Analytical Calculation of the Orbital Spectrum of the Guiding Center Motion in Axisymmetric Magnetic Fields

TL;DR

This paper derives analytical formulas for the orbital spectrum of guiding center motion in axisymmetric magnetic fields, improving understanding of particle dynamics and resonance conditions in plasma physics.

Contribution

It introduces a canonical transformation and provides analytical formulas for orbital frequencies, showing better accuracy than previous pendulum-based models.

Findings

Analytical formulas match numerical frequencies closely.

Significant differences found with standard pendulum-like formulas.

Enhanced understanding of particle resonance conditions.

Abstract

Charged particle motion in axisymmetric toroidal magnetic fields is analyzed within the context of the canonical Hamiltonian Guiding Center theory. A canonical transformation to variables measuring the drift orbit deviation from a magnetic field line is introduced and an analytical transformation to Action-Angle variables is obtained, under a zero drift width approximation. The latter is used to provide compact formulas for the orbital spectrum of the drift motion, namely the bounce/transit frequencies as well as the bounce/transit averaged toroidal precession and gyration frequencies. These formulas are shown to have a remarkable agreement with numerically calculated full drift width frequencies and significant differences with standard analytical formulas based on a pendulum-like Hamiltonian description. The analytical knowledge of the orbital spectrum is crucial for the formulation of particle resonance conditions with symmetry breaking perturbations and the study of the resulting particle, energy and momentum transport.