The Weak Field Limit of the Magnetorotational Instability
  We investigate the behavior of the magneto-rotational instability in the limit of extremely weak magnetic field, i.e., as the ratio of ion cyclotron frequency to orbital frequency (X) becomes small. Considered only in terms of cold two-fluid theory, instability persists to arbitrarily small values of X, and the maximum growth rate is of order the orbital frequency except for the range m_e/m_i < |X| < 1, where it can be rather smaller. In this range, field aligned with rotation (X > 0) produces slower growth than anti-aligned field (X < 0). The maximum growth rate is generally achieved at smaller and smaller wavelengths as |X| diminishes. When |X| < m_e/m_i, new unstable "electromagnetic-rotational" modes appear that do not depend on the equilibrium magnetic field. Because the most rapidly-growing modes have extremely short wavelengths when |X| is small, they are often subject to viscous or resistive damping, which can result in suppressing all but the longest wavelengths, for which growth is much slower. We find that this sort of damping is likely to curtail severely the frequently-invoked mechanism for cosmological magnetic field growth in which a magnetic field seeded by the Biermann battery is then amplified by the magneto-rotational instability. On the other hand, the small |X| case may introduce interesting effects in weakly-ionized disks in which dust grains carry most of the electric charge. 
Julian H. Krolik, Ellen G. Zweibel