Linear Motion of a Shallow-Water Barotropic Vortex as an Initial-Value Problem

Abstract

This paper revisits calculation of motion for a shallow-water barotropic vortex with fixed mean axisymmetric structure. The algorithm marches the linear primitive equations for the wavenumber 1 asymmetry forward intime using a vortex motion extrapolated from previous calculations. Periodically, it examines the calculated asymmetry for the apparent asymmetry due to mispositioning of the vortex center, repositions the vortex to remove the apparent asymmetry, and passes the corrected vortex motion on to the next cycle. This approach differs from the author's earlier variational determination of the steady-state motion after initial transients had died away. The steady-state approach demonstrated that the vortex had normal modes at zero frequency and, when an annulus of weak anticyclonic flow encircled the cyclonic inner vortex, at the most anticyclonic rotation frequency of the mean flow. Forcing of the former model led to too rapid steady-state poleward motion on a beta plane. At least for the linear problem, the key to more realistic simulation of motion and structure is the normal modes' transient response to diverse forcing: environmental potential vorticity gradients, embedded sources and sinks of mass, and initial asymmetries. The beta effect and other environmental potential vorticity gradients excite the normal modes to induce an acceleration of the vortex center toward and to the left of the direction to maximum environmental vorticity. Times ~ 100 days would be required to reach the too fast motions predicted in the earlier work. A rotating mass source-sink pair drives the vortex along a cycloidal track, but does not force the normal modes. A nonrotating source-sink forces a motion from the source toward the sinkand excites the normal modes, leading to motion that persists after the forcing has ceased. Similarly, initial asymmetries that project onto the normal modes maintain themselves for times ≥ 10 days, leading to persistent vortex propagation that evolves as the complex normal-mode frequencies dictate. Understanding of these normal modes can contribute to better tropical cyclone motion forecasts through better initialization of numerical track prediction models.