Academic literature on the topic 'Lagrangian Coherent Structuries'

The subject of this thesis is the detection of Lagrangian Coherent Structures (LCS) in three-dimensional steady flows. LCS are influential material surfaces that act as skeletons of observed mixing patterns in a dynamical system. We review recent variational results for the construction of LCS in two-dimensional flows, and develop our own numerical implementation of these methods. We first test these implementations on two-dimensional examples, in which we obtain some new results and insights. Next, we apply the two-dimensional variational theory to the two-dimensional Poincaré map of a three-dimensional steady flow. For the Poincaré map of the classic ABC flow, we obtain Elliptic and Parabolic LCS at a previously unseen level of detail. Some of these structures, such as twistless KAM curves, have been previously unknown for this flow. Using the flow map, we finally extend the two-dimensional findings to uncover exact transport barrier surfaces in the full ABC flow. This approach applies to transport barrier detection in any three-dimensional, autonomous dynamical system.<br>Le contenu de cette thèse porte sur la détection des Structures Cohérentes de Lagrange (LCS) au sein des flots tri-dimensionnels stationnaires. Les LCS sont des surfaces matérielles de haute importance, qui font office de squelettes parmi les patterns mélangeants observés dans un système dynamique. Nous présentons tout d'abord des résultats variationnels récents sur la construction des LCS dans les flots bi-dimensionnels, puis développons nos propres implémentations numériques de ces méthodes. En premier lieu, nous testons ces implémentations sur des exemples bi-dimensionnels, et obtenons à la fois des résultats inédits et de nouvelles perspectives concernants ces problèmes. Nous appliquons ensuite la théorie du calcul variationnel en deux dimensions à l'application de Poincaré bi-dimensionnelle d'un flot stationnaire tri-dimensionnel. Dans le cas de l'application de Poincaré du flot ABC classique, nous obtenons des LCS elliptiques et paraboliques d'une résolution précédemment inegalée. Certaines de ces structures, telles que des courbes KAM non-tourbillonantes, n'étaient pas connues auparavant comme étant présentes dans ce flot. À l'aide du flot, nous étendons ces résultats bi-dimensionnels et mettons au jour des surfaces de barrière au transport dans l'entièreté du flot ABC. Cette approche s'applique à la détection des barrières au transport dans n'importe quel système dynamique autonome tri-dimensionnel.

In order to understand water quality in the coastal ocean and its effects on human health, the necessity arises to locate the sources of contaminants and track their transport throughout the ocean. Dynamical systems methods are applied to the study of transport of enterococci as an indicator of microbial concentration in the vicinity of Hobie Beach, an urban, subtropical beach in Miami, FL that is used for recreation and bathing on a daily basis. Previous studies on water quality have shown that Hobie Beach has high microbial levels despite having no known point source. To investigate the cause of these high microbial levels, a combination of measured surface drifter trajectories and numerically simulated flows in the vicinity of Hobie Beach is used. The numerically simulated flows are used to identify Lagrangian Coherent Structures (LCSs), which provide a template for transport in the study area. Surface drifter trajectories are shown to be consistent with the simulated flows and the LCS structure. LCSs are then used to explain the persistent water contamination and unusually high concentrations of microbes in the water off of this beach as compared with its neighboring beaches. From the drifter simulations, as well as field experiments, one can see that passive tracers are trapped in the area along the coastline by LCS. The Lagrangian circulation of Hobie Beach, influenced primarily by tide and land geometry causes a high retention rate of water near the shore, and can be used to explain the elevated levels of enterococci in the water.

L’étude des processus physiques d’un système d’upwelling est essentielle pour comprendre sa variabilité actuelle et ses changements passés et futurs. Cette thèse présente une étude interdisciplinaire du système d’upwelling côtier à partir de différentes données acquises par satellite, l’accent étant mis principalement sur le système d’upwelling d’Afrique du Nord-Ouest (NWA). Cette étude interdisciplinaire aborde (1) le problème de l’identification et de l’extraction automatiques du phénomène d’upwelling à partir d’observations satellitaires biologiques et physiques. (2) Une étude statistique de la variation spatio-temporelle de l’upwelling de la NWA tout au long de son extension et de ses différents indices d’upwelling. (3) Une étude des relations non linéaires entre le mélange de surface et l’activité biologique dans les régions d’upwelling. (4) études lagrangiennes de tourbillons cohérents; leurs propriétés physiques et identification automatique. (5) L’étude des transports effectués par les tourbillons lagrangiens de la NWA Upwelling et leur impact sur l’océan<br>Studying physical processes of an upwelling system is essential to understand its present variability and its past and future changes. This thesis presents an interdisciplinary study of the coastal upwelling system from different satellite acquired data, with the main focus placed on the North West African (NWA) upwelling system. This interdisciplinary study covers (1) the problem of the automatic identification and extraction of the upwelling phenomenon from biological and physical satellite observations. (2) A statistical study of the spatio-temporal variation of the NWA upwelling throughout its extension and different upwelling indices. (3) A Study of the nonlinear relationships between the surface mixing and biological activity in the upwelling regions. (4) Lagrangian studies of coherent eddies; their physical properties and automatic identification. (5) The study of transport made by Lagrangian eddies off the NWA Upwelling and their impact on the open ocean. [...]

In many problems in dynamical systems one is interested in the identification of sets which have qualitatively different fates. The finite-time Lyapunov exponent (FTLE) method is a general and equation-free method that identifies codimension-one sets which have a locally high rate of stretching around which maximal exponential expansion of line elements occurs. These codimension-one sets thus act as transport barriers. This geometric framework of transport barriers is used to study various problems in phase space transport, specifically problems of separation in flows that can vary in scale from the micro to the geophysical. The first problem which we study is of the nontrivial motion of inertial particles in a two-dimensional fluid flow. We use the method of FTLE to identify transport barriers that produce segregation of inertial particles by size. The second problem we study is the long range advective transport of plant pathogen spores in the atmosphere. We compute the FTLE field for isobaric atmospheric flow and identify atmospheric transport barriers (ATBs). We find that rapid temporal changes in the spore concentrations at a sampling point occur due to the passage of these ATBs across the sampling point. We also investigate the theory behind the computation of the FTLE and devise a new method to compute the FTLE which does not rely on the tangent linearization. We do this using the 925 matrix of a probability density function. This method of computing the geometric quantities of stretching and FTLE also heuristically bridge the gap between the geometric and probabilistic methods of studying phase space transport. We show this with two examples.<br>Ph. D.

The transport of material in geophysical fluid flows is a problem with important implications for fields as diverse as: agriculture, aviation, human health, disaster response, and weather forecasting. Due to the unsteady nature of geophysical flows, predicting how material will be transported in these systems can often be challenging. Tools from dynamical systems theory can help to improve the prediction of material transport by revealing important transport structures. These transport structures reveal areas of the flow where fluid parcels, and thus material transported by those parcels, are likely to converge or diverge. Typically, these transport structures have been uncovered by the use of Lagrangian diagnostics. Unfortunately, calculating Lagrangian diagnostics can often be time consuming and computationally expensive. Recently new Eulerian diagnostics have been developed. These diagnostics are faster and less expensive to compute, while still revealing important transport structures in fluid flows. Because Eulerian diagnostics are so new, there is still much about them and their connection to Lagrangian diagnostics that is unknown. This dissertation will fill in some of this gap and provide a mathematical bridge between Lagrangian and Eulerian diagnostics. This dissertation is composed of three projects. These projects represent theoretical, numerical, and experimental advances in the understanding of Eulerian diagnostics and their relationship to Lagrangian diagnostics. The first project rigorously explores the deep mathematical relationship that exists between Eulerian and Lagrangian diagnostics. It proves that some of the new Eulerian diagnostics are the limit of Lagrangian diagnostics as integration time of the velocity field goes to zero. Using this discovery, a new Eulerian diagnostic, infinitesimal-time Lagrangian coherent structures is developed. The second project develops a methodology for estimating local Eulerian diagnostics from wind velocity data measured by a fixed-wing unmanned aircraft system (UAS) flying in circular arcs. Using a simulation environment, it is shown that the Eulerian diagnostic estimates from UAS measurements approximate the true local Eulerian diagnostics and can predict the passage of Lagrangian diagnostics. The third project applies Eulerian diagnostics to experimental data of atmospheric wind measurements. These are then compared to Eulerian diagnostics as calculated from a numerical weather simulation to look for indications of Lagrangian diagnostics.<br>Doctor of Philosophy<br>How particles are moved by fluid flows, such as the oceanic currents and the atmospheric winds, is a problem with important implications for fields as diverse as: agriculture, aviation, human health, disaster response, and weather forecasting. Because these fluid flows tend to change over time, predicting how particles will be moved by these flows can often be challenging. Fortunately, mathematical tools exist which can reveal important geometric features in these flows. These geometric features can help us to visualize regions where particles are likely to come together or spread apart, as they are moved by the flow. In the past, these geometric features have been uncovered by using methods which look at the trajectories of particles in the flow. These methods are referred to as Lagrangian, in honor of the Italian mathematician Joseph-Louis Lagrange. Unfortunately, calculating the trajectories of particles can be a time consuming and computationally expensive process. Recently, new methods have been developed which look at how the speed of the flow changes in space. These new methods are referred to as Eulerian, in honor of the Swiss mathematician Leonhard Euler. These new Eulerian methods are faster and less expensive to calculate, while still revealing important geometric features within the flow. Because these Eulerian methods are so new, there is still much that we do not know about them and their connection to the older Lagrangian methods. This dissertation will fill in some of this gap and provide a mathematical bridge between these two methodologies. This dissertation is composed of three projects. These projects represent theoretical, numerical, and experimental advances in the understanding of these new Eulerian methods and their relationship to the older Lagrangian methods. The first project explores the deep mathematical relationship that exists between Eulerian and Lagrangian diagnostic tools. It mathematically proves that some of the new Eulerian diagnostics are the limit of Lagrangian diagnostics as the trajectory’s integration times is decreased to zero. Taking advantage of this discovery, a new Eulerian diagnostic is developed, called infinitesimal-time Lagrangian coherent structures. The second project develops a technique for estimating local Eulerian diagnostics using wind speed measures from a single fixed-wing unmanned aircraft system (UAS) flying in a circular path. Using computer simulations, we show that the Eulerian diagnostics as calculated from UAS measurements provide a reasonable estimate of the true local Eulerian diagnostics. Furthermore, we show that these Eulerian diagnostics can be used to estimate the local Lagrangian diagnostics. The third project applies these Eulerian diagnostics to real-world wind speed measurements. These results are then compared to Eulerian diagnostics that were calculated from a computer simulation to look for indications of Lagrangian diagnostics.

Thesis: Sc. D., Joint Program in Applied Ocean Science and Engineering (Massachusetts Institute of Technology, Department of Earth, Atmospheric, and Planetary Sciences; and the Woods Hole Oceanographic Institution), 2019<br>Cataloged from PDF version of thesis.<br>Includes bibliographical references (pages 207-218).<br>Ocean surface transport is at the core of many environmental disasters, including the spread of marine plastic pollution, the Deepwater Horizon oil spill and the Fukushima nuclear contamination. Understanding and predicting flow transport, however, remains a scientific challenge, because it operates on multiple length- and time-scales that are set by the underlying dynamics. Building on the recent emergence of Lagrangian methods, this thesis investigates the present-day abilities to describe and understand the organization of flow transport at the ocean surface, including the abilities to detect the underlying key structures, the regions of stirring and regions of coherence within the flow. Over the past four years, the field of dynamical system theory has adapted several algorithms from unsupervised machine learning for the detection of Lagrangian Coherent Structures (LCS). The robustness and applicability of these tools is yet to be proven, especially for geophysical flows.<br>An updated, parameter-free spectral clustering approach is developed and a noise-based cluster coherence metric is proposed to evaluate the resulting clusters. The method is tested against benchmarks flows of dynamical system theory: the quasi-periodic Bickley jet, the Duffing oscillator and a modified, asymmetric Duffing oscillator. The applicability of this newly developed spectral clustering method, along with several common LCS approaches, such as the Finite-Time Lyapunov Exponent, is tested in several field studies. The focus is on the ability to predict these LCS in submesoscale ocean surface flows, given all the uncertainties of the modeled and observed velocity fields, as well as the sparsity of Lagrangian data. This includes the design and execution of field experiments targeting LCS from predictive models and their subsequent Lagrangian analysis.<br>These experiments took place in Scott Reef, an atoll system in Western Australia, and off the coast of Martha's Vineyard, Massachusetts, two case studies with tidally-driven channel flows. The FTLE and spectral clustering analyses were particularly helpful in describing key transient flow features and how they were impacted by tidal forcing and vertical velocities. This could not have been identified from the Eulerian perspective, showing the utility of the Lagrangian approach in understanding the organization of transport.<br>by Margaux Filippi.<br>Sc. D.<br>Sc.D. Joint Program in Applied Ocean Science and Engineering (Massachusetts Institute of Technology, Department of Earth, Atmospheric, and Planetary Sciences; and the Woods Hole Oceanographic Institution)

The theory of Lagrangian Coherent Structures (LCS) enables prediction of material transport by turbulent winds, such as those observed in the Earth's Atmospheric Boundary Layer. In this dissertation, both theory and experimental methods are developed for utilizing small fixed-wing unmanned aircraft systems (UAS) in detecting these atmospheric coherent structures. The dissertation begins by presenting relevant literature on both LCS and airborne wind estimation. Because model-based wind estimation inherently depends on high quality models, a Flight Dynamic Model (FDM) suitable for a small fixed-wing aircraft in turbulent wind is derived in detail. In this presentation, some new theoretical concepts are introduced concerning the proper treatment of spatial wind gradients, and a critical review of existing theories is presented. To enable model-based wind estimation experiments, an experimental approach is detailed for identifying a FDM for a small UAS by combining existing computational aerodynamic and data-driven approaches. Additionally, a methodology for determining wind estimation error directly resulting from dynamic modeling choices is presented and demonstrated. Next, some model-based wind estimation results are presented utilizing the experimentally identified FDM, accompanied by a discussion of model fidelity concerns and other experimental issues. Finally, an algorithm for detecting LCS from a single circling fixed-wing UAS is developed and demonstrated in an Observing System Simulation Experiment. The dissertation concludes by summarizing these contributions and recommending future paths for continuing research.<br>Doctor of Philosophy<br>In a natural or man-made disaster, first responders depend on accurate predictions of where the wind might carry hazardous material. A mathematical theory of Lagrangian Coherent Structures (LCS) has shown promise in ocean environments to improve these predictions, and the theory is also applicable to atmospheric flows near the Earth’s surface. This dissertation presents both theoretical and experimental research efforts towards employing small fixed-wing unmanned aircraft systems (UAS) to detect coherent structures in the Atmospheric Boundary Layer (ABL). These UAS fit several “gaps” in available sensing technology: a small aircraft responds significantly to wind gusts, can be steered to regions of interest, and can be flown in dangerous environments without risking the pilot’s safety. A key focus of this dissertation is to improve the quality of airborne wind measurements provided by inexpensive UAS, specifically by leveraging mathematical models of the aircraft. The dissertation opens by presenting the motivation for this research and existing literature on the topics. Next, a detailed derivation of a suitable Flight Dynamic Model (FDM) for a fixed-wing aircraft in a turbulent wind field is presented. Special attention is paid to the theories for including aerodynamic effects of flying in non-uniform winds. In preparation for wind measurement experiments, a practical method for obtaining better quality FDMs is presented which combines theoretically based and data-driven approaches. A study into the wind-measurement error incurred solely by mathematical modeling is presented, focusing on simplified forms of the FDM which are common in aerospace engineering. Wind estimates which utilize our best available model are presented, accompanied by discussions of the model accuracy and additional wind measurement concerns. A method is developed to detect coherent structures from a circling UAS which is providing wind information, presumably via accurate model based estimation. The dissertation concludes by discussing these conclusions and directions for future research which have been identified during these pursuits.

The Lagrangian dynamics of two-dimensional incompressible fluid flows is considered, with emphasis on transport processes in atmospheric and oceanic flows. The dynamical-systems-based approach is adopted; the Lagrangian motion in such systems is studied with the aid of Kolmogorov-Arnold-Moser (KAM) theory, and results relating to stable and unstable manifolds and lobe dynamics. Some nontrivial extensions of well-known results are discussed, and some extensions of the theory are developed. In problems for which the flow field consists of a steady background on which a time-dependent perturbation is superimposed, it is shown that transport barriers arise naturally and play a critical role in transport processes. Theoretical results are applied to the study of transport in measured and simulated oceanographic and atmospheric flows. Two particular problems are considered. First, we study the Lagrangian dynamics of the zonal jet at the perimeter of the Antarctic Stratospheric Polar Vortex during late winter/early spring within which lies the "ozone hole". In this system, a robust transport barrier is found near the core of a zonal jet under typical conditions, which is responsible for trapping of the ozone-depleted air within the ozone hole. The existence of such a barrier is predicted theoretically and tested numerically with use of a dynamically-motivated analytically-prescribed model. The second, oceanographic, application considered is the study of the surface transport in the Adriatic Sea. The surface flow in the Adriatic is characterized by a robust threegyre background circulation pattern. Motivated by this observation, the Lagrangian dynamics of a perturbed three-gyre system is studied, with emphasis on intergyre transport and the role of transport barriers. It is shown that a qualitative change in transport properties, accompanied by a qualitative change in the structure of stable and unstable manifolds occurs in the perturbed three-gyre system when the perturbation strength exceeds a certain threshold. This behavior is predicted theoretically, simulated numerically with use of an analytically prescribed model, and shown to be consistent with a fully observationally-based model.