# Parameterization of Time-Averaged Suspended Sediment Concentration in the Nearshore

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{2}≥ 0.72 above the bottom boundary layer. The time-averaged SSC in the inner surf zone showed good agreement near the bed but poor agreement near the water surface, suggesting that there is a different sedimentation mechanism that controls the SSC in the inner surf zone.

## 1. Introduction

^{3}(bed load transport) and u

^{4}(suspended load transport), where u is the cross-shore horizontal velocity. However, in the surf zone, wave breaking turbulence can have a significant effect on the nearshore hydrodynamics and the resulting sediment suspension and transport (e.g., [4]). Wave breaking turbulence generated near the water surface level penetrates downward and can act to destabilize the sea bed, resulting in large clouds of suspended sediments (e.g., [5,6,7]). The turbulence due to wave breaking also maintains the sediment in suspension and modifies the mean flow field that transports sediments seaward or shoreward [8].

## 2. Observations

^{3}of natural beach sand (d

_{50}= 0.22 mm) in the wave flume. Because details of the experimental conditions and procedures have been presented elsewhere [15,20], only a brief description is given here.

#### 2.1. Experimental Procedures for Erosive and Accretive Morphological Changes

_{s}, observed at the seaward side of the bar crest ranged 51.8 < H

_{s}< 59.1 cm, and the peak periods, T

_{p}, ranged 3.97 < T

_{p}< 4.02 s. The significant wave height was calculated with ${H}_{s}=4.004\sqrt{{m}_{0}}$, where ${m}_{0}$ is the zeroth moment of a wave spectrum. In approximately 90 min, the beach was in quasi-equilibrium, and the bar shape (the ratio of bar height to width) was relatively constant. After that time, the instrument array was moved to five cross-shore locations for runs E7 to E11. Under the accretive beach conditions, the bar trough was filled under a new irregular wave time series. The significant wave heights observed at the bar crest ranged 43.7 < H

_{s}< 46.0 cm, and the peak periods ranged 7.23 < T

_{p}< 7.28 s. In approximately 90 min, the beach reached a quasi-equilibrium condition, and the array was moved to five cross-shore locations for runs A5 to A9. We classified the measurement locations into three categories: bar crest, bar trough, and inner surf zone locations. The bar crest locations were E11, A1, A3, and A5. The bar trough locations were E1–E6, E10, A2, A4, and A6. The inner surf zone locations were E7–E9, A7, and A8. A9 located outside surf zone was not considered here.

**Figure 1.**Bathymetry and cross-shore locations of the instrument array: (

**a**) erosive and unstable (EU); (

**b**) erosive and quasi-equilibrium (EQ); (

**c**) accretive and unstable (AU); and (

**d**) accretive and quasi-equilibrium (AQ). The vertical dashed lines represent the locations of the instrument array [15].

#### 2.2. Velocity and Turbulence Extraction

#### 2.3. Sediment Concentration

#### 2.4. Cross-Shore Variations of Hydrodynamics

**Figure 2.**Cross-shore variation of significant wave height (dots), significant crest and trough levels (horizontal dashed lines), vertical profile of mean current (solid blue lines) and root mean square velocity (dotted red lines) at five transects for erosive and quasi-equilibrium cases. Runs for unstable stages are given in parentheses. The coordinate system is x positive onshore and z positive up, with x = 0 m at the wavemaker and z = 0 m at the still water level.

**Figure 3.**Cross-shore variation of significant wave height (dots), significant crest and trough levels (horizontal dashed lines), vertical profile of mean current (solid blue lines) and root mean square velocity (dotted red lines) at five transects for accretive and quasi-equilibrium cases. Runs for unstable stages are given in parentheses. The coordinate system is x positive onshore and z positive up, with x = 0 m at the wavemaker and z = 0 m at the still water level.

## 3. Vertical Variation of Turbulent Kinetic Energy and Sediment Suspended Concentration

**Figure 4.**Vertical variation of time-averaged turbulent kinetic energy per unit mass ($\overline{k}$) and suspended sediment concentration ($\overline{c}$) at bar crest (

**a**,

**d**); bar trough (

**b**,

**e**); and inner surf zone (

**c**) and (

**f**) for erosive (red) and accretive (blue) conditions. Values are normalized by the depth-averaged value from 0 < $\zeta /h$ < 0.5 and are indicated in angle brackets on each panel.

**Figure 5.**Vertical variation of the ratio of the standard deviation (${\mathrm{\sigma}}_{k}$) to time-averaged turbulent kinetic energy per unit mass ($\overline{k}$) at the bar crest (

**a**); bar trough (

**b**) and inner surf zone (

**c**) for erosive (closed dot) and accretive (open dot) conditions. Vertical variation of the same ratio for the suspended sediment concentration (${\mathrm{\sigma}}_{c}$/$\overline{c}$) at the same cross-shore locations (

**d**–

**f**).

## 4. Relation between Turbulent Kinetic Energy and Suspended Sediment Concentration

^{2}(coefficient of determination) values at the characteristic locations of the barred beach, i.e., the bar crest, trough, and inner surf zone.

^{2}= 0.66 considering the complexity of sediment suspension mechanism in the surf zone. When we applied Equation (4) to the case of $\zeta /h$ = 0.478, which is in the middle of water column, the least-square fit with Equation (4) yields B = 180 with scattered data (R

^{2}= 0.04), as shown in Table 1.

**Figure 6.**Relation between time-averaged turbulent kinetic energy ($\overline{k}$) and suspended sediment concentration ($\overline{c}$) over the bar crest for erosive condition E11 (solid markers) and accretive conditions A1, A3, and A5 (open markers), classified by depth: $\zeta /h$ = 0.016 (cyan circles), 0.079 (blue squares), 0.139 (red diamonds), 0.309 (green triangles), and 0.478 (magenta triangles pointing downward). The solid black line indicates the least-square best fit to Equation (5) for B = 230 (R

^{2}= 0.66) and $\delta /h$ = 0.077, where $\delta $ is the bottom boundary layer thickness (~5 cm). The dashed line indicates the best fit to Equation (4) for $\zeta /h$ = 0.478.

ζ (cm) or δ (cm) | $\zeta /h$ or $\delta /h$ | B | R^{2} |
---|---|---|---|

5.0 (δ) | 0.077 | 230 | 0.66 |

31.0 (ζ) | 0.478 | 180 | 0.04 |

^{2}= 0.72), for all of the data at the elevations of $\zeta /h$ = 0.013, 0.066, 0.119, 0.264 and 0.409 (see Table 2). The tuning parameter B, which accounts for the physics of the sediment suspension mechanism besides $\overline{k}$ and $\zeta /h$, appears to be suitable for use in determining the relationship. The fitting results suggest that $\overline{c}$ could be simply parameterized by the turbulent flow using Equation (4) which is similar to the equation for the turbulent dissipation rate. However, there were some discrepancies at $\zeta /h$ = 0.013 and 0.264 when we applied the “global” fit to all of the elevations.

**Figure 7.**Relation between time-averaged turbulent kinetic energy ($\overline{k}$) and suspended sediment concentration ($\overline{c}$) over the bar trough for erosive conditions E1–E6 and E10 (solid markers) and accretive conditions A2, A4, and A6 (open markers) classified by depth: $\zeta /h$ = 0.013 (cyan circles), 0.066 (blue squares), 0.119 (red diamonds), 0.264 (green triangles), and 0.409 (magenta triangles pointing downward). Solid lines indicate the least-square best fit to Equation (4) for B = 280 (R

^{2}= 0.72) for elevations $\zeta /h$ = 0.066, 0.119, 0.264, and 0.409, except for $\zeta /h$ = 0.013. Dashed lines indicates the best fit to Equation (4) for B determined for each elevation.

ζ (cm) | $\zeta /h$ | B | R^{2} |
---|---|---|---|

1.0 | 0.013 | 530 | 0.23 |

5.0 | 0.066 | 250 | 0.69 |

9.0 | 0.119 | 300 | 0.72 |

20.0 | 0.264 | 480 | 0.90 |

31.0 | 0.409 | 270 | 0.77 |

Global (ζ = 5.0–31.0) | 280 | 0.72 |

^{2}≥ 0.72 with B = 270–480 (average: 350) for the elevations $\zeta /h$ ≥ 0.119. This confirms that SSC is closely related to turbulent flow and can be parameterized using Equation (4) if the intensity of breaking wave (or turbulent intensity) is enough large such as at the bar trough. However, even the “local” fit achieved is not good close to the bottom, as seen in the case of $\zeta /h$ = 0.013 where R

^{2}= 0.23. The proposed relationship yielded good agreement only above the wave bottom boundary layer, which may be because another suspension mechanism (e.g., sheet flow) is predominant close to the bottom except near the bar trough.

^{2}values in the range of 0.44 to 0.49. Note that the coefficient of B in Equation (4) is smaller than at the other locations by almost an order of magnitude, because the suspended sediment concentration is lower in the inner surf zone than at the other locations. At elevations greater than 5 cm above the bed, the agreement rapidly decreases and R

^{2}approaches zero.

## 5. Summary and Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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Yoon, H.-D.; Cox, D.; Mori, N.
Parameterization of Time-Averaged Suspended Sediment Concentration in the Nearshore. *Water* **2015**, *7*, 6228-6243.
https://doi.org/10.3390/w7116228

**AMA Style**

Yoon H-D, Cox D, Mori N.
Parameterization of Time-Averaged Suspended Sediment Concentration in the Nearshore. *Water*. 2015; 7(11):6228-6243.
https://doi.org/10.3390/w7116228

**Chicago/Turabian Style**

Yoon, Hyun-Doug, Daniel Cox, and Nobuhito Mori.
2015. "Parameterization of Time-Averaged Suspended Sediment Concentration in the Nearshore" *Water* 7, no. 11: 6228-6243.
https://doi.org/10.3390/w7116228