Attempt to quantitize non-linear wave particle interactions using multi-pass test particle code

Non-linear plasma waves effects such as phase bunching and phase trapping can cause drastic change of energies of electrons in the magnetosphere in a few seconds. Previous studies of the non-linear wave effects had mainly focused on the energy changes of an electron passing through a plasma wave region, which covers a fraction of the electron's single-trip trajectory between its two mirror points. In this study, we use a test-particle method to study the accumulative non-liner wave effects over many bounces of electrons between their mirror points along the corresponding magnetic field lines. From our simulation results and data analyses, we found that non-linear wave-particle interactions are rather common at L=4 and 2) the non-linear effects by waves can still be treated as diffusion despite that the electron energy diffuses asymmetrically to the energy enhancement side and the corresponding 'asymmetrical' diffusion coefficients can be readily estimated.