N-Vortex Problems: Modeling, Analysis, and Numerics

我们建议研究一个广泛的N-涡问题的基础上奇异或近奇异分解的无粘欧拉方程和高雷诺Navier-Stokes方程的涡量形式。通过这样的离散化，涡量场方程变成粒子相互作用方程，模型可以被看作是N体问题，其长程相互作用由对数哈密顿量控制。这些项目侧重于三个一般性主题：（一）旋转球体上的N-涡旋问题及其在大气流动中的应用;（二）涡旋晶格理论及其在玻色-爱因斯坦凝聚系统中的最新实验中的应用;（三）流线拓扑、分叉和演化研究，重点是将其与混沌平流和粒子输运联系起来。该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新的分析和计算技术，在物理基础良好的模型上测试当前技术，并将模型推向应用，主要是在海洋和大气流动中，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中，其中涉及一些相同的基本问题（尽管有时涉及不同形式的哈密顿）。所有项目的一个压倒一切的主题是开发哈密顿粒子模型结合概率方法来分析流体流动，并开发和应用非线性动力系统理论的新技术。与正在开发的数学模型有关的主要目标是模拟、分析和模拟具体的大气事件，例如2002年9月的南极极涡分裂事件，该事件在南半球表现为分裂的臭氧洞，并导致地球大气层整个平流层的全球粒子传输增强。这种增强的具体原因还不清楚，我们将研究这一重要和前所未有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与在完全不同的物理环境中使用的模型相似，例如超导理论中的Abrikosov晶格研究，最近对玻色-爱因斯坦凝聚晶格进行的一系列实验，以及化学物理中球形分子的建模，例如着名的Buckminsterfullerine分子。我们预计，在三年供资期内，与该提案相关的发展将揭示对这些领域也很重要的许多物理机制。

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.