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

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

This might be a freak wave Freak ?

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

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

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

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

1-D focussing on deep water – exact simulations

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

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

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

Evolution of wavenumber spectrum with time

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

Wave group overtaking – non-linear dynamics on deep water

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, , 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

Focussing of a directional spread wave group

2-D Gaussian group – fully nonlinear focus

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

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

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

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

What about nonlinear Schrodinger equation i u T + u XX - u YY + ½ u c u 2 = 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

NLS modelling – conserved quantities (2-d version) useful for1. checking numerics 2. approx. analytics

Assume Gaussian group defined by A – amplitude of group at focus S X – bandwidth in mean wave direction (also S Y ) gives exact solution to linear part of NLS i u T + u XX - u YY = 0 (actually this is in Kinsman’s classic book)

Assume A, S X, S Y, and T/t are slowly varying 1-D x-direction FULLY DISPERSED FOCUS A - , S X - , T -   A F, S XF, T F =0 similarly 1-D y-direction 2-D (x,y)-directions

Approx. Gaussian evolution 1-D x-long : focussing and contraction A F /A -  = /2 (A -  / S X -  ) 2 + …. S XF /S x-  = /2 (A -  / S X -  ) 2 + …. 1-D y-lateral : de-focussing and expansion A F /A -  = /2 (A -  / S Y -  ) 2 + …. S YF /S Y-  = /2 (A -  / S Y -  ) 2 + ….

Simple NLS-scaling of fully non-linear results

Approx. Gaussian evolution 2-D (x,y) : assume S X-  = S Y-  = S -  A F /A -  =1 + + …. S XF /S -  = (A -  / S -  ) ) 2 + …. S YF /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  )

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 Conclusions based on NLS-type modelling

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

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

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 ?

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

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

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’