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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

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

  • Numerical verification of weak-turbulent theory

  • Freak-waves solitons and modulational instability



Truncated equations:

Normal variables:



where interactions:


Statistical description: interactions:

Hasselmann equation:


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

- empirical dependences


Conservative KE has formal constants of motion interactions:

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


Energy spectrum interactions:


F(x)=0 interactions:, 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


Kolmogorov’s cascades interactions: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



Just a hypothesis to check interactions:

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 S key point of critics of the weak turbulence approach for water wavesnl and Sin. Algorithm by N. Ivenskikh (modified Webb-Resio-Tracy). Young waves, standard JONSWAP spectrum

Mean-over-angle

Down-wind


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 !!!



Quasi-universality of wind-wave spectra dissipation)

Spatial down-wind spectra

w-spectra

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


Numerical solutions for duration-limited case dissipation)vs 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. functions of time (fetch) but experimental wind speed scaling

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 linked to each other. 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 parameter linked to each other. assvs 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 linked to each other.

Hasselmann equation


Dynamical equations : linked to each other.

Hasselmann (kinetic) equation :



Dynamic equations: linked to each other.

domain of 4096x512 point in real space

Hasselmann equation:

domain of 71x36 points in frequency-angle space


  • Four damping terms: linked to each other.

  • Hyper-viscous damping

  • 2. WAM cycle 3 white-capping damping

  • 3. WAM cycle 4 white-capping damping

  • 4. New damping term


WAM Dissipation Function: linked to each other.

WAM cycle 3:

Komen 1984

Janssen 1992

Gunter 1992

Komen 1994

WAM cycle 4:


New Dissipation Function: linked to each other.



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