A simple mathematical model of the greenhouse effect

Let us now construct a simple mathematical model illustrating the greenhouse effect. our purpose in doing so is to see somewhat quantitatively, if approximately, whether the atmosphere might warm the surface up to the observed temperature. Let us make the following assumptions:

1. The surface and the atmosphere are each characterized by a single temperature, Ts and Ta, respectively.

2. The atmosphere is completely transparent to solar radiation.

3. Earth's surface is a blackbody.

4. The atmosphere is completely opaque to infrared radiation, and it acts like a blackbody.

The model is illustrated in figure 1.4, which the reader will appreciate is a very idealized version of figure 1.3.

Figure 1.4. An idealized two-level energy-balance model. The surface and the atmosphere are each characterized by a single temperature, Tsand Ta. The atmosphere absorbs most of the infrared radiation emitted by the surface, but it is transparent to solar radiation.

The parameter e, called the emissivity or the absorptivity, determines what fraction of infrared radiation coming from the surface is absorbed by the atmosphere, and we initially assume that e = 1; that is, the atmosphere is a blackbody and absorbs all the surface infrared radiation. The incoming solar radiation, S0, and the albedo are presumed known, and the unknown temperatures Ts and Ta are obtained by imposing radiative balance at the surface and the atmosphere. At the surface, the incoming solar radiation, S0(1 — a), plus the downward longwave radiation emitted by the atmosphere is balanced by the longwave radiation emitted by the surface, and therefore

Similarly, radiative balance at the top of the atmosphere is

Note that instead of equation 1.10, we could have used the condition that the radiation absorbed by the atmosphere must balance the longwave radiation emitted by the atmosphere so that, looking again at figure 1.4, vT4 = 2vT4a. (1.11)

Using equation 1.9 and either equation 1.10 or 1.11 and just a little algebra, we obtain expressions for Ta and Ts, namely

Note that the equation for Ta, namely equation 1.10 is the same as equation 1.8, and so using S0 = 342 Wm2 and a = 0.3, we find Ta = 255 K. For the surface temperature, we obtain Ts = 303 K = 30°C. This temperature is quite a bit higher than the observed average temperature at Earth's surface (288 K) mainly because we are assuming that Earth's atmosphere is a perfect blackbody, absorbing all the longwave radiation incident upon it. In fact, some of the longwave radiation emitted by the surface escapes to space, so let's try to model that in a simple way.

A leaky blanket

Instead of assuming that the atmosphere absorbs all the longwave radiation emitted by the ground, let us assume that it absorbs just a fraction, e, of it, where 0 < e < 1; thus, an amount vT4 (1 — e) of surface radiation escapes to space. Similarly, we also assume that the atmosphere only emits the (same) fraction e of the amount it would do as a blackbody. In other respects, the model is the same as that described above.

The radiative balance equations corresponding to equations 1.9 and 1.10 are, for the surface, vT 4 S (1 a) + evT 4 (1.13)

and at the top of the atmosphere

S0 (1 a) evT 4 + (1 e) vT 4. (1.14) Solving for Ts, we obtain

If e = 1, we recover the blackbody result (equation 1.12b) with Ts = 303 K. If e = 0, then the atmosphere has no radiative effect and, as in equation 1.8, we obtain Ts = 255 K. The real atmosphere is somewhere between these two extremes, and for e = 3/4, we find Ts = 287 K, close to the observed average surface temperature.

Evidently, the surface temperature increases as e increases, and this phenomenon is analogous to what is happening vis à vis global warming. As humankind puts carbon dioxide and other greenhouse gases into the atmosphere, the emissivity of the atmosphere increases. The atmosphere absorbs more and more of the longwave radiation emitted by the surface and re-emits it downward, and consequently the surface temperature begins to rise. The mechanism is relatively easy to understand. It is much harder to determine precisely how much the emissivity rises and how much the surface temperature rises as a function of the amount of carbon dioxide in the atmosphere. We'll discuss this question more in chapter 7, although there is still no exact answer to it. So let's finish off this chapter by discussing a question to which we do have a reasonable answer, namely, which gases in the atmosphere contribute to the greenhouse effect.

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