On the Emergence of Small and Large Scales in Fluid Motion

Suny At Stony Brook, Stony Brook NY

Investigators

Abstract

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.