Tractable Models of Resonant Wave-Particle Interactions

Abstract:

Analysis of wave-particle interactions often starts with a Hamiltonian formulation that is resonance averaged. A common approximation in this process is to neglect variation of the test particle's adiabatic invariant in the small, wave term H1 of the Hamiltonian. Taking the unperturbed part of the Hamiltonian, H0, to depend on the invariant quadratically leads to a version of the well-studied pendulum Hamiltonian, with phase bunching and (relatively rare) phase trapping particle behavior.

However, this approximation becomes invalid for small initial values of the invariant, and fails to capture the "anomalous phase trapping" seen in recent numerical simulations. [Kitahara and Katoh 2019; Gan et al., 2020]. This phase trapping, rather than being rare, affects all particles in this regime, and leads to pitch angle increase, away from the loss cone, rather than towards it as usual. We have found that retaining the dependence of H1 on the square root of the particle adiabatic invariant leads to what has been termed "a second fundamental model for resonance" [Henrard and Lemaitre, 1983; Neishtadt, 1975]. We find that this model captures the observed numerical behavior, and that analyzing the phase portraits leads to useful analytical estimates for particles with low pitch angle and energy. For other particles, the standard pendulum Hamiltonian and its traditional analysis are recovered.