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

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

Shallow

Deep

Crest

Trough

TroughCrest

Wave kinematics – role of the return flow (2nd 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, 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

Exact non-linear Linear (2+1) NLS

Lo and Mei 1987

2-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. 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

dydxuI 22

dydxuuuI yx

422 2/14

NLS modelling – conserved quantities (2-d version)

useful for 1. checking numerics2. approx. analytics

Assume Gaussian group defined by

A – 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 DISPERSED FOCUS

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 )

• 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) = 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)

Largest crest

2nd 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’