Narrow Operator Models of Stellarator Equilibria in Fourier Zernike Basis

Stellarators are inherently steady-state plasma confinement devices, which is among the key reasons behind their renaissance as promising candidates for fusion power plants. Ideal MHD equilibria are a central part in optimising the complex, three-dimensional plasma shapes which are a necessary condition for steady-state operation of such devices. The equilibrium magnetic field is required not only in optimisation but also plays a role in future real-time control algorithms and simulation frameworks (Schissel et al. 2025). Solving the three-dimensional MHD equations requires numerical approaches, because no analytical solutions throughout the full volume of ideal MHD equilibria with nested magnetic topology exists yet (Bruno & Laurence 1996). Recent work advanced analytical models for Fourier components of the equilibrium magnetic field in a subset of reactor-relevant magnetic fields and analytical expansions close to the magnetic axis are used extensively in research (Nikulsin et al. 2024; Sengupta et al. 2024). These analytical solutions and the following numerical solvers assume nested magnetic topology, or inte-grability throughout the volume, and computation of chaotic regions or magnetic islands takes considerably more effort (Hudson et al. 2012). Accuracy of numerical PDE solutions is inherently connected to the representation which defines gradients, and commonly used ideal MHD equilibrium solvers with nested magnetic field topology can be differentiated accordingly: A widely used finite-difference solver employed in the design of currently operating stellarator devices is VMEC (Hirshman & Whitson 1983), another pseudo spectral solver is DESC (Dudt & Kolemen 2020) and a third example is GVEC (Hindenlang et al. 2025), that abstracts the notion of basis functions, which enabled computation of plasmas with figure-8 shape (Plunk et al. 2025). Email address for correspondence: timo.thun@ipp.mpg.de