The atmospheric height distribution of band-absorbed solar radiation

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Abstract
The band absorption of radiation incident on the atmosphere at any zenith angle χ is considered, when the relation between the absorption coefficient α and the wave-length λ has the form of the error function, so that αλ = α0exp[-(λ-λ0)2/λw2] ≡ α0exp(-l2), where l = (λ - λ0)/λw, and the intensity of the incident radiation is either constant across the absorption band or varies linearly across it. The height distribution of absorbed-energy density, the height of maximum volume absorption, the spectral composition of the energy absorbed at any level, and the proportionate daily variation of absorption density, are first discussed without any special assumption as to the height distribution of the absorbing constituent in the atmosphere. The height distribution of absorbed energy per unit mass of air is shown to depend only on a single function of a single parameter μ, which itself depends on the level, the angle of incidence χ, and the total absorption coefficient A0 of the atmosphere in the centre of the absorption band. The general results are afterwards discussed in relation to the special case of an exponential height distribution of the absorbing constituent, with scale-height H. It is shown that in this case, whatever the angle χ, (i) the height of maximum absorption per unit volume is at a distance 0.616H below the level of maximum monochromatic absorption at the centre λ0 of the band; (ii) the maximum volume density of absorbed energy at angle χ is cos χ times that for vertical incidence, as in the case of monochromatic absorption; (iii) above the level of maximum band absorption the height distribution of absorption differs only slightly from that in the case of monochromatic absorption, but below the maximum level the band-absorption layer extends further down than the monochromatic-absorption layer. If the absorption in the band-fringes where |l| ≥ 2 be neglected, the band-absorption layer is thicker than the monochromatic-absorption layer by approximately 2H, the excess thickness being in the part of the layer below the level of maximum absorption.