Non-axisymmetric instability in thin discs

In a recent important paper, Papaloizou & Pringle (1984) have investigated the dynamical stability of a thick, differentially rotating disc of uniform entropy and uniform specific angular momentum. They find a strong global non-axisymmetric instability which grows on a dynamical time-scale. Here we show that a similar instability exists even in thin discs of arbitrary specific angular momentum. We find that a crucial ingredient for the existence of this instability is a good reflecting boundary at either the inner or outer edge of the disc (or both). We make simple estimates of the growth rate of the instability as a function of the azimuthal wavenumber of the mode, the angular momentum profile, the radial width and thickness of the disc, and the reflectivity of the boundaries. The importance of reflecting boundary conditions to the stability of thick discs/tori is however not completely clear because of the possibility that a KelvinHelmholtz-like instability, independent of boundary conditions, may be common (Papaloizou & Pringle, preprint).