  The non-linear hydrodynamic stability of thin, compressible, Keplerian disks
is studied on the large two-dimensional compressible scale, using a high-order
accuracy spectral method. We show that purely hydrodynamic perturbations, while
developing initially into sheared disturbances or coherent vortices, do not
evolve into a self-sustained turbulence. This is in complete agreement with the
conclusions of Balbus and Hawley (1998). We demonstrate that, temporarily,
energy is being transferred to the largest scale mode (before dissipating). In
the case of an inner reflecting boundary condition it is found that the
innermost disk is globally unstable to non-axisymmetric modes which can evolve
into turbulence. However, this turbulence is not expected to play a major role
in angular momentum transport in the disk.
