Journal cover Journal topic
Natural Hazards and Earth System Sciences An interactive open-access journal of the European Geosciences Union
Nat. Hazards Earth Syst. Sci., 10, 2589-2597, 2010
© Author(s) 2010. This work is distributed
under the Creative Commons Attribution 3.0 License.
Research article
17 Dec 2010
Nonlinear evolution of the modulational instability under weak forcing and damping
J. Touboul1 and C. Kharif2 1Laboratoire de Sondages Éléctromagnétiques de l'Environnement Terrestre, UMR 6017, Institut des Sciences de l'Ingénieur Toulon – Var, Av. G. Pompidou, BP 56, 83162, La Valette Cedex, France
2Institut de Recherche sur les Phénomènes Hors Équilibre, UMR 6594, École Centrale Marseille, Technopôle de Château-Gombert, 49 rue Joliot Curie – BP 146, 13384, Marseille Cedex 13, France
Abstract. The evolution of modulational instability, or Benjamin-Feir instability is investigated within the framework of the two-dimensional fully nonlinear potential equations, modified to include wind forcing and viscous dissipation. The wind model corresponds to the Miles' theory. The introduction of dissipation in the equations is briefly discussed. Evolution of this instability in the presence of damping was considered by Segur et al. (2005a) and Wu et al. (2006). Their results were extended theoretically by Kharif et al. (2010) who considered wind forcing and viscous dissipation within the framework of a forced and damped nonlinear Schrödinger equation. The marginal stability curve derived from the fully nonlinear numerical simulations coincides with the curve obtained by Kharif et al. (2010) from a linear stability analysis. Furthermore, it is found that the presence of wind forcing promotes the occurrence of a permanent frequency-downshifting without invoking damping due to breaking wave phenomenon.

Citation: Touboul, J. and Kharif, C.: Nonlinear evolution of the modulational instability under weak forcing and damping, Nat. Hazards Earth Syst. Sci., 10, 2589-2597, doi:10.5194/nhess-10-2589-2010, 2010.
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