Computing MHD equilibria of stellarators with a flexible coordinate frame

TL;DR

The paper addresses the limitations of using cylindrical (R,Z,φ) coordinates to represent fixed-boundary 3D MHD equilibria in stellarators, especially for highly shaped or non-planar axes. It introduces a general coordinate frame (G-frame) anchored to a guiding curve, and integrates this frame into the GVEC solver to describe cross-sections and boundary geometry with reduced degrees of freedom. Two construction pathways are demonstrated: (i) deriving a G-frame from near-axis expansion (NAE) axis data, untwisting the Frenet frame, and (ii) building a G-frame directly from a boundary surface (e.g., QUASR data), with an explicit rotation function to ensure smoothness. Application to a highly shaped two-field-period QI stellarator shows markedly fewer Fourier modes are needed to represent the boundary and faster convergence (800 vs 16,000 iterations) while maintaining agreement with the NAE reference, underscoring the practical impact for stellarator optimization.

Abstract

For the representation of axi-symmetric plasma configurations, it is natural to use cyl. coordinates (R,Z,$φ$), where $φ$ 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 $φ$. 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.

Figures (17)

• Figure 1: Comparison of $(R,Z,\phi)$ in and Frenet frame for a two field periodic QI-configuration N2-12plunk2024-QI. Top view (left), side view with cross-sections (middle), all cross-sections in respective frame, colored along the toroidal parametrization (right).
• Figure 2: Top view (left) and side view (right) of two configurations. The cross-sections in $R,Z$-planes and in the Frenet frame are shown, one field period is colored green.
• Figure 3: Construction of the local Frenet frame from first and second derivatives of the curve ${\bm{X}}_0(\zeta)$. The normal ${\bm N}^F$ scales with the second derivative, so it must be $\neq 0$ to yield a valid Frenet frame.
• Figure 4: Visualization of the $(N,B)$-frame by an axis-centered rectangle swept along the axis, the surface color marks the directions $+N$ in red, $-N$ in green, $+B$ in blue and $-B$ in yellow. The Frenet frame is undefined at points of zero curvature of the axis, and the normal and bi-normal can flip direction. The flip shown here arises at locations of minimum field strength in quasi-isodynamic stellarators plunk_landreman_helander_2019 (top left). In the $G$-frame, the normal and bi-normal are given functions over the curve parameter, allowing to construct a continuously differentiable frame (bottom left). The Frenet frame can be twisted, here with a self-linking number of $-4$ (top right). The $G$-frame (bottom right) allows to 'untwist', yielding a self-linking number of $0$.
• Figure 5: Signed curvature $\kappa^s$ and torsion for the magnetic axis of the N2-12 configuration. The horizontal axis is arc-length, which varies from $0$ to $\pi$ over the first field period.