Cap Answers it All

Cap tries to explain greenhouse warming. Critiques welcomed - especially from those who know.

From a comment on her blog by Rae Ann

RA -

cip, I'm sorry I still don't really understand how exactly the greenhouse effect on Venus or even Mars can be applied to Earth. You should explain it the way that you would explain it to a child.

I just don't see how you can plug 96% and .003% into the *same* calculations and get any kind of meaningful result.

These are good questions, but unfortunately the answers are complicated, which is why I’ve delayed answering it. Let me say a word about the second question first. One of the most important discoveries in the history of physics was Newton’s realization that the same force that brought the apple from the tree to the Earth dragged the moon in its orbit about the Earth (and the Earth about the Sun, and so on). So it doesn’t surprise a physicist when a single equation explains phenomena differing by many orders of magnitude. Incidentally, it’s not the percentage CO2 that is important, it’s the total amount. Since Venus has about 90 times as much atmosphere as the Earth has, it has about 300,000 times as much CO2 between planet surface and space as the Earth does. So that one thing that needs to be explained is why 300,000 times as much CO2 produces 500 C of greenhouse on Venus, which is a lot, but not nearly 300,000 times the 10 C or so produced by the pre-industrial CO2 in the Earth’s atmosphere.

As first noted by Arrhenius in 1896, the effect of CO2 is approximately logarithmic – a geometric increase in CO2 producing an arithmetic increase in temperature, that is, if doubling the amount of CO2 produces 3 C increase in temperature, 4 times as much CO2 would increase the temperature T by 6 C, 8 times as much CO2 would increase T by 9 C, and so on. The equation isn’t an explanation though, so I will try a simple one.

There is a long term balance between the amount of heat energy being absorbed by the Earth and the amount being emitted (because increasing the temperature causes more heat to be emitted and decreasing it causes less to be emitted, while the Sun’s output is fairly steady). Because the Sun emits mainly in the visible, most of the heat energy absorbed by the Earth is visible radiation, some of which is reflected, but little of which is absorbed by the atmosphere. Consequently, most of the heating of the Earth takes place at the surface, so that the Earth’s atmosphere is heated from below.

Cooling is a more complex process. While a visible light photon has an excellent chance of passing through the atmosphere without being absorbed, the same is not true for the infrared photons the Earth (and its atmosphere) emit. Water vapor, CO2, and some other gases in the atmosphere tend to absorb infrared. Depending on frequency, an infrared photon is likely to travel anywhere from several hundreds of meters to less than one meter before being absorbed. As infrared radiation works its way outward, the atmosphere gradually thins, so the average distance traveled between absorptions (or mean free path) increases until the radiation escapes into space. The average photon emitted from the earth into space come from several kilometers above the surface.

We now need two more facts to complete the argument. We are all familiar with the fact that the Atmosphere cools with height for the first dozen or so kilometers. Consequently, photons emitted from higher up are emitted by a colder atmosphere. Second, it is a fundamental principle of physics that the amount of radiation emitted increases with increasing temperature. Thus, an observer looking down on Earth from space, or near space, can tell the average height from which photons were emitted by measuring the radiation intensity or (equivalently, the so called brightness temperature).

Now for some pictures, from a very cute toy called MODTRAN shown to me by a certain wascally lagomorph:

The red trace is spectral intensity versus wave number (equivalent to frequency) as seen looking down from 70 km up – for an atmosphere with no CO2 whatsoever. The other colored curves are the calculated spectral intensities for a perfectly absorbing (so-called black body) at the temperatures. Notice that on the left, much of the red trace is between the yellow and purple curves, at say 240 or 250 Kelvins (compared to a surface temperature here of 300K), indicating that on average the photons were emitted 7 or 8 km up. On the other hand on the right, and right center, brightness temps are near 300K, indicating emission from the surface or near to it.

The difference between the surface temperature and the mean radiating temperature (the average of those brightness temperatures) is what is responsible for the greenhouse effect. Here is the argument: the Earth adjusts its temperature until the mean radiating temperature is just sufficient to balance absorbed solar radiation. If there were no greenhouse gases, the mean radiating temperature would be close to the surface temperature. Because there are greenhouse gases, and because the Earth’s atmospheric temperature decreases with height (in the troposphere), the temperature at the surface is higher than the mean radiating temperature. Adding more greenhouse gases increases the average height from which photons are emitted, since the average photon encounters more obstruction (more greenhouse gas molecules) before it can make its escape into space. Consequently, the mean radiating height increases, and since the same solar radiation is received, the mean radiating temperature stays about the same, so the surface temperature must increase. I said it was a little complicated.

Compare the previous picture where there was no CO2 with this one. The addition of CO2 at pre-industrial levels (280 ppm) has taken a big bite out of the middle of the radiating spectrum. In order to achieve the same mean radiating temperature the other temperatures, and consequently the surface temperature, would need to be compensatorially increased.

Our model doesn’t allow for Venusian conditions, but we can get a little clue by putting in a very high amount of CO2 (96%), as below.

For various reasons, including the factor of 100 more CO2 in the Venusian atmosphere, pressure broadening, and the presence of other absorbers, this underestimates the magnitude of the effect for Venus.

The cognoscenti (OK, maybe not the *real* cognoscenti, but people like me) might ask: "What's that hump in the middle of the CO2 absorption band in the last picture."

Well, it's warmer in the stratosphere than in the upper troposphere, and when there is this much CO2, even the stratosphere is optically thick.