UFL/COEL99/O1O
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
INTRODUCTION ..... ....................................................... 1
BA CK G RO UND ............................................................. 1
FORMULATION ............................................................ 2
Grid Scheme ............................................................ 2
Equilibrium Beach Profile ................................................. 3
Transport Equation ....................................................... 3
Continuity Equation .... .................................................. 5
Water Levels and Wave Heights ............................................ 5
W ave Runup ..................... ................................... 5
Onshore and Offshore Slopes .............................................. 5
METHOD OF SOLUTION .................................................. 6
TESTS AND RESULTS ....................................................... 6
Test 1 Comparison of Model Results with m = 1 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 ...................... 11
Test 3 Comparison with Average Profile Data from Hurricane Eloise ............. 12
CONCLUSIONS AND RECOMMENDATIONS ................................. 12
C conclusions .... ....................................................... 14
Recommendations .... .................................................. 14
REFERENCES .............................................................. 15
APPENDIX A. RATIONALE FOR m = 3 IN THE TRANSPORT RELATIONSHIP ... 16
ii
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 m113. To Transform to ft'3, Multiply by 1.5 ................ 4
3. Initial Profile Considered in Test 1. Breaking Wave Height = 6 ft .................. 7
4. Maximum Profile Retreat vs. NonDimensional 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, Case 1 .......................................................... 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 (ft) ................................... 4
2. Tests Described in this Report ............................................. 6
3. Comparison of Evolutionary Time Scales for Transport Relationships with m = 1
and m = 3 ............................................................. 8
4. BestFit 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 crossshore 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 peerreviewed 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 loop 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 processbased 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 twodimensional 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 crossshore
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 hn, = 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 + BA h =A 3/2 32
h DA hAAy (1)
where h is the water depth at a distance y from the shoreline, A is the socalled "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 equilibrium 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
D
h = y (2)
Bg
which is a linear relationship between water depth and distance in accord with observations and in
deeper water, Eq. (1) approximates
h =Ay2/3 (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 crossshore sediment transport qy (y) is
qy(y) = K (D D.)m (4)
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 m = 1 whereas it can be shown that the
m = 3 is dimensionally correct and for the laboratory data available, is more appropriate. CROSS
0.01
E 1.0 
LU
0. 0.10 
w
_J
a:
0 0.01
W 0.01
0.
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". To Transform to ft/3, Multiply by 1.5.
Table 1
Summary of Recommended A Values (ft"3)
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 m = 3. Appendix A provides a more detailed rationale for the use of
a transport relationship with m = 3.
Continuity Equation
The continuity equation is
ay_ aqy
at ah
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
in which FR is a runup coefficient, H is the wave height, Lo, 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,
"cascading" occurs with the sand redistributed to the adjacent contours until the maximum allowable
slope is reached.
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 = 1 and m = 3
Prior to comparing the results from models with transport relationships using m=l 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
Test Description
1 Comparison of model results with m = 1 and m = 3
2 Comparison with profile evolution data from Dette and
Uliczka wave tank tests. Three cases
3 Comparison with average profile data from Hurricane Eloise
= 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 onehalf 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 onehalf 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 threequarters evolution than the model
with m = 1. The relative times required to reach onequarter and threequarters 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 556100.0 2 3 4 567101.0 2 3 4 567102.0
NonDimensional Time, t/t5o
Figure 4: Maximum Profile Retreat vs. NonDimensional 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,/ / t2 0.33 0.16
t3/4/tl/2 3.1 15.0
Equilibrium Retrea = 98 feet '
n = ... ..... ........ .... .. ............. .. ...... .... .....
nTheeQarters Equilibrium
:n:
OneHal Equilibrium Retre t
........... OneQuarter Equii rium I
iOneOuarter :E.quilIbrium
.....7 . ... .. .... . ..... . . 7. .. .. .. ..... .. .
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 min1 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=l 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
_J
0
(9
o
.o
< 10
20
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.
o on ... .. .. ... ....... . . . .  . .*. .... . ..... . .. . . .. .. .. ..... ...
0o 0
300 1" :: : :
E .
S 0 0 ........ ........... .... ........... ..... .. ............. . .. ... .
Lo 0
0 I
00
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
J
< 10 .
20
0 100 200 300 400 500 600 700 800
Distance From Baseline (ft)
Figure 7: Initial Profile with Foreshore. Test 2, Case 2.
180
160
140
120
E
6 100
S 80
W
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
BestFit 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"3 8 x 104
2 (Regular Waves) 3.0 x 103 8 x 104
3 (Irregular Waves) 1.0 x 103 1.4 x 104
600 I i
o Measured
 m=3
0
4 00m=1 *
300 .
5 200 ................ .................
0 ......
100
0
0 1000 2000 3000 4000 5000 6000 7000
Number of Waves
Figure 9: Comparison of Measured and Computed Eroded Volumes, Irregular Waves. Test 2,
Case 3.
It is noteworthy that the K values for Case 3 with irregular waves were smaller for both m = 1 and
m = 3 than the K values for Cases 1 and 2. As has been noted, the wave heights were considered to
be Rayleigh distributed; however, it may be that the wavemaker was unable to generate the larger
waves that were predicted by the Rayleigh distribution. Regardless, it is encouraging that there is
generally less variability in the K values with m = 3 than with m = 1.
Test 3 Comparison With Average Profile Data From Hurricane Eloise This test utilized
before and after Hurricane Eloise profiles provided by Dr. T. Y. Chiu (Personal Communication,
1999). The storm tide applied is presented in Figure 10 and a constant wave height of 10 feet was
applied. The K values used were K = 5 x 103 for m = 1 (the average of the two regular wave runs)
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
o 10
10
z
.0
S5
.)
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 cross
shore 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 longterm 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 onequarter
or so of the equilibrium value associated with the maximum storm tide, (3) The model has been peer
reviewed and published in a leading coastal engineering journal. The lack of peerreview 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 recovery better that previous relationships
employed. This is consistent with the slower transport associated with a profile only slightly out of
equilibrium as occurs with m = 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: (1 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. 5384.
Dette, H. and K. Uliczka (1987) "Prototype Investigation on TimeDependent Dune Recession and
Beach Erosion", Proceedings of Coastal Sediments '87, Speciality Conference on Advances in
Understanding of Coastal Sediment Processes, ASCE, pp. 14301443.
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 TimeDependent Beach and Dune
Response", Coastal Engineering, Vol. 9, No. 3, pp. 221246.
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 StormInduced
Beach Change, Report 1: Empirical Foundation and Model Development", U. S. Army Coastal
Engineering Research Center, Waterways Experiment Station, Technical Report CERC899.
Larson, M. and N. C. Kraus (1990) "SBEACH: Numerical Model for Simulating StormInduced
Beach Change, Report 2: Numerical Formulation and Numerical Tests", U. S. Army Coastal
Engineering Research Center, Waterways Experiment Station, Technical Report CERC899.
Swart, D. H. (1974) "Offshore Sediment Transport and Equilibrium Beach Profiles", Publication No.
276, Delft Hydraulics Laboratory.
Zheng, J. (1996) "Improved CrossShore 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. 34, p. 169201.
APPENDIX A
RATIONALE FOR m = 3 IN THE TRANSPORT RELATIONSHIP
Introduction
This appendix presents the rationale for an exponent m = 3 in the transport relationship rather than
m=l 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 Length2/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 m = 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, m = 3, in the transport relationship, but not with m = 1.
