Morphology of two-dimensional MRI in axial symmetry

> In this paper, we analyse the linear stability of a stellar accretion disk having a stratified morphology. The study is performed in the framework of ideal magneto-hydrodynamics and therefore results in a characterization of the linear unstable magneto-rotational modes. The peculiarity of the present scenario consists of adopting the magnetic flux function as the basic dynamical variable. Such a representation of the dynamics allows us to take account of the co-rotation theorem as a fundamental feature of the ideal plasma equilibrium and to evaluate its impact on the perturbation evolution. According to the Alfvenic nature of the magneto-rotational instability, we consider an incompressible plasma profile and perturbations propagating along the background magnetic field. Furthermore, we develop a local perturbation analysis around fiducial coordinates of the background configuration and deal with very small-scale linear dynamics in comparison to the background inhomogeneity size. The main issue of the present study is that the condition for the emergence of unstable modes is the same in the stratified plasma disk as in the case of a thin configuration. Such a feature is the result of the cancellation of the vertical derivative of the disk angular frequency from the dispersion relation, which implies that only the radial profile of the differential rotation is responsible for the trigger of the growing modes.