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.
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