Modelling dust coagulation, dynamical drag and turbulent mixing during star and disc formation

TL;DR

This work presents a unified, three-dimensional framework that couples bin-based dust coagulation with a multigrain SPH dust-dynamics method and a new implicit turbulent diffusion scheme, enabling the simultaneous treatment of dust growth, dynamical drag, and turbulent mixing during star and disc formation. Implemented in the SPH code sphNG, the approach resolves multiple dust species and uses implicit integration to robustly propagate dust evolution without negativity issues, while accounting for turbulence via a Schmidt-number-aware diffusion model. Key findings demonstrate that turbulent diffusion substantially enhances dust growth by supplying small and intermediate grains to high-density growth regions, with diffusion also smoothing and redistributing dust throughout the disc; the gas-dust drag remains important for coupling but does not dominate the evolution on the studied timescales. These results enable realistic, size-resolved dust evolution in 3D star- and disc-formation simulations, providing a framework for direct comparison with ALMA/JWST observations and informing early planet formation scenarios.

Abstract

Planet formation in the discs around young stars involves the coagulation of sub-micron sized dust grains into much larger grains that may be mixed by turbulence and migrate through the disc. In this paper, we describe how we have combined a method for modelling the coagulation of a population of dust grains with the MULTIGRAIN algorithm for modelling the dynamical evolution of a population of dust grains that are subject to strong gas drag. We solve the dynamical evolution of the dust grains due to gas drag using a recently-developed implicit integration method, and we introduce a new implicit method to model the diffusion of the dust due to unresolved hydrodynamic turbulence. The resulting smoothed particle hydrodynamics (SPH) code allows us, for the first time, to model the growth, mixing and migration of dust grain populations during the early stages of star formation and the formation, growth and evolution of a young protoplanetary disc using three-dimensional hydrodynamical simulations. In doing so, we find that including turbulent dust diffusion within the disc provides a substantial enhancement of the rate of dust grain growth due to the fact that the turbulent diffusion provides a source of small and intermediate dust grains to the regions in which the largest dust grains are growing.

Figures (18)

• Figure 1: The vertical distributions of the dust fractions of the 10 dust size bins after 15 orbits of dust settling when calculated using the explicit (left) and implicit (right) multigrain methods. The results are almost identical. The different grain size bins are ordered from 1 mm to 0.1 $\mu$m from top to bottom at $z=0$ (the mid-plane). The largest grains quickly settle towards the mid-plane, while the smallest grains maintain a near uniform dust fraction distribution.
• Figure 2: The vertical distributions of the dust fractions of the 10 dust size bins after 15 orbits of dust settling when calculated using the explicit multigrain method without enforcing that the dust variable should never become negative. Although the solution is correct in regions with a substantial dust fraction, spurious dust appears above and below the dust layers where, in reality, all the dust in that size bin should have settled out.
• Figure 3: The vertical distributions of the dust fractions of the 10 dust size bins after 15 orbits (left column) or 50 orbits (right column; note the change of vertical scale) of the dust settling test but also including grain growth. The different grain size bins are ordered from 1 mm to 0.1 $\mu$m from top to bottom at $z=0$ (the mid-plane). The top panels only include Brownian motions when calculating the relative grain velocities, $v_{kj}$, that are required for grain growth. The centre panels include both Brownian and turbulent motions (with $\alpha_{\rm SS}=10^{-3}$). The lower panels include Brownian and turbulent motions, and the vertical (settling) velocity differences between grains of different sizes, as computed by the multigrain method (i.e., in the terminal velocity approximation). As is the case without grain growth the largest grains quickly settle towards the mid-plane, but with grain growth the smallest grains are depleted. The steps in the abundances of small grains in the centre and lower panels are due to the coarse discretisation with the small number of dust size bins. This means there are large jumps in both the Stokes numbers and vertical velocities between different bins. Using more bins results in smoother distributions.
• Figure 4: The vertical distribution of the dust fraction of 1 mm dust grains from calculations with both dust settling and turbulent stirring after the equilibrium distribution has been attained. The solid black line gives the analytic solution (equations \ref{['eq:analytic2']} and \ref{['eq:epsilon0']}). The red points are from an implicit SPH calculation at our standard particle resolution (25942 particles in $[x,y]=[\pm 0.2,\pm 0.15]$; after 50 orbits). The green points are from an equivalent calculation with 10 times the linear spatial resolution (68080 SPH particles in $[x,y]=[\pm 0.01,\pm 0.0075]$; after 50 orbits). Both simulations match the analytic solution well at high dust fractions near the mid-plane, but the rate at which the dust fraction drops off at large distances from the mid-plane is under-estimated at lower resolution.
• Figure 5: The vertical distribution of the dust fraction from SPH calculations (standard resolution) with both dust settling and turbulent stirring after the equilibrium distribution has been attained. Four calculations are performed, using 1 mm or 100 $\mu$m grains each with $\alpha_{\rm SS}=0.001$ or $0.01$ (see the legend). The solutions for the 1 mm grains with $\alpha_{\rm SS}=0.01$ and 100 $\mu$m grains with $\alpha_{\rm SS}=0.001$ lie on top of each other, as they should since they have the same value of the dimensionless quantity in equation \ref{['eq:dimensionless']}. The white lines give the analytic solutions (equations \ref{['eq:analytic2']} and \ref{['eq:epsilon0']}). Each of the simulations match the analytic solutions well. Cases with larger dust grains and/or less turbulence produce thinner dust layers.