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On diffraction and oblique interactions of horizontally two-dimensional internal solitary waves

Published online by Cambridge University Press:  11 February 2022

Chunxin Yuan Open the ORCID record for Chunxin Yuan [Opens in a new window]  and
Zhan Wang Open the ORCID record for Zhan Wang [Opens in a new window]
Chunxin Yuan
Affiliation:
School of Mathematical Sciences, Ocean University of China, Qingdao 266100, PR China
Zhan Wang*
Affiliation:
Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, PR China School of Engineering Science, University of Chinese Academy of Sciences, Beijing 100049, PR China School of Future Technology, University of Chinese Academy of Sciences, Beijing 100049, PR China
*
†Email address for correspondence: zwang@imech.ac.cn
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Abstract

In the context of three-dimensional oceanic internal waves, taking topographic effects into account, a modified Benney–Luke equation is proposed for describing internal wave–wave interactions on a sloping bottom. The derived equation is characterised by isotropy and bi-directional propagation, which are absent in the widely used Kadomtsev–Petviashvili equation. Indeed, these disparities are confirmed by numerical results of the diffraction of a truncated internal solitary wave and the evolution of a partially bent solitary wave. However, a good agreement between the numerical results of the modified Benney–Luke equation and those of the primitive equations confirms the validity of our simplified model. Because the stratification in a realistic ocean environment is usually continuous, in contrast to the assumption of a sharp density discontinuity used here, to maintain the kinematical equivalence, a layering scheme for determining the density and thickness of each layer from a continuous stratification is proposed. In addition, the occasionally observed but rarely examined X-shaped internal wave–wave interactions are shown to feature novel wave patterns, where topographic effects modulate the propagation speed, amplitude and waveform.

JFM classification

Geophysical and Geological Flows: Internal waves Waves/Free-surface Flows: Solitary waves

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Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 936 , 10 April 2022 , A20
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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References

REFERENCES

Ablowitz, M.J. & Baldwin, D.E. 2012 Nonlinear shallow ocean-wave soliton interactions on flat beaches. Phys. Rev. E 86, 036305.CrossRefGoogle ScholarPubMed
Benjamin, T.B. & Bridges, T.J. 1997 Reappraisal of the Kelvin–Helmholtz problem. Part 1. Hamiltonian structure. J. Fluid Mech. 333, 301–325.CrossRefGoogle Scholar
Benney, D.J. & Luke, J.C. 1964 On the interactions of permanent waves of finite amplitude. J. Math. Phys. (Journal of Mathematical Physics) 43, 309–313.Google Scholar
Biondini, G. & Pelinovsky, D. 2008 Kadomtsev–Petviashvili equation. Scholarpedia 3 (10), 6539.CrossRefGoogle Scholar
Camassa, R., Choi, W., Michallet, H., Rusås, P.-O. & Sveen, J.K. 2006 On the realm of validity of strongly nonlinear asymptotic approximations for internal waves. J. Fluid Mech. 549, 1–23.CrossRefGoogle Scholar
Chakravarty, S. & Kodama, Y. 2008 Classification of the line-soliton solutions of KPII. J. Phys. A 41, 275209.CrossRefGoogle Scholar
Chakravarty, S. & Kodama, Y. 2009 Soliton solutions of the KP equation and application to shallow water waves. Stud. Appl. Maths 123, 83–151.CrossRefGoogle Scholar
Chakravarty, S. & Kodama, Y. 2013 Construction of KP solitons from wave patterns. J. Phys. A 47, 025201.CrossRefGoogle Scholar
Chen, Y. & Philip, L.-F. 1995 The unified Kadomtsev–Petviashvili equation for interfacial waves. J. Fluid Mech. 288, 383–408.CrossRefGoogle Scholar
Choi, W. & Camassa, R. 1996 Weakly nonlinear internal waves in a two-fluid system. J. Fluid Mech. 386, 1–36.Google Scholar
Choi, W. & Camassa, R. 1999 Fully nonlinear internal waves in a two-fluid system. J. Fluid Mech. 313, 83–103.CrossRefGoogle Scholar
Craig, W., Guyenne, P., Nicholls, D.P. & Sulem, C. 2005 Hamiltonian long-wave expansions for water waves over a rough bottom. Proc. R. Soc. Lond. A 461, 839–873.Google Scholar
Craig, W., Schanz, U. & Sulem, C. 1997 The modulational regime of three-dimensional water waves and the Davey–Stewartson system. Annales de I'I.H.P. Analyse non linéaire 14 (5), 615–667.Google Scholar
Funakoshi, M. 1980 Reflection of obliquely incident solitary waves. J. Phys. Soc. Japan 49, 2371–2379.CrossRefGoogle Scholar
Grimshaw, R. & Melville, W.K. 1989 On the derivation of the modified Kadomtsev–Petviashvili equation. Stud. Appl. Maths 80 (3), 183–202.CrossRefGoogle Scholar
Guyenne, P. & Nicholls, D.P. 2008 A high-order spectral method for nonlinear water waves over moving bottom topography. SIAM J. Sci. Comput. 30 (1), 81–101.CrossRefGoogle Scholar
Helfrich, K.R. & Melville, W.K. 2006 Long nonlinear internal waves. Annu. Rev. Fluid Mech. 38, 395–425.CrossRefGoogle Scholar
Hsu, M.-K., Liu, A.K. & Liu, C. 2000 A study of internal waves in the China Seas and Yellow Seas using SAR. Cont. Shelf Res. 20, 389–410.CrossRefGoogle Scholar
Johnson, R.S. 1982 On the oblique interaction of a large and a small solitary wave. J. Fluid Mech. 120, 49–70.CrossRefGoogle Scholar
Kadomtsev, B.B. & Petviashvili, V.I. 1970 On the stability of solitary waves in weakly dispersive media. Sov. Phys. Dokl. 15, 539–541.Google Scholar
Kao, C. & Kodama, Y. 2012 Numerical study of the KP equation for non-periodic waves. Maths Comput. Simul. 82, 1185–1218.CrossRefGoogle Scholar
Li, W., Yeh, H. & Kodama, Y. 2011 On the Mach reflection of a solitary wave: revisited. J. Fluid Mech. 672, 326–357.CrossRefGoogle Scholar
Marshall, J., Adcroft, A., Hill, C., Perelman, L. & Heisey, C. 1997 A finite-volume, incompressible Navier Stokes model for studies of the ocean on parallel computers. J. Geophys. Res. 102 (C3), 5753–5766.CrossRefGoogle Scholar
Melville, W. 1980 On the Mach reflexion of a solitary wave. J. Fluid Mech. 98 (2), 285–297.CrossRefGoogle Scholar
Miles, J.W. 1977 a Obliquely interacting solitary waves. J. Fluid Mech. 79, 157–169.CrossRefGoogle Scholar
Miles, J.W. 1977 b Resonantly interacting solitary waves. J. Fluid Mech. 79, 171–179.CrossRefGoogle Scholar
Milewski, P.A. & Tabak, E.G. 1999 A pseudo-spectral procedure for the solution of nonlinear wave equations with examples from free-surface flows. SIAM J. Sci. Comput. 21 (3), 1102–1114.CrossRefGoogle Scholar
Nakayama, K., Kakinuma, T. & Tsuji, H. 2019 Oblique reflection of large internal solitary waves in a two-layer fluid. Eur. J. Mech. (B/Fluids) 74, 81–91.CrossRefGoogle Scholar
Perroud, P.H. 1957 The solitary wave reflection along a straight vertical wall at oblique incidence. PhD thesis, University of California, Berkeley.Google Scholar
Pierini, S. 1989 A model for the Alboran Sea internal solitary waves. J. Phys. Oceanogr. 19 (6), 755–772.2.0.CO;2>CrossRefGoogle Scholar
Robertson, R. 2001 Internal tides and baroclinicity in the southern Weddell Sea: 1. Model description. J. Geophys. Res. 106, 27001–27016.CrossRefGoogle Scholar
Ryskamp, S., Maiden, M.D., Biondini, G. & Hoefer, M.A. 2021 Evolution of truncated and bent gravity wave soliton: the Mach expansion problem. J. Fluid Mech. 909, A24.CrossRefGoogle Scholar
Schlatter, P., Adams, N.A. & Kleiser, L. 2005 A windowing method for periodic inflow/outflow boundary treatment of non-periodic flows. J. Comput. Phys. 206 (2), 505–535.CrossRefGoogle Scholar
Shimizu, K. & Nakayama, K. 2017 Effects of topography and Earth's rotation on the oblique interaction of internal solitary-like waves in the Andaman Sea. J. Geophys. Res.: Oceans 122, 7449–7465.CrossRefGoogle Scholar
Tanaka, M. 1993 Mach reflection of a large-amplitude solitary wave. J. Fluid Mech. 248, 637–661.CrossRefGoogle Scholar
Vlasenko, V., Sanchez Garrido, J.C., Stashchuk, N., Lafuente, J.G. & Losada, M. 2009 Three-dimensional evolution of large-amplitude internal waves in the Strait of Gibraltar. J. Phys. Oceanogr. 39 (9), 2230–2246.CrossRefGoogle Scholar
Vlasenko, V., Guo, C. & Stashchuk, N. 2012 On the mechanism of A-type and B-type internal solitary wave generation in the northern South China Sea. Deep-Sea Res. (I) 69, 100–112.CrossRefGoogle Scholar
Wunsch, C. & Ferrari, R. 2004 Vertical mixing, energy, and the general circulation of the oceans. Annu. Rev. Fluid Mech. 36 (1), 281–314.CrossRefGoogle Scholar
Xue, J., Graber, H.C., Romeiser, R. & Lund, B. 2014 Understanding internal wave-wave interaction patterns observed in satellite images of the Mid-Atlantic Bight. IEEE Trans. Geosci. Remote Sens. 52 (6), 3211–3219.CrossRefGoogle Scholar
Yeh, H. & Li, W. 2014 Laboratory realization of KP-solitons. J. Phys.: Conf. Ser. 482, 012046.Google Scholar
Yuan, C., Wang, Z. & Chen, X. 2020 The derivation of an isotropic model for internal waves and its application to wave generation. Ocean Model. 153, 101663.CrossRefGoogle Scholar
Yuan, C., Grimshaw, R., Johnson, E. & Wang, Z. 2018 a Topographic effect on oblique internal wave-wave interactions. J. Fluid Mech. 856, 36–60.CrossRefGoogle Scholar
Yuan, C., Grimshaw, R., Johnson, E. & Chen, X. 2018 b The propagation of internal solitary waves over variable topography in a horizontally two-dimensional framework. J. Phys. Oceanogr. 48 (2), 283–300.CrossRefGoogle Scholar
Zakharov, V.E. & Shabat, A.B. 1974 A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. Funct. Anal. Applics. 8, 226–235.CrossRefGoogle Scholar