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

Internal wave generation. 1. Green’s function 9
P' = – ρ00 ∂2 ∂
+ N2 ψ' , (3.12)
∂t2 ∂t
ρ' = ρ00β ∂
∂z
– β
2 ∂
∂t
ψ' . (3.13)
Hereafter v’, P’, ρ’ and ψ’ will implicitly be considered and primes will be omitted. While the
extraction of density factors reduces non-Boussinesq effects upon the propagation of internal
waves to O(β 2 ) (Lighthill 1978 § 4.2), there remain O(β) effects upon their generation
(Appleby & Crighton 1986, 1987). This is exhibited by the non-Boussinesq term in the left-
hand side of (3.10), and the source term in its right-hand side.
3.2. Green’s function and radiation condition
From the mathematical point of view, the Green’s function G(r, t) of internal waves is
defined as the Green’s function of their operator, by
∂ 2
β2
Δ –
∂t2 4 + N2 Δh G (r, t) = δ(r) δ(t) . (3.14)
It physically represents the internal potential generated by an impulsive point mass source
releasing a unit volume of fluid, from which the corresponding velocity, pressure and density
fields are deduced by (3.11)-(3.13). Accordingly, we shall refer to it as an impulsive Green’s
function and also introduce a monochromatic Green’s function G(r, ω), as the internal potential
generated by a monochromatic point source, so that
ω 2 – N 2 Δh + ω
2 ∂2 β2
–
2
∂z
4
G (r, ω) = – δ(r) . (3.15)
They are related to each other by Fourier transformation in time, according to
G(r, ω) = G(r, t) e –i ωt dt ≡ FT G(r, t) , (3.16a)
G(r, t) = 1
2π G(r, ω) eiωt dω ≡ FT –1 G(r, ω) . (3.16b)
In actual fact, neither (3.14) nor (3.15) enable the complete determination of the Green’s
function, in that their solutions contain arbitrary linear combinations of free waves