We will suppose that the properties of the radiation field and the properties of the medium through which it travels are functions of a single coordinate, which we will take to be the pressure in a hydrostatically balanced atmosphere. (Recall that in such an atmosphere there is a one to one correspondence between pressure and altitude). This is the plane-parallel assumption. Although the properties of planetary atmospheres vary geographically with horizontal position within the spherical shell making up the atmosphere, in most cases it suffices to divide up the sphere into patches of atmosphere which are much larger in the horizontal than they are deep, and over which the properties can be considered horizontally uniform. In this case, vertical radiative transfer is much more important than horizontal transfer, and the atmosphere can be divided up into a large number of columns that act independently, insofar as radiative transfer is concerned.
In this section, we will develop an approximate form of the equations of plane parallel radiative transfer. The errors introduced in this approximation are small enough that the resulting equations are sufficiently accurate to form basis of the infrared radiative transfer component of virtually all large scale climate models. These equations will certainly be good enough for addressing the broad-brush climate questions that are our principal concern.
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