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Lagrangian Coherent Structures and Finite-Time Lyapunov Exponents

The transport of a tracer in a fluid is closely related to emergent patterns that are commonly referred to as coherent structures or Lagrangian Coherent Structures (LCS) when they are defined using fluid trajectories. For time- independant dynamic systems, they correspond to stable and unstable manifolds of hyperbolic trajectories [1]. Coherent structures delimit regions of whirls, stretching, or contraction of tracer [2]: contraction is observed along stable manifolds whereas unstable manifolds correspond to divergent directions along which the tracer is stretched. Stable and unstable manifolds are material curves that act as transport barriers and that exhibits locally the strongest attraction, repulsion or shearing in the flow over a finite-time interval [3]. LCSs are usually indentified in a practicall manner as maximizing ridges of Finite-Time Lyapunov Exponents (FTLE) field [4], [5], [6], [7], [8].

FTLE is defined as the largest eigenvalue of the Cauchy-Green strain tensor of the flow map (see below for more details). The corresponding eigenvector is called Finite-Time Lyapunov Vector. FTLE and FSLE are widely used to characterise transport and mixing processes in oceanographic flows where the velocity field is only known as a finite data set [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21]. Recent work also demonstrate the potential use of FTLE in tracer image assimilation in geophysical models [22], [23]. [24] which reviews the use of FTLE for detecting LCSs in other contexts.

FTLE and FSLE: definition and properties

FTLE is a scalar local notion that represents the rate of separation of initially neighboring particules over a finite-time window \([t, t+T]\). FTLE at a point \(x\) is defined as the growth factor of the norm of the perturbation delta started around \(x\) at time \(t\) and advected by the flow after an advection time \(T\). Maximal stretching occurs when delta is aligned with the eigenvector associated to the maximum eigenvalue \(\lambda_{max}\) of the Cauchy-Green strain tensor: \(\Delta = M^T M\) where \(M : x(t) \mapsto x(t+T)\) is the flow map of the advection system. This eigenvector is referred to as the forward Finite-Time Lyapunov Vector. The forward FTLE at the point \(x\) is then defined as: \(FTLE(x) = \frac{log(\lambda max)}{2T}\).

FSLE are FTLE where T is not fixed a-priori but determined so that the particule separate from a given distance. FSLE at a point x and for a final separation distance d is defined as: \(FSLE(x, d) = \frac{log(\lambda max(d))}{2T(d)}\). Seeding a domain with particules initially located on a grid leads to the computation of a discretized FSLE and FSLV fields.

Backward FSLE-Vs are similarly defined, with the time direction being inverted in the advection equation resolution. Ridges of BFTLE fields approximate attracting LCSs [25].

Backward FTLE fields show contours that correspond reasonably well to the main structures such as filaments, fronts and spirals that appear in geophysical and bio-geochemical tracer fields [10], [14], [15], [26]. Also, orientation of the gradient of a passive tracer converges to that of backward FTLVs freely decaying 2D turbulence flow [27] but this behaviour has been observed for realistic oceanic flows and tracers [20].

Furthermore [11], [28] showed, using real data, that properties of FSLE and FTLE remain valid with a mesoscale advection, i.e. when the resolution of the velocity field --- from which FTLE-V are computed --- is much lower than the resolution of the observed tracer field [20] ,[21]_.


[1]Wiggins, S. 1992. Chaotic transport in dynamical systems. Vol. 2 of Interdisciplinary Applied Mathematics. Springer-Verlag, Berlin.
[2]Ottino, J. 1989. The kinematics of mixing: Stretching, chaos, and transport. Cambridge University Press, Cambridge.
[3]Haller, G. and Yuan, G. 2000. Lagrangian coherent structures and mixing in two-dimensional turbulence. Physica D: Nonlinear Phenomena 147 (3-4): 352--370.
[4]Haller, G. 2001. Distinguished material surfaces and coherent structures in three-dimensional fluid flows. Physica D 149 (4): 248 -- 277.
[5]Haller, G. 2001. Lagrangian structures and the rate of strain in a partition of two-dimensional turbulence. Physics of Fluids 13 (11): 3365 --3385.
[6]Haller, G. 2002. Lagrangian coherent structures from approximate velocity data. Physics of Fluids 14 (6): 1851--1861.
[7]Shadden, S.~C., Lekien, F. and Marsden, J. E. 2005. Definition and properties of lagrangian coherent structures from finite-time lyapunov exponents in two-dimensional aperiodic flows. Physica D: Nonlinear Phenomena 212 (3-4): 271--304.
[8]Mathur, M., Haller, G., Peacock, T., Ruppert-Felsot, J. E. and Swinney, H. L. 2007. Uncovering the lagrangian skeleton of turbulence. Phys. Rev. Lett. 98 (14): 144502
[9]Beron-Vera, F. J., Brown, M.~G., Olascoaga, M. J., Rypina, I. I., Koçak, H. and Udovydchenkov, I. A. 2008. Zonal jets as transport barriers in planetary atmospheres. Journal of the Atmospheric Sciences 65 (10): 3316 --3326
[10](1, 2) Beron-Vera, F. J. and Olascoaga, M. J. 2009. An assessment of the importance of chaotic stirring and turbulent mixing on the west florida shelf. Journal of Physical Oceanography 39 (7): 1743--1755.
[11](1, 2) Beron-Vera, F. J., Olascoaga, M. J. and Goni, G. J. 2010. Surface ocean mixing inferred from different multisatellite altimetry measurements. Journal of Physical Oceanography 40 (11): 2466--2480.
[12]Lekien, F., Coulliette, C., Mariano, A. J., Ryan, E. H., Shay, L. K., Haller, G. and Marsden, J. 2005. Pollution release tied to invariant manifolds: A case study for the coast of florida. Physica D: Nonlinear Phenomena 210 (1-2): 1--20.
[13]Olascoaga, M. J. 2010. Isolation on the west florida shelf with implications for red tides and pollutant dispersal in the gulf of mexico. Nonlinear Processes in Geophysics 17 (6): 685--696.
[14](1, 2) Olascoaga, M. J., Beron-Vera, F. J., Brand, L. E. and Kocak, H. 2008. Tracing the early development of harmful algal blooms on the west florida shelf with the aid of lagrangian coherent structures. J. Geophys. Res. 113 (C12014).
[15](1, 2) Shadden, S.~C., Lekien, F., Paduan, J. D., Chavez, F. P. and Marsden, J. E. 2009. The correlation between surface drifters and coherent structures based on high-frequency radar data in monterey bay. Deep Sea Research Part II: Topical Studies in Oceanography 56 (3-5): 161--172.
[16]Aurell, E., Boffetta, G., Crisanti, A., Paladin, G. and Vulpiani, A., 1997. Predictability in the large: an extension of the concept of lyapunov exponent. Journal of Physics A: Mathematical and General 30 (1):1--26
[17]Artale, V., Boffetta, G., Celani, A., Cencini, M. and Vulpiani, A., 1997. Dispersion of passive tracers in closed basins: Beyond the diffusion coefficient. Physics of Fluids 9 (11):3162--3171.
[18]d'Ovidio, F., Fernandez, V., Hernandez-Garcia, E. and Lopez, C. 2004. Mixing structures in the mediterranean sea from finite-size lyapunov exponents. Geophysical Research Letters 31 (L17203).
[19]d'Ovidio, F., Isern-Fontanet, J., Lopez, C., Hernandez-Garcia, E. and Garcia-Ladona, E. 2009. Comparison between eulerian diagnostics and finite- size lyapunov exponents computed from altimetry in the algerian basin. Deep Sea Research Part I: Oceanographic Research Papers 51 (1): 15--31.
[20](1, 2, 3) d'Ovidio, F., Taillandier, V., Taupier-Letage, I. and Mortier, L. 2009. Lagrangian validation of the mediterranean mean dynamic topography by extraction of tracer frontal structures. Mercator Ocean Quarterly Newsletter 32: 24--32.
[21]Lehahn, Y., d'Ovidio, F., Lévy, M. and Heifetz, E. 2007. Stirring of the northeast atlantic spring bloom: A lagrangian analysis based on multisatellite data. J. Geophys. Res. 112 (C08005).
[22]O. Titaud, J.-M. Brankart, and J. Verron. 2011. On the use of finite-time Lyapunov exponents and vectors for direct assimilation of tracer images into ocean models. Tellus A, 63 (5):1038-1051.
[23]L. Gaultier, J. Verron, J.-M. Brankart, O. Titaud, and P. Brasseur. 2012. On the inversion of submesoscale tracer fields to estimate the surface ocean circulation. Journal of Marine Systems 126: 33--42.
[24]Peacock, T. and Dabiri, J. 2010. Introduction to focus issue: Lagrangian coherent structures. * Chaos* 20 (1): 017501.
[25]Haller, G. 2011. A variational theory of hyperbolic lagrangian coherent structures. Physica D: Nonlinear Phenomena 240 (7): 574 -- 598.
[26]Olascoaga, M. J., Rypina, I. I., Brown, M. G., Beron-Vera, F. J., Kocak, H., Brand, L.~E., Halliwell, G.~R. and Shay, L.~K. 2006. Persistent transport barrier on the west florida shelf. Geophys. Res. Lett. 33 (L22603).
[27]Lapeyre, G. 2002. Characterization of finite-time lyapunov exponents and vectors in two-dimensional turbulence. Chaos 12 (3): 688--698.
[28]Beron-Vera, F. J. Mixing by low- and high-resolution surface geostrophic currents. 2010. J. Geophys. Res. 115 (C10027)