The study of transport and mixing in geophysical flows is a vast field, which can include everything from effluent discharging into estuaries to large-scale transport and mixing processes in relation to stratospheric ozone composition. Numerical modeling and modern observational data can provide, for example, essential information about the spread of pollution and hence the potential impact that it has on our environment. Continual advances in the capability of computers have allowed the study of models which are more realistic. However, the computer alone is not sufficient. It is necessary to have a solid grasp of the physics involved with the particular aspect of flow under consideration, and knowledge of advanced mathematics to ensure the correctness of the models and to provide a predictive framework for the questions of interest. A growing community of researchers from diverse fields is now contributing to this approach.

Dynamical systems theory offers an ideal mathematical framework in which to study Lagrangian transport, stirring, and mixing issues in geophysical flows. The mathematical framework of dynamical systems theory and the Lagrangian experimental and observational techniques of oceanography have a natural correspondence. Current-following floats, drifters, and remote-sensing data show numerous localized, coherent structures ranging from major currents like the Gulf Stream and mesoscale phenomena such as rings and associated vortex structures, down to a variety of submesoscale features such as filaments, squirts, and mushroom vortices. It is precisely to study the role of localized structures in governing the motion over extended regions of space that the theoretical tools of dynamical systems have been developed.

There has been much work on applying dynamical systems techniques to the study of mixing and transport issues in general fluid mechanics over the past 10 years. However, most of this work has been done in the context of two-dimensional, time-periodic flows. There is good reason for this. In this situation the equations for fluid particle paths take the form of Hamilton's canonical equations with the streamfunction playing the role of the Hamiltonian function. If the flow is time-periodic, then the study of fluid particle trajectories can be reduced to the study of an associated two-dimensional, area-preserving Poincare map. In this setting many techniques of dynamical systems theory can be immediately applied. Moreover, they have immediate implications for fluid transport issues. For example, KAM tori are barriers to the transport of fluid and Smale horseshoes give rise to chaotic fluid particle paths and rapid mixing. Essentially all of this work has been in the situation where one has an explicit analytical formula for the velocity field. From the point of view of applications, this is a severe limitation, and this is where computational fluid dynamics enters the picture.

Over the past 20 years, computational fluid dynamics has developed into a subject in its own right. Now we have accurate algorithms for solving the Navier-Stokes equations in a variety of physically important settings. However, many problems related to mixing and transport begin once this step has been accomplished. That is, first a solution to the Navier-Stokes equations must be obtained in order to study the transport and mixing properties associated with that velocity field. In the vast majority of situations arising in applications, this solution can only be obtained numerically. Thus, we only have a numerical representation of the vector field.

At the same time, science in general is entering an era of "data wealth". For example, now we are able to measure the velocity field in real time over limited regions using remote-sensing techniques such as high-frequency radar arrays. This capability will only increase in the future. Consequently, we are led to think of a "dynamical system" as not being limted to a representation by standard functions (e.g., trigonometric functions, polynomials, special functions such as Bessel functions, Legendre polynomials, etc.), but being given by a data file, which might either be the output of a numerical simulation, or the result of a remote-sensing experiment. It is when this point of view is adopted that the power of dynamical systems theory really becomes apparent since most dynamical systems results are not dependent on a specific analytical form of the dynamical system under consideration. Rather, they require that only certain geometrical features be present. For example, the existence of stable and unstable manifolds of some invariant set requires only the existence of a hyperbolic invariant set; the existence of Smale horseshoe type chaos requires only the transverse intersection of the stable and unstable manifolds of a hyperbolic periodic orbit; and the existence of KAM tori requires only that the flow be a two-dimensional time-periodic perturbation of an integrable flow that has a region of closed streamlines. If it is known that these structures are present in the flow, then this information, along with information on their geometrical arrangement in the flow, can be used to gain a quantitative understanding of transport. For example, if a flow with periodic boundary conditions contains a KAM torus then, neglecting molecular diffusion, an initial distribution of tracer that is both inside and outside the KAM torus will exhibit asymptotic t² dispersion. If the effects of molecular diffusion are considered, then in the high Peclet number limit the effective diffusivity scales like the square of the Peclet number. The existence of a Smale horseshoe implies the existence of local exponential expansion of fluid line elements and rapid stirring of fluid. The stable and unstable manifolds of hyperbolic periodic orbits may form a template which governs large-scale transport in a flow. A common feature of each of these examples is that a "low dimensional'" geometric feature of the flow can be used to quantify a more global feature of the transport. There are growing opportunities to exploit the dynamical systems point of view to study and analyze geophysical flows. Many of these are described in the following pages.

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