  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 Z (surface density ratio of
dust to gas). Nevertheless, for Z 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 Z 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.
