Spectral up- and downshifting of Akhmediev breathers under wind forcing

Abstract

We experimentally and numerically investigate the effect of wind forcing on the spectral dynamics of Akhmediev breathers, a wave-type known to model the modulation instability. We develop the wind model to the same order in steepness as the higher order modifcation of the nonlinear Schroedinger equation, also referred to as the Dysthe equation. This results in an asymmetric wind term in the higher order, in addition to the leading order wind forcing term. The derived model is in good agreement with laboratory experiments within the range of the facility's length. We show that the leading order forcing term amplifies all frequencies equally and therefore induces only a broadening of the spectrum while the asymmetric higher order term in the model enhances higher frequencies more than lower ones. Thus, the latter term induces a permanent upshift of the spectral mean. On the other hand, in contrast to the direct effect of wind forcing, wind can indirectly lead to frequency downshifts, due to dissipative effects such as wave breaking, or through amplification of the intrinsic spectral asymmetry of the Dysthe equation. Furthermore, the definitions of the up- and downshift in terms of peak- and mean frequencies, that are critical to relate our work to previous results, are highlighted and discussed.Comment: 30 pages, 11 figure

Single-spectrum prediction of kurtosis of water waves in a nonconservative model

“…Figure 1(c) shows the mean up-shifts during forcing and downshifts during damping as discussed in [10]. A similar trend, but in the sense of widening and narrowing is seen for the bandwidth (Figure 1d).…”

“…Our model to propagate the envelope a(x, t) is the space-like version of the forced/damped-Modified NLS (MNLS) equation [10], with higher order terms added to suppress unphysical resonances and improve numerical stability. See Appendix A for a full derivation and notations.…”

“…On the one hand it is to seek for characteristic signatures of wind forcing and dissipation on the wave statistics. From a deterministic point of view, it is known that the spectral mean, bandwidth and steepness are functions of dissipation and wind forcing [10], where the dissipation and forcing have opposite signatures. It is not known to date how this behavior translates to ensemble simulations.…”

“…In contrast to this study, we include high-order nonlinear and dispersion terms of the modified NLS [13]. And, at the same order in steepness as the nonlinear terms, we include effects of wind forcing and dissipation [10]. This allows for the description of broader spectra.…”

Rogue waves and analogies in optics and oceanography

“…In other experiments, the effect of wind on rogue wave dynamics has been studied in specially adapted wind-sea wave tanks [159]. Particular experiments in a one-dimensional wave tank have shown that wind can induce frequency downshifting in the spectrum of initial breather solutions excited by a wavemaker [160,161], while other experiments in an annular wave tank did not use any initial mechanical wave generation, but allowed waves to develop spontaneously from the action of the wind on the water's surface [145]. These results are shown in Fig.…”