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ROGUE WAVES 2004 Workshop

Session 2.1 : Theoretical results, numerical and physical simulations Introductory presentation Rogue Waves and wave focussing – speculations on theory, numerical results and observations Paul H. Taylor University of Oxford. ROGUE WAVES 2004 Workshop. Acknowledgements :.

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ROGUE WAVES 2004 Workshop

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  1. Session 2.1 : Theoretical results, numerical and physical simulationsIntroductory presentationRogue Waves and wave focussing – speculations on theory, numerical results and observationsPaul H. Taylor University of Oxford ROGUE WAVES 2004 Workshop

  2. Acknowledgements : My students : Erwin Vijfvinkel, Richard Gibbs, Dan Walker Prof. Chris Swan and his students : Tom Baldock, Thomas Johannessen, William Bateman

  3. This is not a rogue (or freak) wave – it was entirely expected !!

  4. This might be a freak wave Freak ?

  5. NewWave - Average shape – the scaled auto-correlation function NewWave + bound harmonics NewWave + harmonics Draupner wave For linear crest amplitude 14.7m, Draupner wave is a 1 in ~200,000 wave

  6. 1- and 2- D modelling • Exact – Laplace + fully non-linear bcs • – numerical • spectral / boundary element / finite element • NLEEs • NLS (Peregrine 1983) • Dysthe 1979 • Lo and Mei 1985 • Dysthe, Trulsen, Krogstad & Socquet-Juglard 2003

  7. Perturbative physics to various orders 1st– Linear dispersion 2nd– Bound harmonics + crest/trough ,  set-down and return flow (triads in v. shallow water) 3rd – 4-wave Stokes correction for regular waves, BF, NLS solitons etc. 4th – (5-wave) crescent waves What is important in the field ? all of the time 1st order RANDOM field most of the time 2nd order occasionally 3rd AND BREAKING

  8. Frequency / wavenumber focussing Short waves ahead of long waves overtaking to give focus event (on a linear basis) spectral content how long before focus nonlinearity (steepness and wave depth) In examples – linear initial conditions on ( , ) same linear (x) components at start time for several kd In all cases, non-linear group dynamics

  9. 1-D focussing on deep water – exact simulations

  10. Shallow- no extra elevation Deeper - extra elevation for more compact group Ref. Katsardi + Swan

  11. Crest Trough Shallow Crest Trough Deep Wave kinematics – role of the return flow (2nd order)

  12. 1-D Deepwater focussed wavegroup (kd)Gaussian spectrum (like peak of Jonswap) Extra amplitude 1:1 linear focus

  13. Evolution of wavenumber spectrum with time

  14. 1-D Gaussian group– wavenumber spectra, showing relaxation to almost initial state

  15. Wave group overtaking – non-linear dynamics on deep water

  16. Numerics – discussed here • Solves Laplace equation with fully non-linear boundary conditions • Based on pseudo-spectral G-operator of Craig and Sulem (J Comp Phys 108, 73-83, 1993) • 1-D code by Vijfvinkel 1996 • Extended to directional spread seas by Bateman*, Swan and Taylor (J Comp Phys 174, 277-305, 2001) • *Ph.D. from Dept. of Civil & Environmental Eng at Imperial College, London - supervised by C. Swan • Well validated against high quality wave basin data – for both uni-directional and spread groups

  17. Focussing of a directional spread wave group

  18. 2-D Gaussian group – fully nonlinear focus

  19. 2-D dominant physics is x-contraction, y-expansion Exact non-linear Linear (2+1) NLS Lo and Mei 1987

  20. Extra elevation ? Not in 2-D 1:1 linear focus

  21. In directionally spread interactions – permanent energy transfers (4-wave resonance) – NLEE or Zakharov eqn 2-D is very different from 1-D

  22. Directional spectral changes – for isolated NewWave-type focussed event Similar results in Bateman’s thesis and Dysthe et al. 2003 for random field

  23. What about nonlinear Schrodinger equation i uT + uXX - uYY + ½ uc u2 = 0 NLS-properties 1D x-long group  elevation focussing - BRIGHT SOLITON 1D y-lateral group  elevation  de-focussing - DARK SOLITON 2D group vs.  balance determines what happens to elevation focus in longitudinal AND de-focus in lateral directions

  24. NLS modelling – conserved quantities (2-d version) useful for 1. checking numerics 2. approx. analytics

  25. Assume Gaussian group defined byA – amplitude of group at focusSX – bandwidth in mean wave direction (also SY)gives exact solution to linear part of NLS i uT + uXX - uYY = 0 (actually this is in Kinsman’s classic book)

  26. Assume A, SX, SY, and T/t are slowly varying 1-D x-direction FULLY DISPERSED FOCUS A-, SX -, T- AF, SXF, TF =0 similarly 1-D y-direction 2-D (x,y)-directions

  27. Approx. Gaussian evolution 1-D x-long : focussing and contraction AF /A- = 1 + 2 -5/2 (A- / SX - )2 + …. SXF /Sx- = 1 + 2 -3/2 (A- / SX - )2 + …. 1-D y-lateral : de-focussing and expansion AF /A- = 1 - 2 -5/2 (A- / SY - )2 + …. SYF /SY- = 1 - 2 -3/2 (A- / SY - )2 + ….

  28. Simple NLS-scaling of fully non-linear results

  29. Approx. Gaussian evolution • 2-D (x,y) : assume SX- = SY- = S- • AF /A- = 1 + + …. • SXF /S- = 1 + 2 -3 (A- / S-) )2 + …. • SYF /S- = 1  2 -3 (A- / S-) )2 + …. • focussing in x-long, de-focussing in y-lat, no extra elevation • much less non-linear event than 1-D (0.6 )

  30. Conclusions based on NLS-type modelling • Importance of (A/S) – like Benjamin-Feir index • 2-D qualitatively different to 1-D • need 2-D Benjamin-Feir index, incl.directional spreading • In 2-D little opportunity for extra elevation • but changes in shape of wave group at focus • and long-term permanent changes • 2-D is much less non-linear than 1-D

  31. ‘Ghosts’ in a random sea – a warning from the NLS-equation u(x,t) = 21/2 Exp[2 I t] (1-4(1+4 I t)/(1+4x2+16t2)) t  uniform regular wave t =0 PEAK 3x regular background UNDETECTABLE BEFOREHAND (Osborne et al. 2000)

  32. Where now ? Random simulations Laplace / Zakharov / NLEE Initial conditions – linear random ? How long – timescales ? BUT No energy input - wind No energy dissipation – breaking No vorticity – vertical shear, horiz. current eddies

  33. BUT Energy input – wind Damping weakens BF sidebands (Segur 2004) and eventually wins  decaying regular wave Negative damping ~ energy input – drives BF and 4-wave interactions ?

  34. Vertical shear Green-Naghdi fluid sheets (Chan + Swan 2004) higher crests before breaking Horizontal current eddies NLS-type models with surface current term (Peregrine)

  35. Draupner wave – a rogue-like aspect – bound long waves Set-up NOT SET-DOWN Largest crest 2nd largest crest

  36. Conclusions We (I) don’t know how to make the Draupner wave Energy conserving models may not be the answer Freak waves might be ‘ghosts’

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