HAMILTONIAN MOMENT REDUCTION FOR DESCRIBING VORTICES IN SHEAR

This paper discusses a general method for approximating two-dimensiona l and quasigeostrophic three-dimensional fluid flows that are dominate d by coherent lumps of vorticity. The method is based upon the noncano nical Hamiltonian structure of the ideal fluid and uses special functi onals of the vorticity as dynamical variables. It permits the extracti on of exact or approximate finite degree-of-freedom Hamiltonian system s from the partial differential equations that describe vortex dynamic s. We give examples in which the functionals are chosen to be spatial moments of the vorticity. The method gives rise to constants of motion known as Casimir invariants and provides a classification scheme for the global phase space structure of the reduced finite systems, based upon Lie algebra theory. The method is illustrated by application to t he Kida vortex [5. Kida, J. Phys, Sec. Jpn. 50, 3517 (1981)] and to th e problem of the quasigeostrophic evolution of an ellipsoid of uniform vorticity, embedded in a background flow containing horizontal and ve rtical shear [Meacham et al., Dyn. Atmos. Oceans 14, 333 (1994)]. The approach provides a simple way of visualizing the structure of the pha se space of the Kida problem that allows one to easily classify the ty pes of physical behavior that the vortex may undergo. The dynamics of the ellipsoidal vortex in shear are shown to be Hamiltonian and are re presented, without further approximation beyond the assumption of quas igeostrophy, by a finite degree-of-freedom system in canonical variabl es. The derivation presented here is simpler and more complete than th e previous derivation which led to a finite degree-of-freedom system t hat governs the semi-axes and orientation of the ellipsoid. Using the reduced Hamiltonian description, it is shown that one of the possible modes of evolution of the ellipsoidal vortex is chaotic. These chaotic solutions are noteworthy in that they are exact chaotic solutions of a continuum fluid governing equation, the quasigeostrophic potential v orticity equation. (C) 1997 American Institute of Physics.