Local Simulations of the Magnetorotational Instability in Core-Collapse Supernovae

Bearing in mind the application of core-collapse supernovae, we study the nonlinear properties of the magnetorotational instability (MRI) by means of three-dimensional simulations in the framework of a local shearing box approximation. By systematically changing 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 tot 〉〉) and the shear-vorticity ratio ( g q ) as 〈〈 w tot 〉〉∝ g δ q with an index of δ ∼ 0.5. The MRI-amplified magnetic energy has a similar scaling relative to the turbulent stress, while the Maxwell stress has a slightly smaller power-law index (∼0.36). By modeling the effect of viscous heating rates from MRI turbulence, we show that the stronger magnetic fields, or the larger shear rates initially imposed, lead to higher dissipation rates. For a rapidly rotating PNS with a spin period in milliseconds and with strong magnetic fields of 10 15 G, the energy dissipation rate is estimated to exceed 10 51 erg s −1 . Our results suggest that the conventional magnetohydrodynamic (MHD) mechanism of core-collapse supernovae is likely to be affected by MRI-driven turbulence, which we speculate, on the one hand, could harm the MHD-driven explosions due to the dissipation of the shear rotational energy at the PNS surface; or, on the other hand, its energy deposition might be potentially favorable for the working of the neutrino-heating mechanism.