Wind-wave growth in the laboratory studies S. I. Badulin (1) and G. Caulliez (2) (1) P.P. Shirshov Institute of Oceanology, Moscow, Russia (2) Institut de Recherche sur les Phénomènes Hors Equilibre, Marseille, France

Experimental conditions various fetches ranging from 2 to 26.5 m: X = 2, 4, 6, 9, 13, 18, 26 m various wind speeds U ref ranging from 2.5 m/s to 13 m/s: U 10  2.5 to 17 m/s wavemaker carriage X HR or fast-speed video cameras laser sheet laser slope gauge The large IRPHE-Luminy wind-wave tank water tank: L= 40 m, l = 2.6 m, d = 0.9 m air tunnel: L= 40 m, l = 3.2 m, h = 1.5 m U10  2.5 to 17 m/s

A crazy question Can we reproduce wind-sea growth in the wind-wave tank?

A regular answer NO

The tiny IRPHE-Luminy wind-wave tank Length = O(10 2 ) wavelengths Width = O(10) wavelengths Height = O(10) wavelengths Depth = O(10) wavelengths Problems Scales Capillarity Drift currents Air flow etc

A. We certainly cannot model growth of wind- driven seas in wind-wave channels

Why Wave growth in wave tanks is consistent both qualitatively and quantitatively with wave growth in open sea? Ex.: The Toba 3/2 law (Toba, 1972, 1973) H s =B(gu * ) 1/2 T s 3/2 B=0.061 May be it is just happy chance when formally invalid tool works well

Try to answer within the statistical approach (formally invalid) The kinetic equation for wind-driven seas (the Hasselmann equation) 1.Nonlinear transfer is described from `the first principles’ 2.External forcing is parameterized by empirical formulas

Try to answer within the weakly turbulent self- similar wave growth law (Badulin et al., 2007) The split balance of wind-driven seas Hyp. Nonlinear transfer dominates over wind input and dissipation 1.Conservative Hasselmann equation assures universality (self-similarity) of nonlinear transfer 2.External forcing (spectral fluxes) controls evolution as total quantities. Details of the forcing are of no importance

 Total energy  p - peak frequency  ss - self-similarity parameter Self-similar solutions dictates Kolmogorov-like wave-growth law Badulin, Babanin, Resio & Zakharov, JFM, Integral net wave input is rigidly linked to instantaneous wave parameters: characteristic wave energy and wave frequency; 2.Dependencies of sea wave growth of field experiments are consistent with the law

Measurements were carried out in the Large IRPHE-Luminy wind-wave channel in 2006 with no reference to the problem of growth of wind-driven seas Experimental conditions various fetches ranging from 2 to 26.5 m: X = 2, 4, 6, 9, 13, 18, 26 m various wind speeds U ref ranging from 2.5 m/s to 13 m/s: U 10  2.5 to 17 m/s Tools: wave capacity probes, laser slope gauge

Our data cover wider range of conditions (cf. Toba, 1972 `Traditional’ wave speed scaling gives high dispersion (good in logaritmic axes only) Blue stars – data by Toba (1972) New approach – new knowledge ?

Weakly turbulent scaling (energy-to-flux) Not so bad if locally measured frequency is used (perfect! Axes are linear!)  ss  E  p 4 /g 2 =Steepness 2

dominant wavelength X = 6 m: d  30 cm X = 13 m: d  45 cm X = 26 m: d  80 cm total mean square slope E*  4 /g 2 = mss d  E/  X  2 /2g =  X Capillary and drift effects are included, i.e taken into account in mss d Below d  30 cm, gravity- capillary and capillary- gravity waves: action of T/  and shear drift effects Problems of the new presentation: derivatives and instantaneous quantities (wave heights and frequencies)

Better than perfect !  ss  2 =Steepness 2 Rate of energy=(d(a 2 k 2 )/dt/(2  p )) 1/3

Concl.: We showed consistency of the wind-channel data and the weakly turbulent law (Badulin et al., 2007) The talk is over (?) No, it is just the very begining We are the best !

Try to estimate net wave input and scale it in physically consistent way The weakly turbulent Kolmogorov-like law gives us a chance

Very preliminary results: Wave input vs u * or vs C p (Air flow vs wave dynamics) ~      Different symbols are used for different wind speeds Scaling in wave phase speed looks more attractive

The well-known Toba’s law as a particular case of weakly turbulent wind-wave growth Let One can estimate energy production from instantaneous H s, T s

Very preliminary results Wave input normalized by the Toba input vs C p The scaling is relevant to constant in time production of wave energy

The less-known Hasselmann, Ross, Muller & Sell, 1976 ( “Special solutions” for a parametric wave model) Let - total wave momentum Get See also Resio, Long, Vincent, JGR 2004

Very preliminary results Wave input for scaling Resio et al Relevant to constant in time production of wave momentum

Summary Wind-wave tank data (Toba 1973, Caulliez 2006) are consistent with weakly turbulent scaling – Kolmogorov’s energy-to-flux rigid link The weakly turbulent approach and the new data allow one –to identify qualitatively different physical regimes of wave growth; –to describe quantitatively wind-wave interaction

Summary Wind-wave tanks can give us real physics at unreal conditions Marseille, 07/01/2009

Welcome to Marseille !

Motivation “With a wider perspective and in the long term, we need the wild horse that comes out with unconventional ideas…” L. Cavaleri et al. / Progress in oceanography, 75 (2007) 603–674