![]() Further, the surface statistics follow the Tayfun 32 distribution 32 in agreement with observations 9, 10, 31, 33. His results indicate that large waves (measured as a function of time at a given point) result from the constructive interference (focusing) of elementary waves with random amplitudes and phases enhanced by second-order non-resonant or bound nonlinearities. Tayfun 31 presented an analysis of oceanic measurements from the North Sea. Thus, modulation instabilities may play an insignificant role in the wave growth especially in finite water depth where they are further attenuated 30. Hence, we expect that nonlinear focusing due to modulational effects is diminished since energy can spread directionally 16, 18, 29. Typical oceanic wind seas are short-crested, or multidirectional wave fields. However, wave breaking is inevitable when the steepness becomes larger: ‘breathers do not breathe’ 23 and their amplification is smaller than that predicted by the NLS equation, in accord with theoretical studies 24 of the compact Zakharov equation 25, 26 and numerical studies of the Euler equations 27, 28. Consequently, breathers can be observed experimentally in 1D wave fields only at sufficiently small values of wave steepness 20, 21, 22. For small wave steepness and negligible dissipation, quasi-resonant interactions are effective in reshaping the wave spectrum, inducing large breathers via nonlinear focusing before wave breaking occurs 16, 17, 20, 21. Indeed, in this case energy is ‘trapped’ as in a long wave-guide. The late-stage evolution of modulation instability leads to breathers that can cause large waves 13, 14, 15, especially in 1D waves. Such nonlinear effects cause the statistics of weakly nonlinear gravity waves to significantly differ from the Gaussian structure of linear seas, especially in long-crested or unidirectional (1D) seas 8, 10, 16, 17, 18, 19. In particular, recent studies propose third-order quasi-resonant interactions and associated modulational instabilities 11, 12 inherent to the Nonlinear Schrödinger (NLS) equation as mechanisms for rogue wave formation 3, 8, 13, 14, 15. Several physical mechanisms have been proposed to explain the occurrence of such waves 7, including the two competing hypotheses of nonlinear focusing due to third-order quasi-resonant wave-wave interactions 8 and purely dispersive focusing of second-order non-resonant or bound harmonic waves, which do not satisfy the linear dispersion relation 9, 10. Evidence that such extremes can occur in nature is provided, among others, by the Draupner and Andrea events, which have been extensively studied over the last decade 1, 2, 3, 4, 5, 6. According to the most commonly used definition, rogue waves are unusually large-amplitude waves that appear from nowhere in the open ocean.
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