• TABLE OF CONTENTS
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 Title Page
 Acknowledgement
 Table of Contents
 Abstract
 Introduction
 Background
 Approach
 Average bottom shear stress across...
 Model sensitivity tests
 Results of bar-trough model...
 Conclusions
 Bibliography
 Appendix A: Implicit model for...
 Appendix B: Excerpts from Dally's...
 Biographical sketch






Group Title: UFL/COEL (University of Florida. Coastal and Oceanographic Engineering Laboratory) ; no. 90/007
Title: Prediction of the formation and migration of bar-trough systems
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Permanent Link: http://ufdc.ufl.edu/UF00080976/00001
 Material Information
Title: Prediction of the formation and migration of bar-trough systems
Physical Description: x, 107 leaves : ill. ; 29 cm.
Language: English
Creator: Houston, Samuel H., 1957-
Publication Date: 1990
 Subjects
Subject: Coastal and Oceanographic Engineering thesis M.S   ( lcsh )
Dissertations, Academic -- Coastal and Oceanographic Engineering -- UF   ( lcsh )
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis (M.S.)--University of Florida, 1990.
Bibliography: Includes bibliographical references (leaves 96-99).
Statement of Responsibility: by Samuel H. Houston.
General Note: Typescript.
General Note: Vita.
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.
 Record Information
Bibliographic ID: UF00080976
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: aleph - 001613887
oclc - 23535968
notis - AHN8305

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Table of Contents
    Title Page
        Title Page
    Acknowledgement
        Page i
    Table of Contents
        Page ii
        Page iii
        Page iv
    Abstract
        Page v
        Page vi
    Introduction
        Page 1
        Page 2
        Page 3
    Background
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
        Page 11
        Page 12
    Approach
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
        Page 18
        Page 19
        Page 20
        Page 21
        Page 22
        Page 23
        Page 24
        Page 25
        Page 26
        Page 27
        Page 28
        Page 29
        Page 30
        Page 31
        Page 32
    Average bottom shear stress across the surf zone
        Page 33
        Page 34
        Page 35
        Page 36
        Page 37
        Page 38
        Page 39
        Page 40
        Page 41
        Page 42
        Page 43
        Page 44
        Page 45
        Page 46
        Page 47
        Page 48
        Page 49
        Page 50
        Page 51
        Page 52
        Page 53
        Page 54
    Model sensitivity tests
        Page 55
        Page 56
        Page 57
        Page 58
        Page 59
        Page 60
        Page 61
        Page 62
        Page 63
        Page 64
        Page 65
        Page 66
        Page 67
        Page 68
        Page 69
        Page 70
        Page 71
        Page 72
        Page 73
        Page 74
        Page 75
        Page 76
    Results of bar-trough model predictions
        Page 77
        Page 78
        Page 79
        Page 80
        Page 81
        Page 82
        Page 83
        Page 84
        Page 85
        Page 86
        Page 87
        Page 88
        Page 89
        Page 90
        Page 91
        Page 92
        Page 93
    Conclusions
        Page 94
        Page 95
    Bibliography
        Page 96
        Page 97
        Page 98
        Page 99
    Appendix A: Implicit model for cross-shore sediment transport
        Page 100
        Page 101
        Page 102
        Page 103
        Page 104
    Appendix B: Excerpts from Dally's (1980) Appendix C
        Page 105
        Page 106
    Biographical sketch
        Page 107
Full Text





















PREDICTION OF THE FORMATION AND MIGRATION OF BAR-TROUGH
SYSTEMS



By

SAMUEL H. HOUSTON


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


1990


1















ACKNOWLEDGEMENTS


I sincerely appreciate having been given the opportunity to work under the extremely

helpful guidance given by my advisor and friend, Dr. Robert G. Dean. His insight into

the problems and their solutions made the research presented in this thesis possible. I wish

to thank my committee members, Dr. Ashish J. Mehta and Dr. Y. Peter Sheng, for their

assistance in editing and producing the final copy of this thesis.

The Florida Sea Grant Program provided the sponsorship upon which this research is

based and this support is greatly appreciated.

Many thanks go to Dr. L. H. Lin and Don Mueller for their successful efforts in gathering

field data presented in this thesis at Beverly Beach and Bethune Beach. Helen Tweedell's

assistance in searches for references in the Coastal Engineering Archives and the drafting

of figures by Lillian Pieter are greatly appreciated.

I am thankful to Vernon S., J.J., Chuck B., Danny B., Vic A., Don M., and Roy J. for

endless fun on the Volleyball Court at the lab; my one escape into reality at noon and 3.

Special thanks go to Dr. J. Kirby for giving the Lackadaisical American Students (LAS)

their name; the label was worn with pride during the group's brief time in the Coastal

Engineering Department.

And those whom I thank for their friendship during the good times, as well as the bad:

Sam P., Jeff A., Steve P. (U.F.), Barry D., Steve P. (U.Va.), Steve G., Pat C., and Matt T.


1















TABLE OF CONTENTS


ACKNOWLEDGEMENTS .

LIST OF FIGURES .....

LIST OF TABLES .....

ABSTRACT .........

CHAPTERS

1 INTRODUCTION ....

2 BACKGROUND .....

2.1 Erosion Models ...

2.2 Models with Erosion,

3 APPROACH .......


. ii

. iv


. v


vi

1


Accretion, and Offshore Bar-Trough


Systems . .


3.1 Computational Models for Sediment Transport ................

3.1.1 Governing Equations for Implicit Model ................

3.1.2 Governing Equations for Model with Bar-Trough System ......

3.2 Testing and Calibration of Computational Model using Laboratory Data ..

3.3 Calibration Using Field Data ..........................

3.3.1 The Florida Department of Natural Resources Beach Profile Data

3.3.2 The Coastal Data Network .......................

3.3.3 Beach Profiles Collected by the University of Florida's COE Department

4 BAR-TROUGH EXPLICIT MODEL ........................

4.1 Finite-difference Equations ...........................

4.2 Implementation of Explicit Model . . . ...... . ...









5 THE AVERAGE BOTTOM SHEAR STRESS ACROSS THE SURF ZONE .33

5.1 Profiles with Erosion ............. .................. 33

5.2 Profile with Accretion .................... .......... 36

5.3 Profile with Recovery .................... .......... 44

6 MODEL SENSITIVITY TESTS ............................ 55

6.1 Sensitivity of the Model to the Transport Parameters . . ... 56

6.2 Sensitivity Tests of the Effects of the Lag Weights . . ... 63

6.3 Summary of Conclusions from the Sensitivity Tests . . ... 71

7 RESULTS OF BAR-TROUGH MODEL PREDICTIONS ....... ...... 77

8 CONCLUSIONS ................... ............... 94

BIBLIOGRAPHY .......... ...... ....... .. ... ... .. 96

APPENDICES

A IMPLICIT MODEL FOR CROSS-SHORE SEDIMENT TRANSPORT ..... 100

A.1 Finite-difference Equations ........................... 100

A.2 Using the Model ................... ............. 103

B EXCERPTS FROM DALLY'S (1980) APPENDIX C .............. 105

B.1 Introduction .................... ................ 105

B.2 Problem Formulation ................... ........... 105

BIOGRAPHICAL SKETCH ................... ........... 107














LIST OF TABLES


6.1 First Set of Sensitivity Tests . . . ..... ...... 56

6.2 Lag for T in Sensitivity Tests A1-A4 and B4 . . . ... 57

6.3 Lag for 7 in Sensitivity Tests A5-A7 and B3 . . . ... 62

6.4 Lags Used in Second Set of Sensitivity Tests . . . ... 71

6.5 Lag for T in Sensitivity Test B1 ...................... 71

6.6 Lag for Tb in Sensitivity Test B2 ...................... 76

7.1 Lag Weights Used for prediction of Bar Formation and Migration 78














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

PREDICTION OF THE FORMATION AND MIGRATION OF BAR-TROUGH
SYSTEMS

By

Samuel H. Houston

August 1990

Chairman: Dr. Robert G. Dean
Major Department: Coastal and Oceanographic Engineering

Improvement in the prediction of short- and long-term changes to beaches, such as

recovery following storm erosion, would be useful to coastal managers. Decisions which

those individuals and agencies must make in order to mitigate the impact of erosion caused

by a large storm, as well as the potential impact of sea level rise on the coastline, require a

greatly improved understanding. Attempts have been made in the past to develop numerical

models to predict changes to two-dimensional cross-shore profiles, and generally good results

have been obtained from these for short term erosion events. However, our current predictive

capability of the recovery of eroded profiles is far from adequate. The processes involved in

the recovery phase of beaches occur on a long time scale and may require months or years

to return the shoreline to approximately pre-storm conditions. In addition, the physical

processes involved in the recovery are much more subtle than those found in erosion. An

important feature in some beach profiles is the bar-trough system; this feature normally

forms during storm related erosion events and is involved in the recovery processes of the

beach profile. The model developed as a part of this thesis is an explicit computational

sediment transport model using bottom shear stress parameters to predict the formation

and evolution of bar-trough systems in two-dimensional cross-shore processes.

The computational model is based on two equations: 1) sediment conservation and 2)









sediment transport. One of the parameters in the transport equation is the bottom shear

stress associated with breaking waves. This forcing function is based on the transfer of

the momentum associated with the incoming waves due to breaking. This stress is directed

offshore in the surf zone, and is of a form that tends to generate a bar. Two other important

parameters included in the transport equation are the bottom shear stress due to nonlinear

waves and a slope related term, which represents gravity effects. The bottom shear stress

due to nonlinear waves causes sediment motion to be directed toward the shore and acts not

only to slow the seaward migration of the bar, but is responsible for the observed transport

of sediment from offshore beyond the breaking wave depth. This slope term provides a

stabilizing effect and tends to smooth irregular bottom features. The model utilizes a

procedure developed by Dally to calculate the distributions of wave height and mean water

level setup across the surf zone. The profile elevations are updated at the end of every time

step for each grid across the profile, and then the breaking wave height and setup model is

rerun with this updated profile.

The model is calibrated using large wave tank data in which a number of different tests

were conducted on beach profiles, including cases of erosion resulting in bar-trough system

formation. Examples of the model prediction for several of these cases are presented. The

results of these tests are encouraging for the prediction of the formation and migration of

bar-trough systems. Elements of this computational scheme could likely be used to improve

the performance of predictive models for short- and long-term beach erosion and recovery.














CHAPTER 1
INTRODUCTION



Extensive beach erosion along coastal areas in the central Caribbean and eastern Mexico

by Hurricane Gilbert and to the northeast Caribbean and the Carolinas by Hurricane Hugo

have emphasized the vulnerability of increasingly developed coastlines to storm effects.

Local officials in these areas were often faced with decisions about mitigation of further

losses of property where natural and manmade barriers were often destroyed by severe wind,

wave and storm surge effects caused by these hurricanes. In addition to these immediate

decisions, the officials often had to consider the amount of natural recovery which would

occur after the storm for possible changes in policies such as building set back lines or the

future availability of the beach for recreational uses. For example, in South Carolina after

Hurricane Hugo, there was emergency scraping of sand to the area of the destroyed dunes

using sand from the lower part of the beach profile to protect homes and businesses from

future storms. Additionally, emergency beach renourishment projects were funded for some

of the most heavily eroded and economically important portions of the state's coastline.

The purpose of this study has been to improve the predictive capability of numerical

models which determine long-term changes in beach profiles using wave characteristics and

water level data. If these improved models could be made available to those responsible

for protecting the fragile coastal environment, it would enhance their understanding of

coastal processes, thereby ensuring minimal adverse impact caused by human activity. The

individuals and agencies who make policies relating to coastal construction regulations,

beach nourishment, and shoreline stabilization structures currently base their decisions

on limited direct field observations pertaining to the magnitude and relative frequency of

occurrence of specific erosion events. These local planners and managers who are responsible







2
for maintaining the beaches as a recreational resource, as well as for storm protection,

would benefit from the capability of prediction of short- and long-term erosion effects and

interpretation of specific erosion events. Such a capability would allow a proper perspective

of each erosion event, so the officials could anticipate the timing and magnitude of the

recovery which would occur naturally after any particular storm. Using this information,

they could decide to implement various types of action (including no-action) alternatives to

mitigate the erosion effects on the beaches.

This thesis addresses one of the major problems in computational models for prediction

of storm induced cross-shore erosion effects on beach profiles. Typically, such models have

been capable of only simulating changes to the profile over short time periods on the order

of hours to at most days, and the problem of prediction of profile recovery is not usually

included. One of the most significant features normally found in beach profiles immediately

following a storm erosion event is the bar-trough system, which forms offshore as a result

of increased water level and wave conditions characterized by large wave heights and short

periods. Once these bar-trough systems form, they can dissipate substantial portions of

the wave energy, thereby reducing the erosion of the berm and dune. After the water level

and wave conditions have returned to their normal state following the storm, these bars are

significant elements in the recovery process of the beach profile. The sediment stored in the

bar-trough system is not only available to be returned to the shoreline during the recovery

process, but during the recovery phase the bar dissipates wave energy during subsequent

storm events.

The explicit cross-shore sediment transport model described here predicts bar-trough

system formation based on two equations: 1) sediment conservation and 2) sediment trans-

port. One of the main advantages of this model over many others is that the grid used

is based on constant intervals of distance across the profile rather than constant elevation

increments. The main forcing function in the sediment transport equation is the computed

time-averaged bottom shear stress due to breaking waves. The mean bottom shear stress







3
is calculated using the water depth and wave height at grid points across the profile. This

average bottom shear stress is directed offshore in the surf zone, so that it contributes to the

seaward transport of sand, and bar formation. A second parameter in the transport equa-

tion is the time-averaged bottom shear stress due to nonlinear waves. This forcing function

causes sediment to move toward the shore and tends to balance the tendency for offshore

bar migration, as well as to simulate the observed slow process of shoreward motion of sand

offshore from the breaker zone. There is also a slope related transport term which represents

gravitational effects in the transport equation. This parameter maintains stability in the

system and provides a mechanism to limit the growth of the sand bar. A breaking wave

height and setup model is used to at each time step to calculate the average bottom shear

stress parameters. After each time step the elevation contours of the profile are updated

at each grid point and then the breaking wave height and setup model is rerun to provide

new input values for the transport equation parameters. The migration of the bar-trough

system occurs due to updating the wave model to reflect the change in the location of the

breaker zone as the profile changes after each time step.

The calibration used for this model relies on large wave tank data, in which the erosion

and recovery characteristics of a beach profile were observed for different wave conditions.

Some of these tests included erosion of the profile, which resulted in bar-trough system

formation. Comparisons of the explicit cross-shore transport model profiles and the actual

observed profiles are presented. Field data of bar formation and migration, which were

collected to compare the model results with those observed in nature, are presented. These

comparisons are not included in this thesis.















CHAPTER 2
BACKGROUND



2.1 Erosion Models

In the past, several cross-shore transport models have been proposed; these models are

based primarily on geometrical considerations and will not be discussed here. Examples

of recent work involving onshore-offshore sediment transport include Swart's (1976) empir-

ical method using large-scale wave tank data. This procedure is formulated in terms of

numerous and complex empirical expressions, which can be applied in a relatively straight-

forward manner when programmed on the computer. This method was applied by Swain

and Houston (1984) in the field using storm erosion data collected at Santa Barbara, Cali-

fornia and near Oregon Inlet, North Carolina. They modified the Swart method to include

time-varying tide and wave conditions. Vellinga (1982, 1986) used laboratory wave tank

experiments to simulate erosional effects on Dutch dikes during storm conditions. Vellinga

(1983, 1986) developed a computational model for dune erosion based on empirical results

assuming a constant storm surge level, significant wave height, sediment grain size, and

an initial beach profile to simulate cross-shore sediment transport for a storm of five hour

duration. Distribution functions for parameters related to dune erosion such as maximum

storm surge level, significant wave height, median grain size, and profile shape were used

in the probabilistic approach developed by van de Graff (1983). Hughes' (1983) work was

based on physical models of beach and dune erosion, using a scale relationship between the

equivalence of fall velocity parameter and ratio of inertia to gravity forces in model and

prototype. Model distortion was allowed and a geomorphological time scale was included.

The methods of Vellinga, van de Graff and Hughes used variations in water level over time

and included the appropriate parameters for cross-shore sediment transport, but the meth-


I







5

ods cannot be applied as a general procedure for profiles different than those used in the

laboratory. Balsillie (1986) developed an empirical method that considers the average and

maximum expected erosion caused by a storm based on the time required for the storm

tide rise to occur raised to the 0.8 power and peak storm tide raised to the 1.6 power. This

approach correlated well with numerous field data and provides encouragement for future

use.

A three-dimensional coastal change model developed by Watanabe (1982, 1985) was

based in part on a cross-shore transport rate, which was a function of the Shields param-

eter to the 3/2 power. This model was based on wave and nearshore current effects on

beach profiles and used empirical results to determine transport direction. Moore (1982)

predicted changes in beach profiles using a numerical model, which assumed a transport

rate that was proportional to the energy dissipation from breaking waves per unit volume

above an equilibrium value. Cross-shore sediment transport occurred along the portion of

the profile affected by breaking waves and the predicted profile approached an equilibrium

shape over time if the wave characteristics were not changed. Kriebel and Dean (1985) de-

veloped a numerical method to predict beach and dune erosion, which was time-dependent

and based on time-varying observed or predicted water levels and waves during a storm.

The computational model was based upon the same transport relationship used by Moore

(1982), but Kriebel and Dean solve the equations governing cross-shore sediment transport

by an efficient numerical method. The two-dimensional onshore-offshore transport model

was verified qualitatively with erosion characteristics measured at various scales in labo-

ratory experiments. The numerical model results were found to compare favorably with

the observed beach and dune erosion for Hurricane Eloise, which impacted the panhandle

coast of Florida in 1975. Currently, the Florida Department of Natural Resources is using

a simplified version of this model to developed recommended positions of the Coastal Con-

struction Control Line. These models developed by Moore and Kriebel and Dean assume

profiles have monotonically increasing depths offshore, so bar formation and movement are









not included. Accretion onshore under certain wave conditions and water levels are also not

included in the model simulation.

2.2 Models with Erosion, Accretion, and Offshore Bar-Trough Systems

The erosion models described in the previous paragraphs, have the concept of an equi-

librium profile inherent in their development, so that given a sufficient amount of time,

these models will achieve an equilibrium profile given constant water level and wave con-

ditions. Such models may be termed "closed-loop" models as contrasted to "open-loop"

models which do not converge to a "target" profile (e.g. an equilibrium profile). While the

erosion of two-dimensional beach profiles has been examined extensively by a large number

of investigators, the recovery phase of shoreline development is less well understood. Ero-

sion events generally occur with time scales of several hours to more than several days, and

usually involve deposition of sediment offshore from the dune and berm in a shore-parallel

bar (see Fig. 2.1). During the period following erosion of the beach, recovery of the cross-

shore profile normally begins immediately, but the time required for the beach to return

to approximately pre-storm conditions is highly variable and usually involves a few weeks

or months for most erosion events. Kriebel, et. al. (1986) considered the current state of

knowledge about beach profile recovery and included new data based on laboratory and field

measurements. Their study indicated that even though the processes involved in recovery

appear to be much more simple and orderly than the conditions leading to erosion, there

are in fact many more subtle characteristics in the waves, water level changes, and bed form

conditions associated with profile recovery compared with those found in erosional events.

Hayes (1972) found that the most significant features appearing on the face of beaches

following a storm are berms and ridges, and offshore the main features are longshore or

breakpoint bars. The berm and beach ridge system are normally formed in the intertidal

region which has the runup limit as its highest elevation on recovering beach faces. These

features, which were first described in detail by King and Williams (1949), consist of one or

more ridges formed by accretion of sediment so that the landward slopes are at the angle


















---Normal (summer) profile
-- Storm (winter) profile


Monotonic inshore portion of storm profile


/ N
N


Figure 2.1: Sketch of typical barred or"winter" profile and unbarred or "summer" profile
(Dally, 1980).







8

of repose and the seaward slopes are convex. The sand available for forming these ridge

systems originates below the mean low water level in the inner surf zone near the shoreline

where a large amount of sand is deposited following a storm. The berm and ridge features

are very similar to the swash bar described by King (1972) which forms in the upper limits

of the swash zone where accretion was associated with those waves having small steepness

used in her laboratory experiments. The swash bar usually forms below the water surface,

but can increase in elevation up to the runup limit. The swash bar is different from the

breakpoint bar, which forms in laboratory wave tanks where steep plunging breaking waves

result in a convergence of sand at the breakpoint due to transport of sand toward the shore

from seaward of the breakpoint and in the offshore direction from inside the surf zone. There

is normally a trough near and shoreward of the breakpoint bar where a strong vortex-like

roller scours the bottom. King indicates that the breakpoint bar tends to move offshore for

large breaking waves and shoreward for smaller breakers and the bar height (trough to crest)

increases as the wave steepness increases. The crest of the bar was found in the laboratory

to remain below the water surface, such that the ratio of the water depth above the bar

crest and the height of the bar crest above the initial or smoothed profile is typically 2 to

1. Kriebel, et. al. (1986) note that observations in the field show that the mechanisms of

offshore bar formations and maintenance are more difficult to identify. Breakpoint bars have

been found to form during moderate storms which have plunging waves occurring over long

periods of time, while during the most severe storms these waves may be very steep with

spilling breakers, so no breakpoint bars are formed. Vellinga (1983) confirmed this obser-

vation during large-scale laboratory experiments and field measurements where no offshore

bar formed even though there was offshore deposition of sand. Wright, et al. (1979a) and

Wright and Short (1984) distinguished two types of intermediate breakpoint bars as being

either the longshoree bar-trough" or the "rhythmic bar and beach". Wright, et al. (1986)

observed for various beaches around the world that these two types of bar systems are more

common than either the fully dissipative (Wright, et al., 1982; Wright and Short, 1984) or


1







9

the fully reflective extremes (Wright, et al. 1979a, b; Wright and Short, 1983, 1984). The

fully dissipative beach is characterized by a wide surf zone and a large amount of turbulent

dissipation. Waves usually break 75-300 m offshore from the beach and dissipate much of

their energy across the profile, which normally has an upward concave shape toward the

shore. One or more bars may exist on this type of beach. The fully reflective beach is one

in which most of the incident wave energy is reflected from the beach face, which is usually

steep and linear. Most of the turbulence due to the breaking waves, which are normally

surging to collapsing, is confined to the immediate area of the beach face.

Beaches with the greatest variation over time develop the longshoree bar-trough" or

the "rhythmic bar and beach" state if there is originally a dissipative profile which is in

an accretionary sequence and which has bars that become more prominent as they migrate

toward the shore. There are some beaches in which the profiles have persistent intermediate

bar trough states throughout the year, such that there is an oscillation from one type of

intermediate bar system to the other. The formation and maintenance of bars is not well

understood and considerable research still remains to be done.

Some numerical models have been developed, which include offshore bars in the evo-

lution of beach profiles. Felder (1978), in a Ph.D. dissertation, presented a model which

generated not only monotonic profiles, but barred types as well. His work assumed plung-

ing breakers were present, which is not always valid. A numerical model, which assumed

that suspended transport is dominant in the surf zone was developed by Dally (1980) and

Dally and Dean (1984) to produce two-dimensional profile changes. The distribution of

breaking wave heights over the profile was computed in a numerical model and then used

to estimate the change in the profile. The sediment concentration within the water column

was assumed to have an exponential form. The model was able to approximate bar for-

mation closely using large wave tank data, but was unable to predict bar recovery. Larson

and Kraus (1989) developed a numerical cross-shore sediment transport model, which is

based on extensive correlations of wave, sediment, and profile characteristics. The beach







10
and nearshore is subdivided into four zones having different transport rate properties: (I)

offshore from the breaking zone, (II) breaker transition zone, (III) broken wave zone, and

(IV) swash zone. In Zone I, the net transport rate is approximated by an exponential de-

cay with distance from the break point, with a spatial decay coefficient (average value of

0.18m-') proportional to the ratio of grain size to breaking wave height during erosion.

An exponential decay with distance offshore showed good agreement with transport rate

data for Zone II, and here the spatial decay coefficient was about 0.20 of the value applied

to Zone I. The main part of the surf zone is covered in Zone III, and here the transport

rate is found to be closely related to the energy dissipation per unit volume. The transport

rate in Zone IV is dominated by swash dynamics, and cannot be developed directly due to

the lack of measurements of swash wave properties. The transport rate here is extended

linearly from the edge of the surf zone (arbitrarily set at a depth of 0.3-0.5 m) to the runup

limit. The portion of the profile covered by Zone IV changes if there is erosion and there

is also a simultaneous decrease in the transport rate with time. Steepness of the beach

face in Zone IV during erosion was limited by including an avalanching effect if the slope

exceeded a value of 28 degrees on average. The direction of transport onshore or offshore is

determined by the nondimensional ratio, Ho/wT, which relates, Ho, the deep water wave

height to the product of w,, the fall velocity of the sediment, and T, the wave period.

The model also uses a mass continuity equation to prevent any loss or gain of sediment

from the profile during simulations. Larson and Kraus (1989) determined the parameters

in their model empirically from the large wave tank and did not relate them directly to the

wave processes involved. The model was calibrated using large wave tank data and good

agreement was found between the observed and predicted profiles. The model predictions

were also compared with those predicted by the Kriebel and Dean (1985) model for different

wave and water level conditions. The Larson and Kraus model more realistically described

the profile at the dune toe when no bar was present in the profile and, because it allowed

the formation of bars, there was less erosion of the dune in a case in which offshore bars









were present. Bar movement was simulated in the numerical model and compared with

field data at CERC's Duck, North Carolina Field Research Facility. Here the transport rate

coefficient was assigned a smaller value than in the large wave tank cases. The model was

able to produce troughs or bars, but underestimated their size.

The erosion and recovery effects on beaches may be seasonal in nature. For example,

the "winter" profile shown in Fig. 2.2 generally has no protective berm on the beach face

and a bar system forms offshore as a result of storms causing erosion due to steep waves and

elevated water levels. The frequency of erosion events in the winter season is great enough,

so that any recovery which occurs is reversed as storm conditions destroy any restoration

of the beach profile. The "summer" profile plotted in Fig. 2.2 is normally smooth and

barless with the presence of a berm. This is the result of the typical mild wave activity

resulting in recovery after the winter season's stormy conditions. These seasonal variations

of the beach are well documented by investigators such as Shepard (1950) and Bokuniewicz

(1981). Research relating the beach changes to forcing conditions include the "wave power"

approach of Short (1978), the empirical eigenfunction analysis of Aubrey (1978) and the

empirical correlation approach of Fox and Davis (1971). Because of the descriptive nature

of most of these reports about these investigations of shorelines, rates of shoreline erosion

are not clearly related to wave characteristics or water level data. Other models include

only change in shoreline position without regard to general profile development. One such

model, described by Katoh and Yanagishima (1988), predicts changes in shoreline position

based on the energy flux of incident waves.







12









6 .l i l ail I


_Solid = Mar 1989
Dashed = Aug 1989














00. 200. 300. 00. 0. 600.





-10.
0. 100. 200. 300. 400. 500. 600.

Y (meters)



















Figure 2.2: Examples of observed barred (March, 1989) and unbarred (August, 1989) profiles
measured at Beverly Beach in Flagler County, Florida.


i














CHAPTER 3
APPROACH



3.1 Computational Models for Sediment Transport

3.1.1 Governing Equations for Implicit Model

A computational model, which simulates changes in two-dimensional beach profiles

over time, based on water level fluctuations and variation of wave characteristics has been

developed. The finite difference equations used in the model are described in Appendix

A. The computations in this model are based on an equation of sediment continuity and

a dynamic equation, which govern the movement of sand as a consequence of deviations

of wave energy dissipation levels from equilibrium. An implicit, double-sweep method is

employed to calculate changes in elevation contours in the profile. The model includes the

concept of an equilibrium profile, with a form found originally by Bruun (1954) and later

confirmed by Dean (1977) to be approximated as


h = Ay2/3 (3.1)


where h is the water depth, y is the offshore distance and A is a scale parameter that

is related to an equilibrium value of the energy dissipation per unit volume, D.. This

A parameter has been related empirically to the mean sand grain diameter (Dean, 1977;

Hughes, 1978; Moore, 1982). The numerical model assumes that with a sufficient amount

of time under constant water level and wave conditions, the two-dimensional beach profile

will eventually reach a dynamic equilibrium condition in a manner similar to that developed

by Swart (1974). The expression for offshore transport, Q, at any point in the surf zone is


Q = K(D D.) (3.2)







14

which was the dynamic equation used by Kriebel and Dean (1985). This expression relates

the actual, D, and equilibrium, D., levels of time-dependent wave energy dissipation per

unit volume in the surf zone, while the parameter K is a transport rate. As D increases

at a given point in the surf zone due to an increase in water level, there is a net offshore

transport of sediment. The expression for the conservation of sediment over the profile is

ah OQ
h 9Q (3.3)
9t ay

where y is the seaward directed coordinate. This continuity equation considers no longshore

transport gradients and the boundaries for the active zone are the maximum wave runup

onshore and the breaking depth offshore.

3.1.2 Governing Equations for Model with Bar-Trough System

The development of the berm and offshore bar has been found to be associated with the

dimensionless parameter suggested by Dean (1973) and later tested by Wright and Short

(1984)

Hbl(w,T) (3.4)

where Hib is the wave height, T is the wave period and w, is the fall velocity of the sedi-

ment. Wright et al. (1985a) used over 6 years of observations from a beach at Narrabeen,

N.S.W., Australia to determine empirically the beach states favored by certain values of

this dimensionless parameter. The longshoree bar-trough" state occurred in the range of

4.7 0.93, while the "rhythmic bar and beach" state was found to exist for 3.5 0.76.

The fully dissipative state was found be maintained for values > 5.5 to 6.0. The value for

Equation (2.5) was determined to be 4.0 1.5 at Eastern Beach in Australia by Wright and

Short (1984). Another important factor in the maintenance of the longshoree bar-trough"

and "rhythmic bar and beach" is the tidal range according to Wright, et al. (1985 a, b).

It was found that these bar-trough states exist for tide ranges of 1 m or less and they do

not form if the tide range is greater than 1.5 m. A berm was found likely to form in cases

where values of Equation (3.4) were less than 1.0, but these features, along with bars, could







15

be found in cases where the parameter was in the range from 1.0 to 6.0.

Another important aspect in the shapes of beach profiles, including those with bar-

trough systems, is the type of breaker which exists in the surf zone. The Shore Protection

Manual (1984) classifies breaking waves as spilling, plunging, or surging, according to the

manner in which they break. Spilling breakers break gradually and usually have "white

water" at their crest. The plunging breaker is described as one which curls over at the crest

such that the mass of water in the crest plunges forward into the preceding wave trough.

The surging breaker forms with the same characteristics as the plunging breaker, but the

wave surges up the beach before the plunging of the crest can occur. The breaking wave is

important to changes in the beach profile, because it acts to displace sediment shoreward of

the breakpoint. The scour which occurs under the breaking wave, especially under plunging

breakers, is analogous to the scour found in a sediment bed due to a downward directed

vertical jet (see Fig. 3.1).

Miller (1964) used field data to examine the velocity field under waves breaking onshore.

He found a narrow column of shoreward velocities existed directly under the crest of the

breaker. The size and distinctness of this column of onshore water motion was dependent

on whether the breaker was "Symmetric" (well defined narrow column, likely were plunging

breakers) or "Asymmetric" (less well defined and smaller column, likely were spilling break-

ers). The flow ahead of and behind each breaker was generally directed offshore, but in the

case of "Asymmetric" breakers, there was a layer of shoreward velocities. Dally (1980) notes

that plunging breakers generally dissipate most of their energy in the region just shoreward

of the breaker line, while spilling breakers characteristically have a slower dissipation rate

and a broader area over which dissipation occurs. His breaking wave model was "... at least

qualitatively correct in dealing with ..." these types of breakers. Stive and Wind (1986)

investigated the mean cross-shore flow in a two-dimensional surf zone and developed an

undertow model based on experimental and theoretical considerations. They found that

the result of the strong spatial decay of the wave motion after breaking, was equivalent to

























































Figure 3.1: Progressive scour of a sediment bed by a vertical jet (Brown, 1949).







17
a shear stress at the trough level, which caused a mean offshore flow in the water near the

bottom. Svendsen and Hansen (1988) considered the problem of incorporating cross-shore

circulations into numerical models, which predict wave heights and setup. One of the im-

portant forcing functions they investigated was the determination of the average bottom

shear stress, Tb. Their methods for determining this parameter were found to reproduce

accurately the measured undertow velocity profiles. They concluded that their method of

determining Tb could be used in comprehensive nearshore numerical models.

The approach used in this computational model to determine the location, volume,

and mobility of offshore bar-trough systems is based on the assumption that transport of

sediment along beach profiles is related to the momentum fluxes due to waves. Breaking

waves transfer momentum by exerting a force on the water column directed toward the

shore. The momentum force is defined by

F=- (3.5)
9y

where Sy, is the flux in the y-direction of the y-component of momentum due to waves. The

momentum force is not applied at the centroid of the water column. Instead, the moment

due to the applied momentum force applied about the center of gravity of the water column,

following Boreckci (1982), is

M = (1 + (3.6)
4 2h2

where H is the wave height and E = (1/8)pgH2 is the total energy per unit surface area in

the wave. Figure 3.2 shows a sketch of a water column with the moment due to the applied

momentum force. Balancing the moment with the applied shear stress leads to

1 OM
1 M (3.7)
h/2 0y

in which Tj- is the time-averaged seaward directed bottom shear stress due to the transfer

of wave related moment of momentum. If the applied shear stress is large enough and the

value of Equation (3.4) is greater than 4.0, then a bar would be expected to form in the

profile where there was sediment convergence under the breaking wave. A trough would







18

form in the scour region on the shoreward side of this bar, as a result of the turbulence

generated by the breaking wave. Once the bar forms, if the wave conditions and water level

remain the same, then the location of breaking can move offshore from the bar. This can

result in growth and migration of the bar to some distance offshore, until the turbulence due

to breaking waves no longer mobilizes the sediment or until there is a balance of transport

components thereby resulting in equilibrium.

The computational model used to predict the creation and migration of a bar-trough

system is based on a sediment continuity and transport equation. The conservation of

sediment model is again Eq. (3.3). The transport or dynamic equation for the explicit

model used to calculate changes in the depth contours is

Q = K1~ + K2I + IK3ah/Oy (3.8)

The forcing functions on the right hand side of this equation include Y', described briefly

above, as well as the mean bottom shear stress due to nonlinear waves, To, and the slope

controlling or gravity parameter, Oh/9y. These parameters will be described more fully in

Chapter 4 of this thesis.

3.2 Testing and Calibration of Computational Model using Laboratory Data

Data from beach profiles simulated in large laboratory wave tanks with various water

levels and wave characteristics are used to test and calibrate the model under a number of

conditions. The waves produced in the large wave tanks are monochromatic, thus grouping

of waves and long period wave motion were not present. Larson and Kraus (1989) decided

in their recent sediment transport model tests that this characteristic of large wave tank

data was an asset, by "allowing focus on transport produced solely by short-period incident

waves without ambiguities." These authors also note that these regular waves allow the

investigation of the effects of breaking waves on the beach profile, since they believed the

main cause for changes to the beach profile, including bar formation, were due to breaking

wave conditions. By focusing on the understanding of breaking waves, they believed that

one could begin to understand the "other possible contributing processes, since in nature all
















C

H/2
-- --- Crest level



h2 -- ---- -- Trough level

h/2


IE(1-H)

h/2 i H
2 2h














Figure 3.2: Schematic of the moment due to the wave-related momentum force applied to
the center of gravity of the water column.






20
forcing agents act concurrently and their individual contributions are difficult to distinguish.

Firm knowledge of one will aid in understanding the others."

Large wave tank experiments were conducted by the Beach Erosion Board of the U.S.

Army Corps of Engineers at Dalecarlia Reservation, Washington, D.C. during the periods

1956-1957 and 1962 by conducting movable bed modeling tests (Saville, 1957; Caldwell,

1959; Kraus and Larson, 1988). In these tests the beach profiles normally commenced with

a uniform slope of 1:15 and the change in each profile was monitored for up to 120 hours

for various wave heights, periods, and sediment fall velocities. The two different sediments

used for the tests had median diameters of 0.22 mm and 0.40 mm. There were as many

as 10-15 surveys of the profiles during each large wave tank test and these were made at

various time increments. Normally the time increments were increased in steps from 1 to 5

hours at the beginning of a particular test, but the increments often were set at 10 hours as

the profile approached equilibrium; the distance interval used for the profile surveys was 1.2

m. Wave height measurements were made with a step resistance gage located at the toe of

the beach and the accuracy of these measurements was reported to be about 0.03 m. The

wave period was set very accurately due to the large stroke length used by the wavemaker

and fixed gear ratio.

3.3 Calibration Using Field Data

Field data to be used for testing and calibrating this model were collected at two

locations in Florida. The descriptions of the field data available for these purposes are

presented in the following sections.

3.3.1 The Florida Department of Natural Resources Beach Profile Data

When the Coastal Construction Control Line was adopted by the State of Florida in

1972, The Florida Department of Natural Resources (DNR) began to establish a monu-

mented baseline along those portions of the Florida coastline having sandy beaches. De-

tailed profile measurements were taken at each monument (spaced approximately every 305

m) with every third or fourth profile including offshore bathymetric measurements of the






21
profile; measurements have been repeated approximately every 10 years. If a large erosion

event occurred as a result of a storm over a portion of the monumented coastline, lim-

ited profiles were measured in the affected area. Using this data set, short- and long-term

changes in Florida's beaches could be determined.

3.3.2 The Coastal Data Network

The Department of Coastal and Oceanographic Engineering (COE) at the University

of Florida implemented the Coastal Data Network, which consists of wave and water level

gages at various locations in the coastal waters of Florida. Currently, there are nine of these

measurement stations offshore of the state's coastline. These nine stations are shown in Fig.

3.3.

3.3.3 Beach Profiles Collected by the University of Florida's COE Department

The University of Florida's COE Department collected beach profiles during various

time periods at two locations on the Florida coastline. These survey sites included Beverly

Beach in Flagler County and Bethune Beach in Volusia County.

The profiles at Beverly Beach (see Fig. 3.4) were measured along a portion of the coast-

line with a single 444 m long seawall. The Florida DNR monuments in the vicinity of the

profiles here are R-60 to R-63.The edge of the seawall is the baseline for the beach profiles;

three profiles are surveyed at the sea wall with one at each corner and one approximately

in the center. Additional profiles are measured about 150 m north and south of the seawall.

Dunes on the north side of the seawall are not very large due to this being a commercial

and residential area, while immediately south of the seawall there are large natural dunes.

The beach face slope is about 1:8 below the seawall and dunes. A berm containing cuspate

features periodically exists here and offshore bars following storm conditions are evident

during some portions of the year. The sediment here consists mainly of fine wind-blown

sand in the dunes, while the portion of the profile below the dune line down to the surf zone

consisted of some fine sand mixed with larger amounts and sizes of shell fragments. Offshore,

the ratio of shell fragments to sand decreases rapidly, especially in the region of the bar,





















7 1 OLEY-1


St. Mary's Entrance
S 1983


Marineland
* 1977


Steinhatchee
0


GULF


4,


Cape Kennedy
* 1977


N Clearwater NE) s Vero Beach
1978 0 -, 1.98
A IATE HAROEEI QCHE ST.
q c--' I-- --H ~U ; ILUCIE.
Venice .LAAEUAIII1
SOKEECHIO---
CARt0"n...L" EE
- - LCd4A.3-
LLtL HE Y I PAL U ACH Palm Beach
V\- ; ---/ w 1979
COLL:EA BROIHAU

)*--i g Miami
L.OROI MDo 1977












Figure 3.3: Location of Coastal Data Network Stations maintained by the University of
Florida's COE Department.


44


%
V
























































Figure 3.4: Map of the DNR profile locations for Flagler County in the vicinity of the
Beverly Beach study area.






24

where fine sediments dominate. Profile measurements, including offshore bathymetry to 10

m depth, began in August, 1988 and were conducted at time intervals of approximately

every three months. In March, 1989, following a late winter season storm, a large offshore

bar-trough system was found to have formed. Frequent wading surveys were made every

four to six weeks to determine the migration of this offshore bar toward the shore during

the spring and summer months. This bar was found to have merged with the shoreline in

the August, 1989 profile. Large waves associated in part with Hurricane Gabriel in early

September, 1989 also produced a smaller offshore bar-trough feature. The November 1989

profiles showed that a bar-trough system was still in existence.

Profiles from another study at Bethune Beach (see Fig. 3.5) were also available for

the field tests of the computational model. This seawall, which is approximately 277 m

long, is at a Volusia County park about twelve km south of New Smyrna Beach. The six

profiles measured here use Florida DNR monuments R-201 to R-204 as reference points.

Again three profiles are made at the seawall, with one at each corner and one at the center.

There are two profiles north of the seawall and one south of the seawall. The beach north

of the sea wall is recessed and an abandoned highway is severely eroded, while south of the

seawall boulders form a riprap barrier between the shore and the ocean, and an abandoned

highway also exists. The beach face here is relatively flat with a slope of approximately 1:20

and berms are not normally major features, while offshore bars exist here at various times

during the year. The sediment on the beach face consists of mostly fine sand with some shell

fragments, but offshore there is almost exclusively fine sand. Beach profile measurements

with bathymetric surveys to about 10 m depth began here in May, 1988. Again, as in the

case of Beverly Beach, these surveys were performed approximately every three months.

Intermediate wading surveys were also made at shorter time intervals after May, 1989 to

document the movement onshore of a large bar feature, which probably formed at the same

time as the one at Beverly Beach during the March, 1989 storm. Surveys in August, 1989
























































Figure 3.5: Map of the DNR profile locations in Volusia County in the vicinity of the
Bethune Beach study area.







26

indicated that portions of this bar had also merged with the shoreline, but by November,

1989 a new bar-trough system had formed further offshore.

























TIDE LEVEL

\T11.1 111 7 +1
\ MEAN SEA LEVEL


V h h hl


YI
Y-- 0y + 1

----- ---s^
YS h=Ay23














Figure 4.1: Model representation of beach profile showing depth and transport related to
grid definitions; the cross-shore grid elements are at constant width.







29

This average bottom shear stress term is nonlinear and is difficult to work with due to

the absolute value sign. According to Dean (1987), the shoreward shear stress increases

substantially with shallower water and higher waves. The explicit model uses the following

expression for this average bottom shear stress term:

= pf 0.09Lo (Hi)2
S 8 di T2 )

where Lo = gT2/2ir is the deep water wave length and T is the wave period.

4.2 Implementation of Explicit Model

This explicit model requires a two-dimensional profile to be input initially, with constant

offshore grid size, Ay. This input profile (see Fig. 4.1) must contain a sufficient number of

grids over the entire region from onshore in the dune or berm region to offshore beyond the

closure depth. Before transport computations can be made in any time step, the wave height

and wave setup/setdown model developed by Dally (1980) must be implemented. Using this

breaking wave model, a realistic calculation of the average bottom shear stress parameters

can be made in the explicit model. This model also realistically simulates the reformation

of the wave in the trough on the landward side of a sand bar under certain conditions after

the wave breaks over the bar. The wave setup and setdown, 77i, associated with breaking

waves is also included in this breaking wave model. Therefore the instantaneous value of

depth, di = hi + vii, can be determined for each grid in the profile.

Given these values of Hi and di at each contour at a particular time, the values of

(b)i, (0.)i, and (8h/Oy)i can be calculated for each grid. The momentum induced shear

stress is based on a local balance and thus does not accurately reflect the spreading due to

the breaking wave over some distance toward the shore from the breakpoint. In addition,

the transport of suspended sediment offshore from the breakpoint must be considered in

the computation. The model includes a weighting function which distributes the values

of the average bottom shear stress over adjacent grid cells. The weighting factors are not

symmetrical with the values of (T)i, which weight the onshore values slightly greater than

those for the offshore portion. The value of T-o is decreased across the profile, by including








following equation:

EOH EOh H2 E 2H 2 Oh
-= (- + )(1+ )+ (H H2h ) (4.2)

which can be used to calculate time-averaged forces acting on the sediment in the bed.

Expanding this equation and collecting terms leads to the following simplified form of the

average bottom shear stress:

pg 1(Hi)2 2h rr9 (H )]
(T)i= [= d()2 di ) + Hf i -i(a + (4.3)
8 2 (d7) 9U 09 (di)2

This form of the equation allows the average bottom shear stress to decrease in a uniform

manner toward the shore, because the squared wave height terms in the numerator decrease

at approximately the same rate as the squared depth terms in the denominator. Other

forms of this expression were found to become inappropriate in the portion of the profile

where the depth approached zero as the instantaneous water level intersected the beach

profile.

Dean (1987) observed that in addition to the tendency for sediment motion due to

waves to be offshore, there must be a net shoreward force on the bottom sediment. If not,

there would not be an upward slope in beach profiles in the landward direction. Therefore

the "equilibrium profile" is the result of a balance between landward forces and seaward

forces, including gravity. The average bottom shear stress, which retards the motion of a

fluid in unidirectional open channel flow is expressed in terms of a quadratic friction law

= U2 (4.4)
8

where f is the Darcy-Weisbach friction factor and U is the velocity of the fluid. This

equation was developed by using dimensional analysis and the values for f were derived

empirically. In our case, the flow is oscillatory under waves, so that the fluid reverses

direction, as well as the bottom friction. By introducing an absolute value sign in Eq. (4.4),

the result is

= I U U (4.5)
8














CHAPTER 4
BAR-TROUGH EXPLICIT MODEL



4.1 Finite-difference Equations

The explicit computational model for cross-shore sediment transport uses finite dif-

ference forms of the continuity equation (Eq. 3.3) and the transport equation (Eq. 3.8).

The portion of the profile over which these equations are applied is represented as uniformly

spaced offshore grids. Unlike many previous models, this representation allows the model to

produce offshore bar-trough systems, because it does not require monotonic depth increases

offshore. The two-dimensional profile for this explicit model starts at the berm or dune

and continues offshore to well beyond the maximum breaking depth. The distance offshore

can be represented by the contour location, yi, which is referenced to an arbitrary baseline

located in the dune. To determine the depth at any contour, each elevation contour, hi,

must be considered along with the mean water level, r7i, which includes tide, storm surge,

and wave setup and setdown effects. The total depth can be represented as di = hi + 7i.

The continuity equation in finite difference form using a space-centered finite difference

method, is expressed as
At
Ahi = A(Qi+l Qi) (4.1)

where Qi represents the time-averaged sediment flux. The terms in the transport equation

(Eq. 3.4) can be considered individually. The most important term in the transport equa-

tion used to determine the location, volume, and mobility of offshore bars is based on the

momentum fluxes due to waves. Breaking waves transfer momentum by exerting a force

on the water column directed toward the shore. By substituting Eq. (4.7) into Eq. (4.8),

the average moment can be balanced with the average applied shear stress, yielding the






31
the seaward directed component of bottom shear stress due to nonlinear waves using the

expression

(Qlp)i = K2[{-(m)i + 2w6 sin i} rcr] (4.7)
3

if {} > re; (Qlp)i = 0.0 otherwise. The immersed specific weight of the sediment is w, 6

is the sediment diameter, and Oi is the profile slope. The component of the shear stress

directed toward the shore is


(Qin)i = K2[{3( -)i + 3w6 sin i} + r,] (4.8)

if {} 10 r,; (Qln)i = 0.0 otherwise. These nonlinear waves are responsible for most of the

transport of sediment from offshore, but some offshore sediment transport is possible under

these waves. Therefore, the value of (-)i is multiplied be a factor of "3" in Eq. (4.8) as

opposed to the factor of "-1" in Eq. (4.7). The critical bottom shear stress is

2
r, = -pg(s 1)6 sin cr (4.9)
3

where 0, is the critical slope angle, which is a function of the sediment diameter, and s is

the ratio of the mass density of sediment to the mass density of water (p,/p = 2.65). For

S= 30 and 6 = 0.22mm, the value of -re = 1.185Nm-2.

The volumetric transport over one time step is


Vi = At[Ki(,)i + (0.3(Qip)i + 0.7(Q.);i) + K3(Oh/Oy)i] (4.10)


which when input into Eq. (4.1) yields the equation for the change of depth at any grid in

the profile:

Ahi = v--I (4.11)
Ay

The portion of the profile over which calculations are made is from onshore at the upper

limit of the setup to offshore at the index, imaz 1, where imax is the total number of grid

points in the profile. The two boundary conditions applied to this model are: 1) V, = 0.0,

where is is the index of the instantaneous upper limit along the profile of the water level







32
setup and 2) Vima-i, = 0.0.The elevation contours are updated each time step the explicit

model is run, and in the next time step this new profile is used as input into the Dally (1980)

model to calculate the new wave heights and water level setup/setdown. This model output

is then used to calculate the new values of bottom shear stress for use in the transport

equation again. Stability in the explicit computational model is maintained by limiting the

ratio At/Ay. Equation (3.8) shows that Q oc K3(Oh/Oy). The diffusion equation

ay a2h
= gf3 (4.12)


is found by sustituting Q = K3(oh/9y) into Eq. (3.3). The stability requirement for this

explicit model is IKAt/((Ay)2 < 0.5.


1













CHAPTER 5
THE AVERAGE BOTTOM SHEAR STRESS ACROSS THE SURF ZONE



To demonstrate the effects of the average bottom shear stresses, f and 7, in bar for-

mation and migration, examples of the variations of their values across the profile and in

time are presented. The shear stress depends on changes in beach profile shape, variations

in the wave conditions, and the water level. The examples utilized to show variations in

the calculated average bottom shear stress across the surf zone here are from the large wave

tank tests documented by Kraus and Larson (1988). The breaking wave and water level

setup/setdown computations for each case were determined using the Dally (1980) model.

Four examples are included: 1) three profiles with erosion, 2) a profile with accretion, and

3) a profile with recovery.

5.1 Profiles with Erosion

An example of the large wave tank test with erosion and offshore bar formation is Case

400. This laboratory test used an average breaking wave height of 2.3 m, wave period of

5.6 s, a constant water level, and sediment size of 0.22 mm (w = 0.031 ms-x). Calculations

of the average bottom shear stresses were made at grid intervals of 1.22 m for Case 400 and

the other tests described below. It is stressed that in the following, the results are presented

for the measured (not predicted) profiles. Predictions of the profile evolution are presented

in a later section. Figure 5.1 shows the values of Hb/WsT across the initial profile (planar

with a slope of 1:15), as well as the values of T" and To. The maximum value of Hb/w,T,

located at a distance of 30 m offshore, corresponds to the location of the initial breaking

wave. Note that the maximum value of this ratio is approximately 13, indicating that it

is well above the approximate value of 4, which was found in field experiments by Wright

and Short (1984) to be a critical value of Hb/wsT in the formation of offshore bars-trough







34
systems. The calculated 7 is very small offshore from the breaking wave at about 30 m, but

at that point it increases sharply to 700 Nm-2. Toward the shore, the average bottom shear

stress gradually decreases and approaches zero as the breaking wave diminishes across the

swash zone. The average shear stress due to nonlinear waves, which is always negative, has

the greatest magnitude, -3 Nm-2, at the break point. One hour later, a bar has formed and

the break point has moved offshore to about 37 m, as shown in Fig. 5.2. The maximum T-

has also moved offshore with the breakpoint and now has a maximum value of 800 Nm-2,

while the value of 7o remains approximately -3 Nm-2. After an additional two hours of

wave action, the profile in Fig. 5.3 shows the bar and breakpoint have continued to move

offshore, but at a slower rate, as evidenced by the breakpoint now being at a distance of

42 m. The maximum magnitudes of the average bottom shear stress parameters have also

shifted offshore, following the breakpoint. Figure 5.4 shows that after 10 hours, the Case

400 profile bar crest is located at 42 m and the break point is at 48 m, while the largest

magnitudes of T' and To are both at approximately 45 m. Twenty hours later the offshore

migration of the bar and breakpoint is shown in Fig. 5.5 to have continued to distances of

45 m and 52 m respectively. The maximum average bottom shear stress values are at about

50 m. Note that the offshore slope of the bar has a ratio of approximately 1:7.

The evolution of profiles in Case 400 can be compared with those in Case 401, in which

the average wave conditions were approximately the same, but the sediment size was 0.40

mm (w = 0.055 ms-1). Figure 5.6 shows that the only parameter with a significant change

is the ratio Hb/wT,which has decreased significantly to a maximum value of 7 in this case,

compared with the initial profile data shown in Fig. 5.1. The profile one hour later, shown

in Fig. 5.7, again develops a bar, which is smaller than that found in Case 400. The

bar crest is at about 25 m, while the breakpoint is at 33 m. The maximum T" increased

significantly to 1350 Nm-2 in one hour, and this maximum is located at about 26 m. The

value of T is -4 Nm-2 at 26 m. After a total of 30 hours of waves in the large wave tank

for Case 401, the bar crest is located at 37 m, the breakpoint is at 43 m and the maximum


















-2'
20.

16.

12.

8.
3


0.

-4.
N
1 -8.

12.

-16.

-20.
1.


0.


-1.


5.


-15. -5. 5. 15. 25. 35. 45. 55. 65. 75.


-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.
distance (m)


24.


20.


16.


12.
--4

8.


4.


0.


Figure 5.1: Initial profile and the ratio Hb/w,T for Case 400 shown in lower panel. The
upper panel shows the calculated values of T (units = 100.0 Nm-2), which is always
positive, and T (units = Nm-2), which is always negative.


N
E

-c
o

s


p- I- I- 1 I I I -1I 1 1 1 I I I
Taub












I ,-- I -, I i I l I l i I ,- I- -




......I .i ..........a. .







36

magnitudes of the average bottom shear stresses are at about 38 m (see Fig. 5.8). The
-2
maximum value of 7 is observed to have decreased to 900 Nm-2

The other large wave tank test with erosion is Case 500, in which multiple bars formed

over time. In this trial the average breaking wave height was 1.9 m, the wave period 3.8

s, and the sediment size was 0.22 mm (w = 0.031 ms-'). The initial profile and values of

the ratio Hb/wT are shown in Fig. 5.9, showing that for Case 500 the initial breakpoint

was at about 25 m with the maximum value of the ratio at approximately 15, which is well

above the critical value of 4. The maximum value of T is about 420 Nm-2 and the largest

magnitude of is 2.5 Nm-2. Figure 5.10 shows that a small bar with a crest at 28 m has

formed in the profile, and the breaker location has shifted offshore to 31 m. The maximum

average bottom shear stress is about 700 Nm-2. After thirty hours, the break point has

moved offshore to 48 m and the bar crest is at 46 m(see Fig. 5.11). However at this point

a second bar has formed near the shore at about 13 m. This results in some increasing

bottom shear stress here, but the maximum value of 7 is approximately 400 Nm-2 at 48 m.

Case 500 was run for a total of 100 hours and the resulting profile and shear stress values

are shown in Fig. 5.12. This plot shows that the second bar shoreward of the original

bar, which shifted to 56 m, has developed further and is located at 17 m. Not surprisingly,

two breakpoints are also observed to have formed in the vicinity of each bar. The average

bottom shear stress, T is shown to have become negligible offshore at the first bar, while a

new maximum of about 450 Nm-2 has developed under the second breakpoint.

5.2 Profile with Accretion

The example of the large wave tank data with accretion is Case 600 in which the average

breaking wave height was 1.0 m, the wave period was 16.0 s, and the sediment size was 0.22

mm (w = 0.037 ms-1). The initial profile and ratio Hb/w,T are shown in Fig. 5.13. Again

as in the previous cases the initial profile is planar with a slope of 1:15, but unlike the cases

with erosion, the maximum value of the ratio is about 3 at 25 m. This value of Hb/wsT is

less than the critical value of 4, so a bar would not be expected to form. Figure 5.14 shows

















-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.
20.

N 16. ---- Tou

| 12. -- o
0
o
Z 8.

j 4.
0. --.-



C -4.
N


-12.

-16.

-20. I I i I i I I
1. 24.

S \---- depth -
0. -20.
Hb/T---- H)
-1. 16.
2 -2
S-2 12.
0. -4

-3. 8.


-4. 4.


-5. 0.
-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.
distance (m)


Figure 5.2: Case 400 at 1 hour.

















-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.


-5. 5. 15. 25. 35. 45. 55. 65. 75.
distance (m)


Figure 5.3: Case 400 at 3 hours.


--Toub

--






---- ,i,-







'- t - -


20.

N 16.
S
T 12.
o
o
S8.

O 4.
0.

- -4.
N
I
-8.

i-12.

-16.

-20.
1.


0.


-1.


5-2.
0.

-3.


-4.


24.


20.


16.
C1

12. Z
-4

8.


4.


0.


-5. '- I
-25. -15.

















20.

N 16.
I
E
? 12.
o
Z 8.

1 4.

0.







16.

-20.
1.









5 -4.
0.
-20.






*o

-30.







-4.


-25 -5. 5. 5. 15. 25. 35. 45. 55. fii


-5. I- I -- -
-25. -15. -5.


5. 15. 25. 35. 45. 55. 65. 75.
distance [m)


Figure 5.4: Case 400 at 10 hours.


Tcub

Tou








- I I


24.


20.


16.
0-r

12.
-4

8.


4.


0.


-25. -15. -5.


'*

















i. -15.


-5. 5. 15. 25. 35. 45. 55. 65. 75.


-5. 1 1 1 --I- I I I I
-25. -15. -5. 5. 15. 25. 35.
distance [m)


-25
20.

N 16.
E

1 12.
o
8.



0.

N
i-8.
o
|-12.

-16.

-20.
1.


0.


-1.
E

5 -2.
C)

-3.


-4.


Figure 5.5: Case 400 at 30 hours.


I I I I I I -i I I I i
Toub

Tou






SII I I


I I I 24.
----- depth -

20.
---- Hb/ (wwT)

16.
cr

12. 7
/ "

8.



4.


S 0.
45. 55. 65. 75.


-







41







2-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.
20. I I i ii |

. --- Touo
I 12.
0




- -4.





-16.
-20.



O. 20.


-1. 16.
0. ~ ~ ~ ~ ~ ~ ~ ~ --- de---pth-------------- 20.






5 -2. 12.
a)

-3. 8.


-4. 4.


-5. i0.
-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.
distance [m)


Figure 5.6: Case 401 initially.


















-5. 5. 15. 25. 35. 45. 55. 65. 75.


-5. I I I-'L1,
-25. -15. -5.


5. 15. 25. 35. 45. 55. 65. 75.
distance [m)


Figure 5.7: Case 401 at 1 hour.


. i I i I I- I I- I- I I- I I I I I I I
Toub

I I Tou,



I t
- -=== ^ __:


20.

N 16.
E
I 12.
o
o
: 8.

| 4.

0.

--4.
N
I
1 -8.

I12.

-16.

-20.
1.


0.


-1.


5 -2.
0.

-3.


-4.


24.


20.


16.
crI
12.
,--

8.


4.


0.


-25. -15.






















20.

N 16.
I
E
12.
o
o


4.


0.

-4.
N
I
S-8.

|-12.

-16.

-20.
1.


0.


-1.


5 -2.
0.

-3.


-4.


-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.


-5. -- I I
-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.
distance (m)


Figure 5.8: Case 401 at 30 hours.


24.


20.


16.

cr

12.
-4

8.


4.






44
that in fact no bar has formed in the first hour, but there is an indication of accretion in

the upper portion of the profile, where a berm is beginning to form. The maximum 7

value is approximately 550 Nm-2, while the minimum of T is -2.5 Nm-2. During 30 hours

of waves impacting on the profile in Case 600, accretion of sediment onto the beach face

continued to produce a berm, according to Fig. 5.15, but there is also some indication of

a weak bar feature offshore at the breakpoint. The maximum average bottom shear stress

increased at the breakpoint to 1000 Nm-2. An interesting feature in both the i- and To

values at the upper end of the profile at the base of the berm is that both have increased

in magnitude, unlike the values shown at 1 hour in Fig. 5.14.

5.3 Profile with Recovery

Case 510 is an example of changes in average bottom shear stress in which recovery

occurs in an eroded profile with bar-trough systemss. This case commenced with the final

eroded profile in Case 500 (see Fig. 5.12) to induce shoreward movement of sand by reducing

the average breaking wave height to 1.0 m and increasing the wave period to 16 s (the value

of w changed to 0.037 ms-1 due to temperature effects). Figure 5.16 shows the initial

profile with the associated values of IHb/wT, T and o. The values of these parameters

have diminished significantly from those found at 100 hours in Case 500. After 30 hours

of waves in the large wave tank, the profile in Fig. 5.17 has indications that the offshore

bar-trough system is decreasing in size, while the nearshore bar has nearly vanished. The

maximum bottom shear stress is 400 Nm-2 at 20 m offshore, while the most negative value

of To is 1.5 Nm-2. By the end of 120 hours of testing the recovery of the beach profile, the

maximum value of T has been reduced to a little over 300 Nm-2 at 10 m offshore, while

the bar-trough system centered around 50 m offshore has continued to diminish as a result

of sand in the bar moving shoreward to fill the trough.

















5. 15. 25. 35. 45. 55. 65. 75.


-2
20.

N 16.
I
x 12.
o
0
S8.



0.

N
-4.

r= -8.
a
=I- 12.

-16.

-20.
1.


0.


-1.
E

5 -2.
a

-3.


-4.


-5. '-- I
-25. -15. -5.


5. 15. 25. 35. 45. 55. 65. 75.
distance (m)


Figure 5.9: Case 500 initially.


Taub
Touo
















I I I I I I i r I i I


24.


20.


16.
aC

12.
-4

8.


4.


0.


5. -15. -5.







46








-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.
20. 1

N 16. Touo
Twuo
z 12.
o
0
Z 8.



0.

N
!-8.

12.

-16.

-20.
1. 24.


0. de 20.
S\ ----- Hb/(wTT).

16.
-. 16.


-2. 12.


-3. 8.
) 8.


-4. 4.


-5. I I I I I I I '.
-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.
distance (m)


Figure 5.10: Case 500 at 1 hour.


















-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.
20. 1 V I I I I I


N 16.

12.
o
0
S8.

o 4.

0.

-84.
N

I I
-12.

-16.

-20.
1.


0.


-1.
E

5-2.
0T
P0


-5. I I I I I I I I I I I I I I 1 .
-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.
distance (m)


Figure 5.11: Case .500 at 30 hours.


- I .I b i
Toub
Tau,
















'- -I --I -- --i -I I | f I I f f ( I


24.


20.


16.
Cr

12.
-4

8.


4.




















20.

N 16.
E
i 12.
o
o
8.






I
0.



-8.

I012.

-16.

-20.
1.


0.


-1.


5 -2.
0
-o
-3.


-4.


-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.


-5. 1
-25. -15. -5.


5. 15. 25. 35. 45. 55. 65. 75.
distance (m)


Figure 5.12: Case 500 at 100 hours.


p- I I I-I I- 1 I- 1 I- I I- 1 I I I .


--- TOUb
Tou















I I I I i I I I i I I i I


24.


20.


16.
cI.
-1-

12. "
,--i

8.


4.


0.


















-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.


-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.
distance (m)


4U.

N 16.

a 12.

Z 8.
.0


0.

--4.
N
S"-8.

I012.

-16.

-20.
1.


0.


-1.


S-2.
-5

-3.


-4.


-5.


Figure 5.13: Case 600 initially.


I I I i i I I I -

Touo
















l I l I l I i I I i I I I I-



-- depth

Hb/ V(T)


24.


20.


16.
-r
12.
-4

8.


4.


0.







50










-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.
20. I--
N 16. TOb

12.



4.
0 .





--4.
N
.-8.

I 12.

-16.

-20. I I I I I I I I I I I I I 4


0 ^---- depth -
0. 20.
------ (Hb/(wT):



-2=
-1. 16.


5 -2. 12.


-3.
'3- 3. 8.


-4. 4.


-5. 0.
-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.
distance (m)


Figure 5.14: Case 600 at 1 hour.


















-5. 5. 15. 25. 35. 45. 55. 65. 75.


-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.
distance (m)


20.

N 16.

S12.
o
o



0.
a





-4.
N
9 -8.

m-12.

-16.

-20.
1.


0.


-1.


5-2.
a.

-3.


-4.


-5.


Figure 5.15: Case 600 at 30 hours.


- -- Toub
S- Touo








,- 1 i I I-- i I. i i :b


24.


20.


16.
Cr

12.
-4

8.


4.


0.


-25. -15.




















5. -15.


-5. 5. 15. 25. 35. 45. 55. 65. 75.


-2
20.

N 16.
E
x 12.
o
0
Z 8.



0.

N
'|-8.

I?12.

-16.

-20.
1.


0.


-1.
E

S-2.
0.

-3.


-4.


-5. -'--i I I I I i
-25. -15. -5. 5. 15. 25. 35.
distance (i)


45. 55. 65. 75.


Figure 5.16: Case 510 initially.


Taub


















I I I I I 1 I I I- I I I iI I
Tau,


















I i I I I I I I I I

depth


S Hb/ (WxT)


24.


20.


16.
0-r

12. Z
-0

8.


4.


0.



















-5. 5. 15. 25. 35. 45. 55. 65. 75.


i i I I I I i
L ~------ Touf












r- -I ----- I


0.


-1. -


S-2.


-3.


-4.


-5.
-25. -15. -5. 5. 15. 25. 35.
distance (m)


SI' o0.
45. 55. 65. 75.


Figure 5.17: Case 510 at 30 hours.


5. -15.


-2
20.

N 16.
z
12.
o
: 8.

V 4.
0.

N
|-8.

J12.

-16.

-20.
1.


24.


20.


16.


12. 2
-4

8.


4.








54









-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.


/U.

N 16.
E
12.
0
0
= 8.





N
O-8.

12.

-16.

-20.
1.


0.


-1.


5-2.

-3.
-3.


-5. 1 I I I I I I I I l" I f I 1
-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.
distance (m)


Figure 5.18: Case 510 at 120 hours.


Taub
Tauo


^~ -- --- Ta-

--- ... --- -











I i l i I I I t l I I I l I I .


24.


20.


16.

01
-r

12. ?
-4

8.













CHAPTER 6
MODEL SENSITIVITY TESTS



Sensitivity tests were performed on the explicit model to determine which parameters

caused significant changes in the prediction of the size and location of bar-trough systems.

In addition, the stability of the model was investigated for various coefficients, different

lags of the average bottom shear stress parameter, and changes in wave heights and water

level conditions. The explicit model is based on the finite difference forms of the continuity

equation (Eq. 3.3) and the dynamic or transport equation (Eq. 3.4), which were described

in Chapter 4. The volumetric transport equation in finite difference form is

Vi = At(KI(?)i + (0.3(Qlp)i + 0.7(Ql&)i) + K3(ah/ay)i) (6.1)


which is computed every time step and substituted into

Ahi = vi V- (6.2)
Ay

to calculate the change in depth at each contour. Because the size of the forcing functions

vary considerably, the size of their respective coefficients must be set large enough to permit

them to contribute to the transport at each time step. From the computations of T and

To for the various cases shown in Chapter 5, it was found that 7 is nearly 100 times larger

than To. The slope term is considerably smaller than both of these bottom shear stress

parameters. For each time step the Dally breaking wave model (1980) is run for the given

incident wave conditions to determine the wave heights and setup/setdown conditions across

the profile. The wave characteristics and water depths are used to determine the average

bottom shear stress parameters for each time step, so that new transport volumes can be

estimated repeating the calculation of the volumetric transport by using Eq. (6.1).










Table 6.1: First Set of Sensitivity Tests
TEST FORCING FUNCTION AND COEFFICIENTS USED
Al T, Table 6.2
A2 b, Table 6.2, ah/9y, K3 = 5.80 x 10- m2s-1
A3 b, Table 6.2, Oh/Oy, K3 = 1.16 x 10-3 m2s-1
A4 b, Table 6.2, dh/Oy, K3 = 1.74 x 10-3 ms-1
A5 F, Table 6.3, ah/9y, K3 = 5.80 x 10-4 ms-1
A6 T, Table 6.3, Oh/8y, K3 = 1.16 x 10- m2s-1
A7 -, Table 6.3, Oh/ay, K3 = 1.74 x 10-3 ms-1
A8 9h/Oy, Ki3 = 5.80 x 10-4 m2s-1, 100 time steps
A9 To, 100 time steps
A10 7,, Oh/Oy, K3 = 5.80 x 10- ms-1, 100 time steps
All ~, Oh/ay, K3 = 5.80 x 10-4 ms-1, 500 time steps


6.1 Sensitivity of the Model to the Transport Parameters

The first set of sensitivity tests, listed in Table 6.1, utilize Case 400 of the large wave

tank data (Kraus and Larson, 1988) discussed in the preceding chapter, to determine the

sensitivity of the model using constant incident wave height and period, as well as water

level. Figure 6.1 shows the results of the first test, Test Al, in which the only forcing

function is the average bottom shear stress, lagged as follows:
4
T= > wj1T,+j
j=-4
with the weighting factors, Wj, shown in Table 6.2 and Kz = 2.9 x 10-6 m4(Ns)-1 for 10 time

steps where At = 360 seconds. This lag function is analogous to the scheme in an Appendix

in Dally (1980), in which Dally attempted to model the lateral fluid momentum under

breaking waves (see Appendix B). In his Appendix, Dally explains that he was unsuccessful

in his attempt to develop an analytical expression for this lateral fluid momentum. He found

it was more useful to develop an empirical spreading function, which allows the smoothing

of the shear stress over a grid cell that is of greater lateral extent than the shear stress

driving the lateral fluid momentum in the cell.

The model prediction when compared with the actual observation at 60 minutes (Fig.

6.1) shows that the model produces an unstable profile. The next set of tests was designed









Table 6.2: Lag for T in Sensitivity Tests A1-A4 and B4
i+j WEIGHT,Wi
i+4 0.02
i+3 0.03
i+2 0.06
i+1 0.07
i 0.08
i-1 0.10
i-2 0.11
i-3 0.19
i-4 0.34



to evaluate the sensitivity of the model to the slope or Oh/Oy term, when combined with

the 7 using the Test Al lag and K1 values, while changing the K3 values. The results for

the first of these tests are shown in Fig. 6.2, where K3 = 5.80 x 10-4 m2s-1, Fig. 6.3, where

K3 = 1.16 x 10-3 m2s-1, and Fig. 6.4, where K3 = 1.74 x 10-3. The test with the smaller

of the K3 values, Test A2, shows the model predicted profile is now stable for one hour

(10 time steps). The bar-trough system location is very well predicted, but the bar height

is smaller than was observed in reality. Also, less sediment has been eroded shoreward of

the trough in the model, than was observed in the large wave tank test results. This likely

reduced the amount of sediment available to build the bar. The small secondary bar, which

was observed to be located about 15 m shoreward of the primary bar, is not evident in the

predicted profile. These results are not unexpected, since the Oh/Oy term acts to smooth

perturbations in the profile. Increasing the K3 value as shown in Fig. 6.3, causes the model

to further decrease the bar height compared with Test A2, while the trough is not as deep.

The model predicted profile in Fig. 6.4 after 10 time steps shows that increasing the K3

value to 1.74 x 10-3 m2s-1 causes additional sediment to be transported from the shore.

This causes the sand bar to be slightly higher, even though the trough shoreward of the bar

is less deep

Tests A5, A6, and A7 were used to determine the sensitivity of the model to changes














-25. -15.
1.0 --i--


58





-5. 5. 15. 25. 35. 45. 55. 65. 75.


0.8 Transport

0.6

N 0.4-

" 0.2 -

0.0
Soo -- ----- / ^-----


t -0.2 -
S-0.4
CL
o
- -0.6

-0.8
1.0 I -I- I i I I I I -I-1 i
2.

Original
--- Model result

.---- ."Observed result-
0.

U -1. --

( -2.

-3.

-4.

-5. '
-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.
distance (m)


Figure 6.1: Test Al. Bar-trough system explicit model with T only and lags in Table
6.2. Lower panel shows the original, the observed one hour profile and the beach profile
predicted by the explicit model after 10 time steps (At = 360 s). The upper panel shows
the transport of sediment offshore in the tenth time step.















-5. 5. 15. 25. 35. 45. 55. 65. 75.


-2
1.0

0.8

; 0.6

, 0.4
E
K 0.2
o
(D
S0.0

S-0.2
o
CL
0 -0.4
c
o
S-0.6

-0.8

-1.0
2.


1.


0.


-1.


8- -2. -


-3.


-4.

-5. i I I I I
-25. -15. -5. 5. 15. 25. 35.
distance (m)


45. 55. 65. 75.


Figure 6.2: Test A2. Same as Fig. 6.1 except here the model includes both T and 9h/Oy;
IJ3 = 5.80 x 10-4 m2s-1.


Transport



- I I

- -


- -

- -

- -




i a I a i a I a I i I


!5.


-15.





















5. 15. 25. 35. 45. 55. 65. 75.


II I j I j I
Transport














I A : j I I I I


-15. -5. 5. 15. 25. 35.
distance (m)


45. 55. 65. 75.


Figure 6.3: Test A3. Same as Fig. 6.2 except K3 = 1.16 x 10-3 m2s-1.


-25
1.0

0.8

S0.6

N 0.4

S0.2
o
m 0.0

L -0.2
a0

2 -0.4


-0.8

-1.0
2.

1.


0.


-1.




-3.


-4.


-5. 1-
-25.


i. -15. -5.




















-5. 5. 15. 25. 35. 45. 55. 65. 75.


0.





S -2.


-3.


-4.

-5. : I
-25. -15. -5. 5. 15. 25. 35.
distance (m)


45. 55. 65. 75.


Figure 6.4: Test A4. Same as Fig. 6.2 except K3 = 1.74 x 10-3 ms-1.


-25
1.0

0.8

S0.6
0
N 0.4

" 0.2
o
S0.0

L -0.2

* -0.4
C
-O.
& -0.6

-0.8

-1.0
2. r


I I i I- I I 1 I IT iI a I
Transport


I I I I t I I I I I I I I I


i. -15.


I I

I I






62


Table 6.3: Lag for in Sensitivity Tests A5-A7 and B3
i+j WEIGHT,Wj
i+4 0.00
i+3 0.02
i+2 0.03
i+1 0.05
i 0.10
i-1 0.15
i-2 0.25
i-3 0.40
i-4 0.00



in the lag values for T-, using the weights in Table 6.3. Figure 6.5 shows that in the case

where K3 = 5.80 x 10-4 m2s-1, the model predicted bar-trough system is shifted further

offshore than in the previous tests. The bar is also larger than in the previous three tests.

There appears to be a slight instability in the bar crest in Test A5. Because the model

prediction becomes more stable when K3 is increased to 1.16 x 10-3 m2s-1 (see Fig. 6.6),

the slight instability in the bar crest in Test A5 is likely due to the coefficient for the slope

being too small to adequately smooth the profile in each time step. This also indicates that

the number of weighting factors, Wj, used in the estimation of each 'i, can act to stabilize

the predicted profile. Another important result of reducing this range of weights is that the

width of the bar-trough system has decreased by at least 3 m compared with the results in

Tests A1-A4. Figure 6.7 shows that in Test A7 where K3 = 1.74 x 10-4 m2s-1, the trough

is filled to a greater extent and the bar crest is decreased due to more sediment being shifted

offshore.

Sensitivity test A8 ran the explicit model with only the parameter 9h/ly using K3 =

5.80 x 10-3 m2s-1. Figure 6.8 shows that after 100 time steps there is some erosion of

the upper portion of the profile and deposition at the lower end of the profile. This is

due to the fact that the slope term acts in the offshore direction if depth along the profile

is monotonically increasing. A similar test (Test A9) was used only the average bottom






63

shear stress due to nonlinear waves, 7b, in finite difference form according to Eq. (4.6).

The K2 value in the test was set to 5.8 x 10-5 m4(Ns)-, which resulted in the unstable

profile, shown in Fig. 6.9, after 100 time steps. The next test, Test A10 included both the

parameters o and ah/9y in the explicit model for 100 time steps. The resulting profile,

shown in Fig. 6.10, indicates the model is stable, due to the slope parameter controlling

the profile changes. There is deposition of sand between 0 m and 15 m distance offshore,

unlike the profile in Test A7, shown in Fig. 6.8. This is an indication of the effects of the

shoreward sediment motion produced by the T parameter. Figure 6.11 shows that after

500 time steps of the explicit bar-trough system model using the same conditions for the

combined T and 9h/Oy parameters, there continues to be erosion at the extreme upper end

of the swash zone and deposition of sediment below this region.

6.2 Sensitivity Tests of the Effects of the Lag Weights

The next set of tests of the sensitivity of the explicit model, listed in Table 6.4, analyzed

the impact of changing the lag weight distributions on the predicted profile when all three

forcing functions, 7o, and 9h/By, were included. Each test used 10 time steps, KI =

2.9 x 10-6 m4(Ns)-l, K2 = 5.80 x 10-5 m4(Ns)-, and K3 = 5.80 10-4 m2s-1. The first

set of lags, which are listed in Table 6.5, resulted in the profile shown in Fig. 6.12 (note that

when only one cell was used, the computational model failed before the tenth time step).

This was a very simple case, which gave only a slight lag to the shear stress parameter.

The profile shows that the bar-trough system which is computed is very highly unstable

compared with the one hour large wave tank profile. The sediment transport curve shows

that there is only a small portion of the profile with sand motion. Increasing the spreading

of the weighting factors, Wj, as shown in Table 6.6 begins to produce the bar-trough system

plotted in Fig. 6.13. The bar is very narrow and steep in this case and the predicted profile

appears to be unstable. Using the weights given in Table 6.3, the height of the bar has

been greatly reduced as shown in Fig. 6.14 and the bar-trough system covers an even

larger portion of the profile. The slope term has acted to smooth the bar-trough system




















-25. -15. -5.
1.0 i i

0.8

0.6
a)
0.4
E
1 0.2



L-0.2
0
0-
S-0.4
o
L -0.6

-0.8

-1.0
2.


I.


0.


T -1.




-3.


-4.


5. 15. 25. 35. 45. 55. 65. 75.

Tronsport





/1-


-5. I I I I I I I
-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.
distance (in)


Figure 6.5: Test A5. Same as Fig. 6.2 except using lags in Table 6.3.


S I 1 I I I I I I t I I I ,


















-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.
1.0 i I 1 I

0.8 -Transport

0.6

S0.4

N 0.2 -

o
0.0


0.
S-0.2

' -0.6

-0.8


2.

1. ----- Model result
------ Observed result-
0.



0 -2.


-3.


-4.

-5 I I i i -
-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.
distance (n)


Figure 6.6: Test A6. Same as Fig. 6.5 except K3 = 1.16 x 10-3 m2s-1.

















5. -15. -5.


5. 15. 25. 35. 45. 55. 65. 75.


2.
Original
S---- Model result

S'.--- Observed result-


-C.

-2.


-3.


-4.


-5. '-' I '
-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.
distance (m)


Figure 6.7: Test A7. Same as Fig. 6.5 except K3 = 1.74 x 10-3 ms-1.


-2
1.0

0.8

" 0.6
0C


- 0.2
o
0



:-0.6
o
\ -0.4
o
- -0.6


-0.8
-1 .


---- Tronsport
















- I i I i 1 i I v i I I t I _

















-5. 5. 15. 25. 35. 45. 55. 65. 75.


-5. I I I 1
-25. -15. -5. 5. 15. 25. 35.
distance [mW


45. 55. 65. 75.


Figure 6.8: Test A8. Same as Fig. 6.1, except here only the parameter 9h/9y is used for
100 time steps; K3 = 5.80 x 10-4 m2s-1. Note that the scale of the transport in the upper
graph has changed.


. -15.


Transport
















I I I I I I I I I ,


-25
0.10

0.08
-- a
0.06

S0.04
E
' 0.02
o
, 0.00

u-0.02

0-0.04
o0.06


-0.08

-0.10
2.

1.


0.


Z -1.
t-
-2.


-3.


-4.


a

















-2
0.10

0.08

S0.06

. 0.04

* 0.02
o
(0
S0.00

t-0.02
0
r-0.04



-0.08

-0.10
2.


1 .


0.



-2.

13

-3.


-4.


5.


-15. -5. 5. 15. 25. 35. 45. 55. 65. 75.


----- Transport








'- -I -- I--=LI









- -


-5. I i 1 I I I I I i I I I I 1 1
-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.
distance (n)


Figure 6.9: Test A9. Same as Fig. 6.1, except here only the parameter T is used for 100
time steps; K2 = 5.80 x 10-5 m4(Ns)-1.



















-25
0.10

0.08

S0.06

0.04

S0.02
o
-0.00

S-0.02


C-
i-0. 06

-0.08

-0.10
2.


1.


0.


-5. 5. 15. 25. 35. 45. 55. 65. 75.


-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.
distance (ml


Figure 6.10: Test A10. Same as Fig. 6.1, except here both the parameters T7 and ah/ay
are used for 100 time steps.


Transport
















, I I I I I I I I !


i. -15.



'

I


















-5. 5. 15. 25. 35. 45. 55. 65. 75.


0.


e -I.


c -2.


-3.


-4.


-5. i I I I I
-25. -15. -5. 5. 15. 25. 35.
distance (ml


45. 55. 65. 75.


Figure 6.11: Test All. Same as Fig. 6.10, except for 500 time steps.


-25
0.10

0.08

" 0.06

0.04

S0.02
o
0.00

2-0.02
-0.04
-0.04
0
O o.o0


-0.08

-0.10
2.


Transport


i_/


I~r I I I I I I i I I I


-1--F-


I I I f I I I I I I I I I I I I


I


i. -15.









Table 6.4: Lags Used in Second Set of Sensitivity Tests
TEST LAG TABLE USED
B1 6.5
B2 6.6
B3 6.3
B4 6.2


Table 6.5: Lag for T in Sensitivity Test B1

i+j WEIGHT,Wj
i+4 0.00
i+3 0.00
i+2 0.00
i+1 0.00
i 0.20
i-1 0.80
i-2 0.00
i-3 0.00
i-4 0.00



into a more realistic shape. Finally, using the weighting factors in Table 6.2, the profile in

Fig. 6.15 shows very close agreement in location and shape between the observed one hour

profile and that estimated by the explicit model.

6.3 Summary of Conclusions from the Sensitivity Tests

The tests performed here resulted in an understanding of the manner in which the

forcing functions affect profile evolution and the manner in which they can be used in

a realistic model to predict cross-shore sediment motion, especially when related to bar-

trough system formation. It is obvious from the first test, Al, the T term dominates

the computations in the model. However, even by itself the bottom shear stress does not

adequately describe the transport, and lag weights must be introduced using the analogy

of the shear stress applied to a fluid (See Appendix B). These lag weights act to spread the

effects of the shear stress across a broad area in the same fashion as noted in nature in the

vicinity of breaking waves (or scour due to jets as shown in Fig. 3.1). The transport due


















-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.
1.0 I I I I II

0.8 Tronsport

0.6

N 0.4
E
' 0.2
0.0

-0. 2

o
-0.4
C
- -0.6

-0.8

-1.0 I I i I I I I -- I I I I


S---- Original
1. --- Model result

.------- Observed result-
0.


-1.

- -_


-3.


-4.


-5.
-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.
distance (m)


Figure 6.12: Test B1. All three forcing functions used with lags in Table 6.5.
















-2
1.0

0.8

;0.6

N 0.4

" 0.2


S o.o ----

o .
0


i -0.6

-0.8

-1.0
2. I I I r


5. 15. 25. 35. 45. 55. 65. 75.


I.* --- Model result

------ Observed result-
0.




-2.


-3.


-4.

-5. I I I I I I I
-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.
distance (m)


Figure 6.13: Test B2. Same as Fig 6.12, except lags used are in Table 6.6.


I I I I I I I I I i
Transport



-e

\i


N\/ .


I I I I I I I I I


L


5. -15. -5.








74










-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.
1.0 I I i I

0.8 --- Tronsport

0.6

0.4
S0.2

|0.0 '--- --- ~ --------- ....*

L--0.2
o
CL
a -0.4
o
-0.6 -

-0.8

1.0 I i lI I i I i l i I I I I I
-1.0
2.
Original
i. -M --- Model result

------ Observed result
0.




I- -2.
-C


-3.


-4.


-5.
-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.
distance (ml


Figure 6.14: Test B3. Same as Fig. 6.12, except lags used are in Table 6.3.

















-25. -15.
1.0 ---

0.8

; 0.6

1 0.4

0.2
o
a 0.0

S-0.2 -

o -0.4 -
C
S-0.6 -

-0.8

-1.0
2. '




0.


-5. 5. 15. 25. 35. 45. 55. 65. 75.


-5. 1 I I I I I I I I I I I I I 1 1
-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.
distance (m)


Figure 6.15: Test B4. Same as Fig. 6.12, except lags used are in Table 6.2.






76


Table 6.6: Lag for T in Sensitivity Test B2
i+j WEIGHT,Wj
i+4 0.00
i+3 0.00
i+2 0.00
i+1 0.00
i 0.00
i-1 0.10
i-2 0.30
i-3 0.60
i-4 0.00



to shear stress must next be balanced by the slope or "gravity effect" term to keep the bar

from rising too steeply and prevent the bar from building up to or above the water level. A

term which is important for long-term control of the bar-trough system migration is the 7-

term, which acts to move sediment shoreward due to nonlinear effects under the incident

waves. If all of these terms are not quantified properly, there will be instabilities produced

in the profile and the model will fail. The extent and magnitude of the lag weights are

also very important in the modelling of bar-trough systems. In Chapter 7, the model is

calibrated and tested with the objective of accurately forming the bar-trough system and

then allowing it to migrate and change over a long time period.













CHAPTER 7
RESULTS OF BAR-TROUGH MODEL PREDICTIONS



The predictive bar-trough model was first calibrated using the large wave tank data

in Case 400, which was an experiment resulting in rapid bar-trough system formation and

migration. The rapid changes in the profile were produced by waves with Hb = 2.3 m and

T = 5.6 s. The initial profile was linear as shown in Fig. 5.1, while one hour later a

barred profile is shown in Fig. 5.2. The bar-trough system in Case 400 continued to build

and migrate offshore during the 30 hours the waves were impinging on the profile, which is

shown in Fig. 5.5.

Sensitivity tests were discussed in Chapter 6 using the initial and one hour profiles for

Case 400 to show the impact of the various forcing functions, different coefficients, and

lag weights on the transport rates across the profile. For Case 400 in 1 hour the best

prediction of the bar-trough system location and size by the model was shown in Fig. 6.15.

The computational model has the capability of predicting accurately the bar-trough system

formation, but prediction of the migration of this system is also desired. This migration of

the bar-trough feature is the result of the location of the breaking waves shifting offshore as

the bar builds. The maximum magnitude of the average bottom shear stress parameters, T'

and To, follow the breaking wave, so that the sediment transport model predicts the seaward

shift of the bar-trough system. The simulation of the changes in the bars over time periods

longer than the sensitivity tests reported in Chapter 6 required that different parameters be

used. The coefficients were reset to K1 = 2.4 x 10-6 m4(Ns)-', K2 = 8.4 x 10-5 m4(Ns)-',

and K3 = 4.2 x 10-4 m2s-1 Using the larger 1K2 value in these computations was found to

be necessary to allow the forcing function (T-), which produces shoreward sediment motion

in the model, to compensate for the increasing magnitude of the slope term (9h/Oy) as the






78


Table 7.1: Lag Weights Used for prediction of Bar Formation and Migration

i+j Weight,Wj
i+5 0.01
i+4 0.03
i+3 0.04
i+2 0.05
i+1 0.06
i 0.07
i-1 0.09
i-2 0.10
i-3 0.16
i-4 0.32
i-5 0.07


bar-trough system forms. This is especially important for this model, which is referred to

as an "open-loop" model, as opposed to the "closed-loop" model (e.g. the model described

in Appendix A should converge to an equilibrium profile if run for a sufficient amount of

time with constant wave characteristics and water level) The lag weights, W,, were spread

over a larger range based on the expression
5
Tbi = z wjbi+j
j=-5

using the weights found in Table 7. Note that the lag weight at the last point, i 5 was

reduced to more realistically reflect the reduction of the average bottom shear stress effects

spatially. This allows the computational model to reasonably predict the bar-trough system

found after 10 hours of simulated run time. The time step of the model computations was

also reduced to At = 180 s, which is half the time interval used in the Chapter 6 sensitivity

tests. This had the result of increasing the stability of the model, which requires very

little computational time for the large wave tank beach profiles. After the first hour, the

predictive model produced the profile shown in Fig. 7.1. The location of the bar and

trough system agrees very closely with the observed profile, but the size of the predicted

bar-trough is nearly half of the actual one hour result. This indicates the reduction of the

coefficients on the forcing functions in the transport equation underestimates the size of






79
the bar-trough system during the initial computations. Figure 7.2 shows that the model is

simulating the 5 hour profile reasonably well if the size of the bar and its migration offshore

are compared with the observed profile. A comparison of the transports in both Fig. 7.1

and 7.2 shows that there is a reduction in the magnitude of the transport of sediment with

time. After 300 time steps or 15 hours of simulation, the observed and predicted profiles

in Fig. 7.3 continue to be in good agreement By the end of 30 hours (see Fig. 7.4)

the agreement between the predicted and actual profiles remains quite good. Some of the

small perturbations in the measured profile are likely due to wave reflections. Note that

the maximum value of the transport curve has continued to diminish over time, and it is

very small by the last time step. This indicates that the model has a tendency to converge

toward some sort of stable profile with increasing time.

The next computational model case run was for large wave tank Case 500, which is an

example of a profile that erodes and eventually forms two distinct bar-trough systems. The

same model characteristics used for Case 400 were also applied to Case 500. The one problem

found with the input parameters for this test was that the Dally wave model incorrectly

located the breaking wave, where Hb = 2m and T = 4s, The best agreement between the

observed and modelled breaking wave location was found by changing the dimensionless

parameter, Hb/hb, in the wave model from 1.3 for Case 400 to 1.0 for Case 500. Figure

7.5 shows that there is agreement in the location of the bar-trough system predicted by the

model after one hour. However the size of the model bar is very small compared with that

in the observed profile. The five hour run for the model shown in Fig. 7.6 shows that

the predicted bar-trough system is migrating offshore slower than the observed and the size

of the system is much less than in reality. By the end of 15 hours, the model predicted

profile in Fig. 7.7 shows that the primary bar-trough system migrates close to the correct

location, while the bar height is considerably less than the observed profile. The actual

profile is beginning to build a small bar shoreward of the larger bar. This small bar may

be the result of wave reflection, which is not included in this computational model. The



















L.U


Z 0.6

, 0.4

x 0.2
o
- 0.0


0
C. -0.2

2-0.4
o
- -0.6

-0.8

-1.0
2.


1.


0.


i -i.
-C
u -2.


-3.


-4.


-15. -5. 5. 15. 25. 35. 45. 55. 65.


-5. '-
-25.
-25.


-15. -5. 5. 15. 25. 35. 45. 55. 65. 75.
distance (m)


Figure 7.1: Case 400 profiles initially and one hour observed and predicted by bar-trough
model plotted in lower panel; upper panel is transport of sediment in the most recent time
step.


I I I I I 1 -'-- 1 1 1 I-- I

Tronsport










- I I i i
-




i -I I l i I i l v


-25.



















-25
1.0

0.8

0.6

N 0.4

" 0.2
o
S 0.0

S-0.2
0
- -0.4
o-
-0.6

-0.8


5. 15. 25. 35. 45. 55. 65. 75.


I1.
2.








. -






-4.


-5. 1 I I I I I I I I I I 1
-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.
distance (m)


Figure 7.2: Same as Fig. 7.1, except for 5 hours in Case 400.


p I pI I I p I I I
Transport









-1


I I I I ~ ~ ~~i l l i n I i i


-15. -5.







82









-15. -5. 5. 15. 25. 35. 45. 55. 65. 75.


-5. ' i I I I I
-25. -15. -5. 5. 15. 25. 35.
distance (m)


45. 55. 65. 75.


Figure 7.3: Same as Fig. 7.1, except for 15 hours in Case 400.


.5.


-2
1.0

0.8

S0.6

a 0.4
E
S0.2




0
0.0


I -0.2
o
Q.

-0.6

-0.8

-1.0
2.


1.


0.


-I.
-3.
-2.


-3.


Tronsport
















I I I I I I t I I I


SOriginol
Model result
\ ___ --Observed result
-'








83









-5. 5. 15. 25. 35. 45. 55. 65. 75.


-2
1.0

0.8

^ 0.6
a,
E
S0.2
o
S0.0

-0.2
0.
2-0.4
o
'--0.6

-0.8

-1.0
2.


1.


0.


S-1.
CL
- 2.


-3.


-4.


-5.


-25. -15. -5. 5. 15. 25. 35.
distance [m)


45. 55. 65. 75.


Figure 7.4: Same as Fig. 7.1, except for 30 hours in Case 400.


5.


-15.


Transport





















S- Originol

-\- Model result

i --- Observed resull-
-
-i

:- -- -






84
profile at 30 hours in Case 500 (see Fig. 7.8) shows closer agreement in the location and the

magnitude of the bar-trough system. The further erosion of the upper portion of the profile

bermm), which was not observed to have occurred to the same extent in the actual profile,

may be due to the lack of the secondary bar near the shore to reduce the height of the waves

transmitted over the bar. To examine the changes in the model over an even longer time

period, the model was run for another ten hours to a total of 40 hours. The results shown

in Fig. 7.9 indicate that the model is still eroding the beach face and depositing material

offshore, so that the trough is being filled, while the bar height and width are generally

similar to those of the observed bar. The model was also run for Case 401, which had

nearly the same wave characteristics as Case 400 (JIb = 2.0 m, T = 5.6 s), but the sediment

size in the large wave tank beach profile was 0.4 mm. This changed the rcr value used in

Eqs. (4.7) and (4.8) from 1.19 Nm-2 for 0.22 mm sand to 2.16 Nm-2 for 0.4 mm sand.

This larger sediment size would be expected to result in steeper slopes in the bar-trough

system, especially on the seaward side of the bar, due to the greater fall velocities associated

with sediment having a larger diameter. The profile shown in Fig. 7.10 after the first 20

time steps of the explicit model run for Case 401 shows very good agreement between the

location of the predicted bar-trough system when compared with the actual system. Even

after five hours, the model has predicted correctly the rate of migration of the bar-trough

feature offshore, but the trough predicted by the model is slightly less deep than that of the

observed profile, as shown in Fig. 7.11. The model profile is obviously overestimating the

amount of movement which occurs in the simulations run for longer periods of time (see 15

hours in Fig. 7.12 and 30 hours in Fig. 7.13). The effects of the coarser sediment need to

be included in a realistic manner to slow the migration of the bar- trough system offshore

in the computational model.



















5.


-15. -5. 5. 15. 25. 35. 45. 55. 65. 75.


Tronsport

















i I I I i I I I I I i i I I I
- I I I I I I i I I I I I I I i I -


-5. -'1
-25. -15.


-5. 5. 15. 25. 35. 45. 55. 65. 75.
distance (m)


Figure 7.5: Same as Fig. 7.1, except for Case 500 at 1 hour.


-2
1.0

0.8

; 0.6

N 0.4
E
K 0.2
0



0L


L -0.6

-0.8

-1.0
2.


1.


0.




a-2.


-3.


-4.







86









-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.
1.0 1 1 1

0.8 --- Trnsport

S0.6
a,
0.4

S0.2

0.0

S-0.2
0
0 -0.4
C
S-0.6

-0.8

-1.0 I I I I
2.
Original
--- Model result

--.- Observed result
0.


E-1.


c- -2.


-3.


-4.


-5. L
-25. -15. -5.


5. 15. 25. 35. 45. 55. 65. 75.
distance [m)


Figure 7.6: Same as Fig. 7.1, except for Case 500 at 5 hours.


















i. -15.


-5. 5. 15. 25. 35. 45. 55. 65. 75.


-25
1.0

0.8

0.6

S0.4

* 0.2
o
0
0.0

( -0.2
0
: -0.4
o
--O.B

-0.8

-1.0
2.


1.


0.


S-1.





-3.


-4.


-5.
-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.
distance (m)


Figure 7.7: Same as Fig. 7.1, except for Case 500 at 15 hours.


----- Transport




















Original

Model result

--- Observed result:
NNN



















-2
1.0

0.8

0.6

0.4
E
S0.2
o
- 0.0

S-0.2
0
0
C
-0.4
o
- -0.6

-0.8

-1.0
2.


1.


0.


-.
-J
. -.


-3.


-4.


5. -15.


-5. 5. 15. 25. 35. 45. 55. 65. 75.


Transport

















-\

SOriginal
M--- odel result

---____Observed result:
j -


-5. I I 1 I I 1 i I i
-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.
distance (m)


Figure 7.8: Same as Fig. 7.1, except for Case 500 at 30 hours.


















-15. -5.


5. 15. 25. 35. 45. 55. 65. 75.


Transport


















Original
S- Model result
Observed result
- -,\ -Osre eut


-5. 1 I I I I f I I I I I I I I
-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.
distance (m)


Figure 7.9: Same as Fig. 7.1, except for Case 500 at 40 hours.


5.


-2'
1.0

0.8

" 0.6

- 0.4

" 0.2
o
-0.0

S-0.2
0
0 -0.4
o
.- -0.6

-0.8

-1.0
2.

1.

0.







90







-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.
1.0 I I I iI

0.8 --- Transport

0.6

N 0.4

" 0.2

o
0.0

L -0.2
0
C-
-0.4

S-0.6

-0.8

1.0 i I i I I I I I I, I i I ,1I I- -
2. l- I-- | I-- i I- i I- | I- i-- I-- i I i | -I -- i -

------ Original
1.
1. S ---- Model result

,------ Observed result-
0.




|--2.
-Jr


-3.

-4.

-5. I I 1I I I I I 1 I- I I I
-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.
distance (ml


Figure 7.10: Same as Fig. 7.1, except for Case 401 at 1 hour.

















5. 15. 25. 35. 45. 55. 65. 75.


I I I I I I I I
Transport

















I- i -- --i- i -


0.


-2.


- -2.

-3.


-4.


-5.
-25. -15. -5. 5. 15. 25. 35.
distance (m)


45. 55. 65. 75.


Figure 7.11: Same as Fig. 7.1, except for Case 401 at 5 hours.


-15. -5.


-25
1.0

0.8

S0.6

, 0.4

S0.2
o
0
m 0.0

S-0.2
0
S-0.4

- -0.6

-0.8

-1.0
2. r








92










-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.
1.0 i

0.8 Transport

; 0.6



0
0.4

S0.2
o
0.0

2 -0.2
0
0 -0.4 -


-0.8
-0.8

-1.0
2. : I- i -- I-- | I-- i i- '- I- i I-- i I- | I- | i-- -- -

------ Original
1. : --- Model result
.''------ Observed result
o.










-4.

-5.
-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.
distance (m)


Figure 7.12: Same as Fig. 7.1, except for Case 401 at 15 hours.

















-2
1.0

0.8

S0.6

S0.4
E
N 0.2
o
0o
- 0.0

L -0.2
a
CL
0' -0.4
c
A- -0.6

-0.8


2.


1.


0.


-1.

C-
o -2.
o

-3.


-4.


5. -15.


-5. 5. 15. 25. 35. 45. 55. 65. 75.


I I I -I II II I I I- 1 1 I

Transport





















Original

Model result
.Observed result-







-N -
N.


-5. I I I I I I I I I I I I i
-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.
distance (m)


Figure 7.13: Same as Fig. 7.1, except for Case 401 at 30 hours.




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