Collaborative Research: Mathematical Analysis of the Effects of Rotation, Stratification, and Dissipation in Incompressible Fluid Flows

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The rotation of the planet, the stratification of density, and the generation of friction all play important roles in the motion of geophysical fluids. These mechanisms are always present in our ocean and atmosphere, as well as that of other planets. They are collectively responsible for many well-known phenomena that we observe in nature, e.g., jet streams, zonal jets, the El Niño cycle, and Jupiter's Great Red Spot, to name only a few. These mechanisms typically serve to constrain the motion of the fluid in a very particular way. For instance, it is observed that in a rapidly rotating fluid in three-dimensions, particles that are aligned along a common vertical parallel to the axis of rotation move nearly in unison, thus rendering the overall motion of the flow to be essentially two-dimensional. Despite many experimental and computational efforts to understand the precise development of such phenomena, the mathematical justification for them, that is, from directly studying the equations of motion themselves, remains largely open. This project will systematically address such concerns in various geophysical settings. This project will also provide research and mentorship opportunities for students at the undergraduate and graduate levels, as well as postdoctoral scholars.An overarching goal of this project is to understand various manifestations of finite-dimensionality and its interconnections with the mechanisms of dissipation, rotation, and stratification. The main approach will be through the study of the regularity and long-time behavior of solutions to the associated equations of motion that allow one to obtain precise quantitative relations between the parameters representing the strength of these various mechanisms with the smallest relevant length scales of the fluid flow. The main models of interest will be those that arise naturally in geophysics such as the rotating Navier-Stokes equations and the stably stratified Boussinesq equations. In order to properly quantify the effects carried by rotation and stratification, anisotropic dispersive estimates, and careful analyses of resonance structures inherent in such systems will be carried out. A novelty of this project is the interplay between physical space- and frequency space-based approaches. Although both approaches have seen success in studying the regularity of solutions, the frequency space-based approach is well-suited for quantifying the number of degrees of freedom, while the physical space-based approach is well-suited for exploiting information about the spatial analyticity radius. This project attempts to merge these two approaches in ways that allow one to jointly exploit the reductions in dimensionality in both physical-space and frequency-space that is observed in rotating or stratified fluid flows.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.