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

inverse group velocity with horizontal lines of heights t/r provides, according to (6.18), a
qualitative description of the internal waves received at a fixed point versus time.
Thus, at time
t0 = βr
2
N 2 ω0 sin 2 θ
N 2 – ω 0 2 3/2 ω0 2 – N 2 cos 2 θ 1/2
, (6.19)
the first internal waves of frequency ω 0 given by (4.15) and maximum group velocity c g0 = r/t 0
reach the point under consideration. Kato (1966 b) and Cole & Greifinger (1969) for acoustic-
gravity waves, Mowbray & Rarity (1967) and Tolstoy (1973 § 7.3) for non-Boussinesq
internal waves, investigated the structure of the associated wavefront. It is in fact a caustic:
internal waves gradually build up just prior to its arrival, and subsequently separate into two
components of frequencies ω s1 and ω s2 such that N⏐cos θ⏐ < ω s1 < ω 0 and ω 0 < ω s2 < N.
When t/r → ∞, ω s1 and ω s2 respectively tend toward N⏐cos θ⏐ and N, identifying the
corresponding components as gravity waves and buoyancy oscillations. Ultimately, the
Boussinesq situation is recovered.
Internal wave generation. 1. Green’s function 25
Not only such a qualitative insight into the propagation of internal waves, but also the
related quantitative analysis, are provided by Felsen’s (1969) method. Accordingly, the build-
up of internal waves prior to the arrival of the caustic, and their subsequent decomposition into
gravity waves and buoyancy oscillations, are accounted for by an Airy function; the following
evolution of both components is described by the standard stationary-phase formula.
Unfortunately, these results rely on the graphical determination of the stationary frequencies
ω s1 and ω s2 and cannot be written in explicit form, except in the final stage t/r → ∞, when
analytical expressions are available for ω s1 and ω s2 .
We now focus attention on that stage, when region (ii) is entered: Nt » 1 and βr is held
fixed. To avoid the neglect of evanescent internal waves, we make no further use of Felsen’s
approach and rather remark that, in region (ii), expanding the impulsive Green’s function is
again expanding an inverse Fourier transform for large times. Thus, Lighthill’s (1958)
asymptotic method is relevant again. Gravity waves and buoyancy oscillations arise as,
respectively, the contributions of the pairs of branch points ± N⏐cos θ⏐ and ± N of the
monochromatic Green’s function (5.2). Gravity waves are, from the expansion of G(r, ω) in
terms of (ω – N⏐cos θ⏐) 1/2 and its inversion by (D8),