Nonlinear Motion of a Shallow Water Barotropic Vortex

Abstract

Nonlinear motions of a shallow water barotropic vortex on a ? plane differ substantially from the analogous linear motions. The nonlinear model described here, in which wavenumber 1?3 asymmetries interact with each other and the mean vortex, predicts that an initially completely cyclonic vortex will accelerate toward the NNW, reaching a speed of 2.5 m s?1 at 48 h. During the rest of the 240-h calculation, the speed varies by <0.5 m s?1 as the direction turns from NNW to NW. The vortex accelerations are in phase with temporal changes of vortex-relative angular momentum (LR). The turning of the track coincides with a transition of the wavenumber 1 asymmetry from a single dipole to a double dipole. The latter structure appears to be another orthogonal solution of the second-order radial structure equation for a neutral linear normal mode. The corresponding linear model, in which ? forces only wavenumber 1, shows only the single dipole structure and straight NNW accelerating motion that reaches a speed of 9 m s?1 at 240 h. The slower motion in the nonlinear model stems from wave-induced changes in the axisymmetric vortex and vacillation between the orthogonal modal structures. A nonlinear calculation with zero initial LR on a ? plane follows a curving path dictated by a barotropically unstable linear mode for the first 144 h. Subsequently, the double dipole structure for that mode appears as the track turns toward the NW and the speed accelerates from 1 to 2 m s?1. A spatially uniform geostrophic environmental current on an f plane causes vortex motion by advection and by propagation. The potential vorticity (PV) gradient due to the current acts much as ? does. Although the PV gradient is typically 0.1 of that due to ?, the induced propagation toward high potential vorticity is ½?¼ of that on a ? plane because super-position of the vortex on the geopotential gradient amplifies the PV gradient's effect. In a quiescent environment on an f plane, initial asymmetries that project onto the normal modes induce long-lasting motion that retains about half its speed to 240 h. If the initial speed is ≥2 m s?1, vacillation between orthogonal modal structures may cause dramatic turns and accelerations of the vortex track.