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The magnetorotational instability (MRI) is a promising mechanism to amplifythe magnetic field in fast rotating protoneutron stars. The diffusion ofneutrinos trapped in the PNS induces a transport of momentum, which can bemodelled as a viscosity on length-scales longer than the neutrino mean freepath. This neutrino-viscosity can slow down the growth of MRI modes to such anextent that a minimum initial magnetic field strength of $\gtrsim 10^{12}\,{\rm G}$ is needed for the MRI to grow on a sufficiently short time-scale topotentially affect the explosion. It is uncertain whether the magnetic field offast rotating progenitor cores is strong enough to yield such an initialmagnetic field in PNS. At MRI wavelengths shorter than the neutrino mean freepath, on the other hand, neutrino radiation does not act as a viscosity butrather induces a drag on the velocity with a damping rate independent of thewavelength. We perform a linear analysis of the MRI in this regime, and applyour analytical results to the PNS structure from a one-dimensional numericalsimulation. We show that in the outer layers of the PNS, the MRI can grow fromweak magnetic fields at wavelengths shorter than the neutrino mean free path,while deeper in the PNS MRI growth takes place in the viscous regime andrequires a minimum magnetic field strength.

Neutrino viscosity and drag: impact on the magnetorotational instability in protoneutron stars