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ROGUE WAVES 2004 Workshop

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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

Acknowledgements :

My students : Erwin Vijfvinkel, Richard Gibbs, Dan Walker

Prof. Chris Swan and his students : Tom Baldock,

Thomas Johannessen, William Bateman

Freak ?

NewWave

+ bound harmonics

NewWave + harmonics

Draupner wave

For linear crest amplitude 14.7m,

Draupner wave is a 1 in ~200,000 wave

- 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

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

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

Deeper

- extra elevation for more compact group

Ref. Katsardi + Swan

Crest

Trough

Shallow

Crest

Trough

Deep

Wave kinematics – role of the return flow (2nd order)

Extra amplitude

1:1 linear focus

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, 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

2-D dominant physics is x-contraction, y-expansion

Lo and Mei 1987

1:1 linear focus

2-D is very different from 1-D

Similar results in Bateman’s thesis and

Dysthe et al. 2003 for random field

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

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)

Assume A, SX, SY, and T/t are slowly varying

1-D x-direction

FULLY DISPERSEDFOCUS

A-, SX -, T- AF, SXF, TF =0

similarly 1-D y-direction

2-D (x,y)-directions

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 + ….

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 )

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

‘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)

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)

Draupner wave – a rogue-like aspect – bound long waves

Set-up

NOT SET-DOWN

Largest crest

2nd largest crest

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’