Research Plan: Transport and Mixing in Aperiodic Systems
Background and Significance
Over the past 10 years, much work has been done in applying the approach and methods of dynamical theory to the study of transport and mixing in fluids. For immersed point mass particles, the phase space is the physical space in which the fluid flow takes place. Lagrangian Coherent Structures (LCS's), such as KAM tori and the invariant manifolds of hyperbolic fixed points and periodic orbits divide the phase space of these systems into regions of qualitatively different dynamics [Yuan et al., 2004; Babiano et al., 1994].
The LCS's and their interactions govern the entanglement of the trajectories and generate chaotic mixing and transport phenomena [Haller and Poje, 1998; Wiggins, 1992].
In recent years, Lagrangian transport in the context of two-dimensional, time-periodic flows has been extensively developed. For example, the persistence of invariant curves in the Poincar ´e map (KAM curves) gives rise to barriers in the flow; Chaos and Smale horseshoes are mechanisms for the "randomization" of fluid particle trajectories [Babiano et al., 1994]; Melnikov's method provides flux estimates as well as a description of the parameter regimes where chaotic particle motion occurs [Melnikov, 1963; Guckenheimer and Holmes, 1983]. Lobe dynamics enables efficient computations of transport between qualitatively different flow regimes [Rom-Kedar, 1990; Rom-Kedar and Wiggins, 1991]. The central theme of my research program is the development and the application of this framework for time-chaotic systems [Haller, 2000; Haller, 2001; Ide et al., 2002]. This includes a major effort to study the properties of "moving" LCS's and investigate how deterministic chaos is generated in aperiodic flows.
There are a variety of important applied problems that can be addressed with this methodology. These include space missions, molecular fragmentation, and turbulence in fluids. In recent years, numerous experiments have revealed the presence of LCS's in geophysical fluid dynamics settings [Pierrehumbert, 1991; Samelson, 1992; Voth et al., 2002]. In coastal areas and for global ocean circulation, the identification of dynamical barriers and alleyways provides a better understanding of the mixing processes [Coulliette and Wiggins, 2001; Yuan et al., 2002]. Such studies also permits various optimization problems such as reducing the impact of coastal pollutants, finding energy-efficient routes, or devising adaptive sampling strategies in the ocean.
The following sections detail the challenges and approaches for each aspect of my research program: theoretical developments, numerical implementations, and applications.
A Observation and Control of Exponential Dichotomies
 
Figure 1. An analytical system corresonding to two
rotating eigenvectors (eigenvalues: +1 and 1). For
slow rotation rate (left), a parcel of particles (green
dots) stretches along a moving unstable manifold
(red line). For fast rotations (right), there is not any
exponential dichotomy or unstable manifold.
Figure 2. LCS's (red and blue curves) during the
break-up of the Antarctic polar vortex. Particles
(green parcels) reveal the hyperbolic structure at the
pole and stretch along the attracting LCS.
Autonomous and periodic systems have hyperbolic fixed points (or periodic orbits) and associated stable and unstable manifolds [ Hirsch and Smale, 1974]. An equivalent notion for aperiodic systems are trajectories and invariant manifolds with exponential dichotomies [ Coppel, 1978]. In the vicinity of these structures, particles experience exponential stretching and exponential shrinking [ Ide et al., 2002]. The stretching causes sensitivity to initial conditions, i.e., the exponential separation of two particles initially close to each other. The shrinking is responsible for the folding of the invariant manifolds in a compact region and the existence of a compact chaotic invariant set [ Smale, 1980; Wiggins, 1992]. This description of chaotic transport is based on the observation of particle trajectories (i.e., the Lagrangian dynamics) near the exponential dichotomies [ Ide et al., 2002]. My research plan aims at understanding, detecting and controlling these structures.
As shown in Fig. 1, the relationship between Lagrangian dynamics and observed Eulerian features, i.e., the "frozen"
velocity field, is not straightforward for time-varying systems. The objective is to develop Eulerian criteria for Lagrangian hyperbolicity, i.e., the existence of stable and unstable manifolds. Such criteria are based only on the velocity field and do not require the integration of particles. A necessary criterion [ Haller and Poje, 1998; Poje and Haller, 1999] indicates that fast rotation of Eulerian hyperbolic structures can induce non-hyperbolic flows. This phenomenon is the major source of difficulties in establishing Eulerian rules for Lagrangian hyperbolicity.
Simplified analytical models such as the one depicted in
Fig. 1 reveals typical behaviors and provides approximate
criteria for hyperbolicity that are applicable in more general
settings [ Lekien, 2003].
One such example is the break-up of the Antarctic Polar
vortex. During September 2002, the polar vortex became
unstable and split into two vortices, each holding a smaller
ozone hole and stretching outside the polar area. Figure 2
shows the hyperbolic trajectory at the South pole and its
associated rotating invariant manifolds [ Lekien et al., 2005d].
The 4 point vortex model of the splitting event [ Newton, 2001] is quasi-analytical and has a fixed point at the South pole,
where the dynamics is equivalent to the rotating saddle of
Fig. 1. Similar developments lead to the identification of
vortices [ Haller, 2005].
A.1 Past Accomplishments
Figure 3. Left panel. DLE field and LCS
in high-frequency radar data collected on
the coast of Florida [Lekien et al., 2005b].
Right panel. The estimated flux along the
LCS is negligible with respect to typical currents in the area.
Figure 4. Lagrangian separation at the boundary
of Rayleigh-Bénard convection cells (upside down).
Red triangle: Separation point, Blue triangle: Re-attachment point, Green dots : Particles released in
the vicinity of the boundary.
Lagrangian Coherent Structures. The Lagrangian behavior of the flow during a finite interval of time can be extracted from Finite-Time Lyapunov Exponents (FTLE), a measure of the separation rate. The Direct Lyapunov Exponent (DLE) is a robust method to compute these exponents [ Haller, 2000; Haller, 2001]. Ridges in the DLE field approximate Lagrangian Coherent Structures (LCS's) [ Shadden et al., 2005]. We have shown that these ridges render a geometric description of the flow [ Lekien et al., 2005b].
Figure 3 shows an LCS based on coastal high-frequency
radar data. The ridge is the boundary between two regions
of qualitatively different dynamics: a recirculation zone and
fast Northward current.
Reduced analysis on boundaries: Separation. Hyperbolic stretching near a boundary is best studied in terms of the existence of a separation profile emanating from the boundary [ Sears and Telionis, 1975; Van Dommelen and Cowley, 1990]. A necessary criterion for the occurrence of separation at the boundary of time-chaotic two-dimensional flows is given in [ Haller, 2004]. In recent
work [ Lekien and Haller, 2005], we extended these results to boundaries with a free-slip boundary condition (e.g., geophysical flows, nano-fluids). The separation criterion developed is necessary (based on [ [Mané, 1978]) and sufficient (based on [ Fenichel, 1971]).
A.2 Present and Future Work
  
Figure 5. LCS's for a cylindrical Rayleigh-Bénard convection cell. The two LCS's intersect and form lobes. These lobes carry fluid to and from the core cylindrical cell.
Figure 6 describes transport in a three-layer quasigeostrophic model of the North Atlantic [ Rowley, 1996]. Intergyre transport is governed by the motion of two sequences of lobes moving along the Gulf stream. In particular, the pinching of the invariant manifolds is responsible for the formation of large rings in the ocean [ Haller and Poje, 1998]. I seek global dynamical constraints that provide a geometrical understanding of transport. For example, the uniqueness of trajectory and the continuity of the flow can be used to prove that only anti-cyclonic rings are involved in transport across a meandering jet [ Lekien, 2003]. I am also investigating how lobe dynamics in higher dimensional spaces (based on [ Beigie, 1995]) can be adapted to time-chaotic systems such as the three-dimensional Rayleigh-Bénard cells in Fig. 5.
   
Figure 6. Lobe dynamics and transport in a layered quasigeostrophic (QG) model of the North Atlantic.
Finite-Size Lyapunov Exponent. The coherent structures are defined using Finite-Time Lyapunov Exponents (FTLE) in [ Shadden et al., 2005]. The works of [ Joseph and Legras, 2002] and [ Koh and Legras, 2002] demonstrate the use of Finite-Size Lyapunov Exponent (FSLE) for locating LCS's in atmospheric data. G. Haller and I are currently investigating the relationship between FSLE and FTLE. In particular we analyze the curvature of these fields, a critical quantity determining the properties of the LCS's [ Shadden et al., 2005; Lekien, 2005].
Physical constraints. Early works in dynamical systems theory were mainly applied to kinematically defined velocity fields. Using dynamically evolving fields brings more physical relevance to these transport studies [ Del-Castillo-Negrete and Morrison, 1993; Ngan and Shepherd, 1997]. A next step is to investigate the effect of these dynamical constraints on the existence and properties of LCS's [ Rogerson et al., 1999]. For example, the separation criterion of [ Haller, 2004] is derived using the assumption that the density of the fluid remains bounded. Physical constraints facilitate the development of theorems and methods applicable to physical systems. One of my objectives is to show that Navier-Stokes equations cannot sustain a sufficiently high rotation rate, hence non-hyperbolic rotating saddles (see Fig. 1) do not exist in real fluids.
I am also interested in investigating LCS's for particles with a non-zero volume [ Cartwright et al., 2002]. The motion of neutrally buoyant particles in a two-dimensional aperiodic flow can be analyzed in a 4 dimensional phase space [ Babiano et al., 2000]. Due to the fast action of on-board controllers, the dynamics of complex (non-spherical) underwater vehicles can also be constrained to low dimensional spaces. The differences between the LCS's for point mass particles and immersed solids is an important theoretical question. Furthermore, it is a critical to investigate the effect on the accuracy of the barriers to contaminants and the alleyways for autonomous vehicles.
B Numerical Methods
B.1 Past Accomplishments
Interpolation. Dynamical systems theory typically applies to velocity fields that are sufficiently smooth. This requires a careful interpolation of the data. In [ Lekien and Marsden, 2005], we provide C 1 interpolation for two-dimensional time-dependent flows. Modal analysis [ Lipphardt et al., 2000; Chu et al., 2003] is used to interpolate, extrapolate and filter incomplete datasets. In comparison to orthogonal modes [ Rowley et al., 2004], normal modes do not depend on time or on the measured data and can be used during long real-time experiments. In [ Lekien et al., 2004], we develop Open-Boundary Modes (OMA) that are suitable for assimilating radar data in coastal regions such as Monterey Bay [ Paduan and Rosenfeld, 1996; Paduan and Cook, 1997].
Figure 7. MANGEN: Graphical User Interface.
Coherent Packages. MANGEN (MANifold GENerator) is a collection of tools that I wrote to study mixing and transport in experimental fluid flows. The interface is flexible and permits the computation of Lagrangian coherent structures (LCS's) and separation profiles for a wide range of problems. The abstraction layer between the dynamical systems methods and the physical systems (e.g., high-frequency radar data, model data, Hamiltonian) permits the development of new tools and their direct application to all the systems configured.
MANGEN has been successfully exported to other institutions, including the Monterey Bay Aquarium Research Institute (for Lagrangian studies during the Adaptive Ocean Sampling Network (AOSN-II) project), the Jet Propulsion Laboratory (as a means to offer end-user products based on the ROMS ocean model), the Rosentiel School of Marine and Atmospheric Science (to investigate the properties of the solutions of the parabolic wave equation) and the French Research Institute for Exploitation of the Sea (to study Lagrangian residual circulation in the Iroise Sea).
Concurrent Programming and Real-Time Applications. Real-time applications and high resolution simulations require the use of concurrent algorithms for the computation of LCS's and separation profiles. Parallel code is currently available in MANGEN and is used on clusters at Caltech, JPL, and Princeton University. The real-time capabilities of MANGEN were during the Adaptive Ocean Sampling Network (AOSN-II) experiment in August 2003. High-frequency radar data and ocean model output were assimilated in real-time to display LCS's in the Monterey Bay Aquarium Research Institute's control room.
B.2 Present and Future Work
Lagrangian Coherent Structures. For the purpose of extracting LCS's (as opposed to observing them on a FTLE plot), a precise definition is developed in [ Shadden et al., 2005; Lekien et al., 2005e]. Ongoing work [ Lekien, 2005] reduces the definition and shows that LCS's can be extracted as the level sets of a computable function. The extraction of the LCS's is based on a finite-element method. Consequently, I am also developing more efficient and accurate FTLE computations on finite-element meshes. In particular, the algorithms of MANGEN have been adapted to work on arbitrary manifolds, instead of only Cartesian subspaces. For example, LCS's are computed on a spherical Earth in [ Lekien et al., 2005d].
I am also investigating the relationship between LCS's and almost invariant sets [ Dellnitz and Preis, 2003]. In [ Dellnitz et al., 2005], we combine and compare these two techniques for the restricted three-body problem. The formulation of lobe dynamics and transport problems in terms of almost invariant sets permits the use of highly efficient algorithms based on graph theory [ Dellnitz et al., 2005]. Conversely, I plan to study how the extraction of LCS's can be cast as a set-oriented problem and solved as such. The objective is to replace the stochastic Frobenius-Perron operator in [ Dellnitz and Preis, 2003] by a deterministic transfer function satisfying the objective of [ Shadden et al., 2005].
Optimal Filtering. The recent use of OMA modes [ Lekien et al., 2004] for biological studies in Bodega Bay [ Kaplan et al., 2005] suggested that OMA modes can be generalized and optimized. In particular, a new class of modes are currently developed [ Kaplan and Lekien, 2005] to avoid the artificial vanishing vorticity near the coastline.
Integration with Ocean and Biological Models. Interfaces for MANGEN are developed to interact directly with experimental data and advanced ocean models. In particular, the parallel code of MANGEN has been installed at the Jet Propulsion Laboratory to function directly with the Regional Ocean Modeling System (ROMS). In addition, interfaces are developed for the Harvard Ocean Model and Prediction System (HOPS) allowing the computation of LCS for real-time forecasts of the ocean as well as the direct integration of modeled uncertainties [ Lermusiaux and Lekien, 2005].
C Applications
C.1 Past Accomplishments
Figure 8. Recirculating zone near Fort-Lauderdale, FL..
Pollution Control. Coherent structures in the ocean mark the boundary between regions of qualitatively different dynamics. The impact of contaminants in a coastal environment can be reduced by the use of a release schedule based on the motion of nearby LCS's.
In [ Lekien et al., 2005b], we show that the average concentration of pollutants and the peaks of maximum concentration can be significantly reduced using the LCS's observed in high-frequency radar data collected on the coast of Florida (Fig. 8).
In [ Lekien et al., 2005a], we show the existence of highly sensitive intervals of time in the coastal dynamics of Monterey Bay. Avoiding dangerous operations such as the dismantlement of obsolete oil tanks during these periods lessens the disastrous outcome in the event of an accidental oil spill.
Fuel Efficient Routes. During the Autonomous Ocean Sampling Network ( AOSN-II) experiment in August 2003, autonomous underwater vehicles (SIO and Slocum gliders) were used to sample geophysical processes in Monterey Bay, CA [ Fiorelli et al., 2003]. Coordinated control maintained either specific formations (small-scale sampling) or coordinated arrays (large-scale coverage). Real-time LCS computations permit the optimization of the paths followed by the weakly actuated vehicles ( 25 cm/s) and take advantage of the currents (as high as 100 cm/s). LCS's indicate regions of high stretching and are avoided by vehicles formations [ Bhatta et al., 2005]; Isolated vehicles follow the structures during transient periods to reach their assigned position quickly [ Inanc et al., 2004].
Similar methodologies can be applied to multi-body problems in dynamical astronomy, molecular dynamics, and chemical reaction kinetics. In [ Dellnitz et al., 2005], we focus on the transport rates between two resonance regions for the three body system consisting of the Sun, Jupiter and a third body (such as an asteroid or an autonomous vehicle). The dynamics of the lobes formed by the intersection of the LCS's provide fuel-efficient routes in the solar system.
Control of fluid flows. The recent development of exact criteria for separation in fluid flows [ Haller, 2004; Lekien and Haller, 2005] opens the door to a variety of approaches to actuate the flow and induce separation at specified positions (and with a specified strength or separation angle). Such a method is described in [ Insperger et al., 2004; Lekien et al., 2005c] and permits a rapid control of the lift on a wing or the transport of fuel within the fluid.
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Figure 9. Two wall jets used as actuators and inducing fixed separation in the presence of an unknown background flow.
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Figure 10. A delayed dead-beat controller [Lekien et al., 2005c] produces Lagrangian patterns (green lines) revealing separation at the desired position (blue circle).
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C.2 Present and Future Work
Fragmentation Dynamics. As an example of a time-dependent 4-dimensional system, I am investigating the Van Der Waals complex He-I 2 using the potential of [ Gaspard and Rice, 1989]. An alternative to statistical studies [ MacKay et al., 1984a; MacKay et al., 1984b] of dissociation of the compound is the geometrical approach based on the motion of the lobes between the LCS's. The first 6 steps of the dynamics of the lobes involved in the decay of a restricted two-dimensional (T-shaped) model of the molecule can be found on Fig. 11. Figure 12 shows a section of the LCS's of the full problem where the lobes are bounded by pieces of three-dimensional LCS's [ Gillilan and Ezra, 1991; Beigie, 1995].
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Figure 11. Geometrical description of the fragmentation of He-I2 for the reduced two-dimensional model. An initial ensemble of molecules (shaded area) dissociates progressively as a result of lobes invading the interior of the the quasi-invariant region.
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Figure 12. LCS and lobes in the full 4-dimensional phase space of HeI2
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Figure 13. Atmospheric LCS's based on NCEP/NCAR reanalysis.
Mixing and Transport in the Atmosphere. In collaboration with G. Haller and R.T. Pierrehumbert, I am computing LCS's for the atmosphere using the NCEP/NCAR Reanalysis Project [ Kalnay et al., 1996], an analysis (and forecast) system performing data assimilation using past data from 1948 to the present. The objective is to identify Lagrangian processes and quasi-invariant regions in the reanalysis data (Fig. 13).
Ocean Sampling. The Adaptive Sampling And Prediction ( ASAP) project aim to develop a sustainable, portable, adaptive ocean observing and prediction system for use in coastal environments [ Leonard et al., 2005]. This project employs, among other observation platforms, autonomous underwater vehicles that carry sensors to measure physical and biological signals in the ocean. The measurements from all sensing platforms are assimilated in real-time into advanced ocean models. The objective is to coordinate the mobile assets in order to collect data of highest possible value. To this purpose, optimal sampling arrays are designed based on minimizing the residual error of an objective analysis scheme or advanced ocean assimilation models [ Leonard et al., 2005; Bhatta et al., 2005]. Symmetries in the optimal solutions are broken by properly aligning the arrays with the LCS's.
 
Figure 14. Left panel: FTLE and LCS based on HOPS ensemble average.
Right panel: Corresponding error based on ensemble statistics.
Uncertainties. The impact of uncertainties on the LCS's is an important question for practical applications. Realistic ocean modeling with dense data assimilation provides accurate nowcasts and prediction of ocean flow fields as well as associated uncertainties [ [Jazwinski, 1970; Lermusiaux, 2005]. P.F.J. Lermusiaux and I are investigating the transfer of these uncertainties from the ocean state to the LCS's [ Lermusiaux and Lekien, 2005]. Figure 14 shows that the "upwelling barrier" in Monterey Bay is a robust LCS with minimum uncertainty [ Lermusiaux et al., 2005].
Long Term Objectives
A central objective of my research program is to enhance the understanding and use of Lagrangian Coherent Structures (LCS's) in time-chaotic dynamical systems. Classical results are concerned with autonomous and time-periodic velocity fields. For turbulent flows (or chaotic systems), the velocity field (or state of the system) does not reach an equilibrium and does not follow a periodic pattern in phase space. Following D. Ruelle and F. Takens, I assume that the state of the turbulent flow is wrapping around a strange attractor [ Ruelle and Takens, 1971]. Experimental measurements and analytical examples have shown that dissipative systems such as viscous fluids will not have quasi-periodic motions. The understanding and control of turbulence with dynamical systems methods requires the generalization of these methods to arbitrary time-dependence [ Ruelle and Takens, 1971].
The study of the relationships between the dynamics in phase space and the LCS's in the physical space will provide a deeper understanding of the mechanisms and implications of turbulence, as well as new methods for the control of transport and mixing in turbulent flows. Lagrangian patterns in the physical space are the meaningful observables. The Eulerian velocity field (i.e., the state in phase space) is not measured directly but derived from particle trajectories (e.g., derivative of drifter paths, Digital Particle Image Velocimetry (DPIV) or Doppler effect). In Engineering and Science, the knowledge or the prediction of the velocity is only used as a means to forecast or control the path of actual particles. For example, in a search and rescue mission, the path followed by the drifting body must be forecast, not the currents. My research program approaches turbulence by considering instabilities in the Navier-Stokes equations and observing them from the point of view of a drifting body, not the velocity field. The motivation for this approach is twofold: first, turbulence is studied from a physically relevant coordinate system; chaotic disturbances that do not influence the motion of passive drifters are ignored. Second, it opens the door to a possibly simpler description of turbulence. The apparition of a strange attractor in phase space [ Ruelle and Takens, 1971] is interpreted by using (smooth) moving and bifurcating LCS's in physical space.
My approach is based exclusively on time-dependent dynamical models and is well suited for the direct application to geophysical flows and to experimentally measured velocity fields. In the future, the theory and numerical methods will adapt to particles with a finite volume (pollutants or vehicles) and incorporate statistical models. This will increase the utility of these methods in Science and Engineering and provide a clearer mathematical and physical understanding of hyperbolicty, mixing and separation.
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