Paradox of Velikhov-Chandrasekhar and the ultimate limit for the onset of helical magnetorotational instability

The magnetorotational instability (MRI) triggers turbulence and enables outward transport of angular momentum in hydrodynamically stable accretion discs. By using theWKB 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 new strict limit of the critical Rossby number. This new limit of Roc = −0.802, which appears for a finite Lundquist number of Lu= 0.618, extends the formerly known inductionless Liu limit of Roc = −0.828 valid at Lu = 0.