In this paper, we study the transformation of surface envelope solitons traveling over a bottom step in water of a finite depth. Using the transformation coefficients earlier derived in the linear approximation, we find the parameters of transmitted pulses and subsequent evolution of the pulses in the course of propagation. Relying on the weakly nonlinear theory, the analytic formulas are derived which describe the maximum attainable wave amplitude in the neighborhood of the step and in the far zone. Solitary waves may be greatly amplified (within the weakly nonlinear theory formally, even without a limit) when propagating from relatively shallow water to the deeper domain due to the constructive interference between the newly emerging envelope solitons and the residual quasi-linear waves. The theoretical results are in good agreement with the data of direct numerical modeling of soliton transformation. In particular, more than double wave amplification is demonstrated in the performed simulations.
References
1.
Ablowitz
, M.
, and Segur
, H.
, “On the evolution of packets of water waves
,” J. Fluid Mech.
92
, 691
–715
(1979
).2.
Agafontsev
, D.
, “Bifurcations and the stability of the surface envelope solitons for a finite-depth fluid
,” JETP Lett.
87
, 195
–199
(2008
).3.
Akhmediev
, N. N.
, and Mitzkevich
, N. V.
, “Extremely high degree of N-soliton pulse compression in an optical fiber
,” IEEE J. Quantum Electron.
27
, 849
–857
(1991
).4.
Chabchoub
, A.
, Mozumi
, K.
, Hoffmann
, N.
, Babanin
, A. V.
, Toffoli
, A.
, Steer
, J. N.
, van den Bremer
, T. S.
, Akhmediev
, N.
, Onorato
, M.
, and Waseda
, T.
, “Directional soliton and breather beams
,” Proc. Natl. Acad. Sci. U. S. A.
116
, 9759
–9763
(2019
).5.
Djordjevic
, V. D.
, and Redekopp
, L. G.
, “On the development of packets of surface gravity waves moving over an uneven bottom
,” J. Appl. Math. Phys.
29
, 950
–962
(1978
).6.
Dommermuth
, D. G.
, and Yue
, D. K.
, “A high-order spectral method for the study of nonlinear gravity waves
,” J. Fluid Mech.
184
, 267
–288
(1987
).7.
Ducrozet
, G.
, Bonnefoy
, F.
, Le Touzé
, D.
, and Ferrant
, P.
, “HOS-ocean: Open-source solver for nonlinear waves in open ocean based on high-order spectral method
,” Comput. Phys. Commun.
203
, 245
–254
(2016
).8.
Giniyatullin
, A. R.
, Kurkin
, A. A.
, Semin
, S. V.
, and Stepanyants
, Y. A.
, “Transformation of narrowband wavetrains of surface gravity waves passing over a bottom step
,” Math. Modell. Nat. Phenom.
9
(5
), 73
–82
(2014
).9.
Gouin
, M.
, Ducrozet
, G.
, and Ferrant
, P.
, “Development and validation of a non-linear spectral model for water waves over variable depth
,” Eur. J. Mech.-B/Fluids
57
, 115
–128
(2016
).10.
Grimshaw
, R. H. J.
, and Annenkov
, S. Y.
, “Water wave packets over variable depth
,” Stud. Appl. Math.
126
, 409
–427
(2011
).11.
Kharif
, C.
, Pelinovsky
, E.
, and Slunyaev
, A.
, Rogue Waves in the Ocean
(Springer-Verlag
, Berlin, Heidelberg
, 2009
).12.
Kurkin
, A. A.
, Semin
, S. V.
, and Stepanyants
, Y. A.
, “Transformation of surface waves over a bottom step
,” Izv., Atmos. Oceanic Phys.
51
(2
), 214
–223
(2015a
).13.
Kurkin
, A. A.
, Semin
, S. V.
, and Stepanyants
, Y. A.
, “Transformation of surface and internal waves on a bottom step
,” Fundam. Appl. Hydrophys. (in Russ.)
8
(3
), 3
–19
(2015b
).14.
15.
Li
, Y.
, Adcock
, T. A. A.
, Zheng
, Y.
, Lin
, Z.
, and van den Bremer
, T. S.
, “Surface wavepackets subject to an abrupt depth change. Part 1. Second-order theory
,” J. Fluid Mech.
915
, A71
(2021a
).16.
Li
, Y.
, Draycott
, S.
, Adcock
, T.
, and van den Bremer
, T.
, “Surface wavepackets subject to an abrupt depth change. Part 2. Experimental analysis
,” J. Fluid Mech.
915
, A72
(2021b
).17.
Losada
, M. A.
, Vidal
, C.
, and Medina
, R.
, “Experimental study of the evolution of a solitary wave at an abrupt junction
,” J. Geophys. Res.
94
, 14557
–14566
, (1989
).18.
Mei
, C. C.
, Stiassnie
, M.
, and Yue
, D. K.-P.
, Theory and Applications of Ocean Surface Waves. Part 2: Nonlinear Aspects
(World Scientific
, 2005
).19.
Miles
, J. W.
, “Surface-wave scattering matrix for a shelf
,” J. Fluid Mech.
28
, 755
–767
(1967
).20.
Onorato
, M.
, Proment
, D.
, and Toffoli
, A.
, “Triggering rogue waves in opposing currents
,” Phys. Rev. Lett.
107
, 184502
(2011
).21.
Peregrine
, D. H.
, “Water waves, nonlinear Schrodinger equations and their solutions
,” J. Aust. Math. Soc. Ser. B
25
, 16
–43
(1983
).22.
Satsuma
, J.
, and Yajima
, N.
, “Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media
,” Suppl. Prog. Theor. Phys.
55
, 284
–306
(1974
).23.
Seabra-Santos
, F. J.
, Renouard
, D. P.
, and Temperville
, A. M.
, “Numerical and experimental study of the transformation of a solitary wave over a shelf or isolated obstacle
,” J. Fluid Mech.
176
, 117
–134
(1987
).24.
Sergeeva
, A.
, Pelinovsky
, E.
, and Talipova
, T.
, “Nonlinear random wave field in shallow water: Variable Korteweg-de Vries framework
,” Nat. Hazards Earth Syst. Sci.
11
, 323
–330
(2011
).25.
Shrira
, V. I.
, and Slunyaev
, A. V.
, “Nonlinear dynamics of trapped waves on jet currents and rogue waves
,” Phys. Rev. E.
89
, 041002
(2014
).26.
Slunyaev
, A.
, “Nonlinear analysis and simulations of measured freak wave time series
,” Eur. J. Mech.-B/Fluids
25
, 621
–635
(2006
).27.
Slunyaev
, A.
, “On the optimal focusing of solitons and breathers in long wave models
,” Stud. Appl. Math.
142
, 385
–413
(2019
).28.
Slunyaev
, A. V.
, “A high-order nonlinear envelope equation for gravity waves in finite-depth water
,” J. Exp. Theor. Phys.
101
, 926
–941
(2005
).29.
Slunyaev
, A. V.
, “Persistence of hydrodynamic envelope solitons: Detection and rogue wave occurrence
,” Phys. Fluids
33
, 036606
(2021
).30.
Slunyaev
, A.
, Clauss
, G. F.
, Klein
, M.
, and Onorato
, M.
, “Simulations and experiments of short intense envelope solitons of surface water waves
,” Phys. Fluids
25
, 067105
(2013
).31.
Slunyaev
, A.
, Kharif
, C.
, Pelinovsky
, E.
, and Talipova
, T.
, “Nonlinear wave focusing on water of finite depth
,” Phys. D
173
, 77
–96
(2002
).32.
Slunyaev
, A.
, Klein
, M.
, and Clauss
, G. F.
, “Laboratory and numerical study of intense envelope solitons of water waves: Generation, reflection from a wall and collisions
,” Phys. Fluids
29
, 047103
(2017
).33.
Slunyaev
, A.
, Sergeeva
, A.
, and Pelinovsky
, E.
, “Wave amplification in the framework of forced nonlinear Schrödinger equation: The rogue wave context
,” Phys. D
303
, 18
–27
(2015
).34.
Slunyaev
, A. V.
, and Pelinovsky
, E. N.
, “The role of multiple soliton interactions in generation of rogue waves: The mKdV framework
,” Phys. Rev. Lett.
117
, 214501
(2016
).35.
Sun
, Y.-H.
, “Soliton synchronization in the focusing nonlinear Schrödinger equation
,” Phys. Rev. E
93
, 052222
(2016
).36.
Trulsen
, K.
, Raustøl
, A.
, Jorde
, S.
, and Rye
, L.
, “Extreme wave statistics of long-crested irregular waves over a shoal
,” J. Fluid Mech.
882
, R2
(2020
).37.
Trulsen
, K.
, Zeng
, H.
, and Gramstad
, O.
, “Laboratory evidence of freak waves provoked by non-uniform bathymetry
,” Phys. Fluids
24
, 097101
(2012
).38.
Viotti
, C.
, Dutykh
, D.
, Dudley
, J. M.
, and Dias
, F.
, “Emergence of coherent wave groups in deep-water random sea
,” Phys. Rev. E
87
, 063001
(2013
).39.
Wang
, A.
, Ludu
, A.
, Zong
, Z.
, Zou
, L.
, and Pei
, Y.
, “Experimental study of breathers and rogue waves generated by random waves over non-uniform bathymetry
,” Phys. Fluids
32
, 087109
(2020
).40.
Wang
, L.
, Yang
, C.
, Wang
, J.
, and He
, J.
, “The height of an nth-order fundamental rogue wave for the nonlinear Schrödinger equation
,” Phys. Lett. A
381
, 1714
–1718
(2017
).41.
West
, B. J.
, Brueckner
, K. A.
, Janda
, R. S.
, Milder
, D. M.
, and Milton
, R. L.
, “A new numerical method for surface hydrodynamics
,” J. Geophys. Res.
92
, 11803
–11824
, (1987
).42.
Zakharov
, V. E.
, and Ostrovsky
, L. A.
, “Modulation instability: The beginning
,” Phys. D
238
, 540
–548
(2009
).43.
Zakharov
, V. E.
, and Rubenchik
, A. M.
, “Instability of waveguides and solitons in nonlinear media
,” Sov. Phys. - JETP
38
, 494
–500
(1974
).44.
Zeng
, H.
, and Trulsen
, K.
, “Evolution of skewness and kurtosis of weakly nonlinear unidirectional waves over a sloping bottom
,” Nat. Hazards Earth Syst. Sci.
12
, 631
–638
(2012
).45.
Zhang
, J.
, and Benoit
, M.
, “Wave–bottom interaction and extreme wave statistics due to shoaling and de-shoaling of irregular long-crested wave trains over steep seabed changes
,” J. Fluid Mech.
912
, A28
(2021
).46.
Zheng
, Y. K.
, Lin
, Z. L.
, Li
, Y.
, Adcock
, T. A. A.
, Li
, Y.
, and van den Bremer
, T. S.
, “Fully nonlinear simulations of extreme waves provoked by strong depth transitions: The effect of slope
,” Phys. Rev. Fluids
5
, 064804
(2020
).© 2021 Author(s). Published under an exclusive license by AIP Publishing.
2021
Author(s)
You do not currently have access to this content.
