Shape Dynamics of Vortices: Theory and Numerics

涡流在流体中无处不在。我们观察到河流、海洋和飓风中的漩涡。这个项目的主要目标是了解这些漩涡是如何相互作用的；更具体地说，它们在自然界中可以形成什么样的几何图案。一个简单的例子是由三个飓风中心形成的三角形如何在时间上改变其形状；它是2017年秋季飓风哈维之后不久出现在墨西哥湾附近的三个飓风伊尔玛、何塞和卡蒂亚的数学模型。众所周知，这种相互作用的涡旋在某些材料中也会出现在更小的尺度上，这是理解超导体的关键，超导体具有工程应用，包括强大的电磁铁(用于磁悬浮列车、聚变反应堆和磁共振成像)。这个项目利用数学思想来更好地理解相互作用的涡旋的动力学。该项目将有助于推广活动，PI和研究生还将指导参加数学建模比赛的不同背景的本科生。该项目使用现代几何方法研究普通流体、超流体和超导体中多个涡旋的相互作用，以研究涡旋的形态如何随时间演变。虽然人们很好地了解了只有几个涡旋的形状动力学，但更多涡旋的形状动力学一直是一个挑战。新的几何洞察力通过寻找有助于我们理解可能构型的稳定性的不变量，为旋涡的形状动力学提供了新的线索。该项目利用几何力学中对称性约化的思想，开发了分析旋涡形状动力学的技术。这项技术不仅适用于普通流体中的涡旋，也适用于超流体和超导体中的量子涡旋。该项目还开发了数值方法来准确预测多个旋涡构型的稳定性/不稳定性。涡旋动力学是所谓的不可分哈密顿系统的一个实际例子，许多传统的哈密顿系统的显式方法并不直接适用于这种系统。该项目旨在开发具有良好几何性质的不可分哈密顿系统的显式数值积分器，例如涡旋动力学中不变量的精确/接近守恒。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Vortices are ubiquitous in fluids. We observe swirls in rivers, the ocean, and hurricanes. The main objective of this project is to understand how such vortices interact with each other; more specifically, what kind of geometric patterns they can form in nature. A simple example is how the triangle formed by the centers of three hurricanes changes its shape in time; it is a mathematical model for the three hurricanes, Irma, Jose, and Katia, that appeared near the Gulf of Mexico in the Fall 2017, shortly after the Hurricane Harvey. Such interacting vortices are known to appear in much smaller scales in certain materials as well, and are key to the understanding of superconductors, which have engineering applications including powerful electromagnets (used for maglev trains, fusion reactors, and magnetic resonance imaging). This project exploits mathematical ideas to better understand the dynamics of interacting vortices. The project will contribute to outreach events, and the PI and the graduate students will also mentor undergraduate students with diverse backgrounds participating in a mathematical modeling contest.The project investigates interactions of multiple vortices in ordinary fluids, superfluids, and superconductors using modern geometric methods to study how the configuration of vortices evolves in time. While the shape dynamics of only a few vortices is well understood, the shape dynamics of a larger number of vortices has been a challenge. The new geometric insight sheds a new light on the shape dynamics of vortices by finding invariants that will help us understand the stability of possible configurations. The project develops techniques to analyze the shape dynamics of vortices by exploiting ideas from symmetry reduction in the geometric approach to mechanics. The technique applies not only to vortices in ordinary fluids but also to quantum vortices in superfluids and superconductors. The project also develops numerical methods to accurately predict stability/instability of configurations of many vortices. Vortex dynamics is a practical example of so-called non-separable Hamiltonian systems, to which many conventional explicit methods for Hamiltonian systems do not apply directly. The project aims to develop explicit numerical integrators for non-separable Hamiltonian systems with favorable geometric properties, such as exact/near conservation of invariants in vortex dynamics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.