# Some recent results on the two-layer quasi-geostrophic beta plane equations.

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The two-layer beta-plane quasi-geostrophic (QG) model plays a central role in theoretical studies of atmospheric and oceanic dynamics. It is a pair of coupled non-linear partial differential equations involving functions of two space variables and one time variable (streamfunctions for coupled two-dimensional flows). Solutions represent flows in a sense intermediate between 2-d and 3-d flows: they have a mild form of the ``vortex stretching'' process, absent in 2-d flows, that is at the heart of the difficulty in proving the long-time existence of classical solutions to the

3-d Navier-Stokes equations.

Numerical solutions to these QG equations display analogues of important features of atmospheric and oceanic flow, some of which I will illustrate. As is true of climate models, many interesting features are revealed only by long time averaging of the numerical solutions. The results I will present, on long-time existence of regular solutions and on dissipativity,

are part of an effort to provide a rigorous justification for this averaging, something beyond our reach in the case of the vastly more complicated climate models.

The talk will place the model in the context of other QG models, point out a useful formal similarity to the Kuramoto-Sivishinsky equation, and sketch proofs of the main results. The work is joint with C. Foias, C. Onica, E. Titi, and M. Ziane.