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Modelling of morphological changes due to coastal structures
Coastal Engineering 38 Ž1999. 143–166 www.elsevier.comrlocatercoastaleng
Modelling of morphological changes due to coastal structures I.O. Leont’yev
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Russian Academy of Sciences, P.P. ShirshoÕ Institute of Oceanology, NakhimoÕ Prospect, 36, 117851 Moscow, Russian Federation Received 4 December 1998; accepted 8 June 1999
Abstract Two different concepts are applicable to model the nearshore morphodynamics. The first one takes into account only final consequences of acting mechanisms and is aimed at the prediction of long-term trends in beach development. Another approach implies the modelling of the whole suite of elementary processes responsible for changes in nearshore bottom topography during a given storm, and it is the approach used in the present work. A coastal area model complex is proposed that allows to reproduce the local morphological changes due to both the natural processes and the influence of coastal structures, such as a groin, a detached breakwater and a navigable channel Žunderwater trough.. Consisting of a traditional series of basic components, the model differs from other ones in essential aspects concerning the treatment of transport mechanisms. In particular, the determination of wave-induced near-bed mean flow is based on the hypothesis that the direction and magnitude of bottom drift depend on difference between the actual rate of energy dissipation and its threshold value marking the flow reversal point. This hypothesis is shown to explain a general trend of cross-shore mean flow distributions observed in the nearshore region. Besides, the influence of the wave breaking process on sediment suspension is taken into account and the contribution of the swash zone to total sediment transport is included. Examples of computed morphological response are represented to demonstrate the model capability. A satisfactory agreement of computations with available data is pointed out. q 1999 Elsevier Science B.V. All rights reserved. Keywords: Coastal structure; Morphological changes; Bed deformations; Sediment transport; Wave-induced circulation
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0378-3839r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 3 9 Ž 9 9 . 0 0 0 4 5 - 9
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1. Introduction Coastal structures on sandy beaches usually cause changes in the bottom topography and the shoreline contour. Both overall and local deformations due to such structures can usually be observed. The first ones occur at spatial scales exceeding the structure dimensions and can be interpreted as a long-term trend in morphological evolution due to the disturbance of the shore-parallel net sediment flux. For example, the interruption of longshore sediment movement by a groin field or a navigable channel Žunderwater trench. may result in down-drift beach erosion over hundreds of meters or kilometers. The local deformations are related to changes in wave-induced currents and associated movements of bed grains in the vicinity of structures during a given storm event. Their spatial scales are comparable with the structure dimensions. Mathematical modelling of morphological response under the influence of structures is of great practical interest. Depending on the problem, two different approaches can be applied to model the nearshore morphodynamics. The first one takes into account only the integrated result of morphodynamic processes and does not include the detailed analyses of acting mechanisms. The beach profile in this case is assumed to be close to its equilibrium state and beach development results from changes in the bulk longshore sediment flux. This approach forms the basis of so-called one-line models aimed at the prediction of shoreline evolution under the influence of structures ŽPelnard-Considere, 1956; Hanson, 1989; Leont’yev, 1997.. An alternative approach involves the modelling of the whole suite of elementary processes responsible for the local morphological changes in a given area. A typical coastal area model consists of several modules describing the wave field, the spatial distributions of wave-induced currents and the associated sediment transport, and finally the resulting spatial and temporal changes of the bed level. Using a grid with relatively small cells, the model of this kind can resolve the detailed patterns of local bed deformations. Such an approach is employed, for example, in the model complexes developed by Delft Hydraulics ŽDe Vriend et al., 1993; Roelvink et al., 1995., Danish Hydraulic Institute ŽBroker, 1995; Broker et al., 1995. or HR Wallingford ŽPrice et al., 1995.. The attempts to evaluate the morphodynamic effect of structures by using these models are not yet numerous, but the results obtained look encouraging. The reliability of morphological predictions depends on the adequacy of treatment of each elementary process. The present understanding of phenomena observed in the nearshore region is not sufficient to assess which modelling concept is preferable to simulate the near-bed mean flow and sediment transport mechanisms. Besides, it is a problem to fulfill the requirement of accurate modelling and, at the same time, to provide the computational efficiency necessary for practical use. In the present paper, a coastal area model complex is proposed describing the evolution of 2DH bottom topography during a given storm attacking a beach, both in its natural state and in the presence of structures. As compared with other known models of this kind, the model discussed herein is of a similar general composition, but it differs from them at some essential points. This concerns the treatment of the bottom drift and the sediment transport in shoaling–breaking regions and in the swash zone. First, a brief
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outline of the basic modules is given, and then the model capabilities are illustrated by a series of computations for structures of different types. The cases of a groin field, a detached shore-parallel breakwater and a navigable channel Žunderwater trench. are considered.
2. Waves and currents 2.1. WaÕes The wave field is assumed to be random characterized by a root-mean-square Žrms. height H, spectral peak period T and major direction Q with respect to the shore-normal. Mean current velocities are supposed to be small comparing with wave-induced oscillatory orbital velocities, thus the nonlinear wave–current interaction is ignored. Determination of waves in the nearshore region implies calculations of refraction, diffraction, shoaling and dissipation. Wave incident angles Q Ž x, y . are computed from the equation of the wave-vector conservation ŽPhillips, 1977. E sin Q
E cos Q y
Ex
Ey
C
C
s 0,
Ž 1.
where the axis OX and OY are shore-normal and shore-parallel, respectively, C is the wave celerity. The 2DH distribution of rms heights H Ž x, y . is determined from the energy balance equation E
E
Ž ECg cosQ . q E y Ž ECg sinQ . s yD, Ex
Es
1 8
r gH 2 ,
Ž 2.
where E and Cg are the wave energy per unit area and the rate of its transfer Žgroup velocity., r is the water density, g is the gravity acceleration. The rate of wave energy dissipation D is assumed to be related with fraction Pˆ of the breaking components in the height probability distribution ŽLeont’yev, 1997., E
D s 2 Pˆ , Pˆ s T
H
ž / gˆ h
4
(
, gˆ s 0.5 q H0rL` ,
Ž 3.
where h is the total water depth including the depth from still water level d and wave set-up z : h s d q z ; gˆ is an empirical quantity. The surf similarity parameter ŽIrribarren number. j used below takes the form
(
j s s B r H0rL` , s B s d BrX B ,
Ž 4.
where s B is the mean bed slope in the surf zone, d B and X B are the breaking depth and the surf-zone width with respect to waves of 1% exceedence height H1% , H0 and
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L` s Ž gr2p .T 2 are the input rms height and deep-water wave length. The value of d B is estimated from the breaker index of Larson and Kraus Ž1989. taken as H1% rd B s
½
1.14j 0 .21 , 0.55,
j ) 0.03 j G 0.03
Ž 5.
Calculation of the diffraction coefficient K d in the shaded region behind the structures is based upon the theory of Goda et al. Ž1979. for irregular waves. The theoretical relationship determines K d along the line making angle d to the incident wave direction Q at the tip of structure ŽFig. 1.. The K d value depends on angle d as well as on the concentrating parameter Sm characterizing the degree of directional spread in the wave field. In most practical cases a typical value of Sm falls in the range 25–75 and variations of Sm within these limits have a rather weak influence on the K d value. Thus, to simplify the calculations, we use the uniform value Sm s 50 and approximate changes in K d with d by the following set of equations:
°1
d F ypr6 3
~0.1 Ž d q pr6. q cos Ž d q pr6. ,
Kd s
0.087 Ž 3 y 4drp . , 0
¢
ypr6F d F pr2 pr2 F d F 3pr4 3pr4 F d
Ž 6.
where angle d is treated as positive when measured toward the structure ŽFig. 1.. The boundary of the wave shadow area where K d ™ 1 corresponds to d s ypr6. On the line coinciding with the wave direction, where d s 0, K d is close to 0.7. Wave heights in the area shaded by the groin are determined by multiplying the unaffected height value Žat the shadow edge. K d . In the case of a detached breakwater, the wave energy contributions coming from both tips of the structure Ž1 and 2 in Fig. 1. should be taken into account. The corresponding diffraction coefficients K d1 and K d2 are computed along the basic line BB passing the center of the shaded region. The wave 2 2 energy along this line is estimated as E BB s K d1 E1 q K d2 E2 , where E1 and E2 are the
Fig. 1. Definition sketch of wave diffraction near the structures.
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wave energies at the boundaries of the wave shadow. The values of EBB are limited by the requirement EBB F Ž E1 q E2 .r2. The E quantities between line BB and the shadow boundaries are defined by interpolation. 2.2. Horizontal circulation Wave-driven horizontal circulation in the nearshore region is determined from the depth-integrated equations of motion and water conservation ŽPhillips, 1977. written in terms of mean total water discharges Žincluding steady current and Stokes’ drift. M x s Uh and M y s Vh: EM x
q
EM x2rh
Et EM y
q
EM x M yrh
Ex q
Ey
EM x M yrh
Et
q
EM y2rh
Ex
Ez q Et
EM x
q
Ey
EM y
Ex
Ey
Ez q gh
q
q
r
Ex Ez
q gh
Fx
q
Fy
r
Ey
tbx
y
r q
tby r
1 Et l
r Ey y
1 Et l
r Ex
s 0.
s 0,
s 0,
Ž 7.
Here Fx and Fy represent the forcing terms E Fx s
Ex
E
Ž Sx x q R x x . q
E Fy s
Ey
Ž Sx y q R x y . ,
E
Ž S q R y y . q E x Ž Sy x q R y x . , Ey yy
Ž 8.
caused by the radiation stresses Si j and stresses R i j due to rollers in breaking waves ŽDally and Osiecki, 1994., Sx x s
Sy y s
E 2Cg 2
C
E 2Cg 2
C
Sx y s Sy x s
Ž 1 q cos 2 Q . y 1 y Ž 1 q sin2 Q . y 1 y
E Cg 2 C
sin2Q y
r M W2 x h
r M W2 y
r MW x MW y h
h
,
,
,
R x x s 2 Er cos 2Q , R y y s 2 Er sin2Q , R x y s R y x s Er sin2Q , Er s 4br
C gT
ˆ PE,
where Er is the roller energy per unit area and br ( 0.9.
Ž 9.
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Water discharges M W x , M W y represent the onshore mass flux caused by waves ŽStokes drift. and rollers:
Ž M W x , M W y . s Ž E q 2 Er . Ž cos Q , sinQ . r Ž r C . .
Ž 10 .
The bottom shear t b x , t b y stresses depend on both the oscillatory orbital velocity amplitude u m and mean current velocities U, V ŽLe Blond and Tang, 1974.,
tbxs tbys
2
p 2
p
Cf r u m Ž 1 q cos 2 . U q V sinQ cosQ , Cf r u m U sinQ cosQ q Ž 1 q sin2Q . V .
Ž 11 .
The estimation of the bed friction coefficient Cf is based on the assumption of the logarithmic velocity profile in a steady current: VŽ z. s
V) k
ln
z za
,
Ž 12 .
where k s 0.4 is the Von Karman constant and z a is the apparent roughness due to the effect of wave oscillations. The shear velocity V U may be expressed in terms of depth-averaged velocity V obtained by integrating Eq. Ž12. over the vertical. Since in a pure current V U2 s CfV 2 , it is easy to find Cf s w krŽlnŽ hrz 0 . y 1.x 2 . The presence of waves will change the magnitude of Cf . Besides, the Cf value would depend on the choice of bed roughness parameter. Thus, we adopt Cf s
kX k 2 ln Ž hrz a . y 1
2
, hrz a 4 1
Ž 13 .
where kX is the adjusting constant. The influence of waves on the steady current results in a growth of the effective bed roughness for the current. According to Nielsen Ž1992., the magnitude of z a is evaluated by z a s ey1 l, where l is the thickness of layer with predominantly wave-generated turbulence. The magnitude of l depends on various factors including the bed surface properties and relative current strength. However, it may be expected that for a relatively weak current the scale of l should correspond to the scale of wave boundary layer thickness d w . Therefore, we suppose z a s ey1d w , d w s f w r2 am ,
(
Ž 14 .
where am s u m rv is the amplitude of water particle excursions along the bed and v s 2prT is the angular wave frequency. The wave friction coefficient f w is herein determined from Nielsen’s formula: f w s exp Ž 5.5 Ž rram .
0.2
y 6.3 . ,
Ž 15 .
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where the bed roughness parameter r for movable sand bottom depends on Shields’ parameter C . The magnitude of r under regular wave conditions is about twice as that for random waves with the same significant height ŽKaczmarek et al., 1994.. Hence, in the case of a random wave, Nielsen’s relationship for r may look like r s 85 C 2.5 y 0.05 d s q 4h 2rl ,
(
Ž 16 .
where d S is the mean grain size, h and l are the wave ripple height and length, respectively, Žcomputed from the formulas of Nielsen Ž1992. also. and C 2.5 is defined with respect to conventional roughness r s 2.5d S . Based on the outlined approach to estimate z a , the adjusting coefficient kX in Eq. Ž13. would be about 0.33. This is found by testing the model against the available data on longshore currents. A typical value of Cf for a natural sandy bottom in the surf zone is of the order 10y2 . The Reynolds stress value t l due to turbulent lateral mixing is given by
t l s rn l
ž
EM x Ey
q
EM y Ex
/
, n l s 0.006
ž
gH0 T
n
1r4
/
(
H0 gH0 ,
Ž 17 .
where n s 10y6 m2rs is the molecular water viscosity for 208C. The eddy viscosity n l is represented by the empirical relationship based on available surf-zone mixing data ŽBowen and Inman, 1974; Rodriguez et al., 1997.. The longshore current profiles modeled with this relationship are in satisfactory agreement with the observed ones both in the laboratory ŽBadiei and Kamphuis, 1995. and in the sea during a storm ŽKuriyama, 1994.. Both the shoreline and structure contours are considered as absorbing boundaries, where the current velocities are assumed to be zero. At the liquid boundaries the normal gradients of mean velocities and set-up are taken as zero. The system of Eq. Ž7. is integrated using an economically fully implicit finite-difference scheme based on the method of Maa Ž1990. and adapted to nearshore dynamics. The optimal value of time increment D t was found to be of the order of Ž10y1 –10 0 . X B r gd B . A steady state of circulation is actually achieved after a period of time of the order of 10 2 D t. The dimension scale l of the area studied is measured by hundreds of meters in a typical field case. The size of the cells in the spatial grid employed is of the order 10y2 l.
(
2.3. Near-bed flow Total near-bed mean velocity u determining the net sediment transport is treated as a sum of the near-bed values of both the circulation velocity UB and wave-induced mass ˜ B due to bottom boundary layer mechanisms, transport velocity U
˜B, u s UB q U
Ž 18 .
where u refers to the edge of the wave boundary layer Žto level z B s d w .. ˜ B value. The theory of Different opinions have been brought forward for the U Longuet-Higgins Ž1953. predicts a shoreward drift Žsteady streaming. generated by a phase shift in orbital motions near the bed due to viscosity. Dally and Dean Ž1984. and
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Stive and De Vriend Ž1994. suggested that this trend is also valid in wave breaking regions. However, according to the theory of Svendsen et al. Ž1987. the phase shift mechanism is suppressed by both the developing turbulence and the undertow acting in the middle layer, and so the resulting mass transport should be directed offshore everywhere.
Fig. 2. Cross-shore distributions of wave-induced near-bed mean flow velocity observed in the laboratory ŽOkayasu and Katayama, 1992. and in the sea ŽRodriguez et al., 1994. and computed from Eqs. Ž19. and Ž20..
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The laboratory and field data available indicate that shoreward transport more frequently occurs in the offshore regions, while in the surf zone the seaward near-bed drift predominates of which the strength is maximal in the plunging region. Such a trend shows that the direction and magnitude of mean bottom drift are related with the rate of energy dissipation D due to wave breaking. The increase in D enhances the undertow and causes its displacement into the lower part of the water column closer to the bed ŽSvendsen et al., 1987.. The influence of undertow on the near-bed flow would result in a decay of the onshore steady streaming. At a certain threshold value of D s DU , the undertow penetrates into the wave boundary layer and provokes a reversal of the bottom drift. Based on these speculations, one can assume that both the magnitude and sign of the near-bed mean velocity may depend on the difference between the actual dissipation rate and its threshold value. Hence, keeping in mind that the scale of mass transport velocity is given by quantity u 2m rC ŽLonguet-Higgins, 1953. we may write
ž U˜B ,V˜B / s cX
DU y D u 2m DU
C
Ž cos Q , sinQ . ,
Ž 19 .
where cX should be of the order 1. If breaking is absent Ž D s 0., Eq. Ž19. with cX s 3r4 transforms into the formula of Longuet-Higgins Ž1953. for mean Eulerian bottom velocity. Taking into account that the scale of the dissipation rate is determined by the local gradient of energy flux yEŽ ECg .rE x, we assume that the threshold value DU may be
Fig. 3. Definition sketch of the swash zone.
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characterized by the mean gradient of the energy flux on a distance l c from a given point to the shoreline: DU s ECgrl c ,
Ž 20 .
Eqs. Ž19. and Ž20. were tested using available observations of mean near-bed flow in ˜ B are compared with wave random waves field. In Fig. 2, the computed velocities U flume ŽOkayasu and Katayama, 1992. and field data ŽRodriguez et al., 1994.. The laboratory measurements were performed very close to the bed Žabout 2 mm from the bed., whereas in the field experiment the lowest gauge was placed at 0.1 m height above the bottom. It should be noted that outside the breaking region the mean velocity changes rapidly with the distance from the bed and even may reverse just above the boundary layer ŽVillaret and Davies, 1997.. Thus, the velocities at the 0.1 m level may be inappropriate to characterize the near-bed mean flow. However, in the surf zone an excess mixing created by breakers should result in a more uniform profile of the mass transport in the bottom layer. With these remarks, one can conclude that, in general, the predicted and observed trends are similar. Furthermore, it was found that the depth where the threshold value DU is achieved is close to the breaking depth given by criterion Ž5.. Hence, the condition D s DU may serve as a mark of the surf-zone boundary Žwith respect to waves of 1% exceedence height.. Assuming a logarithmic velocity profile in the circulation current ŽEq. Ž12.., one can express the near-bed circulation velocity UB in terms of depth-averaged value U. Since the reference level for UB is z B s d w where according to Eq. Ž14. d w s e z a , we obtain: UB s Ur Ž ln Ž hrz a . y 1 . , hrz a 4 1.
Ž 21 .
3. Sediment transport 3.1. WaÕe shoaling and breaking regions Sediment transport rates are computed using the energetic-based formula of Bailard Ž1981. where small values proportional to the bottom slope are neglected. An additional term B is inserted representing the contribution of wave breaking in the sediment suspending process ŽLeont’yev, 1997.. Thus, the total immersed-weight sediment transport rate per unit width is given by ´b ´s 1 qs f w ru 3 q f ru 3 < u < q Bu , Ž 22 . 2 tan F WS 2 w
ž
/
where ´ b and ´s are the bedload and suspended-load transport efficiency factors, Fig. 4. Comparison of the observed and predicted temporal changes in beach profiles due to a random wave attack in the case of a gentle beach Ža: data from Roelvink and Stive, 1989, wave height H s 0.123 m, period T s 2 s, sand grain size d S s 0.1 mm, duration of wave action t w s12 h. and in case of a steep beach with dune Žb: data from Hedegaard et al., 1992, H s1.1 m, T s6 s, d S s 0.22 mm, t w s6.3 h.. Also, the computed distributions of cross-shore sediment transport rates q x are shown for initial and final beach profiles.
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respectively, tan F is the grain-to-grain friction coefficient, WS is the grain fall velocity and the overbar denotes averaging in time. Total velocity u is a sum of wave oscillatory and steady components, where the first one may be represented at least by two harmonics: u s u m cos v t q u 2 m cos Ž 2 v t q w . q u, u 2 m , u < u m
Ž 23 .
Herein w is the phase shift and u the mean velocity includes the contributions of both the horizontal circulation and boundary layer mass transport according to Eq. Ž18.. Calculations of the velocity moments u 3 and u 3 < u < yield u3 s
3 4
ž
u 3m 2
u q um
u2 m um
/
cos w , u 3 < u < s
16 15p
ž
u 4m 5
u um
q3
u2 m um
cos w
/
Ž 24 .
The secondary harmonic evidently contributes only in positive Žonshore. transport. In the surf zone where the transport rate is maximal, this contribution is thought to be relatively insignificant as in saw-tooth breaking waves w ™ pr2 and cos w ™ 0. In the offshore region, the role of higher harmonics would be more important. However, it appears that in the case of random waves the main features of sediment transport are to a greater extent determined by the mean flow mechanisms ŽRoelvink and Stive, 1989.. Thus, to simplify the problem we neglect the latter terms in the right-hand sides of the relationships Ž24.. Designating F s Ž 2r3p . f w r u 3m , K b s Ž 9pr8 . Ž ´ brtanF . ,
Ž 25 .
where F is the power expenditure due to bed friction, and separating the abovementioned contributions in u, we can express the cross-shore q x s qcos u and the longshore q y s qsin u sediment transport rates from Eq. Ž22. as q x s G Ž UB q U˜B . , q y s G Ž VB q V˜B . , G s K b Fru m q ´s Ž 4 F q B . rWS ,
Ž 26 .
where G characterizes the immersed weight of moving solid grains per unit area. As seen from Eq. Ž26., in case of a straight uniform beach where the total shore-normal water flux is zero ŽU s 0. the cross-shore net sediment flux is only determined by the ˜ B. mass transport velocity U The term B in Eqs. Ž22. and Ž26. representing the dissipation rate due to excess near-bed turbulence generated by breakers is parameterized in the form ŽLeont’yev, 1997. B s 0.4 j Dexp y3 Ž 1rPˆ y 1 . .
'
Ž 27 .
Fig. 5. Initial bottom topography Ža. and storm-generated streamline distributions in the presence of a groin field Žb., a detached breakwater Žc. and a navigable channel protected by jetty Žd.. Directions of waves and currents are indicated by arrows. Bottom contours and water discharges are given in m and m3 rs, respectively.
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3.2. Swash zone It is supposed that in the swash region the bedload transport predominates and the relationships Ž26. transform into
'
q R x s G R UR , q R y s G R VR , G R s K b Fru R , u R s 2 gR ,
Ž 28 .
where u R is the scale of amplitude of oscillating run-up flow depending on the run-up height R. The power losses due to friction F are estimated from Eq. Ž25. where u m is replaced by u R and a typical value of friction coefficient is taken to be about 0.08 for a sandy beach. The cross-shore net transport velocity is evaluated by ŽLeont’yev, 1996. UR s u R Ž tan beq y tan b . , tan beq s
20WS
Ž gT .
1r2
Ž gH0 .
1r4
,
Ž 29 .
where tan beq and tan b are the equilibrium and actual mean bottom slopes, both taken for the section between the run-up upper limit x s X R and point x s X W , where the depth is equal to the input wave height H0 ŽFig. 3.. The empirical relationship for tan beq is obtained by fitting the model to available laboratory data on a sandy beach profile development during a random-wave attack ŽUliczka, 1987; Roelvink and Stive, 1989; Hedegaard et al., 1992; Larson, 1994.. For obliquely incident waves the water particles in the run-up flow move on saw-tooth trajectories with net longshore displacement l y per one wave period. It is seen from Fig. 3 that l y s l R sin u , where l R is the distance passed by particles during the up-rush. The length of l R in turn depends on the run-up height as l R s RrŽsin br cos Q .. Hence, the mean longshore transport velocity at the shoreline, proportional to l yrT, is determined by VR s cX Ž RrT . Ž tan Qrtan b . ,
Ž 30 .
where sin b f tan b is adopted implying the smallness of b . The value of coefficient cX is found to be about 0.01 if the angle Q is taken in point x s X W . The run-up height R is computed from the modified version of the well-known formula of Hunt Ž1959.
(
R s 1.5HsW tan br Hs 0rL` ,
Ž 31 .
where Hs0 and HsW are the significant wave heights in both initial point and point x s X W ŽFig. 3.. Significant height is related with rms quantity as Hs s 62 H. The relationships Ž28. determine the maximal swash-zone sediment discharges attributed to still-water shoreline x s XC ŽFig. 3.. Approaching to boundaries x s X R and x s X W , the transport rates q R x and q R y decay tending to become zero. Total rates in the section between points x s X W and x s XC are evaluated by a sum of q R x , q R y and qx , qy. Fig. 6. Morphological changes due to groins of various lengths Ž Lg . under the action of swell Ža. and stormy waves Žb, c, d.. Angle of wave incidence is 308 in all cases.
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The efficiency factors in the transport formulas are adopted in accordance with the ‘‘standard’’ of Soulsby Ž1995. as ´ b s 0.1 and ´s s 0.02. Predicted transport rates using these values are in agreement with both the available longshore sediment transport data ŽKamphuis, 1991. and the observed changes in beach profile Ždiscussed below.. 3.3. Bed deformations Temporal changes in bed level are determined from the equation of sediment mass conservation Ed Eq x Eq y s ay1 q , a s r g Ž rsrr y 1 . Ž 1 y s . , Ž 32 . Et Ex Ey
ž
/
where rs and s are the density and porosity of sediments. This equation is integrated by using the Lax–Wendroff numerical scheme where the morphological time step D t M includes two iterations. Sediment transport rates and depth changes are computed at both iterations while the wave field is determined only at the one first. To save the computation time, the horizontal circulation is calculated with greater time steps including several tens of morphological steps. This implies that mean depth-averaged velocities do not change significantly with relatively small changes in depth. The order of magnitude of D t M is evaluated by quantity Ž10 d SrH0 rtan2b .T, which, in case of a gently sloping sandy beach, is about 10 2 T. To provide the stability of the numerical procedure a downslope gravitational transport component is added to the sediment transport rates Žoutside the swash zone. in accordance with recommendations of Rakha and Kamphuis Ž1995.:
(
qXx s q x y 2 < q < s x , qXy s q y y 2 < q < s y , < q < s q x2 q q y2
(
Ž 33 .
where s x , s y are the local cross-shore and shore-parallel bottom slopes. In order to demonstrate the model capabilities in a one-dimensional case, the observed and predicted temporal changes in beach profiles of different kinds are compared in Fig. 4. Also, the initial and final computed distributions of cross-shore sediment transport rates are shown. The observed gentle and steep beach profiles were taken from works of Roelvink and Stive Ž1989. and Hedegaard et al. Ž1992., respectively. It is seen that the swash-zone transport is comparable with that in the surf zone only for a steep beach Žb. and negligible over a gentle slope Ža.. In both cases, the sediment transport rates decay with time indicating that predicted profiles tend to equilibrium state. 4. Results of modelling Some examples represented below demonstrate the computed morphological response due to coastal structures of different kinds for the initially uniform beach with straight Fig. 7. Morphological changes around a detached breakwater placed on various depths Ž d br . under the action of swell Ža. and stormy waves Žb, c, d.. Angle of wave incidence is 08 in cases a, b and c and it is 158 in case d.
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parallel bottom contours ŽFig. 5a.. A slightly convex-o-concave cross-shore profile typical for closed non-tidal sea beaches ŽLeont’yev, 1985. is in a given case similar to the beach profile observed by the author in the East Korean Gulf Žthe Japan Sea.. The mean sand grain size was taken as 0.2 mm. The modeled wave conditions are a relatively weak swell Žrms initial height H0 s 0.5 m, spectral peak period T s 6 s, steepness H0rL` s 0.009, breaking depth d B s 1.7 m., and a moderate storm Ž H0 s 1.0 m, T s 6 s, H0rL` s 0.018, d B s 3.0 m.. To demonstrate the main trends of the morphological changes clearly enough, the duration of the wave attack was taken to be equal to 10 days for stormy waves and 20 days for a swell. In Fig. 5b,c and d the patterns of storm-generated horizontal circulation are displayed for cases of groins, breakwater and a navigable channel. The directions of waves and currents are indicated by arrows. The streamline distributions were computed for the bottom topography existing at the moment of when wave action finished Žcorresponding bottom contours are shown in Figs. 6d–8d.. The morphodynamic effect of a pair of groins is reproduced in Fig. 6. b, c and d characterize storm-generated bed deformations for a groin length Lg s 66, 111 and 156 m, and case a refers to swell when Lg s 66 m. The angle of wave incidence is equal to 308 in all runs. Both the stormy waves and swell cause sediment accretion at the up-drift side of the structures Žwhich is true for the inter-groin space also.. The shore in this section moves out and the bottom contours turn toward the sea. The properties of the accretion process are evidently controlled by the structure length. The influence of relatively short groins Žb. is limited by a narrow beach area, but changes in shoreline contour are maximal in this case. For the longer groins Žc, d. the shoreline displacement decreases while the up-drift shoal is expanded offshore significantly. In the lee of the structures the beach is eroded. This is indicated by a shift of the shoreline and bottom contours towards the land. Maximal erosion is observed at the groin tips Žc, d.. Morphological changes caused by low swell and relatively high stormy waves in general are similar when the proportion between the wave height and the groin length remains approximately uniform Žcases a and d.. Fig. 7 demonstrates the morphodynamics in the vicinity of a detached shore-parallel breakwater placed at various depths under conditions of shore-normal Ža, b, c. and obliquely incident waves Žd.. The influence of structure disposed within the surf zone of stormy waves Žb. is relatively weak as compared with its offshore position Žc, d.. The beach section in the lee of a breakwater advances towards the sea. Sediment accumulation is especially visible at the edges of a wave shadow area where the tombolos are formed. At the ends of the structure, the bottom is actively excavated. Obliquely incident waves Žd. result in more rapid seaward movement of the up-drift section of the beach, and so the effects of breakwaters and a groin look analogous in this case. Down-drift from the structure an obliquely oriented trough is developed.
Fig. 8. Initial contours of a navigable channel bed Ža. and their deformations caused by shore normal Žb. and obliquely incident stormy waves Žc, d.. In case d, the channel is protected by jetty.
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Stormy waves and swell produce similar changes in bottom topography Žexcept for sediment accretion at the outer side of breakwater during a swell.. Fig. 8 represents the bed deformations of a navigable channel exposed to an attack of shore-normal Žb. and obliquely incident stormy waves Žc.. In the latter case, the up-drift slope of the bed channel is filled up by sediments while its down-drift slope is eroded. As a result the channel fairway deviates from the shore-normal and shifts down-drift. Such a trend is visibly suppressed if the channel bed is protected by a jetty Žd.. It is seen from Fig. 5d that in the lee of the structure a stagnation zone is created. However, the beach section located just behind the jetty is actively eroded ŽFig. 8d.. It should be noted that because of relatively small beach slopes in the above examples the contribution of swash transport was rather small except for the case of a channel.
5. Model verification Fig. 9 reflects an attempt to model the small-scale laboratory test NT2 of Badiei et al. Ž1994. where the effect of a groin was studied. An initially plane sandy beach with mean
Fig. 9. Comparison of the observed Ža. and predicted Žb. morphological changes around a groin after 12-h wave attack on initially plane sandy beach Ž d s s 0.12 mm.. Data were taken from the laboratory test NT2 performed by Badiei et al. Ž1994. with irregular waves Ž Hs s 0.08 m, T s1.15 s, Q sy108..
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sand size s 0.12 mm and slope s 0.1 was exposed to irregular waves of 0.08 m significant height, 1.15 s spectral peak period and y108 incidence angle. The groin was installed after 4-h wave attack when a clear convex-o-concave bottom profile was formed. Total duration of the wave action was 12 h. The final bottom contours both observed and predicted are depicted in Fig. 9a and b, respectively Žcontours are smoothed to simplify the comparison.. The laboratory test conditions were characterized by low Shields’ parameter magnitudes ŽC 2.5 - 0.4.. The energetic based model overestimates sediment flux in such a case as the actual transport efficiency should decrease approaching to threshold of sediment movement ŽC 2.5 f 0.05.. It was found that appropriate magnitudes of the longshore sediment flux at low transport stage may be predicted with modified efficiency factors:
(
(
´ b s 0.1 C 2.5 y 0.05 , ´s s 0.02 C 2.5 y 0.05 ,
Ž 34 .
where C 2.5 - 1 is implied. However, the computed cross-shore transport rates still remain too high. This discrepancy is difficult to explain. Perhaps the wave ripples, of which the expected height Žabout 0.01 m. should be only one order less than water depth, could disturb the near-bed flow. Villaret and Davies Ž1997. conclude that over the rippled bed the boundary layer mean flow may be three times weaker as compared to the smooth bed flow. To adjust the temporal changes in beach profile in a given case the computed cross-shore transport velocities Žoutside the swash zone. were reduced by a constant factor of about 0.25. The limit of the observed morphological changes due to a groin roughly corresponds to a depth contour of 0.1 m ŽFig. 9a.. The predicted effect of a groin as a trap for sediments is more significant. Up-drift accretion is traced up to 0.2 m depth ŽFig. 9b. and so the model does not reproduce a score hole seen in Fig. 9a at the groin head. However, the observed and computed changes in shoreline contour agree well. It should be mentioned that in a given case the swash-zone transport noticeably influenced the shoreline displacements in numerical procedure.
6. Conclusions The most essential features of the morphodynamic model proposed in the present work can be outlined as follows. Although the effect of currents on a wave field is ignored, the inverted influence is actually inserted implicitly by means of a bed friction mechanism. The apparent roughness for a current is determined taking into account the friction coefficient for wave orbital motion. The latter characteristic depends on movable bed roughness which is in turn related with the sediment transport regime. The wave-induced nearshore circulation is actually treated as a quasi-3D phenomenon. Total mass transport at the bed is defined as the sum of the near-bed component of horizontal circulation and the bottom drift generated by wave boundary layer mechanisms. The determination of the latter value is based on the hypothesis that the direction and magnitude of the bottom drift depend on the difference between the
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actual rate of energy dissipation and its threshold value marking the flow reversal point. This hypothesis helps to explain a general trend of cross-shore mean flow distributions observed in the nearshore region. The sediment transport treatment is based on the energetic approach formulated by Bailard Ž1981.. It is assumed that net sediment motion at a relatively high transport stage is mostly determined by the near-bed mean flow while the part of wave asymmetry related with higher harmonics plays a secondary role. An additional term is inserted in the transport formula to include the contribution of wave breaking in sediment suspension. Reasonable transport rate magnitudes are obtained with standard values of efficiency factors and the movable bed roughness defined by Nielsen Ž1992.. To improve the predictions of the energetic based model at low transport stage modified efficiency factors are proposed. The contribution of the swash zone transport is shown to be important and included in the total sediment flux. The structure boundaries are considered as non-reflective ones. Due to a simplified treatment of certain nearshore phenomena Žin particular, of wave diffraction behind the structures., the computational efficiency is relatively high. This is important for practical use of the model. If the horizontal circulation in a given morphological step is not determined, then the computations within this step take only about 10 s even for PC with 100 MHz processor using the TURBO-BASIC code. When comparing the predictions of the model proposed and other known models, the general trends look similar. In particular, the double-picked tombolos or an obliquely oriented trough at the down-drift end of a breakwater are noted in the works of Broker et al. Ž1995. and Roelvink et al. Ž1995.. Also, the modelling results given above qualitatively correlate with the available morphological data. Moreover, temporal changes in the shoreline contour near a groin, both predicted and observed in the laboratory ŽBadiei et al., 1994. are shown to be in quantitative agreement. The model testing for other types of structures is subject of further study. In the case of a uniform beach, the computed morphological changes due to a wave attack decay with time in agreement with observations and the cross-shore profile tends to equilibrium ŽFig. 4.. The typical time scale of this process is of the order of 10 h for a natural beach. Bottom topography formed under the influence of structures should also tend to equilibrium. But such a trend is possibly manifested at considerably greater time scales corresponding to overall long-term changes in coastal morphology. Acknowledgements The research described in this publication was made possible in part by Grant No. 97-05-64209 from the Russian Foundation of Fundamental Research. References Badiei, P., Kamphuis, J.W., 1995. Physical and numerical study of wave induced currents in wave basins of various sizes. In: Coastal Dynamics ’95. ASCE, Gdansk, pp. 377–388. Badiei, P., Kamphuis, J.W., Hamilton, D.G., 1994. Physical experiments on the effects of groins on shore morphology. In: 24th Int. Conf. on Coastal Eng. Kobe, pp. 1782–1796.
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Bailard, J.A., 1981. An energetics total load sediment transport model for a plane sloping beach. J. Geophys. Res. 86 ŽC 11., 10938–10954. Bowen, A.J., Inman, D.L., 1974. Nearshore mixing due to waves and wave-induced currents. Rapp. P.-v. Reun. Cons. Int. Explor. Mer 167, 6–12. Broker, I., 1995. Coastal area modelling. MAST 68-M Final Workshop, Gdansk, pp. 2-86–2-90. Broker, I., Johnson, H.K., Zyserman, J.A., Ronberg, J.K., Pedersen, C., Deigaard, R., Fredsoe, J., 1995. Coastal profile and coastal area morphodynamic modelling. MAST 68-M Final Workshop, Gdansk, pp. 7-12–7-16. Dally, W.R., Dean, R.G., 1984. Suspended sediment transport and beach profile evaluation. J. Waterway Port Coast Ocean Eng. ASCE, 110, 15–33. Dally, W.R., Osiecki, D.A., 1994. The role of rollers in surf zone currents. In: 24th Int. Conf. on Coastal Eng. Kobe, pp. 1895–1905. De Vriend, H.J., Zyserman, J., Nicholson, J., Roelvink, J.A., Pechon, P., Southgate, H.N., 1993. Medium-term 2DH coastal area modelling. Coastal Engineering 21, 193–224. Goda, Y., Takayama, T., Suzuki, Y., 1979. Diffraction diagrams for directional random waves. In: 16th Intern. Conf. On Coastal Eng. Hamburg, pp. 628–650. Hanson, H., 1989. GENESIS: a generalized shoreline change numerical model. J. Coastal Res. 5 Ž1., 1–27. Hedegaard, I.B., Roelvink, J.A., Southgate, H., Pechon, P., Nicholson, J., Hamm, L., 1992. Intercomparison of coastal profile models. In: 23rd Int. Conf. On Coastal Eng. Venice, pp. 2108–2121. Hunt, I.A., 1959. Design of seawalls and breakwaters. J. Waterw. Harbors Div. 85, 123–152. Kaczmarek, L.M., Harris, J.M., O’Connor, B.A., 1994. Modelling movable bed roughness and friction for spectral waves. In: 24th Intern. Conf. on Coastal Eng. Kobe, pp. 300–314. Kamphuis, J.W., 1991. Alongshore sediment transport rate. J. Waterw., Ports, Coastal and Ocean Eng. ASCE 117 Ž6., 624–640. Kuriyama, Y., 1994. Numerical model for longshore current distribution on a bar-trough beach. In: 24th Int. Conf. on Coastal Eng. Kobe, pp. 2237–2251. Larson, M., 1994. Prediction of beach profile change at mesoscale under random waves. In: 24th Int. Conf. On Coastal Eng. Kobe, pp. 2252–2266. Larson, M., Kraus, N.C., 1989. SBEACH: numerical model for simulating storm-induced beach change. Tech. Rep. CERC-89-9. Coastal Eng. Res. Center. US Army Corps of Engineers. Le Blond, P.H., Tang, C.L., 1974. On energy coupling between waves and rip currents. J. Geophys. Res. 79 Ž6., 811–816. Leont’yev, I.O., 1985. Sediment transport and beach equilibrium profile. Coastal Engineering 9, 277–291. Leont’yev, I.O., 1996. Numerical modelling of beach erosion during storm event. Coastal Engineering 29, 187–200. Leont’yev, I.O., 1997. Short-term shoreline changes due to cross-shore structures: a one-line numerical model. Coastal Engineering 31, 59–75. Longuet-Higgins, M.S., 1953. Mass transport in water waves. Philos. Trans. R. Soc. London A 245 Ž903., 535–581. Maa, J.P.-Y., 1990. An efficient horizontal two-dimensional hydrodynamic model. Coastal Engineering 14, 1–18. Nielsen, P., 1992. Coastal bottom boundary layers and sediment transport. Advanced series on ocean engineering, Vol. 4. World Scientific, 324 pp. Okayasu, A. and Katayama, H., 1992. Distribution of undertow and long-wave component velocity due to random waves. In: 23rd Int. Conf. On Coastal Eng. Venice, pp. 883–893. Pelnard-Considere, R., 1956. Essai de theorie de l’evolution des formes de rivage en plages de sable et de galets. Quatr. J. Hydraulique. Paris. Question 3. Rapp. No. 1. Phillips, O.M., 1977. The Dynamics of the Upper Ocean. Cambridge Univ. Press, 336 pp. Price, D.M., Chesher, T.J., Southgate, H.N., 1995. PISCES. A morphological coastal area model. HR Wallindford. Report SR 411. Rakha, K.A., Kamphuis, J.W., 1995. A morphology model to predict erosion near a seawall. In: Coastal Dynamics ’95. ASCE, Gdansk, pp. 879–890. Rodriguez, A., Sanchez-Arcilla, A., Collado, F.R., Gracia, V., Coussirat, M.G., Prieto, J., 1994. Waves and
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currents at the Ebro delta surf zone measurements and modelling. In: 24th Int. Conf. On Coasnal Eng. Kobe, pp. 2442–2556. Rodriguez, A., Bahia, E., Diez, M., Sanchez-Arcilla, A., Redondo, J.M., Mestres, M., 1997. Experimental study of mixing process using images. In: Int. Conf. Coastal Dynamics ’97. Plymouth, pp. 395–404. Roelvink, J.A., Stive, M.J.F., 1989. Bar-generating cross-shore flow mechanisms on a beach. J. Geophys. Res. 94 ŽC4., 4785–4800. Roelvink, J.A., Reniers, A.J.H.M., Walstra, D.J.R., 1995. Medium-term morphodynamic modelling. MAST 68-M Final Workshop, Gdansk, pp. 7-3–7-6. Soulsby, R.L., 1995. The ‘‘Bailard’’ sediment transport formula: comparisons with data and models. MAST 68-M Final Workshop, Gdansk, pp. 2-46–2-51. Stive, M.J.F., De Vriend, H.J., 1994. Shear stresses and mean flow in shoaling and breaking waves, in: 24th Int. Conf. Coastal Eng., Kobe, pp. 594–608. Svendsen, I.A., Schaffer, H.A., Buhr Hansen, J., 1987. The interaction between the undertow and the boundary layer flow on a beach. J. Geophys. Res. 92 ŽC 11., 11845–11856. Uliczka, K., 1987. Wave run-up on dunes with and without foreshore at prototype scale. Leichtweiss-Inst. Wasserbau Tech Univ. Braunschweig Mitt. 98, 67–88. Villaret, C., Davies, A.G., 1997. Effect of bed roughness on net sediment transport by assymetrical waves. In: Int. Conf. Coastal Dynamics ’97. Plymouth, pp. 235–244.
Modelling of morphological changes due to coastal structures I.O. Leont’yev
)
Russian Academy of Sciences, P.P. ShirshoÕ Institute of Oceanology, NakhimoÕ Prospect, 36, 117851 Moscow, Russian Federation Received 4 December 1998; accepted 8 June 1999
Abstract Two different concepts are applicable to model the nearshore morphodynamics. The first one takes into account only final consequences of acting mechanisms and is aimed at the prediction of long-term trends in beach development. Another approach implies the modelling of the whole suite of elementary processes responsible for changes in nearshore bottom topography during a given storm, and it is the approach used in the present work. A coastal area model complex is proposed that allows to reproduce the local morphological changes due to both the natural processes and the influence of coastal structures, such as a groin, a detached breakwater and a navigable channel Žunderwater trough.. Consisting of a traditional series of basic components, the model differs from other ones in essential aspects concerning the treatment of transport mechanisms. In particular, the determination of wave-induced near-bed mean flow is based on the hypothesis that the direction and magnitude of bottom drift depend on difference between the actual rate of energy dissipation and its threshold value marking the flow reversal point. This hypothesis is shown to explain a general trend of cross-shore mean flow distributions observed in the nearshore region. Besides, the influence of the wave breaking process on sediment suspension is taken into account and the contribution of the swash zone to total sediment transport is included. Examples of computed morphological response are represented to demonstrate the model capability. A satisfactory agreement of computations with available data is pointed out. q 1999 Elsevier Science B.V. All rights reserved. Keywords: Coastal structure; Morphological changes; Bed deformations; Sediment transport; Wave-induced circulation
)
Tel.: q7-095-124-63-94; fax: q7-095-124-59-83; E-mail: [email protected]
0378-3839r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 3 9 Ž 9 9 . 0 0 0 4 5 - 9
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1. Introduction Coastal structures on sandy beaches usually cause changes in the bottom topography and the shoreline contour. Both overall and local deformations due to such structures can usually be observed. The first ones occur at spatial scales exceeding the structure dimensions and can be interpreted as a long-term trend in morphological evolution due to the disturbance of the shore-parallel net sediment flux. For example, the interruption of longshore sediment movement by a groin field or a navigable channel Žunderwater trench. may result in down-drift beach erosion over hundreds of meters or kilometers. The local deformations are related to changes in wave-induced currents and associated movements of bed grains in the vicinity of structures during a given storm event. Their spatial scales are comparable with the structure dimensions. Mathematical modelling of morphological response under the influence of structures is of great practical interest. Depending on the problem, two different approaches can be applied to model the nearshore morphodynamics. The first one takes into account only the integrated result of morphodynamic processes and does not include the detailed analyses of acting mechanisms. The beach profile in this case is assumed to be close to its equilibrium state and beach development results from changes in the bulk longshore sediment flux. This approach forms the basis of so-called one-line models aimed at the prediction of shoreline evolution under the influence of structures ŽPelnard-Considere, 1956; Hanson, 1989; Leont’yev, 1997.. An alternative approach involves the modelling of the whole suite of elementary processes responsible for the local morphological changes in a given area. A typical coastal area model consists of several modules describing the wave field, the spatial distributions of wave-induced currents and the associated sediment transport, and finally the resulting spatial and temporal changes of the bed level. Using a grid with relatively small cells, the model of this kind can resolve the detailed patterns of local bed deformations. Such an approach is employed, for example, in the model complexes developed by Delft Hydraulics ŽDe Vriend et al., 1993; Roelvink et al., 1995., Danish Hydraulic Institute ŽBroker, 1995; Broker et al., 1995. or HR Wallingford ŽPrice et al., 1995.. The attempts to evaluate the morphodynamic effect of structures by using these models are not yet numerous, but the results obtained look encouraging. The reliability of morphological predictions depends on the adequacy of treatment of each elementary process. The present understanding of phenomena observed in the nearshore region is not sufficient to assess which modelling concept is preferable to simulate the near-bed mean flow and sediment transport mechanisms. Besides, it is a problem to fulfill the requirement of accurate modelling and, at the same time, to provide the computational efficiency necessary for practical use. In the present paper, a coastal area model complex is proposed describing the evolution of 2DH bottom topography during a given storm attacking a beach, both in its natural state and in the presence of structures. As compared with other known models of this kind, the model discussed herein is of a similar general composition, but it differs from them at some essential points. This concerns the treatment of the bottom drift and the sediment transport in shoaling–breaking regions and in the swash zone. First, a brief
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outline of the basic modules is given, and then the model capabilities are illustrated by a series of computations for structures of different types. The cases of a groin field, a detached shore-parallel breakwater and a navigable channel Žunderwater trench. are considered.
2. Waves and currents 2.1. WaÕes The wave field is assumed to be random characterized by a root-mean-square Žrms. height H, spectral peak period T and major direction Q with respect to the shore-normal. Mean current velocities are supposed to be small comparing with wave-induced oscillatory orbital velocities, thus the nonlinear wave–current interaction is ignored. Determination of waves in the nearshore region implies calculations of refraction, diffraction, shoaling and dissipation. Wave incident angles Q Ž x, y . are computed from the equation of the wave-vector conservation ŽPhillips, 1977. E sin Q
E cos Q y
Ex
Ey
C
C
s 0,
Ž 1.
where the axis OX and OY are shore-normal and shore-parallel, respectively, C is the wave celerity. The 2DH distribution of rms heights H Ž x, y . is determined from the energy balance equation E
E
Ž ECg cosQ . q E y Ž ECg sinQ . s yD, Ex
Es
1 8
r gH 2 ,
Ž 2.
where E and Cg are the wave energy per unit area and the rate of its transfer Žgroup velocity., r is the water density, g is the gravity acceleration. The rate of wave energy dissipation D is assumed to be related with fraction Pˆ of the breaking components in the height probability distribution ŽLeont’yev, 1997., E
D s 2 Pˆ , Pˆ s T
H
ž / gˆ h
4
(
, gˆ s 0.5 q H0rL` ,
Ž 3.
where h is the total water depth including the depth from still water level d and wave set-up z : h s d q z ; gˆ is an empirical quantity. The surf similarity parameter ŽIrribarren number. j used below takes the form
(
j s s B r H0rL` , s B s d BrX B ,
Ž 4.
where s B is the mean bed slope in the surf zone, d B and X B are the breaking depth and the surf-zone width with respect to waves of 1% exceedence height H1% , H0 and
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L` s Ž gr2p .T 2 are the input rms height and deep-water wave length. The value of d B is estimated from the breaker index of Larson and Kraus Ž1989. taken as H1% rd B s
½
1.14j 0 .21 , 0.55,
j ) 0.03 j G 0.03
Ž 5.
Calculation of the diffraction coefficient K d in the shaded region behind the structures is based upon the theory of Goda et al. Ž1979. for irregular waves. The theoretical relationship determines K d along the line making angle d to the incident wave direction Q at the tip of structure ŽFig. 1.. The K d value depends on angle d as well as on the concentrating parameter Sm characterizing the degree of directional spread in the wave field. In most practical cases a typical value of Sm falls in the range 25–75 and variations of Sm within these limits have a rather weak influence on the K d value. Thus, to simplify the calculations, we use the uniform value Sm s 50 and approximate changes in K d with d by the following set of equations:
°1
d F ypr6 3
~0.1 Ž d q pr6. q cos Ž d q pr6. ,
Kd s
0.087 Ž 3 y 4drp . , 0
¢
ypr6F d F pr2 pr2 F d F 3pr4 3pr4 F d
Ž 6.
where angle d is treated as positive when measured toward the structure ŽFig. 1.. The boundary of the wave shadow area where K d ™ 1 corresponds to d s ypr6. On the line coinciding with the wave direction, where d s 0, K d is close to 0.7. Wave heights in the area shaded by the groin are determined by multiplying the unaffected height value Žat the shadow edge. K d . In the case of a detached breakwater, the wave energy contributions coming from both tips of the structure Ž1 and 2 in Fig. 1. should be taken into account. The corresponding diffraction coefficients K d1 and K d2 are computed along the basic line BB passing the center of the shaded region. The wave 2 2 energy along this line is estimated as E BB s K d1 E1 q K d2 E2 , where E1 and E2 are the
Fig. 1. Definition sketch of wave diffraction near the structures.
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wave energies at the boundaries of the wave shadow. The values of EBB are limited by the requirement EBB F Ž E1 q E2 .r2. The E quantities between line BB and the shadow boundaries are defined by interpolation. 2.2. Horizontal circulation Wave-driven horizontal circulation in the nearshore region is determined from the depth-integrated equations of motion and water conservation ŽPhillips, 1977. written in terms of mean total water discharges Žincluding steady current and Stokes’ drift. M x s Uh and M y s Vh: EM x
q
EM x2rh
Et EM y
q
EM x M yrh
Ex q
Ey
EM x M yrh
Et
q
EM y2rh
Ex
Ez q Et
EM x
q
Ey
EM y
Ex
Ey
Ez q gh
q
q
r
Ex Ez
q gh
Fx
q
Fy
r
Ey
tbx
y
r q
tby r
1 Et l
r Ey y
1 Et l
r Ex
s 0.
s 0,
s 0,
Ž 7.
Here Fx and Fy represent the forcing terms E Fx s
Ex
E
Ž Sx x q R x x . q
E Fy s
Ey
Ž Sx y q R x y . ,
E
Ž S q R y y . q E x Ž Sy x q R y x . , Ey yy
Ž 8.
caused by the radiation stresses Si j and stresses R i j due to rollers in breaking waves ŽDally and Osiecki, 1994., Sx x s
Sy y s
E 2Cg 2
C
E 2Cg 2
C
Sx y s Sy x s
Ž 1 q cos 2 Q . y 1 y Ž 1 q sin2 Q . y 1 y
E Cg 2 C
sin2Q y
r M W2 x h
r M W2 y
r MW x MW y h
h
,
,
,
R x x s 2 Er cos 2Q , R y y s 2 Er sin2Q , R x y s R y x s Er sin2Q , Er s 4br
C gT
ˆ PE,
where Er is the roller energy per unit area and br ( 0.9.
Ž 9.
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Water discharges M W x , M W y represent the onshore mass flux caused by waves ŽStokes drift. and rollers:
Ž M W x , M W y . s Ž E q 2 Er . Ž cos Q , sinQ . r Ž r C . .
Ž 10 .
The bottom shear t b x , t b y stresses depend on both the oscillatory orbital velocity amplitude u m and mean current velocities U, V ŽLe Blond and Tang, 1974.,
tbxs tbys
2
p 2
p
Cf r u m Ž 1 q cos 2 . U q V sinQ cosQ , Cf r u m U sinQ cosQ q Ž 1 q sin2Q . V .
Ž 11 .
The estimation of the bed friction coefficient Cf is based on the assumption of the logarithmic velocity profile in a steady current: VŽ z. s
V) k
ln
z za
,
Ž 12 .
where k s 0.4 is the Von Karman constant and z a is the apparent roughness due to the effect of wave oscillations. The shear velocity V U may be expressed in terms of depth-averaged velocity V obtained by integrating Eq. Ž12. over the vertical. Since in a pure current V U2 s CfV 2 , it is easy to find Cf s w krŽlnŽ hrz 0 . y 1.x 2 . The presence of waves will change the magnitude of Cf . Besides, the Cf value would depend on the choice of bed roughness parameter. Thus, we adopt Cf s
kX k 2 ln Ž hrz a . y 1
2
, hrz a 4 1
Ž 13 .
where kX is the adjusting constant. The influence of waves on the steady current results in a growth of the effective bed roughness for the current. According to Nielsen Ž1992., the magnitude of z a is evaluated by z a s ey1 l, where l is the thickness of layer with predominantly wave-generated turbulence. The magnitude of l depends on various factors including the bed surface properties and relative current strength. However, it may be expected that for a relatively weak current the scale of l should correspond to the scale of wave boundary layer thickness d w . Therefore, we suppose z a s ey1d w , d w s f w r2 am ,
(
Ž 14 .
where am s u m rv is the amplitude of water particle excursions along the bed and v s 2prT is the angular wave frequency. The wave friction coefficient f w is herein determined from Nielsen’s formula: f w s exp Ž 5.5 Ž rram .
0.2
y 6.3 . ,
Ž 15 .
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where the bed roughness parameter r for movable sand bottom depends on Shields’ parameter C . The magnitude of r under regular wave conditions is about twice as that for random waves with the same significant height ŽKaczmarek et al., 1994.. Hence, in the case of a random wave, Nielsen’s relationship for r may look like r s 85 C 2.5 y 0.05 d s q 4h 2rl ,
(
Ž 16 .
where d S is the mean grain size, h and l are the wave ripple height and length, respectively, Žcomputed from the formulas of Nielsen Ž1992. also. and C 2.5 is defined with respect to conventional roughness r s 2.5d S . Based on the outlined approach to estimate z a , the adjusting coefficient kX in Eq. Ž13. would be about 0.33. This is found by testing the model against the available data on longshore currents. A typical value of Cf for a natural sandy bottom in the surf zone is of the order 10y2 . The Reynolds stress value t l due to turbulent lateral mixing is given by
t l s rn l
ž
EM x Ey
q
EM y Ex
/
, n l s 0.006
ž
gH0 T
n
1r4
/
(
H0 gH0 ,
Ž 17 .
where n s 10y6 m2rs is the molecular water viscosity for 208C. The eddy viscosity n l is represented by the empirical relationship based on available surf-zone mixing data ŽBowen and Inman, 1974; Rodriguez et al., 1997.. The longshore current profiles modeled with this relationship are in satisfactory agreement with the observed ones both in the laboratory ŽBadiei and Kamphuis, 1995. and in the sea during a storm ŽKuriyama, 1994.. Both the shoreline and structure contours are considered as absorbing boundaries, where the current velocities are assumed to be zero. At the liquid boundaries the normal gradients of mean velocities and set-up are taken as zero. The system of Eq. Ž7. is integrated using an economically fully implicit finite-difference scheme based on the method of Maa Ž1990. and adapted to nearshore dynamics. The optimal value of time increment D t was found to be of the order of Ž10y1 –10 0 . X B r gd B . A steady state of circulation is actually achieved after a period of time of the order of 10 2 D t. The dimension scale l of the area studied is measured by hundreds of meters in a typical field case. The size of the cells in the spatial grid employed is of the order 10y2 l.
(
2.3. Near-bed flow Total near-bed mean velocity u determining the net sediment transport is treated as a sum of the near-bed values of both the circulation velocity UB and wave-induced mass ˜ B due to bottom boundary layer mechanisms, transport velocity U
˜B, u s UB q U
Ž 18 .
where u refers to the edge of the wave boundary layer Žto level z B s d w .. ˜ B value. The theory of Different opinions have been brought forward for the U Longuet-Higgins Ž1953. predicts a shoreward drift Žsteady streaming. generated by a phase shift in orbital motions near the bed due to viscosity. Dally and Dean Ž1984. and
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Stive and De Vriend Ž1994. suggested that this trend is also valid in wave breaking regions. However, according to the theory of Svendsen et al. Ž1987. the phase shift mechanism is suppressed by both the developing turbulence and the undertow acting in the middle layer, and so the resulting mass transport should be directed offshore everywhere.
Fig. 2. Cross-shore distributions of wave-induced near-bed mean flow velocity observed in the laboratory ŽOkayasu and Katayama, 1992. and in the sea ŽRodriguez et al., 1994. and computed from Eqs. Ž19. and Ž20..
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The laboratory and field data available indicate that shoreward transport more frequently occurs in the offshore regions, while in the surf zone the seaward near-bed drift predominates of which the strength is maximal in the plunging region. Such a trend shows that the direction and magnitude of mean bottom drift are related with the rate of energy dissipation D due to wave breaking. The increase in D enhances the undertow and causes its displacement into the lower part of the water column closer to the bed ŽSvendsen et al., 1987.. The influence of undertow on the near-bed flow would result in a decay of the onshore steady streaming. At a certain threshold value of D s DU , the undertow penetrates into the wave boundary layer and provokes a reversal of the bottom drift. Based on these speculations, one can assume that both the magnitude and sign of the near-bed mean velocity may depend on the difference between the actual dissipation rate and its threshold value. Hence, keeping in mind that the scale of mass transport velocity is given by quantity u 2m rC ŽLonguet-Higgins, 1953. we may write
ž U˜B ,V˜B / s cX
DU y D u 2m DU
C
Ž cos Q , sinQ . ,
Ž 19 .
where cX should be of the order 1. If breaking is absent Ž D s 0., Eq. Ž19. with cX s 3r4 transforms into the formula of Longuet-Higgins Ž1953. for mean Eulerian bottom velocity. Taking into account that the scale of the dissipation rate is determined by the local gradient of energy flux yEŽ ECg .rE x, we assume that the threshold value DU may be
Fig. 3. Definition sketch of the swash zone.
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characterized by the mean gradient of the energy flux on a distance l c from a given point to the shoreline: DU s ECgrl c ,
Ž 20 .
Eqs. Ž19. and Ž20. were tested using available observations of mean near-bed flow in ˜ B are compared with wave random waves field. In Fig. 2, the computed velocities U flume ŽOkayasu and Katayama, 1992. and field data ŽRodriguez et al., 1994.. The laboratory measurements were performed very close to the bed Žabout 2 mm from the bed., whereas in the field experiment the lowest gauge was placed at 0.1 m height above the bottom. It should be noted that outside the breaking region the mean velocity changes rapidly with the distance from the bed and even may reverse just above the boundary layer ŽVillaret and Davies, 1997.. Thus, the velocities at the 0.1 m level may be inappropriate to characterize the near-bed mean flow. However, in the surf zone an excess mixing created by breakers should result in a more uniform profile of the mass transport in the bottom layer. With these remarks, one can conclude that, in general, the predicted and observed trends are similar. Furthermore, it was found that the depth where the threshold value DU is achieved is close to the breaking depth given by criterion Ž5.. Hence, the condition D s DU may serve as a mark of the surf-zone boundary Žwith respect to waves of 1% exceedence height.. Assuming a logarithmic velocity profile in the circulation current ŽEq. Ž12.., one can express the near-bed circulation velocity UB in terms of depth-averaged value U. Since the reference level for UB is z B s d w where according to Eq. Ž14. d w s e z a , we obtain: UB s Ur Ž ln Ž hrz a . y 1 . , hrz a 4 1.
Ž 21 .
3. Sediment transport 3.1. WaÕe shoaling and breaking regions Sediment transport rates are computed using the energetic-based formula of Bailard Ž1981. where small values proportional to the bottom slope are neglected. An additional term B is inserted representing the contribution of wave breaking in the sediment suspending process ŽLeont’yev, 1997.. Thus, the total immersed-weight sediment transport rate per unit width is given by ´b ´s 1 qs f w ru 3 q f ru 3 < u < q Bu , Ž 22 . 2 tan F WS 2 w
ž
/
where ´ b and ´s are the bedload and suspended-load transport efficiency factors, Fig. 4. Comparison of the observed and predicted temporal changes in beach profiles due to a random wave attack in the case of a gentle beach Ža: data from Roelvink and Stive, 1989, wave height H s 0.123 m, period T s 2 s, sand grain size d S s 0.1 mm, duration of wave action t w s12 h. and in case of a steep beach with dune Žb: data from Hedegaard et al., 1992, H s1.1 m, T s6 s, d S s 0.22 mm, t w s6.3 h.. Also, the computed distributions of cross-shore sediment transport rates q x are shown for initial and final beach profiles.
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respectively, tan F is the grain-to-grain friction coefficient, WS is the grain fall velocity and the overbar denotes averaging in time. Total velocity u is a sum of wave oscillatory and steady components, where the first one may be represented at least by two harmonics: u s u m cos v t q u 2 m cos Ž 2 v t q w . q u, u 2 m , u < u m
Ž 23 .
Herein w is the phase shift and u the mean velocity includes the contributions of both the horizontal circulation and boundary layer mass transport according to Eq. Ž18.. Calculations of the velocity moments u 3 and u 3 < u < yield u3 s
3 4
ž
u 3m 2
u q um
u2 m um
/
cos w , u 3 < u < s
16 15p
ž
u 4m 5
u um
q3
u2 m um
cos w
/
Ž 24 .
The secondary harmonic evidently contributes only in positive Žonshore. transport. In the surf zone where the transport rate is maximal, this contribution is thought to be relatively insignificant as in saw-tooth breaking waves w ™ pr2 and cos w ™ 0. In the offshore region, the role of higher harmonics would be more important. However, it appears that in the case of random waves the main features of sediment transport are to a greater extent determined by the mean flow mechanisms ŽRoelvink and Stive, 1989.. Thus, to simplify the problem we neglect the latter terms in the right-hand sides of the relationships Ž24.. Designating F s Ž 2r3p . f w r u 3m , K b s Ž 9pr8 . Ž ´ brtanF . ,
Ž 25 .
where F is the power expenditure due to bed friction, and separating the abovementioned contributions in u, we can express the cross-shore q x s qcos u and the longshore q y s qsin u sediment transport rates from Eq. Ž22. as q x s G Ž UB q U˜B . , q y s G Ž VB q V˜B . , G s K b Fru m q ´s Ž 4 F q B . rWS ,
Ž 26 .
where G characterizes the immersed weight of moving solid grains per unit area. As seen from Eq. Ž26., in case of a straight uniform beach where the total shore-normal water flux is zero ŽU s 0. the cross-shore net sediment flux is only determined by the ˜ B. mass transport velocity U The term B in Eqs. Ž22. and Ž26. representing the dissipation rate due to excess near-bed turbulence generated by breakers is parameterized in the form ŽLeont’yev, 1997. B s 0.4 j Dexp y3 Ž 1rPˆ y 1 . .
'
Ž 27 .
Fig. 5. Initial bottom topography Ža. and storm-generated streamline distributions in the presence of a groin field Žb., a detached breakwater Žc. and a navigable channel protected by jetty Žd.. Directions of waves and currents are indicated by arrows. Bottom contours and water discharges are given in m and m3 rs, respectively.
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3.2. Swash zone It is supposed that in the swash region the bedload transport predominates and the relationships Ž26. transform into
'
q R x s G R UR , q R y s G R VR , G R s K b Fru R , u R s 2 gR ,
Ž 28 .
where u R is the scale of amplitude of oscillating run-up flow depending on the run-up height R. The power losses due to friction F are estimated from Eq. Ž25. where u m is replaced by u R and a typical value of friction coefficient is taken to be about 0.08 for a sandy beach. The cross-shore net transport velocity is evaluated by ŽLeont’yev, 1996. UR s u R Ž tan beq y tan b . , tan beq s
20WS
Ž gT .
1r2
Ž gH0 .
1r4
,
Ž 29 .
where tan beq and tan b are the equilibrium and actual mean bottom slopes, both taken for the section between the run-up upper limit x s X R and point x s X W , where the depth is equal to the input wave height H0 ŽFig. 3.. The empirical relationship for tan beq is obtained by fitting the model to available laboratory data on a sandy beach profile development during a random-wave attack ŽUliczka, 1987; Roelvink and Stive, 1989; Hedegaard et al., 1992; Larson, 1994.. For obliquely incident waves the water particles in the run-up flow move on saw-tooth trajectories with net longshore displacement l y per one wave period. It is seen from Fig. 3 that l y s l R sin u , where l R is the distance passed by particles during the up-rush. The length of l R in turn depends on the run-up height as l R s RrŽsin br cos Q .. Hence, the mean longshore transport velocity at the shoreline, proportional to l yrT, is determined by VR s cX Ž RrT . Ž tan Qrtan b . ,
Ž 30 .
where sin b f tan b is adopted implying the smallness of b . The value of coefficient cX is found to be about 0.01 if the angle Q is taken in point x s X W . The run-up height R is computed from the modified version of the well-known formula of Hunt Ž1959.
(
R s 1.5HsW tan br Hs 0rL` ,
Ž 31 .
where Hs0 and HsW are the significant wave heights in both initial point and point x s X W ŽFig. 3.. Significant height is related with rms quantity as Hs s 62 H. The relationships Ž28. determine the maximal swash-zone sediment discharges attributed to still-water shoreline x s XC ŽFig. 3.. Approaching to boundaries x s X R and x s X W , the transport rates q R x and q R y decay tending to become zero. Total rates in the section between points x s X W and x s XC are evaluated by a sum of q R x , q R y and qx , qy. Fig. 6. Morphological changes due to groins of various lengths Ž Lg . under the action of swell Ža. and stormy waves Žb, c, d.. Angle of wave incidence is 308 in all cases.
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The efficiency factors in the transport formulas are adopted in accordance with the ‘‘standard’’ of Soulsby Ž1995. as ´ b s 0.1 and ´s s 0.02. Predicted transport rates using these values are in agreement with both the available longshore sediment transport data ŽKamphuis, 1991. and the observed changes in beach profile Ždiscussed below.. 3.3. Bed deformations Temporal changes in bed level are determined from the equation of sediment mass conservation Ed Eq x Eq y s ay1 q , a s r g Ž rsrr y 1 . Ž 1 y s . , Ž 32 . Et Ex Ey
ž
/
where rs and s are the density and porosity of sediments. This equation is integrated by using the Lax–Wendroff numerical scheme where the morphological time step D t M includes two iterations. Sediment transport rates and depth changes are computed at both iterations while the wave field is determined only at the one first. To save the computation time, the horizontal circulation is calculated with greater time steps including several tens of morphological steps. This implies that mean depth-averaged velocities do not change significantly with relatively small changes in depth. The order of magnitude of D t M is evaluated by quantity Ž10 d SrH0 rtan2b .T, which, in case of a gently sloping sandy beach, is about 10 2 T. To provide the stability of the numerical procedure a downslope gravitational transport component is added to the sediment transport rates Žoutside the swash zone. in accordance with recommendations of Rakha and Kamphuis Ž1995.:
(
qXx s q x y 2 < q < s x , qXy s q y y 2 < q < s y , < q < s q x2 q q y2
(
Ž 33 .
where s x , s y are the local cross-shore and shore-parallel bottom slopes. In order to demonstrate the model capabilities in a one-dimensional case, the observed and predicted temporal changes in beach profiles of different kinds are compared in Fig. 4. Also, the initial and final computed distributions of cross-shore sediment transport rates are shown. The observed gentle and steep beach profiles were taken from works of Roelvink and Stive Ž1989. and Hedegaard et al. Ž1992., respectively. It is seen that the swash-zone transport is comparable with that in the surf zone only for a steep beach Žb. and negligible over a gentle slope Ža.. In both cases, the sediment transport rates decay with time indicating that predicted profiles tend to equilibrium state. 4. Results of modelling Some examples represented below demonstrate the computed morphological response due to coastal structures of different kinds for the initially uniform beach with straight Fig. 7. Morphological changes around a detached breakwater placed on various depths Ž d br . under the action of swell Ža. and stormy waves Žb, c, d.. Angle of wave incidence is 08 in cases a, b and c and it is 158 in case d.
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parallel bottom contours ŽFig. 5a.. A slightly convex-o-concave cross-shore profile typical for closed non-tidal sea beaches ŽLeont’yev, 1985. is in a given case similar to the beach profile observed by the author in the East Korean Gulf Žthe Japan Sea.. The mean sand grain size was taken as 0.2 mm. The modeled wave conditions are a relatively weak swell Žrms initial height H0 s 0.5 m, spectral peak period T s 6 s, steepness H0rL` s 0.009, breaking depth d B s 1.7 m., and a moderate storm Ž H0 s 1.0 m, T s 6 s, H0rL` s 0.018, d B s 3.0 m.. To demonstrate the main trends of the morphological changes clearly enough, the duration of the wave attack was taken to be equal to 10 days for stormy waves and 20 days for a swell. In Fig. 5b,c and d the patterns of storm-generated horizontal circulation are displayed for cases of groins, breakwater and a navigable channel. The directions of waves and currents are indicated by arrows. The streamline distributions were computed for the bottom topography existing at the moment of when wave action finished Žcorresponding bottom contours are shown in Figs. 6d–8d.. The morphodynamic effect of a pair of groins is reproduced in Fig. 6. b, c and d characterize storm-generated bed deformations for a groin length Lg s 66, 111 and 156 m, and case a refers to swell when Lg s 66 m. The angle of wave incidence is equal to 308 in all runs. Both the stormy waves and swell cause sediment accretion at the up-drift side of the structures Žwhich is true for the inter-groin space also.. The shore in this section moves out and the bottom contours turn toward the sea. The properties of the accretion process are evidently controlled by the structure length. The influence of relatively short groins Žb. is limited by a narrow beach area, but changes in shoreline contour are maximal in this case. For the longer groins Žc, d. the shoreline displacement decreases while the up-drift shoal is expanded offshore significantly. In the lee of the structures the beach is eroded. This is indicated by a shift of the shoreline and bottom contours towards the land. Maximal erosion is observed at the groin tips Žc, d.. Morphological changes caused by low swell and relatively high stormy waves in general are similar when the proportion between the wave height and the groin length remains approximately uniform Žcases a and d.. Fig. 7 demonstrates the morphodynamics in the vicinity of a detached shore-parallel breakwater placed at various depths under conditions of shore-normal Ža, b, c. and obliquely incident waves Žd.. The influence of structure disposed within the surf zone of stormy waves Žb. is relatively weak as compared with its offshore position Žc, d.. The beach section in the lee of a breakwater advances towards the sea. Sediment accumulation is especially visible at the edges of a wave shadow area where the tombolos are formed. At the ends of the structure, the bottom is actively excavated. Obliquely incident waves Žd. result in more rapid seaward movement of the up-drift section of the beach, and so the effects of breakwaters and a groin look analogous in this case. Down-drift from the structure an obliquely oriented trough is developed.
Fig. 8. Initial contours of a navigable channel bed Ža. and their deformations caused by shore normal Žb. and obliquely incident stormy waves Žc, d.. In case d, the channel is protected by jetty.
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Stormy waves and swell produce similar changes in bottom topography Žexcept for sediment accretion at the outer side of breakwater during a swell.. Fig. 8 represents the bed deformations of a navigable channel exposed to an attack of shore-normal Žb. and obliquely incident stormy waves Žc.. In the latter case, the up-drift slope of the bed channel is filled up by sediments while its down-drift slope is eroded. As a result the channel fairway deviates from the shore-normal and shifts down-drift. Such a trend is visibly suppressed if the channel bed is protected by a jetty Žd.. It is seen from Fig. 5d that in the lee of the structure a stagnation zone is created. However, the beach section located just behind the jetty is actively eroded ŽFig. 8d.. It should be noted that because of relatively small beach slopes in the above examples the contribution of swash transport was rather small except for the case of a channel.
5. Model verification Fig. 9 reflects an attempt to model the small-scale laboratory test NT2 of Badiei et al. Ž1994. where the effect of a groin was studied. An initially plane sandy beach with mean
Fig. 9. Comparison of the observed Ža. and predicted Žb. morphological changes around a groin after 12-h wave attack on initially plane sandy beach Ž d s s 0.12 mm.. Data were taken from the laboratory test NT2 performed by Badiei et al. Ž1994. with irregular waves Ž Hs s 0.08 m, T s1.15 s, Q sy108..
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sand size s 0.12 mm and slope s 0.1 was exposed to irregular waves of 0.08 m significant height, 1.15 s spectral peak period and y108 incidence angle. The groin was installed after 4-h wave attack when a clear convex-o-concave bottom profile was formed. Total duration of the wave action was 12 h. The final bottom contours both observed and predicted are depicted in Fig. 9a and b, respectively Žcontours are smoothed to simplify the comparison.. The laboratory test conditions were characterized by low Shields’ parameter magnitudes ŽC 2.5 - 0.4.. The energetic based model overestimates sediment flux in such a case as the actual transport efficiency should decrease approaching to threshold of sediment movement ŽC 2.5 f 0.05.. It was found that appropriate magnitudes of the longshore sediment flux at low transport stage may be predicted with modified efficiency factors:
(
(
´ b s 0.1 C 2.5 y 0.05 , ´s s 0.02 C 2.5 y 0.05 ,
Ž 34 .
where C 2.5 - 1 is implied. However, the computed cross-shore transport rates still remain too high. This discrepancy is difficult to explain. Perhaps the wave ripples, of which the expected height Žabout 0.01 m. should be only one order less than water depth, could disturb the near-bed flow. Villaret and Davies Ž1997. conclude that over the rippled bed the boundary layer mean flow may be three times weaker as compared to the smooth bed flow. To adjust the temporal changes in beach profile in a given case the computed cross-shore transport velocities Žoutside the swash zone. were reduced by a constant factor of about 0.25. The limit of the observed morphological changes due to a groin roughly corresponds to a depth contour of 0.1 m ŽFig. 9a.. The predicted effect of a groin as a trap for sediments is more significant. Up-drift accretion is traced up to 0.2 m depth ŽFig. 9b. and so the model does not reproduce a score hole seen in Fig. 9a at the groin head. However, the observed and computed changes in shoreline contour agree well. It should be mentioned that in a given case the swash-zone transport noticeably influenced the shoreline displacements in numerical procedure.
6. Conclusions The most essential features of the morphodynamic model proposed in the present work can be outlined as follows. Although the effect of currents on a wave field is ignored, the inverted influence is actually inserted implicitly by means of a bed friction mechanism. The apparent roughness for a current is determined taking into account the friction coefficient for wave orbital motion. The latter characteristic depends on movable bed roughness which is in turn related with the sediment transport regime. The wave-induced nearshore circulation is actually treated as a quasi-3D phenomenon. Total mass transport at the bed is defined as the sum of the near-bed component of horizontal circulation and the bottom drift generated by wave boundary layer mechanisms. The determination of the latter value is based on the hypothesis that the direction and magnitude of the bottom drift depend on the difference between the
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actual rate of energy dissipation and its threshold value marking the flow reversal point. This hypothesis helps to explain a general trend of cross-shore mean flow distributions observed in the nearshore region. The sediment transport treatment is based on the energetic approach formulated by Bailard Ž1981.. It is assumed that net sediment motion at a relatively high transport stage is mostly determined by the near-bed mean flow while the part of wave asymmetry related with higher harmonics plays a secondary role. An additional term is inserted in the transport formula to include the contribution of wave breaking in sediment suspension. Reasonable transport rate magnitudes are obtained with standard values of efficiency factors and the movable bed roughness defined by Nielsen Ž1992.. To improve the predictions of the energetic based model at low transport stage modified efficiency factors are proposed. The contribution of the swash zone transport is shown to be important and included in the total sediment flux. The structure boundaries are considered as non-reflective ones. Due to a simplified treatment of certain nearshore phenomena Žin particular, of wave diffraction behind the structures., the computational efficiency is relatively high. This is important for practical use of the model. If the horizontal circulation in a given morphological step is not determined, then the computations within this step take only about 10 s even for PC with 100 MHz processor using the TURBO-BASIC code. When comparing the predictions of the model proposed and other known models, the general trends look similar. In particular, the double-picked tombolos or an obliquely oriented trough at the down-drift end of a breakwater are noted in the works of Broker et al. Ž1995. and Roelvink et al. Ž1995.. Also, the modelling results given above qualitatively correlate with the available morphological data. Moreover, temporal changes in the shoreline contour near a groin, both predicted and observed in the laboratory ŽBadiei et al., 1994. are shown to be in quantitative agreement. The model testing for other types of structures is subject of further study. In the case of a uniform beach, the computed morphological changes due to a wave attack decay with time in agreement with observations and the cross-shore profile tends to equilibrium ŽFig. 4.. The typical time scale of this process is of the order of 10 h for a natural beach. Bottom topography formed under the influence of structures should also tend to equilibrium. But such a trend is possibly manifested at considerably greater time scales corresponding to overall long-term changes in coastal morphology. Acknowledgements The research described in this publication was made possible in part by Grant No. 97-05-64209 from the Russian Foundation of Fundamental Research. References Badiei, P., Kamphuis, J.W., 1995. Physical and numerical study of wave induced currents in wave basins of various sizes. In: Coastal Dynamics ’95. ASCE, Gdansk, pp. 377–388. Badiei, P., Kamphuis, J.W., Hamilton, D.G., 1994. Physical experiments on the effects of groins on shore morphology. In: 24th Int. Conf. on Coastal Eng. Kobe, pp. 1782–1796.
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Bailard, J.A., 1981. An energetics total load sediment transport model for a plane sloping beach. J. Geophys. Res. 86 ŽC 11., 10938–10954. Bowen, A.J., Inman, D.L., 1974. Nearshore mixing due to waves and wave-induced currents. Rapp. P.-v. Reun. Cons. Int. Explor. Mer 167, 6–12. Broker, I., 1995. Coastal area modelling. MAST 68-M Final Workshop, Gdansk, pp. 2-86–2-90. Broker, I., Johnson, H.K., Zyserman, J.A., Ronberg, J.K., Pedersen, C., Deigaard, R., Fredsoe, J., 1995. Coastal profile and coastal area morphodynamic modelling. MAST 68-M Final Workshop, Gdansk, pp. 7-12–7-16. Dally, W.R., Dean, R.G., 1984. Suspended sediment transport and beach profile evaluation. J. Waterway Port Coast Ocean Eng. ASCE, 110, 15–33. Dally, W.R., Osiecki, D.A., 1994. The role of rollers in surf zone currents. In: 24th Int. Conf. on Coastal Eng. Kobe, pp. 1895–1905. De Vriend, H.J., Zyserman, J., Nicholson, J., Roelvink, J.A., Pechon, P., Southgate, H.N., 1993. Medium-term 2DH coastal area modelling. Coastal Engineering 21, 193–224. Goda, Y., Takayama, T., Suzuki, Y., 1979. Diffraction diagrams for directional random waves. In: 16th Intern. Conf. On Coastal Eng. Hamburg, pp. 628–650. Hanson, H., 1989. GENESIS: a generalized shoreline change numerical model. J. Coastal Res. 5 Ž1., 1–27. Hedegaard, I.B., Roelvink, J.A., Southgate, H., Pechon, P., Nicholson, J., Hamm, L., 1992. Intercomparison of coastal profile models. In: 23rd Int. Conf. On Coastal Eng. Venice, pp. 2108–2121. Hunt, I.A., 1959. Design of seawalls and breakwaters. J. Waterw. Harbors Div. 85, 123–152. Kaczmarek, L.M., Harris, J.M., O’Connor, B.A., 1994. Modelling movable bed roughness and friction for spectral waves. In: 24th Intern. Conf. on Coastal Eng. Kobe, pp. 300–314. Kamphuis, J.W., 1991. Alongshore sediment transport rate. J. Waterw., Ports, Coastal and Ocean Eng. ASCE 117 Ž6., 624–640. Kuriyama, Y., 1994. Numerical model for longshore current distribution on a bar-trough beach. In: 24th Int. Conf. on Coastal Eng. Kobe, pp. 2237–2251. Larson, M., 1994. Prediction of beach profile change at mesoscale under random waves. In: 24th Int. Conf. On Coastal Eng. Kobe, pp. 2252–2266. Larson, M., Kraus, N.C., 1989. SBEACH: numerical model for simulating storm-induced beach change. Tech. Rep. CERC-89-9. Coastal Eng. Res. Center. US Army Corps of Engineers. Le Blond, P.H., Tang, C.L., 1974. On energy coupling between waves and rip currents. J. Geophys. Res. 79 Ž6., 811–816. Leont’yev, I.O., 1985. Sediment transport and beach equilibrium profile. Coastal Engineering 9, 277–291. Leont’yev, I.O., 1996. Numerical modelling of beach erosion during storm event. Coastal Engineering 29, 187–200. Leont’yev, I.O., 1997. Short-term shoreline changes due to cross-shore structures: a one-line numerical model. Coastal Engineering 31, 59–75. Longuet-Higgins, M.S., 1953. Mass transport in water waves. Philos. Trans. R. Soc. London A 245 Ž903., 535–581. Maa, J.P.-Y., 1990. An efficient horizontal two-dimensional hydrodynamic model. Coastal Engineering 14, 1–18. Nielsen, P., 1992. Coastal bottom boundary layers and sediment transport. Advanced series on ocean engineering, Vol. 4. World Scientific, 324 pp. Okayasu, A. and Katayama, H., 1992. Distribution of undertow and long-wave component velocity due to random waves. In: 23rd Int. Conf. On Coastal Eng. Venice, pp. 883–893. Pelnard-Considere, R., 1956. Essai de theorie de l’evolution des formes de rivage en plages de sable et de galets. Quatr. J. Hydraulique. Paris. Question 3. Rapp. No. 1. Phillips, O.M., 1977. The Dynamics of the Upper Ocean. Cambridge Univ. Press, 336 pp. Price, D.M., Chesher, T.J., Southgate, H.N., 1995. PISCES. A morphological coastal area model. HR Wallindford. Report SR 411. Rakha, K.A., Kamphuis, J.W., 1995. A morphology model to predict erosion near a seawall. In: Coastal Dynamics ’95. ASCE, Gdansk, pp. 879–890. Rodriguez, A., Sanchez-Arcilla, A., Collado, F.R., Gracia, V., Coussirat, M.G., Prieto, J., 1994. Waves and
166
I.O. Leont’yeÕr Coastal Engineering 38 (1999) 143–166
currents at the Ebro delta surf zone measurements and modelling. In: 24th Int. Conf. On Coasnal Eng. Kobe, pp. 2442–2556. Rodriguez, A., Bahia, E., Diez, M., Sanchez-Arcilla, A., Redondo, J.M., Mestres, M., 1997. Experimental study of mixing process using images. In: Int. Conf. Coastal Dynamics ’97. Plymouth, pp. 395–404. Roelvink, J.A., Stive, M.J.F., 1989. Bar-generating cross-shore flow mechanisms on a beach. J. Geophys. Res. 94 ŽC4., 4785–4800. Roelvink, J.A., Reniers, A.J.H.M., Walstra, D.J.R., 1995. Medium-term morphodynamic modelling. MAST 68-M Final Workshop, Gdansk, pp. 7-3–7-6. Soulsby, R.L., 1995. The ‘‘Bailard’’ sediment transport formula: comparisons with data and models. MAST 68-M Final Workshop, Gdansk, pp. 2-46–2-51. Stive, M.J.F., De Vriend, H.J., 1994. Shear stresses and mean flow in shoaling and breaking waves, in: 24th Int. Conf. Coastal Eng., Kobe, pp. 594–608. Svendsen, I.A., Schaffer, H.A., Buhr Hansen, J., 1987. The interaction between the undertow and the boundary layer flow on a beach. J. Geophys. Res. 92 ŽC 11., 11845–11856. Uliczka, K., 1987. Wave run-up on dunes with and without foreshore at prototype scale. Leichtweiss-Inst. Wasserbau Tech Univ. Braunschweig Mitt. 98, 67–88. Villaret, C., Davies, A.G., 1997. Effect of bed roughness on net sediment transport by assymetrical waves. In: Int. Conf. Coastal Dynamics ’97. Plymouth, pp. 235–244.