A Third Look at Radiation versus Temperature

Guest post by KEVIN KILTY

Introduction

In a post on June 8, 2019, Willis Eschenbach showed an interesting plot of monthly average surface temperature against total irradiance absorbed at the ground surface. He has updated this original post. His original figure is at the top of this post. He made no hypothesis about the meaning of the plot. In the thread which followed no one suggested a reason for the relationship. However, it did lead a number of commenters to suggest that it demonstrated a doubling of atmospheric CO₂ would produce only 0.38^°C to 0.5^°C of surface warming. I am going to provide a tentative explanation for the relationship. My tools will be some simple physics such as Newton‘s law of cooling or the Stefan-Boltzmann law, known to many who post here, and a block model of heat transfer at Earth’s surface.

1. A Word on Equilibrium

Energy balance is my primary analytical tool in this post. I will use a block model of Earth’s surface known in engineering as a control volume. In this case my control volume is a thin layer at the surface known as the skin. There is material above and below this layer, and so to do a complete analysis we will have to account for energy transfer between this skin and materials above and below. Often, to simplify such an analysis we resort to postulating equilibrium. However, at Earth’s surface we observe: evaporation, rainfall, temperatures that rise and fall, ice and snow melt and so forth. Each of these observations imply or demand non-equilibrium. Thus, the Earth is practically never in equilibrium. However, over a suitably long time period Earth keeps passing through nearly the same state, and we speak of it being in steady state. If we use average values over a suitable time period, our steady state quantities obey a sort of equilibrium even though we are out of equilibrium at any instant. Willis averaged data over each month to produce the graph.

2. Our Block Model

Figure 1. Control volume and heat flows for the present model.

Figure 1 shows the block model with all energy flows. Arrows indicate the typical direction of transport. The control volume is Earth’s skin characterized by a temperature T. Surface absorbed energy is a combination of solar irradiance plus longwave irradiance from the atmospheric greenhouse effect. It is total irradiance. I will use the symbol (I_a) to represent it. The other heat transfer mechanisms are:

(1) Emitted power is blackbody radiation from the surface which we can conveniently calculate because we know that materials at the skin surface are nearly black at infrared wavelengths.

(2) Heat transfer to the substrate. Over the oceans this will occur through bulk mixing or convective transfer. Over land it will occur primarily through thermal conduction. It goes back and forth.

(3) Heat transfer to the overlying air. This occurs as convective transfer through a film. Engineers would model it using Newton’s law of cooling. I think it is primarily away from the skin although there are times and places it goes the other way.

(4) Heat transfer to the overlying air through evaporation, both as sensible and latent heat, and through precipitation which lands on the skin with a different temperature. This is just about exclusively outward.

One might also consider heat stored within the skin. However, on the sea surface the skin is something like 10 micrometers thick, and in soil it is about 5 centimeters thick. This represents very little thermal capacity to consider.

3. Willis’s Data

The data came from the CERES compilation. A number of posts on the original thread were wondering if the temperature involved is at the Earth’s surface or 1.5m above it. The CERES surface temperatures are a skin value, derived from GEOS data. The method of producing these temperatures is complex, and involves both parameterization and maximum likelihood inverse methods. Comparison against independent temperature data, satellite and ground, suggests a precision of about 0.5^°C. The amount of analysis and scientific work that goes into these efforts is well described in a series of technical documents, available online, that are well worth reading for a background in the subject.

What is pertinent about Willis’s plot are two things. First, the slope of the regression line, which is 0.38^°C per 3.77W/m² absorbed energy. Second, one will notice that the range of temperature is small, which suggests that linearizing any non-linear relationships involving temperature is appropriate.

4. Analysis

I collect all of the heat transfer mechanisms 2-4 in my list and call them Q. Willis would likely refer to them as parasitics. They reduce the temperature variation that would occur in their absence. I treat them as a lump because I can’t calculate them from first principles, and I don’t need to for what I plan to do. The energy balance for my control volume is:

(1) I_a = σT⁴ + Q

Variations in temperature are very small. As a result I will linearize Q and write it as h·(T −T_s) where h is an over all transfer coefficient, and T_s is temperature of the environment surrounding the skin. The surroundings might be at different temperatures in different directions, but what is important is that we are just transferring through a small temperature gradient and transfer is proportional to rising T. The slope of Willis’s graph is:

(2)

To compare how the model explains this slope we need a partial derivative with respect to I_a from our model. Unfortunately our model has I_a as a function of T. However, the relations are monotonic in T and so we can just invert a partial derivative of I_a with respect toT to obtain what we need. From our model:

(3)

(4) = 4σT³ + h

At 288K, the term from blackbody radiation is about 5.4W/m²K, and the coefficient h is unknown, but available from a comparison between Willis’s slope and ours. This amounts to:

(5) 0.1 = 1/(5.4 + h)

This is an equality if h is about 4.6W/m²K. What this says in plain terms is that Willis’s graph implies that added irradiance gets divided between emitted power from the surface and heat transfer by other mechanisms in about equal amounts. Seems reasonable.

What about climate sensitivity? The effect of CO₂ is in the down welling longwave (LW) radiation which is implicit in the total irradiance that Willis uses. There is no way, short of introducing new models, assumptions, and measurements to separate the effect and find climate sensitivity.

5. Conclusion

Our discussion demonstrates that the relationship between skin temperature and absorbed radiation from Willis’s post has no direct bearing on climate sensitivity. Instead it shows something very interesting about the relative magnitude of various heat transfer mechanisms which must operate at the skin of our planet. It implies that mechanisms I collectively called Q, are roughly as important to maintaining a skin temperature as is blackbody radiation. There is no way to take Willis’s diagram and determine how much warming a doubling of CO₂ will produce.

Many people who post comments here seem fond of Richard Feynman. I have a quote of his that is not only appropriate to the topic at hand, but is appropriate to all analyses, scientific in particular. In his series The Feynman Lectures on Physics he devoted one chapter to the phenomena of para- and diamagnetism. Here he says

If you start a [classical] argument in a certain place and don’t go far enough, you can get any answer you want.

I have placed brackets around the word classical to make his quotation apply more broadly than the context he was using it in. People understandably thought the slope pertained to climate sensitivity but just didn’t take their argument far enough.

6. Notes:

One reference I found useful for explaining the recovery of sea surface temperature is: GOES-R Advanced Baseline Imager (ABI) Algorithm Theoretical Basis Document for Sea Surface Temperature, Alexander Ignatov, NOAA/NESDIS/STAR. ( https://www.goes-r.gov/products/ATBDs/baseline/baseline-SST-v2.0.pdf)

One can find the Feynman quotation in The Feynman Lectures on Physics, Volume II, Chapter 34, Section 6, first paragraph.