Linear Wave Theory fundamental description: L - wave length H - wave height T - period
Linear and Nonlinear Rogue Wave Statistics in the Presence ...lkaplan/Kaplan Rogue11.pdfFocus on...
Transcript of Linear and Nonlinear Rogue Wave Statistics in the Presence ...lkaplan/Kaplan Rogue11.pdfFocus on...
![Page 1: 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](https://reader033.fdocuments.net/reader033/viewer/2022042322/5f0cc5587e708231d4370c6d/html5/thumbnails/1.jpg)
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
October 11 2011 2 Rogue11 Dresden
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 )()(
October 11 2011 3 Rogue11 Dresden
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)
October 11 2011 4 Rogue11 Dresden
How caustics form refraction from current eddies
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
October 11 2011 5 Rogue11 Dresden
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
October 11 2011 6 Rogue11 Dresden
Refraction from weak random currents
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)
October 11 2011 7 Rogue11 Dresden
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)
October 11 2011 8 Rogue11 Dresden
Microwave scattering experiments
Experimentally observed branched flow
Ray simulation
October 11 2011 9 Rogue11 Dresden
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
October 11 2011 11 Rogue11 Dresden
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
October 11 2011 12 Rogue11 Dresden
Smearing of caustics for stochastic
incoming sea
Competing effects of focusing
and initial stochasticity
kx = initial wave vector spread
kx = wave vector change due to
refraction
October 11 2011 13 Rogue11 Dresden
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
October 11 2011 14 Rogue11 Dresden
Typical ray calculation for ocean waves
= 10ordm = 20ordm
October 11 2011 15 Rogue11 Dresden
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
October 11 2011 16 Rogue11 Dresden
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
Calculation Rogue Waves
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
Analytics limit of small freak index
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)
October 11 2011 21 Rogue11 Dresden
f (xy) (xy)u
Wave height distribution for ocean waves
October 11 2011 22 Rogue11 Dresden
Numerical test of N vs γ
October 11 2011 23 Rogue11 Dresden
Probability enhancement over
Rayleigh predictions
urms = 05 ms v=78 ms
October 11 2011 Rogue11 Dresden 24
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
October 11 2011 25 Rogue11 Dresden
October 11 2011 Rogue11 Dresden 26
Extension to nonlinear waves
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
October 11 2011 27 Rogue11 Dresden
October 11 2011 Rogue11 Dresden 28
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
October 11 2011 29 Rogue11 Dresden
N as function of incoming frequency spread for fixed steepness and angular spread
October 11 2011 30 Rogue11 Dresden
N as function of steepness for fixed incoming angular spread and frequency spread
October 11 2011 31 Rogue11 Dresden
Finally combine currents and nonlinearity
October 11 2011 Rogue11 Dresden 32
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
October 11 2011 33 Rogue11 Dresden
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
October 11 2011 Rogue11 Dresden 34
![Page 2: 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](https://reader033.fdocuments.net/reader033/viewer/2022042322/5f0cc5587e708231d4370c6d/html5/thumbnails/2.jpg)
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
October 11 2011 2 Rogue11 Dresden
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 )()(
October 11 2011 3 Rogue11 Dresden
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)
October 11 2011 4 Rogue11 Dresden
How caustics form refraction from current eddies
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
October 11 2011 5 Rogue11 Dresden
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
October 11 2011 6 Rogue11 Dresden
Refraction from weak random currents
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)
October 11 2011 7 Rogue11 Dresden
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)
October 11 2011 8 Rogue11 Dresden
Microwave scattering experiments
Experimentally observed branched flow
Ray simulation
October 11 2011 9 Rogue11 Dresden
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
October 11 2011 11 Rogue11 Dresden
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
October 11 2011 12 Rogue11 Dresden
Smearing of caustics for stochastic
incoming sea
Competing effects of focusing
and initial stochasticity
kx = initial wave vector spread
kx = wave vector change due to
refraction
October 11 2011 13 Rogue11 Dresden
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
October 11 2011 14 Rogue11 Dresden
Typical ray calculation for ocean waves
= 10ordm = 20ordm
October 11 2011 15 Rogue11 Dresden
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
October 11 2011 16 Rogue11 Dresden
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
Calculation Rogue Waves
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
Analytics limit of small freak index
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)
October 11 2011 21 Rogue11 Dresden
f (xy) (xy)u
Wave height distribution for ocean waves
October 11 2011 22 Rogue11 Dresden
Numerical test of N vs γ
October 11 2011 23 Rogue11 Dresden
Probability enhancement over
Rayleigh predictions
urms = 05 ms v=78 ms
October 11 2011 Rogue11 Dresden 24
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
October 11 2011 25 Rogue11 Dresden
October 11 2011 Rogue11 Dresden 26
Extension to nonlinear waves
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
October 11 2011 27 Rogue11 Dresden
October 11 2011 Rogue11 Dresden 28
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
October 11 2011 29 Rogue11 Dresden
N as function of incoming frequency spread for fixed steepness and angular spread
October 11 2011 30 Rogue11 Dresden
N as function of steepness for fixed incoming angular spread and frequency spread
October 11 2011 31 Rogue11 Dresden
Finally combine currents and nonlinearity
October 11 2011 Rogue11 Dresden 32
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
October 11 2011 33 Rogue11 Dresden
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
October 11 2011 Rogue11 Dresden 34
![Page 3: 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](https://reader033.fdocuments.net/reader033/viewer/2022042322/5f0cc5587e708231d4370c6d/html5/thumbnails/3.jpg)
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 )()(
October 11 2011 3 Rogue11 Dresden
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)
October 11 2011 4 Rogue11 Dresden
How caustics form refraction from current eddies
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
October 11 2011 5 Rogue11 Dresden
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
October 11 2011 6 Rogue11 Dresden
Refraction from weak random currents
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)
October 11 2011 7 Rogue11 Dresden
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)
October 11 2011 8 Rogue11 Dresden
Microwave scattering experiments
Experimentally observed branched flow
Ray simulation
October 11 2011 9 Rogue11 Dresden
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
October 11 2011 11 Rogue11 Dresden
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
October 11 2011 12 Rogue11 Dresden
Smearing of caustics for stochastic
incoming sea
Competing effects of focusing
and initial stochasticity
kx = initial wave vector spread
kx = wave vector change due to
refraction
October 11 2011 13 Rogue11 Dresden
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
October 11 2011 14 Rogue11 Dresden
Typical ray calculation for ocean waves
= 10ordm = 20ordm
October 11 2011 15 Rogue11 Dresden
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
October 11 2011 16 Rogue11 Dresden
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
Calculation Rogue Waves
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
Analytics limit of small freak index
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)
October 11 2011 21 Rogue11 Dresden
f (xy) (xy)u
Wave height distribution for ocean waves
October 11 2011 22 Rogue11 Dresden
Numerical test of N vs γ
October 11 2011 23 Rogue11 Dresden
Probability enhancement over
Rayleigh predictions
urms = 05 ms v=78 ms
October 11 2011 Rogue11 Dresden 24
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
October 11 2011 25 Rogue11 Dresden
October 11 2011 Rogue11 Dresden 26
Extension to nonlinear waves
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
October 11 2011 27 Rogue11 Dresden
October 11 2011 Rogue11 Dresden 28
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
October 11 2011 29 Rogue11 Dresden
N as function of incoming frequency spread for fixed steepness and angular spread
October 11 2011 30 Rogue11 Dresden
N as function of steepness for fixed incoming angular spread and frequency spread
October 11 2011 31 Rogue11 Dresden
Finally combine currents and nonlinearity
October 11 2011 Rogue11 Dresden 32
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
October 11 2011 33 Rogue11 Dresden
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
October 11 2011 Rogue11 Dresden 34
![Page 4: 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](https://reader033.fdocuments.net/reader033/viewer/2022042322/5f0cc5587e708231d4370c6d/html5/thumbnails/4.jpg)
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)
October 11 2011 4 Rogue11 Dresden
How caustics form refraction from current eddies
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
October 11 2011 5 Rogue11 Dresden
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
October 11 2011 6 Rogue11 Dresden
Refraction from weak random currents
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)
October 11 2011 7 Rogue11 Dresden
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)
October 11 2011 8 Rogue11 Dresden
Microwave scattering experiments
Experimentally observed branched flow
Ray simulation
October 11 2011 9 Rogue11 Dresden
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
October 11 2011 11 Rogue11 Dresden
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
October 11 2011 12 Rogue11 Dresden
Smearing of caustics for stochastic
incoming sea
Competing effects of focusing
and initial stochasticity
kx = initial wave vector spread
kx = wave vector change due to
refraction
October 11 2011 13 Rogue11 Dresden
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
October 11 2011 14 Rogue11 Dresden
Typical ray calculation for ocean waves
= 10ordm = 20ordm
October 11 2011 15 Rogue11 Dresden
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
October 11 2011 16 Rogue11 Dresden
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
Calculation Rogue Waves
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
Analytics limit of small freak index
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)
October 11 2011 21 Rogue11 Dresden
f (xy) (xy)u
Wave height distribution for ocean waves
October 11 2011 22 Rogue11 Dresden
Numerical test of N vs γ
October 11 2011 23 Rogue11 Dresden
Probability enhancement over
Rayleigh predictions
urms = 05 ms v=78 ms
October 11 2011 Rogue11 Dresden 24
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
October 11 2011 25 Rogue11 Dresden
October 11 2011 Rogue11 Dresden 26
Extension to nonlinear waves
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
October 11 2011 27 Rogue11 Dresden
October 11 2011 Rogue11 Dresden 28
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
October 11 2011 29 Rogue11 Dresden
N as function of incoming frequency spread for fixed steepness and angular spread
October 11 2011 30 Rogue11 Dresden
N as function of steepness for fixed incoming angular spread and frequency spread
October 11 2011 31 Rogue11 Dresden
Finally combine currents and nonlinearity
October 11 2011 Rogue11 Dresden 32
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
October 11 2011 33 Rogue11 Dresden
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
October 11 2011 Rogue11 Dresden 34
![Page 5: 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](https://reader033.fdocuments.net/reader033/viewer/2022042322/5f0cc5587e708231d4370c6d/html5/thumbnails/5.jpg)
How caustics form refraction from current eddies
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
October 11 2011 5 Rogue11 Dresden
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
October 11 2011 6 Rogue11 Dresden
Refraction from weak random currents
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)
October 11 2011 7 Rogue11 Dresden
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)
October 11 2011 8 Rogue11 Dresden
Microwave scattering experiments
Experimentally observed branched flow
Ray simulation
October 11 2011 9 Rogue11 Dresden
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
October 11 2011 11 Rogue11 Dresden
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
October 11 2011 12 Rogue11 Dresden
Smearing of caustics for stochastic
incoming sea
Competing effects of focusing
and initial stochasticity
kx = initial wave vector spread
kx = wave vector change due to
refraction
October 11 2011 13 Rogue11 Dresden
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
October 11 2011 14 Rogue11 Dresden
Typical ray calculation for ocean waves
= 10ordm = 20ordm
October 11 2011 15 Rogue11 Dresden
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
October 11 2011 16 Rogue11 Dresden
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
Calculation Rogue Waves
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
Analytics limit of small freak index
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)
October 11 2011 21 Rogue11 Dresden
f (xy) (xy)u
Wave height distribution for ocean waves
October 11 2011 22 Rogue11 Dresden
Numerical test of N vs γ
October 11 2011 23 Rogue11 Dresden
Probability enhancement over
Rayleigh predictions
urms = 05 ms v=78 ms
October 11 2011 Rogue11 Dresden 24
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
October 11 2011 25 Rogue11 Dresden
October 11 2011 Rogue11 Dresden 26
Extension to nonlinear waves
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
October 11 2011 27 Rogue11 Dresden
October 11 2011 Rogue11 Dresden 28
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
October 11 2011 29 Rogue11 Dresden
N as function of incoming frequency spread for fixed steepness and angular spread
October 11 2011 30 Rogue11 Dresden
N as function of steepness for fixed incoming angular spread and frequency spread
October 11 2011 31 Rogue11 Dresden
Finally combine currents and nonlinearity
October 11 2011 Rogue11 Dresden 32
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
October 11 2011 33 Rogue11 Dresden
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
October 11 2011 Rogue11 Dresden 34
![Page 6: 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](https://reader033.fdocuments.net/reader033/viewer/2022042322/5f0cc5587e708231d4370c6d/html5/thumbnails/6.jpg)
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
October 11 2011 6 Rogue11 Dresden
Refraction from weak random currents
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)
October 11 2011 7 Rogue11 Dresden
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)
October 11 2011 8 Rogue11 Dresden
Microwave scattering experiments
Experimentally observed branched flow
Ray simulation
October 11 2011 9 Rogue11 Dresden
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
October 11 2011 11 Rogue11 Dresden
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
October 11 2011 12 Rogue11 Dresden
Smearing of caustics for stochastic
incoming sea
Competing effects of focusing
and initial stochasticity
kx = initial wave vector spread
kx = wave vector change due to
refraction
October 11 2011 13 Rogue11 Dresden
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
October 11 2011 14 Rogue11 Dresden
Typical ray calculation for ocean waves
= 10ordm = 20ordm
October 11 2011 15 Rogue11 Dresden
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
October 11 2011 16 Rogue11 Dresden
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
Calculation Rogue Waves
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
Analytics limit of small freak index
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)
October 11 2011 21 Rogue11 Dresden
f (xy) (xy)u
Wave height distribution for ocean waves
October 11 2011 22 Rogue11 Dresden
Numerical test of N vs γ
October 11 2011 23 Rogue11 Dresden
Probability enhancement over
Rayleigh predictions
urms = 05 ms v=78 ms
October 11 2011 Rogue11 Dresden 24
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
October 11 2011 25 Rogue11 Dresden
October 11 2011 Rogue11 Dresden 26
Extension to nonlinear waves
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
October 11 2011 27 Rogue11 Dresden
October 11 2011 Rogue11 Dresden 28
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
October 11 2011 29 Rogue11 Dresden
N as function of incoming frequency spread for fixed steepness and angular spread
October 11 2011 30 Rogue11 Dresden
N as function of steepness for fixed incoming angular spread and frequency spread
October 11 2011 31 Rogue11 Dresden
Finally combine currents and nonlinearity
October 11 2011 Rogue11 Dresden 32
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
October 11 2011 33 Rogue11 Dresden
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
October 11 2011 Rogue11 Dresden 34
![Page 7: 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](https://reader033.fdocuments.net/reader033/viewer/2022042322/5f0cc5587e708231d4370c6d/html5/thumbnails/7.jpg)
Refraction from weak random currents
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)
October 11 2011 7 Rogue11 Dresden
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)
October 11 2011 8 Rogue11 Dresden
Microwave scattering experiments
Experimentally observed branched flow
Ray simulation
October 11 2011 9 Rogue11 Dresden
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
October 11 2011 11 Rogue11 Dresden
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
October 11 2011 12 Rogue11 Dresden
Smearing of caustics for stochastic
incoming sea
Competing effects of focusing
and initial stochasticity
kx = initial wave vector spread
kx = wave vector change due to
refraction
October 11 2011 13 Rogue11 Dresden
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
October 11 2011 14 Rogue11 Dresden
Typical ray calculation for ocean waves
= 10ordm = 20ordm
October 11 2011 15 Rogue11 Dresden
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
October 11 2011 16 Rogue11 Dresden
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
Calculation Rogue Waves
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
Analytics limit of small freak index
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)
October 11 2011 21 Rogue11 Dresden
f (xy) (xy)u
Wave height distribution for ocean waves
October 11 2011 22 Rogue11 Dresden
Numerical test of N vs γ
October 11 2011 23 Rogue11 Dresden
Probability enhancement over
Rayleigh predictions
urms = 05 ms v=78 ms
October 11 2011 Rogue11 Dresden 24
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
October 11 2011 25 Rogue11 Dresden
October 11 2011 Rogue11 Dresden 26
Extension to nonlinear waves
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
October 11 2011 27 Rogue11 Dresden
October 11 2011 Rogue11 Dresden 28
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
October 11 2011 29 Rogue11 Dresden
N as function of incoming frequency spread for fixed steepness and angular spread
October 11 2011 30 Rogue11 Dresden
N as function of steepness for fixed incoming angular spread and frequency spread
October 11 2011 31 Rogue11 Dresden
Finally combine currents and nonlinearity
October 11 2011 Rogue11 Dresden 32
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
October 11 2011 33 Rogue11 Dresden
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
October 11 2011 Rogue11 Dresden 34
![Page 8: 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](https://reader033.fdocuments.net/reader033/viewer/2022042322/5f0cc5587e708231d4370c6d/html5/thumbnails/8.jpg)
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)
October 11 2011 8 Rogue11 Dresden
Microwave scattering experiments
Experimentally observed branched flow
Ray simulation
October 11 2011 9 Rogue11 Dresden
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
October 11 2011 11 Rogue11 Dresden
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
October 11 2011 12 Rogue11 Dresden
Smearing of caustics for stochastic
incoming sea
Competing effects of focusing
and initial stochasticity
kx = initial wave vector spread
kx = wave vector change due to
refraction
October 11 2011 13 Rogue11 Dresden
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
October 11 2011 14 Rogue11 Dresden
Typical ray calculation for ocean waves
= 10ordm = 20ordm
October 11 2011 15 Rogue11 Dresden
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
October 11 2011 16 Rogue11 Dresden
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
Calculation Rogue Waves
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
Analytics limit of small freak index
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)
October 11 2011 21 Rogue11 Dresden
f (xy) (xy)u
Wave height distribution for ocean waves
October 11 2011 22 Rogue11 Dresden
Numerical test of N vs γ
October 11 2011 23 Rogue11 Dresden
Probability enhancement over
Rayleigh predictions
urms = 05 ms v=78 ms
October 11 2011 Rogue11 Dresden 24
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
October 11 2011 25 Rogue11 Dresden
October 11 2011 Rogue11 Dresden 26
Extension to nonlinear waves
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
October 11 2011 27 Rogue11 Dresden
October 11 2011 Rogue11 Dresden 28
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
October 11 2011 29 Rogue11 Dresden
N as function of incoming frequency spread for fixed steepness and angular spread
October 11 2011 30 Rogue11 Dresden
N as function of steepness for fixed incoming angular spread and frequency spread
October 11 2011 31 Rogue11 Dresden
Finally combine currents and nonlinearity
October 11 2011 Rogue11 Dresden 32
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
October 11 2011 33 Rogue11 Dresden
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
October 11 2011 Rogue11 Dresden 34
![Page 9: 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](https://reader033.fdocuments.net/reader033/viewer/2022042322/5f0cc5587e708231d4370c6d/html5/thumbnails/9.jpg)
Microwave scattering experiments
Experimentally observed branched flow
Ray simulation
October 11 2011 9 Rogue11 Dresden
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
October 11 2011 11 Rogue11 Dresden
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
October 11 2011 12 Rogue11 Dresden
Smearing of caustics for stochastic
incoming sea
Competing effects of focusing
and initial stochasticity
kx = initial wave vector spread
kx = wave vector change due to
refraction
October 11 2011 13 Rogue11 Dresden
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
October 11 2011 14 Rogue11 Dresden
Typical ray calculation for ocean waves
= 10ordm = 20ordm
October 11 2011 15 Rogue11 Dresden
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
October 11 2011 16 Rogue11 Dresden
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
Calculation Rogue Waves
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
Analytics limit of small freak index
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)
October 11 2011 21 Rogue11 Dresden
f (xy) (xy)u
Wave height distribution for ocean waves
October 11 2011 22 Rogue11 Dresden
Numerical test of N vs γ
October 11 2011 23 Rogue11 Dresden
Probability enhancement over
Rayleigh predictions
urms = 05 ms v=78 ms
October 11 2011 Rogue11 Dresden 24
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
October 11 2011 25 Rogue11 Dresden
October 11 2011 Rogue11 Dresden 26
Extension to nonlinear waves
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
October 11 2011 27 Rogue11 Dresden
October 11 2011 Rogue11 Dresden 28
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
October 11 2011 29 Rogue11 Dresden
N as function of incoming frequency spread for fixed steepness and angular spread
October 11 2011 30 Rogue11 Dresden
N as function of steepness for fixed incoming angular spread and frequency spread
October 11 2011 31 Rogue11 Dresden
Finally combine currents and nonlinearity
October 11 2011 Rogue11 Dresden 32
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
October 11 2011 33 Rogue11 Dresden
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
October 11 2011 Rogue11 Dresden 34
![Page 10: 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](https://reader033.fdocuments.net/reader033/viewer/2022042322/5f0cc5587e708231d4370c6d/html5/thumbnails/10.jpg)
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
October 11 2011 11 Rogue11 Dresden
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
October 11 2011 12 Rogue11 Dresden
Smearing of caustics for stochastic
incoming sea
Competing effects of focusing
and initial stochasticity
kx = initial wave vector spread
kx = wave vector change due to
refraction
October 11 2011 13 Rogue11 Dresden
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
October 11 2011 14 Rogue11 Dresden
Typical ray calculation for ocean waves
= 10ordm = 20ordm
October 11 2011 15 Rogue11 Dresden
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
October 11 2011 16 Rogue11 Dresden
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
Calculation Rogue Waves
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
Analytics limit of small freak index
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)
October 11 2011 21 Rogue11 Dresden
f (xy) (xy)u
Wave height distribution for ocean waves
October 11 2011 22 Rogue11 Dresden
Numerical test of N vs γ
October 11 2011 23 Rogue11 Dresden
Probability enhancement over
Rayleigh predictions
urms = 05 ms v=78 ms
October 11 2011 Rogue11 Dresden 24
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
October 11 2011 25 Rogue11 Dresden
October 11 2011 Rogue11 Dresden 26
Extension to nonlinear waves
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
October 11 2011 27 Rogue11 Dresden
October 11 2011 Rogue11 Dresden 28
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
October 11 2011 29 Rogue11 Dresden
N as function of incoming frequency spread for fixed steepness and angular spread
October 11 2011 30 Rogue11 Dresden
N as function of steepness for fixed incoming angular spread and frequency spread
October 11 2011 31 Rogue11 Dresden
Finally combine currents and nonlinearity
October 11 2011 Rogue11 Dresden 32
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
October 11 2011 33 Rogue11 Dresden
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
October 11 2011 Rogue11 Dresden 34
![Page 11: 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](https://reader033.fdocuments.net/reader033/viewer/2022042322/5f0cc5587e708231d4370c6d/html5/thumbnails/11.jpg)
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
October 11 2011 11 Rogue11 Dresden
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
October 11 2011 12 Rogue11 Dresden
Smearing of caustics for stochastic
incoming sea
Competing effects of focusing
and initial stochasticity
kx = initial wave vector spread
kx = wave vector change due to
refraction
October 11 2011 13 Rogue11 Dresden
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
October 11 2011 14 Rogue11 Dresden
Typical ray calculation for ocean waves
= 10ordm = 20ordm
October 11 2011 15 Rogue11 Dresden
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
October 11 2011 16 Rogue11 Dresden
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
Calculation Rogue Waves
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
Analytics limit of small freak index
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)
October 11 2011 21 Rogue11 Dresden
f (xy) (xy)u
Wave height distribution for ocean waves
October 11 2011 22 Rogue11 Dresden
Numerical test of N vs γ
October 11 2011 23 Rogue11 Dresden
Probability enhancement over
Rayleigh predictions
urms = 05 ms v=78 ms
October 11 2011 Rogue11 Dresden 24
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
October 11 2011 25 Rogue11 Dresden
October 11 2011 Rogue11 Dresden 26
Extension to nonlinear waves
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
October 11 2011 27 Rogue11 Dresden
October 11 2011 Rogue11 Dresden 28
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
October 11 2011 29 Rogue11 Dresden
N as function of incoming frequency spread for fixed steepness and angular spread
October 11 2011 30 Rogue11 Dresden
N as function of steepness for fixed incoming angular spread and frequency spread
October 11 2011 31 Rogue11 Dresden
Finally combine currents and nonlinearity
October 11 2011 Rogue11 Dresden 32
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
October 11 2011 33 Rogue11 Dresden
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
October 11 2011 Rogue11 Dresden 34
![Page 12: 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](https://reader033.fdocuments.net/reader033/viewer/2022042322/5f0cc5587e708231d4370c6d/html5/thumbnails/12.jpg)
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
October 11 2011 12 Rogue11 Dresden
Smearing of caustics for stochastic
incoming sea
Competing effects of focusing
and initial stochasticity
kx = initial wave vector spread
kx = wave vector change due to
refraction
October 11 2011 13 Rogue11 Dresden
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
October 11 2011 14 Rogue11 Dresden
Typical ray calculation for ocean waves
= 10ordm = 20ordm
October 11 2011 15 Rogue11 Dresden
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
October 11 2011 16 Rogue11 Dresden
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
Calculation Rogue Waves
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
Analytics limit of small freak index
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)
October 11 2011 21 Rogue11 Dresden
f (xy) (xy)u
Wave height distribution for ocean waves
October 11 2011 22 Rogue11 Dresden
Numerical test of N vs γ
October 11 2011 23 Rogue11 Dresden
Probability enhancement over
Rayleigh predictions
urms = 05 ms v=78 ms
October 11 2011 Rogue11 Dresden 24
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
October 11 2011 25 Rogue11 Dresden
October 11 2011 Rogue11 Dresden 26
Extension to nonlinear waves
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
October 11 2011 27 Rogue11 Dresden
October 11 2011 Rogue11 Dresden 28
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
October 11 2011 29 Rogue11 Dresden
N as function of incoming frequency spread for fixed steepness and angular spread
October 11 2011 30 Rogue11 Dresden
N as function of steepness for fixed incoming angular spread and frequency spread
October 11 2011 31 Rogue11 Dresden
Finally combine currents and nonlinearity
October 11 2011 Rogue11 Dresden 32
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
October 11 2011 33 Rogue11 Dresden
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
October 11 2011 Rogue11 Dresden 34
![Page 13: 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](https://reader033.fdocuments.net/reader033/viewer/2022042322/5f0cc5587e708231d4370c6d/html5/thumbnails/13.jpg)
Smearing of caustics for stochastic
incoming sea
Competing effects of focusing
and initial stochasticity
kx = initial wave vector spread
kx = wave vector change due to
refraction
October 11 2011 13 Rogue11 Dresden
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
October 11 2011 14 Rogue11 Dresden
Typical ray calculation for ocean waves
= 10ordm = 20ordm
October 11 2011 15 Rogue11 Dresden
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
October 11 2011 16 Rogue11 Dresden
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
Calculation Rogue Waves
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
Analytics limit of small freak index
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)
October 11 2011 21 Rogue11 Dresden
f (xy) (xy)u
Wave height distribution for ocean waves
October 11 2011 22 Rogue11 Dresden
Numerical test of N vs γ
October 11 2011 23 Rogue11 Dresden
Probability enhancement over
Rayleigh predictions
urms = 05 ms v=78 ms
October 11 2011 Rogue11 Dresden 24
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
October 11 2011 25 Rogue11 Dresden
October 11 2011 Rogue11 Dresden 26
Extension to nonlinear waves
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
October 11 2011 27 Rogue11 Dresden
October 11 2011 Rogue11 Dresden 28
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
October 11 2011 29 Rogue11 Dresden
N as function of incoming frequency spread for fixed steepness and angular spread
October 11 2011 30 Rogue11 Dresden
N as function of steepness for fixed incoming angular spread and frequency spread
October 11 2011 31 Rogue11 Dresden
Finally combine currents and nonlinearity
October 11 2011 Rogue11 Dresden 32
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
October 11 2011 33 Rogue11 Dresden
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
October 11 2011 Rogue11 Dresden 34
![Page 14: 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](https://reader033.fdocuments.net/reader033/viewer/2022042322/5f0cc5587e708231d4370c6d/html5/thumbnails/14.jpg)
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
October 11 2011 14 Rogue11 Dresden
Typical ray calculation for ocean waves
= 10ordm = 20ordm
October 11 2011 15 Rogue11 Dresden
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
October 11 2011 16 Rogue11 Dresden
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
Calculation Rogue Waves
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
Analytics limit of small freak index
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)
October 11 2011 21 Rogue11 Dresden
f (xy) (xy)u
Wave height distribution for ocean waves
October 11 2011 22 Rogue11 Dresden
Numerical test of N vs γ
October 11 2011 23 Rogue11 Dresden
Probability enhancement over
Rayleigh predictions
urms = 05 ms v=78 ms
October 11 2011 Rogue11 Dresden 24
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
October 11 2011 25 Rogue11 Dresden
October 11 2011 Rogue11 Dresden 26
Extension to nonlinear waves
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
October 11 2011 27 Rogue11 Dresden
October 11 2011 Rogue11 Dresden 28
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
October 11 2011 29 Rogue11 Dresden
N as function of incoming frequency spread for fixed steepness and angular spread
October 11 2011 30 Rogue11 Dresden
N as function of steepness for fixed incoming angular spread and frequency spread
October 11 2011 31 Rogue11 Dresden
Finally combine currents and nonlinearity
October 11 2011 Rogue11 Dresden 32
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
October 11 2011 33 Rogue11 Dresden
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
October 11 2011 Rogue11 Dresden 34
![Page 15: 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](https://reader033.fdocuments.net/reader033/viewer/2022042322/5f0cc5587e708231d4370c6d/html5/thumbnails/15.jpg)
Typical ray calculation for ocean waves
= 10ordm = 20ordm
October 11 2011 15 Rogue11 Dresden
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
October 11 2011 16 Rogue11 Dresden
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
Calculation Rogue Waves
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
Analytics limit of small freak index
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)
October 11 2011 21 Rogue11 Dresden
f (xy) (xy)u
Wave height distribution for ocean waves
October 11 2011 22 Rogue11 Dresden
Numerical test of N vs γ
October 11 2011 23 Rogue11 Dresden
Probability enhancement over
Rayleigh predictions
urms = 05 ms v=78 ms
October 11 2011 Rogue11 Dresden 24
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
October 11 2011 25 Rogue11 Dresden
October 11 2011 Rogue11 Dresden 26
Extension to nonlinear waves
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
October 11 2011 27 Rogue11 Dresden
October 11 2011 Rogue11 Dresden 28
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
October 11 2011 29 Rogue11 Dresden
N as function of incoming frequency spread for fixed steepness and angular spread
October 11 2011 30 Rogue11 Dresden
N as function of steepness for fixed incoming angular spread and frequency spread
October 11 2011 31 Rogue11 Dresden
Finally combine currents and nonlinearity
October 11 2011 Rogue11 Dresden 32
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
October 11 2011 33 Rogue11 Dresden
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
October 11 2011 Rogue11 Dresden 34
![Page 16: 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](https://reader033.fdocuments.net/reader033/viewer/2022042322/5f0cc5587e708231d4370c6d/html5/thumbnails/16.jpg)
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
October 11 2011 16 Rogue11 Dresden
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
Calculation Rogue Waves
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
Analytics limit of small freak index
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)
October 11 2011 21 Rogue11 Dresden
f (xy) (xy)u
Wave height distribution for ocean waves
October 11 2011 22 Rogue11 Dresden
Numerical test of N vs γ
October 11 2011 23 Rogue11 Dresden
Probability enhancement over
Rayleigh predictions
urms = 05 ms v=78 ms
October 11 2011 Rogue11 Dresden 24
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
October 11 2011 25 Rogue11 Dresden
October 11 2011 Rogue11 Dresden 26
Extension to nonlinear waves
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
October 11 2011 27 Rogue11 Dresden
October 11 2011 Rogue11 Dresden 28
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
October 11 2011 29 Rogue11 Dresden
N as function of incoming frequency spread for fixed steepness and angular spread
October 11 2011 30 Rogue11 Dresden
N as function of steepness for fixed incoming angular spread and frequency spread
October 11 2011 31 Rogue11 Dresden
Finally combine currents and nonlinearity
October 11 2011 Rogue11 Dresden 32
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
October 11 2011 33 Rogue11 Dresden
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
October 11 2011 Rogue11 Dresden 34
![Page 17: 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](https://reader033.fdocuments.net/reader033/viewer/2022042322/5f0cc5587e708231d4370c6d/html5/thumbnails/17.jpg)
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
Calculation Rogue Waves
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
Analytics limit of small freak index
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)
October 11 2011 21 Rogue11 Dresden
f (xy) (xy)u
Wave height distribution for ocean waves
October 11 2011 22 Rogue11 Dresden
Numerical test of N vs γ
October 11 2011 23 Rogue11 Dresden
Probability enhancement over
Rayleigh predictions
urms = 05 ms v=78 ms
October 11 2011 Rogue11 Dresden 24
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
October 11 2011 25 Rogue11 Dresden
October 11 2011 Rogue11 Dresden 26
Extension to nonlinear waves
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
October 11 2011 27 Rogue11 Dresden
October 11 2011 Rogue11 Dresden 28
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
October 11 2011 29 Rogue11 Dresden
N as function of incoming frequency spread for fixed steepness and angular spread
October 11 2011 30 Rogue11 Dresden
N as function of steepness for fixed incoming angular spread and frequency spread
October 11 2011 31 Rogue11 Dresden
Finally combine currents and nonlinearity
October 11 2011 Rogue11 Dresden 32
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
October 11 2011 33 Rogue11 Dresden
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
October 11 2011 Rogue11 Dresden 34
![Page 18: 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](https://reader033.fdocuments.net/reader033/viewer/2022042322/5f0cc5587e708231d4370c6d/html5/thumbnails/18.jpg)
Calculation Rogue Waves
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
Analytics limit of small freak index
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)
October 11 2011 21 Rogue11 Dresden
f (xy) (xy)u
Wave height distribution for ocean waves
October 11 2011 22 Rogue11 Dresden
Numerical test of N vs γ
October 11 2011 23 Rogue11 Dresden
Probability enhancement over
Rayleigh predictions
urms = 05 ms v=78 ms
October 11 2011 Rogue11 Dresden 24
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
October 11 2011 25 Rogue11 Dresden
October 11 2011 Rogue11 Dresden 26
Extension to nonlinear waves
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
October 11 2011 27 Rogue11 Dresden
October 11 2011 Rogue11 Dresden 28
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
October 11 2011 29 Rogue11 Dresden
N as function of incoming frequency spread for fixed steepness and angular spread
October 11 2011 30 Rogue11 Dresden
N as function of steepness for fixed incoming angular spread and frequency spread
October 11 2011 31 Rogue11 Dresden
Finally combine currents and nonlinearity
October 11 2011 Rogue11 Dresden 32
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
October 11 2011 33 Rogue11 Dresden
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
October 11 2011 Rogue11 Dresden 34
![Page 19: 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](https://reader033.fdocuments.net/reader033/viewer/2022042322/5f0cc5587e708231d4370c6d/html5/thumbnails/19.jpg)
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
Analytics limit of small freak index
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)
October 11 2011 21 Rogue11 Dresden
f (xy) (xy)u
Wave height distribution for ocean waves
October 11 2011 22 Rogue11 Dresden
Numerical test of N vs γ
October 11 2011 23 Rogue11 Dresden
Probability enhancement over
Rayleigh predictions
urms = 05 ms v=78 ms
October 11 2011 Rogue11 Dresden 24
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
October 11 2011 25 Rogue11 Dresden
October 11 2011 Rogue11 Dresden 26
Extension to nonlinear waves
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
October 11 2011 27 Rogue11 Dresden
October 11 2011 Rogue11 Dresden 28
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
October 11 2011 29 Rogue11 Dresden
N as function of incoming frequency spread for fixed steepness and angular spread
October 11 2011 30 Rogue11 Dresden
N as function of steepness for fixed incoming angular spread and frequency spread
October 11 2011 31 Rogue11 Dresden
Finally combine currents and nonlinearity
October 11 2011 Rogue11 Dresden 32
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
October 11 2011 33 Rogue11 Dresden
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
October 11 2011 Rogue11 Dresden 34
![Page 20: 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](https://reader033.fdocuments.net/reader033/viewer/2022042322/5f0cc5587e708231d4370c6d/html5/thumbnails/20.jpg)
Analytics limit of small freak index
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)
October 11 2011 21 Rogue11 Dresden
f (xy) (xy)u
Wave height distribution for ocean waves
October 11 2011 22 Rogue11 Dresden
Numerical test of N vs γ
October 11 2011 23 Rogue11 Dresden
Probability enhancement over
Rayleigh predictions
urms = 05 ms v=78 ms
October 11 2011 Rogue11 Dresden 24
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
October 11 2011 25 Rogue11 Dresden
October 11 2011 Rogue11 Dresden 26
Extension to nonlinear waves
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
October 11 2011 27 Rogue11 Dresden
October 11 2011 Rogue11 Dresden 28
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
October 11 2011 29 Rogue11 Dresden
N as function of incoming frequency spread for fixed steepness and angular spread
October 11 2011 30 Rogue11 Dresden
N as function of steepness for fixed incoming angular spread and frequency spread
October 11 2011 31 Rogue11 Dresden
Finally combine currents and nonlinearity
October 11 2011 Rogue11 Dresden 32
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
October 11 2011 33 Rogue11 Dresden
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
October 11 2011 Rogue11 Dresden 34
![Page 21: 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](https://reader033.fdocuments.net/reader033/viewer/2022042322/5f0cc5587e708231d4370c6d/html5/thumbnails/21.jpg)
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)
October 11 2011 21 Rogue11 Dresden
f (xy) (xy)u
Wave height distribution for ocean waves
October 11 2011 22 Rogue11 Dresden
Numerical test of N vs γ
October 11 2011 23 Rogue11 Dresden
Probability enhancement over
Rayleigh predictions
urms = 05 ms v=78 ms
October 11 2011 Rogue11 Dresden 24
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
October 11 2011 25 Rogue11 Dresden
October 11 2011 Rogue11 Dresden 26
Extension to nonlinear waves
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
October 11 2011 27 Rogue11 Dresden
October 11 2011 Rogue11 Dresden 28
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
October 11 2011 29 Rogue11 Dresden
N as function of incoming frequency spread for fixed steepness and angular spread
October 11 2011 30 Rogue11 Dresden
N as function of steepness for fixed incoming angular spread and frequency spread
October 11 2011 31 Rogue11 Dresden
Finally combine currents and nonlinearity
October 11 2011 Rogue11 Dresden 32
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
October 11 2011 33 Rogue11 Dresden
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
October 11 2011 Rogue11 Dresden 34
![Page 22: 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](https://reader033.fdocuments.net/reader033/viewer/2022042322/5f0cc5587e708231d4370c6d/html5/thumbnails/22.jpg)
Wave height distribution for ocean waves
October 11 2011 22 Rogue11 Dresden
Numerical test of N vs γ
October 11 2011 23 Rogue11 Dresden
Probability enhancement over
Rayleigh predictions
urms = 05 ms v=78 ms
October 11 2011 Rogue11 Dresden 24
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
October 11 2011 25 Rogue11 Dresden
October 11 2011 Rogue11 Dresden 26
Extension to nonlinear waves
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
October 11 2011 27 Rogue11 Dresden
October 11 2011 Rogue11 Dresden 28
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
October 11 2011 29 Rogue11 Dresden
N as function of incoming frequency spread for fixed steepness and angular spread
October 11 2011 30 Rogue11 Dresden
N as function of steepness for fixed incoming angular spread and frequency spread
October 11 2011 31 Rogue11 Dresden
Finally combine currents and nonlinearity
October 11 2011 Rogue11 Dresden 32
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
October 11 2011 33 Rogue11 Dresden
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
October 11 2011 Rogue11 Dresden 34
![Page 23: 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](https://reader033.fdocuments.net/reader033/viewer/2022042322/5f0cc5587e708231d4370c6d/html5/thumbnails/23.jpg)
Numerical test of N vs γ
October 11 2011 23 Rogue11 Dresden
Probability enhancement over
Rayleigh predictions
urms = 05 ms v=78 ms
October 11 2011 Rogue11 Dresden 24
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
October 11 2011 25 Rogue11 Dresden
October 11 2011 Rogue11 Dresden 26
Extension to nonlinear waves
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
October 11 2011 27 Rogue11 Dresden
October 11 2011 Rogue11 Dresden 28
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
October 11 2011 29 Rogue11 Dresden
N as function of incoming frequency spread for fixed steepness and angular spread
October 11 2011 30 Rogue11 Dresden
N as function of steepness for fixed incoming angular spread and frequency spread
October 11 2011 31 Rogue11 Dresden
Finally combine currents and nonlinearity
October 11 2011 Rogue11 Dresden 32
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
October 11 2011 33 Rogue11 Dresden
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
October 11 2011 Rogue11 Dresden 34
![Page 24: 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](https://reader033.fdocuments.net/reader033/viewer/2022042322/5f0cc5587e708231d4370c6d/html5/thumbnails/24.jpg)
Probability enhancement over
Rayleigh predictions
urms = 05 ms v=78 ms
October 11 2011 Rogue11 Dresden 24
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
October 11 2011 25 Rogue11 Dresden
October 11 2011 Rogue11 Dresden 26
Extension to nonlinear waves
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
October 11 2011 27 Rogue11 Dresden
October 11 2011 Rogue11 Dresden 28
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
October 11 2011 29 Rogue11 Dresden
N as function of incoming frequency spread for fixed steepness and angular spread
October 11 2011 30 Rogue11 Dresden
N as function of steepness for fixed incoming angular spread and frequency spread
October 11 2011 31 Rogue11 Dresden
Finally combine currents and nonlinearity
October 11 2011 Rogue11 Dresden 32
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
October 11 2011 33 Rogue11 Dresden
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
October 11 2011 Rogue11 Dresden 34
![Page 25: 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](https://reader033.fdocuments.net/reader033/viewer/2022042322/5f0cc5587e708231d4370c6d/html5/thumbnails/25.jpg)
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
October 11 2011 25 Rogue11 Dresden
October 11 2011 Rogue11 Dresden 26
Extension to nonlinear waves
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
October 11 2011 27 Rogue11 Dresden
October 11 2011 Rogue11 Dresden 28
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
October 11 2011 29 Rogue11 Dresden
N as function of incoming frequency spread for fixed steepness and angular spread
October 11 2011 30 Rogue11 Dresden
N as function of steepness for fixed incoming angular spread and frequency spread
October 11 2011 31 Rogue11 Dresden
Finally combine currents and nonlinearity
October 11 2011 Rogue11 Dresden 32
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
October 11 2011 33 Rogue11 Dresden
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
October 11 2011 Rogue11 Dresden 34
![Page 26: 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](https://reader033.fdocuments.net/reader033/viewer/2022042322/5f0cc5587e708231d4370c6d/html5/thumbnails/26.jpg)
October 11 2011 Rogue11 Dresden 26
Extension to nonlinear waves
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
October 11 2011 27 Rogue11 Dresden
October 11 2011 Rogue11 Dresden 28
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
October 11 2011 29 Rogue11 Dresden
N as function of incoming frequency spread for fixed steepness and angular spread
October 11 2011 30 Rogue11 Dresden
N as function of steepness for fixed incoming angular spread and frequency spread
October 11 2011 31 Rogue11 Dresden
Finally combine currents and nonlinearity
October 11 2011 Rogue11 Dresden 32
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
October 11 2011 33 Rogue11 Dresden
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
October 11 2011 Rogue11 Dresden 34
![Page 27: 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](https://reader033.fdocuments.net/reader033/viewer/2022042322/5f0cc5587e708231d4370c6d/html5/thumbnails/27.jpg)
Extension to nonlinear waves
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
October 11 2011 27 Rogue11 Dresden
October 11 2011 Rogue11 Dresden 28
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
October 11 2011 29 Rogue11 Dresden
N as function of incoming frequency spread for fixed steepness and angular spread
October 11 2011 30 Rogue11 Dresden
N as function of steepness for fixed incoming angular spread and frequency spread
October 11 2011 31 Rogue11 Dresden
Finally combine currents and nonlinearity
October 11 2011 Rogue11 Dresden 32
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
October 11 2011 33 Rogue11 Dresden
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
October 11 2011 Rogue11 Dresden 34
![Page 28: 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](https://reader033.fdocuments.net/reader033/viewer/2022042322/5f0cc5587e708231d4370c6d/html5/thumbnails/28.jpg)
October 11 2011 Rogue11 Dresden 28
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
October 11 2011 29 Rogue11 Dresden
N as function of incoming frequency spread for fixed steepness and angular spread
October 11 2011 30 Rogue11 Dresden
N as function of steepness for fixed incoming angular spread and frequency spread
October 11 2011 31 Rogue11 Dresden
Finally combine currents and nonlinearity
October 11 2011 Rogue11 Dresden 32
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
October 11 2011 33 Rogue11 Dresden
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
October 11 2011 Rogue11 Dresden 34
![Page 29: 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](https://reader033.fdocuments.net/reader033/viewer/2022042322/5f0cc5587e708231d4370c6d/html5/thumbnails/29.jpg)
N as function of incoming angular spread for fixed steepness and frequency spread
October 11 2011 29 Rogue11 Dresden
N as function of incoming frequency spread for fixed steepness and angular spread
October 11 2011 30 Rogue11 Dresden
N as function of steepness for fixed incoming angular spread and frequency spread
October 11 2011 31 Rogue11 Dresden
Finally combine currents and nonlinearity
October 11 2011 Rogue11 Dresden 32
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
October 11 2011 33 Rogue11 Dresden
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
October 11 2011 Rogue11 Dresden 34
![Page 30: 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](https://reader033.fdocuments.net/reader033/viewer/2022042322/5f0cc5587e708231d4370c6d/html5/thumbnails/30.jpg)
N as function of incoming frequency spread for fixed steepness and angular spread
October 11 2011 30 Rogue11 Dresden
N as function of steepness for fixed incoming angular spread and frequency spread
October 11 2011 31 Rogue11 Dresden
Finally combine currents and nonlinearity
October 11 2011 Rogue11 Dresden 32
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
October 11 2011 33 Rogue11 Dresden
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
October 11 2011 Rogue11 Dresden 34
![Page 31: 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](https://reader033.fdocuments.net/reader033/viewer/2022042322/5f0cc5587e708231d4370c6d/html5/thumbnails/31.jpg)
N as function of steepness for fixed incoming angular spread and frequency spread
October 11 2011 31 Rogue11 Dresden
Finally combine currents and nonlinearity
October 11 2011 Rogue11 Dresden 32
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
October 11 2011 33 Rogue11 Dresden
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
October 11 2011 Rogue11 Dresden 34
![Page 32: 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](https://reader033.fdocuments.net/reader033/viewer/2022042322/5f0cc5587e708231d4370c6d/html5/thumbnails/32.jpg)
Finally combine currents and nonlinearity
October 11 2011 Rogue11 Dresden 32
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
October 11 2011 33 Rogue11 Dresden
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
October 11 2011 Rogue11 Dresden 34
![Page 33: 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](https://reader033.fdocuments.net/reader033/viewer/2022042322/5f0cc5587e708231d4370c6d/html5/thumbnails/33.jpg)
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
October 11 2011 33 Rogue11 Dresden
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
October 11 2011 Rogue11 Dresden 34
![Page 34: 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](https://reader033.fdocuments.net/reader033/viewer/2022042322/5f0cc5587e708231d4370c6d/html5/thumbnails/34.jpg)
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
October 11 2011 Rogue11 Dresden 34