session 2.1 : theoretical results, numerical and physical simulations introductory presentation...
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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 :
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’