  The magnetorotational instability (MRI) is a promising mechanism to amplify
the magnetic field in fast rotating protoneutron stars. The diffusion of
neutrinos trapped in the PNS induces a transport of momentum, which can be
modelled as a viscosity on length-scales longer than the neutrino mean free
path. This neutrino-viscosity can slow down the growth of MRI modes to such an
extent 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 to
potentially affect the explosion. It is uncertain whether the magnetic field of
fast rotating progenitor cores is strong enough to yield such an initial
magnetic field in PNS. At MRI wavelengths shorter than the neutrino mean free
path, on the other hand, neutrino radiation does not act as a viscosity but
rather induces a drag on the velocity with a damping rate independent of the
wavelength. We perform a linear analysis of the MRI in this regime, and apply
our analytical results to the PNS structure from a one-dimensional numerical
simulation. We show that in the outer layers of the PNS, the MRI can grow from
weak magnetic fields at wavelengths shorter than the neutrino mean free path,
while deeper in the PNS MRI growth takes place in the viscous regime and
requires a minimum magnetic field strength.
