Velocity fluctuations are calculated near an interface between a turbulent region and a stably stratified layer, and in the absence of mean shear. Based on the observation that the energy dissipation rate is finite in the turbulent region, a linear theory is constructed to match the given Eulerian (not Lagrangian) spectra of the turbulence in the turbulent region to the wave motion in the stable layer.

The theory shows that eddies with frequency of the same order as the buoyancy frequency N of the stratified layer are least affected by the stratification. The mean square vertical velocity $\overline{\omega^2}\rightarrow 0$ when N → ∞, while ω2 is greatest at the interface when NLH/uH ≈ 2 uH and LH are respectively the velocity scale and longitudinal integral scale in the interior of the turbulent layer. In the stratified layer, since waves with frequency ω > N decay rapidly with distance z from the interface, the high-frequency parts of the spectra fall-off sharply, a striking feature of the atmospheric measurements of Caughey & Palmer (1979). In this inviscid model waves with frequency ω < N propagate in the stratified region without decay. The vertical integral scale Lx(w) is found to vary significantly with N, reaching a maximum at the interface when. NLH/uH ≈ 1. The wave energy flux (Fw) is a maximum when NLH/uH ≈ 6, a value frequently observed in the atmosphere.

In the limit of large stratification (N → ∞), the theory shows that the effect of the stable layer on the turbulent region is the same as that of a rigid surface moving with the flow at the same mean velocity (i.e. the solution of Hunt & Graham 1978). Then. Fw → 0 and, at the interface, Lx(w) → 0. In the limit of small stratification (N → O) the vertical motion in the turbulent region decreases in intensity near the interface and irrotational motions are induced in the slightly stable layer (i.e. the same result as Phillips 1955).