N-Vortex Problems

将分析控制流体力学的欧拉方程的离散模型，目的是了解高雷诺数湍流。这种综合方法将结合哈密顿动力系统的数学分析、基于欧拉方程离散粒子表示的物理建模以及高性能计算。数学问题是 N 体类型，因此这些系统的数值算法开发和动力系统可视化技术将是该奖项下开展的工作的重要组成部分。提案中描述的四个主要项目旨在开发动力系统理论中的新分析技术，在物理基础良好的模型上测试现有技术，开发用于粒子相互作用问题的新数值算法和可视化方法，并使模型更接近应用。该项目涉及使用数学工具和高性能计算相结合来开发流体流动的离散模型。一个主要重点是研究球体表面的这些模型，包括旋转和非旋转，这一问题直接应用于行星大气中的流动，其规模如此之大，以至于行星的曲率在动力学中发挥着作用。这个问题对于理解地球大气和海洋的环境过程以及在木星大气中观察到的大涡旋（包括红斑）可能非常重要。

Discrete models of the Euler equations governing fluid mechanics will be analyzed with the goal of understanding high Reynolds number turbulent flows. The comprehensive approach will involve a combination of mathematical analysis of Hamiltonian dynamical systems, physical modeling based on discrete particle representations of the Euler equations, and high performance computing. The mathematical problem is of N-body type, thus numerical algorithm development for these systems and dynamical systems visualization techniques will be an essential component of the work carried out under this award. The four main projects described in the proposal are each designed to develop new analytical techniques in dynamical systems theory, test current techniques on models that are physically well grounded, develop new numerical algorithms and visualization methods for particle interaction problems, and push the models closer towards applications.This project involves the development of discrete models for fluid flow, using a combination of mathematical tools and high performance computing. One major emphasis will be on studying these models on the surface of a sphere, both rotating and non-rotating, a problem which has direct applications to the flow in a planetary atmosphere on such large scales that the curvature of the planet plays a role in the dynamics. This problem is potentially of great importance to the understanding of environmental processes in the atmospheres and oceans of Earth and to the large vortices, including the Red Spot, observed in the atmosphere of Jupiter.