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Page 1: Numerical Computation of Ocean Surface W aves in a Coastal ... · 1 Introduction 4 2 GeoSim Activity 1997 5 2.1 Coastal Region Simulations 5 2.2 River Flood Simulations 7 3 Details

Classification: Unrestricted

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January 1998

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GROUP 2

SELECTED BY AUTHOR(S)

SINTEF Applied Mathematics,

Numerical Simulation

Numerical Computation of Ocean SurfaceWaves in a Coastal Zone,

Simulation Activity in the DynaMap Project

E. NøstandG. Rindahl

NorgesForskningsra˚d (TheNorwegianResearchCouncil)

Unrestricted

Open 42300602 27

DynaMapRep.tex

031 1998-01-30

This reporttreatsresultsandexperiencesfrom theSimulationActivity in theDynaMapprojectduring the fall of 1997.A numericalsimulatorfor investigationof theeffectsofseabedbathymetryon oceansurfacewavesin a coastalregionhasbeendeveloped,andresultsandfacilities of this tool areheredescribed.Experienceswith geodataasa basisfor numericalsimulationsarediscussed,leadingto a descriptionof future requirementson geodatainformationfor usein coastalzonesimulations.Thereportalsodiscussesex-tensionsandfurtherdevelopmentof thesimulator, andtestcasesfor futureactivitiesaresuggested.

CoastalZoneGeodata geodatalangskysten

Oceanwaves havbølger

NumericalSimulations numerisksimulering

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Contents

1 Introduction 4

2 GeoSim Activity 1997 52.1 CoastalRegionSimulations ����������������������������������� 52.2 RiverFloodSimulations �������������������������������������� 7

3 Details of the Coastal Region Simulations 83.1 OceanSurfaceWaves ��������������������������������������� 83.2 TheSimulator ��������������������������������������������� 9

3.2.1 BasicEquations �������������������������������������� 103.2.2 BoundaryConditions ��������������������������������� 113.2.3 TheFEM-SimulationUnit ������������������������������ 133.2.4 Generationof ComputationalGridsandHandlingof Geodata���� 16

3.3 Resultsfrom theHasvikSimulations ���������������������������� 18

4 Conclusions for Further Activities 204.1 GeodataRequirements��������������������������������������� 204.2 ExtensionsandImprovementson theSimulator. ������������������� 22

4.2.1 TheFEM-SimulationUnit ������������������������������ 224.2.2 TheGrid Generator ����������������������������������� 23

4.3 RelevantCaseStudies ��������������������������������������� 244.3.1 Hasvik,Sørøyain Finnmark ����������������������������� 244.3.2 Fedjeosen- MongstadandSture �������������������������� 244.3.3 WavePowerPlants ����������������������������������� 25

5 Bibliography 26

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1 Introduction

DynaMapaddressesdevelopmentof models,methodsandsoftwaretools for the OpenGeodataMarket.Thepresentreportcontainsdescriptionsof theDynamicSeaChartandtheRiverFloodscaseapplications,wherethefirst, fromnowoncalledCoastal region sim-ulations, hasbeenselectedfor a moredetailedinvestigationwith thepurposeto producecaseapplicationrequirements.(Fordetails,seetheDynaMapProjectPlan[SINTEF1996]).

Hence,thetwo testcasesconsideredfor thesimulationactivity in theDynaMapproject,GeoSim,Section2, are

1: Coastalregionsimulationswhereonewouldusenumericalsimulationasa tool for in-vestigatingwaveandcurrentconditionsin a costalzone.Onetypical applicationof thistool wouldbein theplanningof constructionsalongthecoast.Thenecessarygeodatain-formationwould beseabedbathymetry, coastlineinformation,landtopographyclosetotheshoreregionandinformationonmanmadeconstructions(harbours,breakwaters,wavepowerplantsetc.).

2: Riverfloodsimulations,whereonewoulddonumericalsimulationsof riversandwatercourseswith a view to flood forecastingandmanagement.Relevantgeodatain this casewouldbetopographyof river valleysanddrainagearea,rockandsoil characteristics,pre-cipitation,snowcover, groundwaterbasinsetc.

A simplifiedversionof acoastalregionsimulatorhasbeendeveloped,investigatingoceansurfacewaves(i.e. wind generatedwavesandswell) over a seabedin the coastalareasurroundingHasvikonSørøyain Finnmark,Norway(Detailsin Section3).Thesimulatorcanbegeneralizedto includetidal wavesandcurrents,in orderto facethechallengeoftime dependentcoastlineinformation.For the river flood simulationssomeresearchonexistingsoftwareandpossiblecontributionshasbeenmade.For thetime being,we havenosuchsimulatorto demonstrateuseof geodata.

Variousaspectson furtheractivitieswill bediscussedin section4.

SINTEFAppliedMathematics(SAM) hasa long historyin oceanwavesmodelling.Theapplicationareascoverbothwaveinducedforceson marineconstructionsanddesignofpowerplants.TheNorwegiancompanyNorwaveASandSAM havehadalongandfruitfulcooperation.SAM ispresentlysupplyingNorwavewith themathematicalcompetenceandsoftwarenecessaryto conductthecomplicatedanalysisinvolvedin thedesignof powerplants,andNorwavehaspreviouslydevelopedsomeof theunderlyingequationsandnu-mericalmethods.Experiencesandresultsfrom theseprojectshavebeenessentialfor thedevelopmentof theHasviksimulations.

Due to the complexityof the bathymetryof coastalregions,finite elementmethodsareparticularlysuitablefor this problem,sincetheelementscanbefreely variedto discreti-sizethe regionof irregularity. Thesoftwareis basedon thenumericallibrary Dif fpack,[Dif fpack].

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2 GeoSim Activity 1997

Theideaof thesimulationactivity in theDynaMapprojectis todemonstrateuseof geodatain computersimulationof naturalphenomena(DynaMapProjectPlan[SINTEF1996]).A desiredoutcomeis anadaptivecasesimulatorbasedon geodatainformation.Anotherwould be a closerspecificationof the geodatarequirementsfor oneor severalkinds ofapplications.For thework duringthefall of 1997,two caseshavebeenconsidered.

2.1 Coastal Region Simulations

Computersimulationof diffractionandrefractionof oceanwaves/swellsenteringthecoastalzonewith harbourareas,headlands,islands,rock awashandsunkenrocks,etc.,is maintopicof thiscasestudy.

An applicationdemonstratinghowseabedandshoreconditionsmayeffect thebehaviourof wind generatedsurfacewaveshasbeendeveloped.TheFinnmarkseaareaLopphavetandthecoastalregionsurroundingHasvikharbourwaschosenasa specificcase(Figure1).

TROMSØ

� HasvikLopphavet

SØRØYA

FINNMARK

Figure 1: Hasvik on Sørøya in Finnmark, Norway

TheHasvikharbouriscarryingalot of traffic,andhasalwaysbeenverysensitivetoweatherconditions.A newbreakwaterwasconstructedin 1996,andhasmadetheharbourmoresta-ble.Duringastormonthe31stof October1997however, thisbreakwaterwasdestroyedby

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waves(Figure2). Roughlyestimatedspectrafor incomingwavesin thesimulationscon-tain severalwavelengthsanddirectionswith significantwaveheightof about10 meters.Thesearebasedon wind measurementstakenat Torsvag LighthouseandHasvikAirportOctober31st1997.Theharbourareais simulatedbothwith andwithout thenewbreak-water.

Figure 2: Newspaper cuttings from November 1st, 1997.

An interestingaspectof coastalzonewavebehaviour, andan importantreasonfor thischoiceof testcase,is thefocusingeffectof banksandunderwaterslopes.Theseareactingas“lensesandprisms”,changingthedirectionof thewavesandthusthelocalwaveheightamplificationssignificantly. In this particularsimulation,the incomingdirectionsof thewavesarebetweenWestandNorthWest,while in theharbourareathedirectionischangedto SouthEast,directly onto thequayworks.Thesimulatedwaveheightindicatesstrong

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reflectionfrom, andthusheavyloadon thebreakwaters.This probablyaccountsfor thedamageduringthestormof October31st.Detailsof theHasviksimulationsaregiveninSection3.3.

Thenumericalmethodchosenfor thesesimulationsis afinite elementmethod.An essen-tial part of theequationsolvingprocessis thecreationof computationalgrids.Thegridis refinedasthedepthdecreasestowardstheshore.Thenodesmustbesampleddenselyenoughto resolveeachpresentwavelengthover therelevantareas,andsparselyenoughfor thenumericalcomputationsto bemanageableon theentireregionof interest.In orderto achievethis, it is necessaryto adaptthegrid with respectto depthandwaveperiods.A significantpartof thesimulationis thereforethetransitionfrom geodatato thecompu-tationalgrid (seeSection3.2.4for furtherdetails).In orderto obtaina correctpictureofbathymetryeffectson surfacewaves,high quality geodatawith respectto resolutionandconsistencyis crucial,andbetterdatawill ensuremorereliablecalculations.Thenotionofthetermhigh quality geodata in thiscontextis discussedin Section4.1.

An importantfeatureof thesimulator, is theincorporationof aninfiniteoceansurroundingtheactualsimulationarea.Outsidethefinite elementsolutionarea,a finite differencede-scriptionisused,andasymptotictheoryisapplied.Thematchingbetweensthesetwoareasincorporatesradiationconditionsandothernecessaryrequirementsin theinfinite exterior,yielding a far morerealisticresultthana sequentialsolverwould beableto produce.Asequentialsolverwould in thiscasefirst computeasolutionin theouteropenoceanarea,andthenusethisasaboundaryconditionfor theinnersolution,whereasthesimulatorheredescribedletsthesolutionsin thetwo areasinteractmutuallythroughoutthesimulation.Physically, this meansthatall wavereflectioneffectsfrom the innerareaonto theouterregionareincludedin theboundarywavespectra.

2.2 River Flood Simulations

Someresearchhasbeendoneonpossiblecontributionsto floodsimulations.Themainchallengein improvingsignificantlyon currentflood forecastingandmanage-menttools,is toobtaingeodatathatissampleddenselyenoughfor theterrainandthephys-icalphenomenatobeproperlyresolved.Highresolutionsimulationscanalsoleadtoadatahandlingchallenge.

Currently, floodsimulatorsareusuallybasedononedimensionalmodels.Thatis, thesim-ulationisperformedalongtheriveronly.Therefore,aninterestingcontributionmightbetosolvetheequationsin twodimensions,alongandacrossthewatercourse,onasmallerpartof theriver. Theconsequencesof simplificationsfrom a two to a onedimensionalmodelmaythenbeinvestigatednumerically.Accordingto our investigationsin theGeoSimproject,themainhydrologicalsimulatorsappliedin forecastingandmanagementtodaycontainasequenceof variouscomputationalunitsfor eachphysicalprocessor phenomenon.Onemightconsidera tool thatworksnotonly in sequence,but allowsinteractionandfeedbackbetweenthedifferentmodulesbyincorporatingtheminto thesametwo or threedimensionalsimulator.

Thefull designor developmentof suchasimulatorwouldrequireclosecollaborationwithexistinghydrologicalresearchgroups,andisbeyondthescopeof thisprojectandourpresentresources.

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3 Details of the Coastal Region Simulations

Manytypesof wavesinvolvingdifferentphysicalfactorsexistin theocean.A crudeclassi-ficationof oceanwavesaccordingtotherestoringforceisgivenin Table1(from[Mei 1989]).In nature,severalrestoringforcescanbein effectat thesametime,hencethedistinctionbetweenvariouswaveslistedin Table1 is notalwaysverysharp.

Table 1: Types of waves involving different physical factors in the ocean, classifiedaccording to restoring forces

WaveType PhysicalMechanism TypicalPeriod

Sound Compressibility� �������� ����

s

Capillaryripples Surfacetension � � ���� s

Wind wavesandswell Gravity�������

s

Tsunami Gravity�

min.���

h

Internalwaves Gravity, densitystratification�����

h

StormSurges Gravity, Earthrotation�����

h

Tides Gravity, Earthrotation� ����

h

Planetarywaves Gravity, Earthrotation,variationof latitudeor oceandepth O(100days)

3.1 Ocean Surface Waves

Thepresentcoastalzonesimulationscontainnumericalcomputationsof wind generatedwavesandswell in acoastalregionwith seabedinformation.Thetypicalperiodsof thesewavesarefrom1to25secondsandtheyareactingontheseasurface.Thesesurfacegravitywavesarecharacterizedby thedispersionrelation! ��"$#�%�& ' (�)�#+* (1)

whichrelatesthewaveperiod , " ��-/. ! to thelocal wavelength 021 354 6�7 " ��-/. # . Here! denotestheangularwavefrequency,#8"9# 1 ! 4 * 1 354 6�7 7 thewavenumberand

%theac-

celerationof gravity. Thespatialvariationof waterdepthis denoted*:"$* 1 354 6�7 .

In orderto handlethephysicalphenomenaaccuratelyenoughto obtainreliablenumeri-cal results,the local wavelengthmustbe resolvedproperly. For thesamewaveperiod,thelocalwavelengthis shorterin shallowwaterregionscomparedto deeperwaterareas.Therefore,shallowwatersrequirefiner spatialresolutionof theseabedtopography, andhence,densersamplingof theseabed.Regardingthedispersionrelation(1), wecanalsoseethatat thesamewaterdepth,the local wavelengthis shorterfor smallwaveperiodsthanfor longwaveperiods.Therefore,whenperformingnumericalcalculationsof surfacewavesfor variouswaveperiodsoverthesamesimulationarea,oneshoulddiscretisizetheregionwith respectto thesmallestwaveperiodof thecalculations.Table2 indicatessome

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relationsbetweenwaveperiodandwavelengthondifferentwaterdepthsaccordingto thedispersionrelation(1).

A train of wind generatedwavesandswellscanpropagateover largeoceanareasunaf-fectedby the seabed.But, whenenteringthe coastalzoneof shallowwaters,the localwavelengthchangeswith depth,resultingin agradualchangein wavepropagationveloc-ity. In addition,with presenceof physicalobstaclessuchasrockawash,headlands,harbourentrances,etc.,theresultingdiffractionphenomenamaycausea major increasein waveamplitude.

In very deepwaters,thesurfacewavespropagateindependentlyof waterdepth.This oc-curswhenthewavelengthis very largecomparedto thewaterdepth,thatis thedeepwa-ter limit ( ;+<>==@? ). Therefore,in openoceandeepwaterregionsthewavepropagationis unaffectedof diffractionphenomenadueto varyingseabed,andthewavelengthde-pendsonly on thewaveperiod.Thewavelength, A�B , on infinite depth,<+B is alsogivenin Table2 for somewaveperiods.

Table 2: Some relevant wave lengths, A , in meters derived from wave period andwater depth by the dispersion relation (1). The deep water limit, C�D5<2E�A:==9?

WavePeriodF�G ? H sec.F�G ? I sec.

F$G ? J sec.F$G ? K sec.< G 10m 92 123 144 164

Water < G 30m 134 196 234 271Depth < G 70m 156 249 311 372< G 100m 156 259 335 410<+B9L 100 156Deep <+B9L 150 263water <+B9L 200 351limit <+B9L 350 451

3.2 The Simulator

Reliableestimatesof waveheightscanbeobtainedwhentheoceanwavefield is repre-sentedby directionalwavespectra,i.e.,incident(incoming)waveswith variousdirectionsandfrequencies.Theopenoceanareahasinfinite extension,andwhentreatingoceanwavecalculationsnumericallyonehasto dealwith finite calculationareasandintroduceartificial bound-ariesin theopenocean.Figure3.1showsasketchof asimulationareathatcontainsopenoceanboundaries(boldstraightlines)handlingtheradiationconditions.Thesimulatorin-corporatestheseeffectsof an infinite oceanaroundthesimulationareaandneedsinputinformationon theoceanwavefield far awayfrom themajorareaof interest.

Theproblemcanbetreatedwithin well knownwavetheory, andthewavesimulatorcon-sistsof aboundaryvalueproblemsolvedwith numericalcalculationtechniques.Eventhoughthegoverningequationsarewell known,it isamajorresearchtaskto develop

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Incomingcoastline

LAND

waveSimulationarea,M

...

...Scatteredwave

NArtificial openoceanboundary

O:PQR:STUWVXY:Z[\:]

^ ^_ _ ` `a a b bc cd de e

Figure 3: A sketch of a simulation area in a coastal region.

efficient andaccuratesolversin orderto treatsufficiently largeoceanareasandproducefastpredictions.

3.2.1 Basic Equations

Thesimulatorisbasedonthecombinedrefraction-diffractionequation,theso-calledMild-slopeequation[Mei 1989],whichisderivedwithin non-friction(inviscid)linearwavethe-oryandslowlyvaryingseabed.Thismeansthatweconsiderwavesandbathymetrywithinthescales

wave slope f$g�h i j9kk$lbed slope f j�mWng�h n kk9l�o

thatis,smallfreesurfacewaveamplitude,i

, andhorizontallengthscaleof depthvariation,mWn2p n, comparedto thewavelength,

j. Thebedslopeis mild whenbed slope qsr�t l . In

shallowwaterregionsthecrucialsimplificationis the linearassumption.Anotherfactornotincludedin shallowwaterregionsis theseabedfriction, whichshouldbeimplementedin anextendedversionof thesimulator.

Thelinearityof theproblemallowsseparationof thetimefactorandFouriersuperpositionof solutionsfor differentfrequenciesanddirections.Thetotalsurfaceelevation,u�v w5o x+o y z ,

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from theundisturbedsealevelasa functionof spacialposition { |5} ~�� andtime � , canthenbewritten � { |5} ~+} � ����� �� � �+�

���� � �+��� �� { |5} ~���� ��� � � � (2)

where� indicatestherealpart,and � �� { |5} ~�� is thecomplexwaveheightfor frequency� ,� � , anddirection � .

TheMild-slopeequationfor thecomplexwaveheight � � � �� { |5} ~�� for angularfrequency� � � � reads ��� { � � � �5� ���+� � ��� (3)

Giventhevariationof theseabed,thewaterdepthdistribution, ���@�5{ |5} ~�� , thecoeffi-cientsin (3) aregivenby� { |5} ~�� � � ¡ � ¢ �¤£ �¥�¦ � { §�� ¥�¦ �¨ © ª�« ¥�¦ � � (4)

� { |5} ~�� � � ¢� ¡ � §¥ { §�� ¥�¦ �¨ © ª�« ¥�¦ � � (5)

where � ¡ � � ¬ and � ¢ �®­ �­ ¬ arethelocal phaseandgroupvelocity, respectively, and £ istheaccelerationof gravity. Thelocalwavenumber

¦ � ¦ { |5} ~���� ¥�¯/°�± is determinedbythedispersionrelation(1).

3.2.2 Boundary Conditions

In the currentsimulations,threetypesof boundaryconditionsareconsidered;the openboundarytowardsan infinite ocean,the shoreline,anda total reflectingbordertowardsmanmadeconstructions.

Open Ocean Boundary. Along theartificial openoceanboundaries,specialtreatmentofscatteredwavesis introduced(Figure3.1). Usually the solutionin the outerareais de-scribedby analyticformulasfor theincidentandscatteredwave.(Foranexample,seethe“superelement”in Mei (1989)).But, by findingdiscretisizedversionsof theseoneavoidssuddenstepsin approximationmethodsbetweenthesolutionsbeyondandwithin theareaof interest.

In theopenoceanareawith constantdepththewavesolutionis describedby a sumof anincident(incoming)anda scatteredwave.In orderto takecareof theradiationconditionacrosstheartificial boundary, Huygens principle is applied.

This meansthat the scatteredwaveis representedby a superpositionof a largenumberof point sources² , ³µ´ ; ¶®�·§�} ¥ } ¸ ¸ ¸ ¹ , alongtheartificial openboundary. An arbitraryº . Theideaof a point sourcemaybeillustratedroughlyby thecircularwaveinducedby droppinga stoneinto the water. In this picture,Huygensprinciple correspondsto how droppingseveralstonesmomenta-neouslyalonga straightline would resultin a wavebecomingmoreandmorecloseto a planewavethedenserthesestonesaredropped.

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1 2 3 4 5 6 7 8km

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3

4

km

1 2 3 4 5 6 7 8km

1

2

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km

1:

2:

2:

2:

3:

3:

3:

Figure 4: The finite element computation area for the simulations; area 1: the main areaof interest with realistic bathymetry (here shown with the visualization grid), area 2: anadded region where the water depth varies to the constant open ocean depth, area 3: anadded computational area on land handling the wave reflection along the coastline. Thebreakwaters are holes in the computation grid.

wavecanbedescribedasthesumof weightedpointsources,givenby adiscretesetof »pointswith spatialcoordinates¼ ½�¾À¿ Á ¾� for àÄ@Å�¿ Æ�¿ Ç Ç Ç » . Themathematicaldescrip-tion of asuchpoint sourceis aHankelfunction.

Foranincidentwave,È É , with directionÊ , thecomplexwaveheightfor onewavefrequencyandonewavedirectionin theopenoceanareabecomes

Ë open Ä9È É ¼ ½5¿ Á�Â2ÌÎÍϾ�Ð+Ñ�Ò ¾�Óµ¾À¼ ½ÕÔ�½�¾À¿ ÁÔ�Á ¾� (6)

whereÈ É and Óµ¾ containbothanimaginaryandarealpart.Therefore,theessentialbound-aryconditionsalongtheopenoceanboundaryaregivenby complexfunctions.To ensurecontinuityof thesolutionwithin andbeyondtheopenoceanboundaries,thestrengths,Ò ¾ ,of thepointsourcesaredeterminedsuchthatthewaveheightandits derivativearecontin-uousacrosstheopenboundary. Thiscorrespondsto demandingthattheMild-slopeequa-tion mustbesatisfiedovertheboundary. Theformulasfor È É and Óµ¾ arefinite differencesolutionsof (3) with constantdepth,takenfrom [Norwave1993].

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Thefinite differenceforms of Ö × and ص٠leadto specificrequirementson the finite el-ementfield describingwaterdepthandalsoon the computationalmeshalongthe openoceanboundaries.

Thedepthin theouteroceanareaisdescribedasconstant,Ú:Û�Ú+Ü , sinceit is largeenoughtohaveanegligibleeffectonthepropagationof surfacewaves.A transitionzoneensuresasmoothchangefrom thisconstantdepthto thefinite elementrepresentationin thecoastalregionwith varyingdepth.The waterdepthwithin this transitionzonemustslowly in-creaseto theconstantdepthin theinfinite openoceanarea,andin thejoint, thefinite ele-mentgrid mustmatchthelatticegrid, from whichtheformulasof Ö × and ص٠arederived.

Coastline. Anotherchallengingboundaryof thesimulationareais thecoastline.In ordertoachievewavesimulationsthatarephysicallyrealistic,thereflectionandtransmissionofthewavesalongthecoastmustvary alongthecoastlineasthenatureof thecoastvaries.Wemayhavebeacheswheretheshoreslantseasilydownhill into thewaterandthewaveenergy is nearlycompletelyabsorbed,while in contrasta lot of the wavesenergy is re-flectedfrom theprecipitousseaclif fs. Theseeffectsmustbeincludedin thesimulatorinordertogiveasrealisticwavepredictionsaspossible.Thesewavereflectioncharacteristicsaremodelledby avaryingartificial stepin waterdepthintroducinga shallowunderwaterplateauareaoverwhich thewaveenergy is absorbed.In general,all depthvariationsleadto a degreeof wavereflection,andtheseeffectsareincorporatedin themodelequations(3). Theabsorptionof waveenergy overtheaddedlandarea,is representedby animagi-narywavenumber, andthecoefficientsof theMild-slopeequationbecomecomplex.

Constructions. Thebreakwatersof theHasviksimulationsareholesin thecomputationareawith total wavereflectionalongtheir oceanboundary. Thesamekind of conditionswith anadditionalradiationconditionin theoutletwouldbeusedfor thewallsandeventualisletsin simulationsfor designof Tapchanwavepowerplants,[Norwave1983].

Thevariousboundaryconditionsarerelatedto partsof thecomputationalarea,seeFig-ure4, andthegrid generatingprocedureis closelyrelatedto theboundaryconditions.

3.2.3 The FEM-Simulation Unit

Themainsolutionmethodin thesimulatoris theFiniteElementMethod(FEM).Becauseof theirregularityof acoastalregion,FEM is a well suitabletool. Thiscoastalregionap-plicationis a first testof thecomplexversionof theDif fpack’s FEM routines,on whichthis softwareis based,sinceboththecoefficientsof partial differentialequationandthefunctionsgivenalongtheopenoceanboundariesarecomplexnumbers.

Numerical Computation Algorithm. WhenapplyingthefiniteelementmethodtotheMild-slopeequation(3) thewaveheightis approximatedby

Ý+Þ ß5à á�â/ã®äÝ+Þ ß5à á�â Ûæåç�è Ý è éÀè Þ ß5à á�â (7)

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andthesystemof equationsdeterminingthe ê unknowncoefficients ë�ì is givenby

í ì�î ìïð�ñòôó õ ö ÷5ø ù�ú û ê�ü�ý û êÀìÿþ�� � � ö ÷5ø ù�ú ê�ü êÀì ������ ��� (8)

where�

is thecomputationalareaand êÀì ö ÷5ø ù�ú areknownpiecewiselinearfunctions.Theessentialboundaryconditionsgivenby(6)appearasright-hand-sideof thesystemof equa-tions(8).Thelinearityof themodelproblemleadstosolvingthissystemof equationsmanytimeswith variousright-hand-sidesdueto the the incomingandscatteredwavesgivenalongtheopenoceanboundary. Therefore,thefinal waveheightwithin thecomputationalareacanbewritten asa sumof finite elementsolutionsof theMild-slopeequationwiththeincomingwaveandthepoint sourcesgivenby (6) asright-hand-sidesof (8). The �point sourcestrengths,��� ø � � ø � � �5ø ��� aredeterminedsuchthat thewaveheightanditsderivativearecontinuousacrosstheopenoceanboundary.

Thecostof of this interactionbetweentheFEM-solutionareaandtheinfinite oceanareais the largenumberof similar computationsof theequationsystem.This computationalproceduremustberepeatedseveraltimesfor thevariousfrequenciesof incomingwavesrepresentingthephysicaloceanwavefield.

FortheHasviksimulationexample,thenumberof unknowns,ê , i.e. thenumberof nodesin theFEM-mesh,andthenumberof pointsources(openoceanboundarynodes)aregivenin Table3.

Table 3: Parameters of the Hasvik simulations.

Constantopenoceanboundarydepth ��� � 150m

Waveperiodfor grid generation ��� � 14sec.

Numberof grid pointsperwavelength ��� � 10

Constantopenoceanboundarygrid spacing � � 30m

Sizeof computationarea 4.74km � 8.16km

Numberof nodesin computationalgrid ê � 72000

Numberof nodesin visualizationgrid ê"! ü #$� 17000

Numberof point sourcesalongtheopenoceanboundary � � 642

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Figure 5: The various steps in the construction of the computational depth surface. a) Thesoundings and coastline data, b) the plateau depth, and c) the Delauney grid used to createthe preliminary bathymetry surface in d). In e) the frame depth in the transition zone isadded, and f) shows the final computational depth surface where non-interesting parts onland and outside the considered computational area are removed.

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3.2.4 Generation of Computational Grids and Handling of Geodata

Themainrequirementonthegridgeneratorin thisspecificcaseapplicationisafineenoughresolutionof thephysicalphenomenaconsidered,heretheoceansurfacewaves.Theseabedandtheincomingwaveperioddefinethespatialdistributionof thelocalwavelengthandarethusthemainfactorsfor settingthegrid resolution.Thatis, thelocalwavelengthmustberepresentedby enoughgrid points.The couplingto the infinite openoceanandthe variouswavereflectioncharacteristicsalongthecoastintroduceadditionalrequirementsto theboundaryof theconsideredcom-putationalarea.(Section3.1and 3.2.2)

First, theinputdataof thegrid generationalgorithmmustbeavailable,theseare

1. Scattereddepthdata:(Figure5a).2. Coastdata:positionanddepthgivenonacounterclockwiseclosedpolygon.Herethe

depthdatais theartificial plateaudepthtakingcareof thewavereflectionalongthecoast(theorigin of theadditionalcomputationareaon land,Area3 in Figure4).

3. Islet data:Outlineof breakwatersandpossiblesmallislands.4. Boundingboxdefiningthecomputationarea.5. Numberof grid pointsperwavelength, %�& . Themainparameterto determinetheres-

olutionof thecomputationalgrid.6. Theconstantdepth '�& at theopenoceanboundary.7. Relevantperiodsfor theincomingwavespectra,(�& .

Thegrid generationalgorithmroughlyconsistsof thefollowing four steps.

Initialization of the bathymetry surface. The bathymetrysurfaceconsistsof a plateaupart,anoceanpartanda frame/transitionzonepart.

Theplateaupart,shownin Figure5b, takescareof thevariouswavereflectioncharacter-isticsalongthecoastcorrespondingto thecoastalboundarycondition.

Themainoceanpartwherethedetailedoceanwavesimulationsarecarriedout, is repre-sentedbyaconstrainedDelauneytriangulationof soundingsandcoastpoints(Figure5a,c),i.e.,thecoastlinesegmentsarerequiredto beedgesin thetriangulation.

Thethird part,thetransitionzone,is addedto theoceanpartat theopenoceanboundaries(seeFigure5e)to decreasetheinfluenceof theartificial constantdepthin theopenocean.Thedepthin the transitionzoneconsistsof a regionwith constantdepthalongtheopenoceanthena zonewherethedepthis phasedfrom theconstantvalueto thedepthin theinterior. Thelower, left, right anduppertransitionzonescanhavedifferentsizes.

Construction of an initial grid. A coarseinitial computationgrid with breakwatersasholesin thegrid (Figure6a)is constructed.A two-dimensionalmeshgeneratoris applied(Thepublic domainprogramtriangle by Shewchuk(1996))to constructthe initial grid.Thisgrid generatorappliesthescatteredsoundingsandcoastpointsof theinputdataandaddspointsin theinteriorof themeshto ensurea meshwith agivenminimumangleandmaximumtrianglearea.A uniform grid is addedin thedepthtransitionzone(Figure6b)asaninterfaceto thelatticegrid representationin theopenocean.

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Figure 6: Parts of the computational grid. a) The breakwaters are holes in the grid andthe plateau grid beyond the coastline varies in width and is coarser. b) The transition zonein lower right corner of the frame which have structured grid.

Theconstantboundarygrid spacing) of this frameis determinedfrom theinputdataandroundeddownto anintegervaluewhichgive higheraccuracyin thecomputations.

Finally, markersaresetonlandandboundarynodesfor specialtreatmentin therefinementprocessandin theFEM-simulationunit.

Refinement of computational grid. Firsttwopreliminarylevelsof refinementareset.Thefirstcorrespondingto thecoarseresolutionof theland/plateaugridarea,andthesecondfora coarsevizualizationgrid. We canallow coarserefinementof thelandgrid sincewe arenotinterestedin theactualwaveresultshere.Thelandgrid isonly theretohandlethewavereflectioncharacteristicsalongthecoast.Thevisualizationgrid hasreducedaccuracyandis without theartificial transitionzoneandthelandpart.A partof thevisualizationgrid isshownin Figure7.

Thedepthfield iscreatedontheinitial grid,withoutthenon-interestingpartsof thebathymetryon landandoutsideframe(Figure5f). A nestedrefinementof thegrid is conducted.Iter-atively, if agrid elementhasedgeslargerthanthelocalwavelength,theedgesaresplit intwo.

Preparing computational grid for the simulator Thegrid is renumberedin ordertomini-mizethebandwidthof theelementmatrix.Thisisnecessaryin ordertouseadirectsolutionmethodwith LU-factorizationin theFEM numericalsolution.Thisrathertimeconsumingprocesscanbeavoidedby implementinganiterativesolver. SeeSection4.2.1.

Theparametersfor theHasvik-simulationsaregivenin Table3.

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5.5 6 6.5 7 km

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Figure 7: Segment of a finite element grid with 17 900 nodes, used for visualization pur-poses. The grid used in the calculations is refined to contain about four times as manynodes, including the nodes in the visualization grid.

3.3 Results from the Hasvik Simulations

DuringastormonOctober31st1997,thenewerof thethreeHasvikbreakwaterswasde-stroyedby waves.Roughlyestimatedspectrafor incomingwavesin thesimulationscon-tainthreedifferentwaveperiods,13,15and17seconds,andthreedifferentdirectionsforeachof these,givenas0, 10 and20 degreeswith point of compassasin Figure9. TheseestimatedincomingwavespectraarebasedonwindmeasurementstakenatTorsvagLight-houseandHasvikAirport ontheactualday. Thenineincidentwaveshavebeenweighted,with main weight on the 15 secondperiodsandthe 10 degreesdirection,resultingin asignificantwaveheightof about10 meters.

Technicallythishasbeendoneby meansof superposition.Theequationsystembeinglin-ear, onemaysolvefor eachof theninecasesseparately, andthenweightandaddthedif-ferentsolutionsafterwards.Theharbourareais simulatedbothwith andwithout thenewbreakwater. Fromthesetimeseriesdevelopments,asdescribedby eqn.( 2),havebeende-rived,andaredemonstratedin theDynaMapGeoSimvideo,[HasvikVideo].

Theimmediateoutputisacomplexfiniteelementfielddescribingthecomplexwaveheightin eachgrid point.Studyingtheabsolutevalueof thefield will give indicationson maxi-mumwaveheightin thedifferentareas.Otherquantitiesthatmaybederivedin apostpro-cessingunit arevelocitypotential,energyflux, pressuredistributionandloadonshoreandconstructions.Theseareall explicit functionsof depthandsurfaceelevation[Mei 1989].

Thesimulationsconductedillustrateseveralphysicalphenomena.Oneis howbanksmayactas“lensesandprisms”,focusingthewaves.Figure9demonstratestheeffectof thethree

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Hagrunnan Skipperneset

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st.Haen

Figure 8: The bathymetry (water depth in meters)

banks,Silstø,AndottskallenandKnottskallen,combinedwith thesteepslopeto theSouthEast;thewavesaredefractedandtrappedto thecoastlinearoundSkipperneset.Further,oneseesstrongreflectionsfromthebreakwatersandfromcoastlineareaswheresteepclif fsgodirectly into thesea.

Insidethe harbourarea,seeFigure 10, the wavesseemto be somewhatmoredampedwith thethird breakwaterintact.Eventhoughthis is only avaguetendency, onemayalsoobservea slightharbourresonancecloseto thequays.

With thecurrentdatabasis,thesimulatorproducesqualitativephenomenaasillustratedinplotsandin thevideo,which seemto bequite in accordancewith physicalobservationsandexpectations.However, in ordertoachievequantitativeresultsthatarereliableasfore-castsor for planningof constructions,onewould needhigh quality geodataasdescribedin section4.1.

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Figure 9: Snapshot of Wave Height in meters.

4 Conclusions for Further Activities

In ordertospecifyproperlywhatkindof geodatainformationandgeodatatreatmentisnec-essaryor advantageousfor thiskind of numericalcoastalzonesimulations,agreatdealoffurthersoftwaretestingmustbeperformed.Furthersimulationwork will haveto includecomputationsbasedon variousdataandtransitionsfrom datato computationalgridsandfields,in orderto knowmoreonhowsensibleasuchsimulatorwill beto “bad” or “good”descriptionsof bathymetryandshore.Thisisnecessaryfor decidingonbasisof givenspec-ificationsfor a casesimulationwhatlevelsof precisionin dataandcomputationsarefea-sibleandnecessary.

Still, someexperienceshavealreadybeenmade,andseveralconclusionsmaybedrawnfrom theoreticalconsiderations.Sofar, thecriteriastatedin Section4.1 seemevidentasrequirementsto highqualitygeodatain thiscontext.

4.1 Geodata Requirements

Thesimulationsof theHasvikharbourandsurroundingseaareasshownin thevideotape[HasvikVideo],havebeenmadeonaminimaldatabasis.Althoughthequalitativebehav-ior demonstratedis closeto whatonecanobservein nature,to achievequantitativeinfor-mationthatisaccurateenoughfor planninganddesignof constructionswouldrequiredatawith higherquality. For simulationslike these,high quality includesthefollowing.

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Figure 10: Snapshot of Wave Height in the coastal region of Hasvik harbour a) with origi-nal breakwaters and b) with the additional breakwater constructed in 1996 and destroyedby wind generated waves October 31, 1997.

Land data: In asurfacewavesimulationwith theactualperiods,1-25seconds,it is vitalto havelanddatawith hightopographicalresolutioncloseto theshorein orderto estimaterealisticreflectioncoefficientsalongthecoastline.SeeSection3.2.2aboutboundarycon-ditions.Therelevantlanddatawould betopography, geometryandmanmadeconstruc-tions,andotherfeaturesthatmightaffectabsorptionandreflection,suchassandhills andbeachesvsslopesof rock.

Sea bed data: As describedin Section3, thevariationsin theseabedat deepwater, inthis caseat depthsfrom about150 m anddeeper, arenot influencingthesurfacewavessignificantly,andsohighresolutionsamplingsarenotnecessary. Onshallowwaterorcloseto islands,coastlineor constructions,adensesamplingrateof bathymetrydatais oftenofvital importance.Whatis requiredis thereforea databasewhereonecanfind denserseabeddataoverbanksor otherareaswith shallowwater, thanwhatis normallya sufficientresolutionin deeperregions.

The coastline area: Inclusionof tidal effectsrequiresoverlapin topographiclandandseadata,sinceit is of greatimportancethat theseareconsistentin orderto find the correctshapeof thecoastlineatdifferenttide levels.Closeto land,thetidal elevationwill besig-nificantfor thedepth,anddepthinformationmustthereforealsoincludethetidal stateatthetime of measurement.This kind of informationwasnot availableduring thesimula-tionsconductedfor Hasvik.

For furthersimulations,tidal elevationdataareavailablefrom tidal tables,but exactco-

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ordinatesandmomentof time for samplingwill benecessaryin orderto obtaintherightaccuracy.

Additional geodata information: The influenceof the tidal currentwill improvewaveheightpredictions.Waveamplificationsdueto interactionwith theslowly varying tidalcurrentmay turn out to bemoreintensethat thosecausedby seabedinteraction.Whenextendingthesimulatorto includetidal currenteffects,dataaccessis crucial.Thesedataareavailablefrom currentmeasurements.In orderto considerspatialvariationof thetidalcurrentthough,resultsfrom numericaltidal simulationswill benecessaryto achievethedesireddatainformation.

Further, measurementsof waveheightsandcurrentsmustbeavailableif quantitativein-formationis desired.

Onemayalsoimaginea forecastingtool thatperformactualwaveheightpredictions,notonly qualitativeor estimatingvariousworstcasescenarios.Instantwind datain theouteroceancanbeconvertedto a waveforecastapplicableasincomingwavespectra,andthecoastalzonesimulatormaythenpredicttheeffectsin coastnearwaterssomehourslater.Theaccuracyof sucha forecastingtool wouldbedependingquitestronglyon thequalityof obtainablewind measurements.

4.2 Extensions and Improvements on the Simulator.

4.2.1 The FEM-Simulation Unit

Thescopeof thesesimulationsis of courserestrictedby computermemorycapacityandby theprocessorspeedavailable.In severalapplicationcasesit wouldbedesirableto per-form computationsoverlargerareasoronshortertimethanwhatis currentlyfeasible.Theprocessingtime andstoragerequiredpercomputationareacouldbe reducedby severalmeans.

The simulationshavebeenperformedusinga direct solver. A complexiterativesolverwould requireboth lessprocessingtime, andlessmemory, by enablingfasterand lessmemoryconsumingsolutionmethodsfor theactualsystemof equations.Further, band-width reductionon thegrid matrixwould no longerbenecessary, which would savepre-processingtime,andfacilitatetheincorporationof preprocessingtoolsinto thesimulator.Savingruntimeanduseof memorywill againextendthepossiblesolutionarea.An itera-tive algorithmfor solvingmatrixsystemsof complexnumbershasasyetnotbeenimple-mentedin theDif fpackenvironment,butwouldbehighly relevantasafirst improvementon thesimulator.

Anotherpossibleimprovementis a so-called“zoom-solver”. This would bea devicetoenablere-solvingonafinergrid oversmallerregionsof specialinterest,usingtheoriginalsolutionasboundaryconditionson thenewborder.

Thenatureof theproblemalsomakesit quiterelevantto considerparallelcomputations.

Thesimulatorcanalsobeextendedtoincorporatenewphysicalphenomena.Forthepresent,refractionanddeflectionof surfacewaves(period2-20sec)dueto varyingbathymetryisstudied.A highly relevantadditionwouldbetherefractionanddeflectionof surfacewaves

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dueto slowly varyingtidal currentof period12hours,andalsoa couplingbetweenthesetwo phenomena.Further, overshallowareasonewouldachievea morephysicallyrealis-tic resultby addingabedfriction componentto themodel.Thismaybedoneby meansofcomplexwavenumbersasfor theartificial plateaustakingcareof wavereflectionsalongtheshore,describedin Section3.2.2.

Thesimulationoutputis acomplexfinite elementfield,describingwaveheightasa func-tion of spaceatsometime 2 . By postprocessingtheseresults,severalderivedparametersareavailable.As mentionedearlier, thelinearizationsenableusto performthetimestep-pingseparately. Thecomputationof currents,energy flux or pressurecanlikewisebeper-formedin aseparateprogram,takingtheoutputfinite elementfield from theoriginalsim-ulatorasinput.Thepressuredistribution,yieldingtheloadon constructions,is describedby theBoussinesquepressureequation,seeMei (1989).

Constructionof breakwatersis themostcommonwayto protectharboursthatareexposedtoextremeweatherconditions.Thestudiesof focusingeffectsof banksandunderwaterseamountains,mayleadto try otherlinesof actiontoprotectexposedareas.Onenewsolutionis theconstructionof submergedartificial banksdesignedto act as“lensesandprisms”anddeflectthedestructivewavesin othernotsocritical directions.Numericalsimulationscanbeappliedto find theoptimalshapeof suchartificial banks,andfor this purposethesimulatorwill beextendedto includeanoptionfor usermodificationof theseabed.

4.2.2 The Grid Generator

Thegrid generatingprogramshouldbereorganizedto includea menusystemin ordertoenablemoreflexible useof its facilities andto makeit moregeneralwith respectto datainput.Thiswill alsomakeit easierto includethegrid generatordirectly in largeranduserfriendly applications.More flexible usewill of courserequiredetaileddatacheckingal-gorithms,andthat the menuchoicesarenot mutually exclusive.The programwill alsohaveto checkthatthegrid generationalgorithmsucceedsin its varioussteps,andthatthefinal computationalgrid fulfills thenecessaryrequirements.An importanttaskof thefur-therwork would beto acchievea strict definitionof whatis therequirementsto a ’good’computationalgrid in thiscontext.

A newfacility thatwould berelevantto offer is theautomaticchoiceof frame,i.e., tran-sition zone(area2: in Figure4). If theoceanareais sufficiently largeor theouteroceandeeperthanwhatis correspondingto infinite depth,theframe/transitionzonecouldbere-dundant.

Theunit wouldalsobecomemoreuserfriendly by automatizingor simplifying thechoiceof deriveddata.Thiswould includeenteringreflectioncoefficientsat thecoastinsteadofpre-computedplateaudepth,lettingtheprogramcalculatethecorrespondingplateau.

Couplingthegriddingunit to thesolverwouldenableuseof adaptivityto controlcompu-tationalerroror choosinga morecarefulmethodto determinetheadaptivegrid.

Finally,aninterestingextensionwouldbethegenerationof asequenceof gridsfittedfor ef-ficientiterativesolutionmethods,usingmulti-grid-facilitiesthatarealreadyimplementedin theDif fpacklibrary.

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4.3 Relevant Case Studies

Thechoiceof testcasesfor furthersimulationactivity in theDynaMapprojectwill dependon severalaspects.It is vital to haveaccessto high quality geodatafrom the regionsofinterest,andthedecisionwill havetodependonwhatcanbemadeavailableatpresent.Thecasesshouldalsobeinterestingfrom a marketpoint of view, andfor the1998activities,importanceis attachedto theapproachtowardsindustrialandpublicproblemowners.

4.3.1 Hasvik, Sørøya in Finnmark

Whatis documentedhereandontheDynaMapGeoSimvideotapemustbeconsideredasapreliminaryversionof thesimulations.Bettergeodata,improvementsonthehandlingofgeodataandextensionsof thesimulatorwouldproducephysicallymoreexactandreliableresults.

It isalsohighly relevanttoconductfurtherparametersensitivitytestsin thisarea.Onewayof calibratingthesimulatorwould be to compareto physicalmodelsandmeasurementsfrom Hasvikmadeby SINTEFCoastalEngineering,for theNorwegianCoastalAuthori-ties.

4.3.2 Fedjeosen - Mongstad and Sture

Onepossibletestcaseareafor furthersimulationscouldbethewatersaroundFedjeandØygardenwith theentrancesto theMongstadRefineryandto theStureTerminal,seemapin Figure11. Thiswouldbeaninterestingcasefor severalreasons.

In thesefairways,quiteafew majoraccidentshaveoccurredandseveralothershavebeenjust avoideddueto difficult weatherandcurrentconditions.Still, this hasbecomeoneof the mostimportantsealanesfor oil tankersandpetroleumindustryequipmentalongtheNorwegiancoast,andis thereforea highly relevanttestcasealsofrom ancommer-cial/environmentalpointof view.

In acaselike this, it will no longerbesufficient to studybathymetryeffectsonly, sinceinthis areathe tidal effectsareof vital importance.At thesametime, the incorporationofslowly varyingtidal elevationandcurrentinto themathematicalmodelsandthusinto thesimulatoris oneof themaintaskspreviewedfor furthergeodatasimulations.

On thesimulationside,theeffectsof tidal currentswill addcomplexityandrichnesstothemathematicalmodel,whereastakingtidal elevationinto accountwill bea challengein gridding,sincetheshapeandpropertiesof thecomputationareawill haveto varywiththetide.This againinvolvesseveralrequirementson thegeodata.Onewill needdataontidal elevationandcurrent,andaccurateinformationon how thecoastlinechangeswithtidal elevation.This meansthathigh qualityandconsistencyin landandseadatawill becrucial.

Simulationsof theconditionsin fairwaysandharbourmouthsarerelevantasaforecastingdevicein traffic controlandtransportationof heavyconstructions.Along with this it canbeanimportanttool for decidingwhetherto try to changetheactualconditionswith manmadeconstructionslike breakwatersor artificial banks.

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3BERGEN

4 Mongstad

4 Sture

FEDJE

Figure 11: The Fedje island and the entrances to Mongstad and the Sture terminal, Norway

4.3.3 Wave Power Plants

A typicalexampleof constructionalongthecoastwheresimulationsongeodataareusedarewavepowerplants.A maincomponentin a taperedchannelpowerplantlike theonesconstructedby NorwaveAS [Norwave1983], is thewavecollector, a largefunnelwhichis blastedout into therockshore.In obtainingoptimaldesignandlocation,simulationsofsurfacewaveson a realisticseabedwith shoreline andrelevantshapesof thecollectorareessential.Thesesimulationswill haveto becorrelatedto waveheightmeasurementsin orderto estimatetheeffectobtainablefrom thepowerplant.

A powerplantsimulationprojectisnowbeingcarriedoutatSAM (SINTEFAppliedMath-ematics),andthefirst constructionsitescheduledis closeto thevillageof Baron,locatedon thesoutherncoastof Java,Indonesia.

Thereareseveralinterestingsitesfor furthersimulationsof thiskind,butinclusionof tidaleffectswill in thiscasenotbeinitially relevant.A wavepowerplantof thiskind is afixedconstruction,andit wouldusuallybenaturalto choseasitewheretidal elevationis mini-mal in orderto obtainaslittle variationaspossiblein thepowersupply.

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5 Bibliography

References

[SINTEF1996] DynaMapProjectPlan,SINTEFAppliedMathematics,October1996.[Norwave1983] TAPCHAN wavepowerplants,NorwaveAS, October1983.[Norwave1993] Dif fraksjon2D-bunn;NTNF-prosjektEMS28978;Dokumentasjonav

matematikkogprogram.NorwaveAS, February23rd1993.[Mei 1989] C.C.Mei, TheAppliedDynamicsof OceanSurfaceWaves,AdvancedSeries

onOceanEngineering,Volume1, World ScientificPublishing,1989.[Dif fpack] Dif fpack,http://www.nobjects.com/prodserv/diffpack/[HasvikVideo] WaveSimulationsovera realisticSeaBed,SINTEFAppliedMathemat-

ics,December1997.

Acknowledgements Thanksto KlasSamuelsson,EvenMehlumandGunnarMisundforvaluablecontributions.