Exact semi-geostrophic flows in an elliptical ocean basin*

A new family of exact solutions is analysed, which models two-dimensional circulations of an ideal fluid in a uniformly rotating elliptical tank, under the semi-geostrophic approximation from meteorology and oceanography. The fluid pressure and stream function remain quadratic functions of space at each instant in time, and their fluctuations are described by a single degree of freedom Hamiltonian system depending on two conserved parameters: domain eccentricity and the constant value of potential vorticity. These parameters determine the presence or absence of periodic orbits with arbitrarily long periods, fixed points of the dynamics, and aperiodic homoclinic orbits linking hyperbolic saddle points. The energy relative to these parameters selects the frequency and direction in which isobars nutate or precess, as well as the steady circulation direction of the fluctuating flow. The canonically conjugate variables are the moment of inertia and angle of inclination of an elliptical inverse-potential-vorticity patch evolving in dual coordinates. Mathematics Subject Classification: 35J60, 76B60, 76U05, 86A10 (Some figures in this article are in colour only in the electronic version)