Rotationally symmetric spontaneous swirling in MHD flows

The stability of steady axisymmetric MHD flows of an inviscid, incompressible, perfectly conducting fluid with respect to swirling–perturbations of the azimuthal components of the velocity field–is studied in a linear approximation. It is shown that for flows similar to a magnetohydrodynamic Hill–Shafranov vortex, the problem reduces to a one-dimensional problem on a closed streamline of the unperturbed flow (the arc length of the streamline is the spatial coordinate). A spectral boundary-value eigenvalue problem is formulated for a system of two ordinary differential equations with periodic coefficients and periodic boundary conditions. Sufficient conditions under which swirling is impossible are obtained. Numerical solution of the characteristic equation shows that, under certain conditions, for each streamline there is a real eigenvalue that yields monotonic exponential growth of the initial perturbations.