Effects of an optically thin Rayleigh atmosphere on the absorption of solar radiation by a Lambert surface

Abstract

Absorption of the solar radiation by a Lambert surface under an optically thin plane-parallel scattering layer is analyzed. Point-symmetry of the scattering phase function is assumed and therefore the study applies directly to a Rayleigh atmosphere. The scatterers produce two opposite effects: a decrease in the absorption results from backscattering to space out of the solar beam, but an increase results from scattering back to the surface of the fluxes reflected from it. The hemispheric reflectivity Rf (spectral albedo) of the surface atmosphere system is formulated as:

R_f (a₀, μ₀, τ^s) = a₀ + (1 – a₀)|(1/μ₀) – 2a₀ |)τ^s/2)

where a0 is the Lambert Law spectral albedo of the surface, μ0 is the cosine of the solar zenith angle, τ^s is the vertical optical thickness, and where only the first power terms are retained in a series expansion in the optical thickness. Applying this equation, we analyze how the Rayleigh atmosphere mitigates the change in the absorption at the surface which follows from a change in the surface spectral albedo. For a small change in the surface albedo, the change in the daylong absorption at equinox, at latitude z. is smaller by a fraction

τ^s | (π/4 cos z) + 1 – 2a₀ |

than the comparable change in the absence of an atmosphere. This fraction can be termed the mitigating factor, and is discussed as such.

Analysis of the mitigating effects when a large change in the albedo occurs, that from snow to vegetation, indicates that an adequate specification of a Lambert surface for climate studies requires at least two parameters: the visible and the infrared albedo.