The classical instability of a differentially rotating fluid takes place
according to Rayleigh's criterion, i.e. when there is a radially outward
decrease of angular momentum (per unit mass). In 1978, however, I showed
that a tiny radial decrease of angular velocity is enough to give
instability if a small azimuthal magnetic field is present.
This rather unusual instability arises when the magnetic energy is
small compared with the rotational energy, provided that the electrical
conductivity of the fluid or gas is not too weak. Non-axisymmetric waves
then grow and propagate around the rotation axis, and the fastest-growing
modes amplify very fast indeed, at a rate determined solely by the
differential rotation.
Phil. Trans. Roy. Soc. A Vol 289, pp 459-500, 1978