Citation
Cross-shore sediment transport relationships

Material Information

Title:
Cross-shore sediment transport relationships
Series Title:
UFLCOEL-94018
Alternate title:
Cross shore sediment transport relationships
Creator:
Dean, Robert G ( Robert George ), 1930-
Zheng, Jie
University of Florida -- Coastal and Oceanographic Engineering Dept
Place of Publication:
Gainesville FL
Publisher:
Dept. of Coastal and Oceanographic Engineering, University of Florida
Publication Date:
Language:
English
Physical Description:
33 p. : ill. ; 28 cm.

Subjects

Subjects / Keywords:
Sediment transport -- Mathematical models ( lcsh )
Beach erosion -- Mathematical models ( lcsh )
Genre:
bibliography ( marcgt )
non-fiction ( marcgt )

Notes

Bibliography:
Includes bibliographical references (p. 32-33).
General Note:
"June 1994."
Funding:
This publication is being made available as part of the report series written by the faculty, staff, and students of the Coastal and Oceanographic Program of the Department of Civil and Coastal Engineering.
Statement of Responsibility:
by Robert G. Dean and Jie Zheng.

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University of Florida
Holding Location:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
32872743 ( OCLC )

Full Text
UFL/COEL-94/018

CROSS-SHORE SEDIMENT TRANSPORT RELATIONSHIPS
by
Robert G. Dean and
Jie Zheng

June, 1994




CROSS-SHORE 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 cross-shore and long-shore components. It appears under a number of coastal engineering scenarios of interest, the transport is dominated by either the cross-shore or long-shore component. The cross-shore 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 cross-shore sediment transport modeling is relatively recent (about 20 years) and uncertainty in predicting effects of all variables thus may be considerably greater.
Cross-shore 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. Cross-shore transport is caused by variation of a beach profile from the equilibrium. This variation may result from a profile initially in disequilibriumn 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 1956-1957 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: Sa--(ECG) = D. (1)
hay
With shallow water and spilling wave breaker assumptions, Eq.(1) can be integrated to: h(y) = A(D) y13 (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 so-called sediment scale
(D)= 5 1
parameter, A(D P 3/23 -, 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, m0,
y = h +(h)3/2 (3)
m0 A
Since the scale parameter, A, is only a function of sediment size, wave conditions are not include




Shorerlse

5- X2 Xi + Mark
0- MSL -3 B
T-hl
f" 5- :.. ."...
S/ Wave-Cut Terrace
10 ., lllll,,
Shorezone
15
Fig. 1. Definition sketch for Inman's curve fitting. Crosses denote the origin for bar-berm (dotted) and shorerise (dashed) curves, with x coordinates, x, x2, and vertical
coordinates, hl, 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 (bar-berm) and the outer (shorerise) portions. The two portions are matched at breakpoint-bar and fit by curves of the form h = Bx" (4)
The coordinates of Inman's curve fitting are shown in Fig.1 with the subscripts 1 and 2 corresponding to bar-berm 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, x1, z1, BI, MI, x2, B2 and M2, 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.




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
so-called fall velocity parameter, -. By examining small scale wave tank data with only the wT
deep water reference wave height H0 values available, the following relationship for net seaward sediment transport and bar formation was found H >, 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
LHO < 0.0007 o (6)
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 = A0(D)[ 1+a[log(-) P-log(~.)1 (7
where Ao(D) is the value evaluated at the reference fall velocity parameter, H0, and is
a coefficient to account for the influence of fall velocity parameter.




DISCUSSION OF CROSS-SHORE 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 cross-shore 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(D-D,) (8)
In which, D represents the actual total destructive force,
D =.5 "g' ah3/2 (9)
24 aThe transport parameter, K, is considered as a dimensional constant. The following scaling relationship is established from Eq.(8),
Qr (D-D*)m&t = (B-D*)r (10)
(D-D*)prototype
For an undistorted model, according to the definition of D, the disequilibrium scale (D-D,)r can be expressed by the length scale, Lr,
(D-D,)r = Fr (11)
On the other hand, the Froude relationship gives the time scale, T,




T = VIT (12)
Several model studies [Kriebel, et al. 1986 and Hughes, et al. 1990] have conformed that HwT
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, = /L. Two examples are presented in Figs.2 and 3. Following the relationship of Eq.(12), the cross-shore sediment transport, Q, should be scaled as: = 2 (13)
Qr=- r
T.
This equation provides a basis for evaluating transport models. Obviously, Eq.(8) in which Qr = L"2 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
370 'WAV ES Model -Solid
- O C L V = P o t o D a s h e d
u4- RMS 0iF = 0.484 m
LU4
0 0 10 20 30 40 So
RANGE (m)
2 =f ,^ Model Solid
~65 WA ,O qVES Prato D ashed
-,12 i -F=-RMS = 0.444 m
LU
-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 (H0/wT)r= 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(D-D.)ID-D, In-1 (14)
Which results in the following scaling relationship
7




(15)

Qr = kr(D -D.)r ID -D I -

By equating Eq.(13) and (15), we have,
n 3
n-1 (16
Q, = kr(D-D*)rID-D*Ir = kr Lr = 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 L1 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
3 \L2-L,=6OS7-2.956t'o9mat-098-ocoat-O.801e-cooss
REMARKS: W.6.057m Sy.O.S55xiO' m/hr500 4000 4500 2000 2500 2800
time t (hours) Fig.5. (l2-L1) versus time for Swart's flume B case with 81 =0.15m, 82=0.10m.
In an attempt to understand the causes of the different time scale, we examine the following equation

dx k (x-x,) Ix-x, In-1 dt

(19)

Where x, is the equilibrium value of x. Nondimensionizing with x' = x/x, and t'= t/kx-1, we have

dx' dx'- -(x-l) Ix/- 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 +

-1

for n* 1.

(XI-1)
Fig.6 shows the results of (x/-1) versus t' for n= 1, 2 and 3, and xO =2 and 10 respectively.
It appears that the time scale of the linear system (n = 1) is independent of the initial conditions and the two lines in Fig.6(a) are coincident. However, for the non-linear systems, the initial conditions do affect the time scale by the factor Ix- 11 As n increases, this factor becomes

more and more significant.
The proposed cross-shore 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 nonlinearity 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.
t(
(a) n=1

6. 7. 8. 9. 10.

1.0
0.8 x'(t'=o)=2
,0... x'(t'=o)=1o
0.4
0.2 '
0.0 -----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'=o)=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=1 and n=3, is applied. Time dependent profile response is then determined by the numerical solution of transport equation and continuity equation, which is a0y Q (21)
at Ah
Where y is offshore distance from a reference baseline. For each case, k is obtained through best-fitting 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, H = 5. Results from Eq.(14) are compared wT
with experiments. For n=3, the combined best-fit k value is 1xl0-4ft's2/lb3. For n=1, the "dune without foreshore" has best-fit k value of 4 x 10-3 ft/lb, while the "dune with foreshore" has that of 2.2x 10-3 ft4/lb.




1. 2. 3. 4. 5.
time [hours]
(a) Eroded volume versus time

5.
time [hours]

(b) Qmax versus time

-~

-2.0~

o Measured
- equation 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
=

r LJ LJ

I I




o 2.
-3. Initial profile
Observed after 470 waves Predicted after 470 waves
-5.
-6. iI I IiiI
0. 10. 20. 30. 40. 50. 60. 70. 80.
3.
2. i",
1-.
-2.
- 3. Initial profile
- -4. Observed after 1000 waves "
----Predicted after 1000 waves
-5.
- 6. 1I 1 I 1 1 t
0. 10. 20. 30. 40. 50. 60. 70. 80.
3.
2.
0. SWL
o -2.
-3. --Initial profile
-4. -Observed after 2100 waves
---. Predicted after 2100 waves ".
I I 1 I t
0. 10. 20. 30. 40. 50. 60. 70. 80.
3.
2.
1.
-0.
-2.
40 -3. Initial profile
-4. -Observed after 3700 waves
----Predicted after 3700 waves
-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




.o -2.
-3. ..Initial profile
S 4Observed after 470 waves
-4. Predicted after 470 waves
-5.
-6. i i I
0. 10. 20. 30. 40. 50. 60. 70. 80.
3.
2. ;0. SWL
-2.
> -3. Initial profile
Observed after 1000 waves ..---- Predicted after 1000 waves
-5.
-6. -1I I i I I
0. 10. 20. 30. 40. 50. 60. 70. 80.
3.
SWL
1.
o -2.
CU -3. Initial profile
S-4.Observed after 2100 waves ....- Predicted after 2100 waves
-5.
-6. -j I I-j
0. 10. 20. 30. 40. 50. 60. 70. 80.
3.
SWL
0-.
-2.
--3-.... Initial profile
-4. Observed after 3700 waves
Predicted after 3700 waves N-6. i i i iii
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. 1
time [hours]

(b) Qmax versus time

I. 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.
-I.(
-2.
-2.
-3.
-3.
-4.(
-4.
-5.(

5 03()G G G (3 G OGGG 0 0
0
0
5- CG Measured
-Equation (14), n=3
0 ---Equation (14), n=1
5 Swart's model
0I I I 1 1

0

v0




3.
2.
SWL
o -2.
-3. ... Initial profile
Q Observed after 270 waves
Predicted after 270 waves
-5.
-6. I I I
-10. 0. 10. 20. 30. 40. 50. 60. 70. 80. 90. 100.
3.
2.
1.%
0_ SWL
S-2.
-3. --. Initial profile
-4. Observed after 760 waves
.... Predicted after 760 waves
-5.
-6. I I
-10. 0. 10. 20. 30. 40. 50. 60. 70. 80. 90. 100.
3.
2.
1.
0. SWL
o -2.
-3. -Initial profile
V Observed after 1620 waves
----Predicted after 1620 waves
-5.
-6. I I I I
-10. 0. 10. 20. 30. 40. 50. 60. 70. 80. 90. 100.
3.
2.
Z -1 .
0 -2.
e -3. .. Initial profile
(V-4. Observed after 2730 waves
----Predicted after 2730 waves
-5.
-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.
SWL
o -2.
~U -. --Initial profile
Observed after 270 waves Predicted after 270 waves
-5.
-6. I I I I
-10. 0. 10. 20. 30. 40. 50. 60. 70. 80. 90. 100.
3.
2.
1. l
-2.
-3. I-- initial profile
-4. --Observed after 760 waves
----Predicted after 760 waves
-5.
- 6. 1 1 1 1 1 1 1 I
-10. 0. 10. 20. 30. 40. 50. 60. 70. 80. 90. 100.
3.
2.
1.
SWL
-- -- -- -. -2.
-3. Initial profile
-4. Observed after 1620 waves
Predicted after 1620 waves
-5.
-6. I I I I I
-10. 0. 10. 20. 30. 40. 50. 60. 70. 80. 90. 100.
3.
2.
SWL
0 2.
.2 -2.
0 -3. -- Initial profile
-4. Observed after 2730 waves
Predicted after 2730 waves
-6. I I i
-10. 0. 10. 20. 30. 40. 50. 60. 70. 80. 90. 100.
offshore distance [in]
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= 1, 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 = 1. 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 201C 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 best-fit k value increases and the off-shore 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 best-fit 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. Best-fit 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 H0 [in] 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
H0/wT 13.52 9.64 4.96 3.38
best-fit k value for n=3 2x X o1-5 3.8 x 10-5 1X 104 2 X10-5
[ft~s2/1b3] _________ _________________best-fit k value for n =1 2.2 x10-4 6.5 x10-4 9 X10-4 2.2 x10-4
[ft4/lb] _ _ _ _ _ _ __ _ _
eroded vol. at 30 hour 141.6 265.8 277.8 286.4
[ft/ft] _________ _________ _________1bar height at 30 hour[ft] 1 1.65 1 1.58 1.22 1 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 non-linear cross-shore 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 cross-shore 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.
350...
.~300.
250. G
200.
> 150.
. () Measured
Q 00. -C-Equation (14j, n=3 ,,o 5.9"/'/Equation 14,n=1
0
0
0. 5. 10. 15. 20. 25. 30. 35. 40. 45. 50.
time [hours]
(a) Eroded volume versus time
100.
90. .
90. e3 Measured
80. -Equation (14), n=3
70. Equation (14), n=1
60. Swart's model
50.
40.
30. "
20.
10. '
0. -
0. 5. 10. 15. 20. 25. 30. 35. 40. 45. 50.
time [hours]
(b) Qmax versus time
0.0
-3.0 G
x 0 0
-6.00
0 Measured
o -Equation (14), n=3
-9.0 ---Equation (14), n=1
.- ----- Swart's model
Ij>-12.0
- 15.0 I I j I
0. 5. 10. 15. 20. 25. 30. 35. 40. 45. 504
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.




1 U.
0.
> 10. ... Initial profile
-Observed at 5 hours
Predicted at 5 hours
-20. 1 1 1 1
-100. -50. 0. 50. 100. 150. 200. 250.
10.
e 0.
> -10. Initial profile
-Observed at 10 hours VJ .... Predicted at 10 hours
- 20. I I I ---00. -50. 0. 50. 100. 150. 200. 250.
10.
> -10. Initial profile
-Observed at 30 hours .... Predicted at 30 hours
-20. 1 1 1 1 --100. -50. 0. 50. 100. 150. 200. 250.
10.
0.
> -10. Initial profile
-Observed at 50 hours
Predicted at 50 hours
-20. I I 1 1
-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.
10.
10. -- Iitia profile
Observed at 10 hours
...Predicted at 10 hours

-20.L

100.

150.

200.

250.

-10. -

-20.'
-1

100.

-20.1
-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.

. ...- .-.--Initial prfl
-Observed at 5 hours ...Predicted at 5 hours

I I I I I I

SWL
----.- 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. 0 Measured
- Equation (14), n=3 70. --- Equation (14), n=l
60. ----- Swart's model
50.
40.
30. A
20.
10. ----- 0
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

G G G G G 0G
0 Measured
-Equation (14), n=3
--- Equation (14), n=l
----- Swart's model
I I I i I l i

--15.

01

0.

IJ




0. F

-10. -

-20. 1 I I 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.

" z....SWL
--Initial profile' .
Observed at 5 hours
---- Predicted at 5 hours




0 .
-- Initial profile
- Observed at 5 hours
Predicted at 5 hours
-20. I 1
-100. -50. 0. 50. 100. 150. 200. 250.
10.
S0.
0
> -10. Initial profile
-Observed at 10 hours
Predicted at 10 hours
-20. 1 1 1 1 1 1
-100. -50. 0. 50. 100. 150. 200. 250.
10.
-10. Initial profile
-Observed at 30 hours
Predicted at 30 hours
-20. 1 1 1 1 1
-100. -50. 0. 50. 100. 150. 200. 250.
10.
0 -> -10. Initial profile
- Observed at 40 hours V .... 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

-3.0
-6.0
-9.0
-12.
-15.0
-18.0

0

0 G Measured
-Equation (14), n=3
--- Equation (14), n=1
----- ......Swart's model
'G
G 0
- C0 0
-I-- --- -- --- -- -

60. 70. 60. 90. 100.

1. 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.

GGG G G G G
00
0
G Measured Equation (14). n=3
------...... Swart's model --- Equation (14), n=1
0 I I I I I




1U.
0.
-10. --- Initial profile
-Observed at 10 hours
Predicted at 10 hours
-20. I 1 I I
-100. -50. 0. 50. 100. 150. 200. 250.
10.
0.
. .0 .
> -10. Initial profile
-Observed at 30 hours
Predicted at 30 hours
-20. I 1 1 I I
-100. -50. 0. 50. 100. 150. 200. 250.
10.
SWL
S-10. Initial profile
-Observed at 60 hours
Predicted at 60 hours
-20.1 I
-100. -50. 0. 50. 100. 150. 200. 250.
10.
*.1L
0.
o~~- ---- -------------- ...
> -10. .....Initial profile
- -Observed at 100 hours
Predicted at 100 hours
-20. I 9
-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.




-10.1-

-20. I 10.

0. -

-10.-

-20. I 1 I I I
-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.

- Initial profile Observed at 10 hours .--- Predicted at 10 hours

I I I I I I
100. -50. 0. 50. 100. 150. 200. 25

-- Initial profile
-Observed at 30 hours
---- Predicted at 30 hours

0.




- 400.
350.
300.
V 250. C
200.
0G
>s 15.-( Measured
100. /--Equation 14), n-3
- ---Equation 14 n=1
50. "
0. 10. 20. 30. 40. 50. 60. 70. 80. 90. 100.
time [hours]
(a) Eroded volume versus time
50.
40. 0C Measured
-Equation (14), n=3 Equation (14), n=1
N.30. Swart's model
20.
10. "
O. ~ ~ ~ ~ - --------. --..." -... --...--...---"'"------- - '-TF
0. 10. 20. 30. 40. 50. 60. 70. 80. 90. 100.
time [hours]
(b) Qmax versus time
0.0
-5.0 o 0
o" 0 Measured
S0 Equation (14), n=3
15.0- -- Equation (14), n=1
----- Swart's model
-20.0
-25.0I i
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.




ASW.
.0 -0
0-----'- -- -10. Initial profile
-- Observed at 10 hours
---- Predicted at 10 hours
-20. 1 _j jr
-100. -50. 0. 50. 100. 150. 200. 250.
10.
0.
-10. -- Initial profile
- Observed at 30 hours
- Predicted at 30 hours
-20. I I I I
-100. -50. 0. 50. 100. 150. 200. 250.
10.
0
> -10. -- Initial profile
- Observed at 60 hours
---. Predicted at 60 hours
-20. 1 -1 j -1 _j
-100. -50. 0. 50. 100. 150. 200. 250.
10.
0.
-10. ---Initial profile-- Observed at 100 hours "
---- Predicted at 100 hours
-20. 1 -1 _j _j
-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
.,,, 0.
0.
-10. Initial profile
- Observed at 10 hours
- Predicted at 10 hours
-20.1 1 1 1 1 ) 1
-100. -50. 0. 50. 100. 150. 200. 250.
10.
0.
> -10. -- Initial profile
- Observed at 30 hours
- Predicted at 30 hours
-20.
-100. -50. 0. 50. 100. 150. 200. 250.
10.
0.
> -10. Initial profile
- Observed at 60 hours
- Predicted at 60 hours
-20. 1 1 1 1
-100. -50. 0. 50. 100. 150. 200. 250.
10.
0.
> -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 non-linear 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 best-fit 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. 1-24.
Dette, H. and Uliczka, K. 1987. "Prototype investigation on time-Dependent dune recession and




beach erosion". Coastal Sediment'87, Specialty Conference on Advances in Understanding of Coastal Sediment Processes, ASCE, New Orleans, Louisiana pp. 1430-1443.
Hughes, S.A. and Fowler, J.E. 1990. "Validation of movable-bed modeling guidance," Proceedings of the 22nd International Conference on Coastal Engineering, ASCE, pp.2457-2470.
Inman, D.L., Elwany, M.H.S. and Jenkins,S.A. 1993. "Shorerise and bar-berm profiles on ocean beaches". J. of Geophysical Research, Vol. 98, No. dlO, pp. 18,181-18,191.
Kraus, N. C. 1988. "Beach profile change measured in the tank for large waves 1956-1957 and 1962". Technical Report of CREC-88-6.
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. 1296-1310.
Kriebel, D. L. 1986. "Verification study of a dune erosion model". Shore and Beach, Vol.54, No.3, pp. 13-20.
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. 1584-1599.
Swart, D.H. 1974. "Offshore sediment and equilibrium beach profiles," Delft Hydraulics Laboratory, Publication No. 131.