Analytical results for phase bunching in the pendulum model of wave-particle interactions - Albert , Jay M.; Artemyev , Anton; Li , Wen; Gan , Longzhi; Ma , Qianli (Frontiers in Astronomy and Space Sciences, 2022)
Radiation belt electrons are strongly affected by resonant interactions with cyclotron-resonant waves. In the case of a particle passing through resonance with a single, coherent wave, a Hamiltonian formulation is advantageous. With certain approximations, the Hamiltonian has the same form as that for a plane pendulum, leading to estimates of the change at resonance of the first adiabatic invariant I, energy, and pitch angle. In the case of large wave amplitude (relative to the spatial variation of the background magnetic field), the resonant change in I and its conjugate phase angle xi are not diffusive but determined by nonlinear dynamics. A general analytical treatment of slow separatrix crossing has long been available and can be used to give the changes in I associated with "phase bunching," including the detailed dependence on xi, in the nonlinear regime. Here we review this treatment, evaluate it numerically, and relate it to previous analytical results for nonlinear wave-particle interactions. "Positive phase bunching'" can occur for some particles even in the pendulum Hamiltonian approximation, though the fraction of such particles may be exponentially small.