Shear instabilities in the dust layer of the solar nebula I. The linear analysis of a non-gravitating one-fluid model without the Coriolis and the solar tidal forces

As dust aggregates settled toward the midplane of the solar nebula, a thin dust layer was formed. The rotational velocity was a function of the distance from the midplane in this layer, and the shear induced turbulence might occur, which prevented the dust aggregates from settling further toward the midplane. Thus, it was difficult for the dust density on the midplane to exceed the critical density of the gravitational stability. In this paper, the linear analysis of the shear instability is made under the following assumptions: The self-gravity, the solar tidal force (thus the Keplerian shear), and the Coriolis force are neglected; the unperturbed state has a constant Richardson’s number in the dust layer; further we restrict ourselves to the case where dust aggregates are small enough, and a mixture of dust and gas is treated as one fluid. Numerical results show that the growth rate of the most unstable mode is much less than the Keplerian angular frequency, as long as the Richardson number is larger than 0.1.