ave Breang Energy n Coastal Regon Ray-Qng Ln and Lwa Ln Dept. of Seaeepng Davd Taylor Model Basn NSCCD U.S. Army Engneer Researc and Development Center. INTERODUCTION Huang 006 suggested tat wave breang n te coastal regon s a major energy source for te ocean crculaton based on te observatonal data. Because te area of a coastal regon s generally 5% or less of te ocean te energy loss from wave breang n te coastal regon must be sgnfcantly greater tan n te open ocean. Two major dscoveres n te past ave supported Huang s suggeston. In 980 Herterc and Hasselmann ndcated tat rapd nonlnear wave-wave nteractons could occur and ncrease as water dept decreased based on te Boltzmann ntegral metod. In 998 Reso and Tracy also Ln and Perre proved te penomenon usng ter numercal models frst usng Boltzmann metod and later usng te Hamltonan varaton prncple as presented at te Internatonal Base Enancement ave Predcton Conference csburg Msssspp. Ln and Perre 997 furter sowed tat tree wave nteractons cannot satsfy te resonant condton n sallow water. Instead four-wave nteractons consstng of tree local wnd waves nteractng wt one long wave are requred for te resonant n sallow water. Based on tese two dscoveres ger frequency and smaller wave ampltude waves no longer domnate n te coastal regon. Te long wave propagatng from te ocean to te coastal regon can absorb te wave energy from local wnd waves and grow faster tan n te deep ocean. Ts fast nonlnear nteracton n coastal regon wll transfer te energy from ger frequency to lower frequency and allow te wave to grow faster and cause more energy loss to breang tan n te deep ocean. Fnally accordng to te mass conservaton law te wave steepness n te coastal regon sall be greater tan tat n te ocean. Ts ncreases te amount of wave breang as te waves approac te coast mang wave breang more sgnfcant n te coastal regon tan n te open ocean. In Secton a regonal coastal wave model Ln and Huang 996a and b; Ln and Perre 997 and 999; Ln and Kuang 00; Ln and Ln 004a and b s descrbed. Secton sows examples of numercal smulaton of te nonlnear wave energy transfer. Secton 4 sows examples of wave spectral deformaton based on feld data. Secton 5 s for summary and conclusons.. COASTAL AE MODEL Te base equaton for coastal wave model s te acton conservaton Ln and Huang 996a and b: A [ c t u A] [ c x gx v A] [ cθ A] [ cσ A] = S y θ σ gy n S ds S nl
were A s te acton densty equal to N / σ N s te energy densty σ s te ntrnsc frequency t s te tme θ s te propagaton angle C gx C gy C θ and Cσ are group veloctes for x y θ and σ coordnates respectvely S n s te wnd nput functon S ds s te dsspaton functon wave breang bottom dsspaton etc. and S nl s te nonlnear wave-wave nteracton. Te wnd nput functon for wave generaton and dsspaton functon for wave decay were descrbed by Ln and Ln 004a and b. Te nonlnear wave-wave nteracton wc does not cange te total energy can transform te energy dstrbuton wt respect to frequences and propagaton drectons. As te energy transfers from g to low frequency ndrect cascades te waves wll grow; as te energy transfers from low to g frequency drect cascades waves are dsspated. Often te ndrect cascades domnate. Te nonlnear source functon s A = 4π T δ δ τ A A [A A ] A A [A A ] d d d Equaton can be smplfed to a quas-lne ntegraton. Beng ntegrated along te resonant orbt Tracy and Reso 98 and usng te resonant solutons tat ave a gly nonlnear dstrbuton te strong nonlnear solutons are concentrated n a small area and resonant solutons are small n most areas Ln and Perre 999. Equaton tus becomes A OB s n = 4π d d [ ds ] T θ τ n A A ' ' [A A ] A were T s expressed n Appendx A. θ θ θ θ A [A A ] d d d. NUMERICAL SIMULATION RESULTS Usng quas-lne ntegraton Eq. examples of numercal smulaton of te nonlnear wave energy transfer at dfferent wave depts based on JONSAP and Person-Mosowtz P-M spectra are sown n Fgs and respectvely Ln and Perre 999. Tese numercal model results clearly sow wave-wave nteractons ncrease as water dept decreases. Te nonlnear energy transfer s stronger n a wder frequency range for te P-M spectrum tan JONSAP because te ntal P-M spectrum contans more energy tan JONSAP n tese examples. 4. FIELD OBSERATIONS ave spectral data collected from tree Pacfc Coast buoys mantaned by te Coastal Data Informaton Program CDIP ttp://cdp.ucsd.edu and Natonal Data Buoy Center NDBC ttp://www.ndbc.noaa.gov were used to sow te nonlnear wave energy transfer. Fgure sows te locaton map for Buoys 46005 4609 and 46. Bascally Buoy 46005 collects te deepwater wave data wle 4609 and 46 measure ntermedate and sallow water waves. Te surface wnd s collected at
NDBC Buoys 46005 and 4609. Table presents coordnates water dept and data collecton perod for tese buoys. Fgure. One-dmensonal nonlnear transfer rate for JONSAP spectra: lnes A B C and D represent = 6..58.0 and 0.8 Ln and Perre 999 Fgure. One-dmensonal nonlnear transfer rate for P-M spectra: lnes A B C and D represent = 6..58.0 and 0.8 Ln and Perre 999
Fgure. Locaton Map of Pacfc Coast buoys Table Buoy Staton Informaton Staton Coordnates Dept m Data collecton perod NDBC 46005 46 o 0 00 N o 0 85 September 976 December 004 NDBC 4609 46 o 07 00 N 4 o 0 6 8 Marc 984 September 006 CDIP 46 46 o 5 4 N 4 o 4 40 9 November 98 September 006 To llustrate te nonlnear wave energy transfer from te data t s convenent to select te case wt small or mld wnd condton so tat te wnd nput nterference wll be mnmal. However te ntal wave needs to be large enoug tat te energy dsspaton s relatvely small compared to te nonlnear transfer rate. Fgure 4 sows tme seres of measured waves and wnds at tree buoys for October 997. ave energy transfer rates were calculated for one-dmensonal frequency spectra collected around t and 6 t as bot wave and wnd condtons seem deal for te llustraton of nonlnear wave-wave nteractons. On October 997 te sgnfcant wave egt was approxmately m and spectral pea perod was 9 sec at te tree buoys. Te domnant wave drecton measured at Buoys 4609 and 46 was approxmately from N. Te wnd was moderate at 6 to 7 m/sec from. Fgures 5 troug 7 sow representatve one-dmensonal transfer rates estmated at tree buoy locatons. In tese fgures te sallow water wave range s defned as < 0.46 and te deepwater range s defned as >.57 Dean and Dalrymple 984. Terefore te majorty of te wave populaton at Buoy 46005 lely occupes te deepwater range. On te oter and waves at Buoys 46 and 4609 are more lely n sallow and ntermedate waters respectvely. Te deepwater wave spectrum at Buoy 46005 s close to a broad P-M formaton. However measured wave spectra n te ntermedate dept Buoy 4609 and n sallow water Buoy 46 ave sown multple peas Fgs 6 and 7 as a result of nonlnear energy
transfer and wave-bottom nteracton. Te data sow tat te spectral pea perod s longer n te ntermedate dept Buoy 4609 tan n te deep water Buoy 46005 and te longest perod s n te sallow water condton Buoy 46. Te sape of te nonlnear transfer rate n te frequency doman s smlar to te teoretcal predcton Fgs and. In sallow water Buoy 46 because te ntal spectrum as multple peas te observed nonlnear transfer rate s seen as te result of combnng several sngle pea spectral energy transfers Fg 7. Te magntude of nonlnear transfer rate s greater n sallow water tan n ntermedate and deepwater ranges. Ts also agrees wt te teory Secton. Fgure 4. Tme seres of wave and wnd data at Buoys 46005 4609 and 46 for October 997 two tme perods for te llustraton of nonlnear energy transfer are sown n crcles
On 6 October 997 te sgnfcant wave egt was approxmately equal to m at te deepwater buoy 46005 and.5 m n te ntermedate water 4609 and sallow water 46. Te spectral pea perod was between 8 and 9 sec at te tree buoys. Te domnant wave drecton measured at Buoys 4609 and 46 was from. Te surface wnd averaged 0 m/sec n te deepwater buoy and m/sec n ntermedate and sallow water buoys. Te wnd drecton was from N. Fgures 8 to 0 sow representatve one-dmensonal transfer rates estmated at tree buoys. Te ntal deepwater spectrum at Buoy 46005 resembles a broad P-M spectrum. However wave spectra collected from Buoys 4609 and 46 n ntermedate and sallow waters appear bmodal Fgs 9 and 0 as a result of nteracton of two wave systems from dfferent drectons n addton to te nonlnear energy transfer. Te nonlnear transfer rate n te frequency doman s also smlar to te teoretcal predcton Fgs and. In te sallow and ntermedate water depts because te ntal spectra contan multple peas te nonlnear transfer rate becomes more complcated tan tat n a sngle pea spectral energy transfer Fg 0. Te magntude of nonlnear transfer rate s overall greater n sallow and ntermedate waters tan n te deepwater range. Fgure 5. Measured one-dmensonal frequency spectra at 0:00 and :00 GMT t October 997 and spectral energy cange rate estmaton at Buoy 46005
Fgure 6. Measured one-dmensonal frequency spectra at 0:00 and 04:00 GMT t October 997 and spectral energy cange rate estmaton at Buoy 4609 Fgure 7. Measured one-dmensonal frequency spectra at 5:00 and 6:00 GMT t October 997 and spectral energy cange rate estmaton at Buoy 46
Fgure 8. Measured one-dmensonal frequency spectra at 4:00 and 5:00 GMT 6 t October 997 and spectral energy cange rate estmaton at Buoy 46005 Fgure 9. Measured one-dmensonal frequency spectra at :00 and :00 GMT 6 t October 997 and spectral energy cange rate estmaton at Buoy 4609
Fgure 0. Measured one-dmensonal frequency spectra at 06:00 and 07:00 GMT 6 t October 997 and spectral energy cange rate estmaton at Buoy 46 5. SUMMARY AND CONCLUSIONS Huang 006 suggested tat wave breang n te coastal regon s a major energy source for te ocean crculaton based on observatonal data. Snce te coastal regons account for 5% or less of te ocean area te wave breang energy n te coastal regon sall be muc greater tan n te open ocean n addton to relatvely larger wave steepness n te sallower water. Te feld observatonal data collected from tree Pacfc Coast buoys were nvestgated for te nonlnear energy transfer n dfferent water dept. ave spectra measured around te t and 6 t of October 997 were used to calculate te energy transfer rate n te frequency doman. Te sape and relatve magntude of calculated transfer rates were compared for te nonlnearty of wave-wave nteractons n deepwater ntermedate and sallow water ranges. Te energy transfer rate calculated n te frequency doman from data appears to agree well wt te teory Fgs and. Fgures 5 troug 0 are evdence tat nonlnear nteractons ncrease wt water deep decrease. Te maxmum transfer rate from g to low frequency equal to.7 m sec/r occurred at te sallow water buoy 46 and te mnmum transfer rate from g to low frequency 0.65 m sec/r occurred at te deep water buoy 46005. Te effect of stronger nonlnear nteracton n te sallow water regon s te transfer of more energy from sorter waves nto longer waves n te coastal regon. Tese large longer waves brea n te sallow water and a small fracton of te energy can be reflected bac to te ocean resultng n stronger currents along te coast.
Acnowledgements Te autors would le to tan Mr. Terry Applebee for provdng many useful comments and suggestons to ts study. e also tan Dr. Jon Baryoumb for supportng te numercal modelng project under te Offce of Naval Researc s In-House Laboratory Independent Researc ILIR Program at NSCCD. References Dean R.G. and R.A. Dalrymple. 984: ater ave Mecancs for Engneers and Scentsts. Prentce-Hall Inc. Englewood Clffs New Jersey. Herterc K. and K. Hasselmann 980: A Smlarty Relaton for te Nonlnear Energy Transfer n a Fnte Dept Gravty-wave Spectrum J. of Flud Mec. 97 5-4. Huang R.-X. 006: Energy Source of Ocean Crculaton submtted to J. of Pys. Ocean. Ln L. and R.-Q. Ln 004a: ave Breang Functon te 8t Internatonal orsop on ave Hndcastng and Predcton. Nort Sore Oau Hawa. November 4-9. Ln R.-Q. and N. Huang 996a: Te Goddard Coastal ave Model Part I: Numercal Metod. J. Pyscal Oceanograpy 6 8-847. Ln R.-Q. and N. Huang 996b: Te Goddard Coastal ave Model Part II: Numercal Metod. J. Pyscal Oceanograpy 6 848-86. Ln R.-Q. and. Kuang 00: Fnte Ampltude ave-wave Interactons n a Global Statstcal ave Predcton Model. Recent Res. Devel. Pys Oceanograpy 45-59 ISBN 45-59. Ln R.-Q. and L. Ln 004b: nd Input Functon te 8t Internatonal orsop on ave Hndcastng and Predcton. Nort Sore Oau Hawa. November 4-9. Ln R.-Q. and. Perre 997: A New Coastal ave Model Part III: Nonlnear ave-wave Interacton J. Pyscal Oceanograpy 7 8-86. Ln R.-Q. and. Perre 999: ave-wave Interactons n Fnte Dept ater J. of Geopyscal Researc 04 No. C5 9-. Reso Don and Barbara Tracy 998: Nonlnear ave-wave Interacton Source Functon n Fnte ater. Internatonal Base Enancement ave Predcton Conference csburg Msssspp. Tracy B.A. and D.T. Reso Teory 98: Teory and Calculaton of te Nonlnear Energy Transfer Between Sea aves n Deep ater IS Rep. 50 pp. U.S. Army Eng. aterways Exp. Staton csburg Msssspp.
APPENDIX A. T = A ] tan tan tan ][ tan tan [ ] tan tan tan ][ tan tan [ ] tan tan tan ][ tan tan [ 8 π ± ± = ± A and - - - - - - = A tan tan tan tan tan tan [ ] tan tan tan tan [ 64 x = π r A4