N-Vortex Problems: Modeling, Analysis, and Numerics

N 涡问题：建模、分析和数值

• 批准号: 0804629
• 负责人: Paul Newton
• 金额: $ 25万
• 依托单位: University of Southern California
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-09-15 至 2014-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0804629&HistoricalAwards=false
• 关键词: Vortex; Problems; Modeling; Analysis; Numerics

We propose to study a wide range of N-vortex problems based on singular or near singular decompositions of the inviscid Euler equations and high Reynolds Navier-Stokes equations written in vorticity form. With such a discretization, the vorticity field equations become particle interaction equations and the models can be viewed as N-body problems with long-range interactions governed by a logarithmic Hamiltonian. The projects focus on three general themes: (i) the N-vortex problem on a rotating sphere with applications to atmospheric flows, (ii) vortex lattice theory with applications to recent experiments in Bose-Einstein condensate systems, (iii) the study of streamline topology, bifurcations, and evolution with the a focus on connecting this with chaotic advection and particle transport. There are seven detailed projects described in the proposal, each is designed to develop new analytical and computational techniques in dynamical systems theory, test current techniques on models that are physically well grounded, and push the models closer towards applications, mainly in oceanographic and atmospheric flows, but also in molecular modeling and the study of Bose-Einstein condensate lattices and vortices in thin film superconductors, where some of the same underlying issues pertain (albeit sometimes with Hamiltonians of a different form). An overriding theme of all of the projects is to develop Hamiltonian particle models in conjunction with probabilistic methods to analyze fluid flows and to develop and apply new techniques in nonlinear dynamical systems theory. The main goals associated with the mathematical models being developed are to model, analyze, and simulate specific atmospheric events, such as the Antarctic pole vortex splitting event of September 2002, which manifested itself in the form of a split ozone hole in the Southern Hemisphere and led to enhanced global particle transport over the full stratospheric layer of the Earth's atmosphere. The specific reasons for this enhancement are not well understood and we will investigate several low dimensional models of this important and unprecedented event. Interestingly, the mathematical models developed in this context are similar to those used in completely different physical settings, such as the study of Abrikosov lattices in superconductivity theory, the recent flurry of experiments done on Bose-Einstein condensate lattices, and the modeling of spherical molecules in chemical physics, such as the famous Buckminsterfullerine molecule. We expect that developments associated with this proposal over the three year funding period will shed light on many physical mechanisms important to these areas as well.

我们建议基于无粘欧拉方程和高雷诺N-S方程的奇异或近奇异分解来研究广泛的N-涡旋问题。经过这样的离散化，涡量场方程就变成了粒子相互作用方程，模型可以看作是由对数哈密顿量控制的具有长程相互作用的N体问题。这些项目集中在三个一般主题上：(I)旋转球面上的N-涡旋问题及其在大气流动中的应用，(Ii)涡旋格子理论与最近在玻色-爱因斯坦凝聚系统中的实验的应用，(Iii)流线拓扑、分叉和演化的研究，重点是将其与混沌平流和粒子输运联系起来。提案中描述了七个详细的项目，每个项目都旨在开发动力系统理论中的新分析和计算技术，在物理上站稳脚跟的模型上测试当前的技术，并推动模型更接近应用，主要是在海洋和大气流动方面，但也在分子建模和薄膜超导体中的玻色-爱因斯坦凝聚晶格和涡旋研究方面，其中一些相同的潜在问题涉及(尽管有时涉及不同形式的哈密顿理论)。所有项目的一个压倒一切的主题是发展哈密顿粒子模型与概率方法相结合来分析流体流动，并开发和应用非线性动力系统理论中的新技术。正在开发的数学模型的主要目标是模拟、分析和模拟特定的大气事件，例如2002年9月的南极极地涡旋分裂事件，它以南半球臭氧空洞的形式表现出来，导致地球大气层整个平流层上的全球粒子传输得到加强。这种增强的具体原因尚不清楚，我们将研究这一重要且史无前例的事件的几个低维模型。有趣的是，在这种背景下开发的数学模型类似于在完全不同的物理环境中使用的那些模型，例如超导理论中的Abrikosov晶格的研究，最近在玻色-爱因斯坦凝聚晶格上进行的一系列实验，以及化学物理中的球形分子的建模，例如著名的Bakminsterfullerine分子。我们预计，在三年筹资期间与这项提议有关的事态发展也将阐明对这些领域很重要的许多实际机制。