Temperature, heat flow, opacity: Deeper analogies

Earlier, we looked at a fluid flow analogy for radiative heat transport. Changing the radiative opacity was found to be analogous to changing the cross-sectional area of a pipe. If the area is lowered and the amount and flow rate of the fluid are unchanged, the fluid pressure has to increase. It's like high blood pressure in mammals: if the amount of blood and heart rate don't change, then narrowing of arteries leads to higher pressure.

There's an electrical analogy, too, also mentioned briefly. If temperature is like voltage (V) and heat flow like current (I), then opacity is like resistance (R). The analogue to radiative transport in that case is Ohm's law: V = IR. For a given I, raising R means raising V. Likewise, for a given flow of radiative energy in (and out), raising the opacity means raising the temperature.*

The electrical case immediately leads to an interesting conclusion. The power dissipated as heat by the current flowing through a resistance is P = VI = I²R. So more power is dissipated, for a fixed current I, linearly as R is increased. If the electrical circuit has a definite temperature T, we can also conclude that the rate of entropy generation by this conversion of electrical energy to heat is P/T = I²R/T. This rate also increases linearly with R.

The analogy is not perfect. In the radiative case, it's all energy of radiation - what's flowing and what's being dissipated are the same, and "voltage" is temperature now. Instead of conversion of one form of energy to another, radiative energy at higher frequency (shorter wavelength) is "degraded" by being subdivided into more and more smaller pieces, each piece with a lower frequency (longer wavelength).** The radiative energy becomes more randomized, and the entropy of the radiation field increases.

If the opacity is increased, the heat energy flow doesn't change, and the heat isn't "redirected" or "trapped." The atmosphere's "resistance to flow" goes up, and so does the "voltage" as well.

There is an analogous, but more complex, formula for the rate of entropy generation. It's best expressed in terms of optical depth, rather than opacity: optical depth = opacity times length.† We still have a current-voltage-resistance-type relation: radiative heat flow ~ T⁴/(optical depth). But the entropy generation rate is

   (optical depth)*(radiative heat flow)2    (radiative heat flow)3/4
 ~ -------------------------------------   ~ --------------------     ,
                 T5                           (optical depth)1/4

which just goes to show how funky radiation is compared to matter.

And note the curious implication: when opacity and optical depth increase and the radiative heat flow remains constant, the rate of entropy generation decreases! The power dissipated is just (radiative heat flow) itself.
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* It's more subtle than it seems at first. The "current" (heat flow) in the atmosphere is set by the effective "atmosphere top" temperature, which is a function of incoming solar radiation and cloud albedo alone. Increasing "resistance" (opacity) increases the "voltage" (temperature) difference between the surface and the atmosphere top. But since the latter is fixed, the surface temperature is what rises.

** The "pieces" of radiation are just photons; their number is not conserved. The energy of a photon is Planck's constant times its frequency. Because energy is conserved, a given amount of radiative energy in one energetic photon can be converted into many more but less energetic photons, by multiple absorptions and re-emissions in matter.

Entropy is increased by reorganizing the invariant radiative energy flow into smaller bits - a larger number of lower energy photons still headed in the same direction.

† If opacity is mass-specific (as it usually is), then a factor of density is needed as well. The formula given is actually an integral over the depth.