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

H(t)
Gg (r, t) ~ –
2π 3/2Nr sin θ
cos Nt – β2 r 2
8Nt sin 2 θ cos θ – π 4
Nt cos θ
. (6.20)
Similarly buoyancy oscillations are reducible to the inverse transform (D9), of which (D15)
provides an asymptotic expansion, so that
Gb(r, t) ~ –
– 3 3
4
+ e
H(t)
2π 3/2 3 Nr sin θ
βr sin θ
2
2/3
Nt 1/3
sin Nt – 3
2
βr sin θ
2
sin Nt + 3
4
Nt
2/3
βr sin θ
2
Nt
Nt 1/3 – π
4
2/3
Nt 1/3 – π 4
. (6.21)
As in the Boussinesq case an alternative expression of gravity waves may be proposed; its
derivation is outlined in appendix C.
Buoyancy oscillations are now waves, to which non-Boussinesq effects have given
some propagation. Contrary to gravity waves, they comprise both propagating and
evanescent internal waves. However, as long as the Boussinesq approximation is not made,
evanescent waves decay exponentially with time and stay negligible. In § 7.3 we shall carry
the interpretation of gravity and buoyancy waves further, by calculating their characteristics
(frequencies and wavevectors).
Ultimately both waves become Boussinesq, as
for the former, and
Internal wave generation. 1. Green’s function 26
βr
2
« Nt sin θ (6.22)
βrh
2
« Nt (6.23)
for the latter. Instead of the “classical” near-field argument β⏐z⏐/ 2 « 1 (Hendershott 1969), two
surfaces, respectively toroidal and cylindrical, define the validity of the Boussinesq
approximation for gravity and buoyancy waves. They expand along the horizontal at velocity
2N/β and are represented in figure 8. Their existence simply reflects the complex structure of