The modulational instability in deep water under the action of wind and dissipation

Abstract

The modulational instability of gravity wave trains on the surface of water acted upon by wind and under influence of viscosity is considered. The wind regime is that of validity of Miles' theory and the viscosity is small. By using a perturbed nonlinear Schrödinger equation describing the evolution of a narrow-banded wavepacket under the action of wind and dissipation, the modulational instability of the wave group is shown to depend on both the frequency (or wavenumber) of the carrier wave and the strength of the friction velocity (or the wind speed). For fixed values of the watersurface roughness, the marginal curves separating stable states from unstable states are given. It is found in the low-frequency regime that stronger wind velocities are needed to sustain the modulational instability than for high-frequency water waves. In other words, the critical frequency decreases as the carrier wave age increases. Furthermore, it is shown for a given carrier frequency that a larger friction velocity is needed to sustain modulational instability when the roughness length is increased.

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

“…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.…”

“…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.…”

“…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].…”

“…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 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.…”