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Large mode-2 internal solitary waves in three-layer flows

Published online by Cambridge University Press:  16 December 2022

A. Doak Open the ORCID record for A. Doak [Opens in a new window] ,
R. Barros Open the ORCID record for R. Barros [Opens in a new window]  and
P.A. Milewski Open the ORCID record for P.A. Milewski [Opens in a new window]
A. Doak*
Affiliation:
Department of Mathematical Sciences, Univesity of Bath, Bath BA2 7AY, UK
R. Barros
Affiliation:
Department of Mathematical Sciences, Loughborough University, Loughborough LE11 3TU, UK
P.A. Milewski
Affiliation:
Department of Mathematical Sciences, Univesity of Bath, Bath BA2 7AY, UK
*
†Email address for correspondence: add49@bath.ac.uk
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Abstract

In this paper, we investigate mode-2 solitary waves in a three-layer stratified flow model. Localised travelling wave solutions to both the fully nonlinear problem (Euler equations), and the three-layer Miyata–Choi–Camassa equations are found numerically and compared. Mode-2 solitary waves with speeds slower than the linear mode-1 long-wave speed are typically generalised solitary waves with infinite tails consisting of a resonant mode-1 periodic wave train. Herein, we evidence the existence of mode-2 embedded solitary waves, that is, we show that for specific values of the parameters, the amplitude of the oscillations in the tail are zero. For sufficiently thick middle layers, we also find branches of mode-2 solitary waves with speeds that extend beyond the mode-1 linear waves and are no longer embedded. In addition, we show how large amplitude embedded solitary waves are intimately linked to the conjugate states of the problem.

JFM classification

Geophysical and Geological Flows: Stratified flows Nonlinear Dynamical Systems: Bifurcation Waves/Free-surface Flows: Solitary waves
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 953 , 25 December 2022 , A42
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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References

REFERENCES

Amick, C.J. & Turner, R.E.L. 1986 A global theory of internal solitary waves in two-fluid systems. Trans. Am. Math. Soc. 298 (2), 431–484.CrossRefGoogle Scholar
Baines, P.G. 1997 Topographic Effects in Stratified Flows. Cambridge University Press.Google Scholar
Barros, R. 2016 Remarks on a strongly nonlinear model for two-layer flows with a top free surface. Stud. Appl. Maths 136 (3), 263–287.CrossRefGoogle Scholar
Barros, R., Choi, W. & Milewski, P.A. 2020 Strongly nonlinear effects on internal solitary waves in three-layer flows. J. Fluid Mech. 883, A16.CrossRefGoogle Scholar
Benney, D.J. 1966 Long non-linear waves in fluid flows. J. Maths Phys. 45 (1–4), 52–63.CrossRefGoogle Scholar
Brandt, A & Shipley, K.R. 2014 Laboratory experiments on mass transport by large amplitude mode-2 internal solitary waves. Phys. Fluids 26 (4), 046601.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
Carr, M., Davies, P.A. & Hoebers, R.P. 2015 Experiments on the structure and stability of mode-2 internal solitary-like waves propagating on an offset pycnocline. Phys. Fluids 27 (4), 046602.CrossRefGoogle Scholar
Champneys, A.R., Malomed, B.A., Yang, J. & Kaup, D.J. 2001 Embedded solitons: solitary waves in resonance with the linear spectrum. Physica D 152, 340–354.CrossRefGoogle Scholar
Champneys, A.R., Vanden-Broeck, J.M. & Lord, G.J. 2002 Do true elevation gravity-capillary solitary waves exist? A numerical investigation. J. Fluid Mech. 454 (1), 403–417.CrossRefGoogle Scholar
Choi, W. 2000 Modeling of strongly nonlinear internal waves in a multilayer system. In Proceedings of the Fourth International Conference on Hydrodynamics (ed. Y. Goda, M. Ikehata & K. Suzuki), pp. 453–458.Google Scholar
Choi, W. & Camassa, R. 1999 Fully nonlinear internal waves in a two-fluid system. J. Fluid Mech. 396, 1–36.CrossRefGoogle Scholar
Davis, R.E. & Acrivos, A. 1967 Solitary internal waves in deep water. J. Fluid Mech. 29 (3), 593–607.CrossRefGoogle Scholar
Deepwell, D. & Stastna, M. 2016 Mass transport by mode-2 internal solitary-like waves. Phys. Fluids 28 (5), 056606.CrossRefGoogle Scholar
Dias, F. & Il'ichev, A. 2001 Interfacial waves with free-surface boundary conditions: an approach via a model equation. Physica D: Nonlinear Phenomena 150 (3–4), 278–300.CrossRefGoogle Scholar
Dubreil-Jacotin, M.L. 1932 Sur les ondes de type permanent dans les liquides hétérogenes. Atti Accad. Naz. Lincei Rend 6, 814–819.Google Scholar
Duda, T.F., Lynch, J.F., Irish, J.D., Beardsley, R.C., Ramp, S.R., Chiu, C.-S., Tang, T.Y. & Yang, Y.-J. 2004 Internal tide and nonlinear internal wave behavior at the continental slope in the Northern South China Sea. IEEE J. Ocean. Engng 29 (4), 1105–1130.CrossRefGoogle Scholar
Evans, W.A.B. & Ford, M.J. 1996 An integral equation approach to internal (2-layer) solitary waves. Phys. Fluids 8 (8), 2032–2047.CrossRefGoogle Scholar
Gao, T. 2016 Nonlinear flexural-gravity free-surface flows and related gravity-capillary flows. PhD thesis, UCL (University College London).Google Scholar
Gavrilov, N.V., Liapidevskii, V.Y. & Liapidevskaya, Z.A. 2013 Influence of dispersion on the propagation of the internal waves in the shelf zone. Fundam. Appl. Hydrophys. 6 (2), 25–34.Google Scholar
Grimshaw, R. 2003 Internal solitary waves. In Env. Strat. Flows (ed. R. Grimshaw), pp. 1–27. Springer.CrossRefGoogle Scholar
Grimshaw, R. & Joshi, N. 1995 Weakly nonlocal solitary waves in a singularly perturbed Korteweg–de Vries equation. SIAM J. Appl. Maths 55 (1), 124–135.CrossRefGoogle Scholar
Grue, J., Jensen, A., Rusås, P.-O. & Sveen, J.K. 1999 Properties of large-amplitude internal waves. J. Fluid Mech. 380, 257–278.CrossRefGoogle Scholar
Honji, H., Matsunaga, N., Sugihara, Y. & Sakai, K. 1995 Experimental observation of internal symmetric solitary waves in a two-layer fluid. Fluid Dyn. Res. 15 (2), 89–102.CrossRefGoogle Scholar
Kao, T.W. & Pao, H.-P. 1980 Wake collapse in the thermocline and internal solitary waves. J. Fluid Mech. 97 (1), 115–127.CrossRefGoogle Scholar
Khimchenko, E. & Serebryany, A. 2016 Mode-2 internal waves: observations in the non-tidal sea. In Int. Symposium on Stratified Flows, vol. 1. https://escholarship.org/uc/sio_iod_issfs.Google Scholar
King, S.E., Carr, M. & Dritschel, D.G. 2011 The steady-state form of large-amplitude internal solitary waves. J. Fluid Mech. 666, 477–505.CrossRefGoogle Scholar
Koop, C.G. & Butler, G. 1981 An investigation of internal solitary waves in a two-fluid system. J. Fluid Mech. 112, 225–251.CrossRefGoogle Scholar
Lamb, K.G. 2000 Conjugate flows for a three-layer fluid. Phys. Fluids 12 (9), 2169–2185.CrossRefGoogle Scholar
Lamb, K.G. 2005 Extreme internal solitary waves in the ocean: theoretical considerations. In Proceedings of the 14th ‘Aha Huliko’ a Hawaiian Winter Workshop, pp. 109–117. http://www.soest.hawaii.edu/PubServices/2005pdfs/TOC2005.html.Google Scholar
Liapidevskii, V.Y. & Gavrilov, N.V. 2018 Large internal solitary waves in shallow waters. In The Ocean in Motion (ed. M.G. Velarde, R.Y. Tarakanov & A.V. Marchenko), pp. 87–108. Springer.CrossRefGoogle Scholar
Long, R.R. 1953 Some aspects of the flow of stratified fluids: I. A theoretical investigation. Tellus 5 (1), 42–58.CrossRefGoogle Scholar
Miyata, M. 1988 Long internal waves of large amplitude. In Proceedings of the IUTAM Symposium on Nonlinear Water Waves (ed. H. Horikawa & H. Maruo), pp. 399–406. Springer.CrossRefGoogle Scholar
Ramp, S.R., Yang, Y.J., Reeder, D.B. & Bahr, F.L. 2012 Observations of a mode-2 nonlinear internal wave on the Northern Heng-Chun Ridge South of Taiwan. J. Geophys. Res. 117 (C3), C03043.CrossRefGoogle Scholar
Rusås, P.-O. & Grue, J. 2002 Solitary waves and conjugate flows in a three-layer fluid. Eur. J. Mech. B/Fluids 21 (2), 185–206.CrossRefGoogle Scholar
Shroyer, E.L., Moum, J.N. & Nash, J.D. 2010 Mode 2 waves on the continental shelf: ephemeral components of the nonlinear internal wavefield. J. Geophys. Res. 115 (C7), C07001.CrossRefGoogle Scholar
Stamp, A.P. & Jacka, M. 1995 Deep-water internal solitaty waves. J. Fluid Mech. 305, 347–371.CrossRefGoogle Scholar
Sun, S.M. 1999 Non–existence of truly solitary waves in water with small surface tension. Proc. R. Soc. Lond. A 455 (1986), 2191–2228.CrossRefGoogle Scholar
Sun, S.M. & Shen, M.C. 1993 Exponentially small estimate for the amplitude of capillary ripples of a generalized solitary wave. J. Math. Anal. Appl. 172, 533–533.CrossRefGoogle Scholar
Sveen, J.K., Guo, Y., Davies, P.A. & Grue, J. 2002 On the breaking of internal solitary waves at a ridge. J. Fluid Mech. 469, 161–188.CrossRefGoogle Scholar
Talipova, T.G., Pelinovsky, E.N., Lamb, K., Grimshaw, R. & Holloway, P. 1999 Cubic nonlinearity effects in the propagation of intense internal waves. Doklady Earth Sciences. 365, 241–244.Google Scholar
Tung, K.-K., Chan, T.F. & Kubota, T. 1982 Large amplitude internal waves of permanent form. Stud. Appl. Maths 66 (1), 1–44.CrossRefGoogle Scholar
Turner, R.E.L. & Vanden-Broeck, J.-M. 1986 The limiting configuration of interfacial gravity waves. Phys. Fluids 29 (2), 372–375.CrossRefGoogle Scholar
Vanden-Broeck, J.-M. & Turner, R.E.L. 1992 Long periodic internal waves. Phys. Fluids A 4 (9), 1929–1935.CrossRefGoogle Scholar
Wang, Z., Părău, E.I., Milewski, P.A. & Vanden-Broeck, J.-M. 2014 Numerical study of interfacial solitary waves propagating under an elastic sheet. Proc. R. Soc. Lond. A 470 (2168), 20140111.Google ScholarPubMed
Yang, J., Malomed, B.A. & Kaup, D.J. 1999 Embedded solitons in second-harmonic-generating systems. Phys. Rev. Lett. 83 (10), 1958.CrossRefGoogle Scholar
Yang, Y.J., Fang, Y.C., Chang, M.-H., Ramp, S.R., Kao, C.-C. & Tang, T.Y. 2009 Observations of second baroclinic mode internal solitary waves on the continental slope of the Northern South China Sea. J. Geophys. Res. 114 (C10), C10003.CrossRefGoogle Scholar