linear and nonlinear rogue wave statistics in the …lkaplan/egu.pdf · questions: a) is...

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


Questions

A) Is nonlinearity necessary for extreme waves No



unified framework Yes



parameters Yes


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


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


Predicts low probability of extreme waves

119875119877119886119910119897119890119894119892ℎ 22 119878119882119867 = 119875119877119886119910119897119890119894119892ℎ 88 120590 = 63 ∙ 10minus5

119875119877119886119910119897119890119894119892ℎ 3 119878119882119867 = 119875119877119886119910119897119890119894119892ℎ 12 120590 = 15 ∙ 10minus8



Basic idea Waves refracted by

currents (or non-uniform bottomhellip)

Refraction of plane wave leads to

focusing (too many rogue 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



Typical ray dynamics calculation

= 10ordm = 20ordm


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


= (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)




= 10ordm = 20ordm



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


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)


f (xy) (xy)u

Wave height distribution for γ=315


Numerical test of N vs γ


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




Steepness ε = k (mean crest height)

Δθ = 26ordm

Δk k = 01

u = 0

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


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


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


Bringing together these results

Dependence of N parameter on steepness angular spread and frequency spread

119873 asymp 02 120576minus3 Δ119896

1198960 Δθ


4 Finally combine currents and nonlinearity


Summary

Linear refraction of stochastic Gaussian sea produces lumpy energy density

Skews formerly Rayleigh distribution of wave heights

Importance of refraction quantified by freak index

Spectacular effects in tail even for small

Very similar wave height distribution in the presence of moderate nonlinearity

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

Refraction may serve as trigger for full non-linear evolution


Thank you

Ying Zhuang Heller and Kaplan

Nonlinearity 24 R67 (2011)


Questions

A) Is nonlinearity necessary for extreme waves



unified framework



parameters


Questions







parameters Yes


Talk Outline










5 Conclusions









119875119877119886119910119897119890119894119892ℎ ℎ = 119890minusℎ281205902




119875119877119886119910119897119890119894119892ℎ 22 119878119882119867 = 119875119877119886119910119897119890119894119892ℎ 88 120590 = 63 ∙ 10minus5

119875119877119886119910119897119890119894119892ℎ 3 119878119882119867 = 119875119877119886119910119897119890119894119892ℎ 12 120590 = 15 ∙ 10minus8




















= 10ordm = 20ordm






Fluctuations






= (urms v)23

Most dangerous 1


More realistic le 1





= 10ordm = 20ordm





Sea Waves

119891 119868 = 1205942119873 119868 =119873

2

1198732 119868(119873minus2)2

Γ(1198732)119890minus1198731198682



Where 119868 = 1 and 119868 minus 1 2~1

119873asymp

1205742

45≪ 1





K-distribution

2 4

2

2( )(2 )

( 2)

n

n

nxP(h x SWH) K nx

n

2



Sea Waves



correlation =20 km


ltlt 1 (ray limit)





f (xy) (xy)u










NLS with current

where

21 1 1| | 0


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

x y t A x y t e





Δθ = 26ordm

Δk k = 01

u = 0









119873 asymp 02 120576minus3 Δ119896

1198960 Δθ




Summary









Thank you




Questions







parameters Yes


Talk Outline










5 Conclusions









119875119877119886119910119897119890119894119892ℎ ℎ = 119890minusℎ281205902




119875119877119886119910119897119890119894119892ℎ 22 119878119882119867 = 119875119877119886119910119897119890119894119892ℎ 88 120590 = 63 ∙ 10minus5

119875119877119886119910119897119890119894119892ℎ 3 119878119882119867 = 119875119877119886119910119897119890119894119892ℎ 12 120590 = 15 ∙ 10minus8




















= 10ordm = 20ordm






Fluctuations






= (urms v)23

Most dangerous 1


More realistic le 1





= 10ordm = 20ordm





Sea Waves

119891 119868 = 1205942119873 119868 =119873

2

1198732 119868(119873minus2)2

Γ(1198732)119890minus1198731198682



Where 119868 = 1 and 119868 minus 1 2~1

119873asymp

1205742

45≪ 1





K-distribution

2 4

2

2( )(2 )

( 2)

n

n

nxP(h x SWH) K nx

n

2



Sea Waves



correlation =20 km


ltlt 1 (ray limit)





f (xy) (xy)u










NLS with current

where

21 1 1| | 0


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

x y t A x y t e





Δθ = 26ordm

Δk k = 01

u = 0









119873 asymp 02 120576minus3 Δ119896

1198960 Δθ




Summary









Thank you




Talk Outline










5 Conclusions









119875119877119886119910119897119890119894119892ℎ ℎ = 119890minusℎ281205902




119875119877119886119910119897119890119894119892ℎ 22 119878119882119867 = 119875119877119886119910119897119890119894119892ℎ 88 120590 = 63 ∙ 10minus5

119875119877119886119910119897119890119894119892ℎ 3 119878119882119867 = 119875119877119886119910119897119890119894119892ℎ 12 120590 = 15 ∙ 10minus8




















= 10ordm = 20ordm






Fluctuations






= (urms v)23

Most dangerous 1


More realistic le 1





= 10ordm = 20ordm





Sea Waves

119891 119868 = 1205942119873 119868 =119873

2

1198732 119868(119873minus2)2

Γ(1198732)119890minus1198731198682



Where 119868 = 1 and 119868 minus 1 2~1

119873asymp

1205742

45≪ 1





K-distribution

2 4

2

2( )(2 )

( 2)

n

n

nxP(h x SWH) K nx

n

2



Sea Waves



correlation =20 km


ltlt 1 (ray limit)





f (xy) (xy)u










NLS with current

where

21 1 1| | 0


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

x y t A x y t e





Δθ = 26ordm

Δk k = 01

u = 0









119873 asymp 02 120576minus3 Δ119896

1198960 Δθ




Summary









Thank you











119875119877119886119910119897119890119894119892ℎ ℎ = 119890minusℎ281205902




119875119877119886119910119897119890119894119892ℎ 22 119878119882119867 = 119875119877119886119910119897119890119894119892ℎ 88 120590 = 63 ∙ 10minus5

119875119877119886119910119897119890119894119892ℎ 3 119878119882119867 = 119875119877119886119910119897119890119894119892ℎ 12 120590 = 15 ∙ 10minus8




















= 10ordm = 20ordm






Fluctuations






= (urms v)23

Most dangerous 1


More realistic le 1





= 10ordm = 20ordm





Sea Waves

119891 119868 = 1205942119873 119868 =119873

2

1198732 119868(119873minus2)2

Γ(1198732)119890minus1198731198682



Where 119868 = 1 and 119868 minus 1 2~1

119873asymp

1205742

45≪ 1





K-distribution

2 4

2

2( )(2 )

( 2)

n

n

nxP(h x SWH) K nx

n

2



Sea Waves



correlation =20 km


ltlt 1 (ray limit)





f (xy) (xy)u










NLS with current

where

21 1 1| | 0


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

x y t A x y t e





Δθ = 26ordm

Δk k = 01

u = 0









119873 asymp 02 120576minus3 Δ119896

1198960 Δθ




Summary









Thank you





119875119877119886119910119897119890119894119892ℎ 22 119878119882119867 = 119875119877119886119910119897119890119894119892ℎ 88 120590 = 63 ∙ 10minus5

119875119877119886119910119897119890119894119892ℎ 3 119878119882119867 = 119875119877119886119910119897119890119894119892ℎ 12 120590 = 15 ∙ 10minus8




















= 10ordm = 20ordm






Fluctuations






= (urms v)23

Most dangerous 1


More realistic le 1





= 10ordm = 20ordm





Sea Waves

119891 119868 = 1205942119873 119868 =119873

2

1198732 119868(119873minus2)2

Γ(1198732)119890minus1198731198682



Where 119868 = 1 and 119868 minus 1 2~1

119873asymp

1205742

45≪ 1





K-distribution

2 4

2

2( )(2 )

( 2)

n

n

nxP(h x SWH) K nx

n

2



Sea Waves



correlation =20 km


ltlt 1 (ray limit)





f (xy) (xy)u










NLS with current

where

21 1 1| | 0


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

x y t A x y t e





Δθ = 26ordm

Δk k = 01

u = 0









119873 asymp 02 120576minus3 Δ119896

1198960 Δθ




Summary









Thank you





















= 10ordm = 20ordm






Fluctuations






= (urms v)23

Most dangerous 1


More realistic le 1





= 10ordm = 20ordm





Sea Waves

119891 119868 = 1205942119873 119868 =119873

2

1198732 119868(119873minus2)2

Γ(1198732)119890minus1198731198682



Where 119868 = 1 and 119868 minus 1 2~1

119873asymp

1205742

45≪ 1





K-distribution

2 4

2

2( )(2 )

( 2)

n

n

nxP(h x SWH) K nx

n

2



Sea Waves



correlation =20 km


ltlt 1 (ray limit)





f (xy) (xy)u










NLS with current

where

21 1 1| | 0


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

x y t A x y t e





Δθ = 26ordm

Δk k = 01

u = 0









119873 asymp 02 120576minus3 Δ119896

1198960 Δθ




Summary









Thank you







Fluctuations






= (urms v)23

Most dangerous 1


More realistic le 1





= 10ordm = 20ordm





Sea Waves

119891 119868 = 1205942119873 119868 =119873

2

1198732 119868(119873minus2)2

Γ(1198732)119890minus1198731198682



Where 119868 = 1 and 119868 minus 1 2~1

119873asymp

1205742

45≪ 1





K-distribution

2 4

2

2( )(2 )

( 2)

n

n

nxP(h x SWH) K nx

n

2



Sea Waves



correlation =20 km


ltlt 1 (ray limit)





f (xy) (xy)u










NLS with current

where

21 1 1| | 0


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

x y t A x y t e





Δθ = 26ordm

Δk k = 01

u = 0









119873 asymp 02 120576minus3 Δ119896

1198960 Δθ




Summary









Thank you




= (urms v)23

Most dangerous 1


More realistic le 1





= 10ordm = 20ordm





Sea Waves

119891 119868 = 1205942119873 119868 =119873

2

1198732 119868(119873minus2)2

Γ(1198732)119890minus1198731198682



Where 119868 = 1 and 119868 minus 1 2~1

119873asymp

1205742

45≪ 1





K-distribution

2 4

2

2( )(2 )

( 2)

n

n

nxP(h x SWH) K nx

n

2



Sea Waves



correlation =20 km


ltlt 1 (ray limit)





f (xy) (xy)u










NLS with current

where

21 1 1| | 0


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

x y t A x y t e





Δθ = 26ordm

Δk k = 01

u = 0









119873 asymp 02 120576minus3 Δ119896

1198960 Δθ




Summary









Thank you





= 10ordm = 20ordm





Sea Waves

119891 119868 = 1205942119873 119868 =119873

2

1198732 119868(119873minus2)2

Γ(1198732)119890minus1198731198682



Where 119868 = 1 and 119868 minus 1 2~1

119873asymp

1205742

45≪ 1





K-distribution

2 4

2

2( )(2 )

( 2)

n

n

nxP(h x SWH) K nx

n

2



Sea Waves



correlation =20 km


ltlt 1 (ray limit)





f (xy) (xy)u










NLS with current

where

21 1 1| | 0


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

x y t A x y t e





Δθ = 26ordm

Δk k = 01

u = 0









119873 asymp 02 120576minus3 Δ119896

1198960 Δθ




Summary









Thank you






Sea Waves

119891 119868 = 1205942119873 119868 =119873

2

1198732 119868(119873minus2)2

Γ(1198732)119890minus1198731198682



Where 119868 = 1 and 119868 minus 1 2~1

119873asymp

1205742

45≪ 1





K-distribution

2 4

2

2( )(2 )

( 2)

n

n

nxP(h x SWH) K nx

n

2



Sea Waves



correlation =20 km


ltlt 1 (ray limit)





f (xy) (xy)u










NLS with current

where

21 1 1| | 0


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

x y t A x y t e





Δθ = 26ordm

Δk k = 01

u = 0









119873 asymp 02 120576minus3 Δ119896

1198960 Δθ




Summary









Thank you








K-distribution

2 4

2

2( )(2 )

( 2)

n

n

nxP(h x SWH) K nx

n

2



Sea Waves



correlation =20 km


ltlt 1 (ray limit)





f (xy) (xy)u










NLS with current

where

21 1 1| | 0


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

x y t A x y t e





Δθ = 26ordm

Δk k = 01

u = 0









119873 asymp 02 120576minus3 Δ119896

1198960 Δθ




Summary









Thank you






correlation =20 km


ltlt 1 (ray limit)





f (xy) (xy)u










NLS with current

where

21 1 1| | 0


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

x y t A x y t e





Δθ = 26ordm

Δk k = 01

u = 0









119873 asymp 02 120576minus3 Δ119896

1198960 Δθ




Summary









Thank you













NLS with current

where

21 1 1| | 0


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

x y t A x y t e





Δθ = 26ordm

Δk k = 01

u = 0









119873 asymp 02 120576minus3 Δ119896

1198960 Δθ




Summary









Thank you







Δθ = 26ordm

Δk k = 01

u = 0









119873 asymp 02 120576minus3 Δ119896

1198960 Δθ




Summary









Thank you












119873 asymp 02 120576minus3 Δ119896

1198960 Δθ




Summary









Thank you






Summary









Thank you




Thank you




linear and nonlinear rogue wave statistics in the …lkaplan/egu.pdf · questions: a) is...

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