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
- empirical dependences
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
Kolmogorov’s constants are expressed in terms of F(y), where
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
Nonlinear transfer dominates!
Snl >> Sinput , Sdiss
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 !!!
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 !!!
Spatial down-wind spectra
Dependence of spectral shapes on indexes of self-similarity is weak
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
Something more than an idealization?
Thanks to Paul Hwang
Total energy and total frequency
Energy and frequency of spectral “core”
Hasselmann (kinetic) equation :
domain of 4096x512 point in real space
domain of 71x36 points in frequency-angle space
WAM cycle 3:
WAM cycle 4: