A hybrid Eulerian-Lagrangian Vlasov method for nonlinear wave-particle interaction in weakly inhomogeneous magnetic field

We present a hybrid Eulerian-Lagrangian (HEL) Vlasov method for nonlinear resonant wave-particle interactions in weakly inhomogeneous magnetic field. The governing Vlasov equation is derived from a recently proposed resonance tracking Hamiltonian theory. It gives the evolution of the distribution function with a scale-separated Hamiltonian that contains the fast-varying coherent wave-particle interaction and slowly-varying motion about the resonance frame of reference. The hybrid scheme solves the fast-varying phase space evolution on Eulerian grid with an adaptive time step and then advances the slowly-varying dynamics by Lagrangian method along the resonance trajectory. We apply the HEL method to study the frequency chirping of whistler-mode chorus wave in the magnetosphere and the self-consistent simulations reproduce the chirping chorus wave and give high-resolution phase space dynamics of energetic particles at low computational cost. The scale-separated HEL approach could provide additional insights of the wave instabilities and wave-particle nonlinear coherence compared to the conventional Vlasov and particle-in-cell methods.