Gravity Wave Turbulence in Wave Tanks

S Lukaschuk1 , S Nazarenko2

1 Fluid Dynamics Laboratory, University of Hull2 Mathematics Institute, University of Warwick

Solitons, Collapses and Turbulence, Chernogolovka, 2009

Motivation

Knowledge of statistical characteristics of wave turbulence is important for

- weather and wave forecasting

- prediction of climate change

- atmosphere-ocean gas exchange

- absorption of solar energy

- pollutant transport

At the same time there is no clear understanding about mechanisms of

wave energy transport within universal interval and dissipation:

- different theory predict different energy spectral slope

- experimental and statistical results are not sufficient

Experiment

M

8-panel Wave Generator

C1

C2

Laser

CCD

Max water depth 0.9m

C1, C2:Fs=100 Hz

1.3 Mp CCD:Fs=8 Hz

1k

12 m

6 m M

C1 C2

2k

Typical spectra E for small and large wave amplitudes

A=1.85 cm (=0.074)

mk

Ak

tA

m

m

6.1,m4

21.0,052.0,1-

2

A=3.95 cm (=0.16)

400 405 410 415 420 425 430

-5

0

5

Ele

vatio

n, c

m

Elevation as function of time: Ch 1(red), Ch 2(blue), (file 81)

400 405 410 415 420 425 430

-5

0

5

ch 2

time, [s]

Ele

vatio

n, c

m

400 405 410 415 420 425 430

-5

0

5

Ele

vatio

n, c

m

Elevation as function of time: Ch 1(red), Ch 2(blue), (file 81)

400 405 410 415 420 425 430

-5

0

5

ch 2

time, [s]

Ele

vatio

n, c

m

1. Poin-like braking events (Phillips)

sharp wave crests

strong nonlinearity

dimensional analysis

2. Propagating braking waves (Kuznetsov)

slope breaks occurs in 1D lines wave crests are propagating with a preserved shape

3. Weak turbulence theory (Zakharov

random phase (or short correlation length) spatial homogeneity stationary energy flow from large to small scale

4. Finite size effects (Zakharov, Nazarenko)the wave intensity should be strong enough so thatnon-linear resonance broadening is much greater than the spacing of the k-grid (2/L ).

Theoretical prediction for spectra of surface gravity waves

353 kIgI k

k 44 kII k

krdtrxtxeItdttxtxeI rki

k

ti ,,;,,

gk

gk2

743

12 ,

kIgI k

2962127 , kILgI k

411 kL

Spectrum slopes vs the wave spectral density Ef

(f is from the inertial interval)

Inset:spectral density Ef

vs the energy dissipation rate

fE f

=0 “avalanches”and also Phillips

=1/3 WTT

x-domain measurements

Yag Laser

Dye: Rhodamine 6GFs=8 Hz ; N=1200 images; Tacq~ 20 minImage size: 1151 x 476 mmResolution: 0.9 mm/pix

532 nm120 mW

k- and -spectra =0.2

• Weak turbulence theory

• Breaks (Kuznetsov)

• Finite size effects

k- and -slopes

2

5

2

7

kkEk

44 kkEk

2

7

2

9

kkEk

Increments and Structure Functions - definitions

tttDlxxlxD

ttDxlxD

l

l

22 )2()2(

)1()1(

pjjpjl

jl DpSDpS ,

Different types of incoherent and coherent structures (breaks) may lead to the same spectra => to distinguish them we should considerhigh order correlations - SF.

By analogy with hydrodynamic turbulence we introduce differences:

The moments of the increments (SF) are defined as

which are sensitive to singularities of different order

)()()()( )( and )( pjpjl pSlpS

for small l or correspondingly

Asymptotic of the Structure functions

kEkFor Gaussian statistics and 12,)(

12,)()(

2/)1()(

jlpS

jlpSpjj

l

pjl

12,)(

12,)()(

2/)1()(

jpS

jpSpjj

pj

E

apDapDpjjl lNllpS 22)( )(

For singular coherent structure with the cross-section

a

a xCx 0)(

For j>1, 0<D<2, 0<a<1 and l ->0

Now suppose that the wave field is bi-fractal and consists of Gaussian incoherent waves and coherent (braking) structures

apDpjapDpjl pSllpS 22/122/1 ~,~

If a<(-1)/2 we expect to see the scaling assosciated with Gaussian rabdom phasewaves at small p and coherent structures at larger p.

PDF of 2nd difference in the t- and x-domains

=30 ms=60 ms=130 ms

=290 ms

=580 ms

l=0.54 cml=1.1 cml=2.5 cm

l=6.2 cm

l=15 cm

l=36 cm

02.3 kEk5.4E

Smaller deviation from Gaussian in the t-domain might be due to low propagationspeed of the coherent structures leading to their infrequent occurrence in t-domain.

Structure functions in t- and x-domains

p=8

. . .

p=1

SFs as a function of x or demonstrate power-law scaling in the gravity wave range

SF exponents for t- and x-domains

(t) (x)

Fit at high p: D=1, a=1.05 D=1.3 , a=0.5Ku- structures vertical splashes(propagting breaking waves) formed by wave interference

Conclusion

•Our experimental data show that spectral exponents are not universal in both -and k –domains and depend on the amplitude of forcing.

•We showed that the gravity wave field in the wave tank consists coexisting random and coherent components.

•The random waves are captured by the PDF cores and low-order SFs, whereas the coherent wave crests leave their imprints on the PDF tails and on the high-order SFs.

•The wave crests themselves consists of structures of different shapes: slow propagating splashes and propagating Ku-type breaks (seen in the t-domain SFs).

•We hope that our technique based on SF scalings can be useful in future for analysing the open-sea data, as well as to the future WT theory describing the dynamics and mutual interactions of coexisting random –phased and coherent wave components.