# A Wave Input-Reduction Method Incorporating Initiation of Sediment Motion

^{1}

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## Abstract

**:**

## 1. Introduction

- Input reduction, which is based on the principle that the long-term effects of smaller-scale processes can be obtained by applying models of those smaller-scale processes forced with “representative” inputs able to reproduce the aforementioned long-term effects accurately [4].
- Model reduction, in which details of the smaller scale processes are omitted while the model simulation is performed at the scale of interest. The most commonly used acceleration technique of this type in 2-D area models is the morphological acceleration factor (Morfac, [5], which multiplies the bed level change at each time step by this factor, reducing the simulation time while simultaneously predicting the long-term evolution of the morphology.

## 2. Materials and Methods

#### 2.1. Proposed Method of Wave Schematization Based on the Sediment Pick-Up Rate

#### 2.1.1. Theoretical Aspects

_{cr}can be calculated by the following expression proposed by [23]:

^{2}] is the acceleration of gravity, $\mathrm{s}=\frac{{\rho}_{\mathit{s}}}{{\rho}_{\mathit{w}}}$ [-] is the ratio of the sediment $({\rho}_{\mathit{s}})$ to the water $({\rho}_{\mathit{w}})$ density, ν [m

^{2}/s] is the kinematic viscosity of the water and d

_{50}[m] is the median sediment diameter.

_{s}, with ${\mathit{H}}_{\mathit{s}}$ denoting the significant wave height) [21], generate an oscillatory velocity at the sea-bed which is the main factor setting the sand grains into motion. The amplitude of the wave orbital velocity above the bed for the case of a monochromatic wave can be approximated through the linear wave theory as:

_{rms}) can be computed according to [24] by summing the velocity contributions from each frequency (derived from the linear wave theory) over the whole frequency range. Soulsby et al. [25] proposed the following approximate formula to compute U

_{rms}:

_{rms}[m/s] is the root-mean square signal of the orbital velocity near the bed, ${\mathrm{T}}_{\mathrm{n}}=\sqrt{\frac{\mathrm{h}}{\mathrm{g}}}$ [s] is the natural scaling wave period, H

_{s}[m] is the significant wave height, and A [-] and t [-] are non-dimensional quantities defined as:

_{z}[s] is the zero-up crossing wave period. The above approximation is valid in the range of 0 ≤ t ≤ 0.54.

_{w}(or U

_{rms}when referring to spectral wave conditions) at the top of the boundary layer. It can be derived that the most important hydrodynamic property of waves contributing to sediment transport is the bed shear stress they produce. This stress is usually dependent on the wave orbital velocity U

_{w}at the bottom and the wave friction factor f

_{w}and is computed via the following relationship:

^{2}] is the bed shear stress due to the wave effect, ρ [kg/m

^{3}] is the water density and f

_{w}[-] is the wave friction factor.

_{w}and the relative roughness r. The latter quantity is calculated by:

_{s}[-] is the Nikuradse equivalent sand grain roughness.

_{w}[21,26,27]. They are all a faction of the relative roughness r and the wave orbital velocity excursion at the sea bed. Swart’s formulation [26] reads:

_{s}[kg/m

^{3}] is the sediment density.

_{cr}calculated through Equation (1). This forms the basis of the input-reduction method discussed in the present paper, as waves with rather small wave orbital velocities unable to initiate sand grain motion near the bed are disposed of, since it is considered that these waves have a very small contribution in the medium or long-term shaping of the morphological bed evolution.

_{D}incorporating all the additional effects on sediment movement in high velocities, the most important being the damping of turbulence (turbulence collapse) in the near-bed area where sediment concentrations are rather large. Ultimately, the new pick-up rate function reads:

^{2}/s] is the sediment pick-up rate, and ${\mathrm{f}}_{\mathrm{D}}=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\mathsf{\theta}$}\right.$ is a damping factor for high velocity conditions (θ > 1.0). It becomes apparent from Equation (11) that E is zero if θ < θ

_{cr.}

#### 2.1.2. Layout of the Wave Schematization Method

- Wave characteristic time-series (either by buoy measurements, or hindcast/forecast simulations) are obtained for a desirable time range T
_{tot}on a single point offshore coinciding with the open boundary of the computational domain. The minimum wave characteristics that are required by the input-reduction method are H_{s}, T_{p}(or another characteristic wave period) and MWD (mean wave direction). - The wave time-series are then filtered by disposing of wave data that do not contribute in shaping the bed evolution, namely wave components exiting the computational domain.
- Calculation of the critical Shields parameter θ
_{cr}through Equation (1). - Wave characteristics at a characteristic depth (at around h = 8–10 m, set as h = 8 m at the present study) are obtained. For this purpose, either a wave ray model (e.g., [29], a spectral wave model (e.g., [30,31,32], or a mild slope wave model [33,34] can be used. Here we use the parabolic mild slope model with non-linear dispersion characteristics MARIS-PMS. The reason for utilizing this model is the accuracy in prescribing the wave field in mildly sloping beaches due to incorporation of non-linearity and the saving of considerable computational time relatively to the time-dependent formulations of the aforementioned categories of models. After obtaining the wave climate in the nearshore area a “1-1” correspondence between each wave component offshore (H
_{s}, T_{p}, MWD) and the wave characteristics at the characteristic depth (H_{in}, T_{in}, MWD_{in}), is established. - Calculation of the depth of closure (h
_{in}) for the particular time-series through the following equation, which is defined as the seaward limit of the littoral zone [35]:$${\mathrm{h}}_{\mathrm{in}}=8.9\overline{{\text{}\mathrm{H}}_{\mathrm{s}}}$$_{in}[m] is the depth of closure and $\overline{{\mathrm{H}}_{\mathrm{s}}}$ [m] is the mean significant wave height at a characteristic depth h = 8–10 m, utilizing the wave characteristics calculated at step 4. The depth of closure was considered for the purpose of this research the critical depth after which no net sediment movement takes places. Consequently this depth will be later set for the calculation of the wave orbital velocity since the larger proportion of sediment transport takes place between h_{in}and the shoreline. - Calculate the wave orbital velocity signal near the bed through Equation (3) for monochromatic or Equation (4) for spectral waves setting h = h
_{in}. - For each wave component (H
_{in}, T_{in}, MWD_{in}) the friction factor f_{w}(Equation (9)), the bed shear stress due to waves τ_{b,w}(Equation (7)) and ultimately the Shields parameter θ (Equation (10)), are calculated - If the θ < θ
_{cr}the wave component is eliminated since it does not contribute in sediment motion. Through the “1-1” correspondence established at step 4, dispose the relative wave condition of the offshore time-series. The total number of wave components offshore N is thus reduced using the criterion of the initiation of motion at a total of N_{s}(with N_{s}≤ N) - Calculation of the sediment pick-up rate E
_{in}through Equation (11) for each wave component at the depth of closure. Also the cumulative pick-up rate E for the aforementioned wave conditions is determined. - The number of representative wave conditions N
_{r}that will replace the full wave climate (e.g., 12 representative conditions) are determined. The number of representative conditions is based on discretion, however it is advised that a number between 6 and 30 conditions is chosen for sufficiently accurate model results regarding yearly wave climates [12]. Then, the wave components are divided in classes with respect to wave direction and wave height. The boundaries of each class in both direction and wave heights are determined the same wave as the energy-flux wave schematization method (see Section 2.2 for details). Each representative class is characterized by an equal fraction of the cumulative pick-up rate E (E/N_{r}) and can be described by a set of wave characteristics (H_{r,in}, T_{r,in}, MWD_{r,in}). Thus, it can be derived that each class consists of a different number of wave components, N_{cl}. - Utilizing again the “1-1” correspondence of wave characteristics offshore and nearshore, we can obtain a set of representative conditions (H
_{r}, T_{r}, MWD_{r}) in the offshore wave boundary by considering that the bounding limits of each representative class in the depth of closure coincide with the respective ones in deep water. A small numerical extrapolation error stems from the fact that each representative class in the offshore boundary might not be characterized by exactly equal fraction of sediment pick-up rate, since the pick-up rate was calculated for the corresponding wave conditions at shallower water. However, since the proposed input-reduction method concerns medium to long-term morphological bed changes, this error is considered to have a very small effect in shaping the ultimate bed evolution and thus can be neglected. - The frequency of occurrence $\mathrm{f}=\frac{{\mathrm{N}}_{\mathrm{cl}}}{\mathrm{Ns}}$ for each representative class is calculated, based on the wave components of each class relatively to the full set of conditions.
- Finally the simulation is executed with a 2D morphological area model using the representative wave conditions as forcing input. The total model run-time T
_{tot,r}is a fraction of the full time series, denoted as ${\mathrm{T}}_{\mathrm{tot},\mathrm{r}}=\frac{{\mathrm{N}}_{\mathrm{r}}}{\mathrm{N}}{\mathrm{T}}_{\mathrm{tot}}$, since wave components unable to initiate sediment movement are eliminated in step 8 and have little to no contribution in shaping the bed evolution.

#### 2.2. The Energy Flux Wave Schematization Method (Benchmark Reduction Method)

- Calculation of the wave energy flux $\left({\mathit{E}}_{\mathit{f}}\right)$ for each wave component of a time-series,$${E}_{f}=\frac{1}{8}\rho g{H}_{s}^{2}{C}_{g}$$
^{3}] is the water density, ${H}_{s}$ [m] is the significant wave height and ${C}_{g}$ [m/s] is the wave group celerity in deep water, - Calculation of the total wave energy flux of the full wave time-series through:$${E}_{tot}={{\displaystyle \sum}}^{\text{}}{E}_{f}$$
- Division of the wave components in wave direction bins. For a predefined number of directional bins $({N}_{d})$ the time series are separated in bins, each consisting of an equal fraction of the total energy flux $({E}_{tot}/{N}_{d}).$ Further division of the data in wave height bins. Separation is carried out for a predefined number of wave height bins $({N}_{h})$ with each bin characterized by an equal fraction of the total energy flux $({E}_{tot}/({N}_{d}\xb7{N}_{h}).$
- A representative wave height for each bin is derived from the mean energy flux of the bin along with a mean energy-flux direction. The representative wave period is then defined as the mean period of the bin.

#### 2.3. Theoretical Background of Numerical Models

#### 2.3.1. The MIKE21 Coupled Model FM Suite

- MIKE21 SW, a 3rd generation spectral wave model based on the conservation of the wave action balance, suited for the propagation and transformation of waves in the coastal zone.
- MIKE21 HD, a depth-averaged hydrodynamic model based on the Reynolds-averaged Navier–Stokes equations of motion (RANS), for the description of the nearshore circulation.
- MIKE21 ST, a sand transport and morphology updating model, used to calculate sediment transport rates and ultimately the morphological bed evolution.

^{3}] is the water density, ${\mathrm{S}}_{\mathrm{xx}}$, ${\mathrm{S}}_{\mathrm{yy}}$, ${\mathrm{S}}_{\mathrm{xy}}$, are components of the radiation stress tensor, ${\mathrm{p}}_{\mathrm{a}}$ [N/m

^{2}] is the atmospheric pressure, $\mathrm{S}$ [m

^{3}/s] being the magnitude of point sources, with ${\mathrm{u}}_{\mathrm{s}}$, ${\mathrm{v}}_{\mathrm{s}}$. [m/s] being the velocity vectors of the point discharge and ${\mathrm{T}}_{\mathrm{xx}}$, ${\mathrm{T}}_{\mathrm{yy}}$, ${\mathrm{T}}_{\mathrm{xy}}$ [N/m

^{2}] denoting lateral stresses including viscous, turbulent friction and differential advection.

^{2}/s] is the total load sediment transport rates in the x and y direction respectively and $\mathsf{\Delta}\mathrm{S}$ [m/s] is a sediment source or sink term. The new bed level is then obtained by a forward in time differential scheme.

#### 2.3.2. The MARIS-PMS Wave Model

## 3. Method Implementation

#### 3.1. Study Area

#### 3.2. Mesh Generation

#### 3.3. Offshore Wave Data

- A simulation consisting of the full time series at the offshore boundary, hereafter denoted as Reference simulation
- A simulation using 12 representatives as forcing parameters calculated with the pick-up rate method, hereafter called pick-up rate simulation
- A simulation using 12 representatives calculated with the energy-flux input-reduction method, hereafter denoted as energy-flux simulation, to assess how the pick-up rate method fares against a well-established wave schematization technique.

_{mo}= 3.2 m) while the dataset of 20 days was characterized by milder wave conditions with a maximum wave height H

_{mo}= 2.35 m. Regarding the dataset covering the extend of the year, the wave climate was more diverse, with a minimum wave height of 0.09 m and a maximum wave height of 4.66 m. From the initial filtration of the waves that exit the computational domain and therefore have no effect on the morphological bed evolution, the dataset was reduced from 8762 hourly changing wave records to 8219. The vast majority (over 70 %) of the incident waves are entering the computational domain from the north sector, as shown in Figure 4.

#### 3.4. Obtained Representative Wave Conditions

## 4. Results and Discussion

#### 4.1. Morphological Bed Evolution for the Dataset of Seven Days

#### 4.2. Morphological Bed Evolution for the Dataset of 20 Days

#### 4.3. Morphological Bed Evolution for the Dataset of a Year

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**The island of Crete (

**left**) and the study area of the coastal zone of Rethymno (

**right**) where the MIKE21 Coupled Model FM was implemented. Adapted from [54], with permission from Google Earth, 2020.

**Figure 4.**Wave height rose plot for the dataset containing records of 7 days (

**a**), 20 days (

**b**) and a year of data (

**c**).

**Figure 5.**Scatter plot showing the full dataset of the 20 days (blue markers) along with the bins (red rectangles) and representative wave conditions (red markers) using the pick-up rate method (

**a**) and the energy-flux method (

**b**).

**Figure 6.**Scatter plot showing the full dataset of a year of data (blue markers) along with the bins (red rectangles) and representative wave conditions (red markers) using the pick-up rate method (

**a**) and the energy-flux method (

**b**).

**Figure 7.**Finite element mesh showing the area (within the closed polygon) where the morphological model results will be evaluated.

**Figure 8.**Total load magnitude and direction at the end of the pick-up rate simulation for the area of interest.

**Figure 9.**Bed-level changes obtained by the reference simulation (

**a**) and implementation of the pick-up rate method (

**b**) and energy-flux method (

**c**) for the set covering a year of data.

**Table 1.**Representative wave conditions from the time series of 7 days, using the pick-up rate and the energy-flux wave schematization methods.

Class | Pick-up Rate Method Representatives | Energy Flux Method Representatives | ||||||
---|---|---|---|---|---|---|---|---|

H_{mo} (m) | T_{p} (s) | MWD (°) | Frequency (%) | H_{mo} (m) | T_{p} (s) | MWD (°) | Frequency (%) | |

1st | 1.78 | 6.96 | 1.30 | 14.94 | 1.23 | 5.87 | 357.57 | 10.84 |

2nd | 2.77 | 8.20 | 1.65 | 4.60 | 2.55 | 8.09 | 1.46 | 3.01 |

3rd | 2.51 | 7.63 | 2.35 | 5.75 | 2.53 | 7.63 | 2.31 | 3.61 |

4th | 2.77 | 8.01 | 2.01 | 4.60 | 2.78 | 8.01 | 1.93 | 2.41 |

5th | 2.30 | 7.53 | 3.27 | 8.05 | 1.96 | 7.09 | 3.69 | 6.02 |

6th | 2.94 | 8.39 | 3.24 | 3.45 | 2.76 | 8.09 | 2.99 | 3.01 |

7th | 1.74 | 6.99 | 4.85 | 19.54 | 1.44 | 6.38 | 5.14 | 10.24 |

8th | 2.65 | 8.01 | 4.73 | 4.60 | 2.88 | 8.20 | 5.35 | 2.41 |

9th | 1.70 | 7.12 | 8.47 | 18.39 | 1.44 | 6.69 | 8.86 | 10.24 |

10th | 3.19 | 8.39 | 8.50 | 2.30 | 2.90 | 8.01 | 8.62 | 2.41 |

11th | 1.88 | 6.89 | 12.78 | 10.34 | 0.52 | 4.77 | 21.63 | 43.37 |

12th | 3.16 | 7.63 | 12.49 | 3.45 | 3.08 | 7.63 | 14.37 | 2.44 |

BSS | |
---|---|

Excellent | 1.0–0.5 |

Good | 0.5–0.2 |

Reasonable/fair | 0.2–0.1 |

Poor | 0.1–0.0 |

Bad | <0.0 |

**Table 3.**Statistical parameters obtained in the area of interest for both the pick-up rate simulation and the energy-flux simulation-Dataset of 7 days.

Pick-Up Rate Method | Energy Flux Method | |
---|---|---|

Bias | −0.0054 | −0.0101 |

MAE(Y,X) | 0.0218 | 0.0233 |

MSE(Y,X) | 0.0018 | 0.0020 |

RMSE(Y,X) | 0.0429 | 0.0450 |

BSS | 0.9300 | 0.9200 |

**Table 4.**Obtained statistical parameters in the area of interest for both the pick-up rate simulation and the energy-flux simulation dataset of 20 days.

Pick-Up Rate Method | Energy Flux Method | |
---|---|---|

Bias | −0.0177 | −0.0058 |

MAE(Y,X) | 0.0633 | 0.0535 |

MSE(Y,X) | 0.0096 | 0.0066 |

RMSE(Y,X) | 0.0980 | 0.0813 |

BSS | 0.8300 | 0.8800 |

**Table 5.**Obtained statistical parameters in the area of interest for both the pick-up rate simulation and the energy-flux simulation-Dataset of a year.

Pick-Up Rate Method | Energy Flux Method | |
---|---|---|

Bias | −0.2316 | −0.1441 |

MAE(Y,X) | 0.2661 | 0.1885 |

MSE(Y,X) | 0.1070 | 0.0614 |

RMSE(Y,X) | 0.3272 | 0.2478 |

BSS | 0.7445 | 0.8535 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Papadimitriou, A.; Panagopoulos, L.; Chondros, M.; Tsoukala, V.
A Wave Input-Reduction Method Incorporating Initiation of Sediment Motion. *J. Mar. Sci. Eng.* **2020**, *8*, 597.
https://doi.org/10.3390/jmse8080597

**AMA Style**

Papadimitriou A, Panagopoulos L, Chondros M, Tsoukala V.
A Wave Input-Reduction Method Incorporating Initiation of Sediment Motion. *Journal of Marine Science and Engineering*. 2020; 8(8):597.
https://doi.org/10.3390/jmse8080597

**Chicago/Turabian Style**

Papadimitriou, Andreas, Loukianos Panagopoulos, Michalis Chondros, and Vasiliki Tsoukala.
2020. "A Wave Input-Reduction Method Incorporating Initiation of Sediment Motion" *Journal of Marine Science and Engineering* 8, no. 8: 597.
https://doi.org/10.3390/jmse8080597