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Mathematical model of wave transformation over radial sand ridge field on continental shelf of South Yellow Sea

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Mathematical model of wave transformation over radial sand ridge field on continental shelf of South Yellow Sea

    Mathematical model of wave

    transformation over radial sand ridge field

    on continental shelf of South Yellow Sea WaterScienceandEngineering,2010,3(1):3646

    doi:l0.3882~.issn.1674-23702010.01.004

    http://www.waterjourna1.cn

    e-mail:wse2008@vip.163.com

    Mathematicalmodelofwavetransformationoverradial

    sandridgefieldoncontinentalshelfofSouthYellowSea

    YaozhongYANG,WeibingFENG

    CollegeofHarbo~CoastalandOffshoreEngineering,HohaiUniversity,Nanjing210098,PR.China

    Abstract:Accordingtoadeformedmild?slopeequationderivedbyGuang

    wenHongandan

    enhancednumericalmethod,awavereaction

    diffractionnonlinearmathematicalmodelthattakes

    tidallevelchangeandthehigh

    orderbathymetryfactorintoaccounthasbeendeveloped.The

    deformedmild

    slopeequationisusedtoeliminatetherestrictionofwavelengthoncalculationsteps Usingtheharddisktorecorddataduringthecalculationprocess,theenhancednumericalmethod

    cansavecomputermemoryspacetoacertainextent,sothatalargescaleseaareacanbe

    calculatedwithhigh

    resolutiongrids.Thismodelwasappliedtowavefieldintegralcalculation overaradialsandridgefieldintheSouthYellowSea.Theresultsdemonstratesomefeaturesof

thewavefield:(1)thewaveheightcontour1inesarearc

    shapedneartheshore;f2wavesbreak

    manytimeswhentheypropagatetowardtheshore;(3)wavefieldcharacteristicsonthenorthemand

    southernsidesofHuangshayangaredifferent;and(4)thecharacteristicsofwavedistributionmatch

    theterrainfeatures.Theapplicationofthismode1intheregionoftheradialsandridgefieldsuggests

    tl1atitisafeasiblewaytoanalyzewaverefractiondiffractioneffectsundernatura1seaconditions.

    Keywords:wavetransformation,"mathematicalmodel:radialsandridgefield,"SouthYellowSea

    1lntroduction

    Asandridgefieldisalargescalegroupofradialsandbanksabovethewaterlineand

    underwatersandyridges,includingtidalchannels.TheSouthYellowSeasandridgefieldis 1ocatedoffthenorthernJiangsucoastonthesoutherncontinenta1shelfoftheYellowSea. from

    SheyangEstuarytoHaozhigang,whichissouthoftheYangtzeRiver.Ithasaradialfanshape withJianggangasthecentenThereliefmapwasshowninFig.1.Thelengthofthesandridge fieldis200krnfromnorthtosouth,andthewidthis140kmfromeasttowest.Theridgefield fanningouttotheseaconsistsofmorethan70sandridgesandtidalchannels.Thewaterdepth ofthearearangesfrom0to25m.ThemaintidalchannelsthatcutsandridgesareXiyang, Xiaobeicao,Chenjiawucao,Caomishuyang,Kushuiyang,Huangshayang,Lanshayang,and

    Xiaomiaohong.Theaveragewaterdepthoftheselargetidalchannelsismorethan10mand increaseswithdistancefromtheshore.WiththedevelopmentofJiangsuProvince,manyports ThisworkwassupposedbythePh.D.ProgramsFoundationoftheMinistryofEducationofChina(GrantNo

    20070294026).

    Correspondingauthor(e-mail:yyz_520@hhu.edu.cn)

ReceivedOct.26,2009;acceptedJan.16,2010

    willbebuiltinthisregion.Calculationsofwavefieldsareessentialinportconstruction. However,theareaunderconsiderationislarge,andtheterrainchangesrapidly.Therehasbeen littleresearchofwavefields'characteristicsinthisregion.Therefore,anewwavemodelhas beenestablishedinthisstudy.

    Inashallowseaarea,refraction,

    diffraction,andbreakingarethemainwave

    phenomena.Inordertostudywave

    transformation,researchershavedeveloped

    alargenumberofwavemodels.Generally

    speaking,thenumericalmodelscanbe

    dividedbytheircomputationaldomaininto

    threetypes:smallscale,middlescale,and

    large.scalemodels.Thethreedimensional

    Navier-Stokes(NS)equationisoneofthe

    typicalsmallscalemodels(WuandYuan

    2007).Itcanbeusedtocalculatelinearand

    nonlinearwavetransformationover

    complicatedterrain.Atthesametime,itcan

    Fig.1Reliefmapofradialsandridgefield

    providetheverticaldistributionofvelocity.However,duetothelimitationsofcomputer capacityandspeed,itismainlyusedinsmallscaleareas.Themiddlescalemodelcanbe

    representedbytheBoussinesqequationandtheBerkhoffmildslopeequation.Inthe

    Boussinesqequation,thewavetransformationisdescribedbywaterlevelandvelocity,sothe refractionanddiffractionofwavetransformationcanbecalculatedaccurately.Asthe Boussinesqequationdepictsthewavesurfaceelevationwithtime,thetimestepisusually1/30 to1/24thelengthofthewaveperiod,andthespacestepis1/12to1/8thewavelength.Thus, thissortofequationisonlysuitableformiddlescaleareas(Lieta1.2005).Forexample,itcan

    beusedtocomputewavefieldsinaharborbasin.Astherequiredsolutionitemisavelocity

potentialfunctionintheBerkhoffmild

    slopeequation,witheighttotencalculationpointsfor

    onewavelength,thecomputationaldomainisconstrainedbyfinitecomputermemory(Zheng

    eta1.2009).Large

    scalemodelscanberepresentedbytheactionbalanceequation(Zhengetal 2008).Thiskindofequationcanbeusedtocalculatelargescalewavefields.However,the

    preconditionisanincreasedspacestepandreducedgridresolution.Theradialsandridgefield oftheSouthYellowSeaislarge.Thetopographyoftheradialsandridgefieldfluctuatesso dramaticallythatamethodwithverylargespacestepscannotsimulatetherealtopography. Furthermore,withthiskindofmodel,itishardtocalculatethewavediffraction.Considering thesefactors,alargescalefine

    meshmathematicalmodelforwaverefractionanddiffraction

    wasproposed,andusedinthisstudytoanalyzetheoveralldistributioncharacteristicsofthe wavefield.

    YaozhongYANGeta1.WaterScienceandEngineering,Mar.2010,Vo1.3,No.1.36-4637 2Extendedmild-slopeequations

    2.1Governingequations

    Thegoverningequationsofthismodelarebasedonthemildslopeequationsfurther

    developedbyHong(1996).Heassumedthatthewavenumbervectorwasirrotationa1. and

    transformedthemildslopeequationintoawave.activityequationandeikonalequatiOn.In thisway,thesolutionofthevelocitypotentialfunction,whichvariesrapidlyinspace,canbe changedintothesolutionofawaveactionandwavenumberthatvaryslowly ,SOthatthespace

    step1Snotconstrainedbywavelengthbutbytheleveloftheacutetopographychange underwater,whichenlargesthecalculationdomainofthemathematicalmode1.Equationsare

    asfolows:

    Thenonlineardispersionequationis

tanh+H

    +

    ?F,H(1)

    whereistheangularfrequency,gistheaccelerationofgravity,kisthewavenumber,his

    thewaterdepth,Histhewaveheight,and,,andarethenonlineardispersion

    coefficientsderivedbyLiandLee(2000). Thewavenumbervectorirrotationalequationis 8Ksinof8KCOSof

    0

    whereKisthewavenumbervector.andistheangle normalvectoroftheboundary.

    (2)

    betweenwavedirectionandthe

    Thewaveactionconservationequationis

    

    0x

    …箜

    d-

    lJl}(

    wnere=

    ;t-anh;e=;=];

    (3)

    represents

    aissipanonauetomett.mect,=(Ko)];andistherricti.n coefficient.Inthismathematicalmodelthewavebreakingindexis0.78.

    Theeikonalequationis

    ]+

    where=,andGisthehigh-orderbamymetryvariationderivedbyPaneta1.

    (2000).

    2.2Initialconditionsandboundawconditions

    TheinitialwaveheightH0,waveperiodandwavedirectionoftheopenseaare 38gao?zhongYANGeta1.WaterScienceandEngineering,Mar.2010,Vo1.3,No.1,36-46 givendirectly.ThewaveheightboundaryconditionderivedbyPaneta1.(2008)is :BH

    On

    wnereextemnterra…一direction,osis

    theanglebetweentheincidentwaveandinterfacenormaldirection,isthereflection coefficient,andEristhephasedifference.

    ThewavedirectionboundaryconditionderivedbyFengandHong(2000)is =

    0+arcsin(BR/K)(6)

    whereistheangleofreflection,0istheinterfacetangentdirectionangle,and R1

    BR=Kcosa——.Inthecaseoftotalreflection,R

    r

    =1andgr=0.Inthe

    12R+COS,+'

    caseofanopenboundary,=0.

    3Calculationmethodforlarge-scaleseaarea

    3.1Newlydevelopedbookkeepingprocedure

    Thewavenumberirrotationalvectorequationandwaveactionconservationequationare nonlinearhyperbolicpartialdifferentialequations,SOafinitedifferenceschemeshouldbe used,Inthisstudy,thebackwarddifferencesemi.implicitscheme,whichcansavethe computermemorybyusingaharddisktorecordjntermediatedata,wasadoptedtowrite sourcecodes.Thismethodcannotonlyenlargethecalculationdomainbutalsoenhancethe gridresolutionatthesametime.Thisisanewlydevelopedbookkeepingprocedure.Itis suitableforcalculatingtheseaareaofcomplicatedlargescaleunderwatertopography.

    Eq.(2)isusedasanexampleheretoindicatetheimplementationprocessofthe developedbookkeepingprocedure,andthe

    backwarddifferencesemiimplicitschemei

    maxJ

    K

    1J

    sina

    |.|

    Ki?cOs

    ,J

    

    +f+

    l_'1j+

    KijKi

    .COSG

    i.i

    solutionofotherequationsarethesame.Afterthe

    K..|Kl

    i

    Eq.(2)canbediscretizedasfollows maxf

    sinOc

    if

    Kiisinat..

    ,.]

    (7)

    whereAxandAyarethespacesteps,andthesubscriptsiand,indicatevariablesforthe

    ithrowandjthcolumninthegridcomputingdomain.Theiterativemethodisusedtosolvethe

    nonlinearhyperbolicequations.Fromthediscretizationscheme,weknowthevaluesof,

    canbecalculatedwith,while

?

    isnotinvolvedinthefollowingcomputations.Thus,

    thevaluesoff

    1.,canberecordedontheharddisk,andthecomputationmemoryspacecan besaved.Then,thenewlydevelopedbookkeepingprocedureisrealized.Usingthismethod wecancalculatethelarge-rangewavefieldwithahighresolutiongrid. Yao-zhongYANGeta1.rScienceandEngmeering

    ,

    Mar.2010,Vo1.3,No1,364639

    3.2Waterdepthdatavariationwithtidallevel

    Itisnecessarytoconsidertheimpactofthewaterlevelonthewavefield,becausethe areaoftheradialsandridgefieldislarge.Ifwetake8sasthewaveperiodtocalculatethe velocityinadeepwaterarea,thenitwil1requireat1east3hforwavestoreachtheshore fromthedeepsea.Iftheincidentwavesareatalowtidallevelinadeepwaterarea,thesea surfacemayreachthemeansealevelwhenwavespropagatetotheshore.Thus,thevariation intidallevelcannotbeignored.

    Themodelinthisstudyuseswaterdepthdatavaryingwiththewaterleveltoreflectits impact.Firstofall,thewholeareaisdividedintothreesmallregions:northern,central,and southernregions.Inthece,nter,apointisselected,thewaterlevelprocessofwhichisusedto approximatelyrepresentthesituationthroughoutthearea.Secondly,thetimeofwave propagationfromtheithrowtothe(i+1)throwinthecalculationdomainiscalculatedas wavesmovefromdeeptoshallowwaters.Finally,interpolationisperformedbasedonthe existingtidalprocesslines.Wecandeterminethechangingtidevalueatcorresponding moments,andaddittotheoriginalwaterdepthdata.Waterdepthdatachangewiththetide, incorporatingtheeffectsofchangingwaterlevelonthewavefield.

    4VerificationofmathematicaImodeI

    TheBerkhoffphysicalmodelwasusedtoevaluatethecalculationaccuracyofthe n1achematicalwavemode1.Inthephysicalmodel,theslopecoefficientwas50,andtheangle betweenthegradientdirectionoftheslope

    andwave.crestlineoftheincidentwave

was70..Therewasanellipticshoalon

    theslope,whosecenterpointwas(x0,YO)

    (1om.10m,.Themajorhalfaxisand

    minorhalfaxiswere,respectively,4m;

    and3m.Here,H0=0.0464m,T=1.0s,

    thedirectionoftheincidentwavemoved

    alongthe+axis.andthespacestepsIn

    thexandYdirectionswere,respectively,

    0.05inand0.11TI.Fig.2isasketchofa

    submergedshoalandlocationsofthe

    measurementsectionsfrom1through8.

    Thewaterdepthinthephysicalmodel

    was0.0.4m.

    (in)

    Fig.2BottomtopographyforBerkhoffmodeland

    measurementsections

    Fig.3showsthecomparisonofmathematicalmodelresultswithBerkhoffphysical mOde1testresults.Overall,theymatchwel1.Itshouldbenotedthatbehindtheshallowareas

    wherethewaveenergywasconcentrated,themeasuredwaveheightonbothsideswasclose 40YaozhongYANGeta1.WaterScienceandEngineering,Mar.2010,Vo1.3,No1,36-46

topography,highresolutiongridsareneeded.

    1andformwillbesmoothed.Throughtentative

    Ifthespacestepistoolarge,acomplicated

    calculation,thispaperarguesthata30mgrid

    hasbeenabletobetterreflectthetopographyofthearea.Theareaoftheradialsandridgeisso

    largethatahugeamountofmemoryisrequiredifweusefinegrids. A32.bitFortrancompiler

    cannotprocessit.Therefore,weinventedabookkeepingprocedure. 5.2Analysisofwavefieldcharacteristicsinradialsandridgefield

    Accordingtostatisticalobservationofthewavesinthisarea,theprobabilityofwaves comingfromthenortheastishigh.Wecalculatedthewavefielddistributionunderthe conditionofyearlymaximumaveragewindvelocityandtheaveragetidalleve1.Theaverage annualmaximumwindspeedoverthesandridgefieldis17.74m/s.andthecorresponding significantincidentwaveheightatoffshoreboundariesis2.64m:thesecanbeconsideredthe normalconditions(Zhangeta1.1999).Duringthecalculation,wcconsideredanincidentwave

    directionOfNE,awaveperiodof8s,andawaveheightof2.64mtobetheinitia1conditions. 5.2.1Arcshapedwaveheightcontourlines

    ble1showsthecalculatedresultsofZhangeta1.f1999)andthemodelinthispaper. ThecalculationpointsareplottedinFig.4.Fromthetablewecanseethatthereareonlythree points(10,16,and19)thatshowacertaindifference;othersmatchwel1.Thecomparison showsthatthecalculatedwaveheightsarereliable.

    Table1Comparisonofcalculatedresultsofwaveheight

    PointZhang'swaveheightCalculatedwaveheightPointZhang'swaveheightCalculatedwaveheight

    No.(m)(m)No.(m)(m)

    12.482.15132.462.27

    22.252.26142.422.89

    31.220.90l52.452.62

    40.440.40162-250.98

    25O.000.00172.561.99

    52.592.45182.551.8O

    62_362.63191.10O52

    72.632.99202.512.34

    82572.69211.881.79

    92.6l2.59222.572.46

    102.281.06232.802.46

    l1O.61067241.942.09

    122.542.69

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