The effect of the Coriolis force on Kelvin-Helmholtz-driven mixing in protoplanetary disks

We study the stability of proto-planetary disks with vertical velocity gradients in their equilibrium rotation rates; such gradients are expected to develop when dust settles into the midplane. Using a linear stability analysis of a simple three-layer model, we show that the onset of instability occurs at a larger value of the Richardson number, and therefore for a thicker layer, when the effects of Coriolis forces are included. This analysis also shows that even-symmetry (midplane-crossing) modes develop faster than odd-symmetry ones. These conclusions are corroborated by a large number of nonlinear numerical simulations with two different parameterized prescriptions for the initial (continuous) dust distributions. Based on these numerical experiments, the Richardson number required for marginal stability is more than an order of magnitude larger than the traditional 1/4 value. The dominant modes that grow have horizontal wavelengths of several initial dust scale heights, and in nonlinear stages mix solids fairly homogeneously over a comparable vertical range. We conclude that gravitational instability may be more difficult to achieve than previously thought, and that the vertical distribution of matter within the dust layer is likely globally, rather than locally, determined. Subject headings: hydrodynamics — instabilities — turbulence — planets and satellites: formation — planetary systems: protoplanetary disks