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 :.
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
My students : Erwin Vijfvinkel, Richard Gibbs, Dan Walker
Prof. Chris Swan and his students : Tom Baldock,
Thomas Johannessen, William Bateman
+ bound harmonics
NewWave + harmonics
For linear crest amplitude 14.7m,
Draupner wave is a 1 in ~200,000 wave
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
Short waves ahead of long waves
overtaking to give focus event (on a linear basis)
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
- extra elevation for more compact group
Ref. Katsardi + Swan
Wave kinematics – role of the return flow (2nd order)
1:1 linear focus
Numerics – discussed here function
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 functionnonlinear Schrodinger equation
i uT + uXX - uYY + ½ uc u2 = 0
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 by functionA – 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)
FULLY DISPERSED FOCUS
A-, SX -, T- AF, SXF, TF =0
similarly 1-D y-direction
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 function
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
(Osborne et al. 2000)
Laplace / Zakharov / NLEE
Initial conditions – linear random ?
How long – timescales ?
No energy input - wind
No energy dissipation – breaking
No vorticity – vertical shear,
horiz. current eddies
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 function
Green-Naghdi fluid sheets (Chan + Swan 2004)
higher crests before breaking
Horizontal current eddies
NLS-type models with surface current term
2nd largest crest
We (I) don’t know how to make the Draupner wave
Energy conserving models may not be the answer
Freak waves might be ‘ghosts’