MHD simulations of the magnetorotational instability in a shearing box with zero net flux. I. The issue of convergence

We study the properties of MHD turbulence driven by the magnetorotational instability (MRI) in accretion disks. We adopt the local shearing box model and focus on the special case for which the initial magnetic flux threading the disk vanishes. We employ the finite difference code ZEUS to evolve the ideal MHD equations. Performing a set of numerical simulations in a fixed computational domain with increasing resolution, we demonstrate that turbulent activity decreases as resolution increases. We quantify the turbulent activity by measuring the rate of angular momentum transport through evaluating the standard alpha parameter. We find alpha=0.004 when (N_x,N_y,N_z)=(64,100,64), alpha=0.002 when (N_x,N_y,N_z)=(128,200,128) and alpha=0.001 when (N_x,N_y,N_z)=(256,400,256). This steady decline is an indication that numerical dissipation, occurring at the grid scale is an important determinant of the saturated form of the MHD turbulence. Analysing the results in Fourier space, we demonstrate that this is due to the MRI forcing significant flow energy all the way down to the grid dissipation scale. We also use our results to study the properties of the numerical dissipation in ZEUS. Its amplitude is characterised by the magnitude of an effective magnetic Reynolds number Re_M which increases from 10^4 to 10^5 as the number of grid points is increased from 64 to 256 per scale height. The simulations we have carried out do not produce results that are independent of the numerical dissipation scale, even at the highest resolution studied. Thus it is important to use physical dissipation, both viscous and resistive, and to quantify contributions from numerical effects, when performing numerical simulations of MHD turbulence with zero net flux in accretion disks at the resolutions normally considered.