Introduction

Since their discovery by Papaloizou & Pringle (1984) non-axisymmetric instabilities in accretion tori have been discussed by many authors. It has been found that the instabilities are driven by shear and operate – depending on flow and perturbation parameters – through sound waves, surface waves or Kelvin-Helmholtz type modes. The spectrum of internal gravity waves which is associated with finite entropy gradients has not yet been studied and will be described in a forthcoming paper (Glatzel 1989). A brief summary of the main results is given here.

Basic assumptions

In order to allow for an analytical treatment we adopt cylindrical geometry and consider the limit of thin shells which rotate differentially in their own or an external gravitational field. The entropy distribution is required to guarantee a parabolic density stratification. Maximum density occurs when the effective gravity vanishes – its zeros determine the boundaries of the configuration. We assume incompressibility and neglect the self-gravity of the perturbations. Using an additional technical approximation, which has qualitatively no consequence for the modal structure, the perturbation equation is reduced to Whittaker's equation and the dispersion relation can be written in terms of confluent hypergeometric functions.

The modal structure

In a medium at rest a two-fold infinite set of gravity modes is found moving parallel to the boundaries in opposite directions. Modes occur in pairs corresponding to a symmetric and an anti-symmetric eigenfunction, where the symmetric mode owes its existence to the non-monotonic density stratification.