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

ω
v (r, ω) ~ U(ω)
2 ω2 – N2 1/2
N ω2 2 1/2
– Σ+ ω2 2 1/2
– Σ–
1
ξ a r r r
. (8.18)
The motion of fluid particles is radial as if, again, the waves emanated from the origin.
8.2. Monochromatic far field
The sphere is supposed to pulsate monochromatically, at the frequency 0 < ω < N. If
U(t) = U 0 e iωt , and the time dependence e iωt is omitted in all variables, the internal wave field is
given by (8.16)-(8.18) with U(ω) replaced by U 0 . According to the different values of the now
complex coordinate ξ the space must be divided into six regions (figure 10), separated by
the characteristic cones, of vertical axis and semi-angle θ 0 = arc cos (ω/N), tangent to the
sphere (Hendershott 1969, Appleby & Crighton 1987).
In regions II, IV and VI ξ is either real or imaginary, the phase of internal waves does
not vary and no energy is radiated. More precisely ξ reduces in the far field r » a to
so that
ψ(r) ~
ξ ~ ω2 – N 2 cos 2 θ 1/2
N
r a
, (8.19)
aU0
ω 2 – N 2 1/2 ω 2 – N 2 cos 2 θ 1/2 a r = 4πa2 U0 G(r, ω) . (8.20)
The pressure and velocity are similarly given by (7.3)-(7.4), with m 0 replaced by 4πa 2 U 0 .
Regions II, IV and VI are those where the point source model is valid; in agreement with §
7.1 they also are those where no internal waves are found.
On the other hand, in regions III and V ξ is complex, implying phase variation and
energy radiation. Since ⏐cos θ⏐ → ω/N as r/a → ∞, (8.19) is invalidated. It must be taken into
account that transverse distances, represented by Hurley’s (1972) characteristic coordinates
(cf. figure 10)
σ ± = r sin θ + – θ0 = ω
N rh + – N2 – ω2 N
z , (8.21)
remain as the far field is approached finite and nonzero. In region III for instance, as r » a >
⏐σ + ⏐,
Internal wave generation. 1. Green’s function 37
rh ~ N2 – ω 2
N
r + ω N σ+ and z ~ ω N r – N2 – ω 2
N
σ+ , (8.22)