Near-Axis Expansion of Stellarator Equilibrium at Arbitrary Order in the Distance to the Axis

TL;DR

This paper develops an analytical near-axis expansion method for stellarator equilibria at arbitrary order, providing insights into flux surface properties and comparing with existing formalisms, validated by a W7-X equilibrium example.

Contribution

It introduces a novel asymptotic expansion framework for stellarator magnetic fields near the axis, extending previous methods to arbitrary order and enabling detailed shape analysis.

Findings

The framework agrees with Garren-Boozer at lowest order.

It provides analytical expressions for magnetic field and flux surfaces.

Numerical comparison with W7-X shows good agreement at lowest order.

Abstract

A direct construction of equilibrium magnetic fields with toroidal topology at arbitrary order in the distance from the magnetic axis is carried out, yielding an analytical framework able to explore the landscape of possible magnetic flux surfaces in the vicinity of the axis. This framework can provide meaningful analytical insight on the character of high-aspect-ratio stellarator shapes, such as the dependence of the rotational transform and the plasma beta-limit on geometrical properties of the resulting flux surfaces. The approach developed here is based on an asymptotic expansion on the inverse aspect-ratio of the ideal MHD equation. The analysis is simplified by using an orthogonal coordinate system relative to the Frenet-Serret frame at the magnetic axis. The magnetic field vector, the toroidal magnetic flux, the current density, the field line label and the rotational transform are derived at arbitrary order in the expansion parameter. Moreover, a comparison with a near-axis expansion formalism employing an inverse coordinate method based on Boozer coordinates (the so-called Garren-Boozer construction) is made, where both methods are shown to agree at lowest order. Finally, as a practical example, a numerical solution using a W7-X equilibrium is presented, and a comparison between the lowest order solution and the W7-X magnetic field is performed.