Phys Today-LCS

www.physicstoday.org February 2013 Physics Today 41

In April 2010, fine, airborne ash from a volcaniceruption in Iceland caused chaos throughout European airspace. The same month, the explo-sion at the Deepwater Horizon drilling rig in theGulf of Mexico left a gushing oil well on the seafloor that caused the largest offshore oil spill in UShistory. A year later the Tohoku tsunami hit the coastof Japan, causing great loss of life, the Fukushimanuclear-reactor disaster, and the release of substan-tial amounts of debris and radioactive contamina-tion into the Pacific Ocean.

Those three globally significant events, de-picted in figure 1, share a common theme. In eachcase, material was released into the environmentfrom what was essentially a point source, and pre-dicting where that material would be transportedby the surrounding oceanic or atmospheric flowwas of paramount importance.

To predict the outcomes of such events, thestandard approach is to run numerical simulationsof the atmosphere or the sea and use the resultingvelocity-field data sets to forecast pollutant trajec-tories. Although that approach does predict the fu-ture of individual fluid parcels, the predictions arehighly sensitive to small changes in the time and lo-cation of release. Attempts to address the excessivesensitivity to initial conditions include running sev-eral different models for the same scenario. But thattypically produces even larger distributions of ad-vected particlesthose transported by the fluid

flowand thus hides key organizing structures ofthat flow.

Furthermore, traditional trajectory analysis fo-cuses on full trajectory histories that yield convolutedspaghetti plots that are hard to interpret. Improvedunderstanding and forecasting therefore requiresnew concepts and methods that provide more insightinto why fluid flows behave as they do.

Lagrangian coherent structuresRecently, ideas that lie at the interface between non-linear dynamicsthe mathematical discipline thatunderlies chaos theoryand fluid dynamics havegiven rise to the concept of Lagrangian coherentstructures (LCSs), which provides a new way of un-derstanding transport in complex fluid flows.

Although advances have been made in the de-tection of LCSs in fully three- dimensional flows,this article focuses primarily on the many advancesthat have been made for 2D flows. There, LCSs takethe form of material linescontinuous, smoothcurves of fluid elements advected by the flow. Theyare conceptually simpler than the 2D material sur-faces required for LCSs in 3D flows. Furthermore,2D flows are particularly relevant for studies of

New techniques promise better forecastingof where damaging contaminants in the

ocean or atmosphere will end up.

Thomas Peacock and George Haller

Thomas Peacock is a professor of mechanical engineering at the MassachusettsInstitute of Technology in Cambridge. George Haller is a professor of nonlineardynamics at ETH Zrich in Switzerland.

Lagrangiancoherent structures

The hidden skeleton of fluid flows

Downloaded 01 Feb 2013 to 195.176.113.187. Redistribution subject to AIP license or copyright; see http://www.physicstoday.org/about_us/terms

pollution transport on the ocean surface and on sur-faces of constant density in the atmosphere.

Generally speaking, the LCS approach pro-vides a means of identifying key material lines thatorganize fluid-flow transport. Such material linesaccount for the linear shape of the ash cloud in figure 1a, the structure of the oil spill in 1b, and thetendrils in the spread of radioactive contaminationin 1c. More specifically, the LCS approach is basedon the identification of material lines that play thedominant role in attracting and repelling neighbor-ing fluid elements over a selected period of time.Those key lines are the LCSs of the fluid flow. To de-velop an understanding of them, we must first con-sider several ideas.

Lagrange versus EulerThere are two different perspectives one can take indescribing fluid flow. The Eulerian point of viewconsiders the properties of a flow field at each fixedpoint in space and time. The velocity field is a primeexample of an Eulerian description. It gives the in-stantaneous velocity of fluid elements throughoutthe domain under consideration. The identity andprovenance of fluid elements are not important; atany given point and instant, the velocity field sim-ply refers to the motion of whatever fluid elementhappens to be passing.

By contrast, the Lagrangian perspective is con-cerned with the identity of individual fluid ele-ments. It tracks the changing velocity of individualparticles along their paths as they are advected bythe flow. Its the natural perspective to use when

considering flow transport because patterns such asthose in figure 1 arise from material advection.

Another driving force behind the developmentof the LCS approach is the concept of objectivity, orframe invariance. Characterizations of flow struc-tures in terms of the properties of Eulerian fieldssuch as the velocity field tend not to be objective;they dont remain invariant under time- dependentrotations and translations of the reference frame.For instance, a common way to visualize flow fieldsis to use streamlines, which are Eulerian entities thatfollow the local direction of the velocity field at agiven instant.

Traditionally, vortices in fluid flows have beenidentified as regions filled with closed streamlines.But velocity fields, and hence their streamlines,change when viewed from different referenceframes. So what looks like a domain full of closedstreamlines in one frame can appear completely dif-ferent when viewed from another frame. For exam-ple, an unsteady vortex flow may look like a steadysaddle-point flow in an appropriate rotating frame.

For unsteady flows, which are the rule ratherthan the exception in nature, there is no obvious pre-ferred frame of reference. So any conclusion abouttransport-guiding dynamic structures should holdfor any choice of reference frame. With regard to an

42 February 2013 Physics Today www.physicstoday.org

Lagrangian structures

a b

c

Figure 1. Large-scale contaminant flows. (a) A 150-km-wide view of theash cloud from the 2010 Icelandic volcano eruption. (b) A 300-km-wideview of the 2010 Deepwater Horizon oil spill in the Gulf of Mexico. (c) A prediction of the eastward spread of radioactive contaminationinto the Pacific Ocean from the 2011 Fukushima reactor disaster in Japan.

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oil spill, for example, interpretations of the organi-zation of material transport cannot depend onwhether the data are processed in the referenceframe of an onshore radar observation station, a re-connaissance plane, or an orbiting satellite. Reliableforecasting of material transport calls for frame- invariant techniques.

Material linesA detailed understanding of Lagrangian transportalready exists for time-independent flows such asthe steady saddle-point flow in figure 2a. Fluidparcels approaching a saddle point along a promi-nent line of flowing material that serves as a repul-sive transport barrier (the so-called stable manifold)are ultimately drawn away from it toward an or-thogonal material line that constitutes an attractivetransport barrier (the unstable manifold) and carriesthem away from the saddle point. One manifoldlooks like the other with time reversed. The unstablemanifold, despite its name, acts as a core organizingstructure in the vicinity of the saddle point, attract-ing all nearby fluid particles, which then stretch outto adopt its shape.

Prominent material lines are known to exist inperiodic and quasi periodic flows, where they serveas skeletons of observed tracer patterns. An example,shown in figure 2b, is suggested by the organizedcloud features in the wake created by steady windblowing past the Mexican island of Guadalupe. Butfinding them is rarely that easy. The identification ofdynamical skeletons for material patterns in flowswith complex spatial and temporal structure pre -sents an ongoing challenge.

Thats because the mathematical methods usedto identify key material lines in steady, periodic, andquasi periodic flows rely on knowing the flow fieldfor all time. But the flows that most need to be un-derstood are typically aperiodic, and the associatedvelocity-field information is known only in the formof observational or numerical-simulation data setsfor finite time intervals. As a result, even elementary

concepts such as stable and unstable manifolds andsaddle points are ill defined for aperiodic flows.

A modern characterization of repelling and at-tracting material lines has been emerging in fluiddynamics to facilitate the understanding of materialtransport by aperiodic, finite-time flows. The start-ing point is a 2D flow field,

dx/dt = u(x, t),

with position vector x = (x, y) and velocity vectoru = (u,v) in the x,y plane.

Assuming that the velocity field is observed fortimes t ranging over the finite interval [t0 , t1] , theLCSs of the flow during that interval are the mate-rial lines that repel or attract nearby fluid trajecto-ries at the highest local rate relative to other materiallines nearby. As shown in figure 3, the attraction andrepulsion are orthogonal to the flow lines. Overall,the repelling and attracting LCSs play similar rolesto the stable and unstable manifolds, respectively, of the saddle point in figure 2a. As illustrated in figure 3c, the repelling LCSs direct particles to dif-ferent domains of the attracting LCSs.

A simple exampleThe relevance of the LCS approach to understand-ing fluid transport is nicely illustrated by the so-called double-gyre problem,1 presented in figure 4.Even though, as a time-periodic flow, its amenableto more traditional analysis, the double-gyre prob-lem has become a canonical flow field for testingLCS ideas.

In figure 4, a circular blob of black dye is re-leased at time t0 in a flow comprising two oppo-sitely swirling gyres (vortices) whose strengthsand locations vary periodically in time. The arrowsin figures 4a and 4b indicate the velocity field at t0and later at t1.

By t1, the dye has been stretched and trans-ported throughout the fluid. But the new dye dis-tribution is noticeably different from the shape of


Stable manifold

Fluid parcel

Saddle point

Unstable manifold

a b

Figure 2. Prominent lines of advected material form transport barriers near a saddle point. (a) A fluid parcel approachingthe saddle point astride one material line (the repelling stable manifold) eventually becomes drawn out and away from thesaddle point along the orthogonal material line (the attracting unstable manifold). (b) Unstable manifolds (red curve) inferredfrom stretching cloud patterns in a time-periodic atmospheric flow generated by winds blowing past Guadalupe, a volcanic island off Mexicos Baja California.

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the velocity field at t1. Much of it is stretched alongthe outer boundaries, and two dye streaks cutacross to the right side of the domain. But theresno dye drawn clockwise around the left vortex,even though that vortex is a strong feature of thevelocity field.

Recall that if the flow were viewed from, say, arotating reference frame, the form of the instanta-neous velocity field would change significantlywhile the shape of the dye streak would remain thesame. So the key question is, What frame-invariantstructures are responsible for organizing the shapeof the dye streak between t0 and t1?

Figure 4c presents a candidate LCS, the convo-luted white line that cuts the initial dye patch intotwo parts, shown red and blue. That line is thestrongest repelling structure at t0. Bisecting the ini-tial dye blob, it reveals that the blob is about to beseparated by the flow field. Figure 4d reveals thatby t1, the two half-blobs have been drawn out alongopposite sides of another candidate LCS (the blackcurve), the strongest attracting structure.

While the two LCS candidates are notably dis-tinct from the features of the velocity fields, theyclearly shape the transport of the dye blob. Howdoes one find those LCSs?

The finite-time Lyapunov exponentA pioneering insight into Lagrangian features invelocity-field data was provided 20 years ago byRaymond Pierrehumbert and Huijin Yang at theUniversity of Chicago.2 They considered plots ofthe so-called finite-time Lyapunov-exponent(FTLE) field. The Lyapunov exponent is a measureof the sensitivity of a fluid particles future behav-ior to its initial position in the flow field.

To determine the FTLE field, one lets fluid par-ticles flow under the action of the velocity field fromt0 and determines how much initially adjacent parti-cles from a given location have separated by t1. Re-gions of high separation have high FTLE values; theyare locally the regions of most strongly divergingflow. Performing the same procedure in backwardtime, one identifies regions with high backward-timeFTLE values. Those regions of strongest divergencein backward time are the regions of strongest localconvergence in ordinary forward time.

In 2001 the complex patterns of FTLE distribu-tions for physical flow fields were connected to LCSsby one of us (Haller),3 who proposed that ridges ofmaxima in the FTLE field are, in fact, indicators ofrepelling LCSs in forward time and of attractingLCSs in backward time. The ridges were initially be-lieved to be almost Lagrangianthat is, the flux ofmaterial across them was thought to be small.1

Two practical early examples of the FTLE ap-proach were applications to pollution control off thecoasts of Florida4 and California.5 In both cases,ocean-surface velocity fields, obtained over timefrom coastal high-frequency radar stations, wereused to determine appropriate time windows forthe necessary release of pollutants from coastalpower stations. Since then, the FTLE approach hasbeen applied to a great variety of problems such asblood flow in arteries, air traffic control, and flowseparation by airfoils.6

Equating LCSs with FTLE ridges provides anattractively simple computational tool. But it raisessome fundamental mathematical questions thatwere initially overlooked. In particular, FTLEridges can yield both false negatives and false pos-itives in LCS detection.7 Furthermore, the ridges areoften far from being Lagrangian; the flux acrossthem can be large.

StrainlinesTo address the shortcomings of the FTLE approachfor identifying LCSs, Haller and Mohammad Faraz-mand have now shown that repelling LCSs are, infact, material lines whose initial positions are locallythe most repelling strainlines for the time windowin question.8 As described in the box on page 46,strainlines are curves that are everywhere tangentto the eigenvector field of the CauchyGreen straintensor computed over the time window.

For 2D fluid flow, the CauchyGreen tensor isa 2 2 symmetric, positive-definite matrix calcu-lated for each initial position in the fluid. As such,it has positive eigenvalues (0 < 1 < 2), and its twoeigenvectors (1 and 2) are orthogonal. If the fluidis incompressible, 2 = 1/1.



Figure 3. Lagrangian coherent structures in the time interval [t0 , t1].(a) An attracting LCS is a material line (blue) that attracts fluid onto itselfmore strongly than does any nearby material line (gray). (b) Similarly, arepelling LCS is a material line (red) that repels fluid more strongly thanany other nearby line. (c) A repelling LCS acts as the boundary betweendomains of attraction for an attracting LCS. Because LCSs cannot becrossed by material, they bound and shape the regions labeled 14. Theintersection between the repelling and attracting LCSs is a generalizedsaddle point.


For 2D flows, the eigenvectors give the direc-tions, in an infinitesimal sphere released at x0 , thatwill be mapped into the major and minor axes of theellipsoid into which the sphere has formed at timet. The diameter of the initial sphere will be stretchedand compressed by the ratio of the eigenvalues. Bydefinition, strainlines are trajectories of the 1 field.And the initial positions of LCSs are extracted as thelocally strongest repelling or attracting strainlines.One gets later LCS positions by advecting the initialpositions according to the flow map (see the box).

The LCSs thus obtained are truly Lagrangianentities with no material flux across them. Theysolve simple first-order, ordinary differential equa-tions, and hence are smooth, parameterized curves.By contrast, extracting ridges from FTLE calcula-tions has proven to be a challenging image-process-ing problem with no strict mathematical foundation.

The scenario from the Deepwater Horizon oilspill presented in figure 1b is a good example of thestrainline approach. The satellite image, taken on17 May 2010, reveals a large tendril of oil that ex-tends southeast from the body of the main spill; thatfeature became known as the tiger tail. We haverecently applied the strainline method of LCS detec-tion to data from a numerical simulation of the Gulfof Mexico for that time period.

To expose the attracting LCSs responsible forshaping the tiger tail, we calculated the Cauchy-Green strain tensor for the backward-time windowfrom 17 May to 14 May. From that information, the1 and 2 vector fields were determined and arepresented in figure 5a. In general, any point in thedomain is a starting point of a strainlinethat is,a trajectory of the 1 field.

Several such strainlines are marked in figure 5b. The strongest attracting strainlines (thosewith the largest averaged values of 2) are high-lighted in red as the LCSs responsible for shapingthe tiger tail. The same procedure can be carried outin forward time (from 14 May to 17 May) to identify

the repelling structures that played a key role in dis-rupting the original shape of the oil spill.

LCS-based decision makingThe results shown in figure 5, revealing the attract-ing LCSs responsible for shaping the Gulf oil spillbetween 14 May and 17 May, can be called a hind-cast. They provide an explanation of somethingthat happened, based on data gathered beforehand.Nonetheless, the analysis yields valuable insight be-cause it provides a framework for explaining whythings behaved as they did. Building on that new-found knowledge, however, one must ask whetherLCS methods can help forecast features such as thetiger tail.

A first step toward forecasting is called now-casting, the accurate determination of the presentstate of the system from available information. Theability to nowcast LCSs means that the current loca-tion of key transport barriers in the ocean or atmos-phere would be known, which in itself would be asignificant achievement. To accomplish that, theway forward is to use the ever more comprehensive,reliable, and up-to-the-minute data availablethrough satellite measurements and local monitor-ing stations (for example, high-frequency radar andocean drifters) in combination with large-scale nu-merical simulations.

Returning to the Deepwater Horizon spill as ademonstration of the benefits of accurate nowcasting,a recent analysis9 has revealed that a single LCSpushed the oil spill toward the coast of Florida forabout two weeks in June 2010, as shown in figure 6a.Had that information been available at the time, itwould likely have lent greater confidence to decision-making strategies for the Gulf Coast.

The logical extension of nowcasting is the anti -cipation that as the accuracy of numerical simulationsimproves, the velocity-field data they generate cansupport increasingly reliable LCS predictions. Moresignificant, however, is the discovery that so-called


a

c

b

t t= 0 t t= 1

d

Figure 4. Flow in a double gyre. (a) A circularblob of black dye is released at time t0 in atime- periodic flow fieldwith two vortices (gyres).The velocity field at thatinstant is indicated by themagnitude and directionof the blue arrows. (b) After being trans-ported by the time-dependent velocity field,the dye and the field areshown at t1. (c) A candidatefor the strongest repellingLagrangian coherent struc-ture (LCS) (white line) at t0bisects the initial dye blob.The lightest backgroundshading indicates thebiggest positive finite-time Lyapunov exponents (FTLEs, described in the text). (d) A candidate for the strongest attracting LCS (black line) at t1 is responsible for the shape of the blob of dye at that time. The darkest background shading indicatesbiggest negative FTLEs.


hyperbolic cores of LCSs can be used to forecast strongevents such as the tiger tail from nothing but informa-tion obtained up to the present9 (see figure 6b).

A hyperbolic core of an attracting LCS is a shortsegment of the LCS that has uniformly strong attrac-tionthat is, where 2 throughout the segment iswithin the top 1% for the whole domain. If the flowbehaves in a reasonably 2D manner, then volumeconservation requires a strongly repelling LCS tostretch significantly. Thus the region behaves like asaddle point. The identification of a hyperbolic coreprovides predictive capability because it indicates adeveloping transport event, like the tiger tail, thatstoo strong to be halted by short-term future events.

Outlook Yielding profound insight about transport in com-plex, time- dependent flows, the study of LCSs is nowa vibrant research field. Applications abound. In thecoming years, the LCS approach may well prove tobe crucial, for example, in the planned response tothe large quantity of debris from the Tohoku tsunamithat is approaching the US West Coast. In the longerterm, LCS methods are expected to yield improvedpollution monitoring and search-and-rescue strate-gies along seacoasts. Ultimately, LCS applicationsshould also improve our understanding of transportin industrial and biological flows.

Although the mathematical theory of attractingand repelling LCSs is now well established, impor-tant practical challenges remain, such as acquisitionof the requisite velocity data. For coastal regions, thepresence of high-frequency radar stations that canprovide the necessary data is becoming increasinglycommon. And the fast numerical processing of suchdata sets is now within reach, given the broad avail-ability of parallel-computing platforms.

A recent advance in LCS theory provides a gen-eral extraction tool for all key Lagrangian structuresin unsteady flows. Such structures include attractingand repelling LCSs as well as coherent vortex-typepatterns (called elliptic LCSs) and jet-type patterns(called shear LCSs). Via that approach, LCSs can beunmasked by their telltale property of stretching lessthan neighboring material lines do.10

A challenging computational task will be to fol-low the evolution of LCSs as they are advected inforward or backward time by the fluid flow. Thatsmore than simply locating a repelling or attractingLCS, respectively, at the beginning or end of a timewindow. It requires the development of effectivenumerical approaches to the advection of stronglyunstable material lines.

It also remains to connect LCS analysis to otherrecent approaches to Lagrangian coherence such asthe set-theoretical approach of Michael Allshouseand Jean-Luc Thiffeault11 and the probabilisticmethods of Gary Froyland and Katherine Padberg.12



a b

Figure 5. The developing tiger tail in the Deepwater Horizon oil spill (see figure 1b). (a) The eigenvector fields 1 and 2 for thebackwards time window 17 May to 14 May 2010 are shown, respectively, by the yellow and red arrows. (b) Several attractingstrainlines (trajectories of the 1 field) are plotted in black. The dominant strainlines, highlighted in red, are the attracting Lagrangian coherent structures responsible for shaping the tiger tail in figure 1b.

Lagrangian coherent structures (LCSs) are locally the most repelling or attacting strainlines in a flow field. One can obtain them by the followingcomputational steps:8

Step 1: Given a time-dependent velocity field u(x, t) over the time interval [t0 , t1], compute the flow map

the mapping that takes the initial position x0 = (x0 ,y0) of any fluid elementto its final position x1 = (x1,y1) due to the flow.

Step 2: From derivatives of the flow map with respect to variations ofinitial position, compute the deformation-gradient tensor:

Step 3: The CauchyGreen strain tensor is then defined as

Step 4: Strainlines are tangent to the eigenvector field 1 of theCauchyGreen tensors smallest eigenvalue. LCS positions at time t0 aregiven by strainlines with the locally highest averaged values of theCauchyGreen tensors largest eigenvalue.

Ft1

t0( )x0 = ( , , ),x x1 1 0 0t t

Ft1

t0( )x0 = .

x1 x1

y1 y1

x0 y0

x0 y0

C F Ft1 t1 t1t0 t0 t0

( )x0 ( )x0 ( )x0T

[ [[ [.

Lagrangian coherent structures as strainlines


The development of rigorous and efficient LCSmethods for 3D flows is under way. Such methodswill reveal the key 2D material surfaces that act astransport barriers in 3D. That will yield better un-derstanding of flow transport in a great variety ofphysical systems.

References1.S. C. Shadden, F. Lekien, J. E. Marsden, Physica D 212,

271 (2005).2.R. T. Pierrehumbert, H. Yang, J. Atmos. Sci. 50, 2462

(1993).

3.G. Haller, Physica D 149, 248 (2001).4.F. Lekien et al., Physica D 210, 1 (2005).5.C. Coulliette et al., Environ. Sci. Technol. 41, 6562 (2007).6.T. Peacock, J. Dabiri, Chaos 20, 017501 (2010).7.G. Haller, Physica D 240, 574 (2011).8.M. Farazmand, G. Haller, Chaos 22, 013128 (2012). 9.M. J. Olascoaga, G. Haller, Proc. Natl. Acad. Sci. USA

109, 4738 (2012).10.G. Haller, F. J. Beron-Vera, Physica D 241, 1680 (2012).11.M. R. Allshouse, J.-L. Thiffeault, Physica D 241, 95 (2012).12.G. Froyland, K. Padberg, Physica D 238, 1507 (2009).

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a bFigure 6. The Deepwater Horizondrilling rig exploded 20 April 2010about 100 miles south of AlabamasGulf coast (black triangle in bothpanels). (a) A hindcast analysis ofthe oil spill (brown) reveals the evo-lution of an attracting Lagrangiancoherent structure (blue) thatpushed the oil eastward towardFloridas west coast between 9 and19 June.9 Also shown, for reference,are some additional strainlines (red)on the latter date. (b) Analysis ofnumerical data reveals the hyper-bolic core (red circle; described inthe text) of an LCS close to the spill site on 15 May. The oil spill on that date is shown in green. That hyperbolic core forecaststhe later formation of the tiger tail (yellow) by 17 May.9


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