Theoretical and Computational Problems in Fluid Mechanics and Climatology

Temam 9705229 流体力学和湍流方程的理解，以及这些方程的解决方案的有效的数值代码的发展是相当重要的挑战性问题，特别是在工业或在气象学或全球气候变化的研究。 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，使用动力系统方法湍流。 更具体地说，研究人员和他们的同事打算：（i）开发新的有效的计算算法，考虑到湍流的物理学，并很好地适应当前的大规模计算向并行计算的演变;（ii）特别关注与气候和全球变化研究有关的问题。 对于（i），在过去的几年中，已经引入了与吸引子和近似惯性流形的概念相关的新的多级算法。 这些算法的分析开发，其性能得到改善，其应用范围扩大，特别是对实际相关的问题。 在此期间，将出版一本关于这一主题的书，描述理论方面的最新技术，并包括在并行计算机上实际实现的许多方面。 对于（二）项，该项目的一个重要部分是专门研究大气、海洋及其耦合的模型。 所涉及的模型的基础上的原始方程被认为是，以及更简单的模型，如多层或准地转模式。 这项研究包括模型的发展以及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，可以追溯到伦纳德达芬奇，皮埃尔西蒙拉普拉斯，或者在这个世纪，二战后，约翰冯诺依曼。 在一个更温和的水平上，研究人员正在进行一项几年前启动的研究计划，旨在发展地球科学和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，对数学家和其他科学家非常有用（例如，科学家们从E。 Lorenz in chaos）。 相反，数学家可以帮助地球科学界的科学家选择他们的模型，确定它们是否是适定的。 从长远来看，它们也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）是经过很长一段时间编写的非常复杂的代码，但最终会编写新的代码来应对新的需求或新的机会，新的见解可能会变得有用;希望有助于这项艰巨的任务。

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.