Coastal Engineering 63 (2012) 48–61

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Coastal Engineering

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Overwash threshold for gravel barriers

Ana Matias a, Jon J. Williams b,⁎, Gerhard Masselink b, Óscar Ferreira a

a CIACOMAR/CIMA, Universidade do Algarve, Campus de Gambelas, 8005-139 Faro, Portugalb School of Marine Science and Engineering, University of Plymouth, Plymouth, PL4 8AA, UK

⁎ Corresponding author. Tel.: +351 289800969; fax:E-mail addresses: ammatias@ualg.pt (A. Matias), jon

(J.J. Williams), g.masselink@plymouth.ac.uk (G. Masseli(Ó. Ferreira).

0378-3839/$ – see front matter © 2011 Elsevier B.V. Alldoi:10.1016/j.coastaleng.2011.12.006

a b s t r a c t

a r t i c l e i n f o

Article history:Received 21 November 2011Accepted 9 December 2011Available online 23 January 2012

Keywords:BARDEXOvertoppingOverwashThresholdRunupCoastal hazards

This paper uses results obtained from the large-scale BARDEX experiments undertaken in the Delta flume toinvestigate the morphological response of a prototype gravel barrier to wave and tidal forcing during over-wash conditions. Gravel barrier behaviour depends upon a number of factors such as sediment properties(porosity, permeability, grainsize), geological setting and wave climate. Since overwash processes areknown to control short-term gravel barrier dynamics and long-term barrier migration, the development ofa robust quantitative method to define the critical conditions leading to gravel barrier overwashing is impor-tant both for scientific and practical management purposes. It is known that when wave runup exceeds thebarrier crest elevation, three outcomes are possible: 1) insignificant morphological change, when waverunup just overtops the barrier crest and flow velocities are very weak; 2) overtopping, resulting in accretionon the barrier crest region and barrier stabilisation; and 3) overwashing, resulting in erosion, lowering of thecrest region and ultimately to barrier inundation. This study provides an insight into the critical conditionsthat distinguish these two possible outcomes and to the different mechanisms that provide the required positiveand negative feedbacks to the sediment dynamics. In order to define the overwash threshold condition, and topredict the morphological outcome of particular overwash events, use is made here of the Overwash Potential(OP), defined as the difference between the wave runup and the barrier crest elevation. To make effective useofOP it is necessary to identify a reliable runup predictor. Following tests using 12 runup equations the Stockdonet al. [Stockdon, H.F., Holman, R.A., Howd, P.A., Sallenger, A.H., 2006. Empirical parameterization of setup, swash,and runup. Coast. Eng., 53, 573–588] approach has been identified as the best predictor of runup conditions nec-essary to generate positive values of OP, with overtopping and overwashing predicted to occur for average OPvalues of 0.2 m and 0.5 m, respectively. The use of OP values provides a practical means by which to identify po-tential coastal hazards associated with gravel barrier overwash processes and is considered to have a range ofpractical coastal management applications.

© 2011 Elsevier B.V. All rights reserved.

1. Introduction

Gravel barrier beaches are widespread on the wave-dominatedcoastlines of Northern Europe (especially Russia, UK and Ireland),Canada, USA, Japan, New Zealand and Latin America (Buscombe andMasselink, 2006). Gravel-dominated coastal deposits occur in a varietyof settingswhere sediment supply andwave energy favour the accumu-lation of coarse debris in the shore zone (Orford et al., 2002). These in-clude sites with local sources in active erosion of rock cliffs andoutcrops, producing cliff base and pocket beaches in various lithologicaland tectonic settings around the world (Davies, 1972). Gravel beachsediments are spatially differentiated in terms of both size and shapeto a greater degree (Bluck, 1967) than sand beaches. The step, cusphorns, strandlines and berms are composed of coarser sediment than

+351 289800069..j.williams@plymouth.ac.uknk), oferreir@ualg.pt

rights reserved.

foreshores, although a number of levels of textural zonation withinthis general case may be discernible as sediments are distributed con-tinually (Buscombe and Masselink, 2006). Overwash processes playan important role in the evolution of these beaches, which can causethem to migrate onshore over time by the “rollover” mechanism (e.g.,Carter and Orford, 1993; Orford and Carter, 1982; Orford et al., 1995;Orford et al., 2002). This mechanism involves onshore-directed sedi-ment transport driven by storm waves from the front of the barrier,across the barrier crest and deposition at the back of the barrier in theform of washover deposits. By controlling the rate and spatial patternof gravel barrier rollover, storm waves have been regarded as drivingshort-term (annual to decadal) gravel barrier migration (Orford et al.,1995). When the storm-induced sea level is sufficient to completelysubmerge a barrier island, the flows over the barrier are no longer sim-ple overwash, and inundation regime is set (Sallenger, 2000). Overwashcan also contribute to other patterns of barrier evolution, such asbreaching (Bray and Duane, 2001), barrier breakdown (e.g., Pye andBlott, 2009), outlet formation (Hart, 2007) and outlet closure (e.g.,Orford et al., 1988). Despite the importance in determining the dynamic

Table 1BARDEX programme for overwash series, with (+) on tests with overtop and over-wash. hS=mean sea level; hL=mean lagoon level.

Series Run hS (m) hL (m) Hs (m) Tp (s) Overtop Overwash

E1 E1A 3.000 2.50 1.00 4.5E1B 3.125 2.50 1.00 4.5 +E1C 3.250 2.50 1.00 4.5 +E1D 3.375 2.50 1.00 4.5 +E1E 3.500 2.50 1.00 4.5 +E1F 3.625 2.50 1.00 4.5 +

E2 E2A 3.250 2.50 1.05 4.5E2B 3.250 2.50 1.10 4.5E2C 3.250 2.50 1.15 4.5E2D 3.250 2.50 1.20 4.5

E3 E3A 3.250 3.25 1.00 4.5E3B 3.375 3.25 1.00 4.5E3C 3.500 3.25 1.00 4.5E3D 3.625 3.25 1.00 4.5 +E3E 3.750 3.25 1.00 4.5 +

E4 E4A 3.625 3.25 1.00 4.5 +E4B 3.500 3.25 1.00 4.5 +E4C 3.375 3.25 1.00 4.5 +E4D 3.250 3.25 1.00 4.5E4E 3.375 3.25 1.00 4.5E4F 3.500 3.25 1.00 4.5 +E4G 3.625 3.25 1.00 4.5 +E4H 3.750 3.25 1.00 4.5 +

E5 E5A 3.500 3.25 0.80 4.5E5B 3.500 3.25 0.90 4.5E5C 3.500 3.25 1.00 4.5 +E5D 3.500 3.25 1.10 4.5 +E5E 3.500 3.25 1.20 4.5E5F 3.500 3.25 1.30 4.5

E6 E6A 3.250 3.25 1.00 6.0 +E6B 3.375 3.25 1.00 6.0 +E6C 3.500 3.25 1.00 6.0 +E6D 3.625 3.25 1.00 6.0 +

E7 E7A 3.000 3.25 1.00 7.0 +E7B 3.125 3.25 1.00 7.0 +

E8 E8A 2.500 3.25 1.00 8.0E8B 2.625 3.25 1.00 8.0E8C 2.750 3.25 1.00 8.0E8D 2.875 3.25 1.00 8.0 +E8E 3.000 3.25 1.00 8.0 +E8F 3.125 3.25 1.00 8.0 +E8G 3.250 3.25 1.00 8.0 +E8H 3.375 3.25 1.00 8.0 +E8I 3.500 3.25 1.00 8.0 +

E9 E9A 3.000 3.25 0.80 8.0 +E9B 3.125 3.25 0.80 8.0 +E9C 3.250 3.25 0.80 8.0 +E9D 3.375 3.25 0.80 8.0 +E9E 3.500 3.25 0.80 8.0 +E9F 3.625 3.25 0.80 8.0 +E9G 3.750 3.25 0.80 8.0 +

E10 E10A 3.750 3.25 0.80 8.0 +… … … … … … …

E10K 3.750 3.25 0.80 8.0 +

49A. Matias et al. / Coastal Engineering 63 (2012) 48–61

behaviour of gravel beaches,fieldmeasurements of overwash processesare much less common for gravel than for sandy barriers, and the over-wash threshold condition on gravel beaches is not well-established. Im-portant studies of this subject from thefield are reported by Orford et al.(1999), Orford et al. (2003), Bradbury et al. (2005) and Lorang (2002),and in the laboratory by Obhrai et al. (2008).

The occurrence of overwash in areas of human occupation repre-sents a hazard that can lead to damage to coastal properties and infra-structure, intrusion of salt and sand into agriculture soils, interferencewith back-barrier channel navigation, and loss of human life (Matiaset al., 2009). Although several approaches to predict overwash occur-rence have been proposed, most apply only to sandy coasts. The mostused method for predicting overwash occurrence in sandy shores isSallenger's (2000) “Storm Impact Scale”. This method has been testedand discussed by others, namelyWetzell et al. (2003) and Stockdon etal. (2007). Modelling efforts in overwash prediction for sandy shoreshave dealt mainly with profile response to overwash (e.g., SBEACH,Larson et al., 2005; LOBARM, Tuan et al., 2007; XBEACH, Roelvink etal., 2009), and overwash sediment transport rates (e.g., Kobayashi etal., 1996; Nguyen et al., 2006). An empirical method to predict thethreshold for breaching of gravel barrier based on extensive fieldand laboratory data has been developed by Bradbury (2000). Not-withstanding the usefulness of this approach, it neglects to definethe overwash threshold condition. As overwashing has a higher like-lihood of occurrence than breaching, and has potentially damagingeffects upon human occupation behind the barrier, it is helpful for en-gineering and coastal management purposes to define this conditionfor a wide range of forcing conditions that include waves, tides andsurges.

Overwash mainly occurs during storms and accurate field measure-ments of the key parameters are therefore hazardous and difficult to ob-tain. Since large-scale flume experiments allow manipulation of thephysical forcing, they can provide a valuable complement to field data-sets of overwash processes. In this paper, the overtopping/overwashingsimulations of Test Series E in the BARDEX experiments (Williams et al.,2011-this issue) are described and discussed with the aim of improvingour knowledge and understanding of overwash processes on gravelbeaches, and to develop and test quantitative methods for definingthe overwash threshold.

2. Methods

2.1. Overwash experiments

Experiments to study gravel barrier overwash processes were un-dertaken at proto-type scale in the Delta flume (The Netherlands) dur-ing the BARDEX project (Williams et al., 2011-this issue). A gravelbarrier (35 m long, 5 m wide and 4 m high, with seaward and lagoonfacing slopes of 1V/8H and 1V/4H, respectively) composed of sub-rounded gravel (D50=11mm) was emplaced in the flume with themid-barrier crest located at a distanceX=c. 95 m from thewave paddle(Williams et al., 2011-this issue). Theflume volume between the barrierand thewave paddle (hereafter called ‘sea’) was filled to a depth hS andthe region behind the barrier (hereafter called ‘lagoon’) was filled to adepth hL. The barrier was permeable, therefore seepage through thebarrier occur whenever hS and hL were different (Turnerand Masselink, this issue).

The morphological response of the barrier due to overwash wasstudied by exposing the barrier to variable wave and sea-level condi-tions. Test Series E consisted of 10 test sequences (E1 to E10; Table 1),each comprising a number of 15-min wave runs. To achieve over-washing conditions, either hS was gradually increased in steps of0.125 m (Tests E1, E3, E4, E6, E7, E8 and E9), or the significant waveheight Hs was increased in small increments (0.05 m for Series E2;0.1 m for Series E5) until overwash occurred. Hs values ranged from0.8 m to 1.3 m and peak wave period Tp ranged from 4.5 s to 8.0 s

(Table 1), and all wave conditions conformed to a standard JONSWAPspectrum. In Test Series E10, sea level and waves were kept constantto study the morphological response of the barrier under fully devel-oped overwash conditions. At the end of Test Series E2, with hL at2.5 m and hS at 3.0 m, significant outflow through the back-barrierwas observed. Owing to the instability this promoted hL was increasedto 3.25 m for all subsequent overwash tests. In common with all otherBARDEX tests, the barrier was not reshaped between tests.

Barrier morphology was surveyed before and after each run, usinga roller and actuator which followed the bed profile from an overheadcarriage, thereby allowing profile measurement of the subaerial andsubmerged part of the beach. The subaerial back-barrier was alsocontinuously monitored at 4 Hz using acoustic sensors deployed at0.5 m spacing that measure water depth and bed level (Masselinkand Turner, 2011-this issue; Turner et al., 2008). In this paper, only

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the bed-level sensor measurements collected at X=99.5 m (locatedaround the barrier crest during the final run of Test Series E) arereported. For all Test Series E experiments, the position and elevationof the barrier crest (hc) were determined at the end of the runs,whereby the crest was defined as the location of the profile withthe maximum elevation (Fig. 1a). Beach slope was calculated forthe barrier section between mean water level and the base of thebarrier crest. For the purpose of this study, all beach area abovemean water level was considered the area of significant swash activ-ity for beach slope computation. Overtopping and overwash weresimulated, thus larger waves reach the barrier crest. The landwardupper point for beach slope computations was the slope break seawardof the steep barrier top. Overwash sedimentation was calculated basedon the difference between consecutive profiles landward of the barriercrest. The hydraulic gradientwasdefined as the gradient between the hSand hL (Fig. 1a). Freeboard (Rc) was defined as the vertical distance be-tweenmean sea level and the barrier crest (cf.Obhrai et al., 2008; Pullenet al., 2007).

To detect any obliquity of overwash transport across the barrier,sediment tracers were used in Test Series E6. These comprised three10 kg samples of barrier gravel dyed with fluorescent paint usingthree colours (orange, green and blue). Before overwash tests, thesetracers were emplaced level with the barrier surface at three span-wise locations on the barrier crest. Following the image-collectionmethods described in Buscombe (2008) for grain-size analysis, pho-tographs of the bed sediments (13 cm×10 cm visible area) wereobtained after overwash events along three streamwise profiles

Fig. 1. a) Definition sketch showing variables referred in the text; b) Barrier overtopping duwere taken from the barrier looking to the paddle (top centre of the photographs). Bed-lev

spanning the region from the emplaced tracers to the lagoon. Thenumber of painted grains on each photograph was counted.

During the tests a series of pumps were used to maintain the seaand lagoon level at the desired elevation (Williams et al., 2011-thisissue), and the pump rates into and out of the lagoon were recorded.During Test Series E9 and E10, sea level was higher than lagoon leveland the pumping rates were used to estimate the water dischargefrom the sea into the lagoon. These discharge estimates includeboth water flow into the lagoonby overwash, aswell as seepage throughthe barrier (Turner and Masselink, this issue).

2.2. Data analysis

The laboratory observations were used to investigate the applica-bility of a large number of wave runup predictors listed in theAppendix 1. The equations were assessed by their capacity to distin-guish between swash acting on the seaward face of the barrier andoverwashing of the barrier crest. Since prediction of swash or over-wash is a binary classification, confusion matrices (Kohavi andProvost, 1998) can be used to assess performance. In binary frame-works, accuracy, relates to the capability of a given runup equationto distinguish between overwash and swash situations, and isexpressed as:

accuracy ¼ 100Nþ

swash þ Nþoverwash

Nþswash þ N−

swash þ Nþoverwash þ N−

overwashð1Þ

ring Test Series E1; and c) Barrier overwash during tracer Test Series E10. Photographsel sensors (BLS) were attached to the rig shown in the photographs.

Fig. 2. Barrier cross-shore profiles fromseveral Test Serieswith overtopping. a) Series E1; b)Series E5; and c) Series E8. Water levels on the ‘sea’ side (paddle side, to the left) and ‘la-goon’ side are also represented in dash-lines. Note: not all profiles from each Test Seriesare represented.

51A. Matias et al. / Coastal Engineering 63 (2012) 48–61

where N+ is the number of true predictions and N− is the number offalse predictions. In addition, sensitivity and specificity quantify theability of a given equation to predict observed overwash and swashevents, respectively, and are expressed by:

sensitivity ¼ 100Nþ

overwash

Nþoverwash þ N−

swashð2Þ

specif icity ¼ 100Nþ

swash

Nþswash þ N−

overwash: ð3Þ

Accuracy, sensitivity and specificity values are used here to identifythe most suitable runup equation for the BARDEX tests examinedhere.

3. Results

3.1. Morphological response and sediment transport

In Test Series E both overwash and overtopping conditions wereexamined (Table 1; Fig. 1). Overtopping dominated during the earliertests (Figs. 1b and 2) and overwashing became fully developed duringthe Test Series E10 (Figs. 1c and 3). During the 62 runs in Test SeriesE, overtopping was recorded in 22 runs and overwashing in 20 runs(Table 1). During the remaining 20 runs, wave runup did not reachthe barrier crest and swash action only affected the beach face (17runs) or inundation took place (3 runs).

The displacement of tracer sediments during Test Series E6 wasvariable (Fig. 4) with most of green and blue coloured pebblesremaining close to the emplacement point. The number of orangecoloured pebbles retrieved from the central profile was lower, probablydue to burial. Although these results indicate some span-wise variationin sediment transport rates, nomixing of sediment tracer particles fromadjacent sources was detected. This indicates that during overwash thebulk of sediment transport occurred in a cross-shore direction. Changesin barrier profile inferred from the measured central profile of thebarrier are thus considered representative of other alongshore locations.

Fig. 5 shows the changes in barrier crest elevation, beach face gra-dient and volume of overtopping/overwash deposition during TestSeries E. In general, crest elevation increased from Tests E1 to E9,and sharply decreased during Test E10 (Fig. 5a). The beach face wassteepest (>0.2) at the start of the Test Series E and showed an overalldecrease during the overwash tests, especially during Test E10(Fig. 5b). The reduction in beach gradient during overwashing is attrib-utable to the transport of sediment from the beach face and crest re-gions towards the back-barrier, thus making the barrier flatter. Incontrast, during typical overtopping conditions, the beach becamesteeper (cf.Fig. 2) as sediments from the lower beach face were trans-ferred to the upper beach face and barrier crest. Concurrent with thesteepening of the beach face and barrier crest build-up, the submergedbeach step migrated landward. The morphological changes duringovertopping on the one hand decreased overtopping likelihood by in-creasing the crest elevation; on the other hand, the steepening of thebeach face and the landwardmigration of the beach step are likely to in-crease the runup. The crest build-up resulting from overtopping is animportant self-regulatory process that may contribute to delaying theonset of overwashing. In later runs of Test Series E1 and E8, overtoppingand beach slope reduction occurred (Fig. 5b), with more sedimentsbeing carried landwards of the barrier crest (Fig. 5c). These situationsrepresent a transitional step between typical overtopping and typicaloverwash.

During Test E10, when full overwash occurred, sediment wastransported to the back-barrier region and was deposited (Fig. 3b).These deposits created back-barrier slope instabilities which periodi-cally failed and avalanched down the submerged rear-side of the

Fig. 3. Barrier cross-shore profiles from several Test Series with overwash. a) Series E9;and b) Series E10. Water levels on the ‘sea’ side (paddle side, to the left) and ‘lagoon’side are also represented in dash-lines. Note: not all profiles from each Test Seriesare represented.

Fig. 4. Distribution of the three coloured tracers through the barrier during ExperimentE6. Lagoon is located at the top of the figure. Injection locations of the three colourgravel tracers (orange, green, blue) are immediately landward of barrier crest, thatwas located at approximately 92.5 m from the paddle. The rectangles that symbol thethree injection places are at scale.

52 A. Matias et al. / Coastal Engineering 63 (2012) 48–61

barrier forming a steep prograding surface approximately parallel tothe original slope. This test demonstrated the importance of the la-goon water level in controlling the geometry of the back-barrier de-posit, particularly at the interception between the subaerial back-barrier deposits and those below hL. The rate of barrier loweringand widening was relatively constant during Test E10 (Fig. 3b) andthe average sediment transport rate across the barrier was O(0.1 m3/m/min). By the end of Test E10, the volume of the washoverdeposit had increased by 13 m3/m. Approximately 68% of the wash-over sediment originated from the beach face, with the remaining32% coming from sediments in the crest region that were depositedby overwash in the earlier test stages of Test E10.

Pump discharge rates required to maintain the lagoon water levelat 3.25 m during Test Series E9 increased from 5.5 lm−1 s−1 to35 lm−1 s−1 (Fig. 6c), reflecting increased seepage through the barrierdue to the increase in head difference between hS and hL (Fig. 6a). Directflow into the lagoon attributable to overwash did not occur until theend of the E9 Test. During Test E10, the head difference between seaand lagoon was kept constant (hS and hL were not changed; Fig. 6a)

and the seepage rate was expected to have remained constant. The in-crease in pump discharge rates during Series E10 from 35 to almost60 lm−1 s−1 (Fig. 6c) therefore occurred due to enhanced overwashflows across the top of the barrier as a consequence of lowering of thebarrier crest (Fig. 6b). Test runs E10D, E10E and E10F had unexpectedlysmaller pump discharges, and this is attributed to the fact that theseruns were undertaken at the start of the second day of Test E10 andgroundwater gradients were not yet well-established.

The change in overwash characteristics during Test E10 was alsoinvestigated using bed-level sensor data collected near the barriercrest. The data were used to determine the percentage of time thatoverwash flow occurred underneath the bed-level sensor. During TestE9 overwashing was detected for only 15% of the time (Fig. 6d). In con-trast, overwash occurred formore than 60% of the time during Test E10,clearly demonstrating the increased inundation due to lowering of thebarrier crest.

At the very end of the Test E10, the overwash discharge into thelagoon exceeded the capacity of the pumps to maintain the sea andlagoon at their required levels. The consequent drop in sea level andrise in lagoon level rapidly eliminated the 0.5 m sea-lagoon head dif-ference, resulting in almost continuous barrier inundation. Occasion-ally, short-lived seaward return flows occurred across the submergedbarrier crest. This washout process has been noted to occur in nature,and involve channel erosion across the beach and foredunes (Mortonand Sallenger, 2003). However, owing to the infrequency of this pro-cess in the present study it is not considered further here.

3.2. Estimation of the overwash threshold

The computation of wave runup requires consideration of beachslope, wave height, wave period and tidal level, and provides a useful

Fig. 5. Variation of a) crest elevation, b) beach face slope, and c) volume of overtopping/overwash sedimentation on top of the barrier through BARDEX Series E. Runs with over-topping or overwash are marked. Runs with no symbol information had only swash processes.

53A. Matias et al. / Coastal Engineering 63 (2012) 48–61

parameter to define the threshold of coastal impacts driven by astorm (beach erosion, dune retreat, overwash, etc.). Using such ap-proach, Sallenger (2000) proposed a Storm-Impact Scaling modelbased on four regimes: swash, collision, overwash and inundation.In the present study, crest scarping and barrier submergence, repre-senting the collision and inundation regime, respectively, were notobserved, so the focus here is on the swash and overwash regime.The latter is sub-divided into overtopping and overwashing, basedon Orford and Carter (1982): overtopping is characterised by crestaccretion and an increase in crest elevation; and overwashing ischaracterised by lowering of the crest by erosion and the formationof washover deposits landward of the barrier crest. Inundation canbe seen as an extreme form of overwash (sluicing overwash follow-ing Orford et al., 2003) and occurs when the sea level is at or abovethe elevation of the barrier crest.

Following Sallenger (2000), the highest elevation of the landwardmargin of swash relative to a fixed vertical datum Rhigh correspondsto:

Rhigh ¼ tideþ surgeþ runup ¼ ηþ R2 ð4Þ

where η is sea level, including astronomical tides and storm surge,and R2 is the 2% exceedence level for vertical runup, including setupand swash. Thresholds associated with the different regimes definedin Sallenger's (2000) Storm-Impact Scaling model are determinedusing runup predictors. There are numerous equations to predictwave runup and twelve of these, summarised in Appendix 1, havebeen used here. It is noted that the majority of these equations havebeen derived for sandy beaches (Guza and Thornton, 1982, Eq. (A4);Holman, 1986, Eq. (A5); Nielsen and Hanslow, 1991, Eqs. (A8) and

Fig. 6. Variation during Test Series E9 and E10 of a) Seawater level (hS) and lagoon water level (hL); b) Barrier crest elevation; c) Discharge into the lagoon, and d) Percentage of timewith overwash on the back-barrier.

54 A. Matias et al. / Coastal Engineering 63 (2012) 48–61

(A9); Komar, 1998, Eq. (A12); Ruggiero et al., 2001, Eq. (A13); Stockdonet al., 2006, Eq. (A18)), or engineering structures (Hunt, 1959, Eq. (A2);Mase, 1989, Eq. (A6); van der Meer and Stam, 1992, Eqs. (A10) and(A11); Hughes, 2004, Eqs. (A16) and A17) and only the equations ofPowell (1990, Eq. (A7)) and Lorang (2002, Eq. (A15)) have been specif-ically developed for gravel beaches. The twelve runup predictors wereused to obtain estimates of wave runup R2 for the data collected duringTest Series E. Not all equations were developed to predict R2: Guza andThornton (1982) predict significant runup, and Powell (1990) andLorang (2002) predict barrier crest elevation (directly related to verticalelevation reached by the runup). However, all estimates are useful forcomparison with barrier crest elevation. The wave conditions and sealevel during each of the Tests are listed in Table 1 and the relevantbeach gradient is shown in Fig. 5b.

Comparisons between predicted Rhigh and measured hc values areshown in Fig. 7. It is shown lines of unity that indicate overtoppingconditions. Data points below this line should correspond to theswash regime and those above the line should correspond to theoverwash regime. The equations of Mase (1989; Fig. 7d) andHughes (2004; Fig. 7k) clearly overestimate observed runup, whereasthose of Guza and Thornton (1982; Fig. 7b) and Lorang (2002; Fig. 7j)underestimate runup. In contrast, the equations of Powell (1990;Fig. 7e) and Stockdon et al. (2006; Fig. 7l) seem to be performingvery well by correctly predicting the occurrence of swash and over-washing (cases plot below and above the line of unity, respectively).

During overtopping, the difference between predicted Rhigh andmeasured hc should be close to zero or slightly positive. Rhigh−hcvalues for each overtopping run were determined (Fig. 8), and thevalues closest to zero provide the best estimates. The greatest precisionsare obtained using estimates of R2 from Holman (1986), Powell (1990),

Komar (1998), Van der Meer and Stam (1992), and Stockdon et al.(2006, Fig. 8). Most equations developed for or based on data from en-gineering structures overestimated runup (A10/11 and especially A2,A6, A16/17; Fig. 8). This is expected since on impermeable slopes, inthe absence of infiltration, runup reaches higher elevations. Amore var-iable behaviour was recorded for equations developed for sandy bea-ches, which either underestimated (A4, A8, A14) or overestimated(A5, A12) runup.

The ability of the runup equations to distinguish between the occur-rence of overwash and swash was assessed using the binary classifica-tion approach described in Section 2.2. Accuracy, sensitivity andspecificity values were computed for all runup equations and the resultsare presented in Table 2. The estimates with lower accuracy were thoseof Guza and Thornton (1982), Mase (1989), Lorang (2002), and Hughes(2004). Some equations are very sensitive (e.g. Hughes, 2004 and Hunt,1959), because they predict all overwash cases. However, they havepoor specificity as a high percentage of swash cases were predicted asoverwash. For the present study the equations that appear to performbest are those of Holman (1986), Powell (1990), Komar (1998) andStockdon et al. (2006). Inmore than 90% of tests, the four equations cor-rectly predict observed overwash, but the equations of Holman (1986)and Komar (1998) predict overwash in 56% of cases when only swashprocesses were observed. Owing to a great accuracy in estimating theswash regime the equation of Stockdon et al. (2006) is the most specific(95%; Table 2) of four equations. However, whereas this equationmoreaccurately predicts the overwash regime, the equations ofHolman(1986), Komar (1998) and Powell (1990) provide estimatesthat are more sensitive (100%; Table 2). The most accurate predictionsof R2 are provided by the equations of Powell (1990) and Stockdon etal. (2006), with 87% and 92% accuracy, respectively (Table 2).

Fig. 7. Barrier crest elevation versus runup elevation (Rhigh=R2+tidal level), with R2 calculated according to several tested equations. For all plots vertical axis is runup elevationand horizontal axis is barrier crest elevation. Equality line in all plots is represented with a dashed-line. More precise estimates have overtopping situations on top of equality line,and more accurate estimates have overwash situations above and swash situations below equality line.

55A. Matias et al. / Coastal Engineering 63 (2012) 48–61

4. Discussion

4.1. Barrier evolution and feedback processes

The barrier morphology varied significantly during Test Series E.The barrier crest retreated by c. 3 m during Tests Series E1 to E8.This was accompanied by an increase in the barrier crest elevationof almost 0.5 m (Fig. 5a) and a decrease in the width of the barrierby 0.8 m (at an elevation of 3.5 m). The barrier crest was graduallyraised by overtopping during these runs and progressively more

wave power and/or a raised sea level was required to initiate over-wash. Thus, although overtopping did occur under lower energy con-ditions (smaller waves or lower sea level) during the early Tests (e.g.,E1B; Table 1) it was suppressed in later tests (e.g., E5F; Table 1) dueto the elevated barrier crest. As has been noted previously, overtop-ping provides a negative feedback on morphological evolutionthrough raising the crest of the barrier, thereby inhibiting furtherovertopping or overwashing. In the field, various combinations ofweak/moderate storms, storm surge and high tidesmaypromote barrierovertopping, thereby decreasing the likelihood of overwashing. Because

Fig. 8. Difference between barrier crest elevation and maximum elevation of runup (Rhigh) for overtopping regime, estimated by the several test equations. Differences are repre-sented as box-plots with maximum, minimum, and quartiles Q1 and Q3. A band including Rhigh−hc=0±30 cm is also represented.

56 A. Matias et al. / Coastal Engineering 63 (2012) 48–61

barrier morphology, and in particular the crest elevation, is the cumula-tive product of the antecedent conditions (during BARDEX representedby the sequence of overtopping episodes), these antecedent conditionsare of great significance in affecting the future behaviour of the barrierunder extreme storms. The BARDEX results thus demonstrate the im-portance of coastal system “memory” (or “inertia” defined by Orford etal., 2003) provided either by the barrier evolution prior to extremestorm activity, or even by the dynamic interaction between the mor-phology and the swash regime during early stages of a storm. The im-portance of antecedent morphology in coastal evolution is a wellestablished concept and has been widely demonstrated for differentcoastal environments (e.g., Backstrom et al., 2009; Cooper et al., 2004;Dillenburg et al., 2000; Quartel et al., 2008). During the BARDEX over-wash experiment, this concept was further refined through confirma-tion of the importance of morphological feedback and the role ofsequencing of events during a storm, especially the evolution of thebarrier crest.

The BARDEX gravel barrier was based on Slapton Sands, Devon,England (Austin and Masselink, 2006). Apart from a difference inphysical scale (Slapton Sands has awidth of 100 m and a crest elevationof 6–7 m), the overall shape is similar (Slapton Sands has tanβ=0.15and a fresh water lagoon behind the barrier with the lagoon levelaround spring high tide level) and the grain size is also similar (SlaptonSandsD50=6mm). Also along Slapton Sands, significant damage has oc-curred over the last decades during storms due to flooding and barrieroverwash (Ruiz de Alegria-Arzaburu and Masselink, 2010). The natural

Table 2Accuracy of several equations in prediction of swash and overwash regimes.

Sensitivity Specificity Accuracy

Hunt, 1959 (Eq. (A2)) 100% 19% 66%Guza and Thornton, 1982 (Eq. (A4)) 27% 100% 58%Holman, 1986 (Eq. (A5)) 100% 44% 76%Mase, 1989 (Eq. (A6)) 100% 0% 58%Powell, 1990 (Eq. (A7)) 100% 69% 87%Nielsen and Hanslow, 1991 (Eq. (A8)) 45% 100% 68%van der Meer and Stam, 1992 (Eq. (A10)/11) 91% 50% 74%Komar, 1998 (Eq. (A12)) 100% 44% 76%Ruggiero et al., 2001 (Eq. (A14)) 50% 100% 71%Lorang, 2002 (Eq. (A15)) 14% 100% 50%Hughes, 2004 (Eq. (A16)/(A17)) 100% 0% 58%Stockdon et al., 2006 (Eq. 21) 95% 88% 92%

barrier is affected by storm activity which has significantly higher ener-gy (storm Hs≈2–4 m; Ruiz de Alegria-Arzaburu and Masselink, 2010)than the storm conditions simulated during the BARDEX experiments(Hs=0.8–1.2 m). Therefore, the morphological processes observedduring BARDEX experiment attempt to reproduce field conditions, al-beit on a smaller scale (a reduction factor of 2–3).

Despite hydrodynamic conditions being kept the same (Table 1),overwashing during Test E10 resulted in a progressive reduction ofthe barrier crest height (Fig. 6b), and an increase in both the over-wash flow discharge and duration (Fig. 6c and d, respectively). Thisin turn, increased the frequency of overwash events. Therefore, over-wash provides positive feedback until either the barrier becomes in-undated (Fig. 3b) or runup height decreases due to a fall in waterand/or wave energy level. Although a corresponding increase in sed-iment transport would be expected during barrier crest lowering, therate of morphological change did not increase (Fig. 3b). This may beexplained in part by increased dissipation of overwash flow on theflatter back-barrier, thus promoting deposition, and also the reduc-tion in the beach face gradient enhancing wave energy dissipationat the front of the barrier. Both processes are expected to increasein importance as the size of the barrier increases and account forthe fact that the barrier cross-sectional area is an important factorcontrolling the occurrence of barrier breaching (Bradbury et al.,2005). Test E10 provides a clear illustration of gravel barrier responseto storm waves and shows the first stage of barrier rollover describedas a fundamental process in gravel barrier evolution by Carter andOrford (1993).

Had Test Series E continued with a reduction in sea level and/orwave energy level to simulate post-storm conditions, overtoppingwould have replaced overwashing, resulting in crest build-up and asteepening of the beach face. The barrier profile would graduallyhave adjusted towards its pre-storm configuration; however, therewould have been an irreversible 10-m landward migration of thewhole barrier since no mechanism exists to transport the sedimentscomprising the washover deposits back to the beach face.

4.2. Overwash threshold definition

Orford et al. (2002, 2003) observed that the elevation of a gravelbarrier reflects a balance between runup sufficient to deposit materialat the beach crest (overtopping) and runup sufficient to exceed the

57A. Matias et al. / Coastal Engineering 63 (2012) 48–61

crest and move material onto the back-barrier slope (overwash).Overtopping is restricted to a limited range of energy conditions(sea level and wave energy) and only occurs when the runup eleva-tion is slightly higher than that of the barrier crest. On the otherhand, overwashing occurs across a much wider range of energy

Fig. 9. (a) Definition sketch showing Overwash Potential (OP), water level (including tide anand freeboard (Rc). (b) Overwash Potential histograms for swash, overtopping and overwash us(OP) versus volume of overwash sedimentation for overtopping and overwash situations, com

conditions, starting with a relatively modest difference between therunup and the crest elevation at the onset of overwash, to full-blown barrier crest inundation with a negative freeboard.

Either overtopping or overwash is likely when Rhigh−hc>0(Fig. 9). This elevation difference is a measure of the likelihood of

d surge, η), 2% exceedence runup (R2), runup with water level (Rhigh), barrier crest (hc),ing R2 estimated with Stockdon et al. (2006) equation (Eq. (A18)). (c) Overwash Potentialputed over 15 min runs.

58 A. Matias et al. / Coastal Engineering 63 (2012) 48–61

overwash occurrence, and is given the name overwash potential, OP,(Fig. 9).

OP ¼ Rhigh−hc ¼ R2 þ η−hc: ð5Þ

To compute OP, η can be estimated with high precision (tidalgauge), and hc elevations can be obtained by topographic surveys.As described above, the equation used to predict R2 should be one thatcombines both a high precision (given by runup values close to barriercrest for overtopping conditions) and a high accuracy (given by the ca-pacity to correctly predict the occurrence of swash and overwash re-gimes). In this respect Powell (1990, A7) had an average precision of0.22±0.23 m (Fig. 8), sensitivity of 100%, and an accuracy of 87%(Table 2), and Stockdon et al. (2006, A21) had an average precision of0.20±0.34 m (Fig. 8), sensitivity of 95%, and an accuracy of 92%(Table 2). Bothmethods had good precisions and accuracies, significantlyhigher than random chance (50%).

The estimates of Powell (1990) and Stockdon et al. (2006) havesimilar precision and accuracy values (Fig. 8 and Table 2), but it shouldbe pointed out that the equations are based on quite different ap-proaches (field versus lab; physics-based versus empirical) and thatboth have some shortcomings. Notably, the Stockdon et al. (2006) ex-pression was developed for sandy beaches whereas here it is appliedto a gravel setting. More importantly, the Powell (1990) equationdoes not actually estimate runup height, and therefore the potentialfor overtopping and overwashing, but is designed to predict the bar-rier crest height. In fact, Powell (1990) explicitly states that his equa-tion should not be applied to overtopping and overwashing. Asdemonstrated by the difference between predicted runup (i.e. pre-dicted crest height) and the barrier crest elevation in Fig. 7e, thePowell (1990) equation is not able to predict the barrier crest heightunder these conditions.

Present results suggest that the Stockdon et al. (2006) equation isthe best predictor of run-up in the present experiments for the fol-lowing reasons: 1) it is more accurate; 2) it was developed usingdata compiled from ten field experiments ranging from dissipative(ξb0.3) to reflective (ξ=2.2) beaches; and 3) its application rangeis broader, being suitable for the present study case as well as forsandy coats (e.g. Baldock et al., 2008; Stockdon et al., 2007). Accord-ingly, based on BARDEX experiment results, the proposed equationfor estimate overwash potential is given by

OP ¼ 1:1 0:35 tan βffiffiffiffiffiffiffiffiffiffiHoLo

pþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiHoLo 0:563 tan βð Þ2 þ 0:004

� �� �q2

0@

1A

þ η− hc:

ð6Þ

Results of OP using Stockdon et al. (2006) (Eq. (A18)) show that88% of swash cases have OPb0 m, and that 65% of overtopping and95% of overwash have OP>0 m (Fig. 9b). The equation of Stockdonet al. (2006) was developed for and based on sandy beaches; howev-er, it provided good estimates of runup for the gravel barrier withoutadjustments. On the one hand, runup is reduced in gravel because in-filtration is higher; on the other hand, gravel barriers are steeper (dueto the enhanced infiltration) leading to increased runup. It can, per-haps, be argued that the enhanced hydraulic conductivity on gravelbeaches is indirectly accounted for through the steep beach gradients.The data base of Stockdon et al. (2006) includes sandy beaches rang-ing from dissipative beaches in the Netherlands (Irribaren numberξ=0.1±0.004) to reflective beaches in California (ξ=2.2±0.3).During Series E of the BARDEX experiment ξ=1.9±0.65; therefore,on average ξ was well within the range of characteristics used for pa-rameterization deriving the Stockdon et al. (2006).

To transport sediments efficiently to the back-barrier overwashmust have sufficient depth and velocity. Additionally, overwash must

overcome fluid loss through infiltration into the barrier sediments.This loss is highest on the leading edge and during the latter stages ofa wave up-rush event (Horn et al., 2003). In this respect OP should ex-ceed zero. However, OP computed with R2 from Stockdon et al. (2006)equation do not show this lag effect (Fig. 9b), with overwash startingreadilywhenOP≈0 m. Therefore, based on BARDEXexperiment resultsthe threshold for overwash occurrence is OP>0, and therefore negativeOP predicts swash.

There were situations during Test Series E that were poorly repre-sented irrespective of the runup equation used. This is the case ofovertopping during runs in Test Series E8 (OP>0.5 m for overtoppingshown in Fig. 9b and c). As discussed previously, these overtoppingsituations represent a transitional phase between overtopping andoverwash where OP is large (approximately the same as that for theTest Series dominated by overwash, i.e. E9 and E10, Table 1). Thesecases of transitional overtopping, with barrier crest raising andbeach slope decreasing (particularly noticeable in Series E8, Fig. 5),make predictions more complex. Because the wave period is large(Tp=8 s), Stockdon et al. (2006) equation overestimate runup, andOP is larger than required for overtopping.

BARDEX overwash experiments show that regardless of approachthere is no method that allows a clear and sharp distinction betweenovertopping and overwash (Figs. 7 and 9). For small OP (b0.5 m),overtopping and overwash are almost indistinguishable, with verysmall volume variations for both situations (Fig. 9c). In these casesof shallow overtopping/overwash water depth, infiltration is likelyto be important. Enhanced infiltration over gravel sediments preventslong water intrusion and therefore morphological changes are small.For OPb0.5 m, overtopping (80%) is more likely than overwashing(20%), while the reverse is true for OP>0.5 m (24% overtopping and76% overwashing) The water depth and flow velocities are higher,thus promoting erosion of the crest, deposition further inland, andconsequently larger morphological changes. Infiltration is still a sig-nificant process, but its role becomes decreased in importance asoverwash flows increase in depth and strength, and sediment trans-port rates increase accordingly. From a coastal management point ofview, small-scale overwash or overtopping has similar implications.In developed areas, both processes may imply property damage andinfiltration of saltwater due to flooding, and thus demands hazardmanagement. This does not mean that there is no difference betweenthe two processes: rather it demonstrates that the occurrence of ei-ther overtopping or overwash is very sensitive to the forcing condi-tions. A very small change in water level or wave energy conditionscan cause a change from overtopping to overwash or vice-versa.Owing to the feedback mechanisms identified above, transition be-tween one regime and another may simply depend on a few centi-metres difference in runup (Fig. 7). These findings provide supportfor the assertion based on field observations that alongshore differ-ences in storm impact can be attributed to small nearshore, foreshoreor/and dune differences that became enhanced during the stormlandfall (e.g. Leatherman, 1976; Orford and Carter, 1984; Smith etal., 2008; Wang et al., 2006).

For sandy barriers overwash occurrence prediction is based in twotypes of models: (1) models that predict runup, that when comparedto dune crest elevation provide estimates of overwash possibility (e.g.Stockdon et al., 2007); and (2) models that predict profile morphologi-cal changes during storms and account for overwashwhen dune crest islowered (e.g. Roelvink et al., 2009). From the user point of view, themajor difference is that the application of the second type of models re-quires a higher amount and complexity of variables to be considered,in-depth knowledge of modelling, and longer computation-time. Thefirst type of models provides more limited results but can be used bya broader audience of coastal experts (both science and management).The same is valid for gravel barriers; in spite its application is morelimited (e.g. Williams et al., 2011-this issue). The computation ofOverwash Potential proposed in this work follows the first type of

59A. Matias et al. / Coastal Engineering 63 (2012) 48–61

approach that strongly relies in the quality of runup predictions. Thismethod doesn't provide a post-storm profile, rather the possibility ofoverwash occurrence for gravel barriers, but can be used by re-searchers and managers that have limited databases and that arenot familiar with modelling.

For management purposes, Overwash Potential is accurate in dis-tinguishing swash and overwash but it does not permit a definitionof the threshold for the transition between overtopping and over-wash. It must be stressed that transitional overtopping situationsmay present a hazard owing to the considerable intrusion of sedi-ments and seawater associated with them. Therefore, in reality, this‘misclassification’ may be helpful from safety and protective coastalmanagement purposes.

5. Conclusions

The paper has reported the morphological response of a prototypegravel barrier to carefully controlled wave and tidal forcing duringovertopping and overwashing conditions. The observed morphologi-cal evolution of the barrier during the experiments has revealed acontinuum between overtopping and overwashing, and the identifi-cation of a transitional overtopping process. Transitional overtoppinghas not only the distinctive morphological characteristic of overtop-ping (i.e. barrier crest elevation), but also overwash characteristics(i.e. a decrease of beach slope and washover deposition landwardsof pre-existing crest position). Negative and positive feedback pro-cesses have been identified for overtopping and overwashing, respec-tively, which in turn influence subsequent vulnerability to overwash.

Wave runup estimates have been comparedwith barrier crest eleva-tion to define overwash threshold. A critical assessment of the precisionand accuracy of existing equations for runup has been undertaken usingdata from overwash experiments in the BARDEX project. The overwashthreshold has been defined in this study as the condition necessary togenerate positive values of Overwash Potential, OP. OP is defined asthe difference between the highest elevation of the runup (includingthe water level; with runup computed with Stockdon et al., 2006 equa-tion) and the barrier crest elevation. It is considered that OP provides apractical means by which to assist the definition and identification ofcoastal hazards associated with gravel barriers. Shortcomings inthe application of OP may arise from local inherent variability infield conditions, such as porosity, grain size, nearshore or foreshorefeatures. Also, because field and laboratory conditions differ in scale,OP defined thresholds (overwash/overtopping for OP>0±0.2 m; over-wash dominant for OP>0.5 m) may be slightly different. OP has appli-cation in a variety of coastal management issues, such as emergencybeach re-profiling works, beach recharge schemes, regional flood map-ping, insurance valuations and in the implementation of appropriateconservation measures for sensitive back-barrier environments.

Acknowledgements

The data reported here were collected in the Delta flume(Netherlands) as part of the EU-funded BARDEX project (HYDRALABIII Contract no. 022441 (RII3), Barrier Dynamics Experiments). AnaMatias was supported by the Fundação para a Ciência e a Tecnologia,grant reference SFRH/BPD/18476/2004.

Appendix 1. Description of wave runup equations

Based on laboratory model tests with protection structures (sea-walls and rock-rubble structures), Hunt (1959) proposed an empiri-cal relation for vertical wave up-rush, R (in units of ft/s) in the form

R ¼ 2:3tan βffiffiffiffiffiffi

HT2

r ðA1Þ

where tan β is beach slope, H is wave height (Hunt assumesH≈meanwave height, Ho) and T is wave period. Battjes (1974) changed the no-tation of the equation and included Hs and the Irribaren Number ξ

Rmax

Hs¼ ξ ðA2Þ

where Rmax is the maximum elevation of the water line, and ξ is de-fined as

ξ ¼ tan βffiffiffiffiffiffiHs

Lo

s ðA3Þ

where Lo is the deep-water wave length. Using measurements on adissipative sandy beach, Guza and Thornton (1982) found that thesignificant vertical runup excursion Rs was related linearly to Hs sothat

Rs ¼ 0:035þ 0:71Hs: ðA4Þ

For a sandy beach that was dissipative at low tide and more reflec-tive during high tide, Holman (1986) found that the 2% exceedencelevel for swash height, R2%, was

R2%

Ho¼ 0:83ξþ 0:2: ðA5Þ

Using data from a wave flume for random waves on gentle,smooth and impermeable uniform slopes, Mase (1989) found that

R2%

Ho¼ 1:86ξ0:71o : ðA6Þ

The highest point reached by waves relative to water level wasestablished by Powell (1990) using random wave tests on a gravelbeach in a flume. This parabolic relation between the normalisedrunup and wave steepness (Hs/Lo) states that

Rmax

Hs¼ 2:86−62:69

Hs

Lo

� �þ 443:29

Hs

Lo

� �2: ðA7Þ

Using measured runup on many different sandy beaches (rangingfrom highly reflective to dissipative), and on measurements and oninterpretation of data from Holman (1986), Nielsen and Hanslow(1991) proposed two relationships for runup for: a) beaches withslopes ≥0.1; and b) for beach slopes b0.1)

R ¼ 0:6ξ for tan β≥ 0:10 ðA8Þ

R ¼ 0:05ffiffiffiffiffiffiffiffiffiffiffiffiffiffiHrmsLo

pfor tan β b 0:10 ðA9Þ

where Horms is the deep-water root mean square wave height(Horms=Hs /√2). Using runup data for smooth and rock slopes ofcoastal structures van der Meer and Stam (1992) proposed runup rela-tionships that are similar to Eqs. (A2) and (A7) and take the form

R2%

Ho¼ 0:96ξo for x b 1:5 ðA10Þ

R2%

Ho¼ 1:17 ξ0:46o for x > 1:5: ðA11Þ

Komar (1998) suggested a runup equation that includes wavesetup

RT2% ¼ 0:36

ffiffiffig

ptan β

ffiffiffiffiffiffiHo

pT ðA12Þ

60 A. Matias et al. / Coastal Engineering 63 (2012) 48–61

and Ruggiero et al. (2001) develops independent expressions for dis-sipative and reflective beaches

R2% ¼ 0:5Ho þ 0:22 for dissipative beaches ðA13Þ

R2% ¼ 0:27ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffitan β HoLo

pfor reflective beaches: ðA14Þ

Lorang (2002) relates runup to inequalities in the balance be-tween the wave forces acting to transport sediment up the beachface and gravity in the form

R ¼ 12

ρs−ρw

ρw

� �gTD50 tan β

CdUmax

� �ðA15Þ

where ρs is mass density of grain, ρw is mass density of water, D50 ismedian grain-size, the drag coefficient, Cd, is a constant (= 0.027)and Umax is the maximum swash velocity, given by

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig hþ Hsbð Þp

where h is water depth at breaking and Hsb is significant wave heightat breaking. In a re-examination of existing wave runup data onsmooth, impermeable plane slopes, Hughes (2004) defines runupusing the wave momentum flux (MF) that represents the maximumdepth-integrated wave momentum. For irregular waves:

R2%

h¼ 1:75 1−e− 1:3 cotα½ � ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

MF

ρwgh2

s" #ðA16Þ

for HoL b 0.0225 and 1

4≤ tanα ≤ 1:

R2%

h¼ 4:4 tan βð Þ0:7

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiMF

ρwgh2

s" #ðA17Þ

forHo

Lo> 0:0225 and

15≤ tan α ≤ 2

3and for any value of

Ho

Loand 1

30 ≤ tan α ≤ 15 where g is gravitational acceleration

MF

ρgh2

� �¼

A0h

gT2

−A1 , A0 ¼ 0:6392Hh

� �2:0256and A1 ¼ 0:1804

Hh

� �−0:391:

Finally, Stockdon et al. (2006) used data compiled from ten fieldexperiments from sandy beaches ranging from dissipative (ξ b0.3)to reflective (ξ=2.2) to express runup as

R2% ¼ 1:1 0:35 tan βffiffiffiffiffiffiffiffiffiffiHoLo

pþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiHoLo 0:563 tan βð Þ2 þ 0:004

� �� �q2

0@

1A:

ðA18Þ

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