Suspended sand transport in surf zones

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 110,, doi: /2004jc002853, 2005 Suspended sand transport in surf zones Nobuhisa Kobayashi, Haoyu Zhao, and Yukiko Tega Ocean Engineering Laboratory, Center for Applied Coastal Research, University of Delaware, Newark, Delaware, USA Received 24 December 2004; revised 28 June 2005; accepted 14 September 2005; published 13 December [1] Three tests were conducted in a wave flume to investigate time-averaged suspended sediment transport processes under irregular breaking waves on equilibrium beaches consisting of fine sand. Free surface elevations were measured at ten locations for each test. Velocities and concentrations were measured in the vicinity of the bottom at 94 elevations along 17 vertical lines. The relations among the three turbulent velocity variances are found to be similar to those for the boundary layer flow. The vertical variation of the mean velocity, which causes offshore transport, is fitted by a parabolic profile fairly well. The vertical variation of the mean concentration C is fitted by the exponential and power-form distributions equally well. The ratio between the concentration standard deviation s C and the mean C varies little vertically. The correlation coefficient g UC between the horizontal velocity and concentration, which results in onshore transport, is of the order of 0.1 and decreases upward linearly. The offshore and onshore transport rates of suspended sediment are estimated and expressed in terms of the suspended sediment volume V per unit area. A time-averaged numerical model is developed to predict V as well as the mean and standard deviation of the free surface elevation and horizontal velocity. The bottom slope effect on the wave energy dissipation rate D B due to wave breaking is included in the model. The computation can be made well above the still water shoreline with no numerical difficulty. Reflected waves from the shoreline are estimated from the wave energy flux remaining at the shoreline. The numerical model is in agreement with the statistical data except that the undertow current is difficult to predict accurately. The measured turbulent velocities are found to be more related to the turbulent velocity estimated from the energy dissipation rate D f due to bottom friction. The suspended sediment volume V expressed in terms of D B and D f can be predicted only within a factor of about 2. The roller effect represented by the roller volume flux does not necessarily improve the agreement for the three tests. Citation: Kobayashi, N., H. Zhao, and Y. Tega (2005), Suspended sand transport in surf zones, J. Geophys. Res., 110,, doi: /2004jc Introduction [2] Cross-shore sediment transport on beaches has been investigated extensively. For the beach at Duck, North Carolina, alone, a large number of studies have been performed. Trowbridge and Young [1989] used a sheet flow model based on the time-averaged onshore bottom shear stress to explain the onshore movement of a bar on the beach observed during low-energy wave conditions. Thornton et al. [1996] and Gallagher et al. [1998] used the energetics-based total load model of Bailard [1981] to explain the offshore movement of the bar observed during storms. The energetics model could not predict the slow onshore bar migration observed during low-energy wave conditions. Hoefel and Elgar [2003] included the skewed accelerations in the energetics-based sediment transport model to successfully simulate both onshore and offshore bar migration. On the other hand, Henderson et al. [2004] Copyright 2005 by the American Geophysical Union /05/2004JC developed a wave-resolving, eddy-diffusion model of water and suspended sediment motion in the bottom boundary layer. This model also predicted the onshore and offshore bar migration events successfully. However, no model could predict another event for the offshore bar migration. In these studies, it was not possible to assess the accuracy of the sediment transport models because no measurement was made of suspended sediment and bed load. This is normally the case with other numerical studies for beach profile changes [e.g., Roelvink and Stive, 1989; Karambas and Koutitas, 2002; van Rijn et al., 2003]. [3] Suspended sediment transport under breaking waves is examined here in detail in order to improve our capability in predicting the cross-shore sediment transport on beaches. Three tests were conducted in a wave flume on equilibrium beaches consisting of fine sand under irregular breaking waves with the spectral peak periods of 4.8, 1.6 and 2.6 s. The equilibrium beaches allowed us to measure velocities and sand concentrations at 94 elevations along 17 vertical lines. The velocity measurements are used to obtain the three turbulent velocity variances in the vicinity of the 1of21

2 Table 1. Wave Conditions at Wave Gauge 1 for Tests 4.8, 1.6, and 2.6 Reflection Coefficient T p, s Duration, s d, cm h, cm H rms,cm (H rms ) i,cm Data IROLL = 0 IROLL = bottom in the surf zone. The synchronous measurements of the horizontal velocity and concentration allowed us to estimate the onshore suspended sediment transport rate due to their positive correlation. A time-averaged numerical model is developed and compared with the three tests. This model is an extension of existing models with and without the effect of a roller. The bottom slope effect on the rate of wave energy dissipation due to wave breaking is included in the model. This extension allows one to continue the landward marching computation well above the still water shoreline with no numerical difficulty and estimate the degree of wave reflection from the shoreline. [4] This paper is organized as follows. Section 2 describes the experimental procedure for the three tests. Section 3 shows the analyses of the measured velocities and concentrations and the estimation of the onshore and offshore suspended sediment transport rates. Section 4 presents the extended time-averaged model with and without the roller volume flux. Section 5 compares the numerical model with the data and discusses the importance and uncertainty of bottom friction for the turbulent velocities and sand suspension. Section 6 summarizes the findings of this study. 2. Experiments [5] Experiments were performed in a wave tank that was 30 m long, 2.4 m wide, and 1.5 m high. A divider wall was constructed along the centerline in the tank to reduce the volume of sand in a 115-cm-wide flume [Kobayashi and Lawrence, 2004]. The water depth in the flume was 0.9 m. A piston-type wave paddle was used to generate irregular waves, based on the TMA spectrum. The spectral peak period, T p, was 4.8, 1.6 and 2.6 s for the following three tests as listed in Table 1. A fine sand beach with a slope of 1/12 was constructed in the wave flume. The sand was well sorted and its median diameter was 0.18 mm. The measured fall velocity, specific gravity and porosity of the sand were 2.0 cm/s, 2.6 and 0.4, respectively. The sand beach was exposed to the specified irregular waves generated in a burst until the sand beach became quasi-equilibrium as explained below. The burst duration was 900, 300 and 400 s for tests 4.8, 1.6 and 2.6, respectively, where the value of T p is used to identify each test. The initial transition of 60, 10 and 20 s was removed from the time series of 900, 300 and 400 s, respectively, for the subsequent data analyses. It is noted that tests 4.8 and 1.6 were reported by Giovannozzi and Kobayashi [2001, 2002] and Tega et al. [2004]. The report of Zhao and Kobayashi [2005] tabulates the data used in this paper. [6] Beach profiles were measured along three cross-shore transects. The measured profiles were essentially uniform alongshore and the average profile is used in the following. The bottom elevation change was less than 1 cm for the 42, 40 and 12 bursts generated on the equilibrium beaches for tests 4.8, 1.6 and 2.6, respectively, during the measurements of free surface elevations, velocities, and concentrations at a rate of 20 Hz. Figure 1 shows the measured equilibrium beach profiles where x is the onshore coordinate with x =0 at wave gauge 1 and z is the vertical coordinate. The slopes of the foreshore and offshore zone were of the order of 0.1 and relatively steep in these small-scale experiments. [7] Ten wave gauges were used to measure the time series of the free surface elevation h above the still water level (SWL). The still water depth, d, at the most seaward wave gauge 1, is listed in Table 1 where wave gauge 1 was located well outside the surf zone. The measured time series were repeatable within approximately 1% differences. The mean h and standard deviation s h of h are calculated for each wave gauge where the overbar indicates time averaging for the measured time series after the removal of the initial transition. The mean depth h is the sum of h and the still water depth. Thep root-mean ffiffiffi square wave height H rms is defined as H rms = 8 sh in the following. Table 1 lists the values of h and H rms at wave gauge 1 where the wave set-down was negligible and the wave height H rms was in the range of cm. The differences of the three equilibrium profiles were caused mostly by the difference of the spectral peak period T p. The wave gauges 1 3 located outside the surf zone were also used to separate the incident and reflected waves using linear wave theory [Kobayashi et al., 1990]. The separated time series are used to obtain the root-mean square wave heights (H rms ) i and (H rms ) r of the incident and reflected waves, respectively. Table 1 lists the values of (H rms ) i and the reflection coefficient, (H rms ) r /(H rms ) i, where H rms includes both incident and reflected waves but is approximately the same as (H rms ) i. The computed reflection coefficients will be explained in section 5. [8] The wave gauges A G in Figure 1 were placed at the cross-shore locations of the velocity and concentration measurements except for wave gauges B 0,C 0, D and E for test 2.6 due to instrument malfunction. The incident irregular waves on the terraced beach in test 4.8 did not break at gauge A, were breaking frequently at gauge B, broke intensely sometimes at gauges C and D, and became bores at gauges E G. The height and cross-shore wavelength of ripples in the vicinity of gauge A were approximately 2 cm and 11 cm, respectively. No ripples were visible landward of gauge B inside the surf zone. For test 1.6, large incident waves broke at gauges A C over the bar crest and became bores at gauges D G in the bar trough region. The intensity of wave breaking was less on the barred beach and ripples were present in the surf zone. The height and cross-shore length of these ripples were approximately 1.3 cm and 6 cm, respectively. For test 2.6, the equilibrium profile was intermediate between the terraced and barred profiles in tests 4.8 and 1.6. The majority of incident waves broke at 2of21

3 Figure 1. Cross-shore and vertical gauge locations above equilibrium beaches for three tests. gauge A and above the bar crest and became bores at gauges B C. Wave breaking was reduced considerably at gauge D. All waves broke on the steep slope of approximately 0.3 seaward of gauge E buried in the swash zone. The height and length of ripples in the region between gauges A and D were approximately 0.7 cm and 8 cm, respectively. [9] For each burst of the wave generation, the velocities and concentrations were measured at one particular location indicated by a dot in Figure 1. The number of dots in Figure 1 are 42, 40 and 12 for tests 4.8, 1.6 and 2.6, respectively. The measurement elevation was 1, 2,..., n cm above the terraced beach in test 4.8 where the upper limit n along the vertical line at each gauge location was decided by the reliable velocity measurement or negligible sand concentration. The reliability was judged from the measurements at two alongshore locations as discussed below. On the other hand, the measurement elevation z m was 2, 4,...,2n above the barred beaches in tests 1.6 and 2.6 except that the elevations of 3, 5, 7 and 9 cm were added for line 1.6D located at the deepest point of the bar trough. The different vertical lines in Figure 1 are identified by the numeral of each test followed by the letters A to G. [10] Two acoustic-doppler velocimeters (ADV) with a 3D down-looking probe and a 2D side-looking probe were used to measure the velocities at the two alongshore symmetric locations, 13.5 cm from the flume centerline as shown in Figure 2. The wave gauge was placed 22.5 cm laterally from the centerline when the velocities and concentrations were measured at the same cross-shore location. Figure 2. Alongshore gauge locations. 3of21

4 The sampling volume of both ADVs is approximately 0.1 cm 3. The velocity measurements at a sampling rate of 20 Hz in the absence of waves indicated noise less than 1 cm/s for the horizontal velocities and less than 0.2 cm/s for the vertical velocity [Giovannozzi and Kobayashi, 2001]. The measured horizontal velocities U 1 and U 2 were essentially in phase apart from high-frequency oscillations associated with turbulence. The average horizontal velocity U = (U 1 + U 2 )/2 is used here to calculate the mean U and standard deviation s U at each measuring elevation. To estimate the time-averaged turbulent velocity, use is made of the method proposed by Trowbridge [1998]. The measured velocities U 1 and U 2 are assumed to be expressible in the form U 1 ¼ U 1 þ u w þ u 0 1 U 2 ¼ U 2 þ u w þ u 0 2 ; where the wave component u w is assumed to be the same and the prime indicates the turbulent component. Assuming that u u u 0 2 and u 0 1 u0 2 0, the turbulent velocity variance u 0 2 is estimated as ð1þ u : U 1 U 1 U2 U 2 ð2þ 2 [11] The measured alongshore velocities V 1 and V 2 did not contain any wave component but their frequency spectra were dominated by the low-frequency components due to intermittent irregular wave breaking [Giovannozzi and Kobayashi, 2002]. The turbulent velocity variance v 0 2 is estimated as the average of the two velocity variances v 0 2 ¼ 1 h 2 i 2 V 1 V 1 þ V2 V 2 ; ð3þ 2 where the measured mean V 1 and V 2 were practically zero. [12] The vertical velocity W was measured by the 3D probe only. It is noted that the use of the 2D and 3D probes was intended for the comparison of the horizontal velocity measurements in the vicinity of the bottom. The measured vertical velocity was relatively small and appeared to correspond to the velocity of very fine material, which may be assumed equal to the fluid velocity, in the region of very small sand concentrations and an intermediate velocity between the fluid and sand velocities in the region of high sand concentrations. The mean velocity W is compared with the sand fall velocity w f = 2.0 cm/s measured in quiet water. The measured values of W/w f were in the range of 0.88 to 0.30 with the average being The variance s W 2 of the measured vertical velocity increased upward due to the increase of the wave-induced vertical velocity with the elevation z m above the local bottom. Assuming no correlation between the wave and turbulent components, the turbulent variance w 0 2 is crudely estimated as w 0 2 s 2 W 2ps h z 2 m ; ð4þ T p h where linear long wave theory [e.g., Dean and Dalrymple, 1984] is used to estimate the wave velocity variance using the measured values of s h and h at each line and the wave period represented by the peak period T p. The turbulent velocity variances estimated using equations (2) (4) are presented in section 3. [13] A fiber optic sediment monitor (FOBS-7) with two sensors was used to measure the sand concentrations at the two alongshore symmetric locations, 4.5 cm from the flume centerline as shown in Figure 2. The sampling volume located at a distance of 1.0 cm from each sensor is approximately 10 mm 3. The error or fluctuation of approximately 20% was observed in well-mixed samples of known concentrations due to this small sampling volume [Giovannozzi and Kobayashi, 2001]. The measured concentrations C 1 and C 2 were not always in phase, but intermittent high concentration events occurred simultaneously in both time series. The average concentration C =(C 1 + C 2 )/2 is used here to calculate the mean C and standard deviation s C at each measuring elevation. The synchronous measurements of U and C are used to obtain the correlation coefficient g UC defined as g UC ¼ U U 3. Velocity and Concentration Data C C = ð su s C Þ: ð5þ [14] The cross-shore variations of the measured h, s h and s U are presented in Section 5 in comparison with the numerical model which is one-dimensional in the crossshore direction. The vertical variation of s U was small as was observed previously [e.g., Guza and Thornton, 1980]. The vertical variations of the velocity and concentration are analyzed in the following for the purpose of the development of a depth-integrated suspended sediment transport model. [15] The vertical variation of the mean cross-shore velocity U is normally predicted using the time-averaged crossshore momentum equation which includes the terms related to the correlation between the measured velocity U and W [Deigaard and Fredsøe, 1989]. The correlation coefficient g UW defined in the form of equation (5) with C being replaced by W is calculated and plotted as a function of z m to examine the reliability of the measured g UW in comparison with the data of Cox and Kobayashi [1996] who conducted an experiment on a fixed bottom. The comparison indicates that the measured g UW is not reliable because the relatively small velocity W was affected by the suspended sand. [16] Alternatively, the measured U at each line is fitted to the parabolic profile [Svendsen, 1984] U ¼ a U z 2 m þ b U z m þ c U ; where a U, b U and c U are the coefficients determined to minimize the mean square error. The fitted coefficients for each line are listed in Table 2 where x indicates the line location in Figure 1 and h is the mean water depth. The correlation coefficient between the measured and fitted values, denoted as CC hereinafter, is partly because of the limited number of the velocity measurements at each line. Figure 3 shows the measured and fitted parabolic profiles of U at the seven lines for test 1.6 as an example. The undertow current U is small and varies little vertically at line 1.6A of infrequent wave breaking. The vertical ð6þ 4of21

5 Table 2. Fitted Coefficients for Mean Velocity U and Turbulent Velocity k 0.5 Line x, m h, cm a U,cm 1 s 1 b U,s 1 c U, cm/s u0, 0 cm/s t,cm 4.8A B C D E F G A B C D E F G A B C variation of U is large at lines 1.6B and 1.6C of frequent wave breaking and is reduced at lines 1.6D 1.6G in the zone of bores. However, the vertical variations of U in the breaker zone for tests 4.8 and 2.6 are smaller due to more intense wave breaking. The cross-shore and vertical variations of U are qualitatively similar to those of previous laboratory and field data [e.g., Cox and Kobayashi, 1998]. [17] Equation (6) may be used to extrapolate the measure U to the range of 0 < z m < h 0 where U =0atz m = h 0 and U < 0 for 0 < z m < h 0. The vertical distance (h h 0 ) below the mean water level is of the order of 2s h except for lines 1.6 A C and 2.6 A B near the bar crest where (h h 0 ) is of the order of 0.5 s h or zero where h 0 h is imposed. The time-averaged offshore volume flux q 0 and corresponding velocity U 0 are defined as q 0 ¼ Z h0 0 U dzm ð7þ U 0 ¼ q 0 =h 0 ; where equation (6) is used to find the positive values of q 0 and U 0 at each line. [18] The time-averaged turbulent kinetic energy k per unit mass is defined as k ¼ 1 2 u0 2 þ v 0 2 þ w 0 2 ; ð8þ Figure 3. test 1.6. Measured and fitted parabolic profiles of mean horizontal velocity U at lines A G for 5of21

6 Figure 4. Measured turbulent velocity variances u 0 2, v 0 2 and w 0 2 as a function of 2k =(u v w 0 2 ). where the turbulent velocity variances are obtained using equations (2) (4). Figure 4 shows the relations between the turbulent velocity variances which are approximated as u 0 2 = ð2kþ 0:6; v 0 2 = ð2kþ 0:3; ð9þ w 0 2 = ð2kþ 0:1: The value of CC for each empirical equation is shown in Figure 4 and subsequent figures. It is noted that equation (4) yields w 0 2 <0atz m = 6 and 8 cm for line 1.6A and at z m =6, 8 and 10 cm for line 1.6G. These five points plotted as w 0 2 = 0 in Figure 4 are in the range of k <12cm 2 /s 2. The relations in equation (9) are similar to those associated with the boundary layer flow in the inner region [Svendsen, 1987]. Equation (9) is applicable only to the region of the velocity measurement in Figure 1 where no velocities were measured near the free surface. [19] The time-averaged turbulent velocity k 0.5 has been assumed to decay exponentially downward [Svendsen, 1987; Roelvink and Stive, 1989]. The measured values of k 0.5 at each line are fitted to the exponential distribution k 0:5 ¼ u 0 0 exp ð z m= t Þ; ð10þ where u 0 0 is the turbulent velocity extrapolated to the bottom at z m = 0 and t is the vertical length scale of k 0.5. The fitted values at each line are listed in Table 2. Figure 5 shows k 0.5 /u 0 0 as a function of z m /j t j where use is made of the fitted values of u 0 0 and t at each line. At lines 4.8A, 1.6A, 1.6G and 2.6A, where wave breaking was infrequent or weak, t < 0 and k 0.5 decreased upward in the region of the velocity measurement. The length scale is approximately given by j t j h with CC = 0.81 except that t /h = 2.8 and 5.8 at lines 1.6B and 1.6C, respectively. At these two lines above the bar crest, the upward increase of k 0.5 was very small. Figure 6 shows the measured and fitted exponential profiles of k 0.5 at the seven lines for test 1.6 as an example. The vertical variations of k 0.5 are relatively small in the region of the velocity measurement for tests 4.8 and 2.6 as well. The fitted values of u 0 0 and u 0 0 exp(h/ t )at each line are compared with the computed turbulent velocities due to bottom friction and wave breaking in section 5. [20] The vertical distribution of C is examined using the conventional approach based on the vertical balance of suspended sediment fluxes expressed as w f C =( v dc/dz) on an equilibrium beach [Nielsen, 1992]. The vertical mixing coefficient v includes the effects of the waveinduced and turbulent velocities. The assumption of constant v results in C ¼ C b expð z m = C Þ C ¼ v =w f ; ð11þ where C n is the sediment concentration extrapolated to the bottom z m = 0, and C is the vertical length scale of C. The exponential distribution of C has been shown to represent the measured concentration profiles in surf zones [e.g., Peters and Dette, 1999] and over rippled beds in a water tunnel [Ribberink and Al-Salem, 1994]. On the other hand, the assumption of v = z m v C with v C being the velocity scale of v yields C ¼ C a ðz a =z m Þ n ð12þ n ¼ w f =v C ; 6of21

7 Figure 5. Vertical distribution of turbulent velocity k 0.5 expressed as k 0.5 = u 0 0 exp (z m / t ) at each line where t < 0 at lines 4.8A, 1.6A, 1.6G, and 2.6A. where C a is the reference concentration at z m = z a and the reference elevation z a is taken as z a = 1 cm which was the lowest elevation of the concentration measurement in Figure 1. The distribution of C in power form has been shown to represent the measured concentration profiles in sheet flow in a water tunnel [Ribberink and Al-Salem, 1994] and on the inner shelf [Lee et al., 2004]. Dohmen- Janssen and Hanes [2002] obtained n 0.55 under Figure 6. Measured and fitted exponential profiles of turbulent velocity k 0.5 at lines A G for test of21

8 Figure 7. Vertical distribution of mean concentration C expressed as C = C b exp ( z m / C ) at each line where the exponential distribution did not fit well at lines 4.8A and 4.8B. Figure 8. Vertical distribution of mean concentration C expressed as C = C a (z a /z m ) n with z a = 1 cm at each line where the distribution in power form did not fit well at lines 4.8A and 4.8B. 8of21

9 Table 3. Fitted Coefficients for Mean Concentrations C and Correlation Coefficient g UC Line C b,g/l c,cm C a,g/l n a g b g,cm 1 4.8A B C D E F G 0.45 NA 0.45 NA A B C D E F G A B C monochromatic nonbreaking waves in comparison with n 2.1 in the water tunnel [Ribberink and Al-Salem, 1994] and discussed the limitations of the water tunnel experiments. [21] The measured distributions of C(z m ) are fitted to the exponential and power-form distributions given by equations (11) and (12) as shown in Figures 7 and 8 where C b, C, C a and n are fitted at each line and listed in Table 3. It is noted that for line 4.8G, C = 0.40 and 0.49 g/l at z m = 1 and 2 cm, respectively. The average of these values is listed as C b and C a in Table 3 but line 4.8G is excluded from the subsequent sediment transport analysis. Both distributions represent the measured profiles fairly well except for lines 4.8A and 4.8B perhaps because of the limited number of the concentration measurements at the other lines. Figure 9 shows the measured and fitted profiles of C at the seven lines for test 1.6 as an example where the exponential and power-form distributions are plotted for comparison. The mean concentration C decreases rapidly upward at lines 1.6A of infrequent wave breaking. The concentration C is larger at lines 1.6B and 1.6C of frequent wave breaking than at lines 1.6D 1.6G in the zone of bores. The spatial variation of C in Figure 9 is similar to that presented by Peters and Dette [1999]. [22] For the present surfzone data, C and n are related empirically as n 0.2 h/ C with CC = 0.97 for the range of h/ C = The parameter n is the ratio between the sediment fall velocity w f and the velocity scale v C which Figure 9. Measured and fitted profiles of mean concentration C at lines A G for test of21

10 Figure 10. Vertical variation of ratio a = s C /C normalized by the vertically averaged value hai at each line where the ratio varied more at lines 4.8A, 1.6A, and 2.6A. may be related to the turbulent velocity u 0 0 in equation (10). The correlation between n and w f /u 0 0 turns out to improve when (u 0 0 w f ) is used instead of u 0 0 to account for the initiation of sediment suspension [e.g., Lee et al., 2004]. This regression analysis yields n 0.8w f /(u 0 0 w f ) with CC = 0.75 for w f = 2.0 cm/s and u 0 0 = cm/s. Equating the two empirical equations for n above, ( C /h) 0.25 (u 0 0/w f 1) with CC = Alternatively, C may be expressed simply as C 2 s h with CC = In short, it is difficult to estimate C and n accurately. [23] As for C b and C a, Dunkley et al. [1999] showed that empirical formulas could predict only the order of magnitude of the reference concentration under breaking waves. The time-averaged suspended sediment volume V per unit horizontal area is used and predicted in the following analysis of suspended sediment transport. The value of V at each line is obtained by integrating equations (11) and (12) from z m =0orz a to z m = h where equation (12) cannot be extrapolated to z m = 0. The integration yields V b ¼ C b C 1 exp h C " ð13þ V a ¼ C # 1 n az a h 1 ; 1 n where V b and V a are the values of V based on the exponential and power-form distributions and are of the order of 0.01 cm. The difference between V b and V a is less then 17%. [24] The standard deviation s C of the sediment concentration is related to the mean concentration C. The measured values of a = s C /C at each line are approximately constant z a as shown in Figure 10 where hai is the average value at each line. The differences between a and hai are less than about 20% except for lines 4.8A, 1.6A and 2.6A of no or infrequent wave breaking. To simplify the following data analysis, the average value hai without the brackets is used with the assumption of constant a at each line. [25] The measured correlation coefficient g UC between U and C decreases upward gradually because the concentration C of the sediment suspended from the bottom becomes less correlated with the fluid velocity U with the increase of the vertical distance z m. The measured values of g UC at each line are expressed as g UC ¼ a g b g z m ; ð14þ where a g is the extrapolated value of g UC at z m = 0 and b g is the inverse of the length scale associated with the upward decrease of g UC at each line. The fitted coefficients are listed in Table 3. Figure 11 shows the measured values of (g UC a g ) plotted as a function of b g z m for all the lines where a g = except for a g = 0 at line 1.6D above the bar trough and b g 1 is of the order of 100 cm. The measured upward decrease of g UC is well represented by equation (14) except for line 4.8A of no wave breaking. The fitted a g at each line is related to the measured values of s * = s h /h and a at the same line for the convenience of the subsequent sediment transport analysis. The fitted values of a g are of the order of s * /a where a g s * /a with CC =0.24fors * = and s * /a = The value of CC is low partly because of the narrow ranges of a g and s * /a. Plant et al. [2001] observed the increase of g UC with the increase of s * for s * <0.09ona natural beach. 10 of 21

11 Figure 11. Vertical distribution of correlation coefficient g UC expressed as g UC =(a g b g z m ) at each line where the linear distribution did not fit well at line 4.8A. [26] The cross-shore suspended sediment transport is analyzed using uc =(U C + uc) with uc = s U s C g UC from equation (5) where u =(U U) and c =(C C). The offshore suspended sediment transport rate q off due to the negative mean velocity U is calculated as q off ¼ Z h0 z 0 U Cdzm ; ð15þ where U is given by equation (6) and U < 0 for z m < h 0 as explained in relation to equation (7), and the mean concentration C is expressed by equation (11) with the lower limit z 0 = 0 or equation (12) with z 0 = z a = 1 cm as explained in relation to equation (13). Equation (15) can be integrated analytically. To estimate the onshore suspended sediment transport rate q on, use is made of cu = as U g UC C where a and s U are the averaged values at each line. The offshore volume flux q 0 given by equation (7) requires the same onshore volume flux above z m = h 0 in this twodimensional experiment. The onshore sediment flux above z m = h 0 may be estimated as q 0 C h where C h = C at z m = h. As a result, q on is calculated as q on ¼ as U Z h0 z 0 g CU Cdz m þ q 0 C h ; ð16þ where g CU is given by equation (14) and C is expressed by equation (11) with z 0 = 0 or (12) with z 0 = z a. Equation (16) can be integrated analytically. The term q 0 C h turns out to be important in the breaker zone where the extrapolated concentration C h is not negligible. [27] Figure 12 shows the estimated values of q off and q on using the exponential and power-form distributions of C for tests 4.8, 1.6 and 2.6. The average difference between the two estimates using equations (11) and (12) is approximately 10 and 23% for q off and q on where q on is more uncertain because of the uncertainty of the extrapolated value of C h at the mean water level. For both distributions, q on > q off for test 4.8 but q on < q off for test 1.6, whereas for test 2.6, q on > q off at line 2.6A but q on < q off at lines 2.6B and 2.6C. The net cross-shore sediment transport rate should be zero on the equilibrium beach. The difference (q on q off ) is caused by the measurement and extrapolation errors, minor profile changes and the bed load transport rate in the region of z m < z a = 1 cm for the power-form distribution or the transport rate which is not accounted for in the exponential distribution extrapolated to z m = 0. The analysis of the bed load transport rate is beyond the scope of this study. It is noted that the bed load formula of Meyer- Peter and Mueller may be limited to the median sand diameter d mm and wave periods larger than 3 s due to the assumption of quasi-steadiness [Ribberink, 1998]. For the present experiment, d 50 = 0.18 mm and bed load sediment did not move in a layer under breaking waves even in the absence of ripples. [28] Equations (15) and (16) are not convenient for the prediction of q off and q on. If ( U) in equation (15) is approximated by the averaged offshore mean velocity U 0 given by equation (7), q off V U 0 where V is the suspended sediment volume per unit area given by equation (13). If g UC in equation (16) is approximated by a g with g UC a g in equation (14) and q 0 C h is neglected for simplicity, q on aa g s U V, which is simplified as q on s * s U V using a g s * /a as discussed in relation to equation (14). These simplified equations are evaluated in Figures 13 and 14 for the exponential and power-form 11 of 21

12 Figure 12. Estimated onshore and offshore suspended sand transport rates q on and q off using the exponential and power-form distributions of C at each line. Figure 13. Offshore transport rate q off in comparison with offshore return current U 0 multiplied by suspended sediment volume V per unit area. 12 of 21

13 Figure 14. Onshore transport rate q on in comparison with s * s U V. distributions of C. The fitted equations in these figures are expressed as q off ¼ 0:9 V U 0 q on ¼ 0:8 s * s U V: ð17þ It is noted that the regression analysis between q on and (aa g s U V ) does not improve the agreement in Figure 14. Equation (17) involves V, U 0, s * = s h /h and s U which may be predicted by a cross-shore one-dimensional model. 4. Numerical Model [29] The time-averaged model developed here is an extension of the Dutch models by Battjes and Stive [1985], Stive and DeVriend [1994], and Ruessink et al. [2001]. The major improvement is the extension of the model to the lower swash zone. The time-averaged crossshore momentum and energy equations are expressed as ds xx dx ¼ rgh dh dx t b df dx ¼ D B D f ; ð18þ where x is the cross-shore coordinate, positive onshore, S xx is the cross-shore radiation stress, r is the fluid density, g is the gravitational acceleration, h is the mean water depth given by h =(h z b ) with z b being the bottom elevation, h is the mean free surface elevation, t b is the bottom shear stress, F is the wave energy flux, and D B and D f are the energy dissipation rates due to wave breaking and bottom friction, respectively. The terms t b and D f are normally neglected but included here because of the importance of bottom friction for sediment transport. Linear wave theory for onshore progressive waves is used to estimate S xx and F S xx ¼ rgs 2 2C g h 1 þ rc p q r C p 2 ð19þ F ¼ rgc g s 2 h ; where C g and C p are the group velocity and phase velocity in the mean water depth h corresponding to the spectral peak period T p, and q r is the volume flux due to the roller on the steep front of a breaking wave. [30] The roller effect has been represented by its area or energy [Svendsen, 1984] but the roller volume flux is used here because the roller effect is the most apparent in the increase of undertow current. Furthermore, irregular plunging waves in test 4.8 did not exhibit identifiable roller areas. The term rc p q r in S xx is the roller momentum flux due to the roller propagating with the speed of C p and causes the landward shift of h in the breaker zone. The roller effect is not included in F in the Dutch model [e.g., Ruessink et al., 2001] but is included by Svendsen et al. [2003], who related the roller area to the wave height squared. The two different approaches depend on the interpretation of F and D B in the energy equation where F and D B in equation (18) include the wave-related energy only. The practical reason of the adoption of the Dutch approach is that the energy equation without pany ffiffiffi roller predicts the cross-shore variation of H rms = 8 sh well if the breaker ratio parameter g is calibrated as explained later. In the Dutch approach, the dissipated wave energy is converted to the roller energy 13 of 21

14 which is assumed to be governed by [Stive and DeVriend, 1994] d dx rc2 p q r ¼ D B rgb r q r ; ð20þ where the roller dissipation rate, rgb r q r, is assumed to equal the rate of work to maintain the roller on the wave front slope b r of the order of 0.1 [Deigaard, 1993]. [31] Linear shallow-water wave theory has been used to find the approximate local relationships between h and U [Guza and Thornton, 1980]. The standard deviation of U is estimated as 0:5 s U ¼ s * gh ð21þ s * ¼ s h =h: The depth-integrated continuity equation of water on the beach, which is assumed impermeable, is expressed as (s h s U + U h + q r ) = 0 where s h s U is the onshore flux due to linear shallow-water waves [Kobayashi et al., 1998] and U is regarded as the depth-averaged return current. Hence, U is estimated as U ¼ s 2 * gh 0:5 qr =h: ð22þ [32] The time-averaged bottom shear stress and dissipation rate are expressed as t b ¼ 1 2 r f bjuju D f ¼ 1 2 r f bjuju 2 ; ð23þ where f b is the bottom friction factor. To express t b and D f in terms of U and s U, the equivalency of the time and probabilistic averaging as well as the Gaussian distribution of U are assumed [Guza and Thornton, 1985; Kobayashi et al., 1998]. with t b ¼ 1 2 r f b s 2 U G 2 U * ; D f ¼ 1 2 r f b s 3 U G 3 U * ; ð24þ U * ¼ U s U ; G 2 ðþ¼ r 1 þ r 2 r rffiffiffi 2 erf p ffiffiffi þ 2 p r exp r2 2 ð25þ G 3 ðþ¼ r 3r þ r 3 r rffiffiffi 2 erf p ffiffi þ r 2 r 2 þ 2 exp ; ð26þ 2 p 2 where erf is the error function and r is an arbitrary variable with r = U * in equation (24). The functions G 2 and G 3 for the range jrj < 1 can be approximated as G r and G 3 ( r 2 ). [33] Finally, the energy dissipation rate D B due to wave breaking is estimated using the formula by Battjes and Stive [1985]. D B ¼ rgaqh B 2 ; 4T p Q 1 ln Q ¼ H 2 rms ; ð27þ H m H m ¼ 0:88 tanh gk ph ; k p 0:88 where a is the empirical coefficient suggested as a =1;Q is the fraction of breaking waves with Q = 0 for no wave breaking and Q = 1 when all waves break; H B is the wave height used to estimate D B with H B = H m in their formula; H m is the local depth-limited wave height; and g is the breaker ratio parameter with H m = gh in shallow water. The wave number k p is given by k p =2p/(C p T p ). The requirement of 0 Q 1 implies H rms H m but H rms becomes larger than H m in very shallow water. When H rms > H m, use is made of Q = 1 and H B = H rms instead of H B = H m. [34] The choice of a = 1 in equation (27) was based on the assumption of the energy dissipation of a bore distributed uniformly over one wavelength. This assumption may not be reasonable in the region where the energy dissipation is more concentrated locally. The coefficient a is hence taken as the ratio of the wavelength estimated as T p (gh) 0.5 to the horizontal length scale (bh/s b ) with b being an empirical factor, imposed by the small depth h and the local bottom slope S b = dz b /dx, a ¼ b 1 0:5 S b T p g=h 1; ð28þ where a 1 is imposed so that a = 1 in the region of large h and small S b. Use is made of b = 3, which was the value calibrated by N. Kobayashi et al. (Irregular wave transmission over submerged porous breakwaters, submitted to Journal of Waterway, Port, Coastal, and Ocean Engineering, 2005) to increase D B due to intense wave breaking on the steep slope of a submerged porous breakwater. The computed value of a increases rapidly from unity to about ten near the shoreline for the computed results in section 5. It is noted that D B could also be increased by increasing g because H m = gh in shallow water where g was observed to increase with the beach slope [Raubenheimer et al., 1996]. The increase of D B due to the slope effect results in the increase of q r in equation (20). To offset this increase, use is made of b r =(0.1+S b ) 0.1, which implies that the wave front slope increases on the upward slope. In short, the slope effects on D B and b r have been examined very little. [35] In the region of H rms > H m and Q =1,s * = s h /h in equations (21) and (22) becomes large and the computed absolute values of s U and U are too large and vary too rapidly with the mean depth h. To remedy this shortcoming caused by the local use of linear shallow-water wave theory near thepshoreline, ffiffi use is made of s * =(s *c s h /h) 0.5 if s * > s *c = g/ 8 which corresponds to Hrms = g h. This empirical correction reduces the dependency on the mean water depth h. For example, s U =(s *c gs h ) 0.5 if s * > s *c. 14 of 21

15 Figure 15. Measured and computed mean free surface elevation h for test 1.6 where IROLL = 1 and 0 indicate the computed results with and without the roller volume flux q r. [36] Equations (18) and (20) are solved using a finite difference method with constant grid spacing Dx of the order of 1 cm here. The measured bottom elevation z b (x) is specified in the computation domain x 0 where x =0at pffiffi wave gauge 1. The measured values of T p, h, H rms = 8 sh and q r = 0 at wave gauge 1 located outside the surf zone are specified as the seaward boundary conditions. The landward marching computation is continued until the computed value of h or s h becomes negative. This landward limit corresponds to the mean water depth h of the order of 0.1 cm in the present computation. No numerical difficulty is encountered near the shoreline after the introduction of the parameter a given by equation (28). The computation is made with and without the roller effect, corresponding to IROLL = 1 and 0. For the option of IROLL = 0, the roller volume flux q r = 0 and equation (20) is not used. Reflected waves are neglected in the time-averaged model but an attempt is made to estimate the degree of wave reflection. The computed onshore energy flux F decreases landward due to wave breaking and bottom friction. The residual energy flux F sws at the still water shoreline is assumed to be reflected and propagate seaward. This assumption neglects the fact that the landward marching computation is made without regard to wave reflection. The root-mean square wave height (H rms ) r due to the reflected wave energy flux is estimated as ðh rms 0:5; Þ r ¼ 8F sws = rgc g ð29þ where the group velocity C g is assumed to be the same for the incident and reflected waves. [37] After the landward marching computation, the suspended sediment volume V per unit area is estimated using the sediment suspension model by Kobayashi and Johnson [2001] V ¼ e B D B þ e f D f ; ð30þ rgðs 1Þw f where s and w f are the specific gravity and fall velocity of the sediment, and e B and e f are the suspension efficiencies for D B and D f, respectively. For the present experiment, s = 2.6 and w f = 2.0 cm/s. Use is made of e B = and e f = 0.01 calibrated by them to predict beach profile changes observed in large-scale laboratory experiments. Equation (30) is valid for the equilibrium beach on which the sediment suspension rate equals the sediment settling rate. Kobayashi and Johnson [2001] did not include the roller effect in their model. When the roller effect is included, D B in equation (30) is replaced by (rgb r q r )in view of equation (20) because the latter represents the rate of the roller energy dissipation. 5. Comparison With Experiments [38] The breaker ratio parameter g in equation (27) and the bottom friction factor f b in equation (24) are varied to examine the sensitivity of the computed results to these parameters. The cross-shore variation of s h related to the wave energy equation is used to calibrate g where g = 0.6, 0.7 and 0.8 are tried. The increase of g shifts the zone of wave breaking landward. The calibrated value is g = 0.6 for 15 of 21

16 Figure 16. Measured and computed standard deviation s h of free surface elevation for test 1.6. Figure 17. Measured and computed mean horizontal velocity U for tests (top) 4.8, (middle) 1.6, and (bottom) 2.6, where the square denotes the measured value at each elevation and the dot indicates the value of U 0 at each line. 16 of 21

17 Figure Measured (square) and computed standard deviation s U of horizontal fluid velocity for test tests 4.8 and 1.6 and g = 0.8 for test 2.6. The value of f b = has been calibrated and used in the previous timedependent computations for wave runup [Raubenheimer et al., 1995] and sand suspension [Kobayashi and Johnson, 2001; Kobayashi and Tega, 2002]. The computed results by this time-averaged model are found to be insensitive to f b and f b = is used here as well. The computed results for the three tests reported by Zhao and Kobayashi [2005] are presented concisely in the following. [39] Figure 15 shows the measured and computed crossshore variations of the mean free surface elevation h for test 1.6 where q r = 0 in equations (19) and (22) for IROLL = 0. The roller represented by q r > 0 delays the rise of h in the breaker zone and increases h near the still water shoreline. The computed rapid increase of h near the shoreline is consistent with the field data of Raubenheimer et al. [2001]. The degree of the agreement with the data is similar for tests 4.8 and 2.6. The roller effect does not necessarily improve the agreement for these three tests. Figure 16 shows the measured and computed cross-shore variations of the standard deviation s h for test 1.6. The difference between IROLL = 0 and 1 is essentially limited to the swash zone because of the adopted wave energy equation (18) with the wave energy flux F = rgc g s h 2 without any effect of q r. The agreement with the data is better for tests 4.8 and 2.6 because no data point deviates noticeably from the computed s h. The good agreement is partially due to the calibration of g for each test. The reflected wave height is estimated using equation (29) where use is made intuitively of the value of F at the still water shoreline. Since s h decreases rapidly near the shoreline, the estimated reflected wave height is sensitive to the selected location of wave reflection. The measured and computed reflection coefficients are compared in Table 1. This simple and intuitive method appears to be useful in estimating the order of magnitude of wave reflection using the time-averaged model. [40] Figure 17 shows the measured and computed crossshore variations of the mean horizontal velocity U for the three tests. The measured values of U at the different elevations and the average velocity U 0 at each line defined in equation (7) are compared with the computed depthaveraged U given by equation (22). These comparisons are approximate because the numerical model does not predict the vertical variation of U. For test 4.8 with mostly plunging breakers, the agreement is better for IROLL = 0 with q r =0. For test 1.6 with mostly spilling breakers, the agreement is better for IROLL = 1 with the roller volume flux. The largest undertow current on the shoreward side of the bar, which was also observed on a natural barred beach [Garcez Faria et al., 2000], cannot be predicted by the present roller model. Test 2.6 is intermediate between tests 4.8 and 1.6. The computed large offshore mean velocity in the swash zone is qualitatively consistent with the field data of Raubenheimer [2002] and the mean velocities computed by Tega et al. [2004] for tests 4.8 and 1.6 using the timedependent model by Kobayashi and Wurjanto [1992]. Figure 18 shows the measured and computed cross-shore variations of the standard deviation s U for test 1.6 where the measured s U at the different elevations did not vary much. The computed increase of s U near the shoreline before its rapid shoreward decrease is consistent with the field data of Raubenheimer [2002]. The agreement with the data is similar for test 4.8 but the numerical model overpredicts s U for test of 21

18 Figure 19. Measured turbulent velocity k 0.5 (square) and fitted u 0 0 (dot) and u 0 0 exp (h/ t ) (solid triangle) at each line in comparison with computed u 0 f =(D f /r) 1/3 and u 0 B =(D B /r) 1/3 for test 1.6. [41] Figure 19 shows the measured turbulent velocity k 0.5 at the different elevations at the seven lines for test 1.6 together with the extrapolated values of u 0 0 and u 0 0 exp(h/ t ) using equation (10) where t < 0 at lines 1.6A and 1.6G. The computed turbulent velocities u 0 f =(D f /r) 1/3 and u 0 B = (D B /r) 1/3 are those associated with the energy dissipation rates due to the bottom friction and wave breaking, respectively, where D B is replaced by (rgb r q r ) for IROLL = 1 as discussed in relation to equation (30). The use of D B to estimate k near the bottom was proposed by Roelvink and Stive [1989]. This procedure does not work at the lines where k decreases upward. The measured k 0.5 is better represented by u 0 f but is smaller than u 0 f in the outer breaker zone. Since u 0 f is proportional to f b 1/3 in view of equation (24), the bottom friction factor f b = needs to be reduced to the order of in the outer breaker zone. For longshore currents on a natural beach, Feddersen et al. [1998] estimated their drag coefficient, c f = f b /2, to be and within and seaward of the surf zone, respectively. The seaward decrease of c f is qualitatively consistent with the required seaward reduction of f b to improve the agreement between u 0 f and k 0.5, although the precise value of f b is uncertain. [42] The comparisons of the computed u 0 f and u 0 B with the measured k 0.5 for tests 4.8 and 2.6 are similar to those shown in Figure 19. The roller effect causes the spatial lag between the wave breaking and energy dissipation and shifts u 0 B shoreward. The measured and extrapolated turbulent velocities in comparison with u 0 B indicate the additional spatial lag which may be related to the downward transfer of eddies generated by wave breaking. Infrequent but intense turbulent events of short durations, which are important for sediment suspension, are not represented well by the timeaveraged turbulent velocity k 0.5 [Cox and Kobayashi, 2000]. Relatedly, the wave friction factor f w based on the maximum bottom shear stress appears to be much larger than the bottom friction factor f b associated with the time-averaged bottom shear stress and turbulent velocities [Cox and Kobayashi, 1997]. The formula for nonbreaking waves by Madsen and Salles [1999] together with the ripple height and length measured in this experiment predicts f w to be of the order of 0.1 in comparison with f b = used here. Similarly, Feddersen et al. [2003] found no relationship between the bottom friction factor for longshore currents and the bottom roughness (bed forms) inside the surf zone on a natural beach. The time-averaging masks the intermittent turbulence caused by wave breaking and ripples. [43] Figure 20 shows the measured and computed suspended sediment volume V per unit area for tests 4.8, 1.6 and 2.6 where use is made of equations (13) and (30). The exponential and power-form distributions of the mean concentration yield the similar values of V. Consequently, the disagreement between the measured and computed values of V indicates the shortcoming of equation (30). The roller effect causes the shoreward shift of V but does not necessarily improve the agreement. No measurement was made in the swash zone but visual observations indicated intense sediment suspension due to wave breaking and swash oscillations in manners similar to the field measurement by Puleo et al. [2000]. The agreement within a factor of about two is similar to the accuracy of the time-dependent model by Kobayashi and Tega [2002] which requires much more computation time than the present time-averaged model. The offshore and on- 18 of 21