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

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or equivalently cg = N k 2 k 2 + β 2 /4 cg h = N2 – ω 2 ω 3/2 kz kh kh k 2 + β 2 /4 k k × k k × ez + N and cgz β 2 /4 k 2 + β 2 /4 kz 3/2 kh kh = – ω k2 + β 2 /4 , (4.10) . (4.11) It is no more perpendicular to the wavevector. The trajectories of fluid particles become ellipses, studied by McLaren et al. (1973). A point monochromatic source consequently radiates non-Boussinesq internal waves into the whole region ⏐cos θ⏐ < ω/N situated out of the characteristic cone. There, each point receives a wave whose group velocity is directed toward it and wavevector satisfies, according to (4.11), kh = β 2 kz = – β 2 The associated phase is of the form Φ = ωt – k.r = ωt – βr 2 ω 2 N 2 – ω 2 ω 2 – N 2 cos 2 θ sin θ rh rh , (4.12a) N2 – ω2 ω2 – N2 cos θ . (4.12b) cos2θ ω 2 – N 2 cos 2 θ N 2 – ω 2 ≡ ωt – ξ(ω)r . (4.13) Waves are locally plane, with a radial wavenumber ξ(ω), a phase velocity ω/ξ(ω), and a group velocity Internal wave generation. 1. Green’s function 14 cg = 1 ξ'( ω) = 2 β N2 – ω 2 3/2 ω 2 – N 2 cos 2 θ 1/2 N 2 ω sin 2 θ , (4.14) which vanishes at the limits N⏐cos θ⏐ and N of the internal wave spectrum and reaches a maximum c g0 at the frequency (figure 4) ω0 = N cos θ cos θ + 3 – cos2 θ 3 . (4.15)
5. EXACT GREEN’S FUNCTION 5.1. Monochromatic Green’s function For ω > N, rescaling the coordinates according to rh' = ω N rh and z' = ω2 – N2 1/2 N z (5.1) reduces equation (3.15) for the monochromatic Green’s function G(r, ω) to a Helmholtz-like equation, whose Green’s function is given for instance by Bleistein (1984 p. 177). Returning to the original coordinate system described in figure 5 yields G(r, ω) = 1 4πr 1/2 – e βr 2 ω2 – N2 cos2θ ω2 – N2 ω2 – N2 1/2 ω2 – N2 cos2θ 1/2 . (5.2) As implied by the Pierce-Hurley radiation condition, the analytic continuation of this result over the lower half of the complex ω plane provides the value of the Green’s function on the whole real ω axis. The branch cuts emanating from the branch points ± N, ± N⏐cos θ⏐ are taken as extending vertically upwards, as shown in figure 6. Thus, the phase of complex square roots of the form (ω 2 – ω 0 2 ) 1/2 has the following behaviour on the real axis: ph ω 2 – ω 0 2 1/2 = 0 ω > ω0 , (5.3a) = – π/2 ω < ω0 , (5.3b) = – π ω < – ω0 . (5.3c) Denoting from now on complex square roots, defined by (5.3), as powers 1/2 and real square roots by a square root symbol, we obtain for the Green’s function at every real frequency G(r, ω) = 1 4πr e – βr ω 2 2 – N2 cos2θ ω2 – N2 ω2 – N2 ω2 – N2 cos2θ = i Internal wave generation. 1. Green’s function 15 sgn ω 4πr e – i βr 2 ω 2 – N 2 cos 2 θ N 2 – ω 2 N 2 – ω 2 ω 2 – N 2 cos 2 θ sgn ω ω > N , (5.4a) N cos θ < ω < N , (5.4b)
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: the source, but disappear as soon a
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 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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