Slow manifold reduction for plasma science

TL;DR

This paper reviews the theory of slow manifold reduction in plasma physics, illustrating its use in deriving reduced models, understanding model breakdown, and developing implicit integrators for multi-timescale plasma problems.

Contribution

It develops the slow manifold reduction theory tailored for plasma physics, including applications to model breakdown, implicit integrators, and Hamiltonian structure inheritance.

Findings

Derived the slow manifold for kinetic quasineutral plasma dynamics.

Demonstrated slow drift effects due to deviations from charge neutrality.

Illustrated the breakdown of reduced models over long timescales.

Abstract

The classical Chapman-Enskog procedure admits a substantial geometrical generalization known as slow manifold reduction. This generalization provides a paradigm for deriving and understanding most reduced models in plasma physics that are based on controlled approximations applied to problems with multiple timescales. In this Review we develop the theory of slow manifold reduction with a plasma physics audience in mind. In particular we illustrate (a) how the slow manifold concept may be used to understand \emph{breakdown} of a reduced model over sufficiently-long time intervals, and (b) how a discrete-time analogue of slow manifold theory provides a useful framework for developing implicit integrators for temporally-stiff plasma models. For readers with more advanced mathematical training we also use slow manifold reduction to explain the phenomenon of inheritance of Hamiltonian structure in dissipation-free reduced plasma models. Various facets of the theory are illustrated in the context of the Abraham-Lorentz model of a single charged particle experiencing its own radiation drag. As a culminating example we derive the slow manifold underlying kinetic quasineutral plasma dynamics up to first-order in perturbation theory. This first-order result incorporates several physical effects associated with small deviations from exact charge neutrality that lead to slow drift away from predictions based on the leading-order approximation $n_e = Z_i \,n_i$.