In view of its importance to astrophysical problems involving magnetized accretion disks and outflows in stars, we analyze the stability of incompressible, magnetized Couette flow to axisymmetric perturbations. We use an energy variational principle, the so-called interchange or Chandrasekhar's method, to derive the relevant stability criteria. This method is equivalent to the free-energy formalism that we have recently introduced to describe hydrodynamical instabilities in rotating, self-gravitating systems. In its implementation, all the applicable conservation laws are explicitly taken into account during the variations of the free-energy function. Thus we show that a purely toroidal magnetic field B_φ, which does not harm the conservation of circulation by imposing the additional conservation of azimuthal magnetic flux, leads to structural stability in Couette flow: the stability properties of the unmagnetized flow are recovered in the limit B_φ→0. In contrast, an axial-field component B_z, however small, destroys the conservation laws of circulation and azimuthal magnetic flux by imposing isorotation and conservation of the axial current along field lines. This radical change leads to a different stability criterion that implies structural instability, i.e., the stability properties of the flow with B_z ≡ 0 are not recovered in the limit B_z → 0 irrespective of the presence of rotation and/or a toroidal-field component. We discuss the relevance of our results for magnetized accretion flows and for outflows around stars and compact objects in active galactic nuclei. We also provide an application to thin accretion disks in Keplerian rotation.
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