Going to High Order in the Magnetic Near-Axis Expansion

The magnetic near-axis expansion has become increasingly popular due to its ability to quickly explore the space of potential stellarator configurations. This is due to a combination of factors, including the interpretability of the input data, the ease of enforcing quasisymmetry and other targets, and the speed of computation. Unfortunately, the near-axis expansion has a major drawback: it tends to diverge at higher orders in the distance from the axis. This reduces the near-axis expansion's ability to describe important physics, such as corrections to the magnetic shear and curvature.

In this presentation, we explore both a cause and a solution to the divergence of the near-axis expansion. For the cause, we show that the magnetic near-axis expansion is ill-posed for vacuum fields, meaning small perturbations to the on-axis input data cause large high-frequency errors far from the axis. For the solution, we show how the problem can be regularized using a ``viscosity'' term. This regularization removes high-frequency errors, resulting in accurate solutions to Laplace's equation far from the axis. Finally, we show how we can use this to obtain high-order corrections to flux coordinates from the near-axis expansion, giving corrections to the magnetic shear.