  In this note we address the issue of hydrodynamical instabilities in
Astrophysical rotating shear flows in the light of recent publications focused
on the possibility for differential rotation to trigger and sustain turbulence
in the absence of a magnetic field. We wish to present in a synthetic form the
major arguments in favor of this thesis along with a simple schematic scenario
of the transition to and self-sustenance of such turbulence. We also propose
that the turbulent diffusion length scale scales as the local Rossby number of
the mean flow. A new prescription for the turbulent viscosity is introduced.
This viscosity reduces to the so-called beta-prescription in the case of
velocity profiles with a constant Rossby number, which includes Keplerian
rotating flows.
