The Nonlinear Quasi-Geostrophic Equation. Part II: Predictability, Recurrence and Limit Properties of Thermally-Forced and Unforced Flows

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Abstract
Properties of nonlinear quasi-geostrophic flow in unforced and in thermally-forced, dissipative modes are compared. The article is based on the philosophy that precise versions of the important problems of predictability and the theory of climate can be studied analytically with the quasi-geostrophic equation, regardless of whatever deficiencies it may have in representing atmospheric motion. The main result and contrast is that the trajectories of unforced flow are almost always recurrent in their spectral representation, returning infinitely often to a neighborhood of their initial points, and that all trajectories of forced, dissipative flow proceed eventually to the same limit set of measure zero in phase space. The basic quasi-geostrophic model is extended in a number of ways. First, a global model is developed by making appropriate sign changes for the Southern Hemisphere, which produces a satisfactory spectral model even though discontinuities may appear at the equator. The efficacy of the model is illustrated with a generalization of the Rossby wave theory that gives a latitudinally variable wave speed depending on a latitudinally variable basic velocity. Second, the boundary condition on the upper and lower surfaces is generalized considerably, so that temporally varying patterns of potential temperature perturbations and associated wind shears can be included at the boundaries. The changes at the boundaries are internally controlled or forced by the imposed heating field. The statistical properties of the usual unforced quasi-geostrophic flow are considered in phase space, and it is shown that the Poincaré recurrence theorem applies and that long-term averages along the trajectories exist even though the flows are not ergodic. The thermally-forced model is developed by adding a Newtonian heating term to the First Law and by adding a dissipative term to the vorticity equation. In this model every initial set is mapped into a set of vanishing measure as t → ∞. Moreover, it is shown that all trajectories are eventually trapped in a region of phase space specified by the rates of heating and dissipation. The limit properties of the trajectories are examined and it is shown that each has at least one limit point, so that all trajectories not asymptotic to a stationary point are repetitive. However, it is also shown that cycles can occur only in the limit set of measure zero. Some errors in Part I are corrected in an Appendix.

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