On the Asymmetric Spectral Broadening of a Hydrodynamic Modulated Wave Train in the Optical Regime

Takuji Waseda,Wataru Fujimoto,Amin Chabchoub

doi:10.3390/fluids4020084

Takuji Waseda, Wataru Fujimoto + Show 1 more

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https://doi.org/10.3390/fluids4020084

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Abstract

Amplitude modulation of a propagating wave train has been observed in various media including hydrodynamics and optical fibers. The notable difference of the propagating wave trains in these media is the magnitude of the nonlinearity and the associated spectral bandwidth. The nonlinearity and dispersion parameters of optical fibers are two orders of magnitude smaller than the hydrodynamic counterparts, and therefore, considered to better assure the slowly varying envelope approximation (SVEA) of the nonlinear Schrödinger equations (NLSE). While most optics experiment demonstrate an NLSE-like symmetric solutions, experimental studies by Dudley et al. (Optics Express, 2009, 17, 21497–21508) show an asymmetric spectral evolution in the dynamics of unstable electromagnetic waves with high intensities. Motivated by this result, the hydrodynamic Euler equation is numerically solved to study the long-term evolution of a water-wave modulated wave train in the optical regime, i.e., at small steepness and spectral bandwidth. As the initial steepness is increased, retaining the initial spectral bandwidth thereby increasing the Benjamin–Feir Index, the modulation localizes, and the asymmetric and broad spectrum appears. While the deviation of the evolution from the NLSE solution is a result of broadband dynamics of free wave interaction, the resulting asymmetry of the spectrum is a consequence of the violation of the SVEA.

Highlights

Freak waves or rogue waves in the ocean have been a subject of research for physicists, oceanographers, and engineers for several decades
While the deviation of the evolution from the nonlinear Schrödinger equation (NLSE) solution is a result of broadband dynamics of free wave interaction, the resulting asymmetry of the spectrum is a consequence of the violation of the slowly varying envelope approximation (SVEA)
The dynamics governing the evolution of such wave groups is a narrow-banded process represented by the nonlinear Schrödinger equation (NLSE) [15,16]

Summary

Introduction

Freak waves or rogue waves in the ocean have been a subject of research for physicists, oceanographers, and engineers for several decades. Marine accidents may have been related to encounters of ships with freak waves [1,2,3] and several freak wave incidents were reported from offshore platforms [4,5]. Whether to take the freak waves into consideration in the design criterion of ships and offshore platforms relates to the occurrence probability of freak/rogue waves. Recent studies of freak/rogue waves in realistic sea-states show that the probability is well explained by a second order theory [6,7,8] because the modulational instability (MI) is suppressed due to the broadness of the directional spectrum [9,10,11,12]. Even in directional seas, coherent nonlinear wave groups exist and persist for a prolonged lifetime [13,14]. The dynamics governing the evolution of such wave groups is a narrow-banded process represented by the nonlinear Schrödinger equation (NLSE) [15,16]

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