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Wave-particle interaction plays a crucial role in the dynamics of the Earth’s radiation belts. Cyclotron resonance between coherent whistler mode electromagnetic waves and energetic electrons of the radiation belts is often called a coherent instability. Coherent instability leads to wave amplification/generation and particle acceleration/scattering. The effect of wave on particle’s distribution function is a key component of the instability. In general, whistler wave amplitude can grow over threshold of quasi-linear (linear) diffusion theory which analytically tracks the time-evolution of a particle distribution. Thus, a numerical approach is required to model the nonlinear wave induced perturbations on particle distribution function. A backward test particle model is used to determine the energetic electrons phase space dynamics as a result of coherent whistler wave instability. The results show the formation of a phase space features with much higher resolution than is available with forward scattering models. In the nonlinear regime the formation of electron phase space holes upstream of a monochromatic wave is observed. The results validate the nonlinear phase trapping mechanism that drives nonlinear whistler mode growth. The key differences in phase-space perturbations between the linear and nonlinear scenarios are also illustrated. For the linearized equations or for low (below threshold) wave amplitudes in the nonlinear case, there is no formation of a phase-space hole and both models show features that can be characterized as linear striations or ripples in phase-space. 

Abstract A Van Allen Probes observation of a high‐density duct alongside whistler‐mode wave activity shows several distinctive characteristics: (a)—within the duct, the wave normal angles (WNA) are close to zero and the waves have relatively large amplitudes, this is expected from the classic conceptualization of ducts. (b)—at L‐shells higher than the duct's location a large “shadow” is present over an extended region that is larger than the duct itself, and (c)—the WNA on the earthward edge of the duct is considerably higher than expected. Using ray‐tracing simulations it is shown that rays fall into three categories: (a) ducted (trapped and amplified), (b) reflected (scattered to resonance cone and damped), and (c) free (non‐ducted). The combined macroscopic effect of all these ray trajectories reproduce the aforementioned features in the spacecraft observation.