiation belt, based on the significant correlation obtained in 2001–2017 between time‐integrated electron flux measured by satellites and a measure of the preceding time‐integrated homogenizedaa_Hgeomagnetic index. We show that this correlation is likely due to a stronger cumulative chorus wave‐driven acceleration of relativistic electrons and a stronger cumulative inward radial diffusion of such electrons during periods of higher time‐integrated geomagnetic activity. Return levels of 2‐MeV electron flux are provided based on Extreme Value analysis of time‐integrated geomagnetic activity over 1868–2017, in rough agreement with estimates based on 20‐year data sets of measured flux. A high correlation is also found between our measure of time‐integrated geomagnetic activity averaged over each solar cycle and averaged sunspot numbers, potentially paving the way for forecasts of time‐integrated relativistic electron flux during future solar cycles based on predictions of solar activity.

We review comprehensive observations of electromagnetic ion cyclotron (EMIC) wave-driven energetic electron precipitation using data collected by the energetic electron detector on the Electron Losses and Fields InvestigatioN (ELFIN) mission, two polar-orbiting low-altitude spinning CubeSats, measuring 50-5000 keV electrons with good pitch-angle and energy resolution. EMIC wave-driven precipitation exhibits a distinct signature in energy-spectrograms of the precipitating-to-trapped flux ratio: peaks at >0.5 MeV which are abrupt (bursty) (lasting ∼17 s, or$\Delta L\sim 0.56$$\Delta L\sim 0.56$) with significant substructure (occasionally down to sub-second timescale). We attribute the bursty nature of the precipitation to the spatial extent and structuredness of the wave field at the equator. Multiple ELFIN passes over the same MLT sector allow us to study the spatial and temporal evolution of the EMIC wave - electron interaction region. Case studies employing conjugate ground-based or equatorial observations of the EMIC waves reveal that the energy of moderate and strong precipitation at ELFIN approximately agrees with theoretical expectations for cyclotron resonant interactions in a cold plasma. Using multiple years of ELFIN data uniformly distributed in local time, we assemble a statistical database of ∼50 events of strong EMIC wave-driven precipitation. Most reside at$L\sim 5-7$$L\sim 5-7$at dusk, while a smaller subset exists at$L\sim 8-12$$L\sim 8-12$at post-midnight. The energies of the peak-precipitation ratio and of the half-peak precipitation ratio (our proxy for the minimum resonance energy) exhibit an$L$$L$-shell dependence in good agreement with theoretical estimates based on prior statistical observations of EMIC wave power spectra. The precipitation ratio’s spectral shape for the most intense events has an exponential falloff away from the peak (i.e., on either side of$\sim 1.45$$\sim 1.45$MeV). It too agrees well with quasi-linear diffusion theory based on prior statistics of wave spectra. It should be noted though that this diffusive treatment likely includes effects from nonlinear resonant interactions (especially at high energies) and nonresonant effects from sharp wave packet edges (at low energies). Sub-MeV electron precipitation observed concurrently with strong EMIC wave-driven >1 MeV precipitation has a spectral shape that is consistent with efficient pitch-angle scattering down to ∼ 200-300 keV by much less intense higher frequency EMIC waves at dusk (where such waves are most frequent). At ∼100 keV, whistler-mode chorus may be implicated in concurrent precipitation. These results confirm the critical role of EMIC waves in driving relativistic electron losses. Nonlinear effects may abound and require further investigation.

Short and intense lower‐band chorus wave packets are ubiquitous in the Earth's outer radiation belt. In this article, we perform various Vlasov hybrid simulations, with one or two triggering waves, to study the generation of short chorus packets/subpackets inside long rising tone elements. We show that the length of the generated short wave packets is consistent with a criterion of resonance non‐overlap for two independent superposed waves, and that these chorus packets have similar characteristics as in Van Allen Probes observations. We find that short wave packets are mainly formed near the middle/end of long rising tones for moderate linear growth rates, and everywhere for stronger linear growth rates. Finally, we analyze an event characterized by Time History of Events and Macroscale Interactions during Substorms spacecraft measurements of chorus rising tones near the equator and simultaneous measurements by low altitude ELFIN CubeSats of precipitating and trapped electron fluxes in the same sector. The measured precipitating electron fluxes are well recovered by test particle simulations performed using measured plasma and wave properties. We show that short chorus wave packets of moderate amplitudes (160–250 pT) essentially lead to a more diffusive‐like transport of 50–200 keV electrons toward the loss cone than long packets. In contrast, long chorus packets are found to produce important nonlinear effects via anomalous trapping, which significantly reduces electron precipitation below 150 keV, especially for higher wave amplitudes.

Energetic electron precipitation into Earth's atmosphere is an important process for radiation belt dynamics and magnetosphere‐ionosphere coupling. The most intense form of such precipitation is microbursts—short‐lived bursts of precipitating fluxes detected on low‐altitude spacecraft. Due to the wide energy range of microbursts (from sub‐relativistic to relativistic energies) and their transient nature, they are thought to be predominantly associated with energetic electron scattering into the loss cone via cyclotron resonance with field‐aligned intense whistler‐mode chorus waves. In this study, we show that intense sub‐relativistic microbursts may be generated via electron nonlinear Landau resonance with very oblique whistler‐mode waves. We combine a theoretical model of nonlinear Landau resonance, equatorial observations of intense very oblique whistler‐mode waves, and conjugate low‐altitude observations of <200 keV electron precipitation. Based on model comparison with observed precipitation, we suggest that such sub‐relativistic microbursts occur by plasma sheet (0.1 − 10 keV) electron trapping in nonlinear Landau resonance, resulting in acceleration to ≲200 keV energies and simultaneous transport into the loss cone. The proposed scenario of intense sub‐relativistic (≲200 keV) microbursts demonstrates the importance of very oblique whistler‐mode waves for radiation belt dynamics.

Most lower‐band chorus waves observed in the inner magnetosphere propagate under the form of moderately intense short wave packets with fast frequency and phase variations. Therefore, understanding the formation mechanism of such short wave packets is crucial for accurately modeling electron nonlinear acceleration or precipitation into the atmosphere by these waves. We compare chorus wave statistics from the Van Allen Probes with predictions from a simple model of short wave packet generation by wave superposition with resonance nonoverlap, as well as with results from Vlasov Hybrid Simulations of chorus wave generation in an inhomogeneous magnetic field in the presence of one or two simultaneous triggering waves. We show that the observed moderate amplitude short chorus wave packets can be formed by a superposition of two or more waves generated near the magnetic equator with a sufficiently large frequency difference.