Mathematical Problems in Geophysical Dynamics

地球物理流动涉及广泛的相互作用的空间和时间尺度，这使得它们无法进入最强大和最先进的计算机。 理解地球物理流动以及气候预测的巨大挑战在于，控制海洋和大气动力学的数学方程太难分析研究，而且计算成本仍然高得令人望而却步。 该项目的主要重点是开发计算和理论工具，以进一步了解气候系统及其可预测性。 本项目的第一个方面是推导和证明严格简化的大气和海洋模式，并证明存在性，唯一性和连续依赖于初始数据的无粘原始方程和快速旋转下的Boussinesq流体静力近似，以及两层锋面地转模式，包括行星球度和可变地形。 第二方面是对旋转浅水模式和两层连续分层原始方程组的地转调整的非线性理论进行解析研究，目的是回答任意大气或海洋运动可能分裂成慢分量和快分量，使慢分量长时间不受快分量影响的基本问题。 该建议的方法包括渐近分析，数值计算和理论数学分析的组合。 该项目涉及开发新颖和复杂的数学技术，这将提高我们对地球气候背后复杂系统的理解。

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.