Computing MHD equilibria of stellarators with a flexible coordinate frame

TL;DR

This paper introduces a flexible coordinate frame approach for computing MHD equilibria in stellarators, enabling better representation of complex geometries and reducing computational complexity.

Contribution

It proposes a new boundary representation using a general coordinate frame, improving modeling of non-planar and knotted axes in stellarator equilibria.

Findings

Fewer degrees of freedom needed for high-quality solutions.

Enhanced ability to model complex stellarator geometries.

Implementation in the GVEC solver demonstrates efficiency improvements.

Abstract

For the representation of axi-symmetric plasma configurations, it is natural to use cyl. coordinates (R,Z,$\phi$), where $\phi$ is an independent coordinate. The same cyl. coordinates have also been widely used for representing 3D MHD equilibria of non-axisymmetric configurations (stellarators), with cross-sections, defined in RZ-planes, that vary over $\phi$. Stellarator equilibria have been found, however, for which cyl. coordinates are not at all a natural choice, for instance certain stellarators obtained using the near-axis expansion (NAE), defined by a magn. axis curve and its Frenet frame. In this contribution, we propose an alternative approach for representing the boundary in a fixed-boundary 3D MHD equil. solver, moving away from cyl. coordinates. Instead, we use planar cross-sections whose orientation is determined by a general coordinate frame (G-Frame). This frame is similar to the conventional Frenet frame, but more flexible. As an additional part of the boundary representation, it becomes an input to the equil. solve, along with the geometry of the cross-sections. We see two advantages: 1) the capability to easily represent configurations where the magn. axis is highly non-planar or even knotted 2) a reduction in the degrees of freedom needed for the boundary surface, and thus the equil. solver, enabling progress in optimization of these configurations. We discuss the properties of the G-Frame, starting from the conventional Frenet frame. Then we show two exemplary ways of constructing it, first from a NAE solution and also from a given boundary surface. We present the details of the implementation of the new frame in the 3D MHD equil. solver GVEC. Furthermore, we demonstrate for a highly shaped QI-optimized stellarator that far fewer degrees of freedom are necessary to find a high quality equil. solution, compared to the solution computed in cyl. coordinates.