The OLR and surface back-radiation for an optically thin grey atmosphere are obtained by integrating Eq. 4.27 over all frequencies. The result is
I+^ = (1 - tto)'+(0) + T^aT4 I-,s = (1 - tx)I-,x + T^aT4
where the mean atmospheric emission temperature is given by
The first term in the expression for is the proportion of upward radiation from the ground which escapes without absorption by the intervening atmosphere, while the second is the emission to space added by the atmosphere. In the expression for I- s the first term is the proportion of incoming infrared flux which reaches the surface without absorption, while the second is the downward emission from the atmosphere. Note that for an optically thin atmosphere the atmospheric emission to space is identical to the atmospheric emission to the ground; in this regard the atmosphere radiates like an isothermal slab with temperature T. According to Eq. 4.27, a nongrey atmosphere behaves similarly, if it is optically thin for all frequencies.
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