Surface gravity waves from direct numerical simulations of the Euler equations: A comparison with second-order theory

Toffoli, Alessandro; Onorato, Miguel; Bitner‐Gregersen, Elzbieta M.; Osborne, A. R.; Babanin, Alexander V.

doi:10.1016/j.oceaneng.2007.10.004

Cited by 63 publications

(54 citation statements)

References 36 publications

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“…Mori and Yasuda (2002), Ducrozet et al (2007), Toffoli et al (2008a) and Toffoli et al (2008b). Note that other numerical methods have also been proposed by Annenkov and Shrira (2001), Clamond and Grue (2001), and Zakharov et al (2002).…”

Section: The Modelmentioning

confidence: 99%

“…Such a deviation occurred almost immediately as it was already visible after six peak periods. Because the skewness is weakly dependent upon the dynamics of free waves (see, e.g., Mori and Yasuda, 2002;Mori et al, 2007;Toffoli et al, 2008b), it remained rather constant for the rest of the temporal evolution. As expected, substantial variation in the skewness were observed with the decrease of k p h, as finite depth effects become more pronounced.…”

Section: Skewness and Kurtosismentioning

confidence: 99%

“…Whereas bound modes are still present, resonant and non-resonant interactions between free waves are also possible. In this respect, a number of numerical and theoretical works (see, e.g., Janssen, 2003;Onorato et al, 2001Onorato et al, , 2005Onorato et al, , 2006Mori and Yasuda, 2002;Mori and Janssen, 2006;Mori et al, 2007;Toffoli et al, 2008b) have shown that deep water third-order nonlinearity can lead to focusing of wave energy with the consequent formation of extreme events, provided waves are long crested (i.e. unidirectional), narrow banded and sufficiently steep.…”

Section: Introductionmentioning

confidence: 99%

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

Toffoli

Benoît

Onorato

et al. 2009

Nonlin. Processes Geophys.

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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.

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Section: The Modelmentioning

confidence: 99%

Section: Skewness and Kurtosismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Toffoli

Benoît

Onorato

et al. 2009

Nonlin. Processes Geophys.

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“…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).…”

Section: Introductionmentioning

confidence: 99%

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

Zeng

Trulsen

2012

Nat. Hazards Earth Syst. Sci.

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Abstract.We consider the effect of slowly varying depth on the values of skewness and kurtosis of weakly nonlinear irregular waves propagating from deeper to shallower water. It is known that the equilibrium value of kurtosis decreases with decreasing depth for waves propagating on constant depth. Waves propagating over a sloping bottom must continually adjust toward a new equilibrium state. We demonstrate that weakly nonlinear waves may need a considerable horizontal propagation distance in order to adjust to a new shallower environment, therefore the kurtosis can be notably different from the equilibrium value for each corresponding depth both on top of and beyond a bottom slope. A change of depth can provoke a wake-like spatially non-uniform distribution of kurtosis on the lee side of the slope. As an application, we anticipate that the probability of freak waves on or near the edge of the continental shelf may exhibit a rather complicated spatial structure for wave fields entering from deep sea.

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“…The occurrence of extreme waves (also known as freak or rogue waves) has an important role in many branches of physics and engineering (see, for example, Chabchoub et al, 2011;Onorato et al, 2013a, b;Chalikov, 2009;Babanin et al, 2011;Bitner-Gregersen and Toffoli, 2012;Toffoli et al, 2008b;Solli et al, 2007;Kibler et al, 2010;Bailung et al, 2011, among many others). Apart from a linear superposition of wave modes and the effect of currents on waves (caustic theory), the modulational instability of a plane wave to side band perturbations remains the most plausible mechanism for the formation of rogue waves in deep water (Zakharov and Ostrovsky, 2009;Osborne, 2010;Onorato et al, 2013b;Kharif et al, 2009), i.e.…”

Section: Introductionmentioning

confidence: 99%

Modulational instability and wave amplification in finite water depth

Fernández

Onorato

Monbaliu

et al. 2014

Nat. Hazards Earth Syst. Sci.

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Abstract. The modulational instability of a uniform wave train to side band perturbations is one of the most plausible mechanisms for the generation of rogue waves in deep water. In a condition of finite water depth, however, the interaction with the sea floor generates a wave-induced current that subtracts energy from the wave field and consequently attenuates the instability mechanism. As a result, a plane wave remains stable under the influence of collinear side bands for relative depths kh ≤ 1.36 (where k is the wavenumber of the plane wave and h is the water depth), but it can still destabilise due to oblique perturbations. Using direct numerical simulations of the Euler equations, it is here demonstrated that oblique side bands are capable of triggering modulational instability and eventually leading to the formation of rogue waves also for kh ≤ 1.36. Results, nonetheless, indicate that modulational instability cannot sustain a substantial wave growth for kh < 0.8.

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Surface gravity waves from direct numerical simulations of the Euler equations: A comparison with second-order theory

Cited by 63 publications

References 36 publications

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

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

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

Modulational instability and wave amplification in finite water depth

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