DYNAMICS OF WATER WAVES OVER FRINGING CORAL REEFS

Transcription

1 DYNAMICS OF WATER WAVES OVER FRINGING CORAL REEFS DYNAMICS OF WATER WAVES OVER FRINGING CORAL REEFS YAO YU 1 YAO YU SCHOOL OF CIVIL AND ENVIRONMENTAL ENGINEERING 1

2 Dynamics of Wate Waves ove Finging Coal Reefs Yao Yu School of Civil and Envionmental Engineeing A thesis submitted to the Nanyang Technological Univesity in fulfillment of the equiement fo the degee of Docto of Philosophy 1

3 ACKNOWLEDGEMENT The wok pesented in this thesis is done unde the collaboative supevision of Asst. Pof. Zhenhua Huang at Nanyang Technological Univesity (NTU) and Pof. Stephen G. Monismith at Stanfod Univesity unde Singapoe Stanfod Patneship (SSP) Pogam. Duing the last fou yeas, Pof. Huang and Pof. Monismith have guided me with utmost patience and I was deeply inspied with thei oiginal ideas and pofound insights. The eseach challenges posed by them have been endless in that it aoused my industious wok in pusing fo pogess as well as my elevating the quality of this thesis. Heewith, the autho expesses the geatest gatitude to them. My thanks also go to Asso. Pof. Edmond, Lo. and D. Sivadas whose ealie laboatoy wok acts as a key efeence of expeimental settings in this thesis. Meanwhile, I feel gateful to D. Chunong Liu, D. Zikun Xing, D. Hongtao Nie, M. Zhida Yuan and M. Sim Yisheng Shawn fo thei guidance and assistance duing the laboatoy o numeical expeiments at NTU. As fo technician suppots that fom NTU Hydaulic Modeling Laboatoy staffs of M. Fok Yew Seng, M. Lim Kok Hin and M. Foo Shiang Kim, appeciation is also expessed. Special thanks ae due to Pof. Lynett, P., who made the oiginal numeical code available fo this eseach. Last but not the least, I owe my sincee gatitude to the students, faculty, staff of the Envionmental Fluid Mechanics Lab at Stanfod Univesity fo thei helpful discussions and suggestions. Although my teammates and fiends may not have thei inteests ooted in this aea, thei suppot in many ways is most appeciated. In paticula, I would like to acknowledge M. Conghao Xu and Ms. Yao Yao fo thei poofeading this thesis. Finally, I am geatly indebted with my paents, my elatives and all the dea pesons fo thei constant love, undestanding and suppot to my Ph.D candidatue. ii

4 TABLE OF CONTENTS ACKNOWLEDGEMENT... II TABLE OF CONTENTS... III SUMMARY... VIII LIST OF TABLES... X LIST OF FIGURES... XII LIST OF SYMBOLS... XXIII CHAPTER 1 INTRODUCTION SIGNIFICANCE OF WAVE DYNAMICS OVER CORAL REEFS WAVE DYNAMICS OVER CORAL REEFS AT THE REEF SCALE Wave tansfomation and beaking Wave-induced setup and wave-diven flow Theoetical backgound OBJECTIVES AND SCOPE Objectives Scope OUTLINE CHAPTER BREAKING WAVE CHARACTERISTICS FOR LABORATORY FRINGING REEFS LITERATURE REVIEW EXPERIMENTAL SETUP Facilities, eef model and instumentation Wave conditions and test pogam RESULTS Qualitative desciption of beaking waves Detemination of wave tansmission and eflection coefficients Measued beaking-wave popeties DATA ANALYSIS AND DISCUSSION Classification of measued beaking waves iii

5 .4.. Beake indices Sufzone width Wave tansmission, eflection and enegy dissipation CONCLUDING REMARKS CHAPTER 3 A LABORATORY STUDY OF WAVE-INDUCED SETUP OVER A HORIZONTAL REEF WITH/WITHOUT A RIDGE LITERATURE REVIEW EXPERIMENTAL SETTINGS AND INSTRUMENTS Wave flume and eef models Expeimental pocedues RESULTS Visualization of wave tansfomation ove eef cest and eflection Wave evolution acoss eef pofile Mean wate level acoss eef pofile Wave setup as a function of deep-wate wave height A DETAILED WAVE MEASUREMENT WITH IMPROVED SPATIAL RESOLUTION Definition of the suf zone ove eefs Spatial vaiation of wave height and mean wate level Geneation of highe hamonics Calculation of adiation stess UNDERTOW MEASUREMENTS DISCUSSIONS CONCLUDING REMARKS CHAPTER 4 NUMERICAL STUDY OF WAVE TRANSFORMATION OVER FRINGING REEFS LITERATURE REVIEW DESCRIPTION OF THE NUMERICAL MODEL Govening equations Numeical scheme Bounday and initial conditions Wave beaking (R b ) Bottom fiction (R f )... 1 iv

6 4.3 MODEL CALIBRATION AND VALIDATION Expeimental and numeical settings Gid size Bounday conditions Beaking model Model validation fo apidly vaying bathymety Results afte calibation CASES STUDIES Case selection and numeical input Effects of the idge on wave beaking Wave height, mean wate level and wave eflection Wave tansfomation REVISITS OF OTHER NUMERICAL STUDIES Revisit of Skotne and Apelt (1999) Revisit of Demibilek and Nwogu (7) MODEL APPLICATION TO DIFFERENT FORE-REEF SLOPES AND PROFILES Effect of the inclination angle of plane foe-eef Effect of the shape of foe-eef pofile CONCLUDING REMARKS CHAPTER 5 MODELING WAVE-INDUCED SETUP OVER FRINGING REEFS WITH SELECTED EXISTING ANALYTICAL MODELS INTRODUCTION THEORETICAL CONSIDERATIONS OFFSHORE SCALING PARAMETER COMPARISON OF EXPERIMENTAL DATA WITH THE MODEL OF TAIT (197) COMPARISON OF EXPERIMENTAL DATA WITH THE MODEL OF GOURLAY AND COLLETER (5) DISCUSSIONS CONCLUDING REMARKS CHAPTER 6 WAVE SETUP OVER FRINGING REEFS UNDER CRITICAL FLOW CONDITION: AN ANALYTICAL MODEL BASED ON MASS BALANCE INTRODUCTION v

7 6. THEORETICAL CONSIDERATION Consevation of mass Expession fo Q B Expession fo Q F Expession fo wave setup MODEL VALIDATION Oveview of expeimental setting Visualization of the wave beaking pocess Compaison between model pedictions and measuements EXTENSION TO SPECTRAL WAVES APPLICATIONS TO OTHER PUBLISHED EXPERIMENTAL DATA MORE ON THE SCALING FACTOR COMPARISON WITH THE MODEL OF GOURLAY (1996A) CONCLUDING REMARKS CHAPTER 7 WAVE SETUP OVER FRINGING REEFS: AN ANALYTICAL MODEL BASED ON MOMENTUM BALANCE INTRODUCTION THEORETICAL CONSIDERATION Govening equations Appoximation fo 1D finging eefs Uppe and lowe limits of the model validity Estimation of the beake depth indices MODEL VALIDATION BY EXPERIMENTAL DATA Classification of the investigated eef pofiles The measued fo an idealized eef without idge Compaisons among diffeent appoximations to the vaiation of in the suf zone Compaison between expeimental data and model pedictions Moe on the model paamete MODEL SENSITIVITY ANALYSIS APPLICATIONS TO FIELD DATA vi

8 7.5.1 Backgound Results DISCUSSIONS ON THE 1DH MODEL MODEL EXTENSION TO DH Fomulation Case study CONCLUDING REMARKS... 9 CHAPTER 8 CONCLUTIONS AND FUTURE WORK CONCLUSIONS FUTURE WORK Expeimental wok Numeical simulation Analytical modeling REFERENCES APPENDIX A: EXPERIMENTAL DATA IN CHAPTER... 9 APPENDIX B: EXPERIMENTAL DATA IN CHAPTER APPENDIX C: LABORATORY STUDY ON EFFECTS OF ROUGHNESS ON WAVE- INDUCED SETUP APPENDIX D: SPONGE LAYERS FOR NUMERICAL SIMULATION APPENDIX E: DERIVATION OF SOME EQUATIONS IN CHAPTER APPENDIX F: EVALUATION OF BOTTOM REACTION TERM ΔΠ IN CHAPTER 7 FOR AN IDEALIZED REEF PROFILE WITHOUT A RIDGE APPENDIX G: EVALUATION OF BOTTOM REACTION TERM ΔΠ IN CHAPTER 7 FOR AN IDEALIZED REEF PROFILE WITH A RECTANGULAR RIDGE APPENDIX H: APPLICATION OF SELECTED EXISTING ANALYTICAL MODELS TO SOME FIELD DATA vii

9 SUMMARY This thesis epoted a compehensive study of the wave dynamics ove finging eefs with and without a idge, including laboatoy expeiments, numeical modeling and theoetical analyses. Laboatoy expeiments wee fist pefomed in a wave flume to measue the main aspects of beaking-wave chaacteistics ove an idealized submeged finging eef. Results wee epoted fo diffeent monochomatic waves, eef-flat submegences and foe-eef slopes. Dimensionless analysis showed that the elative submegence on the eef flat is a dominant facto affecting the beaking-wave chaacteistics. Results compaisons with othe coastal pofiles wee conducted and some empiical equations wee poposed. A laboatoy study of wave-induced setup ove an idealized finging eef in the absence/pesence of a steep eef cest (idge) was then conducted. Expeimental esults wee epoted fo a ange of eef-cest submegences unde both monochomatic and spectal waves. The behavios of the wave tansfomation and wave setup in the pesence of the idge wee compaed with those in the absence of the idge. Additional high-esolution wave measuements wee conducted to analyze the coss-shoe evolution of hamonic waves and momentum flux. The undetow measuements wee also supplemented to study the vetical flow stuctues acoss the eef. A numeical study based on the one-dimensional (1D), fully nonlinea and weakly dispesive Boussinesq equations is pesented in this thesis. An empiical eddy viscosity model was adopted to account fo wave beaking and a shock-captuing Finite Volume-based solve was employed with a delicate teatment of bounday conditions. The numeical esults wee validated with seveal scenaios fom the afoementioned wave setup expeiments, epesenting vaious combinations of eef pofiles, eef-cest submegences and wave conditions. The model was then applied to study the effects of foe-eef slopes and pofile shapes on the wave tansfomation and setup ove the finging eefs. viii

10 To gain moe physical insights into wave setup geneation on finging eefs, the analytical modeling appoaches wee attempted. Seveal existing one-dimensional hoizontal (1DH) models wee evaluated and poven to be insufficient to epoduce ou laboatoy data. To seek a pope desciption of wave-induced setup unde vey steep eef cest (idge), two theoetical models wee pesented in this thesis: a kinematic model based on the coss-shoe mass balance and some hydaulics, and a dynamic model based on 1D momentum balance and the adiation stess concept. Both models wee validated against the laboatoy data involving a vaiety of eef pofiles and eef-cest submegences unde both monochomatic and spectal waves. The 1DH dynamic model was also applied to some field eefs. The two-dimensional hoizontal (DH) fomulation of the dynamics model was then deived and validated by a field case study. Both models have the empiical paamete elated only to eef mophology. The main contibutions of this thesis ae: (1) a systematic expeimental study of beaking-wave chaacteistics fo finging eefs; () a compehensive undestanding of the ole of a idge stuctue located on the eef edge in detemining the waveinduced setup ove finging eefs; (3) analytical models fo 1DH non-emegent finging eef (with/without a idge). ix

11 LIST OF TABLES Table.1 Test conditions at laboatoy scale Table. Mean value and standad deivation (std) of beake wave height ( H ) obtained fom N successive waves measued by G4 fo a epesentative wave condition ( h.4 m, H.74 m, T 1.5 s and s 1/6)... Table.3 Measued beake wave height ( H and G b I H b ), beake depth ( h b ) and sufzone width ( L s ) of 1 successive waves fo a epesentative wave condition ( h.4 m, H.74 m, T 1.5 s and s 1/6)... Table.4 Measued beake wave heights of 1 successive waves in thee diffeent uns fo a epesentative wave condition ( h.4 m, H.74 m, b T 1.5 s and s 1/6)... 3 Table.5 Values of m and n obtained by least-squae fitting using Eq. (.1); the uncetainties fo m and n ae 95% confidence inteval Table 3.1 Test conditions a Table 3. Selected measuement locations Table 3.3 The aangement of flow measuement locations Table 4.1 A summay of the fou simulated laboatoy expeiments Table 4. The distances of the wave gauges (G1 - G1) fom the toe of foe-eef fo the fou simulated expeiments (Unit: m) Table 4.3 A summay of model paametes fo all the simulations Table 4.4 The R-squaes ( R ) of mean wate level at all measuement locations, the measued maximum setup ( ) and the pedicted maximum setup ( ) o on the eef flat Table 4.5 The measued ( K ) and pedicted ( K ) eflection coefficients m Table 5.1 Maximum coelation between wave-induced setup ( ) and offshoe m n scaling paamete ( H T ) p p x

12 Table 6.1 The values of in pesent model fo diffeent expeimental data with h c Table 6. The values of K in model of Goulay (1996a) fo diffeent expeimental data with hc Table 7.1 Model paamete fo diffeent appoximations to the vaiation of in suf zone Table 7. Model paamete fo available data Table 7.3 Model paamete fo available data by excluding wave eflection.. 19 Table A.1 Measued data with h.3 m and s 1/ Table A. Measued data with h.5 m and s 1/ Table A.3 Measued data with h.7 m and s 1/ Table A.4 Measued data with h.1 m and s 1/ Table A.5 Measued data with h.5 m and s 1/ Table A.6 Measued data with h.5 m and s 1/ Table A.7 Measued data with h.5 m and s 1/ Table B.1 Measued data with monochomatic waves in the absence of the idge Table B. Measued data with monochomatic waves in the pesence of the idge Table B.3 Measued data with spectal waves in the absence of the idge Table B.4 Measued data with spectal waves in the pesence of the idge xi

13 LIST OF FIGURES Fig. 1.1 A typical coal eef pofile (adapted fom /exploations/ 7twilightzone/backgound/plan/media/eef_diagam.html).3 Fig. 1. Wave tansfomation ove coal eef (adapted fom Monismith, 7) Fig. 1.3 Wave-induced setup and wave-diven flow (adapted fom Monismith, 7)... 7 Fig..1 Reviewed coastal pofiles: (a) emeged plane slope (e.g., Goda, 1); (b) submeged finging eef (e.g., Goulay, 1994); (c) submeged plane slope (e.g., Blenkinsopp and Chaplin, 8); (d) submeged baed beach (e.g., Smith and Kaus, 1991); (e) submeged low-cested beakwate (e.g., Van de Mee et al., 5); Hi - incident wave height; SWL - still wate level Fig.. Expeimental setup Fig..3 Definitions of beaking wave chaacteistics and snapshots of diffeent beake types: (a) definitions; (b) plunging beake on the foe-eef; (c) plunging beake on the eef edge; (d) spilling beake on the eef flat... 1 Fig..4 Wave analysis fo tansmitted waves (fom G8) on the eef flat ( h.4 m, H.74 m, T 1.5 s and s 1/6)... 6 Fig..5 Measued quantities as a function of deep-wate wave height ( H ) with foe-eef slope of s 1/6 at diffeent wave peiods (T ) and eef-flat submegences ( h ): (a) Beake type; (b) Beaking location; (c) Beake height ( H b ); (d) Beake depth ( h b ); (e) Total sufzone width ( L s ); (f) Wave-induced setup ( ); (g): Tansmission coefficient ( K t ); (h) Reflection coefficients ( K ). Solid black makes: T 1. s; Solid gey makes: T 1.5 s; Open makes: T 1.67 s. Squaes: h.3 m; Cicles: h.5 m; Tiangles: h.7 m; Diamonds: h.1 m xii

14 Fig..6 Measued quantities as a function of deep-wate wave height ( H ) with eef-flat submegence of h.5 m at diffeent wave peiods (T ) and foe-eef slopes ( s ): (a) Beake type; (b) Beaking location; (c) Beake height ( H b ); (d) Beake depth ( h b ); (e) Total sufzone width ( L s ); (f) Wave-induced setup ( ); (g): Tansmission coefficient ( K t ); (h) Reflection coefficients ( K ). Solid black makes: T 1. s; Solid gey makes: T 1.5 s; Open makes: T 1.67 s. Squaes: s 1/3; Cicles: s 1/6; Tiangles: s 1/9; Diamonds: s 1/ Fig..7 Beake type as a function of: (a) suf-similaity paamete ( ); (b) elative eef-flat submegence ( h / H ) Fig..8 Beaking location as a function of: (a) elative eef-flat submegence h / H ); (b) modified elative eef-flat submegence ( h / H ) ( Fig..9 Beake height index ( ) as a function of deep-wate wave steepness H ( / L ). Squaes: Exp., s=1/3; Cicles: Exp., s=1/6; Tiangles: Exp., s=1/9; Diamonds: Exp., s=1/1. Dashed line: Ped. by Koma and Gaughan (1973); Dotted line: Ped. by Smith and Kaus (1991) with s=1/6; Dash-dot line: Ped. by Blenkinsopp and Chaplin (8); Solid line: Least-squae fitting of Exp. data with powe law Fig..1 Beake height index ( ) as a function of elative eef-flat submegence h / H ). Squaes: Exp., s=1/3; Cicles: Exp., s=1/6; Tiangles: Exp., ( s=1/9; Diamonds: Exp., s=1/ Fig..11 Beake depth index ( ) as a function of H b / gt. (a): Wave beaking on the eef flat; (b): Wave beaking on the eef edge; (c): Wave beaking on the foe-eef. Squaes: Exp., s=1/3; Cicles: Exp., s=1/6; Tiangles: Exp., s=1/9; Diamonds: Exp., s=1/1. Dashed lines: Ped. by Weggel (197).. 4 Fig..1 Beake depth index ( ) as a function of hb / L. (a): Wave beaking on the eef flat; (b): Wave beaking on the eef edge; (c): Wave beaking on the foe-eef. Squaes: Exp., s=1/3; Cicles: Exp., s=1/6; Tiangles: Exp., s=1/9; Diamonds: Exp., s=1/1. Dashed lines: Ped. by Goda (1) b xiii

15 Fig..13 Beake depth index ( ) as a function of deep-wate wave steepness H ( / L ). (a): Wave beaking on the eef flat; (b): Wave beaking on the eef edge; (c): Wave beaking on the foe-eef. Squaes: Exp., s=1/3; Cicles: Exp., s=1/6; Tiangles: Exp., s=1/9; Diamonds: Exp., s=1/1. Dashed lines: Ped. by Smith and Kaus (1991); Dotted lines: Ped. by Goulay (1994); Dash-dot line: Ped. by Blenkinsopp and Chaplin (8) Fig..14 Beake depth index ( ) as a function of elative eef-flat submegence h / H ). Squaes: Exp., s=1/3; Cicles: Exp., s=1/6; Tiangles: Exp., ( s=1/9; Diamonds: Exp., s=1/1. Dashed lines: Ped. by Johnson (6); Dash-dot line: Ped. by Blenkinsopp and Chaplin (8); Solid lines: Least-squae fitting of Exp. data with linea elationship. Solid black makes: Wave beaking on the foe-eef; Solid gey makes: Wave beaking on the eef edge; Open makes: Wave beaking on the eef flat Fig..15 Relative sufzone width ( W ) as a function of deep-wate wave steepness H ( s / L ). Squaes: Exp., s=1/3; Cicles: Exp., s=1/6; Tiangles: Exp., s=1/9; Diamonds: Exp., s=1/ Fig..16 Relative sufzone width ( W ) as a function of the invese of elative eefflat submegence ( H / s h ). Squaes: Exp., s=1/3; Cicles: Exp., s=1/6; Tiangles: Exp., s=1/9; Diamonds: Exp., s=1/1. Dashed line: Ped. by Goulay (1994); Solid line: Least-squae fitting of Exp. data with linea elationship. Dotted lines epesent the 95% confidence limits of the best fitting line Fig..17 Tansmission coefficient ( K ) as a function of suf-similaity paamete t ( ). Squaes: Exp., s=1/3; Cicles: Exp., s=1/6; Tiangles: Exp., s=1/9; Diamonds: Exp., s=1/ Fig..18 Tansmission coefficient ( K ) as a function of elative eef-flat t submegence ( h / H ). Squaes: Exp., s=1/3; Cicles: Exp., s=1/6; xiv

16 Tiangles: Exp., s=1/9; Diamonds: Exp., s=1/1. Dashed line: Ped. by Van de Mee et al. (5); Solid line: Least-squae fitting of Exp. data with linea elationship; Dotted lines epesent the 95% confidence limits of the best fitting line Fig..19 Reflection coefficient ( K ) as a function of the suf-similaity paamete ( ). Squaes: Exp., s=1/3; Cicles: Exp., s=1/6; Tiangles: Exp., s=1/9; Diamonds: Exp., s=1/1. Dashed line: Ped. by Seelig and Ahens (1981) fo plane beach; Dash-dot line: Ped. by Seelig and Ahens (1981) fo ubble-mound beakwate Fig.. Reflection coefficient ( K ) as a function of elative eef-flat submegence ( h / H ). Squaes: Exp., s=1/3; Cicles: Exp., s=1/6; Tiangles: Exp., s=1/9; Diamonds: Exp., s=1/ Fig..1 Enegy loss coefficient ( K ) as a function of elative eef-flat d submegence ( h / H ). Squaes: Exp., s=1/3; Cicles: Exp., s=1/6; Tiangles: Exp., s=1/9; Diamonds: Exp., s=1/1. Solid line: Least-squae fitting of Exp. data with linea elationship; Dotted lines epesent the 95% confidence limits of the best fitting line Fig. 3.1 The eef model. Left: Top view of foe-eef; Right: Side view of eef flat.61 Fig. 3. The idge model. Left: Top view; Right: Side view of idge on eef flat.. 61 Fig. 3.3 Sketch of the expeimental aangement Fig. 3.4 Snapshots of monochomatic wave tansfomation ove eef cest at diffeent phases, the time inteval between consecutive phases is one quate of a wave peiod ( h.45 m, H.95 m and T 1.5 s) Fig. 3.5 Time-seies wave ecods fom selective wave gauges (G1, G3, G6 and G9) acoss diffeent eef pofiles unde monochomatic waves ( h.45 m, H.95 m and T 1.5 s). Dashed lines - without idge; solid lines - with idge Fig. 3.6 Wave specta ( h.4 m, H.87 m and T 1.67 s) fom selective wave gauges (G1, G5, G7 and G9) acoss diffeent eef pofiles: (a) without idge; (b) with idge xv

17 Fig. 3.7 Mean wate level (MWL) offshoe and acoss the eef pofile unde diffeent wave conditions: (a) monochomatic waves; (b) spectal waves. Open cicles - locations of wave gauges; dotted lines - without idge; solid lines - with idge; dashed lines - Still wate level (SWL)... 7 Fig. 3.8 Maximum wave setup on eef flat as a function of deep-wate wave height fo diffeent wave peiods, still wate depths and incident wave conditions. Open makes - without idge; solid makes - with idge Fig. 3.9 Regions and locations in the suf zone (adapted fom Svendsen et al., 1978) Fig. 3.1 Wave height and mean wate level (MWL) acoss the eef pofile ( h.45 m, H.95 m and T 1.5 s) Fig Hamonic wave amplitudes acoss the eef pofiles: (a) without idge; (b) with idge ( ai( i 1, 5) - the i th hamonic wave amplitude; h.45 m, H.95 m and T 1.5 s) Fig. 3.1 Coss-eef vaiation of: (a) dimensional adiation stess ( S xx ); (b) the dimensionless adiation stess ( P ) ( h.45 m, H.95 m and T 1.5 s) Fig The setting of electomagnetic flow mete (EFM) Fig Time-seies of flow ecods fom selective measuement locations (L1, L5, L9, L13) acoss diffeent eef pofiles. Dash lines - without idge; solid lines - with idge. See Table 3.3 fo the distances between those locations and eef edge Fig Vaiation of time-aveaged hoizontal velocity as a function of depth in the absence of the idge ( c gh is the local shallow-wate wave celeity; Solid lines indicate the eo bas based on the standad deviation of phaseaveaged velocity) Fig Vaiation of time-aveaged hoizontal velocity as a function of depth in the pesence of the idge ( c gh is the local shallow-wate wave celeity; Solid lines indicate the eo bas based on the standad deviation of phaseaveaged velocity) xvi

18 Fig. 4.1 Computational domain fo waves popagating ove a finging eef (the exact locations of the wave gauges fo all elevant laboatoy expeiments ae given in Table 4.) Fig. 4. Vaiation of wave height and MWL acoss the flume with diffeent gid sizes Fig. 4.3 Tansmitted wave height ( H ) and maximum wave setup on the eef flat t ( ) as a function of gid numbe pe incident wave length ( N ) Fig. 4.4 Vaiation of wave height and MWL acoss the flume with diffeent bounday conditions Fig. 4.5 Vaiation of wave height and MWL acoss the flume with diffeent tubulence intensities Fig. 4.6 Vaiation of eflection coefficient ( K ) with the slope width ( b ). Solid line: FEM solution of Suh et al. (1997); Open cicles: pesent model Fig. 4.7 Vaiation of wave height and MWL acoss the flume with calibated numeical settings. Solid lines: pedictions by pesent model; Open cicles: laboatoy measuements Fig. 4.8 Time-seies of suface elevations at six locations (G1, G3, G4, G5, G7 and G9). Dashed lines: laboatoy measuements; solid lines: pedictions by pesent model; mse: oot-mean-squae eo Fig. 4.9 Amplitude specta fom selective wave gauges (G1, G3, G4, G5, G7 and G9). Open ba: obseved esults; solid ba: pedicted esults Fig. 4.1 Snapshots of the beaking waves ove the eef cest: (a) Case 1 without idge; (b) Case with idge Fig Vaiations of the wave height and mean wate level (MWL) ove the eef pofile fo Case 1. Solid lines: pedictions by pesent model; Open cicles: laboatoy measuements Fig. 4.1 Vaiations of the wave height and mean wate level (MWL) ove the eef pofile fo Case. Solid lines: pedictions by pesent model; Open cicles: laboatoy measuements xvii

19 Fig Vaiations of the wave height and mean wate level (MWL) ove the eef pofile fo Case 3. Solid lines: pedictions by pesent model; Open cicles: laboatoy measuements Fig Vaiations of the significant wave height and mean wate level (MWL) ove eef pofile fo Case 4. Solid lines: pedictions by pesent model; Open cicles: laboatoy measuements Fig Time-seies of suface elevations at six locations (G, G3, G5, G7, G9 and G11) fo Case 1. Dashed lines: laboatoy measuements; solid lines: pedictions by pesent model; mse: oot-mean-squae eo Fig Time-seies of the suface elevations at six locations (G, G3, G5, G7, G9 and G11) fo Case. Dashed lines: laboatoy measuements; solid lines: pedictions by pesent model; mse: oot-mean-squae eo Fig Time-seies of the suface elevations at six locations (G, G3, G5, G7, G9 and G11) fo Case 3. Dashed lines: laboatoy measuements; solid lines: pedictions by pesent model; mse: oot-mean-squae eo Fig Wave specta at eight locations (G, G4 - G9, and G11) fo Case 4. Dashed lines: laboatoy measuements; solid lines: pedictions by pesent model Fig Vaiation of the mean wate level (MWL) ove the eef pofile fo the Test 6 of Skotne and Apelt (1999). Dashed line: pediction by Skotne and Apelt (1999); Solid line: pediction by pesent model; Open cicles: laboatoy measuements by Skotne and Apelt (1999) Fig. 4. Vaiations of the wave height and mean wate level (MWL) ove the eef pofile fo the Test 48 of Demibilek and Nwogu (7). Dashed lines: pedictions by Demibilek and Nwogu (7); Solid lines: pedictions by pesent model; Open cicles: laboatoy measuements by Demibilek and Nwogu (7) Fig. 4.1 Vaiations of the wave height and mean wate level (MWL) ove eef pofile with diffeent foe-eef slopes. Light black solid line: V:H=1:1; Dash-dot line: V:H=1:3; Dak black solid line: V:H=1:6; Dotted line: V:H=1:1; Dashed line: V:H=1: xviii

20 Fig. 4. Reflection coefficient ( K ), tansmission coefficient ( K ), beaking location elated to the toe of foe-eef (positive if shoewad), maximum wave setdown ( b ) and maximum wave setup on the eef flat ( ) as a function of suf-similaity paamete ( ) Fig. 4.3 Vaiations of the wave height and mean wate level (MWL) ove eef pofile with diffeent foe-eef pofiles. Light black solid line: concave ac with the cuvatue=.15; Dash-dot line: concave ac with the cuvatue=.75; Dak black solid line: plane slope; Dotted line: convex ac with the cuvatue=.75; Dashed line: convex ac with the cuvatue= Fig. 5.1 Configuation of an idealized finging eef with a ectangle idge and some notations adopted in this chapte Fig. 5. Pedicted wave setup ( ) vs. obseved wave setup ( ) using the model p of Tait (197) fo diffeent eef-cest submegences, eef pofiles and wave conditions ( h c - eef-cest submegence; - empiical paamete in the model; cicles - monochomatic waves; squaes - spectal waves; o t R - R- p o squae; Solid line - ) Fig. 5.3 Pedicted wave setup ( ) vs. obseved wave setup ( ) using the model p of Goulay and Collete (5) fo diffeent eef-cest submegences, eef pofiles and wave conditions ( h c - eef-cest submegence; o K p - empiical paamete in the model; cicles - monochomatic waves; squaes - spectal waves; R - R-squae; Solid line - p ) Fig. 6.1 Backwad flow ove the eef cest which esembles the flow ove a boadcested wei Fig. 6. Snapshots of monochomatic wave tansfomation ove eef cest at diffeent phases: (a) without idge; (b) with idge. The aows indicate the occuence of the fee falls o xix

21 p Fig. 6.3 Pedicted wave setup ( ) vs. obseved wave setup ( ) fo diffeent eefcest submegences ( h c ) unde monochomatic waves: (a) without idge; (b) o p with idge ( - scaling facto in the model; Solid line - ) o p Fig. 6.4 Pedicted wave setup ( ) vs. obseved wave setup ( ) fo diffeent eefcest submegences ( h c ) unde spectal waves: (a) without idge; (b) with o p idge ( - scaling facto in the model; Solid line - ) o p Fig. 6.5 Pedicted wave setup ( ) vs. obseved wave setup ( ) fo diffeent eefcest submegences ( h c ): (a) Dataset fom Seelig (1983); (b) Dataset fom Goulay (1996a); (c) Dataset fom Demibilek et al. (7) ( - scaling p o facto in the model; Solid line - ) Fig. 7.1 Configuation of a finging eef with a idge and some notations adopted in this chapte Fig. 7. Compaison of beaking state tansition of laboatoy obsevations with theoetical pedictions fo diffeent eef-cest submegence: (a) without idge; (b) with idge. Solid lines epesent least-squaes fits of Eq. (7.19); Dotted lines epesent the 95% confidence limits of the best fitting line. 181 Fig. 7.3 Laboatoy eef pofiles investigated in this chapte: (a) expeimental setup in Chapte 3 without idge; (b) expeimental setup in Chapte 3 with idge; (c) Seelig (1983); (d) Goualy (1996a); (e) Demibilek et al. (7) ( h c - eef-cest submegence; s - foe-eef slope; se - equivalent foe-eef slope; sa - aveage foe-eef slope; Hi - incident wave height; SWL - still wate level) Fig. 7.4 The measued as a function of deep-wate wave height ( H ) fo: (a) diffeent eef-cest submegences ( h c ) and (b) diffeent foe-eef slopes ( s ) p Fig. 7.5 Pedicted wave setup ( ) vs. obseved wave setup ( ) fo diffeent eefcest submegences ( h c ): (a) without idge; (b) with idge (Red makes - o o xx

22 monochomatic waves; Blue makes - spectal waves; Solid line - p ) o p Fig. 7.6 Pedicted wave setup ( ) vs. obseved wave setup ( ) fo diffeent eefcest submegences ( h c ): (a) Dataset fom Seelig (1983); (b) Dataset fom Goulay (1996a); (c) Dataset fom Demibilek et al. (7) (Solid line - p ) o Fig. 7.7 Sensitivity of the model esults to the efeence paametes. On the abscissa, the value of each paamete X is divided by its calibated value X. On the odinate, the aveage wave setups ( ave o ) ae nomalized by thei ave coesponding aveage value unde efeence conditions ( ) ( X epesents, m o ; X epesents.54, m 1. o ave.4 ; is obtained by aveaging the data in Fig. 7.5a) Fig. 7.8 Pedicted wave setup ( ) vs. obseved wave setup ( ) fo diffeent field p studies ( - paamete in the poposed model; p o o R - R-squae; Solid line - ; ND - N-deployment; GD - G-deployment) Fig. 7.9 Plan view of an idealized eef-lagoon-channel system, including a shoeline, a eef (below the dashed line) and a channel (above the dashed line). Flow moves fom points A though E.... Fig. 7.1 Time-seies of field obsevations and model pedictions (Open Cicles: Obsevations; Dashed line: Pedictions by L9; Solid line: Pedictions by the pesent model) Fig. C.1 Sketch of the expeimental setup Fig. C. Reef flat models: (a) smooth eef flat (Side view); (b) smooth eef flat (Font view); (c) poous eef flat (Side view); (d) poous eef flat (Font view) Fig. C.3 Wave beaking: (a) smooth eef flat; (b) poous eef flat ( H.1 m, T 1.5 s, h.35 m) xxi

23 Fig. C.4 Maximum (a) and aveage (b) wave setup as a function of deep-wate wave height ( H ) fo diffeent eef flats ( - maximum wave setup; a - aveage wave setup; Open makes - smooth eef flat; solid makes - poous eef flat; h.35 m) Fig. F.1 Sufzone seabed pofile fo an idealized eef and some notations adopted in this appendix Fig. G.1 Sufzone seabed pofile fo an idealized eef with a ectangle idge and some notations adopted in this appendix Fig. H.1 Pedicted wave setup ( ) vs. obseved wave setup ( ) fo diffeent field p studies using the model of Tait (197) ( - paamete in the poposed model; R - R-squae; Solid line - p o o ; ND - N-deployment; GD - G- deployment) Fig. H. Pedicted wave setup ( ) vs. obseved wave setup ( ) fo diffeent field p studies using the model of Goulay and Collete (5) ( - paamete in the poposed model; R - R-squae; Solid line - p o ; ND - N- deployment; GD - G-deployment) o xxii

24 LIST OF SYMBOLS a B wave amplitude coss-shoe eef-cest (idge) width B c coss-shoe cest width of low-cested stuctue b b alongshoe eef-cest (idge) length coss-shoe foe-eef width C d bottom dag coefficient g C d bottom dag coefficient fo channel C d bottom dag coefficient fo eef C Couant numbe c c g g c g ci g ct wave celeity/phase speed goup velocity deep-wate (offshoe) goup velocity goup velocity of incident waves goup velocity of tansmitted waves d eef cest (idge) height D total instantaneous wate depth, D h D c citical flow depth above eef cest D total mean wate depth, D h D mean wate depth on eef flat, D h E E wave enegy density deep-wate (offshoe) wave enegy density E c E i E E t specific enegy of open channel flow enegy density of incident waves enegy density of eflected waves enegy density of tansmitted waves xxiii

25 E fb enegy flux dissipated due to bottom fiction befoe wave beaking F feeboad of low-cested stuctue F co nonlineaity paamete in Goulay (1994) f fiction coefficient f p g peak wave fequency gavitational acceleation H local wave height H deep-wate (offshoe) wave height H b beake wave height G H b I Hb beake wave height measued fom wave gauge beake wave height measued fom ecoded image H i incident wave height H eflected wave height Ht tansmitted wave height H s significant wave height H s deep-wate (offshoe) significant wave height H ms oot-mean-squae wave height Hms h h h b h c hg h p h hs K deep-wate (offshoe) oot-mean-squae wave height still wate depth still wate depth in the deep section of the flume beake wate depth still wate depth on the eef cest (eef-cest submegence) still wate depth in channel epesentative wate depth fo suf zone still wate depth on the eef flat (eef-flat submegence) still wate depth at endpoint of suf zone wave-cest asymmety facto in Goulay (1996a) xxiv

26 K d enegy loss coefficient K p eef-cest shape facto in Goulay and Collete (5) K eflection coefficient p K eflection coefficient pedicted by model m K eflection coefficient measued in the expeiments Kt k L L L e L g L l L L s e L tansmission coefficient wave numbe local wave length deep-wate (offshoe) wave length distance between beaking point and eef edge coss-shoe channel width coss-shoe lagoon width coss-shoe eef-flat width total sufzone width effective coss-shoe eef-flat width N gid numbe pe incident wave length n M Manning coefficient P nomalized coss-shoe adiation stess QB total backwad volumetic flux duing the backwash phase of wave beaking QF total fowad volumetic flux duing the up-ush phase of wave beaking Q F offshoe total fowad volumetic flux duing the fowad phase of wave motion q mean flow ate pe unit width q c R b R f citical flow ate fo open channel flow ad-hoc dissipative tem accounting fo wave beaking in the numeical model ad-hoc dissipative tem accounting fo bottom fiction in the numeical model S deep-wate (offshoe) wave steepness, S H gt / xxv

27 S ij adiation stess tenso S xx coss-shoe adiation stess b S xx coss-shoe adiation stess at beaking point s S xx coss-shoe adiation stess at the endpoint of suf zone s s a s e T T p T s T * T t U U b U g U foe-eef slope aveaged slope fo composite foe-eef equivalent slope fo foe-eef including eef cest (idge) effect wave peiod peak wave peiod significant wave peiod mean zeo-upcossing peiod duation of beaking event in the numeical beaking model the time at which beaking event stats in the numeical beaking model coss-shoe depth-aveaged mean velocity coss-shoe depth-aveaged mean velocity at beaking point coss-shoe depth-aveaged mean velocity in channel coss-shoe depth-aveaged mean velocity on eef flat u coss-shoe time-mean velocity u coss-shoe time-mean velocity at a efeence wate depth ef ut () coss-shoe instantaneous velocity u b coss-shoe instantaneous bottom velocity u u w coss-shoe instantaneous velocity at the wate depth z coss-shoe wave obital velocity W sponge laye width in numeical model W g alongshoe channel length W Ws alongshoe eef-flat length elative sufzone width xxvi

28 w w vetical wave obital velocity wt () vetical instantaneous velocity x b x c x e x x s coodinate of beaking point coodinate of eef cest coodinate of eef edge coodinate of leeside edge of the idge coodinate of endpoint of suf zone scaling facto fo enegy dissipation due to bottom fiction 1,, 3 empiical coefficients in the numeical beaking model empiical paamete in the poposed kinematic model 1 m empiical paamete in the poposed dynamic model beake depth index; beake depth index fo foe-eef beake depth index fo eef flat beake depth index fo plane beaches model b empiical coefficient elated to tubulence intensity in numeical beaking enegy dissipation ate due to wave beaking f enegy dissipation ate due to bottom fiction fee wate suface elevation deviation of mean wate level fom still wate level (wave-induced setup/setdown) a aveaged wave-induced setup ove eef-flat width b L o p wave-induced setdown at beaking point wave-induced setup in lagoon maximum wave-induced setup on eef flat maximum wave-induced setup obseved in the expeiments maximum wave-induced setup pedicted by the model xxvii

29 ave maximum wave-induced setup aveaged fo a dataset efeence maximum wave-induced setup aveaged fo a dataset ave satuated value fo beaking cessation in the numeical beaking model ( F ) t theshold value fo beaking inception in the numeical beaking model ' ( I ) t foe-eef slope angle equivalent foe-eef slope angle enegy eduction coefficient wate density wave length scaling paamete defined by h/ L b x b empiical eddy viscosity in the numeical beaking model coss-shoe bottom shea stess angula fequency suf-similaity paamete in tems of deep-wate wave height suf-similaity paamete in tems of beake wave height i suf-similaity paamete in tems of incident wave height beake height index x gid size t time step xxviii

30 CHAPTER 1 INTRODUCTION 1.1 Significance of Wave Dynamics ove Coal Reefs Coal eefs ae abundant in shallow topical and subtopical coastal egions, whee significant amounts of suface wave enegy can be dissipated though wave beaking and bottom fiction pocesses. Coal eefs povide a wide and vaied habitat that suppots some of the most divese assemblages of living oganisms found anywhee on Eath (Dawin, 184). It has been well known ove decades that eefs epesent islands of enomous poductivity because they ae efficient at tapping nutients, zooplankton, and possibly phytoplankton fom the suounding wates. Coal eefs exist unde a wide ange of physical envionments. Flow on eefs can be diven by waves, tides, wind, and buoyancy. The dominant diving mechanism vaies among diffeent eefs and is a combination of geomophologic, meteoological and oceanogaphic focing condition at a specific site. Although winds, tides and buoyancy focing could individually dominate the hydodynamic chaacteistics of some eefs at cetain times, wave action is being inceasingly ecognized as a significant agent in detemining the eef-flat envionment, island fomation as well as many aspects of eef aea ecology. To date, this has been the pimay focus of neashoe hydodynamic studies. Thee ae sound ecological, envionmental, geomophic as well as engineeing implications fo eseach on wave dynamics ove coal eefs. Fo example, the wave-induced setup can dive mean cuents and build up a ciculation in a lagoon, which ae cucial to the poduction, dispesal as well as etention of laval fish, coals and othe invetebates (Hench et al., 8). Meanwhile, flow-mediated mass tansfe may also facilitate eef zonation and segegation, and pevent coal bleaching, this highlights the pimay connection between eef health and hydodynamic pocesses (Monismith, 7). Moeove, wate movement associated with waves is essential fo sediment tanspot in the eef aea, which ultimately 1

31 contibutes to eef oganism distibution, geomophic development of eef systems and island shoeline stability (Kench and Bande, 6). Additionally, coal eefs and eef-top islands ae becoming inceasingly significant to coastal engineeing woks in topical egion, whee waves popagate ove the eef cest and wate tubulence associated with wave beaking imposes foces on the vaious types of stuctues, such as touist facilities and infastuctue on coal cays, maitime activities in eef wates as well as the locations fo navigation aids and weathe stations (Goulay and Collete, 5). Last but not the least, coal eefs shelte many topical islands fom the flood hazads associated with tsunamis, huicanes, and high suf events (Roebe et al., 1), an impoved undestanding of the physics of wave-induced inundation ove eef systems is equied to develop pedictive models fo huicane/typhoon emegency planning puposes and to assess the huicane/typhoon-induced eosion (Demibilek and Nwogu, 7). 1. Wave Dynamics ove Coal Reefs at the Reef Scale The hydodynamics of coal eefs unde wave effects entails a wide ange of scales of fluid motions: the lagest scale as egional ciculation and lagoon flushing, smalle depth scale tubulent flow evoked by eef mophology, and finally the smallest scale of flow in the benthic oganisms, i.e., coal colonies (Monismith, 7). At a eef-scale of 1 to 1 m, the inteaction between the wave pocesses and the eef pofile is of inceasing inteest fo eseaches and enginees. Seveal studies have examined the inteaction of waves with eef stuctues fo the pupose of engineeing design. Results showed that coal eefs act much like submeged beakwates, bas, o depth-limited coastlines which lead to damatic tansfomations in wave chaacteistics and apid attenuation of wave enegy (Kench and Bande, 6). It is the wave enegy leaking onto coal eef flats that is of citical impotance in detemining the tanspot of sediment aound eef systems and contolling the flow and exchange of oceanic wate thoughout eef systems, this ultimately contibutes to island mophology and eef poductivity.

32 The physical stuctue of coal eefs is notably diffeent fom that of nomal coastal beaches. The geomophology of a typical baie (platfom) eef is divided into fou main egions (Fig. 1.1): (1) a sloping foe-eef; () a steep eef cest (a idge o simila configuation); (3) a eef flat (eef-top) whee the bottom slope is minimal, and (4) a deep open lagoon. If the lagoon is closed o thee is no lagoon, i.e., the eef is backed by a coastline, it is called a finging eef. Depending cucially upon the amount of light eceived fom the sun as well as on the consumption of nutients and lavae, coal eefs commonly gow to about mean low tide level and thei matue fom of a plana eef is essentially hoizontal. Hence tidal modulation usually comes into play. At low tide, waves will beak on the foe-eef, dissipating most enegy though tubulence ove a limited spatial aea, thus no significant wave enegy will popagate acoss the eef flat; at high tide, depth-limited waves ae able to popagate acoss the eef flat, eithe as tansfomed incident waves, efomed waves, o wave boes. Meanwhile, unlike beaches, which typically have mild slopes and elatively smooth bottoms, coal eefs often fom a steep tansition fom elatively deep to shallow wate and geneate a vey ough bottom suface. Moeove, coal eef oganisms ae known to fom some of the oughest sufaces in the coastal ocean, thus fictional dissipation ates can be expected to be highe than sandy o even ocky eef sites located along the continental shelf. Fig. 1.1 A typical coal eef pofile (adapted fom /exploations/ 7twilightzone/backgound/plan/media/eef_diagam.html). 3

33 The pecise chaacte of wave modification and spatial extent of enegy dissipation ae contolled by the mophology of the eef stuctue (elevation, eef slope, and eef flat width) and elative wate depth at the eef cest (Kench and Bande, 6). As waves shoal and beak on the foe-eef slope, the location of the initial beake zone on the seawad edge of the eef ceates a wide suf zone in which waveinduced setup and wave-diven flow ae significant in detemining both the mean wate levels and mass tanspot on the eef flat. The whole pocess can be divided into two subjects (shown below), of which much liteatue is concened with one o both. As fo the enegy balance, the tubulence enegy dissipation on eefs aises fom both the effect of coss-eef cuent and the loss of enegy fom beaking of suface gavity waves Wave tansfomation and beaking A wave tansfomation pocess can involve up to five sub-pocesses, namely wave shoaling, wave efaction, wave diffaction, wave bed fiction damping, and enegy dissipation due to wave beaking (Nelson, 1996). In view of wave enegy dissipation, wave attenuation is involved duing the whole pocess. As waves popagate ove the eef, they undego seveal coss-shoe tansfomations (see Fig. 1. fo an illustation). Waves fist begin to inteact with the foe-eef when thei wavelength becomes compaable to the local wate depth. As they move futhe shoewad and shoal, the waves incease in height while dissipate some of thei enegy due to bottom fiction. Eventually thei heights become some citical faction of the wate depth and the waves become unstable and beak. At a given coss-eef location, a maximum allowable wave height is contolled by the local wate depth accoding to the depth-limited condition H b h (1.1) whee H b is the beake wave height and is a empiical beake index,which is smalle on the eef flat than that often used in engineeing pactice (.78 ); h is the local still wate depth. The waves may beak at foe-eef slope (plunging) o on the oute eef flat (spilling), depending on local wate depth and offshoe wave conditions. Fo wave motion though suf zone, the tansfomation of wave 4

34 continues as they beak, esulting in a moving boe. The suf zone always extends ove a finite distance on the eef flat, stating fom the incipient beaking point to the location whee enegy dissipation by wave beaking ceases, boe disappeas and oscillatoy waves ove the eef flat efom. The efomed waves continue to popagate towads the shoeline, while dissipating some of thei enegy to bottom fiction. Unlike beaches whee the wate depth appoaches zeo towad shoe, the wate depth ove the eef flat is nonzeo and elatively constant. Waves with height smalle than the citical beaking height detemined by Eq. (1.1) ae then fee to pass onto the eef flat. The eef thus seves as a low-pass wave height filte by filteing out waves with heights lage than the depth-limited maximum. Fo spectal waves, wave beaking and accompanying enegy tansfe to both high and low fequency waves may cause a flattening and boadening of the wave spectum afte the waves pass ove the eef flat (e.g., Hady and Young 1996; Jago et al., 7; Péquignet et al., 11). Low-fequency motions may also be geneated due to the so-called suf beat o eef-flat esonance (e.g., Demibilek et al., 7; Péquignet et al., 9). Shaoling Beaking Refomed waves Shoe Reef flat Foe eef Lagoon Fig. 1. Wave tansfomation ove coal eef (adapted fom Monismith, 7). 1.. Wave-induced setup and wave-diven flow Wave-induced setup and wave-diven flows have pobably eceived the most attention. Reefs often suound lagoons and low atolls and povide potection fom incident waves by dissipating the wave enegy though the suf zone on the seawad side of the eef. Wave-induced setup and cuents due to wave beaking ove the 5

35 eef could theefoe have a significant impact on the ciculation and flushing of the lagoon. Most eseaches analyzed two-dimensional (D) eef pofiles with a seawad sloping foe-eef and a hoizontal eef flat (see Fig. 1.3). The waves beak on the sloping foe-eef, dissipating thei enegy and geneating a ise of mean sea level. In egad to wave enegy, this means that not all enegy lost fom the wave field is conveted to heat and tubulence, some enegy is used to maintain a sea suface slope known as wave setup. The maximum setup occus at o nea the eef edge, i.e., at the seawad side of the hoizontal eef flat, because ove the eef flat, whee the depth is constant, thee is no focing due to wave beaking. Consequently, a coss-eef flow diven by the baotopic pessue gadient esulting fom the maximum setup in the suf zone aises since the wate level at the downsteam side of the eef is always assumed to be the same as that seawad of the suf zone (sea level). Then the bottom fiction dissipates the enegy associated with the coss-eef flow, which in tun affects the wave setup and cuent acoss the eef flat and thoughout the lagoon. Fo a eef with an open lagoon (baie eef), the cuent eventually exits the lagoon via gaps o channels between the lagoon and open sea and a egional ciculation always foms in analogy to the ip cuent aound baed beaches o detached beakwates. Fo a eef with a closed lagoon o without lagoon (finging eef), the coss-shoe wave setup on the eef flat is moe o less constant thus the wave-geneated flow is elatively weak. Wate may also fom closed gyes o eddies on the eef flat because the wave setup can vay lateally along the line of the eef cest. Wate also flows lateally along the foe eef. The extent of this lateal flow is contolled by the lateal bathymety and fiction (Hean, 11). 6

36 Waves x Beak point Wave setup ( x) Suf zone Foe eef hx ( ) Cuent Reef flat Lagoon Shoe Fig. 1.3 Wave-induced setup and wave-diven flow (adapted fom Monismith, 7) 1..3 Theoetical backgound Owing to the vaied eef pofiles as well as lage oughness of the eef flats, coal eef hydodynamics at eef scale pesents consideable difficulties fo both expeimental and numeical modeling. Theefoe, expeimental and numeical studies on wave inteaction with coal eefs ae limited. Howeve, field measuements of wave tansfomation and elated topics (wave-induced setup and wave-diven flow) can be found in a lage sum of published papes. Detailed liteatue eviews on the concened topics will be given in subsequent chaptes. In the following paagaphs, a theoetical illustation of coal eef hydodynamics is intoduced fist, which functions as the fundamental physical laws of the pesent study. The basic idea is that incident waves beak on the foe-eef slope of the eef and push wate onto the eef flat and then into the lagoon (if it exists) by wave setup. The theoetical explanation of this pocess is fistly advanced by Longuet-Higgins and Stewat (196) who showed that spatial gadient in wave adiation stess, i.e., the depth-integated momentum flux tem associated with popagating waves, appeas as a body foce acting on the mean flow. The efeence fame with x-axis othogonal to the shoe and y-axis paallel to the shoe is consideed hee. The 7

37 depth-integated wave-aveaged momentum equation govening the hoizontal flow can be witten as (e.g., Mei et al., 5) b U j U j 1 Sij j Ui g t x x ( h) x ( h) i j i (1.) whee U U U is the ith component of the Lagangian mean velocity, E s i i i witten hee as the sum of the depth-aveaged Euleian mean (wave-aveaged) velocity ( U ) and depth-aveaged Stokes dift velocity ( U ) (Monismith, 7), E i is the deviation of the mean wate level (MWL) fom still wate level (SWL), h is the local still wate depth, S ij is the adiation stess tenso and b j is the bottom shea stess. Fo waves popagating in the x diection (nomal to shoe), Eq. (1.) educes to the following hoizontally one-dimensional (1DH) fom in steady state s i b 1 dsxx d q x gh ( ) x dx xh (1.3) The coesponding continuity equation gives q x (1.4) whee q U( h ) is the coss-shoe mean flow ate pe unit width, U is the coss-shoe depth-aveaged mean velocity, S xx is the coss-shoe adiation stess and can be estimated using linea wave theoy, stess. b x is the coss-shoe bottom shea In the suf zone, the decease in wave height due to fictional dissipation is geneally vey slow compaed with the apid decease due to beaking. Even fo a spilling beake that has a wide suf zone than a plunging beake, existing liteatue (e.g., Thonton and Guza, 1983) has shown that the atio of fiction dissipation ate to beaking dissipation ate is vey small (e.g., less than 3%). Thus compaing with wave beaking, the wave decay associated with fictional dissipation is negligible and it contibutes little to the vaiation of S xx and the wave 8

38 setup. The decay of olle indeed contibutes to wave-induced setup as found in the existing liteatue (e.g., Apotsos et al., 7), but this is essentially a contibution fom nonlinea beaking waves (spilling beakes). As it will be discussed in section 3.4.4, this is the pimay eason why using S xx based on linea-shallow wave appoximation in suf zones does not agee with the laboatoy obsevations. Thee ae some empiical models in the liteatue to explicitly account fo this as additional contibution to the adiation stess calculated by using linea wave theoy, see e.g., Svendsen (1984a). Fo a baie eef with an open lagoon, pevious investigations (e.g., Lowe et al., 9a) found that thee was a net ( q ) wave-induced mass tanspot (Mean Euleian flow plus Stokes dift) fom the eef flat to the lagoon and that the Stokes dift may contibute to the wave-induced setup as shown in Eq. (1.3). Howeve, thee is no net wave-induced mass tanspot in a closed flume o fo a finging eef, and the Stokes dift is not impotant fo finging eefs. 1.3 Objectives and Scope Objectives The objective of this study is to investigate expeimentally, numeically, and theoetically the wave dynamics (wave tansfomation, wave-induced setup and wave-diven flow) ove idealized finging eefs. Of paticula impotance is to evaluate the maximum wave-induced setup on the eef flat. Both monochomatic and spectal waves ae consideed. Special attention is given to the coss-shoe inteaction between beaking waves and eef-cests (idges) Scope The scope of expeimental, numeical and theoetical investigations is listed below: Laboatoy expeiments: Two sets of laboatoy expeiments ae conducted in a wave flume. Both expeiments ae concened with 1DH eef pofiles. The fist set of expeiments is a 9

39 diect measuement of the main aspects of beaking-wave chaacteistics ove an idealized finging eef model, including the beake type and location, the beake indices, sufzone width, wave tansmission, eflection and enegy dissipation. Diffeent combinations of wate depths and foe-eef slopes ae examined with a seies of monochomatic wave conditions. The pimay focus of the second set of expeiments is given to the compaison of wave-induced setup in the pesence/absence of a ectangula idge located at the eef cest subjected to both monochomatic and spectal waves. As supplements to the second set of expeiments, a spatially high-esolution wave measuement is conducted with monochomatic waves, followed by flow undetow measuements at diffeent cossshoe locations. The effects of eef-flat suface oughness on the wave tansfomation and setup ae also investigated expeimentally. Numeical expeiments: Among the available neashoe modeling techniques and tools, a Boussinesq-type model is calibated and validated. The numeical expeiments ae compaed with fou epesentative scenaios fom the second set of expeiments above. Special attention is given to the coss-shoe vaiations of wave height and setup. A eview of the published numeical woks on simila topic is also conducted. The model is then applied to study the effects of both foe-eef slope and foe-eef shape on the coss-shoe wave tansfomation and setup. Only 1DH wave modeling is pefomed; simulations on wave-induced cuent o in the pesence of ambient cuent ae not discussed. Analytical models: Data fom the fist set of expeiments above ae used to validate the applicability of available empiical fomulae fo othe coastal pofiles into the finging eefs. Some empiical expessions fo the idealized finging eefs ae also poposed. Fo the second set of expeiments above, the pefomance of existing analytical models is evaluated and thei limitations to epoduce the expeimental data ae discussed. Two analytical models ae then developed in this thesis to intepet the geneation of wave setup: a kinematic theoy based on mass balance, and a dynamic theoy 1

40 based on momentum and enegy balance. While the kinematic model is only employed to account fo the data in the 1DH laboatoy expeiments, the dynamic model is applied to both laboatoy and field studies. A DH fomulation fo the dynamic model is also deived. Both models ae deived with some simplifications. 1.4 Outline This chapte has intoduced seveal geneal aspects of coal eef hydodynamics at eef scale and povides the theoetical backgound as well as the objectives and scope of the pesent study. Chapte epots an expeimental study on the influences of vaying elative eef-flat submegence and foe-eef slopes on the popeties of beaking waves ove an idealized finging eef. Chapte 3 pimaily deals with laboatoy expeiments on the effect of a idge stuctue located at a eef edge on the wave tansfomation, setup and flow on the eef flat. A numeical model based on the depth-integated, fully nonlinea Boussinesq-type equations is validated in Chapte 4 to epoduce some data obtained in Chapte 3 as well as to conduct new numeical expeiments. Subsequently in Chapte 5, some existing 1DH analytical models ae tested against the wave setup data obtained in Chapte 3. In Chapte 6, an analytical model fo wave setup is pesented based on the mass balance and hydaulic theoies, and validated by a seies of laboatoy datasets. In Chapte 7, an altenative analytical model is fomulated using momentum and enegy balances, and validated by the same laboatoy datasets investigated in Chapte 6 as well as some field datasets. The DH fomulation and its application to field data ae also discussed. Chapte 8 summaizes the majo conclusions of the pesent study and suggests seveal avenues fo futue wok. Some oiginal laboatoy data in Chaptes and 3, supplemental expeiments on the effects of eef-flat oughness as well as some equation deivations fo numeical and analytical models can be found in the Appendices. 11

41 CHAPTER BREAKING WAVE CHARACTERISTICS FOR LABORATORY FRINGING REEFS.1 Liteatue Review As intoduced in Chapte 1, wave tansfomation ove a coal eef is simila to the wave tansfomation ove shallow shelves. The ocean waves fist shoal on the foeeef and then beak eithe on the foe-eef o the eef flat. The sufzone width is nomally defined as the hoizontal distance between the location whee waves stat beaking and the location whee wave beaking ceases. A significant amount of wave enegy can be dissipated duing wave beaking. Wave beaking ove coal eefs is esponsible fo many neashoe pocesses, and thus knowledge of wave beaking is an essential pat of coal eef hydodynamics. Fo example, the beaking-induced setup and nea-shoe cuents have a significant impact on the ciculation and flushing of lagoons, and thus the tanspot of oganisms, nutient, sediments (e.g., Hean, 1999; Callaghan et al., 6). Beaking waves may also affect mass tansfe ates fo flow though and ove coals and biological toleance of benthic substates (e.g., Madin et al., 6; Monismith, 7). The beakingwave foce imposing on atificial eefs is a peliminay design consideation fo shoeline potection (e.g., Ahens, 1987; Shilal et al., 7; Camo et al., 11). The hydodynamics of wave beaking ove coal eefs is contolled mainly by the mophology of the eef pofiles (foe-eef slope, eef-flat bathymety and eef-flat width) and the chaacteistics of the incident waves. A majo diffeence between a submeged finging eef and an emeged plane beach is that finging eef usually has a eef flat whee the wate depth is a moe o less constant. The pesence of the eef flat may: (1) delay wave beaking and move the incipient beaking point shoewad (Goulay, 1994; Demibilek et al., 7); () make the wave beaking and the esulting nea-shoe cuents diffeent fom those fo a plane beach (Goualy and Collete, 5); (3) function like a semi-infinite boad wei which may impose a hydaulic contol on the amount of wate leaving the eef flat if the wate above it is vey shallow (Goulay, 1996a). In contast to beaches which 1

42 typically have mild slopes, coal eefs often have a steep tansition fom elatively deep to shallow wates. Recently, thee is also an inceasing inteest in using atificial eefs as the beakwates while seving othe puposes (e.g., pomote maine life, contol eosion o impove sufing). Atificial eefs ae elatively shot-cested compaed to natual eefs which have a typical eef-flat width of seveal hundeds to thousands metes. Ove the past decades, numeous theoetical, expeimental and numeical studies have been done on vaious aspects of wave beaking pocess fo diffeent natual and man-made coastal stuctues such as beaches (plana o baie) and low-cested stuctues (LCS) (e.g., the low-cested beakwate). Howeve, the mechanics of wave beaking is still not fully undestood. The majoity of cuent knowledge about the chaacteistics of beaking waves (e.g., the beake type, beake depth index and beake height index) has been obtained meely fom expeimental measuements of waves beaking on emegent, plana slopes in two-dimensional laboatoy wave flumes. Repesentative studies ae the ealie wok by Ivesen (195), Galvin (1969), Goda (197), Weggel (197), Svendsen (1984a), Nelson (1994) and the ecent wok by Tsai et al. (5), She and Canning (7), Camnenen and Lason (7) and Goda (1). The empiical elationships obtained fom these studies have been used in coastal engineeing designs. Fo LCS, simila studies can be found in e.g., Johnson (6); Van de Mee et al. (5). Not much eseach has been done on the chaacteistics of wave beaking ove coal eefs. Smith and Kaus (1991) pioneeed the study of wave beaking ove eef-like submeged stuctues; they caied out a compehensive laboatoy study of wave inteaction with submeged bas and atificial eefs with vaious geometies. Thei esults demonstated that waves with identical deep-wate chaacteistics boke diffeently on plane beaches and bas/eefs. Significant diffeences wee found in popeties such as beake type, beake height index, beake depth index, plunge distance and splash distance. In paticula, it was found that the tansition values of the suf-similaity paamete (discussed late) used to classify the beake types fo plane slopes wee invalid fo baed/eef pofiles. Goulay (1994) examined the 13

43 tansfomation of monochomatic waves on a finging eef with a steep seawad face and a gadually sloping eef flat. His esults demonstated that beaking conditions could be descibed using a nonlineaity paamete denoted in his pape by F co, which contains both offshoe wave and local bathymety infomation: waves with identical deep-wate steepness (.48 ) boke in a plunging manne when Fco 15 and in a spilling manne when Fco 1. Moe ecently, Blenkinsopp and Chaplin (8) pefomed an expeimental study of the effects of elative eef-cest submegence on wave beaking ove a submeged, tuncated 1/1 slope (i.e., the plane slope is tuncated at a cetain level below the wate suface and thee is no hoizontal pat to mimic the eef flat in thei model). They found that the eef-cest submegence was a dominant facto affecting wave beaking as well as the tansmission and eflection chaacteistics. A sketch of the coastal pofiles eviewed in this section is given in Fig..1. Even though the above mentioned eseaches have caied out some measuements of wave beaking ove submeged eef-like stuctues, each of them used a specific eef pofile with a single foe-eef slope duing the expeiments, and consequently the potential effects of foe-eef slope on beaking waves wee not examined. Since it is well known that the chaacteistics of wave beaking on plane beaches depend cucially on the so-called suf-similaity paamete which epesents the elative steepness of beach slope and wave slope, thee is a need to answe the following questions: whethe o not the same citeia used fo plane beaches can still be used fo finging eefs; if not, what ae the main diffeences between wave beaking on these two diffeent conditions. The answes to these questions ae impotant in the study of wave-induced setup ove finging eefs as wave beaking is the main diving foce. The pesent chapte descibes a seies of laboatoy expeiments designed to investigate the influences of both the eef-flat submegence and the foe-eef slope on the chaacteistics of the beaking of monochomatic waves ove submeged finging eefs. The emainde of this chapte is oganized as follows: the laboatoy setting, instuments and expeimental pocedues ae intoduced in section.; The 14

44 diect measuements of beaking popeties ae descibed in section.3, and the dimensionless data analysis ae pefomed in section.4, whee the main aspects of beaking-wave chaacteistics fo eefs, including the beake type and location, the beake indices, sufzone width, wave tansmission, eflection and enegy dissipation, ae epoted; The main esults ae summaized and key conclusions ae dawn in section.5. Fig..1 Reviewed coastal pofiles: (a) emeged plane slope (e.g., Goda, 1); (b) submeged finging eef (e.g., Goulay, 1994); (c) submeged plane slope (e.g., Blenkinsopp and Chaplin, 8); (d) submeged baed beach (e.g., Smith and Kaus, 1991); (e) submeged low-cested beakwate (e.g., Van de Mee et al., 5); Hi - incident wave height; SWL - still wate level.. Expeimental Setup..1 Facilities, eef model and instumentation Refeing to Fig.., the laboatoy expeiments wee caied out in a glass-walled wave flume (36 m long,.55 m wide and.6 m deep) in the Hydaulics Modeling Laboatoy at the Nanyang Technological Univesity, Singapoe. A sevo-contolled piston type wavemake was placed at one end of the flume to geneate the designed waves. At the othe end, a beach with a gentle slope of 1:8 stated at appoximately 3 m fom the wavemake; the slope was coveed with poous mats of 1 cm thick to educe wave eflection. Pilot tests conducted without the eef model showed that 15

45 the poous mats on the final beach could effectively dissipate the incident waves and the eflection coefficient of the beach was less than 5% fo the tested egula waves, i.e., only.5% of the enegy in the incident waves could be eflected. When the eef model is pesent, most enegy in the incident waves has been dissipated by wave beaking and the tansmitted waves popagate to the final beach is much smalle than the incident waves. As a esult, the wave enegy eflected fom the final beach must be much smalle than.5% of the incident wave enegy and the effects of wave eflection fom the beach on wave beaking can be safely ignoed. Thee ae a vaiety of eef pofiles that have been studied in the liteatues (see Goulay, 1996b); the pesent study focuses on the one-dimensional hoizontal (1DH) eef pofile that has a plane sloping foe-eef and an idealized hoizontal eef flat. This type of finging eef model has been the focus of seveal analytical studies (e.g., Tait, 197; Goulay, 1996a; Symonds et al., 1995; Hean, 1999; Goulay and Collete, 5). The use of the idealized eef pofile is intended to povide a easonable fist appoximation of finging eef configuations and to make efeence to simila studies of wave inteactions with plane beaches/submeged low-cested stuctues. To constuct the idealized finging-eef model, a 1:6 plane slope joins a hoizontal platfom at m fom the wavemake. The hoizontal platfom was 6m wide and its length was the same as the inne width of the flume. Both the slope and the hoizontal platfom wee made with PVC plates, and the entie model was fimly held at.35 m above the flume bottom by seveal stainless steel ods attaching to the two walls of the flume. In this expeimental study, the eef model was constucted in accodance with Foude similaity using a epesentative geometic scale facto of 1:5. Given the eef flat of 6 m in width at the laboatoy scale, the coesponding eef-flat width at the pototype scale is 15 m, which is geneally less than the typical natual eef flat (seveal hunded to thousand metes). Since the focus of these expeiments is on wave beaking ove finging eefs, the effect of flat width should be small as long as the laboatoy eef flat is sufficiently long fo waves to complete the sufzone 16

46 pocess. Since the educed off-eef wate depth in the model did not affect wave tansfomation on the eef flat significantly (Goulay, 1994), the wate depth is designed accoding to the allowable wate depth of the wave flume athe than the geometic scale facto. Indeed, most natual coal eefs ae composed of had calcium cabonate skeletal mateial and coveed by a wide vaiety of benthic oganisms, and the coal community on the eef flat is moe like a laye of poous mateial. In the Appendix C, esults fom a set of expeiments ae epoted fo a eef flat being coveed with a laye of poous mat. The poous mat was used to model the suface oughness and poosity of coal community. The esults showed that the effects of poosity and suface oughness wee mino since both the waves and the mean cuents on the eef flat wee weak. The dominant physical pocess in this study is wave beaking, which occus in a naow egion on eithe the foe-eef slope o the eef flat. Theefoe, the impemeable mateial (PVC) was used to constuct the eef models (with smooth sufaces) in this study. To measue the coss-eef wave tansfomation, 8 esistance-type wave gauges (HR Wallingfod Ltd.) wee used and thei aangement is shown in Fig..: G1 and G wee placed appoximately 4. m seawad of the toe of the foe-eef slope to sepaate the incident waves fom the eflected waves; G3 was placed on the slope to measue the shoaling waves; G5 - G7 wee put in the vicinity of the eef edge to measue the waves in the suf zone; G8 was located aound the middle of the eef flat to captue the efomed waves; G4 was used to validate the measuement of beake wave height fom video visualization and its location was adjusted based on a pilot un fo each tested condition. Fou USB cameas (C1 - C4, Logitec Ltd.) with a esolution of pixels wee employed to visualize the waves in the suf zone; thee of them wee aligned pependicula to the wave flume by adjusting thei locations both hoizontally and vetically accoding to the vaiation of wate level and beake positions (see Fig..). The field of view (FOV) of each camea was 1. m.8 m, hence a maximum 17

47 global view size of 3. m.8 m was achieved, which is sufficient to captue the majo wave tansfomation pocesses such as shoaling, beaking, boe popagation and the efomed waves on the eef flat. The last camea was located above the flume and moved along the flume to detemine the shoewad ending locations of the suf zones. Tanspaent gid sheets with a gid size of 1. cm 1. cm wee attached to the font glass wall as a efeence scale fo detemination of the wave beake height, incipient beaking location and sufzone width. The wave gauges and cameas wee synchonized though a data acquisition system (DeweSoft Ltd.). The sampling ate of all wave gauges was 5Hz and the fame ate of all cameas was 5 Hz. Fig.. Expeimental setup... Wave conditions and test pogam Thee wave peiods (T =1. s, 1.5 s and 1.67 s) wee examined in the expeiments. Fo each wave peiod, the taget incident wave height anged fom.1 m to.13 m. The taget wave heights wee slightly diffeent fom those measued in the expeiments. The actual incident wave heights wee detemined by sepaating the incident waves fom the eflected waves using the waves measued at G1 and G. The measued incident wave heights wee conveted to those in deep wate fo subsequent analysis, see H in Table.1. Refeing to Table.1, fou wate depths ( h = 38 cm, 4 cm, 4 cm and 45 cm) wee examined: whee the wate depths wee measued fom the flume bottom; this 18

48 gives the following eef-flat submegences (the wate depths above the eef flat): h = 3 cm, 5 cm, 7 cm and 1 cm, espectively. To examine the effects of foe-eef slope, in addition to the initial slope of 1/6, thee othe foe-eef slopes (1/3, 1/9 and 1/1) wee also examined fo the eef-flat submegence of 5 cm. These slopes wee chosen because they fall in the epesentative ange of the foe-eef slopes (see e.g., Goulay, 1996b fo a eview of diffeent eef pofiles in the liteatue). Based on Foude similaity, fo a geometic scale facto of 1:5, the peiod scale facto is 1:5, thus the equivalent pototype values of the paametes in Table.1 ae.8 m -.5 m fo h,.5 m m fo H and 5. s s fo T, which ae in ageement with the typical values obseved in field conditions (Bonneton et al., 7; Hench et al., 8; Lowe et al., 9a; Vette et al., 1). Waves of lage steepness boke befoe eaching the toe of the foe-eef slope and thus have been excluded fom the dataset epoted in this study. Waves that wee small and passed ove the eef flat without beaking wee not analyzed. They ae categoized as non-beaking waves in section.4.1. The emaining 3 wave conditions have deep-wate wave steepness anging fom.3 to.88 and elative eef-flat submegence (the atio of the wate depth above the hoizontal eef flat to the deep-wate wave height) anging fom.3 to Table.1 Test conditions at laboatoy scale Type Range Reef-cest submegence: h (m).3,.5,.7,.1 Deep-wate monochomic wave height: H (m) Monochomic wave peiod : T (s) 1.,1.5,1.67 Foe-eef slope: s tan 1/3,1/6,1/9,1/1 In the expeiments, suface elevations wee collected fo 3 s, stating immediately afte the stat of the wavemake. Fom the pilot tests, it was found that waves in the 19

49 flume can each a elatively steady state about 15 s afte stating the wavemake. Thus only the last 1 wave cycles in the wave ecods wee used to calculate the wave chaacteistics (i.e., tansmitted wave height, wave-induced setup). The cameas wee tiggeed at 5 s afte stating the wavemake and the ecoding duation was 15 s fo all fou cameas to ensue that at least ten waves could be extacted fom the ecoded videos. Between two subsequent tests, seveal minutes wee allowed to elapse so that the wate suface could become calm and the effects of esidual cuents wee minimal..3 Results.3.1 Qualitative desciption of beaking waves The incipient beaking point, the sufzone endpoint, the beake wave height ( H ) and the beake wate depth ( h b ), wee detemined by analyzing the video images fo each test un. The incipient beaking point fo plunging beake is defined as the point whee a lage potion of the font face of a wave becomes nealy vetical (see Fig..3(a)). This definition is consistent with the majoity of the studies of beaking-waves (Ivesen, 195; She et al., 1994; Blenkinsopp and Chaplin, 8; etc.). Fo a spilling beake, the incipient beaking point is defined as the point whee the whitewate begins to oll down the font face of a wave. The beake height is defined as the vetical distance between the wave cest at the incipient beaking point and the tough immediately in font of the incipient beaking point within one wave length. The beake depth is defined as the vetical distance between the still wate level (SWL) and seabed at the incipient beaking point. The total sufzone width ( L s ) is defined as the hoizontal distance between the incipient beaking point and the endpoint of wave beaking (popagating boes disappea and oganized waves efom). It is stessed hee that the identification of the endpoint of wave beaking takes some expeience. b Wave beaking is a highly nonlinea pocess, which can be sensitive to vaiations in bottom geomety, incident wave chaacteistics, and the pesence of cuent and

50 eflected waves. In pinciple, all measued mean quantities should be obtained by aveaging an infinite long wave ecod. Howeve, thee is a limitation of the RAM size used to stoe the high-esolution (1 M pe image at a fame ate of 5 Hz) video data fom the fou cameas, and a finite numbe of waves ae used in pactice. The minimum numbe of waves equied to obtain stable mean values was tested using the waves measued at the gauge G4 fo a epesentative wave condition ( h.4 m, H.74 m, T 1.5 s and s 1.6 ); the esults ae listed in Table.. It shows that thee is no significant change in both the mean value and the standad deivation when the total numbe of waves used in the analysis is lage than 1, thus, to minimize the andom eos in the measued values of H b, h b and L s detemined fom the video images, these quantities ae aveaged ove at least 1 waves fo each test un. Fig..3 Definitions of beaking wave chaacteistics and snapshots of diffeent beake types: (a) definitions; (b) plunging beake on the foe-eef; (c) plunging beake on the eef edge; (d) spilling beake on the eef flat. 1

51 Table. Mean value and standad deivation (std) of beake wave height ( H ) obtained fom N successive waves measued by G4 fo a epesentative wave condition ( h.4 m, H.74 m, T 1.5 s and s 1/6) N=1 N= N=3 N=4 mean std mean std mean std mean std b Table.3 Measued beake wave height ( H and G b I H b ), beake depth ( h b ) and sufzone width ( L s ) of 1 successive waves fo a epesentative wave condition ( h.4 m, H.74 m, T 1.5 s and s 1/6) Item a Mean Std G H b (m) I H b (m) h b (m) L s (m) a H is measued at the location of G4, G b ecoded video images. I H b, h b and Ls ae detemined fom the To validate the beake wave heights detemined fom the video images, analysis of the ecoded beaking waves was also tied using the wave gauge G4, which was moved to the beaking location detemined fom the pilot un done fo each test condition. The discepancy between the values of H b detemined by these two appoaches is within ±5% fo all test uns. Fo example, Table.3 shows the measued beake wave heights fom G4 ( H G b ) as well as fom the ecoded image ( H ) of 1 successive waves fo the above epesentative wave condition. It can be I b seen that ( H G H I )/ H I.4 and the standad deviations fo both b b b

52 measuements ae small. Table.3 also listed the measued H b, h b and obtained fom 1 successive waves, and thei standad deviations ae also vey small, expect fo those of L s due possibly to the difficulty in judging the endpoint of a suf zone. Theefoe, in the following analysis, the values of L s H b, h b and L s detemined fom the video images ae used since the suface elevation ecoded by wave gauge might be affected by the ai bubbles entained in wate due to wave beaking. To test the epeatability of the expeiments, the above case was un fo thee times. Table.4 lists the measued beake wave heights fom the video images fo 1 successive waves. The discepancy in the mean values between any two uns in Table.4 is within 7% and the standad deivation fo each un is also small. Othe measued quantities such as h b and Ls have simila degee of uncetainty (not shown). The possible souces of eo in the epeatability tests may come mainly fom the wavemake and eading eo. Theefoe, based on the above eo analysis, the estimated eo fo each measued quantity in this study is believed to be less than 1%. Table.4 Measued beake wave heights of 1 successive waves in thee diffeent uns fo a epesentative wave condition ( h.4 m, H.74 m, T 1.5 s and s 1/6) Item Mean Std Run Run Run Thee ae seveal ways to classify the beake types fo plane beaches, and it is geneally accepted that waves beak in one of the following fou foms: spilling, 3

53 plunging, collapsing, and suging (Galvin, 1968). Howeve, fo the pesent finging eef pofile with elatively steep foe-eef slopes, it is found that plunging beake always occued on the foe-eef fo lage waves. As the incident waves became smalle, the beaking point would shift to a location above the eef flat and the plunging beakes could evolve into spilling beakes. As the incident wave height was futhe educed, wave beaking would cease and the incident waves could feely pass ove the eef flat. Theefoe, the beakes in this study ae eithe plunging beakes o spilling beakes, and wave beaking can occu eithe on the foe-eef o above the eef flat. Illustations of the beake types and the beaking locations, as well as the detemination of H b and h b ae shown in Fig..3 (b) - (d)..3. Detemination of wave tansmission and eflection coefficients Wave tansmission and eflection chaacteistics fo submeged stuctues such as low-cested beakwates ae usually quantified by the tansmission coefficient, and the eflection coefficient, K, which ae defined by K t whee H i, H, K H / H, K H / H (.1) t t i i H t, ae the incident, eflected and tansmitted wave heights, espectively. Stictly speaking, the above definitions ae fo linea monochomatic waves. Accoding to the consevation of wave enegy, an enegy loss coefficient, defined as the atio of wave enegy lost though dissipation to the total incident wave enegy, can be calculated by K 1K K (.) d t The wavemake used in the expeiments does not have the active wave absoption function. Theefoe, of the incident waves geneated in the flume will inevitably be contaminated by the multiple e-eflected waves between the models (both fom foe-eef and fom final beach) and the wavemake paddle, leading to a situation whee stable patial standing waves may be fomed afte unning the wavemake fo seveal minutes. The eflection coefficient seawad of foe-eef could be calculated 4

54 by making use of this patial standing-wave patten. In this study, the two-point wave sepaation method poposed by Goda () is adopted, in which the wave patten in the flume is teated as the supeposition of the ight-going incident waves and the left-going eflected waves. The Fouie coefficients (which can be detemined by eithe data fitting o FFT) of the hamonics fom the two wave ecods (i.e., using the wave ecods at G1 and G) ae needed to calculate H i and H. Those coefficients includes the contibutions fom both the incident waves and the multiple eflective waves, thus this method inheently consides the effect of multiple eflections. A maximum 1% eo was epoted fo this method (see Goda, ). Note that the enegy associated with highe hamonic waves is not consideed in Eq. (.). Howeve, fo highly nonlinea tansmitted waves on the eef flat, the contibution fom highe hamonic waves should be consideed when calculating the enegy loss coefficient. In this study, the goup velocity is diffeent in font of the foe-eef fom that on the eef flat; the consevation of wave powe gives the following definition of the tansmission coefficient K t Ec (.3) Ec g t t g i i whee g ci and g c t ae wave goup velocities in the wates in font of the foe-eef and on eef flat, espectively. The incident wave enegy flux ( Ec ) can be g i i g evaluated by the deep-wate wave enegy flux ( Ec ) based on enegy consevation. Using the linea wave appoximation, the deep-wate enegy density is E, whee H is the deep-wate wave height conveted fom the gh /8 g incident wave height ( H ), and the deep-wate goup velocity is c gt /4. i Howeve, fo the tansmitted wave enegy, E t, a detailed analysis of wave ecods at G8 is equied. Fig..4 is an example of the tansmitted waves fo the case of h.4 m, H.74 m, T 1.5 sand foe-eef slope = 1/6. Fig..4(a) shows that thee ae obsevable seconday peaks in the wave pofiles. Fo this case, the 5

55 atio of the maximum wave-induced setup on the eef-flat ( ) to the eef-flat submegence ( h ) is 1/5, indicating that should be consideed when computing g c t fo tansmitted waves. The geneation of both highe and low fequency components ae shown in Fig..4(b), whee up to the 5 th hamonics can be identified. Fig..4(c) demonstates that the enegy contained in the second hamonic is on the same ode of magnitude as that in the fundamental hamonic waves. Theefoe, E is calculated by integating the enegy spectum obtained at G8 in this study. c g t is estimated with gh ( ) by invoking shallow wate appoximation on the eef flat in view that pesent data. kh anges fom.1 to.68 fo the (m) (a) Time seies of fee suface elevation t(s) 6 x 1-3 (b) Amplitude spectum f(hz) 3 x 1-4 (c) Enegy spectum a(m) S(m.s) f(hz) Fig..4 Wave analysis fo tansmitted waves (fom G8) on the eef flat ( h.4 m, H.74 m, T 1.5 s and s 1/6) 6

56 .3.3 Measued beaking-wave popeties In this section, the following measued popeties ae discussed: beake type, beake location, beake height ( H b ), beake depth ( h b ), total sufzone width ( L s ), the wave-induced setup ( ), tansmission coefficient ( coefficient ( K ). The oiginal data ae given in Appendix A. K t ) and eflection Fistly, the measued popeties fo the dataset with a constant foe-eef slope of 1/6 ae plotted in Fig..5 as a function of deep-wate wave height ( H ). Also shown in the figue ae the effects of wave peiod (T ) (indicated by diffeent make colos) and eef-flat submegence ( h ) (indicated by diffeent make types). Figs..5(a) and.5(b) show that fo both beake type and beaking location, the tends agee well with the obsevations descibed in section.3.1. Fo example, plunging beakes appea moe fequently unde lage H. As fo beaking location, beaking on the eef flat is most likely to occu unde smalle H. Howeve, as shown in Fig..5(b), H alone is not a pope paamete to descibe the beaking location due to the ovelap of diffeent beaking locations within a wide ange of H. The effects of wave peiod on both beake type and beaking location ae not obvious in Figs..5(a) and.5(b). Fig..5(c) shows a monotonic incease of with the incease of H, and Hb H b has a weak dependence on T. The effect of h is not visible in Fig..5(c). Fig..5(d) shows that hb has a stong coelation with H and weak coelation with T, but fo smalle H, hb becomes almost constant and is only elated to h due to the waves beaking on the hoizontal eef flat. Fo L s shown in Fig..5(e), stong dependence on both H and T can be identified, and the dependence on h seems to be insignificant. in Fig.5(f) inceases with the incease of H but deceases with the incease of h, and vice vesa fo in Fig..5(g). The influence of T is negligible fo both and in Fig..5(h), thee is no dependence of K on H, T o h. K t shown K t. Lastly, as shown 7

57 The measued popeties fo the dataset with the constant eef-flat submegence of.5 m ae plotted in Fig..6 as a function of deep-wate wave height ( H ). Also shown in the figue ae the effects of wave peiod ( T ) (indicated by diffeent make colos) and foe-eef slope ( s ) (indicated by diffeent make types). Without the effect of eef-flat submegence, it can be obseved in Fig..6 that oveall, the beake type, beaking location, H b, h b, L s, and K t can be descibed by H alone, except that some scatte due to the effects of T. Again, as shown in Fig..6(e), thee is noticeable dependence of L s on T. The eflection coefficient is also insensitive to eithe H o T as shown in Fig..6(h). Afte caefully examining all plots in Fig..6, it is found that the dependence of each measued popety on foe-eef slope is found to be weak fo this dataset. 8

58 Collapsing Plunging Spilling Non-beaking (a) Beake type Foe-eef Reef Edge Reef flat (b) Beake location H (m). (c) H (m). (d).15 H b (m).1.5 h b (m) H (m) 4 3 (e) H (m).4.3 (f) L s (m) H (m) H (m) 1.8 (g).4.3 (h) K t.6 K H (m) H (m) Fig..5 Measued quantities as a function of deep-wate wave height ( H ) with foe-eef slope of s 1/6 at diffeent wave peiods (T ) and eef-flat submegences h ): (a) Beake type; (b) Beaking location; (c) Beake height ( H ); (d) Beake ( depth ( h b ); (e) Total sufzone width ( Tansmission coefficient ( L ); (f) Wave-induced setup ( ); (g): s K t ); (h) Reflection coefficients ( b K ). Solid black makes: T 1. s ; Solid gey makes: T 1.5 s ; Open makes: T 1.67 s. Squaes: h.3 m ; Cicles: h.5 m ; Tiangles: h.7 m ; Diamonds: h.1 m. 9

59 Collapsing Plunging Spilling Non-beaking (a) Beake type Foe-eef Reef Edge Reef flat (b) Beake location H (m). (c) H (m). (d).15 H b (m).1.5 h b (m) H (m) 4 (e) H (m).4 (f).3 L s (m) H (m) H (m) 1.8 (g).4.3 (h) K t.6 K H (m) H (m) Fig..6 Measued quantities as a function of deep-wate wave height ( H ) with eef-flat submegence of h.5 m at diffeent wave peiods (T ) and foe-eef slopes ( s ): (a) Beake type; (b) Beaking location; (c) Beake height ( H ); (d) Beake depth ( h b ); (e) Total sufzone width ( L s ); (f) Wave-induced setup ( ); (g): Tansmission coefficient ( K t ); (h) Reflection coefficients ( K ). Solid black makes: T 1. s ; Solid gey makes: T 1.5 s ; Open makes: T 1.67 s. Squaes: s 1/3; Cicles: s 1/6; Tiangles: s 1/9; Diamonds: s 1/1. b A moe systematic data analysis will be epoted in the following section, whee dimensionless paametes ae intoduced and discussed. The wave-induced setup is 3

60 not the focus of this chapte and is used only to estimate the eef-flat mean wate level as discussed in section Data Analysis and Discussion The analysis in section.3 has shown that the popeties of beaking waves depend pimaily on deep-wate wave height as well as eef-flat submegence, secondaily on wave peiod, and least on foe-eef slope. The esults in section.3 ae dimensional and not conclusive due to the multiple dependences. In this section, the two afoementioned datasets ae combined and a dimensionless analysis is conducted. Gouping fo the above vaiables in analogy to those commonly used to chaacteize beaking waves ove beaches and submeged beakwates yields the following thee dimensionless paametes: the deep-wate wave steepness, H / L (whee L gt, g is the gavity acceleation), the elative eef-flat submegence, / h / H, and the foe-eef slope, s (whee s tan, is the foe-eef slope angle). The eef-flat oughness and poosity might not be impotant if most of wave enegy is lost though wave beaking in the suf zone adjacent to the eef edge..4.1 Classification of measued beaking waves Beake type Battjes (1974) used the following suf-similaity paamete H s L / o (.4) to descibe beake types fo unifom plane slopes. This suf-similaity paamete epesents the elative steepness of the incident waves and the seabed slope. Battjes (1974) detemined the following tansitional values of the suf-similaity paamete based on the expeiments of Galvin (1968) Suging o Collapsing if 3.3 (.5a) 31

61 Plunging if (.5b) Spilling if.5 (.5c) Fistly, the beake type obseved in the pesent expeiments ae plotted in Fig..7(a) against, whee the tansitions values fo plane beaches, i.e., Eq. (.5) ae also shown. It can be obseved that most of the plunging-type beakes fo the pesent finging eef fall coectly in the plunging egion as specified by Eq. (.5). Howeve, spilling beakes fo the finging eef have values of lage than that fo the plane slope and no clea tansition values of can be identified between plunging and spilling beakes o between spilling beake and non-beaking waves. Theefoe, the suf-similaity paamete is not an appopiate paamete to classify the wave beakes fo finging eefs. The beake types classified by the elative eef-flat submegence ( h / H ) is given in Fig..7(b), which shows that the influence of the elative eef-cest submegence is dominant. This is somewhat expected: when waves beak in most of expeiments, they beak on the hoizontal eef flat, thus, it should be the wate depth above the eef flat athe than the foe-eef that contols wave beaking. Fom these expeiments, the tansition values estimated fo the pesent finging eef pofile ae Plunging if h / H 1.8 (.6a) Spilling if 1.8 h / H.7 (.6b) Non-beaking if h / H.7 (.6c) The ovelap aound tansition values might be due pimaily to the subjectivity in detemining the beake types. The above tansition values ae consistent with those found by Goulay (1994), although his eef model was not the same as the pesent. The eason that the beaking conditions fo finging eefs ae diffeent fom those fo plane beaches might be due to the stong seawad etun flows ove the eef edge duing the backwash phase (Smith and Kaus, 1991; Blenkinsopp and Chaplin, 8). 3

62 Beake type Collapsing Plunging Spilling Non-beaking (a) Plane beach Exp. data Beake type Plunging Spilling Non-beaking (b) h /H =1.8 h /H = h /H Fig..7 Beake type as a function of: (a) suf-similaity paamete ( ); (b) elative eef-flat submegence ( h / H ). Beaking location The obseved beaking locations against h / H ae plotted in Fig..8(a), whee a new categoy eef-edge beaking is poposed, since waves in this categoy can be consideed as beaking eithe on the eef flat o on the foe-eef. The substantial ovelap between the eef-flat beaking and the foe-eef beaking in Fig..8(a) suggests that the eef-flat submegence ( h ) when scaled by the deep-wate wave height ( H ), is not an appopiate paamete fo classifying beaking locations. Rescaling h using the local beake wave height ( H b ) is then consideed. Fig..8(b) shows the beaking locations as a function of the modified elative eef-flat submegences ( h / H ); it can be seen that the ovelap is significantly educed. b The tansition values of h / H fo beaking location can be estimated by b h Beaking on foe-eef if 1. H (.7a) b h Beaking on eef flat if 1. H (.7b) b 33

63 Again, the emaining ovelap in the figue may come mainly fom the difficulty in measuing H b and detemining the exact location of a beake event. (a) (b) Beaking location Foe-eef Reef Edge Reef flat Beaking location Foe-eef Reef Edge Reef flat h /H b = h /H 1 3 h /H b Fig..8 Beaking location as a function of: (a) elative eef-flat submegence h / H ); (b) modified elative eef-flat submegence ( h / H ). ( b Relating the beaking locations to the beake wave height ( H ) is easonable since H b is diectly elated to the sufzone dynamics. Fo example, instead of Eq. (.4), beake types fo plane beaches ae also commonly classified using a suf-similaity paamete ( b ) defined by the beake wave height ( H ) b b s b. (.8) H / L Howeve, using H to define the suf-similaity paamete is easie to use in pactice since the beake wave height is difficult to measue. Thee ae welldocumented empiical equations to tansfom b o H b into H fo plane slopes. Since thee is no such equation available fo the eef pofile, this topic will be addessed in the next section It is emaked hee that the paamete F co = g H T / h as suggested by Goulay (1994) fo chaacteization of the beake types and locations was also 34

64 examined; but this paamete does not seem to wok fo the pesent expeimental setting which involved a vaiety of wave steepness and foe-eef slopes..4.. Beake indices The beake wave height and beake wate depth ae of consideable inteest fo coastal enginees, oceanogaphes and sedimentologists. The beake wate depth is usually equied to compute the beake wave height, and vice vesa. Existing methods use the beake height index and/o the beake depth index to calculate them. Numeous empiical expessions fo wave beake indices have been poposed in the liteatue fo plane beaches and othe coastal pofiles; howeve, only a few of them elevant to the pesent study will be discussed below. This section will investigate the key paametes used to descibe the beake indices fo finging eefs and the suitability of applying empiical expessions fo plane slopes and othe coastal pofiles (the baed beach o submeged slope) to finging eefs. Beake height index The beake height index,, is nomally defined as H b (.9) H Pevious studies on plane beaches show that the beake height index is dependent on both the deep-wate wave steepness and the seabed slope(e.g., Koma and Gaughan, 1973; Sakai and Battjes, 198; Sunamua, 198), and this index follows a powe law of the fom H ms () L ns ( ) (.1) whee the empiical coefficients m( s ) and ns ( ) ae detemined by fitting to laboatoy o field data. Fo example, the fomula poposed by Koma and Gaughan (1973) fo plane beaches, as ecommended by the Coastal Engineeing Manual (USACE 3), uses m.56 and n.. Smith and Kaus (1991) suggested that ms ( ).8.17s 6.s and ns ( ) s 4.85s fo thei submeged 35

65 ba pofile unde.7 H / L.9 and 1/5 s 1/1. Fo the submeged, tuncated slope, Blenkinsopp and Chaplin (8) suggested that m.5 and n. fo.8 H / L.68. The beake height indices fo the finging eef models ae shown in Fig..9 as a function of deep-wate wave steepness; also shown in Fig..9 ae the fou empiical expessions discussed above. The pesent expeimental data show that Smith and Kaus (1991) s model is not vey sensitive to the foe-eef slope when s 1/5, thus only s 1/6 is plotted in Fig..9 fo compaison. The pesent data show that the beake height index ( ) inceases with deceasing deep-wate wave steepness ( H / L ). This featue is typical of waves beaking on a plane slope due to the shoaling effect expeienced by longe waves befoe beaking. Data fo diffeent foe-eef slopes show simila tends (see Table.5 fo the fitting paametes coesponds to diffeent foe-eef slopes, thei slope dependences ae not obvious), implying that the effect of the foe-eef slope on is insignificant. A least-squae fitting of all pesent data to the powe law gives.64 H L.16 (.11) with R.7. Compaisons among the cuves in Fig..9 indicate that the values of m.56 and n. suggested by the USACE 3 fo plane beaches ae geneally sufficient to estimate H b fo these coastal pofiles. Fo completeness, elating to h / H is also tied and esults ae shown in Fig..1. It is found that a weak linea dependence of on elative eef-flat submegence might be identified (with cetain degee of scatte) due to the stong linea dependence of on H (shown in section.3.3). H b 36

66 Table.5 Values of m and n obtained by least-squae fitting using Eq. (.1); the uncetainties fo m and n ae 95% confidence inteval s m n 1/3.64± ±.4 1/6.69± ±.1 1/9.55± ±.4 1/1.5±.6 -.3±.3.5 =H b /H H /L Fig..9 Beake height index ( ) as a function of deep-wate wave steepness H ( / L ). Squaes: Exp., s=1/3; Cicles: Exp., s=1/6; Tiangles: Exp., s=1/9; Diamonds: Exp., s=1/1. Dashed line: Ped. by Koma and Gaughan (1973); Dotted line: Ped. by Smith and Kaus (1991) with s=1/6; Dash-dot line: Ped. by Blenkinsopp and Chaplin (8); Solid line: Least-squae fitting of Exp. data with powe law. 37

67 .5 =H b /H h /H Fig..1 Beake height index ( ) as a function of elative eef-flat submegence h / H ). Squaes: Exp., s=1/3; Cicles: Exp., s=1/6; Tiangles: Exp., s=1/9; ( Diamonds: Exp., s=1/1. Beake depth index Fo shallow wate waves, the beake depth index, is usually defined as H h b (.1) which says that the beake wave height is contolled by the local wate depth alone. This beake wate depth has been widely used in the studies of coal eef hydodynamics (see Lowe et al., 5; 9a and Vette et al., 1 fo field obsevations; Goulay, 1994 and Camo et al., 11 fo laboatoy expeiments; and Tait, 197, Hean, 1999 and Low et al., 9 fo analytical o numeical modeling). Thus, it is woth analyzing the pesent data using the existing empiical expessions fo beake wate depth. b 38

68 Compaison with the emeged plane slopes Based on a theoetical analysis, Miche (1944) fist poposed a theoetical beaking limit fo deep-wate waves as ( H / L) b.14. Quite a few empiical beaking citeia have also been poposed fo shallow wate waves in the liteatue. Fo waves in wate of constant depth, McCowan (1894) suggested fom his expeiments a constant / h. 78, which is still widely used in engineeing pactice. Fo H b waves popagating on sloping sea beds, the dependence of on the seabed slope and the incident wave chaacteistics wee ecognized. Subsequent fomulae fo a plane beach of slope s geneally fall into thee categoies: (1) f ( s), such as Collins (197), Nelson (1987); () f ( Hb / L o Hb / Lb), such as Weggel (197), Koma (1998); (3) f (, shb / Loh b / Lb), such as Svendsen (1987), Goda (1). Most maine designs adopted Eqs. (.13) o (.14), which wee oiginally poposed by Weggel (197) and Goda (1974) (see also USACE, 3; Goda, 1). Weggel s expessions ead Hb H b a H b / L.6, s.1 h gt (.13) b whee a 19s 43.8(1 e ) and b 19.5s 15.6 / (1 e ). Goda s fomula is given by Hb A h hb hb / L L b 4/3 1 exp 1.5 (1 11 ) s (.14) whee A is a fitting paamete and A.17 was suggested fo plane slopes. Eq. (.14) can yield bette pedictions fo mild slopes than fo steep slopes (Goda, 1). Both Wiggle s and Goda s fomulae indicate that the value of deceases as the seabed slope deceases. Figs..11 and.1 povide the beake depth indices of the pesent data as a function of Hb / gt and b h / L, espectively, fo diffeent foe-eef slopes and beaking locations (indicated by diffeent subplots). Pedictions fom Eqs. (.13) and (.14) ae also shown in the two figues fo compaison. 39

69 s=1/9 s=1/6 s=1/3 (a) s=1/9 s=1/6 s=1/3 (b) =H b /h b.9 s=1/1 =H b /h b.9 s=1/ H b /gt H b /gt 1.5 s=1/6 s=1/3 (c) 1. s=1/9 =H b /h b.9.6 s=1/ H b /gt Fig..11 Beake depth index ( ) as a function of H b / gt. (a): Wave beaking on the eef flat; (b): Wave beaking on the eef edge; (c): Wave beaking on the foeeef. Squaes: Exp., s=1/3; Cicles: Exp., s=1/6; Tiangles: Exp., s=1/9; Diamonds: Exp., s=1/1. Dashed lines: Ped. by Weggel (197). 4

70 s=1/9 s=1/6 (a) s=1/9 s=1/6 (b) =H b /h b.9 s=1/1 =H b /h b.9 s=1/ h b /L h b /L 1.5 s=1/9 s=1/6 (c) 1. =H b /h b.9.6 s=1/ h b /L Fig..1 Beake depth index ( ) as a function of hb / L. (a): Wave beaking on the eef flat; (b): Wave beaking on the eef edge; (c): Wave beaking on the foeeef. Squaes: Exp., s=1/3; Cicles: Exp., s=1/6; Tiangles: Exp., s=1/9; Diamonds: Exp., s=1/1. Dashed lines: Ped. by Goda (1). The souce of scatte in both Figs..11 and.1 is substantial, and the dependence of on Hb / gt o b h / L is weak, implying that neithe of these two paametes ae appopiate to descibe fo the pesent poblem. Meanwhile, both Wiggle s and Goda s fomulae ove-pedict fo the majoity of pesent data, fo which wave beaking was found mostly on the eef flat. Fo waves beaking on the foe- 41

71 eef slopes, the pedictions ae in a easonable ageement with the pesent data; slight ove-pediction is believed to be attibuted patly to the unsuitability of Eqs. (.13) and (.14) fo steep slopes since they wee oiginally poposed fo mild slopes ( s 1/1). Physically, wave beaking on a foe-eef is vey simila to wave beaking on a plane beach, thus thee is always a bette ageement between the expeiments and pedictions when waves beak on foe-eefs. Compaison with submeged pofiles Smith and Kaus (1991) found a linea elationship between and the sufsimilaity paamete fo waves beaking ove a ba pofile o in tems of H / L as Hb (.15) h b Hb s H / L (.16) h H / L s s b Note that the maximum value of neve exceeds.55 fo a hoizontal bottom (Nelson, 1994). Goulay (1994) poposed the following empiical elation fo steep finging eefs Hb s H / L (.17) h H / L s s b Moe ecent wok on submeged slopes (Blenkinsopp and Chaplin, 8) found Hb.85.8 H / L.68,.76 h / Ho. (.18) h b The beake depth indices of the pesent data ae plotted in Fig..13 as a function of the deep-wate wave steepness fo diffeent foe-eef slopes and beaking locations (indicated by diffeent subplots). The pedictions based on Eqs. (.16) - (.18) ae also shown fo compaison. 4

72 s=1/1 s=1/9 s=1/6 (a) s=1/1 s=1/9 s=1/6 (b) =H b /h b.9 =H b /h b H /L H /L 1.5 s=1/1 s=1/9 s=1/6 (c) 1. =H b /h b H /L Fig..13 Beake depth index ( ) as a function of deep-wate wave steepness H ( / L ). (a): Wave beaking on the eef flat; (b): Wave beaking on the eef edge; (c): Wave beaking on the foe-eef. Squaes: Exp., s=1/3; Cicles: Exp., s=1/6; Tiangles: Exp., s=1/9; Diamonds: Exp., s=1/1. Dashed lines: Ped. by Smith and Kaus (1991); Dotted lines: Ped. by Goulay (1994); Dash-dot line: Ped. by Blenkinsopp and Chaplin (8). It can be seen fom Fig..13 that thee is no noticeable dependence of on H / L o eef-flat slope fo the pesent data. Howeve, the pesent data compae favoably with the tend of Blenkinsopp and Chaplin (8), although the scatte is still consideable. This may be due to thei slope (1/1), which falls in the slope ange of 43

73 the pesent dataset. Both Eqs. (.16) and (.17) eveal a tend of deceasing with inceasing H / L. The oveall ageements with the pesent data ae simila to those fo plane beaches discussed above: the empiical equations only give easonable esults fo the waves with lage, i.e., those that beak on the foe-eef. Fo waves with smalle steepness (waves that beak on the eef flat), again both equations oveestimate the values of. Johnson (6) conducted a seies of laboatoy expeiments on submeged beakwates and found that the feeboad (in analogy to the eef-flat submegence fo the pesent finging eef) contols the depth-limited wave beaking on the beakwates. The empiical expession poposed by Johnson (6) is 1.55 h / H.5 Hb h h / H 1.5 hb Ho.8 h / H 1.5 (.19) Beake depth indices fo the pesent data ae plotted in Fig..14 as a function of elative eef-flat submegence unde diffeent foe-eef slopes and beaking locations. Also shown in Fig..14 ae the pedictions of Johnson (6) and Blenkinsopp and Chaplin (8). Although thee is scatte at smalle elative submegence, a linea tend simila to that of Eq. (.19) can be obseved: is appoximately constant fo both small and lage h / H and a sloping tansition exists in between. Least-squae fitting of the o pesent data to linea functions yields 1.4. h / H.95 Hb h h / H.15 hb Ho.58. h / H.15 (.) 44

74 =H b /h b h /H Fig..14 Beake depth index ( ) as a function of elative eef-flat submegence h / H ). Squaes: Exp., s=1/3; Cicles: Exp., s=1/6; Tiangles: Exp., s=1/9; ( Diamonds: Exp., s=1/1. Dashed lines: Ped. by Johnson (6); Dash-dot line: Ped. by Blenkinsopp and Chaplin (8); Solid lines: Least-squae fitting of Exp. data with linea elationship. Solid black makes: Wave beaking on the foe-eef; Solid gey makes: Wave beaking on the eef edge; Open makes: Wave beaking on the eef flat. A close examination of the pesent data eveals that when h / H.95, waves boke on the foe-eef, and when h / H.15, waves boke on the eef flat. This o is consistent with the obsevation fo plane beaches that is slope-dependent and deceases with the decease of seabed slope. The scatte in pesent data fo those foe-eef beaking cases (small h / H ) may be due to the dependence of on Hb / L o H / o L as shown in Figs..11 o.1, which is not included in the pesent data egession analysis. Howeve, this tend is not appaent within the pesent slope ange. The lowe limit of (.58) is close to the value of.55 as found by Nelson (1994) fo hoizontal bottom. Johnson s esults show a consistent highe than the pesent esults, which is possibly because of the vey steep 45

75 seaside slope (1/) of his beakwate model. The data of Blenkinsopp and Chaplin (8) appea to be insensitive to h / H ; this may be caused by the naow ange of h / H (.76 -.) conducted in his expeiments. o o.4.3 Sufzone width The spatial and tempoal scales of the suf zone ove coal eefs have eceived consideable attention in field investigations (e.g., Madin et al., 6). Fo plane beaches, the suf zone is taditionally defined as the egion extending fom the seawad bounday of wave beaking to the limit of wave upush. In this section, the sufzone width of beaking waves ove finging eefs is investigated, the width is measued fom the incipient beaking point to the point whee waves cease to beak on the hoizontal eef flat. The infomation of sufzone width is valuable to calibate empiical enegy dissipation models in numeal simulations. Fo example, the eddyviscosity model as poposed by Kennndy et al. (). Following Goulay (1994), the measued sufzone width, L s is scaled by the shallow wate wave length ( T g( h ) ). The elative sufzone width, W L / T g( h ) as a function of deep-wate wave steepness is shown in Fig. s s.15 fo diffeent foe-eef slopes. It can be obseved that W s is geneally within the ange of 1.3 W s 3.3 fo the pesent data. The lowe theshold of H / L fo waves to beak is the citical condition unde which waves ae small enough to pass though the eef flat without beaking. An appoximate paabolic elationship between W s and H / L can be obseved in Fig..15. Also no definite dependence of W s on foe-eef slopes can be found. 46

76 4 3.5 W s =L s /[ T(g(h + )).5 ] H /L Fig..15 Relative sufzone width ( W ) as a function of deep-wate wave steepness H ( s / L ). Squaes: Exp., s=1/3; Cicles: Exp., s=1/6; Tiangles: Exp., s=1/9; Diamonds: Exp., s=1/1. Fig..16 shows o W s as a function of the invese of the elative eef-flat submegence, h / H. A least-squae fitting of the pesent data yields W s h Ho 1 (.1) Also shown in Fig..16 is the linea elationship poposed by Goulay (1994), which eads W s h. 1.1 Ho 1 (.) Eqs. (.1) and (.) eveal that W s can be descibed solely by the elative eef-flat submegence. Again, thee is no consistent coelation of W s with foe-eef slope. The sufzone widths fo the pesent idealized finging eefs ae geneally smalle than those pedicted by Eq. (.), obtained fom a finging eef with a sloping eef 47

77 flat. This is expected since Eq. (.) was developed by Goulay (1994) fo the uppe limit of sufzone width. The lowe theshold of elative submegence fo nonbeaking waves indicates H /.38 h, which geneally agees with the value of H /.4 h as found by Goulay (1994) W s =L s /[ T(g(h + )).5 ] H /h Fig..16 Relative sufzone width ( W ) as a function of the invese of elative eefflat submegence ( H / s h ). Squaes: Exp., s=1/3; Cicles: Exp., s=1/6; Tiangles: Exp., s=1/9; Diamonds: Exp., s=1/1. Dashed line: Ped. by Goulay (1994); Solid line: Least-squae fitting of Exp. data with linea elationship. Dotted lines epesent the 95% confidence limits of the best fitting line..4.4 Wave tansmission, eflection and enegy dissipation When waves inteact with finging eefs, a lage potion of the enegy will be dissipated by wave beaking and bottom fiction; the emaining wave enegy will be eithe eflected fom the foe-eef o tansmitted towads shoe in the fom of efomed waves. The ability of a submeged finging eef to potect the coasts by dissipating wave enegy has been long ecognized in pevious eseach. The knowledge of wave tansmission and eflection povides a way to estimate the 48

78 amount of wave enegy dissipated by wave beaking and bottom fiction. Enegy dissipation in the suf zone is essential fo computing wave-induced setup and cuents as well as the wave foces imposed on the back-eef beaches; it can also be applied to calibate sufzone numeical models. Tansmission coefficient Wave tansmission chaacteistics of both submeged and emegent ubble-mound beakwates have been studied by many eseaches ove decades (e.g., Seelig, 198; Van de Mee and Daemen, 1994; Van de Mee et al., 5). In geneal, the tansmission coefficient is a function of the stuctue s geomety (cest height and width), font slope angle, suface oughness and poosity as well as wave paametes (incident wave height, incident wave peiod, angle of wave attack). Fo low-cested stuctues (LCS), thee is a wealth of empiical fomulae in the liteatue fo design puposes. Van de Mee and Angemond (1991) suggested that K t vaies linealy with the elative cest submegence. Recently, Van de Mee et al. (5) also consideed the effects of cest width and the incident waves. Van de Mee et al. (5) s fomula fo LCS eads K t.31 F B B c.5i c.4.64 (1 e ) 1 Hi Hi H.65 F B B e Hi Hi H c.41i c (1 ) 1 (.3) whee F is the stuctue cest feeboad and should be negative when the cest is submeged, B c is the cest width, H i is the incident wave height at the toe of the stuctue, i is the suf-similaity paamete defined as s/ H / L the slope of the stuctue s font face. i i o,whee s is When the above equation is applied to the pesent finging eefs, the long hoizontal eef flat is taken as the cest in analogy to the beakwate by setting B / H. c i 49

79 Consequently, the eef-flat submegence, h (which is always positive in this study) is used fo F, and Eq. (.3) is educed to K t h.35 (.4) H whee H is used instead of H i, the diffeence is due to the shoaling coefficient which is appoximately 1 fo all the tested cases ( K anges fom.91 to.95 fo pesent data). t K t Fig..17 Tansmission coefficient ( K ) as a function of suf-similaity paamete t ( ). Squaes: Exp., s=1/3; Cicles: Exp., s=1/6; Tiangles: Exp., s=1/9; Diamonds: Exp., s=1/1. The effect of i o on K t has been eliminated fom the above equation due to the vey lage elative eef-cest width. Howeve, the oiginal equation, i.e., Eq. (.3), indicates that K t depends on i. To veify this, Kt against is plotted in Fig..17. It appeas that fo a given foe-eef slope, K t inceases with inceasing. Howeve, K t is slope-dependent and ises moe apidly unde the 5

80 mild slope, suggesting that is not a suitable paamete to combine the effects of both foe-eef slope and incident waves on section.3.3 that K t itself is almost slope-independent). K t fo finging eefs (it has shown in K t h /H Fig..18 Tansmission coefficient ( K t ) as a function of elative eef-flat submegence ( h / H ). Squaes: Exp., s=1/3; Cicles: Exp., s=1/6; Tiangles: Exp., s=1/9; Diamonds: Exp., s=1/1. Dashed line: Ped. by Van de Mee et al. (5); Solid line: Least-squae fitting of Exp. data with linea elationship; Dotted lines epesent the 95% confidence limits of the best fitting line. Subsequently, the tansmission coefficients ae plotted as a function of the elative eef-flat submegence fo diffeent foe-eef slopes in Fig..18, along with Eq. (.4). It can be obseved that K t inceases almost unifomly with inceasing h / H, egadless of the foe-eef slopes. This is quite expected in that lage submegence means lage tansmitted waves in view of the depth-limited beaking condition (see Eq. (.1)). The values of K t collapse appoximately onto a staight 51

81 line except fo cetain scatte at vey lage h / H. A least-squae fitting of the pesent data to a linea function gives K t h (.5) H The values of K t fo the pesent data ae pedominantly lowe than those fo submeged stuctues as given by Eq. (.4), paticulaly fo the cases with lage h / H, suggesting that Eq. (.4) is not applicable to the seabed pofiles (e.g., submeged steps, eefs) with elatively long cest widths. Reflection coefficient It is well studied in the liteatue that the eflection coefficient fo a stuctue can be chaacteized by i. The most commonly used design fomula is the one poposed by Seelig and Ahens (1981) K a (.6) b i i The values of a and b depend pimaily on the stuctue geomety, poosity and suface oughness as well as wave type to cetain extent. USACE (3) ecommends that a 1., b 5.5 fo plane slopes unde monochomatic waves and a.6, b 6.6 fo ubble-mound beakwates. Again, the values of K measued fom the expeiments ae plotted as a function of in Fig..19, togethe with Eq. (.6) fo plane slopes and ubble-mound beakwates. Fig..19 shows that the eflection coefficient is geneally vey small fo finging eefs (less than %). The geat scatte in the data is due patly to the eo of the taditional two-point method used to sepaate the incident waves fom the eflected waves, and patly to the multiple eflections existing in the flume (the eflection fom wavemake peddle, the two edges of the foe-eef and the back-eef beaches, etc.). Simila degee of the scatte in K was also found by Van de Mee et al. (5) when they analyzed diffeent datasets fo LCS, although the eo 5

82 souces may be diffeent fom the pesent). Compaison of the pesent data with the cuve of the ubble-mound beakwates shows that K is in an ageement with the pedictions by Eq. (.6) at smalle, but the ageement becomes unsatisfactoy as inceases. Howeve, the values of than those fo submeged beakwates o finging eefs. K fo plane beaches ae consistently lage K Fig..19 Reflection coefficient ( K ) as a function of the suf-similaity paamete ( ). Squaes: Exp., s=1/3; Cicles: Exp., s=1/6; Tiangles: Exp., s=1/9; Diamonds: Exp., s=1/1. Dashed line: Ped. by Seelig and Ahens (1981) fo plane beach; Dash-dot line: Ped. by Seelig and Ahens (1981) fo ubble-mound beakwate. The dependence of the eflection coefficient on the elative eef-flat submegence is shown in Fig.. fo diffeent foe-eef slopes, fo which no clea tend in be seen, implying that K is almost independent of h / to obtain convincing fomula to paameteize K can H. So fa, it is not possible K based on the pesent data. 53

83 K h /H Fig.. Reflection coefficient ( K ) as a function of elative eef-flat submegence ( h / H ). Squaes: Exp., s=1/3; Cicles: Exp., s=1/6; Tiangles: Exp., s=1/9; Diamonds: Exp., s=1/1. Enegy loss coefficient The above analysis has shown that the elative eef-flat submegence, h / H, is a suitable paamete to pedict the tansmission coefficients. Howeve, a suitable paamete to pedict the eflection coefficients could not be found. In view that is small (less than %), its contibution to the enegy balance can be neglected when detemining the enegy loss coefficient, h / H is also a dominant paamete to descibe d K K d, by Eq. (.). It is anticipated that K. 54

84 K d h /H Fig..1 Enegy loss coefficient ( K d ) as a function of elative eef-flat submegence ( h / H ). Squaes: Exp., s=1/3; Cicles: Exp., s=1/6; Tiangles: Exp., s=1/9; Diamonds: Exp., s=1/1. Solid line: Least-squae fitting of Exp. data with linea elationship; Dotted lines epesent the 95% confidence limits of the best fitting line.. The calculated K d as a function of h / H is shown in Fig..1 fo diffeent foeeef slopes. It can be obseved that moe than 5% of incident wave enegy can be dissipated by wave beaking on finging eefs fo all cases. Lage value of appeas fo plunging beakes ( h / H 1.8, see Eq. (.6)), which is consistent with the well-accepted notion that plunging beakes dissipate moe enegy than spilling beakes. Also shown in Fig..1 ae the enegy loss coefficients calculated by Eq. (.7), which is obtained by combining Eq. (.5) with Eq. (.), with the eflection coefficient being ignoed because of its small contibution to the enegy balance (less than 4%) K d K d H H h h (.7) 55

85 Eq. (.7) suggests that h / H appoaches, Kd appoaches to 1, i.e., if the eef flat is dy and almost all the wave enegy can be dissipated though beaking; this might not be tue if lage wave eflection exists. Since wave beaking may cease if the eef flat is deeply submeged, Eq. (15) is valid only when / H. 7 accoding to Eq. (.6). The non-zeo K found fo h / H. 7 in Fig..1 is due d to the difficulty in the detemination of the citical value of h / H fo non-beaking waves. h.5 Concluding Remaks A seies of laboatoy expeiments wee conducted in a wave flume to examine the effects of vaying elative eef-flat submegence and foe-eef slopes on the popeties of beaking waves ove submeged idealized finging eefs. Peliminay analysis of the data shows that deep-wate wave height and eef-flat submegence ae two key paametes to descibe most of the beaking-wave popeties investigated in this study. The influence of foe-eef slope appeas to be insignificant. Dimensional analyses eveal that: (1) The elative submegence of the eef flat has been found to be the deteminating facto to chaacteize the beake type fo finging eefs. () The atio of the eef-flat submegence to the beake wave height has been found to be an appopiate paamete to classify the beaking locations. (3) The beake height index appeas to be stongly dependent on the deep-wate wave steepness in the same manne as that fo plane beaches. (4) The beake depth index is found to be piecewise linealy elated to the elative eef-flat submegence, esembling what is found fo submeged beakwates. (5) The elative sufzone width is on the same ode of magnitude as the length of local shallow wate waves, and it has a linea coelation with the invese of the elative eef-flat submegence. (6) The tansmitted coefficients incease almost linealy with inceasing elative eef-flat submegence. (7) Moe than 5% of incident wave enegy can be damped by the finging eef. Enegy dissipation deceases with inceasing elative eef-flat submegence. 56

86 (8) Some empiical fomulae based on the findings listed above ae poposed within the expeimental data ange (.3 h / H 3.53 ). Natual eefs ae fa moe complex than those that can be studied in laboatoy; fo a natual eef pofile, the eef-cest submegence may become a pimay contolling paamete on the natue of wave beaking. Spectal waves, coastal cuents, foeeef oughness, etc. might modify some conclusions in this study to a moe o less extent. Compaisons with wave beaking obseved in field conditions wee not attempted in this chapte. 57

87 CHAPTER 3 A LABORATORY STUDY OF WAVE- INDUCED SETUP OVER A HORIZONTAL REEF WITH/WITHOUT A RIDGE 3.1 Liteatue Review The shoaling and beaking of suface gavity waves on coal eefs play a cental ole in shaping the physical stuctue of coal eefs as well as seving biogeochemical function (Atkinson and Bilge, 199; Madin et al., 6). In many espects, the physics of waves on eefs paallels those of waves on beaches, although coal eefs ae notably diffeent fom nomal coastal beaches in many espects including the oughness of the substate (Lowe et al., 5), and the vaiety of geometies that ae possible, e.g., baie eefs vs. finging eefs. As discussed in Chapte 1, many eefs ae chaacteized by an inshoe shallow eef flat with a steep offshoe face (the foe-eef). The incident waves beak on the foe-eef o the eef flat, dissipating thei enegy and geneating a ise of mean sea level known as wave setup, a phenomenon fist descibed by Munk and Sagent (1948). Fo a eef with an open lagoon (baie eef o platfom eef), this setup dives a cuent that exits the lagoon via gaps o channels between the lagoon and open sea, wheeas fo a eef with a closed lagoon o with no lagoon, the wave-induced setup on the eef flat is moe o less constant and the wave-geneated flow is always weak (Lowe et al., 9a). Although the eef-flat bathymety vaies fom site to site, a idge o simila configuation ( eef im o eef cest in some papes) is fequently obseved at the edge of many coal eefs (Seelig, 1983; Kench et al., 6; Jago et al., 7; Hench et al., 8; Péquignet et al., 11). Ridges consisting of coal colonies, ubble algal, etc. might be fomed due to long-tem exposue unde high enegy dissipation at the eef edge whee metabolic ates (nutient uptake, photosynthesis, poduction etc.) and biomechanical toleances of benthic oganisms ae significantly enhanced (Atkinson and Bilge, 199; Madin et al., 6). Howeve, most pevious investigations of waves and eef hydodynamics (Goulay, 1996a; Symonds et al., 58

88 1995; Hean, 1999; Goulay and Collete, 5) have geneally ignoed these idge stuctues, focusing on simple eef mophologies with constant foe-eef slopes and hoizontal eef flats. In this chapte, laboatoy esults will be pesented, showing that these idge stuctues can significantly influence wave dynamics and thus alte the wave-induced setups and wave-diven flows. While a numbe of field obsevations of waves on eefs have been epoted in the liteatue (e.g., Young, 1989; Hady and Young, 1996; Lugo-Fenandez et al., 1998a, 1998b; Bande et al., 4; Lowe et al., 5, 9a; Hench et al., 8), expeimental investigations into wave inteactions with eefs ae scace despite the fact that laboatoy expeiments can be used to isolate paametic dependencies on focing paametes in a contolled envionment. Ealy investigations can be found in Geitsen (198) and Seelig (1983). Notably, Goulay (1994, 1996a) epoted a seies of laboatoy expeiments with idealized two-dimensional (D) eef models, which wee late compaed to existing field obsevations (Goulay, 1996b). He showed that wave setup was lage at both low eef-flat wate depths and lage deep-wate wave heights. Using a finging eef pofile simila to that in Seelig (1983), Demibilek et al. (7) conducted laboatoy investigations on the combined effects of wind and waves on wave setup and unup fo spectal waves. Moe ecently, Yao et al. (9) epoted peliminay laboatoy expeiments including a idge stuctue located at the seawad edge of a hoizontal eef flat, showing that the idge could significantly incease the wave-induced setup, and may contol the eef-flat setup in a way same as a boad-cest wei contols the wate level in open channel flows. Motivated by the field obsevations epoted in Hench et al. (8) and pusuing the hypothesis that the eef idge plays an impotant ole in detemining wave setup, a seies of expeiments wee caied out in a wave flume with a eef model that eplicates a finging eef (closed lagoon) system. The pesent laboatoy expeiments will also be used in the subsequent chaptes to evaluate a numeical model and seveal analytical elations that have been developed to pedict the wave setup on coal eefs. The est of the chapte is oganized as follows. In section 3., 59

89 the laboatoy settings, instuments and expeimental pocedues ae intoduced. In section 3.3, the epesentative expeimental esults unde both monochomatic and spectal waves in the absence/pesence of a idge ae shown. Some nonlinea chaacteistics of coss-shoe wave evolution (hamonic waves and adiation stess) ae investigated in section 3.4 though a wave measuement with impoved spatial esolution. The supplementay measuements of sufzone undetow ae given by section 3.5. Some discussions ae given in section 3.6. Main esults ae summaized and key conclusions ae dawn in section Expeimental Settings and Instuments 3..1 Wave flume and eef models To examine the effects of idges on wave-induced setup, a seies of laboatoy expeiments wee caied out in the Hydaulics Modeling Laboatoy, Nanyang Technological Univesity, Singapoe. All tests wee conducted in a glass-walled wave flume 36 m long,.55 m wide and.6 m deep. A sevo-contolled piston type wavemake which can geneate both egula and iegula sea states was placed at one end of the flume to geneate the designed waves. At the othe end, a beach with a slope of 1:8 stated at appoximately 3 m fom the wave make and was coveed with poous mats of 1 cm thick to educe wave eflection. To constuct an idealized D eef model that eplicated a finging eef, a plane slope of appoximate 1:6 was built with PVC plates at m fom the wavemake and it met the hoizontal platfom which was.35 m above the flume bottom. The hoizontal platfom was 7 m width with its length matching the inne width of the flume (see Fig. 3.1). The entie model was fimly held by stainless ods attaching to the two walls of the flume. A ectangula box 55 cm long, 5 cm wide and 5 cm high was placed on the eef flat with its font face aligned with the eef edge to model a idge as shown in Fig. 3.. The dimension (coss-shoe width to height atio is 1:1) of the idge model was chosen to mimic the eef idge existing on the Mooea eef studied by Hench et al. (8). 6

90 Pobe Senso Foe-eef Reef flat Fig. 3.1 The eef model. Left: Top view of foe-eef; Right: Side view of eef flat. Ridge Ridge Reef edge Fig. 3. The idge model. Left: Top view; Right: Side view of idge on eef flat. Simila to the expeimental setting in Chapte, the dimensions of the idge, the wate depth befoe the eef slope, and incoming wave height wee designed accoding with Foude similaity with a tageted geometic scale facto of 1:. Thus the dimensions of the pototype idge ae 1 m high and 1 m wide, which ae in geneal ageement with what have been obseved at Mooea eef, which is a typical baie eef and includes a lagoon open to the ocean at the back of the eef. The expeiments in this thesis wee designed to povide insight into the effects of the idge on wave tansfomation ove finging eef pofiles athe than to mimic the Mooea eef at an exact model scale. Fo example, given the eef flat of 7 m in the laboatoy setting, the coesponding eef-flat width at the pototype scale is 14 m, which is shote than that of the Mooea eef (about 1 km). Howeve, the width of 61

91 ou eef-flat model is acceptable because the autho will show latte in this chapte that: (1) it is longe than the maximum suf-zone width in ou expeiments; () the waves outside the suf zone is vey small and the wave eflection fom the final beach is weak; (3) the wave-induced setup is almost constant outside the suf zone, and (4) the mean cuent is vey small outside the suf zone. The foe-eef slopes in the expeiments wee also slightly steepe than the aveage foe-eef slope of Mooea eef (1:8). The oughness and poosity of the eef flat have only mino effects on the wave-induced setups in the absence of the idge, as shown in the Appendix C; the effects of oughness of poosity will be even smalle when a idge is pesent since both the waves and mean cuent ae vey small behind the idge. Theefoe, epoducing hydaulic oughness and poosity of natual coal eefs at the model scale was not attempted in the expeiments. 3.. Expeimental pocedues The schematic layout of the expeimental aangement is showed in Fig In the shallow egion ove the eef flat, fou Ultalab sensos (Geneal Acoustics Ltd., see also Fig. 3.1) wee used to measue the wate suface elevation. These sensos tansmit ultasound pulses, which afte eflection fom the wate suface ae eceived by the senso as an echo enabling the distance between the senso and the moving wate suface to be calculated based on the time of tavel. These fou sensos (G9 - G1) wee almost equally spaced on the eef platfom with the fist senso located 1.7 m away fom the eef edge. Howeve, in ode to measue the wave setup in the zone acoss the idge, whee the acoustic sensos could not wok well due to the shap tansfomation of waves ove the idge and ai bubbles in wate, five esistance-typed pobes (G4 - G8) (HR Wallingfod Ltd., also shown in Fig. 3.1) wee installed. In addition, two pobes (G1 and G) wee placed upsteam of the foe-eef to sepaate incident waves fom the eflected waves (Goda, ) and one pobe (G3) was placed on the slope itself to estimate wave shoaling. In pinciple, both types of wave gauges have accuacy up to.1 mm. Pactically, it is estimated that the actual accuacy is ±1 mm due to vaious eos including calibation, weathe conditions (the laboatoy is open to the outdoos), fluctuation 6

92 of initial mean wate level, etc.. All the wave gauges wee sampled at 5 Hz though a pesonal compute based data acquisition system. Wavemake G1 G Ridge G3 G4 G5 G6 G7G8 G9 G1 G11 G Still wate level G: Wave gauge Unit: m h Reef edge Foe-eef Reef flat Beach (1:8) with poous mateials Fig. 3.3 Sketch of the expeimental aangement. The design incident monochomatic wave conditions wee selected fom a combination of five incident wave heights (anging fom.5 m to.13 m) and fou wave peiods (anging fom.83 s to 1.67 s). The taget waves (shown in Table 3.1) wee slightly diffeent fom the actual incident waves in the expeiments, due in pat to the accuacy of the wavemake and in pat to the weak multiple eflections that existed in the wave flume. The actual incident waves, which wee detemined by sepaating the incident waves fom the eflected waves using the wave ecods measued at G1 and G, wee conveted to deep wate waves fo analyzing eef-flat wave setups. Fou wate depths ( h ) wee studied (4 cm, 41 cm, 4 cm and 45 cm) in this study. Those wate depths wee selected so that diffeent submegences of the idge-top ( h c cm, 1 cm, cm and 5 cm) could be investigated. Fo compaison, thee wate depths (35 cm, 4 cm and 45 cm) wee tested without the idge; these wate depths coesponded to the eef-flat /cest submegence of h c cm, 5 cm and 1 cm. 63

93 Table 3.1 Test conditions a Type Without idge Range With idge Reef-cest submegence: h c (m),.5,.1,.1,.,.5 Deep-wate monochomic wave height: H (m) Monochomic wave peiod : T (s) Deep-wate significant wave height: H (m) s Peak wave peiod: T p (s) Reef-cest (idge) height: d (m).5 Coss-shoe eef-cest (idge) width: B (m).5 Foe-eef slope: s tan 1:6 1:5 a ' The italic value denotes the equivalent slope ( tan ) as defined in Fig. 5.1 and the bold value was only tested unde monochomatic waves. The wave conditions fo each wate depth wee vaied so that a measuable mean wate level pofile could be obtained. Duing the expeiments, each wave condition was epeated thee times to ensue the epeatability of the expeimental esults. Wave beaking befoe the eef slope was obseved when the deep-wate wave steepness was vey lage; those cases wee excluded fom the following analysis. Afte the wavemake was stated, suface elevations wee ecoded by wave gauges though a data acquisition system fo analyzing wave chaacteistics and wave setup. Based on the pilot tests, wave multiple eflections in the wave flume could each a quasi-steady state (i.e., the wave amplitude does not have a significant vaiation in time) about 3 min afte the stat of the wavemake. This can be explained by the following agument. Fo a epesentative wate depth h.4 m and wave peiod T 1.5 s, the wave celeity is c 1.6 m/s in the expeiments. The distance between the toe of the foe-eef and the wavemake is about m. It takes about.4 s fo the waves to have a ound tip between the wavemake and the eef model. Afte 3 min, the waves have taveled about 9 ound tips, which is enough fo the waves to each a quasi-steady state in the wave flume. Theefoe, wave gauges wee 64

94 sampled fo 5 min and the last 1 wave cycles in wave ecods wee used fo calculating the wave setups. Between two subsequent tests, seveal minutes wee allowed to elapse so that the wate suface could become calm and the effects of esidual cuents wee minimal. The expeiments wee also extended to spectal wave conditions. The wave specta wee geneated fom JONSWAP specta with peak enhancement facto γ = 3.3. The designed spectal wave condition was a combination of significant wave heights anging fom.3 m to.9 m and peak wave peiods anging fom 1. s to 1.67 s (also shown in Table 3.1). The designed anges fo spectal waves wee naowe compaed with those fo monochomic waves to avoid that steepe individual waves in the spectum beak offshoe. The aangement of wave pobes (G1 - G8) was also slightly diffeent fom that fo monochomic waves. G3 was moved offshoe to fom a thee-pobe aay fo wave spectum sepaation (see Goda, fo details). The significant wave height, H s, was estimated by Hs 4 m, whee m is the zeoth moment of measued spectum, which was then conveted to deep-wate wave height; G4 and G5 wee located on the foe-eef slope to captue the elatively wide suf zone of spectal waves and the emaining pobes (G6 - G8) wee put close to the idges. The exact aangement of all wave gauges (G1 - G1) fo both wave conditions (monochomic and spectal) can be found in Fig The wave gauges wee sampled fo min, wave analysis was pefomed using the data 4 min afte the stat of data collection to allow spectal waves to each quasi-steady state. Unlike monochomic waves, only thee wate depths (4 cm, 4 cm and 45 cm) wee investigated fo spectal waves when the idge was pesent. Accoding to Foude similaity, the scale facto fo wave peiod would be 1:4.5 fo a geometic scale facto of 1:. Fo the test conditions listed in Table 3.1, the pototype scales ae h = m - m, H =.7 m -.7 m and T = 3.7 s s fo monochomatic waves, and h = m - m, H s =.64 m m and T p = 4.5 s s fo spectal wave waves. Theses pototype scales ae in geneal ageement 65

95 with those obseved in field conditions (Bonneton et al., 7; Hench et al., 8; Lowe et al., 9a; Vette et al., 1). 3.3 Results Visualization of wave tansfomation ove eef cest and eflection Videos showing the beaking aea wee taken afte the wave field had eached quasi-steady state. Plunging beakes wee obseved in most of the tested cases; spilling beakes wee also obseved in some of the cases with h.45 m. The beaking point moves fom the foe-eef slope onto the eef flat as wate depth inceases. Fig. 3.4 shows the epesentative featues of wave tansfomation and beaking ove the eef cest in the pesence /absence of the idge fo incident monochomatic waves of H.95 m and T 1.5 s in wate of h.45 m. Fou diffeent phases ae displayed stating fom the moment when the lip of beake hit the wate suface ( t / T ). Fo the eef without the idge, waves plunged onto the eef flat at t / T, while at t / T 1 / 4, the splash-up jet due to plunging beake hit the wate ahead of it, poducing an ai-wate mixtue of foam, bubbles and some subsequent white-capping. Afte half wave peiod (t / T 1 / ), the boken waves popagated acoss the suf zone as a fully tubulent boe. Fo the last phase ( t / T 3 / 4), the boe was mostly dissipated and a tansmitted wave efomed on the eef flat. Stong evese flow could be obseved duing this peiod befoe the next incoming wave aived. When a idge was pesent at the eef cest, the beaking point shifted seawad, the beaking waves stoke the font side o the edge of the idge and then plunged onto the idge-top, esulting in a stonge wave eflection and ai entainment. Howeve, the tansfomation pocess was othewise simila to that fo the eef without the idge unde the same wate depth and wave condition. 66

96 Fig. 3.4 Snapshots of monochomatic wave tansfomation ove eef cest at diffeent phases, the time inteval between consecutive phases is one quate of a wave peiod ( h.45 m, H.95 m and T 1.5 s). Reflection coefficients detemined by the two-pobe method anged fom % to 5% without the idge and fom 6% to 55% with the idge fo monochomatic waves. Fo spectal waves, they anged fom 9% to 37% without the idge and fom % to 66 % with the idge, espectively. Spectal waves have lage values of eflection coefficients than the compaable monochomatic waves because the lowfequency potion of wave specta eflects moe efficiently (Seelig, 1983). The effects of the idge on the wave eflection wee moe evident fo waves of small wave steepness. The enhanced wave eflection is expected since the idge stuctue functions like a submeged beakwate, which is used widely to eflect ocean wave enegy (Yao et al., 9) Wave evolution acoss eef pofile Time- seies suface elevation fo monochomatic waves Sample wave ecods at selected wave gauges fo model eefs with and without idges (adjusted to be in phase) ae given in Fig At the deep section of the 67

97 flume (G1), Stokes waves that have shot, peaky cests and longe shallowe toughs wee obseved in the intemediate depth of wate ( kh 1.16 ). When waves shoaled on the foe-eef slope (G3), they became moe asymmetic and skewed. G6 was located in the suf zone fo both cases; thus beaking waves with saw-tooth shape could be obseved thee. On the eef flat (G9), the efomed waves wee vey small and thee wee second peaks in the wave pofiles due to the geneation of highe hamonics. The phase discepancy between the fundamental hamonics became substantial between the two eefs as waves popagated shoewad. (mm) (mm) (mm) (mm) 5 G t (s) 5 G t (s) 5 G t (s) 5 G t (s) Fig. 3.5 Time-seies wave ecods fom selective wave gauges (G1, G3, G6 and G9) acoss diffeent eef pofiles unde monochomatic waves ( h.45 m, H.95 m and T 1.5 s). Dashed lines - without idge; solid lines - with idge. Wave specta fo spectal waves Wave specta fo the case H.87 m and T 1.67 s in wate of h.4 m s ae shown in Fig. 3.6 to exemplify the behavio of spectal waves. As the waves p 68

98 popagated ove the foe-eef slope (G5) into the shallowe wate, wave enegy was tansfeed fom the peak fequency ( f.61 Hz ) to lowe fequencies in the p spectum (shown by G5) fo eefs both with and without the idge. Wave beaking occued eithe on the foe-eef slope o on the eef flat fo the case without the idge; hence thee was still a consideable amount of wave enegy aound the peak fequency in the incident spectum within the suf zone (G7). Howeve at this location, a lage potion of the wave enegy nea the peak fequency was dissipated when the idge was pesent because the idge-top was initially dy in wate of h.4 m and all waves boke on the foe-eef slope o idge edge. Shoewad of the eef-flat suf zone (G9), thee was no notable diffeence in the wave specta between the two eefs since the local wate depths on the eef flat wee both sufficiently shallow (.5 m) so that most of wave enegy was filteed though wave beaking. x 1-3 (a) 1.5 G1 G5 G7 G9 x 1-3 (b) 1.5 G1 G5 G7 G9 S(m.s) 1 S(m.s) f(hz) f(hz) Fig. 3.6 Wave specta ( h.4 m, H.87 m and T 1.67 s) fom selective wave gauges (G1, G5, G7 and G9) acoss diffeent eef pofiles: (a) without idge; (b) with idge Mean wate level acoss eef pofile Twelve wave gauges (G1 to G1) wee used in the expeiments, enabling us to constuct easonably detailed wave setup pofiles acoss the eef models by a linea 69

99 intepolation. Fo example, two cases epesenting monochomatic and spectal wave conditions ae illustated in Fig As expected, fo both waves, setdown occued at the seawad side of the suf zone. In the suf zone itself, thee was a monotonic incease of mean wate level (MWL) due to wave beaking. The peak setup always appeaed at G9. Afte the peak, the setup slightly deceased and then became almost constant. Unde the same wave condition, the magnitude of setup in the pesence of the idge was significantly lage than that seen in the absence of the idge. Moeove, the idge also caused the lowest point of MWL to move seawad along with the point whee waves boke..48 (a) h =.45m,H =.95m,T=1.5s MWL(m) Distance fom toe of the foe-eef (m) (b) h =.4m,H s =.87m,T p =1.67s MWL(m) Distance fom toe of the foe-eef (m) Fig. 3.7 Mean wate level (MWL) offshoe and acoss the eef pofile unde diffeent wave conditions: (a) monochomatic waves; (b) spectal waves. Open cicles - locations of wave gauges; dotted lines - without idge; solid lines - with idge; dashed lines - Still wate level (SWL). 7

100 3.3.4 Wave setup as a function of deep-wate wave height The above analysis shows that the MWL always eached its maximum aound G9 on the eef flat; thus, the setup at G9 is a good estimate of the maximum setup,. As an example, Fig. 3.8 shows this wave-induced setup as a function of deep-wate wave height, H, fo diffeent wave peiods, T, and fo two wate depths h.4 m and.45 m. The cases with and without the idge unde diffeent wave conditions ae also compaed in this figue. All the measued data in the expeiments can be efeed to Appendix B. A key issue aises that which wave height and wave peiod ae selected as the epesentative wave paametes fo spectal waves. Thee is no intinsic elationship between egula and iegula wave paametes, and accodingly thei equivalence depends on which wave featue is consideed as the pimay concen. Since wave setup is elated to the adiation stess (thus the wave enegy) as indicated in Eqs. (5.) and (5.4), the deep-wate oot-mean-squae wave height ( H ms ) is chosen so that an equivalent enegy between the monochomatic and spectal waves is ensued. Fo wave peiod, the peak wave peiod ( T p ) was used, the discepancy between the peak wave peiod and the significant wave peiod is vey small. Unde simila wave conditions, the idge could cause a noticeable incease in the wave setup on the eef flat, paticulaly fo the cases of longe wave peiod. In some cases, the wave-induced setup was as much as doubled by the idge. With o without a idge, the wave-induced setup on the eef flat inceased almost linealy with inceasing H. The elationship between and T was less obvious. Howeve, by compaing the wave setup unde simila H, it could be found that in geneal inceased with inceasing T, in ageement with the obsevations of Goulay (1996a). Meanwhile, the setup vaiation fo spectal waves has simila tend with that fo monochomatic waves when H ms and T p wee used to chaacteize spectal waves. 71

101 (mm) (a) Monochomatic, h =.4m T=1.67s T=1.5s T=1.s T=.83s (mm) (b) Monochomatic, h =.45m T=1.67s T=1.5s T=1.s T=.83s 1 1 (mm) H (m) (c) Spectal, h =.4m T p =1.67s T p =1.5s T p =1.s H ms (m) (mm) H (m) (d) Spectal, h =.45m T p =1.67s T p =1.5s T p =1.s H ms (m) Fig. 3.8 Maximum wave setup on eef flat as a function of deep-wate wave height fo diffeent wave peiods, still wate depths and incident wave conditions. Open makes - without idge; solid makes - with idge. 3.4 A Detailed Wave Measuement with Impoved Spatial Resolution Definition of the suf zone ove eefs As discussed in Chapte, fo wave beaking ove coal eefs, the suf zone is defined as the hoizontal distance between the incipient beaking point to the point whee popagating boes disappea and oscillatoy waves efom. Fo the convenience of subsequent discussion, the autho follows the division of suf zone ove eefs by Goulay (1994) which is based on the ideas of Svendsen et al. (1978): Oute suf zone In this egion thee is a significant change of wave shape as the suf olle develops. Howeve, thee is no significant change in momentum flux and the apid eduction in wave height is associated with the tansfomation of potential enegy into kinetic 7

102 enegy. Fo plunging beakes, the width of this egion is appoximately identical to the plunging distance (descibed below) of the beake. Inne suf zone In this egion the suf olle developed in the oute egion becomes a boe o moving hydaulic jump which tavels landwad ove the seawad flowing undetow. Dissipation of wave enegy and wave-induced setup begin at the tansition between the inne and oute egions. On a hoizontal o nea hoizontal eef, the inne egion coesponds to the egion whee the boe popagates ove the eef flat and this egion ends whee the boe disappeas and an oscillatoy wave efoms. Unlike waves beaking on plane beaches descibed in Svendsen et al. (1978), thee is no swash zone when waves beak ove coal eefs. The efomed oscillatoy waves will continue to popagate ove the shoewad egion of the suf zone (stating fom the endpoint of suf zone to the lagoon (if it exists) o to the coastline) until they beak on the final beach. Fictional dissipation may be significant duing the whole pocess. Following Galvin (1969), the autho defines the plunge distance as the distance fom the beaking point (see section.3.1 fo the definition) to the plunging point (the location whee the wave cest culs ove and touches the wate suface in font of it). The splashing distance is defined as the distance fom the plunging point to the splashing point (the location whee the wate mass displaced upwad by the plunging wave stikes the wate suface). See also Fig. 3.9 fo all the definitions above. 73

103 Fig. 3.9 Regions and locations in the suf zone (adapted fom Svendsen et al., 1978) Spatial vaiation of wave height and mean wate level Section has compaed the magnitudes of the MWL obtained in the absence /pesence of a ectangle idge fo both monochomatic and spectal waves with limited sampling locations. In this section, to futhe undestand the coss-shoe distibution of wave height, MWL, highe hamonic waves and the momentum flux, a detailed wave measuement coveing 6 sampling locations was caied out with a spatial esolution of.75 m in both the absence and pesence of the idge. The same monochomatic waves same as those in section 3.3 ( h.45 m, H.95 m and T 1.5 s ) wee selected. The studied aea began fom the shoaling zone on the foe-eef (.75 m seawad of eef edge) and ended some distance shoewad of the suf zone on the eef flat (3.675 m shoewad of eef edge). The wave ecods wee obtained by moving one esistance-type wave gauge along the flume within the same un. Wave measuements aound the plunging points wee not included in the analysis due to significant ai-entainment. Measuement locations elevant to subsequent discussions in this chapte ae given in Table 3. fo both cases (with and without the idge). 74

104 Table 3. Selected measuement locations Location name Distance a fom eef edge (m) without idge with idge Beaking point. -.4 Plunging point.5. Splashing point Endpoint of suf zone a positive if shoewad of the eef edge. The measued local wave height and mean wate level (MWL) ae shown in Figs. 3.1(a) and 3.1(b) in both the absence and pesence of the idge. Also indicated in the figue ae the sampling locations and seabed elevation. The oigin of the abscissa is at the eef edge. The local wave height is obtained by aveaging ove about 5 waves detemined by a zeo-upcossing method. As it can be obseved, wave beaking occued almost at the same location (maximum wave height points) seawad of the eef/idge edge fo both pofiles (with and without the idge). Fo shoaling waves on the foe-eef, the wave height fluctuated due to patial wave eflection; the fluctuation was enhanced when the idge was pesent. Howeve, shoewad of the beaking point, the wave height deceased moe apidly fo the eef with the idge than without due to a naowe suf zone caused by the pesence of idge. Shoewad of the eef-flat suf zone, the wave height fo the eef with the idge was still slightly smalle although the still wate depth ( h ) on the eef flat was the same fo both pofiles. The mino fluctuation of wave height on the eef flat might be due to the eflection of the beach at the end of flume. Fo the MWL, no fluctuation could be obseved. MWLs eached the lowest points ight in font of the eef/idge edge due to wave setdown and began to ise in the suf zone. The estimated sufzone widths in the figue wee about.7 m without the idge and 1. m with the idge, espectively, which wee consistent with.9 m and 1.38 m (see Table 3.) estimated by obsevation of the popagating boes. Shoewad of suf zone, the setups became nealy constant on the eef flat, which is typical fo finging eefs. 75

105 H(m) (a) Without idge With idge Measuement location (b) MWL(m) Seabed Elev.(m).4.3 (c) Distance fom eef/idge edge (m) Fig. 3.1 Wave height and mean wate level (MWL) acoss the eef pofile ( h.45 m, H.95 m and T 1.5 s) Geneation of highe hamonics The geneation of highe hamonics when waves popagate ove immesed obstacles (such as submeged steps, bas, beakwates and plates) has been studied ove decades (e.g., Beji and Battjes, 1993; Chistou et al., 8; Bossad et al., 9). Pevious expeimental esults and numeical simulations have shown that up to 6% of the incident wave enegy can be tansfeed downsteam of the stuctue though highe hamonic modes. The finging-eef model esembles a sloping step, hamonic geneation may occu above the foe-eef and ove the eef cest. On eenteing deepe wate on the downsteam side of the eef cest (idge), these highe hamonics may be eleased as fee waves. The pesent expeimental settings could not sepaate the fee waves fom the locked waves. In this section, the autho 76

106 pesents a peliminay hamonic analysis to quantify the enegy tansfe fom the fundamental mode to highe hamonic modes in the measued coss-shoe egion. Figs 3.11(a) and 3.11(b) show the coss-shoe vaiation of hamonic waves in the absence/pesence of the idge, espectively. Up to the fifth hamonics ( a 1 to a 5 ) ae shown, and they ae detemined by a least-squae fitting of the Fouie seies to the measued suface elevations. Also shown in Fig is the seabed elevation. a i (m).8.6 (a).4. a 1 a 4 a a 5 a a i (m).8.6 (b).4. Seabed Elev.(m) (c) Distance fom eef/idge edge (m) Fig Hamonic wave amplitudes acoss the eef pofiles: (a) without idge; (b) with idge ( ai( i 1, 5) - the i th hamonic wave amplitude; h.45 m, H.95 and T 1.5 s). m As it can be estimated in both Figs 3.11(a) and 3.11(b), fo shoaling waves at the beginning of the measued egion, the fundamental hamonic waves ( a 1 ) eached up to 95% of the incident wave height ( H /.48 ) while the height of the second hamonic waves ( a ) was about % of a 1, and the thid to fifth hamonic waves wee vey small in the shoaling zone. As wave shoaling continued on the foe-eef, 77

107 the highe hamonics continued to incease shoewad due to the enegy tansfe fom the fundamental mode to highe hamonic modes. The fluctuation of a 1 on the foe-eef, paticulaly when the idge was pesent, was due again to the stong wave eflection fom the foe-eef. Afte the beaking points, the coss-shoe evolution of hamonic waves was not significantly affected by the pesence of the idge: in the oute suf zone, thee was a apid decease of a 1 due to the enegy dissipation by wave beaking, but beaking dissipation only esulted in a slight decease of high hamonic wave heights ( a to a 5 ). In the inne suf zone and the shoewad egion of the suf zone on eef flat, wave attenuation slowed down, and the second hamonic waves became compaable with the fundamental hamonic waves. The thid to fouth hamonics emained vey small on the eef flat. The spatial oscillations in the amplitudes of the fundamental and highe hamonic waves on the eef flat ae due patly to the coexistence of the locked and fee waves (the bound and fee modes popagating downsteam have the same pulsation but diffeent wave numbes; this may be tue when the idge is pesent, but may not be tue in the absence of the idge) and due patly to a esonant behavio caused by the multieflection between the two ends of the eef flat. The above analysis is based only on one wave condition, and futhe studies ae needed in the futue to examine diffeent wave conditions, diffeent eef-cest submegences and diffeent eef/idge widths. Nevetheless, the above simple analysis does show that the geneation of highe hamonics can have a significant impact on the tansmitted wave enegy ove eefs. This highly nonlinea phenomenon becomes moe significant as waves tansfom fom the foe-eef to the eef flat; as a esult, the validity of the existing analytical models that ae based on linea monochomatic waves needs to be evisited using newly obtained expeimental data Calculation of adiation stess The adiation stess concept as intoduced by Longuet-Higgins and Stewat (196) is defined as the excess flux of hoizontal momentum due to the pesence of waves. 78

108 This concept is impotant in neashoe hydodynamics because the spatial gadients of the adiation stess can geneate a vaiation of mean wate level (MWL) (setup and setdown), cuents alongshoe and long waves (Longuet-Higgins and Stewat, 1964). The coss-shoe component of the adiation stess can be defined as (Svendsen, 6) whee u w and 1 xx ( w w) h S u w dz g (3.1) w w ae the hoizontal (nomal to shoe) and vetical components of wave obital velocity, and the ove-ba indicates a wave-aveage ove one wave peiod. The effects of tubulence and viscosity ae neglected in the above definition. Radiation stess cannot be diectly measued with the existing measuing techniques due to the difficulties in obtaining wave obital velocity infomation in the toughcest egion. Theefoe adiation stess has been taditionally detemined in thee diffeent ways (Toes-Feyemuth et al., 7): (1) using analytical and/o paametic esults fom constant depth wave theoies; () making estimations based on MWL spatial measuements, and (3) by using detailed velocity and fee suface measuements o simulations. Longuet-Higgins and Stewat (1964) appoximated Eq. (3.1) using shallow-wate linea wave theoy to give 3 S gh gh 16 xx.1875 (3.) This expession fo the adiation stess has been widely used in wave-diven cuent models, since it elies only on wave height. Howeve, this expession was obtained fo small-amplitude, non-beaking monochomatic waves popagating ove a mild slope bottom. Since the waves in the suf zone ae highly nonlinea, Svendsen and Putevu (1993) detemined the adiation stesses by analyzing measued wave-height and MWL pofiles fo monochomatic waves, and calculated 79

109 the dimensionless adiation stess, P S / gh xx, which was supposed to contain the effect of both olle and nonunifom velocity and pessue distibution in the suf zone. They obseved a common featue in all calculated P (independent of the beach slope and incident waves): P has a value of about.1 at the beake point, and inceases to a maximum value of about.4 somewhee between the beake point and the shoeline, and eventually educes to.1 again nea the shoeline. The elatively small values of P at o befoe wave beaking is attibuted to the fact that nea-beaking waves have shap cests and elatively flat toughs in compaison with a sine wave of the same height (Svendsen, 1984a). Howeve, Svendsen (1984a) also found that in the suf zone, the contibution fom the olles becomes significant, esulting in elatively lage values of P (above.1875 as shown in Eq. (3.)). A epesentative value of P fo suf-zone waves esembling saw-tooth waves is P. (Svendsen, 6). A common issue with this appoach is that the intepolation eos may lead to eos in pedicted setups (Svendsen and Putevu, 1993). Moe ecently, Johnson and Smith (5) and Toes-Feyemuth et al. (7) have used detailed measued o simulated velocities to evaluate the cossshoe and long-shoe adiation stess components. Howeve, owing to the difficulties in measuing the motion of wate above the wave tough, the velocity field in this egion was obtained by extapolation based on the continuity equation. In this section, S xx is evaluated using the exact definition, i.e., Eq. (3.1) to avoid the afoementioned difficulties and assumptions. Since only the wave measuements ae available, following the convention of analyzing iegula waves, the waves measued at a location ae assumed to be a sum of all hamonic waves ( the waves ae assumed to be nonlinea waves with a discete spectum). In addition, all highe hamonics ae supposed to be phase-locked (locked waves), i.e., fee waves ae not consideed. Since we have obtained both the amplitudes and phase angles fom the hamonic analysis (up to five hamonic waves ae consideed hee), thus the fee suface of each hamonic mode ( ) is known fom the hamonic analysis; accoding to linea wave theoy, the coesponding velocities ( u i w and computed. The total velocity o the suface elevation (, u w and i i w w ) can be w w as appea in Eq. 8

110 (3.1)) is simply the summation of all the hamonic contibutions, i.e., 5, u i1 i w 5 u and w i1 i w w 5 w i1 i w. Finally, S xx can be calculated accoding to Eq. (3.1), fist caying out the integation with espect to z and then caying out the time aveage ove one wave peiod. S xx (N/m ) (a) S xx without idge S xx with idge ) P=S xx /(gh ms (b) Eq.(3.1) without idge Eq.(3.1) with idge Linea shallow-wate appoximation Seabed Elev.(m).4.3 (c) Distance fom eef/idge edge (m) Fig. 3.1 Coss-eef vaiation of: (a) dimensional adiation stess ( S xx ); (b) the dimensionless adiation stess ( P ) ( h.45 m, H.95 m and T 1.5 s). The computed dimensional adiation stess ( S xx ) and the non-dimensional adiation stess ( P ), i.e., S xx nomalized by gh ms ( H ms is the local oot-mean-squae wave height and computed by H ms 5 ( ai ), it is employed hee because H ms i1 is elated to wave enegy, and thus the adiation stess.) ae shown in Figs 3.1(a) 81

111 and 3.1(b), espectively, also shown in the figue is the value of P (.1875) obtained fom taditional shallow-wate appoximation fo linea waves. As depicted in Fig. 3.1(a), fo shoaling waves on the foe-eef, paticula in the pesence of the idge, the coss-shoe values of S xx wee affected by the fluctuation of the amplitudes of those hamonic waves (see Fig. (3.11)), but the incease of S xx befoe the beaking points can still be identified fo both cases with and without the idge, which in fact geneated the wave setdown in Fig. 3.1(b). Afte the beaking points, the value of S xx becomes smalle when the idge was pesent because the height of the tansmitted waves was contolled by the wate depth above the idge top; wheeas, the coss-shoe gadient of S xx was significantly inceased by the idge, focing a lage setup on eef flat (see also Fig. 3.1(b)). Fo the dimensionless value P, Fig. 3.1(b) shows that P geneally emained to be a constant aound.1 within the entie measued egion (fom the shoaling zone on the foe-eef to the shoewad egion of the suf zone on eef flat), which was geneally smalle than the adiation stess ( P.1875) pedicted by linea wave theoy. The value of P close to the.1 nea the beaking point was in ageement with pevious studies on monochomatic waves beaking on plane slopes (Svendsen and Putevu, 1993; Toes-Feyemuth et al., 7). Howeve, unlike pevious studies, an obsevable incease of P value in the suf zone was not found in ou expeiments, indicating that thee is a fundamental diffeence in the sufzone momentum flux between the finging eefs and plane beaches. Meanwhile, thee was almost no diffeence in the values of P with and without the idge, except in the oute suf zone aound the idge whee a slight incease of P could be obseved. 3.5 Undetow Measuements The one-dimensional vetical (1DV) flow stuctue in the suf zone is a subject of geat impotance in undestanding many coastal pocesses and has pobably eceived the most attention in the past. The undetow is the below-suface ush of wate etuning to sea afte the wate comes inshoe as beaking waves. Duing the last two decades, new measuement techniques have become available (e.g., Lase 8

112 Dopple Anemomety (LDA), and Paticle Image Velocimety (PIV)), which have successfully been applied in numeous laboatoy expeiments fo undetow measuements fo both plane beaches (Ting and Kiby, 1994; Govende et al., ; Cowen et al., 3; Huang et al., 9) and baed beaches (Hass and Svendsen, ; Govende et al., 9). See Chistensen et al. () fo a eview. Fo coal eefs, wave-diven cuents have a majo influence on lagoon flushing. The wave-diven flow ove the eef and though the lagoon is also a citical facto in detemining community distibution and poduction ates in coal eef ecosystem by contolling both the supply of nutients and the level of tubulence on the eefs (Hean, 1999). Field studies on this topic ae numeous (e.g., Kaine et al., 1999; Lugo-Fenández et al., 4; Luick et al., 7; Hench et al., 8 ; Lowe et al., 9a). The most compehensive laboatoy measuements of wave-diven cuents ove the eef flat to date wee obtained by Goulay (1996a). He found that wave-geneated flow acoss a eef inceased with inceasing both wave height and wave peiod simila to what was found fo the wave setup. In contast to the wave setup, the wave-geneated flow was small at lowe eef-flat wate level and inceased to its maximum value at highe wate level. Howeve, it can be educed to zeo when the wate level was futhe inceased. Howeve, the velocity measuements in these expeiments wee only taken at a single depth and limited locations. In this section, the goal is to investigate the wave-aveaged mean velocity pofiles (undetow) that descibe the ciculation cuent in the vetical plane of the suf zone. It was pefomed as a supplement to the wave setup expeiments in this chapte. The monochomatic waves with h.45 m, H.95 m and T 1.5 s wee again selected to epesent a typical plunging beake on the eef flat. The vetical vaiation of the flow stuctues unde the beaking waves was measued by an electomagnetic flow mete (EFM) (Kenek, Ltd.). This type of flow mete has been successfully used to measue the flow field aound beakwates (e.g., Moy and Hamm, 1997; Cácees et al., 8; Vicinanza et al., 9). The pobe of EFM used 83

113 in the expeiments is 5 mm in diamete and 55 cm in length with an L-shape tip (See Fig. 3.13), and it could be adjusted both hoizontally and vetically with a tolley seated on the sidewalls of the flume. Two instantaneous velocity components, ut ()(hoizontal and nomal to shoe) and wt ( ) (vetical), wee measued at the tip of the pobe. EFM pobes ae geneally moe flexible than ADV pobes fo flow measuement in vey shallow wate. The measuements could only be pefomed beneath the wave tough level and outside the ai-entainment egion. Detailed vetical pofiles fom.5 cm above the bottom to.5 cm below the wave tough with a vetical inteval of.5 cm wee measued at 13 selected coss-shoe locations along the centeline of the flume (L1 - L3 in the shoaling zone, L4 - L11 in the suf zone, L1 - L13 shoewad of suf zone). The exact positions of those locations ae vaied accoding to the obseved beaking, plunging, splashing points and sufzone widths in the absence/pesence of the idge, see Table 3.3 fo details. Special attention was paid to the vicinity of eef/idge edge whee the flow vaiation was supposed to be lage. Howeve, the egion between the plunging and splashing points was avoided due to stong ai entainment. EFM Tolley L-shape tip Fig The setting of electomagnetic flow mete (EFM). Velocity measuements with a fequency of 5 Hz stated afte the wavemake an fo about 1 min. This was done in ode to ensue that waves and mean cuent wee in a state of equilibium. Flow measuement at one location was conducted within the same un, and then it was stated at anothe location in a new un. At 84

114 least a half hou elapsed between the two uns to eliminate possible esidual cuents. The mean velocity was deduced by aveaging the measued velocity ove a peiod of time. A ecoding time of 1.5 min was usually sufficient to obtain mean velocity beneath the tough of beaking waves, except aound the splashing point whee the velocity pofile was not peiodic (e.g., L9 in Fig.3.14), the sampling duation was extended to 3 min at those locations. Sample ecods of measued flows at selected measuement locations ae given in Fig fo eefs with and without the idge (adjusted to be in phase). The u- velocity component measued at mid-depth is shown fo each location. Table 3.3 The aangement of flow measuement locations Location Distance a fom eef edge (m) Sufzone position No. without idge with idge without idge with idge L shoaling zone shoaling zone L shoaling zone shoaling zone L shoaling zone shoaling zone L4 oute suf zone oute suf zone L oute suf zone oute suf zone L inne suf zone oute suf zone L inne suf zone inne suf zone L inne suf zone inne suf zone L inne suf zone inne suf zone L inne suf zone inne suf zone L shoewad suf zone shoewad suf zone L shoewad suf zone shoewad suf zone L shoewad suf zone shoewad suf zone a positive if shoewad of the eef edge. Fig shows that at each location, the maximum instantaneous velocity within one wave peiod in the pesence of the idge was compaable to that in the absence of the idge: At the location L1 obseved in shoaling zone, the measued velocities had identifiable peiods and simila magnitudes. In the oute suf zone and befoe the plunging point (L5), the instantaneous flows became moe asymmetic and skewed, and a saw-tooth shape could be obseved. In the inne suf zone shoewad 85

115 of splashing points (L9), the flows became somewhat iegula, due patly to the effect of the tubulence geneated by beaking waves on the measuements and patly to the geneation of highe hamonics. Thus thee was a notable mismatch between the two velocity time-seies. The maximum velocities at L9 fo both eef pofiles wee also educed compaed with those at L1 and L5. Moving futhe shoewad of the suf zone (L13), the flows became moe oganized and thee wee seconday peaks because of the highe hamonic waves. Howeve, the velocity magnitudes wee almost the same as those at L9. u (t) (cm /s) u (t) (cm /s) u (t) (cm /s) u (t) (cm /s) 5 L t (s) 5 L t (s) 5 L t (s) 5 L t (s) Fig Time-seies of flow ecods fom selective measuement locations (L1, L5, L9, L13) acoss diffeent eef pofiles. Dash lines - without idge; solid lines - with idge. See Table 3.3 fo the distances between those locations and eef edge. The time-aveaged hoizontal velocities (u component) ae shown as a function of wate depth in Fig fo the eef without the idge and Fig fo the eef with the idge. In the pesent study fo monochomatic waves, the ensemble aveage method was implemented by phase-aveaging the measued ecods at each sampling point ove a numbe of wave cycles. These values wee aveaged to yield 86

116 the time-aveaged u velocity. The standad deviation of these values is also indicated in the figue as an eo ba fo that sampling point. The position of each sampling point is specified by the distance fom the eef edge ( x, positive if shoewad of eef edge) and the elative elevation, z/ h (whee z is the local wate depth of sampling point with z on the still fee suface and h is the local still wate depth). Exact measuement locations may be vaied due to the diffeence in suf zone width fo the two eef pofiles. The undetow velocity is nomalized by the local shallow wave celeity, c gh. The magnitudes of the velocities unde the tough of beaking waves wee geneally in the ange - 5 cm/s in pesent expeiments and the measuement eo (see the eo bas in both figues) is geneally lage at those locations nea the splashing points (e.g., L7 and L8) due to the ai bubbles entained in wate. Fo the hoizontal eef flat without the idge, Fig shows that the undetow was offshoe-diected at the shoaling locations (L1 - L3); it inceased almost linealy fom the bed upwads, which is typical fo non-beaking waves. L4 and L5 wee in the oute sufzone but befoe the plunging point, the velocity pofiles wee not linea, showing values close to zeo nea the bed and being the maximum nea the tough of beaking waves. Maximum velocities both at L4 and L5 wee appoximately.15c nea the tough level. In the inne suf zone (shoewad of splashing egion) whee initial wave beaking was completed with fully developed boes, most of the velocities wee negative at the measuement locations (L6 - L11), i.e., offshoe diected, and thei magnitudes deceased almost linealy fom the bottom to the tough. Close to the wave tough, they may become positive, i.e., onshoe diected, due to the mass tanspot above the wave tough as well as in the boe (Svendsen, 1984b). Fom L6 to L11, wave beaking educed in intensity and thee was a gadual eduction in the boe size by visual obsevation. The eduction in boe size was accompanied by a decease in the magnitudes of undetow. Outside the suf zone (L1 and L13) whee the wave setup was almost constant, the vetical velocity became almost zeo, indicating that thee was no eef-flat ciculation shoewad of the suf zone fo finging eefs. The coelation between the setup and 87

117 undetow at all these locations can be justified by the coss-shoe momentum balance as will be discussed in Chapte 7. When the idge was pesent, the offshoe undetow pofiles at L1 and L wee simila to those in the absence of the idge, see Fig Howeve, the flow stuctue at L3, whee the waves wee nea-beaking, was significantly alteed by the idge. At those locations beneath the beaking waves on the shallow idge top (L4 - L6), the offshoe-diected nonlinea cuve pofiles again could be obseved; thei magnitudes wee almost doubled (aound.3c ) compaed with those without the idge. The maximum velocity occus at the eef/idge edge and was appoximately.45c. L7 was close to the splashing point on the idge-top, the velocity pofile at this location esembled that at L6, but the magnitudes wee significantly smalle. Fom L8 to L11 (inne suf zone), the vetical flow stuctue gadually evolved fom the nonlinea cuve to a linea pofile, followed by a eduction of velocity magnitudes futhe shoewad of the idge. The linea pofiles had a seawad cuent nea the bottom and a shoewad cuent nea the tough. Beyond the suf zone, the magnitudes of undetow at L1 and L13 wee again vey weak. By compaing the oveall flow patten in the pesence of the idge with that in the absence of the idge, it can be concluded that the idge behaves in a simila way to a wei in an open channel flow, which can act to block the flow (undetow) upsteam (leeside of the idge). Fom an enegy point of view, it means that moe kinetic enegy (in the fom of undetow velocity) is conveted to potential enegy (in the fom of wave setup), but the total enegy pumped by beaking waves onto the eef flat might be conseved. 88

118 . L1: x= -.75 m. L: x= -.45 m. L3: x= -.15 m. L4: x= m z/h z/h z/h z/h u/c L5: x=.15 m u/c L6: x=.6 m u/c L7: x=.85 m u/c L8: x= 1.15 m z/h z/h z/h z/h u/c. L9: x= 1.5 m u/c. L1: x= 1.95 m u/c. L11: x=.55 m u/c z/h z/h z/h u/c u/c u/c. L1: x= 3.15 m. L13: x= m z/h z/h u/c u/c Fig Vaiation of time-aveaged hoizontal velocity as a function of depth in the absence of the idge ( c gh is the local shallow-wate wave celeity; Solid lines indicate the eo bas based on the standad deviation of phase-aveaged velocity). 89

119 . L1: x= -.75 m. L: x= -.45 m. L3: x= -.15 m. L4: x= m z/h z/h z/h z/h u/c u/c u/c u/c. L5: x=.75 m. L6: x=.15 m. L7: x=.45 m. L8: x=.6 m z/h z/h z/h z/h u/c. L9: x=.85 m u/c. L1: x= 1.15 m u/c. L11: x= 1.5 m u/c z/h z/h z/h u/c u/c u/c. L1: x= 1.95 m. L13: x=.55 m z/h z/h u/c u/c Fig Vaiation of time-aveaged hoizontal velocity as a function of depth in the pesence of the idge ( c gh is the local shallow-wate wave celeity; Solid lines indicate the eo bas based on the standad deviation of phase-aveaged velocity). 9

120 Geneally speaking, the obsevations without the idge in this section ae in ageement with the measuements fo a plane slope with a plunging beake (Ting and Kiby, 1994). The pedominant offshoe-diected undetow cuents ae lagely because of the closed laboatoy flume, which equies that the wave-induced mass tanspot above wave tough level is compensated by a seawad flow below this level so that the depth-aveaged net flux is zeo. If thee is a ciculating pipe connecting both ends of the flume o a gap in the foe-eef o idge model, the undetow may be significantly smalle o even diects towads the shoe as found fom the hoizontally two-dimensional (DH) flow measuements (e.g., Dønen et al., ). An analytical modeling of the DH flow patten fo the eef system will be given by section 7.7 although the pesent laboatoy expeiments did not go that fa. 3.6 Discussions In this section, some discussions on the possible effects of the laboatoy settings on the expeimental esults is given. In the expeiments, thee is a setdown of the mean wate level seawad of the foeeef esulting fom the mass consevation in a closed flume (see e.g., Fig. 3.7). Howeve, the magnitude of this setdown is geneally less than 1 cm. Fo a typical test condition with a wave peiod of 1.5 s and a wate depth of 4 cm, this only causes of a decease of wave length by 1% accoding to the line dispesion elationship, thus it has a negligible effect on the incident wave chaacteistics and does not change the beaking conditions on the foe-eef. It is stessed that the same still wate level is ensued when compaing wave-induced setups and that the setup is defined as the deviation of the mean wate suface fom the still wate level. Theefoe, the highe setup is not affected by the setdown in font of the eef model. In fact, a lage setdown in font of the shoaling zone is actually the esult of a lage setup, not the othe way aound, as thee is no diving foce in the wate in font of the shoaling zone. 91

121 One may also wonde whethe this seaside setdown leads to a stonge offshoe flow nea the eef cest o not, which again might impact the beaking pocess. In the closed flume, since the offshoe mean cuent (undetow) is balanced by shoewad wave-induced mass tanspot, its magnitude should depend lagely on the beaking-wave chaacteistics athe than the setdown. Within one wave peiod, thee is an inflow phase and an outflow phase; each has duation of about half the wave peiod. When wate ushes up the eef flat, it is the inflow phase, and when wate eteats fom the eef flat, it is the outflow phase. The wave beaking pocess is lagely contolled by the incoming wave meeting with the outflow at the beaking point. When waves beak, the wave obital velocity is of O ( g( ), i.e., the inflow velocity is of O ( g( ), which is of O(1 m/ s ) fo the test case in h b section 3.5. The outflow depth is much smalle than the inflow depth in the expeiments, i.e., the outflow is basically a sheet flow, which has a velocity much lage than O ( g( ). The undetow at the eef edge fo the test case is just h b about.3 m/s even when the idge is pesent. Theefoe, the outflow velocity is much stonge than the mean cuent and the effect of the mean cuent on the wave beaking pocess is seconday. h b Fo the finging eef model in this study, the eef o idge stuctue "block" wate exchange between the back eef (shoewad of eef flat) and the seaside of the foeeef). Wave beaking will dissipate most wave enegy within the suf zone, and the waves outside the suf zone on the eef flat ae vey weak. Fo a finging eef with a closed lagoon, the net flow ate is always zeo with o without the idge, even though a ciculation (a fowad mean flow above the wave tough and a backwad mean flow nea the bottom (undetow)) is possible on the eef flat. Figs and 3.16 show that the undetow is weak in the shoaling zone and shoewad of the suf zone fo both eef models with and without the idge, thus the wate exchange between the seawad side of foe-eef and the lagoon is always weak. Howeve, in the sufzone when the idge is pesent, the undetow velocity significantly is inceased in the vicinity of the idge and deceased on the leeside of the idge, suggesting that thee is indeed a cetain degee of blocking effect intoduced by the 9

122 idge, but it is difficult to quantify how much wate is blocked by the idge. Fo a baie eef without an open lagoon, the wave-induced setup could foce the wate to the back lagoon and the net flow ate is not zeo on the eef flat; the non-zeo flow in tun could affect the magnitude of setup in the suf zone. Goulay (1996a) studied expeimentally the wave-induced setup ove both finging eefs and platfom (baie) eefs (open lagoon), and he found that the magnitude of the waveinduced setup on a platfom eef (with a net mean flow) was smalle than that on a finging eef (with zeo net flow) by an amount equal to the velocity head of the mean flow on the eef platfom (i.e., a mean cuent of.5 m/s on the eef flat may educe the wave-induced setup by about 1. cm). The diffeence between the waveinduced setups fo baie eefs and finging eefs should be the uppe bound of the effects of the mean flow on wave-induced setups. In ou expeiments, the net mean flow ate is zeo and the undetow is weak, thus the influence of the mean cuent should be negligible. 3.7 Concluding Remaks In this chapte, a seies of expeimental esults fo eef pofiles with and without a seawad edge idge and fo both monochomatic and spectal waves have been pesented. The expeimental data show that the behavio of the wave tansfomation and wave beaking in the pesence of a idge ae significantly diffeent fom those in the absence of the idge. In paticula, a idge nea the eef edge can cause an incease in the wave-induced setup ove the eef flat. Futhemoe, the idge dastically inceases the eflection coefficients of the eef model. Expeimental esults also show that the enegy dispassion due to wave beaking esults pimaily fom the fundamental hamonic waves, and that the second hamonic waves may become compaable to the fundamental hamonic waves on the eef flat. The distibution of the momentum flux in suf zone fo finging eefs is diffeent fom that fo plane beaches, and a measued value of adiation stess below that obtained by using linea shallow-wate appoximation is consistently obseved. The cossshoe undetow velocity pofile fo finging eefs in the absence of the idge is simila to that fo plane beaches if plunging beakes occu on both bed pofiles ae simila; the existence of the idge may cause a etention of the flow on the leeside of 93

123 the idge. The geneality of above emaks needs to be futhe veified by moe expeiments with diffeent eef/idge configuations unde vaious wave conditions. 94

124 CHAPTER 4 NUMERICAL STUDY OF WAVE TRANSFORMATION OVER FRINGING REEFS 4.1 Liteatue Review Hydodynamics associated with waves on finging coal eefs is moe complex than that on plane beaches. As intoduced in Chapte 1, a typical finging coal eef involves a tansition of bottom pofile fom deep to shallow wates ove a long distance, and a poous eef suface which may povide high esistance to the waves. Numeical modeling of neashoe coal eef hydodynamics faces seveal challenges such as the steep foe-eef slopes (e.g., Seelig, 1983; Goulay, 1996a; Hench et al., 8), the complex configuations of eef cest and eef flat (e.g., Seelig, 1983; Hench et al., 8), the spatially-vaied oughness of eef suface (e.g., Lowe et al., 5). Also, the wave tansfomation usually needs to be modeled ove long time scales (seveal hunded waves) and lage space scales (ove the entie eef pofile). Ove decades, analytical models fequently deal with the one-dimensional hoizontal (1DH) idealized eef pofiles (a typical idealized eef pofile has a plane sloping foe-eef and a hoizontal platfom eef flat). Conventionally, in analogy to the wave-diven coss-shoe flows and the wave-induced setup/setdown on beaches (e.g., Svendsen, 6), analytical solutions based on the adiation stess concept intoduced by Longuet-Higgins and Stewat (1964) had been used fequently in the past to study 1DH eef hydodynamics (Goulay, 1996a; Symonds et al., 1995; Hean, 1999; Goulay and Collete, 5). In ecent yeas, the effects of complex bathymety and diffeent focing mechanisms have been modeled by using twodimensional hoizontal (DH) and thee-dimensional (3D) models to study both the waves and the mean flows, and usually the adiation stess concept is used to couple the waves and the mean flows (Kaines et al., 1998, 1999; Douillet et al., 1; Luick et al., 7; Stolazzi et al., 11). The modeling esults pesented by Lowe et al. (9b) look vey pomising, even though the pedicted mean wate level was not as accuate as the computed wave heights and cuents. Compaed to field 95

125 studies, fewe numeical models have been applied to well-contolled, small-scale laboatoy investigations in the liteatue. The most advanced appoaches based on Navie Stokes equations and vaious tubulence closues, e.g., the models with RANS-based tubulence closue. (Lin and Liu, 1998; Losada et al., 5; Laa et al., 8; Toes-Feyemuth et al., 1), ae well suited fo simulating beaking waves and wave-stuctue inteactions in small confined egions. Howeve, Navie-Stokes appoaches ae still vey computationally expensive to un, especially fo the neashoe zones whee a lage numbe of gid points and a fine mesh ae needed to accuately captue the fine tubulence stuctues, thus its use is geneally esticted to a small numbe of waves and small egions. Cuently, applying Navie-Stokes models to field scale eef pofiles is not feasible. Anothe type of pevailing model, which is moe computationally efficient, is based on Boussinesq-type equations. This depth-integated modeling appoach employs a polynomial appoximation to the vetical pofile of velocity field, theeby educing the dimensions of a thee-dimensional poblem by one. It has been poven to be able to account fo both nonlinea and dispesive effects at diffeent degees of accuacy. Afte the fist intoduction of Boussinesq equations by Peegine (1967), consideable effots have been made in diffeent ways to extend the Boussinesq equations to deepe wates (e.g., Madsen and Søensen, 199; Nwogu, 1993; Wei et al., 1995; Lynett et al., ) and to suf zones o swash zones (e.g., Madsen et al., 1997; Kennedy et al., ; Veeamony and Svendsen, ). One of the pionee studies of extending Boussinesq-type model to coal eef studies was conducted by Skotne and Apelt (1999), who studied the wave-induced setup by monochomatic waves popagating onto a submeged finging coal eef which consisted of a elatively steepe foe-eef. They found that a Boussinesq-type model could simulate satisfactoily the pattens of mean wate level fo 1DH eef pofiles subjected to small waves, but thee was a tendency to undeestimate the waveinduced setup as the incident wave height was inceased. Demibilek and Nwogu (7) used a diffeent set of Boussinesq equations to study spectal waves 96

126 tansfoming ove a eef pofile that was simila to the one used by Skotne and Apelt (1999). They confimed the ability of thei Boussinesq model in descibing the vaiation of significant wave height, the mean wate level acoss the eef pofile, the evolution of the wave spectum, the geneation of infagavity oscillations and shoeline unups. Moe ecently, Roebe et al. (1) employed a shock-captuing Boussinesq-type model to simulate the solitay wave tansfomation ove finging eefs, which involved enegetic wave beaking, boe popagation and the tansition fom subcitical to supecitical flows unde an initially dy eef cest. The main objective of this study is to implement and validate a weakly dispesive and fully nonlinea depth-integated Boussinesq-type model 1 to help intepet some of the pevious laboatoy wok (Chapte 3, also see Yao et al., 9) and othe simila published wok on wave tansfomation ove finging eefs. The pesent numeical model is chosen because it is capable of simulating a wide ange of long and shot wave poblems (Lynett et al., ; Hsiao et al., 5; Lynett, 6). The model has also been applied to wave ovetopping ove a levee system by Lynett et al. (1). This chapte will fist epot a peliminay validation of the adopted model fo an idealized eef pofile, then pesent compehensive compaisons between numeical simulations and available published data fo vaious wave conditions and diffeent finging eef configuations. The effects of the foe-eef shapes and sloping angles on the mean wate levels and wave heights will also be investigated using the validated numeical model. The emainde of this chapte is oganized as follows. In section 4., the mathematical fomulation, numeical scheme, bounday conditions and enegy dissipation sub-models ae descibed. In section 4.3, calibation and validation of the numeical model ae pefomed. Numeical simulations ae compaed with available expeimental data fo fou epesentative scenaios to show the obustness of the model in section 4.4. In section 4.5, a evisit of two published numeical woks on wave tansfomation ove inging eefs is epoted. In section 4.6, the 1 CoulWave code V... was modified fo this study. See Lynett and Liu (8) fo details of CoulWave. 97

127 validated model is applied to study the effects of the inclination of a plane foe-eef and the shape of foe-eef on the wave dynamics ove the finging eefs. The main conclusions dawn fom this study will be given in section Desciption of the Numeical Model 4..1 Govening equations Let x-coodinate be pointing in the diection of wave popagation with its oigin at the toe of the foe-eef, and z-coodinate pointing upwad with its oigin at the still wate level. The 1DH equations in non-consevative fom ae expessed as t hu h hh z uxx hz hu xx 6 z ut uux g x uxxt z hutxx z z u hu z z xu u xx u u xx xx x u z hu hu hu hu x xx x xx t x x uxhu u xt u hu u x u u xx x xx x x x (4.1) (4.) whee is the wate suface elevation; h is the still wate depth; g is the gavitational acceleation and u x, z z ( x, t) is a efeence hoizontal velocity in the x-diection at a specified depth of z ( x, t). 531h (Nwogu, 1993). The complete deivation of above equations was given by Lynett and Liu (8) and thei consevative fom can be found in Kim et al. (9). Fo 1DH poblems, thee ae two souces of enegy dissipation: bottom fiction and wave beaking. Adding the dissipation tems to the momentum equation gives Du t R R (4.3) f b 98

128 whee D h is the total wate depth, R f and R b ae ad-hoc dissipative tems accounting fo the bottom fiction and the wave beaking, espectively. Sub-models fo R b and R f will be discussed in section 4..4 and 4..5, espectively. 4.. Numeical scheme The numeical solve fo the above equations has been descibed in details in Kim et al. (9). A thid-ode Adams-Bashfoth pedicto and a fouth-ode Adams- Moulton coecto scheme ae used fo time maching. Fo the spatial discetization, a shock-captuing Finite Volume (FV)-based appoach is used. The leading ode tems ae solved with the fouth-ode MUSCL-TVD (monotone upsteam-centeed scheme fo consevation laws-total vaiation diminishing) scheme, while fo the second-ode tems, a cell-aveaged finite volume method is implemented. Compaed with taditional Finite Diffeence (FD)-based methods, FV fomulations in consevative fom ae geneally vey stable and accuate, thus appopiate fo the pesent poblems which may have complex flow conditions and apid bottom 4 vaiations. The numeical scheme is accuate to O( t ) in time and 4 O( x ) o O( x ) in space, whee is the wave length scaling paamete defined by h/ L, with L being the incident wave length. Fo the pesent model, a value of.5 fo the Couant numbe ( C ) will typically yield stability and convegence, but fo simulations with highly nonlinea waves, a value as low as.1 may be equied fo stability (Lynett and Liu, 8) Bounday and initial conditions Numeical bounday conditions Two types of numeical bounday conditions can be applied at the two ends of the computational domain: the eflective (o no-flux) bounday condition and the adiation (o open) bounday condition. Fo the latte, sponge layes ae fequently used to effectively damp the enegy of outgoing waves. The sponge laye is usually applied in a manne simila to that ecommended by Kiby et al. (1998). Fo completeness, the implementation of sponge layes in CoulWave is given in 99

129 Appendix D. The moving bounday can be simulated using the slot technique (Tao, 1983), the Lagangian method by Zelt (1991), o the extapolating bounday algoithm (Lynett et al., ); the latte is used hee to descibe wave unup and undown pocesses. It utilizes a linea extapolation of fee suface and velocity though the wet-dy bounday and into the dy egion, theeby allowing the eal bounday location to exist in-between nodal points. The technique is also simple to implement and numeically stable, does not equie any sot of additional dissipative mechanisms o filteing (Lynett et al., ). The moving bounday algoithm is needed only when the initially dy eef flats ae studied. Wave geneation and initial conditions Intenal souce methods ae fequently employed as efficient and accuate methods fo numeical wave geneations (e.g., Lin and Liu, 1999; Wei et al., 1999; Hsiao et al., 5). The method using a distibuted souce function in the continuity equation as poposed by Wei et al. (1999) is adopted in this study. Iegula waves ae geneated by summing up many egula waves with diffeent fequencies, amplitudes and andom phases fo a given spectum. The initial condition assumes no wave o cuent motion in the computational domain Wave beaking (R b ) It is well-known that the depth-integated Boussinesq-type models cannot descibe the ovetuning of a fee suface and the detailed beaking pocess. Hence seveal empiical models have been poposed fo the wave beaking in suf and swash zones. The most common appoach is to add an ad-hoc dissipation sub-model to the momentum equation. Thee ae two pimay types of beaking models: olle models (e.g., Schäffe et al., 1993; Madsen et al., 1997; Veeamony and Svendsen, ) and eddy viscosity models (e.g., Zelt, 1991; Kaambas and Koutitas, 199; Kennedy et al., ). Even though the two appoaches stem fom diffeent ideas and have diffeent contolling paametes, thei oveall effects in the momentum equation ae simila: both equie an enegy dissipation mechanism and a tigge mechanism fo the initiation of wave beaking. One of the basic equiements on empiical beaking models is that they must ensue consevation of mass and 1

130 momentum as well as peseve some nonlinea wave popeties in suf zones. In this study, the simple eddy viscosity-type fomulation poposed in Kennedy et al. () is used to model the wave enegy dissipation caused by wave beaking. It has been poven that the adopted wave beaking model can adequately pedict the enegy dissipation fo both spilling and plunging beakes (Kennedy et al., ; Lynett, 6; Roebe et al., 1). Fo completeness and the convenience of discussing the numeical esults, the beaking model in CoulWave is summaized below. The 1DH fom expession fo R b is given by R Hu (4.4) b x x whee is an empiical eddy viscosity and given by the following zeo-equation tubulence model H (4.5) whee is an empiical coefficient to coect both the mixing-length and fictionvelocity scales. The paamete accounts fo the tigge mechanism to ensue a smooth tansition between beaking and non-beaking states. The expession fo is given by t * 1 t t * * * t t 1 t t t * t t (4.6) whee t evaluated by detemines the onset and stoppage of the beaking pocess and is ( F ) * t t t T * t ( I ) t t ( F ) ( I ) * (4.7) t t t T * t t T ( ) whee I ( ) is a theshold value at the beaking inception ; F is a satuated value t fo the beaking cessation; t is the time at which the beaking event stats; t t is the age of beaking event; T * is the duation of the beaking event. Detemination t 11

131 ( ) of the fou empiical paametes in the beaking model (i.e.,, I ( ), F *, T ) will be discussed late. t t 4..5 Bottom fiction (R f ) The bottom fiction tem law R f can be calculated by the following quadatic fiction R fu u (4.8) f b b whee u b is the bottom velocity and can be evaluated fom u ; f is an empiical fiction coefficient, which can be elated to Manning coefficient ( n M ) by f gn M (4.9) 1/3 H Values of n M fo sufaces of commonly used mateials can be found in any standad text book fo hydaulics o fluid mechanics. 4.3 Model Calibation and Validation Expeimental and numeical settings Expeimental setting The laboatoy expeiments in Chapte 3 ae e-summaized in this section fo the convenience of the following simulations: the measuements wee conducted in a closed wave flume in the Hydaulics Modeling Laboatoy, Nanyang Technological Univesity, Singapoe. The flume is 36 m long,.55 m wide, and.6 m deep. The expeiments wee designed to study the hydodynamic chaacteistics of two types of eef-cest pofile subjected to both monochomatic and spectal waves: (1) an idealized finging eef without a idge and () a finging eef with an idealized idge. The fist eef model had a elatively steep foe-eef (V:H=1:6) as well as a 7 m-long hoizontal eef flat. Fo the second model, a ectangula box was placed (55 cm long, 5 cm wide and 5 cm high) on the eef flat with its font face aligned with the eef 1

132 edge to mimic an idealized idge (see Fig. 4.1). The dimensions of the idge model wee chosen to mimic the eef idge existing on the Mooea eef studied by Hench et al. (8). The detailed expeimental settings and peliminay data esults fo the expeiments examined in this chapte have been epoted in Chapte 3 as well as in Appendix B. To measue the coss-eef wave tansfomation, 1 wave gauges (G1 - G1) wee used and thei aangement is shown in Fig. 4.1: G9 - G1 wee equally spaced ove the eef flat with the fist gauge being located behind the suf zone. Fo monochomatic waves, G4 - G8 wee located in the vicinity of the eef edge to measue the sufzone waves; G1 and G wee placed seawad of the eef slope to sepaate the incident waves fom the eflected waves and G3 was placed on the slope to measue the shoaling waves. Fo spectal waves, the aangement of wave gauges is slightly diffeent: G3 was moved futhe seawad to fom a thee- pobe aay fo wave spectum sepaation and G4 was located on the foe-eef to obseve wave shoaling; the emaining gauges (G5 - G8) wee put close to the eef edge to captue the spectal waves in the suf zone. Exact locations of the wave gauges will be specified late. At the shoewad end of the flume, waves wee effectively damped by a poous wave absobe to educe the wave eflection. A seies of monochomatic and spectal wave conditions wee tested in the expeiments with a ange of eef-flat submegences. Spectal waves wee geneated fom the widelyused JONSWAP spectum with a peak enhancement facto 3.3. Numeical settings Refeing to Fig. 4.1, the computational domain, in tems of the dimensions of the flume and the location and shape of the idealized eef model, wee designed to epoduce the main aspects of the laboatoy settings. The 1DH numeical domain is also 3 m long. Dissipative sponge layes, typically of a width of 1.5 times the incident wavelength, (i.e., W 1.5L, whee L is the wave length), wee placed at the two ends of the computational domain to ensue that the outgoing waves can be absobed satisfactoily. The incident wave field was geneated by using the afoementioned intenal souce method. The intenal souce was placed close to the seawad sponge laye, i.e., L 1. The foe-eef was b.1 m wide with its toe stating at L m fom the wave geneation point. The eef flat, which was 13

133 extended all the way to the shoewad sponge laye, had a width of L3 9.8 m ; the elevation of the eef flat was fixed at.35 m above the flume bottom fo all expeiments. A epesentative case of monochomatic waves of H.57 m, T 1.5 s and h.4 m tansfoming ove the hoizontal eef flat (i.e., in absence of the idge) is used fo numeical calibations. The chosen wave condition gives kh 1.3, ka.9 and a positive eef-flat submegence of.5 m. Fo this case, the model paametes wee calibated by using the R-squae value ( R ) to minimize (though a tial-and-eo pocedue) the global eo between the measued mean wate level and the pedicted mean wate level at the 1 measuement locations. Fiction was not consideed in the calibations. A steady wave field could be eached about 6 s afte the stat of the simulation, which is shote than 3 min obseved in the laboatoy expeiments epoted in Chapte 3; this is because of the use of sponge laye which educes the multiple eflections in the computational domain. A total numbe of waves wee simulated fo each case to ensue that the last 1 wave cycles in wave ecods could be used to calculate the wave height and mean wate level. The main pupose of this section is to undestand the effects of diffeent numeical model paametes on the computed esults. Fig. 4.1 Computational domain fo waves popagating ove a finging eef (the exact locations of the wave gauges fo all elevant laboatoy expeiments ae given in Table 4.) Gid size The gid size is detemined by a pedefined gid numbe pe incident wave length ( N ). A vaiety of gid numbe anging fom N 5 to 1 wee tested, leading to 14

134 the gid size anging fom x. m to.8 m, Numeical instability may occu fo a fine gid esulting fom highe fequency oscillations. Since the time step duing the tests is fixed at.5 s, the Couant numbe changes with gid size. The coss-shoe vaiations of wave height and mean wate level (which ae the pimay concens of this study) ae calibated. Fig. 4. shows that a value of N 75 seems to be sufficient to discetize the computational domain, significant impovement on the pediction of the wave height o MWL pofile is not found by futhe educing the gid size. The values of the tansmitted wave height ( H t ) as well as the maximum wave setup ( ) on the eef flat as a function of the gid size ae shown in Figs. 4.3(a) and 4.3(b), espectively. It can be obseved that the pedictions convege with inceasing N : both H / H and / H ae about 1% when t inceasing N fom 75 to 1. Thus N 75 is used in the subsequent simulations, which gives a gid size of x.4 m. t t Elevation (m) Wave height obseved MWL obeseved Wave height (N=5) MWL (N=5) Wave height (N=5) MWL (N=5) Wave height (N=75) MWL (N=75) Wave height (N=1) MWL (N=1) Distance fom numeical wavemake (m) Fig. 4. Vaiation of wave height and MWL acoss the flume with diffeent gid sizes. 15

135 16 (a) 8 (b) 14 7 H t (mm) 1 (mm) N N Fig. 4.3 Tansmitted wave height ( H ) and maximum wave setup on the eef flat ( ) as a function of gid numbe pe incident wave length ( N ). t Bounday conditions Sponge layes wee used to educe the wave eflection fom the numeical boundaies. Since waves ae geneated intenally by using an intenal souce method in the pesent model, sponge laye needs to be added to the left bounday to absob the eflective waves. Fo the ight bounday, diffeent settings may be attempted. To implement the sponge laye concept, both the mass and momentum equations need to be modified by adding atificial damping tems. Pilot test also found duing the numeical expeiments that the damping tems in the mass balance could significantly affect the computed MWL, which indicated a change of total mass in the computational domain. This was especially tue fo vey shallow eefflat submegences. Theefoe, the following thee types of bounday conditions have been studied in ode to implement the appopiate numeical bounday conditions fo the poblem: BC1: Fo this type of bounday conditions, sponge layes ae used on both sides of the computational domain, and the damping tems ae used in both the momentum and mass balance equations; 16

136 BC: Fo this type of bounday conditions, sponge layes ae used on both sides of the computational domain, but the damping tem is used only in the momentum equation; BC3: Fo this type of bounday conditions, a sponge laye is implemented on the left bounday but only in the momentum equation, and a 1:8 slope combined with high bottom fiction ( n.1) is used fo the ight bounday to educe the eflection. The gid size fo this test was kept at x.4 m. A compaison of the MWLs and wave heights using these thee bounday conditions is shown in Fig Results show that the simulated MWL agees bette with the measuement unde BC than that unde BC1; the impovement can be substantial fo vey shallow eef-flat submegences (not epoted hee). Howeve, the wave heights obtained using BC1 and BC ae almost the same. BC3 in pinciple should pefom the best since it mimics the actual laboatoy expeiment most closely. Howeve, it is difficult to model wave motion in the poous mats placed on the sloping beach to dissipate wave enegy. Fo example, using a high fiction coefficient fo the final slope to model the poous wave absobe could cause some fluctuation in the wave height on eef flat as indicated by Fig Theefoe, BC, i.e., the sponge layes ae used only in the momentum equation, is selected in the following simulations. This calibation eveals that the damping tem in the continuity equation would behave like a sink/souce tem, causing a significant change of the total amount of wate in the computational domain and esulting in an incoect mean wate level. An explanation can be given in view of Appendix D, Eq. (D.1) could emove mass fom the computational domain at the bounday though ; this may be of no concen fo open bounday poblems. Howeve, using Eq. (D.1) in continuity equation to simulate waves in a closed laboatoy flume will eventually cause an imbalance o loss of mass in the flume. 17

137 Elevation (m) Wave height obseved MWL obeseved Wave height (BC1) MWL (BC1) Wave height (BC) MWL (BC) Wave height (BC3) MWL (BC3) Distance fom numeical wavemake (m) Fig. 4.4 Vaiation of wave height and MWL acoss the flume with diffeent bounday conditions Beaking model It is believed that the unde-pediction of both the eef-flat MWL and the wave height ae attibuted to the empiical beaking model used in the simulation. Numeical expeiments suggested that, which is elated diectly to the tubulence intensity, is the most impotant paamete among the fou model paametes. Values of anging fom 1.4 to 1 can be found in the published liteatue (Kennedy et al., ; Lynett, 6; Lynett and Liu, 8). The value depends on the dispesive and nonlinea popeties of a given numeical model (Lynett, 6). Fo example, Kennedy et al. () used 1. fo the model of Wei et al. (1995) with linea dispesion and nonlinea popeties up to kh 3, and Lynett (6) used 1 fo the model of Lynett and Liu (4) with linea dispesion and nonlinea popeties up to kh 6. Howeve, in pactice, a tail-and-eo pocedue is needed to calibate this value with measuements. Thus was vaied fom to 1 duing the pesent calibation. Fo the tigge-elated paametes in Eq. (4.7), the following values suggested by Lynett and Liu (8) wee adopted in this study 18

138 ( F ) * gh gh ; T H g (4.1) ( I ) t 1 ; t 3 / whee.65,.8, 8 fo plane beaches. Kennedy et al. () used 1 3 the still wate depth, h in place of the instantaneous wate depth, H. Results obtained with diffeent values of ae given in Fig 4.5. The best ageement is obtained with, which is employed to pefom the numeical expeiments in the est of the chapte fo all idealized finging eefs without a idge. Fo finging eefs with the idge, model calibation is needed in ode to find a suitable value of. Howeve, the wave-induced setup is not sensitive to the change of. Elevation (m) Wave height obseved MWL obeseved Wave height (=.5) MWL (=.5) Wave height (=1) MWL (=1) Wave height (=) MWL (=) Wave height (=1) MWL (=1) Distance fom numeical wavemake (m) Fig. 4.5 Vaiation of wave height and MWL acoss the flume with diffeent tubulence intensities Model validation fo apidly vaying bathymety Unlike beaches, which typically have mild slopes, a typical coal eef often foms a steep tansition fom the elatively deep to shallow wates. One majo concen with applying Boussinesq models to finging coal eefs is the elatively steep foe-eef 19

139 slopes; this is because deivatives of the wate depth ae included in the highe ode tems of Boussinesq equations. K FEM Pesent model b (m) Fig. 4.6 Vaiation of eflection coefficient ( K ) with the slope width ( b ). Solid line: FEM solution of Suh et al. (1997); Open cicles: pesent model. As a veification of the capability of the pesent model to deal with apidly vaying bathymety, a tain of monochomatic waves popagating ove a plane shelf ae consideed hee. This poblem was fist studied by Booij (1983), who investigated the accuacy of a mild-slope equation by compaing the pedicted eflection coefficients with Finite Element Method (FEM) solutions. Since then, this poblem has become a benchmak against which the accuacy of a hydodynamic model can be veified. An example fo testing Boussinesq-type models using this benchmak is given in Madsen et al. (6). The bottom pofile fo this benchmak poblem consists of a plane slope connecting two constant-depth egions, which is simila to the idealized finging eef model (shown in Fig. 4.1). The offshoe wate depth is ho.6 m, while the eef-flat wate depth is h. m. A tain of monochomatic waves with a peiod of. s was studied. The width of the foe-eef 11

140 slope vaied fom b.1 m to 1 m, coesponding to the slopes of V:H=4:1-1:5. To esolve the steepest slope numeically, a gid size of.4 m and a constant Couant numbe of.5 fo all simulations wee used. The computed eflection coefficients ae shown in Fig. 4.6, togethe with the FEM solutions by Suh et al. (1997). The figue clealy shows that the pesent numeical esults ae accuate up to the slope width of b.3 m, i.e., a slope of V:H=4:3, but slightly unde-pedicts the eflection coefficients fo vey steep slopes. The fluctuation of eflection coefficient with b is due to the multiple eflections between the two edges of the foe-eef Results afte calibation Wave height and MWL acoss the eef pofile The coss-shoe vaiations of MWL and wave height ae shown in Fig. 4.7, whee the BC type bounday conditions, x.4 m and wee used, also included in the figue is the seabed pofile. Oveall, the numeical model gives a good pediction ( R.93) of the vaiation of MWL acoss the numeical flume: both the off-eef setdown due to shoaling and the beaking points (the location whee the wave height each its maximum) ae easonably captued by the numeical model despite that the measued location is not sufficient fo detailed compaison. Howeve, thee is a slight unde-pediction of the eef-flat setup. The vaiation of the wave height along the numeical flume is also satisfactoily pedicted. The wavy featue of the computed wave height seawad of the eef model is due to the patial wave eflection, which poduces a patial standing wave patten in font of the eef model. Theoetically, the length of standing waves fomed by a supposition of the incident and eflected waves should be one half of the wave length of the incident waves. Fo this case, the incident wave length is about.5 m in the foe-eef egion, the estimated wave length of the patial standing wave patten seawad of foe-eef is 1.3 m, which is in ageement with the obsevation shown in Fig The apid decease of wave height in the suf zone is well captued by the numeical simulation, suggesting that the simple empiical beaking model is appopiate fo 111

141 this study. Thee is still a slight unde-pediction of the heights of the efomed waves on the eef flat; this unde-pediction might be impoved if futhe calibations of othe paametes in the beaking mode wee conducted. The calculated eflection coefficient fo the fist hamonic wave is K.45, which agees easonably well with the measued eflection coefficient K.38. When calculating the eflection coefficients using the measued o calculated suface displacements, 4 th ode Stokes wave theoy was used to find the amplitudes of the fundamental waves. H(m) MWL(m) Seabed Elev.(m) Distance fom numeical wavemake (m) Fig. 4.7 Vaiation of wave height and MWL acoss the flume with calibated numeical settings. Solid lines: pedictions by pesent model; Open cicles: laboatoy measuements. 11

142 Wave tansfomation acoss eef pofile Measued and calculated wate suface elevations ae compaed in Fig. 4.8 at six locations, also indicated in the figue is the oot-mean-squae eo (mse) at each location. It can be obseved that vey good ageements between pedictions and measuements ae obtained at most sampling locations (whee the mse is small): off the eef in the deepe wate section (G1), the waves ae Stokes waves. When waves ae on the slope (measued by G3 and G4), wave shoaling makes the wave fom asymmetic and skewed. Gauges G5 and G7 ae located in the suf zone, thus beaking waves with saw-tooth shape ae obseved thee. These esults suggest that the poposed beaking model could easonably simulate some nonlinea popeties such as asymmety and skewness of the wave fom in the suf zone when the incident wave height is elatively small (.57 m fo the case used in calibation). On the eef flat (G9), both numeical and expeimental esults show that the efomed waves ae vey small. The geneation of the highe hamonic waves can be best seen fom the wave amplitude spectum, which is shown in Fig Up to the fifth hamonics ae shown hee. It shows that at locations G1, G3 and G9, the numeical model unde-pedicts the amplitudes of the highe hamonic components. Howeve, in the suf zone (G4, G5 and G7), thee is a tendency to ove-pedict the amplitudes of the highe hamonic components. 113

143 (m) (m) (m) (m) (m) (m) G1 mse= t (s) G3 mse= t (s) G4 mse= t (s) G5 mse= t (s) G7 mse= t (s) G9 mse= t (s) Fig. 4.8 Time-seies of suface elevations at six locations (G1, G3, G4, G5, G7 and G9). Dashed lines: laboatoy measuements; solid lines: pedictions by pesent model; mse: oot-mean-squae eo. 114

144 Hamonic amplitude(m) Hamonic amplitude(m) Hamonic amplitude(m) G Fequency (hz) G Fequency (hz) G Fequency (hz) Hamonic amplitude(m) Hamonic amplitude(m) Hamonic amplitude(m) G Fequency (hz) G Fequency (hz) G Fequency (hz) Fig. 4.9 Amplitude specta fom selective wave gauges (G1, G3, G4, G5, G7 and G9). Open ba: obseved esults; solid ba: pedicted esults. 4.4 Cases Studies Case selection and numeical input Fou epesentative cases ae simulated in this study and these ae summaized in Table 4.1. Case 1 is identical to the calibation case fo monochomatic waves ove an idealized hoizontal eef flat except that both incident wave height and the eefflat submegence ae inceased. Case is diffeent fom Case 1 in that a ectangula idge is pesent to investigate the model capability to deal with the complex geomety with a eef cest. Since highe ode deivatives of wate depth ae 115

145 included in the highe ode tems of the Boussinesq equations, abupt vaiation in bathymety such as the seaside and leeside vetical faces of the idge would cause some high ode tems in the Boussinesq equations to gow to infinity. Thus even with shock-captuing scheme and FVM method, the ectangula idge still need to be smoothed fo numeical simulation. In the simulations, both seaside and leeside vetical faces of the idge wee modified to V:H=1:1 slopes to ensue numeical stability. Since the height of the idge is just 5 cm, the smoothing does not significantly affect the wave tansfomation in the vicinity of the idge. The application of the model to an initially dy eef flat, which coesponds to the conditions whee a eef flat is exposed to ai at low tides, is tested in Case 3. Fo Case 4, the idealized eef pofile subjected to spectal waves is studied. The wate depth in the deepe section of the flume (i.e., h in Fig. 4.1) vaied fom.35 m to.45 m fo the fou cases studied hee, giving a change of kh fom.84 to 1.54 and a change of kh fom.7 to.53 (see Table 4.1). Numeical wave gauges wee placed at the same positions as thei laboatoy countepats in the flume. Thei exact locations measued fom the toe of the foe-eef fo the fou test cases ae given in Table 4.. Table 4.1 A summay of the fou simulated laboatoy expeiments Case No. T a (s) H a (m) h (m) kho kh ka Wave type Reef cest monochomatic plane monochomatic idge b monochomatic plane spectal plane a Fo spectal wave height, H and T efes to the deep-wate significant wave height and peak wave peiod, espectively; b h coesponds to an initial dy eef flat. 116

146 Table 4. The distances of the wave gauges (G1 - G1) fom the toe of foe-eef fo the fou simulated expeiments (Unit: m) Case No. G1 G G3 G4 G5 G6 G7 G8 G9 G1 G11 G The peviously calibated gid spacing x.4 m was used to discetize the computational domain, leading to a vaiable gid points pe wave length N = 5-1. The numeical model was un fo waves using a Couant numbe of.5 fo monochomatic waves and fo 1 waves with a Couant numbe of.35 fo spectal waves, espectively, which yields a time step anging fom.4 s to.7 s. Also, to ensue numeical stability, all simulations descibed in this section used a 4-point filte to educe the cuvatue of the apidly vaying bathymety. To ensue that the tansient effects ae insignificant in computing the wave height and the mean wate level, the initial 75 wave cycles fo monochomatic waves and the initial 1 significant wave cycles fo spectal waves wee not used in the data analysis. Fo empiical paametes in the sufzone model, the suggested values in section wee used, except 1 fo the eef with the idge (Case ). The value 1 was obtained by calibating the model against the expeimental esults; an explanation fo the inceased value of when a idge exists at the eef cest is given in the next section. Since the wave flume and the eef model wee made of glass and PVC, espectively, a constant Manning coefficient nm.1, as suggested in the liteatue, was used to estimate the fiction coefficient in Eq. (4.9). A summay of the model paametes fo the simulated cases is given in Table

147 Table 4.3 A summay of model paametes fo all the simulations Gid size Case No. x Couant numbe C Sponge laye width W c (m) d e 1 e e S&A a D&N b a S&A stands fo Skotne and Apelt (1999); b D&N stands fo Demibilek and Nwogu (7); c W 1.5L, whee L is the incident wave length; d Definition is given in Eq. (4.5); e Definition is given in Eq. (4.1) Effects of the idge on wave beaking Videos of beaking waves wee taken afte the wave fields had eached steady states. Plunging beakes wee obseved in Cases 1 to 3; spilling beakes wee pedominant in Case 4. Fig. 4.1 shows snapshots when the lips of the beakes hit the wate suface fo Cases 1 and. In the absence of a idge, the localized beakes plunged onto the hoizontal eef flat at cetain location downsteam of the eef edge. In the pesence of a idge, the beaking point shifted seawad: the beaking waves fist stoke the font side of the idge and then plunged onto the idge top, accompanying a stonge wave eflection. The sufzone locations and widths wee estimated fom the video ecodings of the expeiments (see Table 3.). The obseved suf zones extended appoximately fom. m shoewad fom the eef edge to.94 m shoewad fom the eef edge fo Case 1, and fom.4 m seawad fom the eef edge to 1.34 m shoewad fom the eef edge fo Case, giving sufzone widths of.9 m and 1.38 m, espectively. The seawad shift of the beaking point fo Case is due to the alteation of the 118

148 wave field by the idge. The calculated suf zone anges appoximately fom.1 m seawad fom the eef edge to. m shoewad fom the eef edge fo Case 1 and fom.3 m seawad fom the eef edge to.4 m shoewad fom the eef edge fo Case. Theefoe, the sufzone width is geatly educed by the idge. The pedicted eduction of suf zone width by the eef idge can be explained by the beaking model used in the simulations. With a idge, the aveage sufzone wate depth is.5 m, which is just 5% of that without the idge, theefoe accoding to Eq. (4.1), the beaking duation T, which is popotional to the squae-oot of the sufzone wate depth, is educed by appoximately 3%, indicating that the beaking pocess would complete within a shote distance. Fig. 4.1 Snapshots of the beaking waves ove the eef cest: (a) Case 1 without idge; (b) Case with idge. Based on the pesent numeical simulations, both the beake wave heights and efomed wave heights ae almost the same fo these two cases: about.1 m fo beake wave height and.4 m fo the efomed wave height. Consequently, about the same amount of wave enegy in the incident waves needs to be dissipated within a elatively naowe suf zone when a idge exists. Theefoe, when a idge is pesent at the edge of a eef cest, the coection facto needs to be inceased to adjust the incoect mixing length and fiction velocity scales in the eddy viscosity model. Of couse, othe factos such as flow sepaation and votex shedding at both the shap edges of the idge may also contibute to a lage, but it is believed that these contibutions ae not substantial. 119

149 4.4.3 Wave height, mean wate level and wave eflection The main concen in this chapte is the mean quantities such as wave heights and mean wate levels. The coss-eef vaiations of wave height and mean wate level, togethe with the bottom pofiles fo the fou expeiments listed in Table 4.1, ae displayed in Figs to 4.14 fo the convenience of the following discussion. The R-squae values ( R ) of mean wate level as well as the measued and pedicted maximum setups on the eef flat fo all fou cases ae listed in Table 4.4. Wave height The simulated wave heights agee easonably well with the measuements fo all the cases, indicating that the empiical beaking model can simulate the enegy dissipation well. Simulations show that the wave height fist inceases on the foeeef due to shoaling, then eaches a peak (which is the beaking point), and finally deceases apidly in the suf zone due to wave beaking. The peak value of the wave height was not captued by the measuements in Figs and 4.1 because of limited numbe of measuement stations and the esulting coase spatial esolution in the expeiments. A set of esults obtained with a highe spatial esolution is shown in Fig. 3.1, whee a peak in the measued wave height can be clealy seen. The modulation of the simulated wave height seawad is again due mainly to the patial wave eflection fom the eef model, which may poduce a patial standing wave patten in font of the eef model. This phenomenon is enhanced by the pesence of the idge in Case, and was nealy invisible in the Case 4 fo spectal waves due mainly to the fact that thee is no oganized patial standing wave patten that can be fomed in font of the eef model. 1

150 H(m) MWL(m) Seabed Elev.(m) Distance fom the toe of foe-eef (m) Fig Vaiations of the wave height and mean wate level (MWL) ove the eef pofile fo Case 1. Solid lines: pedictions by pesent model; Open cicles: laboatoy measuements. Some mino discepancies in the efomed wave heights ove the eef flat can be obseved fo both Case 1 and Case, although the scatte in the measuements is noticeable. These discepancies may be attibuted patly to the empiical value used to define the satuated beaking cessation in Eq. (4.1). Since the wave enegy density is popotional to the wave height squaed, the mino diffeences found in the efomed wave heights do not have a significant effect on the accuacy of the pedicted enegy dissipation. 11

151 H(m) MWL(m) Seabed Elev.(m) Distance fom the toe of foe-eef (m) Fig. 4.1 Vaiations of the wave height and mean wate level (MWL) ove the eef pofile fo Case. Solid lines: pedictions by pesent model; Open cicles: laboatoy measuements. The eef flat with zeo submegence in Case 3 was aely studied in the existing liteatue. Fo this case, wave beaking fist occued on the foe-eef and then pumped wate onto the eef flat though ovetopping. The vey good ageement between the simulated and measued wave heights fo this case is vey impessive (shown in Fig. 4.13). As the flow on the eef flat might be supecitical (Goulay and Collete, 5), the stability and the accuacy of the numeical simulation wee achieved with the help of shock-captuing FV-based solve and the moving bounday algoithm. 1

152 H(m) MWL(m) Seabed Elev.(m) Distance fom the toe of foe-eef (m) Fig Vaiations of the wave height and mean wate level (MWL) ove the eef pofile fo Case 3. Solid lines: pedictions by pesent model; Open cicles: laboatoy measuements. As shown in Fig. 4.14, the ageement between the numeical and expeimental esults is emakable fo Case 4 (spectal waves), suggesting that the poposed sufzone model also wok well fo spectal waves ove finging eefs. Fo this case, the suf zone is wide and spilling beakes ae pedominant. 13

153 .1 H s (m) MWL(m).1 Seabed Elev.(m) Distance fom the toe of foe-eef (m) Fig Vaiations of the significant wave height and mean wate level (MWL) ove eef pofile fo Case 4. Solid lines: pedictions by pesent model; Open cicles: laboatoy measuements. Mean wate level Oveall, the model gives vey good pedictions of the vaiation of mean wate level fo all cases, with all R being lage than.85 as shown in Table 4.4. The setdowns ove the foe-eef ae due mainly to wave shoaling befoe beaking. The beaking points (locations whee the magnitude of wave setdown eaches its maximum) and the setups ove eef flat ae all easonably pedicted by the simulations (see Table 4.4). The setdowns in the wate in font of the foe-eef ae elated to the equiement on the consevation of mass fo a closed flume: the waveinduced setup ove the eef flat must be balanced by the setdown in the othe pat of the flume. The slight undeestimation of the wave-induced setups ove the eef flat fo Case 1 and Case was diectly elated to the unde-pediction of the local wave heights fo these two cases. A compaison of the pedicted wave-induced setups 14

154 ove eef flat fo these two cases eveals that a idge-like eef cest can incease the wave-induced setup by naowing the suf zone as discussed in the pevious section. Since the coss-shoe wave height evolutions fo both cases ae well epoduced by the pesent model, the excellent ageement between the numeical and expeimental mean wate levels fo Case 3 and Case 4 ae not supising given the adiation stess concept. Table 4.4 The R-squaes ( R ) of mean wate level at all measuement locations, the measued maximum setup ( ) and the pedicted maximum setup ( ) on the eef flat o p Case No. R o (mm) p o (mm) / p Reflection coefficient The calculated and measued eflection coefficients fo the fist hamonic waves ae compaed in Table 4.5. Satisfactoy ageements ae obtained between the numeical and laboatoy esults in view that: (1) the small diffeences between laboatoy and numeical bounday conditions (e.g., the eflection fom paddle of wavemake was not modeled in all numeical simulations), () the mino diffeence in the shapes of the idges used in the expeiments and the simulations (the shap edges of the idge have been smoothed in the numeical simulations fo numeical stability consideations), and (3) the eo aises fom the empiical beaking model which may not eflect the eal physics of sufzone pocess. The influence of the idge on the wave eflection is evident fo Case ; the enhanced wave eflection is expected since the idge stuctue functions like a submeged beakwate, which has been widely used to eflect the wave enegy fo shoe potections. Fo spectal waves, fequency aveaged eflection coefficients as defined by Goda () ae used 15

155 whee E i and specta, espectively. K E / E (4.11) i E ae the total enegy contained in the incident and eflective The esults fo Case 4 show that spectal waves geneally have lage values of eflection coefficient than the equivalent monochomatic waves do; this is because low-fequency wave components tend to have lage eflection coefficients than highe-fequency components (Seelig, 1983). Table 4.5 The measued ( K ) and pedicted ( K ) eflection coefficients m p Case No. m K p K 1.7±.1.3.± ± ±

156 4.4.4 Wave tansfomation Monochomatic waves The computed suface elevations at six locations fo Case 1 and Case ae compaed with the expeimental data in Figs and 4.16, espectively, also indicated in the figues is the oot-mean-squae eo (mse) at each location. It can be obseved that vey good ageements ae obtained at most locations (whee the mse is small) fo these two cases. Wave shoaling makes the wave fom asymmetic and skewed at G3 on the foe-eef; at this location, the simulation fo Case 1 agees bette with expeiments than fo Case because the smoothed idge pofile nea the eef edge may have slightly affected the eflected waves and the wave dispesion in the wate above the idge. G5 and G7 ae located in the suf zone, thus beaking waves with saw-tooth shape ae again obseved thee; a elatively lage eo in the measued waves in the suf zone is expected because of the entained bubbles in the sufzone wate. On the eef flat (G9), both the numeical and laboatoy esults show that the efomed waves ae vey small and esemble cnoidal waves. Fig pesents a compaison of the measued and simulated suface elevations fo Case 3 (dy eef flat). The wave fom is satisfactoily pedicted seawad of the foe-eef (G) and in the shoaling zone (G3); the efomed waves ae quite small (G9 and G11), theefoe, most of the wave enegy has been dissipated by wave beaking and bottom fiction. 17

157 (m) (m) (m) (m) (m) (m) G mse= t (s) G3 mse= t (s) G5 mse= t (s) G7 mse= t (s) G9 mse= t (s) G11 mse= t (s) Fig Time-seies of suface elevations at six locations (G, G3, G5, G7, G9 and G11) fo Case 1. Dashed lines: laboatoy measuements; solid lines: pedictions by pesent model; mse: oot-mean-squae eo. 18

158 (m) (m) (m) (m) (m) (m) G mse= t (s) G3 mse= t (s) G5 mse= t (s) G7 mse= t (s) G9 mse= t (s) G11 mse= t (s) Fig Time-seies of the suface elevations at six locations (G, G3, G5, G7, G9 and G11) fo Case. Dashed lines: laboatoy measuements; solid lines: pedictions by pesent model; mse: oot-mean-squae eo. 19

159 (m) (m) (m) (m) (m) (m) G mse= t (s) G3 mse= t (s) G5 mse= t (s) G7 mse= t (s) G9 mse= t (s) G11 mse= t (s) Fig Time-seies of the suface elevations at six locations (G, G3, G5, G7, G9 and G11) fo Case 3. Dashed lines: laboatoy measuements; solid lines: pedictions by pesent model; mse: oot-mean-squae eo. Spectal waves A compaison of the measued and simulated wave specta is pesented in Fig Good ageement can be obseved at G. Fo spectal waves, individual wave beaking occus eithe on the foe-eef o on the eef flat, thus at G5 (located at the seawad side of the eef edge), thee is still a consideable amount of wave enegy aound the peak fequency. The numeical model slightly ove-pedicts the wave enegy at the peak fequency nea the eef edge (G5) as well as duing the shoaling pocess (G4). Waves fist shoal on the foe-eef and beak in the shallow wate 13

160 above the eef flat, tansfeing pat of the enegy fom the fundamental waves to highe and lowe hamonic waves. The numeical model also satisfactoily captues the enegy tansfe fom the peak fequency to both highe and lowe fequency waves, and the pedicted wave specta at G6 to G9 and G11 ae satisfactoy. Shoewad of the suf zone, thee is no noticeable diffeence between the computed and measued wave specta since the local wate depth on the eef flat is sufficiently shallow to filte out most of shot waves though bottom fiction. The lowfequency motions on the eef flat ae pobably due to infagavity waves (Demibilek and Nwogu, 7). S(m.s) S(m.s) S(m.s) S(m.s) x 1-3 G f(hz) x G5 1 3 f(hz) x f(hz) x G7 G9 1 3 f(hz) S(m.s) S(m.s) S(m.s) S(m.s) 1 x 1-3 G4 1 3 f(hz) x G6 1 3 f(hz) x G8 1 3 f(hz) x G f(hz) Fig Wave specta at eight locations (G, G4 - G9, and G11) fo Case 4. Dashed lines: laboatoy measuements; solid lines: pedictions by pesent model. 131

161 4.5 Revisits of Othe Numeical Studies In this section, two othe published laboatoy and numeical studies on wave tansfomation ove finging eefs ae evisited. The pupose is to investigate whethe highly nonlinea Boussinesq equations can impove the pedictions fom the weakly nonlinea Boussinesq equations and to demonstate the effects of numeical bounday conditions on the mean wate levels Revisit of Skotne and Apelt (1999) Skotne and Apelt (1999) epoted both expeimental data and numeical simulations fo setdowns and setups induced by monochomatic waves popagating onto a submeged finging coal eef as descibed by Seelig (1983). The eef pofile consisted of a steep composite foe-eef with an aveage slope of V:H=1:1, which was followed by a small shap idge-like eef cest, and a 7 m-long hoizontal eef flat. Fo thei numeical simulations, they used the weakly nonlinea Boussinesq model oiginated by Nwogu (1993) with a olle-based sufzone model poposed by Schäffe et al. (1993). Nwogu (1993) s equations could be ecoveed by neglecting the nonlinea dispesive tems in Eqs. (4.1) and (4.) t hux u h h z hz u xxx x (4.1) 3 1 ut uux g x z hz utxx (4.13) Monochomatic waves wee geneated intenally with the sponge layes being used only in the momentum equation on both sides of the computational domain to educe the wave eflection. Six tests wee simulated, coveing a ange of beake types. They found that thei numeical model could pedict the tend of the mean wate level easonably well, but thee was an inceasing discepancy in the magnitude of setup o setdown when the height of incident waves was inceased. Thei Test 6 with a wave height of.79 m and a wave peiod of 1.7 s had the pooest ageement. Hence, it is selected in this study to see if the pesent model can povide an impoved pediction. The same bounday conditions and numeical inputs as theis wee used in the simulation with a gid size of x.4 m, Couant 13

162 numbe C =.5, and fiction coefficient f.15. Fo the coection coefficient in the beaking model, the value 1 gave the best fit to the laboatoy data. Details of the simulation paametes ae given in Table MWL(m) Seabed Elev.(m) Distance fom the toe of foe-eef (m) Fig Vaiation of the mean wate level (MWL) ove the eef pofile fo the Test 6 of Skotne and Apelt (1999). Dashed line: pediction by Skotne and Apelt (1999); Solid line: pediction by pesent model; Open cicles: laboatoy measuements by Skotne and Apelt (1999). A compaison among the esults of laboatoy, numeical simulations of Skotne and Apelt (1999) and the pesent model is given in Fig. 4.19, whee the seafloo pofile is also shown. In geneal, the wave-induced setdowns pedicted by the pesent model ae in ageement with the numeical esults of Skotne and Apelt (1999). The simulation in this study gives a bette pediction of the change of mean wate level in the suf zone. Howeve, both models undeestimate the magnitude of the setup ove the eef flat. Skotne and Apelt (1999) suspected that the unde- 133

163 poductions might esult fom the lack of highly nonlinea tems in thei govening equations, but the pesent esults show that the inclusion of the highe ode nonlinea tems in the Boussinesq model does not necessaily impove the pediction of the mean wate level ove the eef flat Revisit of Demibilek and Nwogu (7) Demibilek and Nwogu (7) pesented a combined laboatoy and numeical study fo a typical finging eef without any idge stuctue. Thei tests wee un fo a wide ange of iegula sea states geneated fom JONSWAP specta. Thei numeical model was based on the weakly nonlinea Boussinesq equations deived by Nwogu (1993). The effect of wave beaking was also paameteized using an eddy-viscosity concept based on a one equation tubulence closue model. Since the intention of this section is to investigate the effects of bounday settings, the weekly nonlinea vesion of the pesent model was used (i.e., Eqs. (4.1) and (4.13)). Only a epesentative case (thei Test 48) with H 7.5 m, T 1.5 s s p and h.31 m was evisited in this section. Following thei epot, the pesent simulations wee caied out using a gid spacing x.5 m a time step t.1s and an equivalent fiction coefficient f.11 which was conveted fom thei Chezy coefficient. The pesent physical simulation time was 9 s, and the initial 1 s suface elevations wee not used in computing the mean wate levels. The majo diffeence in the numeical settings is: they used sponge layes in both the continuity and momentum equations at the seawad bounday and a plane beach at the shoewad bounday while in this study the sponge layes wee used only in momentum equation at both boundaies. Paamete in the pesent beaking model needs to be calibate with expeiments; the value suggested in section was used, i.e., fo this case. A summay of all the simulation paametes is given in Table 4.3. The measued and computed significant wave heights ( H ) and the mean wate level vaiations ae compaed in Fig. 4.. Both simulations pedicted simila wave heights ove the eef. Howeve, the simulation of Demibilek and Nwogu (7) 134 s

164 failed to captue the wave-induced setdown seawad of the foe-eef, which was well pedicted by the pesent simulation. Demibilek and Nwogu (7) attibuted the diffeence between the measuements and thei simulations to the sponge laye damping tem used in thei continuity equation. Fom the compaisons in Fig. 4., it can be concluded that the mean wate level ove the eef is sensitive to the types of the bounday conditions used and that the damping tems should not be included in the continuity equation when simulating the mean wate levels in a closed flume. H(m) MWL(m).1 Seabed Elev.(m) Distance fom the toe of foe-eef (m) Fig. 4. Vaiations of the wave height and mean wate level (MWL) ove the eef pofile fo the Test 48 of Demibilek and Nwogu (7). Dashed lines: pedictions by Demibilek and Nwogu (7); Solid lines: pedictions by pesent model; Open cicles: laboatoy measuements by Demibilek and Nwogu (7). 4.6 Model Application to Diffeent Foe-eef Slopes and Pofiles The influences of offshoe wave conditions and eef-flat submegences on the mean quantities such as wave height and the wave-induced setup ove eef flat have been 135

165 investigated expeimentally by, e.g., Goulay (1994, 1996a) and numeically by, e.g., Skotne and Apelt (1999) and Demibilek and Nwogu (7). Howeve, to the best of the autho s knowledge, neithe existing laboatoy no numeical wok has studied the influence of the inclination angle of a plane foe-eef o the shape of foe-eef. In this section, the numeical model is applied to examine a seies of foeeefs numeically. Case 1 (a plane foe-eef with a slope of V:H=1:6) in section 4.4 is taken as the benchmak against which the compaisons and discussion ae made. Two types of numeical tests wee conducted: (1) plane foe-eefs with slopes vaying fom V:H=1:1 to 1:, and () combinations of two types of cicula ac pofile (convex and concave) and two values of ac cuvatue (.75,.15). Fo the cicula ac pofiles, the chods ae the same as the foe-eef length in Case 1. Fo both types of numeical tests, the numeical settings wee the same as those used fo Case Effect of the inclination angle of plane foe-eef The effects of foe-eef slope on the coss-shoe vaiation of wave height and mean wate level ae summaized in Fig While the computed eflection coefficient, tansmission coefficient, beaking location elative to the toe of foe-eef and maximum setup/setdown ae plotted against the suf-similaity paamete ( ), defined by Eq. (.4), in Figs. 4.(a), 4. (b), 4. (c) and 4.(d), espectively. Fig. 4.1 shows that seawad of the eef model, the fluctuation of wave height caused by the patial standing waves is geneally amplified as the foe-eef becomes steepe, indicating an enhanced wave eflection. Howeve, Fig. 4.(a) shows that when the foe-eef slope becomes elatively gentle (small ), thee is no tends fo the vaiation of eflection coefficient with slope; the same is also found in Fig The fluctuation of the seawad mean wate level fo steepe slopes can also be obseved in Fig Simila phenomenon can also be obseved in the simulations of Skotne and Apelt (1999) and Ranasinghe et al. (9). Cuently the autho cannot povide a convincing explanation fo it. 136

166 MWL(m) Distance fom the toe of foe-eef (m) H(m) Seabed Elev.(m) Fig. 4.1 Vaiations of the wave height and mean wate level (MWL) ove eef pofile with diffeent foe-eef slopes. Light black solid line: V:H=1:1; Dash-dot line: V:H=1:3; Dak black solid line: V:H=1:6; Dotted line: V:H=1:1; Dashed line: V:H=1:. The pedicted wave heights on eef flat ae almost the same as indicated by the tansmission coefficients in Fig. 4.(b), which is contolled mainly by the beaking cessation citeia in the beake model. Wave beaking occus on the seaside of the eef edge and beake wave heights ae almost the same fo all simulations. Howeve, the beaking points ae shifted slightly seawad as the slope vaies fom V:H=1:1 to 1:; this is because an inceasing distance between the beaking location and the toe of foe-eef slope with the inceasing, as shown in Fig. 4.(c). 137

167 . (a).35 (b) K.1 K t Beaking location (m) (c) b & (m) (d) Fig. 4. Reflection coefficient ( K ), tansmission coefficient ( K t ), beaking location elated to the toe of foe-eef (positive if shoewad), maximum wave setdown ( b ) and maximum wave setup on the eef flat ( ) as a function of sufsimilaity paamete ( ). Fig. 4.(d) shows that the wave-induced setdown at the beaking point deceases with the incease of and the foe-eef slope. Howeve, the wave-induced setups ae about the same fo all simulations on the eef flat. This is due to the fact that the total amount of enegy dissipated in the suf zone is about the same fo diffeent slopes. Howeve, the spatial vaiation of the setup in suf zone depends on the sufzone width, which is elated to the slopes of foe-eef. The calculated wave 138

168 setup inceases moe apidly fo steepe slopes due to the eduction in the oveall wate depth in sufzone and in sufzone width. When waves shoal on a milde slope ove a longe distance, the bottom fiction may also affect the wave-induced setup. Howeve, the additional tests by vaying the Manning s coefficient have found that the fictional dissipation is negligible fo you poblems since the foe-eef is not long enough. The foe-eef slope fo Case was also adjusted (in the pesence of the idge), simila obsevations (not epoted hee) wee obtained. It seems that the slope of a plane foe-eef can have noticeable influence on the wave-induced setdown and setup only in the shoaling zone and suf zone Effect of the shape of foe-eef pofile The influences of the shape of foe-eef pofile ae summaized in Fig. 4.3, which eveals that seawad of the foe-eef, thee is no notable diffeence among diffeent simulations in both the wave height and mean wate level. Howeve, the location of the beaking point is affected by the slope pofiles: the concave pofiles (in analogy to mild slopes) move the beaking point seawad, while the convex pofiles (in analogy to steep slopes) move it shoewad. Physically, wave beaking in coastal wates is always depth-limited. When a wave shoals on a foe-eef slope, thee is a atio of the wave height to wate depth beyond which the wave will beak. The atio has been found to be almost slope-independent fo eef pofiles as discussed in Chapte. Since the beake wave height depends only on the local wate depth, the beake wate depth should be simila fo all investigated slopes. Fo a milde slope, achieving such a depth equies that the beaking point moves futhe offshoe if the eef edge is kept at the same location. The wave-induced setdown is lage fo the convex pofiles than fo the concave pofiles o plane pofile. Howeve, the wave setup on the eef flat emains nealy unchanged as the foe-eef pofile vaies fom the concave configuations to the convex configuations: this may be because the wave beaking dissipation is almost the same fo all cases. The nea-constant eef-flat setup indicates that the shape of the foe-eef pofile is not a key paamete contibuting to the wave-induced setup 139

169 ove the finging eef and the esults obtained fo the plane foe-eef ae epesentative of othe shapes. MWL(m) Distance fom the toe of foe-eef (m) H(m) Seabed Elev.(m) Fig. 4.3 Vaiations of the wave height and mean wate level (MWL) ove eef pofile with diffeent foe-eef pofiles. Light black solid line: concave ac with the cuvatue=.15; Dash-dot line: concave ac with the cuvatue=.75; Dak black solid line: plane slope; Dotted line: convex ac with the cuvatue=.75; Dashed line: convex ac with the cuvatue= Concluding Remaks Numeical expeiments based on weakly dispesive, fully nonlinea Boussinesqtype equations with a FV-based solve have been pefomed to study the wave tansfomation ove vaious finging eef pofiles. Compaisons with published laboatoy measuements and othe numeical studies lead to the following conclusions: 14

170 In ode to conseve the mass in a closed wave flume, the numeical damping laye can only be used in the momentum equation. The zeo-equation eddy viscosity model, with its model paametes being calibated using measuements, can easonably simulate the key chaacteistics (wave eflection, wave height, wave-induced setup/setdown) of both monochomatic and spectal waves ove vaious finging eef pofiles and conditions, including the dy eef-fat condition and a idge at the eef edge. The eef cest pofile, especially the existence of a idge at the eef edge, may significantly affect the wave-induced setup/setdown ove the finging eef. The incease of the wave-induced setup ove the finging eef with a idge located at the eef edge is due mainly to the eduction of the sufzone wate depth and the sufzone width. Both the inclination angle of plane foe-eef and the shape of foe-eef pofile have negligible effects on the wave-induced setup/setdown outside the shoaling zone and suf zone. The esults fo plane foe-eef ae epesentative of othe pofiles.. Natual eefs ae fa moe complex than can be studied in the laboatoy. Coastal cuents, foe-eef oughness and gaps in the eef idges might affect the wave tansfomation ove eefs. A numeical study of the effects of eef-suface oughness on the wave dynamics ove finging eefs is unde way and the esults will be epoted elsewhee. 141

171 CHAPTER 5 MODELING WAVE-INDUCED SETUP OVER FRINGING REEFS WITH SELECTED EXISTING ANALYTICAL MODELS 5.1 Intoduction The numeical simulations based on Boussinesq-type equations in Chapte 4 have been poven to be able to adequately epoduce the wave setups unde diffeent eef pofiles (with and without a idge) and wave conditions (monochomatic and spectal). Howeve, the applicability of the numeical model is believed to be elated to the FV-based numeical scheme and the paameteization of wave beaking. Simple paameteizations ae still desied fo use in wave models since it is not pactical to solve the Boussinesq equations fo most poblems of inteest. It is pefeable if some analytical solutions can be devised to descibe the wave setups unde cetain special conditions, paticulaly fo the case in which a idge is located at the eef edge, so that moe insight into the undelying physical pocesses can be evealed. In analogy to wave-diven coss-shoe flows and setup/setdown on beaches (e.g., Svendsen, 6), the hydodynamics of wave setup on eefs has been descibed using the adiation stess concept poposed by Longuet-Higgins and Stewat (1964). Taditionally, two main goups of analytical models ae commonly used in the existing liteatue to pedict the maximum wave-induced setup on a eef flat. The fist goup is to deive an analytical solution by integating the momentum equation in the suf zone, and enegy balance is supplemented to estimate some bounday values. Tait (197) was the fist to achieve such a solution. Simila appoaches can be found in e.g., Symonds et al. (1995), Hean (1999) and Vette et al. (1). Impotantly, this type of model was usually developed fo hoizontal eef flats without the idge. This is fo a good eason: the sudden depth change and shallow wate on the eef idge endes the analytical fomulation nealy intactable. The othe goup is established accoding to the coss-shoe enegy balance; this idea was oiginally poposed by Goulay (1996b) and impoved late by Goulay and Collete 14

172 (5). In this goup of methods, the enegy flux of incident, tansmitted, and eflected waves ae paameteized, sepaately. The coss-shoe momentum balance is supplemented to estimate the enegy flux associated with wave beaking and the sufzone mophologic effects ae also addessed though an empiical coefficient, see section 5. fo details. Field obsevations have often been cast in tems of a simple scaling paamete connecting the maximum wave-induced setup on eef flat,, the offshoe wave height, H, and wave peiod, T, ( ) p ( / ) q m n HT H T HT (5.1) whee m and n can be detemined by using field o expeimental data (Hench et al., 8). The fist paamete ( HT) epesents the deep-wate (offshoe) wave enegy flux towads the eef wheeas the second ( H T ) epesents the deep-wate (offshoe) wave steepness. The adiation-stess type model of Hean (1999) gave m 1 and n, although it was unclea in his model which incident wave height it efeed to, since the wave shoaling was not consideed. In contast, the model of Goulay and Collete (5) gave m and n 1 when the eef-flat wate level was vey high. Fo spectal wave conditions, the offshoe oot-mean-squae (ms) wave height, Hms (1 / ) Hs (whee H s is significant offshoe wave height) and the peak wave peiod, T p, may be used instead of the monochomatic wave height, H, and the monochomatic wave peiod,t, in Eq. (5.1). Hench et al. (8) found that m 3. and n 3 fo the Mooea eef, a natual eef with a idge. In contast, Lowe et al. (9a) epoted that fo the eef system in Kaneohe Bay, a natual eef with no idge, m and n anged between 1.6 to.7 and.6 to.8, espectively. / In this chapte, the pefomances of the offshoe scaling paamete (i.e., Eq. (5.1)) and two afoementioned epesentative 1DH analytical models (the models of Tait (197) and Goulay and Collete (5)) ae investigated using the expeimental 143

173 data fo wave setup pesented in Chapte 3. Thei pefomances fo diffeent eef pofiles (with and without the idge) and wave conditions (monochomatic and spectal) ae compaed. The est of the chapte is oganized as follows: the theoetical backgound of the existing analytical models is descibed in section 5.. The offshoe scaling paamete is validated in section 5.3. The model of Tait (197) is tested in section 5.4 while the model of Goulay and Collete (5) is examined in section 5.5. Some discussions on the model implementations and esults ae given in section 5.6. The conclusions ae dawn in section 5.7. Fig. 5.1 Configuation of an idealized finging eef with a ectangle idge and some notations adopted in this chapte. 5. Theoetical Consideations Although seveal theoies poposed to pedict wave setup ove eefs diffe in many espects (see Lowe et al., 9a), they all ely fundamentally on fou pinciples descibing waves and mean flows: (1) The mean momentum balance can be expessed using the adiation stess theoy of Longuet-Higgins and Stewat (1964), which can be witten fo steady 1D flows as (see Eq. (1.3) afte ignoing the convection tem) b d x 1 dsxx gh ( ) dx dx (5.) 144

174 whee h is the undistubed wate depth, is the wave-induced change in mean wate level which has a maximum value of at the shoewad end of the suf zone, b x is the coss-shoe bottom shea stess and S xx is the coss-shoe adiation stess fo unidiectional long-cested waves. Some elevant paametes ae shown in Fig () Wave evolution is modeled by a simplified fom of consevation of wave enegy (e.g., Svendsen, 6) d ( ce g ) b f (5.3) dx whee c g is the goup velocity, E is the enegy density, b and f ae enegy dissipation ates due to wave beaking and bottom fiction, espectively. (3) The wave-aveaged quantities S xx and E, and goup velocity c g (and implicitly the dispesion elation) can be evaluated using linea wave theoy fo shallow wate waves S xx 3 16 gh E 1 8 gh (5.4) c g gh whee H is the local wave height. (4) In shallow wate, whee waves beak unde the depth-limited condition, the beake height is estimated by H (, skhs, ) h whee is the empiical beake depth index, k is the wave numbe, (5.5) s tan is the foe-eef slope with being the foe-eef slope angle as shown in Fig (Fo 145

175 a eef with the idge, an equivalent foe-eef slope angle can be defined as the angle ' between the toe of the beach and the seawad edge of the idge (see the angle in Fig. 5.1). Note that the mean wate level, ( h ), is used hee instead of the still wate level, h, in Eq. (1.1). The deep-wate wave steepness, S, is witten hee in tems of the deep-wate wave height, H, and eithe the wave peiod, T o the deepwate wavelength, L (see e.g., Raubenheime et al., 1996) H 1 H S gt (5.6) L b The simplest model descibed fist by Tait (197) assumed that: (a) ; (b) is a constant on the foe-eef slope, and (c) the enegy flux is conseved until beaking. In this case, Eq. (5.) can be integated exactly to give x h 1 ( ) / 3 b h b h (5.7) whee h b is the beake depth fo monochomatic waves and h is the still wate depth on the eef flat (eef-flat submegence). See Appendix E fo the deivation of Eq. (5.7). Eq. (5.7) is the oiginal fom of the model poposed by Tait (197). Howeve, Tait (197) did not give any analytical solution of h b. In fact, h b can be estimated based on the coss-shoe enegy balance without consideing fictional dissipation (see also Appendix E). Afte ignoing the fist tem in Eq. (5.7) fo wave setdown, Tait (197) s model in its dimensionless fom eads.4 1 (1 K ) h.4. H 1 8/3 (4 ) S H (5.8) whee K is the eflection coefficient. Simila appoaches have also been used by Symonds et al. (1995), Hean (1999) and Vette et al. (1). 146

176 Altenatively, using consevation of enegy and momentum, Goulay and Collete (5) suggested that the wave-induced setup in the wate above the eef flat (without consideing the eef-flat mean cuent) could be calculated with T 3 h 1 h H Kp 1K 4 gh 64 H T g h 3/ (5.9) whee K p is a fee paamete that descibes the effects of eef pofile shape. The beake depth index,, was supposed to apply on o nea the eef flat and was believed to be smalle than that used in Eq. (5.8) on the foe-eef (see also Appendix E fo the deivation of Eq. (5.9)). It is impotant to note that Goulay and Collete (5) also used a adiation stess concept to model the suf zone. Instead of caying out an exact integation of the momentum equation acoss the suf zone, Goulay and Collete (5) assumed that the change of adiation stess would take place at an effective depth, h (1 / K )( h ). In essence, by assuming a p p constant on the eef flat, they wee able to use K p to paameteize wave beaking. 5.3 Offshoe Scaling Paamete Using Eq. (5.1) to analyze the expeimental data in Chapte 3 obtained with and without the idge fo both monochomatic and spectal wave conditions ( H ms and T p wee used fo spectal waves), the coelation coefficients between data and the scaling paamete fitted fo vaious exponents wee computed, and the values of m and n coesponding to maximum coelation coefficient ae listed in Table 5.1. Table 5.1 shows that the values of m and n fo wave setup expeiments in Chapte 3 fall in the ange of to 3, which is consistent with the ange obseved in field as eviewed in section 5.1. Fo eefs both with and without the idge, no obvious elation can be obseved between m o n and the still wate depth on the eef cest, h c. In geneal, the value of p is one ode of magnitude lage than that of q fo 147

177 most cases, which means that the maximum wave-induced setup on the eef flat has a stong dependence on the offshoe wave enegy. Note also that thee is an incease of q/ p with deceasing h c fo both eefs (with and without the idge) subjected to monochomatic waves, indicating an inceasing dependence on the wave steepness fo those cases. Table 5.1 Maximum coelation between wave-induced setup ( ) and offshoe m n scaling paamete ( H T ) h c (m) m n p * q * q/p Coelation With idge coefficient Wave condition no monochomatic no monochomatic no monochomatic yes monochomatic yes monochomatic yes monochomatic yes monochomatic no spectal no spectal no spectal yes spectal yes spectal yes spectal * Refeing to Eq. (5.1) fo dentitions. Since both the wate depths above the eef flat and the pesence of a idge can significantly change the values of m and n, the dimensional paamete goup H m T n is not sufficient to descibe the pedicted wave setup on the eef flat. Moeove, thee is a clea discepancy in the values of m and n obtained fom the laboatoy obsevations and those fom existing theoies, indicating that the existing analytical solutions, which wee developed fo simple eef models, may not fully account fo 148

178 the effects of idge on the wave beaking and the wave setup. Fo example, casting the Tait (197) s model, i.e., Eq. (5.8), into the offshoe scaling paamete gives m.8 and n.4, values that diffe somewhat fom what have been shown in Table Compaison of Expeimental Data with the Model of Tait (197) The pefomance of Eq. (5.8) can be assessed with the wave setup data in Chapte 3. Fo the expeiments with a idge, the still wate depth on the idge-top should be used instead of the wate depth on the eef flat (Yao et al., 9). Hence, the eefcest submegence, h c (the minimum wate depth acoss the eef flat), is used in Eq. (5.8) instead of h fo the eefs with and without a idge. K is obtained fom measuements since thee is no well-accepted appoach to pedict K fo eefs. Fo a given eef pofile and h c, values of can be obtained fom a least-squae fitting of the laboatoy data to Eq. (5.8) fo both monochomatic and spectal waves. The compaisons of the pedicted and obseved setups ae shown in Fig. 5.. Oveall, the model of Tait (197) povides a seasonable pediction of wave-induced setups up to hc.m. Howeve, when h c is futhe inceased, the values of ae geneally unde-pedicted, paticulaly fo the cases with lage incident wave heights, suggesting that a global value of may be inappopiate fo a given wate depth due to the dependence of on wave chaacteistics. Fo both pofiles, Fig. 5. also shows that falls between.3 and 1.7, i.e., the (elatively wide) ange peviously found fo natual beaches (e.g., Guza and Thonton, 1981; Nelson, 1994; Raubenheime et al., 1996, Péquignet et al., 11). Howeve, even fo a given eef pofile, smalle must be used fo lage h c, i.e., depends on the eef-cest submegence. It also implies that the assumption of a constant value of, as is done in simila existing models (e.g., Symonds et al., 1995; Hean, 1999), may not be appopiate. Note that the autho did not include wave setdown in the above 149

179 esults as Tait (197) oiginally did (see Eq. (5.7)). Peliminay tests have shown that the oveall unde-pediction would be enhanced if it wee included. (a) without idge: h c = m 5 =1.16; R =.96 4 (b) without idge: h c =.5 m 5 =.76; R =.59 4 (c) without idge: h c =.1 m 5 =.38; R =.48 4 o (mm) 3 o (mm) 3 o (mm) p (mm) p (mm) p (mm) 5 4 (d) with idge: h c = m =1.66; R = (e) with idge: h c =. m =1.34; R = (f) with idge: h c =.5 m =.75; R =.58 o (mm) 3 o (mm) 3 o (mm) p (mm) p (mm) p (mm) Fig. 5. Pedicted wave setup ( ) vs. obseved wave setup ( ) using the model p of Tait (197) fo diffeent eef-cest submegences, eef pofiles and wave conditions ( hc - eef-cest submegence; - empiical paamete in the model; cicles - monochomatic waves; squaes - spectal waves; p ). o o R - R-squae; Solid line Compaison of Expeimental Data with the Model of Goulay and Collete (5) In this section, the theoy of Goulay and Collete (5) is used to analyze the wave setup data in Chapte 3. The autho used.4 in the model as suggested by Goulay (1996a). Instead of ignoing K as Goulay and Collete (5) did, included in the pesent analysis since the measued values of K is K can be as lage as 15

180 .6. Again, h in Eq. (5.9) is eplaced by the still wate depth ove the eef cest, h c. By doing so, Eq. (5.9) can be solved fo wave setups. The compaisons between the calculated wave setups and the obseved setups ae shown in Fig Hee the shape-dependent paamete K p is taken as the aveage of all values obtained fo both monochomatic and spectal wave conditions. (a) without idge: h c = m 5 K p =.5; R =.96 4 (b) without idge: h c =.5 m 5 K p =.7; R =.96 4 (c) without idge: h c =.1 m 5 K p =.75; R =.81 4 o (mm) 3 o (mm) 3 o (mm) p (mm) p (mm) p (mm) 5 4 (d) with idge: h c = m K p =.35; R = (e) with idge: h c =. m K p =.55; R = (f) with idge: h c =.5 m K p =.7; R =.98 o (mm) 3 o (mm) 3 o (mm) p (mm) p (mm) p (mm) Fig. 5.3 Pedicted wave setup ( ) vs. obseved wave setup ( ) using the model p of Goulay and Collete (5) fo diffeent eef-cest submegences, eef pofiles and wave conditions ( h c - eef-cest submegence; o K p - empiical paamete in the model; cicles - monochomatic waves; squaes - spectal waves; R - R-squae; p Solid line - ). o Oveall, compaing Fig. 5.3 with Fig. 5., it appeas that the model of Goulay and Collete (5) hindcasts the wave setup data moe accuately than does the model of Tait (197). In the absence of a idge, Goulay and Collete (5) gives good pedictions of the wave-induced setup fo both wave conditions (monochomatic 151

181 and spectal). Fo the eef with a idge, thee is a slight ove-pediction of when the incident wave height is lage and h c is small. K p falls between. and.8 fo both eefs (with and without the idge), in ageement with the ange of values suggested by Goulay (1996b). Howeve, even fo the same eef pofile (with o without the idge), thee is a systematic incease of K p with inceasing h c, indicating that K p is not a constant as oiginally poposed by Goulay and Collete (5). 5.6 Discussions To epoduce the expeimental esults in Chapte 3 using both appoaches of Tait (197) and Goulay and Collete (5), the empiical paamete is equied. In addition to the eef-cest submegence, laboatoy and field studies (e.g., Raubenheime et al., 1996; Goda, 1) also epoted that depends on seabed slope, i.e., foe-eef slope fo coal eefs, and a constant of.4 to.6 has been found to be moe appopiate fo flat bottoms like eef flats (Nelson, 1994; Goulay, 1994; see also the expeimental esults indicated by Eq. (.)). By assuming wave beaking on the foe-eef and ignoing wave setdown, Tait (197) s model (i.e., Eq. (5.7)) aived at the conclusion that the wave setup was contolled by the diffeence of wave depth between the beaking point and the eef cest (i.e., hb hc), modulated by a facto 1/(8/3 1). It has been shown by Eq. (.) that is alteed by h c. When hc / H, the values of at both beaking location and eef-cest wee about the same, esembling the emeged slopes. Theefoe, the ageement between the pedictions and the obsevations was good as shown in Figs. 5.(a) and 5.(d). The fitted value of in the absence of the idge (see Fig. 5.(a)) gave 1.16, which was close to expeimental obsevations of 1.4 (see Eq. (.)) than the engineeing value of.78. Howeve, as h c inceased, the fitted values of deceased because waves boke futhe shoewad whee the seabed was milde (e.g., on the eef flat). is also modified by H as 15

182 indicated by Eq. (.). Smalle waves may beak futhe shoewad, esulting in a smalle. Consequently at a given h c, thee was consideable mismatch between the pedictions and the obsevations when hc as shown in Fig. 5.. Theefoe, diffeent values of should be used to paameteize wave beaking on the foe-eef slope and the eef flat, espectively, as attempted by othe investigatos (e.g., Goulay, 1996a; Hean, 1999). In Goulay and Collete (5), tansmitted waves on the hoizontal eef flat wee assumed to be contolled by the depth-limited beaking condition (i.e., Eq. (5.5)), and a single value of appopiate to a flat bottom was adopted. Goulay and Collete (5) avoided the detemination of beaking locations by intoducing an additional fitting paamete, K p. It can be shown that the elationship between K p and the paamete, which was used in Tait (197) on the foe-eef (fo K ), is 16 ( h ) Kp fn( hc / H, S, ) 3(4 ) c H S hc (5.1) (See Appendix E fo the deivation of Eq. (5.1)). Since Eq. (5.5) indicates that is elated to incident wave conditions, it is possible that K p, as shown in Eq. (5.1), is independent of incident wave conditions fo a given pofile. Theefoe, the model of Goulay and Collete (5) outpefomed that of Tait (197) in view of its ability to epoduce all the expeimental data using a constant K p at a given h c. Howeve, a dependence of K p on h c is evident in Fig Reef poosity and bottom oughness may also affect the esults given by the models of Tait (197) and Goulay and Collete (5). Goulay and Collete (5) have pointed out that thei shape-dependent coefficient K p is also elated to the poosity and oughness of the eef cest. The elative impotance of bottom fiction in the suf zone should depend on the foe-eef slope. Fo a steep foe-eef with waves beaking at the eef-edge, bottom fiction should be negligible, while fo a mild 153

183 foe-eef, fictional dissipation can be compaable with beaking dissipation (Lowe et al., 5). Moe impotantly, Goulay (1996a) epoted that wave-induced cossshoe cuents on the eef flat caused a eduction in the magnitude of wave setup, an effect that should depend on the geomety of the eef system, in paticula the natue of connections of the eef flat and back-eef lagoon to the open ocean. The geological significance of idges may also be impotant. Blanchon pointed out that wave eflection fom high algal idges adjacent to eef-flat islands is likely esponsible fo acceleated ates of eef-flat pogadation though time. The pogadation in tun would incease eef-flat width which would educe wave setup due to lateal flow away fom the baotopic pessue gadient. That would educe the algal idge height and futhe educe setup, and this pocess would continue until setup appoached open o unconfined eef-flat values. A gadual emegence of the eef flat would be poduced ove geological time, which is a common featue of Indo-Pacific eefs. Howeve, the impacts of lateal flow out of the confined system on wave setup need to be consideed in ode to undestand this pocess. An attempt will be made in section 7.7 whee a DH analytical mode is pesented. 5.7 Concluding Remaks The pefomance of selected existing models is evaluated using the expeimental data in Chapte 3. The offshoe scaling paamete shows that the maximum waveinduced setup on the eef flat has a stonge dependency on offshoe wave enegy flux than wave steepness, but it is geneally insufficient to analyze the expeimental data. The model of Tait (197) can epoduce the expeimental esults with cetain success, especially at elatively smalle eef-cest submegences. The model of Goulay and Collete (5) can almost account fo all expeimental data, but with the use of an additional empiical paamete. The pimay dawback in both models is that both have empiical paametes that vay with eef-cest submegence. Pesonal communication with D. Blanchon P. Maine Geosciences Lab., Reef Systems Unit, Institute of Maine Sciences & Limnology, National Autonomous Univesity of Mexico (UNAM). 154

184 CHAPTER 6 WAVE SETUP OVER FRINGING REEFS UNDER CRITICAL FLOW CONDITION: AN ANALYTICAL MODEL BASED ON MASS BALANCE 6.1 Intoduction It has shown in Chapte 5 that thee is a dependence of the shape-elated paamete ( K p ) on eef-cest submegence. This suggests that the model of Goulay and Collete (5) has limited pedictive values if a constant K p is assigned fo a given eef pofile. Similaly, the model of Tait (197) teated (the beake depth index) as an empiical paamete that depends on both wave chaacteistics and eef-cest submegence. One nomally anticipates that fo a given eef pofile (with o without the idge), an analytical model could adequately epoduce the maximum wave-induced setup on the eef flat with a single empiical paamete elating only to eef mophology. To achieve this, a kinematic model based on mass balance will be consideed in this chapte. A moe sophisticated dynamic model will be intoduced in next chapte. The eviewed models (empiical, numeical o analytical) in Chaptes 4 and 5 did not addess the flow conditions when the eef-cest submegence is vey small o the eef cest is even emeged. Unde such conditions, wave beaking fist occus on the foe-eef and then pumps wate onto the eef flat though ovetopping. Motivated by boad-cested wei hydaulics, an analytical model is pesented in this chapte to descibe the setup induced by both monochomatic and spectal waves with vey small eef-cest submegence. The wei-like scheme oiginally poposed by Goulay (1996a) is adopted and impoved. The pesent model does not diectly conside wave beaking o othe enegy dissipation souces. The est of the chapte is oganized as follows: the analytical model is fomulated based on the mass consevation in section 6.. The model is validated in section 6.3 by the expeimental data fo wave setup in Chapte 3 unde monochomatic waves. 155

185 Model extension to spectal waves is discussed in section 6.4, and then applied to othe published data with diffeent eef configuations in section 6.5. A discussion on the model coefficient is given in section 6.6. Model compaison with that poposed by Goulay (1996a) is conducted in section 6.7. Some conclusions ae dawn in section Theoetical Consideation 6..1 Consevation of mass The deivation of the model stats with the equiement of mass balance. Fom the one-dimensional (1D) continuity equation fo monochomatic wate waves, we have q t x (6.1) whee is the fee suface elevation and the ove-ba means time-aveaging ove a wave peiod and q UD is the mean volumetic flux. D h is the total mean wate depth and the depth-aveaged mean velocity U is defined by U 1 uwdz D (6.) h whee h is the local wate depth and u w is coss-shoe wave obital velocity. Eq. (6.1) indicates that when monochomatic waves inteact with 1DH finging eefs (lateally confined conditions as in a closed wave flume), the mass flux associated with beaking waves causes a piling-up of wate (the incease of ) on the leeside of eef cest (idge), the baatopic pessue associated with this pilingup will foce a cuent seawad. Initially, the wate mass tanspoted into the leeside of the idge is not balanced by an equal etun flow. This mass-flux diffeence allows the mean wate level to continuously ise at the leeside of the idge on the eef flat, until an equilibium level is eached when the etuning flow globally equals the shoewad mass flux. In this chapte, we only conside the equilibium state. 156

186 Fo steady state, Eq. (6.1) educes to q x (6.3) Fo finging eef backed by a coastline, Eq. (6.3) equies that the mean volumetic flux though any vetical plane is zeo within a wave peiod, i.e., q UD (6.4) which means that within one wave peiod, the mass flux of the fowad flow must be balanced by the mass flux of the backwad flow Q F Q (6.5) B whee Q F is the total fowad volumetic flux duing the up-ush phase of the wave beaking pocess ove the eef cest and Q B is the total backwad volumetic flux duing the backwash phase ove the eef cest if we impose Eq. (6.4) on the eef cest.. Fig. 6.1 Backwad flow ove the eef cest which esembles the flow ove a boadcested wei. 6.. Expession fo Q B Fo the backwad flow ove the cest of the idge, it is assumed that the citical flow condition (fee fall) exists, i.e., the wate level ove the eef flat is contolled by the citical wate depth above the cest fo a given flow ate. The flow is in analogy to 157

187 the flow ove a boad-cested wei as shown in Fig. 6.1, whee enegy loss is ignoed. Fo a boad-cested wei, it is well-known that the citical flow depth fo the wate flow above the cest is D c q gb c 1/3 (6.6) (See Munson et al., ) whee q c is the citical volumetic flow ate pe unit width, b is the lateal width of the boad-cested wei (idge), and g is the gavitational acceleation. Assuming no enegy losses, the specific enegy E c measued above the cest of the wei is equal to D U /g, whee U is the depth-aveaged velocity in the wate upsteam of the wei and D is the upsteam (mean) wate level above the cest (see Fig. 6.1). It is assumed that U gd o U /g D, thus in tems of the specific enegy, E c, the citical depth is Dc Ec D (6.7) 3 3 Combining Eq. (6.6) and Eq. (6.7) and eaanging yields qc b g D 3 3/ (6.8) In tems of the still wate depth above the cest, h c, the wate depth upsteam of the idge can be witten as D hc, whee is the maximum wave-induced setup in the wate upsteam. Thus the flow ate q c can be witten as qc b g ( hc ) 3 3/ (6.9) Consequently, the total volumetic flux duing the wave backwash phase (half wave peiod) is estimated as 3/ T QB b g ( hc ) 3 (6.1) 158

188 whee T is the wave peiod Expession fo Q F Suppose that deep-wate waves with wave height H and peiod T appoach the eef, and cause a setup ove the finging eef. In deep wate, the flux of the fowad flow ove one half of wave peiod can be calculated by F T / Q b u dzdt (6.11) Using linea wave theoy and taking wave eflection into consideation gives w whee gh T Q b u dzdt K b (6.1) T / F w (1 ) 4 K is the eflection coefficient of the finging eef. A scaling facto is intoduced to account fo the change of fowad mass flux when waves popagate fom the offshoe to the idge-top whee citical depth occus. The fowad flow at the citical wate depth can be expessed as ght QF QF (1 K) b (6.13) 4 Note that all othe factos that the pesent theoy does not conside explicitly (fo example, the enegy loss due to the shap edges of eef cest, the asymmety of the wave pofile nea beaking point, the unsteadiness of the wei flow, the flux contibution fom the boe, etc.) ae implicitly included in the paamete Expession fo wave setup Afte equating Eq. (6.1) to Eq. (6.13), the following non-dimensional wave setup aives /3 hc 3/ [(1 K ) ] (6.14) /3 1/3 H ( ) S 159

189 whee S H gt / is the offshoe wave steepness. Note that the eflection coefficient K may be elated to the suf-similaity paamete fo plane beaches. Howeve, as discussed in section.4.4, the autho is cuently unable to find an empiical expession fo K, thus the measued K is used in this chapte. 6.3 Model Validation Oveview of expeimental setting The model is validated with the laboatoy expeiments epoted in Chapte 3. The expeiments wee designed to study the wave-induced setup ove two types of eef pofiles subjected to both monochomatic and spectal waves: (1) an idealized finging eef model, which had a elatively steep foe-eef slope (1:6) and a hoizontal platfom eef flat (7 m long); () a finging eef with a idge, which was fomed by placing a ectangula box (55 cm long, 5 cm wide and 5 cm high) on the eef flat with its font face aligned to the eef edge to simulate the idge. Fo monochomatic waves, thee ae thee tested eef-flat submegences ( cm, 5 cm and 1 cm) in the absence of the idge and fou tested idge-top submegences ( cm, 1 cm, cm and 5 cm) in the pesence of the idge, espectively. Fo spectal waves, thee ae thee tested eef-flat submegences ( cm, 5 cm and 1 cm) in the absence of the idge and thee tested idge-top submegences ( cm, cm, 5 cm) in the pesence of the idge, espectively (see also Appendix B fo the oiginal expeimental data). Note that in this chapte, the unified definition of eef-cest submegence (denoted as h c ) efe to both the eef-flat submegence and idgetop submegence. Moe details of the expeimental settings can be found in section Visualization of the wave beaking pocess Fig. 6. shows epesentative snapshots of wave tansfomation and beaking ove the eef cest in the absence/pesence of the idge fo monochomatic incident waves of H.1 m and T 1.5 s with the zeo eef-cest submegence ( hc ). Fou diffeent phases ae displayed, stating fom the moment when the lip 16

190 of the beake hits the wate suface ( t/ T ). Fo the eef without the idge, waves stated to beak on the foe-eef at t/ T. At t/ T 1/4, the boken waves wee ushing up the slope, poducing a stong shoewad tubulence flow on the eef flat. At t/ T 1/, wate on both the foe-eef and eef flat eteated. At t/ T 3/4, stong evese flow could be obseved befoe the next wave beaking. The citical flow could occu on the eef edge as indicated by the aows in the figue. When a idge was pesent at the eef edge, the beaking waves stoke the font side of the idge and then plunged onto the idge-top, esulting in stong shoewad flow ove the idge top. Stating fom t/ T 3/4, fee fall seawads was obsevable at the edge of the idge (also indicated by the aows). Fig. 6. Snapshots of monochomatic wave tansfomation ove eef cest at diffeent phases: (a) without idge; (b) with idge. The aows indicate the occuence of the fee falls Compaison between model pedictions and measuements The wave setup pedicted by the pesent analytical model, i.e., Eq. (6.14), is plotted against the measued ones in Figs. 6.3(a) and 6.3(b) fo the eefs in the absence and pesence of the idge, espectively, unde monochomatic waves. The measued 161

191 eflection coefficients ae used in the calculations. The scaling facto is obtained by a least-squae fitting to the expeimental dataset fo each eef-cest submegence, h c. Fo a clea pesentation of the data, only those datasets with good pediction by the model ae shown in the figues h c = m, =.15 (a) h c = m, =. h c =.1m, =.3 h c =.m, =.38 (b) o (m).3 o (m) p (m) p (m) p Fig. 6.3 Pedicted wave setup ( ) vs. obseved wave setup ( ) fo diffeent eefcest submegences ( h c ) unde monochomatic waves: (a) without idge; (b) with o p idge ( - scaling facto in the model; Solid line - ). o It can be seen fom Figs. 6.3(a) and 6.3(b) that fo both eefs, with and without the idge, the pedicted setup agees well with the measuements when the eef-cest submegence, h c, is elatively small. Good fit is obtained fo the initial dy eef cests ( hc ), which is anticipated since the citical flow is moe likely to occu when eef-cest wate level is small enough to allow a fee fall duing the backwash peiod. Howeve, as h c inceases, the ageement becomes less satisfactoy and some unphysical negative setups ae pedicted, indicating the hydaulic contol at the eef/idge edge imposed by citical depth is no longe valid in those cases (not shown in the figues). Compaisons between Figs. 6.3(a) and 6.3(b) show that the analytical model could povide bette pedictions fo the expeiments with the idge, as long as h c is not too lage. This is expected because the vetical edge of the idge inceases the duation of fee fall duing backwad phase and the sloping foe-eef 16

192 may impose a downsteam effect if the idge is absent. It is also obseved that the value of inceases with inceasing of h c. 6.4 Extension to Spectal Waves Extension of the pesent model to the spectal waves tested in Chapte 3 is staightfowad povided that the epesentative wave height and wave peiod ae selected. H ms and T p fo spectal waves, as discussed in Chapte 5, ae used hee. The esults ae shown in Figs. 6.4(a) and 6.4(b), espectively, fo the eefs without and with the idge. It can be seen again that fo both eefs, the pedicted setups agee vey well with the measuements when hc. As h c futhe inceases, model pedictions beak down due to the lack of citical flow condition at the eef cests (not shown in the figues)..3.5 h c = m, =.16 (a).3.5 h c = m, =.18 (b).. o (m).15 o (m) p (m) p (m) p Fig. 6.4 Pedicted wave setup ( ) vs. obseved wave setup ( ) fo diffeent eefcest submegences ( h c ) unde spectal waves: (a) without idge; (b) with idge ( p o - scaling facto in the model; Solid line - ). o 163

193 6.5 Applications to Othe Published Expeimental Data In this section, the pesent model is applied to thee othe published laboatoy wok, i.e., Seelig (1983), Goulay (1996a) and Demibilek et al. (7), which have diffeent finging eef pofiles and wave conditions. Seelig (1983) s eef pofile consisted of a steep composite foe-eef with an aveage slope of 1:9.6, a small shap eef cest, and a hoizontal eef flat. The eef pofile of Demibilek et al. (7) was identical to that of Seelig (1983), except fo the shap eef cest, which was not pesent in thei model. The foe-eef of Goulay (1996a) s pofile was vey steep (1:1) compaed with that of Seelig (1983). It had a sloping eef flat without the eef cest. Seveal eef-cest submegences wee tested in these expeiments. Expeiments conducted by Seelig (1983) and Goulay (1996a) wee unde monochomatic waves while those by Demibilek et al. (7) wee unde spectal waves. The eflection coefficients in Seelig (1983) s expeiments wee not available, thus K was assumed in the calculations. Fo both expeiments of Goulay (1996a) and Demibilek et al. (7), the measued eflection coefficients wee used. The pedicted wave setups using the pesent theoy, togethe with the measued setups of Seelig (1983), Goulay (1996a) and Demibilek et al. (7), ae shown in Figs. 6.5(a), 6.5(b) and 6.5(c), espectively. Again, only those data with good ageement ae pesented in the figues. Fo Seelig (1983) s data shown in Fig. 6.5(a), the ageement between pediction and measuement is satisfactoy when the eef cest is initially dy ( hc ). Fig. 6.5(b) shows excellent ageement between the pedictions and measuements fo Goulay (1996a) s expeiments unde hc ; good ageement is also found when h c eached.5m. These should be attibuted to the vey steep foe-eef slope in his model, which ensued a citical flow condition fo those cases. Unde the spectal wave condition tested by Demibilek et al. (7), good ageement is again found fo cases with smalle h c (see Fig. 6.5(c)). Compaisons among the thee figues show that the pesence of a steep eef cest may significantly change the value of, thus alteing the flow patten. 164

194 3.5 h c = m, =.1 (a).1.1 h c = m, =.3 h c =.5 m, =.49 (b).8 o (m) 1.5 o (m) p (m) p (m).3.5. h c = m, =.11 h c =.16m, =.3 (c) o (m) p (m) p Fig. 6.5 Pedicted wave setup ( ) vs. obseved wave setup ( ) fo diffeent eefcest submegences ( h c ): (a) Dataset fom Seelig (1983); (b) Dataset fom Goulay (1996a); (c) Dataset fom Demibilek et al. (7) ( - scaling facto in the model; o p Solid line - ). o 6.6 Moe on the Scaling Facto Pevious discussion has shown that in the model inceases with inceasing h c until a cetain value at which the model is no longe valid. This is due to the fact that as the eef-cest submegence becomes lage, seawad wate may affect the fee fall and citical flow condition (backwate effect). In this section, the vaiation 165

195 of with diffeent eef pofiles is investigated. The fitted values of fo each dataset at hc is summaized and shown in Table 6.1; also shown in the table ae the values of foe-eef slope ( s ), which is believed to be elated to both the foeeef configuation and the eef-cest shape. Table 6.1 The values of in pesent model fo diffeent expeimental data with hc Dataset Wave type s R e Chapte 3 data without idge monochomatic 1: Chapte 3 data with idge monochomatic 1:5 b..81 Chapte 3 data without idge spectal 1: Chapte 3 data with idge spectal 1:5 b Seelig (1983) monochomatic 1:9.6 a,c,d.1.73 Goualy (1996a) monochomatic 1: Demibilek et al.(7) spectal 1:9.6 a a Aveaged slope of the composite foe-eef slope; b The equivalent slope as defined in Fig. 5.1 in the pesent of the eef cest (idge) ; c The slope without consideing the pesence of the eef cest; d Reflection is ignoed in the calculations; e R - R-squae. Table 6.1 eveals that the values of ae geneally within the ange of to.5. The pimay eason fo the smalle should be the eduction of fowad mass flux associated with wave popagation fom offshoe to the eef cest. Howeve, it may also be affected by the unsteadiness, the application of linea wave theoy, the lack of enegy dissipation, etc.. Futhe investigation on this issue may be needed. Meanwhile, the values of geneally incease with inceasing foe-eef slope 166

196 within the investigated data ange. Though tial and eo, the elationship between s and is found to be best descibed by the following powe law.45.13s.45 fo c h and 1/9.6 s 1/1 (6.15) with R.94. The uppe limit.45 may coespond to the case whee the foe-eef becomes vetical, esembling a submeged step. A lowe limit may also exist, and it should be elated to the eef pofile whose foe-eef slope is so gentle that the backwate can affect the fee fall. Howeve, this conjectue cannot be validated with the available datasets. The pesent appoach shows that wave setup is dependent on both deep-wate wave steepness and foe-eef slope (though ) when the wave depth ove eef-cest is sufficiently small. Howeve, thee may be a tansition egion of h c in which the theoy is not valid. 6.7 Compaison with the Model of Goulay (1996a) To the best of the autho s knowledge, simila kinematic models in existing liteatues can only be found in Goulay (1996a). The pesent model diffes fom that in Goulay (1996a) mainly in the estimation of the fowad flow, Q F. In Goulay (1996a) s theoy, Q F was computed at the eef edge by assuming a sawtooth pofile fo beaking waves. The wave height was conveted to offshoe by a shoaling coefficient which was appoximately equal to one. Wave eflection was not consideed in the fomulation, thus he aived at a moe complicated expession than Eq. (6.14) as 5/ /3.391 h h h h /3 c c 3 K c K c H K H H H H (6.16) whee the wave-cest asymmety facto, K (anging fom.5 to 1.) was intoduced in his model as a fitting paamete. 167

197 The esults obtained fom the Eq. (6.16) fo all the afoementioned datasets with h ae listed in Table 6.. c Table 6. The values of K in model of Goulay (1996a) fo diffeent expeimental data with hc Dataset Wave type s K R e Chapte 3 data without idge monochomatic 1: Chapte 3 data with idge monochomatic 1:5 b.9.6 Chapte 3 data without idge spectal 1: Chapte 3 data with idge spectal 1:5 b.8.97 Seelig (1983) monochomatic 1:9.6 a,c,d..38 Goualy (1996a) monochomatic 1: Demibilek et al.(7) spectal 1:9.6 a.3.8 a Aveaged slope of the composite foe-eef slope; b The equivalent slope as defined in Fig. 5.1 in the pesent of the eef cest (idge) ; c The slope without consideing the pesence of the eef cest; d Reflection is ignoed in the calculations; e R - R-squae. Compaing Table 6.1 with Table 6., it can be found that oveall the pesent model outpefomed that of Goulay (1996a) in view of lage values of R in Table 6.1 than those in Table 6.. Goulay (1996a) only gave equivalently good pedictions fo both the spectal wave data in Chapte 3 with the idge and his data. The pooe pefomances of Goulay (1996a) s model fo most datasets cannot be explained by the missing wave eflection in Eq. (6.16): the eflection coefficients wee not available fo Seelig (1983) s data but the pesent model still woked bette. Meanwhile, the fitted values of K ae geneally in the ange of -.5 which is 168

198 quite diffeent fom his suggested ange of.5-1. as the wave asymmety facto. The autho believes that the ole of K in his model is equivalent to in the pesent model: both include not only the effects of waves, but also othe factos such as foe-eef slope, flow unsteadiness, enegy dissipation, etc.. Howeve, Table 6. shows that a definite vaiation of K with the foe-eef slope ( s ) as that of the in the pesent model cannot be identified. Finally, it is woth mentioning that the pesent theoy should be suitable fo emeged eef cest whee the citical flow condition is moe likely to be satisfied. In such case, wave ovetopping is esponsible fo the fowad flow, Q F. Thee is a wealth of existing empiical fomulae to estimate ovetopping dischage ove emeged seawalls and beakwates (Owen, 198; Van de Mee and Janssen, 1995; Pullen et al., 7; Goda, 9 and othes), most of them ae exponentially elated to the elative feeboad and additional empiical paametes ae always needed in those equations. Howeve, the available data fo finging eefs ae limited to the submeged o zeo-submeged conditions thus the applicability of those fomulae fo wave ovetopping cannot be examined in this chapte Concluding Remaks A kinematic analytical model based on the mass balance is poposed in this chapte to study the wave setup ove finging eef pofiles that can allow citical conditions (fee-fall conditions) to occu. It is found that wave setup can be descibed by deepwate wave steepness and a scaling facto. The pesent model is successful in epoducing the maximum wave-induced setup on eef flats with and without the pesence of the idge as long as a fee-all condition appoximately exists. The pedictions match equivalently well with measuements unde both monochomatic and spectal wave conditions. Model applications to othe published expeiments also show that the model is capable of pedicting the available laboatoy data povided that the eef-cest submegences ae sufficiently small. The scaling facto in the model is found to incease with both inceasing eef-cest submegence and foe-eef slope. The pesent model outweighs Goulay (1996a) s model in tems of 169

199 its simple analytical fom, bette pedictions and the systematic vaiation of the empiical paamete with foe-eef slope. Futhe undestanding on the scaling facto is needed. 17

200 CHAPTER 7 WAVE SETUP OVER FRINGING REEFS: AN ANALYTICAL MODEL BASED ON MOMENTUM BALANCE 7.1 Intoduction Pevious discussions have shown that fo 1DH expeiments in a closed flume without ciculation, expeimental esults may be intepeted eithe by waveaveaged momentum balance and enegy balance (e.g., the models of Tait (197) and Goulay and Collete (5) in Chapte 5) o by mass balance ove one wave peiod (the hydaulic theoy in Chapte 6). These two methods appoach the same poblem fom diffeent theoetical consideations: the fome is a dynamic theoy which studies the diving foces that maintain the equilibium state, while the latte is in fact a kinematic theoy which examines the equilibium state itself. When a eef cest (idge) is pesent, two main foces exist: (1) the adiation stess that dives the mean flow, and () the foces on the idge that eflect waves and esist the etuning mean flow. Both foces contibute to the wave setup ove the eef flat and it is difficult to diffeentiate thei individual contibutions. It is desiable to have an analytical appoach (at least in 1DH fom) that can be devised to account fo the expeimental esults in Chapte 3 as well as othe published laboatoy data discussed in Chaptes 6. Additional benefits could be obtained if it could be applied to field obsevations with acceptable accuacy. The necessity to develop a DH model has been emphasized in the pevious chaptes in that coal eef flows obseved in the field ae essentially DH. The effects of cuents (wave-diven cuents o/and tidal cuents) togethe with the oughness of the eef suface on wave-induced setup may be significant in the context of a DH eef system. The modification of cuents to the wave setup acoss the eef has been epoted in laboatoy expeiments (Goulay, 1996a) and field obsevations (Lugo-Fenández et al., 4; Hench et al, 8; Lowe et al., 9a). Some 1DH analytical models (Symonds et al., 1995; Hean, 1999; Goulay and Collete, 5) have consideed the effects of cuent as well. Analytical models fo 171

201 DH flows aound eefs ae ae. Lowe et al. (9a) is one of the pionees to popose a simple DH model that used the mass and momentum balances to study the wave setup and ciculation in the coss-eef diection (both ove the eef flat and within the lagoon-channel egion); they included in thei model both bottom fiction and eef mophological featues. Howeve, thee is a ich liteatue in baed beaches (Bellotti, 4) and submeged beakwates (Calabese et al., 8; Zanuttigh et al., 8; Vicinanza et al., 9), fom which some useful ideas can be boowed to study DH eef flow poblems. Reef-flat sufaces ae often vey ough compaed to sandy beaches because of the pesence of canopy-foming benthic oganisms and the iegula mophology of the platfoms on which they gow. Field measuements of eef-flat suface oughness showed that coal eefs have vey high dag coefficients, being some odes of magnitude highe than those associated with nomal sandy shelves (Falte et al., 4; Lowe et al., 5; Hench et al., 8). The bottom fiction can play an impotant o even dominant ole in the oveall wave enegy dissipation on the mild slope foe-eefs (e.g., Lowe et al., 5), which could significantly educe the wave setup (e.g., Longuet-Higgins, 5; Dean and Bende, 6; Apotsos et al., 7). Howeve, fo 1DH finging eef models with a vey ough eef flat, the influence of fictional dissipation in wate ove the eef flat can be ignoed as shown by expeimental wok epoted in Appendix C. Motivated by the wok in pevious chaptes, an analytical model based on 1DH coss-shoe momentum balance is deived in this chapte and validated by both laboatoy and field data. The model is extended to DH in analogy to ip cuent systems aound baies o beakwates. The est of the chapte is aanged as follows: the 1DH analytical model is fomulated in section 7.. The model is validated in section 7.3 by the expeimental data pesented in Chapte 3 and othe published expeimental data. An analysis of model sensitivity is conducted in section 7.4. Model applications to some published field data ae pesented in section 7.5. Some discussions on the model ae given in section 7.6. An extension to DH poblem is pefomed in section 7.7 and majo conclusions ae dawn in section

202 7. Theoetical Consideation 7..1 Govening equations The coss-shoe depth-integated momentum equation, i.e., Eq. (1.3), is applied in the suf zone fo a geneal finging eef pofile (a seawad foe-eef followed by a eef flat with/without a eef cest (idge) being located at the eef edge). The eef cest must be submeged ( hc ) o at least initially dy ( hc ) (see Fig. 7.1 fo an illustation). Assuming alongshoe unifom waves and bathymety, we have b 1 Sxx [ U ( h)] x g( h) x x x (7.1) o ( ) b U h g ( h ) 1 S xx x h gh ( ) x x x x (7.) See section fo the definitions of the symbols in the equations above. Some assumptions need to be made in ode to futhe simplify Eq. (7.). The fist assumption is that waves beak on a foe-eef location ( x b ), but the suf zone pocess might cease beyond the eef cest ( x c ). This is consistent with ou laboatoy obsevations whee the inne suf zone might occupy a egion shoewad of the eef cest (see e.g., Fig. 3.4). Integating Eq. (7.) fom the beaking point (the stat of the suf zone), x= x b, to the end of the suf zone, x= x s, gives g g h U hu h h S (7.3) ( s ) s b b ( s ) b whee h b is the beake wave height, h s is the wate depth at the end of the suf zone, U b and U s ae the depth-aveaged mean velocities at the beaking point and s b the end of suf zone, espectively. S ( S S )/ is elated to the wave xx xx 173

203 adiation stess, with S b xx and s S xx being the x components of the adiation stess x s tenso at x b and x s, espectively; b / dx is the fictional foce in the suf xs xb zone. g( h )( dh / dx) dx can be viewed as a eaction of the bottom to the xb wate above (Bellotti, 4). Fig. 7.1 Configuation of a finging eef with a idge and some notations adopted in this chapte. 7.. Appoximation fo 1D finging eefs Fo a closed flume without ciculation, the wave-diven cuent tems in Eq. (7.3) ae appoximately zeo. Fictional foce tem can be ignoed in view that the laboatoy eef pofiles ae elatively smooth and the mean cuent is vey weak as measued in section 3.5. The effects of those tems on wave setup will be discussed late. Afte neglecting the mean cuent and bottom fiction, Eq. (7.3) educes to g (h s ) g h S (7.4) b whee the wave-induced setdown at beaking point ( b ) has been neglected since / h 1(see Eq. (7.8) below). b b 174

204 Fo laboatoy expeiments, the following assumptions can be justified when computing the adiation stesses: (1) simple pogessive waves ae assumed shoewad of the suf zone; () geneation of both high and low fequency waves can be ignoed; (3) a patial standing wave patten (see Fig. 3.1 and Fig. 4.11) exists befoe waves beak. The appoximate expession fo b S xx based on the linea shallow wave appoximation, neglecting the shift of the phase constant between incident and eflected waves (Calabese et al., 8), is b b 3 3 Sxx Sxx (1 K ) ghb(1 K ) g1 hb(1 K ) (7.5) whee S 3 gh /16 is the x-component of the adiation stess tenso at the b xx b beakpoint without consideing wave eflection. K is the eflection coefficient, which needs to be obtained fom measuements since no empiical fomula is available fo finging eefs. The depth-limited beaking condition ( Hb 1hb) is used in Eq. (7.5) to estimate H b, with 1 being the beake depth index fo the foe-eef. Assuming nomal incident waves, the beake wate depth, h b, can be deived fom a simple 1D enegy balance afte neglecting the fictional dissipation (see Appendix E) h b H (1 K ).4 (7.6).4. (4 1 ) S whee S H gt / is the offshoe (deep-wate) wave steepness. Theefoe, b h is a function of both incident wave condition and elative eef-flat submegence (implicitly included in 1 as indicated by Eq. (5.5)). On the eef flat, s S xx is evaluated by s b 3 3 Sxx Sxx Kt ght g ( hs ) (7.7) whee Ht HbKt is the tansmitted wave height and K t is the tansmission coefficient. Unlike submeged beakwates, thee is no well-established empiical 175

205 fomula to estimate K t fo eefs, thus a simila depth-limited beaking condition H ( ) h is employed in the wate ove the eef flat, whee the value of t s is diffeent fom that of 1 since the beake depth index fo a hoizontal bottom is smalle than that fo a slope (see section 5.6). The wave-induced setdown ( b ) can be estimated by (see Longuet-Higgins and Stewat, 1964) Hb 1 hb b (7.8) 16h 16 This setdown is appoximately % of, thus will be kept when appoximating the vaiation of in the suf zone. b xs In pinciple, the integal of g( h )( dh / dx) dx in Eq. (7.4) can be xb computed numeically if detailed infomation in the suf zone ( x b, x s, hx ( ) ) is known. Subsequently, the wave-induced setup can be computed based on Eqs. (7.5) to (7.8). Howeve, in pactice, paticulaly in field conditions, detailed measuements of sufzone aea and the seabed topogaphy undeneath ae nontivial. Thee is also no well-accepted fomula to pedict x s so fa, thus a futhe appoximation of this tem is necessay fo some special cases. (1) Idealized pofile in the absence of a idge Fo an idealized finging eef pofile without a idge (a seawad plana foe-eef followed by a hoizontal eef flat, whee the eef edge coincides with the eef cest, i.e., xe xc and hs h hc), Fig. (3.1) shows that MWL inceased apidly ight afte the beaking point and became asymptotic to a moe o less constant futhe shoewad. Theefoe diffeent expessions (the polynomial and the powe law) that ae possible to appoximate such vaiation of fom x b to x s ae investigated in this study (Note that Bellotti, 4 used a linea vaiation of ove a baie). 176

206 Detailed deivation is given in Appendix F, whee a unified geneal expession fo has been given in the following fom (See Eq. (F.4)) g h h b g b h h b1 g h h b (7.9) whee linea polynomial 1 quadatic polynomial cubic polynomial quatic polynomial 5 1 C C C,3, N powe law 1 C (7.1) and is a dimensionless paamete defined by x x L x x L e b e s b s (7.11) Comments on Eq. (7.1) will be given in section Physically, is the atio of the distance between the beaking point and the eef edge ( L e ) to the total suf zone width ( L s ). Since Le Ls, must be within the ange of to 1 ( indicates that wave beaking occus on the eef edge and 1 educes the pesent model e b appoximately to Eq. (5.8)). Fo a given eef pofile, L ( h h ) / s ( s is foeeef slope) is a function of both wave condition and eef-flat submegence. In section.3.3, it has also been shown that L s depends both on the incident waves and eef-flat submegence. Based on all deivations above, substituting Eqs. (7.5), (7.7) and (7.9) into Eq. (7.4) leads to the following second-ode algebaic equation fo 177

207 b c (7.1) which has a unique positive solution b b 4c (7.13) with bh c 16( hb h ) (8 3 ) (7.14) 3 h 3 h (1 K ) 16 ( h h )(1 ) c 1 b b b 8 3 ( c ) (7.15) whee the paamete is elated to by Eq. (7.1); h b and b ae given by Eqs. (7.6) and (7.8), espectively; K needs to be obtained fom measuements since no well-established empiical expession fo K is available; the model paametes 1, and will be discussed late. The equiement on c is satisfied when h h if both wave eflection and setdown ae ignoed. When waves popagate b ove the eef cest without beaking, the pesent appoach is no longe valid (the setup is always zeo). () Idealized pofile in the pesence of an idealized idge Fo an idealized idge pofile such as the ectangula one tested in Chapte 3 (shown in Fig. 5.1), the idge is low-cested with a wide hoizontal idge-top, thus fo most cases, beaking actually ceases on the idge-top, i.e., h c h s. Meanwhile, both seaside and leeside slopes ae vetical, showing the singula points in dh / dx, thus the expession fo (see Eq. (G.7) in Appendix G) will be slightly diffeent fom Eq. (7.9). Detailed deivations fo this case ae given in Appendix G based on the powe law appoximation of in suf zone (the polynomial appoximations can also be deived in a simila way but the expessions ae moe complicated and thus ae not included in this thesis). Fo finging eefs with an idealized idge, the maximum setup on the idge-top can be calculated by Eq. (7.13) with 178

208 bh c C 1 16( hb h) 16 ( h hc) C (8 3 ) (7.16) C 1 3 h 3 h 1K 16 h h h h C c 8 3 c 1 b b b b c (7.17) 1/ C whee C /(1 C) is fom Eq. (7.1) fo powe law appoximation and defined by Eq. (7.11). is (3) Idealized pofile in the pesence of a geneal idge (extension of the theoy) When thee is a idge with abitay shape on a eef, it is anticipated that the above appoximation, i.e., Eq. (7.9), fo in the absence of the idge can still be used, povided that a suitable epesentative h can be selected since wave beaking may cease at diffeent locations fo diffeent waves and geometies. The wate depth h c has been poven to be the key paamete in contolling the eef-flat hydodynamics thoughout the pevious chaptes (Chaptes, 3, 5 and 6) and othe studies (e.g., Goulay, 1996b; Blenkinsopp, and Chaplin, 8), and h c is also easy to be measued, thus it will be used fo h in Eq. (7.9). Consequently, the value of depends on the choice of h c and the physical intepetation of could be diffeent fom what we undestand fo the idealized eefs with/without the ectangula idge, i.e., defined by Eq. (7.11) Uppe and lowe limits of the model validity Fo a given wate depth ove the eef cest, incident waves of sufficiently small amplitude can popagate onto the eef cest without beaking, hence geneating no setup. Unde such cicumstances, the theoy outlined above (Eqs. (7.3) - (7.17)) will no longe be valid. Theefoe, thee must be an uppe limit fo h c / H above which wave beaking ceases fo a given eef pofile. This tansition can be estimated by 179

209 assuming that the maximum allowable local wave height at the eef cest ( H ) is contolled by the local wate depth, i.e., Ht hc and that the enegy dissipation due to bottom fiction can be ignoed. Consevation of the coss-shoe enegy flux between a station offshoe and a station at the eef cest (using linea deep and shallow-wate wave appoximations at these two locations) gives t gt gh(1 K ) gh ghc ghc ghc (7.18) Reaanging Eq. (7.18) gives an expession fo the elative eef-cest submegence, hc / H, as h H 1 K h c c 4 gt 1/4 (7.19) Note that the maximum value of h c / H is independent of the offshoe wave height. Fo example, fo the laboatoy conditions in Chapte 3 with monochomatic waves, if one uses.4 as ecommended by Goulay (1996a) and since the maximum value of h c is.1 m and the minimum value of T is.83 s in the expeiments, one can aive hc / H. though Eq. (7.19) (afte ignoing K ), which could epesent the uppe bound fo any wave beaking in the expeiments. To futhe validate Eq. (7.19), a limited seies of laboatoy tests wee caied out to identify the bounday between non-beaking and beaking waves fo diffeent offshoe monochomatic waves. These obsevations along with the theoetical cuves ae shown in Fig. 7. fo h.5 m and.1 m in the absence of the idge c and fo h.5 m in the pesence of the idge. Fou wave peiods ( T ) wee c investigated. Fo each pai of wate depth and wave peiod, the citical condition fo which waves began to beak was detemined by vaying the incident wave height by an inteval of 1mm, the coesponding deep-wate wave heights ( H ) wee then obtained by conveting the measued incident wave heights to deep-wate ones. Using this set of data, Eq. (7.19) can be used to estimate via a least-squae fitting. 18

210 As seen in Fig. 7., vitually all of the citical beaking cases lie within the 95% confidence limits of the theoetical cuves, indicating that Eq. (7.19) is adequate to estimate the tansition fom non-beaking waves to beaking waves. The aveage value of fo these thee sets of conditions is about.4, which is consistent with the value suggested by Goulay (1996a)..8.7 (a) non-beaking wave beaking wave.8.7 (b) non-beaking wave beaking wave.6.5 Equation (7.19): h c =.1m, = H (m).4 H (m).4.3. Equation (7.19): h c =.5m, = Equation (7.19): h c =.5m, = T(s) T(s) Fig. 7. Compaison of beaking state tansition of laboatoy obsevations with theoetical pedictions fo diffeent eef-cest submegence: (a) without idge; (b) with idge. Solid lines epesent least-squaes fits of Eq. (7.19); Dotted lines epesent the 95% confidence limits of the best fitting line. As afoementioned, a equiement to use the above model is that the eef cest is submeged, i.e., ( hc ), thus the lowe limit of the poposed model is hc / H in tems of the elative eef-cest submegence. In eality, the eef cests may occasionally emege duing low tides, thus negative eef-cest submegence ( hc ) can occu; unde such conditions, wave ovetopping and un-up may be impotant which ae beyond the scope of this thesis. Howeve, the analytical model in Chapte 6 does show some potential to deal with such situations. 181

211 7..4 Estimation of the beake depth indices In Eq. (7.13), thee ae thee paametes in the poposed analytical solution:, 1 and. The appoach to estimate will be discussed late. Fo on the eef flat, a constant value was found suitable fom the expeiments in section.4. as well as othe laboatoy obsevations (Nelson, 1994; Goulay, 1994), thus.4 veified in the pevious section, will be adopted heein. The value of 1 is moe ticky to choose, since it is elated to both wave conditions and seabed slope. It was found that the eef-flat submegence affected the values of 1 (see both sections.4. and 5.6). Fo given offshoe waves, 1 should appoach the value fo an emeged plane slope (denoted by m ) unde hc / H (zeo submegence); wheeas fo lage enough eef-cest submegence whee waves ae about to cease beaking, 1 equals to when c c, as h / H (1 K ) (4 ) g T h (see Eq. (7.19)). Thus by assuming a linea vaiation of 1 with h / H, one aives h (4 ) h ( ).5.5 c c 1 m m H (1 K ) g T hc when (1 K ) (4 ) g T h H c (7.) Existing empiical fomulae to estimate m ae commonly available fo slopes smalle than 1/1 (see section.4.). Some foe-eef slopes ae elatively steep (> 1/1), thus in the cuent model, to be consistent with the laboatoy obsevations in Chapte, the value of m 1. (see Eq. (.)) will be used as a fist appoximation in the 1DH modeling in this chapte. A model sensitivity analysis will be pesented late in this chapte. 18

212 7.3 Model Validation by Expeimental Data Classification of the investigated eef pofiles The poposed model is tested against the laboatoy datasets investigated in Chapte 6, the eef pofiles used in pevious laboatoy studies have been descibed in Chapte 6 and ae illustated in Fig. 7.3 fo efeence in this chapte. Fig. 7.3 Laboatoy eef pofiles investigated in this chapte: (a) expeimental setup in Chapte 3 without idge; (b) expeimental setup in Chapte 3 with idge; (c) Seelig (1983); (d) Goualy (1996a); (e) Demibilek et al. (7) ( h c - eef-cest submegence; s - foe-eef slope; s e - equivalent foe-eef slope; s a - aveage foeeef slope; Hi - incident wave height; SWL - still wate level). The autho has listed thee categoies of eef pofiles in section 7.. fo which the wave setups ae computed by diffeent equations. To facilitate discussion in this section, the classification of each pofile in Fig. 7.3 will be given and the physical meaning of its will be examined. Theoetically, the values of lie in the ange of to 1. Fo eefs shown in Figs. 7.3(a), 7.3(d) and 7.3(e), a common featue is that the bottom topogaphy nea the eef edge (whee the sufzone pocess occus) is idealized (a plana slope followed by a hoizontal eef flat), thus the physical meaning of fo those pofiles is the atio of the two length scales ( L e and L s ) in the suf zone as indicated by Eq. (7.11). They fall into the categoy of idealized 183

213 pofile without the idge and the wave setup will be computed by Eq. (7.13) with Eqs. (7.14) and (7.15). Fo the eef in Chapte 3 when the idge is pesent (Fig. 7.3(b)), the idge-top is hoizontal and elatively wide (5 cm) so that wave beaking actually completes (see e.g., Fig. 3.1) on the idge-top whee the wate depth is known, esembling the sufzone pocess on the hoizontal eef flat without a idge. Howeve, the seaside vetical slope of the idge has some impact on the model esults (See Appendix G), thus it belongs to the categoy of idealized pofile with an idealized idge and the wave setup will be computed by Eq. (7.13) with Eqs. (7.16) and (7.17). in both Eqs. (7.16) and (7.17) again epesents the atio of L / L. Fo the eef in Seelig (1983) (see Fig. 7.3(c)), the effect of pofile shape e s on the value of may be substantial because his eef cest (idge) was naow and steep, thus it lies in the categoy of idealized pofile with a geneal idge and the wave setup will be computed by Eq. (7.13) with Eqs. (7.14) and (7.15), but the eefcest submegence ( h c ) will be used instead of h as the epesentative wate depth in Eqs. (7.14) and (7.15) The measued fo an idealized eef without idge To estimate the magnitude of, the data in Chapte wee investigated. To calculate, L s was obtained fom measuements (See Appendix A), and L e could be estimated by Le ( hb h) / s ( h b is given by Eq. (7.6) with Eq. (7.) and s is the foe-eef slope). The values of as a function of deep-wate wave height ( H ) ae pesented in Figs. 7.4(a) and 7.4(b) fo diffeent eef-cest submegences ( h ) and foe-eef slopes ( s ), espectively. Fig. 7.4(a) shows that at a given s, the values of incease with deceasing h while Fig. 7.4(b) shows that at a given h, they incease with deceasing s ; both figues indicate a monotonic incease of with inceasing H. Oveall, the measued values of ae within a elatively naow ange ( to.5) compaed to its theoetical ange ( to 1). Fo pactical use of the theoy, it is undesiable if the 184

214 theoy equies measued values of to pedict wave setup. It will be shown that the pedicted is not vey sensitive to small vaiations of. In view of the uncetainties in esulting fom extending the theoy to non-idealized eef pofiles (discussed in section 7..) o fom the uncetainties in the measued o obseved x s in both laboatoy and field data to be investigated in this chapte, a constant epesentative fo a given eef pofile will be used in the model, the validity of this simplification will be shown in the following sections s=1/6 h c =.3m h c =.5m h c =.7m h c =.1m (a) s= 1/3 s= 1/6 s= 1/9 s= 1/1 h =.5m (b) H H Fig. 7.4 The measued as a function of deep-wate wave height ( H ) fo: (a) diffeent eef-cest submegences ( h c ) and (b) diffeent foe-eef slopes ( s ) Compaisons among diffeent appoximations to the vaiation of in the suf zone Diffeent appoximations to the vaiation of in the suf zone (given in Eq. (7.1)) ae compaed in this section using the measued wave setups in Chapte 3 fo the cases in the absence of the idge (both monochomatic and spectal waves ae included and the H ms and T p ae used to chaacteize the spectal waves), since the wave measuements in Chapte did not focus on the maximum setup acoss the eef flat. A single value.44, obtained by aveaging the dataset in Chapte (shown in Fig. 7.4(a)) fo the idealized eef with s 1/6 (simila to the 185

215 expeimental settings in Chapte 3 without the idge), was used as a efeence value. The epesentative fo each appoximation is obtained by a best fit between the measued setups in Chapte 3 and the model pedictions based on Eq. (7.13) with Eqs. (7.14) and (7.15). Diffeent values of C (up to C 8 ) wee tested fo the powe law. All the fitted values of togethe with those fom polynomial vaiations (fom linea to Quatic) ae compaed in Table 7.1. The pefomances of the poposed model ae measued by two paametes (also listed in the table): (1) R-squae ( R ), and () the Bias (the mean of the diffeence between measued and pedicted values of wave setup, which is to measue the tendency of undepedicting (positive bias) o ove-pedicting (negative bias) against the expeimental data). A model will have the best pefomance if it is pactically unbiased and has R nea 1. Table 7.1 shows that vey good ageement ( R.98 and Bias= ) between obsevations and pedictions was obtained fo all appoximations to, indicating that teating as a constant fo a given eef pofile is easonable. All appoximations give the same R and bias because they shae the same expession (i.e., Eq. (7.9)), in which is the sole paamete, but thei coesponding values of ae diffeent because of Eq. (7.1). Oveall, the fitted.54 accoding to the powe law with C 7 has the best ageement with the measued ; thus heeafte, all the esults fo the idealized finging eefs without the idge ae 1/7 pesented based on 7 /8, which is to be used in Eq. (7.9). Note that using othe appoximations listed in the table can also povide the same quality of pediction but may esult in a non-physical (e.g., geate than 1) fo the given eef pofile. 186

216 Table 7.1 Model paamete fo diffeent appoximations to the vaiation of in suf zone a Polynomial Powe law Linea 1.15 C=.75 Quadatic.78 C=4.7 Cubic 1.46 C=6.9 Quatic.94 C=7.54 b C=8.31 a Tests ae pefomed against the dataset in Chapte 3 without the idge; b This is the best ageement with the laboatoy obsevation. Fo the measued setups in Chapte 3 in the pesence of an idealized ectangula idge, Figs. 3.1, 4.1 and 4.11 have shown that when the idge was pesent, wave setups inceased moe apidly in the suf zone and the sufzone width, L s, became naowe, indicating that the vaiation of in the suf zone depends on the geomety of the seabed pofile and the value of fo the eef with the idge is expected to be lage than that without. To achieve this, a smalle C fo the powe law appoximations is expected accoding to Table 7.1. It was found that a best fit between the model esults (using Eq. (7.13) with Eqs. (7.16) and (7.17) as well as the powe law paamete C /4, i.e., 4 /5) and the expeimental data gave, which is easonable and will be applied fo the cases whee a ectangula idge exists at the eef edge Compaison between expeimental data and model pedictions Pedicted and measued wave setups ae compaed in Figs. 7.5(a) and 7.5(b) fo the wave setup expeiments in Chapte 3 in the absence and pesence of the idge, espectively. The pedicted setups wee computed based on Eq. (7.13) with Eqs. (7.14) and (7.15) using the powe law appoximation ( C 7 ) fo the eef pofile without the idge, and on Eq. (7.13) with Eqs. (7.16) and (7.17) using the powe law 187

217 appoximation ( C 4 ) fo the eef pofile with the idge. Fo the othe two model paametes, m 1. and.4 wee adopted fo both pofiles. Fo the eef without the idge as shown in Fig. 7.5(a), vey good ageement is found, with R.98 (see Table 7.). The unde-pedictions fo those cases with vey small setups ae due pimaily to the measuement eo fo smalle waves. Fo the eef with the idge as shown in Fig. 7.5(b), although thee is a consistent slight ove-pediction fo the cases unde spectal waves, the oveall ageement is still satisfactoy ( R.94, see Table 7.). Note that H ms and T p have been used to chaacteize the spectal waves in the simulations. Fig. 7.5 Pedicted wave setup ( ) vs. obseved wave setup ( ) fo diffeent eefcest submegences ( p h c ): (a) without idge; (b) with idge (Red makes - o p monochomatic waves; Blue makes - spectal waves; Solid line - ). o Model applications to two othe published monochomatic wave datasets (Seelig, 1983, Goulay, 1996a) and one spectal wave dataset (Demibilek et al., 7) ae shown in Figs 7.6(a), 7.6(b) and 7.6(c), all model esults wee computed based on Eq. (7.13) with Eqs. (7.14) and (7.15) as well as the powe law appoximation ( C 7 ) with m 1. and.4. The values of fo these datasets will be discussed in the next section. Vey good pedictions can be obseved fo the two 188

218 monochomatic wave datasets ( R.95, see Table 7.). Slightly pooe ageement was found fo the spectal wave dataset ( R.91), the ageement can be impoved if m is educed..5 (a).1 (b).8 o (m) o (m) h c = m. h c = m h c =.5m h c = m p (m) p (m) h c =.1m.3.5 (c). o (m).15.1 h c = m h c =.16m.5 h c =.31m h c =.51m p (m) p Fig. 7.6 Pedicted wave setup ( ) vs. obseved wave setup ( ) fo diffeent eefcest submegences ( h c ): (a) Dataset fom Seelig (1983); (b) Dataset fom Goulay p o (1996a); (c) Dataset fom Demibilek et al. (7) (Solid line - ). o 189

219 7.3.5 Moe on the model paamete The values of the model paamete fo all above laboatoy datasets, obtained by matching obsevations with model pedictions using the powe law appoximation of, ae summaized in Table 7.. Table 7. Model paamete fo available data Dataset Wave type Foe-eef slope (s) Powe law Bias(m) R Chapte 3 data monochomatic without idge and spectal 1:6 C= Chapte 3 data monochomatic with idge and spectal 1:5 b C= Seelig (1983) monochomatic 1:9.6 a,c,d C= Goualy (1996a) monochomatic 1:1 C= Demibilek et al. spectal 1:9.6 a C= (7) a Aveaged slope ( s a in Fig. 7.3) of the composite foe-eef; b The equivalent slope ( s e ) as indicated in Fig. 7.3 in the pesent of the eef cest (idge) ; c The slope without consideing the pesence of the eef cest; d Reflection is neglected in the calculations. Table 7. eveals that geneally, fo most datasets, the values of ae elatively small, close to the physical lowe limit of. Fo those datasets in the absence of a idge, a systematic decease of with inceasing foe-eef slope ( s ) among the investigated datasets can be identified, and this decease is moe evident fo vey steep slopes, in that a steepe slope moves the beaking point shoewad, educing L e and thus. This tend is consistent with the vaiation of the measued fo diffeent values of s as indicated in Fig. 7.4(b). Fo the datasets in the pesence of a idge, simila to the data in Chapte 3, the value of fo Seelig (1983) s data is also elatively lage, which was supposed to be due to the shape effect of his steep 19

220 idge as discussed in section and Appendix G. Howeve, oveall the obustness of the model to each dataset using just a single model paamete ( ) can be validated. The eason fo this will be investigated by the model sensitivity analysis (see section 7.4). Since Table 7. shows that the model is nealy unbiased (bias close to zeo) and has values of R nea one, it can be applied fo pactical use if the chaacteistics of waves and the eef geomety ae known and the eef cest is submeged ( hc ). 7.4 Model Sensitivity Analysis The sensitivity analysis in this section is caied out in two steps. The fist step was to conside the sensitivity of the model to paametes that can be measued, and thus may be used as input paametes such as the eflection coefficient, wee e-conducted by setting K. Model uns K to zeo fo all datasets except fo Seelig (1983), and the esults ae given in Table 7.3. Compaing Table 7.3 with Table 7., which has been obtained with the measued in view of the model pefomances (see the Bias and the effect of K, no significant diffeence can be obseved R in both tables). Howeve, K on the empiical paamete is evident: Table 7.3 shows that the values of wee slightly educed when wave eflection was excluded fom the model, paticulaly fo the data in Chapte 3 with the idge whee the values of can be as lage as.55 fo monochomatic waves and.66 fo spectal waves, was educed by about 1% (fom.93 to.84). Theefoe, due to the lack of eliable equations fo calculating K K at pesent, neglecting wave eflection in the model may be possible in that its consequence can be compensated by diffeent value of. The second step of the analysis is to check the sensitivity of the model to othe thee paametes (, m and ) that have been calibated o empiically detemined. Fig. 7.7 shows the aveage of all the wave setups ( ave ) pedicted by the poposed 191

221 model divided by the aveage of all the wave setups ( ave ) obtained with efeence paametes (.54 as given in Table 7. fo the best fit, 1. and.4 as suggested in section 7..4) fo the data in Chapte 3 without the idge (see Fig. 7.5a). Each model un was caied out by changing only the value of the selected paamete. The paametes ae vaied with the following constaints: -.3, based on the measuements shown in Fig. 7.4; - m based on a numbe of expeimental and field data fo plane beaches (e.g., Camenen and Lason, 7; Goda, 1); -..7 based on laboatoy o field obsevations fo hoizontal eef flat (i.e., Goulay, 1994; Nelson, 1994; Lowe et al., 9a; the esults in sections.4. and 7..3). m Table 7.3 Model paamete fo available data by excluding wave eflection Dataset Wave type Powe law Bias(m) R Chapte 3 data monochomatic and without idge spectal C= Chapte 3 data monochomatic and with idge spectal C= Goualy (1996a) monochomatic C= Demibilek et al.(7) spectal C= It can be obseved that the model is vey sensitive to m and the pedicted setups vay almost linealy with m ; howeve, the model esults ae less sensitive to : ave ave vaies moe o less within ±% of fo the tested ange of. Most impotantly, the model is the least sensitive to. Fo example, when ises to 5 times of o educes to 1/5 of its efeence value, ave deceases o inceases by only about 1%. This should be why can be teated as a constant in the model as in 19

222 the pevious sections. Fig. 7.7 also indicates that both a decease of and an incease of m will esult in an ove-pediction of wave setup, and vice vesa. Thus even if the pevious selection of m 1. might not be exact fo a specific finging eef, its consequence on the pedictions could be compensated by adjusting ; howeve, the adjusted value of may no longe eflect its eal physical meaning. Note that the ability to change the pedicted wave setup by adjusting o m does not mean that m can be used as the sole fitting paamete instead of in the pesent model. Peliminay tests evealed that adjusting m alone could not yield a global satisfactoy pediction if was not popely chosen. ave ave / m X/X Fig. 7.7 Sensitivity of the model esults to the efeence paametes. On the abscissa, the value of each paamete X is divided by its calibated value X. On the odinate, the aveage wave setups ( ave ) ae nomalized by thei coesponding ave aveage value unde efeence conditions ( ) ( X epesents, m o ; X ave epesents.54, 1. o.4 ; is obtained by aveaging the data in Fig. 7.5a). m 193

223 7.5 Applications to Field Data Backgound In contay to the 1DH flow in a closed laboatoy flume, the flow pattens at natual eef sites ae essentially DH: cuents may be diven by diffeent focing mechanisms such as waves, winds, tides o even buoyancy (Monismith, 7). As fo waves, the wave-induced setup ceates a mean pessue gadient ove the eef flat and thoughout the lagoon, which can dive a cuent that eventually exits the lagoon via channels in the eef (Hean, 1999). This kind of eef-lagoon-channel system is often temed as baie eef. Meanwhile, many natual finging eefs, which fom adjacent to coastal landmasses, have enclosed lagoons in which wate exchange with the suounding ocean is often esticted by fiction though naow, ough, and sometimes totuous gaps, thus a weak coss-shoe cuent and DH ciculation may still exist. The eduction of wave setup due to the pesence of such cuents has been epoted in the liteatue (e.g., Goulay, 1996a). Moeove, the alongshoe cuents on the eef flat also esult in an alongshoe distibution of the wave setup (e.g., Jago et al., 4). The oveall dynamics contolling wave-diven flows on coastally bounded eefs may be simila to those, fo example, ip cuents that ae fomed aound submeged bas on sandy beaches o aound detached submeged low-cested beakwates. In this case, spatially non-unifom wave setup can dive a flow ove a shallow ba/beakwate cest that etuns to the ocean though naow channels. Howeve, natual eefs may have much wide eef-flat, lagoon and channel geometies than those of baed beach o beakwate systems. The bottom fictional foce in the pesence of waves and cuents is usually paameteized by a quadatic fiction law Cu u (7.1) b x d ef ef whee C d is a epesentative dag coefficient, u ef is a efeence velocity. Fo diffeent selections of u ef, C d has diffeent values. Note that in wave-dominated 194

224 coastal egions (i.e., uw uef, whee u w is the magnitude of wave obital velocity), a linea dag law may be moe appopiate (Hean et al., 1). Finally, it is wothy to mention that although the 1DH simulations based on linea wave theoy could povide satisfactoy pedictions fo the laboatoy data, an altenative wave model may be needed fo field conditions to ecast coss-shoe vaiation of S xx in Eq. (7.1) whee the fiction dispassion may be impotant. A numbe of 1D wave models have been developed to pedict wave tansfomation on beaches (e.g., Thonton and Guza, 1983; Svendsen, 1984a); these models ae moe appopiate fo eefs with elatively mild foe-eef slopes (e.g., less than 1:1). Howeve, it is not the focus of the pesent study to conduct a detailed evaluation of such wave models Results In this section, fou ecent field studies on the wave setup ove coal eefs coveing a vaiety of eef pofiles and wave conditions ae examined. Noting that aside fom waves, the mean wate level may also be affected by tides and winds at a specific site. By consideing that wind effects on the mean wate depth ae epoted to be negligible fo most datasets, the eef-cest submegence ( h c ) in the pesent model epesents the mean wate depth plus the tide modulation in field conditions. The details fo each dataset elated to model application ae listed below. Bonneton et al. (7) The measuements wee conducted at a baie eef, New Caledonia. The eef pofile consisted of a foe-eef with an aveage slope of 1:1, a elatively hoizontal eef flat and a lagoon. The offshoe wave conditions ( H ms, T p ) wee conveted fom thei measuements ( H s, T, which is the mean zeo-upcossing peiod) at thei station A on the foe-eef; the eef-cest submegence ( h c ) and wave setup ( ) wee obtained fom thei station P 1 on the eef flat. 195

225 Hench et al. (8) The measuements wee conducted at a baie eef, Paopao Bay, Mooea. The eef pofile consisted of a foe-eef with an aveage slope of 1:8, a eef cest (idge), a hoizontal eef flat and a lagoon, The values of H ms and T p wee conveted fom thei measued H s and T s at the foe-eef station B, while h c and wee obtained fom the eef-cest station C. The obseved tides at Mooea wee emakably small (on the ode of.m at sping tide), thus tide effect was ignoed and a constant h m (the mean wate depth at the station C ) was applied in the model. c Lowe et al. (9a) The measuements wee conducted at a baie eef in Kaneohe Bay, Hawaii. The eef pofile consisted of a foe-eef with an aveage slope of 1:6, a hoizontal eef flat and a lagoon. Thei H ms and T p wee measued at a foe-eef station, and then conveted to offshoe values. Values of h c and wee extacted fom the measuements at the eef-flat station A. Since tide modulation was found to be small and the epoted data had been filteed to sub-tidal fequencies, tide effect was ignoed and a constant hc m (the mean wate depth at the station A ) was applied in the model Vette et al. (1) The measuements wee conducted at a finging eef, Ipan, Guam. The eef pofile consisted of a foe-eef with an aveage slope of 1:15, a eef cest and a hoizontal eef flat backed diectly by the shoeline. The H ms and T p wee measued at a foe-eef station S 8. was obtained fom thei measuements nea the eef-cest station S 6. h c was obtained fom the measued mean wate depth at S 8 minus the elative seabed elevation between S 6 and S 8. Two sub-datasets wee epoted: a nomal swell event (denoted as N-deployment) and a stom event (denoted as G- deployment) 196

226 Wave diections wee not consideed in the model fo all fou datasets because they wee measued fom emote offshoe stations, which may not be accuate estimation of the wave diections at those foe-eef stations whee the obseved H ms and T p came fom. Finally, eflection coefficients wee not measued in field obsevations thus eflection was not included in the model computations; its influence will be incopoated into as discussed in section Bonneton et al.(7) =, R = Hench et al.(8) = 5.e+4, R = o (m).15 o (m) p (m) p (m).15.1 Lowe et al.(9a) =.3e+6, R = Vette et al.(1) ND: =.97, R =.96 GD: =.8, R =.79 o (m).9.6 o (m) p (m).4 ND GD p (m) Fig. 7.8 Pedicted wave setup ( ) vs. obseved wave setup ( ) fo diffeent field p o studies ( - paamete in the poposed model; ND - N-deployment; GD - G-deployment). R - R-squae; Solid line - p ; o 197

227 Fo the field applications, all model esults wee again computed based on Eq. (7.13) with Eqs. (7.14) and (7.15) as well as the powe law appoximation ( C 7 ), using the model paametes m 1. and.4. The pedicted setups ae plotted against the obseved setups in Fig. 7.8 fo the fou datasets. It can be seen that the best ageement appeas in the N-deployment fo the dataset of Vette et al. (1), with R.96. This is expected since the eef in Vette et al. (1) is a finging eef and the cuent modification to the wave setup is supposed to be small. Moeove, the chaacteistics of obseved wave beaking on the foe-eef ae consistent with the theoetic hypothesis of the pesent model. In fact, Vette et al. (1) also epoduced thei data using a dynamic model simila to Eq. (5.8). Howeve, as fo the G-deployment (the stom event), the pedictions ae not as good ( R.79 ) compaed with those fo the N-deployment; this is possibly due to the cuent and non-wave-elated sea level changes caused by winds duing the stom (Péquignet et al., 11). Consequently, the obseved wave setups at the eef cest wee educed, which esulted in a damatic incease of the fitted value of in the model (see section 7.4 fo the sensitivity analysis). Fo this dataset, an incease of fom.97 (N-deployment) to.8 (G-deployment) could be found. The magnitude of fo N-deployment was consistent with the laboatoy obsevations in Table 7.. The pedictions fo Bonneton et al. (7) s data ae easonable ( R.4 ). Bonneton et al. (7) found that thei obsevations wee mainly contolled by the diffeence between the wate depth at the beakpoint and the wate depth ove the eef, which is in fact patly embodied in Eq. (7.9) of the pesent model. The model failed to epoduce the data in Hench et al. (8), but a tend can still be identified in Fig. 7.8 and this tend could be best descibed by Eq. (5.1) with m 3. and n 3 as found by Hench et al. (8). The eef-lagoon-channel system existed at both eef sites, in which the setup eduction due to the cuent (which was typically on the magnitude of. m/ s fo those sites) may be substantial, thus non-physical values of lage than the uppe limit of 1. wee obtained fo both datasets. 198

228 Model application to the data of Lowe et al. (9a) is unsuccessful. This might be explained by the obsevations that the mild foe-eef slope of Kaneohe Bay educed setup though a combination of fictional wave damping and its elatively wide suf zone compaed to steep eefs. Lowe et al. (9a) had shown that the obseved wave setup was coelated to the offshoe wave powe. The effect of the mean cuent at this eef site was elatively weak (typically less than. m/ s) due to the pesence of significant wave setup inside its coast-bounded lagoon, which educed coss-eef setup gadients by 6% - 8%. Theefoe, vey lage value of was found by the model. The influences of cuent, fictional dissipation and lagoon setup will be included in the DH fomulation of the pesent model in section 7.7. Theefoe, unlike the model applications in the pevious laboatoy data, fo which the foe-eef slope as well as eef-cest shape contibute the value of, the incease of fo field datasets seems to be contolled by the eduction of obseved wave setups caused by the afoementioned factos (cuent, bottom fiction, lagoon setup, etc.). The level of the amplification of depends on the degee of the setup eduction at a specific eef site. Thus among the thee baie eefs in Fig. 7.8, the wave setups in Lowe et al. (9a) deviated fom those fo finging eef most, while the eef in Bonneton et al. (7) was most like the finging eef. Adjusting m o in the model can slightly impove the pedictions fo some datasets. In geneal, the pesent 1DH model woks quite well not only fo the field finging eefs on which its theoetical foundation elies, but also woks easonably well fo some baie eefs whee the setup eduction is elatively weak. Fo completeness of this section, the applicability of the two selected existing models in Chapte 5 is also evaluated against the fou field datasets above, see Appendix H fo details. 7.6 Discussions on the 1DH Model In this section, some comments ae made hee fo the model implementations and paametes: 199

229 (1) The appoach to estimate L s in the model is based on the powe law vaiation of in suf zone satisfying the bounday conditions ( ) and ( ), which x b b x s may be diffeent fom visual obsevation method in Chapte fo the measued L s. Thee is no detailed wave measuements othe than the one shown in Fig. 3.1, but the numeical esults in Chapte 4 indicate that wave setdown is unlikely to occu on the eef flat, thus the calculated h b (always on the foe-eef slope, which is consistent with the model equiement that wave beaking should occu on the foeeef) athe than the measued h b (some may be on the eef flat as obseved in Chapte, depending on the beaking citeia used) was adopted in section 7.3. to estimate L e. Theefoe, the measued in section 7.3. based on above L s and L e may deviate somewhat fom the defined in the model; () The autho has calibated a powe law based on the expeimental data in this thesis. Howeve, it has been shown in section that othe elations such as the polynomials listed in Table 7.1 could also povide the same quality of pedictions as long as was teated as a fitting paamete, which has less obvious physical meaning; (3) In pincipal, obtained fom the measued in the suf zone should be used in the model to pedict wave setup, although doing so is undesiable. As a fist appoximation, a epesentative can be selected by efeing to the values of listed in Table 7. accoding to the eef mophology (foe-eef slope and eefcest configuation) to be investigated, since geneally falls in a naow ange ( -.5) and is insensitive to the model esults. Altenatively, a fitted epesentative value of may be obtained fom the model based on some pilot measuements of the wave setup; (4) Both model paametes m and ae empiical and they may affect the esults in a moe o less same degee as shown by the model sensitivity analysis. Compaed to which is almost constant, m is in fact elated to the foe-eef slope (see section.4.) thus it may be moe appopiate to assign a value accoding to the slope of the given eef pofile athe than using a constant as did in pevious

230 sections. Using a diffeent m equies ecalibating as well as the powe law paamete C. An example will be shown by a case study fo the DH model in section Simila to pevious models (Tait,197; Goulay and Collete, 5), the poposed model is essentially a dynamic model. Howeve, it diffes fom the model of Tait (197), i.e., Eq. (5.8), and the model of Goulay and Collete (5), i.e., Eq. (5.9), in the following aspects: (1) It can accommodate all the investigated eef pofiles even if thee is a idge at the eef edge; () It does not equie that the suf-zone pocess ends at the eef cest, which is moe consistent with the laboatoy obsevations; (3) In contast to both in Eq. (5.8) and K p in Eq. (5.9), the model paamete has a clea physical meaning; (4) In contast to both and K p, a epesentative constant, which is independent of the eef-cest submegence and offshoe waves, can be used in the model applications fo a given eef pofile; (5) The effect of eef-cest submegence on is consideed (see Eq. (7.)), and diffeent values of ae employed fo the foe-eef and eef flat, espectively; (6) It includes the shoaling-induced setdown on the foe-eef athe than teating it sepaately as Tait (197) did. The pesent model diffes fom the kinematic model in the Chapte 6 in the following espects: (1) It explicitly includes wave shoaling, beaking, eflection and tansmission; () It is applicable to all laboatoy data used in the pevious chaptes. 1

231 7.7 Model Extension to DH Fomulation The aim of this section is to pesent a simplified model based on the appoach of Bellotti (4) fo an easy estimate of the DH hydodynamic chaacteistics of natual eefs. An idealized eef-lagoon-channel cell (baie eef) simila to Lowe et al. (9a) is consideed whee setup geneated by wave beaking dives flow acoss a shallow eef flat, though a deepe lagoon, and finally exits though a channel (see Fig. 7.9). Note that the consideed domain is half of complete eefchannel topogaphy; fo symmety easons, it is sufficient to model an aea extending fom the middle of the channel to the middle of the eef. The coss-shoe eef pofile is assumed to be idealized as discussed in section 7... Fig. 7.9 Plan view of an idealized eef-lagoon-channel system, including a shoeline, a eef (below the dashed line) and a channel (above the dashed line). Flow moves fom points A though E. The momentum balance Eq. (7.3) in the suf zone (fom A to B) is ecalled. In contast to the pevious 1DH fom, both flow-elated tems and the fiction tem ae etained hee to give

232 g g Ub U ( h ) U hu b b ( h ) hb S Cd Ls (7.) whee U b and U ae the depth-aveaged mean velocities at the beaking point and on the eef flat, espectively. the velocity ( U U ) / aveaged between the beaking point and the end of the suf zone is employed in the quadatic fiction law fo a shoewad cuent; h is the still wate depth along the hoizontal eef flat; C d is a epesentative dag coefficient fo the eef pofile and L s is the total sufzone width which vaies with incident waves and can be estimated by the definition of as Ls ( hb h)/( s) (see section 7..), whee s is the foe-eef slope. Both S and ae detemined in the same way as did fo the 1DH poblems. Intepetation of the ole of bottom fiction in Eq. (7.) is impotant. Fom a mathematical point of view, this tem induces a decease of wave setup if the depth-aveaged velocity is shoewad diected. b The integated coss-section continuity equation in the suf zone (A to B) eads U ( h ) U h (7.3) b b Fo this eef-channel geomety, an alongshoe continuity equation can be deived by equiing that the inflow ove the baie (eef flat) must be balanced by the outflow though the channel UhWUhW (7.4) g g g whee h g is the still wate depth along the channel; eef and channel widths, espectively, and channel. The above equation also implies that both W and W g ae the alongshoe U g is the depth-aveaged velocity in the U and U g ae unifom cossshoe. Howeve, in pactice, the channel may be fed locally by the alongshoe (lateal) cuent on the eef flat. This is moe likely to occu when the value of W / W o L / L is lage. When this happens, an alongshoe momentum balance g l may be included to pedict the alongshoe setup gadient that dives this flow. Nevetheless, the pesent model assumes that flow oiginating nea the eef cest 3

233 cosses the entie eef flat befoe enteing the lagoon since most eefs have elatively wide eef flat (on the ode of magnitude of 1-1m) and naow channels (on the ode of magnitude of 1 m). If W / W appoaches zeo, the model is expected to appoach its 1DH fom. g Diffeent fom the simple baie-gap system (e.g., Bellotti, 4), an additional momentum equation is intoduced hee to descibe the coss-eef flow on the eef flat, which is diven mainly by the baotopic pessue gadient esulting fom the setup in the suf zone. Since the wave focing is vey small on the eef flat (i.e., S ), Eq. (7.3) can be integated acoss the hoizontal eef flat ( ) fom the endpoint of suf zone (B) to the lagoon (C) to give g g (7.5) e ( L ) U ( h L) ( h ) CdUL whee L is the wave setup in the lagoon. As shown by Lowe et al. (9a), L may not be zeo as commonly assumed in existing 1DH models (Tait 197; Hean, 1999; Goulay and Collete, 5), L e is the effective coss-shoe eef-flat width e (distance between B and C in Fig. 7.9) and can be estimated by L L L (1 ) s since that pat of the suf zone is located on the eef flat. L is the pactical cossshoe eef-flat width as shown in Fig The above equation indicates the impotance of the bottom fiction on the eef flat. The enegy equation fo the lagoon-channel etun flow (D to E in Fig. 7.9) elates the wate head (setup) in the lagoon to the channel cuent. It is assumed that: (1) the lagoon can be teated as a esevoi, i.e., L is constant in the deep lagoon; () the wave focing is negligible in this deep channel egion (i.e., S ), and (3) just offshoe of the channel (i.e., ignoing wave setdown). Following the expession by Lowe et al. (9a) gives 4

234 U L g g Cd (7.6) Lg ghg whee g C d is a epesentative dag coefficient fo the channel path. Standad methods in the open channel hydaulics may be used to estimate g C d. To conside the lateal flow and setup distibution in the lagoon, the flow dynamics fom C to D can be teated as pat of the channel dynamics (Lowe et al., 9a) o be solved by intoducing additional alongshoe momentum balance in the lagoon and imposing mass balance between the lagoon and the channel. Eqs. (7.), (7.3), (7.4), (7.5) and (7.6) epesent a closed set of equations, and can be used to solve fo the five unknown vaiables (, L, U b, U, U g ) fo given wave chaacteistics ( H ms, T ), mophological popeties of the eef-lagoon- g channel system ( W L, W g, L, L g, h, h g, s ), oughness of the system ( C d, C d ) and the afoementioned model paametes ( m,, ). Paametes such as C, C and ae difficult to be measued thus may be obtained by a best fit between the field data and model pedictions. d g d Some featues of the pesent model ae: (1) Bellotti (4) did not conside the baie (coesponding to the eef flat) width, although it is believed to have an influence on the ip cuent system as pointed out by Dønen et al. (). Both the eef flat dimensions and the lagoon setup ae accounted fo in the pesent model; and () Lowe et al. (9a) assumed in thei model / h 1 fom the offshoe to the lagoon, which may not be valid when eefs ae exposed to vey low tides; thee is no such assumption fo the pesent model. Meanwhile, the model of Lowe et al. (9a) needs to be solved numeically fom offshoe to the lagoon which pevents fom deivation of simple analytical solutions Case study The field data of the southen ciculation at Kaneohe Bay (Lowe et al., 9a, heeafte L9) is evisited by the poposed DH model in this section. The 5

235 mophological popeties in L9 wee epoted with a ange of values in view of the eo in the field estimation. The aim of the study is not to seek optimized model inputs to achieve a best model pefomance, but to validate the model s ability to pedict the field data. Theefoe, the autho used thei estimated mean values, which ae: W /( W W ).71, h 5m, L 5m, L 15m, s 1/6, g g g Cd. and Cd.. Fo the model paametes, a m smalle than 1. may be moe appopiate in view of vey gentle foe-eef slope at Kaneohe Bay (1:6). The value of m.78, which is widely adopted fo wave beaking in constant wate depth (USACE, 3), will be used togethe with the powe law given in Eq. (7.1). Recalibating the powe law in the same pocedues as shown in sections 7.3. and based on the laboatoy eef without the idge in Chapte 3 (the coss-shoe eef pofile fo this case study has been assumed to be idealized without the idge by L9) gives C 4 and.85. Pilot tests have shown that the esults wee not vey sensitive to when vaied in the ange of to 1. Fo, the value of.3 was adopted accoding to L9 s measuements. The paametes h, H ms and T p wee obtained in the same way as descibed in section g g The autho has emphasized in section the impotance of choosing a suitable 1D wave model to calculate the adiation stess tem in the model. Both L9 and Péquignet et al. (11) used the model of Thonton and Guza (1983), which needs to be solved numeically. L9 has epoted a 4% enegy eduction esulting fom the fictional dispassion pio to the beaking point due to the vey gentle foe-eef at Kaneohe Bay. Fictional dissipation could be futhe inceased in the shallowe suf zone. Meanwhile, the wave eflection was not measued in thei obsevations. To estimate h b, the oiginal fom of Eq. (7.6), i.e., Eq. (E.4), should be used. Fo simplicity, the autho intoduced an enegy eduction coefficient to account fo both fictional dissipation and eflection in Eq. (E.4), which gives K (see Appendix E). Peliminay simulations have shown that is the most sensitive paamete in the pesent case study, thus it will be used as a fitting paamete instead of in this section. 6

236 A total of 56 days (fom Jan to 19 Ma.) obseved wave setup ( ), lagoon setup ) and eef-flat cuent ( U ) wee epoduced by the pesent model. The esults ( L togethe with the obsevations and the model esults of L9 based on quadatic fiction law ae shown in Fig (m) t (day) L (m) U (m/s) t (day) t (day) Fig. 7.1 Time-seies of field obsevations and model pedictions (Open Cicles: Obsevations; Dashed line: Pedictions by L9; Solid line: Pedictions by the pesent model)..94 was obtained fom a best fit between the obsevations and pedictions, indicating the combined effect of fictional dissipation and eflection at Kaneohe Bay was significant. Fig. 7.1 shows that the poposed DH model adequately 7

237 pedicted the oveall patten of the time-seies field obsevations. The model skill values ae estimated by (Wilmott, 1981) skill 1 X model X obs X X X X mod el obs obs obs (7.7) Pefect ageement will yield a skill of one, wheeas complete disageement will yield a skill of zeo. The compaison gives.93 (pesent model) and.94 (L9) fo ;.91 (pesent model) and.9 (L9) fo L, and.89 (pesent model) and.84 (L9) fo U. Coesponding R values wee.6 (pesent model) and.94 (L9) fo ;.54 (pesent model) and.8 (L9) fo L, and.6 (pesent model) and.8 (L9) fo U. The skill values of the pesent model ae compaable to those of L9, but R values of the pesent model ae lowe. The R values can indeed to be inceased if the input values ae futhe calibated. Fo example, if one uses h 5.5m which is still in L9 s suggested ange, the g U can ise to.69 and.6, espectively, without affecting the R values fo L and R value fo. Howeve, it is not the focus of this section to seek the best pefomance of the model by adjusting the input paametes. Oveall, the poposed DH model could impove the pedictions of the field data in L9 ove its 1DH fom. Though this case study, the impotance of the fictional dispassion has been ecognized when the model in its pesent fom is used fo some field eef with a mild foe-eef. Although adding a damping coefficient can account fo dissipation in a simple way, it will intoduce the additional fee paamete. A wave model may be moe favoable to estimate the wave tansfomation acoss the foe-eef and in the suf zone even though the model computation is time-consuming. The pesent model needs to be futhe validated by well-contolled DH laboatoy expeiments. Nevetheless, it povides a useful tool fo illustating how the wave-diven cuent, eef mophology and bottom oughness cucially affect the setup and ciculation in coastally bounded eef-lagoon-channel systems. 8

238 7.8 Concluding Remaks A 1DH analytical model based on the coss-shoe momentum balance is poposed in this chapte to study the wave-induced setup ove submeged finging eefs. A powe law is found to sufficiently appoximate the wave setup pofile in the suf zone. The model explicitly accounts fo wave tansfomation pocesses ove eefs such as shoaling, beaking, eflection and tansmission. It allows sufzone pocess to extend shoewad of the eef cest. The effect of eef-cest submegence on the beake depth index is consideed. Compaisons with diffeent laboatoy data show that the model is capable of epoducing the maximum wave-induced setup on the eef flat unde a vaiety of eef pofiles (with and without a idge at the eef edge), a ange of eef-cest submegences (fom zeo to the uppe limit whee waves cease to beak), and two types of wave conditions (monochomatic and spectal). The 1DH model is also successfully applied fo some field eefs. The defined model paamete ( ) is physically elated to sufzone length scales and it has been poven to be the most suitable fee paamete in the model. The value of, typically within the ange of to.5, is found to decease with inceasing foe-eef slope in the laboatoy expeiments, but the detailed eef-cest configuation is also impotant. also accounts fo the eduction of wave setup commonly obseved at field eefs. The model fomulation has been extended to DH in analogy to the existing conceptual appoaches fo ip cuent and validated by a field case study. The DH fom equies futhe validation with moe laboatoy and field data. 9

239 CHAPTER 8 CONCLUTIONS AND FUTURE WORK 8.1 Conclusions This thesis has pesented a compehensive study of the suface gavity wave dynamics ove finging coal eefs with/without a idge, including expeimental studies, numeical modeling and theoetical analyses. In Chapte, a seies of laboatoy expeiments have been conducted in a wave flume to examine the effects of vaying eef-flat submegence and foe-eef slopes on the popeties of beaking waves (the beake type and location, beake height and depth indices, sufzone width, as well as wave eflection and tansmission) ove submeged idealized finging eefs subjected to monochomatic waves. Dimensionless analysis has shown that the elative submegence of the eef flat is the detemining facto to chaacteize most beaking chaacteistics ove finging eefs. The influence of foe-eef slope appeas to be insignificant. The idealized eef pofile investigated in this chapte seves as a fist appoximation to natual finging eefs. Thei hydodynamic esponse to waves has been poven to esemble submeged stuctues (e.g., beakwates) athe than plane beaches. Some empiical fomulae have been poposed within the expeimental data ange. The findings obtained in this chapte ae efeed to in subsequent chaptes In Chapte 3, the laboatoy wok was diveted to the effects of a eef cest (a idge stuctue located at the eef edge as is fequently obseved in the field) on the wave-induced setup. Expeimental esults fo eef pofiles with and without a ectangula idealized idge subjected to both monochomatic and spectal waves have been compaed. It is shown that the behavio of the wave tansfomation in the pesence of a idge is significantly diffeent fom that in the absence of the idge. In paticula, a idge nea the eef edge can cause an incease in the wave-induced setup on the eef flat as well as wave eflection seawad of the foe-eef. Additional wave measuements with highe spatial esolution allow the investigation of the coss-shoe evolution of hamonic waves and momentum flux. These measuements 1

240 show that the second hamonics become compaable to the fundamental hamonics on the eef flat afte dissipation due to beaking. The momentum flux, with a value consistently lowe than the theoetical value based on the linea shallow-wate appoximation, was found to be diffeent fom that fo plane beaches in existing studies. Simple coss-shoe undetow measuements wee also obtained. It has been found that the coss-shoe undetow pofile in the absence of a idge is identical to that fo plane beaches unde simila plunging beakes and the pesence of a idge blocks the flow in the leeside of the idge in a way simila to how boad-cested wei contols the wate level and flow in open channel flows. Numeical expeiments based on fully nonlinea, weakly dispesive, Boussinesqtype equations with a FV-based solve have been pefomed to study the wave tansfomation ove vaious finging eef pofiles in Chapte 4. Model validations by the wave data pesented in Chapte 3 and othe simila studies have shown the necessities of an appopiate teatment of bounday conditions and a fine-tuned eddy-viscosity model. The numeical model can efficiently epoduce the key chaacteistics (wave eflection, geneation of highe and lowe fequency waves, wave height vaiation, wave-induced setup/setdown) of both monochomatic and spectal waves ove vaious finging eef pofiles and eef-cest submegences, including a idge at the eef edge and the dy eef-flat condition. Numeical tests have also been extended to eef topogaphies with which existing laboatoy expeiments ae not concened. Both the slope and the shape of foe-eef have been examined, and they wee found to have notable effects on the vaiation of waveinduced setup inside the suf zone. To gain moe insight into the undelying physical pocesses, some analytical modeling appoaches wee attempted. The pefomance of diffeent existing models to simulate the measued wave setups have been evaluated in Chapte 5. The offshoe scaling paamete, commonly used fo field data, has been found to be insufficient to descibe the expeimental data. It has also been shown that the model of Tait (197) is able to epoduce the expeimental esults with some success, especially at elatively small eef-cest submegences. The model of Goulay and 11

241 Collete (5) can account fo most of the expeimental data with the addition of an empiical paamete. The inadequacies of existing 1DH analytical models have been discussed. It was pointed out that the pimay dawback of both models of Tait (197) and Goulay and Collete (5) is that both have fee paametes that vay with eef-cest submegence. An analytical model based on the mass balance has been pesented in Chapte 6 to study the wave setup ove finging eef pofiles whee fee fall (citical flow) conditions exist at the cest of the eef. It was based on the kinematics of the flow aound the eef cest. The model has indicated that wave setup can be descibed by the deep-wate wave steepness and a flux adjustment facto. The kinematic model has been poven to epoduce the maximum wave-induced setup on eef flat with/without the pesence of a idge subjected to both monochomatic and spectal waves. Model applications to othe published expeiments wee also successful. The scaling facto in the model has been found to be a function of both the eef-cest submegence and the foe-eef slope. The model is eady to be extended to the emeged eef-cest conditions which occasionally occu fo field eefs duing low tides. A key equiement fo this model is that the eef-cest submegence must be sufficiently small o emeged so that a citical flow condition exists. A dynamic appoach has been intoduced in Chapte 7. The 1DH analytical model based on the coss-shoe momentum balance is fomulated to study the waveinduced setup ove finging eefs. Validation with vaious laboatoy data has shown that the model is capable of epoducing the maximum wave-induced setup on a eef flat unde a vaiety of eef pofiles (with and without the pesence of a idge) and eef-cest submegences (fom zeo submegence to the uppe limit whee waves cease to beak) as well as unde both monochomatic and spectal conditions. The 1DH model has also been successfully applied fo some field eefs, especially fo the finging type. The model paamete is elated to some physical length scales in the suf zone and it has been found to be dependent on both the foe-eef slope and the eef-cest shape fo the laboatoy data. Fo natual eefs, it also accounts fo factos (cuent, fiction, etc.) that cause eduction of 1

242 wave setup. The model fomulation has been extended to DH to study the idealized baie eef system and validated by a case study. Aside fom supeio model pefomance fo the data investigated, the poposed model has some distinctive featues compaed to existing dynamical appoaches (e.g., Tait (197) and Goulay and Collete (5)) in that: (1) It allows sufzone pocesses on the eef flat; () The modulation of eef-cest submegence on the beake depth index is consideed; (3) The pimay model paamete is independent of eef-cest submegence. Moeove, it diffes fom the kinematic theoy in Chapte 6 in that it explicitly accounts fo the pimay 1DH tansfomation pocesses associated with waves such as wave shoaling, beaking, eflection and tansmission. To summaize, the majo concen of this thesis is 1DH wave inteactions with idealized finging eefs at laboatoy scale. It has shown the similaities and discepancies of impotant aspects of wave dynamics (wave tansfomation, waveinduced setup and flow, etc.) with othe coastal systems in engineeing pespectives. It has eniched the existing data on the wave tansfomation ove finging eefs, especially ove the finging eef with a idge. The laboatoy data povided in this thesis can be used in validation of a vaiety of numeical wave and ciculation models. The ability of the poposed numeical and analytical appoaches in epoducing wave-induced setup is well addessed in this thesis. The analytical models discussed in this thesis ae substitutes fo some diect measuement and expeimentation and povide us a tool to undestand and the obseved phenomena in the laboatoy and undelying physical pocesses in a moe geneal and objective way. Application of the poposed 1DH/DH model to field data also enables us to undestand some fundamental diffeences in wave dynamics between the laboatoy eefs and the natual eefs. 8. Futue Wok Thee possible futue eseach diections can be ecommended hee: 13

243 8..1 Expeimental wok The wave-induced setup and flow ove the eef flat could be investigated in a moe systematic way in a laboatoy wave-cuent flume o wave basin, paticulaly, the fundamental diffeences between the finging eef and baie eef could be studied. Moeove, flow measuements with advanced techniques such as Paticle Image Velocimety (PIV) could povide a high-esolution flow field on the shallowe eef flat to investigate tubulence chaacteistics in the suf zone. Meanwhile, effects of the oughness and poosity of the foe-eef suface on wave-induced setup equie futhe systematic expeimental investigation. 8.. Numeical simulation Natue eel pofiles ae subjected to many othe pocesses in addition to wave focing. The adopted Boussinesq model in Chapte 4 can also be applied to conditions with an ambient cuent; the wave-cuent inteaction as well as wavegeneated cuents can be modeled. The polynomial appoximation of vetical velocity pofile and the multi-laye concept (Lynett, 6) povides a tool to investigate suf zone mean flow quantities such as the mass flux, undetow, etc., although some impovements on the beaking model may be needed. Extension of this appoach to DH laboatoy expeiments o field cases is staightfowad and the inclusion of subgid lateal tubulence diffusion may be necessay. All of these could be attempted with moe detailed flow measuements Analytical modeling The potential of the kinematic theoy in Chapte 6 to deal with emeged eef-cest conditions (negative eef-cest submegence) has not been validated by expeimental data; this could be futhe examined. The magnitude of the scaling facto in the model has been found to be supisingly small (bellow.5), and the undelying easons should be futhe undestood. The dependence of the model paamete in the dynamic model of Chapte 7 on the eef-cest shape is not vey conclusive since a unifom tend has not been obtained due to a lack of data; moe wok on this issue is needed. Futhemoe, an altenative wave model athe than 14

244 using the simple shallow-wate wave appoximation to paameteize the adiation stess could be attempted. Lastly, the validity of DH fomulation of the model equies suppot fom moe laboatoy and field data. 15

245 REFERENCES Ahens, J. P. (1987), Chaacteistics of Reef Beakwates, Technical Repot CERC-87-17, Coastal Engineeing Reseach Cente, U.S. Amy Enginee Wateways Expeiment Station, Vicksbug, MS. Apotsos, A., Raubenheime, B., Elga, S., Guza, R. T. and Smith, J. A. (7), Effects of wave olles and bottom stess on wave setup, J. Geophys. Res., 11, C3, doi:1.19/6jc3549. Atkinson, M. J. and Bilge R. W. (199), Effects of wate velocity on phosphate uptake in coal eef-flat communities, Limnol. Oceanog., 37(), Beji, S. and Battjes, J. A. (1993), Expeimental investigation of wave popagation ove a ba, Coast. Eng., 19, Battjes, J. A. (1974), Suf-similaity, Poc. 14th Int. Conf. Coastal Eng., ASCE, Copenhagen, Denmak, pp Bellotti, G. (4), A simplified model of ip cuents systems aound discontinuous submeged baies, Coast. Eng., 51(4), Blenkinsopp, C. E. and Chaplin, J. R. (8), The effect of elative cest submegence on wave beaking ove submeged slopes, Coast. Eng., 55(1), Booij, N. (1983), A note on the accuacy of the mild-slope equation, Coast. Eng., 7(3), Bonneton, P., Lefebve, J-P., Betel, P., Ouillon, S. and Douillet, P. (7), Tidal modulation of wave-setup and wave-induced cuents on the Aboé coal eef, New Caledonia, J. Coast. Res., SI 5, Bande, R. W., Kench, P. S. and Hat D. E. (4), Spatial and tempoal vaiations in wave chaacteistics acoss a eef platfom, Waabe Island, Toes Stait, Austalia, Maine Geology, 7(1-4), Bossad, J., Peet, G., Blonce, L. and Diedhiou A. (9), Highe hamonics induced by a submeged hoizontal plate and a submeged ectangula step in a wave flume, Coast. Eng., 56(1),

246 Cácees, I., Stive, M. J. F., Sanchez-Acilla, A. and Tung, L. (8), Quantification of change in cuent intensities induced by wave ovetopping aound low-cested stuctues, Coast. Eng., 55(), Calabese, M., Vicinanza, D. and Buccino, M. (8), D wave setup behind submeged beakwates, Ocean Eng., 35(1), Callaghan, D. P., Nielsen, P., Catwight, N., Goulay, M. R. and Baldock,T. E. (6), Atoll lagoon flushing foced by waves, Coast. Eng., 53(8), Camenen, B. and Lason, M. (7), Pedictive fomulas fo beake depth index and beake type, J. Coast. Res., 3(4), Camo, J. A., Neves, G. M. and Voode, M. T. (11), Designing a multifunctional atificial eef: studies on the influence of paametes with most influence in the vetical plane, J. Coast. Consev., 15(1), Chistensen, E. D., Walsta, D. J. and Emeat, N. (), Vetical vaiation of the flow acoss the suf zone, Coast. Eng. 45(1), Chistou, M., Swan, C. and Gudmestad O. T. (8), The inteaction of suface wate waves with submeged beakwates, Coast. Eng., 55(1), Collins, I. A. (197), Pobability of beaking wave chaacteistics, Poc. 1th Int. Conf. Coastal Eng., ASCE, Washington, D.C., USA, pp Cowen, E. A., Sou, I. M., Liu, P. L. -F. and Raubenheime, R. (3), Paticle image velocimety measuements within a laboatoy-geneated swash zone, J. Eng. Mech. ASCE 19 (1), Dawin, C. H. (184), The Stuctue and Distibution of Coal Reefs, London: Smith, Elde and Co. 7 pp. Dean, R. G. and Bende, C. J. (6), Static wave setup with emphasis on damping effects by vegetation and bottom fiction, Coast. Eng., 53(-3), Demibilek, Z., Nwogu, O. G. and Wad D. L. (7), Laboatoy Study of Wind Effect on Runup ove Finging Reefs, Repot 1: Data epot, Coastal and Hydaulics Laboatoy Technical Repot ERDC/CHL-TR-7-4, U.S. Amy Enginee Reseach and Development Cente, Vicksbug, MS. Demibilek, Z. and Nwogu, O. G. (7). Boussinesq Modeling of Wave Popagation and Runup ove Finging Coal Reefs, Model Evaluation Repot, Coastal and Hydaulics Laboatoy Technical Repot, Chapte 4, ERDC/CHL 17

247 TR-7-1, U.S. Amy Enginee Reseach and Development Cente, Vicksbug, MS. Douillet, P., Ouillon, S. and Codie, E. (1), A numeical model fo fine suspended sediment tanspot in the southwest lagoon of New Caledonia, Coal Reefs, (4), Dønen, N., Kaunaathna, H., Fedsøe, J., Sume, B. M. and Deigaad, R. (), An expeimental study of ip channel flow, Coast. Eng. 45 (3-4), Falte, J. L., Atkinson, M. J. and Meifield, M. A. (4), Mass tansfe limitation of nutient uptake by a wave-dominated eef flat community, Limnol. Oceanog., 49(5), Galvin, C. J. (1968), Beake type classification on thee laboatoy beaches, J. Geophys. Res., 73(1), Galvin, C. J. (1969), Beake tavel and choice of design wave height, J. Wtwy. and Hab. Div., ASCE, 95(WW), 175. Geitsen, F. (198), Wave attenuation and wave set-up on a coastal eef, Poc. 17th Int. Conf. Coastal Eng., ASCE, Sydney, Austalia, pp Goda, Y. (197), A synthesis of beake indices, Tans. Japan Soc. Civil Engs. (), Goda, Y. (1974), New wave pessue fomulae fo composite beakwates, Poc. 14th Int. Conf. Coastal Eng., ASCE, Copenhagen, Denmak, pp Goda, Y. (), Random Seas and Design of Maitime Stuctues. Wold Sci. Pess, Singapoe, 443 pp. Goda, Y. (9), Deivation of unified wave ovetopping fomulas fo seawalls with smooth, impemeable sufaces based on selected CLASH datasets, Coast. Eng., 56(4), Goda, Y. (1), Reanalysis of egula and andom beaking wave statistics, Coast. Eng. J., 5 (1), Goulay, M. R. (1994), Wave tansfomation on a coal eef, Coast. Eng., 3(1-), Goulay, M. R. (1996a), Wave set-up on coal eefs. 1. Set-up and wave-geneated flow on an idealised two dimensional eef, Coast. Eng., 7(3-4),

248 Goulay, M. R. (1996b), Wave set-up on coal eefs.. Wave set-up on eefs with vaious pofiles, Coast. Eng., 8(1-4), Goulay, M. R. and Collete, G. (5), Wave-geneated flow on coal eefs: an analysis fo two-dimensional hoizontal eef-flats with steep faces, Coast. Eng., 5(4), Govende, K., Mocke, G. P. and Alpot, M. J. (), Video-imaged suf zone wave and olle stuctues and flow fields, J. Geophys. Res., 17 (C7), 37, 1.19/JC755. Govende, K., Michallet, H., Alpot, M. J., Pillay, U., Mocke, G. P. and Moy, M. (9), Video DCIV measuements of mass and momentum fluxes and kinetic enegies in laboatoy waves beaking ove a ba, Coast. Eng., 56(8), Guza, R. T. and Thonton, E. B. (1981), Wave set-up on a natual beach, J. Geophys. Res., 86, Haas, K. and Svendsen, Ib. A. (), Laboatoy measuements of the vetical stuctue of ip cuents, J. Geophys. Res., 17(C5), 347, 1.19/1 JC911. Hady, T. A. and Young I. (1996), Field study of wave attenuation on an offshoe coal eef, J. Geophys. Res., 11 (C6), Hean, C. J. (1999), Wave-beaking hydodynamics within coal eef systems and the effect of changing sea level, J. Geophys. Res., 14 (C1), Hean, C. J., Atkinson, M. J. and Falte, J. L. (1), A physical deivation of nutient-uptake ates in coal eefs: effects of oughness and waves, Coal Reefs, (4), Hean, C. J. (11), Hydodynamics of coal eefs, In: Hopley D (ed) Encyclopedia of moden coal eefs, Spinge, Belin, pp Hench, J. L., Leichte, J. J., and Monismith, S. G. (8), Episodic ciculation and exchange in a wave-diven coal eef and lagoon system, Limnol. Oceanog., 53(6), Hsiao, S. C., Lynett, P., Hwung, H. H., and Liu, P. L. -F. (5), Numeical simulations of nonlinea shot waves using the multi-laye Model, J. Eng. Mech., 131(3),

249 Huang, Z. C., Hsiao, S. C., Hwung, H. H. and Chang, K. A. (9), Tubulence and enegy dissipations of suf-zone spilling beakes, Coast. Eng. 56(7), Isaeli, M. and Osza, S. A. (1981), Appoximation of adiation bounday conditions, J. Comp. Phys., 41, Ivesen, H. W. (195), Laboatoy study of beakes, Gavity waves, U.S. National Bueau of Standads, Washington, D. C., Cicula 51, 9 3. Jago, O. K., Kench, P. S. and Bande, R. W. (7), Field obsevations of wavediven wate-level gadients acoss a coal eef flat, J. Geophys. Res., 11, C67, doi:1.19/6jc374. Johnson, B. D. and Smith J. M. (5), Longshoe cuent focing by iegula waves, J. Geophys. Res., 11, C66, doi:1.19/4jc336. Johnson, H. K. (6), Wave modeling in the vicinity of submeged beakwates, Coast. Eng., 53(1), Kaambas, Th. V. and Koutitas, C. (199), A beaking wave popagation model based on the Boussinesq equations, Coast. Eng., 18(1-), Kench, P. S. and Bande R. W. (6), Wave pocesses on coal eef flats: Implications fo eef geomophology using Austalian case studies, J. Coast. Res., (1), 9 3. Kennedy, A. B., Chen, Q., Kiby, J. T. and Dalymple, R. A. (), Boussinesq modeling of wave tansfomation, beaking, and unup. I: 1D, J. Wtwy., Pot, Coast. and Ocean Eng., ASCE, 16 (1), Kim, D. H., Lynett, P. and Socolofsky, S. A. (9), A depth-integated model fo weakly dispesive, tubulent, and otational fluid flows, Ocean Modeling, 7(3-4), Kiby, J. T., Wei, G., Chen, Q., Kennedy, A. B. and Dalymple, R. A. (1998), FUNWAVE 1.: Fully Nonlinea Boussinesq Wave Model Documentation and Use s Manual, Cente fo Applied Coastal Reseach, Univesity of Delawae. Koma, P. D. and Gaughan, M. K. (1973), Aiy wave theoy and beake height pediction, Poc. 13th Int. Conf. Coastal Eng., ASCE, Vancouve, Canada, pp

250 Koma, P. D. (1998), Beach Pocesses and Sedimentation, nd Ed., Pentice Hall, Uppe Saddle Rive, N. J. 544 pp. Kaines S. B., Yanagi, T., Isobe, M. and Komiyama, H. (1998) Wind-wave-diven ciculation on the coal eef at Boa Bay, Miyako Island, Coal Reefs, 17(), Kaines, S. B., Suzuki, A., Yanagi, T., Isobe, M., Guo, X. and Komiyama, H. (1999) Rapid wate exchange between the lagoon and the open ocean at Majuo Atoll due to wind, waves, and tide, J. Geophys. Res., 14 (C7), Laa, J. L., Losada, I. J. and Guanche, R. (8), Wave inteaction with lowmound beakwates using a RANS model, Ocean Eng., 35(13), Lin, P. and Liu, P. L. -F. (1998), A numeical study of beaking waves in the suf zone, J. Fluid Mech., 359, Lin, P. and Liu, P. L. -F. (1999), An intenal wave-make fo Navie-Stokes equations models, J. Wtwy., Pot, Coast. and Ocean Eng., ASCE, 15(4), Longuet-Higgins, M. S. and Stewat, R. W. (196), Radiation stess and mass tanspot in gavity waves, with application to suf beats, J. Fluid Mech., 13, Longuet-Higgins, M. S. and Stewat, R. W. (1964). Radiation stesses in wate waves; a physical discussion, with applications, Deep-Sea Res., 11, Longuet-Higgins, M. S. (5), On wave set-up in shoaling wate with a ough sea bed, J. Fluid Mech., 57, Losada, I. J., Laa, J. L., Chistensen, E. D. and Gacia, N. (5), Modelling of velocity and tubulence fields aound and within low-cested ubble-mound beakwates, Coast. Eng., 5 (1-11), Lowe, R. J., Falte, J. L., Bandet, M. D., Pawlak, G. and Atkinson M. J. (5), Spectal wave dissipation ove a baie eef, J. Geophys. Res., 11, C41, doi:1.19./4.jc711. Lowe, R. J., Falte, J. L., Monismith, S. G. and Atkinson M. J. (9a), Wavediven ciculation of a coastal eef-lagoon system, J. Phys. Oceanog., 39(4),

251 Lowe, R. J., Falte, J. L., Monismith, S. G. and Atkinson M. J. (9b), A numeical study of ciculation in a coastal eef-lagoon system, J. Geophys. Res., 114, C6, doi: 1.19/8JC581. Lugo-Fenandez, A., Robets, H. H. and Wiseman W. J. (1998a), Tide effects on wave attenuation and wave set-up on a Caibbean coal eef, Est. Coast. Shelf. Sci., 47(4), Lugo-Fenandez, A., Robets, H. H. and Suhayda, J. N. (1998b), Wave tansfomations acoss a Caibbean finging-baie Coal Reef, Cont. Shelf Res., 18(1), Lugo-Fenandez, A., Robets H. H. and Wiseman, W. J. (4), Cuents, wate levels, and mass tanspot ove a moden Caibbean coal eef: Tague Reef, St. Coix, USVI, Cont. Shelf Res., 4(17), Luick, J. L., Mason, L., Hady, T. and Funas, M. J. (7), Ciculation in the Geat Baie Reef Lagoon using numeical taces and in situ data, Cont. Shelf Res., 7(6), Lynett, P, Wu, T. R. and Liu, P. L. -F. (), Modeling wave unup with depthintegated equations, Coast. Eng., 46(), Lynett, P. and Liu, P. L. -F. (4), A two-laye appoach to wate wave modeling, Poc. R. Soc. London, Se. A, 46, Lynett, P. (6), Neashoe wave modeling with high-ode Boussinesq-type equations, J. Wtwy., Pot, Coast. and Ocean Eng., ASCE, 13(5), Lynett, P. and Liu, P. L. -F. (8), Modeling Wave Geneation, Evolution, and Inteaction with Depth-Integated, Dispesive Wave Equations, Coulwave Code Manual v.., Conell Univ., Ithaca, N.Y. Lynett, P., Melby, J. A. and Kim, D. H. (1), An application of Boussinesq modeling to Huicane wave ovetopping and inundation, Ocean Eng., 37, Madin, J. S., Black, K. P. and Connolly S. R. (6), Scaling wate motion on coal eefs: fom egional to oganismal scales, Coal Reefs, 5, Madsen, P. A. and Søensen, O. R. (199), A new fom of the Boussinesq equations with impoved linea dispesion chaacteistics. Pat : A slowly vaying bathymety, Coast. Eng., 18(3-4),

252 Madsen, P. A., Søensen, O. R. and Schaffe, H. A. (1997), Suf zone dynamics simulated by a Boussinesq-type model. Pat I. Model desciption and cossshoe motion of egula waves, Coast. Eng., 3(4), Madsen, P. A., Fuhman, D. R. and Wang, B. (6), A Boussinesq-type method fo fully nonlinea waves inteacting with a apidly vaying bathymety, Coast. Eng., 53(5-6), McCowan, J. (1894), On the highest wave of pemanent type, Philosophical Magazine, 38, Mei, C. C., Stiassnie, M. and Yue, D. K. P. (5), Theoy and Applications of Ocean Suface Waves, Wold Sci. Pess, Singapoe, 171 pp. Miche, R. (1944), Mouvements ondulatoies de lame en pofondeu ou décoissante, Annles de Ponts et Chaussées, 19, Monismith, S. G. (7), Hydodynamics of coal eefs, Annu. Rev. Fluid Mech., 39, Moy, M. and Hamm, L. (1997), Wave height, setup and cuents aound a detached beakwate submitted to egula o andom wave focing, Coast. Eng., 31(1-4), Munk, W. H. and Sagent, M. C. (1948), Adjustment of Bikini atoll to ocean waves, Tans. Am. Geophys. Union, 9, Munson, B. R., Young, D. F. and Okiishi, T. H. (), Fundamentals of Fluid Mechanics, John Wiley & Sons. Inc., USA., 84 pp. Nelson, R. C. (1987), Design wave heights on vey mild slopes: an expeimental study, Civil. Eng. Tans., Inst. Eng. Aust., 9, Nelson, R. C. (1994), Depth limited design wave heights in vey flat egions, Coast. Eng., 3(1-), Nelson, R. C. (1996), Hydaulic oughness of coal eef platfoms, Applied Ocean Reseach, 18(5), Nwogu, O. (1993), Altenative fom of Boussinesq equations fo neashoe wave popagation, J. Wtwy., Pot, Coast. and Ocean Eng., ASCE, 119(6), Owen, M. W. (198), Design of Seawalls Allowing fo Wave Ovetopping, Technical Repot EX-94, HR-Wallingfod, UK. 3

253 Péquignet, A. C. N., Becke, J. M., Meifield, M. A. and Aucan, J. (9), Focing of esonant modes on a finging eef duing topical stom Man-Yi, Geophys. Res. Lett., 36, L367, doi:1.19/8gl3659. Péquignet A. -C., Becke, J. M., Meifield, M. A. and Boc S. J. (11), The dissipation of wind wave enegy acoss a finging eef at Ipan, Guam, Coal Reefs, 3, Peegine, D. H. (1967), Long waves on a beach, J. Fluid Mech., 7, Pullen, T., Allsop, N. W. H., Buce, T., Kotenhaus, A., Schuttumpf, H. and van de Mee, J. W. (7), EuOtop: Wave Ovetopping of Sea Defenses and Related Stuctues: Assessment Manual, Science Repot: SC559, Envionment Agency, UK. Ranasinghe, R. S., Sato, S. and Tajima, Y. (9), Boussinesq modeling of waves and cuents ove submeged beakwates, Poc. 5th Int. Conf. Asian and Pacific Coasts., APAC, Singapoe. Vol. 3, pp Raubenheime, B., Guza R. T. and Elga S. (1996), Wave tansfomation acoss the inne suf zone, J. Geophys. Res., 11(C1), Roebe, V., Cheung, K. F. and Kobayashi, M. H. (1). Shock-captuing Boussinesq-type model fo neashoe wave pocesses, Coast. Eng., 57(4), Sakai, T. and Battjes, J. A. (198), Wave shoaling calculated fom Cokelet's theoy, Coast. Eng., 4, Schäffe, H. A., Madsen, P. A. and Deigaad, R. (1993), A Boussinesq model fo waves beaking in shallow wate, Coast. Eng., (3-4), 185. Seelig, W. N. (198), Two-dimensional Tests of Wave Tansmission and Reflection Chaacteistics of Laboatoy Beakwates, Technical Repot 8-1, Coastal Engineeing Reseach Cente, U.S. Amy Enginee Wateways Expeiment Station, Vicksbug, MS. Seelig, W. N. and Ahens, J. P. (1981), Estimation of Wave Reflection and Enegy Dissipation Coefficients fo Beaches, Revetments and Beakwates, Technical Pape 81-1, Coastal Engineeing Reseach Cente, U.S. Amy Enginee Wateways Expeiment Station, Vicksbug, MS 4

254 Seelig, W. N. (1983), Laboatoy study of eef-lagoon system hydaulics, J. Wtwy. Pot, Coast. and Ocean Eng., ASCE, 19(4), She, K., Geated, C. A. and Easson, W. J. (1994), Expeimental study of theedimensional wave beaking, J. Wtwy., Pot, Coast. and Ocean Eng., ASCE, 1(1), 36. She, K., and Canning, P. (7), Geometic Study of Monochomatic Wave Beaking on Beaches, J. Wtwy., Pot, Coast. and Ocean Eng., ASCE, 133(5), Shilal, K. G., Rao, S. and Manu (7), Ocean wave tansmission by submeged eef A physical model study, Ocean Eng., 34 (14-15), Skotne, C. and Apelt, C. J. (1999), Application of a Boussinesq model fo the computation of beaking waves. Pat : wave- induced setdown and set-up on a submeged coal eef, Ocean Eng., 6(1), Smith, E. R. and Kaus, N. C. (1991), Laboatoy study of wave beaking ove bas and atificial eefs, J. Wtwy., Pot, Coast. and Ocean Eng., ASCE, 117(4), Stolazzi, C. D., Elias, E., Field, M. E. and Pesto, M. K. (11), Numeical modeling of the impact of sea-level ise on finging coal eef hydodynamics and sediment tanspot, Coal Reefs, 3, Suh, K. D., Lee, C. and Pak, W. S. (1997), Time-dependent equations fo wave popagation on apidly vaying topogaphy, Coast. Eng., 3(-3), Sunamua, T. (198), A laboatoy study of offshoe tanspot of sediment and a model fo eoding beaches, Poc. 17th Int. Conf. Coastal Eng., ASCE, Sydney, Austalia, pp Svendsen, Ib. A., Madsen, P. A. and Hansen, J. B. (1978), Wave chaacteistics in the suf zone, Poc. 16th Int. Conf. Coastal Eng., ASCE, Hambug, Gemany, Vol. 1, pp Svendsen, Ib. A. and Putevu, U. (1993), Suf zone wave paametes fom expeimental data, Coast. Eng., 19(3-4), Svendsen, Ib. A. (1984a), Wave heights and setup in a suf zone, Coast. Eng., 8(4),

255 Svendsen, Ib. A. (1984b), Mass flux and undetow in a suf zone, Coast. Eng., 8(4), Svendsen, Ib. A. (1987), Analysis of suf zone tubulence, J. Geophys. Res., 9, Svendsen, Ib. A. (6), Intoduction to Neashoe Hydodynamics, Wold Sci. Pess, Singapoe, 7 pp. Symonds, G., Black, K. P. and Young I. R. (1995), Wave-diven flow ove shallow eefs, J. Geophys. Res.,1 (C), Tait, R. J. (197), Wave set-up on coal eefs, J. Geophys. Res., 77, Tao, J. (1983), Computation of Wave Run-up and Wave Beaking, Intenal Repot, Danish Hydaulics Institute, Hosholm, Denmak. Thonton, E. B. and Guza, R. T. (1983), Tansfomation of wave height distibution, J. Geophys. Res., 88 (C1), Ting, F. C. K. and Kiby, J. T. (1994), Obsevation of undetow and tubulence in a laboatoy suf zone, Coast. Eng., 4(1-), Toes-Feyemuth, A., Losada, I. J. and Laa, J. L. (7), Modeling of suf zone pocesses on a natual beach using Reynolds-Aveaged Navie-Stokes equations, J. Geophys. Res., 11, C914, doi: 1.19/6JC45. Toes-Feyemuth, A., Laa, J. L. and Losada, I. J. (1), Numeical modeling of shot- and long-wave tansfomation on a baed beach, Coast. Eng., 57(3), Tsai, C. P., Chen, H. B., Hwung, H. H. and Huang, M. J. (5), Examination of empiical fomulas fo wave shoaling and beaking on steep slopes, Ocean Eng., 3(3-4), U.S. Amy Cops of Enginees (USACE) (3), Coastal Engineeing Manual, Chap. II-4 and II-7, U.S. Amy Enginee Reseach and Development Cente, Coastal and Hydaulics Laboatoy, Vicksbug, MS. Van de Mee, J. W. and d Angemond, K. (1991), Wave tansmission at lowcested stuctues, Poc. ICE Confeence Coastal Stuctues and Beakwates, Institution of Civil Enginees, Thomas Telfod, London, U.K. pp

256 Van de Mee, J. W. and Daemen, I. F. R. (1994), Stability and wave tansmission at low cested ubble mound stuctues, J. Wtwy., Pot, Coast. and Ocean Eng., ASCE, 1 (1), Van de Mee, J. W. and Janssen, J. P. F. M. (1995), Wave un-up and wave ovetopping at dikes, In: Kobayashi, N., Demibilek, Z. (Eds.), Wave Foces on Inclined and Vetical Stuctues, Ch. 1, pp Van de Mee, J. W., Biganti, R., Zanuttigh, B. and Wang, B. (5), Wave tansmission and eflection at low cested stuctues: design fomulae, oblique wave attack and spectal change, Coast. Eng. 5 (1-11), Veeamony, J. and Svendsen, Ib. A. (), The flow in suf-zone waves, Coast. Eng., 39(-4), Vette, O., Becke, J. M., Meifield, M. A., Pequignet, A.-C., Aucan, J., Boc, S. J. and Pollock, C. E. (1), Wave setup ove a Pacific Island finging eef, J. Geophys. Res., 115, C166, doi:1.19/1jc6455. Vicinanza, D., Cácees, I., Buccino, M., Gionella, X. and Calabese, M. (9), Wave distubance behind low-cested stuctues: diffaction and ovetopping, Coast. Eng., 56(11-1), Weggel, J. R. (197), Maximum beake height, J. Wtwy., Hab. and Coast. Eng. Div. ASCE, 98(WW4), Wei, G., Kiby, J. T., Gilli, S. T. and Subamanya, R. (1995), A fully nonlinea Boussinesq model fo suface waves. Pat 1. Highly nonlinea unsteady waves, J. Fluid Mech., 94, Wei, G., Kiby, J. T. and Sinha, A. (1999), Geneation of waves in Boussinesq models using a souce function method, Coast. Eng., 36(4), Wilmott, C. J. (1981), On the validation of models, Phys. Geog.,, Yao, Y., Lo, E. Y. M., Huang, Z. H. and Monismith S. G. (9), An Expeimental Study of wave-induced set-up ove a hoizontal eef with an idealized idge, Poc. 8th Int. Conf. Offshoe Mech. and Atic Eng., ASME, Honolulu, Hawaii, USA., Vol. 6, pp Young, I. R. (1989), Wave tansfomations on coal eefs, J. Geophys. Res., 94 (C7),

257 Zanuttigh, B., Matinelli, L. and Lambeti, A. (8), Wave ovetopping and piling-up at pemeable low cested stuctues, Coast. Eng., 55 (6), Zelt, J. A. (1991), The un-up of nonbeaking and beaking solitay waves, Coast. Eng., 15(3),

258 APPENDIX A: EXPERIMENTAL DATA IN CHAPTER The expeimental data fo diffeent combinations of eef-flat submegences ( h ) and foe-eef slopes ( s ) ae shown in Tables A.1 - A.7. Tables A1, A, A3 and A4 ae the souces of dataset 1 (shown in Fig..5) while Tables A, A5, A6, A7 ae the souces of dataset (shown in Fig..6). Table A.1 Measued data with h.3 m and s 1/6 H (m) T (s) Beake type Beake location H b (m) h b (m) L s (m) (m) plunging foe-eef plunging foe-eef plunging foe-eef plunging foe-eef plunging foe-eef plunging eef edge plunging eef flat plunging eef flat plunging eef flat spilling eef flat spilling eef flat plunging foe-eef plunging foe-eef plunging foe-eef plunging foe-eef plunging foe-eef plunging eef edge plunging eef flat spilling eef flat spilling eef flat plunging foe-eef plunging foe-eef plunging foe-eef plunging foe-eef plunging foe-eef plunging eef edge spilling eef flat spilling eef flat K t K 9

259 Table A. Measued data with h.5 m and s 1/6 H (m) T (s) Beake type Beake location H b (m) h b (m) L s (m) (m) plunging foe-eef plunging foe-eef plunging foe-eef plunging foe-eef plunging foe-eef plunging foe-eef plunging eef edge plunging eef flat plunging eef flat spilling eef flat spilling eef flat spilling eef flat non non non plunging foe-eef plunging foe-eef plunging foe-eef plunging foe-eef plunging foe-eef plunging foe-eef plunging eef edge plunging eef flat plunging eef flat plunging eef flat spilling eef flat spilling eef flat spilling eef flat non non plunging foe-eef plunging foe-eef plunging foe-eef plunging foe-eef plunging foe-eef plunging foe-eef plunging eef edge plunging eef flat spilling eef flat spilling eef flat spilling eef flat spilling eef flat K t K 3

260 non Table A.3 Measued data with h.7 m and s 1/6 H (m) T (s) Beake type Beake location H b (m) h b (m) L s (m) (m) plunging foe-eef plunging foe-eef plunging foe-eef plunging eef edge plunging eef flat plunging eef flat plunging eef flat plunging eef flat plunging eef flat spilling eef flat spilling eef flat spilling eef flat non non non plunging foe-eef plunging foe-eef plunging foe-eef plunging eef edge plunging eef flat plunging eef flat plunging eef flat plunging eef flat plunging eef flat spilling eef flat spilling eef flat spilling eef flat spilling eef flat non non plunging foe-eef plunging foe-eef plunging foe-eef plunging foe-eef plunging foe-eef plunging eef edge plunging eef flat plunging eef flat K t K 31

261 spilling eef flat spilling eef flat spilling eef flat spilling eef flat spilling eef flat non non Table A.4 Measued data with h.1 m and s 1/6 H (m) T (s) Beake type Beake location H b (m) h b (m) L s (m) (m) plunging foe-eef plunging foe-eef plunging eef edge plunging eef flat plunging eef flat plunging eef flat plunging eef flat plunging eef flat spilling eef flat spilling eef flat spilling eef flat spilling eef flat non non plunging foe-eef plunging foe-eef plunging foe-eef plunging eef edge plunging eef flat plunging eef flat plunging eef flat plunging eef flat plunging eef flat spilling eef flat spilling eef flat spilling eef flat non non plunging foe-eef plunging foe-eef plunging foe-eef K t K 3

262 plunging foe-eef plunging eef edge plunging eef flat plunging eef flat plunging eef flat plunging eef flat spilling eef flat spilling eef flat non eef flat non non Table A.5 Measued data with h.5 m and s 1/3 H (m) T (s) Beake type Beake location H b (m) h b (m) L s (m) (m) plunging foe-eef plunging foe-eef plunging foe-eef plunging foe-eef plunging foe-eef plunging eef edge plunging eef flat plunging eef flat plunging eef flat spilling eef flat spilling eef flat spilling eef flat non non plunging foe-eef plunging foe-eef plunging foe-eef plunging foe-eef plunging foe-eef plunging eef edge plunging eef flat plunging eef flat spilling eef flat spilling eef flat non non plunging foe-eef K t K 33

263 plunging foe-eef plunging foe-eef plunging foe-eef plunging foe-eef plunging foe-eef plunging eef edge spilling eef flat spilling eef flat spilling eef flat spilling eef flat non Table A.6 Measued data with h.5 m and s 1/9 H (m) T (s) Beake type Beake location H b (m) h b (m) L s (m) (m) plunging foe-eef plunging foe-eef plunging foe-eef plunging foe-eef plunging eef edge plunging eef flat plunging eef flat plunging eef flat spilling eef flat spilling eef flat non plunging foe-eef plunging foe-eef plunging foe-eef plunging foe-eef plunging foe-eef plunging eef edge plunging eef flat plunging eef flat spilling eef flat spilling eef flat non plunging foe-eef plunging foe-eef plunging foe-eef plunging foe-eef plunging foe-eef K t K 34

264 plunging eef edge plunging eef flat spilling eef flat spilling eef flat spilling eef flat non Table A.7 Measued data with h.5 m and s 1/1 H (m) T (s) Beake type Beake location H b (m) h b (m) L s (m) (m) plunging foe-eef plunging foe-eef plunging foe-eef plunging foe-eef plunging eef edge plunging eef flat plunging eef flat plunging eef flat spilling eef flat spilling eef flat non plunging foe-eef plunging foe-eef plunging foe-eef plunging foe-eef plunging eef edge plunging eef flat plunging eef flat plunging eef flat spilling eef flat spilling eef flat non plunging foe-eef plunging foe-eef plunging foe-eef plunging foe-eef plunging eef edge plunging eef flat spilling eef flat spilling eef flat non K t K 35

265 APPENDIX B: EXPERIMENTAL DATA IN CHAPTER 3 The expeimental data fo diffeent combinations of wave conditions (monochomatic and spectal) and eef pofiles (with/without the idge) ae shown in Tables B.1 - B.4. Table B.1 Measued data with monochomatic waves in the absence of the idge H (m) h.35 m h.4 m h.45 m T (s) (m) H K (m) T (s) (m) H K (m) T (s) (m) K 36

266 Table B. Measued data with monochomatic waves in the pesence of the idge H (m) h.4 m h.41m T (s) (m) H K (m) T (s) (m) H (m) h.4 m h.45 m T (s) (m) H K (m) T (s) (m) K K 37

267 Table B.3 Measued data with spectal waves in the absence of the idge H s (m) h.35 m h.4 m h.45 m T p (s) (m) K H s (m) T p (s) (m) K H s (m) T p (s) (m) K Table B.4 Measued data with spectal waves in the pesence of the idge H s (m) h.4 m h.4 m h.45 m T p (s) (m) K H s (m) T p (s) (m) K H s (m) T p (s) (m) K 38

268 APPENDIX C: LABORATORY STUDY ON EFFECTS OF ROUGHNESS ON WAVE-INDUCED SETUP 3 Intoduction In section 3.5, an offshoe-diected flow is identified unde the tough of the suf zone waves popagating ove the idealized platfom eef model (i.e., the eef pofile without the idge). The bottom fiction may dissipate the enegy associated with the undetow, and affect the wave setup (see section 7.5 fo a theoetical desciption). Pevious expeimental investigations (i.e., Goualy, 1996a; Demibilek et al., 7) as well as the expeiments in Chapte 3 focused mainly on the wave setup vaiations with the offshoe wave conditions fo elatively smooth eef flats. Howeve, field measuements of eef flat suface oughness showed that coal eef has a vey high dag coefficient being some odes of magnitude highe than that associated with nomal shelves (Falte et al., 4; Lowe et al., 5; Hench et al., 8). To undestand the effects of eef suface oughness on wave-beaking and waveinduced setup, additional laboatoy expeiments wee conducted unde a seies of monochomatic wave conditions. It has been found in Chapte that most of the suf zone aea was located on the eef flat, thus only the vaiation of the oughness on eef flat was studied. Two eef-flat models wee examined: one with a smooth eef flat and anothe with a poous eef flat. Wate suface elevations at diffeent locations wee measued to obtain the vaiations of wave-induced setup along the eef flat. Expeimental esults ae epoted hee fo one wate depth and eighteen wave conditions. The wave-induced setups ove a smooth eef flat ae compaed with those ove poous eef flat. Expeimental Setup Wave flume and eef models The expeimental settings ae identical to that epoted in section 3. (also see Fig. C.1), except fo the aangement of the eef flat. Fo the smooth eef flat model, 3 This pat of wok has been pesented at the 5th Intenational Confeence on Asian and Pacific Coasts, Singapoe, 9. 39

269 both the slope and the hoizontal platfom wee made of PVC plates to simulate a smooth suface as shown in Figs. C.(a) and C.(b). Poous mats with 3 cm in height and 95.5% in poosity wee paved unifomly on the eef flat to simulate the poous bottom. The top elevation of the poous mats is the same as that of the smooth eef flat (see Figs. C.(c) and C.(d)) so that the wave setups ove these two models can be compaed. Wavemake G1 G G3 G4 G5 G6.3.4 G7 G8 G9 G Still wate level G: Wave gauge Unit: m h Foe-eef (1:6). Reef flat with poous mats Beach (1:8) with poous mateials Fig. C.1 Sketch of the expeimental setup. Aangement of instuments and expeimental pocedues In vey shallow wate ove the eef flat, fou Ultalab sensos (G7 - G1, Geneal Acoustics Ltd.) wee used to measue the fee suface elevation. The distances between the adjacent wave gauges wee optimized by some pilot tests to captue the location whee the maximum setup may occu. G7 was 1. m away fom the eef edge to ensue that this senso was used outside the suf zone. Thee esistance-type wave pobes (G4 - G6, HR Wallingfod Ltd.) wee used to measue the setup in the suface zone on the eef flat. Befoe the eef model, two pobes (G1 and G) wee used to sepaate the incident waves fom the eflected waves. One moe pobe (G3) was placed on the foe-eef slope to estimate wave shoaling. The taget incident wave conditions wee selected fom a combination of five incident wave heights (anging fom.5 m to.13 m) and fou wave fequencies (anging fom.6 Hz to 1. Hz). To compae the wave setups ove the smooth eef flat and ove the poous eef flat, simila wave conditions wee used in the expeiments. 4

270 To guaantee that the eef-flat was shallow enough and fictional dissipation might be dominant when waves came, the wate depth ( h ) was fixed at.35 m in this study, i.e., the still wate level coincides with the eef-flat elevation. The tests wee fist caied out fo the smooth eef flat model unde total 18 wave conditions, and then the same wave conditions wee epeated fo the poous eef flat model. Fig. C. Reef flat models: (a) smooth eef flat (Side view); (b) smooth eef flat (Font view); (c) poous eef flat (Side view); (d) poous eef flat (Font view). Results and Discussion Wave tansfomation ove the eef flat Wave beaking commonly occued on the slope o on the eef flat closed to the eef edge as a plunging beake fo all cases (e.g., Fig. C.3). Both the beaking point and beake type wee almost the same fo these two eef models unde the same incident wave condition. Fo the smooth eef flat, the suf zone might extend to one o two metes into the eef flat befoe wave beaking ceased and the oscillatoy waves efomed. The efomed waves could popagate ove the eef flat and each 41

271 the beach at the end of the flume. Howeve, fo the poous eef flat, the suf zone became naowe and waves efomed ealie, the boe disappeaed vey fast due to the inceased bottom fiction. The efomed waves afte wave beaking attenuated quickly due to the enegy loss caused by the flows inside the poous mat. Consequently, the magnitude of wave-induced setup should be educed accoding to the adiation stess theoy in the suf zone descibed by Eq. (1.3). This conjectue will be veified in the following section. It could also be obseved that fo the poous eef flat, the efomed wave only popagated ove a shot distance befoe it damped away at the ea of the eef flat by fictional dissipation, hence waves could not each the beach at the end of the flume. The eflection coefficient is small fo most cases (less than %) and should have little effect on the esults. (a) (b Fig. C.3 Wave beaking: (a) smooth eef flat; (b) poous eef flat ( H.1 m, T 1.5 s, h.35 m). Wave-induced setup with diffeent eef flat models The maximum wave-induced setup as a function of the deep-wate wave height and the wave peiod is shown in Fig. C.4(a) fo the two eef flat models. Fo the smooth eef flat, the maximum setup was obtained fom G7; fo the poous eef flat, the wave data fom G wee used to obtain the maximum wave setup. The wave setup aveaged ove the width of the eef flat was calculated by aveaging the measuements fom the 4 sensos (G7 - G1) and 3 pobes (G4 - G6) on the eef flat. The vaiation of the aveaged wave setup with the deep-wate wave height and the wave peiod ae shown in Fig. C.4(b) fo the two eef-flat models. 4