## Abstract

This work extends asymmetric balance (AB) theory to the beta plane (*β*-AB). The physical problem examined is the motion of a coherent vortex on a beta plane in a finite depth fluid in the absence of an environmental steering flow. A useful attribute of the *β*-AB formulation is that it allows one to separate the linear and nonlinear balance contributions to the vortex motion when the standard Rossby number is not small compared to unity. It is therefore well suited for testing the hurricane-motion paradigm proposed by Willoughby for equivalent barotropic dynamics.

The *β*-AB model is formulated first for linear shallow water dynamics on a circular vortex forced by the meridional gradient of planetary vorticity (the beta effect) in an earth-based coordinate system. The linear dynamics precludes wave–wave and wave–mean flow interactions. From incipient vortices to hurricanes, the *β*-AB model correctly develops the wavenumber-one asymmetries (the “beta” gyres) necessary for vortex self-advection. Cyclonic vortices move in a northwestward direction consistent with their relative strengths. In contrast to Willoughby’s predictions of a persistent acceleration in the linear problem, numerical simulations with the linear *β*-AB model suggest that finite drift speeds are always attained in a finite depth fluid. The present findings extend the theoretical predictions of a finite linear drift speed for stable quasigeostrophic vortices by Sutyrin and Flierl to the case of stable vortices in gradient wind balance. No evidence of a translating normal mode of zero frequency (other than the pseudo mode) is found when the beta forcing is switched off.

Nonlinear dynamics are considered next by adding the nonlinear quasigeostrophic terms to the linear balance system so that wave–wave and wave–mean flow interactions are included. Consistent with other works, asymptotic drift speeds are reduced from their linear values. Vortices develop an anticyclonic circulation in the vortex periphery and shed a Rossby wave wake in their environment. The sensitivity of vortex tracks with respect to azimuthal-wavenumber truncation is also investigated for the purposes of determining the minimal number of azimuthal modes required to make accurate long-time motion forecasts. While total relative angular momentum and eddy potential vorticity fluxes are known to be useful aids for interpreting changes in the vortex structure and intensity, the authors show in contrast to Willoughby that they give little insight into the behavior of vortex tracks at long times.

*Corresponding author address:* Dr. Michael T. Montgomery, Department of Atmospheric Science, Colorado State University, Fort Collins, CO 80523.