  Whether the magnetorotational instability (MRI) can amplify initially weak
magnetic fields to dynamically relevant strengths in core collapse supernovae
is still a matter of active scientific debate. Recent numerical studies have
shown that the first phase of MRI growth dominated by channel flows is
terminated by parasitic instabilities of the Kelvin-Helmholtz type that disrupt
MRI channel flows and quench further magnetic field growth. However, it remains
to be prop- erly assessed by what factor the initial magnetic field can be
amplified and how it depends on the initial field strength and the amplitude of
the perturbations. Different termination criteria leading to different
estimates of the amplification factor were proposed within the parasitic model.
To determine the amplification factor and test which criterion is a better
predictor of the MRI termination, we perform three-dimensional shearing-disc
and shearing-box simula- tions of a region close to the surface of a
differentially rotating proto-neutron star in non-ideal MHD with two different
numerical codes. We find that independently of the initial magnetic field
strength, the MRI channel modes can amplify the magnetic field by, at most, a
factor of 100. Under the conditions found in proto-neutron stars a more
realistic value for the mag- netic field amplification is of the order of 10.
This severely limits the role of the MRI channel modes as an agent amplifying
the magnetic field in proto-neutron stars starting from small seed fields. A
further amplification should therefore rely on other physical processes, such
as for example an MRI-driven turbulent dynamo.
