JOURNAL OF GEOPHYSICAL
RESEARCH, VOL. 104, NO. C4, PAGES 7683-7693, APRIL 15, 1999
Directional spreading of waves in the nearshore
T. H. C. Herbers
Department of Oceanography,Naval PostgraduateSchool,Monterey, California
Steve Elgar
Schoolof Electrical Engineeringand Computer Science,WashingtonState University,Pullman
R. T. Guza
Center for CoastalStudies,ScrippsInstitution of Oceanography,La Jolla, California
Abstract. Observationsof surfacegravitywavesshoalingbetween 8-m water depth and
the shorelineon a barred beach indicate that breaking resultsin an increasein the
directionalspreadof wave energy,in contrastto the directionalnarrowingwith decreasing
depthpredictedby refractiontheory(Snell'slaw). During low-energywave conditions,
when breaking-inducedwave energylossesover the instrumentedtransectare small, the
observedmean propagationdirection and spreadabout the mean both decreasewith
decreasingdepth, consistentwith the expectedeffectsof refraction.Nonlinearity causes
high-frequencycomponentsof the spectrumto becomedirectionallyalignedwith the
dominantincidentwaves.During high-energywave conditionswith significantwave
breakingon the sandbar, the observedmean directionsstill decreasewith decreasing
depth.However,the observeddirectionalspreadsincreasesharply(nominallya factor of 2
for valuesintegratedover the swell-seafrequencyrange)betweenthe outer edgeof the
surf zone and the crest of the sand bar, followed by a decreasetoward the shoreline.
Observationson a nonbarredbeach also showdirectionalbroadening,with spreads
increasingmonotonicallyfrom the outer edge of the surf zone to a maximumvalue near
the shoreline.Although the mechanismis not understood,these spatialpatternsof
directionalbroadeningsuggestthat wave breakingcausessignificantscatteringof incident
wave energyinto obliquelypropagatingcomponents.
1.
ll q-12(gh) 1/2
Introduction
The propagationdirectionsof shoalingsurfacegravitywaves
changeowingto refractionby spatialvariationsin water depth.
If alongshoredepthvariations,currents,dissipation,and nonlinear effectsare neglected,then both the wave frequencyf
and alongshore
wavenumberl are conserved(Snell'slaw [e.g.,
Kinsman,1965]). As the water depth h decreases,the wave
incidenceangle • decreases,and in the shallowwater limit
l
0=• (gh)
1/2.
(1)
where # is accelerationof gravity.Directionalpropertiesof
shoalingwavesare alsoaffectedby nonlinearenergyexchanges
betweentriads of wave componentsthat obey the interaction
rules [e.g.,Freilichet al., 1990;Herbersand Burton,1997]:
fl + f2 - f3 = 0
(2a)
l• + 12- 13= 0
(2b)
In shallowwater the suminteractionof two wavecomponents
(fl, ll) and (f2, /2) with small incidenceangles01 and 02
drivesa higher-frequency
(f3 = fl + f2, 13 = ll q- /2) wave
with a small angle 03 that is equal to the frequency-weighted
averageof 01 and 02 [equations(1) and (2)]:
Copyright1999 by the American GeophysicalUnion.
Paper number 1998JC900092.
0148-0227/99/1998JC900092509.00
fl
f2
03--flq-f2 2= --fl q-f201q-fl q-f202
(3)
Field [Freilichet al., 1990] and laboratory[Elgaret al., 1993]
measurements
confirmthe effectsof refraction[equation(1)]
and nonlinear interactions[equation (3)] on nonbreaking
shoalingwaves.
Little is known about the effectsof wave breaking on the
directionalpropertiesof shoalingwaves.Video observations
show that incidenceangles decreaseas broken bore fronts
approachthe shoreline,qualitativelyconsistentwith refraction
[Lippmann and Holman, 1991]. However, initially straight
wavecrestssometimesappearto bend suddenlynear locations
separatingbreaking and nonbreakingportions of the wave
crest.In a natural surf zonewith irregularspatialfluctuations
in wave-breakingpatterns,theseapparentlyrandom direction
changesmay causea broadeningof the directionaldistribution
of wave energy, even on beacheswith straight and parallel
depth contours.Velocity measurementsin the surf zone of a
nearly plane beach indeed indicate a significantlystronger
alongshorecomponentof the waveorbitalmotionthan is predicted by Snell's law [Guza and Thornton,1985], consistent
with a broadeningof the directional spectrum.Analysis of
arrayobservations
showsthat a shore-parallelsandbar located
within an energetic surf zone traps obliquely propagating
wavesin the incidentswellfrequencyband [Bryanand Bowen,
1996].However,the generationmechanismand importanceof
thesebar-trappededgewavesare unknown.
In the presentstudythe transformationof directionalwave
7683
7684
HERBERS
ET AL.: DIRECTIONAL
SPREADING
OF NEARSHORE
WAVES
expectedeffectsof refractionof incidentwavestowardnormal
_ (o)
incidenceand the nonlinearexcitationof similarlyaligned
[equation (3)] higher-frequencyharmonics.Inside the surf
zone, mean incidenceanglesremain small while directional
spreadsincreaseat all frequencies,
suggesting
that wavebreaking causesdirectionalscatteringof wave energy.The role of
ß 50 September
-D
12
the barredbeachtopography
in the directionalbroadening
of
wavesin the surfzoneis discussed
by comparison
with obser-
October
200
500
400
vations from a nonbarred beach in section 5. The results are
summarized
in section 6.
(b)
•'
103
2.
102
Wave measurements
were collectedduringSeptemberand
October 1994 on a sandybeach near Duck, North Carolina,
that is exposedto swellarrivingfrom the Atlantic Ocean.A
,I
200
300
400
3o
•
25
_
Field
Data
collocatedpressuretransducer,
bidirectionalelectromagnetic
currentmeter, and sonaraltimeter(to determinethe seafloor
location)were mountedapproximately
50 cm abovethe seafloor at 14 positionsalonga 350-m-longcross-shore
transect
extending
fromneartheshoreline
(minimumdepthabout1 m)
to approximately
6 m depth(Figurela [Elgaret al., 1997]).
Incident wave propertieswere estimatedfrom data obtained
with a coherentarray of 15 pressuregaugeslocatedin 8 m
depth [Long,1996],about400 m directlyoffshoreof the seaward end of the instrumented
-lO
200
.300
400
50
•
•
28
(d)
26
•
24
•
22
•
20
•
18
o16
•
14
transect.
Pressureand velocitydata were collectednearly continuouslywith a 2-Hz samplefrequency.
Spectraandcross-spectra
were estimatedat hourlyintervalsin the swell-seafrequency
rangeof 0.05-0.25 Hz from 51.2-min-longdata segments.
A
relativelywide frequencybandwidth of 0.029Hz wasusedto
ensurestability(about180degreesof freedom)of directional
momentestimatesand to reducevariationsin spectrallevels
resultingfrom phasecouplingbetweenincidentand reflected
wave components.
SurfaceelevationspectraE(f) were estimatedfrom the pressurespectrausinglineartheoryto account
for attenuation
over the water column. The data collection
periodincludesseveralstormsseparatedby calmperiods.At
the offshorearray,significant
waveheightsrangedfrom 0.2 to
4 m and mean wave directions varied between -40 ø and 50 ø.
The beachprofileis characterized
by a shore-parallel
sand
bar that is submergedabout 1.5-2.5 m below the mean sea
200
.300
400
surface(Figure la). The bottomslopeis approximately
1:100
seawardof the sandbar. Shorewardof the sandbar, the depth
cross-shore disfence x (m)
increasesby about 0.5 m into a relativelyflat trough that
Figure 1. Three examplesof the evolutionof bulk-integrated extendsto a steep(1:10)beachface.The crestof the sandbar
(over swell-seafrequencies)directionalwave propertiesobof the shorelineduring
served along a cross-shoretransect on a barred beach near waslocatedabout120-150m seaward
the
first
45
days
of
these
observations
(e.g.,cross-shore
locaDuck, North Carolina. (a) Water depth h. (b) Sea surface
heightvariance.(c) Mean propagationdirection0.... . (d) tion 220-250 m in Figure la) and then migratedabout50 m
Directional spread(ro.
farther offshoreduringa stormin mid-October[Gallagheret
al., 1998].
•
12
10
propertiesacrossa naturalbeachis investigated
with field data
from a densecross-shore
transectof collocatedpressuresensorsand currentmetersdescribedin section2. Althoughdetails of the directionalwavespectrumare not resolved,a mean
propagationdirection 0.... and a measureof the directional
spreadingof waveenergy(rocanbe estimated.Outsidethe surf
zone the frequency-integrated(section 3) and frequencydependent(section4) estimatesof 0.... and (ro showthe
3.
Directional Properties
A mean propagationdirection0.... and a measureof the
directionalspreadingof wave energy(ro (both functionsof
frequency)can be definedin termsof low-orderFourier momentsof the frequency-directional
wavespectrumE( f, O)
[equations(A3a) and (A3b)] [seealsoKuik et al., 1988, and
referencestherein]:
HERBERS
tan [20 .... (f)] •
ET AL.:
DIRECTIONAL
SPREADING
=dO
sin
20E(f,
O)
•dO
cos
20E(f,
O)
OF NEARSHORE
WAVES
7685
averagesof the mean propagationdirection 0....
directionalspreado-o(4b)
(4a) and
(4a)
•0.25Hz
f_nff0.25Hz
f_•df
d0.05Hz
tan [2 0.... ] --
dO sin 20E(f, O)
•'
df
dO cos 20E(f, O)
_•dO
sin
2[0-Omean(f)]E(f,
O)
o-2ø(f
)=
E(f)
(4b)
•0.25Hz
I_rr
d0.05Hz
df
Theseparameterscanbe estimatedfrom the autospectra(
C,,,) and cospectrum(C,,,) of the horizontalvelocitycomponentsu andv [see(Alc) and(Aid)] andhavea simplephysical
interpretation.The mean angle 0mean
, where
2C,v(f)
tan[20....(f)] = C,,(f)- Cv•(f)'
(5)
representsthe orientation(relative to shore normal) of the
major principalaxisof the fluctuatingvelocityvector (u, v)
[e.g.,Higginset al., 1981].The polarizationof the velocityfield
is givenby the corresponding
directionalspreadvariable o-o:
o-2ø(f
) = C,,,,(f
) + C•,•,(f
)
(6)
where u', v' are the velocity componentsrotated into the
principalaxescoordinateframe [e.g.,Batties,1972].For directionallynarrowwavefields(C,,,,,, << C,,,,), o-ois smalland
approximatelyequalto the half width of the directionaldistribution of wave energy(see appendix).For directionallybroad
wave fields, o-ois generallysmallerthan the directionalhalf
d0.05Hz
(7a)
•-
dO sin2 (0 - 0.... )E(f, {9)
•-
o-20
--
(7b)
f0
ø"-s•4z
dfE(f)
.05Hz
Three representativeexamplesof the cross-shoreevolutionof
{9mean
and o-oare shownin Figureslc and ld, respectively.The
beachprofilesin thesecaseswere similarwith a well-developed
sandbar at x • 250 m (Figure la). On September27, lowamplitudeincidentwaves(significantheightin 8-m depthH s =
0.7 m) resulted in nearly constantwave energy acrossthe
instrumentedtransect (Figure lb) and a narrow surf zone
confinedto the steepbeachface shorewardof the shallowest
instrument.Along the entire transect,{9mean
is closeto normal
incidence(Figure lc). The spreadso-oare alsosmall,decreasing slightly(from 16øto 12ø) towardthe shore(Figure ld). On
September30 at a lower tide stage (Figure la), waveswith
approximatelythe same offshore variance broke on the bar
crestand were attenuatedsignificantlyat the shorewardend of
the transect(Figure lb). The mean directiondecreasesowing
to refraction
from 27 ø at the most offshore sensor to 16 ø on the
width,with a maximum
valueof 2-•/2 rad (or 40.5ø) for an bar crestand is approximatelyconstantin the surf zone. The
isotropicdirectionalspectrum[equation(6) and Figure A1]. correspondingo-odecreasesslightlyfrom 16ø to 14ø between
Note that {9mean
and o-odescribethe directionalityof the flow x = 480 and 250 m, followedby a sharpincreaseover the bar
crest to a maximum value of 22 ø at x = 220 m and a decrease
field without assuminglinear wave motion and thus can be
used to characterizedirectionalwave propertiesboth inside to 16øatx = 160 m. Dramatic directionalbroadeningofwaves
in the surf zone was also observed on October 12 with more
and outsidethe surfzone.Equations(5) and (6) were usedto
estimate{9mean(f)and o-o(f) at all currentmeter locations. energetic(H s = 2 m), nearly normallyincidentwaves.While
iswithin 5øof normal incidencealongthe entire transect,
Estimatesfarther offshorein 8 m depth were obtained from 0mean
o-oincreasesfrom 18ø at the outer edge of the surf zone to a
the pressurearray measurementsusinga different t•chnique
maximumvalue of 27øon the bar crest,decreasingagainto 18ø
[Elgaret al., 1994,AppendixA].
at the shoreward end of the transect.
The largestsourceof error in the currentmeter estimatesof
The correspondence
betweensurf zone wave breaking and
{9mean
is uncertaintyin the probe orientationrelative to the
directional
broadening
is
demonstrated
in Figure2 by comparing
beach.Estimatesof this biaswere obtainedby comparingthe
the observedbulk directionalspreadwith the bulk cross-shore
currentmeter estimatesof {9mean
at a frequencyof 0.1 Hz (a
locations.The energyflux Fx
band that is usuallydominatedby incidentswell)with predic- energyflux at four cross-shore
tions obtainedby transformingthe 8-m depth pressurearray
estimatesof 0mean
with Snell'slaw to all current meter loca(8)
Fx = p#
dfcg(f)
dO cosOE(f, O)
tions. The differences
between the current meter estimates and
the predictionsaveragedover several-week
periods(when the
currentmeterswere in a fixedposition)indicatethat rotation
biasesof the different current meters are randomlyscattered
between_ 10ø.All resultspresentedhere incorporatebiascorrectionsbasedon theseaverageddifferences.Subsequentobservationswith accuratelyorientedcurrent meters(to be discussedelsewhere)confirmthe validityof thisprocedure,andin
anyeventthe biasesare smallcomparedwith the largerangeof
observed wave directions.
Current
meter orientation
errors do
f0.25Hz
I_nd0.05Hz
,r
where 9 is the densityof seawater,# is the accelerationof
gravity,andca is the groupvelocity,
wasestimated
frompressuresensor-currentmeter cospectrawith (Ala) and from the
offshorepressurearray data followingElgar et al. [1994]. The
fluxPx and spreadOovaluesshownin Figure2 are both
normalizedby the offshore array values and thus provide an
indicationof the fraction of energylost throughbreaking seaward of the observation
location
and the associated directional
not affect estimatesof the directionalspreado-o.
broadening
of thewavespectrum.
WhenPx > 0.7, i.e.,obVariationsof bulk directionalwave propertiesin the swell- servationson the outer edge or seawardof the surf zone, Oo
seafrequencyrangeare examinedhere with energy-weighted generallyvariesbetween0.5 and 0.9, indicatingrefractivenar-
7686
HERBERS
ET AL.: DIRECTIONAL
SPREADING
1.6-X
-- 0m
OF NEARSHaRE
.6x- 24m,
1.2
ß4
WAVES
.•.,•,
I
I
I I
II
I
I
I
I
I
10 -1
I
I I I III
•, * .
I
I
I
I I I II
10 -1
1
10-1
10 0
1.6
• 12
•
10
6
4
10-1
10 0
normalized energy flux
normalized energy flux
Figure 2. Frequency-integrated
directionalspreadversuscross-shore
energyflux (both normalizedby the
offshorex = 900 m values)at cross-shore
positions
x = 370 m (seawardof the bar crest),240m (on the bar
crest), 190 m (slightlyinshoreof the bar crest), and 145 m (at the toe of the beachface). Each asterisk
representsan estimatebasedon a 1-hour-longdata record.
rowingof the wavefield. In contrast,directionalbroadeningis
estimatesdecreasesgraduallyfrom 0.8 at x - 480 m to 0.7 at
evidentfor Px < 0.3 observations
wellinsidethesurfzone, x = 160 m near the toe of the beach face, followedby an
with frotypicallyrangingfrom 0.9-1.4. A trend of increasing increaseto 0.8 at the most shorewardsite, which is usually
directionalspreadwith decreasingenergyflux is clearestin the within
thesurfzone.WhenPx,20Sm
< 0.3,energy
losses
dueto
vicinityof the sandbar crest(x = 190 and 240 m) but is also wave breakingon the bar are large and the mean value of fro
observedwith more scatterat x = 370 m (usuallyoutsidethe estimates increases over the bar to a maximum value of 1.1 at
surf zone) andx = 145 m (often well insidethe surfzone). x - 190-240 m, decreasingto 0.9 near the shoreline.The
Differences
in the cross-shore evolution
of directional
standarddeviationsof the fro estimatesare smaller than the
spreadswith nonbreakingandbreakingwaveconditionson the differencesbetweenmean valuesin nonbreakingand breaking
sandbar are illustratedin Figure 3a. Breakingconditionsare conditions.
characterized
herebytheobserved
normalized
energy
fluxPx
The velocitymeasurements(upon which the cr0 estimates
at x - 205 m just inshoreof the crestof the sandbar. When are based)maycontainnongravity
wavemotionssuchaswaveturbulence.The ratio R betweenpressureand
Px,20Sm
> 0.7,waves
propagate
overthebarwithlittleorno breaking-induced
breaking and intense breaking usually does not occur until velocityfluctuations,integratedover the swell-seafrequency
near the beach face. In these casesthe mean value of fro range and normalizedby the linear theory transferfunction
HERBERS
ET AL.:
DIRECTIONAL
SPREADING
OF NEARSHORE
WAVES
7687
1/2
0.25Hz
dfCpp(
f)
[] nonbreaking conditions
1.4
.05Hz
R•
-* breaking conditions
(9)
¸
[Cuu(f) + Cvv(f)]
d 0.05Hz
03
is theoreticallyequalto 1 for gravitywaves.For both nonbreaking and breakingwave conditionsthe mean value _+1 standard
deviation of the R estimatesfall within the range 0.82-1.14
(Figure 3b). These small [O(10%)] deviationsfrom 1 are
comparablewith expecteduncertaintiesin the flowmeter response[Guzaet al., 1988]andslightnonlineardeviationsof the
wavenumberk from the dispersionrelationship[e.g.,Freilich
andGuza,1984]not accountedfor in (9). Estimatesof velocitypressure transfer functions at individual frequencies (not
shown)are similarto the bulk valuesshownin Figure 3b. The
estimatedR do not deviatenotablyfrom 1 on the bar crest in
breakingwaveconditionswhere a large increasein directional
spreadis observed(Figure 3a), confirmingthat the observed
directionallybroad motionsare dominatedby gravitywaves.
1.2
¸
c-
o
(D
(1.) 1.0
[,
,
N
o_
¸
.8
E
o
i,,
'
i:
,
lOO
havea minimumvaluenear the spectralpeakfrequency
fp
(Figures 4g-4i), similar to observationsof the directional
spreadingof wind wavesin deep water [e.g., Mitsuyasuet al.,
1975;Hasselmannet al., 1980].On September27 the nonlinear
transformation
of normally
incident,
nonbreaking
waves(fp =
0.1 Hz) over the bar resultsin a pronouncedharmonicpeak
(2fp = 0.2 Hz) atx = 205 m (Figure4a)withmeandirection
and spread approximatelyequal to that of the primary peak
"
,
Frequency Dependence
The frequency dependenceof the directional properties
{9
.... (f) and go(f) observedwell seawardof the bar crestat
x = 480 m and slightlyinshoreof the crest at x = 205 m are
shown in Figure 4 for the same three example casesas in
Figure1. Atx = 480 m the go(f) estimatesfor all threecases
iiE• i '
i
,
I
4.
-)(-
,
I
I
I
I
I
I
(o)
200
300
400
5•)0
200
300
400
soo
120
115
110
1 05
1 O0
95
90
85
80
cross-shore distance x (m)
directionalspread(nor(fp), consistent
withthe theoretical
interaction
rule (3). On Figure 3. (a) Frequency-integrated
September
30, higher-frequency
waves(fp = 0.2 Hz) are malized by the offshore array value) versuscross-shoredispartially dissipatedon the sandbar (Figure 4b). These presumablylocallygeneratedseasare directionallynarrow at x =
tancex. Squaresrepresentestimatesfrom 551 one-hour-long
data recordscollectedin low-energ•v
waveconditions
with
480 m [cr0(fp)= 6ø,Figure4h]owingto thestrongrefractive smallenergylosseson the sandbar (Fx > 0.7 atx = 205 m).
Asterisksrepresentestimatesfrom 304 data recordscollected
collimation
ofwavesarrivingat largeobliqueangles
[{9
.... (fp)
in high-ene•rgy
waveconditions
withintensebreakingon the
= 30ø, Figure4e; see(10a) and (10b), discussed
below].Wave
sandbar (F x < 0.3 at x = 205 m). Symbolsandbarsindicate
the averagevalue +_1 standarddeviationof the estimates.(b)
Estimatedratio R betweenpressureand velocityfluctuations,
normalizedby the theoreticaltransferfunctionfor linear waves
(equation(9)), versuscross-shore
distancex (sameformat as
(fp - 0.13 Hz) wereattenuated
strongly
by wavebreaking Figure 3a).
over the bar (Figure 4c). Whereas {9
.... (f) changeslittle
betweenx = 480 and 205 m (Figure 4f), go(f) increases,
especiallynear the spectralpeak frequency.Breakingand nonCx(f) cos[0...... d(f)]
linear interactionsapparentlyhaveproduceda wave field with
crø'x(f)
=
Cxd(f
) COS
[0......(f)] O'o,xd(f
) (10b)
nearlyuniform(overa widefrequencyrange)energylevelsand
directionalspreading.
whereCxdenotesthe wavephasespeedat cross-shore
location
For directionallynarrowspectraon a beachwith straightand x. Predictions
of {9
...... and oro,x at the spectralpeak freparallel depth contours,the shoalingtransformationof {9
....
quency
fp wereobtainedat all instrument
locations
by substiand cr0 between two cross-shorelocationsxd and x is given tutingthe estimatesfrom the offshorearray (xd = 900 m) in
(10a) and (10b). Comparisons
with observedvaluesare shown
approximatelyby Snell'slaw [e.g.,Kinsman,1965]
in Figure 5 for both nonbreakingand breakingconditionsover
cx(f)
breakingon the sandbar is accompaniedby strongdirectional
broadeningof the entire spectrum,nearly doubling cr0 at all
frequencies,including the weak residual 0.1-Hz swell peak
(Figure 4h). On October 12, energetic,normallyincidentseas
sin[0......(f)] = Cx•(f--•
sin[0......•(f)]
thebar(using
thesame
energy
losscriteria
10x,205m
> 0.7and
(10a)<0.3, respectively,
asin Figure3). Predictions
of {9
......
(fp)
7688
HERBERS
ET AL.'
DIRECTIONAL
27 SEPTEMBER
SPREADING
OF NEARSHORE
30 SEPTEMBER
..•
12 OCTOBER
(b),,,,,,,
,],(c)
c'•10
4
1
•
105
1
1
co 102
1
1
.00
.10
.20
$
.00
.10
1
.20
.00
4O
20-
c-
10-
.10
.20
•o
(d)
50-
(•
WAVES
50
•_o
,o[
o
o
._
-u
-10
-
-IO
o -20 -
-20
-30
-5o
c-
I
-40
.00
I
I
.10
I
-40
.00
.20
35
.10
.2o
.oo
55
I
I
I
.10
.20
55
(h)
,
(i)
50
50-
25
25-
20-
20-
i
i
i
i
/
/
_/
II
II
co 20
/
/
/
/
/
,,
15-
o15
l,I
._
•
/
\/
\\
/
15-
V
/
/
c)
._
L
\
10--
10-
/
\
/
\
/
\
\
O0
.1 0
.20
frequency (Hz/
5
.00
.lO
10-
/
..--
.2o
frequency
5
.00
.10
.20
frequency (Hz)
Figure 4. Observed(a-c) spectrumE (f), (d-f) meanpropagationdirection0.... (f), and(g-i) directional
spreado'o(f) versusfrequencyfor the threecasestudiesshownin Figure1. Dashedandsolidcurvesindicate
estimateswell offshoreof the bar crest(at x = 480 m in about5 m depth) and slightlyinshoreof the bar crest
(atx: 205 m in about2-2.5m depth),respectively.
Thespectral
peakfrequencyfp
isindicated
withanarrow.
agreewell with observations
at all locationsfor both low- and
high-energywave conditions(Figure 5a), indicatingthat mean
propagationdirectionsat the spectralpeak frequencyare not
affectedstronglyby wavebreaking,evenin the inner surfzone,
where most of the wave energyis dissipated.The correspond-
ing observed
andpredictedO'o,x(fp
) (Figure5b) are alsoin
good agreementfor x > 350 m, usuallyoffshoreof the surf
zone. In nonbreakingconditionsthe directionalspreadsobserved closer to shore are slightlylarger than the predicted
values.In contrast,the directionalspreadsobservedon and
inshoreof the bar crest in breaking conditionsare a factor of
2-2.5 larger than the predictedvalues.The closeagreementof
observedwith predicted0..... despitetheselarge differences
betweenobservedand predictedo-o, suggests
that the scattering of waveenergyinto obliqueanglesinducedby wavebreaking is roughlysymmetricwith respectto the mean propagation
direction.
Thecross-shore
evolution
of 0mean
(2fp), themeandirection
HERBERS
ET AL.:
DIRECTIONAL
SPREADING
of waveswith the harmonicfrequencytwicethat of the spectral
OF NEARSHORE
WAVES
7689
20
NONBREAKING
peak(fp), is shownin Figure6. Observed0.... (2fp) in a
/
/
/
/
18-
0.06-Hz bandwidthare comparedwith a linear theory prediction based on Snell's law [equation (10a) initialized at the
offshore array] and with the observeddirection at the peak
/
.... (2fp)--/9
.... (fp). .''
/
frequency
0.... (fp). In nonbreaking
conditions
(Figure6a)
theobserved
2fpdirections
diverge
slightly
fromtheSnell'slaw
e
prediction during shoaling (solid curve) but become more
e
/
14-
/
/
12-
/
/
/
closely
aligned
withthefp direction
(dashed
curve).Thisevo-
e
L
lution showsthe expectedtransitionof motionsat frequency
10-
2fpfromfreelocallygenerated
seasat theoffshore
array(with
directionsunrelated to the directionsof the lower-frequency
swell)to nonlinearlyforcedharmonicwavesat shallowerloca-
03
6-
obs-pred8mean(2fp)
4-
2-
lO
-[] nonbreaking conditions
(o)
0
100
ß breaking conditions
200
300
5OO
400
20
BREAKING
_
, , I
-o
18
_
o
"14
,,
e
14-
e
12-
:
¸
e
10-
•
8-
03
6-
_
I
I
1 O0
I
200
I
I
300
I
I
(o)
400
500
4-
/
/
2-
4.0
0
100
3.5
.... (2fp)-/9
.... (fp)
i
200
300
400
5OO
cross-shore distance x (m)
Figure 6. Root-mean-squaredifferencesbetween observed
and predicted(basedon Snell'slaw) mean directionsof waves
2.5
with frequency
2fp (solidcurves)andbetweenobserved
2fp
directions
andobserved
fp directions
(dashedcurves)versus
2.0
cross-shoredistancefor (a) 247 low-energydata recordswith
nonbreakingconditionson the bar and (b) 136 high-energy
recordswith breakingconditionson the bar (usingthe same
1.5
energylosscriteriaasin Figure3). Onlydatarecords
with2fp
<
i
1.0
.5-
I
I
100
200
Hz are included.
tions[withdirections
aligned
withtheprimaryfpwaves;
(3)].In
breakingconditions
(Figure6b) the meandirections
of 2fp
(b)
.0
0.25
'
I
I
500
I
waves observedat all instrument locationsare more closely
I
5OO
400
cross-shore distance x (m)
Figure 5. Comparisonof observedwith predicted(basedon
Snell's law) directionalpropertiesat the spectralpeak fre-
quency
fp versuscross-shore
distance.
(a) Averagevalueand
alignedwithwaveswithfrequency
fp (dashed
curve)thanwith
the Snell'slaw prediction(solid curve), suggesting
that directionality at harmonicfrequenciesis governedby nonlinearity
along the entire transect.
Directional spreadsobservedin breakingwave conditionsat
the peakfrequency
[O'o(fp)]and twicethe peakfrequency
[(r0(2fp)]
are
compared
in Figure7. Offshoreof thebarcrest
predicted
0mean
(fp). (b) Average
valueandstandard
deviation
observed
at 2fp are about20-80%
of the ratiobetweenobserved
andpredicted(r0(fp). Squares (x > 250 m), spreads
and asterisksrepresentresultsfor nonbreakingand breaking largerthanthe spreads
observed
atfp. Shoreward
of the bar
standard
deviation
of the difference
between
observed
and
conditionson the bar, respectively(sameformat as Figure 3).
crest, these differences decrease, and at the shallowest sitesthe
7690
HERBERS
ET AL.:
DIRECTIONAL
SPREADING
OF NEARSHORE
WAVES
tions than the barred beach observations, the similar results
2.0
(compareFigure 9 with Figure 2) indicatethat the increased
directional spreadinginitiated by wave breaking is not extremelysensitiveto the detailedbeachtopography.The barred
beachobservationsshowmaximumdirectionalspreadingnear
the crest of the sandbar (Figures l d and 3a), and the non1.4
barred beachobservationsshowmaximum directionalspread1.2_
ing near the shoreline(Figure8c). This differencemaybe the
resultof differentwave-breakingpatterns.After breakingon a
1.¸
sand bar, waves may reform in the slightly deeper trough
shorewardof the sandbar [Lippmannand Holman, 1990]. In
contrast, once breaking begins on a monotonically sloping
.6
beach,strongdissipationcontinuesuntil the shoreline.
Although the presentresultsindicate a significantbroaden.4
ing of the directionalwave spectrumin the surfzone, the cause
of this phenomenonis not understood.A possiblemechanism
.2
for scatteringwave energyinto obliquepropagationdirections
is nonlineartriad interactions.During the initial shoalingevo.0
100
200
300
400
500
lution of the wave spectrum,energy is transferredprimarily
through sum interactionsto higher frequencies.Two compocross-shore distonce x (m)
nentswith frequenciesand directions(f•, 0•) and (f2, 02)
Figure7. Ratioof directional
spreads
at frequencies
2fpand transferenergyto a sum-frequency
(f3 = f• + f2) wavewith
fp observed
in high-energy
waveconditions
withbreakingon a propagationdirection 03 that falls within the range [0•, 02]
the bar. Averagevalues +1 standarddeviation(basedon 136
and thus doesnot directionallybroadenthe wave field. During
observations)are shownversuscross-shore
distance.
breakingthe stronglyenhancedhigh-frequency
componentsof
the spectrummay transfersomeenergyback to lower frequenfp and2fp spreads
differbylessthan+20%.Thecorrespond-ciesthrough difference interactions.Accordingto the interacwith frequenciesand
ing meandirections
at frequencies
fp and2fp converge
in a tion rules[see(1)-(3)], two components
directions
(f2,
02)
and
(f3,
03)
can
transfer
energyto a
similarfashion(dashedcurvein Figure 6b) and differ by less
difference-frequency
(f• = f3 - f2) componentwith a propthan a few degreesat the shallowestinstrumentlocations.
agation direction
1.8
_
_
_
_
5.
Discussion
=f3--f203 f3--f202
To investigateif the barred beachtopographyplaysa critical
role in the observeddirectionalbroadeningof wavesin the surf
zone, observationsfrom collocatedpressuresensorsand bidirectional current meters obtained at nonbarred Torrey Pines
Beach, near La Jolla, California, were analyzed. Data were
collectedat sixlocationsalonga cross-shore
transectextending
from the shorelineto about 4 m depth (Figure 8a), with a
samplefrequencyof 2 Hz betweenSeptember13 and October
25, 1996 (B. Raubenheimeret al., Tidal water table fluctuations in a sandyocean beach, submittedto WaterResources
Research,1998). Two examplesof the cross-shore
evolutionof
(]])
o
(o)
4
5
1000
frequency-integrated
directionalspreads[equation(7b)] are
shownin Figure 8 for a high- and a low-tide data record with
nearly identical incident wave conditions(normally incident
swell with a significantwave height of 0.8 m and a peak frequencyof 0.08 Hz). Frequency-integrated
mean directionestimates(not shown)from all sensors
varybetween-3 øand 10ø.
During the high-tide run, rro decreasesslightlybetween the
deepestsensorand the outer edgeof the surf zone (x • 200
m), followedby an increase(abouta factor2) to a maximumat
the shallowestsensor.During low tide, when more instruments
were in the surf zone, rroincreasesgraduallyalong the entire
transect
to a maximum
near the shoreline
that is a factor
3
larger than the offshore estimate.
Comparisonof all directionalspreadestimatesat x = 206 m
(in the middleof the transect)andx = 126 m (the shallowest
sensor)with the local energyflux (Figure 9) demonstratesa
clear correspondencebetween surf zone wave breaking and
increased directional spreading.Although these nonbarred
beachobservationsspana more limited range of wave condi-
lOO
lO
•- 2.5
(c)
-o 2..0
.--- 15
E 10
•
o
c
0.5
100
150
200
250
300
cross-shore diMonce x (m)
Figure 8. (a) Water depth, (b) sea surfaceheightvariance,
and (c) frequency-integrated
directionalspreadversuscrossshoredistanceat Torrey PinesBeachon September29 (triangles,high tide) and September30 (squares,low tide), 1996.
Data recordsare approximately3 hours long.
HERBERS
ET AL.'
DIRECTIONAL
SPREADING
OF NEARSHORE
WAVES
7691
3.5
•
3.0
•
2.5
o
•
2.0
•
1.$
._
o10
c-
ß
0.5
0.1
1.0
normalized energy flux
Figure 9. Frequency-integrated
directionalspreadversuscross-shore
energyflux (both normalizedby estimatesfrom the mostoffshoreinstrument)observedat Torrey PinesBeach.Estimatesare basedon 3-hourlongrecordsfrom instruments
at cross-shore
positions
x = 206 m (asterisks)and 126 m (circles).
that is outsidethe range [02, 03] and thus causesa directional
broadeningof the wavefield.For example,considerthe simple
caseof two incidentswellcomponents
with the samefrequency
f and slightlydifferent incidenceangles0 - A0 and 0 4- A0.
Suminteractions[equations(2a) and (3)] excitethree doublefrequency(2f) components
with directions0 - A0, 0, and 0 4.
A0. Subsequentdifferenceinteractions[equations(2a) and
(11)] betweenthe new 2f and the originalf componentsnot
onlytransferenergybackto the two originalf componentsbut
alsoexcitetwo newf components
(f, 0 - 3 A 0) and (f, 0 4.
3A 0) with a largerspreadingangle.Further interactionsthat
involvethesenewlyformedwavesmay transferenergyto even
larger spreadingangles.Although the effectivenessof this
mechanismhas not been established,multiple triad interactionspotentiallycan scatterenergyfrom directionallynarrow
incidentwavesinto obliquelypropagatingcomponents.
6.
Summary
well, and more detailed observations are needed to resolve the
directionalwaveproperties(e.g.,contributions
of edgewaves).
The observedincreasesof directionalspreadsin the surfzone
are large (nominallya factor of 2) and may have important
implicationsfor the dynamicsof nearshorewave-drivencurrents.
Appendix: Estimation of Directional Moments
The crossspectraof a collocatedpressuresensorand bidirectional current meter yield low-resolutiondirectional wave
informationequivalentto that obtainedfrom measurementsof
commonly used surface-following heave-pitch-roll buoys
[Longuet-Higgins
et al., 1963;Bowdenand White,1966;Long,
1980].The normalizedcospectraof pressurep, and the hori-
zontal(x, y) velocitycomponents
u and v yieldthe lowestfour
Fouriermomentsof the directionaldistributionof waveenergy
s(o) [=
Mean wave propagationdirectionsat the spectralpeak fre-
quency
fv observed
bothinsideandoutsidethesurfzoneon a
a•(f) --
dO cos OS(O;f)
barred beachagreewell with predictionsbasedon Snell'slaw
for wave refractionover straightand parallel depth contours
(Figure 5a). During shoaling,mean directionsof waveswith
{Cvv(f)[C..(f ) + Cvv(f)]} ua
frequency
2fvbecome
alignedclosely
withthedirections
of the
dominant
fv waves(Figure6a),consistent
withthetheoretical
interactionrules for nonlinear energy transfersto harmonic
dO sin OS(O;f)
components.Whereasmean directionsdo not appear to be b•(f) -affected stronglyby the onset of wave breaking on the sand
bar, the corresponding
directionalspreadof waveenergyabout
Cvv(f)
the mean direction increasesdramaticallywhen wavesbreak
{Cvv(f
)[C..(f
) + Cvv(f)]} •/•
over the bar (Figuresld, 2, 3a, and 5b). Well insidethe surf
zone, the observedmean directionand directionalspreadare
nearlyindependentof frequency(Figures4i, 6b, and 7).
Observationsof wave shoalingon a nonbarredbeach(Figures8 and 9) showa similarincreasein directionalspreading
in the surfzone,suggesting
that the broadeningis primarilythe
result of wave breaking and is not sensitiveto the detailed
beach topography.The scatteringprocessis not understood
(Ala)
(Alb)
a2(f)=
f) = C•,•,(f)
•dOcos20S(0;
C•,•,(f
)+- Cvv(f)
Cvv(f
)
b2(f)--
f) = C•,•,(f)
+ Cvv(f)
•dOsin20S(0;
2C•,v(f)
(Alc)
(Aid)
7692
HERBERS
ET AL.: DIRECTIONAL
SPREADING
OF NEARSHORE
WAVES
9O
8O
12
7O
•
60
-100
0
•
100
-100
(deg)
0
•
100
(o•g)
--o 50
4o
o
._
(D
©
3o
._
2O
10
0
0
10
20
30
40
50
holf-width
I
•
70
80
9O
AO
Figure A1. Directionalspreado-0 versusthe half width A0 of the directionaldistributionS(0) at half the
maximumpower,for top-hat(squares)and cosine-power
(asterisks)shapesof S(0). The solidline denotes
a 1:1 correspondence,
and the dotted line indicatesthe maximumvalue of o-0 for a directionallyisotropic
wave-orbital velocity field.
where 0 = 0 correspondsto onshorepropagation.Note that
estimates
of these directional
moments
are insensitive
to in-
strumentgain errors,so long asthe errors are the sameon both
flowmeter
axes.
For narrowS(0), a meanpropagationdirection0.... and a
root-mean-squaremeasure of the directional spreading of
wave energy o-0 can be defined in terms of the first-order
momentsa • and b, [e.g.,Kuik et al., 1988]
_=d
0sin
(0- 0.....
)S(0)
=blCOS
0.......
--a1sin
0....--0
(A2a)
d0211 - cos (0 - 0.... )IS(0)
energy) at sea frequencies (>0.1 Hz), often significant
[O(10%) ] at swellfrequencies(0.05-0.1 Hz) and consistently
strongat infragravityfrequencies(<0.05 Hz). Whereaso-0 is
equalto zerofor unidirectionalwaves,o-0 is increasedto 36øby
a directionallyopposingcomponentwith only 10% of the total
wave energy [equation (A2b)]. Thus weak reflection from
shoremay be misinterpretedas a directionallybroad incident
wave field.
= 211 - (a• cos 0.... -']-'
b• sin 0.... )]
For small 0-
dent wave energyusuallyis dissipatedin the surf zone, significant reflectionfrom shoremay occurif the incidentwavesare
small in amplitude and the beach is relativelysteep [Miche,
1951]. Measurementsof wave reflectionfrom natural sandy
beaches[Elgar et al., 1994, 1997; Raubenheimerand Guza,
1996] show a strongfrequencydependence,with reflection
levelsthat are barelydetectable(lessthan3% of the totalwave
(A2b)
Alternatively, 0.... and o-0 can be defined in terms of the
second-ordermomentsa 2 and b2:
0....
1
sin(0- 0.... )• (0- 0.... )[1 -•(0-
dO• sin[2(0 - 0.... )IS(0)
0.... )2]
- • (b cos20.... -- a2sin20.... ) -- 0
211- cos(0- 0....)] • (0- 0....)211
- 1-•(0- 0....)2]
and thus 0.... and o-0 represent a mean value and standard
deviationof the directionaldistributionof wave energywith an
rr2o
=
O(0 -- 0.... )2bias(seeKuiketal. [1988]forfurtherdiscussion).
A seriousdrawbackof thisformulationis the sensitivityof o-0
to reflectionof wavesfrom shore.Although most of the inci-
1
(A3a)
•dOj{[1
1 - cos[2(0- 0....)]}S(0)
= 511-- (a2cos20.... -']-'
b2sin20.... )]
(A3b)
HERBERS
For small 0-
ET AL.:
DIRECTIONAL
SPREADING
OF NEARSHORE
WAVES
7693
Freilich, M. H., R. T. Guza, and S. Elgar, Observationsof nonlinear
effectsin directionalspectraof shoalinggravitywaves,J. Geophys.
0....
sin[2(0 - 0mean)]
• (0 -- 0mean)J1
2 0-- 0mean)
2]
--X(
1
Res., 95, 9645-9656, 1990.
Gallagher, E. L., S. Elgar, and R. T. Guza, Observationsof sand bar
evolutionon a naturalbeach,J. Geophys.
Res.,103, 3203-3215, 1998.
Guza, R. T., and E. B. Thornton, Velocity momentsin nearshore,J.
Waterw.Port CoastalOceanEng., 111,235-256, 1985.
Guza, R. T., M. C. Clifton, and F. Rezvani,Field intercomparisons
of
electromagneticcurrent meters, J. Geophys.Res., 93, 9302-9314,
•¾{1
-cos[2(0- 0mean)]
} • (0- 0....)211
--•'(0- 0mean)
2]
andthusthe second-order
approximations
(A3a) and (A3b) of
the mean value and standarddeviationof S(0) have larger
biasesthan the first-orderapproximations
(A2a) and (A2b).
However,(A3a) and (A3b) are muchlesssensitivethan (A2a)
and (A2b) to wave reflectionsfrom shore at small incidence
anglesbecausethe double-angleargument2(0 - 0.... ) is the
samefor wavespropagatingin opposingdirections.
The interpretationof rr0 estimatesbasedon (A3b) is illustratedin FigureA1 with simplecosine-power
andtop-hatS( 0 )
functions. For both distributions,rr0 is roughly comparable
with the half width A0 with a negativebias that increaseswith
increasingA0. The maximumrr0 value of 40.5ø for A0 = 90ø
correspondsto wave motion with an isotropicvelocity field
: C.,.,).
1988.
Hasselmann,D. E., M. Dunckel, and J. A. Ewing, Directional wave
spectraobservedduring JONSWAP 1973, J. Phys. Oceanogr.,10,
1264-1280, 1980.
Herbers, T. H. C., and M. C. Burton, Nonlinear shoalingof directionally spreadwaveson a beach,J. Geophys.Res.,102, 21,101-21,114,
, 1997.
Higgins,A. L., R. J. Seymour,and S.S. Pawka,A compactrepresentation of ocean wave directionality,Appl. OceanRes., 3, 105-112,
1981.
Kinsman, B., Wind Waves.'Their Generationand Propagationon the
OceanSurface,676 pp., Prentice-Hall,EnglewoodCliffs, N.J., 1965.
Kuik, A. J., G. P. van Vledder, and L. H. Holthuijsen, A method for
routine analysisof pitch-and-rollbuoy data, J. Phys.Oceanogr.,18,
1020-1034, 1988.
Lippmann,T. C., and R. A. Holman, The spatialand temporalvariability of sandbar morphology,J. Geophys.Res.,95, 11,575-11,590,
1990.
Acknowledgments.This researchwas supportedby the Coastal
DynamicsProgram of the Office of Naval Research and the Ocean Lippmann, T. C., and R. A. Holman, Phasespeedand angle of breakSciencesDivisionof the National ScienceFoundation(CoastalOcean
ing wavesmeasuredwith video techniques,in CoastalSediments'91,
Processes
program).The instrumentswere deployedand maintained
pp. 542-556, Am. Soc.Civ. Eng., New York, 1991.
by staff from the ScrippsInstitution of OceanographyCenter for Long, C. E., Index andbulk parametersfor frequency-direction
spectra
CoastalStudies.B. Raubenheimerand E. L. Gallagher made valuable
measured at CERC Field Research Facility, June 1994 to August
contributionsto the field experiments.Logisticalsupportand wave
1995, Misc. Pap. CERC-96-6, U.S. Army Eng. Waterw. Exp. Stn.,
data in 8 m depthwere providedby the Field ResearchFacilityof the
Vicksburg,Miss., 1996.
U.S. Army Engineer WaterwaysExperiment Station'sCoastal Engi- Long, R. B., The statisticalevaluationof directional spectrumestineeringResearchCenter. Permissionto use thesedata is appreciated.
mates derived from pitch/roll buoy data, J. Phys. Oceanogr.,10,
944-952, 1980.
References
Battjes, J. A., Radiation stressesin short-crestedwaves,J. Mar. Res.,
30, 56-64, 1972.
Bowden, K. F., and R. A. White, Measurements of the orbital velocities of seawavesand their use in determiningthe directionalspectrum, Geophys.J. R. Astron.Soc., 12, 33-54, 1966.
Longuet-Higgins,M. S., D. E. Cartwright,and N. D. Smith, Observations of the directionalspectrumof seawavesusingthe motionsof
a floatingbuoy, in Ocean WaveSpectra,pp. 111-136, Prentice-Hall,
EnglewoodCliffs, N.J., 1963.
Miche, M., Le pouvoirreflechissant
desouvragesmaritimesexposesa
l'action de la houle, Ann. Ponts Chaussees,121,285-319, 1951.
Mitsuyasu,H., F. Tasai, T. Suhara, S. Mizuno, M. Ohkusu, T. Honda,
and K. Rikiishi, Observationsof the directionalspectrumof ocean
Bryan,K. R., and A. J. Bowen,Edgewavetrappingand amplification
wavesusinga cloverleafbuoy,J. Phys.Oceanogr.,5, 750-760, 1975.
on barred beaches,J. Geophys.Res.,101, 6543-6552, 1996.
Elgar, S., R. T. Guza, and M. H. Freilich, Observationsof nonlinear Raubenheimer,B., and R. T. Guza, Observationsand predictionsof
run-up,J. Geophys.Res., 101, 25,575-25,587,1996.
interactionsin directionallyspreadshoalingsurfacegravitywaves,J.
Geophys.Res., 98, 20,299-20,305, 1993.
S. Elgar, Schoolof Electrical Engineeringand Computer Science,
Elgar, S., T. H. C. Herbers, and R. T. Guza, Reflection of ocean
surfacegravitywavesfrom a natural beach,J. Phys.Oceanogr.,24, Washington State University, Pullman, WA 99164-2752.
R. T. Guza, Center for Coastal Studies, Scripps Institution of
1503-1511, 1994.
Oceanography,La Jolla, CA 92093-0209.
Elgar, S., R. T. Guza, B. Raubenheimer,T. H. C. Herbers, and E. L.
T. H. C. Herbers, Department of Oceanography,Naval PostgraduGallagher,Spectralevolutionof shoalingand breakingwaveson a
ate School, Code OC/He, Monterey, CA 93943-5122. (herbers@
barred beach,J. Geophys.Res., 102, 15,797-15,805, 1997.
Freilich, M. H., and R. T. Guza, Nonlinear effectson shoalingsurface oc.nps.navy.mil)
gravitywaves,Philos. Trans.R. Soc.London, Set. A, A311, 1-41,
1984.
(ReceivedDecember9, 1997;acceptedMay 22, 1998.)