Linear Longwave Instability of a Single Class of Steady-State Jet Flows of an Ideal Fluid in the Field of a Self-Electric Current

A necessary and sufficient condition of linear stability of a certain two-parameter class of cylindrical steady-state shear jet MHD flows of an inviscid incompressible ideally-conducting fluid with a free boundary is obtained by the direct Lyapunov method. The magnetic field is induced by a direct current flowing along the jet so that the field linearly depends on the radius. The stability with respect to small axisymmetric longwave perturbations is considered. The perturbations conserve leave the ratio of the distance between a fluid particle and the jet axis to the azimuthal vorticity component unchanged in each fluid particle. Two-sided exponential estimates of the perturbation growth are derived in the case of violation of the stability condition obtained.