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
In collaboration with:
-- average steepness
Canonical transformation - eliminating three-wave interactions:
Statistical description: interactions:
- empirical dependences
Conservative KE has formal constants of motion interactions:
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
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
Phillips, O.M., JFM. V.156,505-531, 1985. interactions:
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
The approximation procedure splits wave balance into two parts when Snl dominates
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 dissipation)
Spatial down-wind 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
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
Self-similarity parameter linked to each other. assvs exponent pcfor 24 experimental fetc-limited dependencies
Numerical verification of the linked to each other.
Dynamical equations : linked to each other.
Hasselmann (kinetic) equation :
Dynamic equations: linked to each other.
domain of 4096x512 point in real space
domain of 71x36 points in frequency-angle space
WAM Dissipation Function: linked to each other.
WAM cycle 3:
WAM cycle 4:
New Dissipation Function: linked to each other.
Freak-waves solitons and modulational instability linked to each other.