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

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。大气和海洋在密度上是分层的，背景气流经常被剪切。密度的变化使内波——在流体内部的波——而不是在其表面传播成为可能。这些波在传播的过程中会变形和破裂，当它们这样做的时候，它们会混合不同密度的流体，改变它们传播的介质。另一种竞争效应是由剪切不稳定性引起的混合。该项目将开发一种数学方法，用于研究完全非线性剪切分层流动的演变，目的是理解和模拟导致流体混合的动力学。在有限和无限多守恒定律的系统理论中，将用工具来处理波浪和剪切产生的混合。我们将计算和研究精确的非线性解，特别是系统的简单波，混合型系统的非线性稳定性的一般准则，以及基于粗粒度混合熵最大化的不完全系统的统计描述。本研究关注的是大气和海洋流动中波浪的传播和切变动力学，以及这些现象如何与它们所在的流体混合。地球物理流动中的混合是气候动力学的一个重要组成部分。例如，海水的混合速率决定了地表温度，而地表温度又直接与大气交流，影响着我们的天气和气候。另一方面，化学物质、二氧化碳和水在大气中的分散和混合对辐射平衡有着深远的影响，从而导致对气候变化的预测。波也是大气遥相关的一个因素：它们可以产生远距离影响，因为它们在不传输流体本身的情况下远距离携带能量。由于波浪和混合过程发生的长度尺度太小、太快，无法与行星尺度的流动一起计算，因此现在它们是由使用简单准则的少量聚合参数（参数化）来表示的。对动力学和混合过程的更好理解将改进这些参数化和使用它们的气候模式的可靠性。

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.