# How well can VMEC predict the initial saturation of external kink modes in near circular tokamaks and $l=2$ stellarators?

## TL;DR

This study evaluates VMEC's ability to predict the initial saturation of external kink modes in near-circular tokamaks and $l=2$ stellarators, highlighting its limitations and potential for future nonlinear MHD modeling.

## Contribution

The paper investigates VMEC's applicability to nonlinear MHD effects and identifies the constraints needed for accurate physical modeling of kink mode saturation.

## Key findings

- VMEC does not converge to a single solution with increasing spectral resolution.
- Saturated kink states decrease with increasing external rotational transform, but bifurcated states appear.
- VMEC captures expected toroidal mode coupling and aligns with JOREK in energy redistribution.

## Abstract

The equilibrium code, VMEC, is used to study external kinks in low $\beta$ tokamaks and $l=2$ stellarators. The applicability of the code when modelling nonlinear MHD effects is explored in an attempt to understand and predict how the initial saturation of the MHD mode depends on the external rotational transform. It is shown that helicity preserving, free boundary VMEC computations do not converge to a single perturbed solution with increasing spectral resolution. Additional constraints are therefore applied to narrow down the numerical resolution parameters appropriate for physical scans. The dependence of the modelled (4, 1) kink mode on the external rotational transform and field periodicity is then studied. While saturated states can be identified which decrease in amplitude with increasing external rotational transform, bifurcated states are found that contradict this trend. It was therefore not possible to use VMEC alone to identify the physical dependency of the nonlinear mode amplitude on the magnetic geometry. The accuracy of the VMEC solutions is nevertheless demonstrated by showing that the expected toroidal mode coupling is captured in the magnetic energy spectrum for stellarator cases. Comparing with the initial value code, JOREK, the predicted redistribution of poloidal magnetic energy from the vacuum to plasma region in VMEC is shown to be physical. This work is a first step towards using VMEC to study MHD modes in stellarator geometry.