M3D-$C^1$ simulations of plasma response in ELM-mitigated ASDEX Upgrade and DIII-D discharges

The extended magnetohydrodynamics (MHD) code M3D-$C^{1}$ is used to study the time-independent, linear response of tokamak equilibria to applied, 3D magnetic perturbations. In doing so, we seek to develop a more complete understanding of what MHD phenomena are responsible for the mitigation and suppression of edge-localized modes (ELMs) and to explain why the success of ELM suppression experiments differs both within a single tokamak and across different tokamaks. We consider such experiments on ASDEX Upgrade and DIII-D. We examine how resonant and non-resonant plasma responses are affected by varying the relative magnitude and phase of sets of magnetic coils. The importance of two-fluid effects, rotation profiles, plasma $\beta$, collisionality, bootstrap current profiles, and various numerical parameters are explored. The results are verified against other MHD codes (e.g., IPEC, MARS), correlated to observations of ELM mitigation or suppression, and validated against observed magnetic responses.