Contact Transformations and Hamiltonian Dynamics in Generalized Semigeostrophic Theories

Abstract

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 manifestly 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.