Under which conditions the Benjamin-Feir instability may spawn an ...

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Within the framework of the fully nonlinear water waves equations, we consider a Stokes wavetrain modulated by the Benjamin-Feir instability in the presence of ...
Eur. Phys. J. Special Topics 185, 159–168 (2010) c EDP Sciences, Springer-Verlag 2010  DOI: 10.1140/epjst/e2010-01246-7

THE EUROPEAN PHYSICAL JOURNAL SPECIAL TOPICS

Regular Article

Under which conditions the Benjamin-Feir instability may spawn an extreme wave event: A fully nonlinear approach C. Kharif1,a and J. Touboul2 1 2

IRPHE, 49 rue F. Joliot-Curie, 13384 Marseille Cedex 13, France LSEET, Universit´e du Sud-Toulon-Var, France Received in final form and accepted 15 June 2010 Published online 23 August 2010 Abstract. Within the framework of the fully nonlinear water waves equations, we consider a Stokes wavetrain modulated by the Benjamin-Feir instability in the presence of both viscous dissipation and forcing due to wind. The wind model corresponds to the Miles’ theory. By introducing wind effect on the waves, the present paper extends the previous works of [6] and [7] who neglected wind input. It is also a continuation of the study developed by [9] who considered a similar problem within the framework of the NLS equation. The marginal stability curve derived from the fully nonlinear numerical simulations coincides with the curve obtained by [9] from a linear stability analysis. Furthermore, it is found that wind input goes in the subharmonic mode of the modulation whereas dissipation damps the fundamental mode of the initial Stokes wavetrain.

1 Introduction Since [1], it is well known that the potential water wave problem admits as solution uniform trains of two-dimensional progressive waves. The stability of this Stokes’ wave solutions began with the paper by [2] who provided a geometric condition for wave instability. Later on, [3] showed analytically that Stokes waves of moderate amplitude are unstable to small long wave perturbations travelling in the same direction. This instability is called the Benjamin-Feir instability (or modulational instability). [4] derived the same result independently by using an averaged Lagragian approach, which is explained in his book. At the same time, [5] using a Hamiltonian formulation of the water-wave problem obtained the same instability result and derived the nonlinear Schr¨ odinger equation (NLS equation). The evolution of a two-dimensional nonlinear wavetrain on deep water, in the absence of dissipative effects, exhibits the FermiPasta-Ulam (FPU) recurrence phenomenon. This phenomenon is characterized by a series of modulation-demodulation cycles in which initially uniform wavetrains become modulated and then demodulated until they are again uniform. Modulation is caused by the growth of the two dominant sidebands of the Benjamin-Feir instability at the expense of the carrier. During the demodulation, the energy returns to the components of the original wavetrain. Recently within the framework of the NLS equation, [6] revisited the Benjamin-Feir instability when dissipation is taken into account. The latter authors showed that for waves with narrow bandwidth and moderate amplitude, any amount of dissipation stabilizes the modulational instability. In the wavenumber space, the region of instability shrinks as time increases. This means that any a

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initially unstable mode of perturbation does not grow for ever. Damping can stop the growth of the sidebands before nonlinear interactions become important. Hence, when the perturbations are small initially, they cannot grow large enough for nonlinear resonant interaction between the carrier and the sidebands to become important. The amplitude of the sidebands can grow for a while and then oscillate in time. [6] have confirmed their theoretical predictions by laboratory experiments for waves of small to moderate amplitude. Later, [7] developed fully nonlinear numerical simulations which agreed with the theory and experiments of [6]. From the latter study we could conclude that dissipation may prevent the development of the Benjamin-Feir instability (or modulational instability). This effect questions the occurrence of modulational instability of water wavetrains. [8] speculated about the effect of dissipation on the early development of rogue waves and asked the question: Can the Benjamin-Feir instability spawn a rogue wave? Nevertheless, this study did not include wind effect. What is the role of wind upon modulational instability when dissipative effects are considered? This situation was considered by [9] within the framework of the forced and damped version of nonlinear Schr¨ odinger equation (NLS). Following [10], they assume the atmospheric pressure at the interface due to wind and the water wave slope to be in phase. The result is to produce an exponential growth of the amplitude of the envelope. In the presence of dissipation and wind they found that carrier waves of given frequency (or wavenumber) may suffer modulational instability when the friction velocity is larger than a threshold value. Conversely, for a given friction velocity it is found that only carrier waves whose frequency (or wavenumber) is less than a threshold value are unstable. Otherwise dissipation prevents instability to develop in time. Following [7] who investigated within the framework of the fully nonlinear equations the effect of dissipation on the Benjamin-Feir instability we can question the validity of the results of [9] for more general conditions such as large amplitude and broad band spectrum. The goal of this study is to extend the results of [9] to higher order of nonlinearity and larger spectrum. Among the many physical mechanisms generating rogue waves is the modulational instability. Hence, the question about the conditions under which this instability may spawn an extreme wave event is important. For more details on rogue waves one can cite the paper by [11] and the book by [12]. In Sect. 2 we remind the main results obtained by [9]. In Sect. 3 the governing water wave equations and the numerical method are briefly presented. Results of the numerical simulations and comparison with [9] are given in Sect. 4 and a conclusion is provided in Sect. 5.

2 The forced and damped NLS equation Recently [9] used a forced and damped version of the nonlinear Schr¨ odinger equation i(ψt + V ψx ) −

Ω0 WΩ0 k0 ψ − 2iνk02 ψ ψxx − 2Ω0 k02 |ψ|2 ψ = i 8k02 2gρ

(1)

to investigate both damping and amplification effects on the Benjamin-Feir instability. Herein, W = ρa κβ2 u2∗ represents wind effect and ν water viscosity. Parameters k0 and ω0 are the wavenumber and frequency of the carrier wave, satisfying the linear dispersion relation ω02 = gk0 , g is the gravity, V = ω0 /2k0 is the group velocity of the carrier wave, ρ and ρa are the water and air densities respectively, κ is the von Karman constant, u∗ is the friction velocity of the wind and β is the Miles’ coefficient depending on the carrier wave age. Equation (1) describes the spatial and temporal evolution of the envelope, ψ, of the surface elevation, η, of weakly nonlinear and dispersive gravity waves on deep water when dissipation due to viscosity and amplification due to wind are considered. [9] found that the stability of the envelope depends on the frequency of the carrier wave and the friction velocity of the wind over the waves and plotted the critical curve separating stable envelopes from unstable envelopes. Namely, they showed that for a given friction velocity u∗ , only carrier wave of frequency Ω0 which satisfies the following condition are unstable to modulational perturbations 4νκ2 Ω0 < 1. (2) βsu2∗

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This condition can be rewritten as follows A2 Ω (M + 1) × kmax . In the absence of wind and damping, the unperturbed initial condition leads to the steady evolution of the Stokes’ wavetrain, whereas the perturbed initial condition leads to the well known Fermi-Pasta-Ulam recurrence. We propagated these initial conditions under various conditions of wind and dissipation, to analyze the behavior of the modulational instability of the Stokes wavetrain.

4 Results and comparison To discuss of the combined effect of wind and dissipation on the modulational instability of a Stokes wavetrain evolving in deep water we consider first the evolution of the unperturbed Stokes waves in the presence of wind and dissipation and then its evolution when the BenjaminFeir instability is added. Herein the reference flow is the Stokes wave under wind and dissipation effects. Then, the nonlinear evolution of the Stokes wave perturbed by a modulational instability in the presence of wind and dissipation is considered and a comparison to the reference flow is achieved. In that way one can measure the impact of both wind and dissipation on the nonlinear evolution of the modulational instability. Within the framework of the dissipative NLS equation [6] have considered formally the stability of the solution corresponding to the damped Stokes wave and showed that this solution is linearly and nonlinearly stable to infinitesimal and small but finite perturbations respectively. For the fully nonlinear system of equation the stability analysis cannot be undertaken analytically as it was developed by [6]. For a given set of parameters (A, Ω), numerical simulations are run with and without the initial modulational instability. In figures 1 and 2 is plotted the time evolution of the amplitude of the fundamental mode k = 5 and subharmonic mode k = 4 for two different situations corresponding to initial conditions of case A and case B respectively. Figure 1 shows the time evolution of the normalized amplitudes a(t) of the fundamental mode k = 5 and subharmonic mode k = 4 with and without modulational instability. For both cases, the simulations correspond to a wind parameter A = 3 and to a viscosity parameter Ω = 0.69. For these values of A and Ω, we known from the results of [9] that the wavetrain is modulationally unstable to infinitesimal perturbations (note that these results were obtained within the framework of the NLS equation). We see from this figure that both wavetrains evolve in a similar manner during the first one hundred periods T of propagation, where T is the fundamental wave period. Then, the development of the modulational instability modifies strongly the behavior of the fundamental mode. For the unperturbed case (case A), the fundamental component increases until the spontaneous occurrence of the modulational instability at t = 900T . For the initially perturbed case (case B), the development of the modulational instability is responsible for the frequency downshift observed at around t = 600T , already discussed by [14] in the presence of only molecular viscosity and by [15] in the presence of wind and eddy viscosity. One can see that

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1.2

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1000

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Fig. 1. Time evolution of the normalized amplitudes of the fundamental mode (k = 5) and subharmonic mode (k = 4) for (A, Ω) = (3, 0.69). Fundamental mode amplitude (—) and subharmonic mode amplitude (– –) for an initially unperturbed Stokes’ wave. Fundamental mode amplitude (– · –) and subharmonic mode amplitude (· · ·) for an initially perturbed Stokes’ wave. (T is the fundamental wave period).

the subharmonic component increases continuously whereas the fundamental component decreases. Hence, wind energy goes to the subharmonic mode whereas dissipation reduces the fundamental component. On figure 3 one can observe the persistence of the modulational instability. Figure 2 corresponds to (A, Ω) = (3, 0.73). Wind condition is similar to the previous numerical simulation, on the other hand the dissipative effect is stronger. Within the framework of the NLS equation [9] have shown that under these conditions the wavetrain is linearly stable to modulational perturbations. In this case dissipation prevails over amplification due to wind and [6] 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 in time so that every wavenumber becomes stable. 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. For our problem, a norm will be defined later. For case A, as expected we can 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 case B, the first maximum of modulation that occurs at t = 218T is followed by partial damped modulation/demodulation cycles. As it can be seen in figure 4 the modulation instability has disappeared at long time evolution. The result of this numerical simulation agrees with that of [6] and [7] who considered only dissipation. Even in the presence of wind the Benjamin-Feir instability restabilises when the dissipative effect dominates wind effect. As it can be seen on figure 2, the evolutions of the fundamental mode k = 5 of case A and case B remain close to each other and the gap between them goes to zero for t → ∞ whereas the amplitude of the subharmonic mode becomes constant. Assuming that the dominant mode describes the main behaviour of a wavetrain, we introduce a norm measuring the distance between the fundamental modes of the unperturbed and perturbed Stokes wave corresponding to case A and case B respectively. Let this norm be

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a(t)/a0

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0

0

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700

800

900

1000

t/T

Fig. 2. Time evolution of the normalized amplitudes of the fundamental mode (k = 5) and subharmonic mode (k = 4) for (A, Ω) = (3, 0.73). Fundamental mode amplitude (—) for an initially unperturbed Stokes’ wave. Fundamental mode amplitude (– · –) and subharmonic mode amplitude (· · ·) for an initially perturbed Stokes’ wave. (T is the fundamental wave period).

defined as  EN (t) =

a0perturbed (t) − a0unperturbed (t) a0unperturbed (0)

2 ,

(10)

where a0unperturbed is the fundamental wavenumber amplitude of the initially unperturbed wavetrain, and a0perturbed the fundamental wavenumber amplitude of the initially perturbed wavetrain. One may assume that the value of this norm will 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) = (3, 0.69) and (A, Ω) = (3, 0.73). The first case exhibits oscillations around an averaged value growing exponentially, whereas the second case exhibits the same oscillations around a value presenting an exponential decay. We can claim that the norm, EN , presents globally exponential growth and exponential decay 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. Furthermore, we have checked that the respective envelopes of the wavetrain are close to each other. Many numerical simulations have been run for various values of the parameters A and Ω. Figure 6 shows a stability diagram which compares the present numerical results with those of [9] 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. We have extended the results derived by [9] within the framework of the NLS equation to the fully nonlinear equations. As it can be seen in figure 6 a rather good agreement is obtained between both approaches.

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t/T = 001

0.08 0.06 0.04

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t/T = 186

0.08 0.06 0.04

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−0.02 −0.04 −0.06 −0.08 −0.1 0

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t/T = 232

0.08 0.06 0.04

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−0.02 −0.04 −0.06 −0.08 −0.1

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X

t/T = 701

0.08 0.06 0.04

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0.02 0

−0.02 −0.04 −0.06 −0.08 −0.1 0

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t/T = 799

0.08 0.06 0.04

η

0.02 0

−0.02 −0.04 −0.06 −0.08 −0.1 0

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t/T = 919

0.08 0.06 0.04

η

0.02 0

−0.02 −0.04 −0.06 −0.08 −0.1 0

1

2

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X

Fig. 3. Surface wave profiles at different times, obtained while propagating initial condition corresponding to case B with (A, Ω) = (3, 0.69). From top to bottom t/T = 1, 186, 232, 701, 799, 919.

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0.1 0.08 0.06 0.04

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X t/T = 218

0.1 0.08 0.06 0.04

η

0.02 0

−0.02 −0.04 −0.06 −0.08 −0.1

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t/T = 302

0.08 0.06 0.04

η

0.02 0

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3X

t/T = 709

0.1 0.08 0.06 0.04

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0.02 0

−0.02 −0.04 −0.06 −0.08 −0.1

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X 0.1

t/T = 804

0.08 0.06 0.04

η

0.02 0

−0.02 −0.04 −0.06 −0.08 −0.1

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t/T = 908

0.08 0.06 0.04

η

0.02 0

−0.02 −0.04 −0.06 −0.08 −0.1

0

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X

Fig. 4. Surface wave profiles at different times, obtained while propagating initial condition corresponding to case B with (A, Ω) = (3, 0.73). From top to bottom t/T = 1, 218, 302, 709, 804, 908.

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Fig. 5. Time evolution of the norm En for (A, Ω) = (3, 0.69) (—) and (A, Ω) = (3, 0.73) (– · –).

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A Fig. 6. Theoretical (—) and numerical (– · –) marginal stability contour lines. The theoretical curve corresponds to the figure 1 of [9].

5 Conclusion This study was aimed at extending the work of [9] to the fully nonlinear case. Within the framework of the NLS equation the latter authors considered the modulational instability of Stokes wavetrains suffering both effects of wind and dissipation. They found that the modulational instability depends on both frequency of the carrier wave and strength of the wind velocity and plotted the curve corresponding to marginal stability in the (A, Ω)-plane. To distinguish stable solutions from unstable solutions we have introduced a norm based on the fundamental

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mode of the wavetrain. 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 [9]. 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 wavetrain.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.

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