Theoretical Studies in Fluid Dynamics and Geophysical Fluid Dynamics

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The phenomena of the atmosphere and ocean are extremely rich in their organization and complexity; many of them cannot be produced by experiments. These phenomena involve a broad range of temporal and spatial scales. The understanding of these complicated phenomena and the relevant scientific issues is one of the grand challenges of contemporary science, and has huge societal impacts. The investigator proposes to study the nature and theory of fluid dynamics and geophysical fluid dynamics, focusing on the structure, its formation, its robustness/stability and transitions of fluid flows and geophysical fluid flows. The study is centered around novel strategies for low-frequency climate variabilities, including in particular specific topics in three interrelated areas: (A) dynamic bifurcation theory and its applications to fluid mechanics,(B) bifurcation and stability issues in geophysical fluid dynamics, and(C) a geometrical theory of incompressible fluid flows and geophysical fluid flows.The study focuses on both fundamental fluid dynamics problems and geophysical fluid dynamics problems including the thermohaline circulation and the tropical circulations associated with the El Nino prediction. The investigator uses a combination of physical modeling, rigorous mathematical theory, and large-scale computing to yield new insights into physical phenomena. The proposed study involves on the one hand applications of existing mathematical theory to the understanding of the underlying physical problems, and on the other hand the development of new mathematical theories. In particular, the proposed work is based on two new mathematical theories which are developed recently by the PI in collaboration with Tian Ma. These theories are motivated by the study in geophysical fluid dynamics, and have additional benefit for other disciplines in science and engineering. The proposed project involves challenging problems with important practical implications for geophysical efforts to quantify predictability, analyze error growth in dynamical models, and develop efficient forecast methods. These investigations will be of great benefit in improving predictions on weather and climate changes, which are of central importance to our economy.