It is known that all two-dimensional, inviscid vortices are exponentially stable to disturbances of azimuthal wavenumber one. However, an exact linear solution for wavenumber one perturbations was found by Smith and Rosenbluth (1990) which demonstrates a persistent O(t1/2) growth in streamfunction amplitude and O(t) growth in energy, provided certain conditions on the mean azimuthal velocity and the initial conditions are met. The necessary and sufficient requirement for secular growth in this solution is that there exist at least one local maximum in the angular velocity of the mean flow other than at the axis.
In this talk we examine the role vortex-Rossby waves play in supporting this secular growth of the disturbances. The local maximum in the angular velocity of these vortices results in a localized region of diminished shear damping. We show how vortex-Rossby waves accumulate in this region, resulting in a constant growth in disturbance energy for long times. An example of such growth is shown in the Figure below. The initial peak is due to a short period of transient growth, and the oscillations are due to periodic phasing of vorticity anomalies at different radii.
The results raise some interesting possibilities for the barotropic dynamics of hurricanes and other geophysical vortices. If realized, the growth in amplitude of the wavenumber one oscillations in the vortex core (observed as a wobble in the hurricane track) would eventually lead to the appearance of secondary instabilities at smaller scales. These secondary instabilities could be associated with mesocyclones in or near the eyewall and would generally enhance the mixing processes in the core of the storm.
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12th Conference on Atmospheric and Oceanic Fluid Dynamics