A Conservative Galerkin Solver for the Quasilinear Diffusion Model in Magnetized Plasmas

Arising from averaging and linearizing over the original Vlasov-Maxwell system for magnetized plasmas, the quasilinear theory describes the resonant interaction between particles and waves. Such a model reduction in weak turbulence regime results in a kinetic diffusion process in momentum space for the particle probability density function(pdf ), where the diffusion coefficients are determined by the wave spectral energy density(sed). Meanwhile, a reaction equation in spectral space governs the time dynamics of the wave sed, with growth rates linearly dependent on the particle pdf. We propose a conservative Galerkin scheme for the quasilinear diffusion model in three-dimensional momentum space and three-dimensional spectral space, with cylindrical symmetry. The conservation laws are preserved by adopting the conservative discrete integro-differential operators and a consistent quadrature rule. We introduce a semi-implicit time discretization, and the stability condition is discussed. Numerical examples with applications in the electron runaway problem are provided, they show that the particle-wave interaction results in a strong anisotropic diffusion effect on the particle pdf.