On the Emergence of Small and Large Scales in Fluid Motion

在流体系统中，运动的大小尺度无处不在，而且是自发的。一个典型的例子是，当奶油被搅拌到早餐咖啡中时，奶油折叠并形成不断减少的细丝，而咖啡本身的运动似乎从这些细丝中形成了一个巨大的漩涡。在地球物理和天体物理系统中，通常可以观察到自发地自组织成大尺度急流(例如，墨西哥湾流、木星急流)和涡旋(例如，大洋旋涡、飓风)。对这些相干结构的振幅、内部结构和长期行为的预测对于天气预报、气候科学和天体物理等领域都是至关重要的。该项目的主要目标是促进对这种粗细尺度特征如何在流体运动中出现的理解。这项研究的最终目的是给出在无粘和轻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景。该项目还将为研究生提供参与研究的机会。流体结构可能比强迫的规模大得多，通常以嵌入小尺度波动海洋的“层流状态”共存。这些特征的详细形式和行为以高度非线性的方式取决于流动中运动的全部范围。该项目将定量研究由Navier-Stokes方程和Euler方程支配的微粘性不可压缩2D流体中小尺度和大尺度结构的出现。对于大尺度的形成，将构造由(粘性无关)力维持并连接到无粘欧拉态的定常Navier-Stokes解的分支。所考虑的强迫要么是通过整体(科尔莫戈罗夫问题)，要么是通过移动的边界。选择“正确的”无粘性欧拉解的问题将被解决，在这个意义上，它们可以随着粘度的消失而获得。关于小规模的创造，该项目将研究涡旋的直接级联，特别是定态附近的涡度扰动如何因背景大尺度剪切而变得细丝。这一奖项反映了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。

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.