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We analyze a linear perturbation scheme for a two-dimensional backgroundplasma, which is rotating with a differential frequency and is embedded in apoloidal magnetic field. The main two assumptions of the present study, whichin turn are related, are that the plasma profile is axially symmetric, both inthe background and the perturbation approximation, where the azimuthal magneticfield is requested to vanish identically and secondly the angular frequency onthe magnetic surface function only still holds in the non-stationary regime,which, on the steady background equilibrium, is ensured by the validity of thecorotation theorem (Ferraro 1937). Indeed, such a restriction of the model israther natural and it implies that the azimuthal component of the linear plasmashift is reabsorbed in the expression for the non stationary electric field (inprinciple, at any order of approximation) and can no longer provide a non-zeroazimuthal component of the magnetic tension field. As a result, theMagneto-rotational instability is suppressed and the magnetic field has theeffect to stabilize the plasma configuration with respect to the purehydrodynamical case.

Counterexample of the magnetorotational instability in two-dimensional axial symmetry