1991
DOI: 10.1017/s0022112091003191
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Resonant generation of finite-amplitude waves by the flow of a uniformly stratified fluid over topography

Abstract: The forced Korteweg-de Vries equation is now established as the canonical equation to describe resonant, or critical, flow over topography. However, when the fluid is uniformly and weakly stratified, this equation degenerates in that the quadratic nonlinear term is absent. This anomalous, but important, case requires an alternative theory which is the purpose of this paper. We derive a new evolution equation to describe this case which, while having some similarities to the forced Korteweg-de Vries equation, c… Show more

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Cited by 50 publications
(96 citation statements)
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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%
See 1 more Smart Citation
“…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%
“…The topography is a Gaussian bump located at x = 0 with length scale L. The solutions are shown at the same time after initiation of the forcing and for the same stratification and other parameters as in figure 3 of Melville & Helfrich (1987). Grimshaw & Yi (1991) addressed the case of a uniform weakly stratified Boussinesq flow over topography, in which case the quadratic nonlinearity goes to zero. The resulting solutions have some similarity to those of the fKdV equation but now weak forcing produces an O(1) response and the wave amplitudes are limited by breaking, defined as an incipient flow reversal.…”
Section: Generationmentioning
confidence: 99%
“…In the resonant response, moreover, we see the emergence of upstream-wave propagation along with a disturbance of opposite sign forming on the downstream side of the topography. This behavior, which is characteristic of resonant flow in a channel of finite depth (Grimshaw and Smyth [13], Grimshaw and Yi [12]), was also found in Prasad and Akylas [20] and must be attributed to the trapping effect caused by the interaction of the induced disturbance with the background buoyancy-frequency oscillations. In the asymptotic theory, the response was tracked in terms of the scaled time T = /2t, corresponding to significantly larger values of t than those in Fig.…”
Section: Uniform Mean Buoyancy-frequency Profilementioning
confidence: 55%
“…In Prasad and Akylas [20], the parameter F plays the role of a Froude number as it defines the critical-flow regime in which resonance occurs, in analogy with resonant flow over topography in a channel of finite depth (Grimshaw and Smyth [13], Grimshaw and Yi [12]). Specifically, the theory of Prasad and Akylas [20] applies to the nearly hydrostatic response (t <« 1) in the case that small sinusoidal oscillations of wavelength X are superposed on a uniform mean buoyancy profile:…”
Section: Xn0"mentioning
confidence: 99%
“…This can lead to a fragmentation of the density current, as seen in laboratory experiments by Simpson (1987) and open-ocean observations by Soloviev and Lukas (1997). A buoyancy-driven current propagating into a two-layer stratified ambient environment in the coastal ocean may result in a resonant generation of an upstream undular bore (Grimshaw and Yi, 1991;Nash and Moum, 2005;White and Helfrich, 2012). In the presence of wind stress, an asymmetry may develop in the sea surface signature of low-density plumes .…”
Section: Introductionmentioning
confidence: 96%