Theoretical Studies in Fluid Dynamics and Geophysical Fluid Dynamics

大气和海洋现象的组织性和复杂性极其丰富；它们中的许多不能通过实验产生。这些现象涉及广泛的时间和空间尺度。对这些复杂现象和相关科学问题的理解是当代科学的重大挑战之一，并具有巨大的社会影响。拟研究流体动力学和地球物理流体动力学的性质和理论，重点研究流体流动和地球物理流体流动的结构、形成、鲁棒性/稳定性和过渡。该研究围绕低频气候变化的新策略展开，包括三个相关领域的特定主题：(A)动力分岔理论及其在流体力学中的应用，(B)地球物理流体动力学中的分岔和稳定性问题，以及(C)不可压缩流体流动和地球物理流体流动的几何理论。研究的重点是基础流体动力学问题和地球物理流体动力学问题，包括与厄尔尼诺预测有关的热盐环流和热带环流。研究者将物理建模、严格的数学理论和大规模计算相结合，以产生对物理现象的新见解。所提出的研究一方面涉及应用现有的数学理论来理解潜在的物理问题，另一方面涉及发展新的数学理论。特别值得一提的是，所提出的工作是基于PI最近与田马合作开发的两个新的数学理论。这些理论受到地球物理流体动力学研究的推动，并对其他科学和工程学科有额外的好处。拟议的项目涉及具有重要实际意义的具有挑战性的问题，对地球物理量化可预测性，分析动力学模型中的误差增长以及开发有效的预测方法具有重要的实际意义。这些调查将对改善天气和气候变化的预测有很大的好处，这对我们的经济至关重要。

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.