# Modelling the Impact of Climate Change on Coastal Flooding: Implications for Coastal Structures Design

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{t}= H

_{t}/H

_{i}[24], where H

_{i}and H

_{t}are the incident and transmitted wave heights, respectively.

_{t}through various approaches, others validate or seek to extend existent formulae through experimental/numerical investigations [25,26,27,28], while an emerging research field combines data analysis with Artificial Neural Network (ANN) techniques in order to provide tools able to predict wave–structure interactions [29,30]. Regarding specifically design formulae, reference should be made to: the fundamental works of [31,32]; the work of [33], who proposed a novel formulation using as basis the theoretical treating of the physical phenomena that govern wave transmission (i.e., breaking/overtopping/energy transfer); the work of [34], who proposed a formulation based on the summation of wave energy transmitted over and through LCS (following [35] and [36], respectively), as well as the update of this last formulation by [37].

## 2. Model Description

#### 2.1. The Large-Scale Wave Propagation Model (WAVE_LS)

_{x}, c

_{y}, c

_{θ}are the x, y, θ components of the group velocity c

_{g}, respectively, according to:

_{m}is the maximum wave height according to [41]; ρ is the water density; and Q

_{b}is the probability of a wave breaking at a certain depth, expressed as (1 − Q

_{b})/(lnQ

_{b}) = (H

_{rms}/H

_{m})

^{2}according to [42], where H

_{rms}is the root-mean-square wave height. Wave diffraction is incorporated to the model by replacing c

_{x}, c

_{y}and c

_{θ}with C

_{x}, C

_{y}and C

_{θ}according to [40]:

#### 2.2. The Storm-Induced Circulation Model (SICIR)

_{h}is the horizontal eddy viscosity coefficient and f

_{c}is the Coriolis coefficient. The terms τ

_{sx}, τ

_{sy}are the shear stress components at the water surface along the x- and y- directions respectively, which represent the vertical boundary condition, expressed as:

^{3}, typically of the order of 10

^{−6}; here we assume k = 10

^{−6}÷ 3∙10

^{−6}), and W

_{x}, W

_{y}are the wind speed components along the x- and y- directions (in m/s, at 10 m above sea level) respectively. The bed friction terms (τ

_{bx}, τ

_{by}) are calculated based on the formulae proposed by [45]:

_{b}is the bottom friction factor, σ

_{T}is the standard deviation of the oscillatory horizontal velocity, and |U| = (U

^{2}+ V

^{2})

^{0.5}. The horizontal eddy viscosity coefficient is expressed by the well-known Smagorinsky model, used for the representation of the damping by eddies smaller than the computational grid size, as:

_{cr}is a terminal depth below which drying is assumed to occur (here this depth is set to h

_{cr}= 0.001 m).

#### 2.3. The Advanced Nearshore Wave Propagation Model (WAVE_BQ)

**M**is defined as:

_{u}**U**is the horizontal velocity vector

**U**= (U,V);

**τ**= (τ

_{b}_{bx}, τ

_{by}) is calculated from Equations (11) and (12), with the wave-current bottom friction factor calculated as in [56]; δ is the roller thickness, determined geometrically according to [57];

**E**is the eddy viscosity term, calculated according to [58]; and

**u**is the bottom velocity vector

_{o}**u**= (u

_{o}_{o}, v

_{o}), with u

_{o}and v

_{o}being the instantaneous bottom velocities along the x- and y- directions respectively. Following [56], wave breaking is initiated using breaking angle φ

_{b}= 30°, which then gradually changes to its terminal value φ

_{b}= 10°.

^{2},εσ

^{2},σ

^{4}) [59] and their numerical solution is based on the accurate higher-order numerical scheme of [60]. Coastal inundation is simulated as in SICIR (see Section 2.2, Equation (16) and [47]). The model is capable of simulating the phenomena of shoaling, refraction, breaking, diffraction, reflection and wave-structure interaction, as well as nonlinear wave-wave interaction. Regarding model capabilities in simulating the non-linear evolution of unidirectional or multidirectional wave fields in the nearshore, one can refer to [61] (see also [62] on the issue); regarding wave-structure interaction and energy transmission, one can refer to [63]. Further details on the model and its implementation to diverse coastal engineering applications can be found in [56,64,65].

## 3. Model Validation

#### 3.1. Coastal Flooding

#### 3.2. Wave Transmission at Coastal Structures

_{t}. Furthermore, and considering that due to the interplay between wave overtopping and wave infiltration energy reduction is expected to be lower behind impermeable structures than behind permeable ones, the former type of LCS is examined in this work so as to test the lower bound of their effectiveness.

_{t,over}and K

_{t,thru}, respectively, with the summation concept applied for the derivation of K

_{t}using the approach of [66]. The formulation was validated using a versatile set of experimental datasets (851 tests in total), yielding a determination coefficient of r

^{2}= 0.865.

_{t}for impermeable structures reduces to K

_{t,over}, which was estimated through empirical fitting to the design diagram of [35] and can be expressed as:

_{c}is the crest freeboard and a is expressed as:

_{0}is the deep water wavelength and B

_{eff}is the effective width of the structure, measured at still water level for emerged structures, at the level of 10% below the crest for zero freeboard structures and at the level of 20% below the crest for submerged structures [34].

_{i}and H

_{t}; the peak period T

_{p}; the wave steepness s

_{op}= 2πH

_{i}/g(T

_{p})

^{2}; the crest freeboard and width, R

_{c}and B; the structure height and seaward slope, h

_{c}and tanα; and the breaker parameter ξ

_{op}= tanα/(s

_{op})

^{0.5}. Model runs were performed for the series of input parameters presented in Table 1 testing combinations of two incident waves and four different crest freeboards, which resulted in relative freeboards R

_{c}/H

_{i}ranging from 0.13 to 1.00 and relative effective structure widths B

_{eff}/L

_{0}ranging from 0.075 to 0.150 (tanα = 0.4 for all runs). The discretization steps used in model (WAVE_BQ) runs were dx = 0.125 m in space and dt = 0.005 s in time. Figure 4 shows the comparison between K

_{t}values as estimated using Equation (21) and as resulted from model runs, for the Tests of Table 1. Data are in good agreement overall, with Test 4 representing a liming case that results in K

_{t}= 0. The model generally predicts lower transmission coefficients than Equation (21). The divergences are mainly observed for higher waves and lower relative freeboard values, a result that is expected considering the concept behind the formulation of [34] and the physics behind the phenomenon.

## 4. Model Applications

#### 4.1. Large-Scale Applications

_{s}= 1.58 m and peak period T

_{p}= 4.60 s. The second scenario (LS2) envisaged the same wave combined with a storm surge of height SSH = 0.30 m (see Table 2). The discretization steps used in model runs were dx = 10.0 m in space and dt = 0.05 s in time for WAVE_LS, and dx = 10.0 m in space and dt = 0.125 s in time for SICIR. It should be noted that dx in large-scale wave propagation models is generally of the order of a wavelength L or less [39,40]. However, the choice of dx also depends on bathymetry and model domain characteristics; rapid bathymetric changes or diffraction effects would require smaller values, such as dx = L/5 [40]. In a modular modelling approach, the spatial step of the circulation model would have to be the same as previously selected due to model interoperability. Finally, the temporal step dt is controlled by the Courant number criterion Cr < 1 (Cr = cdt/dx, where c = (gh)

^{1/2}). In this work, the choices for the spatial and temporal discretization of WAVE_LS and SICIR followed the above rules.

#### 4.2. Applications for Coastal Structures Design Evaluation

_{s}= 5.0 m and peak period T

_{p}= 8.0 s. The second scenario (CS2) envisaged the same wave combined with a storm surge of height SSH = 0.30 m (see Table 4). The discretization steps used in model runs were dx = 0.125 m in space and dt = 0.005 s in time for WAVE_BQ (see Section 4.1 for WAVE_LS and SICIR).

## 5. Results and Discussion

#### 5.1. Large-Scale Applications

^{2}. For scenario LS2 the flooding extends to the low-lying coastal areas west of the port’s 6th pier as well, with the flooded area approximately equal to 2.7 km

^{2}(increase of approximately 125%). The AGET Terminal appears to be mostly impacted in the case of LS2, while impacts on the operations of both the AGET and liquid fuels jetties are to be expected in either scenario.

#### 5.2. Applications for Coastal Structures Design Evaluation

#### 5.3. General Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Surface profiles of solitary wave transformation over a dry reef flat with A/d = 0.5 and a 1:5 slope. Solid lines denote the results of the presented advanced nearshore wave propagation model (see Section 2.3) and circles denote the measurements of [38].

**Figure 2.**Surface profiles of solitary wave transformation over an exposed reef crest with A/d = 0.3 and a 1:12 slope. Solid lines denote the results of the presented advanced nearshore wave propagation model (see Section 2.3) and circles denote the measurements of [38].

**Figure 3.**Governing parameters involved in wave transmission at emerged low-crested structures (LCS).

**Figure 4.**Wave transmission at emerged LCS: comparison between between K

_{t}values as estimated following [34] and as resulted from the presented advanced nearshore wave propagation model runs (see Section 2.3).

**Figure 5.**Geographic location and satellite image of the study area ([68]; privately processed).

**Figure 6.**Model domain and bathymetry (background image from [68]; privately processed).

**Figure 7.**Geographic location and satellite image of the East Macedonia and Thrace Region ([68]; privately processed).

**Figure 8.**Model results indicating the flooded area for scenarios: (

**a**) LS1 and (

**b**) LS2 (background image from [68]; privately processed).

**Figure 9.**Model results for the combinations of layouts L1, L2, L3 and scenarios CS1 and CS2. Coloured “x” symbols denote the landward limit of the flooding extent.

**Figure 10.**Flooding extent for the combinations of layouts L1, L2, L3 and scenarios CS1 and CS2. Flooding extent is measured from the shoreline at SWL (horizontal distance).

**Table 1.**Values of the parameters in the wave transmission Tests (s

_{op}= wave steepness; tanα = slope; ξ

_{op}= breaker parameter; R

_{c}/H

_{i}= relative freeboard; B/H

_{i}= relative crest width; B

_{eff}/L

_{0}= relative effective structure width).

Test | s_{op} | tanα | ξ_{op} | R_{c}/H_{i} | B/H_{i} | B_{eff}/L_{0} |
---|---|---|---|---|---|---|

1 | 0.02 | 0.4 | 2.83 | 0.25 | 2.5 | 0.075 |

2 | 0.50 | 0.100 | ||||

3 | 0.75 | 0.125 | ||||

4 | 1.00 | 0.150 | ||||

5 | 0.04 | 0.4 | 2.00 | 0.13 | 1.3 | 0.075 |

6 | 0.25 | 0.100 | ||||

7 | 0.38 | 0.125 | ||||

8 | 0.50 | 0.150 |

Scenario | Wave | Storm Surge | ||
---|---|---|---|---|

H_{s} (m) | T_{p} (s) | Dir (deg) | SSH (m) | |

LS1 | 1.58 | 4.60 | 0 | - |

LS2 | 1.58 | 4.60 | 0 | 0.30 |

Layout | Structure Characteristics | |||
---|---|---|---|---|

Type | tanα (-) | Β (m) | R_{c}^{1} (m) | |

L1 | Unprotected beach | |||

L2 | LCS | 0.4 | 5.0 | 1.0 |

L3 | Dike | 0.2 | - | 2.0 |

^{1}Measured from SWL.

Scenario | Wave | Storm Surge | ||
---|---|---|---|---|

H_{s} (m) | T_{p} (s) | Dir (deg) | SSH (m) | |

CS1 | 5.0 | 8.0 | 0 | - |

CS2 | 5.0 | 8.0 | 0 | 0.30 |

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## Share and Cite

**MDPI and ACS Style**

Samaras, A.G.; Karambas, T.V.
Modelling the Impact of Climate Change on Coastal Flooding: Implications for Coastal Structures Design. *J. Mar. Sci. Eng.* **2021**, *9*, 1008.
https://doi.org/10.3390/jmse9091008

**AMA Style**

Samaras AG, Karambas TV.
Modelling the Impact of Climate Change on Coastal Flooding: Implications for Coastal Structures Design. *Journal of Marine Science and Engineering*. 2021; 9(9):1008.
https://doi.org/10.3390/jmse9091008

**Chicago/Turabian Style**

Samaras, Achilleas G., and Theophanis V. Karambas.
2021. "Modelling the Impact of Climate Change on Coastal Flooding: Implications for Coastal Structures Design" *Journal of Marine Science and Engineering* 9, no. 9: 1008.
https://doi.org/10.3390/jmse9091008