Gyrokinetics from variational averaging: existence and error bounds

TL;DR

This paper reviews the gyrokinetic paradigm using variational averaging, establishing existence, deriving higher-order maps, and providing error bounds to assess approximation quality in plasma physics models.

Contribution

It introduces variational averaging as a new method for gyrokinetics, proving existence of coordinate maps and deriving error bounds for the approximation.

Findings

Existence of a phase space coordinate map for gyrokinetics.

Explicit third-order map computation independent of electromagnetic gauge.

Error bounds for gyrokinetic solutions relative to Vlasov equation.

Abstract

The gyrokinetic paradigm in the long wavelength regime is reviewed from the perspective of variational averaging (VA). The VA-method represents a third pillar for averaging kinetic equations with highly-oscillatory characteristics, besides classical averaging or Chapman-Enskog expansions. VA operates on the level of the Lagrangian function and preserves the Hamiltonian structure of the characteristics at all orders. We discuss the methodology of VA in detail by means of charged-particle motion in a strong magnetic field. The application of VA to a broader class of highly-oscillatory problems can be envisioned. For the charged particle, we prove the existence of a coordinate map in phase space that leads to a gyrokinetic Lagrangian at any order of the expansion, for general external fields. We compute this map up to third order, independent of the electromagnetic gauge. Moreover, an error bound for the solution of the derived gyrokinetic equation with respect to the solution of the Vlasov equation is provided, allowing to estimate the quality of the VA-approximation in this particular case.