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

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where GB(r, t) ~ – H(t) 4πNr αn = (–1)n π ∑ j +k = n ∞ ∑ n = 0 αn Γ j+1/2 Γ k+1/2 j! k! Nt 2 n + 1 2n+1 ! , (6.5) cos 2 k θ . (6.6) The series (6.5) converges for all finite t, so constituting an exact representation of the impulsive Green’s function rather than a mere expansion. To leading order, GB(r, t) ~ – H(t) t 4πr . (6.7) In agreement with Batchelor’s (1967 § 6.10) general discussion, and Morgan’s (1953) discussion of the analogous case of unstratified rotating fluids, the Boussinesq fluid initially ignores its stratification. Thus its motion, described by (6.7), is irrotational. Further details about the question of initial conditions for stratified or rotating fluids, which has been a controversial subject for several years, are given in appendix B. For large times Nt » 1, gravity waves and buoyancy oscillations have become separated. They respectively correspond to the contributions of the pairs of branch points ± N⏐cos θ⏐ and ± N. Gravity waves are, from the expansion of G B(r, ω) in powers of (ω – N⏐cos θ⏐) 1/2 near N⏐cos θ⏐ and its Fourier inversion by (D3), with GBg (r, t) ~ – βn = (–1)n π 3/2 H(t) 2π 3/2 Nr sin θ ∑ j +k +m = n Re ei Nt cos θ – π/4 Nt cos θ Γ j+1/2 Γ k+1/2 Γ m+1/2 j! k! m! ∞ ∑ n = 0 in Γ n+1/2 π (–1) m cos θ βn Nt cos θ n k + m 2 j 1+ cos θ k 1– cos θ , (6.8) m . (6.9) The conjugate contribution of – N⏐cos θ⏐ has been incorporated by taking two times the real part of the result. We similarly have for buoyancy oscillations with GBb (r, t) ~ – γn = (–1)n π 3/2 Internal wave generation. 1. Green’s function 22 ∑ j +k +m = n H(t) 2π 3/2 Nr sin θ Im ei Nt – π/4 Nt Γ j+1/2 Γ k+1/2 Γ m+1/2 j! k! m! ∞ ∑ n = 0 in Γ n+1/2 π 1 γn Nt n 2 j 1+ cos θ k 1– cos θ , (6.10) m . (6.11)
To better exhibit the angular dependence of the coefficients β n and γ n it may be preferred to express them in terms of generating functions, in which case where Internal wave generation. 1. Green’s function 23 βn , γn = 1 – 2 μ tan 2 θ 1 + 2 μ sin 2 θ ∑ j + k = n (–1) j j! – μ2 tan 2 θ + μ2 sin 2 θ Γ j+1/2 π – 1/2 – 1/2 ∞ ∑ fk , gk 2 j = fk μ k k = 0 ∞ ∑ = gk μ k k = 0 , (6.12) , (6.13) . (6.14) Of crucial importance is the leading-order term of the large-time expansion, GB(r, t) ~ – H(t) 2π 3/2 Nr sin θ cos Nt cos θ – π/4 Nt cos θ + sin Nt – π/4 Nt , (6.15) which extends Lighthill’s (1978) equations (255)-(258) to the Green’s function of internal waves, by simultaneously including gravity waves and buoyancy oscillations. Gravity waves are verified to have the phase expected from the consideration of group velocity. Buoyancy oscillations are confirmed not to propagate, but also to be present everywhere. For the moment we just note these points, and postpone the full interpretation of (6.15) until § 7.2. Another asymptotic expansion of the Green’s function, alternative to (6.15), may also be derived; this is the matter of appendix C. When the vertical line passing through the source (θ = 0 or π) or the horizontal plane containing it (θ = π/2) are approached, (6.15) is invalidated. Near to the vertical, gravity waves and buoyancy oscillations may no more be separated and combine into the persistent oscillation accounted for by (6.3). Near to the horizontal, (6.4) shows gravity waves, whose frequency tends toward zero, to vanish. There only remain buoyancy oscillations, still described by the corresponding terms of (6.15). A more precise study of the range of validity of large-time expansions of internal waves has been achieved by Zavol’skii & Zaitsev (1984). Its consideration is, again, postponed until § 7.2. 6.2. Non-Boussinesq Green’s function Non-Boussinesq effects induce in the phase of the impulsive Green’s function (5.13) an additional dependence upon position r and frequency ω which significantly alters the way that its asymptotic expansions are obtained. First, the calculation of a general form for all terms
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: 6. ASYMPTOTIC GREEN’S FUNCTION 6.
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 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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