A Boussinesq system for two-way propagation of interfacial waves

Nguyen, Hai Yen; Dias, Frédéric

doi:10.1016/j.physd.2008.02.020

Cited by 42 publications

(56 citation statements)

References 31 publications

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“…In order to study the stability of the Boussinesq MCC system in the presence of small disturbances, (16) and (17) are linearized around constant states h 0 and w 0 , to obtain…”

Section: Linear Stability and The Stable Modelmentioning

confidence: 99%

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A Model for Strongly Nonlinear Long Interfacial Waves with Background Shear

Boonkasame

Milewski

2014

Stud Appl Math

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Please cite only the published version using the reference above.See http://opus.bath.ac.uk/ for usage policies.Please scroll down to view the document. A Model for Strongly Nonlinear Long Interfacial Waves with Background Shear By Anakewit Boonkasame and Paul A. MilewskiThe Miyata-Choi-Camassa (MCC) system of equations describing long internal nonhydrostatic and nonlinear waves at the interface between two layers of inviscid fluids of different densities bounded by top and bottom walls is mathematically ill-posed despite the fact that physically stable internal waves are observed matching closely those of MCC. A regularization to the MCC equations that yields a computationally simple well-posed system for time-dependent evolution is proposed here. The regularization is accomplished by keeping the full hyperbolic part of MCC and exchanging spatial and temporal derivatives in one of the linearized dispersive terms. Solitary waves of MCC over a wide range of parameters are used as a benchmark to check the accuracy of the model. Our model includes the possibility of a background shear, and we show that, contrary to the no shear case, solitary waves can cross the midlevel between the top and the bottom walls and may have different polarity from the case with no background shear. Time-dependent solutions of the regularization stable model are presented, including interactions of its solitary waves, and classical and modified Korteweg-de Vries equations for small amplitude waves with the inclusion of background shear are derived. Throughout the paper, the Boussinesq approximation is taken, although the results can be extended to the non-Boussinesq case.

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“…In order to study the stability of the Boussinesq MCC system in the presence of small disturbances, (16) and (17) are linearized around constant states h 0 and w 0 , to obtain…”

Section: Linear Stability and The Stable Modelmentioning

confidence: 99%

“…Both regularizations can be used to describe general flows with nonzero mean (or far-field) shear. (See [17] for another approach at deriving quadratic and cubic Boussinesq equations modeling interfacial waves. )…”

Section: Introductionmentioning

confidence: 99%

A Model for Strongly Nonlinear Long Interfacial Waves with Background Shear

Boonkasame

Milewski

2014

Stud Appl Math

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“…This is automatically satisfied thanks to the definition of G µ [εζ]ψ 1 . Indeed, applying Green's identity to (15), one obtains…”

Section: Asymptotic Expansion Of the Interface Operatormentioning

confidence: 99%

“…Weakly nonlinear models in two-dimensions have been derived by Camassa and Choi [6]. Nguyen and Dias [15] have derived and studied a Boussinesq-type system in a weakly nonlinear regime. Fully nonlinear models were obtained in the two-dimensional case by Camassa and Choi [7].…”

Section: Introductionmentioning

confidence: 99%

Asymptotic behaviors of internal waves

Bona¹,

Lannes²,

Saut³

2010

Journées Équations Aux Dérivées Partielles

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We present here a systematic method of derivation of asymptotic models for internal waves, that is, for the propagation of waves at the interface of two fluids of different densities. Many physical regimes are investigated, depending on the physical parameters (depth of the fluids, amplitude and wavelength of the interface deformations). This systematic method allows us to recover the many models existing in the literature and to derive some new models, in particular in the case of large amplitude internal waves and two-dimensional interfaces. We also provide rigorous consistency results for these models. We refer to [5] for full details.

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“…Therefore, in the last decades, it has been studied by many physicists and mathematicians, with regard to both well-posedness [23,24,2,3,32] and asymptotic models [16,17,11,19,14,6,29,30]. A significant step in the theory of internal waves was made in 2008 by Bona, Lannes and Saut [12].…”

Section: Introductionmentioning

confidence: 99%

Derivation and well-posedness of Boussinesq/Boussinesq systems for internal waves

Anh

2009

Ann. Polon. Math.

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Abstract. We consider the propagation of internal waves at the interface between two layers of immiscrible fluids of different densities, under the rigid lid assumption, with the presence of surface tension and with uneven bottoms. We are interested in the case where the flow has a Boussinesq structure in both the upper and lower fluid domains. Following the global strategy introduced recently by Bona, Lannes and Saut [J. Math. Pures Appl. 89 (2008)], we derive an asymptotic model in this regime, namely the Boussinesq/Boussinesq systems. Then using a contraction-mapping argument and energy methods, we prove that the derived systems which are linearly well-posed are in fact locally nonlinearly well-posed in suitable Sobolev classes. We recover and extend some known results on asymptotic models and well-posedness, for both surface waves and internal waves.

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A Boussinesq system for two-way propagation of interfacial waves

Cited by 42 publications

References 31 publications

A Model for Strongly Nonlinear Long Interfacial Waves with Background Shear

A Model for Strongly Nonlinear Long Interfacial Waves with Background Shear

Asymptotic behaviors of internal waves

Derivation and well-posedness of Boussinesq/Boussinesq systems for internal waves

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