On the realm of validity of strongly nonlinear asymptotic approximations for internal waves

Camassa, Roberto; Choi, Woo-Young; Michallet, Hervé; Rusås, Per-Olav; Sveen, Johan Kristian

doi:10.1017/s0022112005007226

Cited by 113 publications

(110 citation statements)

References 24 publications

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“…These "table top" solitary waves and conjugate states do not occur in the KdV equation, but do appear when the cubic nonlinearity is included in the KdV-type models (Kakutani and Yamasaki 1978;Helfrich and Melville 2006), though with a different limiting amplitude. The MCC solitary wave properties, including the limiting conjugate state amplitude, agree quite well with fully nonlinear and nonhydrostatic theories, numerical calculations, laboratory experiments, and oceanic observations (e.g., Choi and Camassa 1999;Michallet and Barthélemy 1998;Ostrovsky and Grue 2003;Camassa et al 2006). The MCC equations, unlike their weakly nonlinear counterparts, do not filter out Kelvin-Helmholtz instability (Jo and Choi 2002).…”

Section: ͑13͒supporting

confidence: 67%

Nonlinear Disintegration of the Internal Tide

Helfrich

Grimshaw

2008

Journal of Physical Oceanography

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The disintegration of a first-mode internal tide into shorter solitary-like waves is considered. Since observations frequently show both tides and waves with amplitudes beyond the restrictions of weakly nonlinear theory, the evolution is studied using a fully nonlinear, weakly nonhydrostatic two-layer theory that includes rotation. In the hydrostatic limit, the governing equations have periodic, nonlinear inertiagravity solutions that are explored as models of the nonlinear internal tide. These long waves are shown to be robust to weak nonhydrostatic effects. Numerical solutions show that the disintegration of an initial sinusoidal linear internal tide is closely linked to the presence of these nonlinear waves. The initial tide steepens due to nonlinearity and sheds energy into short solitary waves. The disintegration is halted as the longwave part of the solution settles onto a state close to one of the nonlinear hydrostatic solutions, with the short solitary waves superimposed. The degree of disintegration is a function of initial amplitude of the tide and the properties of the underlying nonlinear hydrostatic solutions, which, depending on stratification and tidal frequency, exist only for a finite range of amplitudes (or energies). There is a lower threshold below which no short solitary waves are produced. However, for initial amplitudes above another threshold, given approximately by the energy of the limiting nonlinear hydrostatic inertia-gravity wave, most of the initial tidal energy goes into solitary waves. Recent observations in the South China Sea are briefly discussed.

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Section: ͑13͒supporting

confidence: 67%

Nonlinear Disintegration of the Internal Tide

Helfrich

Grimshaw

2008

Journal of Physical Oceanography

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“…The steady solitary waves of the new stable model are compared to those of the Boussinesq MCC, since, in the case of zero far-field shear, MCC solutions compare well with experiments [12]. The comparisons show that the agreement is generally excellent, and several new qualitative features of large amplitude internal waves with background shear are captured.…”

Section: Introductionmentioning

confidence: 89%

“…We now make the Boussinesq approximation, by replacing ρ 1 and ρ 2 in inertia terms by the mean (unit) density in (12)- (13), and consider only cases in which far-field conditions fix Q = 0 (a constant Q can then be removed by a Gallilean transformation). Thus, (12) and (13) becomē…”

Section: Formulationmentioning

confidence: 96%

“…A well-known fully nonlinear long-wave model, which contains dispersive terms arising from nonhydrostatic pressure corrections, is the Miyata-Choi-Camassa (MCC) system [7,9]. While solitary-wave solutions of this system show good agreement with observations [10][11][12], it is of limited use as a time-dependent model, because it is mathematically ill-posed [13], despite the fact that physically stable internal waves of large amplitude have been observed [14].…”

Section: Introductionmentioning

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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“…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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On the realm of validity of strongly nonlinear asymptotic approximations for internal waves

Cited by 113 publications

References 24 publications

Nonlinear Disintegration of the Internal Tide

Nonlinear Disintegration of the Internal Tide

A Model for Strongly Nonlinear Long Interfacial Waves with Background Shear

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

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