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

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so that Internal wave generation. 1. Green’s function 38 ξ 2 ~ ω N2 – ω 2 N 2 Then the internal wave field is given by ψ(r) ~ i r a e– i arc cos σ+/a . (8.23) aU0 ω 1/2 N 2 – ω 2 3/4 a r e i/2 arc cos σ+/a , (8.24) P(r) ~ ρ0aU0 ω 1/2 N 2 – ω 2 1/4 a r e i/2 arc cos σ+/a , (8.25) v (r) ~ U0 2 ω 1/2 N 2 – ω 2 1/4 a r a a 2 – σ+ 2 r r e i/2 arc cos σ+/a . (8.26) This expression makes Appleby’s & Crighton’s (1987) result for a pulsating sphere more explicit, and is similar to Lighthill’s (1978 § 4.10) result for a distributed mass source. The phase Φ = ωt + (1/2) arc cos (σ + /a) varies transversely, in agreement with the experiments of McLaren et al. (1973). From its differentiation with respect to σ + (cf. (7.10)) we deduce the transverse wavelength λ = 4π (a 2 – σ + 2 ) 1/2 ,and the phase and group velocities cg = N2 – ω 2 k c φ = ω k = 2 ω a2 – σ+ 2 , (8.27) = 2 N 2 – ω 2 a 2 – σ+ 2 , (8.28) which are zero at the edges σ + = ± a of region III and maxima halfway between them. The definition of λ is, however, of purely academic interest, since not even a single oscillation of the phase takes place between these edges. Only a continuous monotonic variation of π/2 in Φ is observed, which replaces the phase jump obtained in § 5.1 between regions II and IV. Interpreting it as a quarter of an oscillation we rather introduce, as did Appleby & Crighton (1987), an effective wavelength “ λ” = 8a. The pressure and velocity oscillate in phase and verify, consistently with (4.5), v ~ P 2 ρ0 N 2 – ω 2 a 2 – σ+ 2 r r . (8.29) Their decrease as 1/√r, conformable to the conservation of energy for conical waves, coincides with the experiments of McLaren et al. (1973) again. The velocity singularity at the edges of
egion III remains integrable, and the energy flux consequently finite. 8.3. Impulsive far field We now consider an impulsive pulsation U(t) = U 0 δ(t) of the sphere, for which U(ω) = U 0 , and investigate by the asymptotic method of § 6.1 the radiated internal waves. For small times Nt « 1, from the high-frequency expansion of (8.16) (ξ ~ (ω/N) (r/a)) and its Fourier inversion by (D2), we have ψ(r, t) ~ – H(t) a2 U0 r t ~ 4πa 2 U0 G(r, t) . (8.30) Similarly the pressure and velocity are given by (7.5)-(7.6), with m 0 replaced by 4πa 2 U 0 . As expected the initial motion is irrotational, and the point source model is then valid. For large times Nt » 1, internal waves separate again into gravity waves (the contribution of the singularities ± Σ +, ± Σ –) and buoyancy oscillations (the contribution of the singularities ± N). For both of them, although for different reasons, the far-field assumption r » a is contradictory to the use of Lighthill’s (1958) method. Thus, we relax it at first. The procedure we shall use instead closely follows that of Bretherton (1967) and Grimshaw (1969). For gravity waves, as long as r/a remains finite, Σ + and Σ – remain separated and Lighthill’s method still applies. After some algebra we find in the vicinity of Σ ± , omitting a regular part which makes no contribution to the large-time expansion, ln ξ – i ξ + i ~ +– i 1 – a2 r2 1/4 2 sin θ± sin θ Inverting then (8.11) by (D3) we finally obtain for r h > a ψg(r, t) ~ H(t) 2π aU0 N × 1 – a2 r 2 1/4 1 sin θ cos θ+ sin θ+ 3/2 sin Σ+t – π/4 Σ+t 3/2 and for r h < a with (z > 0, z < 0) Internal wave generation. 1. Green’s function 39 – cos θ– 1/2 ω – Σ± Σ± 1/2 sin θ– 3/2 sin Σ– t – π/4 Σ– t 3/2 sgn cos θ± . (8.31) , (8.32a)
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: ω v (r, ω) ~ U(ω) 2 ω2 - N2 1/2
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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