laboratory investigation of beach scarp and dune recession

LUND UNIVERSITY

PO Box 117221 00 Lund+46 46-222 00 00

Laboratory Investigation of Beach Scarp and Dune Recession Due to Notching andSubsequent Failure

Erikson, Li; Larson, Magnus; Hanson, Hans

Published in:Marine Geology

DOI:10.1016/j.margeo.2007.04.006

2007

Link to publication

Citation for published version (APA):Erikson, L., Larson, M., & Hanson, H. (2007). Laboratory Investigation of Beach Scarp and Dune Recession Dueto Notching and Subsequent Failure. Marine Geology, 245(1-4), 1-19.https://doi.org/10.1016/j.margeo.2007.04.006

Total number of authors:3

General rightsUnless other specific re-use rights are stated the following general rights apply:Copyright and moral rights for the publications made accessible in the public portal are retained by the authorsand/or other copyright owners and it is a condition of accessing publications that users recognise and abide by thelegal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private studyor research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal

Read more about Creative commons licenses: https://creativecommons.org/licenses/Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will removeaccess to the work immediately and investigate your claim.

https://doi.org/10.1016/j.margeo.2007.04.006

https://portal.research.lu.se/en/publications/85a37680-c0d4-45f0-a015-9364a67647fa

https://doi.org/10.1016/j.margeo.2007.04.006

(2007) 1–19www.elsevier.com/locate/margeo

Marine Geology 245

Laboratory investigation of beach scarp and dune recessiondue to notching and subsequent failure

Li H. Erikson ⁎, Magnus Larson, Hans Hanson

Department of Water Resources Engineering, Lund University, Box 118, 221 00, Lund, Sweden

Received 9 May 2006; received in revised form 31 March 2007; accepted 9 April 2007

Abstract

Analytical models to calculate notch development and subsequent mass failure of dunes are presented. The notch evolution modelis based on a transport equation for sediment from the dune and the sediment volume conservation equation, whereas the models ofmass failure are derived using elementary engineering statics and soil mechanics. An empirical transport coefficient in the modeldescribing the notch growth rate is found to be related to the hydrodynamic forcing at the dune normalized by geotechnical parametersdescribing the resistive strength of the dune. Two modes of mass failure are modeled whereby the overhang generated by the removalof material from the dune foot (notching) slides downward or topples over following the development of a tensile crack some distanceshoreward of the maximum notch depth. The accuracy of the notch evolution and mass failure models are assessed by comparingcalculated recession distances against measurements from a small-scale laboratory experiment.© 2007 Elsevier B.V. All rights reserved.

Keywords: dunes; beach scarps; tensile strength; notching; wave impact

1. Introduction

The aim of this study was to investigate erosion me-chanisms of beach sand formations such as scarps anddunes undergoing wave attack. Although the physicalscale of scarps and dunes can differ by an order ofmagnitude, the mechanisms are often similar. Recessiontypically occurs as a result of a series of steps (Carteret al., 1990). As waves begin to attack a scarp or dune,erosion causes an increased face steepness. Continuedwave attack is focused at the foot of the formation andtypically causes undercutting or removal of the scarp or

⁎ Corresponding author. 2601 Hambleton Lane, Santa Cruz, CA95065, USA. Tel.: +1 831 476 8582.

E-mail address: [email protected] (L.H. Erikson).

0025-3227/$ - see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.margeo.2007.04.006

dune foot (notching). The undercutting leads to a state oftension in the upper part with visible tension cracksparallel to the face. This can readily be observed in beachscarps undergoing attack by swash waves, but is also acommon mechanism of dune recession as pointed out byCarter and Stone (1989). The resulting overhang gene-rated by the tensile crack alters the balance of forces andeventually succumbs to a mass failure whereby a slab ofsand slides downward or topples over leaving a verticalface. The sediment from the mass failure lands at the footof the scarp or dune and provides temporary protection.If wave attack continues, the failed slab material is re-moved by wave action and further undercutting occursfollowed by mass failure, continuing the cycle.

Because dunes are of greater importance than beachscarps, much experimental and theoretical work has

mailto:[email protected]

http://dx.doi.org/10.1016/j.margeo.2007.04.006

Fig. 1. Conceptual sketch of shear- and beam-type failure mechanisms.

2 L.H. Erikson et al. / Marine Geology 245 (2007) 1–19

focused on dune, as opposed to scarp, recession. Mostpredictive models of dune recession focus on stormsurges in conjunction with assumed post-storm profiles.An alternative approach, whereby no post-storm profileis assumed, but where erosion from the impact ofindividual waves is used to determine the amount ero-ded, was proposed by Fisher and Overton (1984) andNishi and Kraus (1996). The frequency and intensity ofwave impacts, multiplied by an empirical transport coef-ficient, determine the total amount eroded. Wave impacttheory has been empirically validated and successfullyemployed to simulate dune erosion in small and largewave tank studies and in the field (Fisher and Overton,1984; Fisher et al., 1986; Overton and Fisher, 1988;Overton et al., 1994; Nishi and Kraus, 1996); however,little has been done in developing a predictive equationfor the transport coefficient. Furthermore, the modelinghas focused on describing the gross features of duneerosion and no attempt has been made to resolve thedetails of the process.

A small-scale wave tank experiment was conductedin this study to obtain a better understanding of theshort-term time-evolution of dunes during erosion byindividual waves. The experimental data should be di-rectly applicable to scarps at natural scale, and assumingthat the scaling model described later is valid, the resultsshould also hold for small dunes. Recession of theexperimental dunes was observed to occur as the resultof a two-step process described by notching and slum-ping or notching and toppling, similar to the descriptionof dune recession presented by Carter et al. (1990). Inthis paper, a distinction is made between dune erosionand recession, such that erosion refers to removal ofmaterial from the lower part of the dune (notching), andrecession refers to the landward movement of the entiredune face (mass wasting). Two modes of mass wastingwere observed and are described as shear- and beam-type failures (Fig. 1). Shear failure is hypothesized tooccur when the weight of the overhang (resulting fromnotching) exceeds the shear strength of the sediment andit slides downward as depicted in Fig. 1a–b. Beam-typefailures occur when tension cracks develop some dis-tance landward of the dune face and the pending failureblock either rotates or slides downward (Fig. 1c–d).

Development of the governing equations and asso-ciated analytical solutions are described in the followingsection. The first equation estimates the rate of sedimentremoval (erosion) from the dune foot by undercutting ornotching, allowing for a description of the temporalgrowth of the notch. Laboratory observations regardingthe shape of the notch at the dune foot were incorporatedinto the model and, thus, may only be representative

of toe erosion from the impact of fully broken bores. Thesecond set of equations estimates the maximum reces-sion distance based on two different observed massfailure modes. The model shows that the recession dis-tance and time at which failure occurs is related to thenotch depth. Following a description of the laboratoryexperiment, a summary of the experimental data is pre-sented after which model calibration and validation arediscussed.

2. Theoretical development

2.1. Dune toe erosion by notching

2.1.1. Governing equationsAssuming that the geometry of the notch can be

specified at any instant in time, notch progression canbe calculated if the volume eroded due to the impact ofwaves can be predicted. Fisher and Overton (1984)proposed a model for dune erosion where the impactforce of individual waves is linearly related to thevolume of sand eroded or, as presented by Nishi andKraus (1996), linearly related to the weight of erodedsand,

DW ¼ C E F ð1Þ

3L.H. Erikson et al. / Marine Geology 245 (2007) 1–19

where CE is an empirical coefficient, F the forceexerted by the impact of waves on the dune, andΔW=ΔVρs (1−p)g, the weight of the eroded volumeΔV, ρs, the specific soil density, p porosity, and gthe acceleration due to gravity. The impacting forceis derived by considering a bore propagating upslopeand changing its rate of momentum as it impacts thedune face (Cross, 1967),

Fo ¼ qu2inchinc ð2Þwhere ρ is the water density, uinc the incident bore speed,and hinc the bore height. The incident bore speed in Eq. (2)can be estimated with (Cross, 1967; Miller, 1968),

uinc ¼ Cu

ffiffiffiffiffiffiffiffiffighinc

pð3Þ

where Cu is an empirical coefficient between 1 and2 related to the frictional resistance. CombiningEq. (2) with Eq. (3) and adding the term Δt/T torepresent the number of bores with wave period Timpacting the dune during a given time, Δt, yieldsa total swash force proportional to uinc

4 :

F ¼ qðu4inc=ðgC2uÞÞðDt=TÞ: ð4Þ

Combining Eqs. (1) and (4) and doing some rear-ranging, and letting Δt approach zero, gives the rate oferosion in terms of the incident bore velocity,

dVE

dt¼ CE

C2u

11� p

qqs

u4incg2T

¼ Kd

1� pqqs

u4incg2T

ð5Þ

where VE is the eroded volume and Kd was introducedto replace the empirical relationship, CE/Cu

2.In order to analytically predict the notch depth with

Eq. (5), a description of notch geometry is necessary.Cross-section profiles and evolution of the notches inthe laboratory experiments are shown in Fig. 2. Theprofile following the impact of every other bore isshown in Fig. 2a whereas the profile following everyother wave train is shown in Fig. 2b. A summary of theexperimental cases, including wave conditions, is fur-ther discussed in Section 3. The profile plots are dis-cussed and presented here to clarify the assumptionsmade regarding the notch shape.

The plots in Fig. 2 show that the slopes along thebottom of the notch and in front of the dune are approx-imately constant and more or less parallel to the initialforeshore slope (tan β). This similarity in slope suggeststhat the transport mechanisms controlling the bottomprofile are the same in these two regions, that is, shear-stress-related transport. Schematically, the upper portion

of the notch may be described by a linear or a powerfunction, so that following the impact of each bore(Fig. 2a) or set of waves (Fig. 2b), the entire notchmaintains a triangular shape, or an upper part that iseither convex or concave. Referring to the definitionsketch in Fig. 3 and assuming a power- or triangular-shaped notch, the geometry is given by,

xðzÞ ¼ b1ðz=BÞm 0 V z V BxðzÞ ¼ b1ð�z=ða� BÞÞ 0 z z z B� a

ð6Þ

where a is the height of the notch at the dune face, b1 themaximum width of the notch, B the notch height mea-sured from the elevation of the maximum notch depth,m an exponent describing the shape of the upper part ofthe notch (mb1 gives a convex shape whereas mN1gives a concave shape), and x and z are coordinates withtheir origin at the maximum notch depth, located atelevation zo. Note that for m=1, the upper part of thenotch is linear so that the notch shape is triangular. In thisschematization, notch evolution will always progress sothat the lower part of the notch has a slope in agreementwith the foreshore slope. In a predictive mode, a isspecified from thewave properties and b1 is the unknownvariable employed to describe the notch evolution.

2.1.2. Notch evolutionThe volume per unit length in the transverse direction

of a triangular-shaped notch is,

VE ¼ 0:5ab1 ¼ 0:5CaðR� zinÞððzo � zinÞ=tanbÞ ð7Þ

where R, zo, and zin are referenced from the still waterline (SWL) or still-water shoreline (SWS), zo is theheight of the most landward point of the notch, zin theforeshore elevation corresponding to the dune face, andb1= (zo− zin/tan β) from geometry. It was assumed thatthe top of the notch corresponds to the runup limitincluding additional height from the reflected boreimpacting the dune face such that a=Ca(R− zin), whereR− zin is the estimated incident bore height, and Ca is anempirical constant greater than one introduced to ac-count for the increased contact area between the boreand dune face (the validity of this assumption is dis-cussed in the results section). The more general case of apower-shaped notch is not discussed here, although ananalytical solution for this case has also been derived.

The rate of change of VE can be expressed as:

dVE

dt¼ dVE

d zo

dzodt

¼ 12CaðR� zinÞ

tanbfs

d zodt

: ð8Þ


Combining Eqs. (5) and (8), along with an estimateof the bore velocity at the dune face based on ballisticstheory uinc

2 =us2−2gzo (us=bore velocity at the SWS),

and a first-order estimate of the maximum runup heightexcluding friction (Shen and Meyer, 1963), R=us

2/2g,yields a governing equation describing the evolution ofzo:

dzodt

¼ 8Kd

1� pqqs

tanbfCaðR� zinÞ

ðR� zoÞ2T

: ð9Þ

This equation is separable and may be solvedemploying the initial condition that zo= zin at timet=0. With some rearranging and noting the geometricrelationship between b1 and zo, previously discussed,the growth of the maximum notch width, assuming thatCe t/T is small, may be expressed as a linear growth withtime such that,

b1 ¼ tðCaðR� zinÞÞ=tanbf bd ½Cet=T � ð10Þwhere

Ce ¼ ½ð8KdÞ=ð1� pÞ�d ½q=qs�d tanbf : ð11Þ

2.2. Mass failure

2.2.1. Shear-type mass failureShear failure occurs when the weight of the over-

hang (resulting from notching) exceeds the shearstrength of the sediment and it slides downward asschematized in Fig. 1a–b. A definition sketch is shownin Fig. 4a, where the shaded part of the dune representsthe pending failure slab. Assuming a triangular-shapednotch the critical case is examined so that the forcesare balanced and the block is in a state of static equi-librium immediately before failure (Thorne and Tovey,1981). The forces are resolved vertically, and assum-ing a unit width, the weight, Ws, of the failure slab atfailure is,

Ws ¼ gb2shrðA� B=2Þ ð12Þ

where γ is the weight of soil at natural moisturecontent, b2shr is the width of the failure slab equalto the critical notch depth, b1cr, and, assuming noaccumulation of sand on the foreshore, A=Do− b1 tanβ, where Do is the initial dune height, and B=A−Ds orB= a− b1 tan β.

The force Fs generated by the shear stress that ba-lances W just before collapse is given by,

Fs ¼ scrðDs þ B=3Þ ð13Þ

where τcr is the critical shear stress when mass fail-ure occurs, and Ds +B/3 is the length over which theshear strength, maintaining the overhang, acts. Thelast term, B/3, reflects the assumption that the shearstress of the triangular-shaped part of the overhang actsalong the length from the top of the said shape to itscenter of gravity. If the critical shear stress is assumedto equal the shear strength of the soil at failure, theoverhang is expected to fall when its weight exceedsthe shear strength of the soil, so that from Eqs. (12) and(13):

b2shrzðscrðDs þ B=3ÞÞ=gðA� B=2Þ: ð14Þ

The shear strength at failure in a non-cohesivesoil can be described by the revised Coloumb equation(Öberg, 1997),

scr ¼ rntan/ Vþ Sðua � uwÞtan/ V ð15Þ

where σn is the normal stress on the failure plane, ϕ′ theinternal friction angle, (ua−uw) the soil suction, and Sthe saturation level (0bSb1). On a vertical failureplane, the normal stress (σn) is zero leaving a shearstrength that is only a function of the matrix suction,saturation level, and internal friction angle. Soil suctionis often ignored in many engineering slope stabilityanalyses and instead accounted for by using a fictitiouscohesion. In this study, soil suction is explicitlyaccounted for and it is this parameter that enableseroding dunes to exhibit the frequently observed verticalfronts (Carter et al., 1990).

Soil suction consists of two components: matrixand osmotic suction. Matrix suction is a function of thepore volume between sediment grains and amount ofwater contained in the soil matrix. In the case of awetted dune, the pores are occupied by both water andair. At the interface between air and water, the dif-ference between the inward attraction results in aninterfacial tension which provides strength and sta-bility to the dune. The strength and stability increaseswith increasing water content (by weight) up to about10 to 15%, and decreases as increased water contentreduces, and ultimately eliminates, the surface tensioneffect of the pore water. Osmotic suction is related tothe presence of chemicals, such as salt, in the porewater, which should be accounted for in the case ofdunes in contact with saltwater. In this study, modelswere developed and tested against laboratory studieswith freshwater so that osmotic suction was not likelypresent and not accounted for. Soil suction and ma-trix suction are used interchangeably throughout thispaper.


2.2.2. Beam-typemass failure or failure by tensile crackingThe second mode of mass failure observed in the

laboratory experiments was beam failure. This type offailure appears to occur by initial tensile cracking at

Fig. 2. a. Notch development of experimental dunes subjected to fully brokeshown. b. Notch development of the experimental dunes subjected to impac

the top of the dune, followed by either shear failurealong an internal failure plane or overturning of thepending failure block due to the moment generated bythe overhang (Fig. 1c–e).

n solitary waves. Profiles following the impact of every other bore areting wave groups. Profiles following every wave packet are shown.

Fig. 2 (continued).


It is hypothesized that tensile failure (vertical tensioncrack that forms some distance back from the dune faceand landward of the notch depth, b1) occurs due tolateral earth pressures and the weight of the overhang.The overhang causes a moment and a bending stress atthe top of the dune (with the bending stress beinganalogous to that generated by a cantilever beam). Whenthe total stress exceeds the tensile strength of the soil,cracking begins and migrates downward until itintersects with an internal failure plane (Fig. 1c) that istypically considered to be at the angle α≈45°+ϕ′/2from the horizontal in non-cohesive soils (Terzaghi and

Peck, 1948). Here, failure continues if the force gene-rated by the gliding weight of the failing slab exceedsthe shear strength of the soil, τcr along the failure planeso that the failing block slides downward and lands withthe top surface pointing upward as depicted in Fig. 1d. Ifthe shear strength is sufficient to keep the pendingfailure block from sliding downward, failure might stilloccur if the overturning moment of the failing blockexceeds the resisting moment and the block topples overas depicted in Fig. 1e.

Tensile cracking is well recognized in the field ofslope stability analysis. Lohnes andHandy (1968) applied

Fig. 3. Definition sketch of dune erosion by notching.


Rankine–Bell earth-pressure theory along with equationsdeveloped by Terzaghi (1943) and presented a model topredict the recession distance of steep cohesive slopeswithout notching. Clean sands however, are cohesionlessand as such a new approach based on the methodpresented by Lohnes and Handy (1968) was developed topredict failure of non-cohesive steep slopes affected bynotching.

The geometric shape of a pending failure block isshown in Fig. 4b along with required dimensions. Theslope of the internal failure plane is estimated to be at αfrom the horizontal (αc=45°−ϕ′/2 from the vertical)and is assumed to start at the maximum notch depth. Inthe figure, it can be seen that the recession distance dueto beam failure, b2beam, can be found from geometry,

b2beam ¼ b1 þ b3 ð16Þ

where b3= (A−Zc) tan αc and Zc=depth of the tensilecrack. The notch depth, b1, can be found with Eq. (10)and assuming no foreshore accumulation, A=Do−b1 tanβ. Predicting the recession distance due to beam failureusing Eq. (16), then reduces to finding a tensile crack, Zc,long enough to intersect with the internal failure plane.

The depth of tensile cracking is estimated by assu-ming that the tensile stress decreases linearly with depthfrom the top surface and that the total tensile stressexceeds the soil tensile strength so that,

Zc ¼ Ztð1� ðrt=rssÞÞ for rssNrt ð17Þ

where σt is the soil tensile strength, σss the tensile stressat the surface, and Zt the depth of soil tension.

2.2.3. Tensile strengthThe tensile strength of the soil is assumed to be

uniform throughout the dune. Several theoretical andexperimental studies have shown that soil tensilestrength is closely related to water content and thatit increases non-linearly with increasing matrix suc-tion (e.g., Kim and Hwang, 2003). The equationsused to calculate tensile strength for this study arebased on the approach presented by Pierrat andCaram (1997) and relations developed by Schubert(1972) so that,

rt ¼ rpzððSc � SÞ=ðSc � Sf ÞÞþrczððS � Sf Þ=ðSc � Sf ÞÞ

where rpz ¼ 0:26½ð1� eÞ=ðeD50Þ�and rcz ¼ S½7ððð1� eÞ=eÞð0:0727=D50ÞÞ�

ð18Þ

and e is the porosity, D50 the median grain sizediameter, S the in situ saturation level, Sc (=90%) theupper saturation limit for the “funicular state”, and Sfthe upper saturation limit for the pendular state whenliquid bridges form (34%, Flemmer, 1991).

2.2.4. Tensile stress and depth of tension due to lateralearth pressures

The tensile stress at the top surface, σss, is estimatedto be the sum of the lateral earth pressure and thebending stress caused by the overhang

rss ¼ jrLEPs þ rbsj ð19Þ

Fig. 4. Definition sketch of (a) shear-type and (b) beam-type massfailure.


where σLEPs is the tensile stress due to lateral earthpressure at the surface, and σbs the bending stress at thesurface.

In a non-cohesive, unsaturated soil matrix, with orwithout an unbraced cut but without an overhang, thestress due to lateral earth pressures at any depth, d, is (Luand Likos, 2004),

rLEP ¼ ðgdÞKu � Sðua � uwÞð1� KuÞ ð20Þ

where Ku is a coefficient of earth pressure, and the firstterm on the right-hand-side is the compressive pressuredue to the self weight of the overlying material (usingRankine earth pressure theory), and the second termrepresents tension caused by matrix suction. Assumingthat the suction stress is constant with depth and the soilis homogeneous, a conceptual diagram of the two com-ponents is shown in Fig. 5.

The Rankine earth pressure (shown as a stressincreasing linearly with depth) is similar to fluids inthat the vertical pressure due to a soil layer of depth dcan be estimated with γd. Unlike fluids, however, soilcan support shear loads, and thus, the horizontal pres-

sure is not the same as the vertical pressure. A coef-ficient of earth pressure, Ku, dependent on the angle ofinternal friction ϕ′ where Ku=tan

2 (45°−ϕ′/2), is usedto calculate the horizontal pressure from the verticalpressure. The coefficient of earth pressure is used in thecalculation of the tensile stress due to matrix suction aswell as in Eq. (20).

The combined effect of the Rankine earth pressureand suction stress, shown in Fig. 5, results in a linearlateral-earth-pressure profile with the upper portion intension and the lower in compression. It is within thezone of tension that tensile cracking is expected. Thedepth of the tension zone can be found by normalizingEq. (23) with the vertical stress. Assuming that thevertical stress at depth d can be represented by γd, acoefficient of active earth pressure for unsaturated soil,Kau, can be defined (Lu and Likos, 2004),

Kau ¼ rLEP=gd ¼ Ku � ½ðSðua � uwÞÞ=gd�d ½1� Ku�: ð21Þ

The resulting earth pressure profile changes fromtension to compression when Kau is zero and d=Zt sothat from Eq. (25),

Zt ¼ ½ðSðua � uwÞÞ=g�d ½1=Ka � 1�: ð22Þ

It is also clear from Fig. 5 and Eq. (23) that the lateralearth pressure at the surface (d=0) is solely due to thematrix suction so that the tensile stress due to the lateralearth pressure at the surface is,

rLEPs ¼ �Sðua � uwÞð1� KaÞ: ð23Þ

In addition to the tensile stress caused by the matrixsuction, the bending moment of the overhang con-tributes to the total stress at the top dune surface. It isassumed that the soil is homogeneous and behaves in alinear-elastic manner, so that the dune is in tension at thetop of the overhang and in compression at the bottom.Based on simple beam theory, the maximum tension dueto bending at the top surface is,

rbs ¼ �ðMb AÞ=ð2IoÞ ð24Þ

where Mb is the per unit-length bending moment of theoverhang, and Io the moment of inertia of the lengthwisecross-section (defined by A times the unit length; Io=A3/12). The bending moment (Mb) of the overhang is(per unit length) Mb=γAcXc, where Ac is the cross-section area of the overhang, and Xc the centroid of theoverhang. With the assumption of a triangular notch, thearea of the cross-section is Ac=b1(Ds+1/2(A−Ds)), and

Fig. 5. Conceptual diagram of active earth pressure components of unsaturated non-cohesive soil (adapted from Lu and Likos, 2004).


the horizontal center of gravity with respect to an originat the maximum notch depth is:

Xc ¼ ðb21ð2Ds þ AÞÞ=6Ac: ð25Þ

3. Experimental design and procedure

The experiment was conducted in a 27 m section of atank, equipped with a piston-type wave paddle, at theEngineering Research and Development Center, U.S.Army Corps of Engineers in Vicksburg,Mississippi. Thetank measured 0.91 m in width, and 0.91 m at maximumdepth. The set-up of the experiment is shown in Fig. 6.Uniform sand with a median grain size diameter (D50) of0.13 mm, a fall speed of 1.4 cm/s, and an internal (un-compacted) friction angle of 33° was used in theexperiment. Specific soil density was back-calculatedfrom the fall speed parameter reported in Hughes andFowler (1990) and an estimated kinematic viscosity atT=15 °C (ν=1.12E−6 m2/s) yielding ρs=2690 kg/m3.The beach profile beneath the still water line was gradedto represent an equilibrium profile (Dean, 1977). Thebeach slope from just below the still water line to thedune was 1:15. An idealized shape, in the form of avertical face (with a height ofDo≈0.21 cm), was used torepresent a dune undergoing erosion (e.g., Edelman,1968). Foreshores and dunes were constructed withwetted sand, and placed and compacted with hand tools.

For the case of scaling the experiment to a prototypedune condition, the design of the experiment was de-pendent on two criteria. First, the experimental set-uphad to be configured so that solitons or other confusedwave forms were not generated.

This was done by ensuring a depth to wavelengthratio greater than 0.09 as suggested by Galvin (1972).Secondly, the scaling criteria presented by Hughes andChiu (1981) and Hughes and Fowler (1990) had to bemet. The scaling criteria was specifically developed forphysical modeling of coastal dune erosion in movable-

bed models based on inertial forces, represented byturbulent shear stress, and gravity in the direction of theprincipal flow. Two criteria are simultaneously satisfied.The first is the Froude number representing the inertia togravity forces of the hydrodynamics. The second is thefall speed parameter describing the morphologicalresponse to the hydrodynamics. For an undistorted sca-ling model, the resulting relationship is Nw ¼ Nt ¼

ffiffiffiffiffiNl

pwhere N represents the prototype-to-model ratio of thesubscribed parameter, w is the fall speed parameter, t istime, and l is length. With a length scale factor of 10:1(time-scale factor of 3.2:1), the corresponding prototypeis a small dune on the order of 2 m in height with a sandsize of D50≈0.41 mm.

Parameters and wave conditions employed in the ex-periments are listed in Table 1a and b. Two wave con-ditions were employed: single solitary waves (Table 1a)and wave trains (Table 1b). Offshore wave heights, Ho,ranged from 6 to 10 cm for the solitary wave experiments(0.6 m to 1 m prototype). The wave trains were generatedusing two closely spaced harmonics of equal amplitude,resulting in wave trains with increasing and subsequentlydecreasing wave heights. The wave trains consisted of tenwaves ranging in offshore wave heights from about1.5 cm–2.5 cm to a maximum of 10 cm, 15 cm, or 20 cm,(1 m to 2 m prototype) depending on the test case. Thewave period was 2.2 s for all wave trains (7 s prototype).Two experimental phases (A and B) were conducted foreach set of wave conditions whereby the water level wasraised enabling higher bores to impact the dunes. All thesolitary waves broke as plunging breakers (misprint inErikson et al., 2003) while the waves of the wave trainsbroke as either spilling or plunging waves before propa-gating upslope and hitting the dune face.

Incident bore heights (hinc) were measured with themost landward swash gauge located 21 cm and 19 cmfrom the dune face for the solitary and wave train expe-riments, respectively. Measured average values are listedin the last columns of Table 1a and b. Video and still

Fig. 6. Laboratory set-up. A cross-section of the tank is shown in the upper plot and a plan view is shown in the lower plot.

10L.H.Erikson

etal.

/Marine

Geology

245(2007)

1–19

Table 1bExperimental design and wave conditions of wave group experiments

Case Foreshore length(cm)

Water level(cm)

H oa

(cm)Do

(cm)h̄ inc

(cm)

A19 40 56 9 20.7 2.5A17 14 20.8 2.8A18 18 22.7 4.1B12 10 58 9 20.8 3.6B15 14 22.4 4.4B14 18 22.3 4.5a Of the highest wave in the group.


camera images were used to measure profile changes.The cameras were positioned about 2 m from the targetso that nominal pixel resolution was better than 2 mmwith a horizontal field of view less than 1.5 m for eachcamera. A grid of control points was marked on theoutside of the glass tank wall (target plane) with an origindefined at the still water shoreline to allow post-pro-cessing rectification.

Dune and sediment properties were obtained withstandardized soil tests. A cylindrical corer with an 8 cmdiameter and height of 11 cm was constructed frommetal sheeting to collect in situ samples of the dunefollowing completion of nearly each experimental case.Samples were collected from the approximate middleof the remaining dune in the vertical and horizontaldirections.

Mass failure was observed to usually occur at thecenter of the dune or the far facing glass wall. In mostcases the dune adhered to the sidewall on the cameraside. This was probably due to the lighting which wasrequired for the cameras, but heated the glass wall andenabled the wet sand to form a strong bond. Notchevolution was quite even across the tank however, anddid not seem to be affected by the sidewalls. Initially,adherence of sediments to the sidewall at mass failurewas viewed as a drawback, however the occurrenceworks to the advantage of data collection in that it was,thus, possible to obtain the depth of the notch that formedat failure.

4. Experimental results

Experimental dunes were subjected to impactingwaves until they failed by mass wasting. The number ofbores or wave packets required for mass failure is listedin Table 2a and b along with the measured height of theoverhang prior to or at mass failure, Ds, measuredlengths from the top surface to the maximum notch

Table 1aExperimental design and wave conditions of solitary waveexperiments

Case Foreshore length(cm)

Water level(cm)

Ho

(cm)Do

(cm)h̄ inc

(cm)

A13 150 53 8 20.5 3.0A14 9 21.2 3.8A15 10 20.8 3.5B1 100 56 6 22.9 3.0B5 7 21.4 3.9B2 8 22.8 4.3B4 9 22.0 4.6B3 10 21.0 5.2

depth, A, maximum notch depth, b1cr, and total recessiondistance, b2. As was done in Table 1a and b,experimental cases are listed in order of increasingoffshore wave heights for each test phase. With theexception of A14 or A15, the number of bores requiredfor collapse decreases with increasing offshore waveheight, as would be expected if the dune properties weresimilar. The recession distance, b2, is equal to the criticalnotch depth, b1cr, for cases B5 and B3; these cases arehypothesized to have recessed by shear-type massfailure (Fig. 6a).

Calculated geotechnical parameters obtained from soilcores are also listed in Table 2a and b. Water contents (bymass, w) ranged from 14% to 23% and should thusprovide strengthening in the form of matrix suction. Thetotal weight of core samples was unfortunately not ob-tained for the experimental cases where dunes weresubjected to wave trains and therefore, the unit weight atnatural moisture content, γ, porosity, p, degree of satu-ration, S, and related matrix suction are not available forthese experiments. These values are listed for the solitarywave cases in the last four columns of Table 2a. Duneporosities (volume of voids over the total sample volume)were quite similar in the range of 0.38 to 0.41 suggestingthat the dunes were slightly to moderately compacted.Saturation levels, listed in the next to last column, werequite high (57%bSb89%) so that the effects of matrixsuction could be ascertained. Although not verified, therelatively high levels possibly represent storm conditionswhere elevated groundwater levels, intense rain, andwave splash cause increased water contents and saturationlevels of moderately compacted or consolidated sandformations.

In this study, the matrix suction was estimated withthe method described by Öberg (1997) whereby thewater retention curve (a plot of matrix suction vs. satu-ration levels) of a reference soil from Andersson andWiklert (1972) was employed. Approximate matrixsuction values, as read from the said curve, are listed inthe last column of Table 2a. The matrix suction varies

Table 2aGeometric and geotechnical parameters at failure for solitary wave tests

Case Bore # at collapse Ds

(cm)A(cm)

b1cr(cm)

b2(cm)

w(%)

γ(g/cm3)

p S(%)

(ua−uw)(kPa)

A13 38 14.7 15.6 5.1 8.8 15 1.85 0.39 64 2.57A14 23 14.7 16.0 4.9 9.9 23 1.92 0.41 89 2.07A15 26 13.2 15.8 5.5 7.8 15 1.84 0.39 62 2.58B1 25 14.3 18.4 3.9 8.0 19 1.80 0.43 70 2.35B5 18 11.4 16.5 5.4 5.4 20 1.96 0.38 88 2.19B2 15 14.3 17.6 5.2 8.0 14 1.82 0.39 57 2.67B4 13 8.8 16.0 5.1 8.9 20 1.87 0.41 78 2.27B3 10 9.1 17.2 5.0 5.0 16 1.79 0.41 60 2.57


only slightly (2.07 kPa to 2.67 kPa) as is typical of cleansands within the given saturation range.

5. Model results

Analytical models describing dune recession by ero-sion of the dune foot (notching) and subsequent massfailure, were developed in Section 2. In this section, thenotching model is first calibrated with experimental data.The accuracy of the mass failure model is then assessedwith data where dune erosion was measured on a wave-by-wave basis.

5.1. Notching model

Referring back to Fig. 2, an interesting result of theexperiment is that the notches are all quite similar inshape. Close inspection indicates that the lower parts ofthe notches are linear whereas the upper portions havemostly a concave curvature. Employing the power func-tion in Eq. (6), the m-exponent was found for each mea-sured notch shape using least-squares regression. Theresults are plotted against offshorewave steepness (Ho/Lo)in Fig. 7, where the deepwater significant wave heightwas employed for the wave trains, and the deep-waterwave lengths were approximated with Lo=1.56T

2 for thewave train data, and Lo ¼ ð4:36Þ= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

3Ho=4hop

for thesolitary wave data (ho=water depth, Synolakis, 1986).

Table 2bGeometric parameters at failure for the wave group experiments

Case Packet #at collapse

Ds

(cm)A(cm)

b1cr(cm)

b2(cm)

w(%)

A19 13 16.0 18.1 2.7 10.0 NDA17 11 12.2 17.2 3.4 10.9 18A18 6 15.6 20.5 3.3 8.4 21B12 6 14.7 18.1 4.8 9.0 22B15 3 15.0 19.8 5.3 8.0 18B14 3 13.0 18.5 6.0 8.2 19

ND: core sample not collected.

The coefficient of determination was 0.72 or better foreach individual notch. The overall mean value of the m-exponent (m̄) for the cases with solitary waves was 1.02(standard deviation (σ2)=0.25), whereas the overall meanfor the dunes subjected to wave trains was 1.55 (σ2=0.21). The notch shapes for the solitary wave cases thusclosely resemble triangles whereas the shape seems to bebetter represented with a power-type formulation for thecases with wave trains. The difference may be due to thetype of offshore waves although once the waves impactedthe dune face, they were all fully broken bores. Oneparameter that appears to affect the notch shape is thewave steepness as can be seen in Fig. 7 where the m-exponent increases with increasing offshore wave steep-ness, particularly for the dunes subjected to wave trains.

5.1.1. Inferred notch heightsThere are three unknowns that need to be determined

before Eqs. (10) and (11) can be used in a predictivemode to determine the notch depth, b1: the empiricalcoefficient, Ca, the runup height, R, and the transportcoefficient, Kd. The coefficient Ca was introduced toaccount for the area of physical contact between the duneface and incident as well as reflecting bores. Followinginitial contact between the leading edge of the swashwith the dune face, the bore front spouts upward in a jet-like manner while the additional volume of water offeredby the remaining incident bore continues to pile upagainst the dune face increasing the area of contact untilthe bore reverses direction. With the aid of still imagesobtained from video, incident bore heights immediatelybefore impact were obtained and compared to measurednotch heights. The results for the solitary waves areshown in Fig. 8 where it is apparent that the notch heightsare about 40% greater than incident bores in front of thedune face (Ca=1.37; R

2 =0.98). Data from the wavetrains were omitted in Fig. 8 because of the difficulty inmeasuring a single incident bore height in front of thedune while there are several bores present in the swashzone and interacting with one another.

Fig. 7. Mean values of the m-exponents describing the shape of the upper part of the notches plotted against the offshore wave steepness. Standarddeviations are shown with error bars.


The runup heights were predicted with the ‘ballisticsmodel’ described by Erikson et al. (2005), including theeffects of swash interaction for the wave trains. Therunup model requires initial velocity at the shoreline, us,which was calculated with Eq. (3) at the SWS but withhinc replaced with measured wave heights at the SWS.Inputs to the model were wave heights and arrival timesat the SWS. For the solitary wave data, there were onlysingle waves and thus only one wave height was re-quired. The coefficient Cu in Eq. (3) was set to 1.83 forthe wave trains (e.g., Cross, 1967) but increased to thetheoretical maximum of Cu=2.0 for the solitary waves.Increasing Cu from 1.83 to 2.00 for the solitary wavesincreased the runup height by about 18% and wasnecessary to better represent the inferred runup heightsof the solitary waves. Although not clear and in need offurther consideration, it is speculated that the greatercoefficient reflects the lower resistance subjected on thesolitary waves as compared to the oscillatory waveswhen they break and transition from propagating wavesto translating bores.

Fig. 8. Measured notch heights plotted against measured incident boreheights in front of the dune face for the solitary wave experiments.

Inferred notch heights with Ca=1.37, are comparedto measured notch heights as shown in Fig. 9. Thecoefficient of determination is 0.93 with a slope of 0.92,excluding case A19 for which the calculated runup didnot reach the dune. Only notch heights following impactof the first bore or wave packet were used as it could notbe assumed that the top of the notch from subsequentbores resulted from the most recent bore as opposed toprevious ones.

5.1.2. Transport coefficientIn order to describe the evolution of the notch depth

on a wave-by-wave basis, Eqs. (10) and (11) were mo-dified to yield,

b1n ¼ Kd8n½ðan=ð1� pÞÞðq=qsÞ� ð26Þ

where Ca (R− zin) was replaced with measured values,an, and n was introduced to represent the bore number.Measured notch depths, b1, were then plotted againstthe right-hand-side of Eq. (26) (excluding Kd) and Kd-

Fig. 9. Inferred versus measured notch heights, a.

Fig. 11. Notch erosion rates.


values were determined from the slopes of least-squareregression lines forced through the origin (Fig. 10).Because Eq. (26) is a function of p which was notdetermined for the wave train data, results from thesolitary wave data only have been analyzed. Measurednotch areas following bore 20 were discarded for caseA15 (see below). Fig. 11 plots the approximate erodedareas (volume per unit width, that is, 1/2ab1) against thetotal number of impacting bores. At bore 22, the erodedarea of case A15 decreases as a result of slumped ma-terial from the notch roof and thus is not related toerosion caused by the impact of bores. Similarly, onlythe first 12 (out of a total of 38) bores were utilized in theanalysis for case A13, where beyond that point the notcharea remained fairly constant (lower line in Fig. 11) andappeared to close up as a result of accumulated sedi-ments originating from seaward of the dune. This wasalso evident for the other cases with relatively lowoffshore wave heights, A14, B1 and B5, but not to sucha significant degree (Fig. 2).

The inferred Kd-values and associated coefficients ofdetermination are listed in Table 3. The coefficients ofdetermination are quite good for all cases except B1(R2 =0.67). Inspection of Fig. 10 shows that at thebeginning of bore impact for case B1, the notch depthincreased quite quickly and then slowed down as b1approached 2.5 cm. One possible explanation may bethat the dune properties were inhomogeneous such thatcompaction and water content were not the same at thefront of the dune compared to further back.

The transport coefficient (Kd) obtained with thesolitary wave data ranges from 4.24E−3 to 10.4E−3(Table 3). The variation is likely due to dependence onsome geotechnical and other physical parameters notalready included in the analytical model. The transportcoefficient describes the rate at which notching occurs,

Fig. 10. Notch depth plotted against the right-hand-side of Eq. (26) forthe solitary wave data. Resulting slopes are Kd-values.

and might be considered to be related to the ratiobetween the forcing at the dune and dune properties thathinder erosion, as suggested by Sunamura (1983).

Overton et al. (1994) measured dune erosion on aper-wave basis in a small wave tank with two differentsediment grain sizes and compaction levels. Theyconcluded that although dune density was statisticallysignificant with respect to the force–erosion relation-ship, sediment grain size was even more important,particularly for the finer grain sands. Larson et al. (2004)found that their transport coefficient (from a slightlydifferent version of the force–erosion relationship) wasrelated to the ratio between the offshore wave height andsediment grain size. The dependence on sediment grainsize may be related to the strengthening that matrixsuction is capable of providing. Although only one grainsize was used for the experiments in this study, indirectmeasurements of matrix suction were obtained by themethod described in Section 4.

In keeping with the idea that Kd is related to the ratiobetween the forcing at the dune and dune properties thathinder erosion, representative swash forces weremultiplied by representative compaction levels anddivided by the shear strength and median grain size.By combining Eqs. (3) and (4), the swash force for asingle bore may be expressed as:

Fo ¼ C2uqgh

2inc ð27Þ

In Fig. 12, the Kd coefficients obtained from Eq. (26)are plotted against the swash force (using mean

Table 3Derived Kd-values and coefficients of determination

A13 A14 A15 B1 B5 B2 B4 B3

Kd (⁎E−3) 4.92 5.57 4.65 4.24 7.64 8.74 10.2 10.4R2 0.96 0.97 0.99 0.67 0.99 0.94 0.95 0.99

Fig. 12. Derived Kd-coefficients plotted against hydrodynamic forcingand geotechnical parameters.

Table 4Comparison of tensile strengths and stresses (kPa) computed for thebeam-failure model

Case σt σLEPs σb σss

A13 1.09 1.15 0.85 2.00A14 1.58 1.31 0.86 2.17A15 1.02 1.13 0.74 1.87B1 0.90 1.19 0.38 1.57B5 2.05 1.36 0.00 1.36B2 0.93 1.08 0.69 1.77B4 1.20 1.24 0.67 1.91B3 0.74 1.08 0.00 1.08


measured incident wave heights in front of the dunes(Table 1a and b)) multiplied by the porosity over thevoid ratio (p/e), and divided by the shear strength andmedian grain size. The ratio p/e is equivalent to thevolume of solids divided by the total volume of a givensample, and is considered to represent the compactionlevel. The shear strength was calculated at depth A sothat from Eq. (15) τf,A=γA tan ϕ′+S(ua−uw) tan ϕ′where τf,A is the soil strength provided by both theweight of overlying material and soil suction. A least-squares regression with Cu=1.83 suggests a linear rela-tionship passing through the origin (R2 =0.84):

Kd ¼ 0:97d 10�4 C2uqg h̄

2

incðp=eÞsf ;AD50

!: ð28Þ

The sediment grain size, D50, was introduced inEq. (28) to ensure a dimensionlessKd. The resulting fittedlinear line has a positive slope suggesting that if D50 isallowed to increase, Kd will decrease. However, Overtonet al. (1994) found that their transport coefficient decrea-sed with decreasing grain size. Hence, Eq. (28) is prob-ably only valid for a given grain size (i.e.,D50=constant),although this was not tested with the data obtained forthis study as only one grain size was employed in theexperiments. In addition, care should be taken in using Eq.(28) as it stems from a limited data set and is only valid forthe approximate range 20b (Cu

2ρgh¯inc2 (p/e))/(τf,AD50)b

140.

5.2. Mass failure model

Equations presented in Section 2.2 were applied to theexperimental data to predict the dune recession distance,b2. Analysis of the data from the wave train experimentsis not possible as the saturation levels (S) and void ratios(e) are unknown. Although a tensile crack formed land-ward of the overhang for all experiments, mass wastingof the overhang only (Figs. 1(a–b) and 4a) occurred,leaving the remaining dune intact for both cases B5 andB3. The shear-type failures occurred when b1cr (=b2shear)was 5.4 cm and 5.0 cm, for cases B5 and B3,respectively. Predicted values using Eq. (14) in conjunc-tion with calculated shear strengths (Eq. (15)) are off bynomore than 10%with 5.1 cm and 5.9 cm, for B3 and B5respectively.

Recession of the remaining experimental dunesoccurred by beam-type failures. The landward locationof the tensile crack (b2beam) was estimated by determin-ing the length of a tensile crack that might be expected todevelop given the tensile strength of the soil and stressesgenerated by lateral earth pressures and bending mo-ments. Calculated tensile strengths were computed withEq. (18), range from 0.74 to 2.05 kPa, and are listed inthe second column of Table 4. Stresses at the surface dueto lateral earth pressures, σLEPs, and bending moments(σb) are listed in the third and fourth columns, while thefinal column lists the (combined) total stress at thesurface, σss.

It is interesting to note that for the majority of cases(A13, A15, B1, B2, B3, and B4) σLEPsNσt suggestingthat notching was not necessary in order for tensilecracking to occur. Although tensile cracking may haveoccurred due to the lateral earth pressure alone, the smalldifference between σt and σLEPs results in very smalltensile crack depths (Zc, Eq. (17)) so that by geometry(Eq. (16)), b2beam becomes about twice as large asmeasured. Alternatively, one might consider that thelateral earth pressure alone is responsible for the tensile

Table 5bOverturning and resisting moments with and without bending stresses

Case Mo (N) Resisting moment, Mr (N)

σLEPs only and with α frommax notch depth

σss and with α atnotch depth

A13 3.5 11.0 1.3A15 3.8 10.7 1.1B1 2.2 11.7 3.4B2 3.5 13.7 2.7B4 2.9 13.6 2.1B3 2.8 7.6 4.1

Only cases for which σLEPsNσt are listed.


crack reaching down to an internal failure plane startingfrom the initial dune foot prior to any notching. However,this too over-predicts the recession distance but only byabout 2 cm. It is speculated that for these cases whereσLEPsNσt, failure did not occur because the shear stresson the internal failure plane did not exceed the shearstrength. This was tested by computing the shear stress onthe inclined failure plane,

F3 ¼ Wbsinð45-þ / V=2Þ ð29Þ

whereWb is the weight of the failure block. The opposingforce was estimated with,

F2 ¼ scrL ð30Þ

where L is the length along the internal failure plane thatremains attached after a tensile crack has formed (=b3/cosα). The calculated forces are compared in Table 5a for thecases where σLEPsNσt and for the described scenarios.The last three columns compare the shear stress andstrength with b3 predicted by accounting for stresses dueto both the lateral earth pressures and bending moment. Inthe first two scenarios, where the bending moment is notaccounted for, the total shear strength (F2) exceeds thegliding force (F3) generated by a block defined by thepredicted geometry, thus supporting the notion that failuredid not occur due to lateral earth pressures alone. In allcases where the bending moment is accounted for, F3NF2

and thus predicts that failure would occur. With theexception of caseB3, this is further supported by themuchbetter comparison between measured and calculated b3values as compared to the other two scenarios. The beam-typemodel predicts that the dune of caseB3 should fail bytensile cracking and subsequent shear failure along theinternal failure plane with a resulting b3=5.8 cm(b2=10.8 cm). Development of a tensile crack wasobserved at about 8 cm although failure at this point didnot occur. Instead, the overhang alone failed with the

Table 5aSliding and resisting forces for scenarios with and without bending stresses

Case Msrdb3 (cm)

σLEPs only at t=0 and with α frominitial dune foot

σLEPs onnotch de

Predicted b3(cm)

F3

(N/m)F2

(N/m)Predicted(cm)

A13 3.7 5.4 185 299 8.5A15 2.3 4.7 188 289 8.6B1 4.1 5.7 216 295 10.0B2 2.8 5.7 224 306 9.6B4 3.8 6.4 216 353 8.7B3 0.0 2.9 171 225 9.3

remaining material landward of the maximum notchdepth intact. The discrepancy for this is uncertain butmight simply be because the shear failure occurred priorto the beam-type failure.

Besides the possibility of shear failure along theinternal failure plane following tensile cracking, there isa potential for failure by overturning as depicted inFig. 1e. This was assessed by calculating the momentsabout the pivot point assumed to coincide with the ma-ximum notch depth. The overturning moment was thusdefined by the moment of the overhang, while theresisting moment was calculated by multiplying theweight of the failing block material landward of thenotch depth by its horizontal center of gravity (neglect-ing any tensile strength offered by the lower part of thepending failure block). The results are shown in Table 5bfor the scenario where only the stress due to the lateralearth pressure is accounted for and second, for thescenario where both lateral earth pressures and bendingmoments contribute. For the scenario where only thelateral earth pressure is accounted for, the resistingmoment exceeds the overturning moment supporting thenotion that the block did not fail with the given geometry.For the case where the bendingmoment is included in thetensile stress analysis, the overturning moment exceeds

ly and with α from maxpth

σss and with α at notch depth

b3 F3

(N/m)F2

(N/m)Predicted b3(cm)

F3

(N/m)F2

(N/m)

238 272 3.3 189 139252 277 2.9 188 129292 332 4.9 211 187283 299 4.4 223 171228 289 4.1 190 166282 295 5.8 216 198

Fig. 13. Measured versus predicted recession distances.


the resisting moment in all cases except one, case B1.Hence it appears that (1) the dune of case B1 failed bytensile cracking and shear failure along the internalfailure plane, and (2) the dunes of cases A13, A15, B2,and B4 failed by tensile cracking and shear along theinternal failure plane or overturning or a combination ofboth. For case A14, for which σssNσt but σLEPbσt, andtherefore not listed in Table 5a and b, the gliding force(F3=228 N/m) exceeds the shear strength (F2=24 N/m)along the predicted internal failure plane and the over-turning moment (Mo=368 Nm/m) exceeds the resistingmoment (Mr=345 Nm/m) so that failure by tensile cra-cking and one or a combination of the aforementionedmechanisms is expected.

Results for recession distances due to mass wastingare shown in Fig. 13. The predicted b3 values obtainedwith the beam-type failure model were added to thenotch depths (b1cr) to give the full recession distances,b2, and compared to measured values. Results for theshear-type failures are also given, and together theyshow a good fit with R2 =0.90 and a slope equal to 1.07(slope of 1 denotes perfect fit). Beam-type failure oc-curred in six of the eight cases. Shear failure occurredfor cases B5 and B3 and are the smaller recessiondistances toward the lower end of the plot. Laboratoryobservations largely confer with this finding in that forthe shear-failures, the top of the failure block remainedfacing upward as it slid bodily downward, while themass failures of the beam-type failures either toppledover so that the top of the failure block faced toward thepaddle or slid downward so the top remained facingupward.

6. Discussion and conclusions

A small wave tank study was conducted to gain furtherunderstanding and to quantify the mechanisms involved

in dune recession. Idealized small-scale experimentaldunes were subjected to two sets of wave conditions atseparate occasions: fully broken single solitary waves andwave trains with increasing and subsequently decreasingwave heights. Solitary waves were employed to investi-gate the effects of dune erosion on a wave-by-wave basis.Recession occurred as a two-step process whereby theimpact of waves formed a notch at the dune foot resultingin a mass instability that caused mass failure. Two modesof mass failure were observed: shear- and beam-type. Inshear-type failures, the weight of the overhang exceedsthe shear strength of the sediments so that the failure blockslides downward or topples over. Beam-type failure,whereby a tensile crack forms some distance landward ofthe maximum notch depth, was observed in the majorityof the experiments.

Three models were developed to describe the dunerecession: first, a notching model predicting the amounteroded at the dune foot, second, a model predicting massfailure by shearing, and lastly mass-failure whereby thesoil tensile strength is exceeded. The upper portion ofthe notches was shown to be strongly linear for caseswhere the dunes were subjected to fully broken solitarywaves, whereas the upper shapes were predominantlyconcave when impacted by wave trains of increasing andsubsequently decreasing wave heights. Although non-definitive, the data suggests that concavity increases withincreasing offshore wave height. Based on a generalizedtriangular-shaped notch, a model describing erosion atthe toe of the dune was developed. A coefficient des-cribing the erosion rate was found to be linearly related tothe ratio of the forcing at the dune (incident wave height)and level of compaction normalized by the soil shearstrength and median grain size diameter.

Equations describing shear- and beam-type mass was-ting were developed for eroding dunes assuming tri-angular-shaped notches. Of the two experimental casesthat resulted in a shear-type failure, the maximum errorbetween the measured and predicted critical notch depthwas less than 0.9 cm (10%). The mechanism by whichbeam-type failure occurs is hypothesized to occur bytensile cracking at the top surface of the dune followed byshear failure along an internal failure plane or overturningdue to the weight of the overhang. It is speculated that avertical tension crack forms some distance back from thedune face and landward of the maximum notch depth as aresult of the lateral earth pressures and instability causedby the overhang (generated by erosion of the foot). Whenthe stress exceeds the soil tensile strength, cracking beginsand migrates downward until it intersects with an internalfailure plane. Failure is then expected to continue if theforce generated by the gliding weight of the failing slab


exceeds the shear strength of the soil along the inclinedinternal failure plane or if the overturning moment ex-ceeds the resisting moment. The beam-type failure modelis driven by soil tensile strength and geometric considera-tions. For the given data set the mean absolute errorbetween measured and calculated recession distances dueto beam-failure was 0.7 cm (9%) with a maximum of1.6 cm (20%). In an overall sense, calculated recessiondistances using the two models to describe mass wasting(i.e., shear-type and beam-type failures) compare wellwith measured values resulting in R2=0.90 and a slope of1.07 with a fitted least-squares linear regression.

The models presented here provide methods to pre-dict dune erosion at the toe and recession distances ofthe dune face knowing incident wave height, initial dunegeometry, and a few geotechnical parameters (porosity,unit weight, and saturation level). Although the mea-sured and inferred geotechnical parameters providesound physical explanations for the erosion and reces-sion mechanisms in addition to calculated values thatcompare well with measured quantities, there was littlevariation in the data. Field conditions are likely to bequite more complex than those simulated in the la-boratory experiment and as such, the model should beseen as a basis for possible future investigations of morecomplex situations (e.g., variations in sand formationsdue to bedding configurations, mineral leaching andorganic matter content, and variable moisture content).

Acknowledgements

Drs. Ryuichiro Nishi at Kagoshima University andAtilla Bayram of Han-Padron Assoc. are both gratefullyacknowledged for their assistance in the laboratory andfor valuable discussions. Financial support for LE andHH was provided by Vinnova (Swedish Agency forInnovation Systems, contract No. 1998-0188), and bythe Swedish Research Council (VR) for ML. Support forthe laboratory experiment and part of the work wasprovided by the Coastal Inlets Research Program, InletChannels and Geomorphology Work Unit. Dr. NicholasC. Kraus at the ERDC, Coastal and Hydraulics Labo-ratory made the laboratory experiments possible andcontributed with guidance, comments, and suggestions.Comments from manuscript reviewers were very helpfuland are greatly appreciated.

References

Andersson, S., Wiklert, P., 1972. Investigation of physical properties ofcultivated soils. J. Agric. Land Improve. 25 (2–3), 53–143 (Uppsala,Sweden).

Carter, R., Stone, G., 1989. Mechanisms associated with the erosion ofsand dune cliffs, Magilligan, Northern Ireland. Earth Surf. ProcessesLandf. 14 (1), 1–10.

Carter, R., Hesp, P., Nordstrom, K., 1990. Erosional landforms incoastal dunes. In: Nordstrom, K., Psuty, N., Carter, B. (Eds.),Coastal Dunes: Form and Process. John Wiley and Sons, Inc.,pp. 217–250.

Cross, R.H., 1967. Tsunami surge forces. J. Waterw. Harb. Div. ASCWE94 (WW4), 201–231.

Dean, R.G., 1977. Equilibrium beach profiles: U.S. Atlantic and GulfCoasts. Ocean Engineering Report, vol. 12. Department of CivilEngineering, University of Delaware, Newark.

Edelman, T., 1968. Dune erosion during storm conditions. Proc. 11thCoastal Engineering Conference, pp. 719–722.

Erikson, L., Larson, M., Hanson, H., Kraus, N.C., Nishi, R., 2003.Prediction of notching and mass failure of dunes. Proc. of CoastalSediments '03, ASCE, CD-ROM, World Scientific PublishingCorp. and East Meets West Productions.

Erikson, L., Larson, M., Hanson, H., 2005. Prediction of swash motionand run-up including the effects of swash interaction. Coast. Eng. 52,285–302.

Fisher, J., Overton, M., 1984. Numerical model for dune erosion due towave uprush. Proc. 20th Coastal Engineering Conference. ASCE,pp. 1553–1558.

Fisher, J., Overton, M., Chrisholm, T., 1986. Field measurements ofdune erosion. Proc. 20th Coastal Engineering Conference. ASCE,pp. 1107–1115.

Flemmer, C., 1991. On the regime boundaries of moisture in granularmaterials. Powder Technol. 66 (2), 191–194.

Galvin, C.J., 1972. Wave breaking shallow water. In: Meyer, R.E. (Ed.),Waves onBeaches andResultingSediment Transport.AcademicPress.

Hughes, S.A., Chiu, T., 1981. Beach and dune erosion during severestorms. Report UFL/COEL-TR/043. Coastal and OceanographicEngineering Department, University of Florida, Gainesville,Florida.

Hughes, S.A., Fowler, J., 1990. Midscale physical model validation forscour at coastal structures. Technical Report CERC-90-8. CoastalEngineering Research Center, Department of the Army, WaterwaysExperiment Station.

Kim, T.-H., Hwang, C., 2003. Modeling of tensile strength on moistgranular earthmaterial at lowwater content. Eng. Geol. 69, 233–244.

Larson, M., Erikson, L., Hanson, H., 2004. An analytical model topredict dune erosion due to wave impact. Coast. Eng. 51, 675–696.

Lohnes, R., Handy, R., 1968. Slope angles in friable loess. J. Geol. 76(3), 247–258.

Lu, N., Likos, W., 2004. Unsaturated Soil Mechanics. J. WileyPublishers, Hoboken, New Jersey. 556 pp.

Miller, R.L., 1968. Experimental determination of run-up of undularand fully developed bores. J. Geophys. Res. 73 (14), 4497–4510.

Nishi, R., Kraus, N., 1996. Mechanism and calculation of sand duneerosion by storms. Proc. 25th Coastal Engineering Conference.ASCE, pp. 3034–3047.

Öberg, A.-L., 1997. Matrix suction in silt and sand slopes:Significance and practical use in stability analysis. Ph.D. Thesis,Chalmers University of Technology, Göteborg, Sweden.

Overton, M., Fisher, J., 1988. Laboratory investigation of duneerosion. J. Waterw. Port Coast. Ocean Eng. 114 (3), 367–373.

Overton, M., Pratikto, W., Lu, J., Fisher, J., 1994. Laboratoryinvestigation of dune erosion as a function of sand grain size anddune density. Coast. Eng. 23, 151–165.

Pierrat, P., Caram, H., 1997. Tensile strength of wet granular materials.Powder Technol. 91, 83–93.


Schubert, H., 1972. Investigation and determination of capillary pressureand suction of moist granular material (in German), Ph.D. Disser-tation, Karlsruhe University, Germany.

Shen, M.C., Meyer, R., 1963. Climb of a bore on a beach: 3 run-up.J. Fluid Mech. 16, 113–125.

Sunamura, T., 1983. Processes of sea cliff and platform erosion. In:Komar, P. (Ed.), Handbook of Coastal Processes and Erosion. CRCPress, Boca Raton, pp. 233–265.

Synolakis, C., 1986. The run-up of long waves, Ph.D. Thesis, CaliforniaInstitute of Technology.

Terzaghi, K., 1943. Theoretical Soil Mechanics. John Wiley and Sons.510 pp.

Terzaghi, K., Peck, R., 1948. Soil Mechanics and EngineeringPractice. John Wiley and Sons, Inc. 566 pp.

Thorne, C.R., Tovey, N.K., 1981. Stability of composite river banks.Earth Surf. Processes Landf. 6, 469–484.

laboratory investigation of beach scarp and dune recession

Documents