**6. Results and comparisons**

**Figure 1.** Time evolution of the normalized amplitudes of the fundamental mode (*k* = 5), subharmonic mode (*k* = 4) and superharmonic mode (*k* = 6) for (A, Ω)=(4, 0.59). Fundamental mode amplitude (**—**), subharmonic mode amplitude (**—**) and superharmonic mode amplitude (**—**) for an initially unperturbed Stokes' wave (unseeded case). Fundamental mode amplitude (**—**), subharmonic mode amplitude (**—**) and superharmonic mode amplitude (**—**) for an initially perturbed Stokes' wave (seeded case). *T* is the fundamental wave period.

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.

8 Will-be-set-by-IN-TECH

conditions of wind and dissipation, to analyze the behavior of the modulational instability of

0 100 200 300 400 500 600 700 800 900 1000

t/T

**Figure 1.** Time evolution of the normalized amplitudes of the fundamental mode (*k* = 5), subharmonic mode (*k* = 4) and superharmonic mode (*k* = 6) for (A, Ω)=(4, 0.59). Fundamental mode amplitude (**—**), subharmonic mode amplitude (**—**) and superharmonic mode amplitude (**—**) for an initially unperturbed Stokes' wave (unseeded case). Fundamental mode amplitude (**—**), subharmonic mode amplitude (**—**) and superharmonic mode amplitude (**—**) for an initially perturbed Stokes' wave (seeded

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

the Stokes wavetrain.

**6. Results and comparisons**

0

case). *T* is the fundamental wave period.

perturbation is thus obtained.

0.2

0.4

0.6

0.8

a(t)/a0

1

1.2

**Figure 2.** Time evolution of the normalized amplitudes of the fundamental mode (*k* = 5), subharmonic mode (*k* = 4) and superharmonic mode (*k* = 6) for (A, Ω)=(4, 0.61). Fundamental mode amplitude (**—**), subharmonic mode amplitude (**—**) and superharmonic mode amplitude (**—**) for an initially unperturbed Stokes' wave (unseeded case). Fundamental mode amplitude (**—**), subharmonic mode amplitude (**—**) and superharmonic mode amplitude (**—**) for an initially perturbed Stokes' wave (seeded case). *T* is the fundamental wave period.

Figures 1 and 2 present the time evolution of the amplitudes of three components of the water waves' spectrum. The mode *k* = 5 is the fundamental mode, while modes *k* = 4 and *k* = 6 are sidebands, respectively the subharmonic and the superharmonic. Each of these figures present to two kinds of initial conditions, namely the unseeded and the seeded cases. The two figures correspond to two different conditions of wind forcing and damping.

Figure 1 shows the time evolution of the normalized amplitudes *a*(*t*)/*a*<sup>0</sup> of the fundamental mode *k* = 5, subharmonic mode *k* = 4 and superharmonic mode *k* = 6 with and without perturbations for the modulational instability. For both cases, the simulations correspond to a wind parameter A = 4 and to a viscosity parameter Ω = 0.59. Within the framework of the NLS equation, [4] showed that the wave train is unstable to modulational instability for these values of A and Ω. From this figure, it appears that both wavetrains (unseeded and seeded cases) present a similar evolution during the first hundred periods of propagation, *T* being the fundamental wave period. Then, the behavior of the wavetrain is strongly affected by the development of the modulational instability. For the unperturbed case (unseeded case), the fundamental component increases, since no occurrence of the modulational instability is expected. However, due to the accumulation of numerical errors, the spontaneous occurrence of the modulational instability cannot be avoided, but not before *t* = 900*T*. On figure 3 one can observe the persistence of the modulational instability through the evolution of the free surface. Indeed, it is observed that the wave packet is alternatively modulated, leading to the formation of a large wave, and demodulated, corresponding to a state which is closer from the origin. This is an expression of the Fermi-Pasta-Ulam quasi-recurrence.

Figure 2 corresponds to (A, Ω)=(4, 0.61). Wind condition is similar to the previous numerical simulation, but the dissipative effect considered is stronger. This case correspond to a linearly stable case of the modulational instability, as obtained by [4] in the framework of the NLS equation. From this figure, one can see that wind energy goes to the subharmonic mode whereas dissipation reduces the fundamental and superharmonic components, as previously observed. However, modulation of modes decrease, and they present a monotonic behavior. For unseeded case, as expected, we observe an exponential decay of the fundamental mode. Note that there is no natural occurrence of the subharmonic mode of the modulational instability as it was found in figure 1. For seeded case, the first maximum of modulation that occurs at *t* = 410*T* is followed by partial damped modulation/demodulation cycles. Figure 4 illustrates the disappearance of the modulational instability through the evolution of the free surface. In this case dissipation prevails over amplification due to wind and [2] have obtained linear and nonlinear stability of modulational perturbations within the framework of the dissipative NLS equation. More specifically they showed that dissipation reduces the set of unstable wavenumbers as time increases. Consequently every mode becomes stable. The result of this numerical simulation agrees with that of [2] and [3] who considered only dissipation. In their approach, a solution is said to be stable if every solution that starts close to this solution at *t* = 0 remains close to it for all *t >* 0, otherwise the solution is unstable. To include nonlinear stability analysis they introduced a norm and considered stability in the sense of Lyapunov.

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

$$E\_N(t) = \frac{\int\_{-\infty}^{\infty} (a\_{k\_S}(t) - a\_{k\_{US}}(t))^2 dk}{\int\_{-\infty}^{\infty} a\_{k\_{US}}^2(0) dk},\tag{24}$$

0 1 2 3 4 5 6 −0.1

0 1 2 3 4 5 6 −0.1

0 1 2 3 4 5 6 −0.1

0 1 2 3 4 5 6 −0.1

0 1 2 3 4 5 6 −0.1

0 1 2 3 4 5 6 −0.1

**Figure 3.** Surface wave profiles at different times, obtained while propagating initial condition

corresponding to seeded case with (A, Ω)=(4, 0.59). From top to bottom

X

X

t/T = 846

X

t/T = 793

X

t/T = 596

X

t/T = 436

X

t/T = 291

t/T = 001

Numerical Simulations of Water Waves' Modulational Instability Under the Action of Wind and Dissipation 385

−0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1

−0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1

−0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1

−0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1

−0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1

−0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1

η

η

η

η

η

η

*t*/*T* = 1, 291, 436, 596, 793, 846.

where *akUS* (*t*) is the amplitude of the component of water elevation *η* of wave number *k*, for the initially unperturbed wave train (unseeded case), and *akS* (*t*) is the amplitude of the component of water elevation *η* of wave number *k*, for the initially perturbed wave train (seeded case). This norm corresponds to the potential energy of the perturbation. Its value characterizes the deviation of the perturbed solution from the unperturbed solution. Figure 5 shows the time evolution of this norm for two sets of parameters (A, Ω)=(4, 0.59) and (A, Ω)=(4, 0.61). For the two cases we can observe two regimes. The first regime corresponds to the development of the modulational instability and shows that it is the nonlinear interaction between the fundamental mode and its sidebands which dominates with a weak effect of the wind forcing and the dissipation. The second regime corresponding to the oscillatory evolution of the norm is dominated by the competition between wind forcing and dissipation. The nonlinear interaction between the fundamental mode and the sidebands is 10 Will-be-set-by-IN-TECH

surface. Indeed, it is observed that the wave packet is alternatively modulated, leading to the formation of a large wave, and demodulated, corresponding to a state which is closer from

Figure 2 corresponds to (A, Ω)=(4, 0.61). Wind condition is similar to the previous numerical simulation, but the dissipative effect considered is stronger. This case correspond to a linearly stable case of the modulational instability, as obtained by [4] in the framework of the NLS equation. From this figure, one can see that wind energy goes to the subharmonic mode whereas dissipation reduces the fundamental and superharmonic components, as previously observed. However, modulation of modes decrease, and they present a monotonic behavior. For unseeded case, as expected, we observe an exponential decay of the fundamental mode. Note that there is no natural occurrence of the subharmonic mode of the modulational instability as it was found in figure 1. For seeded case, the first maximum of modulation that occurs at *t* = 410*T* is followed by partial damped modulation/demodulation cycles. Figure 4 illustrates the disappearance of the modulational instability through the evolution of the free surface. In this case dissipation prevails over amplification due to wind and [2] have obtained linear and nonlinear stability of modulational perturbations within the framework of the dissipative NLS equation. More specifically they showed that dissipation reduces the set of unstable wavenumbers as time increases. Consequently every mode becomes stable. The result of this numerical simulation agrees with that of [2] and [3] who considered only dissipation. In their approach, a solution is said to be stable if every solution that starts close to this solution at *t* = 0 remains close to it for all *t >* 0, otherwise the solution is unstable. To include nonlinear stability analysis they introduced a norm and considered stability in the

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

> ∞ <sup>−</sup><sup>∞</sup> *<sup>a</sup>*<sup>2</sup>

where *akUS* (*t*) is the amplitude of the component of water elevation *η* of wave number *k*, for the initially unperturbed wave train (unseeded case), and *akS* (*t*) is the amplitude of the component of water elevation *η* of wave number *k*, for the initially perturbed wave train (seeded case). This norm corresponds to the potential energy of the perturbation. Its value characterizes the deviation of the perturbed solution from the unperturbed solution. Figure 5 shows the time evolution of this norm for two sets of parameters (A, Ω)=(4, 0.59) and (A, Ω)=(4, 0.61). For the two cases we can observe two regimes. The first regime corresponds to the development of the modulational instability and shows that it is the nonlinear interaction between the fundamental mode and its sidebands which dominates with a weak effect of the wind forcing and the dissipation. The second regime corresponding to the oscillatory evolution of the norm is dominated by the competition between wind forcing and dissipation. The nonlinear interaction between the fundamental mode and the sidebands is

<sup>−</sup>∞(*akS* (*t*) <sup>−</sup> *akUS* (*t*))2*dk*

*kUS* (0)*dk* , (24)

perturbation, as it was stated in [19]. Thus, another norm can be introduced as

<sup>∞</sup>

*EN*(*t*) =

the origin. This is an expression of the Fermi-Pasta-Ulam quasi-recurrence.

sense of Lyapunov.

**Figure 3.** Surface wave profiles at different times, obtained while propagating initial condition corresponding to seeded case with (A, Ω)=(4, 0.59). From top to bottom *t*/*T* = 1, 291, 436, 596, 793, 846.

0 100 200 300 400 500 600 700 800 900 1000

Numerical Simulations of Water Waves' Modulational Instability Under the Action of Wind and Dissipation 387

t/T

0 1 2 3 4 5 6 7 8 9 10 11

 *A*

**Figure 6.** Theoretical (**—**) and numerical (**–** · **–**) marginal stability contour lines. (◦) correspond to numerical results obtained in the framework of the equations suggested by [3]. The theoretical curve

**Figure 5.** Time evolution of the norm *En* for (A, Ω)=(4, 0.59) (**—**) and (A, Ω)=(4, 0.61) (**—**).

10−4

0

corresponds to the figure 1 of [4].

0.2

0.4

0.6

0.8

Ω

1

1.2

1.4

1.6

10−3

10−2

En

10−1

100

101

**Figure 4.** Surface wave profiles at different times, obtained while propagating initial condition corresponding to seeded case with (A, Ω)=(4, 0.61). From top to bottom *t*/*T* = 1, 410, 496, 601, 676, 885.

12 Will-be-set-by-IN-TECH

0 1 2 3 4 5 6 −0.1

0 1 2 3 4 5 6 −0.1

0 1 2 3 4 5 6 −0.1

0 1 2 3 4 5 6 −0.1

0 1 2 3 4 5 6 −0.1

0 1 2 3 4 5 6 −0.1

**Figure 4.** Surface wave profiles at different times, obtained while propagating initial condition

corresponding to seeded case with (A, Ω)=(4, 0.61). From top to bottom

X

X

t/T = 885

X

t/T = 676

X

t/T = 601

X

t/T = 496

X

t/T = 410

t/T = 001

−0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1

−0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1

−0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1

−0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1

−0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1

−0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1

η

η

η

η

η

η

*t*/*T* = 1, 410, 496, 601, 676, 885.

**Figure 5.** Time evolution of the norm *En* for (A, Ω)=(4, 0.59) (**—**) and (A, Ω)=(4, 0.61) (**—**).

**Figure 6.** Theoretical (**—**) and numerical (**–** · **–**) marginal stability contour lines. (◦) correspond to numerical results obtained in the framework of the equations suggested by [3]. The theoretical curve corresponds to the figure 1 of [4].

#### 14 Will-be-set-by-IN-TECH 388 Numerical Simulation – From Theory to Industry Numerical Simulations of Water Waves' Modulational Instability Under the Action of Wind and Dissipation <sup>15</sup>

not the dominant mechanism. The magenta curve exhibits oscillations around an averaged value growing exponentially, whereas the yellow curve exhibits the same oscillations around a constant value. We can claim that the norm, *EN*, presents globally exponential growth or asymptotical saturation corresponding to instability and stability respectively. Herein, the stability can be interpreted in terms of asymptotic stability. The first case is said to be unstable whereas the second case corresponds to a stable solution. In the latter case we expect that the solution will remain close to the unperturbed solution. In other words, nonlinear interactions are affected by the non conservative effects that are wind and dissipation, leading to a long time disappearance of these interactions.

Many numerical simulations have been run for various values of the parameters A and Ω. Figure 6 shows a stability diagram which presents comparison between the present numerical results and those of [4] obtained theoretically. The marginal curve corresponding to the fully nonlinear equations is very close to the theoretical marginal curve obtained within the framework the NLS equation. The region above the critical curve corresponds to stable cases, whereas the region beneath corresponds to unstable cases. Bars in figure 6 correspond to uncertainty on stability or instability. Numerical results obtained in our previous work [18] within the framework of equations suggested by [3] are plotted for the sake of reference (◦). The way of introducing damping effect into the kinematic boundary condition has little influence on the results, especially for young waves. The present numerical simulations demonstrate that the results derived by [4] within the framework of the NLS equation are correct in the context of the fully nonlinear equations.

0 1 2 3 4 5 6 7 8 9 10 11

Numerical Simulations of Water Waves' Modulational Instability Under the Action of Wind and Dissipation 389

 *A*

**Figure 7.** Nondimensional time *τdownshif t* = *t*/*T* for which the permanent frequency downshift is observed, plotted as a function of the wave age parameter A. Ω is chosen here to be on the marginal

K = 0 (see equation (7) for the definition of K). This might be interpreted in terms of balance between wind forcing and damping, corresponding to an equilibrium state. Furthermore, in these conditions, the forced and damped nonlinear Schrödinger equation reduces to the canonical NLS equation. It becomes obvious that the weakly nonlinear theory does not predict any variation in behavior, since the NLS equation remains unchanged for each values of A and Ω. However, the results of the numerical simulations are different. Indeed, figure 7 presents the nondimensional time *τdownshif t* after which the permanent frequency downshift is observed, as a function of A. From this figure, it seems obvious that this time is strongly dependant on the wind speed. The youngest the waves are, the fastest is the downshift.

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 (A, Ω)-plane. Here, a numerical verification is performed, by means of a fully nonlinear approach. The long term behavior of a wave group propagating under both actions of wind and dissipation was obtained thanks to this method. To distinguish stable solutions from

0

stability curve presented on figure 6.

**7. Conclusion**

500

1000

1500

2000

τdownshift

2500

3000

3500

4000

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 *t* = 500*T*. Indeed, one can see that the subharmonic component increases continuously whereas the fundamental and superharmonic component decrease. The superharmonic component decreases faster than the fundamental component. Hence, wind energy goes to the subharmonic mode whereas dissipation reduces the fundamental and superharmonic components. During the modulation process, a broadening of the spectrum is observed, even if not presented here for the sake of clarity. Beyond *t* = 500*T*, the subharmonic mode *k* = 4 is dominant in the spectrum. To investigate the effect of wind and damping, another series of simulations is performed. Namely, the values of A and Ω where chosen to fulfill the condition

**Figure 7.** Nondimensional time *τdownshif t* = *t*/*T* for which the permanent frequency downshift is observed, plotted as a function of the wave age parameter A. Ω is chosen here to be on the marginal stability curve presented on figure 6.

K = 0 (see equation (7) for the definition of K). This might be interpreted in terms of balance between wind forcing and damping, corresponding to an equilibrium state. Furthermore, in these conditions, the forced and damped nonlinear Schrödinger equation reduces to the canonical NLS equation. It becomes obvious that the weakly nonlinear theory does not predict any variation in behavior, since the NLS equation remains unchanged for each values of A and Ω. However, the results of the numerical simulations are different. Indeed, figure 7 presents the nondimensional time *τdownshif t* after which the permanent frequency downshift is observed, as a function of A. From this figure, it seems obvious that this time is strongly dependant on the wind speed. The youngest the waves are, the fastest is the downshift.

#### **7. Conclusion**

14 Will-be-set-by-IN-TECH

not the dominant mechanism. The magenta curve exhibits oscillations around an averaged value growing exponentially, whereas the yellow curve exhibits the same oscillations around a constant value. We can claim that the norm, *EN*, presents globally exponential growth or asymptotical saturation corresponding to instability and stability respectively. Herein, the stability can be interpreted in terms of asymptotic stability. The first case is said to be unstable whereas the second case corresponds to a stable solution. In the latter case we expect that the solution will remain close to the unperturbed solution. In other words, nonlinear interactions are affected by the non conservative effects that are wind and dissipation, leading to a long

Many numerical simulations have been run for various values of the parameters A and Ω. Figure 6 shows a stability diagram which presents comparison between the present numerical results and those of [4] obtained theoretically. The marginal curve corresponding to the fully nonlinear equations is very close to the theoretical marginal curve obtained within the framework the NLS equation. The region above the critical curve corresponds to stable cases, whereas the region beneath corresponds to unstable cases. Bars in figure 6 correspond to uncertainty on stability or instability. Numerical results obtained in our previous work [18] within the framework of equations suggested by [3] are plotted for the sake of reference (◦). The way of introducing damping effect into the kinematic boundary condition has little influence on the results, especially for young waves. The present numerical simulations demonstrate that the results derived by [4] within the framework of the NLS equation are

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 *t* = 500*T*. Indeed, one can see that the subharmonic component increases continuously whereas the fundamental and superharmonic component decrease. The superharmonic component decreases faster than the fundamental component. Hence, wind energy goes to the subharmonic mode whereas dissipation reduces the fundamental and superharmonic components. During the modulation process, a broadening of the spectrum is observed, even if not presented here for the sake of clarity. Beyond *t* = 500*T*, the subharmonic mode *k* = 4 is dominant in the spectrum. To investigate the effect of wind and damping, another series of simulations is performed. Namely, the values of A and Ω where chosen to fulfill the condition

time disappearance of these interactions.

correct in the context of the fully nonlinear equations.

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 (A, Ω)-plane. Here, a numerical verification is performed, by means of a fully nonlinear approach. The long term behavior of a wave group propagating under both actions of wind and dissipation was obtained thanks to this method. To distinguish stable solutions from unstable solutions, a norm based on the potential energy of the perturbations was introduced. A nonlinear stability diagram resulting from the numerical simulations of the fully nonlinear equation has been given in the (A, Ω)-plane which coincides with the linear stability analysis of [4]. In the presence of wind, dissipation and modulational instability it is found that wind energy goes to the subharmonic sideband whereas dissipation lowers the amplitude of the fundamental mode of the wave train yielding to a permanent frequency-downshifting. This permanent frequency downshift is strongly influenced by the wind and dissipation parameter. If the wave group is at equilibrium in energy input and dissipation, the fdNLS equation reduces to the classical NLS equation, and predict no influence of the wind. However, by considering the asymmetry between wave components, induced by strong nonlinearity (higher than fourth order), a strong influence of the wind and dissipation is observed.

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