Effect of Sea Level Rise and Offshore Wave Height Change on Nearshore Waves and Coastal Structures

Abstract

In 1994, Townend proposed a method to calculate the relative changes in various wave characteristics and structure-related parameters due to sea level rise for regular waves. The method was extended to irregular waves by Cheon and Suh in 2016. In this study, this method is further extended to include the effect of future change in offshore wave height and the sea level rise. The relative changes in wavelength, refraction coefficient, shoaling coefficient, and wave height in nearshore area are presented as functions of the relative changes in water depth and offshore wave height. The calculated relative changes in wave characteristics are then used to estimate the effect of sea level rise and offshore wave height change on coastal structures by calculating the relative changes in wave run-up height, overtopping discharge, crest freeboard, and armor weight of the structures. The relative changes in wave characteristics and structure-related parameters are all expressed as a function of the relative water depth for various combinations of the relative changes in water depth and offshore wave height.

Appendix

Worked example for calculation of relative change in wave height

An example is presented for the calculation of the relative change in wave height in several different water depths. We assume m = 1/50 and T = 10 s so that H₀ = (T/3.85)² = 6.75 m, L₀ = gT²/(2π) = 156 m, C₀ = gT/(2π) = 15.6 m/s, ${\beta}_0=0.028{S_0}^{-0.38}{e}^{20{m}^{1.5}}=0.098$, β₁ = 0.52e^4.2m = 0.566, and β_max = max {0.92, 0.32S₀^−0.29e^2.4m} = 0.92. Both the sea level rise and the offshore wave height increase are assumed to be 0.5 m so that H₀^′ = 7.25 m, h₀ = H₀^′/H₀ = 1.074, T^′ = 3.85/H₀^′0.5 = 10.37 s, L₀^′ = gT^′2/(2π) = 167.76 m, and C₀^′ = 16.18 m/s.

• 1.
  $$ D=2\ \mathrm{m} $$

• 1) D^′ = 2.5 m; d = D^′/D = 1.25; D/L₀ = 0.0128; D^′/L₀^′ = 0.0149

• 2) $\frac{H_0}{L_0}=0.0432>\frac{\beta_1}{\beta_{\mathrm{max}}-{\beta}_0}\frac{D}{L_0}=8.8\times {10}^{-3}$

∴ Surf zone before sea level rise

• 3) $\frac{{H_0}^{\prime }}{{L_0}^{\prime }}=0.0432>\frac{\beta_1}{\beta_{\mathrm{max}}-{\beta}_0}\frac{D^{\prime }}{{L_0}^{\prime }}=0.0102$

  ∴ Surf zone after sea level rise.

We can confirm from Fig. 7 that the location belongs to the surf zone (indicated by the dotted line) both before and after the sea level rise.

• 4) Using Eq. (24), h = h₀ + (d − h₀)[β₁D/(β₀H₀ + β₁D)] = 1.185

• 2.
  $$ D=10\ \mathrm{m} $$

• 1) D^′ = 10.5 m; d = D^′/D = 1.05; D/L₀ = 0.064; D^′/L₀^′ = 0.0625

• 2) $\frac{H_0}{L_0}=0.0432<\frac{\beta_1}{\beta_{\mathrm{max}}-{\beta}_0}\frac{D}{L_0}=0.044$; L = 92.37 m;

${C}_g=\frac{1}{2}\left[1+\frac{4\uppi D/L}{\sinh \left(4\uppi D/L\right)}\right]\left(\frac{gT}{2\uppi}\tanh \frac{2\uppi D}{L}\right)=8.07\kern0.50em \mathrm{m}/\mathrm{s}$; ${K}_{si}=\sqrt{C_0/\left(2{C}_g\right)}=0.984$;

K_s = K_si + 0.0015(D/L₀)^−2.87(H₀/L₀)^1.27 = 1.657 > β_max = 0.92

∴ Transition zone before sea level rise

• 3)
  $$ \frac{{H_0}^{\prime }}{{L_0}^{\prime }}=0.0432>\frac{\beta_1}{\beta_{\mathrm{max}}-{\beta}_0}\frac{D^{\prime }}{{L_0}^{\prime }}=0.0430 $$

∴ Surf zone after sea level rise.

• 4) Using Eq. (24), h = (β₀h₀ + β₁d(D/L₀)S₀⁻¹)/β_max = 1.071.

• 3.
  $$ D=25\ \mathrm{m} $$

• 1) D^′ = 25.5 m; d = D^′/D = 1.02; D/L₀ = 0.160; D^′/L₀^′ = 0.152; L = 130.38 m; L^′ = 137.94 m; l = L^′/L = 1.058.

• 2) ${C}_g=\frac{1}{2}\left[1+\frac{4\uppi D/L}{\sinh \left(4\uppi D/L\right)}\right]\left(\frac{gT}{2\uppi}\tanh \frac{2\uppi D}{L}\right)=9.37\kern0.50em \mathrm{m}/\mathrm{s};$${K}_{si}=\sqrt{C_0/\left(2{C}_g\right)}=0.913$;K_s = K_si + 0.0015(D/L₀)^−2.87(H₀/L₀)^1.27 = 0.918 < β_max = 0.92 ∴ shoaling zone before sea level rise.

• 3) C_g^′ = 9.71 m/s; K_si^′ = 0.913; K_s = 0.919 < β_max = 0.92.

∴ Shoaling zone after sea level rise.

• 4) ${k}_{si}={\left(\frac{h_0}{l}\frac{1+\frac{4\uppi D/L}{\sinh \left(4\uppi D/L\right)}}{1+\frac{4\uppi D/(lL)}{\sinh \left(4\uppi D/(lL)\right)}}\right)}^{1/2}=1$; Γ = 0.0015K_s⁻¹(D/L₀)^−2.87(H₀/L₀)^1.27 = 0.0058

• 5)
  $$ {k}_s={k}_{si}+\Gamma \left[{d}^{-2.87}{h_0}^{2.87}-{k}_{si}\right]=1 $$

• 6) Using Eq. (24), h = h₀k_s = 1.074.