Calculation of OLR is one of the most fundamental steps in determining a planet's climate. Now that we are equipped with an ability to compute the OLR for real gases, we can revisit some of our old favorite problems - Snowball Earth, the Faint Young Sun, Early Mars, and so forth -but this time relate the results to the actual atmospheric composition. In this section we present results for the all-troposphere model introduced in Section 4.3.2, occasionally limiting the upper air temperature drop by patching the adiabat to an isothermal stratosphere.

The homebrew exponential sum radiation model described in the preceding sections has the advantage of simplicity, generality and understandability. We will use it wherever it is sufficiently accurate to capture the main phenomena under discussion. However, professionally written terrestrial radiation codes are the product of a great deal of attention to detail, particularly with regard to temperature scaling and the simultaneous effects of multiple greenhouse gases. They can give highly accurate results provided one does not stray too far from the Earthlike conditions for which they have been optimized. In the following, and at various places in future chapters, we will have recourse to one of these standard radiation models, produced by the National Center for Atmospheric Research as part of the Community Climate Model effort. We'll refer to this model as the ccm radiation model. Although it uses a good many special tricks to achieve accuracy at high speed, and a lot of detailed bookkeeping to deal with the properties of a half dozen different greenhouse gases of interest on Earth, what is going on inside this rather massive piece of code is not fundamentally different from the Malkmus type band models and the exponential sum model described previously.

A detailed discussion of surface back-radiation for real gases will be deferred to Chapter 6. Some aspects of the infrared cooling profile for real gas atmospheres will be touched on in Section 4.7. The effect of clouds on OLR and on shortwave albedo will be discussed in Chapter 5.

First we'll compute the OLR for a mixture of CO2 in dry air, with the temperature on the dry air adiabat T(p) = Tg • (p/ps)2/7 and ground temperature equal to surface air temperature. This is a real-gas version of the calculation leading to Eq. 4.33; it amounts to a canonical OLR computation which serves as a simple basis for intercomparison of different radiation models. Performed for other greenhouse gases, it also can provide a basis for comparing the radiative effects of the gases. The results presented here are carried out with Earth gravity and 1 bar of air partial pressure, but can easily be scaled to other conditions. The CO2 path is inversely proportional to gravity, so 100ppmv of CO2 on Earth is equivalent to 1000ppmv on a planet with ten times the Earth's surface gravity or 10ppmv on a planet with a tenth of the Earth's surface gravity. For fixed CO2 concentration, surface pressure has a quadratic effect on the path, since the mass of CO2 in the atmosphere (given fixed concentration) increases in proportion to pressure, but one gets an additional pressure factor in the equivalent path from pressure broadening. Thus, 100ppmv of CO2 in 1bar of air is equivalent to 1ppmv of CO2 in 10bars of air.

In Figure 4.29 we show how OLR varies with CO2 concentration for a fixed surface temperature of 273K. This curve gives the amount of absorbed solar radiation needed to maintain the

Figure 4.29: The OLR vs CO2 concentration (measured in ppmv) for CO2 in a dry air atmosphere with temperature profile given by the dry adiabat. The surface air temperature and the ground temperature are both 273K, and the acceleration of gravity is 9.8m/s2. Results are shown for a simple exponential sum radiation code without temperature weighting, and for the comprehensive ccm radiation code.

surface temperature at freezing. The CO2 amount is expressed as molar concentration in ppmv, but for the range of concentrations considered, the difference between concentration and mixing ratio is not very significant. Results are shown for both the ccm model and the simplest form of the homebrew exponential sums radiation code. The homebrew calculation employed 10-term sums with coefficients computed at 260K. The path was pressure-weighted to reflect collisional broadening, but temperature weighting was neglected. Over the range of CO2 covered, only the principal absorption region centered on 650cm-1 needs to be taken into account. The homebrew calculation deviates by up to 10 W/m2 from the more comprehensive ccm calculation, but this is quite good agreement in view of the fact that the homebrew code takes up barely a page and generalizes easily to any greenhouse gas, in contrast to the rather Earth-specific ccm code, which involves several thousand lines of rather unpretty FORTRAN. Most of the mismatch arises from the neglect of temperature scaling in the homebrew code. The homebrew code slightly overestimates OLR for low CO2 where the radiating level is low in the atmosphere where warmer temperatures ought to increase the infrared opacity. It underestimates OLR because the higher, colder atmosphere is assumed more optically thick than it should be. If one complicates the homebrew code very slightly to incorporate a band-independent temperature weighting of the form exp — (T*/T — T*/To) in the path computation, then one can reduce the mismatch to under 2W/m2 with T* = 900K.

As anticipated from the shape of the CO2 absorption spectrum, the OLR goes down approximately in proportion to the logarithm of the CO2 concentration. Between 10ppmv and 1000ppmv each doubling of CO2 reduces OLR by about 4W/m2 based on the ccm model. At large CO2 concentrations, the logarithmic slope becomes somewhat greater, as the weaker absorption bands begin to come into play. At 10000ppmv, a doubling reduces OLR by 6.8W/m2. CO2 becomes a somewhat more effective greenhouse gas at high concentrations, but never approaches the potency of a grey gas, for which each doubling would more than halve the OLR once the optically thick limit is reached. Were it not for the relatively gentle dependence of OLR on CO2 caused by the highly frequency-selective nature of real-gas absorption, modest fluctuations in the atmosphere's CO2 content would lead to wild swings of temperature and almost certainly render the planet uninhabitable.

Calculations show that for fixed CO2 concentration the OLR increases very nearly like the fourth power of temperature, just as in the grey gas result in Eq. 4.33. In effect, the radiating pressure remains nearly fixed as temperature varies. This makes it easy to do planetary temperature calculations. For example, let's compute what temperature the Earth would have if the CO2 concentration were at the pre-industrial value of 280ppmv, but there were no other greenhouse gases in the atmosphere. The OLR for a surface temperature of 273K is 267W/m2. Balancing against an absorbed solar radiation of 240'W/m2, the temperature is determined by 267 • (T/273)4 = 240, yielding T = 263K. Without the additional greenhouse effect of water vapor, Earth would be a very chilly place. According to Figure 4.29, for a dry Earth CO2 would have to be increased all the way to 24000ppmv just to bring the temperature up to the freezing point.

Exercise 4.5.1 What temperature would Venus have if it had a 1 bar air atmosphere mixed with 280ppmv of CO2? Assume that the planetary albedo is 30%, like that of Earth. How does this temperature compare with the temperature Venus would have without any greenhouse gases in its atmosphere?

One can't get the Earth's temperature right without water vapor, but one can still make a decent estimate of the amount of CO2 increase needed to offset the reduction of Solar forcing in the Faint Young Sun era, or due to a global Snowball Earth glaciation. A 25% reduction of Solar absorption at the Earth's orbit during the Faint Young Sun amounts to 60 W/m2, assuming an albedo of .3. This is equivalent to the radiative forcing caused by increasing CO2 from 100 ppmv to 105ppmv (about 100mb CO2 partial pressure, or 10% of the atmosphere). From this we include that it's likely that it would take somewhat over 100mb of CO2 to keep Earth unfrozen when the Sun was dim. Next let's take a look at what it takes to deglaciate a Snowball Earth. Let's suppose that for a Neoproterozoic solar constant, the tropics freeze over when the CO2 is reduced to 100ppmv (this is not far off estimates based on comprehensive climate models). Icing over the Earth increases the albedo to about .6, leading to a reduction of almost 100W/m2 in absorbed solar radiation. Hence, restoring the Tropics to the melting point would require well in excess of 100mb of CO2 in the atmosphere; calculations with the homebrew model at higher CO2 concentrations than shown in Figure 4.29 indicate that fully a bar of CO2 mixed with a bar of air would be needed. Deglaciation might not require restoring the full 100W/m2 of lost Solar forcing, but it is still clear that a great deal of CO2 is needed to deglaciate a Snowball.

These estimates are crude, but they do get across one central idea: that because of the logarithmic dependence of OLR on greenhouse gas concentration, it takes a huge increase in the mass of CO2 to make up for rather moderate changes in albedo or solar output. Aside from neglect of overlapping water vapor absorption, these estimates somewhat overstate the effect of CO2 on OLR because they employ the dry adiabat rather than the less steep moist adiabat. We'll revisit the estimates shortly, after we bring water vapor into the picture.

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