Evidence for an elliptical instability in rotating fluid bars and ellipsoidal stars

Riemann S-type ellipsoids are non-axisymmetric equilibrium configurations with non-trival internal motion for homogeneous/incompressible self-gravitating systems. Their linear stability has been studied by Chandrasekhar (1969). Lebovitz \& Lifschitz (1996) found that these S-type ellipsoids are subject to an elliptical strain instability that are related to the noncircular streamlines within the ellipsoid.
Recent work of Ou (2006) has made it possible to build three-dimensional exact solutions for incompressible Riemann S-type ellipsoid and their compressible counter parts that are in quasi-equilibrium. Since these compressible models share exactly the same elliptical flow field as Riemann ellipsoids, we are allowed to use them to study the dynamical elliptical strain instability discoverd by Lebovitz et. al. (1996).
Here we present movies from 3D hydrodynamical evolutions of selected models.
Results for direct and adjoint configurations
model b/a c/a \omega \lambda T/|W| stability plots movies comments
D105 0.59 0.49 0.911 0.114 0.105 stable modes,
top view,
This model is stable in the sense that odd modes do not grow, the ellipsoidal figure nicely holds together throughout the simulation. The gradual decay of the m=2 mode is due to the small violation of continuity equation.
A010 0.74 0.82 -0.610 -0.875 0.0098 stable modes, top view ,
This model is stable in the sense that no indication of growth of odd modes. The amplitudes of all the modes remain to be roughly constant.
A134 0.74 0.49 -0.008 -0.911 0.134 marginally stable modes, top view, top view II,
This model is moderately unstable. The ellipsoidal configuration holds together for quite a long time, but from the mode analysis, we found that odd modes grow slowly.
A100 0.59 0.49 -0.114 -0.911 0.100 unstable modes, top view ,
This model is unstable to the elliptical strain instability. The bar-like structure is suddenly destroyed between t=4-7 dynamical times. Odd modes grow rapidly from the beginning and surpass m=2 mode quickly.
Note: models starting with "D" belong to direct configurations, whereas those starting with "A" belong to adjoint configurations.

References:
Chandrasekhar, S. 1969, Ellipsoidal Figures of Equilibrium, New Haven, CT: Yale Univ. Press
Lebovitz, N. R., \& Lifschitz, A. 1996, ApJ, 458, 699
Ou, S., 2006, ApJ, 639, 549