INTENSE VORTEX MOTION ON THE BETA-PLANE - DEVELOPMENT OF THE BETA-GYRES

An analytical theory is presented for the self-induced translation of an intense vortex relative to a uniform background flow on the beta pl ane. The equivalent barotropic approximation is used to formulate the initial value problem within a polar coordinate frame translating with the vortex center. A contour dynamical model of the vortex is melded with the regular beta-plane model of the residual flow. Evolution of v ortex asymmetries for azimuthal mode number one, the so-called beta gy res, which are responsible for the relative vortex motion, is consider ed for a period of time while the Rossby wave radiation is not importa nt. It is shown for an initially axisymmetric vortex that the beta gyr es and corresponding vortex translational velocity consist of two part s. The first one is generated by advection of the background potential vorticity gradient and rotates differentially because of the symmetri c vortex circulation. The second part arises due to distortion in the vortex shape represented by displacements of the piecewise constant po tential vorticity contours relative to the vortex center. The distorti on of the vortex shape is described by the sum of normal modes generat ed by the first part. Explicit solutions for both parts are obtained, and approximate expressions for different stages of the vortex motion are presented. For a vortex with a uniform potential vorticity core (s ingle contour), the beta gyres are found to consist only of the first part so that the vortex translation depends on the ratio of the core s ize to the radius of deformation. A small core corresponds to the geos trophic point vortex limit with initially predominantly meridional mot ion. Asymptotically, after a large number of fluid revolutions at a ra dial distance on the order of the radius of deformation, the westward translation dominates: the meridional velocity and the deviation of zo nal velocity from the maximum linear Rossby wave speed decay linearly with time. This tendency is explained to be a result of effective symm etrization of the potential vorticity due to differential rotation of fluid around the vortex. The period of initial predominantly meridiona l motion is negligible when the core size is on the order of the defor mation radius. For the vortex with two steps in the potential vorticit y, the normal mode rotates faster than the fluid if the potential vort icities in the core and at the periphery have different signs. The eff ect of the distortion in the vortex shape on the vortex translation in creases with increasing deformation radius relative to the vortex size . In a stationary beta gyre, for a finite vortex, the relative contour shift contributes to the westward translation just up to the long Ros sby wave speed. In the nondivergent limit a universal approximate traj ectory has been found for large outer contour radius. The center of a finite vortex moves northwestward with permanent meridional accelerati on due to degeneracy of a zero-frequency normal mode. The zonal transl ational velocity approaches a limit proportional to the vortex area. T he effect of the distortion in the vortex shape in this nondivergent l imit results in decreasing the westward translation and increasing the meridional one. Applications of the theory to hurricanes in the atmos phere and rings in the ocean are discussed.