Abstract
Water waves have long been a subject of attention of both mathematicians and physicists. The formulation of the problem is simple enough to be considered fundamental, but as of yet many questions still remain unanswered and many phenomena associated with wind-driven turbulence remain puzzling. We consider a “unidirectional” motion of weakly nonlinear gravity waves, i.e., we assume that the spectrum of the free surface contains only nonnegative wavenumbers. We use remarkably simple form of the water wave equation that we named “the super compact equation”. This new equation includes a nonlinear wave term (à la NLSE) together with an advection term that can describe the initial stage of wave breaking. This equation has also very important property. It allows to introduce exact envelope for waves without assumption of narrowness bandwidth.
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References
Bjorkavag M, Kalisch H (2011) Wave breaking in Boussinesq models for undular bores. Phys Lett A 375(14):1570–1578
Craig W, Guyenne P, Sulem C (2010) A Hamiltonian approach to nonlinear modulation of surface water waves. Wave Motion 47(8):552–563
Dyachenko AI, Kachulin DI, Zakharov VE (2013a) Collisions of two breathers at the surface of deep water. Nat Hazards Earth Syst Sci 13:3205–3210
Dyachenko AI, Kachulin DI, Zakharov VE (2013b) On the nonintegrability of the free surface hydrodynamics. JETP Lett 98(1):43–47
Dyachenko AI, Kachulin DI, Zakharov VE (2014) Freak waves at the surface of deep water. J Phys Conf Ser 510:012050
Dyachenko AI, Kachulin DI, Zakharov VE (2015a) Evolution of one-dimensional wind-driven sea spectra. Pis’ma v ZhETF 102(8):577–581
Dyachenko AI, Kachulin DI, Zakharov VE (2015b) Freak-waves: compact equation vs fully nonlinear one. In: Pelinovsky E, Kharif C (eds) Extreme ocean waves, 2nd edn. Springer, Berlin, pp 23–44
Dyachenko AI, Kachulin DI, Zakharov VE (2016a) Probability distribution functions of freak-waves: nonlinear vs linear model. Stud Appl Math 137(2):189–198
Dyachenko AI, Kachulin DI, Zakharov VE (2016b) About compact equations for water waves. Nat Hazards 84(2):529–540
Dyachenko AI, Kachulin DI, Zakharov VE (2017) Super compact equation for water waves. J Fluid Mech 828:661–679
Dyachenko AI, Zakharov VE (1994) Is free-surface hydrodynamics an integrable system? Phys Lett A 190:144–148
Dyachenko AI, Zakharov VE (2011) Compact equation for gravity waves on deep water. JETP Lett 93(12):701–705
Dyachenko AI, Zakharov VE (2012) A dynamic equation for water waves in one horizontal dimension. Eur J Mech B/Fluids 32:17–21
Dyachenko AI, Zakharov VE (2016a) Spatial equation for water waves. JETP Lett 103(3):181–184
Dyachenko AI, Zakharov VE (2016b) Spatial equation for water waves. Pis’ma v ZhETF 103(3):200–203
Dysthe KB (1979) Note on a modification to the nonlinear Schrödinger equation for application to deep water waves. Proc R Soc Lond Ser A 369:105–114
Fedele F (2014) On certain properties of the compact Zakharov equation. J Fluid Mech 748:692–711
Fedele F (2014) On the persistence of breathers at deep water. JETP Lett 98(9):523–527
Fedele F, Dutykh D (2012) Special solutions to a compact equation for deep-water gravity waves. J Fluid Mech 712:646–660
Fedele F, Dutykh D (2012) Solitary wave interaction in a compact equation for deep-water gravity waves. JETP Lett 95(12):622–625
Korotkevich AO, Pushkarev AN, Resio D, Zakharov V (2008) Numerical verification of the weak turbulent model for swell evolution. Eur J Mech B/Fluids 27(4):361–387
Zakharov VE (1968) Stability of periodic waves of finite amplitude on the surface of a deep fluid. J Appl Mech Techn Phys 9(2):190–194
Acknowledgements
This work (except Sect. 5) was supported by Grant “Wave turbulence: theory, numerical simulation, experiment” #14-22-00174 of Russian Science Foundation. Numerical simulation of soliton turbulence (Sect. 5) was supported by the Program of Landau Institute for Theoretical Physics “Dynamics of the Complex Environment”.
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Dyachenko, A.I., Kachulin, D.I. & Zakharov, V.E. Envelope equation for water waves. J. Ocean Eng. Mar. Energy 3, 409–415 (2017). https://doi.org/10.1007/s40722-017-0100-z
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DOI: https://doi.org/10.1007/s40722-017-0100-z