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Most of our knowledge on the nonlinear development of the magneto-rotationalinstability (MRI) relies on the results of numerical simulations employing theshearing box (SB) approximation. A number of difficulties arising from thisapproach have recently been pointed out in the literature. We thoroughlyexamine the effects of the assumptions made and numerical techniques employedin SB simulations. This is done in order to clarify and gain betterunderstanding of those difficulties as well as of a number of additionalserious problems, raised here for the first time, and of their impact on theresults. Analytical derivations and estimates as well as comparative analysisto methods used in the numerical study of turbulence are used. Numericalexperiments are performed to support some of our claims and conjectures. Thefollowing problems, arising from the (virtually exclusive) use of the SBsimulations as a tool for the understanding and quantification of the nonlinearMRI development in disks, are analyzed and discussed: (i) inconsistencies inthe application of the SB approximation itself; (ii) the limited spatial scaleof the SB; (iii) the lack of convergence of most ideal MHD simulations; (iv)side-effects of the SB symmetry and the non-trivial nature of the linear MRI;(v) physical artifacts arising on the too small box scale due to periodicboundary conditions. The computational and theoretical challenge posed by theMHD turbulence problem in accretion disks cannot be met by the SBapproximation, as it has been used to date. A new strategy to confront thischallenge is proposed, based on techniques widely used in numerical studies ofturbulent flows - developing (e.g., with the help of local numerical studies) asub-grid turbulence model and implementing it in global calculations.

On the viability of the shearing box approximation for numerical studies of MHD turbulence in accretion disks