SHOALING OF NONLINEAR INTERNAL WAVES ON A UNIFORMLY SLOPING BEACH

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1 SHOALING OF NONLINEAR INTERNAL WAVES ON A UNIFORMLY SLOPING BEACH Ke Yamasa Taro Kaknuma and Kesuke Nakayama 3 Te nernal waves n e wo-layer sysems ave been numercally smulaed by solvng e se o nonlnear equaons n consderaon o bo srong nonlneary and srong dsperson o waves. Aer e comparson beween e numercal resuls and e BO solons as well as e expermenal daa e nernal waves propagang over e unormly slopng beac are smulaed ncludng e cases o e mld and long slopes. Te nernal waves sow remarkable soalng aer e nerace ouces e crcal level. In e lower layer e orzonal velocy becomes larger an e local lnear celery o nernal waves n sallow waer jus beore e cres peak and e poson s dened as e wave-breakng pon wen e rao o nonlnear parameer o beac slope s large. Te rao o nal wave eg o wave-breakng dep becomes larger as e slope s mlder and e wave nonlneary s sronger. Te wave eg does no ncrease so muc beore e wave breakng on e mldes slope. Keywords: nonlnear nernal wave; wo-layer sysem; varaonal prncple; soalng; wave-breakng pon INTRODUCTION Wen densy sracaon s well developed n a nearsore zone nernal waves propagae aecng e waer envronmen. Especally n sallow waer regons no only nernal long-perod waves e.g. nernal seces and des bu also nernal sor-perod waves are observed w wave nonlneary and dsperson. Inernal waves propagang over a slope ave been suded roug ydraulc expermens (e.g. Helrc 99) as well as varous eores ncludng e Benjamn-Ono (BO) equaon consderng bo weak nonlneary and weak dsperson o waves. In e presen sudy a se o nonlnear nernal wave equaons derved on e bass o e varaonal prncple wou any assumpons on wave nonlneary and dsperson (Kaknuma ) s numercally solved n e vercal wo-dmenson. In e dervaon process o equaons e velocy poenal s expanded no a power seres o vercal poson aer wc e velocy poenal s approxmaed usng only several erms o e power seres n numercal compuaon. I as been conrmed a e numercal model sows good resuls n comparson w expermenal daa over a la seabed (Yamasa e al. ) wen e number o erms or e velocy poenal s sucen. Frs e numercal model o nernal waves s vered: n deep-waer cases compuaonal resuls o nerace proles up o eac order on e vercal leng scale o moon are compared w e eorecal soluons o e BO equaon; on e oer and n sallow-waer cases numercal calculaon resuls obaned roug e presen model are compared w ose roug e ully nonlnear model or long nernal waves (Co and Camassa 999) as well as e exsng expermenal daa (Horn e al. ). Second pyscal varables are evaluaed as nernal waves are approacng o wave-breakng pons o nvesgae caracerscs o nonlnear nernal waves propagang on unormly slopng beaces o mld and long slopes wc are dcul o be represened n a laboraory ank. NORMALIZED EQUATIONS FOR NONLINEAR INTERNAL WAVES In wo-layer densy sracaon o nvscd and ncompressble luds beween wo xed orzonal plaes e sll-waer ckness o e -layer s denoed by (x). None o e luds mx even n moon and e densy ρ (ρ < ρ ) s spaally unorm and emporally consan n eac layer. Surace enson and capllary acon are negleced. Flud moon s assumed o be rroaonal suc a e velocy poenal φ s expanded no a seres n erms o a gven se o vercally dsrbued uncons Z mulpled by er wegngs as φ N ( x z ) { Z ( z) ( x ) } Z () were e sum rule o produc s adoped or subscrp. Nonlnear nernal wave eqaons based on e varaonal prncple are as ollows (Yamasa e al. ): Dvson o Naural Scence Graduae Scool o Scence and Engneerng Kagosma Unversy --4 Kormoo Kagosma Kagosma Japan Do 3 Deparmen o Cvl and Envronmenal Engneerng Kam Insue o Tecnology 65 Koen-co Kam Hokkado Japan.

2 COASTAL ENGINEERING ( ) () ( ) { } ( ) b b (3) ( ) ( ) g g ρ ρ (4) were ( ) x y.e. a paral derenal operaor n e orzonal plane; (x) b(x) and g are nerace dsplacemen seabed poson and gravaonal acceleraon respecvely. Te pyscal varables are normalzed as ( ) σ k k k k a a g b b g g z z y y x x (5) were and a are caracersc waer dep waveleng and wave eg respecvely. Tese equaons are subsued no Eqs. () (3) and (4) resulng n ( ) ( ) { } ( ) σ (6) ( ) ( ) σ b b (7) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ). σ ρ ρ σ (8) NUMERICAL CALCULATION METHOD Two-layer problems are solved n vercally wo-dmensonal cases. Equaons (6) (7) and (8) are rewren o ne derence equaons aer wc e me developmen s carred ou by applyng mplc scemes smlar o a o Nakayama and Kaknuma (). COMPARISON BETWEEN NUMERICAL RESULTS AND THEORETICAL SOLUTIONS Deep-Waer Cases We compare compuaonal resuls o nerace proles w eorecal soluons roug e BO equaon. In e dervaon process o e BO equaon e dep o one layer s assumed o be nnely deep. In s sudy s assumed a e lower layer ckness s muc larger an a o e upper layer as sown n Fg. and a/ O( /) <<. Te BO equaon (Benjamn 966; Ono 975) s

3 COASTAL ENGINEERING 3 Fgure. Two-layer sysem beween wo xed orzonal plaes. Fgure. Inal nerace proles (Te nal wave eg s equal o a.). c x x x c c H were H( /x ) s Hlber ransorm o /x. Te eorecal soluon o a solary wave obaned by e BO equaon.e. a BO solon s a (9) BO ( x ) () ( x - c) 4c ρ ρ 3c c ρ c c ac BO c g c c. () 4 a c ρ ρ BO In numercal compuaon e nal nerace proles are sown n Fg. were e nerace dsplacemen (x) a BO /(x BO ) ; BO s e caracersc waveleng o BO solon; e nal oal dep ckness rao / and densy rao ρ /ρ are. m 99. and. respecvely. Tere s a vercal wall o perec relecon a x m. Te grd wd x and e me-sep nerval are equal o. m and.5 s respecvely. Tme varaon o e rao o wave eg o upper layer ckness ' and e rao o upper layer ckness o waveleng σ' are sown n Fgs. 3(a) and 3(b) were e nal wave eg s equal o.5. and. ; C s celery o lnear nernal wave n sallow waer; e raos ' and σ' are dened by Koop and Buler (98) as roug σ () ( ) dx roug x were roug and x roug are nerace dsplacemen and orzonal poson o e rs roug o nernal wave respecvely. I a.5 and N 3 4 or 5 en ' and σ' ave become seady wen C / > ; on e oer and a. and N 4 or 5 en ' and σ' ave become relavely seady wen C / >

4 4 COASTAL ENGINEERING Fgure 3. Tme varaon o e rao o wave eg o upper layer ckness (a) and e rao o upper layer ckness o waveleng (b) were C s e celery o lnear nernal wave n sallow waer. 3. In e vercaon o e model e saonary waves sould be compared w BO solons. In Fg. 4 numercal resuls o nerace proles a C / 33 are compared w ose o e correspondng BO solons were a.5 and a roug. Wen N < 4 e nerace graden s seeper an a o e BO solon. On e oer and wen N 4 or 5 e nerace prole obaned roug e presen model s n armony w a o e BO solon. Wen N 5 ' σ' and e rao o lower layer ckness o waveleng σ' are equal o.45. and.99 respecvely a C / 33. Te resul roug e lnearzed presen model (lnear model) s remarkably dsnegraed due o e wave dsperson. Fgures 5(a) and 5(b) sow vercal dsrbuons o orzonal velocy n e upper layer above e roug o nernal wave u roug and a n e lower layer below e roug u roug respecvely were a.5 and C / 33. Te dsrbuons o u roug are nearly unorm snce e waveleng s muc larger an e upper layer ckness and e wave nonlneary s no so srong. Wen N > e dsrbuons o u roug sow large curvaure wc means a e nernal waves propagae n raer deep waer. Calculaon resuls o nerace proles a C / 33 are compared w ose o e correspondng BO solons n Fg. 6 were a.. Wen N 5 ' σ' and σ' are equal o..3 and.97 respecvely a C / 33. In s case were e wave nonlnery s sronger (a/.) e wave proles obaned roug e presen model are smlar o ose o e BO solons

5 COASTAL ENGINEERING 5 Fgure 4. Inerace proles a C / 33 were / 99. and a.5. Fgure 5. Vercal dsrbuons o orzonal velocy n e upper layer above e roug o nernal wave u roug and a n e lower layer below e roug o nernal wave u roug. wen N > 3. However e nerace graden roug e presen model s mlder an a o e BO solon as sown n Fg. 7 wc means a e presen model consders wave nonlneary beer an e BO equaon snce e nerace graden o srongly nonlnear wave s mlder an a roug e eores or weakly nonlnear waves as e KdV eory (Nakayama and Kaknuma ). Fgure 8 sows e numercal calculaon resuls o wave proles a C / 33 n comparson w ose o e correspondng BO solons n a deeper case were. m / 999. and e densy rao ρ /ρ s.; e nal nerace dsplacemen (x) a BO /(x BO ) and a.5 ; e grd wd x and e me-sep nerval are equal o. m and.5 s respecvely.

6 6 COASTAL ENGINEERING Fgure 6. Inerace proles a C / 33 were / 99. and a.. Fgure 7. Inerace proles a C / 33 were / 99. and a.. Fgure 8. Inerace proles a C / 33 were / 999. and a.5.

7 COASTAL ENGINEERING 7 (a) Dmensons o e laboraory ank. Fgure 9. Scemac o anks. (b) Inal condon were e nerace s nclned lnearly w e angle θ n e orzonal ank. Fgure. Tme varaon o nerace dsplacemen measured a Poson C n e ydraulc expermen and ose roug e presen model were e number o erms or expanded velocy poenal N s equal o ree and e ully nonlnear model by Co and Camassa (999). Te proles o BO solons are muc deren rom e calculaon resuls were larger number o erms or e velocy poenal are requred o represen e wave dsperson more accuraely. Sallow-Waer Case Horn e al. () perormed ydraulc expermens usng a ank were e leng L dep D and wd W were 6..9 and.3 m respecvely as sown n Fg. 9(a). Tree ulra-sonc wave gauges were se a e Posons A B and C. Te ank was lled w a wo-layer sracaon were /D.8. A e begnnng o expermens e led ank were e l angle was θ around e axs o roaon was reurned o a orzonal poson quckly aer wc nernal waves raveled n e wolayer sysem beween wo xed orzonal plaes. In e nal condon o numercal compuaon e ank s orzonal and e nerace s nclned lnearly as sown n Fg. 9(b); e nal velocy poenal s assumed o be zero roug e compuaonal doman. Te grd wd x and e me-sep nerval are equal o.6 m and. s respecvely. Te nerace dsplacemen a Poson C n Fg. 9(a) measured n e expermen s sown n Fg. (a) wle e correspondng calculaon resul roug e presen model were N 3 and e ully nonlnear model or long nernal waves by Co and Camassa (999) (e CC model) were O( ) and O(σ ) 4 << are sown n Fg. (b) n e case were e densy rao ρ /ρ and l angle θ are.9 and.467 respecvely. Te calculaon resul roug e presen model represens e accurae wave perods aloug e wave eg roug e numercal models s oo large because e numercal compuaon does no consder vscosy o e luds and rcon a e nerace. Wen > 4 s e number o cress roug bo e expermen and e presen model s equal o sx wereas a roug e CC model s ve. Te nerace proles a 8 s obaned usng e presen model were N 3 or 4 e CC model and a Boussnesq-ype model were O( ) O(σ ) << are sown n Fg.. Te wave eg roug e presen and e CC models s larger an a roug e Boussnesq-ype weakly nonlnear model. On e oer and e waveleng roug e presen model s sorer an a roug e CC model and e Boussnesq-ype weak dsperson model.

8 8 COASTAL ENGINEERING Fgure. Inerace proles roug e presen model were N 3 and 4 e ully nonlnear model by Co and Camassa (999) and e Boussnesq-ype model a 8 s. Case A: ρ /ρ. /. a /. and s.3 Fgure. Inal nerace prole over a unormly slopng beac. NUMERICAL SIMULATION OF INTERNAL WAVES OVER A UNIFORMLY SLOPING BEACH Denon o Wave-Breakng Pon Helrc (99) perormed ydraulc expermens on bo breakng and run-up o nernal solary waves on a unorm slope usng a waer basn were e beac slope s was larger an.3. In acual cases e beac slope s or example equal o. n a lake or. on e sea sore. In e presen paper numercal smulaon o nernal solary waves propagang on a unormly slopng beac s conduced or e cases sown n Tab. ncludng cases o mld and long slopes wc are dcul o be represened n a laboraory expermen. Te calculaon doman s sown n Fg.. Te case were ρ /ρ / a / and s are... and.3 respecvely s reered o as Case A. Te ncden waves are e rd order eorecal solary waves. Te condons were e densy rao ρ /ρ.4 correspond o e expermenal condons by Helrc (99). Te laeral boundary condon s e perec relecon a a vercal wall. Te numercal calculaon sops wen e nerace ouces e boom. Te number o erms n expanded velocy poenal N s equal o ree. Te grd wd x and e me-sep nerval are equal o.5 m and.5 s respecvely. Numercal resuls o nerace proles n Case A s sown n Fg. 3(a) were e dased lne sows e crcal level.e. e lowes poson or an nerace o nernal solary wave o be able o appear. Te crcal level z c s deermned by e KdV eory as

9 COASTAL ENGINEERING 9 Fgure 3. Tme varaon o nerace proles were e dased lne sows e crcal level obaned roug e KdV eory and e me varaon o e rao o wave eg H o upper layer ckness (ρ /ρ. /. a /. s.3). Fgure 4. Inerace prole orzonal velocy o waer parcles n e vcny o e nerace n e -layer u and vercal acceleraon o waer parcles n e vcny o e nerace n e -layer Dw /D were C s e local celery o lnear nernal wave n sallow waer (ρ /ρ. /. a /. s.3). z b ( ρ ρ ). c (3) Accordng o Fg. 3(a) wen e neraces reac e crcal level e nernal waves propagang on e slope begn nclned backward aer wc e nernal waves sow remarkable dsnegraon. Tme varaon o e rao o wave eg o upper layer ckness n Case A s sown n Fg. 3(b) were e wave eg H s e vercal dsance beween e nerace levels a e rs roug and e rs cres. Te rao ncreases gradually o ave a peak a g 3s aer wc decreases. Te pyscal varables a 3.5 s ncludng e nerace prole e orzonal veloces o waer parcles n e vcny o e nerace n e -layer u and e vercal acceleraon o waer parcles n e vcny o e nerace n e -layer Dw /D are sown n Fg. 4 were C and are local celery o lnear nernal wave n sallow waer and (ρ ρ )/ρ respecvely. Accordng o e gure u exceeds C jus beore e cres peak were e poson can be deermned as e wavebreakng pon. I Dw /D becomes larger an g beore u exceeds C were e poson were Dw /D becomes larger an g s dened as e wave-breakng pon. On e oer and n e cases were /s < 3.5 e nerace ouces e boom beore e nernal waves break; ese cases are ploed and ndcaed as No breakng n Fg. 5 were e nernal-wave-breakng pons are summarzed usng a nonlnear parameer proposed by Agsaee e al. ():.5a. Te amplude o ndcaes e sreng o e wave nonlneary. Accordng o Fg. 5 nernal waves break beore e nerace ouces e boom wen /s > 3.5 wle nernal waves break aer e (4)

10 COASTAL ENGINEERING nerace ouces e boom wen /s < 3.5. Fgure 5. Condons o nernal-wave breakng. Pyscal Varables near a Wave-Breakng Pon Fgure 6 sows e relaonsp beween e rao o nal wave eg a o wave-breakng dep n e lower layer BP and e rao o nonlnear parameer o slope s were e sux BP ndcaes e varable a a wave-breakng pon. Te calculaon resuls are compared w e expermenal daa by Helrc (99) n Fg. 6 accordng o wc e rao a / BP s underesmaed usng e presen model wc means a e wave-breakng dep n e lower layer BP s overesmaed. In e compuaon e rcon s no consdered a e nerace and e ncden waves are assumed o be e rd order eorecal solary waves wc may leads o suc derence beween e calculaed and expermenal resuls. Te endency o e daa s owever smlar were e larger e rao /s s e larger e rao a / BP becomes. Te relaonsp beween /s and a / BP s sown n Fg. 7 were e numercal resuls compensae e expermenal daa obaned by Helrc (99) n consderaon o nernal waves over e mlder and longer slopes. Accodng o e gure a / BP becomes larger as e slope s smaller and e wave nonlneary s sronger. Te relaon beween e slope and e rao o wave eg H BP o wave-breakng dep BP s sown n Fg. 8 were H BP s e vercal dsance o nerace levels beween a e rs roug and e rs cres a e wave-breakng pon. Te value H BP / BP becomes larger.e..9.8 wen e slope s mlder.e. s.3.. Fgure 9 sows e amplcaon acor o e wave eg a e wave-breakng pon H BP /a. On e seeper slopes nernal waves sow wave breakng beore e wave eg s ampled. On e mldes slope e wave eg does no ncrease so muc n e wave soalng because o e wave dsperson roug e longer dsance ravel were e energy o nernal waves as been provded o e waves n e ollowng wave ran as sown n Fg. were ρ /ρ / a / and s are...3 and. respecvely. On e oer and e wave eg s ampled remarkably wen s s equal o. o.4. Te mnmum nerace graden o e rear ace o e rs roug a e wave-breakng pon BP /x mn s sown n Fg.. On e mldes slope were s. e rear ace o e rs roug a e wave-breakng pon s mlder an a o nernal waves propagang on e seeper slopes. CONCLUSIONS Te nernal waves n e wo-layer sysems were numercally smulaed by solvng e se o nonlnear equaons n consderaon o bo srong nonlneary and srong dsperson o waves. Aer e vercaon o e numercal resuls n comparson w e BO solons as well as e exsng expermenal daa e nernal waves propagang over e unormly slopng beac were smulaed ncludng e cases o e mld and long slopes wc were dcul o be represened n a laboraory

11 COASTAL ENGINEERING Fgure 6. Locaon o wave-brakng pons roug e presen model were N 3 and e expermens by Helrc (99). Fgure 7. Locaon o wave-brakng pons roug e presen model were N 3 and e expermens by Helrc (99). ank. Te nernal waves sowed remarkable soalng aer e nerace ouced e crcal level. In e lower layer e orzonal velocy became larger an e local celery o lnear nernal waves n sallow waer jus beore e cres peak and e poson was dened as e wave-breakng pon wen e rao o nonlnear parameer o beac slope /s was larger an 3.5. Te rao o nal wave eg o wave-breakng dep a / BP became larger as e slope was mlder and e wave nonlneary was sronger. On e mldes slope were e slope was equal o. e wave eg dd no ncrease so muc beore wave breakng because o e wave dsperson roug e longer dsance ravel were e energy o nernal waves ad been provded o e waves n e ollowng wave ran.

12 COASTAL ENGINEERING Fgure 8. Relaon beween e beac slope s and e rao o wave eg a e wave-breakng pon H BP o wave-breakng dep BP. Fgure 9. Relaon beween e beac slope s and e amplcaon acor o wave eg H BP/a. Fgure. Inerace prole a 37.3 s (ρ /ρ. /. a /.3 s.).

13 COASTAL ENGINEERING 3 Fgure. Relaon beween e beac slope s and e mnmum nerace graden o e rear ace o e rs roug a e wave-breakng pon. REFERENCES Agsaee P. L. Boegman and K. G. Lam.. Breakng o soalng nernal solary waves Te Journal o Flud Mecancs Benjamn T. B Inernal waves o permanen orm n luds o grea dep Te Journal o Flud Mecancs Co W. and R. Camassa Fully nonlnear nernal waves n a wo-lud sysem Te Journal o Flud Mecancs Helrc K. R. 99. Inernal solary wave breakng and run-up on a unorm slope Te Journal o Flud Mecancs Horn D. A. L. G. Redekopp J. Imberger and G. N. Ivey.. Inernal wave evoluon n a spaceme varyng eld Te Journal o Flud Mecancs Kaknuma T.. A se o ully nonlnear equaons or surace and nernal gravy waves Proceedngs o 5 Inernaonal Conerence on Compuer Modellng o Seas and Coasal Regons WIT Press Koop C. G. and G. Buler. 98. An nvesgaon o nernal solary waves n a wo-lud sysem Te Journal o Flud Mecancs 5-5. Nakayama K. and T. Kaknuma.. Inernal waves n a wo-layer sysem usng ully nonlnear nernal-wave equaons Inernaonal Journal or Numercal Meods n Fluds Ono H Algebrac solary waves n sraed luds Journal o e Pyscal Socey o Japan Yamasa K. T. Kaknuma and K. Nakayama.. Numercal analyses on propagaon o nonlnear nernal waves Proceedngs o 3nd Inernaonal Conerence on Coasal Engneerng ASCE waves. 4-5.