N-Vortex Problems: Analysis, Computation, and Data Acquisition

该项目的重点是从动力系统理论的N-涡问题，基于不可压缩流体力学的欧拉方程的离散涡表示。 重点将放在三个主题上：（1）旋转球体上的N涡问题及其在大气流动中的应用;（2）全球天气模式的分析和数据获取;（3）N体数值算法的开发。 每一个都旨在开发动力系统理论中新的分析和计算技术，在物理上有良好基础的模型上测试当前技术，开发新的数值算法，以保存我们知道应该保存的数量，并将模型推向主要在海洋学和大气科学中的应用。但也在分子建模中，其中一些相同的基本问题涉及（尽管具有不同形式的哈密顿量）。 在许多情况下，已开发的天体力学背景下的N体问题的技术将被利用和适用于这类离散涡问题。 开发的工具属于哈密尔顿和拉格朗日力学系统范围内的高性能计算的一般类别，这些专题将与大气和海洋全球环流模型的开发和制定有关。 该项目的数据采集部分侧重于了解全球天气模式以及这些模式与环境污染物、海洋生物群和大气臭氧等被动和主动标量的传输和混合之间的联系。

The project focuses on a wide range of N-vortex problems from dynamical systems theory, based on discrete vortex representations of the Euler equations of incompressible fluid mechanics. Emphasis will be placed on three topics: (1) The N-vortex problem on a rotating sphere with applications to atmospheric flows; (2) Analysis and data acquisition of global weather patterns; (3) N-body numerical algorithm development. Each is designed to develop new analytical and computational techniques in dynamical systems theory, test current techniques on models that are physically well grounded, develop new numerical algorithms that conserve quantities we know should be conserved, and push the models closer towards applications mostly in oceanographic and atmospheric sciences, but also in molecular modeling where some of the same underlying issues pertain (albeit with Hamiltonians of a different form). In many cases, techniques that have been developed for N-body problems in the celestial mechanics context will be exploited and adapted for use on this class of discrete vortex problems. Tools developed fall under the general category of high-performance computing in the context of Hamiltonian and Lagrangian mechanical systems and the topics will be relevant in the development and formulation of global circulation models for the atmosphere and oceans. The data acquisition portion of the project focuses on the understanding of global weather patterns and the connection between these patterns and the transport and mixing of passive and active scalars such as environmental pollutants, oceanographic biota, and atmospheric ozone.