Slow manifold reduction for plasma science

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 $n_e=Z_i{n}_i$.

Invited review article for CNSNS; 135 pages, single column

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Cited by in corpus (8)

• Slow manifolds of classical Pauli particle enable structure-preserving geometric algorithms for guiding center dynamics
• General formulas for adiabatic invariants in nearly-periodic Hamiltonian systems
• Normal stability of slow manifolds in nearly-periodic Hamiltonian systems
• Discovering exact, gauge-invariant, local energy-momentum conservation laws for the electromagnetic gyrokinetic system by high-order field theory on heterogeneous manifolds
• Hamiltonian reduction of Vlasov-Maxwell to a dark slow manifold
• Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques
• Geometric Methods for Adjoint Systems
• Analytical solution of the Sommerfeld-Page equation