Physics of spectral and geometric singularities on the WKB thresholds of Standard and Helical MRI

The magnetorotational instability (MRI) triggers turbulence and enables outward transport of angular momentum in hydrodynamically stable rotating shear flows, e.g., in accretion disks. What laws of differential rotation are susceptible to the destabilization by axial, azimuthal, or helical magnetic field? How the standard, helical and azimuthal versions of MRI are related to each other? The answer to these questions, which is vital for astrophysical and experimental applications, inevitably leads to the study of spectral and geometrical singularities on the instability threshold. These singularities provide a connection between seemingly discontinuous stability criteria and thus explain several paradoxes in the theory of MRI that were poorly understood since the 1950s. On the other hand, the singular geometry of the instability threshold is a powerful tool for parametric optimization that predicts, e.g., how close to the Kepler or solid body rotation profiles the instability threshold can be moved by varying the magnetic field configuration, velocity, and material properties of the liquid. Using the local WKB approximation we study the thresholds of standard and helical MRI for axi- and non-axisymmetric perturbations, their extrema and the links between them in the fully viscous and resistive setting. We discuss the connection between standard and helical MRI via a spectral exceptional point as well as the behavior of the helical MRI threshold both in the inductionless approximation when the magnetic Prandtl number (Pm) tends to zero and in case when it is small but finite. We demonstrate that the Liu limit for the Rossby number at the onset of HMRI slightly increases when Pm is not vanishing and find its ultimate value.