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In the recent years, MRI-driven turbulent transport has been found to dependin a significant way on fluid viscosity $\nu$ and resistivity $\eta$ throughthe magnetic Prandtl number $Pm=\nu/\eta$. In particular, the transportdecreases with decreasing $Pm$; if persistent at very large Reynolds numbers,this trend may lead to question the role of MRI-turbulence in YSO disks, whosePrandtl number is usually very small. In this context, the principle objectiveof the present investigation is to characterize in a refined way the role ofdissipation. Another objective is to characterize the effect of linear (channelmodes) and quasi-linear (parasitic modes) physics in the behavior of thetransport. These objectives are addressed with the help of a number ofincompressible numerical simulations. The horizontal extent of the box size hasbeen increased in order to capture all relevant (fastest growing) linear andsecondary parasitic unstable modes. The major results are the following: i- Theincreased accuracy in the computation of transport averages shows that thedependence of transport on physical dissipation exhibits two different regimes:for $Pm \lesssim 1$, the transport has a power-law dependence on the magneticReynolds number rather than on the Prandtl number; for $Pm > 1$, the data areconsistent with a primary dependence on $Pm$ for large enough ($\sim 10^3$)Reynolds numbers. ii- The transport-dissipation correlation is not clearly orsimply related to variations of the linear modes growth rates. iii- Theexistence of the transport-dissipation correlation depends neither on thenumber of linear modes captured in the simulations, nor on the effect of theparasitic modes on the saturation of the linear modes growth. iv- The transportis usually not dominated by axisymmetric (channel) modes

MRI-driven turbulent transport: the role of dissipation, channel modes and their parasites