The High-Order Magnetic Near-Axis Expansion: Ill-Posedness and Regularization

TL;DR

This paper investigates the divergence issues in high-order near-axis expansions for stellarator magnetic fields, identifies ill-posedness as a cause, and introduces regularization techniques to enhance convergence and physical accuracy.

Contribution

It reveals the ill-posed nature of the vacuum near-axis expansion and proposes a regularization method to improve its convergence and applicability.

Findings

Regularization improves the convergence of the near-axis expansion.

Numerical results show better approximation of magnetic fields with higher order expansions.

The radius of convergence correlates with coil distance from the axis.

Abstract

When analyzing stellarator configurations, it is common to perform an asymptotic expansion about the magnetic axis. This so-called near-axis expansion is convenient for the same reason asymptotic expansions often are, namely, it reduces the dimension of the problem. This leads to convenient and quickly computed expressions of physical quantities, such as quasisymmetry and stability criteria, which can be used to gain further insight. However, it has been repeatedly found that the expansion diverges at high orders in the distance from axis, limiting the physics the expansion can describe. In this paper, we show that the near-axis expansion diverges in vacuum due to ill-posedness and that it can be regularized to improve its convergence. Then, using realistic stellarator coil sets, we demonstrate numerical convergence of the vacuum magnetic field and flux surfaces to the true values as the order increases. We numerically find that the regularization improves the solutions of the near-axis expansion under perturbation, and we demonstrate that the radius of convergence of the vacuum near-axis expansion is correlated with the distance from the axis to the coils.