The Weak‐Field Limit of the Magnetorotational Instability

We investigate the behavior of the magneto-rotational instability in thelimit of extremely weak magnetic field, i.e., as the ratio of ion cyclotronfrequency to orbital frequency (X) becomes small. Considered only in terms ofcold two-fluid theory, instability persists to arbitrarily small values of X,and the maximum growth rate is of order the orbital frequency except for therange m_e/m_i < |X| < 1, where it can be rather smaller. In this range, fieldaligned with rotation (X > 0) produces slower growth than anti-aligned field (X< 0). The maximum growth rate is generally achieved at smaller and smallerwavelengths as |X| diminishes. When |X| < m_e/m_i, new unstable"electromagnetic-rotational" modes appear that do not depend on the equilibriummagnetic field. Because the most rapidly-growing modes have extremely shortwavelengths when |X| is small, they are often subject to viscous or resistivedamping, which can result in suppressing all but the longest wavelengths, forwhich growth is much slower. We find that this sort of damping is likely tocurtail severely the frequently-invoked mechanism for cosmological magneticfield growth in which a magnetic field seeded by the Biermann battery is thenamplified by the magneto-rotational instability. On the other hand, the small|X| case may introduce interesting effects in weakly-ionized disks in whichdust grains carry most of the electric charge.Comment: 30 pages, including 4 figures; revised version resubmitted to Ap