# Modelling the Past and Future Evolution of Tidal Sand Waves

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Sites

#### 2.2. Model Description

#### 2.3. Model Configuration

#### 2.4. Design of Experiments

#### 2.5. Analysis of Model Output

## 3. Results

#### 3.1. Calibration

#### 3.2. Validation

#### 3.3. Forecast

**a**)–(

**b**)) and 2 ((

**c**)–(

**d**)) which have the best BSS scores. The black solid lines in Figure 7a,c correspond to the initial bed level measured in 2012 and the red lines denote the prediction for 2024. Along transect 1, the predicted crest migration rate is ${c}_{crest}$ = 0.8 ± 1.0 m yr${}^{-1}$ and for the troughs this is ${c}_{trough}$ = 2.9 ± 0.8 m yr${}^{-1}$. In this period, the crests remain at the same height or become slightly lower. Most of the troughs have deepened with an average value of 12 cm. Similarly to the hindcast, the migration rate deviates slightly in the beginning, but is constant in the rest of the simulation.

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Overview of the four study sites, with panel (

**A**) showing their locations in the North Sea (EMODnet bathymetry 2020). Panel (

**B1**–

**B4**) are zoom-ins of these study areas, with the red line corresponding to the transects that are investigated. Panels (

**C1**–

**C4**) show the bed level ${z}_{b}$ (m) over distance x along the transect (km), with the red lines corresponding to surveys performed in 1994–2001 (used as initial bathymetry) and the black lines to surveys of 2011–2016. The panels (

**B1**–

**B4**) and (

**C1**–

**C4**) were drawn using survey data which are freely available from the Hydrographic Service of the Royal Netherlands Navy [25]. Panels (

**C1**–

**C4**) also show definitions of crest migration ${c}_{crest}$, wavelength $\lambda $ and waveheight h. Depths are relative to Normaal Amsterdams Peil (NAP) and coordinates are in World Geodetic System 84/Universal Transverse Mercator 31${}^{\circ}$ N. Note the different colour bars in all plots.

**Figure 2.**(

**a**) Initial bed level ${z}_{b}$ (m) over distance x (km) along transect 1. (

**b**) Grid spacing $\Delta x$ (m) in the horizontal direction and (

**c**) grid spacing $\Delta z$ in the vertical direction.

**Figure 3.**RMSE (m) of modelled 2012 bed level along transect 1 for different values of sand transport factor f and slope parameter ${\alpha}_{bs}$.

**Figure 4.**(

**a**) Modelled bed level ${z}_{b}$ (m) for 2012 (red) over distance x (km) along transect 1. The black dotted line corresponds to initial bed level (1999) and the black solid line to the measured bed level of 2012. (

**b**) Bed level ${z}_{b}$ (m) as a function of distance x (km) and time T (yr). The black dots and circles mark the position of crests and troughs, respectively, over time. The black crosses and squares correspond to the measured positions in 2012.

**Figure 5.**Modelled bed level ${z}_{b}$ (m) for 2011, 2012 or 2016 (red) over distance x (km) along transects 2–4 (panels (

**a**–

**c**)). Black dotted lines correspond to initial bed levels measured in 1999, 2001 or 1994 and black solid lines to bed levels observed in 2011, 2012 or 2016 used for comparison.

**Figure 6.**Bed level ${z}_{b}$ (m) as a function of distance x (km) and time T (yr) for transects 2–4 (

**a**–

**c**). The black dots and circles mark the position of crests and troughs, respectively, over time. The black crosses and squares correspond to the measured positions in 2011, 2012 or 2016. Note the different colour bars.

**Figure 7.**Predicted bed levels ${z}_{b}$ (m) for 2024 (red lines) over distance x (km) along transects 1 (

**a**) and 2 (

**c**) and as a function of distance x (km) and time T (yr) (panels (

**b**,

**d**) for transects 1 and 2, respectively). The black dots and circles mark the position of crests and troughs, respectively, over time. Note the different colour bars.

**Figure 8.**Horizontal gradient of tidally averaged sand transport $\frac{\partial \langle q\rangle}{dx}$ (m/s) over distance along transect x (km) for slope-related bed load transport (magenta), advective bed load transport (red), total bed load transport (green) and suspended load transport (blue) along transects 1 (

**a**), 2 (

**b**), 3 (

**c**) and 4 (

**d**). The black solid line corresponds to the bed level ${z}_{b}$ (m) along each transect. Note the different y-axes.

**Figure 9.**Evolution of the Brier Skill Score (BSS) of modelled bed levels with respect to observations taken at time T (yr) along transect 4.

**Figure 10.**Evolution of the Brier Skill Score (BSS) of modelled bed levels as a function of morphological spin-up time $\tau $ (yr) along transect 3.

**Figure 11.**Ellipticity (ratio of amplitudes of minor and major axis) of the depth-averaged M${}_{2}$ (

**a**) and M${}_{4}$ tidal velocity (

**b**) along the four transects. Ellipticity is calculated from ZUNO output.

Loc 1 | ${\mathit{R}}_{+}$, $\mathit{x}=-\mathit{L}/2$ | ${\mathit{R}}_{-}$, $\mathit{x}=\mathit{L}/2$ | ||
---|---|---|---|---|

Amplitude | Phase | Amplitude | Phase | |

M${}_{2}$ | 1.02 ms${}^{-1}$ | 73${}^{\circ}$ | 0.38 ms${}^{-1}$ | 73${}^{\circ}$ |

M${}_{4}$ | 0.14 ms${}^{-1}$ | 108${}^{\circ}$ | 0.06 ms${}^{-1}$ | 296${}^{\circ}$ |

M${}_{6}$ | 0.05 ms${}^{-1}$ | 92${}^{\circ}$ | 0.04 ms${}^{-1}$ | 89${}^{\circ}$ |

M${}_{0}$ | 0.02 ms${}^{-1}$ | −0.001 ms${}^{-1}$ |

Parameter | Symbol | Transect 1 | Transect 2 | Transect 3 | Transect 4 | Dimension |
---|---|---|---|---|---|---|

Domain length | L | 50 | 50 | 50 | 50 | km |

Length of area of interest | ℓ | 5 | 5 | 5 | 5 | km |

Undisturbed water depth | ${H}_{0}$ | 23.6 | 26.3 | 21.3 | 30.7 | m |

Roughness length | ${k}_{s}$ | 0.0944 | 0.0945 | 0.0945 | 0.0949 | m |

Median sand grain size | ${d}_{50}$ | 0.275 | 0.305 | 0.305 | 0.390 | mm |

Bed slope parameter | ${\alpha}_{bs}$ | 5.5 | 5.5 | 5.5 | 5.5 | - |

Sand transport scale factor | f | 0.45 | 0.45 | 0.45 | 0.45 | - |

Hydrodynamic time step | $\Delta t$ | 6 | 6 | 6 | 6 | s |

Horizontal grid spacing | $\Delta x$ | 5 | 5 | 5 | 5 | m |

No of $\sigma $-layers | - | 40 | 40 | 40 | 40 | - |

Morphological acceleration factor | MORFAC | 148 | 148 | 148 | 148 | - |

Morphological simulation time | ${T}_{m}$ | 12.3 | 12.3 | 9.7 | 22.4 | years |

Runs | Description |
---|---|

H1–32 | Hindcast runs transect 1 with f varying between |

0.3 and 1 and ${\alpha}_{bs}$ varying between 3 and 6 | |

V2–4 | Hindcast runs transects 2–4 with $f=0.45$ and ${\alpha}_{bs}=5.5$ |

F1–2 | Forecast runs transects 1–2 with $f=0.45$ and ${\alpha}_{bs}=5.5$ |

Transect 1 | ||
---|---|---|

Observations | Model | |

${c}_{crest}$ | 2.9 m yr${}^{-1}$ | 1.1 m yr${}^{-1}$ |

${\sigma}_{c,crest}$ | 1.3 m yr${}^{-1}$ | 1.0 m yr${}^{-1}$ |

${c}_{trough}$ | 1.9 m yr${}^{-1}$ | 2.9 m yr${}^{-1}$ |

${\sigma}_{c,trough}$ | 0.9 m yr${}^{-1}$ | 0.7 m yr${}^{-1}$ |

RMSE${}_{h,crest}$ | 0.13 m | |

RMSE${}_{x,crest}$ | 27.0 m | |

RMSE${}_{h,trough}$ | 0.21 m | |

RMSE${}_{x,trough}$ | 16.8 m | |

RMSE | 0.17 m | |

$\alpha $ | 0.74 | |

$\beta $ | 0.01 | |

$\gamma $ | 0.01 | |

$\u03f5$ | 0.01 | |

BSS | 0.74 |

Transect 2 | Transect 3 | Transect 4 | ||||
---|---|---|---|---|---|---|

Observations | Model | Observations | Model | Observations | Model | |

${c}_{crest}$ | 2.2 m yr${}^{-1}$ | 0.4 m yr${}^{-1}$ | 2.8 m yr${}^{-1}$ | 1.0 m yr${}^{-1}$ | 1.3 m yr${}^{-1}$ | −0.4 m yr${}^{-1}$ |

${\sigma}_{c,crest}$ | 1.2 m yr${}^{-1}$ | 0.6 m yr${}^{-1}$ | 1.2 m yr${}^{-1}$ | 1.2 m yr${}^{-1}$ | 0.2 m yr${}^{-1}$ | 0.5 m yr${}^{-1}$ |

${c}_{trough}$ | 1.1 m yr${}^{-1}$ | 1.3 m yr${}^{-1}$ | 2.3 m yr${}^{-1}$ | 0.3 m yr${}^{-1}$ | 1.0 m yr${}^{-1}$ | 0.6 m yr${}^{-1}$ |

${\sigma}_{c,trough}$ | 1.1 m yr${}^{-1}$ | 1.4 m yr${}^{-1}$ | 0.9 m yr${}^{-1}$ | 0.5 m yr${}^{-1}$ | 0.5 m yr${}^{-1}$ | 0.5 m yr${}^{-1}$ |

RMSE${}_{h,crest}$ | 0.44 m | 0.12 m | 1.04 m | |||

RMSE${}_{x,crest}$ | 28.7 m | 19.7 m | 38.7 m | |||

RMSE${}_{h,trough}$ | 0.42 m | 0.22 m | 0.55 m | |||

RMSE${}_{x,trough}$ | 14.5 m | 22.5 m | 14.3 m | |||

RMSE | 0.35 m | 0.13 m | 0.72 m | |||

$\alpha $ | 0.57 | 0.26 | 0.27 | |||

$\beta $ | 0.03 | 0.02 | 0.01 | |||

$\gamma $ | 0.01 | 0.02 | 0.01 | |||

$\u03f5$ | 0.01 | 0.02 | 0.01 | |||

BSS | 0.54 | 0.24 | 0.26 |

**Table 6.**Global growth rate $\sigma $ (yr${}^{-1}$) and migration rate V (m yr${}^{-1}$ ) for each component of the sand transport.

Transect 1 | Transect 2 | Transect 3 | Transect 4 | |
---|---|---|---|---|

${\sigma}_{b,adv}$ | 0.027 yr${}^{-1}$ | 0.031 yr${}^{-1}$ | 0.0226 yr${}^{-1}$ | 0.054 yr${}^{-1}$ |

${\sigma}_{b,slope}$ | −0.027 yr${}^{-1}$ | −0.033 yr${}^{-1}$ | −0.017 yr${}^{-1}$ | −0.076 yr${}^{-1}$ |

${\sigma}_{b}$ | 0.000 yr${}^{-1}$ | −0.002 yr${}^{-1}$ | 0.006 yr${}^{-1}$ | −0.022 yr${}^{-1}$ |

${\sigma}_{s}$ | 0.002 yr${}^{-1}$ | 0.004 yr${}^{-1}$ | 0.003 yr${}^{-1}$ | 0.006 yr${}^{-1}$ |

${V}_{b,adv}$ | 0.461 m yr${}^{-1}$ | 0.387 m yr${}^{-1}$ | −0.198 m yr${}^{-1}$ | −0.135 m yr${}^{-1}$ |

${V}_{b,slope}$ | 0.598 m yr${}^{-1}$ | 0.755 m yr${}^{-1}$ | 0.447 m yr${}^{-1}$ | 0.844 m yr${}^{-1}$ |

${V}_{b}$ | 1.059 m yr${}^{-1}$ | 1.142 m yr${}^{-1}$ | 0.249 m yr${}^{-1}$ | 0.710 m yr${}^{-1}$ |

${V}_{s}$ | 1.070 m yr${}^{-1}$ | 0.779 m yr${}^{-1}$ | 0.485 m yr${}^{-1}$ | 0.067 m yr${}^{-1}$ |

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**MDPI and ACS Style**

Krabbendam, J.; Nnafie, A.; de Swart, H.; Borsje, B.; Perk, L.
Modelling the Past and Future Evolution of Tidal Sand Waves. *J. Mar. Sci. Eng.* **2021**, *9*, 1071.
https://doi.org/10.3390/jmse9101071

**AMA Style**

Krabbendam J, Nnafie A, de Swart H, Borsje B, Perk L.
Modelling the Past and Future Evolution of Tidal Sand Waves. *Journal of Marine Science and Engineering*. 2021; 9(10):1071.
https://doi.org/10.3390/jmse9101071

**Chicago/Turabian Style**

Krabbendam, Janneke, Abdel Nnafie, Huib de Swart, Bas Borsje, and Luitze Perk.
2021. "Modelling the Past and Future Evolution of Tidal Sand Waves" *Journal of Marine Science and Engineering* 9, no. 10: 1071.
https://doi.org/10.3390/jmse9101071