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Breaking of modulated wave groups: kinematics and energy dissipation processes

Published online by Cambridge University Press:  18 September 2018

Francesco De Vita Open the ORCID record for Francesco De Vita [Opens in a new window] ,
Roberto Verzicco Open the ORCID record for Roberto Verzicco [Opens in a new window]  and
Alessandro Iafrati
Francesco De Vita*
Affiliation:
Dipartimento di Ingegneria Industriale, University of Rome ‘Tor Vergata’, Via del Politecnico 1, Rome 00133, Italy CNR-INSEAN, Via di Vallerano 139, Rome 00128, Italy
Roberto Verzicco
Affiliation:
Dipartimento di Ingegneria Industriale, University of Rome ‘Tor Vergata’, Via del Politecnico 1, Rome 00133, Italy Physics of Fluids Group and Twente Max Planck Center, Department of Science and Technology, Mesa+ Institute, and J. M. Burgers Center for Fluid Dynamics, University of Twente, 7500 AE Enschede, The Netherlands
Alessandro Iafrati
Affiliation:
CNR-INSEAN, Via di Vallerano 139, Rome 00128, Italy
*
†Email address for correspondence: fnc.devita@gmail.com
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Abstract

The two-dimensional flow induced by the breaking of modulated wave trains is numerically investigated using the open source software Gerris (Popinet, J. Comput. Phys., vol. 190, 2003, pp. 572–600; J. Comput. Phys., vol. 228, 2009, pp. 5838–5866. The two-phase flow is modelled by the Navier–Stokes equations for a single fluid with variable density and viscosity, coupled with a volume-of-fluid (VOF) technique for the capturing of the interface dynamics. The breaking is induced through the Benjamin–Feir mechanism, by adding two sideband disturbances to a fundamental wave component. The evolution of the wave system is simulated starting from the initial condition until the end of the breaking process, and the role played by the initial wave steepness is investigated. As already noted in previous studies as well as in field observations, it is found that the breaking is recurrent and several breaking events are needed before the breaking process finally ceases. The down-shifting of the fundamental component to the lower sideband is made irreversible by the breaking. At the end of the breaking process the magnitude of the lower sideband component is approximately 80 % of the initial value of the fundamental one. The time histories of the energy content in water and the energy dissipation are analysed. The whole breaking process dissipates a fraction of between twenty and twenty-five per cent of the pre-breaking energy content, independently of the initial steepness. The energy contents of the different waves of the group are evaluated and it is found that after the breaking, the energy of the most energetic wave of the group decays as $t^{-1}$.

JFM classification

Geophysical and Geological Flows: Air/sea interactions Multiphase and Particle-laden Flows: Multiphase flow Waves/Free-surface Flows: Wave breaking
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 855 , 25 November 2018 , pp. 267 - 298
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Copyright
© 2018 Cambridge University Press 

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References

Agrawal, Y. C., Terray, E. A., Donelan, M. A., Hwang, P. A., Williams, A. J. III, Drennan, W. M., Kaham, K. K. & Krtaigorodskii, S. A. 1992 Enhanced dissipation of kinetic energy beneath surface waves. Nature 359, 219–220.Google Scholar
Babanin, A. V. 2011 Breaking and Dissipation of Ocean Surface Waves. Cambridge University Press.Google Scholar
Benjamin, T. B. & Feir, J. E. 1967 The disintegration of wave trains on deep water. Part 1. Theory. J. Fluid Mech. 27, 417–430.Google Scholar
Bonmarin, P. 1989 Geometric properties of deep-water breaking waves. J. Fluid Mech. 209, 405–433.Google Scholar
Buckley, M. P. & Veron, F. 2016 Structure of the airflow above surface waves. J. Phys. Oceanogr. 46, 1377–1397.Google Scholar
Cavaleri, L. 2006 Wave modelling: where to go in future. Bull. Am. Meteorol. Soc. 87, 207–214.Google Scholar
Cavaleri, L., Fox-Kemper, B. & Hemer, M. 2012 Wind waves in the coupled climate system. Bull. Am. Meteorol. Soc. 93, 1651–1661.Google Scholar
Chen, G., Kharif, C., Zaleski, S. & Li, J. 1999 Two dimensional Navier–Stokes simulation of breaking waves. Phys. Fluids 11, 121–133.Google Scholar
Deane, G. B. & Stokes, M. D. 2002 Scale dependence of bubble creation mechanisms in breaking waves. Nature 418, 839844.Google Scholar
Deike, L., Melville, W. K. & Popinet, S. 2016 Air entrainment and bubble statistics in breaking waves. J. Fluid Mech. 801, 91–129.Google Scholar
Deike, L., Popinet, S. & Melville, W. K. 2015 Capillary effects on wave breaking. J. Fluid Mech. 769, 541–569.Google Scholar
Derakthi, M. & Kirby, J. T. 2014 Bubble entrainment and liquid-bubble interaction under unsteady breaking waves. J. Fluid Mech. 761, 464–506.Google Scholar
Derakthi, M. & Kirby, J. T. 2016 Breaking-onset, energy and momentum flux in unsteady focused wave packets. J. Fluid Mech. 790, 553–581.Google Scholar
Diorio, J. D., Liu, X. & Duncan, J. H. 2009 An experimental investigation of incipient spilling breakers. J. Fluid Mech. 633, 271–283.Google Scholar
Dold, J. W. & Peregrine, D. H. 1985 An efficient boundary-integral method for steep unsteady water waves. In Numerical Methods For Fluid Dynamic II, pp. 671–679. Clarendon.Google Scholar
Dold, J. W. & Peregrine, D. H. 1986 Water-wave modulation. In Coast Engineering, 20th International Conference on Coastal Engineering, Taipei, Taiwan, pp. 163–175. American Society of Civil Engineers.Google Scholar
Dommermuth, D. G. & Yue, D. K. P. 1988 A high-order spectral method for the study of nonlinear gravity waves. J. Fluid Mech. 184, 267–288.Google Scholar
Donelan, M. A., Longuet-Higgins, M. S. & Turner, J. S. 1972 Periodicity in whitecaps. Nature 239, 449–451.Google Scholar
Drazen, A. D. & Melville, W. K. 2009 Turbulence and mixing in unsteady breaking surface waves. J. Fluid Mech. 628, 85–119.Google Scholar
Drazen, A. D., Melville, W. K. & Lenain, L. 2008 Inertia scaling of dissipation in unsteady breaking waves. J. Fluid Mech. 611, 307–332.Google Scholar
Duncan, J. H. 1981 An experimental investigation of breaking waves produced by towed hydrofoil. Proc. R. Soc. Lond. 377, 331–348.Google Scholar
Duncan, J. H. 1983 The breaking and nonbreaking wave resistence of a two-dimensional hydrofoil. J. Fluid Mech. 126, 507–520.Google Scholar
Duncan, J. H. 2001 Spilling breakers. Annu. Rev. Fluid Mech. 33, 519–547.Google Scholar
Galchenko, A., Babanin, A. V., Chalikov, D. & Young, I. R. 2010 Modulational instabilities and breaking strenght for deep-water wave groups. J. Phys. Oceanogr. 40, 2313–2324.Google Scholar
Hwang, P. A. 2009 Estimating the effective energy transfer velocity at air–sea interface. J. Geophys. Res. 114, C11.Google Scholar
Iafrati, A. 2009 Numerical study of the effects of the breaking intensity on wave breaking flows. J. Fluid Mech. 622, 371–411.Google Scholar
Iafrati, A. 2011 Energy dissipation mechanisms in wave breaking processes: spilling and highly aerated plunging breaking events. J. Geophys. Res. 116, C7.Google Scholar
Iafrati, A., Babanin, A. & Onorato, M. 2013 Modulational instability, wave breaking and formation of large-scale dipoles in the atmosphere. Phys. Rev. Lett. 110, 184504.Google Scholar
Iafrati, A., Babanin, A. & Onorato, M. 2014 Modeling of oceanatmosphere interaction phenomena during the breaking of modulated wave trains. J. Comput. Phys. 271, 151–171.Google Scholar
Iafrati, A. & Campana, E. 2005 Free-surface fluctuations behind microbreakers: space-time behaviour and subsurface flow field. J. Fluid Mech. 529, 311–347.Google Scholar
Iafrati, A., De Vita, F., Toffoli, A. & Alberello, A. 2015 Strongly nonlinear phenomena in extreme waves. SNAME Trans. 123, 17–38.Google Scholar
Iafrati, A., Onorato, M. & Babanin, A. 2012 Analysis of wave breaking events generated as a result of a modulational instability. In Proceedings of the 29th ONR Symposium on Naval Hydrodynamics. Office of Naval Research.Google Scholar
Janssen, P. 2009 The Interaction of Ocean Waves and Wind. Cambridge University Press.Google Scholar
Kimmoun, O., Hsu, H. C., Branger, H., Li, M. S., Chen, Y. Y., Kharif, C., Onorato, M., Kelleher, E. J. R., Kibler, B., Akhmediev, N. & Chabchoub, A. 2016 Modulation instability and phase-shifted fermi-pasta-ulam recurrence. Sci. Rep. 6, 28516.Google Scholar
Lake, B. M., Yuen, H. C., Rungaldier, H. & Ferguson, W. E. 1977 Nonlinear deep-water waves: theory and experiment. Part 2. Evolution of a continuous wave train. J. Fluid Mech. 83, 49–74.Google Scholar
Lamarre, E. & Melville, W. K. 1991 Air entrainment and dissipation in breaking waves. Nature 351, 469–471.Google Scholar
Lamont-Smith, T., Fuchs, J. & Tulin, M. P. 2003 Radar investigation of the structure of wind waves. J. Oceanogr. 59, 49–63.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1959 Fluid Mechanics. Pergamon.Google Scholar
Landrini, M., Oshiri, O., Waseda, T. & Tulin, M. P. 1998 Long time evolution of gravity wave system. In Proceedings of the 13th International Workshop Water Waves Floating Bodies; www.iwwwfb.org.Google Scholar
Lubin, P. & Glockner, S. 2015 Numerical simulations of three-dimensional plunging breaking waves: generation and evolution of aerated vortex filaments. J. Fluid Mech. 767, 364–393.Google Scholar
Lubin, P., Vincent, S., Abadie, S. & Caltagirone, J. P. 2006 Three-dimensional large eddy simulation of air entrainment under plungin breaking waves. Coast. Engng 53, 631–655.Google Scholar
Ma, Y., Dong, G., Perlin, M., Ma, X. & Wang, G. 2012 Experimental investigation on the evolution of the modulation instability with dissipation. J. Fluid Mech. 711, 101–121.Google Scholar
McLean, J. W. 1982 Instabilities of finite-amplitude water waves. J. Fluid Mech. 114, 315–330.Google Scholar
Melville, W. K. 1982 The instability and breaking of deep-water waves. J. Fluid Mech. 115, 165–185.Google Scholar
Melville, W. K. 1983 Wave modulation and breakdown. J. Fluid Mech. 128, 489–506.Google Scholar
Melville, W. K. 1994 Energy dissipation by breaking waves. J. Phys. Oceanogr. 24, 2041–2049.Google Scholar
Melville, W. K., Veron, F. & White, C. J. 2002 The velocity field under breaking waves: coherent structures and turbulence. J. Fluid Mech. 454, 203–233.Google Scholar
Perlin, M., Choi, W. & Tian, Z. 2013 Breaking waves in deep and intermediate waters. Annu. Rev. Fluid Mech. 45, 115–145.Google Scholar
Phillips, O. M. 1977 The Dynamics of the Upper Ocean. Cambridge University Press.Google Scholar
Phillips, O. M. 1985 Statistical and spectral properties of the equilibrium range in the spectrum of wind-generated gravity waves. J. Fluid Mech. 156, 505–531.Google Scholar
Phillips, O. M., Posner, F. L. & Hansen, J. P. 2001 High range resolution radar measurements of the speed distribution of breaking events in wind-generated ocean waves: surface impulse and wave energy dissipation rates. J. Phys. Oceanogr. 31, 450–460.Google Scholar
Pizzo, N. E., Deike, L. & Melville, W. K. 2016 Current generation by deep-water breaking waves. J. Fluid Mech. 803, 275–291.Google Scholar
Pizzo, N. E. & Melville, W. K. 2013 Vortex generation by deep-water breaking waves. J. Fluid Mech. 734, 198–218.Google Scholar
Popinet, S. 2003 Gerris: a tree-based adaptive solver for the incompressible Euler equations in complex geometries. J. Comput. Phys. 190, 572–600.Google Scholar
Popinet, S. 2009 An accurate adaptive solver for the surface-tension-driven interfacial flows. J. Comput. Phys. 228, 5838–5866.Google Scholar
Qiao, H. & Duncan, J. H. 2001 Gentle spilling breakers: crest flow-field evolution. J. Fluid Mech. 439, 57–85.Google Scholar
Rainey, R. C. T. & Longuet-Higgins, M. S. 2006 A close one-term approximation to the highest stokes wave on deep water. Ocean Engng 33, 2012–2024.Google Scholar
Rapp, R. J. & Melville, W. K. 1990 Laboratory measurements of deep-water breaking waves. Phil. Trans. R. Soc. Lond. A 331, 735.Google Scholar
Reul, N., Branger, H. & Giovanangeli, J. P. 1999 Air flow separation over unsteady breaking waves. Phys. Fluids 11, 1959–1961.Google Scholar
Reul, N., Branger, H. & Giovanangeli, J. P. 2008 Air flow structure over short-gravity breaking water waves. Boundary-Layer Meteorol. 126, 477–505.Google Scholar
Romero, L., Melville, W. K. & Kleiss, J. M. 2012 Spectral energy dissipation due to surface breaking. Am. Metereol. Soc. 42, 1421–1444.Google Scholar
Skandrani, C., Kharif, C. & Poitevin, J. 1996 Nonlinear evolution of water surface waves: the frequency down-shift phenomenon. In Mathematical Problems in the Theory of Water Waves, vol. 220, pp. 157–171. American Mathematical Society.Google Scholar
Sutherland, P. & Melville, W. K. 2013 Field mesurements and scaling of ocean surface wave-breaking statistics. Geophys. Res. Lett. 40, 3074–3079.Google Scholar
Techet, A. H. & McDonald, A. K. 2005 High speed PIV of breaking waves on both sides of the air–water interface. In Proceedings of the 6th International Symposium on Particle Image Velocimetry, Pasadena, CA, USA, Institute of Physics.Google Scholar
Toffoli, A., Babanin, A., Onorato, M. & Waseda, T. 2010 Maximum steepness of oceanic waves: field and laboratory experiments. Geophys. Res. Lett. 37, L05603.Google Scholar
Tulin, M. P. 1996 Breaking of ocean waves and downshifting. In Waves and Nonlinear Processes in Hydrodynamics (ed. Grue, J., Gjevik, B. & Weber, J. E.), pp. 177–190. Springer.Google Scholar
Tulin, M. P. & Waseda, T. 1999 Laboratory observation of wave group evolution, including breaking effects. J. Fluid Mech. 378, 197–232.Google Scholar
Veron, F. 2015 Ocean spray. Annu. Rev. Fluid Mech. 47, 507–538.Google Scholar
Wang, Z., Yand, J. & Stern, F. 2016 High-fidelty simulations of bubble, droplet and spray formation in breaking waves. J. Fluid Mech. 792, 307–327.Google Scholar
Wanninkhof, R., Asher, W. E., Ho, D. T., Sweeny, C. & McGillis, W. R. 2009 Advances in qunatifying air–sea gas exchange and environmental forcing. Annu. Rev. Mar. Sci. 1, 213–244.Google Scholar
Yuen, H. C. & Ferguson, W. E. 1978 Fermipastaulam recurrence in the two-space dimensional nonlinear Schrödinger equation. Phys. Fluids 21, 2116–2118.Google Scholar