Hamiltonian moment reduction for describing vortices in shear

Abstract

This paper discusses a general method for approximating two-dimensional and quasigeostrophic three-dimensional fluid flows that are dominated by coherent lumps of vorticity. The method is based upon the noncanonical Hamiltonian structure of the ideal fluid and uses special functionals of the vorticity as dynamical variables. It permits the extraction of exact or approximate finite degree-of-freedom Hamiltonian systems from the partial differential equations that describe vortex dynamics. 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 the Kida vortex [S. Kida, J. Phys. Soc. Jpn. 50, 3517 (1981)] and to the problem of the quasigeostrophic evolution of an ellipsoid of uniform vorticity, embedded in a background flow containing horizontal and vertical shear [Meacham et al., Dyn. Atmos. Oceans 14, 333 (1994)]. The approach provides a simple way of visualizing the structure of the phase space of the Kida problem that allows one to easily classify the types of physical behavior that the vortex may undergo. The dynamics of the ellipsoidal vortex in shear are shown to be Hamiltonian and are represented, without further approximation beyond the assumption of quasigeostrophy, by a finite degree-of-freedom system in canonical variables. The derivation presented here is simpler and more complete than the previous derivation which led to a finite degree-of-freedom system that 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 vorticity equation.