Inhomogeneity Ratio for Nearly Field-Aligned Electrons Interacting With Whistler-Mode Waves

Nonlinear interactions between electrons and whistler-mode chorus waves play an important role in driving electron precipitation in Earth's radiation belts. In this letter, we employ the second fundamental model of the Hamiltonian approach to derive the inhomogeneity ratio, assessing nonlinear resonant interactions between nearly field-aligned electrons and parallel propagating chorus waves. We perform test particle simulations by launching electrons from a high latitude to the equator, encountering counter-streaming chorus waves. Our simulations reveal that anomalous scattering, encompassing anomalous trapping and positive bunching, extends the resonant location to the downstream of electrons. The comparison with test particle results demonstrates the efficacy of the inhomogeneity ratio in characterizing nonlinear interactions at small pitch angles. We emphasize the importance of applying this ratio specifically for small pitch angle electrons, as the previously provided inhomogeneity ratio significantly underestimates the impact of nonlinear interactions on electron precipitation.