Generalised action-angle coordinates defined on island chains

TL;DR

This paper extends the concept of action-angle coordinates to complex 3-D magnetic field systems with islands and chaos, introducing a variational method to resolve non-uniqueness in the transformation.

Contribution

It proposes a variational approach to define generalized action-angle coordinates on island chains, addressing non-uniqueness issues caused by relabelling symmetry.

Findings

Demonstrates non-uniqueness due to relabelling symmetry.

Introduces a variational method to achieve unique coordinate transformations.

Provides a framework for representing complex magnetic fields in toroidal plasmas.

Abstract

Straight-field-line coordinates are very useful for representing magnetic fields in toroidally confined plasmas, but fundamental problems arise regarding their definition in 3-D geometries because of the formation of islands and chaotic field regions, ie non-integrability. In Hamiltonian dynamical systems terms these coordinates are a form of action-angle variables, which are normally defined only for integrable systems. In order to describe 3-D magnetic field systems, a generalisation of this concept was proposed recently by the present authors that unified the concepts of ghost surfaces and quadratic-flux-minimising (QFMin) surfaces. This was based on a simple canonical transformation generated by a change of variable $\theta = \theta(\Theta,\zeta)$, where $\theta$ and $\zeta$ are poloidal and toroidal angles, respectively, with $\Theta$ a new poloidal angle chosen to give pseudo-orbits that are a) straight when plotted in the $\zeta,\Theta$ plane and b) QFMin pseudo-orbits in the transformed coordinate. These two requirements ensure that the pseudo-orbits are also c) ghost pseudo-orbits. In the present paper, it is demonstrated that these requirements do not \emph{uniquely} specify the transformation owing to a relabelling symmetry. A variational method of solution that removes this lack of uniqueness is proposed.