The role of resonant wave interactions in the evolution of extreme wave events

1. The role of resonant wave interactions in the evolution of extreme wave events R. Gibson and C. Swan Imperial College London

2. 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

3. 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

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

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

6. Spectral Evolution

7. Unidirectional Rapid evolution of the underlying linear spectrum. Spectral Evolution

8. Unidirectional Third-order resonant interactions. Good agreement with the fully nonlinear results. Resonant Interactions

9. 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

10. Idealised narrow-banded spectrum. Stockwell Transform

11. Instantaneous frequency at the time of the extreme event calculated using Zakharov’s equation. Dispersive Properties

12. Sum of the amplitude components of the underlying freely propagating wave components. Amplitude sum increases by 23% in 80 wave periods. Spectral Characteristics

13. 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. Spectral Characteristics

14. 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.

15. 30º wrapped-normal spreading. Energy is transferred away from the peak. Energy is transferred to high frequencies. Spectrum narrows. Spectral Evolution

16. 5º wrapped-normal spreading. Energy is transferred in a horseshoe pattern. Energy is transferred to high frequencies. Spectrum broadens. Spectral Evolution

17. Changes to the amplitude sum 5º: increases by 20%. 30º: decreases by 4%. Spectral Characteristics

18. 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

19. Unidirectional Sea-states Factors

20. Factors • The effect of directionality

21. Gaussian Spectra • Significant broadening of the spectrum. • Crest Elevation • Linear 12.3m • Second-order 13.5m • Fully-nonlinear 15.3m

22. 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. Conclusions