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Investigation and Modeling of Nonlinear Wave Propagation Over Fringing Reefs

Permanent Link: http://ufdc.ufl.edu/UFE0042259/00001

Material Information

Title: Investigation and Modeling of Nonlinear Wave Propagation Over Fringing Reefs
Physical Description: 1 online resource (76 p.)
Language: english
Creator: Martz, Tracy
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: fringing, linear, modeling, nonlinear, reefs, setup, waves, xbeach
Civil and Coastal Engineering -- Dissertations, Academic -- UF
Genre: Coastal and Oceanographic Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Extensive wave breaking occurs on the wide reefs located offshore of many Pacific islands, due to their steep slope nearshore. These reefs provide protection to the island; however, energetic wave events may cause flooding as the result of a combination of wave setup, water level position, and wave energy. These processes are well understood for a typical mildly sloping beach; however, they are wildly different over a reef. This study focuses on an effort to better understand and model these processes using laboratory data collected at the University of Michigan's wind-wave facility. A 1:64 model of a fringing reef was constructed in the flume. The model is 2-D vertical and uniform across the tank width. Nine capacitance-wire wave gauges were arranged in a cross-shore transect to measure water surface elevation. Several tests were ran modeling extreme weather conditions associated with tropical cyclones typical of the Pacific island of Guam. Significant wave heights ranged from 3-6 m with 8-20 seconds spectral peak periods. These tests were repeated at four different still water depths to simulate the effects of tide. Wave breaking is found to occur both on the reef slope and edge. A considerable amount of energy remains at the peak frequency after breaking at the reef edge. However, the spectrum measured on the reef flat is dominated by low-frequency oscillations. After breaking, the waves reform as bores and propagate across the reef flat toward the beach. The low-frequency energy continues to grow as the waves propagate shoreward. In this study, wave propagation is simulated using Xbeach, a public-domain two-dimensional model, which has been validated in several case studies for mildly sloping beaches. Model results for extreme weather conditions over a complex reef system (consisting of steep slopes and shallow areas) were used to investigate the model capabilities to simulate wave breaking, dissipation, wave setup, and runup. Results from the linear wave model, Xbeach, are compared to those obtained using a nonlinear wave model, Nonlinear Mild Slope Equation (NMSE), to investigate the role of nonlinear interactions in these processes. It is determined that a linear wave model, such as Xbeach, can be used to accurately model wave height evolution and setup if the wave breaking constants are calibrated correctly. However, the true physics of the energy flux spectral evolution described by a linear wave model is incorrect. Linear wave models ignore the presence of nonlinear energy transfer from the peak frequency to higher and lower frequencies and the development of harmonics, which are key processes occurring in the reef environment. A nonlinear wave model must be implemented to correctly describe the energy dissipation and transfers that occur as waves shoal and break over a fringing reef.
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 Tracy Martz.
Thesis: Thesis (M.S.)--University of Florida, 2010.
Local: Adviser: Sheremet, Alexandru.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0042259:00001

Permanent Link: http://ufdc.ufl.edu/UFE0042259/00001

Material Information

Title: Investigation and Modeling of Nonlinear Wave Propagation Over Fringing Reefs
Physical Description: 1 online resource (76 p.)
Language: english
Creator: Martz, Tracy
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: fringing, linear, modeling, nonlinear, reefs, setup, waves, xbeach
Civil and Coastal Engineering -- Dissertations, Academic -- UF
Genre: Coastal and Oceanographic Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Extensive wave breaking occurs on the wide reefs located offshore of many Pacific islands, due to their steep slope nearshore. These reefs provide protection to the island; however, energetic wave events may cause flooding as the result of a combination of wave setup, water level position, and wave energy. These processes are well understood for a typical mildly sloping beach; however, they are wildly different over a reef. This study focuses on an effort to better understand and model these processes using laboratory data collected at the University of Michigan's wind-wave facility. A 1:64 model of a fringing reef was constructed in the flume. The model is 2-D vertical and uniform across the tank width. Nine capacitance-wire wave gauges were arranged in a cross-shore transect to measure water surface elevation. Several tests were ran modeling extreme weather conditions associated with tropical cyclones typical of the Pacific island of Guam. Significant wave heights ranged from 3-6 m with 8-20 seconds spectral peak periods. These tests were repeated at four different still water depths to simulate the effects of tide. Wave breaking is found to occur both on the reef slope and edge. A considerable amount of energy remains at the peak frequency after breaking at the reef edge. However, the spectrum measured on the reef flat is dominated by low-frequency oscillations. After breaking, the waves reform as bores and propagate across the reef flat toward the beach. The low-frequency energy continues to grow as the waves propagate shoreward. In this study, wave propagation is simulated using Xbeach, a public-domain two-dimensional model, which has been validated in several case studies for mildly sloping beaches. Model results for extreme weather conditions over a complex reef system (consisting of steep slopes and shallow areas) were used to investigate the model capabilities to simulate wave breaking, dissipation, wave setup, and runup. Results from the linear wave model, Xbeach, are compared to those obtained using a nonlinear wave model, Nonlinear Mild Slope Equation (NMSE), to investigate the role of nonlinear interactions in these processes. It is determined that a linear wave model, such as Xbeach, can be used to accurately model wave height evolution and setup if the wave breaking constants are calibrated correctly. However, the true physics of the energy flux spectral evolution described by a linear wave model is incorrect. Linear wave models ignore the presence of nonlinear energy transfer from the peak frequency to higher and lower frequencies and the development of harmonics, which are key processes occurring in the reef environment. A nonlinear wave model must be implemented to correctly describe the energy dissipation and transfers that occur as waves shoal and break over a fringing reef.
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 Tracy Martz.
Thesis: Thesis (M.S.)--University of Florida, 2010.
Local: Adviser: Sheremet, Alexandru.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0042259:00001


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INVESTIGATION AND MODELING OF NONLINEAR WAVE PROPAGATION OVER
FRINGING REEFS


















By

TRACY J. MARTZ


A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE

UNIVERSITY OF FLORIDA

2010

































2010 Tracy Martz
































I dedicate this to my Dad.









ACKNOWLEDGEMENTS

I would like to acknowledge my advisor, Dr. Alexandru Sheremet for all his

support, advice, and valuable time along with my classmates Ilgar Safak, Shih-Feng Su,

Uriah Gravois, and Cihan Sahin for their continuous guidance and support. These

individuals have helped me to grow immensely during my time here at the University of

Florida. I'd also like to thank Dr. Jane McKee-Smith and the US Army Corps of

Engineers for their contributions, support, and assistance throughout the study and

committee member, Dr. Don Slinn for his input.

Lastly, I wish to thank my parents, Mike and Sharon and sister, Melissa. Their

support and understanding during my studies is recognized with love and gratitude. I'd

like to especially thank my dad for his exceptional encouragement and appreciation for

my work.









TABLE OF CONTENTS

page

A C K N O W LED G EM E NTS .......... ............... ........................................................... 4

LIST O F TABLES ......... ...... ........ ........................................... ......... 7

LIST O F FIG U R E S ..................................................... 8

ABSTRACT ............... .......................................................................... 10

CHAPTER

1 INTRODUCTION ............... .............................. .................... 12

1.1 General Introduction .. ................................................. ............... 12
1.2 Review of Past Studies........................................... ............... 13
1.3 Study Objective and Approach ...... .............. ............. .... .... 15

2 M ETHODOLOGY ............... ...................................................... 16

2.1 Laboratory Study ............ ............................. 16
2.2 Data Analysis................................ ....... ............... 23
2.2.1 Power Spectral Density Analysis................................................... 23
2.2.2 B ispectral A analysis ...................... ....... ......... .............................. 24
2.3 W ave M modeling ... ................................ .......... ............................... 27
2.3.1 X beach ......................................... 27
2.3.2 NMSE ....................... ....... ..................................... 29

3 RESULTS ............... ........... .. .............................................. 33

3.1 Spectral Analysis ...... .. ...................... .............................................. 33
3.2 Bispectral A analysis .. ................................. ........................................... 37
3.3 Xbeach Results ........................... .......... .... ................. 44
3 .4 N M S E R results ........................................ ............................ 54
3.5 Linear Wave Model vs. Nonlinear Wave Model..................... ............... 56

4 CONCLUSIONS ............... ........... ............... ....... ..... ......... 60

4.1 Reef Processes ............. ................................ ........... .. ......... 60
4.2 W ave M odel C capabilities ...................... ....... ........ ............................. 61
4.3 Applications and Future Work ...... ............. ..................... 64









APPENDIX

A X B EA C H IN P U T FILE : param s.txt........................................................ ............... 67

B XBEACH INPUT FILE: testl9.inp................................................................... 69

C XBEACH INPUT FILE: bathy.dep............................ .............................. 70

REFERENCES ..................................... ............... 74

BIO G RA PH ICA L SKETC H ............................................................. .......................... 76






































6









LIST OF TABLES


Table page

2-1 Sum mary of wave-only test conditions.................................. ................ 19

3-1 Summary of optimized wave breaking parameters for linear model.................45









LIST OF FIGURES

Figure page

2-1 Representative study site: G uam .................................... ....................... 16

2-2 M odeled reef profile in the w ave flum e................................... .................... 17

2-3 W ave gauge locations ...... ........ ........... ........ ....... ................ 18

2-4 U rsell num bers: Tests 26-32........................................ ........................... 21

2-5 U rsell num bers: Tests 15-21 ........................................ ......................... 22

2-6 A average U rsell num bers.......................................................... ................ 23

3-1 Test 19 energy flux spectra during shoaling................................................... 33

3-2 Test 19 energy flux spectra on reef flat .................. .......... .................... 34

3-3 E energy flux spectra gauge 8.................................................... ... .. ............... 35

3-4 Test 19 energy flux ........................ .................................................... 36

3-5 Bispectra: Gauge 2 ..... ........ ................ ........ ...... ............... 38

3-6 Bispectra: Gauge 5 ............. ................ ....... ...... ......... ...... 40

3-7 Bispectra: Gauge 6 ............. ................ ....... ...... ......... ...... 41

3-8 Bispectra: Gauge 8 ............. ................ ....... ...... ......... ...... 43

3-9 Xbeach: Tests 15, 17, & 19 dissipation evolution.............................. 46

3-10 Xbeach: Tests 15 & 19 total energy evolution............... ............ ........... 47

3-11 Xbeach: Test 15 wave force x-direction................. ................... 48

3-12 Xbeach: Comparison of model calculations and experimental data for test
1 5 ......... .. ......... .. .. ......... .. .. ................................................ 4 9

3-13 Xbeach: Comparison of model calculations and experimental data for test
1 7 ......... .. ......... .. .. ......... .. .. .......... .................................... 5 0









3-14 Xbeach: Comparison of model calculations and experimental data for test
18 .............. ............. .. ........ .. .... ........ ............ ......... 51

3-15 Xbeach: Comparison of energy flux spectral density modeled and measured....53

3-16 NMSE: Comparison of energy flux spectral density modeled and measured for
te s t 1 5 ........................................................................ 5 4

3-17 NMSE: Comparison of energy flux spectral density modeled and measured for
te s t 1 9 ........................................................................ 5 5

3-18 Test 19 energy flux at sensors 7 & 8: Linear model results compared with
nonlinear m odel results................................. .. .. ............ .............. 57

3-19 Comparison of mean water level for nonlinear model, linear model, and data....58









Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science

INVESTIGATION AND MODELING OF NONLINEAR WAVE PROPAGATION OVER
FRINGING REEFS

By

Tracy J. Martz

August 2010

Chair: Alexandru Sheremet
Major: Coastal and Oceanographic Engineering

Extensive wave breaking occurs on the wide reefs located offshore of many

Pacific islands, due to their steep slope nearshore. These reefs provide protection to the

island; however, energetic wave events may cause flooding as the result of a

combination of wave setup, water level position, and wave energy. These processes are

well understood for a typical mildly sloping beach; however, they are wildly different

over a reef. This study focuses on an effort to better understand and model these

processes using laboratory data collected at the University of Michigan's wind-wave

facility. A 1:64 model of a fringing reef was constructed in the flume. The model is 2-D

vertical and uniform across the tank width. Nine capacitance-wire wave gauges were

arranged in a cross-shore transect to measure water surface elevation. Several tests

were ran modeling extreme weather conditions associated with tropical cyclones typical

of the Pacific island of Guam. Significant wave heights ranged from 3-6 m with 8-20

seconds spectral peak periods. These tests were repeated at four different still water

depths to simulate the effects of tide.









Wave breaking is found to occur both on the reef slope and edge. A considerable

amount of energy remains at the peak frequency after breaking at the reef edge.

However, the spectrum measured on the reef flat is dominated by low-frequency

oscillations. After breaking, the waves reform as bores and propagate across the reef

flat toward the beach. The low-frequency energy continues to grow as the waves

propagate shoreward.

In this study, wave propagation is simulated using Xbeach, a public-domain two-

dimensional model, which has been validated in several case studies for mildly sloping

beaches. Model results for extreme weather conditions over a complex reef system

(consisting of steep slopes and shallow areas) were used to investigate the model

capabilities to simulate wave breaking, dissipation, wave setup, and runup. Results from

the linear wave model, Xbeach, are compared to those obtained using a nonlinear wave

model, Nonlinear Mild Slope Equation (NMSE), to investigate the role of nonlinear

interactions in these processes. It is determined that a linear wave model, such as

Xbeach, can be used to accurately model wave height evolution and setup if the wave

breaking constants are calibrated correctly. However, the true physics of the energy flux

spectral evolution described by a linear wave model is incorrect. Linear wave models

ignore the presence of nonlinear energy transfer from the peak frequency to higher and

lower frequencies and the development of harmonics, which are key processes

occurring in the reef environment. A nonlinear wave model must be implemented to

correctly describe the energy dissipation and transfers that occur as waves shoal and

break over a fringing reef.









CHAPTER 1
INTRODUCTION

1.1 General Introduction

Fringing reefs are a common type of coral reef distinguished by growth extending

directly from the shoreline. These reefs occur in the tropical regions of the Pacific,

Indian, and Atlantic Oceans. In these environments, significant amounts of wave energy

can be dissipated through wave breaking and bottom friction processes. The shoreline

stability of islands protected by fringing reef systems is primarily controlled by wave

action. An understanding of how waves break on reefs and how they transform as they

propagate across the reef flat is necessary to predict shoreline stability and to design

structures for these environments.

The physical structure of a fringing reef is considerably different than that of

mildly sloping beaches. Mildly sloping beaches have previously been the primary focus

of studies on nearshore hydrodynamics. Unlike these beaches, reefs have an abrupt

change from deep water to shallow water within a short distance. Along with this steep

transition, the bottom surface of a reef is significantly rougher than that of a sandy

bottom beach due to a large amount of growth and reef organisms. The combination of

a drastically steeper slope and a significantly larger bottom roughness, leads to

significantly different dominant processes in these environments. These processes in

reef environments are not yet well understood. Our study aims to better understand

these physical processes that occur on steeply sloping reefs, and investigate the

abilities of wave models to accurately describe the physics of these processes.









Accurately predicting wave propagation on coral reefs is important for several

reasons. First, the vertical and horizontal structure of currents on the reef is established

by breaking waves, which drives reef circulation. This circulation is responsible for

cross-reef transport of nutrients, sediment, plankton, and other sea life. The exposure of

reefs to wave action is also correlated with the destruction or growth of the coral

structure itself (Monismith 2007). Lastly, and most directly studied in this research, is

the process of wave-induced setup nearshore that can cause extreme flooding on

islands fringed with reefs during storm events. All of these should be considered in

order to provide a sound engineering and environmental basis for infrastructure and

development in reef areas. In order to do so, it is necessary to better understand wave

processes on reefs.

1.2 Review of Past Studies

Reef environments often experience large waves with extremely violent breaking

occurring at shallow depths. These energetic environments combined with the locations

of such reefs owes to the minimal existence of experimental data and studies.

Difficulties associated with the physical modeling of reefs in a laboratory lead to a lack

of completely accurate laboratory data. Theoretical descriptions are even more rare.

Without reliable data sets, the advances in modeling such environments are not

significant. Much field and theoretical work is still required to understand the complex

environment of reef systems.

The process of wave propagation over reefs is of particular interest due to

broadening of wave spectra during rapid shoaling, followed by abrupt breaking. This

widening of the spectrum has been seen in field observations of waves breaking on a

13









reef and propagating into a lagoon (Gerritsen 1981; Roberts 1981; Young 1989; Hardy

et al. 1991). It was concluded that the change in spectral energy distribution is

influenced by the reef slope steepness, which governs the extent and type of breaking

that occurs. Wave attenuation due to breaking was shown to be a function of both the

incident wave height and the water depth on the reef flat (Young 1989; Gourlay 1994).

The incident wave height becomes more important at higher water levels, when

breaking is minimal or nonexistent. At lower water levels, when breaking is significant,

the water depth is highly influential on wave attenuation.

Gourlay has completed a comprehensive set of laboratory studies

(1992,1994,1996a, 1996b) on reef hydrodynamics, concentrating specifically on wave

setup and currents driven by this setup. He concluded that both were related to wave

conditions along with water level. During his studies, Gourlay assumed linear waves,

ignoring long-wave oscillations, such as infragravity waves. This has been a common

assumption on planar beaches, but it has been shown (Demirbilek & Nwogu 2007) that

significant infragravity waves can develop over the reef flat. This low-frequency energy

can cause runup limits observed to be significantly higher than would be expected on

planar beaches.

Modeling of wave breaking over reefs has been studied only minimally. The

breaking dissipation model widely used was developed for mildly sloping beaches

(Battjes & Janssen 1978) and is known to underestimate the energy dissipation and

overestimate the wave heights on steep beaches and in unsaturated surf zones (Alsina

& Baldock 2007). Breaker parameters used in the dissipation models have been

calibrated in several past studies for data sets which were collected in environments

14









that show significant variations from reef systems (Battjes & Janssen 1978; Thornton &

Guza 1983). Appropriate parameters are largely unknown for the reef system.

1.3 Study Objectives and Approach

The objective of this study is to analyze the laboratory data collected at the

University of Michigan to better understand the wave processes that occur on a fringing

reef and how they affect wave setup and flooding. This study will use wave modeling as

a tool to investigate these processes while testing the performance of the models in the

situation of a steeply sloping reef profile.

Spectral analysis is performed on the water level time series data. This reveals

the power spectra and the evolution of the energy flux as the waves propagate. From

this, energy dissipation, growth, and transfer can be seen. Based on evidence of the

presence of nonlinear interactions, a bispectral analysis is also performed. These

results more clearly reveal the energy being transferred between frequencies and the

development of higher harmonics.

A linear wave model, Xbeach, is used to simulate the wave setup and wave

evolution over the reef, and Xbeach performance is evaluated. A nonlinear wave model,

NMSE, is also used and compared to the outputs of Xbeach. From this, conclusions are

made about the importance of nonlinearities on the wave processes over fringing reefs.









CHAPTER 2
METHODOLOGY

2.1 Laboratory Study

To study the physical processes of wave propagation over fringing reefs, we

used a data set collected at the University of Michigan's wind-wave flume in Ann Arbor,

Michigan. This experiment was designed with a two-fold objective. Firstly, to quantify

the effects of wind on wave runup and secondly, to obtain detailed wave data along a

complex fringing reef system consisting of steep slopes and a shallow reef flat. The reef

system modeled in the flume was representative of the fringing reefs found off the

southeast coast of Guam (Figure 2-1A).

A










B











Figure 2-1. Representative study site. A) the Pacific Island of Guam, B) is the reef
studied, located off of the southeast coast of the island. Labeled are the three
main components of the reef: Reef flat, reef edge, and reef slope. (Source:
Google Earth)
16









The flume used at the University of Michigan is 35 m long, 0.7 m wide, and 1.6 m

high. A computer-controlled, non-absorbing plunger-type wavemaker was used to

generate irregular sea states with significant wave heights and frequencies

representative of both typical and extreme wave conditions as observed at the site.

The representative reef profile constructed in the wave flume is created out of

polyvinyl chloride (PVC), a smooth and impervious material so that the effects of bottom

friction can be ignored. The structure is characterized by three main regions, which will

be referred to in this paper as the reef slope, reef edge, and reef flat (Figure 2-1B). The

model is 2-dimensional and uniform across the width of the tank. From the shore, the

profile consists of a 1:12 sloped beach followed by a wide reef flat. The region of the

reef slope is characterized by 3 areas with slopes of 1:10.6, 1:18.8, and 1:5 (Figure 2-

2).


50- 1:12


40- 1:10.6

1:18.8
30 -


20- -
1:5

10 Reef Slope


0-

100 0 -100 -200 -300 -400 -500 -600 -700
Distance (m)
Figure 2-2. Modeled reef profile built in the wave flume. Three regions shown: reef
slope, reef edge, and reef flat. Sloping sections labeled with slope steepness.

17









The reef profile is equipped with 9 capacitance-wire wave gauges used to

measure water-surface elevation in the flume (Figure 2-3). Three are positioned in deep

water (1, 2, and 3), three along the reef slope (4, 5, and 6), one on the reef edge (7),

and two along the reef flat (8 and 9). All gauges sampled at 20 Hz for 15 minutes per

test condition. Data collection began shortly after waves began to be generated with

initially calm water conditions. Also in this experiment, a runup gauge was located on

the beach slope to quantify runup. Two anemometers measured the wind speed

induced in the flume (Demirbilek et al. 2007). The data sets with wind are not used

during this study.



50-


Runup
Gauge
#7 #8 #9

30 #6
#5
#4
20-



10


#1 #3
0
#2
100 0 -100 -200 -300 -400 -500 -600 -700

Distance (m)
Figure 2-3. The water-surface elevation is measured using 9 capacitance wire wave
gauges spaced across the reef profile and a capacitance runup gauge measures
runup simultaneously.











The three wave gauges located offshore were positioned in an array such that

they could be used to separate incident and reflected waves. The three gauges on the

reef slope are positioned in an area where extreme shoaling, wave transformation, and

breaking are expected to occur. The remaining three gauges are positioned to help

quantify the amount of setup produced in the reef flat. Preliminary analysis of the raw

data shows that the measured significant wave heights at gauge 4 deviate from the

consecutive gauges 3 and 5 by about 37%. Because of these drastic errors, the data

collected at gauge 4 are not used in this study.

Approximately 80 tests were run with combinations of waves and wind. The test

conditions are scaled using Froude scaling and summarized in Table 2-1. References to

units will be given in Froude scale for the remainder of the study. Significant wave

heights were calculated as four times the square root of the frequency-integrated

spectra calculated at gauge 2 and range from 2.8 m to 5.4 m with spectral peak periods

ranging from 8 seconds to 20 seconds.


Table 2-1. Summary of wave-only test conditions where Hs is significant wave height, Tp
is peak period, hr is still water depth on reef flat. All units are in meters. Test cases used
in this study are shown in yellow.
Test Number Hs (m) Tp (m) hr (m)
15 3.968 8.0 3.264
16 3.328 12.0 3.264
17 4.992 12.0 3.264
18 5.440 16.0 3.264
19 5.312 20.0 3.264
20 3.904 10.0 3.264
21 5.248 14.0 3.264
26 3.712 8.0 1.024
27 3.520 10.0 1.024









Table 2-1. Continued
Test Number Hs (m) Tp (m) hr (m)
29 4.544 12.0 1.024
30 4.864 14.0 1.024
31 5.440 16.0 1.024
32 5.056 20.0 1.024
33 3.584 8.0 0.000
34 2.880 12.0 0.000
35 2.880 12.0 0.000
36 4.352 12.0 0.000
37 4.864 14.0 0.000
38 5.376 16.0 0.000
39 4.928 20.0 0.000
44 2.048 8.0 1.984
45 3.904 8.0 1.984
46 3.776 10.0 1.984
47 3.200 12.0 1.984
48 4.800 12.0 1.984
57 4.928 14.0 1.984
58 5.440 16.0 1.984
59 5.248 20.0 1.984


For this study, we only consider data collected in wave-only tests (i.e. no wind).

Also, four different initial water levels were used ranging from 0 m-3.2 m of water on the

reef flat in order to represent variations in mean water level. In this study, we will

concentrate on the tests ran at the highest water level, 3.2 m on the reef flat (yellow box

in Table 2-1).

The nonlinear wave model used in this study has a limitation based on the Ursell

number. The Ursell number is defined as


U, k 3 (2-1)
k h3











where a is the wave amplitude, k is the wavenumber, and h is the local water depth.


Because of the quasi-Gaussian approximation used in the model, a threshold of Ur<1.5


was proposed by Agnon & Sheremet (1997). It then follows that in very shallow water,


where nonlinearities dominate over dispersion, the Ursell number can easily exceed the


threshold for the model validity. Figure 2-4 shows the calculated Ursell numbers for


tests 26-32 where the still-water depth is 1.024 m on the reef

flat.


2
10



10
10


10o
E
3

10


-2
10


0-3 L
100
800


A I+A





+


QL + 8s, 3.7m
XX X 0 l1s, 3.5m
S* 12s, 3.0m -
++ + X 12s, 4.5m
O 14s, 4.9m
O 16s, 5.4m
20s, 5.1m

900 1000 1100 1200 1300 1400 1500 1600 17


800 900 1000 1100 1200 1300 1400 1500 1600 1700
distance(m)


Figure 2-4. Ursell numbers calculated for tests 26-32 at each gauge for varying wave
conditions. The bottom panel shows the gauge locations and reef bathymetry for
reference.



The dominant variable in the calculation of the Ursell number is the local water


depth, h. As the water depth decreases, the Ursell number grows very fast due to the
21










value being cubed. At the gauges close to the reef edge and on the reef flat, where the

water level is very low, the Ursell numbers are very high.


800 900 1000 1100 1200 1300 1400 1500 1600 1700
distance(m)

Figure 2-5. Ursell numbers calculated for tests 15-21 at each gauge location for varying
wave conditions. The bottom panel shows the reef geometry and gauge locations
as a reference.


As expected, the Ursell values for the shallow water tests 26-32 with a still water

depth of 1.024 m on the reef flat, are an order of magnitude greater than the tests with a

still water level of 3.264 m on the reef flat (Figure 2-5) and far exceeded the suggested

threshold proposed by Agnon and Sheremet (1997) for the nonlinear wave model.

Figure 2-6 shows the average Ursell numbers of all tests at each water level. Because


X L
A F -1


A + +


+ + 8s, 3.9m
0 12s, 3.3m
12s, 4.9m
x 16s, 5.4m
L 20s, 5.3m
0 10s, 3.9m
A 14s, 5.2m
900 1000 1100 1200 1300 1400 1500 1600 1











the deepest water level clearly maintains the lowest Ursell numbers, only these tests


(15-21) are used in this study.


10




10
l-o

E
Z 10




10


10-2
90
0

F -20
0
-8

-40
80


0


1000 1100 1200 1300 1400 1500 1


-



0 900 1000 1100 1200 1300 1400 1500 1600 17C
distance(m)


Figure 2-6. Average Ursell numbers for each water level. Bottom panel shows the reef
geometry and gauge locations as a reference.



2.2 Data Analysis


The collected time series data were analyzed to extract quantities of interest.


This chapter presents two types of data analysis applied to the raw data. Spectral and


bispectral analyses are performed to determine trends and variations in the data.


2.2.1 Power Spectral Density Analysis


The energy spectral density describes how the variance (energy) of the water


surface time series is distributed in the frequency domain. This is most often referred to


hr=1.024m
hr= 1.984m
hr=3.264m









as the power spectral density (PSD). Most commonly, the PSD, P(f), is described as the

Fourier transform of the autocovariance function, R(T), if the signal can be treated as a

stationary random process, where T is some fixed length of time:

P(f) = f R(r)e- "dr (2-2)

R (r) = f[(x(t) x)(x (t + T) x)]dt (2-3)

where p is the mean of the time series and asterisk denotes the complex conjugate.

This ensures that the autocorrelation is positive definite. As the averaging time interval,

T, approaches o, the ensemble average of the average periodogram approaches the

PSD:


E P(f) (2-4)


The power in a given frequency band [fi,f2] can be calculated as:

P= fP(f)df (2-5)

In this study, power spectra were calculated after the first 100 seconds of data

were truncated to allow waves to propagate through the entire array of gauges. The

frequency vector has a lower limit of 0.005 Hz and an upper limit of 0.4 Hz. Data is

divided into 512 records in a sequence. Spectra were estimated from zero-meaned,

Hanning windowed wave records with band averaging. The resulting resolution

bandwidth is 0.0049 Hz and spectral estimates have 61 degrees of freedom.

2.2.2 Bispectral Analysis

Spectral analysis approximates the time series of sea surface elevation to be the

linear superposition of statistically uncorrelated waves. In shallow water, different

24









spectral components interact with each other, i.e., nonlinearities become significant.

Increasing steepness of waves in the nearshore and the resulting breaking are evidence

of these nonlinearities. When these interactions occur, the linear spectral analysis is

incapable of completely characterizing the variability of the signal. For some types of

nonlinear data, the second order spectral analysis properties can be similar to those of

a linear time series because this type of linear analysis is not capable of detecting

deviations from linearity and Gaussianity (Rao & Gabr 1984). More specifically, the

power spectral density lacks phase information. For cases in which nonlinearities are

expected to be significant, higher-order spectral methods (e.g., bispectral analysis) are

used. The fast Fourier transform (FFT) amplitude can give useful hints of nonlinearity,

but the bispectrum gives much firmer proof of the presence of nonlinear three-wave

interactions (Hocke & Kampfer 2008).

If ((t) is a stationary random function of time, the spectrum P(f) and bispectrum

B(fi,f2) of ((t) are defined respectively as the Fourier transforms of the mean second and

third order products:

P(f) = f'R(T)e -2"dr (2-6)

where

R(T) = E{(t)(t + T)} (2-7)


B(f, f2) = ff _S(,,2)e- T-2 2dxd2, (2-8)

where

S(r1,72) = E+(t)O(t + r)(t + r2)} (2-9)

where E{} is the expected value operator. The inverse relations to (2-6) and (2-8) are

25









R(r) = f P(f)e 2fdf


S(,r1,T) = f fB(f, f)e 2if'T1+2d22 2dfidf (2-11)

For real ((t)

P(f) = P(-f)* (2-12)

B(f,,f2) = B(-f,,-f2) (2-13)

From the stationarity of ((t) the known symmetry relations

R(r) = R(-r) (2-14)

S(T,2T2) = S(r,21) = S(-r,31 TO) = S(r1 T2,-T2) = S(-r,32 TO) = S(r2 1,-r1) (2-15)

follow immediately. In terms of spectra and bispectra, (2-12) and (2-13) become

P(f)= P(-f) (2-16)

B(f,,f2) = B(f2,f,) = B(-f,,f, f2) = B(f, f,,-f,) = B(-f,,f, f,) = B(f, f,,-f,) (2-17)

From (2-12), (2-13), (2-16), and (2-17) it follows that the spectrum is real and is

determined by its value on a half line, whereas the bispectrum is determined by its

values in an octant.

In this study, the bispectrum is estimated using the direct (FFT-based) method

with an FFT length of 512. Zero mean input data are segmented with 255 samples per

segment with zero overlapping. Bispectral estimates are averaged across records and a

frequency domain smoother is applied (Rao-Gabr optimal window) with side lengths of

1. The resulting bandwidth is 0.002 Hz. Bispectra are then normalized by a product of

the signal's spectrum.


(2-10)









2.3 Wave Modeling

In this study, the laboratory data were modeled using two different models:

Xbeach (Roelvink et al. 2007) and NMSE (Agnon & Sheremet 1997). The intent and

scope of applicability of the two models are very different. The capabilities of the models

were investigated and compared.

2.3.1 Xbeach

Xbeach is a public domain model (http://www.xbeach.orq) that was developed

with funding and support by the US Army Corps of Engineers, by a consortium of

UNESCO-IHE, Delft Hydraulics, Delft University of Technology, and the University of

Miami. It is a two-dimensional model for wave propagation, sediment transport, and

morphological changes of the nearshore, beaches, dunes, and backbarrier during

storms. Xbeach was originally designed to cope with extreme conditions such as those

encountered during hurricanes with the ability to test morphological modeling concepts

of dune erosion, overwashing, and breaching. The model has been validated against a

number of analytical and laboratory tests, both hydrodynamic and morphodynamic, but

to date, all test beds have been on mildly sloping beaches.

Due to generally short length scales in terms of wavelengths and the probability

of supercritical flow, first order upwind is the main numerical scheme implemented in

combination with a staggered grid. Xbeach uses an explicit model scheme with an

automatic time step using the Courant criterion.

The major function implemented in Xbeach and used for this study is the wave

action equation solver. The wave forcing is obtained from a time dependent version of

the wave action balance. Here, the frequency spectrum is represented by a single mean

27









frequency, but the directional distribution of the action density is taken into account. The

wave action balance is given by:

dA dc A + dcYA + dcoA -D (2-18)
dt dx dy dO o

The formulation for the total wave dissipation according to Roelvink (1993) is:

D = 2af,,E,,Qb (2-19)

where a is the wave amplitude and frep is a representative intrinsic frequency. Here,

1
E = -pgHms (2-20)

The fraction of wave breaking is given by:


Qb =min -e yh ,1 (2-21)


Assuming unidirectionality and stationarity, (2-18) becomes

dF
dx = -KF (2-22)
dx

where K = 2af,,eQb and depends on 3 free calibration parameters: a, y, and n. Here, it

should be noted that the only mechanism of dissipation is due to breaking and depends

completely on the value of K.

Currents and water level are computed using depth-averaged and shortwave-

averaged shallow water equations:

du 9u 9u 1u dy211U Ts T, 9( F)
-+ u-+v-- f -Vh --+- =- g (2-23)
&t dx Iy dx ph ph dx ph









9v 9v 9v b9 v\ r v 9rv F
-+u-+v-+ fu-vh + -g--- Lg + (2-24)


+ + Vhu 8hyvp
i+ + = 0 (2-25)
dt dx dy

Here, h is the water depth, u and v are velocities in the x and y direction, Tbx, Tby are the

bed shear stresses, Tsx, Tsy are the wind stresses, g is the acceleration due to gravity, r

is the water level and Fx, Fy are the wave-induced stresses. Again, assuming

unidirectionality and stationarity, the shallow water equations reduce to:

du dr, S.
u- -g + -- (2-26)
dx dx ph

where Sxx is the radiation stress. Here, the shallow water equation is coupled with the

wave-action balance. In our study, these are the important equations. However, Xbeach

is capable of modeling many other processes such as directional waves and currents,

sediment transport, bottom updating, and multiple sediment classes. Xbeach was

chosen for this study because of its broad range of applications. However, during our

study, we focus on the linear wave model implemented in Xbeach coupled with the

shallow water equations to calculate wave setup.

2.3.2 Nonlinear Mild Slope Equation (NMSE)

Agnon and Sheremet (1997) developed a stochastic wave shoaling model based

on the mild slope equation. Regarding the system as stochastic, evolution is described

by a hierarchy of equations for the statistical moments or the Fourier space cumulants.


dF = -2K ,F + [ (W6) j *b, *b, } 2(W6) j, s{b bb} (2-27)
p,q>0








where Fj is the average modal energy flux, S{z} is the imaginary part of the complex

number z, W is the interactive coefficient (see equation 2-28), and 5 is the delta function

(see equation 2-29). The interaction kernel is defined by Agnon and Sheremet (1997) as

( I ; CT2 2 + 2J rP)q 1
Wj, 8 (Y p(q j r cr P

and the delta function is defined as

j1, jTp-q=OjTp=q
jpq = otherwise

Aj,_pq =Oj T Op Oq (2-29)

The triple products on the right-hand side of equation (2-27) can be solved leading to

the closure problem. That is, the average of double products (bib) (spectrum) depends

on the average triple products (b pbb) (bispectrum), which depends on the average

quadruple products (bb,bpb (trispectrum) and so on. This leads to an infinite set of

equations to be solved. This can be dealt with by using assumptions that discard all of

the cumulants that are of a higher order than a specified value. Commonly, the

Gaussian approximation is assumed, which, in general, for a Gaussian wave field, odd-

order spectra cancel exactly and all even-order spectra are expressed in terms of the

second order spectrum. This is a poor assumption particularly for the shoaling

processes, which are characterized by a very fast evolution. Shallow-water waves show

very strong phase-coupled Fourier modes and cannot be fully described with this

assumption.








Instead, NMSE uses the quasi-Gaussian approximation, which assumes, due to

phase correlations created by nonlinearities, the wave field deviates slightly from

Gaussianity, and modes are no longer "exactly" uncorrelated, i.e.,



(b,b) = Ib, 1b6 +O(e), (2-30)

(bb:b)= O(),

(bAbb;bb) = (b,l' b +( b,,2 b, j)6,, +O(8).

where E is the small parameter of the problem (e.g., wave slope), leaving the equation

for the spectrum as,

dF-= -2K F,+ 1[(W6) jp3b bj*bpb,- 2(W6). 3{bj *bb,}]+O(S). (2-31)
p,q>0

The first term on the right-hand side of equation (2-31) allows for dissipation due only to

breaking, which depends completely on the value K. The second term allows energy to

be transferred from mode j to modes p and q. Unlike the linear wave model

implemented in Xbeach, where the only mechanism effecting the energy flux is

dissipation due to breaking, NMSE allows also for energy to be transferred between

frequencies.

Unlike Xbeach, NMSE is strictly a wave model, meaning it is not designed to

compute currents, sediment transport, etc. However, a simple calculation can be made

during post processing to calculate the water level. Assuming unidirectionality and

stationarity, the shallow-water equation is described as

dr, 1
S (2-32)
dx pgh









where r and Sxx represent the along-flume (cross-shore) water level and radiation

stresses, respectively.

Since its development, NMSE has been corrected for application on steep slopes

(Agnon & Sheremet 1997).










CHAPTER 3
RESULTS

3.1 Spectral Analysis

Spectral analysis was performed on the data revealing commonalities between

all test conditions. There remains a clearly defined and constant peak frequency

through all gauges until the point of breaking, which occurred most often between

gauges 6 and 7. It can be seen that after breaking (gauge 7), there is still a considerable

amount of energy remaining at the initial peak frequency. By gauge 8, this energy is

mostly dissipated and the reef flat is now dominated by low-frequency oscillations. As

the waves continue to propagate shoreward, this low-frequency energy continues to

grow as it is being transferred from higher frequencies.

1400
-Gauge 2
Gauge 5
1200 Gauge 6

1000

S800-









0
LL 600

400





0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
Frequency (Hz)




-40 L L L L L
1000 1100 1200 1300 1400 1500
distance(m)


Figure 3-1. The top panel shows the power spectrum at gauges 2, 5, and 6 for test 19
(Hs=5.3, Tp=20s). Bottom panel shows reef profile and gauge locations.

33












As waves propagate from deep water over the reef slope and begin to shoal,

there is evidence of nonlinear transfers and dissipation from the peak frequency. Energy

is being transferred from the peak frequency into higher and lower frequencies leading

to a widening of the spectrum that occurs as the waves steepen and propagate

shoreward (Figure 3-1). Energy is distributed over a wider region surrounding the peak

frequency as the waves shoal due to energy from the peak frequency being transferred

to higher and lower frequencies. In test 19 (Figure 3-1), it can be seen that there is more

energy present at frequencies greater than approximately 0.08 Hz and less than 0.03

Hz after shoaling begins. This widening was found to occur in all test cases.

220
-Gauge 7
200 Gauge 8
180 Gauge 9

S160-




-2 100-


S60
LLJ
40

20

0
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
,0! Freauencv (HA
.F: -20-


-40
900 1000 1100 1200 1300 1400 1500 1600
distance(m)


Figure 3-2. The top panel shows the power spectrum at gauges across the reef flat for
test 19 (Hs=5.2m, Tp=20s). The bottom panel shows the cross-shore reef profile
and gauge locations as a reference.










After the waves break near the reef edge, waves reform and continue to

propagate across the reef flat as bores. Significant energy remains in the infragravity

band over the reef flat (Figure 3-2). Between gauges 7 and 8, the spectrum begins to be

dominated by low-frequency oscillations. This was found to occur during all test cases

and can be seen in Figure 3-3.


0.01 0.02 0.03 0.04 0.05 0.06
_0 Freauency (H4)

r--


-40 ''
900 1000 1100 1200 1300 1400 151
distance(m)


0.07





1600
1600


Figure 3-3. The top panel shows the power spectra at gauge 9 for each of 4 tests: 15,
17, 19, and 21. The bottom panel shows the cross-shore reef profile and gauge
locations as a reference.



As waves propagate into shallow water and begin to shoal followed by breaking,

energy is not only dissipated from the peak frequency, but energy increases are also

seen in other areas of the spectrum. These increases are attributed to energy transfer










from the peak frequency to higher harmonics and lower subharmonics. The

development of higher harmonics and increased energy in the infragravity band can be

seen in the energy flux evolution (Figure 3-4).


I
"5-



0.5
0.1



0.05


E
- -20
a0 ,


1000 1100 1200 1300


I I I I


1000 1100 1200 1300 1400 1500
distance(m)
Figure 3-4. The top panel shows the energy flux density (power x group velocity)
evolution of test 19. The bottom panel shows the cross-shore reef profile and
gauge locations as a reference.



In Figure 3-4, the widening of the spectrum can be seen when the waves shoal as they

propagate over the reef slope. After the waves break between gauges 6 and 7, energy

is dissipated rapidly, first from the higher harmonics. Beginning at gauge 8, the

spectrum is dominated by infragravity energy.

36









This spectral analysis indicates the presence of nonlinear interactions especially

in cases of larger, stronger, and longer waves. To further investigate these nonlinear

properties, bispectral analysis was performed.

3.2 Bispectral Analysis

This section describes the bispectra of shoaling waves observed during the

laboratory test. In particular, two tests will be described: test 15 and test 19. Test 15

was performed with the smallest and shortest wave of the test conditions while test 19

was performed with the largest and longest. Test 15 had a significant wave height of 3.9

m and a peak period of 8 seconds while test 19 had a significant wave height of 5.3 m

and a peak period of 20 seconds. The bispectral evolution at gauges 2, 5, 6, and 8 are

studied.

Figure 3-5 shows the bispectra from the two tests at gauge 2, an offshore

deepwater gauge. The distinct peak in both cases occurs at the peak frequency

showing that the peak frequency is interacting with itself producing a second harmonic

of frequency fp+fp. While the dominant interaction occurring in test 15 (Figure 3-5A) is

between the peak frequency and itself, developing a second harmonic of frequency 0.25

Hz, there is also a smaller second peak visible showing that the peak frequency is also

interacting with low frequency infragravity energy. While peaks in the bispectrum occur

in these two distinct frequency pairs, the peak frequency appears to be interacting, to

some degree, with energy of all higher frequencies. Interactions at gauge 2 for test 19

(Figure 3-5B) are more pronounced than the interactions occurring with the smaller and

shorter wave in test 15. The strongest peak represented in the bispectrum is creating a

second harmonic of frequency 0.1 Hz. It can also be seen that there is a small peak

37











A


0.15


3 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5


0.25 0.3
frequency(Hz)


B
0.2


0.051


0 0 5 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
frequency(Hz)

Figure 3-5. Bispectra and spectra at gauge 2. A) shows data from test 15 (Hs=3.9 m,
Tp=8 s) and B) shows data from test 19 (Hs=5.3 m, Tp=20 s). Top panels are
showing the bispectra and bottom panels are showing the power spectra.



occurring between this harmonic and the peak frequency suggesting that a third

harmonic is also being generated through the interactions between the peak frequency,
38


0.05









fp, and the second harmonic, f2, at a frequency of fp+f2=0.15 Hz. Also present in the

bispectrum calculated at gauge 2 is an interaction between the peak frequency and

infragravity energy.

Figure 3-6 shows bispectra of tests 15 and 19 again, calculated at gauge 5,

located on the reef slope. At gauge 5, shoaling has begun, but waves have not yet

begun to break. Waves are changing shape considerable with increased peakedness.

Test 15 (Figure 3-6A) shows little change from the bispectrum at gauge 2, but the peaks

are suggesting the development of a second harmonic, and the interactions between

the peak frequency and infragravity energy are more developed at gauge 5 as the

waves are shoaling. A slight peak can be seen between the second harmonic (f2=0.25

Hz) and the peak frequency (fp=0.125 Hz) creating a third harmonic at 0.375 Hz (fp+f2).

While clear peaks occur at these specific frequencies, the bispectrum reveals that the

peak frequency is interacting with higher frequencies throughout the entire spectrum.

Figure 3-6B shows that the peak frequency in test 19 continues to interact strongly with

itself transferring energy into the second harmonic at 0.1 Hz (fp+fp). At gauge 5, shoaling

has begun, strongly increasing nonlinear interactions. In this figure, interactions

between the peak frequency and the entire spectrum are seen indicated by the array of

increased values seen across the spectrum at the peak frequency. The second

harmonic (f2=0.1 Hz) has began to interact weakly with higher harmonics as well,

indicated by the slightly increased values found across the spectrum at f2=0.1 Hz.

Interactions are increasingly visible between the infragravity band and the peak

frequency, as well as with other higher frequencies developing interactions with the

infragravity band.













A
0.2


0.15


0.1


0.05


01
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
fl (H7)
S20



0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
frequency(Hz)
0.25

B
0.2


0.15


0.1


0. 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5


0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
100 -

50 -


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
frequency(Hz)


Figure 3-6. Bispectra and spectra at gauge 5. A) shows data from test 15 (Hs=3.9 m,
Tp=8 s) and B) shows data from test 19 (Hs=5.3 m, Tp=20 s). Top panels are
showing the bispectra and bottom panels are showing the power spectra.



Figure 3-7 shows the bispectra for tests 15 and 19 calculated at gauge 6 located


on the reef slope. This is the gauge located just before the reef flat. In test 15 (Figure 3-


7A), strong interactions remain between the peak frequency and itself, transferring

40













0.2h


0.15



-r
0.1 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0.05





0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
frequency(Hz)
0.25
B

0.2


0.15


0.1


0.05


0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
M I(Hz

E 50 /

0-"
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
frequency(Hz)


Figure 3-7. Bispectra and spectra at gauge 6. A) shows data from test 15 (Hs=3.9 m,
T,=8 s) and B) shows data from test 19 (Hs=5.3 m, Tp=20 s). Top panels are
showing the bispectra and bottom panels are showing the power spectra.

energy into the second harmonic. A second peak between the second harmonic

(f2=0.25 Hz) and the peak frequency (fp=0.125 Hz) has become more distinct,

generating a third harmonic, while the peak continues to interact with higher


41









frequencies. Also, interactions are seen between the peak and second harmonics with

the infragravity band. In test 19 (Figure 3-7B), there are distinct higher harmonics

interacting with the peak frequency, but it's at gauge 6 that the interactions between the

second harmonic and higher harmonics become most distinguishable. The infragravity

interactions between higher harmonics are also developed.

Figure 3-8 shows the bispectra calculated at gauge 8, on top of the reef flat.

Here, the waves have broken. In test 15 (Figure 3-8A), the interactions are somewhat

random and trends are difficult to distinguish. Strongest visible interactions appear to

occur at the very low frequencies and in the infragravity band, surrounded by noise due

to very low energy levels. In test 19 (Figure 3-8B), there remain some weak interactions

between the peak and higher harmonics; however, the interactions between the

infragravity band and the harmonics dominate on the reef flat. There are no longer signs

of interactions between the second harmonic and higher harmonics.

From the bispectral analysis it is shown that there are significant nonlinear

interactions occurring between harmonics and subharmonics as the waves shoal across

the reef slope and break. The larger and longer waves (test 19) show more distinct

signs of the development of harmonics and the interactions between harmonics than the

smaller and shorter waves (test 15). Test 19 was more of a plunging type breaker

compared to test 15, which was more of a rolling breaker. It can be determined through

this analysis that nonlinear interactions are important in the reef environment,

regardless of the wave conditions. However, the extent to which the nonlinearities

interact differs with incident wave conditions.
































0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
Hft (U-J\


N'
'I
N I
E
-2

0)
0^


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
frequency(Hz)
B 0.251 I


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
p.201 .. fl(lHz)

E -
-10


0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
frequency(Hz)


Figure 3-8. Bispectra and spectra at gauge 8. A) shows data from test 15 (Hs=3.9 m,
Tp=8 s) and B) shows data from test 19 (Hs=5.3 m, Tp=20 s). Top panels are
showing the bispectra and bottom panels are showing the power spectra.


\J,-J









3.3 Xbeach Results

Xbeach was run to simulate flume tests 15-21 (Table 2-1). Grid cell sizes were

10 m in the cross-shore direction and 5 m in the alongshore direction. All capabilities

except wave propagation and dissipation due to breaking were ignored. A Joint North

Sea Wave Analysis Project (JONSWAP) spectral shape was used with peak

enhancement factor y = 3.3 using the random phase method to input wave conditions.

Roelvink (1993) was used as the breaker dissipation model, which includes three free

calibration parameters: y, a, and n, where y controls the fraction of breaking waves, a

controls the level of energy dissipation in a breaker, and n is a third free parameter. The

parameter a is totally unknown for steep slopes, but is generally accepted to be O(1).

Parameterization within wave models is very limited as it relates to reefs and steep

slopes. Finding an optimal combination of these three free parameters will yield the

most accurate model results. These optimal values differ for each wave condition and

every experimental data set.

The optimal values of y and a were determined by minimizing the normalized

root-mean-squared error between measured and computed wave height and spectra

while the free parameter n was held constant at n=5, the default value, to minimize the

degrees of freedom. A more accurate optimization process would allow n to range from

0 to 10. Optimal values for y and a were obtained and used for each model run and

summarized in Table 3-1.

Xbeach requires parameters to be set for the simulation in a file named

params.txt (Appendix A), which signals the model to also look for a wave input file with









the extension .inp (Appendix B) and a bathymetry file with the extension .dep (Appendix

C).


Table 3-1. Summary of optimized breaking parameters for each test wave condition
along with iterations necessary in optimization process and rms error associated with
parameters.
Wave Hs=3.9m Hs=3.9m Hs=3.3m Hs=4.9m Hs=5.2m Hs=5.4m Hs=5.3m
Conditions Tp=8s Tp=10s Tp=12s Tp=12s Tp=14s Tp=16s Tp=20s

Iterations 41 81 93 56 112 55 60
Error 0.0313 0.0387 0.0346 0.0572 0.0381 0.0430 0.0380
a 0.6953 0.6049 0.6168 0.6438 0.6422 0.6835 0.7715
y 0.6895 0.6003 0.5450 0.7572 0.8035 0.8198 0.9720


Figure 3-9 shows the model outputs for energy dissipation. Across the x-axis is

the cross-shore distance and along the y-axis is the time in seconds that spans for a 2-

hour simulation run. The only mechanism of dissipation acting in these model runs is

dissipation due to wave breaking, so these plots show the areas of wave breaking. It

can be seen that in test 15 (Figure 3-9A), where the significant wave height is 3.9 m and

the peak period is 8 s, the waves begin to break much later and with a lower intensity

than test 19 (Figure 3-9B). Test 19, with a significant wave height of 5.3 m and a peak

period of 20 s begins to shoal and break slightly farther offshore and in deeper water

than test 15 and continues breaking as waves propagate up to the reef flat. Because of

the small zone in which breaking is assumed to occur, and because a linear wave

model assumes that breaking is the only mechanism of dissipation, it can be assumed

that the only change in the spectrum should appear in areas where breaking occurs,










which is, at most, between gauges 5 and 8. Elsewhere, there is assumed to be no

spectral energy flux evolution.

A B
7000 4000 4000

600 0 500 0 00 1500

5000 3000 5000
2500 -2500
54000 6400
22000 2000
3000 15 30 0 1500

2000 1000 200 1000

51000 00 500
1I A too0

1000 1100 1200 1300 1400 1500 1600 1000 1100 1200 1300 1400 1500 1600
t=o- r -10-
.-20 -20-

1000 1100 1200 1300 1400 1500 1600 1000 1100 1200 1300 1400 1500 1600
distance (m) distance (m)

Figure 3-9. Dissipation due to breaking with cross-shore distance as it evolves in time.
A) test 15 (Hs= 3.9 m, Tp=8 s ), B) test 19 (Hs= 5.3 m, Tp= 20 s). Below is the
cross-shore reef profile as a reference.


Figure 3-10 shows the evolution of total energy across the reef profile throughout

the 2-hour simulation. There is no loss or transfer of energy accounted for until

dissipation due to breaking occurs at gauges 5 and 6. It can also be seen (Figure 3-

1 A) that in test 15, with a significant wave height of 3.9 m and a peak period of 8 s,

after the breaking point, very little to no energy remains on the reef flat especially

landward of gauge 8. However, in test 19 (Figure 3-10B), where there is a significant

wave height of 5.3 m and a peak period of 20 s, there remains significant wave energy

after breaking and continuing all the way across the reef flat. Test 19 also shows a










significantly higher amount of energy just before breaking indicating that shoaling just

before breaking is significantly greater in this case.


1000 1100 1200 1300 1400 1500 1600

S-30 ,

1000 1100 1200 1300 1400 1500 1600
distance (m)


,400
E
300

200


1000

1000 1100 1200 1300 1400 1500 1600
0-
-r-o .. o .'
.-20-
S.30, ,
1000 1100 1200 1300 1400 1500 1600
distance (m)


Figure 3-10. Total energy as a function of cross-shore distance throughout the
simulation time of 2 hours. A) test 15 (Hs= 3.9 m, Tp= 8 s, B) test 19 (Hs= 5.3 m,
Tp= 20 s). The bottom panels show the cross-shore profile as a reference.



Wave force was calculated using the radiation stress tensors. Figure 3-11 shows

the wave force created in the cross-shore direction by the waves as they propagate

towards the shore. It is seen that the wave forces are strongly dependent on the

bathymetry of the reef. In the regions of steeper slopes, with more rapid shoaling, the

wave force increases until breaking. It can be seen that the force is positive almost

consistently everywhere that there is no dissipation. In the breaking regions, mainly at

the reef edge and at the shoreline, the wave forces become negative and strong.






























1000 1100 1200
0-
--10
5.-20
- -30 ,
1000 1100 1200
dist


f 1



i










1300 1400


1300
ance (m)


1500 1600
---------- i


1400 1500 1600


Figure 3-11. Test 15 (Hs= 3.9 m, Tp= 8 s) wave force in the x-direction with cross-shore
position through time. Below is the reef profile as a reference.




The wave model in Xbeach has been calibrated specifically for setup and wave

height. Because the model is depth averaged, these are typically the desired parameter

outputs aside from morphological updating, which was not used in this application. By

changing the three free calibration parameters, y, a, and n, the model can be calibrated

to match measured data.

Figure 3-12 shows the setup and wave height evolution calculated by Xbeach

compared to the measured data in the wave flume for test 15 (Hs= 3.9 m, Tp= 8 s). The

calculations are in good agreement for the region of setdown, with a small

underestimate of the setup (Figure 3-12A). It is also noted that where the wave forcing

were seen to be strongly negative, is also where the 0.08 m setdown occurs.
48


I


__8. 8.


^-
















-Modeled
0.5 A Measured
0.4
'g 0.3
t 0.2
0.1-
0-
-0.1
900 1000 1100 1200 1300 1400 1500 1600 1700
distance cross shore (m)


3 .5 I I I I I

3- B

2.5

S2-

1.5

1

0.5
900 1000 1100 1200 1300 1400 1500 1600 1700
distance cross shore (m)
Figure 3-12. Test 15 (Hs= 3.9 m, Tp= 8 s). Comparison between the measured and
calculated A) setup, and B) Hrms.





The predicted wave height evolution is not as closely matched to the measured


data as the setup/setdown, but has disagreements mainly in the region of shoaling at


gauges 5 and 6 (Figure 3-12B). Gauge 4 has been assumed to be a malfunctioning


gauge, explaining the significantly low wave height measured at this gauge and should


be ignored.


Figure 3-13 shows the comparison between the predicted values by Xbeach and


the measured values from the laboratory test of setup and root-mean-squared wave


height evolution for test 17 with a significant wave height calculated as 4.9 m and a









peak period of 12 seconds. In this test, setdown is predicted with very good agreement

(Figure 3-13A). The setup is also closely modeled with a slight over-prediction of the

setup on the reef flat possibly due to an over prediction of wave heights prior to

breaking.

The wave height evolution is again over predicted, especially in the region of

shoaling, with an obvious outlier at gauge 4 due to a gauge malfunction (Figure 3-13B).

The data shows almost no wave height growth as the waves shoal over the reef until

just before breaking at gauge 6, however Xbeach begins to predict shoaling almost

immediately shoreward of the deepwater gauges, where waves begin to propagate over

the reef bathymetry.


distance cross shore (m)


1 I I I I I I I
900 1000 1100 1200 1300 1400 1500 1600 1700
distance cross shore (m)
Figure 3-13. Test 17 (Hs= 4.9 m, Tp= 12 s). Comparison between the measured and
calculated A) setup, and B) Hrms.












Test 19 had a significant wave height of 5.3 m and a peak period of 20 seconds.


Xbeach predicted the setup and setdown very accurately for this test case (Figure 3-


14A). The wave height evolution was again over predicted, but still reasonably modeled,


excluding the calculated wave height at gauge 4 (Figure 3-14B).


1300
distance cross shore (m)


5.5
5
4.5

E 4
3.5
3
2.5


1100 1200 1300
distance cross shore (m)


Figure 3-14. Test 19 (Hs= 5.4 m, Tp= 20 s). Comparison
calculated A) setup, and B) Hrms.


between the measured and


Overall, although Xbeach uses a linear wave model, it does a reasonable job of


predicting the setup, setdown, and wave height evolution over reefs. This is not entirely


B



a


I o I


900 1000


1700









surprising. Recall that when assuming unidirectionality and stationery, Xbeach

calculates waves as:

dF,
dx j j (3-1)



Because the wave energy is entirely dependent on the dissipation rate, K, where K is a

function of three free parameters, y, a, and n. These three calibration parameters were

chosen to fit lab runs based on these simple outputs. In this case, as previously

mentioned, n was held at a constant value of 5 for simplicity while y and a were

optimized for best fit to measured data. This fit could be made more accurate still by

allowing the calibration parameter, n, to range from 1 to 10. In effect, the calibration

parameters in Xbeach were used to force a good fit from the model, with little physical

meaning, and could be done with any data set to accurately model the wave processes.

While Xbeach's linear wave module is capable of outputting accurate setup and

wave height calculations when calibrated correctly, the underlying nonlinear physics of

the process is ignored. The only mechanism of dissipation that a linear wave model

recognizes is dissipation due to breaking. Therefore, up until the breaking point, there is

no change in the energy flux. Even as waves shoal, energy flux remains the same. At

the breaking point, energy is dissipated from all frequencies (Figure 3-15A). It can be

seen (Figure 3-15B) that in the true process, the energy flux density begins to vary

when waves begin to shoal over the reef bathymetry. The spectrum widens and energy

is transferred to higher frequencies. Just before breaking, as the wave shape changes

drastically by steepening, and nonlinear interactions are strong, higher harmonics are











created and energy is transferred into the infragravity band. After breaking, energy is


dissipated from higher frequencies first. Linear wave models do not recognize any of


these processes, but concentrate on accurate calculations of setup and wave height


while taking into account dissipation due to breaking.


0.25

- 0.2
o
o 0.15

" 0.1


0.2


S0.15

" 0.1


0.05


-10

S-20

o -30
1


1.3
1.2
1.1 '
E


0.9
0.8 2
0.7 L
0.6 D
0.5
0.4


947968 1241 1276 1313 1400
Figure 3-15. Energy flux evolution for test 19 (Hs= 5.4 m, Tp= 20 s). A) linear wave
model, B) data from the laboratory experiment. The bottom panel shows the reef
cross-section and gauge locations as a reference.


2 3 5 6 7
-e-- r










3.4 NMSE Results

Unlike Xbeach, the nonlinear wave model, NMSE, allows for energy transfer

across the spectrum (between different frequencies). While this difference will not be

seen with simple plots of the predicted setup and wave height values, a drastic

difference can be seen when comparing the energy flux spectral densities.

By allowing energy to transfer to higher and lower frequencies NMSE is able to

describe the development of higher harmonics and energy transfer to the infragravity

band, which are key characteristics of the laboratory data. Although test 15 has a


0.25
N
S0.2

c 0.15

0.1
LU
0.05



0.25

0.2

c 0.15

So0.1
LU


1.3
1.2
1.1

0.9 $
0.8 -
0.7
0.6 .
0.5
0.4


947968 1241 1276 1313 140(
Position (m)
Figure 3-16. Energy flux spectral density for test 15 (Hs= 3.9 m, Tp= 8 s). A) calculated
by NMSE, B) calculated from laboratory data. The bottom panel shows the reef
cross-section geometry and gauge locations as a reference.
54











significant wave height of only 3.9 m, a peak period of 8 seconds, and is not

extraordinarily nonlinear compared to other test runs (refer to Ursell numbers in Figure

2-5), there is still energy transferred and higher harmonics developed (Figure 3-16B).

NMSE is able to describe the development of higher harmonics and the growth of the

infragravity band. Because of the extra terms in NMSE, it is able to describe this

nonlinearity (Figure 3-16A) and to accurately describe the complete physics of the wave

processes occurring in the reef environment.


. 0.

0.1
S0


1.6 E
U-
1.4 0


1.2 .
u,
x
LL


N
0 .

o
2 0.1
0.01,
O.OE


947968 1241 1276 1313 1400
Position (m)
Figure 3-17. Energy flux spectral density calculated for test 19 (Hs= 5.4 m, Tp= 20 s).
A) calculated by NMSE, B) calculated from laboratory data. The bottom panel
shows the reef cross-section geometry and gauge locations as a reference.

55










In the case of an extremely nonlinear wave like test 19, with a significant wave

height of 5.3 m and a peak period of 20 seconds, NMSE is also much better able to

describe the processes occurring (Figure 3-17A). In such a case of violently nonlinear,

plunging breakers, several higher harmonics are developed as the waves shoal (Figure

3-17B). The spectrum widens significantly during this process and even after breaking,

the way the waves dissipate is not limited to breaking. NMSE is able to describe these

processes with good accuracy and can be seen in Figure 3-17. NMSE describes the

development of harmonics and closely resembles the dissipation after breaking found in

the data.

3.5 Linear Wave Model Vs. Nonlinear Wave Model

It appears, based on the outputs of setup and wave height evolution that a linear

wave model is able to correctly describe the wave processes occurring on a reef

system. After more investigation of the physics, it can be seen that there are processes

occurring that a linear wave model ignores and must compensate for in other ways.

It was shown that the wave processes in these laboratory tests become

extremely nonlinear near breaking. Figure 3-18 shows both models' capabilities to

describe the energy flux density in these areas (gauges 7 and 8). It can be seen that the

linear model is not capable of describing the widening of the spectrum that occurs as

energy is transferred from the peak frequency to higher harmonics (Figure 3-18C). The

linear model also predicts that the peak frequency will dominate on the reef top (gauge

8), but measurements conclude that low-frequency energy in the infragravity band

dominates the energy flux spectrum (Figure 3-18D). The nonlinear model does a much











better job of describing the energy flux in higher frequencies by accurately describing

the widening of the spectrum (Figure 3-18A). Only the nonlinear wave model recognizes

higher harmonics and accurately describes the energy found in the infragravity band as

dominating the energy flux density on the reef flat (Figure 3-18B).

Both the linear and nonlinear wave models are in good agreement with the

measured data (Figure 3-19). In fact, the linear wave model is in better agreement with

A B
10210
10 Sensor 7; Sensor 8;
E Hs=4.5m. Hs=2.4 m.

101


LL C
100
0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.05 0.1 0.15 0.2 0.25 0.3
C Freauencv (Hz) D Freauencv (Hz)
100

E 100 10-5 E

Hs= 4.5 m. Hs= 2.4 m. 10-1
X X
L 10_5 10-15 IL


0 0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4
Freauencv (Hzl)auencv (Hz)
- 10-



S--20
2 3 5 6 7
--30
-4
947968 1241 1276 1313 1400
Position (m)
Figure 3-18. Energy flux density modeled compared with measured data from test 19
(Hs= 5.4 m, Tp= 20 s). Blue solid lines show the modeled data and red circles
show the measured data. A) nonlinear wave model at gauge 7, B) nonlinear
wave model at gauge 8, C) linear wave model at gauge 7, D) linear wave at
gauge 8. Below are the reef cross-section geometry and gauge locations as a
reference.










the collected data than the nonlinear model when describing mean water level through

all gauges, except gauge 7, on the reef edge. The disagreement between linear and

nonlinear wave models is not significant, never exceeding 8 cm.

However, the linear wave model's physics are limited to setup and wave heights.

The nonlinear wave model is capable of describing the full spectral evolution (along with

other parameters such as setup and wave heights). This capability is significant

because of the role that infragravity waves on the reef flat play in runup and flooding.


-10

-20 -
S 2 3 5 6 7 8
Q -30 -
-40
947968 1241 1276 1313 1400
Position (m)
Figure 3-19. Mean water level (setup) calculated by the nonlinear wave model in blue,
linear wave model in red, and measured values in black circles. Bottom panel
shows the reef cross-section geometry and gauge locations as a reference.


The presence of infragravity waves can greatly increase the height of runup generated.

Linear wave models do not account for the energy from the peak frequency transferring
58









to low-frequencies, but assumes that it is only dissipated from the peak frequency,

underestimating the energy present on the reef flat.









CHAPTER 4
CONCLUSIONS

4.1 Reef Processes

The wave processes and importance of nonlinearities are considerably different

in the reef environment than a mildly sloping sandy beach owing to the physical

structure of a fringing reef with steep slopes nearshore and a large reef flat with shallow

water. Extensive wave breaking occurs over a reef as waves propagate from deep

water, shoal, and break on a steeply sloping reef over a relatively short distance. Violent

breaking is observed with plunging breakers. In this study, breaking occurs both on the

reef slope and on the reef edge. After initial breaking, waves reform as bores and

propagate across the reef flat towards the beach.

As the waves propagate from deep water and travel over the reef bathymetry,

shoaling begins. As waves shoals, the spectrum immediately widens due to nonlinear

transfers of energy from the peak frequency to higher harmonics and lower

subharmonics. As the waves approach breaking, the shape of the waves change

considerably and become highly nonlinear with significant amounts of energy being

transferred between frequencies. In most cases, the peak of nonlinear transfers

occurred around gauge 6, just before breaking, seen in the bispectra (Figure 3-7). In

this location, energy is transferred from the peak frequency into higher frequencies

developing harmonics. This can be seen in the power density and energy flux spectra.

Investigating these nonlinear interactions with bispectral analysis reveals that the peak

frequency interacts with several higher harmonics. In extremely nonlinear tests, such as









test 19, there are nonlinear interactions noted through the bispectral analysis that the

second harmonic was also interacting nonlinearly with higher harmonics.

During the laboratory experiment, waves were found to be breaking between

gauges 6 and 7. Breaking type was found to be plunging breakers for almost all of the

test cases. After breaking (gauge 7), considerable energy still remained at the peak

frequency. As the broken waves reformed and propagated across the reef flat as bores,

energy was transferred into lower frequencies. On the reef flat, the spectrum was

dominated by infragravity energy. This low-frequency energy increased as the waves

continued to propagate across the reef flat shoreward.

Linear wave transformations were observed during this experiment only in deep

water (gauges 1-3). Waves began to become nonlinear immediately after approaching

the reef slope. As the waves shoaled, energy was transferred from the peak frequency

to higher and lower frequencies developing harmonics and subharmonics. Following

breaking, energy continued to be transferred from the peak frequency into the

infragravity band and was dissipated from higher frequencies first. On the reef flat, after

breaking, low-frequency oscillations dominated. The processes throughout this study

were found to be highly nonlinear with strong nonlinear interactions that are not as

apparent on mildly sloping beaches. These processes are more dominant in the reef

environment and cause wave transformation here, to be, to a large degree,

incomparable to wave transformation over mildly sloping beaches.

4.2 Wave Model Capabilities

Xbeach is an open source, two-dimensional, depth-averaged wave and

circulation model for sediment transport in the nearshore that simulates dune erosion,

61









overwash, and beach transformation. The wave model implemented into Xbeach is a

linear wave model, in which, the only mechanism of dissipation is dissipation due to

breaking. In this study, only the wave module was used, which was not initially

developed for modeling wave propagation over steep slopes such as reefs. This study

was intended to push Xbeach beyond what it was intended for and to evaluate how it

performs. More specifically, to evaluate the abilities of a linear wave model as it relates

to wave propagation over reefs. Specific interest in Xbeach is due to the extensive list of

capabilities of Xbeach (including sediment transport), which were not used in this study,

but of interest for future study.

The linear wave model capabilities are limited to the mean water level

(setup/setdown) and wave height values. Xbeach shows good agreement of these

measured values, which is of no surprise. The wave module found in Xbeach is a

function completely dependent on the dissipation constant, K, which is a function of

three free parameters, a, y, and n. The linear wave model is calibrated to minimize the

errors between the measured and calculated wave heights and mean water levels by

finding optimal values of the parameters. Without a data set for fit, calibrating the model

would be difficult. With a more complete optimization, the fit between the model

prediction and laboratory data in this study could be improved.

Because the only mechanism of dissipation in a linear wave model is dissipation

due to breaking, there is no variation in the spectral flux until breaking occurs (Figure 3-

4). It can be easily seen from the data that there are, in fact, significant variations in the

spectrum long before breaking occurs. The linear wave model is unable to describe the

widening of the spectrum that occurs as waves shoal over the reef before and after

62









breaking. There are no terms in the wave model to account for energy transfer between

modes, but simply that energy can be lost (dissipated) at each mode individually, but

not until breaking (Figure 3-15A). At which time, energy is dissipated proportionally from

all modes. It is easily seen that this is an incorrect description of the true physics of the

reef system.

The predicted values of the linear wave model of Xbeach compares well with the

measured values from the tests of the physical parameters for which it was developed,

namely setup and wave height. If these parameters are the desired outputs for a study

and there exists data to be used in a calibration process, a linear model can be efficient

and effective. However, the wave processes on a reef are known to be nonlinear and

attempting to model them with a linear model is flawed and the underlying physics of a

linear model is lacking in the case of a steeply sloping beach or reef.

NMSE includes an extra term in the wave model that allows energy to be

transferred between modes. As the waves shoal over the reef and nonlinear interactions

become strong, energy is transferred from the peak frequency to higher and lower

frequencies developing higher harmonics and subharmonics (Figure 3-17B). Only a

nonlinear wave model accounts for these nonlinear energy transfers across the

spectrum (Figure 3-17A). Through post-processing, other parameters are attainable,

such as setup using the shallow-water equations (not coupled). In some instances,

Xbeach predicts the setup better than the nonlinear model (Figure 3-19), although the

difference is not significant, never exceeding 8 cm. It can be noted that in the area of

strong nonlinearities during breaking (gauge 7), the nonlinear model does seem to

describe the mean water level more accurately. The overall slightly better fit from the

63









linear wave model could be attributed to two things. Firstly, in the linear wave model, the

wave model is coupled with the shallow-water equations that produce the average water

level values, but with NMSE, the setup is calculated at the end using the shallow-water

equations. Lastly, the linear wave model was calibrated to fit the output values of setup

and wave height to the laboratory data. The accurate prediction of these values is the

ultimate goal of a linear wave model because this is where the physics of a linear wave

model end. However, the nonlinear wave model, NMSE is calibrated based on the

spectrum and is used to correctly describe the full spectral evolution.

The scope of Xbeach is very wide. Here, only the linear wave model was used.

While Xbeach is sufficient for predicting the simple output values of average water level

and wave heights, the capabilities of the linear wave model end here. The effects of

ignored nonlinear interactions can be aliased into other growth/dissipation mechanisms.

The wave processes occurring on a fringing reef are highly nonlinear. In order to

correctly describe the nonlinear transformation of waves over fringing reefs, a nonlinear

wave model is necessary.

4.3 Applications and Future Work

After the experiment that provided data for this study, a second laboratory test

was conducted at the US Army Corps of Engineers, Engineer Research and

Development Center, Coastal and Hydraulics Laboratory in Vicksburg, Mississippi. In

this laboratory experiment, more wave gauges were concentrated in the areas of

interest (just before and during breaking) to give more insight into the complex

processes that occur during shoaling and breaking. Also included were four acoustic

Doppler velocimeters (ADV) and a pulse coherent acoustic Doppler profiler (PC-ADP),

64









all located in the same region. Unlike the data set from the University of Michigan's

wind-wave facility, this data set is more densely instrumented and is also collecting

current data. During this experiment, tests were also run with two different reef slopes,

1:2 and 1:5. Interestingly, the sizeable decrease in significant wave height just before

shoaling (gauge 4) seen in the laboratory study conducted at the University of Michigan

and regarded as a faulty gauge, was also observed during every test run in the flume for

the more recent study, even after several recalibrations of the wave gauges. Further

investigation of this wave height variation should be done using the new data set to

determine if there is a more fundamental physical process related to waves occurring,

or if this is an effect from the flumes themselves. This second set of laboratory data,

including data with different reef slopes, can also be a useful data set to further study

the calibration parameters included in wave models to begin to assign physical meaning

to these parameters that are largely unknown for steep slopes.

There are several possible applications for the nonlinear wave model, NMSE.

Because it has been observed that the low-frequency oscillations dominate the

spectrum on the reef flat, a model that recognizes the energy transfer into the

infragravity band can be used as a tool to study this energy and its role in reef

hydrodynamics. Also, processes that are strongly frequency dependent can be studied

using this frequency dependent model, for example, bottom friction on reefs. Because of

the coral texture itself along with significant growth of organisms on the reef, bottom

friction is a very dominant forcing in a reef system. The two largest influences on reef

hydrodynamics are wave action and bottom friction. The application of a frequency









dependent wave model on frequency dependent processes, such as bottom friction, is

necessary to achieve a better understanding of these hydrodynamics.

It remains that NMSE is a wave model only, solving wave propagation and

transformation without coupling with any other processes. Xbeach is an open-source

model that couples hydrodynamics, sediment transport, and morphodynamics. It was

not intended that this model be used in an application, such as this study, because the

assumption of purely linear waves is invalid in this environment. Integrating the

nonlinear wave model into Xbeach, that is designed to model many other processes,

could validate this model to be used in the context of steep slopes with nonlinear

interactions while simulating complex hydrodynamic and morphodynamic processes

accurately by correctly describing the physics of the system.









APPENDIX A
XBEACH INPUT FILE: params.txt

PARAMS

grid input

nx=86
ny=9
dx=10
dy=5
xori =0
yori=0
al fa=180
posdwn=-l
vardx=0
depfile=bathy.dep

wave input

instat=4
bcfile=testl9.inp
rt=7200
dtbc=4
break=l
gamma=0.9720
alpha=0.7715
n=5

constants

rho=1025
g=9.81

flow input

C=60
1eft=l
right=l

wind

rhoa=1.25
Cd=.002
windv=0










limiters

gammax=5
hmi n=0.01
eps=0.1
umin=0.1
Hwci =0.01

simulation

tstart=0
ti nt=4
tstop=7200

sediment transport

form=l
dico=l
struct=0

morphological updating

morfac=0
wetslp=l
dryslp=2









APPENDIX B
XBEACH INPUT FILE: testl9.inp

Hm0=5.2430
fp=0.05
mai nang=90.
gammajsp=3.3
s=l.
fnyq=0.3









APPENDIX C
XBEACH INPUT FILE: bathy.dep

-35.26 -35.26 -35.26 -35.26 -35.26 -35.26
33.26 -31.26 -29.26 -27.26 -25.26 -23.26
21.26 -19.26 -18.73 -18.2 -17.67 -17.14 -
16.6 -16.07 -15.54 -15.01 -14.48 -13.94 -13.41
-12.88 -12.35 -11.35 -10.35 -9.35 -8.35
-7.35 -6.35 -5.35 -4.35 -3.35 -3.26
-3.26 -3.26 -3.26 -3.26 -3.26 -3.26
-3.26 -3.26 -3.26 -3.26 -3.26 -3.26
-3.26 -3.26 -3.26 -3.26 -3.26 -3.26
-3.26 -3.26 -3.26 -3.26 -3.26 -3.26
-3.26 -3.26 -3.26 -3.26 -3.26 -3.26
-2.43 -1.6 -0.77 0.06 0.89 1.72 2.55 3.38 4.21 5.04
5.87 6.7 7.53 8.36 9.19 10.02 10.85 11.68
12.51 13.34

-35.26 -35.26 -35.26 -35.26 -35.26 -35.26 -
33.26 -31.26 -29.26 -27.26 -25.26 -23.26
21.26 -19.26 -18.73 -18.2 -17.67 -17.14 -
16.6 -16.07 -15.54 -15.01 -14.48 -13.94 -13.41
-12.88 -12.35 -11.35 -10.35 -9.35 -8.35
-7.35 -6.35 -5.35 -4.35 -3.35 -3.26
-3.26 -3.26 -3.26 -3.26 -3.26 -3.26
-3.26 -3.26 -3.26 -3.26 -3.26 -3.26
-3.26 -3.26 -3.26 -3.26 -3.26 -3.26
-3.26 -3.26 -3.26 -3.26 -3.26 -3.26
-3.26 -3.26 -3.26 -3.26 -3.26 -3.26
-2.43 -1.6 -0.77 0.06 0.89 1.72 2.55 3.38 4.21 5.04
5.87 6.7 7.53 8.36 9.19 10.02 10.85 11.68
12.51 13.34


-35.26 -35.26 -35.26
33.26 -31.26 -29.26
21.26 -19.26 -18.73
16.6 -16.07 -15.54 -15.C
-12.88 -12.35 -11.3
-7.35 -6.35 -5.35
-3.26 -3.26 -3.26
-3.26 -3.26 -3.26
-3.26 -3.26 -3.26
-3.26 -3.26 -3.26
-3.26 -3.26 -3.26
-2.43 -1.6 -0.77


-35.26 -35.26 -35.26
-27.26 -25.26 -23.26
-18.2 -17.67 -17.14
)1 -14.48 -13.94 -13.41
5 -10.35 -9.35 -8.35
-4.35 -3.35 -3.26
-3.26 -3.26 -3.26
-3.26 -3.26 -3.26
-3.26 -3.26 -3.26
-3.26 -3.26 -3.26
-3.26 -3.26 -3.26
0.06 0.89 1.72 2.55 3.38 4.21 5.04
70









5.87 6.7 7.53 8.36 9.19 10.02
12.51 13.34


10.85 11.68


-35.26 -35.26 -35.26 -35.26 -35.26 -35.26
33.26 -31.26 -29.26 -27.26 -25.26 -23.26
21.26 -19.26 -18.73 -18.2 -17.67 -17.14
16.6 -16.07 -15.54 -15.01 -14.48 -13.94 -13.41
-12.88 -12.35 -11.35 -10.35 -9.35 -8.35
-7.35 -6.35 -5.35 -4.35 -3.35 -3.26
-3.26 -3.26 -3.26 -3.26 -3.26 -3.26
-3.26 -3.26 -3.26 -3.26 -3.26 -3.26
-3.26 -3.26 -3.26 -3.26 -3.26 -3.26
-3.26 -3.26 -3.26 -3.26 -3.26 -3.26
-3.26 -3.26 -3.26 -3.26 -3.26 -3.26
-2.43 -1.6 -0.77 0.06 0.89 1.72 2.55 3.38 4.21 5.04
5.87 6.7 7.53 8.36 9.19 10.02 10.85 11.68
12.51 13.34

-35.26 -35.26 -35.26 -35.26 -35.26 -35.26 -
33.26 -31.26 -29.26 -27.26 -25.26 -23.26
21.26 -19.26 -18.73 -18.2 -17.67 -17.14 -
16.6 -16.07 -15.54 -15.01 -14.48 -13.94 -13.41
-12.88 -12.35 -11.35 -10.35 -9.35 -8.35
-7.35 -6.35 -5.35 -4.35 -3.35 -3.26
-3.26 -3.26 -3.26 -3.26 -3.26 -3.26
-3.26 -3.26 -3.26 -3.26 -3.26 -3.26
-3.26 -3.26 -3.26 -3.26 -3.26 -3.26
-3.26 -3.26 -3.26 -3.26 -3.26 -3.26
-3.26 -3.26 -3.26 -3.26 -3.26 -3.26
-2.43 -1.6 -0.77 0.06 0.89 1.72 2.55 3.38 4.21 5.04
5.87 6.7 7.53 8.36 9.19 10.02 10.85 11.68
12.51 13.34

-35.26 -35.26 -35.26 -35.26 -35.26 -35.26 -
33.26 -31.26 -29.26 -27.26 -25.26 -23.26
21.26 -19.26 -18.73 -18.2 -17.67 -17.14 -
16.6 -16.07 -15.54 -15.01 -14.48 -13.94 -13.41
-12.88 -12.35 -11.35 -10.35 -9.35 -8.35
-7.35 -6.35 -5.35 -4.35 -3.35 -3.26
-3.26 -3.26 -3.26 -3.26 -3.26 -3.26
-3.26 -3.26 -3.26 -3.26 -3.26 -3.26
-3.26 -3.26 -3.26 -3.26 -3.26 -3.26
-3.26 -3.26 -3.26 -3.26 -3.26 -3.26
-3.26 -3.26 -3.26 -3.26 -3.26 -3.26









-2.43 -1.6 -0.77 0.06 0.89 1.72 2.55 3.38 4.21 5.04
5.87 6.7 7.53 8.36 9.19 10.02 10.85 11.68
12.51 13.34

-35.26 -35.26 -35.26 -35.26 -35.26 -35.26 -
33.26 -31.26 -29.26 -27.26 -25.26 -23.26
21.26 -19.26 -18.73 -18.2 -17.67 -17.14 -
16.6 -16.07 -15.54 -15.01 -14.48 -13.94 -13.41
-12.88 -12.35 -11.35 -10.35 -9.35 -8.35
-7.35 -6.35 -5.35 -4.35 -3.35 -3.26
-3.26 -3.26 -3.26 -3.26 -3.26 -3.26
-3.26 -3.26 -3.26 -3.26 -3.26 -3.26
-3.26 -3.26 -3.26 -3.26 -3.26 -3.26
-3.26 -3.26 -3.26 -3.26 -3.26 -3.26
-3.26 -3.26 -3.26 -3.26 -3.26 -3.26
-2.43 -1.6 -0.77 0.06 0.89 1.72 2.55 3.38 4.21 5.04
5.87 6.7 7.53 8.36 9.19 10.02 10.85 11.68
12.51 13.34

-35.26 -35.26 -35.26 -35.26 -35.26 -35.26 -
33.26 -31.26 -29.26 -27.26 -25.26 -23.26
21.26 -19.26 -18.73 -18.2 -17.67 -17.14 -
16.6 -16.07 -15.54 -15.01 -14.48 -13.94 -13.41
-12.88 -12.35 -11.35 -10.35 -9.35 -8.35
-7.35 -6.35 -5.35 -4.35 -3.35 -3.26
-3.26 -3.26 -3.26 -3.26 -3.26 -3.26
-3.26 -3.26 -3.26 -3.26 -3.26 -3.26
-3.26 -3.26 -3.26 -3.26 -3.26 -3.26
-3.26 -3.26 -3.26 -3.26 -3.26 -3.26
-3.26 -3.26 -3.26 -3.26 -3.26 -3.26
-2.43 -1.6 -0.77 0.06 0.89 1.72 2.55 3.38 4.21 5.04
5.87 6.7 7.53 8.36 9.19 10.02 10.85 11.68
12.51 13.34

-35.26 -35.26 -35.26 -35.26 -35.26 -35.26 -
33.26 -31.26 -29.26 -27.26 -25.26 -23.26
21.26 -19.26 -18.73 -18.2 -17.67 -17.14 -
16.6 -16.07 -15.54 -15.01 -14.48 -13.94 -13.41
-12.88 -12.35 -11.35 -10.35 -9.35 -8.35
-7.35 -6.35 -5.35 -4.35 -3.35 -3.26
-3.26 -3.26 -3.26 -3.26 -3.26 -3.26
-3.26 -3.26 -3.26 -3.26 -3.26 -3.26
-3.26 -3.26 -3.26 -3.26 -3.26 -3.26
-3.26 -3.26 -3.26 -3.26 -3.26 -3.26









-3.26 -3.26 -3.26 -3.26 -3.26 -3.26
-2.43 -1.6 -0.77 0.06 0.89 1.72 2.55 3.38 4.21 5.04
5.87 6.7 7.53 8.36 9.19 10.02 10.85 11.68
12.51 13.34

-35.26 -35.26 -35.26 -35.26 -35.26 -35.26 -
33.26 -31.26 -29.26 -27.26 -25.26 -23.26
21.26 -19.26 -18.73 -18.2 -17.67 -17.14 -
16.6 -16.07 -15.54 -15.01 -14.48 -13.94 -13.41
-12.88 -12.35 -11.35 -10.35 -9.35 -8.35
-7.35 -6.35 -5.35 -4.35 -3.35 -3.26
-3.26 -3.26 -3.26 -3.26 -3.26 -3.26
-3.26 -3.26 -3.26 -3.26 -3.26 -3.26
-3.26 -3.26 -3.26 -3.26 -3.26 -3.26
-3.26 -3.26 -3.26 -3.26 -3.26 -3.26
-3.26 -3.26 -3.26 -3.26 -3.26 -3.26
-2.43 -1.6 -0.77 0.06 0.89 1.72 2.55 3.38 4.21 5.04
5.87 6.7 7.53 8.36 9.19 10.02 10.85 11.68
12.51 13.34









REFERENCES


Agnon Y, Sheremet A. 1997. Stochastic nonlinear shoaling of directional spectra. J.
Fluid Mech. 345:79-99

Alsina JM, Baldock TE. 2007. Improved representation of breaking wave energy
dissipation in parametric wave transformation models. Coastal Eng. 54:765-69

Battjes JA, Janssen JPFM. 1978. Energy loss and set-up due to breaking of random
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Bowen AJ, Inman DL, Simmons VP. 1968 Wave setdown and setup. J. Geophys.Res.
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Dean RG, Dalrymple RD. 1991 Water Waves Mechanics for Engineers and Scientists
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Demirbilek Z, Nwogu OG. 2007. Boussinesq modeling of wave propagation and runup
over fringing coral reefs. Model evaluation report. USACE-ERDC/CHL TR-07-12.

Demirbilek Z, Nwogu OG, Ward DL. 2007. Laboratory study of wind effect on runup
over fringing reefs. Report 1: Data report. USACE-ERDC/CHL TR-07-4.

Elgar S, Guza RT. 1985. Observations of bispectra of shoaling surface gravity waves.
J. Fluid Mech. 161:425-48

Gerritsen F. 1981. Wave attenuation and wave set-up on a coral reef. University of
Hawaii. Look laboratory tech. Rep. No. 48, 416 pp.

Gourlay MR. 1992. Wave set-up, wave run-up and beach water table: interaction
between surf zone hydraulics and groundwater hydraulics. Coastal Eng.
17:93-144.

Gourlay MR. 1994. Wave transformation on a coral reef. Coastal Eng. 23:17-42

Gourlay MR. 1996a. Wave set-up on coral reefs. 1. Set-up and wave-generated flow
on an idealised two dimensional reef. Coastal Eng. 27:161-93

Gourlay MR. 1996b. Wave set-up on coral reefs. 2. Wave set-up on reefs with various
profiles. Coastal Eng. 28:17-55.

Gourlay MR, COLLETER G. 2005. Wave generated flow on coral reefs: an analysis for
two-dimensional horizontal reef-tops with steep faces. Coastal Eng. 52:353-87









Hardy TA, Young IR, Nelson RC, Gourlay MR. 1991. Wave attenuation on an offshore
coral reef. Proc. 22nd Int. Coastal Eng. Conf. Delft. 1990. 330-44.

Hasselman K, Munk W, MacDonald G. 1962. Bispectra of Ocean Waves. Time Series
Analysis. 125-139.

Hocke K, Kampfer N. 2008. Bispectral analysis of the long-term recording of surface
pressure at Jakarta. J. Geophys. Res. 113: D10113

Lowe RJ, Falter JL, Bandet MD, Pawlak G, Atkinson MJ, et al. 2005. Spectral wave
dissipation over a barrier reef. J. Geophys. Res. (Oceans)
110:doi: 10.1029./2004.JC002711

Massel SR, Gourlay MR. 2000. On the modeling of wave breaking and set-up on coral
reefs. Coastal Eng. 39:1-27

Monismith SG. 2007. Hydrodynamics of coral reefs. Annu. Rev. Fluid Mech. 39:37-55

Rao TS, Gabr MM. 1984. An introduction to bispectral analysis and bilinear series
models. Lecture Notes in Statistic 24.

Roberts HH. 1981. Physical processes and sediment flux through reef-lagoon systems.
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Roberts HH, Murray SP, Suhayda JH. 1975. Physical processes in a fringing reef
system. J. Mar. Res. 33:233-60

Roelvink JA. 1993. Dissipation in random wave groups incident on a beach. Coastal
Eng. 19:127-50

Roelvink JA, Reniers A, van Dongeren A, van Theil de Vries J, Lescinkski J, Walstra
DJ. 2007. Xbeach Model Description and Manual. UNESCO-IHE Institute for
Water Education, Delft Hydraulics, Delft University of Technology.

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wave-current flumes. Coastal Eng. 43:149-59

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BIOGRAPHICAL SKETCH

Tracy Martz was born in Johnstown, Pennsylvania in 1986. Shortly after her birth,

she moved to Currituck, NC where she grew up.

After graduating from Currituck County High School, Tracy enrolled at East

Carolina University, where she pursued an undergraduate degree in systems

engineering and was part of the first graduating class of engineers from the university.

While attending East Carolina University, she was fortunate to be accepted as a

summer student intern for the US Army Corps of Engineers Field Research Facility in

Duck, NC where she spent summers being introduced and falling in love with the field of

coastal engineering. From the experiences of the internship, her desire to pursue

higher education in coastal engineering grew.

After obtaining a bachelor's degree in systems engineering, Tracy enrolled at the

University of Florida. Here, she spent two years pursuing and achieving a master's

degree in coastal and oceanographic engineering.

After moving on from UF, Tracy hopes to obtain experience in the field of coastal

engineering and eventually pursue a doctorate degree.





PAGE 1

1 I NVESTIGATION AND MODELING OF NONLINEAR WAVE PROPAGATION OVER FRINGING REEFS By TRACY J. MARTZ A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE D EGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2010

PAGE 2

2 2010 Tracy Martz

PAGE 3

3 I dedicate this t o my Dad

PAGE 4

4 ACKNOWLEDGEMENTS I would like to acknowledge my advisor, Dr. Alexandru Sheremet for all his support, advice, and valuable time along wi th my classmates Ilgar Safak, Shih Feng Su Uriah Gravois and Cihan Sahin for their continuous guidance and support. These individuals have helped me to grow immensely during my time here at the University of Florida. I' d also like to thank Dr. Jane McKee Smith and the US Army Corps of Engineers for their contributions, support, and assistance throughout the study and committee member, Dr. Don Slinn for his input. Lastly, I wish to thank my parents, Mike and Sharon and s ister, Melissa. Their support and understanding during my studies is recognized with love and gratitude. I'd like to especially thank my dad for his exceptional encouragement and appreciation for my work.

PAGE 5

5 TABLE OF CONTENTS page ACKNOWLEDGEMENTS ................................ ................................ ................................ 4 LIS T OF TABLES ................................ ................................ ................................ .............. 7 LIST OF FIGURES ................................ ................................ ................................ ............ 8 ABSTRACT ................................ ................................ ................................ ..................... 10 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ ...... 12 1.1 General Introduction ................................ ................................ ........................... 12 1.2 Review of Past Studies ................................ ................................ ....................... 13 1.3 Study Objective and Approach ................................ ................................ ........... 15 2 METHODOLOGY ................................ ................................ ................................ ..... 16 2.1 Laboratory Study ................................ ................................ ................................ 16 2.2 Data Analysis ................................ ................................ ................................ ...... 23 2.2.1 Power Spectral Density Analysis ................................ ................................ 23 2.2.2 B ispectral Analysis ................................ ................................ ..................... 24 2.3 Wave Modeling ................................ ................................ ................................ ... 27 2.3.1 Xbeach ................................ ................................ ................................ ....... 27 2.3.2 NMSE ................................ ................................ ................................ ......... 29 3 RESULTS ................................ ................................ ................................ ................. 33 3.1 Spectral Analysis ................................ ................................ ................................ 33 3.2 Bispectral Analysis ................................ ................................ ............................. 37 3.3 Xbeach Results ................................ ................................ ................................ .. 44 3.4 NMSE Results ................................ ................................ ................................ .... 54 3.5 Linear Wa ve Model vs. Nonlinear Wave Model ................................ .................. 56 4 CONCLUSIONS ................................ ................................ ................................ ....... 60 4.1 Reef Process es ................................ ................................ ................................ .. 60 4.2 Wave Model Capabilities ................................ ................................ .................... 61 4.3 Applications and Future Work ................................ ................................ ............ 64

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6 APPENDIX A XBEACH INPUT FILE: params.txt ................................ ................................ ............ 67 B XBEACH INPUT FILE: test19.inp ................................ ................................ ............. 69 C XBEACH INPUT FILE: bat hy.dep ................................ ................................ ............. 70 REFERENCES ................................ ................................ ................................ ................ 74 BIOGRAPHICAL SKETCH ................................ ................................ .............................. 76

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7 LIST OF TABLES Table p age 2 1 Summary of wave only test conditions................. ............................... ..............19 3 1 Summary of optimized wave breaking parameters for lin ear model..................45

PAGE 8

8 LIST OF FIGURES Figure p age 2 1 Representative study site: Guam...................................................................... 16 2 2 Modeled reef profile in the wave flume.................................. ........................... 17 2 3 Wave gauge locations........................................................................ ...............18 2 4 Ursell numbers: Tests 26 32 ........................................................................ .....2 1 2 5 Ursell numbers: Tests 15 21 ............................................................................. 22 2 6 Average Ursell numbers................................................................................... .23 3 1 Test 19 energy flux spe ctra during shoaling......... ............................................ 33 3 2 Test 19 energy flux spectra on reef flat........... ................................................. 34 3 3 Energy flux spectra ga uge 8................... ................... ....................................... 35 3 4 Test 19 energy flux........................................................................................... 36 3 5 Bispectra: Gauge 2....................................................................... .................... 38 3 6 Bispectra: Gauge 5........................................................................................... 40 3 7 Bispectra: Gauge 6........................................................................................... 41 3 8 Bispectra: Gauge 8........................................................................................... 43 3 9 Xbeach: Tests 15, 17, & 19 dissipation evolution............................................. 46 3 10 Xbeach: Tests 15 & 19 tota l energy evolution.................................................. 47 3 11 Xbeach: Test 15 w ave force x direction............................ ............................... 48 3 12 Xbeach: Comparison of model calculations and experimental data for test 15...................................................................................................................... 49 3 13 Xbeach: Comparison of model calculations and experimental data for test 17 ...................... ........................... .................................................................. ... 50

PAGE 9

9 3 14 Xbeach: Comparison of model calculations and experimental data for test 18...................................................................................................... ................... 51 3 15 Xbeach: Comparison of e ne rgy flux spectral de nsity modeled and measured....53 3 16 NMSE: Comparison of energy flux spectral density modeled and measured for test 15.............................................................. .................................................... 54 3 17 NMSE: Comparison of energy flux spectral density modeled and measured for test 19.................................................................................... ............................. .55 3 18 Test 19 energy flux at sensors 7 & 8 : Linear model results compared with nonlinear model results ..................... ...................................................... ......... .... 57 3 19 Comparison of m ean water level for nonlinear model, linear m odel, and data.... 58

PAGE 10

10 Abstract of Thesis Presented to the Graduate School o f the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science INVESTIGATION AND MODELING OF NONLINEAR WAVE PROPAGATION OVER FR INGING REEFS By Tracy J. Martz August 2010 Chair: Alexandru Sheremet Major: Coastal and Oceanographic Engineering Extensive wave breaking occurs on the wide reefs located offshore of many Pacific islands, due to their steep slope nearshore. These reefs provide protection to the island; however, energetic wave events may cause flooding as the result of a combination of wave setup, water level position, and wave energy. These processes are well understood for a typical mildly sloping beach; however, they are wildly different over a reef. This study focuses on an effort to better understand and model these processes using laboratory data collected at the University of Michigan's wind wave facility. A 1:64 model of a fringing reef was constructed in the flume. The model is 2 D ver tical and uniform across the tank width. Nine capacitance wire wave gauges were arranged in a cross shore transect to measure water surface elevation. Se veral tests were ran modeling extreme weather conditions associated with tr opical cyclones typical of the Pacific island of Guam. Significant wave heights ranged from 3 6 m with 8 20 seconds spectral peak periods. These tests were repeated at four different still water depths to simulate the effects of tide.

PAGE 11

11 Wave breaking is fou nd to occur both on the reef slope and edge. A consid erable amount of energy remains at the peak frequency after breaking at the reef edg e. However, the spectrum measured on the reef flat is dominated by low frequency oscillations. Af ter breaking, the wave s reform as bores and propagate across the reef flat toward the beach. The low frequency energy continues to grow as the waves propagate shoreward. In t his study, wave propagation is simulated using Xbeach, a public domain two dimensional model, which h as been validated in several case studies for mildly slopin g beaches. Model results for extreme weather conditions over a complex reef system (consisting of ste ep slopes and shallow areas) were used to investigate the model capabilities to simulate wave br eaking, dissipation, wave setup, and runup. Results from the li near wave model, Xbeach, are compared to those obtained using a nonlinear wave model, Nonlinear Mild Slope Equation ( NMSE ) to investigate the role of nonlinear interactions in these processes. It is determined that a linear wave model, such as Xbeach, can be used to accurately model wave height evolution and setup if the wave breaking constants are calibrated correctly. However, the true physics of the energy flux spectral evolution described b y a linear wave model is incorrect. Linear wave models ignore the presence of nonlinear energy transfer from the peak frequency to higher and lower frequencies and the development of harmonics, which are key processes occurring in the reef environment. A n onlinear wave model must be implemented to correctly describe the energy dissipation and transfers that occur as waves shoal and break over a fringing reef.

PAGE 12

12 CHAPTER 1 INTRODUCTION 1.1 General Introduction Fringing reefs are a common type of coral reef d istinguished by growth extending directly from the shoreline. These reefs occur in the tropical regions of the Pacific, Indian, and Atlantic Oceans. In these environments, significant amounts of wave energy can be dissipated through wave breaking and botto m friction processes. The shoreline stability of islands protected by frin ging reef systems is primarily controlled by wave action. An understanding of how waves break on reefs and how they transform as they propagate across the reef flat is necessary to p redict shoreline stability and to d esign structures for these environment s The physical structure of a fringing reef is considerably different than that of mildly sloping beaches. Mildly sloping beaches have previously been the primary focus of studies o n nearshore hydrodynamics. Unlike these beaches, reefs have an abrupt change from deep water to shallow water within a short distance. Along with this steep transition, the bottom surface of a reef is significant ly rougher than that of a sandy bottom beach due to a large amount of growth and reef organisms. The combination of a drastically steeper slope and a significantly larger bottom roughness, leads to significantly different dominant processes in these en vironments. These processes in reef environment s are not yet well understood. Our study aims to better understand these physical processes that occur on steeply sloping reefs and investigate t he abilities of wave models to accurately describe the physics of these processes.

PAGE 13

13 Accurately predicting wave p ropagation on coral reefs is important for several reasons. First, the vertical and horizontal structure of currents on the reef is established by breaking waves, which drives reef circulation. This circulation is responsible for cross reef transport of nu trients, sediment, plankton, and other sea life. The exposure of reefs to wave action is also correlated with the destruction or growth of the coral structure itself (Monismith 2007) Lastly, and most directly studied in this research, is the process of wa ve induced setup nearshore that can cause extreme flooding on islands fring ed with reefs during storm events. All of these should be considered in order to provide a sound engineering and environmental basis for infrastructure and development in reef areas In order to do so, it is necessary to better understand wave processes on reefs. 1.2 Review of Past Studies Reef environments often experience large waves with extremely violent breaking occurri ng at shallow depths. These energetic environment s combined with the locations of such reefs owes to the minimal existence of experimental data and studies Difficulties associated with the physical modeling of reefs in a laboratory lead to a lack of completely accurate laboratory data. Theoretical descriptions ar e even more rare. Without reliable data sets, the advances in modeling such environments are not significant. Much field and theoretical work is still required to understand the complex environment of reef systems. T he process of wave propagation over ree fs is of particular interest due to broadening of wave spectra during rapid shoaling, followed by abrupt breaking. This widening of the spectrum h as been seen in field observations of waves breaking on a

PAGE 14

14 reef and propagating into a lagoon (Gerritsen 19 81; Roberts 19 81; Young 19 89; Hardy et al 19 91). It was concluded t hat the change in spectral energy distribution is influenced by the reef slope steepness, which governs the extent and type of breaking that occurs. Wave attenuation due to breaking was shown to be a function of both the incident wave height and the water depth on the reef flat (Young 19 89; Gourlay 19 94). The incident wave height becomes more important at higher water levels, when breaking is minimal or nonexistent. At lower water levels, when breaking is significant, the water depth is highly influential on wave attenuation. Gourlay has completed a comprehensive set of laboratory studies ( 19 92, 19 94, 19 96a, 19 96b) on reef hydrodynamics, concentrating specifically on wave setup and currents driv en by this setup. He concluded that both were related to wave conditions along with water level. During his studies, Gourlay assumed linear waves, ignoring long wave oscillations such as infragr avity waves. This has been a common assumption on planar beac hes, bu t i t has been shown (Demirbilek & Nwogu 200 7 ) that significant infragravity waves can develop ov er the reef flat. This low frequency energy can cause runup limits observed to be significantly higher than woul d be expected on planar beaches. Modelin g of wave breaking over reefs has been studied only minimally. The breaking dissipation model widely used was developed for mildly sloping beaches (Battjes & Janssen 1978) and is known to underestimate the energy dissipation and overestimate the wave heig h ts on steep beaches and in un saturated surf zones (Alsina & Baldock 2007). Breaker parameters used in the dissipation models have been calibrated in several past studies for data sets which were collected in environments

PAGE 15

15 that show significant variations fr om reef systems ( Battjes & Janssen 1978 ; Thor nton & Guza 1983 ). Appropriate parameters are largely unknown for the reef system 1.3 Study Objectives and Approach The objective of this study is to analyze the laboratory data collected at the University of Michigan to better understand the wave processes that occur o n a fringing reef and how they a ffect wave set up and flooding. This study will use wave modeling as a tool to investigate these processes while testing the performance of the models in the situat ion of a steeply sloping reef profile. S pectral analysis is performed on the water level time series data. This reveals the power spectra and the evolution of the energy flux as the waves propagate. From this, energy dissipation, growth, and transfer can be seen. Based on evidence of the presence of nonlinear interactions, a bispectral analysis is also performed. These results more clearly reveal the energy being transferred between frequencies and the development of higher harmonics. A linear wave model, Xbeach, is used to simulate the wave setup and wave evolution over the reef and Xbeach performance is evaluated A nonlinear wave model, NMSE, is also used and compared to the outputs of Xbeac h. From this, conclusions are made about the importance of non linearities on the wave processes over fringing reefs.

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16 CHAPTER 2 METHODO L OGY 2.1 Laboratory Study To study the physical processes of wave propagation over fringing reefs, we used a data set collected at the University of Michigan's wind wave flume in Ann Arbor, Michigan. This experiment was designed with a two fold objective. Firstly, to quantify the effects of wind on wave runup and secondly, to obtain detailed wave data along a complex fringing reef system consisting of steep slopes and a shallow ree f flat. The reef system modeled in the flume was representative of the fringing reefs found off the southeast coast of Guam (Figure 2 1 A ). Figure 2 1. Re presentative study site. A ) the Pa cific Island of Guam, B ) is the reef studied located off of the south east coast of the island. Labeled are the three main components of the reef: Reef flat, reef edge, and reef slope. (Source: Google Earth) A B

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17 The flume used at the University of M ichigan is 35 m long, 0.7 m wide and 1.6 m high. A computer controlled non absorbing plunger type wavemaker was used to generate irregular sea states with significant wave heights and frequencies representative of both typical and extreme wave conditions as observed at the site. The representative reef profile constructed in the wave flume is created out of polyvinyl chloride (PVC), a smooth and impervious material so that the effects of bottom friction can be ignored. The structure is characterize d by three main regions which will be referred to in this paper as the reef slope, reef edge, and reef flat (Figure 2 1B). The model is 2 dimensional and uniform across the width of the tank. From the shore, t he profile consists of a 1:12 sloped beach followed by a wide reef flat. The region of the reef s lope is characterized by 3 areas with slo pes of 1:10.6, 1:18.8, and 1:5 (Figure 2 2). F igure 2 2. Modeled reef profile built in the wave f lume. Three regions shown: reef slope, reef edge, and reef flat. Sloping sections labeled with slope steepness. 1:10.6 1:5 1:18.8 Reef Flat Reef Slope Reef Edge 1:12 Distance (m)

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18 The reef pr ofile is equipped with 9 capacitance wire wave gauges used to measure water surface elevation in the flume (Figure 2 3) Three are positioned in deep water (1, 2, and 3) three along the reef slope (4, 5, and 6) one on the reef edg e (7) and two along the reef flat (8 and 9). All gauges sampled at 20 Hz for 15 minutes per test condition. Data collection began shortly after waves began to be generated with initially calm water conditions Also in this experiment, a runup gauge was located on the beach slope to quant ify runup. Two anemometers measured the wind speed induced in the flume (Demirbilek et al. 2007) The data sets with wind are not used during this study. Figure 2 3. The water surface elevation is measure d using 9 capacitance wire wave gauges spaced across the reef profile and a c apacitance runup gauge measures runup simultaneously. Runup Gauge #9 #8 #7 #6 #5 #4 #3 #2 #1 Distance (m)

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19 The three wave gauges located offshore were positioned in an array such that they could be used to separate inc ident and reflected waves. The three gauges on the reef slope are positioned in an area where extreme shoaling, wave transformation, and breaking are expected to occur. The remaining three gauges are positioned to help quan tify the amount of setup produced in the reef flat. Preliminary analys is of the raw data shows that the measured significant wave heights at gauge 4 deviate from the consecutive gauges 3 and 5 by about 37%. Because of these drastic errors, th e data collected at gauge 4 are not used in this study. Approximately 80 tests were ru n with combinations of waves and wind. The test conditions are scaled using Froude scaling and summarized in Table 2 1. References to units will be given in Froude scale for the remainder of the study. Significant wave heights were calculated as four ti mes the square root of the frequency integrated spectra calculated at gauge 2 and range from 2.8 m to 5.4 m with spectral peak periods ranging from 8 seconds to 20 seconds. Table 2 1. Summary of wave only test conditions where H s is significant wave heig ht, T p is peak period, h r is still water depth on ree f flat. All units are in meters. Test cases used in this study are shown in yellow. Test Number H s (m) T p (m) h r (m) 15 3.968 8.0 3.264 16 3.328 12.0 3.264 17 4.992 12.0 3.264 18 5.440 16.0 3.264 1 9 5.312 20.0 3.264 20 3.904 10.0 3.264 21 5.248 14.0 3.264 26 3.712 8.0 1.024 27 3.520 10.0 1.024

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20 Table 2 1. Continued Test Number H s (m) T p (m) h r (m) 29 4.544 12.0 1.024 30 4.864 14.0 1.024 31 5.440 16.0 1.024 32 5.056 20.0 1.024 33 3.584 8. 0 0.000 34 2.880 12.0 0.000 35 2.880 12.0 0.000 36 4.352 12.0 0.000 37 4.864 14.0 0.000 38 5.376 16.0 0.000 39 4.928 20.0 0.000 44 2.048 8.0 1.984 45 3.904 8.0 1.984 46 3.776 10.0 1.984 47 3.200 12.0 1.984 48 4.800 12.0 1.984 57 4.928 14.0 1.98 4 58 5.440 16.0 1.984 59 5.248 20.0 1.984 For this study, we only consider data collected in wave on ly tests (i.e. no wind). Also, four different initial water levels were used ranging from 0 m 3.2 m of water on the reef flat in order to represent var iations in mean water level In this study, we will concentrate on the tests ran at the highest wate r level, 3.2 m on the reef flat (yellow box in Table 2 1). The nonlinear wave model used in this study has a limitation based on the Ursell number. The Urs ell number is defined as U r = a k 2 h 3 (2 1)

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21 where a is the wave amplitude k is the wavenumber and h is the local water depth. Because of the quasi G aussian approximation used in the mo del, a threshold of U r <1 .5 was proposed by Agnon & Sheremet ( 1997). It then follows that in very shallow water, where nonlinear ities dominate over dispersion the Ursell number can easily exceed the threshold for the model validity. Figure 2 4 shows the calcul ated Ursell numbers for tests 26 32 wher e the still water depth is 1.024 m on the reef flat. Figure 2 4. Ursell numbers calculated for tests 26 32 at each gauge for varying wave conditions The bottom panel shows the gauge lo cations and reef bathymetry for reference. The dominant variable in the calculation of the Ursell number is the local water depth, h. As the water depth decreases, the Ursell number grows very fast due to the

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22 value being cubed. At the gauges close to the reef edge and on the reef flat, where the water level is very low, the Ursell numbers are ve ry high. Figure 2 5 Ursell numbers calculated for tests 15 21 at each gauge location for varying wave conditions The bottom panel shows the re ef geometry and gauge locations as a reference. As expected, the Ursell values for the shallow water te sts 26 32 with a still water depth of 1.024 m on the reef flat, are an order of magnitude greater than the tests with a still water level of 3.264 m on the reef flat (Figure 2 5) and far exceeded the suggested threshold proposed by Agnon an d Sheremet (1997 ) for the nonlinear wave model. Figure 2 6 shows the average Ursell numbers of all tests at each water level. Because

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23 the deepest water level clearly maintains the lowest Ursell numbers, only these tests (15 21) are used in this study. Figure 2 6. Averag e Ursell numbers for each water lev el. Bottom panel shows the reef geometry and gauge locations as a reference. 2.2 Data Analysis The collected time series data were analyzed to extract quan tities of interest. This chapter presents two types of data an alysis applied to the raw data. Spectral and bispectral analyses are performed to determine trends and variations in the data. 2.2.1 Power Spectral Density Analysis The energy spectral density describes how the variance (energy) of the water surface time series is distributed in the frequency domain. This is most often referred to

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24 as th e power spectral density (PSD). Most commonly, the PSD P (f), is described as the Fourier transform of the autocovariance function, R( ), if the signal can be treated as a s tationary random process, where is some fixed length of time: P ( f ) = R ( ) e # 2 $ if #% % & d (2 2 ) R ( ) = [( x ( t ) # x #$ $ % )( x ( t + ) # x )] dt (2 3 ) where is t he mean of the time series and asterisk denotes the complex conjugate. This ensures that the autocorrelation is positive definite As the averaging time interval, T, approaches the ensemble average of the average periodogram approaches the PSD: E F ( f ( t )) 2 T # $ % % & ( ( ) P ( f ) (2 4 ) The p ower in a given frequency band [ f 1 f 2 ] can be calculated as: P = P ( f ) df f 1 f 2 (2 5 ) In this study, power spectra were calculated af ter the first 100 seconds of data were truncated to allow waves to propagate through the entire array of gauges. The frequency vector has a lower limit of 0.005 Hz and an upper limit of 0.4 Hz. Data is divided into 512 records in a sequence. Spectra we re e stimated from zero meaned, H anning windowed wave records with band averaging. The resulting resolution bandwidth is 0 .0049 Hz and spectral estimates have 61 degrees of freedom. 2.2.2 Bispectral Analysis S pectral analysis approximates the time series of se a surface elevation to be the linear superposition of statistically uncorrelated waves. In shallow water, different

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25 spectral components interact with each other i.e., nonlinearities become significant. Increasing steepness of waves in the nearshore and th e resulting breaking are evidence of these nonlinearities When these interactions occur, the linear spectral analysis is incapable of completely characterizing the variability of the signal. For some types of nonlinear data, the second order spectral anal ysis properties can be similar to those of a linear time series because this type of linear analysis is not capable of detecting de viations from linearity and Gau ssianity (Rao & Gabr 1984). More specifically, the power spectral density lacks phase informat ion. For cases in which nonlinearities are expected to be significant, higher order spectral methods (e.g., bispectral analysis) are used. The fast Fourier transform ( FFT ) amplitude can give useful hints of nonlinearity, but the bispectrum gives much firme r proof of the presence of nonlinear th ree wave interactions (Hocke & Kampfer 2008). If # (t) is a stationary random function of ti me, the s pectrum P (f ) and bispectrum B(f 1 ,f 2 ) of # (t) are defined respectively as the Fourier transforms of the mean second a nd third order products: P ( f ) = R "# # $ ( % ) e 2 & if % d % (2 6 ) where R ( ) = E # ( t ) # ( t + ) } { (2 7 ) B ( f 1 f 2 ) = S "# # $ $ ( % 1 % 2 ) e 2 & if 1 % 1 2 & if 2 % 2 d % 1 d % 2 (2 8 ) where S ( 1 2 ) = E # ( t ) # ( t + 1 ) # ( t + 2 ) } { (2 9 ) where E{} is the expected value operator. The inverse relations to ( 2 6) and (2 8 ) are

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26 R ( ) = P #$ $ % ( f ) e 2 & if df (2 10 ) S ( 1 2 ) = B #$ $ % % ( f 1 f 2 ) e 2 & if 1 1 + 2 & if 2 2 df 1 df 2 (2 11 ) For real # (t) P ( f ) = P ( f ) (2 12 ) B ( f 1 f 2 ) = B ( f 1 f 2 ) (2 1 3 ) From the stationarity of # (t) the known symmetry relations R ( ) = R ( # ) (2 14 ) S ( 1 2 ) = S ( 2 1 ) = S ( # 2 1 # 2 ) = S ( 1 # 2 # 2 ) = S ( # 1 2 # 1 ) = S ( 2 # 1 # 1 ) (2 15 ) follow immediately. In terms of spectra and bispectra, (2 12) and (2 13 ) become P ( f ) = P ( f ) (2 16 ) B ( f 1 f 2 ) = B ( f 2 f 1 ) = B ( f 2 f 1 f 2 ) = B ( f 1 f 2 f 2 ) = B ( f 1 f 2 f 1 ) = B ( f 2 f 1 f 1 ) (2 17 ) From (2 12), (2 13), (2 16), and (2 17 ) it follows that the spectrum is re al and is determined by its value on a half line, whereas the bispectrum is determined by its values in an octant. In this study, the bispectrum is estimated using the direct (FFT based) method with an FFT length of 512. Zero mean i nput data are segmented with 255 samples per segment with zero overlapping. Bispectral estimates are averaged across records and a frequency domain smoother is applied (Rao Gabr optimal window) with side lengths of 1. The resulting bandw idth is 0.002 Hz. Bispectra are then norma lized by a product of the signal's spectrum.

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27 2.3 Wave Modeling In this study, t he laboratory data were modeled using two different models: Xbeach (Roelvink et al. 2007 ) and NMSE (Agnon & Sheremet 1997 ). The intent and scope of applicability of the two models are very different. The capabilities of the models were investigated and compared. 2.3.1 Xbeach Xbeach is a public domain model ( http://www.xbeach.org ) that was developed with funding and support by the US Army C orps of Engineers, by a consortium of UNESCO IHE, Delft Hydraulics, Delft University of Technology, and the University of Miami. It is a two dimensional model for wave propagation, sediment transport, and morphological changes of the nearshore, beaches, du nes, and backbarrier during storms. Xbeach was originally designed to cope with extreme conditions such as those encountered during hurricanes with the ability to test morphological modeling concepts of dune erosion, overwashing, and breaching. The model h as been validated against a number of analytical and laboratory tests, both hydrodynamic and morphodynamic, but to date, all test beds have been on mildly sloping beaches. Due to generally short length scales in terms of wave lengths and the probability o f supercritical flow, first order upwind is the main numerical scheme implemented in combination with a staggered grid. Xbeach uses an explicit model scheme with an automatic time step using the Courant criterion. The major function implemented in Xbeach and used for this study is t he wave action equat ion solver The wave forcing is o btained from a time dependent version of the wave action balance. Here, the frequency spectrum is represented by a single mean

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28 frequency, but the directional distribution of t he action density is taken into account. The wave action balance is given by: A t + c x A x + c y A y + c # A # = $ D % (2 18 ) The formulation for the total wave dissipation according to Roelvink (1993) is : D = 2 f rep E w Q b (2 19 ) where a is the wave amplitude and f rep is a representative intrinsic f requency. Here, E w = 1 8 gH rms 2 (2 20 ) The fraction of wave breaking is given by: Q b = min 1 e H rms # h $ % & ( ) n 1 $ % & & ( ) ) (2 2 1 ) Assuming unidirectional ity and stationarity (2 18 ) becomes dF j dx = # j F j (2 22 ) where = 2 # f rep Q b and depends on 3 free calibration paramete rs: $ % and n Here, it should be noted that the only mechanism of dissipation is due to breaking and depends completely on the value of & Currents and water level are computed using depth averaged and shortwave averaged shallow water equations: u t + u u x + v u y # fv # v h 2 u x 2 + 2 u y 2 $ % & ( ) = sx + h # bx + h # g ", x + F x + h (2 23 )

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29 v t + u v x + v v y + fu # v h 2 v x 2 + 2 v y 2 $ % & ( ) = sy + h # by + h # g ", y + F y + h (2 24 ) "# dt + hu x + hv y = 0 (2 25 ) Here, h is th e water depth, u and v are velocities in the x and y direction, bx by are the bed shear stresses, sx sy are the wind stresses, g is the acceleration due to gravity, is the water level and F x F y are the wave induced stresses. Again, assuming unidir ectionality and stationarity, the shallow water equations reduce to: u du dx = g d # dx + S xx $ h (2 26 ) where S xx is the radiation stress. Here, the shallow water equation is coupled with the wave action balance. In our study, these are the important equations. However, Xbeac h is capable of modeling many other processes such as directional waves and currents, sediment transport, bottom updating, and multiple sediment classes. Xbeach was chos en for this study because of its broad range of applications How ever, during our study we focus on the linear wave model implemented in Xbeach coupled with the shallow water equation s to calculate wave setup. 2.3.2 Nonlinear Mild Slope Equation (NMSE) Agnon and Sheremet (1997) developed a stochastic wave shoaling model based on the mild slope equation. Regarding the system as stochastic, evolution is described by a hierarchy of equations for the statistical moments or the Fourier space cumulants. dF j dx = 2 # j F j + W ( ) j p q # b j b p b q { } $ 2 W ( ) j $ p q # b j b p b q { } [ ] p q > 0 % (2 27 )

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30 where F j is the average modal energy flux, z { } is the imaginary part of the com plex number z, W is the interact iv e coefficient (see equation 2 28 ), and ( is the d elta function (see equation 2 29 ). The interaction kernel is defined by Agnon and Sheremet (1997) as W j p q = 1 8 g j p q C j C p C q ( ) 1 2 2 k p k q + p j k q 2 + q j k p 2 + p 2 q 2 # j 2 p q $ % & ( ) (2 28 ) and the delta function is defined as j p q = 1 j p # q = 0 j p = q 0 otherwise $ % & j p q = # j # p $ # q (2 29 ) The triple products on the right hand side of equation (2 27 ) can be solved leading to the closure problem. That is, the average of double products b j b j (spectrum) depends on the average triple products b j b p b q (bispectrum), which depends on the a verage quadruple products b u b v b p b q (trispectrum) and so on. This leads to an infinite set of equations to be solved This can be dealt with by using assumptions that discard all of the cumulants that are of a higher order than a specified value. Commonly, the Gau ssian approximation is assumed which, in general, for a Gaussian wave field, odd order spectra cancel exactly and all even o rder spectra are expressed in terms of the second order spectrum This is a poor assumption particularly for the shoaling processes which are characterized by a very fast evolution. Shallow water waves show very strong phase coupled Fourier modes and cannot be fully described with this assumption.

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31 Instead, NMSE uses the quasi Gaussian approximation which assumes, due to phase corr elations created by nonlinearities, the wave field deviates slightly from Gaussianity, and modes are n o longer "exactly" uncorrelated, i.e. (2 30 ) where ) is the small parameter of the problem (e.g., wave slope), leaving the equation for the spectrum as dF j dx = 2 # j F j + W ( ) j p q # b j b p b q { } $ 2 W ( ) j $ p q # b j b p b q { } [ ] p q > 0 % +O( ) ). (2 31 ) The first term on the right hand side of equation (2 31 ) allows for dissipation due only to breaking, which depends completely on the va lue & T he second term allows energy to be transferred from mode j to modes p and q. Unlike the linear wave model implemented in Xbeach, where the o nly mechanism effecting t he energy flux is dissipation due to breaking, NMSE allows also f or energy to be transferred between frequencies Unlike Xbeach, NMSE is strictly a wave model, meanin g it is not designed to compute currents sediment transport, etc. However, a simple calculation can be made during post processing to calculate the water level Assuming unidirectionality and stationarity, the shallow water equation is described as d dx = 1 # gh S xx (2 32 ) b u b q = b u 2 uq + O ( # ), b u b v b p = O ( # ), b u b v b p b q = b u 2 b v 2 up vq + b u 2 b v 2 uq vp + O ( # ).

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32 where and S xx rep resent the along flume (cross shore) water level and radiation stresses, respectively. Since its devel opment NMSE has been corrected for application on steep slopes (Agnon & Sheremet 1997).

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33 CHAPTER 3 RESULTS 3.1 Spectral Ana lysis S pectral analysis was performed on the data revealing commonalities between all test conditions. There remains a clearly defined and constant peak frequency through all gauges until the point of breaking which occurred most often between gauges 6 a nd 7. It can be seen that after breaking (gauge 7), there is still a considerable amount of energy remaining at the initial peak frequency. By gauge 8, this energ y is mostly dissipated and the reef flat is now dominated by low frequency oscillations. As th e waves continue t o propagate shoreward, this low frequency energy continues t o grow as it is being transferred from higher frequencies. Figure 3 1. The top panel shows the power spectrum at gauges 2, 5, and 6 for test 19 ( H s =5 .3, T p =20 s). Bottom panel shows reef profile and gauge locations.

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34 As waves propagate from deep water over the reef slope and begin to shoal, there is evidence of nonlinear transfers and dissipation from the peak frequency. Energy is being transferred from the peak frequenc y into higher and lower frequencies leading to a widening of the spectrum that occurs as the waves steepen and propagate shoreward (Figure 3 1). Energy is distributed over a wider region surrounding the peak frequency as the waves shoal due to energy from the peak frequency being transferred to higher a nd lower frequencies. In test 19 (Figure 3 1), it can be seen that there is more energy present at frequencie s greater than approximately 0.08 Hz and less than 0.03 Hz after shoaling begins. This widening was found to occur in all test cases. Figure 3 2. The top panel shows th e power spectrum at gauges across the reef flat for test 19 (H s =5.2m, T p =20s) The bottom panel shows t he cross shore reef profile and gauge locations as a reference.

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35 After t he waves break near the reef edge, waves reform and continue to propagate across the reef flat as bores. Significant energy remains in the infragravity band over the reef flat (Figure 3 2). Between gauges 7 and 8, the spectrum begins to be dominated by low frequency oscillations This was found to occur du r ing a ll test ca ses and can be seen in Figure 3 3. Figure 3 3. The top panel sh ows the power spectra at gauge 9 for each of 4 tests: 15, 17, 19, and 21. The bottom panel shows the cro ss shore reef pro file and gauge locations as a reference. As waves propagate into shallow water and begin to shoal followed by breaking, energy is not only dissipated from the peak frequency, but energy increases are also seen in other areas of the spe ctrum. These incr eases are attributed to energy transfer

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36 from the peak frequency to higher harmonics and lower subharmonic s The development of higher harmonics and increased energy in the infragravity band can be seen in the energ y flux evolution (Figure 3 4). Figur e 3 4. The top panel shows the energy flux d ensity (power x group velocity) evolution of test 19. The bottom panel shows t he cross shore reef profile and gauge locations as a reference. In F igure 3 4 the widening of the spectrum can be seen when the waves shoal as they propagate over the reef slope. After the waves break between gauges 6 and 7, energy is dissipated rapidly, first from the higher harmonics. Beginning at gauge 8, the spectrum is dominated by infragravity energy.

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37 This spectra l analysis indicates the presence of nonlinear interactions especially in cases of larger, stronger, and longer waves. To further investiga te these nonlinear properties, bispectral ana l ysis was performed. 3.2 Bispectral Analysis This section describes the bispectra of shoaling waves observed during the laboratory test. In particular, two tests will be described: t est 15 and test 19. Test 15 was performed with the smallest and shortest wave of the test conditions while test 19 was performed with the largest and longe st. Test 15 had a significant wave height of 3.9 m and a peak period of 8 seconds while test 19 had a significant wave height of 5.3 m and a peak period of 20 seconds. The bispectral evolution at gauges 2, 5, 6, and 8 are studied. Figure 3 5 shows the bis pectra from the two tests at gauge 2, an offshore deepwater gauge. The distinct peak in both cases occurs at the peak frequency showing that the peak frequency is interacting with itself producing a second harmonic of frequency f p +f p While the dominant in teraction occurring in test 15 (Figure 3 5A) is between the peak frequency and itself, developing a second harmonic of frequency 0.25 Hz, there is also a smaller second peak visible showing that the peak frequency is also interacting with low frequency inf ragravity energy. While peaks in the bispectrum oc cur in these two distinct frequency pairs the peak frequency appears to be interacting, to some degree, with energy of all higher frequencies. Interactions at gauge 2 for test 19 (Figure 3 5B) are more pro nounced than the interactions occurring with the smaller and shorter wave in test 15. The strongest peak represented in the bispectrum is creating a second harmonic of frequency 0.1 Hz. It can also be seen that there is a small peak

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38 Figure 3 5. Bi spectra and spectra at gauge 2. A ) shows data from test 15 (H s =3.9 m, T p =8 s) and B ) shows data from test 19 (H s =5.3 m, T p =20 s). Top panels are showing the bispectra and bottom panels are showing the power spectra occurring between this harmonic an d the peak frequency suggesting that a third h armonic is also being generated through the interactions between the peak frequency, A B

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39 f p and the second harmonic, f 2 at a frequency of f p +f 2 =0.15 Hz. Also present in the b ispectrum calculated at gauge 2 is an interaction between the peak frequency and infragravity energy Figure 3 6 shows bispectra of tests 15 and 19 again, calculated at gauge 5, located on the reef slope. At gauge 5, shoaling has begun, but waves have not yet begun to break. Waves are changi ng shape considerable with increased peakedness. Test 15 (Figure 3 6A ) shows little change from the bispectrum at gauge 2 but the peaks are suggesting the development of a second harmonic, and the interactions between the peak frequency and infragravity e nergy are more developed at gauge 5 as the waves are shoaling. A slight peak can be seen between the second harmonic (f 2 =0.25 Hz) and the peak frequency (f p =0.125 Hz) creating a third harmonic at 0.375 Hz (f p +f 2 ) While clear peaks occur at these specific frequencies, the bispectrum reveals that the peak frequency is interacting with higher freque ncies throughout the entire spectrum. Figure 3 6B shows that the peak frequency in test 19 continues to interact strongly with itself transferring energy into the second harmonic at 0.1 Hz (f p +f p ) At gaug e 5, shoaling has begun, strong ly increasing nonlinear interactions. In this figure, interactions between the peak frequency and the entire spectrum are seen indicated by the array of increased values seen across t he spectrum at the peak frequency The second harmonic (f 2 =0.1 Hz) has began to interact weakly with higher harmonics as well indicated by the slightly increased values found across the spectrum at f 2 =0.1 Hz I nteractions are increasingly visible between the infragrav ity band and the peak frequency, as well as with other higher freque ncies developing interactions with the infragravity band.

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40 Figure 3 6. Bispectra and spectra at gauge 5. A ) shows data from test 15 (H s =3.9 m, T p =8 s) and B) shows data from test 19 (H s =5.3 m, T p =20 s). Top panels are showing the bispectra and bottom panels are showing the power spectra Figure 3 7 shows the bispectra for tests 15 and 19 calculated at gauge 6 located on the reef slope This is the gauge located just b ef ore the reef flat. In test 15 (F igure 3 7A), strong interactions remain between the peak frequency and itself, transferring A B

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41 Figure 3 7. Bispectra and spectra at gauge 6. A ) shows data from test 15 (H s =3.9 m, T p =8 s) and B) shows data from test 19 (H s =5.3 m, T p =20 s). Top panels are showing the bispectra and bottom panels are showing the power spectra energy into the second harmonic. A second peak between the second harmonic (f 2 =0.25 Hz) and the peak frequency (f p =0.12 5 Hz) has become more distin ct, generating a third harmonic, while the peak continues to interact with higher A B

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42 frequencies. Also, interactions are seen between the peak and second harmonics with t he infragravity band In test 19 (F igu re 3 7B), there are distinct higher harmonics inter acting with the peak frequency, but it's at gauge 6 that the interactions between the second harmo nic and higher harmonics become most distinguishable. The infragravity interactions between higher harmonics a re also developed Figure 3 8 shows the bispect ra calculated at gauge 8, on top of the reef flat. Here, the waves have broken. In test 15 (Figure 3 8A), the interactions are somewhat random and trends are difficult to distinguish. S trongest visible interactions appear to occur at the very low frequenci es and in the infragravity band, surrounded by noise due to very low energy levels In test 19 (Figure 3 8B), there remain some weak interactions between the peak and higher harmonics; however, the interactions between the infragravity band and the harmoni cs dominate on the reef flat. There are no longer signs of interactions between the second harmonic and higher harmonics. From the bispectral analysis it is shown that there are significant nonlinear interactions occurring between harmonics and subharmoni cs as the waves shoal across the reef slope and break. The larger and longer wave s (test 1 9) show more distinct signs of the development of harmonics and the interactions between harmonics than the smaller and shorter wave s (test 15). Test 19 was more of a plunging type breaker compared to test 15, which was more of a rolling breaker. It can be determined through this analysis that nonlinear interactions are impor tant in the reef environment, regardless of the wave conditions. However, the extent to which t he nonlinearities interact differs with incident wave conditions.

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43 Figure 3 8. Bispectra and spectra at gauge 8. A ) shows data from test 15 (H s =3.9 m, T p =8 s) and B) shows data from test 19 (H s =5.3 m, T p =20 s). Top panels are showing the bispectra a nd bottom panels are showing the power spectra B A

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44 3.3 Xbeach Results Xbeach was run to simulate flume tests 15 21 (Table 2 1) Grid cell sizes were 10 m in the cross shore direction and 5 m in the alongshore direction. All capabilities except wave propag ation and dissipation due to breaking were ignored. A Joint North Sea Wave Analysis Project (JONSWAP) spectral shape was used with peak enhancement factor % = 3.3 using the random phase method to input wave conditions. Roelvink (1993) was used as the breaker dissipation model which include s three free calibration parameters : % $ and n, where % controls the fraction of breaking waves, $ controls the level of energy dissipation in a breaker, and n is a third free parameter. The parameter $ is totally unknown for steep slopes, but is generally accepted to be O(1). Parameteriz ation within wave models is very limited as it relates to reefs and steep slopes. Finding a n optimal combination of these three free parameters will yield the most accurate model results. These optimal values differ for each wave condition and every experi mental data set. The optimal values of % and $ were determined by minimizing the normalized root mean squared error between measured and computed wave height and spectra while the free parameter n was held constant at n=5, the default value, to minimize the degrees of fre edom. A more accurate optimization process would allow n to range from 0 to 10. Optimal values for % and $ were obtai ned and used for each model run and summarized in Table 3 1. Xbeach requires parameters to be set for the simulation in a file named param s.txt (Appendix A), which signals the model to also look for a wave input file with

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45 the extension .inp (Appendix B) and a bathymetry file with the extension .dep (Appendix C). Table 3 1. Summary of optimized breaking parameters for each test wave conditi on along with iterations necessary in optimization process and rms error associated with parameters. Wave Conditions H s = 3.9m T p =8s H s =3.9m T p =10s H s =3.3m T p =12s H s =4.9m T p =12s H s =5.2m T p =14s H s =5.4m T p =16s H s =5.3m T p =20s Iterations 41 81 93 56 112 55 60 Error 0 .0313 0 .0387 0 .0346 0 .0572 0 .0381 0 .0430 0 .0380 $ 0 .6953 0 .6049 0 .6168 0 .6438 0 .6422 0 .6835 0 .7715 % 0 .6895 0 .6003 0 .5450 0 .7572 0 .8035 0 .8198 0 .9720 Figure 3 9 shows the model outputs for energy dissipation. Across the x axis is the cross s hore distance and along the y axis is the time in seconds that spans for a 2 hour simulation run. The only mechanism of dissipation acting in these model runs is dissipation due to wave breaking, so these plots show the areas of wave breaking. It can be se en that in test 15 (Figure 3 9A) where the significant wave height is 3.9 m and the peak period is 8 s, the waves begin to break much later and with a lower intensity than test 19 (Figure 3 9B) Test 19, with a significant wave height of 5.3 m and a peak period of 20 s begins to shoal and break slightly farther offshore and in deeper water than test 15 and continues breaking as waves propagate up to the reef flat. Because of the small zone in which breaking is assumed to occur, and because a linear wave mo del assumes that breaking is the only mechanism of dissipation, it can be assumed that the only change in the spectrum should appear in areas w h ere breaking occurs,

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46 which is, at most, between gauges 5 and 8. Elsewhere, there is assumed to be no spectral en ergy flux evolution. Figure 3 9. D issipation due to breaking with cross shore distance as it evolves in time. A ) test 15 (H s = 3.9 m T p = 8 s ), B) test 19 (H s = 5.3 m T p = 20 s ). Below is the cross shore reef profile as a reference. Figure 3 10 shows the evolution of total energy across th e reef profile throughout the 2 hour simulation. There is no loss or transfer of energy accounted for until dissipation due to b reaking occurs at gauges 5 and 6. It can also be seen (Figure 3 10 A) that in test 1 5, with a significant wave height of 3.9 m and a peak period of 8 s, after the breaking point, very little to no energy remains on the reef flat especially landward of gauge 8. However, in test 19 (Figure 3 10B), where there is a significant wave height of 5.3 m and a peak period of 20 s, there remains significant wave energy after breaking and continuing all the way across the reef flat. Test 19 also shows a A B A

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47 significantly higher amount of energy just before breaking indicating that shoaling just before bre aking is significantly greater in this case. Figure 3 10. Total energy as a function of cros s shore distance throughout the simulation time of 2 hours. A) test 15 (H s = 3.9 m, T p = 8 s B ) test 19 (H s = 5.3 m T p = 20 s). The bottom panels show the cross shore profile as a reference. Wave force was calculated using the radiation stress tensors. Figure 3 11 shows the wave force created in the cross shore direction by the waves as they propagate towards the sh ore. It is seen that the wave forces are strongly dependent on the bathymetry of the reef. In the regions of steeper slopes, with more rapid shoaling, the wave force increases until breaking. It can be seen that the force is positive almost consistently ev erywhere that there is no dissipation. In the breaking regions, mainly at the reef edge and at the shoreline, the wave forces become negative and strong. B

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48 Figure 3 11. Test 15 (H s = 3.9 m, T p = 8 s) wave force in the x direction wi th cross shore position through time. Below is the reef profile as a reference. The wave model in Xbeach has been calibrated specifically for setup and wa ve height. Because the model is depth averaged, these are typically the desired parameter outputs aside from morphological updating, which was not used in this application. By changing the three free calibration para meters, % $ and n, the model can be calibrated to match measured data. Figure 3 12 shows the setup and wave height evolution calculated by Xbeach compared to the measured data in the wave flume for test 15 (H s = 3.9 m, T p = 8 s). The calculations are in g ood agreement for the region of setdown, with a small underestimate of the setup (Figure 3 12A). It is also noted that where the wave forcings were seen to be strongly neg ative, is also where the 0.08 m setdown occurs.

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49 Figure 3 12. Test 15 (H s = 3.9 m, T p = 8 s). Comp arison between the measured and calculated A) setup, and B) H rms The predicted wave height evolution is not as closely matched to the measured data as the setup/setdown, but has disagreements mainly in the region of shoaling at gauges 5 and 6 (Figure 3 12B). Gauge 4 has been assumed to be a malfunctioning gauge, explaining the significantly low wave height measured at this gauge and should be ignored. Figure 3 13 shows the comparison between the predicted values by Xbeach and t he measured values from the laboratory test of setup and root mean squared wave height evolution for test 17 with a significant wave height calculated as 4.9 m and a A B

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50 peak period of 12 seconds. In this test, setdown is predicted with very good agreement (Fi gure 3 13A). The setup is also closely modeled with a slight over prediction of the setup on the reef flat possibly due to an over prediction of wave heights prior to breaking. The wave he ight evolution is again over predicted, especially in the region of shoaling, with an obvious outlier at gauge 4 due to a gauge malfunction (Figure 3 13B). The data shows almost no wave height growth as the waves shoal over the reef until just before breaking at gauge 6, however Xbeach begins to predict shoaling almost im mediately shoreward of the deep water gauges where waves begin to propagate over the reef bathymetry. Figure 3 13. Test 17 (H s = 4.9 m, T p = 12 s). Comparison between the measured and calculated A) setup and B) H rms A B

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51 Test 19 had a significant w ave height of 5.3 m and a peak period of 20 seconds. Xbeach predicted the setup and setdown very accurately for this test case (Figure 3 14A). The wave height evolution was again over predicted, but still reasonably modeled, excluding the cal culated wave h eight at gauge 4 (Figure 3 14B). Figure 3 14. Test 19 (H s = 5.4 m, T p = 20 s). Comp arison between the measured and calculated A) se tup, and B) H rms Overall, although Xbeach uses a l inear wave model, it does a reasonable job of predicting the s etup, setdown, and w ave height evolution over reefs. This is not entirely A B

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52 surprising. Recall that when assuming unidirectionality and stationery, Xbeach calculates waves as: dF j dx = # j F j (3 1) Because the wave energy is entirely dependent on the dissipatio n rat e, & where & is a function of three free parameters, % $ and n. These three calibration parameters were chosen to fit lab runs based on these simple outputs. In this case, as previously mentioned, n was held at a constant value of 5 for simplicity while % and $ were optimized for best fit to measured data. This fit could be made more accurate still by allowing the calibration parameter, n, to range from 1 to 10. In effect, the calibration parameters in Xbeach were used to force a good fit from the m odel, with little physical meaning, and could be done with any data set to accurately model the wave processes While Xbeach's linear wave module is capable of outputting accurate setup and wave height calculations when calibrat ed correctly, the underlyin g nonlinear physics of the process is ignored. The only mechanism of dissipation that a linear wave model recognizes is dissipation due to breaking. Therefore, up until the breaking point, there is no change in the energy flux. Even as waves shoal, energy flux remains the same. At the breaking point, energy is dissipated from all frequencies (Figure 3 15A). It can be seen (Figure 3 15B) that in the true process, the energy flux density begins to vary when waves begin to shoal over the reef bathymetry. The s pectrum widens and energy is transferred to higher frequencies. Ju st before breaking, as the wave shape changes drastically by steepening and nonlinear interactions are strong, higher harmonics are

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53 created and energy is transferred into the infragravity b and. After breaking, energy is dissipated from higher frequencies first. Linear wave models do not recognize any of t hese processes, but concentrate on accurate calculations of setup and wave height while taking into account d issipation due to breaking Figure 3 15. Energy flux evolution for test 19 (H s = 5.4 m, T p = 20 s) A) linear wave model, B ) data from the laboratory experiment The bottom panel shows th e reef cross section and gauge locations as a reference.

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54 3.4 NMSE Results Unlike Xbeach, the nonlinear wave m odel, NMSE, allows for energy transfer across the spectrum ( between different frequencies ) While this difference will not be seen with simple plots of the predicted setup and wave height values, a drastic difference can be seen when compar ing the energy flux spectral densities. By allowing energy to transfer to higher and lower frequencies NMSE is able to describe the development of higher harmonics and energy tra nsfer to the infragravity band which are key characteristics of the laborat ory data Although test 15 has a Figure 3 16. E nergy flux spectral density for test 15 (H s = 3.9 m, T p = 8 s). A) calculated by NMSE, B) calculated from laboratory data. The bottom panel shows the reef cross section geometry and gauge locations as a re ference.

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55 significant wave height of only 3.9 m, a peak period of 8 seconds, and is not extraordinarily nonlinear compared to other test runs (refer to Ursell numbers in Figure 2 5) there is still energy transferred a nd higher harmonics developed (Figure 3 16B). NMSE is able to describe the development of higher harmonics and the growth of the infragravity band. Because of the extra term s in NMSE, it is able to describe this nonlinearity (Figure 3 16A) and to accurately describe the complete physics of the wave processes occurring in the reef environment. Figure 3 17 E nergy flux spectral density calculated for test 19 (H s = 5.4 m, T p = 20 s). A) calculated by NMSE, B) calculated from laboratory data. The bottom panel shows the reef cross section geome try and gauge locations as a reference.

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56 In the case of an extremely nonlinear wave like test 19, with a significant wave height of 5.3 m and a peak period of 20 seconds, NMSE is also much better able to describe the processes occurring (Figure 3 17A). In such a case of violently nonlinear, plunging breakers, several higher harmonics are developed as the waves shoal (Figure 3 17B). The spectrum widens significantly during this process and even after breaking, the way the waves dissipate is not limited to b reaking. NMSE is able to describe these processes with good accuracy and can be seen in Figure 3 17. NMSE describes the development of harmonics and closely resembles the dissipation after breaking found in the data. 3.5 Linear Wave Model Vs. Nonlinear Wav e Model It appears, based on the outputs of setup and wave height evolution that a linear wave model is able to correctly describe the wave proce sses occurring on a reef system. After more investigation of the physics, it can be seen that there are proces ses occurring that a linear wave model ignores and must compensate for in other ways. It was shown that the wave processes in these laboratory tests become extremely nonlinear near breaking. Figure 3 18 shows both models' capabilities to describe the ener gy flux density in these areas (gauges 7 and 8). It can be seen that the linear model is not capable of describing the widening of the spectrum that occurs as energy is transferred from the peak frequency to higher harmonics (Figure 3 18C) The linear mode l also predicts that the peak frequency will dominate on the reef top (gauge 8), but measurements conclude that low frequency energy in the infragravity band dominates the energy flux spectrum (Figure 3 18D) T he nonlinear model does a much

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57 better job of d escribing the energy flux in higher frequencies by accurately describing the widening of the spectrum (Figure 3 18A) Only the nonlinear wave model recognizes higher harmonics and accurately describes the energy found in the infragravity ba nd as dominating the energy flux den sity on the reef flat (Figure 3 18B ). Both the linear and nonlinear wave models are in good agreement with the measured data (Figure 3 19). In f act, the linear wave model is in better agreement with Figure 3 18. Energy flux density m odeled compared with measured data from test 19 (H s = 5.4 m, T p = 20 s). Bl ue solid lines show the modeled data and red circl es show the measured data. A ) nonlinear wave model at gauge 7, B ) nonlinear wave model at gauge 8, C ) linear wave mo del at gauge 7, D ) linear wave at gauge 8. Below are the reef cross section ge ometry and gauge locations as a reference. A C D B

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58 the collected data than the nonlinear model wh en describing mean water level through all gauges, except gauge 7, on the reef edge. The disagreement between linear and nonlinear wave models is not significant, never exceeding 8 cm. However, the linear wave model's physics are limited to setup and wave heights. The nonlinear wave model is capable of describing the full spectral evoluti on (along with other parameters such as setup and wave heights). This capability is significant because of the role that infragravity waves on the reef flat play in runup and flooding. Figure 3 19. Mean water level (setup) calculated by th e nonline ar wave model in blue, linear wave model in red, and measure d values in black cir cles. Bottom panel shows the reef cross section geometry and gauge locations as a reference. The presence of infragravity waves can greatly increase the height of runup gen erated. Linear wave models do not account for the energy from the peak frequency transferring

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59 to low frequencies, but assumes that it is only dis sipated from the peak frequency, underestimating the energy present on the reef flat.

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60 CHAPTER 4 CONCLUSIONS 4.1 Reef Processes The wave processes and importance of nonlinearities are considerably different in the reef environment than a mildly sloping sandy beach owing to the physical struct ure of a fringing reef with steep slopes nearshore and a large reef fl at with shallow water Extensive wave bre aking occurs over a reef as waves propagate from deep water, shoal, and brea k on a steeply sloping reef over a relatively short distance V iolent breaking is observed with plunging breakers. In this study, breaking occurs both on the reef slope and on the reef edge. After in itial breaking, waves reform as bores and propagate across the reef flat towards the beach. As the waves propag ate from deep water and travel over th e reef bathymetry, shoaling begi n s. As waves s hoals the spectrum immediately widens due to nonlinear transfers of energy from the peak frequency to higher harmonics and lower subharmonics As the waves approach breaking the sh ape of the waves change considerably and beco me highly nonlinear with sign ificant amounts of energy being transferred between frequencies. In most cases, the peak of nonl inear transfers occurred around gauge 6 just before breaking seen in the bispectra (Figure 3 7) In this location energy is transferred from the peak frequen cy into higher frequencies developing harmonics. This can be seen in the power density and energy flux spectra Investigating these nonlinear interactions with bispectral analysis reveals that the peak frequency interacts with several higher harmonics. In extremely nonlinear te sts, such as

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61 test 19, there are nonlinear interactions noted through the bispectral analysis that the second harmonic was also interacting nonlinearly with higher harmonics During the laboratory experiment, waves were found to be br eaking between gauges 6 and 7. Breaking type was found to be plu n ging breakers for almost all of the test cases After breaking (gauge 7), considerable energy still remained at the peak frequency. As the broken waves reformed and propagated across the reef flat as bores energy was transferred into lower frequencies. On the reef flat, the spectrum was dominated by infragravity ener gy. This low frequency energy increased as the waves continued to propagate across the reef flat shoreward. Linear wave transfo rmations w ere observed during this experiment only in deep water (gauges 1 3). Waves began to become nonlinear immediately after approaching the reef slope As the waves shoaled, energy was transferred from the peak frequency to higher and lower frequencie s developing harmonics and subharmonics Following breaking, energy continued to be transferred from the peak frequency into the infragravity band and was dissipated from high er frequencies first. On the reef flat, after breaking, low freque ncy oscillation s dominated. The process es throughout this study were found to b e highly nonlinear with strong nonlinear inter actions that are not as apparent on mildly sloping beaches These processes a re more dominant in the reef environment and cause wave transformatio n here, to be to a large degree, incomparable to wave transformation over mildly sloping beaches 4.2 Wave Model Capabilities Xbeach is an open source, two dimensional, depth averaged wave and circulation model for sediment transport in the nearshore th at simulates dune erosion,

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62 overwash, and beach transformation. The wave model implemented into Xbeach is a linear wave model, in which, the only mechanism of dissipation is dissipation due to breaking. In this study, only the wave module was used, which wa s not initially developed for modeling wave propagation over steep slopes such as reefs This study was intended to push Xbeach beyond what it was intended for and to evaluate how it performs. More specifically, to evaluate the abilities of a linear wave m odel as it relates to wave propagation over reefs. Speci fic interest in Xbeach is due to the extensive l ist of capa bilities of Xbeach (including sediment transport), which were not used in this study, but of interest for future study. The linear wave mod el capabilities are limited to the mean water level (setup/setdown) and wave height values. Xbeach shows good agreement of these measured values, which i s of no surprise. The wave module found in Xbeach is a function compl etely dependent on the dissipation const ant, & which is a function of three free parameters, $ % and n. The linear wave model is calibrated to minimize the errors between the measured and calculated wave heights and mean water levels by finding o ptimal values of the parameters Without a data set for fit, calibrating the model would be difficult. With a more complete opti mization, the fit between the model prediction and laboratory data in t his study could be improved Because the only mechanism of dissipation in a linear wave model is dissipation due to breaking, there is no variation in the spectral flux until breaking occurs (Figure 3 4). It can be easily seen from the data that there a re, in fact, significant variations in the spectrum long before breaking occurs. The linear wave mo del is unable to describe the widening of the spectrum that occurs as waves shoal over th e reef before and after

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63 breaking. There are no terms in the wave model to account for energy trans fer between modes, but simply that energy can be lost (dissipated) at each mode individually, but not until breaking (Figure 3 15A). At which tim e, energy is dissipated proportionally from all modes. It is easily seen that this is an incorrect description of the true physics of the reef system. The predicted values of the linear wave model of Xbeach compares well with the measured values from the tests of the physical parameters for which it was developed, namely setup and wave height. If these parameters are the desired outputs for a study and there exists data to be used in a calibration process, a linear model can be efficient and effective. However, the wave processes on a reef are known to be nonlin ear and attempting to model them with a linear model is flawed and the underlying physics of a linear model is lacking in the case of a steeply sloping beach or reef. NMSE includes an extra te rm in the wave model that allow s energy to be transferred between modes. As th e waves shoal over the reef and nonlinear interactions become strong energy is transferred from the peak f re quency to higher and lower frequencies developing higher harmonics and subharmonics (Figure 3 17B) Only a nonlinear wave model accounts for these nonlinear energy transfers across the spectrum (Figure 3 17A). Through post processing, other parameters ar e attainable, such as setup using the shallow water equations (not coupled). In some instances, Xbeach predicts the setup better than the nonlinear model (Figure 3 19), although the difference is not significant never exceeding 8 cm It can be noted that in the area of strong nonlinearities during breaking (gauge 7), the nonlinear model does see m to describe the mean water level more accurately. The overall slightly better fit from the

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64 linear wave model could be attributed to two things. Firstly, in the li near wave model, the wave mo del is coupled with the shallow water equations that produce the av erage water level values, but with NMSE, the setup is calculat ed at the end using the shallow water equations. Lastly, the linear wave model was calibrated to fi t the output values of setup and wave height to the laboratory data The accurate prediction of these values is the ultimate goal of a linear wave model because this is where the physics of a linear wave model end. However, the nonlinear wave model, NMSE i s calibrated based on the spectrum and is used to correctly describe the full spectral evolution. The scope of Xbeach is very wide. Here, only the linear wave model was used. While Xbeach is sufficient for predicting the simple output values of average wa ter level and wave heights, the capabilities of the linear wave model end here. The effects of ignored nonlinear interactions can be aliased into other growth/dissipat ion mechanisms The wave processes occurring on a fringing reef are highly nonlinear. In order to correctly describe the nonlinear transformation of waves over fringing reefs, a nonlinear wave model is necessary. 4.3 Applications and Future Work After the experiment that provided data for this study a second laboratory test was conducted at the US Army Corps of Engineers Engineer Research and Development Center, Coastal and Hydraulics Laboratory in Vicksburg, Mississippi. In this laboratory experiment, more wave gauges were concentrated in the areas of interest (just before and during break ing) to give more insight into the complex processes that occur during shoaling and breaking. Also included were four acoustic Doppler velocimeters (ADV) and a pulse coherent acoustic D oppler profiler (PC ADP),

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65 all located in the same region. Unlike the da ta set from the University of Michigan's wind wave facility, this data set is more densely instrumented and is also collecting current data. During this experiment, tests were also run with two different reef slopes, 1:2 and 1:5. Interestingly, the sizeabl e decrease in significant wave height just before shoaling (gauge 4) seen in the laboratory study conducted at the University of Michigan and regarded as a faulty gauge, was al so observed during every test ru n in the flume for the more recent study even a fter several recalibrations of the wave gauges. Further investigation of this wave height variation should be done using the new data set to determine if there is a more fundamental physical process related to waves occurring or if this is an effect from the flumes themselves. This second set of laboratory data including data with different reef slopes can also be a useful data set to further study the calibration parameters included in wave models to begin to assign physical meaning to these parameters that are largely unknown for steep slopes. There are several possible applications for the nonlinear wave model NMSE. Because it has been observed that the low frequency oscillations dominate the spectrum on the reef flat, a model that recognizes the ene rgy transfer into the infragravity band can be used as a tool to study this energy and its role in reef hydrodynamics. Also, processes that are strongly frequency dependent can be studied using this frequency dependent model, f or example, bottom friction o n reefs. Because of the coral texture itself along with significant growth of organisms on the reef, bottom friction is a very dominant forcing in a reef system The two largest influences on reef hydrodynamics are wave action and bottom friction. The appl ication of a frequency

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66 dependent wave model on frequency dependent processes, such as bottom friction, is necessary to achieve a better understanding of these hydrodynamics. It remains that NMSE is a wave model only, solving wave propagation and transfor mation without coupling with any other processes. Xbeach is an open source model that couples hydrodynamics, sediment transport, and morphodynamics. It was not intended that this model be used in an application such as this study because the assumption o f purely linear waves is invalid in this environment Integrating the nonlinear wave model into Xbeach that is designed to model many other processes, could validate this model to be used in the context of steep slopes with nonlinear interactions while si mulating complex hydrodynamic and morphodyn a mic processes accurately by correctly describing the physics of the system.

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67 APPENDIX A XBEACH INPUT FILE: params.txt PARAMS grid input nx=86 ny=9 dx=10 dy=5 xori=0 yori=0 alfa=180 posdwn= 1 vardx=0 depfile=bathy.dep wave input instat=4 bcfile=test19.inp rt=7200 dtbc=4 break=1 gamma=0.9720 alpha=0.7715 n=5 constants rho=1025 g=9.81 flow input C=60 left=1 right=1 wind rhoa=1.25 Cd=.002 windv=0

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68 limiters gammax=5 hmin=0 .01 eps=0.1 umin=0.1 Hwci=0.01 simulation tstart=0 tint=4 tstop=7200 sediment transport form=1 dico=1 struct=0 morphological updating morfac=0 wetslp=1 dryslp=2

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69 APPENDIX B XBEACH INPUT FILE: test19.inp Hm0=5.2430 fp=0.05 mainang=90. gammajsp=3.3 s=1. fnyq=0.3

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70 APPENDIX C XBEACH INPUT FILE: bathy.dep 35.26 35.26 35.26 35.26 35.26 35.26 33.26 31.26 29.26 27.26 25.26 23.26 21.26 19.26 18.73 18.2 17.67 17.14 16.6 16.07 15.54 15.01 14.48 13.94 13.41 12.88 12.3 5 11.35 10.35 9.35 8.35 7.35 6.35 5.35 4.35 3.35 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 2.43 1.6 0.77 0.06 0.89 1.72 2.55 3.38 4.21 5.04 5.87 6.7 7.53 8.36 9.19 10.02 10.85 11.68 12.51 13.34 35.26 35.26 35.26 35.26 35.26 35.26 33.26 31.26 29.26 27.26 25.26 23.26 21.26 19.26 18.73 18.2 17.67 17.14 16.6 16.07 15.54 15.01 14.48 1 3.94 13.41 12.88 12.35 11.35 10.35 9.35 8.35 7.35 6.35 5.35 4.35 3.35 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 2.43 1.6 0.77 0.06 0.89 1.72 2.55 3.38 4.21 5.04 5.87 6.7 7.53 8.36 9.19 10.02 10.85 11.68 12.51 13.34 35.26 35.26 35.26 35.26 35.26 35.26 33.26 31.26 29.26 27.26 25.26 23.26 21.26 19.26 18.73 18.2 17.67 17.14 16.6 16.07 15.54 15.01 14.48 13.94 13.41 12.88 12.35 11.35 10.35 9.35 8.35 7.35 6.35 5.35 4.35 3.35 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 2.43 1.6 0.77 0.06 0.89 1.72 2.55 3.38 4.21 5.04

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71 5.87 6.7 7.53 8.36 9.19 10.02 10.85 11.68 12.51 13.34 35.26 35.26 35.26 35.26 35.26 35.26 33.26 31.26 29.26 27.26 25.26 23.26 21.26 19.26 18.73 18.2 1 7.67 17.14 16.6 16.07 15.54 15.01 14.48 13.94 13.41 12.88 12.35 11.35 10.35 9.35 8.35 7.35 6.35 5.35 4.35 3.35 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 2.43 1.6 0.77 0.06 0.89 1.72 2.55 3.38 4.21 5.04 5.87 6.7 7.53 8.36 9.19 10.02 10.85 11.68 12.51 13.34 35.26 35.26 35.26 35.26 35.26 35.26 33.26 31.26 29.26 27.26 25.26 23.26 21.2 6 19.26 18.73 18.2 17.67 17.14 16.6 16.07 15.54 15.01 14.48 13.94 13.41 12.88 12.35 11.35 10.35 9.35 8.35 7.35 6.35 5.35 4.35 3.35 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 2.43 1.6 0.77 0.06 0.89 1.72 2.55 3.38 4.21 5.04 5.87 6.7 7.53 8.36 9.19 10.02 10.85 11.68 12.51 13.34 35.26 35.26 35.26 35.26 35.26 35.26 33.26 31.26 29.26 2 7.26 25.26 23.26 21.26 19.26 18.73 18.2 17.67 17.14 16.6 16.07 15.54 15.01 14.48 13.94 13.41 12.88 12.35 11.35 10.35 9.35 8.35 7.35 6.35 5.35 4.35 3.35 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26

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72 2.43 1.6 0.77 0.06 0.89 1.72 2.55 3.38 4.21 5.04 5.87 6.7 7.53 8.36 9.19 10.02 10.85 11.68 12.51 13.34 35.26 35.26 35.26 35.26 35.26 35.26 33.26 31.26 29.26 27.26 25.26 23.26 21.26 19.26 18.73 18.2 17.67 17.14 16.6 16.07 15.54 15.01 14.48 13.94 13.41 12.88 12.35 11.35 10.35 9.35 8.35 7.35 6.35 5.35 4.35 3.35 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 2.43 1.6 0.77 0.06 0.89 1.72 2.55 3.38 4.21 5.04 5.87 6.7 7.53 8.36 9.19 10.02 10.85 11.68 12.51 13.34 35.26 35.26 35 .26 35.26 35.26 35.26 33.26 31.26 29.26 27.26 25.26 23.26 21.26 19.26 18.73 18.2 17.67 17.14 16.6 16.07 15.54 15.01 14.48 13.94 13.41 12.88 12.35 11.35 10.35 9.35 8.35 7.35 6.35 5.35 4.35 3.35 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 2.43 1.6 0.77 0.06 0.89 1.72 2.55 3.38 4.21 5.04 5.87 6.7 7.53 8.36 9.19 10.02 10.85 11.68 12.51 13.34 35.26 35.26 35.26 35.26 35.26 35.26 33.26 31.26 29.26 27.26 25.26 23.26 21.26 19.26 18.73 18.2 17.67 17.14 16.6 16.07 15.54 15.01 14.48 13.94 13.41 12.88 12.35 11.35 10.35 9.35 8.35 7.35 6.35 5.35 4.35 3.35 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26

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73 3.26 3.26 3.26 3.26 3.26 3.26 2.43 1.6 0.77 0.06 0.89 1.72 2.55 3.38 4.21 5.04 5.87 6.7 7.53 8.36 9.19 10.02 10.85 11.68 12.51 13.34 35.26 35.26 35.26 35.26 35.26 35.26 33.26 31.26 29.26 27.26 25.26 23.26 21.26 19.26 18.73 18.2 17.67 17.14 16.6 16.07 15.54 15.01 14.48 13.94 13.41 12.88 12.35 11.35 10.35 9.35 8.35 7.35 6.35 5.35 4.35 3.35 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 2.43 1.6 0.77 0.06 0.89 1.72 2.55 3.38 4.21 5.04 5.87 6.7 7.53 8.36 9.19 10.02 10.85 11.68 12.51 13.34

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74 REFERENCES Agnon Y, Sheremet A. 1997. Stochastic nonlinear shoaling of directional spectra. J. Fluid Mech. 345:79 99 Alsina JM, Baldock TE. 2007. Improved represe ntation of breaking wave energy dissi pation in parametric wave transformation models. Coastal Eng. 54:765 69 Battjes JA, Janssen JPFM. 1978 Energy loss and s et up due to breaking of random waves. Proc. ICCE, ASCE 569 87 Bowen AJ, Inman DL, Simmons VP. 1968 Wave setdown and setup. J. Geop hys. Res. 73:2569 77 Dean RG, Dalrymple RD. 1991 Water Waves Mechanics for Engineers and Scientists (2 nd ed). Singapore: World Sci. 353 pp. Demirbilek Z, Nwogu OG. 2007. Boussinesq modelin g of wave propagation and runup over f ringing coral reefs. Model evaluation report. USACE ERDC/CHL TR 07 12. Demirbilek Z, Nwogu OG, Ward DL. 2007. Laborator y study of wind effect on runup over fringing reefs. Report 1: Data report. USACE ERDC/CHL TR 07 4. Elgar S Guza RT. 1985. Observations of bispectra of shoaling surface gravity waves. J. Fluid Mech. 161:425 48 Gerritsen F. 1981. Wave attenuation and wave set up on a coral reef. University of Hawaii. Look laboratory tech. Rep. No. 48, 416 pp. Gourlay MR. 1992. Wave set up, wave run up and beach water table: int eraction between surf z one hydraulics and groundwater hydraulics. Coastal Eng. 17: 93 144. Gourlay MR. 1994. Wave transformation on a coral reef Coastal Eng 23:17 42 Gourlay MR. 1996a. Wave set up on coral reefs. 1. Set up and wave generated flow o n an idealised two dimensional reef. Coastal Eng 27:161 93 Gourlay MR. 1996b. Wave set up on coral reefs. 2. Wave set up on reefs with various profiles. Coastal Eng. 28:17 55. Gourlay MR, COLLETER G. 2005. Wave generated flow on coral reefs: an analysis for two dimensional horizontal reef tops with steep faces. Coast al Eng. 52:353 87

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75 Hardy TA, Young IR, Nelson RC, Gourlay MR. 1991. Wave attenuation on an offshore coral reef. Proc. 22 nd Int. Coastal Eng. Conf. Delft. 1990. 330 44. Hasselman K, Munk W, MacDonald G. 1962. Bispectra of Ocean Waves. Time Series Analysis 125 139. Hocke K, Kampfer N. 2008. Bispectral analysis of the long term recording of surface pressure at Jakarta J. Geophys. Res. 113: D10113 Lowe RJ, Falter JL, Bandet MD, Pawlak G, A tkinson MJ, et al. 2005. Spectral wave dissipation over a barrier reef J. Geophys. Res. (Oceans) 110:doi:10.1029./2004.JC002711 Massel SR, Gourlay MR. 2000. On the modeling of wave breaking and set up on coral reefs Coastal Eng 39:1 27 Monismith SG. 2007. Hydrod ynamics of coral reefs. Annu. Rev. Fluid Mech. 39:37 55 Rao TS, Gabr MM. 1984. An introduction to bispectr al analysis and bilinear series models. Lecture Notes in Statistic 24. Roberts HH. 1981. Physical processes and sediment flux through reef lagoon systems. Proc. of 17 th International Conf. on Coastal Eng. Sydney. March 1980. 946 62. Roberts HH, Murray SP, Suhayda JH. 1975. Physical processes in a fringing reef system. J. Mar. Res 33:233 60 Roelvink JA. 1993. Dissipation in random wave groups incident on a beach. Coastal Eng. 19:127 50 Roelvink JA, Reniers A, van Dongeren A, van Theil de Vries J, Lescinkski J, Walstra DJ. 2007. Xbeach Model Description and Manual. UNESCO IHE Institut e for Water Education, Delft Hydraulics, Delf t University of Technology. Suh KD, Park WS, Park BS. 2001. Separation of incid ent and reflected waves in wave current flumes. Coastal Eng. 43:149 59 Thornton EB, Guza RT. 1983. Transformation of wave height distribution J. Geophys. Res. 88:5925 38. Young, IR. 1989. Wave transformation over reefs J. Geophys. Res 94:9779 89

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76 BIOGRAPHICAL SKETCH Tracy Martz was born in Johnstown, Pennsylvania in 1986. Shortly after her birth, she moved to Currituck, NC where she grew up. After graduating from Cu rrituck County High School, Tracy enrolled at East Carolina University, where she pursued an undergraduate degree in systems engineering and was part of the first graduating class of engineers from the university. While attending East Carolina University, she was fortunate to be accepted as a summer student intern for the US Army Corps of Engineers Field Research Facility in Duck, NC where she spent summers being introduced and falling in love with t he field of coastal engineering From the experiences of the internship her desire to pursue higher education in coastal engineering grew. After obtaining a b achelor's degree in systems engineering, Tracy enrolled at the University of Florida. Here, she spent two years pursuing and achieving a m aster's degr ee in coastal and oceanographic engineering. After moving on from UF, Tracy hopes to obtain experience in the field of coastal engin eering and eventually pursue a d octorate degree.