The effect of third-order nonlinearity on statistical properties of random directional waves in finite depth

Abstract

Abstract. It is well established that third-order nonlinearity produces a strong deviation from Gaussian statistics in water of infinite depth, provided the wave field is long crested, narrow banded and sufficiently steep. A reduction of thirdorder effects is however expected when the wave energy is distributed on a wide range of directions. In water of arbitrary depth, on the other hand, third-order effects tend to be suppressed by finite depth effects if waves are long crested. Numerical simulations of the truncated potential Euler equations are here used to address the combined effect of directionality and finite depth on the statistical properties of surface gravity waves; only relative water depth kh greater than 0.8 are here considered. Results show that random directional wave fields in intermediate water depths, kh=O(1), weakly deviate from Gaussian statistics independently of the degree of directional spreading of the wave energy.

Evolution of skewness and kurtosis of weakly nonlinear unidirectional waves over a sloping bottom

“…Short-crested wave fields on deep water have nearGaussian statistics despite the nonlinearity of the wave field (Onorato et al, 2002;Socquet-Juglard et al, 2005;Gramstad and Trulsen, 2007;Onorato et al, 2009). In shallower water, however, Toffoli et al (2009) suggested that when directional components are taken into account, the probability of freak waves is notably increased compared to long-crested waves. In this paper we do not consider the effect of shortcrestedness.…”

“…In deep water it is known that third-order nonlinearity can cause focusing of long-crested and narrow-banded waves, and can be responsible for the occurrence of freak waves (Onorato et al, 2001(Onorato et al, , 2005Mori and Yasuda, 2002;Janssen, 2003;Mori and Janssen, 2006;Toffoli et al, 2008Toffoli et al, , 2009. As nonlinear modulation is reduced for long-crested waves on shallower depths, it may be anticipated that there is a corresponding reduction in the probability of freak waves Janssen and Onorato, 2007;Toffoli et al, 2009;Janssen, 2009).…”

“…We take the opposite approach, in the sense that we let the incoming wave fields have non-vanishing bandwidths. Indeed, in an analysis of wave records from the North Sea, a likely region for application of the present results, Trulsen and Dysthe (1997) observed that wave fields of overall steepness O( ) could have a larger bandwidth of O( 1/2 ), thus linear dispersion is expected to dominate over nonlinearity also away from the critical depth kh = 1.363.…”

“…As nonlinear modulation is reduced for long-crested waves on shallower depths, it may be anticipated that there is a corresponding reduction in the probability of freak waves Janssen and Onorato, 2007;Toffoli et al, 2009;Janssen, 2009). When the depth is less than the critical depth kh < 1.363, the loss of modulational instability may lead to reduced probability of freak waves .…”

Modulational instability and wave amplification in finite water depth

“…Beyond this threshold, nevertheless, oblique perturbations still remain unstable and numerical simulations of the 2 + 1 NLS equation, in this respect, confirm that such modes can still trigger very large amplitude waves (Slunyaev et al, 2002;Didenkulova et al, 2013). Furthermore, direct numerical simulations of the Euler equation also indicate that directional components can sustain the formation of extreme waves in random directional wave fields in water of finite depth, leading to substantial deviations from standard second-order based statistics (Toffoli et al, 2009).…”