modeling the formation of undulations of the coastline...

Continental Shelf Research 27 (2007) 2014–2031

Modeling the formation of undulations of the coastline:The role of tides

M. van der Vegta,1, H.M. Schuttelaarsb,2, H.E. de Swarta,!

aInstitute for Marine and Atmospheric research (IMAU), Utrecht University, Princetonplein 5, 3584 CC Utrecht, The NetherlandsbEnvironmental Hydrogeology Group, Department of Earth Sciences, Budapestlaan 4, 3584 CS Utrecht, The Netherlands

Received 13 February 2006; received in revised form 13 April 2007; accepted 23 April 2007Available online 18 May 2007

Abstract

An idealized model is developed and analyzed to investigate the relevance of tidal motion for the emergence ofundulations of a sandy coastline. The model describes feedbacks between tidal and steady flow on the inner shelf, sandtransport in the nearshore zone and an irregular coastline. It is demonstrated that an initially straight coastline can becomeunstable with respect to perturbations with a rhythmic structure in the alongshore direction. The mechanism causing thegrowth of these perturbations is explained in terms of vorticity concepts. The relative importance of tide-related and wave-driven sediment fluxes in generating undulations of the coastline is investigated for the Dutch coast. Using parametervalues that are appropriate for the Dutch coast it is found that tides can render a straight coastline unstable. The modelpredicts a fastest growing mode (FGM) with a wavelength that is in the order of the observed length of barrier islands. Themode grows on a time scale of 50 yr and it migrates 200m per year. The wavelength of the FGM decreases with increasingamplitude of the tidal currents. This result is consistent with data of tides, waves and the lengths of barrier islands that arelocated along the Dutch and German Wadden coast.r 2007 Elsevier Ltd. All rights reserved.

Keywords: Coastal morphology; Barrier Island; Tidal currents; Wadden Sea

1. Introduction

A large part of the world’s sandy coastlines showsalongshore rhythmic variations on a wide range oflength and time scales (Ehlers, 1988; Komar, 1998).

This paper focuses on rhythmic mesoscale varia-tions of sandy coasts, i.e., with a characteristiclength scale in the alongshore direction of a fewkilometers to tens of kilometers. Such mesoscalevariations include shoreline sand waves (Bruun,1954; Thevenot and Kraus, 1995; Ruessink andJeuken, 2002) and sequences of barrier islands alongthe Dutch, German and Danish Wadden coast(Fig. 1). Here, the typical length of the barrierislands ranges between a few kilometers (Rottumer-oog) and 30 km (Texel). It has been noted that thetypical length of the barrier islands is inverselyrelated to the tidal range (Oost and de Boer, 1994).

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!Corresponding author. Tel.: +31 30 2533275;fax: +3130 2543163.

E-mail address: [email protected] (H.E. de Swart).1Now at Section of Hydraulic Engineering, Delft University of

Technology, Stevinweg 1, 2628 CN Delft, The Netherlands.2Now at Department of Applied Mathematical Analysis, Delft

University of Technology, Mekelweg 4, 2628 CD Delft, TheNetherlands.

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The tidal range increases when moving from theDutch Wadden Sea towards the German Bight. Asimilar relation between tidal range and the lengthof the barrier islands is found for other barriercoasts, e.g., that of Georgia Bight at the eastAtlantic coast of the USA (FitzGerald, 1996).

The general objectives of the present study aretwofold. The first is to gain fundamental knowledgeabout the origin of the observed rhythmic mesoscalevariations of the coastline. The second is to derive aquantitative relationship between the characteristiclength of these undulations and physical controlparameters (like tidal, wave and shelf character-istics). In the past, several models were developed toanalyze the dynamics of coastlines which areinfluenced by waves. They are all one-line models,i.e., the complex three-dimensional dynamics isparameterized, resulting in an equation for thecoastline position only. A widely used one-linemodel is the one described by Komar (1998)(originally from Pelnard-Considere, 1956). It is

based on the idea that obliquely incident wavesbreak in the surf zone and drive a current whichtransports sediment. Alongshore variations in thissediment transport result in changes of the positionof the coastline. Upon assuming that the width ofthe surf zone is constant and by only consideringsmall deviations of the position of the coastline withrespect to its alongshore mean, a diffusion equationfor the position of the coastline is obtained. Thediffusion coefficient is a function of the wavecharacteristics at the breaker line and the anglebetween the direction of the wave rays at breakingand the normal of the local (unperturbed) coastline.A positive (negative) diffusion coefficient impliesthat a small initial perturbation of the coastline willdecay (grow). A negative diffusion coefficient isobtained when the angle between wave rays atbreaking and the normal of the local coastline ismore than 45!.

The traditional one-line model was extended byAshton et al. (2001) and Falques (2003). They

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Fig. 1. Map showing the barrier islands along the Dutch, German and Danish Wadden coast.

M. van der Vegt et al. / Continental Shelf Research 27 (2007) 2014–2031 2015

assumed that a change in the position of thecoastline results in a change of the entire bathy-metric profile. Hence, even far offshore waveproperties will depend on the actual position ofthe coastline. The diffusion coefficient is nowcalculated as a function of the wave characteristicsfar offshore. The results of Falques (2003) show thatthe diffusion coefficient in the model of Komar(1998) is always positive because the waves refractsuch that the wave rays at breaking will have anangle of less than 45! with respect to the normal ofthe local coastline. Nevertheless, in these models astraight coastline can be unstable because theyaccount for alongshore variations in the wave heightat the breaker line. It turns out that perturbations ofthe coastline can grow if wave rays in deep waterhave an angle of more than 42! with respect to thenormal of the local coastline (high-angle waveinstability).

The models of Ashton et al. (2001) and Falques(2003) yield interesting new insights, but they havethe drawback that they assume that even far fromthe coast large variations of bathymetric contoursexist that are related to undulations of the coastline.Falques and Calvete (2005) modified this byassuming that a perturbation in the position of thecoastline results in a perturbation of the bottomprofile with finite cross-shore extent. This model stillincludes the high-angle wave instability mechanism.However, for the Dutch coast waves are hardly ableto trigger undulations of the coastline (Ashton et al.,2003; Falques and Calvete, 2005). Only by using aspecific shape function of the topographic perturba-tions induced by the undulations of the coastline(case C in Falques, 2006) growing undulations ofthe Dutch coastline are found.

The considerations above motivate the develop-ment and analysis of a process-based one-line modelto investigate whether tides can cause the initialformation of undulations of the coastline. If so, thelength scale of the undulations could set the spacingbetween different inlets along a barrier coast. Thisidea elaborates on knowledge (see Bruun andGerritsen, 1960; FitzGerald, 1996) that inlets formduring severe storms when a barrier and backbarrierarea are completely inundated and channels are cut.It seems plausible that along an undulating barrierthe channels will form at locations where the barrieris retreated.

This idea will be explored in the present study andcompared with another explanation for inlet spa-cing (FitzGerald, 1996). The latter is based on the

assumption that, for a given lagoon system with afixed distance between barrier and the coast, eachinlet has the same tidal prism (which is proportionalto tidal range and area of the backbarrier basin).Hence, a larger tidal range implies smaller distancebetween inlets. The assumptions made in that modelare justifiable for the east coast of the US, butthey are not valid for the Dutch and GermanWadden coast.

In Section 2, a model is formulated in whichcoastline undulations develop as free instabilities ofan alongshore uniform coastline. The new aspect ofthis model is that it explicitly accounts for theinfluence of tidal currents on the stability of thecoastline. In Section 3, the basic state is describedand the linear stability analysis is discussed.The model calculates the growth rate and phasespeed of the coastline perturbation for differentalongshore wavelengths of the perturbation. Theresults of this linear stability analysis are presentedin Section 4. The wave–induced sand fluxes areadopted from Falques (2006). In Section 5, aphysical interpretation of the results (focusing ontidal effects) is given. Section 6 discusses whetherthe results of the combined wave–tide modelprovide an explanation for the observed spacingbetween successive inlets along the Dutch andGerman Wadden coast. Finally, in Section 7 theconclusions are given.

2. Model formulation

In this study, a Cartesian coordinate system isadopted with the x-axis pointing in the cross-shoredirection, the y-axis coinciding with the alongshoremean position of the coastline and the z-axispointing in the upward vertical direction. Thedomain of the model consists of three regions, viz.the surf zone, the nearshore zone and the inner shelf(Fig. 2). The surf zone is the area where the wavesbreak and is located between the coastline x " xcand the breaker line x " xb. The nearshore zone isthe area where bottom changes occur due tochanges in the position of the coastline and islocated between the coastline x " xc and thetransition line x " xt, with xt4xb. Here, thetransition from the nearshore zone to the innershelf occurs. The inner shelf is the region that islocated between the nearshore region and the outershelf. The time scale of bathymetric changes in thisarea is large compared to that of the bottomchanges in the nearshore zone.

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2.1. Hydrodynamics

The tidal hydrodynamics at the inner shelf aregoverned by the depth-averaged, shallow waterequations. The water motion is forced by thesemi-diurnal lunar #M2$ tide, which has frequencys%1:4& 10'4 s'1. The characteristic tidal wave-length is Lg%2p

ffiffiffiffiffiffiffiffiffigH0

p=s%300 km, where g is the

acceleration due to gravity and a characteristicwater depth H0 " 5m is used. The spatial scales ofthe phenomena that are the focus of this study(typically Lbarrier%10 km) are small compared to thewavelength of the tidal wave. Hence, the Froudenumber (Fr%Lbarrier=Lg) is very small. This allowsfor a rigid lid approximation: the sea level variationsthemselves can be neglected, but their spatialgradients are important and result in pressuregradients in the momentum equations (see e.g.,Calvete et al., 2001). The hydrodynamic equationsare

q~uqt

( #~u ) ~r$~u( f~ez &~u " 'g~rz'~tbrH

, (1a)

~r ) #H~u$ " 0. (1b)

Here, ~u is the horizontal velocity vector, t is time, ~rthe horizontal gradient operator, f the Coriolisparameter,~ez the unity vector in vertical direction, zthe elevation of the free surface, H the water depthwith respect to z " 0, ~tb the bed shear-stress and rthe density of water. Usually, the bed shear-stress istaken to depend quadratically on the local velocity.In this study a linearized bed shear-stress formula-tion is used:

~tb " rr~u; r "8

3pCdU , (2)

with Cd%0:0025 the drag coefficient and U themean tidal velocity amplitude at the transition line.The friction coefficient r is taken such that thetidally averaged dissipation of energy at the transi-tion line due to the linearized bottom stress equalsthat of the quadratic bottom stress. A discussion onthe derivation of the linearized bed shear-stress isgiven in Zimmerman (1992).

Currents in this model are forced by a prescribedalongshore pressure gradient. Periodic boundaryconditions in the alongshore direction are imposed(with an as yet unspecified length scale). Further-more, the shore-normal velocity component mustvanish at the transition line and far offshore thevelocity is assumed to have a vanishing cross-shorecomponent (no exchange of water between innershelf and nearshore zone),

u " vqxtqy

at x " xt; u ! 0 for x ! 1, (3)

where u; v are the cross-shore and alongshorecomponent of the velocity vector ~u, respectively.

2.2. Alongshore volumetric sediment flux

The tidal hydrodynamics and sediment flux in thenearshore zone are not explicitly calculated in themodel. Instead, the tidal currents are calculated onthe inner shelf and the velocities at the transitionline are taken as representative for the entirenearshore zone. They are used to calculate thevolumetric flux of sediment q through an x' z-section across the nearshore zone (Fig. 2). This fluxconsists of two parts

q " q#wave$ ( q#curr$, (4)

where q#wave$ is the part due to waves only and q#curr$due to the joint action of tides and waves. It isassumed that q#curr$ involves the stirring of sedimentfrom the bed by waves, which is subsequently

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Fig. 2. (a) Side view of the geometry of the model; (b) top view ofthe model. The positions of the perturbed coastline, breaker lineand transition line are indicated by dashed curves. It is assumedthat the width of the nearshore zone is constant. The volumetricsediment flux in the alongshore direction is denoted by symbol qand occurs over the entire nearshore zone.


transported by a tide-driven residual current. This isparameterized as

q#curr$ " bhvki. (5)

Here, b is a constant and hvki is the tidally averagedshore-parallel component of the velocity at thetransition line:

hvki "1

T

Z T

0vk dt, (6)

where T is the tidal period. The constant b in Eq. (5)accounts for the fact that the sediment is trans-ported in the complete nearshore zone and has theunit m2. A physical interpretation of b is theavailable volume of sediment in the nearshore zoneper unit length:

b "Z xt

xc

Z 0

'H

c

rsdzdx. (7)

In this expression c is the tidally averaged sedimentmass concentration and rs the density of the grains.The mean wave height along the Dutch and GermanWadden coast is in the order of 1m Sha (1989).Using observations performed by Grasmeijer andKleinhans (2004) of sediment concentrations atdifferent levels in the vertical in the nearshore zonenear Egmond aan Zee (the Netherlands) yields anestimate of

R#c=r$dz " O#5& 10'521& 10'3$m.

Assuming that the width of the nearshore zoneis in the order of 500m, this yields that b "O#0:0520:5$m2.

Finally, values of the alongshore sediment fluxq#wave$ are adopted from the model of Falques(2006).

2.3. Evolution of the coastline

To obtain equations that govern the evolutionof the coastline and the transition line we assumethat the depth H0 equals the average activewater depth, which is defined in Falques andCalvete (2005). Mass conservation of sediment thenyields

qxtqt

" '1

H0

qqqy

. (8)

Here, for simplicity, it is assumed that #xt ' xc$ "constant.

Next, the magnitude of the morphological timescale Tm is estimated on which the coastline evolvesdue to alongshore variations in q#curr$. From Eqs. (5)

and (8) it follows:

*xt+Tm

"b*hvi+H0l

,

with *xt+ the cross-shore amplitude of the undula-tion of the coastline, l its alongshore wavelengthand *hvi+ the magnitude of the residual current at thetransition line (depth H0). Typical values for theDutch coast are *xt+%500m, l%10 km, b%0:3m2,*hvi+%0:05ms'1 and H0 " 5m. Substitution of thesevalues in the expression given above, the morpho-logical time scale Tm%50 yr. Hence, the position ofthe coastline may be considered as constant duringone tidal cycle.

3. Basic state and linear stability analysis

3.1. Basic state

The model solutions are denoted by vectorC " #~u; ~rz;xt; q#wave$$

T. This state vector can besplit into a part which describes a basic state withalongshore uniform conditions and a part whichdescribes deviations from this basic state, C "Ceq (C0. Here, Ceq " #~U ; ~rZ;X t;Q#wave$$

T con-tains the basic state variables: ~U " #U ;V $ is thevelocity vector with components U and V in thex- and y-direction, respectively, and ~rZ is thegradient of the sea surface. Furthermore, X t "constant is the position of the transition line in thebasic state (straight coast) and Q#wave$ is the undis-turbed wave-driven sediment flux in the breaker zone.

In the basic state, a spatially uniform alongshorepressure gradient is prescribed that consists of aresidual component (S0) and the main tidal (M2)component with amplitude S2 and frequency s,

qZqy

" 'S2 cos st( S0. (9)

No higher harmonics of the tide are taken intoaccount. The free surface elevation Z is linear in thealongshore direction. The velocity in the basic statehas an alongshore component that varies only in thecross-shore direction, i.e., ~U " #0;V #x; t$$. Thisvelocity component is determined by a balancebetween inertia, alongshore sea surface gradient anddepth-dependent friction,

qVqt

" 'gqZqy

' rV

H. (10)

The solution of this equation is

V #x; t$ " V 2#x$ cos#st( F#x$$ ( V0#x$, (11)


with

V 2#x$ "HS2gffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2H2 ( r2

p ; F#x$ " arctan'sHr

" #,

V 0#x$ "'gS0

rH. (12)

The depth-dependent friction term in the along-shore momentum balance causes a phase lagbetween currents and sea surface gradient (in caseof time dependent pressure gradient). It also resultsin an increasing magnitude of the alongshorevelocity with increasing depth, i.e., cross-shoredistance. Hence, the basic state velocity containsvorticity, defined by O#x; t$ " qV #x; t$=qx. Further-more, a characteristic tidal velocity amplitude U isdefined as

U " V 2#x " X t$. (13)

Upon substitution of the basic state variables inEq. (5) it follows that in the basic state the sedimentflux due to currents reads

Q#curr$ " bV 0. (14)

3.2. Stability analysis

The stability of the alongshore uniform coastlineis studied by considering the dynamics ofalongshore rhythmic perturbations. Hence, inEqs. (1)–(8) C " Ceq (C0 is substituted, with Ceq

as defined in Eqs. (9), (11). Furthermore, C0 "#~u0; ~rz0;x0t; q

0#wave$$

T denotes the state vector with theperturbed variables. The magnitudes of the latterare assumed to be much smaller than those of thecorresponding variables in the basic state.

Linearizing the equations of motion with respectto the small variables yields

qu0

qt( V

qu0

qy' fv0 " 'g

qz0

qx'

ru0

H, (15a)

qv0

qt( u0

qVqx

( Vqv0

qy

" #( fu0 " 'g

qz0

qy'

rv0

H, (15b)

u0dH

dx(H

qu0

qx(H

qv0

qy" 0. (15c)

The linearized boundary conditions (3) are

u0 " Vqx0tqy

at x " X t; u0 ! 0 for x ! 1. (16)

The next step is to find the perturbed sedimentfluxes q0#curr$ , q#curr$ 'Q#curr$ and q0#wave$ , q#wave$'

Q#wave$. Using Eq. (5) and the definition vk " ~u )~s,with ~s the tangent vector at the transition line, thealongshore sediment flux due to the joint action oftides and waves reads

Q#curr$ ( q0#curr$ "bhV ( v0i

*1( #qx0t=qy$2+1=2

$$$$$x"X t(x0t

. (17)

Subsequently, a Taylor expansion of the variousvariables at x " X t is made and only contributionsare kept that are linear in the perturbed quantities.Finally, using Eq. (14), the result is

q0#curr$ " b hv0i( x0tdhV0idx

% &

x"X t

. (18)

The values for the perturbed sediment flux q0#wave$are adopted from Falques (2006). Finally, thelinearized equation for the evolution of the transi-tion line reads

qx0tqt

" 'bH0

qqy

hv0i( x0td

dxhV 0i

% &$$$$x"X t

'1

H0

qq0#wave$qy

.

(19)

Recall that in this model x0t " x0c, so Eq. (19) alsogoverns the evolution of the perturbation of thecoastline.

3.3. Solution procedure

The model equations have solutions of the form

C0 " R*#u#x; t$; v#x; t$; ~rz#x; t$; xt; qwave$eikyeGt+T,

(20)

where R stands for taking the real part. Notethat all variables behave exponentially in time.Besides, the flow variables also show oscillatorybehavior on the tidal time scale which is muchsmaller than the time scale on which the transitionline is evolving. The alongshore wave number isdenoted by k and can be chosen arbitrarily.Furthermore, G " Gre ( iGim is the complexgrowth rate, with a real part (Gre) that describesthe growth of the perturbations, and an imaginarypart (Gim) that determines the phase speed,c " 'Gim=k.

The aim is to determine u#x; t$; v#x; t$; ~rz#x; t$;xt; qwave and G as a function of wavenumber k andmodel parameters. The interest is in perturbationsthat have positive growth rates (Gre40). The modewith wave number k " kp that has the largestgrowth rate will dominate the dynamics after some

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time. Therefore, it is called the most preferred mode.The growth rate is calculated as follows. First, theperturbation xt with given wave number k ischosen. The hydrodynamic problem, described byEqs. (15a)–(15c) and boundary conditions (16), hasto be solved. Because the alongshore dependence ofthe variables is known a priori, v is known as afunction of u (Eq. (15c)), and reads

v " 'u#dH=dx$ (H#qu=qx$

ikH. (21)

Next, a vorticity equation is derived by taking the x-derivative of the alongshore momentum equation(15b) and subtracting the y-derivative of Eq. (15a).When substituting for v, one equation for thecomplex cross-shore velocity u is found:

U12q3uqtqx2

(U11q2uqtqx

(U10quqt

(U02q2uqx2

(U01quqx

(U00u " 0. #22$

The coefficients Uij are identical to those given inCalvete et al. (2001). In this problem one boundarycondition is different, viz. at x " X t variable u isprescribed. Furthermore, for x ! 1 it is requiredthat u ! 0. Solving Eq. (22) together with theboundary conditions yields u, and using Eq. (21)yields v. Substituting Eq. (20) into Eq. (19) yields anexpression for growth rate G:

Gxt " 'bH

ikhvi( ikxtdV 0

dx

" #$$$$x"X t

'ikqwaveH

. (23)

3.4. Numerical implementation

Eq. (22) is solved using a pseudospectral method.The spatial variables are expanded in Chebyshevpolynomials (see Boyd, 2001 for details). Inprevious morphodynamic modeling studies theseChebyshev polynomials have been successfully usedin resolving spatial patterns (Falques et al., 1996).Employing the method discussed in Calvete et al.(2001), for the time-dependent part a Galerkinapproach is adopted. The velocity component u andv are expanded in their harmonic agents M0, M2,M4 and so on. In this study the series is truncatedafter the M2 components, so nonlinear tides are notaccounted for. Hence, the cross-shore velocitycomponent is expanded as

u#x; t$ "XN'1

j"0

*u0j ( u'1j e'ist ( u1j e

ist+Tj# ~x$ (24)

and a similar expansion for v is used. In theexpression above, x " Lx#1( ~x$=#1' ~x$ with Lx astretching parameter and Tj are Chebyshev poly-nomials. Furthermore, N is the number of colloca-tion points in the x-direction and u0j ; u

'1j ; u1j are the

Chebyshev coefficients of the residual component,the e'ist Fourier component and the eist Fouriercomponent of the cross-shore velocity, respectively.The expansion (24) is substituted into Eq. (22)and is evaluated at N collocation points in thex-direction. This results in a system of 3N by 3Nlinear algebraic equations with 3N variables of theform Aw " B, which is solved by standard numericaltechniques. Here, A is a complex 3N by 3N matrixdescribing the model equations. The vector wcontains the coefficients of the expansions ofEq. (24). The 3N elements of complex vector Bare all zero, except those which are linked to valuesof u at the collocation point which represents thetransition line. At that location u is prescribed(boundary condition (16)).

4. Results

4.1. Reference case

The focus of this study is on the influence of tidalcurrents on the stability of the equilibrium statewith respect to perturbations in the position of thecoastline. Thus, q0#wave$ " 0 is assumed in Eqs. (19),implying that qwave " 0 in Eq. (23). Experiments areperformed for parameter values which are repre-sentative for the Dutch coastal area. The followingprofile has been used to represent the depth profileof the inner shelf:

H#x$ " H0 ( #Hs 'H0$#1' e'#x'X t$=L$. (25)

Here, H0 " H#x " xt$ and far offshore, where theinner shelf connects to the outer shelf, the depth isHs. Parameter L is an e-folding length scale which isa measure of the width of the inner shelf. Typicalvalues have been determined by fitting this expres-sion to observed profiles along the Dutch coast(Fig. 3). This yields estimates of H0%O#1' 10$m,Hs%O#15' 25$m and L%O#8' 15$km. In thereference experiment H0 " 5m, Hs " 20m andL " 10 km. The alongshore sea surface gradient ischosen such that the tidal velocity amplitude at thetransition line is U " 0:5ms'1. No net flow isconsidered in the basic state (S0 " 0). The Coriolisparameter is f " 1:14& 10'4 s'1, the drag coeffi-cient Cd " 2:5& 10'3 and b " 0:3m2. The number


of collocation points is N " 100 and the stretchingparameter Lx " 10 km.

The dependence of the growth rate on thealongshore wave number is shown in the top panelof Fig. 4. The growth rate has been madedimensionless such that a value of 1 correspondsto an e-folding time scale of 50 yr. The variablealong the horizontal axis is the non-dimensionalalongshore wave number kL " 2pL=l, where l isthe wavelength and L " 10 km is the width of theinner shelf. The basic state is stable with respect toperturbations having small values of the dimension-less wave number, corresponding to wavelengthswhich are long compared to the width of the innershelf. Perturbations with these wavelengths havenegative growth rates and hence decay exponen-tially. For kL43 the growth rate increases withincreasing kL and for kL%8:2 the growth rate iszero. The wavelength of the perturbation with zerogrowth rate is called l0. In this case l0 " 7:6 km.For kL48:2 the growth rate is positive. Hence,there is a positive feedback between the coastlineperturbation and the tidal flow. The growth ratecurve does not show a maximum, so there is nofastest growing mode (FGM).

The model also yields the phase speed of theperturbations. This phase speed is shown as afunction of the dimensionless alongshore wavenumber in the bottom panel of Fig. 4. It has aminimum for kL%8 and increases for larger kL. Anondimensional phase speed of '0:1 is in dimen-sional values '20m=yr. A negative sign meansthat the perturbation is migrating in the negativey-direction, that is to the right when viewing in theseaward direction.

The perturbed sediment flux q0#curr$ is linear in theresidual currents (Eq. (18)). The left panel of Fig. 5shows a vector plot of the residual currents on theinner shelf. Because the model does not yield aFGM, a wavelength is chosen that has a positivegrowth rate, l " 3 km (kL " 20:9). Furthermore,because the model is linear in x0t an amplitude ischosen of *x0t+ " 500m. Two residual circulationcells per alongshore wavelength appear. In this casethe typical magnitude of the residual currents is0:01ms'1. The structure of the residual circulationcells is such that the maximum convergence of thevelocity at the transition line (and hence theconvergence of the net sediment flux) is slightlyout of phase (to the right when viewing in the

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0 0.5 1 1.5 2 2.5 3

x 104

-35

-30

-25

-20

-15

-10

-5

0

5

Cross-shore direction (m)

De

pth

(m

)

North HollandTexelTerschellingSchiermonnikoog

Fig. 3. Four cross-shore profiles along the Dutch coast. The profile of ‘North Holland’ was measured 20 km south of Den Helder. Thelocations of Texel, Terschelling and Schiermonnikoog can be found on Fig. 1.


seaward direction) with the coastline perturbation.The perturbation will therefore grow and migrate.The migration is forced by the Coriolis force. Whenf " 0, there is no phase difference between theresidual velocities at the transition line and thecoastline perturbation (right panel of Fig. 5). Hence,the perturbation does not migrate.

4.2. Sensitivity of results to tidal current amplitude

Observations show that the shore-parallel tidalcurrent amplitudes on the transition line varybetween approximately 0.1 and %1ms'1 in regionswhere barrier islands are observed. In the referencecase the tidal velocity amplitude at the transition

line was 0:5ms'1. In the next experiment themagnitude of the basic state tidal velocity at thetransition line is varied between 0.1 and 1:0ms'1.A change in the magnitude of the basic statevelocity causes a change of the magnitude of thefriction parameter r, defined in Eq. (2). The frictionparameter scales linearly in the magnitude of thetidal velocity. All other parameters have the samevalue as in the reference experiment.

The influence of the magnitude of the tidalcurrents at the transition line (U) on the growthrate is shown in the top panel of Fig. 6. Largervalues of U result in larger growth rates for largekL. Furthermore, a larger velocity magnituderesults in a larger l0. Hence, coastline perturbationswith larger length scales grow exponentially intime. For U " 0:1ms'1 l0 " 4:5 km, while for U "1ms'1 l0 " 10:3 km.

A change of the magnitude of the tidal currents atthe transition line (U) leads to only a moderatechange of the phase speed (bottom panel of Fig. 6).The plot reveals that a change in U will not influencethe phase speed of the coastline perturbation forsmall wavelengths. For larger wavelengths the tidalcurrent amplitude does influence the phase speed. Asmaller tidal current results in a smaller phase speed.However, a doubling of the tidal current amplitudewith respect to the reference experiment hardlyaffects the phase speed for small kL.

In the next experiment, a time-invariant pressuregradient S0 is prescribed. Hence, the basic statevelocity has a residual component on top of thetidal component. Its profile is described by Eq. (12).The residual pressure gradient is chosen such thatV0#x " X t$ is varied between 0.02 and '0:02ms'1.This implies that the residual currents far offshoreare varied between '0:1 and 0:1ms'1 for thereference bathymetry. All other parameters havetheir reference values. Results (not shown) indicatethat only the phase speeds, not the growth rates,of perturbations of the coastline are stronglyaffected by this basic state residual current. ForV0#x " X t$ " '0:02ms'1 all perturbations are mi-grating to the right (when viewing in seawarddirection). For smaller wavelengths (larger kL) thephase speed is larger. For V 0#x " X t$ " 0:02ms'1

the perturbations are all migrating to the right.

4.3. Sensitivity of results to bathymetric parameters

The bathymetry of the inner shelf influences thebasic state velocity profile. In this section the

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Fig. 4. Top: growth rate as function of wave number for thereference case. A growth rate of 1 corresponds to an e-foldingtime scale of 50 yr. Wave numbers are scaled with the width of theinner shelf, 10 km. Bottom: phase speed as a function of wavenumber for the reference case. A phase speed of '0:1 correspondsto a dimensional phase speed of '20m=yr.

M. van der Vegt et al. / Continental Shelf Research 27 (2007) 2014–20312022

sensitivity of the results to the bathymetric para-meters L and H0 is studied. First, the length of theinner shelf is varied with all other parameters havingtheir reference values. Results (not shown) revealthat a steeper profile results in smaller values of thewavelength of the perturbation with zero growthrate l0. The dependence of l0 on L is almost linear:l0%3:8 km for L " 5 km and l0%12:3 km forL " 20 km. Not only l0 changes, but also thegrowth rate. For small values of L, growth ratesare larger for small wavelengths compared to thereference case. For relatively large values of L,growth rates are smaller for small wavelengthscompared to the reference case. For a fixedwavenumber k the magnitude of the phase speedincreases with decreasing L. For L " 5 km the phasespeed has a maximum of 32m=yr for l " 6:8 km(kL " 9).

Next, the dependence of the results on the valueof H0 is investigated. When H0 is changed the valueof b has to be changed as well. Parameter b can beinterpreted as the volume per unit length inthe alongshore direction of sediment that is insuspension in the nearshore zone (see Eq. (7)). It is

assumed that the steepness of the nearshore zoneis constant and that the average concentrationin the nearshore zone does not change withchanging H0. From this it follows that b%H2

0.Furthermore, when H0 is changed and the along-shore sea surface gradient S2 is kept fixed themagnitude of the bottom friction coefficient rshould be changed as well. However, it is assumedthat the profile of the basic state velocity is notchanged, which implies that the velocity profile issuch that the tidal current amplitude is 0:5ms'1 at5m depth. The friction coefficient r and theamplitude of the M2 pressure gradient S2 keeptheir reference values.

The results of Fig. 7 show that a smaller H0

results in a shift of l0 to smaller wavelengths. Forsmall wavelengths a smaller H0 results in largergrowth rates, compared to the reference case (toppanel of Fig. 7). The influence of H0 on the phasespeed is shown in the bottom panel of Fig. 7. Anincrease of H0 with respect to the referenceexperiment results in an increase of the phasespeed. A decrease of H0 results in a decrease ofthe phase speed.

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Fig. 5. Top: vector plot of residual currents for a perturbation of the coastline with alongshore wavelength l " 3 km; reference case. Thecoastline perturbation is shown at the left-hand side. An amplitude of the coastline perturbation of *x0t+ " 500m has been chosen. Thisyields that *hu0i+%*hv0i+ " 10'2 ms'1. The residual flow is such that the perturbation is amplified. In addition, the Coriolis force induces amigration of the perturbation. Bottom: same as top panel, but now the residual currents in the case of no Coriolis. In this case the coastlineperturbation only grows and is not migrating.


4.4. Influence of waves on the instability mechanism

In all the previous experiments, i.e. with signifi-cant tidal currents and q#wave$ " 0, no FGM wasobtained. Hence, the model cannot explain whyundulations of the coastline have preferred lengthscales, as field data indicate. Furthermore, thegrowth rates are largest for the smallest wave-lengths, which implies that perturbations with aninfinitesimal small length scale will grow fastest.Hence, the model is missing a mechanism thatresults in decay of the perturbations with thesmallest length scale. The results of Falques(2003), Falques and Calvete (2005) indicate that

when a net volumetric sand flux due to obliquelyincident waves (q0#wave$) is accounted for, a selectionmechanism for the growing perturbations isintroduced.

Results of the full eigenvalue problem (23) werecomputed for three different sectors of the Dutchcoast: the southern delta coast, the central Hollandcoast and the northern Wadden coast. For the tidalmodel all parameters have their reference value,except that a value of H0 " 9m has been used,which is consistent with the value that has been usedin Falques (2006). Recall that b scales quadraticallyin the value of H0. Values for the perturbed waveflux q0#wave$ were adopted from Falques (2006). Notethat in the latter study different shape functions of

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Fig. 6. Top: Growth rate curves for different values of themagnitude of the basic state tidal M2 flow at the transition line.The magnitude of U is 0.1, 0.2, 0.5 and 1:0ms'1, respectively. Adimensionless growth rate of 1 corresponds to time scale of 50 yrand alongshore wave number is scaled with 10km. Bottom: sameas top panel, but now the phase speed as a function of the wavenumber. A non-dimensional phase speed of '0:1 is in dimen-sional values '20m=yr.

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Fig. 7. Top: growth rate curves for different values of H0. Adimensionless growth rate of 1 corresponds to time scale of 50 yrand alongshore wave number is scaled with 10 km. Bottom: Sameas top panel, but now the phase speed as a function of the wavenumber. A dimensionless phase speed of '0:1 is in dimensionalvalues '20m=yr.

M. van der Vegt et al. / Continental Shelf Research 27 (2007) 2014–20312024

topographic variations induced by undulations ofthe coastline are assumed than in the present flowmodel. This means that results only give a firstindication of the effect of the sediment fluxes q#curr$and q#wave$ on the characteristics of undulations ofthe coastline.

In the top panel of Fig. 8 the growth rate curvesand migration curves are shown for the northernWadden coast. The solid curves represent thesolutions of the full eigenvalue problem. Here thevalues of case C of Falques (2006) are used, forwhich the topographic shape function has a local

maximum at 300m from the coast. It has beenselected because this shape function results in thelargest growth rates due to waves only. When usingother shape functions growth rates were negative forall wavelengths and waves damp the instabilitymechanism due to tides. It appears that withouttides there is a preferred mode with has the largestgrowth rate, but it grows very slowly (e-folding timescale is about 50 yr). Furthermore, also the pertur-bations of the coastline with very long wavelengthscan grow. When tides are accounted for the scaleselection becomes more pronounced: the growthrates of long waves becomes negative and themaximum growth rate increases by approximately50%. Thus, it seems that tides enhance theinstability of the northern Dutch coast. Further-more, the wave number of the preferred mode shiftsto larger values (the preferred wavelength reducesfrom 7.8 km to about 7.5 km). The migration speedsof the perturbations slightly increase with respect tothe case that only a wave flux is considered and arein the order of 200m/yr.

Sensitivity experiments show that the wavelengthof the preferred mode is a function of the magnitudeof tidal currents at the transition line (Fig. 9). WhenU " 0:1ms'1 the preferred wavelength is 7.85 km,while for U " 0:8ms'1 the preferred wavelength is7.15 km. Hence, the model results predict that thewave length of the preferred mode decreases withincreasing magnitude of the tidal currents.

The model has also been applied to the centralHolland coast and the southern delta coast (results

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Fig. 8. Top: dimensionless growth rate Gr versus dimensionlesslongshore wavenumber k as calculated with the combinedwave–tide model for the northern Dutch coast (solid line).Parameters of the tidal model have their default values, exceptthat H0 " 9m (results shown by dotted line); for the wave modelthe data of Falques (2006) for case C are adopted (results shownby dashed line). A value Gr " 1 corresponds to an e-folding timescale of 50 yr. Bottom: as top panel, but dimensionless migrationspeed c versus k. A value c " 0:1 corresponds to a speed of20myr'1.

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Fig. 9. Growth rate curves calculated with the combinedwave–tide model for different values of tidal current amplitudeat sea. Other parameters have their default values for the Dutchand German Wadden coast.


not shown). For the latter case the effects of tides onthe dynamics of undulations of the coastline are notso large and waves overwhelm the positive feedbackmechanism due to tides. For the Holland coast tidesonly have a marginal effect on the dynamics of theundulations of the coastline in case profile A, C orD (Falques, 2006) is chosen. However, if a differenttopographic shape factor is selected (case B ofFalques, 2006) tides cause positive growth rates forwave lengths larger than 9.5 km when U " 0:7ms'1.Because no data is available of the wave model ofFalques (2006) for wave lengths larger than 10 kmthe presence of a FGM could not be studied.

5. Physical interpretation

It was shown that for the default settings of themodel parameters, perturbations with wavelengthsl48 km have negative growth rates, while forlo8 km the growth rate is positive. The perturba-tions not only grow but also migrate to the rightwhen viewing in the seaward direction. When waveeffects are included in the calculation of thealongshore sediment flux, a fastest growingmode is obtained. In this section these results areexplained in terms of basic physical mechanisms.The emphasis is on the role of tides in thismechanism. The influence of wave-induced along-shore sediment fluxes on the evolution of perturba-tions of the coastline has already been discussed inFalques and Calvete (2005).

The growth and migration of the perturbationsare due to the divergence of the volumetric sedimentflux q0#curr$ as defined in Eq. (18). This flux is linearlyrelated to the tidally averaged velocity at thetransition line, which is related to circulationpatterns (see Fig. 5) on the inner shelf. The physicscausing the generation of residual circulation cellswill be analyzed using vorticity concepts in a similarway as described in Zimmerman (1981). Thevorticity balance for the perturbed variables,retaining only linear terms, is obtained by takingthe x-derivative of Eq. (15b) and subtracting they-derivative of Eq. (15a),

qo0

qt

z}|{#a$

(q#u0O$qx

zfflffl}|fflffl{#b$

(qqy

#Vo0 ( v0O$

zfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflffl{#c$

'r

H2

dH

dxv0

zfflfflfflfflffl}|fflfflfflfflffl{#d$

( fqu0

qx(

qv0

qy

" #zfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflffl{#e$

"'ro0

H:

zfflffl}|fflffl{#f $

#26$

Here, O " qV=qx is the vorticity of the basic stateflow and o0 " qv0=qx' qu0=qy is the vorticity of theperturbed flow. In Eq. (26), term (a) models thelocal time evolution of the perturbed vorticity, term(b) the cross-shore gradient of the perturbedvorticity flux in the cross-shore direction, term (c)the alongshore gradient of the perturbed vorticityflux in the alongshore direction, term (d) thegeneration of perturbed vorticity by the frictionaltorque, term (e) the generation of perturbedvorticity due to vortex stretching of the planetaryvorticity and term (f) the dissipation of perturbedvorticity due to bottom friction.

The model results reveal that the residual currentsare organized in cells with cyclonic (anticyclonic)circulation. These cells are areas where the tidallyaveraged perturbed vorticity is positive (negative).To obtain the relation between the perturbation ofthe coastline and residual current at the transitionline, it is thus important to study the tidallyaveraged vorticity balance. The latter follows fromaveraging Eq. (26) over the tidal period and reads

'qqx

u0O( )

(qqy

Vo0 ( v0O( )% &zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{#1$

(r

H2

dH

dxv0( )

zfflfflfflfflfflfflffl}|fflfflfflfflfflfflffl{#2$

(f

H

dH

dxu0( )

zfflfflfflfflfflffl}|fflfflfflfflfflffl{#3$

"r

Ho0( )zfflfflffl}|fflfflffl{#4$

. #27$

Here, h i denotes an average over the tidal cycle, seeEq. (6) for its definition. Also, the continuityequation (15c) has been applied here. The threesource terms of tidally averaged vorticity are on theleft-hand side of Eq. (27). Term (1) describes theconvergence of the tidally averaged perturbedvorticity flux, (2) the frictional torque due toalongshore currents over cross-shore sloping bathy-metry and (3) the Coriolis torque due to vortexstretching. The right-hand side of Eq. (27), term (4),describes the dissipation of residual perturbedvorticity due to bottom friction.

Now, the different terms are estimated for atypical coastline perturbation of l " 3 km, tidalcurrents at the transition line with a magnitude ofU " 0:5ms'1, absence of residual currents at sea(V 0 " 0ms'1) and f " 0. The perturbation wasalready shown in Fig. 5. It turns out that in this casethe magnitude of term #2$ is much smaller than thatof term #1$ because the generated residual currentsare much smaller than the M2 component of u0 andv0. The main balance in Eq. (27) is between the


production of tidally averaged perturbed vorticitydescribed by term (1) and the dissipation of itdescribed by term (4).

So, to understand the generation of residualcirculation cells the focus should be on term (1),which describes the gradient of three fluxes ofvorticity: hu0Oi is the mean flux of basic statevorticity by the perturbed cross-shore currents. Thesecond one is ho0Vi and represents the mean flux ofperturbed vorticity by the basic state alongshorecurrents. The third one is hv0Oi and describes themean flux of basic state vorticity by the perturbedalongshore currents.

First, consider the cross-shore gradient of hu0Oi.During flood the basic state tidal currents are in thenegative y-direction: V 2 is negative, see left panel ofFig. 10. Furthermore, qV 2=qxo0 and thereforeOo0. The perturbed cross-shore velocity u0 duringflood is 180! out of phase with the alongshoregradient of the coastline perturbation (left panel ofFig. 10, which shows the perturbed velocity vector~u0). The vorticity flux u0O is therefore in phase withthe alongshore gradient of the coastline perturba-tion. Because the magnitude of u0O is decreasing inthe cross-shore direction, the cross-shore gradient ofu0O during flood results in areas on the inner shelfwhere vorticity is accumulating with a same sign asthat of the alongshore gradient of the coastline

perturbation. During ebb the value of V 2, O and u0

change sign. This results in the same sign of u0O andin the same cross-shore gradient of u0O. Hence, thecross-shore gradient of the mean cross-shore vorti-city flux results in areas on the inner shelfcharacterized by mean perturbed vorticity whichhas the same sign as that of the alongshore gradientin the coastline perturbation. In other words,residual circulation cells as sketched in the middlepanel of Fig. 10 are generated. The mean currentstransport the sediment from areas where the coast-line has accreted to areas where the coastline haseroded. This results in a decay of the perturbationof the coastline.

Next, consider the alongshore gradient of hv0Oiand hVo0i. During flood V2o0 and Oo0 (seeabove) and the perturbed alongshore velocity v0 is180! out of phase with the coastline perturbation(left panel of Fig. 10). The perturbed vorticity flux isv0O is therefore in phase with the coastlineperturbation. The alongshore gradient of v0O is180! out of phase with the alongshore gradient inthe coastline perturbation and therefore leads toaccumulation of vorticity with a sign that isopposite to that of the gradient of the coastlineperturbation. In this case the residual circulationcells are as sketched in the right panel of Fig. 10.During ebb V 2, O and v0 change sign. This results in

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Fig. 10. Left: perturbed velocity induced by coastline at maximum flood. The basic tidal flow is from north to south (big arrows). Thecoastline perturbation has a cosine structure. Middle: small arrows represent perturbed cross-shore vorticity flux u0O. The cross-shoregradient of the vorticity in the cross-shore direction generates two residual circulation cells (denoted by the two counter rotating cells) thatcause a decay of the coastline perturbation. Right: same as middle panel, but now the alongshore gradient of the alongshore vorticity fluxis visualized. Small arrows represent perturbed alongshore vorticity flux v0O and Vo0. The alongshore gradient of the alongshore vorticityflux generates residual circulation cells that cause a growth of the coastline perturbation.


the same sign of v0O and in the same alongshoregradient of v0O. Hence, averaged over one tidal cyclethe alongshore gradient of v0O results in meanperturbed vorticity with a sign opposite to that ofthe alongshore gradient in the coastline perturba-tion. The mean currents transport the sedimentfrom areas where the coast erodes to areas wherethe coast progrades. This causes the perturbation ofthe coastline to grow.

The last contribution to the convergence of themean vorticity flux is the alongshore gradient ofhVo0i. During flood, the perturbed vorticity o0 is inphase with the coastline perturbation of the coast-line and the perturbed vorticity flux is 180! out ofphase with the coastline perturbation becauseV2o0. Hence, following a similar argumentationas for the alongshore gradient of hv0Oi, thealongshore gradient of hVo0i results in residualcirculation cells that cause a growth of theperturbation of the coastline.

Whether the residual circulation cells are locatedalong the coast in such a way that the coastlineperturbation is growing or decaying depends on themagnitudes of the cross-shore gradient of hu0Oi andof the alongshore gradient of hv0Oi and hVo0i. Forsmall wave numbers the magnitude of the cross-shore gradient of hu0Oi is larger than the alongshoregradient of hv0Oi( hVo0i and therefore the loca-tions of the residual circulation cells are such thatthe perturbations of the coastline decay. Forincreasing wave numbers the magnitude of thealongshore gradient of hv0Oi and hVo0i increasesstronger than that of the cross-shore gradient ofhu0Oi. When the magnitude of the alongshoregradient of hv0Oi( hVo0i is larger than thecross-shore gradient of hu0Oi, the locations ofthe generated residual circulation cells alongthe coast are such that the perturbations of thecoastline grow.

When the Coriolis force is nonzero (fa0) theperturbations also migrate. The Coriolis torque(term (e) in Eq. (26)) is a source of tidal vorticityand causes a phase shift of the tidal vorticity withrespect to the coastline perturbation. While in thecase that f " 0 the perturbed vorticity o0 is in phasewith the coastline perturbation during flood, in thecase that fa0 the maximum of the perturbedvorticity o0 is shifted in the negative y-directionwith respect to the coastline perturbation due toplanetary vortex stretching. The perturbed vorticityis transferred by the basic state velocity and thealongshore gradient of Vo0 results in the accumula-

tion of perturbed vorticity which is slightly shiftedin the negative y-direction compared to the case thatf " 0. During ebb a similar argumentation holdsand consequently, residual circulation cells occurof which the centers are shifted in the negativey-direction compared to those obtained in the casethat f " 0. Hence, the residual circulation cells inthe case that fa0 cause both growth and migrationof the coastline perturbation.

6. Discussion

In this section the results of the combinedwave–tide model are compared with field data ofundulations of bathymetric contours along theDutch coast. Falques (2006) already showed thathis wave model yields wavelengths and migrationspeeds of the FGM that are consistent with those ofobserved shoreline sand waves along the southernand central Dutch coast. However, his model onlyyields very small growth rates when applied to thenorthern Wadden coast. It was shown in Section 4that in this area the effect of tides on the initialformation of undulations of bathymetric contours issignificant, hence we focus on this area.

Note that the present model only describeslocations along the barrier where initially the coasttends to be more subject to erosion. It is hypothe-sized that during a severe storm at these locationschannels are cut through the barrier and inlets form.An important finding of the model is that the lengthscale of the FGM decreases with increasingmagnitude of the tidal currents U , and all othervariables being constant. The predicted migrationrates are in the order of 200myr'1, depending onthe specific parameter values, and perturbations aremigrating to the east. To test the validity of theseresults observed lengths of barrier islands as well aswave and tidal characteristics along the Dutch andGerman Wadden coast were analyzed.

The length of the barrier islands along this coastvaries between a few kilometers (Simonszand andRottumeroog) up to 30 km (Texel) (Ehlers, 1988;Oost and de Boer, 1994). The typical migrationspeed of the islands is in the order of tens of metersper year and migration is from west to east (Luck,1975). According to Sha (1989) the wave climate isalmost constant in this area. Unfortunately, thereare not many observations of the magnitude of thealongshore tidal currents at sea. Instead, the resultsof a numerical model were used. This is the ZUNOmodel from WL—Delft Hydraulics (Roelvink et al.,


2001), a model that has been developed to predicttidal heights and depth-averaged tidal currents inthe southern North Sea. Fig. 11 shows the majoraxis of the M2 tidal current ellipse in the region ofthe Frisian Islands. In the regions close to the tidalinlets currents are strongly influenced by theinteraction between the backbarrier basin and theebb-tidal delta. Therefore, the magnitudes of themajor axis far offshore are considered. It is assumedthat these are representative for the situation thatthe coastline is straight and no tidal inlets arepresent. Two transects (see the two lines in the toppanel of Fig. 11) of the major axis of the M2 tidal

current ellipse are shown in the bottom panel ofFig. 11. These transects show that the long axisincreases from Texel to Schiermonnikoog, where itattains a maximum, after which it decreases up tothe island of Spiekeroog. The length of the barrierislands in this region is slightly larger than it is in theregion just after Schiermonnikoog. These data seemto confirm our model prediction that the lengthscale of the barrier islands is inversely related to themagnitude of the shore-parallel tidal currents.

We have not compared the results of ourcombined wave–tide model with field data ofGeorgia Bight. One reason is that the wave climateis quite variable in this area and no results for q#wave$are available. Second, a detailed comparison be-tween model results and observations is not possibleat this stage, because the wave model and tide modelassume different cross-sectional shape functionsof bathymetric contours. This aspect deservesfurther study.

An important implication of our model hypoth-esis is that the spacing between successive inlets isdetermined by the wave and tidal current character-istics at sea (and not by the tidal range). The tidalcurrents and waves render the coastline unstableand determine the spots where the coastline mightbreach during storms. Note that our model is notdesigned to explain the existence of sequences ofbarrier islands. That would require the study of amodel of a fully developed inlet system and usingconcepts that were already discussed in Bruun andGerritsen (1960). In fact, whether the inlet systemformed during a storm preserves on the long termdepends on the competition between wave-drivenlittoral drift (that tends to close the channel) andtidal currents in the inlet that tend to widen itscross-sectional area.

7. Conclusions

In this paper, a simple model has been developedto study the evolution of alongshore periodicundulations of bathymetric contours. The newaspect of this study is the role of tides in thepossible generation of such undulations. The modeldescribed in this study is a natural extension of theone-line models of Komar (1998), Ashton et al.(2001), Falques (2003), Falques and Calvete (2005)and Falques (2006), which considered only theinfluence of waves on the initial evolution of thecoastline. A linear stability analysis has been carriedout, so no results are obtained that describe the

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Fig. 11. Top: gray-scale plot of the Major axis of the M2 tidalcurrent ellipse along the Dutch and German Wadden coast.Model results are obtained with a numerical model (ZUNO).Bottom: two cross sections along which the magnitude of themajor axis of the M2 tidal current ellipse is shown (for position ofthe barrier islands see Fig. 1). The two cross sections are shown inthe top panel. White diamonds in the top panel show the seawardcross section while black diamonds show the landward crosssection.


finite-amplitude behavior of undulations of thebathymetric contours.

The model shows that in the case that the waveinduced sediment flux is zero the feedback betweenthe location of the coastline and the alongshore sandflux is such that a straight coastline is unstable withrespect to alongshore rhythmic perturbations withlength scales longer than the shelf width. Theseperturbations migrate along the coast to the right(when viewing in the seaward direction). The physicalmechanism resulting in growth or decay of coastlineperturbations and their migration rates can beunderstood with vorticity dynamics. There is acompetition between the cross-shore gradient of themean cross-shore vorticity flux, which acts stabilizing,and alongshore gradient of the mean alongshorevorticity flux, which acts destabilizing. The width ofthe inner shelf determines the cross-shore length scaleover which the cross-shore vorticity fluxes vary, whilethe wavelength of the perturbation determines thelength scale over which the alongshore vorticity fluxesvary. If the length scale in the alongshore direction issmaller (larger) than the length scale in the cross-shore direction, the growth rate is positive (negative).The Coriolis force is responsible for the migration ofthe perturbations. This implies that on the southernhemisphere the perturbations will migrate to the left(when viewing in the seaward direction).

The tidal model has been combined with the wavemodel of Falques and Calvete (2005), Falques(2006) and has been applied to the Dutch coast.Effects of tides on the formation of undulations ofthe coastline turn out to be small for the southernDelta coast and for the central Dutch coast, butthey are significant for the northern Wadden coast.Tidal currents and waves cause damping of undula-tions with large and small wavelengths, respectively,while both cause growth of undulations withintermediate wavelengths. Field data confirm themodel prediction that for a constant wave climatethe preferred length scale of undulations of thecoastline is smaller in regions where the alongshoretidal currents are stronger. The present model thussuggests that it is the magnitude of the tidal currentat sea that determines the length of barrier islands.This is an important difference with the conceptualmodel of FitzGerald (1996) which predicts that thelength of barrier islands is inversely proportional tothe tidal range. It should be kept in mind thoughthat the present model only predicts locations alongthe barrier where breaching will probably occurduring a storm.

Acknowledgements

This work has been conducted in the frameworkof the NWO-ALW Outer Delta Dynamics pro-gramme and was supported by NWO-ALWGrant No. 810.63.12 and 810.63.15. We thankProf. A. Falques for kindly providing the resultsof his wave model.

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