Alexander Haywood, Fatima Ebrahimi

The linear stability of global non-axisymmetric modes in differentially rotating, magnetized, non-ideal plasma is crucial for understanding turbulence and transport phenomena. We investigate the competition between the local Magneto-Rotational Instability (MRI) and the Magneto-Curvature Instability (MCI)--a distinct non-axisymmetric low-frequency curvature-driven global branch--by developing and applying a non-ideal global spectral method, validated against NIMROD code simulations, and an extended effective potential formalism. Our analysis reveals that the global, low-frequency MCI persists at low magnetic Reynolds numbers (Rm), whereas the localized, high-frequency MRI is stabilized by diffusive broadening of its structure around its Alfv\'enic resonances. Consequently, we identify the global MCI as the primary onset mechanism for magnetohydrodynamic instability in systems with finite curvature, e.g., astrophysical rotators. We establish distinct parameter regimes for mode dominance: MCI prevails in geometrically moderate-thickness disks with high curvature and intermediate radial gaps, while MRI dominates in thin, low-curvature disks with large radial gaps. Mode competition is also highly sensitive to the flow profile, particularly vorticity and its gradient, with non-uniform shear profiles exhibiting more robust instability due to flow-curvature and shear contributions. A key outcome is the development of "spectral diagrams" derived from the global spectral method. These diagrams comprehensively map dominant instabilities and their characteristics, offering a predictive tool for critical onset parameters (i.e., flow curvature, magnetic field, and Rm) and facilitating the interpretation of experimental and simulation results. Notably, these diagrams demonstrate that the global MCI is generally the sole unstable mode at the initial onset of instability.