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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Theory of wind driven sea

by V.E. Zakharov

S. Badulin

A.Dyachenko

V.Geogdjaev

N.Ivenskykh

A.Korotkevich

A.Pushkarev

Theory of wind-driven sea

In collaboration with:


Theory of wind driven sea

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


Theory of wind driven sea

- Green function of the Dirichlet-Neuman problem

-- average steepness


Theory of wind driven sea

Truncated equations:

Normal variables:


Theory of wind driven sea

Canonical transformation - eliminating three-wave interactions:


Theory of wind driven sea

where


Theory of wind driven sea

Statistical description:

Hasselmann equation:


Kinetic equation for deep water waves the hasselmann equation 1962

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

- empirical dependences


Theory of wind driven sea

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


Theory of wind driven sea

Energy spectrum


Theory of wind driven sea

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


Theory of wind driven sea

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


Theory of wind driven sea

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


Theory of wind driven sea

Just a hypothesis to check

Nonlinear transfer dominates!

Snl >> Sinput , Sdiss


Theory of wind driven sea

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 !!!


Theory of wind driven sea

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


Theory of wind driven sea

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


Theory of wind driven sea

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 !!!


Theory of wind driven sea

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


Theory of wind driven sea

Quasi-universality of wind-wave spectra

Spatial down-wind spectra

w-spectra

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


Theory of wind driven sea

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

*


Theory of wind driven sea

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


Theory of wind driven sea

Experimental power-law fits of wind-wave growth.

Something more than an idealization?

Thanks to Paul Hwang


Theory of wind driven sea

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”


Theory of wind driven sea

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


Theory of wind driven sea

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


Theory of wind driven sea

Numerical verification of the

Hasselmann equation


Theory of wind driven sea

Dynamical equations :

Hasselmann (kinetic) equation :


Theory of wind driven sea

  • 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


Theory of wind driven sea

Dynamic equations:

domain of 4096x512 point in real space

Hasselmann equation:

domain of 71x36 points in frequency-angle space


Theory of wind driven sea

  • Four damping terms:

  • Hyper-viscous damping

  • 2. WAM cycle 3 white-capping damping

  • 3. WAM cycle 4 white-capping damping

  • 4. New damping term


Theory of wind driven sea

WAM Dissipation Function:

WAM cycle 3:

Komen 1984

Janssen 1992

Gunter 1992

Komen 1994

WAM cycle 4:


Theory of wind driven sea

New Dissipation Function:


Theory of wind driven sea

Freak-waves solitons and modulational instability


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