Attractor Sets and Quasi-Geostrophic Equilibrium

Abstract

The attractor set of a forced dissipative dynamical system is for practical purposes the set of points in phase-space which continue to be encountered by an arbitrary orbit after an arbitrary long time. For a reasonably realistic atmospheric model the attractor should be a bounded set, and most of its points should represent states of approximate geostrophic equilibrium. A low-order primitive-equation (PE) model consisting of nine ordinary differential equations is derived from the shallow-water equations with bottom topography. A low-order quasi-geostrophic (QG) model with three equations is derived from the PE model by dropping the time derivatives in the divergence equations. For the chosen parameter values, gravity waves which are initially present in the PE model nearly disappear after a few weeks, while the quasi-geostrophic oscillations continue undiminished. The states which are free of gravity waves form a three-dimensional stable invariant manifold within the nine-dimensional phase space. Points on this manifold are readily found by an algorithm based on the separation of time scales. The attractor set consists of a complex of two-dimensional surfaces embedded in this manifold. The geostrophic equation is a good approximation on most of the attractor, while the balance equation is better. The attractors of the PE and QG models are qualitatively similar. Some speculations regarding the invariant manifold and the attractor in a large global circulation model are offered.