## 1. Introduction

The seek of uniform, propagative wave train solutions of the fully nonlinear potential equations has been a major topic for centuries. [5] was the first to propose an expression of such waves, the so called Stokes’ waves. However, pioneer works of [6] emphasized that such waves might be unstable, providing a geometric condition for this stability problem. Later on, [1] showed analytically that Stokes’ waves of moderate amplitude are unstable to long wave perturbations of small amplitude travelling in the same direction. This instability is named the Benjamin-Feir instability (or modulational instability). This result was derived independently by [7] in an averaged Lagrangian approach, and by [8] who used an Hamiltonian formulation of the water wave problem. Using this approach, the latter author derived the nonlinear Schrödinger equation (NLS), and confirmed the previous stability results.

Within the last fifty years, the study of this instability became central for fundamental and applied research. The modulation instability is one of the most important mechanisms for the formation of rogue waves [9]. A complete review on the various phenomena yielding to rogue waves can be found in the book of [10]. In the absence of forcing and damping, Stokes’ waves of specific initial steepness are submitted to this instability, when they encounter perturbations of specific wave numbers [11, 12]. In this case, they encounter a nonlinear quasi-recursive evolution, the so called Fermi-Pasta-Ulam recurrence phenomenon ([13]). This phenomenon corresponds to a series of modulation - demodulation cycles, during which initially uniform wave trains become modulated, leading possibly to the formation of a huge wave. Modulation is due to an energy transfer from the wave carrier to the unstable sidebands. In the wave number space, these unstable sidebands are located in a finite narrow band centered around the carrier wave number. During the demodulation, the energy returns to the fundamental component of the original wave train. Using the Zakharov equation, [14] questions the relevance of the Benjamin-Feir index to indicate the intensity of modulational instability. Indeed, this index is often used to quantify the intensity of interactions between a carrier wave and the finite amplitude sidebands. However, [14] emphasized that nonlinear interactions occur also for sidebands located beyond the Benjamin-Feir instability domain.

A damped nonlinear Schrödinger equation (dNLS) was derived by [15] who revisited the Benjamin-Feir instability in the presence of dissipation. They studied numerically the evolution of narrow bandwidth waves of moderate amplitude. More recently [2] investigated theoretically the modulational instability within the framework of the dNLS equation and demonstrated that any amount of dissipation stabilizes the modulational instability in the sense of Lyapunov. Namely, they showed that the zone of unstable region, in the wavenumber space, shrinks as time increases. As a result, any initially unstable mode of perturbation will finally become stable. [2] have confirmed their theoretical predictions by laboratory experiments for waves of small to moderate amplitude. Later, [3] developed fully nonlinear numerical simulations which agreed with the theory and experiments of [2].

From the latter study we could conclude that dissipation may prevent the development of the Benjamin-Feir instability. This effect questions the occurrence of modulational instability of water wave trains in the field. [16] speculated about the effect of dissipation on the early development of rogue waves and raised the question whether or not the Benjamin-Feir instability was able to spawn a rogue wave.

Nevertheless, these authors did not take the effect of wind into account. When considering the occurrence of modulational instability in the field, the role of wind upon this instability in the presence of dissipation needs to be addressed. Based on this assumption, [4] derived a forced and damped nonlinear Schrödinger equation (fdNLS), and extended the analysis of [2] when wind input is introduced. The influence of wind was introduced through a pressure term acting at the interface, in phase with the wave slope, accordingly to Miles’ theory [17]. This quasi-laminar theory of wind wave amplification is based on the Miles’ shear flow instability. This mechanism of wave amplification is a resonant interaction between water waves and a plane shear flow in air which occurs at the critical height where the wind velocity matches the phase velocity of the surface waves. Stokes waves propagating in the presence of such a forcing, when not submitted to modulational instability, encounter an exponential growth. They demonstrated, within the framework of fdNLS equation, that Stokes’ waves were unstable to modulational instability as soon as the friction velocity is larger than a threshold value. Conversely, for a given friction velocity it was found that only carrier waves presenting frequencies (or wavenumbers) lower than a threshold value are subject to Benjamin-Feir instability. Otherwise, due to dissipation, modulational instability restabilizes in the sense of Lyapunov.

As it was mentioned, this physical result is based on the solution of an approached model, the fdNLS equation. Thus, a proper verification is required. However, the phenomenon at hand is based on the long-time behavior of the modulated wave train when propagating in the presence of wind and dissipation. This remark explains the difficulty to provide an experimental verification of the theory. This physical problem is then especially well adapted for a numerical verification. This verification was performed in a first time by [18], who investigated the development of the modulational instability under wind action and viscous dissipation within the framework of fully nonlinear potential equations. This work is an extension of that of [3] when wind input is considered. Later on, [19] emphasized that the equations empirically introduced by [3] were not completely representative of the dispersion relation in the presence of damping, and corrected the equations to overcome this problem, in accordance with the demonstration of [20] and [21].

This work aims to emphasize how numerical simulations can provide useful information to validate long term results based on weakly nonlinear theory. Furthermore, the numerical approach presented here constitutes an extension of the results of [4] to higher orders of nonlinearity and larger band spectra, too. The long time evolution of modulated wave trains can be investigated in a way not allowed by fdNLS equation. The numerical simulations enable to produce results concerning the long time behavior of the modulated wave train. Especially, the phenomenon of permanent frequency downshift will be investigated.

In section 2, the governing equations of the problem are presented. Section 3 presents the weakly nonlinear model obtained by [4], and summarizes their results. The numerical model used to investigate the long time behavior of the modulated wave train is developed in section 4. The initial conditions used to support the numerical strategy for validating the theory introduced by [4] is presented in section 5. Finally, the results obtained are described in section 6.

## 2. Governing equations of the problem

The approach used in this study is based on the potential flow theory. The fluid is assumed to be incompressible, inviscid, and animated by an irrotational motion. Thus, the fluid velocity derives from a potential

The wind has already been introduced in the dynamic boundary condition through a pressure term acting at the free surface in several numerical potential models. Among them, one may cite [22], [23] and [24] who introduced and discussed this approach for BIEM methods and [25], [26], and [27] who extended it to HOS methods. The pressure term used here is based on the Miles’ theory [17], accordingly to the approach of [4]. The viscosity was introduced heuristically by [3] who used the HOS method to address the question raised in [2] on the restabilisation of the Benjamin-Feir instability of a Stokes wave train in the presence of dissipation. The introduction was made through the addition of a damping term in the dynamic boundary condition. However, a proper derivation of the kinematic and dynamic boundary condition in the presence of viscosity was made by [20], and later on by [21]. A modification of both kinematic and dynamic condition was found, resulting in a slight difference in the dispersion relation, as it was discussed by [19]. Finally, the system of equations corresponding to the potential theory, in the presence of wind and viscous damping reads

where

where

## 3. Weakly nonlinear approach: The nonlinear Schrödinger equation

The Nonlinear Schrödinger equation can be obtained from the fully nonlinear potential theory by using the multi-scale method. The equations are expanded in Taylor series, around a small parameter,

to investigate both damping and amplification effects on the Benjamin-Feir instability. Herein,

[4] found that the stability of the envelope depends on the sign of the constant

This condition can be rewritten as follows

where

## 4. Fully nonlinear approach: The High Order Spectral method

Within the framework of two-dimensional flows, a High-Order Spectral Method is used to solve numerically the basic partial differential equations corresponding to equations (1 - 4). The lateral conditions correspond here to space-periodic conditions. The horizontal bottom condition corresponds to infinite depth. The velocity potential is expanded in a series of eigenfunctions fulfilling both these lateral and bottom conditions. A spectral treatment is well adapted to investigate numerically the long time behavior of periodic water waves encountering the modulational instability.

### 4.1. Mathematical formulation

We first introduce the following dimensionless variables into equations (1), (2), (3) and (4):

where

Following [8], we introduce the velocity potential at the free surface

with

The main difficulty in this approach is the computation of the vertical velocity at the free surface,

The term

At a given instant of time,

The boundary conditions, together with the Laplace equations

Note that

### 4.2. Computation of the vertical velocity

Substitution of equation (20) into the set of equations (18 - 19) provides an expression of the modes

This expression might be substituted into the kinematic and dynamic boundary conditions (13) and (14), yielding to the evolution equations for

where

In fact, the version of [29] differs from the version of [28] not only in the expression of the approximated vertical velocity at the surface, but also in its subsequent treatment in the free surface equations. According to [29], the surface equations must be truncated at consistent nonlinear order if they are to simulate a conservative Hamiltonian system. This requires to treat carefully all nonlinear terms containing

## 5. Initial conditions for the numerical simulations

From a numerical point of view, one part of the initial condition is obtained by considering a Stokes wavetrain

## 6. Results and comparisons

One of the difficulties involved in this study is to define clearly the stability. Indeed, since Stokes’ waves are propagating under the action of wind and viscosity, this flow cannot be considered stationary nor periodic. Discussing of the combined influence of wind forcing and damping on the modulational instability, however, implies to define a reference flow. In order to do so, we first consider the evolution of the unperturbed Stokes’ waves in the presence of forcing and dissipation (unseeded case). It is checked that the instability does not develop spontaneously in the laps of time considered. Afterwards, we consider the evolution of the initially perturbed Stokes’ wave train under the same conditions of wind forcing and damping (seeded case). The nonlinear evolution of the Stokes’ wavetrain perturbed by the modulational instability in the presence of wind and dissipation is then compared to that of the reference flow. In that way, the deviation from the reference flow can be interpreted in terms of modulational instability, and the influence of wind forcing and dissipation can be analyzed. Following our previous works [18, 19], the evolution of the energy of the perturbation is thus obtained.

Figures 1 and 2 present the time evolution of the amplitudes of three components of the water waves’ spectrum. The mode

Figure 1 shows the time evolution of the normalized amplitudes

Figure 2 corresponds to

In our previous work [18], we assumed that the dominant mode describes the main behavior of a wave train, and we introduced a norm measuring the distance between the fundamental modes of the unperturbed and perturbed Stokes wave corresponding to unseeded case and seeded case respectively. However, it is more consistent to consider the energy of the perturbation, as it was stated in [19]. Thus, another norm can be introduced as

where

Figure 4 shows the time evolution of this norm for two sets of parameters

Many numerical simulations have been run for various values of the parameters

This result provides a validation of the weakly nonlinear theory obtained in the framework of nonlinear Schrödinger equation. However, the numerical approach allows to investigate the long time evolution of the wave train, taking into account the strongly nonlinear behavior of water waves. One phenomenon especially illustrates this nonlinear behavior: the permanent frequency downshift. This phenomenon was discussed by [32] and [33] within the framework of gravity waves. These authors considered that dissipation due to breaking wave was responsible for this permanent downshift. [34] modeled the phenomenon in the presence of wind and eddy viscosity, and latter on [35] in the presence of only molecular viscosity. All these works are based on equations valid up to fourth order in nonlinearity, or higher. Indeed, it is well known that the frequency downshift cannot be observed in the framework of nonlinear Schrödinger equation, which preserves the symmetry between subharmonic and superharmonic components. In fact, [36] concluded that in the absence of wind and dissipation, it was not possible to observe the phenomenon even with higher order equations.

If going back to figure 1, for the initially perturbed case (seeded case), the development of the modulational instability is responsible for the frequency downshift observed at around

To investigate the effect of wind and damping, another series of simulations is performed. Namely, the values of

## 7. Conclusion

In this work, it was evidence how numerical simulations can provide a good demonstration of a weakly nonlinear theory that cannot be achieved by means of experimental demonstration. In this study, an extension of the work of [4] to the fully nonlinear case was suggested. Within the framework of the NLS equation the latter authors considered the modulational instability of Stokes wave trains suffering both effects of wind and dissipation. The results they obtained show that the modulational instability depends on both frequency of the carrier wave and strength of the wind velocity. They plotted the curve corresponding to marginal stability in the