Problems of Geophysical Fluid Mechanics: Modeling, Theory and Computing

该项目将解决与天气和气候预测有关的数学海洋和大气科学问题。将研究两类问题。第一个是有限区域天气模式的边界条件，物理模式不提供自然边界条件。这个问题与数值天气预报本身一样古老，预计不适当的边界条件将引入虚假模式，特别是随着计算能力的提高，空间和时间分辨率将得到提高。其目的是推导出边界条件，导致数学上正确的问题，并且在计算上令人满意，因为它们让波在计算域内外自由移动。PI将继续朝此方向进行调查。第二类问题是湿对流、降水和云。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之一。这些困难部分是由于尺度上的巨大差异，从数百公里的云到几微米的颗粒（水滴，冰或气溶胶）。有一个类似的时间尺度范围。该项目将使用凸分析和变分不等式的数学工具来研究这个物理问题。数值多级方法也将被用来研究云和赤道附近地形的相互作用;最后，凸分析的工具也将被用来模拟云的相位变化。这些研究将与物理学家密切合作进行。该项目将涉及研究生和博士后助理谁将在这一重要的跨学科领域的培训。 该研究项目将研究天气和气候数值预报的基本问题，以及云和蒸发的数学建模问题。与数值天气预报有关的项目组成部分解决了将来自粗略的全球天气数据的信息用于精细的地方和区域预测的问题。如果粗略的数据没有正确地耦合到具有较高分辨率的区域模型，那么可能会出现虚假的系统误差（例如“幽灵”天气锋）。该项目将研究这个问题的数学基础。关于云和蒸发建模的项目组成部分将使用先进的数学工具来帮助理解天气和气候研究中的一系列基本开放问题：云有助于冷却地球（因为它们反射阳光）还是有助于变暖（因为它们可以保持热能）？天气预报和气候建模是具有重要的人类、社会和经济利益的目标。该项目还将在数学、地球物理流体力学和科学计算的界面上培训年轻科学家。

This project will address problems in mathematical ocean and atmosphere sciences, related to weather and climate prediction. Two classes of problems will be investigated. The first one is that of the boundary conditions for limited area weather models for which physical models do not provide natural boundary conditions. This problem is as old as numerical weather prediction itself, and it is expected that inappropriate boundary conditions will introduce spurious modes, in particular as improved computing capabilities allow better spatial and temporal resolutions. The aim is to derive boundary conditions that lead to mathematically correct problems, and that are computationally satisfactory in the sense that they let the waves move freely inside and outside the domain of computation. The PI will continue his investigations in this direction. The second class of problems is that of moist convection, precipitations and clouds. Our limited understanding of the physics of clouds is one of the primary causes of uncertainty in the current weather and climate predictions. The difficulties are in part due to the large differences in scales, ranging from cloud sizes of hundreds of kilometers to particles (droplets of water, ice or aerosols) of a few microns. There is a similar range of time scales. The project will study this physical problem using mathematical tools from convex analysis and variational inequalities. Numerical multilevel methods will be used as well to study the interaction of clouds and topography near the equator; and finally the tools of convex analysis will also be used for the very modeling of the changes of phase in the clouds. These studies will be conducted in close collaboration with physicists. The project will involve graduate students and post-doctoral associates who will be trained in this important interdisciplinary area. This research project will study fundamental problems in the numerical prediction of weather and climate, as well as problems in the mathematical modeling of clouds and evaporation. The project component related to numerical weather prediction addresses the problem of using information from coarse featured global weather data for fine grained local and regional predictions. If the coarse data are not coupled correctly to the regional model that has higher resolution, then spurious systematic errors (e.g. "ghost" weather fronts) may arise. The project will investigate the mathematical foundations of this problem. The project component on the modeling of clouds and evaporation will use advanced mathematical tools to contribute to the understanding of a set of basic open problems in weather and climate research: Do clouds help cool the planet (because they reflect sunlight) or do they contribute to warming (because they can hold thermal energy)? Weather prediction and climate modeling are objectives with important human, social and economic interest. The project will also train young scientists at the interface of mathematics, geophysical fluid mechanics, and scientific computing.