# Characterization of Overtopping Waves on Sea Dikes with Gentle and Shallow Foreshores

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## Abstract

**:**

_{max}(maximum overtopping flow depth) and u

_{max}(maximum overtopping flow velocity), which led to overestimation of the risk. The time dependent u and h are strongly influenced by the dike configuration, namely by the promenade width and the existence of a vertical wall on the promenade: the simulation shows that the vertical wall induces seaward velocity on the dike which might be an extra risk during extreme events.

## 1. Introduction

_{max}of 600 l/m in combination with H

_{m0}= 1–3 m is a limit for overtopping for people standing at dikes with clear view of the sea but it does not give further detailed explanation. Those are important indications but the applicability, for example, to the gentle and shallow foreshore cases is still not very clear [15]. Moreover, it is of interest how a fixed criterion (e.g., 1 l/s/m or 10 l/s/m) can be linked to the overtopping characteristics such as V

_{max}(maximum individual volume), and time dependent h (overtopping flow depth) and u (overtopping flow velocity). As Altomare et al. [6] indicated, the combination of u and h is linked to the hazard rather than the single maximum values of one of these parameters. According to Suzuki et al. [16], gentle and very shallow foreshore will result in flatter spectrum at the toe of the dike and thus spectral wave period T

_{m-1,0}is much longer than ones in deep water conditions due to infragravity waves contribution [17]. In such a situation, the waves have been transformed into bores and therefore overtopping characteristics, namely, flow pattern on dikes /promenades, might be also different from one which toe is at deep water. However, not so many studies have been conducted on the flow characteristics on dikes with gentle and shallow foreshores and discussed the associated risk.

_{max}, h

_{max}, u

_{max}, V, h, u) on the dike with and without a vertical structure (i.e., a sea wall or a building) with a gentle and shallow foreshore, and eventually to discuss the proper assessment method for overtopping waves. To this end, a non-hydrostatic wave-flow modelSWASH (an acronym of Simulating WAves till SHore) [18] is employed in this study. The model has been validated for the case of wave overtopping over the impermeable dikes with gentle and shallow foreshore configuration [16]. To ensure the applicability of the model to this study, relevant physical model test results from the Climate Resilient Coast project (CREST, http://www.crestproject.be/) are employed for further validation. Using the validated model, flow characteristics on a wide range of different hydraulic and topographic conditions are further investigated. Note that SWASH can provide not only time dependent wave surface elevation but also velocity field. By post-processing, it is possible to calculate individual overtopping volumes and average overtopping discharge too. Obtaining such outputs, especially the velocity fields on the dike, is not an easy task in physical models since the velocity measurement points are exposed to wet and dry conditions (when overtopping happens the bottom becomes wet while in the other moments the bottom is in general dry) which is often a problem for velocimeters and thus numerical simulation is a good alternative to study overtopping hazard.

## 2. Methods

#### 2.1. SWASH

#### 2.2. Model Settings

^{−1/3}s is employed to represent bottom friction for the entire domain, both for sandy beach and the dike. Note that 0.019 m

^{−1/3}s is the recommended value for wave simulations in the user manual. This must be due to the fact that the Manning’s coefficient for sand (e.g., the grain size of 0.3–0.4 mm) is around this value. For the dike it is assumed that the bottom of the promenade is often like unfinished concrete, and which Manning’s coefficient is around 0.014–0.020 m

^{−1/3}s and thus 0.019 m

^{−1/3}s should be an acceptable choice. Standard wave breaking control parameters, alpha = 0.6 and beta = 0.3, are used for wave breaking, and those values are also used in [16].

#### 2.3. Test Matrix

#### 2.4. Bathymetry

#### 2.5. Post-Processing

_{max}, h

_{max}and u

_{max}, respectively. The values are the maximum ones, so it is sensitive to the exceedance probability: when a lower number of waves are applied, the maximum value will be lower.

#### 2.6. Physical Model

## 3. Results

#### 3.1. Validation

_{max}(data only limited to long crested and second order wave generation cases) are further processed and linked to the average overtopping discharge, see Figure 3. See [24] for further details of the data processing.

#### 3.2. Overtopping Flow Characteristics on a Promenade (without a Vertical Wall)

#### 3.2.1. q-V_{max} Relationship

_{max}(maximum individual volume) for the cases with promenade width 0 and 20 m is shown in Figure 4.

_{max}around 2000 l/m, and 10 l/s/m gives V

_{max}around 6000 l/m for both promenade cases: there is no significant difference between the two promenade widths. From this result it can be concluded that V

_{max}is determined by q in the gentle and shallow foreshore case and the promenade width does not make significant difference on q-V

_{max}relationship for the wide range of the input hydraulic conditions and bathymetries. Even though Allsop et al. [5] indicated that the maximum individual overtopping volumes are more suitable hazard indicators, yet in this case V

_{max}and q both give the same information. This might be due to the fact that the incident significant wave height in this shallow foreshore case is not significantly different at the toe of the dike (toe depth is 0.5 m) for different offshore wave conditions: wave height is limited by the shallow water depth.

#### 3.2.2. q-h_{max} and q-u_{max} Relationships

_{max}(maximum overtopping flow depth) and q-u

_{max}(maximum overtopping flow velocity) for the cases with promenade width 0 and 20 m are shown in Figure 5.

_{max}, the difference of h

_{max}between promenade width 0 m and 20 m is significant: to up ~100 l/s/m the ratio is almost 2. Looking at the figure of q-u

_{max}, the difference of the maximum velocity is not significant unless the highest overtopping discharges around ~100 l/s/m.

#### 3.2.3. Time Evolution of Overtopping Flow Characteristics

_{max}gives very similar relationship between different promenade widths and on the other hand q-h

_{max}shows a strong influence of the promenade width. In order to understand these differences, the time series of flow properties (time dependent overtopping flow depth h, velocity u and acceleration) under an overtopping event of similar V (both case around 1000 l/m, see Table 2) is visualized in Figure 6. Note that the V in this specific example is not V

_{max}(maximum overtopping volume) in each case. In addition to u and h, the drag and inertia force acting on a person standing on the promenade is also calculated using the Morison equation since time evolution of the forces will be more relevant to the stability of a person standing on a promenade. In this case, two times of a cylinder with the diameter of 0.1 m are used to representing a person with two legs. Due to the nature of the equation, importance of u is higher than h (cfr. F is proportional to u

^{2}and h). As can be seen, the drag force is dominant and the inertia force is somewhat smaller in this case.

#### 3.2.4. Overtopping Flow Characteristics and Stability

_{max}and h

_{max}do not occur at the exactly same moment (there is a time-lag). If one wants to check stability properly, then it is advised to use a model which can describe the combination of u and h in a time series. The red line shows the time series of the u and h obtained at the end of the 20 m promenade from the SWASH model. Since it is based on 1000 waves, the line goes the same trajectory many times. The case shown in the figure corresponds to 16.2 l/s/m with V

_{max}= 6491 l/m and the highest part the of the time dependent u-h line is located at the edge of the stability curve. In case the stability is evaluated by stand-alone h

_{max}(horizontal red dotted line) in combination with stand-alone u

_{max}(vertical red dotted line), then the hazard is overestimated as can be seen in the figure.

_{max}(q = 15.0 l/s/m and V

_{max}= 6535 l/m) but the promenade width is 0 m. In this case the h-u line exceeds the stability curve clearly and thus the risk is higher. This is the same observation as described in Section 3.2.3: the overtopping hazard is not always a function of the overtopping discharge nor maximum individual overtopping volume, but on the u and h (in the gentle and shallow foreshore case at least).

#### 3.3. Overtopping Flow Properties in Front of a Vertical Wall

## 4. Discussion

#### 4.1. On Accuracy of the Model

_{max}, which shows an excellent match to the physical model test data. Note that the accuracy of h and u in time domain has not been confirmed yet in the present study.

_{max}are compared for promenade 0 m and 20 m cases. However, the influence of the directional spreading is expected to be limited because the relation between q and V

_{max}is similar. The green points depicted in Figure 11 show the q-V

_{max}relationship from the CREST physical model where directional spreading is greater than 0 degree (i.e., 12, 16, 20, and 31.5 degrees). As can be seen, the green points are shown in the cloud of the black points (i.e., directional spreading is zero) while the majority of the green points are located in the lower part of the entire cloud. Strictly speaking directional spreading effect on q-V can be different for the case of 20 m promenade since the oblique wave can make the overtopping trajectory longer than one of perpendicular attack, namely the effective promenade width will be longer. When the width of the promenade is longer, the V can be slightly smaller.

#### 4.2. On the Overtopping Parameters

_{max}are still very important parameters to have the first idea to estimate how severe will be the overtopping event. However, time dependent value u and h are more relevant to understand the risk on the dike in details as shown in this study. These parameters have a direct link to the stability of a person. On top of these parameters, acceleration (a) and overtopping event duration (t

_{o}) can be also parameters to give extra information to characterize the overtopping flow on a dike: acceleration in combination with h can give extra forcing of inertia and duration is also a useful parameter to understand how the overtopping is distributed in time.

#### 4.3. On the Risk

_{max}, in order to understand the risk. Apparently the influence of the promenade is positive in a sense that it reduces not only q but also h, as also indicated in [6]. Eventually, the forcing on pedestrians, vehicles, and structures will be reduced due to the effect of the promenade. The effect can be strengthened if extra obstacles are placed on top of the promenade, for instance sea walls and vegetation. The key will be how to reduce q and V

_{max}and also make overtopping event duration (t

_{o}) longer, so that h-u line stays in a small range.

_{max}is known. One of the effective evacuation strategies is the vertical evacuation, however in order to make sure this evacuation method is safe, first the stability of the building needs to be guaranteed. That is the reason why estimation of force acting on a building is necessary. Suzuki et al. [25] indicated that SWASH is also capable to estimate F on a vertical structure, but it is not explored in this study. In case detailed hydrodynamics needs to be obtained for complex structures, detailed hydrodynamic modelling (e.g., [29,30]) will be an alternative.

## 5. Conclusions

_{max}. However, it becomes clear from this study that overtopping risk is not only characterized by q and V: time dependent h and u are also useful and even better parameters to characterize risks on dikes more in details. For instance, two cases in the example of this study show different h, even though the two cases show very similar q and V

_{max}. This was due to the influence of the promenade which made h smaller and the duration longer. It is noted that the combination of stand-alone h

_{max}and u

_{max}can lead an overestimation of the hazard and therefore time dependent h and u are better for the proper assessment.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Bathymetries (type Q and W) and measurement points (black points: beginning, blue point: middle, red points: end of the promenade).

**Figure 2.**Comparison of the cross section of foreshore and dike profile between the SWASH model and the CREST physical model.

**Figure 3.**Comparison between SWASH and physical model on q-V

_{max}(Average overtopping discharge - maximum individual overtopping volume) for the case of promenade width 0 m.

**Figure 4.**Comparison between promenade width 0 m and 20 m on q-V

_{max}(Average overtopping discharge - maximum individual overtopping volume).

**Figure 5.**Comparison between promenade width 0 m and 20 m on q-h

_{max}(

**upper figure**) and q-u

**(**

_{max}**lower figure**).

**Figure 6.**Time series of h, u, acceleration, F_drag and F_inertia for the case which V is around 1000 l/m (the red lines are slightly shifted to be able to compare with the black lines easily).

**Figure 7.**Time series of h and u (the first and second figures) and h-u relationship calculated in SWASH versus stability curve of a tall adult (the third figure): RSK_Q_8_3_12_69_95_20 00 (a case in which V is around 6500 l/s/m, and the promenade width of 0 m).

**Figure 8.**Time series of h and u (the first and second figures) and h-u relationship calculated in SWASH versus stability curve of a tall adult (the third figure): RSK_Q_7_4_12_65_90_00 (a case in which V is around 6500 l/s/m, and the promenade width of 0 m).

**Figure 9.**Time series of h and u (the first and second figures) and h-u relationship calculated in SWASH versus the stability curve of a tall adult (the third figure): RSK_W_8_3_12_69_95_20 (a vertical wall case corresponding to the case in which V is around 6500 l/s/m, and the promenade width of 20 m).

**Figure 10.**Time series of h and u (the first and second figures) and h-u relationship calculated in SWASH versus the stability curve of a tall adult (the third figure) at 3 output points: RSK_W_8_3_12_69_95_20 (a vertical wall case corresponding to the case in which V is around 6500 l/s/m, and the promenade width of 20 m).

**Figure 11.**Comparison between DSPR = 0 and DSPR > 0 of physical models on q-V

_{max}(Average overtopping discharge - maximum individual overtopping volume) for the case of promenade width 0 m.

Name [-] | Bathymetry [-] | Water Level [m] | H_{m0}[m] | T_{p}[m] | Toe Level [m] | Dike Crest Level [m] | Promenade Width [m] |
---|---|---|---|---|---|---|---|

RSK | Q | 22 (7) ^{1} | 3 | 12 | 21.5 (6.5) ^{1} | 23.5 (8.5) ^{1} | 0 |

W | 23 (8) ^{1} | 4 | 21.9 (6.9) ^{1} | 24.0 (9.0) ^{1} | 20 | ||

5 | 24.5 (9.5) ^{1} | ||||||

25.0 (10.0) ^{1} |

^{1}The value inside the brackets is based on [m TAW] (Tweede Algemene Waterpassing; Belgian standard datum level, situated near MLLWS) and the value is reflected in the case name.

Case [-] | Promenade Width [m] | V [l/m] |
---|---|---|

RSK_7_5_12_69_00_00 | 0 | 1043 |

RSK_7_5_12_65_95_20 | 20 | 1109 |

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## Share and Cite

**MDPI and ACS Style**

Suzuki, T.; Altomare, C.; Yasuda, T.; Verwaest, T.
Characterization of Overtopping Waves on Sea Dikes with Gentle and Shallow Foreshores. *J. Mar. Sci. Eng.* **2020**, *8*, 752.
https://doi.org/10.3390/jmse8100752

**AMA Style**

Suzuki T, Altomare C, Yasuda T, Verwaest T.
Characterization of Overtopping Waves on Sea Dikes with Gentle and Shallow Foreshores. *Journal of Marine Science and Engineering*. 2020; 8(10):752.
https://doi.org/10.3390/jmse8100752

**Chicago/Turabian Style**

Suzuki, Tomohiro, Corrado Altomare, Tomohiro Yasuda, and Toon Verwaest.
2020. "Characterization of Overtopping Waves on Sea Dikes with Gentle and Shallow Foreshores" *Journal of Marine Science and Engineering* 8, no. 10: 752.
https://doi.org/10.3390/jmse8100752