WATER RESOURCES RESEARCH,
VOL. 35, NO. 8, PAGES 2313-2320, AUGUST
1999
Tidal water table fluctuations in a sandy ocean beach
B. Raubenheimer
• and R. T. Guza
Center for CoastalStudies,ScrippsInstitutionof Oceanography,La Jolla, California
SteveElgar•
ElectricalEngineeringand ComputerScience,WashingtonState University,Pullman
Abstract. Tidal water table fluctuationsobservedfor 27 daysin a gentlyslopedocean
beachare predictedwell by numericalmodelsbasedon the Boussinesq
equationdriven
with the observed10 min-averagedshoreline(ocean-beach
intersection)motion.Diurnal
and semidiurnalwater table fluctuationsare almostcompletelydamped100 rn landwardof
the mean shorelinelocationon this fine-grainedsandbeach,but fluctuationsat springneap periods(•14 days)are attenuatedless.Comparisonof the observations
with the
predictionssuggests
that the asymmetriesin the water table level time seriesmeasuredin
this studyresultfrom nonlinearityowingto the large (relativeto the wavelength)
horizontalshorelineexcursions,
rather than from nonlinearityowingto finite-amplitude
water table fluctuations.Cross-shore
variationsof the aquiferdepth are predictedto have
a smalleffecton the landwarddecayrate of the watertablefluctuations.
The seepageface
width is predictedaccuratelyand dependson the nonplanarbeachprofile. In general,the
developmentof a seepageface is predictedto have little effect on the water table level
landwardof the intertidal region.
dition are/• = A/D and e = kA cot/3, respectively,
whereA
and k are the vertical amplitude and wavenumberof the
The elevation of the groundwaterin a sandybeach is be- groundwater
fluctuations,
D isthe (constant)aquiferthickness,
lievedto be importantto erosion[Grant, 1948;Duncan, 1964; and/3 is the (planar) beachslope.
Eliot and Clarke,1988],saltwaterintrusion[Nielsen,1999],and
Nielsen [1990] used analyticsolutionsof the Boussinesq
biologicalactivity [Pollockand Hummon, 1971;McArdle and equationsto examinethe effectsof the movingshorelineon
McLachan, 1991]. Ocean tides drive fluctuationsin the beach water table fluctuations.To simplifythe equations,the aquifer
watertablethat decaylandwardfrom the shoreline(the beach- thicknessand beach slopewere assumedto be constant,and
oceanintersection),are asymmetric
and skewed(e.g.,pitched the water table was assumed to intersect the beachface at the
forward and peaky) in time, and have averagelevelshigher shoreline(e.g., no seepageface). When terms of order
(overheight)than mean sea level [Emeryand Foster,1948; (O(e)) are negligible(e.g.,a steeplyslopedbeach),perturbaIsaacsand Bascom,1949;Lanyonet al., 1982].When the tide tion expansionof the nonlinear(finite-amplitude)Boussinesq
falls more rapidlythan the beachdrains,a seepageface forms equationgives(to O(/•))
1.
Introduction
between
the shoreline
and the intersection
of the water table
with the beach(the "watertableoutcrop"),possiblyenhancing r•(x, t) = A cos(rot- kx)e-•
sedimenttransport[Turner,1993].
Numericalmodelsbasedon the nonlinear(finite-amplitude)
+ •-[1 - e + cos
(2rot
Boussinesq
equation(appropriatefor shallowaquifers)and
tzA -2•
initialized
with water table fluctuations
observed inland of the
intertidalregion,the area of the beachfacealternatelycovered
and uncoveredby the oceantides,accuratelypredict the amplitudeand phaseof tidal water tablefluctuationsfarther landward [Harrisonet al., 1971;Dominicket al., 1971;Fang et al.,
1972]. Recent studieshave concernedprocesseswithin the
intertidal region, which can be wide on beacheswith gentle
slopesor large tides. When the horizontal excursionof the
shorelinelocationovera tidal periodis significant,nonlinearity
introduced
at the shoreline
- cos(2rot- 2kx)e-2*x]
x/•kx)
e-v•x
(1)
whererois the radianfrequencyof the monochromatic
(singlefrequency)oceantide and r•(x, t) representsdeviationsof
the water tableheightat cross-shore
positionx and time t from
the still heightD (e.g.,the total water tableheighth (x, t) =
r/(x, t) + D). In contrast,if termsof O(/•) are negligible,
perturbation solutions of the linear (small-amplitude)
Boussinesq
equationare (to O(e))
affects the inland water table fluc-
tuations. The nonlinearity parameters associatedwith the
Boussinesq
equationand the movingshorelineboundarycon•NowatWoodsHoleOceanographic
Institution,
WoodsHole,Mas-
r/(x, t) = A cos(rot- kx)e-•
[1 xf•
+•A •+•-cos(2rot+z'/4-
x/•kx)el(2
)
sachusetts.
Thusthe meanoverheightof the water tablecouldresultfrom
finite-amplitude
tidalfluctuations
(termsof O(/•)) or from the
movingshoreline(termsof O(e)). The fluctuations
of O(e) at
harmonicfrequencies(e.g.,2ro)in (2) are associated
with the
Copyright1999by the AmericanGeophysicalUnion.
Paper number 1999WR900105.
0043-1397/99/1999WR900105509.00
2313
2314
RAUBENHEIMER
ET AL.: SANDY
OCEAN
BEACH
WATER
TABLE
FLUCTUATIONS
simulationsto assessdifferent model terms and to studyharmonicgenerationin the intertidal regionare describedin section 5.
'•
-0'
-0-0-0'0
....
2.
Observations
Waves, tides, and water table fluctuationswere sampled
nearlycontinuously
at 2 Hz (real-timefrequency)from September 18 to October 14, 1996, at Torrey Pines Beach, California (Figure 1). Surf zone waveswere measuredwith 16
pressuresensors
locatednearthe sandsurface,andwatertable
E
HIGH
fluctuationswere measuredwith 22 buried pressuresensors.
c
0
Wave and water table sensorswere stackedverticallyto esti.o
•........_.._
ß __•
,ow
mate verticalinfiltrationand to detectcapillaryeffects[Gill•
-2
ham, 1984; Turnerand Nielsen,1997]. Beach surfaceprofile
changes,calculatedfrom daily surveysto -2-m water depth,
weresmall(<10 cm)bothin the intertidalregion(-40 -< x <
1 O0
50
0
-50
-1 O0
-150
39 m in Figure lb) and landward.At the beginning,middle,
Cross-shere Dislance(rn)
and end of the experimentthe profilewasmeasuredlandward
of
-5-m water depth.The averagebeachslopeoverthe interFigure 1. (a) Experimentplan view showinglocationsof
tidal
regionwas -0.03, with slopesof 0.05 landwardand 0.02
pressuresensors(solid circles)and beachelevationcontours
(in metersrelativeto meansealevel).The cross-shore
transect seaward of x = 0 m, defined as the intersection of the beach
offshoresea level. Off(dottedline) corresponds
to the modeltrace.(b) Sensorloca- profile and the experiment-averaged
tions,Torrey PinesBeachprofile, and the envelopeof water shorewind wave heightsmeasuredin 10-m water depth -35
table and tidal levels (dashedcurves).The thick solid and km north of the experimentsite rangedfrom 40 to 90 cm, and
dotted curves at the bottom represent the measuredupper peak periodsrangedfrom 8 to 20 s. No rainfallwasrecorded.
surfaceof the Ardath Shale and the (unmeasured)offshore
At 15 locations between cross-shoredistances10 and 100 m,
shaleposition,respectively,usedin the finite-amplitudenu- 8-m deep holeswere drilled to collectsedimentcoresand to
v
IIIlllllllll
I
merical model.
moving shoreline and propagate landward as damped free
waves,satisfyingthe linear dispersionrelation
= 2:z>
determinethe beachstratigraphy.Sieveanalysis
wasconducted
on 11 coresto determinegrain sizes,and five coreswere used
to determinethe porosityand permeabilityof the beachsedimentarylayers.The analysissuggested
that the beachis composedof -2-4 m of uniformfine to mediumsandwith traces
of silt (also• 0.23 mm, porosity•37%, and hydrauliccon-
-1 m of Scripps
formation
(3) ductivity•0.07 cms-•) overlying
where K is the hydraulicconductivityin the saturatedbeach
andne is the effectiveporosity.In contrast,the termsof O(•)
in (1) resultin harmonicsat frequency2towith wavenumbers
both of boundwaves(2k) not satisfying
the free wavedisper-
composedof dense,siltyfine sandwith tracesof gravel(dso•
0.15mm, porosity•33%, andhydraulicconductivity•0.03 cm
s-•) on top of low-permeability
ArdathShale(also• 0.004
mm, porosity•32%, and hydraulicconductivity•0.0005 cm
s-•). The averagehydraulicconductivity
of the unconfined
sionrelation
(3) andof freewaves
(X/•k). Thereisa similar aquifer, estimatedfrom slug tests [Bouwerand Rice, 1976;
generationof free harmonicsby a movingboundaryand of
bound harmonicsby nonlinearity away from the boundary
whenwater wavesare producedby sinusoidaloscillations
of a
wavemakerin a flume [Madsen,1971].
The overheight,harmonicgeneration,and landwarddecay
of fluctuationspredictedby the analyticsolutionfor small
amplitude(2) agreeroughlywith observations
collectedovera
tidal period [Nielsen,1990].Improvedagreementis obtained
Brown and Narasimhan,1995] conductedat cross-shoredis-
tances
60and91m,was0.07cms-•. All pressure
sensors
were
locatedabovethe Scrippsformation.
Experimentmeanwater table levels(Figure 2a) are higher
than mean sea level at all locations
and increased
landward
(becauseof an inland water sourceand nonlinearlydriven
overheight,e.g.,(1) and (2)). Offshoretidal amplitudesranged
from -50 cm duringneaptidesto 100 cm duringspringtides
with numericalsolutions
of the fully nonlinearBoussinesq(Figure 2b). Diurnal and higher-frequency
tidal water table
equations(and extensions)
[Kangand Nielsen,1996;Li et al., fluctuationsdecaylandward(analogousto the conductivepen1997a;Baird et al., 1998].However,the relativeimportanceto etrationof surfaceheat) and are almostcompletelydamped
tidal water table fluctuations of spatially variable aquifer 98 m onshore of the mean shoreline, but fluctuations at the
depth, nonplanarbathymetry,developmentof the seepage frequencyof the spring-neaptidal cycleare attenuatedless
face, and nonlinearityowingto finite amplitudeand the mov- (Figure 2a). Spring-neapwater table fluctuationshave not
beenobservedpreviously,probablybecausemostobservations
ing shorelineare not understoodwell.
Here small-amplitude(• • 0.2) tidal water table fluctua- spannedonly a few days.Oceanwater level and beachwater
tionsand seepagefacelocationsmeasuredcontinuously
for 27 table fluctuationsare dominatedby tideswith 25- and 12-hour
days(e.g.,onelunarcycle)withina gentlyslopednaturalbeach periods(Figures2b and 3). The water table spectraalsoconharmonics(e.g., at multiples
with largeshorelineexcursions
(e >- 1) are usedto testfurther tain peaksat higher-frequency
of the diurnaland semidiurnalfrequencies).
the Boussinesqequations.The observationsand models are and combinations
describedin sections2 and 3, respectively,and model predic- The magnitudesof water table fluctuationsat all frequencies
tionsare comparedwith observations
in section4. Numerical decaylandward(x > 39 m) of the intertidal region.Within
RAUBENHEIMER
ET AL.' SANDY
OCEAN
BEACH
WATER
TABLE
FLUCTUATIONS
2315
the intertidalregion,energywith 6-, 8-, 12-, and 25-hourperiods reaches a maximum near the mean shoreline location,
and energyat 4- and 5-hour periodsreachesa maximumfarther landwardnear cross-shore
distance39 m (Figure3).
3.
Numerical
Models
The Boussinesq
equationfor finite-amplitude(nonlinear)
watertablefluctuationsand spatiallyvariableaquiferthickness
D(x) is
105 -
12
hr
25 hr
Offshore_
6hr
•-• lO
4
E
103
ß•
lO2
8 hr
For small-amplitude
(linear)watertablefluctuationsand constantaquifer depth the Boussinesq
equationbecomes
O•l(x, t)
Ot
KD 02•l(x, t)
ne
OX2
x=
4m
x=59m
.....
x=46m
•
x = 60rr•
!'iir"r ,.r-
>- 101
o•
= ne
' t)]
OX } (4)
Oxl(x,t)
KOx
0{[D(x)
+xl(x
Oxl(x,t)
........
1oø
10-1
0.05
(5)
0.10
0.15
0.20
0.25
0.50
Frequency(hr-')
FollowingLiu and Wen[1997],(4) and (5) weresolvednumer- Figure 3. Energydensityspectra(12 degreesof freedom)of
observed offshore water level fluctuations and water table flucicallyusinga centeredfinite differencemethodin spaceand a
tuations at cross-shoredistances4, 39, 46, and 60 m.
fourth-orderRunga-Kuttaintegrationtechniquein time. The
seawardboundaryconditionfor the numericalsolutionsis
•(Xs, t) = Zs
(6)
whereXs and Zs are the horizontal location and the elevation,
respectively,of the shorelineat time t. Perturbationsolutions
to (4) and(5), assuming
constantbeachslope,constantaquifer
depth,and monochromatic
oceantides,are givenby (1) and
(2), respectively.
The Boussinesq
equationis basedon the assumptions
that
the sanddrainsinstantaneously
(e.g.,no capillaryfringe),density gradientsare negligible,and horizontal flows are much
larger than verticalflows.Tensionheadsobservedwith pressure sensorsnear the sandsurfacesuggestthat the capillary
fringe extended-50 cm abovethe water table level. However,
capillaryeffectsare believedto have little influenceon water
table fluctuationsat tidal frequencies[Barryet al., 1996;Li et
al., 1997b].Densitygradientsowingto freshwaterlandwardof
the beachberm and salineoceanwater in the intertidalregion
havebeen observedpreviously[Harrisonet al., 1971;Nielsen,
1999].Densitieswere not measuredin the presentexperiment,
but the goodagreementof the observedtidal water table levels
with the predictionsdescribedbelow suggests
that densityeffectswere small.Horizontalflowsusuallyare muchlargerthan
verticalflows(seethe appendix)on the 10-minto 14-daytimescalesconsidered
here,consistent
with the modelassumptions.
In the finite-amplitude
variable-aquiferdepthmodel(4) the
bottomboundarywas at the top of the Ardath Shale(Figure
lb). In the small-amplitude
constant-aquifer
depthmodel(5)
the bottomboundarywas at the average(over the model domain) level of the shalelayer (z = -3.5 m), and D wasset
equal to 4.7 m, the time- and space-averaged
observedwater
table height.Above the impermeableboundarythe beachmaterial was assumedto be homogeneous
and isotropic,with an
averagehydraulicconductivity
K equalto 0.07 cm s-•. The
100
effectiveporosityne equaled0.215, determinedusinga least
squaresfit of the finite-amplitudemodel predictionsto the
measuredwater level changes.The inlandpropagationof water table fluctuationsand the seepageface width predicted
using(4) and (5) dependon the ratio K/he [e.g.,Baird et al.,
1998],sodifferentvaluesof K changedthe corresponding
best
fit value of n• but did not alter the agreementbetweenobservationsand predictions.The modelswere initialized with the
watertablelevelobservedat the startof the experiment.Initial
-
levels onshore of the most landward
i:
•
sensor were estimated
using a natural cubic spline extrapolation.Predicted water
tablefluctuations
were negligiblysmallat the inlandboundary
o
-50
-100
Sep 22
Sep 27
Ocr 02
Date
Ocr 07
Ocr 12
(located250 m onshoreof the meanshoreline),wherea constant seawaterhead (Dirichlet) conditionwas imposed.Increasing(decreasing)
the head(2.55m abovemeansealevel)
at the modelinlandboundaryby 40% resultsin an 8% increase
(5% decrease)of the seepagefacewidth and a 14% decrease
(14% increase)of the water table fluctuationamplitudesinland of the maximum
shoreline
location.
Figure 2. Observed(a) 10 min-averagedwater table fluctuThe locationsof the shorelineandthe watertableoutcropat
ations(relativeto meansealevel) at cross-shore
distances
98,
madewith
60, and 39 m, and (b) offshore34 min-averagedsea surface the sandsurfacewere estimatedusingobservations
levels versus time.
the closelyspacedsensorsin the intertidal region and the
2316
RAUBENHEIMER
ET AL.: SANDY
OCEAN
BEACH
WATER
TABLE
FLUCTUATIONS
the seawardmodelboundarycondition(6) waslocatedat the
6O
observed
low-passed
(f -< 0.11 h-•) shoreline
location.
o
40
.o
4.
>o 20
0
• -20
....
Sep 22
100
observations
I
..........
I
b ,-•
x:..
E
i
finite-amplitude predictions
small-amplitude predictions
....
i
Sep 27
t •
-.
....
Oct 02
-'
t."
i
....
i
Oct 07
.
Oct 12
,- • .,.'..,
,..r.,
t:".• v
-.
"
x=4m
v
50
.o
ß
0
'"
.
Q)
shoreline
, , :,, :,, :,,/!:
o
"0
ii
I tI I
I
I
-
,,:
I i
I t I I II tt II II II tI _
I 'I II II I t I I I i
:s0"
,,
',,
'''111'lll'l''l'l''l'''l''ll'll'll':llll''llllll
Oct 08
Oct 09
I I
Oct 10
Oct 11
Oct 12
'
I I
,,,
I
Oct 15
Model-Data Comparisons
The water table observationsare predictedwell by both
numericalmodels(Figure4), suggesting
that for the hydrologic
conditionsconsideredhere, nonlinearitiesare unimportantinland of the movingshoreline.The average(over 27 days)
errorsof the small-and finite-amplitudemodelpredictionsare
<10 cm. The largestmean errorsare underpredictionof the
averagewatertablelevelsin the intertidalregion.The standard
deviations(e.g.,meanerrorsremoved)of differencesbetween
the observationsand the small- and finite-amplitudemodel
predictionsare <4 and 3 cm, respectively.
The magnitudesof the spring-neap
tidal cyclenear the berm
(x = 46 m, Figure 4a) are similarto a time-modulatedtidal
overheight.However,the observedand predictedspring-neap
water table fluctuationamplitudesdecreaselandwardwith an
increasing
phaselag (comparex= 46 withx = 98 m in Figure
4a). The landwardamplitudedecayalsois predictedby perturbation solutions(not shown) to (5) for multifrequency
shoreline
excursions.
When
ocean tides are assumed to be
bichromaticwith energyat frequencies6oand 6o+ 156o
(e.g.,
shorelinefluctuationsat semidiurnalsolar (12.00 hour) and
lunar (12.42 hour) tidal periods),groundwaterfluctuationsof
O(e) includefree wavesat the differencefrequency156o
(e.g.,
spring-neapperiods)as well as at the sum frequencies26o,
26o+ /56o,and 26o+ 2/56o.
The numericalmodelsaccuratelypredict the water table
fluctuations
at diurnalandhighertidalfrequencies
(Figure4b),
Figure 4. Water tab]eand shore]inefluctuationsversustime:
including
the
observed
frequency-dependent
landward
decay
(a) 25 hour-averagedfor 24 daysat cross-shore
distances46
phaselags(Figure5). The observedslowdrainand 98 m and (b) 10 rain-averagedfor 5 daysat cross-shore andincreasing
distances4, 46, and 60 m. Observationsand small- and finite- ageof the beachat low tide,whichcausesskewed(e.g.,peaky)
(e.g.,pitchedforward)waterlevelfluctuation
amp]itudepredictionsat eachlocationare re]ativeto an arbi- and asymmetric
trary datum.
time series,is predictedwell (e.g.,x = 4 and 46 m in Figure
4b). The observedand predictedskewness
and asymmetryof
Dote
the water table fluctuations
increase across the intertidal
re-
gion,reachingmaximumvaluesnear the berm beforedecreasmeasuredbeachprofile.The cross-shore
structureof the mean ing landward(Figure 6).
On the fallingtide the formationof a seepagefacebetween
water surfacewas estimatedby fitting a cubic spline to 10
min-averagedwater levels.Sandlevel was determinedby fit- the water tableoutcropand the shorelineis predictedby both
ting a cubic spline to the initial beach profile. Becauseof numericalmodels(Figure7). The watertableoutcroplocation
by the finite-amplitude
modelbut is too
inaccuracies
in the mean water levelsand beachprofiles,the is predictedaccurately
shoreline location was defined as the most offshore location
far seawardin the small-amplitudemodel, so the observed
wherethe water depthabovethe beachprofilewaslessthan a seepageface width (averagevalue 10 m) is underestimated.
betweenobserved
small number 15
s. The water table outcroplocationwas esti- Average(+_standarddeviation)differences
matedasthe mostlandwardpositionwherethewatertablewas and predictedseepageface widthsare 0.3 _+4.4 and -1.8 _+
models,respectively.
lessthan 15
w below sandlevel. The resultswere insensitiveto 4.3 m for the finite- and small-amplitude
Previousstudieshavesuggested
that the seepagefacewidth
variationof 15
s and15
wbetween1 and5 cm andto the O(10 cm)
changesin the beachprofile that occurredduring the experi- on macrotidalbeachescan be estimatedby assumingthat the
ment. In the resultsdiscussed
below, 15
s - 15
w - 2 cm. At low water table outcroppositionis independentof the pressure
tide,whenthebeachwassaturated,
the10min-•/verage
shore- distributionwithinthe beachanddependsonlyon the rate that
line wasnear the maximumwaverunuplocationin that 10-min the tide falls,the beachslope,andgirl e [Dracos,1963;Turner,
period,whereasat high tide the shorelinewasnear the water 1993]. However, this simple model and the presentoutcrop
table outcropbecausewavesextendingfartherlandwardinfil- observationsagree only if the ratio girl e is increasedunrealtratedinto the dry sand.This shorelinedefinition,suggested
by istically(by a factorof 8) relativeto that usedin the numerical
Nielsen [1989], resulted in a skewingof the shorelinetime Boussinesqmodel. Furthermore, model simulations(not
seriesandthereforein spectralpeaksat harmonicfrequencies. shown)suggestthat the seepagefacewidth is sensitiveto the
However, water table predictionsobtained by driving the head (or hydraulicgradient)at the model inland boundary.
modelwiththetotal(f -< 3.00 h-•, containing
allharmonics) Consistentwith previousobservations[Turner,1993],the preandwith low pass-filtered
(f -< 0.11 h-•, no harmonics) dictedseepagefacewidthdependson the beachgeometry.The
shoreline time series were similar. In the results shown below
averageseepagefacewidth predictednumericallyon a planar
RAUBENHEIMER
ET AL.'
SANDY
OCEAN
BEACH
WATER
beachwith a slope equal to the averageobservedintertidal
slope(0.031)is <-1 m (not shown),approximately
one-tenth
that predictedusingthe observedprofile.Nielsen[1990]speculated that underpredictionof the asymmetryby (2) resulted
from the neglectof a seepageface.However,inland (x > 39
m) of the intertidal region,the amplitudedecayrate, phase
lags,skewness,
and asymmetryof water table fluctuationspredictedon planarandmeasuredprofilesare insensitive(differencesare lessthan 1, 1, 3, and 6%, respectively)to the substantiallydifferent seepageface widths.
TABLE
FLUCTUATIONS
2317
1.5
(A)
1.0
="
0.5
0.0
..........
flnife-ampllfude
1:3
-0.5
observations
1.5
1.0
5.
Discussion
5.1.
E
Nonlinearity and Aquifer Depth
E
0.5
........
[] 0
'....
The predictions of the finite-amplitude variable-aquifer
0.0
depthand small-amplitudeconstant-aquiferdepthmodelsare
-0.5
similar (Figures4-7), suggesting
that the effectsof variable
1 O0
80
60
40
20
0
aquiferthickness(the deviationof the shalelayer from horiCross-shore
Dislance
(rn)
zontal,Figure lb) andnonlinearityin the Boussinesq
equation
are not large for conditionssimilar to those observedhere. Figure 6. (a) Skewness
and (b) asymmetryof measuredand
Althoughsmall,the effect of variableaquifer thicknessin the small- and finite-amplitude-model-predicted
water table flucmodel
simulations
is detectable.
Landward
of the intertidal
tuations versus cross-shore distance.
region,the small-amplitudemodelpredictsthat all frequency
componentspropagateas free waves
•7(x, t) = Ae -kRx
cos(•ot - k/x)
(7)
where kR and k/ are the real and imaginaryparts of the
wavenumber,
respectively,
andare givenbykR = k/= k (e.g.,
(3)). Valuesof k givenanalyticallyby (3) differ by <1% from
valuesof kn and k• obtainedby fitting (7) to the numerical
small-amplitudemodel (5) predictions,confirmingthe accuracy of the numericalsolution.The kn and k• estimatedby
fitting (7) to the observationsand to the finite-amplitude
model (4) predictionsare similarto the analytic(and equivalent small-amplitude
model) values(compareopen and solid
symbols,respectively,
with the solidcurvein Figure8a). How-
kR = 0.085 and0.099m-• andk/= 0.077 and0.087m-• for
frequencies
f = 0.12 and 0.16 hr-•, respectively.
At lower
(A)
frequencies,where the effect of finite aquifer depth is small
(and the agreementbetweenthe observations
andpredictions
E 10.00
25hr
'o
1.00
E
0.10
ever, at fixedfrequencythe observedand finite-amplitudekn
are largerthanthe k/. Retainingfinite amplitudebut assuming
constantaquifer depth in the finite-amplitudemodel yields
kn • k/• k (Figure 8b), indicatingthat differencesbetween
the finite- and small-amplitudemodel predictionsare caused
by the variableaquiferdepth,rather than by nonlinprimari•
earity.
Differencesbetweenthe observations
andmodelpredictions
at 6- and 8-hourperiodsmay be causedby either an intermediate-depth(0.2 < ne•oD/K < 1.0) aquifer [Parlangeet al.,
1984;Nielsenet al., 1996;Liu and Wen,1997]or the capillary
fringe [Barryet al., 1996; Li et al., 1997b]. Small-amplitude
intermediatedepth-aquifersolutions[Nielsenet al., 1996]yield
' •. •"
3O
•
......•....
.•'- 8hr
0.01
finite-amplitude
ß • observations
lO
-100
o
0
1 O0
-20
200
.
1 O0
80
60
40
20
0
Cross-shoreDislance(rn)
- 20
___ . observed
shoreline•
--.,•0
................
Oct Og
Oct
_•
finite-amplitude
outcrop
•
observed outcrop
.s.rn.a•!
:a.m.
p,,!u.d.e
.o.u!c.ro.p.
•I
10
Oct
11
Oct
12
Oct
13
Figure 5. (a) Amplitudesand (b) phaselags(with respectto
Date
observationsat x = 46 m) of water table fluctuationswith
periods of 25 and 8 hours versuscross-shoredistance.Ob- Figure 7. Cross-shore location of the observed 10 minservedvaluesand small-and finite-amplitude-model-predicted averagedshorelineand water table outcropand of the water
valuesare calculatedusingcrossspectrawith 36 degreesof table outcroppredictedwith the finite- and small-amplitude
freedom(dof). Resultsfor 12- and6-hourperiods(not shown) numerical modelsversustime. The seepageface is located
are similar.
betweenthe shorelineand the water table outcrop.
2318
RAUBENHEIMER
0.11
ET AL.: SANDY
OCEAN
0.06
.o
0.05
.•X
ß
o
dora
model
ß
a
kt
ß
o
•
E 0.04
•:
FLUCTUATIONS
(0.02 -<f -< 0.10 h-•) shoreline
fluctuations.
Differences
in
0.08
0.07
TABLE
predictions driven with combined diurnal and semidiurnal
0.09
,-•
WATER
0.06 h- •) or semidiurnal
(0.06 -<f -< 0.10 h- •) periodwith
(A)
0.10
7'
BEACH
(•)
0.11
0.10
0.09
0.08
0.07
0.06
•
0.05
12hr
. 25 hr
0.04
0.04
0.06
0.08
0.10
the modelpredictions
at tidalharmonicperiods(e.g.,8, 6, 5, 4,
and3 hours)are ascribedto the (nonlinear)effectof energyat
the periodneglectedin the shorelineboundarycondition.Consistentwith the analyticalsolution(2) for a slopingbeach,
water table fluctuationsare generatedat harmonics(e.g., integermultiples)of the frequenciesof the peak shorelinefluctuations(Figure9). The magnitudeof spectrallevelsat 6- and
4-hour periodswith semidiurnalforcing alone are similar to
thosepredictedwith combinedforcing(comparedashedwith
solidcurvesin Figure 9), suggesting
that theseenergypeaks
are harmonicsof the semidiurnaltide. However, the energy
levelsat 8-, 5-, and 3-hourperiodspredictedwith onlydiurnal
forcing are reducedsubstantiallyrelative to thosepredicted
with combinedforcing(comparedottedwith solidcurvesin
Figure9), suggesting
that energyat theseperiodsis generated
by nonlinearsuminteractionsbetweendiurnaland semidiurnal
0.12
0.14
0.16
shoreline
fluctuations.
Frequency(hr")
Figure 8. Real (kR) and imaginary(ki) part of the wave- 6. Summary
numberestimatedusing(a) observations
andfinite-amplitude Tidal water table levels were observed on an ocean beach
variable-aquiferdepth predictionsand (b) finite-amplitude withlarge(relativeto thewavelength,
e > 1) horizontalshoreconstant-aquiferdepth predictionsversusfrequency.Vertical line excursions.Similar to previousobservations,
water table
bars representthe 99% confidencelimits.
fluctuations
decreaselandwardwith increasing
phaselagsand
are asymmetricand skewedin time (Figures2-6). Although
diurnal
is good),variableaquiferdepth likely contributesto the differencesbetweenobservedkR and ki.
5.2.
Moving Shoreline
The generationby the movingshorelineof energyat tidal
harmonicswas investigatedby comparingnumerical model
predictionsdriven with shorelinetime seriesband passedto
includeenergyhavingonlya diurnal(25 hour,e.g.,0.02 -<f -<
104
l .... ' .... '"'1""
I ....
25hr ,2 h?
•
.• •
.•,
E
ß
c•
•
',kJr;ll'n;
0.02 < f < 0.10 hr•
/
I ......... 0.02• f
I-- -- -- 0.06• f < 0.10hr•
8hr'6h• - -•
102
1
10
and semidiurnal
water
table fluctuations
are almost
completelydamped 100 m landward of the mean shoreline
location,fluctuationsat spring-neapperiods(•14 days) are
attenuatedless(Figures2 and 4). The observedfluctuations
are described
well (Figures4-6) bybothfinite-amplitude
variable-aquiferdepth (4) and small-amplitude
constant-aquifer
depth(5) numericalBoussinesq
models.The movingshoreline,
includedin both models,resultsin the generationof harmonic
fluctuations(Figure 9). For the hydraulicconditionsconsidered here, nonlinearitiesare unimportantinland of the intertidal region, and water table fluctuationspropagateas free
wavesapproximatelysatisfyingthe dispersionequation(3)
based on the linear (small-amplitude)Boussinesq
equation
with a constantaquifer depth (Figure 8). Cross-shore
variationsof the aquiferdepth,neglectedin (3), causesmallbut
detectabledifferencesbetweenthe predictedreal and imaginarypartsof the wavenumber.Developmentof a seepageface
(Figure 7) and nonplanarbeachbathymetryare predictedto
O0
'.
,:•
•,
.. '
":
have little effect on inland water table fluctuations. Additional
observations
and modelingare neededto generalizetheseresultsto beacheswith properties(e.g.,slope,hydraulicconductivity,porosity,aquiferdepth,andhydraulicgradient)andtidal
rangesdifferent than thoseconsideredhere.
-2
Appendix: Horizontal and Vertical Flows
.
The horizontalu andverticalw flowsthroughsaturatedsand
beneaththe berm and intertidalzone are estimatedusinggraFr•qu•no•
dientsof observed10 min-averagedpressureheads(Pw +
Figure 9. Energy density spectra of finite amplitude- z•p#, wherepw is measuredpressure,z• is the verticalsensor
predictedwater table fluctuationsat a cross-shore
distanceof locationrelativeto a fixeddatum,p is the oceanwater density,
andDarcy'slawfor laminar
35 m. The modelwasdrivenwith band-passed
observedshore- and# isgravitationalacceleration)
line time series. Small-amplitudemodel predictions (not flow,
shown)are similar,so onlyfinite-amplitudemodelresultsare
K O(pw/pg- zR)
shown.Numbersaboveeachspectralpeakindicatethe approxu = +•
Ox
(A1)
imate period.
0.0•
O. 10
O. 1 •
0.20
0.2•
0.•0
RAUBENHEIMER
ET AL.: SANDY
0.03
OCEAN
BEACH
WATER
TABLE
FLUCTUATIONS
2319
tx nonlinearityparameter.
ne effectiveporosity.
u/K
0.02
Pw
0.01
pressure.
p density.
t
0.00
-O.Ol
-0.02
Sep 22
Sep 27
Ocr 02
Ocr 07
time.
u
horizontal
w
vertical
flow.
flow.
to wavefrequency(radians).
Ocr 12
x
Dale
Xs
z
zs
Zr
cross-shore
distance.
horizontal shorelinelocation.
depth below sandsurface.
shorelineelevation.
sensorelevation.
Figure A1. Horizontal (u) and vertical (w) velocitiesnormalized by hydraulicconductivityK versustime. Velocities
were estimatedusing10 min-averagedpressuregradientsmeasuredwith horizontally(x = 39 and53 m at z = - 1.1 m) and
vertically(z = 0.6 and -2.2 m atx = 46 m) separatedsensors
(Figure lb). Negativehorizontaland verticalflowsare landAcknowledgments.This researchwas supportedby the Office of
ward and downward,respectively.
Naval Research,the National ScienceFoundation,the Mellon Foun-
w=
K O(pw/pg- zR)
N
Oz
(A2)
wherez is vertical distancepositiveupward.Horizontal flows
are usuallymuch larger than verticalflows,consistentwith the
model assumptions.
For instance,beneaththe berm (where
pressuregradientscan be estimatedat all tidal stagesusing
deeplyburied stackedsensors,
x = 46 m in Figure lb) the
horizontalvelocitiesare typically--•10 times larger than the
verticalvelocities(FigureA1). Relativelylargeverticalvelocitiesoccurduringspringhightides(e.g.,seedottedline, Figure
A1, September23) whenthe sensors
were beneaththe swashzone (the area of the beachfacealternatelycoveredand uncoveredby waves),but horizontalflowsare greaterthan ver-
dation,and the Naval ResearchLaboratory.Staff from the Center for
CoastalStudiesdeployedand maintainedthe nearshoreinstruments.
Offshorewave informationwasprovidedby the CoastalData Information Program.The cooperationof the staff at Torrey Pines State
Park is gratefullyacknowledged.
The authorsthank the anonymous
reviewersand R. Hansonfor their helpfulcomments.
References
Baird, A. J., T. Mason,and D. P. Horn, Validation of a Boussinesq
modelof beachgroundwater behavior,Mar. Geol.,148, 55-69, 1998.
Barry, D. A., S. J. Barry, and J.-Y. Parlange,Capillarycorrectionto
periodicsolutionsof the shallowflow approximation,in Mixingin
Estuaries
and CoastalSeas,CoastalEstuadneStud.,vol. 50, editedby
C. B. Pattiaratchi,pp. 496-510, AGU, Washington,D.C., 1996.
Bouwer,H., and R. C. Rice, A slug test for determininghydraulic
conductivity
of unconfinedaquiferswith completelyor partiallypenetratingwells, WaterResour.Res.,12, 423-428, 1976.
tical flows more than 95% of the time.
Brown, D. L., and T. N. Narasimhan, An evaluation of the Bouwer and
Offshore of the berm and onshore of the shoreline, where
only two sensorsare stackedvertically,vertical flows are esti-
Dominick, T. F., B. Wilkins, and H. Roberts, Mathematical model for
matedonlywhenthewatertablelevelishigherthantheupper
"wave"sensorelevation(locatednear sandlevel,Figure lb).
Thus these intertidal
flow estimates are restricted to times
when the upper sensorwas beneaththe seepageface or the
swashzone.Correspondinghorizontal flows are estimatedusing the lower water table and adjacentonshoresensor.Althoughverticalflowsare expectedto be largestfor this subset
of the observations
(e.g.,FigureA1), the geometricaverageof
the flow ratio (u/w) neveris <1 and usuallyis >4. Vertical
velocitieswere greaterthan horizontalvelocitiesin 42% of the
estimatesat one swashzone
location(x = 32 m), but at most
locations,
relativelylargeverticalflowsoccurredin <7% of the
estimates.
Notation
A
D
D (x)
e
f
#
r•(x, t)
h (x, t)
K
k
shorelinefluctuationamplitude.
constantaquifer thickness.
spatiallyvariableaquiferthickness.
shorelineparameter.
frequency.
gravitationalacceleration.
water table elevation.
total water table height(r• (x, t) + D (x)).
hydraulicconductivity.
wavenumber.
kR real part of wavenumber.
k/ imaginarypart of wavenumber.
Ricemethod
ofslugtestanalysis,
Water
Resour.
Res.,
31,1239-1246,
1995.
beachgroundwaterfluctuations,WaterResour.Res., 7, 1626-1635,
1971.
Dracos, T., Ebene nichtstationare Grundwasserabflusse mit freier
Oberflache,
Rep.57, 114pp., SwissFed.Tech.Lab. Hydraul.Res.
Soil Mech., Ziirich, Switzerland, 1963.
Duncan, J. R., The effectsof watertableand tide cycle on swashbackwashsedimentdistributionand beach profile development,
Mar. Geol., 2, 186-197, 1964.
Eliot, I. G., and D. J. Clarke, Semi-diurnal variation in beachface
aggradationand degradation,Mar. Geol., 79, 1-22, 1988.
Emery, K. O., and J. F. Foster,Watertablesin marine beaches,J. Mar.
Res., 7, 644-654, 1948.
Fang, C. S., S. N. Wang, and W. Harrison, Groundwater flow in a
sandytidal beach,2, Two-dimensional
finite elementanalysis,Water
Resour. Res., 8, 121-128, 1972.
Gillham, R. W., The capillaryfringe and its effect on watertableresponse,J. Hydrol.,67, 307-324, 1984.
Grant, U.S., Influenceof the watertableon beachaggradationand
degradation,J. Mar. Res., 7, 655-660, 1948.
Harrison, W., C. S. Fang, and S. N. Wang, Groundwaterflow in a
sandytidal beach,1, One dimensional
finite elementanalysis,Water
Resour.Res., 7, 1313-1322, 1971.
Isaacs,J. D., and W. N. Bascom,Water table elevationsin some Pacific
coast beaches,Eos Trans.AGU, 30, 293-294, 1949.
Kang, H.-Y., and P. Nielsen, Watertable dynamicsin coastalareas,
paperpresentedat 25th InternationalConferenceon CoastalEngineering,Am. Soc.of Civ. Eng., Orlando, Fla., 1996.
Lanyon, J. A., I. G. Eliot, and D. J. Clarke, Groundwater variation
during semi-diurnalspringtidal cycleson a sandybeach,Aust. J.
Mar. FreshwaterRes., 33, 377-400, 1982.
Li, L., D. A. Barry, and C. B. Pattiaratchi,Numerical modelingof
tide-inducedbeachwater table fluctuations,CoastalEng., 30, 105123, 1997a.
Li, L., D. A. Barry, J.-Y. Parlange,and C. B. Pattiaratchi,Beach
2320
RAUBENHEIMER
ET AL.: SANDY OCEAN BEACH WATER TABLE FLUCTUATIONS
watertablefluctuations
dueto waverun-up:Capillarityeffects,Water Pollock,L. W., andW. D. Hummon,Cyclicchanges
in interstitial
water
content,atmospheric
exposure,and temperaturein a marinebeach,
Liu, P. L.-F., andJ. Wen,Nonlineardiffusivesurface
wavesin porous
Limnol. Oceanogr.,
16, 522-535, 1971.
media, J. Fluid Mech,, 347, 119-139, 1997.
Turner, I. L., Watertableoutcroppingon macrotidalbeaches:A simMadsen,O. S., On the generationof longwaves,J. Geophys.
Res.,76,
ulation model,Mar. Geol., 115, 227-238, 1993.
8672-8683, 1971.
Turner, I. L., and P. Nielsen,Rapidwater tablefluctuations
withinthe
McArdle, S. B., and A. McLachan,Dynamicsof the swashzone and
beachface:Implications
for swash
zonesediment
mobility?,
Coastal
effluentlineon sandybeaches,
Mar. Ecol.Prog.Ser.,76,91-99, 1991.
Eng., 32, 45-59, 1997.
Nielsen,P., Wave setupand runup:An integratedapproach,Coastal
Resour.Res., 33, 935-945, 1997b.
Eng., 13, 1-9, 1989.
S.Elgar,Electrical
Engineering
andComputer
Science,
Washington
Nielsen,P., Tidal dynamicsof thewatertablein beaches,WaterResour. State University,Pullman,WA 99164.
Res., 26, 2127-2134, 1990.
R. T. Guza and B. Raubenheimer,Centerfor CoastalStudies0209,
Nielsen,P., Groundwaterdynamicsand salinityin coastalbarriers,J. ScrippsInstitutionof Oceanography,
La Jolla, CA 92093.(britt@
CoastalRes.,in press,1999.
whoi.edu)
Nielsen,P., R. Aseervatham,J. D. Fenton,and P. Perrochet,Groundwaterwavesin aquifersof intermediatedepth,Adv. WaterRes.,20,
37-43, 1996.
Parlange,J.-Y., F. Stagnitti,J. L. Starr, and R. D. Braddock,Freesurfaceflow in porousmedia and periodicsolutionof the shallow (ReceivedOctober15, 1998;revisedMarch 25, 1999;
flow approximation,
J. Hydrol.,70, 251-263, 1984.
acceptedMarch 29, 1999.)