On the non-linear hydrodynamic stability of thin Keplerian disks
  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. 
Patrick Godon and Mario Livio