Global axisymmetric magnetorotational instability with density gradients

We examine global incompressible axisymmetric perturbations of a differentially rotating MHD plasma with radial density gradients. It is shown that the standard magnetorotational instability (MRI) criterion drawn from the local dispersion relation is often misleading. If the equilibrium magnetic field is either purely axial or purely toroidal, the problem reduces to finding the global radial eigenvalues of an effective potential. The standard Keplerian profile including the origin is mathematically ill-posed, and thus any solution will depend strongly on the inner boundary. We find a class of unstable modes localized by the form of the rotation and density profiles, with reduced dependence on boundary conditions.