(2007) 1–18www.elsevier.com/locate/margeo

Marine Geology 241

A numerical–empirical approach for evaluating morphodynamicprocesses on gravel and mixed sand–gravel beaches

Adrián Pedrozo-Acuña a,⁎, David J. Simmonds a, Andrew J. Chadwick a, Rodolfo Silva b

a University of Plymouth, Centre for Coastal Dynamics and Engineering, School of Engineering,Drake Circus, PL4 8AA, Plymouth, United Kingdom

b Instituto de Ingeniería, Universidad Nacional Autónoma de México, Cd. Universitaria, 04510 D.F., México

Received 30 September 2006; received in revised form 19 February 2007; accepted 22 February 2007

Abstract

Pedrozo-Acuña et al. [Pedrozo-Acuña, A., Simmonds, D.J., Otta, A.K. and Chadwick, A.J., 2006. On the cross-shore profilechange of gravel beaches. Coastal Engineering, 53(4): 335–347] presented a numerical–empirical investigation of the processesthat control sediment transport in the swash zone on steep gravel beaches. This was based on a sensitivity analysis of a sedimenttransport/profile model driven by a highly non-linear Boussinesq model [Lynett, P., J., Wu, T.-R. and Liu, L.-F., P., 2002.Modelling wave run-up with depth-integrated equations. Coastal Engineering, 46: 89–107] which was compared to near full-scalemeasurements performed in the GWK flume in Hanover. In this paper we have extended our analysis to compare these earlierresults with those relating to a mixed sediment (gravel and sand) beach. The parametric sensitivity analysis also incorporates adiscussion of the effects of acceleration about which there is much debate. The sensitivity analysis suggests that fluid accelerationcan contribute to the onshore movement of sediment that causes steepening of initially flat beach faces composed of coarsesediment. However acceleration alone cannot be the cause of the observed berm growth during the GWK tests. Instead, a complexbalance of processes is responsible for the profile evolution of coarse-grained beaches with no single dominant process.© 2007 Elsevier B.V. All rights reserved.

Keywords: swash; Boussinesq; gravel beach; mixed beach; sediment transport; friction; acceleration; infiltration

1. The role of swash on steep beaches

Gravel and mixed sediment beaches are much ignoredyet comprise important coastal features which, inlocations such as the South Coast of the UK, have greatsignificance for the protection of coastal communities andenvironmental and agricultural resources (Mason andCoates, 2001). Sediment composed of gravel or gravel

⁎ Corresponding author. Tel.: +44 1752 233686; fax: +44 1752 232638.E-mail address: apedrozoacuna@plymouth.ac.uk

(A. Pedrozo-Acuña).

0025-3227/$ - see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.margeo.2007.02.013

and sand mixture forms natural shoreline units thatinclude barrier beaches and spits that protect estuaries andhinterland from flooding, and steep terraces that providetoe protection to soft cliffs. Also of importance are com-posite beaches that comprise a gravel berm atop a sandylower foreshore and engineered beaches that have been re-nourished with mixed sediment. Notwithstanding theirprevalence comparatively little research effort has beenfocussed on studies of these important coastal defences asopposed to that of sandy environments.

Coarse beach environments have a gradient of reposethat is typically in the range 1:12 to 1:2. This steepness

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is a function of both the physical characteristics of thebeach material and its porosity which permits significantinfiltration and absorption of incoming waves resultingin a comparatively weak backwash. The size of thesediments for the gravel varies between 2 to 64 mm,however sometimes cobbles ranging from 64 to 256 mmcan be found. Thus, sediment movement is only likelyclose to the shoreline inside the narrow energetic surfzone that is itself a function of the steep profile.

Indeed, most of the morphological developmentappears to occur in the swash zone, and results in anupper beachface that is generally steeper comparedwith the mean beach face. This was demonstrated in arecent large scale experiment with coarse sedimentsreported by López de San Roman-Blanco et al. (2006).This investigation took place at the Large Wave Flume(GWK) in Hanover, Germany, and focussed on thestudy of the morphological response of two types ofcoarse-grained beach, one gravel and the other amixture of gravel and sand. López de San Roman-Blanco (2003) described many aspects of this research.In particular a conceptual model was constructed inorder to summarise the physical processes that mayhave an important role in shaping these beaches.However, although detailed comparisons of processessuch as the drainage characteristics, hydraulic gradientsand setup at the shoreline for both the beaches weredrawn, a discussion of specific swash zone processeswas omitted. Indeed, from the profile measurements it isclear that for all wave conditions the majority of profilechange occurred within the swash region, withsignificant changes occurring above the mean waterlevel to the point of maximum run-up. It is in this regionthat understanding of the morphodynamics of coarse-grained beaches needs to be advanced.

Swash zone hydrodynamics and sediment transporthas been an active topic of research over the past decade.Yet until the recent GWK experiments, most field andlaboratory experiments have been restricted to observa-tions of sand beaches with mild slopes (Masselink andHughes, 1998; Butt and Russell, 2000; Kobayashi andJohnson, 2001; Butt et al., 2002; Puleo et al., 2003;Masselink and Russell, 2006). It might be argued thatswash processes on sandy beaches should be less of aconcern to the coastal engineer, given that swash sed-iment mobilisation represents only a fraction of the totalamount mobilised on sandy beaches, but the majority oncoarse-grained beaches (Austin and Masselink, 2006).Recently Van Wellen et al. (2000) presented an empir-ical model for swash transport that predicted well thelongshore transport of sediment within the accuracy offield measurements. More recently Pedrozo-Acuña et al.

(2006) have reported on investigations of the sensitivityof profile development to specific swash parameters, inrelation to the GWK gravel beach experiments. It isnotoriously difficult to measure sediment transportparameters directly, especially in the swash. The ap-proach that was adopted was to carry out a sensitivityanalysis of a suitable numerical modelling framework.This permitted a discussion of the sensitivities of pro-cesses that might be shown to control the cross-shoreprofile development in terms of those parameters thatare widely acknowledged as serving as aggregateddescriptions of the immeasurable micro-scale processes.The modelling framework comprised the coupling of ahighly non-linear Boussinesq model with a sedimenttransport formulation and a morphology module. In thestudy, special attention was paid to discussing the rolesof the infiltration and bottom friction parameters on theskill (Brady and Sutherland, 2001) with which predic-tions of the GWK gravel profiles could be achieved. Inaddition, Buscombe and Masselink (2006) presented areview which summarises some of the aspects thatrequire further study in order to understand gravel beachdynamics.

It should be borne in mind that a universal quanti-fication of sediment transport derived from the funda-mental physics has not been established owing to theenormous complexity of the phenomenon (Van Rijn,1993). This is largely due to the inherent difficulties inmaking accurate but non-invasive measurements of ki-nematics and sediment fluxes, especially in transient,aerated and shallow flows. This has, thus far, proven anintractable problem, despite advances in optical andacoustic measurement technology.

Thus a myriad of competing formulae for sedimenttransport under waves and currents have been developedbased on a macroscopic approach to oscillatory and steadyfluid motion. Many of these are presented as universal intheir application yet have mostly been developed for sandsized material. It is desirable that a bottom–up descriptionof processes on the grain–grain interaction scale should bedeveloped, but this is currently unavailable. Indeed it canbe argued that at the micro-scale, the discriminationbetween or validation of some of the more advancedmodelling approaches (Deigaard et al., 1986) is beyond thecapability of current measurement technology.

Therefore, in Pedrozo-Acuña et al. (2006), and in thework presented here a modelling framework of appropriatecomplexity has been adopted for comparison with themacro-scale observations of profile change alone. The“appropriate complexity” is defined after our discussion ofswash transport processes on steep beaches and modellingstate of art.

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This present work extends Pedrozo-Acuña et al.(2006) in two directions, again concentrating on profiledevelopment due to swash processes. Firstly, a newanalysis of profile development over the GWK mixedbeach is presented and compared with this earlier studyof the gravel beach. Both beaches were subjected to thesame series of wave conditions permitting comparisonsto be drawn regarding the influence of beach material onswash processes. A priori it might be expected thatmixed sand and gravel, typical of a dredged beachrecharge, would behave more like a sand beach becauseof the improved sediment packing and correspondingreduction in porosity.

Secondly, an investigation into the role that fluidacceleration plays in coarse-grained beach profileevolution is presented. The latter is motivated by theconclusions put forward by Nielsen (2002) and Puleoet al. (2003) who suggested that the strong onshoredirected acceleration and backwash deceleration arelikely to affect sediment transport in the swash zone, andthus beach profile evolution.

2. Present understanding of swash transport onsteep beaches

One of the most striking features of swash motion onsteep beaches is the asymmetry of flow velocity andwater volume between the uprush and backwash. Thiscan create a corresponding asymmetry in the cross-shoresediment fluxes that depends on the phase of the flowand the steepness of the beachface (Masselink andHughes, 1998). In order to make any progress in theprediction and modelling of swash transport, a carefulassessment of the complex interaction between thecharacteristics of the beach material, groundwater andthe hydrodynamic inputs is required. Amongst theprocesses that are reported to affect sediment transportin the swash zone are: bottom friction, infiltration andexfiltration through the beach face, acceleration effects,and turbulence generation following bore collapse.

2.1. Friction

Pedrozo-Acuña et al. (2006) have reported that thereis considerable debate in the literature concerning valuesof quadratic friction parameters for sediments. Whilstsome researchers suggest a phase dependant value forthe effective quadratic friction parameter is appropriate,others, including Raubenheimer et al. (2004) suggestthat values in both phases are equivalent. Furthermorewhilst Cox et al. (2000) suggest from laboratory exper-iments that friction is higher in the uprush phase, Puleo

and Holland (2001) deduced the opposite from theirfield experiments. It must be borne in mind that thesediscrepancies can be attributed to measurement andanalysis differences and accuracies.

2.2. Groundwater effects

Interaction between the flow above the beachfaceand the groundwater flows, has been a matter of researchin several studies. Its effect on the direction of sedimenttransport has been highlighted by Turner (1995), Turnerand Nielsen (1997), Turner and Masselink (1998), Buttet al. (2001).

Downward and upward pressure gradients created bythe pressure loading of the swash lens, during run-upand run-down result in infiltration and exfiltration ofwater through the beach face. The effects of this onsediment transport in the swash zone were summarisedby Elfrink and Baldock (2002) as follows:

• Reduction of backwash volume and duration.• Increase and decrease of the effective weight ofsediment particles.

• Increase and decrease of the shear force on sedimentparticles.

The reduction in the backwash volume, which isdependent on the beach permeability, leads to both lowermean and maximum velocities in the offshore direction.This changes the flow asymmetry and therefore thesediment deposition patterns. The effect is expected to beof lower magnitude for sandy beaches where permeabil-ity is usually neglected, due to the small vertical flux thatis present. However, in beaches with coarser sedimentlike gravel beaches, this effect may become important.

The amount of water infiltrated into the beach willdepend on its permeability, and thus on both the meangrain size and the grain size distribution. In a recentstudy, analysing time series of velocity and concentrationwith a modified Shields parameter, Butt et al. (2001)found that there is a critical grain size at which theinfluence of infiltration–exfiltration changes the direc-tion of sediment transport from offshore to onshore. Thishas since been supported by Karambas (2003) innumerical simulations using a Boussinesq modelcoupled with a porous flow model. He identified thesame critical grain size that determined the dominance ofthese effects. This investigation also concluded thatrelatively small changes in friction factor might changethe direction of the apparent influence of infiltration–exfiltration, also mentioning that the wave conditionshave no influence on this process.

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Conley and Inman (1994) analysed the flow and theturbulence characteristics of ventilated boundary layers inthe laboratory, which is analogous to a porous beachface.They found that turbulence in the case of infiltration isconfined to a compact layer close to the bed, whereas it ismore evenly distributed across the water column duringexfiltration. The thinning of the boundary layer duringinfiltration leads to enhanced flow velocities and shearstress in the boundary layer, whereas the opposite occursin exfiltration.

Infiltration and exfiltration effects are also associatedwith additional pressure forces on sediment particles.Turner and Masselink (1998) showed that the criticalshear parameter can vary significantly due to the alteredeffective weight. Nevertheless, they found that the effectof the enhanced bed-shear stress was more importantthan the altered effective weight and concluded thatinfiltration/exfiltration processes support onshore sedi-ment transport in the swash zone. These effects areexpected to increase for increasing sediment grain size,since vertical flow velocities and the resulting additionalbed-shear stress become larger for coarser sediments,and a relatively larger part of the sediment transportoccurs close to the bed.

2.3. Acceleration effects

In early investigations, the effect of fluid accelerationwas normally assumed to produce delayed boundarylayer growth and hence higher velocity near the bed thanflows with weaker acceleration after the same durationof boundary layer development (Nielsen, 1992). Severalworks have shown that acceleration has a direct effecton sediment transport, mainly as a further suspensionmechanism (King, 1991; Ribberink and Al-Salem,1995). Moreover, in recent studies, fluid accelerationshave also been related to sandbar morphology in the surfzone, showing that the peak in acceleration skewness ofsurf zone flows was well correlated with onshore barmotion (Elgar et al., 2001; Hoefel and Elgar, 2003).

In the swash zone, the acceleration effects are morepronounced with a strong onshore acceleration under thepropagating bore and weak acceleration during thebackwash. Recent works presented by Nielsen (2002)and Puleo et al. (2003) have concluded that the strongonshore directed acceleration and backwash decelerationare likely to affect sediment transport. However, HughesandBaldock (2004) have suggested that the appearance ofstrong onshore accelerations in some field data may be anartefact, related to the difficulty of measuring velocitiesaccurately in very shallow water. Moreover, results foundin a recent investigation by Terrile et al. (2006) show that

fluid acceleration is indeed important for the initiation ofsediment motion. Nevertheless, further studies arenecessary in order to identify and describe the effects ofacceleration-dependent transport.

2.4. Bore collapse

The leading edge of the swash (collapsing bore)could be expected to entrain or maintain high concen-trations of sediments because of the associated highturbulence levels reaching the bed (Puleo et al., 2000).During the uprush phase Yeh et al. (1989) showed thatturbulence is advected with the bore front and can act onthe dry beach face.

The level of turbulence depends on the beach slope.Steeper beaches produce plunging breakers with moreconcentrated turbulence than on wide dissipative surfbeaches (Butt et al., 2004).

This observation was also put forward by Ting andKirby (1994) in laboratory experiments. Indeed, in theshallower area of the swash zone, turbulent vortices mayreach the seabed and should be expected to entrain ormaintain high concentrations of sediments (Puleo et al.,2000). Jackson et al. (2004) have also observed in thefield that sediment entrainment during bore collapse(seaward of the base of the swash zone) is an importantmechanism for swash transport. It is thus evident thatbore collapse is important for swash transport in theuprush phase (Masselink and Russell, 2006).

2.5. Swash processes—conclusion

The problem that is now addressed is “how is itpossible to simulate swash transport to an accuracythat allows comparison with measured profile devel-opment?” First, it is necessary to choose a model of theappropriate complexity to match the problem. Thisshould comprise a coupled hydrodynamic model witha sediment transport formulation and profile evolutionmodel. The model should also include parameters thatrepresent friction, acceleration and groundwater.Presently, an investigation of the role of turbulencefollowing bore collapse is beyond the scope andcomplexity of this current work and we focus on theremaining processes. The sediment transport predic-tions are critically dependent on the prediction of wave-induced velocities. The asymmetry between the on-shore and offshore velocities plays an essential role indetermining the magnitude and direction of the wave-induced sediment transport. Thus a phase-resolvingmodel of swash transport driven from outside the surfzone is required.

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3. A modelling framework for swash sedimenttransport

A good starting point for this framework would appearto be the Boussinesq approximation. Indeed, Ozanne et al.(2000) have demonstrated that the phase-resolvingBoussinesq equation shows great promise for the pre-diction of those velocitymoments inside the surf zone thatare necessary to drive sediment transport. They used aweakly non-linear model (Madsen et al., 1991) with asurface roller breaking description (Schäffer et al., 1993)to simulate velocitymoments of data reported by Ting andKirby (1994).

Other researchers have used the Boussinesq ap-proach to estimate transport directly. Karambas et al.(1995) used a set of Boussinesq equations of weak non-linearity to estimate sediment transport rates in the surfzone. Rakha et al. (1997) used a similar set of waveequations to estimate morphological evolution in thenearshore zone. Both these studies were carried outlooking at the morphological changes in the regionbelow the still water line (SWL), focusing on bed levelchanges in the surf zone with the final aim of inves-tigating bar formation. The swash zone was ignored inboth studies.

Karambas and Koutitas (2002) presented the ear-liest attempt to predict morphological changes over theentire beach profile including surf and swash zones.Until this work, most of the existing models were aimedat the prediction of only the submerged beach profilechanges. In their investigation, they used a weakly non-linear Boussinesq approximation to study beach profilechanges on sandy beaches.

Additionally, a similar approach has been recentlyused by Long and Kirby (2003) and Long et al. (2004).They employed a highly non-linear version of the equa-tions to improve predictions in deeper water, focusingon the surf zone hydrodynamics. All these models uti-lised averaging of the instantaneous quantities to obtainthe statistical moments needed to drive wave-averagedtransport models, and their application has been mainlydirected at the investigation of bar migration inside thesurf zone. Therefore, this work has been restricted tosandy beaches with mild slopes.

At the edge of the swash it has been demonstrated thatthe moving shoreline boundary, implemented for instancein Lynett et al. (2002), performs well in relation to the slotboundary presented by Kennedy et al. (2000). The latterneeds special attention to achieve accurate results close tothe shoreline (Otta and Pedrozo-Acuña, 2004). Addition-ally this boundary should enable a direct representation ofthe swash hydrodynamics to be achieved.

3.1. Specification of modelling framework

The approach adopted by Pedrozo-Acuña et al.(2006) and in this investigation was motivated by thesuccess of these modelling studies mentioned above.They have established that this type of process-basedmodel is useful in describing surf zone hydrodyna-mics and the associated bar migration. In particular theframework is similar to that presented by Karambas andKoutitas (2002); both numerical models are able todescribe the surf and the swash hydrodynamics andcalculate the resulting morphological changes. Howev-er, the application of both numerical models is different.Karambas and Koutitas (2002) focused mainly onbeaches with mild slopes (1:20) and small grain diam-eters within the mm range. Due to these beach materialcharacteristics, all their simulations showed most of themorphological changes within the surf zone, even forthe steepest case (1:10). Conversely, the work presentedhere focuses its attention on coarse-grained beaches,which are characterised by steep slopes (1:8) composedof larger grain sizes in the range 21 mm to 17 mm. Thusthe importance of suspended load in comparison to themajority bed load is reduced. In consequence, the in-vestigation presented in this paper focuses on swashzone hydrodynamic and sediment transport processes.The Karambas and Koutitas (2002) results were pre-sented without any sensitivity analysis to, for instance,efficiency coefficients or friction. Here the aim is also toassess these sensitivities to give insight into the processes.

Pedrozo-Acuña et al. (2006) is based on the numer-ical code of the Boussinesq equations developed byLynett and Liu (2002). This was then coupled with asediment transport and the morphology module de-scribed below. In order to cope with predictions of wavebreaking in shallow water, the highly non-linear, weaklydispersive Boussinesq equations, as presented by Lynettet al. (2002), were employed for calculating the hydro-dynamics. These equations estimate the horizontalvelocity at a chosen elevation, given by za=−0.531 h.The mechanisms of dissipation used in these equationsare wave breaking and bottom friction. The breakingcriterion is derived from an eddy viscosity approach asdescribed in detail by Kennedy et al. (2000), and bottomfriction is incorporated with the quadratic drag law.

From the sediment point of view there are someassumptions that could be made for this type of beach.Clearly, due to the large grain sizes involved, thedominant mode of transport can be assumed to be bedload (Soulsby, 1997). Moreover, as a result of the steepforeshores usually found on gravel beaches, the effect ofslope on the alteration of the critical shear stress is

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emphasised and must be considered. To determine thesediment transport rate, the relationship (Eq. (1))developed by Madsen (1991) based on the originalwork by Meyer-Peter and Müller (1948) was used. Thisequation accounts for additional effects, e.g. the bedslope, the friction angle of a moving grain. This ex-pression is compatible with evaluation of instantaneousfluxes from velocity time series. The final form of theexpression for the sediment flux, q(t), is given by:

qbðtÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðs−1Þgd3p ¼0 for hbhbcr

C

1þ tanbtan/

ðh−hbcrÞ3=2 ubjubj for hzhbcr

8>><>>:

ð1Þwhere

h ¼ 0:5qfu2bqgd ðs−1Þ ;

hbcr ¼ hcrqgdðs−1Þ ;

C is the efficiency value, g the acceleration of gravity,s is the specific weight of the sediment, d the sedimentdiameter, β the local beach slope (tan β=dh / dx, whereh is the bottom elevation and x a cross-shore coordinate)and ϕ the friction angle for a moving grain. At a givenx, the transport is onshore during the uprush phase andoffshore during the backwash phase.

This bed-load sediment transport equation is drivenby the bed-shear stress and includes a critical thresholdvalue of movement. These stresses are evaluated withthe wave-induced velocities obtained from the Boussi-nesq equations. Therefore, the transport rates arecalculated at the same hydrodynamic time step givenby Δt (i.e. instantaneous hydrodynamic time step).However, bed level changes are calculated every N stepssuch that Δtmorph=NΔt. Here, N=6 typically. Once thecumulative time (t+Δt) is equal to the morphologicaltime step, the sediment transport rates are time-averagedto determine the net transport rate for this morphologicaltime step, over the entire beach profile. This is achievedby numerical solution of the conservation of sedimentequation as described by Rakha et al. (1997).

It seems evident that it is necessary to combinecarefully conducted experiments and systematic mod-elling in order to advance in the understanding of thedynamics of these beaches. From that perspective, thispaper is focused on the identification of the basic pro-cesses governing the morphological response of boththe beaches studied in the GWK.

3.2. Extended framework

The framework in this work has been extendedin comparison to that of Pedrozo-Acuña et al. (2006)by the incorporation of acceleration in the sedimenttransport formulation. In their recent work, Drake andCalantoni (2001) investigated the role of fluid accelera-tions in nearshore bed-load transport. They modified aBagnold type of formula with an additional termdepending on a function of the near-bed fluid acceleration.The expression that they proposed has the general form:

qbðtÞ ¼ Chu3i þ f ðaÞ: ð2ÞThe inclusion of an acceleration term in a time-

domain morphology model was recently used by Hoefeland Elgar (2003) and Long and Kirby (2003) to capturethe onshore migration of a bar located in the surf zone.They showed that the use of the modified Bagnoldformula enabled better predictions. In the light of theseresults, we modify the bed-load formulation to includean acceleration term like the one proposed by Drake andCalantoni (2001). Following their results, the form ofthe equation used in this section is

qbðtÞ¼0 for hbhbcr

C

1þ tanbtan/

ðh−hbcrÞ3=2 ubjubj þ Kaða−acritÞ for hzhbcr :

8>><>>:

ð3ÞThe two new parameters that are included in this added

term are the acceleration threshold given by acrit and theefficiency given by Ka which has units of kg s m−2.

Acceleration is calculated by differentiating thevelocity time series from the Boussinesq hydrodynamicmodel (a=∂u /∂t). Sediment transport is then calculatedwith Eq. (3).

3.3. Overview of GWK experimental data

In this work we use an extension of the dataset usedby Pedrozo-Acuña et al. (2006). This was also collectedduring the GWK experiments (López de San Roman-Blanco et al., 2006).

Fig. 1 shows the experimental setup in the 309 mlong, 5 m wide and 7 m deep facility. The scale of theexperiments avoids scaling uncertainties present insmaller laboratory tests.

Two types of beaches were tested, a gravel beach anda mixed beach comprised of 70% gravel and 30% sand.Beach porosity was estimated from 6 sediment samplesand found to be 0.4 for the gravel beach and 0.2 for the

Fig. 1. Experimental setup at the GWK (SWL = still water line).

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mixed beach (López de San Roman-Blanco, 2003). Themedian diameter (d50) of the gravel was 21 mm, whereasfor mixed beach the median diameter was 17 mm.

The wave conditions that are considered in this studyare defined by a JONSWAP spectrum with Hs=0.6 m,Tp=3.22 s and H /L=0.05. Each test was further dividedinto three sequenced runs with different duration inorder to investigate profile development with time.Beach profiles were measured between tests using theGWK mechanical profiler, a 7.5 m mechanical arm witha roller at the end. For the gravel beach, we will considerthe early profile changes in relation to the initial, flattenedprofile with 1:8 slope. In the mixed beach case, it wasfound necessary to ignore the profile changes thatoccurred over approximately the first 1000 wave periods.This was due to the apparent compaction of the mixedbeach over this time (Pedrozo-Acuña, 2005). Thus themixed beach profile changes are referenced to the profilemeasured after these first 1000 waves.

Fig. 2 shows the evolved profiles for both gravel andmixed beaches under the same wave conditions. Twofeatures are clear. Firstly, the evolution rate of the profileis slower for the mixed beach, for the same waveconditions and almost identical flattened profiles.Secondly, the formation of the berm in the mixedbeach is located two meters further up the profile thanthe berm on the gravel beach, thus indicating that theactive swash zone is much longer for the mixed beach.

The key issue explored here is the differences in thedevelopment of the profiles for the two sediments. Animmediate illustration of the differences in the flowsabove both beaches is shown in Fig. 3a and b. Run-down over the gravel (left-hand panel) and the mixedbeach (right-hand panel) is captured at almost identicalphases of a swash cycle at the point of collapse of thefollowing bore. For the gravel beach it is evident thatmost of the water from the preceding backwash hasinfiltrated the beachface (represented by the arrows inthe picture). However, over the mixed beach, there stillremains a considerable volume of water in the backwash

as the next bore arrives. This closer packed material withlower porosity shows a stronger degree of swash–swashinteraction which appears to mitigate the effects of theplunging bore in comparison with the more poroussediment where the backwash is significantly weaker.

4. A numerical–empirical investigation of swashtransport: results

A sensitivity analysis of the model predictions wasthen performed with respect to the friction and acceler-ation parameters for identical wave forcing. Thesepredictions were then compared against the experimentalprofile measurements. This has enabled a heuristic dis-cussion of the processes that control the beach evolutionfor the coarse-grained beaches to be made, in a similarway to that presented by Pedrozo-Acuña et al. (2006). Thecomparison is “empirical” in that the modelling is com-pared with experimental results and in that the model isused to conduct numerical experiments.

From the observation of the different beaches in thelaboratory to the same wave conditions, it is clear thatboth beaches responded in a similar manner, but tovarying degrees. That is, a berm is built above the stillwater level and sediment is excavated from below thestill water level near the break point. It is thus likely thatthe same processes are responsible for the observedbehaviour on both, but in different proportions. Byapplying our modelling approach to both beaches, weaim to discuss differences in this balance of processesand test the usefulness of the modelling.

Since it can be seen (Fig. 2) that the initial profilesof both beaches were almost identical and that thewave sequences were the same, we assume that thehydrodynamic conditions over both beaches were verysimilar. Indeed the differences in the model runs in thiswork only differed essentially in the values of the sed-iment transport parameters chosen to describe the twobeach sediments. The default parameters for frictionfactor and the chosen mean grain diameter (d) of both

Fig. 2. Beach profile evolution. Top panel for gravel beach; (…) initial profile, (.-.-.) measured profile after 50 waves, (- - -) measured profile after 100waves, (__) measured profile after 500 waves. Bottom panel for mixed beach; (…) measured profile after 1000 waves, (__) measured profile after1500 waves, (-.-.-) measured profile after 3000 waves.

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beaches are summarised in Table 1. Bottom friction isincluded in the numerical model through a quadraticdrag law specified by a friction factor, f. These friction

Fig. 3. Bore collapse on the beaches at the GWK. (left panel) Gravel beach

values are based upon the best consensus from the lit-erature (Conley and Griffin, 2004; Raubenheimer et al.,2004).

; (right panel) mixed beach. Arrows represent the rate of infiltration.

Table 1Standard sediment parameters

Beach f d(mm)

Mixed 0.004 17Gravel 0.02 21

Table 2Parameters for friction investigation

Case C-value fuprush fbackwash

Gravel Mixed Gravel Mixed

1 12 0.04 0.008 0.02 0.0042 12 0.03 0.006 0.02 0.0043 12 0.025 0.005 0.02 0.004

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It must be borne in mind that we also employ themean grain diameter to represent the mixed beachsediment. During the experiment it was observed that athin top layer of the beachface became armoured due tothe wave action washing out sand and leaving gravel.Despite this, we assume that the main effect of the graindiameter is to control the porosity of the sediment, andhence infiltration into the beach. Thus, the use of themedian grain diameter is appropriate.

4.1. Swash phase dependence of friction factor

In Pedrozo-Acuña et al. (2006) the hypothesis thatthe apparent friction factor varies with the swash phase,was tested numerically and the results for the gravelbeach were found to support the conclusion by Cox et al.(2000) and Conley and Griffin (2004). That is, the useof a higher friction factor f during uprush improves thesimulations of morphological changes.

Now we extend this work to compare the role andeffects of bottom friction between both the gravel andmixed beaches in terms of the morphological responsesobserved in the GWK. Again the friction factor valuesused are based on published empirical values.

The initial profile of the mixed beach is that mea-sured after an interval of 1000 Tp. This correspondsclosely to the master profile of slope 1:8 used for thegravel beach comparisons.

Three numerical experiments were performed (seeTable 2) with the friction coefficients set greater in theuprush. For these experiments, the sediment transportefficiency, C=12 in Eq. (1), is kept constant and sym-metric (Nielsen, 1992).

The top-left panel of Fig. 4 shows the computedprofiles for the gravel beach after the action of the wavesfor approximately 50 Tp. This indicates that for smallerasymmetry in friction (Cases 2 and 3) a poor corres-pondence is achieved. Below the SWL, there is netaccretion of sediment with generation of a bar. Thisindicates a net offshore movement of sediment to thatposition. This was not observed in the GWK experi-ments. Above the SWL there is little accumulation ofmaterial to form the berm observed in the experiments.

However, once the friction is made twice as bigduring the uprush as during the backwash (Case 1), the

predicted profile begins to show erosion below the SWLalthough a bar is still incorrectly predicted. Above theSWL, accretion can be seen to form a berm, which isoverpredicted. The bottom-left panel of Fig. 4 illustratesthe differential changes for these tests. The results showthat although the height of the berm is well predicted byone of the cases (Case 2) the volume change is not. Theanomalously generated bar, not present in the GWKdata, appears to form close to the breakpoint. This pointsto the fact that asymmetry in friction alone is not suf-ficient to explain the observed profile evolution.

The top-right panel of Fig. 4 shows comparisons ofcomputed profiles for the mixed beach after 500 Tp. Themild cases (Cases 2 and 3) show very similar resultswith no erosion below the SWL. Even for Case 1, whenthe friction in the uprush is set twice as large as that inthe backwash, very little improvement in the predictedprofile is achieved, especially in the area below the SWLwhere some accretion appears. Differential changes inbottom-right panel of Fig. 4 confirm that the berm hasbeen formed a little bit further down the profile. Forthese friction values, agreement is poorer than in thecase of the gravel beach. Thus, we conclude that mod-ification of the friction factor within the range ofreported values is not capable of reproducing the ob-served profiles on the mixed beach. Indeed, over thegravel beach where high infiltration was observed, wemight expect, a priori, that asymmetric friction valuesfor uprush and backwash would be more appropriate.

It should be noted that despite the observed im-provement on the prediction of beach profiles with thishypothesis, the use of asymmetric friction factors,within the bounds of published values, did not yieldsatisfactory agreement between the measured and com-puted profiles for both beaches. The variation of fric-tion by itself was not able to reproduce the measuredprofiles. This must be attributed to the fact that othermechanisms apart from those encapsulated in theconcept of friction have a significant role in the overallresponse of coarse-grained beaches.

Notwithstanding the above, the results again supportthe general conclusion of Cox et al. (2000) and Conleyand Griffin (2004) that the apparent friction factor is

Fig. 4. Numerical experiments on friction (higher friction during the uprush) Case 1 (-.-.-); Case 2 (…); Case 3 (- - -). Left panels: results for gravelbeach; Right panels: results for mixed beach. (top — profile evolution; bottom — differential changes.

10 A. Pedrozo-Acuña et al. / Marine Geology 241 (2007) 1–18

greater in the uprush than the backwash for both thegravel and mixed sediments.

4.2. The role of acceleration in swash transport

The left hand panel of Fig. 5 shows the measuredprofile and a simulated profile without acceleration inthe gravel beach case. A fixed efficiency factor of C=12is again used in all runs. Two simulated profiles thatinclude acceleration are shown. These were calcula-ted using the same coefficient Ka=0.01 kg s m−2 butdifferent values of acrit, given by acrit =0.09 m/s2 andacrit =0.07 m/s2. The right panel of Fig. 5 illustrates thedifferential changes for these profiles.

Both acceleration runs show the same features, withan onshore movement of the bar in the neighbourhoodof the breaking point. This bar was also present in thesimulation when no acceleration term was included.

Notably, a small berm is generated at the top of theprofile. These features are a consequence of the highertransport that is generated by the contribution of theacceleration term. These results suggest that accelerationmay be part of the processes that emphasise the onshoretransport in this type of beach. Nevertheless, it is clearfrom this result that acceleration alone could not be thecause of the berm growth observed during the GWK tests.

To illustrate further the contributions of the accelera-tion on the sediment transport along the profile, Fig. 6

shows the non-dimensional sediment transport rate asestimated by the numerical model at three coordinates inthe cross-shore direction, given at chainages of 265 m,266 m and 266.75 m from the wave paddle. The threepanels show the time series of the calculated sedimenttransport with Eq. (3). The solid line presents the sedimenttransport rate without the acceleration, the dotted line isthe contribution of acceleration, and the dashed line istheir sum which then gives the total transport.

The above figure suggests that for this particularcase, the contribution of acceleration is mainly onshoredirected, this being stronger in the region close to theSWL (top panel).

In the mixed beach case, the acceleration effects arealso included by using the modified bed-load equationgiven by (3). For all simulations presented here, theefficiency factor is kept constant and symmetric.

Table 3 presents the tested cases where the value ofKa is considered constant. Smaller values of this co-efficient were not considered because the effect of theacceleration in the overall behaviour of the equationwould be very small. On the other hand, bigger valueswere tested and all of them reproduced higher (unreal-istic) evolution rates than those observed in the mea-surements. Therefore it was considered better not toinclude them in the analysis.

Top panel of Fig. 7 presents the predicted profiles forthese cases in comparison with the measured profile at

Fig. 5. Inclusion of acceleration: comparison of profiles (top) and differential changes (bottom) after 50 Tp for the numerical experiments includingfluid acceleration. (__) Measured; (…) simulation with no acceleration term and C=12(- -) simulation with Ka=0.01 and acrit =0.07; (-.-.-.)simulation with Ka=0.01 and acrit =0.09. (BP=Breaker point).

11A. Pedrozo-Acuña et al. / Marine Geology 241 (2007) 1–18

the GWK. From observation of this figure it is clear thatto get results close to those measured in the GWK, it isnecessary to keep a small value of acrit. The dash–dotline (Case Ma3—acrit =0.2 m/s2) gives a better des-cription for the size of the crest in the measured profile,although the rate of erosion below the SWL is notrepresented. It is interesting that for the simulations in-cluding the acceleration term in the sediment transportequation, there is a better agreement to be found thanwhen the same modification was utilised for the gravelbeach case. Despite the agreement found here being notentirely satisfactory, it shows that acceleration couldindeed be playing a role in the shaping of themixed beach.

To further confirm these results, the differentialchanges for these simulations are introduced in thebottom panel of Fig. 7. There, it becomes clear that forthe cases Ma1, Ma2 and Ma3, with a proportionate rateof change (in comparison with the measured profile), theinclusion of the acceleration term in the sediment trans-port equation improves the description of the crest at thetop of the beachface. However, this modification is not

useful to describe the rate of erosion between the co-ordinates 264 and 266 in the cross-shore direction.

A possible mechanism that is thought of as beingresponsible for the erosion at the coordinates 264–266 m, is bore collapse. During the experiments a clearplunging breaker was observed collapsing around thesecoordinates. Clearly, this phenomenon is not modelled inthe present approach. This must be another mechanismenhancing onshore transport.

The inclusion of an acceleration term in the bed-loadtransport equation shows that the influence of the ac-celeration in the swash zone of the mixed beach may beimportant. It was shown that when using this term inboth swash phases the comparison was improved, al-though the improvement was not entirely satisfactory.

5. Sensitivity to asymmetry in C

The bed-load sediment transport equation utilised inthe modelling approach, and given in Eq. (3), has anotherparameter that is possible to vary; the efficiency factor

Fig. 6. Non-dimensional sediment transport rate at three locations on the beach profile. Top panel — at 265 m at SWL; middle panel — at 266 m;bottom panel — at 266.75 m. (__) q without acceleration; (- - -) q with acceleration; (…) q due to the acceleration term only — gravel beach.

Table 3Selected acceleration thresholds acrit for each case

Case C-value f acrit(m/s2)

Ka

(kg s m−2)

Ma1 12 0.004 0.01 0.002Ma2 12 0.004 0.09 0.002Ma3 12 0.004 0.2 0.002

12 A. Pedrozo-Acuña et al. / Marine Geology 241 (2007) 1–18

given by theC-value. Indeed, the use of differentC-valuesfor each phase of the swash for the gravel beach case waspresented by Pedrozo-Acuña et al. (2006). They assumeddifferent empirical constants (C) on the basis that differentvalues are necessary for linking measured velocities andsediment transport in the two phases of the swash. Thiswas attributed to inherent differences in bed-shear stressesin the presence of accelerated flows in the uprush and theeffect of infiltration, as they are likely to be important inmodifying the flow above the beachface.

Moreover, a violent bore collapse from the plungingbreaker was also evident close to the shoreline. Thismechanism is believed to be an important factor in in-creasing the uprush sediment carrying capacity. Wesuggest here that the C-value can be used also toparameterise these effects. This section presents asensitivity analysis of the morphological changes to theC parameter for the mixed beach case, where infiltration

effects on the beachface are greatly reduced. The numer-ical tests are summarised in Table 4.

Fig. 8 shows the simulated profiles for the mixedbeach. These results are compared with the measuredprofile in the GWK after 500 Tp. The figure clearlyshows that increasing the value of the Cuprush does notimprove the agreement between model and experimentevolution of the beach profile, as was seen in the gravelbeach case (Pedrozo-Acuña et al., 2006). Moreover, allthe simulations show accretion above the SWL. This is

Fig. 7. Inclusion of acceleration: comparison of profiles (top) and differential changes (bottom) after 500 Tp for the numerical experiments includingfluid acceleration. (__) measured; (…) simulation Ma1; (- - -) simulation Ma2; (-.-.-.) simulation Ma3 — mixed beach.

Table 4Numerical tests for the sensitivity analysis of the asymmetry of C-valuesduring swash (mixed beach results)

Case Cuprush Cbackwash f

4 8 4 0.0045 12 4 0.0046 19 4 0.004

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further confirmed in the differential changes (Fig. 8,bottom). The lack of a berm at the top of the profile isclearly identifiable. Again, the physical mechanismbelieved to be responsible for this transport is the borecollapse on the beachface.

6. Discussion of swash investigation

Although the ad hoc approach taken here is not assatisfactory as investigating the individual processesdirectly, we believe that such a pragmatic approachovercomes the inherent difficulties in observing theseprocesses. Indeed we argue that the identified processesthemselves such as friction and infiltration are aggre-gated macro-scale representations of still smaller andmore complex processes. We argue that the use of amodel of the appropriate level of complexity to theavailable measurements has greater validity.

This paper extends the work of Pedrozo-Acuña et al.(2006) to consider a comparison of the relative effects offriction and the sediment mobility factor C on the

evolution of beach profiles of both gravel and mixedbeach experiments. The justification for using differentC-values for each phase of the swash for the gravelbeach is that this encapsulates several processes: theeffects of infiltration; the strong asymmetry in acceler-ation in the run-up compared to the backwash (as statedby Nielsen); the differences in the effective friction foreach phase of the swash; and the effect of the plungingbreaker on striking the beachface. It was found that theasymmetry in the C-value is less able to mimic the pro-cesses that are taking part in the mixed beach evolution.Therefore, it is less useful for this case than for the gravelbeach, which showed a better agreement.

Fig. 8. Numerical experiments on C-value asymmetry (higher C during the uprush) Case 4 (-.-.-); Case 5 (- - -); Case6 (…). Results for mixed beach.(top — profile evolution; bottom — differential changes).

14 A. Pedrozo-Acuña et al. / Marine Geology 241 (2007) 1–18

It was found that acceleration may be one of theprocesses that emphasises onshore transport on coarse-grained beaches. Nevertheless, it is clear that within themodelling framework used in this work, accelerationalone could not be the cause of the observed berm growthduring the GWK tests.

Therefore, on gravel beaches there may be more thanone process having an influence on the predominance ofonshore sediment transport.

At this stage it might be necessary to adopt a differentand possibly more complex modelling approach. Alter-natively it may still be appropriate to pursue modelling atthis level of sophistication but include other significantprocesses in the framework. Several other such processescan be identified.

The wave impact on the beachface from a plungingbreaker is just such a process that was evident during theexperiments. The effects of this are clear in the erosionclose to the still waterline in Figs. 4 and 5. During theexperiments a plunging breaker was consistently ob-served at these coordinates.

Clearly, this process will increase sediment mobilisa-tion and transport in the onshore direction. Interestingly,from the observations at the GWK, it could be establishedthat the “impact point” for the approaching waves isexactly where the bar is generated by the model. This maysuggest that it is necessary to modify the sedimenttransport equation to consider the impact of a plungingbreaker as a mechanism for stirring and mobilising thesediment. It could be argued that this process, in com-bination with the strength reduction of the backwash flow(due to the high infiltration) contributes to the bermgrowth at the top of the profile. For the gravel beachwheremore infiltration is present, it was found that the crestheight is higher than that observed in the mixed beachcase with less permeability.

This reasoning is supported by the sediment sortingthat was observed after the wave action on the mixedbeach. Fig. 9 shows the sediment characteristics alongthe profile. It can be seen that the generated berm wascomprised of only coarse material, whereas a layer of adifferent sediment mixture was formed above the

Fig. 9. Sediment redistribution on the mixed beach profile following wave action.

15A. Pedrozo-Acuña et al. / Marine Geology 241 (2007) 1–18

original mixture that was utilised for the beachconstruction. The observed layer of sediment wasfound to be present all along the profile. This can beseen in Fig. 10 which shows the trench that was exca-vated in order to observe the new sediment distribution.It was observed that the material below this layer was

Fig. 10. Excavated trench along the mixed beach profile.

firmly compacted; this is believed to be the reason whythe mixed beach profile was less mobile.

Another process that was neither simulated nor dis-cussed is the influence of low-frequency (long-wave)motion on swash transport. It iswidely held that interactionbetween gravity and infra-gravity waves can yield somecomplex feature of the ensuing sediment transport in thesurf and swash zones of sandy beaches. An example is thepossibility of bar formation due to bound long-waves(O'Hare and Huntley, 1994). Recently Karunarathna et al.(2005) have demonstrated the importance of infra-gravitywaves on swash motion, stating that on steep beaches theswash motion is forced in both infra-gravity and gravitybands. The implications of this for sediment transport onsteep beaches need further examination.

7. Conclusions

There are several observations that can be drawnfrom this investigation. These are:

We have shown that it is possible to use a higher-orderBoussinesq model with a moving shoreline boundarycoupled with a bed-load transport model to investigate theprocesses that control coarse-grained beach evolution.

Bottom friction was identified as a significant factoraffecting the predicted profiles. It was shown that theuse of different friction factors in each phase of theswash (uprush and backwash) improves prediction ofthe beach profiles. Numerical results in both beacheswere found to support the conclusion by Cox et al.(2000) and Conley and Griffin (2004). Therefore, theuse of a higher friction factor f during uprush, improvesthe simulations of morphological changes.

The variation of friction by itself was not able toreproduce the main features of the measured profiles. A

16 A. Pedrozo-Acuña et al. / Marine Geology 241 (2007) 1–18

plausible reason to explain this is that more mechanismsapart from friction are having an important role in theoverall response of coarse-grained beaches.

With regards to the study of the acceleration effectson simulated profiles, different results were obtained foreach of the cases studied. For the gravel beach, thenumerical results showed that including an accelerationterm in the sediment transport equation, did not producea better prediction of beach evolution as observed in theGWK data. This result was reasoned to be a conse-quence of the limitations in the modelling approach,thus the wave equations are derived for impermeablebeds, which neglects the infiltration effects on the flowabove the beachface.

On the other hand, in the mixed beach case, sim-ulations including the acceleration term in both swashphases, showed a better prediction than for the gravelbeach. This confirms that infiltration and its consequentreduction of the backwash phase, is not as significant inthe mixed beach as it was in the gravel beach. Thus for acoarse-grained beach which is not very permeable, theuse of an acceleration term on the sediment transportformulation improves prediction of the beach profileresponse to the wave action.

The use of dissimilar values of the sediment transportefficiency (C-value) in the uprush and backwash wasalso attempted for the mixed beach case. The numericalresults obtained by the adjustment of the C-value werefound to be less significant than those observed in thegravel beach. By comparing the numerical results fromboth beaches, it was possible to further discuss theinfiltration effects on coarse-grained beaches.

In the case of the gravel bed, differences between thepredicted profiles from setting non-identical C-valuesand friction factors for the swash phase are believed tobe linked jointly to the infiltration effects on the flowabove the beachface, the extra sediment mobilised in theuprush by the breaking process, and the accelerated flowin the uprush.

In the case of the mixed sediment this asymmetry inC is important because infiltration is less, the bed is lessmobile because of armouring and the horizontal extentover which the profile changes is greater.

Other processes are believed to be significant but are notspecifically encapsulated by the asymmetry in theC-value,the change of friction factor during the swash or theinclusion of the acceleration term in the sediment transportequation. These include the impact of the plunging breakerand its role in stirring up sediment from the bed to be carriedonshore. This was observed in both the beaches in theGWK.Moreover, this limitation of the model was manifestin the simulated profiles for all the mixed beach results,

where the observed erosion below the SWL was notreproduced by any of the modifications implemented. Inthe gravel beach case, this limitation was not evident as thechange in the C-value for the swash phase seems todescribe all the beach change. This could be due to thenarrowness of the swash zone observed on this beach,whichmakes infiltration and the bore collapse effects closerto each other.

Acknowledgments

Adrian Pedrozo-Acuña extends grateful acknowledge-ment to theMexican Government for the support receivedunder the scholarships programme from Consejo Nacio-nal de Ciencia y Tecnología (CONACYT)— ScholarshipNo. 161279. The large scale tests in the Large WaveChannel (GWK) of the Coastal Research Centre (FZK) inGermany were supported by the European Communityunder the Access to Research Infrastructures action of theHuman Potential Programme (contract HPRI-CT-1999-00101). The authors are very grateful to Dr. Ashwini K.Otta for the several useful discussions during the initialstage of this work. The authors would also like toacknowledge the assistance and support provided by staffof the FZK (GWK flume) in Hanover. In particular thanksare due to Dipl.-Ing. Joachim Grüne.

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