An Unexpected Limit to Climate Sensitivity

Guest Post by Willis Eschenbach

[Update: I have found the problems in my calculations. The main one was I was measuring a different system than Kiehl et al. My thanks to all who wrote in, much appreciated.]

The IPCC puts the central value for the climate sensitivity at 3°C per doubling of CO2, with lower and upper limits of 2° and 4.5°.

I’ve been investigating the implications of the canonical climate equation illustrated in Figure 1. I find it much easier to understand an equation describing the real world if I can draw a picture of it, so I made Figure 1 below.

Be clear that Figure 1 is not representing my equation. It is representing the central climate equation of mainstream climate science (see e.g. Kiehl ). Let us accept, for the purpose of this discussion, that the canonical equation shown at the bottom left of Figure 1 is a true representation of the average system over some suitably long period of time. If it is true, then what can we deduce from it?

Figure 1. A diagram of the energy flowing through the climate system, as per the current climate paradigm. I is insolation, the incoming solar radiation, and it is equal to the outgoing energy. L, the system loss, is shown symbolically as lifting over the greenhouse gases and on to space. Q is the total downwelling radiation at the top of the atmosphere. It is composed of what is a constant (in a long-term sense) amount of solar energy I plus T/S, the amount of radiation coming from the sadly misnamed “greenhouse effect”. T ≈ 288 K, I ≈ 342 W m-2. Units of energy are watts per square metre (W m-2) or zetta-joules (10^21 joules) per year (ZJ yr-1). These two units are directly inter-convertible, with one watt per square metre of constant forcing = 16.13 ZJ per year.

In the process of looking into the implications this equation, I’ve discovered something interesting that bears on this question of sensitivity.

Let me reiterate something first. There are a host of losses and feedbacks that are not individually represented in Figure 1. Per the assumptions made by Kiehl and the other scientists he cites, these losses and feedbacks average out over time, and thus they are all subsumed into the “climate sensitivity” factor. That is the assumption made by the mainstream climate scientists for this situation. So please, no comments about how I’ve forgotten the biosphere or something. This is their equation, I haven’t forgotten those kind of things. I’m simply exploring the implications of their equation.

This equation is the basis of the oft-repeated claim that if the TOA energy goes out of balance, the only way to re-establish the balance is to change the temperature. And indeed, for the system described in Figure 1, that is the only way to re-establish the balance.

What I had never realized until I drew up Figure 1 was that L, the system loss, is equal to the incoming solar I minus T/S. And it took even longer to realize the significance of my find. Why is this relationship so important?

First, it’s important because  (I – Losses)/ I is the system efficiency E. Efficiency measures how much bang for the buck the greenhouse system is giving us. Figure 1 lets us relate efficiency and sensitivity as E = (T/I) / S, where T/I is a constant equal to 0.84. This means that as sensitivity increases, efficiency decreases proportionately. I had never realized they were related that way, that the efficiency E of the whole system varies as 0.84 / S, the sensitivity. I’m quite sure I don’t yet understand all the implications of that relationship.

And more to the point of this essay, what happens to the system loss L is important because the system loss can never be less than zero. As Bob Dylan said, “When you got nothin’, you got nothin’ to lose.”

And this leads to a crucial mathematical inequality. This is that T/S, temperature divided by sensitivity, can never be greater than the incoming solar I. When T/S equals I, the system is running with no losses at all, and you can’t do better than that. This is an important and, as far as I know, unremarked inequality:

I > T/S

or

Incoming Solar I (W m-2) > Temperature T (K) / Sensitivity S (K (W m-2)-1)

Rearranging terms, we see that

S > T/I

or

Sensitivity > Temperature / Incoming Solar

Now, here is the interesting part. We know the temperature T, 288 K. We know the incoming solar I, 342 W m-2. This means that to make Figure 1 system above physically possible on Earth, the climate sensitivity S must be greater than T/I = 288/342 = 0.84 degrees C temperature rise for each additional watt per square metre of forcing.

And in more familiar units, this inequality is saying that the sensitivity must be greater than 3° per doubling of CO2. This is a very curious result. This canonical climate science equation says that given Earth’s insolation I and surface temperature T, climate sensitivity could be more, but it cannot be less than three degrees C for a doubling of CO2 … but the IPCC gives the range as 2°C to 4.5°C for a doubling.

But wait, there’s more. Remember, I just calculated the minimum sensitivity (3°C per doubling of CO2). As such, it represents a system running at 100% efficiency (no losses at all). But we know that there are lots of losses in the whole natural system. For starters there is about 100 W m-2 lost to albedo reflection from clouds and the surface. Then there is the 40 W m-2 loss through the “atmospheric window”. Then there are the losses through sensible and latent heat, they total another 50 W m-2 net loss. Losses through absorption of incoming sunlight about 35 W m-2. That totals 225 W m-2 of losses. So we’re at an efficiency of E = (I – L) / I = (342-225)/342 = 33%. (This is not an atypical efficiency for a natural heat engine). Using the formula above that relates efficiency and sensitivity S = 0.84/E, if we reduce efficiency to one-third of its value, the sensitivity triples. That gives us 9°C as a reasonable climate sensitivity figure for the doubling of CO2. And that’s way out of the ballpark as far as other estimates go.

So that’s the puzzle, and I certainly don’t have the answer. As far as I can understand it, Figure 1 is an accurate representation of the canonical equation Q = T/S + ∆H. It leads to the mathematically demonstrable conclusion that given the amount of solar energy entering the system and the temperature attained by the system, the climate sensitivity must be greater than 3°C for a doubling of CO2, and is likely on the order of 9°C per doubling. This is far above the overwhelming majority of scientific studies and climate model results.

So, what’s wrong with this picture? Problems with the equation? It seems to be working fine, all necessary energy balances are satisfied, as is the canonical equation — Q does indeed equal T/S plus ∆H. It’s just that, because of this heretofore un-noticed inequality, it gives unreasonable results in the real world. Am I leaving something out? Problems with the diagram? If so, I don’t see them. What am I missing?

All answers gratefully considered. Once again, all other effects are assumed to equal out, please don’t say it’s plankton or volcanoes.

Best wishes for the New Year,

w.