Limitations of empirical sediment transport formulas for shallow water and their consequences for swash zone modelling

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1 Limitations of emirical sediment transort formulas for shallow water and their consequences for swash zone modelling WEI LI, Lecturer, Ocean College, Zheiang University, Hangzhou, Peole's Reublic of China PENG HU, Associate Professor, Ocean College, Zheiang University, Hangzhou, Peole's Reublic of China (author for corresondence) THOMAS, PAHTZ, Professor, Ocean College, Zheiang University, Hangzhou, Peole's Reublic of China ZHIGUO HE, Professor, Ocean College, Zheiang University, Hangzhou, Peole's Reublic of China ZHIXIAN CAO, Professor, State Key Laboratory of Water Resources and Hydroower Engineering Science, Wuhan University, Peole's Reublic of China Running Head: Limitations of emirical sediment formulas

2 Limitations of emirical sediment transort formulas for shallow water and their consequences for swash zone modelling ABSTRACT Volumetric sediment concentrations comuted by hase-resolving swash morhodynamic models are shown to exceed unity minus orosity (i.e., the maximally hysically ossible concentration value) by u to factor 10 5 when using standard exressions to comute the sediment transort rate. An ad hoc limit of sediment concentration is introduced as a means to evaluate consequences of exceeding hysically realistic concentration by standard exressions. We find that imlementation of this ad hoc limit strongly changes the quantitative and qualitative redictions of hase-resolving swash morhodynamic models, suggesting that existing swash redictions are unreliable. This is because standard exressions inaroriately consider or ignore the fact that the shallow swash water deth limits the storage caacity of transorted sediment. Key words: hase-resolving swash model; sediment transort; shallow water; standard exressions; swash zone 1 Introduction The swash zone is characterized by very shallow water and successively covered and uncovered by water due to wave run-u/backwash. In numerical and analytical studies of swash sediment transort, standard exressions are used to comute the sediment transort rate (e.g. Briganti, Dodd, Pokraac, & O'Donoghue, 01; Kelly, & Dodd, 010; Postacchini, Brocchini, Mancinelli, & Landon 01; Postacchini, Othman, Brocchini, & Baldock, 014; Zhu, Dodd, & Briganti, 01; Zhu, & Dodd, 013a, 013b, 015), which, however, are not derived or calibrated for very shallow swash conditions. In fact, most, if not all, standard exressions for the sediment transort rate are either indeendent of or inversely roortional to the water deth (Zhu, & Dodd, 013a). For very small water deths, this may lead to sediment transort rates so large that the corresonding sediment concentration may be larger than it can hysically be. As an attemt to overcome this otential issue, Pritchard, & Hogg (005) multilied the standard Grass exression by the water deth (termed as PH exression below). However, whether the PH exression resolves the issue with small water deth remains unexamined. Moreover, some studies avoided this issue by hase-averaging the urush and backwash of the swash (e.g. Masselink, & Hughes, 1998; Hughes, Masselink, & Brander, 1997). However, imlementation of hase-averaging revents models from

3 redicting detailed morhological evolution rocesses. Here existing hase-resolving analytical and numerical studies of swash morhydynamics are revisited. In articular, we examine the magnitudes of sediment concentration, which aears imlicitly in standard sediment transort formulas. We then imose an ad-hoc uer limit on the sediment concentration and thereby limit the sediment transort rate. It is strongly emhasized here that the introduction of this ad-hoc limit is NOT meant to lead to a better "model". Instead it is only a means to test whether the exceeding hysically realistic sediment concentrations has serious consequences on the overall redictions of hase-resolving swash morhydynamic models. Analytical investigations We revisit the theoretical analysis of swash beach evolution by Kelly, & Dodd (010). They used the swash solution derived by Shen, & Meyer (1963) to reresent the swash hydrodynamics, which reads: [4 h( x, t) gh t gt 0 36gt tan x] (1a) [ u( x, t) gh0 t gt tan 3t x] (1b) where h is the water deth; u is the swash velocity; x is the horizontal distance (ositive landwards), t is time; g is gravitational acceleration, tan is the beach sloe, h 0 is the initial wave height. The Exner equation that governs beach deformation reads: z qb ( 1 ) 0 t x () where q b is the bed load transort rate, z is the beach elevation, is the sediment orosity. Beach deformation deth over a swash cycle at a given osition x is obtained by integrating Eq. (), which, using the standard Grass exression velocity, results in: q b 3 AGu and Eq. (1b) for swash td td t d t 1 q d b 3AG u 8A [ gh0 t gt tan x z( x ) z dt ( u ) dt 3 t 1 t x 1 t x 9( 1) t t i i i ] i dt (3) where t i and t d are the inundation and denudation times, which can be derived by setting h = 0 and x x in Eq. (1a). The resulted final exression reads:

4 1 8AG z( x ) [3g 4h0 h0 x tan ( gh0 x g tan)ln( td / ti )] (4) 1 9 For comarison uroses, a modified exression for sediment transort rate is introduced by imosing an uer limit on the sediment concentration, which reads: 3 qb min( hucuer, AG u ) (5) where uer c is the maximally hysically ossible concentration value defined as 1. This definition means that the entire flow is effectively filled with sediment. We wish to strongly emhasize that this technical note does NOT roose Eq. (5) as a better 'model'. Instead, it is only introduced to determine which redictions of the standard Grass exression (and further standard exressions that we test later) are due to exceeding the maximally hysically ossible sediment concentration. In fact, any difference between model redictions when using the standard Grass exression on the one hand and the modified Grass exression (Eq. (5)) on the other hand MUST be a result of exceeding this maximum and thus has no hysical ustification. It is worthy ointing out that limiting the maximal sediment concentration has been done before: Garcia, & Parker (1991) derived an emirical relation for susended sediment concentration with an uer limit of 0.3 [aroximately half of ( 1 ) ]. When the modified exression Eq. (5) is used, the beach deformation is obtained by numerically integrating Eq. (), which gives: c uer z( x ) 1 n qb ( x x, ti t) qb ( x, ti t) t (6) 1 1 x In Eq. (6), the eriod between t i and t d is divided into n intervals of length t ; x is a sufficiently small distance. We use the same arameter values as in Kelly, & Dodd (010): 0. 4 ; A G = s /m; h 0.65 m; tan = 0.1. The eriod T 4 gh0 /( g tan ) 10.3 s. Figure 1 illustrates the 0 final beach deformation deth (Fig. 1a) and beach rofiles at three instants (Fig. 1b, Fig. 1c and Fig. 1d). In Fig. 1, Run 1 reresents the analytical solution by Eq. (4), Run is the numerical integration of Eq. (6) fed by Eq. (3), and Run 3 (limited sediment transort rate) is the numerical integration of Eq. (6) fed by Eq. (5). From Fig. 1a, Run 1 and Run agree satisfactorily with each other, indicating the high accuracy of the numerical integration. Thus Run 1 and Run are combined later in Fig. 1b, Fig. 1c and Fig. 1d. Big differences are seen between Run 1/ and Run 3. Run 1/ comutes bed degradation everywhere in the swash zone, consistent with Kelly, & Dodd (010). A so-called "bed ste" is comuted at the swash

5 front (Run 1/, Fig. 1b, Fig. 1c and Fig. 1d), but it is washed out at the end of the swash (Run 1/, Fig. 1a). In contrast, the bed ste is absent for Run 3. Transient bed aggradations can be seen in Run 3 in some regions (Fig. 1b, Fig. 1c and Fig. 1d). These differences between Run 1/ and Run 3 are exlained using Fig., which shows the cross-shore variation of the swash deth (Fig. a) and velocity (Fig. c) and concentrations (Fig. b) comuted with the standard Grass exression. From Fig. b, the comuted concentrations at the end of the urush (at about 5.15 s) are within the hysically reasonable range. This is because, at the reverse of the swash, the swash is characterized by the smallest velocities (Fig. c). However, concentrations at both the urush and backwash exceed the maximal hysically ossible concentration value ( s reresents urush and 8 s reresents backwash). As the swash deth decreases u to the beach, the concentration increases raidly 4 and attains values as high as 5 10, which is more than 10 5 times larger than the hysically reasonable range. This is because the standard Grass exression does not account for the effects of the very shallow swash water deth. Figure 1. Cross shore variations of (a) beach deformation deth, and (b, c, d) beach rofiles. Run 1 and corresond to using the standard Grass exression to comute the sediment transort rate and Run 3 to using the Grass exression modified by the uer concentration limit. Also included in Fig. 1b, Fig. 1c and Fig. 1d is the original beach rofile.

6 Figure. Cross-shore variations of (a, c) the swash hydrodynamics, and (b) the volumetric sediment concentration. Sediment transort rates are comuted using the standard Grass exression. 3 Numerical Investigations The couled hase-resolving swash morhological model of Hu, Li, He, Pahtz, & Yue (015) is alied. The governing equations read h hu t x 0 (7) hu ( hu 0.5gh z ) gh c t x x D u u (8) (1 ) z qb 0 (9) t x where c D is the drag coefficient. Eqs. (7, 8. 9) are solved by a shock-caturing finite volume method along with a well-balanced version of the Sloe Limited Centered scheme (Hu et al., 015). The numerical cases are designed following Zhu, & Dodd (013a). On a beach of uniform sloe tan = 0.1, beach elevation z( x) x tan. The region with x 0 is initially

7 dry. On the seaside ( x 0 ): h 0 seaside to freely flow u and down the beach. The satial ste = 0.65 m. The swash is simulated by allowing the water on the x = m. Figure 3 resents the non-dimensional beach deformation deth ( z /( h 0 tan ) ) in the sace-time lane comuted with the standard (Fig. 3a) and modified (Fig. 3b) Grass exressions. When the standard Grass exression is alied (Fig. 3a), the beach degrades in the lower swash region and aggrades in the uer swash region during run-u, whereas it degrades in the whole swash zone during backwash. A bed ste is comuted during the urush, but it is washed out at the end of the swash. These results are consistent with Zhu, & Dodd (013a). While bed stes do occur in nature, the exlanation of their formulation obviously cannot rely on exressions that comute sediment transort rates much larger than maximally ossible values. When the modified Grass exression is imlemented (Fig. 3b), the following oints are noted. First, the magnitudes of both deosition and erosion are reduced, which follows naturally from limiting the sediment transort rate. Second, the area with deosition is wider, and the final bed rofile is characterized by some deosition. This indicates some extent of onshore sediment transort, which is consistent with field studies (Masselink, Evans, Hughes, & Russell, 005; Kelly, & Dodd, 010). In contrast, using the standard Grass exression always leads to net offshore transort everywhere on the beach (bed degradation everywhere after a swash cycle), which is at odds with field studies (Kelly, & Dodd, 010). Third, a bed ste is absent during the urush. Instead, a bed hum is left on the beach at the end of the swash (see the contours of -0.1, 0.005, 0.08 in Fig. 3b). These results indicate strong limiting effects of the small swash water deth on beach morhological changes. To evaluate the influence of the small swash deth more comrehensively, extensive numerical cases studies are conducted, covering a wide range of bed frictions, bed mobility and sediment emirical exressions (the modifications of these formulas with an uer concentration limit are done analogous to the modification of the Grass exression). Comarisons are made between the cases with and without modifications through a relative overall discreancy R defined as: z( xi, t ) z( xi, t ) i, R( z) z( x, t ) i, i (10) where z( x i, t ) and z ( x i, t ) are the comuted beach deformation deth at osition x i and time t using the standard and modified exressions, resectively. The higher the R value, the larger the effects of exceeding the uer limit of concentration are.

8 Figure 3. Comuted non-dimensional beach deformation deth in the sace-time lane with 3 3 (a) qb AG u, and (b) qb min[ hu(1 ), AG u ]. A G =0.004 s/m and c D = 0.0. Figure 4 shows the variation of the R-value with the bed mobility in relation to (a) the Grass exression, and (b, c, d) the PH exression ( q b A PH k 3 h u ) with the exonent k = 0.8 (Fig. 4b), 1.0 (Fig. 4c) and 1.3 (Fig. 4d). The bed mobility g( 1 ) for the Grass exression, k and A g( 1 ) h for the PH exression. The comarisons in relation to the Bagnold, PH MPM, Van Rin, and Bailard exressions (Zhu, & Dodd, 013a) are similar to Fig. 4a and thus are not shown. From Fig. 4a, the R-value is consistently around 50% for a frictionless beach ( c D 0.0). This means the standard exressions roduce significant errors for a frictionless beach, which unfortunately has been a common assumtion in most theoretical swash studies. When the beach friction is considered, these standard exressions still roduce significant errors for a wide range of bed mobility and friction conditions. A decreasing trend of R with lower bed mobility is clear from Fig. 4a. This suggests that reducing the coefficient (i.e. A G for the standard Grass exression) through calibration may ensure hysically realistic sediment transort rates. However, in some cases, the coefficient would have to be reduced by a magnitude of This would imly negligible or even no sediment motion and thus be of A G

9 no ractical interest. Kelly, & Dodd (010) has imlied for sand beaches, which rohibits a strong reduction of the coefficient. When the PH exression is used, the extent of exceeding hysically reasonable sediment concentrations is largely reduced, as reflected by the smaller R-value (Fig. 4b, Fig. 4c and Fig. 4d). This is due to the ositive correlation between water deth and sediment transort rate in the PH exression. However, the R-value is not vanishing at all. Note that a larger k in the PH exression would lead to a smaller R- value, which means that a sufficiently large k could give hysically reasonable concentration. However, this kind of mathematical lay is hysically meaningless. Figure 4. Relative discreancy of the comuted beach deformation deth in relation to (a) the Grass exression, (b, c, and d) the PH exression with different exonents. 4 Conclusions Beach deformation in the swash zone is studied using hase-resolving analytical (Section ) and numerical (Section 3) aroaches. Two distinct cases were comared: the standard cases that consider standard exressions for the sediment transort rate, and the modified cases in which an uer limit of the sediment concentration is imosed. The modified cases yielded results quantitatively and even qualitatively different from the standard cases. In articular, our study indicates that the bed ste comuted by many morhodynamic swash zone models

10 might be an artifact of exceeding hysically realistic sediment transort rates. Indeed, the main reason why the maximal hysically ossible values of the sediment transort rate are exceeded so often (by u to factor 10 5 in some cases) is the small swash deth, of which the effect is not accounted for in standard exressions for the sediment transort rate. Though revious studies have noted many issues with the standard exressions (Bagnold, 1966; Pritchard, & Hogg, 005), only a few of them have ever considered the issue of the vanishing water deth and its effect on sediment transort rates. One excetion is Pritchard, & Hogg (005), in which the Grass exression was multilied by the swash deth. Yet, a detailed examination of the PH exression shows that it does not account for vanishing water deth in a satisfactory manner. Finally, we would like to note that, when other swash analytical solutions, which for examle are affected by seaward boundary conditions (Guard, & Baldock, 007), were imlemented, the resent findings should hold true because the very small swash deth is a common feature (Shen, & Meyer, 1963). Acknowledgements The authors are very grateful to several anonymous reviewers for their constructive and critical comments that greatly imroved the aer. Funding This research was artly suorted by the Research Fund for Doctoral Program of Higher Education of China (grant ) and the National Natural Science Foundation of China (grants , and ). Notation A G A PH = emirical coefficients in sediment transort relations (-) c D = drag coefficient (-) c uer = maximally hysically ossible concentration value (-) g = gravitational acceleration (m s - ) h = water deth (m) h 0 = initial wave height (m) k = emirical exonent in the PH exression (-) = sediment orosity (-)

11 q b = unit-width volumetric bed load transort rate (m s -1 ) R = relative overall discreancy (-) t = time (s) t i, t d = inundation and denudation times (s) T = eriod of a swash cycle (s) u = swash velocity (m s -1 ) x = horizontal distance (m) z = beach elevation (m) tan = beach sloe (-) = bed mobility number x = satial ste (m) z = beach deformation deth (m) References Bagnold, R. A. (1966). An aroach to the sediment transort roblem from general hysics. Professional Paer 4-1, Washington, D.C. Retrieved from USGS website: htt://ubs.usgs.gov//04i/reort.df Briganti, R., Dodd, N., Pokraac, D., & O Donoghue, T. (01, July). Numerical and exerimental descrition of the flow, boundary layer and bed evolution in bore-driven swash on a coarse sediment beach. Paer resented at Proceedings of the 33rd Conference on Coastal Engineering, Santander, Sain. Garcia, M., & Parker, G. (1991). Entrainment of bed sediment into susension. Journal of Hydraulic Engineering ASCE, 117(4), Guard, P. A. & Baldock, T. E. (007). The influence of seaward boundary conditions on swash zone hydrodynamics. Coastal Engineering, 54(4), Hu, P., Li, W., He, Z., Pähtz, T., & Yue, Z. (015). Well-balanced and flexible modelling of swash hydrodynamics and sediment transort. Coastal Engineering, 96(), Hughes, M. G., Masselink, G., & Brander, R. W. (1997). Flow velocity and sediment transort in the swash one of a stee beach. Marine Geology, 138(1-), Kelly, D. M., & Dodd, N. (010). Beach face evolution in the swash zone. Journal of Fluid Mechanics, 661,

12 Masselink, G., & Hughes, M. (1998). Field investigation of sediment transort in the swash zone. Continental Shelf Research, 18(10), Masselink, G., Evans, D., Hughes, M. G., & Russell, P. (005). Susended sediment transort in the swash zone of a dissiative beach. Marine Geology, 16(3), Postacchini, M., Brocchini, M., Mancinelli, A., & Landon, M. (01). A multi-urose, intrawave, shallow water hydro-morhodynamic solver. Advances in Water Resources, 38(3), Postacchini, M., Othman, I. K., Brocchini, M., & Baldock, T. E. (014). Sediment transort and morhodynamics generated by a dam-break swash urush: couled vs uncouled modeling. Coastal Engineering, 89(7), Pritchard, D., & Hogg, A. J. (005). On the transort of susended sediment by a swash event on a lane beach. Coastal Engineering, 5(1), 1-3. Shen, M. C., & Meyer, R. E. (1963). Climb of a bore on a beach, III: run-u. Journal of Fluid Mechanics, 16, Zhu, F. F., Dodd, N., & Briganti, R. (01). Imact of a uniform bore on an erodible beach. Coastal Engineering, 60(), Zhu, F. F., & Dodd, N. (013a). Net beach change in the swash zone: a numerical investigation. Advances in Water Resources, 53(3), 1-. Zhu, F. F., & Dodd, N. (013b, June). The influence of bed friction on the creation of a bedste by a solitary wave in the swash. Paer resented at the 7th International Conference on Coastal Dynamics, Arcachon Convention Centre, France. Zhu, F. F., & Dodd, N. (015). The morhodynamics of a swash event on an erodible beach. Journal of Fluid Mechanics, 76,