On a simplified radiative-conductive heat transfer equation

Summary A simplified equation purported to represent the joint influence of radiative and turbulent transfers of heat in the atmosphere is derived by dividing the absorption spectrum of terrestrial radiation into strongly and weakly absorbed regions, classified according to the local scale of variation or to the local heating rate, and introducing two mean absorption coefficients for these two groups of regions. Assurning the validity of theK-theory of turbulent diffusion of heat, it is found that the temperature of the atmosphere is governed by a sixth-order partial differential equation in the heightz. This equation can be simplified to the second order if the mean absorption coefficient of the strongly absorbed regions is much larger while that of the weakly absorbed regions is much smaller than the local scale variation, and the influence of the former is equivalent to an added diffusion while that of the latter is a newtonian cooling, and these two influences are present simultaneously. The values of the two coefficients and their dependencies on the concentration of the absorbing material have been obtained. The equation has been applied to the problem of the thermal interaction between the atmosphere and the underlying earth, as related to the diurnal heat wave, and it was found that the temperature changes within the first few hundred meters from the earth can be predicted accurately by the model when the partition of the two groups is adjusted to the heating rate and the eddy transfer coefficient is allowed to increase very rapidly within the lowest 10 m.