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

ωg = ∂Φg
∂t
= N cos θ , (7.9)
kg = – ∇Φg = Nt
r r r × r r × ez sgn z . (7.10)
The pressure and velocity oscillate in phase and verify, consistently with (4.5),
vg ~ t r Pg
ρ0
r r
. (7.11)
Note, however, that the velocity grows with time as √t and ultimately diverges (Zavol’skii &
Zaitsev 1984), the pressure simultaneously decreasing as 1/√t so as to maintain the radiated
energy flux finite. Meanwhile, the wavelength λ g given by (4.8) decays as 1/t and may
eventually become smaller than intermolecular distances (Sobolev 1965 p. 203), vitiating the
continuous medium model. Clearly, a cutoff eliminating the ultimate dominance of the smaller
wavelengths is needed (Lighthill 1978 p. 359). As above, the consideration of the finite
dimensions of real sources will provide it.
The phase Φ b = Nt – π/4 of buoyancy oscillations confirms that they do not propagate,
whereas their amplitude shows them to be present everywhere and to induce a radial motion
of fluid particles, in contrast to the predictions of the Boussinesq group velocity theory. This
testifies to their non-Boussinesq origin. The pressure and velocity are π/2 out of phase,
implying a zero energy flux. Compared with gravity waves buoyancy oscillations decrease
with time as t –3/2 and remain negligible. This is contradictory to the ultimate dominance of
oscillations of the fluid at the buoyancy frequency, experimentally found by McLaren et al.
(1973) for transient internal waves. The explanation for this phenomenon lies, again, in the
finite size of real sources.
Exact integral expressions of the pressure and velocity fields follow from the same
method that has been used to derive the spectral decompositions (5.12) and (5.16) of the
Green’s function. Amendments are nevertheless required, since (7.4) is not integrable at the
singularities ± N⏐cos θ⏐. A method to circumvent this difficulty, which turns out to be equivalent
to that proposed by Zavol’skii & Zaitsev (1984), may be found in Voisin (1991). It yields for
the pressure
Internal wave generation. 1. Green’s function 30
PB(r, t) = ρ0 m0
2π 2 r π 2
N
δ'(t) + H(t) ω
N cos θ
N2 – ω2 ω2 – N2 cos ωt dω
cos2θ , (7.12)