Linear and Nonlinear Rogue Wave Statistics in the Presence of Random Currents

Transcription

1 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) 1

2 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? 2

3 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. 3

4 Talk Outline 1. Introduction: Longuet-Higgins random waves 2. Linear rogue wave formation a) Mechanism: Refraction from current eddies b) The freak index c) Implications for extreme wave statistics 3. Nonlinear rogue wave formation a) Wave statistics in 4 th order nonlinear equation b) Scaling with parameters describing sea state 4. Can linear and nonlinear effects work in concert? 5. Conclusions 4

5 1. Longuet-Higgins random waves If surface elevation is sum of plane waves with random directions, amplitudes and phases Surface elevation distribution is Gaussian with standard deviation σ Given narrow-banded spectrum, wave height distribution is Rayleigh Cumulative wave height probability is P Rayleigh h = e h2 /8σ 2 5

6 1. Longuet-Higgins random waves Predicts low probability of extreme waves: P Rayleigh 2.2 SWH = P Rayleigh 8.8 σ = P Rayleigh 3 SWH = P Rayleigh 12 σ =

7 2. Linear rogue wave formation Basic idea: Waves refracted by currents (or non-uniform bottom ) Refraction of plane wave leads to focusing (too many rogue waves!) 7

8 2. Linear rogue wave formation Assume incoming random sum of plane waves mean wavenumber k 0, speed v = (g / 4 k 0 ) 1/2 wave number spread Δk << k 0 angular spread Δθ << 2π Refracted by random current field Gaussian distributed, Gaussian correlations current speed u rms << v correlated on scale (typical eddy size) >> 1/ k 0 8

9 Typical ray dynamics calculation = 10º = 20º u rms = 0.5 m/s, v = 7.8 m/s 9

10 Random scattering yields fluctuations in local energy density, even though incoming waves are spatially uniform hot spots (focusing) and cold spots (defocusing) Fluctuations The freak index Grow with increased current strength u rms / v Smeared out by increasing angular spread Define freak index = (u rms / v) 2/3 / 10

11 The freak index = (u rms / v) 2/3 / Most dangerous: 1 (long-crested sea impinging on strong random current field) More realistic: 1 ( e.g. u rms = 0.5 m/s, v = 7.8 m/s, = 10º = 0.92) 11

12 Typical ray dynamics calculation = 10º = 20º u rms = 0.5 m/s, v = 7.8 m/s 12

13 Calculating Probabilities For << 1, local ray density (local energy density) follows χ 2 distribution with N degrees of freedom: f I = χ 2 N I = N 2 N/2 I (N 2)/2 Γ(N/2) e NI/2 Where I = 1 and I 1 2 ~ 1 N γ April 27, EGU 2012: Extreme Sea Waves

14 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: P(h x SWH) exp( 2 x / I) f(i ) di 2 K-distribution: 2 n/4 2( nx ) P(h x SWH) Kn/2(2 nx) ( n / 2) April 27, EGU 2012: Extreme Sea Waves

15 Numerical Simulations Incoming sea with v=7.8 m/s (T=10 s, =156 m) Random current with rms velocity u rms = 0.5 m/s and correlation =20 km Dimensionless parameters: / << 1 (ray limit) ~ (u rms / v) 2/3 << 1 (small-angle scattering) = spreading angle = 5 to 25 = / = freak index ( =3.5 to 0.7) u (x,y) f (x,y) 15

16 Wave height distribution for γ=

17 Numerical test of N vs γ 17

18 Probability enhancement over Rayleigh predictions u rms = 0.5 m/s v=7.8 m/s 18

19 4. Nonlinear extreme waves NLS with current: where t xx yy x 0 ia A A A A U A ( x, y, t) ~ A( x, y, t) e ik 0 x i 0 t 19

20 20

21 Δθ = 2.6º Δk / k = 0.1 u = 0 Steepness ε = k (mean crest height) April 27, 2012 EGU 2012: Extreme Sea Waves 21

22 N as function of incoming angular spread for fixed steepness and frequency spread 22

23 N as function of incoming frequency spread for fixed steepness and angular spread 23

24 N as function of steepness for fixed incoming angular spread and frequency spread 24

25 Bringing together these results: Dependence of N parameter on steepness, angular spread, and frequency spread: N 0.2 ε 3 Δk k 0 Δθ 25

26 4. Finally, combine currents and nonlinearity! 26

27 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 27

28 Thank you! Ying, Zhuang, Heller, and Kaplan, Nonlinearity 24, R67 (2011) 28