Nonlinear Saturation and Metastability of Ballooning Modes in Stellarators

We investigate the nonlinear MHD stability and saturation of ideal ballooning modes. We assume that the nonlinear solution is an isolated flux tube that moves radially[1]. We extend previous results by Ham et al.[2] on the stability and saturation of such flux tubes in tokamaks to stellarators. Tools based on the DESC[3] equilibrium solver were developed to calculate saturated states of perturbed flux tubes along with their energy in general stellarator equilibria. Benchmarks against linear dynamics in both tokamaks and stellarators were performed.

Systematic convergence analysis showed that the direct calculation of energy of a displaced flux tube is very sensitive to the force error of the numerical equilibrium. First order contribution in energy due to force error makes the computed energy dependent on the simulation box size, even when the flux tube radial displacement is converged. A modified algorithm for calculating energy of a flux tube by comparing neighbouring piece-wise $C^1$ flux tubes is proposed to reduce the influence of force error. Using the tools developed, nonlinear saturated flux tubes are calculated on multiple stellarators, including NCSX and ESTELL. Generically the equilibrium is metastable when it is linearly ballooning stable but near the linear marginal stability boundary.

[1] S. C. Cowley, et al., Proc. R. Soc. A 471, 20140913 (2015)

[2] C. J. Ham, et al., Plasma Phys. Control. Fusion 60, 075017 (2018)

[3] D. Panici, et al., 2023 J. Plasma Phys. 89, 955890303