  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.
