In this paper, we investigate, by three-dimensional hydrodynamical simulations, the role that vertical convective motions play in providing angular momentum transport in a Keplerian disk. We begin by deriving simple and general analytic constraints upon the correlated radial and azimuthal velocity fluctuation tensor, critical to the direction of energy and angular momentum transport. When azimuthal pressure gradients are small, as is often the case for incompressible turbulence in a shearing disk, the constraints are particularly straightforward and striking: they imply there can be no net outward transport in a steady flow. (More precisely, any steady transport that is present must be due to the explicit forcing by azimuthal pressure gradients.) Furthermore, numerical simulations show inward transport even in disks characterized by solid body rotation, which are quite far from axisymmetric. If the kinetic energy of rotational velocity fluctuations increases with time because of coupling with the mean flow (as in an instability), the relationship suggests (and our numerical simulations confirm) that in a Keplerian disk the Reynolds stress is negative, i.e., that the angular momentum and energy transport is inward. The analogous relationship for shearing but nonrotating (Cartesian) flow displays the opposite sign for the fluctuation tensor, i.e., an increase in the "streamwise" velocity fluctuations is associated with outward (from higher momentum to smaller momentum) transport. Although Cartesian shear flows are known to be extremely sensitive to disruption by nonlinear secondary instabilities, hydrodynamical calculations presented here demonstrate that Keplerian disk flows show no such inclination. We suggest that the key to understanding this critical difference is the very different nature of the interaction between the mean flow and the transport in each system. We provide a physical interpretation of our findings in terms of the role epicyclic oscillations play in mediating angular momentum transport. We base our convection simulations upon a reproducible analytic expression for the vertical profile of an unstable equilibrium state in a stratified disk. The nonlinear evolution of the convective cells is then followed after the initial profile is perturbed. Convection can be sustained only if an ad hoc source of heating is added to the disk midplane. The net transport associated with steady convection is small and on average inward. A comparison between the volume-averaged Reynolds stress and the time rate of change of the azimuthal kinetic energy associated with fluctuations in the rotational velocity shows remarkable agreement with our simple analytic predictions. Taken as a whole, these results offer little hope that convection-or any other form of incompressible hydrodynamic turbulence-is likely to be a significant source of angular momentum transport in nonmagnetic disks. Coherent pressure forcing by, e.g., spiral density waves, remains a viable option.
All Science Journal Classification (ASJC) codes
- Astronomy and Astrophysics
- Space and Planetary Science
- Accretion, accretion disks