  Most of our knowledge on the nonlinear development of the magneto-rotational
instability (MRI) relies on the results of numerical simulations employing the
shearing box (SB) approximation. A number of difficulties arising from this
approach have recently been pointed out in the literature. We thoroughly
examine the effects of the assumptions made and numerical techniques employed
in SB simulations. This is done in order to clarify and gain better
understanding of those difficulties as well as of a number of additional
serious problems, raised here for the first time, and of their impact on the
results. Analytical derivations and estimates as well as comparative analysis
to methods used in the numerical study of turbulence are used. Numerical
experiments are performed to support some of our claims and conjectures. The
following problems, arising from the (virtually exclusive) use of the SB
simulations as a tool for the understanding and quantification of the nonlinear
MRI development in disks, are analyzed and discussed: (i) inconsistencies in
the application of the SB approximation itself; (ii) the limited spatial scale
of 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 periodic
boundary conditions. The computational and theoretical challenge posed by the
MHD turbulence problem in accretion disks cannot be met by the SB
approximation, as it has been used to date. A new strategy to confront this
challenge is proposed, based on techniques widely used in numerical studies of
turbulent flows - developing (e.g., with the help of local numerical studies) a
sub-grid turbulence model and implementing it in global calculations.
