Mathematical Problems of Geophysical Fluid Mechanics: Uncertainties, Modeling, Theory, and Computing

这项研究项目解决了海洋和大气科学中与天气或气候预测有关的重要问题。这些研究在其他科学领域也有潜在的应用，但这个项目只涉及地球物理问题。所考虑的两类问题是在有限区域内进行预报的边界条件的研究，以及与理解云的行为有关的湿对流的研究。人们普遍认为，我们对云层物理的有限理解是当前天气和气候预测不确定的一个主要原因。开发改进天气预报和气候模拟的方法是一个重要的人类、社会和经济利益的目标。在数学、地球物理和工业流体力学以及科学计算方面对研究生和博士后的培训是该项目的另一个更广泛的影响。发展地球物理流动计算模型的一个主要问题是，当流动延伸到较大区域但特别感兴趣的区域较小时，当物理不提供自然边界条件时，为有限区域模型制定合适的边界条件。冯·诺伊曼和查尼已经提到了这个老问题，预计这个问题将变得非常重要。事实上，当新的计算机将允许使用精细的空间分辨率时，据预测，不适当的边界条件将引入虚假模式。这个问题也出现在当代的其他发展中，比如使用多层数值方法或建模。对于线性和非线性的二维方程，研究者得到了一些相关的结果。在这里，他的目标是继续为无粘性原始方程和其他相关方程建立健全的数学理论。关于湿对流、降水和云的突出问题，需要解决的问题涉及理论、计算和模拟。在理论方面，利用凸分析和变分不等式的工具讨论了湿大气饱和方程的解的存在唯一性问题，这些方程是与已有的不连续偏微分方程组没有直接联系的非线性间断偏微分方程组。在建模方面，他再次使用凸分析工具，通过考虑对饱和状态下的微观物理的更精细描述，对与冰冻水平以下的水云相对应的潮湿大气的方程进行建模，这将引入额外的状态参数。在计算方面，他使用有限体积和多层方法研究了在地形存在的情况下赤道附近发生的非常复杂的对流现象的超参数化。这些现象的重要性是众所周知的，这些现象涉及热带降雨，并影响到中纬度地区的气候。最后，结合对上述现象中出现的许多不确定因素的建模，讨论了某些概率问题。研究生和博士后从事该项目的工作。

This research project addresses problems of great significance in ocean and atmosphere sciences, related to weather or climate predictions. The studies have also potential applications in other areas of sciences but this project only addresses the geophysical problems. The two classes of problems considered are the study of boundary conditions for predictions made in a limited domain, and the study of moist convection, which is related to the understanding of the behavior of clouds. It is generally considered that our limited understanding of the physics of clouds is a major cause of uncertainty in current weather and climate predictions. Developing methods that improve weather predictions and climate modeling is an objective of important human, societal, and economical interest. The training of graduate students and postdoctoral students at the interface of mathematics, geophysical and industrial fluid mechanics, and scientific computing is another broader impact of the project. A major problem in developing computational models of geophysical flows, when the flows extend over a large area but the region of particular interest is smaller, is to develop suitable boundary conditions for limited-area models when the physics does not provide natural boundary conditions. This old problem, already mentioned by von Neumann and Charney, is expected to become very significant. Indeed, when new computers will allow the use of fine spatial resolutions, it is predicted that inappropriate boundary conditions will introduce spurious modes. The problem arises also in other contemporary developments like the use of multilevel numerical methods or modeling. The investigator has obtained a number of relevant results for linear and nonlinear two-dimensional equations. Here he aims to continue building a sound mathematical theory for the inviscid primitive equations and other related equations. Concerning the outstanding problem of moist convection, precipitation and clouds, the the problems to be addressed relate to the theory, computations, and modeling. On the theoretical side, the tools of convex analysis and variational inequalities are used to address questions of existence and uniqueness of solutions for the equations of the humid atmosphere with saturation, which are nonlinear discontinuous partial differential equations (PDEs) not directly connected to any existing PDE theory. On the modeling side, using again tools of convex analysis, he models the equations of the humid atmosphere corresponding to water clouds below freezing levels by considering a more refined description of the micro physics at saturation, which would introduce additional state parameters. On the computational side, he studies numerically the superparameterization of the very complex convection phenomena occurring near the equator in the presence of topography, using finite volumes and multilevel methods. The importance of these phenomena, which involve tropical rains and affect the climate far into the mid-latitude regions, is well known. Finally, certain questions of probability are addressed in connection with the modeling of the many uncertainties occurring in the phenomena described above. Graduate students and postdoctoral students are engaged in the work of the project.