Lagrangian coherent structures in n-dimensional systems

Numerical simulations and experimental observations reveal that unsteady fluid systems can be divided into regions of qualitatively different dynamics. The key to understanding transport and stirring is to identify the dynamic boundaries between these almost-invariant regions. Recently, ridges in finite-time Lyapunov exponent fields have been used to define such hyperbolic, almost material, Lagrangian coherent structures in two-dimensional systems. The objective of this paper is to develop and apply a similar theory in higher dimensional spaces. While the separatrix nature of these structures is their most important property, a necessary condition is their almost material nature. This property is addressed in this paper. These results are applied to a model of Rayleigh-Bénard convection based on a three-dimensional extension of the model of Solomon and Gollub

Lagrangian Structures in Very High-Frequency Radar Data and Optimal Pollution Timing

Very High-frequency (VHF) radar technology produces detailed surface velocity maps near the surface of coastal waters. The use of measured velocity data in environmental prediction, however, has remained unexplored. In this paper, we uncover a striking flow structure in coastal radar observations along the coast of Florida. This structure governs the spread of organic contaminants or passive drifters released in the area. We compute the Lyapunov exponents of the VHF radar data to determine optimal release windows in which contaminants are advected efficiently away from the coast and we show that a VHF radar-based pollution release scheme using the hidden flow structure reduces the effect of industrial pollution in the coastal environment

Optimal Pollution Mitigation in Monterey Bay Based on Coastal Radar Data and Nonlinear Dynamics

The article of record as published may be found at http://dx.doi.org/10.1021/es0630691High-frequency (HF) radar technology produces detailed velocity maps near the surface of estuaries and bays. The use of velocity data in environmental prediction, nonetheless, remains unexplored. In this paper, we uncover a striking flow structure in coastal radar observations of Monterey Bay, along the California coastline. This complex structure governs the spread of organic contaminants, such as agricultural runoff which is a typical source of pollution in the bay. We show that a HF radar-based pollution release scheme using this flow structure reduces the impact of pollution on the coastal environment in the bay. We predict the motion of the Lagrangian flow structures from finite-time Lyapunov exponents of the coastal HF velocity data. From this prediction, we obtain optimal release times, at which pollution leaves the bay most efficiently.Office of Naval Research grant N00014-01-1-0208ASAP MURI N00014-02-1-0826Office of Naval Research grant N00014-01-1-0208ASAP MURI N00014-02-1-082

Collective motion, sensor networks, and ocean sampling

Author Posting. © IEEE, 2007. This article is posted here by permission of IEEE for personal use, not for redistribution. The definitive version was published in Proceedings of the IEEE 95 (2007): 48-74, doi:10.1109/jproc.2006.887295.This paper addresses the design of mobile sensor networks for optimal data collection. The development is strongly motivated by the application to adaptive ocean sampling for an autonomous ocean observing and prediction system. A performance metric, used to derive optimal paths for the network of mobile sensors, defines the optimal data set as one which minimizes error in a model estimate of the sampled field. Feedback control laws are presented that stably coordinate sensors on structured tracks that have been optimized over a minimal set of parameters. Optimal, closed-loop solutions are computed in a number of low-dimensional cases to illustrate the methodology. Robustness of the performance to the influence of a steady flow field on relatively slow-moving mobile sensors is also explored

Computational Method for Phase Space Transport with Applications to Lobe Dynamics and Rate of Escape

Lobe dynamics and escape from a potential well are general frameworks introduced to study phase space transport in chaotic dynamical systems. While the former approach studies how regions of phase space are transported by reducing the flow to a two-dimensional map, the latter approach studies the phase space structures that lead to critical events by crossing periodic orbit around saddles. Both of these frameworks require computation with curves represented by millions of points-computing intersection points between these curves and area bounded by the segments of these curves-for quantifying the transport and escape rate. We present a theory for computing these intersection points and the area bounded between the segments of these curves based on a classification of the intersection points using equivalence class. We also present an alternate theory for curves with nontransverse intersections and a method to increase the density of points on the curves for locating the intersection points accurately.The numerical implementation of the theory presented herein is available as an open source software called Lober. We used this package to demonstrate the application of the theory to lobe dynamics that arises in fluid mechanics, and rate of escape from a potential well that arises in ship dynamics.Comment: 33 pages, 17 figure

Time-Dependent Dynamical Systems and Geophysical Flows

This thesis presents a dynamical systems approach to transport and mixing in geophysical flows. First, new algorithms are developed that allow one to study a dynamical system that is described in a variety of ways such as by means of observational data or numerical simulations of differential equations. Next, methods available to study non-autonomous systems, such as hyperbolic trajectories and Lagrangian coherent structures, are developed. These concepts are applied to examples of interests: Monterey Bay, the coast of Florida and the circulation in the North Atlantic. Combining accurate current measurements and recent developments in dynamical systems theory provides new and original answers to many problems, such as the minimization of the impact of released contaminants in a coastal area or the optimization of the coverage by a group of drifters. The appendices give details about MANGEN, a software package developed to produce the numerical results of this thesis. Some projects that make use of its algorithms, such as the dissociation rate of a molecule and efficient space mission design, are also described.</p