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

We present here both analytical and numerical results of hydrodynamic stability investigations of rota-tionally 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 long-term dynamical activity are found and manifest episodic but recurrent TG behavior and these are associated with the formation and long-term survival of coherent vortices. The lifetime of this spatio-temporal complexity depends on the Re number and the strength and nature of the initial disturbance. The dynamical activity in finite Re solutions ultimately decays with a characteristic time increasing with Re. However, for large enough Re and appropriate initial perturbation, a large number of TG episodes recur before any viscous decay begins to clearly manifest itself. In cases where Re = ∞ nominally (i.e. any dissipation resulting only from numerical truncation errors), the dynamical activity persists for the entire duration of the simulation (hundreds of box orbits). 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 approximation, and if recurrent TG behavior can still persist in three dimensions as well-possibly giving rise to a subcritical transition to long-term spatio-temporal complexity.