Linear Ideal MHD Stability Codes
Ideal MHD stability codes test whether an equilibrium is stable to small perfectly conducting perturbations by solving an eigenvalue problem.
The ideal stability question
Given an equilibrium, the first stability question is whether small ideal-MHD perturbations grow. Ideal MHD assumes zero resistivity, so magnetic field lines are frozen into the plasma and cannot reconnect. A linear ideal MHD stability code linearizes the force-balance equation about the equilibrium and solves for the perturbation displacement that extremizes the potential energy functional, delta-W.
If any allowed displacement lowers the potential energy (delta-W negative), the equilibrium is ideally unstable and would disrupt on the fast Alfven timescale. Ideal stability is therefore a hard operating boundary, not a soft degradation.
Eigenvalue formulation
The problem reduces to a generalized eigenvalue problem where the eigenvalue is the growth rate squared and the eigenvector is the mode structure. Codes expand the displacement in Fourier harmonics of the poloidal and toroidal angles and in radial finite elements, yielding a large matrix eigenproblem.
Modes analyzed
Ideal codes assess internal and external kink modes, ballooning modes, and the pressure- and current-driven limits that set the achievable normalized pressure. External modes may be stabilized by a conducting wall, which motivates the resistive-wall analysis handled by resistive stability codes.
Design relevance
For the Hyperion breeder, ideal stability analysis maps the pressure and current limits of the negative-triangularity spherical-tokamak equilibrium, confirming in simulation that the intended operating point sits inside the ideal-stable region before construction begins Q2 2027.
- Assumes zero resistivity, frozen-in field
- Solves a delta-W eigenvalue problem
- Negative delta-W means fast instability
- Sets pressure and current operating limits