Dust Diffusion in Protoplanetary Disks by Magnetorotational Turbulence

Abstract

We measure the turbulent diffusion coefficient of dust grains embedded in magnetorotational turbulence in a protoplanetary disk directly from numerical simulations and compare it to the turbulent viscosity of the flow. The simulations are done in a local coordinate frame comoving with the gas in Keplerian rotation. Periodic boundary conditions are used in all directions, and vertical gravity is not applied to the gas. Using a two-fluid approach, small dust grains of various sizes (with friction times up to Ω0τf = 0.02) are allowed to move under the influence of friction with the turbulent gas. We measure the turbulent diffusion coefficient of the dust grains by applying an external sinusoidal force field acting in the vertical direction on the dust component only. This concentrates the dust around the midplane of the disk, and an equilibrium distribution of the dust density is achieved when the vertical settling is counteracted by the turbulent diffusion away from the midplane. Comparing with analytical expressions for the equilibrium concentration, we deduce the vertical turbulent diffusion coefficient. The vertical diffusion coefficient is found to be lower than the turbulent viscosity and to have an associated vertical Schmidt number (vertical diffusion Prandtl number) of about 1.5. A similar radial force field also allows us to measure the radial turbulent diffusion coefficient. We find a radial Schmidt number of about 0.85 and also find that the radial turbulent diffusion coefficient is around 70% higher than the vertical. As most angular momentum transport happens through magnetic Maxwell stresses, both the vertical and the radial diffusion coefficients are found to be significantly higher than suggested by the angular momentum transport by Reynolds stresses alone. We also find evidence for trapping of dust grains of intermediate friction time in turbulent eddies.