Collaborative Research: Analytical Study of Certain Turbulence and Large--scale Geophysical Models

即使是最强大的最先进的计算机，对控制气候动态的精确方程进行可靠的长时间数值模拟仍然遥不可及。这是因为这些模型的动态涉及广泛的空间和时间尺度。利用大气层和海洋的浅度以及地球的自转，气候学家引入了更简化的模型--所谓的原始方程及其方差--用于全球气候模型的模拟。本项目将严格证明其中一些简化模型，并证明其解的存在性、唯一性和对初始数据的连续依赖性;并研究这些模型的长期行为。这是证明这些模型的推导及其与相关空间和时间尺度的物理观测一致性的第一步，也是最关键的一步。该项目涉及开发新颖和复杂的数学技术，用于分析研究这些模型和新的海洋湍流次网格尺度模型。后者被提议作为海洋环流动力学的新的参数化分析模式。此外，我们建议研究一类新的海洋和大气动力学的无粘正则化方案/模型的定性和统计近似行为，并计算实现它们。这些新格式具有全局（时间）正则性，其优点是不需要附加边界条件，而标准的超粘性正则化常用于地球物理计算中以抑制数值的、非物理的、小尺度的不稳定性。以及对海洋和大气动力学模型的复杂数学分析，我们只能预测几天的天气。即使是最强大的最先进的计算机，对控制气候动态的精确方程进行可靠的长时间数值模拟仍然遥不可及。地球物理学家已经引入了更简化的气候动力学模型，以便通过计算实现。因此，必须严格证明这些模型的有效性。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将是该项目的主要重点。该项目的方法涉及数学，数值和统计工具的组合。

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.