Titi, DMS-1109640Cao, DMS-1109022 Numerical ocean modeling is computationally very demanding. Traditionally, the hydrostatic approximation balance, in the vertical momentum equation, has been applied to reduce the computational burden. Recently, the investigators have made significant progress in the rigorous understanding of the properties of the solutions to these models. However, it is still impossible to use present computing resources to simulate this large system for the wide range of spatial and temporal scales. Therefore, there is a need to further simplify these models to focus on understanding and simulating certain localized smaller-scale phenomena, such as the geostrophic adjustment of frontal anomalies in a rotating continuously stratified fluid of strictly rectilinear fronts and jets, and rapidly rotating convection which, according to the Taylor-Proudman constraint, takes place in tall columnar structures. The development of fast, accurate, and reliable numerical models of the climate system is key to understanding and predicting the magnitude and distribution of future climate variability. The focus of this project is to provide rigorous justification to these simplified models of the dynamics of the atmosphere and the oceans, and to investigate their long-term behavior. The investigators study the questions of existence, uniqueness, and continuous dependence on the initial data of the solutions to these models. This is the first, and the most crucial step, in justifying the derivation of these models and their consistency with the physical observations. There are numerous governing equations to fully describe the dynamics of climate. It is impossible to use present computing resources to simulate this large system for the wide range of spatial and temporal scales. Indeed, in order to understand, for example, the mechanism of atmospheric and oceanic turbulence, and to properly predict the climate, it is necessary to provide simplified and reliable models that contain the main features of the underlying physical phenomena. The investigators focus on the development of novel, theoretical and computational, mathematical tools for investigating these simplified models, and on justifying their derivation in order to enhance our understanding of fluid dynamics models and the climate system and its predictability. The training of undergraduate and graduate students with analytical and computational skills is also achieved through this study.