Vertical Shearing Instabilities in Radially Shearing Disks: the Dustiest Layers of the Protoplanetary Nebula

Gravitational instability of a vertically thin, dusty sheet near the midplane of a protoplanetary disk has long been proposed as a way of forming planetesimals. Before Roche densities can be achieved, however, the dust-rich layer, sandwiched from above and below by more slowly rotating dust-poor gas, threatens to overturn and mix by the Kelvin-Helmholtz instability (KHI). Whether such a threat is real has never been demonstrated: the Richardson criterion for the KHI is derived for 2-D Cartesian shear flow and does not account for rotational forces. Here we present 3-D numerical simulations of gas-dust mixtures in a shearing box, accounting for the full suite of disk-related forces: the Coriolis and centrifugal forces, and radial tidal gravity. Dust particles are assumed small enough to be perfectly entrained in gas; the two fluids share the same velocity field but obey separate continuity equations. We find that the Richardson number Ri does not alone determine stability. The critical value of Ri below which the dust layer overturns and mixes depends on the height-integrated metallicity Σd/Σg (surface density ratio of dust to gas). Nevertheless, for Σd/Σg between one and five times solar, the critical Ri is nearly constant at ∼0.1. Keplerian radial shear stabilizes those modes that would otherwise disrupt the layer at large Ri. If the height-integrated metallicity is at least ∼5 times greater than the solar value of 0.01, then midplane dust densities can approach Roche densities. Such an environment might be expected to produce gas giant planets having similarly super-solar metallicities. Subject headings: hydrodynamics — instabilities — planets and satellites: formation — planetary systems: protoplanetary disks — turbulence