ALBERT

Notes: Chaotic advection and, more generally, ideas from dynamical systems, have been fruitfully applied to a diverse, and varied, collection of mixing and transport problems arising in engineering applications over the past 20 years. Indeed, the "dynamical systems approach" was developed, and tested, to the point where it can now be considered a standard tool for understanding mixing and transport issues in many disciplines. This success for engineering-type flows motivated an effort to apply this approach to transport and mixing problems in geophysical flows. However, there are fundamental difficulties arising in this endeavor that must be properly understood and overcome. Central to this approach is that the starting point for analysis is a velocity field (i.e., the "dynamical system"). In many engineering applications this can be obtained sufficiently accurately, either analytically or computationally, so that it describes particle trajectories for the actual flow. However, in geophysical flows (and we concentrate here almost exclusively on oceanographic flows), the wide range of dynamically significant time and length scales makes the justification of any velocity field, in the sense of reproducing particle trajectories for the actual flow, a much more difficult matter. Nevertheless, the case for this approach is compelling due to the advances in observational capabilities in oceanography (e.g., drifter deployments, remote sensing capabilities, satellite imagery, etc.), which reveal space-time structures that are highly suggestive of the structures one visualizes in the global, geometrical study of dynamical systems theory. This has been pursued in recent years through a combination of laboratory studies, kinematic models, and dynamically consistent models that have all been compared with observational data. During the course of these studies it has become apparent that a new type of dynamical system is necessary to consider in these studies (i.e., a finite time, aperiodically time-dependent velocity field defined as a data set), which requires the development of new analytical and computational tools, as well as the necessity to discard some of the standard ideas and results from dynamical systems theory. In this article we review a number of the key developments to date in this young, but rapidly developing, area at the interface between geophysical fluid dynamics and applied and computational mathematics. We also describe the wealth of new directions for research that this approach unlocks.