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A 2‐D GCPIC simulation in a dipole field system has been conducted to explore the excitation of oblique whistler mode chorus waves driven by energetic electrons with temperature anisotropy. The rising tone chorus waves are initially generated near the magnetic equator, consisting of a series of subpackets, and become oblique during their propagation. It is found that electron holes in the wave phase space, which are formed due to the nonlinear cyclotron resonance, oscillate in size with time during subpacket formation. The associated inhomogeneity factor varies accordingly, giving rise to various frequency chirping in different phases of subpackets. Distinct nongyrotropic electron distributions are detected in both wave gyrophase and stationary gyrophase. Landau resonance is found to coexist with cyclotron resonance. This study provides multidimensional electron distributions involved in subpacket formation, enabling us to comprehensively understand the nonlinear physics in chorus wave evolution. Plain Language Summary: Subpackets are a series of wave packets within chorus waves, characterized by wave amplitude modulation. In this study, we investigate the electron distributions in various phase spaces associated with subpacket formation, by performing a two‐dimensional simulation in a dipole field. It is found that the electrons can be trapped in the wave phase space through both cyclotron and Landau resonances. These two resonance interactions can also produce the "bump" and "plateau" shapes in momentum space, as well as the fine density structures in spatial space. Therefore, both cyclotron and Landau resonances play an important role in subpacket formation. Our study provides new inspiration for the nonlinear theory of chorus subpackets. Key Points: Oblique chorus subpackets are generated in the 2‐D GCPIC simulation modelElectron hole associated with the inhomogeneity factor oscillates with time during subpacket formationCyclotron and Landau resonances coexist during subpacket formation [ABSTRACT FROM AUTHOR]

Electromagnetic ion cyclotron (EMIC) waves are known to exhibit frequency chirping occasionally, contributing to the rapid acceleration and precipitation of energetic particles in the magnetosphere. However, the chirping mechanism of EMIC waves remains elusive. In this work, a phenomenological model of whistler mode chorus waves named the Trap‐Release‐Amplify (TaRA) model is applied to EMIC waves. Based on the proposed model, we explain how the chirping of EMIC waves occurs, and give predictions on their frequency chirping rates. For the first time, we relate the frequency chirping rate of EMIC waves to both the wave amplitude and the background magnetic field inhomogeneity. Direct observational evidence is provided to validate the model using previously published events of chirping EMIC waves. Our results not only provide a new model for EMIC wave frequency chirping, but more importantly, they indicate the potential wide applicability of the underlying principles of TaRA model. Plain Language Summary: Rapid change of wave frequency, or frequency chirping, can frequently be observed in space and laboratory plasmas. Chirping waves generally appear as discrete and narrowband elements and can cause rapid acceleration or scattering of energetic particles through nonlinear interactions. Correspondingly, the physical mechanism of frequency chirping has attracted considerable research interest. This work aims to explain how electromagnetic ion cyclotron (EMIC) wave frequency chirping arises, which remains an open question. We apply a previously developed model of whistler mode chorus waves named the Trap‐Release‐Amplify (TaRA) model to chirping EMIC waves, and make a series of chirping rate predictions. In particular, we find that after taking into account all the different stages of wave excitation, chirping rate of EMIC waves should be related to both wave amplitude and background magnetic field inhomogeneity. A comparison between the theoretical chirping rates and observations using previously published EMIC wave chirping events shows good consistency. The results provide both a new explanation for EMIC wave frequency chirping and an indirect test of the underlying principles of the TaRA model. Key Points: We propose and validate a model of electromagnetic ion cyclotron (EMIC) wave frequency chirping based on the Trap‐Release‐Amplify model of chorus wavesThe model relates EMIC wave frequency chirping to both wave intensity and background magnetic field inhomogeneity for the first timeObservations demonstrate a clear decrease in EMIC wave chirping rate with increasing L‐shell, consistent with the model prediction [ABSTRACT FROM AUTHOR]

Abstract A 2‐D GCPIC simulation in a dipole field system has been conducted to explore the excitation of oblique whistler mode chorus waves driven by energetic electrons with temperature anisotropy. The rising tone chorus waves are initially generated near the magnetic equator, consisting of a series of subpackets, and become oblique during their propagation. It is found that electron holes in the wave phase space, which are formed due to the nonlinear cyclotron resonance, oscillate in size with time during subpacket formation. The associated inhomogeneity factor varies accordingly, giving rise to various frequency chirping in different phases of subpackets. Distinct nongyrotropic electron distributions are detected in both wave gyrophase and stationary gyrophase. Landau resonance is found to coexist with cyclotron resonance. This study provides multidimensional electron distributions involved in subpacket formation, enabling us to comprehensively understand the nonlinear physics in chorus wave evolution.

Abstract Electromagnetic ion cyclotron (EMIC) waves are known to exhibit frequency chirping occasionally, contributing to the rapid acceleration and precipitation of energetic particles in the magnetosphere. However, the chirping mechanism of EMIC waves remains elusive. In this work, a phenomenological model of whistler mode chorus waves named the Trap‐Release‐Amplify (TaRA) model is applied to EMIC waves. Based on the proposed model, we explain how the chirping of EMIC waves occurs, and give predictions on their frequency chirping rates. For the first time, we relate the frequency chirping rate of EMIC waves to both the wave amplitude and the background magnetic field inhomogeneity. Direct observational evidence is provided to validate the model using previously published events of chirping EMIC waves. Our results not only provide a new model for EMIC wave frequency chirping, but more importantly, they indicate the potential wide applicability of the underlying principles of TaRA model.

Electron fluxes in Earth's radiation belts are significantly affected by their resonant interaction with whistler‐mode waves. This wave‐particle interaction often occurs via first cyclotron resonance and, when intense and nonlinear, can accelerate subrelativistic electrons to relativistic energies while also scattering them into the atmospheric loss cone. Here, we model Electron Losses and Fields INvestgation's (ELFIN) low‐altitude satellite measurements of precipitating electron spectra with a wave‐electron interaction model to infer the profiles of whistler‐mode intensity along magnetic latitude assuming realistic waveforms and statistical models of plasma density. We then compare these profiles with a wave power spatial distribution along field lines from an empirical model. We find that this empirical model is consistent with observations of subrelativistic (<200 keV) electron precipitation events, but deviates significantly for relativistic (>200 keV) electron precipitation events at all MLTs, especially on the nightside. This may be due to the sparse coverage of wave measurements at mid‐to‐high latitudes which causes statistically averaged wave power to be likely underestimated in current empirical wave models. As a result, this discrepancy suggests that intense waves likely do propagate to higher latitudes, although further investigation is required to quantify how well this high‐latitude population can account for the observed relativistic electron precipitation. Plain Language Summary: Whistler‐mode waves, the most prevalent type of plasma wave in Earth's magnetosphere, often interact with electrons by resonating with them, causing them to be accelerated and lost into Earth's atmosphere (in other words, precipitated). These waves are generated at the equator and typically stay constrained to within 20° in latitude; however, they can sometimes propagate to greater than 30° where they can accelerate electrons to relativistic energies. It is difficult to quantify how large of a contribution these mid‐to‐high‐latitude waves have on radiation belt electrons due to a lack of off‐equatorial spacecraft wave measurements. However, previous studies have shown that the energy spectra of precipitating electron fluxes may be used to infer the latitudinal extent of whistler‐mode waves. We therefore compare measurements of relativistic precipitation from NASA's Electron Losses and Fields INvestgation (ELFIN) mission (a pair of CubeSats built and operated by UCLA) with large ensemble test‐particle simulations informed by current empirical models of waves. Discrepancies in this comparison suggest that this elusive population of mid‐to‐high‐latitude whistler‐mode waves is most apparent at Earth's nightside and may even help explain some of the more intense precipitation events observed by the ELFIN CubeSats. Key Points: We evaluate the role of whistler‐mode waves on relativistic electron precipitation by modeling ELFIN case‐studies and statisticsObserved precipitation >200 keV exceeds results obtained with empirical models of latitudinal wave power distributions, notably at nightThe discrepancy may be explained by mid‐to‐high‐latitude intense whistler‐mode waves which are missing from current empirical models [ABSTRACT FROM AUTHOR]

The frequency chirping of chorus waves is commonly observed in the Earth's inner magnetosphere, but its generation remains an open question. Recently, Liu et al. (2021), https://doi.org/10.1029/2021JA029258 reported two unusual rising‐tone (upward chirping) chorus elements. Although the central frequency of constituent subpackets rises, the frequency of a single subpacket is surprisingly downward chirping. With a gcPIC‐δf $\delta f$ simulation in the dipole field, we successfully reproduce this kind of substructure, which contains alternating signs of chirping. Interestingly, both hole and hill structures are formed around the theoretical resonant velocities in the electron phase space, no matter whether the chirping is upward or downward. However, during each chirping interval, only one structure (either a hole or a hill) is associated with wave excitation: the upward chirping is related to the hole, while the hill contributes to the downward chirping. Our study provides a fresh perspective on the theory of frequency chirping in chorus waves. Plain Language Summary: The frequency chirping is a typical feature of chorus waves in the Earth's inner magnetosphere, which generally contain either rising‐tone (upward chirping) elements or falling‐tone (downward chirping) elements. Previous theory has suggested that the chirping is due to the nonlinear wave‐particle interaction, where the hole or hill structure is formed in the electron phase space. Recently, Liu et al. (2021), https://doi.org/10.1029/2021JA029258 have observed the upward chirping elements with their subpackets of downward chirping. What electron structure is associated with these elements becomes a puzzle. With a one‐dimensional (1D) general curvilinear particle‐in‐cell (gcPIC) δf simulation in the dipole magnetic field, we successfully reproduce this kind of chorus element, whose frequency contains alternating upward and downward chirping. Interestingly, both the hole and hill structures are formed during a chirping interval, but only one of the two structures is responsible for wave excitation and frequency chirping. The structure of hole‐hill combination provides an important clue into the theory of the frequency chirping in chorus waves. Key Points: With a gcPIC‐δf $\delta f$ simulation in the dipole field, we reproduce the upward chirping chorus element, whose subpackets are downward chirpingBoth hole and hill structures can be formed in the ζ−v‖ $\zeta -{v}_{\Vert }$ phase space, no matter whether the frequency is upward or downward chirpingThe time evolution of the hole and hill structures in the phase space leads to the alternating frequency chirping [ABSTRACT FROM AUTHOR]

Abstract The frequency chirping of chorus waves is commonly observed in the Earth’s inner magnetosphere, but its generation remains an open question. Recently, Liu et al. (2021), https://doi.org/10.1029/2021JA029258 reported two unusual rising‐tone (upward chirping) chorus elements. Although the central frequency of constituent subpackets rises, the frequency of a single subpacket is surprisingly downward chirping. With a gcPIC‐δf simulation in the dipole field, we successfully reproduce this kind of substructure, which contains alternating signs of chirping. Interestingly, both hole and hill structures are formed around the theoretical resonant velocities in the electron phase space, no matter whether the chirping is upward or downward. However, during each chirping interval, only one structure (either a hole or a hill) is associated with wave excitation: the upward chirping is related to the hole, while the hill contributes to the downward chirping. Our study provides a fresh perspective on the theory of frequency chirping in chorus waves.

Electron losses from the outer radiation belt are typically attributed to resonant electron scattering by whistler‐mode waves. Although the quasi‐linear diffusive regime of such scattering is well understood, the observed waves are often quite intense and in the nonlinear regime of resonant wave‐particle interaction. Such nonlinear resonant interactions are still being actively studied due to their potential for driving fast precipitation. However, direct observations of nonlinear resonance of whistler‐mode waves with electron distributions are scarce. Here, we present evidence for such resonance with high‐resolution electron energy and pitch angle spectra acquired at low‐altitudes by the dual Electron Losses and Fields INvestgation (ELFIN) CubeSats combined with conjugate measurements of equatorial plasma parameters, wave properties, and electron energy spectra by the Time History of Events and Macroscale Interactions during Substorms and Magnetospheric MultiScale missions. ELFIN has obtained numerous conjunction events exhibiting whistler wave driven precipitation; in this study, we present two such events which epitomize signatures of nonlinear resonant scattering. A test particle simulation of electron interactions with intense whistler‐mode waves prescribed at the equator is employed to directly compare modeled precipitation spectra with ELFIN observations. We show that the observed precipitating spectra match expectations to within observational uncertainties of wave amplitude for reasonable assumptions of wave power distribution along the magnetic field line. These results indicate the importance of nonlinear resonant effects when describing intense precipitation patterns of energetic electrons and open the possibility of remotely investigating equatorial wave properties using just properties of precipitation energy and pitch angle spectra. 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. Key Points: Intense near‐equatorial whistler‐mode waves are observed in conjunction with energetic electron precipitation measured by Electron Losses and Fields INvestgationTest particle simulations are employed to directly compare observations with theory of nonlinear wave‐particle resonant interactionsPrecipitating electrons up to relativistic energies are likely due to resonant scattering extending up to 40° in latitude [ABSTRACT FROM AUTHOR]

Abstract In this study, we analyze an electromagnetic ion cyclotron (EMIC) wave event of rising tone elements recorded by the Van Allen Probes. The pitch angle distributions of relativistic electrons exhibit a direct response to the two elements of EMIC waves: at the intermediate pitch angle, the fluxes are lower; and at the low pitch angle, the fluxes are higher than those when no EMIC was observed. In particular, the observed changes in the pitch angle distributions are most likely to be caused by nonlinear wave‐particle interaction. The calculation of the minimum resonant energy and a test‐particle simulation based on the observed EMIC waves support the role of the nonlinear wave‐particle interaction in the pitch angle scattering. This study provides direct evidence for the nonlinear pitch angle scattering of electrons by EMIC waves.

In this study, we analyze an electromagnetic ion cyclotron (EMIC) wave event of rising tone elements recorded by the Van Allen Probes. The pitch angle distributions of relativistic electrons exhibit a direct response to the two elements of EMIC waves: at the intermediate pitch angle, the fluxes are lower; and at the low pitch angle, the fluxes are higher than those when no EMIC was observed. In particular, the observed changes in the pitch angle distributions are most likely to be caused by nonlinear wave‐particle interaction. The calculation of the minimum resonant energy and a test‐particle simulation based on the observed EMIC waves support the role of the nonlinear wave‐particle interaction in the pitch angle scattering. This study provides direct evidence for the nonlinear pitch angle scattering of electrons by EMIC waves. Key Points: Direct evidence of EMIC‐induced pitch angle scattering of relativistic electrons is observedThe nondiffusive pitch angle distributions of electrons indicate the importance of nonlinear wave‐particle interactionThe calculation of minimum resonance energy and a test‐particle simulation support the observations [ABSTRACT FROM AUTHOR]

Electromagnetic ion cyclotron (EMIC) waves in duskside plasmasphere and plasmaspheric plume scatter megaelectron volt electrons into the loss cone and are considered a major loss mechanism for the outer radiation belt. Wave‐particle interaction between energetic electrons and EMIC waves has been studied extensively by the quasi‐linear diffusion theory. However, EMIC waves are typically strong enough to trigger nonlinear wave‐particle interaction effects and transport electrons in very different ways from quasi‐linear diffusion. New mathematical method is therefore in demand to study the evolution of energetic electron distribution in response to nonlinear wave‐particle interaction. In this work, we present a Markov chain description of the wave‐particle interaction process, in which the electron distribution is represented by a state vector and is evolved by the Markov matrix. The Markov matrix is a matrix form of the electron response Green's function and could be determined from test particle simulations. Our modeling results suggest that electron loss rate is not significantly affected by phase bunching and phase trapping, but for strong EMIC waves, electron distribution is more saturated near loss cone than quasi‐linear theory prediction, and negative electron phase space density slope develops inside loss cone. Key Points: A method is developed for modeling PSD evolution under nonlinear EMIC wave‐particle interactionNonlinear transport does not significantly change loss rate as compared to quasi‐linear theoryOur simulations predict peculiar negative electron loss cone PSD slopes in strong EMIC wavefield [ABSTRACT FROM AUTHOR]

Abstract: Resonant interactions between electrons and chorus waves are responsible for a wide range of phenomena in near‐Earth space (e.g., diffuse aurora and acceleration of > 1 MeV electrons). Although quasi‐linear diffusion is believed to be the primary paradigm for describing such interactions, an increasing number of investigations suggest that nonlinear effects are also important in controlling the rapid dynamics of electrons. However, present models of nonlinear wave‐particle interactions, which have been successfully used to describe individual short‐term events, are not directly applicable for a statistical evaluation of nonlinear effects and the long‐term dynamics of the outer radiation belt, because they lack information on the properties of intense (nonlinearly resonating with electrons) chorus waves. In this paper, we use the Time History of Events and Macroscale Interactions during Substorms and Van Allen Probes data sets of field‐aligned chorus waveforms to study two key characteristics of these waves: effective amplitude ℬ w (nonlinear interaction can occur when ℬ w > 2) and wave packet length β (the number of wave periods within it). While as many as 10–15% of chorus wave packets are sufficiently intense ( ℬ w > 2–3) to interact nonlinearly with relativistic electrons, most of them are short (β < 10) reducing the efficacy of such interactions. Revised models of nonlinear interactions are thus needed to account for the long‐term effects of these common, intense but short chorus wave packets. We also discuss the dependence of ℬ w, β on location (MLT and L‐shell) and on the properties of the suprathermal electron population. [ABSTRACT FROM AUTHOR]

Using results of test particle simulations of electrons interacting with whistler mode chorus emissions, we numerically obtain Green's functions to model evolution of the electron distribution function after all of the possible interactions with the waves. In both wave models with and without subpackets, electrons undergoing the cyclotron resonance with the waves are efficiently accelerated by nonlinear wave trapping. Since the strong modulation of the wave amplitude (in the case of the waves with subpackets) affects dynamics of resonant electrons, the electrons are detrapped from the wave potential or entrapped into it more frequently than those interacting with the waves without subpackets. As a result of interaction with the waves with subpackets, the number of electrons undergoing the acceleration is increased, while the acceleration efficiency in energy is decreased because of shorter interaction time. Modifying the numerical Green's function method with the simplified model of chorus waves uniform in longitude (Omura et al., 2015, https://doi.org/10.1002/2015JA021563), we compute the formation process of the outer radiation belt electron fluxes induced by the interaction with the chorus waves localized in longitude occurring for a time scale of 1 hr. The formation of MeV electron fluxes is characterized by large acceleration rates and butterfly pitch angle distributions, which are found in satellite observation results. Key Points: We have developed a new numerical Green's function method for radiation belt modeling including MLT dependency of chorus emissionsNonlinear interaction with chorus emissions localized in longitude causes rapid acceleration and precipitation of relativistic electronsRapid formation of radiation belt electron fluxes can be reproduced by the numerical Green's function method [ABSTRACT FROM AUTHOR]

Abstract: We analyze large‐amplitude upper‐band chorus emissions measured near the magnetic equator by the Electric and Magnetic Field Instrument Suite and Integrated Science instrument package on board the Van Allen Probes. In setting up the parameters of source electrons exciting the emissions based on theoretical analyses and observational results measured by the Helium Oxygen Proton Electron instrument, we calculate threshold and optimum amplitudes with the nonlinear wave growth theory. We find that the optimum amplitude is larger than the threshold amplitude obtained in the frequency range of the chorus emissions and that the wave amplitudes grow between the threshold and optimum amplitudes. In the frame of the wave growth process, the nonlinear growth rates are much greater than the linear growth rates. [ABSTRACT FROM AUTHOR]

Kinetic wave description is used to study the nonlinear influence of background Farley Buneman (FB) modes on the Gradient Drift (GD) modes in the equatorial electrojet ionosphere. The dominant nonlinearity is mediated through the electron flux term in the governing fluid equation which further invokes a turbulent current into the system. Electron dynamics reveals the modification in electron collision frequency and inhomogeneity scale length. It is seen that the propagation and growth rate of GD modes get modified by the background FB modes. Also, a new quasimode gets excited through the quadratic dispersion relation. Physical significance of coupled dynamics between the participating modes is also discussed. [ABSTRACT FROM AUTHOR]

Chorus waves have been suggested to be effective in acceleration of radiation belt electrons. Here we perform gyro-averaged test-particle simulations to calculate the bounce-averaged pitch angle and energy diffusion coefficients for parallel-propagating monochromatic chorus waves, and perform a comparison of test-particle (TP) model with quasi-linear (QL) theory to evaluate the influence of nonlinear processes. For small amplitude chorus waves, the diffusion coefficients of TP and QL models are in good agreement. As the wave amplitude reaches a threshold value, two nonlinear processes (phase trapping and phase bunching) start to occur, especially at large equatorial pitch angles. Phase trapping yields rapid increases in pitch angle and kinetic energy. In contrast, phase bunching causes overall decreases in pitch angle and kinetic energy. For the waves with amplitudes slightly above the threshold value, the average behavior is dominated by the phase trapping, and TP diffusion coefficients are larger than QL ones. As wave amplitude increases, TP diffusion coefficients become smaller than QL ones, indicating that phase trapping gradually reduces the dominance over phase bunching. [ABSTRACT FROM AUTHOR]

Abstract: A theoretical investigation on amplification of electrostatic ion acoustic wave in magnetically confined plasma has been presented in this paper. This investigation considers nonlinear wave–particle interaction process, called plasma maser effect, in presence of drift wave turbulence supported by magnetically confined inhomogeneous plasma. The role of associated nonlinear dissipative force in this effect in a confined plasma has been analyzed. The nonlinear force, which arises as a result of resonant interaction between electrons and modulated fields, is shown to drive the instability. Using the ion fluid equation and the ion equation of continuity, the nonlinear dispersion relation of a test ion acoustic wave has been derived, and the growth rate of ion acoustic wave in presence of low frequency drift wave turbulence has been estimated using Helimak data. [Copyright &y& Elsevier]

Abstract: Energy exchange between waves and resonant (usually energetic) particles is an essential feature of wave–particle interactions in plasma. If the resonant interaction in a magnetized unstable plasma involves two or more cyclotron resonances, the wave excitation can, at the same time, mediate the energy transfer between different populations of energetic particles. This effect, which is particularly pronounced in a marginally unstable plasma, is discussed by example of whistler–electron interactions in connection with particle energization in the Earthʼs radiation belts. The process under discussion is similar, although not identical, to alpha channeling known in the physics of tokamak plasma. [Copyright &y& Elsevier]

In this paper, the results of numerical analysis based on the theoretical analysis of the self-focusing effect when a microwave beam propagates through the ionospheric plasma from the solar power station (SPS) to the ground are shown. The self-focusing effect of the microwave beam is either caused by the plasma heating by the microwave beam or caused by the spatial gradient of the electromagnetic beam intensity. In this paper, the latter case is discussed. Since the microwave frequency used in the SPS is six orders of magnitude higher than the maximum collision frequency in the ionosphere, the theory used in this paper assumes a collisionless plasma and is derived from Maxwell's equations and the equation of motion. The self-focusing effect of the microwave beam predicted by the theory is determined by five parameters: plasma density and temperature; microwave frequency; intensity; and the spatial gradient of intensity. In this paper, based on the results of the numerical analysis of the theoretical equations, the self-focusing effect under the parameters related to the microwave beam used in the energy transmission of a future SPS is described. [ABSTRACT FROM AUTHOR]