Transport, Mixing and Coherent Structures in Chaotic Flows

In this talk, I will describe a dynamical systems framework for studying Lagrangian transport in time-chaotic systems. Numerous experiments have revealed the presence of coherent structures that govern transport and mixing in such systems. These mixing templates are invisible to the naked eye but can be extracted, for example, by computing finite-time Lyapunov exponents. Their presence indicates alleyways and barriers to transport and provide a geometric description of the mixing processes in the system. I will discuss the properties of these Lagrangian coherent structures and the numerical methods used to compute them.

Applications range from space missions to molecular fragmentation. In recent years, there has been much effort in applying this approach to the study of mixing in fluids. I will describe fluid transport near the Atlantic coast of Florida using a velocity field observed experimentally from high frequency radar measurements, a new technology that produces detailed velocity maps near the surface of coastal waters. Recent numerical methods permits the computation of dynamical barriers and alleyways in this complex system and reveal the existence of optimal release windows in which contaminants can be efficiently advected away from the coast, reducing their negative impact on the marine environment.

Identification of Lagrangian coherent structures can also enhance the performance of adaptive sampling networks. I will introduce the Autonomous Ocean Sampling Network II project that coordinates the effort of high-frequency radar, ships, airplanes, satellites, buoys and underwater vehicles for the purpose of improving the observation and the prediction of ocean processes in Monterey Bay, CA. Lagrangian coherent structures can be used to locate biological and sampling ``hot spots.'' They aalso determine efficient routes for vehicles and highlight regions of high stretching that may adversely affect the stability of a vehicle formation.

Transport and mixing near the surface of an airfoil or a coastline can also be studied in terms of separation and re-attachment near the boundary. I will discuss the recent development of exact criteria to detect and extract separation points and related separation profiles. The existence of such analytic criteria permits the development of efficient algorithms to detect and control the separation points in a fluid. I will describe how separation can be controlled in jet-actuated systems. Such a mechanism permits a fine control of the lift on an airfoil or the efficient transport of fuel along separation profiles.