Measuring and modelling longshore sediment transport

Estuarine, Coastal and Shelf Science 83 (2009) 47–59

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Measuring and modelling longshore sediment transport Luciana S. Esteves a, *, Jon J. Williams a, Maria A. Lisniowski b a b

School of Geography, University of Plymouth, Plymouth, Devon PL4 8AA, UK Postgraduate Marine Applied Science, University of Plymouth, Devon PL4 8AA, UK

a r t i c l e i n f o

a b s t r a c t

Article history: Received 28 October 2008 Accepted 17 March 2009 Available online 27 March 2009

Field measurements of longshore sediment transport (LST) was undertaken on barred and non-barred beaches composed of fine, medium and coarse sands in Brazil, Denmark and Portugal. Measurements and predictions of vertical suspended sediment concentration profiles (C-Profiles) and cross-shore hydrodynamic parameters were then combined in a new semi-empirical model for prediction of LST (LTMOD). Instantaneous LST predictions from LT-MOD and well-known bulk LST formulae were compared. Tests using LT-MOD to simulate measured changes in shoreline position in southern Brazil for periods of c. two years showed that LT-MOD gave more accurate predictions than existing bulk LST formulae. Results indicate that LT-MOD may have practical utility at sites where access to equipment is limited and where reliable estimates of LST are required over extended periods. Ó 2009 Elsevier Ltd. All rights reserved.

Keywords: longshore sediment transport suspended load sediment traps field measurements prediction concentration profile Brazil, Parana´, Rio Grande do Sul Portugal, Faro Denmark, Skallingen

1. Introduction Accurate estimates of longshore sediment transport (LST) are required to predict the morphological evolution of sandy coasts at a range of temporal and spatial scales and in a range of practical engineering and beach management applications (cf. Bayram et al., 2007). LST can be estimated: (a) from direct measurements of longshore sand transport flux; (b) from empirical formulae using hydrodynamic and sediment data acquired in the field; or (c) by inferring net LST from observed large-scale changes in shoreline position and/or beach accretion and erosion. Approaches (a) and (c) require considerable resources to acquire the necessary data. Similarly, data required by LST formulae are normally only available at a single or few cross-shore locations, and beach profile surveys are normally restricted to a few kilometres at best with repeat surveys usually spanning only a short period. Furthermore, investigators wishing to quantify LST do not always have access to specialist equipment to measure all the hydrodynamic and sediment parameters required by the formulae. For this reason the majority of LST studies rely only on empirical predictions based on

* Corresponding author. E-mail addresses: [email protected] (L.S. Esteves), jon.j.williams@ plymouth.ac.uk (J.J. Williams), [email protected] (M.A. Lisniowski). 0272-7714/$ – see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ecss.2009.03.020

scant data and empirical calibration constants that may not always be site applicable. As a consequence, predicted rates of LST often have large errors especially when the time- and/or spatial-scales considered are large. In an attempt to develop a more robust methodology to quantify total LST when only limited field data are available, use is made here of data from field experiments undertaken on different coastlines: two located in southern Brazil, one in Portugal and one in Denmark. The experiments employed a range of well-known measurement techniques to acquire sedimentological and hydrodynamic data. Where required, theory is used to derive some key environmental parameters from the measured data to allow prediction of the observed suspended sand concentration profiles. These are then used to estimate the total longshore sand flux in a LST model (LT-MOD) that necessarily makes a number of simplifying assumptions about the cross-shore distributions of sediment and wave properties. Results are then compared with some widelyused bulk LST formula and the performance of LT-MOD over extended time- and spatial-scales assessed using field data from southern Brazil. It is acknowledged from the outset that many of the components in the approach described here are not new. Here the aim is simply to demonstrate that LT-MOD, constructed using robust and well-tested methods, and requiring simple, easily obtained input data, can out-perform some well-known LST formulae, especially for predictions over longer periods.

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2. Background Attempts to simulate LST in large-scale laboratory tests have had mixed success owing to scaling problems (e.g. Wang et al., 2002; Okayasu et al., 2004). Numerical modelling approaches (e.g. Castelle et al., 2006; Ellis and Stone, 2006; Falque´s, 2006) are not yet sufficiently advanced to accurately predict LST over time and spatial scales required by practical engineering or morphological studies. The measurement of sand transport in the nearshore region also presents many challenges. In the surf zone, air bubbles preclude the use of the acoustic instruments frequently employed in studies of suspended sediments, and optical techniques are better suited to fine sediments. Further, instruments designed for work in deeper water are relatively delicate and unable to withstand the energetic conditions in breaking waves. Methods employed in the past to measure LST include optical, interception and impoundment, tracer techniques and inference from measured changes in morphology (e.g. Knoth and Nummedal, 1977; Dean et al., 1983; Bodge, 1986; Kraus, 1987; Ciavola et al., 1997; Wang et al., 1998; Houser and Greenwood, 2005; Tonk and Masselink, 2005; Ari et al., 2007; Silva et al., 2007). Much of the available field data is subject to errors (cf. Bayram et al., 2007) and most studies highlight considerable discrepancies between measured and predicted LST. Bulk formulae used to predict LST make simplifying assumptions regarding hydrodynamics and sediment processes, and most do not consider any beach parameter (slope, grain size, morphodynamic or other). In spite of these apparent deficiencies, the CERC formula (USACE, 1984) is used widely to predict LST and is believed to have an accuracy of 30–50% in ideal conditions (Wang et al., 2002). A second widely-used approach (Kamphuis, 1991) follows closely the CERC formulation and includes additional terms expressing the influence of median grain size (D50), wave period (T) and beach slope (m). Most recently, Bayram et al. (2007) have developed a new bulk LST formula based upon a transport coefficient validated and calibrated against six high quality data sets of LST, including laboratory and field data. The formula is given as

Q ¼

3 ðrs  rÞð1  pÞ gws

FS

(1)

where 3 is a dimensionless transport coefficient expressing sediment diffusivity, rs is the sediment density, r is the water density, p is the sediment porosity, g is the acceleration due to gravity, ws is the sediment settling velocity, F is the flux of wave energy towards the shore and S is the mean longshore current velocity across the surf zone.

3. Proposed method In a new semi-empirical model (LT-MOD), predicted suspended sediment concentration profiles (C-Profiles) obtained at cross-shore locations are integrated to obtain an estimate of LST over the required time-period. The approach set out below accounts for sediment properties, the bed roughness (including bedforms), wave-current interactions and tidal level.

3.1. Bed roughness In the surf zone, where there may be bedforms and active sediment transport, the bed roughness (z0) comprises: (a) roughness due to the bed sediments defined here by the bed roughness length (z0G) as D50/12 (Soulsby, 1997); (b) roughness due to bedforms (form drag, z0R); and (c) roughness due to sediment transport (z0Q). z0R can be defined as

h2r lr

z0R ¼ Q

(2)

(Soulsby, 1997), where hr and lr are the height and wavelength of bed ripples and Q ¼ 4 as suggested by Madsen et al. (2007). A widely used equation for z0Q that does not require knowledge of z0 2:25 (Raudkivi, 1990), where U a priori is z0 ¼ 0:00533Uw w is the wave orbital speed defined below. Thus, from Soulsby (1997)

zo ¼ z0G þ z0R þ z0Q ¼

h2 D50 2:25 þ Q þ 0:00533Uw l 12

(3)

Bedforms, if present, contribute appreciably to the hydraulic roughness of the bed and influence significantly the nature of the CProfile. In the absence of direct field measurements, there are no well-established methods for calculating the dimensions of current-generated and wave-generated bedforms in the surf zone. In cases where it was not possible to measure bedform dimensions directly, the predictive equations of Grant and Madsen (1982) and Van Rijn (1984) are used to estimate hr and lr for wave-generated (hrw and lrw) and current-generated (hrc and lrc) bedforms, respectively.

3.2. Bed shear stress Time-averaged cross-shore profiles of the longshore current were computed in LT-MOD using a 1-D time- and depth-averaged longshore momentum balance approach between forcing terms (waves, wind and longshore slope), bottom stress and lateral mixing (cf. Ruessink et al., 2001). Cross-shore changes in wave height were also obtained using the well-known wave energy balance and included the momentum equation for wave-induced setup (Van Rijn et al., 2003). These models are appropriate for barred and unbarred beaches, and the present field sites meet approximately the model requirement of homogeneous alongshore waves and bathymetry. Laboratory experiments with broken and unbroken waves of the same height show that wave-induced shear stresses under breaking and shoaling waves are not on average significantly different from unbroken waves (cf. Nielsen, 1992, p. 219). This is further supported by Deigaard et al. (1991). It is appropriate therefore to estimate the cross-shore distribution of wave-induced bed shear stresses using the model data. Using predicted values of Hs and Tp, estimates of the peak wave induced oscillatory flow close to the bed (Uw) were obtained using Matlab routines1 to solve Deans stream function wave theory. Although this method gives accurate estimates of wave properties in shallow water conditions, it does not account for wave breaking. Estimates of the peak waves w ) were obtained using induced bed shear stress (b

b s w ¼ 0:5fw Uw2

(4)

where the wave friction factor (fw) is defined in terms of the relative roughness (r) by Swart (1974) as fw ¼ 0:3 for r  1.57 and fw ¼ 0:00251 expð5:21 r0:19 Þ for r > 1.57. Here r ¼ Ao =ks where A0 is the orbital amplitude of the waves (Uw Tp =2p). For rough turbulent, flat bed conditions, the Nikuradse equivalent sand grain s w is the peak wave-only skin roughness (ks) is given by 2.5D50, and b friction bed shear stress. In cases where ripples are present on the bed, and sediment is transported as bedload, ks is defined by the total bed roughness z0 so that ks ¼ 30z0 (Eq. (3)). It is noted here

1 Available at http://faculty.gg.uwyo.edu/borgman/DSF/dsfwav.html, accessed 4 December 2007.

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that instantaneous bed shear stresses may exceed greatly those encountered in non-breaking waves. Also, additional turbulence can be injected into the upper part of the water column during the breaking process, which is likely to impact on the form of the vertical suspended sediment concentration profile. No account of this is taken in the simple approach used here. Using predicted values of bed roughness, S and ab, the algebraic approximation to the w-c model of Fredsøe (1984) from Soulsby b wc ) and the timeet al. (1993) was used to estimate the peak ( U* averaged (U *wc ) bed shear velocity in the combined wave-current conditions across the surf zone. Given the limitations in the model and the field data, the use of more advanced approaches was not considered to be justified. 3.3. Suspended sediment Typically, C-Profiles in combined w–c conditions are estimated using a two-layer model (e.g. Soulsby et al., 1993). However, in previous studies predicted C-Profiles from a simpler single layer model in the form

CðzÞ ¼ Co



zþL aþL

b

(5)

(Williams et al., 1999a,b) were found to compare favourably with measured C-Profiles from field and laboratory studies and thus for simplicity the expression has been adopted here. In Eq. (5), C(z) is the suspended sediment concentration at height z, Co is a ‘‘referb Þ, is a transport coefficient expressence’’ C value, L ¼ 3=kðU * þ U * ing sediment diffusivity and b is a Rouse-type parameter ¼ ws/ b þ U ). The term U defines an apparent sediment mixing k( U * * * coefficient and is used to parameterise the quasi-periodic vortex b is analogous to the standard ejection process over ripples and U * turbulent diffusion mechanism. An expression used to obtain Co shown to be well-suited to surf zone conditions (cf. Williams et al., 2007) is

" Co ¼ rs

0:331ðqw  0:045Þ

1:75 1:75

1 þ 0:720ðqw  0:045Þ

# (6)

(Zyserman and Fredsøe, 1994) at z ¼ 2.5D50, where qw is a wave s w =grðs  1ÞD50 . Shields parameter defined as b 3.4. LST flux Using the predicted cross-shore distributions of wave and longshore current, the C-Profile model is invoked at multiple crossshore locations in LT-MOD and total LST flux is obtained through integration over the height range 2.5D50 < z < zMax, where zMax is the height where C values in the predicted C-Profile are <106 g/l. The model was used to predict instantaneous rates of LST and net LST over specified periods and includes tidal modulations and changes in the wave conditions as measured during the experiments. 4. Field sites and measurements 4.1. Description of field sites Experiments were undertaken at: (1) six sites along the southern shoreline of Rio Grande do Sul, Brazil (RS); (2) three sites along Praia do Leste, Parana´, Brazil (PR); (3) two sites at Skallingen, Denmark (SK); and (4) one site on Praia de Faro, Portugal (PF). Fig. 1 shows the study site locations and Table 1 summarises the main physical characteristics relevant to LST processes for all the

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sampling locations in this study. In all cases wave heights were low to moderate and wave approach angles to the shore range from nearly normal to <5 . 4.2. Meteorology, beach profiles, suspended sediments and bedforms In the cases examined below wind speed and direction data were obtained from meteorological stations situated close to the field sites. All beach profiles were measured at high tide to a maximum possible working depth of 2 m using a total station. Samples of suspended sediment transported by the longshore current were collected within a half-hour period either side of high-water using streamer traps (Kraus, 1987; Williams et al., 2007). The traps comprise a series of nets made from polyester monofilament sieve cloth (mesh size of 63 mm) held open by a metal aperture (9  14 cm) in an equally spaced vertical array up to 1.5 m above the bed on a sampling frame. Frames were carried horizontally to the chosen measurement location and were deployed facing into the longshore current normally for a period of 10 min at known locations on the beach profile. Laboratory tests have shown the streamer trap design to have hydraulic efficiency close to unity (Rosati and Kraus, 1988). Care was exercised to ensure people deploying the frames stood inshore and downstream of the nets to prevent disturbance to the bed upstream of the traps. Accurate measurements of the lower trap height above the bed were made and samples of bottom sediment were collected at streamer trap deployment sites. Where possible, bedforms that were present in the vicinity of the sampling frames were measured by divers equipped with a mask and snorkel. In some cases, reduced visibility by suspended fine sediment prevented this and divers simply felt the bed with their hands and feet and noted if bedforms were present or not. 4.3. Hydrodynamics At the RS and PR sites, the longshore current speed (S) was measured by timing a drogue float between two fixed points 200 m apart and equidistant from a given frame location and at the same offshore location. The float was weighted to ride as low in the water as possible. The longshore current direction (4) was measured by observing the float track and taking a compass bearing. Low wind speeds during the experiments and drag from the drogue ensured the floats travelled as closely as possible to the speed of the surface longshore current. This method was evaluated at the Cassino site using a millimetre wave-radar to track the drifters (Bell et al., 20062) and at the SK and PF sites using a Valeport Midas electromagnetic current meter3 (ECM) deployed at 0.3 m above the bed. Typically S values estimated from the float tracking were at worst 8% larger than S values measured by the ECM and more typically <5% larger (Table 2). Although these differences are all positive, and can be attributed to vertical velocity gradients and to possible wind effects on the floats and to timing errors, no single cause can be identified. Differences between drifter derived values for 4 and values from the ECM were less than 10 (Table 2). At RS and PR sites, the average angle of breaking waves relative to the beach (ab) was estimated by taking a compass bearing at the wave break point. The presence of regular swell waves at the RS, PR and PF sites made this observation relatively easy and accurate. In

2 See also http://www.nav-tech.com/Oceanographic.htm, accessed 4 December 2007. 3 See http://www.valeport.co.uk/currentmeters.htm#midas_ecm, accessed 4 December 2007.

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Fig. 1. Location of the study sites: (a) various, Rio Grande do Sul (RS), Brazil; (b) Praia do Leste, Parana´ (PR), Brazil; (c) Skallingen (SK), Denmark; and (d) Praia de Faro (PF), Portugal. Negative latitude and longitude values identify south and west, respectively.

some cases the accuracy of ab observations was tested at the Cassino site (RS) using x-band radar images. These revealed approaching wave patterns over a broad area spanning 4 km and 3 km longshore and cross-shore, respectively. Typically, differences between observed and measured ab values were 5 or less (Table 2) and less than the variance in wave approach angle. The time-average breaking wave height (Hb) and period (T) of approximately 150 incident waves were estimated by reference to a fixed calibrated marker close to the streamer trap frame. Estimates obtained by individual observers were cross-checked for consistency and were found to agree to within 5% of each other. Additionally, in some cases, estimated values of Hb and T were checked and verified using an In Situ Level Troll 300 pressure sensor4 (PS) deployed at c. 0.1 m above the bed at the SK and PF sites at or close to the wave break point and at streamer trap frame locations. Differences between observed Hb and T values and those derived from the PS data are entirely attributable to observational errors. At best values of Hb could only be measured by observation to the nearest decimetre and T is also subject to observational error in the breaking wave conditions of the surf zone. Increasing the number of waves in a sample by 100% was found not to improve the accuracy. Errors associated with wave data collected by the methods outlined above are also summarised in Table 2. The consequences of errors in the observational data in subsequent estimations of LST are considered further below.

4

http://www.in-situ.com/, accessed 4 December 2007.

Accurate hydrodynamic data were also obtained during most of May 2005 at approximately equidistant locations across the surf zone at the Cassino site using an array of moored instruments. This included: (a) four PUV sensors; and (b) a multi-PUV tower that recorded data continuously at 15 Hz. Although most of these data are coincident only with the experiments on 27 May 2005, they span a wide range of wave directions, heights and periods and on several occasions are closely similar to those observed at Cassino during other LST experiments. Here they are used as a proxy to define cross-shore profiles of several key hydrodynamic variables and to validate predictions from the numerical model described below. 5. Data analysis Samples of bed sediments and suspended sediments washed from the nets were oven dried at 110  C before being weighed to determine the total mass of sediment at a given vertical sampling height z, M(z). Samples were then sieved at 0.25 phi intervals. The

Table 1 Summary of field site characteristics during experiments. Location D50s

RS PA SK PF

D50

Beach slope

Mean Hs Mean Tp Mean ab

(mm)

(mm)

(–)

(m)

(s)

(degrees)

0.15 to 0.21 0.08 to 0.42 0.10 to 0.18 0.25 to 0.50

0.18 to 0.22 0.12 to 0.64 0.10 to 0.20 0.40 to 0.60

1/13 to 1/30 1/20 to 1/25 1/180 1/10 to 1/20

1.4 1.5 1.0 0.9

7 6 4 8

5–40 10–30 5–40 10–40

to 9 to 14 to 6 to 12

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Table 2 Quantification of measurement differences (Diff.) absolute errors for S, Hb, T and ab. Location

Cassino Cassino Skallingen, Frame B Skallingen, Frame C Faro, Frame A Faro, Frame B

Date

27/05/2005 28/05/2005 12/09/2006 13/09/2006 24/10/2007 25/10/2007

S (m s1)

T (s)

Observed

ECM

Diff. %

Observed

PS

Diff. %

– – 0.22 0.24 0.39 0.49

– – 0.21 0.22 0.37 0.42

– – 5 9 5 7

– – 6.0 6.0 6.5 8.0

– – 6.2 5.7 7.3 7.7

– – 3 5 12 4

Observed

PS

Diff. %

Observed

x-Band radar

Error

– – 0.4 0.4 0.6 0.5

– – 0.42 0.34 0.65 0.48

– – 5 15 8 4

5 10 15 15 20 10

9 12 – – – –

4 2 – – – –

ab (degrees)

Hb (m)

Cassino Cassino Skallingen, Frame B Skallingen, Frame C Faro, Frame A Faro, Frame B

27/05/2005 28/05/2005 12/09/2006 13/09/2006 24/10/2007 25/10/2007

median grain size for the bed (D50) and suspended (D50s) sediments were determined for each sample using GRADISTAT software (Blott and Pye, 2001). Estimates of the threshold bed shear stress for incipient grain motion (ws) and sediment mobility (j) required by LT-MOD were calculated using approaches detailed in Soulsby (1997).

The predicted cross-shore distribution of significant wave height (Hs) peak wave period (Tp), ab and S were tested against data from PUV instruments located at distances from the shoreline at Cassino Beach. In Fig. 2 an example of model output shows crossshore profiles of Hs, Tp, ab and S for a single test case at 12h00 on 27 May 2005. Here the errors associated with the model predictions for the 90% confidence interval are indicated by the grey shading. Also shown are measured values of these quantities from the PUVs at five locations. In general the agreement is good demonstrating that this simple model reproduces with reasonable accuracy the principal characteristics of surf zone hydrodynamics relevant to LST. During the periods 20–27 May and 8–22 June 2005, measured and predicted values of Hs, Tp, ab and S usually differed by 10% for ‘‘small’’ incident waves (Hs c. 1 m; 20 cm s1 < S < 50 cm s1). These errors increased as incident wave height increased, reaching 20% at the maximum observed Hs value of 2 m. Since M(z) is an integrated measure of the total suspended sediment flux at a given height above the bed, determination of the C-Profile requires knowledge of the vertical longshore current profile. In the absence of vertical current profile data, it is necessary here to assume that in water depths h in the range 2.5D50 < z < 0.5 h, the vertical current profile takes a logarithmic form. For z > 0.5 h, S(z) is assumed to be constant up to the water surface. This type of vertical velocity profile is a reasonable approximation to longshore current profiles measured in the laboratory (e.g. Reniers and Battjes, 1997; Wang et al., 2002) and in the field (e.g. Garcez Faria et al., 1998). It conforms also to velocity profile measurements obtained from the multi-PUV tower located within the zone of maximum longshore flow (see below). The suspended sediment concentration in g l1 (C) was computed at each sampling height using C(z) ¼ M(z)/[S(z)At] where A is the cross-sectional trap area (0.14  0.09 m) and t is the sampling time (normally 600 s). 6. Results and discussion

Fig. 2. Example of cross-shore modelling output for Cassino, Rio Grande do Sul (RS), Brazil for 12h00, 27/5/2005 showing: (a) Hs, (b) Tp, (c) ab; (d) S; and (e) the smoothed beach profile showing PUV locations. This was measured using a precision echosounder from a jet ski.

The following section comprises two parts. In the first part measured C-Profiles are compared with C-Profiles predicted by Eq. (5). In the second part, the work focuses on the Cassino data set where the cross-shore hydrodynamic environment predicted by the models can be verified by the field measurements. It establishes field verified cross-shore profiles for S, 4, ab, Hs, and Tp using the models and for D50 and D50s using bed and suspended sediment samples. These data are then used to predict C-Profiles at locations across the region of longshore flow and to estimate the total LST.

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Fig. 3. Smoothed beach profiles and streamer trap frame locations for: (a) Cassino and (b) various sites along the Rio Grande do Sul (RS) coast, Brazil.

6.1. Beach profiles

6.2. Measured and predicted bedforms

The cross-shore location of measured C-Profiles are shown for: (a) Cassino, RS (Fig. 3a); (b) the other five sites along the RS shoreline (Fig. 3b); (c) Praia do Leste, PR (Fig. 4a); (d) SK (Fig. 4b); and (e) PF (Fig. 4c). The location of ECM and PS deployments at SK and PF sites are also shown in Fig. 4b and c, respectively. These beach profiles have been smoothed and normalised to enable quick comparisons and extend from the waterline to distances offshore (x) in the range c. 40 m < X < c. 220 m. Profiles at RS sites normally encompass the first (inner) trough and bar system while profiles at SK span the first and second bars. At PR and PF sites a nearshore bar is not present.

Differences between predicted and measured hrw and lrw values were normally c. 15% and 20%, respectively, and thus in the absence of field measurements, predicted hrw and lrw values are used to define the bed roughness (Eq. (3)). Although the predicted low amplitude, long wavelength current-generated bedforms were not detected in the surf zone during the experiments, these were evident at some locations at the SK site at low water. The wavelength of these bedforms conformed approximately to Van Rijn (1993) by 18%, but their amplitude was around half the predicted value. This is thought to reflect the reworking of the bedforms by

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Fig. 4. Smoothed beach profiles and streamer trap frame locations for: (a) Praia do Leste (PL), Parana´, Brazil; (b) Skallingen (SK), Denmark; and (c) Praia de Faro (PF), Portugal. Also shown in (b) and (c) are locations of the ECM and PS instruments.

wave action during the ebbing tide. However, some doubt remains about whether or not these bedforms are present in the field and since their hydraulic roughness is usually less than 20% of that attributable to the wave-generated bedforms, bed roughness is defined here by the wave-generated ripples only. Owing to the problems in obtaining data in the energetic surf zone, only a few studies have reported bedforms in breaking wave conditions (e.g. Gallagher, 2003). The present observations of wave-

generated bedforms are therefore valuable and are considered further below. 6.3. Measured and predicted C-Profiles Measured C values depend critically on the assumption that the time-averaged vertical longshore current profile conforms to a loglaw. This assumption was tested using data from the PUV tower for

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very similar wave and wind conditions as those pertaining on 7/12/ 2004 (i.e. S ¼ 0.46 m s1; Hs ¼ 0.46 m; Tp ¼ 6 s, Test Case A) and 27/ 5/2005 (i.e. S ¼ 0.66 m s1; Hs ¼ 0.50 m; Tp ¼ 8 s, Test Case B) at Cassino, RS. Fig. 5 shows the following time-series: (a) S at z ¼ 0.38 m; (b) the product moment correlation coefficient (R2) expressing the goodness of fit of the log-law to the S profile measured at the three lowest PUVs on the tower at z ¼ 0.38, 0.74 m and 1.21 m; (c) U*; (d) drag coefficient (Cd) derived from zo values using Cd ¼ ½0:4=ð1 þ lnðzo =hÞÞ2 ; (e) Hs and (f) Tp values derived from pressure data using the spectral and zero down-crossing methods. Also indicated by the grey shaded boxes are periods

during which the measured hydrodynamic conditions approximate to Test Case A and Test Case B. R2 values during these test times are close to 1.0 and are significant at the 99.9% confidence level demonstrating that the vertical longshore current profile conforms to a log-law in the region zo < z < 1.2 m. The same results were obtained for other test cases examined thus supporting the assumptions made when computing C-Profiles using the streamer trap data. Obtaining representative samples of suspended sediment using the streamer traps at heights less than 10 cm above the bed was problematic for several reasons. Firstly, the greatly reduced

Fig. 5. Time-series plots of longshore currents speed (S), product moment correlation coefficient for a log fit to velocity profiles (R2), shear velocity (U*), drag coefficient (Cd), statistically-based significant wave height (Hs) and peak wave period (Tp). Data derived from PUV tower measurements for the period 20–27 May 2005. The grey shaded area highlight Test Case A and B.

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alongshore current velocity near the bed was insufficient to hold the net open. Secondly, oscillatory cross-shore wave-induced flows acted at times to close the net and resulted in undersampling of the suspended sediment. At other times, the close proximity of a bedform crest had the opposite effect leading to over-sampling of the suspended sediment concentration with respect to the spatially averaged values measured by nets further away from the bed. For these reasons we rejected suspended sediment concentration values measured closer than 10 cm from the bed. Typical examples of C-Profiles measured at Cassino RS and SK sites between the height range 0.1 cm < z < 0.8 cm, and CProfiles predicted by Eq. (6) between the height range 2.5D50 < z < 0.8 m are shown in Fig. 6. Fig. 6 shows a reduction in C values by around 4 orders of magnitude between Co and C at z ¼ 0.5 m. Infrequent data outliers in the full data set show no systematic trend and are thought to reflect both the inherent inaccuracy of the streamer traps and natural variability of suspended sediment in the surf zone. For the C-Profiles illustrated in Fig. 6, and for almost all other C-Profiles, the correlation between measured and predicted C-Profiles was statistically significant at the 95% confidence level with R2 values typically 0.85 over the range 0.1 m < z < 1 m. Although there is no independent way of verifying the validity of predicted Co values (Eq. (6)), the reduction of C with z is predicted independently of the measured C-Profiles and thus the fit to the data supports the view that predicted Co values are realistic. We argue here that a good fit to the C-Profiles measured in the present study over the height range 10 cm < z < 80 cm must reflect an accurate representation of

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the concentration values closer to the bed. If this was not the case, or if the reference concentration was wrong, the resulting curves would deviate significantly from the measured C-Profiles. Fig. 6 shows that this is clearly not the case. In common with the other field observations, the present approach to prediction of C-Profiles is also subject to a number of errors which, in the absence of detailed field data, are difficult to quantify. However, there are a number of potential sources of error that require comment. The first concerns the definition of bed roughness discussed above which relies on knowledge of the bed sediment grain size and on empirical prediction of bedform dimensions. Although the former is readily established by sampling, the latter is more difficult to measure in the field, and thus errors here could potentially have serious consequences. If the bed roughness is defined simply by the bed sediment grain size (i.e. z0G) when calculating C-Profiles using Eq. (5), measured C-Profiles are underestimated significantly by this and other well-known formulae (e.g. Nielsen, 1992). In all cases, statistically significant agreement between measured and predicted C-Profiles can only be achieved if the total bed roughness term includes bedforms (z0R). Since Eq. (5) was validated and tested using well-constrained data sets, its ability to predict with skill the present C-Profile data demonstrates clearly that the hydrodynamic conditions and bed morphology predicted by the approaches outlined above are reasonably accurate. In this respect it is useful to find that empirical formulae developed to predict wave-generated ripples for nonbreaking wave conditions also function reasonably well in the surf zone and predict the presence of bedforms with sufficient

Fig. 6. Measured and predicted vertical suspended sediment concentration profiles for stated Cassino RS and SK experiments.

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Fig. 7. Cross-shore profiles of water depth (h), significant wave height (Hs), longshore currents speed (S), peak wave orbital speed (Uw), ripple height (h), and wavelength (l), and predicted total sediment transport (QT) from QMOD and suspended transport (Qs) and bedload (Qb) (from Soulsby, 1997) for: (a) Test Case A; and (b) Test Case B.

roughness to give a good agreement between measured and predicted C-Profiles. 6.4. Modelling LST with LT-MOD Data presented so far demonstrate that it is possible to predict with reasonable accuracy the measured C-Profiles at two or more cross-shore locations. Assuming that this model is valid at other cross-shore locations, it can then be invoked across the surf zone in LT-MOD and total LST can be calculated by integration. This semiempirical model requires as input parameters a number of essential site specific data including one or more measurements of: (a) S and 4; (b) H or Hb, T or Tp and ab; (c) beach profile; (d) D50 and D50s; (d) l

and h for wave- and current-generated bedforms; and (e) C-Profile. It is helpful also to know the water temperature (for n and r). In the present study, the cross-shore distribution of these parameters is derived from the field measurements and the field-validated models outlined above. Following the steps described above, C-Profiles were calculated using validated cross-shore distributions of hydrodynamic and sediment parameters over the range 0.3 m < X < 500 m. The total LST flux was then obtained by integration. Selected LT-MOD results for Test Case A and Test Case B are shown in Fig. 7a and b, respectively. In each case these plots show: measured cross-shore profiles of h, Hs and S (including streamer trap frame locations); and crossshore profiles of Uw, mean and peak U*wc, hrw, lrw and predicted

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Fig. 8. Plots of predicted QMOD against predicted LST from USACE (1984), Kamphuis (1991), Bayram et al. (2007), QT, Qs and Qb for all test cases in this study.

rates of sediment transport from LT-MOD and QT and Qb (Soulsby, 1997). For simplicity the other terms used to derive the LST flux in the model are not illustrated. There are a number of assumptions made in these demonstration simulations: (1) D50 is assumed to be invariant across the surf zone; (2) D50s is defined by the mean D50 of sediments in the streamer traps and takes no account of vertical changes in suspended sediment grain size; and (3) current generated bedforms are not considered when defining the bed roughness. In spite of these simplifications, Fig. 7 demonstrates broad agreement between the present model and QT(x). It shows also that QB(x) makes only a relatively small contribution to QT(x) since, in the present conditions, S values are not greatly in excess of the entrainment threshold and sediment dynamics are dominated by wave re-suspension and subsequent advection by the longshore current. Both Test Case A and B exhibit similar characteristics with maximum LST predicted in the troughs. In Test Case B, wave conditions allowed measurements to be obtained at a greater distance from the shoreline. In this case the influence of the second bar is evident and gives rise to two peaks in LST in the troughs. In this case Fig. 7b shows that difference between Q(x) from LT-MOD and predicted QT(x) values are greater than for Test Case A. This may in part be attributable to the use in Eq. (5) of a constant drag coefficient making it unresponsive to possible changes in bed roughness associated with changing cross-shore hydrodynamic

conditions. In this case, peak QT(x) values are correlated with the approximate cross-shore location of maximum Uw and S values. A comparison between instantaneous LST from LT-MOD across the computational domain (QMOD) and from the CERC (K ¼ 0.39), Kamphuis (coefficient ¼ 6.5  104) and Bayram et al. (3 ¼ 1.5  103) formulae are shown in Fig. 8 and span a range of approximately 2 kg s1 < Q < 90 kg s1. Here fairly good agreement is found between values of QMOD and Q1, Q2 and Q3, with predictions from the Bayram et al. formula having the highest correlation with QMOD (R2 ¼ 0.95). It is noted that better agreement between measured LST and bulk formulae predictions might be achieved through optimisation of the coefficients. However, no site-specific method to do this is currently available. In tests using data from all the field sites, LT-MOD was found to be robust, and in most cases gave predicted rates of suspended sediment transport that matched approximately Q predicted by the bulk formulae. However, QMOD predictions are still subject to errors. For example, on a given cross-shore profile, an under-estimation of ws by c. 20% can result in a fourfold increase QMOD. Similarly, failure to correctly parameterise zo results in significant over- or underestimation of C-Profiles leading to large errors in QMOD. This is a particular concern in the surf zone, where relationships between hydrodynamic forcing and the bedforms are largely unknown and where temporal changes in bed roughness may occur over short

Fig. 9. Measured versus predicted change in shoreline position from LT-MOD and Bayram et al. (2007), RS, Brazil: (a) location of the study site; (b) observed and modelled change in shoreline position from November 1997 to November 1998; and (c) observed and modelled change in shoreline position from June 2000 to April 2002. More details on the methods used to derive these plots are given by Williams and Esteves (2005).

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time interval commensurate with wave groups and infragravity modulations in water level. Although the field measurements and data proxies have defined as effectively as possible all the terms in LT-MOD, these errors cannot be eliminated entirely. Thus far, the data presented do not consider the cumulative differences in predicted LST rates likely to arise when applying LTMOD and other LST formulae over extended periods at different locations, where both the magnitude and longshore directions are likely to be highly variable. As a result, over periods of weeks or months, difference in net LST predicted by each method will be much larger. Given the differences between LT-MOD and bulk formulae predictions discussed above, the question now arises as to which formulae or approach gives the most accurate prediction of LST. To investigate this further, the simple continuity model used by Williams and Esteves (2005) to examine changes in position of the almost continuous 600-km long Rio Grande do Sul shoreline, Brazil (Fig. 9a) was re-examined using LT-MOD and the bulk formula of Bayram et al. (2007). Selected results from this study for the periods November 1997 to November 1998 and June 2000 to April 2002 are shown in Fig. 9b and c, respectively. In both cases the shoreline position data exhibit oscillations at a wide range of spatial and temporal scales and thus present a challenge to any model attempting to reproduce the observed changes. Fig. 9b and c demonstrate that LT-MOD reproduces the measured annual oscillations in shoreline positions with more accuracy than by the Bayram et al. (2007) formula (c. 30%). In particular, a wide range of time and spatial-scale changes in the shoreline are all represented well in the new modelling results. This demonstrates that in this specific case LT-MOD out-performs the bulk formulae. 7. Conclusions This study adds high-quality field measurements of suspended sediments in the surf zone to a relatively sparse database. The data span a range of wave conditions, sediment types and beach profiles. The ability of LT-MOD to predict C-Profiles with reasonable accuracy at a number of different cross-shore locations has been verified with field data. Although shear velocity values and bedform dimensions obtained from imposed hydrodynamic gradients across the model domain are at best approximations, and may not be accurate across the whole model domain, the use of more advanced approaches cannot be supported by the present sparse field data. LST-MOD is based on formulae developed primarily for wave-only, uni-directional flows or weakly interacting w þ c situations. It is therefore useful to find that even in the complex hydrodynamics of the surf zone, the relationships they define between the applied bed shear stress and sediment mobility remain reasonably robust. Furthermore, the predicted rates of sediment transported across the surf zone from LT-MOD are in fair agreement with some wellestablished bulk formulae and in the test case presented here provide better estimates of LST operating at large spatial and temporal scales. With this in mind, it may now be possible with further refinements to investigate the temporal and spatial behaviour of LST and thereby improve our present ability to predict accurately LST over extended periods of time. In spite of the uncertainties, it is considered that LT-MOD offers an effective method to predict LST when accurate and very site specific information is required and when LST predictions are required over extended periods. Acknowledgements The authors gratefully acknowledge the help of Raphael Pinotti, Heitor Perotto and Clecio Quadros in the field work in Brazil and the technicians Neusa and Glo´ria from the Laborato´rio de Oceanografia

Geolo´gica (LOG/FURG) for their assistance with the sieving of samples from the Brazilian campaigns. We appreciate the contribution from Ed Thorton, Mark Orzek and Jamie MacMahan of the Department of Oceanography/Naval Postgraduate School in supplying the PUV data. Thanks are also given to School of Geography students at the University of Plymouth who assisted with fieldwork and sediment analysis in Denmark and Portugal.

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