Formulation of PDF of overtopping volume on the deck

The overtopping volume, $Q$, can be defined as the volume of water column above the deck level, shown schematically in Fig.3. The overtopping volume $Q$ only depends on spatial surface profile, if the deck is thin compared to the incident wave amplitude. Therefore, if the spatial wave profile is known, it is possible to calculate the overtopping volume $Q$ accurately. However, there is some difficulty to estimate the spatial profile of random waves. Thus, following Mori and Cox (2003), the overtopping volume $Q$ is calculated by the following assumptions and procedure. First, the profile of the individual wave crest is assumed to be a sinusoidal wave locally. Then, if the deck level $z=z_c$ is given and the crest amplitude $A$ is known, the volume of water column above deck level, $Q$, is simply calculated by

$$Q(A) = \frac{2A}{k}\sqrt{1-\left(\frac{z_c}{A}\right)^2}$$ (12)

where $k$ is the local wave number and $A$ is local amplitude of the wave. Eq.(12) can be roughly simplified for $z_c/A \ll 1$ as

$$Q(A) \simeq \left\{ \begin{array}{l@{\ :\ }l} \display... ..._c)}{k} } & A \ge z_c \\ \ 0 & A < z_c \end{array} \right.$$ (13)

Fig.6 shows an example of Eq.(12) and (13) for the cases of $z_c/A_{max}$ = 0.1 and 0.5. The transformation of the wave amplitude $A$ into overtopping volume $Q$ by the approximate Eq.(13) becomes worse as the height of the deck level $z_c$ increases. The overtopping volume $Q$ decreases as $z_c$ increases as expected. Obviously, the assumptions of the local sinusoidal wave crest profile and local wave number are only valid for narrow banded spectra. The overtopping volume $Q$ can be normalized by the rms value of surface elevation $\eta_{rms}$ and wavenumber $k$ as

$$Q' = \frac{kQ}{\eta_{rms}} = \frac{2(A-z_c)}{\eta_{rms}}$$ (14)

Figure 6: Relationship between wave crest amplitude $A$ and overtopping volume $Q$ for the cases of $z_c=0.1 A_{max}$ and $0.5 A_{max}$.

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Figure 7: Experimental results of the relationship between zero-crossing wave height $H$ and wave period $T$.

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Figure 8: Model/data comparison of relationship between normalized wave crest amplitude $A'$ and overtopping volume $Q'$ with experimental results (solid circle), exact solution: Eq.(12) (dash) and approximated solution: Eq.(13) (solid line) for (a) Case 1 and (b) Case 2.

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(a) Case 1.

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(b) Case 2.

Before formulate the PDF of overtopping volume model, it is necessary to check some of the basic assumptions used though Eq.(12) to Eq.(14). The local wave number $k$ was regarded as a constant above formulation. Empirically, the assumptions of a locally sinusoidal wave crest profile and local wave number are applicable for narrow banded spectra. Evidently from Eq.(13), the wavelength or period is important in estimating the overtopping for extreme waves. Generally, the relationship between wave period and wave height for random waves has a linear trend for small value of wave height, and the wave period becomes relatively constant for large amplitude waves. Therefore, the constant local wave number assumption is more crude for small amplitude waves but it is reasonably acceptable for large amplitude waves (e.g. Goda, 1985). Fig.7 shows experimental results of the relationship between $H$ and $T$. For large wave height ( $H/\eta_{rms}\ge3$ roughly), $T$ can be regarded as approximately constant, $T \simeq T_{1/3}$. Thus, the theory of overtopping is more accurate for larger amplitude waves and it is less valid for small amplitude waves. In this study, local wave number is assumed constant and $k$ is regarded as $k \simeq k_{1/3}$ calculated by the linear dispersion relationship with $T_{1/3}$. Of course, the assumption of $k \simeq k_{1/3}$ is not a necessary condition for the assumption of sinusoidal wave profile.

To check the local sinusoidal wave profile assumption, Fig.8 compares the measured relationship between normalized wave crest amplitudes $A'$ and overtopping volume $Q'$ with exact solution, Eq.(12), and approximated solution, Eq.(13). The approximated solution, Eq.(13), gives better agreement with the measured data than the exact solution, Eq.(13). It means that the spatial profile of mechanically generated waves in the flume was slimmer than sinusoidal wave. Eq.(13) shows better agreement with high deck level case (Case 2) than low deck level case (Case 1). The correlation coefficients between Eq.(13) and the experimental results are 0.75 for Case 1 and 0.85 for Case 2. As a result, the assumptions of the local sinusoidal wave profile and constant wave number are acceptable for large amplitude waves, especially relatively high deck level condition.

From Eq.(14), the inverse relationship between $Q$ and $A$ is given by

$$A' = \frac{1}{2}Q' + z_c'$$ (15)

The PDF of the wave height or wave crest distribution was assumed as the Weibull distribution by Eq.(9), and the overtopping volume $Q$ per wave was assumed to follow Eq.(15). Substituting Eq.(15) into Eq.(9) gives the PDF of overtopping as a function of the normalized overtopping volume $Q'$

$$p_Q(Q',z_c')dQ' = \frac{\alpha}{4} \left( \frac{1}{2}Q'... ...Q' + z_c'\right)^{\alpha} - z_c'^\alpha \right] \right\}dQ'$$ (16)

Integrating Eq.(16) between $[0, \infty)$, we have the exceedance probability of $Q$ on the deck given by

$$P_Q(Q',z_c') = \exp\left\{ -\frac{1}{2} \left[ \left(\... ...{2}Q' + z_c'\right)^{\alpha} - z_c'^\alpha \right] \right\}$$ (17)

Eq.(16) and (17) represent the overtopping distribution based on a modified form of the Weibull wave amplitude distribution and is simply referred to as the ``modified Weibull overtopping distribution" hereafter for clarity. The truncated Weibull distribution for wave crest amplitude $A$ by Eq.(9) is normalized by the rms value of $A$, and the normalized wave amplitude $A'$ has a linear relationship with the overtopping volume $Q'$ by Eq.(14). As a result, the overtopping volume $Q$ in Eq.(16) and (17) is normalized by the rms value of $Q$. It is worth noting that the number of statistical events (overtopping events) depends on the deck level due to the transformation of variable from $A$ to $Q$ using Eq.(15). Physically, this means that the probability of overtopping volume $Q$ has zero value below $A<z_c$. Therefore, number of overtopping events is different from number of incident wave events.

Thus, the PDF of the modified Weibull overtopping distribution has a similar profile compared to the Rayleigh distribution for wave amplitude. This implies that wave nonlinearities (or spectrum band width) directly affects the overtopping volume via $\alpha$ in Eq.(16) to (17).

For the special case of $\alpha$=2 and $\beta$=1/2 which is the Rayleigh condition for the wave amplitude distribution, Eq.(16) and (17) are simplified as

$$p_Q^R(Q',z_c')dQ = \frac{1}{4} \left( Q' + 2z_c' \right) \exp\left[ - \frac{1}{8} Q'(Q' + 4z_c') \right] dQ'$$ (18)

$$P_Q^R(Q',z_c') = \exp\left[ - \frac{1}{8} Q'(Q' + 4z_c') \right]$$ (19)

Following the same manner, Eq.(18) and (19) represent the overtopping distribution based on a modified form of the Rayleigh wave amplitude distribution and is simply referred to as the ``modified Rayleigh overtopping distribution" hereafter for clarity. Having developed a theoretical framework for the statistical modeling of the overtopping via Eq.(16) to (19), the qualitative tendency of the distributions can now be explained.

Figure 9: Dependence of the exceedance probability of overtopping volume $Q'$ on $\alpha$ with $z_c'$=1.0.

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Figure 10: Dependence of exceedance probability of overtopping volume $Q'$ on deck level $z_c'$ with $\alpha$=2.0.

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Fig.9 shows the sensitivity of the exceedance probability of the modified Weibull overtopping distribution on $\alpha$ for the case of $z_c'$=1.0. With a change in $\alpha$ from $\alpha$=1.5 to $\alpha$=2, the exceedance probability of $Q$ decreases one order of magnitude at $Q' \simeq 4.3$ and two orders of magnitude at $Q' \simeq 6.4$. Fig.10 shows the sensitivity of the exceedance probability of $Q$ to the deck level $z_c$ for the case of $\alpha$=2.0. With a change of $z_c'$ from $z_c'$=0 to $z_c'$=2, the exceedance probability decreases one order of magnitude at $Q'\simeq 2.3$ and two orders of magnitude at $Q'\simeq 4.6$.

Figure 11: Model/data comparison of PDF of wave overtopping volume $Q$ for (a) Case 1 and (b) Case 2 with experimental data (solid), modified Rayleigh overtopping distribution (dash), and modified Weibull overtopping distribution (chain).

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(a) Case 1.

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(b) Case 2.

Figure 12: Model/data comparison of exceedance probability of wave overtopping volume $Q$ for (a) Case 1 and (b) Case 2 with experimental data (solid), modified Rayleigh overtopping distribution (dash), and modified Weibull overtopping distribution (chain).

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(a) Case 1.

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(b) Case 2.

Fig.11 compares the PDF of the normalized wave overtopping for the experimental results with the modified Rayleigh overtopping distribution Eq.(18), and the modified Weibull overtopping distribution Eq.(16). The modified Weibull overtopping distribution shows qualitatively better agreement with the experimental data than the modified Rayleigh overtopping distribution, particularly for larger values of $Q$. Fig.12 compares the exceedance probability of wave overtopping for the experimental results with the modified Rayleigh overtopping distribution Eq.(19) and the modified Weibull overtopping distribution Eq.(17). The modified Rayleigh overtopping distribution shows fair agreement with the experimental data for small values of $Q$, and the modified Weibull overtopping distribution shows particularly good agreement with the experimental data for larger values of $Q$. This is reasonable result because the main purpose to use the Weibull distribution is to estimate accurately large wave events. However, the number of overtopping event is insufficient to compare quantitatively, but the qualitative trends of the overtopping volume can be estimated by Eq.(16) and (17).