Spectra of Gravity Wave Turbulence in a Laboratory Flume S Lukaschuk 1, P Denissenko 1, S Nazarenko 2 1 Fluid Dynamics Laboratory, University of Hull 2 Mathematics Institute, University of Warwick WTS workshop, Warwick- Hull, September

1.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 Theoretical prediction for energy spectra of surface gravity waves

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

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 δk ~2  /L:

Numerical experiments: 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. 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

Field experiments: P.A. Hwang, D.W.Wang, Airborne Measurements of surface elevation k-spectra, (2000)

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

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

Small amplitude

Large amplitudes

Typical spectra E  for small and large wave amplitudes A=1.85 cm (  =0.074) A=3.95 cm (  =0.16)

Spectrum slopes vs the wave spectral density E f (f is from the inertial interval) Inset: spectral density Ef vs the energy dissipation rate  =0 “avalanches” and also Phillips  =1/3WTT

Estimation of the Dissipation Rate

PDF of the wave crests Tayfun M.A. J Geophys. Res. (1980)

PDF of the spectral intensity band-pass filtered at f = 6 Hz with  f = 1 Hz

PDF of the spectral intensity E f (f=6 Hz,  f=1Hz)

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

Cross-section images water boundary detection

Boundary detection

k-spectrum