where the transmissivity tr = 1 — Rr, gr is the asymmetry factor set to zero (symmetric scattering). For the atmosphere the flux-mean optical depth over the UV-visible range of the incoming solar radiation is 0.187. The flux-mean rayleigh scattering optical depth can be computed from the incoming solar flux spectral distribution, normalized to a solar constant of 1366 W m~2. For a solar zenith angle corresponding to p = 0.5 the rayleigh reflectivity of the atmosphere is about 16%, according to the above simple expression.

6.4 Clouds absorption and scattering

6.4.1 Cloud types

The total fraction of the sky covered by clouds can be represented by Ac and taken to comprise three non-overlapping: low-level cumulus (cu), stratocumu-

Table 6.3 Typical values of the visible (0.6 ¡m) scattering optical depth, Ts, and corresponding approximate albedo, for cloud types belonging to low-level, middle-level and high-level clouds, based on the ISCCP classification.

Cloud type

Cloud-top level




>680 mbar, <3.2 km




440-680 mbar, 3.2-6.5 km




< 440 mbar, >6.5 km

< 3.55

< 0.35

lus (Sc), stratus (St); middle-level altostratus (As), altocumulus (Ac), nimbostratus (Ns); and high-level cirrus (Ci), cirrostratus (Cs), deep convective (Dc); with cloud components, Acl, Acm and Ach, respectively. Cloud radiative property climatologies, spanning decades, (3-hourly on 2.5°x2.5° latitude-longitude resolution) for the globe are given by the International Cloud Climatology Project (ISCCP), accounting for 15 different cloud types, including both liquid and ice phase (§8.2).

6.4.2 Visible scattering

Typical scattering optical depths in the visible (0.6 ¡m) are given in Table 6.3 for each cloud type. The cloud-top pressure and visible scattering optical depth is used to classify the various cloud types.

The reflectivity, Rswc, and transmissivity, tswc, of each cloud type can be analytically estimated based on expressions derived using the modified two-stream approximation (Sagan and Pollack 1967, Irvine 1968). For the solar UV-visible radiation we use the result for pure scattering

and tswc = 1 — Rswc, where ts is the scattering optical depth and gc the cloud scattering asymmetry factor with a typical value of 0.85. Note that the scattering optical depth of clouds (Mie scattering) is approximately independent of wavelength over the solar spectrum as they have a particle-size distribution of sufficiently large radii compared with the radiation wavelength (Fig. 6.12). In Table 6.3 an approximate reflectivity is given for the different cloud types based on an incident angle corresponding to i = 0.5 using the two-stream approximation. As the scattering optical depth increases, the reflectivity of the cloud approaches unity. Given that Mie scattering is primarily in the forward direction this might appear paradoxical. However, the explanation is that some of the incident photons on the cloud are scattered backwards out of the cloud as they have a smaller optical depth (larger mean free path) to traverse and so escape backwards with a smaller probability of undergoing a scattering event. Those travelling forwards have a higher probability of being scattered and are gradually all reflected backwards and so the reflectivity rises to unity.

6.4.3 Near-infra-red absorption and scattering

For the solar near-infra-red radiation we also include cloud absorption and so


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