Multifractal representation of breaking waves on the ocean surface

JOURNAL OF GEOPHYSICAL Multifractal ocean RESEARCH, VOL. 99, NO. C8, PAGES 16,179-16,196, AUGUST 15, 1994 representation of breaking waves on the surface Bryan R. Kerman AtmosphericEnvironment Service,Canada Centre for Inland Waters, Burlington, Ontario, Canada Lucie Bernier Department of Computer Science,McGill University, Montreal, Quebec, Canada Abstract. It is established for the first time that the spatialdistributionof breaking waveson the ocean surfaceis a multifractal process. This result is based on an analysisof airbornevisibleand near-infraredimageryof the oceansurfaceunder a limited rangeof wind speedand fetch. A detailedstudy of the optical spectraof the imagesand the cumulativeprobability structureof the prevalentbackgroundshows that the lower-intensityreflectiveareasfollow a Rayleighprobability distribution. By contrast, the higher-intensitypixels associatedwith the scatteredlight from foam and breaking wavesdemonstratescalingcharacteristicsin both the optical spectra and the cumulative probability distributions. It is demonstratedthat the degreeto whichthe whitecapsare singularitieson a dark backgroundis describedby a Lipschitzexponentc•, whichuniquelytags eachbreakingwave. This identification process,called "fractal" or "singularityfiltering", leadsto a critical conditionc• = 1 tentatively associatedwith the crossoverfrom active entrainingwhitecapsto passivelydissipatingfoam. The multifractal representationassociatedwith the degreeof singularityis simply a restatementthat the imagery is composedof a continuumof sets,whereeachset consistsof thosebreakingwavesat a particular phasein their existence.The fractal spectrumof the image abovea thresholdis shownto be representable by a fractal generator.Physically,the fractal generator modelsthe energyexchangein a breakingwavefield as a flux of energyinput from the atmosphereto the wave field cascadedover scalesof the order of a kilometer to meters. If the energyflux is further parameterizedin terms of the receivingarea, an assumptionsimilar to closuretechniquesused in classicalturbulence models, the empiricalresultssymmetricallyspan Phillips'sbasicargumentsfor the energyflux terms controlling a wind-driven sea. Introduction 1963, 1983; Srokosz,1986]assumethat the occurrence statistics of breaking waves arise from a joint GausKnowledgeof the spatial distribution of breaking sian distribution of wave height and slope. While such waveson the ocean surfaceis required in a number of models have not been examined for their predictions applicationsas variedas gastransferand soundgener- of spatial statistics, it has remained for modelers to, ation at the air-sea interfaceto representsomephysical somewhat arbitrarily, assumea Poissonprobability dis- or chemicalactivity occurringlocallyin terms of larger tribution [Hollell and Hellmeyer, 1988] when dealing spaceaverages.While someimportant characteristics with the distribution of soundgeneratedunderwater by of breakingwavesrelevantto theseapplications,such breaking waves. Such a model is inadequateto explain as fractional coverage,their numberfor a given image the observedintermittency and "groupiness"of breakarea, or their lifetime, have been studied by many au- ing waves, as discussedbelow. Recently, Huang el al. thors,mostprincipallyMonahah[1993]and Hollhuisen [1992]haveshownthat the intermittent,randomphase and Herbets[1986],there is limited experimentalev- of surface gravity waves follows a fractal distribution. idenceto representtheir spatial geometry[Snyderel They show that rms differencein phase over a time inal., 1983]. Sometheoreticalmodels[Longuel-Higgins,terval At is identical within a scalingoperation to a differenceover eat, where e is somearbitrary multiple. Copyright1994 by the AmericanGeophysicalUnion. Paper number945C00590. 0148-0227/94/945C-00590505.00 It is therefore natural to consider such techniques to describethe strong nonlinearity and intermittency of a breaking wave field. Before introducing the so-calledfractal approach to brea.kingwaves,it is appropriateto attempt to state in a 16,179 16,180 KERMAN AND BERNIER: MULTIFRACTAL BREAKING WAVES ! . Ii . i Figure 1. (a), Field of breakingwavesobserved usinga line scanneron boardan aircraftflying at about500m overthe Bayof Fundy.Fieldof viewabout250m2, resolution wasabout0.5m,andoptical wavelength was0.95/•m. (b), Imagethresholded to removereflective Rayleighcomponent. (c), Imageof asubsets (a) 1) identifiedwith nonentraining (foam)areas.(d), Imageof a subsets, a < 1, identified with entraining(breakingwave)areas. qualitative way what is observedabout breakingwaves, at the locationsof the breaking waves. The localized even at the risk of stating in some casesthe obvious. nature of thesepeaks,akin to islandsof white light in classical,globalrepreWhen the field is imaged either by standard photog- a blackbackground,discourage sentations such as a Fourier representation and suggest raphy [Rossand Cardone,1974;Smith,1981;Kennedy andSnyder,1983]or with side-scanning techniques from underneaththe surface[Thorpeand Hall, 1983;Thorpe andHumphties,1980]or fromthe air (Figurela) [Kerman and Szeto, 1994], severalfeaturesof the spatial field of breakingwavesare evident. The first feature of breaking wavesis their distinct isolated nature: the fact that they are born, mature, and die almostalwayswithout affectingor beingmodulated by other breaking-waveevents. If a line were drawn arbitrarily acrossthe imageof Figure l a, the local light densitywouldbe decidedlyintermittent,that is, almosteverywherenearlyuniform,with narrowpeaks a local topologyand singularitiesto characterizethe process.Two basiccharacteristics of individualwaves, thelightaccumulated withinthemandtheirsize,define a uniquemeasureof the singularity that a breaking waverepresents in an otherwise smoothlyvaryingbackground. The secondfeature of breakingwaveswhich can be demonstratedadequatelyby laboratory measurements [Melvilleand Rapp,1988;Papanicolau and Raichlen, 1988]istheself-similarity oftheevolution process. Each breakingwaveappearsto be identicalto everyotherexceptfor sizeand duration. In this evolutionall waves KERMAN AND BERNIER: MULTIFRACTAL follow a path from their creationas a bright, energetic, linear bubbling froth to ultimate dissipationas a contorted patch of foam, albeit at different sizes. Consequently,any breaking wavefield has within it individual breaking waveswhich can be characterizedbroadly by two properties' initial turbulent energy and age. The challengeis to estimate these propertiesfrom imagery. Of importance to later discussionsis the concept that within a snapshotof an oceanicfield of breaking waves lie a number of self-similar identities, each at a particular phase in its lifetime and each at its own transient energy state. Another feature of a field of breaking waves is the nonuniformity of the occurrenceof the waves. They are not uniformly distributed in space,and their occurrence is not predictablein time. The reasonsfor the clustering and intermittency are related in part to the modulation of wave growth and steepeningwithin a wave group's envelop[Donelanet ai., 1972]and,webelieve,to the inherent intermittency of the energyexchangedriven by the atmosphere. A Poissondistribution fails to capture the essenceof intermittency becauseit assumesa constant densityof eventsin space,a fact clearly violated in the observations.It is necessaryto work with a relaxed set of assumptionsin which the density of occurrences BREAKING WAVES 16,181 equivalentto assumingthat the singular structuresof all breaking waveswhich are basicto the fractal result have a single, common property. This result disagrees with our intuitive sensethat a bright new breakingwave is not geometricallyand dynamicallysimilar to one dissipating at the foam stage. Evidence for a violation of such a commonbut restrictive singularstructure was found in the variation of the apparent fractal dimension as a function of the level of light thresholdthat one imposed. If the field was truly monofractal, it would not matter what subset of the field was examined, as only one singularity property was present at all scales. As mentioned in the conclusion of Kerman and Szeto [1994],it is necessaryto consideran escalationof conceptual difficulty, that is, that a breaking wave field is associatedwith a multifractal processrequiring many or a continuum of fractal dimensionsfor its representation. The multifractal approach has much in common with numerous other geophysicalprocesses,especially in atmosphericdynamics[$cherlzerandLovejoy,1991]. What will become clear is that the breaking-waverepresentation provides one of the clearest examples of a multifractal structure and one perhaps easier to motivate physically. It was our initial purpose to examine whether the is itself some function of scale. process was indeed multifractal. As interesting and Yet another distinguishingfeature of a breaking wave utilitarian as such results may be, the intriguing quesfield is the size range of eventsfrom the obviouslarge tion posedby Kadanoff [1986] (Fractals-Where'sthe waveswith dimensionscomparableto the energy-contain-physics?) remained. Accordingly,this paper has reing wavelengthsof the gravity wavefield to weakly en- focused on the basic premisesof a multifractal repretraining ripplets that are barely visible,the so-calledmi- sentation from which has arisen the conceptof singucrobreakers [PhillipsandBanner,1974].Phillips[1985] larity filtering to isolate individual breaking wavesand has argued that there exists within the wave field a to distinguishtheir turbulent states betweenactive enrangeof scales,the so-calledequilibriumrange, in which trainment and ultimate dissipation. It is shown that the energy available from the atmosphereat a given the singularity strength assignedto eachisolatedevent, scaleis convertedto dissipationin the turbulent break- discernibleabove the reflective background,acts as an ing wave at the same scale. Whereas a Kolmogorov identifying label, much like frequencyor wavenumberin cascade model cannot represent the separate centers a Fourier representation. It is argued that the sudden of dissipationinvolved in a breaking wave field, it is changeobservedin breakingwavesbetweenentrainment possiblethat an atmosphericcascadeinvolving transfer and runout is associated with a critical state characterstrength. from larger connectedactiveareasto apparently disjoint izedby a criticalsingularity A theme running through the fractal conceptualizaimbeddedsubareascarriesboth the spirit of an energy cascade and the existence of multiscale identities. As tion in the sections Fractal Concepts and Analysis is mentioned above, a spatial representation of the im- that a basic fractal generatorof the Cantor type conage as a Poissonprocesspresupposesa spatial density veys most of the relevant structure of breakingwave of independent events. Accordingly, the Poissonrep- fieldsas we now understandthem. In a companionparesentation, whose average distancebetween eventsis per [Kerman,19934],a modelof an energycascade is independent of the scale of representation,that is, the offered to reproduce aspectsof the multifractal strucresolution, is inappropriate to a processwhich is ca- ture of a breakingwavefield basedon sucha generator. pable of generatingprogressivelysmaller eventsas one reducesthe scale of inquiry. Fractal Concepts All these features of breaking wave field-localized singularities,self-similarshapes,intermittency, and an Rather than present a detailed mathematical view evolutionary cascadeare the hallmarks of a fractal pro- of a multifractal process,we prefer to offer an intuitive cess.In Kerman and$zeto[1994]it wasarguedthat the viewbasedon a realisticand reusableexample,the Candistribution of breaking waveson the ocean surfaceis fractal. However, the conceptsand approachusedthere were for what is known as a monofractalrepresentation in which one parameter, the fractional dimension", tor process. Our reasonsare more tutorial; that is, for geophysicists who have not been exposedto suchmeth- ods, than developmental.A reader wishingto study how suchtechniquesare appliedmay wishto commence [Mandlebrot,1983],solelyrepresents the field. This is with the Data section;which describesthe experiment, 16,182 K••AN AND BERNIER: MULTIFRACTAL BREAKING WAVES for a discussionof the separationof breaking wavesfrom the background or with the Analysis section for a discussionof the multifractal properties of the breaking sentinga (fractional)numberof objectsN and strictly equatehis iterative scalereductionfactor 1/r with that here(1/3 eachstep),we haveby his definitionoffractal waves themselves. dimension Monofractal D Model InN D- Considerinitially a line segmentof length L, of length lnr (3) [Feder,1988, p. 62]. Superimpose on L a uniformly which when N - 1/2 and r - 3 are substitutedinto it, leadsto D - In 2/ln 3 - c•. distributed weight w whoseintegral over L is To summarize,a Cantor processhas led to a sequence of identical amplitude and width singularities,characterized by one value of the singularity strength c• to whichcorresponds onefractal dimensionD. Sucha proConsider next a Cantor partition of L whereby the cessis called monofractal. Unfortunately, it is limited singleinterval is split into three equal length subinterin scopeto describinghow the spaceis redistributedin a vals of length, I - 1/3. Let us call the originalsegment particular cascade.Next we considera cascadeprocess the "mother" interval and the subintervals"daughter" wherea measure,say,local light density,is redistributed intervals. Consider next a different partition of the in a Cantor-like cascade. weight w so that the weight of the mother interval is equally placed on, say, the left- and rightmost daughter Model intervals, with no weight on the center daughter. We Multifractal interpret the assignmentof 0 weight as a death process Considernext a generalizationof the above Canand withdraw further consideration of it. We intend tor process. Let the two surviving daughtersubinterto repeat the process, so let us redefine the partition o•wdx -1 (1) vals,expressed as a fractionof the motherinterval,be weightw• - 2-• and the lengthof the daughterinter12,i = 1,2 rather than (1/3) n as above.Similarly,the val wehavegenerated as 1• - 3-1. weight,whichwe preferto call the measurehereafter, Next considereachsurviving daughterinterval one at is partitionedto w•, i - 1,2. We againinsiston cona time. Consider the Cantor processapplied to each servingthe measure,that is, daughter in turn: first a split to three daughtersof lengthle - 3- e andthenan assignment of thesurviving weightfrom the previousstep,that is, we - 2-e. When the processis carried to n iterations, the sur- E w?- 1 (4) viving intervals1• are of length 3-• with equal weights It can be shown[Halseyel al., 1986; Feder, 1988, we - 2-•. We next define the "strength" of each of p. 85] that after n iterations,thereare C•m segments these isolated, thin, packed intervals indexed arbitrar(where C is thecombinatorial symbol) of lengthl•• 12 n-• ily in essence"singularities",by the relationship n-• , k - 0, 1, ..n. Forexample, at the andmeasure Wl• W2 (2) first iteration,n -- 1, thereis onesegmentof lengthl• and onesegmentof length12. At the seconditeration, From the Cantor process,wi - 2-n, li - 3-n the sin- n - 2, thereareonesegment eachof length1• 2 and1• wi - l• gularitystrengtha is In 2/ln 3. FromMandlebrol[1983, andtwosegments ofl• l•. Thecorresponding singularity p. 37], if we considermomentarilythe weightsasrepre- strength after n iterations is found from Figure 2. Realization of a fluxcascade (Besicovitch-Cantor) fractalgenerator (w•=0.6, w2=0.4,l• •-0.5, •2=o.2). KERMAN AND w•w• - (l•l•-•)• BERNIER: MULTIFRACTAL BREAKING WAVES 16,183 cept is evensimpler: computethe singularitystrength (5) for each connected member or "island" defined over its When (5) is simplifiedby substituting• = k/n, k = support(spatiallocationof the set), crossreferenceto 0, 1, ..n, which representsan indexing of the total num- like groupswithin an "a-window",and then examine ber of resulting subsets, the differentsingularitysubsetsseparately. Estimation it can be seen that the singularity strength of the subset is not necessarilyconstant acrossthe different su•sets. Similarly, the fractal dimension f of each of the su•sets formed by different combinationsof I• and l• and of w• and w• can •e expected to vary with We do not carry the analysisto the calculationof •(•) of Fractal Spectrum The singularitystrengtha of (7) is a measureof how much one has to stretch the spacein the vicinity of a non-differentiable function to achieve bounded behav- ior. The largerthe singularity,the smallera is, and the greateris the necessary stretching.Accordingly, in and f(•) or equivalentlyf(•(•)). What we wish to indexingvariousintervalsin the previoussection,the stressis that a generalizediterative procedureh• led singularities can be reconsidered in orderof their sinto imbeddedsubsetsof different fractal properties. Such gularbehavior;that is, monotonically with increasing a processis called multifractal. An example of such a c•cade is given in Figure Distinctive features of the graph include the isolated peaks and their intermittency. Not so obviousmay •e the existenceof commonamplitudesand widths to sets of singularities. These geometric features and the fact that they stem from a c•cade processare common to the qualitative •pects of breakingwaves• discussed in the Introduction. An equivalentprocessis to stretchthe function,not its support,by raisingthe functionprogressively to a givenexponent q,referredto astheqthmoment.(Considerq to be positiveinitially in accordance with our implicit assumptiona < i whendiscussing stretching asopposed to contraction.)As thefunctionis stretched vertically,the singularityfor the ith supportinginterval originallydescribed by o•i(q-- 1) is nowdescribed by ai(q), where Fractal Singularity Filtering o•i(q)• ai(1), (q • 1) (S) Forthe verticalstretching,considera uniformchange The conceptof fractal singularity filtering follows directly from the conceptof imbedded subsets. Consider ofscale I•(q)ateach ofthennontrivial intervals where a positive-definite measure/• supported on an interval v is uniquelyselected to conserve the originalmeasure, of length i at N nontrivial subintervals. We first nor- that is, malize the measureso that its integral over the domain •..•Yi'i -- 1 (9) is unity and then compute an integral of the measure E•li --• q'•'(q) overeachisland,say,/•i, and relateit to the island(interval) lengthli by the relationship i=1 i=1 For an analytical process, such as the Cantor pro- cess,where /•i and li are known and countable,v(q) The definitionof singularitystrengthin (7) can be better understood in terms of local density, say, the amountof light emanatingfrom a remotesensor'sfoot- can be found by solution of a transcendentalrelationship, called the partition function. It remains to relate • to the fractal dimension and singularity strength of the various subsets. Consider by printor pixel,represented as,say,yii•-1. Forareasof [Feder,1988]somefractalsubsets$a characterized constantdensity,ai -• 1. However,for local peaksof a and their union S = t3,•S,• (10) density,if the apparentdiscontinuityis largeenoughin the limit, it is necessaryfor any li to have ai --• O. In Consider the measure relabeled to a number of sets other words, the senseof the singularitystrength a is N(a) for a betweena and a 4-5a. At someinterval inverseto the commonnotion of density: the larger the length 5 (• all li) usedto coverthe sets,one expectsa (discontinuityin) local density,the smallerthe singu- relationship larity strength. In the above Cantor example there would be numer- ous,in fact, C• (wheren >> 1) subintervals, each with a common value of a. We then consider all those N(o•,5) - N(o•)5 -.t(•) (11) wheref(a) is the fractaldimension of the subset.Con- memberswith a singularitystrengthsa betweena and sider a redefinition of a based on the available resolution a + 5a and relabel them • the a subset. Such a pro- in the form cedure is equivalent to Fourier filtering in which we /•i = 5• (12) identify one frequencyor wavenumberband of interBecausethe set elementsare countedonly at locaest, excludeall others, and examinea narrow band representation of the function by sliding our filter along tions where/•i • 0,necessarily •NmN(a,5)[.]- •[.] the frequencyaxis. The fractal singularityfilter con- Nm is the maximumnumberof setsat a givenresolu- 16,184 KERMAN AND BERNIER: MULTIFRACTAL BREAKING WAVES tion). Thereforeupon substituting(12)into (9) rewrit- of the Canada ten for countability, we have nine sorties over the Atlantic and Bay of Fundy. The Centre for Inland Waters was flown on linescancameraestimatedthe light in a pixel footprint N,• E N(a)5•q-•(•)+•'(q) =1 somewhat less than 0.8m 2 over a swath of about 570 (13) m along a path about 30 km long. The best contrast between the breaking waves and the background was This sum, now over all subsets,receivesmany of its found at 0.95/•m. The treatment of the imagery and contributionsfrom terms belongingto a specialsubset more details of the experiment are given by Kerman in which5'•q-I+•' isa maximumfora givenqor in which and $zelo, [1994]. An exampleof the imagerytaken over the Bay of Fundy is given in Figure la for an area 0 o•qa-f(a)+r(a(q)))•-•- 0 about 400 m on a side. (14)Background In (14), a,• succinctlydefinesthat specialsubset in which the maximumoccurs. To have (13) remain Process To examine the fractal propertiesof the image, we boundedrequiresthat •qa.•-](a.•)+r(q)remainfinite must first eliminate thoseaspectswhich are nonfractal. The method usedin this sectionto accomplishthis separation utilizes the optical scatteringspectrumof images and nontrivial as 5 -• 0, which requires r(q) = f(a,•) - qa,• (15) at different thresholds and the statistics of the weak but spatially extensivebackgroundprocess. The meaningof r(q) followsfrom (9) and (12), in The first problem in the identification sequenceis how whichwe note that N(q, 5) is givenby to separate areas of clearly different reflective properties; that is, specularreflectionarisingfrom tilted waves and scatteringfrom bubble-infestedpatches. As menN(q,5)- •• - 5-•(q) (16) tioned above, we have utilized two propertiesof the imi=1 ages' their optical spectra and their cumulative stathat is, the number of boxes required to cover the qth tistical structures. In the first case, various images momentof the me•ure y at spacing 5 varies• 5•(q). [Kermanand $zelo, 1994]of the samesceneat each We therefore•sociate v(q) with the (covering)dimen- of seven optical wavelengths,were analyzed at differsion of the qth moment of the measure and derive it in the customary box-countingmethod of fractal anal- ent percentliesof the cumulative probability functions (cpf) corresponding to differentareal extent or coverysis. We emph•ize that N(q, 5) relatesto the covering ageof the total scene,rangingfrom 25 to 0.05%. The of a measure and not its support, which is restricted to intensityof eachwavelengthfor a giventhresholdcorreq = 0. It alternatively is the moment of the measureof spondingto a given areal extent is presentedin Figure order q evaluated at resolution 5. 3 as a function of optical wavelength. From (14) and (15), For each threshold the intensity decreaseswith optical wavelength. However, the rate of decreaseis less a• = Or(q) Oq (17) whichprovidesa methodto extract am givenv(q) by meansof (16). The fractal dimensionf(am) is then calculatedfrom (15). It is convenient to consider an extension of the above >.5.0• method to the coveringof are• rather than a line • discussedabove. By redefining 5 to be an incremental • ,3.0 area and reconsidering a(q) and v(q)in (12) and (16), • 2.0- 0 1.0- -r _ the b•ic result is preserved. Such a structure which is more natural in definingthe singularity propertiesof a spatial area, also allows us to considera set of points distributed over an area in the same way one considers a Cantor set distributed on a line. The similarity also allows below for the construction of a c•cade model in i- z 4.0 Coverage from •,25 25% to 0.0488 % (byfactor of2) a singledimension,whichis actually area. channels I to 6 run 7 Data I o.5 Experiment 0.6 i 0.7 I i o.8 o.9 1.0 WAVE LENGTH (lam) The imagery to be utilized in this study was gath- Figure 3. Intensity of image at different optical waveered in March 1984 over the ocean near Nova Scotia, lengths. The image has beenthresholdedto achievedifferent Canada. A linescanner mounted on the DC-3 aircraft areal extent and averagedlogarithmically. KERMAN AND BERNIER:MULTIFRACTAL BREAKINGWAVES with increasingoptical wavelength,at least aboveabout 0.67 #m, and with increasing threshold. Clearly, as the image becomes brighter, it also demonstrates an increasedsensitivity at larger optical wavelengths. If the processwas entirely reflective, one would expect the spectra to continue to mirror the product of 16,185 its cumulativeprobability function (cpf). An example of the cpf from flight 7 is providedin Figure 4. As discussed in Kerman and Szeto,[1994],the well-defined linearsubrange of the cpf,herebetweenabout0.1 (10%) and 0.005 (0.5%), can be shownto be statisticallyinvariant to averagingat different length scalesand has the imposedlightingspectrum(directsolarand skyra- been identified as fractal. The question arises as to what, if any, structure the diance)and the surfacereflectioncoefficient.In sucha scenario,as the image'saveragebrightnessis increased, weaker intensity region possesses.When the cpf of an entire image is replotted linearly against intensity I on subelements maintaintheir similarity(hereparallelism) a standard "probability graph", that is, as a test for a with the weakest(herethe mostextensive)image. Such Gaussianstructure, the curvesare invariably straight is not the case. We conclude that there is an additional up to 70 to 85% accumulation,suggestingthat some distribution(i.e., positiveintensityonly) sourceof light exiting the breakingwavepreferentially quasi-normal might be present. On the other hand, if the cpf is replotwith increasingwavelength. The uppermostfour or five curves,ranging in extent ted linearly againstIn I, the resultinglocusis againlinfrom 0.05 to about 0.78% are virtually parallel. In other ear, suggestingthat a log-normaldistribution might be words,as the image elementschosenare brighter in the applicable. However,neither distributionis meaningful senseof total light overall wavelengths,the effectivere- for the process. The Gaussian distribution, conceived flectances of different size elements remain unchanged of as many additive subeffectscombinedin a singleobat eachwavelength.Also the fact that the optical spec- servable effect, implies no limitation either positive or tra at the longestwavelengthsessentiallymaintain their negative to the observedvariable, whereasintensity is parallelismleads us to suspectthat even at such large clearly positive definite. The log-normal process,concoverageas 25%, the imageat longwavelengthsis dom- ceived of as many multiplicative subeffects,is usually applied only in the large variablerange. We believeit is inated by a nonreflective process. We ascribe the apparent source of light at longer not justified in the lower-intensity process,which more wavelengths to the scatteringprocessof bubbles[Davis, likely arises from an additive rather than a multiplica1955]both underand abovethe watersurface.We here- tive effect. Also, even though a log-normaldistribution after refer interchangeablyto the reflective processas appearsto be a good qualitative fit to the cpf in another the background processand the scattering processas subrangeat muchlarger values,say,for the largest 10% the bubble process. or so,we preferto ignoreits relativelygoodperformance The second element of the method to discriminate at larger intensitiesin favor of a fractal representation. betweenthe reflective and scatteringprocessesinvolves We hypothesizefor the weak-intensityregion a ranthe curvesof the rarer but presumably more reflective -1 w -2 -5 -6 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 LOG10 INTENSITY Figure 4. Cumulative probability function forpixelintensity (wavelength, 0.95/zm)forflight7. 16,186 K••AN 2.5 AND BERNIER: I I MULTIFRACTAL I I I BREAKING I I WAVES I I 2.0 1.5 -- -- -- • 1.o n- OBSERVED CUMULATIVE •..• FUNCTION • PROBABILITY n- 0.5 - • -- z OI •©• BESTFIT -1.0 1 2 3 4 5 6 7 8 9 10 INTENSITY Figure 5. Inverse Rayleigh function transformationof cumulativeprobability function versuspixel intensity. Deviationfrom a linear relationshipat weakestintensityinvolves< < 0.1% of area of the image and resultsfrom digital round-offand dropouts. domreflectiveprocess whosevectorphasorhasrandom, Gaussian-distributed amplitudesandrandom,uniformly distributedphases. We associatethe Gaussianaspect of the descriptionwith the nearly Gaussiandistribution of waveheights[Longuet-Higgins, 1983]in sucha mannerthat the smallestimagepixel hasmany additive The questionarisesasto wherethe relationship fails. We havearbitrarilydefinedan equivalentsignalto noise ratio for the method by consideringthe ratio of the actual cpf to the best fit Rayleighdistributionbased on the weightedlowest90% intensities.When the ratio reachesa prescribedvalue(usually2), we selectthe subresolution sources of reflection associated with tilted nearesttabulated intensity correspondingto the nearest facets. The uniformly distributed phasesare thought to arisefrom the original wavefacetswhich are both weakly correlatedspatially and of indeterminateand nonpreferentialphase. The resultingprobabilitydensitydistributionof am- 20 18- plitude follows aRayleigh distribution given by pdf(I) =I ezp() 16- (lS) • 12- whose cumulative function cpf(I.),is cpf(I) - 1- ezp(--•-)(19) 12 and whose inverse function is 4- I -- {-2/n(1- cpf(I)}1/2 (20) 2- Accordingly, if any cpfis indeedRayleighin formand 0 I I I i i the functionalform on the right hand side of (20) is -6 -5 -4 -3 -2 -1 0 plotted againstI, a linear relationshipis expected.An LOG1oCUMULuLATIVE exampleof such a graph of the lower-intensityregion of an imagefromflight 7 [Kermanand$zeto,1994]is Figure 6. Ratio of measuredcumulativeprobabilityto inpresentedin Figure5. Indeed,the relationshipis linear verseRayleighfunctionbasedon structurein 90% of image in all casesfor at least 90% of the image. areaand plottedagainstaccumulated area(coverage). KERMAN AND BERNIER: MULTIFRACTAL 1(•ø 16,187 The enhanced lighting is visualized as arising from the multiple scattering of photons within the bubble cloud on the breaking wave'sface. For a given illumination, the deeper the cloud and the more concentratedits bubbles,the more scatteringis possibleand the brighter the wave appears. Accordingly,the sampledlight level representsa natural vertical integral of reflectanceover >- lff 1_ <• 1(•20 the bubble Cumulative 4_ cloud above the local function Run 7261 ld 5_ rate of turbulence I I I I I 10.-2 Illl "undisturbed" water level. As shown later, the depth, the horizontal dimensions, and the bubble density vary systematically throughout the lifetime of a breaking wave. It is to be expected that the scattered light also varies during the wave'slifetime and momentarily reflects the production probability • BREAKING WAVES I I I ld I I I Illl I ld ø and bubbles. We begin our analysis of light intensity and scale by consideringthe accumulationof light and the area of the islands(patches)left after the thresholdoperation against the background. Each island was formed by joining all neighboring(eight-connectedness) pixels Figure 7. Cumulative probability function showingbest INTENSITY fit Rayleighregion(opencircles)and fxactalregion(closed with intensity above the threshold. Each island's accumulated light and area are computedin terms of the circles). respectivesum over the total ensembleof islands. The fractional light and area of an individual islandare then coverage and useit as a thresholdof the imageagainst comparableto the measureover unit interval as discussedin the Fractal Conceptssection. We denote as Ii the backgroundRayleigh process. The procedure is demonstratedin Figure 6. For the fractional light arisingfrom the fractional area Ai of about 90% of the imagefrom an areal extent of I to 0.1, the ith island. The correspondingsingularitystrength the ratio of cpf valuesis unity and then risesrapidly. ai is then defined by We selectthe arealextent(coverage)nearestto the ratio of 2 and find the associatedthreshold from the cpf. The same result is displayedin Figure 7 for the cpf itself. The changefrom weak reflectivebackgroundto stronger,scattered,fractal processoccursat the boundary of the two charactersets. In hindsight a weak inflection point in the cpf may be noted which was not obvious a priori. We conclude that a self-consistent method exists to Ii - A•' (21) The c•i valuesrange from about 0.85 to 1.05 for the variousflights of the experiment[Kerman and Szelo, 1994].The probabilitydensityof c•is presented in Figure 8 for severalof the flights. Clearly, there are many more larger c• eventsthan smaller ones. The question as to the physical differencebetween different c• islands naturally arises. extract the fractal (scattering)processfrom the background(reflective)process. Our method is basedon first identifying in the optical spectrum two inherently different(reflectiveandscattering)propertiesof the surface. Second,the method relies on a sharp changein To examinethat question,considera reorderingof the c•i from smallestto largest. Figure9 demonstrates how the light and area are accumulatedprogressively from c•minto C•ma•.The two curvesshowcomparablebehavior' risinglessrapidly initially overthe sparsec•islands the detailed inverseof the cpf structure comparedto an and then accumulatingrapidly overthe densesets. The extensiveRayleigh distribution to identify the lowest light accumulationexceedsthat of area, gainingmost intensitiesof the fractal processwhich are statistically of its advantage for severallargebut abnormallybright identifiableagainst the background. events. The advantageis progressively overcomeas the large a events are considered. The difference between Analysis Fractal Singularities The most characteristic feature of a breaking wave field is the pattern of bright patches against an otherwise dark background. While there is evidence of shading within the background,very possibly associated with bubble cloudsconsistingof sufficientlysmall bubbles with terminal velocities comparable to or less than the turbulent velocities left in the breaker's wake, it is the isolated patches which offer good definition comparedto the reflective process. the fractionalaccumulatedlight and the areais alsodisplayedin Figure9. If the light originatingfrom a patch was always proportional to its area, the curve would be identically 0. Initially, with eachnew area at small a, the light accumulatesfaster until it reachesa maximum disparity. Thereafter light accumulatesslower than area. The questionnow is, what is the difference betweenc• subsetsto the left of the maximum, which contributemorelight than area,and thoseto the right, which have the oppositetendency? It is usefulto considersomelaboratoryexperiments on the growth and decayof breakingwavesto try to establishwhat interpretationto give to theseresults 16,188 KERMAN AND BERNIER: MULTIFRACTAL BREAKING WAVES 0.3 c 0.1 o i 0.9 0.95 I 1 Singularity Strength Figure 8. Probability densityof c•subsets. (Figure 9). Replottedin Figure 10a are somedata of This normalization leads to the maximum area occurPapanicolaou and Raichlen[1988]for the growthof the ring at aboutwt - 0.75;that is, at an agefor about0.75 sideview area of a surf breaking wave for various tank of the prebreakingwaveperiod. It alsoimpliesthat the configurationsrangingfrom a beachslopeof 0.6 to 2% effectivephasespeedof the breakingwaveis about 0.8c, and an initial wave height to a water depth ratio be- in which c is the prebreakingphasespeedin accordance tween and 0.4. In their analysis those authors demon- with the measurementsreported by Melville and Rapp strate a similarity in the size of the bubble cloud with [1988](Figure 10). Melvillealsoreportsa well-defined structurein the scalingbehaviorfor the initial entraintime after initial breaking. The geometric similarity apparent in the original ment ending near wt = 1. Thereafter the turbulence plots has been extended to nonshoalingwavesby not- is found to slowly relax to the intermediate asymptotic momentumless wake[Tening the critical relationshipin suchshoalingexperiments regimefor a two-dimensional between depth of initiation hb and the wave'samplitude nekesand Lurnley,1972,p. 124]. Hb' Hb--0.8hb Kerrnan[1988;1993b]haspresented datawhichclearly (22)show that the maximum of the integrated sound field [Galvin,1972]basedon numerousexperiments, and the under a breakingwave occursat a time after initiation relationshipinvolving the wavelengthat breakingAb Hb_ 0 14tanh hb - ' about 0.4 of the waveperiod. It is thought that the rate of bubble entrainment and hence the rate of increase of the bubble cloud size are directly proportional to the soundlevel and that suchproductionwill thereforepre- for shoalingwaves. After a transformationof variables cede the observed time of maximum bubble cloud size. It was also noted that the sound radiation into the air comparedto that into the water noticeablybeginsto increaselocally at about wt = 1. This result is ascribed the solution of the transcendental equation is approxi- to the presenceof bursting bubbleson the surfacein mately y - 0.5, or the aspectratio hb/Abis about0.082. the runout stagewhich radiated only into the air. y- 2•rA• KERMAN AND BERNIER: MULTIFRACTAL BREAKING WAVES 16,189 -0.12 -0.1 0,7-0.08 0.6- o o -0.06 0.5- o o 0.4- o -0.04 0.3- 0.2- -0.02 0.1- alpha_crit 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 Singularity Strength Figure 9. Accumulationof light and area and their differencewith collectionof subsetsfrom c•mi,•to From these various experimentswe concludethe following. 1. The changeof scalingand of the differencein the acousticgenerationin air and water are clear evidence of an entraining and a dissipating(wake) stage within a breaking wave. The end of the entrainment stage occurs near cot - 1, with evidence of the asymptotic decaystage after cot- 5. 2. The maximum bubble cloud area viewed laterally occursat a time cot= 0.75 which, within the potential errors associated with systematic differences between the experiments, is essentially the time of the entraining/dissipatingcrossover. to broken-uppuddlesof foam at larger a. Conceptually, for entrainmentwherethe densityand thicknessof bubblesis above averagefor the whitecap transient as a whole,the light densityis aboveaverageand the sin- gularitystrengthis belowaverage.Conversely, in the dissipativerunout, asfoamholesbeginto appearwhen the wave becomesspatially thinner, the light densityis reducedand the singularitystrengthis increased.At the critical crossoverpoint, for ac - i the fractional light is directly proportionalto its apparenthorizontal area,indicatingthat the processes determininglight and area are balanced, as also seen in Figure 9 previously. Hencethe critical value of a is a consequence of the balance between entrainment and dissipation. We therefore associate the maximum difference in Such a division is the basis of the conceptual differlight accumulationover area accumulationdiscussedin ences betweenwhitecapsand foam as expressedin the conjunctionwith Figure 9, with the critical stagewhen literature. The importanceof the developmenthere is the breaking wave has built up its maximum volume that the distinction can now be expressedquantitatively of bubbles,which impliesa maximumin the light scattered. Thereafter the area continues to increase, but in terms of reflectance contributions over the area conthe volume and the amount of light scattereddecrease. tributions of connected fractal subsets. The individualThis criticalentraining/dissipating crossover is unique- ized identificationof the stageof developmentof each ly referencedby the value at the maximum light and breakingwaveallowsfor individualextractionand maarea disparity, which is defined as c•c. From Figure 9 nipulation of wavesof comparableages,albeit of difthe maximum occurs at ac • 1. Further, for a < ac, all ferent sizesdependingon the initial energyavailableat a subsetsare primarily entraining, and for a > ac, all their creation. We refer hereafter to such a processas subsetsare dissipatingmore than entraining. The sin- singularityor a-filtering. The processis outlined in Figures l a to l d for flight gularity strength is a monotonicfunction of the breaking wave'sage, from fresh youngbreakerswith small c• 7 [Kerrnanand Szeto,1994]. The first transformation 16,190 K••• AND BERNIER: MULTIFRACTAL BRE•NG WAVES INCIPIENT 2O BREAKING P1 • 10- ,. X/hb=O.O ••191 0 P1 hb= 13.50cm Hb=16.50cm X/hb= 2.0 FRAME 43 -10 i -60-50-40-30 '-20-10 0 10 2o 3o 4o -20-10 i i 0 10 I x (cm) ENTRAINMENT 2ø1 ! 10-1 • 0 10 I BUBBLE • e•,•'"PLUME [•:¾•:'.:.:::.%. 20 .,,,, .......... I 30 20 / P1 x/h•,=4.0 10 I 40 r.. 201 1F1 -J E '" [ i 60 I i I I I 50 60 70 80 90 1co 50 P1 xlhb=7.0 I 60 I 70 I 80 I 90 I I I I I 100 110 120 130 140 X (cm) DISSIPATION DOMINATED X/hb P=120'0 I x/ hb=10.0 •:'?..'•._...?•?'•'(':•i•!i!!•':.'%'"•,.-. _,, FRAME 234 "•••'""•' .... -'•••• '"'•..• '.'. N 0 lO-• ...;..., ..•.•.., ...... , , , , , ,•, , •0 8O ' FRAME 172 P1 •00 70 -10 -.,,,,--. i .,........:•::.:• 0 90 i 50 ..,,•.. ..:.':'• •:.;•,:,.%•.• I 20- o •"-• • • 1 X (cm) • i 40 x (cm) "•'L,..• FRAME 92 ........ I I 30 DOMINATED • -10 i 20 •20 •30 •40 •50 •0 •70 •80 x (cm) ,•'.•:.:•..':•:..;•• •.• -•.....:• FRAME483 1 I -10 210 2•0 2•0 2•0 250 ' 260 ' 270 ' 280 ' 2•0 300• ' 310 x (cm) Figure 10. Growth of sideviewarea of breakingwavein laboratory tank with scaledage after breaking (wavephase). From Pavanicolaouand Raichlen(1988). of the image (Figure lb), consistsof a thresholdto removethe reflectiveRayleigh backgroundand is presentedwithout regard to its singularitystructure. The probability distribution of number, area, and brightnessof all alpha categoriesis presentedin Figure 11. As noted with Figure 8, the numberof large a islands greatly exceedsthe number of small a islands. The distributions of area and brightnessbetweenthe minimum of about 0.85 and the maximum of 1.05 are essentially tensive whitecap coverageexperimentssummarizedby Monahah and O'Muirchear'laigh [1980]. In Monahah [1993,Figure la, equations(1) and (2)] the measured partition betweenactive breaking wavesand the total whitecap coverageis about 0.14 comparedto the estimate here of 0.2. The best estimate based on a collec- tion of numerous experiments is that the active white- capswill occupyabout 0.85% of the area comparedto the aboveestimateand 0.68% of the area whena _• ac. identical and indicate a consistent functional structure The agreementbetweenthe techniquesis remarkably at least for a _• a•. However,they displaya crossover good consideringthe independenceof the methods. The near a - a• - 1 that is associatedwith the arguments commontechniqueof making whitecap coverageestiof the entrainment/dissipationbalanceabove. Clearly, mates is to establisha thresholdfor a given image at the dominantcontributionto the arealextent and light an intensity related to, at best, a critical reflectivity. scattered by actively breaking wavesis from the more This procedure is essentiallywhat has been achieved in our study, albeit from a differentperspectiveinvolvmature events near etc. The connectedsubsetsfor a • 1 are presentedin ing the statisticalstructureof the backgroundand inFigure lc. The entraining breakingwavesfor the con- tense regions. Where the difference occurs is in the dition a ( 1 are presentedin Figure ld. The brightness treatment of the individual patches where our discrimcontribution and areal extent of total fractal coverage inator utilizes a nonlinear parameter, the singularity are 8.7 and 3.4% and 2.0 and 0.68%, respectively,for strength, basedon integral propertiesof the patch; that the entraining, whitecap subsetsalone. The estimates is, its total light and spatial support. Our estimate for total areal coverageare consistentwith other esti- of the whitecap/foam ratio is most critical to the esmates (6.3%) for the samewind speed,basedon ex- timate of the threshold of fractal activity comparedto KERMAN AND BERNIER: MULTIFRACTAL BREAKING WAVES 16,191 0.3 0.1 i I 0.95 1 SingularityStrength Number • Area• Light Figure 11. Probabilitydensityfunctionof number(solidtriangle),area(solidbox) andintensity(open box) of connectedsubsetsfor givena categories. the Rayleighbackground.It wouldbe usefulto arrive algorithm to select the best fit partition parameters at an estimate of the threshold in terms of local con- finds the case of minimum nectivity rather than imposinga global threshold. We neverthelessconcludethat the singularityfiltering procedureallowsfor a systematicevaluationof individual breaking-waveeventsin terms of their turbulentstate with an accuracycomparableto that of existingmeth- the calculatedf(a) and a and their exact valuesfor a givenorderof (positive)moment.The error in fitting the fractal spectrumto • Besicovitch-Cantorprocessis typically small (0.019-4-0.007). ods. Fractal Spectrum accumulated error between The results of a secondtest to evaluate the accuracy of the recoveredli and wi are also presentedin Table 1. The estimated I and w are comparedto the exact solution. Again the averageerror is small, typically 4 to 8%. Accordingly,it was concludedthat the comBefore estimating and analyzingthe fractal proper- puter codewas accurateto better than 10% and that ties of the image data [Kerrnanand Szeto, 1994], a any structurefoundby subsequentprocessingof the imcalibration of the computer code to calculate the frac- age data to vary by more than that amount was likely tal dimensions of subsets with a known result was arto be dominated by the processand not the numerical ranged. An analytical expressionfor the fractal spec- analysis. trum of a cascaded flux (Besicovitch-Cantor (BC)) proThe best test of a multifractal processis probably cess,describedin the Multifractal Model section, has whether the successive momentshave the scalingstrucbeenfoundby Halseyet al. [1986]. It thereforere- ture aspredictedby (16). Severalexamplesof the summained to generaterandom realizationsof the BC pro- mations over different moments from -2 to +4 for differcesswith knownpartitionproperties(li, wi, i = 1, 2), to ent boxsizesfor a givenflight (7) arepresentedin Figure calculateits fractal spectrum, and to solvefor its I and 12. The log-logplot of N(q, •) is quite linear for small w parameters. positive q for a range of box size from about I to 512 Table I presentsthe resultsof an error study for fixed pixels(400 m). As the order of the momentincreases li andwi for 128sections of 6562(3s) pointseach.The positively, a perturbation occursnear a scaleof about 16,192 KER]V[AN AND BERNIER: MULTIFRACTAL Table 1. Results of Fractal Parameter Estimated Mean Variable Multfractal Spectrum Extraction Estimated Standard Deviation _ Exact 0.019 0.007 Wl w2 0.629 0.374 0.068 0.068 0.600 il 12 0.482 0.225 0.150 0.112 0.500 Estimation BREAKING WAVES Error 0- 0.400 -1- 0.200 4-2- Error analysis of estimated spectral spectrum and Besicovitch-Cantor parameters based on 128 independent realizationsof length3s. -6- 32 m and increasesin amplitude as q increases. This -7 deviation is believedto arise from a periodicity in the -•' • ' • 4 (• • 10 12 -4 data imposedby swell,with a wavelengthof about 65 m ORDER OF MOMENT modulatingthe localizationof the breakingwaves.The analysis for negative q is less convincing. The source Figure 13. The coveringdimensionr(q) of the various ' of error in this case is believed to be the truncation of order unity integersof the sampleddata so that there is almost no dynamicrange in the successively compacted data implied by negativeq. The scaling exponentsof the coveringsfor different momentswere then computedfrom the best log-logfit to each moment and are plotted in Figure 13 for the total range of q used, as discussedbelow. Becausethe accumulationwasoveran area and not an intervalalong a line as implied in (16), the box length was replaced with box area for consistencybetweenimbeddingdimensions. To safeguard against meaningless,nonphysicalderived parameters, the subsequentderivation of c• and f(c•) wasconstrainedto acceptonly that rangeof q for which (1) a(q+6q)<a(q)(2) f>0(3) 0•_<0. These ]imitations were alsoimposedby Meneveauand I ' ' ' I ' moments associatedwith summationsas in Figure 12. tifractal spectrumderivedfrom r(q) by the methoddescribedby (15) and (17) is presentedin Figure 14 for the acceptablerangeof q (4 •_ q •_ -2). The spectrum's main feature is its maximum(f = 0.8), whichoccurs near c• - 0.9. The fact that the spectral maximum is less than unity results from the thresholdingprocess, whichin essencenullifiesthe existenceof the (breaking wave-foaming)processin distinct regionsof the image. Qualitatively this result agreeswith our intuitive notion that the breaking of waves is sparse, that is that the breaking occurs "almost nowhere". In a similar analysis for the dissipationwithin Kolmogorovturbulence, Meneveau and Sreenivisan old for their data did not establish and arrived a thresh- at the conclusion that Sreenivisan[1987a]in their analysisof time sequences their estimate of dissipationwas spacefilling. For us to of estimated turbulent dissipation. The resultingmul0.8 2O q4 z Z o 3 15- G 0.6- z • 1o- z 5- o • o- 2 :• _ • O.4n- • o.2- o I I 2.00 I I 4.00 I I 6.00 I I 8.00 LOG2 BOX LENGTH Figure from-2 - i o I 10.00 i 0.4 i 0.6 i i 0.8 I i 1.0 i i 1.2 i i 1.4 i 1.6 ALPHA 12. Summations associatedwith order of moments Figure 14. The fractal dimensionf(c•) of the supportof to 4 at different box sizes. different subsetsof singularity strength c•. KERMAN AND BERNIER: MULTIFRACTAL hypothesizealternatively that breaking occurs almost everywhereand that we are incapableof sensingthe many smallerand weakerbreakingwaves,includingthe so-calledmicrobreakers,ignoresthe resultsof the Data section that there are two distinct intensity regions, both physicallybased. We also note that in Figure 14, BREAKING WAVES 16,193 like generator. This aspect is not required in the bino- mial multiplicativeprocess[MeneveauandSreenivisan, 1987b]usedto representturbulence.Further, because the processis multifractal we require a method to generate a distributionof subsetswith differentsingularity strengths after many iterations. Accordingly,we need a(q = 0) corresponding to fma• is lessthan unity in- to considera generalizedfractional split of the origidicatingthat the dissipationconnectedwith turbulence nal motherinterval(seethe Multifractal Modelsection) in the breaking wavesis distributed over noncontinu- and the assignmentof a fractional transferof the meaousspace(fractalsubsets)and that thereis no concern sure from one generation to the next. This brings us to about the finitenessof the energeticsas discussedby the lowestorder of complexityfor the fractal generator Meneveau and Sreenivisan. which will have a multifractal spectrum. Because the fractal spectrum of such a process is Several other significant dimensionsare included in Figure 14. The correlationdimensionfor q - 2 is 0.67, known analytically we can attempt to fit the observed while the informationdimension(q - 1) is 0.74. The spectrum as describedin the Fractal Spectrum section. former dimensionis important in describingnearest- It turns out that the fitting processis realistic and acneighbor distancesand pairing, while the latter leads curate, particularly if one ignoresthat part of the specto an estimateof the informationalentropyS [Feder, trum for q where the data quality makes the derivation dubious. In arriving at the parametersof the BC 1988,p. 78] givenby the relationship process, it is useful to reconsider an interval as an area which is being partitioned and the assignmentof weight to a daughterinterval (area) as a fractionof a flux exof S = 4.6. (f• = f(q = 1)). Also as demonstrated isting at the mother level. We hereafter redefine wi and by Feder, most of the subsetstend to concentratenear li of the Multifractal Model section as Fi and Ai to c• = 0.74 for q = 1. The subset density in this re- represent fractional flux and area. The nature of a BC gion may be related to the estimated critical singular- processallows for 2 subprocesses.Without recourseto ity strength for crossoverfrom active entrainment to a physical model, the significanceof thesesubprocesses runout as describedin the Fractal Singularitiessection. is unknown. We simply refer to them as subprocesses Another distinct feature of the fractal spectrum is I and 2 and arrange them so that subprocess1 is that the rapidly decreasingfractal dimensionof the support processfor which most of the flux is carried. As such it of subsetswith c•(q) < c•(0). A minimumsingularity might be called the flux-dominant subprocess. strength of about 0.59 is predicted from the projection Table 2 summarizes the results of a best fit of the BC of Figure 14 to f = 0. (The projected maximum of processto four flights of the aircraft at a range of wind about 1.6 is believed to be subject to experimental limspeeds. Flight 12 was taken relatively closer to shore itations and thereforepoorly resolved.)If one accepts than the other flights, and its statistics may relate to the hypothesisthat the evolution of a breaking wave an effectivewind speedlessthan the 19 m/s listed. The is describedby /••- c•,•i, [Kerman,1993b],the overall trend is that the flux and area partition of subincreasingfractal dimensionwith Ac• may also be as- process1 increasewith wind speed,while the partition sociatedwith the increasedprobability of samplinga parameters of subprocess2 compensateand decreaseas breakingwave(i.e., they covermore space)as the rate the wind speed increases. In other words, as the wind of evolution of the breaking processdecreases. speed increases the flux balance shifts in favor of the flux-dominant path and its larger areal fraction. Flux Cascade Model A cascademodelbasedon Phillips's[1985]concepts of input, dissipation,and energyflux by wave-waveinIt is instructive to enquire whether a simple model teractionshas beenpresentedby Kerman [1993a]. In suchasthat stemmingfrom the pioneerwork of Besicov- his developmentof an equilibrium subrange in waves itch and Cantor[Feder,1988]mightbe applied.Clearly with length scalesdistinctly smaller than those of the the processwhich we wish to mimic exists at distinct most-energeticwaves and larger than those for which locations and is 'dead' over most of the area, necessi- the drift current exceedstheir phase velocity, Phillips tating the inclusionof a trivial interval in a Cantor- considersa local balance in wavenumberspacebetween $ = -flUff1 Table Flight 2. Flux and Area Cascade Parameters Wind Speed, m/s 05 06 07 12 12 17 17 19 F1 0.621 0.854 0. 786 0.771 Cascade 1 A1 fl• 0.738 0.904 0.859 0.848 1.56 1.56 1.58 1.58 Cascade F2 0.379 0.146 0.214 0.229 A2 0.176 0.039 0.063 0.061 2 fl• 0.55,q 0.593 0.558 0.527 Estimatedflux and areacascade parameters for severalflights. 16,194 KER]V[AN AND BERNIER: MULTIFRACTAL BREAKING WAVES direct energy input from the wind, transfer to neigh- Conclusions boring wavenumbers,and direct dissipation. The order of magnitude of each term in the spectralbalancewas An image of a field of breakingwaveson the ocean given by surface itself contains two well-defined subprocesses' the reflective,low-intensity,nonbreakingRayleighback3 1 • --* u,ak -• --* u,A (26) groundand the scattered,moreintensefoam and white- caps constitutinga fractal process. Their separation In (26), k is the wavenumber and u. is the friction within the image is relatively easily achievedusing a velocityassociatedwith the (predominant)fractionof signal to noise ratio based on the ratio of the meathe integratedmomentumflux, whichlocally variesas sured cumulative probability function to an empirical Rayleighdistributionbasedon the bulk (90%)of the r --• u.ak -3/2 --• image. The thresholdintensity aboveis connectedin isolated islands whoseindividual pixels, without regard for their For our purposeswe have reinterpreted radial waveconnectedness,demonstrate a statistical invariance to length when squaredas an area A and havethus passed averagingindicative of a fractal process.When from a local spectral representationto a local spatial spatial these islands are connected into subsets and the islands representation. reordered by their singularity strength a, defined in of Mandelbrot[1974],Frischel ai. [1978],Schertzerand terms of their fractional contributionto the total light and area of the set above threshold, the accumulation of Lovejoy[1984, 1987], and Meneveauand $reenivasan light and area is smooth. This contraststo the 'devil's The fractal casacademodelis similar in spirit to those [1987b]in characterizing intermittencyand turbulence. staircase' Considera BC processwhich conservesan appropriate flux F (e.g. energy,momentum)with successive partitions from a mother interval, representingarea on the surface,to two subintervalor daughterareas. Consider a fraction fi of the flux from the previousstagecarried to the surviving daughter intervals of nontrivial fractional area ai. Successive iterationsonto the surviving subregions lead to a uniquerealization,eachfor a given random set of permutations, to establish the relative placementat each stage. The final result of many iterationsis a division of the original flux onto highly localizedregions,in fact, the supportof the dissipative process. The generalizedBC processdescriptioncontainsfour variables(fi and ai(i -- 1,2)) and the constraintthat the flux is conserved.We are motivated by Phillips's argumentsfor energy (and possiblyequivalentlymomentum) flux as a function of scaleto proposethat the fractionalfluxesand receivingareasfrom onestage of the partition processto the next are related. We accordingly proposea generalization (26), that is, - a, The representationis somewhatmore generalthan wouldresultfrom the arguments of Phillips(•/i = 1), structure that would occur had the islands been summedas encountered,say, by scanningorthogonally accordingto the axesof the image. The process in effect producesthe smoothestacumulation function possiblebasedon the Lipschitzconditionimplicit in the definitionof the singularitystrength(21). A closer examination of the relative accumulation of brightnessand size demonstratesthat for small a, light accumulatesfaster than area. It was argued that this result can be attributed to those connected sub- sets in which the entrainment processleads to an increased vertical extent of bubbles and hence more mul- tiple light scattering. Converselythe runout stage of breaking wavesin which the vertical extent is confined to the interface,has reducedmultiple-scatteringopportunities. As well, it containsholeswhere the scattering is nonexistentand therefore,will make a larger contribution to apparent area and a smaller contribution to light. This conclusionis consistentwith the often-cited divisioninto active(whitecap)and passive(foam)contributions to the total extent of breaking waves. The predicted split between active entrainment and passivefoam usinga critical c•c- i drawnfrom a point of balancebetweenthesetendenciesagreeswell with existing estimatesbasedon an examination of the intensity abovethresholdin individual photographsor video records based on the concept of a critical reflectivity to separate foam from whitecaps. Additional study is requiredto better achievethe basicfractal/nonfractal separationof the image elements. The analysis in the Fractal Spectrum section has shownthat spatial distributionof breakingwaveson the as it allows for two different subprocesses to coexist throughout the iterated partition operation. It is also similar in spirit to classicalturbulence closureswhich relate energeticsand fluxes to scale. What is most remarkable in Table 2 is the very limited range of •/i over the differentflights. The standard Proof of this contention deviation of estimatescomparedto their averagevalue ocean surface is multifractal. is 4 and 0.6% for i=l,2. Their consistency, in view of followsfrom the power law structure of the estimatesof the number of degreesof freedom,leadsus to conclude various moments between 4 and -2 as a function of the tentatively that the observedimagesare well described spatial scaleof the averaging.It appearsthat the agreeby a BC processand that the parameterizationlinking ment has been disturbed by the presenceof swell and the flux fraction to the areal subdivision is invariant to the limited digital precisionin the positive and negathe wind speed and fetch. tive moment estimations,respectively.The significance KERMAN AND BERNIER: MULTIFRACTAL of the main conclusionof this paper, that the breaking wave field is indeed multifractal, lies beyond the relatively simple description of the obvious statistical geometry.Classicalmodelsof forcingof the seasurface have not includedintermittencyuntil recently [Glazman, 1986; $hen and Mei, 1993]. The contributionof this work is that the centersof dissipation,the breaking waves, are clearly multifractal. Despite such assumptions in models, no measurementsexist as to whether the centersof energyinput from the wind are alsofractal. A useful study would therefore attempt to link the centersof energyinput and dissipation.A model which essentiallyassumesthat the centers are indistinguishable has been presented. It is based on the simplest, nontrivial representation of the whitecapping process in the form of a Besicovitch-Cantorfractal generator. Preliminary analysisindicatesthat two subprocesses exist that maintain a fixed relationship between flux and area. This parameterization is similar in spirit to classical turbulence closureapproximations. The subprocess carrying the majority of the flux apparently increases with increaseswith wind speed. However, the limited rangeof wind speedand the rather coarseimageresolution necessarilyweaken our conclusions. It remains to repeat the experiment with finer spatial resolution imagery,directly measuredinsitu near-surfacewinds and fluxes, and refined data processing. Acknowledgments. The authors would like to thank Shaun Lovejoy for his suggestionto examine the imagery for its possiblemultifractal structure. The work has benefited from the interest of many colleaguesat the University of Cambridge, the Atmospheric Environment Service, the Canada Centre for Inland Waters, and elsewhere, and we wish to thank them. References Davis, G. E. Scattering of hght by an air bubble in water. J. Opt. Soc. Am., •5, 572-581, 1955. Donelan, M.A., M.S. Longuet-Higginsand J.S. Turner. Periodicity in whitecaps. Nature, œ39,449-451, 1972. Feder, J. Fractals, Plenum, New York, 1980. Frisch, U., P.L. Sulem and M. Nelkin. A simple dynamical model of intermittency in fully developed turbulence, J. Fluid Mech., 87, 719-736. Galvin, C.J. 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