Reliable long time numerical simulations of the exact equations that govern the climate dynamics are still out of reach, even for the most powerful state-of-the-art computers. This is because of the wide range of spatial and time scales involved in the dynamics of these models. Taking advantage of the shallowness of the atmosphere and the oceans, as well as the rotation of the earth, geophysicists have introduced more simplified models - the so-called primitive equations and their variance - to be implemented in the simulations of global climate models. This project is to justify rigorously some of these simplified models, and to prove existence, uniqueness and continuous dependence on the initial data of their solutions; and to investigate the long-term behavior for these models. This is the first and the most crucial step in justifying the derivation of these models and their consistency with physical observations for the relevant spatial and time scales. This project involves the development of novel and sophisticated mathematical techniques for the analytical study of these models and of new sub-grid scale models of oceanic turbulent flows. The latter are being proposed as new analytical parametrization models for ocean circulation dynamics. Furthermore, we propose to investigate the qualitative and statistical approximation behavior of a new class of inviscid regularizing schemes/models for oceanic and atmospheric dynamics, and to implement them computationally. The advantage of these new schemes, which are globally (in time) regular, is that they do not require addition boundary conditions, unlike the standard hyper-viscous regularization, which is commonly used in geophysical computations to suppress the numerical, non-physical, small scale instabilities.Due to the developments in modern technology, the ever increasing computational power, and the sophisticated mathematical analysis of the ocean and atmosphere dynamics models we are able predict the weather only for a few days. Reliable long time numerical simulations of the exact equations that govern the climate dynamics are still out of reach, even for the most powerful state-of-the-art computers. Geophysicists have introduced more simplified models for climate dynamics to be implemented computationally. It is therefore essential to justify, rigorously, the validity of these models. Development of computational and theoretical tools to further understanding of the climate system and its predictability will be the major focus of this project. The approach of this project involves a combination of mathematical, numerical and statistical tools.
