The role of resonant wave interactions in the

evolution of extreme wave events

R. Gibson and C. Swan

Imperial College London

Evolution of large ocean waves

• Dispersive focusing.• Resonant interactions.

Results• The evolution of

unidirectional and directional wave spectra.

• The consequences of this evolution.

• The identification of sea-states in which rogue waves are more likely to occur.

Introduction

Bateman et al. 2001• Based upon the unidirectional formulation of Craig and

Sulem.• Fully nonlinear.• Realistic directionally spread sea-states.• Efficiency the result of a Dirichlet-Neumann operator.• Limited to modelling waves in a periodic domain of

constant depth up to the breaking limit.

Wave Models

Zakharov 1968• Nonlinear evolution equation.• Derived to 4th order by Krasitskii 1994.• Possible to separate the ‘bound’ and the ‘resonant’

interactions.

Wave Models

Unidirectional Surface Profile

Unidirectional• JONSWAP spectrum.• Linear crest elevation 9m.• Second-order elevation 9.9m.• Third-order crest elevation 10.1m.• Fully nonlinear crest elevation 11.9m.

Spectral Evolution

Spectral Evolution

Unidirectional• Rapid evolution of the underlying linear spectrum.

Resonant Interactions

Unidirectional• Third-order resonant interactions.

• Good agreement with the fully nonlinear results.

• JONSWAP spectrum• Spectrum in wave-number and frequency at the

time of the extreme event.• A ‘spread’ of energy that doesn’t satisfy the linear

dispersion relationship.

Stockwell Transform

Stockwell Transform• Idealised narrow-banded spectrum.

Dispersive Properties

• Instantaneous frequency at the time of the extreme event calculated using Zakharov’s equation.

Spectral Characteristics

• Sum of the amplitude components of the underlying freely propagating wave components.

• Amplitude sum increases by 23% in 80 wave periods.

Spectral Characteristics

• Changes to the ‘amplitude sum’ of the spectrum.

• Changes to the dispersive properties of the wave group > changes to the focal quality of the wave crest.

Directional Surface Profile

Directional• JONSWAP spectra TP = 12.8s, peak enhancement = 5.

• Linear 8m.

• Second-order 8.8m.

• Fully nonlinear = 5º: 8.5m = 30º: 8.6m.

• 30º wrapped-normal spreading.• Energy is transferred away from the peak.• Energy is transferred to high frequencies.• Spectrum narrows.

Spectral Evolution

Spectral Evolution• 5º wrapped-normal spreading.• Energy is transferred in a horseshoe pattern.• Energy is transferred to high frequencies.• Spectrum broadens.

Spectral Characteristics

• Changes to the amplitude sum– 5º: increases by 20%.– 30º: decreases by 4%.

Factors

• Maximum crest elevation dependent upon four factors:– A0: the initial amplitude sum of the spectrum.

– F0: changes to the amplitude sum owing to resonant interactions.

– F1: the nonlinear amplification owing to bound interactions.

– F2: the focal quality of the event.

Factors

• Unidirectional Sea-states

Factors

• The effect of directionality

Gaussian Spectra

• Significant broadening of the spectrum.

Crest Elevation• Linear 12.3m• Second-order

13.5m• Fully-nonlinear

15.3m

Conclusions

• Spectra can evolve rapidly during the formation of a focused wave-event.– Third-order resonant interactions.– Changes to amplitude and dispersive properties of

wave components.

• In unidirectional sea-states:– Large nonlinear increases in crest elevation.– The phasing of the wave components is relatively

unimportant.

• In directional sea-state:– Balance between the effects of dispersion and the

resonant interactions.

• Swell dominated sea-states– Disperse slowly.– Large nonlinear increases in crest elevation.