Joural of Coastal Research Special Issue 5 5-5 Florida, USA ISS 79-8 Modelig ear-shore Currets Iduced b Irregular Breakig Wave Ju Tag, Yog-Mig She ad Lei Cui State Ke Laborator of Coastal ad Offshore Egieerig, Dalia Uiversit of Techolog, Dalia 1163, P. R. Chia, jtag@dlut.edu.c ABSTRACT ear-shore surface water waves ad wave-iduced currets are importat hdrodamic factors i coastal zoes. Propagatio of irregular water waves ad irregular breakig-wave iduced ear-shore currets have bee umerical studied based o parabolic mild slope equatio ad ear-shore currets model. Based o the JOSWAP wave spectrum, the parabolic mild slope equatio icorporatig irregular ad wave-breakig effects have bee applied to model water waves. The wave radiatio stresses exerted o currets have bee calculated based o variables i the parabolic mild slope equatio, ad ear-shore wave-iduced currets have bee umericall simulated based o these. The umerical results have also bee validated ad aalzed. It is believed that the preseted umerical models are capable of adaptatio to umerical simulatig wave-iduced ear-shore circulatio. ADDITIOAL IDEX WORDS: irregular waves, wave breakig, wave spectrum, ear-shore currets ITRODUCTIO Scietific ivestigatios of water waves ad currets alog the coast have demostrated that ear-shore currets are likel preset o most beaches as a compoet of the complex patter of ear-shore circulatio. As waves travel from deep to shallow water, the udertake a obvious trasformatio due to the combiatio effects of refractio, diffractio ad shoalig, ad the evetuall break ear the shorelie. As waves break alog the coast, the geerate currets that flow i both the offshore ad alogshore directios. Complex iteractios betwee waves, currets, water levels, ad ear-shore bathmetr result i the geeratio of ear-shore curret sstems that iclude both the alogshore ad cross-shore water motio. Alog all coastlies, ear-shore circulatio ma develop whe waves break strogl i some locatios ad weakl i others (JAMIE et al., 6). The ear-shore currets are importat hdrodamic factors i coastal zoes, ad the ivolve the complex iteractios betwee waves, currets, water levels, which ifluece the ear-shore trasport of sad ad cotamiats etc. A accurate descriptio of ear-shore hdrodamics is ver importat, ad umerical simulatio of ear-shore processes is a effect wa that ca provide useful iformatio o a wide rage of coastal problems. All damical models of wave-iduced ear-shore currets are forced b alogshore or cross-shore variatios of wave height that result i alogshore or cross-shore variatios i wave-iduced mometum flux, termed radiatio stress b LOGUET-HIGGIS ad STEWART (196). A coveiet startig poit is the depth-itegrated, horizotal ad mometum balace equatios sice water depth i coastal zoes is relative shallow ad the 3D models are computatioall ver expesive (e.g., LOGUET-HIGGIS, 197; EBERSOLE ad DALRYMPLE, 198; VA ad WIJBERG, 1996; BORTHWICK et al., 1997; PARKER ad BORTHWICK, 1; GRASMEIJER ad RUESSIK, 3; YU ad DOALD, 3; JAMIE et al., 6). Research progress o umerical simulatio of surface water waves, wave-iduced currets have bee greatl improved i recet ears. However, due to the complicated hdrodamic structure i surf zoes, the accurate predictio of irregular water waves ad wave-iduced currets, such as log-shore currets, rip currets etc, are still importat topics for further research. I this paper, umerical models for irregular wave iduced ear-shore currets have bee developed, as well as irregular water waves ad wave-iduced ear-shore currets are ivestigated b developed models. I the umerical model herei, the icidet irregular water waves are divided ito diverse compositive regular water waves with water wave spectrum, ad parabolic mild slope equatio icorporatig irregular ad wave-breakig effects has bee applied to model water waves. The, wave radiatio stresses from the excess mometum flux of the waves which produce mea water level chages ad ear-shore currets are calculated based o the
6 ear-shore Currets b Breakig Wave variables i parabolic mild slope equatio, ad the ear-shore currets model are geerall described b depth-itegrated coservatio equatios for mass ad mometum. The log-shore currets ad rip-currets, i which obliquel icidet water wave climate ad ear-shore bathmetr are resposible for them, are umericall simulated ad ivestigated based o the developed umerical models. Wave Spectrum UMERICAL MODELS I coastal zoes, the water waves from offshore areas are radom water waves. To ivestigate the relatios betwee irregular ad regular wave parameters coveietl, the distributio of water wave parameters i irregular waves have bee described b wave spectrums. With special wave spectrum, the irregular water wave parameters ca be divided ito serious regular wave parameters. The Joit orth Sea Wave Project (JOSWAP) spectrum (GODA, 1999) herei has bee used to divide the irregular waves: 5 5 exp( f fp1 ) S( f ) jh1 3Tp f exp( Tp f ) (1) i which.638 (1.9.1915l ) j.3.336.185 1.9 1 T H 13 p.559 T 1.13..7 f f.9 f f p p Where f is wave frequec; fp is wave top frequec; Tp is wave top period; H1/3 is effective wave height; is wave spectrum factor, ad =3.3 is used herei. Parabolic Mild Slope Equatio The problem of water waves propagatig over irregular bathmetr i arbitrar directios ivolvig combiatio effects of refractio, diffractio ad reflectio is a three-dimesioal problem ad ivolves complicated oliear boudar coditio. To simplif the problem, BERKHOFF (197) developed the mild slope equatio, i which the properties of liear progressive water waves are predicted b a weighted verticall itegrated model. For the liear mild slope equatio, researchers have developed parabolic models, which have efficiet solutio advatages over the elliptic form preseted b BERKHOFF. KIRBY (1986) has developed a extesio parabolic mild slope equatio based o a miimax priciple. For the irregular water waves, the parabolic mild slope equatio icorporatig wave-breakig effect ca be described as follows: A ' 1 g b1 A Cg E A CCg 1 x 1 x 1 k x ' F CCg 1 i which C A ic ' g 1 i g E k a k C D A ' i k b 1 ( k ) C x g x F a1 b 1 k k k C g the oliear term icludes D, which is C kh kh 8sih cosh 8 tah 3 D k C g kh () where, x ad are coordiates defiig the horizotal plae, ad x is the wave propagatio pricipal directio; i is imagiative uit; a, a 1 ad b 1 are coefficiets defied accordig to the aperture width chose to specif the miimax approximatio, ad the correspodig coefficiet values herei are: a =.997333, a 1=-.896831 ad b 1=-.516568; h is still water depth; A is the th wave imagiative amplitude; k is the th wave umber; k is the wave umber averaged i directio; is the th wave agular frequec; C =ω /k ad C k are the th wave velocit ad wave group g velocit respectivel; ε is wave eerg dissipatio factor due to wave breakig ad give as (KIRBY ad DALRYMPLE, 1986; VA ad WIJBERG, 1996; ZHEG, et al., ): 1 b 1 H (3) T H where a 11.; the breakig wave height H b is defied herei b (GRASMEIJER ad RUESSIK, 3): Hb mi( h,.1 Ltah kh) () where h is local still water depth; L is wave legth; is the wave-breakig ratio ad govered i the paper b (GODA, 1975):.exp(.ta ) where is the plae slope. B usig the dissipatio model ad the breakig wave height to determie the oset of breakig, Eq. () is able to represet waves both outside ad iside of a surf zoe. For the parabolic mild slope equatios, ol the lateral boudar coditios are eeded besides the icidet boudar. The lateral boudar coditio for Eq. () is prescribed as follows: A icr A k si (5) where c r1. is wave reflectio coefficiet o the lateral
ear-shore Currets b Breakig Wave 7 boudaries, is wave agle alog the boudaries. The irregular icidet wave has bee divided accordig the wave spectrum ito part regular waves i umerical simulatio of water wave, ad the root mea wave height H has bee guided b the followig equatio: i1 H A (6) Eq. () ca be discretized b a fiite differece method with C- scheme ad solved b tri-diagoal mathematic algorithm. Wave-Iduced ear-shore Curret Equatio I coastal zoes, the wave-iduced ear-shore curret motio ca be guided b followig depth-itegrated, horizotal mometum balace equatios sice water depth is relative shallow ad the 3D models are computatioal expesive: U h t x V h U U U 1 S S xx x U V g t x x h x 1 ηx bx Am x h V V V 1 Sx S U V g t x h x 1 η b Am h where, U=(U,V) is wave-iduced curret velocit vector; is mea water level; S xx, S x, S x ad S are wave radiatio stresses; x, are surface frictio stresses i waves ad currets; bx, b are bottom frictio stresses i waves ad currets; A mx, A m are mixig stresses term. For air waves, wave radiatio stresses ma be derived from the wave potetial parameters (BETTESS ad BETTESS, 198), i which the ukow wave propagatig directios are ot ivolved. The followig expressios derived from parameters i parabolic mild slope equatio have bee used to calculate the wave radiatio stresses herei (ZHEG, ): S g A 1 k 1 h xx ik A 1 x k sih kh A k h k h tah k h 1 sih kh k A A ik A k A x (7) (8) (9) (1) S S xx x g A 1 k 1 h 1 k sih kh A k h k h tah k h 1 sih kh k A A ik A k A x (11) g A A 1 kh Re ika 1 1 x k sih kh (1) where, A is the cojugative complex of A. Surface frictio stresses i waves ad currets are igored herei, ad bottom frictio stresses are defied as follows (LOGUET-HIGGIS, 197): bx cfuu (13) b cfuv (1) where u is the bottom wave velocit amplitude, which ca be give b the liear wave theor as u =a wb /T, ad a wb=h/(sikh) is the bottom horizotal trajector amplitude, c f is the empirical wave frictio coefficiet i the presece of currets (tpicall i the order of 1 - ). The lateral mixig stresses are defied as follows: A A mx m U U x x V V x x (15) (16) where is the lateral mixig coefficiet, which is defied herei as: =u H, ad =.85 is a o-dimesio coefficiet (ÖZKA-HALLER ad KIRBY, 1998). The iitial ad boudar coditios for the ear-shore curret model are specified as follows. Here, U is set to be the ormal velocit to the beach boudar, ad V is the velocit parallel to the beach boudar. I all applicatios of the model, the iitial coditios are assumed to be at the state of rest, ad U, V ad are set to be zero iitiall. The boudar coditios are prescribed as follows: at the offshore ope sea boudar, o-flow coditio is imposed b settig the velocit compoets ad the wave-iduced water level chage compoet equal to zero, assumig that the offshore ope sea boudar is far eough from the wave surf zoe, ad o curret other tha the ear-shore curret flows ito the computatioal regios. At the oshore boudar ad alog the lateral ope sea boudaries, the slip boudar coditio is used o assumig a getle bottom slope o these boudaries: (17) U (18)
S (m s) S (m s) 8 ear-shore Currets b Breakig Wave V (19) A fiite differece scheme has bee used to solve the problem formulated above. The ADI (alteratig directio implicit) scheme is applied to discretize the equatio (7) to (9), i which a oe-time step is divided ito two half-steps, ad the explicit method is applied i the x-directio i the first half-time step, while the implicit method is applied i the -directio, ad this is followed b applicatio of the explicit method i the -directio, ad the implicit method i the x-directio i the secod half-time step. Hece, a tri-diagoal matrix for is formed, which ca be solved readil i each half-time step. For computatioal stabilit, the bottom frictio ad lateral mixig terms are lagged o half-step i time i each advacemet through the half-time step. The oliearit of the problem makes it impossible to deduce a aaltical stabilit criterio i terms of the time step t that is required i umericall solvig the ear-shore curret model. Here, the Courat stabilit criterio is emploed to guide the choice of a time step t, which accordig to EBERSOLE ad DALRYMPLE (198) is give b: x t () gh max where x ad are the spatial discretizatios i the x- ad -directios, respectivel. The wave-iduced ear-shore curret model is ru util the approximate a stead state. VALIDATIO OF THE UMERICAL MODEL The test of the umerical models ivolves umerical simulatig of the wave-iduced log-shore currets ad rip-currets, i which obliquel icidet water wave climate ad the log-shore topograph are resposible for them respectivel. Log-Shore Currets Validatio The log-shore currets experimet has bee performed at State Ke Laborator of Coastal ad Offshore Egieerig i Dalia uiversit of techolog. The experimets were repeated 3 times i each case ad the experimetal data are provided for all 3 times i each case without average (WAG, 1). Log-shore currets are geerated sice the icidet waves break i the plae. The parameters used i the umerical model i the paper are listed i Table 1, where h is the water depth i flat regios, is the agle betwee the icidet wave ad plae slope, H 1/3 is the icidet effective wave height, T 1/3 is the icidet effective wave period. The JOSWAP spectrum herei has bee adopted as iput wave spectrum, ad compariso betwee iput ad theoretic wave spectrum i log-shore currets experimet are show i Fig. 1. It is also show that a good agreemet as a whole betwee the iput ad theoretic wave spectrum has bee achieved, which makes sure that the good agreemet betwee the iput ad theoretic wave parameters ca be achieved at icidet wave boudar. Table 1 Computatioal parameters i log-shore currets experimet case plae slope h / m θ / ( o ) H 1/3 / m T 1/3 / s c f case 1 1 :.5 3.5..9 case 1 : 1.18 3.3 1..65.15.1.5..6 1 1. 1.8 f (Hz)..1 (a) T 1/3=.s, H 1/3=.5m.5 1 1.5.5 3 3.5 f (Hz) (b) T 1/3=1.s, H 1/3=.3m Iput wave spectrum Theoretic wave spectrum Fig. 1 Compariso betwee iput simulated ad theoretic wave spectrum i log-shore currets case The comparisos betwee the experimetal ad umerical results of irregular waves, ad irregular waves iduced mea water level chages ad log-shore currets are show i Fig. ad Fig. 3, where x is the distace to the shore ad it set positive i off-shore directio. It ca be see from Fig. ad Fig. 3 that the umericall simulated wave heights are i good accordace with the experimetal results i the whole for all cases. It is also show that the wave heights are almost liear with the water depths i the surf zoe, ad it is appeared that the wave height i surf zoe ma be govered b the liear relatio with local water depth, which is coveietl ad computatioall ecoomical i wave models. The experimetal data of waves-iduced mea water level chages ad log-shore curret velocit collected i experimet seems some dispersed, which ma be caused b the turbulet water motio ad limitatios to measuremet i surf zoe. It is show that the shape of the measured ad umerical computed distributio of log-shore curret are differs at wave
S (m s) H (m) V (m/s) H (m) V (m/s) ear-shore Currets b Breakig Wave 9 breakig locatios, ad the measured maximum log-shore velocit is located more shoreward. SVEDSE (198) accouted for these with surface roller ifluece i the eerg balace, i which part of the orgaized wave eerg is first coverted ito forward mometum flux to the roller at wave breakig (GRASMEIJER ad RUESSIK, 3). As a whole, the umericall simulated waves-iduced mea water level chages ad log-shore currets are all show good tedec accordace with the experimetal data..1.8.6.. 8 1 16 (m)..16.1.8. -. 8 1 16..1 8 1 16 (a) wave height (b) waves-iduced mea water level chage (c) waves-iduced log-shore curret velocit umerical results experimetal results Fig. Compariso betwee measured ad calculated results for case 1 i log-shore currets case.5..3..1 8 1 16 (m).1.8.6.. -. 8 1 16..1 8 1 16 (a) wave height (b) waves-iduced mea water level chage (c) waves-iduced log-shore curret velocit umerical results experimetal results Fig.3 Compariso betwee measured ad calculated results for case i log-shore currets case Rip Currets Case directio. The shape of the shorelie ad ear-shore bathmetr ma ifluece rip curret developmet. I regios where the coastlie is characterized b cuspate features, rip curret ma be foud betwee the cusps. The rip currets test ivolved the modelig of wave-iduced currets at a idealized multi-cusped beach as reported b BORTHWICK et al (1997). The mea water level was.5m, ad three cusps were located o a 1: plae beach as show i Fig. 5 (a). The still water depth distributio at the cups is give b (PARKER ad BORTHWICK, 1): 3 xl x h.5 xl x.75si 1 si x l 6 where, x is the distace from the cup s toe i the cross-shore directio, ad is the distace from the edge of the cups i alogshore directio, x l is the cup s legth i cross-shore.75.5.5.5 1 1.5.5 3 3.5 f (Hz) Iput wave spectrum Theoretic wave spectrum Fig. Compariso betwee iput simulated ad theoretic wave spectrum i rip currets case I preset model, it is assumed that the icidet wave with
.9 (m) (m) -.1 -.35.7 -.5 -. -.3 -.5 (m) (m) -.5 -.35 -.5.7.1 5 ear-shore Currets b Breakig Wave effective wave period of 1.s ad effective wave amplitude of chages ad currets. It is show that the icidet wave.5m are ormal to the beach. Sice the JOSWAP spectrum i accumulates ad breaks before the cusped shore, ad the wave Fig. has bee adopted as iput wave spectrum herei, the iput height ad water level gradiet alog log-shore or cross-shore wave parameters here ma differs from those i experimet. directios have bee produced. Hece, rip currets have bee Hece, ol the umerical results of waves ad wave-iduced iduced. The tedecies of umerical simulated water waves ad log-shore currets are show. wave-iduced currets here are also agree well with the results from other researches (PARKER ad BORTHWICK, 1). It is Fig. 5 (b, c, d) show the umerical simulated spatial appeared that the preset model is capable of adaptatio to variabilit of wave height, waves-iduced mea water level complicated coastal flow ad bathmetric features. Frame 1 7 Sep 7 WAV_H Frame 1 7 Sep 7 WAV_H 1 1 8 -.15 -. -.1 -.5 -.1 1 1 8.7.7.5.6.3.6.. 6 -.15 6.5. -.5 -. -.15 -.1 -.1 -.5 -.5.7.6.5..6..3..1 Frame 1 7 Sep 7 6 8 1 1 1 (a) Still water depth for multi-cusped beach Frame 1 7 Sep 7 6 8 1 1 1 (b) umerical simulated spatial variabilit of wave height 1 1 -.1 1.5 1 8 6 -.1 -.3.1.5.7.9 8 6.3 (m/s) -.1.3 -.3 -.1.5.9 6 8 1 1 1 (c) umerical simulated spatial variabilit of waves-iduced mea water level chage 6 8 1 1 1 (d) umerical simulated spatial variabilit of wave-iduced curre Fig. 5 umerical simulated rip currets for multi-cusped beach case (uit: m) COCLUSIOS ear-shore currets are likel preset o most beaches as a compoet of the complex patter of ear-shore circulatio. Complex iteractios betwee waves, water levels, ad ear-shore bathmetr result i the geeratio of ear-shore curret sstems that iclude both the alogshore ad cross-shore water motio. The ear-shore currets are importat hdrodamic factors i coastal zoes, ad the ivolve the complex iteractios betwee waves, currets, water levels, which ifluece the ear-shore trasport of sad ad cotamiats etc. umerical models for irregular wave iduced ear-shore currets have bee developed, i which wave radiatio stress exerted o currets have bee calculated based o the variables i the parabolic mild slope equatio. The wave-iduced log-shore currets ad rip-currets, i which obliquel icidet water wave climate ad ear-shore bathmetr are resposible for them respectivel, are umericall simulated based o the developed umerical models. It is believed that the preset umerical models are capable of adaptatio to umerical
51 ear-shore Currets b Breakig Wave simulatig wave-iduced ear-shore circulatio. b obliquel icidet sea waves, 1 ad. Joural of Geophsical Research 75(33), 6778-6789. ACKOWLEDGMETS LOGUET-HIGGIS, M.S., STEWART, R.W., 196. Radiatio This research was fiaciall supported b the atioal Basic Research (973) Program of Chia uder Grat o. stress i water waves, a phsical discussio with applicatios. Deep-Sea Res. 11 (), 59 563. 6CB33 ad b the atioal atural Sciece Foudatio of Chia uder Grat os. 579 ad 57796. ÖZKA-HALLER, H.T. ad KIRBY, J.T., 1998. Spectral model for wave trasformatio ad breakig over irregular bathmetr. LITERATURE CITED Joural of Waterwa, Port, Coastal, ad Ocea Egieerig 1(), 189-198. BERKHOFF, J.C.W., 197. Computatio of combied refractio-diffractio. Proceedig of the 13th Coferece o Coastal Egieerig. Vacouver, Caada, pp. 71-9. PARKER, K.Y. ad BORTHWICK, A.G.L., 1. Quadtree grid umerical model of earshore wave-curret iteractio. Coastal Egieerig, 19-39. BETTESS, P. ad BETTESS, J., 198. A geeralizatio of the radiatio stress tesor. Applied Math. Modelig 6, 16-15. SVEDSE, I.A., 198. Mass flux ad udertow i a surf zoe. Coastal Egieerig 8, 37-365. BORTHWICK, A.G.L., FOOTE, Y.L.M, RIDEHALGH, A., 1997. earshore measuremets at a cusped beach i the UK Coastal Research Facilit. Coastal Damics 97, Plmouth. Pp. 953-96. VA, R.L.C. ad WIJBERG, K.M., 1996. Oe-dimesioal modelig of idividual waves ad wave-iduced logshore currets i the surf zoe. Coastal Egieerig 8, 11-15. DOG, P. ad AASTASIOU, K., 1991. A umerical model of the vertical distributio of logshore currets o a plae beach. Coastal Egieerig 15, 79-198. EBERSOLE, B.A. ad DALRYMPLE, R.A., 198. umerical Modellig of earshore Circulatio. Proceedigs of the 7th Coastal Egieerig, 71-75. GODA, Y., 1975. Irregular wave deformatio i the surf zoe. Coastal Egieerig i Japa 18, 13-6. GODA, Y., 1999. A comparative review o the fuctioal forms of directioal wave spectrum. Coastal Egieerig Joural 1(1), 1-. GRASMEIJER, B.T. ad RUESSIK, B.G., 3. Modelig of waves ad currets i the earshore parametric vs. probabilistic approach. Coastal Egieerig 9, 185-7. WAG, S.P., 1. Stud o the Log-shore Currets. Master s thesis, Dalia Uiversit of Techolog (i Chiese). YU, J. ad DOALD,.S., 3. Effects of wave-curret iteractio o rip currets. Joural of Geophsical Research 18 (C3), 1-19. ZHEG, Y.H., SHE, Y.M. ad QIU, D.H.,. Calculatio of wave radiatio stresses coected with the parabolic mild-slope equatio. Acta Oceaologica Siica (6), 11~116 (i Chiese). ZHEG, Y.H., SHE, Y.M. ad WU, X.G. et al.,. Determiatio of wave eerg dissipatio factor ad umerical simulatio of wave height i the surf zoe. Ocea Egieerig 31, 183 19. JAMIE, H.M., ED B.T., ad AD, J.H.M.R., 6. Rip curret review. Coastal Egieerig 53, 191-8. KIRBY, J.T. ad DALRYMPLE, R.A., 1986. Modelig waves i surf zoes ad aroud islads. Joural of Waterwa, Port, Coastal ad Ocea Egieerig 11, 78-93. KIRBY, J.T., 1986. Ratioal approximatios i the parabolic equatio method for water waves. Coastal Egieerig 1, 355-378. LOGUET-HIGGIS, M.S., 197. Logshore currets geerated
5 ear-shore Currets b Breakig Wave