Mathematical Sciences: Theoretical and Computational Problems in Turbulence and Climatology

小行星9400615 该项目将侧重于研究流体力学和湍流方程，以及开发求解这些方程的有效数值代码。 该项目的一部分将涉及基于动力系统方法的湍流研究;根据动力系统理论，湍流的永久状态在数学上由相空间中的紧凑集吸引子表示。 人们希望，这种吸引子的研究可以提高湍流物理的理解。 吸引子的近似光滑有限维流形称为近似惯性流形（AIM）是这项研究的一部分。 一方面，这些流形与气象学中用于数据同化的慢流形概念有关。 从不同的角度来看，在计算流体动力学，他们导致，另一方面，新的多分辨率算法和方法的数据压缩在科学计算。 因此，人们相信，这种吸引子的近似可以导致有效的新算法的数值模拟湍流。 最后，这项工作的很大一部分将用于解决与气候和全球变化有关的问题。 将从数学和计算的角度研究各种气候模型，以了解它们是如何联系在一起的，以及它们的有效范围是什么。 这类模式包括大气原始方程、具有垂直粘性的原始方程、地转方程和准地转方程。 这类模式将分别针对大气和海洋进行研究，也将针对大气和海洋耦合系统进行研究。 对流体力学和湍流方程的理解以及为求解这些方程开发高效的数值代码是具有挑战性的问题，特别是在工业或气象学或全球气候变化研究中具有相当重要的意义。 该项目的目标有三个方面：（i）从理论和计算的角度，利用湍流的动力系统方法，探索可以了解这些问题的内容;（ii）开发新的有效的计算算法，考虑到湍流的物理学，并很好地适应当前大规模计算向并行计算的演变;（iii）特别关注与气候和全球变化研究有关的问题。

9400615 Temam This project will focus on the study of the equations of fluid mechanics and turbulence as well as on the development of efficient numerical codes for the solution of these equations. Part of the project will involve the study of turbulence, based on a dynamical systems approach; according to dynamical systems theory the permanent regime for a turbulent flow is mathematically represented by an attractor that is a compact set in the phase space. It is hoped that the study of this attractor can improve understanding of the physics of turbulent flows. The approximation of attractors by smooth finite dimensional manifolds called Approximate Inertial Manifolds (AIMs) is part of this study. On the one hand these manifolds are related to the concept of slow manifold used in meteorology for data assimilation. Used from a different point of view in computational fluid dynamics they lead, on the other hand, to new multi-resolution algorithms and to methods of data compression in scientific computing. Thus, it is believed that approximations of this attractor can lead to efficient new algorithms for the numerical simulation of turbulence. Finally, a significant part of the effort will be devoted to problems related to climate and global changes. Various models for climate will be investigated from the mathematical and computational points of view in order to see how they are connected and what are their domains of validity. Such models include the primitive equations of the atmosphere, the primitive equations with vertical viscosity, and the geostrophic and quasi geostrophic equations. Models of this type will be studied for the atmosphere and for the ocean separately and also for the Coupled system of Atmosphere and Ocean (CAO). 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 p articular in industry or for studies in meteorology or global climate change. The aims for this project are three fold: (i) to explore what can be learned about these problems from the theoretical and computational viewpoints, using the dynamical systems approach to turbulence; (ii) 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; and (iii) to devote special attention to the problems related to the study of climate and global changes.