According to this expression, an optically thin atmosphere acts precisely like an isothermal slab with temperature Tv and (small) emissivity tto. It is only in the optically thin limit that the radiative effect of the atmosphere mimics that of an isothermal slab.
Substituting the approximate form of the fluxes into the expression for radiative heating rate, we find
This is small, because k is small in the optically thin limit. The first pair of terms are always positive, and represent heating due to the proportion of incident fluxes which are absorbed in the atmosphere. The second term is always negative, and represents cooling by blackbody emission of the layer of air at pressure p. In contrast to the general case or the optically thick case, the cooling term is purely a function of the local temperature; radiation emitted by each layer escapes directly to space or to the ground, without being significantly captured and re-emitted back by any other layer.
Typical greenhouse gases are optically thin in some spectral regions and optically thick in others. We have seen that the infrared heating rate becomes small in both limits. From this result, we deduce the following general principle: The infrared heating rate of an atmosphere is dominated by the spectral regions where the optical thickness is order unity. If an atmosphere is optically thick throughout the spectrum, the heating is dominated by the least thick regions; if it is optically thin throughout, it is dominated by the least thin regions.
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