Questions about mixing and transport in fluid flows raise some fundamental issues in dynamical systems. A fluid particle follows a Lagrangian trajectory of the velocity field which, in turn, is obtained from solving an appropriate partial differential equation. To assess global reorganizations of the fluid particles, structures such as invariant manifolds of hyperbolic trajectories, invariant tori and other dynamic structures can be used to great effect.
The challenges to dynamicist arise in the context of large-scale flows such as occur in the ocean. Much activity in the ocean is mediated by mesocale ($\sim$ 100 km) features such as eddies and jets. As these structures move through ocean waters, fluid is exchanged between them and the ambient water, as well as between their different internal parts. Quantifying the extent of such transport and understanding the underlying physical mechanisms responsible for it are important issues for oceanographers.
All but the simplest model flows are given, however, as numerical solutions of the relevant PDEs. This immediately puts the dynamicist into new territory as techniques are needed that are adapted to velocity fields given as numerical data. Moreover the flows are rarely periodic and usually only known for finite periods of time.
In this contribution, the basic theory of transport due to ``chaotic advection'' will be presented. Recent advances will be described that extend the basic theory to the case of ``finite-time'' flows and applications of these techniques to specific ocean scenarios will be described.
May 18 1998