# Effects of Wave Height, Period and Sea Level on Barred Beach Profile Evolution: Revisiting the Roller Slope in a Beach Morphodynamic Model

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## Abstract

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## 1. Introduction

## 2. Methodology

#### 2.1. Process-Based Numerical Model

_{w}means the wave energy density, c

_{g}is the wave group velocity, D

_{b}is the wave breaking energy dissipation, D

_{f}is the energy dissipation due to bottom friction. The wave breaking energy dissipation D

_{b}was estimated by using the model of Baldock et al. [24] in this study, though other parametric wave breaking models can be used as alternatives [25,26]. Compared with the original version, the present model updates the representation of the breaker index in the parametric wave breaking dissipation module [27]. The updated breaker index formula exhibits composite dependence on both offshore wave steepness and local normalized water depth; it reads as follows:

_{0}represents the offshore wave steepness, k is the wave number, h means the local water depth. The new breaker index formula was chosen because it considers both the breaking intensification mechanism and the breaking resistance mechanism. Zhang et al. [27] found that implementation of this new γ formula in the parametric wave model of Baldock et al. [24] can systematically reduce the median percentage error of wave height prediction by 10–24%.

_{r}means the roller energy density, c

_{p}means the wave celerity, D

_{r}is the roller energy dissipation. The roller energy dissipation D

_{r}is estimated as

_{rms}L

^{2}/h

^{3}, L is wavelength calculated with the wave peak period. The Ursell number is an important indicator for the nonlinearity of surface gravity waves [30,31,32]. More recently, Rafati et al. [13] modified the formula for roller slope on the basis of Walstra et al. [14] and found that the modified, cross-shore varying roller slope formula is more precise in predicting undertow under energetic wave conditions than the constant one. The modified formula is written as follows:

_{v}, phase-shift angle φ and the Prandtl/Schmidt number σ

_{p}. Details of the numerical schemes and iterative algorithms can be found in Zheng et al. [11].

#### 2.2. Field Data

## 3. Results and Discussion

#### 3.1. Model Calibration with Constant Roller Slope

_{t}is the predicted final beach profile (or RMS wave height), O

_{t}is the observed final beach profile, O

_{0}is the initial beach profile. Following van Rijn et al. [40], BSS = 0.1–0.3 means a poor fit, BSS = 0.3–0.6 means a fair fit, BSS = 0.6–0.8 means a good fit and, finally, BSS = 0.8–1.0 equals an excellent fit.

_{v}, phase-shift angle φ and the Prandtl/Schmidt number σ

_{p}were set as 0.025, 30° and 1, respectively. As can be seen in Figure 3, good agreement was found between predicted and observed RMS wave height and beach profile evolution. The RMSE for the comparison of the RMS wave height at the end of the model was 0.11 m. The BSS for the comparison of the beach profile evolution was 0.92, implying that the performance of the model was excellent [37].

#### 3.2. Model Calibration with Varying Roller Slope

_{0}, T

_{0}and η

_{0}be wave heights, wave periods and sea levels in the original case. These values are time-varying rather than constant during the studying period. Case 1 and Case 2 increase or decrease the wave heights to 1.25H

_{0}or 0.8H

_{0}, with wave periods and sea level unchanged. Case 3 and Case 4 increase or decrease the wave periods to 1.25T

_{0}or 0.8T

_{0}. The wave parameters of Case 5 and Case 6 are same as the original case, but the sea level is decreased or increased by 10 cm, respectively.

#### 3.3. Effect of Wave Height on Sandbar Evolution

_{0}, while the sandbar only migrated 10 m offshore and the bar crest elevation remained almost unchanged for the smaller wave height 0.8H

_{0}. By comparing Figure 5a and Figure 5b, using the roller slope formula of R2021 can lead to a more pronounced sandbar. The bar crest elevation either remained unchanged or even increased by 0.07 m, as shown in Figure 5b,c, respectively. When using the roller slope formula of Z2017, the sandbar was found to be more flattened and decaying during its offshore migration. As can be seen in Figure 5c,f, the bar crest elevation decreased by 0.83 m and 0.33 m for Case 1 and 2, respectively. The differences in sandbar migration using R2021 and Z2017 were mainly because R2021 predicted a smaller roller slope shoreward of the sandbar than Z2017 (as provided in Figure 4a and, thus, a larger undertow carrying the sediment offshore to reinforce the sandbar, i.e., to promote a more pronounced sandbar). A large undertow plays an important role in maintaining the shape of the sandbar; this finding has also been discussed [21,42].

_{0}and 0.8H

_{0}, respectively. Using the roller slope formula of R2021 under wave heights of 1.25H

_{0}, the sandbar migrated offshore with a rate of 5 m/d for the first two days, and the migration rate increased to 15 m/d in the following two days, as can be seen in Figure 6b. By comparing Figure 6d and Figure 6c, the predicted sandbar spatial–temporal evolutions under smaller wave height using the constant roller slope and R2021 were similar, although the latter had a more significant bar shape. While using the roller slope formula of Z2017, the bar shape began to decay, and the span of the sandbar increased in the last two days under the larger wave height of 1.25H

_{0}, as can be seen in Figure 6c.

_{0}, the suspended sediment transport was offshore-directed in the sandbar area and tended to 0 seaward of the sandbar. While under the smaller wave height of 0.8H

_{0}, the magnitude of the suspended sediment transport decreased in the sandbar area and even became positive (i.e., onshore-directed) seaward of the sandbar, as can be seen in Figure 7d. This implies that the current-related component of suspended sediment transport dominates in the sandbar area while the wave-related component dominates seaward of the sandbar. Using the roller slope of R2021 predicts a larger offshore-directed suspended sediment transport rate, i.e., a larger current-related component. It corresponds well with the finding that R2021 predicts a smaller roller slope in the sandbar area and thus a more intense undertow, as can be observed in Figure 7b,e.

#### 3.4. Effect of Wave Period on Sandbar Migration

_{0}and 0.8T

_{0}, respectively. A decrease in the wave period tends to promote offshore sandbar migration when using the constant roller slope in the model. Although the sandbar was more flattened, the prediction was similar using the roller slope formula of Z2017, the sandbar migrated 30 m and 40 m offshore under the wave period of 1.25T

_{0}and 0.8T

_{0}, respectively. However, the prediction using R2021 was different. When the wave period decreased, the sandbar became more pronounced, with the bar position remaining almost unchanged.

_{0}and 0.8T

_{0}, respectively. Comparing Figure 9b,c, it can be found that the bar positions were similar when using R2021 and Z2017, but the latter always predicted a more flattened and decaying bar shape.

_{0}, onshore sediment transport could be observed seaward of the bar crest while the offshore sediment transport occurred shoreward of the bar crest. This is because the wave breaking was weak shoreward of the sandbar because, as can be observed in Figure 3a, the wave height changed insignificantly in this area. Therefore, the undertow was weak, and the wave-related component (caused by the wave nonlinearity) dominated in the area. When the wave period decreased to 0.8T

_{0}, the onshore sediment transport seaward of the sandbar became 0, implying that the quasi-equilibrium was achieved by the wave-related component (onshore) and the undertow-related component (offshore), as can be seen in Figure 10d,e. However, using Z2017 would also predict an onshore sediment transport rate seaward of the sandbar, as can be observed in Figure 10f; it is because Z2017 predicts a larger roller slope and thus a weak undertow, which cannot balance the onshore-directed, wave-related component. It can also be interestingly observed that the position of the offshore sediment transport rate is more shoreward in Figure 10e, i.e., more sediments can be carried from the shore to reinforce the sandbar. Therefore, the sandbar morphology will be more pronounced when using R2021 under the smaller wave period 0.8T

_{0}, as can be seen in Figure 8e and Figure 9e.

#### 3.5. Effect of Sea Level on Sandbar Migration

## 4. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Time series of (

**a**) RMS wave height; (

**b**) wave peak period; (

**c**) wave angle; (

**d**) water level; (

**e**) measured initial and final profiles.

**Figure 3.**Comparison of (

**a**) RMS wave height and (

**b**) beach profile using a constant roller slope of 0.08.

**Figure 7.**Spatial–temporal evolution of the suspended sediment transport rate predicted by constant roller slope (

**a**,

**d**) and the roller slope of R2021 (

**b**,

**e**) and Z2017 (

**c**,

**f**) under two different wave heights. Figure 7a–c are calculated with H = 1.25H

_{0}, and Figure 7d–f are calculated with H = 0.8H

_{0}.

**Figure 10.**Spatial–temporal evolution of the suspended sediment transport rate predicted by constant roller slope (

**a**,

**d**) and the roller slope of R2021 (

**b**,

**e**) and Z2017 (

**c**,

**f**) under two different wave periods. Figure 10a–c are calculated with T = 1.25T

_{0}, and Figure 10d–f are calculated with T = 0.8T

_{0}.

**Figure 13.**Spatial–temporal evolution of the suspended sediment transport rate predicted by constant roller slope (

**a**,

**d**) and the roller slope of R2021 (

**b**,

**e**) and Z2017 (

**c**,

**f**) under two different sea levels. Figure 13a–c are calculated with η

_{0}− 10 cm, and Figure 13d–f are calculated with η

_{0}+ 10 cm.

Case ID | Wave Height | Wave Period | Sea Level |
---|---|---|---|

1 | 1.25H_{0} | T_{0} | η_{0} |

2 | 0.8H_{0} | T_{0} | η_{0} |

3 | H_{0} | 1.25T_{0} | η_{0} |

4 | H_{0} | 0.8T_{0} | η_{0} |

5 | H_{0} | T_{0} | η_{0} − 10 cm |

6 | H_{0} | T_{0} | η_{0} + 10 cm |

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**MDPI and ACS Style**

Wang, G.; Li, Y.; Zhang, C.; Wang, Z.; Dai, W.; Chi, S. Effects of Wave Height, Period and Sea Level on Barred Beach Profile Evolution: Revisiting the Roller Slope in a Beach Morphodynamic Model. *Water* **2023**, *15*, 923.
https://doi.org/10.3390/w15050923

**AMA Style**

Wang G, Li Y, Zhang C, Wang Z, Dai W, Chi S. Effects of Wave Height, Period and Sea Level on Barred Beach Profile Evolution: Revisiting the Roller Slope in a Beach Morphodynamic Model. *Water*. 2023; 15(5):923.
https://doi.org/10.3390/w15050923

**Chicago/Turabian Style**

Wang, Guangsheng, Yuan Li, Chi Zhang, Zilin Wang, Weiqi Dai, and Shanhang Chi. 2023. "Effects of Wave Height, Period and Sea Level on Barred Beach Profile Evolution: Revisiting the Roller Slope in a Beach Morphodynamic Model" *Water* 15, no. 5: 923.
https://doi.org/10.3390/w15050923