Linear & Nonlinear Motion of a Barotropic Vortex

It is generally accepted that the beta gyres contribute to observed tropical cyclone propagation (as distinct from advection). In the 1980s and 90s, beta gyre dynamics was the focus of intense modeling and observational efforts. Semispectral, shallow-water barotropic linear and nonlinear models were used to simulate vortex motion on a beta plane. In the linear model, the vortex accelerated toward the NW without limit, ostensibly through the resonant growth of a free linear mode. In the analogous nonlinear model, wave-wave interaction limited the westward and poleward motion to reasonable speeds of 1-2 m/s (consistent with the observed beta drift). A subsequently developed Asymmetric Balance Model (AB) was unable to replicate the linear result. Here, we revisit the problem in a Barotropic Non-Divergent (BND) context, both to resolve the question of the free mode's existence and role and to clarify the nonlinear dynamics.

The linear BND model is time-dependent and uses a rotating beta-like forcing that yields an increasing magnitude and a phase reversal in the neighborhood of a low (period of ~100 days) cyclonic frequency. It was able to correctly reproduce the beta gyres as WN1 streamfunction dipoles of opposite polarity with a northeast-southwest orientation such that the counter-flow between them produces a uniform southeasterly current (i.e., ventilation flow) across the vortex center. This causes the simulated storm to accelerate to the NW linearly. The theory behind this behavior is that the beta gyres appear to be a normal mode that gets resonantly excited by the beta effect. Thus, the results are consistent with the original work and inconsistent with the AB model, where it produced finite linear drift speeds.

The nonlinear version of the BND model accounts for wave-wave interaction such that linear wavenumber-1 interacts with itself to force wavenumber-2 nonlinearly and nonlinear wavenumber-2 interacts with linear wavenumber-1 to force nonlinear wavenumber-1. The most important result from the complete solution is the nonlinearly-forced wavenumber-1 streamfunction gyres have opposite phase to the linearly forced gyres. The ventilation flow from these asymmetries counteracts the uniform current of the linear gyres to reduce the overall vortex speed. Therefore, these anti-beta gyres provide a mechanism for limiting vortex motion in the nonlinear framework.

The mean vortex is bounded at some finite radius where the circulation approaches zero. As a result, there is a reversal of sign for the relative vorticity and radial vorticity gradient, as required by the Circulation Theorem. Thus, the beta gyres are downstream propagating Vortex Rossby Waves (VRWs) that propagate on the reversed peripheral mean vorticity gradient. They are confined in a narrow band of frequencies between zero and the 1-dimensional VRW cutoff frequency. Thus, VRWs have two waveguides: the inner one supports propagation against the cyclonic mean flow while the outer has the waves propagate faster than the mean flow. The latter supports very low frequency waves that correspond to beta-gyre mode described previously.