Mathematical Study of Certain Geophysical Models

The basic problem faced in geophysical fluid dynamics is that a mathematical description based only on fundamental physical principles, which are called the ``Primitive Equations'', is often prohibitively expensive computationally, and hard to study analytically. In this talk I will present a formal derivation of more manageable shallow water approximate models for the three dimesional Euler equations in a basin with slowly spatially varying topography, the so called ``Lake Equation" and ``Great Lake Equation", which should represent the behavior of the physical system on time and length scales of interest. These approximate models will be shown to be globally well-possed. I will also show that the Charney-Stommel model of the gulf-stream, which is a two dimensional damped driven shallow water model for ocean circulation, has a global attractor. Whether this attractor is finite or infinite dimensional is still an open question. Other results concerning the global well-posedess of three dimensional viscous planetary geostrophic models will be presented.