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Advancement and Validation of Numerical Storm Surge Modeling on Coral Reefs Using Laboratory Comparisons

Permanent Link: http://ufdc.ufl.edu/UFE0042548/00001

Material Information

Title: Advancement and Validation of Numerical Storm Surge Modeling on Coral Reefs Using Laboratory Comparisons
Physical Description: 1 online resource (241 p.)
Language: english
Creator: Hesser, Tyler
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: breaking, coral, hurricanes, models, ocean, stwave, waves
Civil and Coastal Engineering -- Dissertations, Academic -- UF
Genre: Coastal and Oceanographic Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Present numerical storm surge models were developed and tested on gentle sloping coastlines common in the continental United States. Islands with fringing coral reefs have complex small scale bathymetry with steep front slopes which diminish confidence in the accuracy of present storm surge models. Two laboratory experiments were performed using detailed bathymetry based on SHOALS measurements from the coast of Guam. A two dimensional wave flume with front slopes of 1:2.5 and 1:5 was used to test numerical approaches to energy loss due to breaking waves on steep slopes. A three dimensional wave basin was used for updating and validating the coupled STWAVE-ADCIRC storm surge model for the coral reef environment. Results indicated the need for greater dissipation of energy during breaking for the steep slopes than is currently the standard. Through testing in a 1D energy flux model, greater accuracy was found when the energy loss due to breaking was approximated with a bore type energy dissipation model. The wave set-up was calculated using the STWAVE-ADCIRC coupled storm surge model using both the present STWAVE breaking and the bore type breaking functions. The use of the bore type energy dissipation model was found to better match the measured data in the laboratory than the present STWAVE breaking model.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Tyler Hesser.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Slinn, Donald N.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0042548:00001

Permanent Link: http://ufdc.ufl.edu/UFE0042548/00001

Material Information

Title: Advancement and Validation of Numerical Storm Surge Modeling on Coral Reefs Using Laboratory Comparisons
Physical Description: 1 online resource (241 p.)
Language: english
Creator: Hesser, Tyler
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: breaking, coral, hurricanes, models, ocean, stwave, waves
Civil and Coastal Engineering -- Dissertations, Academic -- UF
Genre: Coastal and Oceanographic Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Present numerical storm surge models were developed and tested on gentle sloping coastlines common in the continental United States. Islands with fringing coral reefs have complex small scale bathymetry with steep front slopes which diminish confidence in the accuracy of present storm surge models. Two laboratory experiments were performed using detailed bathymetry based on SHOALS measurements from the coast of Guam. A two dimensional wave flume with front slopes of 1:2.5 and 1:5 was used to test numerical approaches to energy loss due to breaking waves on steep slopes. A three dimensional wave basin was used for updating and validating the coupled STWAVE-ADCIRC storm surge model for the coral reef environment. Results indicated the need for greater dissipation of energy during breaking for the steep slopes than is currently the standard. Through testing in a 1D energy flux model, greater accuracy was found when the energy loss due to breaking was approximated with a bore type energy dissipation model. The wave set-up was calculated using the STWAVE-ADCIRC coupled storm surge model using both the present STWAVE breaking and the bore type breaking functions. The use of the bore type energy dissipation model was found to better match the measured data in the laboratory than the present STWAVE breaking model.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Tyler Hesser.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Slinn, Donald N.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0042548:00001


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1 ADVANCEMENT AND VALIDATION OF NUMERICAL STORM SURGE MODELING ON CORAL RE EFS USING PHYSICAL MODEL COMPAR I S ONS By TYLER J. HESSER A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2010

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2 2010 Tyler J. Hesser

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3 To Ryan and our wonderfully helpful puppies, Ace and Bella

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4 ACKNOWLEDGMENTS I would like to thank my committee for their guidance during my Ph.D. work. I would like to thank the USACE ERDC CHL for helping me to develop and grow while working in Vicksburg, MS. I would especially like to thank Jane Smith and Ernie Smith for their guidance through the entire project. I would also like to thank the SMART fellowship for funding my work the last three years. I would like to thank my Mom, Dad, and brother, Scott for being so supportive of me. Finally I could not have survived this process without my wonderful wife Ryan. This goal would never have been reached without your love and support.

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5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ...............................................................................................................4 LIST OF TABLES ...........................................................................................................................8 LIST OF FIGURES .........................................................................................................................9 ABSTRACT ...................................................................................................................................18 CHAPTER 1 INTRODUCTION ..................................................................................................................19 Coral Reef Laboratory ............................................................................................................20 Numerical Wave Models ........................................................................................................21 Maximum Wave Height Equation ...................................................................................23 Energy Dissipation Due To Breaking ..............................................................................28 Wave Set Up ...........................................................................................................................36 Present Study ..........................................................................................................................37 2 WAVE FLUME LABORATORY .........................................................................................39 Introduction .............................................................................................................................39 M ethodology ...........................................................................................................................39 Experiment Facilities .......................................................................................................39 Laboratory Conditions .....................................................................................................40 Instrumentation ................................................................................................................41 Test Conditions ................................................................................................................42 Testing Procedure ............................................................................................................43 Post Processing .......................................................................................................................43 Test Cases ........................................................................................................................43 Reflection Analysis .........................................................................................................44 Spe ctral Analysis .............................................................................................................45 Statistical Wave Measurements .......................................................................................47 Wave Gauge Confidence I nterval ...................................................................................47 Smooth Surface Results ..........................................................................................................49 Spectral Wave Energy .....................................................................................................49 Low Frequency Spectral Wave Energy ...........................................................................51 Zero Moment Wave Height ............................................................................................53 W ave Set up ....................................................................................................................55 Wave Run up ...................................................................................................................56 Rough Surface Comparisons ..................................................................................................56 Wave Height ....................................................................................................................56 Summary .................................................................................................................................56

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6 3 WAVE BREAKING ...............................................................................................................96 Introduction .............................................................................................................................96 Methodology ...........................................................................................................................96 One Dimensional Energy Flux Balance ..........................................................................96 Error Calculation .............................................................................................................98 Test Cases ........................................................................................................................98 Results .....................................................................................................................................99 Energy Dissipation ..........................................................................................................99 B Coefficient vs. Gamma ..............................................................................................103 Breaking Wave Height ..................................................................................................107 Summary ...............................................................................................................................108 4 WAVE BASIN LABORATORY .........................................................................................133 Introduction ...........................................................................................................................133 Methodology .........................................................................................................................133 Exp eriment Facilities .....................................................................................................133 Laboratory Conditions ...................................................................................................133 Instrumentation ..............................................................................................................135 Testing Procedure ..........................................................................................................136 Post Processing .....................................................................................................................137 Test Cases ......................................................................................................................137 Reflection Analysis .......................................................................................................137 Spectral Analysis ...........................................................................................................138 Statistical Analysis ........................................................................................................139 Wave Gauge Confidence Intervals ................................................................................139 Results ...................................................................................................................................140 Spectral Energy .............................................................................................................140 Low Frequency Spectral Energy ...................................................................................141 Wave Heights ................................................................................................................142 Wave Set up ..................................................................................................................143 Summary ...............................................................................................................................144 5 WAVE BASIN MODELING ...............................................................................................172 Introduct ion ...........................................................................................................................172 Methods ................................................................................................................................172 Results ...................................................................................................................................173 B Coefficient versus Gamma .........................................................................................173 Maximum Breaking Equations ......................................................................................175 First Order B ..................................................................................................................177 Summary ...............................................................................................................................178

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7 6 1D WAVE SET UP ..............................................................................................................191 Introduction ...........................................................................................................................191 Methods ................................................................................................................................191 Results ...................................................................................................................................192 Wave Flume Laboratory Wave Set up ..........................................................................192 Wave Basin Laboratory Wave Set up ...........................................................................193 Summary ...............................................................................................................................195 7 RADIATION STRESS CORRECTION ..............................................................................204 Background ...........................................................................................................................204 Friction ..................................................................................................................................205 Wind Input ............................................................................................................................207 8 COUPLED SET UP MODEL ..............................................................................................217 Introduction ...........................................................................................................................217 Methods ................................................................................................................................217 STWAVE ......................................................................................................................217 ADCIRC ........................................................................................................................219 Coupling System ...........................................................................................................219 Results ...................................................................................................................................220 Summary ...............................................................................................................................221 9 CONCLUSIONS ..................................................................................................................229 LIST OF REFERENCES .............................................................................................................235 BIOGRAPHICAL SKETCH .......................................................................................................240

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8 LIST OF TABLES Table page 21 Location of gauges in flume laboratory. The location of x = 0 is at the wave maker. .....59 22 Designed wave conditions for three water levels in prototype (50:1) scale. .....................60 23 Laboratory tests for the 1:2.5 front reef slope. Tests on the not painted surface have a Top of s and tests on the painted surface have a Top of r. ..............................................61 24 Laboratory tests for the 1:5 front reef slope. Tests on the not painted surface have a Top of s and tests on the painte d surface have a Top of r. .................................................62 25 The 95% confidence interval for each gauge in meters. The mean of the CI was calculated for all tests at each water level. .........................................................................63 31 Mean rms error for three breaking models in 1D model .................................................110 32 Mean rms error for maximum breaking wave height equations in 1D model .................111 41 Location of capacitance wire gauges in 3D basin. The location of (0,0) was at the wave maker on the left side of the basin when looking offshore. ....................................146 42 Test conditions measured from the reflection analysis at gauges 4, 5, and 6. .................147 43 The 95% confidence interval for capacitance wave gauges in meters. ...........................148 51 Mean RMS percent error for the zero moment wave heights in the wave basin laboratory comparisons along with the best fit B coefficient. .........................................180 61 Mean RMS percent error for the wave set up in the wave flume laboratory comparisons. ....................................................................................................................197 62 Mean RMS percent error for the wave set up in the wave basin laboratory comparisons. ....................................................................................................................198

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9 LIST OF FIGURES Figure page 21 Laboratory set up for both the 1:2.5 and 1:5 slopes. The reef profile was built on an existing concrete slope which is the red section on the lower frame. ................................64 22 Picture of the 1:5 reef face slope. The 1:5 slope was placed on top of the 1:2.5 slope. ...65 23 Picture from just off the front edge of the 1:5 reef face slope. The blue paint is the cut acrylic glass. The divider on the left side of the picture was put in place to decrease the testing width and to dampen reflection. ........................................................66 24 Contours of the acrylic reef top. The axis is distorted to fit the page. ..............................67 25 Picture of the cut acrylic glass from on top of the flume. Each piece was 2 feet x 2 feet. .....................................................................................................................................68 26 The capacitance wave gauges were placed in the flume. The gauges were connected to a motor for calibration. ..................................................................................................69 27 Bathymetry for both the 1:5 slope tests ( ) and the 1:2.5 slope tests ( ). The three water levels used in test are displayed relative to the bathymetry. ....................................70 28 The reflection coefficient calculated using the Goda analysis for all test cases against the surf similarity parameter. .............................................................................................71 29 Example of full time series from Test 7s at gauge 2. The time series consisted of 20 minutes or 1200 seconds of measurements. A) Time from 0300 seconds. B) Time from 301600 seconds. C) Time from 601900 seconds. D) Time from 9011200 seconds. ..............................................................................................................................72 210 Energy density spectra from Test 7s, gauge 2. A) S(f) for the raw data without filtering. B) The low frequency S(f). C) The high frequency S(f). The low frequency cut off was at 1 5 .........................................................................................73 211 Time series of the high frequency waves ( ) and the low frequency waves ( ). A) Time from 0 300 seconds. B) Time from 301600 seconds. C) Time from 601900 seconds. D) Time from 9011200 seconds. ......................................................................74 212 The profile of the zero moment wave hei ght for Test 7s with the 95% confidence interval error bars calculated for the full experiment. ........................................................75 213 Energy density spectra for Test 7s in low water on 1:2.5 slope. A) The transformation of the energy density spectrum from offshore. B) The front edge of the reef top. C) The back section of the reef top. ..............................................................76

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10 214 Energy density spectra for Test 21s in mid water on 1:2.5 slope. A) The transformation of the energy density spectrum from offshore. B) The front edge of the reef top. C) The back section of the reef top. ..............................................................77 215 Energy density spectra for Test 36s in high water on 1:2.5 slope. A) The transformation of the energy density spectrum from offshore. B) The front edge of the reef top. C) The back section of the reef top. ..............................................................78 216 Energy density spectra for Test 50s in mid water on 1:5 slope A) The transformation of the energy density spectrum from offshore. B) The front edge of the reef top. C) The back section of the reef top. ..............................................................79 217 Energy density spectra for Test 64s in mid water on 1:5 slope. A) The transformation of the energy density spectrum from offshore. B) The front edge of the reef top. C) The back section of the reef top. ..............................................................80 218 Energy density spectra for Test 79s in mid water on 1:5 slope. A) The transformation of the energy density spectrum from offshore. B) The front edge of the reef top. C) The back section of the reef top. ..............................................................81 219 The low frequency energy density spectra from Test 7s in low wate r on 1:2.5 slope. A) The transformation of the energy density spectrum from offshore. B) The front edge of the reef top. C) The back section of the reef top. .................................................82 220 The low frequency energy density spectra from Test 21s in mid water on 1:2.5 slope. A) The transformation of the energy density spectrum from offshore. B) The front edge of the reef top. C) The back section of the reef top. .................................................83 221 The low frequency energy density spectra from Test 36s in high water on 1:2.5 slope. A) The transformation of the energy density spectrum from offshore. B) The front edge of the reef top. C) The back section of the reef top. ........................................84 222 T he low frequency energy density spectra from Test 50s in low water on 1:5 slope. A) The transformation of the energy density spectrum from offshore. B) The front edge of the reef top. C) The back section of the reef top. .................................................85 223 The low frequency energy density spectra from Test 64s in mid water on 1:5 slope. A) The transformation of the energy density spectrum from offshore. B) The front edge of the reef top. C) The back section of the reef top. .................................................86 224 The low frequency energy density spectra from Test 79s in mid water on 1:5 slope. A) The transformation of the energy density spectrum from offshore. B) The front edge of the reef top. C) The back section of the reef top. .................................................87 225 The profile of the low frequency zero moment wave height ( ) and the highfrequency zero moment wave height ( ) on the 1:2.5 slope. A) Low water, Test 7s. B) Mid water, Test 21s. C) High water, Test 36s. ............................................................88

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11 226 The profile of the low frequency zero moment wave height ( ) and the highfrequency zero moment wave height ( ) on the 1:5 slope. A) Low water, Test 50s. B) Mid water, Test 64s. C) High water, Test 79s. ............................................................89 227 The highfrequency () with the mean water level (*) measured at each gauge in the profile on the 1:2.5 slope. A) Low water, Test 7s. B) Mid water, Test 21s. C) High water, Test 36s. ....................................................................................................90 228 The highfrequency () with the mean water level (*) measured at each gauge in the profile on the 1:5 slope. A) Low water, Test 50s. B) Mid water, Test 64s. C) High water, Test 79s. .........................................................................................................91 229 Maximum mean water level measured on the reef top for all smooth surface test cases compared with offshore wave power. ......................................................................92 230 The 2% run up measured on the runup gauge on the back slope of the laboratory bathymetry for all test cases with the s mooth surface compared against the offshore wave power. .......................................................................................................................93 231 Non dimensional compared at each gauge for the smooth surface tests () and the rough surface tests (o) for all the 1:2.5 tests. A) Low water. B) Mid water. C) High water. .........................................................................................................................94 232 Non dimensional compared at each gauge for the smooth surface tests () and the rough surface tests (o) for all the 1:5 tests. A) Low water. B) Mid water. C) High water. .........................................................................................................................95 31 Bathymetry for the numerical model simulations of the wave flume. The model was tested on both the 1:2.5 ( ) and the 1:5 ( ) slopes. Three water levels were tested in the model: low water wi ...........................................................112 32 Comparison ), with B = 1.2, and JBAB07 ( ), with B = 4.0, energy dissipation functions. A) Typical wave height profile from the 1:2.5 slope (x), Test 7s. B) Change in error for each breaking model with different B coefficients. ................................................................................113 33 Comparison of the STWAVE breaking wave height equation tested in the threshold breaking function ) with B = 1.2, and JBAB07 ( ) with B = 2.3. A) Zero moment wave height comparisons with measured results from Test 7s with the 1:2.5 slope (x). (B) Change in error for each breaking model with different B coefficients. ......................................................................................................................114 34 ), with B = 1.2, and JBAB07 ( ), with B = 4.0, energy dissipation funct ions. A) Typical wave height profile from the 1:5 slope (x), Test 50s. B) Change in error for each breaking model with different B coefficients. ................................................................................115

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12 35 Comparison of the STWAVE breaking wave height equation tested in the threshold ) with B = 1.2, and JBAB07 ( ) with B = 2.3. A) Zero moment wave height comparisons with measured results from T est 50s with the 1:5 slope (x). (B) Change in error for each breaking model with different B coefficients. ......................................................................................................................116 36 Analysis of the mean rms error based on the coefficient of the maximum breaking wave height ( ) and B on the 1:2.5 slope. A) The TG83 energy dissipa tion function. B) The JBAB07 energy dissipation function. ..................................................................117 37 The mean rms error based on changing B and in TG83 for different segments of the wave breaking profile on the 1:2.5 slope. A) Offshore region. B) Breaking region. C) Dissipation region. D) Reef top. ...................................................................118 38 The mean rms error based on changing B and in JBAB07 for different segments of the wave breaking profile on the 1:2.5 slope. A) Offshore region. B) Breaking region. C) Dissipation region. D) Reef top. ...................................................................119 39 The mean rms error based on changing B and in TG83 for different wave periods on the 1:2.5 slope. A) T = 1.0 seconds. B) T = 1.4 seconds. C) T = 1.8 seconds. D) T = 2.2 seconds. E) T = 2.8 seconds. ..............................................................................120 310 The mean rms error based on changing B and in JBAB07 for different wave periods on the 1:2.5 slope. A) T = 1.0 seconds. B) T = 1.4 seconds. C) T = 1.8 seconds. D) T = 2.2 seconds. E) T = 2.8 seconds. .........................................................121 311 The mean rms error based on changing B and in TG83 for different wave heights on the 1:2.5 slope. A) Initial wave heights below 0.10 meters. B) Initial wave heig hts between 0.1 and 0.12 meters. C) Initial wave heights greater than 0.12 meters. ..............................................................................................................................122 312 The mean rms error based on chang ing B and in JBAB07 for different wave heights on the 1:2.5 slope. A) Initial wave heights below 0.10 meters. B) Initial wave heights between 0.1 and 0.12 meters. C) Initial wave heights greater than 0.12 meters. ..............................................................................................................................123 313 Analysis of the mean rms error based on the coefficient of the maximum breaking wave height ( ) and B on the 1:5 slope. A) The TG83 energy dissipation function. B) The JBAB07 energy dissipation function. ..................................................................124 314 The mean rms error based on changing B and in TG83 for differe nt segments of the wave breaking profile on the 1:5 slope. A) Offshore region. B) Breaking region. C) Dissipation region. D) Reef top. ................................................................................125 315 The mean rms error based on changing B and in JBAB07 for different segments of the wave breaking profile on the 1:5 slope. A) Offshore region. B) Breaking region. C) Dissipation region. D) Reef top. ................................................................................126

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13 316 The mean rms error based on changing B and in TG83 for different wave periods on the 1:5 slope. A) T = 1.0 seconds. B) T = 1.4 seconds. C) T = 1.8 seconds. D) T = 2.2 seconds. E) T = 2.8 seconds. .................................................................................127 317 The mean rms error based on changing B and in JBAB07 for different wave periods on the 1:5 slope. A) T = 1.0 seconds. B) T = 1.4 seconds. C) T = 1.8 seconds. D) T = 2.2 seconds. E) T = 2.8 seconds. .........................................................128 318 The mean rms error based on changing B and in TG83 for different wave heights on the 1:5 slope. A) Initial wave heights below 0.10 meters. B) Initial wave heights between 0.1 and 0.12 meters. C) Initial wave he ights greater than 0.12 meters. ...........129 319 T he mean rms error based on changing B and in JBAB07 for different wave heights on t he 1:5 slope. A) Initial wave heights below 0.10 meters. B) Initial wave heights between 0.1 and 0.12 meters. C) Initial wave heights greater than 0.12 meters. ..............................................................................................................................130 320 ), and Ru03 ( ) breaking equations were tested on the 1:2.5 slope. A) TG83 wave height profile from Test 7s in low water. B) Error with TG83 for values of B. C.) JBAB07 wave height profile from Test 7s in low water. D) Error with JBAB07 for values of B. ..........................................................................131 321 ), and Ru03 ( ) breaking equations were tested on the 1:5 slope. A) TG83 wave height profile from Test 7s in low water. B) Error with TG83 for values of B. C.) JBAB07 wave height profile from Test 50s in low water. D) Error with JBAB07 for values of B. ................................................................................132 41 The Directional Spectral Wave Generator Basin (DSWG) layout. The wave generator was located on the left side at x = 0. Wave guides w ere located on each side of the reef study area, and wave absorbers were located behind the study area. .....149 42 Designed bathymetric profile of reef top. The reef top was designed by Dr. Ernie Smith with units of meters. ..............................................................................................150 43 Pictur e of 2 ft x 2ft acrylic pieces with bolts on all four corners for adjustability. .........151 44 Picture of front edge of reef struct ure. The front slope is a steel platform which has marine plywood on top to ensure a smooth transition to the front edge of the acrylic glass. .................................................................................................................................152 45 Picture taken standing on the back slope of reef platform. The back of the designed acrylic reef was smoothed back to the back slope of the steel platform with marine plywood. ...........................................................................................................................153 46 The actual bathymetry of the acrylic reef top. The section was scanned using a LIDAR system to accurately measure the detailed vertical change. ...............................154

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14 47 The difference between the actual reef top contours and the designed reef top contours. Positive values of elevation rela te to higher actual elevations than designed. ..........................................................................................................................155 48 Bathymetric profile for the laboratory basin test. The three water levels tested were the low water, h = 0.418 (m) ( ), the mid water, h = 0.442 (m) ( ) and the high water, h = 0.493 (m) ( ). ..................................................................................................156 49 The reflection coefficient values for all tests calculated using the Goda analysis. .........157 410 Example of an entire time ser ies measured from gauge 8 during Test 36. The test consisted of 20 minutes or 1200 seconds of continuous waves. A) Time from 0300 seconds. B) Time from 301600 seconds. C) Time from 601900 seconds. D) Time from 9011200 seconds. ...................................................................................................158 411 Example of the energy density spectrum for gauge 8 during Test 36. A) The full energy density spectrum. B) The low frequency spectrum. C) The highfrequency spectrum. ..........................................................................................................................159 412 The raw data after the low frequency cut off was applied to separate the low frequency( ) and highfrequency ( ) oscillations for gauge 8 during Test 36. A) Time from 0 300 seconds. B) Time from 301600 seconds. C) Time from 601900 seconds. D) Time f rom 9011200 seconds. ....................................................................160 413 The zero moment wave height from Test 11 with error bars based on the total 95% CI of the laborat ory experiment. ......................................................................................161 414 The energy density spectrum transformation along the cross shore profile for the low water, Test 10. A ) Offshore to wave breaking. B) Reef top gauges. ............................162 415 The energy density spectrum transformation along the cross shore profile for the mid water, Test 22. A) Offshore to wave breaking. B) Reef top gauges. .............................163 416 The energy density spectrum transformation along the cross shore profile for the high water, Test 36. A) Offshore to wave breaking. B) Reef top gauges. .....................164 417 Low frequency energy density spectrum for Test 10 in low water. A) Offshore to wave breaking. B) Reef top gauges. ...............................................................................165 418 Low frequency energy density spectrum for Test 22 in mid water. A) Offshore to wave breaking. B) Reef top gauges. ...............................................................................166 419 Low frequency energy density spectrum for Test 36 in high water. A) Offshore to wave breaking. B) Reef top gauges. ...............................................................................167 420 The low frequency ( ) and highfrequency ( ) zero moment wave heights along the basin profile. A) Low water, Test 11. B) Mid water, Test 24. C) High water, Test 36......................................................................................................................................168

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15 421 The zero moment wave height laterally across the reef top from gauge 12, far left, to gauge 16, far right. The low frequency wave heights (o) and the highfrequency wave heights () are shown. A) Low water. B) Mid water. C) High water. ................169 422 The mean water level (*) and the high frequency zero moment wave height () measu red at the gauges located on the centerline of the basin. A) Low water. B) Mid water. C) High water. ..............................................................................................170 423 The maximum mea n water level measured on the reef top for all test cases. .................171 51 Bathymetry profile for the 1D energy flux model tests on the wave basin. ....................181 52 The wave height profile for Test 36 in the basin laboratory (x) with the results form the energy flux model run with STWAVE breaking criteria ( ). The thin black line is the wave height results from the energy flux model run with STWAVE breaking criteria. .............................................................................................................................182 53 The mean rms error for model comparisons made to the high water laboratory tests. A) Results from the model run with TG83 energy dissipation. B) Results from the model run with JBAB07 energy dissipation. ...................................................................183 54 The mean rms error for model comparisons made to the mid water laboratory tests. A) Results from the model run with TG83 energy dissipation. B) Results from the model run with JBAB07 energy dissipation. ...................................................................184 55 The mean rms error for model comparisons made to the low water laboratory tests. A) Results from the model run with TG83 energy dissipation. B) Results from the model run with JBAB07 energy dissipation. ...................................................................185 56 The best fit B coefficient for the high water model tests comparisons using the ), and Ru03 ( ) breaking equations. A) The wave heights for the TG83 model. B) The rms error for each B c oefficient in TG83. C) The wave heights for the JBAB07 model. D) The rms error for each B coefficient in JBAB07. ...186 57 The best fit B coefficient for the mid water model tests comparisons using the OM79 ), and Ru03 ( ) breaking equations. A) The wave heights for the TG83 model. B) The rms error for each B coefficient in TG83. C) The wave heights for the JBAB07 model. D) The rms error for each B coefficient in JBAB07. ...............187 58 The best fit B coefficient for the low water model tests comparisons using the OM79 ), and Ru03 ( ) breaking equations. A) The wave heights for the TG83 model. B) The rms error for each B coefficient in TG83. C) The wave heights for the JBAB07 model. D) The rms error for each B coefficie nt in JBAB07. ...............188 59 The error results of testing both TG83 and JBAB07 with B to the first power. A) TG83 error. B) JBAB07 error. ........................................................................................189

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16 510 ), and Ru03 ( ) in the TG83 and JBAB07 with the first order B. A) Wave heights from TG83 model. B) Change in error with B for TG83. C) Wave heights from JBAB07 model. D) Change in error with B for JBAB07. .............................................................190 61 The wave set up comparison between the wave model tests and the laboratory test on the 1:2.5 slope. The experimental data shown (x) or (*) is from Test 21 at mid water. The test cases shown are STWAVE ( ), and Ru03 ( ). A) The wave height profile for TG83. B) The wave set up calculated from the model tests compared with the measure data for TG83. C) The wave height profile JBAB97. D) The wave set up calculated from the model tests compared with the measure data for JBAB07. ...............................................................................................199 62 The wave set up comparison between the wave model tests and the laboratory test on the 1:5 slope. The experimental data shown (x) or (*) is from Test 64 at mid ), and Ru03 ( ). A) The wave height profile for TG83. B) The wave set up calculated from the mo del tests compared with the measure data for TG83. C) The wave height profile JBAB97. D) The wave set up calculated from the model tests compared with the measure data for JBAB07. ...............................................................................................200 63 The wave set up comparison between the wave model tests and the basin laboratory test at high water. The experimental data shown (x) or (*) is from Test 36. The test cases shown are STWAVE ), and Ru03 ( ). A) The wave height profile for TG83. B) The wave set up calculated from TG83. C) The wave height profile JBAB97. D) The wave set up from JBAB07. ................................201 64 The wave set up comparison between the wave model tests and the basin laboratory test at mid water. The experimental data shown (x) or (*) is from Test 36. The test cases shown a ), and Ru03 ( ). A) The wave height profile for TG83. B) The wave set up calculated from TG83. C) The wave height profile JBAB97. D) The wave set up from JBAB07. ................................202 65 The wave set up comparison between the wave model tests and the basin laboratory test at low water. The experimental data shown (x) or (*) is from Test 36. The test cas ), and Ru03 ( ). A) The wave height profile for TG83. B) The wave set up calculated from TG83. C) The wave height profile JBAB97. D) The wave set up from JBAB07. ................................203 71 The change in wave height between grid points. The red lines are the wave heights that would be calculated without friction.........................................................................209 72 The impact on wave set up of removing the energy loss due to friction from the equation. A) The change in wave height between a test with friction and without friction. B) The wave set up previously calculated with energy loss due to friction in the radiation stress and without. ..................................................................................210

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17 73 The change in the impact of the removal of friction from the radiation stress with increasing wave heights. A) The three wave heights tested. B) The growth in the difference between the old wave set up and the new wave set up as the i nitial wave height increased. ..............................................................................................................211 74 The impact of slope on the change in wave set up with the removal of energy loss by friction. A) The three slopes tested with initial wave heights of 1 meter and period of 5 seconds. B) The difference between the wave set up decrease with steeper slopes................................................................................................................................212 75 Energy loss due to friction while using JBAB07 energy dissipation model for breaking. A) The wave height decreases due to friction. B) Difference in wave set up when energy lost due to friction is removed. ..............................................................213 76 Bathymetry for wind stress tests in STWAVE. The slope on each side was set to 1:2000. .............................................................................................................................214 77 The impact of 60 m/s wind on the wave height and wave set up across the basin. A) The wave height growth and decay across the basin. B) The wave set up at the shoreline was increased by the removal of the wind stress from the radiation stress calculation. .......................................................................................................................215 78 The impact of three different winds on the wave height and wave set up across the basin. A) The wave height growth and decay across the basin. B) The difference in the wave set up at the shoreline between the old (solid lines) and new (dashed lines) wave set up values was increased with increasing wind speed. ......................................216 81 Mean water level for high water Test 38. ........................................................................223 82 Mean water level for low water Test 10. .........................................................................224 83 The coupled model results with TG83 comparisons with the measured data for Test 38 in high water. A) The wave height profile. B) The profile of the wave set up on the centerline of the basin. C) The wave set up along shore on the reef top. .................225 84 The coupled model result s with JBAB07 comparisons with the measured data for Test 38 in high water. A) The wave height profile. B) The profile of the wave set up on the centerline of the basin. C) The wave set up along shore on the reef top. .......226 85 The coupled model results with TG83 comparisons with the measured data for Test 10 in low water. A) The wave height profile. B) The profile of the wave set up on the centerline of the basin. C) The wave set up along shore on the reef top. .................227 86 The coupled model result s with JBAB07 comparisons with the measured data for Test 38 in high water. A) The wave height profile. B) The profile of the wave set up on the centerline of the basin. C) The wave set up along shore on the reef top. .......228

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18 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy ADVANCEMENT AND VALIDATION OF NUMERICAL STORM SURGE MODELING ON CORAL REEFS USING PHYSICAL MODEL COMPARISONS By Tyler J. Hesser December 2010 Chair: Donald N. Slinn Major: Coastal and Oceanographic Engineering Present numerical storm surge models were developed and tested on gentle sloping coastlines common in the continental United States. Islands with fringing coral reefs have complex small scale bathymetry with steep front slopes which diminish confidence in the accuracy of present storm surge model s Two laboratory experiments were performed using detailed bathymetry based on SHOALS measurements from the coast of Guam. A two dimensional wave flume with front slopes of 1:2.5 and 1: 5 was used to test numerical appr oaches to energy loss due to breaking waves on steep slopes. A three dimensional wave basin was used for updating and validating the coupled STWAVE ADCIRC storm surge model for the coral reef environment. Results indicated the need for greater dissipation of energy during breaking for the steep slopes than is currently the standard. Through testing in a 1D energy flux model, greater accuracy was found when the energy loss due to breaking was approximated with a bore type energy dissipation model. The wav e set up was calculated using the STWAVE ADCIRC coupled storm surge model using both the present STWAVE breaking and the bore type breaking functions. The use of the bor e type energy dissipation model was found to better match the measured data in the labo ratory than the present STWAVE breaking model.

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19 CHAPTER 1 INTRODUCTION Since Hurricane Katrina hit New Orleans in 2005, there has been an increased interest in the predictive capabilities of numerical storm surge models. The destruction caused by Katrina showed the need for the development of more advanced models to better predict the areas impacted by storms. The use of coupled wind w ave numerical models such as STWAVE ADCIRC introduced the possibility of examining all the forcing s affecting storm surge impacts on a coastline. The coupled windwave modeling system has been used to forecast and hind cast hurricane storm surge along t he coasts of the continental U.S. However, the vulnerability of the U.S. population currently living on islands in the Pacific Ocean and Caribbean Sea has received less attention The windwave coupled models were developed and tested on gentle sloping bathymetries such as the U.S. Gulf of Mexico and the East coast These islands often have complex bathymetry consisting of coral reef structures as well as steep slopes on the reef face. Common features to coral reefs have been described by Gourlay (1994, 1996a, 1996b) to consist of steep reef face slopes followed by a reef crest which can front an expansive reef top The reef top usual consists of complex small scale b athymetry created by coral organisms. The wave breaking and flow in a nd around coral reef structures have been found to be site and reef geometry specific which create difficulties in the development of comprehensive predictive numerical models (Demirbilek et al. 2009). The present research is focused on the analysis of generic reef geometry in both physical and numerical models to better understand the impacts of hurricanes or typhoons on island communities fronted by coral reefs.

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20 Coral Reef Laboratory W hen examining t he accuracy of numerical models on distinct environment s, it is important to have field or laboratory measurements for comparisons. L aboratory studies of the coral reefs are site specific with wave conditions and bathymetric profiles particular to a study environment. Nelsen and Lesleighter (1984) study of an idealized platform reef examined the affect of a vertical reef face on wave attenuation. Wave dynamics and set up were measured to better understand the Hydrographers Passage on the Great Barrier Re ef for navigational safety Jensen (1991) and Nielson and Rasmussen (1990) conducted a laboratory experiment designed to study a platform reef in Mali which was susceptible to flooding during strong ocean swells. Wave attenuation and set up were measured for a better understanding of the dynamics on this reef. Gerritsen (1980, 1981) examined wave attenuation and energy dissipation on a laboratory cross section of Ala Moana Reef in Hawaii. As a fringing reef, Ala Moana was a conducive environment for de tailed analysis of wave spectral transformation and wave set up because of the relatively gentle slope when compared to many other reef studies Seelig (1982, 1983) studied the effects of measured wave attenuation on ponding or lagoon wave set up in an idealized Guam reef. The lagoon behind the reef crest adds a dampener to wave energy in front of the beach slope. Gourlay (1994, 1996a ) conducted a laboratory experiment on both platform and fringing reefs. Wave transformation and set up were measured f or the steep reef face sloped platform reef as well as the more gentle reef face sloped fringing reef. Jensen (2004) examined the wave transformation on steep slopes in coral reefs focusing on the reflection and transmission of wave energy over the reef crest. The tests were performed in a wave flume with the reef top made out of concrete blocks. The wave heights were measured on four different fronts slopes: 1:2, 1:1, 1:0.5, and an S shaped slope. The test results were

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21 used in numerical model comparisons of wave transformation and wave energy transmission over the front reef edge. Tsai et al. (2005) focused on the shoaling and maximum wave height prior to breaking of waves o n steep slopes of 1:3, 1:5, and 1:10. The experiment design was based on the steep slopes common on the east coast of Taiwan. The tests were performed in a long wave flume, and only regular waves were examined. The measured wave height results were test ed against popular shoaling and maximum wave height equations. The comparison of wave height transformation for tests with slopes greater than 1:10 did not have good agreement which was likely due to the loss of energy due to the reflected wave. Demirb ilek et al. (2007) performed a laboratory study examining wave transformation and set up on a reef patterned after Guam. A wind generator and wave maker were utilized to create both wind wave as well as swell conditions. The study aimed to examine the ef fects of wind on beach set up, as well as observe wave transformation over a complex reef system. The scaled reef section was constructed using relatively smooth plastic material. All of the previously described laboratory projects were conducted in narr ow wave flumes and with flat surfaces for the reef top. In the present project, we built a reef top structure with small scale fluctuations based on a field survey of a reef top in Guam. Also, a three dimensional laboratory experiment was performed to cr eate a more realistic flow pattern on the reef top. Numerical Wave Models The complexities in the wave shoaling and breaking process in many of the laboratory and field studies previously described have revealed inaccuracies in the numerical modeling of wave transformation with commonly used wave models. Most wave models were developed to calculate the wave transformation over gentle slopes. The MildSlope equation was used by

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22 Massal and Gourlay (2000) with steep slope modi fication to compare against wave breaking measurements from previous coral reef laboratory studies. The model was combined with momentum equations to examine the wave set up on fringing coral reefs. The energy loss due to wave breaking and friction were parameterized using empirical relationships. The Mild Slope equation was used again by Jensen (2004) with modifications to better fit experimental data. The breaking parameter was slightly altered from the form presented by Massal and Gourlay (2000) to fi t with the wave heights measured on the steeper slopes. In both models, the laboratory and numerical results had reasonably good visual agreement. Tsai (2005) modeled the wave shoaling on the steep slopes from experimental tests using the K dV equati on. The equation calculates the wave shoaling prior to breaking with different equations based on the value of the Ursell number = (1 1) The shoaling equation compared well with the measured data, and was used in a time dependent Mild Slope equation model. T he parameterized breaking dissipation used a modified form of the Goda (1974) equation, which is described in a later section, to match the wave shoaling and breaking on the steep coral reef slopes. A spectral wave model was developed b y Lowe et al. (2005) to compare with field measurements from Kaneohe Bay, Hawaii. The spectral model included both dissipation of energy by friction and by breaking. The tests showed that the frictional dissipation was equivalent to the breaking dissipat ion because of the large coral structu res in the reef. Friction was found to play a large role at the reef face where wave breaking is usually considered the dominant form of dissipation.

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23 A comprehensive examination of the ability of one dimensional wave models to compute the wave transformation and wave set up on coral reefs was examined by Demirbilek et al. (2009). Three one dimensional (1D) wave models were used to compare with four laboratory experiments conducted on coral reef bathymetries, Gourlay (1994), Seelig (1983), Thompson (2005), and Demirbilek et al. (2007). The BOUSS 1D numerical wave model, which is based on the Boussine sq equat ion, provided a fully nonlinear approach to the examination of the wave transformation and run up The numerical results matched the experimental results very well for all the cases tested. RBREAK2 is a vertically averaged shallow water equation model for wave run up on impermeable slopes. The model did not match the laboratory data as well as the other models. The final model tested was WAV1D which is a 1D energy flux model. The energy flux model had parameterized wave breaking and friction, and the r esults matched the laboratory data very well. The fully nonlinear BOUSS 1D and the linear energy flux model WAV1D were both found to fit the laboratory data with similar accuracy Maximum Wave H eight E quation Using the assumption that waves near breaking behave as solitary waves, McCowan (1894) derived a limit for the breaking wave height. The theory proposed that the maximum breaking wave height, is directly related to the breaking depth, by = 0 78 (1 2) Miche (1944) developed an equation for the maximum breaking height based on wave transformation on gently sloping beaches to be = 0 142 ( ) (1 3) where is the wave length at breaking. The coefficient 0 142 was later changed to 0 12 by Danel (1952).

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24 Monk (1949) derived the maximum breaking height formula based on a solitary wave breaking with the controlling mechanism being the offshore wave steepness, = 0 3 (1 4) Weggel (1972) advanced the work of Monk (1949) by incorporating a dependence on the bottom slope, into Equation 15, = ( ) ( ) (1 5) where ( ) = 43 8 1 exp ( 19 ) (1 6) ( ) = 1 56 ( 1 + exp ( 19 5 ) ) (1 7) K omar and Gaughan (1973) used linear wave theory to derive an empirical formula for th e maximum breaking height ( Equation 18). The constant, 0.56, in Equation 18 was determined by comparison with three laborator y data sets (Komar and Simmons, 1968; I verson, 1951; Galvin, 1968). The maximum brea king equation was also tested with the field measurements collected by Munk (1 949). = 0 56 (1 8) The empirical formulation presented by Goda (1974) was based on laboratory data from a variety of sources including Iversen (1951), Mitsuyasu (19 62), and Goda (1964). In Equation 19, the breaker height is estimated using the depth from 5 toward the offshore, so the depth of breaking was calculated in Equation 110.

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25 = 0 17 1 1 5 ( 1 + 15 ) ) (1 9) = + 5 (1 10) Battjes an d Janssen (1978) modified Equation 12 by Miche (1944), including an adjustable parameter, to allow for the effects of beach slope and the transformation of random waves ( Equation 111). = 0 142 0 88 (1 11) Through laboratory testing performed by Battjes and Janssen (1978), the coefficient was found to be equal to 0.8. Another modified version of the Miche (1944) breaking equation, Equation 12, was developed by Ostendorf and Madsen (1979). A relationship between the breaking height and the bottom slope was added for slopes less than 1 10 with a cap on the influence of bottom slope for bottom slopes greater than 1 10 = 0 14 ( 0 8 + 5 ) 2 1 10 = 0 14 ( 0 8 + 5 ( 0 1 ) ) 2 > 1 10 (1 12) Singamsetti and Wind (1980) conducted laboratory testing and combined the results with previous laboratory studies by Iverson (1951) and Galvin (1968) to develop a slope dependent maximum breaking height relationship. A relationship b etween maximum breaki ng height and the breaking water depth was developed. = 0 937 ( ) (1 13)

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26 Equation 113 was tested against laboratory conditions ranging from 0 02 0 065 and 1 40 1 5 Previous studies have shown the maximum breaking wave height is affected by the offshore wave steepness and the bottom slope. Battjes and Stive (1985) tested these parameters and found no significant variation in by changing the bottom slope, but var iation was found when changing the offshore wave steepness. The constants in Equation 114 were calibrated by testing against laboratory and field data sets. = ( 0 5 + 0 4 tanh ( 33 ) (1 14) In Equation 114, the deepwater wave height is the root mean square wave height in deep water, and is the deep water wave length associated with the peak period. The breaking parameter presented by Battjes and Stive (1985) was further developed by Nairn (1990). Equation 114 was compa red against a more extensiv e data set, and the result was similar in form bu t with different coefficients ( Equation 115). = ( 0 39 + 0 56 tanh ( 33 ) (1 15) where = Kamphuis (1991) developed a maximum breaking wave height formula based on a laboratory study by Kamphuis and Kooistra ( 1990). The laboratory study included both regular and irregular waves. Equation 116 was derived to represent for the breaking of irregular waves. ( ) = 0 095 exp ( 4 ) tanh 2 (1 16) In Equation 116, ( ) is the maximum breaking significant wave height.

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27 Nelsen (1993) developed the maximum breaking height relationship based on a nonlinear parameter, presented by Swart and Loubser (1979). The nonlinear parameter is a relationship between H, T, and such that a given value of from Equation 113 is related to a certain shape of wave. = . (1 17) By comparing results of against laboratory data sets, Nelsen (1993) found a trend between and given by = 22 + 1 82 (1 18) In Equation 114, the largest ratio between and is 0.55 when 500 Massal and Gourlay (2000) developed a maximum breaking height relationship for complex bathymetry often associated with coral reefs. Eq uation 112 from Singamsetti and Wind (1980) was used for bottom slopes steeper than 1 40 On the reef top where slopes become more gentle a relationship was derived based on Nelson (1993), = 1 + 0 1504 1 0 1654 > 1 40 (1 19) where = (1 20) A comprehensive description for many of the maximum breaking wave height equations previously described was presented by Smith and Kraus (1990) and Jensen (2004).

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28 Energy D issipation D ue T o B reaking The energy flux equation for the one dimensional propagat ion of waves in the horizontal cross shore direction with the only energy loss due to breaking waves is = (1 21) where is the average wave energy flux per unit area, and is the average rate of energy dissipation per unit area. Based on linear wave theory, the energy flux can be calculated by = = 1 8 (1 22) where is the root mean square wave height, and is t he wave group celerity. Equat ion 122 can be integrated across the surf zone to find the total change in wave energy by breaking. Battjes and Janssen (1978), BJ78, followed the approach of LeMehaute (1962) in solving for the energy dissipation rate per unit span for a single, solitary wa ve breaking in a shallow depth. The bore method is based on observations that a breaking wave behaves similar to a bore connecting two regions of uniform flow. The power dissipated in the bore per unit span, was found by Lamb (1932) to be calculated by = 1 4 ( ) ( + ) 2 (1 23) where is the depth on the deeper side of the bore, and is the depth on the shallower side of the bore. Order of magnitude assumptions were made to relate the bore dissipation equation to wave heights. Two assumptions made by BJ78 are that ~ (1 24) and

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29 ( + ) 2 ~ (1 25) By using the se assumptions ( Equation 124 and 125) in Equation 119, an equation for the energy dissipated per solitary broken wave per unit span is ~ 1 4 (1 26) I f the waves are periodic with a frequency, the energy dissipated per unit area (assuming a unit width of beach) can be solved by = = 1 4 (1 27) where is the height of the brea king waves. By averaging Equation 127 over all breaking waves, the average energy dissipation per unit area is = 1 4 (1 28) where is the spectral mean frequency of a random set of waves, and the indicate the ensemble average over all b reaking waves. BJ78 assumed the breaking wave heights were narrowly distributed around the height described earlier as the incipient breaking height of periodic waves, ( Equation 129). (1 29) Because of the assumption that the breaking wave heights are distributed around there is a probability that waves will be both larger than (breaking waves) and smaller than (nonbreaking waves). The percentage of all waves assumed to be breaking, can be described by

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30 1 = (1 30) based on a truncated Rayleigh distribution of wave heights, where ranges from 0 (deep water) to 1 (shallow water ). When is equal to 1, a saturated wave condition exists in which is equal to the incipient breaking height of waves. By using the percentage of breaking waves with Equation 124 the average energy dissipation per unit area as found by Bat tjes and Janssen (1978) is = 4 (1 31) where is a proportionality constant. The value of used in Battjes and Janssen (1978) was described earlier as Battjes and Janssen (1978) maximum breaking wave height, and was later changed by Battjes and Stive (1985) to Equation 115. A modification to the BJ78 method for solving the energy flux equation was proposed by Thornton and Guza (1983), TG83. Through field observations, TG83 observed that wave heights were approximately Rayleigh distributed even in the surf zone. Longuet Higgins (1975) showed the use of a Rayleigh distribution was accurate given the waves are a narrow banded and linear Gaussian process. The Rayleigh wave height probability density function is ( ) = 2 (1 32) Even though, waves near breaking are generally not narrow banded, or linear Gaussian, pre vious research lead TG83 to approximate the breaking wave heights using a Rayleigh distribution. By using Equation 132, the distribution of wave heights both above and below can be approximated by a weighted Rayleigh distribution, ( ) = ( ) ( ) (1 33)

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31 where ( ) is the weighting function. The weighting function with the best fit to field data collected by TG83, was found to be, ( ) = 1 (1 34) where is the ratio of wave height to water depth at breaking and is a variable which was determined by observations to equal 2. The value of was found by TG83 to be equal to 0.42 for a best fit to the field observations. The average energy dissipation per unit area is found by = ( ) ( ) ( ) (1 35) where ( ) is the same as Equation 127 except is replaced by The variable B is a tunable coefficient related to the foam region on the face of a breaking wave. By plugging in the changes in Equation 127 into Equation 135, the equation is now, = 4 ( ) ( ) (1 36) Including the formula for the proba bility density function ( Equation 132) of the weighted Ra yleigh distribution into Equation 136 and integrating gives = 3 16 1 1 + (1 37) where B was found to be equal to 1.5 when evaluated with field data. Whitford (1988), Wh88, followed the model presented by TG83, but derived a new weighting function. The weighting function proposed by Wh88 is

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32 ( ) = 1 + tanh 8 1 1 (1 38) which leads to an average energy dissipation per unit area of = 3 16 1 + 8 1 1 1 + (1 39) Lippmann et al. (1995), Li95, proposed a roller model instead of a bore model to account for the dissipation of breaking waves. The surface wave roller defines the internal dynamics of the turbulent, aerated water in front of a breaking wave (Svendsen, 1984a,b) which is different than the bore models which are based on a hydraulic jump. The roller model breaks the energy into two terms: = + (1 40) where is the contribution of the wave, and is the energy in the roller. Equation 121 is then changed to cos + ( cos ) = (1 41) where cos is the wave direction. The wave celerity is used in the roller ter m because it is associated with the individual wave The w ave contribution term in Equation 140 follow s the method described by Wh88, including use of the same weighting function. The energy of the roller was found to be = 1 8 tan (1 42)

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33 where is the slope of the wave front, which was held constant at 12.5 for the Li95 study. The maximum breaking wave height for Li95 was = 3 4 1 + 8 1 1 1 + (1 43) The equation for the average energy dissipation per unit area was then found by Li95 to be = cos (1 44) The result of Equation 1.44 is very similar to results by Wh88, especially when analyzing a 1D problem where cos = 1 Baldock et al. (1998), Ba98, further developed the BJ78 model to account for steep slopes. BA98 used a full Rayleigh distribution similar to TG83. In the model presented by BJ78, equals 0 when waves are in deep water and 1 when = Ba98 formulated the proportion of broken waves, by integrating the Rayleigh distribution Equation 132 for all waves when such that = (1 45) where = This formulation gives = exp (1 46) such that = 0 for deep water and = 0 4 for = This result is similar to the values found by TG83 where = 0 5 for =

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34 Using the approximation similar to BJ78 that the average energy dissipation per unit area is solved by = 4 (1 47) where B is a tunable parameter of order 1. The explicit expression for is = 4 ( 1 + ) exp ( ) (1 48) where = Massel and Gourlay (2000) MG00, presented an adaptation of the BJ78 model for the complex bathymetry observed in coral reefs. The equation used for the average energy dissipation per unit area was developed by Massel and Belberova (1990) to be = 8 (1 49) where is a free empirical constant and is the angular frequency The MG00 model is related to the nonlinear coefficient, proposed by Nelsen (1993) by the constant Gourlay (1994) showed that the equation for Equation 113, could be based on deepwater wave height, and a representative water depth, to classify wave transformation on a coral reef. Because of the addition of the new variables, will be denoted by = (1 50) The relationship between in Eq 149 and the nonlinear coefficient, is

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35 = 0 ( ) = ( ) > ( ) (1 51) where and are tunable parameters, and ( )is a threshold value of at which energy dissipation and set up are small. The value of ( ) was equal to 100 for the three test cases performed by MG00. Ruessink et al. (2003), Ru03, followed the Ba98 model for energy dissipation. Ru03 used Equation 148 to solve for the average energy dissipation per unit area, but the maximum wav e height was found using Equation 110 which was proposed by BJ78. By comparing the model with laboratory and fie ld data, Ru03 developed an empirical equation for given by = 0 76 + 0 29 (1 52) The model proposed by Ba98 and modified by Ru03, was found to have wave height results that diverge near the water line in very shallow regions. The divergence is due to the approximation that This assumption does not all ow for all possible wave heights given by the full Rayleigh dist ribution. Janssen (2007), a nd Alsina and Baldock (2007), JBAB07, fixed the problem in Ba98 by removing the approximation and integrating over the full cubed wave height and depth. The new eq uation is = 4 (1 53) and the explicit expression for is = 3 16 1 + 4 3 + 3 2 exp ( ) erf ( ) (1 54)

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36 Apo t s os et al. (2008), Ap08, tested and calibrated many of the bore models described above as well as the roller model by Li95 against f ield experiments. The models BJ 78, TG83, Wh88, Li95, Ba98, Ru03, and JB AB 07 were test against field data. Ap08 ran all the models with the coefficients of breaking prescribed by each approach along with a best f it for breaking. A review of the bore model approach by Battjes and Janss en (2008) provided a guide for the basics of the BJ 78, TG83, Ba98, and JB AB 07 methods, and gives a detailed analysis of how the results from each model are different near the water line. Wave Set U p The initial work in radiation stress theory was presented by Longuett Higgins (1953), and Longuett Higgins and Stewert (1962, 1963, 1964). The work examined the m omentum transferred from waves to the water column during the shoaling and decay of wave heights. The wave radiation stress are broken into three components : the cross shore flux in the cross shore direction = ( ( 1 ) 0 5 ) (1 55) the alongshore flux in the alongshore direction, = ( ( 1 ) 0 5 ) (1 56) and the cross shore flux in the alongshore direction = 2 sin (1 57) where is the energy of a wave, is the angle of wave propagation towards shore, and is the ratio of wave celerity to the wave group celerity = = 1 2 1 + 2 sinh 2 (1 58)

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37 The 1D gradients in the wave radiation stress as waves are transformed due to bathymetric changes in the cross shore direction were related to the change in mean water level by = 1 ( + ) (1 59) where is the mean water level. The relationship has been observed to create a set down when t he energy of a wave increases due to shoaling near breaking, and a set up when the energy decreases caused by a reduction in wave height by breaking close to shore. This 1D theory was found by Dean and Dalrymple (1991) to calcul ate an increase in the mean water level by 19% of the breaking wave height for a gently sloping beach. In the analysis of wave set up on coral reefs, Gourlay (1994) found relationships bet ween the maximum wave set up and the offshore wave height, the wave period, and the water depth over the reef crest. A relationship between the wave set up and the wave power given by, was also examined by Seelig (1983) and Jensen (1991) and corro borated by Gourlay (1994). The wave set up was found to increase with increasing wave power The wave set up was also found to increase with decreasing depth on top of the reef crest. Present S tudy In the present research, two laboratory experiments ar e performed to advance the previous laboratory study by Demirbilek et al. (2007). A wave flume is used to examine the wave breaking on steep slopes greater than 1:10. The second laboratory experiment utilizes a large threedimensional (3D) wave basin to provide a more complete analysis of the relationship between wave breaking and the static wave set up on a coral reef. The focus of the laboratory experiments are on fringing reef systems with long reef tops with complex small scale bathyme try.

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38 Existing numerical models are tested against the detailed measurements from both laboratory tests. The 1D energy flux model is used to test commonly used energy dissipation models and maximum breaking equations on ste ep slopes. Particular attent ion was paid to the TG83 and JBAB07 models because TG83 is unique in the comprehensive usage for many coastal environments, and JBAB07 was developed as a model capable of calculating the energy dissipation on steep slopes. Three maximum breaking wave heig ht equations are tested in the energy dissipation models to determine the best combination. The OM79, Ka91, and Ru03 equations were chosen because all the equations focus on local wave characteristics in the calculation of the maximum breaking wave height s. The wave set up is computed using both the 1D equation presented by Dean and Dalrymple (1991) as well as the use of the more complex Advanced Circulation (ADCIRC) model. The ADCIRC model is coupled with the near shore wave model (STWAVE) to calculate the two dimensional (2D) wave breaking and wave set up for comparisons with the large wave basin laboratory results.

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39 CHAPTER 2 WAVE FLUME LABORATOR Y Introduction Steep slopes and complex bathymetry create difficulties for laboratory studies and numerical models observing wave dynamics on coral reefs (Massal 2000) Due to these challenging conditions, laboratory and field studies are not common. However, safety concerns for residents on coral reef barrier islands as well as increased tourism t o these areas have amplified the need to understand this complex environment by engineers and scientists (Gourlay 1996). A laboratory experiment was conducted to measure the impact of steep reef face slopes on wave breaking and wave set up during large h urricane type waves. The small scale of the wave flume provides the opportunity for detailed measurements with small distances between gauges. The control in the measurements as well as the ability to control all wave signals provides an opportunity to obtain measurements which would be difficult in the field Methodology Experiment Facilities The experiment was conducted at th e United States Army Engineer Research and Development Center (ERDC) in Vicksburg, MS. All tests were performed in the 3ft flume which is 45 meters long, 0.91 meters wide, and 0.91 meters deep. Waves were generated with the computer controlled electro hydraulic piston wave maker located at one end of the flume. The wave maker is capable of generating maximum wave heights around 0.25 meters and wave periods between 0.50 10.0 seconds.

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40 Laboratory Conditions The length of the flume was shortened from 45 meters to 22 meters. A concrete beach was previously constructed in the flume, so the reef platform was built on top of the concrete beach (Figure 2 1). The 7.3 meter flat reef top was built with marine plywood starting at 11 meters from the wave maker. Two front slopes were constructed; a permanent 1 on 2.5 reef slope, and a removable 1 on 5 slope (Figure s 22, 23) The toe of the 1 on 2.5 slope and 1 on 5 slope were located at 10.1 and 9.3 meters from the wave maker, respectively. The back slope or beach of the reef system was built with a slope o f 1 on 10. A partition was built to create a 0.61 meter test section of the flume from the slope toe to the shoreward edge of the reef top. Acrylic glass cut into 0.61 x 0.61 meter squares and molded using a detailed reef bathymetry was placed on the con structed flat reef top. The acrylic glass was cut to represent a previously surveyed Guam reef at a 50:1 prototype to model scale (Figure 2 4) The Guam reef was surveyed using the Scanning Hydrographic Operational Airborne Lidar Survey (SHOALS) system w ith measurement spacing of 5 meters. The slope of the measured data was removed and the reef structures were interpolated to a spacing of 0.46 meters. The interpolated data was scaled down to model size and cut into the acrylic glass using a platform ro uter. Each acrylic piece was positioned at exact horizontal and vertical locations and secured with water resistant silicone. Initial testing was performed on the clean cut acrylic glass. A second set of tests were performed with a textured paint applied to the acrylic glass. The reef top was painted with a mixture o f blue paint and a small diameter sediment (Figure 2 5) The grain size was selected to resemble a scaled coral reef polyp. The goal of applying the paint and sediment mixture was to be tter simulate the textures commonly seen in coral reefs.

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41 Instrumentation Twelve capacitance wire gauges were spread along the length of the test area to measure the water surface elevation. The gauges were programmed to measure at a 20 Hz or a measurement every 0.05 seconds. Capacitance wire gauges are constructed with a thin wire that is held in place by a steel L shaped rod (Figure 2 6) The capacitance of the wire changes depending on the water surface elevation. The changes in the capacitance ar e then related to the water surface elevation by calibration. The gaug es were statically calibrated so a measured voltage corresponded to a water surface elevation in unit meters by raising and lowering the gauge to know locations and recording the capaci tance. The calibration was performed in the laboratory using electric computer controlled motors whic h raised/lowered the gauges to 20 vertical locations, and voltage was measured at each known elevation. S tatic calibration was done on all the gauges once a week, and a quadratic regression was used to relate the water elevation to the measure voltage. Gauges were numbered 1 12 where gauge 1 was located closest to the wave maker, and gauge 12 was furthest from the wave maker or closest to the beach The location s of all the gauges are in Table 2 1 where x is the cross shore distance measured from the wave maker. Gauges 1 3 were spaced to measure the reflection created as the waves impact the steep fron t slope a nd some of the energy is reflected back offshore. These gauges will commonly be referred to as the Goda array. Gauge 4 was located just off shore of both of the front slopes to measure the wave heights before the slope induced steepening occurred Gauges 5 8 were located close together to capture the steepening of the wave on the front slope as well as the depthinduced breaking and initial dissipation of the wave s as they reach the reef crest, or the front edge of the acrylic glass. Gauges 9 12 were located on top of the acrylic glass to

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42 me asure the wave attenuation from the front to the back of the reef top. All e ffect of the detailed reef bathymetry was captured in gauges 9 12. All twelve of the capacitance wire gauges measured both the continuous water surface change as well as the mean water level through the simulation. The twelve gauges could be coupled with the capacitance run up gauge, located on the back beach, to measure the mean water elevation increase on the beach, or wav e set up. The run up gauge was a 2.1 meter long capacitance gauge that was secured to the back beach and was calibrated prior to testing. Because the run up gauge was secured to the floor, calibration was performed by raising and lowering the water level s to known elevations. The capacitance was recorded at these levels to develop the relationship between the voltage and the water elevation. Test Conditions The designed wave characteristics in prototype scale are shown in Table 22. Three water levels were tested (Figure 2 7) with prototype peak wave periods between 7 20 seconds and prototype wave heights between 3.9 8.0 meters. The prototype to model scale was 50:1, so the model peak periods were found from = 50 (2 1) where is the model peak period, and is the prototype peak period. The model scale wave heights were found from = 50 (2 2) where is the model zero moment wave height, and is the prototype zero moment wave height. All w aves were generated using a TMA spectral shape and random phase.

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43 Testing Procedure The s equence of events for testing in the laboratory was repeated for each test. First, t he wave conditions and data collection parameters were defined. The wave conditions were random waves generated with a TMA spectrum based on a target zero moment wave height, H, and peak period, T. The peakedness parameter, used in the TMA generate d waves was 3.3. The sampling rate was defined as 20 Hz for all runs, and the run length was 20 minutes. Next, t he water level was checked using a point gauge with an accuracy of 0.001 ft. All of the capacitance wave gauges were balanced to zero for add ed accuracy. The run up gauge was not balanced. The waves were started and data collection commenced. One computer was used to control the wave generator and collect data. The data collection commenced when the wave generation started, and it ended whe n the waves ceased. During each run, pictures and movies were taken of both the wave breaking on the front slope, and the wave motion on the reef top. Lastly, preliminary post processing was performed on the data after each run to ensure all gauges wo rked proper ly, and the results were reasonable Notes were taken at this time to record any problems with the wave maker or gauges. Post Processing Test Cases A total of 172 laboratory simulations were run in the flume. Measured initial values during e ach simulation are shown in Table 23 and Table 24. Table 23 is the test cases for the 1:2.5 slope with both the smooth reef top (143 s) and the rough reef top (1 43 r). Table 24 is the test cases for the 1:5 slope with the smooth reef top (4486 s) and the rough reef top (4486 r). In Table 23 and Table 24, the incident zero moment wave height, Hmo i, the peak period, Tp and the stillwater depth in meters, h, are displayed

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44 Reflection Analysis Wave reflection analysis was performed to separate the incident wave from waves reflecting off the reef structure. Recorded wave data from a laboratory experiment often has reflected waves interacting with the incident waves (Hughes 1993). In this laboratory study, reflection analysis was performed to provide more accurate incident wave characteristics for numerical modeling. Goda and Suzuki (1976) developed a separation technique using two gauges separated by an exact dis tance. The incident and reflected wave components were separated using a Fast Fourier Transform. The method developed by Goda and Suzuki (1976), comm only called the Goda method, is based on linear wave theory, so nonli near interactions were excluded In the present research the Goda method was employed using three gauges which allows for more overlap of values. The time series of water levels for gauges 1, 2, and 3 were used to separate the incident and reflected waves The value of the reflection coefficient, has previously been found to be e ffected not only by the bathymetric profile of the laboratory set up, but by the wave height and wave period of the generated waves. For example, a reflection coefficient of close to zero would be calculated if the flume had no reflecting surface, and a reflection coe fficient of close to 100% would be calculated if a vertical wall was present (Hughes 1993). A previous laboratory experiment on a coral reef by Demirbilek et al. (2007) found reflection coeffic ients no larger than 10% The refl ection coefficient of 10% implies that 10% of the incident wave was reflected off the reef structure in the offshore direction. In Figure 2 8, the reflection coefficient found for al l tests at each water level is shown against the deep water surf similarity parameter. The surf similarity parameter,

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45 = (2 3) where is the slope of the bottom, and and are deep water wave height and wave length, respectively, was found by Battjes (1974) to be a correlating factor in the wave reflection of f a beach. This relationship was based off of slopes more gentle than the 1:5 and 1:2.5 slopes tested in the present study. In the present study the surf similarity parameters ranged from ~ 1 to 5, but the reflection coefficient was never greater than 0.35. The reflection coefficients are low given the surf similarity pa rameters when comparing with Battjes (1974), but a simila r result was found by Smith and Kraus (1990). Spectral Analysis All laboratory simulations were 20 minutes long. An example of a time series from an offsh ore gauge is shown in Figure 29. In post processing, the first 30 seconds of the time series was r emoved to allow the generated waves to reach all the gauges. All the gauges were zeroed to remove drift prior to starting the tests, but it was difficult to get all the gauges to exactly zero. In order to remove any drift from zero, the mean of the first 10 seconds of measured data was subtracted from the rest of the raw data. The spectral analysis method used was the Welch method. The Welch method was derived from the Bartlett method in which the time series is broken into sections. In the present c ase each section consisted of 4096 measurement points. E ach section is overlapped by 50 % and a Hamming window was used to remove any impact from the boundary points. A Fast Fourier Transform was p erformed on all the sections and then all the sections were averaged The raw data were filtered to separate the low frequ ency oscillations from the highfrequency (incident wave) oscillations. First a filter was run with a cut off at 2 Hz to remove high frequency noise. Next, a low frequency cut off was applied at 1 5 which was

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46 determined to be the frequency that separates the incident waves from the low frequency waves. An example of the full energy density spectrum, and the low and high frequency spectra, is shown in Figure 2 10. In Figure 211, t he high frequency and low frequency time series data are shown. The energy density spectrum is an important factor in describing the characteristics of the waves The peak wave period, Tp, is defined as the period associated with the maximum energy in the energy density spectrum. T he zero moment wave height, Hmo, is found from = ( ) (2 4) where the energy density spectrum as a function of frequency, S(f), is integrated from 0 to the nyquest frequency, and is the zeroth moment of the energy spectrum. Hmo is then found from = 4 (2 5) After cutting the data sets into low frequency oscillations and incident or high frequency oscillations, the equations to solve for low and high are slightly di fferent. Solving for the zeromoment of the energy density spectrum for the low frequency energy is given by = ( ) (2 6) and the equation for the zeromoment of the energy density for the incident or high frequency energy is

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47 = ( ) (2 7) where is the peak frequency at the offshore gauges. Statistical Wave Measurements The mean water level, was found by taking the mean over the complete sample size minus the first 30 seconds of the measured time series. The mean water level is associated with the wave set up which results from the moment transfer from the waves to the water column during breaking. A zero upcrossing approach was used to separate individual waves for statistical analysis of run up. Each time a wave crossed zero going in the positive (upward) direction it was defined as a wave. The run up analysis was performed on the time series of the run up gauge located on the beach slope. The common practice is to find the 2% run up on the back slope which is found by taking the mean of the waves which are the highest 2% of the run ups measured. Wave Gauge Confidence Interval Althoug h, the measurements in the laboratory tests were performed with an at tention to detail, some error i s unavoidable. The error of each gauge during a simulation was determined by examining the differences of measured wave heights in segments. Each simulation was initialized with a wave distribution determined from a TMA spectrum, so, in theory; Hmo should be the same over a large sample size of waves. The largest peak period simulated was 2.8 seconds, so a 3minute record was determined to be a sufficient sample size for consistency in the measured Hmo. The 3 minute segment is a sample size of around 60 waves for the 2.8 second period.

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48 The time series of water surface elevations measured at each gauge was cut into 16 3 minute segments. The first segmen t was between 30 seconds and 3 minutes and 30 seconds. The second segment was between 1 minute 30 seconds and 4 m inutes and 30 seconds A Fast Fourier Transform analysis was performed on all 16 segments, and Hmo was calculated from the spectrum from each segment. A 95% confidence interval was calculated using the equation, = 1 96 (2 8) where is the average wave height found using the Welch method previously described, is the standard deviation of the sample, and the sample size. The value of the 95% confidence interval does not decrease the importance of the measurements, b ut it provides limitations for comparisons with the data. The 95% confidence interval was found for all tests and at every gauge. In Tabl e 2 5, the mean of the 95% confidence interval was calculated for all the gauges by taking the mean of all the tests for each water level. In all cases the highest value for the 95% CI was at gauges 5 and 6 which were located on the front slope. This loc ation had the highest amount of reflection and most waves were breaking near these gauges. The total 95% CI for the laboratory experiment is also in Table 2 5. Any variations in wave measurements inside of the 95% CI were assumed to be error in the wave gauges. The total 95% CI is shown with an example of the for a low water test case (Figure 2 12) The largest error bars are located at gauges 4 and 5, and on the reef top (gauges 9 12) the error bars are small. The small error bars give added co nfidence to the laboratory measurements.

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49 Smooth Surface Results Spectral Wave Energy The transformation of wave energy as the waves travel from offshore ( near the wave maker ) to onshore (on the reef top) was measured by the array of water level gauges. Change s in the energy density spectra were measured from the offshore gauges ( gauge 2) across the entire bathymetric profile for all water levels and both front slopes and are shown in Figure s 213218. In Figures 2 13A 218A the energy density change is shown from the offshore gauge to gauge 5 (located in the middle of the front slope), and gauge 6 (the point of breaking for most waves on the front slope). In Figures 2 13B 2 18B the changes in the energy density spectrum from just after breaking (gau ge 7, located on the front edge of the acrylic) through gauges 8 and 9 which are on front part of the acrylic reef top. I n Figures 2 13C 218C, the energy density spectrum from the three gauges located closest to the beach slope (gauge 10, 11, 12). In the gauges close to the beach slope, the change in the peak frequency was very noticeable in most test cases. There were some differences observed in the energy density spectrum transformation between the three different water levels on the 1:2.5 slope (Fig ure s 213215). All of the cases shown were initialized with similar designed wave heights and peak periods. In the top window of all three figures, the amount of energy located at the peak frequency had different growth amounts between gauge 2 and gauge 6 for the different water levels. The low water had the greatest increase in energy at the peak frequency, and the high water had the lowest (Figure 2 13) The low water tests also had a larger amount of energy in the lower freq uencies than the mid (Figure 2 14 ) an d high water levels (Figure 2 15). Again, the high water had the lowest amount of energy at the lower frequency. The growth in the energy in the lower frequencies was

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50 caused by reflection and seiching off the front slope. The low frequency oscillations will be discussed further later. In Figure s 2 13B 2 15B the dissipation of energy at the peak frequency was measured for all three water levels. The high water case (Figure 2 15), had the heights amount of energy after breaking at gau g e 7. The low water (Figure 2 13) and the mid water (Figure 214 ) had more energy decreased between gauge 6 and gauge 7. The lower waters in the low and mid water caused less wave energy to transfer on to the reef top so the energy in the system was decreased at all frequencies The high water had deeper water at the top of the front slope and on the reef top which allowed more energy to be transferred to the gauges on the front part of the reef top. The decrease in energy at the peak frequency between gauge 7 and 9 was the result of a combination of depth limited breaking and friction. The shallower depths increased the dissipation of energy at the peak frequency for the low water cases compared to high water cases. In Figure 2 13C 215C the continue d dissipation of energy at the peak frequency on the reef t op is shown as well as the growth of low frequency energy. In the low water tests (Figure 213C ), there was no energy remaining at the initial peak frequency, so all the energy was located at the lower frequencies. Gauge 12 was not always in the water for the low water tests, so the energy density spectrum at all frequencies was close to z ero. The mid water (Figure 2 14C ) had a very small amount of energy remaining at the initial peak frequency a t gauge 10, but the energy was dissipated between gauge 10 and gauge 11. All the energy at gauge 11 and gauge 12 i n the mid water level test was in the lower frequencies. Gauge 12 was in the water for most of the mid water tests, but the layer o f water was very thin. The low frequency energy did increase between gauge 10 and gauge 11, but the water level was too low for a large amount of

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51 wave energy to reach gauge 12. In t he high water tests (Figure 2 15C ), the energy at the initial peak frequency was completely d issipated by gauge 11. The low frequency energy increased from gauge 10 to gauge 12. Similar trends in the transformatio n of the energy density spectra across the flume profile were measur e d in the 1:5 tests (Figure s 2162 18 ) as the 1:2.5 tests. The biggest difference observed in the measurements between the 1:5 slope and the 1:2.5 slope was in the transformation of wave energy at the peak frequency at gauge 6. Gauge 6 had the most energy at the peak frequency for the 1:2.5 tests, but i n the 1:5 tests the amount of energy at the peak frequency did not increase as much from gauge 2 to gauge 6 due to differences in depth at gauge 6 for 1:2.5 versus 1:5. This result was consistent for all three water levels on the 1:5 slope tests. The amo unt of low frequency energy i n Figure s 216A 218A was less than the amount measured in the 1:2.5 slope. Overall, the changes in the transformation of wave ene rgy between the low (Figure 216), mid (Figure 2 17), and high (Figure 2 18) water levels was s imilar to the trends described about the 1:2.5 tests. Low Frequency Spectral Wave Energy The frequency range considered in the present study to be the low frequency range is show for select gauges through the laboratory profile for both the 1:2.5 (Figures 219221) and 1:5 (Figure s 2222 24) slopes. The test case shown in all figures is representative of all the runs for the three water levels and the two front slopes. The initial peak period of the test case shown was 1.8 seconds, so the peak frequency was 0.56 Hz. Based on the method described in the post processing section for separating the low and high frequency oscillations, the low frequency cut off was set at 0.37 Hz for the demonstration case. The peaks in the low frequency spectra observed in all the water levels and front slopes can be explained by seiching. The shape of the offshore section of the flume had a flat

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52 bathymetry for approximately 10 meters for the 1:2.5 slope and 8 meters for the 1:5 slope. Both of t he slopes are considered very steep, so the shape of the offshore region was approximated as a rectangular basin. The seiching equation for a rectangular basin as given by Dean and Dalrymple (1984) is = 2 (2 10) where T is the period of the seiching, is the length of the basin (in this case assumed to be 9.5 meters as an approximation of the middle of the slope), is the depth of the basin (assumed to be the height of the front edge of the acrylic and top of the slope equal to 0.341 meters), and is the number of oscillations of the wave in the basin. The fundamental frequency of the seiching ( = 1) was 0.09 H z. This is consistent with the peaks measured at a similar frequency i n t he top window of Figures 219 224. The second peak was measured at 0.18 Hz which is similar to the value of the 2nd mode of oscillation from Equation 3.5 ( = 2). The third and f ourth modes of oscillation were only measured with in the low water tests of the 1:2.5 slope, and the values of 0.27 and 0.37 Hz are consistent with the observed values (Figure 219 ). The seiching which occurred in the offshore section of the tests could have affected the wave height measurement s on gauges 27 due to the location of the nodes and antinodes. The peak located at a frequency of 0.03 Hz in the high water tests was caused by seiching on the reef top. The shape of the reef top can be approxima ted by an open boundary system. The equation for the period of seiching for an open boundary system is = 4 ( 2 1 ) (2 11) where the depth was assumed to be the depth plus the mean water level due to waves at gauge 11 in the middle of the reef top, and the length of the basin, was assumed to be 7.31 meters

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53 which is the distance from the front edge of the acrylic glass to the front edge o f the beach slope. The fundamental frequency ( = 1) in Equation 211 was 0.03 which is consistent with the measured results on both slopes ( Figure s 2 22C 224C ). Zero Moment Wave Height The zero moment wave heights separated by the low frequency cut off are shown in Figure 2 25 for the 1:2.5 slope tests and in Figure 226 for the 1:5 slope tests. The and for each of the water levels is represented with the bathymetric profile with a 1/10 scaling on the depth. The low, mid, and high water levels are in A, B, and C respectively for both slopes. In Figure 2 26, gauge 3, which was 5.74 meters from the wave maker, was not functioning properly for the low and mid water tests, so both of the wave height values are slightly high at this gauge. In Figure 2 25, the value in the offshore region of all three water levels was caused by the seiching previously described. In all three water levels, there does not appear to be much of a decrease in the wave height between gauges 1, 2, 3 and 4. This result could mean that friction was not a factor in the offshore region. The increase in wave height on the front slope was very steep for the low (Figure 2 25A) and mid (Figure 2 25B) water levels, but the high (Figure 2 25C) water level had a more gradual increase in wave height. The peak wave height for the highfrequency was clearly located at gauge 6 for the low and mid water levels, in the high water gauge 6 and gauge 7 have similar wave heights. The higher water level o n reef top allowed more transmission of wave energy, so the wave height maintained its form further along the bathymetric profile. A similar peak to the one measured in the low and mid water profiles could have occurred in the high water tests between ga uges 6 and 7.

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54 The wave height attenuation after breaking was measured between gauge 7 and gauge 10 before the wave height stabilized. The wave height measured at gauge 10 was much lower for the low water (Figure 2 25A) t ests than for the high water (Fi gure 2 25C) tests. The wave height measured at gauge 10 was controlled by the water depth on the reef top, and the high water levels on the reef top caused a larger wave height in this region. The decay of the wave height along the reef top was consisten t for all the water levels because the process was controlled by depth dependent breaking. The individual waves in this region were in the form of a travelling bore with constant dissipation. The low frequency wave height on the 1:2.5 slope was controll ed by the magnitude of the seiching occurring i n each tests. The low water (Figure 2 25A) tests had more energy in the low frequencies at gauge 6 than the other two water levels which attributed to the peak in the low frequency zero moment wave height. T he transition of energy from the higher frequencies to the lower frequencies was measured in the wave heights on the reef top. The highfrequency wave heights decrease while the low frequency wave heights grow as the energy was propagated along the ree f t op. The high water tests (Figure 225A) had the most continual growth in the wave height at low frequencies because the water level was always well above gauge 12 (the last gauge toward shore). The low water tests had the low frequency wave height gr ow j ust after breaking, but decay due to the lack of water on the back end of the reef top. The difference between the high frequency zero moment wave height on the 1:2.5 and the 1:5 slope was the shoaling process on the front slope (Figure 2 26) The meas urements of the wave heights show that the gentler 1:5 slope creates a longer shoaling process up to the peak wave height compared to the 1:2.5 wave height results. The peak wave heights measured were lower for the 1:5 slope tests than the 1:2.5 slope tests, but the peak wave heights on the 1:5 tests

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55 could have occurred between gauges. The wave height attenuation measured in all three water levels on the 1:5 tests was very similar to the process observed in the 1:2.5 tests. The changes in the low frequency zero moment wave height on the 1:5 slope was very similar to the wave height values in the 1:2.5 tests. Wave Set up The profile of the mean water level measur ed at each gauge for all three water levels are in Figure 2 27 for the 1:2.5 slope and Figure 228 for the 1:5 slope. The high frequency was included along with the bathymetry profile to provide prospective on the changes in the mean water level. All water levels show a decrease in the mean water level close to the breaking point of the waves which is commonly called the set down. After breaking the mean water level increased across the reef top which is the set up due to the wave breaking. The s et up for th e low water level Figures 2 27A 228A increased the mean water level from the original still water level in the middle of the acrylic reef top to the back of the reef top. This trend was common for most of the low wat er tests The mid water ( Figures 2 27B 2 28B) and the high water (Figures 2 27C 228C ) tests showed increased water level on top of the reef during testing. The profiles of the set up for the 1:5 tests (Figure 2 28) had very little observable difference compared to the 1:2.5 tests. The mean water level measurements from the reef top gauges showed an increasing value of set up for the lower water levels versus the higher water levels (Fi gure 2 29). In Figure 229, the highest mean water level measurement on the r eef top (gaug es 10 12), are shown for each water level and fro nt slope compared with the deepwater wave power, The high water tests had the lowest amount of set up on the reef top which is consistent with previous results. Gourlay (1996) observed similar trends in the mean water level measured during testing of coral

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56 reef structures with various water levels. Another similarity to results presented by Gourlay (1996) is the increasing trend in the mean water level with increasing wave power Wave Run up In Figure 2 30, the 2% run up is shown against the deepwater wave power. There is an observable trend in which the height of the run up increases with increasing wave power. There does not appear to be differences in the wave run up for different water levels, and the different front slopes produce similar values of run up. Rough Surface Comparisons Wave Height The zero moment wave height measured at each gauge was non dimensionalized by the deep water wave height for each test to compare the smooth reef top tests with the r ough reef top tests (Figures 2 312 32). The deep water wave height was calculated using linear wave theory to in verse shoal the average wave height from gauges 1,2, and 3. In Figure 231, the nondimensional wave heights from the rough tests ( o) appear to be similar to the wave heights from the smooth tests ( x ) for the 1:2.5 slope. The 1:5 slope had similar results as well for the wave heig hts on the reef top (Figure 2 32). The wave heights measured at gauge 3 had large differences between the rough and the smooth tests for the 1:5 slopes, but this difference could have been caused by a bad gauge The gauge was later changed out which resulted in more consistent measurements in the rough tests. Summary L aboratory tests to examine the wave breaking and resulting wave set up on steep slopes commonly associated with coral reefs were performed. Measurements were taken along the cross shore profile of the wave flume. Complexity was added to the bathymetry in the form of

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57 sculpted acrylic glass which was designed to mimic the detailed small scale bathymetry created by coral reef structures. The wave height measurements between the two slopes had the greatest difference at breaking. The wave heights from the 1:2.5 tests were higher at or near the breaking point than the 1:5 tests The greater wave heights at breaking are contrary to previous belief that wave heights maxed at breaking at slopes of 1:10. However, the wave height at breaking could have been hi gher for the 1:2.5 slope because of the location of the nearly flat reef top just after breaking. The wave heights were previously assumed to be capped at 1:10 slopes because steeper slopes generally cause more of a surging breaking which usually has a lo wer wave height than plunging breakers. In this laboratory, the form of the waves breaking on both the 1:2.5 and 1:5 slopes were plunging breakers. The resulting wave set up from the large breaking waves was measured in the laboratory for both front s lopes. Similar to previous theory, the largest wave set up was measured in the low water tests, and the smallest wave set up was measured in the high water level tests. There was an increase in wave set up with increasing wave power which also was similar to result by Gourlay (1996) on coral reefs. The wave set up measured on the 1:5 slope was slightly lower than the measured values on the 1:2.5 slope. This was cause d by the differences in the peak wave heights prior to breaking. The higher wave heights in the 1:2.5 slope produced consistently higher wave set up. The 2% run up was measured, and showed little difference between the different water levels or the different front slopes. An increase in the 2% run up was found with increasing wave power w hich is consistent with values measured by Demirbilek et al. (2007). Wave height comparisons between the smooth and rough reef top surfaces showed little change. All three water levels had wave heights similar throughout the wave profile. The added

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58 ro ugh su rface did not result in a significant change in the friction factor. T he result means the skin friction is not a large component of the overall friction on the reef top. The results of all cases give an opportunity for detailed comparisons of both breaking and wave set up on the steep slopes with the added complex bathymetry on the reef top.

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59 Table 2 1. Location of gauges in flume laboratory. The location of x = 0 is at the wave maker. Gauge 1:2.5 slope 1:5 slope 1 4.82 4.82 2 5.13 5.13 3 5.74 5.74 4 9.72 8.77 5 10.64 10.19 6 10.84 10.63 7 11.02 11.02 8 11.35 11.35 9 11.66 11.66 10 12.83 12.83 11 14.68 14.68 12 17.73 17.73

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60 Table 2 2. Designed wave conditions for three water levels in prototype (50:1) scale. Tp (sec) Hmo (m) h (m) Tp (sec) Hmo (m) h (m) Tp (sec) Hmo (m) h (m) 7 4.18 20.75 7 3.93 21.95 7 4.05 24.5 7 5.15 20.75 7 5.18 21.95 7 5.21 24.5 10 4.48 20.75 10 3.78 21.95 10 4.57 24.5 10 5.73 20.75 10 6.03 21.95 10 6.10 24.5 10 6.86 20.75 10 6.83 21.95 10 6.89 24.5 13 5.49 20.75 13 4.42 21.95 13 4.60 24.5 13 6.10 20.75 13 6.00 21.95 13 6.10 24.5 13 7.47 20.75 13 7.47 21.95 13 7.68 24.5 16 5.46 20.75 16 4.57 21.95 16 4.60 24.5 16 6.10 20.75 16 6.00 21.95 16 6.10 24.5 16 7.62 20.75 16 7.62 21.95 16 7.74 24.5 20 5.18 20.75 20 4.21 21.95 20 4.39 24.5 20 6.19 20.75 20 6.00 21.95 20 6.10 24.5 20 7.99 20.75 20 6.92 21.95 20 7.83 24.5 20 8.02 21.95

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61 Table 2 3. Laboratory tests for the 1:2.5 front reef slope. Tests on the not painted surface have a Top of s and tests on the painted surface have a Top of r. Run Top Slope Hmoi Tp h (m) Run Top Slope Hmoi Tp h (m) 1 s 1:2.5 0.081 0.963 0.415 1 r 1:2.5 0.081 0.963 0.415 2 s 1:2.5 0.098 0.963 0.415 2 r 1:2.5 0.098 0.963 0.415 3 s 1:2.5 0.081 1.340 0.415 3 r 1:2.5 0.081 1.340 0.415 4 s 1:2.5 0.107 1.464 0.415 4 r 1:2.5 0.107 1.464 0.415 5 s 1:2.5 0.126 1.477 0.415 5 r 1:2.5 0.126 1.477 0.415 6 s 1:2.5 0.100 1.834 0.415 6 r 1:2.5 0.101 1.834 0.415 7 s 1:2.5 0.110 1.834 0.415 7 r 1:2.5 0.110 1.834 0.415 8 s 1:2.5 0.135 1.834 0.415 8 r 1:2.5 0.135 1.834 0.415 9 s 1:2.5 0.099 2.229 0.415 9 r 1:2.5 0.100 2.250 0.415 10 s 1:2.5 0.111 2.250 0.415 10 r 1:2.5 0.111 2.250 0.415 11 s 1:2.5 0.140 2.241 0.415 11 r 1:2.5 0.141 2.241 0.415 12 s 1:2.5 0.095 2.882 0.415 12 r 1:2.5 0.096 2.882 0.415 13 s 1:2.5 0.112 2.882 0.415 13 r 1:2.5 0.113 2.882 0.415 14 s 1:2.5 0.148 2.882 0.415 14 r 1:2.5 0.149 2.882 0.415 15 s 1:2.5 0.078 0.963 0.439 15 r 1:2.5 0.080 0.963 0.439 16 s 1:2.5 0.101 0.963 0.439 16 r 1:2.5 0.101 0.963 0.439 17 s 1:2.5 0.073 1.294 0.439 17 r 1:2.5 0.074 1.294 0.439 18 s 1:2.5 0.112 1.477 0.439 18 r 1:2.5 0.114 1.477 0.439 19 s 1:2.5 0.129 1.477 0.439 19 r 1:2.5 0.131 1.477 0.439 20 s 1:2.5 0.083 1.931 0.439 20 r 1:2.5 0.084 1.931 0.439 21 s 1:2.5 0.112 1.739 0.439 21 r 1:2.5 0.114 1.739 0.439 22 s 1:2.5 0.140 1.739 0.439 22 r 1:2.5 0.142 1.739 0.439 23 s 1:2.5 0.090 2.229 0.439 23 r 1:2.5 0.091 2.229 0.439 24 s 1:2.5 0.117 2.250 0.439 24 r 1:2.5 0.118 2.250 0.439 25 s 1:2.5 0.145 2.229 0.439 25 r 1:2.5 0.147 2.229 0.439 26 s 1:2.5 0.082 2.882 0.439 26 r 1:2.5 0.083 2.882 0.439 27 s 1:2.5 0.117 2.882 0.439 27 r 1:2.5 0.119 2.882 0.439 28 s 1:2.5 0.135 2.882 0.439 28 r 1:2.5 0.137 2.882 0.439 29 s 1:2.5 0.153 2.882 0.439 29 r 1:2.5 0.155 2.882 0.439 30 s 1:2.5 0.081 0.963 0.490 30 r 1:2.5 0.084 0.963 0.490 31 s 1:2.5 0.104 0.963 0.490 31 r 1:2.5 0.107 0.963 0.490 32 s 1:2.5 0.092 1.340 0.490 32 r 1:2.5 0.095 1.340 0.490 33 s 1:2.5 0.125 1.464 0.490 33 r 1:2.5 0.126 1.477 0.490 34 s 1:2.5 0.140 1.477 0.490 34 r 1:2.5 0.141 1.477 0.490 35 s 1:2.5 0.093 1.834 0.490 35 r 1:2.5 0.093 1.834 0.490 36 s 1:2.5 0.123 1.834 0.490 36 r 1:2.5 0.124 1.834 0.490 37 s 1:2.5 0.153 1.834 0.490 37 r 1:2.5 0.155 1.834 0.490 38 s 1:2.5 0.092 2.208 0.490 38 r 1:2.5 0.097 2.208 0.490 39 s 1:2.5 0.126 2.229 0.490 39 r 1:2.5 0.127 2.208 0.490 40 s 1:2.5 0.157 2.203 0.490 40 r 1:2.5 0.158 2.203 0.490 41 s 1:2.5 0.091 2.641 0.490 41 r 1:2.5 0.092 2.641 0.490 42 s 1:2.5 0.126 2.641 0.490 42 r 1:2.5 0.126 2.641 0.490 43 s 1:2.5 0.163 2.882 0.490 43 r 1:2.5 0.164 2.882 0.490

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62 Table 2 4. Laboratory tests for the 1:5 front reef slope. Tests on the not painted surface have a Top of s and tests on the painted surface have a Top of r. Run Top Slope Hmoi Tp h (m) Run Top Slope Hmoi Tp h (m) 44 s 1:5 0.085 0.963 0.415 44 r 1:5 0.081 0.963 0.415 45 s 1:5 0.103 0.963 0.415 45 r 1:5 0.097 0.963 0.415 46 s 1:5 0.090 1.294 0.415 46 r 1:5 0.081 1.294 0.415 47 s 1:5 0.115 1.464 0.415 47 r 1:5 0.107 1.408 0.415 48 s 1:5 0.137 1.477 0.415 48 r 1:5 0.126 1.477 0.415 49 s 1:5 0.110 1.834 0.415 49 r 1:5 0.100 1.834 0.415 50 s 1:5 0.121 1.834 0.415 50 r 1:5 0.111 1.834 0.415 51 s 1:5 0.150 1.834 0.415 51 r 1:5 0.136 1.834 0.415 52 s 1:5 0.108 2.208 0.415 52 r 1:5 0.100 2.208 0.415 53 s 1:5 0.121 2.208 0.415 53 r 1:5 0.111 2.208 0.415 54 s 1:5 0.153 2.104 0.415 54 r 1:5 0.140 2.104 0.415 55 s 1:5 0.102 2.882 0.415 55 r 1:5 0.094 2.882 0.415 56 s 1:5 0.121 2.882 0.415 56 r 1:5 0.111 2.882 0.415 57 s 1:5 0.160 2.882 0.415 57 r 1:5 0.147 2.882 0.415 58 s 1:5 0.079 0.963 0.439 58 r 1:5 0.078 0.963 0.439 59 s 1:5 0.104 0.963 0.439 59 r 1:5 0.100 0.963 0.439 60 s 1:5 0.075 1.340 0.439 60 r 1:5 0.073 1.340 0.439 61 s 1:5 0.118 1.408 0.439 61 r 1:5 0.114 1.408 0.439 62 s 1:5 0.136 1.477 0.439 62 r 1:5 0.131 1.477 0.439 63 s 1:5 0.087 1.834 0.439 63 r 1:5 0.084 1.834 0.439 64 s 1:5 0.119 1.739 0.439 64 r 1:5 0.114 1.834 0.439 65 s 1:5 0.149 1.739 0.439 65 r 1:5 0.143 1.834 0.439 66 s 1:5 0.094 2.229 0.439 66 r 1:5 0.091 2.229 0.439 67 s 1:5 0.123 2.208 0.439 67 r 1:5 0.118 2.208 0.439 68 s 1:5 0.153 2.229 0.439 68 r 1:5 0.147 2.229 0.439 69 s 1:5 0.085 2.882 0.439 69 r 1:5 0.082 2.882 0.439 70 s 1:5 0.122 2.882 0.439 70 r 1:5 0.118 2.882 0.439 71 s 1:5 0.142 2.882 0.439 71 r 1:5 0.136 2.882 0.439 72 s 1:5 0.161 2.882 0.439 72 r 1:5 0.154 2.882 0.439 73 s 1:5 0.081 0.963 0.490 73 r 1:5 0.080 0.963 0.490 74 s 1:5 0.103 0.963 0.490 74 r 1:5 0.103 0.963 0.490 75 s 1:5 0.092 1.340 0.490 75 r 1:5 0.093 1.340 0.490 76 s 1:5 0.123 1.477 0.490 76 r 1:5 0.123 1.477 0.490 77 s 1:5 0.138 1.477 0.490 77 r 1:5 0.138 1.477 0.490 78 s 1:5 0.091 1.834 0.490 78 r 1:5 0.092 1.845 0.490 79 s 1:5 0.121 1.834 0.490 79 r 1:5 0.123 1.834 0.490 80 s 1:5 0.152 1.834 0.490 80 r 1:5 0.153 1.834 0.490 81 s 1:5 0.094 2.229 0.490 81 r 1:5 0.096 2.229 0.490 82 s 1:5 0.123 2.208 0.490 82 r 1:5 0.126 2.229 0.490 83 s 1:5 0.153 2.229 0.490 83 r 1:5 0.157 2.229 0.490 84 s 1:5 0.089 2.882 0.490 84 r 1:5 0.091 2.882 0.490 85 s 1:5 0.122 2.882 0.490 85 r 1:5 0.125 2.882 0.490 86 s 1:5 0.159 2.882 0.490 86 r 1:5 0.163 2.882 0.490

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63 Table 2 5. The 95% confidence interval for each gauge in meters. The mean of the CI was calculated for all tests at each water level. 1:2.5 1:5 Low Mid High Low Mid High Total Gauge 1 0.0041 0.0042 0.0045 0.0038 0.0040 0.0044 0.0042 Gauge 2 0.0039 0.0041 0.0069 0.0038 0.0039 0.0041 0.0045 Gauge 3 0.0039 0.0044 0.0046 0.0048 0.0047 0.0040 0.0044 Gauge 4 0.0035 0.0037 0.0040 0.0035 0.0036 0.0041 0.0037 Gauge 5 0.0051 0.0050 0.0050 0.0041 0.0042 0.0071 0.0051 Gauge 6 0.0060 0.0063 0.0064 0.0046 0.0048 0.0049 0.0055 Gauge 7 0.0023 0.0029 0.0046 0.0019 0.0025 0.0040 0.0030 Gauge 8 0.0020 0.0021 0.0029 0.0020 0.0021 0.0028 0.0023 Gauge 9 0.0020 0.0021 0.0020 0.0016 0.0017 0.0018 0.0019 Gauge 10 0.0020 0.0021 0.0016 0.0018 0.0018 0.0014 0.0018 Gauge 11 0.0029 0.0028 0.0019 0.0029 0.0027 0.0018 0.0025 Gauge 12 0.0034 0.0038 0.0039 0.0031 0.0034 0.0037 0.0036 Gauge 13 0.0073 0.0092 0.0082 0.0055 0.0068 0.0065 0.0073

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64 Figure 2 1. Laboratory set up for both the 1:2.5 and 1:5 slopes. The reef profile was built on an existing concrete slope which is the red section on the lower frame.

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65 Figure 2 2. Picture of the 1:5 reef face slope. The 1:5 slope was placed on top of the 1:2.5 slope.

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66 Figure 2 3. Picture from just off the front edge of the 1:5 reef face slope. The blue paint is the cut acrylic glass. The divider on the left side of the picture was put in place to decrease the testing width and to dampen reflection.

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67 Figure 2 4. Contours of the acrylic reef top. The axis is distorted to fit the page.

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68 Figure 2 5. Picture of the cut acrylic glass from on top of the flume. Each piece was 2 feet x 2 feet.

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69 Figure 2 6. The capacitance wave gauges were placed in the flume. The gauges were connected to a motor for calibration.

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70 Figure 2 7. Bathymetry for both the 1:5 slope tests ( ) and the 1:2.5 slope tests ( ). The three water levels used in test are displayed relative to the bathymetry.

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71 Figure 2 8. The reflection coefficient calculated using the Goda analysis for all test cases against the surf similarity parameter.

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72 Figure 2 9. Example of full time series from T est 7s at gauge 2. The time series consisted of 20 minutes or 1200 seconds of measurements A) Time from 0 300 seconds. B) Time from 301600 seconds. C) Time from 601900 seconds. D) Time from 9011200 seconds.

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73 Figure 2 10. Energy density spectra from T est 7s, gauge 2. A) S(f) for the raw data without filtering. B) T he low frequency S(f). C) T he high frequency S(f). The low frequency cut off was at 1 5 A C B

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74 Figure 2 11. Time series of the high frequency waves ( ) and the low frequency waves ( ). A) Time from 0 300 seconds. B) Time from 301600 seconds. C) Time from 601900 seconds. D) Time from 9011200 seconds.

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75 Figure 2 12. The profile of the zero moment wave height for Test 7 s with the 95% confidence interval error bars calculated for the full experiment.

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76 Figure 2 13. Energy density spectra for Test 7s in low water on 1:2.5 slope. A) The transformation of the energy density spectrum from offshore. B) T he front edge of the reef top. C) T he back section of the reef top

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77 Figure 2 14. Energy density spectra for Test 21s in mid water on 1:2.5 slope. A) The transformation of the energy density spectrum from offshore. B) The front edge of the reef top. C) T he back section of the reef top.

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78 Figure 2 15. Energy density spectra for Test 36s in high water on 1:2.5 slope. A) The transformation of the energy density spectrum from offshore. B) The front edge of the reef top. C) T he back section of the reef top.

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79 Figure 2 16. Energy density spectra for Test 50s in mid water on 1:5 slope. A) The transformation of the energy density spectrum from offshore. B) The front edge of the reef top. C) T he back section of the reef top.

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80 Figure 2 17. Energy density spectra for Test 64s in mid water on 1:5 slope. A) The transformation of the energy density spectrum from offshore. B) The front edge of the reef top. C) T he back section of the reef top.

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81 Figure 2 18. Energy density spectra for Test 79s in mid water on 1:5 slope. A) The transformation of the energy density spectrum from offshore. B) The front edge of the reef top. C) T he back section of the reef top.

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82 Figure 2 19. The low frequency energy density spectra from Test 7s in low water on 1:2.5 slope. A) The transformation of the energy density spectrum from offshore. B) The front edge of the reef top. C) T he back section of the reef top

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83 Figure 2 20. The low frequency energy density spectra from Test 21s in mid water on 1:2.5 slope. A) The transformat ion of the energy density spectrum from offshore. B) The front edge of the reef top. C) T he back section of the reef top.

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84 Figure 2 21. The low frequency energy density spectra from Test 36s in high water on 1:2.5 slope. A) The transformation of the energy density spectrum from offshore. B) The front edge of the reef top. C) T he back section of the reef top.

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85 Figure 2 22. The low frequency energy density spectra from Test 50s in low water on 1:5 slope. A) The transformation of the energy density spectrum from offshore. B) The front edge of the reef top. C) T he back section of the reef top

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86 Figure 2 23. The low frequency energy density spectra from Test 64s in mid water on 1: 5 slope. A) The transformation of the energy density spectrum from offshore. B) The front edge of the reef top. C) T he back section of the reef top

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87 Figure 2 24. The low frequency energy density spectra from Test 79s in mid water on 1:5 slope. A) The transformation of the energy density spect rum from offshore. B) The front edge of the reef top. C) T he back section of the reef top

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88 Figure 2 25. The profile of the low frequency zero moment wave height ( ) and the high frequency zero moment wave height ( ) on the 1:2.5 slope. A) L ow water, Test 7s B) M id water, Test 21s C) H igh water, Test 36s

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89 Figure 2 26. The profile of the low frequency zero moment wave height ( ) and the high frequency zero moment wave height ( ) on the 1:5 slope. A) L ow water, Test 50s B) M id water, Test 64s C) H igh water, Test 79 s.

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90 Figur e 2 27. The high frequency ( ) with the mean water level (*) measured at each gauge in the profile on the 1:2.5 slope. A) L ow water, Test 7s B) M id water, Test 21s C) H igh water, Test 36s

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91 Figure 2 28. The high frequency ( ) with the mean water level (*) measured at each gauge in the profile on the 1:5 slope. A) L ow water, Test 50s B) M id water, Test 64s C) H igh water, Test 79s

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92 Figure 2 29. Maximum mean water level me asured on the reef top for all smooth surface test cases compared with offshore wave power.

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93 Figure 2 30. The 2% run up measured on the run up gauge on the back slope of the laboratory bathymetry for all test cases with the smooth surface compared against the offshore wave power.

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94 Figure 2 31. Non dimensional compared at each gauge for the smooth surface tests ( ) and the rough surface tests ( o ) for all the 1:2.5 tests. A) Low water. B) Mid water. C) High water.

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95 Figure 2 32. Non dimensional compared at each gauge for the smooth surface tests ( ) and the rough surface tests ( o ) for all the 1: 5 tests. A) Low water. B) Mid water. C) High water.

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96 CHAPTER 3 WAVE BREAKING Introduction Present numerical models were tested on beaches with slopes less than 1:10, but the slope of the reef face on coral reefs can be much steeper. The steep slopes obs erved on these coral reefs create problems for numerical models. It was shown in the previous chapter that the front slope can cause large changes in the maximum wave heights prior to breaking. Also, the rapid energy dissipation after breaking is very different than the slower dissipation rates commonly seen on more gradual sloping beaches. T o test the affect of the steep slopes on breaking waves a 1D energy flux model was used to compare against the laboratory data. The p resent STWAVE breaking functions as well as breaking functions that incorporate different wave characteristics were tested. T he model is also used to test the differences between using the present STWAVE saturated breaking function versus a bore type energy dissipation function. Method ology One Dimensional Energy Flux Balan ce A simple 1D shoaling model was used to test different breaking parameters such as maximum wave heights and energy dissipation models. The conservati on of energy flux equation Equation 31 was used with dissipation for wave breaking. = (3 1) The total average wave energy, was found from = 1 8 (3 2)

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97 where is the root mean square (rms) wave height, is the density of the fluid, and is the acceleration due to gravity. The value of is assumed to be 0.71 T he group speed of waves is based on linear wave theory = 2 1 + 2 2 (3 3) where is the wave celerity, is the water depth, and is the wave number found by solving th e dispersion relationship ( Equation 34). = tanh (3 4) Two half steps were used to numerically solve Equation 31. The first half step was the energy flux equation without dissipation. The equation was discritized using a first order forward finite difference scheme ( Equation 35). = (3 5) In Equation 35, is the previous spatial cell, + 1 is the present spatial cell, and + 1 2 is the half step for energy. The next step was to dissipate the energy of the waves by breaking. The discritization of the next half step was a first order forward finite difference scheme. = (3 6) Friction and reflection were neglected in the 1D energy flux model. If available the mean water level measured during data collection for each experim ent was added to the bathymetry, = + (3 7) where is the still water depth and is the mean water level.

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98 Error Calculation Quantifying the discrepancies between the model calculations and observations was done using two error calculations. To determine if the model results are high or low compared to the laboratory observations, the error calculation in Equation 38 is used. = (3 8) The rms error in Equation 39 is used to determine the total amount of error. The rms error is not dependent on the sign of the error. = (3 9) The error is calculated at all observations points. Averaging was done differently depending on the laboratory case. Test Cases The wave flume laboratory study previously described was used for comparison of steep reef face slopes. Both the 1:2.5 and 1:5 slopes were simulated for all three water levels. The bathymetry of the 1D domain is shown in Figure 31. The bathymetry was t aken from the center cross section of the laboratory study in the wave flume. The three waters levels tested are the same as described in the flume tests (Chapter 2) The grid spacing for the bathymetry input into the model was 0.01 meters in the x (hori zontal) and z (vertical) directions. The offshore boundary of the domain was located at 5.13 meters from the wave maker, which was the location of gauge 2 in the laboratory testing. Comparisons were made between the calculated from the 1D model and the measured f rom the laboratory for the highpass spectrum. The low frequency oscillations were not used in comparisons with the model. The input wave height and peak period were

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99 based on the mean energy density spectrum of the high frequency waves at gauges 1 3, which were the furthest offshore gauges Results Energy Dissipation The present wave breaking in STWAVE is a saturated breaking criteria where = > (3 10) The breaking wave height is calculated based on Smith et al. (1997) formul ation, where the minimum from Equation 311 is used at each grid point. = 0 64 = 0 2 ( ) (3 11) All wav e characteristics used in ( Equation 311) are local values not deep water. An alternative to the saturation method presently used in STWAVE is the bore model or wave breaking energy dissipation model. Two energy dissipation models were tested in comparisons with the wave flume laboratory data set. The di ssipation models by TG83 ( Equation 312) and JBAB 07 ( Equation 313) emplo y different techniques than the present STWAVE breaking function for model ing the energy loss at wave breaking, but th e both models were found by Apot s os et al. (2008) to be capable of simulating a broad range of wave cases = 3 16 1 1 + (3 12) = 3 16 1 + 4 3 + 3 2 exp ( ) erf ( ) (3 13)

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100 The two energy dissipation equations along with the STWAVE breaking equation were tested on the laboratory tests for all three water levels. The two dissipation models using the recommended breaking wave heights were evaluated with B values ranging from 1 to 4 to find the best fit. The be st fit was determined from the mean rms error for all the runs at a give water level. T he mean rms error was calculated using eight comparison points. The three offshore gauges were removed b ecause each simulation was initialized with an average wave height and period between the three gauges. In the 1:5 test, each run was initialized with just the first two gauges because of a bad Gauge 3. Gauge 12, the most onshore gauge, was removed because the low and mid water levels did not always have water on the gauge, so for consistency the error calculation was not performed at this gauge. In Figure 3 4, the STWAVE breaking is shown along with the TG83 and JBAB07 best fit results. The profile of the measured results is a typical profile from the 1:2.5 slope at low water (h = 0.415 m) In Figure 3 4A, the peak wave height for all three models was a low compared to the data, but the wave height attenuation from TG83 and JBAB07 model results were similar to the measured data. In Table 3 1, the mean rms error results for all three models are shown (STWAVE, TG83, and JBAB07). The mean rms error of the present STWAVE saturated breaking equation was higher than both the fitted TG83 and J BAB07 (Table 31 and Figure 3 4B ). The B value used as the best fit for both of the ene rgy dissipation models was the lowest point in Figure 3.4B as well as the value in parenthesis in Table 3 1. The mean rms error of TG83 was lower than the JBAB07, and the B val ue for TG83 was much closer to the commonly recommended value of 1 by TG83 and JBAB07. In Figure 3 4B, the shape of the mean rms error for each dissipation model with the array of B coefficients was different. The TG83 model had a distinct point with the lowest mean rms

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101 error, which means the model is sensitive to changes in B because the error changes rapidly for small c hanges in B. Howeve r, the JBAB07 model results had a flatter response near the lowest error. The values of B were much larger, but once the B coefficient was close then the model accuracy was consistent. Further analysis was perfor med on the two energy dissipation models by using the breaking wave height from STWAVE in the models. In TG83, the equation had to be adapted to account for the difference in breaking notation. The present STWAVE breaking equation solves for the breaking wave height without using a gamma value related to depth. To account for this, all values of were replaced with The energy dissipation for TG83 then becomes, = 3 16 1 1 + (3 14) In Figure 3 2A the maximum wave height before breaking was lower with the breaking wave height from STWAVE than with the recommend values for each dissipation model. The mean error and the fitted B value for the T G83 ST were similar to the results from TG83 The mean error for JBAB07 ST was s maller than JBAB07 (Table 31) and the fitted B value w ere closer to 1.0. This value of B was still larger than what is commonly used, but it is closer than JBAB07 results The smaller B value is likely due to smaller peak wave heights from the STWAVE breaking model. All the tested model formulations were also run for the test cases at mid and high water. The mean rms error s are shown in Table 31. The mean rms error of the present STWAVE breaking function decreased with in creasing water level The lowest mean rms error was at high water. The rms error was most likely smallest for the high water because deeper water depths ha d more water over the reef crest, so more wave breaking occurred closer to the reef and even

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102 on the reef top. This could result in a larger area of wave energy dissipation which is similar to more gentle sloping bathymetry In the TG83 and TG83 ST models, the best fit B coefficient did not change much for all three water levels (Table 3 1). However, in JBAB07 and JBAB07 ST the B coeffici ent was reduced from low water to high water from 3.1 to 2.8 in JBAB07 and from 2.3 to 1.4 in JBAB07 ST for the 1:2.5 tests This result shows a relationship in the JBAB07 dissipation model that is affected by the water depth, and the change in B from JBAB07 to JBAB07 ST demonstrates the impact that the maximum breaking wave height equation c an have on the energy dissipation. The TG83 model had less variation in the B coefficient both with changing the water depth and the maximum breaking wave height equation. The same comparisons that were made for the 1:2.5 slope were made for the 1:5 slope laboratory tests. In Figure 34A, an example of the best fit STWAVE, TG83, and JBAB07 wave breaking models were compared with a typical wave height profile from the laboratory for the 1:5 slope. The peak wave height before breaking was observed to be smaller on the 1:5 slope than it was on the 1:2.5 slope, so the peak wave height s in the models were closer to the measured results. The mean rms errors for the 1:5 tests were generally less t han the mean rms error from all the 1:2.5 tests. However, t he low water level tests had a similar mean rms error on both slopes with the JBAB07 and JBAB07 ST tests. The reason for this has not be en determined When comparing the results from both dissipation models using the STWAVE breaking wave height equation, the peak wave height in the models were low in relation to the observed wave height values in the laboratory. This result could be causing some extra error in the STWAVE breaking model runs, but the model results appear to match the wave height attenuat ion measured after breaking (Figure 35A)

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103 B Coefficient vs. Gamma The relationship between B and the maximum breaking wave height in energy dissipation model s is a complex problem ( Apo t s os et al. 2007; Thor nton and Guza, 1983) TG83 performed tests using B values between 0 and 2 when comparing the TG83 dissipation model to different field data sets The divergence of B from 1 is physically described by TG83 as a difference between how the wave is breaking and how a typical bore would respond under t he same conditions. The greater the value of B, the faster energy is dissipated from the system. In the TG83 dissipation model the B coefficient is cubed compared to JBAB07 where the B is of order 1. In most circumstances the value of B is assumed to be equal to 1, but in the previous section it was shown that the value of B must be larger than 1 in the present study to account for the breaking of waves on steep slopes. The gamma value is related to the maximum breaking wave height by the McCowen (1894) equation, = (3 15) This form of the maximum breaking wave height is similar to the present value recommended in TG83 where is equal to 0.43. The influence of changing the maximum breaking wave height was previously observed by altering the energy dissipation functions to include the STW AVE maximum breaking equation. A series of tests were performed using the 1D energy flux model to better understand the relationship between B and on the present laboratory study. In t he tests all the 1:2.5 and 1:5 slope laboratory tests were simulated with B values ranging from 0.1 to 4.0 and values ranging

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104 from 0.1 to 2.0. Both of the variables had a spacing of 0.1 between values. Each model was run 741 times to determine the best fit values for the specific test conditions. T he contours of the mean rms error for both TG83 and JBAB07 are shown for all the tested B and values in Figure 36A and Figure 36B, respectively Only errors bet ween 0 and 1 are presented but larger error did occur for certain test parameters. In order to achieve the lowest rms error, the JBAB07 model need ed B values larger than 1.8 for all In the TG83 tests, the error was more dependent on a relationship between B and than a distinct set of values for either. To get a more detailed look at how B and impact the accuracy of the dissipation models, Figure 37 shows the mean rms error from TG83 o ver segments of the experiment profile In Figure 37A, the error was found at gauge 4 which was located on the offshore edge of the front slope. For all tested there appears to be B values for which the TG83 dissipation model would result in a low rms error at this point. However, small er values of must have a small value of B in order to avoid becoming too dissipative. At the breaking point the wave heights measured at gauges 5 and 6 were used for comparisons with the models (Figure 3 7B) The mean rms er ror at the breaking point had lower error as the value of increased. This makes sense given the largest wave heights did occur at these points, so the largest maximum breaking wave heights would provide a better match to the data. The TG83 wave height dissipation region afte r breaking used gauges 7 and 8 to match the wave attenuation. The lowest mean rms err or was found with B values ranging from 1 to 3 and corresponding values between 0.4 and 1.3 (Figure 37C). A linear relationship was apparent at this point for the T G83 results where low maximum breaking height required low B coefficient and high maximum breaking wave height required a larger B. The final segment of

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105 the wave height profile was the reef top where gauges 9 11 were used for comparisons. The mean rms error for all B and in TG83 was not below 30% (Figure 3 8D). The reef top region was the most sensitive to the addition of the measured mean water level to the bathymetry. The interpolation of this data could play a role i n the accuracy in this region. In Figure 3 8, the results of the B and tests using the JBAB07 dissipation model were very different than what was observed in TG83. In the offshore region (Figure 38A), the lowest rms error was almost independent of B given a value close to 0.4. The breaking wave region (Figure 3 8B) was similar to TG83 because larger values of result ed in low rms errors with no dependency on B. The dissi pation region (Figure 3 8C), had a very small region where the mean rms error was small. In order to result i n a small error, the B value had to be larger than 2, and was greater than 0.38. T he maximum values for B and that resulted in a low rms error were 4 and 0.78 respectively. Both of these results are higher compared to t he typical values prescribed to these coefficients. On the reef top (Figure 3 8D) the mean rms error possible was smaller than f or TG83. The value of B was greater than 1.0 for the smallest errors and given a value of larger than 0.3, B must increa se rapidly with increasing to maintain a low rms error The accuracy of both TG83 and JB AB 07 with different values of B and was also broken into the five different wave periods. In Figure 39, the rms error for TG83 was high for the lower wave periods, 1.0 and 1.4 seconds (Figure 3 9A,B), but t he mean rms error was lower for the 1.8, 2.2, and 2.8 second wave periods ( Figure 39C,D,E). The best fit values of B and were similar for the higher wave periods. The JB AB 07 model had similar results to the TG83 for the 1.0 second wave period tests. The rms error was high for all values of B and However, the 1.4 second tests had a lower possible mean rms error for the JB AB 07 tests than for

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106 the TG83 tests. The shape of the contours in Figure 3 10B,C,D,E show that the possible values of resulting in a low mean rms error were smaller for JBAB07 than TG83, and the values of B for a low rms error were always approximately 1.5. The tests with B and were separated into three groups of initial wave heights. The initial wave heights less than 0.10 meters were classified as low wave heights, the initial wave heights between 0.10 and 0.12 meters were mid wave heights, and the initial wave heights greater than 0.12 meters were high wave heights. In Figure 3 11, the TG83 model was compared against the three different wave height groupings. The low wave heights had a small region with low mean rms error (Figure 3 11A). The s maller initial wave heights had a smaller peak wave height, so values of the maximum breaking wave height were on the lower end. The high er peak wave heights observed in the mid and high initial wave heights resulted in a larger range of possible values (Figure 3 11B,C). The result of the separated wave height tests were similar between the JBAB07 and TG83 (Figure 3 12). The low initial wave height tests ha d a much smaller region with low mean rms erro rs than the mid and high water. All values of B with low mean rms error remain ed above 1.5, and in the low initial wave height tests the B value was 2.0. In Figure 3 13, the same evaluation of B and was made for all the simulations of the 1:5 slope. The result of the mean rms error analysis was very similar between the different slopes. The greatest difference between the different slopes occurred when comparing results in the breaking region (Fi gures 3 14B 315 B). On the 1:2.5 slope, the mean rms error did not change much given large maximum breaking coefficients but on the 1:5 slope, there was a distinct selection of values for B and that resulted in low rms errors The lowest rms errors occurred for JBAB07 model on the 1:5 s lope when was around 0.4, and B was bel ow 1.0. This

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107 result implied that for the 1:5 slope, adjusting B provides a better results for the breaking section than adjusting in JBAB07. In contrast, on the 1:2.5 slope, the value of controlle d the accuracy of the results. The relationship between B and was similar to the 1:2.5 slope for the separated wave periods (Figures 3 163 17) and wave heights (Figures 3 18319). Breaking Wave Height The previous analysis demonstrated the importance of the relationship between B and on the accuracy of the numerical models compared with the laboratory results A constant value for was unable to closely match the measured data at all segments of the wave height profile. Three breaking wave heigh t parameters which are variable through the wave profile were tested in the 1D energy flux model with both the TG83 and JBAB07 energy dissipation. The maximum wave height parameters were all based on local wave characteristics. The maximum breaking wa ve height equations from Ostendorf and Madsen (1979), OM79, = 0 14 ( 0 8 + 5 ) 2 1 10 = 0 14 ( 0 8 + 5 ( 0 1 ) ) 2 > 1 10 (3 16) Kamphuis (1991), Ka91, ( ) = 0 095 exp ( 4 ) tanh 2 (3 17) and Ruessink et al. (2003), Ru03, = ( 0 76 + 0 29 ) (3 18) were t ested for both of the laboratory slope cases The OM79 and Ka91 breaking w ave height equations were selected because both included the bottom slope, m, in the determination of the maximum breaking wave height. Ru03 breaking equation was included because of its simplicity

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108 and it s development by using comparisons with complex field data. All three of the equatio ns had similarities to the breaking formulation present ly used in STWAVE, so the two dissipation models with the STWAVE breaking formulation was used as a comparison in these tests. The results of the mean rms error for both front slopes and three water levels for each slope are sh own in Table 32. The values of the B coefficient which produce d the smallest mean rms error for each breaking wave height equation are shown in parenthesis in Table 3 2. In Figure 3 20 and Figure 3 21 as well as in Table 3 2, the tested breaking equations had reduced error compared to the STWAVE values because of the increased wave heights just before breaking. All three of the new breaking equations had lower mean rms errors than the present STWAVE breaking equation at all water levels and for both slopes. In comparisons between the two dissipation functions, it became apparent that the breaking equation was a large contributor to the mean rms error. The breaking equation from Ka91 was found to have overall lower mean rm s errors when incorporated in the TG83 energy dissipat ion model. The other two breaking equations, OM79 and Ru03, had lower mean rms errors in the JBAB07 energy dissipation model for the low and mid water, but for the high water the TG83 model had lower m ean rms errors for all breaking wave equations. However, the mean rms errors for both dissipation models at high water were be low 10% for the three new breaking equations which is considered a good fit in the present study. Summa ry The numerical modeling of wave breaking on steep slopes greater than 1:10 was shown to be very difficult for a linear wave model The present STWAVE breaking function had mean rms errors near 30% in comparisons with all laboratory tests. The use of a bore type energy dissipation model was shown to decrease below 20% when the coefficient B was calibrated to fit the data.

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109 The relationship between the B coefficients and the maximum breaking wave height coefficient was tested to determine the best fit values for both. It was found that values of both coefficients were affected by different wave characteristics. By examining the relationship at different segments of the wave profile, it was determined that the breaking coeffi ci ent should be variable in the cross s hore profile. However, B also showed variation given different dissipation equations as well as different wave periods and initial wave heights. The variable breaking coefficient was accounted for by testing three breaking functions OM79, Ka91, and Ru03. Each breaking function was different than the present STWAVE breaking function, but each is still dependent on local wave conditions. Each breaking function was run with both dissipation models, and the best fit was found. The OM79 and Ru03 were found to have lower mean rms errors than the present STWAVE breaking and the breaking equation from Ka91. Between OM79 and Ru03, it was determined that OM79 was a more robust breaking model at matching the laboratory measurements on both of the front slopes t ested with mean rms errors of near 20% for low water, 12% for mid water, and 8% for high water The difference between the two dissipation models was less about the best fit accuracy and more about the differences in the B coefficient. The B coefficien t affects the rate at which energy is dissipated from the breaking wave. Examining the best fit B coefficient for both TG83 and JBAB07 with the breaking equation from OM79 shows a more consistent value for B in the TG83 model. The best fit B varies more in the JBAB07 model. In most previous studies using both energy dissipation models the values of B was default at 1.0, but this study has shown that the value of B has more variation depending not only on the slope of the beach, but on the wave period a nd depth of the water.

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110 Table 3 1. Mean rms error for three breaking models in 1D model STWAVE TG83 TG83 ST JBAB07 JBAB07 ST 1:2.5 Low 39.8 23.0 (1.1) 23.3 (1.1) 29.8 (3.1) 19.5 (2.3) Mid 37.1 17.6 (1.2) 18.4 (1.2) 21.5 (2.9) 15.1 (2.1) High 23.0 10.2 (1.1) 11.4 (1.0) 14.2 (2.8) 14.2 (1.4) 1:5 Low 35.4 22.6 (1.1) 22.8 (1.1) 30.2 (2.6) 19.8 (2.0) Mid 33.8 16.2 (1.1) 16.9 (1.1) 20.8 (2.6) 14.1 (1.9) High 20.4 7.2 (1.1) 9.4 (1.1) 11.5 (2.8) 12.0 (1.5) The mean rms error was found at every gauge for all runs at each water level. The mean at each gauge was found over all the test cases at each water level. Then the mean rms error was found for all the gauges.

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111 Table 3 2. Mean rms error for maximum breaking wave height equations in 1D model OM79 Ka91 Ru03 1:2.5 Low TG83 19.5 (1.5) 22.6 (1.4) 27.5 (1.4) JBAB07 20.0 (3.2) 20.5 (3.2) 21.0 (2.7) Mid TG83 14.2 (1.5) 14.9 (1.3) 21.7 (1.4) JBAB07 12.1 (2.8) 14.5 (2.6) 14.8 (2.6) High TG83 11.1 (1.3) 11.3 (1.2) 14.5 (1.5) JBAB07 10.2 (2.4) 13.2 (2.0) 12.3 (2.5) 1:5 Low TG83 19.5 (1.5) 21.0 (1.4) 28.8 (1.3) JBAB07 20.2 (2.9) 19.1 (2.7) 21.3 (2.4) Mid TG83 12.6 (1.5) 13.1 (1.3) 20.5 (1.4) JBAB07 11.1 (2.6) 12.8 (2.3) 14.0 (2.4) High TG83 7.1 (1.4) 8.3 (1.2) 11.0 (1.5) JBAB07 7.3 (2.7) 10.0 (2.2) 8.6 (2.4) The mean rms error was found at every gauge for all runs at each water level. The mean at each gauge was found over all the test cases at each water level. Then the mean rms error was found for all the gauges. All the breaking equations were run in bot h the TG83 and JBAB07 dissipation models. The B value is in the parenthesis next to the mean rms error.

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112 Figure 3 1. Bathymetry for the numerical model simulations of the wave flume. The model was tested on both the 1:2.5 ( ) and the 1:5 ( ) s lopes. Three water levels were tested in the mode l: low water with a depth of 0.419 m ( ), mid water with a depth 0.439 m ( ), and high water with a depth of 0.49 m ( ).

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113 Figure 3 2. Comparison of ) with B = 1.2, and JBAB07 ( ), with B = 4.0, energy dissipation functions A) T ypical wave height profile from the 1:2.5 slope (x), Test 7s B) Change in error for each breaking model with different B coefficients.

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114 Figure 3 3. Comparison of t he STWAVE breaking wave height equation tested in the threshold breaking function TG83 ( ) with B = 1.2, and JBAB07 ( ) with B = 2.3. A) Zero moment wave height comparisons with measured results from Test 7s with the 1:2.5 slope (x). (B) Change in error for each breaking model with different B coefficients.

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115 Figure 3 4. Comparison of ), with B = 1.2, and JBAB07 ( ), with B = 4.0, energy dissipation functions A) T ypical w ave height profile from the 1: 5 slope (x) Test 50s B) Change in error for each breaking model with different B coefficients.

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116 Figure 3 5. Comparison of t he STWAVE breaking wave height equation tested in the threshold TG83 ( ) with B = 1.2, and JBAB07 ( ) with B = 2.3. A) Zero moment wave height comparisons with measured results from Test 50s with the 1:5 slope (x). (B) Change in error for each breaking model with different B coefficients.

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117 Figure 3 6. A nalysis of the mean rms error based on the coefficient of the maximum breaking wave height ( ) and B on the 1:2.5 slope A) T he TG83 energy dissipation function. B) The JBAB07 energy dissipation function.

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118 Figure 3 7. The mean rms error based on changing B and in TG83 for different segments of the wave breaking profile on the 1:2.5 slope A) O ffshore region B) Breaking region C) Dissipation region. D) Reef top.

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119 Figure 3 8. The mean rms error based on changing B and in JBAB07 for different segments of the wave breaking profile on the 1:2.5 slope A) O ffshore region B) Breaking region C) Dissipation region. D) Reef top.

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120 Figure 3 9. The mean rms error based on changing B and in TG83 for different wave periods on the 1:2.5 slope A) T = 1.0 seconds. B) T = 1.4 seconds. C) T = 1.8 seconds. D) T = 2.2 seconds. E) T = 2.8 seconds.

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121 Figure 3 10. The mean rms error based on changing B and in JBAB07 for different wave periods on the 1:2.5 slope A) T = 1.0 seconds. B) T = 1.4 seconds C) T = 1.8 seconds. D) T = 2.2 seconds. E) T = 2.8 seconds.

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122 Figure 3 11. The mean rms error based on changing B and in TG83 for differen t wave heights on the 1:2.5 slope. A) I nitial wave heights below 0.10 meters B) Initial wave heights between 0.1 and 0.12 meters. C) Initial wave heights greater than 0.12 meters.

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123 Figure 3 12. The mean rms error based on changing B and in JBAB07 for different wave heights on the 1:2.5 slope. A) I nitial wave heights below 0.10 meters B) Initia l wave heights between 0.1 and 0.12 meters. C) Initial wave heights greater than 0.12 meters.

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124 Figure 3 13. A nalysis of the mean rms error based on the coefficient of the maximum breaking wave height ( ) and B on the 1: 5 slope A) T he TG83 energy dissipation function. B) The JBAB07 energy dissipation function.

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125 Figure 3 14. The mean rms error based on changing B and in TG83 for different segments of the wave breaking profile on the 1: 5 slope A) O ffshore region. B) Breaking region. C) Dissipation region. D) Reef top.

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126 Figure 3 15. The mean rms error based on changing B and in JBAB07 for different segments of the wave breaking profile on the 1: 5 slope A) O ffshore region B) Breaking region C) Dissipation region. D) Reef top.

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127 Figure 3 16. The mean rms error based on changing B and in TG83 for different wave periods on the 1: 5 slope A) T = 1.0 seconds. B) T = 1.4 seconds. C) T = 1.8 seconds. D) T = 2.2 seconds. E) T = 2.8 seconds.

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128 Figure 3 17. The mean rms erro r based on changing B and in JBAB07 for different wave periods on the 1: 5 slope A) T = 1.0 seconds. B) T = 1.4 seconds. C) T = 1.8 seconds. D) T = 2.2 seconds. E) T = 2.8 seconds.

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129 Figure 3 18. The mean rms error based on changing B and in TG83 for different wave heights on the 1:5 slope. A) I nitial wave heights below 0.10 meters B) Initial wave heights between 0.1 and 0.12 meters. C) Initial wave heights greater than 0.12 meters.

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130 Figure 3 19. The mean rms error based on changing B and in JBAB07 for different wave heights on the 1:5 slope. A) I nitial wave heights below 0.10 meters B) Initial wave heights between 0.1 and 0.12 meters. C) Initial wave heights greater than 0.12 meters.

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131 Figure 3 20. The OM79 ( ), Ka91 ( ), and Ru03 ( ) breaking equations were tested on the 1:2.5 slope. A) TG83 wave height profile from Test 7s in low water. B) Error with TG83 for values of B. C.) JBAB07 wave height profile from Test 7s in low water. D) Error with JBAB07 for values of B

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132 Figure 3 21. The OM79 ( ), Ka91 ( ), and Ru03 ( ) breaking equations were tested on the 1: 5 slope. A) TG83 wave height profile from Test 7s in low water. B) Error with TG83 for values of B. C.) JBAB07 wave height profile from Test 50s in low water. D) Error with JBAB07 for values of B.

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133 CHAPTER 4 WAVE BASIN LABORATOR Y Introduction Based on a similar premise of the wave flume coral reef laboratory experiment presented previously in Chapter 2, t he goal of the study was to examine wave transformation and set up over a coral reef system The wave basin provided a unique opportunity to examine the effect of the detailed bathymetry in a large three dimensional (3D) wave basin. The larger basin all ows for a n impact by wave refraction on the overall wave set up measured during the experiments. Methodology Experiment Facilities The experiment was conducted at th e United States Army Engineer Research and Development Center (ERDC) in Vicksburg, MS. The concrete basin is 27.43 meters wide and 47.86 meters long and 0.76 me ters high on all sides (Figure 41). Waves were generated using the Directional Spectral Wave Generator (DSWG). The DSWG is 27.4 meters long instal led on one end of the wave basin (Figure 4 1). The DSWG is capable of generating wave heights of 30 cm with a stroke length of + / 36 cm. The wave generator consists of four modules, but for this laboratory experiment only three were in operation. Wave guides were positioned at the beginning of module 2 and at the end of module 4 of the wave generator to direct waves onto study area. Passive wave absorbers were positioned behind the study area to decrease reflection in wave basin. Laboratory Conditions A steel platform was constructed to provide the underlying framework of the coral reef bathymetry. The steel platform was uniform across the basin width and the along basin ba thymetry The platform front edge consisted of two slopes; a toe of 1:4.3, and a second slope

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134 of 1:13.6. The front slope was followed by a flat reef top which was 25 feet long from front to back A simulated beach with slope 1:9 was located on the backside of the flat reef top. A reef top was designed to replicate a section of Guam reef by Dr. Ernie Smith. A field survey was taken using SHOALS with an accuracy of 5 meters The profile was interpolated to provide a higher resolution. After reducing to model scale the reef profile was cut to an accuracy of a hundredth of a f oot A router was used to cut 2 foot x 2 foot x 2 inch pieces of acrylic glass to the bathymetric profile given from the interpolation of the Guam reef. This allows for very small bathymetric structures such as a coral reef to be tested in a laboratory environment. The contour of the designed reef top bathymetry is depicted in Figure 42 with the vertical distance in meters In total 240 acrylic pieces were cut with a total area of 40 feet x 24 feet or 12.19 meters x 7.31 meters (Figure 4 3) Each piece was marked with four points with exact model elevation for more accurate deployment. The elevations were shot with a level based on the datum of zero feet across the front edge of the acrylic reef profile. Adjustments were made with four bolts in the c orners of each acrylic piece to account for any non uniformity in the steel flat top platform. Each piece of acrylic glass was secured in pl ace using 920 glue. This created a water proof seal between the bolts on the four corners of each piece and the steel platform. A plywood front slope was added to create a smooth transition between the front slope of the steel platform and the front edge of acrylic glass (Figure 4 4) The plywood was 3.67 feet or 1.12 meters with a slope of 1:7.2. Plywood was al so added on the back of the acrylic reef to smooth the transition back to the steel beach (Figure 4 5) The back piece of plywood was 50.2 feet or 15.3 meters with no slope. Pea gravel size rocks were placed on the sides of the reef top to dampen waves a nd reduce the affect of a drop off in elevation to the sides of the acrylic.

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135 Some problems occurred during the running of the wave basin which needed fixing to create a more functional and repeatable experiment. A fter running waves, the rocks at the sides of the acrylic were building into large bars at the breaking point. The bars created inconsistencies in the flow on top of the acrylic glass leading to nonuniformities in wave measurements. A thin layer of concrete was placed on top of the r ocks to give a bond at the breaking point. The concrete allowed some transmission of energy into the rocks to dampen flows outside of the experimental area, but created a more stable environment. Secondly, during larger wave tests the bond between the ac rylic pieces and the steel platform became loose and allowed pieces to pop out of place. The reason for this was found to be old paint on the steel platform being peeled back by the strength of the 920 glue. The paint was sanded off at the contact poin ts where the bolts from the corner of the acrylic pieces were being glued to the steel platform. By removing the paint a stronger bond was created to hold the acrylic pieces in place during all wave conditions tested. During installation of the acrylic reef, the pieces were placed with as much accuracy as possible. However, the detail of the pieces made it very difficult to exactly match the designed profile. A survey was taken after construction using a ground lidar system which measured the exact bat hymetry. The contour of the reef top profile which was installed i n the basin is shown in Figure 4 6. The difference between the designed profile and t he actual profile is in Figure 47. The bathymetry measured by the ground Lidar survey provides a bett er contour map for numerical studies which utilize the basin laboratory data. Instrumentation Nineteen capacitance wire wave gauges were used to measure water surface elevation in the basin (Figure 41). Seven of the gauges were positioned in deep water on the flat floor of the basin. Two gauges were located on the front slopes of the reef with one on the steel platform

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136 slope and the other on the plywood slope. Ten gauges were on top of the acrylic reef structure. A ll gauge locations are given in Table 41. The gauges wer e constructed at USACE ERDC CHL, and were calibrated using the dynamic calibration described in Chapter 2 with a maximum error of 0.002% every morning laboratory tests were performed. Five capacitancewire run up gauges were in stalled on the steel back s lope. Three run up gauges were 2.1 m eters and two were 3 meters long. Tape was used to smooth the transition from plywood to steel The tape was also useful for reducing any interference between the gauge and the steel platform. Cal ibration was performed with three speci fied water levels for both the 2.1 meter and 3 meter gauges. The run up gauges were spaced 8 meters apart on the back slope of the basin. During the experiment run up gauge 4 which is a 3 meter run up gauge malfunc tioned, so some tests have only 4 run up gauges Testing Procedure The s equence of events for testing was repeated prior to every run. First, t he wave conditions and data collection parameters were defined similar to the wave conditions in the flume. T he wave conditions were random waves generated with a TMA spectrum based on a target zero moment wave height, H, and peak period, T. The peakedness parameter, used in the TMA generated waves was 3.3. The sampling rate was defined as 20 Hz for all r uns, and the run length was 20 minutes. Next, t he water level was checked using a point gauge with an accuracy of 0.001 ft. All of the capacitance wave gauges were balanced to zero for added accuracy. The run up gauges were not balanced. Then, t he waves were started and data collection commenced. Two computers were used to control the wave generator and collect data. The data col lection commenced when the wave generation started, and it ended when the waves ceased. During

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137 most of the test runs, pic tures and movies were taken of both the wave breaking and the wave transformation on the reef top. A fter each run was finished, the reef top was checked to make sure all the acrylic pieces had stayed in place. Also, prior to the placement of concrete on the rocks, the pea gravel was flattened in between runs. P reliminary post processing was performed on the data after each run to ensure all gauges worked properly, and the results were reasonable. Notes were taken at this time to record any problems with the wave maker or gauges. Post Processing Test Cases All laboratory tests were initialized with a wave spectrum generated from TMA equation with a designed zero moment wave height peak period, and peak enhancement factor of 3.3. Peak period s of approximately 1.0, 1.4, 1.9, 2.2, and 2.8 seconds were tested given a variety of significant wave heights, ranging 0.06 0.18 meters. Three wate r levels were evaluated where h is the water level in the basin (Figure 4 8 ). Descriptions of a ll te st cases run are given in Table 4 2, where Hmoi is the incident zero moment wave heigh t, Tp is the peak period, and h is the water level Reflection Analysis The time ser ies of water levels for gauges 4,5, and 6 were used to separate the incident and reflected wave signals observed in the laboratory set up using the Goda analysis described in Chapter 2 The similar bathymetry between the present laboratory study and Demirbilek et al. (2007) make comparison of re flection coefficients useful. Demirbilek et al. (2007) had three front slopes with values of 1:5, 1:18.8, and 1:10.6. In the present study, the front sl opes were 1:4.3, 1:13.6, and 1:7.2. In the calculation of the surf similarity parameter an average slo pe of near 1:6.5 was used.

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138 The value of the reflection coefficient for all but one test case is below 0.2 or 20% reflection (Figure 4 9) There does not appear to be a change in the reflection coefficient with changing surf similarity or water depth The outlier with reflection coefficient greater than 25% was likely caused by a bad gauge. The reflection values were higher than the reflection coefficients from D emirbilek et al. (2007), but the greater values were likely caused by the i ncrease in average slope from 1: 9 in D emirbilek et al. (2007) to 1: 6.5 in the present study. Spectral Analysis The dynamic water surface of the wave basin was recorded in terms of water surface elevation versus time (Figure 4 10). All laboratory runs w ere prepared for 20 minutes or 2400 seconds. For consistency, all analysis was perfo rmed for measured data between 3 0 seco nds and 20 minutes. In the example shown in Figure 4 10 the wave period, wave height, and water depth were 2.8 seconds, 0.098 meters and 0.418 meters respectively. To complete the spectral analysis the data was detrended to remove the mean. A Welch method for spectral analysis was performed in Matlab using the pwelch command to analyze the e nerg y density spectrum (Figure 4 11A ) The W elch method uses a Hamming window with 50% overlap. The data set was separated into 4096 data point sections for the analysis. The Welch method calculates the spectrum for each section and then averages to get the final spectrum. A filter was r un on the data to remove all energy greater than 2.0 Hz. This was pe rformed to remove the very high frequency oscillations from the data. A second filter was run to separate the low and high frequency oscillations. The cut off for the low frequency oscillations was set at 1 5 or 0.667 times the peak frequency. This value was obtained by observations of the full energy density spectrum. T he full energy density spectrum ( Figure 4 -

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139 11A) along with the low frequency ( Figure 4 11B) and highfrequen cy ( Figure 4 11C) spectra show the distribution of energy over sections of frequencies The low frequency cut off separates the raw data into low frequency oscillations and high frequency oscillations (Figure 4 12). The low frequencies are generally asso ciated with reflection or seiching, and the high frequencies are the generated wave signal. Wave parameters such as the zero moment wave height, and the peak period, were derived from the spectral analysis. The equations for were d iscussed in Chapter 2, Equations 2427. Statistical Analysis The mean water level, was found by computing the mean of the raw wave surface data. Prior to calculating the mean water level, the first 5 seconds of the raw data was averaged and subtracted from the full raw data. This step was performed to remove any float in the gauge that was not fixed by zeroing the gauges prior to testing. The value of the mean water level on gauges which were both wet and dry during a test needs to be used with care. The inconsistency of waves at these gauges creates difficulties in interpreting the computations. The 2% run up was calculated using the run up gauges by performing an upcrossing analysis on the raw data. The upcrossing analysis counts a wave each time the measured water elevation crosses zero traveling in the positive (upward) direction. The 2% run up was determined as the mean of the highest 2% of the run up waves measured. Wave Gauge Confidence Intervals The 95% confidence interval was found b y splitting the raw data into 3 minute sections similar to the procedure performed in the flume laboratory (Chapter 2). A spectral analysi s was performed on all of the 3 minute segments, and the zero momen t wave height was calculated. Using all of the wave heights, the 95% CI was found from Equation 28 for all tests.

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140 The mean 95% CI for all tests at each water level are available in Tab le 4 3. The largest values of CI were calculated for the offshore gauges which are more impacted by reflection off the front slope and the lowest values were on the reef top gauges. However, the error associated with the 95% CI is dependent on the magnitude of the mean wave height. Smaller wave heights measured on the reef top have larger relative errors associated with the 95% CI. The 95% CI for all the gauges over the entire laboratory project was assumed to be the average of the 95% CI for all water level s, and was 3.3 mm The highest 95% CI for the total project at a particular gauge was 6.7 mm which means the 95% CI was between +/ 6.7 millimeters of the aver age wave height measured at that gauge. In Figure 4 13, the 95% CI error bars are shown with the zero moment wave height from Test 11 for gauges located on the centerline of the laboratory basin Results Spectral Energy The transformation of the wave energy across the bathymetric profile tested in the laboratory experiment can be seen by observing changes in the e nergy density. In Figures 414 416, the energy density, S(f), is shown at gauges along the centerline of the basin progressing from offshore to the reef top. The gauges used for the analysis were gauge 5, 8, 9, 10, 11, and 14. Gauge 5 was located in the offshore region of the basin. G auges 8 and 9 were on the front slope of the reef structure, and gauges 10, 11, and 14 were located on top of the reef. At the pe ak frequency, the change in energy density between gauges 5 and 9 was similar for all three water levels. The peak of the en ergy stayed at the same frequency, and the amount of energy at the peak deceased as the wave energy approached the top of the front slope. The dissipation of energy from the offshore through the front slope was caused by a combination of friction and wave breaking After breaking, usually between gauges 9 and 10, the wave attenuati on could be seen in the rapid decrease of energy density at the peak frequency. In the

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141 low water tests (Figure 4 14), the energy at the peak frequency was completely gone by the time the waves arrived at gauge 11 However, the mid (Figure 415) and high (Figure 416) water tests had a small amount energy remaining at the peak frequency at gauge 14. At all water levels, the majority of the energy was located at low frequency at gauge 14. The still water level of the low wa ter was barely past to gauge 11 so the water level at gauge 1 1 was very small for these tests. The mid and high water level tests had more water at gauge 11 which allowed for a greater amount of wave energy t o remain at the peak frequency. Low F requency Spectral Energy The low frequency energy spectral transformation at all three water levels is shown in Figure s 4 174 19 for similar incident waves with wave height close to 0.14 m and period near 2.2 s The energy peaks in the low frequencies in the offshore gauges (the top panels) were caused by reflection and seiching in the basin. The offshore region of the basin could be assumed to be in the shape of a rectangular basin. The period of oscillations for seiching in a rectangular basin is = 2 (4 1) The length of the wave basin, was assumed to be 30 meters and the depth of the basin, was assumed to be 0.354 meters. The first harmonic of the seiching, = 1, has a period of oscillation equal to 30 seconds. This corresponds to a fundamental frequency of 0.03 Hz. There was no peak measured at 0.03 Hz in any of the water levels. However, the 2nd harmonic of seiching, = 2, was equal to 15 seconds which corresponds to a frequency of 0.07 Hz. There was a peak measured in the energy spectrum at 0.07 Hz in all three water levels. The 3rd harmonic had a period of 10 seconds which is a frequency of 0.1 Hz. The second peak measured in the low frequency spectrum was lo cated close to 0.1 Hz.

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142 In Figure s 4 17C 419C the transformation of the low frequency energy spectrum on the reef top is shown. During the laboratory experiments in the wave flume, seiching was measured on the reef top. However, in the basin experiment the peaks associated with seiching on the reef top were no apparent. The lack of energy peak s at the centerline of the basin could be caused by the increased lateral dimension in the basin The reflected and seiching oscillations have greater lateral freedom which could create greater spreading of the energy across the lower frequencies which would minimize the peaks at the expected seiching frequencies. In the high water tests (Figure 4 19), a peak of energy in the low frequency was measured in a gauge on the reef top. The transmission of low frequency energy was more apparent in the high water tests because of the deeper water over the reef crest. Wave Heights The zero moment wave heights measured at the wave gauges along the centerline p rofile of the basin are shown for al l three water lev els in Figure 4 20 for incident conditions of wave heights near 0.14 m and periods near 2.2 s The transformation of the wave height across the profile was similar for all test cases at a given water depth. The wave heights measured offshore and on the front slope had similar dissipation up to breaking for all water depths. The high frequency wave height was dissipated by both friction and wave breaking between gauges 4,5, and 6, which were considered offshore, and gauges 7, 8, and 9 on the front slope. Just after gauge 9, which was near the top of the front slope, the wave heights for the high frequencies attenuated rapidly for the low ( Figure 4 20A) and mid ( Figure 4 20B) water test s The waves app ear to retain their shape through gauge 10 for the high water tests which is likely caused by the deeper depths over the reef crest which induce less breaking. On the reef top the wave heights appear to be controlled by the water level on the reef top. The higher water level

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143 ( Figure 4 20C) had the largest high frequency wave heights, and the low water level ( Figure 4 20A) had the smallest wave heights. As stated before, the low frequency wave height maintained a steady wave height along the basin profile. In the low water ( Figure 4 20A), the low frequency wave heights were greater than the high frequency wave heights on the reef top. The dissipation of the energy at the peak frequency o n the reef top occurs rapidly which leaves only long p eriod oscillations remaining. In the high water ( Figure 4 20C), the high frequency wave heights were still larger than the low frequency wave heights because of the larger transmission of wave energy at the peak frequency for the deeper depths. The zero moment wave heights measured laterally across the basin showed similar relationship between the high frequency wave heights and the low frequency wave heights to the measured values at the centerline gauge 14 (Figure 421). The gauges located to the side of the centerline of the basin measured smaller wave heights on the reef top than the centerline gauges because the bathymetry was deeper in the center than on the sides. The alongshore variation in depth on the reef top in the location of the alongshore gauge array is the black line in Figure 421. The high frequency wave heights measured in the low water tests ( Figure 421A) were very small laterally across the reef top (< 0.02 m) The gauges on both sides of the centerline were dry initially for the low water tests, so all wave heights evolved due to wave set up at these locations. In both the low ( Figure 4 21A) and mid ( Figure 4 21B) water tests, the low frequency wave heights were larger than the high frequency wave heights on the reef top, but in the high water ( Figure 4 21C), the low frequency and high frequency wave heights were similar. Wave Set up The mean water level was measured at each gauge using the statistical approach described previously. The mean water level of gauges which were initial dry during testing was

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144 taken based on the recorded data, but the result should be viewed with caution. T he mean water level measurements for the gauges on the centerline profile of the basin are shown in Figure 4 22A,B,C for the low water, mid water, and high water, respectively. The test cases shown are Test 10 in low water, Test 22 in mid water, and Test 34 in high water. In previous laboratory tests by Gourlay (1996a ), De mirbilek et al. (2007), and the previously described flume tests, the maximum mean water level relative to the still water level decreased with increasing water depth on reef tops. In t he basin study, the lowest water level had the highest mean water level or wave set up compared to t he deeper water levels (Figure 4 23). One test case in the low water level tests had low values of wave set up which is seen in Figure 4 23. All other tes t cases follow the similar trend from the previous labo ratory experiments. In Figure 4 23, the mean water level increased with increasing wave power, which is consistent with results obtained during laboratory testing of coral reef bathymetries by Seelig (1983) and Gourlay (1996b) Summary The results from a complimentary laboratory basin experiment to the wave flume experiment previously described were presented. The 3D basin was used to provide a more detailed experiment by allowing lateral motion to d evelop. Precisely sculpted acrylic glass was used for the 12.2 x 7.1 meter reef top to better mimic the intrusive nature of coral reef structures on the wave and flow patterns observed in the shallow water system. The wave heights measured on the centerline profile of the basin showed the general trend of waves breaking on the front slope and the propagation and dissipation of wave heights on the reef top. The high frequency zero moment wave heights had a large amount of dissipation along the profile of the basin, but the low frequency zero moment wave heights had an almost constant value along the profile of the basin. On the reef top the high frequency wave

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145 heights were measured to be less than the low frequency wave heights for the low and mid water levels, but the high water allowed for the transmission of high frequency energy further along the reef top profile. The results of mean water level change relative to the still water level of each tests was consistent with previous laboratory experiments. The lower water level tests had a greater wave set up on the reef top than higher water level tests because of greater wave dissipation on the reef The maximum increase in the mean water level measured on the reef top was found to be proport ional to the offshore wave power. The amount of wave set up increased with increasing offshore wave power.

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146 Table 4 1. Location of capacitance wire gauges in 3D basin. The location of (0,0) was at the wave maker on the left side of the basin when looki ng offshore. Gauge x (m) y (m) 1 14.02 15.7 2 14.02 15.7 3 18.5 15.7 4 22.98 15.7 5 23.32 15.7 6 23.93 15.7 7 26.79 15.7 8 28.19 15.7 9 29.23 15.7 10 30.45 15.7 11 31.61 15.7 12 33.4 9.6 13 33.4 12.65 14 33.4 15.7 15 33.4 18.75 16 33.4 21.79 17 33.58 15.39 18 33.58 16 19 35.91 15.7

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147 Table 4 2. Test conditions measured from the reflection analysis at gauges 4, 5, and 6. Test Hmo (m) Tp (s) h (m) 1 0.083 0.978 0.418 2 0.097 1.043 0.418 3 0.083 1.477 0.418 4 0.111 1.477 0.418 5 0.115 1.867 0.418 6 0.136 1.867 0.418 7 0.153 1.867 0.418 8 0.109 2.309 0.418 9 0.136 2.206 0.418 10 0.144 2.309 0.418 11 0.098 2.796 0.418 12 0.127 2.878 0.418 13 0.132 2.878 0.418 14 0.084 0.970 0.442 15 0.100 1.043 0.442 16 0.070 1.477 0.442 17 0.119 1.477 0.442 18 0.140 1.477 0.442 19 0.092 1.903 0.442 20 0.159 1.867 0.442 21 0.112 2.309 0.442 22 0.146 2.149 0.442 23 0.156 2.149 0.442 24 0.083 2.757 0.442 25 0.122 2.757 0.442 26 0.080 1.041 0.493 27 0.102 1.009 0.493 28 0.090 1.390 0.493 29 0.129 1.390 0.493 30 0.145 1.477 0.493 31 0.157 1.477 0.493 32 0.096 1.785 0.493 33 0.156 1.796 0.493 34 0.169 1.867 0.493 35 0.093 2.228 0.493 36 0.151 2.228 0.493 37 0.171 2.228 0.493 38 0.085 2.757 0.493 39 0.043 2.841 0.493 40 0.159 2.871 0.493

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148 Table 4 3. The 95% confidence interval for capacitance wave gauges in meters. Gauge Low Water Mid Water High Water Total 1 0.0045 0.0050 0.0057 0.0051 2 0.0047 0.0049 0.0053 0.0050 3 0.0050 0.0054 0.0056 0.0053 4 0.0085 0.0057 0.0059 0.0067 5 0.0056 0.0058 0.0057 0.0057 6 0.0052 0.0053 0.0054 0.0053 7 0.0045 0.0046 0.0048 0.0046 8 0.0045 0.0043 0.0044 0.0044 9 0.0031 0.0036 0.0041 0.0036 10 0.0016 0.0017 0.0023 0.0019 11 0.0012 0.0012 0.0012 0.0012 12 0.0018 0.0018 0.0014 0.0017 13 0.0018 0.0016 0.0014 0.0016 14 0.0018 0.0015 0.0012 0.0015 15 0.0019 0.0017 0.0029 0.0022 16 0.0016 0.0018 0.0013 0.0016 17 0.0017 0.0015 0.0012 0.0015 18 0.0017 0.0015 0.0012 0.0015 19 0.0017 0.0019 0.0013 0.0016

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149 Figure 4 1. The Direction al Spectral Wave Generator Basin (DSWG) layout. The wave generator was located on the left side at x = 0 Wave guides were located on each side of the reef study area, and wave absorbers were located behind the study area.

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150 Figure 4 2. Designed bathymetric profile of reef top. The reef top was designed by Dr. Ernie Smith with units of meters.

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151 Figure 4 3. Picture of 2 ft x 2ft acrylic pieces with bolts on all four corners for adjustability.

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152 Figure 4 4. Picture of front edge of reef structure. The front slope is a steel platform which has marine plywood on top to ensure a smooth transition to the front edge of the acrylic glass.

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153 Figure 4 5. Picture taken standing on the back slope of reef platfo rm. The back of the designed acrylic reef was smoothed back to the back slope of the steel platform with marine plywood.

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154 Figure 4 6. The actual bathymetry of the acrylic reef top. The section was scanned using a LIDAR system to accurately measu re the detailed vertical change.

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155 Figure 4 7. The difference between the actual reef top contours and the designed reef top contours. Positive values of elevation relate to higher actual elevations than designed.

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156 Figure 4 8. Bathymetric profile for the laboratory basin test The three water levels tested were the low water, h = 0.418 (m) ( ), the mid water, h = 0.442 (m) ( ) and the high water, h = 0.493 (m) ( )

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157 Figure 4 9. The reflection coefficient values for all tests calcu lated using the Goda analysis.

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158 Figure 4 10. Example of an entire time series meas ured from gauge 8 during Test 36 The test consisted of 20 minutes or 1200 seconds of continuous waves A) Time from 0 300 seconds. B) Time from 301600 seconds. C) Time from 601 900 seconds. D) Time from 9011200 seconds.

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159 Fi gure 4 11. Example of the energy density spectrum for gauge 8 during Test 36 A) The full energy density spectrum. B) The low frequency spectrum. C) The highfrequenc y spectrum. C B A

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160 Figure 4 12. The raw data after the low frequency cut off was applied to separate the low frequency ( ) and high frequency ( ) oscilla tions for gauge 8 during Test 36 A) Time from 0 300 seconds. B) Time from 301600 seconds. C) Time from 601900 seconds. D) Time from 9011200 seconds.

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161 Figure 4 13. The zero moment wave height from Test 11 with error bars based on the total 95% CI of the laboratory experiment.

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162 Figure 4 14. The energy density spectrum transformation along the cross shore profile for the low water, Test 10. A) Offshore to wave breaking. B) Reef top gauges.

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163 Figure 4 15. The energy density spectrum transformation along the cross shore profile for the mid water Test 22 A) Offshore to wave breaking. B) Reef top gauges.

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164 Figure 4 16. The energy density spectrum transformation along the cross shore profile for the high water Test 36 A) Offshore to wave breaking. B) Reef top gauges.

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165 Figure 4 17. Low frequency energy density spectrum for Test 10 in low water. A) Offshore to wave breaking. B) Reef top gauges.

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166 Figure 4 18. Low frequency ene rgy density spectrum for Test 22 in mid water. A) Offshore to wave breaking. B) Reef t op gauges.

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167 Figure 4 19. Low frequency ene rgy density spectrum for Test 36 in high water. A) Offshore to wave breaking. B) Reef top gauges.

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168 Figure 4 20. The low frequency ( ) and highfrequency ( ) zero moment wave heights along the basin profile A) L ow water, Test 11 B) Mid water, Test 24. C) H igh water, Test 36.

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169 Figure 4 21. The zero moment wave height laterally across the reef top from gauge 12, far left, to gauge 16, far right. The low frequency wave heights ( o) and the high frequency wave heights ( ) are shown A) Low water. B) Mid water. C) High water.

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170 Figure 4 22. The mean water level (* ) and the high frequency zero moment wave height ( ) measured at the gauges located on the centerline of the basin. A) L ow water B) Mid water. C) High water.

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171 Figure 4 23. The maximum mean water level measured on the reef top for all test cases.

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172 CHAPTER 5 WAVE BASIN MODELING Introduction The wave basin laboratory experiment previously described provides unique challenges when attempting to compare with numerical model results. The added space in the lateral direction creates a opportunity to compare the numerical results with realistic flow patterns. In the two dimensional wave flume the lateral flow is controlled by the side walls, but in the wave basin, the lateral motion is relatively unrestrained. This leads to different patterns in the wave breaki ng and wave set up on the reef top. Methods The 1D energy flux model was used for comparisons of the wave shoaling and energy dissipation due to breaking. The bathymetry for the wave basin was used to cre ate a 1D bathymetric profile at the centerline o f the basin. The location of the bathymetry profile in the lateral direction was determined to correspond with the wave gauges in the laboratory. An example of the bathymetry profile is shown in Figure 51. The offshore boundary for all tests was at 23 .316 meters w hich was the location of g auge 5 The tests were initialized with the average wave height from the three gauges lo cated in the Goda array: Gauges 4, 5, 6. The initial wave period was determined from the reflection analysis to be the inciden t wave period. The mean water level measured at each gauge along the centerline profile was linearly interpolated onto the 1D grid because the wave set up was not calculated during initial testing. The high frequency zero moment wave heights measured i n the basin laboratory were used for comparison (see Chapter 4) The compa ris ons between measured and calculated results w ere made at all gauges using the rms error. Th e rms error was calculated as

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173 = 1 (5 1) for each simulation. All tests at a water level were averaged to find the total error at each water level. The gauges on the centerline of the basin were used for comparisons with the model. The gauge numbers were 7, 8, 9, 10, 11, 14, and 19. The same numerical model cases examin ed in comparisons with the wave flume experiment were tested on the wave basin. The three breaking models: STWAVE, TG83, and JBAB07; along with the three breaking wave height formulas: OM79, Ka91, and Ru03; were tested In order to better understand the characteristics of breaking in the wave basin tests, the TG83 and JBAB07 energy dissipation models were tested to find the B coefficient and maximum breaking wave height which resulted in the lowest error over all tests Results Initially the 1D energy flux model was tested on the wave basin bathymetry using the present STWAVE breaking function. In Figure 52, the wave model results had breaking occur earlier than the observed location in the measured results. The w ave heights on the reef top were close to the measured results, which minimizes the error of the model to the observed data. The percent rms error for the 1D energy flux model with STWAVE breaking was 16.0%, 27.2%, and 30.2% for high, mid, and low water t ests respectively (Table 5 1) The large error could be caused by the lack of application of friction in the model T he data did show dissipation in the offshore region of the lab oratory but the largest error was likely caused by the early breaking in the wave model. B Coefficient versus Gamma The two energy dissipation models, TG83 and JBAB07, were tested in the 1D model in an attempt to reduce the error of the numerical model compar ed with the wave basin results.

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174 The relationship between the B coefficient and the maximum breaking wave height coefficient, was previously discussed in comparisons for the wave flume experiment. The B coefficient was found to play a larger rol e tha n expected in the overall accuracy of the numerical model when compared aga inst the results from the steep sloped wave flume laboratory. The wave models were run with B coefficients ranging from 0 to 4, and values between 0 and 2. The rms error was calculated for all the runs t o determine the best fit for the tests at a particular water level. Figure 5 3A shows the rms error for all values of B and tested for the TG83 dissipation model on the high water level. For the TG83 tests, a specific val ue of B corresponded to a value of which would produce the smallest rms error. Thus, B and are interdependent The same error resulted from linearly increasing both B and The JBAB07 dissipation model results showed a more independent relationship between B and than was observed with the TG83 model (Figure 5 3B) The best fit values for ranged from 0.5 to 0.9, but the value of B ranged from 1 to 4. The larger change in B with a small change of signifies that the model is very sensitive to small changes in This result differs from the TG83 model where small changes in require small changes in B to maintain accuracy. In the mid and low water tests, the trends in the relationship between B and wer e very similar to the high water tests for both the TG83 and JBAB07 models (Figure 5 4, 55 ) The biggest change between the low and mid water tests and the high water tests, was the limit on the maximum B values allowed for the smallest error in both TG83 and JBAB07. In the TG83 tests, the maximum B that resulted in the smallest error at the mid water was close to 3, and in the low water the maximum B was closer to 2. The maximum B coefficient with the smallest error for the JBAB07 tests decreased from 4 in high water to 3 in both mid and low water. The reason for a decrease in B with decreasing water level needs to be examined further.

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175 Maximum Breaking Equations The maximum breaking wave heig ht equations previously described in tests from the wave flume laboratory experiment were used in comparisons with the wave basin results. The slope dependent breaking wave height equations from OM79 and Ka91 as well as the equation from Ru03 showed improvement in the agreement between model and m easured wave height results compared with the present STWAVE breaking equation or the constant value for the breaking equation. In the high water tests, the rms error was reduced for all three wave breaking models compared to the present STWAVE breaking The three maximum breaking equations in the TG83 energy dissipation model had errors of 9.4%, 9.0%, and 10.2% for the OM79, Ka91, and Ru03 respectively (Table 5 1) These rms errors were down from the 16% for the present STWAVE breaking equation. Figure 5 6A is an example of the wave height profile using the best fit value of B for the three maximum breaking equations along with the TG83 breaking equation. The TG83 model breaks further up the slope which is more consistent with the measured wave heig hts. However, all three maximum breaking models had higher peak wave heights just before breaking than the measured results In Figure 56B, the rms error for all three breaking equations in TG83 show the sensitivity of the results in TG83 to the value of B for each breaking equation. The results for all three maximum breaking wave height equations in JBAB07 dissipation model were similar to the TG83 results. The rms error was reduced from 16% for the present STWAVE breaking equation to 8.8%, 11.1%, and 10.0% for OM79, Ka91, and Ru03. The wave height profile for each of the tests in Figure 5 6C, show the dissipation of the wave height in the offshore region for OM79 and Ka91 perform better than Ru03. The peak wave heights before breaking were smaller f or the JBAB07 results than in the TG83 results. The smaller wave

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176 heights were due to the higher values of B coefficient needed in the JBAB07 tests than in the TG83. The B coefficient was larger for all three cases in the JBAB07 tests than in the TG83 tests (Table 5 1). This result is consistent with the results previously discussed on the wave model comparisons in the wave flume. The best fit values of B in tests using the maximum breaking coefficients were below 1.0 during all tests (OM79 and Ka91 wer e compared by dividing the maximum wave height by the depth to get the value of ). The low values of relate to values of B below 2 for the lowest errors (Figure 53A) for TG83. The same values of could have B coefficients between 1.5 and 4 depending on the exact value of for JBAB07 (Figure 5 3B). The best fit values of B found during the analysis were 1.0, 1.0, and 1.1 for OM79, Ka91, and Ru03 in the TG83. The values are all close to 1.0 whi ch i s consistent with previous tests described by TG83 using the TG83 energy dissipation model. The recommended values for B by TG83 were between 0.8 and 1.4, and these results fall in this range. The best fit B values for the tests with JBAB07 were 1.7, 1.5, and 1.3. The B values were higher for the JBAB07 model than the TG83 model. In TG83, the B c oefficient is cubed which explain s the higher values of B in JBAB07 than in TG83. The mid water level had a thin layer of water covering the entire reef top. T he less water on the back of the re ef top created explains the larger errors in the numerical model comparisons. The rms error for the present STWAVE breaking equation was 27.2% which is greater than the error for the high water tests. Adding the TG83 energy dissipation model and using the three maximum breaking wave height equations reduced the rms error to 14.9%, 14.9%, and 18.7% for OM79, Ka91, and Ru03. The best fit B coefficient was equal to 0.9 for all three of the maximum breaking wave he ight equation tests. The B coefficients were lower for the mid water

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177 than the high water which is opposite of the results from the wave flume comparisons. The mean rms error for the tests with JBAB07 was 14.7%, 16.8%, and 19.4%. The best fit values for B were 1.4, 1.3, and 1.1. The mean rms error for the present STWAVE model on the low water tests was 30.2%. The low water had the highest errors compared with the mid and high water level tests. The low water had a thin layer of water on the front edge of the reef and no water initially on the back of the reef top. The wave transformation in the shallow water was difficult for the numerical model to calculate. The mean rms error s for the three breaking equations in T G83 were 20.4%, 20.0%, 23.7% for OM79, Ka91, a nd Ru03, respectively. The best fit B values were 0.9, 0.9, and 0.8 for the three models. All three water levels had B values close to 1.0 for the TG83 tests. The mean rms errors for the breaking equations in JBAB07 were 16.9%, 18.7%, 22.2%. The best fit B values were 1.6, 1.5, and 1.5, which is consistent with the previous water levels B 1.5. The B value for JBAB07 did fluctuate more than the B value results from TG83 tests. The more variable B values in JBAB07 were shown in Figure s 53, 4, 5, where the TG83 had a smaller range of B values for each value of First Order B In order to compare TG83 and JBAB07 on even ground, it was important to examine how the models respond with similar values of B. The B value in TG83 is histor ically cubed which acts as a different variable then the one described by JBAB07. The TG83 model was run for all the low water tests with a first order B coefficient, and compared with the laboratory results (Figure 5 9A). The best fit B value was found to encompass a wider range of values which is similar to the JBAB07 results, but the minimum error does not seem to change very much. The shape of the lowest error region when relating the B coefficient and looks similar to the

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178 JBAB07 results (Figure 59). A smaller region of values for a large range of B values can calculate the lowest errors. The TG83 model with first order B was then tested with the three maximum breaking equations to determine the impact on results. Again, the B values had a la rger range which could calculate low errors for each breaking model (Figure 5 10). The shape of the error compared with different B values looks much more similar between TG83 (Figure 510B) and JBAB07 (Figure 5 10D). More research is needed to determine the impact of altering the B value in TG83 on a wide range of wave characteristics. Summary The present STWAVE breaking model was tested in the 1D energy flux model on the wave basin laboratory results. The model calculated the wave breaking offshore compared to the measured wave heights. Even though the model calculated the wave height attenuation due to breaking to o far offshore the wave heights on the reef top were very close for the model and the laboratory. The two energy dissipation models, TG83 and JBAB07, were tested in the 1D energy flux model. The variability of the wave stability parameter, B, and the maximum breaking wave height parameter, was tested to determine the impact on the mean rms error for all three water levels. The TG83 model had less variability in B given a value of This result means the B value in TG83 will tend to have a smaller range of values for different wave conditions. The TG83 dissipation model had lower errors for a large range of values, so the mode l is forgiving on the choice of as long as the appropriate value of B is used. The value of for JBAB07 which resulted in low est errors was a much smaller range than the results from TG83. Also, the

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179 range of B values was much larger for a given The lowest rms errors result from the selecti on of the correct value of because the value of B appears to be more flexible. The variable maximum breaking wave height equations, OM79, Ka91, and Ru03, were tested on the wa ve basin experiment. In the m id and high water tests, the results of the mean rms error for the OM79 breaking equation was lower in the JBAB07 energy dissipation equation. The mean rms error for Ka91 and Ru03 were lower in the TG83 dissipation model. In the low water, the results fo r all the breaking models in the JBAB07 dissipation model had smaller rms errors than the breaking models in the TG83 dissipation model. However, the small differences could be affected by the particular dynamics in each laboratory test because both dissi pation models were generally within a 3% difference. The improvement of the results of both energy dissipation models compared with the results of the present STWAVE model was significant. The best fit B values determined during the tests were very similar for all three water levels with the TG83 dissipation model. The results were close to 1.0 for all three water levels. The JBAB07 tests had a wider range of best fit B values for all three water levels, and with different breaking equations. This is consistent with the flume tests in which the B value in JBAB07 had more variability given a certain maximum breaking wave height coefficient.

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180 Table 5 1. Mean RMS percent error for the zero moment wave heights in the wave basin laboratory comparisons along with the best fit B coefficient. STWAVE OM79 Ka91 Ru03 Low TG83 30.2 20.4 (0.9) 20.0 (0.9) 23.7 (0.8) JBAB07 16.9 (1.6) 18.7 (1.5) 22.2 (1.5) Mid TG83 27.2 14.9 (0.9) 14.9 (0.9) 18.7 (0.9) JBAB07 14.7 (1.4) 16.8 (1.3) 19.4 (1.1) High TG83 16.0 9.4 (1.0) 9.0 (1.0) 10.2 (1.1) JBAB07 8.8 (1.7) 11.1(1.5) 10.0 (1.3)

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181 Figure 5 1. Bathymetry profile for the 1D energy flux model tests on the wave basin.

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182 Figure 5 2. The wave height profile for Test 36 in the basin laboratory (x) with the results form the energy flux model run with STWAVE breaking criteria ( ) The thin black line is the wave height results from the energy flux model run with STWAVE breaking criteria.

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183 Figure 5 3. The mean rms error for model comparisons made to the high water laboratory tests. A) Results from the model run with TG83 energy dissipation. B) Results from the model run with JBAB07 energy dissipation.

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184 Figure 5 4. The mean rms error for model comparisons made to the mid water laboratory tests. A) Results from the model run with TG83 energy dissipation. B) Results from the model run with JBAB07 energy dissipation.

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185 Figure 5 5. The mean rms error for model comparisons made to the low water laboratory tests. A) Resu lts from the model run with TG83 energy dissipation. B) Results from the model run with JBAB07 energy dissipation.

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186 Figure 5 6. The best fit B coefficient for the high water model tests comparisons using the OM79 ( ), Ka91 ( ), and Ru03 ( ) breaking equations. A) The wave heights for the TG83 model. B) The rms error for each B coefficient in TG83. C) The wave heights for the JBAB07 model. D) The rms error for each B coefficient in JBAB07.

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187 Figure 5 7. The best fit B coefficient for the mid water model tests comparisons using the OM79 ( ), Ka91 ( ), and Ru03 ( ) breaking equations. A) The wave heights for the TG83 model. B) The rms error for each B coefficient in TG83. C) The wave heights for the JBAB07 model. D) The rms error for each B coefficient in JBAB07.

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188 Figure 5 8. The best fit B coefficient for the low water model tests comparisons using the OM79 ( ), Ka91 ( ), and Ru03 ( ) breaking equations. A) The wave heights for the TG83 model. B) The rms error for each B coefficient in TG83. C) The wave heights for the JBAB07 model. D) The rms error for each B coefficient in JBAB07.

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189 Figure 5 9. The error results of testing both TG83 and JBAB07 with B to the first power. A) TG83 error. B) JBAB07 error.

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190 Figure 5 ), and Ru03 ( ) in the TG83 and JBAB07 with the first order B. A) Wave heights from TG83 model. B) Change in error with B for TG83. C) Wave heights from JBAB07 model. D) Cha nge in error with B for JBAB07.

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191 CHAPTER 6 1D WAVE SETUP Introduction The wave set up comparisons between the laboratory measurements from flume and basin tests and the one dimens ional energy flux model provide an initial test of the ability of the model to calculate the radiation stress tensors in the cross shore direction. The radiation stress calculated from the change of momentum as the wave travels from offshore through the surf zone was used to predict the amount of mean water level change during gi ven wave conditions. The increase in mean water level is important in the overall understanding of the wave transformation because of the increased wave heights observed due to the increase in water level close to the shoreline. Wave set up is also a cri tical part of the surge for islands. Methods The 1D energy flux model was used for comparisons between the observed wave set up in both the wave flume and wave basin laboratory experiments. The wave set up was calculated based on the equation derived through linear wave theory to relate the gradients in radiation stress to the change in mean water level. The equation for 1D wave set up is = 1 ( + ) (6 1) where is the mean water level, and is the radiation stress in the cross shore dire ction. The radiation stress is calculated using = 2 1 2 (6 2) where is the wave energy, and found from = = 1 2 1 + 2 sinh 2 (6 3)

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192 In the 1D energy flux model, all wave energy was assumed to be in the cross shore direction, so no directional components of radiation stress were calculated. The model was iterated until convergence for calculated wave set up values. The final wave hei ghts were calculated on top of the steady state wave set up after convergence. The observational results from both of the laboratory studies were described in the chapters on each study (Chapter 2 and 4) The set up was found using statistical post processing of the raw data. All points which did not have water on the gauges initially were excluded from numerical comparisons. Initial comparisons made between the 1D energy flux model and the laboratory results were done with the gauges on the reef t op of the wave flume laboratory (gauge 9, 10, 11, 12) and the centerline gauges on t he reef top of the wave basin (gauge 11, 14, 19). The model tests were performed with all the maximum breaking equations tested in the previous chapters. The model was run with STWAVE breaking criteria as well as the TG83 and JBAB07 energy dissipation models. The B coefficients used in the energy dissipation runs were obtained from the best fit results from the tests previously described in Chapter 3 and Chapter 5. Results Wave Flume Laboratory Wave Set up The wave set up measured in the laboratory flume was much larger than the calculated set up in the numerical model. By calculating the rms error, the sign of the difference is lost, but in all comparisons the model results underestimated the flume laboratory data. The mean rms error became larger with decreasing water level. In both the laboratory and model results, the decreasing water level resulted in increasing wave set up. However, the calculated magnitude of the increase wave set up with water depth observed in the laboratory was not matched by the model.

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193 The mean rms error for the model tests with the STWAVE breaking criteria were 61.2%, 81.2%, and 86.1% for the high, mid, and low waters on the 1:2.5 sl ope (Table 61). The large error is significant, but the small values of measured set up also contribute to large errors because small differences become large percent errors. The mean rms error for the 1:5 slope was less than the 1:2.5 slope. The perce nt error was 60.9%, 70.8%, and 77.7% for the high, mid, and low waters respectively. The increase in error between high and low water tests was not as large for the 1:5 slope as the 1:2.5 which is probably the result of the lower measured set up in the la boratory. The tests with the energy dissipation models all had significantly less error than the tests with the STWAVE breaking criteria which is expected because wave height decay is more accurately modeled In the high water, all tests with the energy dissipation models had very similar results of 3040% error. The mean rms error for the dissipation models increased around 5060% for low and mid water on the 1:2.5 slope. The differences between the TG83 and JBAB07 with different values of were small compared to the difference between these models and the STWAVE breaking criterion. The peak wave height in all the energy dissipation model tests occurred later in the profile than the model runs with STWAVE breaking criteria. The later break ing resulted in the larger amount of momentum transfer occurring in shallower water, so the wave set up was greater for these tests. Examples of the wave height profile with the resulting wave set up for the 1:2.5 and 1:5 slopes are shown in Figure 61 and 62, respectively The differences in errors between the two slopes are minimal, but the rms error was found to be lower for the model comparisons with the 1:5 slope tests than with the 1:2.5 slope tests. Wave Basin Laboratory Wave Set up The rms erro r of the wave set up from the 1D energy flux model in the basin with the STWAVE breaking criteria was between 23%, 39%, and 47% for all the tests at the three water

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194 levels, high, mid, and low respectively (Table 6 2). The high percentage errors were par tly due to the small values of measured wave set up. An example of the wave height profile along with the resulting wave set up is in Figure 6 3 for the high water tests. The early wave height attenuation relative to the measure data appears to produce w ave set up in the model that was similar to the measured set up. The wave set up tests with the two energy dissipation models along with the three maximum breaking wave height equations had higher errors than the STWAVE breaking tests. The higher rms errors during the high water tests were the result of over estimating the wave set up compared with the measured results. The tests using the JBAB07 energy dissipation equations had smaller mean rms errors than the TG83 tests for all maximum breaking equations except for Ru03. The smaller errors at the high water level are likely due to the smaller peak wave heights when using the JBAB07 model compared with the TG83 mode l (Figure 6 3A,C). The smaller peak wave heights resulted in smaller wave set up which was closer to the measured values at the high water level (Figure 6 3B,D). T he measured wave set up in the basin laboratory during the mid water tests was higher than the measured values at the high water tests. The wave model with the STWAVE breaking criteria did not increase the wave set up calculated as much as was measured, s o the mean rms error increased as the water level dropped to the mid value The mean rms error for the tests with both TG83 and JBAB07 decrease d in error from around 50% for most of the tests at high water down to around 20% error for tests at mid water ( Table 6 2). The wave set up values calculated using the TG83 model were still over predicting most of the measured values because of the late breaking seen in Figure 6 4A. The tests with JBAB07 had similar results to the

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195 TG83 tests. The wave set up re sults are slightly over predicted but the mean rms error of using any the wave breaking equations was lower than the present STWAVE breaking criteria. The low water level tests used only two gauges on the reef top for comparisons between the model runs and the laboratory measurements. Gauge 19, located furthest away from the wave maker was left out because of the lack of water on the gauge during most simulations. Also, some of the lower wave energy runs were left out of the comparisons because the sm all amount of wave set up measured was difficult to get a proper error calculation. The measured wave set up in the low water was higher than the measured values in both the mid water and high water. The larger wave set up resulted in the largest mean rms error for the model runs with the STWAVE criteria. The 1D energy flux model with the STWAVE criteria calculated values of wave set up that were low compared with the measured wave set up (Figure 6 5). The model tests with the energy dissipation model s TG83 and JBAB07 had the lowest mean rms errors of all the water levels. The rms errors for the three different breaking equations were all in the low teens which were much lower than the results from the model tests with STWAVE breaking criteria (Table 6 2). The peak wave heights calculated later in the profile by the two energy dissipation models contributed to the higher wave set up for all the test cases which matched the measured results from the low water tests (Figure 6 6). Summary The 1D energy flux model wave set up results were not found to match the observed values for the wave flume experiment. The observed values for all laboratory tests were greater than the calculated values for the wave set up. However, the model tests with the energy dissipation models included had a lower error than the model tests with the STWAVE breaking criteria because of the added wave set up caused by calculating the wave breaking later in the bathymetric profile.

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196 The re asons for the high errors in comp arisons between the model and the laboratory are not clear The underlying assumptions for linear wave theory could impact the measured errors. Also, the role of friction in the wave transformation and set up was not modeled. The constriction caused by the small lateral area of the wave flume could affect the wave set up by reducing the return flow. The model wave set up comparisons to th e wave basin laboratory data showed a much better fit than in the wave flume data. The wave model with the STWAVE b reaking criteria still under predicted the wave set up in the low and mid water level tests, but the model had a good match in the high water tests. Although the breaking wave height variation with cross shore distance was not good. TG83 and JBAB07 energ y diss ipation models appeared to over predict the wave set up at the high and mid water levels, but matched the data well in the low water levels. The later breaking on the bathymetric profile caused a larger wave set up to be calculated for the model tes ts with the se energy dissipation models. There was not a significant difference between the accuracy of the TG83 or the JBAB07 energy dissipation models. Both had similar mean rms errors given a specific maximum breaking wave height equation. The additi on of the energy dissipation model appears to b e important in the over predic tion of the wave set up on a lateral expansive bathymetry. However, the choice of TG83 or JBAB07 does not make a large difference given that the models were tuned to fit the spec ific conditions being tested.

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197 Table 6 1. Mean RMS percent error for the wave set up in the wave flume laboratory comparisons. STWAVE OM79 Ka91 Ru03 1:2.5 Low TG83 86.0 61.4 51.0 62.2 JBAB07 60.0 53.2 52.4 Mid TG83 81.2 56.1 49.1 54.9 JBAB07 55.0 51.3 48.4 High TG83 61.4 32.9 35.6 33.4 JBAB07 36.4 40.0 30.4 1:5 Low TG83 77.7 56.6 54.4 54.5 JBAB07 57.3 56.9 51.0 Mid TG83 71.8 49.3 44.6 48.7 JBAB07 50.3 49.6 45.3 High TG83 60.9 33.0 32.3 33.3 JBAB07 36.7 38.6 30.5

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198 Table 6 2. Mean RMS percent error for the wave set up in the wave basin laboratory comparisons. STWAVE OM79 Ka91 Ru03 Low TG83 47.0 14.1 12.5 11.8 JBAB07 13.5 15.6 11.0 Mid TG83 39.0 21.8 19.8 18.8 JBAB07 22.7 22.4 19.5 High TG83 23.2 56.9 53.3 53.5 JBAB07 42.1 36.9 54.8

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199 Figure 61. The wave set up comparison between the wave model tests and the laboratory test on the 1:2.5 slope. The experimental data shown (x) or (*) is from Test 21 at mid water. The test cases shown are ), Ka91 ( ), and Ru03 ( ). A) The wave height profile for TG83. B) The wave set up calculated from the model tests compared with the measure data for TG83. C) The wave height profile JBAB97. D) The wave set up calculated from the model tests compared with the measure data for JB AB07.

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200 Figure 62. The wave set up comparison between the wave model tests and the laboratory test on the 1:5 slope. The experimental data shown (x) or (*) is from Test 64 at mid water. The test cases shown are ), Ka91 ( ), and Ru03 ( ). A) The wave height profile for TG83. B) The wave set up calculated from the model tests compared with the measure data for TG83. C) The wave height profile JBAB97. D) The wave set up calculated from the model tests compared with the measure data for JBAB07.

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201 Figure 63. The wave set up comparison between the wave model tests and the basin laboratory test at high water. The experimental data shown (x) or (*) is from Test 36. The test cases shown are ), Ka91 ( ), and Ru03 ( ). A) The wave height profile for TG83. B) The wave set up calculated from TG83. C) The wave height profile JBAB97. D) The wave set up from JBAB07.

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202 Figure 6 4. The wave set up comparison between the wave model tests and the basin laborat ory test at mid water. The experimental data shown (x) or (*) is from Test 36. The test ( ), Ka91 ( ), and Ru03 ( ). A) The wave height profile for TG83. B) The wave set up calculated from TG83. C) The wave height profile JBAB97. D) The wave set up from JBAB07.

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203 Figure 6 5. The wave set up comparison between the wave model tests and the basin laboratory test at low water. The experimental data shown (x) or (*) is from Test 36. The test cases shown are STWAVE ), Ka91 ( ), and Ru03 ( ). A) The wave height profile for TG83. B) The wave set up calculated from TG83. C) The wave height profile JBAB97. D) The wave set up from JBAB07.

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204 CHAPTER 7 RADIATION STRESS COR RECTION Background The linear wave theory based equations for cross shore wave set up due to radiation stress forcing are presented in Chapter 6 ( Equations 61, 62, 63). In present spectral wave models, the change in radiation stress in the cross shore direction is calculated by fi nding the total radiation stress at each grid cell. The change in radiation stress includes all the energy gained or lost between two cells. Wave set up is calculated in Equation 61 when the change in radiation s tress is negative, or the gradient in of radiation stress between two grid points is negative in the positive x direction. The change in water level induced by the radiation stress is the mathematical representation of the physical transfer of momentum from a wave field to the water column. As a wave shoals prior to breaking due to a sloping bathymetry, the wave height increases which increases the amount of energy in the wave. After a wave breaks the value of is generally constant on a constant sloping bathymetry because the wave is considered in shallow water so equals 1 .0. With constant, the only change in radiation stress is due to the change in energy. After breaking, the wave height decreases caus ing a decrease in the total energy of a wave. The energy loss at breaking is transferred through radiation stress to the water column creating an increase in the mean water level or wave set up. The formulation of radiation stress and the subsequent wave set up by present spectral wave models is correct for conditions with no input of energy from winds in shallow water, and no decrease of wave energy from friction. These two problems with the formulation of radiation stress value s are presented with alo ng a solution to account for the changes in radiation stress needed to correctly handle these situations in STWAVE. We will show both problems using

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205 numerical simulations of wave transformation on gradual slopes with friction and in a lake domain with str ong wind forcing. Friction The 1 D energy flux model is used with the inclusion of a reduction of total wave energy by friction. The amount of energy loss due to friction is calculated using the Manning friction formulation = 1 / (7 1) where is the horizontal bottom orbital velocity, is the manning coefficient of friction, and is the energy dissipation due to friction. All the bathymetries tested where constant sloping beds starting from a depth of 20 meters Tests were performed on slopes from 1:2000 to 1:500 to determine the amount of energy loss due to friction for a variety of slopes. To separate the difference between the energy loss due to friction versus the energ y loss due to breaking, the radiation stress was calculated at each grid point twice. The total energy at each grid point and the energy that would have been present without the loss due to friction were used to calculate radiation stress. An example of the difference in wave height at grid points created by in loss of energy due to friction and without the loss due to friction is in Figure 71. This small selection of wave heights was taken after breaking where the friction loss due to energy was decreasing the wave height along with the wave breaking. The decrease in wave height between two grid points was increased with the added loss of energy by friction. The difference between the wave heights is approximately equal to the change in radiation stress between the grid points. The change in radiation stress would be more negative with the inclusion of energy loss due to friction, but the energy lost due to friction does not transfer to the

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206 water column in most cases. Including this energy loss by fri ction results in a greater wave set up calculated than is physically correct. In Figure 7 2, the transformation of a 1 meter wave with 5 second period was calculated with the present STWAVE breaking criterion on the 1:1000 slope. The wave height tra nsformation was altered by the added loss of energy due to friction given a Manning coefficient of n = 0.2 (Figure 7 2A). The resulting wave set up (including the energy loss due to friction in the radiation stress calculation) calculated using Equation 61, was not very different between the simulation with and without friction (~10%). T he s imulation with friction had very little set down because the shoaling of the wave prior to breaking was reduced. However, when the energy lost due to friction was removed from the radiation stress, the wave set up was greatly reduced. The resulting wave set up up was almost equal to zero because a large part of the energy lost in the waves was due to friction. The amount of energy loss due to friction increases wi th decreasing depths. Also in initial testing the energy lost due to friction was removed at all grid points. However when the energy was removed, the wave height did not meet the criteria for breaking so a positive change in radiation stress was calculated. A set down was calculated where it should not have occurred. In order to remove this issue, the removal of energy loss due to friction from the radiation stress was only performed when the mean water level was greater than zero. This was a t emporary fix which will be examined further. The impact of removing the energy lost due to friction was tested for different initial wave heights (Figure 7 3) and different bathymetric slopes (Figure 74). In Figure 7 3, increasing the initial wave hei ght, increased the resulting wave set up. It appears that removing energy lost due to friction from the radiation stress resulted in a greater decrease in wave set up

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207 for the larger initial wave heights because frictional energy loss is proportional to the wave height squared. The impact of removing the energy loss by friction from the radiation stress on the wave set up calculated decreased with steeper slopes (Figure 7 4). On the steeper slopes, wave sho aling occurs over a shorter spatial region. F ric tio n has a shorter distance to act, so frictional dissipation is reduced (frictional dissipation is linearly related to distance) This result s in less impact of friction on the wave set up for steeper slopes The extreme case of energy loss due to fri ction occurs for long period waves (T = 10 s) and a small wave height (H = 1.0 m) while using the JBAB07 energy dissipation model. While running the test, the wave height is decreased by friction prior to the wave breaking model (Figure 7 5A). The result in the past would give a set up of about 6 cm, but the wave never actually broke. All the energy in the wave was lost to friction prior to breaking, so the actual wave set up is equal to zero (Figure 75B). This case would only occur on long gentle slopes with large friction and long period waves, but it is a good test case for possible outcomes in numerical wave models. Wind Input In a location like Lake Pontchartrain where there is a large body of water with shallow depths the wave growth due to wind input can become an important factor in storm surge. In present models, the increase of wave heights by the added wind stress creates a decrease in the mean water level. This set down is calculated because linear wave theory assumes all wave height growth was caused by shoaling, but in the wave generation case the wave height growth is due to an external input of energy. The removal of the energy gained by wind stress should result in higher mean water levels at the shoreline. The relationship between wind speed and wave height growth is a complex processes, so STWAVE was used for testing instead of the 1D energy flux model. A bathymetry with 1:2000

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208 sloping sides and constant depth of 5 meters across the middle was used in STW AVE. The domain was 40 km long which is similar to the dimensions of Lake Pontchartrain. The model was run for three grid points in the alongshore direc t ion which is the smallest number of cells that STWAVE is capable of running. The bathymetry for th ese tests is shown in Figure 7 6Type equation here Wind stresses were assumed to be constant across the entire domain, and only waves generated by the wind were included in the analysis. Wind speeds between 40 m/s and 100 m/s were tested. A similar approach was applied to the radiation stress calculation as described in the friction section. The radiation stress was calculated at each grid point for both the total energy, and the energy minus the input wind energy. The result of the removal of the input wind energy from the radiation stress can be seen in Figure 7 7 A set down still occurs as the wave grows but the set down is due to the shallow water at the location of wave growth. The set down for the test with the wind energy removed had a smaller set down, so the overall set up was higher. The present test with a 60 m/s wind resulted in an increase in the set up of close to 5 cm (~40%). In Figure 7 8, the change in the wave set up caused by the removal of the wind energy from the radiation stress calculation for three different wind conditions. It is apparent that the faster the wind speed, the larger the discrepancy between the set up w ith wind energy and the set up without wind energy. The 70 m/s wind test appears to have a difference in wave set up of 0.1 meters. The removal of the wind energy from the radiation stress does have an impact on the overall wave set up calculated in a system, but it decreases the set down, so the total water level at the shoreline is increased.

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209 Figure 7 1. The change in wave height between grid points. The red lines are the wave heights that would be calculated without friction.

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210 Figure 7 2. The impact on wave set up of removing the energy loss due to friction from the equation. A) The change in wave height between a test with friction and without friction. B) The wave set up previously calculated with energy loss due to friction in the radiation stress and without. A

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211 Figure 7 3. The change in the impact of the removal of friction from the radiation stress with increasing wave heights. A) The three wave heights tested. B) The growth in the difference b etween the old wave set up and the new wave set up as the initial wave height increased.

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212 Figure 7 4. The impact of slope on the change in wave set up with the removal of energy loss by friction. A) The three slopes tested with initial wave heights of 1 meter and period of 5 seconds. B) The difference between the wave set up decrease with steeper slopes. A B

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213 Figure 7 5. Energy loss due to friction while using JBAB07 energy dissipation model for breaking. A) The wave height decreases due to friction. B) Difference in wave set up when energy lost due to friction is removed.

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214 Figure 7 6. Bathymetry for wind stress tests in STWAVE. The slope on each side was set to 1:2000.

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215 Figure 7 7. The impact of 60 m/s wind on the wave height and wave set up across the basin. A) The wave height growth and decay across the basin. B) The wave set up at the shoreline was increased by the removal of the wind stress from the radiation stress calculation.

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216 Figure 7 8. The impact of three different winds on the wave height and wave set up across the basin. A) The wave height growth and decay acr os s the basin. B) The difference in the wave set up at the shoreline between the old (solid lines) and new (dashed lines) wave set up values was increased with increasing wind speed.

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217 CHAPTER 8 COUPLED SETUP MODEL Introduction The large wave basin pr ovides the opportunity to examine the 3D flow patterns which occur in the complex coral reef environment. Previously the 1D energy flux model was used to compare the numerical wave breaking and wave set up with the measured wave breaking and wave set up on the centerline of the wave basin. Advancing on the 1D model requires the use of 2D models which are capable of calculating both the wave transformation and the resulting wave set up. A newly developed tightly coupled storm surge model provides the abi lity to examine both the wave breaking and the wave set up in the basin. Methods A coupled wave model system was used to numerical ly model the 2D wave breaking and wave set up in the wave basin laboratory. In the coupled system STWAVE was used along with a circulation model, ADCIRC. All tests were performed in prototype scale which was a 50:1 ratio from model scale. STWAVE STWAVE is a wave transformation and generation model. The model solves the steady state conservation of spectral wave energy to calculate refraction, shoaling, and breaking of waves (Smith et al. 2001) The model includes wind wave generation, bottom friction, and wavewave interactions. The input files need to run a simulation are a depth file, initial wave spectrum file, simulation file, and options file. The depth file consisted of a rectangular grid with equal spacing in the x and the y direction. The depth file was built based off the bathymetry studied in the laboratory. The grid spacing is i mportant in the resolution of the model results, so a grid spacing of 0.01 meters was

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218 used for model scale tests. The grid spacing was set to 0.5 for prototype testing. The offshore extent of the grid was located at gauge 5 in the laboratory which was at 23.3161 meters from the wave maker. The initial wave conditions file was constructed from the post processing of the basin laboratory data. The measured water surface elevation from gauges 4, 5, and 6 were post processed to get the energy density spect rum. The spectrum was then averaged to find a consistent wave height near gauges 4, 5, and 6. A reduction in the number of frequency bands was performed by smoothing the averaged energy density spectrum. The size of each frequency bin was 0.01 Hz in model scale or 0.001 Hz in prototype scale. To convert the initial wave file from model scale to prototype scale, the energy density spectrum was multiplied by fifty twice and then multiplied by square root of fifty to account for the meters squared divided by hertz. The simulation file has the name of all the input files and output files. T he simulation file also gives the state plane coordinates of the STWAVE grid, so a relationship can be made between the STWAVE grid and the ADCIRC mesh. The options file was used to provide STWAVE with the proper model features to be functional. In the case of this project, the radi ation stress terms were turned on as well as the input of spatially variable water levels The options file was altered to account for the added breaking functions. Two new variables were added with btype specifying what type of energy dissipation functi on, and hssmax specifying the maximum wave height equation. The new options allowed for tests to be run with all of the breaking functions previously tested in the 1D energy flux model. A few modifications were made in STWAVE in order to code TG83 and JBAB07 into the model. As previously discussed, the two new options in the options file provide a selection in the type of breaking function. A value for btype of 1 gives the present STWAVE breaking

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219 equation, 2 uses TG83, and 3 uses JBAB07. The hss max has three options with 1 being OM79, 2 is Ka91, and 3 is Ru03. The dissipation equations were altered slightly to account for the energy density spectrum. The dissipation function was calculated as previously described in Chapter 3, but the relations hip between the amount of energy dissipated and the spectrum of energy was similar to that described by Booij et al. ( 1999) = ( ) (8 1) where is the energy dissipation, is the total energy in the spectrum, and ( ) is the frequency and direction components of the energy spectrum. ADCIRC The Advanced Circulation Model (ADCIRC ) (Luettich et al. 1992) was used as the circulation portion of the c oupled system. The ADCIRC grid was made with equilateral triangles on a rectangular grid. The mesh was r efined once to get node spacing of 10 meters in the offshore and 5 meters on the reef top. These nodal spacings were equal to 0.2 and 0.1 meters in model scale. The mesh was built to be similar in size to the laboratory basin with approximately 10 meters model scale on each side of the constructed reef. Three sides of the basin had closed boundary conditions with tangential slip allowed. The offshore boundary was an open boundary wit h zero water surface elevations. The tests were run with wet/dry terms on and advective terms on. The water surface elevation was recorded for the locations where the gauges were located in the laboratory. Coupling System The commonly used storm surge coupling system uses a near shore wave model and circulation model similar to the models previously described. A wind model and offshore wave model are often included in the coupling system as described by Weaver (2007), but for the

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220 present project only S TWAVE and ADCIRC were used. Wave set up calculations using the coupled system traditionally were performed by running the wave mode l for all wave conditions. T he radiation stress values from the wave model were then input into the circulation model to calculate the wave set up and then the wave model could be run again to get the wave heights on the new water level. In the tightly coupled storm surge model, the transfer of information between the wave model and the circulation model occurs more frequently. The circulation model is initialized, and then the wave model is run. The radiation stress values are input into the circulation model, and it is run until it is time for the next wave condition. The surge values are passed to the wave model prior to running, and the process is repeated until all wave conditions are finished. Results The coupled system was tested against the basin laboratory data for the present STWAVE breaking criteria and the two dissipation equations with the three options for maximum breaking wave height. An example of the water surface elevation change due to the introduction of wave radiation stress from the model test with the present STWAVE breaking criteria in the high water shows the increase in mean water level in the center of the laboratory test region (Figure 8 1) The low water test using STWAVE breaking criteria has a similar increase in mean water level at the center of the test area, but the increase is localized closer to th e location of th e original shoreline (Figure 8 2 ). The high water test case was R un 38 with initial wave height of 0.14 meters model scale or 7.0 meters in protot ype scale and wave period of 2.26 seconds or 16 seconds prototype The low water test case w as R un 10 with initial wave height of 0.13 meters model scale or 6.0 meters in prototype scale and wave period of 2.26 seconds or 16 seconds prototype.

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221 The wave height profile comparisons for the TG83 tests (Figure 8 3) and JBAB07 (Figure 84) were s imilar to the previous 1D energy flux comparisons. This result is consistent with theory because the wave breaking functions should result in the same wave height profile in both models. The wave set up calculated using the STWAVE breaking criteria was low compared with the measured data. Ho wever, both the TG83 (Figure 8 3B) and JBAB07 (Figure 8 4B) dissipation models over predicted the wave set up on the reef top (~10%). The same result was observed with comparing the measured wave set up and the cal culated wave set up in the longshore direction on the reef top (Figure s 83C, 84C). The comparisons between the coupled model and measured results show the best agreement between the model run with JBAB07 energy dissipation function and the OM79 maximum breaking wave height equation. The wave height profile s for the low water test were similar to the 1D tests in Chapter 6 for both the TG83 (Fi gure 8 5A) and JBAB07 (Figure 86A). The wave set up calculated using the ADCIRC model was lower than the results of the 1D linear equation (Figures 85B, 8 6 B). The lower values of wave set up are likely caused by the added lateral flow which removes some of the build up on the coastline. The alongshore wave set up was close for all the energy dissipation cases tested for the middle two gauges. However, the model did not spread the wave se t up as far to the outside of the reef domain as was measured in the laboratory (Figures 8 5C, 86C) The model test using the STWAVE breaking criteria did not calculate any setup for the outside gauges. The values of wave set up in the alongshore direct ion were measured from initially having no water on the gauges, so there is not a high level of confidence in these values at low water. Summary The tests using the coupled STWAVE and ADCIRC model showed good agreement between the numerical and physical models. The wave height profile was accurately calculated

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222 for both the high water and low water tests. The wave set up was overestimated in the high water test and underestimated in the low water test. This result is consistent with the tests using the 1D energy flux model and the 1D wave set up equation. In both the high water and low water tests, both TG83 and JBAB07 appeared t o be accurate in calculating both the wave height profile and the cross shore and alongshore wave set up profiles. The tests with JBAB07 seem to fit the wave height profile better than the TG83 tests, but the value of B for the JBAB07 tests was set to 1.5 which is a slightly tuned value while B was set to 1.0 for TG83. In all cases, the maximum breaking equations from OM79 see ms to get the best results for the wave height profile and the wave set up.

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223 Figure 8 1. M ean water level for high water T est 38.

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224 Figure 8 2. Mean water level for low water T est 10.

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225 Figure 8 3. The coupled model results with TG83 comparisons with the measured data for Test 38 in high water A) The wave height profile. B) The profile of the wave set up on the centerline of the basin. C) The wave set up along shore on the reef top. B A C

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226 Figure 8 4. The coupled model results with JBAB07 comparisons with the measured data for Test 38 in high water A) The wave height profile. B) The profile of the wave set up on the centerline of the basin. C) The wave set up along shore on the reef top. A B C

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227 Figure 8 5. The coupled model results with TG83 comparisons with the measured data for Test 10 in low water. A) The wave height profile. B) The profile of the wave set up on the centerline of the basin. C) The wave set up along shore on the reef top. A B C

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228 Figure 8 6. The coupled model results with JBAB07 comparisons with the measured data for Test 38 in high water. A) The wave height profile. B) The profile of the wave set up on the centerline of the basin. C) The wave set up along shore on the reef top. A B C

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229 CHAPTER 9 CONCLUSIONS T wo laboratory experiments designed to study the wave breaking and wave set up on coral reefs were presented The laboratory projects used cut acrylic glass for the reef top which simulated the small scale bathymetric fluctuations created on coral reefs In the wave flume tests, rapid shoaling and breaking of waves on slopes of 1:5 and 1:2.5 were examined T he wave heights measured in the laboratory tests showed a trend of increased wave heights pri or to breaking for the 1:2.5 slope than the 1:5 slope. The increase in wave height does not agree with past theory where it was assumed that the highest peak wave heights were achieved for slopes of 1:10, and slopes greater than 1:10 had decreasing maximu m wave heights. The examination of the maximum wave s et up measured on the reef top showed similar trends to past research on coral reefs The wave set up increased with increasing wave power, and the largest amount of set up was measured for the lowest water level and the least amount of set up was measured for the highest water level. Both of these results were consistent with previous research from Gourlay (1996b) and Seelig (1983) The laboratory tests performed in the large wave basin provided an opportunity to examine the wave breaking and wave set up in a 3D setting which is more comparable with the field. The wave basin experiment was constructed similar to the wave flume but with a width of 12.2 meters on the reef top. The front reef slope w as more complex than the tests in the wave flume with an average slope close to 1:10. The wave heigh t attenuation due to breaking occurred very quickly The rapid loss of energy after breaking signifies a very narrow surf zone which is common on steep sl oping beaches. The wave heights continued to decrease across the reef top, but the majority of the energy was lost in the surf zone

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230 The wave setup measured on the reef top for the wave basin was much smaller than the values measured in the wave flume. The three dimensionality of the wave basin likely caused the lower wave set up because there was flow away from the reef platform due to gradients in water level. The piled up water associated with the wave set up was not held only on the reef, but could flow laterally. The wave shoaling and breaking in the wave flume laboratory was examined using a 1D energy flux model. The wave breaking in the energy flux model was parameterized using an energy dissipation function based on the bore model. The Thor nton and Guza (1983) and the Janssen and Battjes (2007), Alsina and Baldock (2007) dissipation models were both tested on the laboratory results from the steep slope tests. Both models required modification to represent breaking on reefs. Testing showed that the TG83 dissipation model had a more flexible dependence on the maximum breaking wave height as long as the B coefficient changed. The JBAB07 dissipation model had a smaller selection of values for the maximum breaking wave height which produced low errors compared to the data, but the value of B was always greater than the recommended value of 1.0. Further testing of the energy dissipation functions was performed using variable equations for the maximum breaking wave height instead of the constant values previously tested. Three options were tested based on the use of local wave characteristics to determine the maximum wave height: Ostendorf and Madsen (1979), Kamphuis (1991), and Ruessink (2003). All three maximum wave height equations showed a good fit to the experimenta l data with some tuning The value for B of all three breaking equations remained close to the recommended range by TG83 of between 0.8 and 1.4, but the B value was much higher than the recommended value for JBAB07. The higher values of B were caused by the rapid dissipation of energy at the

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231 point of breaking. The energy dissipation functions were designed for a slower loss of energy which is common on more gradual slopes, so the dissipation rate needed to be increased with a larger value of B to match the wave breaking measured on the steeper slopes. The 1D energy flux model was compared with the wave height measurements from the centerline of the wave basin laboratory tests. The less steep slope in the wave basin allow ed for smaller values of B in both of the energy dissipation models Comparisons with the TG83 model showed a B value of close to 1.0 for all water levels which is the recommended value for testing without tuning. The JBAB07 model required a value of B closer to 1.5 for the lowest error. The relationship between the B coefficient and the maximum breaking wave height had similar trends for the wave basin comparisons as was shown in the wave flume comparisons The three maximum breaking equations all had good results based on the measured data. The OM79 equation had the lowest rms error for all water levels and in both energy dissipation models, but the dif ferences between the three maximum wave height equations was minimal. The 1D wave setup calculated for the wave flume was very low for all the numerical models tested with the measured data. The higher values in the laboratory are likely due to the laterally confined space in the wave flume. Possible reasons are mass transport due to wave nonlinearit y that isnt included in the 1D flux model, underestimation of radiation stress due to wave nonlinearity and the role of low frequency waves which are removed in the analysis, but exist in the laboratory. In the wave basin, the added space provided more l ateral flow which resulted in lower wave set up values. The calculated wave set up by the 1D model was much closer to the measured values in the wave basin than in the wave flume. T he three dimensionality o f the wave basin is more representative of field conditions.

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232 The wave radiation stress was further examined to determine the impact of friction and wind energy input on wave set up. Theoretically the energy lost due to friction and the energy gained by wind input should not be included in the radiat ion stress calculation. The decrease in wave height due to friction does not cause a momentum transfer to the water column, so the energy lost due to friction was removed from the radiation stress calculation in STWAVE. The removal of ener gy must be perf ormed at each spa t ial grid cell in order to obtain the correct values for radiation stress. A similar situation was examined with the input of wind energy over a large lake. The added wind energy should not produce a set down which is present ly calculated. Instead the water level should stay close to the still water level until breaking which would lead to a larger amount of setup than present ly calculated. Finally, all of the previous work was used in a coupled 2D model using STWAVE and ADCIRC to com pare with the measured data from the wave basin tests. A newly developed tightly coupled STWAVE (including the added breaking equations to increase accuracy on steep slopes) and ADCIRC model was used for testing. Tests were performed at both the high wat er and low water levels to get the extremes in the wave set up calculations. In both the high and the low water tests, the wave set up calculated using the present STWAVE breaking criteria was low compared with the measure set up. However, the wave set up using both of the energy dissipation functions were very close to the measured values for the high water test. The calculated set up for the low water test was low for all cases which were likely caused by the complexity added in by the very shallow water on the reef top. The shoreline for the low water test was more variable than in the high water test. The more accurate wave set up from the tests using the energy dissipation functions with the variable maximum wave heights compared to th e present STWAVE breaking criteria is

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233 attributed to the delayed dissipation of energy in the bore model tests. The threshold breaking equation used by STWAVE was found to calculate the decrease in wave height caused by breaking to occur further offshore t han what was observed in the laboratory. The energy dissipation functions were designed to allow for a more gradual breaking The delayed breaking creates more wave set up because the majority of the momentum is transferred to the water column in smaller water depths. However, the downfall of the energy dissipation functions is the tuning required to produce accurate comparisons. The TG83 model was more consistent with less tuning for the variety of slopes tested than the JBAB07 model. The tuned JBAB07 model was capable of producing smaller errors in comparisons with the laboratory data, but the large amo unt of tuning needed makes it a more difficult option for a wide range of coral reef bathymetries. More examination needs to go into the variation of the B coefficient with increasing reef face slope if the JBAB07 energy dissipation function is to be used as a breaking parameterization in studies of coral reefs. The TG83 performs better on the slopes greater than 1:10 without a lot of tuning which makes it more ideal for tests on a wide range of coral reefs. The variable maximum breaking wave height parameters were not found to cause significantly different results. It would seem that the OM79 and Ka91 models performed similarly because both are variations on Miche (1944) type breaking equations. The performance of the Ru03 maximum breaking wave height equations is promising because of the simplicity of the equation. The laboratory results seemed to show that a slope dependent function was needed, but the Ru03 equation was accurate without a direct dependence on slope. The same relationships implemented in STWAVE need to be evaluated over a large range of slopes and wave conditions for general application.

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234 Overall, the use of the couple d spectral wave model and circulation model for the examination of wave breaking and wave set up on coral reef s seems appropriate. The model does not account for specific characteristics of the breaking wave or the infragravity waves on the reef top, but the calculation of the bulk parameters seem to be acceptable in testing of a broad range of wave conditions and bathymetric slopes. An added accuracy would be available with the use of the bore type parameterization of breaking, but the added need for tuning on the steeper slopes is not ideal. A better idea of the variability in B would make the bore type models much more functional in STWAVE for use on a range of coral reef environments.

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235 LIST OF REFERENCES Alsina J.M., B aldock, T.E., 2007. Improved representation of breaking wave energy dissipation in parametric wave transformati on models. Coastal Engineering 54 (10), 765769. Apo t s os, A., Raubenheimer, B., Elgar, S., Guza, R.T., 2008. Testing and calibrating parametric wave transformati on models on natural beaches. Coastal Engineering 55, 224 235. Baldock, T.E., Holmes, P., Bunker, S., Van Weert, P., 1998. Cross shore hydrodynamics within an unsaturated surf zone. Coastal Engineering 34, 173196. Battjes, J.A., Janssen, J.P.F.M., 1978. Energy loss and set up due to breaking of random waves. Proceedings of the 16th International Conference on Coastal Engineering American Society of Civil Engineering Reston VA, USA, pp. 569587. Battjes, J.A., Janssen, T.T., 2008. Random wave breaking models history and discussion. Proceedings of the 31st International Conference on Coastal Engineering, Hamburg, American Society of Civil Engineers 1, 2537. Battjes, J.A., Stive M.J.F., 1985. Calibration and verification of a dissipation m odel for random breaking waves. Journal of Geophysical Research 90 ( C5 ), 91599167. Booij, N., Ris, R.C., Holthuijsen, L.H., 1999. A third generation wave model for coastal regions 1. Model description and validation. Journal of Geophysical Research 104 (C4), 76497666. Danel, P., 1952. On the limiting clapotis. Gravity Waves, U.S. Department of Commerce, National Bureau of Standards, 521, 3545. Dean, R.G., Dalrymple, R.A., 1991. Water Wave Mechanics for Engineers and Scientists. World Scientific Press, River Edge, New Jersey. Demirbilek, Z., Nwogu, O.G., Ward, D.L., 2007. Laboratory study of wind effect on runup over fringing reefs, report1: data report. ERDC/CHL TR 074, Coastal and Hydraulics Laboratory. Demirbilek, Z., Nwogu, O.G., 2007. Boussinesq model ing of wave propagation and runup over fringing coral reefs, model evaluation report. ERDC/CHL TR 0712, Coastal and Hydraulics Laboratory. Demirbilek, Z., Nwogu, O.G., Ward, D.L., Sanchez, A., 2009. Wave transformation over reefs: evaluation of one dimen sional numerical models. ERDC/CHL TR 091, Coastal and Hydraulics Laboratory. Galvin, C. V ., 1968. Breaker type classification on three laboratory beaches. Journal of Geophysical Research 73 ( 12) 36513659.

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236 Gerritsen, F., 1980. Wave attenuation and wave set up on a coastal reef. Proceedings of the 17th International Conference on Coastal Engineering., Sydney. American Society of Civil Engineers, New York 1, 448461. Gerritsen, F., 1981. Wave attenuation and wave set up on a coastal reef. University of Hawaii, Look Laboratory, Technical Report, 48, 416. Goda Y., 1964. Wave forces on a vertical circular cylinder: experiments and proposed method of wave force computation. Port and Harbour Technical Research Institute 8, Ministry of Transport, Japan. Goda Y., 1974. New wave pressure formula for composite breakwater. Proc eedings of the 14th International Conference on Coastal Engineering American Society of Civil Engineering, Reston VA, USA, pp. 17021720. Gourlay, M.R., 1994. Wave transformation on a coral reef. Coastal Engineering 23 (1 2), 1742. Gourlay, M.R., 1996a. Wave set up on coral reefs. 1. Set up and wave generated flow on an idealized two dimensional horizontal reef. Coastal Engineering 27, 161193. Gourlay, M.R., 1996b. Wave set up on coral reefs. 2. Set up on reefs with various profiles. Coastal Engineering 28 (14), 1755. Hughes, S.A. 1993. Physical models and laboratory techniques in coastal engineering. World Scientific Press, River Edge, New Jersey. Iverson, H.W., 1952. Laboratory study of breakers. Gravity Waves, Circular 52, U.S. Bureau of Standards, 932. Janssen, T.T., Battjes, J.A., 2007. A note on wave energy dissipation over steep beaches. Coastal Engineering 54, 711716. Jensen, M.S. 2004. Breaking of waves over a steep bottom slope. Ph.D. Thesis, Aalborg University, Denmark. Jensen, 0., 1991. Waves on coral reefs. Procedings from 7th Symposium on Coastal and Ocean Management Coastal Zone 1991, Long Beach. American Society of Civil Engineers, New York 3, 2668 2680. Kamphuis J.W., 1991. Wave transformation. Coastal Engineering 15 (3), 173184. Kamphuis, J.W., Kooistra, J., 1990. Three dimensional mobile bed hydraulic model studies of wave breaking, circulation and sediment transport processes. Proceedings of the Canadian Coastal Conference. National Resource Council of Canada, pp. 405418.

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237 Komar, P.D., Gaughan, M.K., 1973. Airy wave theories and breaker height prediction. Proceedings of the 13th International Conference on Coastal Engineering. American Society of Civil Engineers, Reston VA, USA, pp. 405418. Lamb, H., 1932. Hydrodynamics. Dover Publishing, New York. LeMahaute, B., 1963. On nonsaturated breakers and the wave run up. Proceedings of the 8th Conference on Coastal Engineering, Ame rican Society of Civil Engineers, 7792. Lippm ann, T.C., Brookins, A.H., Thornton, E.B., 1995. Wave energy transformation on natural profiles. Coastal Engineering 27 (1 2), 120. Longuett Higgins, M.S., Stewart, R.W., 1962. Radiation stress and mass tr ansport in gravity waves with application to surf beats. Journal of Fluid Mechanics 13, 481 504. Longuett Higgins, M.S., Stewart, R.W., 1963. A note on wave set up. Journal of Marine Research 21, 138159. Longuett Higgins, M.S., Stewart, R.W., 1964. R adiation stresses in water waves; a physical discussion with applications. DeepSea Research 11, 529562. Longuett Higgins, M.S., 1975. On the joint distribution of the periods and amplitudes of sea waves. Journal of Geophysical Research 80, 2688 2694. Lowe, R.J., Falter, J.L., Bandet, M.D., Pawlak, G., Atkinson, M.J., Monismith, S.G., Koseff, J.R., 2005. Spectral wave dissipation over a barrier reef. Journal of Geophysical Research 110. Luettich, R.A., Westerink, J.J., Sheffner, N.W., 1992. Adcirc: An advanced three dimensional circulation model for shelves, coasts, and estuaries. report 1: theory and methodology of adcirc 2ddi and adcirc 3dl with applications. Tech. Rep. DRP 926, Department of the Army, Washington, D.C. Massel, S.R., Belberova, D.Z., 1990. Parameterization of the dissipation mechanisms in surface waves induced by wind. Archives of Mechanics 42, 515530. Massel, S.R., Gourlay, M.R., 2000. On the modeling of wave breaking and set up on coral reefs. Coastal Engineering 39 (1), 127. McCowan, J., 1984. On the highest waves of permanent type. Philosophical Magazine Edinburgh 38 (5) 351358. Miche, R., 1944. Movements ondulatoires des mere en profondeur constant on decroissante. Ann. Des Ponts et Chaussees 114, 131164, 270292, and 369406 [In French].

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238 Mitsuyasu, H. 1962. Experimnetal study on wave force against a wall. Report of the Transportation Technical Research Institute, 47, 39. Munk, W., 1949. The solitary wave theory and its applicat ion to surf problems. Annuls of New York Acadamy of Sciences, 51, 376. Nairn, R.B., 1990. Prediction of cross shore sediment transport and beach profile evolution. Ph.D. Thesis, Department of Civil Engineering, Imperial College, London, 391. Nelson, R.C ., Lesleighter, E.J., 1985. Breaker height attenuation over platform coral reefs. Australian Conference on Coastal and Ocean Engineering. Christchurch, N.Z. 2, 916. Nelson, R.C., 1994. Depth limited design wave heights in very flat regions. Coastal Engineering 23, 4359. Nielsen, P., Rasmussen, A., 1990. Waves on coral reefs. Thesis report, Danish Engineering Academy. Ostendorf D.W., Madsen, O.S., 1979. An analysis of longshore current and associated sediment transport in the surf zone. Massachusetts I nstitute of Technology, Department of Civil Engineering, Rep. 241, p. 169. Ruessink, B.G., Walstra, D.J.R., Southgate,H.N., 2003. Calibration and verification of a parametric wave model on barred beaches. Coastal Engineering 48 (3), 139 149. Seelig, W. N., 1982. Wave induced design conditions for reef/lagoon system hydraulics. Unpublished report for U.S. Army Corps of Engineers, Pacific Ocean Division. Seelig, W.N., 1983. Laboratory study of reef lagoon system hydraulics. Journal of Waterways, Ports, Coastal, and Ocean Engineering. American Society of Civil Engineers, 109, 380391. Singamsetti, S.R., Wind, H.G., 1980.Breakign waves; characteristics of shoaling and breaking periodic waves normally incident to plane beaches of constant slope. Delft Hydra ulics Laboratory Report M 1371, p. 142. Smith, E.R., Kraus, N.C., 1990. Laboratory study on macrofeatures of wave breaking over bars and artificial reefs. Technical Report CERC 9012, WES, U.S. Army Corps of Engineers, 232. Smith, J.M., Resio, D.T., Vincent, D.L., 1997. Current induced breaking at an idealized inlet. Proceedings of Coastal Dynamics. American Society of Civil Engineers, 9931002. Smith, J.M., Sherlock, A.R., Resio, D.T., 2001. STWAVE: Steady State Wave model users manual for STWAVE, Version 3.0. ERDC/CHL SR011, Vicksburg, MS: U.S. Army Engineer Research and Development Center.

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239 Svendsen, I.A., 1984a. Wave heights and set up in a surf zone. Coastal Engineering 8 (4), 303 329. Svendsen, I.A., 1984b. Mass flux and undertow in a sur f zone. Coastal Engineering 8 (4), 347 365. Swart, D.H., Loubser, C.C., 1978. Vocoidal theory for full nonbreaking waves. Proceedings of the 16th International Conference on Coastal Engineering, Hamburg, 1, 467486. Thompson, E.F., 2005. A physical mode l study for investigation of wave process over reefs. Coastal and Hydraulics Lab unpublished letter report. Vicksburg, MS: U.S. Army Engineer Research and Development Center. Tsai, C.P., Chen, H.B., Hwung, H.H., Huang, M.J., 2005. Examination of empirica l formulas for wave shoaling and breaking on steep slopes. Ocean Engineering 32, 469482. Weaver, R.J., 2008. Storm surge: influence of bathymetric fluctuations and barrier islands on coastal water levels. Ph.D. Thesis, University of Florida, Gainesville, FL. Weggel, R.J., 1972. Maximum breaker heig ht. Journal of Waterways, Harbors and Coasta l Engineering Division 98 (4), 529548. Whitford, W.G., 1988. Wind and wave forcing of longshore currents across a barred beach. In: Ph.D.Thesis, Naval Postgraduate School, Monterey, CA, p.205.

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240 BIOGRAPHICAL SKETCH Tyler was born in Southern California in 1982 and lived there until 1993, when his family made the move across the co untry to Alpharetta, Georgia. His parents, along with his older brother, were always a constant support system for Ty, giving him the freedom to try everything. Ty was always on adventures, never took no for an answer, and never stayed still long enough to let it sink in anyways. Ty discovered his love for the ocean at a very early age from playing in the pounding surf of the be aches of Southern California. The obsession grew once he turned 14 and was able to become SCUBA certified. Ty and his dad, David, would travel to Panama City, Key West, and the Bahamas to go diving. It was on these trips that Ty learned that the ocean was not only fascinat ing, but also a dangerous place. From these experiences, Ty soon realized that he wanted to go to college and study marine science. Once in college at Coastal Carolina University, Ty focused on becoming a marine biologist. His plans quickly changed in tha t first biology class, when he realized biology did not come naturally to him. At the same time, Ty was taking two physics courses that were taught by two very innovative teachers, Dr. Louis Kiener and Dr. T heresa Burns. They introduced him to an exciting world where equations were used to answer the physical si tuations occurring all around. The combination of this upstart passion for physics and an i nterest in marine science led Ty to the field of coastal engineering. The University of Florida was Tys top choice for graduate school because of the quality of education available. He was able to spend his first year of graduate school working as a teaching assistant under the guidance of Dr. Robert Theike in hydrod ynamics class. In May of 2005, Ty started working for Dr. Donald Slinn, performing numerical simulations of small scale

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241 sediment transport. In December of 2007, Ty graduated with a masters degree and became a Ph.D. candidate. Ty applied for and received the SMART fellowship whic h provided fundi ng for all of his Ph.D. studies. He also had the opportunity to work with the U.S. Army Corps of En gineers in the development of research. The Ph.D. journey has been full of ups and downs, but the love of his beautiful wife, Ryan was the support that Ty needed to achieve his dream. Ty is now starting a new chapter in life, working for the USACE in Vicksburg, MS. He is working with the Coastal and Hydraulics Branch as a research hydraulic engineer.