On the behaviour of the magnetorotational instability when the Rayleigh criterion is violated


  • Steven A. Balbus

    Corresponding author
    1. Laboratoire de Radioastronomie, École Normale Supérieure, 24 rue Lhomond, 75231 Paris Cedex 05, France
    2. Institut Universitaire de France, Maison des Universités, 103 blvd. Saint-Michel, 75005 Paris, France
    3. Princeton University Observatory, Peyton Hall, Princeton University, Princeton, NJ 08540, USA
    Search for more papers by this author

E-mail: steven.balbus@lra.ens.fr


An elementary but revealing analysis of the dispersion relation of the magnetorotational instability (MRI) in the Ralyeigh-unstable regime is described. The defining properties of the MRI – its maximum growth rate and the direction of the associated eigenvector displacement – remain unchanged as the Rayleigh discriminant passes from positive to negative values. At sufficiently negative discriminant values, however, the spectrum of unstable modes becomes dominated by zero wavenumber disturbances, and the problem loses its local character. These results may be relevant to understanding the level of turbulent fluid stress near the innermost stable circular orbit (ISCO) in an accretion disc around a black hole, since the Rayleigh discriminant changes sign at this location. Our conclusions are consistent with numerical simulations that find finite stress at the ISCO, some dependence of the stress magnitude with scale height, and a plunging region much closer to the event horizon than the ISCO radius.