Figures of merit for stellarators near the magnetic axis

TL;DR

This paper develops rapid computational tools for stellarator design near the magnetic axis, enabling optimization of coil shapes, assessing field complexity, and estimating physical limits on configuration parameters.

Contribution

It introduces new formulas for quickly evaluating magnetic field derivatives, stability bounds, and error fields in near-axis stellarator configurations.

Findings

Computed $ ablaoldsymbol{B}$ and $ abla ablaoldsymbol{B}$ tensors for optimization.

Identified a lower bound on aspect ratio based on flux surface singularities.

Quantified second-order error fields in magnetic configuration design.

Abstract

A new paradigm for rapid stellarator configuration design has been recently demonstrated, in which the shapes of quasisymmetric or omnigenous flux surfaces are computed directly using an expansion in small distance from the magnetic axis. To further develop this approach, here we derive several other quantities of interest that can be rapidly computed from this near-axis expansion. First, the $\nabla\vec{B}$ and $\nabla\nabla\vec{B}$ tensors are computed, which can be used for direct derivative-based optimization of electromagnetic coil shapes to achieve the desired magnetic configuration. Moreover, if the norm of these tensors is large compared to the field strength for a given magnetic field, the field must have a short length scale, suggesting it may be hard to produce with coils that are suitably far away. Second, we evaluate the minor radius at which the flux surface shapes would become singular, providing a lower bound on the achievable aspect ratio. This bound is also shown to be related to an equilibrium beta limit. Finally, for configurations that are constructed to achieve a desired magnetic field strength to first order in the expansion, we compute the error field that arises due to second order terms.