Abstract. In this paper we develop analytical and numerical methods for finding special hyperbolic trajectories that govern geometry of Lagrangian structures in time-dependent vector fields. The vector fields (or velocity fields) may have arbitrary time dependence and be realized only as data sets over finite time intervals, where space and time are discretized. While the notion of a hyperbolic trajectory is central to dynamical systems theory, much of the theoretical developments for Lagrangian transport proceed under the assumption that such a special hyperbolic trajectory exists. This brings in new mathematical issues that must be addressed in order for Lagrangian transport theory to be applicable in practice, i.e. how to determine whether or not such a trajectory exists and, if it does exist, how to identify it in a sequence of instantaneous velocity fields. We address these issues by developing the notion of a distinguished hyperbolic trajectory (DHT). We develop an existence criteria for certain classes of DHTs in general time-dependent velocity fields, based on the time evolution of Eulerian structures that are observed in individual instantaneous fields over the entire time interval of the data set. We demonstrate the concept of DHTs in inhomogeneous (or "forced") time-dependent linear systems and develop a theory and analytical formula for computing DHTs. Throughout this work the notion of linearization is very important. This is not surprising since hyperbolicity is a "linearized" notion. To extend the analytical formula to more general nonlinear time-dependent velocity fields, we develop a series of coordinate transforms including a type of linearization that is not typically used in dynamical systems theory. We refer to it as Eulerian linearization, which is related to the frame independence of DHTs, as opposed to the Lagrangian linearization, which is typical in dynamical systems theory, which is used in the computation of Lyapunov exponents. We present the numerical implementation of our method which can be applied to the velocity field Correspondence to: S. Wiggins (s.wiggins@bristol.ac.uk) given as a data set. The main innovation of our method is that it provides an approximation to the DHT for the entire time-interval of the data set. This offers a great advantage over the conventional methods that require certain regions to converge to the DHT in the appropriate direction of time and hence much of the data at the beginning and end of the time interval is lost.

Abstract. In this paper, we describe a unified algorithm for locating and computing hyperbolic trajectories, and then computing their stable and unstable manifolds, for finite time velocity fields in the form of a data set defined on a spacetime grid. The algorithm is applied to a turbulent regime of a quasigeostrophic wind-driven double gyre in a rectangular domain.

Here we report on an effort to describe in detail the evolution of surface water particles in Monterey Bay from the time they first enter until the time they leave. The data used for this study are objective mappings from hourly surface currents obtained from high frequency (HF) radar measurements in Monterey Bay for the period 2 June through 4 August 1999. The basic concept is simple: compute the origin and fate of a large number of particles for every hour during the analysis period. However, analyzing and displaying the enormous amount of computed trajectory information required a new data compression technique: synoptic Lagrangian maps produced by representing each trajectory by its origin/fate and its residence time. The results show unexpected complexity and variability not apparent in the Eulerian current archive. For example, the fraction of particles that escaped to the open ocean during this period varied from about 17 to more than 92 percent. Mean particle residence times ranged from 4.5 to 11 days. The distribution of particle residence times and transport pathways varied over time scales from hours to weeks, and space scales from 2 to 40 km. The wide range of variability in particle properties reported here shows that surface transport studies in Monterey Bay require detailed wind and tidal current information over the entire bay, as well as information about the flow along the open ocean boundary.

The analytical tec . . . iiques described in the previous talk are used to construct synoptic Lagrangian maps (SLMs) kom HF radar measurements in Monterey Bay forAugust 1994 andAugustthroughOctober 1999. The velocity fields are dominated by diurnal and semidiumal fluctuations due to combined wind and tide effects, so that the statistical characteristics of the flow change little from day to day. However, the spatial pattems in the SLMs and incomingkxiting regions show remarkable variability over time scales ranging kom hours to years.. .Hourly changes correlate well with semi-diurnal and diurnal velocity fluctuations. Asequence oftwelve weekly snapshots shows marked weekly changes in the large scale SLM structure, with no apparent repetition. At monthly scales, the period August through Octc ~ x 1999 showed dramatic short period shifts in the percentage of particles that escaped to the open ocean. In early September there was a five-day period when virtually no particles left the bay. Yet less than two weeks later approximately 60 % of all particles left the bay. Dramatic interannual variability also exists. A comparison of the synoptic Lagrangianmaps for August 1994 andAugust 1999 shows vast differences in the fate of surface particles, although the wind patterns for these months were quite similar. InAugust 1994 most ofthe particles to the open ocean, while in 1999 most encountered the coast. These results provide new insight and analysis tools for studytng the role of surface transport variability in ecology and pollution management.

In the past five years there have been si@cant advances in dynamical systems theory to the point where the framework can now be utilized in the context of "real" problems. In this talk we will briefly describe the dynamical systems framework for Lagrangian transport and the new analysis tools it gives us for obtaining synoptic summaries of particle motion. Our focus will be on transport in a coastal system (i.e., Monterey Bay) using a velocity field obtained from high frequency (I-E) radar measurements.In this talk we describe two new analyt~cal techniques for studying transport: synoptic Lagrangian maps (SLMs) and incoming/exiting regions. SLM's is a method for compressing the information contained in millions of trajectories and, at the same time, reveals detailed, timedependent, L a p g i a n structures. The incominglexiting regions reveal the time varying, geometrical structures in the flow responsible for controlling access to the bay. Both methods also provide novel ways to compute certain statistical quantities related to transport. 0-933957-30-0 2044