Explosive instability of dust settling in a protoplanetary disc

It is shown that gas-dust perturbations in a disc with dust settling to the disc midplane exhibit the non-linear three-wave resonant interactions between streaming dust wave (SDW) and two inertial waves (IW). In the particular case considered in this paper, SDW at the wavenumber $k^\bullet = 2\kappa / (g_z t_s)$, where $\kappa$, $g_z$ and $t_s$ are, respectively, epicyclic frequency, vertical gravitational acceleration and particle's stopping time, interacts with two IW at the lower wavenumbers $k^\prime$ and $k^{\prime\prime}$ such that $k^\prime < k_{\rm DSI} < k^{\prime\prime} < k^\bullet$, where $k_{\rm DSI} = \kappa / (g_z t_s)$ is the wavenumber of the linear resonance between SDW and IW associated with the previously discovered linear dust settling instability. The problem is solved analytically in the limit of the small dust fraction. As soon as the dynamical dust back reaction on gas is taken into account, ${\bf k}^\bullet$, ${\bf k}^\prime$ and ${\bf k}^{\prime\prime}$ become slightly non-collinear and the emerging interaction of waves leads to simultaneous explosive growth of their amplitudes. This growth is explained by the conservative exchange with energy between the waves. The amplitudes of all three waves grow because the negative energy SDW transfers its energy to the positive energy IW. The product of the dimensionless amplitude of initially dominant wave and the time of explosion can be less than Keplerian time in a disc. It is shown that, generally, the three-wave resonance of an explosive type exists in a wide range of wavenumbers $0 < k^\bullet \leq 2\kappa / (g_z t_s)$. An explosive instability of gas-dust mixture may facilitate the dust clumping and the subsequent formation of planetesimals in young protoplanetary discs.