1. Turbulence of Gravity Waves in Laboratory Experiments S Lukaschuk 1, P Denissenko 1, S Nazarenko 2 1 Fluid Dynamics Laboratory, University of Hull 2 Mathematics Institute, University of Warwick

2. Plan Introduction Experimental set-up and methods Measurements of the frequency spectra and PDF for wave elevation Comparison with numerical results and discussion Further experiments

3. Theoretical prediction for surface gravity wave Kolmogorov spectra Kinetic equation approach for WT in an ensemble of weakly interacted low amplitude waves (Theory and numerical experiment - Hasselman, Zakharov, Lvov, Falkovich, Newell, Hasselman, Nazarenko … 1962 - 2006) Assumption: weak nonlinearity random phase (or short correlation length) spatial homogeneity stationary energy flow from large to small scales Kolmogorov spectra for gravity waves in infinite space

4. Phillips Spectrum Surface elevation  -space: asymptotic of sharp wave crests or dimensional analysis.  -space: either dimensional analysis or using Dissipation is determined by sharp wave crests (due to wave breaking) Strong nonlinearity

5. Finite size effects Most exact wave resonances are lost on discrete k-space (Kartashova’1991) “Frozen turbulence” (Pushkarev, Zakharov’2000) Recent numerics by Pokorni et al & Korotkevich et al (2005). To restore resonant interaction, their nonlinear brodening δ  must be greater than the  -grid spacing 2  /L Which in our case means  >1/(kL) 1/4 (Nazarenko, 2005), In numerics, this means 10000x10000 resolution for Intensity  ~0.1.

6. Discrete scenario ( Nazarenko’2005 ) Ineficient cascade at small amplitudes Accumulation of spectrum at the forcing scale until δk reaches to the k-grid spacing 2  /L Excess of energy is released via an avalanche Mean spectrum settles at a critical slope determined by δk ~2  /L, i.e. E ~  -6.

7. Numerical experiments Convincing claims of numerical confirmation of ZF: A.I. Dyachenko, A.O. Korotkevich, V.E. Zakharov, (2003,2004) M. Onorato, V Zakharov et al., (2002). N. Yokoyama, JFM 501 (2004) 169–178. Lvov, Nazarenko and Pokorni (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: 1) nonlinearity truncation at cubic terms 2) artificial numerical dissipation at high k to prevent numerical blowups.

8. N. Yokayama (JFM, 2004) direct numerical simulations Wave action spectra

9. A.Dyachenko, O.Korotkevich, Zakharov (JETP Lett. 2003)

10. Y. Lvov, Nazarenko, Pokorni: numerical experiment: Physica D, 2006

11. Experiments Airborne Measurements of surface elevation k-spectra P.A. Hwang, D.W.Wang (2000)

12. 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. Goals: Long-term: to study transport and mixing generated by wave turbulence Short-term: to characterize statistical properties of waves in a finite system

13. Total Environmental Simulator The Deep, Hull 6 x 12 x 1.6 m water tank 8 panels wave generator 1 m 3 / s – flow rain generator PIV & LDV systems

14. 6 metres 12 metres 90 cm 8 Panel Wave Generator Laser Capacity Probes Rain Generator

15. Wave generation and measurements 2 capacitance probes at distance 40 cm\ Sampling frequency - 50-200 Hz each channel Acquisition time 2000 s

16. Small amplitude

17. Medium amplitude

18. Large amplitudes

20. Energy (elevation) spectrum small amplitude (file 80)

21. PDF of  and  tt PDF of  is close to the Gaussian distribution around the mean value and differs at tail region, s>0, corresponds to the waves with steep tops and flat bottom. PDF of  tt more sensetive to the large wavenumbers and also displays the vertical asymmetry of the wave.

22. N. Yokayama (JFM, 2004) direct numerical simulations PDF of the elevation  and   2 

24. Skewness and Kurtosis for PDF of 2 nd derivative of elevation

25. PDF of filtered envelope: small, medium and high amplitudes

26. Squared amplitude of surface elevation at 6 ± 1 Hz, wire probes Probe 1 Probe 2

27. Numerical results – S.Nazarenko et al (2006)

28. Conclusion Random gravity waves were generated in the laboratory flume with the inertial interval up to 1m - 1cm. The spectra slopes increase monotonically from -7 to -4 with the amplitude of forcing. At low forcing level the character of wave spectra is defined by nonlinearity and discreteness effects, at high and intermediate forcing - by wave breaking. PDFs of surface elevation and its second derivative 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. Questions: Which model should be used to describe our spectra?

29. Acknowledgement The work is supported by Hull Environmental Research Institute