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

地球自转、密度分层和摩擦的产生都在地球物理流体的运动中起着重要作用。这些机制总是存在于我们的海洋和大气中，以及其他行星的大气中。它们共同为我们在自然界中观察到的许多众所周知的现象负责，例如喷流、纬向喷流、厄尔尼诺循环和木星大红斑，仅举几例。这些机制通常用于以一种非常特殊的方式限制流体的运动。例如，人们观察到，在三维快速旋转的流体中，沿着与旋转轴平行的共同垂直方向排列的颗粒几乎一致地运动，从而使流动的整体运动基本上是二维的。尽管进行了许多实验和计算努力来了解这种现象的精确发展，但对它们的数学证明，即直接研究运动方程本身，在很大程度上仍然是开放的。该项目将系统地解决各种地球物理环境中的这种关切。这个项目还将为本科生和研究生以及博士后奖学金提供研究和指导的机会。这个项目的总体目标是了解有限维的各种表现及其与耗散、旋转和分层机制的相互联系。主要方法将通过研究相关运动方程的解的规律性和长期行为，使人们能够获得代表这些不同机构的强度的参数之间的精确定量关系，以及流体流动的最小相关长度尺度。令人感兴趣的主要模型将是那些在地球物理学中自然产生的模型，如旋转的Navier-Stokes方程和稳定分层的Boussinesq方程。为了适当地量化旋转和分层带来的影响，将进行各向异性色散估计，并仔细分析这种系统固有的共振结构。该项目的一个新奇之处是物理空间方法和频率空间方法之间的相互作用。虽然这两种方法在研究解的规律性方面都取得了成功，但基于频率的方法非常适合于量化自由度数，而基于物理空间的方法非常适合于利用关于空间分析半径的信息。该项目试图将这两种方法融合在一起，使人们能够共同利用在旋转或分层流体流动中观察到的物理空间和频率空间的降维。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.