Coexistence of coagulation and streaming instabilities in protoplanetary discs

The streaming instability is considered one of the leading candidates for the formation of planetesimals, due to its ability to overcome the bouncing and fragmentation barriers. The formation of dense dust clumps through this process, however, is possible provided it involves solids with dimensionless stopping times $\sim 0.1$ in standard discs, which typically corresponds to 1-10 cm-sized particles. This implies that dust coagulation is required for the SI to be an efficient process. Here, we employ unstratified, shearing-box simulations combined with a moment equation for solving the coagulation equation to examine the effect of dust growth on the SI. In dust-rich discs with a dust-to-gas ratio $ε\gtrsim 1$, coagulation is found to have little impact on the SI; while in dust-poor discs with $ε\sim 0.01$, we observe the formation of vertically extended filaments through the action of the coagulation instability (CI), which is triggered due to the dependence of coagulation efficiency on dust density. For moderate dust-to-gas ratios $ε\sim 0.1$ and Stokes numbers $St \lesssim 0.1$, we find onset of the SI within these filaments, with a linear growth rate significantly higher compared to standard SI. We refer to this regime as coagulation-assisted SI. The synergy between both instabilities in that case leads to isotropic turbulence and dust concentrations that are increased by a factor of $30-40$. As dust continues to grow, SI tends to overcome the effect of the CI such that the nonlinear saturation phase is similar to pure SI. Our results suggest that coagulation, by simply increasing dust size, may facilitate the formation of dense clumps through the SI; even though it has only little effect on its nonlinear evolution.

💡 Research Summary

This paper investigates how dust coagulation interacts with the streaming instability (SI) in protoplanetary disks, using high‑resolution 2.5‑D axisymmetric shearing‑box simulations that incorporate a single‑size approximation for the coagulation process. The authors first derive a set of governing equations for gas and dust, adding a mass‑growth term obtained from the first moment of the Smoluchowski equation. The growth rate depends on the local dust density, particle size, and turbulent relative velocity (parameterized by α_coag = 10⁻⁴). A constant sink term equal to the initial growth rate ensures that the initial particle mass is an equilibrium solution.

Three dust‑to‑gas ratios (ε = 0.02, 0.2, 3) and an initial Stokes number St_i = 10⁻³ are explored; runs without coagulation keep St fixed at 0.01, 0.1, or 1 for comparison. The pressure gradient parameter η is set to 0.005, and the box spans 0.5 H in the radial direction with 0.1 H vertically, resolved by 2048 × 512 cells to capture the fastest growing modes of both instabilities.

The results fall into three regimes. (1) In dust‑rich disks (ε ≥ 1) the coagulation instability (CI) is essentially absent; the system behaves like the classic monodisperse SI, producing filaments and clumps with growth rates and saturation levels comparable to previous studies. (2) In very dust‑poor disks (ε ≈ 0.02) the CI dominates. A positive density perturbation enhances the local coagulation rate, which increases particle size, accelerates radial drift, and further amplifies the density perturbation—a positive feedback loop. This generates vertically extended filaments that are largely independent of SI, and the SI growth is suppressed. (3) In intermediate dust‑to‑gas ratios (ε ≈ 0.1) with St ≲ 0.1, both instabilities coexist. The CI first creates elongated filaments; within these structures the SI grows with a linear growth rate 2–3 times larger than the standard SI. The combined action, termed “coagulation‑assisted SI,” leads to isotropic turbulence and dust concentration factors of 30–40, far exceeding the enhancement seen in pure SI.

As coagulation proceeds and particles grow toward St ≈ 1, the SI becomes the dominant driver, and the nonlinear saturation phase resembles that of a pure SI simulation; the CI’s influence wanes. The authors therefore conclude that coagulation primarily assists the early linear growth of SI and expands the parameter space (ε, St) over which strong clumping can occur, but it does not fundamentally alter the saturated turbulent state once particles reach St ≈ 1.

The study advances the field by dynamically coupling coagulation to the SI rather than prescribing static size distributions. By making the coagulation rate density‑dependent, the work captures a realistic feedback loop that could operate in real disks. The findings suggest that regions of protoplanetary disks with moderate dust enrichment (ε ~ 0.1) are especially conducive to rapid planetesimal formation: CI can seed the environment, and the ensuing SI can quickly concentrate dust to the levels required for gravitational collapse. This synergy may help explain how centimeter‑sized aggregates, which are otherwise hindered by bouncing and fragmentation barriers, can transition into the planetesimal regime. The paper also clarifies that in highly dust‑rich environments coagulation adds little to SI, while in dust‑poor disks CI alone can generate structures but may not achieve the densities needed for planetesimal formation without additional mechanisms. Overall, the work provides a comprehensive, self‑consistent picture of how dust growth and collective drag instabilities cooperate to bridge the gap between micron‑sized grains and kilometer‑scale planetesimals.