Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows. (English) Zbl 1161.76487

Summary: This paper develops the theory and computation of Lagrangian Coherent Structures (LCS), which are defined as ridges of Finite-Time Lyapunov Exponent (FTLE) fields. These ridges can be seen as finite-time mixing templates. Such a framework is common in dynamical systems theory for autonomous and time-periodic systems, in which examples of LCS are stable and unstable manifolds of fixed points and periodic orbits. The concepts defined in this paper remain applicable to flows with arbitrary time dependence and, in particular, to flows that are only defined (computed or measured) over a finite interval of time.
Previous work has demonstrated the usefulness of FTLE fields and the associated LCSs for revealing the Lagrangian behavior of systems with general time dependence. However, ridges of the FTLE field need not be exactly advected with the flow. The main result of this paper is an estimate for the flux across an LCS, which shows that the flux is small, and in most cases negligible, for well-defined LCSs or those that rotate at a speed comparable to the local Eulerian velocity field, and are computed from FTLE fields with a sufficiently long integration time. Under these hypotheses, the structures represent nearly invariant manifolds even in systems with arbitrary time dependence.
The results are illustrated on three examples. The first is a simplified dynamical model of a double-gyre flow. The second is surface current data collected by high-frequency radar stations along the coast of Florida and the third is unsteady separation over an airfoil. In all cases, the existence of LCSs governs the transport and it is verified numerically that the flux of particles through these distinguished lines is indeed negligible.


76F20 Dynamical systems approach to turbulence
37B55 Topological dynamics of nonautonomous systems
37N10 Dynamical systems in fluid mechanics, oceanography and meteorology
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[1] Barreira, L.; Pesin, Y., Lyapunov exponents and smooth ergodic theory, () · Zbl 1195.37002
[2] Coppel, W.A., Dichotomies in stability theory, () · Zbl 0376.34001
[3] Cottet, G.H.; Koumoutsakos, P., Vortex methods: theory and practice, (2000), Cambridge University Press
[4] Coulliette, C.; Wiggins, S., Intergyre transport in a wind-driven, quasigeostrophic double gyre: an application of lobe dynamics, Nonlinear process. geophys., 7, 59-85, (2000)
[5] Doerner, R.; Hübinger, B.; Martienssen, W.; Grossmann, A.; Thomae, S., Stable manifolds and predictability of dynamical systems, Chaos solitons fractals, 10, 11, 1759-1782, (1999) · Zbl 0982.37012
[6] Eldredge, J.D.; Colonius, T.; Leonard, A., A vortex particle method for two-dimensional compressible flow, J. comput. phys., 179, 371-399, (2002) · Zbl 1130.76393
[7] J.D. Eldredge, Efficient tools for the simulation of flapping wing flows, AIAA Paper 2005-0085
[8] Guckenheimer, J.; Holmes, P., Nonlinear oscillations, dynamical systems and bifurcations of vector fields, () · Zbl 0515.34001
[9] Haller, G.; Poje, A.C., Finite-time transport in aperiodic flows, Physica D, 119, 352-380, (2000) · Zbl 1194.76089
[10] Haller, G., Lagrangian coherent structures and mixing in two-dimensional turbulence, Chaos, 10, 1, 99-108, (2000) · Zbl 0979.37012
[11] Haller, G.; Yuan, G., Lagrangian coherent structures and mixing in two-dimensional turbulence, Physica D, 147, 352-370, (2000) · Zbl 0970.76043
[12] Haller, G., Distinguished material surfaces and coherent structures in 3d fluid flows, Physica D, 149, 248-277, (2001) · Zbl 1015.76077
[13] Haller, G., Lagrangian structures and the rate of strain in a partition of two-dimensional turbulence, Phys. fluids A, 13, 3368-3385, (2001)
[14] Haller, G., Lagrangian coherent structures from approximate velocity data, Phys. fluids A, 14, 1851-1861, (2002) · Zbl 1185.76161
[15] Haller, G., Exact theory of unsteady separation for two-dimensional flows, J. fluid mech., 512, 257-311, (2004) · Zbl 1066.76054
[16] Hartman, P., Ordinary differential equations, (1973), John Wiley and Sons, Inc. Baltimore · Zbl 0125.32102
[17] T. Inanc, S.C. Shadden, J.E. Marsden, Optimal trajectory generation in ocean flows, in: Proc. of 24th American Control Conference, Portland, USA, June 2005
[18] Jones, C.K.R.T.; Winkler, S., Invariant manifolds and Lagrangian dynamics in the Ocean and atmosphere, (), 55-92 · Zbl 1039.86002
[19] Joseph, B.; Legras, B., Relation between kinematic boundaries, stirring, and barriers for the antarctic polar vortex, J. atmospheric sci., 59, 1198-1212, (2002)
[20] Koh, T.-Y.; Legras, B., Hyperbolic lines and the stratospheric polar vortex, Chaos, 12, 382-394, (2002) · Zbl 1080.86002
[21] F. Lekien, N. Leonard, Dynamically consistent Lagrangian coherent structures, in: American Institute of Physics: 8th Experimental Chaos Conference, vol. 742, 2004, pp. 132-139
[22] Lekien, F.; Marsden, J., Tricubic interpolation in three dimensions, Internat. J. numer. methods engrg., 63, 3, 455-471, (2005) · Zbl 1140.76423
[23] Lekien, F.; Coulliette, C.; Mariano, A.J.; Ryan, E.H.; Shay, L.K.; Haller, G.; Marsden, J.E., Pollution release tied to invariant manifolds: A case study for the coast of florida, Physica D, 210, 1-2, 1-20, (2005) · Zbl 1149.86302
[24] F. Lekien, C. Coulliette, G. Haller, J. Paduan, J.E. Marsden, Optimal pollution release in Monterey Bay based on nonlinear analysis of coastal radar data, Environ. Sci. Technol. 2005 (under review)
[25] Liapunov, A.M., Stability of motion, (1966), Academic Press New York
[26] Malhotra, N.; Mezić, I.; Wiggins, S., Patchiness: A new diagnostic for Lagrangian trajectory analysis in time-dependent fluid flows, Internat. J. bifur. chaos, 8, 1073-1094, (1998)
[27] Mancho, A.M.; Small, S.; Wiggins, S.; Ide, K., Computation of stable and unstable manifold of hyperbolic trajectories in two-dimensional, aperiodically time-dependent vector fields, Physica D, 182, 188-222, (2003) · Zbl 1030.37020
[28] Marsden, J.E.; Hoffman, M.J., Elementary classical analysis, (1993), W.H. Freeman and Company New York · Zbl 0777.26001
[29] Mezic, I.; Wiggins, S., A method for visualization of invariant sets of dynamical systems based on the ergodic partition, Chaos, 9, 1, 213-218, (1999) · Zbl 0987.37080
[30] O’Neill, B., Elementary differential geometry, (1997), Academic Press San Diego · Zbl 0974.53001
[31] Oseledec, V.I., A multiplicative ergodic theorem: ljapunov characteristic numbers for dynamical systems, Trans. Moscow math. soc., 19, 197-231, (1968) · Zbl 0236.93034
[32] Ottino, J.M., The kinematics of mixing: stretching, chaos, and transport, (1989), Cambridge University Press Cambridge · Zbl 0721.76015
[33] Peters, H.; Shay, L.K.; Mariano, A.J.; Cook, T.M., Current variability on a narrow shelf with large ambient vorticity, J. geophys. res.-oceans, 107, C8, (2002), art. no.-3087
[34] Pierrehumbert, R.T., Large-scale horizontal mixing in planetary atmospheres, Phys. fluids A, 3, 5, 1250-1260, (1991)
[35] Pierrehumbert, R.T.; Yang, H., Global chaotic mixing on isentropic surfaces, J. atmospheric sci., 50, 2462-2480, (1993)
[36] Poje, A.C.; Haller, G., Geometry of cross-stream mixing in a double-gyre Ocean model, J. phys. oceanogr., 29, 1649-1665, (1999)
[37] Prandle, D., The fine-structure of nearshore tidal and residual circulations revealed by HF radar surface current measurements, J. phys. oceanogr., 17, 231-245, (1987)
[38] Provenzale, A., Transport by coherent barotropic vortices, Annu. rev. fluid mech., 31, 55-93, (1999)
[39] Rogerson, A.; Miller, P.D.; Pratt, L.J.; Jones, C.K.R.T.J., Lagrangian motion and fluid exchange in a barotropic meandering jet, J. phys. oceanogr., 29, 10, 2635-2655, (1999)
[40] Rom-Kedar, V., Transport rates of a class of two-dimensional maps and flows, Physica D, 43, 229-268, (1990) · Zbl 0706.58065
[41] Rom-Kedar, V.; Leonard, A.; Wiggins, S., An analytical study of transport, mixing, and chaos in unsteady vortical flow, J. fluid mech., 214, 347-394, (1990) · Zbl 0698.76028
[42] Rom-Kedar, V.; Wiggins, S., Transport in two-dimensional maps: concepts, examples, and a comparison of the theory of rom-kedar and wiggins with the Markov model of mackay, meiss, ott, and percival, Physica D, 51, 248-266, (1991) · Zbl 0741.76077
[43] Shay, H.C.; Graber, L.K.; Ross, D.B.; Chapman, R.D., Mesoscale Ocean surface current structure detected by HF radar, J. atmos. Ocean. technol., 12, 881-900, (1995)
[44] Shay, L.K.; Cook, T.M.; Haus, B.K.; Martinez, J.; Peters, H.; Mariano, A.J.; An, P.E.; Smith, S.; Soloviev, A.; Weisberg, R.; Luther, M., VHF radar detects oceanic submesoscale vortex along the florida coast, EOS trans. am. geophys. union, 81, 19, 209-213, (2000)
[45] Shay, L.K.; Cook, T.M.; Peters, H.; Mariano, A.J.; Weisberg, R.; An, P.E.; Soloviev, A.; Luther, M., Very high frequency radar mapping of the surface currents, IEEE J. oceanogr. engin., 27, 155-169, (2002)
[46] Stewart, R.H.; Joy, J.W., HF radio measurements of surface currents, Deep-sea res., 21, 1039-1049, (1974)
[47] Truesdell, C.A., The kinematics of vorticity, (1954), Indiana University Press Bloomington · Zbl 0056.18606
[48] Verhulst, F., Nonlinear differential equations and dynamical systems, (1996), Springer-Verlag New York · Zbl 0854.34002
[49] von Hardenberg, J.; Fraedrich, K.; Lunkeit, F.; Provenzale, A., Transient chaotic mixing during a baroclinic life cycle, Chaos, 10, 1, 122-134, (2000) · Zbl 1072.86507
[50] Voth, G.A.; Haller, G.; Gollub, J.P., Experimental measurements of stretching fields in fluid mixing, Phys. rev. lett., 88, 25, 254501.1-254501.4, (2002)
[51] K.C. Wang, On current controversy of unsteady separation, in: Symposium on Numerical and Physical Aspects of Aerodynamic Flows, Long Beach, CA, January 19-21, 1981
[52] Wiggins, S., Chaotic transport in dynamical systems, (1992), Springer-Verlag New York · Zbl 0747.34028
[53] Wiggins, S., The dynamical systems approach to Lagrangian transport in Ocean flows, Annu. rev. fluid mech., 37, 295-338, (2005) · Zbl 1117.76058
[54] Yuan, G.C.; Pratt, L.J.; Jones, C.K.R.T, Barrier destruction and Lagrangian predictability at depth in a meandering jet, Dyn. atmos. oceans, 35, 1, 41-61, (2002)
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