The linearized Vlasov and Vlasov-Fokker-Planck equations in a uniform magnetic field

TL;DR

This paper analyzes the linearized Vlasov and Vlasov-Fokker-Planck equations in a uniform magnetic field, revealing mode decompositions, Landau damping, and collisional relaxation behaviors in weakly collisional regimes.

Contribution

It provides a novel decomposition into Bernstein modes for the transverse modes and establishes uniform Landau damping and relaxation rates in the presence of collisions.

Findings

Decomposition into Bernstein modes for transverse modes

Landau damping for non-Bernstein modes

Enhanced collisional relaxation at O(ν^{-1/3}) time scale

Abstract

We study the linearized Vlasov equations and the linearized Vlasov-Fokker-Planck equations in the weakly collisional limit in a uniform magnetic field. In both cases, we consider periodic confinement and Maxwellian (or close to Maxwellian) backgrounds. In the collisionless case, for modes transverse to the magnetic field, we provide a precise decomposition into a countably infinite family of standing waves for each spatial mode. These are known as Bernstein modes in the physics literature, though the decomposition is not an obvious consequence of any existing arguments that we are aware of. We show that other modes undergo Landau damping. In the presence of collisions with collision frequency $\nu \ll 1$, we show that these modes undergo uniform-in-$\nu$ Landau damping and enhanced collisional relaxation at the time-scale $O(\nu^{-1/3})$. The modes transverse to the field are uniformly stable and exponentially thermalize on the time-scale $O(\nu^{-1})$. Most of the results are proved using Laplace transform analysis of the associated Volterra equations, whereas a simple case of Yan Guo's energy method for hypocoercivity of collision operators is applied for stability in the collisional case.