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Strongly nonlinear effects on internal solitary waves in three-layer flows

Published online by Cambridge University Press:  25 November 2019

Ricardo Barros Open the ORCID record for Ricardo Barros [Opens in a new window] ,
Wooyoung Choi Open the ORCID record for Wooyoung Choi [Opens in a new window]  and
Paul A. Milewski Open the ORCID record for Paul A. Milewski [Opens in a new window]
Ricardo Barros*
Affiliation:
Department of Mathematical Sciences, Loughborough University, LoughboroughLE11 3TU, UK
Wooyoung Choi
Affiliation:
Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ07102-1982, USA
Paul A. Milewski
Affiliation:
Department of Mathematical Sciences, University of Bath, Claverton Down, BathBA2 7AY, UK
*
†Email address for correspondence: r.barros@lboro.ac.uk
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Abstract

We consider a strongly nonlinear long wave model for large amplitude internal waves in a three-layer flow between two rigid boundaries. The model extends the two-layer Miyata–Choi–Camassa (MCC) model (Miyata, Proceedings of the IUTAM Symposium on Nonlinear Water Waves, eds. H. Horikawa & H. Maruo, 1988, pp. 399–406; Choi & Camassa, J. Fluid Mech., vol. 396, 1999, pp. 1–36) and is able to describe the propagation of long internal waves of both the first and second baroclinic modes. Solitary-wave solutions of the model are shown to be governed by a Hamiltonian system with two degrees of freedom. Emphasis is given to the solitary waves of the second baroclinic mode (mode 2) and their strongly nonlinear characteristics that fail to be captured by weakly nonlinear models. In certain asymptotic limits relevant to oceanic applications and previous laboratory experiments, it is shown that large amplitude mode-2 waves with single-hump profiles can be described by the solitary-wave solutions of the MCC model, originally developed for mode-1 waves in a two-layer system. In other cases, however, e.g. when the density stratification is weak and the density transition layer is thin, the richness of the dynamical system with two degrees of freedom becomes apparent and new classes of mode-2 solitary-wave solutions of large amplitudes, characterized by multi-humped wave profiles, can be found. In contrast with the classical solitary-wave solutions described by the MCC equation, such multi-humped solutions cannot exist for a continuum set of wave speeds for a given layer configuration. Our analytical predictions based on asymptotic theory are then corroborated by a numerical study of the original Hamiltonian system.

JFM classification

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

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Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 883 , 25 January 2020 , A16
Copyright
© 2019 Cambridge University Press

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