UFL/COEL94/018
CROSSSHORE SEDIMENT TRANSPORT
RELATIONSHIPS
by
Robert G. Dean
and
Jie Zheng
June, 1994
CROSSSHORE SEDIMENT
TRANSPORT RELATIONSHIPS
June, 1994
By:
Robert G. Dean
and
Jie Zheng
Department of Coastal and Oceanographic Engineering
University of Florida
Gainesville, FL 32611
INTRODUCTION
The increasing use of the coastal zone has made a working understanding of nearshore
and beach process an increasingly challenging aspect of coastal studies. Accurate estimates of
beach profile evolution in response to tides, storms and beach nourishments are required for a
variety of regulatory and design purposes. Due to the complexities of beach profile, sediment
characteristics and concentrations, wave and water level conditions, an analytical treatment is
difficult and recourse to numerical modeling is required, which includes a continuity equation
and a sediment transport equation.
Sediment transport at a point in the nearshore zone has crossshore and longshore
components. It appears under a number of coastal engineering scenarios of interest, the transport
is dominated by either the crossshore or longshore component. The crossshore component
determines profile evolutions primarily for beaches far away from structures and inlets and under
cases of water level increasing, storms and beach nourishment. In contrast to longshore sediment
transport modeling, which has been studied for about five decades, a focus on crossshore
sediment transport modeling is relatively recent (about 20 years) and uncertainty in predicting
effects of all variables thus may be considerably greater.
Crossshore sediment transport models can be broadly classified into two groups: "open
loop" and "closed loop" models. A "closed loop" model converges to a target (equilibrium)
profile, while an "open loop" model is not a priori constrained to the final profile. "Closed loop"
sediment transport models assume the existence of equilibrium profiles, which are conceptually
the results of the balance of destructive (offshore directed) versus constructive (onshore directed)
forces. Changes in a beach profile will diminish and finally cease if the beach is exposed to the
same conditions for a long time. Crossshore transport is caused by variation of a beach profile
from the equilibrium. This variation may result from a profile initially in disequilibrium or
changing of hydrodynamic conditions within the nearshore zone which results in an imbalance
of forces.
In the present study, only "closed loop" models are investigated. The proposed transport
relationship and Swart's model[1974] are studied and compared with two sets of large wave tank
data. One set is from experiments performed in a large German wave flume in Hannover [Dette
1987]. Another is from large wave tank experiments conducted by Saville in 19561957 and
1962 [Kraus 1988].
EQUILIBRIUM BEACH PROFILE
According to the balance of destructive and constructive forces and assuming wave
energy dissipation per unit water volume as the dominate destructive force, Dean[1977] has
proposed the following equilibrium condition for beach profiles as:
(ECG) = D. (1)
h ay
With shallow water and spilling wave breaker assumptions, Eq.(1) can be integrated to:
h(y) = A(D) y213 (2)
Approximately 500 profiles from the east coast and Gulf shorelines of the United States were
examined and provided reasonable support to this equilibrium form. The socalled sediment scale
(D) 24 D 12
parameter, A(D) = 3 3, in Eq.(2) depends on sediment size, D, or equivalently fall
velocity, w.
Two disadvantages of Eq.(2) are the infinite beach slope at the water line and the
monotonic form of profile. The first shortcoming is overcome by including gravity as a
significant destructive force when a profile becomes steep. In this case, Eq.(2) is modified with
beach face slope, mo,
y = +( )3/2 (3)
m0 A
Since the scale parameter, A, is only a function of sediment size, wave conditions are not include
relationship and Swart's model[1974] are studied and compared with two sets of large wave tank
data. One set is from experiments performed in a large German wave flume in Hannover [Dette
1987]. Another is from large wave tank experiments conducted by Saville in 19561957 and
1962 [Kraus 1988].
EQUILIBRIUM BEACH PROFILE
According to the balance of destructive and constructive forces and assuming wave
energy dissipation per unit water volume as the dominate destructive force, Dean[1977] has
proposed the following equilibrium condition for beach profiles as:
(ECG) = D. (1)
h ay
With shallow water and spilling wave breaker assumptions, Eq.(1) can be integrated to:
h(y) = A(D) y213 (2)
Approximately 500 profiles from the east coast and Gulf shorelines of the United States were
examined and provided reasonable support to this equilibrium form. The socalled sediment scale
(D) 24 D 12
parameter, A(D) = 3 3, in Eq.(2) depends on sediment size, D, or equivalently fall
velocity, w.
Two disadvantages of Eq.(2) are the infinite beach slope at the water line and the
monotonic form of profile. The first shortcoming is overcome by including gravity as a
significant destructive force when a profile becomes steep. In this case, Eq.(2) is modified with
beach face slope, mo,
y = +( )3/2 (3)
m0 A
Since the scale parameter, A, is only a function of sediment size, wave conditions are not include
 Shorerlse
5 2" l  Mark
, .. BB_
a i WaveCut Terrace
150
(dotted) and shorerise (dashed) curves, with x coordinates, x,, x2, and vertical
coordinates, hi, h2.
in Eq.(2). Although sediment size appears to be the predominant factor defining equilibrium
beach profiles, wave characteristics do have some effect. With the same sediment size, higher
and steeper waves move sand offshore and cause a milder beach and vice versa for milder
waves.
Inman et al.[1993] developed an equilibrium beach profile that treats a profile as two
parts, the inner (barberm) and the outer (shorerise) portions. The two portions are matched at
breakpointbar and fit by curves of the form
h = Bx" (4)
The coordinates of Inman's curve fitting are shown in Fig.l with the subscripts 1 and 2
corresponding to barberm and shorerise curves respectively. In addition to B and m in Eq.(4),
the origins of two curves must be determined from the profile data. In total, seven variables, x,e
zi, Bi, mi, X2, BA and im, are required for fitting a profile. This method is diagnostic and
generally useful for a beach with measured data available. Comparatively, The method described
earlier is prognostic and needs only a description of the sediment size.
earlier is prognostic and needs only a description of the sediment size.
Based on field observations, Dean[1973] hypothesized that sediment was suspended
during the wave crest phase position and if the fall time were less or greater than one half wave
period, the net transport would be landward or seaward, resulting in a bar formation for the
latter case. This mechanism would exist in the region of wave breaking. Considering the height
of suspended sediment to be proportional to the breaking wave height, resulted in identification
Hb
socalled fall velocity parameter, . By examining small scale wave tank data with only the
wT
deep water reference wave height Ho values available, the following relationship for net seaward
sediment transport and bar formation was found
Ho 0.85 (5)
wT
Later, Kriebel et al.[1986] examined only prototype and large scale laboratory data and found
a constant of 2.8 instead of 0.85 in Eq.(5). Kraus, Larson and Kriebel[1988] examined only
large tank data and proposed the following relationship for bar formation
Ho < 0.0007 Ho (6)
Lo wTr.
Since the fall velocity parameter plays a critical role in transport pattern and bar
formation, it may more or less affect the scale parameter, A. A modified model for the scale
parameter could be
A = A(D) 1+a[log( log (H)o (7)
where Ao(D) is the value evaluated at the reference fall velocity parameter, (H0), and a is
a coefficient to account for the influence of fall velocity parameter.
DISCUSSION OF CROSSSHORE TRANSPORT EQUATION
A beach which is steeper than equilibrium has a smaller volume of water over which a
given incident wave energy is dissipated. This causes the actual energy dissipation per unit
volume to be greater than the equilibrium value. As a result, the total destructive force is greater
than the constructive force. The profile will respond to the imbalance of forces through
redistribution of the sediment. Over time, sand will be carried from onshore to offshore and
deposited near the breakpoint. Similar to this process, for a beach with milder slope than
equilibrium sediments will be moved from offshore to onshore.
Based on these concepts, Kriebel and Dean[1985] proposed that the crossshore sediment
transport rate per unit beach width, Q, could be approximated according to the deviation of
actual wave energy dissipation per unit volume from the equilibrium at each location across surf
zone as:
Q = K(DD,) (8)
In which, D represents the actual total destructive force,
D = .5 pg3 ah3/ (9)
24 ay
The transport parameter, K, is considered as a dimensional constant. The following scaling
relationship is established from Eq.(8),
Qr (DD )mde& = (DD *) (10)
(DD*)prototype
For an undistorted model, according to the definition of D, the disequilibrium scale
(DD.), can be expressed by the length scale, L,,
(DD,)r = (11)
On the other hand, the Froude relationship gives the time scale, T,,
T, = V (12)
HT
Several model studies [Kriebel, et al. 1986 and Hughes, et al. 1990] have conformed that H
wT
is a valid modelling parameter such that if this parameter is the same in model and "prototype",
they are scaled versions of each other and the fall velocity is scaled by the length scale as
w, = F/. Two examples are presented in Figs.2 and 3. Following the relationship of Eq.(12),
the crossshore sediment transport, Q, should be scaled as:
L2 3
Q, L L2 (13)
Tr
This equation provides a basis for evaluating transport models. Obviously, Eq.(8) in which
Qr = La does not provide a valid scaling of the transport.
It is of interest to develop and test a transport model which can ensure convergence to
S370 WAUES Model Solid
S k U lHY Photo washed
2 RMS 0sPF = 0.484 m
10 0 10 '20 30 40 50
RANGE (m)
S160 WA Model Solid
1 0 WAVE U LH S Proto Ooshed
L2  RMS OfF = 0.444 m
4 
10 0 10 20 30 40 50
RANGE (m)
Fig.2. Comparison of beach profiles from medium and large scale wave tank, scaling
according to (Ho/wT),= 1 (from Hughes, et.al. 1990).
Fig.3. Comparison of beach profiles form medium and large scale wave tank, scaling
according to (Ho/wT),=1 (from Kriebel, et al. 1986).
the target(equilibrium) beach profile and also satisfy the scaling relationship given by Eq.(13).
One approach is to consider the following form for the transport model
Q = k(DD.,)IDD, n" (14)
Which results in the following scaling relationship
7
(15)
Qr = k,(DD.), IDD, I1
By equating Eq.(13) and (15), we have,
n 3
Q, = k,(DD)r, DDr = kr L = L (16)
If k is only function of the fall velocity parameter, k, is independent of the length scale and
equals unity and n=3 is determined such that both scaling relationship and convergence to the
equilibrium profile can be satisfied. Otherwise, if kr is related to the length scale as
k, = Lrm (17)
The following relationship must hold to satisfy Eq.(13),
m +! 3 (18)
2 2
In this case, any combination of m and n would satisfy the scaling requirement.
EVALUATION OF TRANSPORT RELATIONSHIP
Sediment transport in the nearshore region is a complicated process. Under different
conditions, some beaches reach equilibrium very fast, but others may vary slowly. The time
scale of beach evolution may vary from decades to thousands of hours for different experiments.
Fig.4 presents the results from the German "dune without foreshore" case and Saville's Case
300. In which, the eroded volume at any time is determined as the cumulative volume of
material displaced between initial and current profiles. It is noticed that the eroded volume of
Case 300 calculated from the observed profiles [Kraus 1988] is different from the one given by
Kriebel[1986]. The beach evolution of flume B case in Swart's experiments[1974] is shown in
Fig.5. Where L, and L2 are schematized profile lengths at water depth 8, and (81+82)
respectively. Comparing these three cases in Fig.4 and Fig.5, the question arises as to the
causes of the greatly different profile response times.
600.
500.
400.
300.
200.
100.
time [hours]
Fig.4. Eroded volumes versus time for the German "dune without foreshore" case and
Saville's Case 300.
6
S3 \L2L,=6.OS72.956t'oo9't0.98oo"ot0.8e1ess
/ REMARKS: W 6.057m
S SyO.S55x 10' m/hr
500 4000 4500 2000 2500 2800
time t (hours)
Fig.5. (L2L1) versus time for Swart's flume B case with 61 =0.15m, 8,=0.10m.
In an attempt to understand the causes of the different time scale, we examine the
following equation
dx = k (xx,) Ixx, I1
dt
(19)
Where x, is the equilibrium value of x. Nondimensionizing with x' = x/x, and t'= t/kx1, we
have
dx'
dx/ (xl) I/ 1"1
dtl
(20)
With the initial condition x'(t'=0) =Xo, the solution of Eq.(20) is given by
x'= 1+(xo 1) e
for n= 1,
x= 1+
xo1
[(nl) '1"11 +1]" n1
for n* 1.
(x'1)
Fig.6 shows the results of (x1) versus t' for n= 1, 2 and 3, and xo =2 and 10 respectively.
(xo1)
It appears that the time scale of the linear system (n= ) is independent of the initial conditions
and the two lines in Fig.6(a) are coincident. However, for the nonlinear systems, the initial
conditions do affect the time scale by the factor Ix'I "1. As n increases, this factor becomes
more and more significant.
The proposed crossshore transport
model is similar to Eq.(20). As
demonstrated above, a nonlinear transport
equation can affect, at least to some
degree, the variety of time scales in beach
evolution. In such nonlinear systems, the
greater variation of initial condition from
the equilibrium corresponds to the smaller
time scale of profile response. Returning
to the question arisen about Fig.4 and
Fig.5, it appears the time scale difference
in the three cases may be caused by the
different initial conditions and non
linearity of transport relationship. Among
three cases, the "dune without foreshore"
had the sand size of 0.33mm and the
initial slope of 1:4 with the wave height
of 1.5m and period of 6s, and its initial
condition is farthest from its equilibrium
0. 1. 2. 3. 4. 5.
(a) n
(a) n=1
6. 7. 8. 9. 10.
1.0
0.8  x'(t'=)=2
0.6  x'(t'=)=10
0.4
0.2 \
o.0
0.0  r   i
0. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
t'
(b) n=2
(c) n=3
Fig.6. Solution of Eq.(20)
x'(t'=0)=2
 x'(t'=O)=10
I I I I I I
and the beach takes the shortest time to approach equilibrium. For Case 300, the sand size was
0.22mm and the initial profile was 1:15 with a wave height of 1.68m and period 11.33s.
Comparing to them, Swart's flume B case had a sand size of 0.17mm and an initial slope of 1:10
with a wave height of 0.07m and period of 1.04s, its initial condition is closest to the
equilibrium and takes the longest time to arrive at the equilibrium.
In the following detailed study, the transport relationship, Eq.(14) with n=l and n=3,
is applied. Time dependent profile response is then determined by the numerical solution of
transport equation and continuity equation, which is
ay =_ (21)
at ah
Where y is offshore distance from a reference baseline. For each case, k is obtained through
bestfitting of eroded volumes overall the time.
1. German Big Wave Flume
The German big wave flume is 324 meters long, 7 meters deep and 5 meters wide. Two
experiments were carried out with the same wave conditions and different initial profiles.
Regular waves were generated at a water depth of 5 meters with wave height of 1.5 meters and
period of 6 seconds. The sand used for both experiments had a mean diameter of D50=0.33mm,
which corresponds to a fall velocity of 5cm/s at a temperature of 200C. Two initial profiles were
termed as "dune without foreshore" and "dune with foreshore". The "dune without foreshore"
had a dune crest of 2 meters above still water level(SWL) and a seaward slope of 1:4 down to
the channel floor. The "dune with foreshore" had a slope of 1:4 from the dune crest of 2 meters
above SWL to 1 meter below SWL and following a slope of 1:20 down to the channel floor.
Two cases have the same fall velocity parameter, 5. Results from Eq.(14) are compared
wT
with experiments. For n=3, the combined bestfit k value is 1x10ft's2/lb3. For n=l, the
"dune without foreshore" has bestfit k value of 4 x 103 ft/lb, while the "dune with foreshore"
has that of 2.2x 103ft4/lb.
1. 2. 3. 4. 5.
time [hours]
(a) Eroded volume versus time
5. I
time [hours]
(b) Qmax versus time
3(
2.0 
3.0  ..... 
3.0 
G Measured
 equation(14), n=3
 equation(14), n=1
 Swart's model
0. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
time [hours]
(c) Elevation of Qmax versus time
Fig.7. German "dune without foreshore". Comparisons of predicted to observed eroded
volumes, maximum transport rates and the corresponding elevations.
0.0
0.5
1.0
1.5
3.5
4.0
4.5
=
V VJ_ L
o 2.
S3. Initial profile 
4  Observed after 470 waves
S  Predicted after 470 waves
5.
0. 10. 20. 30. 40. 50. 60. 70. 80.
3.
S3. Initial profile
S  Observed after 1000 waves '
.3 Predicted after 1000 waves
5.
6 iI I I 1 11 
0. 10. 20. 30. 40. 50. 60. 70. 80.
3.
0. S
31.  
o 2.
S3. Initial profile
4.  Observed after 2100 waves 
 Predicted after 2100 waves
5.
6.I I I I 
0. 10. 20. 30. 40. 50. 60. 70. 80.
3.
2.
1.
 .
2.
4 3.  Initial profile
S Observed after 3700 waves
34  Predicted after 3700 waves
5. 
6. 1 1
0. 10. 20. 30. 40. 50. 60. 70. 80.
offshore distance [m]
Fig.8. German "dune without foreshore". Comparisons of predicted from Eq.(14) with
n=3 to observed profiles at different times.
13
Z 1.
S2.
S3.  Initial profile
S Observed after 470 waves
 Predicted after 470 waves
5. 
6. i i i
0. 10. 20. 30. 40. 50. 60. 70. 80.
3.
3.  Initial profile
 Observed after 1000 waves
. ~  Predicted after 1000 waves
5. 
6. I I I
0. 10. 20. 30. 40. 50. 60. 70. 80.
3.
1.
3. .
2.
S3.  Initial profile
S4. Observed after 2100 waves
4.  Predicted after 2100 waves
5.
6. I i
0. 10. 20. 30. 40. 50. 60. 70. 80.
3.
SWL
0.
2.
a 3.  Initial profile
4. Observed after 3700 waves
4 Predicted after 3700 waves
5.
6. I I I I I
0. 10. 20. 30. 40. 50. 60. 70. 80.
offshore distance [m]
Fig.9. German "dune without foreshore". Comparisons of predicted from Eq.(14) with
n= 1 to observed profiles at different times.
time [hours]
(a) Eroded volume versus time
5. I
time [hours]
(b) Qmax versus time
. 1. 2. 3. 4. 5.
time [hours]
(c) Elevation of Qmax versus time
6. 7. 8. 9. 10.
Fig. 10. German "dune with foreshore". Comparisons of predicted to observed eroded
volumes, maximum transport rates and the corresponding elevations.
15
0.0
0.
1.
2.
2.
3.
3.:
4.(
4.
5.(
0 A  C.  ^  ,
5 G(3 (3 ( ( 0 (OG (3G 0
0
5   
0 
5 Measured
5 Equation (14), n=3
0  Equation (14), n=l
5 . Swart's model
0t II I 1 1
)r
0
3.
2.
SWL
S2.
3.  Initial profile
4 Observed after 270 waves
S  Predicted after 270 waves
5.
6. I I
10. 0. 10. 20. 30. 40. 50. 60. 70. 80. 90. 100.
3.
2.
03'. SWL
02.
S3.  Initial profile
S Observed after 760 waves
4.
S Predicted after 760 waves
5.
6. I IIIi
10. 0. 10. 20. 30. 40. 50. 60. 70. 80. 90. 100.
3.
2.
1.
0. SWL
2.
S3.  Initial profile
S4 Observed after 1620 waves
 Predicted after 1620 waves
5.
6. I I I
10. 0. 10. 20. 30. 40. 50. 60. 70. 80. 90. 100.
3.
2.
1. .
3. i 
S2.
S3. .. Initial profile
S4 Observed after 2730 waves
 Predicted after 2730 waves
5.
6. 1i I I I
10. 0. 10. 20. 30. 40. 50. 60. 70. 80. 90. 100.
offshore distance [m]
Fig. 11. German "dune with foreshore". Comparisons of predicted from Eq.(14) with
n=3 to observed profiles at different times.
3.
2.
SYL
S2.
S3.  Initial profile
Observed after 270 waves
S4. . Predicted after 270 waves
5.
6.
10. 0. 10. 20. 30. 40. 50. 60. 70. 80. 90. 100.
3.
2.
1. sL
0.
2.
S3.  Initial profile
S Observed after 760 waves
4.  Predicted after 760 waves
5.
6. 1 1 i i i 
10. 0. 10. 20. 30. 40. 50. 60. 70. 80. 90. 100.
3.
2.
1.
SWL
0. .
3. t.
. 2.
S3. Initial profile
4. Observed after 1620 waves
S Predicted after 1620 waves
5.
6. I I I I I
10. 0. 10. 20. 30. 40. 50. 60. 70. 80. 90. 100.
3.
1.
SWL
3.~~~~~~~ 
0.
2 2.
S3. Initial profile
S Observed after 2730 waves
S4.  Predicted after 2730 waves
5. 
6. IiI
10. 0. 10. 20. 30. 40. 50. 60. 70. 80. 90. 100.
offshore distance [m]
Fig.12. German "dune with foreshore". Comparisons of predicted from Eq.(14) with
n= 1 to observed profiles at different times.
Figs.7 and 10 present comparisons of predicted to measured eroded volumes, maximum
transport rates and their corresponding elevations for cases "dune without foreshore" and "dune
with foreshore" respectively. The transport rate is defined as positive in the offshore direction
and negative for onshore direction. Predictions include Swart's model and Eq.(14) for the
maximum transport rates and their elevations. It appears that Eq.(14) gives much better
prediction for the transport rate than Swart's model. Moreover, the eroded volume appears to
be better fit by n=3 compared to n=l, especially for the "dune without foreshore" case. The
profiles predicted by Eq.(14) and the observed are compared for different times in Figs. 8, 9,
11 and 12. Overall, the model of n= 3 provides a better prediction for profile evolution than that
of n=l Note that the numerical model predicts a smooth monotonic profile form and can not
represent the bar formation.
2. Saville's Experiments
Among Saville's experiments, Cases 300, 400, 500 and 700 are investigated here. The
four cases had the same initial beach slope of 1:15 and the mean sand diameter of 0.22mm. The
corresponding fall velocity for the sand at a temperature of 200C is 3cm/s. Regular waves were
run for all cases. The experimental conditions and some observed results are presented in Table
1, which shows that Cases 500, 400 and 300 follow a trend. As the fall velocity parameter
decreases, the bestfit k value increases and the offshore bar becomes less significant. Case 700
deviates from the trend exhibited by the other tests. With a smaller fall velocity parameter than
Case 300, it has a smaller bestfit k value and larger offshore bar height.
Comparisons of predicted and observed eroded volumes, maximum transport rates and
the corresponding elevations are shown in the Figs 13, 16, 19 and 22 for the four cases. In the
prediction of Eq.(14), n= 1 and n=3 are used respectively. Bestfit k values are shown in Table
1. The agreement between the prediction of n=3 and the observed data is reasonably good for
Case 300 and Case 400, and acceptable for Case 500 and Case 700. It appears that the variation
of measured eroded volume with time is not consistent in Case 700. Although the wave
conditions were the same during the whole running time, the eroded volume reach a maximum
at 30 hours instead of increasing monotonically. Figures 14, 15, 17, 18, 20, 21, 23, and 24
present the comparisons of the observed profiles to the predictions of Eq.(14) with n= 1 and n=3
Table 1. Wave Conditions and Observed Results
case number 500 400 300 700
wave height Ho [m] 1.52 1.62 1.68 1.62
wave period T [s] 3.75 5.60 11.33 16.00
water depth [m] 4.57 4.42 4.27 4.11
Ho/wT 13.52 9.64 4.96 3.38
bestfit k value for n=3 2x1 05 3.8x 105 1x104 2x105
[fts2/lb3]
bestfit k value for n= 1 2.2 x 104 6.5 x 104 9 x 104 2.2 x 104
[ft4/lb]
eroded vol. at 30 hour 141.6 265.8 277.8 286.4
[ft3/ft]
bar height at 30 hour[ft] 1.65 1.58 1.22 1.31
for the four cases at different times respectively. The agreement between
observed is acceptable.
the predicted and the
SUMMARY AND CONCLUSION
The modified nonlinear crossshore transport equation is investigated with the scaling
relationship presented herein. Two sets of large wave tank experiments are compared against the
proposed transport equation. One set is from the German large wave flume in Hannover and
includes two cases with different initial beach profiles, the same wave conditions and sand size.
Another set is from Saville's large wave tank data and has four cases with the same initial beach
profile, different wave conditions and the same sediment size.
According to the scaling relationship, n=3 in the transport equation Eq.(14) is
determined, which provides better overall predictions than the linear relationship of n= 1. This
nonlinear crossshore sediment transport equation can reasonably explain the significant time
scale difference of profile evolution between different beaches. An analytical analysis of a
similar process demonstrates that the initial condition causes considerable differences in the time
400.
N 350. 
300. ..
250. 
E e
S200.
> 150. 
S(3 C Measured
100. / Equation (14), n=3
S50. Equation 14 n=l
50. g
0. I
0. 5. 10. 15. 20. 25. 30. 35. 40. 45. 50.
time [hours]
(a) Eroded volume versus time
100.
90. ,
80. G Measured
870. Equation (14), n=3
70.  Equation (14), n=l
60.  Swart's model
S50.
40.
320. 
10. 0 !
r 0.
10. " ....... 
0. 5. 10. 15. 20. 25. 30. 35. 40. 45. 50.
time [hours]
(b) Qmax versus time
0.0
0.0~~~ ~~ 0___0________
3.0 
0x 00
S6.0
S0 Measured
0 o Equation (14), n=3
9.0  Equation (14), n=l
.2  Swart's model
S12.0

15.0 i I i I I \_
0. 5. 10. 15. 20. 25. 30. 35. 40. 45. 50.
time [hours]
(c) Elevation of Qmax versus time
Fig.13. Case 300 from Saville's Tests. Comparisons of predicted to observed eroded
volumes, maximum transport rates and their corresponding elevation.
10.
1U.  
S10.  Initial profile ~
Observed at 5 hours
. Predicted at 5 hours
20.
100. 50. 0. 50. 100. 150. 200. 250.
10.
S10. Initial profile
Observed at 10 hours
. Predicted at 10 hours
20. I Il
100. 50. 0. 50. 100. 150. 200. 250.
10.
10.~~~ ~ ~~~~~~ 
S0. Initial profile
S Observed at 30 hours
. Predicted at 30 hours
20. I
100. 50. 0. 50. 100. 150. 200. 250.
10.
0o
10.~ ~ ~ ~ ~ ~ ~ ~~Z: 
> 10.  Initial profile
Observed at 50 hours
... Predicted at 50 hours
20. i I i
100. 50. 0. 50. 100. 150. 200. 250.
offshore distance [ft]
Fig.14. Case 300 from Saville's Tests. Comparisons of predicted from Eq.(14) with
n=3 to observed profiles at different times.
10. 
100. 50. 0. 50. 100. 150. 200. 250.
10.
0.
10.  Initial profile
Observed at 10 hours
 Predicted at 10 hours
20.L
1
100.
150.
200.
250.
10.1
20. 1
1
100.
20.
100.
100. 150.
200.
250.
50. 0. 50. 100. 150. 200. 250.
offshore distance [ft]
Fig.15. Case 300 from Saville's Tests. Comparisons of predicted from Eq.(14) with
n= 1 to observed profiles at different times.
SWL
 Initial profile
 Observed at 5 hours
. Predicted at 5 hours
I I I I I I
^^ ~ ~ ~~~ ~~~  _____________gfL_____
 Initial profile
Observed at 30 hours
 Predicted at 30 hours
I I I I I I
0.
10.
400.
350.
300.
250.
200.
150.
100.
50.
0.
time [hours]
(a) Eroded volume versus time
100.
90.
80. o Measured
Equation (14), n=3
70.  Equation (14), n=l
60.  Swart's model
50.
40. .
30. \
20. 
10. 
o.  ___ . . .,
0. 5. 10. 15. 20. 25. 30. 35. 40. 45. 50.
time [hours]
(b) Qmax versus time
0.0
5. 10. 15. 20. 25.
time [hours]
30. 35. 40. 45. 50.
(c) Elevation of Qmax versus time
Fig.16. Case 400 from Saville's Tests. Comparisons of predicted to observed eroded
volumes, maximum transport rates and their corresponding elevation.
3.0
6.0
9.0
12.0
0 Measured
Equation (14), n=3
 Equation (14), n=l
 Swart's model
I I I i I i i
15.1
'I
0.
I
0. F
10. 
20. 1 I I t I I I
100. 50. 0. 50. 100. 150. 200. 250.
10.
20.
250.
offshore distance [ft]
Fig.17. Case 400 from Saville's Tests. Comparisons of predicted from Eq.(14) with
n=3 to observed profiles at different times.
 _SWnL

 Initial profile
Observed at 5 hours
 Predicted at 5 hours
1Vt.
SWL
0.
. 
S10.  Initial profile
Observed at 5 hours
 Predicted at 5 hours
20. i 1
100. 50. 0. 50. 100. 150. 200. 250.
10.
O. 
S10.  Initial profile
Observed at 10 hours
 Predicted at 10 hours
20. i i i i i
100. 50. 0. 50. 100. 150. 200. 250.
10.
0.
    
S10. Initial profile
Observed at 30 hours
. Predicted at 30 hours
20. i
100. 50. 0. 50. 100. 150. 200. 250.
10.
0. ,
10. ~ ~ ~ ~ ~ ~ ~ ~ ~ ~  
S10.  Initial profile
Observed at 40 hours
. Predicted at 40 hours
20. i i
100. 50. 0. 50. 100. 150. 200. 250.
offshore distance [ft]
Fig.18. Case 400 from Saville's Tests. Comparisons of predicted from Eq.(14) with
n= 1 to observed profiles at different times.causes considerable differences in the time
400.
350.
300.
250.
200.
150.
100.
50.
0.(
time [hours]
(a) Eroded volume versus time
3.(
6.C
9.0
12.(
15.(
18.(
0
0 G Measured
Equation (14), n=3
Equation (14), n=l
 Swart's model
'
G 0
.....__ .. o___ 0 0

60. 70. 80. 90. 100.
). 10. 20. 30. 40. 50.
time [hours]
(b) Qmax versus time
10. 20. 30. 40. 50.
time [hours]
(c) Elevation of Qmax versus time
60. 70. 80. 90. 100.
Fig.19. Case 500 from Saville's Tests. Comparisons of predicted to observed eroded
volumes, maximum transport rates and their corresponding elevation.
)
, U o o o e 0
"G G G G G G
0
   .. .. . 
G Measured  Equation (14), n=3
 Swart's model  Equation (14). n=1
0) I I i I I I
10.
1 0  
S10.  Initial profile
Observed at 10 hours
 Predicted at 10 hours
20. I I I I
100. 50. 0. 50. 100. 150. 200. 250.
10.
0.
10. ,  
S10. Initial profile
Observed at 30 hours
 Predicted at 30 hours
20. I I I
100. 50. 0. 50. 100. 150. 200. 250.
10.
O. 
. 0 ..... 
10.  Initial profile
Observed at 60 hours
. Predicted at 60 hours
20. I
100. 50. 0. 50. 100. 150. 200. 250.
10.
0.
o 2............ .
S 10.  Initial profile
Observed at 100 hours
. Predicted at 100 hours
20. I
100. 50. 0. 50. 100. 150. 200. 250.
offshore distance [ft]
Fig.20. Case 500 from Saville's Tests. Comparisons of predicted from Eq.(14) with
n=3 to observed profiles at different times.
10.
20.
10.
0. 
10. 
20. I i I I 11
100. 50. 0. 50. 100. 150. 200. 25
0.
10.
20.
10.
0.
10.
20.
100.
250.
offshore distance [ft]
Fig.21. Case 500 from Saville's Tests. Comparisons of predicted from Eq.(14) with
n=1 to observed profiles at different times.
SWL
^"^ SS: ^ _______ SW
 Initial profile
Observed at 10 hours
 Predicted at 10 hours
I I I I I I
100. 50. 0. 50. 100. 150. 200. 25
SWL
^ ^ '  ^ ^_____________ ____
 Initial profile
 Observed at 30 hours
 Predicted at 30 hours
0.
 400.
S350.
300.
S250. 0
S200.  '
G0
S10. Measured
100. / '  Equation 14), n=3
o / . Equation 14), n=l
50. /
0.
0. 10. 20. 30. 40. 50. 60. 70. 80. 90. 100.
time [hours]
(a) Eroded volume versus time
50.
S40. Measured
.0 Equation (14), n=3
 Equation (14), n=1
30. J Swart's model
10. 0
ee e e
O.I I ..........".... ......"..   'TF
0. 10. 20. 30. 40. 50. 60. 70. 80. 90. 100.
time [hours]
(b) Qmax versus time
0.0
o.o
1I~~~ ~ (I It. 
O 3
10.0 10. 20. 30. 40. 50. 60. 70. 0. 90. 100.
0ti Measured
S 0  Equation (14), n=3
c15.0 )  Equation (14), n=l
Sm t Swart's model
g 0 " ""  "     "      
25.0
0. 10. 20. 30. 40. 50. 60. 70. 80. 90. 100.
time [hours]
(c) Elevation of Qmax versus time
Fig.22. Case 700 from Saville's Tests. Comparisons of predicted to observed eroded
volumes, maximum transport rates and their corresponding elevation.
AS.

10. Initial profile
 Observed at 10 hours
 Predicted at 10 hours
20. i 1 _j
100. 50. 0. 50. 100. 150. 200. 250.
10.
. 0 _______
S10.  Initial profile
S Observed at 30 hours
 Predicted at 30 hours
20. i I I I
100. 50. 0. 50. 100. 150. 200. 250.
10.
So. SWL
0

> 10.  Initial profile
Observed at 60 hours
 Predicted at 60 hours
20. i i
100. 50. 0. 50. 100. 150. 200. 250.
10.
7 o. >SWL
0.
> 10.  Initial profile
S Observed at 100 hours
 Predicted at 100 hours
20. 1 _i i_
100. 50. 0. 50. 100. 150. 200. 250.
offshore distance [ft]
Fig.23. Case 700 from Saville's Tests. Comparisons of predicted from Eq.(14) with
n=3 to observed profiles at different times.
10.
SWL
,g 0.
0.
S10.  Initial profile
Observed at 10 hours
 Predicted at 10 hours
20.
100. 50. 0. 50. 100. 150. 200. 250.
10.
0.
10. ^
> 10.  Initial profile
Observed at 30 hours
. Predicted at 30 hours
20.
100. 50. 0. 50. 100. 150. 200. 250.
10.
0.
10. ,
> 10.  Initial profile
Observed at 60 hours
 Predicted at 60 hours
20. 1 1 
100. 50. 0. 50. 100. 150. 200. 250.
10.
1 ^?r^"^~____~~~_________
0o
> 10. Initial profile
Observed at 100 hours
. Predicted at 100 hours
20. I 1
100. 50. 0. 50. 100. 150. 200. 250.
offshore distance [ft]
Fig.24. Case 700 from Saville's Tests. Comparisons of predicted from Eq.(14) with
n= 1 to observed profiles at different times.
scale of response for nonlinear system but does not affect that of the linear system. This
behavior is similar to that observed for some wave tank experiments of profile evolution. A
beach with an initial condition farther from the equilibrium will approach the finial profile faster,
while a beach closer to equilibrium will take longer time to reach equilibrium.
The proposed nonlinear transport relationship(n=3) is supported by the comparisons of
numerical predictions to two sets of large wave tank data. The error between eroded volume and
the observed is less than 1 percent for German "dune without foreshore" and 8 percent for "dune
with foreshore". For Saville's data, Case 300 and Case 400 are fit better than Case 500 and
Case 700. The agreement of eroded volume is within 8 percent error for the first two cases and
within 20 percent error for the latter two cases. The bestfit k value increase as the fall velocity
parameter decreases. The detailed relationship between them needs additional study.
REFERENCE
Dalrymple, R.A. 1991. "Prediction of storm/normal beach profiles," Journal of Water Ways,
Port, Coastal and Ocean Engineering, American Society of Civil Engineers.
Dean, R.G. 1973. "Heuristic models of sand transport in the surf zone," Proceedings,
Conference of Engineering Dynamics in the Surf Zone, Sydney, Australia.
Dean, R.G. 1977. "Equilibrium beach profiles: U.S. Atlantic and Gulf coasts," Department of
Civil Engineering, Ocean Engineering Report No.12, University of Delaware, Newark,
Delaware.
Dean, R.G. 1987. "Coastal sediment processes: Toward engineering solution," Coastal
Sediment'87, Specialty Conference on Advances in Understanding of Coastal Sediment
Processes, ASCE, New Orleans, Louisiana, pp.124.
Dette, H. and Uliczka, K. 1987. "Prototype investigation on timeDependent dune recession and
beach erosion". Coastal Sediment'87, Specialty Conference on Advances in Understanding of
Coastal Sediment Processes, ASCE, New Orleans, Louisiana pp. 14301443.
Hughes, S.A. and Fowler, J.E. 1990. "Validation of movablebed modeling guidance,"
Proceedings of the 22nd International Conference on Coastal Engineering, ASCE, pp.24572470.
Inman, D.L., Elwany, M.H.S. and Jenkins,S.A. 1993. "Shorerise and barberm profiles on
ocean beaches". J. of Geophysical Research, Vol. 98, No. cl0, pp. 18,18118,191.
Kraus, N. C. 1988. "Beach profile change measured in the tank for large waves 19561957 and
1962". Technical Report of CREC886.
Kriebel, D.L., Dally, W.R. and Dean, R. G. 1986. "Undistorted Froude model for surf zone
sediment transport," Proceedings of the 20th International Conference on Coastal Engineering,
ASCE, pp. 12961310.
Kriebel, D. L. 1986. "Verification study of a dune erosion model". Shore and Beach, Vol.54,
No.3, pp. 1320.
Kriebel, D. L. and Dean, R. G. 1984. "Beach and dune response to severe storms". Proceeding,
19th International Conference on Coastal Engineering, Houston, Texas, pp. 15841599.
Swart, D.H. 1974. "Offshore sediment and equilibrium beach profiles," Delft Hydraulics
Laboratory, Publication No. 131.
scale of response for nonlinear system but does not affect that of the linear system. This
behavior is similar to that observed for some wave tank experiments of profile evolution. A
beach with an initial condition farther from the equilibrium will approach the finial profile faster,
while a beach closer to equilibrium will take longer time to reach equilibrium.
The proposed nonlinear transport relationship(n=3) is supported by the comparisons of
numerical predictions to two sets of large wave tank data. The error between eroded volume and
the observed is less than 1 percent for German "dune without foreshore" and 8 percent for "dune
with foreshore". For Saville's data, Case 300 and Case 400 are fit better than Case 500 and
Case 700. The agreement of eroded volume is within 8 percent error for the first two cases and
within 20 percent error for the latter two cases. The bestfit k value increase as the fall velocity
parameter decreases. The detailed relationship between them needs additional study.
REFERENCE
Dalrymple, R.A. 1991. "Prediction of storm/normal beach profiles," Journal of Water Ways,
Port, Coastal and Ocean Engineering, American Society of Civil Engineers.
Dean, R.G. 1973. "Heuristic models of sand transport in the surf zone," Proceedings,
Conference of Engineering Dynamics in the Surf Zone, Sydney, Australia.
Dean, R.G. 1977. "Equilibrium beach profiles: U.S. Atlantic and Gulf coasts," Department of
Civil Engineering, Ocean Engineering Report No.12, University of Delaware, Newark,
Delaware.
Dean, R.G. 1987. "Coastal sediment processes: Toward engineering solution," Coastal
Sediment'87, Specialty Conference on Advances in Understanding of Coastal Sediment
Processes, ASCE, New Orleans, Louisiana, pp.124.
Dette, H. and Uliczka, K. 1987. "Prototype investigation on timeDependent dune recession and
