The 1-Hz whistler wave precursor attached to shock-like structures are often observed in foreshock. Using observations from the Magnetospheric Multiscale mission, we investigate the interactions between 1-Hz waves and ions. Incoming solar wind ions do not gyro-resonate with the wave, since typically the wave is right-handed in their frame. We demonstrate that solar wind ions commonly exhibit 180 gyro-phase bunching from the wave magnetic field, understanding it with a reconciled linear picture for non-resonant ions and non-linear trapping theory of anomalous resonance. Along the longitudinal direction, solar wind ions experience Landau resonance, exhibiting either modulations at small wave potentials or trapping in phase-space holes at large potentials. The results also improve our understanding of foreshock structure evolution and 1-Hz wave excitation. Shock-like structures start with having incoming solar wind and remotely-reflected ions from further downstream. The ion-scale 1-Hz waves can already appear during this stage. The excitation may be due to shock-like dispersive radiation or kinetic instabilities resonant with these remotely-reflected ions. Ions reflected by local shock-like structures occur later, so they are not always necessary for generating 1-Hz waves. The wave leads to ion reflection further upstream, which may cause reformation. In one event, locally-reflected ions exhibit anomalous resonance in the early stage, and later approach to the gyro-resonant condition with gyro-phases ~270 . The latter is possibly due to nonlinear trapping in regions with an upstream-pointing magnetic field gradient, linked to reformation. Some additional special features like frequency dispersions are observed, requiring better explanations in the future.

Energetic electron accelerations and precipitations in the Earth’s outer radiation belt are highly associated with wave-particle interactions between whistler mode chorus waves and electrons. We perform test particle simulation to investigate the electron behaviors interacting with both parallel and obliquely propagating chorus emissions at L=4.5. We build up a database of the Green’s functions, which are treated as results of the input electrons interacting with one emission, for a large number of electrons interacting with whistler mode chorus emissions. The loss process of electron fluxes interacting with consecutive chorus emissions in the outer radiation belt are traced by applying the convolution integrals of distribution functions and the Green’s functions. Oblique chorus emissions lead to more electron precipitation than that led by parallel chorus emissions. By checking the resonance condition and resonant energy at loss cone angle, we find that electrons are hardly dropped into the loss cone directly by Landau resonance. The nonlinear scattering via cyclotron resonance is the main process that pushes energetic electrons into the loss cone. We propose a 2-step precipitation process for oblique chorus emissions that contributes to more electron loss: (1) During the first chorus emission, the nonlinear trapping of Landau resonance moves an electron near the loss cone. (2) During the second emission, the nonlinear scattering of cyclotron resonance scatters the electron into the loss cone. The combination of Landau resonance by oblique chorus emissions and cyclotron resonance results in the higher precipitation rate than the single cyclotron resonance by purely parallel chorus emissions.

We present a test particle study of perturbations of a weakly relativistic electron distribution interacting with a rising-tone lower band chorus emission. Trajectories of interacting electrons are traced back in time from the equator to reconstruct the perturbed velocity distribution. The wave field is precalculated from a model based on the nonlinear growth theory and features a realistic subpacket structure. The perturbed distribution reveals a series of stripes of increased and decreased phase space density. This perturbation is associated with the electromagnetic hole structure which exists along the resonance velocity curve of each subpacket. Time-averaging of the final distribution shows that rising-tone lower band chorus emissions produce a sharp decrease in density at low parallel velocities, which might be detectable by spacecraft particle instruments.

We conduct electromagnetic particle simulations in a uniform magnetic environment to verify the nonlinear wave growth process of plasmaspheric hiss in the equatorial plasmasphere. The satisfaction of the separability criterion for coexisting multiple frequency waves in the initial stage of wavenumber-time evolution declares that wave packets are coherent and capable of growing nonlinearly. Spatial and temporal evolutions of two typical modes located in wavenumber-time evolution demonstrate the consistency among wave growths, frequency variations, and inhomogeneity factor $S$ in coherent wave packets, showing that rising and falling tones occur at negative and positive $S$ values, respectively, and an obvious wave growth happens in a reasonable range of $S$ satisfying the second-order resonance condition. Wave packets extracted from wave fields in space and time by band-pass filter confirm good agreement between nonlinear theory and simulation results. The nonlinear growth rates of the extracted wave packets posses similar magnitudes to the growth rates of wave packets in the simulation, and they are much greater than the theoretical linear growth rate, indicating that the nonlinear process is essential in the generation of plasmaspheric hiss.

The nonlinear growth theory of chorus emissions is used to develop a simple model of the subpacket formation. The model assumes that the resonant current, which is released from the source to the upstream region, radiates a new whistler mode wave with a slightly increased frequency, which triggers a new subpacket. Saturation of the growth in amplitude is controlled by the optimum amplitude. Numerical solution of advection equations for each subpacket, with the chorus equations acting as the boundary conditions, produces a chorus element with a subpacket structure. This element features an upstream shift of the source region with time and an irregular growth of frequency, showing small decreases between adjacent subpackets. The influence of input parameters on the number of subpackets, the shift of the source, the frequency sweep rate and the maximum amplitude is analyzed. The model well captures basic features of instantaneous frequency measurements provided by the Van Allen Probes spacecraft. The modeled wave field can be used in future particle acceleration studies.