dp dp where
Hence, in the optically thick limit, the heating and cooling caused by radiative transfer acts just like a thermal diffusion in pressure coordinates, with the diffusivity given by D(v,p). Since dB/dT > 0, the radiative diffusivity is always positive. It becomes weak as k becomes large. Note that the diffusive approximation to the heating is is only valid when one is not too close to the top and bottom of the atmosphere. Near the boundaries, the neglected boundary terms contribute an additional heating which is exponentially trapped near the top and bottom of the atmosphere. The effect of the boundary terms is explored in Problem ??
Consider an atmosphere which is transparent to solar radiation, and within which heat is redistributed only by infrared radiative transfer. Eq 4.15 then requires that the net upward flux I+ -1_ must be independent of altitude when integrated over all wavenumbers. This constant flux is nonzero, since the infrared flux through the system is set by the rate at which infrared escapes from the top of the atmosphere - namely, the OLR. Integrating Eq. 4.22 over the infrared yields an expression for dT/dp in terms of the OLR and the frequency-integrated diffusivity; because both OLR and diffusivity are positive, it follows that dT/dp > 0 for an optically thick atmosphere in pure infrared radiative equilibrium - that is, the temperature decreases with altitude. The more optically thick the atmosphere becomes, the smaller is D, and hence the stronger is the temperature variation in equilibrium. Pure radiative equilibrium will be discussed in detail in Sections 4.3.4 and 4.7, and the optically thick limit is explored in Problem ??.
The optically thin limit is defined by ^ 1. Since tv < and t7 < t in Eq. 4.9, all the exponentials in the expression for the fluxes are close to unity. Moreover, the integral is carried out over the small interval [0, tv], and hence is already of order or less. It is thus a small correction to the first term, and we may set the exponentials in the integrand to unity and still have an expression that is accurate to order tto. The boundary terms are not integrated, though, so we must retain the first two terms in the Taylor series expansion of the exponential to achieve the same accuracy. With these approximations, the fluxes become
In this case, the upward flux is the sum of the upward flux from the boundary (diminished by the slight atmospheric absorption on the way up) with the sum of the unmodified blackbody emission from all the layers below the point in question. The downward flux is interpreted similarly.
In order to discuss the radiation escaping the top of the atmosphere and the back-radiation into the ground, we introduce the mean emission temperature Tv, defined by solving the relation
With this definition, the boundary fluxes are
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