The High-Order Magnetic Near-Axis Expansion: Ill-Posedness and Regularization

TL;DR

This paper analyzes the high-order vacuum near-axis expansion used for stellarator design and shows that the expansion is ill-posed and divergent at high order. It introduces a high-order regularization term, proving convergence of the regularized near-axis expansion under suitable Sobolev-analytic inputs and demonstrating, with coil-based examples, numerical convergence of the vacuum magnetic field and flux surfaces near the axis. The radius of convergence is shown to correlate with the distance from the axis to the coils, and regularization improves stability and accuracy under perturbations. The work provides a practical, mathematically grounded path toward reliable, fast near-axis computations that can inform coil design and optimization, with potential extensions to pressure-driven equilibria via a fictitious-current regularization.

Abstract

When analyzing stellarator configurations, it is common to perform an asymptotic expansion about the magnetic axis. This so-called near-axis expansion is convenient for the same reason asymptotic expansions often are, namely, it reduces the dimension of the problem. This leads to convenient and quickly computed expressions of physical quantities, such as quasisymmetry and stability criteria, which can be used to gain further insight. However, it has been repeatedly found that the expansion diverges at high orders in the distance from axis, limiting the physics the expansion can describe. In this paper, we show that the near-axis expansion diverges in vacuum due to ill-posedness and that it can be regularized to improve its convergence. Then, using realistic stellarator coil sets, we demonstrate numerical convergence of the vacuum magnetic field and flux surfaces to the true values as the order increases. We numerically find that the regularization improves the solutions of the near-axis expansion under perturbation, and we demonstrate that the radius of convergence of the vacuum near-axis expansion is correlated with the distance from the axis to the coils.

Figures (8)

• Figure 1: Schematic of the direct near-axis Frenet-Serret coordinate frame.
• Figure 2: A schematic of the process of finding straight field-line coordinates. On the left, we plot the surfaces of the magnetic field $(h^x,h^y)$ on a cross-section for fixed $s$. Moving one plot to the right, the leading correction transforms to a coordinate frame where the main elliptic component is eliminated. Going one further, the next correction accounts for the most prominent triangularity. This process continues until, in $(\xi,\eta)$ coordinates, the magnetic surfaces are nested circles.
• Figure 3: Coil sets for the rotating ellipse and Landreman-Paul examples. The color indicates the normalized $\bm{B} \cdot \bm N$ error on the outer closed flux surface.
• Figure 4: Plot of (markers) the coefficient norm $\lVert\phi^{(\text{IC})}_m\rVert_{H^2}$ versus the order $m$ and (lines) best-fit lines $A \sigma_{\mathrm{coil}}^{-m}$ where $\sigma_{\mathrm{coil}}$ is the axis-to-coil distance.
• Figure 5: (a-b) The error \ref{['eq:B-error']} as a function of the normalized distance from axis $\sigma/\sigma_{\mathrm{coil}}$ for varying orders of approximation $N_\rho$. (c-d) The error \ref{['eq:B-error']} as a function of $\sigma/\sigma_{\mathrm{coil}}$ for varying values of the regularization parameter $K$ ($K=\infty$ is unregularized).

Theorems & Definitions (33)

• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Definition 3.3
• Proposition 3.4
• proof
• Corollary 3.5
• proof
• Theorem 3.6