Polydisperse streaming instability – II. Methods for solving the linear stability problem

Abstract

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.

The Acoustic Resonant Drag Instability with a Spectrum of Grain Sizes

“…A2, where, despite the presence of resonant grains the growth rate is unconverged in N d (recall that the non-constantdrift simulation in the main text has grains from w s, j ≈ 1 to 12; see § 2.4). 12 Further discussion of similar issues can be found in Krapp et al (2019Krapp et al ( , 2020; Paardekooper et al (2021). However, towards the end of the range of resonant angles, as cos θ k approaches w −1 s,max , there no longer exist resonances at higher θ k , which suggests the existence of a converged, resonant growth region around the resonance of the largest grains cos θ k w s,max = 1.…”

“…Finally, it is also worth mentioning that while GIZMO seems to be able to capture the linear growth rates of the polydisperse acoustic RDI relatively accurately (see, e.g., Fig. B3), exploring the detailed convergence to linear predictions in different regimes with different grain-size distributions is a complex task beyond the scope of this work (see, e.g., Paardekooper et al 2021;Zhu & Yang 2021). While unlikely to affect our results here, given the dominance of the large-scale modes in the non-constant-drift simulation, there may be important effects at smaller scales and/or smaller µ, and a more detailed study of numerical convergence and/or comparison to other codes would be important for exploring such cases.…”

“…Although we use an MRN dust-mass distribution in all cases dµ/d ln grain ∝ 1/2 grain , we use an equal number of numerical super-particles randomly sampled across each logarithmic size range (i.e., the super-particles change in their "total" super-particle mass proportionally to 1/2 grain ) in order to not under-resolve the small particles. We note that, by spreading the grain-size distribution across the full range of particles, this method appears to avoid some of the convergence problems that are well known in the numerical computation of polydisperse dust instabilities (Paardekooper et al 2021). We study this issue further in App.…”

“…We then take the limit as N d → ∞ with the total mass of grains fixed to obtain results valid in the continuous limit. This is by no means the most efficient approach in all cases (see Paardekooper et al 2021), but is physically enlightening for understanding the physical causes of different behaviors. We thus define the properties of discrete grain species j: the dust-to-gas-mass ratio µ j (with µ 0 = j µ j ), the relative drift velocity w s, j = |w s, j |/c s , and the stopping time t s, j , which become functions of grain size grain in the continuous limit.…”

“…A2, and useful analytic expressions for the growth rates are much harder to obtain. Further, in some regimes the growth rate scales with the number of dust species N d , and care is required even for the interpretation of numerically computed growth rates (this also occurs in many regimes for the streaming instability in protoplanetary disks, see Krapp et al 2019;Paardekooper et al 2021). For this reason, we provide only a cursory survey of different modes and regimes here, pointing out some key features of how the growth rate varies with wavelength, but forgoing a full detailed analysis.…”