Linear and nonlinear dynamic instability of rotating polytropes

A three-dimensional hydrodynamic computer program is used to study the growth of nonaxisymmetric structures in rapidly rotating, self-gravitating polytropes. Models with polytropic index n = 0.8, 1.0, 1.3, 1.5, and 1.8 are studied. The initially axisymmetric equilibria are constructed by the Ostriker-Mark self-consistent-field method. The nonaxisymmetric pattern that develops out of low-amplitude random noise is a two-armed spiral with a well-defined pattern speed and growth rate which closely match properties of the toroidal mode predicted from the linear, second-order tensor-virial equation. A Fourier analysis of each polytrope's azimuthal density distribution shows that, even in the linear amplitude regime, higher-order angular patterns also develop exponentially in time. The higher-order patterns ultimately move in synchronization with the broad two-armed spiral, creating a narrow two-armed spiral. As the polytropic index is decreased, a more open and centrally more barlike pattern develops.