Hamiltonian structure with contact geometry for the semi-geostrophic equations

A canonical Hamiltonian structure for the semi-geostrophic equations is presented and from this a reduced non-canonical Hamiltonian structure is derived, providing a fully nonlinear version of the approximate linearized vorticity advection representation. The structure of this model is described naturally within the framework of contact geometry. A Hamiltonian approach leading to a symplectic algorithm for calculating solutions to the equations of motion is formulated. Basic necessary functional methods are introduced and the Lagrangian and Eulerian kinematic structures are discussed, together with their relevance to symplectic integrating algorithms.