Counterexample of the magnetorotational instability in two-dimensional axial symmetry

We analyze a linear perturbation scheme for a two-dimensional background plasma, which is rotating at a differential frequency and is embedded in a poloidal magnetic field. The main two assumptions of the present study, which in turn are related, are (i) that the plasma profile is axially symmetric, both in the background and in the perturbation approximation, where the azimuthal magnetic field is requested to vanish identically, and (ii) that the angular frequency depends on the magnetic surface function only, still holds in the nonstationary regime, which, in the steady background equilibrium, is ensured by the validity of the corotation theorem [V. C. A. Ferraro, Mon. Not. R. Astron. Soc. 97, 458 (1937)]. Indeed, such a restriction of the model is rather natural and it implies that the azimuthal component of the linear plasma shift is reabsorbed in the expression for the nonstationary electric field (in principle, at any order of approximation) and can no longer provide a nonzero azimuthal component of the magnetic tension field. As a result, the magnetorotational instability is suppressed and the magnetic field has the effect to stabilize the plasma configuration with respect to the pure hydrodynamical case.