Bed Drag Coefficient Variability under Wind Waves in a ... Stream/Bricker friction... · Bed Drag Coefﬁcient Variability under Wind Waves in a ... Despite the recent application

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Bed Drag Coefficient Variability under Wind Wavesin a Tidal Estuary

Jeremy D. Bricker1; Satoshi Inagaki2; and Stephen G. Monismith, A.M.ASCE3

Abstract: In this paper we report the results of a study of the variation of shear stress and the bottom drag coefficientCD with sea statand currents at a shallow site in San Francisco Bay. We compare shear stresses calculated from turbulent velocity measuremmodel of Styles and Glenn reported in 2000. Although this model was formulated to predict shear stress under ocean swcontinental shelf, results from our experiments show that it accurately predicts these bottom stress under wind waves in an estuup in the water column, the steady wind-driven boundary layer at the free surface overlaps with the steady bottom boundarycalculating the wind stress at the surface and assuming a linear variation of shear between the bed and surface, however, thbe extended to predict water column shear stresses that agree well with data. Despite the fidelity of the model, an examinaobserved stresses deduced using different wave–turbulence decomposition schemes suggests that wave–turbulence interacportant, enhancing turbulent shear stresses at wave frequencies.

DOI: 10.1061/~ASCE!0733-9429~2005!131:6~497!

CE Database subject headings: Drag coefficient; Hydraulic roughness; Turbulence; Tidal currents; Waves; Estuaries.

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Introduction

The erosion, transport, and deposition of sediments play a crole in many geological, biological, geochemical, and ecologprocesses operant in estuaries~Shrestha and Orlob 1996!. Accord-ingly, engineering studies done to assess how large-scalesuch as constructing airport runways on new landfill, restotidal wetlands, or changes in the disposal of dredge matemight degrade or improve the environment of the estuarwhich these actions would take place often make use of cohydrodynamic/sediment transport models that are designed tdict sediment dynamics~Blumberg et al. 1999!. A key feature oall these models is the way in which bed friction is modeled,how it depends on bed roughness, mean currents, and as wcuss in this paper, how it is affected by wind-wave orbitaltions ~Perlin and Kit 2002!.

In most three-dimensional~3D! circulation models, bottomfriction is represented by a quadratic drag law based on a bodrag coefficientCD

1Research Associate, Dept. of Civil Engineering, Kobe Univ.,Rokkodai-cho, Nada-ku, Kobe 657-8501, Japan. E-mail: brickstanfordalumni.org

2Senior Research Engineer, Environmental Engineering Dept., KTechnical Research Institute, 2-19-1, Tobitakyu, Chofu-shi, Tokyo0036, Japan. E-mail: [email protected]

3Professor, Environmental Fluid Mechanics and Hydrology, DepCivil and Environmental Engineering, Stanford Univ., Stanford,94305-4020. E-mail: [email protected]

Note. Discussion open until November 1, 2005. Separate discusmust be submitted for individual papers. To extend the closing daone month, a written request must be filed with the ASCE ManaEditor. The manuscript for this paper was submitted for review andsible publication on March 4, 2003; approved on September 13,This paper is part of theJournal of Hydraulic Engineering, Vol. 131,
No. 6, June 1, 2005. ©ASCE, ISSN 0733-9429/2005/6-497–508/$25.00.
JO

J. Hydraul. Eng. 2005

-

tc = rCDuUcuUc s1d

where tc=steady shear stress at the bed;r=water density; anUc=velocity of the mean current at heightzr ~Signell et al. 1990!,usually taken as 1 m above the bed~mab!. In 3D modeling,tc isthen used as a bottom boundary condition on the velocity.

tc = − rnc

]U

]zs2d

wherenc=eddy viscosity.It is commonly assumed, and often found to be true in pra

~Nezu and Nakagawa 1993; Lueck and Lu 1997!, that the wellknown law of the wall~e.g., Tennekes and Lumley 1972; Neand Rodi 1986!

uUcu =uu*cu

klnS z

z0D s3d

applies to unsteady estuarine and coastal flows. Herek<0.41 isvon Kármán’s constant;z=positive upwards~and zero at the bed!;the roughness lengthz0=kb/30, wherekb=sand grain roughnessthe bed~Nikuradse 1932!; and the shear velocity is

u*c =Îtc

rs4d

Using these relations,CD, referenced to some heightzr, can beinferred from the roughness length

CD = F k

lnszr/z0dG2

s5d

@see, e.g., Gross et al. 1999#. Generally, the value ofCD dependupon bed sediment grain size and bed-form geometry.

However, in the shoals of many estuaries, areas that areof particular environmental and engineering concern, currentsediment dynamics can be strongly influenced by wind w~Schoellhammer 1996!. As discussed by Grant and Mads
~1979! and others~e.g., Fredsøe 1984; Perlin and Kit 2002!, bot-
URNAL OF HYDRAULIC ENGINEERING © ASCE / JUNE 2005 / 497

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tom drag is enhanced when surface waves are long enoureach the bed. In this case, a thin, oscillatory, wave bounlayer @also known as a Stokes layer~Kundu 1990!# develops neathe bed that is more strongly sheared and experiences pdependent stresses that are larger than those operant in the tsteady current bottom boundary layer. It is the rectification operiodic stress that alters the overlying flow.

Grant and Madsen~1979! modeled the inner wave boundalayer using an eddy viscosity based on a phase-dependentvelocity, u*cw, defined by the phase-dependent bottom s~which is due to both waves and currents! to determine velocitstructure. Outside the wave layer, Grant and Madsen detethe effect of this inner wave layer on the main part of the boary layer by requiring the outer flow stress and velocity to mthe time averaged stresses and velocities of the inner wavyThe result is that enhanced turbulence within the wave layesults in greater drag on the mean flow, an effect that camodeled as an enhanced apparent roughness. The originalMadsen model has been extended by various writers, mocently Styles and Glenn~2000, hereinafter SG2000!, who incor-porated the effects of stratification as well as adjustingassumed profile of eddy viscosity upon which the theory is ba

Fredsøe’s~1984! model is similar to that of Grant and Madsalthough he assumed logarithmic profiles rather than solvinplicitly for the velocity structure. At the other end of the sptrum, Perlin and Kit~2002! make no assumptions about edviscosity profiles, instead using a turbulence closure that sfor the phase-dependent turbulent kinetic energy~TKE! fromwhich phase-dependent eddy viscosities, etc. can be compua similar fashion, Groeneweg and Klopman~1998! more formallyevaluated the average effects of waves on mean currents uK-« model and generalized Lagrangian mean~GLM! averaging.

Most circulation models use a drag coefficient to parametbottom drag~e.g., Casulli and Cattani 1994; Blumberg et1999; Gross et al. 1999!. Since these models resolve neitherdividual waves nor the thin wave bottom boundary layer,effects of waves are then included via a wave-dependentCD orwave-dependent roughness~Davies and Lawrence 1994, 199Signell and List 1997; Xing and Davies 2003!. Kagan et al~2003! found that inclusion of ocean-swell-enhanced roughnea circulation model of Cadiz Bay brought tidal stage and currpredicted by the model into better agreement with observatAll of these studies have found that wave-enhanced roughcan have significant effects on tidal and wind-driven circulatflushing rates and residence times of contaminants, channelasymmetry, and sediment transport.

Despite the recent application of Grant and Madsen’shanced roughness theory to a variety of circulation models,experiments to test the validity of this theory have been caout only under ocean swell on the continental shelf~Cacchione eal. 1994; Drake and Cacchione 1992; Green and McCave 1Lacy et al., unpublished!, and not under wind waves on the shoof an estuary. For ocean swell on the continental shelf, the ssurface wind-driven boundary layer and the steady current boboundary layer are separated by an inviscid core~Grant and Madsen 1986!; these are the conditions for which Grant and Madsmodel was formulated. The accuracy of the drag coefficientsstresses predicted by models like that of Grant and Madsennot been tested for shallow flows on the shoals of an estuathe presence of wind waves, an environment where thesesteady boundary layers overlap.

In this paper we investigate the variation of shear stress
CD under combined tidal currents and wind waves~generated by
498 / JOURNAL OF HYDRAULIC ENGINEERING © ASCE / JUNE 2005


-r,

r

–

l

the local diurnal sea breeze! on a shoal in South San FrancisBay. Our results show that the model presented by StylesGlenn ~2000, hereinafter SG2000!, while formulated for oceaswell on the continental shelf, is capable of reasonably accpredictions of the enhancement of shear stress and the becoefficient over estuarine shoals by wind waves.

Experiments

To investigate variations of water column shear stresses anbed drag coefficient under wind waves on the shoals of aestuary, we ran two experiments at Coyote Point in SouthFrancisco Bay~SSFB, Fig. 1!. This location was chosen becaof ease of access and because of an interest in understcirculation and sediment processes in this part of the SSFlight of proposals to build new runways nearby on fill that wobe placed on the local shoals. Our experiments took place inof 2000 and June-July of 2002. Each experiment was ru2 weeks to capture a full spring-neap tidal cycle. Coyote Pexperiences mixed semidiurnal/diurnal tides, a diurnal sea b~in summer!, and frequent spilling whitecaps. The bathymetrthe study site is relatively flat, with a depth varying between 14 m during a near-solstice spring tide. The bed at the studconsists of silts and fine sands~a representative value ofkb

=0.01 cm!, and bed forms were not present. Shoreward~south! ofthe site is a gently sloping sandy beach, which causes incwaves to break as spilling and plunging breakers, and prereflections.

Instruments

During the first~June 2000! experiment we deployed two SonTfield acoustic Doppler velocimeters~ADVs! mounted on a masitting approximately 90 m north of the beach at high water~seeFigs. 1 and 2! and in 1 m of water~mllw—mean lower lowwater!. The ADV measuring volumes were located 2062 mab~centimeters above bed!. These instruments sampled thcomponents of velocity at 25 Hz, had an accuracy of ±3 mand were cabled to computers on the shore for data acquisApproximately 200 m further north~and thus further offshore! wedeployed an upward-looking 1.5 MHz NorTek high-resoluacoustic Doppler profiler~ADP! mounted on a small gimbaleframe. Operating in a pulse-to-pulse-coherent mode~Lohrman eal. 1990!, it recorded 2-min averages of all three componenvelocity in 3 cm bins covering a range of depths fr43 to 133 cmab. Time-averaged measurements of tidal stagsea state were made with a SeaBird SBE26 absolute presensor~accuracy greater than 1 mm in free surface elevation!. AnOcean Sensors OS200 CTD made conductivity and tempemeasurements. To record local wind speed and directionmounted an anemometer on a 3 m high mast. However, bethe shore site where the anemometer was deployed was pashielded by trees, for our analysis below we used an averalocal winds as measured by the anemometer and those recnearby at San Francisco International Airport~SFO!, locatedabout 2 mi upwind~northwest! of the study site.

During the second experiment~June-July 2002! we made amore comprehensive set of measurements: Along with 3 SoADVs measuring velocities at 20, 53, and 153 cmab we alsployed a NorTek Vector ADV measuring velocities at 95 cmabaddition to the SBE26 used previously, we also used a capa
wave gauge~1 cm accuracy; 1 mm precision! to measure surface
.131:497-508.

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waves. For wave–turbulence decomposition purposes~seebelow!, the wave gauge was synchronized with the ADVsorder to get a more complete velocity profile than we had in 2the body of the ADP was buried, so the transducer facenearly flush with the bed, giving us good data starting at 15 cand extending to a maximum of 200 cmab~at high tide!. Finally,in addition to the OS200 CTD, temperature was also measura vertical array of six high-precisions±0.01°Cd SeaBird SBE3temperature loggers. Unfortunately, during the second experour anemometer failed and so when needed for analysis, wewind data from San Francisco International Airport’s~SFO!weather station.

Observations of Conditions at Coyote Point

During both experiments, winds were nearly westerly in direcand diurnal in strength, reaching 12 m/s each [email protected]~a!# which corresponds to a stress of approximately 0.2 [email protected]~b!#. Waves showed the same diurnal trend, increasing to a mmum heightHs=50 cm and a period,Tw=2 s in the afternoo@Figs. 3~c and d!#. These periods and amplitudes are in gagreement with what might be predicted using formulas presin the U.S. Army Corps of Engineers Shore Protection Ma~Bricker 2003!.

Tidal velocities reached a maximum of 30 cm/s during pflood with a favorable wind@Fig. 4~a!#. Flood currents were stroger than ebb currents~maximum ebb velocity was about 5 cm/!,

Fig. 1. Coyote Point~South

and flood usually lasted longer than ebb, due to both the topog-

JO


raphy of the site and the winds over the site. Due to local batetry, the flood tide~coming from the north! hit the study site othe north side of Coyote Point with full force, while during etide, the study site was in the lee of the Point, and was thuswake. During the afternoons, near-bed wave-induced orbitalocities averaged 12 cm/s, and was thus comparable in strenthe tidal current although they were nearly orthogonal to thecurrent@Fig. 4~b!#. Bottom stresses~determined as below! rangedfrom 0 to 0.13 Pa.

Fig. 2. Schematic of instrument arrangement~see text for details!

rancisco Bay, Calif.! field site
San F

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Fig. 3. Conditions existing in June 2002:~a! wind speed at 10 m measured at San Francisco International airport~SFO!; ~b! wind strescalculated per Eq.~11! from the SFO winds;~c! significant wave height at the instrument array;~d! dominant wave period at the instrument arand ~e! tidal variations in water depth at the instrument array

Fig. 4. Velocities and stresses measured in June 2002 by ADV located atz=20 cmab:~a! long-shore~—! and cross-shore~--! mean velocity;~b!long-shore~—! and cross-shore~--! wave orbital velocity; and~c! long-shore turbulent shear stress inferred by the phase method~…!, theBenilov–Filyushkin method~—!, and by the Shaw–Trowbridge method~-.!


J. Hydraul. Eng. 2005.131:497-508.

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Analysis Methods

Determination of Shear Stress via Time Series of PointVelocity Measurements

In principle, turbulence data from the vertical array of ADVsbe used to determineCD by assuming a constant-stress stebottom boundary layer and directly calculating Reynolds strefrom the fluctuating velocities. However, because varianceciated with the waves is much larger than that associatedturbulence, some form of wave–turbulence decomposscheme must be used~Jiang and Street 1991; Thais and Magndet 1995; Trowbridge 1998!. As we will discuss below severapproaches are possible and it remains an open questionwhich is most appropriate in a given situation.

In a flow with both waves and currents, the instantaneoulocity can be written as

u = U + u + u8 s6d

whereU=mean velocity;u=wave velocity; andu8=turbulent velocity. After Reynolds averaging the mean momentum equausing Eq.~6!, the Reynolds stress becomes@see, e.g., Jiang anStreet~1991!#

−t

r= uw + uw8 + u8w + u8w8 s7d

For irrotational waves~Dean and Dalrymple 1991!, the first termon the right-hand side~RHS! of Eq. ~7! is zero. Furthermorewhen waves and turbulence coexist, the latter can be definmotions that do not correlate with waves~Jiang and Street 199Thais and Magnaudet 1995!, so the second and third terms onRHS of Eq.~7! are also zero. Thus, under these conditionsReynolds stress is the same as that which is found for sflows

−t

r= u8w8 s8d

As shown by Trowbridge~1998!, small uncertainties in instrument orientation or a gently sloping bed can bias velocity msurements such that in practiceuw may not be exactly zero.

In analyzing our data, we used three methods of waturbulence decomposition to remove the waves from our tulence data. The first was that of Benilov and Filyushkin~1970!, inwhich motions that correlate with displacement of the free suare considered to be due to the waves. The second was tTrowbridge~1998!; and Shaw and Trowbridge~2001!, which usestwo velocity measurements spaced farther apart than the laturbulence scale@approximately1

4 the water depth, accordingShaw and Trowbridge~2001!#, but are well within one surfacwave wavelength of each other. Motions that correlate betwthe sensors are waves, while motions that are uncorrelatedefined as turbulence. The third method, which we call the “PLag” method, is described in Bricker~2003!. Assuming equilibrium turbulence and no wave–turbulence interaction, inmethod the phase lag between theu andw components of surfacwaves is used to interpolate the magnitude of turbulence uthe wave peak within the inertial subrange of the spectral domwhich is otherwise removed. In essence, in this approone removes not only the waves, but also the local enhanceof turbulence at or near wave frequencies~see Lumley and Terra
1983!.
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f

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t

After applying the various wave decompositions, we calated u8w8 using blocks of 213 samples~about 51

2 min of data!.This period was chosen because it is significantly longer thawave period, yet shorter than the time over which the sea sttidal regime would change. We assumed that the lowest ADsz=20 cmabd measurements came from within the constant-sinner layer of the steady bottom boundary, and thus that valuu8w8 measured there are equivalent to the bottom stress eon the mean current. Lastly, using this value ofu8w8, CD can befound in the usual fashion from Eqs.~1! and ~8!.

Determination of Theoretical Shear Stress via SG2000and the Overlap Method

Shear stress was also calculated at the bed via SG2000,assumes bottom-boundary-layer turbulence only. Stress wacalculated at the height of each ADV via the overlap method~seebelow!, which takes into account the wind-driven surface bouary layer. To use SG2000, we needed to supply the model wifollowing inputs:fc, the angle between waves and the meanrent;Uc, a reference current a heightzr above the bed;ub andAb,the near-bottom wave orbital velocity and excursion, respectiandkb, the physical roughness scale. The reference heightzr wastaken as the height above the bed of each ADV. The otherparameters were determined via the methods below.

Fine-grain sand dominates the near-shore subtidal zoCoyote Point. For typical values ofu* at Coyote Point, a physicroughness ofkb=0.01 cm, appropriate for fine sand, meansthe flow was hydraulically smooth. As a result, the equivalez0

for a hydraulically smooth bed isz0=0.11n /u*c, resulting inz0

=0.001 cm. Rather than use a variable value ofz0, we used thiconstant value ofz0 as an input to SG2000.

Maximum wave-induced near-bed orbital velocityub, orbitalexcursionAb, and the angle between waves and currentsfc weredetermined spectrally from linear wave theory~Dean and Darymple 1991! using ADV and SBE26 data. With all these paraeters determined, SG2000 then calculated the steady shearnear the bed, and Eq.~1! was used to determine the drag coecient based on this shear stress.

Overlap with Surface Wind-Driven Layer—The OverlapMethod

Since the wind-driven surface boundary layer can overlapbottom boundary layer in shallow flows, we further modifiedapproach given by SG2000 by assuming a linear variatiostress between the bed and the surface. Neglecting nonlinecelerations and assuming pressure to be hydrostatic, themomentum equation in thex direction is

]U

]t= − g

]h

]x+

1

r

]t

]zs9d

whereh=free surface deflection. The pressure term has nodependence, and the unsteady term is observed to vary farwith time than with depth~Fig. 5!. Therefore, to a first approxmation

]t

]z= fstd s10d

i.e., we can make the approximation that stress varies lin
with depth. From a practical standpoint, a linear variation is the

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most complex one we can predict, given knowledge onlstresses at the bed and the surface.

Each component of the wind stress at the surface takeform

twind = rairCD,windU102 s11d

resulting in a shear velocity at the top of the water column o

Fig. 5. Acceleration measured by the ADP in June 2002z=100 cmab as a function of the acceleration atz=60 cmab. The linshown marks the case where the two accelerations are equal.

Fig. 6. Stresses at 20 cmab in June 2002 as functions of stressdata using the Shaw and Trowbridge method~“ST” !; ~b! stresses coGlenn ~“SG”!; ~c! stresses computed from wave and mean flowsstress over depth; and~d! stresses computed using a constant va



u*wind = Îtwind/rwater s12d

with CD,wind determined from Yelland and Taylor’s~1996! empiri-cal relations. The stress throughout the water column wasassumed to vary linearly from its~vector! value at the bedsu*cd toits ~vector! value at the surfacesu*windd. Finally, in our “overlapmethod,” the predicted stress at the height of each ADV is clated using a linear interpolation between the calculated bostress and calculated surface stress.

Discussion

Effect of Waves on Bottom Stress

In what follows, we assume that the observed stresses arrepresented by stresses determined via the phase method.6 we plot the June 2002 data; the June 2000 data is esseidentical and is not shown. We note that near bottom strecalculated via Shaw and Trowbridge’s method@Fig. 6~a!# agreewell with those calculated by the phase method. We also notbecause the surface stress has little effect on the stress nebottom, predictions of the bottom stress made using only SGare essentially identical to those made using the more comoverlap method@Figs. 6~b and c!#. In general either predictiobased on SG2000 gives a reasonable prediction of the effewaves on bottom stress. However, by comparison, the conCD case, at least if based on a physically plausible value oCD,consistently underestimates the stress. Clearly, in terms of ua numerical model for cases where winds and hence wave

erred using the phase lag method~“Phase”!: ~a! stresses inferred from ADed from wave and mean flow conditions using the model of Styg the overlap method based on Styles and Glenn and a linearnd measured mean currents

es infmputusin

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relatively repeatable~for example, in the case of diurnal sbreezes!, CD could be adjusted upwards in the calibration proto match observation as was done by, e.g., Gross et al.~1999!.

The difference between the constantCD case and the observtions can be clearly seen in Fig. 7, where we have plottedCD as afunction of the ratio of the mean velocity to maximum orbvelocity @as was done by Fredsøe~1984! or Perlin and Ki~2002!#. It is clear that bottom stresses we measured were sicantly affected by waves, as Coyote Point is shallow enougwaves to “feel” the seabed. Shear stresses obtained througADVs, and the model of SG2000, all agree in trend. All dcoefficients converge to a value comparable to the canonicacoefficient at 1 m of 0.0025~Dronkers 1964! in the limit wheremean current velocity is much greater than the maximum nbed wave-induced orbital velocity. All methods also reveaincrease inCD of an order of magnitude over the canonical vawhen mean current velocity is much less than the near-bed ovelocity. Results from the 2000 experiment were identical.note that this variation inCD is due to the use of a drag law basentirely on the mean current, i.e., Eq.~1!. Were we to use a dralaw that explicitly included wave motions~cf. Hearn et al. 2001!,CD would not need to be as large.

While in general the comparison of SG2000 to the obsetions is quite good, stresses calculated by SG2000 asymptostrong currents to a value calculated using the canonical valCD. However, the stresses calculated from the ADV datasmaller. Therefore it appears that even in the current-domincase, the water column cannot be modeled as a constantbottom boundary layer, since the stress reduces to nearly zthe surface on a calm day. Stresses calculated via SG2000 athe measurements to be within a constant-stress bottom boulayer only, thus one should expect the model to overestimatshear stress observed at points a finite height above the b~asmuch as 20% of the depth at low tide! during times of minimawind.

Overall, comparison of predictions of bottom stresses withservations for both experiments shows that SG2000 prestresses better than one can do using a constant drag coefTable 1 presents the normalized mean square error~Bevington

Fig. 7. Measured bottom drag coefficient~s! and predictions obottom drag coefficient based on the SG model~—! and on the overlap method~--!, all as functions of the ratio of mean velocity to waorbital velocity. The measured values include error bars baseobserved variability of stresses for each wave condition.

and Robinson 1992! between two-week-long time series of shear

JO


ste

t.

stresses obtained via theory versus observations. Even wconsidering wave state, these values reinforce the viewSG2000 predicts stresses better than does a constant dragcient. In situations where wave-induced orbital velocities wlarger than mean currents~Table 2!, the disparity with the constantCD case is even greater. This agreement between theorobservation shows us that, despite the fact that SG2000’s mwas developed to predict enhanced roughness on the contishelf under ocean swell, it can also be effectively applied toshallows of an estuary under the action of wind waves.

Overlap of Steady Surface- and Bottom-BoundaryLayers

While SG2000’s predictions of stress agree with wave–turbudecomposed near-beds20 cmabd ADV data, as one moves highin the water column, the quality of the prediction~which neglectsurface stress! often underestimates the actual stress~Fig. 8!. Thereason for this is the overlap of the steady wind-sheared suboundary layer with the steady bottom boundary layer~driven byboth wind and tides!. SG2000’s model was formulated for the bof the continental shelf, on which shear stress is constanoriginates from bed friction only. In our data, however, strwaves were accompanied by strong winds. During wind evinstruments high in the water column were affected by the sstress and turbulence generated at the free surface as well abed. Thus, very close to the bottom, bed shear stress is domand thus predictions of the SG2000 model agree well with ovations. Further up in the water column, though, the sheaturbulence in the free surface boundary layer becomes prosively stronger, and the model inaccurately predicts shearstress obtained via assuming an overlap of the bed and sboundary layers, however, agrees well with wave–turbulenccomposed stress at all elevations@Figs. 8~a, c and e!#. To betteshow the quality of the prediction, we have plotted time serie

Table 1. Normalized Mean Square Errors«2d between Two-Week TimSeries of Shear Stresses Observed via the Phase Lag DecompMethod, and Shear Stress Predicted via the Overlap Method, SG20a ConstantCD

Instrument

cmabovebed

«2

phaseoverlap

«2

phaseSG

«2

phaseconstant

Coyote lower ADV 20 0.54 0.54 1.03

Coyote2 lower ADV 20 0.33 0.38 0.60

Coyote upper ADV 62 0.41 0.57 0.98

Coyote2 vector 95 0.73 0.91 0.92

Coyote2 upper ADV 153 0.79 1.12 1.50

Table 2. Same as Table 1, Except Only Considering Times WhenMaximum Wave-Induced Orbital Velocity Exceeded the Mean CuConsidered in the Error Computation

Instrument

cmabovebed

«2

phaseoverlap

«2

phaseSG

«2

phaseconstant

Coyote lower ADV 20 0.84 0.85 1.68

Coyote2 lower ADV 20 0.49 0.61 0.81

Coyote upper ADV 62 0.43 0.76 1.43

Coyote2 vector 95 1.13 1.12 2.31

Coyote2 upper ADV 153 1.03 1.37 2.61


.131:497-508.

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the stresses at 153, 95, and 20 cmab from the 2002 experalong with the corresponding predictions of the overlap mebased on SG2000~Fig. 9!. Excepting spikes in the observstresses, most likely associated with instrument noise, the oagreement is excellent, lending support to the conclusion thaSG2000 model and the simple dynamics embodied in the lstress distribution are reasonably accurate.

Wave–Turbulence Interaction

Stresses determined at 20 and 62 cmab were relatively inddent of the method used to do wave–turbulence decomposHowever, closer to the water surface, the phase lag methodsistently deduces stresses that are as much as an order oftude smaller than that calculated via the methods of BenilovFilyushkin or Shaw and Trowbridge~Fig. 10!. This result shoulbe contrasted with the result that at all elevations stress dmined by the phase lag method agrees remarkably well withdetermined by the overlap method.

The dependence of the inferred stress on the wave decosition method suggests that while wave–turbulence interamay be negligible near the bed, it is more important nearsurface where wave-induced orbital velocities, and the strainthey generate, intensify. The phase lag assumes no interbetween waves and turbulence, an assumption that may nvalid in the upper water column, where shear-generated tlence is stretched by the wave-induced strain field~Teixeira andBelcher 2002! his modulated turbulence therefore overlaps wthe wave field in the spectral domain, yet is still consideredbulence by Benilov and Filyushkin’s and Shaw and Trowbrid

Fig. 8. Measured stresses compared to predictions at several hez=95 cmab June 2002~c! overlap method and~d! SG2000;z=62 cm

methods of wave–turbulence decomposition~Thais and Magnau-



t

i-

det 1995!. In contrast, any spectrally local enhancement of tulence by waves is rejected as a wave by the phase lag met

Using a simplified form of rapid distortion theory~RDT!, Mo-nismith and Magnaudet~1998! showed that this type of wavturbulence interaction is essentially described by the same mthat predicts Langmuir circulations, i.e., the interaction ofStokes drift with the vorticity field of the mean flow. Since wamotions increase in strength further up in the water columndegree to which turbulence is strained is also enhanced clothe surface, and the disparity between the phase lag methothe others grows larger.

The importance of wave strains on turbulence can be gaby the rapidity parameterR, which following the methods of RD~Townsend 1976!, is defined as the ratio of wave-induced straiturbulence-induced strain~Monismith and Magnaudet 1998!

R=S ]u

]zD

wave

S ]u

]zD

turbulence

s13d

When R!1, turbulence-induced strain is much strongerwave-induced strain, and thus wave strain has little effect oturbulence. It is in this region that the assumptions of the plag method are valid. In contrast, whenR.1, wave-inducestrain field strongly modulates the turbulence field~cf., Texeiraand Belcher 2002!. In this case, because the phase lag medoes not account for any periodic variation in turbulence insity, etc., any additional periodic turbulent stress is attributedwave bias instead of turbulence, potentially causing the phas

above the bed:z=153 cmab June 2002~a! overlap method and~b! SG2000une 2000~e! overlap method and~f! SG2000

ightsab J

method to underestimate turbulent stresses. However, even in this

.131:497-508.

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Fig. 9. Measured~—! and predicted~s! stresses during June 2002 at~a! z=153, ~b! z=95, and~c! z=20 cmab

et variable.

Fig. 10. Stresses inferred using different methods at several heights above the bed:z=153 cmab June 2002~a! Benilov–Filyushkin~“Benilov” !method and~b! Shaw–Trowbridge~“ST” !; z=95 cmab June 2002~c! Benilov method;z=62 cmab June 2000~d! ST method;z=20 cmab Jun2002 ~e! Benilov method and~f! ST method. In all cases we have used stresses inferred via the phase lag method as the independen

JOURNAL OF HYDRAULIC ENGINEERING © ASCE / JUNE 2005 / 505

J. Hydraul. Eng. 2005.131:497-508.

ffects

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case, the phase lag method still accounts for any rectified eof the periodic wave straining on the turbulence.

From linear wave theory~Dean and Dalrymple 1991!, thewave-induced strain can be shown to be

S ]u

]zD

wave=

Hs

2kS2p

TwDS sinhskzd

sinhskhdD s14d

whereHs=wave amplitude andTw=wave period. The turbulestrain scales as

S ]u

]zD

turbulence,

u*

l<

u*

kzs15d

For the conditions at Coyote Point,Hs,50 cm,Tw,2 s, 2pi /k=l,10 m, h,2 m, andu* ,0.01 m/s. Atz=20 cmab, this results inR=0.05 whereastz=1 mab,R=5.0. Thus we predict thclose to the bed, turbulence is not affected by the wave-indstrain field ~Fredsøe 1984!, and the phase lag method separwave and turbulent stresses very well. Higher up in the wcolumn, however, wave-induced strain is as strong as turbstrain, and the phase lag method attributes any coherent mtion of the turbulent stress by wave strain to waves insteaturbulence.

Notwithstanding the good agreement between the predictiSG2000 and our measurements, these observations of wturbulence interaction are important in light of the theoreticalderpinnings of models like that of SG2000 or Perlin and~2002!. For example, Perlin and Kit assume that wave sheartributes instantaneously to TKE production. In essence, theimulation includes waves as part of the mean flow despite thethat in frequency space they overlap considerably with thequencies at which energetic turbulence is found. In contrast,RDT view of wave–turbulence interaction, the wave strain molates the turbulence field at wave frequencies such that theduction term can be positive or negative depending on wphase~Texeira and Belcher 2002!, an effect observed in labortory experiments by Pidgeon~1999!. The Grant–Madsen stymodels like SG2000~also Fredsøe 1984! are simpler in principleonly assuming a wave enhancement in the bottom stressbottom friction factor. However, their model implicitly assumno wave–turbulence interaction since it assumes that theviscosity that acts on the mean flow also acts on the wavesimportance of this neglect of the rectified effects of wave string on the turbulence has yet to be quantified.

Conclusions

Our study shows that wind waves and the overlap of bottomsurface boundary layers clearly have a large effect on shearand the drag coefficient on the shoals of an estuary. The enhsteady shear stresses and drag coefficients predicted by Sagreed well with near-bottom~20 cm high! observations of shestress in the steady bottom boundary layer under wind wavshoals. Further up in the water column, however, SG2000curately estimated the shear stress because the steady windfree surface boundary layer overlapped with the steady boboundary layer. At these higher locations, shear stress in thsurface boundary layer was as important as that in the boboundary layer, even in the absence of wind. In this case it
necessary to calculate the shear stress at both the bed and th


-

0

n

surface, and then to assume a linear variation between thesthroughout the water column. Shear stresses obtained in thiagreed well with data.

Water column shear stresses were underestimated by SGwhen the wave-induced strain field was strong. In this situawave–turbulence interaction stretched turbulence and enhReynolds stresses above those predicted by bottom boulayer theory. This effect had greater influence higher in the wcolumn, where the wave-induced strain field was stronger, ainfluence near the bed.

Overall our results show that surface waves should be exitly included in circulation models that are used to predict fland sediment dynamics in shallow estuaries. As discussed ilin and Kit ~2002!, this requires also including some form of wamodel, either one that is empirical such as the formulas givthe U.S. Army Corps of Engineers Shore Protection Manuone that is based on wave dynamics such as SWAN@SimulatingWaves Nearshore—see, e.g., Booij et al.~1999!#. In either casefrom the standpoint of engineering practice, it appears that aorder description of the effects of surface waves affect on bostress can be had through use of the model described in SGNonetheless, it appears that improvement in our understandhow waves and turbulence interact in shallow estuarine flwould improve our ability to predict those flows.

Acknowledgments

The writers thank William Shaw, Richard Styles, and Scott Gfor providing us with the codes for their models. Further thankthe staff of Coyote Point Marina and Coyote Point County Pfor providing support for our experiments. Thanks also to FLudwig and Doug Sinton for providing wind data. Fundingthis work was provided by the UPS foundation, by a grant fthe Ecosystem Restoration Program of the California Bay DAuthority ~ERP 02-P22!, and by the Office of Naval Resear~ONR Grant No. N00014-99-1-0292-P00002 monitored byLouis Goodman!. S.I. gratefully acknowledges support providto him by the Kajima Corporation.

Notation

The following symbols are used in this paper:Ab 5 near-bed orbital excursion;CD 5 bottom drag coefficient;

CD,wind 5 drag coefficient for wind blowing over water;fstd 5 general function of time;

g 5 gravitational acceleration;Hs 5 significant wave height;h 5 water column depth;k 5 wave number;

kb 5 physical roughness length;l 5 turbulence length scale;

R 5 rapidity;r2 5 square of correlation coefficient;t 5 time;

Tw 5 wave period;U10 5 wind velocity at 10 m above free surface;

Uc,U 5 mean flow velocity;u 5 total instantaneous horizontal velocity;

e ub 5 near-bed wave-induced orbital velocity;

.131:497-508.

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u*b 5 shear velocity near the bed, above the wave bottoboundary layer;

u*c 5 shear velocity above the wave bottom boundarylayer;

u*cw 5 maximum shear velocity inside the wave bottomboundary layer;

u*wind 5 shear velocity in water at free surface;u*s 5 shear velocity in water at free surface;

u 5 wave-induced orbital velocity in the horizontal;u8 5 turbulent velocity fluctuation in the horizontal;V 5 mean transverse velocity;vb 5 near-bed wave-induced orbital velocity iny

direction;w 5 total instantaneous vertical velocity;w 5 wave-induced orbital velocity in the vertical;

w8 5 turbulent velocity fluctuation in the vertical;x,y 5 horizontal axes;

z 5 vertical coordinate, increases from 0 at bed;zr 5 reference height;z0 5 physical roughness lengths=kb/30d;«2 5 normalized mean square error;h 5 deflection of free surface from mean sea level;k 5 von Kármáns’s constant<0.41;

lwave 5 surface wave wavelength;nc 5 eddy viscosity above the wave bottom boundary

layer;r 5 density of water;

rair 5 density of air;t ,tc 5 steady-current shear stress;twind 5 wind stress at free surface; and

fc 5 angle between waves and mean current.

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