Internal wave generation in uniformly stratified fluids. Part 1. Green's function and point sources

Abstract

In both Boussinesq and non-Boussinesq cases the Green's function of internal gravity waves is calculated, exactly for monochromatic waves and asymptotically for impulsive waves. From its differentiation the pressure and velocity fields generated by a point source are deduced. by the same method the Boussinesq monochromatic and impulsive waves radiated by a pulsating sphere are investigated.Boussinesq monochromatic waves of frequency ω < N are confined between characteristic cones θ = arccos(ω/N) tangent to the source region (N being the buoyancy frequency and θ the observation angle from the vertical). In that zone the point source model is inadequate. For the sphere an explicit form is given for the waves, which describes their conical 1/r½ radial decay and their transverse phase variations.Impulsive waves comprise gravity and buoyancy waves, whose separation process is non-Boussinesq and follows the arrival of an Airy wave. As time t elapses, inside the torus of vertical axis and horizontal radius 2Nt/β for gravity waves and inside the circumscribing cylinder for buoyancy waves, both components become Boussinesq and have wavelengths negligible compared with the scale height 2/β of the stratification. Then, gravity waves are plane propagating waves of frequency N cos θ, and buoyancy waves are radial oscillations of the fluid at frequency N; for the latter, initially propagating waves comparable with gravity waves, the horizontal phase variations have vanished and the amplitude has become insignificant as the Boussinesq zone has been entered. In this zone, outside the torus of vertical axis and horizontal radius Nta, a sphere of radius a [Lt ] 2/β is compact compared with the wavelength of the dominant gravity waves. Inside the torus gravity waves vanish by destructive interference. For the remaining buoyancy oscillations the sphere is compact outside the vertical cylinder circumscribing it, whereas the fluid is quiescent inside this cylinder.

Internal Waves Radiation by a Turbulent Jet Flow in a Stratified Fluid

“…7 shows that at later times (Nt = 60, 90) amplitude ' z U decreases by the order of magnitude as compared to the time moment Nt = 15, and the regions of local maxima of '( , ) z Uy z are getting aligned with the vertical axis. This picture is similar to the one predicted by a linear theory for the IW radiated by an impulsive point source (Zavolski & Zaytsev 1984, Voisin 1991. According to this theory, amplitude of IW emitted impulsively by a point source has its maximum along the iso-phase line oriented at 0 arctan 2 55   with respect to the www.intechopen.com vertical.…”

“…At sufficiently late times (Nt > 60) there remain only small-amplitude oscillations of the velocity and density with the buoyancy frequency N. These smallamplitude oscillations are similar to the buoyancy oscillations which supercede internal waves, which are mutually cancelled due to the destructive interference as their wavelengths become of the order of the source diameter (i.e. of order one in the considered case) (Voisin 1991). is also equivalent to the average fluid particle vertical displacement amplitude i.e.…”

“…The figure shows also that at time Nt = 90 the vertical velocity amplitude is diminished by the order of magnitude and its maxima and minima are arranged in columns with the same spatial periodicity. These maxima and minima can be associated with smallamplitude buoyancy oscillations which supercede impulsively radiated IW at that stage of the flow development (Voisin 1991). U , in the horizontal (x, y)-plane at the distance z = 3 above the jet streamwise axis obtained in DNS at time moments Nt = 12, 15, 18.…”

Schlieren visualisation and measurement of axisymmetric disturbances

“…Axisymmetric waves and, in particular, internal waves generated by an oscillating sphere have been studied both because this geometry allows one to reconstruct the wave-field from a single camera's perspective and because the experimental results may be compared with the prediction of linear theories (Appleby and Crighton, 1987;Voisin, 1991) Indeed, good agreement is found between the experimental results and a theory that includes the effects of viscous attenuation .…”

Spatial structure of internal tidal waves generated on weak topographies

“…In practice, many efficient methods have been proposed for dealing with a steady-state internal gravity wave problem (e.g., Hurley, 1972;Lighthill, 1967Lighthill, , 1978Voisin, 1991). The objective of this study is therefore to develop a more systematic and generalized procedure using Lighthill's theory (1967Lighthill's theory ( , 1978, solve the Fourier integrals involved in the aforementioned problem, and find the asymptotic far-field expressions for these waves.…”