On the Stability of Vortex Motions in the Presence of Magnetic Fields.

摘要

Stability analyses are performed on a general type of vortex flows with density inhomogeneity and with both the axial and azimuthal magnetic fields. A sufficient condition for stability, subject to arbitrary perturbations, is derived. The condition, however, involves the complex eigenfrequency. Nevertheless, we are able to examine how the flow profile influences the flow characteristics by studying three general types of perturbations. Analytical solutions are obtained to support the resultant arguments. Unlike the axial velocity which always destabilizes the flow, the azimuthal velocity can convey either stabilizing or destabilizing effects depending on the gradient it possesses and the centrifugal force field it creates. This characteristic is shown in all the three general types of perturbations to the flow. For flow subject to azimuthal perturbations, an upper bound on the complex phase velocity, which is reminiscent of the semi-ellipse theorem in the non-hydromagnetic case, is found for some flow profiles satisfying a constraint. Even though it cannot be observed directly from the sufficiency conditions or from the bound on the unstable waves, we are able to show that the presence of the magnetic field, regardless of the detailed profile, always stabilizes azimuthal perturbations. Such an argument is also supported by exact solutions to the governing stability equations. For uniform rotation and constant Alfven velocities, azimuthal instabilities can only occur when negative density gradients exist within the flow domain. Furthermore, the phase velocity for such instabilities must lie on a semi-circle in the complex phase velocity plane independent of the detailed density profile.