Analytical and numerical analysis of self-consistent whistler wave Hamiltonian

The nonlinear evolution of a self-consistent Hamiltonian model for the interaction of resonant electrons with a parallel propagating whistler wave is investigated. Two cases are analyzed. The first case involves the perfectly resonant interaction between the whistler and electrons and exhibits features similar to previously considered single wave models. The second case involves a slightly off resonant interaction which leads to an amplitude modulation of the wave that resembles the amplitude modulation observed of whistler mode chorus in the radiation belts. Simple empirical models for the periodic evolution of the amplitude and phase of the wave are developed so that test-particle motion can be analyzed. It is shown that the formation in phase space of two clumps of particles in the modulated mode is due to a second order island chain in phase space. A single macro-particle approximation is developed to self-consistently model the resonant mode evolution and a two macro-particle approximation is developed for the modulated mode.