Is the Beer-Lambert law partially named after the same guy for which we owe the Lambert-W function, the multi-valued function for which is satisifed ?

**Solution to Puzzle 2: **

The Beer-Lambert law is separable, isn’t it ? If so, it isn’t to hard to show the relationship between the intensity of light at the two positions and . I can do this, but it almost seemed to easy !

**Guess to Puzzle 4:**

I have very little training in physics, so this is more of a guess than a solution. Please correct me if I am wrong! My guess is that it has to do with the air density as altitude increases. Using the ideal gas law as a model, , one can see that the temperature is directly proportional to pressure and volume. Then as altitude increases, pressure and volume decrease, meaning temperature decreases. This wouldn’t explain, however, the increase in temperature as altitude increases in the stratosphere, though, would it ?

**Guess to Puzzle 5:**

My guess is that it has something to do with water vapor… the effect of gravity on water vapor. The idea is that the air above the tropopause has a different chemical makeup then the air below the tropopause, perhaps less water vapor ? One reason why I am thinking in this direction is that it seems clouds do not form above the tropopause.

Thanks Tim for your consideration.

The stumbling block to me is the evaporative cooling. We might be able to get the emission spectra correct but then can’t balance the latent heats between evaporation and condensation properly.

Maybe a thought experiment could apply. Find a visibly transparent glass material with the same radiative properties as water. Ask how much does this heat up if placed on the surface of a pool of water on a bright summer day.

]]>I said that when I was computing how much infrared radiation was emitted by a tiny parcel of air. The fact that some air *somewhere else* is much hotter or colder does not affect the amount of radiation emitted absorbed by this little parcel of air:

What about emission? Air doesn’t just absorb infrared light, it also emits significant amounts of it! As mentioned, a blackbody at temperature emits light with a monochromatic energy flux given by the

Planck distribution:But a gas like air is far from a blackbody, so we have to multiply this by a fudge factor. Luckily, thanks to Kirchoff’s law of radiation, this factor isn’t so fudgy: it’s just the absorption rate

Here are we generalizing Kirchoff’s law from a surface to a column of air, but that’s okay because we can treat a column as a stack of surfaces; letting these become very thin we arrive at a differential formulation of the law that applies to absorption and emission rates instead of absorptivity and emissivity. (If you’re very sharp, you’ll remember that Kirchoff’s law applies to thermal equilibrium, and wonder about that. Air in the atmosphere isn’t in perfect thermal equilibrium, but it’s close enough for what we’re doing here.)

So, when we take absorption

and also emissioninto account, Beer’s law gets another term:where is the temperature of our gas at the position

In short, the amount of radiation emitted at wavelengh and position is very close to

because this small parcel of air is, *taken by itself*, close to being in thermal equilibrium. It’s not *exactly* in thermal equilibrium, so this is just an approximation, but it’s close.

None of this is saying that the climate system as a whole is close to thermal equilibrium.

]]>You’re welcome, and thanks for these references! I was *hoping* that nonlinear systems would still display some form of reciprocity, since mixed partial derivatives are equal even for functions that aren’t quadratic. (In other words, all functions are quadratic to second order at a local minimum.) I’ll have to read your work—the great thing about self-citations is that you can be sure the author actually read those references.

If many forms of reciprocity boil down to equality of mixed partial derivatives, it would be nice to teach this to Wikipedia, and thus the world. Right now the articles there don’t emphasize a single unifying principle. The article on reciprocity in electromagnetism is fascinating, but it says all forms of reciprocity there boil down to the self-adjointness of a certain operator. That’s probably correct as far as it goes, but I’d like to show the self-adjointness of that operator follows from equality of mixed partial derivatives of the action, or something like that.

]]>This is an interesting question which has been puzzling me for quite a while now. The problem is that books about physical oceanography that I know don’t say anything about radiative transfer, but concentrate on fluid dynamics. And the books about radiative transfer for climate science concentrate on the atmosphere. So the question about radiative energy transfer in the oceans seems to fall in one of those dreary nasty black spots between established academic disciplines :-)

There is one book that could help us our here that I know about (unless there is an expert who would like to help us out, which would be most welcome), which is:

• Gary E. Thomas and Knut Stamnes, *Radiative Transfer in the Atmosphere and Ocean*, Cambridge Atmospheric and Space Science Series, Cambridge University Press, Cambridge, 1999.

It has a lot of interesting fromulas, but browsing through it, I did not find a simple rule-of-thumb explanation explaining the most important effects one has to understand.

]]>If engineering fails, call in a chemist. When something is dropped, he uses his foot not his hands. Never play catch-up.

]]>