Two evolutional paths of an axisymmetric gravitational instability in   the dust layer of a protoplanetary disk
  Nonlinear numerical simulations are performed to investigate the density evolution in the dust layer of a protoplanetary disk due to the gravitational instability and dust settling toward the midplane. We assume the region where the radial pressure gradient at equilibrium is negligible so that the shear-induced instability is avoided, and also restrict to an axisymmetric perturbation as a first step of nonlinear numerical simulations of the gravitational instability. We find that there are two different evolutional paths of the gravitational instability depending on the nondimensional gas friction time, which is defined as the product of the gas friction time and the Keplerian angular velocity. If the nondimensional gas friction time is equal to 0.01, the gravitational instability grows faster than dust settling. On the other hand, if the nondimensional gas friction time is equal to 0.1, dust aggregates settle sufficiently before the gravitational instability grows. In the latter case, an approximate analytical calculation reveals that dust settling is faster than the growth of the gravitational instability regardless of the dust density at the midplane. Thus, the dust layer becomes extremely thin and may reach a few tenth of the material density of the dust before the gravitational instability grows. 
Fumiharu Yamoto and Minoru Sekiya