NONLINEAR STABILITY, HYDRODYNAMICAL TURBULENCE, AND TRANSPORT IN DISKS

By a combined analytical and numerical approach, we investigate the th ree-dimensional hydrodynamical nonlinear stability of both differentia l rotation and pure shear flows. Since the high Reynolds number instab ilities in question are in essence inviscid, a Navier-Stokes code is g enerally not necessary for their elucidation. Although our numerical c ode has little difficulty finding nonlinear instability in shear layer s and in rotationally supported disks with constant specific angular m omentum (the latter appears to be a new result), there is no evidence of any kind that Keplerian disks are nonlinearly unstable. This stabil ity is hardly unique to a Keplerian rotation law, but is a property of any velocity profile with angular velocity decreasing outward and ang ular momentum increasing outward. A simple analytic analysis (from fir st principles) suggests that the key to nonlinear stability in rotatio nally supported disks is the interaction of correlated velocity fluctu ations (which determine the nature of turbulent transport) with the ba ckground mean flow. In Rayleigh-unstable disks, outward transport extr acts both energy and angular momentum from the mean how gradients and imparts them to the velocity fluctuations; in a Keplerian disk outward transport interacts with the mean angular momentum gradient to act as a sink for azimuthal velocity fluctuations. The first case is linearl y unstable; the second is both linearly and nonlinearly stable. In a s hear layer, the velocity gradient is a source of free energy, and ther e are no fluctuation interactions with the mean flow to create a dynam ical sink. These flows are nonlinearly unstable, since turbulence, onc e created, is sustainable. Our formalism is also extremely useful in u nderstanding why nonlinear numerical simulations have been yielding in ward transport in convectively unstable disks. When there is overlap, our findings are consistent with well-established results of other inv estigations, both numerical and laboratory. We discuss the effective R eynolds number of the numerical code. We note, however, that the class ical Taylor investigation already suggests that to simulate even linea r (Rayleigh) instability, and effective Reynolds number in excess of 1 0(3) is required. We believe that the results of this investigation al l but rule out any kind of self-generated hydrodynamical turbulence as a source of anomalous accretion disk transport. In an unmagnetized di sk, global nonaxisymmetric instabilities and spiral waves remain, in p rinciple, viable mechanisms.