  The magneto-rotational instability (MRI) is considered to be a promising
mechanism to amplify the magnetic field in fast rotating protoneutron stars. In
contrast to accretion disks, radial buoyancy driven by entropy and lepton
fraction gradients is expected to have a dynamical role as important as
rotation and shear. We investigate the poorly known impact of buoyancy on the
non-linear phase of the MRI, by means of three dimensional numerical
simulations of a local model in the equatorial plane of a protoneutron star.
The use of the Boussinesq approximation allows us to utilise a shearing box
model with clean shearing periodic boundary conditions, while taking into
account the buoyancy driven by radial entropy and composition gradients. We
find significantly stronger turbulence and magnetic fields in buoyantly
unstable flows. On the other hand, buoyancy has only a limited impact on the
strength of turbulence and magnetic field amplification for buoyantly stable
flows in the presence of a realistic thermal diffusion. The properties of the
turbulence are, however, significantly affected in the latter case. In
particular, the toroidal components of the magnetic field and of the velocity
become even more dominant with respect to the poloidal ones. Furthermore, we
observed in the regime of stable buoyancy the formation of long lived coherent
structures such as channel flows and zonal flows. Overall, our results support
the ability of the MRI to amplify the magnetic field significantly even in
stably stratified regions of protoneutron stars.
