An integral transform technique for kinetic systems with collisions

Abstract

The linearized Vlasov-Poisson system can be exactly solved using the G-transform, an integral transform introduced in Morrison and Pfirsch [Phys. Fluids B 4, 3038–3057 (1992)] and Morrison [Phys. Plasmas 1, 1447 (1994); Transp. Theory Stat. Phys. 29, 397 (2000)] that removes the electric field term, leaving a simple advection equation. We investigate how this integral transform interacts with the Fokker-Planck collision operator. The commutator of this collision operator with the G-transform (the “shielding term”) is shown to be negligible. We exactly solve the advection-diffusion equation without the shielding term. This solution determines when collisions dominate and when advection (i.e., Landau damping) dominates. This integral transform can also be used to simplify gyro-/drift-kinetic equations. We present new gyrofluid equations formed by taking moments of the G-transformed equation. Since many gyro-/drift-kinetic codes use Hermite polynomials as base elements, we include an explicit calculation of their G-transform.