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Leaving flatland: diagnostics for Lagrangian coherent structures in three-dimensional flows. (English) Zbl 1356.76108

Summary: Finite-time Lyapunov exponents (FTLE) are often used to identify Lagrangian Coherent Structures (LCS). Most applications are confined to flows on two-dimensional (2D) surfaces where the LCS are characterized as curves. The extension to three-dimensional (3D) flows, whose LCS are 2D structures embedded in a 3D volume, is theoretically straightforward. However, in geophysical flows at regional scales, full prognostic computation of the evolving 3D velocity field is not computationally feasible. The vertical or diabatic velocity, then, is either ignored or estimated as a diagnostic quantity with questionable accuracy. Even in cases with reliable 3D velocities, it may prove advantageous to minimize the computational burden by calculating trajectories from velocities on carefully chosen surfaces only. When reliable 3D velocity information is unavailable or one velocity component is explicitly ignored, a reduced FTLE form to approximate 2D LCS surfaces in a 3D volume is necessary. The accuracy of two reduced FTLE formulations is assessed here using the ABC flow and a 3D quadrupole flow as test models. One is the standard approach of knitting together FTLE patterns obtained on adjacent surfaces. The other is a new approximation accounting for the dispersion due to vertical \((u,v)\) shear. The results are compared with those obtained from the full 3D velocity field. We introduce two diagnostic quantities to identify situations when a fully 3D computation is required for an accurate determination of the 2D LCS. For the ABC flow, we found the full 3D calculation to be necessary unless the vertical \((u,v)\) shear is sufficiently small. However, both methods compare favorably with the 3D calculation for the quadrupole model scaled to typical open ocean conditions.

MSC:

76F20 Dynamical systems approach to turbulence
37M25 Computational methods for ergodic theory (approximation of invariant measures, computation of Lyapunov exponents, entropy, etc.)

Software:

ODEPACK
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References:

[1] Aref, H., Stirring by chaotic advection, J. Fluid Mech., 143, 1-21 (1984) · Zbl 0559.76085
[2] Jones, C.; Winkler, S., Invariant manifolds and Lagrangian dynamics in the ocean and atmosphere, (Fiedler, B., Handbook of Dynamical Systems. Vol. 2 (2002), North-Holland: North-Holland Amsterdam), 55-92 · Zbl 1039.86002
[3] Branicki, M.; Wiggins, S., An adaptive method for computing invariant manifolds in non-autonomous, three-dimensional dynamical systems, Physica D, 238, 16, 1625-1657 (2009) · Zbl 1179.37113
[4] Haller, G., Finding finite-time invariant manifolds in two-dimensional velocity fields, Chaos, 10, 1, 99-108 (2000) · Zbl 0979.37012
[5] Haller, G.; Yuan, G., Lagrangian coherent structures and mixing in two-dimensional turbulence, Physica D, 147, 3-4, 352-370 (2000) · Zbl 0970.76043
[6] Shadden, S. C.; Lekien, F.; Marsden, J. E., Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows, Physica D, 212, 3-4, 271-304 (2005) · Zbl 1161.76487
[7] Branicki, M.; Wiggins, S., Finite-time Lagrangian transport analysis: stable and unstable manifolds of hyperbolic trajectories and finite-time Lyapunov exponents, Nonlinear Process. Geophys., 17, 1, 1-36 (2010)
[8] Haller, G., A variational theory of hyperbolic Lagrangian coherent structures, Physica D, 240, 7, 574-598 (2011) · Zbl 1214.37056
[9] Farazmand, M.; Haller, G., Erratum and addendum to “A variational theory of hyperbolic Lagrangian coherent structures”, Physica D. Physica D, Physica D, 241, 4, 439-441 (2012) · Zbl 1242.37059
[10] Farazmand, M.; Haller, G., Computing Lagrangian coherent structures from their variational theory, Chaos, 22, 1, 013128 (2012) · Zbl 1331.37128
[11] Madrid, J. A.J.; Mancho, A. M., Distinguished trajectories in time dependent vector fields, Chaos, 19, 1, 013111 (2009)
[12] Mendoza, C.; Mancho, A. M., Hidden geometry of ocean flows, Phys. Rev. Lett., 105, 3 (2010)
[13] Mendoza, C.; Mancho, A., Review article: “The Lagrangian description of aperiodic flows: a case study of the Kuroshio Current”, Nonlinear Process. Geophys., 19, 4, 449-472 (2012)
[14] Rypina, I. I.; Scott, S. E.; Pratt, L. J.; Brown, M. G., Investigating the connection between complexity of isolated trajectories and Lagrangian coherent structures, Nonlinear Process. Geophys., 18, 6, 977-987 (2011)
[15] Wiggins, S., The dynamical systems approach to Lagrangian transport in oceanic flows, Annu. Rev. Fluid Mech., 37, 295-328 (2005) · Zbl 1117.76058
[16] Berger, B. S.; Rokni, M., Lyapunov exponents and continuum kinematics, Internat. J. Engrg. Sci., 25, 8, 1079-1084 (1987) · Zbl 0614.73005
[17] Lau, Y.-T.; Finn, J. M., Dynamics of a three-dimensional incompressible flow with stagnation points, Physica D, 57, 3-4, 283-310 (1992) · Zbl 0758.34028
[18] Mezic, I.; Wiggins, S., On the integrability and perturbation of three-dimensional fluid flows with symmetry, J. Nonlinear Sci., 4, 1, 157-194 (1994) · Zbl 0796.76021
[19] Lomelí, H. E.; Meiss, J. D., Heteroclinic primary intersections and codimension one Melnikov method for volume-preserving maps, Chaos, 10, 1, 109-121 (2000) · Zbl 0979.37009
[20] Balasuriya, S.; Mezić, I.; Jones, C. K.R. T., Weak finite-time Melnikov theory and 3D viscous perturbations of Euler flows, Physica D, 176, 1-2, 82-106 (2003) · Zbl 1006.76020
[21] Speetjens, M. F.M.; Clercx, H. J.H.; van Heijst, G. J.F., A numerical and experimental study on advection in three-dimensional Stokes flows, J. Fluid Mech., 514, 77-105 (2004) · Zbl 1066.76022
[22] Haller, G., Distinguished material surfaces and coherent structures in three-dimensional fluid flows, Physica D, 149, 4, 248-277 (2001) · Zbl 1015.76077
[23] Lekien, F.; Shadden, S. C.; Marsden, J. E., Lagrangian coherent structures in \(n\)-dimensional systems, J. Math. Phys., 48, 6, 065404 (2007) · Zbl 1144.81374
[24] Garth, C.; Gerhardt, F.; Tricoche, X.; Hagen, H., Efficient computation and visualization of coherent structures in fluid flow applications, IEEE Trans. Vis. Comput. Graphics, 13, 6, 1464-1471 (2007)
[25] Branicki, M.; Malek-Madani, R., Lagrangian structure of flows in the Chesapeake Bay: challenges and perspectives on the analysis of estuarine flows, Nonlinear Process. Geophys., 17, 2, 149-168 (2010)
[26] Branicki, M.; Mancho, A. M.; Wiggins, S., A Lagrangian description of transport associated with a front-eddy interaction: application to data from the North-Western Mediterranean Sea, Physica D, 240, 3, 282-304 (2011) · Zbl 1217.86005
[27] Branicki, M.; Kirwan, A. D., Stirring: the Eckart paradigm revisited, Internat. J. Engrg. Sci., 48, 11, 1027-1042 (2010) · Zbl 1231.86003
[28] Bettencourt, J. H.; López, C.; Hernández-García, E., Oceanic three-dimensional Lagrangian coherent structures: a study of a mesoscale eddy in the Benguela upwelling region, Ocean Model., 51, 73-83 (2012)
[29] Sulman, M. H.M.; Huntley, H. S.; Lipphardt, B. L.; Kirwan, A. D., Out of flatland: three-dimensional aspects of Lagrangian transport in geophysical fluids, (Lin, J.; Brunner, D.; Gerbig, C.; Stohl, A.; Luhar, A.; Webley, P., Lagrangian Modeling of the Atmosphere. Lagrangian Modeling of the Atmosphere, AGU Monograph (2013), AGU: AGU Washington, DC), 77-84 · Zbl 1356.76108
[30] Hindmarsh, A. C., ODEPACK, a systematized collection of ODE solvers, (Stepleman, R. S.; Ames, W. F.; Vichnevetsky, R.; Carver, M.; Peskin, R., Scientific Computing (1983), North-Holland: North-Holland Amsterdam), 55-64
[31] Childress, S., Solutions of Euler’s equations containing finite eddies, Phys. Fluids, 9, 5, 860-872 (1966) · Zbl 0148.20603
[32] Eremeev, V. N.; Ivanov, L. M.; Kirwan, A. D., Reconstruction of oceanic flow characteristics from quasi-Lagrangian data, 1. Approach and mathematical methods, J. Geophys. Res., 97, C6, 9733-9742 (1992) · Zbl 1147.82302
[33] Dombre, T.; Frisch, U.; Greene, J. M.; Hénon, M.; Mehr, A.; Soward, A. M., Chaotic streamlines in the ABC flows, J. Fluid Mech., 167, 353-391 (1986) · Zbl 0622.76027
[34] Lewis, J. K.; Kirwan, A. D.; Forristall, G. Z., Evolution of a warm-core ring in the Gulf of Mexico: Lagrangian observations, J. Geophys. Res., 94, C6, 8163-8178 (1989)
[35] Sanderson, B. G., Structure of an eddy measured with drifters, J. Geophys. Res., 100, C4, 6761-6776 (1995)
[36] Brassington, G. B., Estimating surface divergence of ocean eddies using observed trajectories from a surface drifting buoy, J. Atmos. Ocean Technol., 27, 4, 705-720 (2010)
[37] Trefethen, L. N.; Bau, D., Numerical Linear Algebra (1997), SIAM · Zbl 0874.65013
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