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.

行星的自转、密度的分层和摩擦力的产生都在地球物理流体的运动中发挥着重要作用，这些机制始终存在于我们的海洋和大气中，以及其他行星的共同作用中。对于我们在自然界中观察到的许多众所周知的现象，例如急流、纬向急流、厄尔尼诺周期和木星大红斑，仅举几例，这些机制通常用于限制流体的运动。非常特别例如，可以观察到，在三维快速旋转的流体中，沿着与旋转轴平行的共同垂直线排列的颗粒几乎一致地移动，从而使流体的整体运动本质上是两个。尽管为了解此类现象的精确发展进行了许多实验和计算工作，但其数学论证（即直接研究运动方程本​​身）仍然在很大程度上是开放的，该项目将系统地解决各种地球物理学中的此类问题。该项目还将提供研究和指导机会。适合本科生和研究生以及博士后学者。该项目的首要目标是了解有限维的各种表现形式及其与耗散、旋转和分层机制的相互关系。对相关运动方程解的规律性和长期行为的研究，使人们能够获得代表这些不同机制的强度的参数与感兴趣的流体流动的最小相关长度尺度之间的定量精确关系。是地球物理学中自然产生的方程，例如旋转纳维-斯托克斯方程和稳定分层布辛斯克方程。为了正确量化旋转和分层所带来的影响，需要进行各向异性色散估计，并对此类系统固有的共振结构进行仔细分析。该项目的一个新颖之处是基于物理空间和基于频率空间的方法之间的相互作用，尽管这两种方法在研究解决方案的规律性方面都取得了成功，但基于频率空间的方法是。非常适合量化自由度数，而基于物理空间的方法非常适合利用有关空间分析半径的信息，该项目试图以允许共同利用缩减的方式合并这两种方法。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维度。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。