  We present here both analytical and numerical results of hydrodynamic
stability investigations of rotationally supported circumstellar flows using
the shearing box formalism. Asymptotic scaling arguments justifying the
shearing box approximation are systematically derived, showing that there exist
two limits which we call small shearing box (SSB) and large shearing box (LSB).
The physical meaning of these two limits and their relationship to model
equations implemented by previous investigators are discussed briefly. Two
dimensional (2D) dynamics of the SSB are explored and shown to contain
transiently growing (TG) linear modes, whose nature is first discussed within
the context of linear theory. The fully nonlinear regime in 2D is investigated
numerically for very high Reynolds (Re) numbers. Solutions exhibiting sustained
dynamics are found and manifest episodic but recurrent TG behavior and these
are associated with the formation and long-term survival of coherent vortices.
The life-time of this spatio-temporal complexity depends on the Re number and
the strength and nature of the initial disturbance. We show results for a case
in which the dynamical activity persists for the entire duration of the
simulation (hundreds of box orbits). Some combinations of the Re number and the
initial perturbation spectrum (and strength) show the transiently growing
phenomenon to ultimately fade away or, in some instances, to be absent
altogether. Because the SSB approximation used here is equivalent to a 2D
incompressible flow, the dynamics can not depend on the Coriolis force.
Therefore, three dimensional (3D) simulations are needed in order to decide if
this force indeed suppresses nonlinear hydrodynamical instability in
rotationally supported disks in the shearing box.
