Wave‐Current Interaction Models for Rip Currents - CiteSeerX

VOL. 83, NO. C12

JOURNAL

OF GEOPHYSICAL

RESEARCH

DECEMBER

20, 1978

Wave-CurrentInteractionModels for Rip Currents ROBERT A. DALRYMPLE

Departmentof Civil Engineering and Collegeof Marine Studies,Universityof Delaware,Newark, Delaware19711 CARLOS J. LOZANO

Instituteof MathematicalSciences andCollegeof Marine Studies,Universityof Delaware,Newark,Delaware19711

Two analyticmodelsaredeveloped to describe rip currentcellson an opencoastline with slopingplanar foreshoreand flat offshorebathymetry.Both modelsextendthe work of LeBlondand Tang (1974) to includethe refractionof the normallyincidentwave field by the nearshorecirculation.The first model includesthe effectof the currentson the incidentwavelength,but no changein wavedirection,and shows that no rip currentsare possible.The second,morecompletemodelconsiders the refractionof thewaves as well and predictsrip currentcells.The spacingof the rip currentsand their associated variablesare shownto be a functionof onedimensionless parameter.Comparisonof the predictedspacingwith some

published fielddataas well aswith measurements obtainedon the:Isleof Sylt,WestGermany,shows reasonableagreementwith the secondmodel.

I INTRODUCTION

rip currentson the incidentwave field, and later he speculated that this interactionmay be importantfor the dynamicsof rip ductedto explainwhy rip currents,the narrow currentswhich currents [Arthur, 1962]. In model studies on rip currents causedby normally incident waves, Harris [1967] noted the flowrapidlyseaward fromthe surfzone,oftenhavevery regular spacingalong the beach. The first major mechanism slowingof the incident wavesby the outflowingrip currents and also the refraction of the wave fronts that ensues. was proposedby Bowen[1969] and Bowenand Inman [1969] The first theoreticalattempt to show that rip currentsmay andindependently by Harris [1967].Thisinvolvedtheconcept of synchronousedge wavemincident wave interaction. The be supportedby wave-currentinteraction was made by Leproblem which ariseswith the proposedmechanismis the Blond and Tang [1974]; later, Iwata [1976] used a similar causeor existenceof the synchronous edgewave,whichtravels approach. Unfortunately, both analysesneed revision, as we parallel to the coast.Unlessthe beachesare flankedby head- will point out later. Finally, Mizuguchi [1976] introduced a lands, as in the study of Huntley and Bowen [1975], where variable bottom friction coefficientwhosefunctionaldepenNumerous

studies of nearshore circulation

have been con-

significantwaveenergycan be trappedby the headlands,the genesisof edgewavesis difficultto explain.The field experimentsof Bowenand Inman [ 1969]showthat the field data on rip current spacingis best fit by mode 1 edgewaves,yet a wave-waveinteractionmodel developedby Guza and Davis [1974] only accountsfor the generationof subharmonic edge waves,whichdo not causerip currents.The modeltheyproposedassumestotal reflectionfrom the beach,corresponding to 'surging'waves,and, in fact, Guzaand Inman [1975] and Guza and Bowen [1976] have shown in the laboratorythat wavesbreakingstronglydamp out edgewavemotions.Therefore the edgewave model for rip currentsis apparentlyrestrictedto steepbeachesand surgingincidentwaves. The regular spacingof rip currentscan be inducedby the presenceof regularlyvaryingbottom topographyas shownby Bowen [1969], Sonu [1972], Noda [1973], Birkemeierand Dalrymple [1975], and Mei and Liu [1977]. Coastlinescharacterized by lar.ge offshore bars (often occurring after major storms) presentan extremeexampleof this case.Hite [1925] notedthe apparentsetupof water behindthe barswhich.then flowed seawardthrough rip channels.Bruun [1963] and Dalrympleet al. [1976] have presenteddifferentmechanismsto explain this case.Also, Dalrymple [1975] has shownthat synchronoustrains of incident wavescan interact to create rip currentswhich are spacedaccordingto the deep water wavelengthand directionsof the waves. The possibleimportanceof the wave-currentinteractionin nearshorecirculationhas been discussedfor many years. As early as 1950,Arthur [1950] discussedthe refraction effectsof Copyright¸

1978 by the AmericanGeophysicalUnion.

Paper number 8C0556. 0148-0227/78/128C-0556501.00

6063

dencewithdepthwasdesigned to ensuretheexistence of rip current

cells.

The presentstudy showsthat steadynearshorecirculation cellsexistfor normal waveincidenceon a prismaticbeachand for any dissipationmechanismin the surf zone for which the

relationship between thewaveheight/-/andthetotaldepth• + • is wellapproximated by therelationship/-] = K(h • + •/), when K is a breaking index of the order of unity. For conveniencein the presentationand in order to relate this work more easily to previous efforts, we discusstwo models.In the first model the incident wave energyis affected by the waves,as in the paperby LeBlondand Tang[ 1974],and the local wavelengthis changedby the presenceof the currents.This model extendspreviousstudies,yet no rip currents occur.

In a second model

an additional

wave-current

interaction

effect, the refraction of the waves on the current, is also in-

cluded, and steady longshoreperiodic nearshorecirculation cells are generated.The refraction of the wavesby the outgoing rip currents causesthem to impinge on the beach

obliquely,thusgenerating longshore currentflowingtoward the location of the rip. It is of importance to note that in these models the dissipationis introducedindirectlyby its effects.More precisely, in the surf zone the energy equation is substitutedby the

relationship between the waveheight/-]and thetotal depth h• + • givenabove.Sincethiscondition is homogeneous in the unknowns,we may obtain the rip current cells but not the intensity of the dynamic variablesand mean water level. We assumethat shallowwater waveconditions(smalldepth to wavelengthratio) hold over the surf zone and a flat bottom

6064

DALRYMPLE ANDLOZANO: RIP CURRENT MODELS u

offshore region. These are not unduly strong restrictions,as the nearshorecirculation is confined to a very small region near the shore, extendingonly severalsurf zone widths off-

O xv•_•

shore. BASIC FORMULATION

////t0/////////// Beach

The time- and depth-averagedequationsof continuity of massandmotionhavebeendeveloped elsewhere [e.g.,Phillips, Fig. 1. Planformdefinitionsketchfor wavenumberand horizontal velocities. 1966]. The steadystatecontinuityequationis

[•0(• + •)]/•

+ [•V(h• + •)]/•:

0

(•)

shallowwateris equalto C = [•(/; + •)]•/•',the wavephase and/5 istheenergy dissipation dueto wavebreaking in wherehAis the still waterdepth,(0, I2) are the horizontal speed, of thewave vertically averagedvelocitiesin the (•,)) directions,and • is the surfzone.The (/•,•,•y) are the components and k is the the time-meanchangein the water leveldueto the presenceof numbervector[t onto the (•,)) directions, of l•. the wavesand currents.0 is referredto as setup or setdown, magnitude The conservationof wave numberin the steadystateis given depending on its sign.The transport streamfunction½, deby

fined by the relations

O(h • + •) = - •½1•

(a•/a))

•(h• + •) :

is now introducedto satisfythis equationexactly.SeeFigure 1 for the coordinatesystem. The steadystateequationsof horizontalmotion are

- (a•/•/a•) = 0

(5)

wherethe wavenumbervector,k• =/•xi +/•yj (wherei, j are unit vectorsin the •, f directions,respectively),is relatedto the (constant)angularfrequencyof the waves• and the water depththroughthe local dispersionrelation,

8 = •[g(h• + •)]-•/' + [t.(Oi + ffj)

t•(h •+•) co•+ col+/•

(2a)

(6)

SCALING AND PERTURBATION SCHEME

The problemis facilitatedby makinguseof nondimensional

variables.We use5,(= 2•r/œ•,whereE• is the rip current spacing),the longshorewavenumberfor the nearshorecircu-

lation,as thehorizontallengthscaleand $,/m,m, the beach

- g(•/+ 0)xa• +•

slope,as the vertical lengthscale.Then the nondimensional quantitiesidentifiedby the absenceof circumflexesare

+g

where• isthesymmetric radiationstress tensorintroduced by

x = X•

y = X,P h = Xm-•h ' • = Xm-•

L•nguet-Higgins andStewart [e.g.,1964],/{,• and/•yarethe k = (o•m;•-•)•/"•-•f a = (•12)Xm-•d anisotropicresistancetermsdue to bottom friction,• is the gravitationalacceleration, andf is thefluid density.Following Longuet-Higgins[1970] and lwata [1976],

1{, = fFK[•(h • + •/)]•/"•r-•[(l+ cos"0)0 + [(sin20)/2]I2] (3a)

sin"0)I2+ [(sin20)/2]0] (3b)

where0 is the angleof incidenceof the waves(cf. Figure 1),

F = 1.41[g/2œ/•e] -"/3• 1.41[(•d-/b)•/"/2dk7e] -"/3 (3c)

e = [t•f•K•'(mJ•-•)•']-•/•• = (•m•-•)-•/"•"m-•½

S = [Ifg•"(mX-•)"]-• (C•, U, V) = (o•mX-•)-•/"(O•, O, i = [l•oog"(m•, -•)"]- •[oOmX•]- •/"J•-•/J f = gF/m•- Ao = rhg"/Sf In the above relationsthe amplitudeof the wave • relatesto

theenergyby theequation œ= l•ood", andin shallow water, disregarding the Reynoldsstresses, thenondimensional radiation stresses are [Phillips, 1966,p. 51]

[fromKao'ura,1958],•, is the equivalent roughness of the

Sx, = le( 1 + 2 cos"0)

bottom material, • is the breaking index [Galvin, 1969], and

S,•y= Sy, = e sin0 cos0

the subscriptb denotesvaluestakenat the breakingline. In our calculationswe approximateF by its valueat the breaking line. Notice that for small angle of incidence0, the onshore

frictionterm is twicethe longshore frictiontermRy. This is becauseof the influenceof the wave-inducedwater particle motion, whichis primarily in the onshoredirection.

The steadystateconservation of waveenergyequationis

(7)

(8)

Syy= }e(l + 2sin"0) We insert(7) and (8) into (1)-(6) to obtain

(h+ •) co--• = 16cox [e(l+ 2cos"0)] K2

- -- -- (e sin0 cos0] +f(h+ 8 COy

(4)

whereC• isthegroupvelocity of theincident waves, whichfor

.Ico•k/coy(l+cos,.O) CO• sin"0 t cox

- -•-fy•

(h+•) coy

2

D^LRYMPLE ANDLOZ^NO: RIPCURRENT MODELS

0x 0y

(h+O)

0•o

(h + •o)

(9)

0y

6065

aO- K a [esin0cos0] (•+0)7-•y 8•-•x

oy

0eo

: -•'

oy

(16b)

(0 / 0x)[eo(h+ •o)•/•']= Do

(16c)

Oko/Oy = 0

(16d)

1 = ko(h + •o)•/•'

(16e)

Offshore,Do -- 0, and (16c) impliesthat eo(h+ •0)1/2is a functionof y, -but(16d) impliesthat ko = ko(x);usingthis informationsuccessively in (16e)andthenin (16b)yields•o= •o(X),eo= eo(x).In theoffshore, then,thesystem (16)reduces to the systemof ordinarydifferentialequations d

deo

(h+ •o)•xx•o= _•2 dx

(lO)

(17a)

d

0-• e

dx[eo(h + •o)TM] =0

(h+O)0y+cøs0(h+o)•/

ko = (h + •o)-•/•'

ß• -

(h+0)-•

(a•/ax)

dOo

+ (sin0cos 0)

(h+ •o)•xx= -•

- 3--$x

(l+2sin'0) 2 e•0[(h+0)-• - (a•/ay)

=o

k (-•i+•J a, 1= k(h+•)•/•+(h+•)'

-D

deo

dx

(18a)

too= (h + •o)--•-

(18b)

eo= (h + •o):

(18c)

(11) We use(18c) in (18a) to obtain •o = -{]tc:(h+ •o) + c•

(•2)

(19a)

Forabeach ofconstant slope mandshoreline atx = 0when

(13)nowavesarepresent,thebottomdepthish = x (//= m)?),and

Offshore,(9)-(13) providefive equationsfor the five unknowns, 0, e, •, 0, k, asin (11), D = 0, whereas in thesurf zonetheenergy balance equation (11)isreplaced bytherelationship

e = (h + 0)'

(17c)

On theotherhand,in thesurfzone,(15) and(14) togetherwith (16a), (16b), (16d), and (16e) yield

+ '•y e (h+ •) Ox+ sinO(h + o)•/ 2 e •0[(h+ •)-1 _(l +2cos•'0.)

(17b)

(14)

•o-

I + b½: + c•

(19b)

We now translateour systemof coordinatesso that in the newcoordinates, h + •o = 0 at x = 0, and hencec• = 0. From hereon we usethelattersystemof coordinates, andfor conve-

It is wellknownthatwavesincident perpendicular to the niencewereferto thenewoffshore distance simplyby x. We

shoreinducea meanwaterlevelsetdownoffshoreanda mean cast(19b)in the form h + •o = rhx, with

waterlevelsetupin thesurfzonewithnolongshore variations.

• = •/(• + •')

In thisstudywe will hypothesize sucha basicstateandusea

formal perturbation procedure toexplore thepossibility ofrip

currentcellsgeneratedby the nonlinearwave-currentinteraction.

(• 9c)

If we use(19c) in (16c) we find a posteriorithat the dissipation in the surf zone is to first order

Accordingly, weseeksolutions ofourequations (8)-(13)in theoffshore zoneand(8)-(10)and(12)-(14)inthesurfzoneof the form

Do = rh(h+ •o)a/:

Offshore,Dois assumed to bezero,andthe bottomis assumed

e = eo + •e•

to be flat;therefore eo= e•oand•o = O(x0),bothconstants,

• = •o + • =

0= r+•0• k = (-ko- •k•)i- •koOd

(15)

k = ko + •k• D = Do + •D•

wherex0 is the breakerline distancefrom shore.The mean depthoffshore(h + •o) is a constantand coincideswith the

meandepthdoat thebreakerline.Theposition of thebreaker linehasbeengivenbyDalrymple etal.[1977]in termsof wave period,beachslope,andbreakerheight.

where• is an orderingparameter. Z•o-O•D•

(20)

FIRST-ORDEREQUATIONS

EQUaTiONS

We insert

We insert(15)in (9)-(13)andcollect theleading termsto

obtain

(h+ •o)aOo ax = -•

aeo ax

d=h + •o

(21)

themeanzero-order totaldepth,into(9)-(14)andcollectthe (16a) first-order terms toobtain fortheoffshore region thefollowing

DALRYMPLEAND LOZANO: RIP CURRENTMODELS

6066

At theshorex = 0, (25)hasa regular singularity, andthe rootsof the indicialequation arezeroand•. The solution associated withtherootzeroyieldsanunbounded energy g'at

equations'

da-• +•

rg•0

3K2 c•e•

rgx

16

rgx t•2 t9 - -• 8 •y

x = 0;hence •k• Axa/•asx -, 0,andit isnotdifficult toshow that• andgareO(x•/•)asx -, 0.Atinfinity if h'--,O,thenthe solutions •k• C exp[-X0x],where X0= lB + [B• + (eoO•) +2fd -•/•g•k•(22a)bounded gy

(õ)]x/•}/2, B = 4do,/3f,andd• = limd(x)asx

Ripcurrent cells willexistif (25)hasbounded solutions at

drg•_ • r9 (eoO•)•16 ge• fd-•/••4'• (22b)x = 0 andat x = ooandif thereexistsx0positivesuchthat c•y c•x

gx

gy._l•

lim•'(x)_ lim½'(x)

eød-1 c3x l

x>x b

• [dt/2e• +«d-•/•eo•] • d+•/•0•] rgx •yyteo = ]}eo•xd-••y

3

-«e0•yL

•x 3

-D•

Ok(x)

x xoandcontinuous at x = xo. widthsoffshore, thecirculation, or ½,is negligible.

Solutiontechnique.Offshore,becauseof the flat bottom Theinfluence of • hascaused thesolution for• to havea assumption,the solutionfor the four variablestakesthe form discontinuous derivative atx = xo,which causes thelongshore of a set of exponential terms,{qeX•'}. Associated with the velocityto be discontinuous at the breakerline. This behavior

matrix equation (31),there arefoureigenvalues (X(i),i = 1,4) is notintrinsic to themodel,for it is caused by theabrupt andassociated eigenvectors (qn•, i = 1, 4) whichspanthe changeof bottomslope,thusmakingdO/dxdiscontinuous at spaceof all possible solutions offshore. For valuesof Ao = thebreaker line.Another manifestation ofthisdiscontinuity is •:rh/8franging from1to6thematrix Mzzhasonepositive real the presenceof two circulationcells.As thereis a zero value eigenvalue (agrowing solution offshore) andthreeeigenvalues for½ withinthesurfzone,thereisonecirculation system TABLE 1. Eigenparameters for Offshore Variables qH = (r•,•, 1•,g) qH•

qH:

q•8

qH4

A o = 1.O, xo = 3. 796

Eigenvalues

Eigenvectors*

10.108

-0.0638

-0.219 -0.00731 1.000

Ct

0

-5.215

-0.314+ 10.211i

-0.174 0.00969 1.000

0.0694 + 0.0118i 0.0436 + 0.00316i 1.000 + 0i

0.000378

6.06X 109

0.000244+ 0.00365i

qH4- (q•8)

-47.3 - 81.56i

Ao = 3.0, xo = 3.124

Eigenvalues Eigenvectors

Ct

22.375 -0.0802

- 12.844 0.000092

-0.0889 -0.00534 1.0

0.0736 0.00666 1.000

0

-8.996 X 10•?

- 1.230 0.000067

-0.149 0.001037

-0.031 -0.0318 1.0

0.0316 - 2.477 1.0

48.368

3.155

*Notethattheeigenvectors areknownonlyto withinanarbitrary multiplicative constant.

6068

DALRYMPLEAND LOZANO:RIP CURRENTMODELS _

i

i

i

becomes small. The wave breaker type during the measurement period was also recorded.Fifty-nine percentof the rips occurredduringtypesof spillingbreakerscorrespondingto the assumptionmade in the derivation, while 36% occurredwhen the waveswere classifiedas spilling-plunging.Only one observation of a rip currentwas made during surgingwaves,which would correspond to the edge wave model of rip current

i

I.•

formation.

From November 10 to 16, 1976, observations were made at E

the North Sea at the Isle of Sylt, West Germany, by the first author in conjunction with a study being conducted by the

0.6

Leichweiss Institut fi•r Wasserbau, Technische Universit•it -

o.z-• -o.z

-o.4

o

• i

i 2

i 3

• 4

5

6

Braunschweig. During this time, detectable rip currents occurred twice; each time when the wave angle at the breaker line 00 was less than or equal to :1:5ø. The data for the measurements are shown in Table 2, and their average values are plotted in Figure 3. For the other days the wave angle exceeded15ø to the beachnormal, and no periodicrips occurred. In comparing the average spacingof the rip currentsto their predictedspacing,the presenttheory yieldsan averageerror of 4.1%, while the edgewave model of Bowenerrs by over 50%. The observedrip currentsstronglyinteractedwith the beach profile during each occurrence.The profile was, in fact, a composite profile consistingof a steep foreshorewhere the waves broke and a fiat berm over which the swash occurred.

Fig. 2. Variation in the dimensionlessoffshoredirection of the

variablesq, 3, g, and• for Ao = 1.0.The breakerline occursat x = 3.79.

whollycontainedwithin the surf zoneanda countercirculating cellwhichcrosses the breakerline. It is expectedthat for a uniformlyslopingoffshorebottom profile this discontinuity will be eliminated. FIELr) DATA COM•'ARISON

There are very few publishedsourcesof field data to compare with the rip current spacingpredictedby theory. Two publishedsources wereusedherein.The firstwasthefielddata taken at El Moreno Beach,Mexico, publishedby Bowenand Inman. Despite extreme values of breaker angles,ranging from l0 ø to 21ø, the comparisonwith theory, Figure4, is not bad; however,the comparisonwith the edgewaveprediction for spacinggivesan averageerror of 3.5%, while the present theoryhas an averageerror of 63.3%.The seconddata source was that of Balsillie [1975], which lists data taken at the CoastalEngineeringResearchCenter'sPrototypeExperimental Groin at Point Mugu, California. The data consistof wave period, breakingwaveheight,and breakinganglesas well as surf zone width and rip currentspacing.The beachslopewas estimatedby dividingthe approximatebreakingdepthby the surfzonewidth,or m = l•o/K.io,whereK = 0.8. Of all the data tabulatedonly thosetaken 1000ft (4305 m) north of the pier

The swashdivided at the limit of uprushand returnedseaward through small swash channels in the berm and then further seaward past the breaker line as a rip current. Figure 5 shows these rip currents. CONCLUSIONS

The two modelsfor steadynearshorecirculationinducedby normally incidentwaveshave beenexploredfor a planar foreshore of constant slope and a fiat offshore bathymetry. The flow pattern in the nearshorezone consistsof two cells,one wholly contained within the surf zone and one extendingoffshoreof the breakerline. The presenceof two counterrotating cellsinsteadof one is presumedto be introducedby the discontinuity in the beach slope which occurs at the breaker line, although the laboratory experiment of Mizuguchi and Horika•,a [1976, case 15] showsthe samegeneralcirculationfor a i

i

,

,

i

_

were used in order to avoid undue interferencefrom the pier.

-•

Further, any data for dayswhen the wind speedexceeded10 knots(5 m/s) or the breakingangleexceeded+5 ø of the beach normal were neglected.Approximately40 individualobservationsremainedduringthe periodMay 1972to April 1973.The

'½.05

•

[

0

observationshave a significantrange of error, as the measurements are made visually by observerswho estimatedwave characteristics and rip currentspacing.It is anticipated,however, that owingto the largeamountof data usedthe variabil-

ity in Ao and•0 will, in themean,cancelout. As canreadily

,

.0 5

0

I

I

I

2

4

6

I

I

I

8

I0

12

x

Fig. 3. Variation in the dimensionless offshore directionof the

be seenin Figure4, the data tend to confirmthe theory quite

variables •, •, g, and• for Ao = 3.0.The breakerlineoccursat x =

well,evenfollowing thepredicted curveof •œ0versus Ao asAo

3.12.

DALRYMPLE

AND LOZANO:

RIP CURRENT

MODELS

6069

mild beachslopeß The longshore dependencies of thevariables TABLE2. Rip CurrentObservations at Isleof Sylt,November1976 and the computedresultsindicate that the offshoredirected flows in both cells causethe waves to be refracted into these

November 10,

November 13,

1976

1976

currents.However,because of the work doneby the currents againsttheradiationstresses, thewaveenergyisreducedat the Wave period T, s waveheight•o, cm locationof the rip currents.The longshoreflow is therefore Breaking

5.3

9.1

30-40

30-40

fromregions of highwaveenergyandsetupto regions of low Breakingangle0o,deg Surf zone width œo,m energy,in the samemanneras thatof Bowen[1969]. Beachslopem, in surfzone

+5 15 0.07

- 5 15 0.1

This analysis,while showingthat wave-currentinteraction Ripcurrentspacing œr,m canin factsupportsteadystaterip currentcirculation,hasnot

treatedthe initiationmechanism, that is, the originaltimedependent instabilitywhichleadsto thissteadystate. Anotherimportantfactorin the rip currentsystemis the convective acceleration (lasttermsin (9) and(10))whichplays no role in our analysisbut is mostlikely responsible for the

35.1

44.6

34.8 24.1 19.6

46.2 30.8 20.1

16.6 24.1 17.8 34.8

22.0 20.3 25.2 27.8 24.3 22.9 26.3 29.0 27.0 17.7

strong offshorejetlike currents observedin the circulation cells.

Finally,a betterunderstanding of thedissipation and frictionmechanisms isnecessary to improveourknowledge ofthe nearshore circulation.

Averagespacing

25.8

27.4

APPENDIX

The matching conditionsat the breaker line are derived

fromtheequations of motionthemselves. The procedure will be to integrate an equation overa smallregioncontaining a

portionof thebreaker line.Forexample, consider theintegral of the continuityequation(1) whichholdsin the inshoreand offshoreregions

, • x•-a

O(h+O),)dxdy=O

c9x

•y

where1• and1: are arbitrary.Integrating the firsttermwith

io

respect to x we find

o

o

o

o

andintegrating thesecond termwithrespect to y wefind