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The slow manifold and the persisting gravity waves

The slow manifold consists of states which are defined to be free of high-frequency gravity waves. Those solutions of primitive equations which consist of slow synoptic-scale motion with the complete absence of gravity waves must be contained in the slow manifold. The low-order, nonlinear primitive equation (PE) model of Lorenz is studied to determine whether its solutions are embedded in a stable invariant slow manifold. The PE model consists of nine dependent variables and is derived from shallow water equations with bottom topography, forcing, and dissipation. The forcing in the model can be interpreted to be the Rossby number. An arbitrary trajectory of the model eventually approaches points of a bounded set called the attractor. The points on the three-dimensional slow manifold are determined by an algorithm based on complete separation of quasi-geostrophic (slow) and gravity waves frequencies. A low-order quasi-geostrophic (QG) model and a low-order balance equation (BE) model derived from the PE model are also studied. The QG and BE models consist of three equations which are obtained by replacing the divergence equations with the diagnostic geostrophic relation, or the classical balance equation, respectively. All models have been studied as a function of forcing, or equivalently the Rossby number, for different values of topography. For the chosen values of the dissipation parameters, the attractors of the models are steady states for very weak forcing. As the forcing is increased, the attractors are found to be alternately chaotic and periodic. When the forcing is weak or slightly moderate, the attractors of the PE model consist of slow motion, varying on the order of a few days, with no apparent sign of the presence of gravity waves. The attractors of all the models are qualitatively similar in this range of forcing. The BE model reproduces the behavior of the PE model in a remarkably close manner for weak and slightly moderate forcing. The QG model, however, is not found to be a good approximation. When the forcing is strong, the attractors of the PE model consist of persisting gravity waves, varying on a time scale of about six hours, 1__1__ CII~III^I~Llli~CLLII-L_ -. )-l-i lX^I~(~-ll~EIIY-n