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

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7. INTERNAL W AVE FIELD OF A POINT MASS SOURCE As the internal potential generated by a point mass source, the Green’s function has no direct physical significance and its interpretation requires the consideration of the associated pressure and velocity fields. In the present section, for an impulsive source m(r, t) = m 0 δ(r) δ(t) releasing a volume m 0 of fluid, we deduce these fields from the Green’s function, by multiplying it by m 0 and derivating it according to (3.11)-(3.12). There may, however, be instances where the Green’s function has a meaning in itself, such as the Cauchy problem for internal waves (Sekerzh-Zen’kovich 1982); see appendix B. 7.1. Monochromatic waves Internal wave generation. 1. Green’s function 28 Continuing the approach of section 5, we first deal with a monochromatic source, and replace in (3.11)-(3.12) ∂/∂t by iω. Then, the differentiation of G(r, ω) yields P(r, ω) = i ρ00m0 4πr ω v (r, ω) = m0 ω2 4πr 2 ω 2 – N 2 ω 2 – N 2 cos 2 θ 1/2 ω2 – N2 1/2 ω2 – N2cos2θ 3/2 βr – e × 1 + βr 2 ω2 – N 2 cos 2 θ ω 2 – N 2 1/2 βr – e 2 ω2 – N2 cos2θ ω2 – N2 r r 2 ω2 – N 2 cos 2 θ ω 2 – N 2 1/2 + βr 2 ω2 – N 2 cos 2 θ ω 2 – N 2 1/2 , (7.1) ez . (7.2) Non-Boussinesq effects add a vertical component to the radial motion of the fluid. In the frequency range N⏐cos θ⏐ < ⏐ω⏐ < N of propagating internal waves the radial and vertical velocities are out of phase, for the term in square brackets in (7.2) is complex. Thus, the trajectories of fluid particles are elliptical, in agreement with the group velocity theory. Under the Boussinesq approximation the pressure and velocity fields become PB(r, ω) = i ρ0 m0 4πr ω vB(r, ω) = m0 ω2 4πr 2 ω 2 – N 2 ω 2 – N 2 cos 2 θ ω 2 – N 2 1/2 1/2 ω 2 – N 2 cos 2 θ 3/2 r r , (7.3) . (7.4) Consistently with the neglect of density variations ρ 00 has been replaced by ρ 0 in (7.3). Fluid
particles move as expected along radial trajectories. Out of the characteristic cone the pressure and velocity are π/2 out of phase implying that no energy flux, proportional to Re [Pv *] where * denotes a complex conjugate (see Lighthill 1978 § 4.2), is radiated. Energy, and thus internal waves, are confined on the cone, where they diverge. There, point sources are no more an adequate model of real sources of internal waves. Only the consideration of the finite extent of the latter will account for the radial decrease as 1/√r, and the transverse phase variations, experimentally exhibited by McLaren et al. (1973). 7.2. Boussinesq impulsive waves Passing to impulsive internal waves we commence as in § 6.1 by studying the Boussinesq pressure and velocity fields. For small times Nt « 1, either differentiating (6.7) according to (3.11)-(3.12) or asymptotically inverting (7.3)-(7.4) by (D1), we obtain PB(r, t) ~ ρ0m0 4πr vB(r, t) ~ m0 4πr 2 r r δ'(t) , (7.5) δ(t) . (7.6) The pressure and velocity impulses necessary to set the fluid into motion are recovered (cf. appendix B). In similarly differentiating the expansion of the Green’s function for large times Nt » 1, care must be taken to retain the first two orders of (6.8) and (6.10) so as to recover the presence of buoyancy oscillations. Lighthill’s (1958) method may also be applied to (7.3)- (7.4), in which cases it involves (D3)-(D4). Both procedures yield PB(r, t) ~ – H(t) ρ0 N 2 m0 2π 3/2 r vB(r, t) ~ – H(t) Nm0 2π 3/2 r 2 r r sin θ cos θ sin Nt cos θ – π/4 Nt cos θ – 1 sin θ sin θ Nt cos θ sin Nt cos θ – π/4 + 1 sin 3 θ sin Nt – π/4 Nt 3/2 cos Nt – π/4 Nt 3/2 , (7.7) . (7.8) The phase Φ g = Nt⏐cos θ⏐ – π/4 of gravity waves is, apart from a phase lag of π/4, identical to that deduced from the consideration of group velocity. Thus, gravity waves propagate as the locally plane waves described in § 4.1, with the group velocity c g = r/t and the frequency and wavevector Internal wave generation. 1. Green’s function 29
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: time-dependent internal wave fields
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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