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Super compact equation for water waves

Published online by Cambridge University Press:  12 September 2017

A. I. Dyachenko Open the ORCID record for A. I. Dyachenko [Opens in a new window] ,
D. I. Kachulin  and
V. E. Zakharov
A. I. Dyachenko*
Affiliation:
Landau Institute for Theoretical Physics, 142432, Chernogolovka, Russia Novosibirsk State University, 630090, Novosibirsk-90, Russia
D. I. Kachulin
Affiliation:
Novosibirsk State University, 630090, Novosibirsk-90, Russia
V. E. Zakharov
Affiliation:
Landau Institute for Theoretical Physics, 142432, Chernogolovka, Russia Novosibirsk State University, 630090, Novosibirsk-90, Russia Department of Mathematics, University of Arizona, Tucson, AZ 857201, USA Physical Institute of RAS, Leninskiy prospekt, 53, Moscow, 119991, Russia Space Research Institute of RAS, 84/32 Profsoyuznaya Str, Moscow, 117997, Russia
*
†Email address for correspondence: [email protected]
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Abstract

Mathematicians and physicists have long been interested in the subject of water waves. The problems formulated in this subject can be considered fundamental, but many questions remain unanswered. For instance, a satisfactory analytic theory of such a common and important phenomenon as wave breaking has yet to be developed. Our knowledge of the formation of rogue waves is also fairly poor despite the many efforts devoted to this subject. One of the most important tasks of the theory of water waves is the construction of simplified mathematical models that are applicable to the description of these complex events under the assumption of weak nonlinearity. The Zakharov equation, as well as the nonlinear Schrödinger equation (NLSE) and the Dysthe equation (which are actually its simplifications), are among them. In this article, we derive a new modification of the Zakharov equation based on the assumption of unidirectionality (the assumption that all waves propagate in the same direction). To derive the new equation, we use the Hamiltonian form of the Euler equation for an ideal fluid and perform a very specific canonical transformation. This transformation is possible due to the ‘miraculous’ cancellation of the non-trivial four-wave resonant interaction in the one-dimensional wave field. The obtained equation is remarkably simple. We call the equation the ‘super compact water wave equation’. This equation includes a nonlinear wave term (à la NLSE) together with an advection term that can describe the initial stage of wave breaking. The NLSE and the Dysthe equations (Dysthe Proc. R. Soc. Lond. A, vol. 369, 1979, pp. 105–114) can be easily derived from the super compact equation. This equation is also suitable for analytical studies as well as for numerical simulation. Moreover, this equation also allows one to derive a spatial version of the water wave equation that describes experiments in flumes and canals.

JFM classification

Mathematical Foundations: Hamiltonian theory Waves/Free-surface Flows: Surface gravity waves Waves/Free-surface Flows: Wave breaking
Type
Papers
Information
Journal of Fluid Mechanics , Volume 828 , 10 October 2017 , pp. 661 - 679
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Copyright
© 2017 Cambridge University Press 

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References

Annenkov, S. Y. & Shrira, V. I. 2011 Evolution of wave turbulence under ‘gusty’ forcing. Phys. Rev. Lett. 107, 114502.Google Scholar
Annenkov, S. Y. & Shrira, V. I. 2013 Large-time evolution of statistical moments of wind-wave fields. J. Fluid Mech. 726, 517–546.Google Scholar
Crawford, D. E., Yuen, H. G. & Saffman, P. G. 1980 Evolution of a random inhomogeneous field of nonlinear deep-water gravity waves. Wave Motion 2 (1), 1–16.Google Scholar
Debnath, L. 1994 Nonlinear Water Waves. Academic.Google Scholar
Dyachenko, A. I., Kachulin, D. I. & Zakharov, V. E. 2013a Collisions of two breathers at the surface of deep water. Nat. Hazards Earth Syst. Sci. 13, 1–6.Google Scholar
Dyachenko, A. I., Kachulin, D. I. & Zakharov, V. E. 2013b On the nonintegrability of the free surface hydrodynamics. J. Expl Theor. Phys. Lett. 98 (1), 43–47.Google Scholar
Dyachenko, A. I., Kachulin, D. I. & Zakharov, V. E. 2014 Freak waves at the surface of deep water. J. Phys. Conf. Ser. 510, 012050.CrossRefGoogle Scholar
Dyachenko, A. I., Kachulin, D. I. & Zakharov, V. E. 2015a Evolution of one-dimensional wind-driven sea spectra. Pis’ma v Zh. Eksp. Teor. Fiz. 102 (8), 577–581.Google Scholar
Dyachenko, A. I., Kachulin, D. I. & Zakharov, V. E. 2015b Freak-waves: compact equation versus fully nonlinear one. In Extreme Ocean Waves, 2nd edn. (ed. Pelinovsky, E. & Kharif, C.), pp. 23–44. Springer.Google Scholar
Dyachenko, A. I., Kachulin, D. I. & Zakharov, V. E. 2016 Probability distribution functions of freak-waves: nonlinear versus linear model. Stud. Appl. Maths 137 (2), 189–198.Google Scholar
Dyachenko, A. I., Lvov, Y. V. & Zakharov, V. E. 1995 Five-wave interaction on the surface of deep fluid. Physica D 87 (1–4), 233–261.Google Scholar
Dyachenko, A. I. & Zakharov, V. E. 1994 Is free-surface hydrodynamics an integrable system? Phys. Lett. A 190, 144–148.Google Scholar
Dyachenko, A. I. & Zakharov, V. E. 2012 A dynamic equation for water waves in one horizontal dimension. Eur. J. Mech. (B/Fluids) 32, 17–21.Google Scholar
Dyachenko, A. I. & Zakharov, V. E. 2016 Spatial equation for water waves. Pis’ma v Zh. Eksp. Teor. Fiz. 103 (3), 200–203.Google Scholar
Dysthe, K. B. 1979 Note on a modification to the nonlinear Schrödinger equation for application to deep water waves. Proc. R. Soc. Lond. A 369, 105–114.Google Scholar
Fedele, F. & Dutykh, D. 2012a Special solutions to a compact equation for deep-water gravity waves. J. Fluid Mech. 712, 646–660.Google Scholar
Fedele, F. & Dutykh, D. 2012b Solitary wave interaction in a compact equation for deep-water gravity waves. J. Expl Theor. Phys. Lett. 95 (12), 622–625.Google Scholar
Fedele, F. 2014a On certain properties of the compact Zakharov equation. J. Fluid Mech. 748, 692–711.Google Scholar
Fedele, F. 2014b On the persistence of breathers at deep water. J. Expl Theor. Phys. Lett. 98 (9), 523–527.Google Scholar
Korotkevich, A. O., Pushkarev, A. N., Resio, D. & Zakharov, V. 2008 Numerical verification of the weak turbulent model for swell evolution. Eur. J. Mech. (B/Fluids) 27 (4), 361–387.Google Scholar
Krasitskii, V. P. 1990 Canonical transformation in a theory of weakly nonlinear waves with a nondecay dispersion law. Sov. Phys. JETP 98, 1644–1655.Google Scholar
Petviashvili, V. I. 1976 Equation for an extraordinary soliton. Sov. J. Plasma Phys. 2, 257–258.Google Scholar
Zakharov, V. E. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9 (2), 190–194.Google Scholar
Zakharov, V. E. 1998 Nonlinear waves and wave turbulence. Am. Math. Soc. Transl. 2 182, 167–197.Google Scholar
Zakharov, V. E. 1999 Statistical theory of gravity and capillary waves on the surface of a finite-depth fluid. Eur. J. Mech. (B/Fluids) 18 (3), 327–344.Google Scholar
Zakharov, V. E. & Faddev, L. D. 1971 KdV equation: a completely integrable Hamiltonian system. Funct. Anal. Applics. 5 (4), 18–27.Google Scholar
Zakharov, V. E., Lvov, V. S. & Falkovich, G. 1992 Kolmogorov Spectra of Turbulence I. Springer.Google Scholar

Dyachenko et al. supplementary movie 1

Collision of two breathers

Download Dyachenko et al. supplementary movie 1(Video)
Video 21.1 MB

Dyachenko et al. supplementary movie 2

Freak wave pre-breaking

Download Dyachenko et al. supplementary movie 2(Video)
Video 6.9 MB

Dyachenko et al. supplementary movie 3

Freak wave pre-breaking zoomed

Download Dyachenko et al. supplementary movie 3(Video)
Video 5.3 MB

Dyachenko et al. supplementary movie 4

Breathers in a flume

Download Dyachenko et al. supplementary movie 4(Video)
Video 37.6 MB