Recurrent solutions of the Alber equation initialized by Joint North Sea Wave Project spectra

Abstract

Linear instability of two-dimensional wave fields and its concurrent evolution in time is here investigated by means of the Alber equation for narrow-banded random surface waves in deep water subject to inhomogeneous disturbances. The probability of freak waves in the context of these simulations is also discussed. The instability is first studied for the symmetric Lorentz spectrum, and continued for the realistic asymmetric Joint North Sea Wave Project (JONSWAP) spectrum of ocean waves with variable directional spreading and steepness. It is found that instability depends on the directional spreading and parameters α and γ of the JONSWAP spectrum, where α and γ are the energy scale and the peak enhancement factor, respectively. Both influence the mean steepness of waves with such a spectrum, although in different ways. Specifically, if the instability stops as a result of the directional spreading, increase of the steepness by increasing α or γ can reactivate it. A criterion for the instability is suggested as a dimensionless 'width parameter', Π. For the unstable conditions, long-time evolution is simulated by integrating the Alber equation numerically. Recurrent evolution is obtained, which is a stochastic counterpart of the Fermi-Pasta-Ulam recurrence obtained for the cubic Schrödinger equation. This recurrence enables us to study the probability of freak waves, and the results are compared to the values given by the Rayleigh distribution. Moreover, it is found that stability-instability transition, the most unstable mode, recurrence duration and freak wave probability depend solely on the dimensionless 'width parameter', Π .

Modulational instability and wave amplification in finite water depth

“…Although the most unstable modes remain collinear in water of infinite depth, oblique perturbations tend to dominate the modulational instability for conditions of arbitrary water depths when k 0 h < ε −1 (Trulsen and Dysthe, 1996). This is also confirmed by laboratory experiments in a relatively wide long wave flume (Trulsen et al, 1999), where a plane wave without any initial seeding of unstable modes was observed to transfer energy towards a lower oblique side band (see also Babanin et al, 2011;Ribal et al, 2013). Direct numerical simulations of the 2+1 NLS equation, furthermore, substantiate that not only can oblique disturbances sustain modulational instability, but they are also capable of triggering the formation of rogue waves (Osborne et al, 2000;Slunyaev et al, 2002).…”

“…3a, b and c). It is interesting to note, in this respect, that collinear disturbances also induce a recurrence in the phenomenon with a sequence of modulation and demodulation of the input surface (cf., for example, Ribal et al, 2013). When seeded with oblique side band perturbations, on the other hand, no significant evidence of recurrence can be detected.…”

Localized instabilities of the Wigner equation as a model for the emergence of Rogue Waves

“…In the context of water waves, Alber (1978) used the Wigner spectrum to study the directional stability of solutions of the two-dimensional Davey-Stewartson equations in a narrowband setting. The Alber equation was derived under a Gaussianity assumption, and it has since been studied in the context of directional stability, as well as numerically (Ribal et al 2013;Crawford et al 1980;Stiassnie et al 2008;Regev et al 2008;Dysthe et al 2003). …”

“…We can use this scheme to approximate the solution set P of the Penrose condition for any background spectrum. It must be noted that an analogous scheme based on the approximation of a general spectrum by a sum of δ functions, i.e., working with P(k) = N j=1 P j δ(k − k j ) was presented in Ribal et al (2013). …”

On the strength of the weakly nonlinear theory for surface gravity waves

“…Some efforts towards progress on the homogeneous case include the works of Annenkov & Shrira (2006) and Gramstad & Stiassnie (2013). Those for the inhomogeneous wave fields have been, so far, limited to narrow spectra only, see Alber (1978) and Ribal et al (2013). One advantage of the latter approach is that it has a good chance of predicting the probability of occurrence of extremely high waves, known as freak or rogue waves.…”