JOURNAL OF GEOPHYSICAL
RESEARCH, VOL. 96, NO. C7, PAGES 12,781-12,795, JULY 15, 1991
Wave VariancePartitioningin theTroughof a BarredBeach
PETER
A. HOWD,
JOAN
OLTMAN=SHAY,
1 ANDROBERT
A. HOLMAN
Collegeof Oceanography,
OregonStateUniversity,
Corvallis
Thewave-induced
velocity
fieldinthenearshore
iscomposed
ofcontributions
fromincident
windwaves
(f
> 0.05Hz),surface
infragravity
waves
(f< 0.05Hz,I}cl
< (o•/g•)andshear
waves
(f< 0.05Hz,I}cl
> •2/g•),
where
fis thefrequency,
• = 2•f, }cistheradialalongshore
wavenumber
(2re/L,
L beingthealongshore
wave-
length),[• is thebeachslope,andg is theacceleration
dueto gravity.Usinganalongshore
arrayof current
meters
located
in thetroughof a nearshore
bar(meandepth= 1.5m), weinvestigate
thebulkstatistical
behaviorsof thesewavebandsovera widerangeof incidentwaveconditions.
The behaviorof eachcontributing
wavetypeisparameterized
in termsof commonly
measured
oreasilypredicted
variables
describing
thebeach
profile,windwaves,
andcurrent
field.Overthe10-day
period,
themeancontributions
(tothetotalvariance)
of
theincident,
infragravity,
andshearwavebandswere71.5%,14.3%and13.6%for thealongshore
component
of flow(mean
rmsoscillations
of 44,20,and19cms-1,respectively),
and81.9%,10.9%,and6.6%forthe
cross-shore
component
(mean
rmsoscillations
of92,32,and25cms-1,respectively).
However,
thevalues
variedconsiderably.
Thecontribution
to thealongshore
(cross-shore)
component
of flowranged
from44.888.4%(58.5-95.8%)for the incidentband,to 6.2-26.6%(2.5-32.4%)for theinfragravityband,and3.433.1%(0.6-14.3%)for theshearwaveband.Incidentwaveoscillations
werelimitedby depth-dependent
satu-
rationovertheadjacent
barcrestandvariedonlywiththefide.Theinfragravity
wavermsoscillations
onthis
barredbeacharebestparameterized
by theoffshore
waveheight,consistent
withprevious
studies
onplanar
beaches.
Comparison
withdatafromfourotherbeaches
of widelydifferinggeometries
showstheshoreline
infragravity
amplitude
tobea near-constant
ratioof theoffshore
waveheight.
Themagnitude
of theratiois
found
tobedependent
ontheIribarren
number,
•0= [•(H/Lo)
-1/2.Shear
waves
are,asprevious
observation
and
theorysuggest
(Oltman-Shay
et al., 1989;BowenandHolman,1989),significantly
correlated
with a
prediction
of theseaward
facingshearof thelongshore
current.
GalvinandEagleson[ 1965]found? to havea valuerangingfrom
INTRODUCTION
0.7 to 1.2. For field data,Thorntonand Guza [ 1982] found? to be
Motivation
muchlower,approximately
0.42.Many researchers
havereported
For nearly two decades,one of the primary interestsin the
studyof nearshoreprocesses
hasbeenthe characterizationof two
frequencybandsof gravitywaves: incidentwind waves(0.33 Hz
> f > 0.05 Hz) and infragravitywaves (f < 0.05 Hz). The recent
discovery of a third "band," shear waves [Oltman-Shay et al.,
difficult problemof quantifyingwave energyin the nearshore.
This paper has two primary objectives: (1) to provide an
? to be a functionof beachslopeand/orwave steepness
[Bowenet
al., 1968; Weishat and Byrne, 1978; Sallenger and Holman,
1985b]. Statisticalrepresentationsof random waves from field
experimentsincludebothbrokenandunbrokenwaves,onereason
why the saturationvalue of ? is significantlylower for field data
than for monochromaticlaboratorywaves.
Infragravity waves are traditionally consideredto be those
surfacegravitywaveswhichresultfrom the second-order
interac-
overview
tion of incident wind waves. Under most conditions in surf zones
1989a' Bowen and Holman, 1989], has added a new twist to the
of the variance contributions
of these different waves in
the trough of a barredbeach (the contributionof shearwaves,
which hasnot beenpreviouslyquantified,is of particularinterest)
and (2) to examine the lowest-order controls of variance on a
barredbeachand to compareour findingswith thoseof previous
studies. In the remainder
of this section we will review
some of
the relevantliteratureconcerningthe three wave typesand their
behaviorsin the surf zone. We then discussthe acquisitionand
analysistechniquesfor the presentdata,followedby thepresentation of our findingsand a comparisonwith {.)Lilt31 11UIU Ui::tti;I,.
on coastsopento the ocean,thesewaveshave frequenciesranging from 0.005 to 0.05 Hz. The free infragravitywaves can be
broken into two distinct groups: a discrete set of edge wave
modes,which are trappedto the shoreline,and a continuumof
leaky waves,which reflect from the shorelineandradiateenergy
back out of the nearshore zone. Bounded waves, the forced dis-
placementof the free water surfaceby the structureof wave
groups[Longuet-Higginsand Stewart,1964], alsocontributeto
infragravitywave energy [Guza et al., 1985; Elgar and Guza,
1985].
Previous
Work
Edgewaveson a planebeachof slope[• havea discretesetof
It has been shown consistentlythat statistical measuresof
incidentwind wave heightbecomedepth-limitedin the inner surf
zone of natural beaches,and thus independentof the offshore
wave height,accordingto the linearrelation
Hrms= Th
(1)
where
Hrms
= (8s2)
1/2,s2 istheincident
wavevariance,
andh is
the local water depth. For monochromaticlaboratory waves,
possible
alongshore
wavenumbers
(r), in therangeo2/g< I•cl<
o2/g• satisfying
therelationship
o2 = girl (2n+l)[• [Eckart,
1951],whileforleakywaves
thereisacontinuum
withI•cl< o2/g
[Suhayda,1974;Guza and Bowen, 1976]. Boundwaveshave no
constrainton r. Analytic solutionsto the linear, shallow-water
equationsof motionon a plane-sloping
beachgive the free wave
velocity potential(•) as
ß = ag/o cos(ry-•t) q•(x)
1NowatQuest
Integrated,
Kent,Washington.
Copyright1991by the AmericanGeophysicalUnion.
Papernumber91JC00434.
0148-0227/91/91 JC-00434505.00
(2)
wherex is thecross-shore
coordinate,
y is thealongshore
coordinate,anda is the wave amplitudeat the shoreline[Eckart, 1951;
Suhayda,1974; Guza and Bowen, 1976]. The cross-shorenodal
structure,q•(x),is as follows
12,781
12,782
HOWDET AL.:WAVE VARIANCEPARTITIONINGIN TROUGHOFBARREDBEACH
{(x) = e-rXLn(2rx)
edgewaves
(3)
/4•:x• 1/2
•)(x)
=J0•,-•I
K=0leaky
wave
with Ln beingthe Lague•e polynomialof ordern (wheren is the
modenumberof the edgewave) andJ0 is the zeroth-orderBessel
function. The cross-shorestructureof the higher-modeedge
waves (n > 2) •d the no•ally incident leaky wave are very
similar near the shoreline and can be representedby the J0
solutionfor approximatecalculations[Holman, 1981; Sallenger
and Holman, 1985a].
The cross-shorestructureof infragravity waves has made frequency spectra of surf-zone cu•ents difficult to inte•ret and
comparebetweenbeaches.Wavenumber-frequencyspectraesti-
mated from cross-shorecu•ents are also difficult to inte•ret,
being dominatedby someunresolvedcombinationof high-mode
edge waves, leaky waves, phase-lockededge waves, and bound
waves [Elgar and Guza, 1985; Oltman-Shay and Guza, 1987;
Huntley, 1988; Haines and Bowen, 1988]. However, wavenumber-frequency spectraof the alongshorecomponentof cu•ents
have beenshownto be panicularlyusefulin dete•ining the progressivelow-mode edge wave contentof the infragravity wave
field [Huntleyet al., 1981; Oltman-Shayand Guza, 1987].
o oo
Pastfield studieshave showna dependencebetweenthe magnitudeof thelocalinfragravitywaveoscillations,
U/Ganda/G (the
cross-shore
oscillationand the wave amplitude),and the offshore
wind wave height [Holman, 1981; Guza and Thornton, 1985].
Holman and Sallenger[1985] andSallengerand Holman [ 1985a]
concludedthat the dependencewas also function of the deep-
waterIribarren
number,•0 = [•/(H/Lo)
]/2, where
H andL0 are
representativeof the deep-waterheight and wavelengthof the
incidentwaves,andof thedimensionless
cross-shore
distance,Z
= c•2x/g[•,
tothelocation
ofthemeasurements
WG= mu(Z, •0)H
(4)
alG= ma(X, •0)H
(5)
wherem is the slopeof thelinearregression.
Shearwaves,a newclassof nearshore
wave,aredistinguished
by large wavenumbers,well outsidethe wavenumberrange of
gravitywaves,Irl > c•2/g[•
[Oltman-Shay
etal., 1989a].Onthe
onebeachstudiedto date,a typicalenergeticperiodis 200 s, with
an alongshorewavelengthof 200 m. This distinctivesignature
permitstheircontribution
to totalcurrentvarianceto be separated
in wavenumber-frequencyspace.Bowen and Holman [1989]
showtheoreticallythat thesewavesmay be an instabilityof the
meanlongshorecurrentwhichconservespotentialvorticity.The
-;:• oo
- 9OO
SUPERDUCK
EXPERIMENT
SITE
- 800
O Pressure
- 700
•
E
I Current
Sensor
Meter
0
z
600 •
500 nO
400 0
0
300
200
I
I
I
I
glll
ß
1
...... ....:........................:.:.:.:.:.:.:.:.:...:................................
..... ...............
......
Fig. 1. Planview of theSUPERDUCKfield siteat theCoastalEngineering
Research
Center'sField Research
Facilityin Duck,
North Carolina.Shownare the nearshorecurrentmeter array,the nestedpressuresensor,and the offshorewind wave directional
arrayrelativeto thelocationof thepier.All measurements
weremadeoutsidetheknownregionof thepier'sinfluence.
HOWD ET AL.: WAVE VARIANCE PARTITIONING IN TROUGH OF BARRED BEACH
VARIANCE
Cross-shore
a
PARTITIONING
SCHEME
Current
b
0.25 _
12,783
Longshore Current
0.25
o
0.225
0.225
_
[]
[]
0.2
0.2
0.175
[]
[]
0.175
_
INCIDENT
0.15
oo
BAND
INCIDENT
0.15
0.125
BAND
0.125
0.1
0.1
E]
[]
0.075
0.075
0.05
0.05
n
INFRAGRAV
0.025
0.025
SHEAR
0.0
SHEAR
I
I
0 ....
o,
o,'
0
0
0
o,
o,
o,
.-,
0
Cq
0
cs
cs
I
I
cs
cs
•'•
0
d
CyclicAlongshore
Wavenumber
(m")
0.0
SHEAR
I
I
I
o.
o.
o.
SHEAR
I
o
o
i
o
o
CyclicAlongshore
Wavenumber
(m-')
I I .......
:•:::::•
!•!•:i•i•!•
"••
"•"
•'
'"•'
;'•.............
':':'
:'':"-'
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
..,
"iS•i•::::-:.
0
0
0
0
0
% Power
% Power
Fig.2. Cyclicalongshore
wavenumber
(k = l/L) versus
frequency
(f= l/T) spectra,
withlinesshowing
thebounds
of eachof the
threewavetypes.Theboxesrepresent
peaksin S(kj')withthedarkness
indicating
thepercent
of variance
in thatfrequency
band
contained
in thatpeak.Thewidthof theboxindicates
thehalf-power
wavenumber
bandwidth.
Thearraywasdesignrd
forinfragravityfrequencies,
thustheincident
bandis shown
forreference
only.(a).Estimate
fromcross-shore
(u)current.
(b).Estimate
from longshore(v) current.
cross-shoreshearof the mean longshorecurrentprovidesthe
Coastal Engineering ResearchCenter (CERC) Field Research
background
vorticity(therole of Coriolisin larger-scale
flows). Facility (FRF) in Duck, North Carolina, during October, 1986
Usinga simplemodel,theyshowthatthereis a frequency
range (Figure 1). The beachis locatednear the centerof a 100-km-long
wherea perturbation
to the meancurrent(the shearwave) will barrier island. The mean slope is approximately 1 in 10 on the
growexponentially
at a ratewhichdepends
onthemagnitude
of foreshore,decreasingoffshoreto 1 in 100. Sandbarsare consistently present,most commonlyin a 3-dimensionalconfiguration
theshearontheseaward
faceof thecurrent(•)V/•)xlseaward).
While
onlythegrowthrateis predicted,
we will testtheassumption
that which becomeslinear during most storm events [Lippmann,
1989]. The extremetide rangeduringthe experimentwas 137 cm.
An array of 10 Marsh-McBirneybidirectionalelectromagnetic
currentmeterswas deployedapproximately55 m seawardof the
Usw
=msw
•xx
(6) mean shorelinepositionin the troughof a nearshorebar system.
seaward
The depth rangedfrom ---0.5to ---2.5m over the courseof this
study. Only the seven southernmostsensorswere used in this
METHODS
studyto minimize spatialinhomogeneity.The sampledarray was
In thissectionwe brieflydescribethe field site,theinstrumen- thus290 m in length,sufficientto resolvetypical wavelengthsat
tationusedin this study,andthe datasampling.Thenwe present this site [Oltman-Shayet al., 1989a]. Sensorswere orientedin
the analysistechniques
andthe sensitivityof thesetechniques
to sucha mannerthat their axes coincidedwith the longshore(+v
currentsflow north) and cross-shore(+u currentsflow offshore)
differentprocessing
options.
The data used for this study were collectedas part of the directions.All gagesremainedsubmergedat low fide.
the rms velocities of the shear waves also scale as the maximum
seawardshearof the longshorecurrent
SUPERDUCK experiment[Crowsonet al., 1988]hostedby the
Complementary
datawerecollectedby otherinvestigators.
A
12,784
HOWD ET AL.: WAVE VARIANCEPARTITIONINGIN TROUGHOF BARREDBEACH
TABLE
1. Basic Environmental
Conditions
Run
Day*
Time,
EST
Tide
H,
cm
T,
s
ot
(o)
HiNC,
cm
<U>,
cms-1
<V>,
cms-1
1
2
3
9
9
10
0930
2200
1030
high
high
high
57.8
35.8
184.2
6.4
5.8
7.3
-31.9
-29.9
24.3
19.6
53.5
-4.5
2.8
31.5
16.3
3.6
-159.3
4
10
1700
low
205.5
7.9
16.0
36.4
12.9
-114.4
5
10
2320
high
216.5
8.9
8.9
55.8
22.2
6
11
0540
low
216.2
9.7
4.2
35.0
7.5
-52.7
7
11
1200
high
213.7
9.7
4.9
69.4
23.6
-41.3
0.3
-13.0
-88.1
8
11
1820
low
190.3
10.0
35.7
3.6
9
10
11
12
12
13
14
15
0040
0130
1500
0330
high
high
high
high
172.6
126.2
57.1
110.5
11.9
12.0
9.7
5.5
-3.7
-10.4
-25.3
36.1
61.3
57.2
49.6
63.3
14.4
16.4
4.8
8.9
13
15
0945
low
96.2
6.0
25.0
23.3
5.9
-91.1
14
15
1600
high
71.3
6.4
22.7
49.8
5.0
-35.5
30.5
29.1
21.4
-87.1
15
15
2200
low
71.4
4.4
25.8
20.2
2.8
-64.2
16
17
16
16
0400
1630
high
high
79.3
75.8
4.6
5.2
27.9
24.4
50.5
45.3
5.0
4.9
-40.0
-35.3
18
16
2240
low
69.1
5.5
16.5
21.4
3.8
-52.8
19
17
0450
high
75.5
5.1
18.4
48.5
5.2
-29.5
20
17
1100
low
70.1
5.8
6.6
21.4
3.7
-40.7
21
18
0530
high
91.3
9.7
20.4
51.8
9.6
-45.3
H, incidentroot meansquarewave heightin 8-m depth;T, peak spectralperiodin 8-m waterdepth,
medianspectraldirectionin 8-m waterdepth(CCW fromshorenormal);HiNC,incidentrootmeansquarewave
height in the surf zone (from pressure);<U>, mean cross-shore
currentaveragedover the array of current
meters;<V>, meanlongshorecurrentaveragedover the arrayof currentmeters.
*Day of the monthOctober 1986.
pressuresensor(maintainedby Asbury Sallenger,Jr., U.S.
GeologicalSurvey)was located3 m southof the centralcurrent
meter.Thewindwaveclimatewassampled
in 8-mdepth,usinga
255-m-long
arrayof bottom-mounted
pressure
sensors
aspartof
sI•
(9)
-•cNy
2 Ioø'ø5I+rNYS(K,f)dKdf2
SSW=
where
s2 istheintegrated
variance
foreachof thebands,
•CS•
is
theNyquistradialwavenumber
of thearray,_+r•
0 aretheestimated
theroutinemonitoring
programof theFRF. The surf-zonemormode zero wavenumbers,•5is a constantwavenumberoffset used
phology was measuredusing the FRF Coastal Research
to account for the wavenumber bandwidth of the mode zero
Amphibious
Buggy(CRAB) [Birkemeier
andMason,1984].
All currentmetersandpressuresensorswerehard-wiredto the
data collectionsystem.Gageswere sampledat 2 Hz for 4 hours
centered on high and low tides. High-quality data from the
majorityof thegageswerecollectedfor 10 days,with theexception of approximately22 hourslost owing to a power failure.
Eachtime serieswasexhaustivelycheckedfor reliabilityin both
time andfrequencydomains.Suspectgageswereexcludedfrom
further analysis,as were time periodsmarked by large spatial
inhomogeneities
or temporalnonstationarity.From a total of 36
collections,21 wereprocessedfurther.
The currentmeterarray datawere usedto calculatethe alongshore wavenumber-frequencyspectrafor both the cross-shore
and alongshorecomponentsof flow. First, each of the 21 time
serieswas dividedinto 13 ensembleswith 50% overlap,2048 s in
length, demeanedand then detrended(using a least squares
quadraticfit) prior to beingtaperedwith a Kaiser-Besselwindow.
Wavenumber-frequency
spectrawere estimatedusingthe iterative maximum likelihood estimator(IMLE) developedby Pawka
[ 1982, 1983] andpreviouslyappliedto surf-zonedataby OltmanShayand Guza [1987] and Oltman-Shayet al. [1989a].
Variancefor each4-hourrun wasthenpartitionedbetweenthe
three bands in wavenumber-frequency space (Figure 2).
Integratedvariancesfor eachbandweregivenby
TABLE
Variable
Mean
S.D.
Maximum
Minimum
Offshore
Hrm
s,crn
T,s
ix,ø
118.4
61.1
216.5
35.8
7.5
-8.3
2.3
19.1
12.0
6.1
4.4
-31.9
Surf-Zone
H•Nc, cm
43.5
15.4
69.4
19.6
U•N
c, cms-1
UtG,
cms-l
usw,
cms-•
V•N
c,cms-l
v•o,cms-•
Vsw,
cms-l
92.0
32.1
25.1
44.4
20.2
19.5
11.0
16.0
10.6
8.8
7.8
7.3
108.5
62.5
40.6
60.8
32.6
34.8
74.2
15.0
6.0
27.5
8.4
8.4
% ue4
c
% UtG
% USW
% v•$C
% vtG
% VSW
81.9
10.9
6.6
71.5
14.3
13.6
11.8
8_5
4.1
11.2
6.0
6.7
95.8
32.4
14.3
88.4
26.6
33.1
58.5
2.5
0.6
48.8
6.2
3.4
Mean indicatesmeanvalueof variablefor the 21 dataruns,S.D. is
standarddeviation about mean value, Maximum is maximum value of
variable for these 21 runs, Minimum is minimum value of variable for
2 = Jf0.05
0.33
f +•:NY
S(K,
f)drdf
SINC
j-•cNy
(7)
2 I O0'05
I (_tO_6)
(+rO+-b)
S(K,f)
dK
df
(8)singlepressuregage.
sIG=
2. Basic Statistics
these
21runs.Themeanvalueforeachrunistheresultof anaverage
of
manysensors,
with theexception
of HiNC, whichis calculated
froma
HOWDETAL.:WAVEVARIANCE
PARTITIONING
IN TROUGHOFBARREDBEACH
250.00
--
Offshore Wave Height
---
--
200.00
H
--
--
LHLH
12,785
LH
150.00
--
HLHLH
HLHL H
--
100.00
_
_
H
50.00
-
.
I
9
0.00
8
H
H
I
10
I
11
I
12
I
13
I
14
I
15
I
16
80.00
40.00
L
L
H
H
HH H H
H
L L
i
g
8
i
10
i
11
LH
i
12
i
13
I
14
i
15
L
i
16
I
9
0.00
8
i
17
i
18
L L
L L
I
11
20
H
H H H HH H
I
10
i
19
H
HH
20.00
H
U Infragravity
H
HLH
40.00
20
L L
80.00
60.00
I
19
L
20.00
0.00
I
18
TroughWave Height
H
H H
60.00
I
17
I
12
I
13
I
14
I
15
I
16
80.00
I
17
i
i
18
19
20
V Infragravity
60.00
40.00
20.00
H H
I
0.00
9
8
HLHLHL
H H
H HLHLH HLHL
H
I
I
I
I
I
I
I
I
I
I
10
11
12
13
14
15
16
17
18
19
80.00
20
U Shear Wave
60.00
HLHLHLH
40.00
H
L
H H H LH HI. HL H
20.00
0.00
8
I
9
H H
I
10
I
11
I
12
I
13
I
14
I
15
I
16
80.00
I
17
I
18
I
19
20
V Shear Wave
60.00
40.00
L
20.00
HLHLHLH
H
•
0.00
8
H
H H Lu
n HL•4
'-L H
I
I
I
I
I
I
I
I
I
I
I
9
10
11
12
13
14
15
16
17
18
19
20
DAY - OCTOBER, 1986
Fig.3. Timeseries
of4-hour
mean
statistics
forwave
parameters
attheoffshore
array
pressure
sensors
andforthecurrent
meters/pressure
sensor
inthesurf
zone.
H andL refer
tothetidestage.
Seethetextfordetails
ofhowthevariances
were
partitioned betweenwave types.
of the offshorepressuregage
spectral
peak(setat0.0094m-! in radialunits;thisaccounts
for basedon the spectralcharacteristics
the variationin •cfrom the bottomof the frequencybin to the top,
seeOltman-ShayandGuza [1987]) andS0cd')is the spectraldensity.The subscripts
referto theincident(INC), infragravity(IG),
and shearwave (SW) bands,respectively.
The incidentbandintegration
limits(in frequency)werechosen
arraydata.The upperlimit, 0.33 Hz, wasthe highestfrequency
for which the depth-attenuated
wave signalsreliably exceeded
instrumentnoise.The lower limit (0.05 Hz) was chosensuchthat
all wind wave variance would be excluded from estimates of the
infragravitymotions(but not the opposite).Examinationof
12,786
HOWDETAL.:WAVE VARIANCEPARTITIONING
IN TROUGHOFBARREDBEACH
Cross-shore Current Variance Partitioning
lOO%
90%
80%
70%
60%
5O%
4O%
co
3O%
0
20%
10%
0%
Run Number
• %Uig
""•%Usw
AlongshoreCurrentVariancePartitioning
lOO%
90%
I=
80%
70%
60%
50%
0
•z
40%
30%
0
20%
10%
0%
Run Number
• %Vig
':":•%Vsw
Fig.4. The percentcontributions
madeby eachbandto thetotalvarianceof eachflow component
(theincidentbandcontribution bringsthe totalto100%).The incidentbandis clearlydominantat thiscross-shore
locationfor boththe alongshore
and
cross-shore flows.
HOWD ET AL.: WAVE VARIANCE PARTITIONING IN TROUGH OF BARRED BEACH
TABLE 3. RegressionResults
12,787
and infragravityregimes.The resultswere not sensitiveto reasonable choices for 6.
Y
X
m + 95% Confidence
b
r2'
HiNC
HiNC
Hrm
s
tide
0.100 + 0.111
0.261+ 0.063
31.3
30.0
0.158
0.798
ltlG
ViG
% ltlG
% ViG
HnT
•
HnT
•
Hnn
s
Hrrm
0.224 + 0.064
0.110 + 0.030
0.001+ 0.0004
0.059+ 0.001
5.6
7.1
-0.023
0.007
0.729
0.746
0.641
0.526
Usw
Vsw
% usw
% Vsw
[<V'A
[<V'A
[<V'A
[<V'A
0.143 + 0.117
0.130 + 0.071
0.0007+ 0.0004
0.0009+ 0.0007
17.7
12.7
0.032
0.088
0.245
0.419
0.342
0.260
usw
VSW
% USW
% VSW
3VM•
3VM•
3VM•
3V•t/3x
868.7+ 388.9
637.7+ 249.2
3.491+ 1.457
4.172+ 2.981
13.9
11.2
0.021
0.083
0.521
0.587
0.555
0.300
Here (19 DOF) Y = mX + b + error.
*Ther2isgreater
than0.185correlation
different
from0 at95%level,
andr2isgreater
than0.303correlation
different
from0 at99%level.
The wavenumberlimits for the infragravityband calculation
are basedon estimatesof +r 0, the largestwavenumberspossible
for free surfacegravity waves (plus and minus signsindicate
directionof propagation).We have assumed+•c0 can be closely
approximatedby the plane beachsolutionwith a simplecorrection for the mean longshorecurrent
+%=
[o - (+%) vl2
g[3
(10)
Kenyon [1972] demonstratedthat the dispersioncurve for the
mode zero edgewave, in the presenceof a currentwith constant
shear, behavesas if it were Doppler-shifted, as above, by the
longshorecurrent at an offshore distance of L/4n, which is
approximately33 m for the "typical"modezeroedgewave in this
study.This behaviorhas alsobeenreportedby Oltman-Shayand
Guza [ 1987] for field data.Values chosenfor the beachslopeand
meanlongshorecurrentat x = 33 m were basedon averagevalues
overthelengthof thealongshore
array.A constant
[• waschosen
spectrafrom the 8-m arrayshowedno energeticswellat frequencies below
0.05
Hz.
The 0.05-Hz
cutoff
results in a small
underestimate
of the infragravityband variance.
Varying the
cutofffrom0.04 to 0.06 Hz resultedin typicalchangesof +5% of
the total variance,with the tradealwaysbeingbetweenincident
to matchthe observedmode zero dispersionlines.
Testswere conductedconcerningthe sensitivityof the analysis
to different data windows and ensemblelength, detrending,and
changes in the wavenumber-frequencypartitioning scheme.
There were no statisticallysignificantdifferencesfrom varying
the ensembleand window characteristics.
As expected,detrended
Trough RMS Wave Height vs. Tide
80.00
70.00
x
x
x
60.00
x
50.00
x
x
x
x
x
x
>
40.O0
30.00
20.00
x
lO.OO -
-
-
.
0.00
I
-40.O0
I
I
I
I
-20.00
I
I
I
I
0.00
''''
I''''
I''''
I''''
I''''
20.00
40.00
60.00
80.00
I''
100.00
Tide (cm, NGVD)
Fig. 5. The magnitudeof the incidentband rms wave height measuredin the troughplotted versusthe tide elevation.This,
combinedwith the lack of correlationbetweenoffshorewaveheightandrmsoscillation,indicatesthe incidentbandis depthlimited duringthe majorityof the experiment.The one outlierrepresentsour lowestwave conditions(2200 EST on October9),
whenoscillationswere not depth-limited.
I
I
120.00
12,788
HOWDETAL.:WAVEVARIANCE
PARTITIONING
IN TROUGH
OFBARRED
BEACH
80.00
a
70.00
60.00
50.00
40.00
30.00
x
X
x
x
20.00
1 o.oo
o.oo
0.00
50.00
100.00
150.00
200.00
250.00
OffshoreWaveHeight(crn)
50.00
b
40.00
-
_
30.00
20.00
-
X
_
_
X
-
10.00
-
-
0.00
I
0.00
....
I
50.00
100.00
....
I
....
150.00
I
200.00
'
'
'
'
250.00
Offshore Wave Height (crn)
Fig.6. Values
of(a)tl!Gand(b)¾IG
plotted
versus
theoffshore
wave
height,
Hrm
s.Bothshow
alinear
relationship
withincreasing waveheight,ashasbeenseenon otherbeaches.
datacontainedlesstotalvariancethandatawith trends,but the
Thedatacovera widerange
of conditions,
withHrm
sin 8-m
waterdepthranging
from30 cmto over215cm (significant
heights
inexcess
of 300cm),thepeakperiod
varying
from4.4to
RESULTS
12.0
s,
and
the
mean
longshore
current
averaged
over
thecurrent
Statistics
meterarray,<V>,ranging
from-159cms-1 (southward)
to +31
Tables
1and2 andFigures
3 and4 present
a summary
ofbasic cms-1 (northward).
waveandcurrent
statistics.
In all cases,
therootmeansquare Statistics for the rms oscillations in each band are found in
(rms)statistics,
calculated
as(852)
1/2,arepresented
for wave Table2. Meanvalues(plusor minusonestandard
deviation)over
differences
occurred
onlyin thelowest-frequency
bin.
heights
andcurrent
oscillations.
Because
oftheprocessing
tech- theexperiment
for/linCandViNCare92 + 11cms-1 and44 + 9
niques,
rmscurrent
oscillations
represent
theaverage
overthe cms-1,withmaxima
of 108and61cms-1,respectively.
Mean
array.Percent
contributions
always
referto thepercent
of total values
of theu andv infragravity
waveoscillations
(U/GandV/G)
were32.1+ 16.0cms-1 and20.2+ 7.8cms-1, withmaximaof
variance
contributed
by theband.
HOWD ET AL.:WAVE VARIANCEPARTITIONINGIN TROUGHOF BARREDBEACH
TABLE 5. DirectionalDependence
TABLE 4. IribarrenRegressionResults
Y
X
rn+ 95% Confidence
12,789
b
r2'
Y
U/G
(• <0.40) •
ItlG(•-o0
> 0.40)
•
0.157
+0.054 8.26 0.808
0.256+ 0.076
6.14
0.872
v/G(• < 0.40)
V•G(•0 > 0.40)
•
•0
0.093+ 0.046
0.121+ 0.043
8.00
7.00
0.677
0.819
X
ItlG
Hrms
I•l
0.22 + 0.06
shore variance and from 2.5% to 32.4% of the cross-shore vari-
P arame te rizatio n
Incidentbandrmscurrentoscillations,/,lin
C andVlNC,andwave
5.62
0.73
76.70
0.85
43.38
0.75
17.08
0.76
16.74
+ 0.38
28.80
Hrmsalone
Hrms
anceat the surf-zonelocationof the alongshorearray.Shearwave
oscillationscontributedup to 33.1% of the total longshorecurrent
variance,but only up to 14.3% of the cross-shore
variance.
Error
rIG
62.5and32.6cms-1,respectively.
Themeanshear
waveoscillationswere25.1+ 10.6cms-] forUswand19.5+ 7.3cms-] for
Vsw.
Maximawere41 and35cms-1,respectively.
infragravitybandwavesrangedfrom6.2% to 26.6% of thelong-
MeanSquare
0.14 + 0.06
-0.74
Here (8 DOF) Y = mX + b + error.
The percentagecontributionsof eachbandto the total variance
are shownin Figure4. The incidentbandoscillations,on average,
provided71.5% of the alongshorecurrentvarianceand 81.9% of
the cross-shorecurrentvariance.The percentcontributionof the
r2
Confidence
Hrmsalone
*Ther2isgreater
than0.400correlation
different
from0 at95%level,
andr2isgreater
than0.585correlation
different
from0 at99%level.
b
rn+ 95%
Ic•l
0.11 + 0.02
7.14
0.10 + 0.04
-0.14
+ 0.24
11.44
Here Y = m•X• + m2X2 + b + error.
(0.20 + 0.03) wasconsiderably
lowerthansaturation
value(0.30
+ 0.01) previouslypredictedfor this beach at the bar crest
[SallengerandHolman, 1985b].
Valuesof UiG,VIG,andtheirpercent
contributions,
werefound
to be significantlycorrelatedwith the offshorewave height
(Figures6a and6b), in agreement
withcurrentmeterdatafrom
similarwaterdepthsonthreeotherbeaches
[Holman,1981'Guza
and Thornton,1985]. Comparisonwith thesedata setswill be
presented
in thediscussion.
Holmanand Sallenger[1985]and
Guza et al. [1985] alsofoundthat for the infragravityband,the
of theregression
linesfor swash
amplitude
versus
offshore
ited, but not saturated,in agreementwith the previousobserva- slopes
ontheIribarrennumber(largerslopesfor
tionsof Wrightet al. [1986]. However,therewas evidencefor waveheightdepended
forthevelocityoscillations,
we
saturation
seawardof thearray,astherewasa lack of statistically larger•0). To testthisdependence
heights,HINC, in thebartroughwereobserved
to be depth-lim-
significant
correlations
between
theincident
waveheights
in the
troughandtheoffshore
waveheight(Table3). Theoscillations
were,however,significantly
correlated
with the tide (Table3,
Figure5), clearlyshowing
thattheincident
bandwastypically
limitedbydepth,presumably
atthebarcrest.Giventhatthesurfzoneinstruments
werein thetroughof thenearshore
barfor much
oftheexperiment,
it isnosurprise
thatthelocalmeanvalueof7
havedividedthedatasetintotwo subsets
(•,0< 0.40 and•,0>
0.40) andcalculatedthe regression
slopefor eachgroup(Table
4). Whilethereisanincrease
in slopeforthehigher•,0,thedifferenceis not statistically
significantfor thissmallsample.
The large range of conditions experienced during
SUPERDUCK also allowed examination of the impact of the
wind waveincidentangleon infragravitybandvariancelevels.
Directional Dependence (H > 125 cm)
80.00
70.00
D
60.00
DID
50.00
o
o
n
40.00
x x
30.00
x
x
x
x
x
20.00
10.00
0.00
ß
0.00
'
'
'
'
I
5.00
....
I
....
10.00
I
15.00
....
I
20.00
....
25.00
Absolute value wind wave direction (degrees)
X v IG
FI U IG
Fig.7. Thedependence
ofrmsinfragravity
oscillations
ontheabsolute
valueoftheincident
direction,
I•l, forcases
withHrm
s>
125 cm.
12,790
HOWD ET AL.: WAVE VARIANCE PARTITIONING IN TROUGH OF BARREDBEACH
50.00
a
45.00
_
_
40.00
-
x
_
_
x
x
_
35.00
x
_
x
_
30.00
-
25.00
-
20.00
-
o
o
xx x
x
x
15.00
-
10.00
-
5.00
__
_
0.00
0.00
20.00
40.00
60.00
80.00
100.00
120.00
140.00
160.00
180.00
Absolute value mean longshore current (cm/s)
50.00
b
45.0O
-
40.00
-
35.00
-
30.00
-
25,00
-
20.00
-
15.00
-
x
o
o
x
x
x
x
x
x
x
x
x
.
10.00
5.0O
0.00
0.00
20.00
40.00
60.00
80.00
100.00
120.00
140.00
160.00
180.00
Absolutevalue mean longshorecurrent(cm/s)
Fig.8. Valuesof (a) Uswand(b) Vswplottedversus
theabsolute
valueof themeasured
meanlongshore
current,I<V>I.
Multiple regression
analysisshowsa significantnegativerelation
Shear waves are hypothesizedto scale with the longshore
of the longbetweenU/GandI•xl,while thereis not a significantrelationship currentshear(equation(5)), but no measurements
betweenV/GandI•xl(Table5). The directionaldependence
is most
shorecurrentshearwere obtainedfor extendedperiodsof the
clearly illustrated (Figure 7) during the passageof the storm
(October 10-12), when the direction of wave approach was
highlyvariable,but the waveheightwasnearlyconstant.The WG
oscillationsclearly decreaseas I•xl increases,while the
oscillationsare essentiallyconstant.Preliminaryanalysissuggests
that U/Goscillationsin the presenceof a large I•xl(low WG) are
dominatedby high-modeedge waves, while the low I•xl
oscillations are dominated by more energetic leaky or bound
waves [Elgar et al., 1989; Oltman-Shayet al., 1989b].The
oscillationsare expected to be dominatedby low-mode edge
wavesregardless
of the leaky/high-mode
contributions
to WG'
experiment.Thusbasedon the observations
of Oltman-Shayet al.
[1989a], we assumethat the absolutevalue of the measuredmean
longshore
current,I<V>l,is a proxyfor thedynamicallyimportant
shear(Figures8a and 8b). The meancurrent,<V>, wasestimated
as the mean of the 4-hour averagesof the currentmetersusedin
the wavenumber-frequency
spectralanalysis.The UswandVsw
were significantlycorrelatedwith I<V>l, but with considerable
scatter.
In an attemptto better parameterizethe shearwaves,we have
modeledthe longshorecurrent, and its theoreticallyimportant
shear, based on the work of Thornton and Guza [1983, 1986].
HOWDET AL.:WAVE VARIANCEPARTITIONING
IN TROUGHOFBARREDBEACH
Waveheights
in thesurfzonewereassumed
to be Rayleigh-
sin[cz(x)]
V•4(x)
= Ou(x)
C(x)
c•} <œ>
distributed
with a probability(p) of
p(U)
=•
12,791
(13)
cfisanempirical
dragcoefficient,
cz(x)
isdefined
asby
(11) where
Thorntonand Guza [ 1986],C is thephasevelocityof the incident
exp
This distributionis modifiedby a weightingfunctionto providea
statisticalestimateof thosewaveswhich are breaking.The mean
waves,andu is givenfor a Rayleighdistribution
by
u(x)=4•;h(x)
Hrms
(x)
g ] 1/2
dissipation
dueto wavebreaking,<œ>(integratedthroughthe
waveheightdistribution),is givenby
(14)
We have assumedthe bottom stressto be linear, ignored the
turbulentReynold's stresses,
andassumed
constant
valuesfor 7 =
0.45,B = 1.0,andcf= 0.009.
lm
A typicaloutputof the modelis shownin Figure9. The pre(Hrms'•2)
5/2 dicted
maximumlongshorecurrentwas significantlycorrelated
X,•'rmsh
]
with theobserved
current(r2 = 0.942).We assume
themodel
wherep is thedensityof water,B is an empiricalcoefficient doesa reasonable
job of predictingtherelativechanges(between
representing
theportion
of theborefaceactively
breaking,
andf
the 21 data collections)in the cross-shorestructureof the longisthepeakincident
frequency.
It isthensimpleto stepshoreward shorecurrentin the absenceof the mixing inducedby the shear
from the inputconditions
at the 8-m array,calculatingwave
wavesandReynold'sstresses.
No attemptwas madeto further
heightbasedon energyflux balance,with the incidentangle tune the model to match conditions observed in the surf zone.
variedaccording
to Snell'slawof linearwaverefraction.
The dependence
of shearwavemagnitude
onthemaximumof
The waveheightanddissipation
profilesacross
thesurfzone
the modeledshearon the seawardfaceof the longshorecurrentis
werethenusedto predictlongshore
currentfollowingThornton
and Guza, [1986]
10
OCT
86
1030
shownin Figure 10. The correlationsare not significantlyimprovedoverthatwith the meanvelocitymeasured
in the trough
(Table3). Clearly,morefield dataareneededto betterdefinethe
forcingof this new phenomenon.
0.00
DISCUSSION
-0.10
The SUPERDUCK data, collectedin the trough of a barred
-0.20
beach,havea "process
signature"
whichqualitatively
agrees
with
theresultsof Wrightand Short[1983] andWrightet al. [1986].
They presenteddistinctiveratiosbetweenthe differentcompo-
-0.30
0
200
400
600
800
(long-shore
barandtrough/rhythmic
barandbeach)statebetween
fully dissipativeandfully reflective.The signatureis characterizedby low-frequency
oscillations
in thesurfzoneapproximately
half themagnitude
of theincidentbandoscillations.
Four large data setsof lQG oscillationsand incidentwave
heights are available for quantitativecomparisonwith the
1.21.0-
•'g• 0.8
0.4-
0.2
0.0
I
0
I
200
400
600
200
400
600
SUPERDUCK data. Guza and Thornton [1985] presenteddata
I
800
• 1.8•
"•
1 2
EO.6
-r- 0.0
nentsof the flow field basedon the morphodynamic
stateof the
beach.The SUPERDUCK data clearly fall into the intermediate
•
0
•
I
I
0
I
I
200
I
I
4o0
I
I
60o
I
800
I
800
Distonce Offshore (rn)
Fig.9. Modelresults
forH, VM,and3VM/3X
onthemeasured
profile.
The
inputvaluesforH and{• werethosemeasured
attheoffshore
arrayatthe
offshoreboundary
of themodel.Twojetsareseenin thecurrent,oneover
the bar, andoneat the shoreline.The shorelinejet is largeowingto the
assumption
thatall incidentenergyis dissipated
(no reflection).Shear
wavetheorypredicts
thattheshearontheseaward
faceof thebaris the
dynamicallyimportantvariable.
fromtwo experiments
at two nonbarred
beaches
of verydifferent
slopes,Torrey Pines(TP) and SantaBarbara(SB). Thesedata
represent
thevarianceintegrated
overthefrequency
range0.0050.05 Hz and averagedover a variablenumberof currentmeters
positionedat depthsbetween1 and2 m. The authorsnotethat
lessthan 10% of the energylay at frequencies
below0.005 Hz.
Wright et al. [1986] reportedstatisticsfor eight moderate-to
high-energy
(H > 160cm)datarunsof two-foursensors
between
1- and 2-m depth(with highly variablecross-shore
locations)
from a barredsectionof NinetyMile Beachin southeast
Australia
(AU). We choseone representativesensorfor each data run
(rather than average different cross-shorelocations) and
combined the subharmonic and infragravity bands (their
infragravity
bandwasf < 0.03Hz). Holman[ 1981] presented
data
from a singlesensorat 2-m depth,integratedfrom 0 to 0.05 Hz,
on MartiniqueBeach(MA). The SUPERDUCK(SD) variances
for thiscomparison
includecontributions
from all wavenumbers
in thefrequencyband0-0.05 Hz. All datawereconverted
to rms
oscillations.
Figure1la showsthesefive datasetsof UiGoscillations
plotted
versus offshore H. The four North American locations show sta-
12,792
HOWDETAL.:WAVEVARIANCEPARTITIONING
IN TROUGHOFBARREDBEACH
a
50.00
45.00
-
40.00
-
35.00
-
30.00
-
A
x
x
x
x x
x
x
x
_
_
_
o
25.00
_
_
x
20.00 -_
_
x x
15.00
-
10.00
_
5.00
-
-X
-
_
0.00
,
'
'
0.00
'
I
....
I
0.01
....
I
0.02
'
0.03
'
'
I
0.04
.05
0.04
0.05
Model I dV/dx I
b
50.00
45.00
-
40.00
-
35.00
-
A
3O.OO -
25.00
x
-
2O.OO
- X
X
_
_
X
x
x x
X
X
X
>
_
15.0o
x
x
_
x•x
_
10.00
_
5.00 - X
-
0.00
,
0.00
0.01
0.02
0.03
Model I dV/dx I
Fig.10.Values
of(a)Usw
and(b)Vsw
plotted
versus
I•)VM/•XI,
theabsolute
valueofthemodeled
maximum
current
shear
sea-
wardofthebarcrest.
Thescatter,
ofunknown
origin,
isnotsignificantly
reduced.
Compare
withFigure
11.
tisticallysignificant
positiveregression
slopes(Table6) even
whiletheseslopes
varybetween
thedifferent
beaches
(except
betweenTP andSB). Thisis reflectedin thelow skill (r2) of
linearregression
modelsbetweenU/Gandoffshore
H for the
siGp
=aft; (0.05)
[•xL• 0
'
(15)
wherexi is the instrument
locationand the point at whichthe
derivativewith respectto x is evaluated.This approach
is very
Theregression
slopefortheAustralian
datais negative,
a result
similar
to
that
of
$allenger
and
Holman
[1985a].
Assuming
a
of the highly variablecross-shore
locationof the chosensensors
white
spectrum
allows
•21G
to
be
taken
outside
the
integrals.
We
(no onesensorwasoperational
for all runs).
assume
thatthecross-shore
shape,q•(x),canberepresented
by the
Theseresults
arenotunexpected
because
infragravity
waves
leaky wave solutiongiven in (3). Substitutionand solutionfor
havean offshoredecaydependent
on [5.In an effortto remove
beaches
asa group,
butthehighr2foreachbeach
individually.
complications
arisingfromthecross-shore
decayandvarying
instrument
positionsfor the four datasets,we havecalculatedan
"equivalent
shoreline
amplitude,"
d/G,by scaling
thedatafrom
eachbeach
bythepredicted
WGvariance,
S2iGP,
contributed
bya
whiteshoreline
spectrum
of normally
incident
standing
waves
•21Gyields
^2
alG =
•0
2•(ø-ø5)
S(o)
do
Jo21'02X
1122
fo2X(ø'ø5)fo2'{•x[gC
(C-•)
)]}dtdo
(16)
HOWD ET AL.: WAVE VARIANCEPARTITIONINGIN TROUGHOF BARREDBEACH
I NFRAGRAVITY
a
T=T.
BAND
PINES
S=S.
BARB
12,793
U OSCILLATIONS
El=DUCK
M=M 'QUE
A=AUS
140
130
-
120
-
110
-
100
-
90BOA
70-
M
60-
50-
M
D
D
O M
A
D
40-
q•
A
AD
A
A
i
I
302010-
0
'
0
i
120
i
i
!
i
160
i
200
240
i
280
OFFSI.-13•E H i•dSCcm)
PREDICTED
b
SHORELINE
T=T.PINE$
S=S.BAR8
I NFRAGRAVlTY
D=DUCK
HEIGHT
M=M 'QUE A=AUS
8O
D
60-
D
M
T
D
O
A
AD
50-
s
•J
S5 T$
40-
T,•
s'•S•s
• S
M
30-
STSMO
MSs•TT!•$
M
20-
OA
10-
M
•s
D
M
M
OFFSHORE H I:lvlS Ccm•)
Fig.11. (a) Comparison
of U/Goscillations
fromfivebeaches
plotted
versus
theoffshore
waveheight.
Thereis considerable
scatter
in thedataaga whole,yeteachbeachtakenby itselfis quitewellbehaved
(Table5). (b) Thesamedatascaledto estimate
the"equivalent
shoreline
height"
if-IiG
- 2•iGoftheinfragravity
oscillations
(assuming
a whitespectrum).
Whilethedata
collapse
considerably,
theyarenotstatistically
better
correlated
withoffshore
waveheightatthe95%level(Table5).
cross-shore
space,Z = •2•/•13,forthese
measurements.
This approach
is a particularlyusefulmethodto removethe site- sionless
(Figure12), however,that thereis a •0
specificnatureof pointmeasurements
in thesurfzone,thusfacili- Thereare indications
tating the comparisonof data collectedeither on different dependencefor the regressionslopes(equation(4)) in agreement
with the predictionof Holman and Sallenger [1985] that infragravityenergylevelsdependon the Irabarrennumber.
It is importantto note that thesenew measurements
of infra(Figure1lb), theimprovement
in r2 from0.521to0.646isnot
significant
at the95% level.This wouldseemto implythatto the gravity and shearwave variancesare representativeof only one
distance,that of the array. It is clear that theseresults
lowestorder,the magnitudeof cross-shore
infragravityoscilla- cross-shore
tions are much less sensitive to the details of the incident wave
are influencedby the arraylocation,sinceSallengerand Holman
spectrumor the beachtopographythan they are to the wave [ 1985a]haveshownU/Gto be a strongfunctionof x on thissame
height.The lack of significantdependence
on the cross-shore beach. The shearwave measurements,while clearly showingan
alsoneedto be takenin the
measurement
locationmay be dueto the limitedrangein dimen- importantandenergeticphenomenon,
beaches,or at different cross-shorelocationson the samebeach.
While thereis visualimprovement
in the clusteringof the data
12,794
HOWDETAL.-WAVEVARIANCE
PARTITIONING
INTROUGH
OFBARRED
BEACH
REGRESSION
SLOPE
vs.
I RIBARREN
NUMBER
T--T. P I NES S=S. BArB D=DOCK IVI=M 'QUE A=AUS
0.B
0.5
-
0.5
-
0.2
-
0.1
-
0O
'
i
i
0.2
I
I
0.4
I
I
O.E
I t:l I BAI:IREN
I
I
I
0.8
I
I
I
I
1.
12
NUMBER
Fig. 12. Regressionslopesfor the five beaches(with 95% errorbars)as a functionof the meanIribarrennumber(plusor minus
one standarddeviation)for that beach.There is a significant increasein regressionslopewith an increasein Iribarrennumber,
confirmingobservations
that infragravitywavestend to be largeron dissipativebeachesthan on reflectivebeachesfor a given
offshorewave height.
contextof some(presentlyunknown)cross-shore
variancestructure. Recent measurementsmade during the 1990 DELILAH
experimentshouldprovidea clearerframeworkfor understanding
the cross-shore
structureof theselow-frequencymotions.
Theinfragravity
waveoscillations
averaged
20 cms-] and32
cms-] forVlGandtriG,respectively,
andarebestparameterized
CONCLUSIONS
Twenty-one4-hour data runs collectedover a wide rangeof
incident wave conditionshave been used to quantify the bulk
statistical behaviors of the alongshoreand cross-shoreflow
components
of threeclassesof wavesin thetroughof a nearshore
bar. The incident band oscillations,while not locally saturated,
were limited by the presenceof a depthminimumat the offshore
bar. Thus the incident band, while not significantly correlated
m + 95% Confidence
b
r2
DOF
Surf-Zone
U•GrmsOscillations
VersusOffshore
IncidentH*
All beaches
0.174 _+0.068
TorreyPines
0.847_+0.153
0.24
0.924
11
Santa Barbara
Duck
0.839 _+0.162
0.232 + 0.027
7.35
7.80
0.751
0.776
33
19
0.759
25
Martinique
Australia
0.419+ 0.095
--0.339 + 0.315
31.26
3.73
127.69
0.521
0.479
102
6
EquivalentShorelineHeightVersusOf/•horeIncidentH*
13.92
All beaches
0.215 + 0.031
TorreyPines
0.554+ 0.100
0.15
0.924
11
Santa Barbara
Duck
0.459 + 0.089
0.237 + 0.057
4.02
7.77
0.751
0.790
33
19
Martinique
0.323+ 0.073
0.759
25
Australia
0.185 + 0.157
2.87
by the offshorewave height. During stormsthey provided up to
33% of the total variance,with an averageof approximately12%.
Thesepercentagesare expectedto be higherat the shoreline.We
have alsofoundevidencethat UiG,the cross-shore
componentof
infragravitywave oscillation,was, to a lesserdegree,sensitiveto
the directional character of the incident waves.
Shear waves were found to be an importantcontributorto the
energeticsof this beach,typicallycomprisingapproximately10%
of the total variance,but at times contributingup to 35%. The
magnitudes
of theoscillations
exceeded
40 cms-l in cross-shore
and30 cms-l alongshore.
Shearwavemagnitudes
werecorre-
TABLE 6. Intercomparison
of Beaches
Beach
with the offshorewave height,was correlatedwith the tide elevation. The incidentbandat this location,approximately55 m from
the shorelinein the bar trough,is the largestcontributorto total
variance,typically providing70-80%.
0.646
102
lated with the seaward facing shear of the longshorecurrent
(which was modeled,not measured),as expectedfrom theory. It
is difficult to place thesemeasurementsin cross-shorecontext
with this data, owing to the lack of adequatemeasurement
of the
cross-shore
structureof the longshorecurrent.
Comparisonof the cross-shorecomponentof the infragravity
bandwith datacollectedon otherbeachesandpreviouslyreported
in the literature showsthat while each experiment'sdata scales
approximatelylinearlywith offshorewaveheight,the trendof the
relationshipvariesas a functionof the Iribarrennumber.For these
data, all collectedin a similar depthof water, correctingfor the
cross-shorelocation of the sensorprovides only a minor improvementin the correlationwith the offshorewave height.
6
Acknowledgments.
We wouldlike to thank(onceagain)the staff of
CERC's Field ResearchFacility duringthe SUPERDUCK experiment.
*Model:
u•o= mH+ b+ error
orequivalent
shoreline
height
(//to)=
Not onlydidtheyassistwhenever
andwherever
needed,theyoftendidso
withouthavingto be asked.In additionto the manpowersupport,CERC
mH + b + error.
14.26
0.525
HOWD ET AL.: WAVE VARIANCE PARTITIONINGIN TROUGH OF BARREDBEACH
fundedthedeployment
of thegagesandthecollection
of thedatawhich
form the bulk of this paper.P.A.H. wasemployedby, andJ.O.S.was
undercontract
to, CERCduringtheperiodof theexperiment.
R.A.H. and
thesubsequent
analysis
of thedatahavebeenfundedby grantsfromthe
National ScienceFoundation(OCE87-11121) and the Office of Naval
Research,
CoastalSciences
Program
(N00014-87-K0009).
Thispaperhas
beenimproved
by thecomments
of TonyBowen,BobGuza,andthetwo
anonymous
reviewers.
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(Received October 31, 1989;
acceptedFebruary22, 1990.)