Theory and simulations of two-dimensional vortex motion driven by a background vorticity gradient

Abstract

This paper examines two-dimensional vortex motion in a shear-flow with nonuniform vorticity. Typically, a vortex travels to an extremum in the background vorticity distribution. In general, the rate of this migration increases with the magnitude of the background vorticity gradient; however, a retrograde vortex, which rotates against the local shear, moves orders of magnitude faster than a prograde vortex of equal strength. Retrograde and prograde vortices travel at different speeds because they perturb the background vorticity differently. Linearized equations accurately describe the background vorticity perturbation that is created by a weak retrograde vortex, whereas nonlinear effects dominate for a prograde vortex of any strength. An analytic theory is developed for the velocity of a retrograde vortex, based on the linearized equations. The velocity of a prograde vortex is obtained by a simple “mix-and-move” estimate, which takes into account the nonlinear trapping of fluid around the vortex. Both velocity predictions are verified by vortex-in-cell simulations. If the ratio of background shear to background vorticity gradient exceeds a critical level, there is no vortex motion up or down the background vorticity gradient. Estimates of the critical shear are obtained for both prograde and retrograde vortices. These estimates compare favorably to vortex-in-cell simulations.