On the stability of elliptical vortex solutions of the shallow-water equations

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Abstract
The one-layer reduced gravity (or ‘shallow water’) equations in the f-plane have solutions such that the active layer is horizontally bounded by an ellipse that rotates steadily. In a frame where the height contours are stationary, fluid particles move along similar ellipses with the same revolution period. Both motions (translation along an elliptical path and precession of that orbit) are anticyclonic and their frequencies are not independent; a Rossby number (R0) based on the combination of both of them is bounded by unity. These solutions may be taken, with some optimism, as a model of ocean warm eddies; their stability is studied here for all values of R0 and of the ellipse eccentricity (these two parameters determine uniquely the properties of the solution).Sufficient stability conditions are derived from the integrals of motion; f-plane flows that satisfy them must be either axisymmetric or parallel. For the model vortex, the circular case simply corresponds to a solid-body rotation, and is found to be stable to finite-amplitude perturbations for all values of R0. This includes R0 > ½, which implies an anticyclonic absolute vorticity.The stability of the truly elliptical cases are studied in the normal modes sense. The height perturbation is an n-order polynomial of the horizontal coordinates; the cases for 0 ≤ n ≤ 6 are analysed, for all possible values of the Rossby number and of the eccentricity. All eddies are stable to perturbations with n ≤ 2. (A property of the shallow-water equations, probably related to the last result, is that a general finite-amplitude n-order field is an exact nonlinear solution for n ≤ 2.) Many vortices - noticeably the more eccentric ones - are unstable to perturbations with n ≥ 3; growth rates are O(R02f) where f is the Coriolis parameter.