On the Two Approaches to Incorporate Wave-Particle Resonant Effects Into Global Test Particle Simulations

Energetic electron dynamics in the Earth's radiation belts and near-Earth plasma sheet are controlled by multiple processes operating on very different time scales: from storm-time magnetic field reconfiguration on a timescale of hours to individual resonant wave-particle interactions on a timescale of milliseconds. The most advanced models for such dynamics either include test particle simulations in electromagnetic fields from global magnetospheric models, or those that solve the Fokker-Plank equation for long-term effects of wave-particle resonant interactions. The most prospective method, however, would be to combine these two classes of models, to allow the inclusion of resonant electron scattering into simulations of electron motion in global magnetospheric fields. However, there are still significant outstanding challenges that remain regarding how to incorporate the long term effects of wave-particle interactions in test-particle simulations. In this paper, we describe in details two approaches that incorporate electron scattering in test particle simulations: stochastic differential equation (SDE) approach and the mapping technique. Both approaches assume that wave-particle interactions can be described as a probabilistic process that changes electron energy, pitch-angle, and thus modifies the test particle dynamics. To compare these approaches, we model electron resonant interactions with field-aligned whistler-mode waves in dipole magnetic fields. This comparison shows advantages of the mapping technique in simulating the nonlinear resonant effects, but also underlines that more significant computational resources are needed for this technique in comparison with the SDE approach. We further discuss applications of both approaches in improving existing models of energetic electron dynamics. We discuss two approaches to incorporate resonant effects into test particle simulation models Detailed approaches have been shown for continuous stochastic differential equations (SDEs) and the mapping technique, respectively In contrast to continuous SDE, the mapping technique allows one to simulate nonlinear resonant effects