Contact Transformations and Hamiltonian Dynamics in Generalized Semigeostrophic Theories

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This paper is concerned with the fundamental role played by contact transformations and their corresponding generating functions in determining the structure and dynamical properties of a very general class of semigeostrophic theories possessing a Hamiltonian of the kind discovered by Salmon. It is shown that each member of this class of theories is associated with a self-adjoint tendency equation. To illustrate the utility, of the contact transformation concept, a new vortex theory is constructed by imposing upon the generating function a radial scaling symmetry, equivalent dynamically to the very reasonable constraint that small balanced perturbations about any state of solid-body rotation are simulated accurately. The new theory is a consistent generalization of the existing axisymmetric-vortex form of semigeostrophic dynamics and preserves the important conservation laws of mass, energy, potential vorticity, and angular momentum. In a sensitive idealized test, the new theory is shown to give reasonably accurate simulations of barotropic instability. 36 refs., 5 figs., 2 tabs.