### 1. Introduction

^{−4}and a −6% to +7% difference at the POE level of 10

^{−5}.

### 2. Numerical Wave Model

### 2.1 High Order Spectral Method

*L*in the

*x*direction and a depth

*h*in

*z*direction. The sea bottom was assumed to be flat. The mean free surface was located at

*z*= 0, and the

*z*-axis was orienting upward. The perfect fluid and irrotational flow were assumed to introduce the velocity potential

*Φ*. The velocity potential satisfied the Laplace’s equation in the fluid domain

*D*as given in Eq. (1). The impermissible condition given in Eq. (2) was imposed on the sea bottom at

*z*=−

*h*. The periodic boundary condition shown in Eq. (3) is imposed at the lateral boundary.

*z*=

*η*(

*x*,

*t*), two free surface boundary conditions were imposed. The dynamic and kinematic free surface boundary conditions are given in Eqs. (4) and (5), respectively.

*Φ̃*(

*x*;

*t*)=

*Φ*(

*x*,

*z*=

*η*(

*x*);

*t*) is the free surface velocity potential, and

*w*(

*x*)=

*∂Φ*/

*∂z*is the vertical velocity at the free surface

*z*=

*η*(

*x*,

*t*). The periodic boundary condition enables the free surface velocity potential and wave elevations with modal functions to be expressed as Eqs. (6) and (7).

*B*(

_{n}*t*) are modal amplitudes of free surface potential and wave elevations, respectively.

*N*is the number of modes.

_{x}*k*=

_{n}*nπ*/

*L*is a pseudo wavenumber adjusted to the domain length. The velocity potential defined in the fluid domain and the vertical fluid velocity are expressed with the perturbation series as follows: with where the superscript

*q*

^{(}

^{m}^{)}represents a quantity of

*O*(

*∊*),

^{m}*M*is the nonlinear order of the HOS method, where

*A*

_{n}^{(}

^{m}^{)}(

*t*) is the modal amplitude of velocity potential of corresponding order, and the vertical fluid velocity can be given in

*z*=

*η*were evaluated by applying the Taylor series expansion as given in Eqs. (4) and (5), respectively.

### 3. Numerical Set-Up

### 3.1 Wave Condition

*ω*= 2

_{p}*π*/

*T*is the peak angular frequency, and

_{p}*H*is the significant wave height of each sea state. Fig. 1 presents the considered wave spectra for different sea states.

_{s}### 3.2 HOS Computational Setup and Computational Parameters

*L*= 15

_{x}*λ*in the horizontal direction, and the deep water condition was assumed. Two computational parameters in the HOS method are important. The first was the order of nonlinearity, e.g., HOS order (

_{p}*M*), and the second was the number of HOS modes (

*N*) corresponding to the number of spatial discretizations. Four HOS orders and three levels of HOS modes were adopted, as summarized in Table 2. Each HOS simulation was conducted for the three-hours, and 100 simulations with different random seeds were performed for each simulation setup. The HOS waves were measured at

_{x}*L*= 5

_{x}*λ*. For example, Fig. 2 shows a HOS wave time series of sea state 7 with the HOS order

_{p}*M*= 4 and the HOS modes

*N*= 512.

_{x}### 4. Results and Discussions

### 4.1 Spectral Analysis

*T*. The averaged wave spectrum can be taken from 100 wave spectra and compared with the target wave spectrum. Fig. 3 gives an example of normalized wave spectra of sea state 7. The solid black line is a reference wave spectrum, and each colored marked lines are the wave spectrum with different combinations of HOS parameters. The wave spectra with HOS modes

_{p}*N*= 128 clearly showed an overestimated spectrum compared to the target spectrum. While with other computational setups, the difference between each setup was minimal, and the difference between each spectrum quantitatively is challenging.

_{x}*f*∈[0.75

*f*; 1.5

_{p}*f*], where

_{p}*f*is the peak frequency of the target sea state. Note that this spectral qualification criterion was adopted from previous studies (Canard et al., 2020; Canard et al., 2022). The red-colored region presents the target tolerance zone. First, the wave spectra obtained with

_{p}*N*= 128 were overestimated compared with the target spectra for all sea states regardless of the HOS order. For all combinations of the HOS parameters except for the HOS modes

_{x}*N*= 128, the difference ratio of wave spectrum lie in the ±5% tolerance within the target frequency range. A comparison of the difference ratio with respect to the HOS order revealed a relatively large variation as the HOS order increases.

_{x}*N*= 128 have maximum and average absolute difference ratios larger than 5%. The difference ratio decreased as the number of modes increased. In addition, the difference ratio tends to increase as the HOS order increases, but the increment is relatively small.

_{x}*N*= 512 and HOS orders

_{x}*M*= 1. The largest maximum difference ratio with HOS modes

*N*= 256 was 4.00% (sea state 4,

_{x}*N*= 256,

_{x}*M*= 4), and the largest maximum difference ratio with HOS modes

*N*= 512was 3.82% (sea state 5,

_{x}*N*= 512,

_{x}*M*= 3). The largest average absolute difference ratio with HOS modes

*N*= 256 was 1.58% (sea state 5,

_{x}*N*= 256,

_{x}*M*= 3), and the largest average absolute difference ratio with HOS modes

*N*= 512 was 1.58% (sea state 5,

_{x}*N*= 512,

_{x}*M*= 3). The average absolute difference ratio with HOS modes

*N*= 256 and

_{x}*N*= 512 was suppressed by under 2%. In addition, approximately 1% of the average absolute difference ratio is measured with HOS mode

_{x}*N*= 512 and HOS order

_{x}*M*= 2. Therefore from the spectral analysis, the HOS modes

*N*= 512 is recommended for the HOS wave generation.

_{x}### 4.2 Wave Crest Probability Distribution Analysis

*η*). In Eq. (20)

_{c}*U*is the Ursell number,

_{r}*S*

_{1}is the steepness parameter,

*k*

_{1}is the wavenumber,

*T*

_{1}is the mean wave period, and

*d*is the water depth. The Huang PDSR and PDER coefficients in Eq. (19) are summarized in Table 4.

*N*= 128, the change of HOS order did not influence the mean PDSR curves significantly, and the mean PDSR curves mostly follow the Rayleigh distribution regardless of the wave condition. With HOS modes

_{x}*N*= 256 and

_{x}*N*= 512, a meaningful difference was measured with the change in the HOS order. When the first-order (

_{x}*M*= 1) was applied, the mean PDSR curves followed the Rayleigh distribution regardless of the number of HOS modes. When the second-order (

*M*= 2) was used, the mean PDSR curves followed the Forristall distribution. This trend was presented more distinctly for the higher sea states.

*M*= 3) and the fourth-order (

*M*= 4) HOS simulations with HOS modes

*N*= 256 and

_{x}*N*= 512 were compared for sea states 5 and 7. The mean PDSR curves mostly lied between the Forristall distribution and the Huang mean distribution. The mean PDSR curve showed a significant increase as the number of HOS mode increased. Moreover, the variations of the PDSR markers from the mean PDSR curve has changed as the number of HOS mode increased. For example, in the case ‘sea state 7/

_{x}*M*= 4/

*N*= 256’ and ‘sea state 7/

_{x}*M*= 4/

*N*= 512’, the number of PDSR markers under the Huang 99% lower bound decreased, and the number of PDSR markers close to the Huang 99% upper bound increased. This result suggests that HOS mode

_{x}*N*= 512 is required to generate extreme wave events. One reason for this variation of PDSR markers is the increase in the Nyquist frequency and the increase in HOS mode. The increase in Nyquist frequency leads to more proper modeling of high-frequency wave components and the nonlinearity of waves.

_{x}^{−2}and compares them with the Huang bounds. Table 5 lists the occurrence rate of the measured wave crest height POE not being inside the Huang bounds for POE lower than 10

^{−2}for each sea state and each HOS computational parameter. The computational setup with an occurrence rate higher than 10% is colored red, higher than 5% is colored yellow, and lower than 5% is colored green (Table 5).

*M*= 1) HOS simulations were lower than the Huang 99% lower bound. The occurrence rate gradually decreases as the HOS order increases from

*M*= 1 to

*M*= 3. While the difference between

*M*= 3 and

*M*= 4 was minimal. Similarly, the occurrence rate decreases as the number of HOS modes increases. The lowest occurrence rate for each sea state was commonly found with HOS setup

*M*= 3/

*N*= 512 or

_{x}*M*= 4/

*N*= 512, and those values were retained under 2%.

_{x}^{−5}.

*N*= 256. The third- and fourth-order PDER results usually lie between the Forristall distribution and the Huang PDER distribution, while the difference between the two HOS orders is minimal. The

_{x}*M*= 3/

*N*= 512 and

_{x}*M*= 4/

*N*= 512 setups showed the best compromise among all HOS computational setups with the Huang ensemble distribution. Within the POE level between 10

_{x}^{−2}and 10

^{−4}, the HOS PDER curves follow the −5% bound of Huang ensemble distribution. Within the POE level between 10

^{−4}and 10

^{−6}, HOS PDER curves show overestimated wave crest probability compared to the Huang ensemble distribution. On the other hand, the number of realizations is insufficient to conclude the PDER curve’s accuracy at the POE level of 10

^{−5}.

### 5. Conclusions

*N*) and four levels of HOS orders (

_{x}*M*), twelve combinations of HOS computational parameters were applied to the HOS simulations to find the recommended HOS parameters for irregular wave generation. Therefore, 100 three-hour realizations of HOS waves were made for five sea states, and the quality waves were verified by following qualification procedures.

The results with HOS modes

*N*= 128 show overestimated wave spectrum and underestimated wave crest height regardless of the HOS order. Consequently, the HOS mode was larger than_{x}*N*= 128 is required for the current HOS computational setup, regardless of the sea state._{x}The minimum spectral difference (averaged absolute difference ratio) was found when the first order was applied. The spectral difference increases as the HOS order increases. On the other hand, regardless of the HOS order and the sea state, the maximum difference ratio was maintained below 5%, and the averaged absolute difference ratio was kept below 2% with HOS modes

*N*= 256 and_{x}*N*= 512._{x}The wave crest height PDSR and PDER for different combinations of HOS parameters were compared with the reference distributions.

When the first-order HOS order was used, the wave crest height was significantly underestimated even for sea states 3 and 4. This underestimation clearly showed that it is important to check both the wave spectrum and the wave crest height POE for the irregular wave analysis.

Comparing the PDSR of the second-, third-, and fourth-order HOS simulation with the Hunag 99% bounds, the HOS wave crest generated using two combinations of HOS parameters

*M*= 3/*N*= 512 and_{x}*M*= 4/*N*= 512were considered adequate for all sea states. For sea state 3, other HOS setups, such as_{x}*M*= 2/*N*= 256 or_{x}*M*= 2/*N*= 512 were considered appropriate._{x}

The HOS setups,

*M*= 3/*N*= 512 and_{x}*M*= 4/*N*= 512, satisfy the ±5% tolerance in the spectral analysis. The amount of nonlinearity of waves measured using the two HOS setups was verified by comparing the wave crest height POE with the Huang PDSR and PDER distributions. The difference in wave properties between the two HOS setups was small. In conclusion, considering that the HOS domain length is 15_{x}*λ*, the suggested HOS computational parameter would be_{p}*λ*/_{p}*Δx*≧ 35 with*M*= 3.