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Turbulence of capillary waves forced by steep gravity waves

Published online by Cambridge University Press:  10 July 2018

M. Berhanu Open the ORCID record for M. Berhanu [Opens in a new window] ,
E. Falcon  and
L. Deike Open the ORCID record for L. Deike [Opens in a new window]
M. Berhanu*
Affiliation:
Matière et Systèmes Complexes (MSC), Université Paris Diderot, CNRS (UMR 7057), 75013 Paris, France
E. Falcon
Affiliation:
Matière et Systèmes Complexes (MSC), Université Paris Diderot, CNRS (UMR 7057), 75013 Paris, France
L. Deike
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, NJ 08544, USA Princeton Environmental Institute, Princeton University, NJ 08544, USA
*
†Email address for correspondence: michael.berhanu@univ-paris-diderot.fr
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Abstract

We study experimentally the dynamics and statistics of capillary waves forced by random steep gravity waves mechanically generated in the laboratory. Capillary waves are produced here by gravity waves from nonlinear wave interactions. Using a spatio-temporal measurement of the free surface, we characterize statistically the random regimes of capillary waves in the spatial and temporal Fourier spaces. For a significant wave steepness (0.2–0.3), power-law spectra are observed both in space and time, defining a turbulent regime of capillary waves transferring energy from the large scale to the small scale. Analysis of temporal fluctuations of the spatial spectrum demonstrates that the capillary power-law spectra result from the temporal averaging over intermittent and strong nonlinear events transferring energy to the small scale in a fast time scale, when capillary wave trains are generated in a way similar to the parasitic capillary wave generation mechanism. The frequency and wavenumber power-law exponents of the wave spectra are found to be in agreement with those of the weakly nonlinear wave turbulence theory. However, the energy flux is not constant through the scales and the wave spectrum scaling with this flux is not in good agreement with wave turbulence theory. These results suggest that theoretical developments beyond the classic wave turbulence theory are necessary to describe the dynamics and statistics of capillary waves in a natural environment. In particular, in the presence of broad-scale viscous dissipation and strong nonlinearity, the role of non-local and non-resonant interactions should be reconsidered.

JFM classification

Waves/Free-surface Flows: Capillary waves Waves/Free-surface Flows: Waves/Free-surface Flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 850 , 10 September 2018 , pp. 803 - 843
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Copyright
© 2018 Cambridge University Press 

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References

Aubourg, Q., Campagne, A., Peureux, C., Ardhuin, F., Sommeria, J., Viboud, S. & Mordant, N. 2017 Three-wave and four-wave interactions in gravity wave turbulence. Phys. Rev. Fluids 2, 114802.Google Scholar
Aubourg, Q. & Mordant, N. 2015 Nonlocal resonances in weak turbulence of gravity–capillary waves. Phys. Rev. Lett. 114, 144501.Google Scholar
Aubourg, Q. & Mordant, N. 2016 Investigation of resonances in gravity–capillary wave turbulence. Phys. Rev. Fluids 1, 023701.Google Scholar
Bedard, R., Lukaschuk, S. & Nazarenko, S. 2013 Non-stationary regimes of surface gravity wave turbulence. JETP Lett. 97 (8), 529–535.Google Scholar
Berhanu, M. & Falcon, E. 2013 Space–time resolved capillary wave turbulence. Phys. Rev. E 87, 033003.Google Scholar
Brazhnikov, M. Y., Kolmakov, G. V., Levchenko, A. A. & Mezhov-Deglin, L. P. 2002 Observation of capillary turbulence on the water surface in a wide range of frequencies. Europhys. Lett. 58, 510–516.Google Scholar
Bréon, F. M. & Henriot, N. 2006 Spaceborne observations of ocean glint reflectance and modeling of wave slope distributions. J. Geophys. Res. 111, C6.Google Scholar
Campagne, A., Hassaini, R., Redor, I., Sommeria, J., Viboud, S. & Mordant, N. 2018 Impact of dissipation on the energy spectrum of experimental turbulence of gravity surface waves. Phys. Rev. Fluids 3, 044801.Google Scholar
Caulliez, G. 2013 Dissipation regimes for short wind waves. J. Geophys. Res. 118, 672–684.Google Scholar
Choi, Y., L’vov, V., Nazarenko, S. & Pokorni, B. 2005 Anomalous probability of large amplitudes in wave turbulence. Phys. Lett. A 339, 361.Google Scholar
Cobelli, P., Petitjeans, P., Maurel, A., Pagneux, V. & Mordant, N. 2009 Space–time resolved wave turbulence in a vibrating plate. Phys. Rev. Lett. 103, 204301.Google Scholar
Cobelli, P., Przadka, A., Petitjeans, P., Lagubeau, G., Pagneux, V. & Maurel, A. 2011 Different regimes for water wave turbulence. Phys. Rev. Lett. 107, 214503.Google Scholar
Connaughton, C., Nazarenko, S. & Newell, A. C. 2003 Dimensional analysis and weak turbulence. Physica D 184, 86–97.Google Scholar
Crapper, G. D. 1957 An exact solution for progressive capillary waves of arbitrary amplitude. J. Fluid Mech. 2, 532.Google Scholar
Deike, L., Bacri, J.-C. & Falcon, E. 2013 Nonlinear waves on the surface of a fluid covered by an elastic sheet. J. Fluid. Mech. 733, 394–413.Google Scholar
Deike, L., Berhanu, M. & Falcon, E. 2012 Decay of capillary wave turbulence. Phys. Rev. E 85, 066311.Google Scholar
Deike, L., Berhanu, M. & Falcon, E. 2014a Energy flux measurement from the dissipated energy in capillary wave turbulence. Phys. Rev. E 89, 023003.Google Scholar
Deike, L., Fuster, D., Berhanu, M. & Falcon, E. 2014b Direct numerical simulations of capillary wave turbulence. Phys. Rev. Lett. 112, 234501.Google Scholar
Deike, L., Miquel, B., Gutiérrez, P., Jamin, T., Semin, B., Berhanu, M., Falcon, E. & Bonnefoy, F. 2015a Role of the basin boundary conditions in gravity wave turbulence. J. Fluid. Mech. 781, 196–225.Google Scholar
Deike, L., Popinet, S. & Melville, W. K. 2015b Capillary effects on wave breaking. J. Fluid. Mech. 769, 541–569.Google Scholar
van Dorn, W. G. 1966 Boundary dissipation of oscillatory waves. J. Fluid Mech. 24, 769.Google Scholar
Dudok de Wit, T. 2003 Spectral and statistical analysis of plasma turbulence: beyond linear techniques. In Space Plasma Simulation. Springer.Google Scholar
Dulov, V. A. & Kosnik, M. V. 2009 Effects of three-wave interactions in the gravity–capillary range of wind waves. Izv. Atmos. Ocean. Phys. 45 (3), 380–391.Google Scholar
Duncan, J. H. 2001 Spilling breakers. Annu. Rev. Fluid Mech. 33, 519–547.Google Scholar
Duncan, J. H., Qiao, H., Philomin, V. & Wenz, A. 1999 Gentle spilling breakers: crest profile evolution. J. Fluid Mech. 379, 191–222.Google Scholar
Düring, G. & Falcón, C. 2009 Symmetry induced four-wave capillary wave turbulence. Phys. Rev. Lett. 103, 174503.Google Scholar
Falcón, C., Falcon, E., Bortolozzo, U. & Fauve, S. 2009 Capillary wave turbulence on a spherical fluid surface in low gravity. Europhys. Lett. 86, 14002.Google Scholar
Falcon, E., Laroche, C. & Fauve, S. 2007 Observation of gravity–capillary wave turbulence. Phys. Rev. Lett. 98, 094503.Google Scholar
Fedorov, A. V. & Melville, W. K. 1998 Nonlinear gravity–capillary waves with forcing and dissipation. J. Fluid. Mech. 354, 1–42.Google Scholar
Fedorov, A. V., Melville, W. K. & Rozenberg, A. 1998 An experimental and numerical study of parasitic capillary waves. Phys. Fluids 10 (6), 1315.Google Scholar
Hammack, J. L. & Henderson, D. M. 1993 Resonant interactions among surface water waves. Annu. Rev. Fluid Mech. 25, 55–97.Google Scholar
Haudin, F., Cazaubiel, A., Deike, L., Jamin, T., Falcon, E. & Berhanu, M. 2016 Experimental study of three-wave interactions among capillary–gravity surface waves. Phys. Rev. E 93, 043110.Google Scholar
Henderson, D. M. & Segur, H. 2013 The role of dissipation in the evolution of ocean swell. J. Geophys. Res. 118, 5074.Google Scholar
Henry, E., Alstrom, P. & Levinsen, M. T. 2000 Prevalence of weak turbulence in strongly driven surface ripples. Europhys. Lett. 52, 27.Google Scholar
Herbert, E., Mordant, N. & Falcon, E. 2010 Observation of the nonlinear dispersion relation and spatial statistics of wave turbulence on the surface of a fluid. Phys. Rev. Lett. 105, 144502.Google Scholar
Humbert, T., Cadot, O., Düring, G., Josserand, C., Rica, S. & Touzé, C. 2013 Wave turbulence in vibrating plates: the effect of damping. Europhys. Lett. 102, 30002.Google Scholar
Hwang, P., Burrage, D. M., Wang, D. W. & Wesson, J. C. 2013 Ocean surface roughness spectrum in high wind condition for microwave backscatter and emission computations. J. Atmos. Ocean. Tech. 30, 2168–2187.Google Scholar
Issenmann, B., Laroche, C. & Falcon, E. 2016 Wave turbulence in a two-layer fluid: coupling between free surface and interface waves. Europhys. Lett. 116, 64005.Google Scholar
Jamin, T., Gordillo, L., Ruiz-Chavarría, G., Berhanu, M. & Falcon, E. 2015 Experiments on generation of surface waves by an underwater moving bottom. Proc. R. Soc. Lond. A 471, 245–259.Google Scholar
Janssen, P. 2004 The Interaction of Ocean Waves and Wind. Cambridge University Press.Google Scholar
Kosnik, M. V., Dulov, V. A. & Kudryavtsev, V. N. 2010 Generation mechanisms for capillary–gravity wind wave spectrum. Izv. Atmos. Ocean. Phys. 46 (3), 369–378.Google Scholar
Lamb, H. 1932 Hydrodynamics. Springer.Google Scholar
Lin, H. J. & Perlin, M. 2001 The velocity and vorticity fields beneath gravity–capillary waves exhibiting parasitic ripples. Wave Motion 33, 245–257.Google Scholar
Longuet-Higgins, M. S. 1963 The generation of capillary waves by steep gravity waves. J. Fluid. Mech. 16 (1), 138–159.Google Scholar
Longuet-Higgins, M. S. 1992 Capillary rollers and bores. J. Fluid. Mech. 240, 659–679.Google Scholar
Longuet-Higgins, M. S. 1995 Parasitic capillary waves: a direct calculation. J. Fluid. Mech. 301, 79–107.Google Scholar
Melville, W. K. & Fedorov, A. V. 2015 The equilibrium dynamics and statistics of gravity–capillary waves. J. Fluid. Mech. 767, 449–466.Google Scholar
Michel, G., Petrelis, F. & Fauve, S. 2016 Acoustic measurement of surface wave damping by a meniscus. Phys. Rev. Lett. 116, 174301.Google Scholar
Miquel, B., Alexakis, A. & Mordant, N. 2014 Role of dissipation in flexural wave turbulence: from experimental spectrum to Kolmogorov–Zakharov spectrum. Phys. Rev. E 89, 062925.Google Scholar
Miquel, B. & Mordant, N. 2011 Nonlinear dynamics of flexural wave turbulence. Phys. Rev. E 84, 066607.Google Scholar
Mordant, N. 2010 Fourier analysis of wave turbulence in a thin elastic plate. Eur. Phys. J. B 76, 537–545.Google Scholar
Nazarenko, S. 2011 Wave Turbulence. Springer.Google Scholar
Nazarenko, S. & Lukaschuk, S. 2016 Wave turbulence on water surface. Annu. Rev. Condens. Matt. Phys. 7, 61.Google Scholar
Nazarenko, S., Lukaschuk, S., McLelland, S. & Denissenko, P. 2010 Statistics of surface gravity wave turbulence in the space and time domains. J. Fluid Mech. 642, 395.Google Scholar
Newell, A. C. & Rumpf, B. 2011 Wave turbulence. Annu. Rev. Fluid Mech. 43, 59.Google Scholar
Pan, Y.2017 Understanding of weak turbulence of capillary waves. PhD thesis, Massachusetts Institute of Technology.Google Scholar
Pan, Y. & Yue, D. K. P. 2014 Direct numerical investigation of turbulence of capillary waves. Phys. Rev. Lett. 113, 094501.Google Scholar
Pan, Y. & Yue, D. K. P. 2015 Decaying capillary wave turbulence under broad-scale dissipation. J. Fluid. Mech. 780, R1–1.Google Scholar
Pan, Y. & Yue, D. K. P. 2017 Understanding discrete capillary-wave turbulence using a quasi-resonant kinetic equation. J. Fluid. Mech. 816, R1–1.Google Scholar
Perlin, M., Jiang, L. & Ting, C.-H. 1993 On parasitic capillary waves generated by steep gravity waves: an experimental investigation with spatial and temporal measurements. J. Fluid. Mech. 255, 597–620.Google Scholar
Perlin, M. & Schultz, W. W. 2000 Capillary effects on surface waves. Annu. Rev. Fluid Mech. 32, 241–274.Google Scholar
Punzmann, H., Shats, M. G. & Xia, H. 2009 Phase randomization of three-wave interactions in capillary waves. Phys. Rev. Lett. 103, 064502.Google Scholar
Pushkarev, A. N. & Zakharov, V. E. 1996 Turbulence of capillary waves. Phys. Rev. Lett. 76, 3320.Google Scholar
Pushkarev, A. N. & Zakharov, V. E. 2000 Turbulence of capillary waves – theory and numerical simulation. Physica D 135, 98.Google Scholar
Schultz, W. W., Den-Broeck, J.-M. V., Jiang, L. & Perlin, M. 1998 Highly nonlinear standing water waves with small capillary effect. J. Fluid. Mech. 369, 253–272.Google Scholar
Snouck, D., Westra, M.-T. & van de Water, W. 2009 Turbulent parametric surface waves. Phys. Fluids 21, 025102.Google Scholar
van Staveren, H. J., Moes, C. J. M., van Marie, J., Prahl, S. A. & van Gemert, M. J. C. 1991 Light scattering in intralipid-10 % in the wavelength range of 400–1100 nm. Appl. Opt. 30, 4507.Google Scholar
Stiassnie, M. 1996 On the equilibrium spectrum of gravity-capillary waves. J. Phys. Oceanogr. 26 (6), 1093–1098.Google Scholar
Thielicke, W. & Stamhuis, E. J. 2014 PIVlab towards user-friendly, affordable and accurate digital particle image velocimetry in MATLAB. J. Open Res. Softw. 2 (1), e30.Google Scholar
Tsai, W.-T. & Hung, L.-P. 2010 Enhanced energy dissipation by parasitic capillaries on short gravity–capillary waves. J. Phys. Oceanogr. 40, 2435–2450.Google Scholar
Watson, K. M. 1999 Interaction of capillary waves with longer waves. Part 2. Applications to waves in two surface dimensions and to waves in shallow water. J. Fluid Mech. 397, 99–117.Google Scholar
Watson, K. M. & Buchsbaum, S. B. 1996 Interaction of capillary waves with longer waves. Part 1. General theory and specific applications to waves in one dimension. J. Fluid Mech. 321, 87–120.Google Scholar
Watson, K. M. & McBride, J. B. 1993 Excitation of capillary waves by longer waves. J. Fluid Mech. 250, 103.Google Scholar
Whitham, G. B. 1999 Linear and Nonlinear Waves. Wiley-Interscience.Google Scholar
Wright, W. B., Budakian, R., Pine, D. J. & Putterman, S. J. 1997 Imaging of intermittency in ripple-wave turbulence. Science 278, 1609.Google Scholar
Wright, W. B., Budakian, R. & Putterman, S. J. 1996 Diffusing light photography of fully developed isotropic ripple turbulence. Phys. Rev. Lett. 76, 4528.Google Scholar
Xia, H., Maimbourg, T., Punzmann, H. & Shats, M. 2012 Oscillon dynamics and rogue wave generation in Faraday surface ripples. Phys. Rev. Lett. 109, 114502.Google Scholar
Xia, H., Shats, M. G. & Punzmann, H. 2010 Modulation instability and capillary wave turbulence. Europhys. Lett. 91, 14002.Google Scholar
Yokoyama, N. & Takaoka, M. 2014 Single-wave-number representation of nonlinear energy spectrum in wave turbulence of Föppl–von Kármán equation: energy decomposition analysis and energy budget. Phys. Rev. E 90, 063004.Google Scholar
Zakharov, V. E. 2010 Energy balance in a wind-driven sea. Phys. Scr. T 142, 014052.Google Scholar
Zakharov, V. E. & Filonenko, N. N. 1967a Energy spectrum for stochastic oscillations of the surface of a liquid. Sov. Phys. Dokl. 11, 881.Google Scholar
Zakharov, V. E. & Filonenko, N. N. 1967b Weak turbulence of capillary waves. J. Appl. Mech. Tech. Phys. 8, 37.Google Scholar
Zakharov, V. E., L’vov, V. & Falkovich, G. 1992 Kolmogorov Spectra of Turbulence. Springer.Google Scholar
Zhang, X. 2002 Enhanced dissipation of short gravity and gravity capillary waves due to parasitic capillaries. Phys. Fluids 14, 81.Google Scholar
Zhang, X. & Cox, C. S. 1994 Measuring the two-dimensional structure of a wavy water surface optically: a surface gradient detector. Exp. Fluids 17, 225.Google Scholar
Zonta, F., Onorato, M. & Soldati, A. 2016 Decay of gravity–capillary waves in air/water sheared turbulence. Intl J. Heat Fluid Flow 61, 137–144.Google Scholar
Zonta, F., Soldati, A. & Onorato, M. 2015 Growth and spectra of gravity–capillary waves in countercurrent air/water turbulent flow. J. Fluid. Mech. 777, 245–259.Google Scholar

Berhanu et al. supplementary movie 1

3D Free-surface reconstruction for the highest forcing amplitude σ_h =3.6 mm. The animation corresponds to 2.5 s in real time. Steep gravity waves generate smaller capillary waves.

Download Berhanu et al. supplementary movie 1(Video)
Video 16.8 MB

Berhanu et al. supplementary movie 2

Simultaneous plot of wave-field gradient map and spatial spectrum f wave elevation. σ_h =3.6 mm. The animation corresponds to 2.5 s in real time. Transient generations of capillary wave train are associated with sudden increases of spatial spectrum at small scale.

Download Berhanu et al. supplementary movie 2(Video)
Video 36.2 MB