a1 Institut de Recherche sur les Phénomènes Hors Équilibre, 49, rue F. Joliot-Curie, BP 146, 13384 Marseille CEDEX 13, France
a2 Instituto de Fisica Teorica, UNESP, R. Pamplona 145, 01405-900 São Paulo, Brazil
a3 Laboratoire de Physique Théorique et Astroparticules, CNRS-UMR 5207, Université Montpellier II, Place Eugène Bataillon, 34095 Montpellier CEDEX 05, France
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 water-surface 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.
(Received February 05 2010)
(Revised August 11 2010)
(Accepted August 11 2010)
(Online publication November 01 2010)