A Simplified Consistent Nonlinear Mild-Slope Equation Model for Random Waves Propagation and Dissipation

Abstract

The model for ocean surface wave propagation can be formulated either in the form of deterministic models or stochastic models. The stochastic models appear to be particularly attractive in the global domain due to their computational efficiency. However, in the nearshore region, the phase becomes highly correlated, and the phase information therefore becomes critical. Therefore, a simplified consistent nonlinear mild-slope equation model has been developed in order to take advantage of the deterministic model for handling phase information, as well as the stochastic model for numerical simplicity. We demonstrate the advanced performance of the present model for random waves by comparing it with laboratory data and previous models.

1. Introduction

As ocean surface waves propagate from deep water into shallow coastal areas, they transform due to several physical processes such as refraction, shoaling, nonlinear interactions, and bottom-induced breaking. Wave prediction plays a central role in coastal engineering problems because more accurate wave predictions have a crucial impact on the estimation of sediment transport, sandbar migration, as well as the design of coastal structure and harbors. When the surface waves start to interact with the bottom, wave breaking becomes one of the dominant processes in the shallow water region, leading to energy dissipation. Additionally, changes in the nature of the nonlinear energy exchange between wave components are important for predicting the nearshore wave field.

Quartet (four-wave) interactions are dominant in dispersive deep water; however, the dispersive wave numbers prevent triad interactions from satisfying resonant conditions in deep water. On the contrary, in shallow water, the triad wave-wave interactions, which have an order of $\epsilon$ ($\epsilon =ak$, or wave steepness, where $a$ is the amplitude, $k$ is the wave number) lower than the four wave-wave interactions, become near-resonant due to weakly-dispersive wave numbers (or non-dispersive wave numbers in very shallow water). Consequently, nonlinearity is characterized by the triad interactions in the nearshore area. When two wave components interact via the near-resonance condition of triad wave-wave interactions in shallow water, the difference and sum frequencies between two arbitrary modes lead to transfer energy to lower and higher frequencies, respectively. Among the two distinct interactions, the difference interaction results in the radiation of long-wave motion, which plays a significant role in the modulation of nearshore processes (Janssen et al., 2006). Additionally, energy dissipation through depth-induced wave breaking is as important as the nonlinear effects in the nearshore region (Elgar et al., 1997). Both nonlinearity and wave breaking work in concert to form asymmetric and skewed wave profiles in the surf zone (Kofoed-Hansen and Rasmussen, 1998).

Since Freilich and Guza (1984) (hereafter FG84 in figures and tables) derived a frequency-domain model for nearshore surface waves evolution using the Boussinesq equation (e.g., Peregrine, 1967), phase-resolved models in the frequency domain have been extensively developed to simulate the nonlinear evolution of Fourier amplitudes for free surface elevation. These include Liu et al. (1985), Agnon et al. (1993), Madsen and Sørensen (1993), Kaihatu and Kirby (1995), Tang and Ouellet (1997), Eldeberky and Madsen (1999), Janssen et al. (2006), Toledo (2013), Ardani and Kaihatu (2019), and others.

In this context, Kim and Kaihatu (2021) (referred to as KK21 hereafter) revisited the derivation of the model proposed by Kaihatu and Kirby (1995) (hereafter KK95 in figures and tables) and developed a consistent nonlinear mild-slope equation model in frequency-domain. The model of KK21 takes into consideration a closer correspondence among the order of amplitude (or nonlinearity), modulation scale, and the scaling of horizontal variation in the bottom. By establishing a consistent scaling approach, KK21 introduced several nonlinear terms to the model of Kaihatu and Kirby (1995), allowing for a more comprehensive representation of the triad wave-wave interaction between wave frequency modes. The performance of the KK21 model was confirmed through comparisons with experimental data for a finite number of harmonics and with previous models (i.e., Freilich and Guza, 1984; Liu et al., 1985; Kaihatu and Kirby, 1995; Tang and Ouellet, 1997). For full details, the reader is referred to KK21. Although KK21 brings about notable improvements by incorporating the additional nonlinear terms, the inclusion of such terms can pose computational challenges in modeling random wave propagations. Specifically, an increasing number of frequency components in numerical simulations may lead to numerical divergence issues (Kim, 2022).

As an alternative to the deterministic models, there have been proposals for stochastic evolution equations, which correspond to phase-averaged models, to describe the evolution of surface gravity wave spectrum (e.g., Agnon and Sheremet, 1997; Herbers and Burton, 1997; Kofoed-Hansen and Rasmussen, 1998; Eldeberky and Madsen, 1999; Sheremet et al, 2011; Vrecica and Toledo, 2016, 2019). These stochastic models are developed by averaging the nonlinear phase-resolved equations and employing a closure to truncate the system. These approaches are different from spectral phase-averaged models such as SWAN (Booij et al. 1999), which are based on averaged variables (e.g., spectral density) without considering phase information.

Generally, these stochastic models consist of a set of coupled evolution equations. One equation describes the wave power spectrum and bispectrum, derived by manipulating the deterministic formulations for the complex amplitude. The other equation involves the bispectrum and trispectrum (the next order cumulant of bispectrum). At the core of stochastic model derivation, two significant schemes are employed, similar to those used in turbulence problem: a stochastic closure and an ensemble average. While a Gaussian Sea state with statistically independent waves is required to apply these methods, the stochastic closure can also be valid for weakly nonlinear waves in a non-Gaussian Sea state (Benney and Saffman, 1966). Since the equations are averaged (and thus ensembles do not need to be run through the model), stochastic models calculate wave spectrum and bispectrum with computational efficiency, making them feasible and practical for studying wave behavior over relatively extensive areas in time and space.

The present study aims to simplify the model proposed by KK21 for the random wave field and evaluate its performance in simulating shoaling and breaking waves using experimental data. Section 2 provides a derivation of the equation, while Section 3 outlines the laboratory experiments conducted for random waves. In section 4, the numerical integrations of the present study and previous models are compared with the experimental observations. Summary and conclusions are presented in section 5. For additional details regarding the model derivation, reference is made to Appendix A.

2. Derivation of equations

2.1. Phase-resolved model

Following Smith and Sprinks (1975) and subsequently employing the parabolic approximation method (e.g., Radder, 1979), Kaihatu and Kirby (1995) proposed a two-dimensional parabolic nonlinear frequency-domain mild-slope equation model for waves propagating mainly in the $+x$ direction (cross-shore) direction, with modulation in the y (longshore) direction to account for diffractive effects. To investigate the spectral evolution of unidirectional random waves, the formulation of Kaihatu and Kirby (1995) can be reduced to one dimension considering wave propagating only in the $+x$ direction for the complex amplitude of the nth frequency component $A_n$:

$$A_{nx}+\frac{{\left(kC{C}_g\right)}_{nx}}{2{\left(kC{C}_g\right)}_n}{A}_n=-\frac{i}{8{\left(kC{C}_g\right)}_n}\left[\sum _{l=1}^{n-1}{R}_1{A}_l{A}_{n-l}{e}^{i\theta }+2\sum _{l=1}^{N-n}{S}_1{A}_l^{\ast }{A}_{n+l}{e}^{i\psi }\right]$$ (1)

where $C$ and $C_g$ are wave celerity and group velocity, respectively, an arbitrary frequency mode $l$ and the nth frequency mode interact with the other frequency mode (i.e., $n-l$ or $n+l$), and $k_n$ is the nth wave number satisfying the linear dispersion relation:

$${\omega }_n^2=g{k}_n\tanh k_{n}h$$ (2)

with $g$ is the gravitational acceleration, $h$ is the water depth, and ${\omega }_n$ denotes the nth wave angular frequency. Two types of phase mismatch are given:

$$\theta =\int k_l+k_{n-l}-k_{n}dx$$ (3)

$$\psi =\int k_{n+l}-k_l-k_{n}dx$$ (4)

where these phase mismatches play a crucial role in governing the oscillation of nonlinear terms and resulting detuning effect, causing a deviation from exact resonance. Nonlinear dispersive frequency domain models explicitly account for these mismatches due to the use of wave numbers resulting from the linear dispersion relation. However, in deep water, these mismatches can undergo significant oscillations, which have the potential to violate the assumption of slow spatial amplitude variation inherent in the formulation of these frequency domain models.

As previously mentioned, KK21 developed a nonlinear mild-slope equation model in the frequency domain, with additional nonlinear terms which insured consistency between the scales of bathymetric variation and amplitude modulation. The one-dimensional version of the model of KK21 for varying depth is:

$$\begin{array}{l}{A}_{nx}+\frac{{\left(kC{C}_g\right)}_{nx}}{2{\left(kC{C}_g\right)}_n}{A}_n\\ =-\frac{i}{8{\left(kC{C}_g\right)}_n}\left[\begin{array}{l}\sum _{l=1}^{n-1}\left[R_1{A}_l{A}_{n-l}+R_2{A}_{lx}{A}_{n-l}+R_3{A}_l{A}_{n-lx}\right]e^{i\theta }\hfill \\ +2\sum _{l=1}^{N-n}\left[S_1{A}_l^{\ast }{A}_{n+l}+S_2{A}_{lx}^{\ast }{A}_{n+l}+S_3{A}_l^{\ast }{A}_{n+lx}\right]e^{i\psi }\hfill \end{array}\right]\end{array}$$ (5)

where the interaction coefficients are:

$$R_1=\frac{g}{{\omega }_l{\omega }_{n-l}}\left[{\omega }_n^2{k}_l{k}_{n-l}+\left(k_l+k_{n-l}\right)\left({\omega }_{n-l}{k}_l+{\omega }_l{k}_{n-l}\right){\omega }_n\right]-\frac{{\omega }_n^2}{g}\left({\omega }_l^2+{\omega }_l{\omega }_{n-l}+{\omega }_{n-l}^2\right)$$ (6)

$$R_2=-2i\frac{g{\omega }_n^2{k}_{n-l}}{{\omega }_l{\omega }_{n-l}}$$ (7)

$$R_3=-2i\frac{g{\omega }_n^2{k}_l}{{\omega }_l{\omega }_{n-l}}$$ (8)

and

$$S_1=\frac{g}{{\omega }_l{\omega }_{n+l}}\left[{{\omega }_n}^2{k}_l{k}_{n+l}+\left(k_{n+l}-k_l\right)\left({\omega }_{n+l}{k}_l+{\omega }_l{k}_{n+l}\right){\omega }_n\right]-\frac{{\omega }_n^2}{g}\left({\omega }_l^2-{\omega }_l{\omega }_{n+l}+{{\omega }_{n+l}}^2\right)$$ (9)

$$S_2=2i\frac{g{\omega }_n^2{k}_{n+l}}{{\omega }_l{\omega }_{n+l}}$$ (10)

$$S_3=-2i\frac{g{\omega }_n^2{k}_l}{{\omega }_l{\omega }_{n+l}}$$ (11)

KK21 incorporated the $x$-derivative of amplitude in the newly added nonlinear terms, specifically in the terms involving the interaction coefficients of $R_2$, $R_3$, $S_2$, and $S_3$ (on the right-hand side), which are not present in the model of Kaihatu and Kirby (1995; Eq. 1). Although Eq. (5) solves for the nth component of wave amplitude, which allows us to obtain the x-derivative of amplitude for nth component (i.e., $A_{nx}$), it is also necessary to compute the x-derivative of amplitude for different components from the nth component (i.e., $A_{lx}$, $A_{n-lx}$, ${A_{lx}}^{\ast }$, and $A_{n+lx}$). This requirement leads to a higher number of iterations compared to the model of Kaihatu and Kirby (1995; Eq. 1). Despite the advancement resulting from these additional nonlinear terms in the consistent model of KK21, their inclusion can make the model computationally demanding, particularly in the case of random waves, and may give rise to numerical divergence problems, or computational instability (Davis et al., 2014; Sheremet et al., 2016).

While the amplitudes $A_n$ were expected to simulate the evolution of Fourier amplitudes for free surface elevation, they correspond to the amplitudes of the velocity potential based on the first order dynamic free surface boundary condition. To address this inconsistency in the model of Kaihatu and Kirby (1995), Kaihatu (2001) formulated a second-order relationship between the amplitudes of wave potential function $A_n$ and those of free surface elevation $B_n$ from the second-order nonlinear dynamic free surface boundary condition. Kaihatu (2001) demonstrated that incorporating the second-order correction leads to improved wave shape predictions and slightly better descriptions of spectral density. In this study, we utilize the corrected results obtained through the nonlinear correction to enhance the accuracy of the model and complete consistency within the ordering system:

$$B_n=A_n+\frac{1}{4g}\left[\sum _{l=1}^{n-1}I{A}_l{A}_{n-l}{e}^{i\int \left(k_l+k_{n-1}-k_n\right)dx}+2\sum _{l=1}^{N-n}J{A}_l^{\ast }{A}_{n+l}{e}^{i\int \left(k_{n+l}-k_l-k_n\right)dx}\right]$$ (12)

where the interaction coefficients $I$ and $J$ are

$$I={\omega }_l^2+{\omega }_l{\omega }_{n-l}+{{\omega }_{n-l}}^2-g^2\frac{k_l{k}_{n-l}}{{\omega }_l{\omega }_{n-l}}$$ (13)

$$J={\omega }_l^2-{\omega }_l{\omega }_{n+l}+{{\omega }_{n+l}}^2-g^2\frac{k_l{k}_{n+l}}{{\omega }_l{\omega }_{n+l}}$$ (14)

2.2. Simplified model

For the aforementioned reasons, we simplify the model of KK21 for the problem of random wave propagation, where surface gravity waves can be represented as a combination of multiple waves. First, we modify the interaction coefficients and the phase mismatches by adding subscripts to clearly indicate the interacting frequency modes:

$$A_{nx}+\frac{{\left(kC{C}_g\right)}_{nx}}{2{\left(kC{C}_g\right)}_n}{A}_n=-\frac{i}{8{\left(kC{C}_g\right)}_n}\left[\begin{array}{l}\sum _{l=1}^{n-1}\left[R_{1(l,n-l)}{A}_l{A}_{n-l}+R_{2(l,n-l)}{A}_{lx}{A}_{n-l}+R_{3(l,n-l)}{A}_l{A}_{n-lx}\right]e^{i{\theta }_{(l,n-l)}}\hfill \\ +2\sum _{l=1}^{N-n}\left[S_{1(l,n+l)}{A}_l^{\ast }{A}_{n+l}+S_{2(l,n+l)}{A_{lx}}^{\ast }{A}_{n+l}+S_{3(l,n+l)}{A_l}^{\ast }{A}_{n+lx}\right]e^{i{\psi }_{(l,n+l)}}\hfill \end{array}\right]$$ (15)

with

$$R_{\text{l}(l,n-l)}=\frac{g}{{\omega }_l{\omega }_{n-l}}\left[{\omega }_n^2{k}_l{k}_{n-l}+\left(k_l+k_{n-l}\right)\left({\omega }_{n-l}{k}_l+{\omega }_l{k}_{n-l}\right){\omega }_n\right]-\frac{{\omega }_n^2}{g}\left({\omega }_l^2+{\omega }_l{\omega }_{n-l}+{{\omega }_{n-l}}^2\right)$$ (16)

$$R_{2(l,n-l)}=-2i\frac{g{\omega }_n^2{k}_{n-l}}{{\omega }_l{\omega }_{n-l}}$$ (17)

$$R_{3(l,n-l)}=-2i\frac{g{\omega }_n^2{k}_l}{{\omega }_l{\omega }_{n-l}}$$ (18)

$${\theta }_{(l,n-l)}=\int k_l+k_{n-l}-k_{n}dx$$ (19)

and

$$S_{1(l,n+l)}=\frac{g}{{\omega }_l{\omega }_{n+l}}\left[{\omega }_n^2{k}_l{k}_{n+l}+\left(k_{n+l}-k_l\right)\left({\omega }_{n+l}{k}_l+{\omega }_l{k}_{n+l}\right){\omega }_n\right]-\frac{{\omega }_n^2}{g}\left({\omega }_l^2-{\omega }_l{\omega }_{n+l}+{{\omega }_{n+l}}^2\right)$$ (20)

$$S_{2(l,n+l)}=2i\frac{g{\omega }_n^2{k}_{n+l}}{{\omega }_l{\omega }_{n+l}}$$ (21)

$$S_{3(l,n+l)}=-2i\frac{g{\omega }_n^2{k}_l}{{\omega }_l{\omega }_{n+l}}$$ (22)

$${\psi }_{(l,n+l)}=\int k_{n+l}-k_l-k_{n}dx$$ (23)

The subsequent step involves representing $A_{nx}$ and ${A_{nx}}^{\ast }$ up to third order in $\epsilon$ to eliminate the x-derivative of amplitudes in the nonlinear summation of Eq. (15). This is necessary to alleviate the numerical expense and instability that arise when considering the x-derivate nonlinear terms $A_{x}A$. The $x$-derivative nonlinear terms $A_{x}A$ comprise of $A_x$ and $A$, where $A_x$ is of $O\left({\epsilon }^3\right)$ and $A$ is of $O\left(\epsilon \right)$ according to the derivation of KK21. Since KK21 considered both linear and nonlinear terms up to fourth order in $\epsilon$, we simplify the parabolic approximation of Eq. (15) by keeping only the nonlinear terms of $O\left({\epsilon }^3\right)$ or lower in the expression of $A_{nx}$ and ${A_{nx}}^{\ast }$:

$$A_{nx}=-\frac{1}{2{\left(kC{C}_g\right)}_n}\left[{\left(kC{C}_g\right)}_{nx}{A}_n+\frac{i}{4}{\left(\sum A^2\right)}_n\right]+O\left({\epsilon }^4\right)$$ (24)

$$A_{nx}^{\ast }=-\frac{1}{2{\left(kC{C}_g\right)}_n}\left[{\left(kC{C}_g\right)}_{nx}{A}_n^{\ast }+\frac{i}{4}{\left(\sum A^2\right)}_n^{\ast }\right]+O\left({\epsilon }^4\right)$$ (25)

where ${\left(\sum A^2\right)}_n$ denotes the nonlinear terms in the model of Kaihatu and Kirby (1995; Eq. 1), which are at second order in $\epsilon$:

$${\left(\sum A^2\right)}_n=\left[\sum _{l=1}^{n-1}{R}_{1(l,n-l)}{A}_l{A}_{n-l}{e}^{i{\theta }_{(l,n-l)}}+2\sum _{l=1}^{N-n}{S}_{1(l,n+l)}{A}_l^{\ast }{A}_{n+l}{e}^{i{\psi }_{(l,n+l)}}\right]$$ (26)

and ${{\left(\sum A^2\right)}_n}^{\ast }$ is conjugate complex of ${\left(\text{Σ}{A}^2\right)}_n$:

$${\left(\sum A^2\right)}_n^{\ast }=\left[\sum _{l=1}^{n-1}{R}_{1(l,n-l)}{A}_l^{\ast }{A_{n-l}}^{\ast }{e}^{-i{\theta }_{(l,n-l)}}+2\sum _{l=1}^{N-n}{S}_{1(l,n+l)}{A}_l{A_{n+l}}^{\ast }{e}^{-i{\psi }_{(l,n+l)}}\right]$$ (27)

Substituting Eqs. (24) and (25) into the nonlinear part of Eq. (15), we adjust them to the frequency modes of $A_x$ on the right-hand side of Eq. (15) (i.e., $l$, $n-l$, and $n+l$):

$$\begin{array}{l}{A}_{nx}+\frac{{\left(kC{C}_g\right)}_{nx}}{2{\left(kC{C}_g\right)}_n}{A}_n\\ =-\frac{i}{8{\left(kC{C}_g\right)}_n}\left[\sum _{l=1}^{n-1}{R}_{1(l,n-l)}{A}_l{A}_{n-l}{e}^{i{\theta }_{(l,n-l)}}+2\sum _{l=1}^{N-n}{S}_{1(l,n+l)}{A}_l^{\ast }{A}_{n+l}{e}^{i{\psi }_{(l,n+l)}}\right]\\ -\frac{i}{8{\left(kC{C}_g\right)}_n}\times \left\{\begin{array}{l}\sum _{l=1}^{n-1}\left[\begin{array}{l}-\frac{R_{2(l,n-l)}}{2{\left(kC{C}_g\right)}_l}\left[{\left(kC{C}_g\right)}_{lx}{A}_l+\frac{i}{4}{\left(\sum A^2\right)}_l\right]A_{n-l}{e}^{i{\theta }_{(l,n-l)}}\hfill \\ -\frac{R_{3(l,n-l)}}{2{\left(kC{C}_g\right)}_{n-l}}\left[{\left(kC{C}_g\right)}_{n-lx}{A}_{n-l}+\frac{i}{4}{\left(\sum A^2\right)}_{n-l}\right]A_l{e}^{i{\theta }_{(l,n-l)}}\hfill \end{array}\right]\\ +2\sum _{l=1}^{N-n}\left[\begin{array}{l}-\frac{S_{2(l,n+l)}}{2{\left(kC{C}_g\right)}_l}\left[{\left(kC{C}_g\right)}_{lx}{A}_l^{\ast }-\frac{i}{4}{\left(\sum A^2\right)}_l^{\ast }\right]A_{n+l}{e}^{i{\psi }_{(l,n+l)}}\hfill \\ -\frac{S_{3(l,n+l)}}{2{\left(kC{C}_g\right)}_{n+l}}\left[{\left(kC{C}_g\right)}_{n+lx}{A}_{n+l}+\frac{i}{4}{\left(\sum A^2\right)}_{n+l}\right]A_l^{\ast }{e}^{i{\psi }_{(l,n+l)}}\hfill \end{array}\right]\end{array}\right\}\end{array}$$ (28)

By applying the closure of Benney and Saffman (1966) to the second nonlinear summation in the curly brackets of Eq. (28), we obtain the final simplified version of the model of KK21 (further details in Appendix A):

$$\begin{array}{l}{A}_{nx}+\frac{{\left(kC{C}_g\right)}_{nx}}{2{\left(kC{C}_g\right)}_n}{A}_n\\ =-\frac{i}{8{\left(kC{C}_g\right)}_n}\times \left[\begin{array}{l}\sum _{l=1}^{n-1}\left(R_{1(l,n-l)}-R_{2(l,n-l)}\frac{{\left(kC{C}_g\right)}_{lx}}{2{\left(kC{C}_g\right)}_l}-R_{3(l,n-l)}\frac{{\left(kC{C}_g\right)}_{n-lx}}{2{\left(kC{C}_g\right)}_{n-l}}\right)A_l{A}_{n-l}{e}^{i{\theta }_{(l,n-l)}}\hfill \\ +2\sum _{l=1}^{N-n}\left(S_{1(l,n+l)}-S_{2(l,n+l)}\frac{{\left(kC{C}_g\right)}_{lx}}{2{\left(kC{C}_g\right)}_l}-S_{3(l,n+l)}\frac{{\left(kC{C}_g\right)}_{n+lx}}{2{\left(kC{C}_g\right)}_{n+l}}\right)A_l^{\ast }{A}_{n+l}{e}^{i{\psi }_{(l,n+l)}}\hfill \end{array}\right]\\ -\frac{i}{8{\left(kC{C}_g\right)}_n}\left\{\begin{array}{l}\sum _{l=1}^{n-1}\left(-\frac{i{R}_{2(l,n-l)}{S}_{1(n-l,n)}}{4{\left(kC{C}_g\right)}_l}{\left|A_{n-l}\right|}^2{A}_n-\frac{i{R}_{3(l,n-l)}{S}_{1(l,n)}}{4{\left(kC{C}_g\right)}_{n-l}}{\left|A_l\right|}^2{A}_n\right)\hfill \\ +2\sum _{l=1}^{N-n}\left[\frac{i{S}_{2(l,n+l)}{S}_{1(n,l+n)}}{4{\left(kC{C}_g\right)}_l}{\left|A_{l+n}\right|}^2{A}_n-\frac{i{S}_{3(l,n+l)}{R}_{1(l,n)}}{4{\left(kC{C}_g\right)}_{n+l}}{\left|A_l\right|}^2{A}_n\right]\hfill \end{array}\right\}\end{array}$$ (29)

With this simplification and the incorporation of the closure scheme of Benney and Saffman (1966), the amplitude gradient terms in the nonlinear summations of Eq. (15) do not appear in Eq. (29), and as a result the computational burden associated with them has been eliminated. However, it is still an extension of the deterministic model of Kaihatu and Kirby (1995; Eq. 1) since it does not inherently include ensemble averaging and retains the phase-resolved nonlinear terms, along with the associated phase mismatches. The other terms in the first nonlinear summation represent the interaction between the effects representing water depth change (shoaling, $\frac{{\left(kC{C}_g\right)}_x}{2\left(kC{C}_g\right)}$) and near-resonant quadratic interactions AA. In the second nonlinear summation, the terms in the curly brackets consists of quasi-cubic terms $|A{|}^{2}A$, which have a cubic-like form but are derived from the quadratic terms $A_{x}A$ involving the x-derivate of amplitude. As described in KK21, the potentially-deleterious effect of phase mismatches on the solution is alleviated by the newly-added nonlinear terms $A_{x}A$. This is evident in this resulting equation as it results in the terms within the second summation having zero phase mismatch $|A{|}^{2}A$.

Benney and Saffman (1966) argued that Gaussian closure hypothesis remains valid for weakly nonlinear dispersive waves, even in regions where non-Gaussian waves are present, such as the nearshore region. This application of the Gaussian closure hypothesis is present in several nearshore phase-averaged stochastic models, where it is used to reduce the trispectrum to products of the spectrum (e.g., Herbers and Burton 1997). In this study, we extend the deterministic evolution equations of Kaihatu and Kirby (1995) to include not only the nonlinear terms that retain the phase mismatches but also the additional consistently-ordered terms thorough the closure scheme of Benney and Saffman (1966). Hence, the simplified model combines the strengths of the deterministic model and fully considers triad wave-wave interaction, thus overcoming the limitation of both the stochastic model and the model proposed by Kaihatu and Kirby (1995).

2.3. Wave breaking

While it is feasible to treat random wave dissipation on a wave-by-wave basis in the frequency domain (e.g., Bredmose et al. 2004), statistical approaches are more readily employed in the spectral phase-resolved wave models. The model of Battjes and Janssen (1978), which estimated the spatial decay of energy flux due to the wave breaking in the energy flux balance equation, is widely used for breaking and decay descriptions. Thornton and Guza (1983) extended the model of Battjes and Janssen (1978) to describe more realistic wave height distribution in the surf zone, allowing waves to be temporarily higher than their theoretical limiting height in order to account for the spatial lag between height limit exceedance and the onset of breaking and dissipation.

Whereas Battjes and Janssen (1978) expressed the non-breaking wave distribution as a Rayleigh distribution with a sharp wave height cutoff to replicate the impact of breaking on the probability distribution, Thornton and Guza (1983) redistributed the truncated probability over the remainder of the distribution to mimic the effect of momentary wave height exceedance above breaking. Recently, several efforts have been made to enhance the modeling of dissipation in shallow water, particularly with steeper beach profiles. Baldock et al. (1998) represented broken waves across the surf zone using a Rayleigh distribution with a Heaviside step function, and Janssen and Battjes (2007) extended Baldock et al. (1998) to correct the shoaling law. As a result, the model of Janssen and Battjes (2007) allows for the vanishing of wave height at the shoreline, thus further refining the accuracy of the breaking model.

Several authors have integrated bottom-induced wave breaking dissipation terms into frequency-domain nonlinear models: Mase and Kirby (1993), Kaihatu and Kirby (1995), Chen et al. (1997), Eldeberky and Battjes (1999), Davis et al. (2014), Sheremet et al. (2016), and Ardani and Kaihatu (2019) for deterministic models; Eldeberky and Battjes (1999), Sheremet et al. (2011), and Vrecica and Toledo (2019) for stochastic models.

In the present study, to assess the simplified consistent nonlinear mild slope equation model in simulating the spectral evolution of breaking surface gravity waves, the newly derived frequency-domain model is enhanced by incorporating a lumped parameter for wave breaking dissipation, denoted as ${\alpha }_n$. The numerical integration is performed by employing the Crank-Nicolson method (Crank and Nicolson, 1947), and the solutions are obtained through the iteration procedure with the relative tolerance set to 10^-2. The computer used is a desktop with an Intel(R) Core(TM) i7–8700 CPU 3.20 GHz. The details for numerical implementation can be found in Liu et al. (1985):

$$\begin{array}{l}{A}_{nx}+\frac{{\left(kC{C}_g\right)}_{nx}}{2{\left(kC{C}_g\right)}_n}{A}_n+{\alpha }_n{A}_n\\ =-\frac{i}{8{\left(kC{C}_g\right)}_n}\times \left[\begin{array}{l}\sum _{l=1}^{n-1}\left(R_{1(l,n-l)}-R_{2(l,n-l)}\frac{{\left(kC{C}_g\right)}_{lx}}{2{\left(kC{C}_g\right)}_l}-R_{3(l,n-l)}\frac{{\left(kC{C}_g\right)}_{n-lx}}{2{\left(kC{C}_g\right)}_{n-l}}\right)A_l{A}_{n-l}{e}^{i{\theta }_{(l,n-l)}}\hfill \\ +2\sum _{l=1}^{N-n}\left(S_{1(l,n+l)}-S_{2(l,n+l)}\frac{{\left(kC{C}_g\right)}_{lx}}{2{\left(kC{C}_g\right)}_l}-S_{3(l,n+l)}\frac{{\left(kC{C}_g\right)}_{n+lx}}{2{\left(kC{C}_g\right)}_{n+l}}\right)A_l^{\ast }{A}_{n+l}{e}^{i{\psi }_{(l,n+l)}}\hfill \end{array}\right]\\ -\frac{i}{8{\left(kC{C}_g\right)}_n}\left\{\begin{array}{l}\sum _{l=1}^{n-1}\left(-\frac{i{R}_{2(l,n-l)}{S}_{1(n-l,n)}}{4{\left(kC{C}_g\right)}_l}{\left|A_{n-l}\right|}^2{A}_n-\frac{i{R}_{3(l,n-l)}{S}_{1(l,n)}}{4{\left(kC{C}_g\right)}_{n-l}}{\left|A_l\right|}^2{A}_n\right)\hfill \\ +2\sum _{l=1}^{N-n}\left[\frac{i{S}_{2(l,n+l)}{S}_{1(n,l+n)}}{4{\left(kC{C}_g\right)}_l}{\left|A_{l+n}\right|}^2{A}_n-\frac{i{S}_{3(l,n+l)}{R}_{1(l,n)}}{4{\left(kC{C}_g\right)}_{n+l}}{\left|A_l\right|}^2{A}_n\right]\hfill \end{array}\right\}\end{array}$$ (30)

We used a frequency weighting function first proposed by Mase and Kirby (1993), who introduced the function to address the breaking-induced energy decay proportional to a frequency squared (${f_n}^2$):

$${\alpha }_n=\beta [F+(1-F)\frac{f_n^2\sum {\left|A_n\right|}^2}{\sum f_n^2{\left|A_n\right|}^2}]$$ (31)

where $\beta$ is the dissipation function, $f_n$ is the nth wave frequency, and $F$ is a weight coefficient that weights frequency dependence for ${\alpha }_n$.

Some frequency-domain nonlinear models have adopted a frequency independent model for wave breaking across all frequency modes, represented by $F=1$ (e.g., Eldeberky and Battjes, 1996; Eldeberky and Madsen, 1999). However, Elgar et al. (1997) examined wave spectra developed in the surf zone using field observations and concluded that a form of the damping coefficient should retain frequency dependence. Furthermore, the frequency dependence of wave breaking was confirmed through comparison with the structure of dissipation term in time-domain Boussinesq model (Kirby and Kaihatu, 1997). Kaihatu et al. (2007) further demonstrated that the wave spectra of nearshore shoaling and breaking waves exhibited an $f^{-2}$ spectral shape in the high-frequency range, which is consistent with the concept of an ${f_n}^2$ dependence on ${\alpha }_n$ across the entire frequency range.

Chen et al. (1997) also supported the ${f_n}^2$ weighting of the dissipation term by comparing the spectrum as well as wave-shape-associated quantities (i.e., asymmetry, skewness) with 10 different field and laboratory data. Consequently, many studies have chosen a ${f_n}^2$ weighting wave breaking model (e.g., Mase and Kirby, 1993; Kaihatu and Kirby, 1995; Chen et al., 1997; Mase and Kitano, 2000; Kaihatu, 2001; Ardani and Kaihatu, 2019; Vrecica and Toledo, 2019). It is worth noting that Kim and Kaihatu (2022) proposed a more recent version of the ${f_n}^2$ weighting function for the dissipation term. However, in this study, we employed the model of Mase and Kirby (1993) to focus on inter-model comparisons without the potential influence of additional calibration that the new weighting function might require. Thus, we used $F=0$ in Eq. (31) to apply the ${f_n}^2$ weighting to the entire dissipation term. For the bulk dissipation $\beta$, the “simple” dissipation function of Thornton and Guza (1983) is adopted, along with shallow water approximation for the group velocity $C_g$:

$$\beta =\frac{3\sqrt{\pi }{B}^3}{4{\gamma }^4}\frac{f_p}{\sqrt{gh}}{\left(\frac{H_{rms}}{h}\right)}^5$$ (32)

where $B$ is a breaking-type-related parameter which determines the intensity of wave breaking, a peak frequency of the spectrum $f_p$ is chosen for the characteristic frequency, and $\gamma$ is a wave-height-related parameter presenting the ratio of maximum wave height to water depth, and the root-mean-square (RMS) wave height is:

$$H_{rms}=2\sqrt{\sum _{n=1}^N{\left|A_n\right|}^2}$$ (33)

The inclusion of wave setup using momentum flux balance (Longuet-Higgins and Stewart 1965) was previously considered. However, subsequent testing against laboratory data had shown its contribution to be negligible.

3. Model validation for exchange of energy between a finite number of frequencies

Before verifying the model’s ability to simulate the spectral evolution in random waves, we compare the results of the present simplified model (Eq. 29) with previous models, including the original model of KK21 (Eq. 15) and experimental data.

To quantitatively access the fit to data for each component, we compare the value of index of agreement (IOA; Willmott, 1982) varying from 0 to 1 where 1 indicates the complete agreement:

$$I_a=1-\frac{{\sum }_{j=1}^{N_y}{\sum }_{i=1}^{N_x}{\left(B_n^{i,j}-B_{n,obs}^{i,j}\right)}^2}{{\sum }_{j=1}^{N_y}{\sum }_{i=1}^{N_x}{\left[\left|B_n^{i,j}-{\overline{B}}_{n.obs}^{i,j}\right|+\left|B_{n,obs}^{i,j}-{\overline{B}}_{n,bs}^{i,j}\right|\right]}^2}$$ (34)

where $B_n^{i,j}$ is the computed nth amplitude of free surface elevation from models at $x\left(=i\text{Δ}x,i=1\text{to}{N}_x\right)$ and $y\left(=j\text{Δ}y,j=1\text{to}{N}_y\right)$, and $B_{n,obs}^{i,j}$ is the observed nth harmonic function from experimental data at $x(=i\text{Δ}x)$ and $y(=j\text{Δ}y)$, and over bar indicates an average.

3.1. Chapalain et al. (1992)

Chapalain et al. (1992) carried out a laboratory experiment to study the energy transfer between a limited number of harmonics in a constant depth wave flume (33.54 m-long and 1.3-m deep wave flume). We used this dataset to assess the performances of the present model in simulating nonlinear interactions among a finite number of wave components for weakly nonlinear and dispersive long waves over the flat bottom of the experiment. In all cases, the total numbers of harmonics were set to $N=5$, and we compared the first four harmonics. Table 1 provides the wave parameters and dimensionless parameters (i.e., $Ur=\frac{a}{k^2{h}^3}$ is the Ursell number and $kh$ is the relative water depth) for the experiment of Chapalain et al. (1992). We assigned the initial amplitude at the wave maker location $\left(a_0\right)$ as the amplitudes for the first harmonic, while the initial values for the other harmonics were set to zero. We here used $\text{Δ}x=m$ and $N_x=250$ for all the computations.

Table 1.

Wave parameters and dimensionless parameters of Chapalain et al. (1992).

Case	a0 (m)	T (s)	h (m)	kh	Ur
A	0.042	2.5	0.4	0.530	0.373
C	0.042	3.5	0.4	0.371	0.764
D	0.0355	2.5	0.3	0.454	0.574
H	0.035	3	0.4	0.436	0.460

Figs. 1 and 2 show the numerical results obtained for each case using the simplified consistent nonlinear mild-slope equation model (Eq. 29); the original model of KK21 (Eq. 15); the models of Kaihatu and Kirby (1995; Eq. 1); and Freilich and Guza (1984). All models are also compared to the experimental data of Chapalain et al. (1992). The second order correction (Eq. 12) is applied to all models except for that of Freilich and Guza (1984).

Fig. 1.

Comparison of wave amplitudes between models and data of Chapalain et al. (1992) for Case A: (a) first harmonic; (b) second harmonic; (c) third harmonic; (d) fourth harmonic; for Case C: (e) first harmonic; (f) second harmonic; (g) third harmonic; (h) fourth harmonic (Solid: present model; Dashed: model of Kim and Kaihatu (2021); Dotted: model of Kaihatu and Kirby (1995); Dash-dot: model of Freilich and Guza (1984); Circle: experimental data).

Fig. 2.

Comparison of wave amplitudes between models and data of Chapalain et al. (1992) for Case D: (a) first harmonic; (b) second harmonic; (c) third harmonic; (d) fourth harmonic; for Case H: (e) first harmonic; (f) second harmonic; (g) third harmonic; (h) fourth harmonic (Solid: present model; Dashed: model of Kim and Kaihatu (2021); Dotted: model of Kaihatu and Kirby (1995); Dash-dot: model of Freilich and Guza (1984); Circle: experimental data).

Tables 2 and 3 present the IOA values for each harmonic, along with the average IOA values calculated over the first four harmonics for a comprehensive comparison of models. Overall, the simplified model demonstrates the highest level of agreement (as judged by the averaged IOA values) for all cases except Case C. While the performance of the present model for Case C is not the best of the models, it is not poor by any means (average IOA = 0.920). Notably, the Ursell number of Case C is 0.764, which approaches the strict validity limit of the closure method used. The newly developed model provides higher IOA values compared to the other models for higher-order harmonic waves (i.e., third- and fourth-harmonic amplitudes). These higher-order harmonics have larger values of relative water depth $\left(k_{n}h\right)$ and smaller nonlinearity parameters $\left(A_n/h\right)$, resulting in smaller value of Ursell number $\left(U{r}_n=\frac{A_n}{{k_n}^2{h}^3}\right)$.

Table 2.

Comparison of IOA between models for Case A and C of Chapalain et al. (1992).

Case	Harmonics	Present	KK21	KK95	FG84
A	1	0.891	0.941	0.767	0.756
	2	0.921	0.940	0.852	0.807
	3	0.726	0.708	0.680	0.660
	4	0.737	0.484	0.622	0.380
	Average	0.819	0.768	0.730	0.651
C	1	0.947	0.985	0.924	0.998
	2	0.964	0.981	0.961	0.987
	3	0.879	0.886	0.868	0.961
	4	0.891	0.897	0.851	0.823
	Average	0.920	0.937	0.901	0.942

Table 3.

Comparison of IOA between models for Case D and H of Chapalain et al. (1992).

Case	Harmonics	Present	KK21	KK95	FG84
D	1	0.909	0.960	0.866	0.876
	2	0.965	0.979	0.957	0.908
	3	0.765	0.770	0.762	0.721
	4	0.762	0.623	0.657	0.479
	Average	0.850	0.833	0.811	0.746
H	1	0.911	0.879	0.925	0.923
	2	0.950	0.968	0.916	0.950
	3	0.809	0.839	0.753	0.872
	4	0.871	0.745	0.796	0.612
	Average	0.885	0.858	0.848	0.839

Furthermore, it is evident that the simplified model better predicts all harmonic wave amplitudes in comparison to the deterministic models of Kaihatu and Kirby (1995) in most cases. This is primarily due to the inclusion of $A_{x}A$, which take the form of a quasi-cubic term with zero phase mismatch in the present model (Eq. 29).

Table 4 compares the numerical efficiency (CPU time) and the numerical stability (averaged iteration number) between the numerical wave models, which are averaged over all the cases. Compared to the original model of KK21, the CPU time as well as the iteration number is considerably reduced using the present simplified model.

Table 4.

Comparison of numerical efficiency and stability of Chapalain et al. (1992).

Model	Present	KK21	KK95	FG84
CPU time (s)	1.66	3.01	1.34	1.35
Iteration number	4.25	9.01	4.47	3.21

3.2. Berkoff et al. (1982)

To test the capability of the present model for wave refraction and diffraction over a varying water depth in a two-dimensional horizontal direction $h(x,y)$, we extend the simplified equation (29) by adding a linear refraction-diffraction term and nonlinear diffraction terms of the model of KK21:

$$\begin{array}{l}{A}_{nx}+\frac{{\left(kC{C}_g\right)}_{nx}}{2{\left(kC{C}_g\right)}_n}{A}_n-i\frac{{\left[{\left(C{C}_g\right)}_n{A}_{ny}\right]}_y}{2{\left(kC{C}_g\right)}_n}\\ =-\frac{i}{8{\left(kC{C}_g\right)}_n}\times \left[\begin{array}{l}\sum _{l=1}^{n-1}\left(R_{1(l,n-l)}-R_{2(l,n-l)}\frac{{\left(kC{C}_g\right)}_{lx}}{2{\left(kC{C}_g\right)}_l}-R_{3(l,n-l)}\frac{{\left(kC{C}_g\right)}_{n-lx}}{2{\left(kC{C}_g\right)}_{n-l}}\right)A_l{A}_{n-l}{e}^{i{\theta }_{(l,n-l)}}\hfill \\ +2\sum _{l=1}^{N-n}\left(S_{1(l,n+l)}-S_{2(l,n+l)}\frac{{\left(kC{C}_g\right)}_{lx}}{2{\left(kC{C}_g\right)}_l}-S_{3(l,n+l)}\frac{{\left(kC{C}_g\right)}_{n+lx}}{2{\left(kC{C}_g\right)}_{n+l}}\right)A_l^{\ast }{A}_{n+l}{e}^{i{\psi }_{(l,n+l)}}\hfill \end{array}\right]\\ -\frac{i}{8{\left(kC{C}_g\right)}_n}\left\{\begin{array}{l}\sum _{l=1}^{n-1}\left(-\frac{i{R}_{2(l,n-l)}{S}_{1(n-l,n)}}{4{\left(kC{C}_g\right)}_l}{\left|A_{n-l}\right|}^2{A}_n-\frac{i{R}_{3(l,n-l)}{S}_{1(l,n)}}{4{\left(kC{C}_g\right)}_{n-l}}{\left|A_l\right|}^2{A}_n\right)+\sum _{l=1}^{n-1}{R}_{4(l,n-l)}{A}_{ly}{A}_{n-ly}{e}^{i{\theta }_{(l,n-l)}}\\ +2\sum _{l=1}^{N-n}\left[\frac{i{S}_{2(l,n+l)}{S}_{1(n,l+n)}}{4{\left(kC{C}_g\right)}_l}{\left|A_{l+n}\right|}^2{A}_n-\frac{i{S}_{3(l,n+l)}{R}_{1(l,n)}}{4{\left(kC{C}_g\right)}_{n+l}}{\left|A_l\right|}^2{A}_n\right]+2\sum _{l=1}^{N-n}{S}_{4(l,n+l)}{A}_{ly}^{\ast }{A}_{n+ly}{e}^{i\psi (l,n+l)}\end{array}\right\}\end{array}$$ (35)

where

$$R_{4(l,n-l)}=-2\frac{g{\omega }_n^2}{{\omega }_l{\omega }_{n-l}}$$ (36)

$$S_{4(l,n+l)}=2\frac{g{\omega }_n^2}{{\omega }_l{\omega }_{n+l}}$$ (37)

The experiment, conducted by Berkhoff et al. (1982), aimed to study the wave evolution over the topography, which consists of an elliptic shoal situating on a plane sloping beach with a 1:50 slope, as shown in Fig. 3. The wave field at the initial condition ($kh=1.895$ and $Ur=0.014$ where $h=0.45$ m , $T=1$ s, and $a_0=0.0232$ m) corresponds to a demanding test for some nonlinear wave models, especially for the Boussinesq-type models. We used $N=2$ harmonics, but the first harmonic was compared with a relative amplitude of the experimental data. The reflective lateral boundary conditions are employed along the side walls, and $\text{Δ}x=\text{Δ}y=0.083\text{m}$ and $N_x\times N_y=261\times 243$ were chosen for the numerical domain.

Fig. 3.

Bathymetry and gauge Layout of Berkhoff et al. (1982) (Dashed: gauge transects).

Fig. 4 shows relative amplitudes at the gauge locations, comparing the first harmonics calculated by the present model, as well as the models of KK21 and Kaihatu and Kirby (1995). Although the wave condition at the wave maker is obviously outside the range of validity for Boussinesq-type model, the Kadomtsev-Petviashvili (K-P) model of Liu et al. (1985) (hereafter KP85 in figures and tables) is included in comparison for a comprehensive analysis of the present model in a two-dimensional wave field propagation. Liu et al. (1985) derived evolution equations for two horizontal dimensions based on the Boussinesq model of Freilich and Guza (1984). In Table 5, we summarize a comparison of the IOA values at all the gauges of the experiment. The present mode yields wave descriptions comparable to those provided by the KK21 model, despite the simplification of the amplitude gradient terms in the nonlinear summations. Specifically, both models better describe the reduction in amplitudes observed along gauge transect 6, including an amphidromic point, than the other models. As expected, the Boussinesq-type model of Liu et al. (1985) shows a great discrepancy between prediction and reference values, while the nonlinear mild-slope equation model of Kaihatu and Kirby (1995), retaining the fully-dispersive feature, agrees favorably with the experimental data.

Fig. 4.

Comparison of normalized wave amplitudes between models and data of Berkhoff et al. (1982) for N = 2: (a) gauge 1; (b) gauge 2; (c) gauge 3; (d) gauge 4;(e) gauge 5; (f) gauge 6; (g) gauge 7; (h) gauge 8 (Solid: present model; Dashed: model of Kim and Kaihatu (2021); Dotted: model of Kaihatu and Kirby (1995); Dash-dot: model of Liu et al. (1985); Circle: experimental data).

Table 5.

Comparison of IOA between models at all the gauges of Berkhoff et al. (1982).

Gauge	Present	KK21	KK95	KP85
1	0.923	0.913	0.902	0.356
2	0.949	0.973	0.977	0.704
3	0.983	0.980	0.966	0.783
4	0.968	0.988	0.977	0.756
5	0.936	0.979	0.958	0.694
6	0.968	0.991	0.818	0.583
7	0.879	0.930	0.943	0.575
8	0.861	0.869	0.938	0.381

Table 6 provides a comparison of the numerical efficiency and the numerical stability for all the models. It is found that the application of a simplifying closure method improves model stability and reduce iteration numbers. Nonetheless, the present model remains computationally expensive (albeit generally affordable) compared to the model of Kaihatu and Kirby (1995).

Table 6.

Comparison of numerical efficiency and stability of Berkhoff et al. (1982).

Model	Present	KK21	KK95	KP85
CPU time (s)	4.31	6.79	1.55	1.48
Iteration number	3.15	4.55	2.74	3.09

4. Comparison to random wave data from laboratory experiments

We used six cases from three experimental data sets in the present study. The first one is Case 2 of Mase and Kirby (1993) (hereafter MK93 in figures and tables), who generated random waves propagating over a 1:20 sloping bottom. The shape of the spectra was determined using a Pierson-Moskowitz spectrum with $H_{rms}=4.7\text{cm}$ and $f_p=1\text{Hz}$ as the input. This case is considered a challenging test for many frequency-domain nonlinear wave models since the relative water depth at the spectral peak is well outside the shallow water range $\left(k_{p}h=1.97\right)$. Time series data for the free surface elevation were collected at 12 different gauges located along the sloping bottom. However, we excluded the shallowest gauge ($h=2.5\text{cm}$) due to the poor comparisons shown by most nonlinear wave models considered in this study when $F=0$ for the dissipation mechanism in Eq. (31). This anomalous behavior has been previously noted at the $h=2.5\text{Hz cm}$ gage in prior studies using the same dissipation mechanism and value for $F$ (Mase and Kirby, 1993; Kirby and Kaihatu, 1997; Chen et al., 1997; Kim and Kaihatu, 2022). The experimental data was sampled at 20 Hz. For input and comparison with the models, the data were divided into seven realizations, each containing 2048 points. Each realization was then run through the model. Following Mase and Kirby (1993), in which the optimized values of the free parameters $B$ and $\gamma$ in $\beta$ (Eq. 32) were determined based on the predictions of $H_{rms}$, we set $B=1.0$ and $\gamma =0.6$. A sketch of experimental setup is shown in Fig. 5.

Fig. 5.

Layout of experiment of Mase and Kirby (1993).

The second data set used in this study is from Bowen and Kirby (1994) (hereafter BK94 in figures and tables). From the experimental data set, we specifically chose Case A to investigate the performance of wave propagating models in intermediate water depth $\left(k_{p}h=0.72\right)$. According to Bowen and Kirby (1994), the free surface elevation was measured at a sampling rate of 25 Hz. We divided the data into 24 realizations, with each realization containing 1024 data points. Bowen and Kirby (1994) determined the optimal values of $B$ and $\gamma$ in $\beta$ as 1.15 and 0.6, respectively, and we used these values in this study. Fig. 6 illustrates the experimental layout showing 44 gauges along the 1:35 beach slope, with a total of 47 wave gauges.

Fig. 6.

Layout of experiment Bowen and Kirby (1994).

The last four cases were taken from a data set by Smith and Vincent (1992) (hereafter SV92 in figures and tables), who conducted laboratory experiments where random waves with bimodal spectra were transformed over a bottom rising at a slope of 1:30. To generate the double-peaked spectra, they superimposed two TMA (Texel, MARSEN, and ARSLOE, Bouws et al., 1985) spectra as input to a piston-type wave paddle. Data were collected at 9 different gauges located along the slope beach and sampled at a rate of 10 Hz. The data were then divided into 11 realizations, each containing 1024 data points. The values of $B$ and $\gamma$ in $\beta$ were set to 0.8 and 0.6, respectively. For the cases of Smith and Vincent (1992), we used the mean frequency instead of the peak frequency in $\beta$ due to the bimodal spectral shape. Fig. 7 shows the experimental flume with 8 gauges along the 1:30 beach slope of a total 9 gauges. They used two sets of two double-peak wave periods: one set where the second peak was at a multiple of the first peak ($T_p=2.5\text{s}/1.25\text{s}$) and another set where it was not ($T_p=2.5\text{s}/1.75\text{s}$). They concluded that all cases demonstrated similar trends in spectral evolution, whether or not the two spectral peaks were harmonic multiples. To examine the energy transfer between the frequency modes, particularly the two primary peaks, we selected Cases 2, 5, 8, and 11, where the low-frequency peak was nearly equal in energy to the high frequency. In other cases of Smith and Vincent (1992), they assigned different distributions of energy to the two peaks. However, we obtained similar comparisons between the wave models in those other cases compared to the cases presented in this study. Hence, the results in the other cases are not shown here for the sake of conciseness.

Fig. 7.

Layout of experiment of Smith and Vincent (1992).

Table 7 provides an overview of the wave parameters, including the total number of frequency components $N$ used in the simulation, the dimensionless parameters at the peak and the experiment setup. We chose the total number of frequency components, denoted as $N$ in Table 7, to ensure that it allows for a sufficient energy content retained in the spectrum of each case compared to the Nyquist limit, for example, 1024 for Case 2 of Mase and Kirby (1993). In work unpublished at the time of writing, Kim and Kaihatu reported that these values of $N$, particularly for the original model of KK21, can lead to difficulties in solving the model equation due to numerical divergence problems, or computational instability.

Table 7.

Wave parameters, dimensionless parameters, and setup of experiments

Experiments	Bed slope	$H_{rms}$ (cm)	$f_p$(Hz)	N	k p h	$U{r}_p$
Case 2 of MK93	1:20	4.7	1	400	1.97	0.01
Case A of BK94	1:35	7	0.5	150	0.72	0.15
Case 2 of SV92	1:30	10	0.4/0.8	250	0.67/1.68	0.18/0.29
Case 5 of SV92	1:30	6	0.4/0.8	250	0.67/1.68	0.11/0.02
Case 8 of SV92	1:30	10	0.4/0.57	250	0.67/1.03	0.18/0.08
Case 11 of SV92	1:30	6	0.4/0.57	250	0.67/1.03	0.11/0.05

5. Comparisons with data

To assess the performance of the model, we compare it to the experimental data in this section. As an initial step in calculating the power spectra using numerical models, we discarded a certain number of initial data values in the time series records for both model comparison and initial conditions. This was done to ensure that waves had reached the furthest wave gauge in each experiment prior to analysis. Subsequently, we obtained the complex Fourier amplitudes $A_n$ from the time series data at the wave gauge closest to the wave maker using the Fast Fourier Transform (FFT). Next, the complex Fourier amplitudes $A_n$ at the locations of the wave gauges were obtained by solving the numerical wave propagation models augmented with the breaking-induced dissipation term (e.g., Eq. 30).

Considering the demonstrated improvement achieved by the second-order correction (Eq. 12) in solving permanent form and simulating both narrow-banded waves and random waves (Kaihatu, 2001; Kim and Kaihatu, 2021), we concerned ourselves with the second-order corrected results for all the models, except for the model of Freilich and Guza (1984), the inclusion of which would not benefit the weakly-dispersive formulation (Eldeberky and Madsen 1999). Therefore, we calculated a set of complex amplitudes of surface elevation $B_n$ by substituting the obtained amplitudes of wave potential $A_n$ into the second-order correction, and then the resulting amplitude $B_n$ were used to calculate wave energy spectrum $S\left(f_n\right)$:

$$S\left(f_n\right)=\frac{{\left|B_n\right|}^2}{2\text{Δ}f}$$ (38)

where the frequency resolution is $\Delta f=1/T$ (where $T$ is the record length of one realization). The obtained spectra were averaged across all realizations (Bartlett averaging, Bartlett, 1948). Additional smoothing was accomplished by band averaging over nine adjacent bands for all cases except for Case A of Bowen and Kirby (1994), which had a sufficiently high value of incremental frequency $\Delta f$ that spectral detail would have been lost unless a smaller band averaging width were used. In this case we used three adjacent bands as the averaging band width. The number of degrees of freedom is 126 for Case 2 of Mase and Kirby (1993), and 144 for Case A of Bowen and Kirby (1994) and all examined cases of Smith and Vincent (1992). For Case 2 of Mase and Kirby (1993), the numerical setup is $\text{Δ}x=0.025\text{m}$ and $N_x=337$. For Case A of Bowen and Kirby (1994), it is $\text{Δ}x=0.02\text{m}$ and $N_x=651$. In all cases of Smith and Vincent (1992), the setup is $\text{Δ}x=0.046\text{m}$ and $N_x=361$. The corresponding lengths of the sampled time series are 102.4 s, 40.96 s, 102.4 s, respectively.

We use the RMSPE (Root Mean-Square Percentage-Error) to evaluate the performance of the models over the entire frequency range. This metric ranges from 0 to $\infty$, with 0 corresponding to the ideal model:

$${\text{RMSPE}}_j=\sqrt{\frac{1}{N_{band}}\sum _{i=1}^{N_{band}}{\left(\frac{S_j\left(f_i\right)-S_{j,obs}\left(f_i\right)}{S_{j,obs}\left(f_i\right)}\right)}^2}$$ (39)

where ${\text{RMSPE}}_j$ is evaluated at the $j\text{th}$ gauge ($j=1$ to $N_{gauge}$), $S_j\left(f_i\right)$ is the calculated band-averaged wave energy spectrum of frequency $f_i$ ($i=1$ to $N_{band}$) at the $j\text{th}$ gauge ($j=1$ to $N_{gauge}$) and $S_{j,obs}\left(f_i\right)$ is the observed band-averaged wave energy spectrum of frequency $f_i$ ($i=1$ to $N_{band}$) at the $j\text{th}$ gauge ($j=1$ to $N_{gauge}$).

Figs. 8 and 9 present comparisons between the modeled wave spectra using the present model and the models of Kaihatu and Kirby (1995) and Freilich and Guza (1984) with the corresponding experimental data for Case 2 of Mase and Kirby (1993) and Case A of Bowen and Kirby (1994), respectively. Figs 10 through 13 compare the energy spectra of the surface elevation computed by the models to the laboratory measurements of Smith and Vincent (1992) at selected gauges for Cases 2, 5, 8, and 11, respectively. Tables 8 through 11 compare the RMSPE values between models. For Case 2 of Mase and Kirby (1993), the model of Freilich and Guza (1984), based on the weakly-dispersive characteristics and the corresponding non-dispersive shoaling (i.e., Green’s law shoaling), consistently overpredicts the spectral peaks with the high relative water depth at the first gauge $\left(k_{p}h=1.97\right)$. However, the predicted peaks of two models with the fully-dispersive linear theory for the dispersive and shoaling characteristics (i.e., the present model and the model of Kaihatu and Kirby (1995)) are in better agreement with the measured data.

Fig. 8.

Comparison of wave spectra density using N = 400 for Case 2 of Mase and Kirby (1993): (a) h = 47 cm; (b) h = 30 cm; (c) h = 17.5 cm; (d) h = 12.5 cm; (e) h = 10 cm; (f) h = 5 cm (Solid: experimental data; Dashed: present model; Dotted: model of Kaihatu and Kirby (1995); model of Freilich and Guza (1984)).

Fig. 9.

Comparison of wave spectra density using N = 150 for Case A of BK94: (a) h = 44 cm; (b) h = 33 cm; (c) h = 21 cm; (d) h = 15 cm; (e) h = 13 cm; (f) h = 7 cm (Solid: experimental data; Dashed: present model; Dotted: model of Kaihatu and Kirby (1995); model of Freilich and Guza (1984)).

Fig. 10.

Comparison of wave spectra density using N = 250 for Case 2 of SV92: (a) h = 61 cm; (b) h = 36.6 cm; (c) h = 24.4 cm; (d) h = 18.3 cm; (e) h = 12.2 cm; (f) h = 7.6 cm (Solid: experimental data; Dashed: present model; Dotted: model of Kaihatu and Kirby (1995); model of Freilich and Guza (1984)).

Fig. 13.

Comparison of wave spectra density using N = 250 for Case 11 of SV92: (a) h = 61 cm; (b) h = 36.6 cm; (c) h = 24.4 cm; (d) h = 18.3 cm; (e) h = 12.2 cm; (f) h = 7.6 cm (Solid: experimental data; Dashed: present model; Dotted: model of Kaihatu and Kirby (1995); model of Freilich and Guza (1984)).

Table 8.

Comparison of RMSPE between models for Case 2 of Mase and Kirby (1993).

Case	Depth (cm)	Present	KK95	FG84
Case 2 of MK93	47	-	-	-
	35	0.278	0.389	0.409
	30	0.235	0.330	0.494
	25	0.341	0.412	0.591
	20	0.403	0.491	0.774
	17.5	0.295	0.588	0.823
	15	0.286	0.462	0.842
	12.5	0.298	0.488	0.917
	10	0.395	0.548	0.843
	7.5	0.496	0.558	0.855
	5	0.818	0.816	1.318
	Average	0.384	0.508	0.787

Table 11.

Comparison of RMSPE between models for Cases 8 and 11 of Smith and Vincent (1992).

Case	Depth (cm)	Present	KK95	FG84
Case 8 of SV92	61	-	-	-
	36.6	0.721	1.590	0.435
	24.4	0.399	0.812	0.448
	18.3	0.371	0.548	0.463
	15.3	0.433	0.541	0.497
	12.2	0.572	0.610	0.659
	9.2	0.424	0.434	0.516
	7.6	0.357	0.530	0.368
	6.1	0.413	0.729	0.386
	Average	0.461	0.724	0.472
Case 11 of SV92	61	-	-	-
	36.6	0.547	0.903	0.417
	24.4	0.642	1.158	0.467
	18.3	0.490	0.738	0.530
	15.3	0.480	0.683	0.565
	12.2	0.572	0.720	0.721
	9.2	0.459	0.493	0.567
	7.6	0.396	0.416	0.418
	6.1	0.374	0.427	0.390
	Average	0.495	0.692	0.509

On the other hand, the wave conditions in Case A of Bowen and Kirby (1994) represent intermediate water depth, and thus correspond to shallower water depth compared to Case 2 of Mase and Kirby (1993). The model of Freilich and Guza (1984) provides better predictions for the primary peak at the shallower gauge in this case, as shown in Fig. 9(d), (e), and (f). In contrast, the model of Kaihatu and Kirby (1995) underestimates the energy spectrum of the peak frequency at such gauges, possibly due to problems with the nonlinear behavior moving energy out of the peak at a faster rate than reflected by data. Surprisingly, we notice that, compared to the model of Kaihatu and Kirby (1995), the present model offers better descriptions of the spectral peaks, which is consistent with the slight improvements around the peak by the present model for Case 2 of Mase and Kirby (1993), as shown in Fig. 8(d), (e), and (f). These improvements are likely due to the inclusion of additional quasi-cubic terms.

Furthermore, the present formulation exhibits superior performances for the higher frequencies (deep water part of spectra) at the shallow gauges in both cases. In the higher-frequency range of spectra, the magnitude of mismatches becomes more pronounced, which gives an advantage to the present model as it is less affected by these significantly great mismatches. It is particularly evident at $f>2$ Hz in Case A of Bowen and Kirby (1994), as shown in Fig. 9. The value of Ursell number $U{r}_n\left(\frac{H_{rms}}{2k_n^2{h}^3}\right)$ approaches zero in the deep water range of wave power spectra, which support the validity of the ordering in the present study (i.e., $O(Ur)<<1$) rather than that in Boussinesq type model (i.e., $O(Ur)\sim O(1)$).

The low-frequency wave range $(f<0.5f_p)$ is likely influenced by reflection effects from the beach, as well as re-reflection from the wavemaker, neither of which are taken into account in any of the models. This might explain the deviation of the models from the experimental data in the low-frequency range for all cases compared. Specifically, Bowen and Kirby (1994) determined the seiching frequencies in their wave flume, and determined that a statistically significant proportion of the evolutionary characteristics of wave energy at these frequencies are impacted by these seiching processes. (We note that this analysis made use of work that was later fully detailed in Veeramony and Svendsen (1997)).

The impact of the improvements in the present model is also evident in the comparison with the experimental data of Smith and Vincent (1992). The improvements are most evident in two aspects: (1) at the individual spectral peaks; and (2) over the higher-frequency range of the spectra (e.g., $f>1.2$ Hz in Fig. 12). In addition, the comparison of the average RMSPE values from Tables 8 to 11 quantifies the improved performance of the present model relative to the other models tested here over the entire frequency range.

Fig. 12.

Comparison of wave spectra density using N = 250 for Case 8 of SV92: (a) h = 61 cm; (b) h = 36.6 cm; (c) h = 24.4 cm; (d) h = 18.3 cm; (e) h = 12.2 cm; (f) h = 7.6 cm (Solid: experimental data; Dashed: present model; Dotted: model of Kaihatu and Kirby (1995); model of Freilich and Guza (1984)).

In comparison to the model of Kaihatu and Kirby (1995), the present model therefore demonstrates significantly better agreement at the location farthest from the wave maker in shallower cases of Smith and Vincent (1992), such as Cases 2 and 8 where $U{r}_p=0.18$ and 0.19 at the first peak, respectively. Notably, the Boussinesq-type equation, which retains only nonlinear terms with zero phase mismatch, compares favorably with the present model at these points (see Figs. 8 (f) and 10 (f)). Although the interaction between wave numbers (e.g., $k_l+k_{n-l}-k_n$) is minimal at the shallowest gauge, the phase mismatch has the greatest value at this location. This is because the phase mismatch contains the cumulative effect of the wave number interactions from the offshore location to the shallow gauge location.

In conclusion, the present model better describes wave propagation over long distances with good accuracy, incorporating both shallow and deep water waves. Notably, when applied to the laboratory and field data with a larger number of frequency components and a more dominant offshore part of spectra, the present model may become an even more feasible alternative to previous models. Additionally, in the computation of wave shape statics such as skewness and asymmetry, it is essential to retain a sufficient number of frequency components.

To comprehend the numerical complexity of wave physics associated with enhancements by the present model, Table 12 presents details on CPU time and averaged iteration numbers for all the cases considered in this section. It is noteworthy that the use of a relatively large number of frequency components (as outlined in Table 7) causes the original model of KK21 to blow up in all the cases, whereas the present simplified model yields a converged solution. It is confirmed that the present model significantly increases numerical complexity with its accuracy, albeit generally not a prohibitable extent, for the random wave simulation. Surprisingly, the improved numerical stability, as indicated by the averaged iteration number, is achieved via the inclusion of the additional nonlinear terms. The increase in CPU time can therefore be interpreted as the time increase required to build up the additional nonlinear summation. Ultimately, with the addition of this new nonlinear summation, one can trade-off model accuracy with numerical efficiency.

Table 12.

Comparison of numerical efficiency and stability of experiments

Case	Model	Present	KK95	FG84
Case 2 of MK93	CPU time (s)	355.5	91.0	71.4
	Iteration number	3.4	5.2	3.7
Case A of BK94	CPU time (s)	110.8	32.3	22.7
	Iteration number	3.6	5.8	3.6
Case 2 of SV92	CPU time (s)	128.5	45.4	33.5
	Iteration number	3.0	5.2	3.8
Case 5 of SV92	CPU time (s)	111.7	33.6	29.8
	Iteration number	2.6	4.2	3.4
Case 8 of SV92	CPU time (s)	156.5	70.7	33.0
	Iteration number	3.2	5.9	3.9
Case 11 of SV92	CPU time (s)	118.2	36.1	29.2
	Iteration number	2.7	4.5	3.5

The present model seemingly provides similar wave spectrum descriptions at all the gauges for Case 2 of Mase and Kirby (1993), despite requiring more CPU time than the model of Kaihatu and Kirby (1995). To demonstrate the enhancement brought about by the newly added nonlinearity in the present formulation, we present parts of the temporal evolutions of free surface elevations computed using both the present and previous models, alongside the experimental data in Fig. 14. For compactness of presentation, we present detailed results only for this case. The present model leads to a significant improvement in the resulting surface elevations, ensuring a much more accurate capture of the zero-crossing periods compared to the previous models.

Fig. 14.

Comparison of free surface elevation at h = 5 cm for Case 2 of MK93: (a) Present model; (b) model of KK95; (c) model of FG84 (Solid: measured time series; Dashed: calculated time series).

6. Conclusions

The consistent nonlinear mild-slope equation model of Kim and Kaihatu (2021) was developed as a parabolic equation in the frequency-domain. It establishes a robust ordering system that combines the order of bottom change scales from Kirby and Dalrymple (1983) and the modulation scale from Yue and Mei (1980). The inclusion of additional x-derivative nonlinear terms (i.e., $A_{x}A$), brought about by the reformulated scaling, allows the new model to overcome the problematic impacts of the phase mismatch terms of previous models (e.g., Kaihatu and Kirby 1995). In doing so, the model provides a more complete description of the effects of triad wave-wave interaction compared to earlier models. However, the introduction of these new nonlinear terms may lead to an increase in the number of iterations necessary for numerical integration, thereby resulting in a more computationally demanding and potentially unstable model.

To achieve computational stability in modelling irregular wave processes, we simplified the consistent nonlinear mild-slope equation model using the closure scheme of Benney and Saffman (1966). The closure schemes allowed us to address higher-order terms by truncating them beyond the inclusion of the exact resonant interactions between four waves. As a result, the final form of nonlinear terms in the simplified model includes one part that is proportional to the phase mismatch terms, and another with zero phase mismatch. Specifically, the nonlinear terms with zero phase mismatch are introduced by manipulating the gradient amplitude nonlinear terms $A_{x}A$ in the model equation of Kim and Kaihatu (2021). These nonlinear terms are expected to serve as a buffer against excessive oscillations in the wavefield, thereby mitigating the mismatch effect, as discussed in Kim and Kaihatu (2021).

Numerical investigations were conducted to assess the performance of the new formulations in simulating random wave propagations. The present model outperforms other models in accurately predicting the spatial evolution of wave spectra. Specifically, it is evident that the present model with fully-dispersive linear shoaling characteristics provides better predictions than the Boussinesq-type equation model (i.e., Freilich and Guza, 1984) in cases of deep water. On the other hand, in shallow water cases, the present model is much improved compared to the previous nonlinear mild-slope equation model (i.e., Kaihatu and Kirby, 1995), and it is roughly comparable to the Boussinesq type-equation model. The enhanced performance not only over the high-frequency range but also at the spectral peaks can be attributed to the reduced sensitivity of the new model to strong mismatch. It is likely that the present model can potentially serve as a unified wave propagation model valid for both deep and shallow water.

However, due to the assumptions considered in the derivation of the present model, it may not be applicable for waves on coastal structures or steep beaches. First, the frequency domain model, which considers wave propagation as an initial value problem, lacks the capability to capture backward-scattered wave reflection (Herbers et al., 2003). Additionally, steep bed profiles can violate the assumption of slowly varying bathymetry in the mild-slope equation model. The present model could be extended to address this case by incorporating a bottom reflection source term (Yevnin and Toledo, 2018) and higher-order bottom change terms that were neglected in the original form (Chamberlain and Porter, 1995).

While the present simplified model requires more computational time than Kaihatu and Kirby (1995) (e.g., Table 12), it is also more accurate (Tables 8–10). This is not surprising, as the present model is based on the KK21 model, which has itself been shown to be more accurate than that of Kaihatu and Kirby (1995). However, the present simplified model is more computationally expedient than KK21 for even a small number of frequencies (Table 6) and is also more stable for high numbers of frequencies, as would be the case for broad wave spectra.

Table 10.

Comparison of RMSPE between models for Cases 2 and 5 of Smith and Vincent (1992).

Case	Depth (cm)	Present	KK95	FG84
Case 2 of SV92	61	-	-	-
	36.6	0.385	0.804	0.396
	24.4	0.301	0.571	0.468
	18.3	0.331	0.474	0.417
	15.3	0.424	0.457	0.472
	12.2	0.591	0.612	0.670
	9.2	0.380	0.427	0.545
	7.6	0.282	0.454	0.390
	6.1	0.279	0.598	0.454
	Average	0.372	0.550	0.476
Case 5 of SV92	61	-	-	-
	36.6	0.398	0.572	0.407
	24.4	0.262	0.507	0.510
	18.3	0.233	0.334	0.485
	15.3	0.266	0.352	0.477
	12.2	0.384	0.520	0.604
	9.2	0.330	0.386	0.458
	7.6	0.270	0.304	0.337
	6.1	0.273	0.348	0.344
	Average	0.302	0.415	0.453

Given the rise in use of time-domain Boussinesq models (e.g., FUNWAVE, Wei et al., 1999), it is natural to ask how models like the present model or KK21 would compare to these time domain models. First of all, the point of this paper is to show a simplified, yet still accurate, version of the model of KK21, which has itself been shown to have superior performance to earlier nonlinear frequency domain models. Secondly, a rigorous, meaningful comparison between time domain and frequency domain models – one that demonstrates the advantages of one or the other, using a common foundation – can only be done between time-domain and frequency domain models that are analogues of each other. A good example of this would be the frequency domain model of Kaihatu and Kirby (1998), which is directly based on the time domain model of Nwogu (1993). However, while there are some nonlinear mild-slope equation models in the time domain (e.g., Beji and Nadaoka 1999), they are not the time-domain analogues of either this present model, KK21, or Kaihatu and Kirby (1995). As for comparisons without this analogue relationship, it would be difficult to rigorously pinpoint the causes for differences between models without a common foundation.

Future work will focus on expanding the model to include additional physical effects and consider various scenarios which will serve to expand model capability. These improvements will allow the model to serve as an important part of larger scale modeling systems. Specifically, further investigation into the comparison with field study data over a two-dimensional beach, which may involve a preprocessing step, will be addressed in a separate work (Kim, 2022).

Fig. 11.

Comparison of wave spectra density using N = 250 for Case 5 of SV92: (a) h = 61 cm; (b) h = 36.6 cm; (c) h = 24.4 cm; (d) h = 18.3 cm; (e) h = 12.2 cm; (f) h = 7.6 cm (Solid: experimental data; Dashed: present model; Dotted: model of Kaihatu and Kirby (1995); model of Freilich and Guza (1984)).

Table 9.

Comparison of RMSPE between models for Case A of Bowen and Kirby (1994).

Case	Depth (cm)	Present	KK95	FG84
Case A of BK94	44	-	-	-
	41	0.225	0.421	0.135
	38	0.301	0.557	0.133
	35	0.366	1.063	0.180
	33	0.474	1.437	0.235
	30	0.705	1.782	0.298
	27	0.446	2.102	0.401
	24	0.912	2.786	0.444
	21	0.541	2.060	0.521
	18	0.698	1.792	0.536
	15	0.473	1.274	0.538
	13	0.358	0.802	0.558
	10	0.339	0.409	0.562
	7	0.446	0.516	0.710
	Average	0.483	1.308	0.404

• A simplified consistent nonlinear mild-slope equation model has been developed for random wave propagation by employing a closure scheme.

• The new model demonstrated improved performance in simulating both narrow-banded waves and random waves compared to previous models.

• The inclusion of additional quasi-cubic terms of the new model provides better descriptions at the spectral peaks and over the high-frequency range.

Appendix

In Section 2, we derived the simplified consistent nonlinear mild-slope equation. In this appendix, we will provide detailed derivations regarding the closure scheme of Benney and Saffman (1966).

First, the second nonlinear summation in the curly brackets on the right-hand side of Eq. (28) is

$$\begin{gathered} \sum _{l=1}^{n-1}\left[\begin{array}{l}-\frac{R_{2(l,n-l)}}{2{\left(kC{C}_g\right)}_l}\left[{\left(kC{C}_g\right)}_{lx}{A}_l+\frac{i}{4}{\left(\sum A^2\right)}_l\right]A_{n-l}{e}^{i{\theta }_{(l,n-l)}}\hfill \\ -\frac{R_{3(l,n-l)}}{2{\left(kC{C}_g\right)}_{n-l}}\left[{\left(kC{C}_g\right)}_{n-lx}{A}_{n-l}+\frac{i}{4}{\left(\sum A^2\right)}_{n-l}\right]A_l{e}^{i{\theta }_{(l,n-l)}}\hfill \end{array}\right] \\ +2\sum _{l=1}^{N-n}\left[\begin{array}{l}-\frac{S_{2(l,n+l)}}{2{\left(kC{C}_g\right)}_l}\left[{\left(kC{C}_g\right)}_{lx}{A}_l^{\ast }-\frac{i}{4}{\left(\sum A^2\right)}_l^{\ast }\right]A_{n+l}{e}^{i{\psi }_{(l,n+l)}}\hfill \\ -\frac{S_{3(l,n+l)}}{2{\left(kC{C}_g\right)}_{n+l}}\left[{\left(kC{C}_g\right)}_{n+lx}{A}_{n+l}+\frac{i}{4}{\left(\sum A^2\right)}_{n+l}\right]A_l^{\ast }{e}^{i{\psi }_{(l,n+l)}}\hfill \end{array}\right] \end{gathered}$$ (A.1)

where the index $m$ is used to specify components involved in the triad wave-wave interactions:

$${\left(\sum A^2\right)}_l{A}_{n-l}{e}^{i{\theta }_{(l,n-l)}}=\left[\sum _{m=1}^{l-1}{R}_{1(m,l-m)}{A}_m{A}_{l-m}{e}^{i{\theta }_{(m,l-m)}}+2\sum _{m=1}^{N-l}{S}_{1(m,l+m)}{A_m}^{\ast }{A}_{l+m}{e}^{i{\psi }_{(m,l+m)}}\right]A_{n-l}{e}^{i{\theta }_{(l,n-l)}}$$ (A.2)

$${\left(\sum A^2\right)}_{n-l}{A}_l{e}^{i{\theta }_{(l,n-l)}}=\left[\sum _{m=1}^{n-l-1}{R}_{1(m,n-l-m)}{A}_m{A}_{n-l-m}{e}^{i{\theta }_{(m,n-l-m)}}+2\sum _{m=1}^{N-(n-l)}{S}_{1(m,n-l+m)}{A_m}^{\ast }{A}_{n-l+m}{e}^{i{\psi }_{(m,n-l+m)}}\right]A_l{e}^{i{\theta }_{(l,n-l)}}$$ (A.3)

$${\left(\sum A^2\right)}_l^{\ast }{A}_{n+l}{e}^{i{\psi }_{(l,n+l)}}=\left[\sum _{m=1}^{l-1}{R}_{1(m,l-m)}{A}_m^{\ast }{A_{l-m}}^{\ast }{e}^{-i{\theta }_{(m,l-m)}}+2\sum _{m=1}^{N-l}{S}_{1(m,l+m)}{A}_m{A_{l+m}}^{\ast }{e}^{-i{\psi }_{(m,l+m)}}\right]A_{n+l}{e}^{i{\psi }_{(l,n+l)}}$$ (A.4)

$${\left(\sum A^2\right)}_{n+l}{A}_l^{\ast }{e}^{i{\psi }_{(l,n+l)}}=\left[\sum _{m=1}^{n+l-1}{R}_{1(m,n+l-m)}{A}_m{A}_{n+l-m}{e}^{i{\theta }_{(m,n+l-m)}}+2\sum _{m=1}^{N-(n+l)}{S}_{1(m,n+l+m)}{A}_m^{\ast }{A}_{n+l+m}{e}^{i{\psi }_{(m,n+l+m)}}\right]A_l^{\ast }{e}^{i{\psi }_{(l,n+l)}}$$ (A.5)

In the four “internal” summations, namely Eqs. (A.2) through (A.5), one mismatch of the trial wave-wave interaction is multiplied by another mismatch between three different waves, but one component of each mismatch is canceled out (e.g., the lth component in Eq. A.2). Accordingly, the final arguments of the complex exponential terms correspond to the phase mismatches between four waves. By employing the closure method of Benney and Saffman (1966), we only retain the four-wave interactions with matching indices (i.e., exact resonant interactions), and we neglect the terms involving other interactions (i.e., near resonant interactions). Each internal summation involves two multiplications of mismatches, resulting in a total of eight multiplications. Here, we present the resulting multiplications of mismatches:

• (i)

  $$\begin{gathered} {\theta }_{(m,l-m)}{\theta }_{(l,n-l)}=\int (k_m+k_{l-m}-k_l)+(k_l+k_{n-l}-k_n)dx \\ =\int k_m+k_{l-m}+k_{n-l}-k_{n}dx \end{gathered}$$ (A.6)

  The case of exact resonance does not exist.

• (ii)

  $$\begin{gathered} {\theta }_{(m,n-l-m)}{\theta }_{(l,n-l)}=\int \left(k_m+k_{n-l-m}-k_{n-l}\right)+\left(k_l+k_{n-l}-k_n\right)dx \\ =\int k_m+k_{n-l-m}+k_l-k_{n}dx \end{gathered}$$ (A.7)

  When $m=n-l\text{,}$, exact resonance is satisfied.
• (iii)

  $$\begin{gathered} {\theta }_{(m,n-l-m)}{\theta }_{(l,n-l)}=\int \left(k_m+k_{n-l-m}-k_{n-l}\right)+\left(k_l+k_{n-l}-k_n\right)dx \\ =\int k_m+k_{n-l-m}+k_l-k_{n}dx \end{gathered}$$ (A.8)

  The case of exact resonance does not exist.

• (iv)

  $$\begin{gathered} {\psi }_{(m,n-l+m)}{\theta }_{(l,n-l)}=\int \left(k_{n-l+m}-k_m-k_{n-l}\right)+\left(k_l+k_{n-l}-k_n\right)dx \\ =\int k_{n-l+m}-k_m+k_l-k_{n}dx \end{gathered}$$ (A.9)

  When $m=n$, exact resonance is satisfied.

• (v)

  $$\begin{gathered} -{\theta }_{(m,l-m)}{\psi }_{(l,n+l)}=\int -\left(k_m+k_{l-m}-k_l\right)+\left(k_{n+l}-k_l-k_n\right)dx \\ =\int -k_m-k_{l-m}+k_{n+l}-k_{n}dx \end{gathered}$$ (A.10)

  The case of exact resonance does not exist.

• (vi)

  $$\begin{gathered} -{\psi }_{(m,l+m)}{\psi }_{(l,n+l)}=\int -\left(k_{l+m}-k_m-k_l\right)+\left(k_{n+l}-k_l-k_n\right)dx \\ =\int -k_{l+m}+k_m+k_{n+l}-k_{n}dx \end{gathered}$$ (A.11)

  When $m=n$, exact resonance is satisfied.

• (vii)

  $$\begin{gathered} {\theta }_{(m,n+l-m)}{\psi }_{(l,n+l)}=\int \left(k_m+k_{n+l-m}-k_{n+l}\right)+\left(k_{n+l}-k_l-k_n\right)dx \\ =\int k_m+k_{n+l-m}-k_l-k_{n}dx \end{gathered}$$ (A.12)

  When $m=l$ and $m=n$, exact resonance is satisfied.

• (viii)

  $$\begin{gathered} {\psi }_{(m,n+l+m)}{\psi }_{(l,n+l)}=\int \left(k_{n+l+m}-k_m-k_{n+l}\right)+\left(k_{n+l}-k_l-k_n\right)dx \\ =\int k_{n+l+m}-k_m-k_l-k_{n}dx \end{gathered}$$ (A.13)

  The case of exact resonance does not exist.

Considering all the case of the exact resonances, Eq. (A.1) becomes:

$$\begin{gathered} \sum _{l=1}^{n-1}\left(-R_{2(l,n-l)}\frac{{\left(kC{C}_g\right)}_{lx}}{2{\left(kC{C}_g\right)}_l}-R_{3(l,n-l)}\frac{{\left(kC{C}_g\right)}_{n-lx}}{2{\left(kC{C}_g\right)}_{n-l}}\right)A_l{A}_{n-l}{e}^{i{\theta }_{(l,n-l)}} \\ +2\sum _{l=1}^{N-n}\left(-S_{2(l,n+l)}\frac{{\left(kC{C}_g\right)}_{lx}}{2{\left(kC{C}_g\right)}_l}-S_{3(l,n+l)}\frac{{\left(kC{C}_g\right)}_{n+lx}}{2{\left(kC{C}_g\right)}_{n+l}}\right)A_l^{\ast }{A}_{n+l}{e}^{i{\psi }_{(l,n+l)}} \\ +\left\{\begin{array}{l}\sum _{l=1}^{n-1}\left(-\frac{i{R}_{2(l,n-l)}{S}_{1(n-l,n)}}{4{\left(kC{C}_g\right)}_l}{\left|A_{n-l}\right|}^2{A}_n-\frac{i{R}_{3(l,n-l)}{S}_{1(l,n)}}{4{\left(kC{C}_g\right)}_{n-l}}{\left|A_l\right|}^2{A}_n\right)\hfill \\ +2\sum _{l=1}^{N-n}\left[\frac{i{S}_{2(l,n+l)}{S}_{1(n,l+n)}}{4{\left(kC{C}_g\right)}_l}{\left|A_{l+n}\right|}^2{A}_n-\frac{i{S}_{3(l,n+l)}{R}_{1(l,n)}}{4{\left(kC{C}_g\right)}_{n+l}}{\left|A_l\right|}^2{A}_n\right]\hfill \end{array}\right\} \end{gathered}$$ (A.14)

which is applied in Eq. (29).