Context :

Aims : We study the properties of MHD turbulence driven by the magnetorotational instability ( MRI ) in accretion disks .
To do this we perform a series of numerical simulations for which the resolution is gradually increased .

Methods : 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 .

Results : Performing a set of numerical simulations in a fixed computational domain with increasing resolution , we demonstrate that turbulent activity decreases as resolution increases .
The highest resolution considered is 256 grid cells per scale height .
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 .

Conclusions : 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 .