The On-Axis Magnetic Well and Mercier's Criterion for Arbitrary Stellarator Geometries

TL;DR

This paper derives simplified analytical expressions for the on-axis magnetic well and Mercier's criterion in three-dimensional stellarator geometries, enabling easier stability analysis through one-dimensional integrals.

Contribution

It introduces a novel near-axis expansion method that expresses the magnetic well and Mercier's criterion as one-dimensional integrals, streamlining stability calculations for arbitrary stellarator configurations.

Findings

Derived analytical forms for magnetic well and Mercier's criterion.

Validated expressions numerically on various stellarator configurations.

Enhanced understanding of stability criteria in complex magnetic geometries.

Abstract

A simplified analytical form of the on-axis magnetic well and Mercier's criterion for interchange instabilities for arbitrary three-dimensional magnetic field geometries is derived. For this purpose, a near-axis expansion based on a direct coordinate approach is used by expressing the toroidal magnetic flux in terms of powers of the radial distance to the magnetic axis. For the first time, the magnetic well and Mercier's criterion are then written as a one-dimensional integral with respect to the axis arclength. When compared with the original work of Mercier, the derivation here is presented using modern notation and in a more streamlined manner that highlights essential steps. Finally, these expressions are verified numerically using several quasisymmetric and non-quasisymmetric stellarator configurations including Wendelstein 7-X.