A compact rotating shallow mass of fluid with a free boundary is released on a horizontal plane. Initially it is supposed that horizontal sections through the mass are elliptical, vertical sections are parabolic and the two velocities are linear functions of the horizontal coordinates. Thus four numbers suffice to describe the configuration of the fluid and another four the velocity field. The subsequent motion preserves this simple structure, and so the shallow-water equations reduce to ordinary differential equations in time for the eight parameters required to specify the initial condition.

This eighth-order system can be reduced to quadratures using the well-known invariants of the shallow-water equations. There are five such integrals: volume, energy, enstrophy, angular momentum, and a fifth, which lacks a familiar name. The remaining three degrees of freedom can be related to the shape (i.e. eccentricity), horizontal size (i.e. radius of gyration) and orientation of the mass. In general, all of these are periodic functions of time but with different characteristic timescales. Simplest are size changes, which occur at the inertial frequency f. Shape changes are superinertial, while orientation changes may be subinertial or superinertial.

This solution is used to discuss the unsteady motion of a non-axisymmetric Gulf Stream ring. We argue that size and shape changes excite internal gravity waves in the underlying fluid, while orientation changes generate Rossby waves. While this wave radiation decreases the energy of the ring, and may alter the angular momentum, it cannot lead to a state of no motion because the volume and enstrophy are unaffected.