Citation
Wave-current interaction over a submerged bar field

Material Information

Title:
Wave-current interaction over a submerged bar field
Series Title:
UFLCOEL
Creator:
McSherry, Thomas Richard ( Dissertant )
University of Florida -- Coastal and Oceanographic Engineering Laboratory
Kirby, James T. ( Thesis advisor )
Place of Publication:
Gainesville, Fla.
Publisher:
University of Florida
Publication Date:
Copyright Date:
1989
Language:
English
Physical Description:
ix, 102 leaves : ill. ; 28 cm.

Subjects

Subjects / Keywords:
Shore protection ( lcsh )
Breakwaters ( lcsh )
Ocean waves ( lcsh )
Coastal and Oceanographic Engineering thesis M.S
Coastal and Oceanographic Engineering -- Dissertations, Academic -- UF
Genre:
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

Notes

Abstract:
People are moving to the coast in great numbers that seem to increase each year. Development along the shoreline is therefore also on the uprise, and the public often assumes that the shore environment is stable. Recognizing the appeal of the beach, and understanding that this environment is anything but stable, the coastal engineer must attempt to curb the natural destructive forces from the sea that would normally hinder man's use of the beach and it's surroundings. One present goal in this attempt is to achieve shore stabilization with a minimum impact on the dynamic processes in the nearshore region. One candidate receiving much recent investigation is the shore parallel low profile bar field. The bar field acts with the incoming wave to actually reflect energy back out to sea. This action thus protects the beach from some of the destructive wave energy that often erodes massive amounts of dune material in heavy weather. If the concept could be perfected, the low profile bar field could conceivably be a huge shield against any wave possible. The way the bar field reflects energy is by having bar spacings that are one half the incident wave wavelength. With this satisfied, the incident energy serves to form another wave that propagates opposite to that of the incident wave. Knowing that the bar field would be of finite longshore length, a low spot in energy would then exist shoreward of the bar field. What would then be expected is to have a circulation begin by this energy differential, much in the same manner as circulation into the lee of breakwater, or along groins. It is also expected that the current would flow outward through the middle of the bar field. Clearly, then, the current would alter the character of the incident wave, changing it's wavelength. Then the resonant condition between the incident wave and the immoveable bars would be upset. If the bar field is to considered for practical implementation, the behaviour of the rip current driven by the energy differential will need to be investigated. The most important aspect of the wave-driven current is how it responds to slight shifts off resonant incident wave frequency. The main hypothesis of this paper is that the bar-wave system seeks resonance, and therefore if the wave were to be initially out of resonance a current would be required to tune the wave. By incrementally lowering the initial wave frequency, and observing how the system responds and whether reflection is enhanced or reduced, the hypothesis is proven or repudiated. Another factor in this scheme is the available energy from the incident wave. If the waves are not big enough to create sufficient energy differentials, then the current will never reach the strength required to tune the waves that are out of resonance. The thesis thus looks at how current strength responds to wave heights, and how reflection behaves during the process. This paper studies results from laboratory data taken from a finite length bar field, and also findings from a numerical model that incorporates coupled wave equations with the momentum equations. The numerical scheme iterates between the two parts until a final steady state is achieved. Mean values in surface elevation, lateral depth integrated currents and wave amplitudes are solved for through a semi-implicit algorithm
Thesis:
Thesis (M.S.)--University of Florida, 1989.
Bibliography:
Includes bibliographical references.
Funding:
This publication is being made available as part of the report series written by the faculty, staff, and students of the Coastal and Oceanographic Program of the Department of Civil and Coastal Engineering.
Statement of Responsibility:
by Thomas Richard McSherry.

Record Information

Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
All rights reserved, Board of Trustees of the University of Florida
Resource Identifier:
20312025 ( OCLC )

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UFL/COEL-89/014


WAVE-CURRENT INTERACTION OVER A SUBMERGED BAR FIELD









By


Thomas Richard McSherry 1989


Thesis
























WAVE-CURRENT INTERACTION OVER A SUBMERGED BAR FIELD


By

THOMAS RICHARD MCSHERRY




















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


UNIVERSITY OF FLORIDA


1989














ACKNOWLEDGEMENTS


In the course of completing this thesis I have received abundant help in many forms from a variety of people. The work required the extensive help of my Advisor, Dr. James Kirby, and I am truly grateful for his assistance. I consider myself very fortunate to have had the opportunity to study under Dr. Kirby. I also used the COEL extensively on two seperate occasions, and depended on the help of several people there. I thank Jim Joiner and Sydney Schofield for their patient assistance, and the inputs from the entire group. A great source of help at the COEL was in the way of mental release in the form of the daily volleyball game, and although I jammed a few fingers in the less-than-benevolent net play, I am grateful for the enthusiastic games.

I also thank Dr. Robert Dean for his help in every way, and I thank the secretaries who have told me the logistical requirements for getting out from under the deadline hammer. Also, thanks to the students that offered friendship and release, and provided help when I really needed it. These people I wish the greatest of success.


ii















TABLE OF CONTENTS


ACKNOWLEDGEMENTS .............

LIST OF FIGURES ...............


ABSTRACT ..............

CHAPTERS

1 INTRODUCTION ..........

1.1 Problem Statement ........

1.2 Literature Review ........ 2 THEORETICAL REVIEW ....

2.1 Introduction ..........

3 GOVERNING EQUATIONS ...

3.1 Introduction ..........

3.2 Circulation Model .......

3.2.1 Governing Equations .

3.2.2 Radiation Stress Terms

3.3 Wave Model ..........

3.3.1 Governing Equation

4 FINITE DIFFERENCING ....

4.1 Introduction ...........

4.2 Circulation Model ........

4.2.1 Method of Solution . ,


4.2.2

4.2.3


Boundary Conditions . Radiation Stress Subroutine


iii


ii


iv

v


1

1

2

5

5

12 12 12 12 14 20 20 24 24 27 27 32 33


..........................


. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .








4.3 W ave Model . . . . . . . . . . . . . . . . . . 35

4.3.1 Method of Solution . . . . . . . . . . . . . . 35

4.3.2 Finite Differencing . . . . . . . . . . . . . . 35

4.3.3 Boundary Conditions . . . . . . . . . . . . . 37

4.3.4 Lateral Smoothing . . . . . . . . . . . . . . 37

5 LABORATORY W ORK . . . . . . . . . . . . . . . 48

5.1 Introduction . . . . . . . . . . . . . . . . . . 48

5.2 Equipment . . . . . . . . . . . . . . . . . 48

5.2.1 Basin . . . . . . . . . . . . . . . . . 48

5.2.2 W avemaker . . . . . . . . . . . . . . . . 50

5.2.3 Beach and Cart System . . . . . . . . . . . . 50

5.2.4 Electronics . . . . . . . . . . . . . . . . 52

5.2.5 Bar Field . . . . . . . . . . . . . . . . 54

5.3 Setup, Procedure and Data Analysis . . . . . . . . . . . 55

5.3.1 Session One . . . . . . . . . . . . . . . . 56

5.3.2 Session Two . . . . . . . . . . . . . . . . 62

6 RESULTS AND CONCLUSIONS . . . . . . . . . . . . 76

6.1 Introduction . . . . . . . . . . . . . . . . . . 76

6.2 Laboratory W ork . . . . . . . . . . . . . . . . 76

6.2.1 Reflection Results . . . . . . . . . . . . . . 76

6.2.2 Bathymetry . . . . . . . . . . . . . . . . 86

6.3 Numerical Results . . . . . . . . . . . . . . . . 87

6.4 Conclusions and Status Report . . . . . . . . . . . . 92

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . .. 100

BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . 102


iv















LIST OF FIGURES


2.1 Bar field in the presence of a depth-uniform current. From Kirby (1988). 6 2.2 Contour of T, v. JFJI v. W/Wro. . . . . . . . . . . . 9

2.3 Surface projection of transmission Tr v. frequency and current. .... 10 4.1 Flow chart for model MCSIIERRY. . . . . . . . . . . 25

4.2 Hyperbolic tangent startup function. . . . . . . . . . 26

4.3 Grid for circulation model. . . . . . . . . . . . . 28

4.4 Differencing of radiation stress gradients in x-sweep . . . . . 30

4.5 Differencing of advective acceleration in x-sweep. . . . . . . 32

4.6 Wave calculated from parabolic equation on domain of width W. . 39 4.7 Spectrum of A(y) without smoothing. Cutoff at nA = 9.051. . . . 41 4.8 Spectrum of y-direction driving force before smoothing. . . . . 42

4.9 Spectrum of y-direction driving force after smoothing. . . . . . 43

4.10 JAl without smoothing. Bars are located on the bottom of this domain. 44 4.11 JAl after smoothing. Bars are located on the bottom of this domain. 45 4.12 y-direction forcing without smoothing. . . . . . . . . . 46

4.13 y-direction forcing after smoothing. . . . . . . . . . . 47

5.1 Plan view of large basin. . . . . . . . . . . . . . 49

5.2 Plan view of cart track system. . . . . . ... . . . . 51

5.3 Side view of profiling cart. . . . . . . . . . . . . 52

5.4 Sam ple profile. . . . . . . . . . . . . . . . . 53

5.5 Sample calibration curve for wave gage. . . . . . . . . 54


v









5.6 Bar frame design. . . . . . . . . . . . . . 55

5.7 Gage locations for two-gage reflection method. . . . . . . 57

5.8 Reconstructed bar profile for Session One. . . . . . . . 60

5.9 Plan view of barfield for Session One. . . . . . . . . . 61

5.10 Gage locations for Session One. . . . . . . . . . . . 62

5.11 Three gages inline with x-direction waves . . . . . . . . 63

5.12 Sample envelope, Session Two. Retrieved digitally from one of the three
on-offshore transects using the moving cart gage. . . . . . . 68

5.13 Sample envelope height versus X. . . . . . . . . . . 69

5.14 Sample K,. with offshore distance x from Session Two. . . . . 69

5.15 Plot of K, with transect location, Session Two. . . . . . . 70

5.16 Plan view of barfield, showing data collection locations. . . . . 71

5.17 Sample time series of magnitude of current meter data, Session Two. 73 5.18 Profile for Session Two. . . . . . . . . . . . . . 74

5.19 Instrument setup for Session Two. . . . . . . . . . . 75

6.1 Plot of K, with wave period, Session One data and theoretical prediction
from 1-D model Kirby (1987). . . . . . . . . . . . . 78

6.2 Plot of K,. with period, data and theoretical for the case of no discernable
rip current . . . . . . . . . . ... . . . . . . 81

6.3 Plot of K, with period for the case of a discernable shore rip current.
Data and theoretical . . . . . . . . . . . . . . 83

6.4 Two profiles, dotted line referring to profile used in the second and third
sets of testing, where a weak current formed. . . . . . . . 84

6.5 Plot of K, with period for case of discernable shore rip current, moving
toward resonance. . . . . . . . . . . . . . . . 85

6.6 Bathymetry 1. Shoreward direction with increasing x. . . . . 88

6.7 JA| for initial amplitude of .03 meters and perfectly resonant waves. . 89

6.8 JBI for initial amplitude of .03 meters and perfectly resonant waves,
without currents. . . . . . . . . . . . . . . . 90

6.9 Steady state mean surface, j. . . . . . . . . . . . 91


vi








6.10 No-current wave model, incident amplitude of .02 meters. K, = 0.81. 93 6.11 No-current wave model, incident amplitude of .03 meters. K, = 0.81. 94 6.12 No-current wave model, incident amplitude of .04 meters. Kr = 0.81. 95

6.13 No-current wave model, incident amplitude of .03 meters. K, = 0.81,
and bar half-length is 3 meters. . . . . . . . . . . . 96

6.14 No-current wave model, incident amplitude of .03 meters. Kr = 0.843,
and bar half-length is 2.25 meters. . . . . . . . . . . 97

6.15 No-current wave model, incident amplitude of .03 meters. K, = 0.88,
and bar half-length is 1.75 meters. . . . . . . . . . . 98

6.16 No-current, tapered bars with .03 meter incident wave. K, 774. . 99


vii














Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science WAVE-CURRENT INTERACTION OVER A SUBMERGED BAR FIELD By

THOMAS RICHARD MCSHERRY

December 1989

Chairman: Dr. James T. Kirby
Major Department: Coastal and Oceanographic Engineering

People are moving to the coast in great numbers that seem to increase each year. Development along the shoreline is therefore also on the uprise, and the public often assumes that the shore environment is stable. Recognizing the appeal of the beach, and understanding that this environment is anything but stable, the coastal engineer must attempt to curb the natural destructive forces from the sea that would normally hinder man's use of the beach and it's surroundings.

One present goal in this attempt is to achieve shore stabilization with a minimum impact on the dynamic processes in the nearshore region. One candidate receiving much recent investigation is the shore parallel low profile bar field. The bar field acts with the incoming wave to actually reflect energy back out to sea. This action thus protects the beach from some of the destructive wave energy that often erodes massive amounts of dune material in heavy weather. If the concept could be perfected, the low profile bar field could conceivably be a huge shield against any wave possible.

The way the bar field reflects energy is by having bar spacings that are one half the incident wave wavelength. With this satisfied, the incident energy serves to form another wave that propagates opposite to that of the incident wave. Knowing that the bar field would be of finite longshore length, a low spot in energy would then exist shoreward of


viii








the bar field. What would then be expected is to have a circulation begin by this energy differential, much in the same manner as circulation into the lee of breakwater, or along groins. It is also expected that the current would flow outward through the middle of the bar field. Clearly, then, the current would alter the character of the incident wave, changing it's wavelength. Then the resonant condition between the incident wave and the immoveable bars would be upset. If the bar field is to considered for practical implementation, the behaviour of the rip current driven by the energy differential will need to be investigated.

The most important aspect of the wave-driven current is how it responds to slight shifts off resonant incident wave frequency. The main hypothesis of this paper is that the bar-wave system seeks resonance, and therefore if the wave were to be initially out of resonance a current would be required to tune the wave. By incrementally lowering the initial wave frequency, and observing how the system responds and whether reflection is enhanced or reduced, the hypothesis is proven or repudiated. Another factor in this scheme is the available energy from the incident wave. If the waves are not big enough to create sufficient energy differentials, then the current will never reach the strength required to tune the waves that are out of resonance. The thesis thus looks at how current strength responds to wave heights, and how reflection behaves during the process.

This paper studies results from laboratory data taken from a finite length bar field, and also findings from a numerical model that incorporates coupled wave equations with the momentum equations. The numerical scheme iterates between the two parts until a final steady state is achieved. Mean values in surface elevation, lateral depth integrated currents and wave amplitudes are solved for through a semi-implicit algorithm.


ix














CHAPTER 1
INTRODUCTION





1.1 Problem Statement

The need for effective shore protection seems to be increasing with each year. The worldwide dependence upon the coastline coupled with such chronic problems as rising sea level has placed a responsibility upon coastal engineers to investigate truly effective methods of shore stabilization. Effective means that it must reduce erosion, hold up under severe conditions, be economically feasible, be environmentally sound, and not impede the regular use of the coastline. Historically, structures meant to curb erosion have had some success, but the side effects caused by the structures were oftentimes worse than the original problem. Today, the main thought is to develop a device that works elegantly and efficiently. One of the promising candidates is the low profile bar field that reflects incident wave energy by use of a phenomenon now known in water wave theory as Bragg reflection, which was named after the analogous process in crystallography.

By placing bars along the sea bottom parallel to the shore, and making their spacing one half of the incident wavelength, a resonant condition occurs between these bars and the surface waves. Specifically, a reflected wave is generated through the bottom boundary condition. This reflected wave grows in amplitude with the number of bars placed until theoretically complete reflection can occur if there were an infinite number of bars. The findings to date have been very promising, with upwards of 80% reflection for waves travelling over four sinusoidal bars. Previous laboratory data also confirm the expectations, and these encouraging findings are fostering continued research into the low profile bar field.

These bars would necessarily be of finite longshore length, and if part of the incoming


1






2
energy were reflected, a low spot in energy would exist shoreward of the bar field, with full amounts at the sides of the bars. This differential in the mean surface elevation along the coast could possibly generate a large enough circulation to create a fairly strong rip through the barfield and back offshore. What this means is that the incident wavefield will be distorted over the barfield, and the resonant condition will be upset. An important question that has been answered by Mei (1985), is how far off resonance the waves can be before the bars quit reflecting energy. With the introduction of a current, however, the question is whether reflection will be enhanced or reduced.

This thesis focuses on the wave-current interaction over the bar field, and what happens when initial wave frequency is incrementally reduced. What the thesis tries to answer is what happens to reflection given the current, and how the current changes as wave height changes. Also, the thesis tries to prove that the wave-bar system seeks a resonant condition by tuning off- resonant waves closer to resonance through the rip current. These questions are answered by means of laboratory data and a numerical model. Some theoretical review introduces the main content, and the problem is fully developed. Then the equations necessary to the problem are presented, followed by a numerical solution to the equations. After this, the laboratory work is covered, and the methods of data analysis are reviewed. Finally, the results from the laboratory work and the numerical model are given, through which the hypothesis will be proven or repudiated. Before all of this, however, the past research that this thesis depends upon will be reviewed.



1.2 Literature Review


This paper is a direct extension of various avenues of previous research. The radiation stress concept developed by Longuet-Higgins and Stewart (1964), is obviously important in any study of waves and currents. These concepts were extended to terms involving complex surface amplitude y' by Mei (1972). The expressions in Mei's book (1983) are used here directly,and are the driving forces in the time averaged, depth integrated momentum






3
equations that govern the circulation and mean water surface.

The other terms appearing in the momentum equations have been included in the model, and most of those formulations are based on previous work. The bottom friction as developed by Longuet-Higgins (1970a) is represented in the model as a function of the linear water wave maximum orbital velocity and the mean current U. In a companion paper, Longuet-Higgins (1970b) looked at the transition of longshore current at the breaker line, and developed an expression for the lateral mixing. This term also is included in the present model.

The advective acceleration terms in the momentum equations are included in the model, and are formulated as a four grid-point average of the current values, which is done by Winer (1988). The momentum equations are solved by an Alternating Direction Implicit (ADI) method which was introduced by Sheng and Butler (1982). The matrix set up for this method can be seen also in Winer (1988). Although the matrix implicitly refers to linear terms, the model introduces limitations in time step by including the nonlinear terms explicitly. Iteration between wave model and circulation model was used by Noda et. al. (1974), where the importance of wave current interaction in determining the final current field was noted.

There has been much work on Bragg reflection in the last decade, as elegant and environmentally sound methods are sought for shore stabilization. A non-resonant theory was developed by Davies and Heathershaw (1984), and partially successful comparisons to laboratory data were made. The theory broke down at resonant frequencies, however, and the reflection coefficients predicted were significantly larger than the data. Mei (1985) extended the theory to include resonant frequencies, and successfully predicted the reflections seen in previous experimentation. Kirby (1986a) allowed for the presence of a mild slope in the theory. The theory was further extended to include currents flowing over the bar field, although on a flat bottom by Kirby (1988). Kirby (1986b) attempted to include currents over a mild slope without the bar field. The work on flat topography noted that the pres-






4
ence of a current enhances reflection, and presented the background that this paper uses as a starting point. The presence of a current allows incident waves of frequencies slightly off resonance to tune with the bar field and actually achieve greater reflections than with perfectly resonant waves. Transmission coefficients always drop in the presence of a current. Thus with the joint action of a depressed energy zone driving a rip current that can alter the incident wavelength, and transmission reduced against the current which further reduces the low energy zone, the system may tend to want to go toward perfect tuning. The scheme depends largely, however, on the incoming available energy to drive the current. Laboratory data will demontrate cases where incident wave heights were too small to sufficiently create a rip current strong enough to keep the reflective plateau.

Various methods are used to measure reflection from the laboratory data, and the previous work that made these methods available will be cited in Chapter Five.














CHAPTER 2
THEORETICAL REVIEW



2.1 Introduction


To begin the task of answering some of the questions raised in the last chapter, specific wave theory should be reviewed. It has been conjectured already that reflection will increase when a current flows over the bar field against the waves.. The thesis will seek to prove this, and the more important concept that the entire system will try to reach a resonant condition. The mathematical basis for these hypotheses was developed by Kirby (1988).

Referring to the following figure 2.1 for the case of a current over a sinusoidal bar field of length L, with bar wavenumber A = 27r/Lb, Kirby showed that if one considers incident waves at or very near resonance, the ratio of transmission with a current to transmission without a current is



TA = A(L)urrent cosh QL (2.1) A( L)nocurrent cosh Qc

The subscripts c refer to properties in the presence of a current. The bar field has length L, where A(L) is the value of the transmitted wave amplitude at the back edge of the bars. The lateral bar shape is described as Db. The other terms are defined as




Q =(2.2) Cg

Q = C (2.3)
(C1C2)5


go = gk2Db
4wcosh2kh


5






6


x


-U




Lb=-f 1(y) x=L
Lb=





Figure 2.1: Bar field in the presence of a depth-uniform current. From Kirby (1988).


(CU)2 = (2.5) where C1 = Cgi + U and C2 = C2 U. Qc is defined later by (3.24). Note that when a strong enough current opposes either wave, the wave can be stopped when C.1 + U = 0. To simplify the expressions and to show what happens when the current strength changes, consider a shallow water wave where the expressions reduce to the following; ADb
8h
Qc C 8 h ( F)where the Froude number in the x-direction is F, = U//V~gi. It appears that Qc has a singularity as IFxI -* 1, which corresponds to the stopping Froude number. When IFMj < 1, from the above simplifications it is seen that Qc > Q, and therefore TA is less than unity. What this means is that in the presence of a current, transmission for waves which are perfectly resonant with the bars with an opposing current is less than the transmission






7
for perfectly resonant waves without a current. The current is thus perceived to enhance reflection, and it is interesting to note that this is independent of the direction of the current along the x-axis.

Now for the shallow water limit consider resonant frequencies, where Wr is in the presence of a current, and w,, is without an imposed current. These are




1r = /7A(1 F) (2.6) Wro VghA (2.7)
2

which from the ratio w,/w0r = 1 Fx describes a parabola. Refer to figure 5 in Kirby (1988) for a picture of this ratio for various values of Ah, where Ah = 0 is also shown as the shallow water limit. Now consider any frequency w shifted off the resonant frequency Wr by an amount Q, so that W = Wr + Q. There is a cutoff condition for this shift, above which for a given current, the resonance is lost between the wave and the bar field. This cutoff is described by



Q 2 6C2 ; C = Vg (2.8) cutof f =64C2

The behaviour of the wave field depends-on the relation between Q and Qcutoff. Mei (1985) and Kirby (1988) denoted four cases where the solutions to the coupled equations described different wave fields, and these solutions for A(x) and B(x) can be put in terms of the transmission coefficients, T, = IA(Lb)/A(0)I. These expressions for the four cases are case 1: > Qcutoff



PC1
T, =C / (2.9) [(PC1)2cos2PLb + (S')2Si2PLb]/ ( case 2: 9 = Sctoff







8


T, = C, (2.10) [C2 + ('Lb)211/2

case 3: Q < cutoff




T= QC1 (2.11) [(QC1)2cosh2QLb + (QI)2sinh2QLbl1l/2

case 4: Q = 0 (perfect tuning)



1
T = coshQLb (2.12) These relations may be evaluated close to shore using the approximations:



C1 = v/gh(1+ F)

C2 = v/g(1 F.)
gADb 1
8Vg-h (1 F.)


P' = 2

2gh(l Fx)

Q2 _p2 when Q < Qcutoff These expressions can be put into a program to plot the surface projection of transmission Tr for any combination of w/Wr, and IFI. Figure 2.2 shows the contour of the transmission coefficient, and figure 2.3 is a surface projection plot of the same result.

Considering an incident frequency that is not far off wro, say 85% of it's value, then by no-current resonant theory the transmission coefficient will be 0.8. The mean surface elevation in the surfzone shoreward of the bar field would then be lower than surrounding areas, and the differential could be enough to drive a current. By following this line of constant w/wro along the surface projection, clearly as the current increases the line heads







9


























1.00






H Hg





0.






JR FROM 0.5000BE-01 TO 1.000 CONTOUR INTERVAL OF 2.5909SE-01 PT13.31. 6.99934 LABELS SCALED BY 1960.
JF;
Figure 2.2: Contour of T, v. IFl v. w/wro.









10


02


0.5


0.7.0


0.b






;.O .g









Figure 2.3: Surface projection of transmission T, v. frequency and current.






11
down the bank of the large trench toward the perfect resonance ratio wr /Wro, which again is illustrated in figure 5 of Kirby (1988). As this minimum transmission coefficient is reached, any more current would drive the transmission coefficient up the other side of the trench. This will decrease the energy differential and reduce the current, allowing the system to fall into the base of the trench again.

This assumes that there is enough available energy from the wave to drive the currents necessary to reach the trench base, but clearly shows the process that the system seeks an equilibrium value that is as close to the perfect resonance ratio as it can get. This is the main point of this thesis, and Chapter Six will present results which are aimed at investigating this hypothesis.














CHAPTER3
GOVERNING EQUATIONS





3.1 Introduction


This chapter addresses the governing equations that apply for the case of water waves over mild topography, and the resulting mean currents and mean surface elevation. The problem as stated must be simplified to be solved numerically, and therefore the equations governing momentum balance in the domain between the mean values must be reduced to two dimensions. Also, the wave equations that govern wave amplitude over the bathymetry are presented as a parabolic equation. These simplications inherently take away from the full process, but without them the solution would be very difficult to achieve. First to be presented in this chapter are the equations and terms that apply to the circulation. Then the equations for the wave field will be presented.

3.2 Circulation Model


3.2.1 Governing Equations

The governing equations are presented in equations (3.1)-(3.2). They are the depthintegrated and time averaged equations of motion, where time averaging is done over a wave period. The derivation can be found in a number of references, most recently in Appendix A of Winer (1988).



aU aU oU aW 1 1
+ U- V ++g- -+Tb- sx Ft ax 0y Dx pD pD


12






13
1 (dSo B~N 1 drj + I+ I+ I- = 0 (3.1) pD dx dy pdy dV dV dv Od 1 1
+ x U-+ -+ -+ --rhy rbi~Y
t + x y -Oy p D p D



+ + dS + = 0 (3.2) pD Ox OY pox The continuity equation completes the needed three equations for the three unknowns W, U and V.
d 9 d d + (U D) + -(V D) = 0 (3.3)


These equations are simplified to the spatially two-dimensional case to render a realistically solvable set of equations. The assumption does, however, take away from the complete process which might include strong effects from vertically variable current fields. In these equations, the terms are defined as




U = x component of mean current V = y component of mean current = mean water surface elevation p = vertically constant water density

ho = still water depth

D = total water depth = h, + 71

rT = lateral shear stress

rbx = x component of bottom shear stress rby = y component of bottom shear stress r,, = x component of surface shear stress ry = y component of surface shear stress






14

SoX = x component of flux due to x-propagating wave S ,y = y component of flux due to x-propagating wave Svy = y component of flux due to y-propagating wave


Note that the equations will retain the nonlinear advective acceleration, gradients in the radiation stresses, bottom friction, and lateral mixing. The numerical representation of these terms are the same in this thesis as was presented in Winer (1988). Referring to that dissertation, the bottom friction term is represented by equations (2.15) and (2.16), which is a direct application of Longuet-Higgins (1970a). The lateral mixing term is in equations (2.26) and (2.27), and also in Longuet-Higgins (1970b). The advective acceleration is represented in finite difference form in equations (4.24) and (4.25). The surface shear stress is not included in the paper, but could be added easily enough. It was not included for it remains out of the scope of the present problem dealing with energy differentials due to reflection.



3.2.2 Radiation Stress Terms

The original model by Winer (1988) did not address the reflected wave that is now present. Since a reflected wave will be expected, the radiation stress terms must be altered to include this wave to maintain a proper balance. Also, when considering waves on a current, the incident and reflected wavenumbers will be different. If one looks at how the waves interact with each other, there will be terms that will oscillate spatially at a spatial frequency related to the difference of these wavenumbers. Therefore, deriving radiation stresses would necessarily be involved. For demonstration of what types of terms are present in a two-wave system at zero angle of propagation, consider the situation of zero current and thus equal wavenumbers. To begin, the linear velocity potential for a two-wave system is given by






15


>(X, yZ,)= -i cosh k(ho + z) A(x, y) ei (kx-wt)
2 w cosh kh+
ig cosh k(h + z) B(x, y)e (-kx-wt) + *
2 w cosh kh,


(3.4)


and the instantaneous surface displacement r' is given by


r'(x,y,) = [A(x, y) eik + B(x, y) e-ikx e-iwt + *


(3.5)


where denotes the complex conjugate. It has been assumed that the complex amplitudes relate to a scaling parameter E defined as E = kIA l, the steepness of the wave. The amplitudes vary with x at 0(2), and in y at O(c). In keeping with the parabolic simplification to the wave model, the wave angles are then assumed to be at or very near zero. The wave induced velocities are then, by (3.6-8)


Ox







V' = =
Sy




0
1 9z


g cosh k(ho + z) k -A i(k-wt) i A ei(kx-wt)
2w cosh kh, I
+ kA*e-i(kx-wt) + iA* e i(kx-wt) kBei(-kx-wt)

iBxei(~kx-w) kB*e-i (-k-wt) + i B* e-i(~kx-t) (3.6)



g cosh k(ho + z) -i AY ei(k-Lu) + iA* ei(kx-wt) 2wcoshkho I +
- i By ei(-kx-wt) + i B* e-i(-kx-wt) (3.7)


i g k sinh k(ho + z)-A i(kx-wt) + A* e-i(kx-wt)
2w cosh kh I
B ei(-kx-wt) + B* e-i(-kx-wt) (


These wave induced velocities go in to form the radiation stress terms, along with the mean surface elevation and wave surface displacement. The general expression for each of the radiation stress terms are presented as a review from Mei (1983) as


(3.8)






16


Sx = p '2dz + p g( -z)dz + T p /"a d( dz
K-h K-h ,-h \ /

p w2 dz (h, + )2 + (71)2 (3.9)
-ho 2 2


PjU'v'dz (3.10)



SYY= p v2dz + p g(u- z)dz + L(pJ d() dz
-h0 .-h0 Jho\JO Y

p 2dz P (h, + u)2 + ()2 (3.11) From these the derivation will now be completed for the two-wave system without currents. First note that for any two complex number R and Q, that RQ* + R*Q = 2R[RQ*] and RQ* R*Q = 2iQ{RQ*]. Now consider each term in the radiation stress expressions separately;


TERM ONE

p fV,,T2 dz




pI 22dz {2k2|A|2 + 2k2IB2 4k 2 R[A B* e2ikx + 2|A,|2 + 2|Bx\2 + 4 R[A, B* e2ikx]

4 k[A A*] 4k [A B* e2ikx] 4 kQ4[Ax B* eikx + 4kZ[B B*] } p coshk ho+z)2 dz (3.12) The integral in this expression has the solution,



-h (coshk(h, ) dz = 2k2 (I + sinh2khc,) wCg (3.13)






17

TERM TWO

p fh1go z)dz



pg ('I z) dz = -g-(h, + pg)2= J-ho 2






TERM THREE

PfVh0 f7727-7 d(dz


p d( dz = J-h Jz 1 9x


{ 2 |Ax|2 + 2|BxI2 + 2 R[A A*x] + 4 R[A B e2ikxi + 2 R[[B B* ] + 2 R[A B* e2ikx] 8k Z[A B* e2ikx]

- 8 k '[Ax B* e2ikx] + 2 R[ Axx B* e2ikx]

- 8 k2 R[A B* e2ikxl ff lp)sinh 2k(h, + () d( dz
-h0Jz sinh2kh,
(3.15)


The integral is evaluated as,


-ho sinh 2k(2h + dCdz = [2kh0 coth 2kho 1]


TERM FOUR P fhjH 2 dz


(3.14)


(3.16)






18


P j p2 dz = 1( sinhk2h hO) { A2 + |B12 + 2R[A B* e2ikx]} (3.17)




TERM FIVE

-P(h. + )2




TERM SIX





UI (')2 = .g { Al2 + |B12 + 2,R[A B* e2ikx] (3.18)






Then writing the final expression for the radiation stress component Sx,



Sxx =
pg (1 + 2hk h k 2| A|2 + k 2|B|2 2k 2 R[A B* e2ikx
4k2 \. sinh2k, fIAI2 2k
+ A|2 + |Bx12 + 2Ri[Ax B* e2ikx

2 k Q[A A* ] 2 k Z[A B* e2ikx] k Z[Ax B* e2ikx] + 2 kQ[B B*]} + Pg (2khcoth2kh 1) {|Ax|2 + lB.|2 + R[A A*x] + R[BB*x]

+ R[A B* e2ikx] 4k Z[A B* e2ikx] 4 k 7[4Ax B* e2ikr] + R[Ax, B* e2ikx] 4 k2 R[[A B* e2ikx] + 2 R[Ax B* e2ikx

p sinh2kho) {|A2 + B|2 + 2FR[AB*e2ik}
+ { |A12 + |B12 + 2R[ AB*e2ikx] (3.19)






19
Now for the terms in the S., expression, only the first applies.


p ffhu' V dZ

and the expression for S., follows as




S.Y = 1 + 2h h) { [A A*] + R[A, B* e2ikx + R[B* AY e2ikx] + R[Bx B*] kQ [A A*]

k Q [A B* e2ikx] k Q[Ay B* e2ikx] + k Z[B BY*} (3.20)





For the S.y expression, which is similar to the Sxx term in the integral representations, the final expression is



Syy = 1 1 + 2k h,, { Ay12 + |Byj2 + 2R[AyB*e2ikx]

+ g(2khoth2kh 1) {IAy12 + IByI2 + 2 R[AyB*e2ikx] + R[AA*Y]

+ 4R[B B*y] + + [A B*, e2ikx] + R[ Ayy B* e2ikx]

pg sinh2k ho) {A12 + |B12 + 2R[AB*e2ikx]
4I~ 'B'2 k+h
+ 4 |A\2 + |B12 + 2 R[A B* e2ikx] (3.21)




The terms arising because of the two-wave system couple nicely when not considering currents. After doing some preliminary testing of the model, it was found that the interaction between the two waves in the radiation stress terms would give rise to small currents. Since the only concern for the modelling effort was currents arising from differentials in overall energy, these small disturbances did not matter. Thus it was decided to calculate the wave field for a given current field and let the bar coupling extract the energy from






20
the incident wave, then set the reflected wave field to zero in the grid prior to calculating the radiation stresses. This would not only remove the currents over the bar field due to interplay of the partial wave, but also allow a simple form of the subroutine that included the incident wavenumber. The equations are the same as (3.19)-(3.21), but contain only incident A terms. The final expressions used in the model will be presented in Chapter Four.





3.3 Wave Model

3.3.1 Governing Equation

The wave model emulates the wave equation presented in this section. It governs the energy balance of a wave travelling over a mild bottom with fast undulations, and riding on a current. The parent equations come from two sources, one being Kirby (1988) and the other being a University of Florida Technical Report, Kirby (1986b). The first paper developed coupled equations for waves on a current over a bar field on a flat bottom. The other paper handled such a case, but on a mild slope. The forcing from Kirby (1988) is used in the present thesis, and the main left hand side of Kirby (1986b) is also used to complete the full equation. As a review, the coupled equations for the incident and reflected waves from Kirby (1988) are the refraction approximation,



{TT1 + (C., cosO1 + Uo) Tx, + (C1 sin91 + V) Ty1} = -OR (3.22)



RT + (Cg2 cosd2 + U,) Rx, + (-C92 sin92 + Vo) Ry} = Oc T (3.23) where the following terms are defined;


Qc = Qoc cos(O1 + 92) + (1c,2


(3.24)







21


QC = gklk2Db (3.25) 4w cosh k1h cosh k2h'



Q1, = (4wa coshAh)-' {AU0(120-1 lior2)Db

U[gkik2 cos(01 + 92) + (010-2 + (AUO)2)]Db} (3.26)
9 g



a = AhF2 tanhAh A = Barfield wavenumber = Lb
U0
F = Froude number = Ii = |kicos9ij

-i = intrinsic frequency = Vgki tanh kih

2 = 0r- 2
27r
w = absolute frequency = wave period Db = Uniform bar amplitude The amplitudes in these equations are surface displacement amplitudes divided by intrinsic frequency, or T = A/1 and R = B/-2. From the technical report, the parabolic coupled equations are written as,



2ic-(Cgi + U.)T. + 2oi(Cgi + U0)(k1 k1)T + 2ic-1VTy

+ i {[o-(Cgi + U.)]. + (aV)v} T + {(CCgi V2)(T),}

= coupling with R (3.27)



2io2(C2 UO)R, 2-2(C92 U0)(k2 k2)R 2ia-2VR, + i {[0-2(Cg2 Uo)]. (0i2V)} R {(CCg2 V2)( R)y}






22


= coupling with T (3.28) By dividing by 2i from (3.27-28), and multiplying w to the coupling from Kirby (1988), a set of equations result for the case of a bar field on a mild slope with currents. These are then solved using a Crank-Nicolson method for each wave until convergence is achieved. A term w that handles the wave breaking energy dissipation is also included. This is added to the incident wave only, since the reflected wave will not be expected to break anywhere. This term is obtained from Dally et al. (1985). The concept is that if wave height exceeds a criterion based on the local water depth, written as


D1AI (3.29) then breaking commences, where K = 0.78. The term is then defined as



w = [1 4[2] (3.30) W = Th 41A12I
where = 0.4 and K = 0.15 for the present paper. The full parabolic equations used in the model are then




al(C1 + U0)T. ioi(C.i + U0)(k1 ki)T + olVTY

+ {{[a(Cg1 + U.)]. + (iV),} T ){(CCgi 2

+ -c i(Ci + U0)T

= -wQ R (3.31)



02(C2 U0)R7 + iU2(Cg2 U0)(k2 k2)R 92VRY

+ {[2(Cg2 U.)]. (0i2V)y} R + {(CC2 V2)(R)I,

= -wQ T (3.32)






23

This is recognized as the equations governing waves that are at perfect resonance with the bars. If there were a slight shift Q, then the forcing would have a different form.














CHAPTER4
FINITE DIFFERENCING





4.1 Introduction

The incorporation of the equations developed in Chapter Three involves finite differencing, decisions on initial and boundary conditions, and placement within a coherent algorithm that performs efficiently. The latter requirement is best described by means of a flow chart. So before moving to the details of the coding, an overall view of the model MCSHERRY would be timely. Figure 4.1 provides the logic behind the main program.

After inputs of bathymetry and wave information, the model iterates between the wave model WAVEMOD and circulation model CIRC. A hyperbolic tangent startup function is multiplied to the initial offshore incident amplitude, which slowly increases it's value to the final after a specified number of iterations. This is done so that the circulation model will not be shocked by suddenly large forcing terms. The form of the startup function is



C(t) 0.5[1 + tanh(t/cl c2)]

where the parameters c and c2 are chosen to alter the offset and slope of the curve. For the present case, ci = 3000 and c2 = 3 to render a curve as shown in figure 4.2.

The subroutine FLOOD allows the grid area to fluctuate depending in the mean surface elevation. It is called at every iteration, and alters values at wetted or dried grids to conserve mass flux in those grids. The subroutine CHECK simply checks for convergent values, and sets a flag for the model to quit if the condition is satisfied at each grid point for U,V and


24





25


START


CALL INPUT




CITER STARTUP COUNT FUNCTION


CALL W


AVEMOD


CALL RADI


CALL CIRC CALL SHEA

CALL CALL EXC
CHECK

OUTPUT CALL UP


Figure 4.1: Flow chart for model MCSHERRY.


AT-]

R2 IEF


CALL FLOOD


,


I


I





26


100
time (sec)


150 200


Figure 4.2: Hyperbolic tangent startup function.


-j


1.0 0.8 0.6 0.4 0.2 0.0


0


50








-






27

4.2 Circulation Model


4.2.1 Method of Solution

The equations presented in equations (3.1)-(3.3) are solved using an implicit Alternating Direction method that solves for the new U value and interim 7y value in the x-sweep, and the new i7 and new V'in the y-sweep. This method applies to the matrix setup as suggested by Sheng and Butler (1982). The detailed treatment of this method as used here can be seen in Chapter 4 of Winer (1988).

The two unknowns in each sweep are applied at different places in the grid, where g is at the grid centers and the mean velocities apply to the grid edges. Note that the first offshore value is ij, at i = 1, and the first mean current U is at i = 2. The value of U and V at i = 1 do not exist for the grid edge values, and are not used directly in the code. As stated before, however, the wave model uses averages of the currents and for the i = 1 value uses U, V at i = 2. The grid domain is shown in figure 4.3.

The x-sweep occurs in subroutine XCOEF, which sets up the coefficients, then passes them to the double-sweep solver. At the lateral boundaries, the special subroutines XCOJ1 and XCOJN are called to handle the special coefficients when j = 1 and j = N. The y-sweep is handled by subroutine YCOEF, except at the shoreline, where the variable grid domain determined by FLOOD is deciphered by the subroutine YCVAR.

There is no time restriction for ADI method, but some of the terms are expressed explicitly and added to the knowns of the implicit scheme, and restrictions are introduced. The finite difference form of each of the terms in the momentum equations will now be presented.


The radiation stress forcing is a function of the square of the amplitudes, thus being nonlinear. Also, since the wave amplitudes are determined at grid centers in the wave model, the radiation stresses also apply to grid centers.

The radiation stress gradients are put into the nth time level of the scheme, which





28


offshore


j


shoreline


Figure 4.3: Grid for circulation model.


(1,1)


1* I-


t4 4-


(MN)


i






29
referring back to the last chapter for the x- and y-sweeps are



1 (OSxx + 0S)
pD Ox 0y


1 (OS + OSyy)
pD Ox 4y9

These terms are represented in finite difference form respectively by



1 SXXij SX X _,+ SXYi j+1 SXYij-1 + SXYi-1,j+1 -sXY
SXX~,+ .5 s ,1 2AY- i-1,j-1
pD AX 2AY


1 SYYij SYY,1+ sxyi+1,3 SXYi-1,j + SXY+1,j-1
pD AY 2AX

The symbols in the code for each variable are hopefully clear. The subroutine for the radiation stresses is RADIAT, which is called just after the wave model and prior to the subroutine CIRC.

Referring to figure 4.4, the terms are used to find the mean velocities, and are therefore differenced to straddle the known (i, j) velocity location. Note differencing is done over one grid space, and is accurate to second order.


The bottom friction pFIUIC7 is nonlinear, but uses mean currents from unknown and known time steps, and is thus represented implicitly. The terms are represented based on the work of Longuet-Higgins (1970a) as



Tbx = pF IuorbIU + pF |11U



Tby = pF IuorbIV + pF I||V


These are finite differenced according to the sample x-component,






30


Figure 4.4: Differencing of radiation stress gradients in x-sweep.




2! H
7bx =ue (T sinh kh + f U2+V2) where the linear formulation for maximum orbital velocity has been used, T being the wave period. Since the reflected wave is considered nonexistent when performing the radiation stress subroutine, this also applies for this subroutine. Therefore, the linear formulation of maximum orbital velocity for the one wave applies. The subroutine that calculates this term is called SHEAR2, and is called just prior to CIRC.


The lateral mixing is expressed in the momentum equations as



D 071 a OU
p 19y Dy 09(y


J-1 J +1






I-1 -_______ -- ___I _+_ ___ -- ---






31
D -rj a /0/ p tx Ox O X These are added to the knowns at the nth time level explicitly, and thus introduce a serious timestep limitation. In the complete finite difference form they are expressed as




at C'T= AtEY Uj+1 2Usj + U,5j-1
Oy AY2



-1 = AtEY 1 +1,j 2Uj + Ui1,j 4x AX2 For the solution to remain in front of the fastest travelling error, the timestep must satisfy



At
AX2

These values are calculated in subroutine EXCOEF just prior to CIRC. Note that the coefficient E is set to the highest value of Ex, which occurs at the breakerline.


The terms represented in the momentum equations by UK-- and V2J- in the x direction are represented in finite difference form respectively as,




.i Ui+1,j Ui-1,j
''2AX

and




ij + vii + Vij+ + ,iil+i) Uil~ Ui-l,j
4 2AY The same form applies to the y direction equations. A four point average is used for the cross-derivative of the four surrounding velocities as shown in figure 4.5






32


Figure 4.5: Differencing of advective acceleration in x-sweep.

4.2.2 Boundary Conditions

The solution for the equations governing mean currents and mean surface elevation require boundary conditions for each boundary in the grid. Winer (1988) used a fixed lid condition at the offshore edge, with no flow into the shoreline, and free flow at the lateral boundaries. Each condition had to be reviewed for the present case, and changes were made where appropriate.


The original model constrained the mean elevation to be zero at the offshore grid edge. This was a fine assumption for the one wave system, but was found to cause instabilities if a reflected wave occured. The instabilities were due to a severe mean surface gradient at the first two grid rows which gave rise to large velocities. For the standing wavefield, if the grid edge happened to occur on the envelope antinode, the problem was accentuated. Obviously a radiation condition was required to free that edge to allow disturbances within


J-1 j J+I1






33


the domain to escape. The condition



-+ V/ ( -- + S = 0 (4.1) at ( x pD a x

was applied to the x-sweep of the model, where the interim 7 is solved for at the grid center i = 1. After making the decision to not include the reflected wave, however, the original fixed lid condition was again used and is what is specified now. If the model were to be modified to include the reflected wave, an appropriate radiation condition would need to be developed. Also, if a strong current were to be flowing across the offshore edge, an energy set down condition might suffice.


The lateral conditions are reflective, that is the y-sweep sets V = 0 at j = 1 and j = N. This is to remain consistent with the wave model.


There is no flow into or out of the boundary, so the x-sweep condition sets U = 0 here. Also, there is no longshore current, so the y-sweep specifies that V = 0.


The model MCSHERRY begins by setting all unknowns to zero in subroutine INPUT. Then, as the waveheights are ramped to their full value, the forcing slowly increases in the circulation program. The model iterates until the difference between unknowns in successive time steps is smaller that a prescribed tolerance.



4.2.3 Radiation Stress Subroutine

The finite differencing of the radiation stress components that were developed in Chapter Three will now be presented. The calculation of these terms is performed in the subroutine RADIAT. This immediately follows the wave model and precedes the circulation model. Since the wave model is only accurate to O(c2) in spatial derivatives, there are terms based on derivatives in the radiation stresses that should be dropped as being too small. Referring to the expressions for S.,, S.y and Syy from Chapter Three, the terms






34


that should be dropped are



0X2






The terms that remain are differenced to remain at grid centers where the amplitudes are calculated. As an example, the derivatives are expressed as



Ax- Ai+1,j Ai-,,
2AX
A = Aijj+1 Aij-1 2AY
A -Aij+1 2Aij + A,,j1 YY y2 As was stated in Chapter Two, the reflected waves have been considered nonexisitent, since they do not physically affect the circulation landward of the bar field. To incorporate only the differences in wave energy in the grid, and not wave-wave interaction effects, the final expressions that exist in subroutine RADIAT therefore neglect terms consisting of reflected wave B. The final forms used in the model are



SXX pg + 2(kh) [(k2|A|2)i 2(k(A A*))' 4(k2) sinh2(kh), / pg 2(kh) [(kA (4.3)
4 sinh2(kh,) + [(A A12)i (4.2)



Spg ( +2(kh,)) 2 SXYy 9. 1 + i 4-(k)(A A*)) (4.3) 4(k2) sinh2(kh,)i j SYY =" 1+2(kh )' 4(2 sinh2(kh






35

pg 2(kh )
4 sinh 2(kh,)

+ L[(J|A12i
4 L i
P92(kh,)'
8(k2) tanh 2(kh (iA|2 + R(A A*,)' (4.4)




4.3 Wave Model


4.3.1 Method of Solution

The wave equations have been presented by (3.31) and (3.32). The finite differencing follows a Crank-Nicolson method, where a tridiagonal matrix is formed for the three forwardrow amplitudes based on three known values at the present row. The solution requires only an initial condition at the offshore edge, and lateral conditions to complete the matrix.

For the two waves, two loops make up the subroutine WAVEMOD. The first loop solves for the incident wave A up to the shoreline. Then, the second loop starts at the shoreline and solves for B out to the offshore edge. Due to the coupling through the bottom boundary condition, the two sweeps must be iterated until a convergent solution is obtained.


4.3.2 Finite Differencing

Now the incident equation will be differenced to serve as an example. Again, the amplitudes T and R refer to amplitudes divided by intrinsic frequency. Using a CrankNicolson method leads to the finite-difference approximation of the wave equation of




2AX {[ai(C9i + U)] +' + A(C1 + U)]} {Tj+' i
[o1(Cgi + U)(ki k1)]t+l + [al(Cgi + U)(ki T1)]} {Tj+' + T3}

+ I[aiV] +' + [o-V]'j f[T+ +f + [T-1 T8AY [ + U 1 -1 U Tj+1 + T
+ 1 [ai (Cgi + U)] +' ai(Cgi + )i Ti+ '
4AX 3 j






36

8+zY {k1V]i [+iV]j+i + [oV]1 V1]++ _Tj +1 T}



[(CCg1 V2) ++ + (CC1 V2>1][T;+1 -i1

+ [(CCg1 V2) +1 + (CCg1 V2) ][Tj+1 Tj] [(CCg1 V2)i + (CCg1 V2)i_1][Tj Ti._i + 1 {[0i(Cgi + U)wT] +' + [a(C91 + U)wT]'l

coR++ [cwR]}


The reflected form is similar, with sign changes where appropriate. These are then put into three coefficients for the i and i +1 rows, and put into a complex tridiagonal matrix doublesweep solver. Since currents exists at grid edges, the values to place in the WAVEMOD need to be averaged straddling the grid center. Thus Uj is really (Uj+' + Uj), and likewise for V. The grid domain is set so that the first calculated value of U is at i = 2, so then the averaged value for velocities are set to UJ and V1i when marching from i = 1 to i = 2.

In WAVEMOD, a preamble defines the coded meanings of each variable. Then statement functions are defined for the coefficients in the equations for the two waves. Before the first loop, subroutine WAVNUM is called to determine the incident and reflected wavenumbers, average velocities, celerities, group velocities, intrinsic frequencies, and forcing terms Qo, and Q1, as defined by (3.25) and (3.26). After this is done, the iterations begin, with a maximum of four iterations, but convergence is usually complete within 0.1% after three. If the wave is perceived to meet the breaking criteria and start dissipating energy at some grid point, then w is defined something other than zero, and the row is iterated again, letting the energy out. The wave model uses depths of the uniform bottom, not including the barfield. Therefore, shoaling over the bars is not included in the driving forces.







37


4.3.3 Boundary Conditions

The boundary conditions are that the lateral boundaries behave as walls. This requires that no reflected wave disturbance reach the walls and bounce back into the domain, which would be unrealistic.


4.3.4 Lateral Smoothing

When doing some preliminary testing on the wave model, the radiation stress gradients that would be put into the circulation model would be plotted simply for observation. It was found that the plots exhibited much irregularity in the driving forces, more than was expected. In fact, the circulation model had stability problems, and the irregularities were thought to be at fault. After some review, suggestion was made that the parabolic approximation had altered the nature of wave properties in the transverse y-direction.

Physically, a wave can have a maximum wavelength for given frequency and depth. This is uniform in all directions. But when assuming small incident angles, thus primarially xdirection propogation, which is done to get from a hyperbolic equation to a feasibly solvable parabolic equation, a modification is introduced. Consider a surface described as



r/ = Ae (4.5) where A is the complex wave amplitude. For a = Al, the surface in the y- direction is



r/ = ae (4.6) If one considers this wave in the y-direction, the radiation stresses are proportional to the wavenumbers, with the Sxy term being proportional to k1, and the term S., being proportional to k2. And then when considering the gradients of these terms as they appear in the momentum equations, the term O would be proportional to k. Therefore, there is a large sensitivity to the wavenumber in the transverse direction. By equating (4.5) and (4.6), the amplitude A could be expressed as






38


A-aikyeikx ~)ikx (7 Now consider a general parabolic approximation equation of form



2ikAx + AYY = 0 (4.8) and by making the substitution for A, the ODE is



Zyy + 2k2A = o, (4.9) which has the solution



A = aei\/-ky (4.10) Therefore, a limit has been placed on the maximum physically meaningful wavenumber in a parabolic wave equation for the y-direction. Instead of the normal L" = 2,, the new limit is L, = 2'g. Any wavenumber in the transverse direction that is larger than this will have severe effects on the driving forces in the transverse direction.

By showing this, consider a numerical grid with lateral width W, and some calculated wave field signal along a transect in the y-direction, as shown in figure 4.6.

A(y) can be written as a Fourier series



A(y) = eA(4.11) n=O
where A = g is the base spatial frequency. By taking the FFT of the signal, the spectrum will show energy distribution over each component nA. For the parabolic model, however, any energy that exists beyond frequency nA = lrk is not physically meaningful.

Now for the wavefields calculated by the numerical model WAVEMOD, a transect in y was FFT'd, and was found to have a very small amount of energy beyond what was physical. This is shown in figure 4.7, where the cutoff frequency component was nA = 9.051. Then a





39


y=o A(y)












SHORELINE


Figure 4.6: Wave calculated from parabolic equation on domain of width W.






40

transect of the driving forces (s + ) were likewise decomposed spectrally, and a large amount of energy existed beyond what was allowed. Shown in figure 4.8 for the y-direction, this shows how a small amount of error in amplitudes can expressly alter what the circulation model sees. The leakage of miniscule energy into high transverse wavenumbers in the wavefield translated into major amounts of energy beyond the cutoff frequency component for the driving forces. The process of the radiation stress gradients being proportional to kV shows the drawback when making a parabolic approximation. After taking the wave field and doing the FFT row by row, and truncating the energy beyond nA = 9.051 and inversing, a smoothed wave field was obtained. Figure 4.9 is the spectrum of the y-direction driving after smoothing. By figures 4.10 and 4.11, the slight improvements are seen in the wavefield, and in figures 4.12 through 4.13 the difference is remarkable. The variance in the energy spectrum of the y-direction driving before the smoothing was calculated as 0.00804, while after smoothing the total variance was 0.0030. In comparison to this 60% reduction in variance, the wave field had an unsmoothed variance of 4.065 x 10-7 and a smoothed variance of 3.80 x 10-7, about a 6% reduction.





41


0.000000008 0.000000006


0.000000004 0.000000002 0.000000000


0


IIIl IT 1-f7 1111


- -


lit


' '


' '


5


I I I I


15


20


10


frequency component


Figure 4.7: Spectrum of A(y) without smoothing. Cutoff at nA = 9.051.


C,'
S


I I I I I














0.0015 0.0010


S


0.0005


0.0000 E
0


frequency component


Figure 4.8: Spectrum of y-direction driving force before smoothing.


42


I I IIIIIII III IIIII


20


40


60


80





43


0.0015 0.0010


S


0.0005 0.0000


0


10


20


30


frequency component
Figure 4.9: Spectrum of y-direction driving force after smoothing.


I i I I I I I l i I I I I


40






44























196 00




160 Figure 4.10: JAI without smoothing. Bars are located on the bottom of this domain.







45 160 Figure 4.11: IAI after smoothing. Bars are located on the bottom of this domain.






46


11I IWL 'I U K5 416


0s









!'.














G-)







N)' l


Figure 4.12: y-direction forcing without smoothing.














-.013








.1600









-16
CC)





-.057-----

S 1 l I l it-TTT Illll lrrr r -u an -ai r n m A 1 11 11 11 1 1 11 1!Tl I .














CHAPTER 5
LABORATORY WORK





5.1 Introduction

The laboratory work was done to get a better idea of the process of Bragg reflection, where effects not represented by the wave model were shown. The following chapter treats the laboratory phase of this project specifically. Since the work was completed at two separate times, the procedures and setup unique to each will be presented separately. The results from both sessions, however, will be presented together, with comparisons to a simple numerical model.

The work was done at the Coastal and Ocean Engineering Laboratory (COEL) of the University of Florida. To begin the discussion, the basin and the other necessary equipment will be described.




5.2 Equipment

5.2.1 Basin

The modelling basin is illustrated in figure 5.1. The internal walls are moveable, and were positioned at right angles to the wave crests to minimize side wall reflection. The basin can be filled to a depth of 60 centimeters, and allows for waves up to heights of several centimeters. The main testing region lies inside the internal wave vanes, to minimize intrusion of backscatter from the outer basin walls. A wavemaker sits at one end, with a sloping beach at the other.


48






49














TERMINAL WALKWAY






.. BEACH





10*

WAVE VANES

WAVEMAKER


Figure 5.1: Plan view of large basin.






50


5.2.2 Wavemaker

The wavemaker has 80 paddles, each being 23.5 centimeters in width. Each paddle can be adjusted for stroke and phase. The entire system has a frequency range of 0.5 Hz to 1.25 Hz. Since the beach was oriented at 10 degrees relative to the wavemaker, the paddles were adjusted to generate an oblique wave at a 10 degree angle, which then arrives normally at the beach. This is done by calculating the wavenumber for the wavefield over the desired depth using a Newton-Raphson technique. Then the relation of paddle wavenumber kP to the wavefield wavenumber vector k, is, by (5.1)



kp = km sinO (5.1) Then the paddle wavelength is L = l-. For paddle width Wp, and paddle number n,, the phase of each paddle in degrees from 0 to 360 is, by (5.2)


(360)(np)(Wp)
#, = L(5.2)

5.2.3 Beach and Cart System

The beach gently slopes to a maximum height of 60 centimeters above the floor. A structure has been placed over the beach to aid in various types of testing. It allows a cart to ride on tracks and be placed at any (x, y) coordinate in an 8 meter by 17 meter area on the beach. For a plan view of this track system, see figure 5.2.

The track sits several centimeters above the greatest water elevation allowable in the basin. The cart is positioned in the y-direction by the wire connected to the handcrank at the side of the basin. The variable speed motor fitted onto the cross-member for the cart controls the position of the cart in the x-direction.

The cart can carry wave gages, current meters, and whatever else is necessary in an experiment. See figure 5.3 for a side view of the cart. Since profiling is important in the conduct of moveable bed studies, the cart is fitted with a profiling system. The profiler arm






51


Figure 5.2: Plan view of cart track system.


VINCH
CART






52


AREA FOR HORIZ. INSTRUMENTS POTENTIOMETER







VE EIA
POTENTIOMETER PROFILER ARM
PRD FIL R A RM.............





Figure 5.3: Side view of profiling cart. extends underneath the cart, dragging over the beach surface. A potentiometer produces a signal which is converted to readings of arm angle, while another fitted to the trailing wheel measures in voltage how many revolutions that wheel has made. With these two instruments, the beach (x, z) position can be traced digitally. A sample profile is depicted in figure 5.4.



5.2.4 Electronics

The computer for the basin is a PDP-11 microcomputer. It receives all basin data which is amplified at a box next to the basin. A remote terminal at the basin runs all of the programs. The main program is ATOD.FOR which takes multiple channels simultaneously at any rate for any length of time. The PDP- 11 is located in an instrument room about 200 feet from the basin.

Wave gages were the main source of data. The capacitance gages were built at the






53


60




40




N 20





0

0 200 400 600 800 x (cm)

Figure 5.4: Sample profile.

COEL. Figure 5.5 is a sample of the calibration points transposed on the calculated second order curve. The response appears linear over the operational range. Calibration was done daily, or if the temperature changed drastically. Along with the gages, a strip chart took the signal from the cart gage in session one. It was used to trace wave envelopes from the moving gage.

A Marsh-McBirney current meter took data for the second session. Since the calibration was 10ft/sec = 1 Volt, and since the currents were small, an amplification of twenty provided the necessary range to increase resolution. The data were quite erratic, and will be addressed in a following section.

Lastly, a video camera was mounted above the beach. The camera was fitted with a remote control to zooin, focus, point, and contrast. A taping device was included. Rhodamine dye transport was recorded as a way to show current magnitude. The only drawback was that the camera is black and white, and the dye does not show up as well as it could if the







54


8





C) 4


2


0


-2 I I I I

1000 1500 2000 2500 digital

Figure 5.5: Sample calibration curve for wave gage. camera filmed in color.



5.2.5 Bar Field

The bars were made of wood and metal. The same bars served both sessions, and are shown in figure 5.6 without the metal covering. They were sized according to the local water depths at their position of placement. The planform dimensions were 3.048 meters long and 0.318 meters wide, and each bar was made to be 40% of the local water depth. Once the height h was determined by inspection, the bar radius in centimeters followed equation (5.3).


(19.05)2 + h = (5.3) 2h


M







55


NOTEt
DIMENSIONS ARE IN METERS


0.381


hR

R


11111





S__ __ I __ S I


3.048


Figure 5.6: Bar frame design.

The full pattern was traced on paper, then transferred to the plywood frames and constructed. Since the main material was wood, the bars would float. They were anchored in the first session using wires connected to the concrete floor. In the second session, the undersides were filled with sand which was kept in place with a panel covering. Both methods worked fine, but the first proved to be better in time. Another problem was wave orbital velocities flowing under the bars, scouring large holes. Strips of sheet metal 4 inches wide were extended off the bar edges as aprons sunk into the sand. This effectively stopped the scouring.




5.3 Setup, Procedure and Data Analysis


This next section discusses the actual testing methods. Since different methods were used in each session, each will be discussed individually. The first session will be addressed, with it's purposes and procedures, followed by the second session.







56


5.3.1 Session One

The first session ran from September of 1987 to February of 1988. Since this was the first work done in the lab on the currents attributed to Bragg reflection, the goals were fairly loose. The purposes were specifically:

1. To measure Bragg resonant reflection.

2. To compare this reflection to theory.

3. To describe qualitatively the properties of the wave-induced current field.

4. To observe how reflection and current magnititude varied as wave period varied.

5. To observe sand movement during the experiment.



The first goal received most of the attention, with two methods being developed. The first was to use two stationary gages, in line, and take data simultaneously. The data from each then can be used to determine the reflection coefficient, and this method is from the work of Goda, et. al. (1976). To review that method, for a wavefield which can be described by a surface displacement



= a cos(kx wt + 61) + b cos(-kx Wt + 62) (5.4) the unknowns are a, b, 61, and 62. Four values for y from the data will therefore be needed. Let two gages be located at x = 0 and x = 1, as in the figure 5.7.

If a data value is taken for ?7 from each gage at times t = 0 and t = -r, four values will result. Each one can be expressed through (5.4) according to (5.5-8) as



1,= 71(0,0) = a cos 61 + b cos 62 (5.5) 172 77(l,0) = a cos(kl + 61) + b cos(-kl + 62) (5.6) 173 = 7(0,r) = a cos(b6 wr) + b cos(62 Wr) (5.7) 74 = 7(l, 7) = a cos(61 + k1 w-r) + b cos(62 k1 wr) (5.8)







57


Figure 5.7: Gage locations for two-gage reflection method. Let the following relations hold;



Ci = cos 61 (5.9) C2 = COS 62 (5.10) s, = sin 61 (5.11) S2 = sin 62 (5.12) d = coskI (5.13) e = sin kl (5.14) f = cos Wr (5.15) g = sin Wr (5.16)


Expanding out (5.5-8) and substituting (5.9)-(5.16) gives







58


71 = a c, + b c2 (5.17) 712 = (a ci + b c2)d (a si b S2)e (5.18) 1q3 = (a ci + b c2)f + (a s, + b 2)g (5.19) 174 = (ac,+ bc2)fd+(bs2 asi)fe + (a s, + b s2)g d + (a ci b c2)g e (5.20) By substituting sequentially from (5.17) down to (5.20) and eliminating terms, the remaining expressions are,


2a c1 = A = 71+ 14 -3 72 + 1 f d (5.21) ge

2as1 = B = -2 71 d (5.22)
9 e



2bc2 = C = 11- 74 13 12 + 71 f d (5.23) g e

2bs2 = D = + 2 1 d (5.24)
9 e

From these expressions, the incident and reflected wave amplitudes are, by (5.25) and (5.26),



A2 + B2 (5.25) a = +5.5
4

b = C2D2 (5.26)
4

The reflection coefficient is then K, = b/a. The problem in this method is that if g or e approach zero, the expressions for A, B, C and D blow up. Thus one must not choose

7 = n(j) or 1 = n('y) where T = wave period, L = wavelength, n = integer.

To determine the accuracy of the method, a test was run where the reflection was from the beach. To compare, a moving gage retrieved the partial standing wave envelope. It was discovered that for various time increments r, the reflection coefficients were inconsistent. This inconsistency was enough to scrap the two gage system, and use the envelopes as







59


Table 5.1: Bar heights and positions for Session One.
bar no. I height (cm) [ x-position Jj
1 7.62 420.
2 9.50 495.
3 11.20 570.
4 12.60 645.




the major source of reflection data. This required measuring by hand the minimum and maximum envelope amplitudes, 7min and 1max. Then the reflection coefficient is,



Kr = 77max 77min (5.27) 77max + 77min

Envelope transects across the barfield were taken at four locations during the testing to get a lateral average. The positions of these transects are presented later in this section. The results are then presented and compared to the circulation model in the next session.

A camera was mounted above the testing site which filmed the movement of rhodamine dye across the barfield at each wave period. The film is available for viewing through the author.

The tests were run at several wave frequencies around the designed resonant frequency. Each run was made from a still water condition. With reflection and current observations made at each run, a feel of the dependence was gained. The different runs are listed in the next section for the periods and reflection coefficients.

Observations were made in regard to sand movement during the course of the testing. These observations are listed as part of the results in Chapter Six.

The setup consists of where the bars were oriented. By using the equation for the bars from the local depths, the bars were built according to Table 5.1.

Plots of the testing bathymetry were done as testing commensed, and are available as of this writing, but the actual data that was written onto magnetic tape is not immediately







60


60





40





20





0

0 2 4 6 8

x (m)

Figure 5.8: Reconstructed bar profile for Session One.

available to the author. There is no other recourse than to reconstruct what the bathymetry was at the time from the existing plots. This is the only handicap from losing the data, since the envelopes are printed on strip chart paper. Figure 5.8 shows the lab bathymetry with the bars placed. The depth was 55 centimeters, and the breakerline in the testing was at about x = 3 meters. Figure 5.9 gives an overhead view of the barfield, with four lines in the x-direction offshore which represent the transect lines where envelope data was taken. The profile is available as a test bathymetry in the numerical model, although the bars need to be sinusoidal.

Once the bars were placed, the wave gages were arranged. The barfield would create a reflected wave that would propagate and diffract once beyond the bars. Two gages were mounted on tripods in the center of the bar field longitudinally, and directly offshore of the last bar. As a control, one gage was put off to the side, away from the influence of the barfield. Another gage was on the moveable cart. Figure 5.10 shows a plan of the beach







61


Figure 5.9: Plan view of barfield for Session One.


1365 1300 1175 1110




16.0 16.0 16.0.2.0
?40 i -___ -


-7







62


Figure 5.10: Gage locations for Session One. region with the gage placements.

This concludes the discourse on session one. The data from the experiment will be presented later, with the test run parameters, in Chapter Six. Now the second session will be addressed with the procedures and setup.




5.3.2 Session Two

With the experience from the first session, the second experiment ran from September of 1988 to December of the same year. The goals were more defined in this second session.


1. To repeat Bragg resonant reflection and a resultant current.

2. To extend the reflection to frequencies off the no-current resonant value.

3. To observe differences between moving away from resonance and shifting toward

it.


GAGE ON
CART BARFIELD




0

CONTROL
TWO GAGES GAGE
[N TRIPOD







63


Figure 5.11: Three gages inline with x-direction waves


4. To quantify current values.



In this session, two methods were again employed. The first was from the work of Funke and Mansard (1980). Here, three in line wave gages took simultaneous data, and the reflection coefficient came out of the spectra. The work covered waves travelling over a flat bottom, with no current involved. Their results therefore needed to be revised for the addition of current and variable topography. For the setup in figure 5.11, the algorithm was modified in the following manner;

The wave profile for any probe p = 1,2,3 can be described as a Fourier series sum of each component k by


N 21rkt
rp(t)= E Ap,k sin T + Qp,k) (5.28) k=1


XR1
X13 wavemaker X12 -I Xl







64

where Ap,k is the Fourier coefficient for frequency -, T is the time length of the data run, ap,k is a random phase shift, and N is the desired number of components for the profile. The Fourier coefficient is gotten from the transform of the time series at each probe p, q,(t), which for any probe would be, by (5.29)




fp(t)} = Bp,k = CI,k exp i 2(XL + XlP) + iOk

+ CR,k exp i 27r(Xl + 2XR1 X1P) Fi(Ok + Ok) + Y,k exp Ji(Pp,k)} (5.29) These equations are actually (1) and (10) in Funke and Mansard (1980). They point out that the transforms represent functions of the complex amplitudes for the incident and reflected waves. The three expressions for the FFT's of the three probes then lead to a least squares determination of the incident and reflected amplitudes, ZI and ZR.




B1,k = ZI,k + ZR,k + ZN,1,k (5.30) B2,k = Ks,I,2 ZI,k exp[q I,1,2] lis,R,2 ZR,k exp[-TR,1,2] + ZN,2,k (5.31) B3,k = K8,I,3 ZI,k exp[WI,1,3] + Ks,R,3 ZR,k exp[- IJR,1,3] + ZN,3,k (5.32) The terms in these three equations are defined as follows;




ZI,k = CI,k exp i 27Xl+ iOk (5.33)
1 Lk I

ZR,k= CR,k eXp i 2r(X1 Lk + i(Ok + i3 ) (5.34) ZN,p,k = Yp,k exp {iPp,k} (5.35) The shoaling coefficients in the incident and reflected directions for each probe, and group velocity Cp are,







65


KS,I,2 Cg'i (5.36)


13 Cgj (5.37) C9,3

s,R,2 (5.38) C9,1

Is,R,3 = Cg,3 (5.39)

The phase accumulations are obtained according to


X2
"'I,1,2 = ki dx (5.40)


/X3
'@,1,3 = fx1kIdx (5.41)



qfR,1,2 = Ix2kR dx (5.42)


/X1
'PR,1,3 = kR dx (5.43) X3
(5.44)

These modifications reflect the different case of waves over a variable topography on a current. Funke and Mansard (1980) addressed waves over a flat bottom where the incident and reflected wavenumbers were equal, and wavelength Lk for each frequency was constant in the domain. The reflective and incident wavenumbers k are found from the dispersion relation that includes the current effect, using an iterative technique. Setting the ZN terms to a small error parameter E, the least squares method is followed and the resulting equations for unknowns ZI and ZR are



ZI {2 + 2K,,I,2exp[2iTI,1,2] + 2KI,3exp[2iq'i,1,3]







66

+ ZR {-' s1,I,2A8,R,2exp[i(TII,1,2 'PR,1,2)] + 2KS,I,3Ks,R,3exp[i(JI,1,3 T R,1,3)1} = 2B1 + 2K,,I,2exp[iJI,1,2]B2 + 2K8,i,3exp[i'PI,1,3]B45.45) and then




Zi {2 + 2K,,I,2K,,R,2exp[i('PI,1,2 T R,1,2)] + 2K3,I,3KS,R,3exp[i(TII,1,3 @R,1,3)11 + ZR {2 + 2Ki2,R,2exp[--2iQR,1,2] + 2K.2,R,3exp[-2i PR,1,3]} = 2B1 + 2Ks,R,2exp[-iTIR,1,2]B2 + 2K,i,3 exp[-iQR,1,3]B3 (5.46) These can be solved for the two complex amplitudes, ZI and ZR, and then K, = IZRI

The modifications were done numerically according to the following forms. For the dispersion relation with opposing current,



w = -U, k + Vgk ianh kh (5.47) a Newton-Raphson iterative technique was used to solve for k, given depth, frequency, and current U0. It is assumed here that the incident wave angle 9 = 0, so that k cos 9 = k.

The simple integration scheme of the phase in space is performed by,



W, = (k, + kp-1) P2~"1 + XPP-1 (5.48)
2

where again p = the gage number, either 1, 2, or 3. A xp.._1 is the interval size between gages p and p 1. The intervals were on the order of 15 centimeters, so that the bottom could be considered a plane sloping beach in these intervals, allowing the simple integration.

The group velocity used for the shoaling coefficient K is, for each gage p,



Cg, = (1+ w fh2 (kh)) (5.49) ,,2kP sinh 2(kh),

This method was tested for accuracy for reflection from the beach, being compared to envelope data. It was found that the reflections differed between the two methods by as







67

much as 60%, and the development of the equations thus leaves something in error. Rather than an in-depth search for an accurate form, it was again decided to use the envelope method, as in Session One. Again, partial standing wave envelopes were retrieved from a moving gage across the barfield, but were recorded digitally, and analyzed as per the methods outlined below. The envelopes were taken at seven transects over the barfield. It was found that the three center transects represented the highest consistent reflection for a test. These locations are shown in the plan view in figure 5.16. After determining the reflection from each transect, it was plotted with location in figure 5.15. As is apparent, sections 850, 900, and 950 give a consistent representation of reflection. The method of analyzation was to take each envelope, as shown in figure 5.12, and condense it into envelope height versus x-position. A sample is shown in figure 5.13. The reflection coefficient was calculated by taking a local maximum and immediately following minimum, using equation (5.27). Thus a plot of K, with x was obtained, as seen in figure 5.14. The maximum reflection was taken from each envelope, and the three were averaged. This was done for each wave period. The results are presented in the next chapter and show a repeatability of the process, which is a good sign. Further similarity with theoretical output gives confidence in the values themselves. From figure 5.14, there is a variability in measured reflection along the bars, and the ending minimum should be disregarded since the cart may not have traced the full wave envelope before stopping. The variability within the bar region, however, could be attributed to focusing effects over the bar field.

This data was taken using program ATOD.FOR for a sampling frequency of 26.0485 Ilz, and a run of 2048 points. The gage was the cart mounted gage, moved slowly through the water over the barfield, insuring that the wave excursions stayed in the middle portion of the gage wire.

The second priority was to get quantified results for current induced by Bragg reflection. At each period, a line of current data was taken across the front edge of the barfield. The locations were at the same locations used for the envelope measurements, except they were


I





68


0d


4 2 0


-2


-4
200


, , I , I|I


400


600


800


x (cm)







Figure 5.12: Sample envelope, Session Two. Retrieved digitally from one of the three on-offshore transects using the moving cart gage.





69


400


600


800


x (cm)
Figure 5.13: Sample envelope height versus X.


200


400 600 800


x (cm)
Figure 5.14: Sample K,. with offshore distance x from Session Two.


8


6


4-) Q)


4


7 I I I I I I I I






- I I I


21


0


200


- -


0.5 0.4 0.3

0.2 0.1 0.0





70


800


900


1000 1100


section number
Figure 5.15: Plot of K, with transect location, Session Two.


0.5

0.4 0.3

0.2 0.1 0.0


600


- I I I


700







71


UI tQ M o M a M








16.0- -. 16.0
-- .6.










24. 0 =---__V





Figure 5.16: Plan view of barfield, showing data collection locations.

stationary just inshore of the first bar. Figure 5.16 is a plan view of the session two barfield with the locations used for current meter measurements and envelope data. A side view is shown in figure 5.18, where the water depth was 50 centimeters, and the breakerline was at x = 320 centimeters. The current meter was placed at 750, 800, 850, 900, 950, 1000, and 1050.

Data was taken at 26.0485 Hz for 512 points. The data proved very erratic from the twochannel Marsh-McBirney current meter. The calibration for the instrument was 10 ft/sec =







72

1 Volt, and since the expected currents would be on the order of a few centimeters per second, an amplification was done. An amplifier was built in house to magnify the signal by a factor of twenty. The raw results from such a time series is given in figure 5.17, which represents the velocity magnitude. Because of the nature of the signal, any trustworthiness of discharge calculations would be in doubt. The irregular signal could be due to interference from the metallic bars that were about 10 centimeters away from the probe. Still water readings taken in a container away from the bars showed a fairly stable signal. Other reasons could be excessive low frequency water displacements, but nothing like that was observed. Coupled with a serious time limitation as of this writing, the choice was made to disregard laboratory quantification of the rip current. A major goal of Session Two therefore is yet to be realized, but the reflection results weakly imply the presence of a current, and the dye experiments from Session One that are on film definitely show the rip current. Indeed, observations made at the time were that the currents were very mild, and did not get past the first bar.


Unlike session one, session two was a continuous test between wave period changes. The intention was to extend the reflective peak as long as possible, thus keeping the induced circulation from previous periods going through the enhanced reflection. Once it was observed that circulation had dropped considerably, the runs began at rest from a high nonresonant period, and moved at slow increments toward resonance. Thus a test was made to see if reflection commenced upon return at the same place where it slacked when moving away from resonance. As said, the current magnitudes were so small that it was difficult to see a definite drop in reflection or current strength. Nevertheless, the results are presented in Chapter Six and still offer some useful lessons for future study.




The bars were the same from session one, but some of the bars were buried slightly to accomodate the new conditions. Table 5.2 lists the heights and positions of the bars in this







73


15 1 [ TiT I --I-


Q)






0


0-
0)


10





5





0


(


Figure 5.17: Sample time series of magnitude of current meter data, Session Two. part.

For an overhead view of the barfield, refer back to figure 5.16. A side view of the barfield is shown in figure 5.18.

The instrument setup consisted of everything being put on the cart. The three gages were positioned inline on the cart, along with the current meter. The overhead of the


Table 5.2: Session Two bar heights and positions
bar no. 1 height (cm)! |x-position (cm) J
1 6.25 436 2 7.00 500 3 17.00 565 4 12.50 631


-L













) 5 10 15 20 time (sec)







74

60





40





20





0

0 200 400 600 800

x (cm)

Figure 5.18: Profile for Session Two. instrument array is shown in figure 5.19. With the points made earlier, the three-gage system was dropped in favor of the one moving gage to retrieve envelopes across the barfield in the x-direction.

This concludes the description of Session Two. The next chapter describes the results obtained from the laboratory work.







75


Figure 5.19: Instrument setup for Session Two.


THREE CART
GAGES
BARFIELD














CHAPTER 6
RESULTS AND CONCLUSIONS



6.1 Introduction

The results from the work done in the past year and a half are presented in this chapter. The results are from laboratory data, where the methods that were used to obtain this data are outlined in Chapter Five. The other source of results is the numerical modeling effort. This consists of the wave- current model MCSIIERRY that as of this writing handles a limited set of conditions. The preliminary findings could be construed as inconclusive, but definitely suggest the need for more attention. The model should be running at full speed in the next year (1989).

6.2 Laboratory Work

The data collected consisted of reflection data from wave gages, current meter data, bathymetric data from the profiler, and some data from the camera. The results will be presented in tabular form and pictorally where possible.


6.2.1 Reflection Results

The Session One reflection data is presented in table 6.1. Each run represents a Static test, i.e., startup from a still basin. The averaged data over the middle transects is also plotted in figure 6.1, transposed upon the theoretical prediction for a one dimensional barfield without currents. This one-dimensional model is from Kirby (1987) that excludes currents. This has been placed over the data to simply give an idea of the effect the currents have on the data. The incident wave had a resonant period of 0.98 seconds, and was 7 centimeters in height. The waves arrived normally onto the beach, with a breakerline at x = 3 meters, refering to the profile in figure 5.8.

76


I







77


Table 6.1: K, and comments from Session One. period (sec) 1110 11175 1 1300 ] 1365 comments on rip
0.980 0.357 0.667 0.529 0.362 weak rip current
0.910 0.347 0.189 0.235 0.091 no current 0.963 0.172 0.579 0.368 0.238 no current 1.050 0.368 0.550 0.368 0.538 rip current
1.110 0.294 0.692 0.458 0.514 significant current.
1.180 0.333 0.529 0.300 0.176 no rip current
1.110 0.333 0.474 0.647 0.428 again strong rip repeated.
1.085 0.290 0.412 0.579 0.375 slightly weaker than T=1.11.
1.020 0.474 0.438 0.400 0.405 weak rip, dye is stagnant.
1.000 0.333 0.474 0.538 0.333 weak current
0.942 0.273 0.444 0.350 0.170 no current


The predicted reflection from the one-dimensional model underestimates the reflection for the given bathymetry, for the envelope data describes 50% reflection. The data were not carried to sufficiently large periods so that the reflection clearly died away. From the films of the current, the magnitude increases in strength until a period of 1.11 seconds where it is quite strong, after which it dies off at 1.18 seconds. This would tend to suggest that while the waves are moving increasingly off no-current resonance, the rip current increases to keep the waves at resonance. This supports the hypothesis that the system seeks a resonant condition.

From Session Two, the data were more abundant, but perhaps not as telling. Much of the setup was the same as in Session One, except that the depths over the bar field were less. This is due possibly to the state of the beach when testing began, in that the slope was less than in Session One. To offset this the bars were buried into the sand about 10% of their heights, which was typically about a centimeter. Also, due to the testing that had been going on before this work, some of the sand had been removed from the main testing region, so water depths could not be as high. In Session One, the testing depth was 55 centimeters, and for Session Two the depth was 50 centimeters. To further offset this,






78


0.6





0.4 + ~~0.2





0.0
0.5 1.0 1.5 2.0 period (sec) Figure 6.1: Plot of KT, with wave period, Session One data and theoretical prediction from 1-D model Kirby (1987).







79

wave heights were decreased from 7 centimeters offshore to 5.5 centimeters offshore. This seems to be the main value that has the greates effect on the results, for a rip current never developed as strongly as the one appearing in Session One. The reflections were high, but it is conjectured that there was not enough available energy to drive the circulation.

The reflection data are presented in three tables, where the first set refers to reflection with absolutely no discernible current throughout the course of the runs. The next set was made after digging sand out from the region in between the bars and the land. This allowed a weak current to form in some of the periods. The third set refers to runs that started at low frequencies and slowly approached the no-current resonant case. In all the testing in Session Two, the runs were continuous, without stopping the wavemaker.

The averaged data from the first set with no current are tabulated in table 6.2, and illustrated in figure 6.2, which has the theoretical result transposed. Again, the theoretical values are from the no-current, one- dimensional model of Kirby (1987). Three or less runs were done at each period to get an idea of the repeatability of the experiment results.

Like the first type, the second type started at resonance and moved slowly outward. A weak rip appeared after removing some-sand from the foreshore. The tabulated results are given in table 6.3, with the plot in figure 6.3. The two profiles from set one and two are shown in figure 6.4.

From these plots, the data follows the theoretical fairly well, but the data drops away at later periods. The sand did not move onto the first bar at all, whereas in Session One, the first bar had to be repeatably uncovered.

The third type started off resonance, then moved toward resonance continuously until reflection commensed and a weak current was excited. Table 6.4 gives the results, also plotted in figure 6.5.

The difference between this test and the other is again not clearly evident, and may be due to the lack of dominance of the current.

The one-dimensional algorithm from Kirby (1987) is over a variable mild topography.








80


Table 6.2: Kr. Reflections for case of no discernable rip current, Session Two.
period (sec)_ K,.
1.014 0.419 0.515 0.423
1.020 0.381 0.418 1.030 0.433 0.503 1.050 0.507 0.471 1.065 0.511 0.508 1.080 0.530 0.505
1.100 0.494 0.526 0.525
1.125 0.516 0.481
1.150 0.470 0.454 0.473
1.175 0.518 0.493 1.200 0.425 0.386 1.225 0.323 0.341 1.300 0.385 0.310 1.350 0.374 0.414 1.400 0.396 0.372 1.450 0.386 0.367 1.500 0.352 0.339






81


0.6





0.4 3 ]a





0.2





0.0
1.0 1.5 2.0 period (sec) Figure 6.2: Plot of KC, with period, data and theoretical for the case of no discernable rip current.







82


Table 6.3: K, measurements from case of a discernable shore rip current. Session Two.
period (sec) Kr
1.014 0.425 0.464 1.030 0.391 0.442
1.050 0.521 0.536 0.539 1.080 0.466 0.510 0.483 1.100 0.483 0.503 0.514 1.125 0.482 0.521 0.482
1.135 0.447 0.486 1.150 0.532 0.520 1.160 0.461 0.490
1.200 0.402 1.250 0.349 1.300 0.349
1.350 0.231 0.289
1.400 0.215 1.450 0.257 1.500 0.259 1.550 0.323 1.600 0.310
1.650 0.272 J1










83



















0.6 I


0
0 00
0
0.
00
0

0.4 0 0



000
0 0 00


0.2










0.0

1.0 1.5 2.0 period (sec)



Figure 6.3: Plot of K, with period for the case of a discernable shore rip current. Data and theoretical.







84


0.6 0.5



0.4


N


0.3



0.2


0


2


4


6


x (m)

Figure 6.4: Two profiles, dotted line referring to profile used in the second and third sets of testing, where a weak current formed.




Table 6.4: K, measurements for case of discernable shore rip current, moving toward reso-


nance.


period (sec) K, j
1.650 0.216 1.600 0.330 1.550 0.323 1.500 0.316 1.450 0.316 1.400 0.356 1.350 0.249 1.300 0.302
1.200 0.436 0.460 1.175 0.464 0.459 1.160 0.518 0.512 1.150 0.529 0.520 1.135 0.487 0.556


---


- -*


8






85


0.6





0.4

Ul U



0.2





0.0
1.0 1.5 2.0 period (sec) Figure 6.5: Plot of K,. with period for case of discernable shore rip current, moving toward resonance.







86
It again excludes current effects. The plots do not show the kind of reflection extension that could occur because the currents were not as strong in the second session as the first. Thus, only so much tuning was possible. Some notes about the comparisons of the first and second sessions are worth noting to explain some problems with the execution.


1. The waves were slightly higher in the first session, thus involving more current

driving energy.

2. The anchoring system kept the bars very still in the first session, while there was

slight movement in the second.

3. Not enough time was allowed between period shifts in the second test to allow a

proper current maintainance.


The first reason seems the main cause of the decreased current generation. As is stated in the theory, if a wave field does not have enough available energy, the rip current will not be strong enough to achieve tuning of the nonresonant incident waves. Thus the efficiency of the bar field is comparable to the case with no current, and affects only closely resonant waves. For practical considerations, the most damaging waves will probably carry enough energy to drive a sufficient current, and the bar field will be effective when most needed.



6.2.2 Bathymetry

This last section deals with the movement of sand during the experiments. There are profile data on backup tape from Session One that is presently not available to the author. Observations conclude that the current was sufficient to move sand over the barfield. In fact, it was neccessary to uncover the first bar periodically, which had been buried and which lessened the effectiveness of the barfield. In the second Session, the current was too weak to transport sand, and the profile shown in figure 5.18 was the same throughout the testing.







87

6.3 Numerical Results


After much tinkering, some results are available from the numerical model. As of the writing, however, the model requires another two months of work. Specifically, lateral mixing was left out because the time step limitations would make the model very slow, too slow for the given time allotment. Therefore there are circulation variations in the domain that would probably not exist if the cells were allowed to influence on another. However, it is thought that the overall pattern is similar for both cases.

The results presented now are from the no-current wave model coupled with the circulation model. The process to achieve results was to run the wave model once and get the final radiation stresses. These terms were then multiplied by the startup function and inserted into the circulation model. The tests were run on a common bathymetry which is shown in figure 6.6.

Without the wave-current equations to govern the wave field, the thesis cannot answer the question about whether the system tends toward resonance through numerical results. However, the no-current model can answer questions about how the magnitude of the current is affected by wave height. Therefore, three wave inputs were chosen for the bathymetry, where the period was the no- current resonant period. As a point of interest, the wave field and mean surface elevation that are calculated by the wave model for an input of a .03 meter wave are shown in figures 6.7-6.9.

Then the following plots show how the current pattern varies as the amplitude is increased. Velocity times total depth is plotted in the figures, which describes the discharge across a vertical line. It is seen as expected that the rip current discharge increases as wave height increases. The circular cell over the barfield ends is thought to be attributed to a slow leaking of energy from the incident wave over the bar field. The action stated in Chapters Three and Four of removing the reflected wave probably account for this lost energy. The reflection coefficients remain the same for each wave height, being calculated simply by the maximum reflected wave amplitude in the grid divided by the initial incident





88


I I I I I


2


-I I I I1 F- F I I I I I 1 1 1lj


2


m


I I I I I ~ ~ I lit II ~


4


6


8


x (m)


Figure 6.6: Bathymetry 1. Shoreward direction with increasing x.


0.1 0.0


E


-0.1


N


-0.2


-0.3


0


10


I f I I I I I I I I I I I I I I I I 1 1 1


7








89


.o1-


.02



.01.


Figure 6.7: JAI for initial amplitude of .03 meters and perfectly resonant waves.


P"







90

















.03



























Figure 6.8: IBI for initial amplitude of .03 meters and perfectly resonant waves, without currents.


N




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UFL/COEL-89/014 WAVE-CURRENT INTERACTION OVER A SUBMERGED BAR FIELD By Thomas Richard McSherry 1989 Thesis

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WAVE-CURRENT INTERACTION OVER A SUBMERGED BAR FIELD By THOMAS RICHARD MCSHERRY A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 1989

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ACKNOWLEDGEMENTS In the course of completing this thesis I have received abundant help in many forms from a variety of people. The work required the extensive help of my Advisor, Dr. James Kirby, and I am truly grateful for his assistance. I consider myself very fortunate to have had the opportunity to study under Dr. Kirby. I also used the COEL extensively on two seperate occasions, and depended on the help of several people there. I thank Jim Joiner and Sydney Schofield for their patient assistance, and the inputs from the entire group. A great source of help at the COEL was in the way of mental release in the form of the daily volleyball game, and although I jammed a few fingers in the less-than-benevolent net play, I am grateful for the enthusiastic games. I also thank Dr. Robert Dean for his help in every way, and I thank the secretaries who have told me the logistical requirements for getting out from under the deadline hammer. Also, thanks to the students that offered friendship and release, and provided help when I really needed it. These people I wish the greatest of success. ii

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TABLE OF CONTENTS ACKNOWLEDGEMENTS .................... ............ ii LIST OF FIGURES ..................... ............... iv ABSTRACT ... ... .. ... ... ... ... ... ..... ... ... .... .. v CHAPTERS 1 INTRODUCTION .................... ............... 1 1.1 Problem Statement ..................... ........... 1 1.2 Literature Review .................... ............ 2 2 THEORETICAL REVIEW .............................. 5 2.1 Introduction ..................... ............... 5 3 GOVERNING EQUATIONS .................... ......... 12 3.1 Introduction ..................... .............. .12 3.2 Circulation Model .................... ............ 12 3.2.1 Governing Equations .................... ....... 12 3.2.2 Radiation Stress Terms ........................ .14 3.3 Wave Model ..................... .............. .20 3.3.1 Governing Equation ................... ....... 20 4 FINITE DIFFERENCING ...................... ........ 24 4.1 Introduction ..... ....... ................... ........ 24 4.2 Circulation M odel ................... .............. .27 4.2.1 Method of Solution ............... .. ........... 27 4.2.2 Boundary Conditions ................... .. .... .. .. 32 4.2.3 Radiation Stress Subroutine ................... ..... 33 iii

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4.3 Wave Model ...... ........ ........................ .35 4.3.1 Method of Solution .................... ........ ... ... 35 4.3.2 Finite Differencing .. ........ ................. .. ..35 4.3.3 Boundary Conditions .......................... ..37 4.3.4 Lateral Smoothing ...... ......... ............ .. .37 5 LABORATORY WORK ....... ......... .............. .48 5.1 Introduction .. ... .... ...... ...... .. ... ... ... ...... ..48 5.2 Equipm ent ........... ...... ....... .... ....... .48 5.2.1 Basin ............... ........... ........ ...48 5.2.2 W avemaker .... ............ ............... .50 5.2.3 Beach and Cart System ......................... .. 50 5.2.4 Electronics .. ............................. 52 5.2.5 Bar Field .................... ... .......... .. 54 5.3 Setup, Procedure and Data Analysis ...................... 55 5.3.1 Session One ....... ........... .............. .. 56 5.3.2 Session Two ... ... ... ... ... .. .. .... ... .... .. 62 6 RESULTS AND CONCLUSIONS .......................... 76 6.1 Introduction ..... ......... .... ..... .............. .76 6.2 Laboratory Work ................. ... ..... ........ .. 76 6.2.1 Reflection Results ............ .... ........... .. 76 6.2.2 Bathymetry ...... ......... ..... ....... .... 86 6.3 Numerical Results ........... ... ................. .. 87 6.4 Conclusions and Status Report ......................... 92 BIBLIOGRAPHY ............... ................... .100 BIOGRAPHICAL SKETCH .. ............................. 102 iv

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LIST OF FIGURES 2.1 Bar field in the presence of a depth-uniform current. From Kirby (1988). 6 2.2 Contour of Tr v. JFx v. W/Wro. ...................... .9 2.3 Surface projection of transmission Tr v. frequency and current. ....10 4.1 Flow chart for model MCSERRY. ................... .25 4.2 Hyperbolic tangent startup function. ................... 26 4.3 Grid for circulation model .......................... 28 4.4 Differencing of radiation stress gradients in x-sweep ............ 30 4.5 Differencing of advective acceleration in x-sweep. ............ 32 4.6 Wave calculated from parabolic equation on domain of width W. ...39 4.7 Spectrum of A(y) without smoothing. Cutoff at nA = 9.051. ......41 4.8 Spectrum of y-direction driving force before smoothing. ......... 42 4.9 Spectrum of y-direction driving force after smoothing. .......... 43 4.10 JAl without smoothing. Bars are located on the bottom of this domain. 44 4.11 [Al after smoothing. Bars are located on the bottom of this domain... 45 4.12 y-direction forcing without smoothing. .................. 46 4.13 y-direction forcing after smoothing. ..................... ..47 5.1 Plan view of large basin. ......................... .. 49 5.2 Plan view of cart track system. ........................... 51 5.3 Side view of profiling cart. ......................... 52 5.4 Sample profile. ........ .... ........ ............. 53 5.5 Sample calibration curve for wave gage. .. ................ 54 v

PAGE 7

5.6 Bar frame design. ............................... 55 5.7 Gage locations for two-gage reflection method. .............. 57 5.8 Reconstructed bar profile for Session One. ................ 60 5.9 Plan view of barfield for Session One. ................... 61 5.10 Gage locations for Session One. ................... ...62 5.11 Three gages inline with x-direction waves ................. 63 5.12 Sample envelope, Session Two. Retrieved digitally from one of the three on-offshore transects using the moving cart gage. ............. 68 5.13 Sample envelope height versus X. ................... ..69 5.14 Sample Kr with offshore distance x from Session Two. ......... 69 5.15 Plot of Kr with transect location, Session Two. ............. 70 5.16 Plan view of barfield, showing data collection locations. ......... 71 5.17 Sample time series of magnitude of current meter data, Session Two. .73 5.18 Profile for Session Two ............................ 74 5.19 Instrument setup for Session Two. ................... ..75 6.1 Plot of K, with wave period, Session One data and theoretical prediction from 1-D model Kirby (1987) ......................... 78 6.2 Plot of Kr with period, data and theoretical for the case of no discernable rip current. ....................... ................ 81 6.3 Plot of Kr with period for the case of a discernable shore rip current. Data and theoretical .............................. 83 6.4 Two profiles, dotted line referring to profile used in the second and third sets of testing, where a weak current formed. ............... 84 6.5 Plot of Kr with period for case of discernable shore rip current, moving toward resonance ................... ............ 85 6.6 Bathymetry 1. Shoreward direction with increasing ........... 88 6.7 |A[ for initial amplitude of .03 meters and perfectly resonant waves. .. 89 6.8 IBI for initial amplitude of .03 meters and perfectly resonant waves, without currents.......... ............ .......... .90 6.9 Steady state mean surface, ........................ .91 vi

PAGE 8

6.10 No-current wave model, incident amplitude of .02 meters. K, = 0.81. .93 6.11 No-current wave model, incident amplitude of .03 meters. Kr = 0.81. .94 6.12 No-current wave model, incident amplitude of .04 meters. Kr = 0.81. .95 6.13 No-current wave model, incident amplitude of .03 meters. K, = 0.81, and bar half-length is 3 meters ....................... .96 6.14 No-current wave model, incident amplitude of .03 meters. Kr = 0.843, and bar half-length is 2.25 meters. ................... ..97 6.15 No-current wave model, incident amplitude of .03 meters. Kr = 0.88, and bar half-length is 1.75 meters. ................... ..98 6.16 No-current, tapered bars with .03 meter incident wave. Kr = .774 ..99 vii

PAGE 9

Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science WAVE-CURRENT INTERACTION OVER A SUBMERGED BAR FIELD By THOMAS RICHARD MCSHERRY December 1989 Chairman: Dr. James T. Kirby Major Department: Coastal and Oceanographic Engineering People are moving to the coast in great numbers that seem to increase each year. Development along the shoreline is therefore also on the uprise, and the public often assumes that the shore environment is stable. Recognizing the appeal of the beach, and understanding that this environment is anything but stable, the coastal engineer must attempt to curb the natural destructive forces from the sea that would normally hinder man's use of the beach and it's surroundings. One present goal in this attempt is to achieve shore stabilization with a minimum impact on the dynamic processes in the nearshore region. One candidate receiving much recent investigation is the shore parallel low profile bar field. The bar field acts with the incoming wave to actually reflect energy back out to sea. This action thus protects the beach from some of the destructive wave energy that often erodes massive amounts of dune material in heavy weather. If the concept could be perfected, the low profile bar field could conceivably be a huge shield against any wave possible. The way the bar field reflects energy is by having bar spacings that are one half the incident wave wavelength. With this satisfied, the incident energy serves to form another wave that propagates opposite to that of the incident wave. Knowing that the bar field would be of finite longshore length, a low spot in energy would then exist shoreward of viii

PAGE 10

the bar field. What would then be expected is to have a circulation begin by this energy differential, much in the same manner as circulation into the lee of breakwater, or along groins. It is also expected that the current would flow outward through the middle of the bar field. Clearly, then, the current would alter the character of the incident wave, changing it's wavelength. Then the resonant condition between the incident wave and the immoveable bars would be upset. If the bar field is to considered for practical implementation, the behaviour of the rip current driven by the energy differential will need to be investigated. The most important aspect of the wave-driven current is how it responds to slight shifts off resonant incident wave frequency. The main hypothesis of this paper is that the bar-wave system seeks resonance, and therefore if the wave were to be initially out of resonance a current would be required to tune the wave. By incrementally lowering the initial wave frequency, and observing how the system responds and whether reflection is enhanced or reduced, the hypothesis is proven or repudiated. Another factor in this scheme is the available energy from the incident wave. If the waves are not big enough to create sufficient energy differentials, then the current will never reach the strength required to tune the waves that are out of resonance. The thesis thus looks at how current strength responds to wave heights, and how reflection behaves during the process. This paper studies results from laboratory data taken from a finite length bar field, and also findings from a numerical model that incorporates coupled wave equations with the momentum equations. The numerical scheme iterates between the two parts until a final steady state is achieved. Mean values in surface elevation, lateral depth integrated currents and wave amplitudes are solved for through a semi-implicit algorithm. ix

PAGE 11

CHAPTER 1 INTRODUCTION 1.1 Problem Statement The need for effective shore protection seems to be increasing with each year. The worldwide dependence upon the coastline coupled with such chronic problems as rising sea level has placed a responsibility upon coastal engineers to investigate truly effective methods of shore stabilization. Effective means that it must reduce erosion, hold up under severe conditions, be economically feasible, be environmentally sound, and not impede the regular use of the coastline. Historically, structures meant to curb erosion have had some success, but the side effects caused by the structures were oftentimes worse than the original problem. Today, the main thought is to develop a device that works elegantly and efficiently. One of the promising candidates is the low profile bar field that reflects incident wave energy by use of a phenomenon now known in water wave theory as Bragg reflection, which was named after the analogous process in crystallography. By placing bars along the sea bottom parallel to the shore, and making their spacing one half of the incident wavelength, a resonant condition occurs between these bars and the surface waves. Specifically, a reflected wave is generated through the bottom boundary condition. This reflected wave grows in amplitude with the number of bars placed until theoretically complete reflection can occur if there were an infinite number of bars. The findings to date have been very promising, with upwards of 80% reflection for waves travelling over four sinusoidal bars. Previous laboratory data also confirm the expectations, and these encouraging findings are fostering continued research into the low profile bar field. These bars would necessarily be of finite longshore length, and if part of the incoming 1

PAGE 12

2 energy were reflected, a low spot in energy would exist shoreward of the bar field, with full amounts at the sides of the bars. This differential in the mean surface elevation along the coast could possibly generate a large enough circulation to create a fairly strong rip through the barfield and back offshore. What this means is that the incident wavefield will be distorted over the barfield, and the resonant condition will be upset. An important question that has been answered by Mei (1985), is how far off resonance the waves can be before the bars quit reflecting energy. With the introduction of a current, however, the question is whether reflection will be enhanced or reduced. This thesis focuses on the wave-current interaction over the bar field, and what happens when initial wave frequency is incrementally reduced. What the thesis tries to answer is what happens to reflection given the current, and how the current changes as wave height changes. Also, the thesis tries to prove that the wave-bar system seeks a resonant condition by tuning offresonant waves closer to resonance through the rip current. These questions are answered by means of laboratory data and a numerical model. Some theoretical review introduces the main content, and the problem is fully developed. Then the equations necessary to the problem are presented, followed by a numerical solution to the equations. After this, the laboratory work is covered, and the methods of data analysis are reviewed. Finally, the results from the laboratory work and the numerical model are given, through which the hypothesis will be proven or repudiated. Before all of this, however, the past research that this thesis depends upon will be reviewed. 1.2 Literature Review This paper is a direct extension of various avenues of previous research. The radiation stress concept developed by Longuet-Higgins and Stewart (1964), is obviously important in any study of waves and currents. These concepts were extended to terms involving complex surface amplitude r' by Mei (1972). The expressions in Mei's book (1983) are used here directly,and are the driving forces in the time averaged, depth integrated momentum

PAGE 13

3 equations that govern the circulation and mean water surface. The other terms appearing in the momentum equations have been included in the model, and most of those formulations are based on previous work. The bottom friction as developed by Longuet-Higgins (1970a) is represented in the model as a function of the linear water wave maximum orbital velocity and the mean current U. In a companion paper, Longuet-Higgins (1970b) looked at the transition of longshore current at the breaker line, and developed an expression for the lateral mixing. This term also is included in the present model. The advective acceleration terms in the momentum equations are included in the model, and are formulated as a four grid-point average of the current values, which is done by Winer (1988). The momentum equations are solved by an Alternating Direction Implicit (ADI) method which was introduced by Sheng and Butler (1982). The matrix set up for this method can be seen also in Winer (1988). Although the matrix implicitly refers to linear terms, the model introduces limitations in time step by including the nonlinear terms explicitly. Iteration between wave model and circulation model was used by Noda et. al. (1974), where the importance of wave current interaction in determining the final current field was noted. There has been much work on Bragg reflection in the last decade, as elegant and environmentally sound methods are sought for shore stabilization. A non-resonant theory was developed by Davies and Heathershaw (1984), and partially successful comparisons to laboratory data were made. The theory broke down at resonant frequencies, however, and the reflection coefficients predicted were significantly larger than the data. Mei (1985) extended the theory to include resonant frequencies, and successfully predicted the reflections seen in previous experimentation. Kirby (1986a) allowed for the presence of a mild slope in the theory. The theory was further extended to include currents flowing over the bar field, although on a flat bottom by Kirby (1988). Kirby (1986b) attempted to include currents over a mild slope without the bar field. The work on flat topography noted that the pres-

PAGE 14

4 ence of a current enhances reflection, and presented the background that this paper uses as a starting point. The presence of a current allows incident waves of frequencies slightly off resonance to tune with the bar field and actually achieve greater reflections than with perfectly resonant waves. Transmission coefficients always drop in the presence of a current. Thus with the joint action of a depressed energy zone driving a rip current that can alter the incident wavelength, and transmission reduced against the current which further reduces the low energy zone, the system may tend to want to go toward perfect tuning. The scheme depends largely, however, on the incoming available energy to drive the current. Laboratory data, will demontrate cases where incident wave heights were too small to sufficiently create a rip current strong enough to keep the reflective plateau. Various methods are used to measure reflection from the laboratory data, and the previous work that made these methods available will be cited in Chapter Five.

PAGE 15

CHAPTER 2 THEORETICAL REVIEW 2.1 Introduction To begin the task of answering some of the questions raised in the last chapter, specific wave theory should be reviewed. It has been conjectured already that reflection will increase when a current flows over the bar field against the waves.. The thesis will seek to prove this, and the more important concept that the entire system will try to reach a resonant condition. The mathematical basis for these hypotheses was developed by Kirby (1988). Referring to the following figure 2.1 for the case of a current over a sinusoidal bar field of length L, with bar wavenumber A = 27r/Lb, Kirby showed that if one considers incident waves at or very near resonance, the ratio of transmission with a current to transmission without a current is SA(L)current cosh QL (2.1) A(L)nocurrent cosh QcL The subscripts c refer to properties in the presence of a current. The bar field has length L, where A(L) is the value of the transmitted wave amplitude at the back edge of the bars. The lateral bar shape is described as Db. The other terms are defined as Q = Q (2.2) Cg c, Qc = (2.3) (C1C2) gk2Db o gkDb (2.4) 4wcosh2kh 5

PAGE 16

6 X ---U =Lb=L Figure 2.1: Bar field in the presence of a depth-uniform current. From Kirby (1988). ()2 ala2(2.5) where C1 = Cg1 + U and C2 = Cg2 -U. Qc is defined later by (3.24). Note that when a strong enough current opposes either wave, the wave can be stopped when Cg. + U = 0. To simplify the expressions and to show what happens when the current strength changes, consider a shallow water wave where the expressions reduce to the following; ADb Q-8h ADb 21-l Q c -F)where the Froude number in the x-direction is F, = U/vg-h. It appears that Qc has a singularity as IFxI -1, which corresponds to the stopping Froude number. When IFx\ < 1, from the above simplifications it is seen that Qc > Q, and therefore TA is less than unity. What this means is that in the presence of a current, transmission for waves which are perfectly resonant with the bars with an opposing current is less than the transmission

PAGE 17

7 for perfectly resonant waves without a current. The current is thus perceived to enhance reflection, and it is interesting to note that this is independent of the direction of the current along the x-axis. Now for the shallow water limit consider resonant frequencies, where wr is in the presence of a current, and wro is without an imposed current. These are Wr = A(1 -F) (2.6) 1ro = IVghA (2.7) which from the ratio wr/wro = 1 -Fx describes a parabola. Refer to figure 5 in Kirby (1988) for a picture of this ratio for various values of Ah, where Ah = 0 is also shown as the shallow water limit. Now consider any frequency w shifted off the resonant frequency wr by an amount 0, so that w = Wr + -.There is a cutoff condition for this shift, above which for a given current, the resonance is lost between the wave and the bar field. This cutoff is described by S2 ; C = ffgC= h. (2.8) cutoff = 64C2 The behaviour of the wave field depends'on the relation between Q and Qcutoff. Mei (1985) and Kirby (1988) denoted four cases where the solutions to the coupled equations described different wave fields, and these solutions for A(x) and B(x) can be put in terms of the transmission coefficients, Tr = IA(Lb)/A(O)I. These expressions for the four cases are case 1: 0 > Qcutoff PCI T, = 1 (2.9) [(PC1)2cos2PLb + (S')2sin2PLb]/2 case 2: t = ,,ctoff

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8 C1 T = (2.10) r [C 2 + (fLb)211/2 (2.10) case 3: Q < cutoff T QC (2.11) [(QC1)2cosh2QLb + (QI)2sinh2QLb]1/2 case 4: Q = 0 (perfect tuning) 1 Tr = (2.12) coshQLb These relations may be evaluated close to shore using the approximations: Ci = vgh(l + F) C2 = Vh(-F) gADb 1 8" {8 h(1 -F2) 1Fp2 = 2[s22 2 2(l2) +^? Q2 = -P2 when Q < ,cutoff These expressions can be put into a program to plot the surface projection of transmission Tr for any combination of w/wro and IFxI. Figure 2.2 shows the contour of the transmission coefficient, and figure 2.3 is a surface projection plot of the same result. Considering an incident frequency that is not far off wro, say 85% of it's value, then by no-current resonant theory the transmission coefficient will be 0.8. The mean surface elevation in the surfzone shoreward of the bar field would then be lower than surrounding areas, and the differential could be enough to drive a current. By following this line of constant wlwo along the surface projection, clearly as the current increases the line heads

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9 JR FROM 1.5B000E-1 TO 1.8101 CONTOUR INTERVAL OF 8.S1100E-11 PT13.311.99934 LABELS SCALED BY 196B. IF.1 Figure 2.2: Contour of T, v. IFI v. w/w,,ro.

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10 0 0.25/ r 00. 1.0 0.0. 0.6 o.a 0.0 Figure 2.3: Surface projection of transmission Tr v. frequency and current.

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11 down the bank of the large trench toward the perfect resonance ratio Wr/wro, which again is illustrated in figure 5 of Kirby (1988). As this minimum transmission coefficient is reached, any more current would drive the transmission coefficient up the other side of the trench. This will decrease the energy differential and reduce the current, allowing the system to fall into the base of the trench again. This assumes that there is enough available energy from the wave to drive the currents necessary to reach the trench base, but clearly shows the process that the system seeks an equilibrium value that is as close to the perfect resonance ratio as it can get. This is the main point of this thesis, and Chapter Six will present results which are aimed at investigating this hypothesis.

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CHAPTER 3 GOVERNING EQUATIONS 3.1 Introduction This chapter addresses the governing equations that apply for the case of water waves over mild topography, and the resulting mean currents and mean surface elevation. The problem as stated must be simplified to be solved numerically, and therefore the equations governing momentum balance in the domain between the mean values must be reduced to two dimensions. Also, the wave equations that govern wave amplitude over the bathymetry are presented as a parabolic equation. These simplications inherently take away from the full process, but without them the solution would be very difficult to achieve. First to be presented in this chapter are the equations and terms that apply to the circulation. Then the equations for the wave field will be presented. 3.2 Circulation Model 3.2.1 Governing Equations The governing equations are presented in equations (3.1)-(3.2). They are the depthintegrated and time averaged equations of motion, where time averaging is done over a wave period. The derivation can be found in a number of references, most recently in Appendix A of Winer (1988). aU aU oU 01 1 + U + V + -g -b --s t 9x 9y Dx pD pD 12

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13 1 (S, BS, N 1 drj + X ++ 0 (3.1) p D z x y p dy OV OV dV Or 1 1 + U + v + g + -hy -by 9t 9x 9y 9y pD pD +_1 + SY) + 0 (3.2) pD \ Ox Oy px 0 The continuity equation completes the needed three equations for the three unknowns ?, U and V. + (U D) + -(V D) = 0 (3.3) Tt Ox By These equations are simplified to the spatially two-dimensional case to render a realistically solvable set of equations. The assumption does, however, take away from the complete process which might include strong effects from vertically variable current fields. In these equations, the terms are defined as U = x component of mean current V = y component of mean current S= mean water surface elevation p = vertically constant water density ho = still water depth D = total water depth = ho + 7 rl = lateral shear stress Tbx = x component of bottom shear stress Tby = y component of bottom shear stress r,, = x component of surface shear stress ry = y component of surface shear stress

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14 Sxx = x component of flux due to x-propagating wave Sxy = y component of flux due to x-propagating wave Syy = y component of flux due to y-propagating wave Note that the equations will retain the nonlinear advective acceleration, gradients in the radiation stresses, bottom friction, and lateral mixing. The numerical representation of these terms are the same in this thesis as was presented in Winer (1988). Referring to that dissertation, the bottom friction term is represented by equations (2.15) and (2.16), which is a direct application of Longuet-Higgins (1970a). The lateral mixing term is in equations (2.26) and (2.27), and also in Longuet-Higgins (1970b). The advective acceleration is represented in finite difference form in equations (4.24) and (4.25). The surface shear stress is not included in the paper, but could be added easily enough. It was not included for it remains out of the scope of the present problem dealing with energy differentials due to reflection. 3.2.2 Radiation Stress Terms The original model by Winer (1988) did not address the reflected wave that is now present. Since a reflected wave will be expected, the radiation stress terms must be altered to include this wave to maintain a proper balance. Also, when considering waves on a current, the incident and reflected wavenumbers will be different. If one looks at how the waves interact with each other, there will be terms that will oscillate spatially at a spatial frequency related to the difference of these wavenumbers. Therefore, deriving radiation stresses would necessarily be involved. For demonstration of what types of terms are present in a two-wave system at zero angle of propagation, consider the situation of zero current and thus equal wavenumbers. To begin, the linear velocity potential for a two-wave system is given by

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15 (xytz,) ig coshk(ho + z) A(x,y) ei(kx-wt) 2 w cosh kho ig cosh (ho + z) ig coshk(h + z)B(x, y) ei(-kx-wt) + (3.4) 2 w cosh kho and the instantaneous surface displacement 4' is given by '(x,y,t) [A(x,y) eik + B(x,y)e-ikx] e-it + (3.5) where denotes the complex conjugate. It has been assumed that the complex amplitudes relate to a scaling parameter c defined as E = k lA, the steepness of the wave. The amplitudes vary with x at O(e2), and in y at O(c). In keeping with the parabolic simplification to the wave model, the wave angles are then assumed to be at or very near zero. The wave induced velocities are then, by (3.6-8) i 0_ gcosh k(h, + z) kA ei(kx-wt) -i A ei(k-wt) Ox 2wcosh kh, + kA*e-i(kx-wt) + i A e-i(kx-wt) -kBei(-kx-wt) -iBxei(-kx-wt) -kB*e-i(-kx-wt) + iBe-i-kx-wt)] (3.6) v 0 y g cosh k(ho + z) [-i AY ei(kx-t) + i A* e-i(kx-wt) Sy 2wcosh kho -iBy ei(-k-t) + iB* e-i(-kx-t)] (3.7) WI 0 ig k sinh k(ho + z) -Ai(kx-wt) + A* e-i(kx-wt) Sz 2 w cosh kho -B ei-kwt) + e-i(-kx-wt)1 (3.8) These wave induced velocities go in to form the radiation stress terms, along with the mean surface elevation and wave surface displacement. The general expression for each of the radiation stress terms are presented as a review from Mei (1983) as

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16 = p I '2 dz + p g -z) dz + (p / -d( dz J-ho J-ho -ho \ Jz X p Iw'2 dz -g (h, + j)2 + (71)2 (3.9) -ho 2 2 SXy = p u'v' dz (3.10) 3ho Syy = p v'2dz + pI g (-z) dz + p d( dz .-ho -ho J-ho\ Jz O y p wf dz -Pg (ho + u)2 + ()2 (3.11) -ho 2 2 From these the derivation will now be completed for the two-wave system without currents. First note that for any two complex number R and Q, that RQ* + R*Q = 2R[RQ*] and RQ* -R*Q = 2ij[RQ*]. Now consider each term in the radiation stress expressions separately; TERM ONE PfhoU/2 dz p If 2dz = 2k 2|AI2 + 2k 2BIB2 -4k2R[AB* e2ikx] + 21A 12 + 21Bx12 + 4 R[A B e2i"kx] -4 k[AA*] -4k [ABj e2ikr] -4 k [A B*e2ikx] +4 [B B[ ] p ( )2 ()2 dz (3.12) 2w -h coshkho The integral in this expression has the solution, J (cosh k(ho dz = 2 (+ s2kho -wCg J=coshkhoW + 2kh, 1 (3.13) cosh kho 2 gsinh 2kho gk

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17 TERM TWO pf2ghog(z)dz p ( -z) dz = (ho + )2 = (3.14) J-ho 2 TERM THREE P fh f7'u'W' d dz P j -h o z a x---g p f~ ou dC dz = p 2 Ax2 + 2 1Bx,2 + 2 [A A;] + 4 [A, B e2ikx] + 2 R[B Bl] + 2 R[A ~B e2ikx] -8k .S[A B e2ikx] -8 k A[Ax B* e2ikx] + 2 R[[Axx B* e2ikx] -8 [A B xsinh 2k(ho + d( dz -J-hoJz sinh 2kho (3.15) The integral is evaluated as, 9 [sinh 2k(ho + () 1 nh2k + dC dz = [2kho coth 2kh -1] (3.16) sinh2kh 4k2 TERM FOUR P Jh 2dz

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18 S 2dz = (i2kho) { A2 + Bl2 + 2R[AB*e2ikx]} (3.17) TERM FIVE -((ho + )2 TERM SIX Pg (?,)2 = P.g { A2 + IB12 + 2,[AB* e2ikx]} (3.18) Then writing the final expression for the radiation stress component Sxx, SX= S(1 + sink h, )k2A2 + k2 B12 -2k2R[AB* e2ik] + IA, 2 + iBx12 + 2R[AxxBe2ikx] -2 k S[A A*] -2 k .[A B e2ikx] -k Q[AB B* e2ikx] + 2 k Q[B B] + -g(2khcoth2kh -1) {|Ax12 + IB.12 + R[AAZx] + R[B B*] + [[A BZ e2ikx] -4 k S[A B e2ikx] -4 k bS[A B* e2ikx] + R[Ax B* e2ikx] -4 k2 R[A B* e2ikx] + 2 R[Ax Bj e2ik]} p g ( 1 2_ k h ,_12 + 1 2 2ik x] 4 sinh2kho) {lAI + I B2 + 2b[AB*e2]} + {A|12 + B|12+ 2R[AB*e2ikx]} (3.19)

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19 Now for the terms in the S,, expression, only the first applies. P f h,' v' dz and the expression for Sx follows as S,= 2(1 + sinh2 h2 ) {[ A[ A] + [A, Be2ik] + R[Bx Aye2ikx] + n[BxB ] -kQ [AA;] -k Q[A B e2ikx] -k 9[A B* e2ikx] + k 3[B B*] (3.20) For the Syy expression, which is similar to the Sxx term in the integral representations, the final expression is Syy = + sin2 k h ){Ay2 + By2 + 2 •[AB* e2ikx] + g(2khcoth2kh -1) {IAy2 2 + IBI2 + 2 [AB* e2ikx] + R[AAy] + ~ [B B*;] + R[AB*;e2ikx] + R[Ay, B* e2ikx]} pg -sin2k ho { A2 + IB12 + 2 R[AB*e2ik]} + {jAI2 + IB12 + 2 R[A B* e2ikx] (3.21) The terms arising because of the two-wave system couple nicely when not considering currents. After doing some preliminary testing of the model, it was found that the interaction between the two waves in the radiation stress terms would give rise to small currents. Since the only concern for the modelling effort was currents arising from differentials in overall energy, these small disturbances did not matter. Thus it was decided to calculate the wave field for a given current field and let the bar coupling extract the energy from

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20 the incident wave, then set the reflected wave field to zero in the grid prior to calculating the radiation stresses. This would not only remove the currents over the bar field due to interplay of the partial wave, but also allow a simple form of the subroutine that included the incident wavenumber. The equations are the same as (3.19)-(3.21), but contain only incident A terms. The final expressions used in the model will be presented in Chapter Four. 3.3 Wave Model 3.3.1 Governing Equation The wave model emulates the wave equation presented in this section. It governs the energy balance of a wave travelling over a mild bottom with fast undulations, and riding on a current. The parent equations come from two sources, one being Kirby (1988) and the other being a University of Florida Technical Report, Kirby (1986b). The first paper developed coupled equations for waves on a current over a bar field on a flat bottom. The other paper handled such a case, but on a mild slope. The forcing from Kirby (1988) is used in the present thesis, and the main left hand side of Kirby (1986b) is also used to complete the full equation. As a review, the coupled equations for the incident and reflected waves from Kirby (1988) are the refraction approximation, -{TT, + (C9, cosOe + Uo)Tx, + (Cg, sin10 + V,)Ty,} = -c R (3.22) 2 {RT, + (C92 cos2 + Uo)Rx, + (-C, sin2 + Vo)Ry} = fncT (3.23) where the following terms are defined; Q, = oc cos(0i + 2) + Oc, (3.24)

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21 gklk2Db o= (3.25) S 4wcosh kh coshk2h' fl, = (4wa coshAh)-l {AUo(1201 -llr2)Db U[gkik2 coS(1 + 2) + (02 + (,o)2)]Db} (3.26) g g a = AhF -tanhAh 2r A = Barfield wavenumber = Lb Uo Fx = Froude number = Ii = kicosOi; ai = intrinsic frequency = /gki tanh kih 02 = -02 27r w = absolute frequency = wave period Db = Uniform bar amplitude The amplitudes in these equations are surface displacement amplitudes divided by intrinsic frequency, or T = A/la and R = B/a2.From the technical report, the parabolic coupled equations are written as, 2ia(Cgi + Uo)T + 2oi(Cgi + Uo)(kl -k1)T + 2iiTVTy + i {[l(Cg, + Uo)]. + (aiV),} T + {(CCgi -V2)(T),} =coupling with R (3.27) 2iO2(C,2 -U,)R, -2U2(Cg2 -Uo)(k2 -k2)R -2iC2VR, + i {[2(Cg2Uo)] -(2V))} R -{(CCg2-V2)(R)y}

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22 = coupling with T (3.28) By dividing by 2i from (3.27-28), and multiplying w to the coupling from Kirby (1988), a set of equations result for the case of a bar field on a mild slope with currents. These are then solved using a Crank-Nicolson method for each wave until convergence is achieved. A term w that handles the wave breaking energy dissipation is also included. This is added to the incident wave only, since the reflected wave will not be expected to break anywhere. This term is obtained from Dally et al. (1985). The concept is that if wave height exceeds a criterion based on the local water depth, written as 2[A| 2AI (3.29) D then breaking commences, where K = 0.78. The term is then defined as K [ 72 h2 w = 1 [] (3.30) 2= Th 41A12I where 7 = 0.4 and K = 0.15 for the present paper. The full parabolic equations used in the model are then al(Ci + Uo)T. -io,(C.i + Uo)(ki -k)T + olVT + {[a(Cg1 + Uo)]. + (aV),} T -{(CC, + -a(Ci + Uo)T = -w R (3.31) 02(Cg2 -Uo)Rx + ia2(Cg2 -Uo)(k2 -k2)R -92VRy + [U2(Cg2 -U0)]x -(02V)y} R + (CCg2V2)(R) = -w T (3.32)

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23 This is recognized as the equations governing waves that are at perfect resonance with the bars. If there were a slight shift Q, then the forcing would have a different form.

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CHAPTER 4 FINITE DIFFERENCING 4.1 Introduction The incorporation of the equations developed in Chapter Three involves finite differencing, decisions on initial and boundary conditions, and placement within a coherent algorithm that performs efficiently. The latter requirement is best described by means of a flow chart. So before moving to the details of the coding, an overall view of the model MCSHERRY would be timely. Figure 4.1 provides the logic behind the main program. After inputs of bathymetry and wave information, the model iterates between the wave model WAVEMOD and circulation model CIRC. A hyperbolic tangent startup function is multiplied to the initial offshore incident amplitude, which slowly increases it's value to the final after a specified number of iterations. This is done so that the circulation model will not be shocked by suddenly large forcing terms. The form of the startup function is C(t) = 0.5[1 + tanh(t/cl -c2)] where the parameters cl and c2 are chosen to alter the offset and slope of the curve. For the present case, cl = 3000 and c2 = 3 to render a curve as shown in figure 4.2. The subroutine FLOOD allows the grid area to fluctuate depending in the mean surface elevation. It is called at every iteration, and alters values at wetted or dried grids to conserve mass flux in those grids. The subroutine CHECK simply checks for convergent values, and sets a flag for the model to quit if the condition is satisfied at each grid point for U,V and 24

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25 START ) CALL INPUT ITER STARTUP COUNT FUNCTION CALL VAVEMOD CALL CIRC CCALL RADIAT CALL SHEAR2 CALL CALL EXCDEF CHECK C OUTPUT) CALL UP STOP ) CALL FLOOD Figure 4.1: Flow chart for model MCSHERRY.

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26 1.0 0.80.60.4 -p 0.2 0.0 0 50 100 150 200 time (sec) Figure 4.2: Hyperbolic tangent startup function.

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27 4.2 Circulation Model 4.2.1 Method of Solution The equations presented in equations (3.1)-(3.3) are solved using an implicit Alternating Direction method that solves for the new U value and interim 7Y value in the x-sweep, and the new 7 and new V in the y-sweep. This method applies to the matrix setup as suggested by Sheng and Butler (1982). The detailed treatment of this method as used here can be seen in Chapter 4 of Winer (1988). The two unknowns in each sweep are applied at different places in the grid, where J is at the grid centers and the mean velocities apply to the grid edges. Note that the first offshore value is i7, at i = 1, and the first mean current U is at i = 2. The value of U and V at i = 1 do not exist for the grid edge values, and are not used directly in the code. As stated before, however, the wave model uses averages of the currents and for the i = 1 value uses U, V at i = 2. The grid domain is shown in figure 4.3. The x-sweep occurs in subroutine XCOEF, which sets up the coefficients, then passes them to the double-sweep solver. At the lateral boundaries, the special subroutines XCOJ1 and XCOJN are called to handle the special coefficients when j = 1 and j = N. The y-sweep is handled by subroutine YCOEF, except at the shoreline, where the variable grid domain determined by FLOOD is deciphered by the subroutine YCVAR. There is no time restriction for ADI method, but some of the terms are expressed explicitly and added to the knowns of the implicit scheme, and restrictions are introduced. The finite difference form of each of the terms in the momentum equations will now be presented. The radiation stress forcing is a function of the square of the amplitudes, thus being nonlinear. Also, since the wave amplitudes are determined at grid centers in the wave model, the radiation stresses also apply to grid centers. The radiation stress gradients are put into the nth time level of the scheme, which

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28 1) __offshore -----_ -_ shoreline (M,N) Figure 4.3: Grid for circulation model.

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29 referring back to the last chapter for the xand y-sweeps are pD Ox 9Dy These terms are represented in finite difference form respectively by 1 SXXi,-SX + 5SXY+1 SXY, + SX -SXYI 1,j_ + SXY.5 l2 yX -I pD AX 2AY 1 SYYj -SYYi,j1 .SXYi+j -SXYi-_,j + SXYi+lj -SXYl,j pD AY b 2AX The symbols in the code for each variable are hopefully clear. The subroutine for the radiation stresses is RADIAT, which is called just after the wave model and prior to the subroutine CIRC. Referring to figure 4.4, the terms are used to find the mean velocities, and are therefore differenced to straddle the known (i, j) velocity location. Note differencing is done over one grid space, and is accurate to second order. The bottom friction pFlUIU is nonlinear, but uses mean currents from unknown and known time steps, and is thus represented implicitly. The terms are represented based on the work of Longuet-Higgins (1970a) as bx = pF luoblU + pF UIU Tby = pF luobl V + pF lIV These are finite differenced according to the sample x-component,

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30 J-1 J J+1 I ------__--------------1-1+1 Figure 4.4: Differencing of radiation stress gradients in x-sweep. UneW 2fH V \b = U T sinhkh f/U + ) where the linear formulation for maximum orbital velocity has been used, T being the wave period. Since the reflected wave is considered nonexistent when performing the radiation stress subroutine, this also applies for this subroutine. Therefore, the linear formulation of maximum orbital velocity for the one wave applies. The subroutine that calculates this term is called SHEAR2, and is called just prior to CIRC. The lateral mixing is expressed in the momentum equations as p y y a Oy P 9Y Ony Y 1 9

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31 D Orl a / V\ p O9x Ox \ Ox These are added to the knowns at the nth time level explicitly, and thus introduce a serious timestep limitation. In the complete finite difference form they are expressed as Ot U,,j+1 -2Ui, + Uij-1 Oy AY2 tr AEY +lj -2Uij + Ui-_,j It -AtEY Ox AX2 For the solution to remain in front of the fastest travelling error, the timestep must satisfy At < 1 AX2 These values are calculated in subroutine EXCOEF just prior to CIRC. Note that the coefficient Ey is set to the highest value of Ex, which occurs at the breakerline. The terms represented in the momentum equations by U -and V2J in the x direction are represented in finite difference form respectively as, Uj Ui+l,j -Ui-l,J u'3 2AX and (Vi,j + Vi + *Vi,+1 + V-i-l,j+l) Ui+l -Ui-l,j 4 2AY The same form applies to the y direction equations. A four point average is used for the cross-derivative of the four surrounding velocities as shown in figure 4.5

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32 J-1 j J+I I-1 1+1 Figure 4.5: Differencing of advective acceleration in x-sweep. 4.2.2 Boundary Conditions The solution for the equations governing mean currents and mean surface elevation require boundary conditions for each boundary in the grid. Winer (1988) used a fixed lid condition at the offshore edge, with no flow into the shoreline, and free flow at the lateral boundaries. Each condition had to be reviewed for the present case, and changes were made where appropriate. The original model constrained the mean elevation j to be zero at the offshore grid edge. This was a fine assumption for the one wave system, but was found to cause instabilities if a reflected wave occured. The instabilities were due to a severe mean surface gradient at the first two grid rows which gave rise to large velocities. For the standing wavefield, if the grid edge happened to occur on the envelope antinode, the problem was accentuated. Obviously a radiation condition was required to free that edge to allow disturbances within

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33 the domain to escape. The condition By (07 1 BSc + + 1 -x =0 (4.1) at Ox pD O ) was applied to the x-sweep of the model, where the interim 7 is solved for at the grid center i = 1. After making the decision to not include the reflected wave, however, the original fixed lid condition was again used and is what is specified now. If the model were to be modified to include the reflected wave, an appropriate radiation condition would need to be developed. Also, if a strong current were to be flowing across the offshore edge, an energy set down condition might suffice. The lateral conditions are reflective, that is the y-sweep sets V = 0 at j = 1 and j = N. This is to remain consistent with the wave model. There is no flow into or out of the boundary, so the x-sweep condition sets U = 0 here. Also, there is no longshore current, so the y-sweep specifies that V = 0. The model MCSHERRY begins by setting all unknowns to zero in subroutine INPUT. Then, as the waveheights are ramped to their full value, the forcing slowly increases in the circulation program. The model iterates until the difference between unknowns in successive time steps is smaller that a prescribed tolerance. 4.2.3 Radiation Stress Subroutine The finite differencing of the radiation stress components that were developed in Chapter Three will now be presented. The calculation of these terms is performed in the subroutine RADIAT. This immediately follows the wave model and precedes the circulation model. Since the wave model is only accurate to O(E2) in spatial derivatives, there are terms based on derivatives in the radiation stresses that should be dropped as being too small. Referring to the expressions for S.x, S,, and Syy from Chapter Three, the terms

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34 that should be dropped are 02 8z2 --1 ~ 4 Ox2 4 0 3 51 E3T Ox Oy The terms that remain are differenced to remain at grid centers where the amplitudes are calculated. As an example, the derivatives are expressed as A Ai+l,j -Ai-,l,j 2AX A Aij+j -Ai,j-1 Ay 2AY A Aij+ -2Ai,j + Ai,j-1 dyy As was stated in Chapter Two, the reflected waves have been considered nonexisitent, since they do not physically affect the circulation landward of the bar field. To incorporate only the differences in wave energy in the grid, and not wave-wave interaction effects, the final expressions that exist in subroutine RADIAT therefore neglect terms consisting of reflected wave B. The final forms used in the model are pg [(2)] (4.2)(kh)' SX --1+ (k2A2) -2(AA)) (A A))4.3) S4(k2) sinh2(kho) i P 2(kho)) 12 ) 4 sinh2(kho0) + (|A|12) (4.2) -pg 2(kho)' S 4(k 1 + s-(k(nh2k))h' (4.3) g2(kh)' SYYP9 -1+ [(IAy12)i 4(k2) sinh2(kho)j

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35 pg 2(ko) 4 sinh2(kh) + [(IAl2i] + 8(k2) tanh 2(kh ) [A2 + R(A Ay,))] (4.4) 4.3 Wave Model 4.3.1 Method of Solution The wave equations have been presented by (3.31) and (3.32). The finite differencing follows a Crank-Nicolson method, where a tridiagonal matrix is formed for the three forwardrow amplitudes based on three known values at the present row. The solution requires only an initial condition at the offshore edge, and lateral conditions to complete the matrix. For the two waves, two loops make up the subroutine WAVEMOD. The first loop solves for the incident wave A up to the shoreline. Then, the second loop starts at the shoreline and solves for B out to the offshore edge. Due to the coupling through the bottom boundary condition, the two sweeps must be iterated until a convergent solution is obtained. 4.3.2 Finite Differencing Now the incident equation will be differenced to serve as an example. Again, the amplitudes T and R refer to amplitudes divided by intrinsic frequency. Using a CrankNicolson method leads to the finite-difference approximation of the wave equation of 1 {[ai(Ci9 + U)]+'1 + a(C,1 + U)]} {Tj+1 -T 2AX j Tj {['al-(C1i + U)(ki -l)]+1 + [rl(Cql + U)(k -U( l'1)] fTj+' + Tj} + s V] + [ -+ [T[j -fT_,l] + K [a(Cg + U)]+ -Ci(Ci + U)i T+1 + 4AX 3 j

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36 +16Y { V]i+l -[rlV]t-J + [rlV]i+l -V[ClV]ij_ {T+1 + Tj} 1 {A[(CCg -V21 + -V2 [T+1 -T +1] -[(ccg -2) +1 + (cc --V2> ] 1 T -1 + [( ,g -v2)Il + (CCg -V2)1] -TJ] -[(CC,, -V2)j + (CCgi -V2)-][Ti -TI-i] + [ {[i(Cgi + U)wT]+' + [ai(C,9 + U)wT]}} -I{ R + [Qcw Rl} The reflected form is similar, with sign changes where appropriate. These are then put into three coefficients for the i and i +1 rows, and put into a complex tridiagonal matrix doublesweep solver. Since currents exists at grid edges, the values to place in the WAVEMOD need to be averaged straddling the grid center. Thus Uj is really (Uj+ + UJ), and likewise for V. The grid domain is set so that the first calculated value of U is at i = 2, so then the averaged value for velocities are set to U2 and Vj2 when marching from i = 1 to i = 2. In WAVEMOD, a preamble defines the coded meanings of each variable. Then statement functions are defined for the coefficients in the equations for the two waves. Before the first loop, subroutine WAVNUM is called to determine the incident and reflected wavenumbers, average velocities, celerities, group velocities, intrinsic frequencies, and forcing terms 2oc and 1,c as defined by (3.25) and (3.26). After this is done, the iterations begin, with a maximum of four iterations, but convergence is usually complete within 0.1% after three. If the wave is perceived to meet the breaking criteria and start dissipating energy at some grid point, then w is defined something other than zero, and the row is iterated again, letting the energy out. The wave model uses depths of the uniform bottom, not including the barfield. Therefore, shoaling over the bars is not included in the driving forces.

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37 4.3.3 Boundary Conditions The boundary conditions are that the lateral boundaries behave as walls. This requires that no reflected wave disturbance reach the walls and bounce back into the domain, which would be unrealistic. 4.3.4 Lateral Smoothing When doing some preliminary testing on the wave model, the radiation stress gradients that would be put into the circulation model would be plotted simply for observation. It was found that the plots exhibited much irregularity in the driving forces, more than was expected. In fact, the circulation model had stability problems, and the irregularities were thought to be at fault. After some review, suggestion was made that the parabolic approximation had altered the nature of wave properties in the transverse y-direction. Physically, a wave can have a maximum wavelength for given frequency and depth. This is uniform in all directions. But when assuming small incident angles, thus primarially xdirection propogation, which is done to get from a hyperbolic equation to a feasibly solvable parabolic equation, a modification is introduced. Consider a surface described as r' = Aeikx (4.5) where A is the complex wave amplitude. For a = IA|, the surface in the ydirection is 77 = aeiky (4.6) If one considers this wave in the y-direction, the radiation stresses are proportional to the wavenumbers, with the Sxy term being proportional to k1, and the term S,, being proportional to k2.And then when considering the gradients of these terms as they appear in the momentum equations, the term 0@ would be proportional to k3.Therefore, there is a large sensitivity to the wavenumber in the transverse direction. By equating (4.5) and (4.6), the amplitude A could be expressed as

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38 A = aeike-ikx = (y)e-ik (4.7) Now consider a general parabolic approximation equation of form 2ikAx + Ayy = 0 (4.8) and by making the substitution for A, the ODE is Ayy + 2k2A = 0, (4.9) which has the solution A = aei\Vky (4.10) Therefore, a limit has been placed on the maximum physically meaningful wavenumber in a parabolic wave equation for the y-direction. Instead of the normal L, = 2, the new limit is L, = '-Any wavenumber in the transverse direction that is larger than this will have severe effects on the driving forces in the transverse direction. By showing this, consider a numerical grid with lateral width W, and some calculated wave field signal along a transect in the y-direction, as shown in figure 4.6. A(y) can be written as a Fourier series 00 A(y) = anein (4.11) n=O where A = J is the base spatial frequency. By taking the FFT of the signal, the spectrum will show energy distribution over each component nA. For the parabolic model, however, any energy that exists beyond frequency nA = v/2k is not physically meaningful. Now for the wavefields calculated by the numerical model WAVEMOD, a transect in y was FFT'd, and was found to have a very small amount of energy beyond what was physical. This is shown in figure 4.7, where the cutoff frequency component was nA = 9.051. Then a

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39 y=O y=W SA(y) SHORELINE Figure 4.6: Wave calculated from parabolic equation on domain of width W.

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40 transect of the driving forces ( s + asy) were likewise decomposed spectrally, and a large amount of energy existed beyond what was allowed. Shown in figure 4.8 for the y-direction, this shows how a small amount of error in amplitudes can expressly alter what the circulation model sees. The leakage of miniscule energy into high transverse wavenumbers in the wavefield translated into major amounts of energy beyond the cutoff frequency component for the driving forces. The process of the radiation stress gradients being proportional to k3 shows the drawback when making a parabolic approximation. After taking the wave field and doing the FFT row by row, and truncating the energy beyond nA = 9.051 and inversing, a smoothed wave field was obtained. Figure 4.9 is the spectrum of the y-direction driving after smoothing. By figures 4.10 and 4.11, the slight improvements are seen in the wavefield, and in figures 4.12 through 4.13 the difference is remarkable. The variance in the energy spectrum of the y-direction driving before the smoothing was calculated as 0.00804, while after smoothing the total variance was 0.0030. In comparison to this 60% reduction in variance, the wave field had an unsmoothed variance of 4.065 x 10-7 and a smoothed variance of 3.80 x 10-7, about a 6% reduction.

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41 0.000000008 1 1 i ,iT I ,, I I I_ 0.000000006 S0.000000004 0.000000002 0.000000000'' ' 0 5 10 15 20 frequency component Figure 4.7: Spectrum of A(y) without smoothing. Cutoff at nA = 9.051.

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42 0.0015 0.0010 0.0005 0.0000 0 20 40 60 80 frequency component Figure 4.8: Spectrum of y-direction driving force before smoothing.

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43 0.0015 0.0010 0.0005 0.0000 0 10 20 30 40 frequency component Figure 4.9: Spectrum of y-direction driving force after smoothing.

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44 196 Figure 4.10: IAI without smoothing. Bars are located on the bottom of this domain.

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45 Figure 4.11: after smoothing. Bars are located on the bottom of this domain. Figure 4.11: JAI after smoothing. Bars are located on the bottom of this domain.

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46 0. N) 19". 1 --t-I Z--ri. .< I I I IIn I Figure 4.12: y-direction forcing without smoothing.

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47 I r" I "4 i ,I SIf S It %I 16!!I : / -"'..,. 2 e 4: \ I,, 1 r .' \, Figure 4.13: y-direction forcing after smoothing.

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CHAPTER 5 LABORATORY WORK 5.1 Introduction The laboratory work was done to get a better idea of the process of Bragg reflection, where effects not represented by the wave model were shown. The following chapter treats the laboratory phase of this project specifically. Since the work was completed at two separate times, the procedures and setup unique to each will be presented separately. The results from both sessions, however, will be presented together, with comparisons to a simple numerical model. The work was done at the Coastal and Ocean Engineering Laboratory (COEL) of the University of Florida. To begin the discussion, the basin and the other necessary equipment will be described. 5.2 Equipment 5.2.1 Basin The modelling basin is illustrated in figure 5.1. The internal walls are moveable, and were positioned at right angles to the wave crests to minimize side wall reflection. The basin can be filled to a depth of 60 centimeters, and allows for waves up to heights of several centimeters. The main testing region lies inside the internal wave vanes, to minimize intrusion of backscatter from the outer basin walls. A wavemaker sits at one end, with a sloping beach at the other. 48

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49 TERMINAL WALKWAY ./ BEACH 10* WAVE VANES WAVEMAKER Figure 5.1: Plan view of large basin.

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50 5.2.2 Wavemaker The wavemaker has 80 paddles, each being 23.5 centimeters in width. Each paddle can be adjusted for stroke and phase. The entire system has a frequency range of 0.5 Hz to 1.25 Hz. Since the beach was oriented at 10 degrees relative to the wavemaker, the paddles were adjusted to generate an oblique wave at a 10 degree angle, which then arrives normally at the beach. This is done by calculating the wavenumber for the wavefield over the desired depth using a Newton-Raphson technique. Then the relation of paddle wavenumber kp to the wavefield wavenumber vector k, is, by (5.1) kp = k, sinO (5.1) Then the paddle wavelength is Lp = -. For paddle width Wp, and paddle number np, the phase of each paddle in degrees from 0 to 360 is, by (5.2) S(360)(np)(W,) p = L (5.2) Lp 5.2.3 Beach and Cart System The beach gently slopes to a maximum height of 60 centimeters above the floor. A structure has been placed over the beach to aid in various types of testing. It allows a cart to ride on tracks and be placed at any (x, y) coordinate in an 8 meter by 17 meter area on the beach. For a plan view of this track system, see figure 5.2. The track sits several centimeters above the greatest water elevation allowable in the basin. The cart is positioned in the y-direction by the wire connected to the handcrank at the side of the basin. The variable speed motor fitted onto the cross-member for the cart controls the position of the cart in the x-direction. The cart can carry wave gages, current meters, and whatever else is necessary in an experiment. See figure 5.3 for a side view of the cart. Since profiling is important in the conduct of moveable bed studies, the cart is fitted with a profiling system. The profiler arm

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51 MOTOR SVINCH CART Figure 5.2: Plan view of cart track system.

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52 AREA FOR HORIZ. INSTRUMENTS POTENTIOMETER VERTICAL POTENTIOMETER PROFILER ARM Figure 5.3: Side view of profiling cart. extends underneath the cart, dragging over the beach surface. A potentiometer produces a signal which is converted to readings of arm angle, while another fitted to the trailing wheel measures in voltage how many revolutions that wheel has made. With these two instruments, the beach (x, z) position can be traced digitally. A sample profile is depicted in figure 5.4. 5.2.4 Electronics The computer for the basin is a PDP-11 microcomputer. It receives all basin data which is amplified at a box next to the basin. A remote terminal at the basin runs all of the programs. The main program is ATOD.FOR which takes multiple channels simultaneously at any rate for any length of time. The PDP11 is located in an instrument room about 200 feet from the basin. Wave gages were the main source of data. The capacitance gages were built at the

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53 60 11 -TT I I I 40 N 20 0 j 0 200 400 600 800 x (cm) Figure 5.4: Sample profile. COEL. Figure 5.5 is a sample of the calibration points transposed on the calculated second order curve. The response appears linear over the operational range. Calibration was done daily, or if the temperature changed drastically. Along with the gages, a strip chart took the signal from the cart gage in session one. It was used to trace wave envelopes from the moving gage. A Marsh-McBirney current meter took data for the second session. Since the calibration was 10ft/sec = 1 Volt, and since the currents were small, an amplification of twenty provided the necessary range to increase resolution. The data were quite erratic, and will be addressed in a following section. Lastly, a video camera was mounted above the beach. The camera was fitted with a remote control to zoom, focus, point, and contrast. A taping device was included. Rhodamine dye transport was recorded as a way to show current magnitude. The only drawback was that the camera is black and white, and the dye does not show up as well as it could if the

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54 -2 -1000 1500 2000 2500 digital Figure 5.5: Sample calibration curve for wave gage. camera filmed in color. 5.2.5 Bar Field The bars were made of wood and metal. The same bars served both sessions, and are shown in figure 5.6 without the metal covering. They were sized according to the local water depths at their position of placement. The planform dimensions were 3.048 meters long and 0.318 meters wide, and each bar was made to be 40% of the local water depth. Once the height h was determined by inspection, the bar radius in centimeters followed equation (5.3). (19.05)2 -h2 + h = R (5.3) 2h

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55 NOTE F DIMENSIONS ARE IN METERS 0.381 h /_ S 3.048 Figure 5.6: Bar frame design. The full pattern was traced on paper, then transferred to the plywood frames and constructed. Since the main material was wood, the bars would float. They were anchored in the first session using wires connected to the concrete floor. In the second session, the undersides were filled with sand which was kept in place with a panel covering. Both methods worked fine, but the first proved to be better in time. Another problem was wave orbital velocities flowing under the bars, scouring large holes. Strips of sheet metal 4 inches wide were extended off the bar edges as aprons sunk into the sand. This effectively stopped the scouring. 5.3 Setup, Procedure and Data Analysis This next section discusses the actual testing methods. Since different methods were used in each session, each will be discussed individually. The first session will be addressed, with it's purposes and procedures, followed by the second session.

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56 5.3.1 Session One The first session ran from September of 1987 to February of 1988. Since this was the first work done in the lab on the currents attributed to Bragg reflection, the goals were fairly loose. The purposes were specifically: 1. To measure Bragg resonant reflection. 2. To compare this reflection to theory. 3. To describe qualitatively the properties of the wave-induced current field. 4. To observe how reflection and current magnititude varied as wave period varied. 5. To observe sand movement during the experiment. The first goal received most of the attention, with two methods being developed. The first was to use two stationary gages, in line, and take data simultaneously. The data from each then can be used to determine the reflection coefficient, and this method is from the work of Goda, et. al. (1976). To review that method, for a wavefield which can be described by a surface displacement -= a cos(kx -wt + 61) + b cos(-kx -wt + 62) (5.4) the unknowns are a, b, 61, and 62. Four values for y from the data will therefore be needed. Let two gages be located at x = 0 and x = 1, as in the figure 5.7. If a data value is taken for 17 from each gage at times t = 0 and t = r, four values will result. Each one can be expressed through (5.4) according to (5.5-8) as 71 = r(0,0) = a cos 61 + b cos 62 (5.5) 72 = (1, 0) = a cos(kl + 61) + b cos(-kl + 62) (5.6) 773 = 7(0, r) = a cos(b1 -wr) + b cos(62 -wr) (5.7) 774 = 7(1, r) = a cos(61 + kl -wr) + b cos(62-kl -wr) (5.8)

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57 AFigure 5.7: Gage locations for two-gage reflection method. Let the following relations hold; cl = cos61 (5.9) C2 = coS62 (5.10) s1 = sin 6i (5.11) s2 = sin b2 (5.12) d = coskl (5.13) e = sinkl (5.14) f = cos wr (5.15) g = sin wr (5.16) Expanding out (5.5-8) and substituting (5.9)-(5.16) gives

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58 ?71 = a cl+ b 2 (5.17) 12 = (aci + b C2)d -(asi -b 2)e (5.18) 773 = (acix + bc2)f + (as + b2)g (5.19) 7q4 = (aci -+bc2)f d +(bs2 -as)fe + (a s1 + b 82)g d + (a ci -b c2)g e (5.20) By substituting sequentially from (5.17) down to (5.20) and eliminating terms, the remaining expressions are, 2acl = A = 171i+ 4 --72 1fd (5.21) ge 2asl=B 773 -771f 772 -1 d.22) 2as1 = B (5.22) g e ; 4 -r]3 -712 + 111 fd 2bc2=C = 71 -14-3-72 11fd (5.23) ge 2 bS2 = D = 773 -771 f 772 -71 (5.24) 2bs2=D = +3f7 (5.24) g e From these expressions, the incident and reflected wave amplitudes are, by (5.25) and (5.26), A2 + B2 a A2 B2 (5.25) a C2 + 2 b -C2 D2 (5.26) 4 The reflection coefficient is then Kr = b/a. The problem in this method is that if g or e approach zero, the expressions for A, B, C and D blow up. Thus one must not choose r = n( ) or 1 = n( ) where T = wave period, L = wavelength, n = integer. To determine the accuracy of the method, a test was run where the reflection was from the beach. To compare, a moving gage retrieved the partial standing wave envelope. It was discovered that for various time increments r, the reflection coefficients were inconsistent. This inconsistency was enough to scrap the two gage system, and use the envelopes as

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59 Table 5.1: Bar heights and positions for Session One. bar no. height (cm) [x-position 1 7.62 420. 2 9.50 495. 3 11.20 570. 4 12.60 645. the major source of reflection data. This required measuring by hand the minimum and maximum envelope amplitudes, 7?min and r1max. Then the reflection coefficient is, Kr = ]7max -r7min (5.27) ?7max + l7min Envelope transects across the barfield were taken at four locations during the testing to get a lateral average. The positions of these transects are presented later in this section. The results are then presented and compared to the circulation model in the next session. A camera was mounted above the testing site which filmed the movement of rhodamine dye across the barfield at each wave period. The film is available for viewing through the author. The tests were run at several wave frequencies around the designed resonant frequency. Each run was made from a still water condition. With reflection and current observations made at each run, a feel of the dependence was gained. The different runs are listed in the next section for the periods and reflection coefficients. Observations were made in regard to sand movement during the course of the testing. These observations are listed as part of the results in Chapter Six. The setup consists of where the bars were oriented. By using the equation for the bars from the local depths, the bars were built according to Table 5.1. Plots of the testing bathymetry were done as testing commensed, and are available as of this writing, but the actual data that was written onto magnetic tape is not immediately

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60 40 S -_ C) N 20 0 2 4 6 8 x (m) Figure 5.8: Reconstructed bar profile for Session One. available to the author. There is no other recourse than to reconstruct what the bathymetry was at the time from the existing plots. This is the only handicap from losing the data, since the envelopes are printed on strip chart paper. Figure 5.8 shows the lab bathymetry with the bars placed. The depth was 55 centimeters, and the breakerline in the testing was at about x = 3 meters. Figure 5.9 gives an overhead view of the barfield, with four lines in the x-direction offshore which represent the transect lines where envelope data was taken. The profile is available as a test bathymetry in the numerical model, although the bars need to be sinusoidal. Once the bars were placed, the wave gages were arranged. The barfield would create a reflected wave that would propagate and diffract once beyond the bars. Two gages were mounted on tripods in the center of the bar field longitudinally, and directly offshore of the last bar. As a control, one gage was put off to the side, away from the influence of the barfield. Another gage was on the moveable cart. Figure 5.10 shows a plan of the beach

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61 1365 1300 1175 1110 16.0 16.0 ---16.0S.0 .0.0 "------6 --_4.0 4.0-_____ 32.0 32.0 2.0 Figure 5.9: Plan view of barfield for Session One.

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62 GAGE ON CART BARFIELD CONTROL TVD GAGES GAGE ON TRIPOD Figure 5.10: Gage locations for Session One. region with the gage placements. This concludes the discourse on session one. The data from the experiment will be presented later, with the test run parameters, in Chapter Six. Now the second session will be addressed with the procedures and setup. 5.3.2 Session Two With the experience from the first session, the second experiment ran from September of 1988 to December of the same year. The goals were more defined in this second session. 1. To repeat Bragg resonant reflection and a resultant current. 2. To extend the reflection to frequencies off the no-current resonant value. 3. To observe differences between moving away from resonance and shifting toward it.

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63 XR1 X13 wavemaker 1 ; xl X12 -Figure 5.11: Three gages inline with x-direction waves 4. To quantify current values. In this session, two methods were again employed. The first was from the work of Funke and Mansard (1980). Here, three in line wave gages took simultaneous data, and the reflection coefficient came out of the spectra. The work covered waves travelling over a flat bottom, with no current involved. Their results therefore needed to be revised for the addition of current and variable topography. For the setup in figure 5.11, the algorithm was modified in the following manner; The wave profile for any probe p = 1,2,3 can be described as a Fourier series sum of each component k by N / 2rkt rp(t) = Ap, sin -T + ap,k (5.28) k=l

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64 where Ap,k is the Fourier coefficient for frequency -, T is the time length of the data run, ap,k is a random phase shift, and N is the desired number of components for the profile. The Fourier coefficient is gotten from the transform of the time series at each probe p, rp(t), which for any probe would be, by (5.29) 7p(t)} = Bp,k = C,k exp 2(X ) +iOk + CR,k exp { 27r(Xi 2XR1 -X1P) k i + k)} I Lk + Yp,k exp{i(pp,k)} (5.29) These equations are actually (1) and (10) in Funke and Mansard (1980). They point out that the transforms represent functions of the complex amplitudes for the incident and reflected waves. The three expressions for the FFT's of the three probes then lead to a least squares determination of the incident and reflected amplitudes, Zi and ZR. B1,k = ZI,k + ZR,k + ZN,1,k (5.30) B2,k = Ks,I,2 ZI,k exp[qI,i,2] + Ks,R,2 ZR,k exp[-R,1,2] + ZN,2,k (5.31) B3,k = Ks,I,3 ZI,k exp[~I,1,3] + Ks,R,3 ZR,k exp[-'Ra,,3] + ZN,3,k (5.32) The terms in these three equations are defined as follows; ZI,k = CI,k exp i27r + iOk (5.33) SLk .2X1+2XR1) ZR,k = CRexp { 2r(X1Lk i(O+(k +i) (5.34) ZN,p,k = Yp,kexp {iPp,k} (5.35) The shoaling coefficients in the incident and reflected directions for each probe, and group velocity Cg are,

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65 Ks,1,2 = i (5.36) V C, 2 K,, = Cj (5.37) V C9,3 As,R,3 = C, 13 (5.39) The phase accumulations are obtained according to yX2I ,1,2 = k dx (5.40) JX1 cX3 I,1,3 = k dx (5.41) R,1,2 = kRdx (5.42) JX2 RR,1,3 = 3 kR dx (5.43) JX3 (5.44) These modifications reflect the different case of waves over a variable topography on a current. Funke and Mansard (1980) addressed waves over a flat bottom where the incident and reflected wavenumbers were equal, and wavelength Lk for each frequency was constant in the domain. The reflective and incident wavenumbers k are found from the dispersion relation that includes the current effect, using an iterative technique. Setting the ZN terms to a small error parameter E, the least squares method is followed and the resulting equations for unknowns ZI and ZR are ZI {2 + 2K',I,2exp[2i,,1,2l] + 2Kl,3,aexp[2i~,i,,3]}

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66 + ZR {-' -: /,,,2AKs,R,2exp[i(II,1,2 -R,1,2)] + 2Ks,I,3KS,,R,3exp[i(JI,1,3 -T'R,1,3)]} = 2B1 + 2Ks,I,2exp[i'J,i ,2]B2 + 2Kj,i,3exp[i'I,,3] B5.45) and then ZI {2 + 2Ks,I,2Ks,R,2exp[i(PI,1,2 -TR,1,2)] + 2K,I,,3Ks,,R,3exp[i('I,1,3 -@R,1,3)]} + ZR {2 + 2 ,R,2exp[-2iQR,1,2] + 2 K,R,3exp[-2iR,1,3]} = 2B1 + 2Ks,R,2exp[-i'R,1,2]B2 + 2K',I,3 exp[-iQR,1,3]B3 (5.46) These can be solved for the two complex amplitudes, ZI and ZR, and then Kr = ~IZR The modifications were done numerically according to the following forms. For the dispersion relation with opposing current, w = -Uo k + V/gk tanh kh (5.47) a Newton-Raphson iterative technique was used to solve for k, given depth, frequency, and current Uo. It is assumed here that the incident wave angle 0 = 0, so that k cos 0 = k. The simple integration scheme of the phase in space is performed by, Wp = (kp + kp-)A i --1 + Xp-1 (5.48) 2 where again p = the gage number, either 1, 2, or 3. Ax p-.p-1 is the interval size between gages p and p -1. The intervals were on the order of 15 centimeters, so that the bottom could be considered a plane sloping beach in these intervals, allowing the simple integration. The group velocity used for the shoaling coefficient K, is, for each gage p, Cgp P (1 + sinh2(h)p (5.49) This method was tested for accuracy for reflection from the beach, being compared to envelope data. It was found that the reflections differed between the two methods by as

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67 much as 60%, and the development of the equations thus leaves something in error. Rather than an in-depth search for an accurate form, it was again decided to use the envelope method, as in Session One. Again, partial standing wave envelopes were retrieved from a moving gage across the barfield, but were recorded digitally, and analyzed as per the methods outlined below. The envelopes were taken at seven transects over the barfield. It was found that the three center transects represented the highest consistent reflection for a test. These locations are shown in the plan view in figure 5.16. After determining the reflection from each transect, it was plotted with location in figure 5.15. As is apparent, sections 850, 900, and 950 give a consistent representation of reflection. The method of analyzation was to take each envelope, as shown in figure 5.12, and condense it into envelope height versus x-position. A sample is shown in figure 5.13. The reflection coefficient was calculated by taking a local maximum and immediately following minimum, using equation (5.27). Thus a plot of Kr with x was obtained, as seen in figure 5.14. The maximum reflection was taken from each envelope, and the three were averaged. This was done for each wave period. The results are presented in the next chapter and show a repeatability of the process, which is a good sign. Further similarity with theoretical output gives confidence in the values themselves. From figure 5.14, there is a variability in measured reflection along the bars, and the ending minimum should be disregarded since the cart may not have traced the full wave envelope before stopping. The variability within the bar region, however, could be attributed to focusing effects over the bar field. This data was taken using program ATOD.FOR for a sampling frequency of 26.0485 IIz, and a run of 2048 points. The gage was the cart mounted gage, moved slowly through the water over the barfield, insuring that the wave excursions stayed in the middle portion of the gage wire. The second priority was to get quantified results for current induced by Bragg reflection. At each period, a line of current data was taken across the front edge of the barfield. The locations were at the same locations used for the envelope measurements, except they were

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68 4 l 1 1 l 2 --4 200 400 600 800 x (cm) Figure 5.12: Sample envelope, Session Two. Retrieved digitally from one of the three on-offshore transects using the moving cart gage.

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69 E 6o 6 4> 2 200 400 600 800 x (cm) Figure 5.13: Sample envelope height versus X. 0.5 --------T-------1 0.4 0.3 0.2 0.1 0.0 200 400 600 800 x (cm) Figure 5.14: Sample Kr with offshore distance x from Session Two.

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70 0.5 0.4 0.3 0.2 0.1 0.0 600 700 800 900 1000 1100 section number Figure 5.15: Plot of IK with transect location, Session Two.

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71 la s in tQ i n .&s L I g s 16.0 6 16.0 16.0 1V-.16. -4.0 0._?!--------Figure 5.16: Plan view of barfield, showing data collection locations. stationary just inshore of the first bar. Figure 5.16 is a plan view of the session two barfield with the locations used for current meter measurements and envelope data. A side view is shown in figure 5.18, where the water depth was 50 centimeters, and the breakerline was at x = 320 centimeters. The current meter was placed at 750, 800, 850, 900, 950, 1000, and 1050. Data was taken at 26.0485 Hz for 512 points. The data proved very erratic from the twochannel Marsh-McBirney current meter. The calibration for the instrument was 10 ft/sec =

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72 1 Volt, and since the expected currents would be on the order of a few centimeters per second, an amplification was done. An amplifier was built in house to magnify the signal by a factor of twenty. The raw results from such a time series is given in figure 5.17, which represents the velocity magnitude. Because of the nature of the signal, any trustworthiness of discharge calculations would be in doubt. The irregular signal could be due to interference from the metallic bars that were about 10 centimeters away from the probe. Still water readings taken in a container away from the bars showed a fairly stable signal. Other reasons could be excessive low frequency water displacements, but nothing like that was observed. Coupled with a serious time limitation as of this writing, the choice was made to disregard laboratory quantification of the rip current. A major goal of Session Two therefore is yet to be realized, but the reflection results weakly imply the presence of a current, and the dye experiments from Session One that are on film definitely show the rip current. Indeed, observations made at the time were that the currents were very mild, and did not get past the first bar. Unlike session one, session two was a continuous test between wave period changes. The intention was to extend the reflective peak as long as possible, thus keeping the induced circulation from previous periods going through the enhanced reflection. Once it was observed that circulation had dropped considerably, the runs began at rest from a high nonresonant period, and moved at slow increments toward resonance. Thus a test was made to see if reflection commenced upon return at the same place where it slacked when moving away from resonance. As said, the current magnitudes were so small that it was difficult to see a definite drop in reflection or current strength. Nevertheless, the results are presented in Chapter Six and still offer some useful lessons for future study. The bars were the same from session one, but some of the bars were buried slightly to accomodate the new conditions. Table 5.2 lists the heights and positions of the bars in this

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73 15 U) 10 0 .5 O > 0 0 5 10 15 20 time (sec) Figure 5.17: Sample time series of magnitude of current meter data, Session Two. part. For an overhead view of the barfield, refer back to figure 5.16. A side view of the barfield is shown in figure 5.18. The instrument setup consisted of everything being put on the cart. The three gages were positioned inline on the cart, along with the current meter. The overhead of the Table 5.2: Session Two bar heights and positions bar no. height (cm) x-position (cm) 1 6.25 436 2 7.00 500 3 11.00 565 4 12.50 631

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74 60 I 40 N 20 20 0 i1 1 1 1 i i i I I ,i 0 200 400 600 800 x (cm) Figure 5.18: Profile for Session Two. instrument array is shown in figure 5.19. With the points made earlier, the three-gage system was dropped in favor of the one moving gage to retrieve envelopes across the barfield in the x-direction. This concludes the description of Session Two. The next chapter describes the results obtained from the laboratory work.

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75 THREE CART GAGES BARFIELD Figure 5.19: Instrument setup for Session Two.

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CHAPTER 6 RESULTS AND CONCLUSIONS 6.1 Introduction The results from the work done in the past year and a half are presented in this chapter. The results are from laboratory data, where the methods that were used to obtain this data are outlined in Chapter Five. The other source of results is the numerical modeling effort. This consists of the wavecurrent model MCSIIERRY that as of this writing handles a limited set of conditions. The preliminary findings could be construed as inconclusive, but definitely suggest the need for more attention. The model should be running at full speed in the next year (1989). 6.2 Laboratory Work The data collected consisted of reflection data from wave gages, current meter data, bathymetric data from the profiler, and some data from the camera. The results will be presented in tabular form and pictorally where possible. 6.2.1 Reflection Results The Session One reflection data is presented in table 6.1. Each run represents a Static test, i.e., startup from a still basin. The averaged data over the middle transects is also plotted in figure 6.1, transposed upon the theoretical prediction for a one dimensional barfield without currents. This one-dimensional model is from Kirby (1987) that excludes currents. This has been placed over the data to simply give an idea of the effect the currents have on the data. The incident wave had a resonant period of 0.98 seconds, and was 7 centimeters in height. The waves arrived normally onto the beach, with a breakerline at x = 3 meters, refering to the profile in figure 5.8. 76

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77 Table 6.1: Kr and comments from Session One. period (sec) 1110 1175 1300 1365 comments on rip 0.980 0.357 0.667 0.529 0.362 weak rip current 0.910 0.347 0.189 0.235 0.091 no current 0.963 0.172 0.579 0.368 0.238 no current 1.050 0.368 0.550 0.368 0.538 rip current 1.110 0.294 0.692 0.458 0.514 significant current. 1.180 0.333 0.529 0.300 0.176 no rip current 1.110 0.333 0.474 0.647 0.428 again strong rip repeated. 1.085 0.290 0.412 0.579 0.375 slightly weaker than T=1.ll. 1.020 0.474 0.438 0.400 0.405 weak rip, dye is stagnant. 1.000 0.333 0.474 0.538 0.333 weak current 0.942 0.273 0.444 0.350 0.170 no current The predicted reflection from the one-dimensional model underestimates the reflection for the given bathymetry, for the envelope data describes 50% reflection. The data were not carried to sufficiently large periods so that the reflection clearly died away. From the films of the current, the magnitude increases in strength until a period of 1.11 seconds where it is quite strong, after which it dies off at 1.18 seconds. This would tend to suggest that while the waves are moving increasingly off no-current resonance, the rip current increases to keep the waves at resonance. This supports the hypothesis that the system seeks a resonant condition. From Session Two, the data were more abundant, but perhaps not as telling. Much of the setup was the same as in Session One, except that the depths over the bar field were less. This is due possibly to the state of the beach when testing began, in that the slope was less than in Session One. To offset this the bars were buried into the sand about 10% of their heights, which was typically about a centimeter. Also, due to the testing that had been going on before this work, some of the sand had been removed from the main testing region, so water depths could not be as high. In Session One, the testing depth was 55 centimeters, and for Session Two the depth was 50 centimeters. To further offset this,

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78 0.6 0.4 0.2 0.0 0.5 1.0 1.5 2.0 period (sec) Figure 6.1: Plot of IK with wave period, Session One data and theoretical prediction from 1-D model Kirby (1987).

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79 wave heights were decreased from 7 centimeters offshore to 5.5 centimeters offshore. This seems to be the main value that has the greates effect on the results, for a rip current never developed as strongly as the one appearing in Session One. The reflections were high, but it is conjectured that there was not enough available energy to drive the circulation. The reflection data are presented in three tables, where the first set refers to reflection with absolutely no discernible current throughout the course of the runs. The next set was made after digging sand out from the region in between the bars and the land. This allowed a weak current to form in some of the periods. The third set refers to runs that started at low frequencies and slowly approached the no-current resonant case. In all the testing in Session Two, the runs were continuous, without stopping the wavemaker. The averaged data from the first set with no current are tabulated in table 6.2, and illustrated in figure 6.2, which has the theoretical result transposed. Again, the theoretical values are from the no-current, onedimensional model of Kirby (1987). Three or less runs were done at each period to get an idea of the repeatability of the experiment results. Like the first type, the second type started at resonance and moved slowly outward. A weak rip appeared after removing some sand from the foreshore. The tabulated results are given in table 6.3, with the plot in figure 6.3. The two profiles from set one and two are shown in figure 6.4. From these plots, the data follows the theoretical fairly well, but the data drops away at later periods. The sand did not move onto the first bar at all, whereas in Session One, the first bar had to be repeatably uncovered. The third type started off resonance, then moved toward resonance continuously until reflection commensed and a weak current was excited. Table 6.4 gives the results, also plotted in figure 6.5. The difference between this test and the other is again not clearly evident, and may be due to the lack of dominance of the current. The one-dimensional algorithm from Kirby (1987) is over a variable mild topography.

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80 Table 6.2: Kr. Reflections for case of no discernable rip current, Session Two. Speriod (sec)__ Kr_ 1.014 0.419 0.515 0.423 1.020 0.381 0.418 1.030 0.433 0.503 1.050 0.507 0.471 1.065 0.511 0.508 1.080 0.530 0.505 1.100 0.494 0.526 0.525 1.125 0.516 0.481 1.150 0.470 0.454 0.473 1.175 0.518 0.493 1.200 0.425 0.386 1.225 0.323 0.341 1.300 0.385 0.310 1.350 0.374 0.414 1.400 0.396 0.372 1.450 0.386 0.367 1.500 0.352 0.339

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81 0.6 I I a a 0.0 04 a oD D 0 period (sec) 0 D 0 0.2 -/ 1.0 1.5 2.0 period (sec) Figure 6.2: Plot of Kr with period, data and theoretical for the case of no discernable rip current.

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82 Table 6.3: Kr measurements from case of a discernable shore rip current. Session Two. period (sec) K,r 1.014 0.425 0.464 1.030 0.391 0.442 1.050 0.521 0.536 0.539 1.080 0.466 0.510 0.483 1.100 0.483 0.503 0.514 1.125 0.482 0.521 0.482 1.135 0.447 0.486 1.150 0.532 0.520 1.160 0.461 0.490 1.200 0.402 1.250 0.349 1.300 0.349 1.350 0.231 0.289 1.400 0.215 1.450 0.257 1.500 0.259 1.550 0.323 1.600 0.310 1.650 0.272_

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83 0.6 S0 o 000 0 00 0 0 oo, 0 0 0 0.4 -\ 0.2 1.0 1.5 2.0 period (sec) Figure 6.3: Plot of K, with period for the case of a discernable shore rip current. Data and theoretical.

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84 0.6 0.5 0.4 0.3 0.2 0 2 4 6 8 x (m) Figure 6.4: Two profiles, dotted line referring to profile used in the second and third sets of testing, where a weak current formed. Table 6.4: Kr measurements for case of discernable shore rip current, moving toward resonance, Speriod (sec) IKr 1.650 0.216 1.600 0.330 1.550 0.323 1.500 0.316 1.450 0.316 1.400 0.356 1.350 0.249 1.300 0.302 1.200 0.436 0.460 1.175 0.464 0.459 1.160 0.518 0.512 1.150 0.529 0.520 1.135 0.487 0.556

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85 0.6 0.40.2 0.0 1.0 1.5 2.0 period (sec) Figure 6.5: Plot of K, with period for case of discernable shore rip current, moving toward resonance.

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86 It again excludes current effects. The plots do not show the kind of reflection extension that could occur because the currents were not as strong in the second session as the first. Thus, only so much tuning was possible. Some notes about the comparisons of the first and second sessions are worth noting to explain some problems with the execution. 1. The waves were slightly higher in the first session, thus involving more current driving energy. 2. The anchoring system kept the bars very still in the first session, while there was slight movement in the second. 3. Not enough time was allowed between period shifts in the second test to allow a proper current maintainance. The first reason seems the main cause of the decreased current generation. As is stated in the theory, if a wave field does not have enough available energy, the rip current will not be strong enough to achieve tuning of the nonresonant incident waves. Thus the efficiency of the bar field is comparable to the case with no current, and affects only closely resonant waves. For practical considerations, the most damaging waves will probably carry enough energy to drive a sufficient current, and the bar field will be effective when most needed. 6.2.2 Bathymetry This last section deals with the movement of sand during the experiments. There are profile data on backup tape from Session One that is presently not available to the author. Observations conclude that the current was sufficient to move sand over the barfield. In fact, it was neccessary to uncover the first bar periodically, which had been buried and which lessened the effectiveness of the barfield. In the second Session, the current was too weak to transport sand, and the profile shown in figure 5.18 was the same throughout the testing.

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87 6.3 Numerical Results After much tinkering, some results are available from the numerical model. As of the writing, however, the model requires another two months of work. Specifically, lateral mixing was left out because the time step limitations would make the model very slow, too slow for the given time allotment. Therefore there are circulation variations in the domain that would probably not exist if the cells were allowed to influence on another. However, it is thought that the overall pattern is similar for both cases. The results presented now are from the no-current wave model coupled with the circulation model. The process to achieve results was to run the wave model once and get the final radiation stresses. These terms were then multiplied by the startup function and inserted into the circulation model. The tests were run on a common bathymetry which is shown in figure 6.6. Without the wave-current equations to govern the wave field, the thesis cannot answer the question about whether the system tends toward resonance through numerical results. However, the no-current model can answer questions about how the magnitude of the current is affected by wave height. Therefore, three wave inputs were chosen for the bathymetry, where the period was the nocurrent resonant period. As a point of interest, the wave field and mean surface elevation that are calculated by the wave model for an input of a .03 meter wave are shown in figures 6.7-6.9. Then the following plots show how the current pattern varies as the amplitude is increased. Velocity times total depth is plotted in the figures, which describes the discharge across a vertical line. It is seen as expected that the rip current discharge increases as wave height increases. The circular cell over the barfield ends is thought to be attributed to a slow leaking of energy from the incident wave over the bar field. The action stated in Chapters Three and Four of removing the reflected wave probably account for this lost energy. The reflection coefficients remain the same for each wave height, being calculated simply by the maximum reflected wave amplitude in the grid divided by the initial incident

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88 0.1 0.0 -0.2 -I I I I I I -0.3 0 2 4 6 8 10 x (m) Figure 6.6: Bathymetry 1. Shoreward direction with increasing x.

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89 .Figure 6.7: for initial amplitude of .03 meters and perfectly resonant waves. Figure 6.7: IAj for initial amplitude of .03 meters and perfectly resonant waves.

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90 .03 Figure 6.8: IBI for initial amplitude of .03 meters and perfectly resonant waves, without currents. 1

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91 Figure 6.9: Steady state mean surface, 7.

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92 amplitude. Then considering a constant amplitude, the bars were altered in lateral length. The following plots 6.13-6.15 show that reflection increases as the bars decrease in longshore length, possibly due to focusing of the reflected wave over the bar ends. Finally, consider the bars being a lateral cosine, tapering down to the uniform bathymetry with half-lengths of 3 meters. The plot 6.16 shows that the final circulation pattern is comparable to the case of plot 6.15. The reflection is somewhat less at 0.774 as opposed to 0.88 for the short bars of uniform longshore bar height. 6.4 Conclusions and Status Report This concludes the results that can be presented at this time from the numerical model. What needs to be done is a comparison between the numerical model and the laboratory data by running the no-current model over the laboratory bathymetry. The condition of the wave-current model is that it has instabilities at the shoreline as the model is ramping to the full values. Before the wave is high enough to break onto the beach, the mean current is flowing in and possibly creating some negative terms in the wave model, like C.g + U. The model responds correctly, however, for it was tested over a bathymetry with no shoreline and an imposed current at the i = M grid row. The results from this test show that the wave is behaving as it should in the presence of a current, shoaling against the opposing flow. I regret that this chapter must end in this light, but am confident that the work will continue with more developments by other hands. Again, this project was designed to prove that when a bar field creates reflection and a low zone is created in the nearshore region, acurrent starts that drives the system closer to a maximum reflection condition.

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93 _, -.-4 ^-^^ ^^ -.---^ ---------. >.>> .I I I %%% S *4 % 4 4 4 4 4 4 4_44 4 .-* * e * * *Ifff 44,.*4 I -4 C * g< .. Iy I 'f of 0 .@ I I, -----^,.. ~..~\\ \. t **\4\I***i C 44 *\\. *. .4 C T* ******* ** ** 4^\ \~\\l 111 ** .759E-02 FROM -0.3000BE-01 TO 0.20000 CONTOUR INTERVAL OF 0.10000E-01 PT(3.38.15625 LAB I%% C9R Figure 6.10: No-current wave model, incident amplitude of .02 meters. K, = 0.81.

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94 ._ _.^..._J........l_. ,.. ., f ,, t: ZS'%I% t Z. % %te ute *% % % I t iti _till lee iI I I -lI t .-ttt%%t..t%%er|itl el tele te 1 .---* i i ,--fff^ it 11 1 i . <1 1 .* O -t ---2C T U R ~-.TE RL1T e.1t iL e F g r 6.1 Noer n IIIe I r ?-^-------~: 1,. ,",, a etlI lt It ItCC t g C ; t ., it -nl ill j1 .114E-01 FROM -0.30000E-01 TO 0.20000 CONTOUR INTERVAL OF 0.1000SE-01 PT(3.31= 0.15625. LAB0t 14 C-0 R Figure 6.11: No-current wave model, incident amplitude of .03 meters. K, = 0.81.

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95 *i-. % --%4r -..* --*-a..-> .. J -----^ k i * J 4 I II ____________9t 61 1 e .o ,t o p. ...,,...---.....ss....... ... ----------w*l*lm-a m il I 'a t I I 'o I -I --~----------\\\\\\ ,, • ntata, le.e[l,! I I I ,LI I I t t ti, "a',',,,,, ,, ,, ,, 4, t "* -*,, ., ,....%%S \\\~,~% % \\%\9 S I t I -l o .g I;t t I t I I t\ * ti .. .T'i i I t u t Ill it 9 If -: BI tl Ii t i illt I I Fti r It N w li i t 9a4. o h 0 8 tCtlIf I QI 9 ii i ** iI I l 11 I r lt ttIi t it 8 p il i ,-r::: ..s se .* .. ....zz ---meIfl-ffitt t l*tel4i0 ti t cC il t ::h,:....... ...6 ***.*4%• • a I !% ,,) ,l, S4* 9 ....... .-.. .. 9. 152E-01 FROM -9.30998E-91 TO 0.20909 CONTOUR INTERVAL OF 0.19000E-01 PT(3,31= 0.15625 LABA#:I%%R Figure 6.12: No-current wave model, incident amplitude of .04 meters. K, = 0.81.

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96 mo.| ., *r ^-n^ ^s tm %e * ** ** * * I *S • II itII I. ------. .. ....... .. *4 41 11 Ic 4 -*l (* -" ---------... .I .I *r. ^ --S- x -1 \ei l ., o a *, n *. --* *O -. -I\I 1 1 1 0* 4 tl 4 I i .9 .9 CI eet tett it o a --zz_ 4 • t it 4i t ... ......* t ....... ... t~ I1 I t I .I ...* *. .5 .• ifrrrn-rr rP r: f : 1 1 r 1 0.114E-01 FROM -0.30000E-01 TO 0.20000 CONTOUR INTERVAL OF 0.10000E-01 PT(3.31) 0.15625 LAB 0.114R Figure 6.13: No-current wave model, incident amplitude of .03 meters. K,r = 0.81, and bar half-length is 3 meters.

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97 --------------rrr------------------------------------------, r-------.. --------------------------------_Z:::::l lllj_" l -of I. .* ....%% %V ---. -44 -------% %9 a tl t 1 %4 4 % 4 I II I t I f I -I & itS IS S t. .t I I I I I I I I I I f I I I4% % % % 1 t t f I rrrrr i i 11 ii Ed i e 6 .1 4 N -r e w e m. K bar half-length is 2.25 meitiers .%44%%44~iiit :ii i tii iti i t,**, 0.1 iBE-Ol FROM -I.30000E-01T TO .20000 CONTOUR IHTERY?.L OF 0.10000E-01 PTI.3.3) 1.15625 LAB~f4h\IS~~Qt~Fr~R Filgure 6.14: No-current wave model, incident amplitude of .03 meters. Kr 0 .8437 aiid9 bar half-length is 2.25 meters.

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98 ----------------------------------------**........... .......... .o..*********.....o***eo ***o > > e * .p -* b .. .................... .... ................-, 1, ... ..--.\ ------------------1---< -r-" .................... i ........""............. -...i .1 .a ... F-rr--rr----------r\r\l * *a SS S~ % tt *te***** ,* t%**a SS 4Ittttt, tt* .4.o -, ,.s> llS... 4IIItI I I .. .**q. T m* *** •l I: -t : fl ::: *** : : : S :l 1nffi -H :: lidt f r i e l n I Figure 6.15: No-current wave model, incident amplitude of .03 meters. K, = 0.88, and bar half-length ---------SSS *is 1.7 .*............... .. ...... .,,,,,,,,,, ,,,,,,, e .,, ....e.. .l I. u*I S,,,~%,tt*tgtesWesese,,g1 0 -tt, ji tt i # -,. #. .o t* Figur6.1:No-currentwave modelincident ampitude ofif.03 meters. K, d 0.88, and bar half-engts 1.75meters. O l II l l 1 # m* iPrrPPPPP 11111111' 1 .' 1 1r 11 1 lie f ,I 8.127E-81 FROH -.3800E-81 TO 8.20000 CONTOUR INTERVAL OF 8.19888E-01 PT(3.3)1 8.15625 LABr,IfP-Fb R Figure 6.15: No-current wave model, incident amplitude of .03 meters. K, = 0.88, and bar half-length is 1.75 meters.

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99 .444jjj444 jj4 jjjj4j4j4j44 j4 jjj44 44 444444444444444j j 144444441444 It14 44. it 11, I T I I I ....[------------------.--:.I..... S----------......... ...... 11 -------------------S tt-rrtrtr. tI*tt T ,,,********* *************,.%***4*S***-t,**%*** ** e, f* a * -** ** ** * ** ** *** ** ** ** *k **-** %*&* f* %*t * *** ***t * 4 * * * ....................... .. II 5. ..... -s**%* ** s o .* ..* ..-... --**..* s\ \\\* I I I. I l*t**,.* ^^** ** ^ o ^\I\\* *\\ *I I 1 F -01 1 T 1 0 COTU N O I # 2 I i 1 l A1I E Fiure 6.16: No-curenttap e bas w tte iI i we f f 7 ....... *I .. \.r e ess''ab n' n ''4 \< r<' y ^* ** *" .< o *i "' i Z z is I II o l1I Ill wonw asI :;:*, ,-,t op .4 .0 .. .1 A -"-'',; ..i .. ..*a ^ 1 -/ i tat t 1 1 *. ., ,. -.. .... f P *F 0 t so 1 .110EFigure 6.16: No-current, tapered bars with .03 meter incident wave. K. = .774.

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100 BIBLIOGRAPHY Davies, A.G., and Heathershaw, A.D., 1984, "Surface wave propogation over sinusoidally varying topography.", Journal of Fluid Mech., vol. 144, 419-443. Dally, W.R., Dean., R.G., and Dalrymple, R.A., 1985, "Wave height variation across beaches of arbitrary profile.", Journal of Geophysical Research, vol. 90, 11,917-11,927. Ebersole, B.A., and Dalrymple, R.A.,1979, "A numerical model for nearshore circulation including advective acceleration and lateral mixing.", Ocean Engineering Report No. 21, Department of Civil Engineering, University of Delaware. Funke, E.R., and Mansard, E.P.D., 1980, "The measurement of incident and reflected spectra using a least squares method.", Proceedings of the 17th International Conference on Coastal Engineering. Goda, Y., and Suzuki, Y., 1976, "Estimation of incident and reflected waves in random wave experiments.", Proceedings of the 15th International Conference on Coastal Engineering. 828-845. Kirby, J.T., 1986a, "A general wave equation for waves over ripple beds.",Journal of Fluid Mech., Vol. 162, 171-186. Kirby, J.T., 1986b, "A model for gradual reflection of weakly twodimensional waves in a slowly varying domain with currents.", UFL/COEL-TR/060, Department of Coastal and Oceanographic Engineering, University of Florida. Kirby, J.T., 1987, "A program for calculating the reflectivity of beach profiles.", UFL/COEL87/004, Department of Coastal and Oceanographic Engineering, University of Florida. Kirby, J.T., 1988, "Current effects on resonant reflection of surface water waves by sand bars.", Journal of Fluid Mech., vol. 186, 501-520. Longuet-Higgins, M.S., and Stewart, R.W., 1964, "Radiation stresses in water waves; a physical discussion, with applications.", Deep Sea Research, vol. 11, 529-562. Longuet-Higgins, M.S., 1970a, "Longshore currents generated by obliquely incident seawaves, I.", Journal of Geophysical Research, vol. 75, 6778-6789. Longuet-Higgins, M.S., 1970b, "Longshore currents generated by obliquely incident seawaves, II.", Journal of Geophysical Research, vol. 75, 67906801. Mei, C.C., 1972,"A note on the averaged momentum balance in twodimensional water waves.", Journal of Marine Research, vol. 31, 97-104. Mei, C.C., 1983, The Applied Dynamics of Ocean Surface Waves, Wiley, New York, NY, 453-463. Noda, E.K., Sonu, C.J., Rupert, V.C., Collins, J.I., 1974, "Nearshore circulation under sea breeze conditions and wave-current interactions in the surfzone.", Tetra Tech No. TC 149-4.

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101 Sheng, Y.P., and Butler, H.L., 1982, "ADI procedures for solving the shallow-water equations in transformed coordinates.", ARO Report 82-3, Proceedings of 1982 Army Numerical Analysis and Computers Conference. Winer, H.S., 1988, "Numerical modeling of wave-induced currents using a parabolic wave equation", UFL/COEL-TR/080, Department of Coastal and Oceanographic Engineering, University of Florida.