Nonlinear Alfv\'en eigenmode studies through optimized PIC simulations

A marker optimization technique for $\delta f$ PIC simulations has been developed and implemented in the kinetic/MHD hybrid code M3D-K. The technique removes markers to achieve importance sampling for $\delta f$, so as to save markers as well as computing time. The technique saves about 95\% of markers in a nonlinear simulation of toroidal Alfv\'en eigenmode (TAE) driven by energetic particles. The technique is then applied to studies of nonlinear dynamics of energetic particles, TAE and reversed shear Alfv\'en eigenmode (RSAE). In our single mode study, a marginally unstable $n = 2$ RSAE is excited by energetic particles. The mode exhibits frequency chirping up and down in the nonlinear stage. The down chirping component has higher amplitude than the up chirping component. Such frequency chirping is due to the nonlinearities of the energetic particles, since the background MHD is linear in this simulation. The frequency chirping is probably related to hold-clump formation in energetic particle distribution function as predicted by analytic theory. In our multiple mode study, two TAEs with similar growth rates are excited simultaneously by energetic particles. The nonlinear mode-mode coupling effect is being investigated.