Edge localized linear ideal magnetohydrodynamic instability studies in an extended-magnetohydrodynamic code

A linear benchmark between the linear ideal MHD stability codes ELITE [H. R. Wilson et al, Phys. Plasmas 9, 1277 (2002)], GATO [L. Bernard et al, Comput. Phys. Commun. 24, 377 (1981)], and the extended nonlinear magnetohydrodynamic (MHD) code, NIMROD [C. R. Sovinec et al., J. Comput. Phys. 195, 355 (2004)] is undertaken for edge-localized (MHD) instabilities. Two ballooning-unstable, shifted-circle tokamak equilibria are compared where the stability characteristics are varied by changing the equilibrium plasma profiles. The equilibria model an H-mode plasma with a pedestal pressure profile and parallel edge currents. For both equilibria, NIMROD accurately reproduces the transition to instability (the marginally unstable mode), as well as the ideal growth spectrum for a large range of toroidal modes $(n=1–20)$⁠. The results use the compressible MHD model and depend on a precise representation of “ideal-like” and “vacuumlike” or “halo” regions within the code. The halo region is modeled by the introduction of a Lundquist-value profile that transitions from a large to a small value at a flux surface location outside of the pedestal region. To model an ideal-like MHD response in the core and a vacuumlike response outside the transition, separate criteria on the plasma and halo Lundquist values are required. For the benchmarked equilibria the critical Lundquist values are $108$ and $103$ for the ideal-like and halo regions, respectively. Notably, this gives a ratio on the order of $105$⁠, which is much larger than experimentally measured values using $Te$ values associated with the top of the pedestal and separatrix. Excellent agreement with ELITE and GATO calculations are made when sharp boundary transitions in the resistivity are used and a small amount of physical dissipation is added for conditions very near and below marginal ideal stability.