Collaborative Research: Conservation Laws, Simple Waves and Mixing in Stratified Fluids

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The atmosphere and the ocean are stratified in the density and background flows are often sheared. The variation of density enables the propagation of internal waves - waves in the interior of the fluid - not on its surface. These waves, as they propagate, can deform and break and, when they do, they mix fluid of different densities, changing the medium in which they propagate. A competing effect is mixing engendered by shear instabilities. This project will develop a mathematical methodology for the study of the evolution of fully nonlinear sheared, stratified flows, with the goal to understand and model the dynamics leading to fluid mixing. The wave and shear-generated mixing will be addressed with tools in the theory of systems of finite and infinitely many conservation laws. We shall calculate and study exact nonlinear solutions that break - especially simple waves of the system, general criteria for the nonlinear stability of systems of mixed type, and the statistical description of incomplete systems, based on the maximization of a coarse-grained mixing entropy.This research is concerned with the propagation of waves and the dynamics of shear in atmospheric and oceanic flows and how these phenomena mix the fluid in which they exist. Mixing in geophysical flows is a crucial component of climate dynamics. The rate of mixing of ocean waters, for instance, determines the temperature at the surface, which communicates directly with the atmosphere, affecting our weather and climate. The dispersion and mixing of chemicals, carbon dioxide and water in the atmosphere, on the other hand, has a profound effect on the radiative balance leading to predictions for climate change. Waves are also an element of atmospheric teleconnection: they can have long range effects since they carry energy over long distances without transporting the fluid itself. Since the length scales in which the waves and the mixing processes occur are too small and fast to be computed together with planetary scale flows, they are nowadays represented by a low number of aggregate parameters (parameterized) using simple criteria. A better understanding of the dynamics and mixing processes will improve these parameterizations and the reliability of the climate models that use them.