Climatology’s Startling Error – An Update

By Christopher Monckton of Brenchley – Re-Blogged From WUWT

Well, we sent out our paper On an error in defining temperature feedback to a leading journal for review. The reviewers did not like it at all. “And, gracious! How Lord Lundy cried!”


We are persevering, though, for in our submission nothing the reviewers have said in any way undermines the scientific validity of our result, which I outlined here in a series some months back.

Here, I shall summarize our argument in layman’s terms (for a layman is what I am). If you want a more detailed account of the physics, Anthony has kindly posted a single-sheet scientific summary here:

error-summary (PDF)

After the brief account of our argument that follows, just for fun I shall set out the reviewers’ principal objections, together with our answers. Feel free to comment on whether we or the reviewers are right.

How climatologists forgot the Sun was shining

Climatologists trying to predict global warming forgot the sunshine in their sums. After correction of this startling error of physics, global warming will not be 2 to 4.5 K per CO2 doubling, as climate models imagine. It will be a small, slow, harmless and net-beneficial 1.17 K.

The Climate Model Intercomparison Project (CMIP5: Andrews+ 2012) had predicted that doubling CO2 will warm the world by 1.04 ± 0.1 K (before feedbacks act) and 3.37 ± 1.3 K (after feedbacks have acted). IPCC says 3.0 ± 1.5 K. Some papers (e.g. Murphy 2009) give high-end estimates up to 10 K per CO2 doubling.

Climatologists erred when they borrowed feedback mathematics from control theory without quite understanding it. They used a variant feedback system-gain equation that relied solely on small changes in reference temperature (before feedback) and in equilibrium temperature (after feedback). But the mainstream equation they borrowed from control theory uses entire, absolute temperatures in Kelvin, not just changes in temperature.

Their variant equation is a valid equation, for it constitutes the difference between two instances of the mainstream equation. However, in taking that difference, they effectively subtracted out the term for the 243.3 K emission temperature as it would have been at the Earth’s surface without non-condensing greenhouse gases, driven by the fact that the Sun is shining, as well as the term for the 11.5 K warming from the pre-industrial greenhouse gases.

Because they lost this vital information, their variant equation could not reliably yield the true system-gain factor – the ratio of equilibrium to reference temperature. Instead, they tried to find that factor, the Holy Grail of global warming studies, by hunting for individual feedbacks computer models’ outputs. They were looking for blunt needles in the wrong haystack, when all they needed (if only they had known it) was a pin they already had.

Measurement and observation cannot tell us the magnitudes of individual feedbacks, and cannot help us to distinguish individual feedbacks either from each other from the manmade warmings that triggered them.

Restoring the missing sunshine and pre-industrial greenhouse-gas warming allows anyone to calculate the true system-gain factor. The calculation is direct, swift and accurate. You do not even need to know the magnitude of any individual feedback. All you need are the entire reference temperature (before feedback) and equilibrium temperature (after feedback) in any chosen year.

In 1850, reference temperature – the sum of the 243.3 K warming from the Sun and a further 11.5 K from the pre-industrial non-condensing greenhouse gases – was 254.8 K. The measured equilibrium surface temperature was 287.5 K (HadCRUT4). Therefore, the feedback system-gain factor for that year was 287.5 / 254.8, or 1.13.

Using the variant equation, however, one cannot derive the system-gain factor for 1850 at all.

By 2011, manmade influences had increased reference temperature by 0.68 K to 255.5 K. Measured temperature had risen by 0.75 K, but another 0.27 K that might not yet have come through because of an imagined “radiative imbalance” has to be allowed for, raising equilibrium temperature by 1.02 K to 288.5 K. Therefore, the system-gain factor for 2011 was 288.5 / 255.5, or 1.13.

That 2011 value is just as it was in 1850. It is not difficult to see why. The 254.8 K reference temperature in 1850 that was left out of climatologists’ sums is about 375 times the 0.68 K manmade reference warming from 1850 to 2011. That is why our effect on the system-gain factor is minuscule.

The climate stability evident after correcting climatologists’ striking error of physics should come as no surprise. For more than 800,000 years, according to analyses of air trapped in ancient ice (Jouzel+ 2006), global mean surface temperature has varied by little more than 3 K either side of the average temperature for the period.

Though IPCC (2013) mentions “feedback” 1000 times, feedback can be ignored with very little error. The system-gain factor may be taken as constant at 1.13. The non-linearity in feedbacks that climatologists had imagined makes very little difference.

Using the variant equation, the system-gain factor would be 1.02 / 0.68, i.e, 1.50, and the equilibrium warming from doubled CO2 would thus be 1.50 times the reference warming of 1.04 K in response to doubled CO2: i.e., 1.55 K. Even that value is only half the 3.37 K mid-range estimate in the CMIP5 models.

Using the mainstream equation, though, the true equilibrium warming from doubled CO2 is even smaller. It is 1.13 times the reference warming of 1.04 K: i.e., a harmless 1.17 K. To make sure, ten separate official estimates of manmade radiative forcing were studied. In each case, global warming in response to doubled CO2 was 1.17 K.

A statistical Monte Carlo simulation showed the true range of global warming as 1.08 to 1.25 K.

The control theory underlying the present result was verified on two test rigs, one of them at a government laboratory.

Climatologists had imagined that individual temperature feedbacks would self-cancel, except for water vapor, the largest. The atmosphere can carry 7% more water vapor for each Kelvin of warming. Can, not must. Models had predicted that, if and only if warming were manmade, the tropical upper air would warm at thrice the surface rate. Yet the water-vapor content up there is falling. Therefore, the tropical mid-troposphere “hot spot” does not exist.

Bottom line: global warming is not a problem after all. Enjoy the sunshine climatologists forgot about.

Reviewers’ comments, and our responses

“Simply inserting emission temperature in place of anthropogenic surface warming in the equations, and proceeding as before, is a massive violation of energy conservation.”

Um, no. One of my co-authors, John Whitfield, built a test rig – effectively an analog computer – to verify the control theory underlying our argument. There was certainly no “massive violation of energy conservation”. Instead, the outputs from the rig, in 23 distinct experiments, confirmed our understanding in all respects.

To make assurance doubly sure, we commissioned a government laboratory to build a test rig to its own design and to carry out the same 23 experiments. The results agreed with what the theory had led us to predict, and did so to the equivalent of a tenth of a Kelvin in each case. If there had been any “massive violation of energy conservation”, it would definitely have shown up in the experiments. It didn’t.

Besides, the reviewer had provided no evidence or argument whatsoever to justify the nonsensical assertion that our method was a “massive violation of energy conservation”.

“Instead of feeding in the perturbation temperature and asking what the perturbation in the top-of-atmosphere energy budget is, they shove the whole temperature difference from absolute zero into the equation by fiat and without physical justification. It’s plain rubbish.”

The physical justification is this. Feedback processes, being inanimate, cannot discriminate between a pre-existing temperature and a perturbation of that temperature. They have no means of deciding not to react at all to the former and yet to react vigorously to the latter. Nor are those inanimate processes concerned with what might have been if the Sun were not shining. For the Sun – like it or not – is shining.

Feedback processes simply respond to the temperature as they find it. Let us see why by studying the block diagram for a feedback loop –


The reference temperature (i.e., the temperature before feedbacks act) comes in from top left and is input to the summative input/output node. From that node, the fraction of the output temperature represented by the feedback response goes round the feedback loop and is fed back to the input/output node, where it is added to the original reference temperature to give the equilibrium sensitivity.

Now, increase the reference temperature by some increment. Then the input to the feedback loop is a little larger than before. The feedback processes simply respond to that larger reference temperature. There is self-evidently no physical mechanism by which those processes can “know” that they must not respond to a somewhat larger reference temperature than before.

“The analogy to a Bode amplifier, on which the authors place so much emphasis, is not an identity. If it were a perturbation voltage that were isolated and it was the perturbation voltage on which the feedbacks operated, the analogy could be made more closely.”

To understand why the reviewer sees things this way, let us recall IPCC’s official definition of a “climate feedback” (IPCC, 2013, glossary, p. 1450) –

Climate feedback An interaction in which a perturbation in one climate quantity causes a change in a second, and the change in the second quantity ultimately leads to an additional change in the first. A negative feedback is one in which the initial perturbation is weakened by the changes it causes; a positive feedback is one in which the initial perturbation is enhanced. In this Assessment Report, a somewhat narrower definition is often used in which the climate quantity that is perturbed is the global mean surface temperature, which in turn causes changes in the global radiation budget. In either case, the initial perturbation can either be externally forced or arise as part of internal variability.”

Notice that the word “perturbed” or “perturbation” occurs five times in this short and calculatedly inspissate definition. Let us draw the block diagram for the variant feedback loop imagined by official climatology –


Here, there is scarcely an absolute quantity in the entire diagram. So, what is going on? Well, the mainstream feedback system-gain equation used in official climatology states that the change in equilibrium temperature is equal to the sum of the change in reference temperature and the product of the feedback factor and the change in equilibrium temperature.

Now, climatology’s variant equation is a perfectly valid equation. In effect, it represents the difference between two successive instances of control theory’s mainstream equation, which states that the equilibrium temperature is equal to the sum of the reference temperature and the product of the feedback factor and the equilibrium temperature.

But the variant equation is not useful for finding equilibrium sensitivities, because one cannot reliably derive from it the Holy Grail of global-warming studies – namely, the feedback system-gain factor, which is the ratio of equilibrium to reference temperature.

For present purposes, though, it is necessary only to observe that, since climatology’s variant equation is a valid equation, so is control theory’s mainstream equation, from which the variant equation is derived.

Let us correct the official definition of a “climate feedback” –

“Positive feedback in dynamical systems amplifies the output signal. Negative feedback attenuates it. In climate, the input signal is the global mean surface reference temperature clip_image008 that would obtain without feedback. The output signal is the global mean surface equilibrium temperature clip_image010 after allowing for feedback. The feedback response clip_image012 constitutes the entire difference clip_image014 between equilibrium and reference temperatures, such that the feedback factor clip_image016, which is the fraction of equilibrium temperature that constitutes the feedback response, is equal to clip_image018. The system-gain factor clip_image020 is equal to clip_image022, i.e. clip_image024.”

Note in passing that the feedback-loop block diagrams (a) simplify to the system-gain block diagrams (b). What this means is that all one needs to know to find the system-gain factor clip_image026 for any given year is the reference temperature (before feedback) and the measured equilibrium surface temperature (after feedback) in that year. One does not need to know the value of any individual feedback.

“[Test rigs] are all very well, but simply show that one can construct systems for which the one-dimensional energy-balance equations are exactly true. There is no information contained therein to say whether these models are relevant to the real climate.”

If the feedback mathematics borrowed by official climatology from control theory is as inapplicable as the reviewer suggests, then there is no legitimate basis for climatology’s current mistaken belief that feedback response accounts for at least two-thirds of equilibrium sensitivity. Paper after paper (see e.g. Hansen 1984, Schlesinger 1985, Bony 2006, Roe 2009) uses feedback mathematics, explicitly referring to Bode. But these and suchlike papers use Bode in a fashion that prevents accurate derivation of the system-gain factor. IPCC (2013) mentions the word “feedback” more than 1000 times.

These and numerous other authors have accepted that feedback mathematics is relevant to the derivation of equilibrium sensitivity. Quite right too: for equilibrium temperature is greater than reference temperature, and feedback response constitutes the entire difference between them.

It is interesting to see how ready the reviewers are to ditch the “settled science” that has been in the literature for decades whenever they find it inconvenient.

“The energy-balance equation used by climate science is just a Taylor-series expansion of the difference between the global average top-of-atmosphere energy imbalance and the radiative forcing. Higher-order terms have been dropped. This is why emission temperature does not appear in the zero-dimensional energy-balance equation. I just don’t see any opposing argument that would change this view of the equation.”

Since climatology’s variant equation is a valid equation, there is nothing in itself wrong with it. It is validly derived from the energy-balance equation, and the fact that it is derived via a leading-order Taylor-series expansion does not in any way impugn our argument: for a Taylor-series expension is merely a mechanism for expressing the shape of a curve about a particular point.

But leaving out the sunshine term makes it impossible to derive the feedback system-gain factor accurately from the variant equation.

Nothing in the derivation of the variant equation from the top-of-atmosphere energy-balance equation tells us anything about the magnitude of the system-gain factor. It is precisely for this reason that climate modelers have spent decades futilely attempting to constrain the interval of Charney sensitivities, which, in IPCC (2013), was [1.5, 4.5] K, just as it was four decades ago in Charney (1979).

“The authors would do well to educate themselves on the literature evaluating the linearity or otherwise of feedbacks.”

Yes, some feedback responses are non-linear. The water vapor feedback is the prime example. As the space occupied by the atmosphere warms, it can carry 7% more water vapor per Kelvin. Indeed, close to the Earth’s surface, at a pressure altitude of 1000 mb, it does precisely that:


At 600 mb, however, there is no increase in the specific humidity with warming. And at the crucial mid-troposphere altitude 300 mb, the specific humidity has been falling. Why is this important? Well, official climatology regards all individual feedbacks except water vapor as broadly self-canceling. It is only the water vapor feedback that provides the pretext for the notion that because of feedbacks equilibrium warming is three or four or even ten times reference warming.

Yet the only altitude at which the predicted rate of increase the specific humidity is observed in reality is very close to the surface, where, as Harde (2017) has pointed out, the spectral lines of water vapor are very close to saturation.

Turn to Fig. 9.1c of IPCC (2007). There, the predicted tropical mid-troposphere “hot spot” – I had the honor to name it – is made evident in the fashion with which we are now wearily familiar: lurid colors –


So much for what is predicted. I could show dozens of similar images from various general-circulation models. In reality, however, the predicted “hot spot” is conspicuous by its entire absence –


Now, the U.S. Climate Change Science Program produced its real-world data showing no “hot spot” a year before IPCC persisted in its false claim that the “hot spot” exists. And why would it exist? For the specific humidity that would have to increase to deliver the predicted faster-than-surface warming has actually decreased.

However, using our method of finding the feedback system-gain factor, one does not need to know anything about individual feedbacks. All one needs to know is the reference temperature (before feedback) and the equilibrium temperature (after feedback) in any given year.

And to find out whether nonlinearities in individual feedbacks are varying the system-gain factor with time and temperature, all one needs to do is find the system-gain factor for two different years – one close to the beginning of the industrial era and one close to the end. So we did that. And we even made allowance for the imagined (and probably imaginary) “radiative imbalance” that may have delayed about a quarter of the manmade warming to date.

In both 1850 and 2011, the system-gain factor, to three decimal places, was 1.129. It didn’t change even in the third decimal place. It didn’t change because the combined temperature from the Sun and from the pre-industrial non-condensing greenhouse gases was 375 times bigger than the 0.68 K reference sensitivity between those two dates. Nonlinearity? Schmonlinearity.

“The fact that feedbacks, calculated properly from models, give the right range of climate sensitivity in models probably should have given the authors pause in their conviction it [their analysis] is fundamentally defective.”

And this, gentle reader, is our old friend the circular argument, the argumentum ad petitionem principii, one of the dozen commonest logical fallacies. From this fallacy the only valid conclusion that may be drawn is that the perpetrator is insufficiently educated to know any better.

To demonstrate the utility of the simple system-gain equation in studying equilibrium sensitivities, we had taken climatology’s variant of it and demonstrated that, using the range of feedback factors officially derived from the models by Vial et al. (2013), it would deliver the published interval of equilibrium sensitivities. But that exercise told us nothing of the correct value of the feedback factor, or of its cousin the system-gain factor. To derive the correct values of these variables, one needs to look outside the window, notice that the Sun is shining, and take proper account of that fact by using the mainstream system-gain in one’s calculations.

“The sensitivity of any climate model is what it is – it cannot change due to any post-hoc analysis of its feedbacks. In a model the CO2 level is doubled, the radiative transfer calculation alters, and temperatures, water vapor, circulation, clouds etc. all change. The simulated climate system eventually stabilizes and the resulting net change in surface temperature is the sensitivity of that model.”

And this is the fundamental fallacy of relevance known as the straw-man argument, the argumentum ad ignorationem elenchi. For we had not undertaken any post-hoc analysis of any model’s feedbacks. Instead of adopting the models’ doomed-to-failure bottom-up approach to deriving equilibrium sensitivity by making fanciful guesstimates of the values of individual feedbacks, we had adopted the far simpler and more robust top-down approach of finding the reference and equilibrium temperatures for two well-separated years in the industrial era, discovering that the system-gain factors derived from these values were the same, applying the system-gain factor to the reference sensitivity to doubled CO2 and demonstrating, beyond all reasonable doubt, that the equilibrium sensitivity to doubled CO2 is just 1.17 K, plus or minus less than a tenth of a Kelvin.

The reviewer is, in effect, saying that the models must be right. Well, however elaborate they are, they are not right. They are wrong, as our analysis has demonstrated.

“No physical arguments are given for why the sensitivity should be so small, and accepting this simple estimate as plausible would require rejecting all previous work by scientists to understand the physics of climate change, much of which has been proven beyond doubt. The analysis given is both rudimentary and fundamentally flawed and I cannot recommend publication by a reputable journal.”

See the analysis of the water vapor feedback, earlier in this article. The magnitude of that feedback has not been “proven beyond doubt”: it has been disproven beyond doubt. Consider, for instance, John Christy’s fascinating graph of predicted tropical mid-troposphere temperature change in 73 models from 1979-2012. All 73 models showed tropical mid-troposphere warming at a mean rate about four times the observed rate, and no model’s prediction was below the observed outturn –


It is very likely, therefore, that the chief reason why the corrected value of the system-gain factor, and hence of equilibrium sensitivity, is so much below all official estimates is the overegging of the water-vapor pudding in the models. But we don’t need to know what the models got wrong – it is sufficient to demonstrate – in our submission irrefutably – that wrong they were.

In one respect, though, the reviewer is right. We are indeed rejecting all previous work by scientists to derive equilibrium sensitivity, insofar as that work, however honest and diligent, is incompatible with the correct result which we have reached by a far simpler and more reliable method than theirs.

“Look back at the definition of the feedback factor above, and marvel at what they have done. The perturbation in climate forcing that they use to estimate feedbacks is, quite literally, Switching On The Sun. Start with the Earth at zero Kelvin. Now switch on the Sun, forbid any feedbacks, and we get a reference temperature of 255 K. Now allow feedbacks to perated, and in our current world we actually get to equilibrium temperature 287 K.”

Perhaps all climatologists are Scottish. For it comes as a great surprise to us, whenever we take the road to England – or the boat for the cold coast of Greenland, or the flight to almost anywhere – and we find, to our fascination and delight, that the land is often bathed in the holy radiance of a large, bright, warm, yellow object in the sky. We don’t see it that much in the Gaidhealtachd.

We do not have to Switch On The Sun. For, owing to the bounty of Divine Providence, it has already been Switched On for us (except in Scotland), and the angels – the intergalactic grease-monkeys whose task is keep the Universe unfolding as it should – are doing a splendid job of care and maintenance.

For the Sun, you see, is shining. Are we wrong to take account of that fact? We think not. The feedback processes operating today don’t care what feedback processes operated at zero Kelvin. They simply respond to the temperature as they find it. And that means it is better to take account of the fact that the Sun is shining than to ignore it.

It was not only the reviewers nominated by the journal who reviewed it. Somehow, a copy of our paper reached the Vice-Chancellor of the University of East Anglia, who, on reading the paper, summoned a meeting of all 65 Professors and Doctors of science in his Environmental Sciences faculty and yelled at them as follows –

“Monckton’s paper is a catastrophe for us. If the general public ever gets to hear of Monckton’s paper, there will be hell to pay.”

He ordered the faculty to drop everything and work on trying to refute our paper – which, at that time, was merely a 2000-word outline that has now been developed into a full-length, 6000-word paper. He later denied that the meeting had taken place, but we heard about it directly from one who was present.

Finally, here is a comment from a notoriously irascible skeptical blogger (not, of course, our genial host here):

“No, we’re not going to discuss Monckton’s result here. We don’t do simple.”

My reading in mathematics and physics has led me to imagine – perhaps wrongly – that there is more rejoicing in Heaven at the discovery of a simple method to derive a correct result than at the use of a pointlessly complex method to derive a result that, not least on account of the complexity, is incorrect.

Some final questions for those who have had the persistence to read this far. Are the reviewers correct, or are we correct? And would you like to be kept abreast of developments with occasional pieces here? The paper remains out for review and, in due course, we shall learn whether it has been accepted for publication. We have also been invited to write a book giving an account of our result and how we came by it.

And we have sent to IPCC a formal notice that all of its Assessment Reports are gravely in error. Though we have followed IPCC’s own published protocol for submission of alleged errors, we have been unable to obtain from the Secretariat the acknowledgement which its own rules require. So we are about to put the matter into the hands of the Bureau de l’Escroquerie, the Swiss Fraud Office, via the London Ambassador of Switzerland, the nation where IPCC is headquartered.

Before we call in InterPlod, are we right to think we are correct and the reviewers wrong?

For a 45-minute You-Tube presentation by me explaining our result, follow this link. I’m most grateful to John Charleston for having filmed the presentation in his own studio, and for having edited it and posted it up.

And here is the single slide, from my presentation at next week’s Camp Constitution in Connecticut, that brings the entire global warming foofaraw to an unlamented end –


As my noble friend the Earl of Seafield once put it, “There’s ane end to ane auld sang.”


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