Rogue waves 2004 workshop
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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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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


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

  • 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


Shallow- no extra elevation

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)


1-D Deepwater focussed wavegroup (kd)Gaussian spectrum (like peak of Jonswap)

Extra amplitude

1:1 linear focus


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, 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


Focussing of a directional spread wave group


2-D Gaussian group – fully nonlinear focus


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

Exact non-linear Linear (2+1) NLS

Lo and Mei 1987


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. 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


NLS modelling – conserved quantities (2-d version)useful for1. checking numerics2. approx. analytics


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 + ….


Simple NLS-scaling of fully non-linear results


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’


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