A note on stabilizing the Benjamin–Feir instability

Abstract

In a recent paper, Segur et al. (J. Fluid Mech. vol. 539, p. 229, 2005, hereafter referred to as ${\cal S}$ showed, based on a damped version of the nonlinear Schrodinger equation (NLS), that any amount of dissipation (of a certain type) stabilizes the Benjamin–Feir instability of a modulated Stokes wave train. Their theoretical predictions are confirmed by laboratory experiments for waves of small or moderate amplitude, but not for waves of large amplitude or with relatively large perturbations. ${\cal S}$ left open questions regarding the validity of their theoretical results for these large-amplitude waves, and possibly the validity of the NLS assumptions of weak nonlinearity and narrow-bandedness. We investigate these issues using direct simulations of the primitive equations, incorporating constant and wavenumber-dependent dissipation models. For small or moderate amplitudes, our full simulations agree with the theory and experiments of ${\cal S}$. For large amplitudes, we find that it is primarily the form of the dissipation model, rather than the assumptions of NLS, that is responsible for the failure of ${\cal S}$'s theoretical predictions. Indeed, with an appropriate wavenumber-dependent dissipation model, both the full simulations and NLS obtain the correct evolution behaviour for large-amplitude waves. Finally, using direct and NLS simulations, we confirm the general conclusion of ${\cal S}$ for stabilization of the Benjamin–Feir instability over long-time wave train evolution.