Linear Motion of a Shallow-Water, Barotropic Vortex

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Abstract
A barotropic model of tropical cyclone motion follows from calculation of linear wavenumber-1 perturbations on a moving axisymmetric, maintained vortex. The perturbations are Rossby waves that depend upon the radial gradient of axisymmetric relative vorticity. The vortex has normal modes at zero frequency and at the most anticyclonic orbital frequency; the latter mode is barotropically unstable. The structure of the perturbations is calculable for arbitrary motion of the vortex, but one can select the actual motion in a particular situation because that motion minimizes the Lagrangian of the system. Motion of tropical cyclones may arise from environmental currents, convection, or the beta effect. In an environmental current that turns as time passes, the motion is nearly the same as the current, except when the frequency matches a normal mode. The effect of convection is simulated by an imposed, rotating mass source-sink pair, which excites both the normal modes and a perturbation that depends upon forcing at Rossby-wave critical radii. The latter response seems to correspond to the trochoidal motion of real tropical cyclones. It has the fastest vortex motion when its frequency is the same as the orbital frequency of the axisymmetric flow where the forcing is imposed. On a beta plane, the vortex motion is poleward with speed proportional to the total relative angular momentum of the vortex. Because of the normal mode at zero frequency, the poleward motion is much too fast when the vortex has cyclonic circulation throughout. This physically unreasonable result highlights the importance of nonlinear processes in tropical cyclone motion.

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