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# Applicability and limitations of highly non-linear potential flow solvers in the context of water waves

Guillaume Ducrozet, Félicien Bonnefoy, Yves Perignon

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Guillaume Ducrozet, Félicien Bonnefoy, Yves Perignon. Applicability and limitations of highly non-linear potential flow solvers in the context of water waves. Ocean Engineering, 2017, 142, pp.233-244. 10.1016/j.oceaneng.2017.07.003. hal-01563219

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Submitted on 27 May 2025

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# Applicability and limitations of highly non-linear potential flow solvers in the context of water waves

Guillaume Ducrozet*, Félicien Bonnefoy, Yves Perignon

Ecole Centrale de Nantes, LHEEA Lab. (ECN/CNRS),
1, rue de la Noë - 44321 Nantes, France

## Abstract

This paper presents a general overview of the practical applicability of non-linear potential flow solvers for water wave propagation. Those numerical models are unable to describe explicitly the wave breaking phenomena, including free surface reconnexion, energy dissipation, etc. This reduces their range of application in real sea waves, which are in the most general context irregular and directional wave fields. This study covers the influence of discretization, water depth, directional spreading and spectral peakedness on the limitations in the use of such non-linear potential models. The approach at use relies on a highly non-linear model based on the High-Order Spectral method.

*Keywords:* Water waves, Non-linearity, Potential flow solvers, High-Order Spectral method, Wave breaking

## Introduction

The fine description of the water waves propagation is necessary to ocean engineers, especially in the context of the numerical simulation of wave-structure interactions. More and more efforts are dedicated to the accurate description of extreme waves (and their possible interactions with structures at sea). In this concern, highly and fully non-linear potential flow solvers are

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<footnote>* Corresponding author</footnote>

<footnote>E-mail address: guillaume.ducrozet@ec-nantes.fr</footnote>

<footnote>Tel: +33 240 371 645</footnote>

<footnote>Fax: +33 240 372 523</footnote>
the dedicated models for the description of the wave environment. For wave-
structure interactions, the usual approach, once the surrounding wave-field
is accurately described, is to couple such non-linear wave model with a spe-
cific model dedicated to the solution of the interactions. Depending on the
main physical processes at play, these can use e.g. non-linear potential flow,
Reynolds-Average Navier-Stokes (RANS) equations, Smooth Particle Hydro-
dynamics (SPH), etc. Several ways to perform the coupling exist: spatial,
temporal or mathematical decomposition (the so-called SWENSE approach,
see e.g. [1]). The use of such procedures is still necessary for the practical
solution of wave-structure interactions, primarily due to the computational
effort inherent to the direct solution of the problem thanks to advancedCom-
putational Fluid Dynamics (CFD) solvers.

At the same time, the study of non-linear wave processes is also of in-
terest with an increased concern with the extreme sea-states as well as with
the formation of large waves in a given sea state (the so-called freak or rogue
waves [2]). The accurate description of the corresponding wave fields is ac-
cessible through the non-linear potential flow solvers. Consequently, they are
now popular to answer the previous needs and for practical applications in
the context of wave-structure interactions.

However, the formalism adopted is limited to non-breaking waves, redu-
cing the range of application in real sea waves. This limitation is mainly due
to: i) the irrotational flow assumption which is violated when having free
surface reconnection, air entrapment, etc. (see e.g. [3]) and ii) the inviscid
fluid assumption, which does not allow to take care of the consequent dissipa-
tion of energy. We indicate that some efforts are currently done to model the
main effects of the breaking process on the wave field evolution (see e.g. [4]).
However, we restrict our study to original nonlinear potential flow solvers
without such modeling.

This papers intends to present the practical applicability of such models
with respect to the different wave conditions. These are usually described as
irregular directional wave fields. The possible limitation of the solvers will be
assessed with the equivalent of scatter diagrams, *i.e*. validity with respect to
the significant wave height $H_s$ and peak period $T_p$ of the corresponding sea
state. The proposed study covers the influence of numerical discretization,
water depth, angular and frequency spreading.

The approach used here relies on highly non-linear numerical simulations
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based on the High-Order Spectral (HOS) method initiated by West et al. [5] and Dommermuth & Yue [6]. This method has been widely validated and applied to several configurations, e.g. modulational instabilities [7, 8], nonlinear energy transfers [9], bi-modal seas [10], freak waves [11, 12, 13] among others. This method can consequently be considered as a good example of non-linear potential flow solvers: it is mature and accessible to practical engineering applications. The implementation provided in the open-source HOS-ocean code [14] will be used throughout the study.

The paper will be divided as follows, the first part presents the non-linear wave propagation model based on the HOS method and the different initial conditions accessible within the model. Then, the second part focuses on the methodology chosen to assess the limitations of the non-linear simulations as well as some existing theoretical and experimental results concerning the presence of breaking events in the different configurations. In the end, the last part focuses on the description of the influence of some of the key parameters for regular and irregular sea states applicability ranges. This part will enable an overview of the practical capabilities of the non-linear potential flow solvers.

# 1. Numerical wave model

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The study of large-scale and long-term evolution of non-linear directional wave fields relies on the existence of accurate and efficient numerical methods. In this concern, highly and fully non-linear potential flow solvers are the dedicated models. Indeed, in non-breaking wave conditions, the essential physics are included in the non-linear potential formalism. Within this class of approximation, several different numerical approaches exist that are briefly reminded hereafter. We also indicate that in such models, different methodologies can be used to take into account the effect of the wave breaking on the sea-state evolution. However, this is still an active field of research (e.g. [4]) and a definite model is still needed. In this study, we consequently restrict the field to nonlinear potential flow solvers without any dissipation model.

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The whole numerical domain may be discretized (i.e. in horizontal and vertical directions) leading to volume methods. Usually, after the application of a $\sigma$-transform, such models solve efficiently the problem in a fixed computational domain. We refer for instance to the development of Ocean-Wave3D model based on a finite-difference discretization of the volume problem [15, 16].

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An alternative in potential flow formalism is to solve the problem on its boundaries, reducing the number of unknowns. Those methods are referred as Boundary Element Methods (BEM). Up to now, the main limitation with these is the computational effort which appears as prohibitive for large-scale and long-term computations (even with the latest developments of Fast Multipole Algorithms, see e.g. [17]).

In the end, the problem may be solved on the free surface only usually making use of the Hamiltonian formulation of the free surface problem. Efficient and accurate solution of this surface problem is usually achieved thanks to pseudo-spectral methods, namely High-Order Spectral (HOS) or Dirichlet to Neumann Operator (DNO) approaches (which are equivalent, see [18]). The main limitation is here associated to the geometry that has to be simple (usually rectangular domain in horizontal directions and constant water depth).

In this section, the highly non-linear potential flow wave model at use to assess the range of applicability of this class of method is described. The detailed set-up of the numerical computations using this HOS method is also given with a specific attention paid to the initial conditions.

## 1.1. *High-Order Spectral method*

In this study, all numerical simulations are performed with the HOS method. This method has been initiated by West *et al.* [5] and Dommermuth & Yue [6]. Open-source HOS-ocean code [14] will be used and we refer to the different publications related to this model for examples of validation [19, 11].

We consider a rectangular fluid domain $D$ of horizontal dimensions $L_x \times L_y$ and constant water depth $h$ associated with a Cartesian coordinate system. Its origin $O$ is located at one corner of the domain with (Oxy) representing the horizontal axes and (Oz) the vertical one oriented upward with $z = 0$ located at the mean free surface. We are working under the potential flow theory (assuming the fluid to be incompressible and inviscid and the flow irrotational). Then, continuity equation reduces to the Laplace equation for the velocity potential $\phi$.

Boundary conditions close the system of equations. Periodicity is assumed in horizontal directions $x$ and $y$. These lateral boundary conditions, associated to the bottom one, allow us to define a spectral basis on which the velocity potential in the whole volume will be expanded. At the same
time, surface quantities are also expressed on a spectral basis allowing the use of Fast Fourier Transforms (FFTs). We consider the free surface boundary conditions which are written, following [20], using surface quantities namely the free surface elevation $\eta$ and the free surface velocity potential $\tilde{\phi}(x,y,t) = \phi(x,y,\eta(x,y,t),t)$. $z = \eta(x,y,t)$ describes the free surface position, assuming no wave breaking occurs (i.e. the free surface is a single valued function of $x$ and $y$). The kinematic and dynamic free surface boundary conditions then read

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

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

with $\nabla$ the horizontal gradient and $W(x,y,t) = \frac{\partial \phi}{\partial z}(x,y,\eta,t)$ the vertical velocity at the free surface. This is the only quantity beyond the system (1)-(2) which needs solution in the water bulk. Later on $W$ will be evaluated thanks to the order consistent HOS scheme of West et al. [5]. Once the vertical velocity is evaluated, it is possible to advance in time the two unknowns $\eta$ and $\tilde{\phi}$ thanks to an efficient 4th order Runge-Kutta Cash-Karp scheme with adaptive step size [21].

The HOS procedure relies on a series expansion in wave steepness $\epsilon$ up to the so-called HOS order $M$ of the velocity potential. Expanding a Taylor series around $z = 0$ and collecting terms at each order in wave steepness leads to a triangular system. A similar series expansion for the vertical velocity leads to another triangular system, which is solved iteratively. This enables an arbitrary choice of the order of nonlinearity $M$.

The resulting numerical method is pseudo-spectral and exhibits very interesting convergence properties: exponential convergence rate with respect to the horizontal discretization as well as with the HOS order $M$ [19]. Thus, this HOS model features high efficiency and accuracy compared to other advanced methods for wave propagation, see [22].

## 1.2. Initial conditions

The HOS method allows the accurate and efficient propagation of wave fields with possibly high level of non-linearity. As stated previously, the method has been widely validated in several configurations, e.g. nonlinear energy transfers [9], modulational instabilities [7, 8], bi-modal seas [10] or freak waves [11, 12, 13].
Then, the numerical simulations rely on the definition of relevant initial conditions of interest. Once the free surface elevation $\eta(x,y,t = 0)$ and velocity potential $\tilde{\phi}(x,y,t = 0)$ are specified, the two unknowns are advanced in time with the previous HOS procedure. Within HOS-ocean, different kind of initial conditions are accessible, starting from regular waves up to directional irregular sea states.

The regular waves are initialized thanks to the nonlinear regular wave solution of Rienecker and Fenton [23], based on the stream function theory. This fully non-linear numerical solution is valid over the whole range of existence of unidirectional regular waves (in terms of steepness $H/\lambda$ and relative water depth $h/\lambda$ with $\lambda$ the corresponding wavelength and $H$ the crest-to-trough wave height).

The initialization of an irregular wave field is achieved using a superposition of linear components, each component having its own frequency and direction. The amplitude of each component is determined from a directional spectrum $S(\omega, \theta)$ with $\omega$ standing for the angular frequency and $\theta$ for the direction. Then, the phase of each component is evaluated by a random number so that the phases assume uniform distribution over the range $[0, 2\pi]$. The free surface potential is linearly evaluated from the elevation and the adequate dispersion relation. Finally, the initial wavefield is constructed with an inverse Fourier transform. HOS-ocean provides the possibility to use a directional spectrum form $S(\omega, \theta) = F(\omega)G(\theta)$ defined with:

i) a JONSWAP spectrum, specifying the peak period $T_p = \frac{2\pi}{\omega_p}$, the significant wave height $H_s$ and the shape factor $\gamma$

$$F(\omega) = \alpha_J H_s^2 \omega_p^4 \omega^{-5} \exp \left[ -\frac{5}{4} \left( \frac{\omega}{\omega_p} \right)^{-4} \right] \gamma^{\exp \left[ -\frac{(\omega - \omega_p)^2}{2\sigma^2 \omega_p^2} \right]} \quad (3)$$

with $\sigma = \begin{cases} 0.07 & \text{for } \omega < \omega_p \\ 0.09 & \text{for } \omega \ge \omega_p \end{cases}$ and $\alpha_J$ chosen to obtain the correct significant wave height.

ii) a specific directional spreading characterized with $\beta$ parameter (a large value of $\beta$ induces a large spreading):

$$G(\theta) = \frac{1}{\beta} \left[ \cos \left( \frac{\pi \theta}{2\beta} \right) \right]^2 \quad (4)$$
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It has been demonstrated in [24] that the definition of linear initial wavefields for fully-nonlinear computations can lead to numerical instabilities. In order to overcome this problem, a transition period is introduced through the use of a relaxation scheme, allowing a smooth transition from the linear initial conditions to the fully nonlinear computation. The transition period is described by its duration $T_a$ and an exponent parameter $n$ (for details, see [14]).

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## 2. Methodology

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This section presents the methodology used for the assessment of the applicability of the numerical simulations. In addition, a bibliographic overview of some theoretical and experimental works dealing with the existence of stable and/or non-breaking wave fields in different conditions is given.

### 2.1. *Numerical simulations*

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In the context of irregular sea states, the target practical applicability has to answer the question: is the sea state of interest accessible to simulation with a non-linear potential flow solver? Depending on the initial solution, the way to answer may be different.

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For regular waves, the assessment of practical applicability is quite simple: the parameters of the initial wave field can be changed (i.e. steepness $H/\lambda$ and relative water depth $h/\lambda$). Then, if the numerical simulation allows an accurate evaluation of the wave propagation, this is assumed to be a case accessible by the model. The evaluation of the accuracy of the solution is easily accessible in the case of regular wave solution, where a reference solution exist (which is the one used for the initialization [23]). Then, the possible limitation of non-linear potential flow solvers resides mainly in an issue of the accuracy of the solution.

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As seen previously, we will consider the irregular wave field defined with a JONSWAP spectrum but it is awaited that the results can be extrapolated to other kind of wave spectra.

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For the design of a structure at sea, the encountered sea states are usually described with a scatter diagram which gives the probability of occurrence of a given sea state, at a given location, as a function of the integral parameters of the sea state, for instance the significant wave height $H_s$ and the
peak period $T_p$. In the following, we will consequently present the practical applicability as contour maps with respect to those two parameters (see e.g. Fig. 8).

As stated previously, potential flow solvers are limited to non-breaking waves, due to the irrotational flow assumption which is broken when fa-cing free surface reconnection, air entrapment, etc. (see e.g. [3]) as well as the assumption of inviscid fluid, which neglects the consequent energy dis-sipation. With the proposed HOS model based on a pseudo-spectral for-malism, waves cannot overturn (free surface is a single valued surface defi-ned as $z = \eta(x,y,t)$). If the physical wave breaking would occur during the propagation of a wave field, when the wave steepens (prior to breaking), high-frequency components gains energy (see [25]) up to a point where the simulation breaks down. The range of applicability will consequently simply be assessed by a successful run of the whole numerical simulation. Note that if the wavefield presents marginal breaking very localized in space and time, the simulation may be able to continue. The latter point is dependent on space/modal discretization and will be one of the parameter of interest in the study. However, this approach gives a reasonable estimate of the limi-tations of the numerical model and the inherent possible presence of wave breaking in the sea state of interest.

As detailed in the previous section, the initialization of the irregular wave field is dependent on the choice of phases of each wave component. In order to have an accurate estimation of the possible presence of wave breaking in the simulations, it is then necessary i) to have a domain size sufficient large, ii) to have a sufficiently long duration of the simulation and iii) to set different choices of random initial phases.

As a summary, the numerical parameters for the different simulations operated during the study are the following:

- Size of the domain: $L_x = 51 \lambda_p$ and $L_y = 51 \lambda_p$

- Space discretization: $N_x \times N_y = 512 \times 512$ points free of aliasing errors (total number of points is $1024 \times 1024$)

- Duration of the simulation: $T = 100 T_p$

- HOS nonlinear order: $M = 3$ allowing the solution of all four-wave interactions (see [9])

- Number of phase draws: 20

- Time relaxation of non-linear terms (initialization): $T_a = 10 T_p$ and
$n = 4$

The size of the domain along with the space discretization defines the highest wave number accurately solved (i.e. without any aliasing errors) by the HOS model. This is referred as $k_{max}$ in the following and with the previous choices of parameters it is $k_{max} = 5 k_p$ with $k_p = 2\pi/\lambda_p$ the peak wave number. This is the usual choice that is done with HOS method when studying the propagation of a nonlinear sea state. This allows the accurate solution of the most energetic part of the spectrum. The influence of this parameter will be detailed in Sec. 3.2.1.

## 2.2. Physical limitations

This section introduces reference solutions that are used in the following to compare the limitations obtained numerically to the theoretical or experimental evidence of the presence of wave breaking in similar configurations. At first, the case of regular waves is investigated to establish their limit of stability. Then, some works done on real sea waves and the possible presence of breaking events are reported.

### 2.2.1. Regular waves

Several theoretical, numerical and experimental studies deal with the stability and existence of regular waves. A review of the different models established for regular waves' stability limit (over flat or constant slope) can be found in [26]. The useful parameters are in this concern the steepness $\epsilon = H/\lambda$ and the relative water depth $h/\lambda$ (or the corresponding ratio $H/h$ representing the wave height to water depth ratio).

The highest stable wave has been studied by Miche [27], which provides a simple formula for the highest steepness $\epsilon_{lim} = H_{lim}/\lambda$ (or equivalently the highest wave height $H_{lim}$ for a given wave length $\lambda$), valid for arbitrary water depth $h$. This equation, validated against numerous experimental data, is consequently widely used and reads:

$$ \epsilon_{lim} = 0.14 \tanh (kh) \qquad (5) $$

This formula enables a simple way to assess the possibility to evaluate a given regular wave field. The two limiting cases of deep water and shallow water are also reminded. In deep water, Miche's result is that the ratio $H_{lim}/\lambda$ is 0.14, or equivalently $ka_{lim} = 0.45$. The opposite limit, in which the highest solitary wave has a height $H_{lim}/h = 0.83$ has been detailed in the numerical
285 work of Hunter and Vanden-Broeck [28]. In this concern, note that the limit provided by Eq. (5), namely $H_{lim}/h = 0.89$ may lack of accuracy. However, the shallow water configuration ($h \to 0$) is not the primary concern of the following study.

Figure 1 presents the limit given by Eq. (5) in terms of the non-dimensional numbers at play ($H/h$ and $\lambda/h$) as well as in terms of dimensional quantities wave height $H$ and wave period $T$. Note that the relation between $\lambda$ and $T$ is approximated thanks to the linear dispersion relation. Different water depths are indicated following the studies performed later in irregular sea states. It is seen that in the range of period tested, the effect of water depth is only important for the smaller water depth $h = 35 \ \text{m}$ where the highest possible regular wave height is reduced. For larger water depths, the limit is only function of the wave steepness $\epsilon = H/\lambda$ which in such graphs $H/T$ is a parabola.

<figure><img src="image_3.png" /><figcaption>FIG. 1 – Highest possible regular wave from experiments by Miche [27]. Non-dimensional quantities (left) and dimensional ones in $H/T$ plane (right) - color online</figcaption></figure>

### 2.2.2. Irregular waves - deep water

In the case of real sea states, i.e. irregular and directional wave fields, it appears more complex to define a threshold for the presence of wave breaking in the sea state. The first information at our disposal is, from measurements at sea, the scatter diagrams that will characterize the probabilities of occurrence of the sea-states described with integral parameters, such as $H_s$ and $T_p$. For instance, Fig. 2 presents a scatter-diagram obtained from the data collected in the northern North Sea during the period 1973-2001 [29]. These data
consist of composite measurements from four locations, namely the Brent, Troll, Statfjord and Gullfaks fields. These locations exhibit a water depth $h \in [150; 300]m$. As a consequence, the influence of water depth on the wave field is in a range of period $T > [14; 20]s$, which has only limited effect in the present configuration.

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<figure><img src="image_4.png" /><figcaption>FIG. 2 – Probabilities of occurrence (scatter-diagram) of sea-states in the North Sea. Data for the period 1973-2001 extracted from Haver et al. [29] - color online</figcaption></figure>

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We can observe in this figure that the steepest wave fields observed in the northern North Sea (with a non-negligible probability) follows approximately a line at constant steepness $\epsilon = H_s/\lambda_p \approx 0.06$, $\lambda_p$ being evaluated from $T_p$ thanks to the linear dispersion relation. This results is however dependent on the limit of probability chosen to depict the limit (fixed here as $Pr < 0.0005$). A smaller probability may induced a slightly larger limit. However, it is reminded from this graph that the highest sea states (in terms of wave height) are limited by the wave breaking process and are consequently closely linked to the steepness of the sea-state $\epsilon$. Furthermore, this gives an indication of the order of magnitude of the possible threshold in irregular sea states.

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During a series of experiments in wave basins dealing with the propagation of narrow-banded deep water sea states, it has been reported by Shemer & Alperovitch [30] that wave breaking was observed and possible for every sea states satisfying $\epsilon \ge 0.064$. This is consistent with the previous results.

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### 2.2.3. Irregular waves - limited water depth

As seen previously, the limited water depth has an important influence on the limit of stability of regular waves. The same behavior is expected to occur when dealing with irregular sea states. A review of the different thresholds and models of wave breaking at use in the stochastic models for wave propagation can be found in [31]. However, those models are not adapted to the definition of a simple criteria in deterministic wave models (for instance establishing a threshold with respect to $H_s/h$, $H_s/\lambda_p$, etc.).

In this concern, the reference work appears to be the one initiated by Dally et al. [32]. The starting point of the analysis is to define an integral quantity from the energetic point of view. The different thresholds for the presence of wave breaking are consequently defined as limits from which we observe dissipation of energy. This experimental work allows the definition of the limit of stable energy in a given wave field in small relative water depth as $H_s = 0.4 h$.

Following this work, Rattanapitikon [33, 34] provides an extension of all works done on regular waves to establish limits of stability to irregular wave fields. The idea is the same as the previous one, that is to say using for the description of irregular sea states integral or statistical quantities (called representative in his work): $H_{rms}$, $H_{1/3}$, $H_s$, etc. Dozens of models have been studied and adjusted with respect to different series of experiments/observations. The adjustment procedure relies on the minimization of an error criteria.

Throughout this study, we make use of two models to assess the threshold for wave breaking existence. The first is the simplest one and may be seen as an extension of the original work of Miche [27].

$$H_s = \kappa\lambda_p \tanh(k_p h) \qquad (6)$$

When using this model [33, 34], the author changes the value of the constant $\kappa \in [0.067 ; 0.076]$. Note that the previous deep-water results corresponds to $\kappa \simeq 0.06$, which seems consistent with this model reminding that the threshold from Fig. 2 is only an approximation. Figure 3 presents the example of the corresponding limits for different water depth and two values of $\kappa$. In the case of irregular waves and within the range of the study in terms of $H_s$ and $T_p$ parameters, the influence of the limited water depth is clear for $h = 35 m$ and $h = 75 m$.
<figure><img src="image_5.png" /><figcaption>FIG. 3 – Highest possible irregular wave field $\kappa = 0.067$ (solid line) and $\kappa = 0.076$ (dashed line) - color online</figcaption></figure>

In order to establish the relevance of this simple model, Fig. 4 presents the scatter-diagram obtained from the SEMREV [35] marine test site. This 1 km² restricted area, located 12 nautic miles offshore of Le Croisic on the French Atlantic coast, takes advantage of a whole infrastructure dedicated to the test at sea of marine renewable energy devices and components. Among other sensors, two Datawell Waverider Mk3 buoys are moored on site since 2009 recording displacements nearly 1 km apart (East and West corners of the site). The water depth for those location reaches 34 m and 36 m LAT respectively, with a tidal amplitude up to 6.1 m. The currents on site remain below 0.5 m.s⁻¹ for the vast majority of its environmental conditions and the tidal influence over significant wave height only begins to show for specific sea states during spring tides [36]. The aforementioned scatter diagram is extracted from nearly 6 years measurements (2010-2016), using the vertical displacement of the buoy to evaluate the standard integral parameters $H_s$ and $T_p$.

This test site experiences a mean water depth $h = 37$ m where the limited water depth effects are expected to be significant (referring to Fig. 3). For such water depth, the periods $T > 7$s are expected to be influenced, which covers the most energetic part of the scatter diagram, see Fig. 4. This figure also presents the limitations obtained from Eq. (6) with two values of the pa-
<figure><img src="image_6.png" /><figcaption>FIG. 4 – Probabilities of occurrence (scatter-diagram) of sea-states at the SEMREV test site (mean water depth is h = 37 m). Comparisons with model Eq. (6) - color online</figcaption></figure>

rameter κ. Some deviations are noticed with the values suggested in [33, 34]. However, it is seen that adjusting the value of the parameter κ ≃ 0.5 allows an accurate description of the physical limitations of the sea-state on a quite large range (up to T$_{p}$ ≃ 12 s). For wave fields with larger peak periods, the maximum significant wave height seems to reach a maximum, independent of T$_{p}$ (as observed in [32]). The limitation observed at large peak periods is probably due to the physical process and not to the wave breaking. Sea states with larger steepness are not measured because, at this location, the conditions in North Atlantic may not be sufficiently strong to allow their existence. This simple model (Eq. (6)) is consequently not able to describe the whole upper boundary of this scatter diagram.

The second model used to assess the threshold for wave breaking existence has been introduced in [37]. It is studied in [33] and is an adjustment of the previous one with increased complexity in order to approximate more accurately the threshold of interest. This is defined as

$$
H_s = K_1 \lambda_p \tanh ([K_2 k_p h + K_3] k_p h) \quad (7)
$$

Three parameters can now be adjusted and their original values are K₁ = 0.14, K₂ = 0.86, K₃ = 0.33 in [37] modified to K₁ = 0.14, K₂ = 0.70,
$K_3 = 0.45$ in [33]. Note that the parameter $K_1$ corresponds to the limiting steepness in deep water, while other two coefficients will allow a possible better fit of the water depth influence.

Taking as example the SEMREV test site, it is possible to approximate the whole period range thanks to Eq. (7). We choose the deep-water steepness to be $K_1 = 0.06$, as observed in Fig. 2 and other parameters are chosen as $K_2 = 0.2$ and $K_3 = 0.33$. The resulting limit is depicted in Fig. 5.

<figure><img src="image_7.png" /><figcaption>FIG. 5 – Probabilities of occurrence (scatter-diagram) of sea-states at the SEMREV test site (mean water depth is $h = 37$ m). Comparisons with model Eq. (7) with $K_1 = 0.06$, $K_2 = 0.2$, $K_3 = 0.33$ – color online</figcaption></figure>

This choice clearly allows an accurate definition of the threshold for the wave existence at this location. It is important to note that the calibrated values $\kappa, K_1, K_2$ and $K_3$ appear different from the reference ones [33, 34, 37]. We reiterate that those values extracted from this specific scatter diagram are only an approximation. The description of the limiting significant wave height for a given peak period is highly dependent on the local water depth as well as the location considered.

Furthermore, the need of such complex formula may be attributed to the limits associated to the measurements themselves: i) the water depth is changing with a tidal amplitude up to 6.1 m (i.e. $\pm 8\%$ of the mean water depth), ii) the presence of current, even if it is of low amplitude, may affect
the small wave periods and iii) the wave measurements from buoys always rely on some assumption, which are questionable (effects of anchoring system, bio-fouling, non-linearities, etc.). However, we demonstrate that Eq. (7) with varying parameters $K_1, K_2$ and $K_3$ will be able to approximate the ranges of applicability of our simulations in limited water depth. It will be preferred to Eq. (6) in the following.

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### 3. Results

This section is dedicated to the effective results of the range of applicability of non-linear potential flow solvers in different contexts. At first, some results regarding regular waves are given. Then, the irregular wave fields are studied in details. The influence of different numerical and physical parameters is described:

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- numerical discretization

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- water depth

- directional spreading

- spectral peakedness (frequency spreading)

The corresponding results are depicted as scatter-diagrams of applicability ($H_s/T_p$).

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#### 3.1. Regular waves

In the case of regular waves, it has been seen previously (Sec. 2.2.1) that a simple formula (Eq. (5)) enables the knowledge of the possible existence of a stable non-breaking solution. This equation covers the whole range of variation of the significant parameters for gravity waves (namely wave height, wave length and water depth). This may be used to determine the possible use of non-linear potential flow models to the case of regular waves.

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However, as a matter of correctness, it has to be noted that the practical limit of the numerical solver is usually lower than the previously given limit. Indeed, the highest stable wave exhibits very high-degree of non-linearity and the numerical discretization at play in all non-linear potential flow solvers may cause instabilities close to the wave breaking limit. The accurate and efficient HOS model has been shown to be very accurate in this concern and manage to evaluate accurately, in deep water, the solution up to $\simeq 90\%$ of this breaking limit. In this concern, the initial work of Dommermuth & Yue [6] has been enhanced in the code HOS-ocean [19].

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We provide here an extension of the previous deep water study in order to assess the domain of applicability in limited water depths. Figure 6 presents an estimate of the highest regular wave heights accessible with HOS-ocean for a wide range of relative water depth ($λ/h \in [0.1; 40]$). This includes very shallow water waves which are not the natural target applications of the model. HOS method is indeed a pseudo-spectral method, which looks for the solution with a decomposition on the eigenmodes of the computational domain. It is known that such method is not the most suitable one for very shallow water waves (see e.g. [38]). The results are compared to the Miche's formula (Eq. (5)). Note that this is only an estimate since this limit is dependent on the level of accuracy one wants to achieve in the numerical simulation (a more accurate solution inducing a reduced range of applicability).

<figure><img src="image_8.png" /><figcaption>FIG. 6 – Highest possible regular wave from experiments by Miche [27] and in HOS-ocean simulations.</figcaption></figure>

This figure confirms that, as stated previously, the highly non-linear potential flow solver used here allows, in deep water (i.e. low values of $λ/h$), the accurate solution of the regular wave propagation up to steepnesses (or equivalently wave heights) close to the breaking wave limit (≈ 90%). However, it is also clear in this plot that for reduced water depths, the range of applicability (with respect to the ratio to highest possible wave height) of the model is also reduced.

This feature is inherent to the pseudo-spectral formalism. For small water depth, the regular wave solution becomes a cnoidal wave which departs more and more from a sinusoidal wave when the water depth decreases. Consequently, one needs a very large number of components (modes) to describe
such wave profiles (especially with large amplitudes). This consequently leads to difficulties in the numerical solution of the problem with the possible appearance of instabilities during the wave propagation. These are responsible of the present reduction of the application range of the HOS model. This indicates that such model is not suitable for the study of large waves in very shallow water. In such condition, other numerical models have to be used, see e.g. [39]. However, for the smallest relative water depths tested in Fig. 6, $λ/h = [30-40]$, the solution is still possible over a large range of wave heights, up to ≈ 70% of the limiting wave height extracted from Eq. (5).

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This behavior with respect to the relative water depth may consequently be slightly different with other non-linear potential flow solvers (using different numerical methods for the solution). However, this analysis gives a detailed overview of the application range of one of this model. In addition, we remind here that the main concern for the applicability of the potential flow solvers to regular waves is the accuracy of the solution, which is directly linked to the numerical model at use. Convergence studies should be carried out for every configuration in order to assess the effective accuracy of the method at use.

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## 3.2. Irregular waves

This section provides the range of applicability of the HOS solver for irregular sea states in different configurations. As stated previously, the directional irregular sea-states are defined following a JONSWAP spectrum (Eq. (3)) with a given directional spreading (Eq. (4)). The initial parameters of the wave field are chosen as $γ = 3.3$ and $β = 0.30$ with a numerical discretization $k_{max} = 5k_p$. At first, the choice of the numerical parameters is discussed. Then a detailed study of the influence of the physical parameters of the sea states on the applicability of the nonlinear model is provided.

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### 3.2.1. Influence of space/modal discretization

In this section, we provide an analysis with respect to the space/modal discretization of the numerical simulations and its influence on the applicability of the model. When dealing with a pseudo-spectral method, all surface quantities are represented on a given spectral basis, the number of points used in the space domain being directly linked to the discretization in the transformed Fourier space. As a consequence, an increased number of points results, for a fixed domain length, leads to a larger highest mode solved (with

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a corresponding wavenumber $k_{max}$). This wavenumber cut-off is usually expressed as a function of the peak wavenumber $k_p$ (and can be related to a corresponding frequency cut-off of the spectrum). It is important to recall that in the present numerical simulations, no filtering nor smoothing is used. The only procedure that possibly affects the energy is the dealiasing which only removes the energy transfer (direct energy cascade) resulting from wave turbulence.

Depending on the problem studied, the target accuracy and the non-linearities taken into consideration (the HOS order $M$), this $k_{max}$ can be changed. For practical applications in ocean engineering in which one wants to model the wave field evolution, the induced kinematics, etc. the choice $k_{max} = 5k_p$ is typical. This choice allows the correct evolution of the energetic part of the wave spectrum, taking into account non-linearities up to third-order ($M=3$).

The possible limitations of the HOS model associated to the correct solution of the nonlinear sea states' evolution is detailed. As an example of the approach used, Fig. 7 presents the contour map that depicts the ratio of successful computations as function of significant wave height $H_s$ and peak period $T_p$. It is reminded that 20 different simulations are conducted for each sea-state, with different initial random phases. For each sea-state in the scatter diagram (i.e. each couple $H_s/T_p$) the ability of the model to perform the simulation is tested. Then, the value 0 corresponds to no successful simulation while the value 1 indicates that all choices of random phases enables the computation. Blue zones consequently indicates regions of the scatter diagram accessible to the numerical model and red zones the regions which are not.

Figure 7 presents the results for this maximum wavenumber varied in the range $k_{max} \in [3 k_p; 12 k_p]$. We restrict our study to deep water, the results being similar in limited water depth.

As a matter of analysis, Fig. 7 depicts the upper bound of the blue regions, together with the previous theoretical limits that one can extract from literature, see Sec. 2.2. Then, the present figure indicates clearly, as expected, this frequency cut-off has a large influence on the range of applicability of the HOS solver. Furthermore, this limitation is seen to be driven by the wave breaking phenomena. Indeed, the domain of applicability can be accurately bounded by limits in terms of wave steepness for the different choices of maximum wavenumber. The limitation of the computations is described by
<figure><img src="image_9.png" /></figure>

$$k_{max} \approx 3 k_p$$

<figure><img src="image_10.png" /></figure>

$$k_{max} = 5 k_p$$

<figure><img src="image_11.png" /></figure>

$$k_{max} = 8 k_p$$

<figure><img src="image_12.png" /></figure>

$$k_{max} \approx 12 k_p$$

<caption>FIG. 7 – Domain of validity of HOS nonlinear simulations, influence of frequency cut-off for $\beta = 0.30$, $\gamma = 3.3$ and $h = \infty$ - color online</caption>
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limiting steepnesses with the corresponding values: $k_{max} = 3 k_p, \epsilon = 0.090$; $k_{max} = 5 k_p, \epsilon = 0.060$; $k_{max} = 8 k_p, \epsilon = 0.030$ and $k_{max} = 12 k_p, \epsilon = 0.025$.

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When the shortest wavelengths are not resolved (i.e. small $k_{max}$), the possible breaking events are filtered out and consequently the numerical model can propagate sea states up to very large steepness ($\epsilon = 0.090$). In those sea states, wave breaking is present in a real configuration and the simulation is consequently not highly accurate.

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Similarly, when increasing $k_{max}$, the steep wave fields that possibly contains very steep waves, which modal representation is characterized by energy up to high wavenumbers, are described more accurately at the cost of a reduced robustness. This reduction is inherent to pseudo-spectral methods which are highly accurate without any numerical diffusion (thanks to filtering or local smoothing). The domain of applicability is consequently drastically reduced when increasing $k_{max}$. In this concern, and similarly to any numerical simulation, space convergence has to be achieved for any numerical computation in order to ensure that the considered discretization is appropriate in terms of accuracy of the solution. In the following, the discretization is fixed to $k_{max} = 5 k_p$ which provides, with $M = 3$, an efficient and accurate solution of the propagation of directional irregular sea states for typical ocean engineering applications.

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### 3.2.2. Influence of water depth

The influence of the water depth on the applicability of HOS solver to the simulation of non-linear sea states is now presented. Figure 8 shows the contour map of applicability for different water depths $h \in [35 \text{ m}, 75 \text{ m}, 150 \text{ m}, \infty]$.

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The present figure indicates that the water depth has, as expected, an essential influence on the domain of applicability of non-linear potential flow solvers. Once again, this limitation is seen to be driven by the wave breaking phenomena since the domain of applicability is accurately defined by a constant wave steepness ($\epsilon = 0.060$) for the large water depths.

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When reducing the water depth, this range of applicability also reduces. In the presented examples, for small water depths at the ocean scale ($h = 35 \text{ m}$ and $75 \text{ m}$), the steepness is not the only effect to take into account to establish a comprehensive threshold, especially for large periods. Limited water depth effect are however captured accurately by the model introduced previously (Eq. (7)) with $K_1 = 0.06$, $K_2 = 0.20$ and $K_3 = 0.65$. Since for large water depths Eq. (7) reduces to $\epsilon = K_1$, the proposed equation describes the range

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<figure><img src="image_13.png" /></figure>

$h = \infty$

<figure><img src="image_14.png" /></figure>

$h = 150 \text{ m}$

<figure><img src="image_15.png" /></figure>

$h = 75 \text{ m}$

<figure><img src="image_16.png" /></figure>

$h = 35 \text{ m}$

FIG. 8 – Domain of applicability of HOS nonlinear simulations, influence of water depth $\beta = 0.30$ and $\gamma = 3.3$. Eq. (7) parameterized with $K_1 = 0.06$, $K_2 = 0.20$, $K_3 = 0.65$ - color online
of applicability in conditions ranging from deep to shallow water. The case $h = 75$ m compares a given threshold based on the only steepness to the one taking into account reduced water depth. As expected, the effect is more pronounced at large $T_p$ and tends to reduce the applicability of the model. It is also worth to note that for small relative water depth (i.e. $h = 35$ m and large wave periods $T_p > 20 - 25$ s), this limit of applicability tends to a constant ratio $H_s/h$, as predicted in [32].

In the end, we can also observe in Fig. 8 that the definition of the limit is sensitive to the choice of threshold for its establishment in the interval [0; 1]. The one chosen in the following will corresponds to the limit where at least 70 – 80% of the simulations are successful.

For the conciseness of the analysis provided hereafter with respect to the different wave parameters, the previous figure is simplified into one simple plot depicting the different limits established from the previous contour plots. This allows a more straightforward analysis and overview of the results. Figure 9 presents the corresponding plot for the influence of water depth.

<figure><img src="image_17.png" /><figcaption>FIG. 9 – Applicability limits for HOS simulations. Influence of water depth $\beta = 0.30$ and $\gamma = 3.3$ – color online</figcaption></figure>

### 3.2.3. Influence of directional spreading

In this section, we indicate some results dealing with the influence of directional spreading. As stated previously, the water depth has a large influence
on the possible application of the non-linear simulations. Consequently, we present here this influence for infinite water depth as well as a reduced one. In this analysis, the directional spreading, characterized by the parameter $\beta$ in Eq. (4), is varied in the range $\beta \in [0; 0.60]$. The case $\beta = 0$ actually corresponds to a 2D computation (i.e. only one horizontal direction) while $\beta = 0.60$ represents a short-crested sea state with a very large spreading.

Figure 10 presents, for infinite water depth, the influence of the directional spreading on the range of applicability of HOS model. The limits are in this configuration defined as limiting steepnesses with the corresponding values: $\beta = 0$ (i.e. 2D), $\epsilon = 0.085$; $\beta = 0.15$, $\epsilon = 0.053$; $\beta = 0.30$, $\epsilon = 0.060$ and $\beta = 0.60$, $\epsilon = 0.060$.

<figure><img src="image_18.png" /><figcaption>FIG. 10 – Applicability limits for HOS simulations. Influence of directional spreading $\gamma = 3.3$ and $h = \infty$ – color online</figcaption></figure>

Then, we observe in this figure that for 3D directional sea states, the application range is slightly enhanced with larger directional spreading. In this concern, the 2D computations appears to behave differently. This is not an unexpected behavior since in several non-linear wave phenomena, directional effects are of main importance (e.g. on modulational instability [40]). However, this point allows us to remind that there are fundamental differences between a 2D and a 3D non-linear simulation of water wave evolution.

The influence of the directional spreading on the range of applicability can be analyzed with respect to non-linear phenomena at play. Especially,
the modulational or so-called Benjamin-Feir instabilities are known to be re-
duced with an increase of directional spreading [40]. If those instabilities are
reduced, the evolution of kurtosis during the non-linear wave evolution will
be different and exhibit lower values. Reminding that the kurtosis quantifies
the probability of occurrence of extreme waves in a given sea state, a reduced
kurtosis can be associated to a reduced probability of wave breaking, explai-
ning the observed behavior.

Figure 11 presents for the water depth $h = 35 \ m$ the corresponding
domain of applicability. The limits are this time defined as limited water
depth approximation, Eq. (7). As previously, we define $K_2 = 0.20$, $K_3 =
0.65$ and $K_1$ as the deep water limiting steepness established above: $\beta =
0.15$, $K_1 = 0.053$; $\beta = 0.30$, $K_1 = 0.060$ and $\beta = 0.60$, $K_1 = 0.060$.
This choice allows a unified formula for deep and limited water depth limits of
applicability. The 2D configuration appears to be singular in this case with
$\beta = 0$ (i.e. 2D), $\epsilon = 0.090$.

<figure><img src="image_19.png" /><figcaption>FIG. 11 – Applicability limits for HOS simulations. Influence of directional spreading $\gamma = 3.3$ and $h = 35 \ m$. Eq. (7) is used with $K_2 = 0.20$, $K_3 = 0.65$ – color online</figcaption></figure>

As explained previously, the fundamental difference between 2D and 3D
simulations appears also clearly here. The limiting steepness is still very large
$\epsilon = 0.090$ if compared to the theoretical works and possibly points out the
limitations inherent to the simplification of a real sea state to a unidirectional
one. For this limited water depth ($h = 35 \ m$), the physics of unidirectional
and directional wave fields are probably very different resulting in this large discrepancy. A first attempt of explanation relies on the fact that for large peak periods, the phase velocities of the frequency components with most of the energy are very similar (due to the small water depth). The directional spreading allows the superposition of waves traveling with different directions and the inherent possible wave breaking. This effect is completely absent from 2D simulations where all those frequency components propagate with similar velocities and low probability of superposition. This explains the fundamental differences observed between 2D and 3D sea states as well as in Figs. 10 & 11 (infinite or limited water depth).

For the irregular directional sea-states, the behavior with respect to di-
rectional spreading is similar to the infinite water depth case. A broader
directional spreading results in a slightly larger range of applicability of the
model. Reminding that the modulational instability is reduced in small water
depth (theoretically absent for $kh < 1.36$), the physical processes are at play
may be different in this configuration.

3.2.4. Influence of the spectral peakedness

In this section, we present the influence of the spectral peakedness (or
frequency bandwidth) on the range of applicability for infinite and small wa-
ter depth. In this study, this bandwidth, characterized by the parameter $\gamma$
of the JONSWAP spectrum in Eq. (3), is varied in the range $\gamma \in [1; 10]$. The
case $\gamma = 1$ corresponds to a broad band spectrum while $\gamma = 10$ represents a
wave field with a narrow-banded spectral content.

Figure 12 depicts the results for infinite water depth. The limits are in this
configuration defined as limiting steepnesses with the corresponding values:
$\gamma = 1, \epsilon = 0.060; \gamma = 3, \epsilon = 0.060; \gamma = 5, \epsilon = 0.055$ and $\gamma = 10, \epsilon = 0.050$.

The narrowness of the spectrum appears to have a limited influence on
the range of applicability studied. However, a clear trend is observed in the
simulations which indicates that when the frequency bandwidth of the spec-
trum is reduced (i.e. larger $\gamma$ values), this limits the possible application of
the HOS model. The applicability is once again well described by a steep-
ness threshold. Similarly to the analysis of the influence of the directional
spreading, the effect of the peakness of the spectrum can be discussed with
respect to the modulational instability. This process is indeed enhanced with
the narrowness of the spectrum, resulting in an increase probability of ex-
treme wave occurrence. This is the probable cause of the influence of $\gamma$ on
<figure><img src="image_20.png" /><figcaption>FIG. 12 – Applicability limits for HOS simulations. Influence of the spectral peakedness $\beta = 0.30$ and $h = \infty$ - color online</figcaption></figure>

the domains of applicability observed in Fig. 12.

Figure 13 presents the same study with small water depth $h = 35$ m. The limits are defined thanks to Eq. (7) with $K_3 = 0.65$. In this configuration, $K_2$ is also adjusted together with $K_1$ in order to describe the most accurately possible the limits of applicability of the numerical model. The final parameters are: $\gamma = 1$, $K_1 = 0.053$, $K_2 = 0.20$; $\gamma = 3.3$, $K_1 = 0.060$, $K_2 = 0.20$, $\gamma = 5$, $K_1 = 0.055$, $K_2 = 0.40$ and $\gamma = 10$, $K_1 = 0.050$, $K_2 = 0.50$. Note that as in the previous paragraphs, $K_1$ is chosen as the deep water limiting steepness for consistency in the analysis between deep and limited water depths.

It is observed that the influence of the spectral peakedness is very limited in the range of values tested ($\gamma \in [1; 10]$). A clear trend is difficult to observe even if the narrower wave spectrum seems to define the lower boundary of this applicability graph. This has to be related to the fact that, as noticed previously, the modulational instability is reduced for limited water depth as well as in directional wave fields resulting in this limited effect.

## Conclusion

This study allows to establish in a simple way the limitations in terms of applicability of highly and fully non-linear potential flow solvers. This
<figure><img src="image_21.png" /><figcaption>FIG. 13 – Applicability limits for HOS simulations. Influence of the spectral peakedness $\beta = 0.30$ and $h = 35$ m. Eq. (7) is used with $K_3 = 0.65$ - color online</figcaption></figure>

limitation is a direct consequence of the wave breaking phenomena which can
not be modeled in a deterministic way in such solvers. This study relies on the
use on the High-Order Spectral model, HOS-ocean. Numerous simulations
have been conducted on large domains and over long duration to ensure the
accuracy relative of this range of applicability.

The proposed study characterize the applicability of the numerical model
to treat the regular waves case. In this configuration, simulation are acces-
sible up to configuration close to the physical stability limit of such wave
fields. In deep water, HOS-ocean model manages to accurately solve the re-
gular wave propagation up to 90 % of the stability limit. In reduced water
depth, it is shown that the range of applicability (with respect to the ratio
to highest possible wave height) of the model is reduced up to ≈ 70 % of the
limiting wave height in very shallow water.

In the case of directional irregular sea states, the domains of applicability
are presented as curves presenting the limiting significant wave height $H_s$ as
a function of the peak period $T_p$. This allows, for each configuration, to know
the possible sea states accessible to simulation in a scatter diagram using
non-linear potential flow solvers. The numerical parameters and especially
the discretization are shown to be highly influent on the domain of applica-
bility of the HOS model. The whole study is conducted with a typical choice
of maximum wave number $k_{max}$ that allows good accuracy and efficient numerical solution in the configuration studied. However, it is shown that the applicability is drastically reduced when increasing $k_{max}$.

The influence of the water depth appears also as very important with respect to the limitations of the numerical models. For intermediate and large water depths, the limit is accurately defined as a limiting steepness (which is reduced with the water depth). For small relative water depth, the range of applicability is reduced and well approximated by a simple formula taking into account the limited water effects. Then, the effect of the directional spreading is studied and it appears that a broadening results in a larger range of applicability of the model. The influence of the spectral peakedness (i.e. the frequency bandwidth) appears has limited, especially in small water depths. The range of applicability is reduced with narrower wave spectrum. The latter two observations are consistent with the modulational instability which is essential to understand the formation of extreme waves [40].

Figures 9-13 present a simple overview of the different ranges of applicability. We indicate as a conclusion that non-linear potential flow solvers can be used over a wide range of different sea-states. The main limitation of those solvers resides in the wave breaking phenomena which cannot be simulated in a direct way. The wave breaking in the numerical solver appears at different limits than the physical one. However, it is shown to be governed by similar physics since the thresholds at use (based on steepness or Eq. (7)) are the same as in experiments (with adjusted parameters). Recent works have been carried out with different numerical methods in order to find a model to describe accurately this phenomena and is an active field of research (see e.g. [4, 41, 42]).

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