Long nonlinear waves in stratified shear flows

Maslowe, S. A.; Redekopp, L. G.

doi:10.1017/s0022112080001681

Cited by 120 publications

(91 citation statements)

References 12 publications

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“…Nonlinear internal waves propagating along a shallow thermocline above a weakly stratified deep lower layer may radiate internal waves into the lower layer, thereby damping the nonlinear waves. This problem was formulated by Maslowe & Redekopp (1980), who derived the inhomogeneous Benjamin-Ono equation and found adiabatic solutions for the decay rate of the solitary waves of the homogenous equation. Numerical solutions of the inhomogeneous evolution equation by Pereira & Redekopp (1980) showed that the adiabatic solutions overestimated the damping, a result associated with the fact that the low (horizontal) wave numbers are damped more rapidly than the larger wave numbers.…”

Section: Wwwannualreviewsorg • Long Nonlinear Internal Wavesmentioning

confidence: 99%

Long Nonlinear Internal Waves

Helfrich

Melville

2006

Annu. Rev. Fluid Mech.

693

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: Wwwannualreviewsorg • Long Nonlinear Internal Wavesmentioning

confidence: 99%

Long Nonlinear Internal Waves

Helfrich

Melville

2006

Annu. Rev. Fluid Mech.

693

491

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“…5b and c, respectively. According to Benney (1966), Lee and Beardsley (1974), Pelinovsky and Shavratsky (1977), Maslowe and Redekopp (1980), Grimshaw (1981), Gear and Grimshaw (1983), Apel et al (1997), and Apel (2003), the modal displacement η n is governed by the Kortewegde Vries (K-dV) equation neglecting rotational effects and energy exchange between modes and assuming weakly nonlinear finite-amplitude plane-progressive waves propagating in a specific direction:…”

Section: Analytic Three-layer Ocean Modelmentioning

confidence: 99%

Convex and concave types of second baroclinic mode internal solitary waves

Yang

Fang

Tang

et al. 2010

Nonlin. Processes Geophys.

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Abstract. Two types of second baroclinic mode (mode-2) internal solitary waves (ISWs) were found on the continental slope of the northern South China Sea. The convex waveform displaced the thermal structure upward in the upper layer and downward in the lower layer, causing a bulge in the thermocline. The concave waveform did the opposite, causing a constriction. A few concave waves were observed in the South China Sea, marking the first documentation of such waves. On the basis of the Korteweg-de Vries (K-dV) equation, an analytical three-layer ocean model was used to study the characteristics of the two mode-2 ISW types. The analytical solution was primarily a function of the thickness of each layer and the density difference between the layers. Middle-layer thickness plays a key role in the resulting mode-2 ISW. A convex wave was generated when the middle-layer thickness was relatively thinner than the upper and lower layers, whereas only a concave wave could be produced when the middle-layer thickness was greater than half the water depth. In accordance with the K-dV equation, a positive and negative quadratic nonlinearity coefficient, α 2 , which is primarily dominated by the middle-layer thickness, resulted in convex and concave waves, respectively. The analytical solution showed that the wave propagation of a convex (concave) wave has the same direction as the current velocity in the middle (upper or lower) layer. Analysis of the three-layer ocean model properly reproduced the characteristics of the observed mode-2 ISWs in the South China Sea and provided a criterion for the existence of convex or concave waves. Concave waves were seldom seen because of the rarity of a stratified ocean with a thick middle layer. This analytical result agreed well with the observations.

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“…For the singular modes it appears possible that the denominator can change signs. It can be verified that for the eigenfunctions found by Maslowe and Redekopp (1980) for the singular limit k = 0 and Couette shearing, the denominator is also negative for all Ri. Figure 5 demonstrates weak dependence of ν (k) on kapparently quadratic with small coefficient ∼ − 0.01 in the long wave regime.…”

Section: Perturbation Expansionmentioning

confidence: 82%

“…. , and the singular spectrum (Maslowe and Redekopp, 1980) (ν n ∈ [0, 1]) are known when Ri > 1/4. For 0 < Ri < 1/4, ν n are complex conjugates, implying that one harmonic wave grows without bound and the other decays.…”

Section: Wave Disturbances Of Permanent Formmentioning

confidence: 99%

“…Benney proceeded to expand in small ε and k 2 . There are good reasons to adopt the infinitely long wave as the basis of an expansion procedure: (i) there are closed form solutions for some shear profiles (Maslowe and Redekopp, 1980) and stratifications (Weidman and Velarde, 1992); (ii) the procedure shows that long wave disturbances approximately satisfy the KdV equation and thus have soliton or conoidal character. However, extending the analysis of Benney to a viscous fluid is problematic-awkward questions about the ordering of weak nonlinearity, weak dispersion, and weak dissipation arise.…”

Section: Perturbation Expansionmentioning

confidence: 99%

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Long solitary internal waves in stable stratifications

Zimmerman

Rees

2004

Nonlin. Processes Geophys.

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Abstract.Observations of internal solitary waves over an antarctic ice shelf (Rees and Rottman, 1994) demonstrate that even large amplitude disturbances have wavelengths that are bounded by simple heuristic arguments following from the Scorer parameter based on linear theory for wave trapping. Classical weak nonlinear theories that have been applied to stable stratifications all begin with perturbations of simple long waves, with corrections for weak nonlinearity and dispersion resulting in nonlinear wave equations (Korteweg-deVries (KdV) or Benjamin-Davis-Ono) that admit localized propagating solutions. It is shown that these theories are apparently inappropriate when the Scorer parameter, which gives the lowest wavenumber that does not radiate vertically, is positive. In this paper, a new nonlinear evolution equation is derived for an arbitrary wave packet thus including one bounded below by the Scorer parameter. The new theory shows that solitary internal waves excited in high Richardson number waveguides are predicted to have a halfwidth inversely proportional to the Scorer parameter, in agreement with atmospheric observations. A localized analytic solution for the new wave equation is demonstrated, and its soliton-like properties are demonstrated by numerical simulation.

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Long nonlinear waves in stratified shear flows

Cited by 120 publications

References 12 publications

Long Nonlinear Internal Waves

Long Nonlinear Internal Waves

Convex and concave types of second baroclinic mode internal solitary waves

Long solitary internal waves in stable stratifications

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