On stability of MHD flows located on the surface of axisymmetric torus

Abstract

We consider the problem of the evolution of azimuthal perturbations (spontaneous swirling) in axisymmetric magnetohydrodynamic flows of an ideally conducting fluid with circular streamlines. The fluid is in a toroidal gap between two surfaces with constant values of the stream function. The equations of fluid motion are derived in the approximation of narrow gap. The parameters at which spontaneous swirling is possible are determined numerically, and the properties of secondary swirling flows resulting from instability of the initial steady-state poloidal flow are established. It is shown that for certain parameters of the initial poloidal flow, the energy of the initial flow is almost completely converted to the energy of the azimuthal (rotational) velocity field and magnetic field that arise. In this case, over a wide range of parameters of the initial flow, the time-averaged energies of the rotational motion and magnetic field take identical values for large t. We note that even for a small degree of magnetization, instability can give rise to a considerable magnetic field, which can be treated as the spontaneous occurrence of a magnetic field due to extension of the force lines of the initial weak poloidal magnetic field. The numerical studies of spontaneous swirling revealed some properties of the secondary flow, which is irregular (chaotic) in time and its spatial structure is rather complex with the presence of differential rotation, which is a consequence of the conservation laws for the angular momentum and azimuthal magnetic flux.