Thursday, September 15, 2016

Thermodynamics of climate feedback

I have been describing and responding to blog arguments about climate feedback and circuit analogies here and here. The arguments have continued, and they do provoke ideas. I'm going to write some down in this post.

The usual circuit analogy has surface temperature as voltage, and TOA flux as current. I showed in the first post that the feedback, including Planck, could be regarded as conductances. It's interesting to probe what this might mean. The units are watts/m²/K, which are actually the units of entropy/s/m². Does entropy make sense?

I wrote about entropy and atmospheric fluxes here and here. Sunlight (Q=240 W/m² global average after albedo) arrives, does things, loses the capacity to do work, and eventually leaves as thermal IR. It has accumulated entropy, or if you prefer, lost free energy. You might think that with a heat sink at 3K (space) the heat could go on doing work. But in fact you need a miniumum temperature to radiate that flux to space, which for Earth is about 255K (note 1). That constitutes a resistance. To get Q=240 W/m² to flow to space, you need 255K (voltage).

In our system, that resistance, inverted, is the Planck conductance, or feedback. It represents the entropy flux to space. It's really the maximum or optimal entropy flux for 240 W/m². In fact, emission to space comes from a rather large component at about 225 W/m², from GHGs, and some from the surface at average 288K (atmospheric window). We know uniform blackbody emission exports most entropy for a given flux, because any variation means that more entropy could be generated by transporting heat from the hotter parts to cooler.

This lies behind a supposed failing proclaimed in a WUWT post of Lord Monckton. The Planck feedback calculated for the Earth at 255K, the temperature for uniform BB emission of 240 W/m², is 3.75 W/m²/K, and Lord M thinks they erred by not using it. But its inadequacy has been long known, and I wrote in the previous post how Soden and Held (among others) did a thorough study with GCMs to get a value of about 3.2 W/m²/K. The difference is usually attributed to absorption in the atmosphere, but thermo gives an alternative viewpoint, which I find more useful. It is the entropy export reduced by the non-uniformity (sub-optimality) of apparent emission temperature.

So why is better entropy transport a negative feedback? If more entropy can be removed, then more can (and must) be generated internally. That means larger temperature drops as the Q=240 W/m²/K passes through. In my first post, I noted that the big entropy creation was when sunlight was initially thermalised, mostly at the surface, and it was Q(1/Tₛ - 1/Tₛun) where Tₛun is the related to apparent temperature of sunlight (see post), and Tₛ is surface temperature. So if more entropy can be generated, Tₛ will be lower.

Water vapor

So can we develop this for a positive feedback like water vapor? Yes! Water vapor creates entropy associated with Q. It absorbs IR from a higher temperature, and re-emits at a lower. This is actually a general explanation for the GHE which I will develop in a future post. Creating entropy where the eventual export is capped means that less entropy can be created upstream. That means, by the same logic, that the surface temperature must be higher, so that less entropy is generated there. In the analogy, that appears as a negative conductivity in the feedback path.

Surface albedo

What about surface albedo feedback? That is a bit different, because it's fairly easily understood in any system of interpretation. Reducing albedo increases Q - more heat flowing, so naturally surface temperature increases, no? But what about entropy? By S-B, the optimal exit temperature (255K) must increase. Entropy export increases with Q, but reduces with the higher temperature (net effect increase, but less than proportional). That means at the surface, more entropy can be generated, but again less than proportional to Q. So yes, surface temperature must rise.


Feedbacks have the units of entropy transport (conductance) and they add. Improving conductance (facilitating entropy flux) associated with Tₛ allows more entropy production at the surface, which means a greater temperature drop. So it is a negative feedback on temperature.


  1. Be careful about equating entropy with other measures just because they have the same units. Heat capacity has the same units as entropy even though they are considered different things (although they are of course related).

    Where I think entropy fits in is in understanding the lapse rate. I went a little batty trying to decipher why the Standard Atmosphere lapse rate comes out extremely close to g/(1.5cp). Where does the 1.5 come from? Some think this is the entropy effects of water vaporization and condensation and a kind of average "moistness" of the atmosphere. But then it also could be related to a polytropic atmosphere or application of the virial theorem. The weirdness is in the universality of this effect in different planetary atmospheres that obviously differ from Earth's. I have more of that analysis here that I did a few years ago:

    The usual problem with these thermodynamic problems is that one doesn't have many interacting parameters to fit against. It always boils down to a single number and there are lots of ways to arrive at a single number. That's why I started to look at other planets lapse rate, because that would give more degrees of freedom to work with.

    But I essentially gave up on this because I realized it will eventually drive you nuts and there really is no exit criteria to prove it one way or another.

    More interesting is to work on ENSO because the data is dazzling in its intricacy (many degrees of freedom here) and real progress can be made in mapping the results to known angular momentum variations in the earth's rotation rate.

    1. "Where I think entropy fits in is in understanding the lapse rate."

      I agree. I've written about it, specially here. A temperature gradient generates entropy just by being there, though it depends on the effective conductivity k, here mostly turbulent, but also radiative. The rate per volume is k ∇T·∇Τ Τ⁻². And if you're generating entropy in steady state, it has to be removed. That takes a heat pump. The up-down motion in the atmospheric with near-adiabatic compression works as a heat pump (if the lapse rate is less than DALR, and proportional to the difference). The finite energy available from the wind determines how near DALR it can get. Plus of course the effect of water vapor, which when condensing effectively increases cₚ in the DALR expression.

    2. Oh yea, that was an intense comment discussion with the usual AGW deniers trying to mess things up.

      Why again is the lapse rate almost exactly 2/3 of the adiabatic lapse rate (on other planets with convective atmospheres too)? In that thread, it was suggested that one part is the heat capacity term and 1/2 a part went into an entropy of mixing. Or consider 1/3 of the gravitational energy term goes into a convective kinetic energy, which is energy lost to the heat pump.

      If you can come up with the 1/3 factor from your rate expression, that would be interesting.

    3. WHUT,
      I've noticed too that the moist lapse rate is often said to be about 2/3 of dry, and I've wondered why. Moist LR really depends on where condensation occurs (and how much). But it often seems to come out at about 6 K/km.

      I see it as a contention between a finite wind energy supply, and a pump effect which is proportional to the difference from DALR, vs turbulent and radiative working against the temp gradient. But I can't presently see a reason why it should settle to specifically 2/3.

    4. Condensation of water vapor releases heat. So a rising air parcel will cool less as it rises if it contains moisture that condenses. This results in a lower lapse rate to maintain stability in a moist atmosphere.


  2. Nick, what about clouds? Since they inhibit outgoing IR and can strongly reflect incoming solar radiation, do they fit into this analogy? And of course, water vapor condensation to create cloud droplets releases latent heat, warming the air and increasing convection, and likewise cloud evaporation causes cooling of the air and subsidence. I'm thinking clouds may be as much or more important than water vapor in the atmosphere because of their multiple effects. Oceans and land/sea ice/snow are additional major complicating factors as I'm sure you are well aware and there are also winds and turbulence involved in transporting and dissipating energy. I'm not sure trying to simplify the science does a very good job of replicating what really happens in the extremely complex real world.

    1. Bryan,
      Nick, what about clouds?"
      Yes, you're right, I skipped that one. Too hard. And yes, the feedback may be very important. As I see it clouds, like GHG, add to entropy. They intercept IR that came from a warmer place, and convert to heat at a cooler, which is classic entropy increase. Then they re-emit it (thermal IR), but not all in the direction it was originally going.

      On the other hand, they increase albedo. This isn't really mainly an entropy effect; just reduces sunlight absorbed. Though there is an entropy issue in that they also do absorb sunlight that would have otherwise gone to the warmer surface below. That increases entropy generation.

      "trying to simplify the science" - well, I see it as another way to get a quantitative handle. It may turn out that in the end there are too many gaps for it to be a really good handle, but I think it is worth looking at.

    2. Thanks for your insight Nick. I guess I was a bit harsh on berating simplistic science. Science is all about simplifying things so we can better understand them. The trick is to find simplistic models that work. Sometimes they do have to be more complex to improve results. Maybe this is just my engineering background showing through.