Coexistence of coagulation and streaming instabilities in protoplanetary discs

TL;DR

Coagulation can extend the parameter space for the streaming instability in protoplanetary discs by increasing dust sizes, enabling SI to form dense clumps even when the initial dust-to-gas ratio is modest. The study uses unstratified, 2.5D shearing-box simulations with a single-size coagulation model to examine how the coagulation instability (CI) interacts with the SI. It finds that CI forms vertically extended filaments in dust-poor discs, and that SI can onset within these filaments for $\epsilon \sim 0.1$ and $St \lesssim 0.1$, a regime termed coagulation-assisted SI, leading to isotropic turbulence and enhanced dust diffusion by factors of $30$–$40$; as dust grows, SI ultimately dominates and the nonlinear state approaches that of pure SI. In dust-rich discs, coagulation barely affects SI, while in intermediate regimes coagulation amplifies SI growth rates inside filaments; overall coagulation helps seed dense clumps but is not essential to nonlinear saturation.

Abstract

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.

Figures (15)

• Figure 1: Time evolution of the maximum dust-to-gas ratio $\epsilon_{max}$ and Stokes number for all simulations listed in Table \ref{['table']}. The left, middle, right columns correspond to $\epsilon=0.02, 0.2, 3$ respectively while the upper, middle, and lower panels correspond to ${\rm St}=0.01, 0.1, 1$ respectively. In each panel, we show the evolution of $\epsilon_{max}$ for the case with (blue) and without (green) coagulation. For the evolution of ${\rm St}$, the light colour traces the minimum and maximum values whereas the dark colour traces the box averaged value.
• Figure 2: In the case $\epsilon=0.02$, snapshots of the dust density (left) and Stokes number (right) for model E0v02S0v01c with ${\rm St}=0.01$ (left), and model E0v02S0v1c with ${\rm St}=0.1$ (right). Both models include the effect of coagulation and result in the growth of the coagulation instability.
• Figure 3: In the case $\epsilon=0.02$, time evolution of the vertically averaged dust density for model E0v02S0v01c with ${\rm St}=0.01$ (left), and model E0v02S0v1c with ${\rm St}=0.1$ (right).
• Figure 4: For $\epsilon=0.2$ and ${\rm St}=0.01$ (model E0v02S0v01c) , time evolution of the dimensionless diffusion coefficients $\alpha_{g,i}$ in each direction (top). The lower panel shows expected linear growth rates for the same model in the inviscid case (left) and for $\alpha_{g,i}=5\times 10^{-6}$ (right). These have been obtained from linear theory (see App. \ref{['sec:appA']}) using $D=\nu=\alpha_{g,i}c_s H_g$. Here, $\tilde{K}_{x,z}=k_{x,z} \eta R_0$ are the dimensionless wavenumbers.
• Figure 5: For $\epsilon=0.2$, time evolution of the maximum dust-to-gas ratio $\epsilon_{max}$ and Stokes number for ${\rm St}=0.01$ (top) and ${\rm St}=0.1$ (bottom), in the case where the effect of the dust backreaction onto the gas is discarded.