Abstract
We investigate the dynamical stability of differentially rotating, self-gravitating masses. We obtain two types of shear instability. The first is caused by the ‘principal’ mode which is characterized by a peak growth rate and large azimuthal wavenumbers. This mode disappears for large shear or large flattening. We show that this instability can be described in terms of two parameters. The m = 2 principal mode achieves a non-linear evolution from an initially axisymmetric state towards a uniformly rotating Jacobi ellipsoid. The second type of instability is related to the ‘secondary’ modes. They form a discrete spectrum of corotating modes. They have a smaller growth rate than the principal mode for low values of m. Larger growth rates are obtained for large m. In the case of a flattened disc, we show that these modes can be identified with the unstable acoustic waves which are known to occur in compressible shear flows.
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