CAREER: Order and Disorder in Two-dimensional Fluid Motion

许多地球物理学和天体物理学系统--如洋流、大尺度天气模式和行星大气--都是用二维流体方程来描述的，因为这些系统的垂直范围通常比水平范围小得多。因此，了解二维流体的长期动力学是天气预测、气候科学和天体物理学的基础。本项目旨在从几何学和动力学系统的角度增进我们对二维流体的理解，并通过培训初级研究人员和组织小型课程和会议来弥合这些学科之间的差距。这项工作的重点将是研究所有此类系统的原型，强迫二维不可压缩Navier-Stokes方程及其无粘，非受迫对应物，欧拉方程我们将分析长时间和弱耗散/强迫的物理相关制度，在各种限制的顺序。这个系统的一个相当令人惊讶和神秘的特征是，在一个区域（欧拉）中，它以飓风般的大漩涡的形式描述了秩序的诞生和持久性，而在另一个区域（纳维尔-斯托克斯）中，它描述了一种无序的、看似随机的、湍流的漩涡汤。也就是说，一个理想的欧拉流体倾向于通过涡合并和混合的过程随着时间的推移，而一个非常轻微的粘性和强迫的Navier-Stokes流体倾向于通过一系列的不稳定性，在雅阁长期的Kolmogorov方程的无序。该项目旨在通过偏微分方程、几何学和动力系统的透镜研究接近和远离平衡的溶液的行为，促进我们对流体运动这些基本方面的理解。计算机实验将被用来通知严格的数学分析和框架问题。这个奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

Many geophysical and astrophysical systems – such as oceanic currents, large-scale weather patterns and planetary atmospheres – are described, to good approximation, by two-dimensional fluid equations, since the vertical extent of these systems is typically much smaller than the horizontal. Understanding the long-term dynamics of two-dimensional fluids is thus fundamental to weather prediction, climate science, and astrophysics. This project aims to advance our understanding of two-dimensional fluids from the perspectives of both geometry and dynamical systems, as well as to bridge the gap between these disciplines through the training of junior researchers and the organizing of mini-courses and conferences.The focus of this work will be the study of the prototype of all such systems, the forced two-dimensional incompressible Navier-Stokes equations and their inviscid, unforced counterpart, the Euler equations. We will analyze the physically relevant regimes of long time and weak dissipation/forcing, in various orders of limits. A rather surprising and mysterious feature of this system is that, in one regime (Euler), it captures the birth and permanence of order in the form of large hurricane – like whirls – while in another regime (Navier-Stokes) it describes a disorderly, seemingly random, turbulent soup of eddies. That is, an ideal Euler fluid tends towards order over time through a process of vortex mergers and mixing, whereas a Navier-Stokes fluid with very slight viscosity and forcing tends towards disorder through a cascade of instabilities in accord with longstanding conjectures of Kolmogorov. The project aims to advance our understanding of these fundamental aspects of fluid motion by studying the behavior of solutions both near and far from equilibrium through the lens of partial differential equations, geometry, and dynamical systems. Computer experiments will be used to inform rigorous mathematical analysis and frame questions.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.