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

Appendix A. INTERNAL W AVE GENERA TION BY POINT MASS AND FORCE
SOURCES
When a force source F per unit volume is added to the mass source considered in §
3.1, the horizontal motion of the fluid may become rotational. No internal potential exists any
more, and a direct derivation of the equations verified by the physical variables v, P and ρ
must be performed. In terms of the vertical displacement ζ z and the pressure P, eliminating v h
and ρ from (3.2)-(3.4), then removing the density factors (3.9), we obtain
where
L ζz = ∂
∂z
– β
2 ∂
∂t
L P = – ρ00 ∂2
∂
+ N2
∂t2 ∂t
e – βz/2 m – 1
ρ00
e – βz/2 m + ∂2
ez.∇ × ∇ – β
2 ez × e βz/2 F , (A1)
β
∇ +
∂t2 2 ez + N 2 ∇ h . e βz/2 F , (A2)
L = ∂2 β2
Δ –
∂t2 4 + N2 Δh . (A3)
From these equations, and the definition (3.14) of the Green’s function G(r, t), the
internal wave field generated by the point mass and force sources m(r, t) = m 0 δ(r) δ(t) and F
(r, t) = F 0 δ(r) δ(t) readily follows as
ζz(r, t) = m0 ∂
Internal wave generation. 1. Green’s function 46
∂z
– β
2 ∂
∂t
P(r, t) = – ρ00 m0 ∂2
– 1
ρ00
F0h ∂
∂z
– β
2 – F0z ∇ h .∇ h G(r, t) , (A4)
∂
+ N2 + F0.
∂t2 ∂t
∂2 β
∇ +
∂t2 2 ez + N2 ∇h G(r, t) . (A5)
For the point mass source P(r, ω) has already been calculated in (7.1), while ζ z (r, ω) reduces to
– i/ω times the vertical component of the velocity (7.2). Closely analogous are the pressure