In the optically thick case, the heating rate is nearly zero in the interior of the slab, but there is strong radiative cooling within about a unit optical depth of each surface. In this case the radiation drains heat out of a thin skin layer near each surface, causing intense cooling there. In the optically thin limit, the cooling is distributed uniformly throughout the slab.
It turns out that condensed water is a much better infrared absorber than the same mass of water vapor. Hence, an isolated absorbing layer such as we have just considered can be thought of as a very idealized model of a cloud. The following slight extension makes the connection with low lying stratus clouds, such as commonly found over the oceans, more apparent.
Exercise 4.2.3 Instead of being suspended in an infinite transparent medium, suppose that the cloud is in contact with the ground, and that the ground has the same temperature as the cloud. We still assume that the air above the cloud is transparent to radiation at the frequency under consideration. Compute the upward and downward fluxes, and the radiative heating rate, in this case.
This exercise shows that convection in boundary layer stratus clouds can be driven by cooling at the top, rather than heating from below. This is rather important, since the reflection of sunlight by the cloud makes it hard to warm up the surface. Entrainment of dry air due to top-driven convection is one of the main mechanisms for dissipating such clouds.
We now depart from the assumption of constant temperature. While allowing T to vary in the vertical, we assume the atmosphere to be optically thick at frequency v. This means that a small change in pressure p amounts to a large change in the optical thickness coordinate tv . Referring, to Eq. 4.1, we see that the assumption of optical thickness is equivalent to the assumption that KÓp/g ^ 1, where 5p is the typical amount by which one has to change the pressure in order for the temperature to change by an amount comparable to its mean value. For most atmospheres, it suffices to take 5p to be the depth of the whole atmosphere, namely ps, so that the optical thickness assumption becomes = Kps/g ^ 1 Since k depends on frequency, an atmosphere may be optically thick near one frequency, but optically thin near another.
The approximate form of the fluxes in the optically thick limit can be most easily derived from the integral expression in the form given in Eq. 4.9. Consider first the expression for I+. Away from the immediate vicinity of the bottom boundary, the boundary term proportional to Tv(p,ps) is exponentially small and can be dropped. To simplify the integral, we note that Tv(p,p') is very small unless p' is close to p. Therefore, as long as the temperature gradient is a continuous function of p', it varies little over the range of p' for which the integrand contributes significantly to the integral. Hence dT/dp' can be replaced by its value at p, which can then be taken outside the integral. Likewise, dB/dT can be evaluated at T(p), so that this term can also be taken outside the integral. Finally, if one is not too close to the bottom boundary, fPS Tv(p',p)dp' « r Tv(p',p)dp' = r Tv(T',t)dT' = , (4.18)
Jp Jp k Jt k whence the upward flux in the optically thick limit becomes
Near the bottom boundary, the neglected boundary term would have to be added to this expression. In addition, Eq. 4.18 would need to be corrected to allow for the fact that there is not room for J T to integrate out to its asymptotic value for an infinitely thick layer.
Using identical reasoning, the downward flux becomes
so long as one is not too near the top of the atmosphere. Near the top of the atmosphere, the neglected boundary term becomes significant.
In both expressions the second term, proportional to the temperature gradient, becomes progressively smaller as k is made larger and the atmosphere becomes more optically thick. To lowest order, then, the upward and downward fluxes are both equal to the blackbody radiation flux at the local temperature. In this sense, the optically thick limit looks "locally isothermal." The term proportional to the temperature gradient represents a small correction to the locally isothermal behavior. In the expression for 1+, for example, if dT/dp > 0 the correction term makes the upward flux somewhat greater than the local blackbody value. This makes sense, because a small portion of the upwelling radiation comes from lower layers where the temperature is warmer than the local temperature. Note that the correction term depends on v through the frequency dependence of k, as well as through the frequency dependence of dB/dT.
The radiation exiting the top of the atmosphere (i+,TO) is of particular interest, because it determines the rate at which the planet loses energy. In the optically thick approximation, we find that as long as dT/dp is finite at p = 0, becomes close to nB(v, TTO) as the atmosphere is made more optically thick. Hence, at frequencies where the atmosphere is optically thick, the planet radiates to space like a blackbody with temperature equal to that of the upper regions of the atmosphere - the regions "closest" to outer space.
Similarly, the downward radiation (I_,s(v)) from the atmosphere into the ground - sometimes called the back radiation - is of interest because it characterizes the radiative effect of the atmosphere on the surface energy budget. In the optically thick limit, I_,s(v) = nB(v, Tsa) to lowest order, so that the atmosphere radiates to the ground like a blackbody with temperature equal to the low level air temperature. If dT/dp > 0 at the ground, as is typically the case, then the correction term slightly reduces the downward radiation, because some of the radiation into the ground comes from higher altitudes where the air is colder. Suppose now that the surface temperature Tg is equal to the air temperature Tsa, and that the surface has unit emissivity at the frequency under consideration. In that case, the net radiative heating of the ground is
/ ™ - / \ / ™ - gcos - dB dT, - B(v,Ts) = I_,s(v) - B(v,Tso) = -n^- — |Tsa ^|ps (4.21)
at frequencies where the solar flux is negligible. This is negative when dT/dp > 0, representing a radiative cooling of the ground. The radiative cooling vanishes in the limit of large k. In the optically thick limit, then, the surface cannot get rid of heat by radiation unless the ground temperature becomes larger than the low level air temperature. Remember, though, that the radiative heating of the ground is but one term in the surface energy budget coupling the surface to the atmosphere. Turbulent fluxes of moisture and heat also exchange energy between the surface and the atmosphere, and these become dominant when the radiative term is weak.
In the optically thick limit, the net flux is dT dp whence the radiative heating rate is
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