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## Abstract

An analytical theory is presented for the self-induced translation of an intense vortex relative to a uniform background flow on the *β* plane. 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 vortex asymmetries for azimuthal mode number one, the so-called beta gyres, which are responsible for the relative vortex motion, is considered for a period of time while the Rossby wave radiation is not important.

It is shown for an initially axisymmetric vortex that the beta gyres and corresponding vortex translational velocity consist of two parts. The first one is generated by advection of the background potential vorticity gradient and rotates differentially because of the symmetric vortex circulation. The second part arises due to distortion in the vortex shape represented by displacements of the piecewise constant potential vorticity contours relative to the vortex center. The distortion of the vortex shape is described by the sum of normal modes generated 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 (single 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 size to the radius of deformation. A small core corresponds to the geostrophic point vortex limit with initially predominantly meridional motion. Asymptotically, after a large number of fluid revolutions at a radial distance on the order of the radius of deformation, the westward translation dominates: the meridional velocity and the deviation of zonal velocity from the maximum linear Rossby wave speed decay linearly with time. This tendency is explained to be a result of effective symmetrization of the potential vorticity due to differential rotation of fluid around the vortex. The period of initial predominantly meridional motion is negligible when the core size is on the order of the deformation radius.

For the vortex with two steps in the potential vorticity, the normal mode rotates faster than the fluid if the potential vorticities in the core and at the periphery have different signs. The effect of the distortion in the vortex shape on the vortex translation increases 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 Rossby wave speed.

In the nondivergent limit a universal approximate trajectory has been found for large outer contour radius. The center of a finite vortex moves northwestward with permanent meridional acceleration due to degeneracy of a zero-frequency normal mode. The zonal translational velocity approaches a limit proportional to the vortex area. The effect of the distortion in the vortex shape in this nondivergent limit results in decreasing the westward translation and increasing the meridional one.

Applications of the theory to hurricanes in the atmosphere and rings in the ocean are discussed.

## Abstract

An analytical theory is presented for the self-induced translation of an intense vortex relative to a uniform background flow on the *β* plane. 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 vortex asymmetries for azimuthal mode number one, the so-called beta gyres, which are responsible for the relative vortex motion, is considered for a period of time while the Rossby wave radiation is not important.

It is shown for an initially axisymmetric vortex that the beta gyres and corresponding vortex translational velocity consist of two parts. The first one is generated by advection of the background potential vorticity gradient and rotates differentially because of the symmetric vortex circulation. The second part arises due to distortion in the vortex shape represented by displacements of the piecewise constant potential vorticity contours relative to the vortex center. The distortion of the vortex shape is described by the sum of normal modes generated 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 (single 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 size to the radius of deformation. A small core corresponds to the geostrophic point vortex limit with initially predominantly meridional motion. Asymptotically, after a large number of fluid revolutions at a radial distance on the order of the radius of deformation, the westward translation dominates: the meridional velocity and the deviation of zonal velocity from the maximum linear Rossby wave speed decay linearly with time. This tendency is explained to be a result of effective symmetrization of the potential vorticity due to differential rotation of fluid around the vortex. The period of initial predominantly meridional motion is negligible when the core size is on the order of the deformation radius.

For the vortex with two steps in the potential vorticity, the normal mode rotates faster than the fluid if the potential vorticities in the core and at the periphery have different signs. The effect of the distortion in the vortex shape on the vortex translation increases 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 Rossby wave speed.

In the nondivergent limit a universal approximate trajectory has been found for large outer contour radius. The center of a finite vortex moves northwestward with permanent meridional acceleration due to degeneracy of a zero-frequency normal mode. The zonal translational velocity approaches a limit proportional to the vortex area. The effect of the distortion in the vortex shape in this nondivergent limit results in decreasing the westward translation and increasing the meridional one.

Applications of the theory to hurricanes in the atmosphere and rings in the ocean are discussed.