Spectra of Gravity Wave Turbulence in a Laboratory Flume

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Spectra of Gravity Wave Turbulence in a
Laboratory Flume
S Lukaschuk1, P Denissenko1, S Nazarenko2
1 Fluid Dynamics Laboratory, University of Hull 2
Mathematics Institute, University of Warwick
WTS workshop, Warwick- Hull, 17-21 September
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Theoretical prediction forenergy spectra of
surface gravity waves

• Phillips (JFM 1958, 1985)
• sharp wave crests
• strong nonlinearity
• dimensional analysis
• 1K. Kuznetsov (JETP Letters, 2004)
• slope breaks occurs in 1D lines
• wave crests are propagating with a preserved
  shape

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2. Weak turbulence theory(Theory and numerical
experiment - Hasselman, Zakharov, Lvov,
Falkovich, Newell, Hasselman, Nazarenko 1962 -
2006)

• Kinetic equation approach for WT in an ensemble
  of weakly interacted low amplitude waves
  (Hasselmann)
• Assumptions weak nonlinearity
• random phase (or short correlation length)
• spatial homogeneity
• stationary energy flow from large to small
  scales
• Zakharov Filonenko spectrum for gravity waves
  in infinite space
• is an exact solution of Hasselmann equation
  which describes a steady state with energy
  cascading through an inertial range from large to
  small scale (Kolmogorov - like spectrum) for
  gravity waves in infinite space

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3. Finite size effects (mesoscopic wave
turbulence) Theory Kartashova (1998), Zakharov
(2005), Nazarenko (2006) et al

• For the WTT mechanisms to work in a finite box,
  the wave intensity should be strong enough so
  that non-linear resonance broadening is much
  greater than the spacing of the k-grid (?2?/L ).
  This implies a condition on the minimal angle of
  the surface elevation
• Discrete scenario (Nazarenko, 2005)
• For weaker waves the number of four-wave
  resonances is depleted. This arrests the energy
  cascade and leads to accumulation of energy near
  the forcing interval. Such accumulation will
  proceed until the wave intensity is strong enough
  to the nonlinear broadening to become comparable
  to the k-grid spacing. At this point the
  four-wave resonances will get engage and the
  energy will propagate towards lower k. Mean
  spectrum settles at a critical slope
• determined by dk 2?/L

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Numerical experiments

• Confirmation of ZF spectra
• Zakahrov et al (2002-5),
• Onorato (2002),
• Yokyama (2004),
• Nazarenko (2005).
• Results are not 100 satisfying because no
  greater than 1 decade inertial range

• Phillips spectrum
• could not be expected in direct numerical
  simulations because
• nonlinearity truncation at cubic terms,
• artificial numerical dissipation at high k to
  prevent numerical blowups.

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Field experiments P.A. Hwang, D.W.Wang,
Airborne Measurements of surface elevation
k-spectra, (2000)
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Goals Long-term to study transport and mixing
generated by wave turbulence Short-term to
characterize statistical properties of waves in a
finite system

• Advantages of the laboratory experiment
• Wider inertial interval two decades in k
• Possibility to study both weakly and strongly
  nonlinear waves
• No artificial dissipation natural wavebreaking
  dissipation mechanism.

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Small amplitude
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Large amplitudes
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Typical spectra E? for small and large wave
amplitudes
A3.95 cm (??0.16)
A1.85 cm (??0.074)
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Spectrum slopes vs the wave spectral density
Ef(f is from the inertial interval)
Inset spectral density Efvs the energy
dissipation rate
?0 avalanches and also Phillips ?1/3 WTT
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Estimation of the Dissipation Rate
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PDF of the wave crests
Tayfun M.A. J Geophys. Res. (1980)
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PDF of the spectral intensity band-pass filtered
at f 6 Hzwith ??f 1 Hz
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PDF of the spectral intensity Ef (f6 Hz, ?f1Hz)
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Conclusion
Random gravity waves were generated in the
laboratory flume with the inertial interval up to
1m - 1cm. The spectra slopes are not universal
they increase monotonically from about -6 to -4
with the amplitude of forcing. At low forcing
level the character of wave spectra is defined by
the nonlinearity and discreteness effects, at
high and intermediate forcing - by the wave
breaking. PDFs of surface elevation are
non-gaussian at high wave nonlinearity. PDF of
the squared wave elevation filtered in a narrow
frequency interval (spectral energy density)
always has an intermittent tail.
Acknowledgements Hull Environmental Research
Institute References P. Denissenko, S.
Lukaschuk and S. Nazarenko, PRL, July 2007
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Cross-section images water boundary detection
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Boundary detection
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k-spectrum