Theory of wind-driven sea

by V.E. Zakharov

S. BadulinA.DyachenkoV.GeogdjaevN.IvenskykhA.KorotkevichA.Pushkarev

In collaboration with:

Plan of the lecture:

1.Weak-turbulent theory

2.Kolmogorov-type spectra

3.Self-similar solutions

4.Experimental verification of weak-turbulent theory

5.Numerical verification of weak-turbulent theory

6.Freak-waves solitons and modulational instability

),( zrZ ),( yxr

V 0divV 0

hz |

H

t H

t

UTH

sdsdssssGdzdrTr

)()(),(2

12

),(),( ssGssG - Green function of the Dirichlet-Neuman problem

hz | 0z

z

...210 HHHH432

k -- average steepness

Normal variables:

*

*

||2

2

kkk

k

kkk

k

aak

i

aag

*a

Hi

t

ak

][ˆ]ˆ[])ˆ[ˆ(ˆ]ˆ[ˆ))((ˆ 2212

21 kkkkkkkkt

]ˆ[]ˆ[ˆ]ˆ[])ˆ()[( 2221 kkkkkgt

Truncated equations:

),,,(),,,(

2

1

3213

321

***

321321321

kkkkTkkkkT

bbbbTdkbbH kkkkkkkkkkkkkkk

)( 41233210

*3

*2

*1

)4(012312332103

*2

*1

)3(0123

123321032*1

)2(01231233210321

)1(0123

12210*2

*1

)3(012122102

*1

)2(0121221021

)1(012

00

bOdkbbbBdkbbbB

dkbbbBdkbbbB

dkbbAdkbbAdkbbA

ba

Canonical transformation - eliminating three-wave interactions:

24132

32324141241

23131

42423131231

22121

43432121221

32324141

42423131

43432121

3232414141322

41

4242313131422

31

4343212121432

21

43214

1

4321

21234

)(

))(()(4

)(

))(()(4

)(

))(()(4

))((

))((

))((

)()()(2

)()()(2

)()()(2

12)(

1

32

1

q

qqkkqqkk

q

qqkkqqkk

q

qqkkqqkk

qqkkqqkk

qqkkqqkk

qqkkqqkk

qqkkqqkk

qqkkqqkk

qqkkqqkk

qqqqqqqq

T

|| kq where

Statistical description: )(* kknbb kkk

Hasselmann equation: nlkrkk

k SNt

N

3213213210

310210320321

2

012325

)()(

)(16

dkdkdkkkkk

NNNNNNNNNNNNTgSnl

)4/()()( 2knkN

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

32132103210

3102103213202

0123

)()(

)(||2

kkkkkkk ddd

nnnnnnnnnnnnTSnl

dissS,inputS - empirical dependences

dt

dnknlS inputS dissS

),,,(),,,( 3216

321 kkkkkkkk TT

Conservative KE has formal constants of motion

wave action

energy

momentum

kdnN k

kkkk ndE ;k

kkM dnk

,divQ

nlk St

n

,divP

nlkk St

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

ddndd ||||)()(),( kkkk

))((2),( 2

4 kn

g

311

31

34

)2(623

)2(2

4

31

34

)1(4)1(1

21

)(;~;623

;)(;~;4

when,0)(

QgCknx

PgCknx

,xxxxF

q

p

k

k

F(x)=0, when x=23/6, x=4 – Kolmogorov-Zakharov solutions

Kolmogorov’s constants are expressed in terms of F(y), where

31

3/11

231

4

2

38

;3

8

yqyp y

FC

yF

C

42x-y exponent for

yn ~)()(y

F(y)

Kolmogorov’s cascades Snl=0 (Zakharov, PhD thesis 1966)

4 / 3 1/ 3(1)

4( ) p

g PC

4 / 3 1/ 3(2)

11/ 3( ) q

g QC

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.

Snl >> Sinput , Sdiss

Nonlinear transfer dominates!

Just a hypothesis to check

kdissdiss

kinin

nS

nS

Existence of inertial intervals for wind-driven waves is a key point of critics of the weak turbulence approach for water waves

Wave input term Sin for U10p/g=1

Non-dimensional wave input rates

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

(*)k

nl

kin diss

dn Sdtd n

S Sdt

2;~when

),(

rtn

tbUatNr

k

k

We have two-parametric family of self-similar solutions where relationships between parameters are determined

by property of homogeneity of collision integral Snl

4219

;4/19 ba

and function of self-similar variable Uobeys integro-differential equation

(**))]([ USUU nlStationary 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 !!!

),( 11/120

11/2 tUtn

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

Quasi-universality of wind-wave spectra

Spatial down-wind spectra spectra

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

Numerical solutions for duration-limited case vs non-dimensional frequency U/g

*

1. Duration-limited growth

2. Fetch-limited growth

qpEE 00~~;~~

qpEE 00~~;~~

g

U

U

EgE h

h

~;~

4

2

Time-(fetch-) independent spectra grow as power-law functions of time (fetch) but experimental wind speed scaling

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

2

110;

2 31

100

20

q

pp

Ess

ss – self-similarity parameter

Thanks to Paul HwangExperimental power-law fits of wind-wave growth.

Something more than an idealization?

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”

219

q

p

qpEE 00~~;~~

Exponents p (energy growth) vs q (frequency downshift) for 24 fetch-

limited experimental dependencies. Hard line – theoretical dependence

p=(10q-1)/2

1. “Cleanest” fetch-limited

2. Fetch-limited composite data sets

3. One-point measurements converted to fetch-limited one

4. Laboratory data included

Self-similarity parameter ss vs exponent p for 24 experimental

fetc-limited dependencies

1. “Cleanest” fetch-limited

2. Fetch-limited composite data sets

3. One-point measurements converted to fetch-limited one

4. Laboratory data included

Numerical verification of the

Hasselmann equation

ˆ][ˆ]ˆ[])ˆ[ˆ(ˆ]ˆ[ˆ))((ˆ 2212

21 kkkkkkkkt

ˆ]ˆ[]ˆ[ˆ]ˆ[])ˆ()[( 2221 kkkkkgt

Dynamical equations :

Hasselmann (kinetic) equation :

yxrki

kdkdkekk

2

1ˆ

kkkkkk ndkdkdkkkkknnnnnnnnTt

n

321321321321132

2

123

Two reasons why the weak turbulent theory could fail:

1.Presence of the coherent events -- solitons, quasi - solitons, wave collapses or wave-breakings

2.Finite size of the system – discrete Fourier space:

Quazi-resonances

4321

4321

kkkk

Dynamic equations:

domain of 4096x512 point in real space

Hasselmann equation:

domain of 71x36 points in frequency-angle space

22

Four damping terms:

1. Hyper-viscous damping

2. WAM cycle 3 white-capping damping

3. WAM cycle 4 white-capping damping

4. New damping term

2)1024( kCk

),(~~

~

~

~1),(

4

Ek

k

S

S

k

kCS

PMdsds

totEkS~~

2/13)1002.3(

~ PMS

4,,1036.2 5 P0. Cds

4,,1010.4 5 P0.5 Cds

WAM Dissipation Function:

WAM cycle 3:

WAM cycle 4:

Komen 1984

Janssen 1992 Gunter 1992Komen 1994

),(~~

~

~

~1),( Ek

k

S

S

k

kCS

P

PM

dsds

totEkS~~

2/13)1002.3(~ PMS

12,,1000.1 6 P0 Cds

New Dissipation Function:

Freak-waves solitons and modulational instability