Long Non‐Linear Waves in Fluid Flows

Benney, D. J.

doi:10.1002/sapm196645152

Cited by 377 publications

(248 citation statements)

References 18 publications

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“…Note, the KdV equation is well known to be a suitable physical model for describing weakly nonlinear advective effects and linear dispersion in IWs. It was originally developed by Benney (1966) and extended to second order by Lee and Beardsley (1974). The KdV and eKdV Eq.…”

Section: Two-layer Modelmentioning

confidence: 99%

Simulations and observation of nonlinear internal waves on the continental shelf: Korteweg–de Vries and extended Korteweg–de Vries solutions

O’Driscoll

Levine²

2017

Ocean Sci.

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Abstract. Numerical solutions of the Kortewegde Vries (KdV) and extended Korteweg-de Vries (eKdV) equations are used to model the transformation of a sinusoidal internal tide as it propagates across the continental shelf. The ocean is idealized as being a two-layer fluid, justified by the fact that most of the oceanic internal wave signal is contained in the gravest mode. The model accounts for nonlinear and dispersive effects but neglects friction, rotation and mean shear. The KdV model is run for a number of idealized stratifications and unique realistic topographies to study the role of the nonlinear and dispersive effects. In all model solutions the internal tide steepens forming a sharp front from which a packet of nonlinear solitary-like waves evolve. Comparisons between KdV and eKdV solutions are made. The model results for realistic topography and stratification are compared with observations made at moorings off Massachusetts in the Middle Atlantic Bight. Some features of the observations compare well with the model. The leading face of the internal tide steepens to form a shock-like front, while nonlinear high-frequency waves evolve shortly after the appearance of the jump. Although not rank ordered, the wave of maximum amplitude is always close to the jump. Some features of the observations are not found in the model. Nonlinear waves can be very widely spaced and persist over a tidal period.

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Section: Two-layer Modelmentioning

confidence: 99%

Simulations and observation of nonlinear internal waves on the continental shelf: Korteweg–de Vries and extended Korteweg–de Vries solutions

O’Driscoll

Levine²

2017

Ocean Sci.

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“…Weakly nonlinear theory (henceforth WNL), following the original derivation of Benney (1966), albeit in the notation of Lamb and Yan (1996) (also see this latter article for details of the derivation), considers an expansion in two small parameters, for amplitude and µ for the square of the aspect ratio. The first order, mode-1 streamfunction is given by…”

Section: Descriptions Of Iswsmentioning

confidence: 99%

Spectral methods for internal waves: indistinguishable density profiles and double-humped solitary waves

Dunphy

Subich

Stastna

2011

Nonlin. Processes Geophys.

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Abstract. Internal solitary waves are widely observed in both the oceans and large lakes. They can be described by a variety of mathematical theories, covering the full spectrum from first order asymptotic theory (i.e. Korteweg-de Vries, or KdV, theory), through higher order extensions of weakly nonlinear-weakly nonhydrostatic theory, to fully nonlinear-weakly nonhydrostatic theories and finally exact theory based on the Dubreil-Jacotin-Long (DJL) equation that is formally equivalent to the full set of Euler equations. We discuss how spectral and pseudospectral methods allow for the computation of novel phenomena in both approximate and exact theories. In particular we construct markedly different density profiles for which the coefficients in the KdV theory are very nearly identical. These two density profiles yield qualitatively different behaviour for both exact, or fully nonlinear, waves computed using the DJL equation and in dynamic simulations of the time dependent Euler equations. For exact, DJL, theory we compute exact solitary waves with two-scales, or so-called double-humped waves.

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“…The formulation by Benney (1966) predicts that, in the Boussinesq approximation, the vertical displacement field, ξ , has the form…”

Section: Internal Solitary Wavesmentioning

confidence: 99%

“…In future work we will adapt this idealized study to more realistic oceanographic circumstances in order to predict the behaviour of radially spreading solitary waves generated by river plumes. Weidman and Velarde (1992) rigourously extended the theory of Benney (1966) to derive a formula for the leading order weakly nonlinear evolution of axisymmetric internal solitary waves. Here we outline a similar derivation, but we futher impose that the amplitude of the wave must decrease as r −1/2 as required by energy conservation.…”

mentioning

confidence: 99%

The lifecycle of axisymmetric internal solitary waves

McMillan¹,

Sutherland²

2010

Nonlin. Processes Geophys.

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Abstract. The generation and evolution of solitary waves by intrusive gravity currents in an approximate two-layer fluid with equal upper-and lower-layer depths is examined in a cylindrical geometry by way of theory and numerical simulations. The study is limited to vertically symmetric cases in which the density of the intruding fluid is equal to the average density of the ambient. We show that even though the head height of the intrusion decreases, it propagates at a constant speed well beyond 3 lock radii. This is because the strong stratification at the interface supports the formation of a mode-2 solitary wave that surrounds the intrusion head and carries it outwards at a constant speed. The wave and intrusion propagate faster than a linear long wave; therefore, there is strong supporting evidence that the wave is indeed nonlinear. Rectilinear Korteweg-de Vries theory is extended to allow the wave amplitude to decay as r −p with p = 1 2 and the theory is compared to the observed waves to demonstrate that the width of the wave scales with its amplitude. After propagating beyond 7 lock radii the intrusion runs out of fluid. Thereafter, the wave continues to spread radially at a constant speed, however, the amplitude decreases sufficiently so that linear dispersion dominates and the amplitude decays with distance as r −1 .

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Long Non‐Linear Waves in Fluid Flows

Cited by 377 publications

References 18 publications

Simulations and observation of nonlinear internal waves on the continental shelf: Korteweg–de Vries and extended Korteweg–de Vries solutions

Simulations and observation of nonlinear internal waves on the continental shelf: Korteweg–de Vries and extended Korteweg–de Vries solutions

Spectral methods for internal waves: indistinguishable density profiles and double-humped solitary waves

The lifecycle of axisymmetric internal solitary waves

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