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

We investigate the dynamical stability of a differentially rotating disc (or torus) of fluid of uniform entropy and uniform specific angular momentum. Such a fluid is neutrally stable to axisymmetric perturbations. In this paper we consider non-axisymmetric perturbations and undertake a global stability analysis. We present a general study of the normal mode eigenvalue problem and the explicit analytic solution of a pair of particular limiting cases. We derive the fastest growing eigenmodes by numerical integration of the full linearized equations for more general cases. Our overall result is that the tori are unstable to low order non-axisymmetric modes and that the modes grow on a dynamical time-scale. We argue that 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.

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