Morphodynamic modelling of beach cusp formation: the role of wave forcing and sediment composition

A field of beach cusps formed during a field experiment at Nha Trang Beach, Vietnam, under accretive conditions. The measured data was used to set-up morphodynamic simulations in XBeach, which was able to simulate cusp formation from an initially long-shore uniform beach profile. Several types of simulations were run in order to observe the resulting variation in mean cusp dimensions (length, depth and height), swash flow patterns, and sediment sorting. Both time-constant (JONSWAP) and time-varying (measured) wave forcing conditions were superimposed on the measured tide. In the former, four wave parameters were varied (wave height, period, direction, and spreading), while in the latter, the median sediment size and sediment composition were varied. The wave period was found to primarily influence long-shore length scales, the wave height cross-shore length scales, and obliquely incident waves enhance all these dimensions particularly under narrow-banded conditions. Cusps are not prominent if the wave energy is too low to effect ∗Corresponding Author: christopher.daly@legos.obs-mip.fr Preprint submitted to Geomorphology October 12, 2020 significant onshore transport, if the wave angle of incidence and spreading are too large (effectively smoothing out swash perturbations), or if the sediment is too fine in relation to the wave conditions (dissipative beaches or highly erosive wave conditions). Coarse sediment generally tends to be located on cusp horns above the waterline, but is otherwise variable depending on cross-shore location and tide levels. As the XBeach model results show large agreement with well-established norms, it may therefore be used to more rigorously study processes that help to initiate cusps in future work.

LMS511 2D laser scanner was used to measure surface elevation (both of 89 the bed and water) in the swash along the same transect, from which the 90 swash excursion, swash height and beach slope is determined (Fig. 2c-d). 91 The beach is composed of coarse grained sediment (median grain size, D 50 92 = 0.5 mm) and is located in a diurnal, micro-tidal environment (tide range 93 = 1.6 m). As a result, the beach has a fairly steep (1:8) swash slope and a 94 narrow low tide terrace. Beach topography data was measured using high-  (Panel c) swash excursion, S x , and swash height, S z . The three-day simulation period for Series C is highlighted in grey.     The mean cross-shore profile of the study area on 28 November is used to 172 create a long-shore uniform initial bathymetry for the model (Fig. 3e). When     in the long-shore dimension is computed by removing the long-shore mean 242 profile from each cross-shore transect:

Numerical Model
Subsequently, the root-mean-square (RMS) long-shore bed level variation 244 (∆), which indicates the degree of vertical variability in bed levels and thus 245 prominence of the cuspate features, is computed as: Only data located between 0.5 and 1 m elevation are used Eq. 2, an area in v, respectively) along the central cross-shore transect as: where u and v are fluctuations of the velocity components after removal of 258 the mean over a sample period of 10 minutes.  Finally, the surface sediment composition, P D 50 is computed for case C4

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(having a mixed sediment bed) as: where P c is the percentage of coarse sediment in the surface layer initially  of the tide is low, leaving the upper beach exposed and morphologically inac-  Run H s  It is important to also note in Fig. 4 that for cases where θ is varied, the 315 resulting cusps are saw-toothed shaped due to the asymmetry of the swash 316 trajectory. This is not seen in the cases where σ is varied, as the swash 317 trajectory is symmetrical about the shore normal. For increased σ, L x tends 318 to slightly decrease while L z and ∆ remain fairly stable. For increased θ, L x , 319 L z and ∆ tend to increase.  ever, too small cause any significant variation in H s,y or u y , therefore 344 the bathymetry is slow to respond. Nonetheless, during this initial period, 345 sediment is slowly moved onshore, just below the tide level (Fig. 7f). This 346 subaqueous mass of accreted sediment becomes exposed when the tide turns the top of the swash as the water level recedes, creating a berm (Fig. 7g).

Temporal Evolution and Swash Dynamics
This trail of sediment is slowly sculpted into small cuspate features as sed-   around high tide and more gently sloping around low tide). S x is consistently 385 negatively correlated with the swash slope and tide elevation above MSL (i.e., 386 S x is smallest around high tide, where the beach slope is steepest). In some 387 cases, S x is maximum at low tide while in others S x is maximum just below 388 mid-tide and subsequently decreases towards low tide (Fig. 9c). The latter 389 is due to a berm forming at the low tide level that increases β around that    to show a dependence of L y on elevation above MSL; however, their study 455 site was located in a meso-tidal environment (2.6 m range) exposed to more 456 energetic wave conditions. in Eq. 6 and Eq. 7 following as: where f is a factor generally taken to be 1.63 (but which may range between 467 1 and 3); S x is the swash excursion; m is a factor equal to 1 and 2 for sub-468 harmonic (L y,Sub ) and synchronous (L y,Syn ) edge waves, respectively; β is the 469 beach slope; and T i is the incoming wave period (Coco et al., 1999). Results 470 are shown in Table 4  478 When using Eq. 5 to compute L y,Sun in Table 4, we computed and used 479 the value of A that minimised the root-mean-square error (the best-fit value) 480 between L y,Sun and L y , which was equal to 0.6 -very close to the value of  Table 4 shows that simulated β and f generally increase with H s and T m .

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With regard to f , the simulation results ranges from 1.16 to 2.47, which fits  Run S x β f L y L y,Sub L y,Syn L y,SO L y,Sun  In terms of the sediment sorting pattern around cusps, by looking at the 573 correlation between P D 50 and z b,y in Fig. 10b, we showed that sediment is 574 generally coarser on the horns than in the trough of the cusps. This is true 575 for most field observations, such as Antia (1987) and Sallenger (1979) who 576 also explains that, as swash flow is more powerful than backwash and as flow 577 is generally horn divergent, fine sediment is removed from the horn (leaving 578 coarser sediment behind) and deposited in the trough.

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The effect of varying sediment size, by decreasing D 50 , we obtain slight 580 increases in L y , as noted in Sunamura (2004 The simulations have been done using the non-hydrostatic wave solver 591 in XBeach while enabling sediment transport. This is quite experimental, 592 as the sediment transport equations only account for transport due to flow 593 and wave-averaged orbital motions and therefore do not resolve intra-wave Table 1 with the Kingsday version of XBeach allows bedload transport to be 596 only onshore-directed, which is an unusual result that is repaired in subse-597 quent model releases. Nonetheless, an appropriate balance between onshore 598 and offshore transport fluxes are obtained for our simulations despite these 599 shortcomings. Further development of XBeach is therefore necessary to bet-600 ter and more realistically account for intra-wave and swash sediment trans-601 port processes. One suggestion to the model developers may be, for exam-

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The authors declare no conflict of interest.