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 (Tulane University) In collaboration with Alex Dahlen and Eric Heller (Harvard) Linghang Ying and Zhouheng Zhuang (Tulane) 1

2 Talk Outline Linear rogue wave formation Mechanism: Refraction from current eddies Combining refraction with stochasticity The freak index Implications for extreme wave statistics Introducing wave nonlinearity Wave statistics in 4 th order nonlinear equation Scaling with parameters describing sea state Can linear and nonlinear effects work in concert? 2

3 Rogue Waves: Refraction Picture Focus on linear refraction of incoming wave (velocity v) by random current eddies (current speed u rms << v) For deep water surface gravity waves with current ( r, k ) gk u( r) k dk dt Current correlated on scale (typical eddy size) Ray picture justified if k >> 1 x x dx dt k x 3

4 How caustics form: refraction from current eddies Parallel incoming rays encountering pair of eddies Effect similar to potential dip in particle mechanics Focusing when all paths in a given neighborhood coalesce at a single point (caustic), producing infinite ray density Different groups of paths coalesce at different points ( bad lens analogy) 4

5 How caustics form: refraction from current eddies Generic result: cusp singularity followed by two lines of fold caustics At each y after cusp singularity, we have infinite density at some values of x Phase space picture 5

6 Refraction from weak, random currents First singularities form after d u rms 2/3 Further evolution: exponential proliferation of caustics Tendrils decorate original branches Branch statistics described by single distance scale d v 6

7 Refraction from weak, random currents Of course, singularities washed out on wavelength scale Qualitative structure independent of dispersion relation (e.g. ~ k 2 for Schrödinger vs. ~ k 1/2 for ocean waves) details of random current field Can calculate distribution of branch strengths, fall off with distance, etc (LK, PRL 2002) 7

8 Analogies with other physical systems Electron flow in nanostructures (10-6 m) Topinka et al, Nature (2001) Microwave resonators, Stöckmann group (1 m) Long-range ocean acoustics, Tomsovic et al (20 km) Twinkling of starlight, Berry (2000 km) Gravitational lensing, Tyson (10 9 light years) 8

9 Microwave scattering experiments Höhmann et al (PRL 2010) Experimentally observed branched flow Ray simulation 9

10 Branching pattern for tsunami waves 10

11 Problems with refraction picture of rogue waves Assumes single-wavelength and unidirectional initial conditions, which are unrealistic and unstable (Dysthe) Singularities washed out only on wavelength scale Predicts regular sequence of extreme waves every time No predictions for actual wave heights or probabilities Solution: replace incoming plane wave with random initial spectrum Finite range of wavelengths and directions 11

12 Smearing of caustics for stochastic incoming sea Singularities washed out by randomness in initial conditions: k Hot spots of enhanced average energy density remain as reminders of where caustics would have been 12

13 Smearing of caustics for stochastic incoming sea Competing effects of focusing and initial stochasticity: k x = initial wave vector spread k x = wave vector change due to refraction 13

14 Quantifying residual effect of caustics: the freak index Define freak index = k x / k x Equivalently = / = k x / k = initial directional spread = typical deflection before formation of first cusp ~ (u rms / v) 2/3 ~ / d Most dangerous: well-collimated sea impinging on strong random current field ( 1) Hot spots corresponding to first smooth cusps have highest energy density ( y) 1 2 ( y / d) ( y) d / y 14

15 Typical ray calculation for ocean waves = 10º = 20º 15

16 Implications for Rogue Wave Statistics Simulations (using Schrodinger equation): long-time average and regions of extreme events average >3 SWH >2.2 SWH 2 1 SWH=significant wave height 4 crest to trough 16

17 Calculation: Rogue Waves No currents: Rayleigh height distribution (random superposition of waves) 2 2 P( h) ( h / )exp( h / 2 ) With currents: Superpose locally Gaussian wave statistics on pattern of hot/cold spots caused by refraction Local height distribution I r is position-dependent variance of the water elevation (proportional to ray density; high in focusing regions, low in defocusing regions) Total wave height distribution: P h h I h I 2 2 I ( ) ( / )exp( / 2 ) P( h) PI ( h) f ( I ) di Nov. 11, Rogue11

18 Calculation: Rogue Waves Thus, calculate wave height probability by combining Rayleigh distribution P h h h 2 2 ( ) ( / )exp( / 2 ) Distribution f(i), which describes ray dynamics and depends on scattering strength (freak index) Rogue wave forecasting?? Nov. 11, Rogue11

19 Analytics: limit of small freak index P(h x SWH) exp( 2 x / I) f(i ) di 2 For <<1: f(i) well approximated by χ 2 distribution of n ~ --2 degrees of freedom (mean 1,width ~ ) 2 n/4 2( nx ) P(h x SWH) Kn/2(2 nx) ( n / 2) Nov. 11, Rogue11

20 Analytics: limit of small freak index 2 n/4 2( nx ) P(h x SWH) Kn/2(2 nx) ( n / 2) Perturbative limit: for x 2 <<1 (x << n 1/4 ) 4 n P(h x SWH) [ 1 (x x )] exp( 2x ) Asymptotic limit: for x 3 >>1 (x >> n 3/2 ) ( n 1)/2 ( nx) P(h x SWH) exp( 2 nx) ( n / 2) Nov. 11, Rogue11

21 Numerical Simulations Incoming sea with v=7.8 m/s (T=10 s, =156 m) u (x,y) f (x,y) 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) 21

22 Wave height distribution for ocean waves 22

23 Numerical test of N vs γ 23

24 Probability enhancement over Rayleigh predictions u rms = 0.5 m/s v=7.8 m/s 24

25 Extension to nonlinear 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 25

26 26

27 Extension to nonlinear waves Key result: for moderate steepness, full wave height distribution again reasonably approximated by K-distribution with N degrees of freedom Wave height probability depends on single parameter N ( Rayleigh as N ) How does N depend on wave steepness, incoming angular spread, spectral width, etc? 27

28 Δθ = 2.6º Δk / k = 0.1 u = 0 Steepness ε = k (mean crest height) October 11, 2011 Rogue11: Dresden 28

29 N as function of incoming angular spread for fixed steepness and frequency spread 29

30 N as function of incoming frequency spread for fixed steepness and angular spread 30

31 N as function of steepness for fixed incoming angular spread and frequency spread 31

32 Finally, combine currents and nonlinearity! 32

33 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 33

34 Thank you! Ying, Zhuang, Heller, and Kaplan, Nonlinearity 24, R67 (2011) 34