We demonstrate for the first time that the nonlinear saturation of neoclassical tearing modes (NTMs) can be found directly using a variational principle based on Taylor relaxation, without needing to simulate the intermediate, resistivity-dependent dynamics. As in previous investigations of classical tearing mode saturation (Loizu et al. 2020; Loizu & Bonfiglio 2023), we make use of Stepped Pressure Equilibrium Code (SPEC) (Hudson et al. 2012), an equilibrium solver based on the variational principle of the Multi-Region relaxed MHD, featuring stepped pressure profiles and arbitrary magnetic topology. We work in slab geometry and employ a simple bootstrap current model Jbs = C ▽p to study the bootstrap-driven tearing modes, scanning over the asymptotic matching parameter △′ and bootstrap current strength. Saturated island widths produced by SPEC agree well with the predictions of an initial value resistive Magnetohydrodynamics (MHD) code (Huang & Bhattacharjee 2016) while being orders of magnitude faster to calculate. Additionally, we observe good agreement with a simple analytical Modified Rutherford Equation, without requiring any fitting coefficients. The match is obtained for both linearly unstable classical tearing modes in the presence of bootstrap current, and neoclassical tearing modes, which are linearly stable but nonlinear-unstable due to the effects of the bootstrap current.

We present the first single-stage optimization of islands in finite-$\beta$ stellarator equilibria. Stellarator optimization is traditionally performed as a two-stage process; in the first stage, an optimal equilibrium is calculated which balances a set of competing constraints, and in the second stage a set of coils is found that supports said equilibrium. Stage one is generally performed using a representation for the equilibrium that assumes nestedness of flux surfaces, even though this is not warranted and occasionally undesired. The second stage optimization of coils is never perfect, and the mismatch leads to worse performing equilibria, and further deteriorates if additional constraints such as force minimization, coil torsion or port access are included. The higher fidelity of single-stage optimization is especially important for the optimization of islands as these are incredibly sensitive to changes in the field. In this paper we demonstrate an optimization scheme capable of optimizing islands in finite $\beta$ stellarator equilibria directly from coils. We furthermore develop and demonstrate a method to reduce the dimensionality of the single-stage optimization problem to that of the first stage in the two-stage approach.