The magneto-rotational instability is presently the most promising source of turbulent transport in accretion disks .
However , some important issues still need to be addressed to quantify the role of MRI in disks ; 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 presence of both viscous and resistive dissipation , exhibit simple scalings for all relevant limits , and give them a physical interpretation .
We then reexamine 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 disks close to marginal stability is highly time-dependent and surprisingly efficient .
Second , we bring to light a significant dependence of the global transport on the magnetic Prandtl number , with \alpha \propto Pm ^ { \delta } for the explored range : 0.12 < Pm < 8 and 200 < Re < 6400 ( \delta being in the range 0.25 to 0.5 ) .
We show that the dimensionless transport is not correlated to the dimensionless linear growth rate , contrarily to a largely held expectation .
For large enough Reynolds numbers , one would expect 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 disk transport theory , as the magnetic Prandtl number Pm is expected to vary by many orders of magnitude between the various classes of disks , from Pm \ll 1 in YSO disks to Pm \gtrsim or \gg 1 in AGN disks .
More generally , these results stress the need to control dissipation processes in astrophysical simulations .