CAREER: Order and Disorder in Two-dimensional Fluid Motion

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.