Statistics of long-crested extreme waves in single and mixed sea states

Abstract

Most of the studies on extreme waves are focused on the systems with single-peak wave spectra. However, according to the statistics of occurrence, the bimodal spectral system is also frequent in real sea conditions. In order to summarize the statistics of extreme waves, irregular wave trains under single-peak and bimodal spectra for long durations are simulated in this paper, based on a two-dimensional High Order Spectral (HOS) numerical wave tank. A large number of configurations have been tested under unimodal and bimodal spectra. The investigation on the wave trains under single-peak spectrum indicates that although in conditions often referred as deep water (k_ph > π), the relative water depth has a significant influence on the probabilities of occurrence of extreme waves. A detailed analysis of the combined effect of Benjamin-Feir Index (BFI) and relative water depth is provided. However, the situation is more complex in real sea conditions, which may exhibit multimodal spectra. We focus in this study on long-crested bimodal spectra characterized by the same significant wave height H_s and mean zero-crossing period T_z of the sea states as the single-peak spectrum. The wave conditions under bimodal spectrum present milder extreme wave statistics than those under single-peak spectrum. In addition, mixed ocean systems with equivalent energy distribution (i.e., Sea-Swell Energy Ratio (SSER) is close to 1.0) and larger separation between partitions (i.e., Intermodal Distance (ID) > 0.10) are the less prominent to extreme waves appearance. The comparison of the mixed sea states and the corresponding single independent systems demonstrates that the complexity of the underlying physics of a given sea state (for instance the presence of modulational instability or other nonlinear process) cannot be deduced by an analysis limited to the statistical content of the combined sea state. The wave energy being distributed among frequencies plays a major role. Additionally, Gram-Charlier distribution can accurately predict the probability of large waves (1.5 < H/H_s < 2.0) compared to the MER distribution, but it underestimates the statistics of the wave height distribution when H/H_s is larger than 2.0 for both single-peak and bimodal states.

Appendices

Appendix A: Generation of irregular waves

The wave-maker motion is determined by the target free surface elevation and the linear transfer function, which depends on its geometry. According to the linear superposition theory, the two-dimensional irregular free surface elevation can be represented as a superposition of regular wave components with different frequencies:

$$ \eta \left(x,t\right)=\sum \limits_{i=1}^{N_f}{a}_i\cos \left({k}_ix-{\omega}_it+{\varphi}_i\right) $$

(17)

where the subscript i stands for the ith wave component; N_f is the total number of wave components; a_i, k_i, and ω_i are the wave amplitude, wave number, and wave frequency of each component; and φ_i is the initial random phase, which is uniformly distributed from 0 to 2π. The value of k_i can be obtained through the dispersion equation:

$$ {\omega_i}^2={gk}_i\mathrm{th}{k}_ih $$

(18)

with g the gravity acceleration. The wave amplitude of each component a_i is defined from the wave spectrum:

$$ {a}_i=\sqrt{2S\left({\omega}_i\right)\Delta {\omega}_i} $$

(19)

where Δω_i is the division widths of the frequencies, Δω_i = 2π(f_H−f_L)/(N_f−1). (f_L, f_H) is the considered frequency range. To avoid the periodic time repetition, the actual frequency of the ith wave component ω_i is selected randomly in the frequency spin as

$$ {\omega}_i=\left(i-1+\mathit{\operatorname{rand}}\right)\Delta {\omega}_i $$

(20)

where rand is a random number distributed from 0 to 1.

Appendix B: Convergence analysis

The data from the irregular wave experiment under single-peak spectrum by Li et al. (2013) are used to conduct the convergence analysis of the established numerical model. The experiment was carried out in the State Key Laboratory of Coastal and Offshore Engineering in Dalian University of Technology, China. The wave flume is 69.0 m long, 2.0 m wide, and 1.8 m deep. The water depth for the experiment is set at 1.2 m. Eleven resistive wave gauges are arranged along the length of the wave flume to record the free surface elevations. The experimental setup is displayed in Fig. 15.

Fig. 15

The setup of the experiment

Irregular waves are characterized by the JONSWAP spectrum with random initial phases. Details of the experimental parameters are listed in Table 5. Two cases with different BFI values of 0.51 and 0.87, which represent different stabilities of the wave group, have been chosen by varying the peak period T_p and the peak enhancement factor γ in the JONSWAP spectrum. BFI is defined as deep-water BFI$=\frac{\sqrt{2}\varepsilon }{2\Delta f/{f}_p}$, with ε = k_pH_s/2 the wave steepness and Δf/f_p the relative spectral bandwidth. For the two cases, the significant wave height H_s is the same. However, case II has a deeper relative water depth k_ph, larger wave steepness ε, and larger BFI value, which represents a stronger nonlinear wave condition than case I. Thus, the convergence analysis is performed with the irregular wave trains for case II.

Table 5 Parameters of the physical experiment

The numerical wave tank replicas the experimental setup. Its length is enlarged to 80 m to ensure correct wave absorption. The error is measured thanks to the free surface elevation recorded at the middle of the wave tank on a time window fixed to 100T_p and calculated as:

$$ \mathrm{error}=\frac{\int_t\left|{\eta}_{\mathrm{numerical}}-{\eta}_{\exp \mathrm{eriment}}\right|}{\int_t\left|{\eta}_{\exp \mathrm{eriment}}\right|} $$

(21)

The convergence analysis is carried out with the parameters N_z = N_x/4, M = 5, and the result of the convergence analysis with respect to the number of points per peak wavelength N_Lp is represented in Fig. 16. The figure displays two lines representing the convergence rates of 2 and 3 as a reference. The numerical convergence rate is slightly larger than 2nd order, which is the theoretical expected value (Ducrozet et al. 2012). Considering the numerical simulation of the unidirectional irregular wave trains, the overall error is about 5% with 30 points per corresponding peak wavelength. This value of N_Lp = 30 is chosen as converged parameter for the rest of the study. It ensures the accuracy of the numerical simulation as well as a fast solution. Similar convergence analysis has been conducted for the discretization in time (time-step Δt) as well as the order of nonlinearity of the method M. The final numerical configuration is N_Lp = 30, T_p/Δt = 100, and M = 5.

Fig. 16

Convergence of the numerical simulation with respect to the number of points per corresponding peak wavelength

The comparison of the input target spectrum and numerically generated one at x = 5 m is exhibited in Fig. 17. The comparison ensures the correct wave generation procedure. In addition, the results obtained with a linear solver (M = 1) is added as a reference. It can be observed that the generated spectrum with M = 1 is exactly the target spectrum, which demonstrates the effectiveness of the established numerical model in dealing with random wave trains. Consequently, the discrepancies between the input target spectrum and the generated one in a nonlinear context (M = 5) are only due to the nonlinear effects in the process of the wave propagation, the generation of waves being strictly equivalent.

Fig. 17

Comparison of amplitude spectra input target and numerically generated at x = 5 m

Appendix C: Numerical validation

Complementary to the previous convergence analysis, which demonstrated the accuracy of the HOS numerical wave tank compared to the experiments on the free surface elevation at a single location, this paragraph assesses its relevance for the systematic study presented in Sections 3 and 4. The experimental data listed in Table 5 are further used to validate the accuracy of the established numerical model for long-time simulation and statistical characteristics.

The total simulation duration is 2000 s, i.e., more than 1000 waves are involved in the simulated wave trains. 2D HOS numerical model is adopted to reproduce the two wave trains. Figure 18 compares the free surface elevations between the experimental data and numerical results at some pre-setting locations along the wave tank. The horizontal axis represents the time corrected from the group velocity at the peak of the spectrum c_g. This allows to follow the wave groups in their evolution and possibly identify the large wave events in each case. It can be noted that all the free surface elevations of the numerical simulation remain consistent with the experimental data along the wave tank, irrespective of the value of BFI.

Fig. 18

The comparisons of the free surface elevations between experimental data and numerical results along the wave tank (the black solid lines represent the experimental data, and red solid lines represent the numerical results). a Case I. b Case II

The evolution of the kurtosis along the wave flume for both cases with different BFI values is presented in Fig. 19, where experiments and numerical results are compared. The horizontal axis represents the distance away from the wave-maker normalized by the corresponding peak wavelength. Complementary, the probability distribution of the wave height at different positions of the numerical waves is compared to the experimental data in Fig. 20, in which Naess distribution are plotted as a reference. It can be found that the kurtosis and the wave height distribution of the numerical results have good agreement with the experimental data, even for case II that exhibits the largest waves. These comparisons in Figs. 18, 19, and 20 demonstrate that the established HOS numerical model is able to accurately simulate the irregular wave trains for a long time, even in the presence of significant modulation instability. Furthermore, in Fig. 20, the wave height distributions show different behaviors for the two cases at different locations. This is an awaited behavior from literature: lower k_ph and lower BFI induce weaker modulation instability (Janssen 2003; Janssen and Bidlot 2009; Fedele 2015) and consequently smaller occurrence of extreme waves (characterized by a smaller kurtosis). We are consequently confident in the accuracy of the numerical model to study this phenomenon.

Fig. 19

The comparisons of kurtosis between experimental data and numerical results along the wave tank (L_p is the wavelength corresponding to the peak frequency). a Case I. b Case II

Fig. 20

The comparisons of the probability distribution of the wave height between experimental data and numerical results at different locations (L_p is the wavelength corresponding to the peak frequency). a Case I. b Case II