Physics Reports 437 (2006) 55 – 124 www.elsevier.com/locate/physrep A tutorial on dynamical systems concepts applied to Lagrangian transport in oceanic flows defined as finite time data sets: Theoretical and computational issues Ana M. Mancho a , 1 , Des Small b , Stephen Wiggins b, ∗, 2 a Departamento de Matemáticas, IMAFF, Consejo Superior Investigaciones Científicas, Serrano 121, 28006 Madrid, Spain b School of Mathematics, University of Bristol, UniversityWalk, Bristol, BS8 1TW, UK Accepted 5 September 2006 editor: D.K. Campbell Abstract In the past 15 years the framework and ideas from dynamical systems theory have been applied to a variety of transport and mixing problems in oceanic flows. The motivation for this approach comes directly from 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. In this tutorial, we motivate this approach by showing the relationship between fluid transport in two-dimensional time-periodic incompressible flows and the geometrical structures that exist for two-dimensional area-preserving maps, such as hyperbolic periodic orbits, their stable and unstable manifolds and KAM (Kolmogorov–Arnold–Moser) tori. This serves to set the stage for the attempt to “transfer” this approach to more realistic flows modelling the ocean. However, in order to accomplish this several difficulties must be overcome. The first difficulty that confronts us that any attempt to carry out a dynamical systems approach to transport requires us to obtain the appropriate “dynamical system”, which is the velocity field describing the fluid flow. In general, adequate model velocity fields are obtained by numerical solution of appropriate partial differential equations describing the dynamical evolution of the velocity field. Numerical solution of the partial differential equations can only be done for a finite time interval, and since the ocean is generally not time-periodic, this leads to a new type of dynamical system: a finite-time, aperiodically time-dependent velocity field defined as a data set on a space–time grid. The global, geometrical analysis of transport in such dynamical systems requires both new concepts and new analytical and computational tools, as well as the necessity to discard some of the standard ideas and results from dynamical systems theory. The purpose of this tutorial is to describe these new concepts and analytical tools first using simple ∗ Corresponding author. E-mail address: s.wiggins@bristol.ac.uk (S. Wiggins). 1 Research supported by a Ramón y Cajal Research Fellowship and a grant from the MEC (Spanish Government) reference MTM2004-00797 and a Royal Society-CSIC cooperation agreement reference B2003GB03. 2 Research supported by ONR Grant No. N00014-01-1-0769. 0370-1573/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physrep.2006.09.005