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

该研究项目涉及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，但这个项目只解决地球物理问题。 考虑的两类问题是在有限的域中进行预测的边界条件的研究，以及湿对流的研究，这与云的行为的理解有关。 人们普遍认为，我们对云的物理学的有限理解是当前天气和气候预测不确定性的主要原因。 开发改进天气预测和气候建模的方法是重要的人类，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等方面对研究生和博士后进行培训是该项目的另一个更广泛的影响。在开发地球物理流的计算模型时的一个主要问题是，当流延伸到大面积但特别感兴趣的区域较小时，当物理学不提供自然边界条件时，为有限区域模型开发合适的边界条件。 冯·诺依曼和查尼已经提到的这个老问题，预计将变得非常重要。 事实上，当新的计算机将允许使用精细的空间分辨率时，可以预测，不适当的边界条件将引入虚假模式。 这个问题也出现在其他当代的发展，如使用多级数值方法或建模。 研究者已经获得了一些线性和非线性二维方程的相关结果。 在这里，他的目标是继续建立一个健全的数学理论的无粘原始方程和其他相关方程。 关于湿对流、降水和云的突出问题，需要解决的问题涉及到理论、计算和模拟。 在理论方面，凸分析和变分不等式的工具被用来解决问题的湿大气饱和，这是非线性不连续偏微分方程（PDE）没有直接连接到任何现有的偏微分方程理论的方程的解的存在性和唯一性。 在建模方面，再次使用凸分析工具，他通过考虑饱和时微观物理的更精细描述来模拟与冻结水平以下的水云相对应的潮湿大气的方程，这将引入额外的状态参数。 在计算方面，他研究了在地形存在下赤道附近发生的非常复杂的对流现象的数值超参数化，使用有限体积和多级方法。 这些现象涉及热带降雨并影响到远至中纬度地区的气候，其重要性众所周知。 最后，某些问题的概率解决与上述现象中发生的许多不确定性的建模。 研究生和博士后都在从事该项目的工作。

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.