Geophysical flows involve a broad spectrum of interacting spatial and time scales, which makes them inaccessible to the most powerful and state-of-the-art computers. The grand challenge in understanding geophysical flows, and thus climate prediction, is that the mathematical equations governing the ocean and atmosphere dynamics are too difficult to study analytically, and still prohibitively expensive computationally. Development of computational and theoretical tools to further understanding of the climate system and its predictability is the major focus of this project. The first aspect of this project is to derive and justify rigorously simplified models of the atmosphere and the ocean, and to prove existence, uniqueness, and continuous dependence on the initial data for the inviscid primitive equations and the Boussinesq hydrostatic approximation under fast rotation, and two-layer frontal geostrophic models including planetary sphericity and variable topography. The second aspect is to study analytically the nonlinear theory of geostrophic adjustment for a rotating shallow-water model and the two-layer continuously stratified primitive equations, with the goal of answering the fundamental question of the possible splitting of an arbitrary atmospheric or oceanic motion into slow and fast components in such a way that the slow component will not be influenced by the fast one for long times. The approach of the proposal involves a combination of asymptotic analysis, numerical computation, and theoretical mathematical analysis. The project involves development of novel and sophisticated mathematical techniques, which will enhance our understanding of the complex system underlying the Earth's climate.
