  Possible effects of magnetic fields in core collapse supernovae rely on an
efficient amplification of the weak pre-collapse fields. It has been suggested
that the magneto-rotational instability (MRI) leads to rapid field growth.
Although MRI studies exist for accretion discs, the application of their
results to core collapse supernovae is inhibited as the physics of supernova
cores is substantially different from that of accretion discs. We address the
problem of growth and saturation of the MRI by means of semi-global
simulations, which combine elements of global and local simulations by taking
the presence of global background gradients into account and using a local
computational grid. We analyze the dispersion relation of the MRI to identify
different regimes of the instability. This analysis is complemented by
simulations, where we consider a local computational box rotating at
sub-Keplerian velocity, and where we allow for a radial entropy gradient. We
identify six regimes of the MRI depending on the ratio of the entropy and
angular velocity gradient. Our numerical models confirm the instability
criteria and growth rates for all relevant regimes. The MRI grows exponentially
within milliseconds the flow and magnetic field geometries being dominated by
channel flows. The MRI growth ceases once the channels are disrupted by
resistive instabilities (due to finite numerical conductivity), and MHD
turbulence sets in. From an analysis of the growth rates of the resistive
instabilities, we deduce scaling laws for the termination amplitude of the MRI
which agree well with our numerical models. We determine the dependence of the
development of coherent flow structures in the saturated state on the aspect
ratio of the simulation boxes. [abridged]
