An asymptotic Grad-Shafranov equation for quasisymmetric stellarators*

The MHD equilibrium and stability for high beta quasisymmetric (QS) stellarators is of great interest for designs of stellarator experiments and fusion power plants. Attempts to solve the 3D MHD equilibrium problems have not led to tractable forms of the Grad-Shafranov equation (GSE), which are well-known to be elliptic in axisymmetric geometry. We present an asymptotic high-beta model for QS devices where beta is ordered as a small parameter epsilon, the ratio of the field due to plasma current to the coil field (arXiv:2403.08022). The first-order correction, which involves both coil and plasma current fields, is governed by a GSE and the requirement that flux surfaces exist. Further, we show that the first-order near-axis expansion (NAE) is a subset of our model.

Our analytic derivation of a high-beta GSE shows a way to incorporate fast equilibrium solvers in stellarator optimization codes. This is an improvement over the popular NAE models, which cannot account for magnetic shear easily. Our model also allows more general pressure and current profiles than the NAE. We present analytical and numerical solutions outside the scope of the NAE, demonstrating our model's usefulness for QS stellarators. We also show that high beta requires high aspect ratio. By allowing beta to be of order epsilon squared (low beta), we can model compact devices. Finally, yet another generalization is possible that will allow the modelling of general omnigenous stellarators at the expense of significantly more complicated equations.