  Bearing in mind the application to core-collapse supernovae, we study
nonlinear properties of the magneto-rotational instability (MRI) by means of
three- dimensional simulations in the framework of a local shearing box
approximation. By changing systematically the shear rates that symbolize the
degree of differential rotation in nascent proto-neutron stars (PNSs), we
derive a scaling relation between the turbulent stress sustained by the MRI and
the shear- vorticity ratio. Our parametric survey shows a power-law scaling
between the turbulent stress ($<< w_{\rm tot}>>$) and the shear- vorticity
ratio ($g_q$) as $<<w_{\rm tot}>> \propto g_q^{\delta}$ with its index $\delta
\sim 0.5$. The MRI-amplified magnetic energy has a similar scaling relative to
the turbulent stress, while the Maxwell stress has slightly smaller power-law
index ($\sim 0.36$). By modeling the effect of viscous heating rates due to the
MRI turbulence, we show that the stronger magnetic fields or the larger shear
rates initially imposed lead to the higher dissipation rates. For a rapidly
rotating PNS with the spin period in milliseconds and with strong magnetic
fields of $10^{15}$ G, the energy dissipation rate is estimated to exceed
$10^{51} {\rm erg\ sec^{-1}}$. Our results suggest that the conventional
magnetohydrodynamic (MHD) mechanism of core-collapse supernovae is likely to be
affected by the MRI-driven turbulence, which we speculate, on one hand, could
harm the MHD-driven explosions due to the dissipation of the shear rotational
energy at the PNS surface, on the other hand the energy deposition there might
be potentially favorable for the working of the neutrino-heating mechanism.
