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Ocean Engineering 34 (2007) 1096–1103

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Verification of sediment transport rate parameter oncross-shore sediment transport analysis

U. Turkera,�, M.S. Kabdas-lıb

aNear East University, Department of Civil Engineering, Northern Cyprus, Mersin 10, TurkeybIstanbul Technical University, Department of Civil Engineering, 80626 Maslak, Istanbul, Turkey

Received 15 April 2006; accepted 16 August 2006

Available online 3 January 2007

Abstract

The sediment transport parameter helps determining the amount of sediment transport in cross-shore direction. The sediment

transport parameter therefore, should represent the effect of necessary environmental factors involved in cross-shore beach profile

formation. However, all the previous studies carried out for defining shape parameter consider the parameter as a calibration value. The

aim of this study is to add the effect of wave climate and grain size characteristics in the definition of transport rate parameter and thus

witness their influence on the parameter. This is achieved by taking the difference in between ‘‘the equilibrium wave energy dissipation

rate’’ and ‘‘the wave energy dissipation rate’’ to generate a definition for the bulk of sediment, dislocating within a given time interval

until the beach tends reach an equilibrium conditions. The result yields that empirical definition of transport rate parameter primarily

governs the time response of the beach profile. Smaller transport rate value gives a longer elapsed time before equilibrium is attained on

the beach profile. It is shown that any significant change in sediment diameter or wave climate proportionally increases the value of the

shape parameter. However, the effect of change in wave height or period on sediment transport parameter is not as credit to as mean

sediment characteristics.

r 2006 Published by Elsevier Ltd.

Keywords: Beach profile; Equilibrium; Grain size; Transport rate parameter

1. Introduction

Change in beach profile due to cross-shore sedimenttransport is one of the most important factors while dealingwith nearshore dynamics. Beach profile change is generallytreated in the framework of different approaches depend-ing on the professions, like geologists, oceanographers andcoastal engineers. There are different cross-shore sedimentmodels that are developed for modeling beach profileschanges (Kriebel and Dean, 1985; Watanabe and Dibajnia,1988; Larson and Kraus, 1989; Steetzel, 1990; Turker andKabdas-lı, 2004). Some of these models are based onanalytical grounds while the others on empirical resultsfrom experimental studies. Roelvink and Broker (1993),describe the state-of-the-art in modeling of cross shoreprofile evolution and classified the available models as;

front matter r 2006 Published by Elsevier Ltd.

eaneng.2006.08.002

ng author. Tel. +90392 22 36 464; fax: +90392 22 35 427.

ss: uturker@neu.edu.tr (U. Turker).

descriptive models (Lippmann and Holman, 1990), Equili-brium profile models (Bruun, 1954; Dean, 1977), Empiricalprofile evolution models (Kriebel and Dean, 1985) andProcess Based Models (Broker Hedegaard et al., 1992).Although the fundamental assumptions used in thesemodels are reasonable, the empirical coefficients involvedin them cover so many processes that empirical coefficientsmust be determined through calibration for a given site(Roelvink and Broker, 1993). In order to provide unique-ness to the developed models, a definite characterizationfor empirical coefficients is necessary, rather than calibra-tion. Among all these cross-shore profile evolution modelsthe model proposed by Kriebel and Dean proved to beuseful in cases of sandy beaches and dunes, wherelongshore transport gradients can be neglected, and wherethe profile is able to reach equilibrium. The model,however, uses an empirical coefficient called transport rateparameter, K. This transport rate parameter primarilygoverns the time response of the beach profile. Smaller

ARTICLE IN PRESSU. Turker, M.S. Kabdas-lı / Ocean Engineering 34 (2007) 1096–1103 1097

value gives longer elapsed time before equilibrium isattained, whereas large value produces more rapid evolu-tion. In other words, smaller K-value implies flatterequilibrium beach profile. K-value for Work and Dean(1995), in their solution, describing the response of anevolving beach nourishment project, is given as0.2� 10�6m4/N. Kriebel and Dean (1985) refers to thestudy carried by Moore (1982) suggesting that K may eitherbe a constant or a function of other wave and sedimentparameters. Moore (1982) found K to be constant,equivalent to 2.2� 10�6m4/N. According to Moore(1982), the parameter K should also vary with the lengthscale of the system. For their analysis in SBEACH, Larsonand Kraus (1989) used the average value of the transportrate coefficient as 1.1� 10�6m4/N, which is approximatelyhalf the value originally obtained by Moore (1982). In theirstudies, while comparing 10 cross shore sediment transportmodels Schoonees and Theron (1995) simulated a value of8.75� 10�6m4/N.

As it can be concluded, in all the above mentionedstudies, the transport rate parameter is obtained by fieldcalibration and is not entirely comparable between themodels, in view of the fact that the structures of the modelsare different from each other. In this study, an attempt ismade to generate a ratio, representing difference between‘‘the equilibrium wave energy dissipation rate’’ and ‘‘thewave energy dissipation rate’’ to the bulk of sedimentdislocated in cross shore direction. The result is incorpo-rated with physical model based in the wave energydissipation rates, thus, the more accurate values fortransport rate parameter for any given climatic andsediment condition can be predicted.

2. Theoretical approach

Cross-shore sediment transport relationships are gener-ally categorized into two models, ‘Closed Loop and OpenLoop’. ‘‘Closed Loop’’ models are those where thesediment transport relationships could converge to a targetequilibrium profile, while in ‘‘open loop’’ models there isno priority to achieve final equilibrium profile. One of thefirst closed loop transport relationships proposed was thatof Kriebel and Dean (1985). They have hypothesized thatthe cross-shore transport rate depends on the deviation ofthe dissipation rate per unit volume from its equilibrium,and thus improved a time dependent model that convergesto Dean’s equilibrium profile. This concept was extended tobarred profiles by Larson and Kraus (1989) and Larson(1993). Based on equilibrium profile considerations, it isreasonable to express the offshore transport rate ofsediments at any point in the surf zone, qs, in terms ofthe difference between the actual and equilibrium levels ofwave energy dissipation due to wave break.

qs ¼ KðDw �DeqÞ (1)

in which Dw and Deq are the actual and equilibrium energydissipation rates per unit volume and K is the transport

rate parameter. Theoretically, qs represents the cross-shoresediment transport load resulting from destructive forcesduring breaking process. Namely, the expected occurrenceof sediment transport mechanism depends on the level ofwave energy dissipation, which should be above someequilibrium threshold, where no net transport occurs ifnot overestimated. Thus, above equation can be simpli-fied to reflect the relation between the rate of energyrequired for sediment dislocation due to wave breakingper unit volume, Ds, to the energy difference betweenthe equilibrium wave energy dissipation rate and theactual wave energy dissipation rate, (Turker and Kabdas-lı,2004).

Ds ¼ Dw �Deq. (2)

As the equilibrium in beach profile is attained, Dw ¼ Deq

the net sediment transport is accepted as zero, Ds ¼ 0.Nevertheless, during the evolution of beach profile, theintensive sediment transport in the area enclosed withinbreaking zone and the coastline gives birth to high erosionrates in onshore–offshore direction. One of these changes isdefined as storm profile or winter profile, where dominantsediment transport is in offshore direction. This transportmechanism continues until all the sediment grains accu-mulate some point offshore, forming an offshore bar. Inthe evolution of above physical event, it appears that thereare destructive and constructive forces that tend totransport sand offshore and onshore, respectively. Gravityis one obvious destructive force, and it is evident thatconstructive forces, bottom shear stress etc., exist tocounteract these destructive forces; otherwise; beachescannot cease the sediment transport and thus the beachprofile changes infinitely (Dean, 1995).Consider a unit area on a beach slope, locally inclined

at an angle, b to the horizontal. The intensive wavebreaking process conveying a sediment transport dis-sipates energy. When a certain transitional stage of bedmovement is exceeded the fluid flow is considered asunaffected by the existence of stationary boundary. Thewhole resistance to flow, in the two-dimensional case, isthat exerted by solid phase. Bagnold (1963) demonstratesthe tangential thrust stress required to maintain thesediment in motion.

rs � rrs

gm cos bðtan f� tan bÞ, (3)

where m is the whole mass of the displaced sediment perunit area, rs and r is the density of sediment and pervadingfluid respectively, j is the angle of repose, which is ameasure of the mean angle of contact between thesediments grains.The product of the load of grains and the mean speed u

of their travel is the rate of mass transport of the load perunit width. The multiplication of the tangential thruststress, by mean speed of sediment grains results in the workdone by the fluid per unit time. Hence the necessary fluidpower per unit of bed area in transporting the sediment

ARTICLE IN PRESSU. Turker, M.S. Kabdas-lı / Ocean Engineering 34 (2007) 1096–11031098

load can be given as

rs � rrs

gm cos bðtan f� tan bÞu, (4)

where the mean speed is the ratio between the averagedistance traveled by the sediment particles and the timescale that the event takes place.

u ¼ L=Tn, (5)

L is the length of displacement of bulk of sedimentaryparticles in cross-shore direction. This length is defined interms of wave and sediment characteristics, and the termT* is fictive period representing the time interval starting bythe incipient motion of sediment from foreshore until itsettles at offshore (Turker and Kabdas-lı, 2004). InsertingEq. (5) into (4) represents the energy that is dissipated inunit time per unit of bed area in the process of transportingthe sediment grains:

Es ¼ ðrs � rÞg

rsm cos bðtan f� tan bÞ

LTn

. (6)

The volume of sediment particles (m/rs) is directly relatedwith the fluid energy required to maintain the incipientsediment dislocation.

Es ¼ ðgs � gwÞV g cos bðtan f� tan bÞLTn

. (7)

In Eq. (7), gs and gw are the specific weight of sediment andfluid, respectively, and Vg is the volume of sediment grains.Perhaps, the total volume of sediment grains, transportedin offshore direction, qualifies the volume of the offshorebar. Therefore, the volume of offshore bar Vbar, canreplace the volume of grains. Vbar is the optimum volumeof sand contained within the bar per unit length of beachformed after the storm. In this form, Eq. (7) represents theamount of energy per unit of bed area that the sedimentparticles should overestimate in order to move from oneplace to another to form offshore bar. The representationof the energy dissipation per unit volume in transportingthe sediment grains, Ds, can be therefore defined as

Ds ¼ Vbarðgs � gwÞ cos bðtan f� tan bÞLTn

1

h. (8)

In Eq. (8), h is the local water depth. In the derivation ofEq. (8), it is assumed that the sea state and other boundaryconditions are stationary over the duration of a morpho-logical time step.

Remembering the relation given in Eqs. (1) and (2), theterm representing the energy dissipation per unit volumecan also be defined in terms of offshore sediment transportrate, and transport rate parameter:

Ds ¼ qs=K . (9)

In the above equation, the sediment transport rate is thevolume of sediment, forming the offshore bar, dislocatingwithin a given time interval defined as fictive period:

qs ¼ Vbar=Tn. (10)

By inserting Eq. (10) into Eq. (8), we get,

Ds ¼ qsðgs � gwÞ cos bðtan f� tan bÞL1

h. (11)

Eq. (11) shows that the necessary energy dissipation for thedisplacement of sediment grains is directly proportionalwith the rate of sediment transport. At the same time, asgiven in Eq. (1), the rate of sediment transport is alsoproportional with the difference between actual andequilibrium energy dissipation rates per unit volume.Combining Eqs. (1), (2) and (11) and choosing thesediment transport rate as subject of the formula, themodel lately proposed by Kriebel and Dean (1985) can bewritten as

qs ¼1

ðgs � gwÞ cos bðtan f� tan bÞL 1h

ðDw �DeqÞ. (12)

In the former expression, the first term at the right sideof the equation represents the so called transport rateparameter, K in terms of wave climate and grain sizecharacteristics.

K ¼hb

ðgs � gwÞ cos bðtan f� tan bÞL. (13)

Dally et al. (1984) in their study suggest that waterdepths at the break and at stable conditions are equal.Therefore, using this suggestion, the considered waterdepth at the location of the event can also assumedto be equal to the water depth at the break. The validityof Eq. (13) can be checked by the help of experi-mental studies where the parameters evolved in the aboveequation can be used to find representative values for K

parameter.

3. Experimental testing

Model studies are conducted at Istanbul TechnicalUniversity, Hydraulics Laboratory. The aim is to investi-gate a representative transport rate parameter under theattack of regular and irregular wave climates. Glasswave flume with dimensions 24� 1� 1m is used togenerate irregular waves (Fig. 1). Irregular waves weregenerated by computer controlled wave maker capable togenerate waves practicing Pierson–Moskowitz wave spec-trum. The still water depth channel was 60 cm. During thestudies three different grain sizes were used. The size ofmaterials involved in the test was selected as 0.38, 0.5 and0.7mm. The slope of beach profile prior to each test waskept at 1:5.About 31 experiments for irregular waves were carried

out. A wave monitor amplifier and a resistance typewave electrode were used to record wave data. The waverecords, wave height and its period and average dislo-cation of sediment particles are then measured for eachtest. The results of experiments are summarized in Table 1.The analysis on wave data show that, for the irregularwaves the highest one-tenth of wave height was dominating

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Fig. 1. Experimental setup of irregular wave channel.

Table 1

Laboratory results of irregular wave tests

The code of sediment

particles

Wave height

H1/10 (cm)

Wave period

T1/10 (s)

Length of

displacement L (cm)

The mean diameter

d50 (mm)

No of experiments

accepted

A 12.0–16.0 1.2–1.9 85–112 0.38 11

B 11.0–17.0 1.5–2.0 76–115 0.50 9

C 12.0–16.0 1.3–1.9 75–93 0.70 6

U. Turker, M.S. Kabdas-lı / Ocean Engineering 34 (2007) 1096–1103 1099

the process. Perhaps, the reason for such a result lays in thefact that actually the offshore migration of bars isnecessarily associated with break point processes. Thus,under random waves the small amplitude waves evencannot broke to dissipate their energy to initiate sedimentmotion.

Finally, the experimental results are analyzed andconsidered. Whilst some of the experimental results areaccepted as a good indicator for the sediment transportparameter relationship some did not comply with. Due tothe unfavorable outcomes of five of these experiments,they are rejected from the analyses. Four of themwere rejected from the experiments on 0.7mm sizedsediments whereas one of them was from 0.5mm. Theunfavorable results are mainly due to the defaultshappened during the experimental studies, for example intwo of the experiments carried over on 0.7mm diametersediments the electricity shortcut has shorten the timescaleof the experiments.

4. Discussion and results

Theoretically, an attempt is made to generate a definitionrepresenting the transport rate parameter in cross-shore sediment transport studies. The resultant equation,Eq. (13), depends on specific weight, angle of repose,wave breaking depth and the length of displacement ofsediment particles until the equilibrium conditions aremaintained. The length of displacement of sedimentaryparticles are studied by Turker and Kabdas-lı (2004). Intheir results, it is shown that the average dislocationdepends on both the wave climate and sediment character-istics. However, their theoretical definition involves toomany limitations and variables, which are correlated withempirical definitions. Therefore, herein by using the

experimental results length of displacement of sedimentaryparticles is to be redefined.

4.1. Length of closure effect

It is clear that all the experiments that practicedincrement in wave energy dissipation rate are related withan increase in incoming wave height. As a result,convolution in offshore displacement of sediment particlesconsumes more time. The reason, without a doubt, isincrease in displacement length of total volume ofsediments while displacing offshore to form larger offshorebars. Furthermore, according to stability criteria as the sizeof sediment particles increases, necessary wave energyrequired to dislocate the particles also increases. Perhaps,the initial slope of beach profile also affects the naturalevolution of the displacement length of sediment particlesto form offshore bar. It is also evident from theexperimental studies that the rate of profile evolution isdecreasing consistent with an approach to equilibrium. Theestimation of the length of displacement of sediment grainscan be valid parameter for predicting cross-shore profilechange up to which the seaward limit to which thedepositional front has advanced.According to the previous studies there are too many

recommendations for seaward limit of offshore deposi-tions. Vellinga (1983) proposed the depth of closure as75% of the deepwater significant wave height using linearwave theory. Based on Laboratory and field dataHallermeier (1978, 1981), developed a condition forsediment motion resulting from wave conditions. Accord-ing to his studies the resultant equation is given as

hc ¼ 2:28H � 68:5H2

gT2

� �, (14)

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y = -1.2808 Ln (x) - 8.74

R2 = 0.85

0.0

2.0

4.0

6.0

8.0

10.0

12.0

1.0E-07 1.0E-06 1.0E-05 1.0E-04

Λ/ H

(H/gT2)2.68 (H /wT)-2.05

Fig. 2. Logarithmic variances in the definition of transport rate

parameter.

U. Turker, M.S. Kabdas-lı / Ocean Engineering 34 (2007) 1096–11031100

where H and T are significant wave height and significantwave period, respectively. Birkemeier (1985) evaluatedHallermeier’s relationship creating a simple approximationto the effective depth of closure providing a good fit to thedata given as

hc ¼ 1:57H. (15)

Perhaps, the depth of closure is the best point to define theoffshore limits of sediment disturbance due to wave action.However, during the offshore bar formation the depth ofwater above the offshore bar is not remaining constant.Consequently, the depth of water is not such a preciselocation to define the limiting depth of closure. Thus, it isnecessary to define the depth of closure by anotherreferring, which can be defined as length of closure. Thenecessary length of closure actually defines the overalllength of displacement of sediment particles during theformation of offshore bar.

The findings of the experiments that are carried over inIstanbul Technical University, Hydraulics Laboratory,concluded that in forming the equilibrium beach profile,the seaward limit of beach profile depends on wave height,wave length and its period; the size of sediment grains andthe slope of beach profile. However, since in this study thebeach profile slope is considered as constant, the results ofexperiments brings about uncertainty in the definition ofseaward limit of beach profile. For sure, the considerationof the initial beach slope has unexpectedly failed toimprove the accuracy of the analysis.

Harmonizing the parameter which are effective indefining the seaward limit of beach profile results in threenon-dimensional parameters:

Ho=L; dimensionless dislocation parameter;

Ho=wT ; fall speed parameter;

Ho

�gT2; Froude parameter:

In some sense, fall speed parameter is both a measure ofwave steepness and a sediment-related descriptor, indicat-ing both profile response and geometric properties ofvarious major morphologic features. It is also related tomobilization and suspension of sediment and performs wellindividually and in combination as criteria for predictingdirection of cross-shore sediment transport. The strongrelationship between wave and sand characteristics andmorphologic features suggests the possibility of quantita-tively predicting the beach profile response and evolutionperiod (Larson and Kraus, 1989). The dimensionlessdislocation parameter indicates the dominance of waveheight on the morphological changes in seaward limitationsof beach profile. The Froude parameter is useful in definingthe dominancy of gravity forces, which in turn affects theevolution time (transport rate parameter) of equilibriumbeach profile. The relationship among the dimensionlessdislocation parameter, fall speed parameter and the Froudeparameter for irregular wave analysis is given in the

following equation. The trend between the parameters issimulated in Fig. 2.

LH¼ �1:28 ln

H

gT2

� �2:68H

wT

� ��2:05 !� 8:74. (16)

The logarithmic relation between the dimensionlessdislocation parameter and its empirical definition can berepresented graphically on a semi logarithmic paperresulting with a coefficient of determination equivalent to0.79. The quality of fit analysis is applied to the predictedequation and the experimental data. It is seen that theresultant empirical equation is fitting the data quite well.The quality of fit is determined through the followingequation:

x ¼

PLm � Lp

� �2PL2

m

, (17)

where Lm is the length of closure measured during theexperimental analyses, whereas Lp is the length of closurevalues predicted by Eq. (17). The value x ¼ 0 correspondsto a perfect fit between the two data, and increasing valueof x refers to poorer fit. Eq. (17) proves to have goodresults with x ¼ 0.00345. Fig. 3, on the other hand, plotsthe predicted dimensionless dislocation parameter againstthe observed values for the irregular wave analysis. Thefigure indicates that the predicted method fits the data verywell. The presented method has root mean square error of3% where the root mean square error, erms is defined as

erms ¼

PLm � Lp

� �2N

" #1=2, (18)

where N is the number of observations carried out duringthe experimental work. In effect, the root mean squareerror describes the probability of a predicted positionvariation from the observed location. Therefore, anestimation of the dislocation parameter in natural environ-ments practicing Pierson–Moskowitz wave spectrum willyield 3% error. The results for the irregular wave analysisare outstanding such that, out of the 26 experiments carriedout, the number of errors exceeding 10% is only 1.

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0.0E+00

5.0E-06

1.0E-05

1.5E-05

2.0E-05

2.5E-05

3.0E-05

3.5E-05

0 2 4 6 8 10 12

D = 0.38mmD = 0.7mmD = 0.5mm

tran

spor

t rat

e pa

ram

eter

, K

wave height, H

Fig. 4. Change in wave height with respect to transport rate parameter for

different grain size diameters.

0.0E+00

5.0E-06

1.0E-05

1.5E-05

2.0E-05

2.5E-05

3.0E-05

3.5E-05

0 0.2 0.4 0.6 0.8 1 1.2wave period, T

tran

spor

t rat

e pa

ram

eter

, KD = 0.38 mmD = 0.7 mmD = 0.5 mm

Fig. 5. The trend of the relation between wave period and transport

parameter for different sized sediment particles.

0

10

20

30

40

50

60

70

80

90

0 10 20 30 40 50 60 70 80 90Observed (Λ / H)

Pred

icte

d (Λ

/ H)

Fig. 3. The relation between the observed and predicted dislocation

values.

U. Turker, M.S. Kabdas-lı / Ocean Engineering 34 (2007) 1096–1103 1101

4.2. Transport rate parameter estimation

During the experiments, the behavior of sedimentarygrains was thoroughly observed. The results show that thewave climate parameters are not as effective as thesediment characteristics while defining the transport rateparameter. The transport rate parameter is primarilygoverning the time response of beach profile. The timedependent evolution of offshore bar is one of the observedproofs of this physical event. The experimental investiga-tions helped to correlate the relationship between timeresponse of beach profile and the wave climate andsediment characteristics.

The theoretical proof of sediment transport rate para-meter, K is already given in Eq. (13). Experimentalinvestigations help to find the variables involved inEq. (13) as physical quantities. Thus, the observed resultsare inserted into Eq. (13) to evaluate the sediment transportrate parameter. What is discussed was the change intransport rate parameter with respect to incoming waveheight, wave period and length of displacement of sedimentparticles. The results depicted the independency of trans-port rate parameter to the wave climate parameters such aswave height and period (Figs. 4 and 5). Also, it is worth tomention that, whatever the length of displacement ofsedimentary particles is; the transport rate parameter is notaltering. However, any change in sediment grain size isdirectly affecting the scale of transport rate parameter.

What is concluded after the investigations on experi-mental studies is that, detailed survey must be proceeded todetail the relationship between transport rate parameterand the mean grain size. The earlier observations show thattransport rate parameter considerably increases withincrease in mean grain size. Further increase in mean grainsize, however, keeps altering transport rate parameter, butnot as effective as in preliminary phase; slight changes areobserved. For a specific grain diameter the changes in thetransport rate parameter are considerably small withrespect to sharp changes in wave period and height.

The plotted data given in Fig. 4 shows that for differenttypes of sediment grain sizes (0.38, 0.5 and 0.7mm) anychange in wave height is not significantly altering

magnitudes of transport rate parameter. However, anincrement on grain size from 0.38 to 0.5mm puts up themagnitude of transport rate parameter by 25%. Furtherincrement in grain size of sedimentary particle, 0.5–0.7mm,increases the magnitude of transport rate parameter by10%, indeed, the necessary wave energy requirement forthe mobility of sediment grain size increases. The resultdepicts the asymptotic approach of K-value as much as thegrain diameter approaches to nonerodable (hard) seabottoms.Fig. 5 reclaims the independency of transport rate

parameter on wave period. Different grain sizes posses’approximately same transport rate parameter, whilst waveperiod is changing in between 0.6 and 1.1 s. Likewise, theeffect of wave height, while considering the effects of waveperiod, the rate of change in transport rate parameter isdescending in the order of increasing grain diameter. Allthese results are supporting zero cross-shore sedimenttransport phenomena like in non-erodible sea bottoms. Thelength of sediment displacement described in Eq. (16) canbe defined in terms of both the sediment characteristics andthe wave climate. However, it is not directly affecting themagnitude of transport rate parameter. Fig. 6 repeats theminor importance of length of sediment displacement inorder to dislocate from onshore to offshore.All the above discussions related with Figs. 4–6, also

reveals the important fact that whilst the sediment grain

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0.0E+00

5.0E -06

1.0E-05

1.5E-05

2.0E-05

2.5E -05

3.0E-05

3.5E-05

0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0

D = 0.38mmD = 0.7mmD = 0.5mm

tran

spor

t rat

e pa

ram

eter

, K

average dislocation of sediments, Λ

Fig. 6. The trend of the relation between average distance traveled by

sedimentary particles and transport parameter for different sized sediment

particles.

6.0E-07

1.0E-06

1.4E-06

1.8E-06

0.1 1 10

K p

aram

eter

(N

/m4)

Hs =1mHs = 2mHs = 3mHs = 4mHs = 5m

Gain diameter, (mm)

Fig. 7. The trend of the relation between grain diameter and transport

rate parameter for different wave heights where wave period is 6 s.

6.0E-07

1.0E-06

1.4E-06

1.8E-06

0.1 1 10Grain diameter, (mm)

K p

aram

eter

(N

/m4)

Hs = 1mHs = 2mHs = 3mHs = 4mHs = 5m

Fig. 8. The trend of the relation between grain diameter and transport

parameter for different wave heights where wave period is 8 s.

6.0E-07

1.0E-06

1.4E-06

1.8E-06

0.1 1 10Grain diameter, (mm)

K p

aram

eter

(N

/m4)

Hs = 1mHs = 2mHs = 3mHs = 4mHs = 5m

Fig. 9. The trend of the relation between grain diameter and transport

rate parameter for different wave heights where wave period is 10 s.

U. Turker, M.S. Kabdas-lı / Ocean Engineering 34 (2007) 1096–11031102

size is 0.38mm, whatever the wave height, wave period andthe length of displacement is, the transport rate parameteris not changing significantly. However, an increase insediment size strengthens the beach profile preventing thedislocation of sediment particles under the applied impactforces. Under these circumstances, the displacement ofsedimentary particles requires higher waves that result inlonger elapsed time for the evolution of equilibrium beachprofiles. The sediment characteristics therefore, dominantlydetermine the time scale that a beach profile reaches to anequilibrium conditions.

The outcome of the necessary magnitudes of transportrate parameter versus the sediment grain size, dependingon the changes in wave height is plotted throughFigs. (7–9). Analyzing in more detail clarifies that eachcurve consists of two phases; first phase is enclosed between0.1 and 1mm grain sizes, where as, phase two indicates thechanges in transport rate parameter for grain sizes largerthan 1mm. In the first phase, the change in sedimenttransport rate parameter is steep, whereas the change insecond phase flattens. The results of each figure show thatunder the constant grain size conditions, the rate of changeof transport rate parameter increases, as much as the waveheight increases. The turning point, 1mm diameter grainsize, represents the willingness of beach profile whileapproaching to equilibrium conditions. As the grain sizeof sea bottom increases, the waves require more energy to

destabilize beach profile, reflecting the time response ofbeach profile while attaining equilibrium conditions.Transport rate parameter is the one, which will decide onthe moment of equilibrium of a beach profile.

5. Conclusion

The ratio of unit discharge of sediment moving inoffshore–onshore direction, and the difference in between‘‘the equilibrium wave energy dissipation rate’’ and ‘‘thewave energy dissipation rate’’ results in so called transportrate parameter. This parameter mainly governs the timeresponse of beach profiles. It is compulsory that whereverthe transport rate parameter is used in cross-shoremodeling studies, it is accepted as a calibration value.However, in this study, by means of theoretical andexperimental proofs the effect of wave climate and grainsize characteristics are added to the definition of transportrate parameter. The transport rate parameter observed tobe dependent on specific weight, angle of repose, wavebreaking depth and the average distance traveled bysedimentary particles. After the analysis based on experi-mental studies carried over, the latter one is defined interms of wave climate parameters. This result is achievedonly when the laboratory results are observed as a macro-scale feature. It is found that the distance traveled bysedimentary particles is well governed by dimensionlessFroude and Fall speed parameters.

ARTICLE IN PRESSU. Turker, M.S. Kabdas-lı / Ocean Engineering 34 (2007) 1096–1103 1103

The combination of above variables addressed to areasonable definition for transport rate parameter. Theresults depicted the independency of parameter to the waveclimate parameters. However, any change in sedimentgrain size directly affects the parameter. Also, an asymp-totic relationship between the parameter and grain size isinvestigated, approaching to hard sea bottoms. Therelationship between transport rate parameter and sedi-ment grain size keeps on linearly increasing trend.However, as the sediment grain develops to a size ofgreater than 1mm, the evolution time of equilibrium beachprofile increases and approaches to hard sea bottomsurface profiles.

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