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Simulation of breaking waves using the high-order
spectral method with laboratory experiments:
wave-breaking energy dissipation

Betsy Seiffert, Guillaume Ducrozet

► To cite this version:

Betsy Seiffert, Guillaume Ducrozet. Simulation of breaking waves using the high-order spectral method with laboratory experiments: wave-breaking energy dissipation. Ocean Dynamics, 2018, 68, pp.65-89.
10.1007/s10236-017-1119-3 . hal-01644316

HAL Id: hal-01644316

https://hal.science/hal-01644316v1

Submitted on 27 May 2025

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# Simulation of breaking waves using the High-Order Spectral method with laboratory experiments: Wave-breaking energy dissipation

Betsy R. Seiffert · Guillaume Ducrozet

Received: date / Accepted: date

**Abstract** We examine the implementation of a wave-breaking mechanism into a nonlinear potential flow solver. The success of the mechanism will be studied by implementing it into the numerical model HOS-NWT, which is a computationally efficient, open source code that solves for the free surface in a numerical wave tank using the High-Order Spectral (HOS) method. Once the breaking mechanism is validated, it can be implemented into other nonlinear potential flow models. To solve for wave-breaking, first a wave-breaking onset parameter is identified, and then a method for computing wave-breaking associated energy loss is determined. Wave-breaking onset is calculated using a breaking criteria introduced by Barthelemy et al. (submitted) and validated with the experiments of Saket et al. (2017). Wave-breaking energy dissipation is calculated by adding a viscous diffusion term computed using an eddy viscosity parameter introduced by Tian et al. (2008) and Tian et al. (2012), which is estimated based on the pre-breaking wave geometry. A set of two-dimensional experiments is conducted to validate the implemented wave breaking mechanism at a large scale. Breaking waves are generated by using traditional methods of evolution of focused waves and modulational instability, as well as

<footnote>B.R. Seiffert
Dept. of Ocean & Mechanical Eng., Florida Atlantic Univ., Boca Raton, FL 33431, USA
École Centrale de Nantes, LHEEA, UMR CNRS 6598, Nantes, France
Institut de Recherche Technologique Jules Verne, Bouguenais, France
E-mail: eseiffert@fau.edu</footnote>

<footnote>G. Ducrozet
École Centrale de Nantes, LHEEA, UMR CNRS 6598, Nantes, France</footnote>
irregular breaking waves with a range of primary frequencies, providing a wide range of breaking conditions to validate the solver. Furthermore, adjustments are made to the method of application and coefficient of the viscous diffusion term with negligible difference, supporting the robustness of the eddy viscosity parameter. The model is able to accurately predict surface elevation and corresponding frequency/amplitude spectrum, as well as energy dissipation when compared with the experimental measurements. This suggests the model is capable of calculating wave-breaking onset and energy dissipation successfully for a wide range of breaking conditions. The model is also able to successfully calculate the transfer of energy between frequencies due to wave focusing and wave breaking.

This study is limited to unidirectional waves but provides a valuable basis for future application of the wave-breaking model to a multidirectional wave field. By including parameters for removing energy due to wave-breaking into a nonlinear potential flow solver, the risk of developing numerical instabilities due to an overturning wave is decreased, thereby increasing the application range of the model, including calculating more extreme sea states. A computationally efficient and accurate model for the generation of a nonlinear random wave field is useful for predicting the dynamic response of offshore vessels and marine renewable energy devices, predicting loads on marine structures, and in the study of open ocean wave generation and propagation in a realistic environment.

**Keywords** Wave-breaking · High-Order Spectral method · Laboratory experiments · Numerical wave tank · Wave focusing

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# 1 Introduction

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In nature, a steep wave may grow to surpass some limiting threshold leading to a collapse of the water surface as a broken wave. Energy from the wave is transferred through the generation of currents and turbulence and changes occur in the spectral distribution of energy before and after a wave breaks (Iafrati (2009)), Iafrati et al. (2014), Babanin (2011), Tian et al. (2011), Shuxiu et al. (2015)). Wave breaking is an irregular and rapidly occurring two-phase process, making it a challenge to calculate using numerical models.

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Computations have been performed using some volume of fluid (VOF) solvers to calculate velocities in both air and water phases, vorticity, splash up
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(Chen et al. (1999)), current generation (Pizzo et al. (2016)), bubble entrainment (Deike et al. (2016)), effects of bubble entrainment on energy dissipation (Derakhti and Kirby (2014)), ocean-atmosphere interaction (Iafrati et al. (2014)) and sand transport (Fernandez-Mora et al. (2017)). Models such as REEF3D (Bihs et al. (2016)) and CFDShip-Iowa (Wang et al. (2009)) use a level set method to calculate a sharp interface at the free surface, which allows for the computation of some complex phenomena such as wave breaking and spray. Although wave breaking phenomena and associated physical processes can be computed, the drawback of these solvers is that they are computationally expensive. Consequently, these approaches are not suitable for large-scale wave propagation (on the order of 10 - 100 km²) that we target. In this study, we choose to use a non-linear potential flow formalism allowing a deterministic, highly non-linear and fully dispersive approach to the problem.

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Depending on the intended application of the wave model, it is of interest to determine methods of approximating some wave breaking processes in order to have a more efficient model. Ruvinsky et al. (1991) derived a form of the closed free-surface boundary conditions for weakly damped surface waves which included viscous dissipation by separating potential and vortical components of the flow. Dias et al. (2008) arrived at the same free-surface boundary conditions using the linear approximation of the Navier-Stokes equations. Longuet-Higgins (1992) first suggested the modified free-surface boundary conditions including viscous dissipation could be used to calculate the free surface with energy dissipation due to wave breaking.

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Ma et al. (2012), Derakhti et al. (2016a), and Derakhti et al. (2016b) use the model NHWAVE, a shock-capturing non-hydrostatic model for nonlinear free-surface wave processes. Governing equations are the incompressible Navier-Stokes equations written in surface and terrain-following form using a σ-coordinate and are discretized by a combined finite volume and finite difference scheme. The shock-capturing scheme is based on Godunov-type approach, which can deal with a discontinuous flow and are able to track the location of wave breaking without requiring criterion telling the model when and where the wave is breaking.

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Kennedy et al. (2000) and Chen et al. (2000) use a momentum-conserving eddy viscosity technique to model breaking in a model that solves the Boussinesq-type equations in shallow water. Kurnia and van Groesen (2014) demonstrated success in modeling wave breaking onset and energy dissipation for deep water irregular breaking waves and breaking waves over variable bathymetry using a
high order Hamiltonian model for water waves. They use a kinematic criterion for wave breaking onset and an eddy viscosity model for energy dissipation which is an extension of the model used by Kennedy et al. (2000) but applicable to fully dispersive waves.

Tian et al. (2010) presented an eddy viscosity model to simulate energy dissipation due to wave breaking in a potential flow solver. Breaking onset was calculated based on the local surface elevation slope, i.e. $-\partial\eta/\partial x$, surpassing a critical value of 0.95, where $\eta$ is the surface elevation. The authors developed empirical formulae to calculate the viscous diffusion parameter $\nu_{eddy}$ based on their measurements of pre-breaking wave geometry. Comparisons are made between the model and experiments for plunging and spilling breakers generated by dispersive focusing and modulational instability. An attempt was made to validate their model for irregular breaking waves by increasing the significant wave height of some large amplitude non-breaking irregular waves used in the experiments of Goullet and Choi (2011), to get breaking waves in the numerical model. The model was generally successful in predicting the overall energy dissipation due to wave breaking. However for some of the larger breaking waves, there was significant delay in the prediction of breaking onset, causing the agreement between the numerical model and the experiments in the region of wave breaking to not be quite as good for some smaller breaking waves.

Chalikov and Sheinin (2005) introduced a method of calculating the free-surface of a breaking wave in a potential flow solver which includes a filtering term to reduce spurious energy at high frequencies and a diffusion term to remove wave-breaking associated energy. Wave breaking is determined by the curvature of the front face surpassing a critical threshold. The coefficient of diffusion is also dependent on this surface curvature. The authors argue this method provides a stable solution for most breaking wave cases, but may not be suitable for waves with very high initial steepness or energy.

Several studies have been conducted on understanding the process of energy dissipation and the transfer of energy between frequency components during wave breaking. Some experimental studies on wave breaking dissipation include Rapp and Melville (1990), Lim (2010), and Na et al. (2016). Iafrati (2009) showed using numerical calculations that viscous dissipation sharply grows just after breaking inception and is localized around the entrapped air bubbles during the most energetic phase of breaking, which lasts about one wave period. After this phase, the total dissipation decays as $t^{-2}$. Tian et al.
(2011) studied the spatial evolution of wave frequency spectra for breaking and non-breaking wave groups generated using dispersive focusing. Experimental measurements show a rapid energy transfer to above-peak frequencies just prior to breaking. The energy is then dissipated in these higher frequencies and returns to almost the initial energy spectrum after breaking, with a slight downshift in the peak spectral frequency.

This following study investigates wave-breaking associated energy dissipation calculated using the numerical model HOS-NWT and validated with large-scale experiments. HOS-NWT is a computationally efficient, open source code, which solves for the free surface in a numerical waves tank using the High-Order Spectral (HOS) method. Once the wave breaking model is validated in HOS-NWT, it can be implemented in other nonlinear potential flow models. The implemented wave breaking onset criteria and energy dissipation is validated for unidirectional wave fields first. Although we can generate repeatable breaking waves using energy focusing and modulational instability, there are different physical processes at the possible origin of breaking waves in a realistic environment. When generating irregular waves, we try to recreate these physical processes in the laboratory, thus, the main physical features occurring in the ocean are present and consequently are representative of real conditions. Validating the model for unidirectional irregular breaking waves is the first step towards developing a model which can solve for multidirectional irregular breaking waves over a large domain and in a realistic environment.

# 2 Numerical methods

## 2.1 Nonlinear wave model

The High-Order Spectral (HOS) method was originally introduced by Dommermuth and Yue (1987) and West et al. (1987) to solve for highly nonlinear wave propagation over a flat bottom. This original version has an open fluid domain with periodic boundary conditions. The initial free surface elevation is specified, and then marched in time to get the evolution of the free surface. Inside the domain the fluid is assumed to be incompressible and inviscid with irrotational flow, and therefore is governed by the Laplace equation. The free surface boundary conditions are written in terms of the surface quantities $η$ and the surface potential $\tilde{\phi}(\mathbf{x}, t) = \phi(\mathbf{x}, z = \eta, t)$, where $\mathbf{x}$ is the horizontal
(x, y) vector. The kinematic and dynamic free surface boundary conditions are therefore written as

$$ \frac{\partial \eta}{\partial t} = (1 + |\nabla \eta|^2) \frac{\partial \phi}{\partial z} - \nabla \tilde{\phi} \cdot \nabla \eta \quad (1) $$

$$ \frac{\partial \tilde{\phi}}{\partial t} = -g\eta - \frac{1}{2} |\nabla \tilde{\phi}|^2 + \frac{1}{2} (1 + |\nabla \eta|^2) \left( \frac{\partial \phi}{\partial z} \right)^2 \quad (2) $$

on $z = \eta(\mathbf{x}, t)$, where $\nabla$ is the horizontal gradient.

One such available open-source model which uses the HOS method is HOS-ocean, which has been validated for a wide range of configurations (Ducrozet et al. (2007), Ducrozet et al. (2016a), Ducrozet et al. (2016b)) and has been extended to be used in more complex configurations, such as a varying bathymetry (Gouin et al. (2016) and Gouin et al. (2017)).

The purpose of this study is to validate the implementation of a wave breaking mechanism into a numerical model which solves for the free surface using the HOS method. To do this, a series of laboratory experiments are conducted, and will be discussed in Section 3. To effectively validate the model, numerical calculations must be done in a domain that mimics laboratory conditions. The model HOS-NWT has been developed as a numerical wave tank (Ducrozet et al. (2012), and Ducrozet et al. (2016b)), and it incorporates a wavemaker boundary condition to generate waves, perfectly reflective side walls and an absorbing zone to act as a numerical beach and absorb reflections in the domain. Once the wave breaking mechanism is validated using HOS-NWT, and the presented laboratory experiments, it can be incorporated into other nonlinear potential flow models.

## 2.2 Calculating wave breaking in HOS-NWT

Following Tian et al. (2010), and Tian et al. (2012), an eddy viscosity model is implemented into HOS-NWT to model breaking waves. Breaking onset is determined by the breaking criterion proposed by Barthelemy et al. (submitted). Using this criterion, a wave is considered broken at the moment the ratio of water particle velocity under the crest ($U$) to crest velocity ($C$) exceeds the threshold value of 0.85. Fedele (2014), Fedele et al. (2016) and Barthelemy et al. (submitted) observed a slowdown of the crest velocity just before the wave breaks, and because of this, they suggest the breaking criterion is closer to $U/C = 0.85$, instead of the traditional value of $U/C = 1.0$, given by Wu and
Nepf (2002), for example. The success of the reformulated breaking criterion was validated with the experiments of Saket et al. (2017). Breaking onset time and location calculated by HOS-NWT using this criterion have been validated in Seiffert et al. (2017), with the same large-scale unidirectional wave breaking experiments which will be used in this study to validate energy dissipation. Once breaking is initiated, breaking time and length scales are calculated based on the pre-breaking local wave geometry, and an eddy viscosity parameter is estimated using these time and length scales to dissipate energy from the breaking event.

Eddy viscosity is introduced as a diffusion term in the kinematic and dynamic free surface boundary conditions, therefore Eqs. 1 and 2 are modified as

$$ \frac{\partial \eta}{\partial t} = (1 + |\nabla \eta|^2) \frac{\partial \phi}{\partial z} - \nabla \tilde{\phi} \cdot \nabla \eta + 2\nu_{eddy} \nabla \cdot \nabla \eta \quad (3) $$

$$ \frac{\partial \tilde{\phi}}{\partial t} = -g\eta - \frac{1}{2} |\nabla \tilde{\phi}|^2 + \frac{1}{2}(1 + |\nabla \eta|^2) \left(\frac{\partial \phi}{\partial z}\right)^2 + 2\nu_{eddy} \nabla \cdot \nabla \tilde{\phi}. \quad (4) $$

The eddy viscosity $\nu_{eddy}$ is estimated as in Tian et al. (2010) and Tian et al. (2012) by using the time and length scales associated with the breaking event. A ramp function of length $L_c$ is applied between the inviscid and viscous domain to avoid irregularities, where $L_c$ is the length under the crest between two consecutive zero-crossing points.

## 3 Experimental design

Experiments are conducted in the 2D wave flume in the Hydrodynamics, Energetics & Atmospheric Environment Lab (LHEEA) at École Centrale de Nantes (ECN). The flume is 148 m long, 5 m wide and 3 m deep with a parabolic shaped absorbing beach approximately 8 m long. Breaking waves are generated using traditional methods of energy focusing and modulational instability, as well as by generating some steep irregular waves. Breaking waves in an irregular wave series are also generated by adding amplitude/frequency components of an energy focusing wave to the amplitude/frequency components of the irregular wave series, resulting in one or two large breaking waves. These waves are run at a range of primary frequencies, providing a wide range of experimental measurements of breaking waves of different types, strengths and scales. Waves are generated by a flap-type wavemaker with bottom hinge 0.48 m
above the flume bottom. Surface elevation is measured by 14 resistance-type wave gauges.

To validate our experimental measurements, and to determine the effects of small versus large scale, the experiments of Tian et al. (2012) are reproduced at a 4:1 scale using Froude’s similitude law. Wave gauges are positioned at 7.32, 10.12, 13.28, 16.16, 19.08, 21.52, 23.96, 26.40, 28.88, 31.56, 34.28, 37.68, 40.92, and 44.24 m distance from the mean position of the wavemaker. In Tian’s experiments, waves are generated by a piston-type wavemaker using energy focusing (EF) and modulational instability (BFI) methods. For breaking waves generated using energy focusing, surface elevation at the wavemaker is described as

$$ \eta(0, t) = \sum_{n=1}^{N} a_n \cos(-k_n x_b - \omega_n(t - t_b)) \quad (5) $$

where $a_n$ is the amplitude of the nth wave component, $k_n$ is the wavenumber, $\omega_n = 2\pi f_n$ is the angular frequency, N is the number of frequency components, and $x_b$ and $t_b$ are the input breaking location and time. For waves generated using modulational instability, surface elevation at the wavemaker is described by

$$ \eta(t) = a_0 \cos(\omega_0 t) + b \cos \left( \omega_1 t - \frac{\pi}{4} \right) + b \cos \left( \omega_2 t - \frac{\pi}{4} \right). \quad (6) $$

$a_0$ and $\omega_0$ are the amplitude and angular frequency of the carrier wave, and $b$, $\omega_1$ and $\omega_2$ are the amplitude and frequencies of the sideband perturbations, where $\omega_1 = \omega_0 - \Delta\omega/2$, and $\omega_2 = \omega_0 + \Delta\omega/2$. Input parameters for these generated waves are given in Table 1. Wave probe measurements are collected for 80 s for waves generated using energy focusing and for 100 s for waves generated using modulational instability.

For the large scale experiments conducted at LHEEA, wave gauges are positioned at 7.32, 14.5, 20.00, 21.20, 22.41, 23.60, 24.69, 26.05, 27.94, 30.06, 34.26, 37.88, 43.66 and 50.23 m from the mean position of the wavemaker. Input parameters are given in Table 2 for waves generated using energy focusing by use of Eqn. (5). Table 3 provides parameters input into Eqn. (6) to generate breaking waves by modulational instability. Table 4 provides the peak period ($T_p$) and significant wave height ($H_s$) nondimensionalized by peak wavelength ($L_p$) for irregular waves, along with parameters for the energy focused wave components, if included. The surface elevation for energy focused waves are measured for 100 s and for modulational instability and irregular waves sur-
<table><caption>**Table 1** Input parameters for the Tian et al. (2012) experiments reproduced at a 4:1 scale using Froude scaling. Wave probe measurements are be collected for 80 s for waves generated using energy focusing (EF) and for 100 s for waves generated using modulational instability (BFI). N = 128 for energy focused waves.</caption><thead><tr><th>Wave</th><th>f<sub>c</sub> (Hz)</th><th>Δf (Hz)</th><th>ε<sub>n</sub></th><th>t<sub>b</sub> (s)</th><th>x<sub>b</sub> (m)</th></tr></thead><tbody><tr><td>EF1</td><td>0.85</td><td>0.7</td><td>0.0020</td><td>50.0</td><td>28.0</td></tr><tr><td>EF2</td><td>0.85</td><td>0.7</td><td>0.0032</td><td>50.0</td><td>28.0</td></tr><tr><td>EF3</td><td>0.85</td><td>0.7</td><td>0.0045</td><td>50.0</td><td>28.0</td></tr><tr><td>EF4</td><td>0.85</td><td>0.7</td><td>0.0058</td><td>50.0</td><td>28.0</td></tr></tbody></table><table><thead><tr><th>Wave</th><th>f<sub>0</sub> (Hz)</th><th>Δf/f<sub>0</sub></th><th>ε = k<sub>0</sub>a<sub>0</sub></th><th>b/a<sub>0</sub></th></tr></thead><tbody><tr><td>BFI1710</td><td>0.85</td><td>0.166</td><td>0.10</td><td>0.5</td></tr><tr><td>BFI1712</td><td>0.85</td><td>0.200</td><td>0.12</td><td>0.5</td></tr><tr><td>BFI1716</td><td>0.85</td><td>0.266</td><td>0.16</td><td>0.5</td></tr><tr><td>BFI1720</td><td>0.85</td><td>0.330</td><td>0.20</td><td>0.5</td></tr></tbody></table>

<table><caption>**Table 2** Parameters for waves generated using energy focusing. N = 64 and t<sub>b</sub> = 50 s. * denotes an unbroken wave.</caption><thead><tr><th>Wave</th><th>f<sub>c</sub> (Hz)</th><th>Δf (Hz)</th><th>ε<sub>n</sub> = k<sub>n</sub>a<sub>n</sub></th><th>x<sub>b</sub> (m)</th></tr></thead><tbody><tr><td>EF6-54</td><td>0.6</td><td>0.54</td><td>[0.0047* 0.0048 0.0050 0.0051]</td><td>22</td></tr><tr><td>EF7-64</td><td>0.7</td><td>0.64</td><td>[0.0048 0.0054 0.0060]</td><td>24</td></tr><tr><td>EF8-72</td><td>0.8</td><td>0.72</td><td>[0.0062 0.0070]</td><td>22</td></tr></tbody></table>

<table><caption>**Table 3** Input parameters for waves generated using modulational instability.</caption><thead><tr><th>Wave</th><th>f<sub>0</sub> (Hz)</th><th>Δf/f<sub>0</sub></th><th>ε = k<sub>0</sub>a<sub>0</sub></th><th>b/a<sub>0</sub></th></tr></thead><tbody><tr><td>MI0719</td><td>0.7</td><td>0.317</td><td>0.19</td><td>0.5</td></tr><tr><td>MI0720</td><td>0.7</td><td>0.330</td><td>0.20</td><td>0.5</td></tr></tbody></table>

<table><caption>**Table 4** Input parameters for irregular waves and irregular waves with a superimposed energy focusing wave. Additional input parameters for the focusing component are N = 64, x<sub>b</sub> = 22 m and t<sub>b</sub> = 50 s. *IRR-4 did not contain any breaking waves.</caption><thead><tr><th rowspan="2">Wave</th><th colspan="2">Irregular</th><th colspan="3">Focusing</th></tr><tr><th>T<sub>p</sub></th><th>H<sub>s</sub>/L<sub>p</sub></th><th>f<sub>c</sub></th><th>Δf</th><th>ε<sub>n</sub></th></tr></thead><tbody><tr><td>IRR-4*</td><td>2.1</td><td>0.030</td><td>-</td><td>-</td><td>-</td></tr><tr><td>IRR-1</td><td>2.2</td><td>0.037</td><td>-</td><td>-</td><td>-</td></tr><tr><td>IRR-2</td><td>2.2</td><td>0.040</td><td>-</td><td>-</td><td>-</td></tr><tr><td>IRR-5</td><td>2.1</td><td>0.035</td><td>-</td><td>-</td><td>-</td></tr><tr><td>IRREF-1</td><td>2.2</td><td>0.02</td><td>0.7</td><td>0.64</td><td>0.005</td></tr><tr><td>IRREF-2</td><td>2.2</td><td>0.02</td><td>0.7</td><td>0.64</td><td>0.0055</td></tr><tr><td>IRREF-3</td><td>2.2</td><td>0.03</td><td>0.7</td><td>0.64</td><td>0.0055</td></tr></tbody></table>
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face elevation is measured for 150 s. All measurements are collected at 128 Hz sampling rate. Further details on wave generation and experimental setup can be found in Seiffert et al. (2017).

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# 4 Results and Discussion

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Calculations are performed using a numerical domain of 100 m length discretized with 1025 modes in the x-direction and 65 modes in the z-direction, and with HOS order 5 and wavemaker order of 2, following a sensitivity study, detailed in Seiffert et al. (2017). The numerical wavemaker is a flap-type with hinge 0.48 m from the bottom, to correspond to the wavemaker used in the LHEEA experiments. A spatial interpolation at 5 times the spatial discretization is applied using a cubic spline to the surface elevation, before breaking onset and eddy viscosity is calculated. This allows the breaking and the pre-breaking wave geometry measurements to be calculated at a higher spatial resolution, thus increasing the accuracy. The frequency for output data used for analysis was 20 times the central frequency for energy focused and modulational instability waves, and 17 Hz for irregular waves.

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## 4.1 Tian, et al. (2012) scaled waves

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Figures 1 and 2 compare surface elevation and corresponding frequency/amplitude spectra for the energy focused wave EF4 measured during the LHEEA experiments and Tian experiments with calculations made using HOS-NWT. Figures 3 and 4 compare measurements and calculations for surface elevation and corresponding frequency/amplitude spectra for breaking waves generated using modulational instability, BFI1720. Measurements from the Tian experiments are scaled at a 4:1 scale using Froude's similitude law. The wave gauges selected for Figs. 1 - 4 are the first and last wave gauges, as well as 4 additional wave gauges corresponding to a region of wave breaking for EF4 and BFI1720. For frequency and energy analysis of the wave generated using modulational instability, a moving time window of 100 s is selected to follow the same wave packet from WG1 to WG14.

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There is generally good agreement between the surface elevation and corresponding frequency/amplitude spectra, particularly for the large breaking waves for both the LHEEA and Tian experiments, as well as with the HOS-
<figure><img src="image_3.png" /><figcaption>**Fig. 1** Measured surface elevation at six selected wave gauges for the breaking wave EF4 taken during the LHEEA experiments is compared with measurements made during the Tian experiments and calculations made using HOS-NWT.</figcaption></figure>

NWT calculations. Figure 4 clearly shows the development of the Benjamin-Feir instability with the transfer of energy from the center carrier frequency to the side band frequencies. Figures 1 and 3 show slightly larger surface elevation measured in the Tian experiments for the waves leading and following a large breaking wave. Additionally, there is slightly more energy in the 0.99 Hz sideband frequency measured during the Tian experiments than the LHEEA experiments and HOS-NWT calculations, in Fig. 4.

The long-time integration of the surface variance ($<\eta^2> = \int_0^T \eta(x, t)^2 dt$) is given in Figs. 5 and 6 for waves calculated using HOS-NWT and measurements made during the LHEEA and Tian experiments for each of the waves generated using energy focusing and modulational instability, respectively. According to linear theory $<\eta^2>$ is proportional to the passage of energy at a location and therefore is representative of the reduction of energy due to wave breaking at a location. This is derived from the relationship that total energy at a given location $E(x)$ is equal to the time integration of the energy flux at that
<figure><img src="image_4.png" /><figcaption>**Fig. 2** The frequency/amplitude spectra corresponding to the measured surface elevation for the breaking wave EF4 taken during the LHEEA experiments is compared with measurements made during the Tian experiments and calculations made using HOS-NWT.</figcaption></figure>

location, or $E(x) = \int_0^T F(x,t)dt$. According to the deep water assumption, $F(x,t) = \rho g C_g \eta(x,t)^2$, and therefore $\langle \eta^2 \rangle$ is proportional to $E(x)$. Further discussion can be found in Tian et al. (2010) or Derakhti and Kirby (2016), for example.

During these experiments, each wave was repeated three times and experimental measurements presented for energy are an average taken of the three. The overall reduction of energy due to wave breaking measured in the LHEEA experiments agrees well with the reduction of energy from the Tian experiments and HOS-NWT calculations. Average normalized root-mean-square error (NRMSE) of the 14 wave probes and normalized with the first wave probe gives a percent error of 3.5% between Tian and LHEEA measurements and 3.5% between calculations and LHEEA measurements for the wave EF4. NRMSE for the wave BFI1720 between Tian and LHEEA measurements is 5.9% and is 3.5% between calculations and LHEEA measurements. It was noted in Tian et al. (2010) that for large breaking waves, there were some dif-
<figure><img src="image_5.png" /><figcaption>**Fig. 3** Measured surface elevation at six selected wave gauges for the wave series BFI1720 taken during the LHEEA experiments is compared with measurements made during the Tian experiments and calculations made using HOS-NWT.</figcaption></figure>

ferences between experiments and calculations in the region of wave breaking. Figures 5 and 6 show excellent agreement between calculations and experiments in the region of wave breaking, suggesting both wave breaking onset and energy dissipation in this critical region are calculated correctly. This suggests the breaking criteria used in HOS-NWT may be better suited to calculating surface elevation in the region of wave breaking than the crest-steepness parameter used in the calculations of Tian et al. (2010).

In Fig. 5, $<\eta^2>$ for the LHEEA experiments at the first wave gauges appear to be slightly larger than calculations, particularly for the large breaking wave EF4. However, a comparison of surface elevation calculations and measurements for EF4 at WG1 observed in Fig. 1, suggests small differences in the leading and trailing waves of the larger breaking waves may be compounding to cause the overall energy to be larger.

Differences in measurements made during the LHEEA and Tian experiments are likely due to the waves generated during Tian's experiments not be-
<figure><img src="image_6.png" /><figcaption>**Fig. 4** The frequency/amplitude spectra corresponding to the measured surface elevation for the breaking wave series BFI1720 taken during the LHEEA experiments is compared with measurements made during the Tian experiments and calculations made using HOS-NWT.</figcaption></figure>

ing generated exactly. According to Tian et al. (2012), adjustments were made to the wave amplitude to achieve non-breaking and breaking wave groups. Therefore, the steepness of the wave components of the generated wave may be slightly different than the steepness input to the wavemaker initially if amplitude had been adjusted *in situ*. The waves generated in the LHEEA experiments are generated exactly according the equations for energy focused waves and waves generated using modulational instability, and therefore are easily reproducible experimentally and numerically. Differences can also be explained by viscous effects that could be present in Tian's experiments due to the small scale, as well as reflection from the end of the flume, which was 15 m in length, and 3.96 m from the last wave gauge.

The largest difference between both the Tian and LHEEA experimental measurements and calculations is observed at WG14 in Fig. 3. The agreement between measured and calculated energy at WG14 for the wave BFI1720 in Fig. 6 suggests that this difference is likely due to a phase shift, which ap-
<figure><img src="image_7.png" /><figcaption>**Fig. 5** The long time integration of the surface variance ($<\eta^2> = \int_0^\tau \eta(x, t)^2 dt$) is computed using HOS-NWT and compared with measurements made during the LHEEA and Tian experiments for waves generated using energy focusing.</figcaption></figure>

pears to be less severe for the large-scale experiments. A possible explanation for this could be that too much energy is being removed from the lower frequency components when the eddy viscosity is applied, influencing the phase and/or group velocities. This suggests that a re-evaluation of the way the eddy viscosity is applied may produce more accurate results. Overall, the parameterization of the Tian model for wave-breaking energy dissipation appears to be independent of scale as the exact parameters used for the small scale Tian et al. (2012) experiments were successfully applied at the larger scale. In the next sections, we apply this model to a range of large scale wave-breaking experiments.

## 4.2 Energy focused waves

Surface elevation and corresponding frequency/amplitude are predicted well by the numerical model when compared with the LHEEA experimental measurements for the energy focused wave EF6-54 with $\epsilon_n = 0.0051$, shown in Figs. 7 and 8. There is a plunging breaker at $t_b = 49.9$ s and $x_b = 22.53$ m, where $t_b$ and $x_b$ are the time and location of breaking measured during
<figure><img src="image_8.png" /><figcaption>**Fig. 6** The long time integration of the surface variance ($<\eta^2> = \int_0^\tau \eta(x, t)^2 dt$) is computed using HOS-NWT and compared with measurements made during the LHEEA and Tian experiments for waves generated using modulational instability.</figcaption></figure>

<table><caption>**Table 5** Percentages of average normalized root-mean-square error (NRMSE) between LHEEA experimental measurements and HOS-NWT calculations for selected waves.</caption><thead><tr><th rowspan="2">Wave</th><th colspan="3">NRMSE %</th></tr><tr><th>f/f<sub>c</sub> &lt; 1.0</th><th>f/f<sub>c</sub> = 1.0 - 4.0</th><th>Total</th></tr></thead><tbody><tr><td>EF4</td><td>-</td><td>-</td><td>3.5</td></tr><tr><td>BFI1720</td><td>-</td><td>-</td><td>3.5</td></tr><tr><td>EF6-54 ϵ<sub>n</sub> = 0.0051</td><td>1.5</td><td>18.0</td><td>6.3</td></tr><tr><td>EF7-64 ϵ<sub>n</sub> = 0.0061</td><td>2.5</td><td>12.1</td><td>4.0</td></tr><tr><td>EF8-72 ϵ<sub>n</sub> = 0.0070</td><td>3.0</td><td>11.8</td><td>4.0</td></tr><tr><td>MI0719</td><td>-</td><td>-</td><td>6.1</td></tr><tr><td>MI0720</td><td>-</td><td>-</td><td>6.3</td></tr><tr><td>IRR-5</td><td>3.5</td><td>7.9</td><td>5.1</td></tr><tr><td>IRREF-1</td><td>3.7</td><td>7.4</td><td>3.7</td></tr></tbody></table>

experiments. Some differences in the surface elevation in the region of active wave breaking can be attributed to the approximation of the free surface by HOS-NWT. For example, the wave given in Fig. 7 breaks just after WG5. At WG6, the wave has overturned but there is little reduction in the actual height of the measured water surface. However, the wave-breaking mechanism in HOS-NWT has begun to remove energy from the wave, preventing the in-
<figure><img src="image_9.png" /><figcaption>**Fig. 7** Surface elevation measured during experiments is compared with surface elevation calculated using HOS-NWT for EF6-54 and $\epsilon_n = 0.0051$.</figcaption></figure>

stability that an overturning wave presents in solving for potential flow, and resulting in a smaller calculated surface elevation just after $t = 50$ s. At WG7, the broken wave has resulted in bubbly white water left at the water surface which would not be accounted for in the calculated approximation, therefore resulting in a larger measured surface elevation.

Figure 8 shows a slightly larger amplitude is measured for frequencies between 0.5 and 1 Hz at WG7 and WG14. As these are large plunging waves, the differences could be the result of three-dimensional effects in the wave flume. No downshift of the peak spectral frequency is observed in either the measured or calculated frequency/amplitude spectra, which is in agreement with Shuxiu et al. (2015) but unlike observations made by Tian et al. (2011) or Babanin et al. (2010), for example. It is unclear the reason for this difference, but may be due to some non-linear effects, viscous effects, or the perception of a reduction of spectral peak amplitude as a slight downshift.

The normalized energy distribution for low frequency ($f/f_c < 1.0$) and high frequency ($1.0 < f/f_c < 4.0$) regions as well as total energy ($0.0 < f/f_c < 0.5$)
<figure><img src="image_10.png" /><figcaption>**Fig. 8** Frequency/amplitude spectra corresponding to surface elevation measured during experiments and calculated using HOS-NWT for EF6-54 and $\epsilon_n = 0.0051$.</figcaption></figure>

4.0) is given in Figs. 9 - 11 for each of the wave steepnesses generated with the frequencies EF6-54, EF7-64 and EF8-72, respectively. Energy is calculated by integrating amplitude over the wave frequency spectrum at each wave gauge, and normalizing with the energy at WG1. This method for calculating energy was suggested by Sriram et al. (2015) as a means of obtaining the variation of energy along the flume. Each wave was repeated 5 times during experiments, and the measurements presented for energy are an average of the 5. The model is able to predict well pre- and post-breaking energy at each of the frequencies, and percentages of average normalized RMSE are given in Table 5. In some of the cases, the model calculates less energy at the higher frequencies than measured during experiments, but this is a significantly smaller portion of the total energy than the lower frequencies, so the agreement for total energy is still good. If we examine WG14 in Figs. 7, it appears this difference may be mainly attributed to differences in the smaller trailing waves, rather than the breaking wave itself. A sensitivity study was conducted on the effects of a higher wave breaking onset threshold of 1.0 on energy dissipation for the single
<figure><img src="image_11.png" /><figcaption>**Fig. 9** Normalized energy for low frequency ($f/f_c < 1.0$) and high frequency ($f/f_c = 1.0 - 4.0$) regions, and total energy for EF6-54, $\epsilon_n = 0.0047$, 0.0048, 0.0050, and 0.0051. The solid red line indicates breaking onset measured during experiments and the dashed blue line indicates breaking onset calculated by HOS-NWT.</figcaption></figure>

breaking wave in EF6-54 with $\epsilon_n = 0.0051$. The higher threshold was found to have negligible difference on the overall energy dissipation, because in this case a breaking wave is still calculated, just slightly later. The main difference in using a higher threshold would be cases where breaking waves observed in experiments are not calculated due to the threshold being too high. Further discussion on breaking onset threshold can be found in Seiffert et al. (2017).

To investigate the transfer of energy between frequencies in more detail, Fig. 12 plots the change in amplitude at a given frequency from the amplitude measured at WG1 ($\bar{a}$) at a given location $x$ along the flume. Positive values (warm colors) mean there is more energy at the given frequency than measured or calculated at WG1; negative values (cool colors) mean there is less energy at that frequency than at WG1. The solid white line indicates wave breaking onset and the black dashed lines indicate the region of primary frequencies, or the frequencies that were input to the wavemaker.
<figure><img src="image_12.png" /><figcaption>**Fig. 10** Normalized energy for low frequency ($f/f_c < 1.0$) and high frequency ($f/f_c = 1.0 - 4.0$) regions, and total energy for EF7-64, $\epsilon_n = 0.0048, 0.0054$, and $0.0061$. The solid red line indicates breaking onset measured during experiments and the dashed blue line indicates breaking onset calculated by HOS-NWT.</figcaption></figure>

Some interesting observations can be made regarding the transfer of energy between frequencies in Figs. 9 and 12. As wave energy progresses from $x = 0$ m to approximately $x = 18$ m, Fig. 9 shows a rapid increase in energy in the higher frequencies and Fig. 12 shows a transfer of energy from the higher primary frequencies to the above primary frequencies. Just before the wave focuses, both figures show a transfer of energy to the lower frequencies at approximately $x = 21$ m, followed by a sharp transfer of energy back to the high frequencies just as the wave breaks at approximately $x = 22$ m. A significant portion of the energy loss occurs at the higher primary frequencies, whereas the lower frequencies as well as the above primary frequencies return to close to pre-focusing values, with little energy loss. It is interesting to note that this transfer of energy occurs in the non-breaking case ($\epsilon_n = 0.0047$) as well, therefore the energy transfer can be attributed to nonlinearities as well as wave breaking. A similar transfer of energy followed by dissipation in the
<figure><img src="image_13.png" /><figcaption>**Fig. 11** Normalized energy for low frequency (f/f_c < 1.0) and high frequency (f/f_c = 1.0 - 4.0) regions, and total energy for EF8-72, ε_n = 0.0062 and 0.0070. The solid red line indicates breaking onset measured during experiments and the dashed blue line indicates breaking onset calculated by HOS-NWT.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>**Fig. 12** At a given location along the flume x, $\bar{a}$ plotted by the colored contour represents the difference in amplitude at a given frequency from the amplitude measured at WG1 for EF6-54, $\epsilon_n$ = 0.0047, 0.0048, 0.0050, and 0.0051. The solid white line indicates breaking onset and the black dashed lines indicate the region of primary frequencies.</figcaption></figure>

high frequencies was observed for each of the $\epsilon_n$ for EF7-64 and EF8-72 (not shown). Similar observations have been made by Sriram et al. (2015), Tian et al. (2011), and Rapp and Melville (1990), among others.

Another interesting characteristic that can be observed from Fig. 12 is that HOS-NWT calculates some transfer of energy within the primary frequencies
that are also measured in experiments. For example, there is a significant gain
in energy at the frequency $f = 0.75$ Hz for each of the initial wave steepness
just after wave focusing or breaking, likely due to the initial spectrum having
lower energy at this frequency.

These figures suggest that HOS-NWT is successful in calculating the non-
linear transfer of energy between frequencies for highly nonlinear waves, and
the reduction of energy due to wave breaking. Furthermore, no parameteriza-
tion of the dissipation model is done here, suggesting the wave-breaking model
is robust and accurate.

4.3 Breaking waves generated using modulational instability

Figures 13 and 14 compare surface elevation and corresponding frequency/amplitude spectra for the time series MI0719 with wave breaking generated using modulational instability. For frequency and energy analysis, a moving time window of approximately 100 s is selected to follow the same wave packet from WG1 to WG14. In Fig. 13, a wave in the energy packet P1 breaks between WG10 and WG11, and a wave in the packet P2 breaks between WG11 and WG12. HOS-NWT is able to successfully predict the surface elevation, as well as the primary, sub and superharmonic frequencies pre- and post wave breaking up to WG14. At WG14, more energy is calculated in the center frequency ($f_c = 0.7$) than experiments, and less energy in the lower sideband frequency, and there appears to be a phase shift in the surface elevation. Similar to observations made by Tulin and Waseda (1999), the peak spectral amplitude shifts from the center frequency (0.7 Hz) to the lower frequency sideband (0.59 Hz).

The long time integration of the surface variance ($<\eta^2> = \int_0^\tau \eta(x,t)^2 dt$) calculated using HOS-NWT is compared with measurements made during the LHEEA experiments for waves generated using modulational instability in Fig. 15. This method of comparing wave breaking associated energy loss is chosen instead of integrating over each frequency/amplitude spectrum as the input spectrum for these waves is composed of only three frequency components (the central frequency and the two sidebands). It is noted there is a vertical offset of 0.1 when plotting MI0720 so that it can easily be distinguished from MI0719. The experimental measurements presented are the average values of $<\eta^2>$ taken over the 5 runs. Although there are some differences in energy calculated versus measured, HOS-NWT is able to capture overall energy losses due to wave breaking. NRMSE between measurements and calculations is 6.3%
<figure><img src="image_15.png" /><figcaption>**Fig. 13** Surface elevation measured during experiments is compared with surface elevation calculated using HOS-NWT for MI0719.</figcaption></figure>

for MI0720 and 6.1 % for MI0719 (also given in Table 5.) At WG4 in Fig. 13, the measured surface elevation appears larger everywhere than the calculated surface elevation and there is a significant difference in <η²> at WG3 - WG7 for MI0719 and WG3 and WG4 for MI0720. It is unclear if the cause of this difference is associated with wave breaking and the influence of wave breaking on the nonlinear energy transfer that occurs during modulational instability wave propagation. Any difference in the carrier wave could lead to differences in the transfer of energy to the sideband frequencies. It is possible experimental measurements could be larger due to some three-dimensional effects in the flume, not accounted for in calculations.

The agreement between measured and calculated energy at WG14 for MI0719 suggests differences in the frequency/amplitude spectrum as well as the difference in η at WG14 in Figs. 13 and 14 are likely due to a phase shift. This may be the result of wave breaking or nonlinear effects or both, and similar differences observed in the surface elevation and frequency/amplitude of unbroken waves suggest this is not entirely the result of wave breaking.
<figure><img src="image_16.png" /><figcaption>**Fig. 14** Frequency/amplitude spectra corresponding to surface elevation measured during experiments and calculated using HOS-NWT for MI0719.</figcaption></figure>

However, it is possible that too much energy is removed from the lower frequency components when calculating wave breaking associated energy loss in HOS-NWT, resulting in the phase shift.

A study was done to determine whether a dynamic application of the eddy viscosity or variation of the proportionality coefficient $\alpha$ in would improve the results. In the static application of eddy viscosity, $\nu_{eddy}$ is applied in the domain beginning at the location of the breaking wave crest and over the length $L_{br}$, and remains stationary for the duration of breaking, as was discussed in section 2.2. In the dynamic application of eddy viscosity, $\nu_{eddy}$ is applied over a length $L_{br}$ in the domain, but is centered at the wave crest and moves with the wave as it propagates. Figure 16 shows a comparison of static versus dynamic application of $\nu_{eddy}$ along with experimental measurements for surface elevation and corresponding frequency/amplitude spectra for MI0719 at wave gauges 11, 12 and 14. One can observe there is very little difference in the dynamic application of $\nu_{eddy}$ compared with the static application of $\nu_{eddy}$. Additionally, a small variation in the proportionality constant in $\nu_{eddy}$ was
<figure><img src="image_17.png" /><figcaption>**Fig. 15** The long time integration of the surface variance ($<\eta^2>=\int_0^\tau \eta(x, t)^2 dt$) is calculated using HOS-NWT and compared with measurements made during the LHEEA experiments for waves generated using modulational instability.</figcaption></figure>

found to be negligible, suggesting both the eddy viscosity parameter as well as the static application of $\nu_{eddY}$ is robust.

## 4.4 Irregular waves

For each of the cases presented where breaking waves are generated by irregular waves, multiple breaking waves occur in the time series. Figure 17 shows the full time series for surface elevation measured during the LHEEA experiments compared with surface elevation calculated using HOS-NWT for the irregular time series IRR-5. A vertical offset is included to separate the 14 wave gauges. Figures 18 and 19 show surface elevation and corresponding frequency/amplitude spectra at WG1, 2, 9, 10, 11 and 14. For frequency and energy analysis, a moving time window of approximately 100 s is selected to follow the same wave packet from WG1 to WG14. A Welch's power spectral density estimate is applied to the spectra using a t/3 point symmetric Hanning window with 50% overlap. This reduces noise in the spectrum by averaging the data over several selected overlapping time windows.

Figures 17, 18 and 19 show HOS-NWT is able to accurately calculate surface elevation and corresponding frequency/amplitude spectra for the irregular wave series IRR-5, particularly for breaking waves indicated by the arrows in Fig. 18. Breaking onset for one wave occurs just before WG10 and a second
<figure><img src="image_18.png" /><figcaption>**Fig. 16** The static and dynamic application of $\nu_{eddy}$ are compared with experimental measurements for surface elevation and corresponding frequency/amplitude spectra at wave gauges 11, 12, and 14 for the wave MI0719.</figcaption></figure>

breaking wave onset occurs just before WG11. The model is also successful in calculating surface elevation and corresponding frequency/amplitude spectra for the irregular wave series with superimposed energy focusing wave, IRREF-1, given in Figs. 20, 21 and 22. For this case, breaking onset occurs between WG5 and 6.

Normalized energy is computed for low and high regions of the energy spectra as well as total energy for each of the time series of steep irregular waves and irregular wave with added energy focusing components, given in Figs. 23 and 24, respectively. In Fig. 24, the solid red lines indicate breaking onset measured during experiments and the dashed blue lines indicate breaking onset calculated by HOS-NWT. Location of breaking onset is not included in Fig. 23 as there were several breaking waves located along the domain.

The model is generally successful at predicting energy loss associated with wave breaking, with both experiments and calculations showing the majority of energy loss occurring in the higher frequencies with little to no energy loss
<figure><img src="image_19.png" /><figcaption>**Fig. 17** Surface elevation measured during experiments is compared with surface elevation calculated using HOS-NWT for the irregular wave time series IRR-5.</figcaption></figure>

in the lower frequency components. Percentages of average normalized RMSE for IRR-5 and IRREF-1 are given in Table 5. The non-breaking case IRR-4 shows some energy transfer from the high frequencies to the lower frequencies but with no loss of total energy in calculations, as expected. Although there is energy gain in the lower frequencies, there is no downshifting of the peak frequency. There appears to be a gain in total energy at WG2 through 7 for this case measured in experiments but not calculated in numerical simulations. This is similar to the gain in energy at these wave gauges observed in Fig. 15 for the modulational instability waves. The cause for this is unclear as the system is conservative and there do not appear to be large differences in calculated surface elevation and wave gauge data at these locations. However, the way energy is evaluated is not perfect, and is subject to variability depending on the exact phasing of the chosen time window, and could therefore cause some variation in energy.

In each of the cases with breaking waves, experimental measurements at WG2 through 7 are slightly larger than calculations, as was seen previously in the modulational instability cases in Fig. 15. This appears to be independent of wave breaking as the same trend is seen in the non-breaking case IRR-4. It appears at WG2 in Fig. 19 there is some energy transfer to both the low and high frequencies in experiments that is not calculated in the model, causing
<figure><img src="image_20.png" /><figcaption>**Fig. 18** Surface elevation measured during experiments is compared with surface elevation calculated using HOS-NWT for the irregular wave time series IRR-5 at selected regions of wave breaking.</figcaption></figure>

energy measurements at this wave gauge to be larger than calculations. HOS-NWT also underestimates the energy loss at the lower frequencies, particularly at WG8 through 10. This could be due to both non-linear effects, as well as the way wave breaking energy is removed in calculations. Energy loss during experiments occurs in a very short time period, but in calculations energy is removed before the wave can overturn to prevent instabilities, causing a more gradual reduction of energy. This also explains why calculated energy is lower than measurements for both high frequencies and total energy in the breaking region for the irregular waves in Fig. 24, but agrees well for overall energy loss at locations further down the flume. As the main interest in applying a wave-breaking model in HOS-NWT is to calculate the surface elevation accurately post wave-breaking, as well as to reduce wave-breaking associated numerical instabilities, the present model is considered a success.
<figure><img src="image_21.png" /><figcaption>**Fig. 19** Frequency/amplitude spectra corresponding to surface elevation measured during experiments and calculated using HOS-NWT for the irregular wave time series IRR-5 at selected regions of wave breaking.</figcaption></figure>

## 5 Conclusions

Overall, the numerical model HOS-NWT with added wave breaking parameter is successful at calculating surface elevation, frequency/amplitude spectra and energy dissipation when compared with experimental measurements. Additionally, the model is successful at predicting the transfer of energy between frequencies due to energy focusing, and the reduction of energy due to wave breaking for a wide range of types, strengths and scales of breaking waves. Experiments and calculations demonstrate the reduction of energy associated with wave-breaking is predominantly in the higher frequencies, with little energy loss in the lower frequencies. Measurements and calculations for maximally recurring non-breaking waves suggest the movement of energy between frequencies is the result of nonlinear focusing and de-focusing and therefore some of this energy transfer must be independent of wave breaking. For breaking waves generated using energy focusing and irregular waves, no downshifting
<figure><img src="image_22.png" /><figcaption>**Fig. 20** Surface elevation measured during experiments is compared with surface elevation calculated using HOS-NWT for the irregular wave time series IRREF-1.</figcaption></figure>

of the spectral peak is observed. For breaking waves generated using modu-
lational instability, the spectral peak shifts from the center frequency to the
lower sideband frequency. A comparison of static and dynamic application of
eddy viscosity as well as variation in the proportionality constant show negli-
gible difference, supporting the robustness of the wave-breaking parameter.

This study, has successfully validated the ability of the implemented wave-
breaking model to predict wave-breaking onset and energy dissipation in a
nonlinear potential flow solver in two-dimensions. The robustness of the model
is demonstrated by the application of the model without any adjustments to
the parameters. Therefore, the model can be universally applied regardless of
sea state or scale, without any calibration. Now that the model is validated for
unidirectional waves over a flat bathymetry, the work of Gouin et al. (2016) for
waves over a variable depth can be extended to include breaking waves, and the
model can be further extended to calculate wave breaking in multidirectional
wave fields.

Acknowledgements

The assistance of Félicien Bonnefoy, of the LHEEA at École Centrale de
Nantes, received during the course of the experiments, is greatly appreciated.
<figure><img src="image_23.png" /><figcaption>**Fig. 21** Surface elevation measured during experiments is compared with surface elevation calculated using HOS-NWT for the irregular wave time series IRREF-1 at selected regions of wave breaking.</figcaption></figure>

The authors are grateful for the experimental data provided by Dr. Zhigang (Kevin) Tian.

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