The Saturation of Gravity Waves in the Middle Atmosphere. Part II: Development Of Doppler-Spread Theory

The irregular winds of the middle atmosphere are commonly attributed to an upwardly propagating system of atmospheric gravity waves. Their one-dimensional (in vertical wavenumber m) power spectrum has been reported to exhibit a nearly universal behavior in its “tail” region of large m: both the form (∼m−3) and the intensity of the tail are approximately invariant with meteorological conditions, time, place and height. This universality is often described as resulting from “saturation” of the system, with the physical cause of saturation being left for separate identification and analysis. Here the cause is attributed to nonlinear interaction between the waves of the full spectrum, most specifically to the advective nonlinearity of the Eulerian fluid-dynamic equations. This nonlinearity has the effect of Doppler shifting the local intrinsic frequency of any given wave in the wind field imposed by all waves. Only an approximation to its effects is sought here, the wind field of the full spectrum being taken to be horizontal, horizontally stratified and constant in time, but otherwise that field is taken to have the statistical characteristics that would be expected of a wave-induced spectrum, including a propensity for growth with height. It is found that waves with relatively large vertical wavenumber m exceeding a characteristic value mc (comparable to or greater than N0/2σr, where N0 is buoyancy frequency and σr is rms wind speed) are substantially Doppler spread in vertical wavenumber, most particularly into a large-m tail. At sufficiently large m, the waves are taken to be obliterated by dissipative processes. The net effect is to leave a tail that will be universal and can be identified readily with the observed tail (subject to further approximation in the treatment of the wave obliteration). If the tail extends to a sufficiently large m—a well defined value mMinst—the spectrum as a whole renders itself unstable. The length of the tail is then taken to be limited by the instability, any m values observed beyond mMinst being attributed to turbulence. The implications of these conclusions are built into spectral models of the “modified Desaubies” form for future application to middle-atmosphere modeling. The theory is backed by appeal to observations and is brought briefly into contact with related views in oceanographic studies. The body of the paper ignores the Coriolis force and the contribution of vertical motions to the advective nonlinearity, but their effects are touched on in two Appendices. It also assumes, for the most part, azimuthal isotropy in the propagation of the waves, but the consequences of abandoning this assumption are outlined in a third Appendix.