# The Method for Evaluating Cross-Shore Migration of Sand Bar under the Influence of Nonlinear Waves Transformation

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Data

#### 2.1. Study Site and Morphological Survey Data

#### 2.2. Wave Data

## 3. Results: Sand Bar Migration Assessment Method

#### 3.1. Calculation of Second Wave Harmonic Maximum Position

_{b}) can be calculated by the following Equations [22]:

_{1}, k

_{2}are the first and second harmonic wavenumbers, which can be calculated, for example, using linear dispersion relation:

_{b}are calculated using linear dispersion relation (Equations (5)), then depth of the nearest to the shore maximum as the depth at 1.5 L

_{b}distance from the shoreline is evaluated. Then it is substituted in the mismatch calculation equation, instead of the starting depth value. The process is repeated until the difference in the predicted L

_{b}value between iterations becomes small enough (i.e., 5% of the L

_{b}value, or the criterion to stop the iterative process chosen by the researcher).

#### 3.2. Analysis of the Wave Parameter Series

_{2}) and significant wave height. A histogram is a way to analyze the time series of the wave regimes and to reveal those that affect sand bar migration the most. It shows joint distribution of Hs and distance of the second harmonic maximum location off the shoreline. As the wave characteristics in the reanalysis are available at a 1 h timestep, the number of values that fall in each bin indicates the time of the action of waves with the parameters that lay in the specific bin’s range during the considered period (in our case the period between relief measurements). In this way, the histogram allows determination of where the second harmonic maximum is most often located. Wave height (H) value indicates whether the waves will influence the relief to a greater or lesser degree, as the most repetitive regimes are usually characterized by small wave heights and will not affect the relief as much as higher waves that are not so common. The size of the bins is chosen empirically to be small enough to represent the spatial structure of the wave regime’s distribution (20 bins gives around 5 m spatial resolution) and wave height step 0.2 m. Smaller resolution (more bins with smaller ranges of the considered parameters) will be hard to interpret due to the small number of regimes that will fall into each bin. If, on the other hand, the coarser division is used, the tendencies will be the same, but as the center values of each bin are analyzed, the positions of the most persistent second harmonic maximum locations can be biased.

_{i,j}is the number of counts in the i, j-th bin, H

_{i}is the wave height corresponding to the center of the i-th bin, and d

_{j}is the depth of the second harmonic corresponding to the j-th bin center.

_{b}offshore) and the point with the second harmonic minimum (1 L

_{b}offshore).

## 4. Discussion of Results

#### 4.1. Time Period 1: 24 May–8 November 2019

_{a2}located 100 and 170 m off the shoreline. The sand bar moved onshore only slightly (about 10 m) (Figure 4). This may be due to the fact that the second harmonic maximum point and the corresponding divergence point were located far more shoreward with respect to the sand bar crest location for most of the time (Figure 6). The wave regimes with second harmonic maximum position at 250 m distance off the shoreline (seaward of the sand bar crest) have wave steepness in the range of [0.026, 0.03].

#### 4.2. Time Period 2: 9 November 2019–23 March 2020

#### 4.3. Time Period 3: 24 March–13 November 2020

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Wright, L.D.; Short, A.D. Morphodynamic variability of surf zones and beaches: A synthesis. Mar. Geol.
**1984**, 56, 93–118. [Google Scholar] [CrossRef] - Lippmann, T.C.; Holman, R.A. The spatial and temporal variability of sand bar. J. Geophys. Res.
**1990**, 95, 11575–11590. [Google Scholar] [CrossRef] - Sallenger, A.H.; Holman, R.A.; Birkemeier, W.A. Storm induced response of a nearshorebar system. Mar. Geol.
**1985**, 64, 237–257. [Google Scholar] [CrossRef] - Gallagher, E.L.; Elgar, S.; Guza, R.T. Observations of sand bar evolution on a natural beach. J. Geophys. Res.
**1998**, 103, 3203–3215. [Google Scholar] [CrossRef][Green Version] - Saprykina, Y. The Influence of Wave Nonlinearity on Cross-Shore Sediment Transport in Coastal Zone: Experimental Investigations. Appl. Sci.
**2020**, 10, 4087. [Google Scholar] [CrossRef] - Andreeva, N.; Saprykina, Y.; Valchev, N.; Eftimova, P.; Kuznetsov, S. Influence of Wave Climate on Intra and Inter-Annual Nearshore Bar Dynamics for a Sandy Beach. Geosciences
**2021**, 11, 206. [Google Scholar] [CrossRef] - Van Enckevort, I.M.J.; Ruessink, B.G. Video observations of nearshore bar behaviour. Part 1: Alongshore uniform variability. Cont. Shelf Res.
**2003**, 23, 501–512. [Google Scholar] [CrossRef] - Masselink, G.; Austin, M.; Scott, T.; Poate, T.; Russell, P. Role of wave forcing, storms and NAO in outer bar dynamics on a high-energy, macro-tidal beach. Geomorphology
**2014**, 226, 76–93. [Google Scholar] [CrossRef][Green Version] - Ruessink, B.G.; Pape, L.; Turner, I.L. Daily to interannual cross-shore sandbar migration: Observations from a multiple sandbar system. Cont. Shelf Res.
**2009**, 29, 1663–1677. [Google Scholar] [CrossRef] - Nielsen, P. Selected unanswered challenges in coastal dynamics. In Proceedings of the Coastal Dynamics 2017, Helsingør, Denmark, 12–16 June 2017. [Google Scholar]
- Kuznetsova, O.; Saprykina, Y. Influence of Underwater Bar Location on Cross-Shore Sediment Transport in the Coastal Zone. J. Mar. Sci. Eng.
**2019**, 7, 55. [Google Scholar] [CrossRef][Green Version] - Plant, N.G.; Ruessink, B.G. On cross-shore migration and equilibrium states of nearshore sandbars. J. Geophys. Res.
**2010**, 115, F03008. [Google Scholar] [CrossRef][Green Version] - Pape, L.; Ruessink, B.G.; Wiering, M.A.; Turner, I.L. Recurrent neural network modeling of nearshore sandbar behavior. Neural Netw.
**2007**, 20, 509–518. [Google Scholar] [CrossRef][Green Version] - Roelvink, D.; Reniers, A.; van Dongeren, A.; van Thiel de Vries, J.; McCall, R.; Lescinski, J. Modelling storm impacts on beaches, dunes and barrier islands. Coast. Eng.
**2009**, 56, 1133–1152. [Google Scholar] [CrossRef] - Walstra, D.J.R.; Reniers, A.J.H.M.; Ranasinghe, R.; Roelvink, J.A.; Ruessink, B.G. On bar growth and decay during interannual net offshore migration. Coast. Eng.
**2012**, 60, 190–200. [Google Scholar] [CrossRef] - Roelvink, J.A.; Stive, M.J.F. Bar-generating cross-shore flow mechanisms on a beach. J. Geophys. Res.
**1989**, 94, 4785–4800. [Google Scholar] [CrossRef] - Kuriyama, Y. Process-based one-dimensional model for cyclic longshore bar evolution. Coast. Eng.
**2012**, 62, 48–61. [Google Scholar] [CrossRef] - Rafati, Y.; Hsu, T.J.; Elgar, S.; Raubenheimer, B.; Quataert, E.; van Dongeren, A. Modeling thehydrodynamicsandmorphodynamicsofsandbarmigrationevents. Coast. Eng.
**2021**, 166, 103885. [Google Scholar] [CrossRef] - IFC Documentation—Cy46r1. Part VII: ECMWF Wave Model. European Centre for Medium-Range Weather Forecasts (ECMWF). Available online: https://www.ecmwf.int/en/elibrary/19311-ifs-documentation-cy46r1-part-vii-ecmwf-wave-model (accessed on 26 November 2021).
- Shtremel, M. ERA5 wave data verification with buoy field measurements in the nearshore region of the Baltic Sea. In Proceedings of the 6th IAHR Europe Congress: No Frames No Borders, Warsaw, Poland, 15–18 February 2021; pp. 433–434. [Google Scholar] [CrossRef]
- Boczar-Karakiewicz, B.; Davidson-Arnott, R. Nearshore bar formation by non-linear process—A comparison of model results and field data. Mar. Geol.
**1987**, 77, 287–304. [Google Scholar] [CrossRef] - Kuznetsov, S.; Saprykina, Y. Nonlinear Wave Transformation in Coastal Zone: Free and Bound Waves. Fluids
**2021**, 6, 347. [Google Scholar] [CrossRef] - Saprykina, Y.; Kuznetsov, S.Y.; Andreeva, N.; Shtremel, M. Scenarios of nonlinear wave transfor- mation in coastal zone. Oceanology
**2013**, 53, 422–431. [Google Scholar] [CrossRef] - Salem, A.S.; Jarno-Druaux, A.; Marin, F. Physical modeling of cross-shore beach morphodynamics under waves and tides. J. Coast. Res.
**2011**, 64, 139–143. [Google Scholar] - Saprykina, Y.V.; Shtremel, M.N.; Kuznetsov, S.Y. On the possibility of biphase parametrization for wave transformation in the coastal zone. Oceanology
**2017**, 57, 253–264. [Google Scholar] [CrossRef]

**Figure 1.**Study site, region of the bathymetry measurements (small pink rectangle), ERA5 point and buoy location.

**Figure 2.**Sand bar crest position detected (

**a**) with field measurement (23 May 2019) data (

**b**) in the satellite image (Sentinel-2) captured on 21 May 2019.

**Figure 4.**Bathymetry maps obtained during expeditions in 2019–2020. (

**a**) 23 May 2019, (

**b**) 8 November 2019, (

**c**) 23 March 2020, (

**d**) 13 November 2020.

**Figure 5.**Wave height, peak wave period and ratio between deep water and critical Iribarren number (below the dashed line are regimes with pronounced exchange of energy between wave harmonics). Red circles—wave regimes without pronounced energy exchange between harmonics. Black stars—shore-normal regimes.

**Figure 6.**(

**a**) Bivariate histogram depending on the value of the second harmonic maximum distance off the shoreline weighted by the dimensionless relation between the significant wave height and the depth of the second harmonic maximum; (

**b**) wave impact metric (Equation (6)) in dependence on the distance from the second harmonic maximum to the shoreline.

**Figure 7.**Wave height, peak wave period and ratio between deep water and critical Iribarren number (below the dashed line are regimes with pronounced exchange of energy between wave harmonics). Red circles—wave regimes without pronounced energy exchange between harmonics. Black stars—shore-normal regimes.

**Figure 8.**(

**a**) Bivariate histogram depending on the value of the second harmonic maximum distance off the shoreline weighted by the dimensionless relation between the significant wave height and the depth of the second harmonic maximum; (

**b**) wave impact metric (Equation (6)) dependent on the distance from the second harmonic maximum to the shoreline.

**Figure 9.**Wave height, peak wave period and ratio between deep water and critical Iribarren number (below the dashed line are regimes with pronounced exchange of energy between wave harmonics). Red circles—wave regimes without pronounced energy exchange between harmonics. Black stars—shore-normal regimes.

**Figure 10.**(

**a**) Bivariate histogram depending on the value of the second harmonic maximum distance off the shoreline weighted by the dimensionless relation between the significant wave height and depth of the second harmonic maximum; (

**b**) wave impact metric (Equation (6)) dependent on the distance from the second harmonic maximum to the shoreline.

**Table 1.**Distance from the shoreline to the sand bar crest (location of the sand bar) and incline of the seaward bottom slope of the sand bar.

Date of Measurements | Sand Bar Offshore Distance, m | Bottom Slope |
---|---|---|

23 May 2019 | 210–220 | 0.0185 |

8 November 2019 | 200 | 0.0169 |

23 March 2020 | 170–190 | 0.0161 |

13 November 2020 | 130–180 | 0.0136 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Shtremel, M.; Saprykina, Y.; Ayat, B.
The Method for Evaluating Cross-Shore Migration of Sand Bar under the Influence of Nonlinear Waves Transformation. *Water* **2022**, *14*, 214.
https://doi.org/10.3390/w14020214

**AMA Style**

Shtremel M, Saprykina Y, Ayat B.
The Method for Evaluating Cross-Shore Migration of Sand Bar under the Influence of Nonlinear Waves Transformation. *Water*. 2022; 14(2):214.
https://doi.org/10.3390/w14020214

**Chicago/Turabian Style**

Shtremel, Margarita, Yana Saprykina, and Berna Ayat.
2022. "The Method for Evaluating Cross-Shore Migration of Sand Bar under the Influence of Nonlinear Waves Transformation" *Water* 14, no. 2: 214.
https://doi.org/10.3390/w14020214