Title: MRI turbulence in vertically stratified accretion discs at large magnetic Prandtl numbers

Abstract: The discovery of the first binary neutron star merger, GW170817, has spawned a plethora of global numerical relativity simulations. These simulations are often ideal (with dissipation determined by the grid) and/or axisymmetric (invoking ad hoc mean-field dynamos). However, binary neutron star mergers (similar to X-ray binaries and active galactic nuclei inner discs) are characterised by large magnetic Prandtl numbers, $\rm Pm$, (the ratio of viscosity to resistivity). $\rm Pm$ is a key parameter determining dynamo action and dissipation but it is ill-defined (and likely of order unity) in ideal simulations. To bridge this gap, we investigate the magnetorotational instability (MRI) and associated dynamo at large magnetic Prandtl numbers using fully compressible, three-dimensional, vertically stratified, isothermal simulations of a local patch of a disc. We find that, within the bulk of the disc ($z\lesssim2H$, where $H$ is the scale-height), the turbulent intensity (parameterized by the stress-to-thermal-pressure ratio $\alpha$), and the saturated magnetic field energy density, $E_\text{mag}$, produced by the MRI dynamo, both scale as a power with Pm at moderate Pm ($4\lesssim \text{Pm} \lesssim 32$): $E_\text{mag} \sim \text{Pm}^{0.74}$ and $\alpha \sim \text{Pm}^{0.71}$, respectively. At larger Pm ($\gtrsim 32$) we find deviations from power-law scaling and the onset of a plateau. Compared to our recent unstratified study, this scaling with Pm becomes weaker further away from the disc mid-plane, where the Parker instability dominates. We perform a thorough spectral analysis to understand the underlying dynamics of small-scale MRI-driven turbulence in the mid-plane and of large-scale Parker-unstable structures in the atmosphere.