The Saturation of Gravity Waves in the Middle Atmosphere. Part IV: Cutoff of the Incident Wave Spectrum

Abstract

A Doppler-spread theory for the saturation of middle atmosphere gravity waves was presented in an earlier member of this sequence of papers. It employed a model in which a broad spectrum of waves subject to linear theory is incident from below. The spectral distribution (in vertical wavenumber m) is deformed, as it propagates upward, in response to the growing importance of the Eulerian advective nonlinearity imposed on each wave by the total wave-induced wind. The deformation is such as to statistically spread the spectrum towards larger m, with the largest-m waves being progressively obliterated in quasi-critical-layer interactions. The model invoked a cutoff of the incident spectrum at a vertical wavenumber specified as lying in the range 0.5?1.0 times the local buoyancy frequency divided by the rms wind speed, with the choice 0.5 being adopted tentatively. A qualitative argument for the chosen cutoff wavenumber was presented but was not supported by any more certain quantitative analysis at the time. The present paper derives an analytic form for the cutoff function, illustrates it in application, and provides quantitative support for a value possibly as low as 0.5 in the stratosphere and a value possibly as high as 1.0 in the mesosphere. In addition, it slightly recasts the heuristic approach to the Doppler-spread analysis, and it admits to certain difficulties, associated with the largest-m waves, whose circumvention appears to require a far more detailed analysis of wave-wave interaction through the advective nonlinearity.