Diffusive approximations of nonlinear interaction of electrons with whistler-mode waves

Resonant interactions of radiation belt electrons with intense whistler-mode waves can lead to rapid nonlinear acceleration through phase trapping. The efficiency of this process depends strongly on wave coherence. In the random-phase approximation (fully incoherent), particle transport in velocity space can be described as diffusion with coefficients given by quasilinear theory. However, intense and coherent whistler-mode chorus waves are ubiquitous in the Earth’s radiation belts during geomagnetically active periods, raising the question of whether the diffusive Fokker–Planck equation implemented in state-of-the-art radiation belt models remains applicable.

In this work, we study the phase space density evolution of electrons interacting with narrow-band whistler-mode waves using test-particle simulations and compare the results with solutions of the diffusion equation. A key element of our approach is following particles over many bounce periods between magnetic mirror points, ensuring bounce-phase mixing and a gradual transition toward stochastic behavior. Starting from step-function initial conditions in pitch-angle phase space density, we analyze the broadening and erosion of initially sharp gradients and extract effective diffusion and drift coefficients.

Focusing on regions where quasilinear theory predicts nearly homogeneous diffusion and pitch-angle transport dominates over energy transport, we represent the theoretical solution using a Legendre polynomial expansion and determine transport coefficients via least-squares fitting. We find that the inferred diffusion coefficients agree with quasilinear predictions within a factor of 1.3 over a broad range of energies and pitch angles. A small negative effective drift term is sometimes required to reproduce the observed gradient erosion. This agreement persists even at very low pitch angles, where anomalous phase trapping occurs, suggesting that such nonlinear effects do not preclude a diffusive description of phase space density evolution and do not strongly modify diffusion rates relative to quasilinear expectations. While a wider range of wave parameters needs to be explored, these preliminary results support the continued use of quasilinear diffusion models in radiation belt simulations.