Are ghost surfaces quadratic-flux-minimizing?

TL;DR

This paper compares quadratic-flux-minimizing surfaces and ghost surfaces in magnetic fields, showing they are nearly equivalent under certain conditions and introducing a generalized Hamiltonian framework for their analysis.

Contribution

It provides a unified Hamiltonian and Lagrangian approach to define and compare QFMin and ghost surfaces, demonstrating their equivalence up to second order in perturbed magnetic fields.

Findings

QFMin and ghost surfaces are numerically similar in chaotic magnetic fields.

A perturbative construction shows their equivalence up to second order.

The concepts are applicable to other nonintegrable Hamiltonian systems.

Abstract

Two candidates for "almost-invariant" toroidal surfaces passing through magnetic islands, namely quadratic-flux-minimizing (QFMin) surfaces and ghost surfaces, use families of periodic pseudo-orbits (i.e. paths for which the action is not exactly extremal). QFMin pseudo-orbits, which are coordinate-dependent, are field lines obtained from a modified magnetic field, and ghost-surface pseudo-orbits are obtained by displacing closed field lines in the direction of steepest descent of magnetic action, $\oint \vec{A}\cdot\mathbf{dl}$. A generalized Hamiltonian definition of ghost surfaces is given and specialized to the usual Lagrangian definition. A modified Hamilton's Principle is introduced that allows the use of Lagrangian integration for calculation of the QFMin pseudo-orbits. Numerical calculations show QFMin and Lagrangian ghost surfaces give very similar results for a chaotic magnetic field perturbed from an integrable case, and this is explained using a perturbative construction of an auxiliary poloidal angle for which QFMin and Lagrangian ghost surfaces are the same up to second order. While presented in the context of 3-dimensional magnetic field line systems, the concepts are applicable to defining almost-invariant tori in other $1{1/2}$ degree-of-freedom nonintegrable Lagrangian/Hamiltonian systems.