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

合作研究：不可压缩流体流动中旋转、分层和耗散影响的数学分析

• 批准号: 2206491
• 负责人: Vincent Martinez
• 金额: $ 10.6万
• 依托单位: CUNY Hunter College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-09-01 至 2025-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2206491&HistoricalAwards=false
• 关键词: Collaborative; Research; Mathematical; Analysis; Effects

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.

地球的旋转，密度的分层和摩擦的产生都在地球物理烟道运动中起着重要作用。这些机制始终存在于我们的海洋和气氛中，以及其他行星的机制。他们对我们在自然界中观察到的许多众所周知的现象集体负责，例如喷气流，Zonal Jets，Elniñocycle和Jupiter的Great Red Spot，仅举几例。这些机制通常以非常特殊的方式来限制流体的运动。例如，观察到，在三维迅速旋转的流体中，沿着公共垂直轴平行于与旋转轴平行地对齐的粒子几乎是一致的，从而使流动的总体运动基本上是二维的。尽管有许多实验和计算努力以了解这种现象的精确发展，但它们的数学理由，也就是说，从直接研究运动方程本​​身，仍然在很大程度上开放。该项目将系统地解决各种地球物理环境中的此类问题。该项目还将为本科生和研究生级别的学生以及博士后学者提供研究和心理机会。该项目的总体目标是了解有限维持性的各种表现及其与耗散，旋转和分层机制的互连。主要方法将是通过研究相关运动方程的解决方案的规律性和长期行为，使人可以在代表这些各种机制的强度的参数之间获得精确的定量关系，并具有最小的相关长度尺度。感兴趣的主要模型将是自然出现在地球物理学中的模型，例如旋转的Navier-Stokes方程和稳定的分层BoussinesQ方程。为了正确量化旋转和分层，各向异性分散估计以及对这种系统固有的谐振结构的仔细分析。尽管两种方法在研究解决方案的规律性方面都取得了成功，但基于频率的方法非常适合量化自由度的数量，而基于物理空间的方法非常适合利用有关空间分析性半径的信息。该项目试图以允许人们共同探索物理空间和频率空间的降低的方式合并这两种方法，这些方法在旋转或分层的流体流动中都可以观察到。该奖项反映了NSF的法定任务，并被认为是通过基金会的知识分子的智力和宽广的影响来评估Criteria criteria criteria criteria criteria criteria criteria criteria criteria。