Abstract

The Lagrangian approach can be employed to the dynamics of the linear perturbations of gas-dust mixture in protoplanetary disc with the dust streaming through the gas. It shows that gas-dust perturbations may have negative energy provided that there is a bulk settling of dust to the disc midplane. The most simple variant of the negative energy gas-dust mode is constructed of the dust density perturbations transported by the bulk stream of dust though the unperturbed gas environment. This mode is referred to as the streaming dust wave (SDW). SDW coupled with the positive energy inertial wave (IW) in the gas gives rise to the recently discovered dust settling instability (DSI). Further, SDW by itself exhibits exponential growth as gas-dust dynamics becomes dissipative in the presence of turbulence and provided that the effective turbulent viscosity exceeds the effective turbulent diffusion. This new instability can be referred to as a viscous settling instability (VSI). As compared to DSI , VSI is produced by modes with larger wavelengths, while it vanishes in laminar disc and survives up to the largest value of the effective viscosity in turbulent disc.

Additionally, it can be shown that the three-wave resonance is possible between one negative energy SDW and two positive energy IW’s. A particular case of such a resonance is considered in detail in order to show that it gives rise to the non-linear explosive settling instability (ESI) with a physically reasonable time of explosion. DSI , VSI and ESI may serve as early mechanisms for generation of the dust overdensities in the long-standing problem of planetesimal formation.