linear and nonlinear rogue wave statistics in the …lkaplan/egu.pdf · questions: a) is...
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Linear and Nonlinear Rogue Wave
Statistics in the Presence of Random
Currents
Lev Kaplan and Linghang Ying
(Tulane University)
In collaboration with
Alex Dahlen and Eric Heller (Harvard)
Zhouheng Zhuang (Tulane)
April 27 2012 1 EGU 2012 Extreme Sea Waves
Questions
A) Is nonlinearity necessary for extreme waves
B) Can wave statistics associated with linear and
nonlinear formation mechanisms be considered in
unified framework
C) Can we try to make quantitative predictions of
extreme wave formation probability given sea
parameters
April 27 2012 2 EGU 2012 Extreme Sea Waves
Questions
A) Is nonlinearity necessary for extreme waves No
B) Can wave statistics associated with linear and
nonlinear formation mechanisms be considered in
unified framework Yes
C) Can we try to make quantitative predictions of
extreme wave formation probability given sea
parameters Yes
April 27 2012 3 EGU 2012 Extreme Sea Waves
Talk Outline
1 Introduction Longuet-Higgins random waves
2 Linear rogue wave formation
a) Mechanism Refraction from current eddies
b) The ldquofreak indexrdquo
c) Implications for extreme wave statistics
3 Nonlinear rogue wave formation
a) Wave statistics in 4th order nonlinear equation
b) Scaling with parameters describing sea state
4 Can linear and nonlinear effects work in concert
5 Conclusions
April 27 2012 4 EGU 2012 Extreme Sea Waves
If surface elevation is sum of plane waves with
random directions amplitudes and phases
Surface elevation distribution is Gaussian with
standard deviation 120590
Given narrow-banded spectrum wave height
distribution is Rayleigh
Cumulative wave height probability is
119875119877119886119910119897119890119894119892ℎ ℎ = 119890minusℎ281205902
1 Longuet-Higgins random waves
April 27 2012 5 EGU 2012 Extreme Sea Waves
Predicts low probability of extreme waves
119875119877119886119910119897119890119894119892ℎ 22 119878119882119867 = 119875119877119886119910119897119890119894119892ℎ 88 120590 = 63 ∙ 10minus5
119875119877119886119910119897119890119894119892ℎ 3 119878119882119867 = 119875119877119886119910119897119890119894119892ℎ 12 120590 = 15 ∙ 10minus8
1 Longuet-Higgins random waves
April 27 2012 6 EGU 2012 Extreme Sea Waves
Basic idea Waves refracted by
currents (or non-uniform bottomhellip)
Refraction of plane wave leads to
focusing (too many rogue waves)
2 Linear rogue wave formation
April 27 2012 7 EGU 2012 Extreme Sea Waves
Assume incoming random sum of plane waves
mean wavenumber k0 speed v = (g 4 k0)12
wave number spread Δk ltlt k0
angular spread Δθ ltlt 2π
Refracted by random current field
Gaussian distributed Gaussian correlations
current speed urms ltlt v
correlated on scale (typical eddy size) gtgt 1 k0
2 Linear rogue wave formation
April 27 2012 8 EGU 2012 Extreme Sea Waves
Typical ray dynamics calculation
= 10ordm = 20ordm
April 27 2012 9 EGU 2012 Extreme Sea Waves
urms= 05 ms v = 78 ms
Random scattering yields fluctuations in local energy
density even though incoming waves are spatially uniform
ldquohot spotsrdquo (focusing) and ldquocold spotsrdquo (defocusing)
Fluctuations
Grow with increased current strength urms v
Smeared out by increasing angular spread
Define freak index = (urms v)23
The ldquofreak indexrdquo
April 27 2012 10 EGU 2012 Extreme Sea Waves
= (urms v)23
Most dangerous 1
(long-crested sea impinging on strong random current field)
More realistic le 1
( eg urms= 05 ms v = 78 ms = 10ordm rarr = 092)
The ldquofreak indexrdquo
April 27 2012 11 EGU 2012 Extreme Sea Waves
Typical ray dynamics calculation
= 10ordm = 20ordm
April 27 2012 12 EGU 2012 Extreme Sea Waves
urms= 05 ms v = 78 ms
Calculating Probabilities
April 27 2012 13 EGU 2012 Extreme
Sea Waves
119891 119868 = 1205942119873 119868 =119873
2
1198732 119868(119873minus2)2
Γ(1198732)119890minus1198731198682
For ltlt 1 local ray density (local energy density)
follows χ2 distribution with N degrees of freedom
Where 119868 = 1 and 119868 minus 1 2~1
119873asymp
1205742
45≪ 1
Implications for wave statistics
Now assuming the wave height distribution is locally
Rayleigh the total distribution integrating over hot and
cold spots is given by convolution
K-distribution
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
April 27 2012 14 EGU 2012 Extreme
Sea Waves
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)
April 27 2012 15 EGU 2012 Extreme Sea Waves
f (xy) (xy)u
Wave height distribution for γ=315
April 27 2012 16 EGU 2012 Extreme Sea Waves
Numerical test of N vs γ
April 27 2012 17 EGU 2012 Extreme Sea Waves
Probability enhancement over
Rayleigh predictions
urms = 05 ms v=78 ms
April 27 2012 EGU 2012 Extreme Sea Waves 18
4 Nonlinear extreme 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
April 27 2012 19 EGU 2012 Extreme Sea Waves
April 27 2012 EGU 2012 Extreme Sea Waves 20
April 27 2012 EGU 2012 Extreme Sea Waves 21
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
April 27 2012 22 EGU 2012 Extreme Sea Waves
N as function of incoming frequency spread for fixed steepness and angular spread
April 27 2012 23 EGU 2012 Extreme Sea Waves
N as function of steepness for fixed incoming angular spread and frequency spread
April 27 2012 24 EGU 2012 Extreme Sea Waves
Bringing together these results
Dependence of N parameter on steepness angular spread and frequency spread
119873 asymp 02 120576minus3 Δ119896
1198960 Δθ
April 27 2012 25 EGU 2012 Extreme Sea Waves
4 Finally combine currents and nonlinearity
April 27 2012 EGU 2012 Extreme Sea Waves 26
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
April 27 2012 27 EGU 2012 Extreme Sea Waves
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
April 27 2012 EGU 2012 Extreme Sea Waves 28
Questions
A) Is nonlinearity necessary for extreme waves
B) Can wave statistics associated with linear and
nonlinear formation mechanisms be considered in
unified framework
C) Can we try to make quantitative predictions of
extreme wave formation probability given sea
parameters
April 27 2012 2 EGU 2012 Extreme Sea Waves
Questions
A) Is nonlinearity necessary for extreme waves No
B) Can wave statistics associated with linear and
nonlinear formation mechanisms be considered in
unified framework Yes
C) Can we try to make quantitative predictions of
extreme wave formation probability given sea
parameters Yes
April 27 2012 3 EGU 2012 Extreme Sea Waves
Talk Outline
1 Introduction Longuet-Higgins random waves
2 Linear rogue wave formation
a) Mechanism Refraction from current eddies
b) The ldquofreak indexrdquo
c) Implications for extreme wave statistics
3 Nonlinear rogue wave formation
a) Wave statistics in 4th order nonlinear equation
b) Scaling with parameters describing sea state
4 Can linear and nonlinear effects work in concert
5 Conclusions
April 27 2012 4 EGU 2012 Extreme Sea Waves
If surface elevation is sum of plane waves with
random directions amplitudes and phases
Surface elevation distribution is Gaussian with
standard deviation 120590
Given narrow-banded spectrum wave height
distribution is Rayleigh
Cumulative wave height probability is
119875119877119886119910119897119890119894119892ℎ ℎ = 119890minusℎ281205902
1 Longuet-Higgins random waves
April 27 2012 5 EGU 2012 Extreme Sea Waves
Predicts low probability of extreme waves
119875119877119886119910119897119890119894119892ℎ 22 119878119882119867 = 119875119877119886119910119897119890119894119892ℎ 88 120590 = 63 ∙ 10minus5
119875119877119886119910119897119890119894119892ℎ 3 119878119882119867 = 119875119877119886119910119897119890119894119892ℎ 12 120590 = 15 ∙ 10minus8
1 Longuet-Higgins random waves
April 27 2012 6 EGU 2012 Extreme Sea Waves
Basic idea Waves refracted by
currents (or non-uniform bottomhellip)
Refraction of plane wave leads to
focusing (too many rogue waves)
2 Linear rogue wave formation
April 27 2012 7 EGU 2012 Extreme Sea Waves
Assume incoming random sum of plane waves
mean wavenumber k0 speed v = (g 4 k0)12
wave number spread Δk ltlt k0
angular spread Δθ ltlt 2π
Refracted by random current field
Gaussian distributed Gaussian correlations
current speed urms ltlt v
correlated on scale (typical eddy size) gtgt 1 k0
2 Linear rogue wave formation
April 27 2012 8 EGU 2012 Extreme Sea Waves
Typical ray dynamics calculation
= 10ordm = 20ordm
April 27 2012 9 EGU 2012 Extreme Sea Waves
urms= 05 ms v = 78 ms
Random scattering yields fluctuations in local energy
density even though incoming waves are spatially uniform
ldquohot spotsrdquo (focusing) and ldquocold spotsrdquo (defocusing)
Fluctuations
Grow with increased current strength urms v
Smeared out by increasing angular spread
Define freak index = (urms v)23
The ldquofreak indexrdquo
April 27 2012 10 EGU 2012 Extreme Sea Waves
= (urms v)23
Most dangerous 1
(long-crested sea impinging on strong random current field)
More realistic le 1
( eg urms= 05 ms v = 78 ms = 10ordm rarr = 092)
The ldquofreak indexrdquo
April 27 2012 11 EGU 2012 Extreme Sea Waves
Typical ray dynamics calculation
= 10ordm = 20ordm
April 27 2012 12 EGU 2012 Extreme Sea Waves
urms= 05 ms v = 78 ms
Calculating Probabilities
April 27 2012 13 EGU 2012 Extreme
Sea Waves
119891 119868 = 1205942119873 119868 =119873
2
1198732 119868(119873minus2)2
Γ(1198732)119890minus1198731198682
For ltlt 1 local ray density (local energy density)
follows χ2 distribution with N degrees of freedom
Where 119868 = 1 and 119868 minus 1 2~1
119873asymp
1205742
45≪ 1
Implications for wave statistics
Now assuming the wave height distribution is locally
Rayleigh the total distribution integrating over hot and
cold spots is given by convolution
K-distribution
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
April 27 2012 14 EGU 2012 Extreme
Sea Waves
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)
April 27 2012 15 EGU 2012 Extreme Sea Waves
f (xy) (xy)u
Wave height distribution for γ=315
April 27 2012 16 EGU 2012 Extreme Sea Waves
Numerical test of N vs γ
April 27 2012 17 EGU 2012 Extreme Sea Waves
Probability enhancement over
Rayleigh predictions
urms = 05 ms v=78 ms
April 27 2012 EGU 2012 Extreme Sea Waves 18
4 Nonlinear extreme 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
April 27 2012 19 EGU 2012 Extreme Sea Waves
April 27 2012 EGU 2012 Extreme Sea Waves 20
April 27 2012 EGU 2012 Extreme Sea Waves 21
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
April 27 2012 22 EGU 2012 Extreme Sea Waves
N as function of incoming frequency spread for fixed steepness and angular spread
April 27 2012 23 EGU 2012 Extreme Sea Waves
N as function of steepness for fixed incoming angular spread and frequency spread
April 27 2012 24 EGU 2012 Extreme Sea Waves
Bringing together these results
Dependence of N parameter on steepness angular spread and frequency spread
119873 asymp 02 120576minus3 Δ119896
1198960 Δθ
April 27 2012 25 EGU 2012 Extreme Sea Waves
4 Finally combine currents and nonlinearity
April 27 2012 EGU 2012 Extreme Sea Waves 26
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
April 27 2012 27 EGU 2012 Extreme Sea Waves
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
April 27 2012 EGU 2012 Extreme Sea Waves 28
Questions
A) Is nonlinearity necessary for extreme waves No
B) Can wave statistics associated with linear and
nonlinear formation mechanisms be considered in
unified framework Yes
C) Can we try to make quantitative predictions of
extreme wave formation probability given sea
parameters Yes
April 27 2012 3 EGU 2012 Extreme Sea Waves
Talk Outline
1 Introduction Longuet-Higgins random waves
2 Linear rogue wave formation
a) Mechanism Refraction from current eddies
b) The ldquofreak indexrdquo
c) Implications for extreme wave statistics
3 Nonlinear rogue wave formation
a) Wave statistics in 4th order nonlinear equation
b) Scaling with parameters describing sea state
4 Can linear and nonlinear effects work in concert
5 Conclusions
April 27 2012 4 EGU 2012 Extreme Sea Waves
If surface elevation is sum of plane waves with
random directions amplitudes and phases
Surface elevation distribution is Gaussian with
standard deviation 120590
Given narrow-banded spectrum wave height
distribution is Rayleigh
Cumulative wave height probability is
119875119877119886119910119897119890119894119892ℎ ℎ = 119890minusℎ281205902
1 Longuet-Higgins random waves
April 27 2012 5 EGU 2012 Extreme Sea Waves
Predicts low probability of extreme waves
119875119877119886119910119897119890119894119892ℎ 22 119878119882119867 = 119875119877119886119910119897119890119894119892ℎ 88 120590 = 63 ∙ 10minus5
119875119877119886119910119897119890119894119892ℎ 3 119878119882119867 = 119875119877119886119910119897119890119894119892ℎ 12 120590 = 15 ∙ 10minus8
1 Longuet-Higgins random waves
April 27 2012 6 EGU 2012 Extreme Sea Waves
Basic idea Waves refracted by
currents (or non-uniform bottomhellip)
Refraction of plane wave leads to
focusing (too many rogue waves)
2 Linear rogue wave formation
April 27 2012 7 EGU 2012 Extreme Sea Waves
Assume incoming random sum of plane waves
mean wavenumber k0 speed v = (g 4 k0)12
wave number spread Δk ltlt k0
angular spread Δθ ltlt 2π
Refracted by random current field
Gaussian distributed Gaussian correlations
current speed urms ltlt v
correlated on scale (typical eddy size) gtgt 1 k0
2 Linear rogue wave formation
April 27 2012 8 EGU 2012 Extreme Sea Waves
Typical ray dynamics calculation
= 10ordm = 20ordm
April 27 2012 9 EGU 2012 Extreme Sea Waves
urms= 05 ms v = 78 ms
Random scattering yields fluctuations in local energy
density even though incoming waves are spatially uniform
ldquohot spotsrdquo (focusing) and ldquocold spotsrdquo (defocusing)
Fluctuations
Grow with increased current strength urms v
Smeared out by increasing angular spread
Define freak index = (urms v)23
The ldquofreak indexrdquo
April 27 2012 10 EGU 2012 Extreme Sea Waves
= (urms v)23
Most dangerous 1
(long-crested sea impinging on strong random current field)
More realistic le 1
( eg urms= 05 ms v = 78 ms = 10ordm rarr = 092)
The ldquofreak indexrdquo
April 27 2012 11 EGU 2012 Extreme Sea Waves
Typical ray dynamics calculation
= 10ordm = 20ordm
April 27 2012 12 EGU 2012 Extreme Sea Waves
urms= 05 ms v = 78 ms
Calculating Probabilities
April 27 2012 13 EGU 2012 Extreme
Sea Waves
119891 119868 = 1205942119873 119868 =119873
2
1198732 119868(119873minus2)2
Γ(1198732)119890minus1198731198682
For ltlt 1 local ray density (local energy density)
follows χ2 distribution with N degrees of freedom
Where 119868 = 1 and 119868 minus 1 2~1
119873asymp
1205742
45≪ 1
Implications for wave statistics
Now assuming the wave height distribution is locally
Rayleigh the total distribution integrating over hot and
cold spots is given by convolution
K-distribution
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
April 27 2012 14 EGU 2012 Extreme
Sea Waves
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)
April 27 2012 15 EGU 2012 Extreme Sea Waves
f (xy) (xy)u
Wave height distribution for γ=315
April 27 2012 16 EGU 2012 Extreme Sea Waves
Numerical test of N vs γ
April 27 2012 17 EGU 2012 Extreme Sea Waves
Probability enhancement over
Rayleigh predictions
urms = 05 ms v=78 ms
April 27 2012 EGU 2012 Extreme Sea Waves 18
4 Nonlinear extreme 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
April 27 2012 19 EGU 2012 Extreme Sea Waves
April 27 2012 EGU 2012 Extreme Sea Waves 20
April 27 2012 EGU 2012 Extreme Sea Waves 21
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
April 27 2012 22 EGU 2012 Extreme Sea Waves
N as function of incoming frequency spread for fixed steepness and angular spread
April 27 2012 23 EGU 2012 Extreme Sea Waves
N as function of steepness for fixed incoming angular spread and frequency spread
April 27 2012 24 EGU 2012 Extreme Sea Waves
Bringing together these results
Dependence of N parameter on steepness angular spread and frequency spread
119873 asymp 02 120576minus3 Δ119896
1198960 Δθ
April 27 2012 25 EGU 2012 Extreme Sea Waves
4 Finally combine currents and nonlinearity
April 27 2012 EGU 2012 Extreme Sea Waves 26
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
April 27 2012 27 EGU 2012 Extreme Sea Waves
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
April 27 2012 EGU 2012 Extreme Sea Waves 28
Talk Outline
1 Introduction Longuet-Higgins random waves
2 Linear rogue wave formation
a) Mechanism Refraction from current eddies
b) The ldquofreak indexrdquo
c) Implications for extreme wave statistics
3 Nonlinear rogue wave formation
a) Wave statistics in 4th order nonlinear equation
b) Scaling with parameters describing sea state
4 Can linear and nonlinear effects work in concert
5 Conclusions
April 27 2012 4 EGU 2012 Extreme Sea Waves
If surface elevation is sum of plane waves with
random directions amplitudes and phases
Surface elevation distribution is Gaussian with
standard deviation 120590
Given narrow-banded spectrum wave height
distribution is Rayleigh
Cumulative wave height probability is
119875119877119886119910119897119890119894119892ℎ ℎ = 119890minusℎ281205902
1 Longuet-Higgins random waves
April 27 2012 5 EGU 2012 Extreme Sea Waves
Predicts low probability of extreme waves
119875119877119886119910119897119890119894119892ℎ 22 119878119882119867 = 119875119877119886119910119897119890119894119892ℎ 88 120590 = 63 ∙ 10minus5
119875119877119886119910119897119890119894119892ℎ 3 119878119882119867 = 119875119877119886119910119897119890119894119892ℎ 12 120590 = 15 ∙ 10minus8
1 Longuet-Higgins random waves
April 27 2012 6 EGU 2012 Extreme Sea Waves
Basic idea Waves refracted by
currents (or non-uniform bottomhellip)
Refraction of plane wave leads to
focusing (too many rogue waves)
2 Linear rogue wave formation
April 27 2012 7 EGU 2012 Extreme Sea Waves
Assume incoming random sum of plane waves
mean wavenumber k0 speed v = (g 4 k0)12
wave number spread Δk ltlt k0
angular spread Δθ ltlt 2π
Refracted by random current field
Gaussian distributed Gaussian correlations
current speed urms ltlt v
correlated on scale (typical eddy size) gtgt 1 k0
2 Linear rogue wave formation
April 27 2012 8 EGU 2012 Extreme Sea Waves
Typical ray dynamics calculation
= 10ordm = 20ordm
April 27 2012 9 EGU 2012 Extreme Sea Waves
urms= 05 ms v = 78 ms
Random scattering yields fluctuations in local energy
density even though incoming waves are spatially uniform
ldquohot spotsrdquo (focusing) and ldquocold spotsrdquo (defocusing)
Fluctuations
Grow with increased current strength urms v
Smeared out by increasing angular spread
Define freak index = (urms v)23
The ldquofreak indexrdquo
April 27 2012 10 EGU 2012 Extreme Sea Waves
= (urms v)23
Most dangerous 1
(long-crested sea impinging on strong random current field)
More realistic le 1
( eg urms= 05 ms v = 78 ms = 10ordm rarr = 092)
The ldquofreak indexrdquo
April 27 2012 11 EGU 2012 Extreme Sea Waves
Typical ray dynamics calculation
= 10ordm = 20ordm
April 27 2012 12 EGU 2012 Extreme Sea Waves
urms= 05 ms v = 78 ms
Calculating Probabilities
April 27 2012 13 EGU 2012 Extreme
Sea Waves
119891 119868 = 1205942119873 119868 =119873
2
1198732 119868(119873minus2)2
Γ(1198732)119890minus1198731198682
For ltlt 1 local ray density (local energy density)
follows χ2 distribution with N degrees of freedom
Where 119868 = 1 and 119868 minus 1 2~1
119873asymp
1205742
45≪ 1
Implications for wave statistics
Now assuming the wave height distribution is locally
Rayleigh the total distribution integrating over hot and
cold spots is given by convolution
K-distribution
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
April 27 2012 14 EGU 2012 Extreme
Sea Waves
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)
April 27 2012 15 EGU 2012 Extreme Sea Waves
f (xy) (xy)u
Wave height distribution for γ=315
April 27 2012 16 EGU 2012 Extreme Sea Waves
Numerical test of N vs γ
April 27 2012 17 EGU 2012 Extreme Sea Waves
Probability enhancement over
Rayleigh predictions
urms = 05 ms v=78 ms
April 27 2012 EGU 2012 Extreme Sea Waves 18
4 Nonlinear extreme 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
April 27 2012 19 EGU 2012 Extreme Sea Waves
April 27 2012 EGU 2012 Extreme Sea Waves 20
April 27 2012 EGU 2012 Extreme Sea Waves 21
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
April 27 2012 22 EGU 2012 Extreme Sea Waves
N as function of incoming frequency spread for fixed steepness and angular spread
April 27 2012 23 EGU 2012 Extreme Sea Waves
N as function of steepness for fixed incoming angular spread and frequency spread
April 27 2012 24 EGU 2012 Extreme Sea Waves
Bringing together these results
Dependence of N parameter on steepness angular spread and frequency spread
119873 asymp 02 120576minus3 Δ119896
1198960 Δθ
April 27 2012 25 EGU 2012 Extreme Sea Waves
4 Finally combine currents and nonlinearity
April 27 2012 EGU 2012 Extreme Sea Waves 26
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
April 27 2012 27 EGU 2012 Extreme Sea Waves
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
April 27 2012 EGU 2012 Extreme Sea Waves 28
If surface elevation is sum of plane waves with
random directions amplitudes and phases
Surface elevation distribution is Gaussian with
standard deviation 120590
Given narrow-banded spectrum wave height
distribution is Rayleigh
Cumulative wave height probability is
119875119877119886119910119897119890119894119892ℎ ℎ = 119890minusℎ281205902
1 Longuet-Higgins random waves
April 27 2012 5 EGU 2012 Extreme Sea Waves
Predicts low probability of extreme waves
119875119877119886119910119897119890119894119892ℎ 22 119878119882119867 = 119875119877119886119910119897119890119894119892ℎ 88 120590 = 63 ∙ 10minus5
119875119877119886119910119897119890119894119892ℎ 3 119878119882119867 = 119875119877119886119910119897119890119894119892ℎ 12 120590 = 15 ∙ 10minus8
1 Longuet-Higgins random waves
April 27 2012 6 EGU 2012 Extreme Sea Waves
Basic idea Waves refracted by
currents (or non-uniform bottomhellip)
Refraction of plane wave leads to
focusing (too many rogue waves)
2 Linear rogue wave formation
April 27 2012 7 EGU 2012 Extreme Sea Waves
Assume incoming random sum of plane waves
mean wavenumber k0 speed v = (g 4 k0)12
wave number spread Δk ltlt k0
angular spread Δθ ltlt 2π
Refracted by random current field
Gaussian distributed Gaussian correlations
current speed urms ltlt v
correlated on scale (typical eddy size) gtgt 1 k0
2 Linear rogue wave formation
April 27 2012 8 EGU 2012 Extreme Sea Waves
Typical ray dynamics calculation
= 10ordm = 20ordm
April 27 2012 9 EGU 2012 Extreme Sea Waves
urms= 05 ms v = 78 ms
Random scattering yields fluctuations in local energy
density even though incoming waves are spatially uniform
ldquohot spotsrdquo (focusing) and ldquocold spotsrdquo (defocusing)
Fluctuations
Grow with increased current strength urms v
Smeared out by increasing angular spread
Define freak index = (urms v)23
The ldquofreak indexrdquo
April 27 2012 10 EGU 2012 Extreme Sea Waves
= (urms v)23
Most dangerous 1
(long-crested sea impinging on strong random current field)
More realistic le 1
( eg urms= 05 ms v = 78 ms = 10ordm rarr = 092)
The ldquofreak indexrdquo
April 27 2012 11 EGU 2012 Extreme Sea Waves
Typical ray dynamics calculation
= 10ordm = 20ordm
April 27 2012 12 EGU 2012 Extreme Sea Waves
urms= 05 ms v = 78 ms
Calculating Probabilities
April 27 2012 13 EGU 2012 Extreme
Sea Waves
119891 119868 = 1205942119873 119868 =119873
2
1198732 119868(119873minus2)2
Γ(1198732)119890minus1198731198682
For ltlt 1 local ray density (local energy density)
follows χ2 distribution with N degrees of freedom
Where 119868 = 1 and 119868 minus 1 2~1
119873asymp
1205742
45≪ 1
Implications for wave statistics
Now assuming the wave height distribution is locally
Rayleigh the total distribution integrating over hot and
cold spots is given by convolution
K-distribution
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
April 27 2012 14 EGU 2012 Extreme
Sea Waves
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)
April 27 2012 15 EGU 2012 Extreme Sea Waves
f (xy) (xy)u
Wave height distribution for γ=315
April 27 2012 16 EGU 2012 Extreme Sea Waves
Numerical test of N vs γ
April 27 2012 17 EGU 2012 Extreme Sea Waves
Probability enhancement over
Rayleigh predictions
urms = 05 ms v=78 ms
April 27 2012 EGU 2012 Extreme Sea Waves 18
4 Nonlinear extreme 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
April 27 2012 19 EGU 2012 Extreme Sea Waves
April 27 2012 EGU 2012 Extreme Sea Waves 20
April 27 2012 EGU 2012 Extreme Sea Waves 21
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
April 27 2012 22 EGU 2012 Extreme Sea Waves
N as function of incoming frequency spread for fixed steepness and angular spread
April 27 2012 23 EGU 2012 Extreme Sea Waves
N as function of steepness for fixed incoming angular spread and frequency spread
April 27 2012 24 EGU 2012 Extreme Sea Waves
Bringing together these results
Dependence of N parameter on steepness angular spread and frequency spread
119873 asymp 02 120576minus3 Δ119896
1198960 Δθ
April 27 2012 25 EGU 2012 Extreme Sea Waves
4 Finally combine currents and nonlinearity
April 27 2012 EGU 2012 Extreme Sea Waves 26
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
April 27 2012 27 EGU 2012 Extreme Sea Waves
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
April 27 2012 EGU 2012 Extreme Sea Waves 28
Predicts low probability of extreme waves
119875119877119886119910119897119890119894119892ℎ 22 119878119882119867 = 119875119877119886119910119897119890119894119892ℎ 88 120590 = 63 ∙ 10minus5
119875119877119886119910119897119890119894119892ℎ 3 119878119882119867 = 119875119877119886119910119897119890119894119892ℎ 12 120590 = 15 ∙ 10minus8
1 Longuet-Higgins random waves
April 27 2012 6 EGU 2012 Extreme Sea Waves
Basic idea Waves refracted by
currents (or non-uniform bottomhellip)
Refraction of plane wave leads to
focusing (too many rogue waves)
2 Linear rogue wave formation
April 27 2012 7 EGU 2012 Extreme Sea Waves
Assume incoming random sum of plane waves
mean wavenumber k0 speed v = (g 4 k0)12
wave number spread Δk ltlt k0
angular spread Δθ ltlt 2π
Refracted by random current field
Gaussian distributed Gaussian correlations
current speed urms ltlt v
correlated on scale (typical eddy size) gtgt 1 k0
2 Linear rogue wave formation
April 27 2012 8 EGU 2012 Extreme Sea Waves
Typical ray dynamics calculation
= 10ordm = 20ordm
April 27 2012 9 EGU 2012 Extreme Sea Waves
urms= 05 ms v = 78 ms
Random scattering yields fluctuations in local energy
density even though incoming waves are spatially uniform
ldquohot spotsrdquo (focusing) and ldquocold spotsrdquo (defocusing)
Fluctuations
Grow with increased current strength urms v
Smeared out by increasing angular spread
Define freak index = (urms v)23
The ldquofreak indexrdquo
April 27 2012 10 EGU 2012 Extreme Sea Waves
= (urms v)23
Most dangerous 1
(long-crested sea impinging on strong random current field)
More realistic le 1
( eg urms= 05 ms v = 78 ms = 10ordm rarr = 092)
The ldquofreak indexrdquo
April 27 2012 11 EGU 2012 Extreme Sea Waves
Typical ray dynamics calculation
= 10ordm = 20ordm
April 27 2012 12 EGU 2012 Extreme Sea Waves
urms= 05 ms v = 78 ms
Calculating Probabilities
April 27 2012 13 EGU 2012 Extreme
Sea Waves
119891 119868 = 1205942119873 119868 =119873
2
1198732 119868(119873minus2)2
Γ(1198732)119890minus1198731198682
For ltlt 1 local ray density (local energy density)
follows χ2 distribution with N degrees of freedom
Where 119868 = 1 and 119868 minus 1 2~1
119873asymp
1205742
45≪ 1
Implications for wave statistics
Now assuming the wave height distribution is locally
Rayleigh the total distribution integrating over hot and
cold spots is given by convolution
K-distribution
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
April 27 2012 14 EGU 2012 Extreme
Sea Waves
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)
April 27 2012 15 EGU 2012 Extreme Sea Waves
f (xy) (xy)u
Wave height distribution for γ=315
April 27 2012 16 EGU 2012 Extreme Sea Waves
Numerical test of N vs γ
April 27 2012 17 EGU 2012 Extreme Sea Waves
Probability enhancement over
Rayleigh predictions
urms = 05 ms v=78 ms
April 27 2012 EGU 2012 Extreme Sea Waves 18
4 Nonlinear extreme 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
April 27 2012 19 EGU 2012 Extreme Sea Waves
April 27 2012 EGU 2012 Extreme Sea Waves 20
April 27 2012 EGU 2012 Extreme Sea Waves 21
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
April 27 2012 22 EGU 2012 Extreme Sea Waves
N as function of incoming frequency spread for fixed steepness and angular spread
April 27 2012 23 EGU 2012 Extreme Sea Waves
N as function of steepness for fixed incoming angular spread and frequency spread
April 27 2012 24 EGU 2012 Extreme Sea Waves
Bringing together these results
Dependence of N parameter on steepness angular spread and frequency spread
119873 asymp 02 120576minus3 Δ119896
1198960 Δθ
April 27 2012 25 EGU 2012 Extreme Sea Waves
4 Finally combine currents and nonlinearity
April 27 2012 EGU 2012 Extreme Sea Waves 26
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
April 27 2012 27 EGU 2012 Extreme Sea Waves
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
April 27 2012 EGU 2012 Extreme Sea Waves 28
Basic idea Waves refracted by
currents (or non-uniform bottomhellip)
Refraction of plane wave leads to
focusing (too many rogue waves)
2 Linear rogue wave formation
April 27 2012 7 EGU 2012 Extreme Sea Waves
Assume incoming random sum of plane waves
mean wavenumber k0 speed v = (g 4 k0)12
wave number spread Δk ltlt k0
angular spread Δθ ltlt 2π
Refracted by random current field
Gaussian distributed Gaussian correlations
current speed urms ltlt v
correlated on scale (typical eddy size) gtgt 1 k0
2 Linear rogue wave formation
April 27 2012 8 EGU 2012 Extreme Sea Waves
Typical ray dynamics calculation
= 10ordm = 20ordm
April 27 2012 9 EGU 2012 Extreme Sea Waves
urms= 05 ms v = 78 ms
Random scattering yields fluctuations in local energy
density even though incoming waves are spatially uniform
ldquohot spotsrdquo (focusing) and ldquocold spotsrdquo (defocusing)
Fluctuations
Grow with increased current strength urms v
Smeared out by increasing angular spread
Define freak index = (urms v)23
The ldquofreak indexrdquo
April 27 2012 10 EGU 2012 Extreme Sea Waves
= (urms v)23
Most dangerous 1
(long-crested sea impinging on strong random current field)
More realistic le 1
( eg urms= 05 ms v = 78 ms = 10ordm rarr = 092)
The ldquofreak indexrdquo
April 27 2012 11 EGU 2012 Extreme Sea Waves
Typical ray dynamics calculation
= 10ordm = 20ordm
April 27 2012 12 EGU 2012 Extreme Sea Waves
urms= 05 ms v = 78 ms
Calculating Probabilities
April 27 2012 13 EGU 2012 Extreme
Sea Waves
119891 119868 = 1205942119873 119868 =119873
2
1198732 119868(119873minus2)2
Γ(1198732)119890minus1198731198682
For ltlt 1 local ray density (local energy density)
follows χ2 distribution with N degrees of freedom
Where 119868 = 1 and 119868 minus 1 2~1
119873asymp
1205742
45≪ 1
Implications for wave statistics
Now assuming the wave height distribution is locally
Rayleigh the total distribution integrating over hot and
cold spots is given by convolution
K-distribution
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
April 27 2012 14 EGU 2012 Extreme
Sea Waves
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)
April 27 2012 15 EGU 2012 Extreme Sea Waves
f (xy) (xy)u
Wave height distribution for γ=315
April 27 2012 16 EGU 2012 Extreme Sea Waves
Numerical test of N vs γ
April 27 2012 17 EGU 2012 Extreme Sea Waves
Probability enhancement over
Rayleigh predictions
urms = 05 ms v=78 ms
April 27 2012 EGU 2012 Extreme Sea Waves 18
4 Nonlinear extreme 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
April 27 2012 19 EGU 2012 Extreme Sea Waves
April 27 2012 EGU 2012 Extreme Sea Waves 20
April 27 2012 EGU 2012 Extreme Sea Waves 21
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
April 27 2012 22 EGU 2012 Extreme Sea Waves
N as function of incoming frequency spread for fixed steepness and angular spread
April 27 2012 23 EGU 2012 Extreme Sea Waves
N as function of steepness for fixed incoming angular spread and frequency spread
April 27 2012 24 EGU 2012 Extreme Sea Waves
Bringing together these results
Dependence of N parameter on steepness angular spread and frequency spread
119873 asymp 02 120576minus3 Δ119896
1198960 Δθ
April 27 2012 25 EGU 2012 Extreme Sea Waves
4 Finally combine currents and nonlinearity
April 27 2012 EGU 2012 Extreme Sea Waves 26
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
April 27 2012 27 EGU 2012 Extreme Sea Waves
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
April 27 2012 EGU 2012 Extreme Sea Waves 28
Assume incoming random sum of plane waves
mean wavenumber k0 speed v = (g 4 k0)12
wave number spread Δk ltlt k0
angular spread Δθ ltlt 2π
Refracted by random current field
Gaussian distributed Gaussian correlations
current speed urms ltlt v
correlated on scale (typical eddy size) gtgt 1 k0
2 Linear rogue wave formation
April 27 2012 8 EGU 2012 Extreme Sea Waves
Typical ray dynamics calculation
= 10ordm = 20ordm
April 27 2012 9 EGU 2012 Extreme Sea Waves
urms= 05 ms v = 78 ms
Random scattering yields fluctuations in local energy
density even though incoming waves are spatially uniform
ldquohot spotsrdquo (focusing) and ldquocold spotsrdquo (defocusing)
Fluctuations
Grow with increased current strength urms v
Smeared out by increasing angular spread
Define freak index = (urms v)23
The ldquofreak indexrdquo
April 27 2012 10 EGU 2012 Extreme Sea Waves
= (urms v)23
Most dangerous 1
(long-crested sea impinging on strong random current field)
More realistic le 1
( eg urms= 05 ms v = 78 ms = 10ordm rarr = 092)
The ldquofreak indexrdquo
April 27 2012 11 EGU 2012 Extreme Sea Waves
Typical ray dynamics calculation
= 10ordm = 20ordm
April 27 2012 12 EGU 2012 Extreme Sea Waves
urms= 05 ms v = 78 ms
Calculating Probabilities
April 27 2012 13 EGU 2012 Extreme
Sea Waves
119891 119868 = 1205942119873 119868 =119873
2
1198732 119868(119873minus2)2
Γ(1198732)119890minus1198731198682
For ltlt 1 local ray density (local energy density)
follows χ2 distribution with N degrees of freedom
Where 119868 = 1 and 119868 minus 1 2~1
119873asymp
1205742
45≪ 1
Implications for wave statistics
Now assuming the wave height distribution is locally
Rayleigh the total distribution integrating over hot and
cold spots is given by convolution
K-distribution
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
April 27 2012 14 EGU 2012 Extreme
Sea Waves
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)
April 27 2012 15 EGU 2012 Extreme Sea Waves
f (xy) (xy)u
Wave height distribution for γ=315
April 27 2012 16 EGU 2012 Extreme Sea Waves
Numerical test of N vs γ
April 27 2012 17 EGU 2012 Extreme Sea Waves
Probability enhancement over
Rayleigh predictions
urms = 05 ms v=78 ms
April 27 2012 EGU 2012 Extreme Sea Waves 18
4 Nonlinear extreme 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
April 27 2012 19 EGU 2012 Extreme Sea Waves
April 27 2012 EGU 2012 Extreme Sea Waves 20
April 27 2012 EGU 2012 Extreme Sea Waves 21
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
April 27 2012 22 EGU 2012 Extreme Sea Waves
N as function of incoming frequency spread for fixed steepness and angular spread
April 27 2012 23 EGU 2012 Extreme Sea Waves
N as function of steepness for fixed incoming angular spread and frequency spread
April 27 2012 24 EGU 2012 Extreme Sea Waves
Bringing together these results
Dependence of N parameter on steepness angular spread and frequency spread
119873 asymp 02 120576minus3 Δ119896
1198960 Δθ
April 27 2012 25 EGU 2012 Extreme Sea Waves
4 Finally combine currents and nonlinearity
April 27 2012 EGU 2012 Extreme Sea Waves 26
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
April 27 2012 27 EGU 2012 Extreme Sea Waves
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
April 27 2012 EGU 2012 Extreme Sea Waves 28
Typical ray dynamics calculation
= 10ordm = 20ordm
April 27 2012 9 EGU 2012 Extreme Sea Waves
urms= 05 ms v = 78 ms
Random scattering yields fluctuations in local energy
density even though incoming waves are spatially uniform
ldquohot spotsrdquo (focusing) and ldquocold spotsrdquo (defocusing)
Fluctuations
Grow with increased current strength urms v
Smeared out by increasing angular spread
Define freak index = (urms v)23
The ldquofreak indexrdquo
April 27 2012 10 EGU 2012 Extreme Sea Waves
= (urms v)23
Most dangerous 1
(long-crested sea impinging on strong random current field)
More realistic le 1
( eg urms= 05 ms v = 78 ms = 10ordm rarr = 092)
The ldquofreak indexrdquo
April 27 2012 11 EGU 2012 Extreme Sea Waves
Typical ray dynamics calculation
= 10ordm = 20ordm
April 27 2012 12 EGU 2012 Extreme Sea Waves
urms= 05 ms v = 78 ms
Calculating Probabilities
April 27 2012 13 EGU 2012 Extreme
Sea Waves
119891 119868 = 1205942119873 119868 =119873
2
1198732 119868(119873minus2)2
Γ(1198732)119890minus1198731198682
For ltlt 1 local ray density (local energy density)
follows χ2 distribution with N degrees of freedom
Where 119868 = 1 and 119868 minus 1 2~1
119873asymp
1205742
45≪ 1
Implications for wave statistics
Now assuming the wave height distribution is locally
Rayleigh the total distribution integrating over hot and
cold spots is given by convolution
K-distribution
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
April 27 2012 14 EGU 2012 Extreme
Sea Waves
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)
April 27 2012 15 EGU 2012 Extreme Sea Waves
f (xy) (xy)u
Wave height distribution for γ=315
April 27 2012 16 EGU 2012 Extreme Sea Waves
Numerical test of N vs γ
April 27 2012 17 EGU 2012 Extreme Sea Waves
Probability enhancement over
Rayleigh predictions
urms = 05 ms v=78 ms
April 27 2012 EGU 2012 Extreme Sea Waves 18
4 Nonlinear extreme 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
April 27 2012 19 EGU 2012 Extreme Sea Waves
April 27 2012 EGU 2012 Extreme Sea Waves 20
April 27 2012 EGU 2012 Extreme Sea Waves 21
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
April 27 2012 22 EGU 2012 Extreme Sea Waves
N as function of incoming frequency spread for fixed steepness and angular spread
April 27 2012 23 EGU 2012 Extreme Sea Waves
N as function of steepness for fixed incoming angular spread and frequency spread
April 27 2012 24 EGU 2012 Extreme Sea Waves
Bringing together these results
Dependence of N parameter on steepness angular spread and frequency spread
119873 asymp 02 120576minus3 Δ119896
1198960 Δθ
April 27 2012 25 EGU 2012 Extreme Sea Waves
4 Finally combine currents and nonlinearity
April 27 2012 EGU 2012 Extreme Sea Waves 26
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
April 27 2012 27 EGU 2012 Extreme Sea Waves
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
April 27 2012 EGU 2012 Extreme Sea Waves 28
Random scattering yields fluctuations in local energy
density even though incoming waves are spatially uniform
ldquohot spotsrdquo (focusing) and ldquocold spotsrdquo (defocusing)
Fluctuations
Grow with increased current strength urms v
Smeared out by increasing angular spread
Define freak index = (urms v)23
The ldquofreak indexrdquo
April 27 2012 10 EGU 2012 Extreme Sea Waves
= (urms v)23
Most dangerous 1
(long-crested sea impinging on strong random current field)
More realistic le 1
( eg urms= 05 ms v = 78 ms = 10ordm rarr = 092)
The ldquofreak indexrdquo
April 27 2012 11 EGU 2012 Extreme Sea Waves
Typical ray dynamics calculation
= 10ordm = 20ordm
April 27 2012 12 EGU 2012 Extreme Sea Waves
urms= 05 ms v = 78 ms
Calculating Probabilities
April 27 2012 13 EGU 2012 Extreme
Sea Waves
119891 119868 = 1205942119873 119868 =119873
2
1198732 119868(119873minus2)2
Γ(1198732)119890minus1198731198682
For ltlt 1 local ray density (local energy density)
follows χ2 distribution with N degrees of freedom
Where 119868 = 1 and 119868 minus 1 2~1
119873asymp
1205742
45≪ 1
Implications for wave statistics
Now assuming the wave height distribution is locally
Rayleigh the total distribution integrating over hot and
cold spots is given by convolution
K-distribution
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
April 27 2012 14 EGU 2012 Extreme
Sea Waves
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)
April 27 2012 15 EGU 2012 Extreme Sea Waves
f (xy) (xy)u
Wave height distribution for γ=315
April 27 2012 16 EGU 2012 Extreme Sea Waves
Numerical test of N vs γ
April 27 2012 17 EGU 2012 Extreme Sea Waves
Probability enhancement over
Rayleigh predictions
urms = 05 ms v=78 ms
April 27 2012 EGU 2012 Extreme Sea Waves 18
4 Nonlinear extreme 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
April 27 2012 19 EGU 2012 Extreme Sea Waves
April 27 2012 EGU 2012 Extreme Sea Waves 20
April 27 2012 EGU 2012 Extreme Sea Waves 21
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
April 27 2012 22 EGU 2012 Extreme Sea Waves
N as function of incoming frequency spread for fixed steepness and angular spread
April 27 2012 23 EGU 2012 Extreme Sea Waves
N as function of steepness for fixed incoming angular spread and frequency spread
April 27 2012 24 EGU 2012 Extreme Sea Waves
Bringing together these results
Dependence of N parameter on steepness angular spread and frequency spread
119873 asymp 02 120576minus3 Δ119896
1198960 Δθ
April 27 2012 25 EGU 2012 Extreme Sea Waves
4 Finally combine currents and nonlinearity
April 27 2012 EGU 2012 Extreme Sea Waves 26
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
April 27 2012 27 EGU 2012 Extreme Sea Waves
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
April 27 2012 EGU 2012 Extreme Sea Waves 28
= (urms v)23
Most dangerous 1
(long-crested sea impinging on strong random current field)
More realistic le 1
( eg urms= 05 ms v = 78 ms = 10ordm rarr = 092)
The ldquofreak indexrdquo
April 27 2012 11 EGU 2012 Extreme Sea Waves
Typical ray dynamics calculation
= 10ordm = 20ordm
April 27 2012 12 EGU 2012 Extreme Sea Waves
urms= 05 ms v = 78 ms
Calculating Probabilities
April 27 2012 13 EGU 2012 Extreme
Sea Waves
119891 119868 = 1205942119873 119868 =119873
2
1198732 119868(119873minus2)2
Γ(1198732)119890minus1198731198682
For ltlt 1 local ray density (local energy density)
follows χ2 distribution with N degrees of freedom
Where 119868 = 1 and 119868 minus 1 2~1
119873asymp
1205742
45≪ 1
Implications for wave statistics
Now assuming the wave height distribution is locally
Rayleigh the total distribution integrating over hot and
cold spots is given by convolution
K-distribution
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
April 27 2012 14 EGU 2012 Extreme
Sea Waves
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)
April 27 2012 15 EGU 2012 Extreme Sea Waves
f (xy) (xy)u
Wave height distribution for γ=315
April 27 2012 16 EGU 2012 Extreme Sea Waves
Numerical test of N vs γ
April 27 2012 17 EGU 2012 Extreme Sea Waves
Probability enhancement over
Rayleigh predictions
urms = 05 ms v=78 ms
April 27 2012 EGU 2012 Extreme Sea Waves 18
4 Nonlinear extreme 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
April 27 2012 19 EGU 2012 Extreme Sea Waves
April 27 2012 EGU 2012 Extreme Sea Waves 20
April 27 2012 EGU 2012 Extreme Sea Waves 21
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
April 27 2012 22 EGU 2012 Extreme Sea Waves
N as function of incoming frequency spread for fixed steepness and angular spread
April 27 2012 23 EGU 2012 Extreme Sea Waves
N as function of steepness for fixed incoming angular spread and frequency spread
April 27 2012 24 EGU 2012 Extreme Sea Waves
Bringing together these results
Dependence of N parameter on steepness angular spread and frequency spread
119873 asymp 02 120576minus3 Δ119896
1198960 Δθ
April 27 2012 25 EGU 2012 Extreme Sea Waves
4 Finally combine currents and nonlinearity
April 27 2012 EGU 2012 Extreme Sea Waves 26
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
April 27 2012 27 EGU 2012 Extreme Sea Waves
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
April 27 2012 EGU 2012 Extreme Sea Waves 28
Typical ray dynamics calculation
= 10ordm = 20ordm
April 27 2012 12 EGU 2012 Extreme Sea Waves
urms= 05 ms v = 78 ms
Calculating Probabilities
April 27 2012 13 EGU 2012 Extreme
Sea Waves
119891 119868 = 1205942119873 119868 =119873
2
1198732 119868(119873minus2)2
Γ(1198732)119890minus1198731198682
For ltlt 1 local ray density (local energy density)
follows χ2 distribution with N degrees of freedom
Where 119868 = 1 and 119868 minus 1 2~1
119873asymp
1205742
45≪ 1
Implications for wave statistics
Now assuming the wave height distribution is locally
Rayleigh the total distribution integrating over hot and
cold spots is given by convolution
K-distribution
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
April 27 2012 14 EGU 2012 Extreme
Sea Waves
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)
April 27 2012 15 EGU 2012 Extreme Sea Waves
f (xy) (xy)u
Wave height distribution for γ=315
April 27 2012 16 EGU 2012 Extreme Sea Waves
Numerical test of N vs γ
April 27 2012 17 EGU 2012 Extreme Sea Waves
Probability enhancement over
Rayleigh predictions
urms = 05 ms v=78 ms
April 27 2012 EGU 2012 Extreme Sea Waves 18
4 Nonlinear extreme 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
April 27 2012 19 EGU 2012 Extreme Sea Waves
April 27 2012 EGU 2012 Extreme Sea Waves 20
April 27 2012 EGU 2012 Extreme Sea Waves 21
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
April 27 2012 22 EGU 2012 Extreme Sea Waves
N as function of incoming frequency spread for fixed steepness and angular spread
April 27 2012 23 EGU 2012 Extreme Sea Waves
N as function of steepness for fixed incoming angular spread and frequency spread
April 27 2012 24 EGU 2012 Extreme Sea Waves
Bringing together these results
Dependence of N parameter on steepness angular spread and frequency spread
119873 asymp 02 120576minus3 Δ119896
1198960 Δθ
April 27 2012 25 EGU 2012 Extreme Sea Waves
4 Finally combine currents and nonlinearity
April 27 2012 EGU 2012 Extreme Sea Waves 26
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
April 27 2012 27 EGU 2012 Extreme Sea Waves
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
April 27 2012 EGU 2012 Extreme Sea Waves 28
Calculating Probabilities
April 27 2012 13 EGU 2012 Extreme
Sea Waves
119891 119868 = 1205942119873 119868 =119873
2
1198732 119868(119873minus2)2
Γ(1198732)119890minus1198731198682
For ltlt 1 local ray density (local energy density)
follows χ2 distribution with N degrees of freedom
Where 119868 = 1 and 119868 minus 1 2~1
119873asymp
1205742
45≪ 1
Implications for wave statistics
Now assuming the wave height distribution is locally
Rayleigh the total distribution integrating over hot and
cold spots is given by convolution
K-distribution
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
April 27 2012 14 EGU 2012 Extreme
Sea Waves
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)
April 27 2012 15 EGU 2012 Extreme Sea Waves
f (xy) (xy)u
Wave height distribution for γ=315
April 27 2012 16 EGU 2012 Extreme Sea Waves
Numerical test of N vs γ
April 27 2012 17 EGU 2012 Extreme Sea Waves
Probability enhancement over
Rayleigh predictions
urms = 05 ms v=78 ms
April 27 2012 EGU 2012 Extreme Sea Waves 18
4 Nonlinear extreme 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
April 27 2012 19 EGU 2012 Extreme Sea Waves
April 27 2012 EGU 2012 Extreme Sea Waves 20
April 27 2012 EGU 2012 Extreme Sea Waves 21
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
April 27 2012 22 EGU 2012 Extreme Sea Waves
N as function of incoming frequency spread for fixed steepness and angular spread
April 27 2012 23 EGU 2012 Extreme Sea Waves
N as function of steepness for fixed incoming angular spread and frequency spread
April 27 2012 24 EGU 2012 Extreme Sea Waves
Bringing together these results
Dependence of N parameter on steepness angular spread and frequency spread
119873 asymp 02 120576minus3 Δ119896
1198960 Δθ
April 27 2012 25 EGU 2012 Extreme Sea Waves
4 Finally combine currents and nonlinearity
April 27 2012 EGU 2012 Extreme Sea Waves 26
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
April 27 2012 27 EGU 2012 Extreme Sea Waves
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
April 27 2012 EGU 2012 Extreme Sea Waves 28
Implications for wave statistics
Now assuming the wave height distribution is locally
Rayleigh the total distribution integrating over hot and
cold spots is given by convolution
K-distribution
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
April 27 2012 14 EGU 2012 Extreme
Sea Waves
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)
April 27 2012 15 EGU 2012 Extreme Sea Waves
f (xy) (xy)u
Wave height distribution for γ=315
April 27 2012 16 EGU 2012 Extreme Sea Waves
Numerical test of N vs γ
April 27 2012 17 EGU 2012 Extreme Sea Waves
Probability enhancement over
Rayleigh predictions
urms = 05 ms v=78 ms
April 27 2012 EGU 2012 Extreme Sea Waves 18
4 Nonlinear extreme 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
April 27 2012 19 EGU 2012 Extreme Sea Waves
April 27 2012 EGU 2012 Extreme Sea Waves 20
April 27 2012 EGU 2012 Extreme Sea Waves 21
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
April 27 2012 22 EGU 2012 Extreme Sea Waves
N as function of incoming frequency spread for fixed steepness and angular spread
April 27 2012 23 EGU 2012 Extreme Sea Waves
N as function of steepness for fixed incoming angular spread and frequency spread
April 27 2012 24 EGU 2012 Extreme Sea Waves
Bringing together these results
Dependence of N parameter on steepness angular spread and frequency spread
119873 asymp 02 120576minus3 Δ119896
1198960 Δθ
April 27 2012 25 EGU 2012 Extreme Sea Waves
4 Finally combine currents and nonlinearity
April 27 2012 EGU 2012 Extreme Sea Waves 26
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
April 27 2012 27 EGU 2012 Extreme Sea Waves
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
April 27 2012 EGU 2012 Extreme Sea Waves 28
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)
April 27 2012 15 EGU 2012 Extreme Sea Waves
f (xy) (xy)u
Wave height distribution for γ=315
April 27 2012 16 EGU 2012 Extreme Sea Waves
Numerical test of N vs γ
April 27 2012 17 EGU 2012 Extreme Sea Waves
Probability enhancement over
Rayleigh predictions
urms = 05 ms v=78 ms
April 27 2012 EGU 2012 Extreme Sea Waves 18
4 Nonlinear extreme 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
April 27 2012 19 EGU 2012 Extreme Sea Waves
April 27 2012 EGU 2012 Extreme Sea Waves 20
April 27 2012 EGU 2012 Extreme Sea Waves 21
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
April 27 2012 22 EGU 2012 Extreme Sea Waves
N as function of incoming frequency spread for fixed steepness and angular spread
April 27 2012 23 EGU 2012 Extreme Sea Waves
N as function of steepness for fixed incoming angular spread and frequency spread
April 27 2012 24 EGU 2012 Extreme Sea Waves
Bringing together these results
Dependence of N parameter on steepness angular spread and frequency spread
119873 asymp 02 120576minus3 Δ119896
1198960 Δθ
April 27 2012 25 EGU 2012 Extreme Sea Waves
4 Finally combine currents and nonlinearity
April 27 2012 EGU 2012 Extreme Sea Waves 26
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
April 27 2012 27 EGU 2012 Extreme Sea Waves
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
April 27 2012 EGU 2012 Extreme Sea Waves 28
Wave height distribution for γ=315
April 27 2012 16 EGU 2012 Extreme Sea Waves
Numerical test of N vs γ
April 27 2012 17 EGU 2012 Extreme Sea Waves
Probability enhancement over
Rayleigh predictions
urms = 05 ms v=78 ms
April 27 2012 EGU 2012 Extreme Sea Waves 18
4 Nonlinear extreme 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
April 27 2012 19 EGU 2012 Extreme Sea Waves
April 27 2012 EGU 2012 Extreme Sea Waves 20
April 27 2012 EGU 2012 Extreme Sea Waves 21
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
April 27 2012 22 EGU 2012 Extreme Sea Waves
N as function of incoming frequency spread for fixed steepness and angular spread
April 27 2012 23 EGU 2012 Extreme Sea Waves
N as function of steepness for fixed incoming angular spread and frequency spread
April 27 2012 24 EGU 2012 Extreme Sea Waves
Bringing together these results
Dependence of N parameter on steepness angular spread and frequency spread
119873 asymp 02 120576minus3 Δ119896
1198960 Δθ
April 27 2012 25 EGU 2012 Extreme Sea Waves
4 Finally combine currents and nonlinearity
April 27 2012 EGU 2012 Extreme Sea Waves 26
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
April 27 2012 27 EGU 2012 Extreme Sea Waves
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
April 27 2012 EGU 2012 Extreme Sea Waves 28
Numerical test of N vs γ
April 27 2012 17 EGU 2012 Extreme Sea Waves
Probability enhancement over
Rayleigh predictions
urms = 05 ms v=78 ms
April 27 2012 EGU 2012 Extreme Sea Waves 18
4 Nonlinear extreme 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
April 27 2012 19 EGU 2012 Extreme Sea Waves
April 27 2012 EGU 2012 Extreme Sea Waves 20
April 27 2012 EGU 2012 Extreme Sea Waves 21
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
April 27 2012 22 EGU 2012 Extreme Sea Waves
N as function of incoming frequency spread for fixed steepness and angular spread
April 27 2012 23 EGU 2012 Extreme Sea Waves
N as function of steepness for fixed incoming angular spread and frequency spread
April 27 2012 24 EGU 2012 Extreme Sea Waves
Bringing together these results
Dependence of N parameter on steepness angular spread and frequency spread
119873 asymp 02 120576minus3 Δ119896
1198960 Δθ
April 27 2012 25 EGU 2012 Extreme Sea Waves
4 Finally combine currents and nonlinearity
April 27 2012 EGU 2012 Extreme Sea Waves 26
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
April 27 2012 27 EGU 2012 Extreme Sea Waves
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
April 27 2012 EGU 2012 Extreme Sea Waves 28
Probability enhancement over
Rayleigh predictions
urms = 05 ms v=78 ms
April 27 2012 EGU 2012 Extreme Sea Waves 18
4 Nonlinear extreme 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
April 27 2012 19 EGU 2012 Extreme Sea Waves
April 27 2012 EGU 2012 Extreme Sea Waves 20
April 27 2012 EGU 2012 Extreme Sea Waves 21
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
April 27 2012 22 EGU 2012 Extreme Sea Waves
N as function of incoming frequency spread for fixed steepness and angular spread
April 27 2012 23 EGU 2012 Extreme Sea Waves
N as function of steepness for fixed incoming angular spread and frequency spread
April 27 2012 24 EGU 2012 Extreme Sea Waves
Bringing together these results
Dependence of N parameter on steepness angular spread and frequency spread
119873 asymp 02 120576minus3 Δ119896
1198960 Δθ
April 27 2012 25 EGU 2012 Extreme Sea Waves
4 Finally combine currents and nonlinearity
April 27 2012 EGU 2012 Extreme Sea Waves 26
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
April 27 2012 27 EGU 2012 Extreme Sea Waves
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
April 27 2012 EGU 2012 Extreme Sea Waves 28
4 Nonlinear extreme 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
April 27 2012 19 EGU 2012 Extreme Sea Waves
April 27 2012 EGU 2012 Extreme Sea Waves 20
April 27 2012 EGU 2012 Extreme Sea Waves 21
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
April 27 2012 22 EGU 2012 Extreme Sea Waves
N as function of incoming frequency spread for fixed steepness and angular spread
April 27 2012 23 EGU 2012 Extreme Sea Waves
N as function of steepness for fixed incoming angular spread and frequency spread
April 27 2012 24 EGU 2012 Extreme Sea Waves
Bringing together these results
Dependence of N parameter on steepness angular spread and frequency spread
119873 asymp 02 120576minus3 Δ119896
1198960 Δθ
April 27 2012 25 EGU 2012 Extreme Sea Waves
4 Finally combine currents and nonlinearity
April 27 2012 EGU 2012 Extreme Sea Waves 26
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
April 27 2012 27 EGU 2012 Extreme Sea Waves
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
April 27 2012 EGU 2012 Extreme Sea Waves 28
April 27 2012 EGU 2012 Extreme Sea Waves 20
April 27 2012 EGU 2012 Extreme Sea Waves 21
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
April 27 2012 22 EGU 2012 Extreme Sea Waves
N as function of incoming frequency spread for fixed steepness and angular spread
April 27 2012 23 EGU 2012 Extreme Sea Waves
N as function of steepness for fixed incoming angular spread and frequency spread
April 27 2012 24 EGU 2012 Extreme Sea Waves
Bringing together these results
Dependence of N parameter on steepness angular spread and frequency spread
119873 asymp 02 120576minus3 Δ119896
1198960 Δθ
April 27 2012 25 EGU 2012 Extreme Sea Waves
4 Finally combine currents and nonlinearity
April 27 2012 EGU 2012 Extreme Sea Waves 26
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
April 27 2012 27 EGU 2012 Extreme Sea Waves
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
April 27 2012 EGU 2012 Extreme Sea Waves 28
April 27 2012 EGU 2012 Extreme Sea Waves 21
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
April 27 2012 22 EGU 2012 Extreme Sea Waves
N as function of incoming frequency spread for fixed steepness and angular spread
April 27 2012 23 EGU 2012 Extreme Sea Waves
N as function of steepness for fixed incoming angular spread and frequency spread
April 27 2012 24 EGU 2012 Extreme Sea Waves
Bringing together these results
Dependence of N parameter on steepness angular spread and frequency spread
119873 asymp 02 120576minus3 Δ119896
1198960 Δθ
April 27 2012 25 EGU 2012 Extreme Sea Waves
4 Finally combine currents and nonlinearity
April 27 2012 EGU 2012 Extreme Sea Waves 26
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
April 27 2012 27 EGU 2012 Extreme Sea Waves
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
April 27 2012 EGU 2012 Extreme Sea Waves 28
N as function of incoming angular spread for fixed steepness and frequency spread
April 27 2012 22 EGU 2012 Extreme Sea Waves
N as function of incoming frequency spread for fixed steepness and angular spread
April 27 2012 23 EGU 2012 Extreme Sea Waves
N as function of steepness for fixed incoming angular spread and frequency spread
April 27 2012 24 EGU 2012 Extreme Sea Waves
Bringing together these results
Dependence of N parameter on steepness angular spread and frequency spread
119873 asymp 02 120576minus3 Δ119896
1198960 Δθ
April 27 2012 25 EGU 2012 Extreme Sea Waves
4 Finally combine currents and nonlinearity
April 27 2012 EGU 2012 Extreme Sea Waves 26
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
April 27 2012 27 EGU 2012 Extreme Sea Waves
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
April 27 2012 EGU 2012 Extreme Sea Waves 28
N as function of incoming frequency spread for fixed steepness and angular spread
April 27 2012 23 EGU 2012 Extreme Sea Waves
N as function of steepness for fixed incoming angular spread and frequency spread
April 27 2012 24 EGU 2012 Extreme Sea Waves
Bringing together these results
Dependence of N parameter on steepness angular spread and frequency spread
119873 asymp 02 120576minus3 Δ119896
1198960 Δθ
April 27 2012 25 EGU 2012 Extreme Sea Waves
4 Finally combine currents and nonlinearity
April 27 2012 EGU 2012 Extreme Sea Waves 26
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
April 27 2012 27 EGU 2012 Extreme Sea Waves
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
April 27 2012 EGU 2012 Extreme Sea Waves 28
N as function of steepness for fixed incoming angular spread and frequency spread
April 27 2012 24 EGU 2012 Extreme Sea Waves
Bringing together these results
Dependence of N parameter on steepness angular spread and frequency spread
119873 asymp 02 120576minus3 Δ119896
1198960 Δθ
April 27 2012 25 EGU 2012 Extreme Sea Waves
4 Finally combine currents and nonlinearity
April 27 2012 EGU 2012 Extreme Sea Waves 26
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
April 27 2012 27 EGU 2012 Extreme Sea Waves
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
April 27 2012 EGU 2012 Extreme Sea Waves 28
Bringing together these results
Dependence of N parameter on steepness angular spread and frequency spread
119873 asymp 02 120576minus3 Δ119896
1198960 Δθ
April 27 2012 25 EGU 2012 Extreme Sea Waves
4 Finally combine currents and nonlinearity
April 27 2012 EGU 2012 Extreme Sea Waves 26
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
April 27 2012 27 EGU 2012 Extreme Sea Waves
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
April 27 2012 EGU 2012 Extreme Sea Waves 28
4 Finally combine currents and nonlinearity
April 27 2012 EGU 2012 Extreme Sea Waves 26
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
April 27 2012 27 EGU 2012 Extreme Sea Waves
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
April 27 2012 EGU 2012 Extreme Sea Waves 28
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
April 27 2012 27 EGU 2012 Extreme Sea Waves
Thank you
Ying Zhuang Heller and Kaplan
Nonlinearity 24 R67 (2011)
April 27 2012 EGU 2012 Extreme Sea Waves 28