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 Phys. Plasmas 27 070701; Loizu and Bonfiglio 2023 J. Plasma Phys. 89 905890507), we make use of Stepped Pressure Equilibrium Code (SPEC) (Hudson et al 2012 Phys. Plasmas 19 112502), an equilibrium solver based on the variational principle of the multi-region relaxed magnetohydrodynamics (MHDs), 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 Δ′ and bootstrap current strength. Saturated island widths produced by SPEC agree well with the predictions of an initial value resistive MHDs code (Huang and Bhattacharjee 2016 Astrophys. J. 818 20) 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 NTMs, which are linearly stable but nonlinear-unstable due to the effects of the bootstrap current.

We present single-stage optimization of islands in finite-β stellarator equilibria. Stellarator optimization is traditionally performed as a two-stage process; in the first stage, an optimal equilibrium is calculated that balances a set of competing constraints, and in the second stage, a set of coils is found that supports the 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 β 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.