Atmospheric Mixing

Our onetime blogging mentor and occasional critic Luboš Motl is doing some greenhouse warming calculations. This is good, since Lumo is a much cleverer fellow than the AGW critics that he likes to listen to, and he can hardly help but learn some important things by doing the math himself. In particular, he has now calculated the logarithmic dependence of the temperature change on the CO2 concentration. This is all to the good, as I said, but I would like to focus on one little mistake he made, not because it’s very important, but because it’s symptomatic of the flaws in his reasoning about AGW in general – he simply hasn’t mastered the relevant facts. Another reason I bring it up is that we clashed before on the same subject, and at that point I didn’t bother to understand exactly why he was wrong, and didn't follow up on his mistake.

What he does is assume that he can calculate the distribution with height of CO2 molecules based on the Boltzmann distribution for the CO2 molecules alone. This leads to a biased result, since CO2 molecules are among the heaviest common constituents of the atmosphere, and the atmosphere is in fact well mixed up to about 100 km (the so-called turbopause). The underlying assumptions leading to his result are two: that the atmosphere is a perfect gas, and that it is in thermal equilibrium. Both assumptions are slightly false, and the two conspire to make the atmosphere well-mixed for much of its depth.

If molecular diffusion were the only process occurring, the atmosphere would indeed assort by height, and the heaviest molecules (like CO2) would concentrate more at the bottom than the lightest (H2 and He) would. Really heavy stuff, like dust particles and sand grains wouldn’t get off the ground at all. In fact, the atmosphere is constantly stirred by turbulence, and when the turbulence is vigorous, dust and even sand get lofted.

In a truly perfect gas, molecules don’t notice each other at all. Each species passes through its fellows and other molecules completely unhindered. In the surface level atmosphere, though, the mean free path for an air molecule is about 1 micrometer = 10^-6 meters. This means that two molecules that start out together take along time to wander apart by diffusion. Meanwhile, winds and turbulence, large scale collective motions, sweep whole large chunks of atmosphere from one point to another. Turbulent stirring of the atmosphere, which doesn’t care about individual molecular masses (only the mass of the parcel of which it is a part) dominates until reduced density makes the mean free path of molecules longer than the mixing length (the average distance an air parcel travels while maintaining its identity). This doesn’t happen for Earth until you get to about 100 kilometers up. Above this turbopause, diffusion takes over, and molecules tend to stratify in the heterosphere.