An adjoint-based method for optimising MHD equilibria against the infinite-n, ideal ballooning mode

Abstract

We demonstrate a fast adjoint-based method to optimise tokamak and stellarator equilibria against a pressure-driven instability known as the infinite- $n$ ideal ballooning mode. We present three finite- $\beta$ (the ratio of thermal to magnetic pressure) equilibria: one tokamak equilibrium and two stellarator equilibria that are unstable against the ballooning mode. Using the self-adjoint property of ideal magnetohydrodynamics, we construct a technique to rapidly calculate the change in the eigenvalue, a measure of ideal ballooning instability. Using the SIMSOPT optimisation framework, we then implement our fast adjoint gradient-based optimiser to minimise the eigenvalue and find stable equilibria for each of the three originally unstable equilibria.

Asymptotic quasisymmetric high-beta three-dimensional magnetohydrodynamic equilibria near axisymmetry

“…To calculate the stability, we numerically solve (Gaur et al. 2023) the ballooning eigenmode equation (Connor, Hastie & Taylor 1978; Dewar & Glasser 1983) where is the ballooning eigenvalue, and are the following normalized geometric coefficients: where is the field-line label, is the PEST straight-field-line angle, is the cylindrical toroidal angle and is the ballooning parameter. All lengths in the ballooning equation are normalized using , the effective minor radius (named Aminor_p) in the VMEC output, and magnetic field and plasma pressure are normalized using , where is the toroidal flux enclosed by the boundary (without the factor of ).…”

“…The ideal ballooning mode is another curvature-driven instability that can appear in equilibria with finite magnetic shear. To calculate the stability, we numerically solve (Gaur et al 2023) the ballooning eigenmode equation (Connor, Hastie & Taylor 1978;Dewar & Glasser 1983) where λ is the ballooning eigenvalue, and g, c, f are the following normalized geometric coefficients:…”