Nonlinear magnetohydrodynamic modeling of ideal ballooning modes in high-$\beta$ Wendelstein 7-X plasmas

TL;DR

This paper uses nonlinear MHD simulations to analyze ideal ballooning modes in high-$eta$ Wendelstein 7-X plasmas, revealing that stability is influenced by profile shape and magnetic configuration, and is not solely predicted by linear growth rates.

Contribution

It demonstrates the application of nonlinear MHD modeling to stellarator plasmas, showing that benign saturation is not guaranteed by linear stability and depends on profile and magnetic configuration.

Findings

Increased parallel conductivity reduces linear growth rate but not saturated pressure.

Profile shape significantly affects nonlinear stability, with peaked profiles more susceptible.

Saturation mechanism is not specific to resonant or non-resonant modes.

Abstract

We present nonlinear magnetohydrodynamic (MHD) simulations of high-$\beta$ Wendelstein 7-X plasmas using the stellarator extension of the M3D-$C^1$ code, building on the recent work that shows benign saturation of ideal ballooning modes above the designed $\beta$ limit in the standard configuration [Y. Zhou et al, Phys. Rev. Lett. 133, 135102 (2024)]. First, we examine the results' sensitivity to the parallel thermal conductivity. It is found that while an increased parallel conductivity reduces the linear growth rate, the saturated pressure profile is barely affected. Second, we consider the dependence on the profile shape. It is shown that an equilibrium with a peaked pressure profile and lower $\beta$ is subject to more significant change than a broad profile with higher $\beta$ and a larger growth rate, suggesting that benign saturation, or nonlinear stability, is not guaranteed and not dictated by linear growth. Third, we study the influence of the magnetic configuration, with the equilibrium rotational transform varied by adjusting the planar coil current. With similar growth rates, similar magnitudes of profile change are found regardless of the presence of a low-order resonance, which implies that the saturation mechanism is not specific to a resonant or non-resonant mode. These results indicate that MHD stability should still be treated seriously in stellarator operation and design, for which nonlinear modeling using tools like M3D-$C^1$ can play an instrumental role.