Disks: just the place for a spot (final report GFD 1992 at Woods Hole)

The equations governing a thin accretion disk orbiting a central object are reduced to a shallow water system. The integrals of motion of this system are used to find conditions necessary for the formal stability of azimuthal basic states. Linear perturbation theory is used to show that one of the conditions of stability (that the velocities be sub-gravity-wave speed) is easily violated, but that associated growth rates of gravity wave instabilities may be small. Numerical experiments are performed to investigate the stability of basic states (axisymmetric equilibria) and basic states with superimposed vortices. It is found that the near-Keplerian shear of a thin disk is hostile to vortices, but also that superposition of a vortex is destablizing to the basic state of the disk. Possibilites for a balanced basic state that incorporates a large vortex are investigated analytically and numerically.