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

and
kg h = Nt
r
kgz
ωg = N cos θ 1 + 1 2
sin θ cos θ rh
rh
1 – 1
2
= – Nt
r sin2 θ sgn z 1 – 1 2
kb = – Nt
rh βrh
2Nt
In both cases (4.12) still relates ω and k.
βr
2Nt sin θ
βr
2Nt sin θ
βr
2Nt sin θ
ωb = N 1 – 1 2 βrh
2Nt
2/3
rh
rh
2/3
2
= – β
2 2Nt
βrh
2
2
, (7.16)
3 cos2 θ – 1
sin 2 θ
3 cos2 θ + 1
sin 2 θ
, (7.17a)
, (7.17b)
, (7.18)
1/3
rh
rh
. (7.19)
As expected non-Boussinesq effects involve the small parameters (βr)/(2Nt sin θ)
and (βr h )/(2Nt), and give to buoyancy oscillations a horizontal propagation. To leading order
the wavelength of gravity waves reduces to (4.8) and is constant on tori, whereas the
wavelength of buoyancy waves,
Internal wave generation. 1. Green’s function 32
λb = 2π
kb
= 2π
Nt
rh 2Nt
βrh
2/3
= 4π
β βrh
2Nt
1/3
, (7.20)
is constant on cylinders. Thus, non-Boussinesq transient internal wave fields are ruled by two
surfaces, a torus and a cylinder, represented in figure 8. On them the wavelengths of gravity
and buoyancy waves are comparable with the scale height 2/β of the stratification; well inside
them λ « 2/β and the Boussinesq situation is recovered.
As long as gravity and buoyancy waves remain fully non-Boussinesq, that is near to
the cylinder, (βr h )/(2Nt) is finite and both waves are O((Nt) –1/2 ). As the cylinder is entered the
nature of buoyancy waves changes until eventually, when (6.23) is verified, they become
Boussinesq buoyancy oscillations. Then, although their wavelength is zero and their
wavenumber infinite (as implied by the second form of k b ), the spatial variations of their
phase are negligible compared with its time variations (as implied by the first form of k b ).
Meanwhile, the leading-order term in their pressure vanishes so that they are now O((Nt) –3/2).