On the Emergence of Small and Large Scales in Fluid Motion

运动的尺度，无论大小，在流体系统中无处不在、自发地出现。一个典型的例子是在早晨的咖啡中加入奶油；奶油折叠成细丝，变成越来越小的鳞片，而咖啡本身的运动似乎从这些细丝中形成了一个大漩涡。在地球物理和天体物理系统中，通常可以观察到自发的自组织形成大规模喷流（如墨西哥湾流、木星喷流）和涡旋（如海洋环流、飓风）。预测这些相干结构的振幅、内部结构和长期行为对天气预报、气候科学和天体物理学等领域具有重要意义。该项目的主要目标是促进对流体运动中这些粗尺度和细尺度特征如何出现的理解。本研究的最终目的是给出在无粘性和微粘性流动中长时间出现的持久结构和行为的第一性原理图。该项目还将为研究生提供参与研究的机会。流体结构可以比强迫的尺度大得多，并且常常以嵌入在小尺度波动海洋中的“层流状态”共存。这些特征的详细形式和行为以一种高度非线性的方式取决于流中运动的全部范围。该项目将定量研究由Navier-Stokes和Euler方程控制的微粘性不可压缩二维流体中出现的小型和大型结构。对于大规模地层，将构建由（与粘度无关的）力维持并连接到无粘欧拉状态的固定Navier-Stokes解的分支。所考虑的强迫要么是通过大块（Kolmogorov问题），要么是通过移动边界。选择“正确的”无粘欧拉解的问题，在某种意义上，它们可以在粘度消失时获得，将被解决。关于小规模的创造，该项目将研究熵的直接级联，特别是在稳定状态附近的涡度扰动如何倾向于在背景大尺度剪切下形成丝状。研究了强范数下稳态的非线性不稳定性和无限时间奇点的形成。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Scales of motion, large and small, appear ubiquitously and spontaneously in fluid systems. A typical example arises when cream is stirred into morning coffee; the cream folds and filaments into ever decreasing scales, while the motion of the coffee itself appears to form, out of these filaments, one large swirl. In geophysical and astrophysical systems, spontaneous self-organization into large-scale jets (e.g., gulf stream, Jovian jets) and vortices (e.g., oceanic gyres, hurricanes) is commonly observed. The prediction of the amplitude, internal structure, and long-term behavior of these coherent structures is of fundamental importance for weather prediction, climate science, and astrophysics, among other fields. The main objective of the project is to advance understanding of how such coarse and fine-scale features emerge in fluid motion. The ultimate aim of this research is to give a first-principles picture of the persistent structures and behaviors that emerge at long times in inviscid and slightly viscous flows. The project will also provide opportunities for involvement of graduate students in the research.Fluid structures can be significantly larger than the scale of forcing and often coexist as "laminar states" embedded in a sea of small-scale fluctuations. The detailed form and behavior of these features depends in a highly nonlinear fashion on the full range of scales of motion in the flow. The project will quantitatively study the emergence of small and large-scale structures in slightly viscous incompressible 2D fluids governed by the Navier-Stokes and Euler equations. Regarding large-scale formation, branches of stationary Navier-Stokes solutions that are sustained by (viscosity independent) forces and connect to inviscid Euler states will be constructed. The forcing considered will either be through the bulk (the Kolmogorov problem) or through a moving boundary. The issue of selecting "correct" inviscid Euler solutions, in the sense that they can be attained as viscosity vanishes, will be addressed. Concerning small-scale creation, the project will study the direct cascade of enstrophy and, in particular, how vorticity perturbations near stationary states tend to get filamented by background large-scale shearing. Nonlinear instability of steady states in strong norms and singularity formation at infinite time will be investigated.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.