Lagrangian coherent structures in geophysical flows

Journal of Fixed Point Theory and Applications, 2010

The method of using Finite Time Liapunov Exponents (FTLE) to extract Lagrangian Coherent Structures (LCS) in aperiodic flows, as originally developed by Haller, is applied to geophysical flows. In this approach, the LCS are identified as surfaces of greatest separation that parse the flow into regions with different dynamical behavior. In this way, the LCS reveal the underlying skeleton of turbulence. The time-dependence of the LCS provides insight into the mechanisms by which fluid is transported from one region to another. Of especial interest in this study is the utility with which the FTLE-LCS method can be used to reveal homoclinic and horseshoe dynamics in aperiodic flows. The FTLE-LCS method is applied to turbulent flow in hurricanes and reveals LCS that delineate sharp boundaries to a storm. Moreover, intersections of the LCS define lobes that mediate transport into and out of a storm through the action of homoclinic lobe dynamics. Using the FTLE-LCS method, the same homoclinic structures are seen to be a dominant transport mechanism in the Global Ocean, and provide insights into the role of mesoscale eddies in enhancing lateral mixing.

2006

The Lagrangian trajectories of fluid particles are experimentally studied in an oscillating four-vortex velocity field. The oscillations occur due to a loss of stability of a steady flow and result in a regular reclosure of streamlines between the vortices of the same sign. The Eulerian velocity field is visualized by tracer displacements over a short time period. The obtained data on tracer motions during a number of oscillation periods show that the Lagrangian trajectories form quasi-regular structures. The destruction of these structures is determined by two characteristic time scales: the tracers are redistributed sufficiently fast between the vortices of the same sign and much more slowly transported into the vortices of opposite sign. The observed behavior of the Lagrangian trajectories is quantitatively reproduced in a new numerical experiment with two-dimensional model of the velocity field with a small number of spatial harmonics. A qualitative interpretation of phenomena o...

This work investigates how a Lagrangian perspective applies to models of two oceanographic flows: an overturning submesoscale eddy and the Western Alboran Gyre. In the first case, I focus on the importance of diffusion as compared to chaotic advection for tracers in this system. Three methods are used to quantify the relative contributions: scaling arguments including a Lagrangian Batchelor scale, statistical analysis of ensembles of trajectories, and Nakamura effective diffusivity from numerical simulations of dye release. Through these complementary methods, I find that chaotic advection dominates over turbulent diffusion in the widest chaotic regions, which always occur near the center and outer rim of the cylinder and sometimes occur in interior regions for Ekman numbers near 0.01. In thin chaotic regions, diffusion is at least as important as chaotic advection. From this analysis, it is clear that identified Lagrangian coherent structures will be barriers to transport for long times if they are much larger than the Batchelor scale. The second case is a model of the Western Alboran Gyre with realistic forcing and bathymetry. I examine its transport properties from both an Eulerian and Lagrangian perspective. I find that advection is most often the dominant term in Eulerian budgets for volume, salt, and heat in the gyre, with diffusion and surface fluxes playing a smaller role. In the vorticity budget, advection is as large as the effects of wind and viscous diffusion, but not dominant. For the Lagrangian analysis, I construct a moving gyre boundary from segments of the stable and unstable manifolds emanating from two persistent hyperbolic trajectories on the coast at the eastern and western extent of the gyre. These manifolds are computed on several isopycnals and stacked vertically to construct a three-dimensional Lagrangian gyre boundary. The regions these manifolds cover is the stirring region, where there is a path for water to reach the gyre. On timescales of days to weeks, water from the Atlantic Jet and the northern coast can enter the outer parts of the gyre, but there is a core region in the interior that is separate. Using a gate, I calculate the continuous advective transport across the Lagrangian boundary in three dimensions for the first time. A Lagrangian volume budget is calculated, and challenges in its closure are described. Lagrangian and Eulerian advective transports are found to be of similar magnitudes.

Coastal Engineering Proceedings, 2012

We study the horizontal surface mixing and the transport induced by waves, using local Lyapunov exponents and high resolution data from numerical simulations of waves and currents. By choosing the proper spatial (temporal) parameters we compute the Finite Size and Finite Time Lyapunov exponents (FSLE and FTLE) focussing on the local stirring and diffusion inferred from the Lagrangian Coherent Structures (LCS). The methodology is tested by deploying a set of eight lagrangian drifters and studying the path followed against LCS derived under current field and waves and currents.

Journal of the Atmospheric Sciences, 1999

Statistical diagnostics of mixing and transport are computed for a numerical model of forced shallow-water flow on the sphere and a middle-atmosphere general circulation model. In particular, particle dispersion statistics, transport fluxes, Liapunov exponents (probability density functions and ensemble averages), and tracer concentration statistics are considered. It is shown that the behavior of the diagnostics is in accord with that of kinematic chaotic advection models so long as stochasticity is sufficiently weak. Comparisons with random-strain theory are made.

Journal of Fluid Mechanics, 2007

We use direct Lyapunov exponents (DLE) to identify Lagrangian coherent structures in two different three-dimensional flows, including a single isolated hairpin vortex, and a fully developed turbulent flow. These results are compared with commonly used Eulerian criteria for coherent vortices. We find that despite additional computational cost, the DLE method has several advantages over Eulerian methods, including greater detail and the ability to define structure boundaries without relying on a preselected threshold. As a further advantage, the DLE method requires no velocity derivatives, which are often too noisy to be useful in the study of a turbulent flow. We study the evolution of a single hairpin vortex into a packet of similar structures, and show that the birth of a secondary vortex corresponds to a loss of hyperbolicity of the Lagrangian coherent structures.

2012

Lagrangian transport structures for three-dimensional and time-dependent fluid flows are of great interest in numerous applications, particularly for geophysical or oceanic flows. In such flows, chaotic transport and mixing can play important environmental and ecological roles, for examples in pollution spills or plankton migration. In such flows, where simulations or observations are typically available only over a short time, understanding the difference between short-time and longtime transport structures is critical. In this paper, we use a set of classical (i.e. Poincaré section, Lyapunov exponent) and alternative (i.e. finite time Lyapunov exponent, Lagrangian coherent structures) tools from dynamical systems theory that analyze chaotic transport both qualitatively and quantitatively. With this set of tools we are able to reveal, identify and highlight differences between short-and long-time transport structures inside a flow composed of a primary horizontal contra-rotating vortex chain, small lateral oscillations and a weak Ekman pumping. The difference is mainly the existence of regular or extremely slowly developing chaotic regions that are only present at short time.

Physical Review Letters, 2013

The ocean and the atmosphere, and hence the climate, are governed at large scale by interactions between pressure gradient and Coriolis and buoyancy forces. This leads to a quasigeostrophic balance in which, in a two-dimensional-like fashion, the energy injected by solar radiation, winds, or tides goes to large scales in what is known as an inverse cascade. Yet, except for Ekman friction, energy dissipation and turbulent mixing occur at a small scale implying the formation of such scales associated with breaking of geostrophic dynamics through wave-eddy interactions or frontogenesis, in opposition to the inverse cascade. Can it be both at the same time? We exemplify here this dual behavior of energy with the help of three-dimensional direct numerical simulations of rotating stratified Boussinesq turbulence. We show that efficient small-scale mixing and large-scale coherence develop simultaneously in such geophysical and astrophysical flows, both with constant flux as required by theoretical arguments, thereby clearly resolving the aforementioned contradiction.

Journal of Physical Oceanography, 1997

The aim of this paper is to renew interest in the Lagrangian view of global and basin ocean circulations and its implications in physical and biogeochemical ocean processes. The paper examines the Lagrangian transport, mixing, and chaos in a simple, laminar, three-dimensional, steady, basin-scale oceanic flow consisting of the gyre and the thermohaline circulation mode. The Lagrangian structure of this flow could not be chaotic if the steady oceanic flow consists of only either one of the two modes nor if the flow is zonally symmetric, such as the Antarctic Circumpolar Current. However, when both the modes are present, the Lagrangian structure of the flow is chaotic, resulting in chaotic trajectories and providing the enhanced transport and mixing and microstructure of a tracer field. The Lagrangian trajectory and tracer experiments show the great complexity of the Lagrangian geometric structure of the flow field and demonstrate the complicated transport and mixing processes in the World Ocean. The finite-time Lyapunov exponent analysis has successfully characterized the Lagrangian nature. One of the most important findings is the distinct large-scale barrier-which the authors term the great ocean barrier-within the ocean interior with upper and lower branches, as remnants of the Kolmogorov-Arnold-Moser (KAM) invariant surfaces. The most fundamental reasons for such Lagrangian structure are the intrinsic nature of the long time mean, global and basin-scale oceanic flow: the three-dimensionality and incompressibility giving rise to chaos and to the great ocean barrier, respectively. Implications of these results are discussed, from the great ocean conveyor hypothesis to the predictability of the (quasi) Lagrangian drifters and floats in the climate observing system.

Journal of Physics: Conference Series, 2017

The nonlinear effects that appear in test particle (tracer) transport in twodimensional incompressible velocity fields are discussed. Test particle trajectories show both random and quasi-coherent aspects. The coherent motion is associated with trapping or eddying in the structure of the stochastic field. It generates quasi-coherent trajectory structures. The random motion leads to diffusive transport while the structures determine a microconfinement process. The strength of each of these aspects depends on the parameters of the turbulence. The transport coefficients and the average size of the quasi-coherent structures are determined as function of the characteristics of the turbulence. The transport process is completely different in the presence of structures in the sense that the dependence on the parameters is different. The quasi-coherence of the motion can also be represented by the generation of flows. We show that tracers flows yield in a stochastic potential with spacedependent amplitude.