Resonant drag instabilities for polydisperse dust

Context. Dust grains embedded in gas flow give rise to a class of hydrodynamic instabilities called resonant drag instabilities, some of which are thought to be important during the process of planet formation. These instabilities have predominantly been studied for single grain sizes, in which case they are found to grow fast. Non-linear simulations indicate that strong dust overdensities can form, aiding the formation of planetesimals. In reality, however, there is going to be a distribution of dust sizes, which may have significant consequences. Aims. We aim to study two different resonant drag instabilities – the streaming instability and the settling instability – taking into account a continuous spectrum of grain sizes, to determine whether these instabilities survive in the polydisperse regime and how the resulting growth rates compare to the monodisperse case. Methods. We solved the linear equations for a polydisperse fluid in an unstratified shearing box to recover the streaming instability and, for approximate stratification, the settling instability, in all cases focusing on low dust-to-gas ratios. Results. Size distributions of realistic widths turn the singular perturbation of the monodisperse limit into a regular perturbation due to the fact that the back-reaction on the gas involves an integration over the resonance. The contribution of the resonance to the integral can be negative, as in the case of the streaming instability, which as a result does not survive in the polydisperse regime; or positive, which is the case in the settling instability. The latter therefore has a polydisperse counterpart, with growth rates that can be comparable to the monodisperse case. Conclusions. Wide size distributions in almost all cases remove the resonant nature of drag instabilities. This can lead to reduced growth, as is the case in large parts of parameter space for the settling instability, or complete stabilisation, as is the case for the streaming instability.