1 APPLICATION OF A NONLINEAR WAVE MODEL TO DISSIPATIVE COASTAL ENVIRONMENTS By SHIH-FENG SU A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORID A IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2010
2 2010 Shih-Feng Su
3 To my parents
4 ACKNOWLEDGMENTS I want to thank many exce ptional people for their invalu able contributions helping me to achieve my greatest and most challenging success, the writing of this dissertation. First and foremost I wish to express my sincerest gratitude to my academic advisor and supervisory committee chair, Dr Alexandru Sheremet, for his guidance and assistance throughout this course of study. Special thanks go to Dr. Jane McKee Smith who reviewed my dissertation and gave me many insightful and germane comments. Thanks are also extended to t he other committee members, Dr. Arnoldo Valle-Levinson, and Dr. Kirk Hatfield, and Dr. Sivara makrishnan Balachandar, for their ongoing participation and valuable suggestions. My gratitude is extended to the w hole of the Coastal and Oceanographic Engineering Program fa culty, staff and graduate students fo r their help during my study at the University of Florida. I am particu larly thankful to Ilgar, Luciano, Yang, and Emre for their assistance with my research. Finally, my sincerest appreciation goes to my father, mother, older sister and younger sister, for their unending encouragement and moral support.
5 page ACKNOWLEDGMENTS..................................................................................................4 LIST OF TABLES............................................................................................................7 LIST OF FIGURES..........................................................................................................8 ABSTRACT...................................................................................................................10 CHAPTER 1 INTRODUCTION....................................................................................................12 1.1 Overview...........................................................................................................12 1.2 Fringing Reefs..................................................................................................12 1.3 Muddy Environments........................................................................................15 1.4 Study Objectives...............................................................................................17 2 DESCRIPTION OF NONLINEAR WAVE MODELS................................................19 2.1 Introduction.......................................................................................................19 2.2 Review of The Problem Formulation.................................................................20 2.2 The Deterministic Wave Model.........................................................................22 2.3 The Stochastic Wave Model.............................................................................24 3 WAVE OBSERVATIONS OVER REEFS AND MUDDY SEAFLOORS...................28 3.1 Laboratory Experiments of Waves Over A Fringing Reef.................................28 3.2 Observations On The Muddy Atchafalaya Shelf...............................................31 3.2.1 Experiment..............................................................................................31 3.2.2 Data Analysis..........................................................................................32 3.2.3 Observations...........................................................................................33 4 WAVE GENETATION AND DISSIPATION MECHANISMS....................................40 4.1 Breaking Waves................................................................................................40 4.2 Fluid Mud-induced Wave Damping...................................................................46 4.3 Wind Input and Whitecapping...........................................................................50 4.3.1 Implementation of Source Terms into the Nonlinear Wave Model...........50 4.3.2 Wind Input...............................................................................................52 4.2.4 Whitecapping...........................................................................................54 5 MODEL APPLICATION TO REEF ENVIRONMENT..............................................56 5.1 Model Integration..............................................................................................56 TABLE OF CONTENTS
6 5.2 Calib ration.........................................................................................................56 5.3 Discu ssion........................................................................................................58 6 MODEL APPLICATION TO MUDDY SEAF LOORS...............................................87 6.1 Inverse Modeling of Bottom M ud State.............................................................87 6.2 Discu ssion........................................................................................................90 7 CONCLUSIONS AND RE COMMENDAT IONS.......................................................95 7.1 Conc lusi ons......................................................................................................95 7.2 Recomm endations............................................................................................98 APPENDIX A MODIFIED MILD SLOP EQ UATION.......................................................................99 B THE MODAL ENERGY FL UX DISSIPAT ION RATE.............................................105 LIST OF RE FERENCES.............................................................................................106 BIOGRAPHICAL SKETCH..........................................................................................112
7 LIST OF TABLES Table page 3-1 Experimental wa ve condi tions............................................................................36 5-1 Optimal parameters obtained from linear and nonl inear models........................65
8 LIST OF FIGURES Figure page 3-1 UM fringing reef experiment se tup ....................................................................36 3-2 Bathymetry of the Atchafalaya Bay ...................................................................37 3-3 Wind and wave observations at T1 versus time ...............................................38 3-4 a) Time evolution of significant wave height of seas and swell. b) intensity of PC-ADP acoustic backscatter. c) net swell dissipation rate between T1 and T2. d) normalized frequency distribution of sw ell dissipation rate versus time ...........................................................................................................................39 4-1 Sketch of a single bore ......................................................................................41 4-2 Sketch of two-la yer fluid m ud system.................................................................50 5-1 Optimal versus optimal B ..............................................................................65 5-2 Calibrated versus offshore wave steepness, 0S ...........................................66 5-3 Observed and calibrated versus offshore wave steepness, 0S ....................67 5-4 Calibrated versus the non-dimensional depth kh .........................................68 5-5 Observed and calibrated against surf similarity parameter, 0 ......................69 5-6 Optimal B versus the non linearity parameter, cF ............................................70 5-7 Energy dissipation rate versus frequency (04.9 m H 12 spT ) ...................71 5-8 Cross-shore spec tral evolution (03.9 m H 10 spT ) ...................................72 5-9 Energy flux spectral density at various locations of sensors (03.9 m H 8 spT ) ...........................................................................................................73 5-10 Energy flux spectral density at various locations of sensors (03.9 m H 10 spT ) .........................................................................................................74 5-11 Energy flux spectral density at various locations of sensors (03.3 m H 12 spT ) .........................................................................................................75
9 5-12 Energy flux spectral density at various locations of sensors (04.9 m H 12 spT ) .........................................................................................................76 5-13 Energy flux spectral density at various locations of sensors (05.2 m H 14 spT ) .........................................................................................................77 5-14 Energy flux spectral density at various locations of sensors (05.4 m H 16 spT ) .........................................................................................................78 5-15 Energy flux spectral density at various locations of sensors (05.3 m H 20 spT ) .........................................................................................................79 5-16 Cross-shore evolution of normalized spectral flux (03.9 m H 8 spT ) ........80 5-17 Cross-shore evolution of normalized spectral flux (03.9 m H 10 spT ) ......81 5-18 Cross-shore evolution of normalized spectral flux (03.3 m H 12 spT ) ......82 5-19 Cross-shore evolution of normalized spectral flux (04.9 m H 12 spT ) ......83 5-20 Cross-shore evolution of normalized spectral flux (05.2 m H 14 spT ) ......84 5-21 Cross-shore evolution of normalized spectral flux (05.4 m H 16 spT ) ......85 5-22 Cross-shore evolution of normalized spectral flux (05.3 m H 20 spT ) ......86 6-1 Evolution of normalized (peak value is 1) frequency distribution of the swell dissipation rate between March 10th 12:00 and March 11 th 18:00 ..................92 6-2 Wave dissipation rates for March 10t h 15:00. a) Spectral density of wave variance. b) Frequency distribut ion of dissipa tion ra tes .....................................93 6-3 Wave dissipation rates for March 10t h 21:00. a) Spectral density of wave variance. b) Frequency distribut ion of dissipa tion ra tes......................................94 6-4 Wave dissipation rates for March 11t h 11:00. a) Spectral density of wave variance. b) Frequency distribut ion of dissipa tion ra tes......................................94
10 of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy APPLICATION OF A NONLINEAR WAVE MODEL TO DISSIPATIVE COASTAL ENVIRONMENTS By Shih-Feng Su August 2010 Chair: Alexandru Sheremet Major: Coastal and Oceanographic Engineering A nonlinear wave model is used to investigate the spectral evolution of wave fields in different dissipative coastal environments, fringing reefs and muddy seafloors. Depth-induced wave breaking, mud-induced dissipation, wind generation and whitecapping dissipation are implemented into the nonlinear model. This nonlinear wave model is computationally efficient so that the model can be used in an inverse means in order to estimate the parameters in the related dissipation modulus implemented. For the reef environment, there are two free parameters in the wave breaking modulus, J which is the ratio of wave height to water depth at initial breaking, and B, which is a measure of intensity of breaking. A discrepancy is found between the optimal values J calculated in this study and the results of previous parameterizations. J is found to be correlated to the inverse of offshore wave steepness and dimensionless depth kh. The analysis indicates that these two free parameters are related to the breaker type. Predicted spectral shapes are sensitive to the frequency dependence of wave breaking. Infragravity waves are observed to dissipate in the experiments. Under the assumption of no breaking-induced dissipation in infragravity waves, the nonlinear Abstract of Dissertation Presented to the Graduate School
11 wave model accurately predicts observed wa ve spectra in swell and sea band, but still underestimates the generation of the infragravit y waves. An in-depth investigation of this mechanism has yet to be pursued. For the investigation of mud-induced dissi pation on muddy seafloors, observations of wave and sediment processes collected at two locations on the Atchafalaya inner shelf are used. The observations show that wave dissipation in shallow, muddy environments is strongly coupled to bed-sedi ment reworking by wa ves. In the waning stage of the storm, the contributions of different wave-generation processes are analyzed using an inverse modeling approach. For this, mud-induced dissipation, wind input and whitecapping terms are incor porated into the nonlinear wave model. Numerical results reveal that although wa ve-mud interaction dominates dissipative processes, nonlinear triad wave interacti ons control the frequency distribution of the dissipation rate.
12 CHAPTER 1 INTRODUCTION 1.1 Overview Nonlinear wave energy transfer is a crucia l process in the shoaling evolution of waves propagating toward the shore. Through triad wave interactions, energy flux is being redistributed across the spectrum, resu lting in the development of secondary peaks in the spectral frequency domain as well as the transfer of spectral energy to the infragravity frequency band. This process is a significant contributor to the study of morphological evolutions, hydrodynamics, sedi ment transport, and of importance in the design of coastal structures. Increa sing high-frequency wave energy leads to asymmetric and skewed wave profiles that are characteristic of nearly breaking and broken waves. The generation of low-frequency (infragrav ity) waves increase strong oscillations of water level near the shore line, and may entrain and transport sediments. The evolution of wave spectrum ou tside the surf zone has been reasonably described by nonlinear wave models. Wave pr ocesses are fairly well understood in the surf zone where wave breaking plays a dom inant role in shaping wave spectra. Additionally, it is f ound that seabed friction damps the entire wave spectrum rapidly. In the present study, a nonlinear wa ve model is used to describe the spectral distribution of different dissipative coastal environment s: fringing reefs where wave energy is dissipated due to depth-induced wave break ing, and muddy seafloors where wave energy is damped due to mud-induced dissipation. 1.2 Fringing Reefs Islands surrounded by fringing reefs are abundant in the tropical and subtropical regions. The form of fringing reefs provides natural protection landward. With increasing
13 development of reef coasts, protecting c oastal area from wave attack and providing habitat for marine ecology are a concern. Th is protection can only be achieved through comprehensive understanding of the physica l processes which shape the reef environment and control its ecology. Wave acti on is a significant co ntributor to these processes. The dynamics of wave propagat ion over reefs have been assumed to be dominated by breaking [Young, 1989; Gourlay, 1994]. Moment um transfers associated with wave breaking over the reef slope are dom inant forcings that establish the vertical and horizontal structure of curr ents and water levels on reefs. Many spectral wave models have been developed using linear and nonlinear wave theories for description of the processes of waves propa gating over reefs. Linear spectral models can accurately predict wa ve energy dissipation due to wave breaking and bottom friction [Lowe et al., 2005; Ma ssel and Gourlay, 2000]. However, linear spectral models are unable to predict t he transfer of energy among the different frequency components due to nonlinear wave interactions. There have been many field measurements studies on reefs that demonstrate the importance of spec tral distribution [Young, 1989; Hardy and Young, 1996; Samo sorn and Woodroffe, 2008; Pquignet et al., 2009]. High-harmonic energy increases induce wave asymmetry and skewness, which are specially important for sediment transport predi ction. The generation of lowfrequency oscillations increases the time -varying water level near the shoreline [Pquignet et al., 2009]. The simu lated significant wave height s alone are insufficient to assess complicated wave processes occurring over reefs. Even if predicted wave height values are correct, wave setup and run-up ma y not be predicted accurately [Demirbilek and Nwogu, 2007]. Around the reef island, the development of infragravity waves on
14 wide reef platforms may be capable of entraining and transporting sediments both across the reef flat and around the reef island [Samosorn and Woodroffe, 2008]. Therefore, it has been shown that the pr operties of nonlinear waves should not be neglected in a coastal reef environment. Parametric wave breaking models are co mmonly used to predict wave dissipation across the surf zone. In spite of various atte mpts to improve predictions of wave height on beaches [Baldock et al., 1998; Dally et al ., 1985; Janssen and Battjes, 2007; Kaihatu and Kirby, 1995; Thornton and Guza, 1983; Whi tford, 1988], the accu racy of prediction is sensitive to the m odel parameters (e.g., the ratio of breaking wave height to breaking depth, and B, the intensity of breaking). Exis ting empirical parameterizations of are in good agreement on planar and barred beaches. The value of is found to be dependent on the beach slope [Salleng er and Holman, 1985], offshore wave steepness [Battjes and Stive, 1985; Nairn, 1990] and nondimensional depth kh [Ruessink et al., 2003]. Most formulations were obtained from calibration with field observations and laboratory experiments for mild slopes (e.g., sandy bottom beaches). There are few parameterizati ons of breaking waves on steep slopes. Therefore, a major challenge for modeling wave transformati on on reefs is choosing the appropriate parameters for wave breaking models. The aim of this study is to provide more adequate physical characteristics that determine the free parameters in wave breaking models for predicting wave transformation over reefs. To achieve this aim, parameters in wave breaking models are optimized based on laborator y experiments and compared to existing empirical parameterization from previous field and labo ratory experiments. A deterministic model
15 [Agnon et al., 1993] and a stochastic model [Agnon and Sheremet, 1997] are applied to investigate the evolution of wave spectra over a fringing reef. A parametric wave breaking model [Janssen and Battjes, 2007] is incorporated into the nonlinear wave models to describe the depth-induced wave dissipation. Parameters are calibrated using detailed laboratory ex periments of waves across a fringing reef profile at University of Michigan [Demirbilek et al., 2007]. Existing empirical parameterizations of wave breaking are examined in order to study the adequateness for application to reef profiles. 1.3 Muddy Environments The other wave-dissipative environments st udied here are muddy seafloors. There are many theories for dissipation of waves owi ng to interactions with a mud layer at the seafloor. Although wave-mud interactions do minate dissipation in muddy environments, the frequency-dependent dissipation rate is not well understood. To study the distribution of dissipation with frequency, relative effects such as triad wave interactions, generation by wind and di ssipation by whitecapping, should be considered. The dissipative effect of muddy seafl oors on wave propagation was observed by numerous investigations and is being studied ex tensively. Field observations [Wells and Coleman, 1981; Forristall and Reece, 1985; Jiang and Mehta, 1996a; Mathew et al., 1995] and laboratory experiments [Gade, 1958; Jiang and Mehta, 1996b; De Wit, 1995; Hill and Foda, 1999; Chan and Liu, 2009; Chan and Liu, 2009; Holland et al., 2009] show that up to 80% of incoming waves c an be lost energy due to wave-mud interaction over a distance of just a few wave lengths Theoretical efforts to understand wave-mud interaction have proposed several stressstrain rheologies for mud layers and corresponding dissipation mechanisms. Mud has been described as a viscous
16 Newtonian fluid [De Wit, 1995; Dalrymple and Liu, 1978; Ng, 2000]; visco-elastic solid [Jiang and Mehta, 1996b]; visco-plastic Bin gham material [Chan and Liu, 2009; Mei and Liu, 1987]; or poro-elastic material [Yamam oto et al., 1978; Yamamoto and Takahashi, 1985]. The applicability of most of these models for field data is limited by the relatively simple mud rheologies in the models (i.e., linear, with characteri stic parameters such shear modulus or shear viscosity assum ed independent of strain-rate amplitude). The complexity of proposed nonlinear rheology models [Mei and Liu, 1987; Chou and Hunt, 1993] has precluded their application to fi eld observations. However, there is considerable observational evi dence [Sheremet et al., 2005; Jaramillo et al., 2009], that bed reworking by waves results in significant changes in the bed rheology. Based on analysis of acoustic backscatter records near the seafloor, Jaramillo et al.  identified several distinct stages in the res ponse of bed sediment to wave activity: stiff mud state; bed liquefaction; and resuspension and hindered settling, leading to the formation of a fluid mud la yer of gradually decreasing thickness. The mechanism for fluid-mud formation appears to be dependent on wave frequency and amplitude. While linear mud-rheology models could work for par ticular stages of a storm, capturing the full bed evolution likely requires models capa ble of handling nonlinear transitional, multilayer regimes. Sheremet and Stone  reported unexpected short-wave (frequency higher than 0.2 Hz) dissipation in muddy env ironments, and they hypot hesized that this is due to triad nonlinear coupling between short and long waves, the latter interacting efficiently with the bed. The implication t hat nonlinearities are not suppressed by the strong mud-induced dissipation is support ed by subsequent numerical experiments [Kaihatu et al., 2007], but has not yet to be confirmed by observational data. Net wave
17 dissipation estimates represent coupled e ffects of multiple dissipation/growth mechanisms that can include, in addition to mud-induced dissipation, depth-limited breaking, wind-induced wave growth, whit ecapping, and nonlinear triad interactions. Evaluating the contribut ion of each of these mechanisms is difficult without the use of numerical models. In turn, numerical models are seriously challenged by shallow-water, muddy environments. The aim is to use a nonlinear model to study the frequency-dependent dissipation rate in muddy seafloors. The model used is based on the stochastic model of Agnon and Sheremet , incorporat ing other dissipation/growth processes. An inverse modeling approach to estimate the effectiv e viscosity of the bed sediment was adapted to account for the contribution of triad wave interactions. For simplicity, the analysis presented here focuses on mud st ate in the waning stage of the storm, modeled as a simple viscous fluid with an effective viscosity. 1.4 Study Objectives The major objective of this research is applying the nonlinear wave model to investigate the evolutio n of wave spectra and dissipation ra te distribution in dissipative coastal environments. The primary tasks performed to achieve this objective are to (1) investigate wave spectral energy distribution in reef and mud bed coastal environments, (2) incorporate a wave breaking model in to the nonlinear model to predict wave transformation on a steep beach, (3) investigate if existing parameterizations of wave breaking model for reef topography are suitable,
18 (4) incorporate a mud-induced energy damping mechanism into the nonlinear model to predict wave attenuation over a fluid-mud bed, (5) generation and dissipation in natural coastal waters are implemented into the nonlinear model to increase the predict ion of wave spectra evolution. This dissertation is organized as se ven chapters. Chapter 1 presents the importance of study on nonlinear waves in oceanic water, the background of two dissipative coastal environments, fringing re efs and muddy seafloor s, and the objectives of the present study. In Chapt er 2, a deterministic and a st ochastic formulation of the nonlinear wave model including triad wave interactions are presented. Chapter 3 describes the wave measurements in a one-dimensional fringi ng reef experiment and field observations on muddy seafloor coast, Atchafalaya inner shel f. Chapter 4 presents significant models accounting for other wave processes, wave breaking, mud-induced dissipation, and wind input and whitecapping. In Chapter 5, the numerical model, including wave breaking, is validated against laboratory experiments. Optimal parameters in the wave breaki ng model are investigated with existing parameterizations, and results of numerical simulations are s hown. In Chapter 6, the numerical model including mud-induced dissipation, wind energy input and whitecapping, is used to study the field measurements conducted on the At chafalaya shelf. In Chapter 7, major findings in this study and future recommendations are summarized.
19 DESCRIPTION OF NONLINEAR WAVE MODELS 2.1 Introduction As a wave field propagates toward a shore, the wave spectrum evolves due to many mechanisms including shoaling, refraction, diffraction and nonlinear interactions. In shallow water, wave energy is distributed across spectrum by interaction dominated by near-resonant triad wave interactions. Nonlinear wave models have been extensively used to study effects of nonlinear waves in shallow water. In general, these models can be classified as: deterministic (phase-resolving) models and stochastic (phase-averaged) models. Deterministic models are formulated in terms of wave amplitude and phase. Most of them have been based on the Boussinesq-type equation for the spatial variation of the Fourier amplitudes in the frequency domain [Forristall and Reece, 1985; Madsen and Srensen, 1993; Eldeberky and Battjes, 1996; Madsen et al., 1997]. These models incorporate weak nonlinearity as well as weak dispersion. As an alternative, they can be derived from the irrotational and inviscid governing equations with surface boundary conditions expanded to second order [Agnon et al., 1993; Kaihatu and Kirby, 1995]. These models incorporate weak nonlinearity but full dispersion. Although the deterministic models accurately describe the wave shoaling process, this class of models is computationally time-consuming. To obtain viable results, the deterministic model has to calculate many different initial amplitude and phase variations, then average the simulated results. In order to improve model efficiency, a number of attempts have been made to develop numerical wave models with improved levels describing the nonlinear evolution of frequency spectrum in shallow regions. CHAPTER 2
20 The stochastic (phase-averaged) models predi ct the wave spectrum evolution in shallow water based on an energy balance equati on, similar to spectral models used in deepwater applications. These models solve ev olution equations for statically averaged spectral wave properties. Abreu et al.  developed a model for the nonlinear evolution of the frequency-direct ional wave spectrum, represent ing the triad interactions by a single source term in spectral energy balance. The model is based on a closure for nondispersive waves with only exact reso nance and nonlinear ph ase-coupling between wave triads is neglected. Agnon and S heremet  developed a stochastic directional shoaling model that takes into account the development of phase-correlation in bound waves by solving an additional evolution equation for phase-mismatch. Herbers and Burton , and KofoedHansen and Rasmussen  developed stochastic formulations based on Boussinesq-type evolution equations. In their formulations, the secondand third-order st atistics of random, shoaling waves were described by a coupled set of evolution e quations for the energy spectrum and the bispectrum. Because these models are num erically efficient, they can be used in shallow water with wave spectra obtained from directional measurem ents or predictions of a wave model at offshore boundary. However, models using a statistical closure have the drawback that simulations may yield significant errors over long propagation distances and in the region with strong nonlinearity. 2.2 Review of The Problem Formulation In this section, the formulations of t he nonlinear deterministic and stochastic mild slope equations presented by Agnon et al.  and Agnon and Sheremet  are briefly outlined.
21 The equations governing the irrotational fl ow of an inviscid incompressible fluid with a free surface ar e the Laplace equation: 20, 0zzhz (2-1) bottom boundary condition: 0 zhzh (2-2) free-surface kinematic boundary condition: 0 0tzz, (2-3) free-surface dynamic boundary condition: 2211 0 0 22ttzzgz (2-4) where is the horizontal gradient operator, is the velocity potential, is the freesurface elevation and h is the local water depth. In the following, the surface boundary conditions are expanded in power series about 0 z and retaining terms to 2() O where the nonlinear ity parameter is defined as ka ( k is the wave number, a is the wave amplitude, and 1 ka), and eliminating in equation (2-3) and (2-4), after some algebra, new forms of ki nematic and dynamic free su rface boundary conditions of governing equations are obtained: 22111 0 22ttzzttztgz g (2-5) 22111 0 22ttztzgz g (2-6) The system of equations (2-1), (2-2) and (2-5) is the starting point for the derivation for the velocity potential. Next, a multiple scales expansion in space and time
22 is introduced to this system to divide the fast (the wave movement) and slow (mean flow and set-down induced by waves) variation of the wave field. Then, to obtain fast variations, it is useful to eliminate the variable t by using the Fourier transform to move into the frequency domain. The main task in t he derivation of the evol ution equation is to reduce the system to a single equation. A gnon et al.  in itially included two different kinds of terms: free waves, whic h satisfy the linear dispersion relation and bound waves, which do not satisfy the linear di spersion relation and are forced by the interactions of two free wave s. Additionally, one important step towards the derivation is the elimination of the vertical structure of the velocity pot ential and the vertical structure of bound waves as well as free waves was considered. The m odel derived was based on the vertical structure of a free wave. Agnon et al  defined a detuning parameter bf fkk k (2-7) which gives a measure of the deviation between bound and free wave numbers, where bk is the wave number of the bound wave and fk is the wave number of the free wave. The bound waves within the model ar e described with an error of () O This approach assumes the spatial variation of water dept h, wave numbers, and wave amplitudes to be weak. 2.2 The Deterministic Wave Model The resulting equation of the deterministic model derived by Agnon et al.  is ,,,, ,01 2 2i jj j jpqpq jpqjpq pqdCda aCWaaWaa dxdx, (2-8) and can be expressed by
23 ,,,,2j j jpqpq jpqjpqdb KbWbbWbb dx iip,q>0 (2-9) in which the variable j b is defined as 1/2exp() j jjjbaCi (2-10) where mode j in the Fourier series representation of the free surface is characterized by the complex amplitude j a, group velocity j C, phase of mode j the asterisk denotes complex conjugation, and j K the complex wave number which can be written as j jjKk i (2-11) in which the real part j k satisfies the dispersion relation 2tanh j jjkkh (2-12) with radian frequency /jjg the modal radian frequency 2 j jf the gravitational acceleration g, the local depth h, and with the relation j jd k dx (2-13) The imaginary part j represents the modal dissipation rate which can combine effects of wind input, whitecapping, wave breaki ng and bottom friction. The interaction coefficient W is a function of the tr iad frequencies and wave numbers 22222 ,,1 2 8 jpq j pqpqqppqjpq pqjjg Wkkkk (2-14) and 1 2 ,,,, jpqjpqjpqWWCCC (2-15)
24 The notation ,,... j pq indicates that the quantity is evaluated for the frequency triplet j f, p f and qf. The Kronecker symbol used in the frequency selection criterion is ,,1 0 otherwise.jpq j pq (2-16) The first term in the righ t-hand side of equation (2-9) represents linear shoaling and the second term represents the nonlinea r triad interactions. The linear shoaling term was extended to account for a steep-sl ope topography [Chamberlain and Porter, 1995], keeping terms of order 2h (neglected in the original derivation of Agnon et al. ), but neglecting higher-ord er curvature terms (e.g., 2h and higher). The linear part of equation (2-9) is revised to db b dx mik+k+i, (2-17) where the steep-slope wa ve number correction k given by lnln 2mgPh kCcC C 21dd hkdxdx (2-18) with 2 43 3 2sec 4sinh9sinhsinh2 12sinh 32sinhcosh2cosh3, khkh Phkhkhkhkhkh khkh khkhkhkhkh (2-19) Details of derivation are shown in Appendix A. 2.3 The Stochastic Wave Model On the basis of the deterministic model (2-9), the stochastic model can be derived for the energy spectrum and for the complex bispectrum. The following shows
25 the procedure outlined by Agnon an d Sheremet . First, equation (2-9) is multiplied by j b ***** ,,,, ,>0=2j j jjjjpqjpq jpqjpq pqdb bKbbWbbbWbbb dx ii, (2-20) and the complex conjugate of equation (2-20) is given by ** ,,,, ,>0=2***j qj jjjjpqjp jpqjpq pqdb bKbbWbbbWbbb dx ii, (2-21) Then equation (2-20) is added to equation (2-21) and the ensemble average of the result yields ** ,,,, ,>0=22*j jjjpqjpq jpqjpq pqd WbbbWbbb dx (2-22) where ... denotes the imaginary pa rt of complex number and ... is the ensemble average operator, and the aver age modal energy flux j is defined as 2211 =||=|| 22 j jjjbaC (2-23) The average of the triple products on the right-hand side of equation (2-22) is the discrete bispectrum. To obtain the nonlinear properties of the wave field, one needs to get higher-order moments and evalute the bi spectrum. The spatial derivative of the discrete bispectrum is given by ***= j pq jpqpqjqjpdbdbdb d bbbbbbbbb dxdxdxdx. (2-24) The terms on right-hand side of equation (2-24) include the trispectrum (i.e., uvpqbbbb), and would contain terms involving the higher-o rder moments and so on. This procedure would lead to an infinite set of equations. In order to solve this problem, they introduce
26 the quasi-Gaussian approximati on, a statistical closure hypothesis, so that the trispectrum is expressed as products of the second-order averages. With some manipulations and taking the ensemble average of the equations, they can be written as ***22 ,,=||||iijpqjpqjpqpq jpqd bbbKKKbbbWbb dx 2222 ,,,,2||||2||||, ii qjpj pqjqpjWbbWbbO(2-25) and ***22 ,,=2||||***iijpqjpqjpqpq jpqd bbb KKKbbbWbb dx 2222 ,,,,2||||2||||, ii jqjp pjqqjpWbbWbbO(2-26) with 222211 ||||=4||||=4 22 p qpqpqbbbb, (2-27) which may be integrated with respect to x by assuming that the spectrum varies slowly with x. Integrating by parts, the terms in wh ich the derivatives of the spectrum appear may be neglected, since they are 2O or smaller. For simplicity the bispectrum in deep water are neglected because it is very sma ll in a nearly Gaussian sea state. It can be shown that this does not affect the shoa ling spectrum. However, the contribution of the initial bispectrum should be consider ed if the shoaling computation does not start from deep water. After integration and substitution back into equation (2-22) one obtains the stochastic model 2j j jd dx
27 08,,,,,, ,, j pqpqjqpqjqpjpj jpq pqWGWWW 016,,,,,, ,, j pqpqpjqjqqjpjp jpq pqWGWWW (2-28) where G is the real part of G, which is defined as ,,,,,, xx jpqjpqx iKdsids jpqGxeedu, (2-29) where G satisfies the simple differential equation 1 i d GxKGx dx, (2-30) with the initial value 0 22 ik G k, (2-31) where K has an imaginary part representing dissipation ,,,, jpqjpqjpqjpqKkkkkj,p,qii, (2-32) and the wave number j k satisfies the dispersion relation.
28 WAVE OBSERVATIONS OVER REEFS AND MUDDY SEAFLOORS 3.1 Laboratory Experiments of Waves Over A Fringing Reef Demirbilek et al.  conducted laboratory experiments of shoaling and breaking of irregular waves over a fringing reef at the University of Michigan (UM). The laboratory study was conducted under the Surge and Wave Island Modeling Studies (SWIMS) Program, to investigate the effect of wind on wave processes affecting the inundation of Pacific islands resulting from typhoons. A 1:64 scale profile representative of typical fringing reefs of the Pacific island of Guam was built in the wind-wave flume of 35 m length, 0.7 m width, and 1.6 m height. The corresponding prototype scale of the experimental setup is shown in Figure 3-1. The reef slope consists of three sloping regions, with slopes of 1:5, 1:18.8, and 1:10.6, starting from offshore and ending at the reef edge. The reef top is wide and flat, meeting the beach slope at a slope of 1:12. In the present study, the model scale and Froude scaling are used to convert the measured data in the laboratory to equivalent prototype conditions for all calculations. The length scale factor is 64 and the velocity and time scale factors are 8, by Froude scaling. The incident waves were generated with a JONSWAP spectral shape with peak enhancement factor 3.3, significant wave heights, 0H, ranging from 2.05 to 5.44 m, and peak wave periods, p T, ranging from 8 to 20 sec. These wave tests represent extreme conditions associated with tropical cyclones. These test conditions were ran at four different still-water depths to simulate conditions at different tide and surge levels. Water levels used were 0 m, 1.024m, 1.984m, and 3.2 m on the reef flat. The still-water level used in this study is 3.2 m, which is the highest water-level in the experiment. The incident wave conditions are listed in Table 3-1. We refer to the wave conditions using CHAPTER 3
29 the notation peak wave period_in itial significant wave height _water level, all prototype values. For example the notation 8_39_32 i dentifies the case with peak wave period 8 spT, initial significant wave height 03.9 mH and water level 3.2 m. Values of the corresponding deepwater steepness 00 / rmsSHL where 2 0 / 2pLgT and 01/2rmsHH are listed in the Table. Breaker type is determined by the surf similarity number 0 [Battjes, 1974] which is defined as: 0 1 2 0 0tanH L (3-1) where tan is the slope of the bottom prof ile. Spilling breakers occur when 00.5 and plunging breakers occur when 00.53.3 All wave conditions considered in the present study are the plunging breaker type. Nine capacitance-wire wave sensors we re used to measure the water surface elevation (refer to Figure 31). Three sensors (Sensor 13) installed seaward of the reef face were arranged to allow separation of th e incident and reflected wave trains. Three sensors (Sensor 46) were positioned over the reef slope. One sensor (Sensor 7) was installed at reef edge. The remaining two wave sensors (Sensor 89) were installed on the reef top. The sensors sa mpled at 20 Hz for 900 sec. The first 100 sec of data were neglected to allow waves to propagate through the gauge array. Spectra were estimated from zero-meaned, 10% cosine bell windowed wave records with band averaging. Resulting resolution bandwidth is 0.019 Hz and spectral estimates have a nominal 62 degrees of freedom (DOF). Preliminary analysis of raw data showed that the measured significant wave heights at sens or 4 deviated from t hose measured at the
30 consecutive Sensor 3 and Sensor 5 by about 37%. Therefore, the data collected at Sensor 4 were ignored in the present study.
31 3.2 Observations On The Muddy Atchafalaya Shelf 3.2.1 Experiment Details of the experimental site and inst rumentation are discussed by Jaramillo et al. . Here, we review the essential for t he scope of this study. The experiment site is located shoreward of the 5-m isobath, on the nearly flat (maximum slope less than 1:1000) topset of the Atchafalaya clinoform in southwes tern Louisiana (Figure 3-2). In winter and early spring (November to Apr il), the Atchafalaya Sh elf and the adjacent muddy coast area are swept perio dically (about 3-10 day interv als) by cold atmospheric fronts accompanied in the prefrontal phase by energetic onshore swells (8-10 s period, 1-2 m wave height). These storms usually coincide with rising or high MississippiAtchafalaya River water levels and sedim ent discharge, due to basin-wide influences. On the shelf, wave-induced bottom turbulence resuspends significant quantities of sediment. As wave activity decreases at the end of the storm, and the direction of propagation of short-wave fields rotates to align with post-front al seaward winds, hindered settling of suspended sediment leads to the formation of episodic fluid-mud layers with a duration of about 12 hours, and thickness less than 30 cm [Jaramillo et al., 2009; Allison et al., 2000; Draut et al., 2005]. The experiment site on the inner Atchafalaya Shelf was chosen for the opport unity it provides to observe wave propagation in shallow-wa ter (the 10-m isobath is in some places about 50 km offshore), muddy environments. Observations of wa ve and suspended sediment concentration were collected by two instrumented platform s (T1 and T2), deployed in a cross-shore array near the 5-m isobath on the Atchafalaya Shelf (Figure 3-2) between February 15th and April 1st, 2006. The mean water level wa s approximately 5 m at T1, and 4.3 m at T2, and the platforms were separated by a di stance of 3.8 km. The analysis presented
32 here is based on wave observations (pressure velocities, and acoustic surface track) and acoustic backscatter intensity collected by acoustic velocity profilers. 3.2.2 Data Analysis Pressure observations were processed using standard spectral analysis. Time series segments of 20-min length were detrended and de-meaned, then divided into 128-s blocks with 50% overlap, and tapered using a Hanning window. Resulting spectra have approximately 18 DOF wi th a frequency resolution of 0.0078 Hz. Wave spectra were corrected to account for the mean depth of the sensors using linear wave theory, with a high-frequency cutoff defined by a depth a ttenuation of wave variance larger than 95%. Significant wave height values were ca lculated based on the first spectral moment, 4() Sfdf, where ()Sf is the spectral density of wave variance and f is frequency. The swell band is defined as 0.050.2 Hz f The standard definition of the modal energy flux Ffx dissipation rate is the relative change of the energy flux per unit distance x The definition of frequen cy-dependent dissipation rate f is described in Appendix B. Note that the estimates given in equation (B-4) are measures of net wave growth/dissipation, as they do not distingu ish between different processes such as wind growth, whitecapping, wave breaking, m ud-induced damping, nonlinear interactions, and others. The distance cosoxd is the effective propagati on distance between T1 and T2, where d is distance between tripods, and is the propagation angle with respect to the T1-T2 axis. For the event studied her e, the propagation angle was held steady at approximately 40o 3800 m d yielding 2926 m x The use of the effective
33 distance is justified by the small bottom sl ope and the negligible swell refraction, which resulted in a change of propagati on direction of less than 2o between the two platforms. To reduce noise due to use of low-DOF spectra, f estimates were further smoothed by band-averaging (in 21 frequency bands with 0.0156 Hz frequency width), and timeaveraging using a running mean. 3.2.3 Observations The major event of the 2006 experiment wa s the frontal storm that passed over the experimental site on March 10th, with sustained 10 m/s winds out of the south and seas (waves with frequency 0.2 Hzf) between 0.5-1 m significant wave height (Figure 3-3a,b). These conditions lasted for mo re than 4 days and resulted in swells that peaked 1.5 m significant wave height in 5m water depth (Figure 3-3a,b). Throughout the storm, propagatio n direction of the sea was approxim ately northward (Figure 3-3c). Swell intensity was correlated with the two pea ks in wind intensity. Swells arrived at the T1 site early March 9th, peaked the evening of March 9th (1.5-m significant wave height) and again briefly at midnight March 10th (1.0-m significant wave height). Mean swell direction was N-NNW, Figure 3-3c). The response of bed sediment to wave activity has been discussed in detail in Jaramillo et al. . Near-bed backscatte r and wave conditions (Figure 3-4a,b) suggest that, for strong events, the bed sedim ent responds to swell activity in several distinct stages: starting from a stiff mud state, the bed s ediment liquefies, in this particular case when swell approaches 1-m height; liquefaction is followed by rapid resuspension and the formation of a lutocline at the peak of the st orm (fluid mud event
34 ); hindered settling results in the format ion of a well-defined fluid mud layer of gradually decreasing thickness (fluid-mud event ). Swell dissipation increases steadily durin g the storm (Figure 3-4c), qualitatively consistent with the assumption that dissi pation is mainly driven by bed softening (reworking) by waves [Sheremet and Stone, 2003; Elgar and Raubenheimer, 2008; Rogers and Holland, 2009]. Hourly estimates are correlated to the tidal phase (higher dissipation at low tides), likely due to in creased wave-bottom interaction and increased wave breaking. Remarkably, swell dissipati on continues to increase after the storm, even after any detectable fluid-muds have di sappeared. Because wave energy at this stage is low, whitecapping and wave breaki ng are likely weak and wave-mud interaction should be the dominant mechanism for wave di ssipation. The de-tided evolution of swell dissipation shows no detectable correlation wi th the two fluid mud events [Dalrymple and Liu, 1978; Ng, 2000], possibly due to low suspended sediment concentrations (viscosity), or sparse spat ial distribution of concentrati ons. However, the complex structure of the frequency dist ribution of the dissipation ra te (Figure 3-4d) cannot be explained by mud-induced wave dissipation al one. The peak of t he distribution is strongly correlated with the spectral peak, and its width shows rapid and significant changes. The width of the dissipation band (0 ) is on average 0.2 Hz, however, between noon and midnight March 10t h, it decreases from 0.3 Hz to 0.15 Hz. In general, the dissipation band appears to narrow when wave activity peaks (e.g., evening of March 9th and 10th). In contra st, the Newtonian fluid-mud model (e.g., Dalrymple and Liu ; Ng ) predicts a wide and stable distribution of the dissipation rate. The position of the dissipation peak changes by approximately 0.05 Hz. The Newtonian
35 fluid-mud model is probably a good approx imation for bottom processes before liquefaction and during the fluid-mud events, but not for the end of the settling period and the final soft-mud stage of the storm (Figur e 3-4b). The variability of swell dissipation rate (Figure 3-4d) suggests that non-dominant processes (e.g., wind input, whitecapping, depth-induced br eaking) cannot be neglected. The growth rates (0 ) at high frequencies are probably due to wind i nput. The growth in the infragravity band (0.05 Hzf), the correlation with spectral peak, and the variability of the width of the dissipation band can be explained as effects of nonlinear triad wave interactions as shown in Chapter 6.
36 Table 3-1. Experimental wave conditions Case no. 0()Hm ()pTs 0S 0 8_39_32 3.9 8 0.0276 0.478 10_39_32 3.9 10 0.0177 0.596 12_33_32 3.3 12 0.0106 0.770 12_49_32 4.9 12 0.0156 0.636 14_52_32 5.2 14 0.0120 0.723 16_54_32 5.4 16 0.0099 0.797 20_53_32 5.3 20 0.0057 1.054 35.2 m 1:12 1:10.6 1:18.8 1:5 1006 m 83 m137 m 307 m 86 m 3.2 mWedge wave maker Wave sensors Reef top Reef face Reef edge Beach Figure 3-1. UM fringing r eef experiment setup (Dimensi ons are in prototype scale).
37 Figure 3-2. Bathymetry of the Atchafalaya Bay. a) Q ualitative map of surficial sediments on the Atchafalaya inner s helf (Neill and Allison, 2005). The approximate location of the tripods deploy ed in February 13th to March 14th, 2006 are marked by circles. b) Smoothed es timate of bathymetry profile along a transect through the two experiment sites (T1 and T2). Maximum slope is 0.0008, reached at cross-s hore distance of 5 km, s lightly offshore of the platform T1.
38 Figure 3-3. Wind and wave observations at T1 versus time: a) Wind direction and intensity, significant wave heights of sea (f > 0.2 Hz, red line) and swell (f 0.2 Hz, blue line). b) Evolution of normalized frequency spectra. c) Propagation direction of the peak of the directional spectral density for each frequency band (derived from ADCP measurements). Wind and wave propagation direction are responseted as flow directions (N means propagating northward). Surface wind is numerically simulated using the Coupled Ocean-Atmosphere Prediction System (COAMPS, [Hodur, 1997]).
39 Figure 3-4. a) Time evolution of significant wave height of se as and swell. b) intensity of PC-ADP acoustic backscatter. c) net swell dissipation rate between T1 and T2. d) normalized frequency distribution of swell dissipation ra te, versus time. Panel b): black lines mark the positi on of peak backscatter intensity (upper line), and the position where the PC -ADP records zero velocity (hydrodynamics bottom, lower line). Panel c): red dots are hourly estimates of swell dissipation, and the blue line is a running average; the black line is the mean water level (axis on right). Panel d) : black dots mark the position of the spectral peak. Panel c-d): positive va lues represent dissipation, negative values-growth. The black rectangl e marks the period analyzed using numerical models.
40 WAVE GENETATION AND DISSIPATION MECHANISMS 4.1 Breaking Waves The evolution of wave energy across surf zone can be described by the one-dimensional wave energy flux balance equation ()dECDdx (4-1) where x is the cross-shore coordinate, positive onshore, C is the group velocity and E is the total wave energy per unit surface area given by c2C=12sinh2khkhÂ§Â¨, (4-2) 21=8rmsEgHU, (4-3) where U is the water density, g is the gravitational acceleration, c is the wave phase velocity, k is the wavenumber, h is the water depth, rmsH is the root-mean-square wave height, and D represents power dissipated per unit area, described below. For a single breaking wave the dissipation can be treated as a bore [LeMehaute, 1962]. Consider a bore connecting two regions with depths 1h and 2h (see Figure 4-1). The power dissipated in the bore per unit span is given as Lamb : ff 1/2312'2112142ghhDghhhhU (4-4) CHAPTER 4
41 h2 BH H h1 h2 BH H h1 Figure 4-1. Sketch of a single bore (w ave sign denotes turbulent roller region). For a wave with frequency f, the average power dissipa ted per unit area can be expressed as: 31 4H DBgf h, (4-5) where B is a calibration parameter which cont rols the intensity of the dissipation. B also accounts for the differences in vari ous breaker types, and is considered as a function of the area of the surface rolle r of the breaker. In application to random waves across t he surf zone, Battjes and Janssen  proposed: 21 4brmsDBgfQH, (4-6) where f is the mean frequency of the energy s pectrum, used as a representative value of f, and bQ represents the fraction of broken waves and can be estimated using a Rayleigh wave height probability distribution function truncat ed at some maximum wave height and determined by:
42 21lnbrms bbQH QH (4-7) where bH is the maximum wave height and can be approximated by Miche : 0.88 tanh 0.88bHkh k (4-8) which in the limit of shallow water is bHh (4-9) where the free parameter is equal to the ratio of breaking wave height to depth at which breaking occurs. Consequently, based on field observati ons [Thornton and Guza, 1983], it was shown that wave heights across the surf zone are well described by a full Rayleigh distribution with empirical we ighting functions, given by: 2 3 3 5/2 231 116 1(/)rmsrms rmsHH DgfB hh Hh (4-10) Whitford  included more observations and improved equation (4-10) as: 3 3 5/2 231 1tanh811 16 1(/)rmsrms rmsHH DgfB hh Hh (4-11) For steep beaches where not all waves reach the maximum height and break, Baldock et al.  extended equation (4-6) using a full Rayleigh distribution without the depth limitatio n of nearshore waves, yielding 2 221 4b brms rmsH DgfBHH H (4-12)
43 Later, Janssen and Battjes  and Alsina and Baldock  modified equation (4-12) to enhance the dissipation le vels on steep beaches and non-saturated surf zones: 332133 exp-1424rmsDBgfHRRRerfR h (4-13) where erf represents the error function and /brmsRHH In order to represent the energy spread in the frequency domain, the present study will use the mean frequency of the local spectrum Additionally, equation (4-13) is used in this study to predict wave transformation over a steep slope (i.e., reef environments), Mase and Kirby  and Kaihatu and Kirby  estimated a dissipation rate with a frequency dependence, to accommodate em pirical observations. The result was that energy is lost more strongly at higher frequencies. Wave breaking models described above can be modified to acc ount for a frequency dependent mechanism: 2 1 2 11N j j j N jj ja PFFFGf Gfa (4-14) where 2 j jGff represents the dissipation tr end obtained from the data, and F is a weighting coefficient (01 F ). When 1F ( 1 PF ), the dissipation rate is independent on frequency. Mase and Kirby  and Kaihatu and Kirby  predicted the spectral evolution on laboratory experiments data well using 0.5 F however, the dissipation trend should be examined with field observations. To simulate the spectral distribution of wave breaking, a parameterization has to be implemented in the wave-breaking dissipati on rate in the deterministic model and the stochastic model. As mentioned above, thes e wave breaking models give the total
44 energy dissipation rate due to breaking but not for each discrete frequency. Here this study adopts the approach propo sed by Eldeberky and Battjes . They formulated the wave breaking dissipation rate in the s pectral domain with two hypotheses. The first is that the dissipation does not interact wi th triad interactions and other processes affecting the wave evolution. The second is that the total energy dissipation rate distributed over the spectrum is proportion to the spectral density on each frequency, thus not affecting wave spectral shape. With these hypotheses, Eldeberky and Battjes  presented the quasi-linear dissipation expression of the complex amplitudes j j totda 1D =-a dx2F (4-15) where totF is the total energy flux given by N 2 totjj j=11 F=aC 2. (4-16) The root-mean-squared wave height is calculated as: N 2 2 rmsj j=1H=4a. (4-17) The wave breaking dissipation rate can be expressed as: 2 14 = j N j rms jD gCH. (4-18) Wave breaking models mentioned abov e contain two free parameters, B and Accurate parameterizations can provide good qualitative modelin g predictions. These two parameters are interdepende nt and can be combined into a single parameter, e.g., the value of B is held constant at 1 and the value of is varied with processes of
45 calibration [Roelvink, 1993]. Therefore there is effectively only one degree of freedom in tuning the model to a measured wave height variation. Ther efore, the tuned values of that implicitly take into account variations in B, and are not necessarily comparable with field observations of /rmsHh from laboratory and field observations. Parameterization of was found to correlate with the local slope, the offshore wave steepness and the dimensionless depth kh Battjes and Stive  calibrated equation (4-6) by estimating an optimal value of under a constant 1B and obtained: 0=0.5+0.4tanh(33S) (4-19) Nairn  modified (4-19) as: 0=0.39+0.56tanh(33S) (4-20) Ruessink et al. [ 2003] calibrated equation (4-12) by estimating an optimal value of under a constant 1B and obtained: 0.760.29kh (4-21) As mentioned above, parameter B represents the percent age of aerated water on the face of the wave, which is a measure of the intensity of breaking. The value of B is expected to be function of the breaking wa ve characteristics. The condition 1B corresponds to fully develope d bores. Numerous tests for gentle slopes have shown that this parameter is (1)O. However, B is not well known for steep slopes. For a fully predictive model, it is desirable to define B in terms of wave parameters that can be estimated.
46 Thornton and Guza  determined 0.42 from the data by observations of rmsH at varying depth, and fitting equation (4-10) to the data by adjusting the parameter B. Massel and Gourlay  implem ented the periodic bore equation (4-5) into a extended refraction-diffraction equation to pr edict monochromatic wave transformation on reef face, and investigated B. They assumed on a slope tan1/40 corresponding to the result of laboratory ex periments [Singamsetti and Wind, 1980] 0.13 0 00.937tanH L (4-22) B was found to be related to bottom sl ope, incident wave parameters, and the characteristic water depth at the r eef through the nonlin earity parameter cF [Gourlay, 1994] which is defined as: 220.127.116.11 0 1.75c rgHT F h (4-23) where g is the gravity acceleration, T is the wave period and rh is the still water depth over the reef edge. cF is proposed to be a suitable parameter in classification of wave transformation regimes over r eefs. In particular, when 150cF waves plunge on the reef edge, and the amount of wave energy reaching the shore is small. When 100cF waves spill on the reef top, but the greater part of energy is transmitted over the reef top. 4.2 Fluid Mud-indu ced Wave Damping Theoretical formulation of mud-in duced wave dissipation is based on the assumption that wave motion reaches the bo ttom and interacts with bed. In this study, the formulation of Ng  is used in par ameterization of mudinduced dissipation of
47 wave energy. Ng  formulated the vi scous dissipation mechanism as a boundary layer simplification of the two-layer viscous fluid model of Dalrymple and Liu . The model is based on the assumption that the depth of the mud layer and the Stokes' boundary layer thickness are the same orde r of magnitude as t he wave amplitude, which is much smaller than the wavelength Wave steepness is used to get an ordering parameter and expressed as: 1mmkakdk (4-24) where a is the wave amplitude, md is the depth of the mud layer, 2/m is the Stokes boundary layer thickness, is the fluid kinematic viscosity, and is the angular wave frequency. Fluid mud and overlying water are a ssumed to be Newtonian, laminar, and incompressible. For this twolayer fluid mud system (Figur e 4-2), the following set of equations for continuity and conserva tion of momentum are described: 0jjuw xz (4-25) 2 2 21jjjjj jjuuuPu uwO txzxz (4-26) 21 0j jP O z (4-27) where the subscripts j wm denote water and mud, respectively, P is dynamic pressure, and ,uw are the horizontal and vertical components of fluid velocity, respectively. A local coordinate for the boundary layer is introduced as mnzhd
48 which is positive upward from the botto m of the mud layer. The rigid bottom, 0n noslip condition is set: 0 mmuv. (4-28) The displacement at t he wave-mud interface on mnd can be expressed as ,ikxtxtbe (4-29) where b is the amplitude of the displacemen t. The continuity of velocity and stress components at the interface of wave and flui d mud are expanded as Taylor series about mnd and are written as: 2 wm wmuu uuO nn, (4-30) 2 wm wmww wwO nn, (4-31) 22 2 22 wwmm wwmmuuuu O nnnn, (4-32) 2 wm wwmmPP PgPgO nn. (4-33) The kinematic condition at the interface mnd is jwO t (4-34) First-order analytical solution of the set of equations ((4-28)(4-34)) is based on the asymptotic theory. As a result of this scaling, the wa ve number of a dissipative mode can be written as 1212, with kkkkk, (4-35)
49 where the first order term 1k is real and corresponds to the nondissipative dispersion relation: 2 11tanhgkkh. (4-36) The second order 2k is complex and expressed as 1 2 111sinhcoshBk k khkhkh (4-37) The imaginary part of 2k represents the wave dissipation rate: 1 2 111Im() Im() sinhcoshmBk k khkhkh (4-38) where B is a complex coefficient. The real and imaginary parts of B are: 112 1 3Re() 2m mkBB Bkd B (4-39) 112 3Im() 2mkBB B B (4-40) where 2 222 1221sinhcoshcoshsinh Bdddd 2 22221coshcossinhsin dddd 21coshsinhcos ddd (4-41) 2 2 2221sincos21sinhcoshsin Bddddd (4-42) 22 22 3coshsinhcossinhcoshsin Bdddddd (4-43) in which /wm 1/2//mwmw and / mmdd where is the density.
50 n z x z=0 h dmrigid bottom fluid mud n=0(,) x t (,) x t a n z x z=0 h dmrigid bottom fluid mud n=0(,) x t (,) x t a Figure 4-2. Sketch of tw o-layer fluid mud system. 4.3 Wind Input and Whitecapping 4.3.1 Implementation of Source Terms into the Nonlinear Wave Model The stochastic and deterministic model s can be supplemented with numerical modules for wind input and whitecapping based on the source terms of the spectral wave model SWAN (Simulating Waves Nearshor e). It computes the evolution of wave action density by the spectral acti on balance equation [Booij et al., 1999]: y tot xcN cNcNS cN N txy (4-44) where the action density is defined as ,/NE with ,E the power spectral density, the relative radian frequency and propagation direct ion. Propagation velocities in spectral space are xxxcCU and y yycCU where /xyC is the group velocity in the x and y directions, / xyU is an ambient current, c and c are the
51 propagation velocities in spectral space The term totS represents processes that generate, dissipate or r edistribute wave energy: 34totinwcbotbrknlnlSSSSSSS (4-45) The terms on the right hand side describe transfer of energy from the wind to the waves, inS dissipation of energy due to whitecapping, wcS dissipation due to bottom friction, botS depth-induced breaking, brkS and nonlinear transfer of wave energy due to triad (three wave) interactions, 3nlS which is dominant in shallow water, and nonlinear transfer of wave energy due to quadr uplet (four-wave) interaction, 4nlS which is dominant in deep water. Here, we simplify the spectral wave action equation (4-44) by bringing it from twodimension to one-dimension, steady state and no ambien t current. These yield the following: totECS x (4-46) In order to add source term s from SWAN into the st ochastic and deterministic models, the formulation of the linear part of equation (2-9) is used: db b dx (4-47) Multiplying equation (4-47) by j b and adding the resulting equation to its complex conjugate results in: 222 d bb dx (4-48) which can be written as:
52 222 d aCaC dx (4-49) Therefore, the dissi pation/growth rate j can be written as 1 2tot j j jS EC (4-50) 4.3.2 Wind Input Strong and steady wind conditions over time scales of hours are characteristic of extratropical storms. Two mechanisms for the generation of waves by wind are taken into account. One is a resonance between the free surface and wind-induced pressure fluctuations proposed by Phillips . The pressure distribution induced by wind at the sea surface is random, and propagates in a more or less frozen pattern over the surface with wind speed. This can be Fourier transformed to produce harmonic pressure waves that propagate with wind sp eed. If this harmonic pressure wave remains in phase with a free harmonic surface wave, then the wind energy is transferred from the pressure wave to t he surface wave. The energy input by this mechanism contributes to the initial stages of wave growth and varies linearly with time. The second mechanism is the feedback of wave-induced pressure fluctuations, proposed by Miles . Once the water surface is distributed, it in turn distributes the air flow over it and causes a greater transfer of energy from wind to waves. This results in an exponential growth of wave energy. Sinc e these mechanisms [Phillips, 1957; Miles, 1957] act independently from each other, the wave growth by wind is expressed as ,,inSABE (4-51) where A is the linear growth term,
53 3 4 21.510 max0,cos 2wAuH g (4-52) in which the cut-off function H is 4 *expPMH (4-53) with *0.13 2 28PMg u (4-54) where is the direction of waves, w is the wind direction, H is the filter and *PM is the peak frequency of the fully developed sea stat e according to Pierson and Moskowitz  as reformulated in terms of friction velocity. BE describes the exponential growth by wind. Two expressi ons are available in the SWAN model. This study uses the expression proposed by Komen et al. : *max0,0.2528cos1a w wu B c (4-55) where a and w are the densities of air and water respectively, and *u is the friction velocity of the wind, 22 *10DuCU (4-56) in which 10U is the wind speed at 10 m elevation above the sea surface, DC is the winddrag coefficient [Wu, 1982] determined as: 3 10 3 10101.287510 for <7.5/ 0.80.06510 for 7.5/DUms C UUms (4-57)
54 By obtaining a fit to observed spectr a, through the experim ental datasets of Snyder et al.  and Plant , Yan  proposed: 2 ** 1234coscoscosfitwwwuu B YYYY cc (4-58) where 1Y 2Y 3Y and 4Y are coefficients. Van der We sthuysen et al.  improved the formulation and got 2 14.010 Y 3 25.5210 Y 5 35.210 Y and 4 43.0210 Y Equation (4-50) is used herein to get t he growth term due to wind: 11 22tot jfit jjjjS A ECCE (4-59) 4.2.4 Whitecapping Wave breaking in deep water (e.g., not depth-induced wave breaking) is called whitecapping. It involves highly nonli near hydrodynamics on a wide range of scales, from gravity surface waves to capillary wa ves, then down to turbulence [Holthuijsen, 2007]. The sink term of whitecapping adopt ed the expression of Alves and Banner , which is due to the fact that it c an be applied to mixed sea-swell conditions and in shallow water: 011 2 2 22 4,tanh,p p a wcds rBk SCkdgkE B (4-60) where dsC is a tunable parameter. The density function ()aBk is the azimuthalintegrated spectral saturation which is ca lculated from frequency space variables as follows: 2 33 0,agd BkkEdckE dk (4-61)
55 rB in equation (4-60) is a threshold sa turation level. When ()arBkB waves break and the exponent p is set equal to a calibration parameter 0 p When ()arBkB there is no breaking, but some residual dissipation prov ed necessary. This is obtained by setting 0 p A smooth transition betw een these two situations is achieved by Alves and Banner ; 00() tanh101 22a rppBk p B (4-62) and ** 03tanh0.1 uu p cc (4-63) where is a shape parameter (default value 26 in SWAN). The source term of whitecapping c an be expressed as the general form wcwcSE. The dissipation term due to whit ecapping is obtained through equation (4-50) as: 011 2 22 4() 1 tanh 2p a wcds rBk Ckhgk CB (4-64)
56 MODEL APPLICATION TO REEF ENVIRONMENT 5.1 Model Integration The stochastic model requires knowledge of the power spectrum at the offshore boundary for its integration. The measured time series at the sensor 2 (refer to Figure 3-1) are used as the offshore boundary condition, which is analyzed using a standard fast Fourier transformation algorithm to obtain an initial wave spectrum. The initial wave spectrum is obtained by averaging over 68 sets of wave spectrum, each with 81 frequency modes and a frequency resolution 0.0049 Hz f. The number of frequency modes used in the simulation is chosen based on a high cutoff frequency of 0.4 Hzf The deterministic model requires knowledge of the set of complex amplitude and phases at offshore boundary for its integration. The measured time records at sensor 2 are used as the offshore boundary condition, and the measurements are analyzed using a standard fast Fourier transformation algorithm to obtain 68 sets of amplitude and phase, each with 81 frequency modes and a frequency resolution of 0.0049 Hz' f for a high frequency cutoff at 0.4 Hzf Following numerical integration using the Adams-Moulton-Bashford method, the set of amplitudes and phases is obtained at each location. The numerical integration is needed to execute each set, and the results are ensemble averaged to obtain the spectra. 5.2 Calibration As mentioned above in Chapter 4.1, B and J could be combined into one coefficient (i.e., by setting B=1), however, physical insight would not be observed. Alternatively, distinct values for these two parameters may be appropriate if the surf zone conditions or breaker types differ significantly from those for which they were CHAPTER 5
57 originally proposed. In this study, the linear wave model (i.e., the triad wave interactions term is excluded) is used to calibrate parameters by minimi zing the error between measured and modeled rms wave height: 2 mod 2 11mea N rmsnrmsn mea n rmsnHxHx e N Hx (5-1) where 4 N is the number of data points fo r tuning (sensor 5, 6, 7 and 8), obs rmsH and modrmsH are the measured and m odeled rms wave height, respectively. The rms wave height is defined as 2 18f rms fHSfdf (5-2) where Sf is the energy spectral density. The data are high-pass filtered with a low frequency cutoff 10.5 p ff to exclude infragravity frequen cy, and low-pass filtered with a high frequency cutoff 20.4 fHz Considering the model efficiency, the st ochastic model is chosen to obtain the optimal parameters. Different from the wave height, parameters of the spectral evolution are calibrated by minimizing the rms er ror between the logarithms of measured and modeled spectral densities 2 11/2 2 mod1 log(,)log(,)f meann f neSxfSxfdf N (5-3) where meaS and modS are the measured and modeled energy wave spectral densities, respectively. Dissipation-ra te weighting coefficient F is calibrated with ( B ) pairs to minimize the error at high frequencies.
58 In spectral calculations of sea surfac e elevation data, the low-frequency band is usually truncated, and swell and sea band are integrated to obtain significant wave height. However, shoaling commonly involves energy transfers from the peak frequency to lower and higher frequencies. In the present study, low-frequency waves are assumed to be non-breaking. Therefore, the wave br eaking model excludes the dissipation in the infragravity waves (0.5 p ff ). Two calibration results are shown in Table 5-1. The first is the calibration of the linear model (stochastic model excluding the triad wave interactions term). The rms error is estimated with equation (5-1). The second is the ca libration of the nonlinear model (stochastic model) by tuning both and B two distinct parameters. The first calibrations set 1 F because the linear model is unabl e to account for spectral evolution. Nonlinear stochastic model is calibrated in the second calibration, where B and F are all tuned by equation (5-3). 5.3 Discussion Laboratory and field observations of random wave transformation in the inner surf zone have been predicted accurately with wave breaking models (ment ioned in Chapter 4.1). Many applications of wa ve breaking models evaluate wave transformation in the saturated surf zone (wave height in t he inner surf zone be ing independent on the incident wave height or energy). In contra st, on steep beaches and r eefs, the surf zone is commonly very narrow and there is not su fficient distance for all the incident short wave energy to be dissipated. In reef environm ents, wave breaking occurs much closer to the reef edge. As a result, an increase in t he offshore incident waves will result in an increase in the wave height in the inner surf zone, i.e., the surf zone is unsaturated and
59 wave groups may still be apparent in the inner surf zone [Nairn, 1990; Battjes and Stive, 1985]. Although parametric wave breaking model s were proved to be accurate in describing wave height transformations across saturated zones, there are few comparisons of such models against wave height data from uns aturated surf zone. Figure 5-1 shows optimal values of and B calibrated using t he linear model and the nonlinear stochastic model. It can be seen here that optimum values for the nonlinear model are slightly larger than th ose calculated using the linear model. There are two possible causes for the difference. T he first is due to the different method of calibration. The linear model parameters ar e evaluated based on wave heights, but the nonlinear model parameters are evaluated based on spectr al densities. Higher means wave heights increase further onshor e after breaking, in contrast, small means breaking waves are initiated further offshore. The second is that the nonlinear model with frequency-dependent dissipation is more efficient than the linear model which is frequency-independent. This is tr ue so for the nonlinear model, energy is transferred toward higher frequency spectr al bands with higher dissipation. Figure 5-2 shows the variati on of optimal values of with offshore wave steepness for the data sets. Parameterizations of as a function of offshore steepness [Battjes and Stive, 1985a; Nair n, 1990] are examined to compar e with optimal values of As mentioned above, parameterizations of are based on equation (4-6) [Battjes and Janssen, 1978] for constant 1B In this study, equation (4-13) [Janssen and Battjes, 2007] is used to calibrate There is a clear di screpancy between optimal values of and the existing parameterizations [B attjes and Stive, 1985a; Nairn, 1990]. Especially for small offshore steepness, optimal values of are approximately twice
60 previous parameterizations. It is shown that empirical parameterizations decrease with decreasing offshore wave steepness. By contrast, optimal values of (obtained by both linear model and the nonlinear models) reveal a divergent trend occurs more or less at offshore steepness below 0.015. Figure 5-3 shows the observed and calibrated values of versus offshore wave steepness. A monochromatic-wave experimenta l formulation [Singamsetti and Wind, 1980] is also shown in the figure: 3 20 1 0tan H L (5-4) where 10.937 20.155 30.130 are coefficients, calibrated for deepwater steepness 000.02/0.06 HL and 1/40tan1/5 Calibrated and observed (most waves break as plunging at the reef edge) values of increase with decreasing offshore steepness, corresponding to the trend by Singamsetti and Wind  but with smaller values of Present and previous experiment al trends, indeed, verify that parameterizations based on offshore steepness [Battjes and Stive, 1985a; Nairn, 1990] could induce large errors in wave predicti on. It can be concluded that reef breaking wave conditions can not be characterized onl y by offshore steepness. Therefore, using only 0S is not sufficient to specify Figure 5-4 shows the optimal values against the nondimensional water depth kh The general trend of decreasing with increasing kh is in agreement with the relation proposed by Raubenheime r et al. , but contradi cting the parameterization of Ruessink et al. . The trend was co nsistent with Raubenheimer et al. ,
61 who used a one-dimensional depth-aver aged nonlinear water equation and field observations: 01tan CC kh (5-5) where 0C and 1C are adjustable coefficients. T he coefficients have been fitted to optimal values of by the nonlinear model, yielding the best fit values of 00.431 C 11.032 C with the correlation coefficient is 0. 8524. The reef-face slope is given with tan1/10.6 Based on the relations of with 0S and kh, it can be inferred that wave length (i.e., wave period) is a significant factor. Apotsos et al.  tuned using several wave breaking models [Baldock et al., 1998; Janssen and Battj es, 2007; Thornton and Guza, 1983a; Battjes and Janssen, 1978]. All cases show that increases as the hyperbolic tangent of the deepwater wave height, therefore, offshore wave height is also a significant factor. Therefore, is expected to be a functi on of offshore wave height, offshore wave length and local sl ope that is correlated with the surf similarity number (equation (3-1)). Figure 5-5 shows results of varying with 0 It is reveal that should also be function of 0 As mentioned in Chapter 4.1, Mass el and Gourlay  proposed that B is closely related to t he nonlinearity parameter CF which describes the type of breaking wave. Figure 5-6 reveals the optimal B values is dependent on CF In the present study, 150CF reveals waves start to break on the reef edge at which Sensor 7 is located. In other words, B dominates breaking intensity from S ensor 7 to Sensor 8. Values of B
62 was varied between 1.19 and 1.57 for the nonlinear model and between 1.28 and 2.2 for the linear model. Linear least squares fits through the values of B is 01cBbbF (5-6) where 01.245 b and 5 19.6510 b for the linear model with the correlation coefficient 0.9281 and 01.169 b and 5 14.3110 b for the nonlinear model with the correlation coefficient 0.8517. It can be inferred that triad wave interaction dominates energy transfer, and affects the numerical values of B (e.g., breaking intensity). Figure 5-7 shows measured dissipation ra te versus frequency using the energy flux spectra (04.9 m H ,12 spT ) of consecutive wave sensors of Sensor 6, Sensor 7, Sensor 8, and Sensor 9. Measured dissi pation rate between Sensor 6 and Sensor 7 contains the effects of shoaling, wave br eaking and nonlinear energ y transfer. To study the dissipation rate induced by wave breakin g, sensors located on the reef top (Sensor 79) are utilized because the effect of shoaling can be neglect ed and nonlinear energy transfer are weak on the flat bottom. It is found that the dissipati on rates of Sensor 78 and Sensor 89 increase with increasing frequen cy at higher frequencies (roughly as 2f ). The inferred dissipation rates in the surf zone are shown to increase with increasing frequency, in qualitative agr eement with Kaihatu and Ki rby , Mase and Kirby  and Chen et al. . The optimal values of F are scattered over 0.5 to 0.65 (Table 5-1). Chen et al.  proposed that spectral evolution using a Boussinesq-equation model is nearly insensitiv e to the frequency-depende nt distribution of the dissipation, because increased dissipation at high frequencies is compensated by nonlinear wave energy transfers. However, in the present st udy, the simulated spectra are sensitive to
63 the frequency dependence of the di ssipation. One possible expl anation of this disparity is that plunging breakers induce strong turb ulence, and therefore, dissipate more energy at high frequency. Figure 5-8 shows the cross-shore evol ution of spectral energy flux (03.9 m H 10 spT ). Both model simulations and laboratory observations show that energy is spread from peak frequency band to hi gher and lower frequency bands, as waves propagate over the reef face and begin to break. Figure 5-9 to 5-15 show energy flux spectr a at various locations computed by the stochastic model and the deterministic model. It is shown that spectra in seven cases are well predicted by the stochastic model an d the deterministic model. Not only energy transfers to the higher harm onics over reef face (Sensor 5 and Sensor 6), but also energy dissipation at reef flat (Sensor 7 and Sensor 8) are captured by the model. These results use the breaking parameters calibrated for each run. To test the performance of the determinist ic model, the parameters ( B ) calibrated for the stochastic model are applied. Spectra comp uted by the deterministic model in sea bands are in good agreement with the measur ements as well as computed by the stochastic model in the shoaling region. Ho wever, the deterministic model tends to overestimate the dissipation due to wave br eaking. In the present experiment, only one instrument was located in the middle of reef-top. To investi gate the evolution of spectral energy on the reef, increasing the density of sensors both on the re ef-face and the reeftop (e.g., inside the surf zone) is necessary. Figures 5-16 to 5-22 shows the cross-shor e evolution of the normalized spectral flux at the sea, swell, infragravity bands and integrated over the entire frequency
64 domain for measurements and t he stochastic model. The stochastic model accurately predicts evolution of energy flux in the swell frequency band. The observed energy flux in sea frequency band reaches its maximum at Sensor 6 and starts to decay. The model is able to capture the energy increase with in the sea frequency band due to triad wave interactions over reef face, and the decay induced by wave break ing onshore the reef edge. Additionally, the observed energy flux in the infragravity band increases as waves propagate over the reef face, however, dram atic energy decay occurs around the reef edge. Prediction of infragravity band spectral flux is less accura te due to the limitation of model applicability. The reflection of infr agravity waves is neglected owing to the limitation of both models. T herefore, infragravity waves are represented only crudely in the present model formulation. Numerical si mulation results indicate that infragravity energy loss is partly due to energy transfer from low frequencies to higher frequencies. Numerical simulations of the nonlinear stoc hastic model excluding the wave-breaking dissipation in infragravity band result in the underestimation of infragravity wave energy.
65 Table 5-1. Optimal parameters obta ined from linear and nonlinear models Linear Model Nonlinear Model Case no. B F e B F e 8_39_32 0.578 1.458 1.00 0.031 0.6651.2640.405 3.997 10_39_32 0.551 1.276 1.000. 039 0.6291.1870.525 1.013 12_33_32 0.510 1.482 1.000. 035 0.7251.5230.569 1.453 12_49_32 0.640 1.465 1.000. 053 0.8001.2000.500 4.145 14_52_32 0.671 1.561 1.000. 038 0.8501.4000.500 3.741 16_54_32 0.682 1.723 1.000. 043 0.9001.4000.500 10.26 20_53_32 0.738 2.193 1.000. 038 0.9481.5680.652 9.284 0.2 0.4 0.6 0.8 1 1.2 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 B Figure 5-1. Optimal versus optimal B Values calibrated by linear wave model (cross) and by the nonlinear stochastic wave model (circle).
66 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 S0 Figure 5-2. Calibrated versus offshore wave steepness, 0S Solid line denotes the parameterization of Battjes and Stive ; dashed line denotes the parameterization of Nair n . Calibration by the linear and the nonlinear stochastic models are denoted by crosses and circles.
67 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 S 0 Figure 5-3. Observed and calibrated versus offshore wave steepness, 0S Red squares denote measured values at t he sensor 7 (i.e., reef edge). Crosses and circles denote calibrated values based on the linear model and the nonlinear stochastic model, respective ly. Solid line corresponds to the parameterization of Singamsetti and Wind .
68 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.2 0.4 0.6 0.8 1 1.2 kh Figure 5-4. Calibrated versus the non-dimensional depth kh The dashed line denotes the parameterization by Ruessink et al. . Solid line denotes the relation proposed by Raubenheimer et al. , with tuned coefficients 00.431 C and 11.032 C with the correlation coefficient is 0.8524.
69 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 0 Figure 5-5. Observed and calibrated against surf similarity parameter, 0 Red squares denote values meas ured at the sensor 7 (i .e., reef edge). Crosses and circles denote calibrated values based on the linear model and the nonlinear stochastic wave model, respectively.
70 0 2000 4000 6000 8000 10000 1 1.2 1.4 1.6 1.8 2 2.2 2.4 FcB Figure 5-6. Optimal B versus the nonlinearity parameter, cF Cross and circle points denote calibrated values by the li near wave model and the nonlinear stochastic wave model, respectively. The least squares linear fit of the calibrated values of B using nonlinear model and linear model to cF are 51.1694.3110cBF (red line) with the corre lation coefficient is 0.8517 and 51.2459.65310cBF (blue line) with the correlation coefficient is 0.9281, respectively.
71 0 0.1 0.2 0.3 0.4 0.5 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Frequency (Hz)Dissipation rate (1/m) Sensor 6-7 Sensor 7-8 Sensor 8-9 f2 Figure 5-7. Energy dissipation rate versus frequency (04.9 m H 12 spT ). The red line is proportional to 2f The dissipation rate between two consecutive sensors is plotted with dotted (Sensor 67), dash-dotted (Sensor 78) and dashed (Sensor 89) line, respectively.
72 (a)Frequency (Hz) 0.05 0.1 0.15 0.2 0.25 Flux Spectral Density (m3) 0.7 0.8 0.9 1 1.1 1.2 (b)Frequency (Hz) 0.05 0.1 0.15 0.2 0.25 947 968 1241 1276 1313 1400 -40 -20 0 235678 Position (m)Depth(m) Figure 5-8. Cross-shor e spectral evolution (03.9 m H 10 spT ). Panel (a) is the model simulation, and panel (b) is t he laboratory observation. Bottom panel represents the profile wi th the location of sensors (red circles).
73 100 102 Sensor 2; Hs= 4.0 m. (a) 100 102 Sensor 3; Hs= 4.0 m. (b) 100 102 Sensor 5; Hs= 3.5 m. (c) 100 102 Sensor 6; Hs= 3.6 m. (d) 0 0.05 0.1 0.15 0.2 0.25 0.3 100 101 Sensor 7; Hs= 2.6 m. (e) Frequency (Hz)Flux density (m3) 0 0.05 0.1 0.15 0.2 0.25 0.3 100 101 Sensor 8; Hs= 1.6 m. (f) Frequency (Hz)Flux density (m3) Figure 5-9. Energy flux spectral dens ity at various locations of sensors (03.9 m H 8 spT ). Measured spectra (red circles), computed spectra by the stochastic model (blue curves) and computed spectr a by the deterministic model (black dashed curves).
74 100 102 Sensor 2; Hs= 3.9 m. (a) 100 102 Sensor 3; Hs= 3.8 m. (b) 100 102 Sensor 5; Hs= 3.7 m. (c) 100 102 Sensor 6; Hs= 3.9 m. (d) 0 0.05 0.1 0.15 0.2 0.25 0.3 100 101 Sensor 7; Hs= 3.0 m. (e) Frequency (Hz)Flux density (m3) 0 0.05 0.1 0.15 0.2 0.25 0.3 100 101 Sensor 8; Hs= 1.6 m. (f) Frequency (Hz)Flux density (m3) Figure 5-10. Energy flux spectral dens ity at various locations of sensors (03.9 m H 10 spT ). Measured spectra (red circle s), computed spectra by the stochastic model (blue curves) and co mputed spectra by the deterministic model (black dashed curves).
75 100 102 Sensor 2; Hs= 3.3 m. (a) 100 Sensor 3; Hs= 3.4 m. (b) 100 102 Sensor 5; Hs= 3.3 m. (c) 100 102 Sensor 6; Hs= 3.6 m. (d) 0 0.05 0.1 0.15 0.2 0.25 0.3 100 101 Sensor 7; Hs= 2.9 m. (e) Frequency (Hz)Flux density (m3) 0 0.05 0.1 0.15 0.2 0.25 0.3 100 101 Sensor 8; Hs= 1.5 m. (f) Frequency (Hz)Flux density (m3) Figure 5-11. Energy flux spectral dens ity at various locations of sensors (03.3 m H 12 spT ). Measured spectra (red circle s), computed spectra by the stochastic model (blue curves) and co mputed spectra by the deterministic model (black dashed curves).
76 100 102 Sensor 2; Hs= 5.0 m. (a) 100 102 Sensor 3; Hs= 5.0 m. (b) 100 102 Sensor 5; Hs= 5.0 m. (c) 100 102 Sensor 6; Hs= 5.1 m. (d) 0 0.05 0.1 0.15 0.2 0.25 0.3 100 101 Sensor 7; Hs= 3.6 m. (e) Frequency (Hz)Flux density (m3) 0 0.05 0.1 0.15 0.2 0.25 0.3 100 101 Sensor 8; Hs= 1.8 m. (f) Frequency (Hz)Flux density (m3) Figure 5-12. Energy flux spectral dens ity at various locations of sensors (04.9 m H 12 spT ). Measured spectra (red circle s), computed spectra by the stochastic model (blue curves) and co mputed spectra by the deterministic model (black dashed curves).
77 100 102 Sensor 2; Hs= 5.2 m. (a) 100 102 Sensor 3; Hs= 5.2 m. (b) 100 102 Sensor 5; Hs= 5.6 m. (c) 100 102 Sensor 6; Hs= 5.6 m. (d) 0 0.05 0.1 0.15 0.2 0.25 0.3 100 101 102 Sensor 7; Hs= 4.0 m. (e) Frequency (Hz)Flux density (m3) 0 0.05 0.1 0.15 0.2 0.25 0.3 100 101 Sensor 8; Hs= 2.1 m. (f) Frequency (Hz)Flux density (m3) Figure 5-13. Energy flux spectral dens ity at various locations of sensors (05.2 m H 14 spT ). Measured spectra (red circle s), computed spectra by the stochastic model (blue curves) and co mputed spectra by the deterministic model (black dashed curves).
78 100 102 Sensor 2; Hs= 5.4 m. (a) 100 102 Sensor 3; Hs= 5.3 m. (b) 100 102 Sensor 5; Hs= 6.0 m. (c) 100 102 Sensor 6; Hs= 5.8 m. (d) 0 0.05 0.1 0.15 0.2 0.25 0.3 100 102 Sensor 7; Hs= 4.3 m. (e) Frequency (Hz)Flux density (m3) 0 0.05 0.1 0.15 0.2 0.25 0.3 100 102 Sensor 8; Hs= 2.3 m. (f) Frequency (Hz)Flux density (m3) Figure 5-14. Energy flux spectral dens ity at various locations of sensors (05.4 m H 16 spT ). Measured spectra (red circle s), computed spectra by the stochastic model (blue curves) and co mputed spectra by the deterministic model (black dashed curves).
79 100 102 Sensor 2; Hs= 5.2 m. (a) 100 102 Sensor 3; Hs= 5.2 m. (b) 100 102 Sensor 5; Hs= 6.5 m. (c) 100 102 Sensor 6; Hs= 6.3 m. (d) 0 0.05 0.1 0.15 0.2 0.25 0.3 100 102 Sensor 7; Hs= 4.5 m. (e) Frequency (Hz)Flux density (m3) 0 0.05 0.1 0.15 0.2 0.25 0.3 100 102 Sensor 8; Hs= 2.4 m. (f) Frequency (Hz)Flux density (m3) Figure 5-15. Energy flux spectral dens ity at various locations of sensors (05.3 m H 20 spT ). Measured spectra (red circle s), computed spectra by the stochastic model (blue curves) and co mputed spectra by the deterministic model (black dashed curves).
80 10-1 100 F(x)/F(0) IG Swell Sea Total Data 950 1000 1050 1100 1150 1200 1250 1300 1350 1400 -40 -20 0 235678 Position (m)Depth(m) Figure 5-16. Cross-shore evoluti on of normalized spectral flux (03.9 m H 8 spT ) at upper panel. The curves are and circle s represent model simulations on cross-shore and observations at sensor locations, respectively. The black denotes total spectral flux, the bl ue denotes swell-frequency band spectral flux, cyan denotes sea-frequency band spectral flux and red denotes infragravity-frequency band spectral flux. Th e reef profile with the location of sensors (red circles) is shown in the lower panel.
81 10-1 100 F(x)/F(0) IG Swell Sea Total Data 950 1000 1050 1100 1150 1200 1250 1300 1350 1400 -40 -20 0 235678 Position (m)Depth(m) Figure 5-17. Cross-shore evoluti on of normalized spectral flux (03.9 m H 10 spT ) at upper panel. The curves are and circ les represent model simulations on cross-shore and observations at sensor locations, respectively. The black denotes total spectral flux, the bl ue denotes swell-frequency band spectral flux, cyan denotes sea-frequency band spectral flux and red denotes infragravity-frequency band spectral flux. Th e reef profile with the location of sensors (red circles) is shown in the lower panel.
82 10-1 100 F(x)/F(0) IG Swell Sea Total Data 950 1000 1050 1100 1150 1200 1250 1300 1350 1400 -40 -20 0 235678 Position (m)Depth(m) Figure 5-18. Cross-shore evoluti on of normalized spectral flux (03.3 m H 12 spT ) at upper panel. The curves are and circ les represent model simulations on cross-shore and observations at sensor locations, respectively. The black denotes total spectral flux, the bl ue denotes swell-frequency band spectral flux, cyan denotes sea-frequency band spectral flux and red denotes infragravity-frequency band spectral flux. Th e reef profile with the location of sensors (red circles) is shown in the lower panel.
83 10-1 100 F(x)/F(0) IG Swell Sea Total Data 950 1000 1050 1100 1150 1200 1250 1300 1350 1400 -40 -20 0 235678 Position (m)Depth(m) Figure 5-19. Cross-shore evoluti on of normalized spectral flux (04.9 m H 12 spT ) at upper panel. The curves are and circ les represent model simulations on cross-shore and observations at sensor locations, respectively. The black denotes total spectral flux, the bl ue denotes swell-frequency band spectral flux, cyan denotes sea-frequency band spectral flux and red denotes infragravity-frequency band spectral flux. Th e reef profile with the location of sensors (red circles) is shown in the lower panel.
84 10-1 100 F(x)/F(0) IG Swell Sea Total Data 950 1000 1050 1100 1150 1200 1250 1300 1350 1400 -40 -20 0 235678 Position (m)Depth(m) Figure 5-20. Cross-shore evoluti on of normalized spectral flux (05.2 m H 14 spT ) at upper panel. The curves are and circ les represent model simulations on cross-shore and observations at sensor locations, respectively. The black denotes total spectral flux, the bl ue denotes swell-frequency band spectral flux, cyan denotes sea-frequency band spectral flux and red denotes infragravity-frequency band spectral flux. Th e reef profile with the location of sensors (red circles) is shown in the lower panel.
85 10-1 100 F(x)/F(0) IG Swell Sea Total Data 950 1000 1050 1100 1150 1200 1250 1300 1350 1400 -40 -20 0 235678 Position (m)Depth(m) Figure 5-21. Cross-shore evoluti on of normalized spectral flux (05.4 m H 16 spT ) at upper panel. The curves are and circ les represent model simulations on cross-shore and observations at sensor locations, respectively. The black denotes total spectral flux, the bl ue denotes swell-frequency band spectral flux, cyan denotes sea-frequency band spectral flux and red denotes infragravity-frequency band spectral flux. Th e reef profile with the location of sensors (red circles) is shown in the lower panel.
86 10-1 100 F(x)/F(0) IG Swell Sea Total Data 950 1000 1050 1100 1150 1200 1250 1300 1350 1400 -40 -20 0 235678 Position (m)Depth(m) Figure 5-22. Cross-shore evoluti on of normalized spectral flux (05.3 m H 20 spT ) at upper panel. The curves are and circ les represent model simulations on cross-shore and observations at sensor locations, respectively. The black denotes total spectral flux, the bl ue denotes swell-frequency band spectral flux, cyan denotes sea-frequency band spectral flux and red denotes infragravity-frequency band spectral flux. Th e reef profile with the location of sensors (red circles) is shown in the lower panel.
87 MODEL APPLICATION TO MUDDY SEAFLOORS 6.1 Inverse Modeling of Bottom Mud State The proposed evolutionary sequence of bottom mud rheology (Chapter 3.2) is tested by using the nonlinear wave model in an inverse means. This approach is similar to that proposed by Rogers and Holland . The objective of the inverse modeling approach was to determine the effective viscosity of the bed that results in a best fit between the observed and simulated spectra. However, the analysis presented in Chapter 3.2 suggests that untangling the effects of evolving bed-rheology from other wave growth/dissipation mechanisms is not trivial, in particular due to the possibility of wave nonlinearities modifying the balance of growth/dissipation terms. For example, mud-induced dissipation is typically limited to the swell band, while wind input and whitecapping are active at the high-frequency end of the spectrum [van der Westhuysen et al., 2007]. Despite their relatively well decomposed spectral domains, mud-induced dissipation can partially balance the effect of whitecapping, due to triad interactions transferring energy from high frequencies to the swell band. To account for the effects of triad interaction, the inverse modeling approach was performed using the stochastic model [Agnon and Sheremet, 1997]. While the results show a good agreement with observations, neglecting the directional spread is expected to lead to an overestimation of energy transfers toward low frequencies. To simplify the description of bed rheology, the numerical analysis presented here focuses on the period from March 10th, 12:00 to March 11th, 18:00 (rectangle in Figure 3-3), when the bed state can be identified with some certainty as fluid mud. The model is initialized using spectral observations at T1, and simulated spectra at T2 are compared CHAPTER 6
88 with observations. Tide elevations and surf ace currents are derived from observations (assumed spatially constant over the integr ation domain). Surface winds were produced by the COAMPS model (Coupled Ocean/Atmo sphere Mesoscale Prediction System, [Hodur, 1997]). The stochastic model is supp lemented with numerical modules for wind input, whitecapping and mud-induced dissipat ion as discussed in Chapter 4.2 and 4.3. Depth-induced breaking is not expected to be active during this period. The Newtonian fluid description proposed by Ng  is m odified by introducing an effective kinematic viscosity that depends only on the density of t he fluid mud yielding the dissipation rate. If the mud-layer thickness can be estimated fr om the observations, the dissipation rate is only a function of a singl e mud parameter, the effectiv e viscosity. Because the mud thickness does not appear to change much during the March 10th-11th event, and due to significant uncertainties related to the spatial structure of t he fluid-mud event, the mud layer thickness is assumed constant in time. This implies that density changes in the fluid-mud density/viscosity are mainly the result of hindered settling, consistent with the character of the fluid-mud event. Observations of fluid-m ud layers [Jaramillo et al., 2009] suggest that the characteristic mud thickness values for the March 10th-11th event were between 10-15 cm at T1 and about 5 cm at T2. Based on the observations, the fluid-mud layer was assumed to be const ant in time, and varying linearly in space between 10 cm at T1 to 5 cm at T2. Based on the speed of the FORTRAN impl ementation of the stochastic model (about 10 s on an average desktop computer), the problem was formulated as a constrained nonlinear least-square mini mization (the algorithm coded using the
89 MATLAB fmincon function) seeks the value of the kinematic viscosity that minimizes the deviation 2 mod22 2 1 2(,)(,) 1 (,)N jmeaj j meajFfxFfx e NFfx (6-1) where modF and meaF are the modeled and measured energy flux spectrum at T2, and N is the number of spectral modes used. A best-fit kinematic viscosity value was obtained for each spectral estimate, within a typical by 30 iterations, and with typical s pectral error of 3-5%. The similarity between the observed and the numerically simulated, best-fit frequency distribution of the dissipation rate (Figure 6-1) suggests that the numerical simulations can provide some insight into the structure of the dissipativ e processes. Figures 62, 6-3, and 6-4 show the energy flux spectra and dissipation rates for events and in Figure 6-1. Mud-induced dissipation is the main dissi pative process; wind growth and whitecapping processes modify the high-fr equency end of the spectrum, bending the dissipation curve toward growth; finally, nonlinearities modulate it by further transferring energy from the peak of the spectrum toward the lowe r frequency (infragravity) band and the second harmonic of the spectral peak. Nonlinear effe cts intensify from March 10th 15;00 hrs to March 10th, 21:00 hrs, but decrease signi ficantly on March 11th, 11:00, when the evolution is nearly linear. Because nonlinear ities conserve the energy of the system, their effects are only represented as dissipati on in the net results. The numerical results show that important features of the frequen cy shape of wave dissipation in time are due to nonlinear interactions, particularly the peak at the first harmoni c. The peak of the dissipation distribution follows the spectral peak because of transfers of energy toward
90 both the infragravity band, and high frequencies; narrower distributions of dissipation rate occur when nonlinear intera ction is stronger, e.g., for larger waves. Growth rates in the infragravity band are exclusively due to trans fers of energy from the spectral peak, a nonlinear process. 6.2 Discussion Observations of wave dissipation over the muddy Atchafalaya delta support the hypothesis of an effective coupling bet ween surface waves and bed sediment. The dominant wave dissipation mechanism is wa ve-bottom interaction; the process is triggered by the reworking of bed sediment by waves. Based on backscatter records, several stages can be distinguished in the evolution of the bed: stiff mud at the beginning of the storm, liquefaction and rapid resuspension as the storm intensifies, followed by hindered settling (fluid mud formation). The de-tided net dissipation rate increas es steadily during a storm, with no detectable correlation to fluid mud layers, possibly due to the discontinuous spatial distribution, or low density of the fluid-mud layers. Ne t wave dissipation reaches a maximum of about 50% (over the distance bet ween the two observation stations) in the wake of the storm, when no bed change is detected in the near bed acoustic backscatter data. The evolutionary trend of suspended sedim ent concentration suggests that the maximum dissipation efficien cy corresponds to a soft, underconsolidated state of bed sediment (gelling and slow dewatering). The numerical results suggest that, while mudinduced dissipation dominates wave propaga tion processes in the swell band, nonlinear triad wave interactions determine the shape of the frequency di stribution of the dissipation rate. Nonlinear energy transfers from the spectr al peak toward infragravity
91 waves and toward the high-frequency band regist er in the net dissipation estimates as dissipation at the spectral peak and growth in the infragravit y and high-frequency band. The values of the effective kinematic vi scosity returned by the inverse model are consistent with measurements and with the observed evolution trend of suspended sediment concentration. The analysis supports only in general terms (i.e., triad interacti ons are active) the mechanism proposed by Sheremet and Stone , that explai ns short-wave dissipation as nonlinear energy transfers to ward low frequency waves. However, during the period analyzed here, sustained wind speeds were over 10 m/s and picked up at the end (evening of March 11th, 2006). If this mechanism was active during the event discussed here, the high-frequency damping could have been compensated by windinduced growth. Wave-bottom interaction is the dominant pr ocess, however, triad wave interactions play an integral role in the interpretati on of the frequency distribution of net wave dissipation. Neglecting the effe ct of nonlinearities leads to aliasing nonlinear energy transfers into dissipation effects, and di storts the represent ation of mud-induced dissipation. To fully validate the scenario pr oposed here, further in-situ observations of bed sediment evolution under waves are required.
92 Figure 6-1. Evolution of nor malized (peak value is 1) fr equency distribution of the swell dissipation rate between March 10th 12:00 and March 11th 18:00 (marked with a black rectangle in Figure 3-3). a) Observations. b) Model simulations using the optimal values of fl uid-mud viscosity and density.
93 Figure 6-2. Wave dissipation rates for March 10th 15:00 (ev ent in Figure 6-1). a) Spectral density of wave variance (red circles observations; blue line model simulations). b) Frequency distribut ion of dissipation rates (red-circles net dissipation of observations; blue line net dissipation of model simulations; black dashed line mud i nduced dissipation rate using best-fit fluid-mud viscosity and density; crosse s net dissipation (excluding triadwave interactions) including contribut ion by wind and decay by whitecapping and mud-induced dissipation). Positive va lues correspond to wave dissipation, negative values to wave growth.
94 Figure 6-3. Wave dissipation rates for March 10th 21:00 (ev ent in Figure 6-1). a) Spectral density of wave variance. b) Frequency distribution of dissipation rates. Figure 6-4. Wave dissipation rates for March 11th 11:00 (ev ent in Figure 6-1). a) Spectral density of wave variance. b) Frequency distribution of dissipation rates.
95 CHAPTER 7 CONCLUSIONS AND RECOMMENDATIONS 7.1 Conclusions The purpose of this research is to apply numerical models for studying nonlinear transformation of wave spectra in two dissi pative coastal environm ents, a fringing reef coast and a muddy seafloor coast. The domi nant wave energy dissipation over reefs and muddy seafloor are depth-induced wa ve breaking and viscous mud-induced damping, respectively. In order to describe these complex wave processes, a numerical model is required to account for nonlinear wave interactions, wave breaking, muddy bottom dissipation, and other gener ation/dissipation terms. There are two classes of unidirecti onal nonlinear wave models developed to predict energy transfer among wave spectrum due to triad wave interactions. The deterministic model (phase-resolving) model, which needs more precise input of wave data, both the modal amplitudes and phases, is computationally time-consuming. The stochastic (phase-averaged) model, which on ly needs modal amplitudes with average initial phases, offers a lower computational cost. Results of previous modeling studies compared well with experimental data of non-breaking waves in shoaling region. Therefore, this study is based on the deterministic and stochastic models [Agnon and Sheremet, 1997] which reproduce t he effects of nonlinear triad interactions. Additionally, in order to give more accu rate results on steep profiles the models are modified to account for higher-order bottom effects. Various parametric wave breaking models expressed as total rate of energy dissipation and empirical paramet erizations in wave breaking models are reviewed. Under the hypothesis of total energy dissipat ion without affecting the spectral shape
96 [Eldeberky and Battjes, 1996], the dissipation rate can be formulated in spectral form, which can be incorporated into the nonli near models. Despite of frequency-independent dissipation rate proposed, a dissipati on term dependent on frequency was indeed observed in the surf zone and needs to be considered. A paramet ric wave breaking model [Janssen and Battjes, 2007], predicting significant waves accurately on steep beaches, is chosen to incorporate into the nonl inear models to predict the evolution of wave spectrum over a fringing reef. For the reef study, three calibration parameters (, BF ) in the parameterization are obtained by minimizing the normaliz ed root-mean-squared e rror between measured and modeled total energy and spectral shapes. The stochastic model that accounts for the frequency-dependent dissipation accurate ly captures the peak frequency energy transfer to second harmonic frequency and infr agravity frequency. Wi th the parameters calibrated by the stochastic m odel, the deterministic model is also able to reproduce the nonlinear energy transfer to high frequency. However, the simulations result in underestimation of spectral densities in the swell band. Good comparison of model predictions with observations in extreme wave conditions demonstrates the robustness of pa rameterization of wave breaking effects. The effects of two free parameters ( and B ) in the parametric wave breaking model are investigated. Optimal values are found to be close to observed values that correspond with physical mechanism. A discrepancy is found in between the optimal values in this study and values that we re previously calibr ated using data from laboratory and field experiment s. The optimal parameter is found to be correlated to the reciprocal of offshore wa ve steepness and dimensionless depth kh The optimal
97 value of B is linearly related to the nonlinearity parameter cF The parameter is related to the breaker type. Frequency-depe ndent dissipation term enhances the more accuracy of prediction of spec trum at higher frequencies. In reef experiments, it is observed that infragravity wave energy increases by nonlinear wave interactions as swell and sea propagating over t he reef face, then decreases around the reef edge. The reflection of infragravity waves from beach and the reef face is neglected owing to the limita tion of the models. Therefore, infragravity waves are represented only crudely in the present model formulation. Numerical simulation shows that infragravity energy loss is partly due to energy transfer from low frequencies to higher frequencies. Even numerical simulation excluding the dissipation induced by wave breaking in infragravity band, the results of predicting spectra indicate that the nonlinear models tend to underestimate the content of infragravity wave energy For the muddy seafloor environment study, observations of wave dissipation over the muddy Atchafalaya subaque ous delta support the hypot hesis that there is an effective coupling between surface wave s and bed sediment. The dominant wave dissipation mechanism is found to be wave-bo ttom interaction; the process is triggered by the reworking of bed sediment by wave s. The inverse modeling approach was used to investigate this hypothesis. The numeric al results suggest that, while mud-induced dissipation dominates wave propagation proc esses in the swell band, nonlinear triad wave interactions determine the shape of the frequency distributi on of the dissipation rate. Nonlinear energy transfers from the spectral peak toward infragravity waves and toward the high-frequency band r egister in the net dissipati on estimates as dissipation at the spectral peak and growth in the infragravity and high-frequency band.
98 The analysis supports only in general terms (i .e., triad wave interactions are active) the mechanism proposed by Sheremet and St one , that explains short-wave dissipation as nonlinear energy transfers to ward low frequency waves. However, during the period analyzed, the high-frequency da mping could have been compensated by wind-induced growth. 7.2 Recommendations To develop a better understanding of domains of applicability of nonlinear wave models, and explore the wave breaking a nd mud-induced dissipation. The following recommendations are made for future studies. The applicability of prediction of infr agravity waves by present models is questionable. A more sophisticated m odel that can capture precisely the generation of infragravity waves needs to be developed. Various mechanisms of infragravity energy loss are proposed such as bottom drag, reflection, turbulence, and long-wave breaking need to be explored. The precise dissipation mechanism in reef environment des erves further study. More studies are necessary to narrow down the range of parameters in order to improve the accuracy of wave breaking models. Coupling the nonlinear wave model with a shallow-water equation can offer more accurate prediction. An intriguing question related to wave-m ud interaction in shallow environments is whether characteristics of bottom sedi ment processes can be inferred from surface-wave observations. Wave-bottom interaction is the dominant process, however, triad wave interactions play an in tegral role in the interpretation of the frequency distribution of net wave dissi pation. Neglecting the effect of nonlinearities leads to aliasing nonlinear ene rgy transfers into dissipation effects, and distorts the representat ion of mud-induced dissipation. To fully validate the scenario proposed here, further work is required for in-situ observations of bed sediment evolution under waves.
99 MODIFIED MILD SLOP EQUATION The modified mild-slope equation [Chamberlain and Porter, 1995] has been used to study linear waves in water of variable depth. It was derived by using variational principles and keeping the higher-order terms f2Â’h and 2 Â’ h that were neglected in the original derivation of Berkhoff . In the following study, the hyperbolic form of modified mild-slope equation is derived. The modified mild-slope equation given by Chamberlain and Porter  is written as f f ^ ` 20IIÂ’Â’ cCkcCgrh, (A-1) where I is the velocity potential, c is the phase velocity, C is the group velocity, g is the gravitational acceleration, and k is wavenumber. The function frh is given by f f f f2212rhuhhuhh Â’Â’, (A-2) where the coefficients are defined by f ff f1sechsinhcosh4sinhkhuhKKKKK (A-3) ff ff243232sech4sinh9sinhsinh212sinh 32sinhcosh2cosh3,kkhuh [ [[[[[[[[[[[ (A-4) where the abbreviation 2kh [ has been used. LetÂ’s assume working on the full nonlinear mild slope equation using / ck Z rewrite the equation as fCkCNkZIIÂ§Â’Â’* Â¨, (A-5) APPENDIX A
100 with / gr where is angular frequency, and N repr esenting the nonlinear terms, and substitute ii g Ae (A-6) yields 2iiiCCi AeAekCAeN kkg (A-7) The derivatives of are, with the wave number vector iiiiAeiAeeAiAAe (A-8) 2222 iii A eiAAeAiAiAAe (A-9) Substituting (A-8) and (A-9) into equation (A-7) yields for the first term and second terms of left-hand side iiiCCCC AeiAAeiAAe kkkk (A-10) 222 22 2ii iCC AekAiAiAAe kk CC kCAiCAiAAe kk (A-11) which yields 22iCCi iCAiCAAeN kkg (A-12) which, grouping the terms, becomes 21 2iCC ACCAiAiAiANe kkg (A-13) The substitution
101 1/2ACB kk (A-14) yields 1/21/211 2 22 CCAACAC k (A-15) That is 1/21 22 2CAACCB k (A-16) Equation (A-13) can be written then 1/21/2 1/221 222iiCCC BCAAANe kkkg (A-17) 1/221 2 2iCCi iCBAAANe kkkg (A-18) The nonlinear terms in the right-hand side of the equation are of order ()O Assuming that each ()Os x where ()sO is the slope, with a small parameter (e.g., the wave steepness) the order of magnitude of the te rm in the equation are 21/21/2 1/221 222iiCCC BBCAAANe kkg (A-19) with k If the slop is mild, with ()c O x and ()C O x one obtains the leading order linear term the equation 1/2 21 () 22iC BBNeO g (A-20)
102 For unidirectional propagation, choosing n=1 leads to the equation used for mild slope 1/2 2() 2iBC NeO xg (A-21) With iiBb beeikb xxx (A-22) The equation for quantity ibBe is then 1/2 2() 2 bC ikbNO xg (A-23) For steeper slopes, assume that the slope is ()sO Then ()c O x ()C O x and the orders of the vari ous terms in the equation (A-19) are 1/2 1/221 () 222iiCC BBCAANeO kg (A-24) where we assumed that se cond order derivatives are 2() O and should be neglected. For unidirectional propagation, with 1/2BAC, the equation becomes, as before, 1/2 1/23/2() 222idBidCdAiC CBNeO dxdxkdxCg (A-25) and using the definition of B 1/21/21/21/23/21 2 dAddBddBdC BCCBCCCB dxdxdxdxdxdx (A-26) 1/2 1/21/23/23/21 () 2222idBidCdBdCiC CCCBBNeO dxdxkdxdxCg (A-27) 1/2 3/2 2() 222idBidCdBidCdCiC BBNeO dxCdxkdxCdxkdxCg (A-28)
103 And grouping the terms 1/2 3/211 1() 222iidCdBidCdCC BNeO CdxkdxCCdxkdxg (A-29) Lets make some order here. Denote 1 2 dC Cdxk (A-30) 111 2 dCdC CCdxkdx (A-31) where the coefficient in powers of the order of magnitude of the factors the equation becomes 1/2 3/2 221 1 () 112ii dBiC iBNeO dxg (A-32) which can also be approximated by 1/2 223/21111() 2idBC iiBiNeO dxg (A-33) That is 23/23 1/2 23/233/21 1() 2 idB iiiB dx C iiNeO g (A-34) Now, neglecting all terms 3/2() O leaves the equation 1/2 3/2() 2idBC iBNeO dxg (A-35) i.e., the form 1/2 3/211 () 22idBidCdCC BNeO dxCCdxkdxg (A-36)
104 For the quantity ibBe the equation becomes 1/2 3/2() 2mdbC ikkbNO dxg, (A-37) with the wavenumber correction, mk introduced by the steeper slope defined as 1 lnln 2mdd kCcC kdxdxC (A-38)
105 THE MODAL ENERGY FLUX DISSIPATION RATE The standard definition of modal energy dissipation rate is the relative change of energy flux unit distance x, f ff,2,dFfxfFfxdxN (B-1) where the modal energy flux is f f f,,,FfxSfxCfxdf C is the modal group velocity, and df is the frequency bandwidth. N is the dissipation rate of wave amplitude and is often introduced as an imaginary modal wave number, and was estimated as f ff 21,1ln2,Ffxf x FfxN (B-2) where 1 x f21 x x! are the cross-shore positions of a pair of adjacent sensors, x is the distance of a pair of adjacent sensors. The Taylor expansion of equation (B-2) can be expressed as f ff 21,112,Ffxf x FfxN (B-3) A simple measure of swell dissipation can be derived by applying a finite-difference discretization of equation (B-1), and a mean-value theorem for integrals ff 12,Swell dissipation12,swellswellswellFfx x FfxN Â¦ Â¦ (B-4) where swellN can be regarded as the characteristic (band-averaged) swell dissipation. APPENDIX B
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112 BIOGRAPHICAL SKETCH Shih-Feng Su was born September 7, 1975 in Taipei, Taiwan. He completed his undergraduate degree in the Depa rtment of Hydraulic and Ocean Engineering, National Cheng Kung University (NCKU) in 1997, c ontinued his postgraduate education at the graduate school of NCKU, where he obtained his Masters degree in 1999. Following that, he worked for the Taiwanese military servic es in the Naval Meteorological Center until September of 2001. Subsequently, Shih-F eng worked for three years in the private sector as a harbor and coastal engineer for CE CI Engineering Consul tants, Inc. ShihFeng was admitted to the University of Flor ida in July of 2006 and started his doctorate study in the Department of Civil and Coas tal Engineering. He obtained his PhD in coastal and oceanographic engineeri ng at the University of Flor ida, Gainesville, in 2010.