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

ψg(r, t) ~ H(t)
2π aU0
N
× cos θ+
1 – a2
r 2
1/4
1
sin θ
sin θ+ 3/2 sin, cos Σ+t – π/4
Σ+t 3/2
– cos θ–
sin θ– 3/2 cos, sin Σ– t – π/4
Σ– t 3/2
. (8.32b)
Accordingly the gravity wave field results from the interference between the waves emanating
from the points T + and T – of the sphere.
As the far field r » a is entered, the separation between T + and T – vanishes in the
amplitude of these waves but is still present in their phase; there it reduces to the small but
fundamental difference (8.15) between Σ + and Σ – . Then (8.32a) becomes
ψg (r, t) ~ – H(t) 2
π aU0
N
cos θ
sin 2 θ
sin Nt a r
sin θ cos Nt cos θ – π/4
Nt cos θ 3/2
, (8.33a)
~ 4πa 2 U0 sin kg a
kg a Gg (r, t) . (8.33b)
The interferences between T + and T – take on the familiar form of a factor sin (k g a) / (k g a), with k
g
the wavevector (7.10), multiplying the gravity waves radiated by a point impulsive source
releasing the volume 4πa 2 U 0 . The same is true of the pressure and velocity, which vary as
t –3/2 and t –1/2 , respectively, and remain at any t finite. Where the sphere is small compared
with the wavelength λ g of gravity waves, as
r
a
» Nt sin θ , (8.34)
the interferences are constructive and the point source is equivalent to the sphere. As for the
Boussinesq approximation a torus, hereafter called characteristic, of vertical axis and radius
Nta, defines the validity of the point source model for gravity waves.
Note that the distinction between the two forms (8.32) and (8.33) of gravity waves
corresponds to the separation of space into two regions similar to those encountered in § 6.2:
(i) Nt » 1 with r/t fixed, (ii) Nt » 1 with r/a fixed (Bretherton 1967, Grimshaw 1969). (8.32) is
relevant in region (ii), and (8.33) at the boundary between regions (i) and (ii). A particular
feature of the pulsating sphere is that (8.33) remains valid across the whole of region (i); see
Voisin (1991).
Internal wave generation. 1. Green’s function 40
For buoyancy oscillations ξ vanishes in the vicinity of N for r h < a, according to