On the development of large-scale cuspate features ona semi-re£ective beach: Carchuna beach, Southern Spain

M. Ortega Sa¤nchez �, M.A. Losada, A. BaquerizoGrupo de Puertos y Costas, Universidad de Granada, Granada 18004, Spain

Received 6 August 2002; accepted 25 April 2003

Abstract

Carchuna beach, located on the southeast coast of Spain, has a series of natural permanent horns of time-dependent horizontal dimensions, which do not appear to be periodically spaced, and do not propagate alongshore.Until at present, it has been assumed that the permanency of these forms is related to the existence of bed rocks;nevertheless, analysis of the configuration of the depth contours and the bed sediment composition does not justifythis hypothesis. This paper explores the formation of cuspate features by the effects of the wind wave dynamics andtheir development by nearshore circulation and infragravity waves, including edge waves. Three levels of wave energyflux conditions are considered. In all likelihood, wave refraction of severe storms on submerged fluvial valleys is themain cause for longshore variation of breaking wave height, which seems to be one of the initial conditions necessaryfor the cuspate to initiate. Moderate storms enhance the deepening of the embayments, whereas mild sea states areable to maintain the features by self-organization. The possibility of the occurrence of edge waves due to the non-linear interaction between the components of a gravity wave spectrum approaching the shore under moderate stormconditions is explored. Preliminary analysis of video images taken by an Argus station seems to confirm thishypothesis.3 2003 Elsevier Science B.V. All rights reserved.

Keywords: beach morphodynamics; large-scale cuspate features; coastal evolution; nearshore circulation; edge waves

1. Introduction

Carchuna beach, bounded by Cape Sacratif andCarchuna horn, is ca. 3900 m long with an ap-proximately W^E alignment, showing cuspate fea-tures (Figs. 1 and 2). Its plan form commonlyshows ¢ve cuspates with horns spaced 850, 650,

850, 475, and 1710 m respectively (Fig. 2, H-1 toH-6). Cape Sacratif (west boundary of the beach)represents the ¢rst horn of the cuspate system.The extent of the other four horns seaward de-pends on the wave climate. Analysis of aerial pho-tographs since 1956 and images from a video-camera station installed in November 2002 doesnot reveal a signi¢cant change in the position ofthe horns, although their cross-shore length andwidth may vary in time.Rosen (1975) studied the set of cuspate features

on Nantucket Island, Massachusetts, concluding

0025-3227 / 03 / $ ^ see front matter 3 2003 Elsevier Science B.V. All rights reserved.doi:10.1016/S0025-3227(03)00126-9

* Corresponding author. Fax: +34-958-248314.E-mail address: miguelos@ugr.es (M. Ortega Sa¤nchez).

MARGO 3329 20-6-03

Marine Geology 198 (2003) 209^223

R

Available online at www.sciencedirect.com

www.elsevier.com/locate/margeo

that they are both erosional and depositional, andcaused by reversals in wave direction that result inthe erosion of the embayments and deposition onthe horns of the cusps. Other mechanisms thathave been mentioned in the literature on the topicare eddies, which form on the lee side of the hornsand induce sediment to be transported back tothe horns, or seiches (long-period standing waves;Wilson, 1972) that have nodes or antinodes atwhich the horns can form. Since the ¢rst observa-tions by Munk (1949) and Tucker (1950), the ex-istence of these morphological patterns have oftenbeen related to infragravity waves, particularly toedge waves, which generate beach cusps with aspacing of half their wavelength (Bowen and In-man, 1971; Guza and Inman, 1975).A common and still debated question is whether

the topography induces the circulatory system, orwhether it is the circulatory system that redistrib-utes sediments and builds up the bottom con¢gu-ration, with the consequent interaction betweenthe morphology and the currents. Based on ob-servations made by Sonu (1972), a theory wasderived which argues that an irregular topography

always controls the water circulatory system. As aresult, the circulatory system would be generatedby the existing topography. However, accordingto Noda (1974), there is no explanation for topog-raphy generation if the circulatory system is notassumed to be responsible for it. Moreover, ripcurrents can also be generated on beaches withoutinitial bottom irregularities, for example, due tothe interference of two incident wave ¢elds (Dal-rymple, 1975) or as a result of a bed-£ow insta-bility (Barcilon and Lau, 1973; Hino, 1975).Other authors have analyzed the possibility that

these rhythmic formations are the result of inher-ent instabilities of the morphodynamic equilibri-um, as studied by Werner and Fink (1993), or aredue to the coupling between small perturbationson a reference uniform bottom topography andthe disturbances thereby produced on the watermotions (Deigaard et al., 1999). The previous ex-istence of a rhythmic topography is not necessaryfor any of these explanations, but rather an initialbed or water surface perturbation.Another analysis of the problem focuses on the

relation between the wave angle approach and

Fig. 1. View of Carchuna beach from Cape Sacratif (April 2001). Di¡erent large-scale cuspate features can be distinguished.

MARGO 3329 20-6-03

M. Ortega Sa¤nchez et al. / Marine Geology 198 (2003) 209^223210

the shoreline orientation (Ashton et al., 2001). Itis assumed that alongshore sediment transportsmoothes the coastline, but this is only validwhen angles between waves and the shorelineare small. However, when these angles are su⁄-ciently large, small perturbations of a straightshoreline will grow.The objective of this paper is to analyze the

generation and permanence of the large-scale cus-pate features existing at Carchuna beach. It seemsthat alongshore variation of wave refraction in-duced by submerged £uvial valleys is the mainsource of their initial formation. Nevertheless,other mechanisms may also be important, notonly in their generation but in their maintenanceor reinforcement. The ¢rst section of this paper isa description of the physical environment anddate; next, the methodology is presented: wavepropagation and wave breaking are evaluated asa step to a discussion on nearshore circulationpatterns induced by di¡erent intensities of wavebreaking. Subsequently, the in£uence of wave di-rection on the evolution of features is considered,and ¢nally the selective mechanisms of the gener-ation of infragravity waves and the enhancementof some edge wave modes by the cuspate featuresare discussed.

2. Physical description and data

Carchuna beach is located on the southernslope of the Sierra Nevada, facing the Albora¤nSea. The area of study is de¢ned by the hydro-logical basin, beach, and continental shelf downthe 3200 m isobath.

The geologic setting is formed by two groups ofclearly di¡erentiated materials : pre-orogenic ones,with a high degree of metamorphism that consti-tutes the Alpujarrian nappies (Blumenthal, 1935;Copponex, 1959; Boulin et al., 1970; Aldaya,1969, 1970, 1981) upon which more modern (Ho-locene) materials, constituting alluvial deposits,are placed.The main erosional agents are rain-induced

runo¡ and the resulting sediment yield. Rainfalland the existence of an important surface runo¡are the principal agents that can model the sur-face. Rainfall on the coast of Granada is scarce,averaging 350^400 mm/yr, and corresponds to asemi-arid area (Pulido, 2000). Its temporal distri-bution is irregular, especially during the dry sea-son (July^September) when torrential rains oftenoccur.The short streams which discharge in Carchuna

beach have their sources in the high mountainousrelief of the Alpujarrian complex. The hydrolog-ical basins of these rivers are characterized bysteep relief ; the foothills over Carchuna beachand neighboring areas exceed a 40% slope withnon-cohesive material covering the rock substrate.The hillsides show frequent outcrops of rocks as aconsequence of the high erosion rate.The maximum tidal range is 1.1 m with a mean

value of 0.6 m. South Atlantic and South Medi-terranean storms generate wind waves under lim-ited fetch conditions (approximately 300 km) andaverage wind speed u10 is 18^22 m/s. The mostenergetic wave climate approaches are E, ESE,SW, WSW, and SW with probabilities of occur-rence in a mean year given in Table 1. This tablealso shows the conditional probability to wave

Table 1Probability of occurrence of a sea state in each directional sector (¢rst column) and conditional probability to the wave height(second^fourth columns) at Carchuna beach

Sector Probability Pr[ajHs6 1] Pr[ajHs6 1^2] Pr[ajHss 3.5](a) (%)

E 21.01 34.2 43.23 0.70ESE 2.22 51.6 32.17 0.19SW 6.01 46.43 39.11 0.75WSW 14.37 31.45 41.35 1.45W 26.72 28.02 41.39 2.45Calms 29.67

MARGO 3329 20-6-03

M. Ortega Sa¤nchez et al. / Marine Geology 198 (2003) 209^223 211

height, with a mean period in the range 7^9 s,obtained from the annual sea state regime.Under South Atlantic storm conditions, swell

waves generated in the Gulf of Cadiz propagatethrough the Strait of Gibraltar. These swell wavesimpinge the coast simultaneously with the localwind waves, but at slightly di¡erent angles.The analysis of the bathymetry is signi¢cant

for the objectives of this study (Fig. 2). First ofall, seaward of Cape Sacratif, the existence of asubmerged £uvial valley or submarine canyon,known as Jolucar Canyon, can be identi¢ed.From there, the seabed contour o¡shore Carchu-na beach is straight and convergent towardCarchuna horn, following the plan form of thealluvial plain and providing an insight as to aprevious emplacement of the main stream mouth.As in other alluvial plains in the surroundingarea, this change of location is related to anthro-pogenic action (mainly due to the recent intensiveagricultural activities that need water redistribu-tion). For further applications the following threezones can be distinguished:

Zone 1. Foreshore and nearshore extendingfrom the coast to approximately the 25 isobath.

Four beach pro¢les are shown in Fig. 3; beachslope increases from west to east (from tan= 1/30to tan= 1/8) and beach alignment is approxi-mately west^east. The grain size in the beach isheterogeneous, varying from ¢ne to coarse or verycoarse sand with cobbles (Fig. 4). The sediment is

Fig. 2. Area of study. Distances between horns are given and three zones can be appreciated in the bathymetry.

a

b

Distance (m)

BP-1BP-2BP-3BP-4

Beach profiles

Dep

ht (

m)

0

-1

-2

-3

-4

-5

-6

-7

-8

0 50 100 150 200 250 300 350

Fig. 3. (a) Location of the beach pro¢les. (b) Beach pro¢les.

MARGO 3329 20-6-03

M. Ortega Sa¤nchez et al. / Marine Geology 198 (2003) 209^223212

composed of the clast of metamorphic materialsproceeding from the surrounding relief, mainlyschists and quarzites. The overall morphodynam-ics of the beach is re£ective with waves breakingin collapsing or plunging.

Zone 2. Continental shelf extending from thenearshore down to a water depth of approxi-mately 75 m. A gradual adjustment of contourlines can be seen as water depth increases. Inzones 1 and 2 the sediment over the shelf is char-acterized by massive sands and gravels changingupward to prodeltaic deposits. These depositsform homogenous sequences of heterometricgravels and poorly classi¢ed sands (Herna¤ndez-Molina et al., 1993).

Zone 3. Continental slope extending from 75 mdown to (at least) 500 m water depth (deep waterfor wind waves). The shelf break is between 375and 3100 m water depth. Next to Jolucar Canyonthe gradient of the continental slope is moreabrupt than in the eastern region. The upper slopeis characterized by sediment £ows proceedingfrom the outer shelf region which are channeledthrough the small submarine canyons such as Jo-lucar Canyon (Giermann et al., 1968; Alonso andMaldonado, 1992; Pe¤rez-Belzuz, 1999).The streams £owing into Carchuna beach have

concave longitudinal pro¢les with slopes decreas-

ing toward the mouth. Their main courses extendbelow the mean sea level approximately down to75 m depth (Fig. 5).These pro¢les show three di¡erent regions:

(1) the upper region extending from 300 m alti-tude down to 40 m; (2) the middle region, wherethe stream slows down and discharges water andsolids into the sea, extending to between +40 and375 m; and (3) the o¡shore region where theslope is steep again. As can be seen, the upperand o¡shore regions are quite similar (Fig. 5, re-gions 1 and 2), with slopes of about 20%.As a result of the £uvial and marine processes,

the o¡shore bathymetry opposite Carchuna beachis characterized by straight and convergent con-tours toward Carchuna horn, furrowed trans-versely by several submerged valleys (Fig. 2), Jo-lucar Canyon being the most prominent.As previously described, the area studied con-

stitutes a physiographic unit. The presence of Jo-lucar Canyon and the coarse sediments at Carch-una horn suggests that the sediment exchangethrough the west and east boundaries of theunit is almost negligible. Moreover, the actualsediment supply from the local streams is alsonegligible, because of the construction of a largenumber of greenhouses and the construction ofterraces in their beds to divert water for irrigation.

Fig. 4. Grain size detail at Carchuna beach, varying from very coarse to ¢ne.

MARGO 3329 20-6-03

M. Ortega Sa¤nchez et al. / Marine Geology 198 (2003) 209^223 213

Nowadays, it can be concluded that there is nosediment input to Carchuna beach. Because of thelarge grain sizes, which armour the foreshore, aswell as the mild annual wave energy £ux, thebeach is eroding slowly, but consistently.

3. Methodology

3.1. Wave propagation and wave breaking in thestudy area

A parabolic numerical model (Kirby and Dal-rymple, 1983) which includes wave breaking wasused to analyze wave propagation under di¡erentinitial wave conditions. It is a weakly non-linearcombined refraction and di¡raction model, whichincorporates shoaling, refraction, energy dissipa-tion, and di¡raction e¡ects. It is solved in ¢nitedi¡erence and is applied with two nested gridswith a resolution of 5 m for the one closest tothe shore. The results are shown in Fig. 6, wherevector length is proportional to the wave height atthe point considered, and its direction indicatesthe direction of propagation.Fig. 6 shows the propagation pattern of a sea

state with signi¢cant wave height of 1 m, and awave period of 8 s propagating from the west. Itcan be observed that the presence of the canyonand the submerged £uvial valleys produce localconvergences and divergences of the wave energy£ux; waves converge toward Cape Sacratif, andenergy concentration takes place in the embay-ments. The propagation coe⁄cient, de¢ned asthe ratio between the deep water wave heightand a representative wave breaking depth atCape Sacratif and at the embayments was eval-uated.To analyze the propagation pattern for the

most energetic sea states in the study area, thealongshore evolution of the propagation coe⁄-cient for waves coming from both the west^southand the south^east was obtained. Two levels ofwave energy, 1 and 4 m signi¢cant wave height,were considered. The ¢rst one is representative ofmild and moderate sea states (50% of exceedance).The 4-m wave height in deep water (90% of ex-ceedance) represents storm conditions in the area.Figs. 7 and 8 show the results of the propaga-

tion coe⁄cient (data were averaged alongshore2U2 over 50 m) for west^south and south^east,respectively.

Fig. 5. Pro¢le of river beds at Carchuna beach.

MARGO 3329 20-6-03

M. Ortega Sa¤nchez et al. / Marine Geology 198 (2003) 209^223214

Fig. 7a shows the longshore distribution underwesterly waves with wave height oscillating anddecreasing eastward, getting maximum values atboth sides of Cape Sacratif and westward of thehorns. For a mild wave climate, and irrespectiveof the incoming wave direction, the convergencesof energy £ux take place approximately at thesame locations.Fig. 7b represents the longshore variation of

the dimensionless wave height evaluated at a rep-resentative wave breaking depth of storm condi-tions. It can be seen that these results are similarto the previous ones, but the maximum and mini-mum values are less pronounced. Between H-2and H-3, the values of the propagation coe⁄cientcurves of storm waves are signi¢cantly closer onthe three wave directions analyzed than undermild conditions because waves break inside theembayments. It can also be observed that betweenH-4 and H-5 wave energy concentration is higherunder storm conditions than for mild ones be-cause H-4 provides less protection. Finally, itshould be mentioned that the range of variationof the propagation coe⁄cients between H-1 andH-2 is wider for mild conditions because wavebreaking takes place inside the embayments.The alongshore variation of the propagation

coe⁄cients for easterly waves is represented inFig. 8a,b. It can be observed that the energy£ux concentration takes place eastward of thehorns, with lower di¡erences between maximumand minimum values than those produced by thewesterly waves. Again, those maxima are shorterfor storm conditions than for mild ones.

3.2. Nearshore circulation induced by wavebreaking

Attending to the mechanism generating cuspatefeatures, swash cusps and surf zone cusps are tra-ditionally considered (Inman and Guza, 1982).While the ¢rst ones may be generated by theswash and backswash excursion over the beachface and berm (Guza and Bowen, 1981), the sec-ond ones may be formed by the currents of thenearshore circulation cells (Bowen and Inman,1969), whose typical wavelengths can vary be-tween 1 m on lakes to hundreds of meters onbeaches exposed to the ocean (Inman et al.,1971), which are often called giant cusps (Komar,1971).However, none of these mechanisms is respon-

sible for the generation of Carchuna. Rather, themechanisms involved in the development, mainte-

Fig. 6. Wave propagation at Carchuna beach. Wave energy concentration takes place in the embayments.

MARGO 3329 20-6-03

M. Ortega Sa¤nchez et al. / Marine Geology 198 (2003) 209^223 215

nance, enhancement or attenuation, depending onwave climatic conditions, are related to the surfzone circulation and they are therefore similar tothe one for surf zone cusps.In order to study the development of the cus-

pate features, it is necessary to know whetherthese features can last under the continuous tem-poral variation of the wave climate. Among otherfactors, cuspate evolution has to be related to thenearshore circulation induced by wave breaking.In this section that circulation is analyzed forthree wave energy £ux intensities, representative

of severe (rare), moderate (frequent) and mild(habitual) conditions.

3.2.1. Severe storm conditionsBagnold (1940) described the circulatory system

in a cuspate shoreline for waves approaching thebeach normally, and where wave breaking occursevenly alongshore in a straight line. In his circu-lation model, wave surge piling up on the horns isdivided into two diverging streams towards theadjoining embayments. In the middle of the cus-pate arc, £ows coming from both sides converge,

Fig. 7. Propagation coe⁄cient curve. Sector S^W.

MARGO 3329 20-6-03

M. Ortega Sa¤nchez et al. / Marine Geology 198 (2003) 209^223216

resulting in a stronger return £ow through thecenter of the embayment. Sediment eroded inthe arc is deposited outside the bay where thecentral stream loses strength, causing the bathy-metric lines seaward of the breaker line to adopt acuspate form alongshore symmetric to the shore-line. This new bathymetric con¢guration preventswave breaking to be uniform alongshore.The depth of closure at Carchuna beach can be

estimated to be in the range of h�V6^7 m. Thebathymetric contours are approximately parallel(Fig. 2). Generic beach pro¢les, taken at an em-bayment and at a horn, and identi¢ed respectivelyas B1 and B2 in Fig. 9, are parallel seaward from

the crossing location. Intriguingly enough, thecrossing point is located approximately at thedepth of closure. The constant di¡erence of about0.5 m in the deeper parts of the pro¢le is partlydue to the slightly alongshore convergent bathym-etry.Thus, to have uniform alongshore wave break-

ing, the most energetic sea states have to occur(Hs s 3.5 m, TV7^9 s). Because they generallyoccur with a large wave period, the incidence an-gle is strongly corrected by refraction. Under suchconditions, Bagnold’s model seems to be applica-ble. Since wave breaking approximately occursevenly alongshore (plunging or collapsing break-

Fig. 8. Propagation coe⁄cient curve. Sector S^E.

MARGO 3329 20-6-03

M. Ortega Sa¤nchez et al. / Marine Geology 198 (2003) 209^223 217

ers), the nearshore circulation would then adjustto Bagnold’s scheme: eroding horns and trans-porting material towards the center of the bay.If the storm lasts long enough, the system seemsto arrive at an equilibrium geometry: an along-shore uniform coastline.It should be underlined that this circulation

pattern is almost identical to the swash mecha-nism for beach cusp initiation and maintenance(Dean and Maurmeyer, 1980) and like Bagnold’smodel, only explains the development of the so-called equilibrium geometry. The basic concept ofthis type of mechanism is that the interaction ofthe cusp topography and the breaking-induced£ow causes circulation cells that tend to constructand then perpetuate the cuspate feature. More-over, to initiate the beach cusp, a small perturba-tion of elevation on the beach face and a syn-chrony between the incoming wave period andthe natural swash period are necessary.

3.2.2. Moderate sea state conditionsThis section relates the nearshore circulation to

moderate sea states, Hs =1^2 m and Tp =6^8 simpinging obliquely. As can be checked by therefraction^di¡raction model, westerly waves prop-agate well inside the embayments (tan= 1/50) be-fore breaking in plunging. Higher breaking wavesoccur inside the embayments than on the horns,where waves are breaking in collapsing. Thus, dueto their obliquity and the set-up gradient insidethe embayment, an alongshore current towardsthe horns is generated, transporting sedimentsand reinforcing the submerged topography ofthe horn. The intermediate sea states are able toerode the embayment, increasing the curvature ofthe arc and supplying sediment to the circulationsystem.Similarly, under waves arriving from E^ESE,

the alongshore current £ows from the center ofthe embayments towards the western horns. Forthose directions, the wave height gradients are notso signi¢cant, and the currents are probablyweaker than those generated by the westerlywaves. In other words, easterly waves are not ase¡ective as westerly waves to build up cuspatefeatures.This circulatory scheme, in contrast to Bag-

nold’s model, reinforces and enhances the embay-ment-horn systems. In principle, as long as the seastate characteristics last, this mechanism does notstop since alongshore variation of wave heightincreases as the horn grows. This could be thereason why after the occurrence of intermediatestorms, it can be observed that the horns haveadvanced toward the sea. Only the variation ofthe wave energy £ux, magnitude or direction canstop this cuspate feature from further develop-ment.

3.2.3. Mild sea state conditionsFinally, under mild sea states, (Hs 6 1 m,

T=4^6 s) waves propagate inside the embayment,breaking on the steep shoreface (tan= 1/10), withstrong plunging and collapsing breakers (Iribarrennumber, Ir s 2). On the other hand, waves spilland plunge (tan 1/20, and Ir 6 1) on the horns.At both sides of the horn, an alongshore currenttowards the bay is generated, smoothing thehorns. The sediment transported to the bay pro-duces a seaward movement of the embayment,which provokes a seaward advance of the wavebreaking. This changes the breaking obliquity,and has the e¡ect of reducing the Iribarren num-ber, changing the type of breaking, and generat-ing a current towards the horns.Depending on the temporal sequence of incom-

ing wave energy (height and period), the long-shore £ow moves the sediment into the embay-ment or to the horns, causing cuspate featuresto develop through a combination of positiveand negative feedbacks. When incipient changesoccur on the embayment, more deposition takesplace to form the horns, whereas more fullyformed embayments tend to retard depositionand erosion. In other words, there is a self orga-nization that is quite robust and not strongly af-

Fig. 9. Pro¢les in points and embayments.

MARGO 3329 20-6-03

M. Ortega Sa¤nchez et al. / Marine Geology 198 (2003) 209^223218

fected by small changes in the incoming wave en-ergy £ux, as long as it is mild.

3.3. Cuspate evolution and wave incidence angle

So far, the evolution of the cuspate has beenanalyzed in terms of the magnitude of the waveenergy £ux. This section addresses the dependenceof cuspate evolution on angle of approach.On a periodic coastline, depending on the wave-

length and the angle of wave incidence, oscillatorybehavior can be enhanced or suppressed (Larsonet al., 1987). If the initial shoreline is assumed tobe:

yðx; 0Þ ¼ BcosðV xÞ

where B is the initial amplitude of the shorelineperturbation, V is its alongshore wave number,and the mean beach position is located at y=0,the solution to the one-line equation is periodicalongshore (the x direction) :

yðx; tÞ ¼ Bexpð3GV2tÞcosðV xÞ

where G is referred to as the longshore di¡usivity,showing that the amplitude of the perturbationapproaches zero exponentially with time (t).The di¡usivity coe⁄cient depends on the 5/2

power of the wave height, the cosine of the doubleof the wave incidence angle, db, and inversely onthe total depth (depth of closure plus the bermheight). For a small or moderate angle of inci-dence, the one-line model gives that the cuspateshould be smoothed. However, for dbs 45 thesign of G changes. This results in an exponentialincrease of the periodic shoreline features, ir-respective of their wavenumber and incomingwave energy £ux.Given the orientation of the beach and wave

refraction, it is hard for energetic sea states toimpinge on Carchuna beach with such large an-gles. Only sea states from the west or from theeast, generated by winds parallel to the coast (as-sociated with South Atlantic or South Mediterra-nean storms), can generate the prescribed condi-tions. Additionally, local waves generated underdiurnal sea breeze conditions with small waveperiods can satisfy the above requirements. Suchconditions generally occur in fair weather.

It can be concluded that although the enhance-ment of the cuspate features can occur under suchconditions, these are frequently insu⁄cient to jus-tify the permanence of the features over severalyears. The following section explores the perma-nence mechanisms related to the development ofinfragravity oscillations.

3.4. Infragravity oscillations at Carchuna beach

Hasselmann (1962) showed that the non-linearinteraction between the components of a gravitywave spectrum approaching the shore can forcesecondary infragravity waves. When wave break-ing of the incident swell takes place, these forcedoscillations are released as free waves (Longuet-Higgins and Stewart, 1962), propagating towardsthe shore, and they are re£ected back at theshoreline. For certain angles of incidence, theseshoreward propagating waves can be radiated todeeper waters as leaky waves are re£ected backtowards the shore at a given trapping depth. Inthe latter case, waves are e¡ectively con¢ned atthe shore, as is the case of edge waves.Wave spectra arriving at Carchuna beach dur-

ing moderate storm conditions usually show twopeak harmonic frequencies, f1W0.14 Hz andf2W0.17 Hz, with di¡erent directions of propaga-tion in deep water, a1W245^270‡ and a2W200^230‡, corresponding to di¡erent generation condi-tions. The largest peak belongs to westerly waves,which are usually generated in the Gulf of Ca¤diz,and propagates through the Strait of Gibraltar,whereas the shortest one travels in southwestern

Fig. 10. Typical wave spectrum in the Albora¤n Sea understorm conditions.

MARGO 3329 20-6-03

M. Ortega Sa¤nchez et al. / Marine Geology 198 (2003) 209^223 219

direction, and is associated with waves generatedin the Albora¤n Sea.The non-linear interaction between these two

more energetic components, according to Hassel-man’s theory, would excite a secondary oscillationat a frequency, f= f23f1W0.03 Hz, with a vectorwave number, k

!= k!

23 k!

1, where k!

1 and k!

2

are, respectively, the vector wave numbers of thewave components. For the case in which its mod-ulus approaches the linear theory wave number,the excitation is resonant and the ampli¢cationfactor grows, theoretically, without limits.A scalar buoy located at Cabo de Gata, 40

miles east of Carchuna, has been operating since1997. The recorded data represent the wave cli-mate along the continental shelf of the Albora¤nSea. Moreover, the WAM model has a hind cast-ing point near Carchuna. Fig. 10 shows the wavespectrum of the water surface elevation recordedduring a storm that occurred on the 23 December,2000, at 5:00 a.m. At that time, WAM predicts apeak wave direction of 260‡ and a wind meandirection of 230‡ in deep waters.For those wave components, Fig. 11shows the

cross-shore depth evolution of the linear and theforced oscillation wave numbers. Both curves ap-proach as water depth decreases, converging at adepth of around 1.5 m.In order to estimate the depth at which most of

the waves propagating as incident swell are ex-pected to break, the two-peaked spectrum wassplit into two parts, one from f=0.1 Hz to thefrequency f=0.16 Hz, and the second one fromf=0.16 Hz to f=0.23 Hz. Evaluating at the beachHrms;i, i = 1.2 from the zero-order moment of thesespectra, propagated with their corresponding peak

frequencies, a root mean square wave heightHrms ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiH2

rms;1 þ H2rms;2

q=1.2 m is obtained. Con-

sidering a breaking index Q=0.8, this sea state hasa breaking depth, hbW1.5 m, which coincideswith the converging depth of the wave numbercurves (Fig. 11).Therefore, a forced oscillation with a period

around 33 s would be released at breaking underresonant conditions, with a wavelength of about125 m. The direction of propagation of the re-leased free wave is aW192‡.The trapping depth for such a free wave is

ht =28.8 m. Interestingly enough, this depth isapproximately the same as that at which the shelfpro¢le sharply changes the slope. This wave en-

Fig. 12. Convergent bathymetry used for the analysis of edge wave propagation.

Fig. 11. Evolution of the wave number of the forced oscilla-tion with water depth.

MARGO 3329 20-6-03

M. Ortega Sa¤nchez et al. / Marine Geology 198 (2003) 209^223220

ergy con¢ned to the shore up to depth ht is ex-pected to propagate in the longshore direction asan edge wave.It seems that the necessary circumstances for

edge waves to occur at Carchuna beach are foundunder moderate storm conditions.

3.4.1. Propagation of edge wavesThe propagation of edge waves at Carchuna

beach has been explored. Because of the smallradius of curvature of the contour depths, it canbe assumed that waves are propagating along-shore on a straight beach, with a convergent ba-thymetry (Fig. 12) and are partially re£ected backand forth at Cape Sacratif and Carchuna horn.Fig. 13 shows the evolution along the conver-

gent bathymetry of the wave lengths of the modesfor the secondary wave forced by the two swellcomponents considered above. While the highermodes grow rapidly alongshore, the variation ofthe ¢rst mode is very slow. This result reinforcesthe possibility that low mode edge waves, whichare indeed the likeliest to occur, can smoothlyadapt themselves as they propagate alongshore.Beach materials at Carchuna beach are made of

¢ne sand and coarse gravels. Partial re£ection ofedge waves of wave length L at horns would buildup rhythmic beach lobes with wave lengths ofapproximately L/2 (see Baquerizo and Losada,2001; Baquerizo et al., 2002).On the other hand, the results presented in Sec-

tion 3.2 provide a solid basis for the hypothesisthat at Carchuna beach the convergence of mod-erate wind wave energy £uxes is a mechanism forthe maintenance of the horns. Under such condi-tions they would act like re£ecting structures that

select the modes of the edge waves. Moreover,such waves may be important in the developmentof smaller beach cusps between horns.Fig. 14a presents the shoreline between H-5 and

the middle point (Mp) of the last embayment,obtained from Argus time exposure images takenon January 20 (2003). Smaller beach cusps oflengths around 60, 120 and 200 m show up as isindicated in Fig. 14b, which corresponds to thelinearly detrended shoreline of the stretch of thecoast starting at H-5. In order to assess the pres-ence of such wavelengths in the shoreline, a fastFourier transform was performed.Fig. 14c shows the amplitude of the component

oscillations in terms of their inverse wavelengths.It can be seen that oscillations about 200, 135,115, and 65 m, which are close to the ones ob-served in the shoreline (Fig. 14b), are present witha higher energetic content than other components.The 65-m length is about half of the 125-m lengthof the possible free trapped oscillation analyzed inSection 3.4, which is indeed close to the wave-length of the ¢rst edge wave mode in Fig. 13.Other wavelengths are also in the order of mag-nitude of L/2, where L is the wavelength (H-5,Mp) of the stretch of the coast, as can be seen inFig. 13.Fig. 13. Alongshore evolution of the edge wavelength.

Fig. 14. (a) Shoreline between H5 and the middle point (Mp)of the last embayment. (c) Spectral components of the linear-ly detrended stretch of the coast shown in (b).

MARGO 3329 20-6-03

M. Ortega Sa¤nchez et al. / Marine Geology 198 (2003) 209^223 221

4. Future work

In the previous sections several mechanisms re-lated to large-scale cuspate feature generation andpermanence have been discussed. In fact, one sin-gle process is not su⁄cient to explain the mor-phology. Instead, it seems that there are severalmechanisms which can help to give a coherentand consistent explanation.In order to further explore and explain which

mechanisms are responsible for large-scale cus-pate feature development and permanence, athree-video camera station was installed at CapeSacratif in November 2002 (based on Argus Tech-nique; e.g. Lippmann and Holman, 1989). Theresulting analysis will be focused on shorelinemapping and wave propagation. The analysis isaimed at obtaining: (1) the relationship betweenweather sea state conditions and wave propaga-tion patterns; (2) wave energy concentrationalongshore; and (3) the circulatory system andhorn shape response. The importance of this ¢eldwork is con¢rmed by the preliminary analysis ofsome images as shown in the previous section,Fig. 14.

5. Conclusions

Carchuna beach, bounded by Cape Sacratif andthe Carchuna horn, is ca. 3900 m long with anapproximately west^east alignment, showing cus-pate features (Figs. 1 and 2). Its plan form com-monly presents ¢ve cuspates with horns spaced850, 650, 850, 475, and 1710 m, respectively(Fig. 2, H-1 to H-6). In relation to their genera-tion and permanence, the following conclusionsmay be drawn.(1) It seems that under severe storms (Hs s 3 m,

T=7^9 s) the alongshore variation of wave refrac-tion induced by the submerged £uvial valleys isthe main source of initial formation of the cuspatefeatures. Moderate sea states (Hs =1^2 m, T=6^8s) generate a circulatory system that reinforces thehorn and the embayment shape. Apparently, theonly limit that this mechanism has is the stormduration. Erosion of the embayment supplies thematerial that causes the horns to protrude sea-

ward. Mild and low energy sea states (Hs 6 1 m,T=4^6 s) may develop a self organization that isquite robust and not strongly a¡ected by smallchanges in the incoming wave energy £ux, main-taining the cuspate features in a dynamic equilib-rium. For cusp features to grow or to disappear, asigni¢cant change in the energy £ux intensity isneeded.(2) For this speci¢c case, it seems that wave

direction is not relevant for the development ofthe cuspate system.(3) Other mechanisms may also be important,

not only in their generation but in their mainte-nance or development. Under moderate windwave energy conditions, non-linear interaction be-tween the components of a gravity wave spectrumapproaching the shore can force secondary infra-gravity waves, which would be released at break-ing. There is trapping of free waves at a depth atwhich the shelf pro¢le sharply changes its slope.This wave energy when con¢ned to the shore isexpected to propagate in the alongshore directionas an edge wave. Research results illustrate thepossibility that the lower edge wave modes, whichare indeed the likeliest to occur, can smoothlyadapt themselves as they propagate alongshore.(4) Partial re£ection of edge waves of wave

length L at horns would built up rhythmic beachlobes with wave lengths of approximately L/2,(Baquerizo and Losada, 2001). These results pro-vide a solid basis for the hypothesis that at Car-chuna beach, the convergence of moderate windwave energy £uxes may be a mechanism for themaintenance of the horns. The spectral analysis ofa stretch of coastline con¢rms the existence ofsmaller beach cusps whose wavelength V is aboutL/2 of the lower edge wave modes.

Acknowledgements

The authors are indebted to the Commission ofthe European Community in the framework ofthe Energy, Environment and Sustainable Devel-opment Thematic Programme, for its supportwithin the Project HUMOR EVK3-CT-2000-00037. The ¢rst author has been partially fundedby the Ministry of Education (Spain), Research

MARGO 3329 20-6-03

M. Ortega Sa¤nchez et al. / Marine Geology 198 (2003) 209^223222

Grant AP2000-4121. We thank Dr. A.D. Shortand Dr. G.A. Ma¤lvarez for constructive reviews.Signi¢cant improvements to the original manu-script were suggested by an anonymous referee.

References

Aldaya, F., 1969. Los mantos Alpuja¤rrides al sur de SierraNevada. Acta Geol. Hisp. 4, 126^130.

Aldaya, F., 1970. La sucesio¤n de etapas tecto¤nicas en el do-minio alpuja¤rride. Cuad. Geol. 1, 159^181.

Aldaya, F., 1981. Hoja 20-44, Albun‹ol, Escala 1:50.000. Mapageolo¤gico de Espan‹a. Instituto geolo¤gico y minero.

Alonso, B., Maldonado, A., 1992. Plio^Quaternary margingrowth patterns in a complex tectonic setting: NortheasternAlbora¤n Sea. Geo-Mar. Lett. 12, 137^143.

Ashton, A., Murray, A.B., Arnault, O., 2001. Formation ofcoastline features by large-scale instabilities induced by high-angle waves. Nature 414, 296^300.

Bagnold, R.A., 1940. Beach formation by waves: Some modelexperiment in a wave tank. J. Inst. Civ. Eng. 15, 27^52.

Baquerizo, A., Losada, M.A., Losada, I.J., 2002. Edge wavescattering by a coastal structure. Fluid Dyn. Res. 31, 275^287.

Baquerizo, A., Losada, M.A., 2001. Propagacio¤n de ondas deborde a trave¤s de un sistema de espigones. In: Tejedor, B.,Pascual, J.R. (Eds.), F|¤sica de la Tierra, Oceanograf|¤aF|¤sica: Su investigacio¤n en Espan‹a. Servicio de Publica-ciones de la Universidad Complutense, Madrid, pp. 25^42.

Barcilon, A.I., Lau, J.P., 1973. A model for transverse bars.J. Geophys. Res. 78, 2656^2664.

Blumenthal, M., 1935. Reliefu«berschiebungen in den westli-chen Betischen Cordilleren. Geol. Med. Occid. 8, 3^28.

Boulin, J., Dimpault, P., Leroy, H., 1970. Observations ge¤o-logiques en Sierra de Lu¤jar (Cordille'res be¤tiques internes).Esp. B.S.G.F. 8, 585^591.

Bowen, A.J., Inman, D.L., 1969. Rip currents, part 2. Labo-ratory and ¢eld observations. J. Geophys. Res. 74, 5479^5490.

Bowen, A.J., Inman, D.L., 1971. Edge waves and crescenticbars. J. Geophys. Res. 76, 8662^8671.

Copponex, J.P., 1959. Observations ge¤ologiques sur les Alpu-jarrides occidentales. Bol. Inst. Geol. Min. Esp. 70, 79^208.

Dalrymple, R.A., 1975. A mechanism for rip current genera-tion on an open coast. J. Geophys. Res. 24, 3485^3487.

Dean, R.G., Maurmeyer, E.M., 1980. Beach cusps at PointReyes and Drakes Bay beaches, California. Proc. 17th Int.Conf. Coast. Eng. ASCE, Sydney, pp. 863^885.

Deigaard, R., Droonen, N., Fredsoe, J., Jensen, J.H., Jorgesen,J., 1999. A morphological stability analysis for a longstraight barred coast. Coast. Eng. 336, 171^195.

Giermann, G., Pfannestiel, M., Wimmeauner, W., 1968. Rela-

tion entre morphologie tectonique et volcanism en mer d’Al-boran (Mediterrene¤e occidentale). C.R. Somm. Soc. Geol.Fr. 4, 116^118.

Guza, R.T., Bowen, A.J., 1981. On the amplitude of beachcusps. J. Geophys. Res. 86, 4125^4132.

Guza, R.T., Inman, D.L., 1975. Edge waves and beach cusps.J. Geophys. Res. 80, 2997^3012.

Hasselmann, K., 1962. On the non-linear energy transfer in agravity-wave spectrum, 1. General theory. J. Fluid Mech.12, 481^500.

Herna¤ndez-Molina, F.J., Va¤zquez, J.T., Somoza, L., Rey, J.,1993. Estructuracio¤n sedimentaria de los cuerpos deltaicosholocenos del margen septentrional del mar de Albora¤n.Geogaceta 14, 40^45.

Hino, M., 1975. Theory on formation of rip currents andpointidal coast. Proc. 14th Conf. Coast. Eng., pp. 901^919.

Inman, D.L., Guza, R.T., 1982. The origin of swash cusps onbeaches. Mar. Geol. 49, 133^148.

Inman, D.L., Tait, R.J., Nordstrom, C.E., 1971. Mixing in thesurf zone. J. Geophys. Res. 76, 3493^3514.

Kirby, J.T., Dalrymple, R.A., 1983. A parabolic equation forthe combined refraction^di¡raction of stokes waves bymildly varying topography. J. Fluid Mech. 136, 543^566.

Komar, P.D., 1971. Nearshore cell circulation and distributionof giant cusps. Geol. Soc. Am. Bull. 82, 2643^2650.

Larson, M., Hanson, H., Graus, N.C., 1987. Analytical solu-tions of the one-line model of shoreline change. Tech. Rep.CERC-87^15, US Army Eng. Waterw. Exp. Stn., Vicksburg,MS.

Lippmann, T.C., Holman, R.A., 1989. Quanti¢cation of sandbar morphology: A video technique based on wave dissipa-tion. J. Geophys. Res. 94, 995^1011.

Longuet-Higgins, M.S., Stewart, R.W., 1962. Radiation stress-es and mass transport in gravity waves, with application to‘surf beats’. J. Fluid Mech. 13, 481^504.

Munk, W.H., 1949. Surf beats. Eos Trans. AGU 30, 849^854.Noda, E.K., 1974. Wave-induced nearshore circulation.J. Geophys. Res. 79, 4097^4106.

Pe¤rez-Belzuz, F., 1999. Geolog|¤a del margen y cuenca del Marde Albora¤n durante el Plio^Cuaternario: sedimentacio¤n ytecto¤nica. Instituto Ciencias del Mar 556.

Pulido, P., 2000. Contribucio¤n al conocimiento de los acu|¤feroscosteros complejos: Caso de Castell de Ferro. M.Sc. Thesis,Universidad de Almer|¤a, Almer|¤a.

Rosen, P.S., 1975. Origin and processes of cuspate spit shore-lines. In: Cronin, L.E. (Ed.), Estuarine Research 2, Ameri-can Press, New York, pp. 77^92.

Sonu, C.J., 1972. Field observation of nearshore circulationand meandering currents. J. Geophys. Res. 77, 3232^3247.

Tucker, M.J., 1950. Surf beats: Sea waves of 1 to 5 min.period. Proc. R. Soc. Ser. A. 202, 565^573.

Werner, B.T., Fink, T.M., 1993. Beach cusps as self-organizedpatterns. Science 260, 968^971.

Wilson, B.S., 1972. Seiches. Chow, Van T. (Ed.), Advances inHydroscience. Academic Press, New York.

MARGO 3329 20-6-03

M. Ortega Sa¤nchez et al. / Marine Geology 198 (2003) 209^223 223