MHD simulations of the magnetorotational instability in a shearing box with zero net flux

Aims.We study the influence of the choice of transport coefficients (viscosity and resistivity) on MHD turbulence driven by the magnetorotational instability (MRI) in accretion disks. Methods.We follow the methodology described in Paper I: we adopt an unstratified shearing box model and focus on the case where the net vertical magnetic flux threading the box vanishes. For the most part we use the operator split code ZEUS, including explicit transport coefficients in the calculations. However, we also compare our results with those obtained using other algorithms (NIRVANA, the PENCIL code and a spectral code) to demonstrate both the convergence of our results and their independence of the numerical scheme. Results.We find that small scale dissipation affects the saturated state of MHD turbulence. In agreement with recent similar numerical simulations done in the presence of a net vertical magnetic flux, we find that turbulent activity (measured by the rate of angular momentum transport) is an increasing function of the magnetic Prandtl number Pm for all values of the Reynolds number Re that we investigated. We also found that turbulence disappears when the Prandtl number falls below a critical value $\rm Pm_{c}$ that is apparently a decreasing function of Re. For the limited region of parameter space that can be probed with current computational resources, we always obtained $\rm Pm_{c}>1$. Conclusions.We conclude that the magnitudes of the transport coefficients are important in determining the properties of MHD turbulence in numerical simulations in the shearing box with zero net flux, at least for Reynolds numbers and magnetic Prandtl numbers that are such that transport is not dominated by numerical effects and thus can be probed using current computational resources