Stability of steady-state shear jet flows of an ideal fluid with a free boundary in an azimuthal magnetic field against small long-wave perturbations

The problem of the linear stability of steady-state axisymmetric shear jet flows of a perfectly conducting inviscid incompressible fluid with a free surface in an azimuthal magnetic field is studied. The necessary and sufficient condition for the stability of these flows against small axisymmetric long-wave perturbations of special form is obtained by the direct Lyapunov method. It is shown that if this stability condition is not satisfied, the steady-state flows considered are unstable to arbitrary small axisymmetric long-wave perturbations. A priori exponential estimates are obtained for the growth of small perturbations. Examples are given of the steady-state flows and small perturbations imposed on them which evolve in time according to the estimates obtained.