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SHORT COMMUNICATION
Note on estimating bed shear stress caused by breaking random waves
Dag Myrhaug
a,∗, Muk Chen Ong
baDepartmentofMarineTechnology,NorwegianUniversityofScienceandTechnology(NTNU),Trondheim,Norway
bDepartmentofMechanicalandStructuralEngineeringandMaterialsScience,UniversityofStavanger,Stavanger,Norway
Received 9October2020;accepted26March2021 Availableonline11April2021
KEYWORDS Bedshearstress;
Breakingrandom waves;
Surfsimilarity parameter;
Waveheight;
Individualwaves;
Jointdistributions;
Surfandswashzones
Abstract Thisnotepresentsamethodofhowthebedshearstresscausedbybreakingrandom wavesonslopescanbeestimated.ThisisobtainedbyadoptingtheSumer etal.(2013)bed shearstressformuladuetospillingandplungingbreakingwavesonhydraulicallysmoothslopes combinedwiththeMyrhaugandFouques(2012)jointdistributionofsurfsimilarityparameter andwave heightforindividual randomwavesindeepwater.Theconditionalmeanvalue of themaximaofmeanbedshearstressduringwaverunupgivenwaveheightindeepwateris providedincludinganexampleforspillingandplungingbreakingrandomwavescorresponding totypicalfieldconditions.Anotherexamplecomparesthepresentresultswithonecasefrom ThorntonandGuza(1983)estimatingthewaveenergydissipationcausedbybedshearstress beneathbreakingrandomwaves.
© 2021 Institute of Oceanology of the Polish Academy of Sciences. Production and hosting by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
Thekinematicsanddynamicsofthefluidmotionwithin the wave boundarylayer near the seabed in shallow and intermediate water depth of wave-dominated areas are
∗ Correspondingauthorat:DepartmentofMarineTechnology,Nor- wegianUniversityofScienceandTechnology(NTNU),OttoNielsens vei10,NO-7491Trondheim,Norway.
E-mailaddress:[email protected](D.Myrhaug).
PeerreviewundertheresponsibilityoftheInstituteofOceanology ofthePolishAcademyofSciences.
thedominantmechanism governingtheflowandsediment transport.Asthewavesapproachthesurfandswashzones theflowisintensifiedbywavebreakingleadingtoenhanced turbulenceproduction.Thebed shearstressrepresentsan important flow component playing a crucial role affect- ingthe sediment transportandmorphology and therefore the stability of scour protections in coastal regions. Fur- ther details on the background and complexity together withacomprehensiveliteraturereviewofbreakingwavesin coastalzonesareprovidedintherecenttextbookofSumer andFuhrman(2020).
The purposeof this article is to present a simple ana- lytical method which can be used to give first estimates ofthe bed shearstress caused bybreaking randomwaves
https://doi.org/10.1016/j.oceano.2021.03.004
0078-3234/©2021InstituteofOceanologyofthePolishAcademyofSciences.ProductionandhostingbyElsevierB.V.Thisisanopenaccess articleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).
on slopes. This is achieved by adoptingthe Sumer etal.
(2013) bed shear stress formula for spilling and plunging breaking regularwaves onhydraulicallysmooth slopesas- sumingthatitisvalidforindividualbreakingwaveswithin a seastate ofrandomwaves. Then,thejointstatisticsof bed shearstressandwaveheightin deepwaterisderived bytransformationoftheMyrhaugandFouques(2012)joint distributionofsurfsimilarityparameterandwaveheightfor individualrandomwavesindeepwater.Examplesofcalcu- lating theconditional meanvalue ofthe maximaofmean bed shearstressduring waverunupgiventhewaveheight indeepwatercorrespondingtotypicalfieldconditionsare provided.
The article is organizedasfollows.Thisintroduction is followedbygivingthetheoreticalbackgroundofthepresent approach. Thenthestatisticalpropertiesofthe bedshear stressisderived bycombiningtheSumeretal.(2013)for- mulaandtheMyrhaugandFouques(2012)jointprobability densityfunction(pdf)ofsurfsimilarityparameterandwave height. Two examplesareprovidedby firstdemonstrating applicationofresultsintermsoftheconditionalmeanvalue ofthemaximaofmeanbedshearstressduringwaverunup given waveheight indeep waterfor spillingand plunging breakingrandomwaves correspondingtotypicalfieldcon- ditions, and second by comparing thepresent model pre- dictionswithonecasefromThorntonandGuza(1983)esti- mating thewaveenergydissipationduetobottomfriction beneathbreakingrandomwaves.Finally,asummaryandthe mainconclusionsareprovided.
The Sumer etal. (2013)empirical formulafor the bed shear stress due tospilling and plunging breakingregular wavesis
u∗ gH0
=0.085ξ00.6, 0.19<ξ0<1.42 (1)
wherethebedshearstressisgiventermsofthefrictionve- locityu∗=
max(τ0max)/ρ,i.e.,associatedwiththemaxi- mumvalueofthemeanbedshearstressduetothebreaking wave(seetheirEqs.(16),(17),(18)andFig.9).Hereρisthe fluiddensity,gistheaccelerationduetogravity,H0isthe deepwaterwaveheight,ξ0isthesurfsimilarityparameter definedasξ0=m/√s0wherem=tanβistheslopewithan angleβwiththehorizontal,s0=H0/((g/2π)T2)isthewave steepnessindeepwater(i.e.,usingthelineardispersionre- lation),andTisthewaveperiod.Eq.(1)isbasedonbest fittodataobtainedfromthesmallscalelaboratoryexperi- mentsbyDeigaardetal.(1991)andSumeretal.(2013).The data representflow conditionsfor regular breakingwaves overahydraulicallysmoothbedduringthewaverunup,and thetypesofbreakingwaveswereclassifiedbasedonthose accordingtoGalvin(1968):
ξ0 <0.5: spilling 0.5<ξ0<3.3: plunging ξ0>3.3: surging
(2)
Further details on the background of these results are provided in Sumer et al. (2013) and Sumer and Fuhrman (2020). It shouldbe notedthat the surfsimilarity param- eter was introduced originally by Iribarren and Nogales (1949)andlaterappliedbyBattjes(1974).
Sumeretal.(2013)statedthatcautionmustbeobserved whenEq.(1)isusedoutsidetheindicatedparameterrange.
Thiswasfollowedby:“However,therangeofthesurfsimi- larityparameterissufficientlybroadtocover,forthemost part,mostpracticalsituations,namelyspillingandplunging breakingconditionsnotonlyinthelaboratorybutalsointhe field.” Thus,motivatedbythisandtakenasafirst-orderap- proximation,Eq.(1)isadoptedandassumedtobevalidfor individualbreakingrandomwaveswhichindeepwaterhas thewaveheightH0andwaveperiodT.Then,forindividual breakingrandomwaves,Eq.(1)isrearrangedto
u=√
hξd, u1<u<u2 (3) where
u= u∗ cξrmsd
gH0rms
, (c,d)=(0.085,0.6) (4)
andh=H0/H0rms,ξ=ξ0/ξrms arethenormalized variables usingtheroot-mean-square(rms)valuesH0rmsandξrms,re- spectively. Thus, u1=√
hξ1d with ξ1=0.19/ξrms and u2=
√hξ2dwithξ2=1.42/ξrms.
Here the joint pdf of u∗ and H0 is obtained from the joint pdf of ξ0 and H0 provided by Myrhaug and Fouques (2012)whichwasderivedbytransformationof ajointpdf of wavesteepness s0 and H0 given byMyrhaug andKjeld- sen (1984). This empirically based joint pdf of s0 and H0
wasa resultof best fitto datafrom wavemeasurements withwave rider buoys made at threedifferent deep wa- terlocationsatseaontheNorwegiancontinentalshelf.The Myrhaug and Fouques (2012) joint pdf of the normalized variablesξ=ξ0/ξrmsandh=H0/H0rmsis
p(ξ,h)=p(ξ|h)p(h) (5)
wherethemarginalpdfofh, p(h),andtheconditionalpdf ofξ givenh, p(ξ|h),aregivenasatwo-parameterWeibull pdfandalognormalpdf,respectively,as
p(h)= 2.39h1.39 1.052.39 exp
− h
1.05 2.39
; h≥0 (6) p(ξ|h)= 1
√2πσξξ exp
−(lnξ−μξ)2 2σξ2
(7)
Themeanvalueμξandthevarianceσξ2oflnξaregiven as,respectively,
μξ=
−0.048+0.5105h−0.279h2 for h<1.7
−0.125arctan[4(h−1.7)]+0.0135 forh>1.7 (8)
σξ2=−0.0375arctan[1.75(h−1.20)]+0.05625 (9) Oneshouldnoticethathere thefunctionarctanθ isde- finedforanangleθ intherange−π/2<θ <π/2.Further- more,basedonbestfittodatathermsvaluesH0rmsandξrms
are
H0rms=0.714Hs (10)
ξrms= m
√0.7sm
, sm= Hs
(g/2π)Tz2 (11)
whereHsis thesignificant waveheightandsm isacharac- teristicwave steepnessfor the seastate definedinterms ofHsandthemeanzero-crossingwaveperiodTz.Aplotof p(ξ,h)isgiveninFig.6inMyrhaugandFouques(2012).
Figure1 Isocontoursofp(u,h)withthepeakvaluepmax=2.24locatedatu=0.93andh=0.74.
Thejointpdfuandhisobtainedfromthejointpdfofξ andhbyachangeofvariablesfromξ,htou,hyielding p(u,h)=p(u|h)p(h) (12) where p(h)isgiveninEq.(6).Thischangeofvariablefrom ξ tou only affects p(ξ|h) sinceξ=u1dh−21d,and by using theJacobian|dξ/du|=u1d−1h−21d/d,thisgivesthefollowing conditionallognormalpdfofugivenh
p(u|h)= 1
√2πσuu exp
−(lnu−μu)2 2σu2
(13)
Here μu andσu2 are the mean value andthe variance, respectively,oflnu,givenby
μu= 1
2lnh+dμξ (14)
σu2=d2σξ2 (15)
whereμξandσξ2aregiveninEqs.(8)and(9),respectively.
Figure1showstheisocontoursof p(u,h)withthepeak valuepmax=2.24locatedatu=0.93andh=0.74.Overall, itappearsthatuincreasesashincreasesuptoabouth=1.5 abovewhichthepdfisnearlysymmetricwithrespecttou ofabout1.25.
Figure 2 depicts the conditional pdf of u given h for h=0.5,1.0,1.4and2.1,i.e.correspondingtowaveheights equal to H0rms/2, H0rms, Hs and 1.5Hs, respectively (since H0=hH0rms andbyusingEq.(10)).FromFigure2itis ob- servedthatthepeakvaluesareshiftedtohighervaluesofu withvaryingpeakvaluesashincreases.Thepdfsforh=1.4 and 2.1arenearlysymmetric withrespecttou.The fea- turesobserved hereareconsistentwiththoseobservedin Figure1.
Figure2 p(u|h)versusuforh=0.5,1.0,1.4and2.1.
According toEq.(3),u isdefined withina finiteinter- val,andthus theconditional pdf of u givenh followsthe truncatedlognormalpdf:
pt(u|h)= 1
N1p(u|h), u1<u<u2 (16)
N1=
lnu2−μu
σu −
lnu1−μu
σu
(17)
Figure3 u∗det/
gH0rms,E[u∗|h=1]/
gH0rmsandE[u∗|h=1.4]/
gH0rmsversusξrmsfromthelowertotheupperlines,respec- tively.
where is the standard Gaussiancumulative distribution function(cdf)givenby
(γ)= 1
√2π γ
−∞e−t2/2dt (18)
ItshouldbenotedthattheresultsinFigures1and2are modified for thetruncatedpdf in Eqs.(16)and (17)since the results depend onthe interval limitsu1 and u2. That is, theresultswillvarydependingonthebeachslopeand thewaveconditionsasdemonstratedinthesubsequentex- ample. However,overall Figures1 and2 exhibitthe main featuresofthejointpdfofuandh.
Herethe resultswillbeexemplified byconsidering the conditionalexpectedvalueofugivenhascalculatedfrom thetruncatedpdfinEqs.(16)and(17)as(Bury,1975) E[u|h]=u2
u1 upt(u|h)du
= NN21exp
μu+12σu2 (19)
N2=
lnu2−(μu+σu2)
σu −
lnu1−(μu+σu2) σu
(20)
Analternativetothepresentstochasticmethodistoap- ply Eq.(1) for breaking random waves by using it for an equivalentsinusoidalwave,i.e.byreplacingH0 withH0rms and T with Tz. By referring to this as the deterministic method,theresultis
u∗det=0.085ξrms0.6
gH0rms (21)
Figure 3 shows u∗det/
gH0rms versus ξrms in the range 0.19<ξrms<1.42 according to Eq. (21). The other two lines depict E[u∗|h=1]/
gH0rms=1.1303u∗det and E[u∗|h=1.4]/
gH0rms=2.1228u∗det versus ξrms, which are based on that E[u|h=1]=1.1303 and E[u|h=1.4]= 2.1228, respectively, using the non-truncated pdf, i.e.
Eqs. (19) (with N2/N1=1), (8), (9), (14) and (15). Thus, Figure 3 shows the relative differences between u∗ us- ing the deterministic method and u∗ given the wave heights corresponding to H0rms and Hs, respectively.
AsreferredtoregardingtheresultsinFigures1and2,the resultsinFigure3willalsobemodifiedusingthetruncated pdfinEqs.(19)and(20)dependingonthebeachslopeand thewaveconditions,i.e.alsoaffectingtherangeofvalidity.
Oneshouldnoticethatthebedshearstressisnotafunc- tion of the localwater depth. However, asdemonstrated intheexampleby comparingthepresentresultswithone casefromThorntonandGuza(1983)(hereafterreferredto asTG83),thewave energydissipation duetobottomfric- tionbeneathbreakingrandomwavesgiveninEq.(36)con- tains the local water depth hd. Moreover, another factor not accounted for is the bed roughness. The results are validforhydraulicallysmoothflowforwhichtheroughness Reynoldsnumberu∗ks/ν <5,ks=2.5d50isNikuradse’sbed roughness,d50isthemediangrainsizediameter,andνisthe kinematicviscosityofthefluid(Soulsby,1997).Thisaspect isaddressedfurtherinthesubsequentexample.
The present method should bevalidated by comparing predictions withdatafrommeasurements. Datafrom bed shearstressmeasurementsforspillingandplungingbreak- ing random waves on slopes associated with well-defined random wave conditions in deep water are required for makingapropervalidationofthemethod.Totheauthors’
knowledge suchdataarenot availablein theopen litera- ture.Thus,firstanexampleisincludedtodemonstratethe application of estimating the bed shear stress caused by breakingrandomwavesonslopesoverhydraulicallysmooth beds,andsecondbycomparingthepresent modelpredic- tionswithonecasefromTG83estimatingthewaveenergy dissipationduetobottomfrictionbeneathbreakingrandom waves.
First,resultsareexemplifiedforh=1,i.e.adeepwater wave height corresponding toH0rms. Thus, substitution of thisinEqs.(8),(9),(14)and(15)yields
μu=0.1101 (22)
σu2=0.0248 (23)
Thegivenflowconditionsare:
• Hs=2m, Tz=6.5s.
This gives H0rms=1.4m from Eq. (10) and sm= 0.03, ξrms=6.9mfromEq.(11).
• Beachslopesm=0.05, 0.10.
For these slopes, ξrms=0.345 and 0.69, respectively.
Thus, by taking Eq. (2) tobe valid for random waves re- placingξ0withξrms,thesetwovaluesofξrms correspondto spillingandplungingbreakers,respectively.
Forspillingbreakers(m=0.05)thisgives:
u1=(0.19/0.345)0.6=0.6991 (24)
u2=(1.42/0.345)0.6=2.3371 (25) whichsubstitutedinEqs.(17),(19)and(20)yields
N1=0.9985, N2=0.9991 (26)
E[u|h=1]= 0.9991
0.9985·1.1303=1.1310 (27) Forplungingbreakers(m=0.10)thisgives:
u1=(0.19/0.69)0.6 = 0.4613 (28)
u2=(1.42/0.69)0.6 = 1.5419 (29) whichsubstitutedinEqs.(17),(19)and(20)yields
N1=0.9798, N2=0.9710 (30)
E[u|h=1]= 0.9710
0.9798·1.1303=1.1201 (31) It should be noted that for the non-truncated pdf the resultisE[u|h=1]=1.1303,i.e.theeffectoftruncationis insignificantinthisexample.
Finally,basedonthetruncatedpdf,theconditionalex- pected value of the friction velocity for spillingbreakers usingEqs.(4)and(27)becomes
E[u∗|H0=H0rms]=1.1310·0.085
·0.3450.6√
9.81·1.4=0.188m/s (32) Similarly, theconditionalexpectedvalueofthefriction velocityforplungingbreakersbecomes
E[u∗|H0=H0rms]=1.1201·0.085
·0.690.6√
9.81·1.4=0.282m/s (33) Furthermore, by using Eq. (21) it follows that the stochastictodeterministicratio,E[u∗|H0=H0rms]/u∗det,be- comes1.1310and1.1201forspillingandplungingbreakers according toEqs.(32)and(33),respectively,i.e.ratiosof about1.1inthisexample.
As referredtothepresentmethodisvalidfor hydrauli- cally smooth flow,i.e., for u∗d50/ν <2,and consequently ford50<2ν/u∗.Thus,bytakingν=1.36·10−6m2/s,itfol- lows by substituting for u∗ from Eqs. (32) and (33) that the results are valid for d50 < 15 mm and d50 < 10 mm
for spillingandplungingbreakers, respectively.These up- pervaluesofgrainsizescorrespondtogravelrepresenting pebble(Soulsby,1997).
Second,thewaveenergydissipationduetobottomfric- tionbeneathbreakingrandomwavesisestimatedandcom- paredwithonecasefromTG83.TG83calculatedtheaver- agefrictionalenergydissipationbyfirsttime-averagingand then averaging over all Rayleigh-distributedwave heights yielding(seetheirEq.(40))
EDTG=ρcf 1 16√
π(0.42
ghd)3 (34)
wherecfisthebedfrictioncoefficient,andhdisthewater depth.Thisresultisbasedonassumingshallowwaterwaves andthatHrms=0.42hd.
The present method is used toestimate the wave en- ergydissipationbyfirstconsidering thatforregularwaves asED= T1T
0 τ(t)u(t)dt where τ(t)=ρu∗2u(t)|u(t)|is the time-varyingbedshearstress,tisthetime,u(t)=U0sinωt isthehorizontalregularwave-inducedvelocitywiththeam- plitudeU0, and ω=2π/T is the angularwave frequency.
Substitution of this gives ED=(4/3π)ρu∗2U03. Further, in shallowwaterU0=(H/2)
g/hd,giving ED=ρu∗2 1
6π(H
g/hd)3 (35)
Then,applicationof thisfor breakingrandomwaves in shallowwaterwithH=Hrms=0.42hd,thedeepwaterwave heightasH0=H0rms,andtakingu∗asE[u∗|h=1],yields ED=ρ 1
6π(0.42
ghd)3u∗2det(E[u|h=1])2 (36)
ThepresentresultiscomparedwiththatbyTG83bytak- ingtheratioofEq.(36)(usingEq.(21))andEq.(34)giving
ED
EDTG = g cf
8 3√
π0.0852ξrms1.2H0rms(E[u|h=1])2 (37) Now the wave conditions representative for Torrey Beach, California during November 1978 used by TG83 is adopted: cf=0.01, beach slope β=0.02, H0rms=0.5 m, spectralpeakperiodTp=14s.However,thepresentmethod uses Tz, and by assuming a Pierson-Moskowitz wave am- plitudespectrum(TuckerandPitt,2001)Tz=0.7Tp=9.8s, whichgivesξrms=0.35fromEqs.(10)and(11),i.e.,spilling breakers.BytakingE[u|h=1]=1.13fromthepreviousex- ample,substitutioninEq.(37)givestheratioED/EDTG=1.9. TG83estimatedthefrictionaldissipationtobelessthan3%
ofthedissipationduetobreakingfordepthslargerthan0.2 mwithinthesurfzone,whichwassimilartoresultsforlab- oratorybeaches. Thus,the present methodestimates the frictionaldissipationtobelessthan5%to6%ofthedissipa- tionduetobreaking.
Asummaryandthemainconclusionsofthisworkareas follows:
Asimpleanalyticalmethodforestimatingthebedshear stress caused by breaking random waves on slopes using deep water wave statistics is presented. The results are achieved by adopting the Sumer et al. (2013) bed shear stress formula for spilling and plunging breaking regular wavesonhydraulicallysmoothslopesduringwaverunupas- sumingit tobevalid for breakingindividualwaves within aseastate ofrandomwaves.The statistical propertiesof
thebed shearstressandthewaveheightindeepwateris thenderivedbytransformationoftheMyrhaugandFouques (2012) joint distribution of surf similarity parameter and waveheightforindividualrandomwavesindeepwater.Re- sultsaregivenintermsoftheconditionalmeanvalueofthe maximaofmeanbedshearstressduringwaverunupgiven thewaveheightindeepwater.
Exampleofresultsareprovidedforspillingandplunging breakingrandomwavescorrespondingtotypicalfieldcondi- tions.Forthisparticularexamplethepresentmethodyields bedshearstressvalueswhichareabouttenpercentlarger than thoseobtained byusingthe Sumeretal.(2013) for- mulaforbreakingregularwavesreplacingthewaveheight andsurfparameterwiththeircorrespondingrmsvalues.
Another example compares the present model predic- tionswithonecasefromThorntonandGuza(1983)byesti- mating thewaveenergydissipationduetobottomfriction beneath breakingrandom waves representing field condi- tions.Thepresentmethodestimatesthefrictionaldissipa- tiontobeafactor 1.9largerthanthatgivenby Thornton andGuza.
As stated by Sumeret al. (2013) caution must be ob- servedwhenusingthepresentformulaoutsideitsrangeof validity, whichalsois thecase for thepresented method.
However,comparisonwithdataarerequiredinordertoval- idatethemethod.Meanwhilethisapproachshouldenhance thepossibilitiesofassessingfurtherthebedshearstressfor spilling and plungingbreaking random waves onslopes in laboratoryandfieldconditions.
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