Summary
This paper presents the mathematical background of the derivation of atmospheric gravity-wave spectra from Rayleigh lidar observations of fractional density and/or temperature fluctuations. This is accomplished by first considering the relationships between the observed wave-induced density or temperature change and the quantities that can be derived from this (horizontal wind variance, energy per unit mass measures) for monochromatic waves, using the ‘polarisation relations’ given by Hines (1960). The relationships between the observed and derived quantities for single waves are then generalised to the calculation of gravity wave spectra. Specifically, the relationship between the wave-induced perturbation in density and horizontal wind is derived. Attention is given to the validity of two approximations used to derive this relationship – that of limiting to ‘small vertical scale’ waves (i.e. large vertical wave-number waves only) and the ‘hydrostatic approximation’ (allowable wave frequencies only up to ~0.3 times the Brunt-Vaisala frequency), and also to the conditions necessary for the wave-induced relative density fluctuations to be equal in magnitude to the relative temperature fluctuations. These are shown to be valid for the likely range of waves observable with a Rayleigh lidar covering the stratosphere and mesosphere. The relationship between the temperature (or density) fluctuations and wave energy per unit mass is also presented. The derivation of gravity wave spectra is then considered. An often used technique in the analysis of lidar data is to derive vertical wave number spectra of wave-induced horizontal wind variance calculated from the wave-induced fractional density fluctuations. However, for ‘small-scale waves’ where the only density change is due to vertical motion of air parcels, the density changes are greatest for the higher frequency waves which have the least horizontal velocity motion. A better interpretation of the scaled squared