Theory of ridges in compact quasi-axisymmetric devices

A common feature of most numerically optimized stellarator geometries is the presence of sharp ridges in flux surfaces near the last closed flux surface that could be utilized for constructing non-resonant divertors. In this work, we develop a general theory of these ridges for quasi-axisymmetric (QA) devices. First, motivated by recent work (Henneberg and Plunk, PRR 2024) on compact hybrid devices, we develop a perturbative treatment of nearly axisymmetric quasisymmetric devices by expanding in the deviation from perfect axisymmetry. As a result, we can analytically describe key features of compact QA devices, such as the tendency for ridges to be localized on the inboard side where the Gaussian curvature is typically negative, and the field strength is maximum. To develop a nonlinear theory, we utilize A. Boozer's key physical insight that a magnetic field must follow a sharp ridge to avoid bending. Leveraging key geometrical ideas developed in computer vision, differential geometry, and singularity theory, we show that the magnetic field strength must be approximately constant near the ridges, which necessitates localization typically on the inboard side. Thus, expanding in the deviation from the maximum B, we can provide nonlinear descriptions of the ridges and the singularities that tend to form on the last closed flux surface. Finally, we will discuss the impact of ridges and surface geometry on coil complexity and show correlations with the LgradB metric (Kappel and Landreman PPCF 2024 ).