An Integrated Prediction Model of Wave Groups and their Associated Long Waves in the Coupling Field of Surf and Breaking Zones on Bar- Type Beaches

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1 Joural of Coastal Researh SI ICS9 (Proeedigs Portugal ISSN A Itegrated Preditio Model of Wave Groups ad their Assoiated Log Waves i the Couplig Field of Surf ad reakig Zoes o ar- Type eahes M.R. Akarpour Jaat ad T. Asao Dept. of Marie Egieerig ad Dept. of Oea Civil Tehology Iraia Natioal Egieerig Kagoshima Ceter for Oeaography Tehra Uiversity --4 Korimoto 4889 IRAN Kagoshima JAPAN akarpour@io.a.ir asao@oe.kagoshima-u.a.jp ASTRACT AKARPOUR JANNAT M.R. ad ASANO T. 9. A Itegrated Preditio Model of Wave Groups ad their Assoiated Log Waves i the Couplig Field of Surf ad reakig Zoes o ar-type eahes SI 56 (Proeedigs of the th Iteratioal Coastal Symposium Liso Portugal ISSN This study ivestigates the ouplig field of grouped wid waves ad their assoiated log waves i the surf ad reakig zoes over liear shore parallel ars. Whereas the SYMONDS model o log wave geeratio y time varyig reakpoit has ee eteded to the ar-trough topography. To keep the prolem as simple as possile the model is oe dimesioal preludig edge wave solutios. Stadig wave solutio etwee the shorelie ad the mea reak poit over the ar are otaied ad two possile resoae oditios are idetified. y takig travel time ito osideratio the Symods model has ee modified ad eteded to the ar-trough topography. I additio the respose depeds o two parameters related to the iidet wave forig the mea wave height ad its modulatio are disussed. The results have show that a amplifiatio i the log wave respose ours whe the stadig wave has a atiode i elevatio at the ar est. It was show that early fully modulated waves have importat role o the log wave ru-up height at the shorelie. ased o the alulated wave fields the otriutios of the log waves o the sedimet trasport effiiey are disussed. The Shields parameters uder the assoiated log waves are evaluated to assess the moility of the ottom sedimet. The results show that the log waves have greater sedimet trasport effiiey ear the shorelie. ADITIONAL INDEX WORDS: Wave groups reakig zoe Ifragravity waves Shields parameter INTRODUCTION I a trai of irregular wid waves the high ad low waves usually appear i groups. The eistee of the wave groupigs will idue seodary waves with a period orrespodig to that of the groupigs. The typial period of the log waves is aroud s. The log waves sometimes alled ifragravity waves were first otied i the earshore area y MUNK (949 ad TUCKER (95. Durig a storm of oe or two days eah erosio ours rapidly with sad trasport from the foreshore eah to the offshore. Formerly wid waves were osidered to e the mai eteral fores of eah erosio. Although the larger waves reak further offshore makig the surf zoe wider the wave heights i the ier surf zoe remai the same as those durig o-storm oditios eause the wave heights after reakig are limited y the water depth. Therefore it is diffiult to attriute the arupt eah erosio solely to the wid waves. I otrast ifragravity waves do ot reak i the surf zoe ad reah their maimum height at the shorelie (AKARPOUR JANNAT ad ASANO 7. KOTAH ad YANAGISHIMA (99 ased o their field measuremets reported that the ifragravity waves idued y grouped wid waves play a sigifiat role o the eah erosio. So far a umer of aalytial models have ee developed to desie the geeratio mehaisms of ifragravity waves (LONGUET-HIGGINS ad STEWART 96; SYMONDS et al. 98. O a atural eah however the miture of wave modes; free waves ad oud waves ompliates the disussio of the geeratio mehaism. LONGUET-HIGGINS ad STEWART (96 showed the presee of a set-dow wave or surf eat fored y the radiatio stress variatios due to alteratig high/ low waves. As oud log waves propagate to the shallow water regio they start to lag ehid the wave groups ad the free waves are released whe their short arrier waves reak. The free waves fially reah the shorelie ad reflet ak offshore. SYMONDS et al. (98 eplaied the log wave geeratio y a time-varyig reakpoit mehaism. They osidered that the wave groupig is destroyed y the reakig ad afterward the wave height deays uder the ostraits of the loal water depth. The reakpoit varies over the period of the grouped waves eause higher waves i the group will reak further offshore tha lower waves. The time-varyig reakpoit ats as a wave maker iduig a slow variatio of the set-up ad geeratio of free log waves propagatig oth i offshore ad oshore diretios. METHODS I the preset study results from a theoretial model of log wave geeratio y a time varyig reak poit over liear shore parallel ars are preseted. We are partiularly iterested i the effet of ear-shore ar topography o the geeratio of low Joural of Coastal Researh Speial Issue

2 A Itegrated Preditio Model of Wave Groups ad their Assoiated Log Waves U S g ( t h ( hu t ( Figure. Shemati represetatio of iidet wave height (top over arred topography frequey waves. To keep the prolem as simple as possile the model is oe dimesioal preludig edge wave solutios. Stadig wave solutio etwee the shorelie ad the mea reak poit over the ar are otaied ad two possile resoae oditios are idetified. y takig travel time ito osideratio the Symods model has ee modified. With usig the oepts of travel time ad radiatio stress a Fourier represetatio of the forig have ee derived. The umer of itial parameters suh as the water depth over the ar ad the ratio of water depth at ar est to water depth at ar trough were also osidered. I additio the respose depeds o two parameters related to the iidet wave forig the mea wave height ad its modulatio were disussed. It should e oted that the preset solutios are derived for the free waves fored y the variatio i reaker height. ANALYSIS OF THE GOVERNING EQUATIONS Goverig Equatios A idealized ottom topography of omposite uiform slopes m m m is osidered (Figure. Whe groupig waves iidet to the ottom topography large waves may reak offshore ad small waves may reak oshore. Here ad mi ma are termed as the miimum ad maimum reakig poits of the groupig waves. Suh time variatio of the reakpoit positio a geerate log waves of whih period oiides to the groupig wave period. I Figure is defied as the mea positio of the reakig poit ad L is the offshore distae of the ar est. The aalyzig domai is divided ito five regios orrespodig to the differees of the followig wave dyamis. Regio-5 is defied for the offshore zoe where o wave reaks. Whereas for the zoe etwee ad Regio-4 is set mi ma where the log waves are drive y the time varyig reakig poit mehaism. These reakig poits are assumed to our offshore from the ar est. For the regio eteded from to the ar est positio we all Regio-. After reakig waves are assumed to deay i a way that their amplitude is proportioal to the loal water depth (i.e. saturated wave height hypothesis. Over the trough regio (Regio- ad Regio- aother assumptio is used that the waves ease to reak ad the wave height osequetly remais ostat as show i Figure. The goverig equatios to desie the o-offshore log wave motio are the depth-itegrated liearized shallow water equatios for the flow averaged over the iidet wave period (PHILLIPS 977 as follows mi where is positive offshore with the origi at the shorelie U is the depth itegrated veloity i the diretio is the sea surfae h is the depth is the desity ad g is the gravitatioal aeleratio. S is the radiatio stress term represetig the oshore mometum flu give y S ga 4 where a is the iidet wave amplitude. Iside the reakpoit o the seaward fae of the ar similarity theory idiates that the iidet wave amplitude remais a fied proportio of the depth. Outside the surf zoe we assume the iidet wave amplitude varies siusoidally at the group frequey. I additio it is further assumed that the variatio i wave height is eough small that the regio of varyig wave height the forig regio a e ofied to the seaward fae of the ar. A shemati represetatio of wave height is show i Figure. The smallest waves withi a group reak at mi while the largest reak at.through the surf zoe the amplitude ma remais a fied proportio of the depth. Shoreward of the ar est the iidet wave height at a give loatio is ostat i time the forig at the fudametal ad higher harmois ours etirely over the ar. We odimesioalize Equatios ( ad ( y salig the variales with their mea values as follows; ' ( L' X h' ( L' X h X ( L' X X X ( L' X X a [ h X ] a X t U X U t C where γ is the ratio of wave amplitude to water depth (γ ~.4 σ is the agular frequey of the grouped waves. The asi o-dimesioal depth itegrated time-average shallow water equatios are writte y U h t ( hu t a C L ( otherwise where ( L X ( X X X gm X hm ( h m is the water depth at L X s s ( L X. The forig term i right had side of Equatio (5 due to wave reakig over the ar is the give y ( (4 (5 (6 Joural of Coastal Researh Speial Issue

3 Akarpour Jaat ad Asao a h m L i whih ( is time varyig reakig positio epressed y (7 h h h hc h mi otherwise ma (6 ( a os( t t (8 Here Δa is the horizotal amplitude of the time varyig reak poit ad t is the travelig time required for a wave to propagate from ma =+Δa to =. The followig Fourier series epasio is itrodued for a it it F ( ( e e (9 h The solutio for Regio-5 a e derived osiderig radiatio oditio at the offshore oudary as ( it K 7 H ( k z e (7 i whih k =4 χ(+l/x ad z =[X +(-L]. For Regio-4 we have the followig solutio K p ( ( it 5 H ( k z K6 H ( k z e (8 where t t ( i whih H ( H ( are the Hakel futios of the first ad seod kid respetively. The term p represets the partiular solutio epressed as follows 4 t t e it ( Where t t t t ( os ad t is travelig a time writte as follows L X t ( ma t ( g m Sustitutig those epressios for t ad t ad evaluatig the itegrals i Equatios ( ad ( the Fourier oeffiiets for the regio mi << ma a e epressed as follows: Here p p p ( ( A H ( kz H ( kz (9 i ( ChH ( kz / ( Chz H ( kz d mi ( Ap( mi i ( C hh ( kz / / ( C hz H ( kz / d ( p mi mi ad C =X /(X C +X. The solutios for Regio- ad Regio- are give respetively y K ( ( it H ( k z K4 H ( k z e ( m os a ( ( ( it H ( k' z K H ( k' z e K ( i whih i 4 m e i os t i os t a a e (4 ( ' k 4 ( m / m ( L/ z h / m ( (4 X Aalytial Solutios y Regioal Mathig Water surfae flutuatio of the log wave for is epressed usig a omple amplitude of η ( y it e (5 Elimiatig U i Equatios (5 ad (6 we have For Regio- the solutio is oly omprised y a essel futio J eause a Neuma futio has a sigularity at = as follows. K ( e J k ' it (5 i whih k=4 X(m /m (+L/X. These omple oeffiiets K ~ K 7 are determied from otiuity oditios at the regioal oudaries with respets to ad. Joural of Coastal Researh Speial Issue

4 A Itegrated Preditio Model of Wave Groups ad their Assoiated Log Waves TRANSFORMATION OF WAVE GROUPS To otaiig ma ad mi whih are eeded for the aove developed aalytial model the pheomeo of wave groups a e desied i a simple way where two siusoidal waves ad with the same amplitude a ad losely eighorig agular frequeies ad are superposed as. The groupig waves a e epressed y k ( a os( ( k os( k t ( (6 where is the amplitude modulatio oeffiiet of the primary waves ( =; ostat siusoidal waves =; fully modulated waves partially modulated waves k k k is the differee of the wave umer ad agular frequey of the ompoet waves respetively. Determiatio of the reakig poit is aother importat prolem i wid wave alulatios. The modified reakig ide is used as follows (ISOE 986. u.5.ep d d 5ta ep 45. L L (7 i whih u is the orital veloity amplitude at the still water level L is the deep water wave legth ad d is the still water depth at the reakig poit. oudary Coditios The offshore oudary where iidet waves are presied is treated as a ope oudary i order to let the refleted waves go out of the regio freely. For this a regio with loally ostat depth was set eside the offshore oudary. The oudary oditio for the water surfae elevatio is give y ( ai si( k si k( ( t (8 ( where is the horizotal grid legth ad the susipt deotes the quatities at the offshore oudary. a I ad are the amplitude ad agular frequey of iidet waves respetively. I the preset omputatios the wave heights will deay after reakig ad dimiish to zero at the shorelie therefore the shorelie oudary oditio a e give simply as Q. OTTOM SEDIMENT MOILITY I order to evaluate the time averaged sedimet trasport effiiey aother Shields parameter ( is proposed; tt IGW ( T t (9 IGW i whih T IGW is the first harmois period of the ifragravity * waves ad ( is the istataeous effetive Shields parameter defied y if ( if ( ( sig ( ( ( ( where sig ( idiates a operator to take the positive ad egative of the value ad ( is the istataeous Shields parameter alulated y f uigw ( uigw ( ( ( ( s gds ad is the itial Shields parameter where the value of.5 is adopted. RESULTS AND DISCUSSION A series of model solutios at the group frequey separated i time y is show i Figure to illustrate oth the stadig ad the progressive wave respose. The idealized arred topography has ee used i this aalysis. At the shorelie a ostat slope of. was assumed ad eteded offshore to a depth of. m. The ar est was 5 m offshore i whih the depth over the ar eig. m ( h h. 65. Seaward of the ar est a ostat slope of. was assumed.the asi iput oditios i Figure are as follows; the ompoets wave periods are T 6 s T 7 the iidet group wave height is ad s ad the modulatio oeffiiet is. Shoreward of H. 4m the reakpoit stadig wave solutio were otaied at the group frequey. Seaward of the reak poit the solutio were i the form of out-goig free waves. The amplitude of the outgoig wave is strogly frequey depedet ad usig eergy argumets a e show to go to zero whe the stadig elevatio is i phase with the forig (Figure. For this ase the stadig wave elevatio has a atiode i the forig regio. Figures shows the variatio of elevatio amplitude ( R (solid lie ad outgoig wave amplitude (dashed lie with ( h h at the shorelie for iidet group wave height H. 4 m. The variatio of ( h h was made y hagig h ad m with keepig the other parameters ostat. The arrows idiate the half wave resoae oditios. The results show a imodal distriutio represetig two maima at h h. 5 ad h h. 65 assoiated with H.4 ( m. Figure 4 shows the distriutio of veloity ad elevatio amplitude for iidet wave height H. 4 m depth ratio h h.65 ad modulatio rate. 5. It implies half wave resoat oditio i whih water surfae elevatio has atiode at ar est ( 5m. It implies that the presee of a shoreparallel ar provides resoat oditio that is half wave resoae etwee the shorelie ad the ar est. This oditio ours whe the stadig wave elevatio has a atiode at the ar est. Figure 5 illustrates the effets of the amplitude modulatio rate o the omposite Shields parameter. The Shields parameter at Joural of Coastal Researh Speial Issue

5 Akarpour Jaat ad Asao the shorelie learly respods to the iease of the modulatio rate. It is oluded that the amplitude modulatio muh affets the growth of the ifragravity waves. Figure 5. Effets of amplitude modulatio rate o the Shields parameter for differet iidet wave heights Figure. Sequee of model solutios separated i time y oe quarter of the dimesioless group frequey for h h. 65 Figure. Variatio of elevatio amplitude at shorelie with ( h h Figure 4. Distriutio of veloity ad elevatio amplitude with offshore distae CONCLUSION I the preset study the model of log wave geeratio y time varyig reakpoit preseted y SYMONDS et al. (98 has ee eteded to ilude variale topography shoreward of the forig regio. A umer of itial parameters suh as the offshore positio of the mea reak poit ad the depth over the ar were ivestigated. I additio the respose will deped o two other parameters related to the iidet wave forig the mea wave height ad its modulatio. The results have show that a amplifiatio i the log wave respose ours whe the stadig wave has a atiode i elevatio at the ar est. It was show that early fully modulated waves (. have importat role o the log wave ru-up height at the shorelie. LITERATURE CITED AKARPOUR M.R. ad ASANO T. 7. Eteral Fores of Sedimet Trasport i Surf ad Swash Zoes Idued y Wave groups ad Their Assoiated Log Waves. Coastal Egieerig Joural 49 pp ISOE M A paraoli equatio model for trasformatio of irregular waves due to refratio diffratio ad reakig. Coastal Egieerig i Japa JSCE ( pp KATOH K. ad YANAGISHIMA S. 99. erm erosio due to log period waves. Coastal Egieerig 57 pp LONGUET-HIGGINS M.S. ad STEWART R.W. 96. Radiatio stress ad mass trasport i gravity waves with appliatio to surf eat. Joural of Fluid Mehais pp MUNK W.H Surf eats. Eos Tras. AGU. pp PHILLIPS O.M The Dyamis of the Upper Oea. Camridge Uiversity Press New York 6p. SYMONDS G.; HUNTLEY D.A. ad OWEN A.J. 98. Twodimesioal surf eat: Log wave geeratio y a timevaryig reakpoit. Joural Geophysial Researh 87 pp TUCKER M.J. 95. Surf eats: Sea waves of to 5 miutes. Proeedig R. So. Lodo Ser. A. pp Joural of Coastal Researh Speial Issue