Resonant wave interactions near a critical level in a stratified shear flow

1994 ◽  
Vol 269 ◽  
pp. 1-22 ◽  
Author(s):  
R. Grimshaw
Keyword(s):  
Shear Flow ◽  
Critical Level ◽  
Harmonic Component ◽  
Internal Gravity Waves ◽  
Resonant Interaction ◽  
Wave Interactions ◽  
Incoming Wave ◽  
Resonant Wave ◽  
Background Density ◽  
Stratified Shear Flow

Resonant interactions between internal gravity waves propagating in a stratified shear flow are considered for the case when the background density and shear flow vary slowly with respect to the waves. In Grimshaw (1988) triad resonances were considered, and interaction equations derived for the case when the resonance conditions are met only on certain space-time surfaces, being resonance sites. Here this analysis is extended to include higher-order resonances, with the aim of studying resonant wave interactions near a critical level. It is shown that a secondary resonant interaction between two incoming waves, in which two harmonic components of one incoming wave interact with a single harmonic component of another incoming wave, produces a reflected wave. This result is shown to agree with the study of Brown & Stewartson (1980, 1982a, b) who obtained this same result by a different approach.

Resonant wave interactions in a stratified shear flow

1988 ◽  
Vol 190 ◽  
pp. 357-374 ◽  
Author(s):  
R. Grimshaw
Keyword(s):  
Shear Flow ◽  
Gravity Waves ◽  
Critical Level ◽  
Internal Gravity Waves ◽  
Wave Interactions ◽  
Resonant Wave ◽  
Resonant Interactions ◽  
Stratified Shear Flow ◽  
Weak Resonance ◽  
Background Shear

Resonant interactions between triads of internal gravity waves propagating in a shear flow are considered for the case when the stratification and the background shear flow vary slowly with respect to typical wavelengths. If ωn, kn(n = 1, 2, 3) are the local frequencies and wavenumbers respectively then the resonance conditions are that ω1 + ω2 + ω3 = 0 and k1 + k2 + k3 = 0. If the medium is only weakly inhomogeneous, then there is a strong resonance and to leading order the resonance conditions are satisfied globally. The equations governing the wave amplitudes are then well known, and have been extensively discussed in the literature. However, if the medium is strongly inhomogeneous, then there is a weak resonance and the resonance conditions can only be satisfied locally on certain space-time resonance surfaces. The equations governing the wave amplitudes in this case are derived, and discussed briefly. Then the results are applied to a study of the hierarchy of wave interactions which can occur near a critical level, with the aim of determining to what extent a critical layer can reflect wave energy.

An experiment on third-order resonant wave interactions

1966 ◽  
Vol 25 (3) ◽  
pp. 417-435 ◽  
Author(s):  
M. S. Longuet-Higgins ◽  
N. D. Smith
Keyword(s):  
Gravity Waves ◽  
Response Curve ◽  
Resonant Interaction ◽  
Third Wave ◽  
Theoretical Discussion ◽  
Wave Interactions ◽  
Third Order ◽  
Large Wave ◽  
The Third ◽  
Resonant Wave

An experiment has been carried out to verify the existence of the resonant interaction between trains of gravity waves, predicted by Phillips (1960). As suggested by Longuet-Higgins (1962), two trains of waves in mutually perpendicular directions were generated in a rectangular wave tank. The ratio σ1/σ2of the wave frequencies was varied (1·4 < σ1/σ2< 2·1). When σ1/σ2[eDot ] 1·7357 it was expected that a resonant interaction would take place, generating a wave of frequency (2σ1−σ2). The amplitude of the third wave was expected to increase almost linearly in the direction of wave propagation. The shape of the response curve as a function of σ1/σ2was also predicted.In the present experiments rather large wave amplitudes had to be used, and the theoretical shape of the response curve was distorted by non-linear detuning. Nevertheless the peak amplitude of the resonant wave was found to increase with distance in very nearly the manner predicted.These experiments were carried out in 1961 but publication was deferred pending a similar but more accurate investigation by McGoldrick, Phillips, Huang & Hodgson (1966). Much of the theoretical discussion given in the present paper is relevant to their work.

A variational method for weak resonant wave interactions

1969 ◽  
Vol 309 (1499) ◽  
pp. 551-577 ◽  
Keyword(s):  
Variational Method ◽  
Internal Gravity Waves ◽  
Wave Number ◽  
Local Conservation ◽  
Wave Interactions ◽  
First Order ◽  
Resonant Wave ◽  
Resonant Interactions ◽  
Energy And Momentum ◽  
Phase Angles

Whitham’s variational method is formulated so as to apply to weak second-order resonant interactions among waves whose amplitudes and phase angles vary slowly with position and time. The method is applied in detail to capillary-gravity wave interactions. An internal gravity waves problem is also discussed briefly. The method leads to new and substantial simplifications of the interaction equations. This makes possible the proof of local conservation of total mean wave energy and momentum laws. These, together with another integral of the motion, are found to be of central importance in classifying and characterizing the slow modulations of planewave-like form. Such a classification is given in detail for all initial values of phase angles and relative amplitudes. All progressive uniform waves in the capillary range are found to be unstable with perturbation growth rates which can be of first order in the wave slopes. In this formulation amplitude dependent first-order corrections of classical frequency and/or wave-number arise for all waves participating in a resonance. A few predictions which could be verified by simple experiments are made.

Measurements of third-order resonant wave interactions

1966 ◽  
Vol 25 (3) ◽  
pp. 437-456 ◽  
Author(s):  
L. F. Mcgoldrick ◽  
O. M. Phillips ◽  
N. E. Huang ◽  
T. H. Hodgson
Keyword(s):  
Gravity Waves ◽  
Resonant Interaction ◽  
Experimental Demonstration ◽  
Wave Interactions ◽  
Third Order ◽  
The Third ◽  
Resonant Wave ◽  
Interaction Distance ◽  
At Resonance ◽  
Wave Trains

This paper presents the results of experiments on the resonant interaction of gravity waves. Two mutually-orthogonal primary wave trains are generated in a tank and their interaction products studied at various positions on the surface. Under suitable conditions, the growing resonant third-order interaction product is identified; its amplitude is shown to be a linear function of the interaction distance. The band-width of the response decreases with increasing distance, as is characteristic of the phenomenon of resonance. The ratio of the frequencies of the primary waves at resonance is very close to that predicted theoretically; the growth rate of the third component is close to, though about 20% higher than, the predicted value. Conditions far from resonance are also studied; it is found that the growing tertiary wave is absent in this case.These results offer the first unambiguous experimental demonstration of resonant wave interactions.

Simulation and stability of two-dimensional internal gravity waves in a stratified shear flow

1993 ◽  
Vol 19 (1-4) ◽  
pp. 325-366 ◽  
Author(s):  
C.-L. Lin ◽  
J.H. Ferziger ◽  
J.R. Koseff ◽  
S.G. Monismith
Keyword(s):  
Shear Flow ◽  
Gravity Waves ◽  
Internal Gravity Waves ◽  
Two Dimensional ◽  
Stratified Shear Flow

Resonant gravity-wave interactions in a shear flow

1968 ◽  
Vol 34 (3) ◽  
pp. 531-549 ◽  
Author(s):  
Alex. D. D. Craik
Keyword(s):  
Shear Flow ◽  
Resonance Condition ◽  
Resonant Interaction ◽  
Finite Amplitude ◽  
Wave Interactions ◽  
Primary Flow ◽  
Viscous Forces ◽  
Uniform Shear ◽  
Uniform Shear Flow ◽  
Large Growth

Among a triad of gravity waves in a uniform shear flow, a remarkably powerful second-order resonant interaction may take place. This interaction is characterized by large growth rates of waves which propagate in directions oblique to that of the primary flow, and by a systematic transfer of energy from the primary flow to such waves. Most of the energy transfer takes place in the vicinity of a ‘critical layer’, where viscous forces are dominant.Provided the resonance condition may be satisfied, a uniform shear flow which is perturbed by a two-dimensional wave of small but finite amplitude may be unstable, owing to the growth of two initially infinitesimal oblique waves which complete the resonant triad.

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