Linear Normal Modes of a Moving, Shallow-Water Barotropic Vortex

Abstract

Calculations with a linear semispectral model of a moving tropical-cyclone-like barotropic vortex (Willoughby 1988) show that a vortex with cyclonic circulation throughout exhibits unphysically fast poleward motion on a beta plane, but a vortex with enough anticyclonic circulation at its periphery to make the total relative angular momentum (LR) small moves slowly. The high poleward speed arises because the vortex has a linear normal mode at zero frequency, where the beta effect forces asymmetric perturbations. Advection of planetary vorticity by the axisymmetric circulation forces this normal mode at a rate proportional to LR. Because the governing equations are third-order in time, as many as three wavenumber-one normal modes are possible. A completely cyclonic vortex has three repeated stable normal modes at zero frequency, whereas one with small LR has a single stable mode at zero frequency and a conjugate pair of barotropically unstable modes. The frequency of the unstable modes lies at the most anticyclonic rotation frequency of the axisymmetric circulation, and the growth rate is slow; the e-folding time is typically 75 days. If the fluid is made very shallow, the stable normal mode moves away from zero frequency. In this situation, the beta effect fails to force the resonance, and the vortex propagates westward much as a planetary Rossby wave does. In this model, meridional motion of vortices with LR ≠ 0 always acts to adjust LR toward zero through conservation of absolute angular momentum. Since the asymmetric perturbations are Rossby waves that propagate upon the radial gradient of mean relative vorticity, the mode at zero frequency experiences critical-radius absorption where the mean swirling wind is zero—at the boundary between cyclonic and anticyclonic mean circulation and at the edge of the vortex. Regardless of the sign of LR, the wave momentum convergence is concentrated at these critical radii and weakens the circulation while expanding it spatially. When LR = 0, waves emanating from the cyclonic and anticyclonic circulations interfere destructively, so that the vortex radiates no angular momentum to its environment.