Collaborative Research: Mathematical Analysis of Certain Geophysical and Fluid Dynamics Models

DMS-1109022海洋数值模拟在计算上要求很高。传统上，在垂直动量方程中采用静力近似平衡来减少计算负担。最近，研究人员在严格理解这些模型解的性质方面取得了重大进展。然而，利用现有的计算资源来模拟大范围的空间和时间尺度的大系统仍然是不可能的。因此，有必要进一步简化这些模型，以侧重于理解和模拟某些局部的小尺度现象，例如具有严格直线锋面和喷流的旋转连续分层流体中锋面异常的地转调整，以及根据Taylor-Proudman约束发生在高柱状结构中的快速旋转对流。发展快速、准确和可靠的气候系统数值模式是理解和预测未来气候变率的大小和分布的关键。该项目的重点是为这些简化的大气和海洋动力学模型提供严格的理由，并调查它们的长期行为。研究人员研究这些模型解的存在性、唯一性和对初始数据的连续依赖性问题。这是证明这些模型的推导及其与物理观测的一致性的第一步，也是最关键的一步。有许多控制方程可以完全描述气候的动态。利用现有的计算资源来模拟大范围的空间和时间尺度的大系统是不可能的。事实上，为了理解例如大气和海洋湍流的机制，并适当地预测气候，有必要提供包含基本物理现象的主要特征的简化和可靠的模型。研究人员专注于开发新的、理论和计算的数学工具来研究这些简化的模型，并证明它们的推导是合理的，以加强我们对流体动力学模型和气候系统及其可预测性的理解。通过这项研究还实现了对本科生和研究生的分析和计算技能的培训。

Titi, DMS-1109640Cao, DMS-1109022 Numerical ocean modeling is computationally very demanding. Traditionally, the hydrostatic approximation balance, in the vertical momentum equation, has been applied to reduce the computational burden. Recently, the investigators have made significant progress in the rigorous understanding of the properties of the solutions to these models. However, it is still impossible to use present computing resources to simulate this large system for the wide range of spatial and temporal scales. Therefore, there is a need to further simplify these models to focus on understanding and simulating certain localized smaller-scale phenomena, such as the geostrophic adjustment of frontal anomalies in a rotating continuously stratified fluid of strictly rectilinear fronts and jets, and rapidly rotating convection which, according to the Taylor-Proudman constraint, takes place in tall columnar structures. The development of fast, accurate, and reliable numerical models of the climate system is key to understanding and predicting the magnitude and distribution of future climate variability. The focus of this project is to provide rigorous justification to these simplified models of the dynamics of the atmosphere and the oceans, and to investigate their long-term behavior. The investigators study the questions of existence, uniqueness, and continuous dependence on the initial data of the solutions to these models. This is the first, and the most crucial step, in justifying the derivation of these models and their consistency with the physical observations. There are numerous governing equations to fully describe the dynamics of climate. It is impossible to use present computing resources to simulate this large system for the wide range of spatial and temporal scales. Indeed, in order to understand, for example, the mechanism of atmospheric and oceanic turbulence, and to properly predict the climate, it is necessary to provide simplified and reliable models that contain the main features of the underlying physical phenomena. The investigators focus on the development of novel, theoretical and computational, mathematical tools for investigating these simplified models, and on justifying their derivation in order to enhance our understanding of fluid dynamics models and the climate system and its predictability. The training of undergraduate and graduate students with analytical and computational skills is also achieved through this study.