Magnetorotational instability (MRI) triggers turbulence and enables outward transport of angular momentum in hydrodynamically stable accretion discs. By using the WKB approximation and methods of singular function theory, we resolve two different paradoxes of MRI that appear in the limits of infinite and vanishing magnetic Prandtl number. For the latter case, we derive a strict limit of the critical Rossby number. This limit of ${\mathrm{Ro}}_{\mathrm{c}}=-0.802$, which appears for a finite Lundquist number of $\mathit{S}=0.618$, extends the formerly known inductionless Liu limit of ${\mathrm{Ro}}_{\mathrm{c}}=-0.828$ valid at $\mathit{S}=0$.

• Received 4 April 2011

DOI:https://doi.org/10.1103/PhysRevE.84.036304