Entropy and the greenhouse effect.

Efforts to explain the greenhouse effect are never-ending. Not that it's a hard concept, but the Second Law gets debated as well. In fact, entropy and its transport are very much part of the story.

Some will think that entropy is an abstract notion and can't clarify anything. But its something that can be tracked and budgetted, and that is a useful overlay on regular heat transfer talk. What I'm going to say here isn't any novel physics, and describes no unfamiliar mechanisms. But I found that it helped with my mental accounting.

A while ago I essayed an entropy budget for the Earth. We have a heat flux budget. Every time a heat flux Q arrives at or leaves a region, it carries a flux Q/T of entropy. Here T is absolute temperature at the transfer surface; if it isn't uniform, integration is needed. Entropy is created when sunlight is first absorbed by materials on the Earth, and leaves in the outgoing IR flux. Since it can't be destroyed, and isn't significantly accumulating, the IR must carry it all. That's a constraint worth investigating.

Entropy and heat conservation

We know that energy is conserved, and enters and leaves the Earth as electromagnetic radiation. After allowing for albedo reflection of sunlight, on average (time and space) about 238 W/m2 is absorbed. Since only a tiny fraction can accumulate without causing major warming, it is necessary that about the same flux is emitted as thermal IR. While the heat fluxes are the same, the entropy fluxes are not.

Entropy on entry

Entropy is tied up with free energy - the capacity to do work. Heat engines create entropy by transferring heat from hotter to cooler, increasing 1/T. You can reason backward and identify (not equate) entropy change with the work that could have been done by that heat transfer.

Entropy is created by many irreversible processes in the atmosphere. But the big one is the initial conversion of sunlight. In my earlier budget I attached an estimate of the entropy of the arriving flux. The sun is hot, but not infinitely so, and in principle more work could be extracted from 238 W/m2 arriving from an even hotter source. However, in fact the absorption process would be the same, so it might as well be said that the entropy of the arriving sunlight is zero.

An estimate of the entropy created is then 238/287 W/m2/K, where 287 is the average temperature of the earth. However, this is rough, and really it should be formed as an integral of arriving flux divided by arriving temperature. Note that warmer arrival means less entropy created - this will be important. It means the average estimate is not very good, since sunlight predominantly arrives during summer (and in daylight) - times of above average warmth.

Entropy on exit

That same 238 W/m2 leaves at much lower temperatures, on average. Much is emitted by GHG in the high troposphere, at temperatures about 225K. So it carries much more entropy than was created by the initial thermalisation. That is just as well, because all the happenings that make Earth interesting (eg life) create entropy, and that has to be removed.

Snowball Earth maximizes entropy flux

GHE discussions tend to start with a statement of the uniform temperature at which an Earth-sized spherical black (in IR) body would emit 238 W/m2. That temp is 255°K. This is about 33°K below the average Earth temp, and the difference is attributed to the greenhouse effect. The black body need not be in vacuo - and atmosphere of oxygen and nitrogen, which do not interact with IR, would not affect this calc. But greenhouse gases certainly do.

An interesting, rarely mentioned, significance of this "Snowball Earth", at uniform temperature is that it carries the maximum possible entropy that could be carried by a 238 W/m2 flux (proof appended). Maximum entropy removal. Any variation in space (latitude) or time (diurnal), or emissivity would detract from that.

But Murphy's Law intrudes - no interesting processes can create that entropy. The entropy creation at 255K is also a maximum, and they are equal.

Emerging budget constraints.

Snowball Earth already gives an indication of how an entropy budget affects global warming. Entropy export is bounded from above. Anything that reduces it must be balanced, and that must mean a reduction in entropy generation, ie surface warming. Also anything that generates entropy in other ways must cause warming at the point where sunlight is absorbed.

Inhomogeneous snowball

The very entry-level snowball has uniform irradiation etc which is very unrealistic. vatiation affects entropy transport. But there is one thing that can be said about a snowball Earth that has latitudinal variation, rotates, and even has seasons. None of these create substantial entropy (there is some local conduction, and with a nonb-GHG atmosphere, there may be more). So though export is reduced, export and import still pretty much balance.

There's an implication here if you're trying to maintain a cool Earth. While inhomogeneity on export hurts, on import it's your friend. You get less entropy at the same average temperature. Or at fixed entropy, can run at a lower average temperature.

Introduction of GHG's

GHG's do a number of things to impact the entropy budget. They raise the point of emission, which combines with the lapse rate to make emission from a cooler location. That helps with entropy export per unit flux, but the cold reduces the flux. That is made up from flux in atmospheric window frequencies, direct from the warm surface.

The nett result is a negative for entropy export. Remember, the optimum was uniform temperature, and anything that diverts from that, spatially, temporally, or in this case, spectrally, diminishes total entropy export. And the nett result is less entropy at thermalisation - ie warming.

But GHG's have another effect. A large part of the heat flux must now be transported from the surface to the emission point. In the IR-absorbing wavelength bands, that happens by repeated emission and absorption. Photons are more likely to be absorbed by gas cooler than their emission point - that's how the transport occurs. Each such pairing creates entropy. And again, this has to be compensated by surface warming.

In fact, this can be quantified. At frequencies where the optical depth is substantially greater than 1, the transport process follows a Fourier law - flux proportional to temperature gradient. Then entropy production per unit volume follows the rule:

E_V = (k/T²) ∇T • ∇T

where k is the effective conductivity.

Because entropy export is capped, that entropy production too has to be compensated by reduced entropy creation - ie warmer thermalisation.

Summary

It helps to add entropy accounting to the study of heat transfer underlying the Greenhouse Effect. Entropy export is constrained, and so any effects that limit it further or create extra entropy in other ways require a reduction in the main entropy generation, which is thernalisation of insolation. That means the thermalisation must take place at a higher temperature - ie warming.

Appendix - uniform temperature for black sphere is maximum entropy exporter.

Here's a proof re instantaneous spatial distribution using Lagrange multipliers. Suppose the world is divided up into areas A_i, each at temperature T_i. Then total flux is fixed at
F=ΣσA_iT_i⁴.

The rate of entropy export is
E=ΣσA_iT_i³.

The Lagrange formulation maximises
ΣσA_iT_i³+λΣσA_iT_i⁴-F)
over T_i and λ (the Lagrange multiplier)

This gives
3σA_iT_i²+4λσA_iT_i³=0

or T_i=-3/4λ for all i
ie uniform T.