By Christopher Monckton of Brenchley
In a previous post, I explained that many of the climate-extremists’ commonest arguments are instances of logical fallacies codified by Aristotle in his Sophistical Refutations 2300 years ago. Not the least of these is the argumentum ad populum, the consensus or head-count fallacy.
The fallacy of reliance upon consensus, particularly when combined with the argumentum ad verecundiam, the fallacy of appealing to the authority or reputation of presumed experts, is more likely than any other to mislead those who have not been Classically trained in mathematical or in formal logic.
To the Classicist, an argument founded upon any of the Aristotelian logical fallacies is defective a priori. Nothing more need be said about it. However, few these days are Classicists. Accordingly, in this post I propose to explain mathematically why there can be no legitimate consensus about the answer to the central scientific question in the climate debate: how much warming will occur by 2100 as a result of our sins of emission?
There can be no consensus because all of the key parameters in the fundamental equation of climate sensitivity are unknown and unknowable. Not one can be directly measured, indirectly inferred, or determined by any theoretical method to a precision sufficient to give us a reliable answer.
The fundamental equation of climate sensitivity determines how much global warming may be expected to occur once the climate has settled back to a presumed pre-existing state of equilibrium after we have perturbed it by doubling the atmospheric concentration of CO2. The simplifying assumption that temperature feedbacks are linear introduces little error, so I shall adopt it. For clarity, I have colored the equation’s principal terms:
Climate sensitivity at CO2 doubling (blue) equals the product of the CO2 forcing (green), the Planck parameter (purple) and the feedback gain factor (red).
The term in green, ΔF2x, is the “radiative forcing” that the IPCC expects to occur in response to a doubling of the concentration of CO2 in the air. Measurement and modeling have established that the relation between a change in CO2 concentration and a corresponding change in the net down-minus-up flux of radiation at the top of the climatically-active region of the atmosphere (the tropopause) is approximately logarithmic. In other words, each additional molecule of CO2 exerts less influence on the net radiative flux, and hence on global temperature, than its predecessors. The returns diminish.
To determine the radiative forcing in response to a CO2 doubling, one multiplies the natural logarithm of 2 by an unknown coefficient. The IPCC’s first and second Assessment Reports set it at 6.3, but the third and fourth reduced it by a hefty 15% to 5.35. The CO2 forcing is now thought to be 5.35 ln 2 = 3.708 Watts per square meter. This value was obtained by inter-comparison between three models: but models cannot reliably determine it. Both of the IPCC’s values for the vital coefficient are guesses.
The term in purple, 
, denominated in Kelvin per Watt per square meter of direct forcing, is the Planck or zero-feedback climate-sensitivity parameter. This is one of the most important quantities in the equation, because both the direct pre-feedback warming and separately the feedback gain factor depend upon it. Yet the literature on it is thin. Recent observations have indicated that the IPCC’s value is a large exaggeration.
The Planck parameter is – in theory – the first differential of the fundamental equation of radiative transfer about 3-5 miles above us, where incoming and outgoing fluxes of radiation are equal by definition. The measured radiative flux is 238 Watts per square meter. The radiative-transfer equation then gives us the theoretical mean atmospheric temperature of 255 Kelvin at that altitude, and its first differential is 255 / (4 x 238), or 0.267 Kelvin per Watt per square meter. This value is increased by a sixth to 0.313 because global temperatures are not uniformly distributed. However, it is also guesswork, and the current Lunar Diviner mission suggests it is a considerable overestimate.
Theory predicts that the Moon’s mean surface temperature should be around 270 Kelvin. However, Diviner has now found the mean lunar equatorial temperature to be 206 K, implying that mean lunar surface temperature is little more than 192 K. If so, the theoretical value of 270 K, and thus the lunar Planck parameter, is a 40% exaggeration.
If the terrestrial Planck parameter were similarly exaggerated, even if all other parameters were held constant the climate sensitivity would – on this ground alone – have to be reduced by more than half, from 3.3 K to just 1.5 K per CO2 doubling. There is evidence that the overestimate may be no more than 20%, in which event climate sensitivity would be at least 2.1 K: still below two-thirds of the IPCC’s current central estimate.
If there were no temperature feedbacks acting to amplify or attenuate the direct warming caused by a CO2 doubling, then the warming would simply be the product of the CO2 radiative forcing and the Planck parameter: thus, using the IPCC’s values, 3.708 x 0.313 = 1.2 K.
But that is not enough to generate the climate crisis the IPCC’s founding document orders it to demonstrate: so the IPCC assumes the existence of several temperature feedbacks – additional forcings fn demonimated in Watts per square meter per Kelvin of the direct warming that triggered them. The IPCC also imagines that these feedbacks are so strongly net-positive that they very nearly triple the direct warming we cause by adding CO2 to the atmosphere.
The term in red in the climate-sensitivity equation is the overall feedback gain factor, which is unitless. It is the reciprocal of (1 minus the product of the Planck parameter and the sum of all temperature feedbacks), and it multiplies the direct warming from CO2 more than 2.8 times.
Remarkably, the IPCC relies upon a single paper, Soden & Held (2006), to establish its central estimates of the values of the principal temperature feedbacks. It did not publish all of these feedback values until its fourth and most recent Assessment Report in 2007.
The values it gives are: Water vapor feedback fH2O = 1.80 ± 0.18; lapse-rate feedback flap = –0.84 ± 0.26; surface albedo feedback falb = 0.26 ± 0.08; cloud feedback fcld = 0.69 ± 0.38 Watts per square meter per Kelvin. There is also an implicit allowance of 0.15 Kelvin for the CO2 feedback and other small feedbacks, giving a net feedback sum of approximately 2.06 Watts per square meter of additional forcing per Kelvin of direct warming.
Note how small the error bars are. Yet even the sign of most of these feedbacks is disputed in the literature, and not one of them can be established definitively either by measurement or by theory, nor even distinguished by any observational method from the direct forcings that triggered them. Accordingly, there is no scientific basis for the assumption that any of these feedbacks is anywhere close to the stated values, still less for the notion that in aggregate they have so drastic an effect as almost to triple the forcing that triggered them.
Multiplying the feedback sum by the Planck parameter gives an implicit central estimate of 0.64 for the closed-loop gain in the climate system as imagined by the IPCC. And that, as any process engineer will tell you, is impossible. In electronic circuits intended to remain stable and not to oscillate, the loop gain is designed not to exceed 0.1. Global temperatures have very probably not departed by more than 3% from the long-run mean over the past 64 million years, and perhaps over the past 750 million years, so that a climate system with a loop gain as high as two-thirds of the value at which violent oscillation sets in is impossible, for no such violent oscillation has been observed or inferred.
Multiplying the 1.2 K direct warming from CO2 by its unrealistically overstated overall feedback gain factor of 2.8 gives an implicit central estimate of the IPCC’s central estimate of 3.3 K for the term in blue, 
, which is the quantity we are looking for: the equilibrium warming in Kelvin in response to a doubling of CO2 concentration.
To sum up: the precise values of the CO2 radiative forcing, the Planck parameter, and all five relevant temperature feedbacks are unmeasured and unmeasurable, unknown and unknowable. The feedbacks are particularly uncertain, and may well be somewhat net-negative rather than strongly net-positive: yet the IPCC’s error-bars suggest, quite falsely, that they are known to an extraordinary precision.
It is the imagined influence of feedbacks on climate sensitivity that is the chief bone of contention between the skeptics and the climate extremists. For instance, Paltridge et al. (2009) find that the water-vapor feedback may not be anything like as strongly positive as the IPCC thinks; Lindzen and Choi (2009, 2011) report that satellite measurements of changes in outgoing radiation in response to changes in sea-surface temperature indicate that the feedback sum is net-negative, implying a climate sensitivity of 0.7 K, or less than a quarter of the IPCC’s central estimate; Spencer and Braswell (2010, 2011) agree with this estimate, on the basis that the cloud feedback is as strongly negative as the IPCC imagines it to be positive; etc., etc.
Since all seven of the key parameters in the climate sensitivity equation are unknown and unknowable, the IPCC and its acolytes are manifestly incorrect in stating or implying that there is – or can possibly be – a consensus about how much global warming a doubling of CO2 concentration will cause.
The difficulties are even greater than this. For the equilibrium climate sensitivity to a CO2 doubling is not the only quantity we need to determine. One must also establish three additional quantities, all of then unmeasured and unmeasurable: the negative forcing from anthropogenic non-greenhouse sources (notably particulate aerosols); the warming that will occur this century as a result of our previous enrichment of the atmosphere with greenhouse gases (the IPCC says 0.6 K); the transient-sensitivity parameter for the 21st century (the IPCC implies 0.4 K per Watt per square meter); and the fraction of total anthropogenic forcings represented by non-CO2 greenhouse gases (the IPCC implies 70%).
Accordingly, the IPCC’s implicit estimate of the warming we shall cause by 2100 as a result of the CO2 we add to the atmosphere this century is just 1.5 K. Even if we were to have emitted no CO2 from 2000-2100, the world would be just 1.5 K cooler by 2100 than it is today. And that is on the assumption that the IPCC has not greatly exaggerated the sensitivity of the global temperature to CO2.
There is a final, insuperable difficulty. The climate is a coupled, non-linear, mathematically-chaotic object, so that even the IPCC admits that the long-term prediction of future climate states is not possible. It attempts to overcome this Lorenz constraint by presenting climate sensitivity as a probability distribution. However, in view of the uncertainty as to the values of any of the relevant parameters, a probability distribution is no less likely to fail than a central estimate flanked by error-bars.
If by this time your head hurts from too much math, consider how much easier it is if one is a Classicist. The Classicist knows that the central argument of the climate extremists – that there is a (carefully-unspecified) consensus among the experts – is an unholy conflation of the argumentum ad populum and the argumentum ad verecundiam. That is enough on its own to demonstrate to him that the climate-extremist argument is unmeritorious. However, you now know the math. The fact that not one of the necessary key parameters can be or has been determined by any method amply confirms that there is no scientific basis for any assumption that climate sensitivity is or will ever be high enough to be dangerous in the least.
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I expect most readers of the alarmist variety will be unable to understand the meaning of “cannot” in the title.
Concise and explicit. I’d like to hear the alarmist argument to knock it down?
We cannot repeat it too much that the null hypothesis of weak CO2 response
is well supported by the entire Cenozoic climate record.
Even during the height of the Eocene Optimum, 12C warmer than today,
there was no thermal runaway, no mass extinctions (like those when the optimum ended),
and no ‘Earth crisis’. CO2 was in the thousands, driven as it was out of the warm oceans.
Since the Eocene hot-house, the Earth has been steadily cooling, which should be impossible.
Better yet, when the Earth’s orbital eccentricity goes high (over 5%) every 400 kyr,
the semiannual variation in sunlight is the square of (1+e)/(1-e) or, at e=5%
TWENTY TWO PERCENT! Shouldn’t that light off the greenhouse bomb?
Why didn’t the first such perihelion yield the runaway greenhouse
predicted by the IPCC ‘equation’?
(It’s as much an equation as Hansen is a scientist — not at all.)
Well-done
“demonimated” … surely a typo, but wonderfully evocative.
Lord Monckton cracks me up!
“sins of emission?”
Would those be diurnal or nocturnal?
Brilliant!
You don’t need Classical training. Read the Book of Proverbs, or read any author who wrote before 1960, or listen to your grandmother.
I love this stuff!
“Unknown and unknowable”.
Lord Monckton establishes beyond a shadow of a doubt that climate alarmism is founded on little more than pure dogma — the antithesis of science.
Thank you for a wonderful and brilliantly succinct essay!
Christopher: You say: “Multiplying the feedback sum by the Planck parameter gives an implicit central estimate of 0.64 for the closed-loop gain in the climate system as imagined by the IPCC. And that, as any process engineer will tell you, is impossible. In electronic circuits intended to remain stable and not to oscillate, the loop gain is designed not to exceed 0.1. Global temperatures have very probably not departed by more than 3% from the long-run mean over the past 64 million years, and perhaps over the past 750 million years, so that a climate system with a loop gain as high as two-thirds of the value at which violent oscillation sets in is impossible, for no such violent oscillation has been observed or inferred”
Can I ask if the value of 0.64 in the second line might be ten times too high? That is, should it perhaps be 0.064? The reason I ask is that you go on to say that “a loop gain as high as two thirds of the value at which violent oscillation sets in is impossible”. But aren’t you saying that the latter value is 0.1, thus a number two thirds of 0.1 would be 0.064, not 0.64.
I would appreciate it if you could clarify this point. .
I have a serious problem with this bit.
“This value was obtained by inter-comparison between three models: but models cannot reliably determine it.”
How can comparing models show anything about deriving a constant such as CO2 forcings? This is backward. A model can only illustrate outcomes after such constants have been determined by real world observation and experiment. I think that Monckton may be pulling his punches a bit here. Models by definition cannot prove anything. The most they can hope to do is provide some verification for things discovered elsewhere.
And we see it in the next sentence:
“Both of the IPCC’s values for the vital coefficient are guesses.”
Meaning that the models are utterly worthless as evidence of anything.
This all reminds me of Drake’s equation for the number of civilizations in the galaxy. While interesting speculation in 1960, it was utterly meaningless because none of the constants were known then and are still not known to this day. Moreover this presumes that Drake had included all of the limiting factors which it’s now apparent he had not.
Feynman was right. This is cargo-cult science at its worst.
I admit that I have not yet digested this article, but I observe that it appears that the central equation is untenable. That, in itself, should be grounds to walk away. Of course observations that the values predicted have not come to be should also carry some weight.
So this becomes a political equation expressed in terms that scientists might devourer for more research money?
I thought the holy grail of research was the cause of cancer, these idiots have turned it into finding the cause of anthropogenic warming when the earth is cooling. Lets declare victory and go home.
You can’t get a reliable answer to the standard of a logically sound argument with regards to the climate. But that doesn’t mean you should ignore the information available or that you should not make a judgement call based on unreliable information. Judgement calls are often PROPERLY based on logical falacies and generalizations when there is not enough information to know for certain what is the right thing to do. For example, if a coal fired power plant is only in service for 5 years, the return on investment will not be repaid. Yet despite the fact that we have needed to build many power plants in the past, it is a logical falacy to conclude that we will certainly still need another one in five years from now. A wide variety of energy technology breakthroughs from fusion to super cheap solar and storage could make a coal plant built today obsolete. But that doesn’t mean it is a mistake to build a coal plant today.
It’s logical falacy to believe with certainty that it will freeze tonight just because the current weather patterns have often been followed by freezing, but you still might decide to expend resources to bring your potted plants in to protect them from the freezing, even though it might not freeze.
Probabalistic reasoning often properly relies on logical falacies.
Mondo asks for clarification of how the implicit closed-loop gain of 0.64 in the climate system is determined. I am happy to provide the requested clarification. The loop gain, in the climate, is the product of the feedback sum (the IPCC’s implicit central estimate is approximately 2.06 Watts per square meter per Kelvin) and the Planck parameter (0.313 Kelvin per Watt per square meter: IPCC, 2007, p. 631 fn).
The singularity in the Bode feedback-amplification equation occurs at a loop gain of unity. Process engineers designing electronic circuits that are intended to be stable usually set a design limit of one-tenth of this value – or 0.1 – to make absolutely sure that no operational circumstances sufficient to drive the circuit across the singularity arise. They would certainly not design in a loop gain as high as 0.64. That is far, far too close to the singularity.
In the climate, over at least the past 64 million years, the Earth has not seen the violent oscillations either side of the singularity that would be near-inevitable if the loop gain were anything like as high as 0.64.
Interestingly, the literature (e.g. Roe, 2009) says that at or above the singularity the climate sensitivity becomes undefined. Though the sensitivity is certainly undefined at the singularity itself, it is as certainly defined at loop gains above 1 as at loop gains below 1. In fact, at loop gains above 1 it is as strongly negative as it was strongly positive at loop gains below 1. In effect, the equation has the effect of rotating the climate-sensitivity curve by 180 degrees about the point (1, 0) as the singularity is crossed. It is this reversal that produces the violent oscillations that are sometimes deliberately designed into circuits that are not intended to be stable. Yet global temperature has not fluctuated by more than 3% either side of the long-run mean over the past 64 million years.
The conclusion is that feedbacks are extremely unlikely to be anything like as strongly net-positive as the IPCC’s climate-sensitivity estimates imply. To produce the formidable temperature-stability that the climate has exhibited over the past 64 million years, feedbacks cannot really exceed the process engineers’ limit of 0.1. And that, in turn, implies on this ground alone a maximum global warming of 1.3 Celsius per doubling of CO2 concentration – not the 3.3 Celsius imagined by the IPCC.
Christopher,
Thanks you for your post. Speaking as an electrical engineer who spent many years developing communications amplifiers, your assertion that the loop gain needs to be limited to about 0.1 when the feedback is positive to avoid oscillations is consistent with my experience. In EE-speak, we limit the feedback amplification to 1dB. When I give talks, I tell people that the standard climate model has 10dB of feedback amplification in it, and I have yet to find an EE who thinks that is plausible.
Very good! I always enjoy your writings and agree with most everything except the climate being chaotic part. Yes, I realize that’s basically word for word from the IPCC’s description. I, however, see no compelling evidence that the climate behaves chaotically. It’s certainly massively complicated, but that does not equate to chaotic. Climate models, on the other hand, seem highly likely to behave chaotically.
I have a small nit with the logic.
You assert that there can be no legitimate consensus. It is tautological that a group sharing a common view has a “consensus.” The legitimacy of the consensus hangs only on the nature of the unanimity of the group on the matter — that agreement was unfeigned and not coerced.
However, you prove very nicely indeed there can be no *rational* consensus. You have thus further proven that the group holding this view is, ipso facto, irrational to the degree the consensus is legitimate.
Civilly and Classically Yours,
EC.
@ur momisugly Monckton,
Being an Electronics and Process Control Engineer I can say you are outstandingly correct, anything with a loop gain of 0.64 is going to be unstable, it will ring in both directions, and we would see wild temperature fluctuations in response to changes in temperature or CO2 say the sunspot cycle, or Diurnal ranges. Also temperatures would overshoot, keep rising or falling in response to the feedbacks after the stimulii is removed.
I think however you should look a bit deeper, if you consider the emission (lapse rate) negative feedback is -0.84, then to acheive a total feedback factor of 1.8 (1 for CO2 and 1.8 for Feedbacks = 2.8 overall response) the positive feedback has to first overcome the negatives. If the lapse rate feed back is -0.84 then that equates to a loop gain of -0.455. The Positive feedback contribution is supposed to raise the loop gain to +0.64 – It’s been a while so excuse me if my math is wrong, but doesn’t this imply that the change in gain factor is over unity 0.64- (-0.455) = 1.095. If one could exclude emission in an experiment so the lapse rate feedback is not present but the positive feedbacks were, should not the environment within the experiment represent an oscillator? Clearly the real world doesn’t behave like this!
I also worry about other negative feedbacks that are omitted, for example as water is evaporated and raised to the Troposphere mgh Newtons is absorbed or about 30kN per kg when this inevitably rains out this is expended against the earth as kinetic energy or 1/2 mV(t)^2 at the terminal velocity of the rain. The energy is absorbed by the mass of the earth and is expended into the gravitational system. I would see this kinetic energy as being removed from the thermal system yet it is not represented in your equation. I have not looked at the magnitude. But given the capacity of Hydro power plants generating gigawatts of energy using a very small part of this precipitation one might guess it would be significant. If water represents a 1.8 x feedback then presumably the much larger amount cycling through the climate system must represent a significant negative feedback. This makes our analysis of the feedbacks above much worse.
Do you have any idea what this rain feedback might amount to?
Lord Monckton,
Thank you for far and away the most complete, understandable and convincing refutation of the raditive forcing equation I have read. You may not be a Climate Scientist, but you are an extraordinary climate scientist.
Reblogged this on Climate Ponderings.
Christopher,
Thank you for your polite and informative response to my question. I now understand that point. Overall of course, I think you do an outstanding job of providing evidence and facts. I look forward to some of the “real” climate scientists giving us detailed reasoning as to why your conclusions are not valid. In the absence of receiving such response, I think that we can assume that your conclusions are unchallenged.
Thank you, Lord Monckton! And the Hockey Team laugh at this guy??
I wonder if someone would be kind enough to answer a sort of involved question, one that may be a little stupid. I feel like I must be missing something rather obvious.
In electronics, amplifiers with gain modified by positive feedback are required to have some kind of dividing network which make the input to the amplifier a different (smaller) value than the output of the amplifier. This is where the idea of “loop gain” or “feedback factor” comes from, and why in positive feedback amplifiers the loop gain is always less than 1, the loop is a physical structure that buffers input from output and cuts the magnitude. On the other hand, if the amplifier has any intrinsic gain at all prior to the loop, even the tiniest amount, and the input and the output are ever allowed to touch each other, to become the same quantity, there’s no chance of avoiding runaway. The output of an amplifier with gain can NEVER be fed directly to its input or disaster.
But, true enough, if you feed only a tiny proportion of the output back to the input, then you can increase gain and keep stability. Climate modellers call on this well known fact to back up their claim that the positive feedback conception of climate can remain stable — they seem to just stipulate that the feedback factor is less than 1. And then, I think, they proceed from there not to actually MODEL the feedback, but rather just assume a particular amplification of the direct forcing, that is they plug 3 into the forcing instead of 1 (or whatever) and thell themselves it’s okay, they don’t have to model the feedback to know what the amplification is, and they know it can be positive because, after all, the feedback equation allows it if the feedback factor is below unity. But from the feedback equation, in order for feedback factor to be below unity, the input must be isolated from the output. Output and input are completely different variables with completely different values, and they’re kept apart from each other by a mathematical, and if you want your amplifier to work, an equivalent physical divider.
So what I’d earnestly like to know is, in climate, where the input to the amplifier (evaporation of water, or melting of permafrost, or CO2 coming out of the ocean or whatever) is apparently always atmospheric temperature, and the ouput of the amplifier is ALSO atmospheric temperature, what physical mechanism is thought to provide the “divider” which buffers the output temperature from becoming the input temperature and allowing only a tiny portion of the output temperature from returning to the input of the process? I feel like the basic structure of the atmosphere, which doesn’t have wires or pipes or walls in it to provide isolation from output to input has no way of having a feedback factor that could be (effectively, after maybe time delays) anything other than 1. That would imply that the atomosphere could never have net “gainy” processes, but clearly I’m not understanding something. Help?
Hi Ned,
“I feel like the basic structure of the atmosphere, which doesn’t have wires or pipes or walls in it to provide isolation from output to input has no way of having a feedback factor that could be (effectively, after maybe time delays) anything other than 1.”
The input is the greenhouse forcing (W/m^2) at the top of the atmosphere and the output is surface temperature (K). Does that help?
Dave