High Order Free Boundary MHD Equilibria in DESC

TL;DR

The paper presents a method to compute free boundary equilibria in 3D ideal MHD within the DESC framework by integrating a high‑order singular boundary integral technique based on a partition‑of‑unity approach. It formulates the free boundary problem as an optimization over the plasma boundary shape and a current potential, using a Biot–Savart representation that includes external coils, plasma currents, and surface currents, and avoids solving a large exterior Neumann problem. Comparisons against VMEC and direct field‑line tracing show that DESC achieves higher accuracy, including a reduced residual in the boundary conditions, and can handle vacuum and finite‑beta cases with sheet currents. The work supports straightforward extension to single‑stage optimization (simultaneous plasma and coil evolution) and enhances DESC’s utility for stellarator coil design and equilibrium verification.

Abstract

In this work we consider the free boundary inverse equilibrium problem for 3D ideal MHD. We review boundary conditions for both fixed and free boundary solutions and under what circumstances a sheet current may exist at the plasma-vacuum interface. We develop an efficient and accurate algorithm for computing the residual of these boundary conditions and use it to compute free boundary equilibria in the DESC code both in vacuum and at finite plasma beta, with and without sheet currents.

Figures (4)

• Figure 1: Flux surfaces for the Landreman & Paul QA equilibrium from DESC, overlayed with field line trajectories traced from the coil set.
• Figure 2: Deformation of initially circular boundary shape into helical stellarator.
• Figure 3: Convergence of boundary condition residual as function of spectral resolution for helical stellarator with finite edge pressure.
• Figure 4: Comparison of free boundary DESC and VMEC solutions for W7-X at $\beta=2\%$.