Investigations of the electron phase space dynamics in triggered whistler wave emissions using low noise $\delta f$ method

The evolution of the electron phase space structures during excitation of a triggered emission is investigated using the nonlinear $\delta f$ method. Previous studies suggested that the dynamics of phase space structures due to nonlinear wave particle interactions is critical to the excitation of triggered emissions with frequency chirping. We introduce the use of the nonlinear $\delta f$ method to simulate triggered emissions. Compared with full-f particle-in-cell method, the nonlinear $\delta f$ method significantly reduces numerical noise, therefore making the phase space structures more identifiable. Specific to the simulation of triggered emissions, the nonlinear $\delta f$ method also does not show numerical distortion of the distribution function due to reflecting particle boundary conditions. Using the nonlinear $\delta f$ method, we show that during the main portion of the chirping element, the phase space structure roughly maintains a shape so that the resonant island moves a distance in phase space that is on the same order as its width during one phase space bounce period of deeply trapped particles, supporting that the interaction is non-adiabatic. We also demonstrate the disappearance of the phase space structure near the end of the chirping. Our work suggests that the nonlinear $\delta f$ method could be very useful for the study of excitation of triggered emissions and to understand the mechanism of frequency chirping. (paper)