Nonlinear Processes in Geophysics www.nonlin-processes-geophys.net 1023-5809 1607-7946 17 1 2010 10.5194/npg-17-1-2010 http://www.nonlin-processes-geophys.net/17/1/2010/ http://www.nonlin-processes-geophys.net/17/1/2010/npg-17-1-2010.html http://www.nonlin-processes-geophys.net/17/1/2010/npg-17-1-2010.pdf 1 36 2010-01-05 Finite-time Lagrangian transport analysis: stable and unstable manifolds of hyperbolic trajectories and finite-time Lyapunov exponents M. Branicki m.branicki@bristol.ac.uk S. Wiggins School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK College of Earth, Ocean, and Environment, University of Delaware, Robinson Hall, Newark, USA We consider issues associated with the Lagrangian characterisation of flow structures arising in aperiodically time-dependent vector fields that are only known on a finite time interval. A major motivation for the consideration of this problem arises from the desire to study transport and mixing problems in geophysical flows where the flow is obtained from a numerical solution, on a finite space-time grid, of an appropriate partial differential equation model for the velocity field. Of particular interest is the characterisation, location, and evolution of transport barriers in the flow, i.e. material curves and surfaces. We argue that a general theory of Lagrangian transport has to account for the effects of transient flow phenomena which are not captured by the infinite-time notions of hyperbolicity even for flows defined for all time. Notions of finite-time hyperbolic trajectories, their finite time stable and unstable manifolds, as well as finite-time Lyapunov exponent (FTLE) fields and associated Lagrangian coherent structures have been the main tools for characterising transport barriers in the time-aperiodic situation. In this paper we consider a variety of examples, some with explicit solutions, that illustrate in a concrete manner the issues and phenomena that arise in the setting of finite-time dynamical systems. Of particular significance for geophysical applications is the notion of flow transition which occurs when finite-time hyperbolicity is lost or gained. The phenomena discovered and analysed in our examples point the way to a variety of directions for rigorous mathematical research in this rapidly developing and important area of dynamical systems theory. Acrivos, A., Aref, H., and Ottino, J M. (Eds.): Symposium on Fluid Mechanics of Stirring and Mixing, Part 2, Phys. Fluids A, 3(5), 1009–1469, 1991. Alam, M.-R., Liu, W., and Haller, G.: Closed-loop separation control: An analytic approach, Phys. Fluids, 158, 18(4), 043601, doi 10.1063/1.2188267, 2006. Aref, H. and El Naschie, M S. (Eds.): Chaos Applied to Fluid Mixing, Chaos Soliton. Fract., 4(6), 370~pp., 1994. Arnold, V. and Avez, A.: Ergodic Problems of Classical Mechanics, W.A. Benjamin Inc., 1968. Babiano, A., Provenzale, A., and Vulpiani, A. (Eds.): Chaotic Advection, Tracer Dynamics, and Turbulent Dispersion, in: Proceedings of the NATO Advanced Research Workshop and EGS Topical Workshop on Chaotic Advection, Conference Centre Sereno di Gavo, Italy, 24–28 May 1993, Physica D, vol 76, 329~pp., 1994. Baldoma, I. and Fontich, E.: Stable manifolds associated to fixed points with linear part equal to identity, J. Differ. Equations, 197, 45–72, 2004. Baldoma, I., Fontich, E., de~la Llave, R., and Martin, P.: The parametrization method for one-dimensional invariant manifolds of higher dimensional parabolic fixed points, Discrete Cont. Dyn. S., 17, 835–865, 2007. Batchelor, G K.: An Introduction to Fluid Dynamics, Cambridge University Press, Cambridge, 1967. Berger, A., Son, D T., and Siegmund, S.: Nonautonomous finite-time dynamics, Discrete Cont. Dyn B, 9, 463–492, 2008. Boffetta, G., Lacorata, G., Redaelli, G., and Vulpiani, A.: Detecting barriers to transport: a review of different techniques, Physica D, 159, 58–70, 2001. Bonckaert, P. and Fontich, E.: Invariant manifolds of dynamical systems close to a rotation: Transverse to the rotation axis, J. Differ. Equations, 214, 128–155, 2005. Bowen, R.: Symbolic Dynamics for Hyperbolic Flows, Am. J. Math., 95, 429–460, 1973. Bowen, R. and Ruelle, D.: The ergodic theory of axiom A flows, Invent. Math, 29, 181–202, 1975. Brown, G L. and Roshko, A.: Density effect and large structure in turbulent mixing layers, J. Fluid Mech., 64, 775–816, 1974. Casasayas, J., Fontich, E., and Nunes, A.: Invariant manifolds for a class of parabolic points, Nonlinearity, 5, 1193–1210, 1992. Casasayas, J., Faisca, J., and Nunes, A.: Melnikov method for parabolic orbits, NODEA-Nonlinear Diff., 10, 119–131, 2003. Cicogna, G. and Santoprete, M.: Nonhyperbolic homoclinic chaos, Phys. Lett A, 256, 25–30, 1999. Coddington, E A. and Levinson, N.: Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955. Coppel, W A.: Dichotomies in Stability Theory, in: Lecture Notes in Mathematics, Springer-Verlag, New York, Heidelberg, Berlin, 629, 97~pp., 1978. Dabiri, J O.: LCS MATLAB Kit, available at: http://dabiri.caltech.edu/software.html, last access: 2009. de~Blasi, F S. and Schinas, J.: On the stable manifold theorem for discrete time dependent processes in Banach spaces, B. Lond. Math. Soc., 5, 275–282, 1973. Dieci, L. and Eirola, T.: On smooth decompositions of matrices, SIAM J. Matrix Anal. A., 20(3), 800–819, 1999. Dieci, L. and Vleck, E. S V.: Lyapunov spectral intervals: Theory and computation, SIAM J. Numer. Anal., 40(2), 516–542, 2003. Dieci, L., Russell, R D., and Vleck, E. S V.: On the computation of Lyapunov exponents for continuous dynamical systems, SIAM J. Numer. Anal., 34(1), 402–423, 1997. d'Ovidio, F., Fernández, V., a, E. H.-G., and López, C.: Mixing structures in the Mediterranean Sea from finite-size Lyapunov exponents, Geophys. Res. Lett., 31, L17203, doi:10.1029/2004GL020328, 2007. d'Ovidio, F., Isern-Fontanet, J., Lopez, C., Hernandez-Garcia, E., and Garcia-Ladon, E.: Comparison between Eulerian diagnostics and finite-size Lyapunov exponents computed from altimetry in the Algerian basin, Deep-Sea Res. Pt I, 56, 15–31, 2009. Duan, J. and Wiggins, S.: Lagrangian transport and chaos in the near wake of the flow around an obstacle: a numerical implementation of lobe dynamics, Nonlin. Processes Geophys., 4, 125–136, 1997. Duc, L H. and Siegmund, S.: Hyperbolicity and invariant manifolds for planar nonautonomous systems on finite time intervals, Int. J. Bifurcat. Chaos, 18, 641–674, 2008. Ershov, S V. and Potapov, A B.: On the concept of stationary Lyapunov basis, Physica D, 118, 167–198, 1998. Fontich, E.: Stable curves asymptotic to a degenerate fixed point, Nonlinear Anal.-Theor., 35(6), 711–733, 1999. Garcá-Olivares, A., Isern-Fontanet, J., and Garcá-Ladona, E.: Dispersion of passive tracers and finite-scale Lyapunov exponents in the Western Mediterranean Sea, Deep-Sea Res. Pt I, 54, 15–31, 2007. Geist, K., Parlitz, U., and Lauterborn, W.: Comparison of Different Methods for Computing Lyapunov Exponents, Prog. Theor. Phys., 83(5), 875–893, 1990. Ghosh, S., Leonard, A., and Wiggins, S.: Diffusion of a passive scalar from a no-slip boundary into a two-dimensional chaotic advection field a two-dimensional chaotic advection field, J. Fluid Mech., 372, 119–163, 1998. Goldhirsch, I., Sulem, P.-L., and Orszag, S A.: Stability and Lyapunov Stability of Dynamical Systems: A Differential Approach and a Numerical Method, Physica D, 27, 311–337, 1987. Greene, J M. and Kim, J.-S.: The Calculation of Lyapunov Spectra, Physica~D, 24, 213–225, 1987. Haller, G.: Finding finite-time invariant manifolds in two-dimensional velocity fields, Chaos, 10(1), 99–108, 2000. Haller, G.: Distinguished material surfaces and coherent structure in three-dimensional fluid flows, Physica D, 149, 248–277, 2001a. Haller, G.: Lagrangian structures and the rate of strain in a partition of two-dimensional turbulence, Phys. Fluids, 13, 3365–3385, 2001b. Haller, G.: Lagrangian coherent structures from approximate veolcity data, Phys. Fluids, 14, 1851–1861, 2002. Haller, G.: Exact theory of separation for two-dimensional flows, J. Fluid Mech., 512, 257–311, 2004. Haller, G.: An objective definition of a vortex, J. Fluid Mech., 525, 1–26, 2005. Haller, G. and Poje, A.: Finite time transport in aperiodic flows, Physica D, 119, 352–380, 1998. Haller, G. and Yuan, G.: Lagrangian coherent structures and mixing in two-dimensional turbulence, Physica D, 147, 352–370, 2000. Henry, D.: Geometric theory of semilinear parabolic equations, in: Lecture Notes in Mathematics, Springer-Verlag: New York, Heidelberg, Berlin, 840, 348~pp., 1981. Holland, M. and Luzzatto, S.: Stable manifolds under very weak hyperbolicity conditions, J. Differ. Equations, 221, 444–469, 2006. Ide, K., Small, D., and Wiggins, S.: Distinguished hyperbolic trajectories in time-dependent fluid flows: analytical and computational approach for velocity fields defined as data sets, Nonlin. Processes Geophys., 9, 237–263, 2002. Irwin, M C.: Hyperbolic time dependent processes, B. Lond. Math. Soc., 5, 209–217, 1973. Jones, C. K. R T. and Winkler, S.: Invariant manifolds and Lagrangian dynamics in the ocean and atmosphere, in: Handbook of dynamical systems, North-Holland, Amsterdam, 55–92, 2002. Joseph, B. and Legras, B.: Relation between kinematic boundaries, stirring and barriers for the Antarctic polar vortex, J. Atmos. Sci., 59, 1198–1212, 2002. Ju, N. and Wiggins, S.: On roughness of exponential dichotomy, J. Math. Anal. Appl., 262, 39–49, 2001. Ju, N., Small, D., and Wiggins, S.: Existence and Computation of Hyperbolic Trajectories of Aperiodically Time-Dependent Vector Fields and Their Approximations, Int. J. Bif. Chaos, 13, 1449–1457, 2003. Katok, A. and Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, 1995. Kloeden, P. and Siegmund, S.: Bifurcations and continuous transitions of attractors in autonomous and nonautonomous systems, Int. J. Bif. Chaos, 15, 743–762, 2005. Koh, T Y. and Legras, B.: Hyperbolic trajectories and the Antarctic polar vortex, Chaos, 12(2), 382–394, 2002. Langa, J., Robinson, J C., and Suàrez, A.: Bifurcations in non-autonomous scalar equations, J. Differ. Equations, 221, 1–35, 2006. Langa, J A., Robinson, J C., and Suàrez, A.: Stability, instability, and bifurcation phenomena in non-autonomous differential equations, Nonlinearity, 15, 887–903, 2002. Lapeyre, G.: Characterization of finite-time Lyapunov exponents and vectors in two-dimensional turbulence, Chaos, 12(3), 688–698, 2002. Legras, B. and Vautard, R.: A Guide to Liapunov Vectors, in: Proceedings of the 1995 ECMWF Seminar on Predictability, edited by: Palmer, T., 143–156, 1996. Lekien, F. and Coulliette, C.: Chaotic stirring in quasi-turbulent flows, Philos. T. Roy. Soc A, 365~pp., 2007. Lekien, F., Shadden, S., and Marsden, J E.: Lagrangian coherent structures in n-dimensional systems, J. Math. Phys., 48, 065404, 2007. Mancho, A M., Small, D., Wiggins, S., and Ide, K.: Computation of Stable and Unstable Manifolds of Hyperbolic Trajectories in Two-Dimensional, Aperiodically Time-Dependent Vector Fields, Physica~D, 182, 188–222, 2003. Mancho, A. M., Small, D., and Wiggins, S.: Computation of hyperbolic trajectories and their stable and unstable manifolds for oceanographic flows represented as data sets, Nonlin. Processes Geophys., 11, 17–33, 2004. Mancho, A M., Small, D., and Wiggins, S.: A tutorial on dynamical systems concepts applied to Lagrangian transport in oceanic flows defined as finite time data sets: Theoretical and computational issues, Phys. Rep., 437, 55–124, 2006. Mancho, A M., Hernández-Garc\'ia, E., Small, D., and Wiggins, S.: Lagrangian transport through an ocean front in the North-Western Mediterranean Sea, J. Phys. Oceanogr., 38, 1222–1237, 2008. Mathur, M., Haller, G., Peacock, T., Ruppert-Felsot, J E., and Swinney, H L.: Uncovering the Lagrangian Skeleton of Turbulence, Phys. Rev. Lett., 98, 144502, 2007. McGehee, R.: Stable manifolds theorem for degenerate fixed-points with applications to celestial mechanics, J. Differ. Equations, 14, 70–88, 1973. Miller, P D., Jones, C. K. R T., Rogerson, A M., and Pratt, L J.: Quantifying transport in numerically generated velocity fields, Physica D, 110, 105–122, 1997. Oseledec, V.: A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc., 19, 197–231, 1968. Ottino, J.: The Kinematics of Mixing: Stretching, Chaos, and Transport, Cambridge University Press, Cambridge, 1989. Pierrehumbert, R T.: Large-scale horizontal mixing in planetary atmospheres, Phys. Fluids A, 3(5), 1250–1260, 1991. Pierrehumbert, R T. and Yang, H.: Global chaotic mixing on isentropic surfaces, J. Atmos. Sci., 50, 2462–2480, 1993. Pollicott, M.: Symbolic Dynamics for Smale Flows, Am. J. Math., 109, 183–200, 1987. Samelson, R. and Wiggins, S.: Lagrangian Transport in Geophysical Jets and Waves: The Dynamical Systems Approach, Springer-Verlag, New York, 2006. Sell, G R.: Non-autonomous differential equations and dynamical systems, T. Am. Math. Soc., 127, 241–83, 1967. Sell, G R.: Lectures on Topological Dynamics and Differential Equations, Van-Nostrand-Reinhold, Princeton, NJ, 1971. Shadden, S., Lekien, F., and Marsden, J E.: Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows, Physica~D, 212, 271–304, 2005. Shadden, S., Dabiri, J O., and Marsden, J E.: Lagrangian analysis of fluid transport in empirical vortex ring flows, Phys. Fluids, 18, 047105, 2006. Shadden, S., Katija, K., Rosenfeld, M., Marsden, J E., and Dabiri, J O.: Transport and stirring induced by vortex formation, J. Fluid Mech., 593, 315–331, 2007. Shadden, S C.: Online LCS tutorial: available at: www.cds.caltech.edu/~shawn/LCS-tutorial/, last access: 2009. Shariff, K., Pulliam, T H., and Ottino, J M.: A dynamical systems analysis of kinematics in the time-periodic wake of a circular cylinder, in: Vortex dynamics and vortex methods (Seattle, WA, 1990), Amer. Math. Soc., Lectures in Appl. Math., 28, 613–646, 1991. Sturman, R., Ottino, J M., and Wiggins, S.: The Mathematical Foundations of Mixing, in: Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, vol 22, 281~pp., 2006. Surana, A., Grunberg, O., and Haller, G.: Exact theory of three-dimensional flow separation. Part 1. Steady separation, J. Fluid Mech., 564, 57–103, 2006. Surana, A., Jacobs, G B., and Haller, G.: Extraction of Separation and Attachment Surfaces from Three-Dimensional Steady Shear Flows, AIAA Journal, 45, 1290–1302, 2007. von Hardenberg, J., Fraedrich, K., Lunkeit, F., and Provenzale, A.: Transient chaotic mixing during a baroclinic life cycle, Chaos, 10(1), 122–134, 2000. Wang, Y., Haller, G., Banaszuk, A., and Tadmor, G.: Closed-loop Lagrangian separation control in a bluff body shear flow model, Phys. Fluids, 15, 2251–2266, 2003. Wiggins, S.: The Dynamical Systems Approach to Lagrangian Transport in Oceanic Flows, Annu. Rev. Fluid Mech., 37, 295–328, 2005. Yuster, T. and Hackborn, H W.: On invariant manifolds attached to oscillating boundaries in Stokes flows, Chaos, 7, 769–776, 1997. Zhang, W.: Invariant manifolds for differential equations, Acta Math. Sin., 8, 375–398, 1992.