Initial saturation dynamics and current sheet formation of external kink modes in near circular tokamak and $l=2$ stellarator geometry

Equilibrium and initial value codes are used to study the initial saturation of external kinks in low $\beta$ tokamaks and $l=2$ stellarators. Within the defined parameter space, the saturated states observed with helicity preserving VMEC computations do not converge to a single solution with increasing spectral resolution. This is because the ideal MHD model allows for the unconstrained formation of current sheets at the plasma boundary. Helicity conservation, the incompressibility constraint, and magnitude of the induced plasma current spike are used to constrain the solution space towards a physical solution. Parameter scans of the edge safety factor, and external rotational transform show how the structure of a saturated (4, 1) external kink depends on the magnetic geometry of classical stellarators. Saturated states can be identified at different field periodicities which decrease in amplitude with increasing external rotational transform, following the expected trend informed by the modification of the linear growth rate. For a subset of these perturbed equilibria, the magnetic energy spectrum is used to demonstrate that the nonlinear mode coupling of the perturbation is dominated by members of the toroidal mode family of the observed MHD instability. The nonlinear magnetic energy spectra, poloidal localisation of induced current sheets, and predicted redistribution of poloidal magnetic energy from the vacuum to plasma region are compared with JOREK simulations to further explain observations from VMEC results. This work constitutes a first step towards using both approaches to analyse external modes in advanced stellarators.