Nonlinear and Computational Problems for Geophysical and Classical Fluid Mechanics

该奖项是根据2009年美国复苏和投资法（公法111-5）资助的。研究者和他的同事们利用分析、非线性偏微分方程理论、动力系统理论和计算提供的数学工具，研究气象学、海洋学和流体力学中的非线性问题。除了提高对一些重要问题的理解外，该项目的一个目的是帮助改进具有许多社会和工业应用的现象的数值模拟。本项目与气象学和海洋学相关的部分侧重于有限域模拟的开放边界条件问题，以及在没有粘度的情况下气象学和海洋学的良好定位问题。该项目的其他部分涉及奇异摄动的研究，即在存在强烈影响方程的小系数的情况下的计算，以及湍流中的理论问题。天气预报的数值模拟通常是在目标区域周围的有限区域内进行的；这种计算包括通常的天气预报，或极端情况的预报，如飓风，或沙漠化的研究。类似的地理局部问题也出现在海洋学中，例如对河口的研究。有限区域的一个重要问题是边界的非物理部分的边界条件的选择，这些部分是为了计算而人工创建的。随着更好的计算机允许更精细（更精确）的计算，这个问题变得更加关键。不适当的边界条件可能产生不正确的波浪（或风），这可能导致例如不正确的降水预报。研究者和他的同事通过理论和计算方法来解决这个问题。他们与地球科学家密切合作，以确保项目的实际意义。该项目还包括对地球科学和数学领域的学生和博士后进行强有力的交叉培训。

TemamDMS-0906440 This award is funded under the American Recovery andReinvestment Act of 2009 (Public Law 111-5). The investigator and his colleagues study nonlinear problemsfrom meteorology, oceanography, and fluid mechanics, using themathematical tools offered by analysis, the theory of nonlinearpartial differential equations, dynamical systems theory, andcomputation. Besides improving the understanding of someimportant problems, an aim of the project is to help improve thenumerical simulations of phenomena which bear many societal andindustrial applications. The parts of this project related tometeorology and oceanography focus on the issue of open boundaryconditions for limited domain simulations, and well-posednessissues for meteorology and oceanography in the absence ofviscosity. Other parts of the project relate to the study ofsingular perturbations, that is, calculations in the presence ofsmall coefficients strongly affecting the equations, andtheoretical issues in turbulence. Numerical simulations for weather forecasts are usually donein a limited area around the region of interest; suchcomputations include the usual weather forecasts, or forecasts inextreme situations, like a hurricane, or the study ofdesertification. Similar geographically local questions arise inoceanography, for example, the study of an estuary. An importantissue for limited areas is the choice of the boundary conditionson the nonphysical parts of the boundary that are artificiallycreated for the sake of the computations. This issue becomesmore crucial as better computers allow more refined (moreprecise) calculations. Inappropriate boundary conditions maycreate incorrect waves (or winds), which may result for instancein incorrect precipitation forecasts. The investigator and hiscolleagues tackle this problem by theoretical and computationalapproaches. They work in close collaboration with geoscientiststo ensure the practical relevance of the project. The projectalso includes strong cross-training of students and postdocs ingeosciences and mathematics.