linear and nonlinear rogue wave statistics in the presence ...lkaplan/kaplan rogue11.pdffocus on...

$: Linear and Nonlinear Rogue Wave Statistics in the Presence ...lkaplan/Kaplan Rogue11.pdfFocus on linear refraction of incoming wave (velocity v) by random current eddies (current speed$
Linear and Nonlinear Rogue Wave

Statistics in the Presence of Random

Currents

Lev Kaplan

(Tulane University)

In collaboration with

Alex Dahlen and Eric Heller (Harvard)

Linghang Ying and Zhouheng Zhuang (Tulane)

October 11 2011 1 Rogue11 Dresden

Talk Outline

Linear rogue wave formation

Mechanism Refraction from current eddies

Combining refraction with stochasticity

The ldquofreak indexrdquo

Implications for extreme wave statistics

Introducing wave nonlinearity

Wave statistics in 4th order nonlinear equation

Scaling with parameters describing sea state

Can linear and nonlinear effects work in concert


Focus on linear refraction of incoming wave (velocity

v) by random current eddies (current speed urms ltlt v)

For deep water surface gravity waves with current

Current correlated on scale (typical eddy size)

Ray picture justified if k gtgt 1

Rogue Waves Refraction Picture

x

x

dk dxdt dtx k

krugkkr )()(


How caustics form refraction from current eddies

Parallel incoming rays encountering pair of eddies

Effect similar to potential dip in particle mechanics

Focusing when all paths in a given neighborhood coalesce at a single point (caustic) producing infinite ray density

Different groups of paths coalesce at different points (ldquobad lensrdquo analogy)



Generic result ldquocusprdquo singularity followed by two lines of ldquofoldrdquo caustics

At each y after cusp singularity we have infinite density at some values of x

Phase space picture


Refraction from weak random currents

First singularities form after

Further evolution exponential proliferation of caustics

Tendrils decorate original branches

Branch statistics described by single distance scale d

23

rmsud

v



Of course singularities washed out on wavelength

scale

Qualitative structure independent of

dispersion relation (eg ~ k2 for Schroumldinger vs

~ k12 for ocean waves)

details of random current field

Can calculate distribution of branch strengths fall off

with distance etc (LK PRL 2002)


Analogies with other physical systems Electron flow in nanostructures (10-6 m)

Microwave resonators Stoumlckmann group (1 m)

Long-range ocean acoustics Tomsovic et al (20 km)

Twinkling of starlight Berry (2000 km)

Gravitational lensing Tyson (109 light years)

Topinka et al

Nature (2001)


Microwave scattering experiments

Experimentally observed branched flow

Ray simulation


Houmlhmann et al

(PRL 2010)

Branching pattern for tsunami waves

October 11 2011 Rogue11 Dresden 10

Problems with refraction picture of

rogue waves Assumes single-wavelength and unidirectional initial

conditions which are unrealistic and unstable (Dysthe)

Singularities washed out only on wavelength scale

Predicts regular sequence of extreme waves every time

No predictions for actual wave heights or probabilities

Solution replace incoming plane wave with random

initial spectrum

Finite range of wavelengths and directions


Smearing of caustics for stochastic

incoming sea Singularities washed out by randomness in initial

conditions k

ldquoHot spotsrdquo of enhanced average energy density remain as

reminders of where caustics would have been



incoming sea

Competing effects of focusing

and initial stochasticity

kx = initial wave vector spread

kx = wave vector change due to

refraction


Quantifying residual effect of caustics

the ldquofreak indexrdquo Define freak index = kx kx

Equivalently =

= kx k = initial directional spread

= typical deflection before formation of first cusp

~ (urms v)23 ~ d

Most dangerous well-collimated sea impinging on

strong random current field ( 1)

Hot spots corresponding to first smooth cusps have

highest energy density

ydydyy )()(1)( 2


Typical ray calculation for ocean waves

= 10ordm = 20ordm


Implications for Rogue Wave Statistics

Simulations (using Schrodinger equation) long-time average and regions of extreme events

average gt3 SWH gt22 SWH

2

1 SWH=significant wave

height 4 crest to trough


Calculation Rogue Waves

No currents Rayleigh height distribution (random

superposition of waves)

With currents Superpose locally Gaussian wave statistics

on pattern of ldquohotcold spotsrdquo caused by refraction

Local height distribution

is position-dependent variance of the water

elevation (proportional to ray density high in focusing

regions low in defocusing regions)

Total wave height distribution

2 2( ) ( ) exp( 2 )P h h h

2 2( ) ( )exp( 2 )IP h h I h I

I r

( ) ( ) ( )IP h P h f I dI

Nov 11 2011 17 Rogue11


Thus calculate wave height probability by combining

Rayleigh distribution

Distribution f(I) which describes ray dynamics and

depends on scattering strength (freak index)

Rogue wave forecasting

2 2( ) ( ) exp( 2 )P h h h

Nov 11 2011 18 Rogue11

Analytics limit of small freak index

For ltlt1 f(I) well approximated by χ2 distribution of

n ~ --2 degrees of freedom (mean 1width ~ )

2 4

2

2( )(2 )

( 2)

n

n

nxP(h x SWH) K nx

n

2

exp 2 )P(h x SWH) ( x I) f(I dI

Nov 11 2011 19 Rogue11


Perturbative limit for x2 ltlt1 (x ltlt n14)

Asymptotic limit for x 3 gtgt1 (x gtgt n 32)

4 2 241 exp 2P(h x SWH) [ (x x )] ( x )

n

2 4

2

2( )(2 )

( 2)

n

n

nxP(h x SWH) K nx

n

( 1)2( )exp 2

( 2)

nnxP(h x SWH) ( nx)

n

Nov 11 2011 20 Rogue11

Numerical Simulations Incoming sea with v=78 ms (T=10 s =156 m)

Random current with rms velocity urms = 05 ms and

correlation =20 km

Dimensionless parameters

ltlt 1 (ray limit)

~ (urms v)23 ltlt 1 (small-angle scattering)

= spreading angle = 5 to 25deg

= = freak index ( =35 to 07)


f (xy) (xy)u

Wave height distribution for ocean waves


Numerical test of N vs γ


Probability enhancement over

Rayleigh predictions

urms = 05 ms v=78 ms


Extension to nonlinear waves

NLS with current

where

21 1 1| | 0

4 8 2t xx yy xiA A A A A U A

0 0( ) ~ ( )i k x i t

x y t A x y t e




Key result for moderate steepness full wave height distribution again reasonably approximated by K-distribution with N degrees of freedom

Wave height probability depends on single parameter N ( Rayleigh as N rarr infin )

How does N depend on wave steepness incoming angular spread spectral width etc



Steepness ε = k (mean crest height)

Δθ = 26ordm

Δk k = 01

u = 0

N as function of incoming angular spread for fixed steepness and frequency spread


N as function of incoming frequency spread for fixed steepness and angular spread


N as function of steepness for fixed incoming angular spread and frequency spread


Finally combine currents and nonlinearity


Summary

Linear refraction of stochastic Gaussian sea produces lumpy energy density

Skews formerly Rayleigh distribution of wave heights

Importance of refraction quantified by freak index

Spectacular effects in tail even for small

Very similar wave height distribution in the presence of moderate nonlinearity

Even more dramatic results obtained when random currents and nonlinearity are acting in concert

Refraction may serve as trigger for full non-linear evolution


Thank you

Ying Zhuang Heller and Kaplan

Nonlinearity 24 R67 (2011)


Talk Outline

Linear rogue wave formation

Mechanism Refraction from current eddies

Combining refraction with stochasticity

The ldquofreak indexrdquo

Implications for extreme wave statistics

Introducing wave nonlinearity

Wave statistics in 4th order nonlinear equation

Scaling with parameters describing sea state

Can linear and nonlinear effects work in concert








x

x

dk dxdt dtx k

krugkkr )()(











Phase space picture







23

rmsud

v




scale













Topinka et al

Nature (2001)




Ray simulation


Houmlhmann et al

(PRL 2010)










initial spectrum





conditions k





incoming sea





refraction




Equivalently =



~ (urms v)23 ~ d





ydydyy )()(1)( 2



= 10ordm = 20ordm





2














2 2( ) ( ) exp( 2 )P h h h

2 2( ) ( )exp( 2 )IP h h I h I

I r

( ) ( ) ( )IP h P h f I dI

Nov 11 2011 17 Rogue11







2 2( ) ( ) exp( 2 )P h h h

Nov 11 2011 18 Rogue11




2 4

2

2( )(2 )

( 2)

n

n

nxP(h x SWH) K nx

n

2


Nov 11 2011 19 Rogue11




4 2 241 exp 2P(h x SWH) [ (x x )] ( x )

n

2 4

2

2( )(2 )

( 2)

n

n

nxP(h x SWH) K nx

n

( 1)2( )exp 2

( 2)

nnxP(h x SWH) ( nx)

n

Nov 11 2011 20 Rogue11



correlation =20 km


ltlt 1 (ray limit)





f (xy) (xy)u










NLS with current

where

21 1 1| | 0


0 0( ) ~ ( )i k x i t

x y t A x y t e










Δθ = 26ordm

Δk k = 01

u = 0









Summary









Thank you










x

x

dk dxdt dtx k

krugkkr )()(











Phase space picture







23

rmsud

v




scale













Topinka et al

Nature (2001)




Ray simulation


Houmlhmann et al

(PRL 2010)










initial spectrum





conditions k





incoming sea





refraction




Equivalently =



~ (urms v)23 ~ d





ydydyy )()(1)( 2



= 10ordm = 20ordm





2














2 2( ) ( ) exp( 2 )P h h h

2 2( ) ( )exp( 2 )IP h h I h I

I r

( ) ( ) ( )IP h P h f I dI

Nov 11 2011 17 Rogue11







2 2( ) ( ) exp( 2 )P h h h

Nov 11 2011 18 Rogue11




2 4

2

2( )(2 )

( 2)

n

n

nxP(h x SWH) K nx

n

2


Nov 11 2011 19 Rogue11




4 2 241 exp 2P(h x SWH) [ (x x )] ( x )

n

2 4

2

2( )(2 )

( 2)

n

n

nxP(h x SWH) K nx

n

( 1)2( )exp 2

( 2)

nnxP(h x SWH) ( nx)

n

Nov 11 2011 20 Rogue11



correlation =20 km


ltlt 1 (ray limit)





f (xy) (xy)u










NLS with current

where

21 1 1| | 0


0 0( ) ~ ( )i k x i t

x y t A x y t e










Δθ = 26ordm

Δk k = 01

u = 0









Summary









Thank you













Phase space picture







23

rmsud

v




scale













Topinka et al

Nature (2001)




Ray simulation


Houmlhmann et al

(PRL 2010)










initial spectrum





conditions k





incoming sea





refraction




Equivalently =



~ (urms v)23 ~ d





ydydyy )()(1)( 2



= 10ordm = 20ordm





2














2 2( ) ( ) exp( 2 )P h h h

2 2( ) ( )exp( 2 )IP h h I h I

I r

( ) ( ) ( )IP h P h f I dI

Nov 11 2011 17 Rogue11







2 2( ) ( ) exp( 2 )P h h h

Nov 11 2011 18 Rogue11




2 4

2

2( )(2 )

( 2)

n

n

nxP(h x SWH) K nx

n

2


Nov 11 2011 19 Rogue11




4 2 241 exp 2P(h x SWH) [ (x x )] ( x )

n

2 4

2

2( )(2 )

( 2)

n

n

nxP(h x SWH) K nx

n

( 1)2( )exp 2

( 2)

nnxP(h x SWH) ( nx)

n

Nov 11 2011 20 Rogue11



correlation =20 km


ltlt 1 (ray limit)





f (xy) (xy)u










NLS with current

where

21 1 1| | 0


0 0( ) ~ ( )i k x i t

x y t A x y t e










Δθ = 26ordm

Δk k = 01

u = 0









Summary









Thank you









23

rmsud

v




scale













Topinka et al

Nature (2001)




Ray simulation


Houmlhmann et al

(PRL 2010)










initial spectrum





conditions k





incoming sea





refraction




Equivalently =



~ (urms v)23 ~ d





ydydyy )()(1)( 2



= 10ordm = 20ordm





2














2 2( ) ( ) exp( 2 )P h h h

2 2( ) ( )exp( 2 )IP h h I h I

I r

( ) ( ) ( )IP h P h f I dI

Nov 11 2011 17 Rogue11







2 2( ) ( ) exp( 2 )P h h h

Nov 11 2011 18 Rogue11




2 4

2

2( )(2 )

( 2)

n

n

nxP(h x SWH) K nx

n

2


Nov 11 2011 19 Rogue11




4 2 241 exp 2P(h x SWH) [ (x x )] ( x )

n

2 4

2

2( )(2 )

( 2)

n

n

nxP(h x SWH) K nx

n

( 1)2( )exp 2

( 2)

nnxP(h x SWH) ( nx)

n

Nov 11 2011 20 Rogue11



correlation =20 km


ltlt 1 (ray limit)





f (xy) (xy)u










NLS with current

where

21 1 1| | 0


0 0( ) ~ ( )i k x i t

x y t A x y t e










Δθ = 26ordm

Δk k = 01

u = 0









Summary









Thank you






scale













Topinka et al

Nature (2001)




Ray simulation


Houmlhmann et al

(PRL 2010)










initial spectrum





conditions k





incoming sea





refraction




Equivalently =



~ (urms v)23 ~ d





ydydyy )()(1)( 2



= 10ordm = 20ordm





2














2 2( ) ( ) exp( 2 )P h h h

2 2( ) ( )exp( 2 )IP h h I h I

I r

( ) ( ) ( )IP h P h f I dI

Nov 11 2011 17 Rogue11







2 2( ) ( ) exp( 2 )P h h h

Nov 11 2011 18 Rogue11




2 4

2

2( )(2 )

( 2)

n

n

nxP(h x SWH) K nx

n

2


Nov 11 2011 19 Rogue11




4 2 241 exp 2P(h x SWH) [ (x x )] ( x )

n

2 4

2

2( )(2 )

( 2)

n

n

nxP(h x SWH) K nx

n

( 1)2( )exp 2

( 2)

nnxP(h x SWH) ( nx)

n

Nov 11 2011 20 Rogue11



correlation =20 km


ltlt 1 (ray limit)





f (xy) (xy)u










NLS with current

where

21 1 1| | 0


0 0( ) ~ ( )i k x i t

x y t A x y t e










Δθ = 26ordm

Δk k = 01

u = 0









Summary









Thank you









Topinka et al

Nature (2001)




Ray simulation


Houmlhmann et al

(PRL 2010)










initial spectrum





conditions k





incoming sea





refraction




Equivalently =



~ (urms v)23 ~ d





ydydyy )()(1)( 2



= 10ordm = 20ordm





2














2 2( ) ( ) exp( 2 )P h h h

2 2( ) ( )exp( 2 )IP h h I h I

I r

( ) ( ) ( )IP h P h f I dI

Nov 11 2011 17 Rogue11







2 2( ) ( ) exp( 2 )P h h h

Nov 11 2011 18 Rogue11




2 4

2

2( )(2 )

( 2)

n

n

nxP(h x SWH) K nx

n

2


Nov 11 2011 19 Rogue11




4 2 241 exp 2P(h x SWH) [ (x x )] ( x )

n

2 4

2

2( )(2 )

( 2)

n

n

nxP(h x SWH) K nx

n

( 1)2( )exp 2

( 2)

nnxP(h x SWH) ( nx)

n

Nov 11 2011 20 Rogue11



correlation =20 km


ltlt 1 (ray limit)





f (xy) (xy)u










NLS with current

where

21 1 1| | 0


0 0( ) ~ ( )i k x i t

x y t A x y t e










Δθ = 26ordm

Δk k = 01

u = 0









Summary









Thank you






Ray simulation


Houmlhmann et al

(PRL 2010)










initial spectrum





conditions k





incoming sea





refraction




Equivalently =



~ (urms v)23 ~ d





ydydyy )()(1)( 2



= 10ordm = 20ordm





2














2 2( ) ( ) exp( 2 )P h h h

2 2( ) ( )exp( 2 )IP h h I h I

I r

( ) ( ) ( )IP h P h f I dI

Nov 11 2011 17 Rogue11







2 2( ) ( ) exp( 2 )P h h h

Nov 11 2011 18 Rogue11




2 4

2

2( )(2 )

( 2)

n

n

nxP(h x SWH) K nx

n

2


Nov 11 2011 19 Rogue11




4 2 241 exp 2P(h x SWH) [ (x x )] ( x )

n

2 4

2

2( )(2 )

( 2)

n

n

nxP(h x SWH) K nx

n

( 1)2( )exp 2

( 2)

nnxP(h x SWH) ( nx)

n

Nov 11 2011 20 Rogue11



correlation =20 km


ltlt 1 (ray limit)





f (xy) (xy)u










NLS with current

where

21 1 1| | 0


0 0( ) ~ ( )i k x i t

x y t A x y t e










Δθ = 26ordm

Δk k = 01

u = 0









Summary









Thank you













initial spectrum





conditions k





incoming sea





refraction




Equivalently =



~ (urms v)23 ~ d





ydydyy )()(1)( 2



= 10ordm = 20ordm





2














2 2( ) ( ) exp( 2 )P h h h

2 2( ) ( )exp( 2 )IP h h I h I

I r

( ) ( ) ( )IP h P h f I dI

Nov 11 2011 17 Rogue11







2 2( ) ( ) exp( 2 )P h h h

Nov 11 2011 18 Rogue11




2 4

2

2( )(2 )

( 2)

n

n

nxP(h x SWH) K nx

n

2


Nov 11 2011 19 Rogue11




4 2 241 exp 2P(h x SWH) [ (x x )] ( x )

n

2 4

2

2( )(2 )

( 2)

n

n

nxP(h x SWH) K nx

n

( 1)2( )exp 2

( 2)

nnxP(h x SWH) ( nx)

n

Nov 11 2011 20 Rogue11



correlation =20 km


ltlt 1 (ray limit)





f (xy) (xy)u










NLS with current

where

21 1 1| | 0


0 0( ) ~ ( )i k x i t

x y t A x y t e










Δθ = 26ordm

Δk k = 01

u = 0









Summary









Thank you






conditions k





incoming sea





refraction




Equivalently =



~ (urms v)23 ~ d





ydydyy )()(1)( 2



= 10ordm = 20ordm





2














2 2( ) ( ) exp( 2 )P h h h

2 2( ) ( )exp( 2 )IP h h I h I

I r

( ) ( ) ( )IP h P h f I dI

Nov 11 2011 17 Rogue11







2 2( ) ( ) exp( 2 )P h h h

Nov 11 2011 18 Rogue11




2 4

2

2( )(2 )

( 2)

n

n

nxP(h x SWH) K nx

n

2


Nov 11 2011 19 Rogue11




4 2 241 exp 2P(h x SWH) [ (x x )] ( x )

n

2 4

2

2( )(2 )

( 2)

n

n

nxP(h x SWH) K nx

n

( 1)2( )exp 2

( 2)

nnxP(h x SWH) ( nx)

n

Nov 11 2011 20 Rogue11



correlation =20 km


ltlt 1 (ray limit)





f (xy) (xy)u










NLS with current

where

21 1 1| | 0


0 0( ) ~ ( )i k x i t

x y t A x y t e










Δθ = 26ordm

Δk k = 01

u = 0









Summary









Thank you





incoming sea





refraction




Equivalently =



~ (urms v)23 ~ d





ydydyy )()(1)( 2



= 10ordm = 20ordm





2














2 2( ) ( ) exp( 2 )P h h h

2 2( ) ( )exp( 2 )IP h h I h I

I r

( ) ( ) ( )IP h P h f I dI

Nov 11 2011 17 Rogue11







2 2( ) ( ) exp( 2 )P h h h

Nov 11 2011 18 Rogue11




2 4

2

2( )(2 )

( 2)

n

n

nxP(h x SWH) K nx

n

2


Nov 11 2011 19 Rogue11




4 2 241 exp 2P(h x SWH) [ (x x )] ( x )

n

2 4

2

2( )(2 )

( 2)

n

n

nxP(h x SWH) K nx

n

( 1)2( )exp 2

( 2)

nnxP(h x SWH) ( nx)

n

Nov 11 2011 20 Rogue11



correlation =20 km


ltlt 1 (ray limit)





f (xy) (xy)u










NLS with current

where

21 1 1| | 0


0 0( ) ~ ( )i k x i t

x y t A x y t e










Δθ = 26ordm

Δk k = 01

u = 0









Summary









Thank you






Equivalently =



~ (urms v)23 ~ d





ydydyy )()(1)( 2



= 10ordm = 20ordm





2














2 2( ) ( ) exp( 2 )P h h h

2 2( ) ( )exp( 2 )IP h h I h I

I r

( ) ( ) ( )IP h P h f I dI

Nov 11 2011 17 Rogue11







2 2( ) ( ) exp( 2 )P h h h

Nov 11 2011 18 Rogue11




2 4

2

2( )(2 )

( 2)

n

n

nxP(h x SWH) K nx

n

2


Nov 11 2011 19 Rogue11




4 2 241 exp 2P(h x SWH) [ (x x )] ( x )

n

2 4

2

2( )(2 )

( 2)

n

n

nxP(h x SWH) K nx

n

( 1)2( )exp 2

( 2)

nnxP(h x SWH) ( nx)

n

Nov 11 2011 20 Rogue11



correlation =20 km


ltlt 1 (ray limit)





f (xy) (xy)u










NLS with current

where

21 1 1| | 0


0 0( ) ~ ( )i k x i t

x y t A x y t e










Δθ = 26ordm

Δk k = 01

u = 0









Summary









Thank you





= 10ordm = 20ordm





2














2 2( ) ( ) exp( 2 )P h h h

2 2( ) ( )exp( 2 )IP h h I h I

I r

( ) ( ) ( )IP h P h f I dI

Nov 11 2011 17 Rogue11







2 2( ) ( ) exp( 2 )P h h h

Nov 11 2011 18 Rogue11




2 4

2

2( )(2 )

( 2)

n

n

nxP(h x SWH) K nx

n

2


Nov 11 2011 19 Rogue11




4 2 241 exp 2P(h x SWH) [ (x x )] ( x )

n

2 4

2

2( )(2 )

( 2)

n

n

nxP(h x SWH) K nx

n

( 1)2( )exp 2

( 2)

nnxP(h x SWH) ( nx)

n

Nov 11 2011 20 Rogue11



correlation =20 km


ltlt 1 (ray limit)





f (xy) (xy)u










NLS with current

where

21 1 1| | 0


0 0( ) ~ ( )i k x i t

x y t A x y t e










Δθ = 26ordm

Δk k = 01

u = 0









Summary









Thank you







2














2 2( ) ( ) exp( 2 )P h h h

2 2( ) ( )exp( 2 )IP h h I h I

I r

( ) ( ) ( )IP h P h f I dI

Nov 11 2011 17 Rogue11







2 2( ) ( ) exp( 2 )P h h h

Nov 11 2011 18 Rogue11




2 4

2

2( )(2 )

( 2)

n

n

nxP(h x SWH) K nx

n

2


Nov 11 2011 19 Rogue11




4 2 241 exp 2P(h x SWH) [ (x x )] ( x )

n

2 4

2

2( )(2 )

( 2)

n

n

nxP(h x SWH) K nx

n

( 1)2( )exp 2

( 2)

nnxP(h x SWH) ( nx)

n

Nov 11 2011 20 Rogue11



correlation =20 km


ltlt 1 (ray limit)





f (xy) (xy)u










NLS with current

where

21 1 1| | 0


0 0( ) ~ ( )i k x i t

x y t A x y t e










Δθ = 26ordm

Δk k = 01

u = 0









Summary









Thank you














2 2( ) ( ) exp( 2 )P h h h

2 2( ) ( )exp( 2 )IP h h I h I

I r

( ) ( ) ( )IP h P h f I dI

Nov 11 2011 17 Rogue11







2 2( ) ( ) exp( 2 )P h h h

Nov 11 2011 18 Rogue11




2 4

2

2( )(2 )

( 2)

n

n

nxP(h x SWH) K nx

n

2


Nov 11 2011 19 Rogue11




4 2 241 exp 2P(h x SWH) [ (x x )] ( x )

n

2 4

2

2( )(2 )

( 2)

n

n

nxP(h x SWH) K nx

n

( 1)2( )exp 2

( 2)

nnxP(h x SWH) ( nx)

n

Nov 11 2011 20 Rogue11



correlation =20 km


ltlt 1 (ray limit)





f (xy) (xy)u










NLS with current

where

21 1 1| | 0


0 0( ) ~ ( )i k x i t

x y t A x y t e










Δθ = 26ordm

Δk k = 01

u = 0









Summary









Thank you










2 2( ) ( ) exp( 2 )P h h h

Nov 11 2011 18 Rogue11




2 4

2

2( )(2 )

( 2)

n

n

nxP(h x SWH) K nx

n

2


Nov 11 2011 19 Rogue11




4 2 241 exp 2P(h x SWH) [ (x x )] ( x )

n

2 4

2

2( )(2 )

( 2)

n

n

nxP(h x SWH) K nx

n

( 1)2( )exp 2

( 2)

nnxP(h x SWH) ( nx)

n

Nov 11 2011 20 Rogue11



correlation =20 km


ltlt 1 (ray limit)





f (xy) (xy)u










NLS with current

where

21 1 1| | 0


0 0( ) ~ ( )i k x i t

x y t A x y t e










Δθ = 26ordm

Δk k = 01

u = 0









Summary









Thank you







2 4

2

2( )(2 )

( 2)

n

n

nxP(h x SWH) K nx

n

2


Nov 11 2011 19 Rogue11




4 2 241 exp 2P(h x SWH) [ (x x )] ( x )

n

2 4

2

2( )(2 )

( 2)

n

n

nxP(h x SWH) K nx

n

( 1)2( )exp 2

( 2)

nnxP(h x SWH) ( nx)

n

Nov 11 2011 20 Rogue11



correlation =20 km


ltlt 1 (ray limit)





f (xy) (xy)u










NLS with current

where

21 1 1| | 0


0 0( ) ~ ( )i k x i t

x y t A x y t e










Δθ = 26ordm

Δk k = 01

u = 0









Summary









Thank you







4 2 241 exp 2P(h x SWH) [ (x x )] ( x )

n

2 4

2

2( )(2 )

( 2)

n

n

nxP(h x SWH) K nx

n

( 1)2( )exp 2

( 2)

nnxP(h x SWH) ( nx)

n

Nov 11 2011 20 Rogue11



correlation =20 km


ltlt 1 (ray limit)





f (xy) (xy)u










NLS with current

where

21 1 1| | 0


0 0( ) ~ ( )i k x i t

x y t A x y t e










Δθ = 26ordm

Δk k = 01

u = 0









Summary









Thank you






correlation =20 km


ltlt 1 (ray limit)





f (xy) (xy)u










NLS with current

where

21 1 1| | 0


0 0( ) ~ ( )i k x i t

x y t A x y t e










Δθ = 26ordm

Δk k = 01

u = 0









Summary









Thank you













NLS with current

where

21 1 1| | 0


0 0( ) ~ ( )i k x i t

x y t A x y t e










Δθ = 26ordm

Δk k = 01

u = 0









Summary









Thank you












Δθ = 26ordm

Δk k = 01

u = 0









Summary









Thank you












Summary









Thank you




Thank you




linear and nonlinear rogue wave statistics in the presence ...lkaplan/kaplan rogue11.pdffocus on...

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