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where the minus sign arises since decreases as z increases. From Kirchoff's law, the slab will also absorb radiation already upwelling along this path by an amount d4 (rs, -, 0) = - h (rs, - 0) drv -

and thus the total radiative transfer equation for spectral radiance in the atmosphere at an optical depth rp may be expressed as

d rv

For scattering atmospheres, the equations become more complicated in that particles may scatter light out of a beam and also scatter light into a beam that was initially traveling in other directions. The equation of transfer under these conditions becomes (Hanel et al., 2003)

where ; p, 0; p', 0') is the phase function at the atmospheric level of optical depth rp; and p', 0' are the zenith and azimuth angles of radiation incident upon the scattering particles there. The single scattering albedo ) at a particular level in the atmosphere is the ratio of o-sca/^ext for both aerosols and gas (i.e., it includes the scattering of aerosols and Rayleigh scattering of the gas together with absorption by both the aerosols and gas). This equation clearly reduces to the thermal emission form for nonscattering atmospheres, where to = 0.

The scattering second term of Equation (6.72) contains the contribution of both the scattered diffuse field and the scattering of direct sunlight that has reached a particular optical depth rp. It is very useful to be able to discriminate between the two. Suppose that the incident sunlight (considered to be a beam of collimated radiation) carries a spectral flux F0 (W m -2 (cm-1)-1) normal to its direction, and suppose that the cosine of the zenith angle of the Sun is p0. The magnitude of the flux directly vertically downwards is p0F0 and Equation (6.72) may be re-expressed as

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