Wednesday, October 6, 2010

With 660 ppm CO2

In our In our previous post we extended our atmospheric absorption spectra to cover wavelengths 5 μm to 60 μm. These spectra assume CO2 concentration 330 ppm, which is typical for the Earth today, and temperature and water vapor concentration typical for a 30°N on a clear spring day. We plotted the extended spectra and applied them as input to our total escaping power program. The program says that the total power radiated by the Earth's surface and its atmosphere below the tropopause is 296.0 W.

Today we make one change to our atmospheric conditions: we double the CO2 concentration to 660 ppm. We leave water vapor concentration, temperature, and pressure the same. With the help of the Spectral Calculator, we obtain new absorption spectra and plot them.

We feed our new spectra into our total escaping power program, and obtain the following output. The final column we added by hand. It gives us the change in total escaping power when we raise CO2 concentration from 330 ppm to 660 ppm.
Name Temp BB Layer Escaping Change
Surface 290.0 401.1 388.8 88.2 -1.6
0-km 280.0 348.5 264.0 55.6 -3.3
3-km 270.0 301.3 188.6 47.8 -1.9
6-km 250.0 221.5 115.2 51.0 -0.4
9-km 230.0 158.7 53.4 29.7 +0.3
12-km 220.0 132.8 24.2 11.5 +0.3
15-km 220.0 132.8 16.3 5.7 +0.2
Total: 289.4 -6.6
The total escaping power drops by 6.6 W/m2. The power escaping from the Earth's surface and the first three 3-km layers of the atmosphere decreases by 7.2 W/m2, but the power radiated by the top three layers increases by 0.8 W/m2. In the lower layers, more CO2 causes more radiation to be absorbed on its way into space. In the upper layers, more CO2 allows the atmosphere to radiate its heat more efficiently into space. We see that the increased absorption far outweighs the increased radiation, and the result is a 2.2% reduction in the heat escaping the Earth.

According to Stephan's Law, the amount of power radiated by a black body increases as the fourth power of its temperature. For small changes in temperature, the amount of power radiated by any body will increase as the fourth power of its temperature. A 1% increase in temperature will cause a 4% increase in radiated power. In order to increase total escaping power by 2.2%, we can increase the temperature of the Earth's surface and all the atmospheric layers by one quarter of 2.2%, or 0.55%. We increased all our layer temperatures by 0.55% and applied our total escaping power program to the resulting atmospheric layers. We obtain the following output. Note that the new temperatures are recorded in the first column.
Name Temp BB Layer Escaping
Surface 291.6 410.0 397.5 90.6
0-km 281.5 356.1 269.3 57.0
3-km 271.5 308.1 192.3 48.8
6-km 251.4 226.5 117.4 52.0
9-km 231.3 162.3 54.4 30.2
12-km 221.2 135.8 24.7 11.7
15-km 221.2 135.8 16.7 5.9
Total: 296.1
We see that there is no net change in total radiated power when we combine a doubling of CO2 concentration with a 1.6°C warming of the Earth's surface. That's not to say that the Earth will actually warm up by 1.6°C if we double the CO2 concentration, because the climate might be a chaotic system that defies such simple logic, or a system with feedback that acts in quite the opposite manner to our expectations. Nevertheless, in the absence of such complex behavior, the doubling the CO2 concentration will tend to warm the Earth by 1.6°C. This 1.6°C is our estimate of the Earth's CO2 doubling temperature.

Our analysis considered only the heat radiated by the troposphere, which is the region of the atmosphere below the tropopause. We ignored the effect of increased CO2 upon the stratosphere, which is the region above the tropopause. Even though there is no net heat flow between the troposphere and the stratosphere, an overall warming or cooling of the stratosphere will affect the temperature of the troposphere, and therefore change its total radiated power. Furthermore, we ignored the effect of increased CO2 upon radiation arriving from the Sun. We will consider these effects in future posts. In our next post, however, we will try doubling the water vapor concentration in various atmospheric layers and re-calculating total escaping power. By this means, we will be able to determine whether the effect of CO2 upon total escaping power is significant compared that of water vapor.

UPDATE: See slightly improved calculation of CO2 warming in next post, where we arrive at 1.5°C instead of 1.6°C as the size of the effect.

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