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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
- 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:

where

Statistical description:

Hasselmann equation:

- empirical dependences

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

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

F(y)

exponent for

y

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

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 !!!

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

Quasi-universality of wind-wave spectra

Spatial down-wind spectra

w-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

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