The dynamical stability of differentially rotating discs with constant specific angular momentum

The dynamical stability of a differentially rotating disc (or torus) of fluid of uniform entropy and uniform specific angular momentum is investigated. Such a fluid is neutrally stable to axisymmetric perturbations. Non-axisymmetric perturbations are considered as part of a global stability analysis. A general study of the normal mode eigenvalue problem and the explicit analytic solution of a pair of particular limiting cases are presented. The fastest growing eigenmodes by numerical integration of the full linearized equations for more general cases are derived. The overall result is that the tori are unstable to low order non-axisymmetric modes and that the modes grow on a dynamical time-scale. Because of the strength of the instability, similar unstable modes must exist in tori of non-uniform entropy or of non-uniform specific angular momentum.