  We consider the problem of convergence in homogeneous shearing box simula-
tions of magneto-rotationally driven turbulence. When there is no mean magnetic
flux, if the equations are non dimensionalized with respect to the diffusive
scale, the only free parameter in the problem is the size of the computational
domain. The problem of convergence then relates to the asymptotic form of the
solutions as the computational box size becomes large. By using a numerical
code with a high order of accuracy we show that the solutions become
asymptotically inde- pendent of domain size. We also show that cases with weak
magnetic flux join smoothly to the zero flux cases as the flux vanishes. These
results are consistent with the operation of a subcritical small-scale dynamo
driving the turbulence. We conclude that for this type of turbulence the
angular momentum transport is a proportional to the diffusive flux and
therefore has limited relevance in as- trophysical situations.
