theory of wind driven sea
Download
Skip this Video
Download Presentation
Theory of wind-driven sea

Loading in 2 Seconds...

play fullscreen
1 / 67

Theory of wind-driven sea - PowerPoint PPT Presentation


  • 149 Views
  • Uploaded on

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

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Theory of wind-driven sea' - gerard


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
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:

slide2
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
slide5
Truncated equations:

Normal variables:

slide8
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

slide10
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

slide12
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

slide13
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

slide15
Just a hypothesis to check

Nonlinear transfer dominates!

Snl >> Sinput , Sdiss

slide16
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 !!!

slide17
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

slide18
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
slide19
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 !!!

slide21
Quasi-universality of wind-wave spectra

Spatial down-wind spectra

w-spectra

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

slide22
Numerical solutions for duration-limited casevs non-dimensional frequency w*=wU/g

*

slide23
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

slide24
Experimental power-law fits of wind-wave growth.

Something more than an idealization?

Thanks to Paul Hwang

slide25
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”

slide26
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
slide27
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
slide29
Dynamical equations :

Hasselmann (kinetic) equation :

slide30
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
slide31
Dynamic equations:

domain of 4096x512 point in real space

Hasselmann equation:

domain of 71x36 points in frequency-angle space

slide32
Four damping terms:
  • Hyper-viscous damping
  • 2. WAM cycle 3 white-capping damping
  • 3. WAM cycle 4 white-capping damping
  • 4. New damping term
slide33
WAM Dissipation Function:

WAM cycle 3:

Komen 1984

Janssen 1992

Gunter 1992

Komen 1994

WAM cycle 4:

ad