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 properly 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 simulations 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 magnetic 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.

We propose a simple ansatz for estimating the value of the numerical resistivity and the numerical viscosity of any Eulerian MHD code. We test this ansatz with the help of simulations of the propagation of (magneto)sonic waves, Alfvén waves, and the tearing mode instability using the MHD code Aenus. By comparing the simulation results with analytical solutions of the resistive-viscous MHD equations and an empirical ansatz for the growth rate of tearing modes we measure the numerical viscosity and resistivity of Aenus. The comparison shows that the fast-magnetosonic speed and wavelength are the characteristic velocity and length, respectively, of the aforementioned (relatively simple) systems. We also determine the dependance of the numerical viscosity and resistivity on the time integration method, the spatial reconstruction scheme and (to a lesser extent) the Riemann solver employed in the simulations. From the measured results we infer the numerical resolution (as a function of the spatial reconstruction method) required to properly resolve the growth and saturation level of the magnetic field amplified by the magnetorotational instability in the post-collapsed core of massive stars. Our results show that it is to the best advantage to resort to ultra-high order methods (e.g., 9 th −order Monotonicity Preserving method) to tackle this problem properly, in particular in three dimensional simulations.

The magnetorotational instability (MRI) can be a powerful mechanism amplifying the magnetic field in core collapse supernovae. Whether initially weak magnetic fields can be amplified by this instability to dynamically relevant strengths is still a matter of debate. One of the main uncertainties concerns the process that terminates the growth of the instability. Parasitic instabilities of both Kelvin-Helmholtz and tearing-mode type have been suggested to play a crucial role in this process, disrupting MRI channel flows and quenching magnetic field amplification. We perform two-dimensional and three-dimensional sheering-disc simulations of a differentially rotating proto-neutron star layer in non-ideal magnetohydrodynamics with unprecedented high numerical accuracy, finding that Kelvin-Helmholtz parasitic modes dominate tearing modes in the regime of large hydrodynamic and magnetic Reynolds numbers, as encountered close to the surface of proto-neutron stars. They also determine the maximum magnetic field stress achievable during the exponential growth of the MRI. Our results are consistent with the theory of parasitic instabilities based on a local stability analysis. To simulate the Kelvin-Helmholtz instabilities properly a very high numerical resolution is necessary. Using 9th order spatial reconstruction schemes, we find that at least 8 grid zones per MRI channel are necessary to simulate the growth phase of the MRI and reach an accuracy of ∼ 10% in the growth rate, while more than ∼ 60 zones per channel are required to achieve convergent results for the value of the magnetic stress at MRI termination.