Citation
Wave-current interaction and quasi-three-dimensional modeling in nearshore zone

Material Information

Title:
Wave-current interaction and quasi-three-dimensional modeling in nearshore zone
Creator:
Lee, Jung Lyul, 1959- ( Dissertant )
Wang, Hsiang ( Thesis advisor )
Ochi, Michael K. ( Reviewer )
Sheng, Y. Peter ( Reviewer )
Kurzweg, Ulrich H. ( Reviewer )
Place of Publication:
Gainesville, Fla.
Publisher:
University of Florida
Publication Date:
Copyright Date:
1993
Language:
English
Physical Description:
x, 158 leaves : ill. ; 29 cm.

Subjects

Subjects / Keywords:
Boundary conditions ( jstor )
Coastal currents ( jstor )
Momentum ( jstor )
Shear stress ( jstor )
Three dimensional modeling ( jstor )
Velocity ( jstor )
Wave diffraction ( jstor )
Wave energy ( jstor )
Wave interaction ( jstor )
Waves ( jstor )
Coastal and Oceanographic Engineering thesis Ph. D
Dissertations, Academic -- Coastal and Oceanographic Engineering -- UF
City of Gainesville ( local )
Genre:
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

Notes

Abstract:
This dissertation represents a study of wave-current interaction, surf-zone hydrodynamics, and wave-induced three dimensional current patter. Based on this study, a quasi-three dimensional circulation model has been developed. When waves propagate through a region with varying current, an instability in wave heights has been observed int he tidal inlets. This fact is clarified by both the surface action and energy transformation which arises due to the change in the absolute frequency in the varying current field. consequently, the varying current can excite the wave field to lead the time variation to wave heights. Two new equations governing the excitation of wave-current field are derived, and form the cornerstones of this dissertation. These equations are termed here 'wave action equation' and 'the kinematic conservation of intrinsic angular frequency'. Five representative wave models are used here; two hyperbolic types, two elliptic types, and one parabolic type. At this moment, no single model has proven perfect or has clearly outperformed the others. The governing equation of each model is derived from the mild slope equation of hyperbolic type, with varying degrees of approximation. Asimple surf zone model is presented. the model is based on the consideration of wave energy balance and wave action conservation. It is noteworthy that wave height, surface onshore current and surface longshore current across the surf zone are presented in analytical forms for the two-dimensional gradually-sloped bottoms. this surf zone model provides the current pattern of the vertical circulation model, and consequently, significantly contributes to solving the three-dimensional current pattern for the general topography. A hydrodynamic model for the nearshore zone is developed by combining the circulation models and five wave models; one wave model is selected from them according to the problem of interest. The model is based on the mathematical models of wave-current interaction, and solved by using a fractional step method in conjunction with the approximate factorization techniques leading the implicit finite difference schemes.
Thesis:
Thesis (Ph. D.)--University of Florida, 1993.
Bibliography:
Includes bibliographical references (leaves 153-157).
General Note:
Typescript.
General Note:
Vita.
Funding:
Technical report (University of Florida. Coastal and Oceanographic Engineering Dept.) ;
Statement of Responsibility:
by Jung Lyul Lee.

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Full Text
WAVE-CURRENT INTERACTION AND
QUASI-THREE-DIMENSIONAL MODELING IN NEARSHORE ZONE
By
JUNG LYUL LEE

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA

1993







ACKNOWLEDGMENTS

The author would like to express the sincere appreciation and gratitude to my adviser, Professor Hsiang Wang whose encouragement and guidance with his sage advise and infinite patience over the years have been a source of inspiration without which this work would not have been possible.
Gratitude is extended to the members of my dissertation committee, Professor Michael K. Ochi, Professor Y. Peter Sheng and Professor Ulrich H. Kurzweg for their quidance and suggestions. Professor Robert G. Dean, Professor Donald M. Sheppard and Professor Ashish J. Mehta are also acknowledged for the review of the manuscript.
Special thanks go to Dr. Li-Hwa Lin and Becky Hudson for providing timely advice and comfortable circumstances. The author owes a lot of debt to Mr. ChulHee Yoo, Mrs. Yoo and my fellow students for their hospitality and companionship during my years in Gainesville.
Finally, the author wishes to truly thank my parents for their love and support which sustained me throughout studying abroad.




TABLE OF CONTENTS
ACKNOWLEDGMENTS ............................. ii
LIST OF FIGURES ................................ vi
ABSTRACT .................................... ix
CHAPTERS
I INTRODUCTION ............................... 1
1.1 Literature Review ............................. 3
1.1.1 Wave-Current Interaction Model ................. 3
1.1.2 Nearshore Circulation Model ................... 5
1.2 Summary of Contents ........................... 7
2 DESCRIPTION OF NEARSHORE HYDRODYNAMIC MODEL ..... 10
2.1 W ave M odels ............................... 12
2.2 Circulation Models ............................ 14
2.2.1 Vertical Current Profile ...................... 14
2.2.2 Quasi-3D Model .......................... 15
3 ENERGY PROPAGATION THROUGH VARYING CURRENT FIELD 17
3.1 Governing Equations and Boundary Conditions ............ 21
3.2 Wave Energy Equation .......................... 24
3.2.1 Time-Averaged Wave Energy .................. 26
3.2.2 Time-Averaged Wave Energy Flux ............... 27
3.2.3 Time-Averaged Wave Energy Equation ............. 28
3.3 Wave Action Equation .......................... 28
3.4 Excitation Due to Wave-Current Interaction .............. 29
4 MATHEMATICAL WAVE MODELS .................... 32
4.1 Mild Slope Equation ........................... 33
4.2 Derivations of Governing Equation of Each Model ........... 36
4.2.1 Hyperbolic-Type Model I (HM I) ................ 36
4.2.2 Hyperbolic-Type Model II (HM II) ............... 38
4.2.3 Elliptic-Type Model I (EM 1) .................. 39
4.2.4 Elliptic-Type Model II (EM II) ................. 40
4.2.5 Parabolic-Type Model (PM) ................... 41




5 NUMERICAL SCHEMES AND COMPARISONS OF WAVE MODELS 44
5.1 Numerical Scheme of Wave Models........................44
5.1.1 Hyperbolic Model I............................. 44
5.1.2 Hyperbolic Model II............................ 47
5.1.3 Elliptic Model I................................48
5.1.4 Elliptic Model II............................... 51
5.1.5 Parabolic Model............................... 52
5.2 Comparison of Wave Models.............................53
5.2.1 Computational Difficulty...........................53
5.2.2 Wave Shoaling and Refraction.......................54
5.2.3 Wave Diffraction............................... 56
5.2.4 Wave Reflection................................ 56
5.2.5 Wave-Current Interaction .. .. .. ... ... ... ... ...62
5.2.6 Summary .. .. .. .. .... ... ... ... ... ... ...62
6 SURF ZONE MODEL .. .. .. .. ... ... .... ... ... ... ....70
6.1 Time-Averaged Wave Energy Equation in Surf Zone .. .. .. .. ....71
6.2 Wave Action Equation in Surf Zone .. .. .. .. ... ... ... ....73
6.3 Wave Height Transformation in Surf Zone. .. .. .. ... .... ..74
6.4 Surface Currents in Surf Zone. .. .. .. ... ... ... ... .....80
6.5 Set-Up and Set-Down. .. .. .. .. ... ... ... ... ... .....85
7 MATHEMATICAL MODEL FOR WAVE-INDUCED CURRENTS .. 88
7.1 Turbulence- Averaged Governing Equations. .. .. .. .. .... ....89
7.2 Horizontal Circulation Model.................90
7.2.1 Depth-Integrated and Time-Averaged Equation of Mass 90 7.2.2 Depth-Integrated and Time-Averaged Equations of Momentum 92 7.2.3 Roles of New Surface Advection Terms in Radiation Stress 95
7.3 Vertical Circulation Model. .. .. .. ... ... ... ... ... ....98
7.3.1 Theoretical Undertow Model. .. .. .. .. ... ... ... ..100
7.3.2 Longshore Current Model. .. .. .. ... .... ... ....104
7.3.3 Estimation of Eddy Viscosity. .. .. .. .... ... ... ..105
7.3.4 Theoretical Solutions. .. .. .. ..............107
7.3.5 Model Adoption for General Three- Dimensional Topography .112
7.4 Quasi-3D Model .. .. .. .. .. ... .... ... ... ... ... ..116
7.4.1 Modification of 2D Depth-Integrated Equations. .. .. .. ..116
7.4.2 Estimation of Integral Coefficients for Vertical Velocity Profile 120
8 NUMERICAL SCHEME AND RESULT OF CIRCULATION MODEL 122
8.1 Numerical Scheme of Quasi-3D Circulation Model. .. .. .. .. ...122
8.1.1 Fractional Step Method. .. .. ... ... ... ... ... ..123
8.1.2 Finite-Volume Discretization. .. .. .. .. ... ... ......124
8.1.3 Approximate Factorizations .. .. .. ... ... ... ... ..126
8.1.4 Formation to Tridiagonal Matrix .. .. .. ... ... ... ..130
8.2 Comparison with Gourlay's Experiments .. .. .. .. ... ... ...133




9 CONCLUSION AND FURTHER STUDY .................. 144
9.1 Conclusions ................................ 144
9.2 Further Study ............................... 145
APPENDICES
A WAVE-AVERAGED ENERGY DENSITY ................. 147
B WAVE-AVERAGED ENERGY FLUX .................... 149
C GREEN'S IDENTITIES CONSISTENT WITH ENERGY EQUATION 151 D NUMERICAL DIFFERENCE OPERATORS ................ 152
BIBLIOGRAPHY ................................. 153
BIOGRAPHICAL SKETCH ........................... 158




LIST OF FIGURES

2.1 Structural overview of nearshore hydrodynamic model. .. .. ....11
3.1 Time histories measured at the offshore (upper) and inlet points
(lower) during flood tide and ebb tide. .. .. .. .. .. ... ....18
3.2 Change in absolute frequencies and wave heights measured in
physical model. .. .. .. .. .. .. .... ... ... ... .....19
5.1 Staggered grid system for hyperbolic model I .. .. .. .. .. ....45
5.2 Sub-grid system for elliptic model I. .. .. .. .. ... ... ....49
5.3 Shoal configuration for comparison of CPU time (concentric circular contours of h/Li) .. .. .. .. .. .. ... .. ... ... ...55
5.4 Comparison with the laboratory data of Ito and Tanimoto (1972). 55
5.5 Comparison of wave shoaling .. .. .. .. .. .. ... .... ....57
5.6 Comparison of wave refraction (wave height) .. .. .. .. .. ....58
5.7 Comparison of wave refraction (wave angle). .. .. .. .. .....59
5.8 Comparison of wave diffraction for semi-infinite breakwater (00)
between analytic solutions (dotted line) and numerical solutions (solid contour line of 0.8, 0.6, 0.4 and 0.2 diffraction coeff. from
left). .. .. .. .. .. .. ... .... ... ... ... ... .....60
5.9 Comparison of wave diffraction for semi-infinite breakwater (300). 61
5.10 Wave reflection tests against wall. .. .. .. .. ... ... .....63
5.11 Wave reflection tests against bottom slope .. .. .. .. .. .....64
5.12 Condition of collinear wave-current interaction. .. .. .. .. ....65
5.13 Condition of shearing wave-current interaction. .. .. .. .. ....65
5.14 Comparison of collinear wave-current interaction. .. .. .. .. ...66
5.15 Comparison of shearing wave-current interaction (height). .. ....67




5.16 Comparison of shearing wave-current interaction (angle) ...... ..68
6.1 Effect of a parameter P on the dimensionless wave height in the
surf zone ....................................... 77
6.2 Comparison with laboratory experiments presented by Horikawa
and Kuo (1966) ....... ............................ 78
6.3 Comparison between theoretical and numerical results ....... .. 79
6.4 Effect of a parameter 83 on the dimensionless surface onshore current in the surf zone ................................. 82
6.5 Effect of a parameter P3 on the dimensionless surface longshore
current in the surf zone .............................. 83
6.6 Comparison of longshore current with laboratory experiments presented by Visser (1991) .............................. 84
6.7 Effect of a parameter P on the dimensionless set-up in the surf zone. 87
7.1 Vertical profile of cross-shore currents measured by Hwung and
Lin (1990) ...................................... 99
7.2 Vertical profiles of cross-shore current ..................... 109
7.3 Comparison with experiments presented by Hansen and Svensen
(1984) ....... ................................. 110
7.4 Effect of the advection term in the undertow model ........ ..111
7.5 Vertical profiles of longshore current ..................... 113
7.6 Comparison with experiments presented by Visser (1991)..... ..114
7.7 Combined three-dimensional profiles ..................... 115
7.8 Vertical profiles of cross-shore current by using bottom shear stress. 117
7.9 Comparison with experiments presented by Hansen and Svensen
(1984) ....... ................................. 118
8.1 Computational grid of finite volumetric scheme ........... ..125
8.2 Grid system of circulation model ....................... 131
8.3 Physical layout of Gourlay's experiment (1974) ............. 135
8.4 Wave height and set-up contours represented by Gourlay (1974). 136
8.5 Measured current pattern ............................ 137




8.6 Comparisons of wave height contours . 138
8.7 Comparisons of set-up contours . 139
8.8 Comparisons of depth-mean current pattern resulted from quasi3D m odel 140
8.9 Comparisons of depth-mean current pattern resulted from depthaveraged model . 141
8.10 Comparisons of mean surface current pattern 142
8.11 Comparisons of bottom current pattern 143




Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy WAVE-CURRENT INTERACTION AND QUASI-THREE-DIMENSIONAL MODELING IN NEARSHORE ZONE By
JUNG LYUL LEE
August 1993
Chairman: Dr. ilsiang Wang
Major Department: Coastal and Oceanographic Engineering
This dissertation represents a study of wave-current interaction, surf-zone hydrodynamics, and wave-induced three dimensional current pattern. Based on this study, a quasi-three dimensional circulation model has been developed.
When waves propagate through a region with varying current, an instability in wave heights has been observed in the tidal inlets. This fact is clarified by both the surface action and energy transformation which arises due to the change in the absolute frequency in the varying current field. Consequently, the varying current can excite the wave field to lead the time variation of wave heights. Two new equations governing the excitation of wave-current field are derived, and form the cornerstones of this dissertation. These equations are termed here 'wave action equation' and 'the kinematic conservation of intrinsic angular frequency.'
Five representative wave models are used here; two hyperbolic types, two elliptic types, and one parabolic type. At this moment, no single model has been proven perfect or has clearly outperformed the others. The governing equation of each model is derived from the mild slope equation of hyperbolic type, with varying degrees of approximation.




A simple surf zone model is presented. The model is based on the consideration of wave energy balance and wave action conservation. It is noteworthy that wave height, surface onshore current and surface longshore current across the surf zone are presented in analytical forms for the two-dimensional gradually-sloped bottoms. This surf zone model provides the surface current pattern of the vertical circulation model, and consequently, significantly contributes to solving the three-dimensional current pattern for the general topography.
A hydrodynamic model for the nearshore zone is developed by combining the circulation models and the five wave models; one wave model is selected from them according to the problem of interest. The model is based on the mathematical models of wave-current interaction, and solved by using a fractional step method in conjunction with the approximate factorization techniques leading to the implicit finite difference schemes.




CHAPTER 1
INTRODUCTION
The purpose of this dissertation is to present 1) a connected account of the mathematical theory of gravity waves and current motions in the nearshore region extending into the surf zone where the effects of turbulence due to wave breaking become important, and 2) a numerical model applicable to the nearshore zone.
Wave motion has been one of the more challenging problems among the hydrodynamic phenomena since it is basically unsteady and nonlinear. When waves interact with currents, their interaction induces wave instability as often was observed near entrances of tidal inlets and harbors. The effect is particularly pronounced when incoming waves encounter strong opposing currents, producing unstable but nearly group wave environment. This poses extremely hazardous navigation condition as can be atested by the mariners. The very nature of the dynamic fluid motion also induces active sediment transport creating unpredictable shoals and causing large nearshore morphological changes. Thus, the problem of wave-current interaction in nearshore zone has great engineering significance.
The common assumption that wave energy on a spactially nonuniform current propagate with a constant velocity, which is equal to the vectoral sum of the current and wave group velocity, is shown here to be incorrect. In fact, the excitation of wave-current field has been observed near the region where waves meet a nonuniform current, which might be initiated by shifting of wave frequency. Thus, the wave motion slowly modulates in the presence of a nonuniform current which influences the rate of wave energy transport. This slowing time varying wave energy transport in turn affects the current fields. This interaction eventually could lead to instability




2
as will be shown in this dissertation. Under such circumstance, large radiation stress gradient appears. Under normal circumstance, the primary wave field is unable to compensate for the effect; the excess wave energy radiates toward far field in the form of progressive long waves. The radiation of the forced waves is made possible through surface action, satisfying the surface boundary conditions. In surf zone, the excess wave energy could be disspated due to turbulence, therefore, the flow field could be steady as often observed.. The excess radiation stress is balanced here by the wave set-up and the wave-induced current such as the longshore, undertow, etc.
To solve the wave-current interaction problem two equations governing the dynamics are to be satisfied, the wave action equation and the wave energy equation. They are formally derived first. Another basic equation governing the wave kinematics named here the conservation of intrinsic wave frequency is proposed. The relationships between the newly derived equation and that of the traditional forms are examined.
One of the main objectives of this dissertation is to construct a numerical model that is able to reproduce the three-dimensional features of nearshore zone flow such as surface onshore flow, undertow, longshore current, etc. In the past two decades, we have witnessed remarkable progress in modeling nearshore hydrodynamics by numerical techniques. Commonly, the vertically averaged differential equations have been employed to predict dynamics in coastal waters and shallow seas. This approach appears to yield reasonable results in both circulation patterns and wave field beyond the surf zone but fails to produce useful information inside the surf zone. This is because of the highly three-dimensional flow structure within it. At present, the vertical structure of the flow field inside the surf zone has been investigated only for simple case of two-dimensional profiles. Most of the proposed surf zone models whether theoritical or numerical were also restricted to two-dimensional. The model developed here allows for applications to more general three-dimensional topographies.




3
The surf-zone problem is extremely difficult to formulate although it is a very narrow zone on oceanic scale and may appear to be uncomplicated. In particular, the wave breaking and the consequent turbulent flow are exceedingly complex processes to deal with theoretically. The current pattern in the this zone is certainly three dimensional. The cross-shore component in the surf zone contains a shoreward mass transport and an offshore- directed return flow. This circulation system is considered to be the main cause of shore erosion and profile changes under storm waves and of formation and maintenance. The longshore component is known to be responsible for long term shoreline changes. Therefore, if one is interested in predicting nearshore morphological changes the hydrodynamic model must appropriately describe this three-dimensional current features. Detailed description of the fluid motion including the wave breaking phenomenon and the subsequent turbulent flow field is clearly too complicated. The approach used here is to decompose the motion into different time scales. The large scale motion such as the mean flow can then be obtained by time averaging over smaller scale motions. In this manner, the effects of small scale motions are retained in an integrated manner even though the small scale motions can not be explicitly described. For example, the effects of turbulence on wave motion are manifested by energy dissipation and the influence of wave motion on mean flow is the residue flow, excess radiation stress, etc. The surf zone model is then constructed by the applications of the three fundamental governing equations derived in this dissertation.
1.1 Literature Review
1.1.1 Wave-Current Interaction Model
Wave-current interaction problems have interested a considerable number of mathematicians beginning apparently with Longuet-Higgins and Stewart (1960-1961), and Whitham (1962). Longuet-Higgins and Stewart (1960) introduced the concept of radiation stress against which the variation in wave energy corresponds to work done by




4
the current. This has been a singificant contribution to the advancement of coastal science from both the theoretical and the practicial point of view. The concept of radiation stress has become fundamental to modern the analysis of water waves in nearshore region. Further notable contribution was made by Bretherton and Garrett (1969) who showed that wave energy flux is no longer conserved owing to the nonlinear interaction of the radiation stress with the current and introduced the conservative quantity called wave action. So far, the concept of wave energy flux balance and the conservation of wave action are the primary tools for analyzing wave-current interactions. The problems of wave-current interaction can generally be classified into two categories; the wave field as affected by the presence of current and the current field induced by the waves.
The literature concerning wave models is very extensive. The Boussinesq equation which is based on the time-dependent, depth-integrated equations of conservation of mass and momentum, for instance, has been applied to a number of practical engineering problems. The prediction of nearshore waves, however, took a new dimension with the introduction of the mild slope equation by Berkhoff (1972) which is capable of handling the combined effects of refraction and diffraction. The mild slope equation is formulated by linear monochromatic waves in areas of moderate bottom slope. Since then significant progress has been made in computational techniques as well as model capabilities, notably by Radder (1979), Copeland (1985), Ebersole et al. (1986), Yoo and O'Connor (1986a), and Dalrymple et al. (1989). The approach based on the mild slope equation is presently preferred over the more comprehensive approach based on the Boussinesq equation since the former can be numerically solved with faster and more efficient algorithms.
The combined refraction-diffraction effects of wave-current interaction were included by Booij (1981) and Kirby (1984), who derived wave equations by using variational principle. Kirby (1984) made corrections to the equation of Booij (1981) so




as to yield the wave energy equation suggested by Bretherton and Garrett (1969). Olinaka et al. (1988) extended the equation set of hyperbolic type suggested by Copeland (1985) to wave-current interaction; Yoo and O'Connor (1986b) extended their equation set of hyperbolic type; Kirby (1983) derived the elliptic and parabolic wave equations; and more recently, Jeong (1990) applied the wave-current interaction to Ebersole's model (1986). The main differences of these wave models are their governing wave equations and the associated numerical methods. However, all wave equations can be derived from the mild slope equation given by Kirby (1984), with varying degrees of approximations.
1.1.2 Nearshore Circulation Model
The effort in nearshore circulation modeling can generally be broken down into three related areas, namely, the cross-shore circulation and the long shore current generation, both inside the surf zone and the circulation outside the surf zone. Efforts of combining them, such as the present dissertation, are also appearing.
In the cross-shore surf zone modeling, the usual elements include the fixation of breaking criterion, the energy decay rate across the surf zone, the wave set-up and finally the cross-shore current patterns. Of the various breaking criteria proposed, Miche's criterion (1951) is still the most durable and widely used for its simplicity. His model can also extended into the surf zone for energy decay simply assuming the local wave height to be proportional to the local water depth. This assumption has also been used extensively for the analytic approach to surf zone models. be Mehaute (1962) first suggested the physically appealing approach assuming that the rate of energy dissipation in a breaking wave is the same as a propagating bore, and similarly Battjes and Janssen (1978) developed a bore model inspired in the energetic dissipation produced in a tidal bore. Mizuguchi (1980) introduced a two-parameter eddy viscosity model which satisfies the two requirements for a stable wave height in constant water depth and for an approximately constant ratio of wave height to




6
water depth on plane beaches. The turbulence model was developed by Izumiya and Horikawa (1984), based on the similarity law for the structure of turbulence induced in wave breaking. Dally, Dean, and Dalrymple (1984) presented a wave decay model that takes the wave reformation into consideration. They assumed that the dissipation rate is simply proportional to the difference between the local energy flux and that in the reformation zone, divided by the local water depth. The model also has two empirical parameters. Their model has been widely used.
The vertical profile of cross-shore current pattern has been studied during the past ten years for the simple two-dimensional case on the plane beach. Since the postulation of the driving mechanism by Nielsen and Sorensen (1970), the analytical treatment was first appeared by Dally (1980). Svensen (1984) proposed a theoretical undertow model using the first order approximation technique in describing the breaking waves, and Hansen and Svensen (1984) further considered the effect of the bottom boundary layer in the undertow. More recently, as the study progresses, several ideas have been added. Okayasu et al. (1988) estimated undertow profiles based on the assumed mean shear stress and eddy viscosity, and Yamashita and Tsuchiya (1990) developed the numerical model which consists of surface and inner layers.
Wave-induced longshore currents within the nearshore zone may be generated by a number of mechanisms including an oblique wave approaching the shoreline, a longshore variations in wave breaking height, or the combination of the above. The need for the development of an adequate theory for longshore currents has attracted a large number of contributors since the pioneer paper of Putnam and Arthur (1945). The theory for longshore currents by Longuet-Higgins (1970) was a rather elegant analysis based on the concept of radiation stress. His model and the conept behind it are widely accepted. Most of the later attempts were more or less modifications on the orginal model. The longshore current profile has been proposed by a number of investigators (de Vriend and Stive, 1987; Svensen and Lorenz, 1989); most of




7
them assumed the profile to be equivalent to the logarithmic velocity profile found in uniform steady stream flows.
Prediction of nearshore wave-induced currents beyond the surf zone has also advanved considerably since some of the earlier developments by Noda et al. (1974) and Ebersole and Dalrymple (1979). Both of these earlier models were driven by a wave refraction model but with no current feedback. More recently, Yoo and O'Connor (1986b) developed a wave-induced circulation model based upon what could be classified as a hyperbolic type wave equation; Yan (1987) and Winer (1988) developed their interaction models based upon parabolic approximation of the wave equation. The nearshore circulation, which was predicted by all models listed above, deals primarily with the vertically-averaged longshore current. Recently, however, de Vriend and Stive (1987) improved the nearshore circulation model by a quasi-three dimensional approach, which employed a combined depth-integrated current model and a vertical profile model. They introduced a primary and secondary profile for velocity variation over depth and assumed the primary velocity profile to be the same in the cross-shore and longshore direction. Svensen and Lorenz (1989) assumed that the equations in the cross-shore and longshore directions could be decoupled and solved the cross-shore and longshore motion independently of long coast with straight bottom contours.
1.2 Summary of Contents
It has already been stated that this dissertation presents a set of governing equations for wave and current motions for incompressible fluids with a free surface. The main force is the gravity but forces induced by turbulence due to wave breaking and by bottom friction are also included where they become important such as in surf zone and shallow water, respectively. A numerical model is then developed.
Chapter 2 gives an overview on the main structure of this nearshore hydrodynamic model and briefly introduces the governing equations and numerical schemes




involved. In Chapter 3 the governing equations are derived with physical interpretations on the phenomenon of wave-current interaction. It begins with a brief description of basic wave characteristics and continues with the fundamental equations for estimating energy and energy flux of flows of invicid and incompressible fluids. Two dynamic equations governing wave energy transport and wave action conservation are presented.
Chapter 4 derives the hyperbolic wave equation for the treatment of combined wave refraction- diffraction. Based on this equation, five different versions of wave equation suitable for numerical modeling are given as they have their own merit depending upon the problems to be solved. In Chapter 5, the numerical methods of the five types of mathematical models are described. These models are evaluated through mutual comparisons using a number of bench mark cases.
The theory for wave transformation in the surf zone is introduced in Chapter 6 an analytical wave decay model is presented for the simple plane beach cases. The solution is compared with experiments conducted by Horikawa and Kuo (1966). Based on the proposed wave action equation and the wave decay model, analytic solutions for surface currents in onshore and longshore directions are derived for the case of plane beach. In Chapter 7, the cross-shore and longshore circulation model is established by solving the time averaged mass and momentum equations. Finally, a quasi-3D) model is constructed by combining the horizontal circulation model and the vertical circulation model.
In Chapter 8, The numerical scheme solving the quasi-3D model is breiefly presented. The governing equations are solved implicitly using a fractional step method in conjunction with the approximate factorization techniques. The equation of each step is discretized by the finite volume scheme which yields more accurate and conservative approximations than schemes based on finite differences. Examples of computed nearshore current patterns are presented to demonstrate the applicability of the




9
model for typical situations through comparison with laboratory experimental data. Chapter 9 concludes with the discussion of a few restricted problems overlooked by the assumptions employed in the model.




CHAPTER 2
DESCRIPTION OF NEARSHORE HYDRODYNAMIC MODEL
In the present model, the wave model and circulation model can be combined or separated through the control of the main program. Therefore, the model is basically applicable to problems of shallow water wave propagation and nearshore cirlulations driven by tides, wind and/or waves-induced radiation stress.
This chapter describes the nearshore hydrodynamic model for wave and current field in the nearshore zone. The main model is the circulation model for computing mean water surface and mean currents. The wave model is a sub-model used to determine the radiation stress which is required in the circulation model as the forcing mechanism. The wave model takes offshore wave conditions as input offshore and propagates into nearshore zone while accounts for various nearshore processes including shoaling, refraction, diffraction, reflection, and surf-zone wave breaking. The basic wave model is developed for an irrotational flow field with steady mean currents. Various modifications are incorporated to accommodate for breaking and bottom friction effects. Figure 1 illustrates the main structure of the wave-induced nearshore circulation model.
The governing equations for the nearshore circulation model are divided into three major computational steps; 1) advection, 2) diffusion, and 3) propagation, according to the fractional step method. Each time step is composed of x- and y-directional computations by the alternating direction implicit method which speeds up the algorithm.
The wave field is affected by mean surface elevation and currents which are the outputs of the circulation model. As mentioned earlier, five different wave models




Figure 2.1: Structural overview of nearshore hydrodynamic model.

CIRCULATION MODEL 1) vertical circulation model 2) horizontal circulation model
2.1) advection step 2.2) diffusion step
2.3) propagation step

* MAIN PROGRAM *
e read input
* call wave model selected e call circulation model e write output

WAVE MODELS INCLUDED 1) hyperbolic model I 2) hyperbolic model II 3) elliptic model I 4) elliptic model II 5) parabolic model




12
are tested. They are categorized into 1) hyperbolic model I, 2) hyperbolic model II, 3) elliptic model I, 4) elliptic model II, and 5) parabolic model. At this moment, no single model clearly outperforms the others. Therefore, the wave model should be selected according to the problem of interest. Each of them is programmed separately and can be chosen to couple with the circulation model.
The computational boundaries and the boundary conditions are automatically specified by input depth data. Therefore, the model requires no adjustment from run to run. Detailed description will be given in a separate operation manual. The flooding and drying of beach face are also taken into account in a two-dimensioal sense as will be explained later.
The following sections provide a brief description of each models and the computational methods. All mathematical symbols are to be defined later so that the structural overview is more clearly illustrated.
2.1 Wave Models
Five representative types of wave models are coupled with the quasi-three dimensional circulation model. The main differences of the five models are their governing wave equations and the associated numerical methods. Of the five wave models four of them were selected from existing literature and one is developed by the author.
" Hyperbolic Model I (HM I):
S(Cg 11 + V. + V (CCgV ) 0
0V__+ wg"r/ 7 = 0
atC a
at + wgVY- =0
at~
" Hyperbolic Model II (HM II):
aK + (Cg-K + U) .K + K VU + s k _h [V2()] 0
at k ih 2kh h~~2a a
aa2 K2
a +V. k+ v)- 0




* Elliptic Model I (EM I):
-iw{2U. V + (V U)} + (U. V)(U-. V) + (V. U)(U V)
-V. (CCgV4) + ( w kCCg) = 0
* Elliptic Model II (EM II):
Ka2
V[(cgK + U)-] = 0
CCga K2 V. (CCgVa) k2CCga = 0 VxK=0
* Parabolic Model (PM):
B A' 180
'(Cg., + u)- + i(ko k,)(Cg, + u)A' + 2a x[e(Cg, + u)]A'= i 8 OA' wOv 8A'
-(CCg ) -A' wv2 y -y 2 y -y
Model Unknowns Numerical scheme
HM I 77 and V4 FDM on a staggered grid system
HM II a and K FDM on a staggered grid system
EM I complex Combined Gragg's method-FDM
EM II a, K and 0 Generalized Lax-Friedrich FDM
PM complex A' Crank-Nicholson FDM
The wave breaking model nested in to each wave model can be applied by replacing the Cg-fCgb to the group velocity in a breaking zone, where f can be estimated by the dimensionless wave height arriving to the shoreline on a uniform slope, and a value between 1.1 to 1.4 is suggested depending on the slope, bed condition, etc.




14
2.2 Circulation Models
2.2.1 Vertical Current Profile
A cross-shore circulation model is developed to account for the effect of vertically nonuniform currents. Based upon theoretical solution of simple cases as well as laboratory measurements, the velocity profile may be approximated by a second order parabola.
u = Ciz'2 + C.2z' + C.3 v = C ,z'2 + C,2z' + Cy3 where C1, C2 and C3 are determined in terms of discharge, Q, wave height, H, total depth, h + 1c, turbulent-induced bottom shear stress, TB,tb, and wind stress, Tw as follows:
U7 + h g H Taxtb
P 2 C 1 r~~gH2z 16 8z rw t__~ p p
C.2 = r7, + h g aH2 +w} e6, 16 Ox p
= Q. C.1 C.2
c+ h 3 2
S + h g OH wy TW Bytb
2e 16 Oy p
90+ g8H2 rw1A
C3 -+
ez 16 By pf
C,3 = ~Q Cyl C,2
c+ h 3 2
where,
Ez = N uorbl 77, + h
TB,tb = F. IUorb IY
0 V.(KH/k)
S= 24(1 Y)OL ( Cg/Cgb) F3 V (KH/k) F 4 =
470s (# Cg/Cgb)




where |Uorbl is defined as gH/2C.
2.2.2 Quasi-3D Model
The quasi-3D depth-varying circulation model is governed by the continuity and momentum equations integrated over depth by parameterizing the vertical structure of the currents. The model described herein may be considered as a approximation to the full three-dimensional model. The continuity equation
8c 8 gH2k, 8 gH2k,
-7,+ a(Q+ )+ -(QV,+ 8 0)=o
Wt 82 8o y 8oThe x-directional modified momentum equation
8Q a Q2 Q
- + -[ + (h + )T.] + yy- + (h + rhT)Try]
at Ox h +m zU ay h +17
S 1aS== 1 aS, 8i0 _ws
+ 1 + -- + g(h + 77) + = 0
p ax p ay ox p p
The y-directional modified momentum equation
aQ, oQ [_rQt8QQ, +8( +0'T
- + I + (h + 1c) Ty] + [ + (h +71e)Tyy]
at ax h + o 7,ay h +77c
1 iSX as h877c TWY TB Y
+ + +g(h + oc) w+ =0
p 8 p y ay p p
where,
QX = udz, QY =h vdz
1 us
S = E'[n(cos2 0 + 1) + 2cos 0-]
Scy = Sy, = E'[sin O(cos On + ) + cos 0-SC C
1 V
Sy = E'[n(sin2 0 + 1) + 2 sin 0 ]
2
T. = [4CX1 +C22 C.1C.2]
T 45 12+ + 6
14TCyj+ Cz2Cy2 C.l Cy2 Cx2Cyl T I 45 12 12 12
T = 4C +
45 12 61




16
As noted below, the bottom friction consists of turbulent shear stress at the bottom, 'rB,tb, and bottom friction due to viscous and streaming flows, rB,bf: 7B = TB,tb + rB,bf
= FIUo rblUJ + Fluo ,bl(UB + Ustrm)
where F, is the wave friction factor varying over the wave breaking zone as given earier, while F is the current friction factor assumed constant here.
The lateral shear stress is added to the momentum equations as rau a
The mixing length coefficient, is assumed to be proportional to the distance from the shoreline, lxi, multiplied by the shallow water phase speed as suggested by Longuet-Higgins (1970).
'= N.IlxIVd
It was suggested that a dimensionless constant N, should be less than 0.016. The y-directional mixing length coefficient, ey, is assumed to be a constant everywhere.
The main numerical technique follows that proposed by Rosenfeld et al. (1991) for solving time-dependent, three-dimensional incompressible Navier-Stokes equations in generalized coordinate systems. The governing equations are implicitly solved by using a fractional step method in conjunction with the approximate factorization techniques. The equation of each step is discretized by the finite volume scheme which yields more accurate and conservative approximations than schemes based on finite differences. The use of a finite-volume method allows one to handle arbitrary grids and helps to avoid problems with metric singularities and non-conservative nature that are usually associated with finite-difference methods.




CHAPTER
ENERGY PROPAGATION THROUGH VARYING CURRENT FIELD
A common property of waves is their ability to transport energy without the need of any net material transport. In gravity waves, energy is propagated through the fluid media via the oscillations of the potential and the kinetic energies. When waves propagate through a region with currents, their energy is also transported by the moving fluid. The general appearance of the waves including wave height, length and period will also be altered. It is commonly observed that when the currents and waves are in the same direction, waves' are lengthened but with reduced wave heights. Opposing currents, on the other hand, shorten the waves but with increased wave heights. This latter situation is particularly hazardous for navigation. Moreover, our recent field and laboratory wave measurements near an inlet entrance seemed to indicate that waves could become unsteady, or modulated, in a nonuniform current field. This unsteadiness is more pronounced if waves counter the current. Figure 3.1 shows the time histories of waves measured at offshore and near inlet in the laboratory (from Wang et al., 1992). It can be seen that under ebb tidal conditions, the wave field is heavily modulated. Figure 3.2 shows the changes in wave periods (estimated from crest points) and wave heights under flood and ebb conditions. The reference wave period and wave height are 1.03 sec and 3.36 cm, respectively, in the far field. The phenomena described above have not been previously investigated.
In this chapter, two fundamental dynamic equations governing the behavior of wave-current interactions are derived. They are known as the wave energy equation




Flood Tide

Ebb Tide

Figure 3.1: Time histories measured at the offshore (upper) and inlet points (lower) during flood tide and ebb tide.




Wave Period

Variation

0.0 10.0

20.0 30.0 40.0 50.0 60.0
Wave Height Variation

70.0

0.0 10.0

20.0 30.0 40.0 50.0 60.0 70.0

Time (sec)
Figure 3.2: Change in absolute frequencies and wave heights measured in physical model.




20
and the wave action equation. The wave energy equation has the final form of:
--+ vh. (Cg+O-A k)E] =0
and the wave action equation that of
O9B
-B + Vh.- [VB] =0
at
where E and B represent the wave energy and wave action, respectively, defined as E p H2 B P g H 2
These two equations differ from the commonly accepted forms such as presented in Phillips' monograph (1977). In there, the wave energy equation is given in terms of E' = pgH2/8 as
-E' + Vh" [(Cg + fJ)E'] + Radiation Stress Term = 0
and in terms of B as
-[(Cg + J)B] = 0
In Section 3.1, basic problem formulations and appropriate boundary conditions are given. In Section 3.2, the depth-integrated wave energy equation is derived and the exact forms of wave energy and wave energy flux are presented in wave-averaged quantities. Section 3.3 introduces the wave action equation and the conservation equation of intrinsic frequency. It will be shown that the wave energy equation originates from the conservation of energy whereas the wave action equation is derived from the free surface boundary condtions. The relation between these two new equations and the traditional forms, along with the differences, are also discussed in Section 3.4. These two equations form the cornerstones of this dissertation upon which various specific wave-current interaction models both inside and outside surf-zones are constructed in later chapters.




21
3.1 Governing Equations and Boundary Conditions
A velocity potential 0 is assumed to exist such that the water particle velocities are given by VO where V is the 3-dimensional differential operator V 0 -i + .+ Ok
ax Sy 8z
The kinematic and dynamic boundary conditions to be satisfied at the free surface, z = 7, are, respectively, z = 7, t + Vh' Vh7 = 0 (3.1)
1
Ot + 2(V)2 + gz = C(t) (3.2)
2
where C(t) may depend on t, but not on the space variables. We may take C(t) 0 without any essential loss of generality. The subscripts t and z indicate the differentiations with respect to time and z-axis, respectively. Vh is the horizontal differential operators defined as
a a
V aOx y+
The cartesian coordinate system is used with origin at the still water level, x(x, y) in the horizontal plane and z directed vertically upwards. The velocity vector, U(u, v, w), is related to 4 by = a, V= andW
Sax v and w a
The velocity potential and the free surface displacement are assumed to be composed of current and wave components.
(x, z, t) = c(x;t, z) + (xz, t) (3.3)
n(x,t) = ye(x;t) +6e,(x,t) (3.4)
where e is an undefined factor used to separate the current (such as tidal current, wave-induced current, etc.) from the wave parts of the velocity potential. The '; t, z'




22
in the current part is used to recognize that q, and 70 may vary slowly over time much longer than the wave period and it could also accommodate small vertical variations in currents. Equations (3.1) and (3.2) are then expanded in Taylor series to relate the boundary conditions at the mean water level z = 77,
[,1t + Vh VAh?7 zl=,,?c + e,7 ( ,t + VhO VhA7) + v] +... = 0 z= 77e
[O, + 2(V)2 + gz]z=_e + e7,w [t + -(V)2 + gz]=,. +- = 0
Equations (3.3) and (3.4) are substituted into the above equations to give
[(7c)t + Vhqc Vh?7l.z=, +
e[(,lw),t + Vhc Vh7. + Vhqw Vh,7c (Ow)z + hw l=, + -" = 0 (3.5) [(4OC) + 2(VOc)2 + g7c]Z=n + E[(Ow)t + Vc. VOw + 97w]z=,ne +.. = 0(3.6)
2
The above equations are separated for current and wave parts and then truncated to retain only the significant terms:
(OC)Z -- + Vhc Vhre (3.7)
1 (Vhc)2
g 2
- Dt + (V&)r+ eVh Vhrc (3.9)
1, 1D (3.10)
g Dt
where D/Dt /Ot + Vh Vh.
It should be noted here that the terms retained in the above set of equations are not necessarily of the same order of magnitude for all general conditions. For instance, the last term in Eq. (3.9) is, in general, a higher order term than the first two and only becomes significant when wave diffraction occurs. The (0c)t term in Eq. (3.8), as will be shown later, becomes significant only under counter current condition. For slowly varying water depth, the wave part of the velocity potential may be written as

O,(x,z,t) = f(z) ,(x,t) + Enon-propagating modes

(3.11)




23
where f(z) = cosh k(h + z)/ cosh k(h + 77,) is a slowly varying function of x, k is a real value wave number and q, denotes the velocity potential at the mean water level, termed as 'surface potential.' For progressive waves, the velocity potential and the free surface displacement can be written in terms of the wave-averaged, slowly varying quantities as
q,,(x,z,t) = f(z)A(x;t)ie'O (3.12)
77.(x,t) = a(x;t)e'io (3.13)
where a is commonly defined as wave amplitude. The phase function is defined as 0=(K. x wt), where K is a wave number vector including the diffraction effects owing to the retention of the 3rd term in Eq. (3.9), and w is an absolute frequency. All slowly varying quantities are given here as real numbers. The relation between a and A can be established by the dynamic free surface boundary condition specified in Eq. (3.10), which, after substituting Eqs. (3.12) and (3.13) into it, yields
-g977, DO,
Dt
-gae'io = { 0+U.V}{Aie'}
at
= crdAe' + ie'{- + VA} (3.14)
at
where ad is the intrinsic frequency including the diffraction effects, defined as ad = w U. K, and U defined as VhO,.
This equation states that a and A should have a phase difference unless we impose the condition
-a- + V. A = 0 (3.15)
Then, the relation between A and a can be given by the following familiar form a (3.16)
Ord




24
Similarly, substituting Eqs. (3.12) and (3.13) into the kinematic free surface boundary condition given by Eq. (3.9) yields tr = gktanhk(h + ,e) VA- Vc (3.17)
A
Ba
- + V (a) + AK VlC = 0 (3.18)
at
Again, the last term in the above equations reflects the wave diffraction effect and, under normal circumstanaces, is of a higher order.
3.2 Wave Energy Equation In this section, the Eulerian expression of energy equation will be presented. The Eulerian expression governs the local balance between the rate of change of energy and the divergence of energy flux at a point. The Euler equation for incompressible and inviscid fluid flow is
DU
P- = -V(p + pgz) (3.19)
Dt
Taking the scalar product of U(u, v, w) with the respective terms in Eq. (3.19) and then summing the three components yields D q2 -U. V(p + pgz) (3.20)
Dt 2
where q2 = (u2 + V2 + w2)/2. With the use of continuity equation, the mechanical energy conservation equation becomes
-p2] + V [U(- 2 + p + pgz)] = 0 (3.21)
at 2 2
Integrate over water depth,
] + V U( + p + pgz) dz = 0 (3.22)
The first term in the integrand represents the volumetric rate of energy change and the second term gives the energy flux through the enclosing surface. The depth-integrated




25
energy equation has been derived from Eq. (3.22) by many including Longuet-Higgins and Stewart (1961) Witham (1962). A brief account is given here.
Using Leibnitz's rule (ffh D fdz = D ffh fdz f|,D + fl-hDh), Eq. (3.22) can be written as
7[7 q ]dz + Vh f _[U( + p + pgz)]dz
-[U(L + p + pgz)], V [U(L + p + pgz)]_ Vhh 2q2
+[W( + p + pgz)I" = 0 (3.23)
S2
Substituting the kinematic boundary conditions at the free surface and at z =-h, w| U Vh77 = 0 (3.24)
*0t
WI-h + U Vah = 0 (3.25)
and letting p be zero at the water surface, the above equation becomes
c3 h[-dpq2ld + P977Ltr/"U~q
[ + Vh [U + p + pgz)]dz = 0 (3.26)
Letting the total energy, and the total energy flux, F, as
J= q [ ]dz + 92 (3.27)
S= U( + p + pgz)dz (3.28)
-h 2
Equation (3.26) yields the well-known energy equation given by Longuet-Higgins and Stewart (1961) and Witham (1962), 9- + Vh" .-F = 0 (3.29)
t
Here, E is the energy density in the water column per unit surface area and F is the energy flux through the vertical surface enclosing the water column.




3.2.1 Time-Averaged Wave Energy
The energy per unit surface area given in Eq. (3.27) is expanded in a Taylor series with respect to the mean water level z = 7c,
S = P1 (V )2 + (0z)2] dz + 1P9[(9c + ) h2] + P w [(Vh#)2 + (0z0)2]
6 = P/_ f1 (h- +e7)-h -7w[V
= P- [(V.2 + VM,)2 + (qcz + ek Ow] dz + 1pg[(7c + e,)2 h2]
+P 1 [(Vh + eVh#)2 + (C0 + eO4w)2] (3.30)
Taking time average over the wave period and collecting terms associated with current( 0(1) terms) and wave ( 0(e2) terms) separately, we obtain the following pair of equations,
Ec = p] -[(Vhc) + (&c)2]dz + -2pg[C h2] (3.31)
h 2 2
Sp [(Vh )2 + (0w.)2] dz + -pg9q + p% [VhOb Vh]~n (3.32)
where E, and E are, respectively, the mean values of current and wave energy. The mean wave energy density, which is of primary interest here, can be expressed in terms of the slowly varying quantities by substituting Eqs. (3.12), (3.13) and (3.17); E = pgwH2 (3.33)
8 a
where H is the wave height defined as twice the wave amplitude a. Detailed derivations of Eq. (3.33) from Eq. (3.32) are given in Appendix A.
A few general remarks regarding the definition of wave energy density are made here:
1. E is the energy density directly associated with the 0(e) flcutuation motions
only. It does not take into account contributions associated with mean water level change. The contribution due to mean water level changes simply moves
the reference from the mean water level to that of no wave condition.




27
2. The last term in Eq. (3.31) represents the contribution due to wave-current
interaction. Longuet-Higgins and Stewart (1961) and Phillips (1977) all included this term in the mean flow energy rather than in the mean wave energy.
However, this is truly a O(e2) term from the fluctuation motion.
3. In the absence of current, E reduces to the conventional definition of energy
density in a wave field.
3.2.2 Time-Averaged Wave Energy Flux
By virtue of Bernoulli equation for unsteady flow the energy flux term given in Eq. (3.28) can be written as
F= -PJ adz (3.34)
Introducing the current and wave components defined in Eqs. (3.3) and (3.4) and expanding in Taylor series with respect to the mean water level z = i?,, we have
17a a
P c Vh(o + w)5t(Oc + 60,)dz pe?17[Vh(O. + E (4+ )],
(3.35)
Collecting the terms of O(e2) and taking time averaging over the wave period, we have the mean wave energy flux, F,
F = ,h 0. aoi dz P7.[VhO Vh+ (3.36)
fFh-at at at ,
This mean wave energy flux can be expressed in terms of the slowly varying quantities, utilizing Eqs. (3.12) and (3.13);
F= [Cg+ O- k] E (3.37)
Detailed derivations of Eq. (3.37) are given in Appendix B.
Again, the quantity of mean wave energy flux given here differs from the conventional one in that the mean wave energy density is different and that an additional term, 4Ork/w, is included. It will be shown that this additional term is responsible for the unsteadiness of energy transport under a nonuniform current field.




28
3.2.3 Time-Averaged Wave Energy Equation
By substituting Eq. (3.37) into Eq. (3.29) the time-averaged wave energy equation is obtained
aE
- + Vh (Cg + etk)E = 0 (3.38)
Here the mean wave energy density, E, is as defined in Eq. (3.33).
3.3 Wave Action Equation
In this section, the wave action equation and the conservation equation of intrinsic frequency are derived from the surface boundary conditions using the same definitions given in Eq. (3.12), (3.13) and (3.17). Substracting Eq. (3.10)xpgi, from Eq. (3.9) xpo,, and ignoring the higher order term due to diffraction effect, we obtain
a F .a
N(py,4,) + V (OU.P4,) p z -g, =0 (3.39)
where U, is current velocity of the mean flow at the water surface level. Substituting Eqs. (3.12) and (3.13) into Eq. (3.40), the following equation is obtained:
9 gk tanh k(h + r7e)
(Bie2) + V. (,Bie2") + UBe2 {2 + 1 = 0
where B is defined as
B = pg H2 (3.40)
8o
Expanding and separating the harmonic motions,
ie2i' + V. (O,B) + 'Be2' -2 + gk tanh k(h + 7) = 0
which yields the dispersion relation, o= = gk tanh k(h + r7), and the following wave action equation:
aB+ V [,B] = 0 (3.41)
The above equation can also be derived directly from Eqs. (3.16) and (3.18).




29
Now if we substitute Eq. (3.15) into Eq. (3.18), the following equation is obtained, DuA
-'" + V. [C~o-A] = 0 (3.42)
at
Eliminating A from Eqs. (3.42) and (3.16), we arrive at the final equation that governs the change of the intrinsic wave frequency in a current field: C0,.
- + Vh =0 (3.43)
which is termed here as 'the kinematic conservation equation, or simply the conservation equation, of intrinsic frequency.'
3.4 Excitation Due to Wave-Current Interaction
Three basic equations governing wave-current interactions have been derived in this chapter. They consist of two dynamic conservation equations and one kinematic conservation equation of the following forms: Energy Conservation Equation
-i+ vh. (Cg+fJ-L ) =0
r
Wave Action Conservation Equation
aB
+ Vh. [0,B]= 0
Kinematic Conservation of Intrinsic Frequency
+ Vh.[U~I= 0
where
E = PWH2 B= P 2
8o 8"
The wave energy conservation equation is different from the conventional forms presented by previous investigators. Under the no-current condition, this equation reverts to the familar form of W+ Vh -[CgZ'] = 0 at'




30
where E' = pgH2/8. In the presence of current, it is known that the wave energy as defined by pgH2/8 is not conserved (Longuet-Higgins and Stewart, 1961) but, instead, the wave action, given by pgH2/8o, is conserved (Bretherton and Garrett, 1969). From Eq. (3.37), we can see that this is only true if the absolute wave frequency remains constant. Garrett (1969). Although the wave action conservation equation presented in this chapter deals with a quantity identical to that defined by pgH2/8a as wave action, the meaning of the equation is very different. It is a surface property governed only by the surface current condition.
We now like to demonstrate that the absolute wave freqency does not, in general, remain constant in the presence of nonuniform current. Eliminating B from the wave energy equation and wave action equation, the following equation governing the change of absolute frequency results:
t O+(Cg+U- ck.w+-Vh. [(cg-tk)B] =0 (3.44)
5T + (Cj k w+Wh c
If the absolute frequency is constant over all domains, the first and second terms in the above equation become zero; then we have Vh. [(Cg -ck)B] =0 (3.45)
For the waves propagating in x-direction,
(Cg Lct)B = CgoBo (3.46)
C.
Differentiating with x,
au 0 C
- [ (CgB CgB)]
where Cg, and o are values at the area far from the zone of nonuniform currents. The right hand side in the above equation may not be equal to zero in the nonuniform current zone. Therefore, the assumption of the constant absolute frequency may yield the long-term fluctuation of currents. Consequently, when we consider the Doppler




31
equation, in which the intrinsic frequency and wave number are determined by the kinematic conservation of intrinsic frequency and dispersion relationship, respectively; Lo= o,+ 0-k
we realize the assumption of the constant absolute frequency may yield the inconsistency with the Doppler equation.
Therefore, if the absolute frequency is a variable, we have three unknowns, namely, the wave height, the absolute wave frequency, and the intrinsic wave frequency in the above set of equations. The wave number is determined by the intrinsic wave frequency by use of the dispersion relationship; the vector can be obtained by irrotationality of wave number vector. Therefore, in theory, the wave properties in the current field can be solved with given wave conditions at the boundaries. In practice, the problem is more complicated and simplifying assumptions are to be made as will be presented in later chapters.




CHAPTER 4
MATHEMATICAL WAVE MODELS
In the past two decades, prediction of nearshore waves took a new dimension with the introduction of the mild slope equation by Berkhoff (1972) which is capable of handling the combined effects of refraction and diffraction. Since then significant progress has been made in computational techniques as well as model capabilities, notably by Radder (1979), Copeland (1985), Ebersole et al. (1986), Yoo and O'Connor (1986a), and Dalrymple et al. (1989). The five types of wave equations are concerned here since no single model has been proven to be perfect or has clearly outperformed the others at present; hyperbolic type I by Copeland (1985), hyperbolic type II by Yoo and O'Connor (1986a), elliptic type I by Berkhoff (1972), elliptic type II by Ebersole et al. (1986), and the parabolic type by Radder (1979). Type 'I' implies that the unknown to be solved is expressed in terms of wave mode, while type 'II' in terms of all wave-averaged quantities, and the parabolic type is expressed in terms of wave-averaged quantities for the main propagating axis, and wave mode for the the other axis.
The extension of the mild slope equation to the wave-current interaction has been successfully done by Kirby (1984), and the corresponding equation eventually yielded the energy equation suggested by Bretherton and Garrett (1969):
D-- + (V V)-- V.- (CCgV) + (oa' -2CCg) = 0
where is the complex velocity potential at the water surface. The equation set of hyperbolic type I suggested by Copeland (1985) was extended to wave-current interaction by Ohnaka et al. (1988), the equation set type of hyperbolic type II by




33
Yoo and O'Connor (1986b), the elliptic I and parabolic types by Kirby (1984), the elliptic type II by Cheong (1990). Each can be derived from the mild slope equation revised by Kirby (1984), with varying degrees of approximations.
In Section 4.1, the mild slope equation revised by Kirby (1984) will also be directly obtained from the energy equation, assuming that the steady state condition of wave and current field could be retained for the wave-induced currents which are usually accompanied by the waves. Discussion will be given to the equation revised by Kirby (1984) with the alternative mild slope equation of elliptic type. In Section 4.2, the governing wave equation of each model will be derived from the mild slope equation.
4.1 Mild Slope Equation
The mild slope equation has been derived from Green's second identity. Green's first and second identities coresponding to the mechanical energy conservation equation and the mild slope equation, respectively, are presented in Appendix C. In this section, however, the mild slope equation will be derived directly from the energy equation as described below. Introducing the velocity potentials and free surface displacements composed of current and wave components as given in Eqs. (3.3-3.4), Eq. (3.26) becomes
,a (V7 + eV o)2
- Lh [ eCV )2 ]dz + g(rc + eq) (tc + Eq.)
-Vh. [(Vc + eV,)(c- + eq,),t]dz = 0 (4.1)
Taking Taylor expansion about z = 7c,
0(V + V,) (Vc + eV .)2 a
-I:: [7c ] + (77dz + -[e77 ]-+g (77C + e67
8t -h 2 2 a
-Vh [f (o + eVo,)(oc + e0.)t]dz + eq,[(V c + eV.)(0 + e,)t] = 0 Collecting the terms of O(e 2) and ignoring the long-term fluctuation of current field:
0g n(V.) 2 a197h,
t [( ]dz + -[ ,7vo. V0.1 + g7W
-Vh [t[V0,4,,]dz + V&cqwst] = 0 (4.2)




Substituting Eq. (3.11) into Eq. (4.15),
at f-h ]2 + f d z[ ] + 2[1a V<^ V-V ] +9 a
8- .A 2z -a t2 tB
-Vh .[7~ f2dzV~b + = 0 (4.3)
where
I7f2dz = CCg (4.4)
-a_ g
I7C 2 1 k 2CCg
fh fdz = kCg (4.5)
h g
Therefore, Eq. (4.3) becomes
CCg a (V<$a)2 c2 k2CCg Oa(<^)2 +
I I + 1 1+ a-1wV 2g at 2 2g Ot 2 at Ot
-Vh CCgvw,., + V ewtm = 0
9
Hereafter, we omit the subscript denoting the wave mode, w.
CCgVh. (Vh)t + qt(0o2 k2CCg) + g(qVC V c ) + g27t
-tVh (CCgV ) CCgV5. Vht gVh. (VSltq) = 0
The first and sixth terms offset each other and expanding the third and last terms,
<^$,e"- k 2CCg) + git~Vc V + gVc V4 t +
-,Vh (CCgV4) gVh (Vc,) g9V,0 VhYA = 0 and also offsetting the third and seventh terms,
Substituting the dynamic free surface boundary condition of O(e) given in Eq. (3.10) to q of the third term,
^t(.2 k'CCg)S + g7tVoc V' g(t + V Vc)Vt
-VhqV (CCgV<) gqY Vh(VOO) = 0




35
- k2CCg)S qtVh. (CCgVq) gqS. Vh(V4o7) = 0
Combining the second and fourth terms by use of the kinematic free surface boundary condition of (e) given in Eq. (3.9),
(0"2 k2CCg)4 Vh (CCgV) = 0 (4.6)
Now we consider two kinds of expressions of q.. The first is obtained by substituting the dynamic free surface boundary condition, Eq. (3.10), into the kinematic boundary condition, Eq. (3.9),
t+ (V-U0) (4.7)
and the second expression is simply obtained from Eq. (3.11) as = 01 (4.8)
g
which eliminates the slowly varying motion through the mean surface. With the first relation, we finally get the mild slope equation as same as derived by Kirby (1984);
D2- + (V V)-D V. (CCgV$) + (2 k2CCg)b = 0 (4.9)
Dt2 Dt
and with the second expression, we get the mild slope equation of elliptic type: Vh. (CCgV) + k2CCg = 0 (4.10)
which can be regarded as the wave equation of elliptic type having the same form as the case of no current. This may imply that the wave energy is eventually conserved by the relative group velocity, Cg, in the field of semi-steady state although they may be transported by the absolute group velocity, Cga = Cg + U, causing the unsteady motion of waves and currents. This hypothesis might be examined through the laboratory experiments. If the wave and current field were retained in the steady state, both equations would be consistent with each other by the help of the wave action equation which describes the slowly varying motion at the mean water level. However, we select the mild slope equation of first type as a governing equation of each model, which has been believed to be the one describing the wave motion.




36
4.2 Derivations of Governing Equation of Each Model
In this section, the governing wave equations of the five numerical models selected are derived from the linearized mild slope equation Eq. (4.9) for waves interacting with currents. It is rewritten below with the symbol description; the five governing wave equations include two hyperbolic types, two elliptic types, and one parabolic type.
D-- + (V fI)- -i V. (CCgV ) + (02 k2CCg) 0
Dt2 Dt
where, q is the two dimensional complex potential C is the relative phase velocity (c/k)
Cg is the relative group velocity (Na/Ok)
cr is the intrinsic frequency (02 = gk tanh kh)
w is the absolute frequency
k is the wave number
h is the water depth
V is the steady current velocity vector (u,v)
V= 0/8x + 9/Oy, omitting the subscript h
where o, and k are determined by the Doppler relation, w = o + U k.
4.2.1 Hyperbolic-Type Model I (HM I)
The first governing equation of the hyperbolic-type is given as a pair of first-order equations which constitute a hyperbolic system similar to those used for the solution of the shallow water equations. The basic idea in deriving the equation came from an approach of Ito and Tanimoto (1972) and completed by Copeland (1985) through the mild slop equation of hyperbolic type given by Booij (1984). Ohnaka, Watanabe and Isobe (1988) derived a set of time-dependent mild slope equations extended to a wave and current coexisting field. In the presence of strong currents, however, this governing equation seems to be invalid. Thier equation will be written in the slightly




altered form.
The complex velocity potential at the water surface, q, can be related to the surface elevation ,77, by the dynamic boundary condition at the water surface as follows.
-g7= DO (4.11)
Assuming (x, t) = q(x)e'O where V)(x, t) is a phase function of the wave defined as
= k. x wt, we have obtained
g
io.
Differentiating with t gives a_ g 0r(4.12) Substituting Eqs. (4.11) and (4.12) into Eq. (4.9), we obtain [1 Cg 1)] -7 + O + V7 (CCgV)=0 (4.13)
W C- t g
where V yields another equation given by taking V on Eq. (4.12) and differentiating with t,
+wgV.- = 0 (4.14)
atva
0--- gV =0 (4.15)
Eqs. (4.13-15) are a set of time dependent wave equations for linear water waves interacting with currents which are derived from the mild slope equation and expressed in terms of 17, V,,$ and Vq.
Substituting Eqs. (3.12) and (3.13) to the first order PDE set yields the following wave energy equation, which isn't consistent with the conservation of wave action given by Brethorton and Garett (1968) (see Equation 4.29).
a +Vh(a + V[Cg = 0
This inconsistency is caused by Eq. (3.15) which satisfies a slowly varying property of the dynamic motion.




4.2.2 Hyperbolic-Type Model II (HM II)
The second governing equation of the hyperbolic type is based on the kinematic and dynamic conservation equations which are defined in terms of the wave-period and wave-length average properties of wave motion. While the governing equation of HM1 reflects the effects of wave reflection over the mild slope, this equation isn't able to involve any reflection effect due to the assumptions induced by wave-averaged quantities of progressive waves.
The kinematic conservation of progressive waves can be expressed as 8K
+ Vw = 0 (4.16)
where w is the apparent angular frequency, and K is the wave number vector modified by diffraction effects as approximated by (details are shown in 'Elliptic-type model II'),
K2 = k2 + 2() (4.17)
a 01
ignoring the derivative of CCg. a is the wave amplitude and k is a wave number defined by the Doppler relation which can be approximated by w = d+0.K
2 e+ -K= gktanhkh+ -.K
Substituting this into Eq. (4.16),
8K
- + Vr + K V + 0 U. VK + K x (Vx U) + 0 x (Vx K) = 0 (4.18)
where
Vr Vk + Vh = CgVk + ka Vh (4.19)
- k h sinh2kh
Equation (4.19) is now differentiated with respect to x to obtain, Vk = K VK- C (a)] (4.20)
k 2a 0




39
ignoring the derivative of CCg again. Equation (4.20) is now combined with irrotationality of the modified wave number vector to yield the conservation equation of the wave crest,
4K
- + (Cg + U) VK =-F (4.21)
where
ka CCg a
F = K. VU + K x (Vx ) + ksn Vh CV[V2(a)]
sinh kh 2a
Equation (4.21) states that the rate of change of wave number following a point moving with the group velocity riding on the convective velocity is equal to -F; if F is zero, namely, U and h are constant, then K is constant following such points. The time-dependent dynamic conservation equation is generally given by Oa e a*
(422
S-2V [(Cg + -]-=-0 (4.22)
0t 2a 0
4.2.3 Elliptic-Type Model I (EM I)
The governing equation for the first elliptic-type model can be obtained directly from Eq. (4.9), expressing the two dimensional velocity potential in the linear stationary wave field as follows.
O(x, t) = $(x)e-"t (4.23)
where q(x) is the surface potential in steady state. Substituting the above equation into Eq. (4.9) gives
-iw{2 + (V.U )} + (o.V )(O.V )+(V -)(o.V )
-V. (CCgV) + (o2 w2 k 2CCg)O = 0 (4.24)
The above equation is always valid as long as the wave motion is time-harmonic.




4.2.4 Elliptic-Type Model II (EM II)
If the two-dimensional velocity potential is assumed through use of the waveperiod and wave-length average property, A, as O(x, t) = A(x; t)ie'O the linearized dynamic free surface boundary condition yields, as given in Eq. (3.15),
a
A = -gU~d
where a(x; t) is an amplitude function expressed as 7 = a(x; t)e*'', and ad is the intrinsic frequency including the diffraction effects, defined as ad = w V K. The linearized kinematic free surface boundary condition also yields, as given in Eq. (3.17), a = gk tanh kh AVA. V
A
Substituting Eq. (3.12) into Eq. (4.9) yields the real and imaginary parts shown as follows:
Real part
a a, a a
s ( )+aV-(-) + +V -(Oa)+ N 9dd O t
CCgK V(-~) + K V(CCg-a) + CCgaV K = 0 (4.25)
OUdd Ud
Imaginary part
CCga-K' V (CCgVa) -o'da + (oa2 k2CCg)a = 0 (4.26)
O-Ud o'd
Multiply a/ed to Eq. (4.25) to obtain Sa2 a2
(a) + V. [(Cg + )-] =0 (4.27)
at Ud Ud
This equation can be expressed in terms of energy. Neglecting the diffraction effect on the intrinsic frequency, SE E4.28)
) + V. [(Cg + U)-] = 0 (4.28)




41
It states that the wave action (E/u) riding on the convective velocity is conserved by the group velocity. From Eqs. (4.25) and (4.26), the so called wave energy equation (real part) and Eikonal equation (imaginary part) become finally O (a2)
S+ V- [(Cg + )-] = 0 (4.29)
CCga K2 V. (CCgVa) k 2CCga = 0 (4.30)
The term V. (CCgV!) can be expanded as V. (CCgVa) = CCgV2(a) + VCCg. V(a) (4.31)
The second term in Eq. (4.30) may be negligible in the mild slope approximation. Since we have three unknowns, a, K, and K,, we introduce one more equation which expresses the irrotationality of the gradient of the wave phase function.
Vx K = 0 (4.32)
4.2.5 Parabolic-Type Model (PM)
The parabolic approximation to the elliptic-type equation of harmonic wave motion (Eq. 4.24) is derived by splitting the velocity potential into two components, S= + 5- (4.33)
composed of a transmitted field q+ and a reflected field q-. Eq. (4.24) can be written, through the transformation carried out by Booij (1981), in the following form,
8 1 84I)
-( --) + -y = 0 (4.34)
ax 'Y ax
which is split exactly into two equations, 4+ = +i-/+, = -i- (4.35)
ax ax
where (D can be chosen to be (D = ( Under the assumption that the waves are oriented in x-direction so that k., 0,




expanding Eq. (4.34),
o 1 8
+ (CCg U ) O
822 (CCg U2) 8x ax

+ k 2 1 +

M
k(CCg = 0
k.2(CCg U2))

where

O, 0,q,
M6 = [2wk.u + iw(V- U)I- x(uV-) Oxay
+-[(CCgV) 2 iwUV
and mathching with Eq. (4.34), we get the two relations

O( 8, (uvOX

S kM 1/2
k+k(CCg 2)
= kji2(CCg- u2)1/2 1+ k(1/4
V kg2(CCg U2))

(4.37) (4.38)

Substituting the above relations into Eq. (4.35) yields, for the transmitted field q,

- {k12(CCg u2)1/2
ikkil2(CCg u2)1/2 (
ik~vk (l

By expanding the pseudo-operators in the form

( 1 +

M 1/4
k(CCg u2)}

M )1/2
+
k'(CCg U2),

M
+ 2k2(CCg U2)

results in the simple form

k {kl/2(CCg u')1/2~+} = ikk'(CCg u2)1/2

+' M 1+2k2(CCg u2) 4
(4.40)

where, k,(CCg u ) kCCg u(-u + w) = a(Cgk./k + u) wu. Therefore, expanding Eq. (4.40) and multiplying kl/2(CCg u2)1/2 yields

o(Cg + u) 8x

10
28[z,(c9, +u,)]+ =

(4.36)

+ k(CC+ .U2)
T2M 2) 3/4 + k(CCg -u )

(4.39)

( 1+

M 1/4
k2(CCg u2)




43
ika(Cgs + u)4O + {+-8(uy ) -(uV-) + 8(c 8v -+
+ [(Cg v) + i -qS + 2iwv- } (4.41)
where Cg, = Cgk./k.
The above equation gives the modification to the equation of Kirby (1983) to obtain the more correct energy conservation. Now the equation is modified by introducing an reference wave number which can usually taken as an average wave number, k, as ~+ = A'eix
where the y-directional part of the phase function, e if kydy, and the x-directional phase modulation due to the difference between the local and averaged wave numbers, ei(k-f k.d), are absorbed into the amplitude function of surface potential, A'. Substituting this into Eq. (4.41) and after dropping squares of the components of mean current, the above equation is simplified to
B A' 180
o'(Cgs + u)- + i(k k.)(Cg. + u)A' + -x[o'(Cg + u)]A' = i 0 OA' wOv A'
(CCg ) A' (4.42)
2,9y y 28ay '9y
The above equation admits the instantaneous wave surface to be recovered in the the following form:
77 = Im{-A'eik} (4.43)
g




CHAPTER 5
NUMERICAL SCHEMES AND COMPARISONS OF WAVE MODELS
5.1 Numerical Scheme of Wave Models We have derived the mild slope equation of hyperbolic type in the last chapter, however, in terms of practical applications, the equation is not only accurately solvable, but also attractive since it can only be used for the small domains due to the present computer capacities. In this chapter, four of the five wave models were selected from existing literature and one was developed by the author. These will be described later. The five models have their own numerical methods depending on the type of governing equations. All numerical methods applied here fall under the category of the finite difference method. All spatial finite difference operators shown in this chapter are given in Appendix D.
5.1.1 Hyperbolic Model I
On the staggered grid system as shown in Figure 5.1, the wave displacement is specified at the center of each grid, while the gradient of surface velocity potential is situated at the center of the side of the grid. Eqs. (4.13-4.15) are solved explicitly for the x-y Cartesian coordinate system as shown in Figure 5.1 as follows.
+ uwg() (-) =0 (5.1)
At
- + V n + wgD v(!) = 0 (5.2)
At 0
Equation (4.13) will be solved by the following explicit numerical equation which was suggested by Ohnaka et al. (1988) to avoid the numerical damping on the con-




45
AA A A
> 0> J> J
A A A A
> > > >
A A A A > J> >
-A- A- -A-

w a
>4- V
e s

0: 77 t>: V3,q A: Vy
Figure 5.1: Staggered grid system for hyperbolic model I. vective term.
0_ Og ,9"" ,g
{1+ n 1)} t + D(up) + 9,(vy)
W C At
+-[Dv(CCgVA) + DV(CCgV~v)] = 0 (5.3)
g
9
There are three kinds of boundaries. The first is the upwave boundary which generates the wave field, the sencond is the nonreflective boundary which passes the waves without reflection, and the last is the reflective boundary such as breakwater, jetty, seawall, etc.
The upwave boundary requires some refinement in order that reflected waves traveling back towards the upwave boundary pass out of the model. This boundary condition is given by the incident surface elevation and the reflected surface elevation passing out of the model as follows:

(5.4)

t'(Xo, yo) = T, (Xo, yo) + '-' (xo + Ax, yo)




where,
t(xo, yo) = a, sin(k cos Oxo + k sin 0yo wLt)
r -(zo + Az, yo) = 7t- (xo + AX, yo) + a, sin[k cos O(xo + Ax) + k sin 0yo wt] = k cos 0 A
where ai is an incident wave amplitude and 0 is an angle measured from clockwise from x-axis.
The non-reflective boundary is satisfied either by characteristic method for the downwave boundary or by Neumann condition based on Snell's law for the lateral boundary
7lt(xe,yeM = 77'-'(x-AXI Ye) Dy7 = iky
The three unknowns, 77, V$ and Vy, are given complex to accommodate the application of the lateral boundary condition and calculation of wave amplitude and wave angle. Similarly, the reflective boundary is also satisfied by ,t(x,,ye) = R,7T(x- Ax,y,) Dy77 = iky77
where R is a weighting factor expressed in terms of reflection coefficient, and ky can be specified as zero for the perfect reflection. The subscript e implies the end points along a downwave boundary.
In the wave-current interaction, the determination of wave angle is very important because phase speed, group velocity and intrinsic frequency in the wave equation are determined through the product of wavenumber and current in the dispersion equation. The wave angles are calculated at the center of each grid location by the approximation
0 = tan-1 LTe(VM /)J (5.5)




and the wave height is calculated by H = 2lRe{2} + Im{q} (5.6)
The following relationship between the space interval Ax and Ay and time interval At must be satisfied to perform stable calculations. The stability condition is given as
T
At < [(L AX)2 T (5.7)
(Lmax/z + (Lmx/Ay)2]1/2 where both L/Ax and L/Ay are recommended to be less than O(10) to obtain the accurate solution (Watanabe and Maruyama, 1986).
5.1.2 Hyperbolic Model II
On the staggered grid system as used in hyperbolic model I, the wave amplitude is specified at the center of each grid, while the wave number vector is situated at the center of the side of the grid.
The upstream difference scheme has beeen found to be an excellent choice for the present system of equations (Yoo and O'Connor, 1986a and 1986b). The finite difference form of the kinematic wave equation (4.21) for x component of wave number vector is then represented by
K n+I -n"
- K + (CgI- + u)D *K + (Cg1-C + v)D*K, + K., Du + KD*v
At
kaCCg aD aa ka Vh a D [D~r() + D, ()] =0 (5.8)
Ssinh2kh 2a
and for the y component of wave number vector,
K n+ -n"
- +1A- + (Cg + u)DZK. + (Cg K + v)D Ky + K.D u + KDv
At
kmCCgD a: aa + Dh CcDy,[D(a) + D(-)] = 0 (5.9)
sinh 2kh 2a 0- o
where all finite difference operators indicated by D are given in finite difference form in the Appendix. The K vector along the side boundaries is approximated by the Snell's




48
law. The finite difference form of the wave amplitude equation (4.22) is represented by
(a/o')+1 (a/o)n {D[(Cg + u)--] + [(Cg + v)-} = 0 (5.10)
At 2 a k a k a 0(.0
The stability condition in the small diffraction zone may be determined by
T
At < [(nL AX)2 T (L Ay)2/2 (5.11)
[(naLma/Az)2 + (naLmax/Ay)'V2 where na = Cga/Ca with subscript a indicating the absolute.
5.1.3 Elliptic Model I
Equation (4.24) can be treated as an ordinary differential equation for as given below, so that the y-directional difference operator, D, is explicitly approximated by using either a finite difference method or a fast Fourier transform method. In this section, only the finite difference method is employed.
(u2 CCg).. + {(-2iwu + 2uu, + u + uvy (CCg),} x +2uvD,(.) + {(-2iwv + 2vvx + uvs + u.v (CCg)v}D,,( )
+(v2- CCg)Dy,( ) + {-iw(ux + v) + 2 2 k2CCg} = 0 (5.12) where, the x axis is suggested to be selected for the main direction of wave propagation between x and y axis. We convert the above equation to a pair of first-order equations by the simple expedient of defining the derivative as a second function.
1
i = C [{-2iwu + 2uuz + uv + uvy (CCg)0}b1
CCg U2
+2uvD.,( ) + F(k)
where
F() = {(-2iwv + 2vv, + uv + upv (CCg),}D, + (v2 CCg)DyV
+{-iw(u. + v,) + 0' -2W k'CCg}




sub grids
1-- 4 1 S I i I l l

w
a
V
e
s

Figure 5.2: Sub-grid system for elliptic model I.
In this study, the ordinary differential equations are numerically solved by Gragg's method whose main algorithm for a differential equation O/(x) = f(x, O(x)) is given as

YI = 4i- + hf(xi-l, i-1)
Yj+I = Yji-1 + 2hf(xi-1 + jh, yj)
i (yn + n-i + hf(xi, y.))/2

j = 1,2,..,n- 1

where h is a subgrid space defined as h = Ax/n as shown in Figure 5.2.
The upwave boundary condition is merely the specified complex determined by the incident wave amplitude and wave angle. The lateral boundary conditions are either nonreflective or reflective. The nonreflective boundary condition can be expressed as

, = ikb,

where ky = k sin 0 = ko sin 0,

(5.13)

according to Snell's law in the absence of diffraction effects. While the reflective boundary condition is expressed as = 0 i.e., k. = 0 (5.14)




50
Using the finite central differences on row i yields
- ik, Ay
2+1 2 (J+1 + 6) (5.15)
The lateral boundaries are assumed to be located between the grid columns (1,2) and (NY-1,NY). Therefore, ~ values at the lower lateral boundary can be given as
- =(1 ikyAy/2)
4= (1' (5.16)
1 (1 + ikyAy/2) 2 (5.16)
on row i. The same procedure is used for the upper boundary.
~. (1 + ikyAy/2)
'Y- = (1 ik'NAy/2) -1 (5.17)
Equation (5.13) can also be approximated in the Runge-Kutta method (Dalrymple et al., 1988) by
~6 = [1 ikAy 2(kAy)2 + i6(kAy) + (kAy)4] s
l~(kuAy)3 +5.(8) A= + ikAy 2(k y) i6(kzAy)" + 4(kAY)4 (5.18)
If there is any reflective structure posed in the y-direction, the direction of the reflected waves is the mirror image of that of the incident wave. Since the unknown in this model is the complex surface potential, the reflected wave field can be easily specified as the conjugate by tracing the computation backward.
The wave angle is calculated by
0 = tan-1 (K) (5.19)
K.
where
Kf ==*S (5.20)
where,S = K x = tan-1[Im()/Re()].
The wave height is calculated easily by
H = 2 e{-6}2 +Im{- }2 (5.21)
9 9




51
The following condition for stability is roughly obtained by taking the von Neumann stability analysis on the Helmholtz equation as Ay = Lmaz/7r (5.22)
where L is a wave length (Panchang et al., 1988).
5.1.4 Elliptic Model II
By introducing wave angle, 0, Eq. (4.30) can be expressed as AK2 C = 0
where
A = CCg a, C = DV[CCgD,( a)] + D,[CCgD,(a)] + k2CCga
The solution of K is simply K=
Using the Generalized Lax-Friedrich (L-F) scheme which is identical to that used by Perlin and Dean (1983), 0 can be found by the following expression of Eq. (4.32), Dt [K sin 0] D,[K cos 0] = 0 (5.23)
where
v-t[] (1 r)[]+,j + 0.5r([+l,jl + []i+,j+) []i,j AX
where r is a value between 0 and 1 as a kind of weighting parameter. This parameter is used to enhance the stability of the numerical scheme. However, 7 = 0 is recommended for the most accurate results. The above equation for the solution of 0 is solved row by row using implicit FDM. The 0 value along the side boundaries is approximated by Snell's law.




52
The wave amplitudes can also be found by applying the generalized L-F scheme to Eq. (4.29),
2 a 2 )
D [(CCgK cos 0 + ua) + D, [(CCgK sin 0 + vo) = 0 (5.24)
The above equation for the solution of a is also solved row by row using implicit FDM until a certain accuracy of the wave amplitude is achieved.
5.1.5 Parabolic Model
The parabolic-type wave equation (4.42) is written in the finite difference form using the Crank-Nicolson scheme.
1
u(Cgs + u)DzA' + i(k, ks)(Cg. + u)A' + 2D[a(Cgs + u)]A' =
+-0Dy(CCgDyA') -DvA wvyA' (5.25)
Equation (5.24) is now expressed in the tri-diagonal matrix form which can be easily solved by the double sweep method. The first sweep is only for determining the group velocity of x component.
Presuming Snell's law to hold along the non-reflective lateral boundary
A' = ik A' ,where ky = k sin 0 = ko sin 0o (5.26)
For the lateral reflective boundary ky can also be given according to the ratio of reflection, while for the x-directional reflective boundary this wave equation is inescapable. Using the finite central difference on row i, the expressions for A' and A'Ny are obtained as similarly as done in Section 5.1.3.
The wave angle is calculated by
0 = tan-1' (5.27)
where
K, k 1 (Ky/k)2, K, = D*S (5.28)




53
where S = f Kdx kox + Kvy = tan-'[Im(A')/Tle(A')].
The wave height is calculated easily by
H = 2 'AZe + I ml (5.29)
5.2 Comparison of Wave Models All wave models will be evaluated through the comparison of computational difficulty and capability of handling wave characteristics such as wave shoaling, refraction, diffraction, reflection, wave-current interaction, etc. These characteristics are mainly due to the bottom variation and current, and occur simultaneously. We restrict all comparisons to ideal cases in order to give the mathematical solution or expression if possible.
5.2.1 Computational Difficulty
The degree of computational difficulty is measured in terms of stability as previously provided, as well as CPU time. The stability criteria given below are obtained for ideal cases only. Therefore, they are not general as well as not vigorous.
Table 5.1
Model Stability
HM I At < T/[(Lma/Ax)2 + (Lma/Ay)2]/2
HM II At _< T/[(nLma/AX)2 + (naLmax/Ay)2]"/2 EM I Ay Lmaz/ir for central difference method EM II stable but convergable when Ay > Lma /r
PM stable
n, = Cga/Ca (with subscript a indicating the absolute)
The comparison for the computational time is also not general. Rather, a specific configuration as shown in Figure 5.3 is used as the test bench mark. This configuration is a circular shoal used by Ito and Tanimoto (1972) in their laboratory experiment to study combined diffraction and refraction. This configuration has been cited by many authors for verification purposes. Here, the same grids and same accuracy criteria are used in all models. Wave heights along three cross-sections as shown in




54
Figure 5.4 are compared with the laboratory data of Ito and Tanimoto. It is seen that the wave models of hyperbolic type II and elliptic type II, which took the averaging procedure, don't reproduce local minima and diffraction lobes differently from the other models. Kirby (1986) showed that the absence of the minima and lobes may result from the governing equations of models obtained by taking the time average on periodic motion. The CPU time on a VAX-8350 computer and the values of the agreement parameter are given below.
Table 5.2
Model CPU time d(Sec.1) d(Sec.2) d(Sec.3)
HM I 15 min 0.98 0.97 0.95
HM II 12 min 0.97 0.97 0.96
EM I 24 sec 0.96 0.97 0.94
EM II 5 min 0.98 0.97 0.96
PM 17 sec 0.97 0.97 0.95
The models of ordinary (EM I) and parabolic (PM) types provide the fast solutions showing close agreement when compared with the other models. The agreement is based on an index, d, given here as an agreement parameter (Willmott, 1981): EN(P, 0,)2
d = 1 S IP J+I,- 0)
where Pi is the numerical value, Oi is the theoretical or observed value and 0 is the mean of the variates Oi. The values for d vary between 0 and 1.0, with 1.0 indicating perfect agreement.
5.2.2 Wave Shoaling and Refraction
The phenomenon of wave shoaling and refraction occurs over variable topography or current field similar to that which occurs in optics and acoustics. However, the comparison of these phenomena is restricted over the variable topography. Numerical results were compared with the analytical solution based on the energy conservation equation and Snell's law for waves propagating over a uniform slope. The input data are uniformly given to each model as follows:




SEC.2 SEC.3
D 1 2 3 4 5 a 7
X/U

Figure 5.3: Shoal configuration for comparison of tours of h/Li).

CPU time (concentric circular con-

2 H" I
0 12 17
2x/I E Ct
0 t 2 3 S 5 1 X/Li SEC. I

0 2 2 3 S 6
T/L2 SEC.2H 11 21 EM II
0 2 3 S
T/Ij SEC.2

0 1 2 3 q 5 6
TILl SEC.3

Figure 5.4: Comparison with the laboratory data of Ito and Tanimoto (1972).

4
2

SEC.1




NX NY Ax(m) Ay(m) T(sec)
101 21 0.02 0.14 0.8 )
The time step in the hyperbolic models is fixed at 0.01 sec.
As shown in figures 5.5-5.7, all models except hyperbolic model I produce results of good agreement. Hyperbolic model I, on the other hand, induces periodical fluctuations. The numerical error appears to be related to the ratio of grid size to wave length. As the wave length shortens towards shoreline the error becomes larger and also propagates upwave as time progresses. The numerical results were taken along a center grid line in x- axis.
5.2.3 Wave Diffraction
Wave diffraction is significant in zones causing the height variation, such as behind a semi-infinite breakwater or a groin facing obliquely to the wave approacing. In this section, the model capability of handling wave diffraction was evaluated by comparing wave height with the analytical solution given by Wiegel (1962) for a semi-infinite breakwater. The input data are uniformly given as NX=91, NY=75, ZAx=0.04 m (=0.1 L), Ay=0.08 m for T=0.511 sec except elliptic model II where NY=38 and Ay=0.16 m were used to avoid numerical instability. The time step is 0.01 sec in the hyperbolic models.
Figure 5.8 shows the comparisons for waves approaching normal to the breakwater axis. All models appear to agree well with the analytical result. For the 300 angle to the normal, however, only hyperbolic model I and elliptic model II perform adequately (Fig.5.9). The performance in general can be improved by reducing zy, with the exception of elliptic model II which is almost stable regardless of the size of Ay.
5.2.4 Wave Reflection
Wave reflection was tested for the case of waves approaching a seawall at 00 and 300 in constant deep water depth of 3 m, using the listed input conditions,




57
SHOALING
1.6 [
1.4 Theory
0 HM I
1.2 (D( C
1.0 0
0.8
1.4 Theory
: .01". HM 11 & EMII1
~1.2 ( 1- M1
1.0 0.8
1.4 Theory
o )PM I
1.2
1.0 D
0.8
1.4 --Theory
0 OPMI
1.2
1.0
0.8
0.1 0.2 0.3 0.4 0.5
ko h

Figure 5.5: Comparison of wave shoaling.




58
REFRACTION (height)
1.6
1.4 Theory
D OHM I
1.2
1.0 C 0 (D
0.8
1.4 Theory
D HM I & EM 02
0.
1.2
1.0
0.8
1.4 Theory
o~ OEM I
00
1.2 1.0 0.8
1.4 Theory
o C PM I
1.2
1.0 D
0.8
0.1 0.2 0.3 0.4 0.5
ko h

Figure 5.6: Comparison of wave refraction (wave height).




59
REFRACTION (angle)
40.0 30.0
S20.0
- Theory .,
10.0 O HM I
0.0
30.0
S20.0
- Theory
z
10.0 0 HM II & EM II
0.0
30.0
' 20.0
- Theory
z
10.0 0 EM I
0.0
30.0 20.0
- Theory
z
S10.0 (D PM I
0.0 1 1
0.1 0.2 0.3 0.4 0.5
ko h

Figure 5.7: Comparison of wave refraction (wave angle).




Hyperbolic Model I
8
0
4 3 3
2
I I
0
-4 -3 -2 -1 0 1 2 3 4

y/L

Elliptic Model II
l
/ ~ ?
/'

-4 -3 -2 -1 0 1 2 3

y/L

Elliptic Model I

-4 -3 -2 -1 0 1 2 3 4
y/L

Parabolic Model
70
4
3
0
-4 -3 -2 -1 0 1 2 3 4
y/L

Figure 5.8: Comparison of wave diffraction for semi-infinite breakwater (00) between analytic solutions (dotted line) and numerical solutions (solid contour line of 0.8, 0.6,
0.4 and 0.2 diffraction coeff. from left).

a 7 8 5
4
3 2
0




Hyperbolic Model I

4
3
2
0 0

-4 -3 -2 -1 0 1 2 3 4
y/L

Elliptic Model II

-4 -3 -2 -1 0 1 2 3 4
y/L

-4 -3 -2 -1 0 1 2 3 4
. y/L

Parabolic Model

-4 -3 -2 -1 0 1 2 3 4
y/L

Figure 5.9: Comparison of wave diffraction for semi-infinite breakwater (300).

Elliptic Model I




input data NX NY Ax(m) Ay(m) At(sec) T(sec)
HM 61 21 0.08 0.08 0.02 1.0
EM I 61 21 0.08 0.50 1.0
Owing to the finite grid size and time step a numerical error is also expected. Figure 5.10 shows that the numerical results, on the whole, agree well with theory for both 00 and 300 wave angles. The hyperbolic model tends to yield slightly larger error in wave height, whereas the elliptic model I produces slightly larger phase error.
The wave reflection against the bottom slope was also compared with the 3dimensional numerical solution represented by Booij (1983). Both models run in this study reflect reasonable agreement as shown in Figure 5.11.
5.2.5 Wave-Current Interaction
Wave-current interaction is compared for cases of collinear current (Figure 5.12) and wave refraction due to the shearing current (Figure 5.13), both in constant deep water depth of 3 m. The analytic solution for the shearing current is given by LonguetHiggins and Stewart (1961). The given wave conditions are H,=0.1 m at the upwave boundary and T=1 sec. Waves are allowed to freely pass through the downwave boundaries. The input data are uniform with NX=101, NY=21, Ax=0.1 m Ay=0.6 m for the elliptic models and Ay=0.1 m for the rest. At, whenever applicable, is taken as 0.01 sec. The comparisons with analytical solutions are given in Figures 5.14-16. For the collinear case, all except hyperbolic model I performed adequately. For non-collinear cases, hyperbolic model II and elliptic model II yield good results; the rest all produce varying degrees of inconsistency.
5.2.6 Summary
Each model was evaluated or run on a number of bench mark cases. The final evaluations with assigned rankings are given in the following matrix:




63
REFLECTION AGAINST SEAWALL

4.0 3.0 2.0 1.0 0.0 3.0 2.0 1.0 0.0 3.0 2.0 1.0 0.0 3.0

0.5 1.0

1.5 2.0

2.5 3.0

x/L

Figure 5.10: Wave reflection tests against wall.




0.01

Slope

Figure 5.11: Wave reflection tests against bottom slope.




z
IVI
77 X v
x
0
Figure 5.12: Condition of collinear wave-current interaction.
x
x 1waves

Figure 5.13: Condition of shearing wave-current interaction.




COLLINEAR

3.0
2.5 2.0 1.5 1.0 0.5 0.0 2.5
2.0
* 1.5
1.0 0.5 0.0 2.5
2.0 1.5 1.0 0.5 0.0 2.5
2.0 1.5 1.0 0.5 0.0

-0.3 -0.2

-0.1 -0.0 0.1 0.2 0.3 0.4 0.5

u/Co

Figure 5.14: Comparison of collinear wave-current interaction.




67
SHEARING (height)
3.0
2.5 Theory
2.0
oC OHM I
:= 1.5 1.0 0.5 0.0
2.5 Theory
2.0
2. -(D HM 11 EM 11
:: 1.5 1.0
0.5
0.0
2.5 Theory
2.0
o 0DE1I
p1.5 1.0 0.5
0.0
2.5 Theory
2.0
o 0 PM I
~1.5
1.0 0.5 0.0
-0.3 -0.2 -0.1 -0.0 0.1 0.2 0.3
v/Co
Figure 5.15: Comparison of shearing wave-current interaction (height).




68
SHEARING (angle)

50.0 40.0 30.0 20.0 40.0 30.0 20.0 40.0 30.0 20.0 40.0 30.0 20.0

-0.1 -0.0
v/Co

0.1 0.2

Figure 5.16: Comparison of shearing wave-current interaction (angle).

-0.3

-0.2




69
Table 5.3
Case HMI HMII EMI EMII PM
Governing equation M 0 0 0 M
Programming ease 0 M 0 M 0
Numerical stability M X X X 0
Computational time X X 0 M 0
Shoaling 0 0 0 0 0
Refraction M 0 M 0 M
Diffraction (normal) 0 0 0 0 0
Diffraction (oblique) 0 0 X 0 M
Reflection (vertical) 0 0
Reflection (slope) 0 0
Current (collinear) M 0 0 0 0
Current (refraction) X 0 M 0 X

0: good M: marginal

X: bad -: not applicable




CHAPTER 6
SURF ZONE MODEL
The flow properties in surf zone are utmost complex owing to the strong interactions among motions induced by waves, currents, and turbulence. The present knowledge on surf zone dynmaics is still inadequate and most of the models are rather rudimentary. Most numerous developments in the study of wave breaking have been made by approximation of the wave energy dissipation. These models can be classified into two groups: one is based on the similarity of the wave breaking process with other hydraulic phenomena such as a hydraulic jump (Dally et al., 1984), a tidal bore (Battjes and Janssen, 1978), etc., and the other is based on estimation of the eddy viscosity (Mizuguchi, 1980) or turbulence (Izumiya and Horikawa, 1984).
In this chapter a simple surf zone model is presented. This model is based on the consideration of wave energy balance and wave action conservation so that the wavecurrent interaction is fully taken into account. The model is capable of predicting wave decay and yields quasi-three dimensional mean current profiles both in the across-shore and longshore directions. The model is presented in analytical form for the case of two dimensional gradually-sloped bottoms. Numerical scheme for general topographic applications including the wave diffraction effects is presented in latter chapters in conjunction with the nearshore circulation model.
Section 6.1 modifies the wave energy equation given in Chapter 3 for surf zone application with the addition of an energy dissipation term. In Section 6.2, the validity of wave action equation in surf zone is established. Subsequent sections are devoted to solve decays of wave heights, mean surface currents, and mean set-ups in surf zone. Whenever possible, the results are compared with available experimental




data.
6.1 Time-Averaged Wave Energy Equation in Surf Zone
It is assumed here that the surf zone is coherent in that the essential wave-like periodic motion is retained and is quasi-stationary when time-averaged over wave period. Furthermore, the turbulent motions are of much smaller time scale that it effect on the flow of interest can be simply treated as a dissipative force menifested by an eddy viscosity term. Thus, the familar Navier Stokes' equation of the following form can be applied:
aU ( + E + )= (V X U) x U+vV2U (6.1)
where y is the total viscosity coefficient including the eddy viscosity due to turbulence. Let the surface displacement and the velocity vector, U(u, v, w), be decomposed into mean value, wave and turbulent fluctuations, which are distinguished by subscripts c and w, and prime, respectively; thus, 7= +q'=?lo+liw+1' (6.2)
U = U+U'=U +U,U' (6.3)
where the superscript-is used to denote turbulent averaging. After turbulent-averaging Eq. (6.1) becomes
aU ( + ) =) xU+vtV2U (6.4)
The superscript is omitted hereafter.
Taking the scalar product of U(u, v, w) with the respective terms in the Navier Stokes equation and summing the products give the energy equation with dissipation
Jh{[2]+V. [U(- 2+ + gz)] dz =- vtU V2Udz (6.5)
by applying the kinematic boundary conditions at the free surface and the bottom, Wj7 U.Vh7 = 0
W1h+U.Vhh = 0




72
and letting p equal to zero at the surface, we obtain,
I dz + q [vh U + + gz)ldz + vtU" V2Udz = 0 (6.6)
-h[ 2 at+hUq p f
again, utilizing the Leibnitz' rule of integration. This equation is similar to the wave energy equation given by Eq. (3.26) with the additional disspative term. This disspative term is in the form of a energy flux and can be treated as a head loss term in the context of Bernoulli equation, i.e., E) = VtU. V2Udz = Vh4 J glUdz (6.7)
here I is the head loss due to turbulence. Equation (6.6) can then be expressed as
a i7 q2 7,7 q2 [U( -+gz+ gl)dz=0 (6.8)
Following the same procedures in Chapter 3 by taking time average over wave period, an energy equation similar to that of Eq. (3.38) can be obtained, a- + Vh-(UhE) = 0 (6.9)
Here the transport velocity Uh is the counter part of (Cg + iY dk/u) in Eq. (3.38). Here, the last term can be dropped because of the quasi-steady assumption made here. Clearly, this transport velocity is different from the non-dissipative case and can be represented by
Uh = Cg + + CgD (6.10)
where the first two terms constitute the transport velocity due to non-dissipative forces much the same as in Eq. (3.38) whereas the last term manifests the effect due to dissipative force. This term, in general, should be negative indicating a reduced energy flux due to dissipation. In theory, it can be estimated from the time-averaged energy dissipation term can be estimated following the definition,

Vh- (CgDE) = D

(6.11)




73
where D is the time-averaged dissipation given by D tU" V2Udz = Vh jglUdz hh
6.2 Wave Action Equation in Surf Zone In this section, the wave action equation given in Eq. (3.41) is shown to be also valid even in the presence of strong turbulence. It is assumed here that the dynamic free surface boundary condition given in Eq. (3.10) is still valid with the inclusion of a head loss term. Following the approach by Kirby (1983), a virtual work term is proportional to WD- is introduced to represent the head loss, where W is an positive undefined coefficient indicating the strength of the dissipation. We examine this approach. The dynamic free surface boundary condition then becomes (1 + W) 1(q5)t + VhO5 Vh5] (6.12)
g
which is divided into two secular components that should be separately satisfied, a (6.13)
OA'
-- + VA = 0 (6.14)
where 0D = (1 W)(w U- k). The subscript, s, denotes the mean water surface level. The kinematic free surface boundary condition also yields
0 = (1 + W)gk tanh k(h + 1i7) (6.15)
a
- -- V. 0 (6.16)
Combining Eq. (6.14) and Eq. (6.16), we obtain the following wave action equation which is valid in the surf zone: a pg H2 __-_g_2
- + V (U~-~D g 2) = 0 (6.17)
e t 8 OD 8o
This equation enables us to estimate the surface velocity in the surf zone once the wave height decay rate can be established.




74
6.3 Wave Height Transformation in Surf Zone
Waves break when their height reaches a certain limiting value relative to their length or water depth as a result of wave shoaling on a slope. The broken waves normally keep breaking as the water depth decreases, finally reaching the shoreline. Svensen et al. (1978) divided the breaking zone into inner and outer regions: From the breaking point and for some distance shoreward, it is the outer region where a violent transition of the wave slope takes place large scale vortices are formed in this region. After outer region, the inner region begins as the wave becomes very similar to a tidal bore or a hydraulic jump. However, this wave breaking process not yet been fully clarified since the strong currents and turbulence are generated by the broken waves, and interacted with wave breaking. In this section, we suggest the new approach which takes account of the wave-current interaction in the surf zone. Differently from most of the existing wave breaking models, this new model provides the analytical expression of wave height over the wave breaking zone.
The wave energy equation given in Eq. (6.9), when expressed in terms of wave height, can be written as
O9 wH2, w H2.
-(pgD--) + Vh" [(Cg + f + CgD)pg-- ] 0 (6.18)
,9t 0D 8 OD 8
Now, again we assume that the surf zone retains a quasi-steady state when integrated over wave period, then the slowly varying flow properties become time independant, and the absolute frequency becomes a constant. Accordingly, Eq. (6.18) becomes Vh- [(Cg + f + CgD) H = 0 (6.19)
8 oD
and applying the wave action equation in the steady state, the wave energy equation in the surf zone is reduced to
Vh" [(Cg + CgD)o = 0 (6.20)
8 OUD




75
The quantity corresponding to the dissipative force is assumed here to be proportional to group velocity at the breaking point, Cgb, since wave height is known to decrease steadily within the surf zone; therefore, CgD = -3Cgb (6.21)
where P3 is a positive coefficient. Equation (6.20) now becomes Vh (Cg*L8H2CrD) = 0 (6.22)
8
where the real relative group velocity in the turbulent surf zone is estimated by Cg* = Cg Cgb (6.23)
Equation (6.22) is the final form of the proposed energy transformation model. This model has only unknown coefficient, namely, the dissipation coefficient, fl and is applicable to the general three-dimensional topography and any arbitraly incident wave angles.
An analytical expression can be obtained for two-dimensional beaches of uniform slopes. Equation (6.22) becomes (g- Cgb)- = 0 (6.24)
where x axis is directed onshore. The cross shore component of the above equation gives,
H 2 CD (6.25)
cos 0(#Cgb Cg)
Applying the dynamic free surface boundary condition given in Eq. (6.14) in 2-D steady state condition, we have O'D proportional to the wave height in surf zone. Equation (6.25) can be written as H = PH (6.26)
COS 0(#cgb C9)




76
with il determined later. 0 can be determined by Snell's law as
0 = sin-'(C sin 00/C0) (6.27)
where C, = w/k is an absolute phase speed. Equation (6.26) is non-dimesionized as H 1 (6.28)
Hb H6Cgb cos 0(P Cg')
where Cg' = Cg/Cgb. Now we apply two boundary conditions; H/Hb = 1 at a breaking point d/db = 1, and H/Hb = H' where Cg becomes zero. We then obtain H H' (6.29)
Hb cos 0[1 Cg'(1 H'/ cos Ob)] Applying shallow approximation to Cg' gives
H 1 H'_ _H I_ H' (6.30)
Hb cos 0 1 V(1 H'/cos O()
where d' = d/db and d is defined as a total water depth, (h + ve). Of the three parameters H', f1H and 0 only one is independent If the value of H' is determined by experiement the other two are solved by, f3~~ =cos ObH'
COS ObH, HbCgb (6.31)
COS 0b H,'
cos Os
p = (6.32)
cos Ob HI
Figure 6.1 plots the dimensionless wave height in the surf zone for different 0 values. Figure 6.2 shows the comparison between the present theory and the laboratory data by Horikawa and Kuo (1966). The dimensionless values of the wave height at the shoreline Hi, are determined from the data; they are 0.22, 0.18, 0.14 and 0.14, respectively, for slopes of 1/20, 1/30, 1/65 and 1/80. It was found that the experimental values of H, can be closely approximated by VWana with a being the slope of the beach. Extensive laboratory experiments have shown that the pattern of wave height decay across the surf zone is strongly a function of the beach slope.




Wave Breaking

0.6 0.8

h/hb

Figure 6.1: Effect of a parameter 0 on the dimensionless wave height in the surf zone.




Slope 1/20

0.0 0.2 0.4 0.6 0.8 1.0
d/db
Slope 1/65

0.0 0.2 0.4 0.6 0.8 1.0
d/db

0.0 0.2 0.4 0.6 0.8 1.0
d/db
Slope 1/80

0.0 0.2 0.4 0.6 0.8 1.0
d/db

Figure 6.2: Comparison with laboratory experiments presented by Horikawa and Kuo (1966).

Slope 1/30




Hyperbolic Model

Elliptic Model I

1.0 0.8
0.6
" 0.4
0.2 0.0

0.0 0.2 0.4 0.8 0.8 1.0
d/db
Elliptic Model II

0.0 0.2 0.4 0.8 0.8 1.0
d/db
Parabolic Model

1.0 0.8 0.6 S 0.4
0.2 0.0

0.0 0.2 0.4 0.6 0.8 1.0
d/db

0.0 0.2 0.4 0.8 0.8 1.0
d/db

Figure 6.3: Comparison between theoretical and numerical results.

1.0 0.8
0.6 :: 0.4
0.2 0.0

~zz




80
In the full 3-D model, Eq. (6.22) has to be employed. It is solved numerically either implicitly or explicitly depending upon whether one treats Cg-f3Cgb as a group velocity (implicit) or solves D independently by the explicit expression of Eq. (6.10). It is suggested that the implicit scheme be used for uniform slope cases and the explicit scheme for more complicated topographies.
Figure 6.3 compares the numerical results with the theroretical curves for the case with P=1.28 (or H, = 0.22) corresponding to a uniform slope of 1/20. Since the dimensionless wave weights from the numerical compuatation are expressed as (I"fl 2 1 HSI2
( =) (6.33)
~Hb' = cos 0 1 v/(1 H2/cos Ob) and from the theoretical model based on Eq. (6.30) the following relation has to be used to compare them,
(,)2 (nX Hb
Hb Hb
where the subsripts m and t denote numerical and theoretical values, respectively.
6.4 Surface Currents in Surf Zone
The main mechanism responsible for current generation inside the surf zone is suggested to be due to the excess wave-induced momentum also known as the radiation stresses. Vertical circulations are known to exist as a consequence mass balance to maintain a quasi-steady state. Solution conerning longshore current and its distribution was originally obtained by Longguet-Higgins (1970) and later modified and refined by numerous other investigators. The model was based on the balance between the friction forces and gradients in the radiation stress. Cross-shore current modeling is more recent effort based on such ideas as wave set-up, undertow, etc. In this section, the surface current vectors in the surf zone containing both cross shore and longshore components are solved by the applications of wave action equation and the steady state wave energy equation:
_ H2
v.- (u,-)= 0 (6.34)




81
K H2
V [(PCgb Cg) ] = 0 (6.35)
k oD
Elimilating from the above equations we obtain a pair of simple equations for
CfD
surface currents,
u, = o cos0O(3Cgb Cg) (6.36)
v. = flL sin O(/3Cgb Cg) (6.37)
where us and v, denote, respectively, the cross-shore and longshore components of surface current vector at the mean water level, V,; f3o and PL are constants of proportionarity. When expressed in terms of wave height the above pair of equations become,
, = floH (6.38)
sin 0 aD
v8 = Ay(6.39) Vs -- cos 0 H2 (6.39)
which shows that while both onshore and longshore current components are inversely proportional to the wave height sqaured, only longshore component is a function of wave angle. The surface current equations can be more conveniently expressed in non-dimensional forms,
u/ = 0 os 0(0 Cg') (6.40)
and
v' = /LsinO(1-Cg') (6.41)
where u' = u,/Cgb and Cg' = _. Applying the shallow water condition, we have,

u' = Io cos0(P Vh)

(6.42)




Surface

Onshore

Current

0.4 0.6

h/hb
Figure 6.4: Effect of a parameter 3 on the dimensionless surface onshore current in the surf zone.

\
"". .'.. '
BETA= 1.3 .BETA=1.25 BETA= 1.2
BETA= 1. 15 ........
........ BETA= 1.1




83
Surface Longshore Current

0.0 0.2

0.4 0.6

h/hb
Figure 6.5: Effect of a parameter 0 on the dimensionless surface longshore current in the surf zone.




Slope 1/10

0.2 0.4 0.6 0.8

Slope 1/20

0.2 0.4 0.6 0.8

d/db
Figure 6.6: Comparison of longshore current with laboratory experiments presented by Visser (1991).




85
'L sin .0 V -(1 -- -) (6.43)
Figure 6.4 illustrates the effect of P on the dimensionless surface onshore current given by Eq. (6.42). Unfortunately, no experimental data are available at present.
Figure 6.5 illustrates the effect of /3 on the dimensionless surface longshore current given by Eq. (6.43). Figure 6.6 compares the theory with the laboratory longshore current data measured by Visser (1991). It should be pointed out here that the Visser's data are depth-averaged and theory is for surface current. In the longshore direction, however, one expects the vertical distribution to be rather uniform as will be shown in Chpater 7. The theory appears to fit the data remarkably well. One of the major advantages of the present model is that it requires only one empirical coefficient to control the magnitude and eliminates the troublesome mixing coefficient appeared in most of the existing theories. In order to fit the data one often has to assume large mixing strength without justification.
6.5 Set-Up and Set-Down
The wave set-up in the surf zone is solved here by the equation of 'the kinematic conservation of intrinsic frequency' and the steady state continuity, V. (U8H) = 0
V. [C(h + 77c)] = 0
It is assumed here that the magnitude of the depth-averaged return current beneath the mean water level, V is proportional to the onshore surface current, C., then H = r.(x)(h + c) (6.44)
where ic(x) is, in general, a spatial dependent coefficient. If r.(x) is a constant across the surf zone, the above equation is simply the frequently adopted extension of Miche's criterion. From Eq. (6.44) the set up is solved as
H
7- (x) h (6.45)




or in non-dimensional form,
S C(X=Xb) H' -h' (6.46)
where r/ = 7c/hb and xb indicates the breaking point. Here the definition of the mean water level is limited to the where the bed is at all times covered by water. The set-up or set-down can be computed at once when the wave height within the surf zone is determined. Figure 6.7 illustrates the effect of /3 on the dimensionless set-up given by Eq. (6.46) for a constant K over the surf zone.
There is one notable feature of the model that is generally lacking in most of the existing models. The model predicts rather mild, sometimes near constant, setdown in the transition zone immediately after breaking point where the wave height drops sharply. This phenomenon referred to as transition region 'paradox' has been noted as a significant feature in the transition region (Basco and Yamashita, 1986; Thieke,1988). The conventional theory of balancing the momentum due to radiation stress should produce a jump of set-up in the transition region where wave height reduces sharply. Laboratory data, on the other hand, showed nearly contant set-down across the transition region where the wave height is reduced nearly proportional to the reduction of water depth. This phenomenon will be further clarified in Section
7.2.3 through the momentum balance.




87
Set-Up

0.0
-0.1

0.0 0.2

0.4 0.6

h/hb

Figure 6.7: Effect of a parameter # on the dimensionless set-up in the surf zone.




CHAPTER 7
MATHEMATICAL MODEL FOR WAVE-INDUCED CURRENTS
A prominent feature in the nearshore zone is the wave induced current circulation. It is commonly accepted that the primary driving force is the wave induced radiation stress first introduced by Longuet-Higgins and Stewart (1961). Modelling this circulation has since advanced considerably from the earlier development by Noda et al. (1974) and Ebersole and Dalrymple (1979). Both of these earlier models were driven by a wave refraction model with no current feedback. In recent years, Yoo and O'Connor (1986b) developed a coupled wave-induced circulation model based upon what could be classified as a hyperbolic type wave equation; Yan (1987) and Winer (1988) developed their interaction models based upon parabolic approximation of the wave equation. All these models employed the depth-averaged formulations which have three major deficiencies 1) the surface effects due to wave-current interaction, which generally is very strong, are being neglected; 2) the bottom friction is expressed in terms of the mean velocity which makes the model unrealistic in areas where the current profile is strongly three dimensional but the mean current could be small such as in the surf zone and 3) the convective acceleration terms are also depthaveraged which has the same problem as (2). Recently, de Vriend and Stive (1987) improved the nearshore circulation model by employing a quasi-three dimensional technique. This technique is very attractive to accomodate the surf zone in which the depth-averaged model is no longer valid but the full three-dimensional modelling is currently not attainable. In this chapter, this apprach of quasi-three dimensional modelling is adopted by developing a circulation model combining depth-integrated properties with vertical profiles.




89
In Section 7.1, the fundamental conservation equations of mass and momentum time-averaged over turbulent scale are presented. In Section 7.2, the depth-integrated formulations serving as the basic equations for a quasi-three dimensional circulation model are derived. An amended form of radiation stress is presented and the existence of the surface advection terms is verified through comparisons with wave energy equation. Section 7.3 develops a new model prescribing turbulence-induced vertical current distributions in surf zone. This model employing the surface current boundary conditions given in Chapter 6 yields circulation patterns in both cross-shore and longshore directions. Finally, in Section 7.4, a quasi-three dimensional model suitable for the entire nearshore zone is developed by linking the depth-integrated model to the surf zone model.
7.1 Turbulence-Averaged Governing Equations
The strong presence of turbulence is a prominent feature in surf zone. Consequently, the fundamental equations governing the fluid motion should also include the turbulent effects. This is usually accomplished with the introduction of Reynolds stresses by time averaging over the turbulent fluctuations. Accordingly, the turbulence-averaged governing equations are presented here. Again, the basic equations are: the continuity equation for incompressible fluid, 19u av aw
u +V o (7.1)
the horizontal momentum equations,
Ou Ouu Ouv Ouw 1 Op 1 Or3, Oray Or.
+ + + z- = -- + -(- + --+-- ) (7.2)
at az YPax p Ox O9y az
Ov Ovu Ovv Ovw I 1Op + 1 Ory Or OrZy
Ox + oz o+ Z- ) (7.3)
ax ay P a OY az
and the vertical momentum equation,
Ow Owu Owv Oww 1 O(p + pgz) 1 Or Or Or
T +-5x +Ox oz p Oz + a(-'-- y- + 7- (4)
Oy a PO ax Oy Oz




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