  Using a three-dimensional nonlinear hydrodynamic code, we examine the
dynamical stability of more than twenty self-gravitating, compressible,
ellipsoidal fluid configurations that initially have the same velocity
structure as Riemann S-type ellipsoids. Our focus is on ``adjoint''
configurations, in which internal fluid motions dominate over the collective
spin of the ellipsoidal figure; Dedekind-like configurations are among this
group. We find that, although some models are stable and some are moderately
unstable, the majority are violently unstable toward the development of $m=1$,
$m=3$, and higher-order azimuthal distortions that destroy the coherent, $m=2$
bar-like structure of the initial ellipsoidal configuration on a dynamical time
scale.
  The parameter regime over which our models are found to be unstable generally
corresponds with the regime over which incompressible Riemann S-type ellipsoids
have been found to be susceptible to an elliptical strain instability
\citep{LL96}. We therefore suspect that an elliptical instability is
responsible for the destruction of our compressible analogs of Riemann
ellipsoids. The existence of the elliptical instability raises concerns
regarding the final fate of neutron stars that encounter the secular bar-mode
instability and regarding the spectrum of gravitational waves that will be
radiated from such systems.
