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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Rogue waves 2004 workshop

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

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


This might be a freak wave

This might be a freak wave

Freak ?


Newwave average shape the scaled auto correlation function

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- 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

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

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


Rogue waves 2004 workshop

1-D focussing on deep water – exact simulations


Shallow no extra elevation

Shallow- no extra elevation

Deeper

- extra elevation for more compact group

Ref. Katsardi + Swan


Rogue waves 2004 workshop

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

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

Extra amplitude

1:1 linear focus


Evolution of wavenumber spectrum with time

Evolution of wavenumber spectrum with time


1 d gaussian group wavenumber spectra showing relaxation to almost initial state

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


Rogue waves 2004 workshop

Wave group overtaking – non-linear dynamics on deep water


Rogue waves 2004 workshop

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

Focussing of a directional spread wave group


2 d gaussian group fully nonlinear focus

2-D Gaussian group – fully nonlinear focus


Exact non linear linear 2 1 nls

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

Extra elevation ? Not in 2-D

1:1 linear focus


Rogue waves 2004 workshop

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

Directional spectral changes – for isolated NewWave-type focussed event

Similar results in Bateman’s thesis and

Dysthe et al. 2003 for random field


Rogue waves 2004 workshop

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 for 1 checking numerics 2 approx analytics

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


Rogue waves 2004 workshop

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 s x s y and t t are slowly varying

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

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

Simple NLS-scaling of fully non-linear results


Rogue waves 2004 workshop

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 )


Rogue waves 2004 workshop

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

‘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

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


Rogue waves 2004 workshop

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 ?


Rogue waves 2004 workshop

Vertical shear

Green-Naghdi fluid sheets (Chan + Swan 2004)

higher crests before breaking

Horizontal current eddies

NLS-type models with surface current term

(Peregrine)


Rogue waves 2004 workshop

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

Set-up

NOT SET-DOWN

Largest crest

2nd largest crest


Rogue waves 2004 workshop

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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