Effect of grain size sorting on the formation of shoreface...

Effect of grain size sorting on the formation of

shoreface-connected sand ridges

M. Walgreen and H. E. de SwartInstitute for Marine and Atmospheric Research (IMAU), Utrecht University, Utrecht, Netherlands

D. CalveteDepartment de Fısica Applicada, Universitat Politecnica de Catalunya, Barcelona, Spain

Received 11 April 2002; revised 29 November 2002; accepted 6 December 2002; published 6 March 2003.

[1] Field data of shoreface-connected ridges show persistent spatial variations of meangrain size over the bed forms. In the shore-normal direction, the profiles of bottomtopography and mean grain size are approximately 90� out of phase. To investigate themechanisms responsible for the observed grain size distribution and the influence ofsediment sorting on the temporal and spatial characteristics of shoreface-connected ridges, amodel is developed and analyzed. A linear stability analysis of an alongshore uniform basicstate (describing a storm-driven flow on a microtidal inner shelf) with respect to smallbottom perturbations is carried out. The transport of nonuniform sediment is described byformulations for both bed load and suspended load, both of which account for dynamichiding effects. A one-layer model for the bed evolution and a bottom friction term, whichdepends on the grain size, are used. The initial formation of the ridges is studied for abimodal sediment mixture. The results of the model indicate that the phase shift between bedtopography and mean grain size for shoreface-connected ridges is due to the selectivetransport via suspended load of grains with different sizes. A net stabilizing effect on thegrowth of bed forms and enhanced migration are predicted, caused by the bimodal characterof the sediment. The wavelengths of the bed forms are only slightly affected. Including atidal current or a grain size dependent formulation for the bottom friction has no effect on theresults. A physical explanation for the model results is also given. INDEX TERMS: 3022

Marine Geology and Geophysics: Marine sediments—processes and transport; 4219 Oceanography: General:

Continental shelf processes; 4255 Oceanography: General: Numerical modeling; 4558 Oceanography:

Physical: Sediment transport; 4512 Oceanography: Physical: Currents; KEYWORDS: morphodynamic model,

shoreface-connected ridges, grain sorting, hiding, sediment transport, nonuniform sediment

Citation: Walgreen, M., H. E. de Swart, and D. Calvete, Effect of grain size sorting on the formation of shoreface-connected sand

ridges, J. Geophys. Res., 108(C3), 3063, doi:10.1029/2002JC001435, 2003.

1. Introduction

[2] Shoreface-connected ridges are rhythmic bed formsthat are observed on storm-dominated inner shelves ofcoastal seas, in water depths of 4–20 m. Analysis of fieldobservations [Swift et al., 1978; Antia, 1996; Van de Meeneand Van Rijn, 2000] has revealed that the spacing betweenthe crests is in the order of 5–10 km, with heights varyingbetween 1 and 6 m. Migration occurs in the direction of themean alongshore storm-driven flow, and characteristic phasespeeds are 1–4 m yr�1. Previous model studies have dem-onstrated that the formation of these large-scale bed forms isdue to inherent positive feedbacks between the water motionand the eroding bed [Trowbridge, 1995;Calvete et al., 2001a,2001b]. The combined action of stirring of sediment bywaves and transport by storm-driven currents is necessaryto generate shoreface-connected ridges. Furthermore, these

studies have found that the seaward end of the ridges isalways shifted upcurrent with respect to their attachments tothe shoreface and explanations for this morphological char-acteristic have been put forward. The model developed byCalvete et al. [2001b] provides information on the spatialpattern, evolution timescale and migration speed of the bedforms. These results are in good agreement with availablefield data of many different shelves.[3] A basic limitation of these models is that they assume a

uniform grain size distribution of the sediment. This is notconsistent with field data, which show persistent spatialvariations of the mean grain size over the bed forms.Especially the ridges on the Mid-Atlantic shelf are docu-mented extensively and detailed information on the grain sizecharacteristics has been given in the literature [see, forexample, Swift et al., 1972; Hoogendoorn, 1986; Schwab etal., 2000]. The ridges located in the Mid-Atlantic Bight onthe North American inner shelf [Swift et al., 1978; Swift andField, 1981; Figueiredo et al., 1982] reveal, in the directionnormal to the shore, grain size and topography variations that

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 108, NO. C3, 3063, doi:10.1029/2002JC001435, 2003

Copyright 2003 by the American Geophysical Union.0148-0227/03/2002JC001435$09.00

8 - 1

are approximately 90� out of phase: the coarsest materialoccurs on the landward (upcurrent) flank. This out-of-phaserelationship extends over the entire ridge area (Figure 1). Themean diameter as plotted here is defined in section 2.2.Similar trends in mean grain size are observed for shore-face-connected ridges on the inner shelf of Brazil [Figueiredoet al., 1982] and Argentina [Parker et al., 1982], and forsimilar ridges located in the German Bight of the southernNorth Sea [Antia, 1996].[4] In the present study, the effect of sediment sorting on

the formation of shoreface-connected ridges is investigatedby extending the model by Calvete et al. [2001b] forsediment mixtures. New dynamics related to the presenceof different grain sizes are incorporated in the sedimenttransport formulation, the sediment continuity equation, andin the formulation of the bottom friction in the hydro-dynamic equations. Previous model studies on sedimentsorting have mainly focused on river bars and sea ripples[Ribberink, 1987; Seminara, 1995; Foti and Blondeaux,1995; Lanzoni and Tubino, 1999], i.e., on spatial scalesmuch smaller than those for sand ridges. These studiesindicate that the nonuniform character of sediment has astabilizing effect on the growth of bed forms.[5] The work presented in this paper contains several new

aspects. First, it focuses on sediment sorting in the sandfraction in combination with ridge formation, while previouswork has largely concentrated on gravel and sand–gravelmixtures (representative of river sediments). Second, theinfluence of grain size on the entrainment and deposition ofsuspended sediment is included. Third, the effect of sedimentsorting in the modeling of large-scale bed forms in coastalseas is investigated. The first objective of this paper was toinvestigate the influence of sediment sorting on the temporaland spatial characteristics of shoreface-connected ridges. Thesecond goal was to gain insight into the physical mechanismsresponsible for the observed grain size distribution overshoreface-connected ridges. This paper focuses on the initialformation of shoreface-connected ridges, i.e., small bottom

perturbations are assumed. Therefore, a one-layer model forthe bottom evolution, based on the concept of an activetransport layer overlaying an inactive substrate, is used. Themodel uses a two-size sediment mixture. The motivation forusing a simple model is that it allows for a systematic analysisof the underlying processes.[6] In section 2, the formulation of the model is given,

followed by an outline of the solution procedure in section 3.Results are presented in section 4 and a physical interpreta-tion is given in section 5. A discussion of the model results,including a comparison with field observations, is presentedin section 6, followed by conclusions in the last section.

2. Model Formulation

2.1. Hydrodynamics

[7] Following earlier studies by Trowbridge [1995] andCalvete et al. [2001b], we hypothesize that shoreface-con-nected sand ridges form as an inherent instability of amorphodynamic system, in which there is a feedbackbetween the storm-driven flow and the eroding bed. A highlyidealized model is used to investigate the flow–topographyinteraction on coastal shelves during storm conditions. Theshelf geometry is schematized as a semi-infinite domain,bounded on the landward side by the transition from theshoreface to the inner shelf (see Figure 2). The undisturbedbathymetry (no ridges present) is uniform in the alongshore(y) direction. In the cross-shore (x) direction it consists of aninner shelf (with a linearly sloping bottom) and an outer shelfrepresented by a horizontal bottom. The water depth at thebeginning of the inner shelf (x = 0) is H0, Ls is the inner shelfwidth and Hs is the depth of the outer shelf. Representativevalues for the Long Island inner shelf (Mid-Atlantic Bight,US), which is considered as a prototype storm-driven shelf inthis study, are H0 = 14 m, Hs = 20 m and Ls = 5.5 km.[8] In the model the water motion is described by the

2DH shallow water equations. They read

@~v

@tþ ~v � ~r� �

~vþ~f �~v ¼ �g ~rzs þ~ts �~tb

rDð1Þ

@D

@tþ ~r � D~vð Þ ¼ 0 ð2Þ

Here ~v is the depth-averaged and wave-averaged velocity,f 10�4 s�1 is the Coriolis parameter, ~ts the wind shearstress vector, ~tb the bottom shear stress vector, g the

Figure 1. Measured shore-normal profile of water depth(in meters) and mean grain size (in phi units) for theshoreface-connected ridges on the Long Island inner shelf.Larger phi values imply smaller grain sizes. FromFigueiredo et al. [1982], #Elsevier Science.

Figure 2. Sketch of the geometry of the model, represent-ing the inner shelf (width Ls) and part of the outer shelf of acoastal sea. For further information, see the text.

8 - 2 WALGREEN ET AL.: GRAIN SORTING OVER SHOREFACE RIDGES

acceleration due to gravity, r the density of water, t the timeand ~r the horizontal nabla operator. The local water depthis given by D = zs � zb, where zs is the free surface elevationand zb is the bottom depth, both measured with respect tothe undisturbed water level z = 0.[9] Both observations and model studies indicate that

shoreface-connected ridges mainly develop during storms,tides do not play an important role, and the timescalesinvolved are about 100 years. This appears to justify theneglect of tidal forcing in the model (microtidal shelvesare assumed) and the use of a probabilistic approach. Tworealizations of the system are considered. During storms(which occur during a time fraction m 0.05) largewaves and strong currents cause significant sedimenttransport. In contrast, during fair weather conditions (timefraction 1 � m) the waves and currents are not sufficientlystrong to erode sediment from the bottom. Thus, (1) and(2) are assumed to be representative of conditions duringstorms. The quasi-steady approximation is made in (1)and (2), such that terms involving time derivatives areexcluded. This is because the hydrodynamic timescale ismuch smaller than the timescale on which the bedevolves. Also, the rigid-lid assumption is used, in whichcase the free surface effects in the local water depth areneglected, i.e., D ’ �zb.[10] The main forcing of the water motion is by wind and

an alongshore pressure gradient. During storms the presenceof large waves causes a wave-orbital velocity amplitude uw,which is much larger than the wave-averaged velocityamplitude. This allows for a linearization of the bed shearstress:

~tb ¼ rr~v

with r the bottom friction coefficient which is written interms of a Chezy coefficient Ch [see Ribberink, 1987;Soulsby, 1997]. This results in

r ¼ g juwjh iC2h

Ch ¼ffiffiffig

p

kln

12D

ks

� �ð3Þ

where uw is the near-bed wave-orbital velocity and h. . .idenotes a time average over many wave periods. Further-more, k is the von Karman’s constant and ks is theroughness length, which is proportional to a coarse grainsize [see, for example, Ribberink, 1987; Lanzoni andTubino, 1999]. This formulation of the friction coefficientintroduces a dependence on the grain size in thehydrodynamic equations. The value we used for the frictioncoefficient is r 7 � 10�4 m s�1.[11] Calvete et al. [2001b] found that it is essential for the

growth of shoreface-connected ridges to parameterize thewave-orbital velocity increase in decreasing water depths.The description of the wave-orbital velocity as uw = uw cos(wt) (symmetrical waves with frequency w), with the ampli-tude given by

uw ¼ Uw H0=Hð Þm2 ; ð4Þ

includes this effect. Here H is the undisturbed water depth,Uw 1 m s�1 the amplitude at the shoreface boundary x = 0

and m a coefficient. Runs with a simple wave shoalingmodel indicate that m 1.6.

2.2. Sediment Characteristics

[12] For a sediment mixture it is convenient to use alogarithmic scale (the phi scale) to describe the graindiameters. The definition is

d ¼ 2�f or f ¼ � log2 d

where d is the grain diameter measured in units of mm [seeDyer, 1986]. Accordingly, larger values of f correspond tofiner sizes. A sediment mixture is described by a probabilitydistribution function F as a function of the grain size. Thisis the weight percentage of each grain size, hence F has thefollowing property:

Z 1

�1F fð Þdf ¼ 1

For many sand mixtures F (f) is approximately a Gaussiancurve if plotted on this phi scale. In that case, two statisticalproperties describe the sediment distribution: the mean graindiameter fm and the standard deviation s, defined as

fm ¼Z 1

�1fF fð Þdf s2 ¼

Z 1

�1f� fmð Þ2F fð Þdf

The mean diameter is calculated as dm ¼ 2�fm . A measureof the sorting is given by the standard deviation of thedistribution. Small values of s corresponds to a sharplypeaked curve, representing an almost uniform sample, andis classified as well sorted. A poorly sorted mixture ofsediment has larger values of s.

2.3. Sediment Continuity

[13] The hydrodynamic equations discussed above aresupplemented with a sediment transport formulation, basedon the concepts introduced by Bailard [1981] for the totalload transport on a sloping bed, and the bottom evolutionequation. The evolution of the bottom is a result of diver-gence in the sediment flux and depends on the compositionof the bottom sediment. The simplest models dealing withthe effect of sediment sorting on bed level changes considertwo separate layers in the bottom (Figure 3) [see Ribberink,1987; Seminara, 1995]. The first is the active layer, whichcontains the material available for transport, and F is thecorresponding probability density function for the grainsizes in this layer. Underneath this active layer a substrateis located with a probability density function of F s. Thebottom location is denoted by zb = �H + h, with H theundisturbed water depth and h the bed elevation with respectto this reference level. Furthermore, zh = zb � La is the levelof the interface between the active (surface) layer and thesubstrate. The active layer thickness, La, is in the order of2–3 times d90 (grain size for which 90% of the material isfiner). The thickness of the total sediment column is con-sidered to be so large that modifications of the sedimentcomposition in the substrate, due to exchanges of sedimentwith the upper layer, can be omitted.[14] A well-mixed active layer (F is independent of the

depth) and a time-independent grain size composition in the

WALGREEN ET AL.: GRAIN SORTING OVER SHOREFACE RIDGES 8 - 3

substrate (F s) were assumed. Consequently, the continuityequation of sediment reads

F h@h

@tþ La

@F@t

þ F �F h� � @La

@t¼ � ~r �~qf ð5Þ

Sedimentation :@zh@t

¼ @

@th� Lað Þ > 0 : F h ¼ F

Erosion :@zh@t

¼ @

@th� Lað Þ < 0 : F h ¼ F s

The terms on the left-hand side of (5) represent the bottomchanges, changes in the sediment distribution in the activelayer, and changes in the thickness of the active layer due toexchange of sediment with the substrate, respectively.Furthermore, ~qf is the volumetric flux per unit width ofgrains of size f, including pores. In the initial growth stageof bed forms, sorting can be seen as the rearrangement ofmaterial in the active layer with negligible interactionbetween substrate and active layer [Ribberink, 1987;Seminara, 1995]. This assumption, which implies thatF h ¼ F , is adopted here.[15] We consider a discrete number (N ) of grain sizes,

such that F fð Þ ¼PN

i¼1 F id f� fið Þ, with d the Dirac deltafunction. Furthermore, we use the fact that the morphody-namic timescale is much larger than the hydrodynamictimescale. This implies that the bed evolution for a discretedistribution of grain sizes and grains in class i (and diameterdi) is given by

F i

@h

@tþ La

@F i

@t¼ � ~r � ~qih i ð6Þ

Note that the sediment flux is averaged over the wave cycle.Together with the constraint

PNi¼1 F i ¼ 1 a closed system

of equations is specified if ~qi is known.

2.4. Sediment Transport

[16] Calvete et al. [2001b] demonstrated that both bedload and suspended load fluxes are needed to describe the

growth and migration of the shoreface-connected ridges.The sediment flux, therefore, reads

~qi ¼~qbi þ~qsi

where~qbi and~qsi represent the bed load and suspended loadcontributions, respectively.2.4.1. Bed Load[17] The transport of sediment of class i depends on the

shear stress exerted by the flow on the bed and on the grainproperties. A general formulation for bed load transport ofgrains with size di over a flat bed is [Ribberink, 1987]:

qbi / F i

ffiffiffiffiffiffiffiffiffig0d3i

p�

b2

i . Here g0 = g(rs � r)/r, rs = 2650 kg m�3

is the density of the grains, r =1030 kg m�3 is the waterdensity and b is an exponent. In the case of uniformsediment the Shields parameter �i,u is

�i;u ¼t

rg0di¼

u2*

g0diand �i;u ¼

dm

di�m

In this expression t is the bed shear stress, u* the frictionvelocity, dm the mean grain size of a mixture, and �m theShields parameter corresponding to grains of size dm. In asediment mixture, the effective Shields parameter �i ofsediment of size class i differs from �i,u. This is because thebehavior of a sediment mixture is influenced by the effect ofdynamic hiding: finer grains feel fluid drag less intenselythan larger grains. The effect is modeled by

�i ¼1

xi�i;u ¼

1

xi

dm

di�m ð7Þ

with xi = x(di) a (dynamic) hiding function. According tofield and laboratory data, xi decreases with di/dm. Thus, finesand is less exposed to the shear stress than coarse sand. Theeffect of xi is incorporated in a vectorial form of the bedload transport as follows:

~qbi ¼ F iq

ffiffiffiffiffiffiffiffiffig0d3m

q�m

xi

� �32 ~tj~tj � lb

~rh

�

The exponent b = 3 is chosen consistent with the argumentspresented by Bailard [1981] and q is a constant. Note thatstatic hiding effects related to the presence of a criticalshear stress for erosion [see Ribberink, 1987; Seminara,1995] are not modeled. Furthermore, it is assumed that allgrains are transported in the same direction: ~t=j~tj isindependent of the grain size and the effect of bottomslopes (see the discussion by Fredsøe and Deigaard [1992,and references therein]) are explicitly accounted for. Thebottom perturbations are given by h and lb is a parameterthat accounts for the gravitational effect of sedimentmovement on a sloping bottom. For simplicity we assumelb to be constant (1).[18] The bottom shear stress vector used in the sediment

transport is given by ~t ¼ rcf j~vtj~vt. In this expression ~vt isthe total velocity, which consists of a wave-averagedvelocity ~v, as used in the hydrodynamic equations, and awave contribution uw (~vt ¼~vþ uw). A constant skin friction

Figure 3. Definition of the sediment layer as used in themodel. The thickness of the active layer is denoted by La,and h is the elevation of the bottom with respect to areference level.


coefficient cf 2 � 10�3 is assumed. Application of theseassumptions results in

~qbi ¼ nbF iGbij~vt j3~vtj~vtj

� lb~rh

� Gbi ¼

1

xi

� �32

ð8Þ

where nb = qcfffiffiffifficf

p=g0 2 � 10�5 s2 m�1 and Gbi is the bed

load transport capacity function for sediment of size di. Asimple hiding function is used: xi ¼ dm=dið Þmb . In Figure 4,the dynamic hiding function according to Day (see in thestudy of Ribberink [1987]) and two simplified relations areshown (mb = 0.5 and mb = 1), together with the correspond-ing transport capacity functions. Note that grains of adiameter equal to dm experience no hiding effects (xi = 1).[19] As was discussed in section 2.3, the model for

shoreface-connected ridges requires information about thewave-averaged bed load flux, ~qbih i, during storms. In thatcase the amplitude of wave-orbital motion is much largerthan the mean current, thus uw � j~vj. Furthermore, thewaves are supposed to be almost parallel to the current,leading to

~qbih i ¼ F iGbi~qb ; ~qb ’3

2nb u2w~v�

8

9plbu

3w~rh

� �

Gbi ¼di

dm

� �cb

; cb �3

2mb ð9Þ

We use mb = 0.5, which yields cb = 0.75 for the exponent inthe transport capacity function.2.4.2. Suspended Load[20] During storms it has been observed that the sediment

on the inner shelf is mainly transported as suspended load[Green et al., 1995]. The vectorial formulation of thesuspended load flux reads

~qsi ¼ Cij~vtj~vtj~vtj

� lsj~vtj ~rh

� ð10Þ

In (10), Ci is the depth-integrated volume concentration ofgrains in class i, which includes all grain size dependence,and ls is the bed slope coefficient for suspended loadtransport. Since Ci is a monotonically increasing function ofthe bed shear stress t, this formulation is consistent with that

of Bailard [1981]. However, it is modified for strong forcingconditions, whereas Bailard’s expression was derived forrelative moderate forcing conditions. This adjustment is alsomotivated by the analysis preformed by Bayram et al.[2001], who demonstrated that Bailard’s formulation under-estimates the observed transport during storms. An expres-sion for Ci is derived in Appendix A. In the case of fine sand,an approximate balance between sediment erosion anddeposition near the bed exists, yielding

Ci ¼ F idiEiD ð11Þ

In this expression, Ei is the dimensionless entrainment ofgrains of diameter di and di is the ratio of the thickness of thesuspended load layer of these grains over the total waterdepth D, which is inversely proportional to the settlingvelocity wsi of grains in size class i.[21] Two (out of many) formulations for the entrainment

are considered, of which one is derived for sedimentmixtures and one for uniform sediment. The formulationaccording to Van Rijn [1993] is Ei / j~vtj3 and is testedagainst laboratory experiments for uniform sediment. Inprevious work on shoreface-connected ridges this formula-tion was used [Calvete et al., 2001b]. We use the Garciaand Parker [1991] formulation, which is based on resultsfrom experiments carried out with sediment mixtures,thereby accounting for possible hiding effects. The resultis that Ei / j~vtj5 (for the complete expressions, see Appen-dix A). Similar to the hiding expression for bed load (7), wewrite

diEi ¼ zi diEi½ �u� GsidmEm;u ð12Þ

with zi the hiding function for the entrainment of sedimentand dm is related to the thickness of the suspended load layerfor grains with size dm. The quantity Em,u is the entrainmentof grains of diameter dm only. The hiding function accordingto Garcia and Parker [1991] is

zi ¼ lE

di

dm

� �mE� 5

ð13Þ

The parameter mE defines the importance of hiding for theentrainment of sediment, its default value is mE = 0.2. Astraining parameter, lE = 1 � 0.288s, is used with s thestandard deviation on the f scale, as defined in section 2.2.It models the reduced mobility of the sediment mixture asits standard deviation increases and corrects for the other-wise overestimated entrainment rates. The entrainment ofgrains from all size classes decreases with increasing stand-ard deviation of the sediment mixture in the active layer.Substituting expression (13) in (12), in combination withthe results from Appendix A, yields for the transportcapacity function of suspended load

Gsi ¼ l5E

di

dm

� �cs

; cs � 5mE þ 4:5� 6ew ð14Þ

The first value in the definition of cs incorporates hidingeffects, whereas the second includes the dependence of theentrainment function on the particle Reynolds number. The

Figure 4. Different dynamic hiding functions xi for thebed load sediment transport and the corresponding transportcapacity functions Gbi as a function of di/dm. Here, di is thegrain size class i, dm the mean grain size and mb � 2

3cb. For

further information, see the text.


coefficient ew defines the dependence of the settling velocityon the grain size: wsi / dewi . The formulation of Hallermeier[see Soulsby, 1997] for fine to coarse sand yields ew = 1.1,resulting in cs = �1.1. Another formulation by Van Rijn[1993] for settling of grains in the sand range gives ew = 1.0,resulting in cs = �0.5. The negative value of the exponentphysically means that the depth-integrated concentration forgrains with sizes smaller than the mean is larger than forgrains of sizes larger than the mean grain size. Combining(11)–(14) yields for the concentration

Ci ¼ F iGsiC C ¼ dmEm;uD ¼ dmEm;uj~vt j5D

with Em,u a constant and dm 0.19. The averagedsuspended load flux (10) during storms reads

~qsih i ¼ F iGsi~qs ð15Þ

with Gsi defined in (14) and

~qs ’32

5pdmEm;uD u5w~v�

1

7lsu

7w~rh

� �

3. Basic State, Stability Analysis, and SolutionProcedures

3.1. Basic States

[22] We investigate the possible onset of bed forms asfree morphodynamic instabilities on a planar morphology.The basic state is uniform in the alongshore direction, with ashore-parallel current V(x). The corresponding bottom pro-file is sketched in Figure 2. The grain size distributionfunction Fi for the basic state can have an arbitrary structurein the cross-shore (x) direction without violating the equi-librium conditions. In this paper, we assume that Fi isindependent of this coordinate. The basic state is charac-terized by

u ¼ 0 v ¼ V xð Þ zb ¼ �H xð Þ zs ¼ z x; yð Þ

r ¼ r0 xð Þ F i ¼ Fi fm ¼ �m s ¼ s0

From the alongshore momentum balance (1) for the basicstate, it follows that

V xð Þ ¼ tsy=r� gs0H

r0

The basic state velocity consists of a steady component,which is driven by a prescribed alongshore free surfacepressure gradient s0 � @z/@y 2 � 10�7 and an alongshorewind stress tsy �0.25 N m�2, such that the characteristicbasic state velocity is U = jVj 0.4 m s�1 at x = 0 in thenegative y direction. This is a representative value of thestorm-driven flow on the American Atlantic inner shelf[Niedoroda and Swift, 1981; Lentz et al., 1999].[23] The characteristic magnitude of suspended load

transport Qs ¼ 325pC0U is defined for uniform sediment of

a grain size dm, with a representative value for the depth-integrated volume concentration ofC0 = dmEmuUw

5H0 7.5�

10�4 m. For bed load transport the scale is Qb =32vbUw

2U andthe rate of bed over suspended load transport is Qb/Qs 0.016.

3.2. Stability Analysis

[24] The stability of the basic state is considered bystudying the evolution of small perturbations on this state.The linearized momentum and mass conservation equationsare solved for a fixed bed level to find the perturbed velocityfield as a function of the bottom topography. The flowvariables are substituted in the bottom evolution equation tocompute the changes in the bed.3.2.1. Hydrodynamics[25] We consider solutions of the hydrodynamic equa-

tions (1) and (2) of the following form:

~v ¼ 0;V xð Þð Þ þ u0 x; y; tð Þ; v0 x; y; tð Þð Þzs ¼ z x; yð Þ þ h0 x; y; tð Þzb ¼ �H xð Þ þ h x; y; tð Þ

r ¼ r0 xð Þ þ r0 x; y; tð Þ

The perturbations (indicated by primes) are assumed to besmall. The expressions for r0 and r

0 are given in Appendix B.The linearized versions of the momentum equation (1) are

V@u0

@y� fv0 ¼ �g

@h0

@xþ tsx

rh

H2� r0

u0

Hð16Þ

u0@V

@xþ V

@v0

@yþ fu0 ¼ �g

@h0

@yþ s0

h

Hþ r0

v0

H� r0

V

Hð17Þ

and for mass conservation (2):

H@u0

@xþ @H

@xu0 þ H

@v0

@y� V

@h

@y¼ 0 ð18Þ

3.2.2. Two-Size Mixture[26] As with perturbations in the hydrodynamics, small

perturbations in the probability distribution function F i areassumed, resulting in small perturbations in the mean grainsize and standard deviation:

F i ¼ Fi þ fi x; y; tð Þ fm ¼ �m þ f0m x; y; tð Þ

s ¼ s0 þ s0 x; y; tð Þ

In this paper a two-size sand mixture is considered, with d1and d2 the grain diameters of the fine and coarse sizefraction, respectively (f1 � f2). A mean grain size ofmedium sand, characteristic of inner shelf sediment, is dm =0.35 mm or fm = 1.5. The constraint on the distributionfunction yields:

F1 þ F2 ¼ 1 f1 ¼ �f2 ð19Þ

The expressions for the mean grain size and the standarddeviation (as defined in section 2.2) simplify to

�m ¼ f1F1 þ f2F2 s20 ¼ F1F2 f1 � f2ð Þ2


From this, we derive an expression for f1 and f2 in terms ofthe mean grain size and the standard deviation:

f1 ¼ �m þ s0

ffiffiffiffiffiF2

F1

rf2 ¼ �m � s0

ffiffiffiffiffiF1

F2

rð20Þ

[27] Using the expressions given above, the perturbationsin the mean grain size and standard deviation read

f0m ¼ s0ffiffiffiffiffiffiffiffiffiffi

F2F1

p f1 s0 ¼ s0 F2 � F1ð Þ2F2F1

f1 ð21Þ

3.2.3. Sediment Dynamics[28] The sediment flux consists of a bed load and a

suspended load part: ~qbih i ¼ F iGbi~gb and ~qsih i ¼ F iGsi~qs(see expressions (9) and (15)). In particular,

~qb ¼~qb0 xð Þ þ~q0b x; y; tð Þ Gbi ¼ Gbi þ gbi x; y; tð Þ~qs ¼~qs0 xð Þ þ~q0s x; y; tð Þ Gsi ¼ Gsi þ gsi x; y; tð Þ

In the basic state only an alongshore transport component,which depends on the distance x to the shoreface, is present.Thus ~r � (FiGbi~qb0) = ~r � (FiGsi~qs0) = 0. The transportcapacity functions Gbi, Gsi and the perturbations gbi, gsi forbed load and suspended load are defined in Appendix C.Following the formulation for the roughness length ks weuse La = dm 2s (see Appendix B); the thickness of the activelayer in the basic state corresponds to La0 ¼ 2s0��m . Thelinearized form of (6) is

Fi

@h

@tþ La0

@fi@t

¼ � ~r � ~q0i� �

ð22Þ

where

~q0i� �

¼ FiGbi~q0b þ~qb0 Gbifi þ Figbið Þ

þ FiGsi~q0s þ~qs0 Gsifi þ Figsið Þ

Summation of (22) over all size fractions, combined withthe constraint on the distribution function, leads to anequation that relates the bed evolution to the sum of thesediment flux over all sizes. Back substitution of thisresult into (22) yields the evolution of the probabilityfunction fi. For two-size fractions (i = 1, 2) the finalequations read

@h

@t¼ � ~r � ~q01

� �þ ~r � ~q02

� �h ið23Þ

La0@f1@t

¼ F1~r � ~q02

� �� F2

~r � ~q01� �

ð24Þ

where

~r � ~q01� �

¼ Gb1 F1~r �~q0b þ qb0 F1Tb5 þ 1ð Þ @f1

@y

� �

þ Gs1 F1~r �~q0s þ qs0 F1Ts5 þ 1ð Þ @f1

@y

� �

~r � ~q02� �

¼ Gb2 F2~r �~q0b þ qb0 F2Tb5 � 1ð Þ @f1

@y

� �

þ Gs2 F2~r �~q0s þ qs0 F2Ts5 � 1ð Þ @f1

@y

� �

The quantity f2 is eliminated by using (19) and for a two-size mixture gb1, gs1, gb2, and gs2 are expressed inperturbations of the probability function f1 (Appendix C).Also the expressions for qb0, qs0, ~r �~q0b and ~r �~q0s aregiven in this appendix. Equations (23), (24), and (19) givethe set of equations to be solved for h and f1. Together with(16), (17), and (18), they form a closed system.[29] Boundary conditions are that u0 = 0 and h = 0 at the

transition from the shoreface to the inner shelf (x = 0) andfor x ! 1. Furthermore, periodicity in the alongshoredirection is assumed.

3.3. Solution Procedure

[30] The solutions for the bottom perturbations aretopographic waves, which propagate along the shelf andhave a certain cross-shelf structure. They are of the formh(x, y, t) = Re{h(x)eiky+ �t}: a similar expression holds forf1. Here k is the alongshore wave number and � thecomplex frequency. The stability analysis yields, for eachwave number k, solutions for �; its real part �r being thegrowth rate, with �r

�1 the e-folding timescale. Further-more, the imaginary part �Im is the frequency. Themigration velocity of the perturbation is obtained fromc = ��Im /k. For a fixed value of the alongshore wavenumber k the different values of � correspond to differentcross-shore modes.[31] Of specific interest are growing perturbations, which

have �r > 0. The perturbation for which a maximum in thegrowth rate is found is called the preferred mode. Theperturbed velocity u0 is expressed in the bottom perturbationh, by eliminating the free surface h0 from the momentumequations and using mass conservation to express v0 in u0. Inthis study, the bottom friction is related to the grain size,thus a part of u0 is related to f1. Solving the equation for u0

and substituting this in (23) and (24) results in an eigen-value problem, which determines the cross-shore structureof h and f1.[32] (24) can be simplified, because the term on the left-

hand side is a factor La0 /H0 � 1 smaller than the con-tributions on the right-hand side and can thereforeexcluded. This factor was derived from (23), which definesthe scale for the divergence of the sediment fluxes. Theresult is a decoupled set of equations for the bottom and thefraction of fine grains. Consequently, the eigenvalue prob-lem for h and f1 is reduced to a single eigenvalue problemfor h and f1 is an algebraic function of h. Therefore, the finegrain fraction adapts instantaneously to changes in thebottom. Solutions of these equations were obtained bynumerical methods.

4. Results

[33] In this section, the influence of the nonuniformity ofthe sediment on growth rates, migration velocities andwavelengths of the resulting bed forms is investigated.Furthermore, the resulting spatial variations in mean grainsize and bottom topography are presented. Different experi-ments were performed, with particular emphasis on thesensitivity of the model results to the formulation of thehiding functions and the properties of the sediment. First,the influence of variation in sediment sizes was studied byvarying the standard deviation of the distribution. Next, the


sensitivity of the results to the hiding coefficients for bedand suspended load was investigated.

4.1. Parameter Values

[34] Values for the characteristic length and velocityscales were given in sections 2.1 and 3.1, respectively. Anoverview of the parameter values used for the differentexperiments is presented in Table 1; they are representativeof conditions on Long Island inner shelf and are partlyextracted from the studies of Figueiredo et al. [1982] andSchwab et al. [2000]. Note that the sediment has the samemean grain diameter in all experiments, while the diametersof the fine and coarse grains were allowed to vary.

4.2. Standard Deviation

[35] The influence of the standard deviation of the mix-ture on the characteristics of the ridges was investigated. Fora fixed mean grain size, the standard deviation of thesediment was varied over a range, such that d1 and d2 arein the fine to coarse sand range. In this section, the defaultvalues of cb = 0.75 and cs = �1.1 are used. Figure 5 showsthe changes in the growth rate and migration velocity of themost unstable mode (first cross-shore mode) for each wavenumber and for different values of the standard deviation.Values are scaled by those of the preferred mode in the caseof uniform sediment (s0 = 0). The maximum growth ratefor uniform sediment is �ru = 8.6 � 10�3 yr�1, attained forwave number ku = 1.9 km�1 and the corresponding migra-tion speed is cu = �0.9 m yr�1. The alongshore spacingbetween successive crests is l* = 2pk u

�1 = 3.2 km, and thetimescale for the growth is �ru

�1 = 117 yr. In the compu-tations of the timescale it was assumed that storms prevailduring a time fraction m = 0.05, whereas no growth ofperturbations occurs during the remaining time fraction.Here we use the formulation for the entrainment of sus-pended sediment by Garcia and Parker [1991]. In the caseof s0 = 0, the growth rate and migration curves are similarto those obtained with the Van Rijn [1993] formulation, aswas used by Calvete et al. [2001b].[36] A clear stabilizing effect on growth rates of the

bimodal mixture, as compared to uniform sediment, isfound if the standard deviation is increased (Figure 5). Thisgoes along with a (small) decrease of the wave number, i.e.,increase in wavelength, of the bed forms. The migrationvelocities are enhanced. The maximum growth rate isreduced by 50% for a bimodal sediment mixture with astandard deviation of s0 = 0.5. Migration velocities for thiscase increase to �1.2 m yr�1. The fastest growing pertur-bation in this case has a wavelength l* = 3.4 km, and itsspatial pattern is shown in Figure 6. The elevation of the

bottom is indicated by the dark and light colors. The contourlines are those of the fraction of fine grains (with diameterd1). More fine sediment ( f1 > 0) results in an increase in themean grain size in phi units (f0

m / f1). The results indicatethat f1 is positive on the downcurrent (seaward) flank of theridges, hence the mean grain size becomes finer in this areaand coarser on the upcurrent (landward) flank. The per-turbed bottom topography and the perturbed mean grain sizepatterns are approximately 90� out of phase. Note that thebasic state velocity is directed from the top to the bottom ofthis figure (V < 0), so that the ridges are characterized by anupcurrent rotation.[37] The locations of the maxima and minima in the mean

grain size depend on the value of the standard deviation s0(or sorting index). Figure 7 shows a shore-normal crosssection of the bottom topography and fraction of fine grainsfor two different sorting indices. A decrease in the phasedifference between the two patterns is found for largervalues of the standard deviation. An interpretation of theseresults will be given in section 5.[38] Experiments were conducted to investigate the sen-

sitivity of the model results to different values of theparameter F1. It was found that, if the fraction of fine grainsF1 > 0.5, the influence of the standard deviation on the bedform characteristics (wavelength, growth rate and migrationspeed) becomes stronger. On the contrary, if F1 < 0.5 (morecoarse grains than fine grains), these dependencies becomeweaker. Changing the value of F1 does not affect the bottompattern or the distribution of the mean grain size of Figure 6,but a different sorting pattern is obtained. In the case of F1 =0.5 it follows that s0 = 0 according to (21), hence no

Table 1. Parameter Values for Uniform and Bimodal Sediment

Uniform(Figure 5)

Bimodal(Figures 5 and 7 (t))

Bimodal(Figures 5 and 7 (b))

Bimodal(Figure 6)

Bimodal(Figure 8)

s0 0 0.2 1.0 0.5 0.5F1 1 0.5 0.5 0.5 0.7F2 0 0.5 0.5 0.5 0.3�m 1.5 1.5 1.5 1.5 1.5f1 1.5 1.7 2.5 2.0 1.8f2 1.5 1.3 0.5 1.0 0.7d1 (mm) 0.35 0.31 0.18 0.25 0.28d2 (mm) 0.35 0.41 0.71 0.50 0.60

Figure 5. Growth rates (left) and migration velocities(right) of the first cross-shore mode as a function of thewave number k (scaled by the preferred mode for uniformsediment) for different values of s0. The parameters used forthe bimodal mixtures are F1 = 0.5, cb = 0.75, and cs = �1.1.


changes in the standard deviation occurred. In Figure 8, it isshown that for F1 = 0.7 the sediment located on the seawardflank is finer (fm

0 > 0) and better sorted (s0 < 0) thansediment on the landward flank, which is coarser and morepoorly sorted (s0 > 0).

4.3. Hiding Functions: Bed Load

[39] The characteristics of both the preferred bottommode and the grain size distribution also depend on thecoefficient cb in the hiding function of the bed load trans-port, defined in (9). Therefore, experiments were conductedin which cb was varied. Physically this means that thehiding effects in the bed load transport were reduced orenhanced. The coefficient for the hiding in suspended loadwas kept constant at its default value cs = �1.1.[40] In Figure 9, the characteristics of the preferred

mode are shown as a function of the standard deviationfor different values of cb. The maximum growth rate�rmax, migration velocity cmax and preferred wave numberkmax in the bimodal sediment case are scaled by theircorresponding values for uniform sediment (being ku, �ru

and jcuj). The curves show, for all cases, a reduction in thewave number and growth rates, and an enhancement inmigration rate if s0 is increased. A value of cb = 0 impliesthat there is no hiding, cb = 1.5 corresponds to a value ofmb = 1 in the hiding function (see Figure 4). The newinformation deducted from this figure is that the inclusionof a hiding function in the bed load transport formulationhas little effect on the growth of the shoreface-connectedridges, but increasing hiding effects cause larger migrationspeeds. An interpretation of the results will be given insection 5.[41] In Figure 10, cross sections show the change in the

distribution of the fraction of fine sediment for no hiding

and strong hiding in the bed load flux. In the former casethe selective suspended load transport results in an almost90� out-of-phase relation between topography and meangrain size. An increase in the strength of the hiding in bedload (cb > 0) reduces this phase shift (see Figure 10,bottom).

4.4. Hiding Functions: Suspended Load

[42] The same experiments were done to investigate thedependence of model results on the hiding coefficient cs andthe straining coefficient lE in the transport capacity functionof suspended load transport (defined in (14)). Only one ofthese parameters, cs, can result in differences between thedepth-integrated concentrations of the size classes in asediment mixture, thereby introducing a mechanism forselective transport of suspended load.[43] The value of exponent cs is mainly determined by

two factors. The first is the dependence of the entrainmentand the relative thickness of the suspended load layer on thegrain size (including settling velocity and particle Reynolds

Figure 6. Bottom perturbations (colors; light: bars, dark:troughs) and perturbations in the fraction of fine grains(lines; solid: f1 > 0, dotted: f1 < 0) for s0 =.0.5, F1 = 0.5, cb =0.75, and cs = �1.1. The arrow indicates the direction of thebasic state velocity.

Figure 7. Cross sections through a ridge, normal to shore,of perturbations in bottom, h, and in the fraction of finegrains, f1 (or similar in mean grain size, fm

0 ). Quantities arescaled by their maximum values and shown for F1 = 0.5 anda small standard deviation of the mixture (top: s0 = 0.2) aswell as for a large value (bottom: s0 = 1.0). Furthermore,cb = 0.75 and cs = �1.1.

Figure 8. Cross sections through a ridge, normal to shore,of bed topography h, perturbations f0

m in the mean grainsize and perturbed sorting values (standard deviation) s0.Results are shown for F1 = 0.7, s0 = 0.5, cb = 0.75, and cs =�1.1.


number). The second factor is the strength of the hiding forthe entrainment of sediment in suspension, indicated by thecoefficient mE in the hiding function zi(13). We onlyinvestigated the influence of the second contribution andassumed ew = 1.1 (section 2.4.2), i.e., cs = 5mE � 2.1.Without hiding effects in the entrainment (mE = 0), thedepth-integrated concentration of fine sediment is largerthan that of coarse material. The inclusion of hiding effects(default: mE = 0.2) reduces the entrainment of fine sedimentfrom the bed, nevertheless, it still results in a depth-integrated concentration of fine material that is larger thanthat of coarse material. In the special case of mE = 0.42 itfollows that cs = 0, hence, Gsi is independent of the grainsize. The higher concentration of fine sediment, as aconsequence of, for example, their smaller settling veloc-ities, is counterbalanced by a reduced entrainment flux offine grains from the bed due to hiding effects. These threesituations are shown in Figure 11 as cs = �2.1, �1.1 and 0,including the straining factor and a constant value of cb =

0.75. With higher standard deviations, a decrease in wavenumber and growth rates and an increase in migration ratesis found. The growth rates are most strongly influenced.[44] The straining parameter lE in (14) reduces the

suspended load flux of both size classes in the sedimentmixture, whereas the reduction becomes more important forlarger values of s0. To demonstrate the importance of thisparameter, results are shown without straining (lE = 1) andno hiding in the entrainment (cs = �2.1). Figure 11 indeedreveals a change in the results: instead of a decrease in themaximum growth rate, an increase with s0 is found and themigration speed is decreased. This will be discussed in moredetail in section 5.[45] The bed form and the mean grain size patterns are in

phase if no grain size dependence is used in the suspendedload transport (cs = 0), with the finer sediment on the crestsfor cb > 0 (see Figure 12, top). A phase shift between themean grain size and the bed topography pattern is inducedby the suspended load flux. In fact, the sediment is finer onthe seaward (downcurrent) flank of the ridges for cs < 0.These phase shifts do not change if the straining parameteris excluded.

5. Physical Interpretation

[46] The results presented in the previous section can beexplained in physical terms. The concepts discussed hereare based on mechanisms for the formation of shoreface-connected ridges under the assumption of uniform sedi-ment, as presented earlier by Trowbridge [1995] and byCalvete et al. [2001b]. They have demonstrated that thetransverse slope of the bottom plays an essential role inthe formation of the ridges. An offshore deflection of thecurrent, i.e., u0 > 0, results in a convergence of the sedimentflux and the growth of upcurrent oriented ridges. Thisbehavior is due to mass conservation of both water andsediment. The presence of a ridge causes an enhancedconvergence in both the water and sediment flux (withrespect to that induced by the offshore movement of waterand sand) on the downstream side of the ridge. Likewise,the convergence reduces on the upstream flank of the ridge.The result is that sediment is eroded (deposited) on theupstream (downstream) flank of the ridges. Therefore,

Figure 9. Wave number kmax of the preferred mode(scaled by its value ku for uniform sediment), growth rates,and corresponding migration velocities as a function of thestandard deviation of the mixture s0. Results are shown fordifferent formulations of the hiding function in the transportof bed load, with cs = �1.1 and F1 = 0.5.

Figure 10. As in Figure 7, but for s0 = 0.5, F1 = 0.5, cs =�1.1 and cb = 0 (top), cb = 1.5 (bottom).


downcurrent migration of the ridges takes place. It appearsthat the growth of the shoreface-connected ridges is mainlydetermined by the suspended sediment flux, while bed loadtransport determines the downcurrent migration of the bedforms.[47] In the case of a bimodal sediment mixture growth

rates become smaller, migration rates speed up, the pre-ferred length scale becomes longer, and sorting of sedimentis observed. To understand these new features, we examinedthe effects of dynamic hiding in both bed load andsuspended load in the bottom evolution equations (23)and (24).

5.1. Hiding in Bed Load

[48] For the exponents in the transport capacity func-tions for bed load and suspended load we consider: cb > 0and cs = 0, respectively. These assumptions allowed for aninterpretation of the results shown in Figures 11 and 12(cs = 0). The transport of suspended load is independentof the grain size, therefore only the effect of the strainingparameter for suspended load and hiding in bed load areincluded. As a result, the equations for the evolution of

the bottom and the fraction of fine sediment reduce to:

@h

@t¼ �Tb2 ~r �~q0b � �5

E~r �~q0s � Tb1 þ Tb2Tb5½ �qb0

@f1@y

ð25Þ

0 ¼ �Tb3 ~r �~q0b � �5Eqs0

@f1@y

ð26Þ

where

Tb1 ¼ Gb1 � Gb2 Tb2 ¼ F1Gb1 þ F2Gb2

Tb3 ¼ F1F2 Gb1 � Gb2ð Þ Tb4 ¼ F2Gb1 þ F1Gb2

The expressions for Tb5 and �E are given in Appendix C.The left-hand side of (26) has been set to zero, following thearguments presented in section 3.3. Furthermore, since jqs0j� jqb0j (suspended load dominates over bed load trans-port), the terms proportional to qb0 are omitted in (26).[49] To understand the alongshore variation in the mean

grain size we consider the expression for ~r �~q0b, as given inAppendix C, and neglect slope effects. After substitution ofexpression (4) for the wave-orbital velocity

~r �~q0b / u2w � m

Hm=2� 1

� � 1

H

@H

@xu0 þ V

H

@h

@y

�

This expression is simplified by applying scaling argu-ments. For the parameter values, V �0.4 m s�1, H 14 m, @H/@x 1 � 10�3 and m 1.6, the first term is afactor 100 smaller than the second term and is neglected.Since qs0 / uw

5HV (Appendix C), this yields for (26):

@f1@y

/ � Gb1 � Gb2ð Þ @h@y

This result shows that the fraction of fine sand is either inphase or 180� out of phase with the topography. For bedload the transport capacity function is given byGbi ¼ di=dmð Þcb . If cb > 0 it follows that Gb1 < 1 < Gb2,leading to a reduced transport of fine sediment relative tothe transport of coarse sediment. The relation between the

Figure 11. As in Figure 9, but now for differentformulations for the hiding coefficient cs in the suspendedload transport and cb = 0.75.

Figure 12. As in Figure 7, but for s0 = 0.5, F1 = 0.5, cb =0.75 and cs = 0 (top), cs = �2.1 (bottom).


bottom topography and the fraction of fine grains istherefore given by f1 / h for this situation. A schematicview of this selective transport mechanism is shown inFigure 13. The mean grain size is finest on the crests of theridges, and explains the pattern shown in Figure 12 (cs = 0).[50] The enhanced migration rates and reduced growth

rates for a sediment bed composed of a bimodal mixture(Figure 11, cs = 0) are understood from (25). The suspendedload flux is only modified by the straining factor, whencompared to uniform sediment. This factor is smaller thanunity and determines the reduction of the growth rates by �E

5

0.2 for s0 = 1 (Figure 11, middle). The bed load flux is afactor Tb2 different from the uniform sediment case. As Tb2�1 for bed load transport, hiding effects cause a faster migra-tion compared to uniform sediment, while it hardly changesthe growth. The last term (redistribution of sediment) in (25)could also change the migration, because f1 is related to h.However, experiments indicated that this contribution is onlyof minor importance to the downcurrent migration.

5.2. Hiding in Suspended Load

[51] For the investigation of hiding (and straining) insuspended load we set cb = 0 and cs < 0. This enables aninterpretation of the results shown in Figures 9 and 10 (cb =0). The equations for the evolution of the bottom and thefraction of fine sediment reduce to

@h

@t¼ � ~r �~q0b � Ts2 ~r �~q0s � Ts1 þ Ts2Ts5½ �qs0

@f1@y

ð27Þ

0 ¼ �Ts3 ~r �~q0s � Ts4 þ Ts3Ts5½ �qs0@f1@y

ð28Þ

where

Ts1 ¼ Gs1 � Gs2 Ts2 ¼ F1Gs1 þ F2Gs2

Ts3 ¼ F1F2 Gs1 � Gs2ð Þ Ts4 ¼ F2Gs1 þ F1Gs2

Equation (28) relates alongshore changes in the fining of thebottom material to the convergence of the suspended loadflux, and bed load contributions are neglected. In the secondterm on the right-hand side, [Ts4 + Ts5Ts3] is positive forrealistic values of the parameters. In Figure 14, themechanism for selective suspended load transport issketched.[52] Substitution of the expression for ~q0s (Appendix C)

in (28) yields

�V@f1@y

/ Gs1 � Gs2ð Þ @uw@x

u0

In this expression @uw/@x < 0 (waves induce less stirring ofsediment in larger depth) and V < 0. The momentumequations are used to relate u0 to h. Calvete et al. [2001b]showed that the perturbations in the cross-shore velocity arein phase with the bottom perturbations (u0 > 0 if h > 0).Hence, if the transport in suspension is more pronounced forthe finer grains (Gs1 >Gs2), this leads to @f1/@y/� u0 / � h.Consequently, the pattern of the mean grain size for sus-pended load is approximately 90� out of phase with thetopography, such that the finer sediment is found on thedowncurrent flank of the ridges. This effect is clearly seen inFigure 10 (top) for cb = 0 and cs = �1.1.[53] The reduced growth rates for graded sediment com-

pared to uniform sediment can be understood from (27).Growth is mainly determined by the second term on theright-hand side, where the part related to the grain size is afactor Ts2 different from that found in the case of uniformsediment. For selective suspended load transport, withhiding coefficient cs = �1.1, the factor Ts2 < 1. The presenceof a straining parameter in the transport capacity functionfor suspended load (14) causes the total transport of the twograin sizes in a mixture to be less than the transport ofsediment of a uniform grain size. A stabilizing effect isfound with an increasing standard deviation of the mixture(see Figure 9, cb = 0), whereas the contribution of cs to thetransport capacity function has the opposite effect. Thus, astronger hiding in suspended load (larger values of cs)counteracts the stabilizing effect of the straining. Thisexplains why growth rates increase with increasing valuesof jcsj for a fixed value of s0 (see Figure 11).[54] The bed load contribution to the migration (down-

current) is the same as for uniform sediment, but thecontribution of suspended load (upcurrent migration)decreases with increasing values of s0, resulting in a netenhancement of the migration in the downstream direction.An additional contribution to the bottom evolution is givenby the last term in (27), which turns out to be very small.[55] If selective transport in both suspended and bed load

is included, the pattern of the mean grain size resembles thatfound in the case of only hiding in suspended load. This isbecause the suspended load flux dominates over the bedload flux. The hiding function of the bed load flux slightlymodifies the 90� phase difference (induced by suspendedload) between the mean grain size and bed topography. Thistendency is visible in Figure 10.[56] The standard deviation s0 of the mixture also influ-

ences the phase difference between the mean grain size andbottom topography. Because of the straining parameter lEin the selective suspended load flux, the latter decreases

Figure 13. Schematic view of the selective bed loadtransport mechanism. The downcurrent migration inducedby the bed load flux results in an erosion/deposition pattern,which is shifted with respect to the bed topography.Equation (26) requires the convergence rate of the bed loadflux to be proportional to a changes in the alongshore (i.e.,downstream) fining of the sediment. In a deposition area( ~r �~q0b < 0), therefore qs0 @f1/@y < 0 for the situation wheretransport of coarse material is favored with respect to that offine grains. When moving from the crest to the adjacenttroughs, a coarsening of the sediment is found.


with increasing s0. The bed load flux is not changed by thestraining parameter. This implies that the relative impor-tance of the suspended load transport, with respect to bedload transport, decreases with increasing s0. Hence, thephase shift becomes smaller with increasing standard devi-ations, which explains the results shown in Figure 7.

6. Discussion

[57] The model results are consistent with the field data, asdiscussed in section 1. The data reveal the mean grain sizepattern is approximately 90� out of phase with topography,with the coarsest sand on the landward flank (upcurrent), andthe finest sand on the seaward (downcurrent) flank. Anexception to this very consistent pattern in the mean grainsize, as observed on different shelves, concerns the shore-face-connected ridges along the central Dutch coast (south-ern North Sea) [Van de Meene et al., 1996]. No markedspatial variation in the mean grain size across the ridgetopography is found, except for a weak tendency toward abetter sorting (smaller standard deviations) at the crests ofthe ridges. A potentially important difference between theAmerican Atlantic shelf and the Dutch shelf is the strength ofthe tidal current. Model experiments with both bimodalmixtures and uniform sediment showed that adding tidalcurrents does not change the results as presented here. Thissuggests that tidal currents are not the cause of the differencein the observed patterns for the Dutch coast (strong tides)and North America (weak tides).[58] Besides the mean grain size, another aspect of the

model results to be compared with the field data is the va-riation in the standard deviation over the ridges. For theAmerican shelf, the most pronounced differences in sortingcharacteristics are seen between sediment in the crest andtrough. In general, the values of standard deviation of thesediment are higher in the troughs than on the crests of the

ridges [Swift et al., 1972, 1978; Schwab et al., 2000]. This isconsistent with the sorting pattern found over the ridgesalong the central Dutch coast, in contrast to the lack of avariation in the mean grain size pattern. Data gathered fromthe German Bight [Antia, 1993] show that the surficialsediment found on the seaward flank are best sorted (smalls) and poorest in the troughs and on the landward flank. Inthe model, the variation in the sorting index is determined byrelation (21). Clearly, variations in sorting across the ridgesin a two-size mixture are only present in the model if thefractions of fine and coarse grains are not equal. Choosing F1

> F2 (weight percentage of fine sediment is larger than ofcoarse sediment) yields that in areas where positive pertur-bations in the mean grain size are present the material isbetter sorted. Combined with the effect of hiding in sus-pended and bed load (finer seaward flank), this case providesa good representation of the data.[59] The results should be interpreted with care: the

formulation for the selective transport of suspended andbed load are based on expressions which are found as a bestfit with many different data sets [Zyserman and Fredsøe,1994; Admiraal et al., 2000]. These data are mostly basedon measurements in shallow rivers and flume experimentsand we assumed them to be also applicable to shallowcoastal seas. However, in the sensitivity experiments it wasshown that the general trend is the same for a large range ofvalues of the sediment parameters, such as for the exponentsin the hiding functions and the composition of the sedimentmixture.

7. Conclusions

[60] In this paper a model was developed and analyzed tostudy the initial formation of shoreface-connected ridgesand the corresponding grain size distribution on storm-dominated shelves. The model consists of the shallow waterequations, a sediment transport formulation and a massbalance of sediment. The sediment is represented by twograin size classes. Both bed load and suspended load sedi-ment fluxes are considered, as are dynamic hiding effects.The basic state represents a storm-driven flow on an innershelf with a transversely sloping bottom. The results of themodel presented here indicate that, in the case of a sedimentmixture, there is a positive feedback between storm-drivencurrents and the eroding bottom. This confirms and general-izes earlier findings by Trowbridge [1995] and Calvete et al.[2001b] for a single grain size fraction.[61] The first objective of this research was to investigate

the influence of sediment sorting on the temporal and spatialcharacteristics of shoreface-connected ridges. The stabiliz-ing effect of sediment sorting on the growth of bed forms, asfound earlier in many laboratory experiments and othermodel studies, is also observed within the present model.Based on the experiments that were carried out, we con-clude that the behavior of growth rates is determined by theformulation for the hiding in suspended load. A stabilizedgrowth turns out to be mainly because of the presence of astraining parameter in the hiding function of suspendedload. This accounts for the reduced mobility of grains insuspension with increasing standard deviations of the sedi-ment mixture. If the straining parameter is excluded in thesuspended load formulation, a destabilizing effect with

Figure 14. Schematic view of the selective suspendedload transport mechanism. Suspended load mainly causesthe growth of the ridges (erosion/deposition pattern almostin phase with the bottom topography). The deposition ofsuspended sediment is related to the gradient in thedistribution of fine grains (28). In a deposition area( ~r �~q0s < 0), therefore qs0@f1/@y > 0 for a situation wheretransport of fine material is favored with respect to that ofcoarse sand. The result is a downstream fining in thedeposition area; the finest material is located approximately14wavelength downstream of the crest.


respect to uniform sediment is found. The migration of theshoreface-connected ridges is in the downcurrent directionand enhanced by the bimodal character of the sediment.Despite a suspended load flux, which is much larger thanthe bed load flux, the migration is controlled by the bed loadflux and its hiding coefficient cb. Wavelengths of the bedforms are only slightly affected. A spatial phase shift ofapproximately 90� is found between the topography andmean grain size for shoreface-connected ridges: the coarsestmaterial occurs on the landward (upcurrent) flank. Selectivetransport of sediment in suspension causes this phase shift.[62] The second objective of this study was to gain

knowledge into the physical mechanism responsible forthe observed grain size distribution over shoreface-con-nected ridges. Combining the observations with the modelresults, and the subsequent analysis of the equations, leadsto several conclusions with respect to the physical mecha-nisms which could be responsible for the observed sedi-mentary patterns. First of all, the persistent finerdowncurrent flank of the ridges and the coarser sedimenton the upcurrent flank appears to be in reasonable agree-ment with observed grain size distributions over shoreface-connected ridges. The model results support the assumptionthat the transport of sediment as suspended load cannot beneglected and that the corresponding flux of fine material islarger than of coarse material. The relative importance of thebed load flux can shift the pattern of the mean grain sizemore in phase or out of phase with the topography. Themodel reproduces observed sorting patterns over the ridges(well/poorly sorted sediment on the seaward/landward sideof the ridges) if the weight percentage of fine grains is largerthan that of coarse grains. Another quantity that influencesthe phase shift is the standard deviation of the sediment: if itincreases it causes a reduction in the importance of sus-pended load over bed load. Finally, the model resultsindicate that tidal currents and a grain size dependentformulation for the bottom friction coefficient do not changethe sediment patterns for shoreface-connected ridges.

Appendix A: Suspended Load Concentration

[63] The suspended load flux requires knowledge of thedepth-integrated volumetric concentration of sediment. Thelatter is governed by

@Ci

@tþ ~r � ~vtCið Þ ¼ wsi F iEi � cbið Þ ðA1Þ

The first term on the right-hand side is the flux of sedimentinto suspension, the second term is the deposition flux. Herewsi is the settling velocity of grains of size di, Ei is thedimensionless entrainment of these grains, and cbi is theactual volume concentration near the bed. The entrainmentof a size fraction is multiplied by the probability F i thatsediment of this grain size actually occurs.[64] For sand mixtures on inner shelves the settling period

is much smaller than the hydrodynamic timescale. Thisimplies that (A1) reduces to an approximate balancebetween erosion and deposition flux near the bed:

F iEi �Ci

diD¼ 0

The second term represents the deposition flux of thesegrains and is expressed in terms of the depth-integrated

concentration Ci. Parameter di is the ratio of the thickness ofthe suspended load layer of grains in class i over the totalwater depth D.[65] In the study of Garcia and Parker [1991] an expres-

sion for the entrainment of a mixture of particles is obtainedby analyzing laboratory and field data. They find

Ei ¼ A lEZið Þ5 Zi ¼u*wsi

R0:6pi

di

dm

� �0:2

with A = 1.3 � 10�7 a constant. Hiding effects are coveredby the last factor in the expression for Zi. Furthermore, lE =1 � 0.288s is a straining parameter, Rpi ¼

ffiffiffiffiffiffiffiffiffig0d3i

p=n is the

particle Reynolds number of grains of size di and v = 1.36 �10�6 m2 s�1 is the kinematic viscosity coefficient of water.The entrainment for uniform sediment (grain size dm) reads

Em;u ¼ AZ5m ¼ Em;uj~vtj5 Em;u ¼ A

ffiffiffifficf

p

wsm

R0:6pm

� �5

such that Zm corresponds the value of Zi for di = dm. Fromthis result it follows that the expression for Gsi in (12) iswritten as:

Gsi � zididm

Ei

Em

� �u

¼ ziwsm

wsi

Rpi

Rpm

� �3wsm

wsi

� �5

Appendix B: Bottom Friction Coefficient

[66] The expression for the grain roughness length usedin the formulation for the bottom friction coefficient (see(3)) reads ks = 3 � dmsm, with sm the geometric standarddeviation, defined as sm = 2s. When expressed in terms ofphi (section 2.2) it follows that ks ¼ 3 � 2s�fm (defined inunits of mm). Including small perturbations in the meangrain size and standard deviation in the friction parametergives for the quantities that correspond to the basic state andthe perturbation of the bottom friction parameter:

r0 ¼2

pg uw

C2h0

r0 ’ 2 r0

Ch0

s0 � f0m

� �ln 2þ h

H

� �ðB1Þ

The expression for the Chezy coefficient, as defined insection 2.1 with D = H � h, for the basic state is Ch0 =Ch(H, �m, s0). The perturbations in the bottom frictioncoefficient used in the momentum equations, with r0 asdefined in (B1), is written in terms of the unknowns h and f1by using (21).

Appendix C: Sediment Flux

[67] The transport capacity functions, defined in (8) and(14) for bed load and suspended load, was split intocontributions which correspond to the basic state and theperturbed state, respectively. These results are used insection 3.2.3 and read

Gbi ¼ 2cb fm�fið Þ Gbi ¼ 2cb �m�fið Þ

gbi ¼ cb ln 2 Gbif0m ¼ GbiTb5 f1

Gsi ¼ l5E2

cs fm�fið Þ Gsi ¼ �5E2

cs �m�fið Þ

gsi ¼ Gsi cs ln 2 f0m þ 5

l0E

�E

� ¼ GsiTs5f1

lE ¼ 1� 0:288s �E ¼ 1� 0:288s0l0E ¼ �0:288s0


where

Tb5 ¼ cb ln 2s0ffiffiffiffiffiffiffiffiffiffiF2F1

p

Ts5 ¼ cs ln 2s0ffiffiffiffiffiffiffiffiffiffiF2F1

p � 50:288s0

�E

F2 � F1ð Þ2F1F2

[68] The total load sediment flux in the basic state onlyhas an alongshore component and consists of contributionscaused by bed load and suspended load, given by:

qb0 �~qb0;y ¼3

2nbu2wV qs0 �~qs0;y ¼

32

5pdmEm;uu

5wHV

The divergences of the perturbed sediment fluxes are

~r �~q0b ¼3

2nb u2w

2

uw

@uw@x

� 1

H

@H

@x

� �u0 þ V

H

@h

@y

� �

� 8

9plbu

3w

@2h

@x2þ 3

uw

@uw@x

@h

@xþ @2h

@y2

� ��

~r �~q0s ¼32

5pdmEm;u 5u4wH

@uw@x

u0�

�ls

7u7wH

@2h

@x2þ 7

uw

@uw@x

þ 1

H

@H

@x

� �@h

@xþ @2h

@y2

� )

Here, (18) has been used to eliminate v0.

Notation

cb exponent in bed load transport capacity functioncs exponent in suspended load transport capacity

functionCi depth-integrated volume concentration of grain sizes

in class di, md1 grain diameter of finest sediment fraction, mmd2 grain diameter of coarsest sediment fraction, mmdm mean grain size of a sediment mixture, mmEi entrainment of sediment of diameter di in suspension

Ei,u as Ei, but in the case of uniform sedimentF i probability distribution function for size class i in the

active layerFi weight percent of the grains in the finest (i = 1) or

coarsest (i = 2) grain size class in basic statefi perturbation on Fi (i = 1, 2)

F s probability density function in substrateGbi bed load transport capacity function for sediment of

size diGbi basic state bed load transport capacity functiongbi perturbation on Gbi

Gsi suspended load transport capacity function forsediment of size di

Gsi basic state suspended load transport capacity functiongsi perturbation on Gsi

La active layer thickness, mLa0 basic state active layer thickness, mm exponent in wave stirring functionmb exponent in hiding function bed loadmE exponent in entrainment of sediment in suspension~qi total sediment flux of sediment of grain size di,

including pores, m2 s�1

~qbi bed load flux of grains di, m2 s�1

qb0 basic state alongshore bed load flux, m2 s�1

~q0b perturbation in bed load flux, m2 s�1

~q0si suspended load flux of grains di, m2 s�1

qs0 basic state alongshore suspended load flux, m2 s�1

~q 0s perturbation in suspended load flux, m2 s�1

uw wave-orbital velocity amplitude, m s�1

uw near-bed wave-orbital velocity, m s�1

di ratio of the thickness of the suspended load layer ofgrains of diameter di over the total water depth

dm ratio of the thickness of the suspended load layer ofgrains of diameter dm over the total water depth

zi hiding function for entrainment of sediment insuspension

�i,u Shields parameter for uniform sediment of size di�m Shields parameter for sediment of size dmlE straining parameter�E basic state straining parameterl0E perturbation on �E

xi dynamic hiding function for bed loads standard deviation of the mixture on the phi scales0 standard deviation of the mixture in the basic states0 perturbation on s0t bed shear stress (skin friction), N m�2

~tb bed shear stress vector (form drag + skin friction),N m�2

fm mean grain diameter on phi scale�m basic state mean grain diameterf0m perturbation on �m

[70] Acknowledgments. This research was financially supported bythe Cornelis Lely Foundation and the National Institute of Coastal andMarine Management in Netherlands (RIKZ) and by the EC-MAST 5HUMOR project (contract EVK3-CT-2000-00037).

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��H. E. de Swart and M. Walgreen, Institute for Marine and Atmospheric

Research (IMAU), Utrecht University, Princetonplein 5, NL-3584 CC,Utrecht, Netherlands. ([email protected])D. Calvete, Department de Fısica Applicada, Universitat Politecnica de

Catalunya, E-08034 Barcelona, Spain.


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