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Prediction of the formation and migration of bar-trough systems

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Prediction of the formation and migration of bar-trough systems
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Houston, Samuel H., 1957-
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English
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x, 107 leaves : ill. ; 29 cm.

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Coastal and Oceanographic Engineering thesis M.S ( lcsh )
Dissertations, Academic -- Coastal and Oceanographic Engineering -- UF ( lcsh )
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bibliography ( marcgt )
non-fiction ( marcgt )

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Thesis:
Thesis (M.S.)--University of Florida, 1990.
Bibliography:
Includes bibliographical references (leaves 96-99).
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Typescript.
General Note:
Vita.
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This publication is being made available as part of the report series written by the faculty, staff, and students of the Coastal and Oceanographic Program of the Department of Civil and Coastal Engineering.
Statement of Responsibility:
by Samuel H. Houston.

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University of Florida
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University of Florida
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001613887 ( ALEPH )
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AHN8305 ( NOTIS )

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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




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.




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.........

* v
*. vi

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 Summaxy 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 M odel .............. I ................... 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 T in Sensitivity Tests A5-A7 and B3 ................... 62
6.4 Lags Used in Second Set of Sensitivity Tests ................... 71
6.5 Lag for b in Sensitivity Test BIl ...... ...................... 71
6.6 Lag for T 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 fax from adequate. The processes involved in the recovery phase of beaches occur on a long time scale and may require months or yeaxs to return the shoreline to approximately pre-storm conditions. In addition, the physical processes involved in the recovery axe 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 Daily 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
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 tinting 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 bax-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 transport. 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 equation 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 axe 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) empirical 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 straightforward 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, California 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-




ads 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 parameter to the 3/2 power. This model was based on wave and nearshore current effects on bea ch 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) developed 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 laboratory 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 Construction 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 equilibrium 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 conditions. 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. Erosion 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-paxallel bar (see Fig. 2.1). During the period following erosion of the beach, recovery of the crossshore 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 axe 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 axe 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 observation 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 "longshore 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




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 axe 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 wen understood and considerable research still remains to be done.
Some numerical models have been developed, which include offshore bars in the evolution 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 plunging 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 Daily 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 formation closely using large wave tank data, but was unable to predict bar recovery. Laxson 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 decay with distance from the break point, with a spatial decay coefficient (average value of 0.18m'1) 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 TV 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, H0/IwT, which relates, H,,, 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
2.
0.
S-2. '
-4.
-6. '
-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.




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 = Ay 2/3 (3.1)
where h is the water depth, y is the offshore distance and A is a scale parameter that is related to am 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 h aQ (3.3)
19t 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)
HbI(wT) (3.4)
where Hib is the wave height, T is the wave period and w, is the fall velocity of the sediment. 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 "longshore 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 "longshore 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 bartrough 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 shorewaxd 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 breakers). The flow ahead of and behind each breaker was generally directed offshore, but in the case of "Asymmetric" breakers, there was a layer of shorewaxd 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 important forcing functions they investigated was the determination of the average bottom shear stress, T'. Their methods for determining this parameter were found to reproduce accurately the measured undertow velocity profiles. They concluded that their method of determining 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)
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 = Eh(1 + 2 (3.6)
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 h/2 ~(3.7)
-h/2 0y
in which '- 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 = KTb + K2T. + KC30h/ay (3.8)
The forcing functions on the right hand side of this equation include YK, described briefly above, as well as the mean bottom shear stress due to nonlinear waves, T and the slope controlling or gravity parameter, .9h/l9. 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




- -- -- -- - --- Crest level
E 1+ -- -- -- -- -- -- -- ------ -Mean water level
- -- -- -- - - Trough level
h/2
E(i- H)
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 monumented baseline along those portions of the Florida coastline having sandy beaches. Detailed 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, limited 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 coastline 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,




17,1 -1

St. Mary's Entrance
S 1983

Marineland
* 1977

Steinhatchee
0

GULF

4,

Cape Kennedy
0 1977

N Clearwater NE) Vero Beach
1978 - 1980
AlI O .IA T HA O I OKE -= S MANATEEI HAROEEI tCHEE ST.
q .-' I-- -"l ua% LUCIE.%
Venice a. - 1 ,LAAEUAI.IIII
I r,-" J OKEECAIO .....
- --- -C-AR EEH-LLL 'HotY Au IACH Palm Beach L 197
COL E A I 8 O%'I N,,
S)*Miari
LCNHOL DADE
Figure 3.3: Location of Coastal Data Network Stations maintained by the University of Florida's COE Department.

44 7

%
Q
V




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




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 fiat 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.




CHAPTER 4
BAR-TROUGH EXPLICIT MODEL
4.1 Finite-difference Equations
The explicit computational model for cross-shore sediment transport uses finite difference 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, 77i which includes tide, storm surge, and wave setup and setdown effects. The total depth can be represented as di= hi + tqi.
The continuity equation in finite difference form using a space-centered finite difference method, is expressed as
Ahi = At(Q +i TO (4.1)
where Ti 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 equation 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 r},- ra.] (4.7)
3
if {} r,; (Qlp)i = 0.0 otherwise. The immersed specific weight of the sediment is w, b is the sediment diameter, and Oi is the profile slope. The component of the shear stress directed toward the shore is
(Ql.)i = K2[{3(-)i + 2w6 sin i} + 7,] (4.8)
if] {} 1 r,; (Qin)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
rC, = pg(s 1)6 sin 0, (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 = 30' and 6 = 0.22mm, the value of r,. = 1.185Nm-2.
The volumetric transport over one time step is
Vi = At[Ki(,)i + (0.3(Qip)i + 0.7(Ql.)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 = Vi Vi-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, imax 1, where imax is the total number of grid points in the profile. The two boundary conditions applied to this model are: 1) Vi, = 0.0, where is is the index of the instantaneous upper limit along the profile of the water level




32
setup and 2) Vimax-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
p
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 82h
S= K3 (4.12)
O9t
is found by sustituting Q = K3(oh/9y) into Eq. (3.3). The stability requirement for this explicit model is K3At/(Ay)2 < 0.5.




TIDE LEVEL MEAN SEA LEVEL
y
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: ( =)i pf 0.09L0 (H,)2 (4.6)
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 + tii, 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
(-)i, (T.o)i, and (,9h/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,- is decreased across the profile, by including




following equation:
EOH EOh H2 E 2aH 2 Ah
-H= ( y_ + _2 Ty + + (Hh Hhy) (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:
=pgL 1 (H,)2 d (A hH (4.
)2 7 9-) + 9Y h(l+(4.3)
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 y"o ef 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 UIU (4.5)




CHAPTER 5
THE AVERAGE BOTTOM SHEAR STRESS ACROSS THE SURF ZONE
To demonstate the effects of the average bottom shear stresses, b and -7, in bar formation 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-'). 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 T;. The maximum value of Hb/wT, 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 Tb- 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-f has also moved offshore with the breakpoint and now has a maximum value of 800 Nm-2, while the value of 7; 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'g and T' 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 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.

I04.
0.
N
m-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.
a
12.
,-4
8.
4.
0.

Figure 5.1: Initial profile and the ratio Hb/wT 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 7 (units = Nm-2), which is always negative.

N
I
I.

p I I I I I I I I
Taub
Touo
i....................................




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 NmThe 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/w8T 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 T 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 T 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/wT 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. V I I
N 16. To
s S12. o
0
Z 8.
0 -.
N
I
12.
-16.
-20. I I I I I I
1. i 24.
depth
0. 20.
Hb/(vT)
-1. 16.
5 -2. 12.
//12.
-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
----
-a
I I I l l i s I i I s I I

20. N 16.
I
S12.
o S8.
0.
O -4
N
I
-8.
o
12*
-16.
-20.
1.
0.
-1.
S
-2.
-3.
-4.

24. 20. 16.
1
0*
12. Z
--4
8.
4.
0.

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




20. N 16.
I
E
S12.
o 8.
4.
0.
--4.
N E _8.
I
-16.
-20.
1.
0.
-1.
2
S-2.
-3.
-4.

5. 15. 25. 35. 45. 55. 65 7

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

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

Figure 5.4: Case 400 at 10 hours.

- aub
faua
- I

24. 20. 16.
=
-r
CT" 12. '
8.
4.
0.

-25. -15. -5.

.




i. -15.

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

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

-25
20. N 16.
E
S12.
0
0
a
- 8.
0.
--4.
N
I12.
-16.
-20.
1*
0.
-1.
7
c -2.
0.
-3.
-4.

Figure 5.5: Case 400 at 30 hours.

I I I p i i i p i p i i p I
-- Toub
Tou,
-=I -

S I I 24.
depth "
20.
Hb/ (wiwT)
16.
r
/ 12. 7
J/ E
4.
. 5 75 0.
45. 55. 65. 75.

-




41
-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.
20.
N 16. b
-.Touo
S12.
0
8.7
N
z -8.
:)12.
-16.
-20. I I I I I I I I I
1. 1 1 1 24.
- --- depth 2
0. 20.
SHb/
-1 -. 16.
5-2. 12.
CL
-3. 8.
-4. -4.
-5. I ,- Ii I it i i t i s i l I I 0.
-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. -'1 I
-25. -15. -5.

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

Figure 5.7: Case 401 at 1 hour.

. i IjI I I I I I i
Toub
/ \
l I I I I l I I I t I I

20. N 16.
E
z 12.
0
0
"" 8.
4.
0.
N
-8.
12.
-16.
-20.
1.
0.
-1.
:-2.
-3.
-4.

24.
20. 16. 12. "
,
8.
4.
0.

-25. -15.




20.
N 16.
I
E
1~2.
O
O
z8.
.
0.
N
I
-8.
a12.
-16.
-20.
1.
0.
-1.
E
-2.
-3.
-4.

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

-5. I I I I I 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.
r
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 T 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 T- and T 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 system(s). 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 Jb/w/T, T" and T;. 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 7; 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 1 6.
z
x 12.
0
0
" 8.
0.
-4.
r= -8. =Ig-12.
-16.
-20.
1.
0.
-1.
E
--2.
a-o
-3.
-4.

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

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

Figure 5.9: Case 500 initially.

I I I I I I i I r I i I # Taub
Tauo
' I I I I I I I 0

24. 20. 16.
12.
-,4
8.
4.
0.

25. -15. -5.




46
-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.
20. 1 1
N 16. TOUb
= Twuo
z 12.
Z8.
0.
-4.
N
r" -8.
f12.
-16.
-20. f
1. '24.
0. 20deph .
_ Hb/ (wwT] 0
-1. 16.
r-2. 12
-L 12.4
0.
-3. 8.
-4. 74.
-5. ,0.
-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. I I I I I I

N 16.
E
12.
0
8.
o 4.
0.
-84.
IJI
-16.
-20.
1.
0.
-1.
2
-2.
T2
V

-5. I I I I I I I I I I I 1 0.
-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.
distance (m)

Figure 5.11: Case .500 at 30 hours.

- I I .* | I
SToub
Tau,
2%
I I I I l l I I ( 1 f I f I ( I

24. 20. 16.
Cr
12.
"-4
8.
4.




20. N 16.
E
12.
o
8.
0.
N
I
-.
I12.
E
-16.
-20.
I*
0.
-1.
6
S-2.
CL
-3.
-4.

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

S-5. A I
-25. -15. -5.

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

Figure 5.12: Case 500 at 100 hours.

I p I II
-- TOub
Tou
I I 1 I t I i I I I i I I i I

24. 20. 16. 12. "
g
,--i
8.
4.
0.




-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.
20. I
*N 16.
- Touo
a 12.
o
N
2-8.
.
12.
-16.
-20. I i I i I t I lii
O. "- depth
0.
-1.
-2.
0
-3.
-4.
-5. t I I I ,

-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.
distance (m)
Figure 5.13: Case 600 initially.

24. 20. 16.
r
12.
N -4
8.
4.
0.




50
-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.
20. o
N 16. b
12.
--- -- TOo
0
0
8 .
4.
0.
- -4.
N
I 12.
-16.
-20. I I I I I i I I I I I I I
0 ~dept~h-2..
----Hb/ (wxT)I 116.
z -2. 12. Z
CL4
-3. 8.
-4. 4.
-5. I I 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.
I
12.
0
Z8.
n
0.
- -4.
N
!
z -8.
m-12.
-16.
-20.
1.
0.
-1.
5-2.
a.
-3.
-4.
-5.

Figure 5.15: Case 600 at 30 hours.

Touo
I I m I I t I i I i I ,

24. 20. 16.
r
12.
N
8.
4.
0.

-25. -15.




5. -15.

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

-2
20. N 16.
i
S12.
o
a Z 8.
0.
-8.
N
12.
-16.
-20.
1*
0.
-1.
-2.
0.
-3.
-4.

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

45. 55. 65. 75.

Figure 5.16: Case 510 initially.

Toub
Touo
/--
I I I I I sI I i i
i l I 1 I I I
depth
- Hb/ (wvT)

24. 20. 16.
r
12. Z
8.
4.
0.




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

i i i i I I I i
Tub
Toub
SI i I ,-- I I - I I

-2.
-3.
-4.
-25. -15. -5. 5. 15. 25. 35.
distance (m)

0.
45. 55. 65. 75.

Figure 5.17: Case 510 at 30 hours.

5. -15.

-2
20. N 16.
E
z
m12.
0
0 :. 8.
a4.
0.
N
J12.
-16.
-20.
1.

24. 20.
16.
a 12. "2
N
8.
4.




54
-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.
20.1 1
N 16. Tub
Ea
r=Touo 12.
0
0
:: 8.
O
* .., ...--- ..
12.
-16.
-20. I
depth
0.
Hb/ (wTI
-1.
V
-2.
-3.
-4. -

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

Figure 5.18: Case 510 at 120 hours.

24. 20. 16. CF
1 12.
8.
4.
0.




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(Kj(T)j + (0.3(Qlp)i + 0.7(Ql,)i) + K3(ah/,y)j) (6.1)
which is computed every time step and substituted into Ahi = Vi Vi-1 (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
-- for the various cases shown in Chapter 5, it was found that T 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 Tb, Table 6.2
A2 7, Table 6.2, 0h/,9y, K3 = 5.80 x 10-4 m2s-1 A3 T, Table 6.2, Oh/Oy, K3 = 1.16 x 10-1 m2s-1 A4 T, Table 6.2, Dh/ay, K3 = 1.74 x 10-' m2s-1 A5 F, Table 6.3, ah/9y, K3 = 5.80 x 10-4 m2s-1 A6 T, Table 6.3, ah/8y, K3 = 1.16 x 10-3 m2s-1 A7 T, Table 6.3, Oh/ay, 1K3 = 1.74 x 10-3 m2s-1 A8 9h/Oy, 1K3 = 5.80 x 10-4 m2s-1, 100 time steps
A9 T;, 100 time steps
A10 7, Oh/O, 13 = 5.80 x 10-4 m2s-', 100 time steps All -', Oh/y, K3 = 5.80 x 10-4 m2s-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
= _i: wJT,+j
j=-4
with the weighting factors, Wi, shown in Table 6.2 and K1 = 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 Al-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 T 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 K(3 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

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

0.8 Transport
0.6
d)1
N 0.4
0.2
0.0
-0.4
-j L
a
an -0.4
L
- -0.6
-0.8
-1.0 I I I i I t I I I I 1
2.
Original I. Model result
0.
- t
-3.
-4.
-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 S0.4 S0.2 0.0 ( -0.2
o
Q.
0
CL
S-0.6
-0.8
-1.0
2.

1.
0.
-2.
-3.
-4.
-53.
-5 I 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 Oh/Oy; IC3 = 5.80 x 10-4 m2s-1.

Transport
-
-
-
-
-
-
S I I I I i I I I i I i
i a I a i a I a I i I a

5.

-15.




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

rantI j
Transport
I I tI I I I 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 S0.4 0.2
CC
O
m 0.0
L -0.2
0
0.
2 -0.4
C
1-.s
0
-0.8
-1.0
2.
1o
0.
-2.
-3.
-4.

-5. 1-25.

. -15. -5.




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

0.
f-1
cL0-2.
-3.
-4. :
-5.o I t I i
-25. -15. -5. 5. 15. 25. 35.
distance WI}

45. 55. 65. 75.

Figure 6.4: Test A4. Same as Fig. 6.2 except K3 = 1.74 x 10-3 m2s-1.

-25
1.0
0.8 0.6
0.4 N 0.2 Co
0.0
-0.2
0
Q
0) -0.4
C
-0.6
-0.8
-1.0
2. r

I i Tr an o i
Tronsport -

I I I I t I I I I I I I I

5. -15.




62
Table 6.3: Lag for T" in Sensitivity Tests A5-A7 and B3 i+j WEIGHT,W
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 -, 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 KC3 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 Oh/Oy 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, Tb- 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)-1, which resulted in the unstable profile, shown in Fig. 6.9, after 100 time steps. The next test, Test A10 included both the parameters T; and 9h/ay 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 7; and i9h/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, T 7:, and Oh/Oy, were included. Each test used 10 time steps, K1 = 2.9 x 10-6 m4(Ns)' KC2 = 5.80 x 1i-5 m4(Ns)'I, and K3 = 5.80 x 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, Wi, 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
0.8
a)
0.4
E
1 0.2
0
0.0
L-0.2
0
0
S-0.4
C
0
L -0.6-0.8
-1.0
2.
1.
0.
T-1.
-3.
-4.

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

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

Figure 6.5: Test A5. Same as Fig. 6.2 except using lags in Table 6.3.

1 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 1 1
0.8 Tronsport
0.6
S0.4 N 0.2 o .
0.0
0.
S-0.4
S-0.6
-0.8
-1.0 i I u I t I I I ,1,
2.
Original
1. -Model result
---- Observed result:
0.
x-2.
-3.
-4.
-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.
distance (m)

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
1.Model result
0' -- Observed result-2.
-3.
-4.
-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 m2s-1.

-2
1.0 0.8
" 0.6
CU
0
E
0.2
o
0.0 L -0.2
* -0.4
C
o
: -0.6
-0.8
-1 0

Transport
I I i I I I t I t I i I I t I _




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

-S. I I I a I I I 1 1 1
-25. -15. -5. 5. 15. 25. 35.
distance Wi]

45. 55. 65. 75.

Figure 6.8: Test A8. Same as Fig. 6.1, except here only the parameter 9h/i9y 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 0.06
CU
0.04
E
i 0.02
0
~0.00
u-0.02
0.o
2-0.04
-0. 06
-0.08
-0.10
2.
1.
0.
Z-1.
-2.
-3.
-4.

a




-2
0.10
0.08 S0.06 S0.04
E
* 0.02
o
(0
,0.00
o
"- 0.06
-0.08
-0. 0
1.
C
0.
-.6
-0.0
-0.10
2.
1.
0.
Z -1.
-3.
-4.

5.

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

Transport
I I p I t I i I I I i I I ,
r -

-5. I I 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 6.9: Test A9. Same as Fig. 6.1, except here only the parameter Y is used for 100 time steps; K2 = 5.80 x 10-s m4(Ns)-1.




-25
0.10
0.08
~0.06
0.04
S0.02
o
0.00
E-0.02
0
2-0.04
C
o
cL
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 (m)

Figure 6.10: Test A10. Same as Fig. 6.1, except here both the parameters 7 and ah/y are used for 100 time steps.

Transport
, I I I I I I I I !

5. -15.




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

0.
cL-2.
-3.
-4.
-5. I I I I I m
-25. -15. -5. 5. 15. 25. 35.
distance m]

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
S
N 0.02
(0
OL0.00
S-0.02
0
0.
2-0.04
0
-0.05
-0. 10
2.

' I Trnsp Iort I
Transport

I I t 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

5. -15.
i l




Table 6.4: Lags Used in Second Set of Sensitivity Tests
TEST -LAG TABLE USED
BI 6.5
B2 6.6
B3 6.3
B4 6.2

Table 6.5: Lag for -fb in Sensitivity Test B1 i+j WEIGHT, W1
i+4 0.00
i3 0.00
i+2 0.00
i+1 0.00
1 0.20
i-i 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 bartrough system formation. It is obvious from the first test, Al, the Tb 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 1 I lI I
0.8 Tronsport
0.6
U =
N 0.4
E
' 0.2
0
0.
0 -.
-0.6
-0.8
-1.0
0.
-
-3.
-4.
-5.
-25. -15. -5. 5. 15. 2S. 35. 45. 55. 65. 75.
distance (m)

Figure 6.12: Test B1. All three forcing functions used with lags in Table 6.5.




-25
1.0 0.8 ;0.6
0.4 0.2

0.0
0
C
-0.4
iL-0.6
-0.8
-1.0
2.

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

I. Model result" Observed result0.
-U2. ., %
-3.
-4.
-S I 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 aI I I
Tronsport

IN ./

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 1 1 1 I
0.8 Transport
0.6
0.4 -i .E
S0.2
0.0
-0.2
o
(0
CL
a -0.4
C
0
-0.6
-0.8
-1.0 I I t I i I I i I i I m
------- Observed result-1.
0.
-3..
-4.
-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.
distance (m)

Figure 6.14: Test B3. Same as Fig. 6.12, except lags used are in Table 6.3.




-25. -15.
1.0 ,-0.8 S0.6 N 0.4
a
0.2
o
S0.0 &-0.2
0~
CL
o -0.4
L -0.6
-0.8
-1.0
2. I
0.

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

-5. 1 I I I I I !g 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-i 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 T 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 T;, 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 K, = 2.4 x 10-6 m4(Ns)-', K2 = 8.4 x I0-' m4(Ns)-', and K3 = 4.2 x 10-4 m2s-1 Using the larger I2 value in these computations was found to be necessary to allow the forcing function (-), which produces shoreward sediment motion in the model, to compensate for the increasing magnitude of the slope term (ah/Oy) as the




78
Table 7.1: Lag Weights Used for prediction of Bar Formation and Migration i+j Weight,W
i+5 0.01
i+4 0.03
i+3 0.04
i+2 0.05
i+1 0.06
i 0.07
i-i 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
j=-S
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 Daily wave model incorrectly located the breaking wave, where -Tb =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 (U
, 0.4
E
N 0.2
- 0.0
C. -0. 2
C
0
-0.4
C
'- -0.8
-0.8
-1.0
2.
1.
0.
s-i.

S-2.
-3.
-4.

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

-5.
-25.

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

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.

Transport
i I t I t I i I i I I a l I

-25.




-25
1.0 0.8 ;0.6
0 0.4 J 0.2 0.0
0 0~
-0.4
L
-0.6
-0.8

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

o1.0
2-i.
2.
-3.
-4.

-5. 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 (m)

Figure 7.2: Same as Fig. 7.1, except for 5 hours in Case 400.

I I I ran po r
Tronsport
p | I I I I I I *

I I I I ~ ~~i l l i n I i i




82
-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.
1.0 I I l
0.6 Transport
0.6
0.4
E
K 0.2 z0.0
0- .2
-0.4
0~ I-0.6
-0.8
2 I p I ; I u I I I *
-.
-4.
-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.
distance (in)

Figure 7.3: Same as Fig. 7.1, except for 15 hours in Case 400.




83
-5. 5. 15. 25. 35. 45. 55. 65. 75.

-2
1.0 0.8 "G0.6
a,
N0.4
E
i 0.2
0
- 0.0
-0. 2
-0.4
-0.6
-0.8
-0.0
2.
1.
0.
2-1.
-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
I I i I I i I I I I I I I ,
~Original "
- \ Model result
: -- Observed result-




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 (berm), 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 (JHb = 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 7,, 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.

Transport
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.
dlstonce (m)

Figure 7.5: Same as Fig. 7.1, except for Case 500 at 1 hour.

-2
1.0 0.8
; 0.6 S0.4
E
m 0.2
0
S0.0
-0.2
o
2-0.
0
o.
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.
1.01 1 1 1
0.8 Tronsport
0.6
a
0.4 0.2
0
0.0
S-0.2
0o
CL
0 -0.4
C
S-0.6
-0.8
-1.0
-------Original
1.
* ------Model result.
----Observed result0.
-.
-3.
-4.

-5. 1 1 1 '
-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.




5. -15.

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

-25
1.0 0.8
-O
Uo.
0.4
S0.2
0
-0.0
( -0.2
0
a -0.4
C
o
S-0.6
-0.8
-1.0
2.
1.
0.
-4.
-3.
-4.

-5. 1 1 ,
-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
t I I I p I I I I I i I i I
Model result Observed result:
-N




-2
1.0 0.8 0.6
O.4
E
N 0.2
0.
L -0.2
0
0.
0 -0.4
0
L
- -0.6
-0.8
-1.0
2.
1.
0.
.
-2.
-3.
-4.

5. -15.

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

L I p I I I I I
Tronsport
, I I I I i I I i I I i I
Original
~Model result

-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
i I I I m I i I t I I t I a
j I I I I I j I j I j I
Original Model result Observed resultI

-s 1 f I I I I I I I I I I I
-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.
distance W']

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
0
-0.0
0
-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 j i
0.8 Transport
0.6
N 0.4 E 0.2
-0.0
L -0.2
0
C
0-0.4
C
-0.6
-0.8
-1.0
2.
. - Original
1. Model result
: -Observed result0.
2
c-L 2
-3.
-4.
-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.
distance (m)

Figure 7.10: Same as Fig. 7.1, except for Case 401 at 1 hour.




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

S I I I I I I I p Ior
Transport
I I I I I i I I i I i I I I i

0.
-2.
-3.
-4.
-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 0.6
; 0.6
" 0.4
E
S0.2
0
oL
U 0.0

t-0.2
a
Q.
2-0.4
0
& -0.6
-0.8
-1.0
2. -




92
-25. -15. -5. 5. 15. 25. 35. 45. 55. 65. 75.
1.0 i
0.8 Tronsport
~;0.6
N0.4
E
' 0.2
0.0
-0.2
0
2-0.4
C
L-0.6
-0.8
-1.0 I I I t I I I I *
~-1.0
-3.
- Original
1. \. Model result
0. -- Observed result-2.
-5.
-2S. -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
a)
0.4
E
S0.2
o
0
- 0.0
L -0.2
a
CL 0' -0.4
c
a
- -0.6
-0.8
-1.0
2.
1.
0.
-1.
C
o-2.
-3.
-4.

5. -15.

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

I I I I I I I I I
Transport
I I I I I I eI uI I

-------Originl
\ ------- Model result .
. - - - O b s e r v e d r e s u l t "
-
, i"\\

-5 I I I I I I I I I I I I J
-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.