A Grad-Shafranov model for compact quasisymmetric stellarators

TL;DR

This paper derives a Grad-Shafranov equation for compact quasisymmetric stellarators using an asymptotic expansion, enabling analytical solutions and efficient optimization of magnetic flux surfaces.

Contribution

It introduces a new Grad-Shafranov model for quasisymmetric stellarators, including analytical solutions and a method for approximate equilibrium optimization.

Findings

Derived a Grad-Shafranov equation for quasisymmetric stellarators.

Provided analytical solutions for flux surface existence.

Proposed an optimization approach for equilibria outside hybrid device class.

Abstract

A Grad-Shafranov equation (GSE) valid for compact quasisymmetric stellarators is derived by an asymptotic expansion around a vacuum field carried to first order. We obtain an equation for the existence of flux surfaces leading up to the GSE. The flux surface label must simultaneously satisfy the existence equation and the GSE, which generally leads to an overdetermined problem. We show how the overdetermined problem can be resolved within our model for a class of hybrid devices similar to that studied by Henneberg and Plunk (S. Henneberg and G. Plunk, PRR 2024). We are also able to solve the existence equation for flux surfaces analytically in the most general case by introducing a special coordinate system. This will enable us to carry out an optimization seeking to minimize the error in our GSE while obeying the flux surface existence equation, which will allow us to find solutions outside the class of hybrid devices. This will allow for a coarse-grained approximate search in the space of quasisymmetric equilibria that should be faster than a conventional stellarator optimization. Nevertheless, it would still be necessary to fine-tune the approximate solutions using conventional tools to obtain a more precise optimized equilibrium.