We introduce a polydisperse version of the streaming instability, where the dust component is treated as a continuum of sizes. We show that its behaviour is remarkably different from the monodisperse streaming instability. We focus on tightly coupled particles in the terminal velocity approximation and show that unstable modes that grow exponentially on a dynamical time scale exist. However, for dust to gas ratios much smaller than unity they are confined to radial wave numbers that are a factor $\sim 1/{\overline{\rm St}}$ larger than where the monodisperse streaming instability growth rates peak. Here ${\overline{\rm St}}\ll 1$ is a suitable average Stokes number for the dust size distribution. For dust to gas ratios larger than unity, polydisperse modes that grow on a dynamical time scale are found as well, similar as for the monodisperse streaming instability and at similarly large wave numbers. At smaller wave numbers, where the classical monodisperse streaming instability shows secular growth, no growing polydisperse modes are found under the terminal velocity approximation. Outside the region of validity for the terminal velocity approximation, we have found unstable epicyclic modes that grow on ∼104 dynamical time scales.

Planet formation via core accretion requires the production of km-sized planetesimals from cosmic dust. This process must overcome barriers to simple collisional growth, for which the Streaming Instability (SI) is often invoked. Dust evolution is still required to create particles large enough to undergo vigorous instability. The SI has been studied primarily with single size dust, and the role of the full evolved dust distribution is largely unexplored. We survey the Polydispserse Streaming Instability (PSI) with physical parameters corresponding to plausible conditions in protoplanetary discs. We consider a full range of particle stopping times, generalized dust size distributions, and the effect of turbulence. We find that, while the PSI grows in many cases more slowly with a interstellar power-law dust distribution than with a single size, reasonable collisional dust evolution, producing an enhancement of the largest dust sizes, produces instability behaviour similar to the monodisperse case. Considering turbulent diffusion the trend is similar. We conclude that if fast linear growth of PSI is required for planet formation, then dust evolution producing a distribution with peak stopping times on the order of 0.1 orbits and an enhancement of the largest dust significantly above the single power-law distribution produced by a fragmentation cascade is sufficient, along with local enhancement of the dust to gas volume mass density ratio to order unity.

Occurring in protoplanetary discs composed of dust and gas, streaming instabilities are a favoured mechanism to drive the formation of planetesimals. The Polydispserse Streaming Instability is a generalisation of the Streaming Instability to a continuum of dust sizes. This second paper in the series provides a more in-depth derivation of the governing equations and presents novel numerical methods for solving the associated linear stability problem. In addition to the direct discretisation of the eigenproblem at second order introduced in the previous paper, a new technique based on numerically reducing the system of integral equations to a complex polynomial combined with root finding is found to yield accurate results at much lower computational cost. A related method for counting roots of the dispersion relation inside a contour without locating those roots is also demonstrated. Applications of these methods show they can reproduce and exceed the accuracy of previous results in the literature, and new benchmark results are provided. Implementations of the methods described are made available in an accompanying Python package psitools.

Motivated by the stability of dust laden vortices, in this paper we study the terminal velocity approximation equations for a gas coupled to a pressureless dust fluid and present a numerical solver for the equations embedded in the FARGO3D hydrodynamics code. We show that for protoplanetary discs it is possible to use the baricenter velocity in the viscous stress tensor, making it trivial to simulate viscous dusty protoplanetary discs with this model. We also show that the terminal velocity model breaks down around shocks, becoming incompatible with the two fluid model it is derived from. Finally we produce a set of test cases for numerical schemes and demonstrate the performance of our code on these tests. Our implementation embedded in FARGO3D using an unconditionally stable explicit integrator is fast, and exhibits the desired second order spatial convergence for smooth problems.

Context. Dust grain dynamics in molecular clouds is regulated by its interplay with supersonic turbulent gas motions. The conditions under which interstellar dust grains decouple from the dynamics of gas in molecular clouds remain poorly constrained. Aims. We first aim to investigate the critical dust grain size for dynamical decoupling, using both analytical predictions and numerical experiments. Second, we aim to set the range of validity of two fundamentally different numerical implementations for the evolution of dust and gas mixtures in turbulent molecular clouds. Methods. We carried out a suite of numerical experiments using two different schemes to integrate the dust grain equation of motion within the same framework. First, we used a monofluid formalism (or often referred to as single fluid) in the terminal velocity approximation. This scheme follows the evolution of the barycentre of mass between the gas and the dust on a Eulerian grid. Second, we used a two-fluid scheme, in which the dust dynamics is handled with Lagrangian super-particles, and the gas dynamics on a Eulerian grid. Results. The monofluid results are in good agreement with the theoretical critical size for decoupling. We report dust dynamics decoupling for Stokes number St > 0.1, that is, dust grains of s > 4 μm in size. We find that the terminal velocity approximation is well suited for grain sizes of 10 μm in molecular clouds, in particular in the densest regions. However, the maximum dust enrichment measured in the low-density material - where St > 1 - is questionable. In the Lagrangian dust experiments, we show that the results are affected by the numerics for all dust grain sizes. At St ≪ 1, the dust dynamics is largely affected by artificial trapping in the high-density regions, leading to spurious variations of the dust concentration. At St > 1 , the maximum dust enrichment is regulated by the grid resolution used for the gas dynamics. Conclusions. Dust enrichment of submicron dust grains is unlikely to occur in the densest parts of molecular clouds. Two fluid implementations using a mixture of Eulerian and Lagrangian descriptions for the dust and gas mixture dynamics lead to spurious dust concentration variations in the strongly and weakly coupled regimes. Conversely, the monofluid implementation using the terminalvelocity approximation does not accurately capture dust dynamics in the low-density regions, that is, where St > 1 . The results of previous similar numerical work should therefore be revisited with respect to the limitations we highlight in this study.