MODELLING THE THREE-DIMENSIONAL TIDAL FLOW STRUCTURE IN SEMI-ENCLOSED BASINS. Olav van Duin

Transcription

1 MODELLIG THE THREE-DIMESIOAL TIDAL FLOW STRUCTURE I SEMI-ECLOSED BASIS Olav va Dui

2 Modllig th th-dimsioal flow stuctu i smi-closd basis Olav va Dui MODELLIG THE THREE-DIMESIOAL TIDAL FLOW STRUCTURE I SEMI-ECLOSED BASIS This Mast thsis was witt b: Olav J.M. va Dui as th thsis fo a Mast s dg i: Wat Egiig & Maagmt, Uivsit of Twt ud th supvisio of th followig committ: Supviso: Pof. D. Suza J.M.H. Hulsch (Uivsit of Twt) Dail supviso: D. I. Pit C. Roos (Uivsit of Twt) Extal supviso: D. Hk M. Schuttlaas (TU Dlft) 5 Sptmb 9 Pag of 58

3 Modllig th th-dimsioal flow stuctu i smi-closd basis Olav va Dui ABSTRACT Coastal aas a gall itsl usd aas with high populatio dsit ad coomic activit. O a basi scal th tid dictl dtmis wat lvls ad cuts i a basi. Ths flow chaactistics futhmo dtmi th shap of th basi itslf, fo xampl th fomig ad volutio of tidal sadbaks, which i tu iflucs th flow patt. Bcaus of its impotac fo vaious huma ad atual activitis th modllig of tidal flow has b studid b ma authos i th past. This has lad to dpth-avagd (DH) ad 3D modls amogst oths Th fist aaltical 3D-modl that dscibs tidal flow i a smi-closd basi usig Klvi ad Poicaé mods with patial slip was catd fo this sach. Fo this th mthod dvisd b Mofjld (98) fo 3D tidal flow alog a sigl coast with viscosit ad o-slip was xtdd, thb followig Talo s appoach (9). As a fc situatio th oth Pat of th oth Sa was modld ad th poptis of th Klvi ad Poicaé mods dscibd. Also th flow ad sha stss poptis w studid. Th flow poptis w also compad to a uivalt DH modl but fo this fist valus fo th fictio paamt had to b dtmid. Fo this vaious mthods w adoptd with vaig succss i appoximatig th 3D poptis. It is cla that that 3D stuctu is impotat to b abl to pcisl dtmi th flow poptis. Th valu fo th fictio paamt that givs th bst sults of th mthods mplod was that b foud b fittig th Klvi dissipatio facto of th 3D modl (usig viscosit ad slip paamts) with th DH modl (usig a fictio paamt). Th fittig of th Klvi dissipatio facto lad to a fictio paamt of.7-3 m/s fo th fc cas (th 3D modl had a slip paamt of.5 m/s ad a viscosit paamt of.9 m /s). With this paamt th DH modl sults w compad with th 3D modl sults, showig that 3D stuctu is idd impotat. Evtuall this fictio lad to a avag o i pdictig 3D logitudial bottom sha stss amplitud with a DH modl of 3% whil th thoticall bst sult would hav b 3%. This all lads to th coclusio that cotiud sach i this aa ca futh impov 3D ad DH modlig. 5 Sptmb 9 Pag 3 of 58

4 Modllig th th-dimsioal flow stuctu i smi-closd basis Olav va Dui FOREWORD This thsis was witt as th culmiatio of m Mast stud Wat Egiig & Maagmt at th Uivsit of Twt. A impotat aspct of this stud is latig th tchical kowldg gaid to pactical ad socital ds. Though this sach was ath mathmatical, it still had a solid pactical cotxt which hlpd i makig th sach itslf udstadabl, usful ad mo plasuabl. A scod v impotat pat duig m sach was th guidac of m supvisos, Suza, Hk ad Pit. Thi isight ad as of ic gadig th hadld subjcts was ivaluabl i bigig this sach to a good d ad makig it scitificall ad pacticall valid. Though I hav to admit it was somtims bogglig to ha thm discuss what lid bhid thigs I hadld lightl ad did t lai o lo i dpth, all i all this mat that I lad much mo tha I would hav do i solitud. So, thak ou all! I would spciall lik to thak m dail supviso Pit, who was almost alwas availabl fo guidac ad asws. B doig m sach as a gaduatio it at th Uivsit of Twt itslf I had th oppotuit to wok with a goup of oth Mast studts i th oom svd fo th gaduatio its. This of cous mat that th was ough to discuss, talk about ad laugh duig luch baks, coff baks, cak ad coff baks, cookis ad coff baks, t cta. This was a wlcom lif fom th pimail idividual udtakig th witig of a Mast thsis is. Th vaious activitis that w ogaizd/istigatd lik wadwalkig, babcus ad v a thik-tak-comptitio b ctai idividuals (Wib, Bt, Ewi, Fak, Daiël, I hop I m ot fogttig somo...) also gatl hlpd i makig this a joabl tim. Futhmo I would spciall lik to thak Wib fo big m suppot i all thigs mathmatical ad tidal; it would hav b lol without ou! But of cous, all oth fllow studts also dsv pais fo thi suppot ad good compa! Ad fiall I thak all m fids ad famil fo big th ad thb givig m a lif bsids m dail scitific davous. Most of all I thak Aick, fo big a uctd but dal lovd light i m lif. I hop ou jo adig what I v witt about th dimsioal tidal flow stuctu i smiclosd basis! Olav va Dui, th 5 th of Sptmb 9, Eschd 5 Sptmb 9 Pag 4 of 58

5 Modllig th th-dimsioal flow stuctu i smi-closd basis Olav va Dui COTETS AT A GLACE ABSTRACT...3 FOREWORD...4 ITRODUCTIO...8 MODEL FORMULATIO OF TIDAL FLOW I A SEMI-ECLOSED BASI... 3 FIDIG WAVE MODES I A IFIITE CHAEL WAVE MODES I SEMI-ECLOSED BASIS FLOW PROPERTIES OF THE SOLUTIO BOTTOM SHEAR STRESS MODELLIG I A SEMI-ECLOSED BASI DISCUSSIO COCLUSIO AD RECOMMEDATIOS...45 LITERATURE...47 LIST OF SYMBOLS...49 APPEDICES Sptmb 9 Pag 5 of 58

6 Modllig th th-dimsioal flow stuctu i smi-closd basis Olav va Dui TABLE OF COTETS ABSTRACT...3 FOREWORD...4. ITRODUCTIO...8 PROBLEM DESCRIPTIO...8. RESEARCH COTEXT RESEARCH QUESTIOS....4 APPROACH... MODEL FORMULATIO OF TIDAL FLOW I A SEMI-ECLOSED BASI.... DESCRIPTIO OF THE MODELLED BASI.... MODIFYIG THE GOVERIG EQUATIOS AD BOUDARY CODITIOS GOVERIG EQUATIOS BOUDARY CODITIOS SCALIG THE FLOW EQUATIOS AD BOUDARY CODITIOS SCALED GOVERIG EQUATIOS SCALED BOUDARY CODITIOS LEADIG-ORDER SYSTEM OF EQUATIOS LIEAR SYSTEM OF EQUATIOS FIDIG WAVE MODES I A IFIITE CHAEL...7 THE SOLUTIO I ROTATIG VELOCITY COMPOETS SPLITTIG THE GOVERIG EQUATIOS AD BOUDARY CODITIOS TRASFORMATIO TO ROTATIG VELOCITY COMPOETS SOLVIG THE ROTATIG VELOCITY COMPOETS DETERMIIG WAVEUMBER RELATIOS DISPERSIO RELATIO EAR-COASTAL BOUDARY CODITIOS SOLVIG THE WAVEUMBER RELATIOS SATISFYIG WAVEUMBER RELATIOS WITH KELVI WAVES SATISFYIG WAVEUMBER RELATIOS WITH POICARÉ MODES WAVE MODES I SEMI-ECLOSED BASIS...4 KELVI WAVES THE DEPTH-AVERAGED MODEL DEPTH-AVERAGED VELOCITIES OF A KELVI WAVE DEPTH-AVERAGED VELOCITIES OF POICARÉ MODES SUPERIMPOSIG THE WAVE SOLUTIOS FLOW PROPERTIES OF THE SOLUTIO...8 PROPERTIES OF THE IDIVIDUAL MODES WAVE UMBERS OF KELVI AD POICARÉ MODES PROPERTIES OF THE KELVI WAVE PROPERTIES OF THE POICARÉ MODES TRASLATIO OF WAVE MODE PROPERTIES TO DH FRICTIO PROPERTIES OF THE TOTAL SOLUTIO ELEVATIO AMPHIDROMIC SYSTEM OF THE TOTAL SOLUTIO LOGITUDIAL CURRET AMPHIDROMIC SYSTEM LATERAL CURRET AMPHIDROMIC SYSTEM BOTTOM SHEAR STRESS MODELLIG I A SEMI-ECLOSED BASI...4 BOTTOM SHEAR STRESSES OF THE 3D MODEL BOTTOM SHEAR STRESSES OF THE DEPTH-AVERAGED MODEL VALIDITY OF DH BOTTOM SHEAR STRESSES DISCUSSIO Sptmb 9 Pag 6 of 58

7 Modllig th th-dimsioal flow stuctu i smi-closd basis Olav va Dui 8 COCLUSIO AD RECOMMEDATIOS COCLUSIO RECOMMEDATIOS...46 LITERATURE...47 LIST OF SYMBOLS...49 APPEDICES...5 A. MODIFYIG THE AVIER-STOKES EQUATIOS AD BOUDARY CODITIOS...5 B. RESCALIG OF SEVERAL PARAMETERS AD VARIABLES...53 C. THE DEPTH-AVERAGED MODEL...54 C.I. DISPLACEMET...54 C.II. C.II.I VELOCITY...54 VELOCITY FOR COLLOCATIO D. ELEVATIO AMPHIDROMY WITH VARYIG VISCOSITY AD SLIP PARAMETERS...56 E. AMPLITUDES OF THE KELVI AD POICARÉ MODES Sptmb 9 Pag 7 of 58

8 Modllig th th-dimsioal flow stuctu i smi-closd basis Olav va Dui ITRODUCTIO Coastal aas a gall itsl usd aas with high populatio dsit ad coomic activit. This is patl bcaus of th viciit of th sa, which offs gat oppotuitis fo huma dvlopmt. But th sa dos ot ol off oppotuitis, it ca also b a that. O a basi scal tidal flow dictl dtmis wat lvls ad cuts i a basi. Ths flow chaactistics futhmo dtmi th shap of th basi itslf, fo xampl th fomatio ad volutio of tidal sadbaks, which i tu iflucs th flow patt. It is impotat to udstad flow poptis i ths aas fo fuctios lik colog, saft ad taspot. This sach focuss o a tidal flow i a spcific mai viomt, aml smiclosd basis with widths ad lgths i th od of hudds of kilomts ad dpths i th od of ts of mts. It is impotat to xami th 3D aspcts of flow i ths basis as th fo xampl dtmi th a-bd flow which cotols sdimt taspot (Padl, 997) ad th fomatio of bd fatus lik sad wavs (Hulsch, 996 ad Gkma, ). Diffcs i 3D flow poptis ca b sigificat i smi-closd basis, which is obsvd fo istac i th Chigcto Ba (T, 98). Futhmo th Log Islad Soud Block Islad Soud chal shows tidall iducd sidual cuts (Iaillo, 98), which ca hav a lag ifluc o t sdimt taspot. Bsids th mo gal ctd bhaviou latal flow ciculatios a ctd fo tidal flow which cotols th dispsio of pollutats ad salt wat (Padl, 98 ad Huijts t al., 9). Euatio Sctio I th idal situatio a full 3D modl is availabl that icopoats all phsical pocsss ad ca b usd to stud flow i smi-closd basis. Howv, icopoatig all ths aspcts uis a lag amout of wok ad th sultig modl would b ath complx ad tim-cosumig to us. Also, bcaus of th ihtl complx atu of th mathmatics associatd with this tp of flow th modl would hav to b umical which maks it had to xactl distiguish btw th ffcts of difft aspcts ad poptis. To b abl to stud which pocsss a most impotat to icopoat, it is valuabl to mak (icasigl complx) aaltical modls which allow fo a asi itptatio of th sults. Such a modl ca b usd to dtmi which pocsss a lvat i which situatio ad thfo which a most impotat to iclud i a umical modl. This stud aims to poduc a 3D aaltical modl that is a logical xtsio of th wok do i th past. I th followig paagaphs this will b laid i dtail.. PROBLEM DESCRIPTIO Cosid a shallow sa that is lagl costaid o th sids b coasts. Th tidal patt i such a basi is affctd b th th boudais as wll as divig focs. Th tidal wav ts th basi fom op dp sa ad popagats btw th two paalll coasts util it achs th closd latal bouda. Bcaus of th psc of this bouda th tidal wav tus aoud ad popagats back towads th op d. Bcaus a Klvi wav, a simpl pstatio of a tidal wav, caot mak this tu a additioal tp of wav mods is dd to pst th flow a th latal bouda. Ths additioal mods a calld Poicaé mods. I th figu blow a v ough appoximatio of this tidal wav movmt ca b s. Figu : schmatizatio of th tidal wav movmt 5 Sptmb 9 Pag 8 of 58

9 Modllig th th-dimsioal flow stuctu i smi-closd basis Olav va Dui If th basi is sufficitl lag, Coiolis ffcts will occu. This fictitious foc oigiats fom th Eath s otatio, dflctig flow to th ight o th oth hmisph. Du to cotiuit ad th boudais this dflctio will caus ciculatio i th vtical. This ad th fact that paticls td to cokscw i ad out of th basi (Wiat, 7) idicat that a dpth-avagd (DH) appoach will los impotat poptis of th flow. Futhmo vtical viscosit mas that th tidal wav ics dissipatio whil movig though th basi. I a dpth-avagd modl bd fictio is a paamtizatio of this 3D-ffct. A solid pstatio of th vtical patt of cuts is spciall impotat fo mophological studis wh fo istac th vlocit divativs at th bd a usd fo th calculatio of bottom sha stss which i tu dtmis th mophological dvlopmt of th sa bottom. It is ctd that a combiatio of a icomig ad outgoig Klvi wav ad a tucatd st of Poicaé mods ca dscib th tidal damics i a smi-closd basi sufficitl wll to gt a good ida of th pocsss at wok ad th cosucs fo bottom sha stss. I th followig chapts th applicatio of ad th mtiod wav mods thmslvs will b laid i mo dtail.. RESEARCH COTEXT Bcaus of its impotac fo vaious huma ad atual activitis th modllig of tidal flow has b studid b ma authos i th past. A uick ovviw will b giv of th lvat studis which icopoat som of th pocsss that will b modlld i this stud, though o of thm idividuall cotais all of ths aspcts. Th popagatio of a tidal wav i a smi-closd basi is fd to as th Talo poblm ; this autho ivstigatd how a Klvi wav is flctd i a ctagula smi-closd basi (Talo, 9). Th aaltical solutio pstd dos ot tak dissipatio ito accout ad is dpthavagd. To modl th closd latal bouda Talo foud Poicaé mods to complmt th icomig ad outgoig Klvi wavs. Hdshott & Spaza (97) add o Talo s wok b allowig th bouda at th had of th basi to absob g, thus itoducig a mchaism fo dissipatio. Opposd to dissipatio localizd at th latal bouda, dissipatio thoughout th basi was modlld b Rick & Tub (98) b itoducig fictio tms. Mofjld (98) xamid 3D poptis of tidal flow alog a sigl staight coast with a flat bottom, icopoatig Coiolis ad vtical viscosit ffcts. Fo this th autho usd Klvi wavs to pst th tidal flow. Costat viscosit is assumd ad a o-slip coditio is applid at th bottom. Pdlosk (98) psts a solutio fo th dpth-avagd tidal flow i a ifiit chal (a shallow sttch of sa with two paalll boudais) which lads to Poicaé ad Klvi wav mods. Th basi shap was studid i DH ad a flat bottom was usd. Th autho took Coiolis ffcts ito accout, but o bd fictio (as a DH-pstatio of vtical viscosit). Davis & Jos (995) also studid th basi shap that is studid fo this sach. Th mad a umical modl fo a smi-closd basi with a slopig bottom as wll as a flat bottom. Th authos icopoatd olia tms (advctio), o-costat viscosit ad Coiolis ffcts. Th w th ol os mtiod h that studid a patial-slip coditio as wll as a o-slip coditio. Wiat (7) also lookd at tidal flow i a logatd smi-closd basi, but fo a bottom that is paabolic-shapd i th tasvs. Th autho has laid th latal flow ciculatios i a smi-closd basi with a 3D basi-modl that icopoats Coiolis as wll as vtical viscosit ffcts. Howv, this solutio is ol applicabl to basis that a aow ad it is ucla whth this appoach will wok with a o-paabolic bottom. This sach stivs to mak th fist aaltical 3D-modl that dscibs tidal flow i a smiclosd basi usig Klvi ad Poicaé mods with patial slip. A dawback of this appoach is that it ca ol b applid to flat bottoms, dd viscosit is costat ad that ol lia tms a tak ito accout. O th plus sid this modl has o coditio that th basi must b aow lik that of Wiat (7). Also, th chos aaltical appoach with Klvi ad Poicaé mods maks that th 5 Sptmb 9 Pag 9 of 58

10 Modllig th th-dimsioal flow stuctu i smi-closd basis Olav va Dui fial modl is asil udstood ad th ifluc of vaious paamts ca b asil ivstigatd. Futhmo th assumptio of costat viscosit actuall implis us of a patial slip coditio to gt alistic vlocitis ad sha stsss (Hulsch, 996 ad Hulsch & Va d Bik, ), so us of this combiatio is mo coct tha th oigial wok of Mofjld (98). Th us of a patial slip coditio allows fo a lativl simpl though ffctiv calculatio of bottom sha stss (Gkma, ); it avoids solvig th complx pocsss i th a-bottom bouda la dscibd i Bowd (978)..3 RESEARCH QUESTIOS Th sach ustios that will b ivstigatd i th followig pot a:. I what wa ca Mofjld s (98) aalsis ad th appoach documtd b Pdlosk (98) b combid ad xtdd to fid Klvi ad Poicaé mods i a ifiit chal of fiit width as a solutio to 3D tidal flow icludig a patial slip coditio?.a What a th tpical poptis of ths wav mods?.b Which valus of th fictio paamt i a uivalt dpth-avagd modl should b chos to gt th sam poptis giv a ctai combiatio of th viscosit ad slip paamts?.c How should ths wav mods dscibd abov b combid to simulat tidal flow i a smi-closd basi (i.. th Talo poblm)?. Usig th combid wav mods; what a th 3D tidal flow poptis (lvatio, cuts, bottom sha stsss) i th oth pat of th oth Sa?.a How do ths poptis diff i th vtical? Ad fo vaig slip paamt ad vtical viscosit?.b Usig th fictio paamts foud bfo; how wll dos th dpth-avagd modl cospod to th 3D modl cocig flow poptis? Which mthod poducs th bst sults?.4 APPROACH I shot th goal of this sach is to st up a mathmatical modl to calculat th 3D flow fild i a smi-closd basi, ivstigat its poptis ad s if a uivalt dpth-avagd modl ca achiv simila sults. Th pocsss that a icludd a th pssu gadit, Coiolis foc ad dd viscosit. A patial-slip coditio will b applid at th sa bottom. Th complt modl stup is dscibd i chapt. Th mthod applid b Mofjld (98) fo tidal flow alog a sigl coast is xtdd to a ifiit chal (also followig th mthod dscibd b Pdlosk, 98) ad wav mods a sought that satisf th bouda coditios ad govig uatios (s chapt 3). Th dpth-avagd vsios of ths mods a pstd, as wll as a umical mthod which combis ths wav mods i such a wa that th coditio at th closd latal bouda is satisfid (s chapt 4). Th poptis of th wav mods ad th uasi-aaltical modl which psts th tidal flow of a smi-closd basi a also xamid ad a attmpt is mad to fit a uivalt dpth-avagd modl to ths poptis (s chapt 5). Th uasi-aaltical modl is usd to pdict th bottom sha stss ad this is also compad to th uivalt dpth-avagd modl (s chapt 6). Th fial chapts hadl th discussio, coclusio ad commdatio. Appdics a icludd with additioal ifomatio. 5 Sptmb 9 Pag of 58

11 Modllig th th-dimsioal flow stuctu i smi-closd basis Olav va Dui MODEL FORMULATIO OF TIDAL FLOW I A SEMI-ECLOSED BASI I this chapt th stup of th modl will b dscibd. Fist a impssio will b giv of what basis a modlld. Aft that th govig uatios ad bouda coditios a divd fom th full avi-stoks uatios b makig appopiat assumptios fo th kid of basis dscibd bfo. Aft that th uatios a scald ad th lia vsios a foud. Euatio Sctio. DESCRIPTIO OF THE MODELLED BASI Th basi shap (s Figu ) that will b usd fo this stud is that of a smi-closd smi-ifiit ctagula basi with width B, ad costat udistubd wat dpth H. It should b otd that th is o coditio imposd o th dottd lis, it is just to sigif that th basi xtds ifiitl bod th dottd lis. Th hoizotal vlocit paalll to th alogsho dictio x is calld u, th hoizotal vlocit paalll to th coss sho dictio is calld v ad th upwad vtical vlocit paalll to th vtical dictio z is calld w. Th displacmt of th wat sufac with spct to th udistubd wat dpth is calld η, which dpds o x, ad tim t. Th oigi of th coodiat sstm lis at th bottom of th basi i th low lft co as show i Figu. η(x,,t) z O B η(x,,t) H O.H. f x Figu : sid viw (top) ad top viw (bottom) of a smi-closd basi (ot to scal) This basi will b modlld o th f-pla (a local appoximatio of th sphical Eath) ad lis o th oth Hmisph. Th fom implis that th Coiolis dflctio is costat, whil th latt implis that th tid will otat cout-clockwis. Th tidal flow ts th basi o th ight sid of 5 Sptmb 9 Pag of 58

12 Modllig th th-dimsioal flow stuctu i smi-closd basis Olav va Dui Figu ad xits o th lft sid; this basic tidal movmt will b pstd b a icomig ad outgoig Klvi wav. At th closd latal bouda th tidal wav will hav to tu, this will gat so-calld Poicaé mods. Th icomig Klvi wav will b imposd whil th oth wav mods (th outgoig Klvi wav ad th Poicaé mods) will follow fom th solutio mthod. Ths mods hav o dpth-itgatd omal flow though th logitudial boudais. Th modl will adjust th amplituds of th possibl mods to su th is o omal dpth-avagd flow at th closd latal bouda wh th usd mods a supimposd. km Figu 3: sctio of th oth Sa that will b modld (souc: Th modl will b tailod to basis with a lgth ad width i th hudds of kilomts ad a dpth i th ts of mts. Th latitud will b aoud 5 dgs ad th modlld tidal costitut will b th M-tid. Actual phsical valus a pstd wh th poptis of th modl a ivstigatd as a fc cas. Bcaus th aim of this stud is ot to modl a spcific basi popl, th xact phsical valus a ot lvat ow. Lat i this thsis th modl will b applid to th oth pat of th oth Sa, as ca b s i th figu abov. 5 Sptmb 9 Pag of 58

13 Modllig th th-dimsioal flow stuctu i smi-closd basis Olav va Dui. MODIFYIG THE GOVERIG EQUATIOS AD BOUDARY CODITIOS I this paagaph th govig uatios ad bouda coditios that will b usd to dscib tidal flow i a smi-closd basi a pstd. Th govig uatios follow fom th tp of basi that will b modld ad th oigial full avi-stoks uatios, s appdix A fo dtails. I th followig paagaph ths uatios ad th bouda coditios will b scald... GOVERIG EQUATIOS Ud th coditios sktchd i appdix A th momtum uatios ad th cotiuit uatio a as follows. A astisk dots that th dimsioal vsio of th paamt o vaiabl is mat; i paagaph.3 th o-dimsioal vsios will b itoducd. u u u v u w u f v g t x z x η A u [.] v u v v v w v f u g t x z v η A v [.] z z v z z u v w [.3] x z H th paamt f dots th Coiolis paamt, A v vtical dd viscosit ad g th gavit acclatio. Th subscipts x,, z, t dot th divativ of that vaiabl to th spctiv coodiat... BOUDARY CODITIOS At th walls of th basi a coditio is imposd that th ca b o dpth-itgatd omal flow though th wall. A stog coditio would b that all flow is zo at th coast, this howv mas that th complx flow i th bouda la at th coast ds to b solvd (Mofjld, 98). Mofjld (98) lais that this coditio ca b placd b disgadig ths sid-las ad ol cosidig th gio sawads of thos las, this will thfo also b do h. It should b otd that th fact that b disgadig ths hoizotal sid-las hoizotal viscosit is implicitl glctd (i appdix A hoizotal viscosit was disgadd bfohad). Fom ow o, wh th coastal bouda coditios a mtiod actuall a-coastal bouda coditios a mat. H η v dz at, B [.4] H η u dz at x [.5] Th coditio at th paalll walls will b satisfid b th wav mods idividuall, whil th coditio at th latal wall will b satisfid b th wav mods collctivl (b us of a collocatio mthod). At th f wat sufac a o-stss damic bouda coditio ad a kimatic bouda coditio is applid. Ths coditios will b mt b th wav mods idividuall. It should b otd that th upp ssio is a appoximatio of th actual coditio which uss th divativs alog th sufac omal. Bcaus of th assumptio of log wavs (sufac lvatio is small compad to tidal wav lgth), th sufac omal poits almost xactl upwad (i.. paalll to th vtical axis z) ad this appoximatio ca b usd. u, v z z w η u η t x v η at z H η [.6] 5 Sptmb 9 Pag 3 of 58

14 Modllig th th-dimsioal flow stuctu i smi-closd basis Olav va Dui A damic bouda coditio (patial slip, which appoachs o slip wh th stss paamt s gos to ifiit) is applid at th bd, as wll as a kimatic bouda coditio. Ths coditios will also b mt b th wav mods idividuall. Usig a patial slip coditio maks th assumptio of costat dd viscosit asoabl (Hulsch, 996). It should b otd that th patial slip coditio actuall uss th divativs alog th bottom omal, but bcaus a flat bottom is usd th bottom omal is paalll to th z-axis, so this vsio ca b usd. Av u s u z Av v s v z w at z [.7].3 SCALIG THE FLOW EQUATIOS AD BOUDARY CODITIOS Th dimsios ad vaiabls a scald with ctai scalig paamts which a tpical valus fo th dimsios o vaiabl th a applid to. B doig this th sultig dimsiolss paamts a mo latabl to th sstm ad its flow poptis tha th oigial paamts w. Also, wh th uatios a scald, th od of magitud of th tms ca b mo asil compad thb makig it asi to s which a most sigificat. Th ssios fo th dimsiolss paamts ca b foud blow. t σ t [.8] (, ) K ( x, ) x [.9] η η η [.] ( u, v) ( u, v ) U [.] H th tidal fuc σ, tidal lvatio amplitud z z H [.] w w W [.3] η, maximum hoizotal vlocit U η g H [.4], wav umb K σ g H [.5] ad th vtical vlocit scal W K H U σ η [.6] (this follows fom applig th pvious scalig paamts to th cotiuit uatio [.3]) a usd. Both U ad K a tpical fo iviscid D Klvi wavs (Pdlosk, 98). W is th maximum hoizotal vlocit adjustd fo th lativ diffc i vtical ad hoizotal lgth scal. A additioal itptatio is that W is th amplitud dividd b th tidal piod. This all mas that uit of t uals th tidal piod, uit of x o uals th Klvi wav lgth dividd b π, uit of z uals th wat dpth, uit of η uals th Klvi lvatio amplitud, uit of u o v uals th Klvi vlocit amplitud ad uit of w uals th spd of goig up with a spd of o amplitud p tidal ccl. Aft substitutig th scald paamts w scald vsios of th oigial uatios ad bouda coditios a foud. Ths a usd to div th lia sstm of uatios i th followig paagaph..3. SCALED GOVERIG EQUATIOS Itoductio of th o-dimsioal paamts ad som aagig ilds th followig fo th uatios of motio ad th cotiuit uatio. u v t t δ v [ uu x vu wu z ] fv η x u zz ε [.7] δ v [ uvx vv wvz ] fu η vzz ε [.8] u v w [.9] x z 5 Sptmb 9 Pag 4 of 58

15 Modllig th th-dimsioal flow stuctu i smi-closd basis Olav va Dui H th dimsiolss paamts δ f f σ [.] (atio of itial ad tidal fuc), v H Av σ [.] (sua of th Stoks umb) ad th Foud umbε η H U g H [.] a usd..3. SCALED BOUDARY CODITIOS Th bouda coditios at th coast tasfom to th followig fom. εη εη vdz at, B [.3] udz at x [.4] H th o-dimsioal width sufac is giv b u z, vz w ηt ε [ uη vη ] x Th bouda coditios at th bottom ad s u z u s vz v w wh s A ( H s ) v at B K B is usd. Th dimsiolss bouda coditios at th z εη [.5] at z, [.6] [.7]..4 LEADIG-ORDER SYSTEM OF EQUATIOS Bcaus usuall th Foud umb ε<<, th sstm of uatios ca b dvlopd as a pow sis i ε. Fo this stud th lowst od cotibutio i th Foud umb (i.. at O(ε )) is cosidd, which mas that all o-lia tms a doppd fom th uatios ad th upp vtical domai boudais a gatl simplifid. Fist th govig uatios ad th th bouda coditios will b hadld..4. LIEAR SYSTEM OF EQUATIOS Substitutig ε i th govig uatios it follows that o advctiv cotibutio is foud ad th govig uatios duc to th followig lia sstm of uatios: u v t t δ v fv η x u zz, [.8] δ v fu η vzz, [.9] u v w. [.3] x z Th bouda coditios at th sid walls duc to 5 Sptmb 9 Pag 5 of 58

16 Modllig th th-dimsioal flow stuctu i smi-closd basis Olav va Dui vdz at, B, [.3] udz at x. [.3] At th f wat sufac th kimatic bouda coditio is adjustd bcaus th o-lia cotibutio ducs to zo, whil th damic bouda coditio stas th sam. Th locatio of th bouda ducs to z. u z, vz w ηt at z [.33] Th damic bouda coditio at th flat bd, as wll as th kimatic bouda coditio dos ot chag du to takig ε. s u z u s vz v w at z [.34] 5 Sptmb 9 Pag 6 of 58

17 Modllig th th-dimsioal flow stuctu i smi-closd basis Olav va Dui 3 FIDIG WAVE MODES I A IFIITE CHAEL I this chapt wav solutios will b sought fo a ifiitl log chal. Fist th vlocit will b tasfomd to otatig vlocit compots to aid i solvig th dpth-dpdc of th uatios. Aft that, latios btw th logitudial wav umb ad th displacmt a divd. Ths latios a usd to fid a sigl coditio which lads to two possibl tps of wav solutios, Klvi wavs ad Poicaé mods. Fo ths wav mods th displacmt uatios a foud which a th usd to taslat th otatig vlocit compots back to th omal vlocit compots. Euatio Sctio 3 3. THE SOLUTIO I ROTATIG VELOCITY COMPOETS I this paagaph th dpth-dpdc is foud b solvig uatios fo otatig vlocit compots. Fist th oigial uatios a split to facilitat th solutio mthod, aft that th actual tasfomatio is applid. Th dpth-idpdt pat of th otatig compots is solvd fist aft which th dpth-dpdt pat is also solvd. Th ssios fo th otatig compots still implicitl cotai a dpdc o x ad which will b hadld i paagaph SPLITTIG THE GOVERIG EQUATIOS AD BOUDARY CODITIOS Th oigial uatios of motio (s [.8]-[.9]) a split b dfiig th vlocitis as uu u ad vv v spctivl. Th vlocitis with idx sigif th pat of th vlocit that is assumd ot to va i th vtical dictio, ladig to th followig uatios (aft Dailso & Kowalik, 5): u v ; t fv η x, [3.] ; t fu η. [3.] Th dpth-dpdt uatios with idx ad (aft Dailso & Kowalik, 5): δ, [3.3] v u ; t fv u ; zz δ. [3.4] v v ; t fu v ; zz To solv th uatios fo th otatig vlocit compots (which will b show i th xt paagaph) th damic bouda coditios fo th hoizotal vlocitis at th bottom ad sufac also hav to b split. Th slip coditio bcoms as follows (with s - fo o slip ad s> fo patial slip). u v s s ( u u ) ( v v ) z z u v Th sufac coditios bcom ( u u ) ad ( v ) z z at z [3.5] v at z [3.6] 3.. TRASFORMATIO TO ROTATIG VELOCITY COMPOETS Th followig otatig compots a itoducd which (wh summd) pst th cut llipss fomd b th othogoal vlocitis i a mo covit wa (Mofjld, 98). H is th clockwis compot ad is th cout clockwis compot, which cospod with th distict cout otatig las i th bottom la (Mofjld, 98). ot that [3.7] ad [3.8] ad that th ssios blow hold fo idx ad idx spaatl. u iv, u iv [3.9] 5 Sptmb 9 Pag 7 of 58

18 Modllig th th-dimsioal flow stuctu i smi-closd basis Olav va Dui Fom th abov it follows that u, v [3.] i To pst th piodicit of th tidal flow th tim-dpdc of, ad η is giv b (-it). Th scod uatio of motio ([3.] ad [3.4]) multiplid with th imagia uit i is addd to th fist uatio of motio ([3.] ad [3.3]) to ild th followig: i f ( η iη ) x, [3.] iδ v ; zz ( f ). [3.] Hb th splittig i dpth-dpdt ad dpth-idpdt pats will facilitat solvig th uatios lat (s 3..3). Wh istad of a additio a subtactio is caid out th followig is foud: i f ( η iη ) x, [3.3] iδ v ; zz ( f ). [3.4] Th bouda coditio at th bottom taslats as follows (usig th sam mthodolog as fo th govig uatios). It should b otd that th divativs of ad to z a both zo, which follows fom th uatio fo ad abov. This coditio ad th coditio at th sufac a applid i s s ; z ; z Th sufac coditio bcoms as follows. at z [3.5] ad at z [3.6] ; z ; z Th taslatio of th dpth-itgatd cotiuit uatio [.3] to otatig vlocit compots ilds th followig. Fo this o diffc is mad btw th dpth-dpdt ad th dpthidpdt pats. This is bcaus th tms that do dpd o dpth a itgatd ov dpth. This coditio is usd i subsctio 3... ( i ) ( i ) iη x x dz [3.7] Th taslatio of bouda coditios at th coasts [.3] ito otatig compots ilds th followig. Also this uatio dos ot d to b split i dpth-dpdt ad dpth-idpdt pats. This bouda coditio is usd i subsctio 3... ( ) dz i at, B [3.8] 5 Sptmb 9 Pag 8 of 58

19 Modllig th th-dimsioal flow stuctu i smi-closd basis Olav va Dui 3..3 SOLVIG THE ROTATIG VELOCITY COMPOETS Th ssios fo th dpth-idpdt otatig compots [3.] ad [3.3] a solvd, but th solutios fo th dpth-dpdt compots [3.] ad [3.4] a still dd. Wh ths a foud th total solutio fo th otatig vlocit compots ca b divd (b addig th difft pats). I th followig paagaphs thos ssios will b usd to fid th uatios fo η, u, v ad w. To fid th solutios fo ad fist th gal solutio fo a scod-od lia odia difftial uatio is applid to th uatios [3.] ad [3.4]: ( z) c ( z) c, [3.9] ( z) c ( z) c3 4, [3.] with paamts ad dfid as i ( f ) /, [3.] δ v i ( f ) /. [3.] δ v Applig th sufac bouda coditio at z ilds ( ) c c ( ) c c z z ( ) c3 c4 ( ) c4 c3 z z At th bottom th slip coditio is applid. This ilds ( ) c c s ( c c ) c( s ) c ( s ), [3.3]. [3.4], [3.5] ( ) c3 c4 s ( c3 c4 ) c3 ( s ) c4 ( s ). [3.6] Applig th coditio fo c ad c 4 foud abov givs th followig sult. ( ) [ s sih ] c cosh c [3.7] s [ ] c [ cosh s sih ] ( ) c3 [ cosh s sih ] c3 [3.8] s [ ] c4 [ cosh s sih ] This is put back i th pviousl foud uatios fo ad to ild th fial sults. cosh s [ ( z ) ] cosh [3.9] sih cosh s [ ( z ) ] cosh [3.3] sih Usig [3.7], [3.] ad [3.9] th followig is foud: 5 Sptmb 9 Pag 9 of 58

20 Modllig th th-dimsioal flow stuctu i smi-closd basis ( η iη )( Q) [3.3] with Q( z) i x f Ad aalogous to th abov fo [3.8], [3.3] ad [3.3] th followig is foud. ( η iη )( R) [3.33] with R( z) i x f [ ( z ) ] Olav va Dui cosh [3.3] cosh s sih [ ( z ) ] cosh [3.34] cosh s sih H th w paamts Q ad R dtmi th vtical patt of th otatig vlocit compots. Th ssios fo Q ad R duc to thos foud b Mofjld (98) if s - (o slip), though i o-dimsioal fom. 3. DETERMIIG WAVEUMBER RELATIOS I this paagaph a dispsio latio ad a w vsio of th coastal bouda coditios will b divd which dfi th latio btw th logitudial wav umb ad th displacmt. Ths latios a usd i th xt paagaph. 3.. DISPERSIO RELATIO Th otatig compots ad a substitutd i th dpth-itgatd cotiuit uatio [.3], which, aft som maipulatio, ilds th followig sult. η xx f η η H Fo clait a w paamt H was itoducd, which is show blow. H ( f ) [ cosh s sih ] ( f ) [3.35] sih sih [3.36] [ cosh s sih ] This dispsio uatio is th sam as th iviscid dispsio uatio xcpt that this uatio is coctd with th paamt H which accouts fo viscous ffcts ad th ffcts of th patial slip coditio. As ca b ctd th ssio fo H ducs to th ssio foud b Mofjld (98) wh divd fom th dimsioal fom of th dispsio latio ad with s - (o slip). Euatio [3.35] allows fo solutios of th followig fom, wh η ( ) is a -dpdt fuctio ad k th wav umb of this wav solutio. [ η ( ) ( i( kx t) )] η R [3.37] This wav uatio is substitutd i [3.35] to obtai th dispsio latio. k f H η η 3.. EAR-COASTAL BOUDARY CODITIOS f o η k η [3.38] H Applig th a-coastal bouda coditio ([3.8]) ilds th followig aft som maipulatio. η γ k η at, B [3.39] H H th paamts H ad γ w substitutd ito th uatio to ild th fial sult. H was alad show i th pvious paagaph, whil γ is show blow. 5 Sptmb 9 Pag of 58

21 Modllig th th-dimsioal flow stuctu i smi-closd basis Olav va Dui γ ( f ) [ cosh s sih ] ( f ) sih sih [3.4] [ cosh s sih ] f This paamt is simila to H but is a coctio applid to f ath tha to. As ca b ctd th ssio fo γ ducs to th ssio foud b Mofjld (98) wh mad dimsioal agai ad with s - (o slip). 3.3 SOLVIG THE WAVEUMBER RELATIOS Th gal solutio to [3.38] ads η C si Dcos [3.4] with Substitutig this solutio ito th bouda coditio ([3.39]), f k. [3.4] γ ( C cos Dsi) k ( C si Dcos) at, B. [3.43] H Substitutig,B i [3.43] ilds th followig st of uatios fo C ad D: γ C k D, [3.44] H γ γ C cos B k sib Dk cosb sib. [3.45] H H This sstm of uatios lads to otivial solutios fo C ad D wh th dtmiat uals zo. Takig th dtmiat, substitutig th latio fo [3.4] ad aagig, lads to th followig coditio fo otivial solutios: ( f ) H k H γ sib. [3.46] I th followig paagaphs th two cass will b aalsd fo which th coditio abov is mt SATISFYIG WAVEUMBER RELATIOS WITH KELVI WAVES Th coditio [3.46] is mt wh k (idx sigifis th Klvi wav) satisfis (divd fom [3.46]) ( f ) H k k o k H γ ( f ) H H γ / H. [3.47] Th positiv oot fo k is chos, so that th wav tavls i th positiv x-dictio (it thfo psts th outgoig Klvi wav). With a gativ wav umb th wav would tavl i th gativ x-dictio (ad would b th icomig Klvi wav). Th abov combid with th latio fo ilds th followig. k γ kγ o i [3.48] H H Also h th positiv oot is chos, bcaus th gativ oot givs o xta ifomatio (it would b th sam as choosig a gativ valu fo k abov, also s Pdlosk, 98 ad D Swat, 8). 5 Sptmb 9 Pag of 58

22 Modllig th th-dimsioal flow stuctu i smi-closd basis Olav va Dui To fid th uatio fo th f sufac displacmt, th fist uatio fo C ad D [3.44] is tasfomd. kγ C D [3.49] H Valu of D is fo ow st to uit, as η will b -scald with η vtuall to fid η. Th paamt D thfo svs o al fuctio, as th actual amplitud will b dtmid b η. Th latio fo C is substitutd i th latio fo η [3.4]. Aft substitutig th latio fo [3.48] fo which a Klvi wav is foud ad som fial aagig, th ssio fo th latal dpdc of th sufac displacmt is foud to b ( ) ( k γ ) η, [3.5] H which is usd i uatio [3.37]. To tasfom th otatig vlocit compots back ito th omal vlocit compots th spatial divativs of th sufac displacmt a dd. Ths a substitutd ito th uatios fo th otatig compots [3.3] ad [3.33]. Usig th latio btw th omal ad otatig vlocit compots [3.] th followig is vtuall foud fo th vlocit compots. ikη u R kη v R ( i( k x t) ) ( i( k x t) ) γ Z H γ Z H Zˆ γ H Zˆ γ H H Z ( z) i[ Q( z) ]( f ) [3.53] ad ˆ ( z) i[ R( z) ]( f ) [3.5] [3.5] Z [3.54], with Q ad R dfid i [3.3] ad [3.34] spctivl. To fid th vtical vlocit w th spatial divativs of th vlocitis u ad v a dd (s th cotiuit uatio [.3]). Th bottom ad sufac bouda coditios a usd to fid th fial ssio fo w, which ads: ( ) ( i( k x t) ) k η γ ~ w R Z, [3.55] H with ~ Z ( z) i f sih z [ ( )] z sih i sih[ ( z ) ] sih z. [3.56] sih f cosh s sih cosh s 3.3. SATISFYIG WAVEUMBER RELATIOS WITH POICARÉ MODES Th coditio si B follows fom ssio [3.46] ad is mt wh π,,,3,.... [3.57] B Hi is ot a valid solutio bcaus [3.4] th ducs to ( ) η D, which mas it will hav o latal vaiatio. Th latal vlocit is howv o-zo; both factos combid mas that such a wav caot mt th coditio of vaishig dpth-itgatd omal flow though th walls (Pdlosk, 98). Combiig th latio abov with th pviousl foud uatio fo [3.4] givs th followig. 5 Sptmb 9 Pag of 58

23 Modllig th th-dimsioal flow stuctu i smi-closd basis Olav va Dui f k H π [3.58] B As i th pvious sctio h th positiv oot of is chos. Fo this poblm th tidal fuc is giv (M-tid), but th wavumb is ukow so th latio abov is tasfomd to fid th wavumb k k f ± H π B /. [3.59] As i th pvious sctio h th positiv oot is chos. To fid th uatio fo th f sufac displacmt th latio [3.44] fo is tasfomd. kγ C D [3.6] H Th paamt D is calld η (with η η /η ) fom ow o. Th latio fo C is substitutd i th latio fo η [3.4] (th subscipt is addd fo clait). Aft substitutig th latio [3.58] fo fo which Poicaé mods a foud ad som fial aagig th ssio fo th latal dpdc of th sufac displacmt is foud. Bk γ η [3.6] ( ) cos( ) si ( ) π H To tasfom th otatig vlocit compots back ito th omal vlocit compots th spatial divativs of th sufac displacmt a dd. Ths a substitutd ito th uatios fo th otatig compots [3.3] ad [3.33]. Usig th latio btw th omal ad otatig vlocit compots [3.] th followig is foud fo th vlocit compots. u v iη R η R ( i( k x t) ) k η ( Z Zˆ ) ηˆ ( Z Zˆ ) [ ] ( i( k x t) ) k η ( Z Zˆ ) ηˆ ( Z Zˆ ) π [ ] wh ηˆ ( ) si( ) k cos( ) B H γ [3.6] [3.63] To fid w th spatial divativs of th vlocitis u ad v a dd (s th cotiuit uatio [.3]. Th bottom ad sufac bouda coditios a usd to fid th fial ssio fo w, which ads: w ( i( k x t) ) η f ~ R η Z, [3.64] H with Z ~ as dfid bfo i uatio [3.56]. 5 Sptmb 9 Pag 3 of 58

24 Modllig th th-dimsioal flow stuctu i smi-closd basis Olav va Dui 4 WAVE MODES I SEMI-ECLOSED BASIS I this chapt a solutio fo tidal flow i a smi-closd basi is foud. Th Klvi wav dscibd bfo tavls alog th positiv x-dictio ad is thfo th outgoig Klvi wav. I th fist paagaph th ssios fo th icomig Klvi wav a divd. Aft that, dpth-avagd vsios of th ssios fo th outgoig Klvi wav a pstd (th icomig vsios ca b divd aalogousl). I th fial paagaph ths dpth- avagd ssios will b usd to fid th amplituds of th difft wav mods b applig th latal bouda coditio [.3] with a collocatio mthod. Euatio Sctio 4 4. KELVI WAVES Th ssios that w divd bfo ([3.5], [3.5], [3.5] ad [3.55]) appl to th outgoig Klvi wav i a ifiit chal but could hav b divd fo a icomig Klvi wav as wll. Bcaus this icomig wav is dd as wll th cssa adjustmts to th oigial ssios a dscibd blow. Th icomig Klvi wav ts th sstm fom th ight ad tavls towads th latal bouda at x (s Figu ). Th dictio of th hoizotal vlocitis of th icomig wav must b opposit to that of th outgoig wav. Th ssios fo th outgoig wav ca b asil adjustd b takig th gativs of ths vlocitis. Th icomig wav has its highst amplitud th w it ts th sstm, bcaus du to th viscosit tms i th govig uatios th wav ics g dissipatio ov distac, s paagaph 5... To su that th logitudial dpdc is coct th gativ wav umb k is usd fo th icomig wav (ik x -ik x). Th amplitud of th icomig Klvi wav s lvatio ad vlocitis must dcas with dcasig (i.. wh movig awa fom th upp coast, s Figu ). Th ssios fo th outgoig Klvi wav of cous lt th amplitud dcas with icasig (i.. wh movig awa fom th low coast); thfo a gativ latal coodiat is usd fo th latal dpdc. Futhmo, to su that both th icomig ad outgoig Klvi wavs hav th sam valu fo η at th upp ad low coast spctivl (i.. to su smmt), th agumt of ssio [3.5] fo η of th icomig wav is adjustd ( B-). Ths tasfomatios a applid to th ssios foud i 3.3. ([3.5], [3.5], [3.5] ad [3.55]), sultig i th followig uatios fo th icomig Klvi wav: [ η ( B ) ( i( k x t) )] η, [4.] u v i R i i w i ikη R kη R ( B ) ( i( k x t) ) ( B ) ( i( k x t) ) ( B ) ( i( k x t) ) γ Z H γ Z H Zˆ γ H Zˆ γ H, [4.], [4.3] k η γ ~ R Z. [4.4] H Fo th outgoig Klvi wav th oigial ssios a of cous still coct ad will b usd to dscib tidal flow i a smi-closd basi. It should b otd that du to dissipatio th outgoig wav will hav a low amplitud tha th icomig wav. To mak su this ca b modlld 5 Sptmb 9 Pag 4 of 58

25 Modllig th th-dimsioal flow stuctu i smi-closd basis Olav va Dui accodigl th icomig ssios a multiplid b a flctio cofficit R; th valu of this cofficit will b calculatd with a collocatio mthod (s paagaph 4.3). 4. THE DEPTH-AVERAGED MODEL I this paagaph dpth-avagd uatios fo th hoizotal vlocitis of th Klvi wavs ad Poicaé mods will b divd. To su th is o omal dpth-avagd flow though th latal bouda, th umical appoach pstd i paagaph 4.3 will us th ssio fo th logitudial vlocit. 4.. DEPTH-AVERAGED VELOCITIES OF A KELVI WAVE Th hoizotal alog chal vlocit u of a Klvi wav will b itgatd ov th ti wat colum. Fo this ol th pat btw sua backts i th followig ssio will b itgatd as ol th Z ad Ẑ paamts a dpdt o th vtical positio. u udz R ik η ( i( k x t) ) γ Z H ˆ γ Z H dz Aft itgatio, substitutio of th paamts k, γ ad H ad som aagig th followig sult is obtaid. u η R ( i( k x t) ) k Th dpth-avagd hoizotal latal vlocit v of a Klvi wav is divd i simila ma. kη vdz v R ( i( k x t) ) γ Z H ˆ γ Z H dz Aft itgatio ad som aagig th followig sult is obtaid. v ikη R ( i( k x t) ) f H γ H γ This shows that bouda coditio of zo omal dpth-itgatd flow at th chal boudais is idd satisfid b a Klvi wav. This coditio v lads to th popt that th dpthavagd latal vlocit is zo ov th ti width of th basi. 4.. DEPTH-AVERAGED VELOCITIES OF POICARÉ MODES Th hoizotal alog chal vlocit u of a Poicaé mod is itgatd aalogous to that of th Klvi wav. iη udz u R ( i( k x t) ) k [ Z Zˆ ] ˆ [ Z Zˆ ] η η { } dz Aft itgatio ad som aagig th followig sult is obtaid. u k η R ( i( k ) ) x t ( H ) η γηˆ f Also th hoizotal latal vlocit v of a Poicaé mod is itgatd ov th ti wat colum. [4.5] [4.6] [4.7] [4.8] [4.9] [4.] 5 Sptmb 9 Pag 5 of 58

26 Modllig th th-dimsioal flow stuctu i smi-closd basis Olav va Dui v dz v R η ( i( k x t) ) k η ( Z Zˆ ) ηˆ ( Z Zˆ ) [ ] Aft itgatio ad som aagig th followig sult is obtaid. ( ( )) iη i kx t k γ v R si H ( ) f H [4.] [4.] Th abov shows that th dpth-itgatd latal vlocit of a Poicaé mod is ot zo fo all x, ad t (as opposd to a Klvi wav). Howv, substitutig,b shows that th dpth-itgatd omal flow is idd zo at th boudais fo a Poicaé mod. 4.3 SUPERIMPOSIG THE WAVE SOLUTIOS H a solutio to th tidal flow poblm i a smi-closd basi is pstd as a suppositio of a icomig Klvi wav, a outgoig Klvi wav (with a flctio cofficit R) ad a ifiit umb of Poicaé mods (wh th amplitud dpds o ). It should howv b otd that fo calculatios ot th ifiit st of Poicaé mods is icludd (which would of cous b impossibl), but a tucatd st of Poicaé mods, which mas that a total of mods will b summd. Fo icasig th solutio coms clos to th xact solutio, but it is ctd that a st i th od of ts of Poicaé mods will lad to good sults. Th ssios fo total displacmt ad vlocitis a as follows. ( xzt,,, ) total i out η η η η [4.3] (,,, ) total i out u xzt u u u [4.4] (,,, ) total i out v xzt v v v [4.5] (,,, ) total i out w xzt w w w [4.6] Of th mods th amplitud of th icomig Klvi wav is a giv. Th lativ amplituds of th maiig mods (R ad η ; /η ) will b sought with th mthod dscibd blow. Fo this mthod coditios will b divd which th dpth-itgatd omal vlocit must mt at th latal bouda (at x). This lads to th followig ssio fo u total as fuctio of ad t. u total ( t) R [ ( B ) ( it) R η ( it) ] ( it) ( H η γ ˆ ) k η, η η [4.7] k f Th collocatio mthod fids th appopiat amplituds of th spaat mods b miimisig th dpth-itgatd omal vlocit fo a fixd st of poits at th latal bouda. Th collocatio poits hav th followig (x,)-coodiats. m m with m B fo m,,..., [4.8] (, ) Th dpth-itgatd vlocit i th x-dictio (alogsho) must b zo at ths collocatio poits. Th afomtiod taslats to th followig dsid stat (which should b mt fo all t). ( x, ) (, ) fo m,,..., utotal at m [4.9] 5 Sptmb 9 Pag 6 of 58

27 Modllig th th-dimsioal flow stuctu i smi-closd basis Olav va Dui Bcaus this coditio must b mt all momts duig th tidal ccl a ssio fo <u total > that ol dpds o th latal coodiat is dd. Usig uatios [4.7] ad [4.9] th followig is divd: k k η η. [4.] f [ ( B ) R η ] { H η γηˆ } This must hold at simplicit. m fo m,,...,. Th coditio is witt i th followig fom fo K ; i m K ; out m P m ( ) RA ( ) A (, ) A η at m fo m,,..., [4.] Hb A K;i, A K;out ad A P a dfid as (with ssios fo η, η ad ηˆ substitutd): AKi ; ( ( B ) kγ H ) k AKout ; ( kγ H). [4.] k k k γ γ AP H cos ( ) si( ) si( ) cos ( ) γ f H k H I matix-fom th pvious coditio is as follows: A A A AK K ; out K ; out K ; out ; out K ( ) AP (,) AP (,) M AP (, ) ( ) A (,) A (,) M A (, ) P K P ( ) AP (,) AP (,) M AP (, ) ( ) A (,) A (,) M A (, ) K O R A η A η M A η AK ( ) ( ) K ; i L ( ) ( ) K ; i ; i P P P P K K ; i. [4.3] Th amplitud vcto is th dtmid usig stadad tchius i Th Mathwoks TM MATLAB. 5 Sptmb 9 Pag 7 of 58

28 Modllig th th-dimsioal flow stuctu i smi-closd basis Olav va Dui 5 FLOW PROPERTIES OF THE SOLUTIO Poptis of th wav solutios hav b xamid fo th oth pat of th oth Sa (s Figu 3), as that basi appoximatl is a smi-closd basi ad has dimsios that match th assumptios ad coditios usd to div th govig uatios ad bouda coditios. Th basi is appoximatl 5 km wid ad has a avag dpth of 65 m (tak fom Davis & Jos (995), also s figu blow). Th modl divd uss a costat Coiolis fuc, thfo a avag latitud of 55 is usd. Euatio Sctio 5 Figu 4: bathmt of th oth Sa i mts (souc: Th Coiolis fuc th follows fom f Ω siθ [5.], with Ω th agula fuc of th Eath s otatio ad θ latitud. Th vtical viscosit paamt A v is calculatd with A v.5u H [5.]; this latio tak fom Bowd (967) modls th obsvd maximum dd viscosit i tidal cuts i homogous wat. Th autho actuall uss th dpth-ma amplitud of th cut istad of th tpical Klvi wav vlocit U, with th latt pobabl small tha th fom. It is assumd this is compsatd bcaus th maximum dd viscosit paamt istad of a avag is tak. Fo this cas th picipal smidiual tidal costitut as causd b th gavitatioal pull of th Moo (M tid) is usd as th tidal focig. Th piod T of this costitut is hous ad 5 miuts ad th tidal fuc is dfid as σ π T. Th 5 Sptmb 9 Pag 8 of 58

29 Modllig th th-dimsioal flow stuctu i smi-closd basis Olav va Dui gavit costat g is 9.8m/s fo this pat of th Eath. Th amplitud of tidal lvatio η is a suitabl valu fo that pat of th oth Sa tak fom Siha & Pig (997). Th valu fo th slip paamt s is tak fom Davis & Jos (995). I th tabl blow th dimsioal valus a giv fo th mtiod paamts. Valu H 65 m B 5 km θ 55 f.9e-4 s - A v.947 m /s η.5 m σ.4e-4 s - g 9.8 m/s s.5 m/s Tabl : modl paamts of th oth pat of th oth Sa Th modl is implmtd i Th Mathwoks TM MATLAB ad us with a tucatd st of Poicaé mods (). Poptis of th wav umbs, idividual wav mods ad th total solutio will b pstd i th followig paagaphs. 5. PROPERTIES OF THE IDIVIDUAL MODES I this paagaph poptis of th idividual mods a studid. Fist th wav umbs a hadld, th ctai aspcts of Klvi ad Poicaé mods a xamid ad fiall a taslatio is mad btw th usd 3D fictio modl (with A v ad s) ad a dpth-avagd modl (DH, usig a fictio paamt f ). 5.. WAVE UMBERS OF KELVI AD POICARÉ MODES I th followig wav umb plots th sults fo difft valus of th viscosit A v (with difft maks) a plottd fo th difft wav mods (s Figu 5 ad Figu 6). Th sultig wav umbs hav b mad dimsioal agai; s Appdix B fo how vaious paamts a mad dimsioal agai. A v vais btw.3 ad tims th fc valu mtiod abov, th low limit is chos bcaus MATLAB could ot comput th sults fo low valus. 5 Sptmb 9 Pag 9 of 58

30 Modllig th th-dimsioal flow stuctu i smi-closd basis Olav va Dui Figu 5: wav umbs k i th complx wav umb pla fo difft valus of A v [m /s] I th figu abov th wav umbs k of ol th Klvi wav a plottd, alogsid th iviscid wav umb K ad (as a fc) th wav umb k ;DH of a dpth-avagd Klvi wav (fo vaig fictio paamt f ). It should b otd that fo all dpth-avagd calculatios th modl pstd b Rick & Tub (98) is usd (s Appdix C). Fo th Klvi wav th al ad imagia pat of th wav umb icass with icasig viscosit ad fictio paamt (th fom is also obsvd b Mofjld, 98). This sigifis that th Klvi wav shows icasd dissipatio whil also its popagatig chaact bcoms stog. This ca b laid b lookig at th ssio fo th wav umb [3.47], which uss th paamts H [3.36] ad γ [3.4]. Ths paamts hav a dpth-itgatd cotibutio of th clockwis ad cout clockwis compots Q [3.3] ad R [3.34]. This complx itactio (with hpbolic cosis ad sis) lads to th dvlopmt of th wav umb ad also causs th bump i th gaph abov fo th 3D modl. I paagaph 5.. it will b ivstigatd what th t ffct (mo dissipatio o mo popagatio) of icasig A v is o th 3D Klvi wav. With dcasig A v ad f, th imagia pat of th wav umb of th Klvi wav appoachs zo ad th al pat appoachs th iviscid wav umb K. This is to b ctd, bcaus wh ths valus dcas th dissipatio mchaism is shut dow ad th poptis should smbl that of a iviscid wav. I th followig figu th wav umbs fo all Klvi ad Poicaé mods a plottd i th complx pla fo vaig viscosit (th at i th lgd is multiplid with A v of th fc cas to gt th usd valu). 5 Sptmb 9 Pag 3 of 58

31 Modllig th th-dimsioal flow stuctu i smi-closd basis Olav va Dui Figu 6: wav umbs i th complx wav umb pla fo difft valus of A v [m /s] I th figu abov, p valu of A v (th valu i th lgd multiplid with A v of th fc cas), th mod with (th Klvi wav) has th highst al pat of th wav umb ad th lowst imagia pat (th all plot i th low ight co of th plot). With icasig (icasigl high Poicaé mods) th imagia pat icass, whil th al pat dcass. This sigifis that th dissipativ chaact of high mods is stog tha that of low mods. Th Poicaé mods hav a wav umb that is patl al ad patl imagia. If th wav umb would b pul imagia th Poicaé mods would ol xist dictl at th coast, th would b boud. With a pul al wav umb th would fl popagat thoughout th chal, th would b f. I this cas th a ith compltl f o boud, but bcaus th imagia pat is gall much bigg tha th al pat thi ifluc i th basi is limitd to a small aa ad th ca b pacticall calld boud. 5 Sptmb 9 Pag 3 of 58

32 Modllig th th-dimsioal flow stuctu i smi-closd basis Olav va Dui Figu 7: Poicaé wav umbs (,) i th complx wav umb pla fo vaig A v /A v;f Th followig ca b mo clal obsvd i th figus abov fo two spaat Poicaé mods tha i th plot with all wav umbs. With icasig A v th al pat bcoms lag fo all Poicaé mods whil th imagia pat shows a slight dcas, but ol o th lft sid of th plot (fo th small viscosit). O th ight sid of th plot th is a (slight icas) of th imagia pat with icasig A bfo doppig off wh th viscosit gts v high. This is du to th itactio btw th clockwis ad cout clockwis compots Q ad R, which i dpthitgatd fom ifluc th Poicaé wav umb though H (s ssios [3.3], [3.34] ad [3.36].). 5.. PROPERTIES OF THE KELVI WAVE Th wav lgth of th Klvi wav λ is dtmid b λ π / [5.3] with k th wav umb of th Klvi wav. Th wav lgth idicats th pogssiv chaact of th Klvi wav. Th atio D of th Klvi wav amplituds at two locatios spaatd b o wavlgth i th logitudial dictio shows th lativ ffct of th dissipatio causd b iclusio of viscosit tms. This is π k k [5.4]. I th tabl blow ata v /A v f wh th valus with D ; im / ; giv b ( ) subscipt f a fom th fc cas. Futhmo th sultig valus fo th wav umb k, wav lgth λ ad dca atio D a giv. k ; at.. 5 k (m ) 5.5-8i.76-7i 5.-7i 8.-7i.55-6i.59-6i λ (km) D Tabl : Klvi wav poptis Th tabl shows that fo icasig viscosit (th icasig paamts icas lial with icasig A v ), th wav lgth fist icass ad th dcass, showig that th popagativ chaact i itslf fist bcoms stog ad th wak which is du to th dvlopmt of th Klvi wav umb (s 5..). Howv, as th dca facto D kps gttig small gadlss of whth popagatio icass o dcass, th icas of dissipatio is stog. 5 Sptmb 9 Pag 3 of 58

33 Modllig th th-dimsioal flow stuctu i smi-closd basis Olav va Dui 5..3 PROPERTIES OF THE POICARÉ MODES I th iviscid cas fo B<B cit with ( ) B cit π f all Poicaé mods hav a pul imagia wav umb (ad a thfo boud, also s 5..). Howv du to th additio of viscous ffcts th f mods gai a slightl dissipativ chaact (th wav umb is o log pul al), whil th boud mods gai a slightl popagatig chaact (th wav umb is o log pul imagia), lik mtiod i Roos & Schuttlaas (9) fo hoizotal viscous ffcts. Th authos stat that fo small valus of th hoizotal viscosit ad fictio ths modificatios a small so that th coditio B<B cit usd i th iviscid cas that sas whth Poicaé mods a f o boud ca still b usd. It is assumd that this also applis fo th cut cas (vtical viscosit). I th fc cas B cit is 7 km ad B is 5 km, so all Poicaé mods a boud. I th tabl blow th Poicaé wav umbs as wll as th -foldig dca lgths a giv (dtmid b L k ; im ). I this tabl it ca b s that icasigl high Poicaé mods hav lss ifluc i th chal as th dca ov a small distac. 5 Sptmb 9 Pag 33 of 58

34 Modllig th th-dimsioal flow stuctu i smi-closd basis Olav va Dui k (m - ) L (km) i i i i i i i i i i i i i i i i i i i i 7.96 Tabl 3: Poicaé mod poptis I th figu blow th -foldig dca lgths a giv fo vaig valus of A v, dividd b th maximum dca lgth p Poicaé mod fo th complt ag of valus. This shows that (i gal) th dca lgth icass with icasig viscosit, but that th vaiatio is small (o mo tha 9% fo a at btw. ad ). Som mods achiv a local maximum at a at of about.5 ad th dcas slightl to stat icasig agai fo high ats. That th ifluc of th Poicaé mods xtds futh wh th viscosit icass sms a paadox, but appatl is a popt of ths mods. This ca b obsvd i th wav umb plot of figu Figu 6. Fo high viscosit th al pat of th wav umb gts lativl high (compad to th imagia pat), though it mais lativl small. But bcaus this atio chags its ifluc alog th basi gts lag. 5 Sptmb 9 Pag 34 of 58

35 Modllig th th-dimsioal flow stuctu i smi-closd basis Olav va Dui Figu 8: Poicaé dca lgth L fo vaig viscosit dividd b L ;max p valu of 5..4 TRASLATIO OF WAVE MODE PROPERTIES TO DH FRICTIO Th DH modl pstd b Rick & Tub (98) ca b s as a simplificatio of th 3D modl pstd h. Th ol diffc is that thi modl is dpth-avagd ad thfo dos ot us a slip coditio ad has a fictio tm that is a paamtizatio of th fictio tm usd h. This mas that th is o viscosit paamt o slip paamt but a sigl fictio paamt f. Th ustio is which valu fo f should b chos to appoximat th 3D flow poptis. A fist simpl appoximatio basd o vlocit poptis is giv b 8 ( 3π ) C U [5.5] (th Lotz liaizatio of a uadatic fictio law, s Zimmma 98) which givs f. m/s. This could b a asoabl appoximatio as it is dictl basd o th phsical poptis of th basi that still appl to th dpth-itgatd modl. Aoth wa to fid valus fo f is b tig to appoximat th poptis of th idividual wav mods mtiod abov. Fo this ags of valus fo A v ad s a usd to dtmi th Klvi wav lgth λ (a), Klvi dissipatio facto D (b) ad th -foldig dca-lgth of th fist Poica mod L (c). Ths poptis a also dtmid fo a ag of valus of f. P combiatio of A v ad s a valu fo f is th foud which has th smallst absolut diffc with spct to λ, D ad L. Ths sults ca b obsvd i th plot blow. f d 5 Sptmb 9 Pag 35 of 58

36 Modllig th th-dimsioal flow stuctu i smi-closd basis Olav va Dui Figu 9: sults fo f [m/s] fo combiatios of A v ad s b fittig o (a) λ, (b) D ad (c) L It should b otd that th ags chos fo A v ad s i th figu abov a basd o th ag of tpical valus fo th oth Sa i Bsio t al. (8). This ag was btw. ad.5 m /s, but was xtdd to btw slightl abov zo ad almost. m /s (appoximatl tims A v;f ) to show somthig mo of th dvlopmt of f. Th ag potd fo s was btw.5 ad. m/s, but was xtdd to btw slightl abov zo ad m/s bcaus th dvlopmt of f blow. m/s was uit miimal fo th dissipatio facto ad dca lgth plot. It should also b otd that th hoizotal scal is logaithmic. I th figu abov it ca b s that i gal with icasig A v ad s th fictio paamt f bcoms high. This is bcaus fictio ad viscosit both caus g dissipatio ad bcaus th slip paamt also limits th flow thoughout th wat colum b limitig th flow at th bottom. Th Klvi wav lgth ad dissipatio facto plots show a cotiuous dvlopmt of f which is to b ctd as thos paamts a closl latd to dissipatio ffcts. Fo th dca lgth plot a lag combiatio of slip ad viscosit paamts lad to a valu of zo fo f. This mas that ol a iviscid dpth-avagd Poicaé mod ca appoach th dsid 3D dca lgth. This is pobabl bcaus th imagia pat of th Poicaé wav umb ad thb th dca lgth ol lightl spods to chags i viscosit (s Figu 6 though Figu 7). Ol with high viscositis tha about.5 m /s (about.4 tims A v;f ) th dca lgth of th fist Poicaé mod bcoms low tha it was fo zo viscosit (s Figu 8) ad fo somwhat high valus th is mo dvlopmt of f i th figu abov. This mas that th dca lgth is alwas low fo th 3D modl util viscosit is high ough (lagl gadlss of s). With th fc cas mtiod i th itoductio of this chapt it is foud that f uals 4. -3,.7-3 ad m/s fo th fittig paamts (a), (b) ad (c) spctivl. Ev th lagst of ths valus is still a facto 3 small tha that foud with ssio [5.5] ad th th fittd valus thmslvs also diff sigificatl; this at last sigifis that th 4 mthods a distict, which lads to th bst sults dpds o what is dmd most impotat. I th followig paagaphs ad chapts this is lod futh. 5. PROPERTIES OF THE TOTAL SOLUTIO I this chapt poptis of th total solutio a giv i th fom of lvatio ad cut amphidomic sstms. Th solutio is costuctd b us of th collocatio mthod dscibd i paagaph 4.3; this mthod dtmis th lativ amplituds (R fo th outgoig Klvi wav ad η fo all usd Poicaé mods) i such a wa that th is o dpth-avagd omal flow at th 5 Sptmb 9 Pag 36 of 58

37 Modllig th th-dimsioal flow stuctu i smi-closd basis Olav va Dui latal bouda. Th amphidomic sstms a compad to a uivalt dpth-avagd modl with th valus fo f divd bfo. I gal th amplitud is dtmid b movig th tim-dpdt compot of th ssio (fo u, v, w o η) ad takig th absolut valu (complx modulus). This givs a ssio that givs th amplitud p x,-locatio. This ca b s i th ssio blow fo th fictioal paamt p (which ca b placd with u, v, w o η). p; total, ( x ) p ( x, B ) Rp ( x, ) p ( x ), [5.6] Istad of takig th absolut valu also th phas ca b dtmid which ilds th followig ssio. Coodiats with th sam phas ach thi maximum valu at th sam momt i th tidal piod. θ p; total ( x, ) ag p ( x, B ) Rp ( x, ) p ( x, ) [5.7] 5.. ELEVATIO AMPHIDROMIC SYSTEM OF THE TOTAL SOLUTIO I th figu blow th lvatio amphidomic sstm is giv; i this figu th coloud lis sigif th cotous of a ctai tidal lvatio i m whil th black lis dot th phas at which that poit achs its highst lvatio. I gal this tidal patt looks lik what ca b ctd i a smi-closd basi. Figu : amplitud ad phas of η [m] Th o poducd b Davis ad Jos (995) looks simila. Th phas patt has a simila shap but has a phas shift of about 3 dgs. Th amphidomic poits (wh th amplitud is zo) li at oughl th sam locatio ad th amplitud dvlopmt looks v simila as wll. This stgths th tust i pfomac of th modl as thos authos usd a mo complx umical 3D modl. I th followig plots th lvatio amphidomic sstm of th basi is plottd fo th dpth-avagd modl. Fo this th fou valus of f divd i paagaph 5..4 a usd. 5 Sptmb 9 Pag 37 of 58

38 Modllig th th-dimsioal flow stuctu i smi-closd basis Olav va Dui Figu : amplitud ad phas of η [m] i th dpth-avagd modl Fom th amphidomic plots it ca b s that th dpth-avagd modl has widl diffig sults fo th fou valus of f. Thos foud with th Lotz liaizatio ad Klvi dissipatio facto D com ath clos, but thos basd o th Klvi wav lgth ad th fist Poicaé dca lgth do ot. It should b otd that th appoach with th lagst fictio (basd o Klvi wav lgth) gts th highst amplituds bcaus th amplitud of th lvatio (ad vlocitis) of th icomig Klvi wav is fixd at th latal coast fo all vaiats ad icass with distac fom th latal coast. Bcaus fictio is high, th dcas wh movig towads th bouda is lag ad thfo th icas wh movig awa fom th bouda is lag. Thfo th amplituds a th upp coast ad th basi tac (wh th ifluc of th icomig Klvi wav is stogst) bcom v lag. Plots of th lvatio amphidom fo vaig viscosit ad slip paamt a foud i appdix D. 5.. LOGITUDIAL CURRET AMPHIDROMIC SYSTEM I th figu blow th logitudial cut amphidomic sstm is giv (at th bottom). This shows that th cut amphidomic poits a localizd btw th lvatio amphidomic poits, which is bcaus th lvatio gadits a miimal th ad th vlocitis must bcom v small as wll (this latio divs fom th govig uatios). 5 Sptmb 9 Pag 38 of 58

39 Modllig th th-dimsioal flow stuctu i smi-closd basis Olav va Dui Figu : amplitud ad phas of u [m/s] (at th bottom) I th followig plots th logitudial cut amphidomic sstm of th basi is plottd fo th dpth-avagd modl with valus fo f as foud bfo. Figu 3: amplitud ad phas of u i th dpth-avagd modl This shows a simila compaiso of 3D ad DH as fo lvatio. Agai th appoachs basd o Klvi dissipatio ad Lotz liaizatio pfom bst. To compa th 3D appoach with th DH appoach th amplitud ad phas acoss th wat colum at x354 km ad 44km a plottd blow. This locatio was chos bcaus o amphidomic poit appas th ad th figus a just mat to illustat th diffcs btw 3D ad DH fo a utal locatio. 5 Sptmb 9 Pag 39 of 58

40 Modllig th th-dimsioal flow stuctu i smi-closd basis Olav va Dui Figu 4: compaiso of amplitud ad phas of u acoss th chal This figu shows that th is a cla phas ad amplitud diffc btw th DH ad 3D appoachs, but also alog th vtical fo th 3D appoach as wll. This sigifis th impotac of a 3D appoach i dtmiig th pcis stuctu of flow LATERAL CURRET AMPHIDROMIC SYSTEM I th figu blow th latal cut amphidomic sstm is giv (at th bottom). Th amphidomic poits a v difft fom thos of th logitudial cut sstm ad it ca b s that th ifluc of th Poicaé mods (which appa mostl i th lft pat of th basi) is stog tha that of th Klvi wavs (which domiat i th ight pat of th basi). Figu 5: amplitud ad phas of v (at th bottom) 5 Sptmb 9 Pag 4 of 58

41 Modllig th th-dimsioal flow stuctu i smi-closd basis Olav va Dui 6 BOTTOM SHEAR STRESS MODELLIG I A SEMI-ECLOSED BASI I this chapt bottom sha stss is calculatd fo th 3D modl as wll as th DH modl (with th valus fo f foud bfo). Aft that btt valus fo th fictio paamt a sought b fittig th bottom sha stss amphidom of DH to that of th 3D cas b adjustig th fictio paamt. Euatio Sctio 6 6. BOTTOM SHEAR STRESSES OF THE 3D MODEL To calculat th hoizotal bottom sha stss th followig fomula is applid; τ v τ, τ s u, v [6.]. Th amphidomic plot of ths bottom sha stsss b ( ) ( ) b; x b; ca b s blow. total z total z Figu 6: bottom sha stss (m /s ) amphidom i hoizotal dictio (3D modl) Ths plots of cous hav th sam shap as th hoizotal vlocit plots of bfo. Ol th valus diff with a facto of s. I aalog to th vlocit plots it is obsvd that th stsss i x-dictio a gat tha thos i th -dictio i th majoit of th basi. Ol at th latal bouda th latal stsss a gat, bcaus th th logitudial vlocit at th bottom is limitd b th bouda coditio that th ca b o omal dpth-itgatd logitudial vlocit at th bouda. Th sha stsss i th latal dictio a pobabl lagl domiatd b th Poicaé mods, giv th patt. Bcaus th ifluc of th latal sha stss is gall limitd thoughout th basi it is ot tak ito accout i th followig compaiso with th dpthavagd modl. 6. BOTTOM SHEAR STRESSES OF THE DEPTH-AVERAGED MODEL To fid out how th sults of th 3D cas compa with th DH cas sha stss is also calculatd with th dpth-avagd modl. Hi it is assumd that th sultig vlocitis appl at th wat τ v τ τ u, v [6.]. bottom. Th sha stss is th calculatd with ( ) ( ) DH This lads to th followig sults (with th valus fo f foud bfo). x, DH DH f DH DH 5 Sptmb 9 Pag 4 of 58

42 Modllig th th-dimsioal flow stuctu i smi-closd basis Olav va Dui Figu 7: DH sha stss amphidom i x-dictio fittd o Lotz ad wav mod poptis I th figu abov it ca b s that th sha stsss of th DH modl diff littl to a lot with thos of th 3D modl. Th f basd o Lotz lads to 3 tims too low amplituds, f basd o th Klvi wav lgth lads to 3 tims too high amplituds, f basd o th Klvi dissipatio facto lads to about.3 tims too low amplituds ad f basd o th fist Poicaé dca lgth givs o sha stss at all (bcaus f, s [6.]). I gal th DH modl is wak i coctl pdictig sha stsss, though th appoach basd o dissipatio facto D givs a asoabl sult. 6.3 VALIDITY OF DH BOTTOM SHEAR STRESSES To chck how wll th difft DH appoachs match th 3D sha stsss a basi o fuctio is dfid. This fuctio is show i th followig ssio: BE BL ( τ τ ; ) avg x DH xb dx d 3D /. [6.3] H L is th plottig domai usd bfo, which is somwhat small tha th 3D Klvi wav lgth fo th fc cas (s paagaph 5..). Th itgal is tak ov th ti plottig domai, though it is solvd umicall ath tha aalticall. To hav som fcs valus to compa with th DH sults of idividual wav mods ssio [6.3] fo th bottom sha stsss of th full DH Talo solutio is miimizd b fidig th bst fittig valu fo f. Bcaus pvious sults sm to sigif that th Klvi dissipatio facto is a good masu of th full solutios sha stsss th sam is also do fo just th icomig DH Klvi wav (which is of cous compad with th sha stsss causd b th icomig 3D Klvi wav). It should b otd that both sults a just usd as fcs; th sv as a kid thotical bst match btw DH ad 3D bottom sha stsss. Th sults of ths two optimizatios ca b s i th figu blow. 5 Sptmb 9 Pag 4 of 58

43 Modllig th th-dimsioal flow stuctu i smi-closd basis Olav va Dui Figu 8: DH sha stss (m /s ) amphidom i x-dictio fittd o bottom sha stss I th figu abov it ca b s that both mthods lad to simila sults. Th valus foud fo f diff lss tha % ad th amphidomic patts a almost idistiguishabl. This mas that b just lookig at th icomig Klvi wav valus fo f ca b foud that coms v clos to th o foud wh lookig at th full Talo solutio. Th stsss also compa wll to th bottom sha stsss i x-dictio as calculatd fo th 3D modl (s Figu 6). Th sultig idividual wav mod poptis of th 4 DH appoximatios, th DH basi o fuctio optimizatios ad th 3D modl a pstd blow. Also th maximum amplitud calculatd fo th basi (absolut ad as a pctag of th 3D maximum amplitud) ad th valu fo BE avg a giv (absolut ad as a pctag of th 3D maximum amplitud). Basd o [m/s] λ [km] D [-] L [km] Max τ b;x [m/s] Max τ b;x [%] BE avg [m/s] Lotz.4E E E-4 λ 4.E E E-3 58 D.68E E E-4 3 L E-4 4 Klvi i.7e E E-5 3 Talo solutio.8e E E-5 3 3D E Tabl 4: summa of sults of vaious mthods BE avg [%] Th tabl clal shows that th two fcs cass divd i this paagaph wok bst (as ctd, as th a mat to do just that) ad also b thmslvs th wok uit wll (a avag o thoughout th basi of 3%). Also, th diffc btw th o basd o th full Talo solutio ad just th icomig Klvi wav is v small. This futh sigifis that th icomig Klvi wav is a v impotat masu of th full solutio. A laatio fo this is that th icomig Klvi wav itslf of cous dtmis how th outgoig Klvi wav Poicaé mods tu out. Th ifomatio of that icomig wav thfo stogl dtmis th total solutio. Futhmo th ualit of th Klvi dissipatio facto appoach is oc agai cofimd. Its o is sigificat but still asoabl (3%) ad all idividual wav mod poptis cospod wll with th 3D cas. Aft that th Lot liaizatio givs th bst sult (%), followd fom afa b th Poicaé dca lgth appoach (4%) ad Klvi wav lgth appoach (58%). It should howv b otd that th dca lgth appoach i itslf is v wak, as it dfaults to a valu of f m/s i a lot of cass (s 5..4). Its maig is thfo vituall o-xistt, which is cofimd b th fact that it caot v xactl appoach th paamt it was basd o i th fist plac! It dos hav th bst fit (th umbs a oudd dow), but it dos ot hav a xact fit fo its ow paamt lik th wav lgth ad dissipatio facto appoachs do hav. 5 Sptmb 9 Pag 43 of 58

44 Modllig th th-dimsioal flow stuctu i smi-closd basis Olav va Dui 7 DISCUSSIO Th modl pstd i this pap has ctai small ad gat dawbacks. Th chos appoach ca ol b applid to flat bottoms, whil th ifluc of a slopig bottom ca b sigificat (Davis & Jos, 995). As th is o wa to implmt this i a aaltical modl this caot b asil fixd. Howv, th modl pstd still lads to ctd tidal bhaviou so this ma ot b a big poblm i this cas. This also sigifis that th assumptios (idalizd basi shap, log wavs t cta) udlig th divatio of th lia govig uatios ad bouda coditios (statig fom th avi-stoks uatios) w asoabl fo tidal flow i a smi-closd basi. It could b agud that th tubulc modl is too simpl, though it is still a impovmt compad to Mofjld s wok (98). o vaiatio of viscosit is allowd whil that dos occu. But still, th tubulc modl at last givs 3D ifomatio ad ca b futh twak b adjustig th slip paamt so it is pobabl th bst possibl fo a aaltical modl. Th modls pfomac with oth tidal costituts is ufotuatl ukow. Also th validit of som paamts ad spciall th usd fc slip paamt is ukow ad ma b lativl wak. This is o of th asos it was ot compad i gat dtail to mo complx modls o compad at all to actual basis. It was assumd that a limitd umb of Poicaé mods would still lad to good sults. Fo th fc cas a tucatd st of Poicaé mods was usd. I appdix E th lativ amplituds of th outgoig Klvi wav ad Poicaé mods of th 3D ad DH modl ca b foud. This shows that th high Poicaé mods hav a v small cotibutio compad to th low Poicaé mods, which maks th us of a small st of Poicaé mods asoabl. Futhmo this modl was dvlopd ud th assumptio that th Foud umb is v small. Fo th usd fc cas th Foud umb tus out to b. (with η.5 m ad H65m), so that assumptio was asoabl. Also it was assumd that ol log wavs a dalt with. As th Klvi wav lgth tus out to b lag tha km ad th tidal lvatio is.5 m this assumptio was v asoabl. 5 Sptmb 9 Pag 44 of 58

45 Modllig th th-dimsioal flow stuctu i smi-closd basis Olav va Dui 8 COCLUSIO AD RECOMMEDATIOS 8. COCLUSIO w ssios hav b divd fo th 3D flow fild of a Klvi wav icig vtical dd viscosit i a ifiit chal of fiit width with a patial slip coditio at th bottom. With ths ssios th ti flow fild ca b calculatd, which mas th fist sach ustio is aswd. Th poptis of that Klvi wav hav b ivstigatd b dlvig ito th ssios ad th udlig paamts (wav umb, t cta). Th Klvi wav i gal bhavs as ctd; it ics icasd dissipatio with icasd viscosit ad has o dpth-itgatd latal flow. O uctd sult was that though th Klvi wav i gal has dcasig wav lgth with icasig viscosit it still shows icasig wav lgth somtims. This suggsts its popagatig chaact is icasig istad of dcasig. Howv, it should b otd that bcaus of mo apidl icasig dissipatio th t ffct is still that it ics icasig dissipatio. Mofjld s (98) aalsis ad th appoach documtd b Pdlosk (98) w xtdd ad idd Poicaé mods w foud to dscib tidal flow i a ifiit chal of fiit width. Fo this w ssios w divd which vtuall ld to a coditio (foud b couplig th dispsio latio ad a-coastal bouda coditio) that allowd Poicaé mods. B usig a patial-slip coditio istad of a o-slip coditio as do b th two mtiod authos th solutio chagd with spct to th paamts that dscib th vtical dpdc; this holds fo th Klvi wavs as wll as th Poicaé mods. This ca asil b tasfomd back to th ssios fo o-slip wh th slip-paamt is st to ifiit. Th Poicaé mods also ic dissipatio ad a ol activ i a limitd pat of th basi. Ths mods do cotibut to th dpthitgatd latal flow. High Poicaé mods dca ov a shot distac, but icasig viscosit lads to gat dca lgths. B fittig th idividual wav mod poptis (Klvi wav lgth, Klvi dissipatio facto ad th fist Poicaé dca lgth) of a dpth-avagd modl to thos foud fo th 3D modl valus fo fictio paamts w foud fo ags of viscosit ad slip paamt valus. This was applid to a fc basi (th oth Sa) to fid th spcific fictio paamts fo futh aalsis. Also a valu fo th fictio paamt was calculatd fom Lotz liaizatio of th lia fictio law. Ths w usd i th dpth-avagd modl to compa a 3D appoach with a DH appoach. B usig a collocatio mthod th Klvi ad Poicaé mods w combid to simulat tidal flow i a smi-closd basi. This mthod calculats th amplituds of th outgoig Klvi wav ad Poicaé mods so that o dpth-itgatd omal flow is pst at th latal bouda. This uasi-aaltical solutio compas wll to th umical modl sults of Davis & Jos (995). Btw th 3D ad DH vsio of th modl th a sigificat diffcs i amplitud of th lvatio ad cuts. This sigifis th impotac of a 3D appoach. This mas that (as ctd) fo coct sha stss calculatios a 3D appoach is mo appopiat as stss dpds o th cut poptis. Howv, it is possibl to appoximat th sha stsss of th 3D modl with a DH modl asoabl wll. A thoticall lowst avag o of 3% was foud b fittig th ti sha stss patt as a fc. Th fictio paamt foud with th Klvi dissipatio facto lad to a o of 3% ( f was about.7-3 ), whil th o foud with th Lotz liaizatio lad to a o of % ( f was about. -3 ). This is a pomisig sult that ca b impovd futh with th us o iclusio of mo paamts. Espciall paamts that lat to th (dissipatio of) th icomig Klvi wav will b of gat lvac. All i all th costuctio of th fist aaltical 3D-modl that dscibs tidal flow i a smi-closd basi usig Klvi ad Poicaé mods with patial slip has lad to a valuabl ad pomisig istumt i futh impovig 3D ad DH modllig. 5 Sptmb 9 Pag 45 of 58

46 Modllig th th-dimsioal flow stuctu i smi-closd basis Olav va Dui 8. RECOMMEDATIOS Although th modl sms to pfom wll, it ca b futh validatd b doig mo ssitivit chcks ad mo compaiso with th mtiod ad oth authos. Fo istac Mofjld (98) dscibs mo poptis of th foud Klvi wav, such as th bhaviou of th co-phas lis ud ifluc of chagig tidal fuc. Futhmo th bhaviou i th bouda las, th ffcts of a chagig slip paamt o th idividual wav mods ad th tidal llipss ca b studid. Th ifluc of th amplitud of th tidal lvatio at th op d lativ to th maximum dpth ad th ifluc of basi dimsios is foud to b impotat b Wiat (7). Also th abilit of th modl to coctl iclud viscous ffcts ca b chckd b ivstigatig if it poducs th aticlockwis compot of flow (Davis & Jos, 995). This all was ot do i this sach bcaus it was outsid th iitial scop ad could ot b ivstigatd uickl to add it lat. Som poptis that w ivstigatd ca b studid mo thooughl. Fo istac it is itstig to chck how th (cut) amphidomic plots chag wh amogst oths th viscosit ad tidal fuc (i.. difft tidal costituts) a adjustd. Also th latal bottom sha stsss, tidall avagd bottom sha stsss ca b of itst as wll as th (domiat) dictio of ths stsss. A possibl wakss of th modl is its gatl idalizd basi shap ad paamts that a assumd to b costat (.g. th dd viscosit). It is thfo advisd to ivstigat how impotat ths aspcts a fo this paticula modl. This ca b do b (as mtiod bfo) chckig it with mo oth modls, but also b tig to simulat basis that a clos (Gulf of Califoia, s Cabajal & Backhaus, 997) o futh movd fom th idalizd basi shap. Also it is commdd to fid btt valus fo ctai paamts, spciall th slip paamt. Th uasi-aaltical appoach also allows fo th couplig of spaat smi-closd basis with difft dimsios ad paamts. Couplig th oth pat (Davis & Jos, 995) with th South Bight (Roos & Schuttlaas, 9) is possibl ad ma lad to itstig sults. If this woks out wll th modls applicabilit gatl icass. It ma also b possibl to coupl a smiclosd basi lik a stua with a sigl-coast basi lik a shlf sa as do b Y & Gavi (998), but that uis som xta wok fo th sigl-coast basi as that is ot suppotd b th cut vsio of th modl. Th taslatio of 3D poptis to a wll-pfomig DH modl ca b futh impovd. It follows fom th aalsis pstd h that th Klvi wav ad spciall th Klvi dissipatio a a good masu of th total 3D solutio. Oth paamts that sa somthig about th Klvi wav could b usd as th dissipatio facto has b i this stud to fid a wll-fittig fictio paamt. Fo istac th amplitud of bottom sha stss at th latal bouda could b usd o th Rossb dfomatio adius. Mab a combiatio of difft paamts (dissipatio facto combid with bottom sha stss amplitud) i a mixd fittig algoithm ca lad to v btt sults. Espciall i shallow basis o with gatl vig flow it is commdd to also iclud th latal bottom sha stss i th aalsis as that will icas i magitud. 5 Sptmb 9 Pag 46 of 58

47 Modllig th th-dimsioal flow stuctu i smi-closd basis Olav va Dui LITERATURE BESIO, G. (8), P. BLODEAUX, M. BROCCHII, S.J.M.H. HULSCHER, D. IDIER, M.A.F. KAAPE, A.A. ÉMETH, P.C. ROOS & G. VITTORI. Th mophodamics of tidal sad wavs: A modl ovviw. Coastal Egiig (Vol. 55, o. 7-8), pp BOWDE, K.F. (967). Tubult mixig i tidal cuts. Th Phsics of Fluids (), pp. S78-S8. BOWDE, K.F. (978). Phsical poblms of th bthic bouda la. Gophsical Suvs (3), pp CARBAJAL,. & BACKHAUS, J.O. (997). Simulatio of tids, sidual flow ad g budgt i th Gulf of Califoia. Ocaologica Acta (, o. 3), pp DAIELSO, S. & KOWALIK, Z. (5). Tidal cuts i th St. Lawc Islad gio. Joual of Gophsical Rsach (). DAVIES, A.M. & JOES, J.E. (995). Th ifluc of bottom ad ital fictio upo tidal cuts: Talo s poblm i th dimsios. Cotital Shlf Rsach (5, o. ), pp GERKEMA, T. (). A lia stabilit aalsis of tidall gatd sad wavs. Joual of Fluid Mchaics (47), pp HEDERSHOTT, M.C. & SPERAZA, A. (97). Co-oscillatig tids i log, aow bas; th Talo poblm visitd. Dp-Sa Rsach (8), pp HUIJTS, K.M.H. (9), SCHUTTELAARS, H.M., DE SWART, H.E. & FRIEDRICHS, C.T. Aaltical stud of th tasvs distibutio of alog-chal ad tasvs sidual flows i tidal stuais. Cotital Shlf Rsach (9), pp. 89. HULSCHER, S.J.M.H. (996). Tidal-iducd lag-scal gula bd fom patts i a th-dimsioal shallow wat modl. Joual of Gophsical Rsach (, o. C9), pp.,77-,744. HULSCHER, S.J.M.H. & VA DE BRIK, G.M. (). Compaiso btw pdictd ad obsvd sad wavs ad sad baks i th oth Sa. Joual of Gophsical Rsach (6, o. C5), pp. 937, IAIELLO J.P. (98). Tidall-iducd Rsidual Cuts i Log Islad ad Block Islad Souds. Estuai, Coastal ad Shlf Scic (), pp MOFJELD, H.O. (98). Effct of vtical viscosit o Klvi wavs. Joual of Phsical Ocaogaph (), pp PEDLOSKY, J. (98). Gophsical Fluid Damics. w Yok: Spig-Vlag. PRADLE, D. (98). Th vtical stuctu of tidal cuts ad oth oscillato flows. Cotital Shlf Rsach (, o. ), pp PRADLE, D. (997). Th ifluc of bd fictio ad vtical dd viscosit o tidal popagatio. Cotital Shlf Rsach (7, o. ), pp RIEECKER, M.M. & TEUBER, M.D. (98). A ot o fictioal ffcts i Talo s poblm. Joual of Mai Rsach (38), pp ROOS, P.C. & SCHUTTELAARS M. (9, ACCEPTED). Solvig th viscous Talo poblm. Joual of Fluid Mchaics. SIHA, B. & PIGREE R.D. (997). Th picipal lua smidiual tid ad its hamoics: basli solutios fo M ad M4 costituts o th oth-wst Euopa Cotital Shlf. Cotital Shlf Rsach (7, o. ), pp SWART, H.E. DE (8). Oca wavs. Lctu ots. 5 Sptmb 9 Pag 47 of 58

48 Modllig th th-dimsioal flow stuctu i smi-closd basis Olav va Dui TAYLOR, G.I. (9). Tidal oscillatios i gulfs ad ctagula basis. Pocdigs of th Lodo Mathmatical Socit (), pp TEE, K-T. (98). Th Stuctu of Th-dimsioal Tid-gatig Cuts: Expimtal Vificatio of a Thotical Modl. Estuai, Coastal ad Shlf Scic (4), pp WIAT, C.D. (7). Th-dimsioal tidal flow i a logatd, otatig basi. Joual of Phsical Ocaogaph. (37), pp YE, J. & GARVIE R.W. (998). A modl stud of stua ad shlf tidall div ciculatio. Cotital Shlf Rsach (8), pp ZIMMERMA, J.T.F. (98). O th Lotz liaizatio of a uadaticall dampd focd oscillato. Phsics Ltts A (89, o. 3), pp Sptmb 9 Pag 48 of 58

49 Modllig th th-dimsioal flow stuctu i smi-closd basis Olav va Dui LIST OF SYMBOLS Th smbols usd i th mai txt of this thsis ad thi maig ca b foud blow. Scald Dimsioal Uits Maig A - - amplitud of dpth-avagd vlocit at th coast Ah A h m /s hoizotal dd viscosit Av A v m /s vtical dd viscosit B B m basi width C D BE avg avag o thoughout basis btw 3D ad DH C D - viscosit cofficit f f s - Coiolis paamt / atio of itial ad tidal fuc g g m/s gavitatioal costat H H m udistubd wat dpth H - - coctio o icopoatig vtical stuctu (s γ) K K m - tpical iviscid Klvi wav umb k - - Klvi wav umb k - - Poicaé wav umb L L m basi lgth m - - collocatio umb - - Poicaé umb - - usd umb of Poicaé mods - - clockwis otatig vlocit compot Q - - clockwis otatig dpth-dpdc of lvatio - - cout clockwis otatig vlocit compot R - - cout clockwis otatig dpth-dpdc of lvatio f f m/s DH fictio paamt p - - amplitud of plachold paamt p (placabl with u, v, w o η) s - s m/s atio of slip ad vtical viscosit paamt / slip paamt t t s tim u u m/s alogsho hoizotal vlocit U U m/s tpical iviscid Klvi vlocit v v m/s latal hoizotal vlocit w w m/s upwads vtical vlocit W W m/s vtical vlocit scal x x m alogsho hoizotal coodiat m latal hoizotal coodiat m - latal collocatio coodiats z z m upwads vtical coodiat Z - - clockwis otatig dpth-dpdc of hoizotal vlocit Ẑ ^ - - cout clockwis otatig dpth-dpdc of hoizotal vlocit Z ~ - - otatig dpth-dpdc of vtical vlocit - - Klvi wav umb oot paamt - - Poicaé wav umb oot paamt - - clockwis otatig oot paamt - - cout clockwis otatig oot paamt γ - - coctio o f icopoatig vtical stuctu (s H ) δ h - - atio of hoizotal bouda la ad tpical iviscid Klvi wav lgth 5 Sptmb 9 Pag 49 of 58

50 Modllig th th-dimsioal flow stuctu i smi-closd basis Olav va Dui δ v - - sua of Stoks umb ε - - Foud umb η η m sa lvl lvatio η - - latal dpdc of lvatio amplitud ηˆ - - latal dpdc of Poicaé vlocit amplitud η η m lvatio amplitud facto θ θ latitud θ p - - phas of plachold paamt p (placabl with u, v, w o η) σ σ s - tidal fuc Ω Ω s - agula fuc of th Eath's otatio Tabl 5: smbols usd i this thsis 5 Sptmb 9 Pag 5 of 58

51 Modllig th th-dimsioal flow stuctu i smi-closd basis Olav va Dui APPEDICES A. MODIFYIG THE AVIER-STOKES EQUATIOS AD BOUDARY CODITIOS It is assumd that th flow is icompssibl. Thfo th cotiuit uatio ducs to th followig bcaus dsit ρ ow is costat, th dsit divativ bcoms zo ad th dsit paamt ca b dividd fom th oigial uatio: Euatio Sctio u v w. [App.] x z Icompssibl flow coupld with th assumptio of costat damic viscosit µ ad gavit maks that th momtum uatios duc to th followig: ( u uu vu wu ) p ( u u u t ) x z x xx zz ( v uv vv wv ) p ( v v v t ) x z xx zz ( w uw vw ww ) p ( w w w t ) x z z xx zz ρ µ, [App.] ρ µ, [App.3] ρ µ. [App.4] Bcaus th wat dpth is small compad to tidal wav lgth (log wavs) vtical vlocit w is lativl small, as a its divativs ad th momtum uatio fo w ducs to th followig: ( ) ρ g p p ρ g H η, [App.5] z which is th ssio fo hdostatic pssu. This ca b put back i th momtum uatios fo u ad v to fid th followig (H dops fom th uatio bcaus its divativ is zo): ( u uu vu wu ) g ( u u u t ) x z x xx zz ( v uv vv wv ) g ( v v v t ) x z xx zz ρ ρ η µ, [App.6] ρ ρ η µ. [App.7] B dividig ths uatios with th dsit paamt ad placig µ/ρ with th kimatic viscosit A th followig ssios a foud: ( u ) ( ) uu vu wu g η A u u u t x z x xx zz ( v ) ( ) uv vv wv g η A v v v t x z xx zz, [App.8]. [App.9] Th viscosit must also icopoat additioal supamolcula tubulc du to th atu of tidal flow (dd viscosit). Ths additioal tubulc ffcts a difft fo vtical ad hoizotal movmt bcaus th hoizotal ad vtical scals a v difft, so hoizotal ad vtical viscosit paamts a distiguishd. ( ) ( ) u uu vu wu g η A u u Au, [App.] t x z x h xx v zz v uv vv wv gη A v v Av. [App.] t x z h xx v zz Hoizotal viscosit is tpicall small compad to vtical viscosit ad oth paamts, so th hoizotal viscosit tms hav a limitd cotibutio. Futhmo th lgth scal hoizotal viscosit opats o is v lag whil vtical viscosit opats o a limitd lgth scal (wat dpth). Th vaiatio alog th hoizotal viscosit scal is much small tha that of th vtical 5 Sptmb 9 Pag 5 of 58

52 Modllig th th-dimsioal flow stuctu i smi-closd basis Olav va Dui viscosit scal, th vlocit divativs a small ad ths tms a thfo doppd fom th ssios. u uu vu wu g η Au, [App.] t x z x v zz v uv vv wv gη Av. [App.3] t x z v zz Th flow is modlld o th f-pla which lads to th followig ssios (icludig a Coiolis tm ad a Coiolis paamt which is costat): u u u v u w u f v g t x z x η A u, [App.4] v u v v v w v f u g t x z η A v. [App.5] v z z v z z Expssios [App.], [App.4] ad [App.5] a usd as th statig uatios i paagaph... 5 Sptmb 9 Pag 5 of 58

53 Modllig th th-dimsioal flow stuctu i smi-closd basis Olav va Dui B. RESCALIG OF SEVERAL PARAMETERS AD VARIABLES Som paamts hav to b mad dimsioal agai to suppl dimsioal sults i th mai txt of this thsis. Th ssios fo ths paamts div fom makig th govig ad divd uatios dimsioal agai. [App.6] H [App.7] H H H H [App.8] γ γ H [App.9] k k K [App.] k k K [App.] 5 Sptmb 9 Pag 53 of 58

54 Modllig th th-dimsioal flow stuctu i smi-closd basis Olav va Dui 5 Sptmb 9 Pag 54 of 58 C. THE DEPTH-AVERAGED MODEL Th followig ssios to dscib dpth-avagd tidal flow i a ifiit chal a tak fom Rick & Tub (98) ad a adjustd to most closl smbl th 3D ssios divd fo this sach. Th ssios pstd h a dimsioal, as ca b s b th us of astisks lik i th mai txt of this thsis. Th govig uatios th ssios fo displacmt ad logitudial vlocit a basd o th lia dpth-itgatd shallow wat momtum ad mass uatios dscibd blow: H u g v f u x t η, [App.] H v g u f v t η. [App.3] ( ) t x H u v η. [App.4] Th followig bouda coditios also appl. η H dz v at, B [App.5] η H dz u at x [App.6] Th ssios fo η ad u ad th collocatio appoach fd to i th mai txt a pstd blow. C.I. DISPLACEMET Th total displacmt is calculatd as follows. ( ) ( ) ( ) ( ) ( ) ( ) ( ) si cos B k f B k if t x k i t x k i R B t x k i π π β β σ η σ η σ η η, [App.7] with th wl itoducd paamts: ( ) β β σ π β σ β σ β f H g i k k k B i H g i k k if i H f. [App.8] H f is th ol all w paamt; it is th fictio paamt i m/s. C.II. VELOCITY Th total logitudial vlocit is calculatd with:

55 Modllig th th-dimsioal flow stuctu i smi-closd basis Olav va Dui 5 Sptmb 9 Pag 55 of 58 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) si cos B B k if t x k i k ig B t x k i f g R B t x k i f g U π φ π β σ η σ η σ η, [App.9] with k H g if σ φ. [App.3] C.II.i VELOCITY FOR COLLOCATIO As fo th 3D modl i paagaph 4.3 h th ssios to dtmi th amplituds of th wav mods a giv. ( ) ( ) ( ) si cos B f B B k if k i f R π φ π β η [App.3] at m fo,,..., m [App.3] Th uivalt i matix fom ilds: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ; ; ; ; ; ; ; ;,,,,,,,,,,,, i K i K i K i K P P P out K P P P out K P P P out K P P P out K A A A A R A A A A A A A A A A A A A A A A L M M M K O K K K M M η η η,[app.33] with ( ) ( ) ( ) ; ; si cos B B k if ik A f A B f A P out K i K π φ π β. [App.34] Th amplitud vcto is th dtmid usig stadad tchius i Th Mathwoks TM MATLAB.

56 Modllig th th-dimsioal flow stuctu i smi-closd basis Olav va Dui D. ELEVATIO AMPHIDROMY WITH VARYIG VISCOSITY AD SLIP PARAMETERS I th figu blow th lvatio amphidom ud ifluc of chagig viscosit paamt ca b s (at A v /A v;f ). Figu 9: amphidomic sstm of η [m] fo vaig viscosit This figu clal shows a shift of th amphidomic poits. This is bcaus dissipatio is stog ad thfo th lativ ifluc of th icomig Klvi wav (as opposd to th outgoig) is stog alog th latal dictio, ad th li th amphidomic poits li o otats towads th low coast. I th followig figu th lvatio amphidom ud ifluc of chagig slip paamt ca b s (at s/s f ). 5 Sptmb 9 Pag 56 of 58

57 Modllig th th-dimsioal flow stuctu i smi-closd basis Olav va Dui Figu : amphidomic sstm of η [m] fo vaig slip paamt This figu slows a simila shift of amphidomic poits with icasig slip paamt, though th ffct is lss pooucd tha with a chagig viscosit paamt. This sigifis that th lativ magitud of th viscosit paamt is mo impotat i dtmiig th lvatio patt. 5 Sptmb 9 Pag 57 of 58