# On the nature of the resonant drag instability of dust streaming in protoplanetary disc

## TL;DR

This paper investigates the resonant drag instability in protoplanetary discs, revealing how mode coupling between inertial waves and dust waves leads to dust density perturbations and growth rates dependent on dust mass fraction.

## Contribution

It provides a detailed theoretical analysis of the mode coupling mechanism underlying RDI, including growth rate dependencies and the conditions for resonance and instability.

## Key findings

- RDI growth rate scales as f^{1/2} for mode coupling.
- High wavenumber RDI growth rate scales as f^{1/3} due to triple mode coupling.
- Mode coupling occurs only with combined radial and vertical dust streaming.

## Abstract

The recently discovered resonant drag instability (RDI) of dust streaming in protoplanetary disc is considered as the mode coupling of subsonic gas-dust mixture perturbations. This mode coupling is coalescence of two modes with nearly equal phase velocities: inertial wave (IW) having positive energy and a streaming dust wave (SDW) having negative energy as measured in the frame of gas environment being at rest in vertical hydrostatic equilibrium. SDW is a trivial mode produced by the bulk streaming of dust, which transports perturbations of dust density. In this way, settling combined with radial drift of the dust makes possible coupling of SDW with IW and the onset of the instability. In accordance with the concept of the mode coupling, RDI growth rate is proportional to the square root of the coupling term of the dispersion equation, which itself is proportional to mass fraction of dust, $f\ll 1$. This clarifies why RDI growth rate $\propto f^{1/2}$. When SDW has positive energy, its resonance with IW provides an avoided crossing instead of the mode coupling. In the high wavenumber limit RDI with unbounded growth rate $\propto f^{1/3}$ is explained by the triple mode coupling, which is coupling of SDW with two IW. It coexists with a new quasi-resonant instability accompanied by bonding of two oppositely propagating low-frequency IW. The mode coupling does not exist for dust streaming only radially in a disc. In this case RDI is provided by the obscured mechanism associated with the inertia of solids.