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Dynamic soliton–mean flow interaction with non-convex flux

Published online by Cambridge University Press:  06 October 2021

Kiera van der Sande Open the ORCID record for Kiera van der Sande [Opens in a new window] ,
Gennady A. El Open the ORCID record for Gennady A. El [Opens in a new window]  and
Mark A. Hoefer Open the ORCID record for Mark A. Hoefer [Opens in a new window]
Kiera van der Sande
Affiliation:
Department of Applied Mathematics, University of Colorado, Boulder, CO 80309, USA
Gennady A. El
Affiliation:
Department of Mathematics, Physics and Electrical Engineering, Northumbria University, Newcastle upon Tyne NE1 8ST, UK
Mark A. Hoefer*
Affiliation:
Department of Applied Mathematics, University of Colorado, Boulder, CO 80309, USA
*
†Email address for correspondence: hoefer@colorado.edu
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Abstract

The interaction of localised solitary waves with large-scale, time-varying dispersive mean flows subject to non-convex flux is studied in the framework of the modified Korteweg–de Vries (mKdV) equation, a canonical model for internal gravity wave propagation and potential vorticity fronts in stratified fluids. The effect of large amplitude, dynamically evolving mean flows on the propagation of localised waves – essentially ‘soliton steering’ by the mean flow – is considered. A recent theoretical and experimental study of this new type of dynamic soliton–mean flow interaction for convex flux has revealed two scenarios where the soliton either transmits through the varying mean flow or remains trapped inside it. In this paper, it is demonstrated that the presence of a non-convex cubic hydrodynamic flux introduces significant modifications to the scenarios for transmission and trapping. A reduced set of Whitham modulation equations is used to formulate a general mathematical framework for soliton–mean flow interaction with non-convex flux. Solitary wave trapping is stated in terms of crossing modulation characteristics. Non-convexity and positive dispersion – common for stratified fluids – imply the existence of localised, sharp transition fronts (kinks). Kinks play dual roles as a mean flow and a wave, imparting polarity reversal to solitons and dispersive mean flows, respectively. Numerical simulations of the mKdV equation agree with modulation theory predictions. The mathematical framework developed is general, not restricted to completely integrable equations like mKdV, enabling application beyond the mKdV setting to other fluid dynamic contexts subject to non-convex flux such as strongly nonlinear internal wave propagation that is prevalent in the ocean.

JFM classification

Waves/Free-surface Flows: Solitary waves Nonlinear Dynamical Systems: Pattern formation Geophysical and Geological Flows: Internal waves

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Type
JFM Papers
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Journal of Fluid Mechanics , Volume 928 , 10 December 2021 , A21
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© The Author(s), 2021. Published by Cambridge University Press

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References

REFERENCES

Apel, J.R. 2002 Oceanic internal waves and solitons. Prepared for Office of Naval Research Code 322 PO. Global Ocean Associates.Google Scholar
Apel, J.R., Ostrovsky, L.A., Stepanyants, Y.A. & Lynch, J.F. 2007 Internal solitons in the ocean and their effect on underwater sound. J. Acoust. Soc. Am. 121 (2), 695–722.CrossRefGoogle ScholarPubMed
Baqer, S. & Smyth, N.F. 2020 Modulation theory and resonant regimes for dispersive shock waves in nematic liquid crystals. Physica D 403, 132334.CrossRefGoogle Scholar
Bikbaev, R.F. 1989 Korteweg-de Vries equation with finite-gap boundary conditions, and the Whitham deformations of Riemann surfaces. Funct. Anal. Applics. 23 (4), 257–266.CrossRefGoogle Scholar
Biondini, G. & Lottes, J. 2019 Nonlinear interactions between solitons and dispersive shocks in focusing media. Phys. Rev. E 99 (2), 022215.CrossRefGoogle ScholarPubMed
Boegman, L., Ivey, G.N. & Imberger, J. 2005 The energetics of large-scale internal wave degeneration in lakes. J. Fluid Mech. 531, 159–180.CrossRefGoogle Scholar
Bühler, O. 2009 Waves and Mean Flows. Cambridge Monographs on Mechanics. Cambridge University Press.CrossRefGoogle Scholar
Chanson, H. 2010 Tidal Bores, Aegir and Pororoca: The Geophysical Wonders. World Scientific.Google Scholar
Chanteur, G. & Raadu, M. 1987 Formation of shocklike modified Korteweg–de Vries solitons: application to double layers. Phys. Fluids 30 (9), 2708–2719.CrossRefGoogle Scholar
Chassagne, R., Filippini, A.G., Ricchiuto, M. & Bonneton, P. 2019 Dispersive and dispersive-like bores in channels with sloping banks. J. Fluid Mech. 870, 595–616.CrossRefGoogle Scholar
Choi, W. & Camassa, R. 1999 Fully nonlinear internal waves in a two-fluid system. J. Fluid Mech. 396, 1–36.CrossRefGoogle Scholar
Congy, T., El, G.A. & Hoefer, M.A. 2019 Interaction of linear modulated waves and unsteady dispersive hydrodynamic states with application to shallow water waves. J. Fluid Mech. 875, 1145–1174.CrossRefGoogle Scholar
Congy, T., El, G.A., Hoefer, M.A. & Shearer, M. 2021 Dispersive Riemann problems for the Benjamin–Bona–Mahony equation. Stud. Appl. Math. 147 (3), 1089–1145.CrossRefGoogle Scholar
Congy, T., Kamchatnov, A.M. & Pavloff, N. 2016 Nonlinear waves in coherently coupled Bose-Einstein condensates. Phys. Rev. A 93 (4), 043613.CrossRefGoogle Scholar
Dafermos, C.M. 2016 Hyperbolic Conservation Laws in Continuum Physics, 4th edn. Springer.CrossRefGoogle Scholar
Davis, K.A., Arthur, R.S., Reid, E.C., Rogers, J.S., Fringer, O.B., DeCarlo, T.M. & Cohen, A.L. 2020 Fate of internal waves on a shallow shelf. J. Geophys. Res. 125 (5), e2019JC015377.CrossRefGoogle Scholar
Driscoll, C.F. & O'Neil, T.M. 1975 Modulational instability of cnoidal wave solutions of the modified Korteweg–de Vries equation. J. Math. Phys. 17 (7), 1196–1200.CrossRefGoogle Scholar
El, G., Hoefer, M. & Shearer, M. 2017 Dispersive and diffusive-dispersive shock waves for nonconvex conservation laws. SIAM Rev. 59 (1), 3–61.CrossRefGoogle Scholar
El, G.A. 2005 Resolution of a shock in hyperbolic systems modified by weak dispersion. Chaos 15, 037103.CrossRefGoogle ScholarPubMed
El, G.A. & Hoefer, M.A. 2016 Dispersive shock waves and modulation theory. Physica D 333, 11–65.CrossRefGoogle Scholar
El, G.A. & Smyth, N.F. 2016 Radiating dispersive shock waves in non-local optical media. Proc. R. Soc. A 472 (2187), 20150633.CrossRefGoogle ScholarPubMed
Flaschka, H., Forest, M.G. & McLaughlin, D.W. 1980 Multiphase averaging and the inverse spectral solution of the Korteweg-de Vries equation. Commun. Pure Appl. Maths 33, 739–784.CrossRefGoogle Scholar
la Forgia, G., Ottolenghi, L., Adduce, C. & Falcini, F. 2020 Intrusions and solitons: propagation and collision dynamics. Phys. Fluids 32 (7), 076605.CrossRefGoogle Scholar
Grimshaw, R. 1979 Slowly varying solitary waves. I. Korteweg-de Vries equation. Proc. R. Soc. Lond. A 368, 359–375.Google Scholar
Grimshaw, R.H.J. 2002 Internal solitary waves. In Environmental Stratified Flows (ed. R.H.J. Grimshaw), pp. 1–27. Kluwer.CrossRefGoogle Scholar
Gruzinov, A.V. 1992 Contour dynamics of the Hasagawa-Mima equation. JETP Lett. 55 (1), 76–80.Google Scholar
Gurevich, A.V., Krylov, A.L. & El, G.A. 1990 Nonlinear modulated waves in dispersive hydrodynamics. Sov. Phys. JETP 71 (5), 899–910.Google Scholar
Gurevich, A.V. & Pitaevskii, L.P. 1974 Nonstationary structure of a collisionless shock wave. Sov. Phys. JETP 38 (2), 291–297, translation from Russian of A. V. Gurevich and L. P. Pitaevskii, Zh. Eksp. Teor. Fiz. 65, 590–604 (August 1973).Google Scholar
Hayes, B. & Shearer, M. 1999 Undercompressive shocks and Riemann problems for scalar conservation laws with non-convex fluxes. Proc. R. Soc. Edin. A 129 (4), 733–754.CrossRefGoogle Scholar
Helfrich, K.R. & Melville, W.K. 2006 Long nonlinear internal waves. Annu. Rev. Fluid Mech. 38, 395–425.CrossRefGoogle Scholar
Henyey, F.S. & Hoering, A. 1997 Energetics of borelike internal waves. J. Geophys. Res. 102 (C2), 3323–3330.CrossRefGoogle Scholar
Hoefer, M.A., Smyth, N.F. & Sprenger, P. 2019 Modulation theory solution for nonlinearly resonant, fifth-order Korteweg-de Vries, nonclassical, traveling dispersive shock waves. Stud. Appl. Maths 142 (3), 219–240.CrossRefGoogle Scholar
Holloway, P., Pelinovsky, E. & Talipova, T. 2001 Internal tide transformation and oceanic internal solitary waves. In Environmental Stratified Flows (ed. R.H.J. Grimshaw), pp. 29–60. Kluwer.CrossRefGoogle Scholar
Holloway, P.E., Pelinovsky, E. & Talipova, T. 1999 A generalized Korteweg-de Vries model of internal tide transformation in the coastal zone. J. Geophys. Res. 104 (C8), 18333–18350.CrossRefGoogle Scholar
Horn, D.A., Imberger, J., Ivey, G.N. & Redekopp, L.G. 2002 A weakly nonlinear model of long internal waves in closed basins. J. Fluid Mech. 467, 269–287.CrossRefGoogle Scholar
Iacocca, E., Silva, T.J. & Hoefer, M.A. 2017 Breaking of Galilean invariance in the hydrodynamic formulation of ferromagnetic thin films. Phys. Rev. Lett. 118 (1), 017203.CrossRefGoogle ScholarPubMed
Il'ichev, A.T. & Tomashpolskii, V.Y. 2015 Soliton-like structures on a liquid surface under an ice cover. Theor. Math. Phys. 182 (2), 231–245.CrossRefGoogle Scholar
Ivanov, S.K. 2020 Riemann problem for the light pulses in optical fibers for the generalized Chen-Lee-Liu equation. Phys. Rev. A 101 (5), 053844.CrossRefGoogle Scholar
Ivanov, S.K. & Kamchatnov, A.M. 2017 Riemann problem for the photon fluid: self-steepening effects. Phys. Rev. A 96 (5), 053827.CrossRefGoogle Scholar
Ivanov, S.K., Kamchatnov, A.M., Congy, T. & Pavloff, N. 2017 Solution of the Riemann problem for polarization waves in a two-component Bose-Einstein condensate. Phys. Rev. E 96 (6), 062202.CrossRefGoogle Scholar
Jamshidi, S. & Johnson, E.R. 2020 The long-wave potential-vorticity dynamics of coastal fronts. J. Fluid Mech. 888, A19.CrossRefGoogle Scholar
Kakutani, T. & Yamasaki, N. 1978 Solitary waves on a two-layer fluid. J. Phys. Soc. Japan 45 (2), 674–679.CrossRefGoogle Scholar
Kamchatnov, A.M., Kuo, Y.-H., Lin, T.-C., Horng, T.-L., Gou, S.-C., Clift, R., El, G.A. & Grimshaw, R.H.J. 2012 Undular bore theory for the Gardner equation. Phys. Rev. E 86 (3), 036605.CrossRefGoogle ScholarPubMed
Kamchatnov, A.M., Kuo, Y.-H., Lin, T.-C., Horng, T.-L., Gou, S.-C., Clift, R., El, G.A. & Grimshaw, R.H.J. 2013 Transcritical flow of a stratified fluid over topography: analysis of the forced Gardner equation. J. Fluid Mech. 736, 495–531.CrossRefGoogle Scholar
Kamchatnov, A.M., Spire, A. & Konotop, V.V. 2004 On dissipationless shock waves in a discrete nonlinear Schrödinger equation. J. Phys. A: Mathe. Gen. 37 (21), 5547–5568.CrossRefGoogle Scholar
Kivshar, Y.S. & Malomed, B.A. 1989 Dynamics of solitons in nearly integrable systems. Rev. Mod. Phys. 61, 763–915.CrossRefGoogle Scholar
Kluwick, A., Scheichl, S. & Cox, E.A. 2007 Near-critical hydraulic flows in two-layer fluids. J. Fluid Mech. 575, 187–219.CrossRefGoogle Scholar
Lax, P.D. 1973 Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves. SIAM.CrossRefGoogle Scholar
Leach, J.A. 2012 An initial-value problem for the defocusing modified Korteweg–de Vries equation. J. Differ. Equ. 252 (2), 1032–1051.CrossRefGoogle Scholar
Leach, J.A. 2013 An initial-value problem for the modified Korteweg-de Vries equation. IMA J. Appl. Math. 78 (6), 1196–1213.CrossRefGoogle Scholar
LeFloch, P.G. 2002 Hyperbolic Systems of Conservation Laws. Birkhauser.CrossRefGoogle Scholar
Levermore, C.D. 1988 The hyperbolic nature of the zero dispersion KdV limit. Commun. Part. Diff. Equ. 13 (4), 495–514.CrossRefGoogle Scholar
Li, L., Wang, C. & Grimshaw, R. 2015 Observation of internal wave polarity conversion generated by a rising tide. Geophys. Res. Lett. 42 (10), 4007–4013.CrossRefGoogle Scholar
Lowman, N.K. & Hoefer, M.A. 2013 Dispersive shock waves in viscously deformable media. J. Fluid Mech. 718, 524–557.CrossRefGoogle Scholar
Madsen, P.A., Fuhrman, D.R. & Schäffer, H.A. 2008 On the solitary wave paradigm for tsunamis. J. Geophys. Res. 113, C12012.CrossRefGoogle Scholar
Maiden, M.D., Anderson, D.V., Franco, N.A., El, G.A. & Hoefer, M.A. 2018 Solitonic dispersive hydrodynamics: theory and observation. Phys. Rev. Lett. 120 (14), 144101.CrossRefGoogle ScholarPubMed
Marchant, T.R. 2008 Undular bores and the initial-boundary value problem for the modified Korteweg-de Vries equation. Wave Motion 45 (4), 540–555.CrossRefGoogle Scholar
Mei, C.C., Stiassnie, M. & Yue, D.K.-P. 2005 Theory and Applications of Ocean Surface Waves. Advanced Series on Ocean Engineering, vol. 23. World Scientific.Google Scholar
Miura, R.M. 1968 Korteweg–de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation. J. Math. Phys. 9 (8), 1202–1204.CrossRefGoogle Scholar
Miyata, M. 1985 An internal solitary wave of large amplitude. La Mer 23 (2), 43–48.Google Scholar
Nakamura, Y., Ferreira, J.L. & Ludwig, G.O. 1985 Experiments on ion-acoustic rarefactive solitons in a multi-component plasma with negative ions. J. Plasma Phys. 33 (2), 237–248.CrossRefGoogle Scholar
Nash, J.D. & Moum, J.N. 2005 River plumes as a source of large-amplitude internal waves in the coastal ocean. Nature 437 (7057), 400–403.CrossRefGoogle ScholarPubMed
Nycander, J., Dritschel, D.G. & Sutyrin, G.G. 1993 The dynamics of long frontal waves in the shallow-water equations. Phys. Fluids A 5 (5), 1089–1091.CrossRefGoogle Scholar
Pedlosky, J. 2003 Waves in the Ocean and Atmosphere. Introduction to Wave Dynamics. Springer.CrossRefGoogle Scholar
Pratt, L.J. 1988 Meandering and eddy detachment according to a simple (looking) path equation. J. Phys. Oceanogr. 18 (11), 1627–1640.2.0.CO;2>CrossRefGoogle Scholar
Pratt, L.J. & Stern, M.E. 1986 Dynamics of potential vorticity fronts and eddy detachment. J. Phys. Oceanogr. 16 (6), 1101–1120.2.0.CO;2>CrossRefGoogle Scholar
Ruderman, M.S., Talipova, T. & Pelinovsky, E. 2008 Dynamics of modulationally unstable ion-acoustic wavepackets in plasmas with negative ions. J. Plasma Phys. 74 (5), 639–656.CrossRefGoogle Scholar
Scott, D.R. & Stevenson, D.J. 1984 Magma solitons. Geophys. Res. Lett. 11 (11), 1161–1164.CrossRefGoogle Scholar
Scott, D.R., Stevenson, D.J. & Whitehead, J.A. 1986 Observations of solitary waves in a viscously deformable pipe. Nature 319 (6056), 759–761.CrossRefGoogle Scholar
Shroyer, E.L., Moum, J.N. & Nash, J.D. 2009 Observations of polarity reversal in shoaling nonlinear internal waves. J. Phys. Oceanogr. 39 (3), 691–701.CrossRefGoogle Scholar
Sprenger, P. & Hoefer, M. 2017 Shock waves in dispersive hydrodynamics with nonconvex dispersion. SIAM J. Appl. Maths 77, 26–50.CrossRefGoogle Scholar
Sprenger, P., Hoefer, M.A. & El, G.A. 2018 Hydrodynamic optical soliton tunneling. Phys. Rev. E 97 (3), 032218.CrossRefGoogle ScholarPubMed
Stubblefield, A.G., Spiegelman, M. & Creyts, T.T. 2020 Solitary waves in power-law deformable conduits with laminar or turbulent fluid flow. J. Fluid Mech. 886, A10.CrossRefGoogle Scholar
Whitham, G.B. 1965 Non-linear dispersive waves. Proc. R. Soc. A 283, 238–261.Google Scholar
Whitham, G.B. 1974 Linear and Nonlinear Waves. Wiley.Google Scholar
Zhang, G. & Yan, Z. 2020 Focusing and defocusing mKdV equations with nonzero boundary conditions: inverse scattering transforms and soliton interactions. Physica D 410, 132521.CrossRefGoogle Scholar