Criterion for linear long-wave stability of steady-state shear jet flows of an ideal fluid in an azimuth magnetic field

The linear stability problem for steady-state axisymmetric shear jet MHD flows of an inviscid perfectly conducting incompressible fluid with a free boundary is studied in the following statement. A jet of unlimited length is situated in the boundless space. The azimuth magnetic field is "frozen" in the jet. Along the jet surface the longitudinal direct current flows that generates a quasi-stationary azimuth magnetic field outside the jet. A surface tension on the free boundary of the jet is neglected. A criterion for stability of a particular class of given flows against small axisymmetric long-wave perturbations of a special form is obtained by the direct Lyapunov method. In the case of violation of this stability criterion a priori double-sided exponential estimates for the growth of small perturbations are constructed, and increments are calculated by parameters of steady-state flows under consideration and initial data for perturbations. An example of a steady-state axisymmetric shear jet MHD flow and initial small axisymmetric long-wave perturbations imposed on this flow which evolve according to the constructed estimates is exhibited. It is also shown that the obtained stability criterion is a sufficient condition for instability of more general steady-state axisymmetric shear jet MHD flows against the same perturbations.