E. Linacre |
1/'99 |
Upward terrestrial radiation
The Earth’s net incoming solar radiation (i.e. the total incoming minus the part that is reflected away) is known, and, on average, is matched by an equal amount of longwave radiation outwards. So the Stefan-Boltzmann equation allows calculation of the Earth’s effective black-body temperature (T_{bb}): it is -18° C. However, the average surface temperature is about 15° C, which is 33 K warmer, and the mean temperature lapse rate through the atmosphere is about 6 K/km, so T_{bb} occurs at about 5.5 km in the atmosphere (i.e. 33/6). Therefore this is the mean radiating height, h_{u} ~5.5km. Also it happens to be close to halfway through the atmosphere’s mass. In reality longwave radiation is emitted from the Earth's surface, the tops of clouds, and the entire range of atmospheric layers.
Sky radiation
That effective height for longwave radation outwards from the Earth into space contrasts with the effective height h_{d} for sky radiation R_{ld} downwards to the ground. The latter can be calculated by comparing the empirical expression for clear-sky R_{ldc} given below (1), with the Stefan-Boltzmann equation for the radiation from a ‘black body’ at height h_{d}, where the temperature is Tk degrees Kelvin, as follows -
R_{ldc} = |
208 + 6 T |
W/m^{2} |
= |
5.67 x 10^{-8} Tk^{-4} |
W/m^{2} |
where T is the screen temperature, which will be assumed to be 15ºC. Hence Tk is 269 K, i.e. -4ºC. Thus h_{d} is about 3 km, i.e. (15 + 4)/6, if the lapse rate is 6 K/km. This is below the 5.5 km calculated for radiation to space.
Of course, such calculations are complicated by clouds. Cloud whose base is below 3 km will radiate downwards at a temperature higher than Tk, increasing R_{ld}. In general, it is found empirically that a sky with C oktas of cloud radiates more by a factor of (1 + 0.0034 C^{2}) :
R_{ld_cloudy }= R_{ldc} (1 + 0.0034 C^{2})
according to Linacre p95 (1). This factor equals 1.054 if C is 4 (50% cloudiness), for instance, which would raise Tk by 4 K, so that h_{d} becomes about 2.5 km instead of 3 km. For a completely overcast sky, the height works out at about 1.5 km.
Reference
(1) Linacre, E.T. 1992. Climate Data and Resources (Routledge).