Skip to main content
×
Home

Dromions of flexural-gravity waves

  • Mohammad-Reza Alam (a1)
Abstract
Abstract

Here we show that weakly nonlinear flexural-gravity wave packets, such as those propagating on the surface of ice-covered waters, admit three-dimensional fully localized solutions that travel with a constant speed without dispersion or dissipation. These solutions, that are formed at the intersection of line-soliton mean-flow tracks, have exponentially decaying tails in all directions and are called dromions in contrast to lumps that decay only algebraically. We derive, by asymptotic expansion and assuming multiple scales for spatial and temporal variations, the three-dimensional weakly nonlinear governing equations that describe the coupled motion of the wavepacket envelope and the underlying mean current. We show that in the limit of long waves and strong flexural rigidity these equations reduce to a system of nonlinear elliptic–hyperbolic partial differential equations similar to the Davey–Stewartson I (DSI) equation, but with major differences in the coefficients. Specifically, and contrary to DSI equations, the elliptic and hyperbolic operators in the flexural-gravity equations are not canonical resulting in complications in analytical considerations. Furthermore, standard computational techniques encounter difficulties in obtaining the dromion solution to these equations owing to the presence of a spatial hyperbolic operator whose solution does not decay at infinity. Here, we present a direct (iterative) numerical scheme that uses pseudo-spectral expansion and pseudo-time integration to find the dromion solution to the flexural-gravity wave equation. Details of this direct simulation technique are discussed and properties of the solution are elaborated through an illustrative case study. Dromions may play an important role in transporting energy over the ice cover in the Arctic, resulting in the ice breaking far away from the ice edge, and also posing danger to icebreaker ships. In fact we found that, contrary to DSI dromions that only exist in water depths of less than 5 mm, flexural-gravity dromions exist for a broad range of ice thicknesses and water depths including values that may be realized in polar oceans.

Copyright
Corresponding author
Email address for correspondence: reza.alam@berkeley.edu
References
Hide All
Ablowitz M. J. & Segur H. 1979 On the evolution of packets of water waves. J. Fluid Mech. 92, 691715.
Akylas T. R., Dias F. & Grimshaw R. H. J. 1998 The effect of the induced mean flow on solitary waves in deep water. J. Fluid Mech. 355, 317328.
Benney D. J. & Luke J. C. 1964 On the interactions of permanent waves of finite amplitude. J. Math. Phys. 43, 309313.
Berger K. M. & Milewsky P. A. 2000 The generation and evolution of lump solitary waves in surface-tension-dominated flows. SIAM J. Appl. Maths 61 (3), 731750.
Besse C., Mauser N. J. & Stimming H. P. 2004 Numerical study of the Davey–Stewartson system. ESIAM; Math. Model. Numer. Anal. 38 (6), 10351054.
Boiti M., Leon J. J.-P., Martina L. & Pempinelli F. 1988 Scattering of localized solitons in the plane. Phys. Lett. A 132 (8,9), 432439.
Champagne B. & Winternitz P. 1988 On the infinite-dimensional symmetry group of the Davey–Stewartson equations. J. Math. Phys. 29 (1), 18.
Chen X. J., Jensen J. J., Cui W. C. & Fu S. X. 2003 Hydroelasticity of a floating plate in multidirectional waves. Ocean Engng 30 (15), 19972017.
Clarkson P. A. & Hood S. 1994 New symmetry reductions and exact solutions of the Davey–Stewartson system. I. Reductions to ordinary differential equations. J. Math. Phys. 35 (1), 255283.
Davey A. & Stewartson K. 1974 On three-dimensional packets of surface waves. Proc. R. Soc. Lond. A 338 (1613), 101110.
Djordjevic V. D. & Redekopp L. G. 1977 On two-dimensional packets of capillary-gravity waves. J. Fluid Mech. 79 (4), 703714.
Duan W.-S. 2003 Weakly two-dimensional modulated wave packet in dusty plasmas. Phys. Plasmas 10 (7), 30223055.
Fokas A. S. & Santini P. M. 1990 Dromions and a boundary value problem for the Davey–Stewartson 1 equation. Physica D 44 (1–2), 99130.
Forbes L. K. 1986 Surface waves of large amplitude beneath an elastic sheet. Part 1. High-order series solution. J. Fluid Mech. 169, 409428.
Freeman N. C. & Davey A. 1975 On the evolution of packets of long surface waves. Proc. R. Soc. Lond. A 344 (1638), 427433.
Gilson C. R. & Nimmo J. J. C. 1991 A direct method for dromion solutions of the Davey–Stewartson equations and their asymptotic properties. Proc. R. Soc. Lond. A 435 (1894), 339357.
Groves M. D. & Sun S.-M. 2008 Fully localised solitary-wave solutions of the three-dimensional gravity-capillary water-wave problem. Arch. Rat. Mech. Anal. 188, 191.
Hayashi N. & Hirata H. 1996 Global existence and asymptotic behaviour in time of small solutions to the elliptic hyperbolic Davey Stewartson system. Nonlinearity 9 (6), 13871409.
Hietarinta J. 1990 One-dromion solutions for genetic classes of equations. Phys. Lett. A 149 (2–3), 113118.
Hietarinta J. & Hirota R. 1990 Multidromion solutions to the Davey–Stewartson equation. Phys. Lett. A 145 (5), 237244.
Hogan S. J. 1985 The fourth-order evolution equation for deep-water gravity-capillary waves. Proc. R. Soc. Lond. A 402 (1823), 359372.
Hǎrǎguş-Courcelle M. & Il’ichev A. 1998 Three-dimensional solitary waves in the presence of additional surface effects. Eur. J. Mech. (B Fluids) 17 (5), 739768.
Hizel E., Turgay N. C. & Guldogan B. 2009 The symmetry reductions and new exact solutions of the generalized Davey–Stewartson equation. Intl J. Contemp. Math. Sci. 4 (18), 883894.
Johnson R. S. 1997 A Modern Introduction to the Mathematical Theory of Water Waves. Cambridge University Press.
Kadomtsev B. B. & Petviashvili V. I. 1970 On the stability of solitary waves in weakly dispersive media. Sov. Phys. Dokl. 15 (6), 539541.
Kim B. & Akylas T. R. 2005 On gravity capillary lumps. J. Fluid Mech. 540, 337351.
Korobkin A., Parau E. I. & Vanden-Broeck J.-M. 2011 The mathematical challenges and modelling of hydroelasticity. Proc. R. Soc. Lond. A 369 (1947), 28032812.
Kuznetsov E. A., Rubenchik A. M. & Zakharov V. E. 1986 Soliton stability in plasmas and hydrodynamics. Phys. Rep. 142 (3), 103165.
Leblond H. 1999 Electromagnetic waves in ferromagnets: a Davey–Stewartson-type model. J. Phys. A 32 (45), 79077932.
Leblond H. 2001 The Davey–Stewartson model in quadratic media: a way to control pulses. Soliton-driven Photonics 31 (4), 215218.
Liu A. K. & Mollo-Christensen E. 1988 Wave propagation in a solid ice pack. J. Phys. Oceanogr. 18 (11), 17021712.
Marko J. R. 2003 Observations and analyses of an intense waves-in-ice event in the Sea of Okhotsk. J. Geophys. Res. 108 (C9, 3296).
Miles J & Sneyd A. D. 2003 The response of a floating ice sheet to an accelerating line load. J. Fluid Mech. 497, 435439.
Milewski P. A., Vanden-Broeck J.-M. & Wang Z. 2011 Hydroelastic solitary waves in deep water. J. Fluid Mech. 679, 628640.
Nishinari K. & Yajima T. 1994 Numerical studies on stability of Dromion and its collisions. J. Phys. Soc. Japan 63 (10), 35383541.
Osborne A. R. 2009 Nonlinear Ocean Waves and the Inverse Scattering Transform. Academic.
Parau E. I. & Vanden-Broeck J.-M. 2011 Three-dimensional waves beneath an ice sheet due to a steadily moving pressure. Proc. R. Soc. Lond. A 369 (1947), 29732988.
Radha R. & Lakshmanan M. 1994 Singularity analysis equations and localized coherent structures in inline-graphic $(2+ 1)$ dimensional generalized Korteweg-de Vries. J. Math. Phys. 35 (9), 47464756.
Schultz C. L., Ablowitz M. J. & Bar Yaacov D. 1987 Davey–Stewartson I system: a quantum inline-graphic $(2+ 1)$ -dimensional integrable system. Phys. Rev. Lett. 59 (25), 28252828.
Strathdee J., Robinson W. H. & Haines E. M. 1991 Moving loads on ice plates of finite thickness. J. Fluid Mech. 226, 3761.
Suzuki H. 2005 Overview of megafloat: concept, design criteria, analysis, and design. Mar. Struct. 18 (2), 111132.
White P. W. & Weideman J. A. C. 1994 Numerical simulation of solitons and dromions in the Davey–Stewartson system. Math. Comput. Simul. 37 (4–5), 469479.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords:

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 24 *
Loading metrics...

Abstract views

Total abstract views: 127 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 17th November 2017. This data will be updated every 24 hours.