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Nonlinear processes associated with the generation process of whistler-mode chorus emissions are summarized. The nonlinear dynamics of energetic electrons interacting with a coherent whistler-mode wave and the formation of electromagnetic electron holes or hills in the velocity phase space are described. The condition for resonant electrons to be free from the anomalous trapping at low pitch angles is obtained. In the presence of the inhomogeneity due to the frequency variation and the gradient of the magnetic field, the electron holes or hills result in resonant currents generating rising-tone emissions or falling-tone emissions, respectively. After formation of a coherent wave at a frequency of the maximum linear growth rate, triggering of the nonlinear wave growth takes place when the wave amplitude is above the threshold amplitude. The wave grows to a level close to the optimum wave amplitude as an absolute instability near the magnetic equator. The nonlinear growth rate at a position away from the equator is derived for a subtracted Maxwellian momentum distribution function with correction to the formulas in the past publications. The triggering process is repeated sequentially at progressively higher frequencies in the case of a rising-tone emission, generating subpackets forming a chorus element. With a higher plasma density as in the plasmasphere, the triggering of subpackets takes place concurrently over a wide range of frequency forming discrete hiss elements with varying frequencies. The mechanism of nonlinear wave damping due to quasi-parallel propagation from the equator is presented, which results in the formation of a gap at half the electron cyclotron frequency, separating a long rising-tone chorus emission into the upper-band and lower-band chorus emissions. The theoretical formulation of an oblique whistler mode wave and its interaction with energetic electrons at the n-th resonance is also presented along with derivation of the inhomogeneity facto

Van Allen Probes in situ observations are used to examine detailed subpacket structure observed in strong VLF (very low frequency) rising-tone chorus elements observed at the time of a rapid MeV electron energization in the inner magnetosphere. Analysis of the frequency gap between lower and upper chorus-band waves identifies f[subscript ceEQ], the electron gyrofrequency in the equatorial wave generation region. Initial subpackets in these strong chorus rising-tone elements begin at a frequency near 1/4 f[subscript ceEQ] and exhibit smooth gradual frequency increase across their > 10 ms temporal duration. A second much stronger subpacket is seen at frequencies around the local value of 1/4f[subscript ce] with small wave normal angle (< 10°) and steeply rising df/dt. Smooth frequency and phase variation across and between the initial subpackets support continuous phase trapping of resonant electrons and increased potential for MeV electron acceleration. The total energy gain for individual seed electrons with energies between 100 keV and 3 MeV ranges between 2 and 15%, in their nonlinear interaction with a single chorus element., NASA (Contract NAS5‐01072)

Efficient acceleration of relativistic electrons at Landau resonance with obliquely propagating whistler-mode chorus emissions is confirmed by theory, simulation, and observation. The acceleration is due to the perpendicular component of the wave electric field. We first review theoretical analysis of nonlinear motion of resonant electrons interacting with obliquely propagating whistler-mode chorus. We have derived formulae of inhomogeneity factors for Landau and cyclotron resonances to analyze nonlinear wave trapping of energetic electrons by an obliquely propagating chorus element. We performed test particle simulations to confirm that nonlinear wave trapping by both Landau and cyclotron resonances can take place for a wide range of energies. For an element of large amplitude chorus waves observed by the Van Allen Probes, we have performed detailed analyses of the wave form data based on theoretical framework of nonlinear trapping of resonant electrons. We compare the efficiencies of accelerations by cyclotron and Landau resonances. We find significant acceleration can take place both in Landau and cyclotron resonances. What controls the dynamics of relativistic electrons in the Landau resonance is the perpendicular field components rather than the parallel electric field of the oblique chorus wave. In evaluating the efficiency of nonlinear trapping, we have taken into account variation of the wave trapping potential structure controlled by the inhomogeneity factors., National Aeronautics and Space Administration (NASA)

Rapid radiation belt recovery following storm time depletion involves local acceleration of multi-MeV electrons in nonlinear interactions with VLF chorus waves. Previous studies of an apparent impenetrable barrier at L ~ 2.8 focused on diffusion and precipitation loss mechanisms for an explanation of the sharp reduction of multi-MeV electron fluxes earthward of L ~ 3. Van Allen Probes observations for cases when the plasmasphere is contracted earthward of L ~ 3 indicate that strong coherent signals from VLF transmitters can play significant roles in the suppression of nonlinear chorus wave growth earthward of L ~ 3. As a result, local nonlinear acceleration of hundreds of keV electrons to MeV energies does not occur in this region. During the recovery of the outer radiation belt when the plasmasphere is significantly contracted, the suppression of chorus wave growth and local acceleration by the action of the transmitter waves at the outer edge of the VLF bubble contributes to the sharp inner edge of the new MeV electron population and the formation of the impenetrable barrier at L ~ 2.8., NASA (Contract NAS5‐01072)

We have conducted a one‐dimensional electromagnetic particle simulation with a parabolic magnetic field to reproduce whistler‐mode hiss emissions in the plasmasphere. We assume a bi‐Maxwellian distribution with temperature anisotropy for energetic electrons injected into the plasmasphere and find that hiss emissions are generated with spectrum characteristics typical of those observed by spacecraft near the magnetic equator. The hiss emissions contain fine structures involving rising tone and falling tone elements with variation in frequencies. The amplitude profile of the spectra agrees with the optimum wave amplitude derived from the nonlinear wave growth theory. The simulation demonstrates that hiss emissions are generated locally near the magnetic equator through linear and nonlinear interactions with energetic electrons with temperature anisotropy. The coherent hiss emissions efficiently scatter resonant electrons of 2.5–80 keV into the loss cone.

We perform a one‐dimensional electromagnetic full particle simulation for triggered falling‐tone emissions in the Earth's magnetosphere. The equatorial region of the magnetosphere is modeled with a parabolic magnetic field approximation. The short whistler‐mode waves with a large amplitude are excited and propagate poleward from an artificial current oscillating with a constant frequency and amplitude. Following the excited waves, clear emissions are triggered with a falling frequency. Without the inhomogeneity of the background magnetic field, no triggered emission appears. The falling tone has several subpackets of amplitude and decreases the frequency in a stepwise manner. The positive resonant current formed by resonant electrons in the direction of the wave magnetic field clearly shows that an electron hill is formed in the phase space and causes the frequency decrease. The entrapping of the resonant electrons at the front of the packets and the decrease of the amplitude at the end of packets are essential for the generation of falling‐tone emissions. Each wavefront of the emission has a strongly negative resonant current −JE, which results in the wave growth. In the formation process of the resonant currents, we investigate the inhomogeneous factor S, which controls the nonlinear motion of the resonant electrons interacting with waves. The factor S consists of two terms, a frequency sweep rate and a gradient of the background magnetic field. The resonant current JE in the wave packet changes its sign from negative to positive as the packet moves away from the equator, terminating the wave growth.

We present a statistical study of electromagnetic ion cyclotron (EMIC) waves observed at Antarctic station (geographic 70.7° S, 11.8° E, L=5) on quiet and disturbed days during 2011–2017. The data span a fairly good period of both ascending and descending phases of the solar cycle 24, which has witnessed extremely low activity. We noted EMIC wave occurrence by examining wave power in different frequency ranges in the spectrogram. EMIC wave occurrence during ascending and descending phases of solar cycle 24, its local time, seasonal dependence and durations have been examined. There are total 2367 days for which data are available. Overall, EMIC waves are observed for 3166.5 h (≈5.57% of total duration) which has contributions from 1263 days. We find a significantly higher EMIC wave occurrence during the descending phase (≈ 6.83%) as compared to the ascending phase (≈ 4.08%) of the solar cycle, which implies nearly a twofold increase in EMIC wave occurrence. This feature is attributed to the higher solar wind dynamic pressure during descending phase of solar activity. There is no evident difference in the percentage occurrence of EMIC waves on magnetically disturbed and quiet days. On ground, EMIC waves show marginally higher occurrence during winter as compared to summer. This seasonal tendency is attributed to lower electron densities and conductivities in the ionosphere, which can affect the propagation of EMIC waves through ionospheric ducts. In local time, the probability distribution function of EMIC wave occurrence shows enhancement during 11.7–20.7 LT (i.e., afternoon–dusk sector). Daily durations of EMIC waves are in the range of 5–1015 min and it is noted that the longer duration (240–1015 min) events are prevalent on quiet days and are mostly seen during the descending phase of solar cycle.

We show the regions where nonlinear growth of whistler-mode chorus waves is preferred to occur in the inner magnetosphere. A global magnetohydrodynamics (MHD) simulation was used to obtain large-scale electric and magnetic fields under the southward interplanetary magnetic field condition. With the electric and magnetic fields obtained by the MHD simulation, we ran a comprehensive inner magnetosphere-ionosphere model to solve the evolution of phase space density of electrons. Hot electrons originating from the tail region drift sunward and penetrate deep into the inner region due to a combination of convection and substorm-associated electric fields. Cold electrons also drift sunward, resulting in a contraction of the plasmasphere. We obtained the following results. (1) The whistler waves can first grow due to the linear mechanism (pitch angle anisotropy) in the premidnight-prenoon region outside the plasmapause, followed by rapid, nonlinear mechanism accompanied with rising-tone chorus elements. (2) When the solar wind speed is high, the whistler waves grow more efficiently due to linear and nonlinear mechanisms over a wider area because of deep penetration of hot electrons and the large contraction of the plasmasphere. This is consistent with the observation that the outer belt electrons increase for the fast solar wind. (3) For slow solar wind, the linear growth is mostly suppressed, but the nonlinear growth can still take place when external seed waves are present. This may explain the persistence of dawn chorus and large-amplitude chorus waves that are often observed in the premidnight-postdawn region in relatively weak geomagnetic activities.

In this study we present an inversion method which provides thermal plasma population parameters from characteristics of chorus emissions only. Our ultimate goal is to apply this method to ground‐based data in order to derive the lower‐energy boundary condition for many radiation belt models. The first step is to test the chorus inversion method on in situ data of the Van Allen Probes in the generation region. The density and thermal velocity of energetic electrons (few kiloelectron volts to 100 keV) are derived from frequency sweep rate and starting frequencies of chorus emissions through analysis of wave data from the Electric and Magnetic Field Instrument Suite and Integrated Science on board the Van Allen Probes. The nonlinear wave growth theory of Omura and Nunn (2011, https://doi.org/10.1029/2010JA016280) serves as the basis for our inversion method, assuming that the triggering wave is originated by the linear cyclotron instability. We present 16 consecutive rising‐tone emissions recorded in the generation region between 11 and 12 UT on 14 November 2012. The results of the inversion are compared with density and thermal velocities (parallel and perpendicular) of energetic electrons derived from the unidirectional flux data of the Helium, Oxygen, Proton, and Electron instrument, showing a good agreement: The normalized root‐mean‐square deviation between the measured and predicted values are less than ∼15%. We found that the theoretical amplitudes are consistent with the measured ones. The relation between linear and nonlinear wave growth agrees with our basic assumption; namely, linear growth is a preceding process of nonlinear wave growth. We analyze electron distributions at the relativistic resonant energy ranges.

Efficient acceleration of relativistic electrons at Landau resonance with obliquely propagating whistler‐mode chorus emissions is confirmed by theory, simulation, and observation. The acceleration is due to the perpendicular component of the wave electric field. We first review theoretical analysis of nonlinear motion of resonant electrons interacting with obliquely propagating whistler‐mode chorus. We have derived formulae of inhomogeneity factors for Landau and cyclotron resonances to analyze nonlinear wave trapping of energetic electrons by an obliquely propagating chorus element. We performed test particle simulations to confirm that nonlinear wave trapping by both Landau and cyclotron resonances can take place for a wide range of energies. For an element of large amplitude chorus waves observed by the Van Allen Probes, we have performed detailed analyses of the wave form data based on theoretical framework of nonlinear trapping of resonant electrons. We compare the efficiencies of accelerations by cyclotron and Landau resonances. We find significant acceleration can take place both in Landau and cyclotron resonances. What controls the dynamics of relativistic electrons in the Landau resonance is the perpendicular field components rather than the parallel electric field of the oblique chorus wave. In evaluating the efficiency of nonlinear trapping, we have taken into account variation of the wave trapping potential structure controlled by the inhomogeneity factors.