ABSTRACT

The global climate involves fluid motions that occur over a huge range of interacting length and time scales. The multiscale aspect of the challenge of modeling the global climate summons a unified approach that should have the capability to address a sequence of nested subproblems in fluid dynamics. The approach should be based on fundamental principles and it should have the capability to incorporate physical processes at many different scales. The Euler-Poincaré theorem provides the frameworkfor such an approach. After introducing the global climate problem from the viewpoint of modeling global ocean circulation, we review the Euler-Poincaré theorem and apply it to address a sequence of modeling challenges that ranges from balance equations for geophysical fluid dynamics, to large eddy simulation models for three-dimensional turbulence, to Hamiltonian dynamics of solitons.

The problem of ocean circulation & global climate

Figure 1.1 will help us focus our minds on a serious problem — the problem of the role of ocean circulation in climate modeling and global warming. The ocean and the atmosphere transport heat from the Earth's equator to the poles at about equal rates. This coupled ocean-atmosphere interaction is the basis of the global climate problem. This problem is timely. Recently the “temperate zones” have been experiencing extreme weather (floods in Europe, droughts in America) and the issue of global warming is often in the news.