Numerical simulations of the gravitational instability in the dust layer of a protoplanetary disk using a thin disk model

The growth of the gravitational instability in the dust layer of a protoplanetary disk is investigated. In order to see the effects of only the gravitational instability, we assume a laminar disk which has no radial pressure gradient as an unperturbed state so that the shear and the streaming instabilities do not grow. We neglect the relative velocity between the dust and gas parallel to the disk plane assuming that the dust and gas couple firmly by the mutual friction. However, we take account of the dust settling by using an analytic solution of dust density growth. We construct a two-dimensional thin disk model in which the radial and azimuthal directions in the midplane are taken as independent variables. In order to keep a certain amount of a disturbance, which is considered to exist not only at the beginning but all through the time evolution, we give perturbations repeatedly per Keplerian shear time in a local frame of reference. We find that the gravitational instability grows for the dust particle when the dimensionless gas friction time (the product of the gas friction time and the Keplerian angular velocity) is equal to 0.01. On the other hand, the gravitational instability does not grow sufficiently before the dust layer becomes infinitesimally thin if the dimensionless gas friction time is equal to 0.1. These results are consistent with the axisymmetric study by Yamoto and Sekiya. However, the gravitational instability grows nonaxisymmetrically, and trailing surface density patterns arise.