MRI-driven $\alpha-\Omega$ dynamos in protoneutron stars

Magnetars are highly magnetized neutron stars that can produce X-ray and soft gamma-ray emissions and that have a dipole of $10^{14}$ G to $10^{15}$ G. A promising mechanism for explaining magnetar formation is magnetic field amplification by the MRI in fast-rotating protoneutron stars (PNS). This scenario is supported by recent global models, which showed that small-scale turbulence can generate a dipole with magnetar-like intensity. However, the impact of buoyancy and density stratification on the efficiency of the MRI in generating a dipole is still unknown. We assess the impact of the density and entropy profiles on the MRI dynamo in a global model of a fast-rotating PNS, which focuses on its outer stratified region that is stable to convection. Using the pseudo-spectral code MagIC, we performed 3D Boussinesq and anelastic MHD simulations in spherical geometry with explicit diffusivities. We performed a parameter study in which we investigate the effect of different approximations and of thermal diffusion. We obtain a self-sustained turbulent MRI-driven dynamo. This confirms most of our previous incompressible results once rescaled for density. The MRI also generates a nondominant equatorial dipole, which represents about 4.3% of the averaged magnetic field strength. Interestingly, in the presence of a density gradient, an axisymmetric magnetic field at large scales oscillates with time, which can be described as a mean-field $\alpha\Omega$ dynamo. Buoyancy damps turbulence in the equatorial plane but it has overall a relatively weak influence with a realistic high thermal diffusion. Our results support the ability of the MRI to generate magnetar-like large-scale magnetic fields. They furthermore predict the presence of an $\alpha\Omega$ dynamo in the protoneutron star, which could be important to model in-situ magnetic field amplification in core-collapse supernovae. [abridged]