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

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H(t) Gg (r, t) ~ – 2π 3/2Nr sin θ cos Nt – β2 r 2 8Nt sin 2 θ cos θ – π 4 Nt cos θ . (6.20) Similarly buoyancy oscillations are reducible to the inverse transform (D9), of which (D15) provides an asymptotic expansion, so that Gb(r, t) ~ – – 3 3 4 + e H(t) 2π 3/2 3 Nr sin θ βr sin θ 2 2/3 Nt 1/3 sin Nt – 3 2 βr sin θ 2 sin Nt + 3 4 Nt 2/3 βr sin θ 2 Nt Nt 1/3 – π 4 2/3 Nt 1/3 – π 4 . (6.21) As in the Boussinesq case an alternative expression of gravity waves may be proposed; its derivation is outlined in appendix C. Buoyancy oscillations are now waves, to which non-Boussinesq effects have given some propagation. Contrary to gravity waves, they comprise both propagating and evanescent internal waves. However, as long as the Boussinesq approximation is not made, evanescent waves decay exponentially with time and stay negligible. In § 7.3 we shall carry the interpretation of gravity and buoyancy waves further, by calculating their characteristics (frequencies and wavevectors). Ultimately both waves become Boussinesq, as for the former, and Internal wave generation. 1. Green’s function 26 βr 2 « Nt sin θ (6.22) βrh 2 « Nt (6.23) for the latter. Instead of the “classical” near-field argument β⏐z⏐/ 2 « 1 (Hendershott 1969), two surfaces, respectively toroidal and cylindrical, define the validity of the Boussinesq approximation for gravity and buoyancy waves. They expand along the horizontal at velocity 2N/β and are represented in figure 8. Their existence simply reflects the complex structure of
time-dependent internal wave fields; the significance of (6.22) and (6.23) remains, in agreement with Lighthill (1978 § 4.2), small wavelengths λ « 2/β. For buoyancy waves the Boussinesq approximation is moreover non-uniform. As (6.23) becomes valid evanescent waves become comparable with propagating waves until in the end, when non-Boussinesq terms vanish in (6.21), both contributions are the same. Then the buoyancy oscillations appearing in (6.15) are recovered, but multiplied by 2/√3. Such a non-uniformity is not surprising, since making both β → 0 and ω → N in (5.2) is clearly contradictory. Again we postpone the interpretation of this phenomenon until § 7.3 and just note, as regards its mathematical significance, that it is due to the coalescence of two saddle points with a branch point of both the argument of the exponent and the amplitude (cf. appendix D). To this case the uniform asymptotic expansions of Bleistein & Handelsman (1986 ch. 9) do not seem to apply. Internal wave generation. 1. Green’s function 27
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: inverse group velocity with horizon
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 and 44: 9. CONCLUSION Internal wave generat
Page 45 and 46: Let us finally emphasize that, howe
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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