THE RESPONSE OF A ROTATING ELLIPSE OF UNIFORM POTENTIAL VORTICITY TO GRAVITY-WAVE RADIATION

Kirchhoff's incompressible rotating elliptical vortex solution is extended to the case of weak compressibility in the rotating f-plane shallow water equations by means of matched asymptotic expansion, using the small Froude number F as the expansion parameter. The analysis shows that there is a correction to the shape of the rotating configuration at O(F-2), and a gradual elongation of the shape on a time scale F(4)t. When the aspect ratio of the ellipse is 4.6:1, the O(F-2) perturbation to its boundary shape becomes secular, and the vortex exhibits a tendency to pinch in the middle, breaking into two separate vortices. This behavior is consistent with the weakly nonlinear analysis of Williams (Ph.D. thesis, University of Leeds, 1992), and the numerical work of Chan et al. [J. Fluid Mech. 253, 173 (1993)], for the formally equivalent problem in a two-dimensional compressible gas. When the Coriolis parameter is sufficiently large, the elongation of the ellipse may equilibrate before it reaches an aspect ratio of 4.6:1. The nature of the approach of the ellipse to its equilibrium aspect ratio is discussed in these cases, highlighting an asymmetry between cyclonic and anticyclonic vortices.