Long Internal Waves of Large Amplitude

Miyata, Motoyasu

doi:10.1007/978-3-642-83331-1_44

Cited by 85 publications

(110 citation statements)

References 28 publications

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“…The solitary-wave solutions of the Miyata-Choi-Camassa system have been studied in the original papers of [17,38]. In particular we know that for a given amplitude, or a given velocity, there exists at most one solitary wave (up to spatial translations).…”

Section: Two-layer Settingmentioning

confidence: 99%

“…Such a configuration appears naturally as a model for the ocean, as salinity and temperature may induce sharp density stratification, so that internal solitary waves are observed in many places [27,29,40]. Due to the weak density contrast, the observed solitary waves typically have much larger amplitude than their surface counterpart, hence the bilayer extension of the GreenNaghdi system introduced by [17,35,38], often called Miyata-Choi-Camassa model, is a very natural choice. It however suffers from strong Kelvin-Helmholtz instabilities-in fact stronger than the ones of the bilayer extension of the water waves system for large frequencies-and the work in [23] was motivated by taming these instabilities.…”

Section: F{ϕ}(k) = F(k) ϕ(K)mentioning

confidence: 99%

“…For an efficient and successful outcome of the method, it is important to have a fairly precise initial guess. To this aim, we use the exact solution of the Green-Naghdi model, which is either explicit (in the one-layer situation [42]) or obtained as the solution of an ordinary differential equation (in the bi-layer situation [17,38]) that we solve numerically. Our solutions are compared with the corresponding ones of the full Euler system.…”

Section: Description Of the Numerical Schemementioning

confidence: 99%

See 2 more Smart Citations

Solitary Wave Solutions to a Class of Modified Green–Naghdi Systems

Duchêne

Nilsson

Wahlén

2017

J. Math. Fluid Mech.

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Abstract. We provide the existence and asymptotic description of solitary wave solutions to a class of modified Green-Naghdi systems, modeling the propagation of long surface or internal waves. This class was recently proposed by Duchêne et al. (Stud Appl Math 137:356-415, 2016) in order to improve the frequency dispersion of the original Green-Naghdi system while maintaining the same precision. The solitary waves are constructed from the solutions of a constrained minimization problem. The main difficulties stem from the fact that the functional at stake involves low order non-local operators, intertwining multiplications and convolutions through Fourier multipliers.

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Section: Two-layer Settingmentioning

confidence: 99%

Section: F{ϕ}(k) = F(k) ϕ(K)mentioning

confidence: 99%

Section: Description Of the Numerical Schemementioning

confidence: 99%

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Solitary Wave Solutions to a Class of Modified Green–Naghdi Systems

Duchêne

Nilsson

Wahlén

2017

J. Math. Fluid Mech.

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“…A detailed investigation on the dependence of wavelength of large-amplitude internal solitary waves on depth is given by Vlasenko et al [2000]. In the present study we will refer to the wavelength of an internal solitary wave defined as follows: wavelength relationship for internal solitary waves simulated bv our numerical model using the five different stratifications mentioned above (see Table 1) and derived by the Miyata model, an analytical two-layer model in which the full nonlinearity of the shallow-water theory up to first-order phase dispersion is retained [Miyata, 1988] …”

Section: Model Theorymentioning

confidence: 99%

On the dynamics of internal waves in a nonlinear, weakly nonhydrostatic three‐layer ocean

Rubino¹,

Brandt²,

Weigle³

2001

J. Geophys. Res.

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Abstract. Aspects of the dynamics of internal solitary waves evolving in a three-layer ocean are investigated using a new numerical multilayer model that solves the nonlinear, weakly nonhydrostatic Boussinesq equations and uses high-resolution in situ data. The model applications refer to two different phenomena frequently observed in the real ocean, which can be described using a three-layer model rather than a two-layer model. In the first application the influence of the strength of a shallow seasonal thermocline superimposed on a two-layer permanent stratification on the structure of internal solitary waves is studied. It is found that while for small to medium wave amplitudes a decrease in the strength of the thermocline yields an increase in the simulated wavelengths, for large wave amplitudes this dependence is no longer monotonic. In particular, in the limiting case of a vanishing thermocline, first-mode internal solitary wave solutions of the three-layer numerical model tend to the analytical internal solitary wave solutions of the Miyata equations, a two-layer model, in which the full nonlinearity of the shallow-water theory up to first-order phase dispersion is retained. In the second application that refers particularly to high-resolution

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“…Actually, this basic approach was first suggested by Whitham 58 for surface waves based on the expansion of the Lagrangian; the later work by Green and Naghdi 59 includes a sloping bottom. For internal waves in a two-layer case, Miyata 60 suggested Boussinesq-type long-wave equations for strongly nonlinear, weakly dispersive waves in a two-layer fluid, and constructed a stationary solitary solution of these equations. A comprehensive analysis of this problem for a two-layer fluid was performed by Choi and Camassa.…”

Section: Model Equations For Dispersive Wavesmentioning

confidence: 99%

Beyond the KdV: Post-explosion development

Ostrovsky

Pelinovsky

Shrira

et al. 2015

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Several threads of the last 25 years' developments in nonlinear wave theory that stem from the classical Korteweg-de Vries (KdV) equation are surveyed. The focus is on various generalizations of the KdV equation which include higher-order nonlinearity, large-scale dispersion, and a nonlocal integral dispersion. We also discuss how relatively simple models can capture strongly nonlinear dynamics and how various modifications of the KdV equation lead to qualitatively new, non-trivial solutions and regimes of evolution observable in the laboratory and in nature. As the main physical example, we choose internal gravity waves in the ocean for which all these models are applicable and have genuine importance. We also briefly outline the authors' view of the future development of the chosen lines of nonlinear wave theory. V C 2015 AIP Publishing LLC.[http://dx.doi.org/10.1063/1.4927448]In the aftermath of the revolutionary development in the theory of nonlinear waves in the 1960s-1970s, the intensity of studies in this field does not show signs of decreasing. Here, several lines of studies related to the KdV equation are surveyed. Focusing on generalizations of the KdV equation, we trace the development of main ideas and concepts of nonlinear wave theory yielding qualitatively new solutions such as "fat" and table-top solitons, breathers, and slowly radiating solitons. The reasons underpinning the unmatched universality of the KdV equation as a mathematical model applicable in many physical contexts are quite natural: for long waves the model combines the most typical, small quadratic nonlinearity and weak, small-scale dispersion. The balance between the nonlinearity and dispersion allows the possibility of solitary waves possessing astonishing particlelike properties such as robustness and persistence in collisions not only with each other but also with other perturbations. This particle-like behavior is at the origin of the term soliton introduced to emphasize the affinity of such waves with elementary particles (electrons, protons, etc). As the understanding of nonlinear waves matured, the limitations of the KdV model and necessity to go beyond it became apparent; hence, the trend towards developing more general and rich models which generalize the KdV equation and yield many new and non-trivial results. Here, we briefly discuss their appearance in various mathematical and physical contexts and some of the results which follow, such as qualitatively new types of solitons, their limiting shapes and parameters, and interactions. It is also demonstrated that these features are not mathematical artefacts, which is illustrated by examples mainly related to nonlinear internal gravity waves in the ocean.

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Long Internal Waves of Large Amplitude

Cited by 85 publications

References 28 publications

Solitary Wave Solutions to a Class of Modified Green–Naghdi Systems

Solitary Wave Solutions to a Class of Modified Green–Naghdi Systems

On the dynamics of internal waves in a nonlinear, weakly nonhydrostatic three‐layer ocean

Beyond the KdV: Post-explosion development

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