Citation
Phase 1 development of a "next generation" beach and dune erosion model for CCCL and other beach management applications

Material Information

Title:
Phase 1 development of a "next generation" beach and dune erosion model for CCCL and other beach management applications
Series Title:
UFLCOEL-99011
Alternate title:
Beach and dune erosion model for the Coastal Construction Control Line and other beach management applications
Creator:
Dean, Robert G ( Robert George ), 1930-
Florida -- Dept. of Environmental Protection
University of Florida -- Dept. of Civil and Coastal Engineering
Place of Publication:
Gainesville Fla
Publisher:
Coastal & Oceanographic Engineering Program, Dept. of Civil & Coastal Engineering, University of Florida
Publication Date:
Language:
English
Edition:
Rev. July 26, 1999.
Physical Description:
iii, 16 leaves : ill. ; 28 cm.

Subjects

Subjects / Keywords:
Beach erosion -- Mathematical models ( lcsh )
Coast changes -- Mathematical models ( lcsh )
Genre:
government publication (state, provincial, terriorial, dependent) ( marcgt )
bibliography ( marcgt )
non-fiction ( marcgt )

Notes

Bibliography:
Includes bibliographical references (leaf 15).
Statement of Responsibility:
prepared for Office of Beaches and Coastal Systems, Department of Environmental Protection, Tallahassee ; submitted by Robert G. Dean.

Record Information

Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
43269793 ( OCLC )

Full Text
UFL/COEL-99/010

PHASE 1 DEVELOPMENT OF A "NEXT GENERATION" BEACH AND DUNE EROSION MODEL FOR CCCL AND OTHER BEACH MANAGEMENT APPLICATIONS
by
Robert G. Dean

July 11, 1999
Prepared for: Office of Beaches and Coastal Systems Department of Environmental Protection Tallahassee, FL 32399




PHASE 1 DEVELOPMENT OF A "NEXT GENERATION"
BEACH AND DUNE EROSION MODEL FOR CCCL
AND OTHER BEACH MANAGEMENT APPLICATIONS
July 11, 1999
Prepared for:
Office of Beaches and Coastal Systems
Department of Environmental Protection
Tallahassee, FL 32399
Submitted by:
Robert G. Dean
Department of Coastal and Oceanographic Engineering University of Florida Gainesville, FL 32611




TABLE OF CONTENTS
LIST OF FIGURES .......................................................... iii
LIST OF TABLES ........................................................... iii
INTRO DUCTIO N ............................................................ I
BA CK G R O UND ............................................................. I
FO RM ULATIO N ............................................................. 2
G rid Schem e ............................................................ 2
Equilibrium Beach Profile ................................................. 3
Transport Equation ....................................................... 3
Continuity Equation ...................................................... 5
W ater Levels and W ave Heights ............................................ 5
W ave Runup ............................................................ 5
Onshore and Offshore Slopes .............................................. 5
M ETHOD OF SOLUTION ..................................................... 6
TESTS AND RESULTS ....................................................... 6
Test I Comparison of Model Results with m = I and m = 3 ..................... 6
Test 2 Comparison with Profile Data from Dette and Uliczka Wave Tank Tests ..... 9
Case 1. Beach without Foreshore, Regular Waves ........................ 9
Case 2. Beach with Foreshore, Regular Waves .......................... 9
Case 3. Beach without Foreshore, Irregular Waves ...................... I I
Test 3 Comparison with Average Profile Data from Hurricane Eloise ............. 12
CONCLUSIONS AND RECOMMENDATIONS .................................. 12
C onclusions ........................................................... 14
Recom m endations ...................................................... 14
REFEREN CES .............................................................. 15
APPENDIX A. RATIONALE FOR m = 3 IN THE TRANSPORT RELATIONSHIP ... 16




LIST OF FIGURES

FIGURE PAGE
1. Model Grid Used in Profile Evolution, with y as the Dependent Variable and
Depth, h, and Time, t, as the Independent Variables........................... 2
2. Variation of Sediment Scale Parameter, A, with Sediment Size and Fall Velocity.
Note: Values of A are in in'13. To Transform to t'13, Multiply by 1.5 ................4
3. Initial Profile Considered in Test 1. Breaking Wave Height = 6 ft ................. 7
4. Maximum Profile Retreat vs. Non-Dimensional Time. Test 1 .................... 8
5. Initial Profile without Foreshore. Test 2, Cases 1 and 3 ......................... 9
6. Comparison of Measured and Calculated Eroded Volumes. Regular Waves. Test
2, CaselI.......................................................... 10
7. Initial Profile with Foreshore. Test 2, Case 2............................... 10
8. Comparison of Measured and Computed Eroded Volumes. Test 2, Case 2 .......... 11
9. Comparison of Measured and Computed Eroded Volumes, Irregular Waves. Test
2, Case 3.......................................................... 12
10. Storm Surge Hydrograph Used for Hurricane Eloise Erosion, Test 3 ...............13
11. Comparison of Measured and Computed Beach and Dune Erosion, Hurricane
Eloise............................................................ 13
LIST OF TABLES
TABLE PAGE
1. Summary of Recommended A Values (0'3) ................................ 4
2. Tests Described in this Report........................................... 6
3. Comparison of Evolutionary Time Scales for Transport Relationships with m = 1
andmn= 3.......................................................... 8
4. Best-Fit K Values for the Three Tests of Dette and Uliczka ....................11




PHASE 1 DEVELOPMENT OF A "NEXT GENERATION"

BEACH AND DUNE EROSION MODEL FOR CCCL
AND OTHER BEACH MANAGEMENT APPLICATIONS
INTRODUCTION
Beaches respond to the elevated water levels and high waves associated with storms by beach and dune recession with substantial quantities of sediment transferred from the dry beach and shallow water portions of the profile to the more seaward portions of the active profile. The responsibilities of the Office of Beaches and Coastal Systems (OBCS) of the Florida Department of Environmental Protection (FDEP) include both regulation and identification of the landward limits of a 100 year storm, both of which require a capability to predict profile response to storms. The model for profile (beach and dune) evolution in use by the OBCS was developed in the early 1980's and has served the OBCS well for purposes of both regulation and as an element in the process of setting the Coastal Construction Control Line (CCCL). Since the 1980's considerable progress has been made in the understanding of cross-shore sediment transport including results from the laboratory and field; this progress warrants the development of an improved profile evolution model which incorporates new developments, understanding of processes and data.
The purpose of this report is to develop and provide the results of a limited evaluation of a recently developed profile response model (Zheng and Dean, 1997) for the requirements of the OBCS. This model uses a relatively new and peer-reviewed transport relationship which appears to have inherent merits over those employed previously. Considerable further development and calibration will be required prior to adoption of this model as a broadly applicable and suitable tool for the next two decades or so.
BACKGROUND
The two generic types of profile evolution models can be referred to as "open loop" and "closed loop". The distinguishing feature of these two types of models is that closed loop models specify a target profile to which the evolving profile will converge if the forcing conditions are maintained constant, whereas there is no guarantee that open ioop models will converge. Open loop models require more physics, are the more basic of the two and generally specify the hydrodynamics and sediment transport separately and, on this basis, calculate the profile evolution. At the present time, open loop models are regarded as being more in the research arena whereas closed loop models are much more suited for applications. This situation may change in the future but is not expected to do so for at least the next several decades. The current OBCS profile evolution model is of the "closed loop" type as are "EDUNE" developed by Kriebel (1982) and Kriebel and Dean (1985) and SBEACH developed by Larson (1988) and Larson and Kraus (1989, 1990).
The model being evaluated here has been termed "CROSS" and was developed by Zheng (1996) and Zheng and Dean (1997). This model is one stage more process-based than is the current OBCS




model in that it incorporates a transport model whereas the current model employs a fixed profile (relative to the changing water level) and a time scale to establish the profile evolution. Moreover, as described later, the transport relation utilized in "CROSS" has certain inherent advantages over that in EDUNE and SBEACH.
FORMULATION
As for the case of most profile evolution models, CROSS is a two-dimensional model. That is, sand can be transferred across the profile; however, no sand is lost out of the profile. The most basic formulation includes a sediment transport model and a conservation of sand equation for the underwater portions of the profile. In the paragraphs which follow, the grid scheme employed will first be described followed by the equations and various elements incorporated into the model.
Grid Scheme
The model proposed here (a modification of CROSS) utilizes a grid which specifies the cross-shore displacements y (h,t) at evenly spaced contour elevations, hi and at various times, t. The quantity h represents depth and thus elevations above the vertical datum (NGVD) are negative. Figure 1 presents the grid scheme.

MSL
I h,1,= A(y y..)2/3

Figure 1: Model Grid Used in Profile Evolution, with y as the Dependent Variable and Depth, h, and Time, t, as the Independent Variables.

I -




Equilibrium Beach Profile.
The equilibrium beach profile is represented by the following form
h 3/2+ g-A 3/2h= 3/2
D- y 1
where h is the water depth at a distance y from the shoreline, A is the so-called "Sediment Scale Parameter" and has been found to be a function of sediment size as shown in Figure 2 or as presented in Table 1, D. is the eciuilibrium wave energy dissipation per unit volume, D is the local wave energy dissipation per unit water volume, g is the gravitational constant and B is a constant related to the beach slope at the shoreline. It can be shown that in shallow water, Eq. (1) reduces to
h = D. Y(2)
B g
which is a linear relationship between water depth and distance in accord with observations and in deeper water, Eq. (1) approximates
h-=Ay 213 (3)
which was first proposed by Bruun (1954) and later tested and evaluated by Dean (1977, 1991). It can be shown that Eq. (3) is consistent with uniform wave energy dissipation per unit water volume. This concept is consistent with intuition since: (1) The greater the wave energy dissipation per unit water volume, the greater the turbulence and the forces that would remold the profile such that the turbulence level would be uniform across the profile, and (2) The larger grain sizes can withstand a greater level of turbulence. Equation (3) has the disadvantage that the slope becomes infinite at the shoreline and this disadvantage is removed by the somewhat more complex Eq. (1). To proceed with model formulation, a transport equation and continuity equation are needed.
Transport Equation
The transport equation which has been used extensively for cross-shore sediment transport qY (y) is
in which D is the actual wave energy dissipation per unit water volume and as noted previously, D. is the equilibrium value of this variable. The form of Equation (4) is advantageous since when D = D., there will be no transport and the profile will be of the equilibrium form by definition. The models EDUNE and SBEACH use Eq. (4) with an exponent mn = 1 whereas it can be shown that the mn = 3 is dimensionally correct and for the laboratory data available, is more appropriate. CROSS




0.01 E 1.0
LU
LU
0. 0.10
W ~1 .)
a:
0 0.01 W 0.01
0L

SEDIMENT FALL VELOCITY, w (cmls)
1.0 10.0

0.1 1.0 10.0 100.0

SEDIMENT SIZE, D (mm)

Figure 2: Variation of Sediment Scale Parameter, A, with Sediment Size and Fall Velocity. Note: Values of A are in m"3. To Transform to ft"3, Multiply by 1.5.
Table 1
Summary of Recommended A Values (ft3)
D(mm) 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.0945 0.1008 0.1071 0.1134 0.1197 0.1260 0.1308 0.1356 0.1404 0.1452
0.2 0.1500 0.1545 0.1590 0.1635 0.1680 0.1725 0.1755 0.1785 0.1815 0.1845
0.3 0.1875 0.1905 0.1935 0.1965 0.1995 0.2025 0.2055 0.2085 0.2115 0.2145
0.4 0.2175 0.2199 0.2223 0.2247 0.2271 0.2295 0.2319 0.2343 0.2367 0.2391
0.5 0.2415 0.2433 0.2451 0.2469 0.2487 0.2505 0.2523 0.2541 0.2559 0.2577
0.6 0.2595 0.2613 0.2631 0.2649 0.2267 0.2685 0.2703 0.2721 0.2739 0.2757
0.7 0.2775 0.2789 0.2802 0.2816 0.2829 0.2843 0.2856 0.2870 0.2883 0.2897
0.8 0.2910 0.2922 0.2934 0.2946 0.2958 0.2970 0.2982 0.2994 0.3006 0.3018
0.9 0.3030 0.3042 0.3054 0.3066 0.3078 0.3090 0.3102 0.3114 0.3126 0.3138
1.0 0.3150 0.3162 0.3174 0.3186 0.3198 0.3210 0.3222 0.3234 0.3246 0.3258




uses a transport equation with mn = 3. Appendix A provides a more detailed rationale for the use of a transport relationship with mn = 3.
Continuity Equation
The continuity equation is
ay _aqy
at 8h
Water Levels and Wave Heights
The model can accommodate a specified changing water level with time. The model also can account for wave heights specified by their probability distribution.
Wave Runup
Through wave runup, waves will cause sediment transport at elevations higher than the mean water level. The wave runup, R, is calculated in accordance with the Hunt equation
R=FR Slope
P1ITL
in which FR is a runup coefficient, H is the wave height, L0, is the deep water wave length and "Slope" is a representative slope of the profile. For purposes here, the slope between the runup limit out to the break point is employed and a value of unity is used for the runup coefficient, FR.
Onshore and Offshore Slopes
The program allows specification of onshore and offshore slopes to ensure that they do not exceed some maximum specified value. These exceedances occur at the upper limit of runup where seaward transport can cause the angle of repose of the profile to be exceeded above this elevation. Slopes exceeding the allowed can also occur at the seaward end of the computational domain where the most seaward active contour is displaced seaward, thus creating a locally steep slope. At each time step, the program checks and if it is determined that a local slope exceeds the specified values, cascadingn" occurs with the sand redistributed to the adjacent contours until the maximum allowable slope is reached.




Test Description
1 Comparison of model results with in = 1 and m = 3
2 Comparison with profile evolution data from Dette and
Uliczka wave tank tests. Three cases
F 3 Comparison with average profile data from Hurricane Eloise

METHOD OF SOLUTION

The numerical solution requires three components: (1) Casting the governing equations presented in Section 2 into finite difference forms, (2) Developing a method for solving the finite difference equations, and (3) Programming the finite difference equations for solution.
The details of the solution will not be presented here. Basically, the two types of numerical solutions available are the "explicit" and "implicit" methods. The explicit method solves the transport and continuity equations sequentially whereas the implicit method solves these two equations simultaneously. The primary advantage of the explicit method is simplicity in formulation and programming; however, in order to avoid numerical instabilities, very small time steps can be required. The implicit method allows much larger time steps while maintaining the solution stable. For the model programmed and demonstrated here, the implicit method was chosen for advantages of stability and longer time step features.
TESTS AND RESULTS
Three different tests demonstrating the results of "exercising" the model will be presented to illustrate its characteristics and differences from previous models. These tests/comparisons are listed in Table 2.
Table 2
Tests Described in This Report

Test 1 -Comparison of Model Results With m = I and m = 3
Prior to comparing the results from models with transport relationships using m=1 and m=3, it is useful to discuss what differences and similarities exist for the two models. First, both of the models should converge to the same target profile for unchanging wave conditions and unlimited time. Thus, the only difference that exists is the evolution from the initial profile to the target profile. The principal difference is that the evolution with a transport relationship with m = 3 will evolve more rapidly at first and more slowly later. The forms of this evolution for the two transport relationships are examined in the following paragraphs.
The initial profile used in the simulations here is shown in Figure 3 along with the following wave conditions: Hb = breaking wave height = 6 ft, T = wave period = 6 sec, and S = constant storm surge




= 6 ft. The most instructive way to demonstrate the differences in the form of the evolution is to select a particular feature on the two profiles and to represent the variation of this feature as a function of time. For purposes here, the maximum shoreline recession was selected. This usually occurred at the still water line (ie at an elevation of 6 ft) or, in some cases, at 5 feet. In order to provide a reasonable basis for comparison, the rate constants (K values) in the transport equations were selected to yield the same time at one-half the maximum recession distance to equilibrium which was determined to be 98 ft. Figure 4 presents the results in which the horizontal axis is the ratio of time to the time for one-half equilibration (t50)' It is seen that, as expected, the model with m = 3 reaches the quarter evolution considerably sooner than the model with m = 1. Also, the model with m = 3 requires a much greater relative time to reach three-quarters evolution than the model with m = 1. The relative times required to reach one-quarter and three-quarters of full evolution are summarized in Table 3.

200 400 600 800 1000 1200
Distance From Baseline (ft)
Figure 3: Initial Profile Considered in Test 1. Breaking Wave Height = 6 ft.

1400




100 90
80 70 60 50
40 30 20 10
0
10

-1

.0 2 3 4 5 5 6 100.0 2 3 4 5 6 7 101.0 2 3 4 5 6 7 102.0 Non-Dimensional Time, t/t5o
Figure 4: Maximum Profile Retreat vs. Non-Dimensional Time. Test 1.

2 3

Table 3
Comparison of Evolutionary Time Scales for Transport Relationships With m = 1 and m = 3
Exponent in Transport Relationship Relative Time
m= 1 m=3
t/4 / t12 0.33 0.16
t3/4/tl/2 3.1 15.0

... . . . . . -Z .. ...,...,. .,.. .. ... . . .. ,
E qu Iliribm Retreid = 98 feet
...~~~~..... ... ....... ...... ........ ..... ..... ........... .......... .. ..'" ........ .
Sn 1 =i ThreeQtiartersEquilibriumi
I ... .. ... ...ii ii i i i i:i : : .. ... ....... ......... i. .... .. i ........ ........
nn
one-Hi Equilibrium REiibri m
. .............. .............. ........ .....................e E.......q ........I r...i
. . . . ,. .. .i . . .. . . . . ... . . .i . . . . . . .
i':"i~:'I Eqiibim R, i i Oeurt 'atuli~ru
. ... ... ... '. .'..."." .,........ . . .. ..L ........... I . : .' ........ I . .




Test 2 Comparison With Profile Data From Dette and Uliczka Wave Tank Tests
Dette and Uliczka (1987) presented the results from three large scale wave tank tests. The sand size used in these tests was 0.33 mm resulting in a sediment scale parameter of 0.13 in13 or 0.197 ft13 (Table 1). The profile data published by Dette and Uliczka were analyzed to determine the eroded volumes as a function of time. These volumes were compared with the volumes predicted by the transport equations with m=1 and m=3. In these tests, the value of the sediment transport coefficient, K that provided the best fit to the data for each test was used. Each of these tests is discussed below.
Case 1. Beach Without Foreshore Regular Waves These tests were conducted with the initial profile shown in Figure 5 and with a periodic wave height of 4.9 feet and a wave period of 6 sec. The comparison between the eroded measured and calculated volumes is shown in Figure 6. It is seen that the results with m = 3 provide a considerably better fit that those with m = 1.

10
0
_J
0
M (Ti < -10
.0
-20

100 200

300

Distance From Baseline (ft)
Figure 5: Initial Profile without Foreshore. Test 2, Cases 1 and 3.
Case 2. Beach With Foreshore, Regular Waves The profile for which these tests were applied is shown in Figure 7 and the wave conditions were the same as for Test 1. The measured and calculated eroded volumes are presented in Figure 8.




4 0 0 .... . ........ .......... | i"" .. ..... .. .. .. ....:. .. ... ....... ::..... ......
00 0
0j 200 0
2 ~ ~ 200 .. .... . .. .. .. ....1 1 1- ..... .... .. .. .... . .... ...
LU> 0 : .
o 2 0 . ./ . . . . . . . . .. . ... . . .. .. .. ... . . .. .. . . . .. .. ..... . . ...
I I I I
100 ....... .............................................
0
0 1 2 3 4 5 6 7 8
Time (Hours)

Figure 6: Comparison of Measured and Calculated Eroded Waves. Test 2, Case 1.

Volumes. Regular

10
<., -10 ......... .......
-LJ
-2
f. 0 100 20 300 400 500 600 700 800
Distance From Baseline (ft) Figure 7: Initial Profile with Foreshore. Test 2, Case 2.




180
160
140
120
E
6 100
(D 80 "0
a
2U

1 2 3 4

Time (Hours)
Figure 8: Comparison of Measured and Computed Eroded Volumes. Test 2,
Case 2.
Case 3. Beach Without Foreshore, Irregular Waves The initial profile used in these tests was the same as used in Test 1 as shown in Figure 5. The significant wave height of the irregular waves was 4.9 ft and the period of the spectral peak was 6 sec. For this case, a Rayleigh distribution was applied for the determinations of wave heights. The results are presented in Figure 9 where similar comments apply as for Tests 1 and 2.
The best fit K values in the transport relations for these three tests are presented in Table 4.
Table 4
Best-Fit K Values for the Three Tests of Dette and Uliczka
Best Fit K Values
Test
m = 1, K (ft/pound) m = 3, K (ft s2/pound3)
1 (Regular Waves) 7.0 x 10" 8 x 10-4
2 (Regular Waves) 3.0 x 10" 8 x 10-4
3 (Irregular Waves) 1.0 x 10.1 1.4 x 104




600 I
* 0 Measured- *- m=3
500 ........ ................... ...
0
4~ 00
V0
0
00
00 .. 000 200.00.00.00.60.70
2 ~ ~ ~ ~ ~ ~ ~ ~ N m e of0 W aves ... ....... ....... ................ ......
Fiur9:CmaioofMaueanCoptdEoeVouerrglrWvsTet2
Case 3
Iti oeoty htteKvlesfrCs ih reua aeswr mle0orbt n
m0 ... 3.than.the.K.values.for.Cases..and.. As has...... been..... noed.te.av.higtswee .onidre. t
Testr 39- Comparison Wit Mease Prdoiled aaErome Hourcae Iregls WaThis. Test utlie
befs oewondyafte hiae Evloieoies provs3wiiegbyar. T.ve wer Chiul(erso Cotmunican, 1999). Then torm tidales applies prsneIn ue1 and a cons bentetat wave height on10deetoa
geapld Tesvraiyi he K valuesusdwrK=5x ithfo m = 1 (hnth avrg of th1w.euarwvus
and K = 8 x 10' for m = 3. The results of these calculations are presented in Figure 11. In evaluating these results, it is important to recall that the use of a constant wave height of 10 feet is somewhat arbitrary which may account for some of the differences between the measured and calculated eroded profiles.




0 2 4 6 8 10 12 14
Time (Hours)
Figure 10: Storm Surge Hydrograph Used for Hurricane Eloise Erosion, Test 3.

20
15
a >_ 10
0 .0
.o- 5
.2, 4)
w

100 200 300 400
Distance From Baseline (ft)

Figure 11: Comparison of Measured and Computed Beach and Dune Erosion, Hurricane Eloise.




CONCLUSIONS AND RECOMMENDATIONS

Conclusions
This report describes and provides a preliminary evaluation of a beach and dune erosion model with several advantages over the model currently in use by the Office of Beaches and Coastal Systems.
These advantages include: (1) The model is more physics based and requires the solution of a crossshore sediment transport model and a continuity equation, whereas the current OBCS model simply maintains the specified profile form and allows this to be displaced upward and landward with a rising and/or falling storm tide, maintaining sediment conservation in the process. The evolution of the profile toward equilibrium for any current storm tide and wave height is represented through a specified exponential form in the current OBCS model, (2) The transport relationship included in the new model is more dimensionally correct than other published models of this general type and is more consistent with long-term profile evolution results from large scale wave tank tests. The differences are a rapid evolution early in the process and a slower evolution later as the profile approaches equilibrium. This would seem to be important, for example, under hurricane conditions which, because of their usually short duration storm tides, cause erosion which may be one-quarter or so of the equilibrium value associated with the maximum storm tide, (3) The model has been peerreviewed and published in a leading coastal engineering journal. The lack of peer-review of the model currently in use has been a source of criticism, (4) Because the model is more physics based, employing a transport and continuity equation, adaptation of the model to incorporate other desirable features including the effects of seawalls and overwash should be more readily possible, and (5) Although not presented in this report, observations conducted during this investigation suggest that the new transport relationship may represent profile recoyM better that previous relationships employed. This is consistent with the slower transport associated with a profile only slightly out of equilibrium as occurs with in = 3.
Recommendations
It is recommended that the development of the model examined in this report be continued for eventual adoption by the OBCS for CCCL and other management purposes. The continued development would include, but not necessarily be limited to the following: (I Comparison of model output with as many high quality data sets as possible. Also comparison with other models, (2) Include in a physically realistic manner, the presence of seawalls, (3) Include in a physically realistic manner, the effects of overwash, and (4) Consider the use of a different model grid, one that would not require monotonic profiles so that offshore bars could be represented more readily.




REFERENCES

Chiu, T. Y. (1999) Personal Communication.
Dean, R. G. (1991) "Equilibrium Beach Profiles: U.S. Atlantic Gulf Coasts", Ocean Engineering Technical Report #12, University of Delaware, Newark, DE.
Dean, R. G. (1991) "Equilibrium Beach Profiles: Characteristics and Applications", Journal of Coastal Research, Vol. 7, No. 1, pp. 53-84.
Dette, H. and K. Uliczka (1987) "Prototype Investigation on Time-Dependent Dune Recession and Beach Erosion", Proceedings of Coastal Sediments '87, Speciality Conference on Advances in Understanding of Coastal Sediment Processes, ASCE, pp. 1430-1443.
Kriebel, D. L. (1982) "Beach and Dune Response to Hurricanes", M. Sc. Thesis, Civil Engineering Department, University of Delaware, Newark, DE.
Kriebel, D. L. and R. G. Dean (1985) "Numerical Simulation of Time-Dependent Beach and Dune Response", Coastal Engineering, Vol. 9, No. 3, pp. 221-246.
Larson, M. (1988) "Quantification of Beach Profile Change", Report No. 1008, Department of Water Resources and Engineering, University of Lund, Lund, Sweden.
Larson, M. and N. C. Kraus (1989) "SBEACH: Numerical Model for Simulating Storm-Induced Beach Change, Report 1: Empirical Foundation and Model Development", U. S. Army Coastal Engineering Research Center, Waterways Experiment Station, Technical Report CERC-89-9.
Larson, M. and N. C. Kraus (1990) "SBEACH: Numerical Model for Simulating Storm-Induced Beach Change, Report 2: Numerical Formulation and Numerical Tests", U. S. Army Coastal Engineering Research Center, Waterways Experiment Station, Technical Report CERC-89-9.
Swart, D. H. (1974) "Offshore Sediment Transport and Equilibrium Beach Profiles", Publication No. 276, Delft Hydraulics Laboratory.
Zheng, J. (1996) "Improved Cross-Shore Sediment Transport Relationships and Models", Ph. D. Dissertation, Department of Coastal and Oceanographic Engineering, University of Florida, Gainesville, FL.
Zheng, J. and R. G. Dean (1997) "Numerical Models and Intercomparisons of Beach Profile Evolution", Coastal Engineering, Vol. 30, Nos. 3-4, p. 169-201.




APPENDIX A

RATIONALE FOR mi = 3 IN THE TRANSPORT RELATIONSHIP
Introduction
This appendix presents the rationale for an exponent m = 3 in the transport relationship rather than m=--1 as employed in EDUNE and SBEACH. Refer to Eq. (4).
Rationale
There are two bases which support the use of an exponent m = 3 in Eq. (4), the transport relationship:
(1) Dimensional arguments, and (2) Data from wave tank tests.
(1) Dimensional Arguments The dimensions of the transport rate per unit width are Length 2/time. Considering a Froude model in which gravity and inertia are the dominant forces and time scales as the square root of the length scale, it follows that the sediment transport per unit length should scale as Length /2. Since, the units of the term (D D*) are Length"2, with the exception of some physical constants which are fixed and would be the same in model and prototype, in order for the transport to scale according to the Froude model, the exponent must be mn = 3.
(2) Data From Wave Tank Tests Dean and Zheng (1997) have presented a figure from Swart (1974) in which wave tank tests were conducted for a duration of 2,800 hours and it appears that the system was still evolving toward equilibrium after at least 1,500 hours. This long equilibration time is consistent with an exponent, mn = 3, in the transport relationship, but not with mn = 1.