Theoretical model of the nonlinear resonant interaction of whistler-mode waves and field-aligned electrons

The nonlinear resonant interaction of intense whistler-mode waves and energetic electrons in the Earth’s radiation belts is traditionally described by theoretical models based on the consideration of slow-fast resonant systems. Such models reduce the electron dynamics around the resonance to the single pendulum equation, that provides solutions for the electron nonlinear scattering (phase bunching) and phase trapping. Applicability of this approach is limited to not-too-small electron pitch-angles (i.e., sufficiently large electron magnetic moments), whereas model predictions contradict to the test particle results for small pitch-angle electrons. This study is focused on such field-aligned (small pitch-angle) electron resonances. We show that the nonlinear resonant interaction can be described by the slow-fast Hamiltonian system with the separatrix crossing. For the first cyclotron resonance, this interaction results in the electron pitch-angle increase for all resonant electrons, contrast to the pitch-angle decrease predicted by the pendulum equation for scattered electrons. We derive the threshold value of the magnetic moment of the transition to a new regime of the nonlinear resonant interaction. For field-aligned electrons the proposed model provides the magnitude of magnetic moment changes in the nonlinear resonance. This model supplements existing models for not-too-small pitch-angles and contributes to the theory of the nonlinear resonant electron interaction with intense whistler-mode waves.