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

This paper examines two-dimensional vortex motion in a shear-flow with nonu niform vorticity. Typically, a vortex travels to an extremum in the backgro und vorticity distribution. In general, the rate of this migration increase s with the magnitude of the background vorticity gradient; however, a retro grade vortex, which rotates against the local shear, moves orders of magnit ude faster than a prograde vortex of equal strength. Retrograde and prograd e vortices travel at different speeds because they perturb the background v orticity differently. Linearized equations accurately describe the backgrou nd vorticity perturbation that is created by a weak retrograde vortex, wher eas nonlinear effects dominate for a prograde vortex of any strength. An an alytic theory is developed for the velocity of a retrograde vortex, based o n the linearized equations. The velocity of a prograde vortex is obtained b y 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 backgro und vorticity gradient exceeds a critical level, there is no vortex motion up or down the background vorticity gradient. Estimates of the critical she ar are obtained for both prograde and retrograde vortices. These estimates compare favorably to vortex-in-cell simulations. (C) 2001 American Institut e of Physics.