Breather generation in fully nonlinear models of a stratified fluid

Lamb, Kevin G.; Polukhina, O. E.; Talipova, Tatiana; Pelinovsky, Efim; Xiao, Wen; Kurkin, A. A.

doi:10.1103/physreve.75.046306

Cited by 60 publications

(59 citation statements)

References 16 publications

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“…Similar to the case 1, the characteristics and energy loss of the amplitude-modulated wave packet for the case 5 were also similar with the breather definition. Noted that the breather solution exist only if the cubic nonlinearity coefficients in Gardner equation is positive (Lamb et al, 2007), but the configuration of the stratification in our simulation results a negative cubic nonlinearity coefficient in Gardner equation (Grimshaw et al, 2010;Talipova et al, 2011) and it means the breather is not allowed The oscillating tail was frequently observed in similar studies (Carr et al, 2015;Olsthoorn et al, 2013;Stastna et al, 2015). Its generation was related to the shear, and the tail was sustained by continuous energy input.…”

supporting

confidence: 45%

The evolution of mode-2 internal solitary waves modulated by background shear currents

Zhang¹,

Xu²,

Qun³

et al. 2018

Preprint

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Abstract. The evolution process of mode-2 internal solitary waves (ISWs) modulated by the background shear currents was investigated numerically. The forward-propagating long wave, amplitude-modulated wave packet were generated during the early stage of modulation, where the amplitude-modulated wave packet were suggested playing an important role in the energy transfer 15 process, and then the oscillating tail was generated and followed the solitary wave. Five different cases were introduced to assess the sensitivity of the energy transfer process to the Δ, which defined as a dimensionless distance between the centers of pycnocline and shear current. The forward-propagating long waves were found robust to the Δ, but the oscillating tail and amplitude-modulated wave packet decreased in amplitude with increasing Δ. The highest energy loss rate was observed when Δ = 0. In the 20 first 30 periods, ~36% of the total energy lost at an average rate of 9 W m -1 , it would deplete the energy of the solitary wave in ~4.5 h, corresponding to a propagation distance of ~5 km, which is consistent with the hypothesis of Shroyer et al. (2010), who speculated that the mode-2 ISWs are "short-lived" in the presence of shear currents.

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supporting

confidence: 45%

The evolution of mode-2 internal solitary waves modulated by background shear currents

Zhang¹,

Xu²,

Qun³

et al. 2018

Preprint

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“…25 Meanwhile, direct numerical simulations within the fully nonlinear Euler equations for threelayer water flow suggest the existence of long-lived internal wave breathers. 26 In the context of internal waves in the ocean, the dispersion coefficient b is always positive, whereas the nonlinearity coefficients a and a 1 can be of either sign. The mapping of all coefficients of the Gardner equation was undertaken in Ref.…”

Section: Internal Waves In the Ocean: Soliboresmentioning

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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“…The other branch with −∞ < B < −1, has the opposite polarity and ranges from large waves with a "sech"-profile when B → −∞, to a limiting algebraic wave of amplitude −2µ/µ 1 when B → −1, see the lower panel in Fig. 2 Solitary waves with smaller amplitudes cannot exist, and from the point of view of the associated spectral problem are replaced by breathers, that is, pulsating solitary waves, see, for instance, Pelinovsky and Grimshaw (1997), Grimshaw et al (1999Grimshaw et al ( , 2010, Clarke et al (2000), Lamb et al (2007). When µ 1 → 0, B → 1 and the family reduces to the well-known KdV solitary wave family…”

Section: Constant Depthmentioning

confidence: 99%

Internal solitary waves: propagation, deformation and disintegration

Grimshaw

Pelinovsky

Talipova

et al. 2010

Nonlin. Processes Geophys.

157

139

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Abstract. In coastal seas and straits, the interaction of barotropic tidal currents with the continental shelf, seamounts or sills is often observed to generate largeamplitude, horizontally propagating internal solitary waves. Typically these waves occur in regions of variable bottom topography, with the consequence that they are often modeled by nonlinear evolution equations of the Kortewegde Vries type with variable coefficients. We shall review how these models are used to describe the propagation, deformation and disintegration of internal solitary waves as they propagate over the continental shelf and slope.

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Breather generation in fully nonlinear models of a stratified fluid

Cited by 60 publications

References 16 publications

The evolution of mode-2 internal solitary waves modulated by background shear currents

The evolution of mode-2 internal solitary waves modulated by background shear currents

Beyond the KdV: Post-explosion development

Internal solitary waves: propagation, deformation and disintegration

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