The coast between Cabo de Santa Maria (Portugal)

The coast between Cabo de Santa Maria (Portugal) and Rabat (Morocco): a megasize headland-bay shoreline under control of the North Atlantic swell? Johannes Göttisheim & Burg W. Flemming

Geo-Marine Letters An International Journal of Marine Geology ISSN 0276-0460 Volume 33 Combined 2-3 Geo-Mar Lett (2013) 33:183-193 DOI 10.1007/s00367-012-0308-9

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Author's personal copy Geo-Mar Lett (2013) 33:183–193 DOI 10.1007/s00367-012-0308-9

ORIGINAL

The coast between Cabo de Santa Maria (Portugal) and Rabat (Morocco): a mega-size headland-bay shoreline under control of the North Atlantic swell? Johannes Göttisheim & Burg W. Flemming

Received: 27 January 2012 / Accepted: 5 September 2012 / Published online: 22 September 2012 # Springer-Verlag 2012

Abstract Equilibrium headland-bay beach systems have been mathematically described by logarithmic, parabolic and hyperbolic curve functions. The largest system of this type reported to date has a shoreline length of about 62 km. In the present study, an apparent headland-bay system is presented which has a shoreline length of about 500 km. It was discovered on satellite images, and is located between Cabo de Santa Maria in Portugal and the coastal city of Rabat in Morocco. It appears to be controlled by long-period North Atlantic swells diffracting around Cabo São Vicente at the southwestern tip of Portugal, in combination with SW–SE wind wave climates impinging on the northern shoreline of Cádiz Bay. The coast shows two marked departures from the equilibrium shoreline along its central section north and south of the Strait of Gibraltar, which are easily explained. Thus, the promontories to the north of the strait still exist because there has not been sufficient time to erode these back to the equilibrium shoreline since postglacial sea-level recovery. The coastal indentation to the south is explained by an insufficient sediment supply from terrestrial sources to facilitate the required beach accretion. Perfectly adjusted planimetric headland-bay shoreline shapes represent situations where wave orthogonals approach the coast at right angles everywhere, i.e. there is no longer any alongshore Responsible guest editor: I. Montoya-Montes J. Göttisheim Department of Geosciences, University of Bremen, P.O. Box 330440, 28334 Bremen, Germany B. W. Flemming (*) Senckenberg Institute, Suedstrand 40, 26382 Wilhelmshaven, Germany e-mail: [email protected]

sediment transport. Equilibrium shorelines form independently of the grain size of the beach sediment, whereas morphodynamic beach states are indirectly affected by the shoreline shapes because the latter are modulated by wave period and breaker height which also control the morphodynamic response of the beach in combination with the local grain size.

Introduction About 50 % of the world’s coastline consists of sandy beaches (Short and Masselink 1999). Many of these are interrupted by rocky headlands around which ocean swells are diffracted and refracted. As a consequence, the attached beach systems take on characteristic curved shapes which have variously been called zeta-curved bays (e.g. Halligan 1904; Silvester et al. 1980), logarithmic-spiral beaches (e.g. Krumbein 1944; Yasso 1965; Bremner and LeBlond 1974; Bremner 1991; Terpstra and Chrzastowski 1992), crenulate shaped bays (e.g. Silvester and Ho 1972; Gonzales and Medina 2001; Wang et al. 2008), headland-bay beaches (e.g. Yasso 1965; LeBlond 1979; Phillips 1985; Quevauviller 1988; Iglesias et al. 2009), parabolic shaped bays (e.g. Hsu and Evans 1989) or hyperbolic shaped shorelines (Moreno and Kraus 1999). Already de la Bèche (1833) and later de Beaumont (1845) recognized that a relationship existed between the angle of wave approach and the pattern of shoreline curvature (cf. Lewis 1938). However, this assessment was critically received at the time, and alongshore currents were instead favoured for a long time as the main control factors of asymmetrically curved shorelines (Gulliver 1899; Wheeler 1902). It was Krumbein (1944) who first suggested that the shapes of such shorelines could be mathematically described by logarithmic spirals, a classical example

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being Halfmoon Bay located along the coast of California south of San Francisco (Yasso 1965). Headland-bay beaches have not only attracted scientific interest because of their conspicuous shapes. In anticipation that such shorelines, be they logarithmic, parabolic, hyperbolic or otherwise shaped, were in a state of equilibrium, they aroused the interest of coastal engineers for the purpose of designing stable shorelines in the rear of breakwaters and other coastal structures (e.g. Silvester and Ho 1972; Walton 1977; Hsu and Evans 1989; Moreno and Kraus 1999; Gonzales and Medina 2001; Klein et al. 2003). The largest headland-bay beach system described to date is that of Algoa Bay along the south coast of South Africa (Bremner 1983, 1991), which stretches over an alongshore distance of about 62 km. In the present paper, a very large apparent headland-bay coastal system is described between Cabo de Santa Maria in southern Portugal and Rabat in Morocco, with a total length of almost 500 km (Fig. 1). The purpose of the study was to test whether the shape of this extraordinary large headland-bay coastal system could be mathematically described by a parabolic curve shape, and to investigate what relationships, if any, existed between the shoreline shape, the local beach sediment and beach morphodynamics.

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equation for the parabolic shape of a headland-bay beach in static equilibrium has the general form of Rn =RQ ¼ C0 þ C1 ðb =θn Þ þ C2 ðb=θn Þ2 in the case where θ≥ß, or Rn =RQ ¼ sin b =sinθ in the case where θ≤ß, with Rn being the radius to a point P along the curve at an angle θ, Rß the radius to the control point at angle θ to the direction of the predominant wave front, ß the angle defining the parabolic shape, θ the angle between the line from the focus to a point P along the curve and the direction of the predominant wave front, and C0, C1 and C2 the coefficients determined as functions of ß. To simplify the fitting procedure, the MEPBAY software of Klein et al. (2003) was applied on the basis of a Mercator projection which corresponds to that of standard nautical charts, the best fit being achieved with waves approaching from the WNW. In the present study, the parabolic fit was merely used to mathematically describe the conspicuously curved shoreline without assuming that the curve centre also represented the point of diffraction. The best fit was determined by a trial and error procedure. Sediment sampling and analysis

Materials and methods Shoreline shape analysis In the present case, the plan shape of the headland-bay shoreline has been mathematically described by fitting a parabolic curve as suggested by Hsu and Evans (1989). The model had been developed using the mean of a second-order polynomial fit to 27 headland-bay beaches assumed to be in static equilibrium. In the model, the curve centre is located at the tip of the headland, which is assumed to correspond with the physical location of the wave diffraction point. The

Fig. 1 The mega-sized headland-bay coastal system between Cabo de Santa Maria (Portugal) and Rabat (Morocco) as seen on a satellite image

To assess the grain-size characteristics of the beach sediments at various locations along the coast, a total of 54 sediment samples were collected between Cabo de Santa Maria in Portugal and Rabat in Morocco in the summer months of 2009 (Fig. 2). The spacing between samples was partly dictated by the nature of the coast (rocky or sandy) and partly by its accessibility from land, the latter having been particularly limiting in Morocco where the coast was often inaccessible over long distances. In all cases, the sediment samples were taken at mid-tide level to a depth of 1 cm at most, the sample positions being recorded by means of a portable GPS.

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the beach slope was to estimate the prevailing morphodynamic beach state at each sample location in relation to the mean grain size. The close correspondence between grain size, beach slope and degree of wave exposure (very sheltered to very exposed) had already been demonstrated by Bascom (1951, 1964) and Wiegel (1964). The original grain size and slope scaling was later rescaled by Flemming and Fricke (1983) from mm- to phi-values and from slope ratios to degrees respectively. The positions of the trend lines demarcating sheltered (reflective) and exposed (dissipative) conditions in beach-slope versus grain-size plots follow the beach morphodynamic classification of Short (1979) and Wright et al. (1985). The trend lines have recently been corrected and mathematically expressed by Flemming (2011). The beach state classification is based on the non-dimensional, socalled Dean parameter (Dean 1973) defined as Bs 0Hb / (ws T), where Bs is the morphodynamic beach state, Hb the breaker height (m), ws the settling velocity of the beach sand (ms–1), and T the wave period (s). Beach states of Bs 01 and 6 as dissipative (high energy), and Bs 02–5 as intermediate energy states. Wave diffraction/refraction modelling Fig. 2 Locations of beach sediment samples and beach slope measurements. Note that the sample density per unit shoreline length was dependent of the nature of the coast (rocky or sandy) and its accessibility from land. The symbol and colour schemes serve to identify the different shoreline sections in Figs. 7 and 8

The sediment samples were subsequently washed in fresh water to remove the salt, dried and split down to subsamples of about 1 g with a mechanical sample splitter in preparation for settling tube analysis. The grain-size distributions of the sample splits were measured in a self-recording, highresolution settling tube (Macrogranometer™, Brezina 1979; Flemming and Ziegler 1995), the settling velocities being converted to equivalent settling diameters before textural grain-size parameters were calculated on the basis of percentile statistics (Brezina 1979; Blott and Pye 2001). In the case of coarse sediments, several splits were settled in succession and the individual results summed before statistical processing, in order to avoid distortions in the grain-size distribution curves caused by overall insufficient grain numbers in the samples (cf. Brezina 1979).

Regional wave data, in particular long-period swells from the north-westerly sector of the North Atlantic Ocean, were extracted from the Wavewatch3 “hindcast reanalysis” wave climate dataset, which is based on a 12-year observational period (NWS 2009). The boundary conditions were defined by median values for the primary peak period (Tp 09.9 s), significant wave height (Hs 02.15 m), peak wave direction (Dp 0313°) and directional spreading (Ds 092°), the diffraction point being Cabo São Vicente at the south-western tip of Portugal. The extracted wave data were then modelled by means of the Delft 3D software (Delft Hydraulics 2007) using a grid resolution of 5 km for the regional model, and 1.667 km for the nested grid focusing on the northern Cadiz embayment using the same boundary conditions as for the regional model. Physical process calculations included depthinduced breaking and bottom friction after Hasselmann et al. (1973) and diffraction/refraction processes after Holthuijsen et al. (1994). The local bathymetry was extracted from the Gebco08 bathymetric dataset (BODC 2012), which has a grid resolution of 926 m.

Beach slope measurements Results At each sampling point, at least five beach slope measurements were carried out using a so-called Bevel-Box (ELV, Germany), which is an electronic inclinometer capable of measuring slope angles at a resolution of 0.1°. After discarding any extraneous values, the remaining ones were in each case summed and averaged. The purpose of measuring

Shoreline shape analysis The parabolic curve fitted to the shoreline between Cabo de Santa Maria and Rabat is illustrated in Fig. 3. The centre for this best fit parabolic curve is located at


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Fig. 3 Mega-sized parabolic curve describing the planimetric shape of the headland-bay shoreline between Cabo de Santa Maria (Portugal) and Rabat (Morocco). Note the excellent fits along the northern and southern sections but the divergence along the central section dominated by rocky promontories in the north, the Strait of Gibraltar in the centre, and some structural offsets in the south. The areas outlined in yellow and labelled a–e are addressed in the main text. Also marked is the location of wave recordings (yellow star), and the mean winter wave period and significant wave height measured at this location

36°34.24 N and 07°24.42 S at a water depth of about 600 m, and remote from any headland which could be construed to act as a diffraction point. The total shoreline distance along the curve amounts to about 500 km. Although the overall fit is quite striking, there are two major departures from the equilibrium shoreline trend. Besides the gap caused by the Strait of Gibraltar, the most pronounced deviations are located north and, to a lesser extent, south of the strait, as outlined by the yellow ellipses labelled c and d in Fig. 3. The coastline protrusions to the north of the strait (area c in Fig. 3) are caused by the rocky promontories north and south of Cádiz, whereas the coastal indentation to the south of the strait (area d in Fig. 3) comprises a stretch of sandy shoreline in NW Morocco which is displaced towards the east (landwards) relative to the theoretical equilibrium shoreline, the recess increasing stepwise at major structural offsets at Larache and about halfway between Larache and Cape Espartel, which marks the entrance to the Strait of Gibraltar. Another peculiarity of the coastline is the fact that the more sheltered, up-coast section of the shoreline does not end in the rear of a rocky headland but is instead formed by

the Ria Formosa barrier-island chain, which ends in a cuspate sandy headland known as the Cabo de Santa Maria (area b in Fig. 3). Whereas the eastern part of this barrier shoreline accurately follows the inner part of the parabolic curve, it begins to depart from it as Cabo de Santa Maria is approached. The sandy cape of this enigmatic depositional feature evidently overshoots the equilibrium shoreline, which suggests that the forces which control the equilibrium shape of the shoreline are unable to move the sediment eastwards at the same rate as it is being supplied to the cape from the west. Also indicated on Fig. 3 is Cabo São Vicente (headland in area a) around which the North Atlantic swell is initially diffracted and refracted before impinging on various parts of the shoreline to the east and southeast. To investigate the degree to which the actual shoreline follows the parabolic curve, the lengths of the radial vectors between the curve centre point and the shoreline were determined at 5° intervals, beginning at 0° along the baseline in the northwest and ending at 180° near Rabat in the southeast (37 vectors in all). In a plot of radial vector length versus radial angle, all points representing the theoretical


equilibrium shoreline would form a highly correlated progression representing the parabolic curve, whereas those exceeding or falling short of the equilibrium shoreline would plot above or below this progression. In the present case, distinctly offset vector points lie along the 120-kmlong section between the mouth of the Guadalquivir River and northern Morocco, which represents ~31 % of the total shoreline length. To calculate the best fit correlation, the offset vectors (11 of the 37) were omitted from the regression analysis (Fig. 4). The resulting correlation between the remaining 26 points is almost perfect, the coefficient of determination r2 of 0.9997 testifying to the excellent fit. An important point to note here is that seven of the vector points, which together cover an alongshore distance of about 180 km or ~36 % of the total shoreline length between Cabo de Santa Maria and Rabat, lie along the Moroccan coast. The other highly correlated section, stretching between Cabo de Santa Maria and the mouth of the Guadalquivir, covers a distance of 165 km or ~33 % of the total shoreline length. The highly correlated shoreline sectors together therefore cover 345 km or 69 % of the total shoreline length. To test the goodness of fit between the theoretical equilibrium shoreline defined by the parabolic curve and the actual shoreline, the distance between the centre point and the curve was plotted against the distance between the centre point and the actual shoreline (Fig. 5) using the same 5° intervals as in the previous plot. Again, the points of obvious departure were excluded from the regression analysis. The remaining 80 % of the points fall onto the parabolic curve with a correlation r of 0.9942 (coefficient of determination r2 of 0.9885). Only a few

Fig. 4 Length of radial vector between the curve centre and the shoreline plotted against the radial angle at 5° intervals beginning at 0° in the northwest of the baseline (Portugal) and ending at 180° in the southeast near Rabat, Morocco. Values obviously departing from the equilibrium shoreline along the central section of the coast (open circles) have been omitted from the regression analysis in order to highlight the best possible fit. Note that the best fit includes the northern (Portuguese/Spanish) and the southern (Moroccan) shoreline

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Fig. 5 Distance from the curve centre to the coastline plotted against the distance from the curve centre to the parabolic curve as a goodness of fit test between the two. Again, the values from the central section have been omitted from the regression analysis. Note the excellent fit to the two spatially separated shoreline sections

vectors can be seen to depart very slightly due to local influences like protruding river deltas or small structurally controlled offsets in the coastline. A few examples of such local departures from the equilibrium shoreline defined by the parabolic curve are illustrated in Fig. 6, the locations being indicated on the regional map in Fig. 6a. Also shown in Fig. 6a are two parabolic curves, the green curve representing the one used in this study, the yellow one being an alternative but very much inferior fit which was consequently abandoned. Figure 6b shows the protruding delta of the Rio Guadiana and the location of a 16th Century watch tower (Torre Canela) which had been erected along a former shoreline. Evidently, the shoreline (broken yellow line) has prograded substantially since that time by sediment accretion probably derived from the river and/or littoral drift from the west. This vividly illustrates the interference by the river and, in more recent times, also by anthropogenic interventions (breakwaters lining the inlet). Figure 6c shows the mouth of the Guadalquivir River. Of interest here are the cuspate spitbar and the structural offset marking the beginning of the rocky promontory north of Cádiz, both features protruding from the equilibrium shoreline trend. Figure 6d depicts the engineered river mouth at Kenitra along the Moroccan coast. As in the case of the Guadiana and Guadalquivir, the mouth has a cuspate shape which protrudes seawards from the equilibrium shoreline. Of interest here is the symmetry of the cuspate shoreline. The absence of any offset in the beaches north and south of the breakwaters lining the inlet suggests that there is no net alongshore sediment transport in any preferred direction, which is in line with an equilibrium shoreline model where wave orthogonals approach the


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Fig. 6 Satellite images of places of special interest discussed in the main text. a Map showing the locations of the places of interest (white arrows). b Guadiana River delta. Note the location of a coastal watch tower (Torre Canela) dating back to the 16th Century (yellow star) and the position of the equilibrium shoreline (broken yellow line). c Mouth of the Guadalquivir River. Note the departures from the equilibrium shoreline (broken yellow line) by the protruding river-mouth spitbar

and rocky promontory. d Mouth region of the estuary at Kenitra, Morocco. Note the cuspate shoreline protruding seawards of the equilibrium shoreline (broken yellow line), and the symmetrical beach alignment north and south of the breakwaters lining the inlet. e Structural offset in the coastline at Larache, Morocco. From this point northwards, the coastline increasingly departs from the equilibrium shoreline (broken yellow line)

shore at right angles everywhere. Finally, Fig. 6e illustrates a small structural offset at the coastal town of Larache situated about 60 km south of Cape Espartel along the Moroccan coast. Although the offset at this location is quite small, the sediment supply to the coast is evidently so small that the beach has not been able to prograde sufficiently to reach the position of the equilibrium shoreline. This offset gradually increases northwards until the departure from the equilibrium shoreline reaches more substantial proportions at another structural offset about halfway to Cape Espartel. The general sediment deficit along this part of the Moroccan coast is explained by the semiarid climate and correspondingly low rainfall and, hence, low river discharge.

in grain size are unclear but probably related to local source effects. Overall, the mean trend suggests a slight fining southwards towards Rabat, although this trend is very subtle and practically negligible if the four extraneous samples and the two samples nearest to Cabo de Santa Maria are omitted.

Sediment characteristics The trend of mean grain size versus distance along the shoreline is illustrated in Fig. 7, the differently coloured symbols coinciding with those of Fig. 2. Of the sampled beach sediments, 77 % fall into the medium sand, 14 % into the fine sand, and 9 % into the coarse sand categories. A few values can be regarded as being extraneous (encircled symbols), i.e. they depart substantially from the trend observed up- and down-coast of their respective sample locations. The jumps

Fig. 7 Plot of mean grain size versus alongshore distance. Symbols and colours correspond to those in Figs. 2 and 8. Extraneous data points are encircled


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The latter two samples are particularly interesting because they were taken close to the sandy cape which was shown to depart from the equilibrium shoreline. They belong to the group of samples (blue squares) taken along the barrier-island shore between the sandy cape and the mouth of the Rio Guadiana, and which reveal a down-coast fining trend (with the exception of one extraneous sample). This would suggest that the much coarser sands near the cape signify a lag effect, finer components having been winnowed and transported eastwards towards the mouth of the Guadiana River. With the exception of such local alongshore size-sorting, the variability in mean grain size along the coast as a whole probably reflects the variability in the regional geology and the composition of local source rocks from which they were derived.

conditions are generally higher than further north where the north-westerly swell has been strongly diffracted/refracted and the energy hence spread over larger alongshore distances. The only exception is the sample location closest to Cabo de Santa Maria, which should strictly be the most sheltered part of the shoreline. It would appear that, because of the coarse beach material at this location, summer waveenergy levels are too low to transform the dissipative beach morphology attained in winter back to a reflective state in summer. This emphasizes the fact that this part of the coast is evidently not in equilibrium.

Beach morphodynamics

In the present study, preference was given to a parabolic fit because the availability of a computer program simplified the fitting procedure considerably, and because a main objective of the study was to test whether the 500-km-long shoreline between Cabo de Santa Maria in Portugal and Rabat in Morocco could, in principle, be described by a mathematical model, especially when omitting the obvious departures from the fitted curve in the central section. The results clearly support the concept. As shown in Fig. 3, the shape of the shoreline between Cabo de Santa Maria in Portugal and the coastal city of Rabat in Morocco is mathematically well represented by a parabolic curve. The departures from this shape north and south of the Strait of Gibraltar are easily explained. North of the strait, the time which has passed since the postglacial sea-level recovery has evidently been too short for the waves to cut the rocky promontories back to the line of equilibrium defined by the parabolic curve. This is documented by the ongoing severe erosion along this coastal sector, which requires frequent artificial nourishment (e.g. Muñoz-Perez et al. 2001). South of the Strait of Gibraltar, along the northernmost structurally recessed coastal sector, sediment supply is too small to facilitate beach progradation up to the theoretical equilibrium shoreline. Instead, the beaches are characterized by neutral sediment budgets, as revealed by their morphodynamic response to the seasonal wave climate (Taaouati et al. 2011). It was intimated above that the log-spiral method failed when applied to the entire length of the coastal system. This is illustrated in Fig. 9 where the parabolic fit (solid line) is contrasted with the corresponding log-spiral fit (broken blue line). In the case of a log-spiral, the shoreline trend should follow the straight-line relationship indicated on the semilog plot. Initially, the log-spiral fit is arguably as good as the parabolic fit but, from an alongshore distance of about 165 km approximately coinciding with the mouth of the Guadalquivir, the log-spiral begins to fall increasingly short of the actual shoreline trend. A similar observation was

The scatter diagram of mean grain size versus beach slope reveals the morphodynamic beach states which prevailed at the different sample locations along the shore at the time of sampling (Fig. 8). The differently coloured symbols correspond to those in Fig. 7 (and in Fig. 2). Overall, reflective (or low-energy) conditions prevailed at 23 sample locations (Bs 01 and 6) at nine locations. The predominance of reflective or almost reflective conditions is not surprising, as the measurements were made in the lower-energy summer season. It is interesting to note, however, that all the dissipative sites are located in the central and southern sectors of the coast, i.e. where energy

Fig. 8 Plot of mean grain size in phi versus beach slope angle in degrees. Symbols and colours correspond to those in Figs. 2 and 7. The diagonal black lines separate the dissipative and reflective domains from the intermediate domain

Discussion


Fig. 9 Length of the radial vector between the curve centre plotted against the radial angle at 5° intervals (cf. Fig. 4): the straight broken blue line in this semi-log plot represents the trend which would be described when fitting a log-spiral to the shoreline. Note the good fit of the log-spiral to the first 150 km or so, and the progressive divergence from the actual shoreline beyond that point

previously made on a different coastal system by Hsu et al. (1987). While the log-spiral approach evidently fails beyond a certain alongshore distance, it is not clear at this stage how well approaches other than the parabolic one would perform at this scale. It would therefore be of considerable interest to also test the performance of the hyperbolic-tangent model of Moreno and Kraus (1999) and the neural network model of Iglesias et al. (2009). This could help to clarify once and for all which of these options is the most flexible under a variety of conditions, keeping in mind that the mathematical expression as such, while being a useful tool for the purpose of artificial shoreline design, does not explain why shorelines take on such shapes. Attempts to elucidate the physical basis were made by May and Tanner (1972) on the basis of a littoral power gradient model, and Walton (1977) who used a wave-height energy distribution model by which shoreline shapes very similar to those described by the mathematical procedures outlined above were generated. The excellent correlation between the shoreline trend and the parabolic fit strongly suggests that the long-period North Atlantic swell climate, characterized by periods between 9 and 15 s, controls the overall shape of the shoreline between Cabo de Santa Maria in the north and Rabat in the south. In Cádiz Bay the diffracted long-period swell is augmented by shorter-period waves generated by south-westerly and south-easterly seasonal winds (Granja et al. 1984). The predominance of long-period swells from the NW sector during the winter months has been confirmed by deepwater wave recordings off the northwest coast of Morocco (Fig. 2; Taaouati et al. 2011). It is interesting to note that the initial point of wave diffraction is not located at the arbitrary parabolic curve centre but evidently at the south-western tip of Portugal (Cabo São Vicente). This raises the question as

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to why the shoreline is nevertheless so well represented by a parabolic curve. To test the hypothesis that Cabo São Vicente is the original diffraction point, a large-scale wave diffraction/refraction exercise was carried out using the Wavewatch3 wave climate dataset in combination with the Delft 3D modelling software. The result of this exercise is illustrated in Fig. 10. It confirms that diffraction and nearshore refraction of north-westerly swells cause wave orthogonals, here represented by the direction of the arrows, to be bent in such a way as to approach the parabolic shoreline more or less at right angles almost everywhere. To highlight this, the inner part of Cádiz Bay was modelled at a higher grid resolution nested within the regional model. For better visualization, a blow-up of this part is reproduced in the inset of Fig. 10. It can be seen that, along this part of the curved shoreline, the orthogonals of the diffracted North Atlantic swell are less well aligned, which calls for additional factors to explain the parabolic fit along this coastal sector. An explanation is provided by local wave modelling, which also throws some light on the enigmatic depositional system of the Ria Formosa. Thus, Granja et al. (1984) have demonstrated that the eastward littoral drift associated with waves approaching from the SW is partly terminated at Cabo de Santa Maria by waves approaching from the SE and S. As a consequence, most of the sand supplied from the west has accreted in the form of the enigmatic cuspate headland characterizing the Ria Formosa. At the time of their study, those authors did not consider any contribution by diffracted ocean swells approaching from the NW Atlantic. Excess material bypasses Cabo de Santa Maria and then continues towards the east, thereby explaining the accretion west of the breakwater lining the mouth of the Rio Guadiana and also the eastward-facing sand spits at the mouths of the Rio Piedras and Rio Odiel (Borrego et al. 1995). It is this ongoing process which causes the disequilibrium of the shoreline close to Cabo de Santa Maria. Furthermore, the downdrift fining of the beach sediment east of the cape recorded in the present study had already been noted by Bettencourt (1988). It can therefore be inferred that the interaction between the diffracted swell and the local wave climate seasonally generated by south-westerly to southeasterly winds is responsible for the parabolic alignment of the shoreline in the northernmost sector of the larger-scale headland-bay coastal system. The superimposition of several wave systems may explain the fact that the centre of the parabolic fit does not represent the associated diffraction point. Nevertheless, the resulting shape of the shoreline closely follows a parabolic curve, the excellent fit mitigating against a fortuitous coincidence. With respect to the mean grain size of the beach sediment along the shore, the relatively large and irregular scatter in the values over the entire distance recorded in Fig. 7


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Fig. 10 Regional wave diffraction/refraction model of the eastern North Atlantic seaboard from southern Portugal to Morocco. Note the more or less orthogonal approach of the diffracted waves to the curved shoreline east of Cabo de Santa Maria

suggests that grain size per se has no influence on the shape of the equilibrium shoreline. This is also confirmed by the observation that even extensive occurrences of cobble-sized material along headland-bay beaches—e.g. along the beach of Bahía Coquimbo in Chile (ca. 20 km in length)—do not alter the trend of the equilibrium shoreline (B. Flemming, unpublished data). The evolution of equilibrium shorelines thus depends predominantly on the wave period (T), the angle of wave approach and the degree of wave diffraction/refraction, the coastal geology playing an additional but secondary role. Correspondingly, the morphodynamic state of a beach at any particular location and time along the shoreline is indirectly dependent on the planimetric shape which, over long time periods, is modulated by the dominant wave periods and heights. As illustrated in Fig. 8, the morphodynamic beach state can range from dissipative to reflective, depending on the local grain size and the shortterm energy input defined by the coincidental wave periods and breaker heights at the time of observation. In this context, it must be remembered that the waves impinging on a beach may, in addition, have been modulated by the offshore seabed morphology, wave orthogonals converging over and behind shoals and banks, causing energy to

increase per unit area, and diverging when crossing valleys or depressions, causing energy to decrease per unit area.

Conclusions From the results of this study the following major conclusions are drawn. 1. The shoreline between Cabo de Santa Maria in Portugal down to Rabat in Morocco shows an essentially perfect fit to the parabolic shape model over 69 % of cumulative distance. The departure from the curve along its central section can be rationally explained by the promontories around Cádiz, the open Strait of Gibraltar, and the structurally recessed, northernmost coastal section of Morocco. 2. Wave diffraction/refraction modelling confirms that the orthogonals of the diffracted NW swells approach the parabolic shoreline more or less at right angles everywhere but that, in the northern sector, the additional superimposition of seasonal SW–SE wind wave climates is required to generate and maintain the observed parabolic shoreline shape.


3. For such a large headland-bay system, the application of a log-spiral fit is evidently inappropriate as it progressively diverges from the equilibrium shape after about 165 km, although it is an acceptable alternative for the coastal sector between Cabo de Santa Maria and the mouth of the Guadalquivir River. 4. Perfect planimetric curve shapes of headland-bay shorelines indicate situations where wave orthogonals approach the coast at right angles everywhere, i.e. there is no longer any alongshore sediment transport. 5. The equilibrium planimetric shape is independent of the grain size of the beach sediment. It mainly depends on the wave period (T), the angle of wave approach and the degree of wave diffraction/refraction, wave (or swell) height additionally modulating the overall energy input. 6. The morphodynamic beach state is indirectly affected by the spiral shape because the latter is modulated by wave period and breaker height, both of which also control the morphodynamic response of the beach in combination with the local grain size, albeit on different time scales. Acknowledgements We thank the Senckenberg Institute of Natural History, Frankfurt (Germany) for providing financial support which facilitated the collection of the beach samples in Portugal, Spain and Morocco. We also gratefully acknowledge constructive comments by two anonymous reviewers.

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