Quasi-axisymmetric magnetic fields: weakly non-axisymmetric case in a vacuum

TL;DR

This paper develops an asymptotic method to construct weakly non-axisymmetric quasi-axisymmetric magnetic fields, identifies solution spaces, and verifies the approach with numerical solutions, advancing magnetic confinement design.

Contribution

It introduces an asymptotic expansion approach to construct quasi-axisymmetric magnetic fields and demonstrates solutions satisfying quasi-axisymmetry and omnigeneity without numerical search.

Findings

Identified a large space of quasi-axisymmetric solutions on a single flux surface.

Proved globally quasi-axisymmetric solutions do not exist, aligning with previous theories.

Validated the asymptotic solutions with numerical methods, confirming quasi-axisymmetry at the appropriate order.

Abstract

An asymptotic expansion is performed to obtain quasi-axisymmetric magnetic configurations that are weakly non-axisymmetric. A large space of solutions is identified, which satisfy the condition of quasi-axisymmetry on a single magnetic flux surface, while (non-axisymmetric) globally quasi-axisymmetric solutions are shown to not exist, agreeing with the conclusions of previous theoretical work. The solutions found are shown to be geometrically constrained at low aspect ratio or high toroidal period number. Solutions satisfying the more general condition of omnigeneity (generalized quasi-axisymmetry) are also shown to exist, and it is found that quasi-axisymmetric deformations can be superposed with an omnigenous solution, while preserving the property of omnigeneity, effectively extending the space of "good" configurations. A numerical solution of the first order quasi-axisymmetry problem is demonstrated and compared with solutions found with a widely used MHD equilibrium solver, independently verifying that quasi-axisymmetry is satisfied at the appropriate order. It is thereby demonstrated that approximately quasi-axisymmetric solutions can be directly constructed, i.e. without using numerical search algorithms.