The Propagation of Long Large Amplitude Internal Waves

Benney, D. J.; Ko, Denny R. S.

doi:10.1002/sapm1978593187

Cited by 55 publications

(33 citation statements)

References 12 publications

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“…This corresponds to α 1 = 0, for which KdV solitary waves do not exist. If the stratification is allowed to be slightly nonuniform, or non-Boussinesq, then steady solitary waves can again be found by balancing weak nonlinearity against weak dispersion (Benney & Ko 1978, Grimshaw & Yi 1991, with the necessary balance achieved by finite-amplitude waves. Solitary waves exist up to an amplitude limited by incipient breaking.…”

Section: Fully Nonlinear Wavesmentioning

confidence: 99%

Long Nonlinear Internal Waves

Helfrich

Melville

2006

Annu. Rev. Fluid Mech.

694

491

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Over the past four decades, the combination of in situ and remote sensing observations has demonstrated that long nonlinear internal solitary-like waves are ubiquitous features of coastal oceans. The following provides an overview of the properties of steady internal solitary waves and the transient processes of wave generation and evolution, primarily from the point of view of weakly nonlinear theory, of which the Korteweg-de Vries equation is the most frequently used example. However, the oceanographically important processes of wave instability and breaking, generally inaccessible with these models, are also discussed. Furthermore, observations often show strongly nonlinear waves whose properties can only be explained with fully nonlinear models.

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Section: Fully Nonlinear Wavesmentioning

confidence: 99%

Long Nonlinear Internal Waves

Helfrich

Melville

2006

Annu. Rev. Fluid Mech.

694

491

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“…(10) transforms to a unit circle nonuniformly with respect to (F 1 , F 2 ). This nonuniformity arises because of the presence of the small parameter σ at the leading power of F 2 in the dispersion relation (9) [with allowance for Eq. (6) for λ] and, correspondingly, in Eq.…”

Section: Long-wave Approximationmentioning

confidence: 99%

“…In the present work, we assume that the ratio σ/μ is small and use the parameter σ as a modeling parameter. Following [9], we consider the long-wave approximation, where the ratio of the vertical and horizontal scales of motion is of the order of √ σ. Using the undisturbed depth h 2 of the upper layer as the vertical scale and the fluid discharge in the jth layer as the scale for the stream function ψ = ψ j in this layer, we introduce the dimensionless variables…”

Section: Long-wave Approximationmentioning

confidence: 99%

Asymptotic models of internal stationary waves

Makarenko

Mal’tseva

2008

J Appl Mech Tech Phy

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Equations of stationary long waves on the interface between a homogeneous fluid and an exponentially stratified fluid are considered. An equation of the second-order approximation of the shallow water theory inheriting the dispersion properties of the full Euler equations is used as the basic model. A family of asymptotic submodels is constructed, which describe three different types of bifurcation of solitary waves at the boundary points of the continuous spectrum of the linearized problem. Introduction.Equations of an inviscid two-layer fluid with a piecewise-constant density experiencing a jump at the interface between the layers are used (see [1,2]) as a mathematical model of internal waves in a pynocline. In this model, solitary waves and smooth bores are described by the equation of the second-order approximation of the shallow water theory, which was derived by Ovsyannikov [3]. Such an equation was obtained in [4] for the case with no slipping of the layers in the main flow. A similar approximation for long waves in a fluid with a piecewise-constant Brunt-Väisälä frequency was considered in [5]. It was noted [6] that asymptotic series for solitary waves are highly sensitive to small perturbations of the density field. The main objective of the present work is to estimate the influence of weak continuous stratification on the parameters of nonlinear waves on the interface. The behavior of the critical parameters of wave motion with stratification vanishing in one of the layers is studied. The density of the fluid in the second layer is assumed to be constant. The basic feature of this limit transition is the concentration of the spectra of higher modes in a narrow band of the boundary-layer type in the plane of bifurcation parameters. The presence of such a layer substantially affects the asymptotic behavior of solitary waves of the leading mode. In particular, a region of parameters is found, where the branching of solutions at the points of the spectrum boundary differs from the bifurcation of solitary waves to plateau-and bore-type waves for the model of a two-layer fluid [7,8].

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“…We use the perturbation method suggested by Benney and Ko (1978) for large amplitude internal waves with linear stratification. We put…”

Section: Linear Density Backgroundmentioning

confidence: 99%

Conjugate flows and amplitude bounds for internal solitary waves

Makarenko

Maltseva

Kazakov

2009

Nonlin. Processes Geophys.

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Abstract. Amplitude bounds imposed by the conservation of mass, momentum and energy for strongly nonlinear waves in stratified fluid are considered. We discuss the theoretical scheme which allows to determine broadening limits for solitary waves in the terms of a given upstream density profile. Attention is focused on the continuously stratified flows having multiple broadening limits. The role of the mean density profile and the influence of fine-scale stratification are analyzed.

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The Propagation of Long Large Amplitude Internal Waves

Cited by 55 publications

References 12 publications

Long Nonlinear Internal Waves

Long Nonlinear Internal Waves

Asymptotic models of internal stationary waves

Conjugate flows and amplitude bounds for internal solitary waves

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