Spectra of the vertical displacement (potential energy) have been observed to be only slightly enhanced at the buoyancy frequency ω = N, whereas spectra of horizontal velocity u, v (kinetic energy) are greatly enhanced at the inertial frequency ω = f (except at equatorial latitudes). Consequently. the former are ignored in certain model spectra, whereas the latter are allowed for explicitly (e.g., by a term (ω2 − f2)−1/2). I have attempted to interpret these observations in terms of the behavior of free wave packets at the turning points. Local resonant generation may also be a factor (Fu, 1980) but is not considered here.
In this tutorial N′ = dN/dz and f′ = df/dy ≡ β are taken as constant in order to make the derivation of the solutions near N and f as simple and as parallel as possible; these turning point solutions (in terms of Airy functions) fail in narrow waveguides, e.g., near a sharp buoyancy peak and at equatorial latitudes. The β-plane approximation fails at polar latitudes. Limit functions are evaluated numerically for a super-position of wave modes with relative energy (j2 + j*2)−1, j = 3, assuming horizontal isotropy. The computed cutoffs are smooth functions of frequency, with a peak just below N and just above f, respectively. The N amplification in the vertical displacement spectrum is by less than 2 (but equals 5 for the spectrum of vertical strain rate). The f amplification in the horizontal velocity spectrum is by a factor of 8 at latitude θ = 30°, and diminishes With latitude as (sinθ tanθ)1/3. In general, the amplification varies with the width of the waveguide (vertical and latitudinal) expressed in units of a characteristic wavelength. Thus the inertial peak is a consequence of linear wave theory and should not be independently imposed on model spectra.