Physica D 210 (2005) 1–20

Pollution release tied to invariant manifolds: A case study for thecoast of Florida

Francois Lekiena,b,∗ , Chad Coulliettea,c, Arthur J. Marianod, Edward H. Ryand,Lynn K. Shayd, George Hallere, Jerry Marsdena

a Control and Dynamical Systems, California Institute of Technology, CA, USAb Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA

c Environmental and Engineering Science, California Institute of Technology, CA, USAd Rosenstiel School of Marine and Atmospheric Science, University of Miami, FL, USA

e Department of Mechanical Engineering, Massachusetts Institute of Technology, MA, USA

Received 27 July 2003; received in revised form 14 June 2005; accepted 16 June 2005Available online 3 August 2005

Communicated by U. Frisch

Abstract

High-resolution ocean velocity data has become readily available since the introduction of very high frequency (VHF) radartechnology. The vast amount of data generated so far, however, remains largely unused in environmental prediction. In this paper,we use VHF data of the Florida coastline to locate Lagrangian coherent structures (LCS) hidden in ocean surface currents. Suchs We use theL the coastale©

K

1

hild

theay. It

vityingns.ows

articleenttheir

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tructures govern the spread of organic contaminants and passive drifters that stay confined to the ocean surface.agrangian structures in a real-time pollution release scheme that reduces the effect of industrial contamination onnvironment.2005 Elsevier B.V. All rights reserved.

eywords:Finite-time Lyapunov exponents; Lagrangian coherent structures; Pollution control

. Introduction

The release of pollution in coastal areas[1–3] mayave a dramatic impact on local ecosystems, especially

f the pollution recirculates near the coast rather thaneaving for the open ocean. Due to the sensitiveependence of Lagrangian particle motion on initial

∗ Corresponding author.E-mail address:lekien@princeton.edu (F. Lekien).

conditions, identical parcels of fluid released atsame time but from two slightly different locations mproduce vastly different contaminant distributionsis commonly believed that this concept of sensitito initial conditions only applies to trajectories startat the same time but at slightly different locatioHowever, time-dependent vector fields generate flon the extended phase space (space-time) and pstarting at the same location but at slightly differtime are also subject to sensitive dependence on

167-2789/$ – see front matter © 2005 Elsevier B.V. All rights reserved.oi:10.1016/j.physd.2005.06.023

2 F. Lekien et al. / Physica D 210 (2005) 1–20

initial conditions (cf. Section7). Except for degeneratecases, trajectories with high sensitivity to the initialreleasepositionare also highly sensitive to the initialreleasetime. As a result, one needs a detailed under-standing of the nonlinear dynamics of surface currentsto devise pollution schemes that result in favorableoutcomes.

Such a detailed understanding may be gained fromvery high frequency (VHF) radar technology (seeSection2), which produces well resolved real-timesurface velocities at select coastal locations. In thispaper, we show how such velocity data can be used touncover hidden invariant manifolds in the Lagrangianparticle dynamics of the ocean surface. We then usethe location of numerically computed Lagrangiancoherent structures (as defined in[4]) to devise anautomated pollution release scheme that minimizesthe harmful recirculation of contaminants near thecoastline.

In our analysis, we use a combination of accuratesurface current measurement[5,6] and recent develop-ments in nonlinear dynamical systems theory[7]. Incontrast to earlier approaches to pollution control insimple flow models[8–12], we rely on real-time ve-locity data collected by coastal Doppler radar systems.Discounting measurement errors (typically of the orderof 5 cm s−1), we consider the VHF velocity data shownin Fig. 1 to be a faithful discretized representation ofthe actual surface velocity field.

We locate time-dependent coherent structures in theV oveLa andt ownt pro-c entea urespa bedD ectt ente -t edo ng al mes

very small with respect to the corresponding Eulerianerror.

Other methods for extracting invariant manifoldsfrom geophysical data include the advection of ju-diciously chosen material lines[14–16], relative andabsolute dispersion[17], finite-size Lyapunov expo-nents[18], and the use of a Lagrangian version of theOkubo-Weiss criterion[19]. Reviews of the differenttechniques can be found in[7,20]. In this work, we fa-vor Lagrangian structures computed using Direct Lya-punov exponent field for their robustness to Eulerianerrors.

Analyzing the location of the extracted coherentstructure relative to a hypothetic pollution release spot,we identify the main frequency components of themanifold’s own motion. We then use these componentsto predict the short-term motion of the structure, with anemphasis on predicting its environmentally friendly po-sitions. Our automated pollution scheme uses the envi-ronmentally friendly time windows to release contami-nants. As we show by simulations, this scheme achievea significant reduction in recirculating pollutants.

The organization of this paper is as follows. The ex-perimental setting used to measure the velocity alongthe coast of Florida is described in Section2. Thisdata is used in Sections3 and 4to integrate numer-ically parcels of pollution. Based on the definition ofLagrangian Coherent Structure given in Section5, a La-grangian barrier is computed for the domain of interestin Section6. Section7 uses the barrier to minimizet nala

2

sur-f astalo (HF)r b-s CR)u ealc st ofF

uth-e MC)f 5 toA from

HF velocity data by employing the Direct Lyapunxponent (DLE) algorithm proposed by Haller[7].ocal maximizing curves or ridges[4] of the DLE fieldre time-dependent manifolds governing mixing

ransport in the fluid flow. These structures are kno be quite robust with respect to measurement andessing errors[13]. In other words, the measuremrror on the velocity field (VHF radar data) doesnotffect much the geometry and position of the structresented here. A formal proof is given in[13] butsimpler, more intuitive justification can easily

erived from the definition given in Section5. TheLE field is the derivative of the flow map with resp

o the initial conditions. As a result, measuremrrors in the velocity field areaveraged along trajecories before it affects the DLE field. For unbiasbservation stations and long trajectories spanni

arge domain, the averaged Lagrangian error beco

he effect of the pollution in coastal Florida and a filgorithm is given in Section8.

. VHF radar data along the coast of Florida

The use of radio frequencies to measure oceanace currents has received attention in recent coceanographic experiments, using high frequencyadar techniques[21,22]. Recent surface current oervations from ocean surface current radar (OSsing the very high frequency (VHF) mode revomplex surface current patterns near the coalorida[5,6].

The OSCR VHF system was deployed for the Sorn Florida Ocean Measurement Center (SFO

our-dimensional current experiment from June 2ugust 25, 1999. Radio waves are backscattered

F. Lekien et al. / Physica D 210 (2005) 1–20 3

Fig. 1. Surface velocity maps obtained by VHF radar along the southeast coast of Florida, near Fort Lauderdale, during the SFOMC 4D CurrentExperiment on June 26, 1999: (a) 01:20 GMT; (b) 02:20 GMT; (c) 04:00 GMT; (d) 05:20 GMT. The sequence shows a northward propagatingsubmesoscale vortex. The translation speed of the vortex is about 30 cm s−1 and its horizontal scale is 2–3 km[5].

the moving ocean surface by surface waves of one-halfof the incident radar wavelength.

This Bragg scattering effect[23] results in twodiscrete peaks in the Doppler spectrum. In the absenceof surface current, spectral peaks are symmetric andtheir frequencies are offset from the origin by anamount proportional to the surface wave phase speedand the radar wavelength. If there is an underlyingsurface current, Bragg peaks in the Doppler spectrumare displaced by the radial component of current alongthe radar’s look direction. Using two radar stationssequentially transmitting radio waves resolves thetwo-dimensional velocity vector[5,6]. The resultant

data set represents coastal ocean surface currentsmapped over a 7 km× 8.5 km domain at 20 minintervals with a horizontal resolution of 250 m at 700grid points. The radars, transmitting at 50 MHz, werelocated in John Lloyd State Park, Dania Beach, Florida(Master) and an oceanfront site in Hollywood Beach,Florida (Slave), which are separated by 7 km.

Given the narrow shelf off Ft. Lauderdale and thestrong Florida Current that intrudes onto the shelf onone-to-three day time scales[6,24], the VHF radardomain is ideally located for purposes of examining awide spectrum of coastal and oceanographic processes.At times, the speed of the Florida Current exceeds

4 F. Lekien et al. / Physica D 210 (2005) 1–20

2 m s−1 just 5 km offshore[6]. The average ambientrelative vorticity is 4f, where f is the local Coriolisparameter, and maximum relative vorticity exceeds10f [24]. The dominant period in the velocity dataincreases from ten hours near-shore to 5 days offshore.There is also significant energy at 27 h, the inertialperiod.

During this experiment, surface current observa-tions (Fig. 2) revealed Florida Current intrusions overthe shelf break, wavelike structures along the inshoreedge of the current and numerous submesoscale vor-tices. One example started at 01:20 GMT on July 20,1999 (Fig. 1) when a submesoscale vortex was locatedalong the southern part of the VHF-radar domain justinshore of the Florida Current. Surface currents withinthe vortex ranged from 20–30 cm s−1 at a diameter ofabout 1–1.25 km from the vortex’s center. The vortex’snorthward displacement of about 6 km occurred over a5-h period. See[5] for a more detailed analysis of thisvortex.

3. Interpolation and velocity field

The discrete data set containing the radar measure-ment does not constitute a velocity field, a vector func-tion giving the velocity at each point and at each time.Experimental data must be interpolated between gridpoints and the resulting interpolating function is useda flowd cityfi thefi e-l mea andte

v

w taid ons o-n no-m lla f

in [25]). In addition, the functions∂2u/∂x2, ∂2u/∂t2,∂2u/∂t2 ∂3u/∂x∂y∂t and the corresponding derivativesfor v are also continuous. As a result, the local tricu-bic interpolator described here is trulyC1 in space andtime. Uniqueness, existence andC1 smoothness of so-lutions for the system is therefore guaranteed (see[26],for example).

Notice that the methods described in this paper areindependent on the model (HF radar data and tricubicinterpolation). For instance, modal analysis[27,28]orempirical modes[29] can be used to interpolate theexperimental data. A smooth model velocity field canalternatively be obtained from high resolution oceanmodeling[30–32]. Recently, data assimilation was ex-tended to use Lagrangian data (floats and drifters),broadening the range of geophysical flow available forLagrangian studies[33,34]. The Lagrangian coherentstructures presented in this paper are robust to errorsand variations in the velocity field[13].

4. Numerical experiments

The complexity of the flow resulting from theintegration of the interpolated HF radar data becomesevident from tracking different realizations of afluid parcel–a model for a spreading contaminant–released at the same time, but at a slightly differentlocation. The results for two such numerical exper-i ea omha d at0 bleh lesa lberga ici foru ltingv

am-i ast,w y tot e lat-t izest , byc ean.

s a velocity field. Since the smoothness of theepends directly on the smoothness of the veloeld and the methods used in this paper involverst derivative of the flow, we require that the vocity field used is differentiable in space and tind that its derivatives are continuous in space

ime, which is usually denoted byv ∈ C1, or morexplicitly:

=(u(x, y, t)

v(x, y, t)

)∈ C1(× R → R

2), (1)

hereΩ ⊆ R2 is the spatial domain of interest. Da

s provided at the vortices of a regular mesh (xi, yj), atiscrete timestk. We use the tricubic local interpolaticheme described in[25] that represents each compent of the velocity as a piecewise third order polyial function. The polynomials areC∞ inside each cend the interpolated function is globallyC1 (see proo

ments are shown inFigs. 3 and 4. The completnimation, along with others, are available frttp://www.lekien.com/∼francois/papers/rsmas. Thenalysis uses two parcels of particles launche9:45 GMT on July 10, 1999. Using the availaigh-resolution VHF velocity data, the fluid particre advected using a 4th order Runge-Kutta-Fehlgorithm (RKF45) combined with 3rd order tricub

nterpolation in both space and time. The motivationsing such a complex interpolator is that the resuelocity field isC1 in extended phase space[25].

Note that large concentrations of the black contnant remain for relatively long times near the cohereas the white parcel exits the domain quickl

he north and are advected into the open ocean. Ther scenario is highly desirable, because it minimhe impact of the contaminant on coastal watersausing it to be safely dispersed into the open oc

F. Lekien et al. / Physica D 210 (2005) 1–20 5

Fig. 2. (Top panel) The velocity pattern obtained by HF radar along the coast of Florida, near Fort Lauderdale, for two different time periods,July 20 and August 3, 1999. A submesoscale vortex is evident in each velocity map. (Middle panel) The corresponding normalized (by the localCoriolis parameter) relative vorticity anomaly fields. The mean vorticity, of order 4f , was removed from each estimate to reveal the anomalies.Large positive relative vorticity values are associated with the vortices that are elongated due to the velocity shear of the Florida Current. Largenegative values are found in the vicinity of a near-shore topographic step. (Bottom Panel). The corresponding horizontal divergence fields arecalculated from spline fits to the velocity data.

6 F. Lekien et al. / Physica D 210 (2005) 1–20

Fig. 3. Two parcels of contaminant released at exactly the same time,but at slightly different initial locations on July 10, 1999 at 09:45GMT. The white parcels leave the domain quickly as they are ad-vected by the northward flowing Florida Current. The black parcelsre-circulate near the coast for more than 36 h.Animation availableat http://www.lekien.com/∼francois/papers/rsmas.

This observation provides an opportunity to understandand predict differing evolution patterns of a fluid par-cel, depending on its initial location and time of release.Such patterns are known to be delineated by repellingmaterial lines or finite-time stable manifolds[35–38],which we compute in the next section.

Notice that the domain studied inFig. 4is not com-pletely closed by a coastline. Particles can leave thedomain through the open-boundary (southern, west-ern and northern edges). Once particles leave the do-main, no information is available to follow their trackand they are disregarded. One might wonder if ourconclusion holds in the real world where the whiteparcel can possibly be pushed back into the domainof Fig. 4 by unknown currents. This would invali-date the reasoning above and the proposed contaminantcontrol scheme. Such a problem is inherent to open-boundaries and must be studied from two differentapproaches:

• Open-boundary flow. An estimate on how likelyparcels can leave the domain and re-enter itin areasonable amount of timecan be derived from astudy of the flow on a section of the open-boundary.

In the case of the white parcel ofFig. 4 (or moregenerally, particles leaving the domain through thenorthern open-boundary), one can notice that the ve-locity vectors are almost always pointingtoward theoutsideof the domain. During the numerical exper-iment (July 10, 1999→ July 12, 1999), less than0.1% of the velocity vectors on the northern edgewere indicating inflow. Moreover, such vectors wereof relatively small magnitude and localized near theshoreline at the beginning of the experiment. It istherefore very unlikely that particles forming thewhite parcel will re-enter the domain through thenorthern edge. Re-entrant particles must travel along distance outside the domain and re-enterthrough the western or southern open-boundary.After such a long travel time, diffusion and dis-persion have destroyed the parcel of contaminantsignificantly and the resulting very low concentra-tions of white contaminant can be ignored. To gen-eralize this notion, we computed the inflow and theoutflow through the northern segment of the open-boundary. For a given segment of open-boundary∂Ω the outflow is defined as

Jout =∫

dt∫∂Ω

U(n · v)dl, (2)

where the unit normal vectorn is pointing outsidethe domain and

0 if z ≤ 0

aveatain

hisnd, then it

• eceure

U(z) =z if z > 0

. (3)

Similarly, we define the inflow as

Jin =∫

dt∫∂Ω

U(−n · v) dl. (4)

For the months of July and August 1999, we hJin/(Jin + Jout) < 0.01%. This justifies the fact thparticles leaving the northern edge of the domdo not typically re-enter the domain through tedge. If enough diffusion of the contaminant adispersion due to sub-scale processes occurwhite parcel can safely be disregarded wheleaves the domain.

Diffusion and dispersion. The argument abovonly holds if there is a reasonable differenbetween the motion of the tracers and the p

F. Lekien et al. / Physica D 210 (2005) 1–20 7

Fig. 4. A time sequence of the motion of two parcels released almost at the same position on July 10, 1999 at 09:45 GMT. The interpolatedvelocity from the radar and the position of the parcels is shown for (a) July 10 09:45 GMT, (b) July 10 13:45 GMT, (c) July 10 16:45 GMT, (d)July 10 23:45 GMT, (e) July 11 11:45 GMT and (d) July 11 20:45.Animation available athttp://www.lekien.com/∼francois/papers/rsmas.

8 F. Lekien et al. / Physica D 210 (2005) 1–20

Lagrangian model. In reality, contaminant are notadvected exactly as fluid particles and their motionis best described by an advection-diffusion equation[39,40], for example. The parcels inFig. 4 onlyrepresent the motion of contaminant for short inte-gration time. For longer integration times, such asparticles leaving through the northern edge and re-entering the domain through the western or southernedge, the advective terms destroy this localizedrepresentation and smooth the distribution of con-taminants. In addition to diffusion, a correct modelof the motion of such contaminant must include sta-tistical processes to model sub-grid processes thatare not resolved in the VHF radar data[41,42]. Forsmall trajectories, the idealized Lagrangian parcelis a reasonable approximation of the concentrationof contaminant[43]. For long trajectories, diffusionand dispersion tend to average the concentrations[44]. The objective of the contaminant controlscheme in this paper is toavoid peaks of largeconcentration. As a result, we are only concernedwith the small time effect of the release (e.g., beforeparcels could re-enter the domain). In the longterm, diffusion and dispersion play in favor of ourobjective.

A recently developed nonlinear technique, called thedirect Lyapunov exponent (DLE) analysis[7], identi-fies repelling or attracting material lines associated toLagrangian coherent structures (LCS) in velocity datab rials pu-t nvi-r

5

sc ei res[ ngt ries.A in[ po-n esa elyd

5.1. Direct Lyapunov exponents

The DLE algorithm starts with the computation ofthe flow map, the map that takes an initial fluid particlepositionx0 at timet0 to its later positionx(t, x0) at timet. To perform this analysis, a uniform grid of 200× 200particles are launched at timet0. Each particle isadvected using a 4th order Runge-Kutta-Fehlberg al-gorithm and a 3rd order interpolation fort − t0 = 25 h.An analysis of the influence of the integration timet − t0 in the computation of DLE maps can be foundin [48].

These particle trajectories are used to approximatethe flow map, which associates current positions to po-sitions at timet0. The coastline is modeled as a free-slipboundary. Particles that cross the open boundaries ofthe domain on the northern, eastern and southern edgesare disregarded. These numerical algorithms have beencompiled into a software package, MANGEN, that isavailable from the authors upon request[49].

The spatial gradient of the flow map (∂x/∂x0) isa 2× 2 matrix and can be computed by using usingfinite differences in the grid of trajectories[37]. Thedirect Lyapunov exponentσt(x0, t0) is defined as thenormalization of the largest singular value of the spatialgradient of the flow map[37]. More specifically,

σt(x0, t0) =ln λmax

([∂x(t,x0)∂x0

] [∂x(t,x0)∂x0

])2|t − t0| , (5)

w a-tt ests s alv ntv lyi eret pedw dis-r nove tiont imed tami-n firsta er-s eryl erm

y means of curves experiencing maximum matetretching. Measured HF radar data allow the comation of such structures and the identification of eonmentally friendly release spots.

. Lagrangian coherent structures

The study of transport and mixing in fluid flowan be highly simplified by the use of finite-timnvariant manifolds or Lagrangian coherent structu14,16,45–47]. This work does not involve discussihe existence or uniqueness of hyperbolic trajecto

preliminary discussion on this topic can be found4]. In this paper we use the Direct Lyapunov Exent algorithm[7] to compute Lagrangian structurnd divide the domain into regions of qualitativifferent dynamics.

ith the superscript referring to the transpose of a mrix. Notice that the denominator of Eq.(5) normalizeshe Lyapunov exponent. The logarithm of the largingular value of the gradient of the flow map hainear dependence in the integration time (t − t0). Di-iding the result by (t − t0) allows for a more constaalue ofσt when (t − t0) changes. This is particularmportant for domains with an open-boundary whhe computation of certain trajectories must be stophen they leave the domain. Such trajectories are

egarded when they exit the domain and the Lyapuxponent must be computed with a smaller integraime. As noted above, such smaller integration to not influence the results presented here. Conants only follow pure Lagrangian advection as approximation. In the long term, diffusion and dispion will naturally distribute the contaminant at a vow concentration. Our goal is to study the short-t

F. Lekien et al. / Physica D 210 (2005) 1–20 9

peaks of concentration, hence Direct Lyapunov expo-nents with short integration times.

5.2. Lagrangian ridges

Repelling material lines are maximizing ridges ofthe scalar fieldσt(x0, t0) [7,13]. By ridge of a scalarfield, we mean a gradient line of this field that has max-imum curvature in the orthogonal direction[4]. Morespecifically, a ridge is aC1 curvec(s), s ∈]a, b[, satis-fying the following conditions:

(1) c(s) is parallel to∇σt(c(s), t0),

(2)dc(s)

ds= 0,

(3) (n,n) = min‖u‖=1(u,u), wheren is a unit vec-tor normal to the curvec(s) and = ∂2σt/∂x2 isthe second derivative ofσ thought of as a bilinearform evaluated at the pointc(s).

Notice that the ridges defined here are not necessarilycontour level sets ofσt . They correspond to “water-dividing lines” of the field. The ridges of the Lyapunovexponent field as defined in this section are not neces-sarily Lagrangian and do not strictly behave as materiallines. They are instantaneous screenshots of the futureLagrangian dynamics. However, recent studies[4] haveshown that the Lyapunov exponentσt is Lagrangianfor large integration times (t − t0) (in the sense that thechange of its value along trajectories varies as the in-v ans r-v stalfl has ar i-ma

po-n aoticp ha ld ist vea ng.R fi ale.St thanc ents

and Lagrangian coherent structures[4,37]provide sucha framework. A fundamental difference is that there isno need for estimating relevant length scales. The algo-rithm described above will extract both small and large-scale structures. Some small-scale structures mighthave high exponents. However, they do not shadow thelarge-scale structures of interest. Lagrangian structuresare identified as ridges[4] and the intensity of theLyapunov exponents is much less relevant than thecurvature along the ridge, for example. Once extracted,the ridges differ by theirlengthand can be classifiedor eliminated based on their length, rather thanintensity.

The same procedure performed backward in time(i.e., for t < t0) renders attracting material lines att0as ridges ofσt(x0, t0). These curves are not apparentto naked-eye observations and are not easily deducedfrom velocity field plots, yet they govern global mixingpatterns in the fluid[55]. Such Lagrangian structures inmeasured ocean data have previously been inaccessibledue to lack of an efficient extraction methods and coarseresolution of the observations.

6. Data analysis

Direct Lyapunov Exponents are used to analyze theLagrangian trajectories in the VHF radar domain. Inparticular, we want to be able to define pollution barri-ers and pathways near the southeast coast of Florida.

themD ableL nearF Thiss weent CS)a CS).R sc Asa rrier.S inF ianb romS t theb

asto ain

erse of (t − t0)). In the definition above, a Lagrangitructure is a gradient line ofσ that has maximum cuature in the transverse direction. For typical coaows such as the one studied here, such a ridgeelatively constant value ofσt and is, in good approxation, a Lagrangian line of the flow (see[4] for proofsnd numerical results).

Relative dispersion and finite-size Lyapunov exents have been successfully used to identify chatterns in fluid flows[50,51]. In small domains sucs the one depicted in this paper, the velocity fie

ypically too small for the diffusion coefficients to giny relevant information about transport in mixiecent extensions[52,53] allow for the extraction o

nformation for a particular length scale or time scuch methods were verified experimentally in[54]. In

his work, we seek precise Lagrangian lines ratherhaotic regions or patterns. Direct Lyapunov expon

Selected frames of the contour level sets ofaximum Lyapunov exponents are shown inFig. 5.uring the experiment, the plot reveals a strong stagrangian structure attached to the coastort Lauderdale, propagating to the southeast.tructure acts as a quasi Lagrangian barrier bethe coastal recirculating zone (southwest of the Lnd the Florida Current (northeast of the same Lecent work [4] shows that the flux of particlerossing the Lagrangian structure is negligible.

result, the LCS also acts as a material bauperimposed onFig. 5 are the two parcels usedigs. 3 and 4. The average lifetime of the Lagrangarrier is about 48 h and parcels typically travels fouth to North in less than 12 h, so we expect thaarrier has a significant effect on the flow.

Analysis ofFig. 3reveals that any particle northef the barrier (white parcel) is flushed out of the dom

10 F. Lekien et al. / Physica D 210 (2005) 1–20

Fig. 5. Level Sets of the maximal Direct Lyapunov Exponentsσ along the coast of Florida on (a) July 10, 1999 09:45 GMT, (b) July 1013:45 GMT, (c) July 10 16:45 GMT, (d) July 10 23:45 GMT, (e) July 11 11:45 GMT and (d) July 11 20:45. The simulation shows repellingmaterial lines attached to the coast near Fort Lauderdale. Superimposed on each figure panel are the respective positions of the two parcelsfrom Fig. 4. Every particle North of the coherent structure flows through the northern open-boundary. It is non-optimal to release contami-nants below the branch of the manifold because it will remain between the coast and the manifold for a long time.Animation available at:http://www.lekien.com/∼francois/papers/rsmas.

F. Lekien et al. / Physica D 210 (2005) 1–20 11

Fig. 6. Level Sets of the maximal Direct Lyapunov Exponentsσ

along the coast of Florida on July 15, 1999 at 9:45 GMT. The dashedline represents future positions of the repelling coherent structure.The parcel North of the manifold flows through the northern open-boundary (white parcel). It is dangerous to release contaminants be-low the branch of the coherent structure because they will persistbetween the coast and the structure for a long time (black parcel).

in only a few hours. In contrast, parcels starting south-west of the barrier (black parcel) typically re-circulateseveral times near the Florida coast before they finallyrejoin the current. Interestingly, such behavior is notobvious from a simple observation of the velocity foot-prints, which is typical of fast varying time-dependentflows. In this mathematical framework, the surface cur-rents are not necessarily influencing particle paths di-rectly, but the currents influence the Lagrangian struc-tures, such as causing transport barriers and pathwaysand the Lagrangian structures act directly on particlepaths (Fig. 6).

7. Minimizing the effect of pollution

The location of the base of the Lagrangian coherentstructure (along the coastline) can be used as a cri-terion to minimize the effect of coastal pollution. We

will refer to the intersection of the coastline and the La-grangian structure as the barrier point. For the regionand time period analyzed here, factories and sewagedischarge pipelines along the coast should not releaseanything if the barrier point is locatedNorth of them.Optimizing the release site by moving the source ofpollution is unrealistic. Optimizing the release timesis a much more implementable operation. The DirectLyapunov exponents typically give the influence of theinitial positionx0 on the final positionx(t; t0, x0) of thetrajectory that started atx0. However, a change in ini-tial time t0 → t0 + δt0 can be interpreted as a changeof initial position x0 → x0 − v(x0, t0)δt0 and wehave

∂x(t; t0, x0)

∂t0= −∂x(t; t0, x0)

∂x0v(t0, x0), (6)

so sensitivity with respect to the initial position can al-ways be interpreted as sensitivity with respect to theinitial condition in the direction of the initial velocityvectorv(x0, t0) (seeFig. 7). Moreover, in this case, weused a slip boundary condition and the velocity fieldis always tangent to the coastline. In other words, foranyx0 on the coastline,v(x0, t0) is tangent to the coast-line, so optimizing therelease site along the coastlineis equivalent to optimizing therelease time. More pre-cisely, we have,

∂x(t; t0, x0(s))

∂t0= ∓∂x(t; t0, x0(s))

∂s‖v(t0, x0(s))‖, (7)

heon

of

,

wheres is the arc-length along the coastline and tsign is determined by the projection of the velocitythe tangent vector∂x/∂s. Eq.(7) shows that sensitivity

Fig. 7. Sensitivity to initial conditions must be studied in termsspatial perturbationsδx0 and temporal perturbationsδt0 of the ini-tial condition (x0, t0). However, the law of motionx = v(x, t) givesdirectlyx(t; t0 + δt0, x0) = x(t; t0, x0 − v(x0, t0) δt0) and sensitivityto initial time can be derived from the dependence inx0.

12 F. Lekien et al. / Physica D 210 (2005) 1–20

Fig. 8. Imaginary source of pollution along the Florida coast.The black spots are the resulting contaminants from a fac-tory releasing at a constant rate. Superimposed on this fig-ure are the white spots of a factory releasing only duringenvironmental friendly time windows.Animations available athttp://www.lekien.com/∼francois/papers/rsmas.

in terms of the release time is equivalent to sensitivity interms of release position along the coastline and we candecide to keep the pollution source fixed and modify therelease schedule, which is much easier to implement inpractice.

To illustrate how an efficient pollution release algo-rithm can be set up, a fixed imaginary source of pol-lution is placed along the coastline. Using the DLEplots ofFig. 5, we identify zones of (green) favorablerelease1 and (red) dangerous release.2

To minimize the effect of coastal pollution, wepropose using a holding tank that stores contaminantsduring dangerous release times. The tank storespollution during the half-period of the barrier pointoscillation, during which contaminants should not bereleased. The contents of the tank are released once thebarrier point moves South of the source of pollutionas shown inFig. 8.

The black spots onFig. 8 are the trace of the pol-lution of a factory releasing at a constant rate. Super-imposed on this figure is the white trace of contami-nants released only during time windows determined

1 When the structure is below the position of the factory.2 When the structure is above the factory.

Fig. 9. Three different release strategies for pollutants at a factory onthe coast. The black line shows the mass of pollutant in the coastalarea for a uniform release in time. The dashed line is for releasingpollutants at times determined by the DLE analysis. The purple lineis a strategy that is based on the DLE analysis and the criterion ofminimizing peak values.

using our DLE algorithm. The total mass of contam-inant in the coastal area for the two modes is shownonFig. 9.

The two sources of pollution release the exactsame mass of pollutant in the ocean. However, byobtaining information from the DLE results, the whitefactory (dashed curve) is able to reduce the effect ofthe pollution in the shallow coastal area by a factor ofthree.

Also shown onFig. 9 are the results of a third nu-merical experiment. In many cases, the damage to theenvironment is a function of the maximum concentra-tion of contaminant. From this viewpoint, the algorithm(releasing nothing during “red” zones and as much aspossible during “green” zones) does not seem to beefficient. The peak of maximum concentration for thewhite factory has only decreased by a small amount.The results reveal that a long “red” zone can lead to theaccumulation of large amount of pollution in the tank.If such a zone is followed by a short “green” zone, thelarge content of the tank is released quickly and createsa peak in the concentration. To set up a more elaboratealgorithm, we release a minimum flux of pollution intothe ocean independently of the type of zone (red orgreen). We define a new degree of freedomα, the per-

F. Lekien et al. / Physica D 210 (2005) 1–20 13

centage of contaminant produced that will be releasedduring “red” zones. Here, the solid curve ofFig. 9corre-sponds toα = 100%, the dashed curve toα = 0%. Thepurple curve ofFig. 9corresponds toα = 33% (i.e., 1/3of the pollution is released at all time).Fig. 9shows thata significant reduction of the peak of maximum con-centration can be obtained using an appropriate partialrelease during zones that are marked dangerous by theDLE algorithm.

8. An automated pollution control algorithm

The purpose of this section is to show that it is pos-sible to implement a pollution release scheme based onthe motion of a LCS in real time. If, at anytime, the DLEfield was available, a simple decision system based onthe position of the LCS at the present time should beable to select the beginning and the end of favorablerelease zones (see previous section). Unfortunately, ifthe velocity is known up to the present time, the DLEfield can only be computed up to a certain time in thepast. As we approach the present time, DLE can onlybe computed over a short amount of time and does notconverge towards the expected Lagrangian structure.Instead, the plot reveals the maximum eigenvalues ofthe linearized flow. We determined empirically that 8 hat least were necessary to obtain a correct picture ofthe LCS. Our algorithm needs to be able to predict theposition of the Lagrangian structure at least 8 h in thef theE thep entt

8

Onem top eldb s oft pre-

dicted directly. We elected a simple prediction scheme.Since we are only interested in the latitude of the mov-ing barrier, for each time, the latitude of the barrier iscomputed up to the latest time in the past where DLEcan be computed (i.e. 8 h before the present time). Thisforms a time sequence that we would like to predict forat least 8 h. We refer to this operation as ashort timeLagrangian predictionbecause the velocity (Eulerianfield) isnever interpolated in time during this process.

The spectrum of each sequence has been com-puted using the last 50 h and a few examples areshown in Fig. 10. A complete animation of thespectrums computed at each time can be foundathttp://www.lekien.com/∼francois/papers/rsmas. Weidentified the different components of the oscillationsof the Lagrangian barrier as the frequency at the max-imum of each peak in the spectrum.Fig. 11shows thesignificant frequencies that were identified at each timestep. Notice that there is very little tidal influence onthe spectrum ofFig. 11. These spectrums are basedon the motion of Lagrangian particles along the coast-line. Tidal oscillations influence the flow mostly in thedirection orthogonal to the coastline for such a linearcoast. Notice that, even for more complicated costlines,non-tidal dominant frequencies usually appear in themotion of Lagrangian structures (see[49] for exam-ple). The weak amplitude of the tidal frequencies inthis example is a direct consequence of the fact that theLagrangian motion studied is parallel to the coastline.Simulations using longer time sequences (e.g.Fig. 12)r sitivet ap-p s notn

ys om-p r isr s

y

a izest

N

i=1

Ai

uture. Notice that we do not attempt to predictulerian velocity field. We propose to extrapolateosition of the Lagrangian barrier up to the pres

ime.

.1. Prediction algorithm

Several predictive methods could be used.ight think of using a model of the area, use itredict the velocity field and compute the DLE fiased on predicted velocities. However, the value

he DLE field are sequences in time and can also be

ε =

√√√√√∑k

(yk − A0 −

∑

eveal that the computed frequencies are not seno the length of the time sequence used. Within anropriate range (50–200 h), using more data doeecessarily provide more accurate predictions.

Based on the relevant frequenciesTi, we use a verimple predictive algorithm based on a Fourier decosition where the motion of the Lagrangian barrieepresented as a finite sequence of Fourier mode

= A0 +N∑i=1

Ai cos

(2π

Tit

)+ Bi sin

(2π

Tit

), (8)

nd, the optimal model is obtained when one minimhe error

cos

(2π

Titk

)− Bi sin

(2π

Titk

))2

, (9)

14 F. Lekien et al. / Physica D 210 (2005) 1–20

Fig. 10. Entropy spectrum of the time sequence of the latitude of the Lagrangian barrier computed using 50 h of data at selected time steps.

F. Lekien et al. / Physica D 210 (2005) 1–20 15

Fig. 11. Spectral peaks in the time sequence of the latitude of theLagrangian barrier computed using 50 h of data.

whereyk are theK positions of the Lagrangian barriermeasured in the last 50 h (at timestk). The error isminimum when

∂ε

∂Ai= 0, (10)

Fig. 12. Spectral peaks of the time sequence of the latitude of theLagrangian barrier computed using 75 h of data.

and

∂ε

∂Bi= 0, (11)

for all i, which leads to the linear system in (2K + 1)unknowns3

(Qij)

...

Aj

...

Bj

...

=

...∑k yk cos2πtk

Tj

...∑k yk sin 2πtk

Tj

...

, (12)

where

Qij =∑k

fi(tk)fj(tk), (13)

and

fi(t) =

1 if i = 1

cos 2πtTi−1

if i = 2,3, . . . , K + 1

sin 2πtTi−K−1

if i = K + 2,

K + +3, . . . ,2K + 1

(14)

In this paper, the linear system given by Eq.(12)was solved using theGNU Scientific Library (GSL)for maximum 13 modes and each mode is a domi-nant frequency automatically extracted from the spec-t s ill-c eacho ars entsA idt atived intoa thea fore thej tr

e -p he

B

rums computed at each time slice. The system ionditioned when some frequencies are close tother (e.g., ifTi ≈ Ti′ , the determinant of the lineystem is close to zero, in which case the coefficii,Bi,Ai′ andBi′ can grow arbitrarily large). To avo

his problem, groups of close peaks, defined as relistances less than 1%, are ignored and combinedsingle resonant frequency. Moreover, we allow

lgorithm to select the optimal number of modesach time step. The linear system is solved with

th most significant modes (3≤ j ≤ 13) and the besesult is kept.

At each time step, once the numberK of modes isstablished and the coefficientsAk andBk are comuted, the sequence in Eq.(8) is used to extrapolate t

3 The (2K + 1) unknowns are A0, A1, A2, · · ·AK and

1, B2, . . . BK.

16 F. Lekien et al. / Physica D 210 (2005) 1–20

position of the barrier during the past 8 h where DLEwas unavailable as well as in the future.Fig. 13showssome of these results for selected time slices.

Sometimes the prediction can be very accurate formany hours in the future, such as on July 12 10:00 GMTor July 14 03:45 GMT. However, large errors can occureven at the present time such as July 12 17:45 GMT.On July 12 21:00 GMT and July 13 11:45 GMT, thesystem automatically switched to a low-mode analysisbecause a better fit of the past data could not be ob-tained with more modes. Results can be significantlyimproved such as on July 13 11:45 or have a strong di-vergence such as July 12 21:00 GMT. This phenomenahas been observed several times and is a consequenceof the presence of close frequencies in the spectrum.Removing lower modes usually provides a sufficientsolution.

While no attempts were made to improve the pre-diction method, the objective was to study the possibleadvantages of a pollution algorithm. Thus, it sometimesused erroneous data as input, even though it may havebeen obvious not to be able to predict the position ofthe barrier accurately.

Our goal is to show that even in the presence oferrors in the prediction of the position of the Lagrangianbarrier, a significant reduction of the concentration ofpollution can be achieved by the algorithm.

To determine favorable release time intervals, onlythe relative position of the predicted barrier withrespect to the latitude of the release site is important.T e onF hent ases hm,t cidew one.T eacht times ea s. Int ther wasu ll oft

ent,t

8.2. Decision algorithm

To account for the high error rate on the raw deci-sion algorithm (favorable release zones), the factory oranother release center is expected to simulate a morecomplex decision algorithm to decide which portion ofthe pollutant is to be released and which is to be storedin the tank. The data from the 4 last predictions (lasthour) is used to increase the accuracy. A persistent lowimpact release time interval occurs when at least 75%of the last 4 points indicate that it is a good releaseinterval. A persistent high impact zone corresponds toat least 75% of the points detecting an unfavorable in-terval. In other cases, the system is in an undeterminedstate.

Based on the type of zone that the factory thinks itis in, the release flux or position of the valve to the tankwas computed as follows:

• If the system is in alow impactzone, the productionof the factory is released in the ocean. If not empty,the tank is emptied at maximum speed;

• If the system is in anhigh impactstate, the produc-tion of the factory is moved to the tank. If the tank isfull, the production is released in the ocean but thetank is not emptied; and,

• If the system is in anundecidablestate, the actiondepends on the amount of liquid stored in the tank.If the tank is less than half full, the production isstored in the tank. If the tank is more than half full,

but

T fromt ther t, iti antsp

am longh temm vor-a max-i ull,i is atl le. Ina rnedi f the

he release site is indicated by a the horizontal linig. 13 and a favorable release interval occurs w

he latitude of the barrier point is below the releite. To assess the quality of the prediction algorithe predicted curve was used at each time to dehether or not the system was in a favorable zhe answer to this question was compared at

ime with the correct answer based on the actualequence of the DLE ridge.4 The predicted curvgrees with the actual curve in 78% of the case

his experiment, the actual curve only stays belowelease site about 40% of the time. The algorithmsed to simulate with different release sites, and a

hem had a success rate ranging from 70 to 80%.

4 The “actual” DLE ridge is computed at the end of the experimhus with all data available.

the pollution produced is emptied in the oceanthe tank is not to be emptied.

he action taken in the undecidable case resultshe following observation: if we are uncertain thatelease will have a low impact on the environmens better not to take any risk and keep the pollutroduced in the tank.

However, if the tank is more than half full, it isuch higher risk to store more pollutant because aigh impact zone could follow. In this case, the sysay be forced to release pollution during the unfable zone, simply because the tank has reached its

mum capacity. When the tank is more than half ft is less risky to release the pollution when thereeast a 50% chance that the release will be favorabddition to the framework presented above, we lea

n the previous section that a small percentage o

F. Lekien et al. / Physica D 210 (2005) 1–20 17

Fig. 13. Predictive motion of the intersection of the coherent structure with the coastline. The prediction is shown for selected times and usesthe last 50 h of data. The black curve represents the actual time sequence that can be computed at the end of the time of the experiment. Theblue vertical line in each figure represents the actual time and the green vertical line represents the time up to which DLE can be computed. Inother words, the position of the Lagrangian structure is known up to the green vertical line. The red curve shows the Fourier sequence that wasused to predict the position of the LCS in the future. It has been fitted to the measured data up to the green line and may or may not make a goodprediction during the 8 h preceding the current time and in the future.

18 F. Lekien et al. / Physica D 210 (2005) 1–20

Fig. 14. Concentration of pollutant. The solid curve shows the con-centration of pollutants in the coastal area of a source releasing ata constant rate. The dashed curve shows the effect of a source ofpollution releasing the same amount of pollutant, but using the DLEalgorithm to reduce its effect on the coastal environment.

pollution has to be released in the ocean at all timesto achieve effective reduction of the peak of maximumconcentration.

The result of such an experiment is presented onFig. 14. The factory was producing 1.2 tonnes of pol-lutant per hour and at least 0.4 tonnes h−1 were to bedumped in the ocean (α = 33%). The tank had a max-imum capacity of 30 tonnes and the maximum releaserate from the tank was set to 3.6 tonnes h−1.

9. Conclusions

We have shown the existence of a set of repellingmaterial lines in the VHF-radar derived surface cur-rent fields acquired along the Hollywood, Florida coastin July 2003. We have also shown how these materiallines can be used to minimize the effect of coastal pol-lution by determining and predicting optimal releasetimes. This approach can be used for simulating tra-jectories of buoyant contaminants or the trajectoriesof nearly Lagrangian tracers.5 The data source can be

5 By nearly Lagrangian tracers, we refer to particles followingalmost exactly the currents. In this case, the Lagrangian structures

VHF radar data or any other current data source, suchas data-assimilated ocean models that approximate thenear-surface velocity field to some reasonable level ofaccuracy. The advantage of using ocean models is thatthe velocity provided is 3D+1, and thus we can ex-plore the Lagrangian structures that develop at variousdepths. We have shown that a real-time experimentalrealization of our pollution release is possible and canefficiently reduce the impact of a polluting source ina coastal area without reducing productivity. Applica-tions of the algorithm are limited to regions where anexperimental setting (such as VHF radars) or an accu-rate model of the flow is available. The cost of such aninstallation is usually high but can be shared by severalsources of pollution in a nearby area.

The algorithm is only efficient when an effective bar-rier exists in the domain,6 attaches to the coastline andoscillates about the pollution sources. In any other case,the algorithm presented here does not reduce the effectof the pollutant better than a constant release scheme.Future improvements include the combination of thisalgorithm with other methods that are not based on La-grangian structures to combine the beneficial effects ofeach method. A major effort is currently furnished toadapt these methods to three-dimensional spaces andcapture three-dimensional effects, such as upwellingand downwelling in coastal areas.

Acknowledgements

valR hiss ands andN byA rantD awnS sug-g ck-a ngt heO ted

c theirt

s oft

The authors are grateful to the Office of Naesearch and particularly to Manuel Fiadero forupport and advice. This research was fundedupported by ONR grants N00014-97-1-007100014-95-0257. G.H. was also supported byFOSR Grant F49620-03-1-0200 and NSF GMS-04-04845. The authors also want to thank Shhadden and Jan Van Walke for their remarks andestion to maintain and improve the software page MANGEN and Katharine Ratnoff for providi

hem with her editorial skills. The acquisition of tSCR data by the University of Miami was suppor

omputed in this paper and the suggested algorithm apply toransport.6 Coastal flows may not have any barriers for certain period

ime or at all.

F. Lekien et al. / Physica D 210 (2005) 1–20 19

by the Office of Naval Research (N00014-98-1-0818,LKS). Tom Cook, Brian Haus and Jorge Martinez keptthe OSCR up and running for the experiments. JoseVasquez of the City of Hollywood Beach and Jim Davisof Broward County Parks and Recreation kindly per-mitted us to the property on the beach in Hollywood.June Carpenter allowed us access to her beach housefor OSCR operations. Renate Skinner and Sid Leve per-mitted us to use the beach at the John U. Lloyd StatePark. Bill Venezia of the US Navy’s South Florida TestFacility provided us with real estate and electricity toconduct our operations.

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