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Home , Deep sea, Iribarren number, Radiation stress, Wave setup, Wind wave

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 106, NO. C3, PAGES 4629-4638, MARCH 15, 2001

Field observationsof -driven setdown and setup

B. Raubenheimer WoodsHole OceanographicInstitution, Woods Hole, Massachusetts

R.T. Guza ScrippsInstitution of ,La Jolla,

SteveElgar WoodsHole OceanographicInstitution, Woods Hole, Massachusetts

Abstract. Wave-drivensetdown and setupobserved for 3 monthson a cross-shoretransect between the shorelineand 5 rn depth on a barred are comparedwith a theoreticalbalance between cross- gradients of the mean water level and the wave .The observedsetdown, the depressionof the mean water level seaward of the , is predictedwell when radiation stressgradients are estimatedfrom the observationsusing linear theory at each location along the transect. The observed setdownalso agreeswith analyticalpredictions based on offshorewave observationsand the assumptionof linear,dissipationless, normally incident shoaling on alongshore homogeneousbathymetry. The observedsetup, the superelevationof the mean water level owing to wave breaking,is predictedaccurately in the outer and middle surf zone, but is increasinglyunderpredicted as the shorelineis approached.Similar to previous studies,setup at a fixed cross-shorelocation increases with increasingoffshore and is sensitiveto tidal fluctuationsin the local water depth and to bathymetric changes.Numerical simulationsand the observationssuggest that setupnear the shoreline dependson the bathymetryof the entire surf zone and increaseswith decreasingsurf zone beach slope,defined as the ratio of the surf zone-averagedwater depth to the surf zone width. A new empiricalformula for shorelinesetup on nonplanarbeaches incorporates this dependence.

1. Introduction whereE isthe wave energy, 0 is the wave direction, and Cg andC arethe group and phase velocities, respectively. Wave setdownand setupare changesin mean water level Thetheoretical balance (1), withSxx given by (2), is con- thataccompany shoaling and breaking surface gravity waves. sistentwith observationsin narrowwave flumes(where 0 With no alongshorevariations in wavesor bathymetryand = 0) with smooth,impermeable, fixed bottoms and planar negligiblewind and bottomstresses, the cross-shorepres- [Bowenet al., 1968], barred[Battjes and Janssen,1978; sure gradient associatedwith the time-averagedwave set- downand setup r/theoretically balances the cross-shoregra- Battjesand Stive, 1985], and other [Gourlay, 1992] depth profiles.Predictions of setupdiffer in detailbecause of dif- dientof thetime- anddepth-averaged onshore wave momen- tum (the waveradiation stress $•) [Longuet-Higginsferent model dependencies of $x• on localwave properties (e.g., linear or nonlineartheory) and differentformulations and Stewart, 1962], of theeffect of waverollers [e.g., Svendsen, 1984; Diegaard os• o• et al., 1991;Sch•iffer et al., 1993]. Ox + pg(v+ h) - 0 (• ) Thereare few fieldtests of the balance(1) usingconcur- rent observationsof both r/ and $• acrossthe surf zone. where x is the cross-shorecoordinate, h is the still water Battjesand Stive[1985] integrated(1) usinga wavetrans- depth, p is the water ,g is gravitationalaccelera- formationmodel to predict$•x andobtained good agree- tion, andfor linear,shoreward propagating, monochromatic ment with setupobserved in 2 and 4 m depthduring a waves, . Assumingthat the depthand setupvary linearly acrossthe beach,Lentz and Raubenheimer[1999] showed - , (2) thatthe setup measured for 3.5years in 2 m waterdepth was modeledqualitatively well by (1) with Sz• estimatedwith (2) usingwave measurements in 8 and2 m depths.However, Copyright2001 by theAmerican Geophysical Union. the assumptionsof lineardepth and setupvariations were Paper number 2000JC000572. shownto leadto integrationerrors as largeas 50% of the 0148-0227/01/2000JC000572509.00 observedsetup. More accuratesetup predictions with $x• 4629 4630 RAUBENHEIMER ET AL.' WAVE-DRIVEN SETDOWN AND SETUP and h estimatedfrom wave and bathymetrydata collected buried pressuregages (setup sensors) located between the alonga cross-shoretransect agreed well with setupobserved shorelineand about 5 m water depth (Figure 1, solid cir- for 2 monthsin 2 m water depth,even thoughthe setupmea- cles). The setuppressure sensors were buriedto avoidflow- surementswere made with unburiedpressure sensors subject induceddeviations from hydrostaticpressure (see Appendix to flow-inducedmeasurement errors (see Appendix A) and A). After correctingfor temporalchanges in water density offset drift. [Lentzand Raubenheimer,1999] with conductivityand tem- Field measurementsshow that surf zone setupdepends on peraturemeasured in 5 m water depth, mean water levels the local waterdepth and the offshorewave height [Nielsen, were calculatedfrom 512 s (8.5 min) recordsby assuming 1988; King et al., 1990]. Field observationsof setupat the hydrostaticpressure. shoreline77shore suggest Setup(setdown) was defined as the increase(decrease) of the mean water level relative to that observed at the most 77shore-- cHs,o, (3) offshoresetup sensor (cross-shore location x = 58 m). The observedshoreline setup was estimatedas the setupwhere whereHs,0 is the offshoresignificant wave height and c is the total water depth was < 0.1 m. Note that 77shorewas a constant between about 0.2 and 0.3 [Hansen, 1978; Guza measuredonly when the shoreline,defined as the intersec- and Thornton, 1981; Nielsen, 1988; Hanslow et al., 1996]. tion of the mean water level with the beach,approximately This result is consistentwith (1) and (2) assuminga mono- coincidedwith a setupsensor location, which occurredat tonic beachslope, normally incidentlong waves,and surf mostonce per rising(and falling) . zone wave heightsthat are a constantfraction of the wa- At all but the shallowest three locations, sensor offset ter depth. However, scatterabout (3) is considerable(of- drifts (typically equivalentto about 0.03 m of water over ten greater than 100% of 77shore),possibly because natu- the 3 monthexperiment) were removedby subtractingfrom ral beachesoften are barred or alongshoreinhomogeneous, eachtime seriesa quadraticcurve fit to setupestimated at wavereflection may be largenear the shoreline,and the ratio 17 times when negligiblesetup or setdownwas expected of waveheight to waterdepth may dependon thebeach slope (H•,0 < 0.35 m andh > 2 m). In shallowerwater (x > 350 and wave conditions. Additionally, observedmean water m), drifts were removedby adjustingthe calculatedmean levelsnear the shoreline(in bothfield andlaboratory studies) waterlevels (using a quadraticfit) sothat setupand setdown canbe sensitiveto the measurementtechnique [e.g., Holland were negligiblefor small nonbreakingwaves (estimated as et al., 1995] and to the definition of setup [e.g., Gourla3; locationsand times when the ratio 78 of significantwave 1992]. heightH8 to total waterdepth h + 77was < 0.2) and so that In contrastto (3), Holman and Sallenger [1985] found the water level equaledsand level when the saturatedsand no correlationbetween video-basedestimates of 77shoreand aboveswash zone sensorsfirst wasexposed during rundown H•,0, butinstead suggested that 77shore/Hs,o increased with [Raubenheimeret al., 1995]. increasingIribarren number •0 = •/v/H•,o/Lo, where• is the foreshorebeach slope and L0 is the offshorewave- ...... I ...... I ...... I ...... length of the spectralpeak . Scatterin this rela-

ß tionshipwas reduced by separatingthe resultsinto low, mid- dle, and high tidal stages,and it was hypothesizedthat the offshorebar morphologyinfluenced the low-tide shoreline setup. However,the observationsof Nielsen [1988] showed little effect of the offshorebarred on the setup, and thus the importanceof barredbathymetry to 77shoreis _ uncertain. Here the balance (1) is tested with field observationsof waves and time-averagedwater levels measuredbetween the -4 shorelineand about5 m water depthon a barredbeach. Wa- ter levels are estimatedwith buried,stable sensors. -6 Setdownand setupup are predictedby integrating(1) with 400 300 200 1 O0 0 Sxx basedon (2) usingthe wave observations.The observed Gross-shore Lo½ofionx(rn) setdownis consistentwith (1). Similar to Lentz and Rauben- heimer[1999], setupis predictedwell in 2 m waterdepth, Figure 1. Locationsof deeplyburied pressure sensors used butthe balance breaks down in depthsshallower than about to measuresetup (solid circles), colocated unburied pres- 1m. Setup near the shoreline isshown tobe sensitive tothe suresensors, meters, and altimeters (open cir- surfzone bathymetry andtidal fluctuations. cles),near-bed pressure sensors (open diamonds), andthe conductivitysensor (asterisk). The most seaward11 setup sensorswere accurateParoscientific gages. All pressure 2. Field Experiment and Data Processing measurementswere corrected for temperatureeffects. The solidcurves are selectedbeach profiles measured between 1 Observationswere acquired from Septemberthrough September and 31 November.The thick black curve is the November1997 on a sandyAtlantic beach near Duck, 13 Septemberprofile. The x axisis positiveonshore with the North Carolina. Bottompressure was measuredwith 12 originat thelocation of theoffshore sensor. RAUBENHEIMER ET AL.' WAVE-DRIVEN SETDOWN AND SETUP 4631

Beach profiles were surveyedevery few days with an Table 1. Least SquaresLinear Fits (With Intercept a and amphibiousvehicle, and seafloorelevations were measured Slopeb) of SetupPredictions to Observations,and Range of nearly continuouslyat 11 cross-shorelocations (Figure 1) ObservedSetup for DifferentTotal Depth (h + •) Ranges with sonaraltimeters [Elgar et al., 2001]. Foreshoresand TotalDepth Tipred-- a q- boobs Range m a, rn b correlation r/ohs,rn levelsat the three shallowestsetup gages (cross-shore loca- 4.0 to 5.0 0.000 0.98 1.00 -0.016 to 0.112 tions 375, 382, and 390 m) were measuredapproximately 3.0 to 4.0 0.000 0.98 1.00 -0.017 to 0.167 daily usingreference rods. 2.5 to 3.0 0.000 0.98 1.00 -0.033 to 0.171 Significantwave heights(4 timesthe standarddeviation 2.0 to 2.5 0.001 0.95 0.99 -0.032 to 0.203 of seasurface elevation fluctuations) and centroidalfrequen- 1.5 to 2.0 0.002 0.94 0.99 -0.033 to 0.236 1.0 to 1.5 0.002 0.93 0.99 -0.030 to 0.248 ciesin the -wavefrequency (f) band(0.05 < f < 0.30 0.8 to 1.0 0.000 0.87 0.98 -0.022 to 0.269 Hz) werecalculated every 512 s usingobservations from the 0.6 to 0.8 0.005 0.69 0.97 -0.019 to 0.275 threemost shoreward setup sensors and from the wave sen- 0.4 to 0.6 0.007 0.64 0.96 -0.025 to 0.304 sors(Figure 1, opensymbols) located between the setupsen- 0.2 to 0.4 0.009 0.57 0.92 -0.021 to 0.372 sors.Attenuation of pressurefluctuations through the water 0.0 to 0.2 0.027 0.45 0.88 -0.013 to 0.547 and the saturatedsand above the buried setup sensorswas accountedfor usinglinear wave and poroelastic theories, re- maximumsetup (0.547 m) was observednear the shoreline, spectively[Raubenheimer et al., 1998]. Orbital velocities and the maximum setdown(-0.033 m) was observedin 1.5- observedwith bidirectionalelectromagnetic current meters 3.0 m water depth(Table 1). were usedto estimate3 hour mean (energy-weightedaver- ageover frequency) wave directions [Kuik et al., 1988,Her- 3. Model bers et al., 1999]. Waves were assumednormally incident onshoreof the shallowestcurrent meter (z > 350 m). Cross-shoreintegration of (1) yields Offshore(Figure 1, x = 0 m) wave heights(Figure 2a), directions,and centroidalfrequencies ranged from 0.20 to (4) 2.85 m, -35ø to 35ø, and0.09 to 0.20 Hz, respectively.The x2• pg(rl+h) 1 t9Sx•c•x dx,

3 (A)

0.25 x = 250rn ...... •bserved (8)_ • predicfed

0.15

0.05

-0.05 I I

0.25 •:.: ti !1!11,$•. i 0.15 x= •75., i: rn ' :•'-';!•i,.... ' •'[: (c)_ i.... •" { ' ' :': --

0.05

-0.0• , •ep O• Oal O• Nov O• Dea O• Date

Figure 2. Observed(a) offshore(x - 0 m) significantwave height and observed(dotted curve) and predicted(solid curve) setupat cross-shorelocations (b) x - 250 and (c) x - 375 m versustime. The horizontaldotted line in Figure2c is the still water level (setupequal to 0.0 m). 4632 RAUBENHEIMER ET AL.' WAVE-DRIVEN SETDOWNAND SETUP

. , where•/is thesea level difference (i.e., therelative setdown E 1.2 v 1.0 or setup)between cross-shore locations a:• and a:2. The inte- .• 0.8 gratedsetup balance (4), withthe wave radiation stress given '• 0.6 eel 0ß LowHigh TideTide by (2), is solvednumerically for •/relativeto a:• = 158m us- -r 0.4 > 0.2 inga fourth-orderRunga-Kutta scheme with an adaptive step o 0.0 size for all 512 s data records.In (2) the wave energyis es- 0.25 timatedas E - pgH•2/16;the wave direction 0 is estimated ß • 0.20 predicted asthe mean direction; and the group and phase velocities C a E 015 I•I observed o. 0.10 and C are estimatedusing linear theory, the centroidalfre- -,- 0.05 quency,and the total water depth (h + •/). Significantwave • 0.00 '.. '...... HighTide(B -0.05 heights,mean wave directions, and centroidal areinterpolated linearly between observation locations. The 0.25 _-' ' LowTide (C 0,20 : . predicfed depthh, calculatedfrom the most recent beach survey and 0.15 0 observed the meanwater level (relativeto meansea level) observedat 0.10 0.05 themost offshore setup sensor (a: = 58 m, Figure1), includes 0.00 tidesand other processes (e.g., wind-driven setup) that affect -0.05 the water level across the entire surf zone. • 1 Setupcontributes to thetotal water depth in (4) andalso E 0 c --1 affectsthe groupvelocity and phase speed. Therefore (4) is .-. - 2 solvediteratively by assumingthat •/at eachshoreward step o --3; initiallyis equalto •/at theneighboring offshore location. Differencesare small(0.00 4- 0.01 m) betweenhourly av- 400 3;00 200 1 co eragedsetup based on Sxx calculatedusing the bulk wave Cross-shore Locationx(m) properties(0 mean,Hs, Ca, and6' describedabove) and Figure 4. (a) Observedsignificant wave heightson 13 hourlysetup estimates based on waveradiation stresses es- September,and observed (open circles) and predicted (solid timatedas f Sx•(f)df, whereS•(f) is calculatedwith a curve)setdown and setup on 13 Septemberat (b) high tide directionalmoment technique [Herbers and Guza, 1990;El- (1600)and (c) low tide(2042), and (d) measuredbeach pro- gar et al., 1994]. file versuscross-shore location (times represent the startof In 8-13 m waterdepth at thisfield site,mean water level each512 s record). The verticaldotted lines in Figures4c and 4d mark the locationsa: = 338,363, and 383 m discussed in the text. The horizontaldotted lines in Figures4b and4c arestill waterlevel. The horizontaldotted lines in Figure4d are tidal elevationsduring the two runs.

changesdriven by the cross-shorewind stressand by the >• o.s associatedwith the alongshoreflow can be comparableto wave-drivenwater level changes[Lentz et al., • O.O 1999]. However for the conditionsconsidered here, the esti- matedwater level changesonshore of a:l (depthabout 5 m) owing to the wind stressand are at leastan orderof magnitudesmaller than those caused by wavesand thereforeare neglected. '5 -0.5 .t 4. Observationsof Setup and Comparisons 0.20 With Predictions 0.15 4.1. Observations 0.10 0.05 Consistentwith empiricalformulas [e.g., Nielsen, 1988; o.oo Gourlay, 1992] and previousobservations, the setupob- -0.05 servedat fixed locationsincreases with increasingoffshore 07 09 11 13 15 17 19 21 Day in Septernber waveheight Hs,o (Figure2). Nearthe shoreline(a: = 375 m; Figures2c and 3c) the measuredsetup also is sensitive Figure3. (a) Observedoffshore (a: = 0 m) waveheights, to changesin the localdepth owing to tidesand bathymetric (b) observedoffshore tidal elevation (8.5 min averagedsea evolution. During each tidal cycle the setupobserved at surfacelevel) (solidcurve) relative to meansea level (hori- a fixed surf zone locationis larger at lower tide when the zontaldotted line), and(c) observed(dotted curve) and pre- dicted(solid curve) setup at cross-shorelocation a: = 375 m observationlocation is closer to the shoreline(Figure 3c; versustime. The horizontaldotted line in Figure 3c is the and comparethe threedata points for :c _> 375m in Fig- still water level (setupequal to 0.0 m). ure 4b with thosein Figure4c). With approximatelyequal RAUBENHEIMER ET AL.' WAVE-DRIVEN SETDOWN AND SETUP 4633 offshorewave heights,the setupobserved at :r = 390 m in- 0.30 creasesfrom 0.03 rn at high tide to 0.23 rn at low tide as 0.25 the tidal elevationdecreases by 0.9 m (Figure4). Consis- .-• 0.20 tentwith previouslaboratory [Bowen et al., 1968]and field E [Nielsen, 1988] studies,the slopeof the mean water level v 0.15 at the mostshoreward setup observations is approximately • O.lO twice as steepas that fartheroffshore (Figures 4 and 5a). •' 0.05 It has been suggested[Nielsen, 1988; Hanslow et al., 1996] that cross-shoreprofiles of setup(normalized by the 0.00 offshorewave height) can be parameterizedas a function -0.05 ...... of the local water depth(normalized by the offshorewave 0.00 0.10 0.20 0.30 0.40 0.50 0.60 height). However,on barredprofiles the setupoften is dif- r/obs(rn) ferentat locationson the outerslope of the bar,in the trough of thebar, and on theforeshore even though the waterdepths Figure6. Predictedversus observed mean (circles) and stan- darddeviation (vertical bars) of 512 s setdownand setupin are identical at the three locations(z = 338,363, and 383 m totalwater depths of lessthan (open circles) and greater than in Figure4c) wherethe depthis 1 m. The observedsetup in (solid circles) 1 m. the troughis consistentlyhigher than that observedin simi- lar depthsoffshore of the bar (Figure5a; comparevalues at :r = 375 m at low tide in Figures2c and3c). With moderate constanth/ m•,0). waveconditions, the heights of nonbreakingshoaling waves 4.2. Comparison With Predictions increaseon the offshoreslope of the sandbar (200 < z < 350m, Figure4a), andsetdown is observedand predicted Setdownand setupare predictedwell (using(4) and (2)) (Figures4b and4c). Fartheronshore, breaking reduces the exceptnear the shorelinewhere setup is underpredicted(e.g., waveheights, and setup is observedand predicted. At low tidethe setup on the foreshore is underpredicted(Figure 4c; and comparesetup observations with setuppredictions for o foreshore '"(a'" thesmallest normalized depths inFigure 5)ß o frough Setupis predictedaccurately in waterdepths > 1 m for 0.30 ßoffshore ofbar theentire 3month long data set (solid symbols inFigure 6), but setupis underpredictedin shallowerwater depths (open 0.20 symbolsin Figure 6). In all depthsthe leastsquares linear fits of the predictionsto the observationshave small inter- 0.10 ceptsrelative to the rangeof setupvalues (Table 1). How- ever,the regressionslopes indicate increasing underpredic- tion of setupin depths• 1 m, reachinga maximumunder- 0.00 predictionof abouta factorof 2 nearthe shoreline(Figure 7 andTable 1). The causeof theunderpredictions is unknown. -0.10 The correlationbetween the observationsand predictions is > 0.88 in all water depths. 0.40 ......

0.30 1.0 ß ß ß - _

_ O.20 0.8

0.10 _

0.00 0.4 -0.10 ...... 0 1 2 3 4 5 6 7 h/Hs,0 0.2 Figure 5. (a) Observedand (b) predictednormalized (by off- shorewave height)setup versus normalized water depthfor all 8.5 min runsbetween 13 September0100 and 14 Septem- 0 1 2 3 4 5 ber 0100. Cross-shore locations are foreshore z > 383 m h + •/ (rn) (diamonds),trough 383 _( z _( 363 m (open circles), and offshore of bar crest z ( 363 m (solid circles). Offshore Figure7. Slopeof leastsquares fit of setuppredictions to wave heightsranged from 0.75 to 1.00 m. observations(b in Table1) versustotal water depth (h + •7)- 4634 RAUBENHEIMER ET AL.' WAVE-DRIVEN SETDOWN AND SETUP

Setdownin the shoalingregion is predictedwell (root- mean-square(RMS) error0.002 m) by thenumerical model 0 ((4) and(2) with $xx estimatedfrom observations along the m 1 transect)(Figure 8a). Assuminglinear shoreward propa- o gating,monochromatic, nondissipative waves, $• canbe 2 predictedanalytically and the setdown is givenby [Longuet- •- 5

Higginsand Stewart, 1964] o 4

, , , rl - -a•kaf (kah), (5) 0.10 (B)- Run 1, Sep 15 . where 0.08

0.06 1 coth2 kh f(kah) - kah = kh tanh kh, 2 2kh + sinh 2kh' 0.04 (6) ...... !*********************** ..... 0.02 where aa and ka are the deep water wave 0.00 and ,respectively. For narrow-banded(in fre- , , , quency)random waves aa - Hs,a/(2x/•)and E 0.10 (cy:' o. 0.08

H2aka _ • 0.06 r/- 8 f (7) 'o 0.04

.• 0.02 o. o.oo 0.000 I .... , .... i .... -0.002 0 50 1 O0 150 DisfenceOffshore of Shoreline(rn) -0.004

E -0 006 Figure 9. Model simulations show the effect of two ß -0.008 bathymetrieson setup.In both simulations,Rbreak is initial- ized with the pressureand velocity time seriesand tidal ele- •'-0.010 vationsobserved on 13 September2140-2148. (a) Observed -0.012 depthprofiles relative to still water level (SWL, the horizon- tal dashedline), (b) Rbreak-predictedwave radiationstress -0.014 Sx•, and(c) simulated(using (4)) setdownand setup versus -0.015 -0.010 -0.005 0.000 the distance offshore of the shoreline. The solid and dotted curvesin Figures9b and 9c are resultsfor the 13 September •7obs(m) 2140 (run 1) and 18 November0450 (run 2) depthprofiles (shownin Figure 9a), respectively.

where ka is now the wavenumberof the spectralpeak. The deepwater values of Hs,a andka areestimated using the ob- servationsat the offshoresensor (H•,0 and k0 at z = 0 m) and linear theory.Similar to laboratoryobservations [Bowen et al., 1968], the observed(and predicted)setdown is mod- eled well by (7) (Figure 8b).

0.01 0.10 5. Discussion kdh 5.1. Importance of Bathymetry and Tidal Elevation Figure 8. (a) Numericallypredicted (with (4)) versusob- servedhourly averaged setdown for datawhere the observed The importanceof bathymetryand tidal fluctuationsto the setdownis largerthan 0.003 m and(to avoidbreaking waves) wave-drivensetup is investigatedby driving the numerical the observed?s is lessthan or equalto 0.2. The soliddiag- setupmodel (4) with wave radiationstresses estimated us- onalline is /lobs-- /}pred.The correlation coefficient and ing(2) and wave properties (Hs, C 9, and C) calculatedfrom RMSerror between observations andpredictions are0.75 seasurface elevation fluctuations predicted bya numerical and0.002 m, respectively. (b)Observed (open circles) and model (Rbreak) based onthe nonlinear shallow water equa- numericallypredicted (solid circles)normalized (by deep waterwave height squared Hs2,aand wavenumber ka)hourly tions [Kobayashi etal., 1989; Raubenheimer etal., 1996]. averagedsetdown versus normalized localwater depth. The To test the wave model, Rbreak isinitialized at:c - 260 dashedline is (7). TheRMS errors between the analytical m (water depth 3.23 m) with pressure and velocity time se- estimates(7) andthe observed and numerically predicted r/ riesmeasured on 13 September from 1917 until 2200. Wave are0.004 and0.003 m, respectively. radiationstresses calculated from Rbreak are within6% of RAUBENHEIMER ET AL.' WAVE-DRIVEN SETDOWN AND SETUP 4635 those estimatedfrom the observations(not shown). 0.14 To examine bathymetric effects, Rbreak model simula- ß Sep 15 tionsare initialized with pressure and velocity time series 0.12 o Nov 18 (H8 = 0.8 m) and tidal level (-0.57 m) measuredon 13 o September2140-2148 atcross-shore location :r - 260m o;0.10 (Figure 1), but are run on two different observedbathyme- • tries(13 September (run1) and 18 November (run2)). For •o 0.08 bothruns the modelis initializedin 3 m water depth' which _c:mO.06 '-. -. 0 0 ß occurs at distances 130 and 125 m offshore of the shoreline •' onthe 13 September and18 November bathymetries, respec- 0.04 tively. In the innersurf zone (within about50 m of the shore- line) the setuppredicted using (4) and the Rbreak-simulated 0.02 wave conditionsis 25% largeron the run 1 bathymetrythan on the run 2 bathymetry(Figure 9). For a given radiation 1 00 stressdecrease, the setupgradient (1) increaseswith decreas- ing water depth [e.g., Battjes and Janssen,1978], so rela- tivelylarge setup occurs on the shallow bar crest (offshore 80 distance30-50 m) of run 1. The effects of changesin tidal elevationare illustratedby comparingsetup predicted using 6O

ß . _ 4O

20 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 Tidol Elevofion(rn)

: i .... i .... i .... Figure 11. (a) Normalized (by offshorewave height) shore- .---, 0.10 , , , line setuppredictions (using (4)) drivenwith observedwave E Run1, Low Tide radiation stressesand (b) inverseof the surf zone-averaged '• 0.08 beachslope (8) versustidal elevationfor all 512 s records

•- 0.06 on 13 September(solid circles)and 18 November(open cir- cles). The least squareslinear fits to the 13 Septemberand c 0.04 o 18 Novemberpredictions are shownby the solid and dot- ted lines, respectively.The range of offshorewave heights • 0.02 was 0.75-1.00 m on 13 Septemberand 0.65-0.95 m on 18

"' 0.00 , , . November.

(c• E 0.10 . n 0.08 (4) and Rbreak wave propertiessimulated with the samein- ß 0.06 cident wave conditionsand the 13 Septemberbathymetry, 'o 0.04 but with different tidal levels (-0.57 (run 1) and 0.57 m (run .2 0.02 3)). The model is initialized in 3 m water depth, which in these cases occurs at distances 130 and 105 m offshore of o_ 0.00 the shorelinefor tidal levels -0.57 and 0.57 m, respectively. o 50 1 oo 150 DistanceOffshore of Shoreline(m) As shownin Figure 10, more setupis predictedat low tide when dissipationis strongover the shallowbar crest(run 1) than at high tide when the bar crestis in deeperwater (run Figure 10. Model simulationsshow the effect of two tidal elevationson setup. In both simulations,Rbreak is 3). initialized with the pressureand velocity time seriesand In all three runs the offshore wave conditions are identical, bathymetryobserved on 13 September2140-2148. (a) Ob- and the foreshorebeach slopes are similar. The differences serveddepth profiles relative to still waterlevel (SWL, the in modeledsetup suggest that some of thescatter of observed horizontaldashed line), (b) Rbreak-predictedwave radiation //shoreabout empirical formulasbased on wave parameters stressSxx, and (c) simulated(using (4)) setdownand setup offshoreof the surf zone and foreshorebeach slopes[e.g., versus the distance offshore of the shoreline. The bar crest Holman and Sallenger,1985; Nielsen, 1988] may be caused in Figure10a is bothdeeper and farther from the shoreline at hightide thanat low tide. The solidand dotted curves in by changingsurf zone bathymetryand water levels. Figures10b and 10c are resultsfor the 13 September2140 The effects of bathymetry and tidal fluctuationsalso are (run 1, tidal level-0.57 m) and 13 September1645 (run 3, apparentin the shorelinesetup (Figure 11) predictedby the tidal level +0.57 m) tidal elevations,respectively. numerical setupmodel (4) driven with the observedwaves 4636 RAUBENHEIMER ET AL.: WAVE-DRIVEN SETDOWN AND SETUP andbathymetry for 24 hours(170 512-sruns) on both13 bathymetryconsidered here,/•v decreaseswith decreasing Septemberand 18 November.Wave conditions and tidal tidal elevation(Figure 1lb). Althoughthe shorelinesetup rangeswere similar during the 2 days,but the bathymetries (Figures 9, 10, and 11) is underpredictedby the numerical differedsubstantially (Figure 9a). The predictedshoreline model(Figures 6 and7), theseresults and those of Hansen setup(normalized by theoffshore wave height) is largeron [1978]suggest that the shoreline setup is sensitiveto thesurf 13 Septemberthan on 18November for almost all tidalel- zonewater depth and surf zone width and may be correlated evations(Figure 1 l a). Furthermore,in contrastto (3) in with •5•v. whichrlshore/Hs,o is constant, on both profiles the normal- ized shorelinesetup increases with decreasingtidal eleva- 5.2. Empirical Formulasfor ShorelineSetup tion. On a planarbeach, tidal fluctuations do notaffect the Consistentwith previousobservations and empirical for- averagesurf zone beach slope mulas(e.g., (3)), the observedshoreline setup T]shore in- creaseswith increasingoffshore wave height Hs,o (Figure •v __ h•vAx' (8) 12a). However, scatterabout the least squareslinear fit (rlshore/Hs,0 = 0.24)is substantial(RMS error0.10; Figure where 12b). Betterpredictions of rlshore/Hs,oresult from a least h•v= Ax squareslinear fit to/5•-v• (RMS error 0.06; Figure 12b) or I / (h+ rl)dx, (9)to a surfzone /Savx/Lo/H,,o (RMS error with Ax the distance from the shorelineto the outer cdgc 0.08; Figure 12c). In contrastto someprevious observations, of the surf zone,defined as the mostoffshore location where thecorrelation of rlshore/Hs,owith Iribarrennumber where ?s _>0.45 [Lentz and Raubenheimer, 1999]. However, on the /• is equalto theforeshore slope is not statisticallysignificant (not shown).The bestfit (RMS error0.06) to the presentob- servationson a barredbeach (0.01 _< /5•v _<0.04) is given 0.60 • 0.50 by E 0.40 •]shore/Hs,o-- 0.019 + 0.003•-v1. (10) • 0.30 o Jr 0.20 6. Conclusions 0.10 0.00 Setup and setdownobserved for 3 monthsbetween the 0.0 0.5 1.0 1.5 2.0 2.5 shorelineand 5 m waterdepth on a barredbeach were com- Hs,0 (rn) paredwith modelpredictions based on a balance(1) between 0.60 ...... B cross-shoregradients of meanwater level andthe waveradi- o 0.50 ß ation stress. The observed wave setdown is consistent with -,- 0.40 the observedradiation stress gradients (estimated using wave '•e•0.$0 o observationsand (2)) and alsowith radiationstress gradients Jr 0.20 predictedanalytically (7) usingoffshore wave observations •' 0.10 0.00 and assumingnormally incident, monochromatic, nondissi- 0 20 40 60 80 100 120 140 patire waves(Figure 8). The observedwave setup is consis- tent with the radiation stresses estimated from observations

0.60 ...... in the outer and middle surf zone, but is underpredictedby o 0.50 roughlya factorof 2 in depthsshallower than • 1 m (Fig- -,- 0.40 ures2, 3, 4, 6, and7). Similarto previousfield studies,setup ß 0.50 ß.•,. ß .-;;-•, ß •. ß o at a fixed cross-shorelocation increases with increasingoff- ,- 0.20 shorewave height and is sensitiveto tidal fluctuationsin the •' O. lO o.oo local waterdepth and to the local bathymetry(Figures 2, 3. o.oo O. 10 0.20 0.50 0.40 0.50 and 4). Consistentwith numerical simulationsthat suggest •avq(Lo/Hs,o) setupnear the shorelinedepends on the bathymetryof the entire surf zone (Figures9, 10, and 11), the observedshore- Figure 12. (a) Observedshoreline setup versus offshore line setupincreases as the surf zone-averagedbeach slope significantwave heightand (b) and (c) observedshore- decreases(Figure 12b). A new empiricalformula (10) for line setupnormalized by offshoresignificant wave height shorelinesetup incorporates this dependence. versus/Sa- • and /5avx/Lo/Hs,o, respectively. The solid line in Figure 12a and the horizontaldashed line in Fig- ure 12b are the least squaresfit to (3) (T]shore = ½Hs,o AppendixA: Flow Noise where c = 0.24 and RMS error = 0.10). The horizontal Errors in the measuredmean water level causedby flow dottedline in Figure 12b is the averagenormalized setup aroundthe unburiedpressure sensors are investigatedby (r/shore/Hs,0)avg= 0.26, RMS error 0.10. The solid lines in Figures12b and 12c are the least squares linear fits given comparingmean water levels estimated using deeply buried byTlshore/Hs,o -- 0.019 + 0.003/•-vx,RMS error 0.06 (Fig- sensorswith thoseestimated using unburiedpressure sen- ure12b), rlshore/Hs,o -- 0.336-0.628•vv/Lo/Hs,o, RMS sors. Pressuresensors were housed in ,-, 8 cm diameter, 30 error0.08 (Figure12c). RMS differencesof 0.02 arcstatis- cmlong cylinders that were oriented vertically with the pres- tically significantat the98% level. sureports pointing upward. From Bernouilli's equation, dif- RAUBENHEIMER ET AL.' WAVE-DRIVEN SETDOWN AND SETUP 4637

0.02 ' ' ' I ' ' ' I ' ' ' I ' ' ' I ' ' ' I ' ' ' I Acknowledgments. This researchwas supported by NSF and ONR. Stafffrom the Center for CoastalStudies deployed and main- 0.00 tainedthe instruments.The staffof the Field ResearchFacility as- sistedwith the instrumentdeployment and provided the bathymet- ric surveys. Woods Hole OceanographicInstitution contribution -0.02 10240.

-0.04 References Battjes,J. A., and J.P. E M. Janssen,Energy loss and setupdue to -0.06 breakingof randomwaves, paper presented at 16th International Conferenceon CoastalEngineering, ASCE, Hamburg,Germany, 1978. -0.08 Battjes, J. A., and M. J. F. Stive, Calibration and verificationof a dissipationmodel for randombreaking waves, J. Geophys.Res., -O. lO 90, 9159-9167, 1985. Bowen, A. J., D. L. Inman, and V. P. Simmons, Wave 'setdown' o.oo 0.02 0.04 0.06 0.08 0.10 0.12 and ,J. Geophys.Res.. 73. 2569-2577. 1968. O.5u;/g(m) Diegaard, R., P. Justesen,and J. Freds0e, Modeling of by a one-equationturbulence model, Coastal Eng., 15, 431-458, Figure A1. Mean (symbols)and standarddeviations (ver- 1991. tical bars) of differencesin mean water level (solid circles) Elgar, S., T. H. C. Herbers, and R. T Guza, of ocean and setup(open circles) estimatedfrom measurementswith surfacegravity wavesfrom a natural beach,J. Phys. Oceanogr., buriedand unburied pressure gages versus 0.$ut 2/g. 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