A global theory of internal solitary waves in two-fluid systems

Amick, C. J.; Turner, R. E.

doi:10.1090/s0002-9947-1986-0860375-3

Cited by 94 publications

(70 citation statements)

References 23 publications

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“…The first study using this approach was, apparently, made in Ref. 53. It was found that there exists maximal possible solitary wave amplitude at which it acquires a flat-top shape and tends to two infinitely separated kinks.…”

Section: Strongly Nonlinear Kdv-type Models a Stationary Waves: mentioning

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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Section: Strongly Nonlinear Kdv-type Models a Stationary Waves: mentioning

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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“…This naturally introduces the notion of conjugate states as introduced by Benjamin (1966), and computed by Turner and Vanden-Broeck (1988) and described in further detail by Amick and Turner (1986) and by Evans and Ford (1996). Values of the limiting solitary wave amplitudes for the models KdV2, KdV2N, KdV3N, KdV3Nβ 1 , KdV3Nβ c and KdVEβ c are presented in Table 2 for a range of depth ratios, and these limiting amplitudes are compared with results of the conjugate state amplitude computed using the theory of Amick and Turner (1986). For the special case of a two-layer stratification with a rigid upper surface, the environmental case under examination here, Amick and Turner (1986) obtained an analytic solution for the conjugate state.…”

Section: Comparison Of Evolution Modelsmentioning

confidence: 99%

“…Values of the limiting solitary wave amplitudes for the models KdV2, KdV2N, KdV3N, KdV3Nβ 1 , KdV3Nβ c and KdVEβ c are presented in Table 2 for a range of depth ratios, and these limiting amplitudes are compared with results of the conjugate state amplitude computed using the theory of Amick and Turner (1986). For the special case of a two-layer stratification with a rigid upper surface, the environmental case under examination here, Amick and Turner (1986) obtained an analytic solution for the conjugate state. When the Boussinesq approximation is invoked, and when their results are transposed in terms of the non-dimensional variables used herein, the conjugate state is defined simply by the relations…”

Section: Comparison Of Evolution Modelsmentioning

confidence: 99%

Models for strongly-nonlinear evolution of long internal waves in a two-layer stratification

Sakai

Redekopp

2007

Nonlin. Processes Geophys.

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Abstract. Models describing the evolution of long internal waves are proposed that are based on different polynomial approximations of the exact expression for the phase speed of uni-directional, fully-nonlinear, infinitely-long waves in the two-layer model of a density stratified environment. It is argued that a quartic KdV model, one that employs a cubic polynomial fit of the separately-derived, nonlinear relation for the phase speed, is capable of describing the evolution of strongly-nonlinear waves with a high degree of fidelity. The marginal gains obtained by generating higher-order, weaklynonlinear extensions to describe strongly-nonlinear evolution are clearly demonstrated, and the limitations of the quite widely-used quadratic-cubic KdV evolution model obtained via a second-order, weakly-nonlinear analysis are assessed. Data are presented allowing a discriminating comparison of evolution characteristics as a function of wave amplitude and environmental parameters for several evolution models. Preliminary considerationsThe wide-spread appearance of packets of long internal waves in the shallow, stratified waters of the coastal ocean and lakes is firmly established (Osborne and Burch, 1980;Apel et al., 1985;Scotti and Pineda, 2004;Ostrovsky and Stepanyants, 1989;Stanton and Ostrovsky, 1998;Antenucci et al., 2000;Duda et al., 2004;Helfrich and Melville, 2006). These long-wave packets in many contexts are decidedly nonlinear, containing waves with amplitudes that are equal to and greater in magnitude than the controlling length scale of the problem, which is typically the scale of the uppermixed layer depth. Furthermore, these long-wave packets are known to stimulate strong benthic dissipation and mixing, having a determining influence on the decay of internal Correspondence to: T. Sakai (tsakai@usc.edu) tidal energy and the resuspension and transport of sedimentary material (Bogucki et al., 1997;Bogucki and Redekopp, 1999;Stastna and Lamb, 2002;Bogucki et al., 2005). Hence, there is increasing interest in the development of models that can yield quantitative, evolutionary descriptions of such wave packets in a wide range of environments. The present work is directed toward proposing such a model, and assessing its potential (along with that of various alternate models) to describe reliable descriptions of packets possessing large amplitude internal waves.Aspects of the propagation of nonlinear internal waves in the long-wave limit are examined in the case of a two-layer model under the Boussinesq approximation. Specifically, uni-directional propagation of a plane wave is considered in the idealized environmental model of a two-layer density stratification. This idealized case represents a convenient and quantitatively relevant model for the lowest internal wave mode in realistic stratifications when the water column possesses a single, prominent, thermoclinic layer whose depth is at most a modest fraction of the fluid depth. In what follows, the characteristics of the environmental model are defined in term...

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“…dy This allows us to solve for y as a function of (x, y/), y = y(x, ^), say. We continue to follow the technique in [8,9] and define w(xx, x2) = y(x,, 4/(x2)) -x2, (x,, x2) £ T+ .…”

Section: Notationmentioning

confidence: 99%