Impact of dimensionless numbers on the efficiency of magnetorotational instability induced turbulent transport

The magnetorotational instability (MRI) is presently the most promising source of turbulent transport in accretion discs. However, some important issues still need to be addressed to quantify the role of MRI in discs; in particular no systematic investigation of the role of the physical dimensionless parameters of the problem on the dimensionless transport has been undertaken yet. For completeness, we first generalize existing results on the marginal stability limit in the presence of both viscous and resistive dissipation, exhibit simple scalings for all relevant limits, and give them a physical interpretation. We then re-examine the question of transport efficiency through numerical simulations in the simplest setting of a local, unstratified shearing box, with the help of a pseudo-spectral incompressible 3D code; viscosity and resistivity are explicitly accounted for. We focus on the effect of the dimensionless magnetic field strength, the Reynolds number and the magnetic Prandtl number. First, we complete existing investigations on the field strength dependence by showing that the transport in high magnetic pressure discs close to marginal stability is highly time dependent and surprisingly efficient. Secondly, we bring to light a significant dependence of the global transport on the magnetic Prandtl number, with alpha proportional to Pm-delta for the explored range: 0.12 < Pm < 8 and 200 < Re < 6400 (delta being in the range 0.25-0.5). We show that the dimensionless transport is not correlated to the dimensionless linear growth rate, contrary to a largely held expectation. For large enough Reynolds numbers, one would expect that the reported Prandtl number scaling of the transport should saturate, but such a saturation is out of reach of the present generation of supercomputers. Understanding this saturation process is nevertheless quite critical to accretion disc transport theory, as the magnetic Prandtl number Pm is expected to vary by many orders of magnitude between the various classes of discs, from Pm 1 in young stellar object discs to Pm greater than or similar to or 1 in active galactic nucleus discs. More generally, these results stress the need to control dissipation processes in astrophysical simulations.