Nonlinear stability, hydrodynamical turbulence, and transport in disks

By a combined analytical and numerical approach, we investigate the three-dimensional hydrodynamical nonlinear stability of both differential rotation and pure shear flows. Since the high Reynolds number instabilities in question are in essence inviscid, a Navier-Stokes code is generally not necessary for their elucidation. Although our numerical code has little difficulty finding nonlinear instability in shear layers and in rotationally supported disks with constant specific angular momentum (the latter appears to be a new result), there is no evidence of any kind that Keplerian disks are nonlinearly unstable. This stability is hardly unique to a Keplerian rotation law, but is a property of any velocity profile with angular velocity decreasing outward and angular momentum increasing outward. A simple analytic analysis (from first principles) suggests that the key to nonlinear stability in rotationally supported disks is the interaction of correlated velocity fluctuations (which determine the nature of turbulent transport) with the background mean flow. In Rayleigh-unstable disks, outward transport extracts both energy and angular momentum from the mean flow 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 linearly unstable; the second is both linearly and nonlinearly stable. In a shear layer, the velocity gradient is a source of free energy, and there are no fluctuation interactions with the mean flow to create a dynamical sink. These flows are nonlinearly unstable, since turbulence, once created, is sustainable. Our formalism is also extremely useful in understanding why nonlinear numerical simulations have been yielding inward transport in convectively unstable disks. When there is overlap, our findings are consistent with well-established results of other investigations, both numerical and laboratory. We discuss the effective Reynolds number of the numerical code. We note, however, that the classical Taylor investigation already suggests that to simulate even linear (Rayleigh) instability, and effective Reynolds number in excess of 10³ is required. We believe that the results of this investigation all but rule out any kind of self-generated hydrodynamical turbulence as a source of anomalous accretion disk transport. In an unmagnetized disk, global nonaxisymmetric instabilities and spiral waves remain, in principle, viable mechanisms.