Proton interaction with quasi-electrostatic whistler mode waves in an inhomogeneous plasma (magnetosphere)

We study the interaction between energetic protons of the Earth’s radiation belts and quasi-electrostatic whistler mode waves. The nature of these waves is well known: whistler waves, which are excited in the magnetosphere due to cyclotron instability, enter the resonant regime of propagation and become quasielectrostatic, while their amplitude significantly increases. Far enough from the equator where proton gyrofrequency and transversal velocity increase the nonlinear interaction between these waves and energetic protons becomes possible. We show that plasma inhomogeneity may destroy cyclotron resonance between wave and proton on the time scale of the order of particle gyroperiod which in fact means the absence of cyclotron resonance; nevertheless, the interaction between waves and energetic particles remains nonlinear. In this case, particle dynamics in the phase space has the character of diffusion; however, the diffusion coefficients are determined by the averaged amplitude of the wave field, but not by its resonant harmonics. For real parameters of the waves and magnetospheric plasma, proton pitch-angle diffusion leading to their precipitation from the magnetosphere becomes essential.