Theory of wind driven sea
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by V.E. Zakharov. S. Badulin A.Dyachenko V.Geogdjaev N.Ivenskykh A.Korotkevich A.Pushkarev. Theory of wind-driven sea. In collaboration with:. Plan of the lecture:. Weak-turbulent theory Kolmogorov-type spectra Self-similar solutions Experimental verification of weak-turbulent theory

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Theory of wind-driven sea

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by V.E. Zakharov

S. Badulin






Theory of wind-driven sea

In collaboration with:

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

- Green function of the Dirichlet-Neuman problem

-- average steepness

Truncated equations:

Normal variables:

Canonical transformation - eliminating three-wave interactions:


Statistical description:

Hasselmann equation:

Kinetic equation for deep water waves (the Hasselmann equation, 1962)

- empirical dependences

Conservative KE has formal constants of motion

wave action



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

Energy spectrum

F(x)=0, when x=23/6, x=4 – Kolmogorov-Zakharov solutions

Kolmogorov’s constants are expressed in terms of F(y), where


exponent for


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

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

Just a hypothesis to check

Nonlinear transfer dominates!

Snl >> Sinput , Sdiss

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 !!!

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



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

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 !!!

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

Quasi-universality of wind-wave spectra

Spatial down-wind spectra


Dependence of spectral shapes on indexes of self-similarity is weak

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


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

Experimental power-law fits of wind-wave growth.

Something more than an idealization?

Thanks to Paul Hwang

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”

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

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

Numerical verification of the

Hasselmann equation

Dynamical equations :

Hasselmann (kinetic) equation :

  • 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

Dynamic equations:

domain of 4096x512 point in real space

Hasselmann equation:

domain of 71x36 points in frequency-angle space

  • Four damping terms:

  • Hyper-viscous damping

  • 2. WAM cycle 3 white-capping damping

  • 3. WAM cycle 4 white-capping damping

  • 4. New damping term

WAM Dissipation Function:

WAM cycle 3:

Komen 1984

Janssen 1992

Gunter 1992

Komen 1994

WAM cycle 4:

New Dissipation Function:

Freak-waves solitons and modulational instability

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