Theory of wind-driven sea

1. by V.E. Zakharov S. Badulin A.Dyachenko V.Geogdjaev N.Ivenskykh A.Korotkevich A.Pushkarev Theory of wind-driven sea In collaboration with:

2. Plan of the lecture: • Weak-turbulent theory • Kolmogorov-type spectra • Self-similar solutions • Experimental verification of weak-turbulent theory • Numerical verification of weak-turbulent theory • Freak-waves solitons and modulational instability

4. - Green function of the Dirichlet-Neuman problem -- average steepness

5. Truncated equations: Normal variables:

6. Canonical transformation - eliminating three-wave interactions:

7. where

8. Statistical description: Hasselmann equation:

9. Kinetic equation for deep water waves (the Hasselmann equation, 1962) - empirical dependences

10. Conservative KE has formal constants of motion wave action energy momentum Q – flux of action P – flux of energy For isotropic spectra n=n(|k|) Q and P are scalars let n ~ k-x, then Snl ~ k19/2-3xF(x), 3 < x < 9/2

11. Energy spectrum

12. F(x)=0, when x=23/6, x=4 – Kolmogorov-Zakharov solutions Kolmogorov’s constants are expressed in terms of F(y), where F(y) exponent for y

13. Kolmogorov’s cascades Snl=0(Zakharov, PhD thesis 1966) Direct cascade (Zakharov PhD thesis,1966; Zakharov & Filonenko 1966) Inverse cascade (Zakharov PhD thesis,1966) Numerical experiment with “artificial” pumping (grey). Solution is close to Kolmogorov-Zakharov solutions in the corresponding “inertial” intervals

14. Phillips, O.M., JFM. V.156,505-531, 1985.

15. Just a hypothesis to check Nonlinear transfer dominates! Snl >> Sinput , Sdiss

16. Existence of inertial intervals for wind-driven waves is a key point of critics of the weak turbulence approach for water waves Non-dimensional wave input rates Wave input term Sin for U10wp/g=1 Dispersion of different estimates of wave input Sin and dissipation Sdiss is of the same magnitude as the terms themselves !!!

17. Term-to-term comparison of Snl and Sin. Algorithm by N. Ivenskikh (modified Webb-Resio-Tracy). Young waves, standard JONSWAP spectrum Mean-over-angle Down-wind

18. The approximation procedure splits wave balance into two parts when Snl dominates • We do not ignore input and dissipation, we put them into appropriate place ! • Self-similar solutions (duration-limited) can be found for (*) for power-law dependence of net wave input on time

19. We have two-parametric family of self-similar solutions where relationships between parameters are determined by property of homogeneity of collision integral Snl and function of self-similar variable Ub(x) obeys integro-differential equation Stationary Kolmogorov-Zakharov solutions appear to be particular cases of the family of non-stationary (or spatially non-homogeneous) self-similar solutions when left-hand and right-hand sides of (**) vanish simultaneously !!!

20. Self-similar solutions for wave swell (no input and dissipation)

21. Quasi-universality of wind-wave spectra Spatial down-wind spectra w-spectra Dependence of spectral shapes on indexes of self-similarity is weak

22. Numerical solutions for duration-limited casevs non-dimensional frequency w*=wU/g *

23. Time-(fetch-) independent spectra grow as power-law functions of time (fetch) but experimental wind speed scaling 1. Duration-limited growth 2. Fetch-limited growth is not consistent with our “spectral flux approach” Experimental dependencies use 4 parameters. Our two-parameteric self-similar solutions dictate two relationships between these 4 parameters For case 2 ass – self-similarity parameter

24. Experimental power-law fits of wind-wave growth. Something more than an idealization? Thanks to Paul Hwang

25. Exponents are not arbitrary, not “universal”, they are linked to each other. Numerical results (blue – “realistic” wave inputs) Total energy and total frequency Energy and frequency of spectral “core”

26. Exponents pc(energy growth) vs qc(frequency downshift) for 24 fetch-limited experimental dependencies. Hard line – theoretical dependence pc=(10qc-1)/2 • “Cleanest” fetch-limited • Fetch-limited composite data sets • One-point measurements converted to fetch-limited one • Laboratory data included

27. Self-similarity parameterassvs exponent pcfor 24 experimental fetc-limited dependencies • “Cleanest” fetch-limited • Fetch-limited composite data sets • One-point measurements converted to fetch-limited one • Laboratory data included

28. Numerical verification of the Hasselmann equation

29. Dynamical equations : Hasselmann (kinetic) equation :

30. Two reasons why the weak turbulent theory could fail: • Presence of the coherent events -- solitons, quasi - solitons, wave collapses or wave-breakings • Finite size of the system – discrete Fourier space: • Quazi-resonances

31. Dynamic equations: domain of 4096x512 point in real space Hasselmann equation: domain of 71x36 points in frequency-angle space

32. Four damping terms: • Hyper-viscous damping • 2. WAM cycle 3 white-capping damping • 3. WAM cycle 4 white-capping damping • 4. New damping term

33. WAM Dissipation Function: WAM cycle 3: Komen 1984 Janssen 1992 Gunter 1992 Komen 1994 WAM cycle 4:

48. New Dissipation Function:

50. Freak-waves solitons and modulational instability