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 ( \langle \langle w _ { tot } \rangle \rangle ) and the shear-vorticity ratio ( g _ { q } ) as \langle \langle w _ { tot } \rangle \rangle \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 } { 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 .