Vacuum magnetic fields with exact quasisymmetry near a flux surface. Part 1: Solutions near an axisymmetric surface

TL;DR

This paper presents the first vacuum surface expansion solutions for exactly quasisymmetric magnetic fields near an axisymmetric surface, including analytical and numerical models across various geometries, revealing localized surface perturbations.

Contribution

It introduces a novel vacuum surface expansion formalism to find exact quasisymmetric solutions, including closed-form and numerical solutions in different geometries.

Findings

Closed-form solutions in cylindrical, slab, and isodynamic geometries.

Numerical solutions in axisymmetric toroidal geometry.

Surface perturbations tend to localize on the inboard side.

Abstract

While several results have pointed to the existence of exactly quasisymmetric fields on a surface (Garren & Boozer 1991a,b; Plunk & Helander 2018), we have obtained the first such solutions using a vacuum surface expansion formalism. We obtain a single nonlinear parabolic PDE for a function $\eta$ such the field strength satisfies $B = B(\eta)$. Closed-form solutions are obtained in cylindrical, slab, and isodynamic geometries. Numerical solutions of the full nonlinear equations in general axisymmetric toroidal geometry are obtained, resulting in a class of quasi-helical local vacuum equilibria near an axisymmetric surface. The analytic models provide additional insight into general features of the nonlinear solutions, such as localization of the surface perturbations on the inboard side.