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.

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.

Theoretical and experimental studies of gravity wave interactions

1967 ◽  
Vol 299 (1456) ◽  
pp. 104-119 ◽  
Keyword(s):  
Gravity Waves ◽  
Experimental Studies ◽  
Internal Gravity Waves ◽  
Stokes Wave ◽  
Wave Interactions ◽  
Principal Characteristics ◽  
Resonant Wave ◽  
Resonant Interactions ◽  
Interaction Distance ◽  
Wave Modes

This paper is concerned with various aspects of the resonant interactions among waves. An experiment was suggested by Longuet-Higgins (1962) to detect this type of interaction among surface waves. This was subsequently performed by Longuet-Higgins & Smith (1966) and by McGoldrick, Phillips, Huang & Hodgson (1966). The results of the two sets of experiments are compared. Together they demonstrate very clearly the principal characteristics of the interaction; the maximum response at resonance and the linear growth with interaction distance, the decrease in band width with interaction distance and the shift of the resonance point that results from the amplitude dispersion. It is shown further that the instability of the Stokes wave, discovered and analysed by Benjamin & Feir, can be described in terms of these interactions and that it is not restricted to purely two dimensional motion. A Stokes wave is unstable to a disturbance containing a pair of wavenumbers defined by any point in the zone just inside the figure-of-eight loop shown in figure 12. Another example of resonant wave interactions is provided by short, internal gravity waves in a stratified fluid with constant Brunt-Väisälä frequency. The interactions among Fourier modes are considered, and it is shown that there arise both free and forced modes. In the latter, the dispersion relation for internal waves is not satisfied; there is no particular relation between wavenumber and frequency. The amplitudes of these are small compared with those of the internal wave modes provided the harmonic mean of the vorticity in the two interacting waves is small compared with the Brunt-Väisälä frequency. The motion then consists of interacting internal gravity waves, whose interaction sets are closed. On the other hand, if the forced components are comparable in magnitude with the wave modes, these interact strongly and indiscriminately; a ‘cascade’, characteristic of turbulence, develops.

scholarly journals Phase Structure of Internal Gravity Waves in the Ocean with Shear Flows

2021 ◽  
Vol 37 (4) ◽  
Author(s):  
V. V. Bulatov ◽  
Yu. V. Vladimirov ◽  
I. Yu. Vladimirov ◽  
◽  
◽  
...  
Keyword(s):  
Shear Flow ◽  
Spectral Problem ◽  
Gravity Waves ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Internal Gravity Waves ◽  
Buoyancy Frequency ◽  
Flow Profile ◽  
Wave Fields ◽  
Background Shear

Purpose. The description of the internal gravity waves dynamics in the ocean with background fields of shear currents is a very difficult problem even in the linear approximation. The mathematical problem describing wave dynamics is reduced to the analysis of a system of partial differential equations; and while taking into account the vertical and horizontal inhomogeneity, this system of equations does not allow separation of the variables. Application of various approximations makes it possible to construct analytical solutions for the model distributions of buoyancy frequency and background shear ocean currents. The work is aimed at studying dynamics of internal gravity waves in the ocean with the arbitrary and model distributions of density and background shear currents. Methods and Results. The paper represents the numerical and analytical solutions describing the main phase characteristics of the internal gravity wave fields in the stratified ocean of finite depth, both for arbitrary and model distributions of the buoyancy frequency and the background shear currents. The currents are considered to be stationary and horizontally homogeneous on the assumption that the scale of the currents' horizontal and temporal variability is much larger than the characteristic lengths and periods of internal gravity waves. Having been used, the Fourier method permitted to obtain integral representations of the solutions under the Miles – Howard stability condition is fulfilled. To solve the vertical spectral problem, proposed is the algorithm for calculating the main dispersion dependences that determine the phase characteristics of the generated wave fields. The calculations for one real distribution of buoyancy frequency and shear flow profile are represented. Transformation of the dispersion surfaces and phase structures of the internal gravitational waves’ fields is studied depending on the generation parameters. To solve the problem analytically, constant distribution of the buoyancy frequency and linear dependences of the background shear current on depth were used. For the model distribution of the buoyancy and shear flow frequencies, the explicit analytical expressions describing the solutions of the vertical spectral problem were derived. The numerical and asymptotic solutions for the characteristic oceanic parameters were compared. Conclusions. The obtained results show that the asymptotic constructions using the model dependences of the buoyancy frequency and the background shear velocities’ distribution, describe the numerical solutions of the vertical spectral problem to a good degree of accuracy. The model representations, having been applied for hydrological parameters, make it possible to describe qualitatively correctly the main characteristics of internal gravity waves in the ocean with the arbitrary background shear currents.

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.

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

scholarly journals Phase Structure of Internal Gravity Waves in the Ocean with Shear Flows

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
V. V. Bulatov ◽  
Yu. V. Vladimirov ◽  
I. Yu. Vladimirov ◽  
◽  
◽  
...  
Keyword(s):  
Shear Flow ◽  
Spectral Problem ◽  
Gravity Waves ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Internal Gravity Waves ◽  
Buoyancy Frequency ◽  
Flow Profile ◽  
Wave Fields ◽  
Background Shear

Purpose. The description of the internal gravity waves dynamics in the ocean with background fields of shear currents is a very difficult problem even in the linear approximation. The mathematical problem describing wave dynamics is reduced to the analysis of a system of partial differential equations; and while taking into account the vertical and horizontal inhomogeneity, this system of equations does not allow separation of the variables. Application of various approximations makes it possible to construct analytical solutions for the model distributions of buoyancy frequency and background shear ocean currents. The work is aimed at studying dynamics of internal gravity waves in the ocean with the arbitrary and model distributions of density and background shear currents. Methods and Results. The paper represents the numerical and analytical solutions describing the main phase characteristics of the internal gravity wave fields in the stratified ocean of finite depth, both for arbitrary and model distributions of the buoyancy frequency and the background shear currents. The currents are considered to be stationary and horizontally homogeneous on the assumption that the scale of the currents' horizontal and temporal variability is much larger than the characteristic lengths and periods of internal gravity waves. Having been used, the Fourier method permitted to obtain integral representations of the solutions under the Miles – Howard stability condition is fulfilled. To solve the vertical spectral problem, proposed is the algorithm for calculating the main dispersion dependences that determine the phase characteristics of the generated wave fields. The calculations for one real distribution of buoyancy frequency and shear flow profile are represented. Transformation of the dispersion surfaces and phase structures of the internal gravitational waves’ fields is studied depending on the generation parameters. To solve the problem analytically, constant distribution of the buoyancy frequency and linear dependences of the background shear current on depth were used. For the model distribution of the buoyancy and shear flow frequencies, the explicit analytical expressions describing the solutions of the vertical spectral problem were derived. The numerical and asymptotic solutions for the characteristic oceanic parameters were compared. Conclusions. The obtained results show that the asymptotic constructions using the model dependences of the buoyancy frequency and the background shear velocities’ distribution, describe the numerical solutions of the vertical spectral problem to a good degree of accuracy. The model representations, having been applied for hydrological parameters, make it possible to describe qualitatively correctly the main characteristics of internal gravity waves in the ocean with the arbitrary background shear currents.

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