THE STABILITY OF ACCRETION TORI .5. UNSTABLE MODES AND AVOIDED CROSSINGS IN EXTENDED, SELF-GRAVITATING ANNULI

We continue a study of the dynamical evolution of two-dimensional, differentially rotating, self-gravitating annuli by investigating systems having arbitrarily large radial thickness and orbiting around a central mass. Such extended annuli are idealized models of thick, pressure-supported accretion disks. The initial axisymmetric equilibria are cylindrical with well-defined inner and outer edges, compressible with polytropic index n = 1, and have constant specific anagular momentum. Linear growth rates and eigenfunction characteristics of the different modes are calculated through a solution of the linearized fluid and Poisson equations. Some model evolutions have also been carried into the nonlinear regime using a two-dimensional, Eulerian hydro code that is second-order-accurate in both space and time. As we abandon the quasi-Cartesian, slender-annulus approximation, several new features are revealed by the stability analysis, most of which were unexpected. (1) In contrast to the results obtained for slender annuli, the I mode instability survives in purely self-gravitating extended annuli without a central mass. Furthermore, this mode is not symmetric across the pressure maximum but has corotation only in the vicinity of the outer edge. These results are in agreement with previous three-dimensional calculations of self-gravitating tori. (2) Neutral acoustic modes and neutral gravity modes undergo deflections called ''avoided crossings'' right outside the outer edge. This type of mode interaction has been previously identified in solar/stellar oscillation theory. (3) Such deflected modes enter the fluid where they intersect with acoustic modes and produce sequences of short, weakly unstable branches different in character from ''conventional'' edge modes and rather similar to the I modes. Because of the presence of a strong I mode with m = 1 (which distorts these extended systems), the weakly unstable modes are not expected to be dominant. (4) Besides the I mode, the P and J modes also exist in these systems. As is already known, the P and J modes dominate for low and high degrees of self-gravity, respectively. In addition, J modes appear only in radially slender systems. (5) As soon as the P mode disappears with self-gravity, ''conventional'' edge modes become dominant before the I mode appears. In contrast to the I and J modes of slender systems, these edge modes as well as the P modes do not appear capable of breaking the annulus. Instead, they only stir up the fluid and result in the appearance of distinct inhomogeneities (edge modes) or spiral shocks (P mode). (6) Surprisingly, all annuli are stable against infinitesimal axisymmetric perturbations. Consequently, cylindrical annuli do not obey Toomre's local stability criterion derived for differentially rotating sheets.