Perturbation analysis of trapped-particle dynamics in axisymmetric dipole geometry

TL;DR

This paper develops a perturbation analysis for trapped particle dynamics in axisymmetric dipole fields, deriving higher-order corrections to bounce action-angle coordinates and frequencies with excellent numerical agreement.

Contribution

It introduces a systematic perturbation method to derive anharmonic corrections to bounce action-angle coordinates and frequencies in dipole magnetic fields.

Findings

Explicit anharmonic corrections for canonical coordinates

Analytical bounce and drift frequencies with anharmonic effects

Excellent agreement with numerical simulations

Abstract

The perturbation analysis of the bounce action-angle coordinates $(J,\zeta)$ for charged particles trapped in an axisymmetric dipole magnetic field is presented. First, the lowest-order bounce action-angle coordinates are derived for deeply-trapped particles in the harmonic-oscillator approximation. Next, the Lie-transform perturbation method is used to derive higher-order anharmonic action-angle corrections. Explicit expressions (with anharmonic corrections) for the canonical parallel coordinates $s(J,\zeta)$ and $p_{\|}(J,\zeta)$ are presented, which satisfy the canonical identity $\{s,\; p_{\|}\}(J,\zeta) \equiv 1$. Lastly, analytical expressions for the bounce and drift frequencies (which include anharmonic corrections) yield excellent agreement with exact numerical results.