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We present here both analytical and numerical results of hydrodynamicstability investigations of rotationally supported circumstellar flows usingthe shearing box formalism. Asymptotic scaling arguments justifying theshearing box approximation are systematically derived, showing that there existtwo limits which we call small shearing box (SSB) and large shearing box (LSB).The physical meaning of these two limits and their relationship to modelequations implemented by previous investigators are discussed briefly. Twodimensional (2D) dynamics of the SSB are explored and shown to containtransiently growing (TG) linear modes, whose nature is first discussed withinthe context of linear theory. The fully nonlinear regime in 2D is investigatednumerically for very high Reynolds (Re) numbers. Solutions exhibiting sustaineddynamics are found and manifest episodic but recurrent TG behavior and theseare associated with the formation and long-term survival of coherent vortices.The life-time of this spatio-temporal complexity depends on the Re number andthe strength and nature of the initial disturbance. We show results for a casein which the dynamical activity persists for the entire duration of thesimulation (hundreds of box orbits). Some combinations of the Re number and theinitial perturbation spectrum (and strength) show the transiently growingphenomenon to ultimately fade away or, in some instances, to be absentaltogether. Because the SSB approximation used here is equivalent to a 2Dincompressible flow, the dynamics can not depend on the Coriolis force.Therefore, three dimensional (3D) simulations are needed in order to decide ifthis force indeed suppresses nonlinear hydrodynamical instability inrotationally supported disks in the shearing box.Comment: Submitted to A+A on April 1, 2004, Accepted August 3, 200

Hydrodynamic stability of rotationally supported flows: Linear and nonlinear 2D shearing box results