Dust Settling in Magnetorotationally-Driven Turbulent Discs I: Numerical Methods and Evidence for a Vigorous Streaming Instability

(Abridged) In this paper we have used the RIEMANN code for computational astrophysics to study the interaction of a realistic distribution of dust grains with gas in a vertically stratified protostellar accretion disc. The disc was modeled to have the density and temperature of a minimum mass solar nebula, and was driven to a fully-developed turbulence via the magnetorotational instability (MRI). We find that the inclusion of standard dust to gas ratios does not have any significant effect on the MRI even when the dust sediments to the midplane of the accretion disc. The density distribution of the dust reaches a Gaussian profile, and the scale heights for the dust that we derive are shown to be proportional to the reciprocal of the square root of the dust radius. The largest dust shows a strong tendency to settle to the midplane of the accretion disc, and tends to organize itself into elongated clumps of high density. The dynamics of these clumps is shown to be consistent with a streaming instability. The streaming instability is seen to be very vigorous and persistent once it forms. Each stream of high density dust displays a reduced RMS velocity dispersion, and the densest clumpings of large dust are shown to form where the streams intersect. We have also shown that the mean free path and collision time for the dust that participates in the streaming instability is reduced by almost two orders of magnitude relative to the average mean free paths and collision times. We show that some of the large dust in our 10 au simulations should have a propensity for grain coalescence.