JOURNAL
Undertow
OF GEOPHYSICAL
over a barred
RESEARCH,
VOL. 105, NO. C7, PAGES 16,999-17,010, JULY 15, 2000
beach
A. F. Garcez Faria,• E. B. Thornton, T. C. Lippmann,2 and T. P. Stanton
Oceanography
Department,Naval Postgraduate
School,Monterey,California
Abstract. The spatialdistributionof the meancross-shore
flow (undertow)over a barred
beachis examinedwith field data obtainedon three energeticwave daysduring the
Duck94 experiment.The verticalstructureof the undertowis modeledusinga turbulent
eddyviscosityclosureand includesthe importanteffectsof wavebreaking(describedusing
the roller concept)and convectiveaccelerationof the current.Other than a more realistic
descriptionof observedturbulencevariations,a depth-dependent
eddyviscosity(compared
with a constant)doesnot improvethe agreementbetweenpredictedand observed
undertowprofiles.The effect of usingdifferent boundaryconditionsis investigatedby
extendingthe formulationsof Stiveand Wind [1986]and Svendsen
et al. [1987]to include
randomwavesby ensembleaveragingover the wave height distribution.The contribution
of breakingwave rollersto the surfacemassflux can be of the sameorder or greaterthan
the contributionassociatedwith the organizedwave motion. The largestdiscrepancies
betweenmodel predictionsand observations
occurover the sandbar,where the mass
transportof the breakingwavesappearsto be underestimated.
1.
Introduction
The local vertical imbalancebetween the wave setup pressure gradient,which is uniform with depth, and the depthvaryingwave radiation stressis conceptuallyresponsiblefor
drivingthe undertow[Dyhr-Nielsen
and SOrensen,
1970].In the
past2 decadesseveraltheoreticalmodelsfor the verticalstructure of the undertow
for two-dimensional
beaches have been
developed[e.g., Svendsen,1984;Dally and Dean, 1984; Stive
and Wind, 1986;Deigaardet al., 1991;Haines and Sallenger,
1994]. All modelsuse an eddy viscosityclosureschemeand
solvefor the depth-dependentundertowby integratingthe
cross-shore
momentumequationtwice over depth, which requires two boundaryconditionsto evaluatethe integration
constants.
There is a generalconsensus
throughoutthe literature of
using local conservationof mass over the vertical as one
boundarycondition.Commonly,the secondboundarycondition is either the stressat the trough level [Stiveand Wind,
1986]or the no-slipconditionat the bottomcombinedwith the
steadystreaminggeneratedby the bottom boundarylayer
(BBL) [Svendsen,
1984]. Despite significantphysicaldifferencesboth approachesreduce to the sameform betweenthe
troughlevel and the top of the bottomboundarylayerwithin
the surf zone as contributionsfrom steadystreamingand bed
shear stressare outweighedby mean water slope and wave
forcinggradients.
The wave-inducedonshoremassflux in the regionbetween
the wave crest and trough is critical to predictingthe magnitudeof the undertow,whichis predictedheuristicallyby adding
the contributionfrom breakingwaverollersto the masstransport givenby an irrotationalwave theory [Svendsen,
1984].
Dally and Brown [1995],found better agreementfor laboratory-generatedregularwavesbreakingover a planarbeachusing
streamfunctiontheory [Dean, 1974]than the linear wave theory, which tended to overpredictthe depth-averagedundertow. On the otherhand,MasselinkandBlack [1995],usingfield
measurements
from experimentsat two planarbeaches,found
good agreementusingshallowwater linear wave theory.
Most existingundertowmodelsshowgood agreementwith
laboratorydata for monochromatic
wavesbreakingoverplanar
beacheswhenthe depth-averaged
meanreturn flow is adjusted
to fit the data (insteadof usingpredictedmassflux) and the
magnitudesof the two largestdynamicalforcingterms (wave
setupandradiationstressgradients)are determinedfrom data.
Validation of thesemodelswith field data hasbeen limited by
the lack of data. Smith et al. [1992] comparedan undertow
modelto data from the 1990Delilah experimentperformedat
the same Duck, North Carolina, beach studied here and found
large discrepancies
over the bar, where the model underpredictedobservedvelocities.This strong"undertowjet" over the
bar was also observed during an earlier field experiment
[Hainesand Sallenger,1994] at the samesite.
Within the surf zone, wave breaking-generatedturbulence
dominatesbottomboundarylayerprocesses,
with the turbulent
shearstressmaximumat the surfaceand decreasingtowardthe
bottom.TingandKirby[1994],usinglaserDoppler velocimetry
(LDV) data from a wave flume experiment,found that the
primary turbulence-generating
mechanismin the surf zone is
due to wave breaking at the wave-roller interface and that
turbulence
intensities decrease with distance from the surface.
Cox and Kobayashi[1997], using laboratory LDV measurements of regular, spillingbreakerson a rough plane slope,
observed that the shear stress distribution
•Nowat Marinha-Directoria
de Hidrografiea
e Navegacao,
Rio de
Janeiro, Brazil.
2Nowat ByrdPolarResearchCenter,Ohio StateUniversity,Columbus.
This paper is not subjectto U.S. copyright.Publishedin 2000 by the
American GeophysicalUnion.
Papernumber2000JC900084.
within the surf zone
varieslinearlywith depthuntil the top of the (BBL) and that
the eddyviscosity•z is smallnear the troughlevel,increases
to
a maximum about one-third of the depth below the trough
level, and then decreasestowardthe bottom.Svendsen[1984]
investigatedthe effect of introducingan exponentiallyvarying
•z to the predictedvertical profile of the undertow.In his
formulation,two free parametersare necessaryto define the
16,999
17,000
GARCEZ FARIA ET AL.: UNDERTOW
OVER A BARRED
BEACH
magnitudeand decayrate with depth of/•z, and despitethe wave, and turbulence contributions. In a Eulerian reference
more realisticdepth variation of/•z, only marginal improve- frame and assumingirrotational flow below the trough level,
ments are obtained for the vertical profile of the undertow there is a net shorewardmean masstransportby waveslimited
to an upper region between the crestand troughthat is given
[Svendsen
and Buhr Hansen,1988].
In this paper, field observationsof vertical profiles of the by
mean
cross-shore
current
obtained
over a barred
beach are
used to test various models. Wave height transformationis
qw=
p•(z) dz,
(2)
basedon an energyflux balanceincludingthe effectsof rollers
to describewave breaking and a probabilisticdescriptionof
wave heights.The surfacemassflux is investigatedusingboth
where the subscriptsc and t refer to the crestand trough.
linear and higher-orderwave theoriesand includesthe contriAn additionalcontributionto the masstransport(per crest
bution from surfacerollers.The influenceof depth-dependent
width)occursabovethe wavetrough(surfacelayer)that arises
formulationsfor the eddy viscosityand different choicesof
from the presenceof turbulent wave rollers [Svendsen,
1984]
boundaryconditionson the verticalstructureof the undertow
and
is
defined
by
are examined.
t
c
2.
qr= PrA L'
Theory
(3)
where Pr is the roller density,A is the roller cross-sectional
line (x positiveoffshore)andz positiveupwardfrom the sea area,L is the wavelength,and c is the advectionvelocityof the
surfaceis used.Solutionsfor the surfacemassflux, the setup, roller, assumedto be givenby the wavephasespeed.The roller
and the verticalprofile of the undertoware describedassuming contribution(3) to the surfacemassflux is basedon the calstationarywave conditions,straight and parallel depth con- culated cross-sectional area of the roller. Earlier models
[Svendsen,
1984;Deigaardet al., 1991]assumedthat thisarea is
tours, and random waves that are narrowbanded in both freproportionalto the rms wave height.As a consequence,
the
quencyand direction.
largestmean return flow would be predictedto occur at the
2.1.
Wave Transformation
breaker point, which is contrary to laboratoryobservations
[NadaokaandKondoh,1982].Dally andBrown[1995]solvefor
Wave transformationis determined using a probabilistic the area of the roller for the case of monochromatic waves
breakingwave model that includesroller energygradientsin
breakingover a planar beach by simultaneously
solvingthe
the energyflux balance [Lippmannet al., 1996]. This model
depth-integratedand time-averagedenergy, continuity,and
assumesthat wave heightsboth inside and outside the surf
cross-shoremomentum equationsand found good agreement
zone can be reasonablydescribedby the Rayleighdistribution
with existinglaboratorydata. However, their model requires
[Thorntonand Guza, 1983] and givesaccurateresultsfor ran- observations of the cross-shore distribution of either the mean
dom wavesbreakingover both planar and barred beaches.The
return flow or the setupto constrainthe model and therefore
model has two free parameters: or, the mean angle of the
cannotbe applied to the presentdata.
wave-roller interface, and % a measureof breaking wave inAnotherrecentroller model(LippmannandThornton,subtensity. Although the model is insensitiveto the interfacial
mittedmanuscript,1998)presentsan independentmethodfor
angle(kept constantat cr = 10ø for all runs)basedon model
calculatingqr and is calibrated using video observationsto
resultsobtainedby Lippmannet al. [1996] at the samelocation determine the cross-shore variation of the fraction of waves
of the Duck94 experiment,adjusting•, to fit the data is unnecthat are breaking. This model is based on the energy flux
essary.
balance and describesenergy dissipationfollowingDeigaard
Wave propertiesare ensemble-averaged
by integratingwave
[1993]. Two free parameters in the model, B, the vertical
heightsthroughthe assumedRayleighdistributionp(H). Folfractionof waveheightcoveredby the roller, and ½,a measure
lowing Thorntonand Guza [1983], the fraction of wavesthat
of the averagewaveface angle,are adjustedto givearms best
are breakingis foundby integratingthroughthe breakingwave
fit to breakingobservations.
distributionp•,(H)= W(H)p(H), whereW(H) is a weighting
The onshoremasstransportin the upper regionis assumed
function (T. C. Lippmann and E. B. Thornton, The spatial
to be balancedlocallyby a mean return flow belowthe trough
distributionof wave breakingon a barred beach,submittedto
(undertow):
Journalof Geophysical
Research,1998, hereinafterreferred to
as Lippmannand Thornton,submittedmanuscript,1998).
A right-handedcoordinatesystemwith origin at the shore-
2.2.
Surface
Mass
qw+ q,.= -
Flux
•t
pU(z) dz = -pUsh,,
(4)
h
The conservationof massfor straightand parallel contours
with the no-flowboundaryconditionthroughthe beachis given where Ur is the depth-averagedreturn flow, andh t is the depth
belowthe trough.The assumptionthat the undertowis limited
by
to the regionbelow the troughwill be shown,in general,to be
a reasonableassumption.
(1)
•p[U(z)
+5(z)
+ti(z)]
dz
=O,
h
where the overbar indicatestime averaging;rt is surfaceelevation; h is the local water depth; p is water density;and
onshorehorizontal velocity has been partitioned into mean,
2.3.
Undertow
The time-averagedcross-shoremomentum equation, neglectingmolecularviscousstressesand assumingstraightand
parallel contoursand stationarywave conditions,is givenby
GARCEZ FARIA ET AL.: UNDERTOW
Opu2
Opuw
+
_ 0/5
--
ox
oz
(5)
o
Ox
The horizontaland verticalvelocities(u and w) are expanded
into mean, turbulent, and wave-inducedcomponents,u =
U + a + 5 and w = v½+ •, where the meanverticalvelocity
is assumednegligible(on a slopingbed, there will be a nearbed mean vertical currentwhen a current is presentto satisfy
the impermeabilitycondition).The time-averagedpressure,
p -- p[g(• -- Z) -- 1•2 -- •2], isobtained
fromthevertically
integratedverticalmomentumequation,after neglectingcontributionsfrom the cross-shore
gradientof the verticallyintegratedwave and turbulentshearstress[Stiveand Wind, 1982].
After substitutingfor the time-averagedpressure,(5) can be
further simplifiedfor the regionbetweenthe top of the BBL
and the troughlevel (middle layer)with the aid of the following assumptions:
(1) wave and turbulentvelocitycomponents
are statisticallyindependent;(2) turbulenceis near isotropic,
ti2 = if22[Stiveand14qnd,
1982];(3) thewaveshearstress
is
OVER A BARRED BEACH
1
1
U(z)
=2• F(X)Z2
q- Cl(X)Z
q-C2(X)(8)
p l•
First, followingStiveand 145nd[1986],C l(X) is determined
by integrating(6) once over depth and solvingfor the shear
stressat the trough level, and C2(x) is found by applying
conservationof massover the vertical (see Appendixfor details) to give
S(z)--Urq-•-•F(x)[Ao
+A•+A2
Z2]
q-•-• zq-h-
where 5•,x is the mean cross-shorebed shear stressthat is
calculated
usinga quadratic
formulation
(Sbx= cfllu,
whereCœ= 0.01 is a constantfrictionfactorand5b is the
near-bottomwave-inducedvelocity).This solutionis a seconddegree polynomialin z with coefficients
1
Ao=5,
1 0
Oz
20xp[•2 1•2]
and (4) a first-ordereddy viscosityclosurefor the turbulent
shearstressis givenby - 9tiv½= pl•z(OU/Oz). Applyingthese
assumptions,
(5) reducesto
O( OU)1O
O•OpUr
2
= F(x),
A2 •
3h2- h•
The solution(9) hasbeenpreviouslyappliedto monochomatic
waves and can be extended to random waves by ensemble
averagingover the wave height distribution:
(6)
where/•z is the time-invariantturbulenteddyviscosityand the
advectionof mean cross-shore
momentumhas been approxi-
matedbypU2(z) • put2. Theforcing
F(x) in (6) isdueto the
cross-shore
gradientsof radiation stress,setup (setdown),•,
and convectiveaccelerationof the depth-averagedundertow.
F(x) can be assumedto be independentof elevationon the
basisof empiricalevidencefrom laboratorystudies[Nadaoka
and Kondoh,1982;Stiveand Wind, 1982, 1986] or to be more
restrictive,limiting the discussionto long waves.
A solution for the vertical distribution
of the mean undertow
can be determinedby integrating(6) twice over depth to give
U(z) = F(x)
,
(9)
givenby [Riveroand Arcilla, 1995]
Op•
17,001
fz
•
pUz
dz + Cl(X)
f dz
p l•z
q- C2(x)
'
(7)
where Cl(X) and C2(x) are integrationcoefficients.Both the
Stiveand 14qnd[1986] and Svendsen
et al. [1987] solutionsare
used,with modificationsto their originaldevelopmentsto improve the derivation. For both solutions,a different solution
for the undertowforcingF wasusedas they wronglyassumed
that the wavestressis smallcomparedwith the Reynoldsstress
(p• << paff2), resultingin an overestimationof the forcing
by [Riveroand Arcilla, 1995]
1
0
20xp[•2_1•2].
Here (6) will be usedinstead.
Although different formulationsfor the eddy viscosityvariation with depth are investigated(appendix),the simplestsolution for a depth-independenteddy viscosity/• is outlined
here. For this case,(7) simplifiesto
(U(z))
: Urq-•-•
(A0q-Aizq-A2z2)F(x)
+5'bx(z+h
•)Ip(H)dH
(lO)
The effect of 5•x is smallcomparedwith wave-breakingdissipation [Thorntonand Guza, 1983; Svendsen,1984;Dally and
Brown, 1995], and hencethe vertical structureof the undertow
is mostly determinedby the quadraticterm. The amount of
curvaturein the verticalvelocityprofile is a functionof both F
(calculatedfrom wave quantities)and the eddy viscosity/•.
Large valuesof F producemore vertical shear,resultingin a
parabolicprofile,whereaslargevaluesof/• reducethe vertical
shear,producinga more uniform velocityprofile with depth.
Differencesfrom Stiveand 14qnd[1986]are the inclusionof the
momentumflux of the mean current term in the forcing, retaining the bottom shear stress,and treating of the waves as
random requiring ensembleaveraging.
The influenceof the boundaryconditionchoiceon the vertical structureof the mean undertowis investigatedby comparing the resultsobtained by (10) with the Svendsenet al.
[1987] model that usesa no-slip conditionat the bottom to
replacethe stressat the trough level as the secondboundary
condition.This no-slip conditionis obtainedby couplingthe
undertowmodel with a bottom boundarylayer model. Within
the BBL, the flow is a combinationof the steadystreaming
induced by the oscillatorymotion and the undertow above,
whichresultsin a meanvelocityat the top of this layer (U•)
that is obtainedby requiring continuityin velocityand shear
stressbetweenthese two regionsto give (see Svendsenet al.
[1987] for details)
17,002
GARCEZ
FARIA
ET AL.: UNDERTOW
+ (5-bx-5¾s)(z+ h)]p(H) dH,
(11)
where•13 = h2/2 and5•,sisrelatedto thesteady
streaming
in
the BBL:
OVER A BARRED
BEACH
with accuracyO(1ø).Measuredtwo-component
velocitieswere
rotated to a shorenormalright-handedcoordinatesystemusing compassdata and addingat eachsledpositionany deviation of the bottomcontourline from a shoreparalleldirection.
Determinationof the rotationangleis importantto avoidcontaminationof cross-shore
velocitiesby the observedstrong
longshore
currents
O(1 m s-•). Shoreparallelwasdetermined
by averagingover _+50m alongshore[seeThorntonand Guza,
1986]with an estimateduncertaintyof _+2ø resultingin errors
of <_+4 cm s-•.
wheref is wave frequencyand •z is the eddyviscosityinside
the BBL). This modelfurther assumes
that •z is muchsmaller
than the eddyviscosiffin the middle layer,whichwasrecently
verifiedin a laborato• experimentusingLDV [CoxandKobayashi, 1997].FollowingPutrevuand Sven&en[1993],the eddy
viscosityinsidethe BBL is estimatedby
h•- 0.32
C•
m for subsequentmeasurementrunsthat are referred to in the
text by sequentialnumberswithin eachday.Each datarun was
nominally1 hour, and sevento eight runswere made acrossa
transect
duringeachdayspanning
thehightideduringthisperiod.
Wavesand meanwater levelwere measuredusingan array
.
An objectiveis to determinevaluesof • appropriatefor the
surf zone by model fitting to the obse•ations.
2.4.
For the first run on each day the sled was towed by the
CoastalResearchAmphibiousBuggy(CRAB) to its farthest
offshorelocationseawardof the bar (-160 m from the shoreline). A forklift on the beachpulled the sledshoreward10-30
Setup
of five pressuresensorsmountedon the sled.Directionalwave
spectrawere alsoacquiredusinga linear arrayof 10 pressure
sensorsin 8 m depth.Additionally,a 13-elementcross-shore
array of pressuresensorswas used to measurewave heights
spanningthe width of the surf zone [Elgaret al., 1998].The
The setupgradientis a dominantdrivingforce for the undertow.The setupis calculatedby depth integratingthe timeaveragedcross-shore
momentumequation(5) from the bottom to the mean water level to give
fixed arraywaslocated-25 m to the north of the sledtransect.
These data were continuously
sampledat 2 Hz. Video observationswere usedto measurethe fraction of wave breaking
along the sametransectusingthe method of Lippmannand
Holman [1991].
OSxx OMr OoU•(• + h)
O•
Ox
Ox
Ox
•+•+
= -p9(h+h) •, (12) Meteorologicalinformationof wind and atmosphericpressurewere recordedsimultaneouslyat the seawardend of the
where Sxx is the wave radiation stress[Longuet-Higgins
and 600 m longFRF pier and atopthe FRF buildingin front of the
Stewart,1964],M, = cq, is the momentumflux associated
with pier. The bathymetrywasmeasureddailyusingthe CRAB, and
wave rollers, and the last term on the left-hand side is the
depth contourswere found to be nearly straightand parallel
momentumflux of the mean current of the depth-averaged for the 3 daysunder consideration.
undertowapproximatedby
4.
Results
and Discussion
dz--+h).
Measuredundertowflowsare maximumon top and on the
shorewardslopeof the bar, with increasingmagnitudefrom
In the derivationof (12) the meanbed shearstress[Longuet- October10 to 12 aswaveforcingincreasedwith the approach
Higginsand Stewart,1964] and the cross-shorewind stress of a storm(Figure1). The verticalstructureoverthisregionis
(owingto smallobservedonshorewindsin the data described the classicparabolic shape associatedwith strong wavebreakingturbulence[Svendsen,
1984;Stiveand Wind,1986].In
later) are assumedto be negligible.
the inner troughthe return flow isweak, and almostno vertical
structureis seen.At the seawardslope of the bar, observed
3.
Data
profileson October 10 (Figure l a) are nearly uniform with
Field measurements
were acquiredas part of the Duck94 depth. Significantbottom roughnessin the trough of the
experiment[e.g., GarcezFaria et al., 1998] conductedat the barredprofilesrelatedto the developmentof lunateandlongU.S. Army Corpsof EngineersField ResearchFacility(FRF) crestedmegaripplesasa resultof the stronglongshorecurrents
in Duck, North Carolina. The data selectedfor analysisare [Thorntonet al., 1998]is foundhere.The bar migratedoffshore
from October 10-12 when strongcross-shore
currents(0.05- (0 (20 m)) betweenOctober10 and 12, whichGallagheret al.
[1998] associatedwith the strongundertow.
0.4 m s-1) associated
witha stormwerepresent.
The vertical structure of the current was measured with a
Model-datacomparisons
are evaluatedby calculatingabsovertical stackof seventwo-componentMarsh-McBirneyelec- lute andrelativermserrors.Absolutepercenterror hasdimentromagneticcurrentmeters(ems)mountedon a mobilesledat sionalunits and is definedby
h
elevations 0.4, 0.7, 1.0, 1.5, 1.8, 2.2, and 2.6 m above the bed.
The emswere horizontallydisplacedat least 1 m from the sled
and oriented suchthat the vertical stackwas in the updrift
directionof the longshorecurrentto minimizeinterferenceby
the sled structure. The ems offsets were determined
in situ and
•gN
Eabs:
• (Pt- Or)2,
(13)
wherePi is modelprediction,0 i is the observedquantity,and
foundto berepeatable
to withinI cms-•. Thesledorientation N is the numberof observations.
The relativepercenterror is
was determinedusinga digital compassmountedon the sled
calculatedby
GARCEZ FARIA ET AL.' UNDERTOW
OVER A BARRED BEACH
17,003
(a)
1.5
(14)
•o• = 100
and weights the difference between model predictionsand
observationsagainsta measureof expectationthat is representedby the observation.This statisticis not well behavedfor
small 0,, which sometimes is the case for observed mean
cross-shore
velocities.
Thus measurements
smaller
..,,•,_•• _•
--
than the in
situdetermined
offsets
(_+1 cms-•) areexcluded
from(14).In
1•0
section4.1, the optimized wave transformationmodel is describedfirst and is then usedto predict the surfacemassflux,
setup, and vertical profilesof the undertow,includingcontributions
4.1.
Model
/
/
2•0
Cubic spline
3•0
4•0
(b) 0-
from surface rollers.
Wave
Transformation
The rmswaveheightis approximated
by Hrms = •,
where 0-2 is the variance calculated from the surface elevation
time series.Surfaceelevationwas calculatedby Fourier trans-5
forming a 1 hour pressurerecord, applyinga linear wave theory transferfunctionto the complexFourier amplitudesin the
-6
1•0
2•0
3•0
4•0
frequencydomain, band-passfiltering by zeroing coefficients
CROSS-SHORE DISTANCE (m)
outsidethe range of interest (0.05 Hz < f < 0.5 Hz), and
inversetransformingto obtainthe surfaceelevationtime series. Figure 2. (a) Lippmann et al.'s [1996] "worst case" model
The sensitivityof the models for surfacemassflux, setup, prediction of Hrms versuscross-shoredistance(dash-dotted
and undertow to errors associated with the use of a random
line) comparedwith cubicsplineinterpolation(solid line) of
wave transformationmodel [Lippmannet al., 1996] is investi- observations
(asterisks)for the firstrun of October11. (b) The
gated by comparingthe model resultswith resultsusingmea- bottom profile.
suredwave heightsinterpolatedwith cubicsplines.The differencebetweenmodel output and data interpolationmethodsis
A plot of measuredversusmodeled Hrms for the 3 days
shownin Figure 2 for the first run of October 11,whichhasthe
analyzed
(October 10-12) showsgoodagreement(Figure 3).
largestsr½
• (7%).
The mean sr½•is 5% for all runs, with largesterrors at any
cross-shorepositionwithin 12%. Best fit valuesfor 7 and
are summarizedin Table 1. In general, the transformation
a) 10 OCT. ,0.5m/•
o - TROUGHLEVEL
model representsmeasuredwave heightswell.
4.2.
i
i
i
Mass
Flux
The inferred surface massflux is obtained by numerically
integratingthe observedcross-shoremean currentsfrom the
bottom to the trough level and is comparedwith model estimates (equation (4)). Here qr is determinedfrom the cali-
-2
-4
Surface
•
b) 11 OCT.
1.4
/'0 0
//o ø
1.3
-4
//
1.2
c) 12 OCT.
1.1
/
// 00 •/•
7 0 //
-2
E0.9
-4
Cross-shore Distance(rn)
Mass flux inferred
from
the measured
0
//
c)//
0.7
Figure 1. Measured (asterisks) and rms best fit modelpredicted(equation(10)) verticalprofilesof meancross-shore
return currents(heavy line) superposedon bottom profiles
with mean troughlevel indicatedby open circlesfor the entire
periodbeingexamined.Modelsrunswere performedfor each
profile on the transect as they were measured at different
times.
/
E0.8
300
undertow
was
usedalongwith optimal coefficientsfor the wave model and
constanteddyviscosityin (10).
0.6
0.5
0.4
.,
0.4
0'.6
0.8
1
1.2
1.4
Hrms mea (m)
Figure 3. Predicted(Lippmannet al.'s [1996] model) versus
observedHrms. The solid line representsperfect agreement,
and the dashedlines representthe _+10%error bounds.
17,004
GARCEZ FARIA ET AL.: UNDERTOW
Table 1. Best Fit Model Parametersfor the Lippmann et
al. and Lippmann and Thornton Models
Wave
Model
Day
Transformation a
Run
7
Error, %
Roller b
B
½
Error, %
10
10
1
2
0.33
0.32
2.3
4.9
0.70
0.60
1.3
1.6
2.0
3.8
10
10
10
10
10
11
11
11
11
11
11
11
11
12
12
12
12
12
12
12
3
4
5
6
7
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
0.32
0.32
0.32
0.31
0.31
0.32
0.33
0.33
0.33
0.32
0.32
0.32
0.31
0.34
0.33
0.34
0.34
0.34
0.35
0.34
0.33 c
4.5
4.9
4.1
4.1
5.0
7.0
4.7
4.3
3.9
4.8
4.1
5.5
6.5
5.9
5.1
5.2
4.1
4.6
5.8
5.8
4.9 c
0.60
0.70
0.65
0.75
0.75
0.85
0.65
0.70
0.65
0.60
0.85
0.70
0.85
0.90
0.80
0.80
0.80
0.75
0.65
0.95
0.75 c
1.2
1.1
1.7
1.7
1.9
1.4
1.7
1.1
1.2
1.7
1.2
1.9
1.8
1.3
1.6
1.3
1.1
1.3
1.7
1.4
1.3 c
3.5
5.2
4.2
4.7
4.3
7.5
6.5
7.3
5.7
3.8
7.3
3.4
5.0
3.8
7.8
9.0
8.2
7.8
5.3
7.0
5.6 c
aLippmannet al. [1986] model.
bLippmann
andThornton(submitted
manuscript,
1998)model.
CMean values.
OVER A BARRED
BEACH
For the 22 profilesexamined,best fit valuesof B range from
0.60 to 0.95 and of qtrangefrom 1.1 to 1.9 (Table 1), and the
error betweenthe observedand modeledpercentageof breakers is Src•= 5.6%. An exampleof the observedand predicted
percentageof breakersis shownin Figure4 for the third run of
October12, whichhasthe largestrms error (9%).
Inferred and predictedsurfacemassfluxesusinglinear wave
theory are compared for the entire ensemblein Figure 5.
Becauseof significantchangesassociatedwith wave-breaking
characteristics,
the data have been divided into four regions:
seawardslopeof the bar, shorewardslopeof the bar, trough,
and foreshore.The Srelbetweenobservations
and predictions
using only q,•, given by linear theory, is 40%. Including the
massflux contributionfrom wave rollers, qr, improvesthe
overallagreement(erc•= 28%). Largestvaluesof qr occurat
the shorewardslopeof the bar (Figure5b), wherewavebreaking is mostintense.In this region,qr is, on average,72% of
but canbe aslargeas 144% (fourthrun of October12). Thisis
in accordance
with earliermodelstudies[Svendsen,
1984;Dally
and Brown, 1995]but contraryto Masselinkand Black [1995],
who contendon the basisof the resultsfrom field experiments
at two near-planarbeachesthat the roller contributionto the
masstransportis of secondaryimportance.
The impact of usingnonlinearstreamfunctionwave theory
[Dean, 1974] to calculateq,• is examinednext as it was shown
to give more accurateresultswith monochromaticlaboratory
wavedata [DallyandBrown,1995].For the field data examined
here, there is little difference between the use of linear or
streamfunction theory,with linear wave theory givingvalues
on average8% larger (Figure 6).
Errors in surfacemassflux predictionscouldalsoarisefrom
brated Lippmannand Thornton (submittedmanuscript,1998)
model, and qw is determinedfirst usinglinear wave theory and the misfit in estimatingwave heightswith a transformation
then usingnonlinear theory for comparison.Predictedqr are model.The rms relativeerror in massflux calculatedusingthe
dependenton the percentageof breakingwavesin this model. Lippmannet al. [1996] model-predictedHrms comparedwith
(a)
(a)
Seaward
Slope
of
the
Bar•
600-
500
o
Shoreward
Slope
of
the
Bar
•
Trough
•
Foreshore • ++ +
+
Data
400
300
o
w0.5
LT97 Model
200
,,•
xX.•
ø"xx
lOO
z
ß
o
0
o
(b) o
1 O0
200
300
400
500
600
(b)
o: 600
-1
-• 500
O
,,+400
+
•
•,•..300
• 200
-5
-6
• 100
a.o
CROSS-SHORE DISTANCE(m)
Figure 4. (a) Lippmann and Thornton (submittedmanuscript,1998)modelpredictions(solidline) andobserved(open
circles)percentageof wavesbreakingversuscross-shore
distancefor the third run of October12. (b) The bottomprofile.
a.
0
0
1•)0
200
300
400
500
600
INFERRED (SLED DATA)
Figure 5. (a) Measured(pUrh,) versuslinear wave theory
predictedsurfacemassflux (q,•, only)for the entire ensemble
of 22 runs(upperpanel). (b) The effectof includingcontributionsfrom wave rollers (q,• + qr)'
GARCEZ FARIA ET AL.' UNDERTOW
usingthe cubic splineHrms measurementsfor all runs is 9%,
with a maximumrelativeerror of 19.5% (third run of October
12). These resultssuggestthat the surfacemassflux is not
overly sensitiveto the choiceof the random wave transformation model. The differencesin observedand predictedundertow are not believedto be associatedwith large-scale,threedimensionalcirculation cells, as measured bathymetry was
essentiallyuniform alongshoreduring the period being analyzedand no qualitativeevidenceof stationaryrip currentswas
observed
4.3.
OVER A BARRED BEACH
("•
501
40f.
i '\
30-•i
./ •
•20•I! /''t / "
10I•
I •, / ,
-I \•
•,.
Setup
'
'-'Radiation
Stress
--Roller
--Setup
Mean
Momentum
Flux
,. ",.
•
i ',
..'-•.........
,,'•'
•-10
in the video.
-3ø1;
•/ ;,'
Setupis calculatedusinga finite centereddifferencemethod
to solve(12) numerically,with the conditionthat the setupis
assumedto be zero at the most offshoregrid point. Contributionsfrom eachterm of (12) are examinedfor the fourth run
of October 12 (Figure 7), which corresponds
to the most energeticperiod.The cross-shore
gradientof the momentumflux
associatedwith wave rollers is calculatedusingthe Lippmann
and Thornton (submittedmanuscript,1998) model,the radiation stressterm is calculatedusinglinear wave theory, and the
-40
5-5050 200 250 3bo
.00 .50 500
(b)ø/'
•-2
••.,•
momentum
fluxof thecurrent(OUr
2) isestimated
byapplying
cubicsplinesto measureddepth-integratedcross-shoremean
return flow. The largestcontributionsare due to the radiation
.'
1'7,005
150
200
as0 a60 ai0
460 4i0
s00
CROSS-SHORE DISTANCE (m)
Figure 7. (a) Termsof the momentumbalanceequation(12)
stress and roller terms. The momentum flux of the current,
versus cross-shore distance for the fourth run of October
althoughgenerally an order of magnitudesmaller than the
otherterms,cannotbe neglected,asit canbe of the sameorder
as the sum of the larger terms.
The effects on setup using contributionsfrom the roller
momentumflux and usingnonlinearwave theory to calculate
the radiationstressare examinedwith one examplefrom each
day (Figure 8). The effect of the roller is to redistributemomentumlaterallyinto the troughof the bar, shiftingthe point
where the setupbeginsonshore[Nairn et al., 1990].This shift
hasa significantimpacton the setup/setdown
profilewithin the
surfzone.Nonlinear streamfunctionwavetheorypredictsless
radiationstressand as a result lesssetdown,althoughthe use
of nonlinearwave theory does not appear to alter the setup
profilesignificantly
(Figure8).
(b) The bottomprofile.
4.4.
12.
Undertow
4.4.1. Cross-shorevariation of the eddyviscosity. An implicit assumptionof the closuremodel used in the undertow
solutionis that the eddyviscositycoefficient/zis proportional
to turbulenceintensity.Conceptually,large cross-shore
varia-
--
0'02ta
)Oct
10
-1240
_• _
LWT
....
STF
+Roller
-- LWT+ Roller
0.01
500
.... -,-,,._.-_
O]
:,•
.... _...............
+Seawa'rdSIopeof'the
Bar
'
0.021
b)
Oct
11
-1202
-0.01
450-
.....
ßShoreward
Slope
of
the
Bar
400350-
m 300o
•- 250+
001'
i,,,200-
0.04
-[!•c)p_c.,t
12-1227
1500.02
100500
,
0
100
200
300
400
500
LWT + ROLLER
Figure 6. Comparisonbetweenpredictedsurfacemassflux
(qw + qr) givenby linear (LWT) and streamfunction[Dean,
1974](STF) wavetheories.The dashedline representsperfect
agreement,and the solid line representsa linear regression
with a slopeof 0.92.
a00 400 s00
760
CROSS-SHORE DISTANCE (m)
Figure 8. Setupcalculatedusinglinearwavetheory(dashed
line), streamfunctiontheorywith roller (dash-dotted
line), and
linear wave theorywith roller (solid line) versuscross-shore
distancefor the fourth run of each day.
17,006
GARCEZ FARIA ET AL.: UNDERTOW
Table 2.
ObservedWave, Tide, Undertow, and Viscosity
Significant
Wave Height
Day
10
10
10
10
10
10
10
11
11
11
11
11
11
11
11
12
12
12
12
12
12
12
OVER A BARRED BEACH
Peak
Peak
Frequency Direction
Run
Ho,m
fp, Hz
1
2
3
4
5
6
7
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
1.70
1.81
1.81
1.68
1.68
1.44
1.44
2.11
1.88
1.75
1.70
1.70
1.70
1.60
1.60
1.91
2.29
2.29
2.32
2.32
2.35
2.35
0.171
0.162
0.162
0.152
0.152
0.162
0.162
0.142
0.142
0.142
0.142
0.142
0.142
0.142
0.142
0.162
0.142
0.142
0.142
0.142
0.142
0.142
a0,deg
44
38
38
36
36
24
24
18
16
17
18
18
18
16
16
18
12
12
10
10
10
10
Depth-Averaged rmsBestFit
Undertow
EddyViscosity
Tide
0.40
0.79
0.98
0.77
0.44
0.04
-0.12
0.22
0.64
0.86
0.92
0.85
0.62
0.32
0.03
0.05
0.34
0.71
0.83
0.84
0.69
0.42
Uobs,
m s-•
/x,mes-•
0.06
0.08
0.16
0.14
0.05
0.05
0.05
0.10
0.12
0.19
0.19
0.05
0.08
0.07
0.15
0.13
0.20
0.30
0.30
0.20
0.07
0.10
0.500
0.500
0.058
0.017
0.500
0.500
0.500
0.240
0.250
0.033
0.049
0.063
0.500
0.500
0.140
0.130
0.240
0.032
0.036
0.035
0.020
0.500
Best-fiteddyviscosity
valuesare basedon usingsplinedHrmsand observedmeanundertowto specify
mass transport.
tion of/x is expectedthroughoutthe surf zone over a natural
(F(x))
U(Z) : Wre
f qS(z),
(15)
barred beach associatedwith significantchangesin wave
breaking-generated
turbulence.Hainesand Sallenger[1994],
in an earlierfield experimentlimited to 3 emsoverthe vertical where Uref is a referencevelocity that could be either the
at this same beach, used a different model for the vertical depth-averaged
undertow[e.g.,Stiveand Wind, 1986] or the
variation of the undertow and found that the best fit/x for each velocityat the top of the BBL [e.g.,Svendsen
et al., 1987].A
locationvariedby more than an order of magnitudeacrossthe nearlyuniformverticalprofileof undertow,asobserved
in the
surfzone(/x = 0.0055- 0.075m2 s-1).Smithetal. [1992],on innertroughandseawardslopeof the bar (Figure1), requires
the otherhand,applieda constant/xacrossthe surfzone(/x = either no forcing(F = 0) or an infiniteeddyviscosityto be
0.05m2s-•) andfoundreasonable
agreement
withtheDelilah reproducedcorrectly.An exampleof the spatialdistributionof
data.
eachterm andof the total F(x) for the firstrun of Octoberis
To investigatethe cross-shore
variationof the eddyviscosity, shownin Figure 9. The dominanceof the setupgradientterm
bestfit/x (Table 2) were calculatedusing(10) by minimizing is evident,andhencethe modelpredictsoffshoredirectedflow
both the /•absand /•relin (U(z)) at eachcross-shore
position throughoutthe entire surf zone, which agreeswith observaoccupiedby the sled(Figure1). Observeddepth-meanunder- tions. The modeled total forcing,althoughsmall offshoreof
tow velocitiesUr and observedsplinedrmswave heightswere the bar and within the troughwhere the undertowwasnearly
used in the calculations to isolate effects due to the choice of
uniform,wasnevernil within the cross-shore
transectoccupied
/x. For all 22 runs,(Gabs)
= 2.2 cm s-1 ((erel)--' 19%) with by the sled,whichwasalsothe casefor all the otherrunsduring
maximum
ea•,,•
- 6.1cms-1 (erc•- 44%)forthefourthrunof the period being analyzed.Minimum rms errors over these
October 12. Here (eab,•)is lessthan the accuracyof the obser- regionsare obtainedfor largeeddyviscosity
values(Table2),
vations.Using modeled rms wave heights[Lippmannet al., witha maximum
cutofflevelarbitrarily
setat/x = 0.5m2 s-•.
1996],rabs-- 2.5cms-• (ertl-- 22%),whichindicates
thatthe Theseunrealisticallylargevaluesof/x are associated
with the
verticalprofile of the undertowis not sensitiveto errorsasso- sensitivityof the eddyviscositymodel to small errorsin the
ciated with the use of a random wave transformation model.
dynamicalforcing,which is calculatedas the differencebeUsing the surfacemassflux model to estimateUr resultsin tweentwolargenumbers(setupandradiationstress
gradients).
(ea•,•)-- 5.1 cms-• and(ertl) - 48%,whichsuggest
thatthe
The parameterizationfor the eddyviscosityis of the form
largesterrors in the vertical profile of mean undertoware tx • l C, where l and C are characteristiclengthand velocity
relatedto the failure of the existingmodelsto predictcorrectly scales[Battjes,1975].Severaldimensionally
consistent
paramthe surfacemassflux in breakingwaves.
etersfor/xarefoundin theliterature,
suchas(H/2) 2f [ThornModels for the vertical structure of the undertow
that use an
ton,1970],h(D/p)•/3[Battjes,
1975],hXf• [Stive
andWind,
eddyviscosityclosureto solvefor the turbulentshearstress 1986],and [Hainesand Sallenger,
1994],
(Reynoldsstress),independentof the choiceof boundaryconditionsand depthdependenceof the eddyviscosity,
resultin a
generalsolutionof the form
GARCEZ
FARIA
ET AL.: UNDERTOW
where D is dissipationassociatedwith wave breaking.An attemptwasmadeto relatebestfit/• with theseparameters,after
excludingfrom the ensemble,values of /• that reached the
cutofflevel (Table 2). No statisticallysignificantcorrelationat
the 95% confidencelevel was obtainedbetweentheseparameters and best fit/•. On the bar crest and shorewardslope of
the bar the mean undertowprofile assumes
a parabolicshape,
and best fit eddy viscosityapproachesa constantvalue (/• •
OVER A BARRED
BEACH
17,007
1
Constant
Quadratic
Linear
0
0.04m2 s-l).
4.4.2.
Vertical
variation
oftheeddy
viscosity.
Here
the
impact
bothof a depthvarying/•z
andof different
boundary
conditionson the verticalstructureof the undertoware inves-
tigated.It isexpected
fromlaboratory
measurements
[Tingand
Kirby,1994;CoxandKobayashi,
1997]that/•z shouldincrease
from the bottomwith a maximumnear the surfaceand with
increasing
levels
ofwave
breaking-generated
turbulence.
Sew •o0
eralmathematical
formulations
forthevertical
variations
of/•
\
1
were investigated,and the three solutionsthat bestagreedwith
data (constant(10), linear (A6), and quadratic(All) /• are
describedin the appendix.A comparisonamong these solutions givingthe smallestoverall rms errors for the entire ensembleis shownin Figure 10 for the two stationswithin each
day that have largestobservedverticalstructureof the undertow. The absoluteand relative percent errors for the entire
ensembleof data for optimally fit valuesare for /• constant
0
0.05
0.1
0
0.05
0.1
Normalized velocity ( m / s)
(0.014m2 s-• and14.2),/•zlinear(0.018m2 s-j, 14.5),and/•z
quadratic
(0.018m2s-•, 14.8).Surprisingly,
thesmallest
errors Figure 10. Comparisonbetweenmeasuredundertow(aster-
are for/• constant,thoughthe differencesbetweenusinglinear
or quadratic/• are small.Theseresultsconfirmearlier modeling comparisons
with laboratorydata showingthat a depthdependenteddy viscositydoes not substantiallyimprove the
descriptionof the vertical structureof the mean undertow
[Svendsen
and Buhr Hansen,1988;Nadaokaet al., 1989].
The influenceof the boundaryconditionchoiceon the ver-
isks)and predictionsusingdifferentformulationsfor the vertical variation of the eddy viscositywith optimally fit coefficients (constant, solid line; linear, dash-dotted line; and
parabolic,dashedline) for the (a) third and (b) fourth runsof
October 10, the (c) third and (d) fourth runs of October 11,
andthe (e) third and (f) fourthrunsof October12.Model runs
usesplinedHrms and observedmeanundertowto specifymass
transport.
(a) 10
I
- Radiation
Stress Gradient
-- Setup Gradient
,, Mean Momentum
•
•
5
Flux Gradient
--Total
[F(x)]
o
•
0
o
z
=• -5
-10
150
200
(b)0
2•0
'
'
360
350
4()0
,
tical structureof the meanundertowin the middlelayer (using
/• constant)is investigatednext. Comparisonbetweenpredictions by (10), which usesthe conservationof massover the
vertical and the stressat the trough level as boundaryconditions,andby (11), whichusesa no-slipconditionat the bottom
to replacethe stressat the troughlevelasthe secondboundary
condition,is shownin Figure 11 for the samestationsused in
Figure 10. Again, no significantimprovementin the total rms
errors betweenobservationsand predictionsby these two solutionsis obtained,althoughthere are noticeabledifferencesin
the predictedcurrentsby each model as a functionof depth.
Equation(10) showsbetteroverallcomparison
with data,while
(11) represents
betterthe structureof the flowin the lowerhalf
of the water column.Although modelscouplingthe middle
layer and BBL flows(equation(11)) are expectedto providea
more realisticdescriptionof the undertowstructurecloseto
the bed, their applicabilityto field conditionsis still limited by
the lack of velocitydata near the bed to constraintheir free
parameters.
I3-5•
150
200
250
300
3•0
CROSS-SHORE DISTANCE (m)
400
5.
Conclusions
The predictedspatialdistributionsof mean cross-shore
curFigure 9. (a) Undertow dynamicalforcing terms (equation
(6)) versuscross-shore
distancefor the firstrun of October10. rent (undertow)over a barredbeachare comparedwith field
(b) The bottom profile. The verticaldashedline indicatesthe observationsduring the Duck94 experimentto quantify the
relativeimportanceof contributionsfrom the varioustermsin
cross-shore
positionoccupiedby the sled.
17,008
GARCEZ
FARIA ET AL.: UNDERTOW
Equation 10
OVER A BARRED
BEACH
wave breakingwas observed,the mean undertowprofile assumesa parabolicshape,and bestfit eddyviscosityapproaches
a constant
value(Ix • 0.04m2 s-•).
Equation 11
The effectof usingdifferentboundaryconditionsto solvefor
the vertical structureof the mean undertowis investigatedby
includingrandomwavesin the formulationsof Stiveand Wind
[1986] and Svendsen
et al. [1987] by ensembleaveragingover
the wavedistribution(equations(10) and (11)). No significant
difference
in the total
rms errors
between
observations
and
predictionsis found between these two solutions.Despite the
better representationof the undertow structurecloseto the
bed obtainedby using(11), whichcouplesthe middlelayerand
BBL flows,its applicabilityto field conditionsis still limited by
the lack of velocity data near the bed to constrainits free
parameters.
In the inner trough and seawardslope of the bar the measurements
show almost no vertical
structure
for the mean un-
dertow, which would require a local balance between setup
gradientand wave forcing (F = 0) or a large eddyviscosity
(p, -• oc)to be properly modeled.Although small, modelpredicted forcing was never nil within these regions,so that
unrealisticallylarge values of Ix are required to model the
observeduniformprofile of the undertow.Theseunrealistically
largevaluesof Ix are associatedwith the sensitivityof the eddy
0
0.05
O. 0
0.05
0.1
viscositymodel to small errors in the dynamicalforcing,which
Normalized velocity ( m / s)
is calculated as the difference between two large numbers
(setupand radiationstressgradients).
Figure ll. Same as Figure 10 but comparing predictions
The absenceof both setupmeasurementsand observations
givenby (10) (solidline) and (11) (dashedline).
of currentscloseto the bed is a major limitationof the present
study.Setupis an integralmeasureof the dynamicresponseof
the mean water level to cross-shoregradientsof momentum
the cross-shoremomentum equation and to identify physical fluxes,and its gradient is an important driving force for the
mechanismsnot yet incorporatedin existingnearshoremodels. undertow within the surf zone. The lack of current measureThe largest discrepanciesbetween model predictionsand mentscloseto the bed preventsa quantitativeevaluationof the
observationsof undertow velocities appear to be associated effect of applyingdifferent boundary conditionsto solvefor
with the failure of existingmodels to predict correctly the the vertical profile of the undertow.
surfacemassflux under breakingwavesjust insidethe bar. The
surfacemassflux modelbasedon potentialflow theorycontri- Appendix
butionsfrom wave rollers is comparedwith field observations
The solution for the case of an eddy viscositythat varies
obtained by integratingthe measuredreturn flow from the
linearly with depth, as suggestedby Okayasuet al. [1988], is
bottom to the wave trough level and assumingconservationof
derivedin more detail. FollowingStiveand Wind [1986], the
mass over the vertical. It was found that the roller contribustressat the trough level and conservationof massover the
tionsto the masstransportcanbe largerthan the contributions
vertical are used as boundary conditions.The final solutions
from the organized wave motion when energetic breaking
for other depth-dependentformulationsfor the eddyviscosiwaves are present. Surface mass flux predicted using linear
ties investigatedin this studyare alsoincluded.
wave theory combinedwith contributionsfrom wave rollers
were 8% larger comparedwith the solutiongivenby nonlinear A1. Linear: Ptz= c• + •z'
stream function wave theory.
In this solution a normalized
vertical coordinate is introSetup (setdown)is a dominantforcingmechanismfor the duced:
undertow and is calculated from the depth-integratedand
(z + h)
time-averagedcross-shore
momentumbalance.It is found that
despitethe dominanceof the setup (setdown)and radiation
ht
'
stressgradientsin this balancethe contributionfrom the convective accelerationof the mean current is significantduring The primes are omitted hereafter for simplicityof notation.
energeticwave events.It is also shownthat the inclusionof Assumingthe undertowforcingto be constantover depth and
usingthe eddyviscosityclosurefor the turbulent shearstress,
contributions
from wave rollers results in an onshore shift of
e) , /i•
f),
,
a solution for the vertical
distribution
of the mean undertow
is
the point where the setupbegins,whichhasa significantimpact
obtainedby integrating(6) over depthto give
on the dynamicalbalanceof forceswithin the surf zone.
The vertical structureof the undertow is modeled using a
oS(z)
turbulent eddy viscosityclosure,and depth-dependenteddy
ptzz Oz = •'(z)= F(x)z+ C•,
(A1)
viscosityformulationsare found not to improvethe agreement
with the field data comparedwith usingconstanteddyviscosity. where C• is an integrationconstantthat is solvedby evaluating
On the bar crestand shorewardslopeof the bar, where strong (A1) at the bottom (z = 0)'
GARCEZ
FARIA
ET AL.: UNDERTOW
C• = ?•,
(A2)
Integrating(A1) a secondtime and using(A2) gives
OVER A BARRED
BEACH
17,009
½
Nl(Z)/3tanh
l(2•z
- ]3
) - 2-In
O [oz+ /3(z- z2)],
(A12)
U(z): F(x)[
•- z-• c•
ln(a+ /3z)
2/3z
- /3) (A13)
+ 9•-In(a+ /3z)+ C2,
(A3)
N• = ½d,- (2a + /3d,)tanh• 2/3d
t - /3)
where C2 is an integrationconstantthat is solvedby applying
conservationof massover the vertical to give
-2atanh-•
(•)- • In[a+•(dt-dt2)]+N4,
C2=U,. F(x)
• (7-•Cø
d,oz) -•C0,
•-(z) (A4)
(A14)
N2(z): 2/3 tanh • --
q,
'
where
C0= • In
+ In(a+ fid,)- 1. (A5)
N4:(2•dt_[3)
tanh-,
(213d'-[3)
•
½ -/3tanh-(•)
½In a+/3(d,-dt
a
'
2)]
Substituting(A4) in (A3), the final solutionfor the vertical
profile of the undertowis givenby
+
+
p•-
- c0]
(A6)
,
M•(z) 4d•/2
15 •2M3
F(x)•
U(z)= U•+ •
M2(z)+•-M3
,
(A7)
where
2
a
5z3/2_
2A 2
2a 3
•/2
In (• + Bz•/2) (AS)
,
M2(z) = 2z•/2- 2a In (a + •z•/2),
M3= • 1 {4•3
••t •3/2
+ a•2dt+
2a
to better
of currents close to
constrain
this value.
Acknowledgments. This researchwas funded by the Office of Naval Research,Coastal SciencesProgram, under contract N00114-95AF-002. The authorswish to expresstheir appreciationto all those
who participated in the Duck94 experiment,particularly the staff of
the U.S. Army Field ResearchFacilityunderthe directionof B. Birkemeier. The cross-shore
array of pressuresensordata was providedby
S. Elgar and R. T. Guza. In addition,specialappreciationis expressed
to Jim Stockeland R. Wyland, Naval PostgraduateSchool,for their
role in the acquisitionand processingof wave and current data. The
authors thank Jurjen Battjes for helpful suggestionsand the anonymous reviewers
for their constructive
criticisms.
(A9)
-
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- a[2
In(a)+1]},
(A10)
where • > 0.
•.
field conditions
The solutionwithin the middle layer doesnot, however,change
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Quadratic:
+
(A16)
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:
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(A15)
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(ReceivedAugust 23, 1998;revisedDecember 14, 1999;
acceptedMarch 22, 2000.)