Comparisons of low-order atmospheric dynamic systems

A. WIIN NIELSEN

Abstract

A low-order, quasi-geostrophic, two-level model is used to investigate the response to external heating. The heating has variations in both south-north and west-east directions. The frictional dissipation is incorporated by using both boundary layer and internal friction. The zonal flow is described by two dependent variables, one for the vertical mean flow and the other for the vertical shear flow. The remaining four dependent variables in the model are the amplitudes of the sine- and cosine-components of a travelling wave in the vertical mean flow and in the vertical shear flow. The low-order model is stable in the sense that trajectories starting outside a certain circle will cross the circle and approach the origin of the six-dimensional space. The model has also the property that the rate of change of a small volume is negative indicating that the small volume will shrink to zero. Any attractor, which may exist, is thus of zero volume. A detailed study of the multiple steady states of the model and their stability is postponed to a later publication. In this study we rely on a number of long-term numerical integrations, which show that the model approaches either a stable, steady state or a periodical time-dependent solution. It appears therefore'that the model does not contain chaotic solutions. Comparisons are made with the three parameter model recently published by Lorenz and with the models developed by Saltzman et al. This Lorenz model, which contain chaotic solutions for sufficiently large south-north external forcing, can be obtained as a special case of the six parameter model. The different behavior of the two models may be explained by the assumptions, which are necessary to obtain the simpler model from the other. It is shown that the Lorenz-model describes the thermal flow of the two-level model, and that the phase difference between the thermal and the mean model flow waves always is a quarter of the wavelength assuring that the south- north transport of sensible heat is at a maximum for given amplitudes. It is also pointed out that large values of the external heating are necessary to obtain chaos in the Lorenz-model. These results are also found in the Saltzman-model, which is a generalization of the Lorenz-model, although both have three dependent variables only. It appears therefore that to explain the inter-annual variations of the atmosphere in terms of chaotic behavior in the cold season and non-chaotic behavior in the warm season will require further investigations using models, which can simulate the cascade processes in the real atmosphere.

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