Internal Wave Generation in Uniformly Stratified Fluids. 1 ... - LEGI

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λg = 2π Nt r sin θ is constant on tori. Boussinesq buoyancy waves are radial oscillations at frequency N, found everywhere in the fluid; in actual fact they are propagating waves, of horizontal wavevector and wavelength λb = 4π β constant on cylinders (r h denotes the horizontal distance from the source and 2/β the scale height of the stratification), whose propagation has vanished as they have become Boussinesq. βrh 2Nt 1/3 (9.1) (9.2) For both gravity and buoyancy waves the Boussinesq approximation implies small wavelengths λ « 2/β. Thus it is valid, for the former when and for the latter when βr 2 « Nt sin θ , (9.3) βrh 2 « Nt , (9.4) that is inside a torus of vertical axis and radius 2Nt/β, and inside the circumscribing cylinder, respectively (figure 8). As these regions are entered buoyancy waves, initially comparable with gravity waves, simultaneously lose their propagation and become less significant. There two separate criteria define, again, the validity of the point source model; for a spherical source of radius a « 2/β they are for gravity waves, and Internal wave generation. 1. Green’s function 44 r a » Nt sin θ (9.5) rh » a (9.6) for buoyancy oscillations. Outside the torus of vertical axis and radius Nta, gravity waves are dominant (figure 11); for them the sphere is compact, since a « λ g . Inside this torus interferences caused by the finite size of the sphere blur gravity waves, and buoyancy oscillations are observed. For them the sphere is compact far from the vertical cylinder circumscribing it. Inside this cylinder the fluid is quiescent.
Let us finally emphasize that, however different from the fields of acoustic or electromagnetic waves internal wave fields may be, their structure is governed by the same underlying principles, with the amendments required by the special form of the wavelengths (9.1) and (9.2) of their two components. To illustrate this fact table 1 compares the different zones of a Boussinesq gravity wave field with those of acoustic and electromagnetic wave fields (cf. Jackson 1975 § 9.1 and Pierce 1981 § 4.7). Acknowledgment Internal wave generation. 1. Green’s function 45 This work is part of a Ph.D. thesis prepared at Université Pierre et Marie Curie and Thomson-Sintra ASM. It has been written during a one-year stay at the Institut de Mécanique de Grenoble, whose hospitality is acknowledged. The author would like to thank Prof. Zarembowitch of UPMC, and Dr Tran Van Nhieu of Thomson-Sintra, for their guidance throughout his Ph.D.; he is also indebted to Profs. Crighton and Hopfinger for helpful discussions and kind encouragements given at decisive steps of the study. Financial support by the ANRT through Convention CIFRE No. 249/87 is acknowledged.
Page 1 and 2: Internal Wave Generation in Uniform
Page 3 and 4: 1. INTRODUCTION Internal gravity wa
Page 5 and 6: 2. BIBLIOGRAPHICAL REVIEW Investiga
Page 7 and 8: 3. STATEMENT OF THE PROBLEM 3.1. In
Page 9 and 10: Internal wave generation. 1. Green
Page 13 and 14: the source, but disappear as soon a
Page 15 and 16: 5. EXACT GREEN’S FUNCTION 5.1. Mo
Page 17 and 18: theorem and the radiation condition
Page 19 and 20: G(r, t) = 1 8π 2 r +∞ - ∞ e i
Page 21 and 22: 6. ASYMPTOTIC GREEN’S FUNCTION 6.
Page 23 and 24: To better exhibit the angular depen
Page 25 and 26: inverse group velocity with horizon
Page 27 and 28: time-dependent internal wave fields
Page 29 and 30: particles move as expected along ra
Page 31 and 32: and for the displacement vector ζ
Page 33 and 34: Thus, buoyancy waves have simultane
Page 35 and 36: and apply Pierce’s (1963) procedu
Page 37 and 38: ω v (r, ω) ~ U(ω) 2 ω2 - N2 1/2
Page 39 and 40: egion III remains integrable, and t
Page 43: 9. CONCLUSION Internal wave generat
Page 47 and 48: P(r, ω) = F0z ω2 4πr 2 ω2 - N2
Page 49 and 50: educes to the Green’s function. N
Page 51 and 52: Appendix D. SOME INVERSE FOURIER TR
Page 57 and 58: CAPTIONS Table 1. Compared structur
Page 59: TYPE OF WAVE acoustic electromagnet

Internal Wave Generation in Uniformly Stratified Fluids. 1 ... - LEGI

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