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Explosive ripple instability due to incipient wave breaking

Published online by Cambridge University Press:  28 January 2019

Alexei A. Mailybaev Open the ORCID record for Alexei A. Mailybaev [Opens in a new window]  and
André Nachbin Open the ORCID record for André Nachbin [Opens in a new window]
Alexei A. Mailybaev*
Affiliation:
Instituto Nacional de Matemática Pura e Aplicada – IMPA, Rio de Janeiro, CEP 22460-320, Brazil
André Nachbin
Affiliation:
Instituto Nacional de Matemática Pura e Aplicada – IMPA, Rio de Janeiro, CEP 22460-320, Brazil
*
†Email address for correspondence: [email protected]
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Abstract

Considering two-dimensional potential ideal flow with a free surface and finite depth, we study the dynamics of small-amplitude and short-wavelength wavetrains propagating in the background of a steepening nonlinear wave. This can be seen as a model for small ripples developing on the slopes of breaking waves in the surf zone. Using the concept of wave action as an adiabatic invariant, we derive an explicit asymptotic expression for the change of ripple steepness. Through this expression, nonlinear effects are described using the intrinsic frequency and intrinsic gravity along Lagrangian (material) trajectories on a free surface. We show that strong compression near the tip on the wave leads to an explosive ripple instability. This instability may play an important role in the understanding of fragmentation and whitecapping at the surface of breaking waves. Analytical results are confirmed by numerical simulations using a potential theory model.

JFM classification

Waves/Free-surface Flows: Surface gravity waves
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 863 , 25 March 2019 , pp. 876 - 892
Copyright
© 2019 Cambridge University Press 

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References

Agafontsev, D. S., Kuznetsov, E. A. & Mailybaev, A. A. 2015 Development of high vorticity structures in incompressible 3D Euler equations. Phys. Fluids 27 (8), 085102.10.1063/1.4927680Google Scholar
Baker, G. R., Meiron, D. I. & Orszag, S. A. 1982 Generalized vortex methods for free-surface flow problems. J. Fluid Mech. 123, 477–501.10.1017/S0022112082003164Google Scholar
Baker, G. R. & Xie, C. 2011 Singularities in the complex physical plane for deep water waves. J. Fluid Mech. 685, 83–116.10.1017/jfm.2011.283Google Scholar
Bretherton, F. P. & Garrett, C. J. R. 1968 Wavetrains in inhomogeneous moving media. Proc. R. Soc. Lond. A 302 (1471), 529–554.Google Scholar
Bühler, O. 2014 Waves and Mean Flows. Cambridge University Press.10.1017/CBO9781107478701Google Scholar
Castro, A., Córdoba, D., Fefferman, C. L, Gancedo, F. & Gómez-Serrano, J. 2012 Splash singularity for water waves. Proc. Natl Acad. Sci. USA 109 (3), 733–738.10.1073/pnas.1115948108Google Scholar
Ceniceros, H. D. & Hou, T. Y. 1999 Dynamic generation of capillary waves. Phys. Fluids 11 (5), 1042–1050.10.1063/1.869975Google Scholar
Dyachenko, A. I., Kuznetsov, E. A., Spector, M. D. & Zakharov, V. E. 1996a Analytical description of the free surface dynamics of an ideal fluid (canonical formalism and conformal mapping). Phys. Lett. A 221 (1), 73–79.10.1016/0375-9601(96)00417-3Google Scholar
Dyachenko, A. I., Zakharov, V. E. & Kuznetsov, E. A. 1996b Nonlinear dynamics of the free surface of an ideal fluid. Plasma Phys. Rep. 22 (10), 829–840.Google Scholar
Dyachenko, S. & Newell, A. C. 2016 Whitecapping. Stud. Appl. Maths 137 (2), 199–213.10.1111/sapm.12126Google Scholar
Grilli, S. T. & Svendsen, I. A. 1990 Corner problems and global accuracy in the boundary element solution of nonlinear wave flows. Engng Anal. Bound. Elem. 7 (4), 178–195.10.1016/0955-7997(90)90004-SGoogle Scholar
Hou, T. Y. 2009 Blow-up or no blow-up? A unified computational and analytic approach to 3D incompressible Euler and Navier–Stokes equations. Acta Numerica 18, 277–346.10.1017/S0962492906420018Google Scholar
Kuznetsov, E. A., Spector, M. D. & Zakharov, V. E. 1994 Formation of singularities on the free surface of an ideal fluid. Phys. Rev. E 49 (2), 1283–1290.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1987 Fluid Mechanics. Pergamon.Google Scholar
Lighthill, J. 2001 Waves in Fluids. Cambridge University Press.Google Scholar
Longuet-Higgins, M. S. 1983 Bubbles, breaking waves and hyperbolic jets at a free surface. J. Fluid Mech. 127, 103–121.10.1017/S0022112083002645Google Scholar
Longuet-Higgins, M. S. 1995 Parasitic capillary waves: a direct calculation. J. Fluid Mech. 301, 79–107.10.1017/S0022112095003818Google Scholar
Longuet-Higgins, M. S. & Cokelet, E. D. 1976 The deformation of steep surface waves on water-I. a numerical method of computation. Proc. R. Soc. Lond. A 350 (1660), 1–26.10.1098/rspa.1976.0092Google Scholar
Peregrine, D. H. 1976 Interaction of water waves and currents. Adv. Appl. Mech. 16, 9–117.10.1016/S0065-2156(08)70087-5Google Scholar
Peregrine, D. H. 1983 Breaking waves on beaches. Annu. Rev. Fluid Mech. 15 (1), 149–178.10.1146/annurev.fl.15.010183.001053Google Scholar
Ribeiro, R., Milewski, P. A. & Nachbin, A. 2017 Flow structure beneath rotational water waves with stagnation points. J. Fluid Mech. 812, 792–814.10.1017/jfm.2016.820Google Scholar
Villermaux, E. 2007 Fragmentation. Annu. Rev. Fluid Mech. 39, 419–446.10.1146/annurev.fluid.39.050905.110214Google Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley.Google Scholar
Wu, S. 1997 Well-posedness in sobolev spaces of the full water wave problem in 2-d. Invent. Math. 130 (1), 39–72.10.1007/s002220050177Google Scholar
Yu, J. & Howard, L. N. 2012 Exact Floquet theory for waves over arbitrary periodic topographies. J. Fluid Mech. 712, 451–470.10.1017/jfm.2012.432Google Scholar
Zakharov, V. E., Dyachenko, A. I. & Vasilyev, O. A. 2002 New method for numerical simulation of a nonstationary potential flow of incompressible fluid with a free surface. Eur. J. Mech. (B/Fluids) 21 (3), 283–291.10.1016/S0997-7546(02)01189-5Google Scholar
Zeff, B. W., Kleber, B., Fineberg, J. & Lathrop, D. P. 2000 Singularity dynamics in curvature collapse and jet eruption on a fluid surface. Nature 403 (6768), 401–404.10.1038/35000151Google Scholar