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.
Hello Willis
Interesting comments again.
You will appreciate how, in control engineering, we usually work with changes from an underlying condition. The relevant parameters relate to the slope of system changes, and not so much the abolute slope from some null state.
I often see discussion about climate sensitivity from the absolute state (of no CO2), and that leads me to the same question. It was one of the first things that crossed my mind as I read your post.
Is is worthwhile looking at how your analysis would change if you were to focus on the incremental response that an assumed forced change of CO2 emissions might have on the system. As you say – not so much the instantaneous dynamic response, but the long-term equilibrium change.
I suspect all of the changes will be concerned with the “slopes” in the inside portion of your diagram. It would also tease open the assertion of amplification by feedback – that’s something which greatly troubles me for an assumed passive system.
shouldn’t you be starting with a non-atmospheric model where Solar =L to get a blackbody temperature (Which based on the Lunar Regolith is around 238K) and then applying the other terms to the difference in temp as T/S and Delta-H? or comparing between a non-atmospheric and atmospheric?
The only other thing I can think of is DeltaH acts as a damper
John Aiken says:
January 3, 2011 at 10:58 pm
John, thanks for the chance to discuss this question, as people bring it up a lot. What you say would be large enough to be relevant if the Earth were small. But the earth is huge. The circumference is 6,380 km. Let’s say we’re up 10 km in the air. The depression of the horizon is about 3.2°, and the distance to the horizon is about 360 km.
So the horizon depression isn’t large, a few degrees. And there’s another factor at work. This is that a photon of IR starting for the horizon is not likely to make it. Compared to a photon going vertically upward, it is much more likely to be absorbed.
So theoretically a photon could take a path slightly below horizontal, graze the earth, and escape. And theoretically that would mean more photons escaping. But in the real world, there’s a huge difference between the length of a path from 10 km straight up to outer space (a few dozen km upwards with decreasing greenhouse gases) and a grazing path below the horizontal to space (six hundred km to get to the horizon and out the other side through increasing and then decreasing GHGs).
As a result, the theoretical difference doesn’t mean a lot. The effect is real, but very small. As a result, it is usually ignored in analyses of this type.
w.
As far as I can tell, everything works out from trivial algebra up to the point CO2 is mentioned.
I cannot see the logical connection between all the preceding info which looks correct, and the assumption inserted at this point that CO2 is relevant in any way. I’m not saying CO2 isn’t relevant, just there is a logical disconnect at this point.
So if there is a problem in this analysis, I think it has to lie in the sentence:
“And in more familiar units, this inequality is saying that the sensitivity must be greater than 3° per doubling of CO2”
which should read
“And in more familiar units, this inequality is saying that the sensitivity must be greater than 3° per change in unknown units of whatever is driving the so-called greenhouse effect”.
JohnH gets it right.
What about NIGHT? That’s the half of the planet that is ONLY radiating. No matter how complicated you want to make the sun side, the night side is ALL about radiating enough energy to maintain the temperature that is overwhelmingly defined by the overall gas mix and pressure.
If it gets too warm on the sun side, extra energy is radiated out even faster from the night side. End of story.
JohnH says:
January 4, 2011 at 1:17 am
All of the models take that into effect, even the simplest ones.
w.
A small question from an ignorant layman… If according to the figure Energy in = Energy out, then there should be no warming at all? I guess I am misinterpreting the figure. It would make sense if Energy out = Energy in – Energy absorbed = System loss (L)?
There are so few CO2 molecules in Earth’s atmosphere that they could be far warmer than they are,adding heat to the atmosphere, and it wouldn’t make a significant change in atmospheric temperatures. The sun giveth (or not), and IR radiation into outer space taketh away.
Molecules of CO2 and other air molecules are NOT “black bodies”, perfect absorbers and perfect radiators of IR. It is long since the time to be treating them as such in “calculations.
Q=T/S is just a 0 dimensional model (not even 1 dimensional) .
It makes some very limited sense for a homogeneous , isothermal body in radiative equilibrium .
There is no energy transport in this “model” per definition , the temperature gradients are zero everywhere .
The real Earth is not a homogeneous isothermal body in radiative equilibrium and the 0 dimensional “model” has no relevance for its dynamics .
You just trivially found out that Q=T/S with S and T constant is wrong and can’t even begin to describe what happens when the spatially highly non homogeneous temperature and radiation fields vary .
For the real Earth S is ill defined and if it is considered as a statistical parameter , then it depends on the chosen time scale and is highly variable with other dynamical parameters (ice cover , cloudiness , albedo , ocean currents etc) .
The usual handwaving about time averages where every inconvenient (but very real !) dynamical process goes away and cancels is just the well known “and here is where the miracle happens” .
oldseadog says:
January 4, 2011 at 1:38 am
As an old seadog myself, I like you prefer observations to theory. In this case, however, their entire claim (that if forcing changes, temperature must perforce change) rests on this theoretical equation above. I think it’s not true, but that’s hard to prove, since we don’t have a model planet so we can’t run laboratory experiments.
So I continue working to show that equation, Q = T/S + ∆H, can’t be true. This is my latest finding in that regard. It’s an odd one, but a good one, since it is both a testable and a mathematically derivable result of their theory.
And as a seadog old enough to have done my share of mathematical calculations to derive my position by sextant, that’s the kind of thing I like.
This whole article is negated by the fact that back radiation CANNOT cause warming of the earths surface. It is impossible under the laws of thermodynamics that a greenhouse effect could work in this way. Therefore there is no need to look at feedbacks, positive or negative as there is nothing for the feedback to feed upon.
read these sites for more information.
http://www.ilovemycarbondioxide.com/
http://claesjohnson.blogspot.com/
chris smith says:
January 4, 2011 at 1:40 am
Their equation relates forcing, temperature, sensitivity, and change in heat content. It says nothing about the source of the forcing, it could be a sudden change in solar. The equation is general.
Willis, you lost me early on. My business school math isn’t enuff to keep up with you. Rather than try to figure it out I watched this video to cheer myself up. Just 2 minutes long, safe for work.
Willis,
I don’t understand why your reply at 12:58 is an answer to gnarf at 11:44. He said – haven’t you calculated the sensitivity to doubling all GHG’s not just CO2?
I see no flaw in your reasoning, so if the “canonical equation” is correct, the obvious conclusion has to be that the expression used for calculating the change of radiative forcing with the change of CO2 concentration is not valid in the real world.
AndrewG says:
January 4, 2011 at 1:48 am (Edit)
The terms “Q’ and “T”, the forcing and the temperature, are absolute quantities. They are not a difference from anything. So there is no need to start at a blackbody.
Next, blackbody temperature for the earth if we assume no clouds is about 6° C. With clouds, it is about -20°C. Take your pick.
Finally, there’s a reason the average temperature of the moon is colder than blackbody. It is a mathematical oddity that has to do with radiation occurring at a rate of T^4, and the slow rotation of the moon. And there is a good, if longwinded, explanation of why that is at ScienceOfDoom’s site.
w.
I’m no mathematician, but given that the GHG properties of CO2 is one of diminishing returns, and that the climate ‘scientists’ claim that the 100 ppm increase in CO2 concentration over the previous 100 years caused the observed 0.7 °C temperature increase, how can they say that the next 100 ppm (2 ppm per year over the next 50 years) will result in a much larger increase of 2 – 4 °C?
Is the IPCC’s “sensitivity factor” really so dominant? Is that why the computer models are so pessimistic, or is it just more pseudo-science?
jimmi says:
January 4, 2011 at 2:39 am
I have calculated the sensitivity in K (W m-2) -1. You can also express it in degrees per doubling of CO2. I have simply calculated the minimum and probably climate sensitivities to any kind of forcing. CO2 forcing, GHG forcing, solar forcing, it’s all the same to the equation shown in Figure 1.
What I have not done is calculate the forcing for either a doubling of all GHGs, or for a doubling of CO2. Those questions are outside the scope of this discussion. I have used the IPCCs numbers for the latter, and I have not discussed the doubling of GHGs at all. It is not relevant to this discussion, which is about any type of forcing.
Looks like the equation is based on sensible heat flows only. It has no way to take into account the latent heat effects that are 2 to 3 orders of magnitude larger (for water/water vapor).
The latent heat of the water vapor need never be released. If there is cooling the release of this energy prevents the temperature from dropping further. It provides a massive heat storage potential (flywheel effect) at the atmosphere’s gas/liquid interface. It occurs with very small temperature changes in either direction and resists any temperature change.
In this case the important variable is the change in mass of water vapor in the low atmosphere, not the change in temperature.
It is also consistent with the outgassing theories for CO2.
I think the explanation for the value of S you derive is simply a misunderstandind: T sensitivity is not usually understood for absolute value – where a linear relation would make no sense anyway. It is defined for (more or less) small perturbation. The idea is to define delta T = S (delta Q – delta H). If you use those relations, your analysis fall appart, imho.
Not that it would make CAGW any more believable, but the basic E balance presented by its supporters really makes sense. It is the feedbacks and also the assumptions relative to the non radiative heat exchanges and thickness of the zone where they take places that are really fragile imho…
Willis Eschenbach says:
January 4, 2011 at 1:03 am
I’d need a citation for that claim, crosspatch. My understanding is that at the temperature of the sun, 5K°C, very little of the radiation is in the IR. Always willing to learn, though.
About half is in the IR region, almost entirely SW IR.
http://en.wikipedia.org/wiki/File:Solar_Spectrum.png also shows SW IR absorption by water vapour, with a little by CO2. This accounts for much of the “absorbed by atmosphere” quantity on the Kiehl & Trenberth budget diagram.
I’ve been through the K&T Radiation Budget papers with a sceptical “fine toothcomb”, and was initially highly sceptical of the magnitude of the downgoing LW IR (~324 W/m²), and the apparent omission of upgoing radiation of equal magnitude. However there have been several experiments to measure the downgoing IR, and I’m satisfied on that score – they find fluxes of 300+ W/m².
The subject of upgoing vs downgoing radiation is a little more complicated, however. Logically, one would expect half of the total GHG radiation to be upwards, and half downwards, yet K&T apparently shows no such thing, with all the emission downwards. K&T have been much criticised in online blogs on this point, and I joined in that criticism. However, the radiation experiments appear to confirm their figure, with the outer envelope of the radiation spectrum following (though a little below) the relevant Planck curve. (That “little below” would appear to reinforce another criticism of K&T, that GHGs do not behave as a perfect “black body”). The spectrum is clearly an emission spectrum, showing the bands from GHGs. However,the critical point is the corresponding spectrum from 20km looking down – this is NOT an emission spectrum, but an absorption spectrum,with GHG absorption bands and the “atmospheric window” all showing up clearly. Most examples of these spectra are contained in papers with limited or no public access, but an illustration can be found at http://mensch.org/5223/IRspectra.pdf
I can offer no simple explanation for this apparent lack of “equal and upward” GHG radiation, but I’m forced to accept it – the evidence is clear. My apologies to K&T on THIS point, at least.
Willis: Aren’t you assuming that the forcing from Total Solar Irradiance (which includes Downward Shortwave Radiation) is the same as the forcing from the Longwave Radiation from greenhouse gases? This may hold true for land but I don’t believe it’s the same for the oceans. The oceans have their own greenhouse effect because DSR penetrates to depth while the oceans can only release heat at the surface.
Assume that the global temperature would be like in 1850 and the CO2 content of 280ppm. Then 280-400ppm barely gave 0.7 C and 400-650ppm (David Archers Modtranprogram) would give 0.7 C if it were not for T4 (energy radiation increases with T ^ 4) … so an increase to 650ppm would result in considerably less than 0.7 C.. Consider that two equally surfaces with 50C and 80C-radiates as if both were not -15C but 10C, average temperature is for idiots … Increased temperature differences with the same average temperature gives greatly increased cooling by radiated energy.
Therefore, the earth probably emanated roughly the same when the Sahara was forests and at least the North Pole 5-6C warmer ..
Sahara was warmer at night and colder during the day, radiating less energy with the same average temperature.
Furthermore, it appears not all atmospheric physicists bring water extremely strong thermostat, when it is available in liquid and gaseous form.
Therefore, the Earth never had a warmer than the maximum life-friendly environment.
When sun has expand all the water will be gasified cease the thermostat, in G years, man must have moved from Tellus long ago.
All about the scientific redacted CO2-threat was only the question of energy system when 4times global population have our buying power, nuclear is all reedy the cheapest, so what´s the problem?
Some wrong over; +50C and -80C radiates like both where +10C not -15C.
T4 is one of the strongest negative forces and whit water in liquid and gas form it will keep earth from warming catastrophe, but not cooling, because when water frees, the thermostat stops working, and albedo even cool it more.
Sorry for my bad English and hasty writing…. ;o)
//gunnar
Dear Willis,
I have enjoyed reading all your previous interesting posts but this one is a complete mystery. Your equations bear no resemblence to the standard formalism, for example in the reference Kiehl07. First of all, the standard equations refer to small deviations from an initial balance between ingoing and outgoing energy flux. The ‘forcing’ delta Q is a change in the balance of these fluxes which must be reestablished by a change in temperature. Only then can the system response be assumed to be linear in the temperature (change). Your loss term L confuses the picture and should not be there. The system response deltaT/S should include both the direct radiation into space from the surface (L) and the indirect radiation from molecules in higher layers of the atmosphere. Otherwise S does not correspond to the normal definition of climate sensitivity.