Theoretical and Computational Problems in Fluid Mechanics and Climatology

对流体力学和湍流方程的理解以及为这些方程的解开发有效的数值代码是相当重要的具有挑战性的问题，特别是在工业或气象学或全球气候变化研究中。这个项目的目的是探索可以从理论和计算的角度来了解这些问题，使用动力系统方法来研究湍流。更具体地说，研究人员和他们的同事打算：(i)开发新的高效计算算法，考虑到湍流的物理特性，并很好地适应当前大规模计算向并行计算的发展；（二）特别关注与气候和全球变化研究有关的问题。对于(i)，在过去几年中引入了与吸引子和近似惯性流形概念相关的新的多层算法。对这些算法的分析得到了发展，它们的性能得到了改进，它们的应用范围得到了扩展，特别是在解决实际相关问题方面。在这项资助期间，将出版一本关于这一主题的书，描述理论方面的艺术状态，并包括它们在并行计算机上的实际实现的许多方面。对于（ii），本项目的很大一部分致力于研究大气、海洋及其耦合的模式。考虑了基于原始方程的相关模型，以及多层或准地转模型等较简单的模型。该研究包括模型的发展和它们提出的数学和数值问题。气象学和数学的相互作用有着悠久的传统，可以追溯到莱纳德·达·芬奇、皮埃尔·西蒙·拉普拉斯等著名人物，也可以追溯到本世纪二战后的约翰·冯·诺伊曼。在一个较为温和的层面上，研究人员正在进行几年前发起的一个研究项目，旨在发展地球科学与数学之间的相互作用。这些互动可以是互利的。气象学和海洋学提出了非常具有挑战性的数学问题，这些问题对数学家和其他科学家非常有用（例如，科学家从E. Lorenz在混沌中的经验中学到了很多）。相反，数学家可以帮助地球科学领域的科学家选择他们的模型，确定模型是否恰当。从长远来看，它们还可能有助于开发新的有效的数值程序；虽然气象学和海洋学中使用的代码（程序）是在很长一段时间内编写的非常复杂的代码，但最终会根据新的需求或新的机会编写新的代码，新的见解可能会变得有用；希望它能为这项艰巨的任务作出贡献。

Temam 9705229 The understanding of the equations of fluid mechanics and turbulence as well as the development of efficient numerical codes for the solution of these equations are challenging problems of considerable importance, in particular in industry or for studies in meteorology or global climate change. The aims for this project are to explore what can be learned about these problems from the theoretical and computational viewpoints, using the dynamical systems approach to turbulence. More specifically the investigators and their colleagues intend: (i) to develop new efficient computational algorithms taking into account the physics of turbulence and well adapted to the current evolution of large scale computing towards parallel computing; (ii) to devote special attention to the problems related to the study of the climate and global changes. For (i), new multilevel algorithms related to the concept of attractor and approximate inertial manifolds have been introduced during the past years. The analysis of these algorithms is developed, their performances improved, and their range of application extended, in particular towards problems of practical relevance. During the period of this grant a book will be published on this subject, describing the state of the art on the theoretical side, and including many aspects of their actual implementation on parallel computers. For (ii), a significant part of this project is devoted to the study of models for the atmosphere, the ocean and their coupling. Involved models based on the primitive equations are considered, as well as simpler models such as multilayer or quasi-geostrophic models. The study includes the development of the models and the mathematical and numerical problems that they raise. The interaction of meteorology and mathematics has very long traditions going back to such famous names as Leonard da Vinci, Pierre Simon Laplace or, in this century, after WWII, John Von Neumann. At a more modest level, the investigators pursue a program of research initiated a few years ago and aimed at developing interactions between geosciences and mathematics. These interactions can be mutually beneficial. Meteorology and oceanography raise very challenging mathematical problems very useful for mathematicians and other scientists (for example scientists have learned much from the experience of E. Lorenz in chaos). Conversely, mathematicians might help scientists of the geoscience communities in choosing their models by determining if they are well posed. In the long range they might help also develop new efficient numerical procedures; although the codes (programs) used in meteorology and oceanography are very involved codes written over a long period of time, eventually new codes will be written responding to new needs or new opportunities, and new insights could become useful; it is hoped to contribute to this daunting task.