Wind-wave growth in the laboratory studies

S. I. Badulin (1) and G. Caulliez (2)

(1) P.P. Shirshov Institute of Oceanology, Moscow, Russia

(2) Institut de Recherche sur les Phénomènes Hors Equilibre, Marseille, France

Experimental conditions

• various fetches ranging from 2 to 26.5 m: X = 2, 4, 6, 9, 13, 18, 26 m• various wind speeds Uref ranging from 2.5 m/s to 13 m/s: U10 2.5 to 17 m/s

Fetch

W IN D

video camera

light box

wavemaker

carriage

XHR or fast-speed

video cameras

laser sheet

laser slope gauge

The large IRPHE-Luminy wind-wave tank

water tank: L= 40 m, l = 2.6 m, d = 0.9 m air tunnel: L= 40 m, l = 3.2 m, h = 1.5 m

U10 2.5 to 17 m/s

A crazy question

•Can we reproduce wind-sea growth in the

wind-wave tank?

A regular answer

•NO

The tiny IRPHE-Luminy wind-wave tank

Length = O(102) wavelengths

Width = O(10) wavelengths

Height = O(10) wavelengths

Depth = O(10) wavelengths

Problems

•Scales

•Capillarity

•Drift currents

•Air flow

•etc

A. We certainly cannot model growth of wind-

driven seas in wind-wave channels

Why Wave growth in wave tanks is consistent both

qualitatively and quantitatively with wave growth in open sea?

Ex.: The Toba 3/2 law (Toba, 1972, 1973)

Hs=B(gu*)1/2Ts3/2

B=0.061

May be it is just happy chance when formally invalid tool works well

Try to answer within the statistical approach(formally invalid)

The kinetic equation for wind-driven seas(the Hasselmann equation)

knl in diss

dN S S Sdt

1. Nonlinear transfer is described from `the first principles’

2. External forcing is parameterized by empirical formulas

Try to answer within the weakly turbulent self-similar wave growth law (Badulin et al., 2007)

The split balance of wind-driven seasHyp. Nonlinear transfer dominates over wind input and dissipation

knl

kin diss

dN Sdtd N

S Sdt

1. Conservative Hasselmann equation assures universality (self-similarity) of nonlinear transfer

2. External forcing (spectral fluxes) controls evolution as total quantities. Details of the forcing are of no importance

31

2

3

2

4

gdt

d

g

p

ssp

Total energy

p - peak frequency

ss - self-similarity parameter

Self-similar solutions dictates Kolmogorov-like wave-growth lawBadulin, Babanin, Resio & Zakharov, JFM, 2007

0.55 0.25ss

1. Integral net wave input is rigidly linked to instantaneous wave parameters: characteristic wave energy and wave frequency;

2. Dependencies of sea wave growth of field experiments are consistent with the law

Measurements were carried out in the Large IRPHE-Luminy wind-wave channel in 2006 with no reference to the problem of

growth of wind-driven seas

Experimental conditions

• various fetches ranging from 2 to 26.5 m:

X = 2, 4, 6, 9, 13, 18, 26 m• various wind speeds Uref ranging from

2.5 m/s to 13 m/s: U10 2.5 to 17 m/s

Tools: wave capacity probes,

laser slope gauge

Our data cover wider range of conditions (cf. Toba, 1972`Traditional’ wave speed scaling gives high dispersion

(good in logaritmic axes only)

Blue stars – data by Toba (1972)New approach – new knowledge ?

Weakly turbulent scaling (energy-to-flux)Not so bad if locally measured frequency is used

(perfect! Axes are linear!)

ss

31

2

3

2

4

gdt

d

g

p

ssp

E

p4 /g2 =

Ste

epne

ss2

dominant wavelength

X = 6 m: d 30 cm

X = 13 m: d 45 cm

X = 26 m: d 80 cm

total mean square slope

E* 4/g2 = mssd

E/X 2/2g = X

Capillary and drift effects are included , i.e taken into account in mssd

Below d 30 cm, gravity-capillary and capillary- gravity waves: action of T/ and shear drift effects

Problems of the new presentation: derivatives and instantaneous quantities (wave heights and frequencies)

Better than perfect !

ss

31

2

3

2

4

gdt

d

g

p

ssp

<ak

>2 =

Ste

epne

ss2

Rate of energy=(d(a2k2)/dt/(2p))1/3

Concl.: We showed consistency of the wind-channel data and the weakly turbulent law (Badulin et al., 2007)

The talk is over (?)

No, it is just the very begining

31

2

3

2

4

gdt

d

g

p

ssp

We are the best !

Try to estimate net wave input and scale it

in physically consistent way

The weakly turbulent Kolmogorov-like law gives us a

chance

Very preliminary results: Wave input vs u* or vs Cp

(Air flow vs wave dynamics)<d/dt> ~

Different symbols are used for different wind speeds

Scaling in wave phase speed looks more attractive

The well-known Toba’s law as a particular case of weakly turbulent wind-wave growth

31

2

3

2

4

gdt

d

g

p

ssp

Let 3

2const ~d H Tdt

One can estimate energy production from instantaneous Hs, Ts

const~ 3* udt

d

3*Toba_input=1.3 a

w ss

u

g

Very preliminary resultsWave input normalized by the Toba input vs Cp

The scaling is relevant to constant in time production of wave energy

The less-known Hasselmann, Ross, Muller & Sell, 1976

(“Special solutions” for a parametric wave model)

31

2

)(3

gdt

dM

g

M pMss

p

Let 5

3

2*

const ~

~ ~ php

dM H Tdt

dMdtd C udt

M - total wave momentum

Get

See alsoResio, Long, Vincent, JGR 2004

Very preliminary resultsWave input for scaling Resio et al. 2004

Relevant to constant in time production of wave momentum

2*Resio_input ~ phC u

Summary• Wind-wave tank data (Toba 1973, Caulliez 2006)

are consistent with weakly turbulent scaling – Kolmogorov’s energy-to-flux rigid link

• The weakly turbulent approach and the new data allow one – to identify qualitatively different physical regimes

of wave growth;

– to describe quantitatively wind-wave interaction

Summary

Wind-wave tanks can give usreal physics

at unreal conditions

Marseille, 07/01/2009

Welcome to Marseille !

Motivation

• “With a wider perspective and in the long term, we need the wild horse that comes out with unconventional ideas…”

L. Cavaleri et al. / Progress in oceanography, 75 (2007) 603–674