Phase space eigenfunctions with applications to continuum kinetic simulations

TL;DR

This paper develops methods to compute and visualize phase space eigenfunctions in plasma physics, enhancing the initialization and analysis of kinetic simulations of plasma instabilities in both unmagnetized and magnetized conditions.

Contribution

It introduces novel representations of the dielectric function and visualizations of eigenfunction phase space fluctuations, advancing the understanding of plasma eigenmodes and their nonlinear behaviors.

Findings

Eigenfunctions have simple analytic forms in unmagnetized plasma.

Truncation of cyclotron-harmonic expansion approximates eigenfunctions in magnetized plasma.

Visualizations clarify phase space structures like Bernstein modes and nonlinear saturation states.

Abstract

Continuum kinetic simulations are increasingly capable of resolving high-dimensional phase space with advances in computing. These capabilities can be more fully explored by using linear kinetic theory to initialize the self-consistent field and phase space perturbations of kinetic instabilities. The phase space perturbation of a kinetic eigenfunction in unmagnetized plasma has a simple analytic form, and in magnetized plasma may be well approximated by truncation of a cyclotron-harmonic expansion. We catalogue the most common use cases with a historical discussion of kinetic eigenfunctions and by conducting nonlinear Vlasov-Poisson and Vlasov-Maxwell simulations of single- and multi-mode two-stream, loss-cone, and Weibel instabilities in unmagnetized and magnetized plasmas with one- and two-dimensional geometries. Applications to quasilinear kinetic theory are discussed and applied to the bump-on-tail instability. In order to compute eigenvalues we present novel representations of the dielectric function for ring distributions in magnetized plasmas with power series, hypergeometric, and trigonometric integral forms. Eigenfunction phase space fluctuations are visualized for prototypical cases such as the Bernstein modes to build intuition. In addition, phase portraits are presented for the magnetic well associated with nonlinear saturation of the Weibel instability, distinguishing current-density-generating trapping structures from charge-density-generating ones.