An integral transform technique for kinetic systems with collisions

TL;DR

This paper explores the use of the $G$-transform to solve kinetic systems with collisions, demonstrating its effectiveness in simplifying equations and deriving new gyrofluid models, with implications for plasma physics simulations.

Contribution

It introduces the interaction of the $G$-transform with the Fokker-Planck collision operator and derives new gyrofluid equations using this transform.

Findings

The shielding term is negligible in the $G$-transform context.

Exact solutions for advection-diffusion equations are obtained.

The $G$-transform simplifies gyro-/drift-kinetic equations.

Abstract

The linearized Vlasov-Poisson system can be exactly solved using the $G$-transform, an integral transform introduced in Refs. 1-3 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 basis elements, we include an explicit calculation of their $G$-transform.