On Planar Perturbations in Vertically Stratified Keplerian Discs in a Uniform Magnetic Field

A local modal analysis of the magnetorotational instability (MRI) in the midplane of a thin Keplerian accretion disc with polytropic vertical density stratification has been carried out. To obtain an analytic solution, the approximation \(\partial/\partial r=0\) for all perturbations is adopted and planar adiabatic perturbations in an incompressible fluid with \(u_{z}=0\) are considered. An analytic closed-form dimensionless dispersion relation has been derived, the unstable intervals of the vertical wavenumber \(k_{z}\) are determined, and the MRI-unstable branch is identified. It is shown how the unstable wavenumber ranges and the extremal values of the dispersion curve depend on the strength of the background magnetic field.